Cycles, Growth and Structural Change

Running head i
Cycles, Growth and Structural

Change
With its unique combination of theoretical models and empirical analyses, this
collection of specialists’ essays contributes to an assessment of our understanding
of the relationship between cycles, growth and structural change. In this light it
also offers a critical evaluation of results in the recent literature on complex and
chaotic dynamics. The book is constructed upon the hypothesis that a re-unification
of the economic analyses of dynamic phenomena can be obtained through
comparison and cross fertilization of their up-to-now specialized approaches.
Recent developments in economic dynamics, one of the exciting frontiers in
current economic research, have been drawing from very diverse research areas
and different specialisms: economic theorists, applied economists and
econometricians, of course, but also mathematicians and researchers in information
theory, time series and dynamical system theories. The variety of approaches in
this book well represents the economists’ views of this research area. They range
from neoclassical and endogenous growth theories, to classical, evolutionary and
neo-Austrian dynamics; from real business cycle and information-based
econometrics to other non-linear techniques related to complex dynamics.
Likewise, modelling is done in the macro-, micro-, multi-agent and large dynamical
system styles. Moreover, the book offers materials for an evaluation of recent
literature which relies on complexity properties to search for alternative
formulations of economic policy issues.
Providing both historical evidence and theoretical formulations, written in a
critical vein by leading specialists in their own areas, Cycles, Growth and Structural
Change offers state-of-the art surveys of a fast expanding field and suggests a
research agenda for the future.
Lionello F. Punzo is Professor of Economics at the University of Siena.

ii The reading of theoretical texts
Routledge Siena Studies in Political Economy
The Siena Summer School hosts lectures by distinguished scholars on topics

characterized by a lively research activity. The lectures collected in this series
offer a clear account of the alternative research paths that characterize a certain
field. Former workshops of the School were printed by different publishers. They
include:
Macroeconomics
A survey of research strategies
Edited by Alessandro Vercelli and Nicola Dimitri
Oxford University Press, 1992
International Problems of Economic Interdependence

Edited by Massimo Di Matteo, Mario Baldassarri and Robert Mundell
Macmillan, 1994
Ethics, Rationality and Economic Behaviour

Edited by Francesco Farina, Frank Hahn and Stefano Vannucci
Clarendon Press, 1996
The Politics and Economics of Power

Edited by Samuel Bowles, Maurizio Franzini and Ugo Pagano
Routledge, 1998
The Evolution of Economic Diversity

Edited by Antonio Nicita and Ugo Pagano
Routledge, 2000
Cycles, Growth and Structural Change

Theories and empirical evidence
Edited by Lionello F. Punzo
Routledge, 2001
The Routledge Siena Studies in Political Economy Series gives a comprehensive

access to the publications of the School, which emphasizes the common
methodology employed in organizing the different workshops.
Running head iii
Cycles, Growth and

Structural Change
Theories and empirical evidence
Edited by
Lionello F. Punzo
London and New York

iv The reading of theoretical texts
First published 2001

by Routledge
11 New Fetter Lane, London EC4P 4EE
Simultaneously published in the USA and Canada
by Routledge
29 West 35th Street, New York, NY 10001
Routledge is an imprint of the Taylor & Francis Group
© 2001 Selection and editorial matter, Lionello F. Punzo;
individual chapters, the contributors
This edition published in the Taylor & Francis e-Library, 2006.
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or in any information storage or retrieval system, without permission in
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British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library
Library of Congress Cataloging in Publication Data

Punzo, Lionello F.
Cycles, growth and structural change : theories and empirical
evidence / Lionello F. Punzo.
p. cm.
Includes bibliographical references and index.
1. Business cycles–Congresses. 2. Economic development–
Congresses. 3.
Economics–Congresses. 4. Business cycles–Research–Methodology–
Congresses. 5. Economic development–Research–Methodology–
Congresses. 6. Economics–Methodology–Congresses. I. Title.
HB3711 .P93 2001

330–dc21
00–051777
ISBN 0-203-16483-0 Master e-book ISBN
ISBN 0-203-25909-2 (Adobe eReader Format)

ISBN 0–415–25137–0 (Print Edition)
Contents v
Contents
List of figures ix
List of tables xi
List of contributor xiii
Preface xvii
PART I
Facts and interpretations of growth and fluctuations 1
1 Economic cycles since 1870 3

SOLOMOS SOLOMOU
2 Growth and fluctuations: the post-war Japanese 27

experience
H I R O S H I Y O S H I K A WA
3 Productivity–investment fluctuations and structural

change 47
B E R N H A R D B Ö H M A N D L I O N E L L O F. P U N Z O
PART II
The macroeconomy and its dynamics 93
4 Qualitative dynamics and macroeconomic evolution

in the very long run 95
R I C H A R D H . D A Y A N D O L E G V. P AV L O V
5 Out-of-equilibrium dynamics 115

M A R I O A M E N D O L A A N D J E A N -L U C G A F F A R D
6 Disequilibrium growth in monetary economies:

basic components and the KMG working model 128
PETER FLASCHEL
vi Contents
7 Schumpeterian dynamics: a disequilibrium theory of
long run profits 169
K A T S U H I T O I WA I
PART III
Dynamics by interaction 201
8 Asymmetrical cycles and equilibrium selection in

finitary evolutionary economic models 203
M A S A N A O AO K I
9 The instability of markets 219

TA D H O G G , B E R N A R D O A . H U B E R M A N A N D
MICHAEL YOUSSEFMIR
10 Heterogeneity, aggregation and capital market

imperfection 229
D O M E N I C O D E L L I G AT T I A N D M A U R O G A L L E G AT I
11 Toward the microeconomics of innovation: growth

engine of market economies 266
W I L L I A M J . BAU M O L
PART IV
Challenges for quantitative methodologies 281
12 Business cycle research: methods and problems 283

E D WA R D C . P R E S C O T T
13 Complexity-based methods in cycles and growth:

any potential value-added? 301
WILLIAM A. BROCK
14 Information, complexity and the MDL principle 339

JORMA RISSANEN
15 The ‘exogenous’ in ‘endogenous’ growth theory 351

FRANK HAHN
Index 367
List of figures vii
Figures
1.1a Kalman filter decomposition of UK GDP cycles,

1870–1913 7
1.1b Kalman filter decomposition of French GDP cycles,
1870–1913 7
1.1c Kalman filter decomposition of German GDP cycles,
1870–1913 8
1.1d Kalman filter decomposition of US GDP cycles,
1870–1913 8
1.2 Nominal effective exchange rates 9
1.3 Sectoral weather effects 12
1.4 Sectoral shares in GDP 13
1.5 Weighted sectoral effects and total weather effect 13
2.1 Growth rate of real GDP 28
2.2 Population flow 1955–80 into and out of Toyko, Osaka and
Nagoya metropolitan areas 31
2.3 Growth rates of households and population, 1956–90 31
2.4 Domestic-demand-led high ecnomic growth of the Japanese
economy, 1955–70 33
2.5 Equipment and investment in plant, 1960–84 40
3.1 Scatter plot of growth rates of investment and value added
per employment 73
3.2 Transition function 1 75
3.3 Transition function 2 75
3.4 Time series graphs of simulated series 76
3.5 Scatter plot of series in levels 76
3.6 Time series of growth rates of y1 and y2 77
3.7 The Framework Space 77
3.A2.1 Italy 87
3.A2.2 France 88
3.A2.3 Germany 89
3.A2.4 USA 90
3.A2.5 Japan 91
viii List of figures
4.1 A simulated population history 107
4.2 Logarithm of simulated population 109
4.3 Details of the population dynamics 109
4.4 Simulated history of structural change 110
4.5 Simulated history of structural change in greater detail 110
4.6 Two extended simulations compared 111
5.1 Scenario 1: a) distribution of real wages, b) distribution of
productivity, c) distribution of final employment 121
5.4 Scenario 1 with increase in saving rate: a) distribution of real
wages, b) distribution of productivity, c) distribution of final
employment 124
6.1 The scope of traditional Keynesian theory 129
6.2 The Goodwin (1967) growth cycle model 133
6.3 The Goodwin growth cycle model in perspective 136
6.4 Implications of the kinked Phillips curve in the case of
steady state inflation 146
7.1 Industry supply curve in the short run and in the long run 170
7.2 Cumulative distribution of capacity shares 173
7.3 Relative form of industry supply curve 174
7.4 Evolution of the state of technology under the pressure of
either economic selection or technological diffusion 177
7.5 Evolution of the state of technology under the joint
pressure of economic selection, technological diffusion and
recurrent innovations 180
7.6 Long-run cumulative distribution of capacity shares 183
7.7 Long-run industry supply curve 185
7.8 Determination of long-run profit rate 186
7.9 Short-run pseudo-aggregate production function 190
7.10 Long-run pseudo-aggregate production function 191
7.A1 The case of a perfectly elastic demand curve 194
7.A2 The case of an absolutely inelastic demand curve 194
7.A3 The general case 195
9.1 Plot showing the growth of the largest eigenvalue of a
hierarchical matrix with branching ratio b = 2 and R = 0.55
as a function of d = log2n 225
10.1 Main features of theoretical frameworks examined 254
List of tables ix
Tables
1.1 International output correlation: pre-1914 11

1.2 Standard deviation (per cent) measures of output volatility 20
2.1 Growth rate of real GDP 27
2.2 Total employment in primary, secondary and tertiary
occupation: Japan, USA, UK and West Germany, 1950 and 1989 29
2.3 The relative contribution of demand components to the
business cycle 34
2.4 Investment in the petrochemical industry, 1956–80 35
2.5 Stability of demand components, 1956–89 39
2.6 Variance decompositions of investment, export and gross
domestic demand 43
3.1 Estimation of NL least squares 75
3.A1 Coding of sectors 86
4.1 A time-line for the major systems 106
4.2 System parameters for the mathematical history 107
11.1 Some leading growth macromodels 269
x List of tables
Contributors xi
Contributors
Mario Amendola is Professor of Economic Analysis at the University of Rome

‘La Sapienza’. He is the author of various books on the theories of capital and
technical progress, on dynamic models, on technological innovation and
industrial change, and of papers published on important international journals,
and also carries out an intense applied research activity. He has recently
published (with J.-L. Gaffard) Out of Equilibrium (Clarendon Press, 1998).
Masanao Aoki is Professor Emeritus at UCLA, and Professor in the Department

of Economics and Information, Gifu Shotoku Gakuen University, Japan. His
main current fields of research are illustrated by his most recent books, both
from Cambridge University Press: New Approaches to Macroeconomic
Modeling: Evolutionary Stochastic Dynamics, Multiple Equilibria and
Externalities as Field Effects (1996) and Modeling Aggregate Behavior and
Fluctuations in Economics: Stochastic View of Interacting Agents (forthcoming,
2001)
William J. Baumol is Professor of Economics at New York University and

Professor Emeritus and Senior Research Scholar at Princeton University. His
main current fields of interest are innovation and growth, public policy related
to monopoly power and international trade in the presence of scale economies.
He is the author of numerous books and articles, including Productivity and
American Leadership: The Long View (with S.A. Batey Blackman and E.N.
Wolff), (MIT Press, 1989).
Bernhard Böhm is Professor for Economics and Applied Econometrics at the

Institute of Econometrics, Operations Research, and Systems Theory of the
University of Technology, Vienna, Austria. He also chairs a postgraduate
programme in applied economics at the Academia Istropolitana Nova in
Slovakia. He specializes in econometric model building, structural change,
and environmental economics.
William A. Brock is Vilas Research Professor of Economics at the University of

Wisconsin, Madison. His interests include general economic dynamics,
xii Contributors
business cycle and monetary theory, heterogeneous agent models, econometrics,
optimal growth theory, and finance. He is author of numerous articles and
several books.
Richard H. Day is Professor of Economics at the University of Southern

California. His research began with simulations of production, investment
and technological change in various agricultural regions and industrial sectors
using a class of recursive programming models based on bounded rationality
and adaptive economizing. Subsequent work has involved the theory of adaption
and economic evolution, business cycles, economic growth and economic
development in the very long run. He is author of Complex Economic Dynamics
vol 1: Introduction to Dynamical Systems and Market Mechanisms; vol 2:
Introduction to Macroeconomic Dynamics (MIT Press, 1994 and 2001).
Domenico Delli Gatti is Economics Professor, Catholic University, Milan, Italy.

He is the associate editor of the JEBO and has published articles on financial
fragility, monetary economics, and non-linear dynamics. He has published a
book on Money, accunulation and Cycles (NIS, 1994) and more recently edited
a book on Interaction and Market Structure (Springer, 2000).
Peter Flaschel is Professor of Economics, Bielefeld University and Visiting

Professor of Economics, University of Technology, Sydney. His research
interests include disequilibrium, cycles and growth, open economy,
macrodynamics, microdynamics, macroeconometric model building, numerical
analysis of high order macrosystems. He is the author of several published
books and articles.
Jean-Luc Gaffard is Professor of Economics at the University of Nice-Sophia

Antipolis and senior member of the Institut Universitaire de France. His major
research interests are in Industrial Economics, Economics of Innovation, and
Dynamic Economics. He has published many papers in scientific journals and
several books including, The Innovative Choice. An Economic Analysis of the
Dynamics of Technology (with Mario Amendola) (Blackwell, 1988), and Out
of Equilibrium (with Mario Amendola) (Clarendon Press, 1998).
Mauro Gallegati is Professor of Economic Dynamics, University of Teramo,

Italy. He has written extensively on financial fragility, business cycle
fluctuations, and non-linear dynamics. He has published a book on Fluctuations
in Italy (Giappichelli, 1998) and has edited two books, Beyond the RA (Elgar,
1999) and Interaction and Market Structure (Springer, 2000).
Frank Hahn is Professor Emeritus at the University of Cambridge, Professore

Ordinario at the University of Siena and Fellow of Churchill College. He is the
author of numerous papers and has published a number of books.
Contributors xiii
Tad Hogg is a member of the research staff at the Xerox Palo Alto Research
Center. His current research includes developing institutional mechanisms
promoting privacy and electronic commerce and investigating the dynamics of
computational ecosystems consisting of processes interacting with an
unpredictable environment.
Bernardo A. Huberman is the Scientific Director of the Hewlett Packard Sand

Hill Laboratories, a research centre recently established by Hewlett Packard to
explore a number of novel issues in the research of distributed systems and its
organizational setting. He is co-winner of the 1990 CECOIA prize in Economics
and Artificial Intelligence and he recently shared the IBM Prize of the Society
for Computational Economics. Dr. Huberman is one of the creators of the field
of ecology of computation, editor of a book on the subject, and an expert on
the dynamics and growth of the internet.
Katsuhito Iwai is Professor of Economics at the University of Tokyo. His works

include: Disequilibrium Dynamics, (Yale University Press, 1981), ‘The
Bootstrap Theory of Money,’ Structural Change & Economic Dynamics, 7,
(4), 1996, and ‘Persons, Things and Corporations,’ American J. of Comparative
Law, 47, (4), 1999.
Oleg V. Pavlov is a postdoctoral fellow at the Boston University of School of

Management. His research concentrated on complex economic dynamics and
computer simulation in such fields as population economics, economic
development and technological evolution. His current research and teaching
focuses on economics of information technology.
Edward C. Prescott is Regents’ Professor at the University of Minnesota and an

advisor to the Federal Reserve Bank of Minneapolis. His current research is
concerned with economic growth and development, which is synthesized and
extended in Parente’s and his monograph Barriers to Riches (MIT Press, 2000).
Lionello F. Punzo is Professor of Economics at Siena University. His current

interests are in the theory and empirics of growth and complex dynamics. Among
his works are The Dynamics of a Capitalist Economy (co-authored with Richard
M. Goodwin) (Polity Press and Westview Press, 1987), Economic Performance
(edited with B. Böhm), (Physika Verlag, 1992), European Economies in
Transition (edited with O. Fabel and F. Farina), (Macmillan, 2000) and Mexico
Beyond NAFTA (edited with M. Puchet), (Routledge, 2001).
Jorma Rissanen is a member of the research staff at IBM Almaden Research

Center. He is Visiting Professor at Royal Holloway, University of London and
Professor Emeritus in the Technical University of Tampere, Finland. His main
research interest is in creating a theory of statistical modeling, as outlined in
his contribution to this book.
xiv Contributors
Solomos Solomou is a Senior Lecturer in the Faculty of Economics at the
University of Cambridge, and a Fellow of Peterhouse. His work includes Phases
of Economic Growth, 1850–1973: Kondratieff Waves and Kuznets Swings
(Cambridge University Press, 1987) and Economic Cycles (Manchester
University Press, 1998).
Hiroshi Yoshikawa is Professor of Economics at the University of Tokyo. His

research interests are in macroeconomics and the Japanese economy. He is the
author of Macroeconomics and the Japanese Economy (Oxford University
Press, 1995).
Michael Youssefmir Michael Youssefmir is Director of Product Management for

ArrayComm, Inc’s wireless Internet product line. Previously he worked in the
Internet Ecologies Group at the Xerox Palo Alto Research Center. He holds a
number of patents in the application of adaptive array technology to wireless
systems.
Preface xv
Preface
This volume presents contributions revolving around the general theme of the
relationship, empirical as well as theoretical, between economic oscillations,
growth and structural change. It is constructed upon a twofold hypothesis. A re-
unification of the analyses of the three phenomena may be obtained through a
comparison and cross fertilisation of the existing specialised approaches. Moreover,
the accomplishment of such a project is perhaps within reach, now. This preface
will discuss such a hypothesis, outlying the structure of the book and the string
connecting individual contributions.
Most chapters are revised versions of papers read at the International Summer
School of the same title, held in Siena in July 1998. Some chapters have been
added with the intent of broadening the view without however any attempt to
reach exhaustiveness. The field, in fact, represents one of the most exciting frontiers
in current economic research, calling for the individual as well as, more often,
joint efforts of specialists from very diverse areas: economic theorists, real life
applied economists, econometricians, of course, but also mathematicians and people
working in information-theory related areas. Cross fertilisation is at one of its
historical peaks, and it is taking place here more than elsewhere in the economic
discipline. The School was concluded by a lively workshop dedicated to the works
of the late Richard M. Goodwin, whose lifetime research has been dedicated to
the themes of this book.
Convergence: the new impetus to growth theory and growth

empirics
Among the factors that brought about the present rich and stimulating, though
still fluid, state of affairs, one should count the new impetus to growth theory and
growth empirics coming from the debate on ‘convergence’ (further supported by
recent quantitative advances in cross-national comparative databases). Indeed,
growth has been traditionally an area where a division of labour between theoretical
and applied economics (and economists) could not be enforced. Similarly, the
actual experience of fluctuations along with growth was the background for the
development of the classical macrodynamics (the set of theories of business
fluctuations of the 1930s to the 1950s) and more recently of new classical
xvi Preface
macrodynamics. So, the intersection between theoretical work and empirical
evidence has been remarkably extensive for fluctuations as well as for growth.
Perhaps it has been greater than in other parts of economics, and in a sense this
very aspect makes these theories so appealing to academic economists, but more
so to policy makers and the general public alike. The questions raised by the
opening of the new millennium, in a scenario of uncertainties about the future
developments of the world as well as the national economies, can only increase
such general interest. Does economics now offer answers to such grand questions
as: how to support long run, smooth (or relatively smooth) as well as even growth?
This is an open question, at present. The recent developments expose a varied
scenario, where high instability is coupled with equally highly, cross-country,
diversified growth performance.
Thus, there is an issue of growth and convergence, but we are also still looking
for a satisfactory answer to the age-old issue of explaining why growth seems to
marry with fluctuations. This is the coupling of trend and oscillation that focussed
the reflection of Schumpeter and inspired the more mathematically inclined thinkers
of classical macrodynamics. This is one of the issues addressed in the chapters of
this book. It is done with diverse approaches. They range from neo-Classical, to
neo-Austrian, to Schumpeterian and Evolutionist, and they vary between macro
and micro. The book shows as well a variety of styles or methodologies: from
historical reconstruction to quantitative studies (cluster analysis-type of approach
to real business cycle econometrics to information-theoretic model theory), to the
modelling of large artificial economies (often of heterogeneous interacting agents/
firms/sectors), and qualitative econometrics. Such twofold variety reflects the
philosophy of the series of International Summer Schools at Siena since their
operation.
The list of factors for the present situation is far from short. It should include
the unending dispute between exogenous and endogenous explanations of long
run growth and fluctuations; the discovery of the possibility, besides multiple
equilibria, of endogenous fluctuations and of chaotic, or non-periodic, behaviours
even in simplest theoretical models, constructed upon the conventional equilibrium
ingredients; finally, the formalization of a non-mainstream tradition, along
Evolutionary, neo-Schumpeterian, or else neo-Austrian lines, to name a few.
I believe all this can be traced back to the key issue anticipated above: the need
for a coordination of the analysis and explanation of short run economic variability
with that of growth as a long run phenomenon and of these two with the analysis
of structural change. It may be useful to spend a few words on how this issue
presents itself now against the background of the history of its conceptual evolution,
and to indicate the Goodwin connection.
Cycles, growth and structural change

The three ‘wings’ of cycles, growth and structural change became part of one and
the same conceptual design, in this century, synthesised by the Schumpeterian
vision of a market economy growing only through fluctuations. In the so-called
Preface xvii
Schumpeterian clock, economic time is irregularly struck by the swarms of
innovations, i.e. clusters of technological and economic shocks erratically but
systematically administered to the markets. This vision proposed the irregular
oscillation as the unifying concept for dynamics but this got lost in the process of
formalization by Hicks, Samuelson, Harrod, Tinbergen, Kalecki and others which
created modern mathematical dynamics. (It only survived in Goodwin’s lifetime
work.) Then, attention shifted towards more orderly dynamics, the existence and
stability of point-equilibria and of regular oscillations (limit cycles and the like).
The eventual pre-eminence of this quest went along with the separation of the two
issues, the explanation of growth from the explanation of the oscillations. In place
of the single exogenously initialised mechanism of fluctuating growth, a dualism
emerged between oscillations and growth, and in parallel the former was identified
with the short run and the latter with the long run. The products of such a separation
were the neo-classical theory of growth and the Hicks–Goodwin theory of self-
sustained oscillations. Such a separation took also the shape of a methodological
(or weltanschauung’s) opposition of an exogenous versus an endogenous
explanation of dynamics, the former associated with Solow’s growth model, and
the latter with the classical macrodynamics just mentioned (in particular,
Goodwin’s own version of it). There are many reasons for this divorce between
two research lines that otherwise both descend from the same Harrod’s model
(Solow inserting a production function, Goodwin going non-linear). One is the
shift in post-war theoretical interests towards equilibrium issues and hence the
related stability. But, probably, a more profound conceptual reason lies in the
historical failure of those who believed in the paradigm of the endogenously
sustained oscillations, to accomplish the self-assigned task of building the grand
theory of the trend cum oscillations. Perhaps, the task was simply overly ambitious
for those times and the mathematics at hand. More realistically, neo-classical growth
theory never played with such a ‘dream’, as acknowledged by Solow himself.
It is too well known how the exogenous theory of growth survived (though
with alternating phases) while at the same time the endogenous theory of
oscillations went out of fashion. From the 1960s till recently, the latter was
superseded by a simpler account of oscillations, sometimes referred to as the
linear econometric model. This rests upon the notion of a fundamentally stable
equilibrium behavior for the economy subjected to exogenous and random
disturbances. Thus, this new dynamic wisdom had both long and short run
explained by fundamental forces that were not modelled by the economist himself:
for growth they were determistic in nature, while they were stochastic for
oscillations, so that they did not exclude each other and on the contrary could be
combined through a sort of division of labour. Such a mechanical view of the
working of an economy producing short lived oscillations as a result of passive
behaviours on the part of economic agents was eventually questioned by both the
first and the second monetarist waves, the latter producing the new classical theory
of business fluctuations. But, although enriching the representation of decision
making processes by taking into account expectation-induced adjustments, the
dynamics of oscillations remained the response to exogenous shocks of an
xviii Preface
otherwise point-stable mechanism. In growth theory, on the other hand, Solow’s
paradigm was challenged by the blossoming of a body of models collectively
going under the name of endogenous models. They are endogenous in the sense
that they account for the possibility of a sustained long run growth on the basis of
economic forces associated with the very basic mechanisms driving a market
economy.
After the middle of the 1980s, (macro-)economists lived in a situation that can
be depicted as the mirror image of the one at the beginning of the 1950s: growth
theory had tendentially become endogenous, while business cycle theory
(Monetarist version) was exogenous! Beyond this interchange of points of view,
basically the two ‘varieties of dynamics’ were still being treated in different parts
of economics; the explanation of one would not be the explanation of the other.
The issue of conjugating trend and oscillation in a unified model was still there to
be addressed. RBC theory, which bravely picked it up, can be seen as a provocatory
but forceful attempt at realising such unification. Likewise, in the same light, one
can appraise the birth and diffusion of Evolutionary and Schumpeterian modelling,
neo-Austrian modelling, and some of the non-label literature on endogenous
fluctuations and chaos that has emerged from the mid-1980s to the present day.
One of the chapters in this volume goes over the issue to recall the RBC proposal
of a (re-)unification of growth and fluctuations dynamics centred upon the
equilibrium theory of growth in a stochastic environment. Other contributions
present alternative viewpoints, with an out-of-equilibrium approach yielding
naturally long run, persistent fluctuations in growth rates (as well as levels, if
needed). The complex dynamics thus obtained does not distinguish between
fluctuations and growth, the latter being but one simple type of the former. Their
proposal of re-unification centred upon a generalised notion of growth cycle in a
fundamentally deterministic framework can be contrasted to the essentially
stochastic equilibrium approach of RBC. Likewise, the chapter on the positiveness
of long-run profits in a evolutionary and Schumpeterian framework, shows such a
possibility (excluded by Schumpeter himself) to be strongly related with the issue
at stake.
In a sense, all the chapters in this book address the same question: how can we
have both sustained growth and sustained oscillations? If the answer is positive,
they should rest upon the same mechanism. Searching for the answer, some theories
and models redefine the very notion of growth and, more often, of oscillations to
accommodate frontier phenomena belonging to so-called complex dynamics. To
see where the common mechanism could be, one ingredient is missing, though.
Dynamics from structural change

In fact, parallel to the theoretical divorce of growth from fluctuations, there was a
separation of formal dynamics from structural change. Taking the economic
structure as given, dynamics became basically a theory of adjustment processes.
Structural dynamics, on the other hand, was left out of a style of modelling too
aggregative to afford a notion of structure richer than the description of some
Preface xix
statistical regularity. Thus, the analysis of structural change migrated to
development theory and it stayed there until recently. As remarked by some of the
contributors to a previous volume in this series, structural change eventually has
re-appeared under disguise in some of the models of the endogenous growth family
where the line between monotonic growth and discontinuous development is never
clearly marked. Even Lucas’ critique can be read as a critique of a modelling
approach that assumed structure and kept it outside theoretical consideration,
treating it as exogenous in the diminutive sense that Hahn points out in this volume.
As a result of the recent historical developments, structural change has re-appeared
on the agenda of the dynamic theorist and empiricist of the First World economies,
too.
The mechanism that may provide a unique explanation of self-sustained
dynamics lies inside the structure of the economy. This hypothesis was present
more than intuitively to the minds of the believers of the endogenous program
when forging classical macrodynamics. This was naturally so. More recent
theoretical developments have forced us to reconsider our understanding of what
makes up an economic structure (to include expectations, decision rules, institutions
and rules of game, etc.), but all the recent dynamic literature accepts the principle
that it is the endogenous structure of the economy, a set of broadly defined
interacting ‘fundamentals’ that determines the dynamics we observe. The
increasing wealth of interesting results obtained in this area springs exactly from
the re-consideration of the endogenous inner network that makes up an economy.
The re-consideration of the relationship between dynamics and structure follows
different lines, some of which are represented in this volume. If we use modelling
styles as classification criterion, we can focus upon the architecture of such models.
In fact, if we think of models as computing devices, then their formal structures
are architectures just like in hardware structures. Hence, we may compare models
built upon a serial with models built upon a parallel architecture. In this light, the
various lines can be grouped into only two views, reflecting distinct modelling
strategies but also different images of the economic world. In one of them, the
economy is seen as a large-scale replica of a ‘standardized’ economic agent, which
is a complex entity by itself and therefore encapsulates whatever is complex in
the economic process. Hence, either the agent is a metaphor for the system, or
else, the latter is a metaphor of the former. This is a serial view, in that the model
of the whole economy is a scaled-up copy of the model of the economic agent (or
vice versa), interaction between agents adding but complications to the basic
mechanism. The alternative, parallel view depicts the economic system as
fundamentally a web of interacting parts (e.g. typically, agents but also institutions,
markets, rules etc.). It is their interaction that determines economic dynamics and
whatever complex features it may exhibit in time and space. The corresponding
strategy simplifies the description of the small component unit, and simple agents/
markets/institutions are made to interact to produce complex dynamic outcome.
These two views have always been simultaneously present in economics. Their
difference becomes relevant as soon as we leave the safe nest of the equilibrium-
only analysis and extend our walk to the frontiers of out-of-equilibrium dynamics.
xx Preface
The dispute between the two views was never settled in favour of one or the
other and this should be no surprise. (The same situation exists in the computing
industry too, with the conflict between supercomputers and the philosophy of the
web.) So far, neither modelling philosophy, too often taken to reflect pre-theoretical
or ideological views, has proved superior. This book puts together examples: the
macroeconomic chapters in Part II reflect mostly the former viewpoint, chapters
in Part III the latter.
The conception of the economies as decentralised parallel structures of decision-
making units has been traditionally considered the gateway to the dynamics of
instability and fluctuations, in general what is now called complex dynamics. The
classical theories of the business cycle as a disequilibrium phenomenon are children
of this very idea, crystallised in R. Frisch’s notions of structure and macrodynamics.
An anticipatory analysis can be found in the contribution of Richard M. Goodwin
on the dynamical coupling between sectors, where the mere phase interlocking
between otherwise regular parallel oscillators is shown to be able to generate
many sorts of complicated sectoral and aggregate time series. It may be useful to
compare it with modern exercises on endogenous fluctuations and chaos where,
instead, complex dynamics is basically the result of a non-linear model of a single-
sector macroeconomy (thus, in the sense above, they show a serial view). But
there is now also a vast literature on dynamics from interacting heterogeneous
components, e.g. fundamentally economic agents, but also markets and sectoral
growth cycles. The contributions appearing in this volume can be seen as a natural
extension of the classical approach. Such models also try to cope with the issue of
co-ordinating the disaggregated framework with aggregate outcomes, re-uniting
micro and multisectoral with macroeconomic analyses. Does interaction per se
generate growth, as well as, or as much as, fluctuations? Some of the contributions
presented hereafter take up this point.
Parts I and II
The interaction between economic fluctuation, growth and structural change is
the main thrust in Part I which is dedicated to a survey of facts and interpretations.
The strength of such an interaction is the fundamental motivation for Solomou’s
rejection of the ‘methodology of stylised facts’ searching for repetitive patterns
and ‘dynamical laws’ of one sort or another. In an essay rich in both historical
reconstruction and methodological reflections, the author highlights the
complications involved: the varying patterns of causality/interdependence relations
between the key factors of shorter term fluctuations, and the interlocking of
oscillations at various frequencies with structural change. He points out that,
contrary to common beliefs, the post-war experience of growth in the developed
countries is fundamentally a unique string in a long time series of events. Thus,
the author promotes a historical perspective as an alternative the methodology of
stylised facts which yields anyway a rather unstable and thus interpretation-
unreliable set.
Yoshikawa focuses upon the exemplary intermingling of growth and structural
change in the post-war history of Japan. He criticizes in particular various existing
Preface xxi
interpretations of the rapid growth era, to propose a two-phase model whereby an
early (1950–60) modernisation process of a Lewis-type dualistic economy is
followed by a dramatic break and re-orientation. The change in household structure
and urbanisation, driving labour force away from traditional agricultural
employment and creating new urban demand, generated the unprecedented
expansion of the earlier phase. This yielded, at the end of the era, to market
saturation creating the need for a new, externally driven expansion. Domestic
demand was substituted by foreign demand as an engine of growth. The chapter
contains also interesting remarks on the prospects of NICS of various generations
and on the experience of countries like Italy sharing with Japan a dualistic structure.
In the chapter by Böhm and Punzo a new framework is introduced where growth
is naturally associated with fluctuations and structural change is defined as a
qualitative change in the growth model, thus spanning a dynamic menu that is
richer than expected. The chapter reviews in this light empirical findings for a set
of European countries, the USA and Japan. It is shown that these went through
repeated structural changes, but also that growth models were different across
countries and were strung differently in each country’s history. Observed high
irregularity and cross-country variability reflect shock responses, of course, but
more deeply they reveal the workings of the countries’ own structures. Looking
for changes in the growth model is one way of capturing structural change, and it
seems to be the most natural one in the formalised setting in the dynamical systems
style: it is a regime shift. The notion of regime dynamics translates naturally into
that of Day’s multi-phase dynamics, whose application to very long run growth is
well illustrated in the chapter jointly written with Pavlov (opening Part II). Their
Generalised Evolutionary Model (GEM) focuses upon the evolution of economic
and social macrostructures through phases of growth following sometimes an
irregular, by no means determined sequence, and it accounts for such an evolution
through the working of an internal instability mechanim. The GEM is calibrated
to reproduce (in the simulation sense) what is known of the history of mankind,
an exercise which illustrates an approach to qualitative analysis aptly termed
qualitative econometrics.
In fact, the output of a dynamic model exhibits scenarios, artificial histories in
other words. In general, what matters in these simulation exercises cannot really
be the quantitative coherence or closeness of the artificial to the actual time series,
so that debate goes on about their uses. One proposal comes from the real business
cycle approach, illustrated in the chapter by Prescott, where some quantitative
criterion is retained to assess the model at hand against evidence. As an alternative,
following Day and Pavlov, artificial histories can be treated as qualitative
descriptions. Their worth is in their capability of reproducing the shifting across
phases (or regimes) following a given known pattern. Thus, the model may explain
to a certain extent features of actual macroeconomic behaviour; adding some
extrapolation exercises, we get to qualitative econometrics.
Clearly, an economy whose history exhibits repeated regime or phase switches
can hardly be described by an equilibrium technique. The analysis of scenarios of
out-of-equilibrium dynamics is the centre of the neo-Austrian approach proposed
by Amendola and Gaffard, highlighting the conditions for the emergence of co-
xxii Preface
ordination problems in the decision processes as well as within the production
processes. Thus, stability towards a given state or attractor (that is the core of the
conventional dynamical modelling) is no longer the key issue and, actually, it
turns out to be not even definable as a property, given the continually shifting
nature of the system’s theoretical equilibrium. Instead, it is the issue of the
sustainability of the dynamic process itself, i.e. the viability of the growth path,
that comes to the fore redefining the very contours of economic policy.
The two chapters concluding Part II review and contribute novel insights into
the out-of-equilibrium approach. Flaschel’s contribution offers a disequilibrium
framework where a monetary economy works according to rules and laws set out
by a composite model built upon the seminal contributions of Keynes, Marx and
Goodwin. This economy’s dynamic outcome shows a strong bias towards
fluctuation rather than steady growth, Flaschel building upon the alternative
tradition that looks at the market economy through the pessimistic eyes of the
classical thinker.
Iwai tackles the time-honoured issue of demonstrating the possibility of
reconciling the notion of long run with the notion of disequilibrium. The test
ground for his effort is a classical issue, the possibility of a long run state where
profits are above their normal levels. The effort proves successful with the re-
definition in statistical terms of the very notion of long run equilibrium, which
now allows for a cross-firm distribution of technologies, as it results from the
continuously on-going process of innovation. Baumol’s contribution, on the other
hand, looks at the long run growth aspects of that same process and offers an in-
depth analysis of some of its microeconomic features. He explains why the evidence
of unprecedented growth record of the market economy contradicts the poor
performance predicted by traditional growth theory. The key argument is centred
on the trade-off between flows of inventions and distribution of benefits, and
therefore on the novel observation that positive spillovers (linked with non-
appropriability) do not necessarily impede the innovation process. As, on the
contrary, a certain amount of spillovers does fuel performance, the author gives a
positive answer to the question previously raised, whether interaction can generate
growth and at the same time fluctuations. This vindicates Schumpeter’s (and
Goodwin’s) viewpoint.
Parts III and IV

Three contributions in Part III, by Masanao Aoki, Hogg et al., and Delli Gatti and
Gallegati, share the common theme that dynamics is produced by interaction between
heterogeneous agents. The first of them shows how the property of an interacting
agents economy of exhibiting multiple equilibria states (with different stability
properties) is essential to generate asymmetric fluctuations, but it also permits the
analysis of equilibrium selection ‘in the limit’, i.e. when the size of the economy,
defined by the number of agents, tends to infinite. Variability and fluctuations are
obtained by Hogg et al. in an analytical context that recalls a classical theme in the
theory of interacting markets. The obvious reference is to the studies on stability of
Preface xxiii
general equilibrium, but also to the already mentioned paper by R.M. Goodwin on
dynamical coupling. It is shown in an original manner that stability depends upon
the degree of interdependence among markets, and the novel result lies in that such
stability decreases with increasing dynamic interaction that results from economic
agents’ learning and broadening the environment in which they operate. Therefore
greater integration, even in presence of locally weak connections, can only increase
the likelihood of the fragility of the whole system. Recent turmoil on some key
financial markets gives the appropriate background to appreciate the relevance of
the conclusion or its ‘realism’.
The dynamics driven by the actions of heterogeneous agents provides the topic
of the contribution by Delli Gatti and Gallegati, which reviews a burgeoning
literature on the structure of financial markets and its link with fluctuations. Here,
the focus is on the heterogeneity among agents, a clear departure from the shortcut
provided by the representative agent hypothesis, as well as on its effects on the
outcomes predicted by a variety of macroeconomic models. Of course, given the
location in the present book (and the inclinations of the authors), this review exhibits
to a line of research where, instead of stabilising the economy towards a unique
equilibrium, agents’ heterogeneity generates complicated dynamics. Fluctuations
can result from the distribution of the agents’ characteristics acting as a powerful
oscillatory propagation mechanism in the presence of even the smallest shock.
One more door is thus opened to an endogenous explanation of observed
fluctuations (the literature on endogenous oscillations, sunspots, complex dynamics
and chaos having opened the first one). Putting all these contributions together,
one seems to ‘sense’ that a new generation of models is now around where growth,
fluctuations and structural change are simultaneously explained by the same
theoretical engine, the structure of the economy.
Part IV moves onto more methodological issues. The opposition made by
Prescott between the inductive approach that would be appropriate only to the
natural sciences with the deductive one of economics (and perhaps other social
sciences) meets the dual position of Rissanen questioning the foundations and
practice of statistical modelling. The two authors, however, address different issues.
Rissanen’s position, which rests upon the notion of useful (and learnable)
information, was not too long ago reflected by Sims, whose interpretation of the
work of science in general as a search for laws as compressed data strings, shares
the philosophy of Rissanen’s Minimum Description Principle and in general of
information theories in the grand tradition of Shannon and Kolmogorov. With a
highly technical argument, which is however made sufficiently palatable to students
versed in modern statistical techniques, Rissanen highlights the relevance of the
notions of stochastic complexity for the analysis of given data strings and the
recovery of candidate generating models. (Data strings are often time series to the
economist, typically so in studies of dynamics.) Hence comes the link with
Prescott’s position where simulation output or predictions of the theory are to be
compared (in a special way) with actual data. Prescott reviews this RBC
methodology and indicates key issues still seeking satisfactory treatment, so that
his chapter ends with a useful and stimulating research agenda.
xxiv Preface
Brock’s chapter critically reviews the literature at the interface between theories
of complex economic dynamics, which are typically non-stochastic, and the
statistics of time series, attempting an assessment of gains from their cross-
fertilisation, a project he himself has contributed to significantly. The impression
is that the record is rather mixed, and that research is still going on (and, on big
themes like this, it takes a long time to obtain new far-reaching results). This also
indicates the nature and the scope of the theoretical research on complexity in
economic dynamics. For its non-stochastic approach and its insistence on
endogenous explanations, the latter has been taken to represent the alternative to
the dominating paradigm, whereby dynamics is explained by fundamentals and
off-equilibrium irregular paths result from stochastic disturbances. On the whole,
the sober message in Brock’s contribution is that a less partisan attitude is still the
best the current researcher can entertain. One can connect this message with the
main point in Hahn’s contribution, that a lot of what was assumed to be endogenous
was in fact exogenous, but a lot of the exogenous was assumed to be endogenous,
too.
Sharp methodological demarcations as these may be useful to label theories
and schools, but all theories should be understood first of all as experiments of a
mental kind. Hence, the endogenous/deterministic explanation of fluctuations
should be taken to show only how far one could go without calling in external
forces and resorting to shocks. The same can be said of the distinction between
exogenous and endogenous factors. Their survival, and recent revival, show the
theoretical strength and appeal of such demarcations. This also reminds us that
the frontiers of our economic knowledge have not yet been reached.
Lionello F. Punzo
Acknowledgement
I wish to thank for their comments (with the usual caveats) J.-L. Gaffard and
M. Puchet Anyul. Research for this preface and work in the final stages in the
production of this book were carried on as part of a project on ‘Recent developments
in qualitative analysis and object-oriented simulation in economic dynamics’,
financed by the Italian MURST as Progetto di ricerca scientifica di rilevante
interesse nazionale, 2000.
References
Goodwin R.M. (1982), Essays in Economic Dynamics, London: Macmillan
Goodwin R.M. (1983), Linear Economic Structures, London: Macmillan
Goodwin, R.M. (1990), Chaotic Economic Dynamics, Oxford: Oxford University Press
Goodwin R.M. and L.F. Punzo (1987), The Dynamics of a Capitalist Economy, Cambridge:
Polity Press, and Boulding, CO: Westview
Coricelli F., M. Di Matteo and F.H. Hahn (eds) (1998), New theories in growth and
development, Basingstoke; New York: Macmillan
Part I
Facts and interpretations of

growth and fluctuations
1 Economic cycles since 1870
Solomos Solomou
1.1 Introduction
Current research on business cycles has focused on the post-war period. The aim
has been to derive the main ‘stylised facts’ of fluctuations. The approach here is
very different. The emphasis is on the need for an historical perspective to business
cycles. History inevitably provides us with a broader empirical basis that will allow
us to formulate more general theoretical and empirical questions. It is argued that to
focus on explaining the stylised facts of any one historical period to gain a general
insight on economic fluctuations will lead to serious errors of interpretation.
An historical perspective is important not because history repeats itself but
because history illustrates the evolutionary nature of business cycle behaviour and
gives us an understanding of the factors that generate change. In economic systems
that entail behavioural, institutional, structural and policy changes we can safely
predict that the nature of business cycles will not be stable over time. The case
for an historical perspective is founded on two important empirical features. First,
the process of economic growth of modern economies has undergone massive
structural change in the past two hundred years. One important aspect of struc-
tural change is the reallocation of production across sectors. For example, with
industrialisation we would expect a change in the role of agricultural cycles in
macroeconomic fluctuations. Given that these type of changes take place over a
very low frequency, we need to observe business cycles over the long-term to gain
insights on the resulting cyclical effects of structural change.
Second, policy regime changes have an observable effect on cyclical fluctua-
tions. Business cycles during the rules-driven policy framework of the pre-1913
gold standard epoch were very different to those of the inter-war period. Similarly,
the fluctuations of the Bretton Woods era were very different to those observed in
the post-1973 era. Major policy regime changes have been few in number, once
again, implying the need for a long-run perspective.
These observations raise serious doubts about the empirical relevance of the
so-called ‘stylised facts’ approach to business cycles. This approach assumes that
regularities exist over time and across countries (Lucas, 1981). However, many
of these empirical regularities are usually derived from studies of very short time
4 Solomos Solomou
periods (often the post-1960 era) and a small selection of countries (usually the
UK and the USA).
Even if we focus on a very limited set of empirical features we observe im-
portant changes. The average cyclical duration has changed over time. During
1870–1913 a number of variables (including aggregate investment, agricultural
production and construction sector output) fluctuated with a long swing duration
averaging about 20 years. During the inter-war period shorter fluctuations were
observed. During the post-war ‘golden age’ the average cyclical period fell to 5
years. During the post-1973 era the average duration has once again lengthened,
averaging approximately 10 years since the late 1970s. Cyclical amplitudes have
also varied significantly over time. Low macroeconomic volatility during 1870–
1913 gave way to high amplitude fluctuations during 1919–38; the stability of
the post-war ‘golden age’ has been followed by the relatively more volatile post-
1973 era. Patterns of co-movement of key variables have not been stable over time.
Much of post-war research on business cycles has noted that prices and output have
fluctuated contra-cyclically and has proceeded to explain this feature in terms of
‘real’ business cycle theory (Danthine and Donaldson, 1993). However, during
the classical gold standard the relationship was not stable and during the inter-
war period price and output fluctuations moved in a pro-cyclical manner (Cooley
and Ohanian, 1991). Assuming universal stylised facts is not a realistic way of
understanding business cycles.
This chapter surveys business cycle features across three historical epochs to
illustrate some of the themes discussed in this introduction. Section 1 considers the
pre-1913 epoch. Section 2 considers the changes observed in the inter-war period.
Section 3 considers the post-war epoch.
1.2 1870 – 1913

The main institutional aspect of the period 1870–1913 is the gold standard. Be-
tween 1879–1913 Britain, France, Germany and America pegged their currencies
to gold at a fixed rate. The relationship between this institutional-policy framework
and economic fluctuations needs careful consideration. In particular, it is important
to evaluate whether the rules-based policy framework modulated aspects such as
the amplitude of fluctuations. The main structural aspect of the pre-1913 period is
the large size of the primary sectors (such as agriculture and mining), as a percent-
age of the labour force and gross domestic product. Such a structure implies that
supply-side shocks (such as weather shocks) are likely to have significant effects
on sectoral and macroeconomic fluctuations. In terms of the world economy, the
period is one of integration in trade, capital and labour flows between the industrial
countries of Europe and the primary producing economies of the world. Such in-
ternational linkages were important to determining the adjustment path to shocks
in the industrial economies.
Economic cycles since 1870 5
1.2.1 Describing cycles
Three different cycles have been identified by economic historians of the period:
the Juglar trade cycle, the Kuznets swing and the Kondratieff wave. The Juglar
cycle has an average period of seven to nine years. A number of studies have argued
that this is the dominant cycle over this period (Aldcroft and Fearon, 1972; Lewis,
1978; Crafts, et al., 1989). For example, industrial production in the UK fluctuated
with peaks in 1873, 1882, 1889, 1899, 1907 and 1913. The average peak to peak
cycle is eight years. However, a number of data and conceptual problems should
be noted before we accept this perspective to pre-1913 cycles. We should note
a number of serious limitations in the historical industrial production series. For
example the series for Britain (produced by Arthur Lewis) has been constructed
using indicator variables and methods of extrapolation and interpolation. For a large
proportion of the industrial production index Lewis has actually imposed a cycle
of nine years on a priori grounds (Lewis, 1978; Solomou, 1994). A Juglar cycle is
imposed on iron and steel products, commercial building, clothing, printing and
chemicals, which account for over 28 per cent of Lewis’ total industrial production
index and 35 per cent of the manufacturing and construction index. Thus, the Lewis
industrial production index provides only limited independent information for the
importance of a Juglar cycle during this period. If we only consider the path of
those industries where the cycle is not imposed a priori, the evidence for a dominant
Juglar cycle is very weak. The construction sector sees long-term fluctuations of
twenty years (Thomas, 1973). Coal production showed variations in trend without
any discernible regular short cycle (Catao and Solomou, 1993). Investment was
dominated by a long cycle of 23 years (Cairncross, 1953). Agricultural production
was dominated by a long cycle of 20 years (Solomou, 1994). This evidence gives
some support to Hicks (1982), who noted that during 1875–1914 business cycles
became far more irregular in duration.
The Kuznets swing refers to a variation in economic growth that is longer than
the Juglar trade cycle. The swings observed are variations either in levels or in rates
of growth. The actual length of the swings found varies with different studies, but
something between 14 to 22 years is representative. This type of fluctuation is most
clear in describing the path of capital formation in the leading economies. British
andAmerican investment fluctuated along a 20–year cycle in the level of investment
whilst French and German investment fluctuated along a long cycle in the rate of
growth (Solomou, 1987). Irregular Kuznets swings can also be observed in the
level and growth of agricultural production, construction output, migration, the
sectoral terms of trade and trade balances (Solomou, 1987; Lewis, 1978; Thomas,
1973; Cairncross, 1953; Rowthorn and Solomou, 1991).
The Kondratieff wave is a cycle of prices and output with an approximate
period of fifty to sixty years. In some of the earlier literature the Kondratieff
wave has been used as a framework for understanding epochs such as the ‘Great
Depression’ of 1873–96 and the Belle-epoch inflation of 1899–1913 (Kondratieff,
1935; Schumpeter, 1939). The empirical evidence suggests that we need to tread
with care in the evaluation of Kondratieff waves. Most macroeconomic variables
6 Solomos Solomou
(such as GDP, investment and industrial production) have not followed a long-
wave growth pattern (Solomou, 1987). However, long cycle adjustments in prices
have been noted, particularly over the period 1873–1913 (Lewis, 1978; Rostow
and Kennedy, 1979).
Given the historical discussion of multiple cycles it is useful to describe the rel-
evance of this perspective using modern time-series methods. Solomou (1998) con-
siders the empirical relevance of multiple cycle models by employing the Kalman
filter to describe cycles in GDP for Britain, France, Germany and America. As
can be observed in Figure 1.1a–1.1d the cyclical path of all these economies is
depicted as the sum of short and long cycles. Although we have only presented
the decomposition for GDP, multiple cycles are relevant at different levels of ag-
gregation. This perspective provides an alternative to Hicks’ (1982) description of
irregular fluctuations. One aspect of irregularity is that the epoch was influenced
by cycles of different average periods. Accepting the idea of multiple cycles as
reality suggests one way of capturing some of the observed irregularity.
1.2.2 Rules-driven policy framework and business cycles

The amplitude of business cycles was significantly lower during the gold standard
period relative to the inter-war era (Sheffrin, 1988). The literature has attributed this
to the rules-driven policy framework of the gold standard (Crafts and Mills, 1992).
Such an idea offers only a partial explanation of this phenomenon. To understand
the relatively low macroeconomic volatility of the period we need to consider
the cyclical adjustment mechanisms operating over this period. Three adjustment
mechanisms were of central importance. First, international labour mobility was
exceptionally high. Second, capital mobility from Britain, France and Germany
to the newly industrialising and primary producing economies created liquidity
in the international economy. Third, capital flows sustained aggregate demand
(by stimulating exports) during episodes of downswing in the capital exporting
countries.
These features resulted in stabilising interactions between the domestic econ-
omy and the international economy. For example, depressed conditions in the
national economy gave rise to a cyclical propagation mechanism that resulted in
an adjustment of the labour market via overseas migration. International migra-
tion played an important role in accounting for cycles over this period (Thomas,
1973). The fall in home investment was also correlated with a rise of overseas
investment and the income transfer overseas effected demand for the exports of
industrial countries. Thus, compensatory equilibrating mechanisms were operat-
ing in the pre-1913 era. A rules-driven policy regime by itself was not sufficient
to generate these cyclical features. It is the combination of historically unique
circumstances with the gold standard policy framework that provided stabilising
adjustment paths. Bayoumi and Eichengreen (1996) provide empirical evidence
to support this conclusion: they find that the relative stability of the gold standard
period cannot be explained by an absence of destabilising shocks; instead they
show that there existed rapid adjustment to disturbances.
0.08
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Figure 1.1a Kalman filter decomposition of UK GDP cycles, 1870–1913
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Figure 1.1b Kalman filter decomposition of French GDP cycles, 1870–1913

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Figure 1.1c Kalman filter decomposition of German GDP cycles, 1870–1913
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Figure 1.1d Kalman filter decomposition of US GDP cycles, 1870–1913

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Figure 1.2 Nominal effective exchange rates
Although the classical gold standard period is often thought of as a prime

example of a rules-based policy framework, the non-universality of the system
introduced significant nominal shocks. During the period 1879–1914, whilst all of
the core industrial countries sustained a fixed exchange rate, the gold standard was
not a global fixed exchange rate system. Alternative exchange rate regimes, such
as the silver standard and paper currencies, prevailed in most primary producing
countries – some of which witnessed considerable exchange rate variability (Ford,
1962; Nugent, 1973; Eichengreen, 1989; Bordo and Rockoff, 1996). Industrial and
primary producing economies were not self-contained economic blocs. A large
proportion of international trade during this period consisted of the exchange of
industrial goods mostly produced in ‘core’ countries, for food and raw materials
produced in ‘periphery’economies outside the gold standard. Given these linkages,
one would expect significant fluctuations in nominal effective exchange rates to
have taken place in the ‘core’ countries.
Solomou and Catao (1998) calculate nominal and real effective exchange rates
during this period to illustrate this effects. The nominal effective exchange rates of
the four core economies are presented in Figure 1.2. The movements of nominal
effective exchange rates in the core countries suggest the existence of stochastic
trends, which would not be expected in a fully-fledged international gold standard.
The magnitudes of the movements appear small when compared to the volatility
of exchange rates during the inter-war period and the post-1973 era, but do stand
out in a period when nominal variables were rather stable.1 During the 1880s all
four countries witnessed relatively low nominal effective exchange rates, relative
to the whole period; this episode was followed by a phase of appreciation in the
1890s; the period from the late 1890s and early 1900s was one of depreciation.
Thus, it is misleading to think of the era as one of rules giving rise to an absence
of nominal shocks.
10 Solomos Solomou
1.2.3 International economic cycles?
One aspect of business cycles over this period that has been discussed in the recent
literature is that the timing of fluctuations differed significantly across the major
industrial countries (Backus and Kehoe, 1992; Eichengreen, 1994). At face value
this suggests that national-specific shocks were more important than international
business cycle transmission mechanisms. Table 1.1 reports the results of Backus
and Kehoe (1992) which show that the only statistically significant correlation
between different national GDP cycles is that between Germany and the USA, and
that is small in magnitude.
The robustness and meaning of this result needs to be considered in the context
of the methodologies used to derive it. The result is dependent on the kind of filter
being used to decompose and describe economic cycles. Backus and Kehoe use the
Hodrick–Prescott (H–P) filter which can be criticised for generating artefact cycles
from the data (Harvey and Jaeger, 1993; Cogley and Nason, 1995). Moreover, the
H–P filter focuses on deriving one short cycle from the data when, in fact, the pre-
1913 data verify the existence of a number of cycles. The relevant question that
needs to be addressed is, did countries share common cyclical paths at particular
cyclical frequencies. The degree of integration of the international economy during
the gold standard era was reflected in the existence of the type of international
adjustments we have discussed above. Depressed investment and consumption
opportunities in industrial countries were reflected in rising overseas investment,
migration flows and a stimulus of the export sectors. This is an era when the long
swing cyclical process is so pervasive across countries that it does not make sense
to analyse business cycles as short fluctuations in GDP. The evidence reported
above suggests that a multiplicity of cycles was the norm in the pre-1913 epoch.
There is enough evidence to suggest that we can reject the idea that pre-1913 cycles
were national-specific. The type of adjustments observed during the gold standard
period resulted in significant interactions across countries. Most of these were
observed over the low frequency fluctuations in migration, international capital
flows and trade flows.
1.2.4 Weather shocks and business cycles

The economic structure of pre-1913 economies suggests that significant fluctuation
is arising from a coupling of the economic system with weather fluctuations as a
source of supply-side shocks. The recent climatology literature has documented the
existence of cyclical behaviour in weather variables such as temperature, rainfall
(Lamb, 1977, 1982) and soil moisture deficits (Wigley and Atkinson, 1977). Such
‘cyclical’ and random weather variations affect the cyclical path of production in
weather sensitive sectors, such as agriculture, construction and energy demand
(Khatri et al., 1998; Solomou and Wu, 1999). What is the aggregate effect of such
weather cycles? The accepted view of economic historians is that adverse weather
ceased to be a significant shock to industrial economies. The formulation and
dismissal of simplistic theories about weather and business cycles have also held
Table 1.1 International output correlation: pre-1914

UK USA Germany Japan
UK –
USA 0.01 –
(0.14)
Germany 0.03 –0.40 –
(0.12) (0.13)*
Japan 0.08 0.22 0.14
(0.17) (0.15) (0.17) –
The entries show the contemporaneous correlation of cyclical variations of output. Numbers in
parenthesis are standard errors: only the German–American correlation is significantly different from
zero. * denotes significant at the 5 per cent level.
Source: Backus and Kehoe, 1992, p. 876.
back research in this area. For example, Jevons (1884, p.235) argued for a sunspot
theory of the trade cycle. In general, modern economists dismiss such theories as
unconvincing and misleading, although they continue to find favour in some of the
literature (Zahorchak, 1983).
The impact of weather shocks will vary from sector to sector as the conditions
favouring one activity may be adverse to another. The extent of such dampening
will also depend on changes in the sectoral structure of the economy over time. The
largest weather-sensitive sectors during this period were agriculture, construction
and energy demand. In Britain, these three sectors accounted for approximately 25
per cent of GDP in the 1870s, falling to 17 per cent of GDP in the early twentieth
century. In Germany and France these ratios were approximately 50 per cent of
GDP in 1870, falling to 30–40 per cent by the early twentieth Century.
Solomou and Wu (2000) show that weather fluctuations had large effects on
all these sectors. Using data from Britain, the effect range of weather shocks on
the growth rate of construction and coal sectors was around ±0.05 during the
period 1870–1913. A similar range is observed for the agricultural sector during
1880–1913, although this was significantly wider during the 1870s. The aggregate
effect of these sector-specific effects depends on two features: first, the pattern of
covariance of effects across different sectors and second, the relative weight of the
different sectors in the macroeconomy and changes in these shares over time.
Consider the pattern of correlation of weather effects on different sectors (Fig-
ure 1.3 plots sector-specific weather effects). A significant positive correlation is
observed between agriculture and construction sector effects (r=0.51)2 . There is
no evidence that weather shocks were having a neutral effect on the macroecon-
omy because adverse shocks to one sector were consistently being compensated
by favourable effects to other sectors.
In order to evaluate the sum of all these sectoral effects the magnitude of
the sectoral shocks are aggregated using GDP shares as weights. Figure 1.4 plots
the shares of the three sectors in GDP. The agricultural sector is dominant in
determining the weather effect on the economy. This arises because of the relative
12 Solomos Solomou
0.100
0.050
0.000
72
74
76
78
80
82
84
86
88
90
92
94
96
98
00
02
04
06
08
10
12
18
18
18
18
18
18
18
18
18
18
18
18
18
18
19
19
19
19
19
19
19
-0.050
-0.100
-0.150
coal construction agriculture
Figure 1.3 Sectoral weather effects
size of the agricultural sector over this period, a pattern of positive co-movement
between the weather effects affecting agriculture and construction and the wider
effect range of weather shocks to agriculture. Figure 1.5 illustrates this result with
a plot of the aggregate weighted weather effect and the weighted sectoral effects.
It is clear that the total weather effect on the economy is dominated by the effect
from agriculture and the co-movement with the weather effect on construction.
The effect from coal simply adds some randomness to the magnitude of the total
effect.
Aggregating sectoral weather effects using GDP shares as weights, suggests
that the impact of weather shocks remained large throughout this period. The effect
range of the sum of the sectoral shocks to GDP ranged from +0. 7 per cent of GDP
to −1. 5 per cent of GDP. These estimates represent the lower bounds given the
economic structure of other European countries compared to Britain. Moreover,
because weather shocks are autocorrelated, we can safely conclude that weather
added significant direct cyclicality to the economy. To dismiss the role of weather
(even as late as the post-1870 period) in business cycles is ahistorical.
Summarising, economic cycles during the pre-1913 period are best depicted
by multiple cycles. The causal structure behind such a process is likely to be
complex. A number of adjustment processes were made possible by the nature of
the world economy during this period. Free capital and labour movements across
national boundaries resulted in long cycles in both these variables. The economic
structure of the period also resulted in a coupling of weather cycles and sectoral
fluctuations. At the macro level the relationship was not a simple coupling of one
cycle with another. Instead, structural change acted as an important filter. Such
structural change meant that agricultural fluctuations had a declining effect on the
macroeconomy during the pre-1913 period, with the largest effects being observed
0.160
0.140
0.120
0.100
Per Cent
0.080
0.060
0.040
0.020
0.000
1872
1874
1876
1878
1880
1882
1884
1886
1888
1890
1892
1894
1896
1898
1900
1902
1904
1906
1908
1910
1912
construction coal agriculture
Figure 1.4 Sectoral shares in GDP
0.010
0.005
0.000
72
74
76
78
80
82
84
86
88
90
92
94
96
98
00
02
04
06
08
10
12
18
18
18
18
18
18
18
18
18
18
18
18
18
18
19
19
19
19
19
19
19
Effect on GDP
-0.005
-0.010
-0.015
-0.020
WCOAL WAGRO WTOTAL WCONSO
Figure 1.5 Weighted sectoral effects and total weather effect

14 Solomos Solomou
in the 1870s. An emphasis on the gold standard as the key feature determining
fluctuations over the period is misleading.
1.3 Inter-war epoch

During 1919–38 the world economy witnessed economic fluctuations that were,
in many ways, distinctly different from those of the past. A study of this period
illustrates that whilst business cycles are a recurrent phenomenon their character-
istics evolve and change over time. The following are some of the more important
changes that emerged in the inter-war period:
Macroeconomic trend-stationarity
During the pre-1913 gold standard era at least some of the major industrial countries
displayed trend-stationarity in their aggregate growth paths, with a tendency to
grow along a stable path in the long run (Mills, 1991; Solomou, 1998). This is the
case for Britain and America over the period 1870–1913 and for Germany during
1880–1913. During the inter-war era these underlying growth paths were displaced
in a persistently downward direction as a result of adverse business cycle shocks
during the early 1920s and early 1930s. The timing of these adverse shocks varied
by country: for example, Germany and Britain experienced persistent adverse
effects in the early 1920s while America saw persistence arising from the adverse
shocks of the early 1930s.
The ‘passing of the kuznets cycle’

The pre-1913 era (particularly during the period 1860–1913) witnessed Kuznets
swings in the trends of a number of important variables, including GDP, migration,
agricultural production, construction output, domestic and overseas investment,
consumption, exports and the balance of trade (Solomou, 1987, 1998). During the
inter-war period shorter cycles in all these variables became the norm, suggesting
a break in the long swing growth process (Abramovitz, 1968).
Business cycle amplitudes

Business cycle volatility was significantly higher in the inter-war than in the pre-
1913 era (Sheffrin, 1988: Backus and Kehoe, 1992). The high variance of GDP
and industrial production during the inter-war period is historically unique.
International cycles
Business cycles showed high levels of international co-movement in the inter-war
era relative to the pre-1913 era (Backus and Kehoe, 1992; Eichengreen, 1994).
This high level of international co-movement stands out even when comparisons
are made with the post-war era.
These new descriptive features raise a number of important questions: first,
is the persistence of shocks influenced by the collapse of the rules-driven pol-
icy regime of the pre-1913 classical Gold standard, and its replacement by more
discretionary policy regimes in the 1920s and 1930s? Second, why did the in-
ternational adjustments of the long swing process (such as the inverse home and
overseas investment swings) come to an end during the inter-war period, and what
implications did this have for economic cycles? Finally, how do we account for
the internationalisation of the business cycle during the inter-war period? These
questions are addressed below.
1.3.1 Persistence of shocks

Many business cycle shocks during the inter-war period had persistent effects on
aggregate macroeconomic performance. This feature is in marked contrast to the
pre-1913 gold standard era, which saw the workings of a number of long-run
adjustment processes resulting in the feature of macroeconomic trend-stationarity.
Examples of shocks that resulted in persistent effects are the 1920–1 depression
in Britain, the German monetary stabilisation of 1923–4 and the Great Depression
of 1929–33, which had long-term effects on a large number of economies.
Two key influences determined this new feature. First, the more discretionary
policy framework of the inter-war period that arose out of the flexible exchange
rate era (1919–25), combined with the attempts to re-establish the gold standard
(in the form of the gold exchange standard) forced economies to deviate from their
long-run paths on a permanent basis. This is very much at the heart of understand-
ing the British and German experiences of the early 1920s (Solomou, 1998). The
maladjusted gold exchange standard has also been given a central role in explana-
tions of the 1929–33 depression (Eichengreen, 1992; Temin, 1989). The defence
of the fixed exchange rate regime by most countries during 1929–31 enforced
excessive monetary and fiscal deflation. In the USA this had a devastating effect
on the stability of the financial system with one third of American banks failing
between 1929–33. The collapse of the financial system, combined with a high
debt overhang from the 1920s prevented a full revival of investment and consumer
durable demand in the 1930s. Bernanke and James (1991) show that this mecha-
nism is applicable to a wide number of countries. Countries that defended the gold
standard for long periods raised the probability of financial crisis. Once financial
crisis occurs, the event is likely to have persistence effects by raising the cost of
credit and limiting its availability (Bernanke, 1983; Calomiris, 1993).
Second, the inter-war epoch did not have an equivalent set of international
adjustment mechanisms to the pre-1913 era. The pre-1913 gold standard survived
for so long partly because there existed viable international adjustment mecha-
nisms to national-specific shocks: including international migration flows, trade
protection and overseas investment (which stimulated the tradable sector). These
adjustments manifested themselves in long swing fluctuations in a number of inter-
national economic variables, such as exports, overseas investment and international
migration. The inter-war era saw an abrupt end of many of these adjustment out-
16 Solomos Solomou
lets. Legislation in the New World prevented mass emigration as a solution to mass
unemployment. The disintegration of world trade, partly due to protection policies
and a collapse of overseas investment during 1928–38, prevented export growth
from stabilising the effect of domestic demand shocks. Instead, severe business
cycle shocks left economies with high unemployment and low output levels. The
‘passing of the Kuznets cycle’ (Abramovitz, 1968) in the inter-war era is of central
importance to business cycle experiences, just as the presence of Kuznets swings
before 1914 represented the workings of various stabilising cyclical adjustment
mechanisms in the international economy.
1.3.2 Business cycle volatility

The evidence of high output volatility raises a number of questions: did the severity
of shocks increase in the inter-war relative to the past; is wage and price flexibility
lower (forcing greater quantity adjustments on the economic system); did the tran-
sition to American economic leadership of the world economy transmit volatility
to the rest of the world?
The nature of shocks did change significantly. Monetary policy shocks were of
far greater importance in the inter-war period than before 1913 (Capie and Mills,
1991, 1992; Eichengreen, 1992). The flexible exchange rate era of 1919–25 was
associated with a number of severe monetary and exchange rate variations; the mal-
functioning gold exchange standard imposed excessive monetary deflation on the
world economy between 1929–33 (Temin, 1989; Eichengreen, 1992); the deval-
ued exchange rates of the 1930s made possible the use of discretionary monetary
policy to stimulate economic recovery (Eichengreen and Sachs, 1985; Solomou,
1996).
Maladjustment and inflexibility in the labour market was emphasised by con-
temporaries of the inter-war period. However, although wage rigidity was present
in the inter-war era it was also a feature of the pre-1913 era (Lewis, 1978). In
an examination of the wage behaviour of Britain, France, Germany, the USA and
Sweden, Phelps-Brown and Hopkins (1950) found that the degree of wage rigidity
was comparable in the two periods in all these economies. More recent evidence
suggests that only the USA provides an example of increased wage rigidity in the
inter-war period (an outcome of the New Deal labour market policies in the 1930s)
relative to the pre-1913 era.
Increased volatility during the inter-war period is an international phenomenon.
Kindleberger (1983) has attributed this to the leadership structure of the world
economy after the World War I. Although the USA emerged from the World War I
as the only major capital exporter, it was not willing to take the lead in stabilising
the international economy. Kindleberger sees the contra-cyclical British overseas
investment before 1913 (captured by an inverse long-run relationship between
home and overseas investment levels) as reflecting stabilising international be-
haviour by Britain. In contrast, the USA followed a pro-cyclical pattern of overseas
investment that destabilised the world economy. While Kindleberger views these
features as a reflection of national leadership qualities, it seems more plausible that
the existence of ‘Frontier’ economies before 1913 made the absorption of capital
by the world economy more likely. Given comparable profitability conditions, it
is most likely that British investors would have responded in similar ways to their
American counterparts during the inter-war period. The outcomes of the pre-1914
era were mainly fortuitous rather than planned; as in the inter-war period, these
were the market outcomes of individuals seeking high rates of return3 .
1.3.3 International business cycles

Much of the modern empirical literature on business cycles suggests that whilst
pre-1913 fluctuations were national-specific, an international business cycle clearly
manifested itself during the inter-war epoch (Backus and Kehoe, 1992; Eichen-
green, 1994). As noted above, this result is sensitive to the definition of busi-
ness cycles that have been employed by these studies. Backus and Kehoe use the
Hodrick-Prescott (H–P) filter to derive cyclical decompositions. The working def-
inition of business cycles that the H–P filter implies is one that emphasises patterns
of high frequency fluctuations. Longer cycles in the data are assigned to the trend
component.
Focusing on the co-variation of high frequency cycles misses relevant business
cycle information. Pre-1913 fluctuations were highly integrated, but the strongest
linkages are observed in the low frequency fluctuations. What changed in the
inter-war period was not the creation of an international business cycle, but a shift
in the period of the cyclical co-variations. The passing of the Kuznets cycle (in
terms of international swings in migration, capital and export growth) modulated
a different type of international business cycle. As large groupings of countries
pursued common policy aims, such as the re-establishment of the gold standard
in the 1920s, the defence of the gold standard parities during the early 1930s
and the exit from gold in 1931–3, patterns of business cycles became conditioned
by common policy regimes. This resulted in significant co-variations in the high
frequency cycles across countries.
The new features of persistence, high volatility and high covariance of short
cycles across countries created an important role for economic policy management.
Whilst the economic environment of the pre-1913 period contained a number of
adjustment mechanisms that activated a mean reverting tendency in the aggregate
path of the economy, during the inter-war period shocks resulted in more persistent
effects. Discretionary policy took on a new role of adjustment to shocks in the
economy. Studies of the 1930s show that countries that devalued and pursued
active expansive monetary policies saw a far better path of economic recovery
during the period than those countries pursuing conventional policies, such as the
gold bloc (Eichengreen, 1992; Temin 1989; Solomou, 1996)
1.4 Post-war cycles

The post-war period has witnessed a number of institutional changes that were
to have a major effect on economic fluctuations. The Bretton Woods arrange-
18 Solomos Solomou
ments linked exchange rates to the dollar with the ‘adjustable peg’ mechanism,
re-establishing rules-driven policy frameworks and breaking from the policy re-
sponses of the inter-war epoch. This system collapsed in the early 1970s and
attempts were made to re-establish exchange rate stability within newly emerging
trading blocs during the 1980s and 1990s. Comparing the period of the post-war
‘golden age’ with the post-1973 era allows us to evaluate the effect of different
policy regimes on business cycle behaviour. The post-war period is also interesting
in the light of two important long-run structural changes. First, the rapid growth
of the government sector in most of the major industrial countries has been noted
as a stabilising influence on aggregate demand, reducing the volatility of business
cycle fluctuations relative to the past (Tobin, 1980; Zarnovitz, 1992). Secondly,
the rapid growth of intra-regional trade has resulted in significant changes to the
structure of world trade, and thus in the international transmission of shocks. These
features have resulted in new business cycle features.
1.4.1 Business cycle amplitudes

Table 1.2 shows that there has been a marked reduction in the variance of macroe-
conomic output fluctuations in the post-war era relative to the inter-war era. The
volatility of the post-war period is also lower than for the pre-1913 era, suggesting
that post-war policies and long run structural changes (such as the growth in the
size of the Government sector) may have generated stabilising effects on modern
economies (Burns, 1960; Tobin, 1980; Zarnovitz, 1992).
Some recent work has, however, questioned the comparability of these long-run
data series. For example, Romer (1986) has argued that the high variance of the pre-
1913 US economy is largely a statistical artefact, resulting from Simon Kuznets’s
estimation methods for US gross national product. Romer has revised the existing
macroeconomic series on the assumption that they have an artificially high variance
for the pre-1913 period. Simon Kuznets estimated an annual series of GDP using
regression analysis. The regression series was constructed by taking the period
1909–38, a period when fairly reliable estimates of GDP can be obtained using the
income–expenditure approach. Kuznets regresses the percentage deviation from
trend of GDP on the percentage deviation from trend for aggregate commodity
production.4 He then uses the estimated coefficient to form an estimate of GDP for
the period 1869–1918. However, the regression for the period 1909–38 is heavily
influenced by the depression of the 1930s. Thus the constructed series may be
biased to generate large cyclical variations. The percentage deviations from trend
of GDP and commodity output move much closer to 1:1 during the 1930s than
for other periods. During 1909–28 the coefficient is only 0.6. Romer (1986) uses
the coefficient for the period 1909–28 as the basis for constructing the GDP series
and this yields a substantially less volatile series before 1929. In the light of the
revised data, Romer argues that the depression of the 1930s is exceptional and
nothing comparable can be found before 1929. In fact the variance of US business
cycles for the period 1870–1929 looks comparable to that for the post-war era,
questioning the idea that demand management has moderated cyclical fluctuations
after World War II.
There are, however, a number of problems to consider. If we assume that the
macroeconomic relationships of the period 1870–1909 are similar to those for the
period 1909–1928 then we observe a significantly lower amplitude than in the
original Kuznets series. However, it is unlikely that these quantitative relation-
ships will remain stable over such long periods. Moreover, the existence of long
swings in the American economy suggests that Romer’s revisions are unlikely to
be correct because the period 1909–1928 is mainly capturing an upswing phase
in the American economy; the behaviour of the 1930s, may be more relevant to
depicting depression phases such as 1890s. Romer’s assumptions are likely to have
introduced artificial stability in the output data for the pre-World War I period.
Accepting the need to revise the US national income data Balke and Gor-
don (1989) have constructed a GDP series using new information on distribution,
construction and consumer prices. Their results suggest that the variance of the pre-
1914 economy is 1.77 times greater than for the post-war. Although this is lower
than the original Kuznets data the results suggest that the pre-1914 economy was
far more volatile than the post-war era, reinforcing the conventional picture.
Table 1.2 also shows that the phenomenon of a low variance during the post-war
period is observed in a large number of countries, with very different historical
data construction methods. It would be difficult to argue that all these long run
changes are a statistical artefact (Backus and Kehoe, 1992; Sheffrin, 1988). Thus,
although measurement errors make it difficult to compare long run historical data,
the evidence is consistent with the idea of a structural shift towards relatively low
business cycle volatility after World War II.
1.4.2 Cyclical duration

Cyclical duration averaged 3 to 5 year growth cycles during the golden age of the
1950s and 1960s (Zarnovitz, 1992). Although short inventory cycles have been ob-
served for the pre-war period, such a short duration for the major economic cycle
is historically unique. The average length of the cycle has increased significantly
since the 1970s. Analysing data for 1960–86 using the statistical methodology
of the maximum entropy spectrum to determine cyclical duration for the OECD
economies Hillinger (1992) found that cycles have increased in duration, although
average duration varies significantly across countries. A large number of countries
were influenced by a medium-term cycle of 12 to 15 years in duration.5 The ev-
idence suggests that cyclical durations have not been stable even within the post
war era, with a discernible shift in average duration since the late 1960s. Although
most business cycle theories do not seek to explain a regular periodic cycle, the ob-
servation of shifting mean cyclical duration suggests that the adjustment processes
to shocks and/or the nature of shocks undergo significant change over time.
1.4.3 International synchronization

The post-war business cycle was not a world economic cycle; instead bilateral
cyclical linkages can be seen across a variety of countries. Strong bilateral co-
20 Solomos Solomou
Table 1.2: Standard deviation (per cent) measures of output volatility
(standard errors in parenthesis)
Country Prewar Inter-war Post-war

Australia 6.30 4.85 1.93
(0.72) (0.75) (0.19)
Canada 4.47 9.80 2.22
(0.43) (1.40) (0.23)
Denmark 3.02 3.41 1.88
(0.22) (0.64) (0.20)
Germany 3.35 10.19 2.30
(0.32) (1.61) (0.28)
Italy 2.52 3.59 2.05
(0.24) (0.46) (0.17)
Japan 2.42 3.13 3.11
(0.24) (0.44) (0.32)
Norway 1.85 3.49 1.76
(0.16) (0.65) (0.17)
Sweden 2.43 3.74 1.45
(0.37) (0.59) (0.12)
United Kingdom 2.12 3.47 1.62
(0.24) (0.37) (0.21)
United States 4.28 9.33 2.26
(0.38) (1.27) (0.18)
Source: Backus and Kehoe (1992)
movements were observed between the UK and the USA, Japan and the USA,
Canada and the USA, Germany and the UK and Canada and the UK. Insignificant
linkages are observed between Japan and the UK, Japan and Germany and Japan
and Canada. Such strong bilateral cyclical linkages are in marked contrast to the
inter-war era when the business cycle is far more of an international phenomenon
(Backus and Kehoe, 1992).
Using less formal statistical methods Zarnovitz (1981) compared the business
cycle timings of the US, Canada, the UK, Germany and Japan within the National
Bureau reference cycle framework during the period 1948–80. He also finds strong
co-movements between different sets of countries, such as the US and Canada and
the UK and Germany, rather than an international business cycle that affects all
the major countries simultaneously.
Using quarterly output data for a larger cross-section of countries for the pe-
riod 1959–89 Danthine and Donaldson (1993) document both bilateral and inter-
national business cycle linkages. At the bilateral level there are some very strong
cyclical co-variations. For example, Germany shows significant co-movements
with Austria, France, Italy, Switzerland and the UK; the UK has strong bilateral
linkages with France and Germany. Aggregating the EC countries into one bloc
we can also compare how each individual EC member varies with the EC bloc.
The co-movements are very much weaker, suggesting that it is strong bilateral
ties that are important in the intra-European cyclical linkages, not bloc behaviour.
Comparing the individual EC members with the USA, their linkages have been far
stronger than with the EC bloc. The data also allow us to compare inter-bloc cycli-
cal influences: comparing the cyclical variations of the EC, the USA and Japan
we see strong positive co-movements across these three major economic zones.
At this level of aggregation there is some evidence of international business cycle
linkages.
1.4.4 Price-output cyclical co-movements

In the light of developments in the monetary theory of business cycles (Lucas,
1977) and the real business cycle theory (Kydland and Prescott, 1982) the cyclical
relationship between prices and output is central to business cycle research. Lucas
simply assumed that prices move pro-cyclically with output as a stylised fact.
Analysis of the post-war data suggests the very opposite, with price and output
fluctuations moving contra-cyclically (Danthine and Donaldson, 1993).
However, evidence from other historical eras does not suggest stability of this
relationship in the long run. During the classical gold standard period price–output
fluctuations do not show any consistent positive or negative pattern; the inter-war
period is chiefly characterised by pro-cyclical movements; the post-war phase up
to the 1970s has been dominated by contra-cyclical movements. The most recent
period suggests further changes. Pro-cyclical movements have been observed in
the USA and the European economies in the 1980s and 1990s; as an example, the
depression of the early 1990s in the major industrial economies resulted in the
lowest inflation rates since the 1960s.
Pro-cyclicality, contra-cyclicality and non-cyclicality are all possible ‘stylised
facts’, depending on the historical period being observed. The aims of theoretical
frameworks should be to determine the reasons for the observed shifts in this
relationship across different historical periods. Theoretical discussions that have
sought to explain a universal stylised fact are, thus, misleading.
The evidence raises a number of theoretical and empirical questions that need
to be considered further. For example, pro-cyclical price–output fluctuations arise
in the inter-war and in some phases of the post-1973 epoch; both of these phases
are periods of monetary policy discretion and unsettled exchange rate experiences.
In contrast during the classical gold standard period and the Bretton Woods era
the leading economies sustained fixed exchange rates, limiting national policy dis-
cretion. Thus, a prima-facie case can be established that the shifts in price–output
movements are, at least partly, determined by the monetary-exchange rate policy
regime. This does not mean that the price–output relationship is not influenced by
real business cycle influences, such as the effects of technology and other supply-
side shocks. What it does mean is that in some periods these effects are of secondary
importance to the impact of the policy regime.
22 Solomos Solomou
1.4.5 Disaggregated business cycle volatility
Investment, consumption and aggregate output co-move over the business cycle.
This, in itself, may seem a rather intuitive and uninteresting result. However, a
further stylised fact has a central role in business cycle discussions: it is often
argued that there is a regular variance structure to the disaggregated data, with
investment volatility being greater than aggregate volatility. The phenomenon of
consumption smoothing by individuals, as predicted by the permanent income and
life cycle hypotheses, implies that consumption is expected to be the least volatile
component of aggregate demand.
Empirical studies generally accept this description of the structure of the Dis-
aggregated data (Hillinger, 1992; Lucas, 1977; Kydland and Prescott, 1982). How-
ever, a number of important ‘outliers’ to this generalisation need to be noted. The
US Great Depression of the 1930s cannot be understood without an explanation
of consumption volatility far in excess of investment fluctuations (Temin, 1976
and 1989; Calomiris, 1993). Similarly, to account for the depression of the early
1990s in both the USA and Europe we need to be able to explain the observed
consumption shifts (Blanchard, 1993; Hall, 1993). These two events in history
carry much weight in terms of their impact on the economics and politics of mar-
ket economies. The depression of the 1930s is historically unique in terms of its
severity and the depression of the early 1990s is the most severe downturn for fifty
years. To focus exclusively on investment volatility as an explanation of business
cycle fluctuations is incomplete in that there is a selection bias towards neglecting
information from major depressions. Thus, although consumption smoothing is an
important empirical feature that needs to be integrated into business cycle research,
episodic consumption volatility also needs to be integrated into a broader picture.
Explaining these consumption shifts, or at the very least recognising their eco-
nomic implications, is essential if we are to understand some of the major business
cycle fluctuations of the twentieth century. Moreover, the financial deregulation
which has been taking place since the 1980s means that consumption behaviour is
likely to play an increasingly important role in business cycle fluctuations in the
future.
1.5 Implications for business cycle research

This survey suggests that an historical perspective is an essential ingredient to an
analysis of economic fluctuations. The ‘stylised facts’ approach to business cycles
runs the risk of generating misperceptions. For example, in defining business cycles
as high frequency annual fluctuations we risk neglecting relevant information with
regard to the workings of cyclical adjustment mechanisms in the past (and in the
future). The historical evidence suggests that long swings were a central feature
of business cycle fluctuations in the past. To neglect this information will result in
incomplete analyses. The long swings of the pre-1913 epoch created adjustment
mechanisms that help us understand some of the key features of macroeconomic
fluctuations within the period. Similarly the ‘passing of the Kuznets cycle’ helps
us understand some of the changed features of the inter-war period. Utilising the
total information set on fluctuation in the economy will result in a more informed
analysis of economic cycles than if we simply use a working definition of business
cycles that focuses only on high frequency fluctuations. The evidence we have
suggests that we need to think of (at least) a three level decomposition: stochastic
trend, low frequency long cycles and high frequency short cycles. Although we
may choose to focus on cycles of a particular frequency (for whatever reason) we
should always bear in mind that when a number of cycles may co-exist, the total
cyclical information will also be useful.
Irregularity in business cycles seems to be a normal historical feature. Whether
we focus on period, amplitude or patterns of co-movements we do not observe sta-
ble long run regularities that can be depicted as ‘stylised facts’. Explaining irregu-
larity opens up a number of challenges in theoretical research on business cycles,
within both the impulse and propagation perspectives to business cycles. The ex-
amples considered in this survey suggest that policy regimes, long-run sectoral
structural change, changing international inter-relatedness have had important ef-
fects. Understanding this will help in understanding that the future will be different
to the present.
Notes
1 Given the stochastic trends of nominal effective exchange rates (EER) we investigated
the existence of a common stochastic trend using the Johansen co-integration frame-
work. The nominal EER movements of the core countries are found to be co-integrated
over this period.
2 The correlations between construction and coal and agriculture and coal are −0. 14
and 0.001 respectively
3 One aspect of inter-war international capital flows, which may have destabilised the
world economy, is the link between capital flows and the ‘recycling’ of reparation
payments (Solomou, 1998).
4 Commodity production is a composite index of industrial and agricultural production.
5 Most countries were also influenced by a short cycle 5 to 10 years and a shorter
inventory cycle.
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2 Growth and fluctuations
The post-war Japanese
experience
Hiroshi Yoshikawa
2.1 Introduction
The century-long experience of the Japanese economy provides economists with
a great opportunity to study growth, business cycles and structural change. Japan,
so poorly endowed with natural resources, has grown to become one of the major
industrial nations. Figure 2.1 shows annual growth rates of real GDP for the pre-
war period from 1885 to 1940 and the post-war period from 1955 to 1995. The
pre-war Japan kept 3 per cent growth on average for more than half a century, and
the growth rate accelerated to almost 6 per cent in four decades after World War II.
The standard Ramsey model cannot satisfactorily explain this postwar acceleration
of growth.
Her growth has not been necessarily smooth. A glance at the figure reveals that
the pre-war Japanese economy was on the whole more unstable than the post-war
economy. The standard deviation of the growth rate is, in fact, 3.5 per cent for the
whole post-war period whereas it is 3.7 per cent for the pre-war period (Table 2.1).
The standard deviation for the post-war period is, however, inflated by the fall of
growth rate in the early 1970s. If we divide the sample period taking into account
this break, we find that the standard deviation declines to 2 per cent as against 3.7
per cent for the pre-war period. Why was the post-war Japanese economy more
stable than the pre-war economy? It is an interesting question. But in this chapter,
I simply draw attention to this important fact, and focus on the post-war period. A
good general reference for the post-war Japanese economy is Nakamura (1981).
The Japanese economy started from ashes after the Second World War. Real
GDP fell to a half of the pre- war peak level, and the economy suffered from
hyperinflation. Most Japanese economists believe that the economy completed the
Table 2.1 Growth rate of real GDP (%)
1886-1940 1956-95 1956-70 1970-90 1990-95

(1) Mean 3.2 5.7 9.2 4.4 1.9
(2) Standard Deviation 3.7 3.5 2.1 2.3 2.0
(3) (2)/(1) 1.16 0.61 0.23 0.52 1.05
28 Hiroshi Yoshikawa
14
12
10
6
Percent
–2
–4
–6
1885
1889
1893
1897
1901
1905
1909
1013
1917
1921
1925
1929
1933
1937
1941
1945
1949
1953
1957
1961
1965
1969
1973
1977
1981
1985
1989
1993
Figure 2.1 Growth rate of real GDP
War Reconstruction by 1955, and, therefore, I take 1955 as a starting point of

my study. Readers interested in the turbulent decade after the war are referred to
Yoshikawa and Okazaki (1933) and references cited therein. It is convenient and
standard to divide the post-war period into the following three sub-periods:
1. 1955–73: The average growth rate ten per cent
2. 1975–90: The average growth rate four per cent
3. 1990– : The average growth rate below two per cent
It is a central theme of this chapter to explain what mechanism generated extraordi-
narily high growth for such a long period as almost two decades after 1950. The era
of high growth ended in the early 1970s. This ‘structural change’ of the Japanese
economy is clearly seen in Figure 2.1. An obvious question is what caused this
‘structural change’. A standard explanation is the first oil shock in 1973–74 (Bruno
and Sacks (1985), for example). In this chapter, I underplay the role of the first oil
shock, and instead emphasize the importance of domestic factors. For the 1970s
and 1980s, I emphasize the role played by exports and the exchange rate. Finally
I briefly touch on the long recession in the 1990s.
2.2 The period of rapid economic growth and its end

The Japanese economy enjoyed an average of 10 per cent growth for almost two
decades beginning in the mid-1950s. There are several factors which are believed
to have contributed to the rapid economic growth of the Japanese economy during
Growth and fluctuations 29
the 1950s and 1960s. The abundance of importable foreign technology is often
mentioned as such a factor; Lincoln (1988) underlines the importance of this factor.
However, it is not obvious that the stock of available foreign technology was much
greater in the 1950s and 1960s than in the 1920s and 1930s or for that matter, in
the nineteenth century. One might plausibly expect that such a stock was greatest
when Japan opened her door to the West in the late nineteenth century.
A sharp decline in natural resource costs is also often mentioned. Technical
progress in marine transportation is believed to have contributed to this effect. A
sharp decline in marine transport costs has made natural resources commodities
rather than part of the ‘factor endowment’ of individual countries in the post-war
era. It naturally gave a leverage to the resource-poor Japanese economy. Total factor
productivity in the industry, in fact, grew at the annual rate of 10 per cent in the
post-war era as against 2–3 per cent in the pre-war period. Granted that these factors
are very important I suggest below that (i) demographic trends, and (ii) diffusion of
consumer durables, namely ‘catch up of demand’ played a particularly important
role in the process of rapid economic growth.
2.2.1 The mechanism generating rapid economic growth

Recall that the Japanese economy in the 1950s and 1960s was a two-sector econ-
omy consisting of a rural agricultural sector and an urban manufacturing sector. As
of 1950, nearly half of the total labour force was still engaged in agriculture (Table
2.2). In this respect, only Italy was comparable to Japan. Population continuously
flowed from the former into the latter in the process of economic growth. The
dual structure of the economy enabled the manufacturing sector to hire enough
labour at the level of real wages which, determined in the agricultural sector with
‘disguised’ unemployment, were lower than the marginal product in the industrial
Table 2.2 Total employment in primary, secondary, and tertiary occupation: Japan, USA,
UK and West Germany, 1950 and 1989 (%)
Primary Secondary Tertiary

(agricultural (manufacturing (services)
and mining) and construction)
1950
Japan 48.3 21.9 29.7
USA 12.4 35.3 49.7
UK 5.1 47.5 47.0
Germany 23.2 42.2 32.4
Italy 46 27 27
1989
Japan 7.6 33.8 58.1
USA 2.9 25.9 71.1
UK 2.4 29.0 68.6
Germany 4.9 38.4 55.
Note: For Italy, 1951, UK and Germany, 1987.

sector. Growth of the manufacturing sector, therefore, entailed high profits in the
same sector rather than an increase in real wages. The high profits in turn were
supposed to induce high investment. All this is, of course, what Lewis (1954)
describes as a typical process of growth of the underdeveloped dual economy.
The Lewisian model has been indeed successfully applied to the century long de-
velopment of the Japanese economy by a number of economists (see, for example,
Ohkawa and Rosovsky (1973), Minami (1968), and Inada, Sekiguchi and Shoda
(1992)). In the Lewisian model, however, population flow between two sectors is
taken solely as a result of the growth of the modern manufacturing sector. Minami
(1968, 1970) and others demonstrate that internal migration was in fact quite sen-
sitive to growth of the manufacturing sector; More people left rural agricultural
areas for urban industrial cities in booms and vice versa. Yearly fluctuations of
population flow was therefore a result of industrial growth. In the Lewisian model,
the key factor behind this industrial growth is low real wages made possible by the
existence of disguised unemployment in the agricultural sector.
In contrast to the Lewisian model, however, in what follows I argue that in
the case of the post-war Japanese economic growth (1955–70), population flow
between the two sectors was in fact the major factor in generating high demand
for products of the industrial sector. In my view, population flow was a cause as
well as a result of economic growth.
According to the Lewisian theory, population is supposed to continuously flow
from the rural agricultural sector to the urban industrial sector, as actually happened
in Japan. Among Asian developing countries, however, this Lewisian population
flow occurred to a substantial extent only in a few NIES countries. The basic
problem of the theory is that the growth of a modern industrial sector is sustained
by high profits which are guaranteed by repressed real wages while demand for
products in such a modern industrial sector is assumed to automatically emerge as
production grows. In reality, demand does not emerge automatically. Demand is
precisely the point which divides pre-war and post-war Japan.
The post-war Japanese growth during the period 1955–70 was led by domestic
demand. For example, the contribution of net export to growth was on average
–0.2 per cent for the high growth period. In the process of domestic demand-led
growth, population flow and household formation played a crucial role. Because of
the large-scale population flows (see Figure 2.2), household formation dramatically
accelerated during the period of high economic growth, 1955–70 (Figure 2.3). I
underline the fact that in this period, the growth rate of households forms a hump
shape at a high level parallel to the growth of real GDP while the growth rate of
population was quite stable at a much lower level of about 1 per cent . Population
growth or the growth of the labour force which plays such an important role in
the standard growth theory has little explanatory power for the high growth of the
Japanese economy during the 1950s and 1960s.
As one might expect, traditional three generation merged households hardly
increased during this period. Instead, the ‘core’ households consisting of a married
couple, possibly with unmarried children, and an unmarried adult dramatically
increased, particularly in urban industrial areas. When three generations of family
members kept a traditional single household in rural villages, they would have
700
600
Inflow
400
200
0
Outflow
–100
1955 1960 1965 1970 1975 1980
Figure 2.2 Population flow 1955–80 into and out of Tokyo, Osaka and Nagoya metropoli-
tan areas.
Source: Annual Report on Internal Migration Statistical Bureau, Management and Co-
ordination Agency.
4.0
3.5
Growth rate of households
Growth rate of population

3.0
2.5
Percent
2.0
1.5
1.0
0.5
0.4
1956 1960 1965 1970 1975 1980 1985 1990
Figure 2.3 Growth rates of households and population, 1956–90
Note: The figure for 1970 is an outlier; Okinawa prefecture (Ryuku Island) was returned to
Japan by the United states in 1970.
needed only one of each consumer durable such as a refrigerator, television set,
washing machine and car. In fact, young people giving up agriculture left rural
villages for urban industrial areas where they formed new households. Household
formation necessarily generated additional demand for houses, consumer durables,
and electricity. In this way, population flow sustained high domestic demand in
the period of high economic growth, 1955– 70.
Along with the creation of a large number of households, the high growth
period was also the diffusion process of newly available consumer durables. The
diffusion of consumer durables was facilitated by a steady decline in prices of
those products over time on the one hand, and an increase in income on the other.
Electric washing machines, for example, first appeared in the Japanese market in
1949. At the time, a machine cost 54 thousand yen while the average annual labour
income was about 50 thousand yen. Understandably, only twenty machines were
sold per month! By 1955, only six years later, however, the price of a washing
machine had been reduced to 20 thousand yen while the average annual income
had risen to above 200 thousand yen. About a third of households could afford to
own a washing machine in 1955. The same story holds for other consumer durables
as well. Since it was urban cities that led this diffusion process, urbanization not
only created new households but also in itself sustained high demand for those
consumer durables. By the end of the 1960s, however, most of the then available
consumer durables saw saturation in the domestic market.
This whole process of domestic demand-led high economic growth (1955–70)
is schematically summarized in Figure 2.4. Channels 1, 2 in the diagram have been
well recognized: capital accumulation in the industrial sector raising labour de-
mand brings about population flow from rural agricultural areas to urban cities. In
addition to these well-recognized channels, I emphasize the oft-neglected and yet
very important fact that such population flow in turn, creating new households and
raising demand for consumer durables and electricity, ultimately sustained prof-
itability of investment in manufacturing industry (channels 3, 4, 5). I underline that
the role of newly available consumer durables was not confined to demand for those
products themselves. Through an input-output interrelationship, they augmented
demand for intermediate goods such as steel and electricity, and accordingly high
investment in those sectors.
In this virtuous circle for high economic growth, low real wages were not so
instrumental as Lewis (1954) emphasized. Rather it was growth of domestic de-
mand that sustained profitability of investment. And for growth of consumption
demand, a steady rise in real wages, rather than low repressed wages, is a con-
tributing factor. In fact, in the pre-war period, except for 1920–21 and 1929–30,
real wages saw little increase, while in the post-war period they enjoyed steady
growth. A steady rise in real wages sustained effective demand for the post-war
Japanese economy because the key product was consumer durables which not yet
being international competitive, had to find a domestic market.
It is important to note that the pre-war Japanese economy was, like today’s
NIES, a typical export-led economy (see Shinohara (1961)). The key industry was
cotton, and low real wages were in fact instrumental for international competi-
tiveness. The export/GNP ratio in pre-war Japan was, for example, 20 per cent
(Embodied technology)
Investment Technical progress Capital accumulation
Lower price of Increase in labour income

1 5 manufactural product in the industrial sector
More households can afford consumer durables

4
High consumer demand particularly
for consumer durables
Increase in households in urban cities
Increase in labour demand 2 Population flow from rural agricultural areas

in the industrial sector to urban industrial cities
Figure 2.4 Domestic-demand-led high economic growth of the Japanese economy, 1955–
70.
(1931–40) as against 10 per cent in the 1955–70 period. Chenery, Shishido and
Watanabe (1962), using the input–output analysis, also drew a similar conclusion
on the difference between the pre-war and post-war Japanese economies.
The domestic demand-led virtuous circle for economic growth based on the
Lewisian dual structure is not unique to the post-war Japanese economy. Kindle-
berger (1967, 1989), for example, discusses the post-war growth of the European
economy in a Lewisian model; See also Boltho (1982). The relation between house-
hold formation and economic growth is also well-known. The so-called building
cycle has a long tradition (See, for example, Hickman (1974)). However, in the
case of the pre-war US economy, particularly prior to the restrictive immigration
law of 1924, it was immigration rather than domestic population flow that endoge-
nously responded to the growth of income and fueled household formation. And
in the building cycle, it is residential construction that brings about a close relation
between household formation and economic growth.
In the post-war Japanese economy, it was firm’s fixed investment rather than
residential construction that played a crucial role in growth and fluctuations. In the
1960s, residential construction was less than half of total private construction. Its
share rose to above 70 per cent in the mid-1970s when the growth rate sharply fell.
Household formation by way of diffusing consumer durables rather than residential
construction, ultimately sustained high investment in manufacturing industry. In
this sense, my explanation for the Japanese economic growth during the period
of 1955–70 is more similar to Gordon (1951) than to the building cycle theory.
Table 2.3 The relative contribution of demand components to the business cycle (%)
GNP Inventory Housing Fixed Consumption Exports Imports Government

Investment Investment Investment Expenditures
Japan 100 15.8 16.6 58.2 45.0 27.8 –51.8 –11.6

US 100 24.3 20.8 25.1 35.5 11.5 –17.5 0.4
Source: Yoshikawa (1993). See the text for details.
Gordon argued that high investment, both residential and nonresidential, in the US
in the 1920s was fueled by urbanization and diffusion of the automobile.
In the Japanese economy during 1955–70, population flows and the consequent
household formation by way of diffusing newly available consumer durables con-
tinuously stimulated economy-wide investment demand. Fluctuations in invest-
ment demand were in turn the most important generating force of business cycles
from the late 1950s through the 1960s. To demonstrate this point, I show the extent
to which different demand components have accounted for different shares of the
change in GNP (Table 2.3). Since the Japanese economy has been growing rapidly,
almost all variables increase in absolute terms even in recessions I therefore first
calculated the change in each variable measured from trough to peak in case of a
recovery, and from peak to trough in case of a growth recession. I then subtracted
the latter from the former to obtain the difference. Table 2.3 reports the relative
contribution of each demand component to this cyclical difference in the change
in real GNP. It is the average for the eight cycles during the period beginning
February 1957 through March 1990. For the sake of comparison, I also present
results for the United States.
In Japan throughout the whole period, the relative contribution of fixed invest-
ment has been the greatest of all the demand components: 60 per cent of GNP on
average. In contrast, in the United States fixed investment accounts for only 25 per
cent on average of the change in real GNP. The relative contribution of inventory
and housing investments is greater in the United States than in Japan. Changes
in housing investment in Japan are not really systematic over the business cycle.
On the other hand, until the mid-1960s, inventory investment had a large impact
on the Japanese business cycle: a 60–70 per cent contribution. A substantial por-
tion of the inventory investment was, however, raw materials – which were also
imports. Therefore, the contribution of inventory investment and imports almost
canceled each other out. As a result, fixed investment retained its importance. As
a long-term trend, the role of inventory investment in the business cycle seems to
have diminished in both Japan and the United States.
Net exports have been counter-cyclical in Japan’s business cycle except for the
years 1977–85, in which economic growth was export-led as we will see shortly.
In particular, imports have been very counter- cyclical: the fraction of output was
–52 per cent on average, compared to –17 per cent in the United States. Until very
recently, the bulk of Japanese imports consisted of raw materials and therefore
moved very mechanically in parallel with the level of aggregate economic activity.
The contribution of consumption to GNP seems to be in large part similar in
the two countries. As for government expenditures, we find them counter-cyclical
for Japan (–12 per cent of GNP on average) but neutral (0.4 per cent) for the United
States. In sum, the major differences between Japan and the United States lie in
the facts that fixed investment plays a much larger role in the business cycle in
Japan than in the United States.
Coming back to the high growth era, we find that the situation changed dramat-
ically around 1970. By then the pool of the so-called disguisedly unemployed in
the agricultural sector had been largely exhausted. Therefore the population flow
from the rural sector and the associated urban household formation both sharply
decelerated. At the same time, the then available consumer durables saw saturation
in the domestic market. In this way, the domestic demand-led virtuous circle for
high economic growth was lost. Judging from Figures 2.2 and 2.3, we note that
this structural change occurred around 1970, a few years in advance of the first
oil embargo in 1973 (see also Horie et al, 1987).
The structural change is clearly seen for individual industry as well as the
macroeconomy. Table 2.4 shows the time series of capacity and investment in the
petrochemical industry. By the end of the 1960s, the industry had faced the major
turning point. The situation of the industry was typical, not exceptional. The first
oil shock hit the Japanese economy which had already seen the structural change
Table 2.4 Investment in the petrochemical industry, 1956–80
Real investment (Index: 1970=100)

1956 3.3
1957 9.5
1958 9.9
1959 11.5
1960 15.9
1961 27.2
1962 23.3
1963 25.4
1964 37.3
1965 45.0
1966 30.6
1967 42.5
1968 78.3
1969 81.8
1970 100.0
1971 92.5
1972 55.6
1973 44.1
1974 57.7
1975 65.3
1976 50.0
1977 41.9
1978 24.7
1979 24.3
1980 35.4
caused by domestic factors explained above. Before we turn to the post high growth
period beginning in the 1970s, I will critically review other explanations for the
end of high economic growth.
2.2.2 Other explanations for the end of rapid economic growth

I explained the end of rapid growth during the period of 1955–70 by the domestic
structural change of the Japanese economy which had occurred by the end of the
1960s prior to the first oil embargo in 1973–74. A popular view, on the other
hand, attributes the end of high growth to the first oil shock. There are two slightly
different explanations. Note that our problem is to explain a permanent fall in the
growth rate of the economy, not just a temporary decline in the level of output.
Bruno and Sachs (1985) explicitly introduce raw materials and energy into a
gross production function. In their framework, the oil shock can be identified with
an unanticipated permanent increase in the real price of raw materials/energy. It can
be shown that the oil shock is equivalent to a Hicks neutral technical regress. For
capital and labour to be fully employed, both real wages and the marginal product
of capital must decrease. A decrease in the marginal product of capital becomes
larger if for any reason, a decline in real wages did not realize to a full extent.
In any case, a decline in the marginal product of capital caused by an increase in
the real raw material/energy price entails a decline in Tobin’s q, and accordingly a
decline in the growth rate, as long as the real interest rate did not proportionately
decline. This is the Bruno/Sachs explanation for deceleration of growth in terms
of the oil shock.
I note that in this explanation, the basic reason for growth deceleration lies in the
inflexibility of the real interest rate. If the real interest rate declined proportionately
with the marginal product of capital, then an increase in the real material/energy
price would leave the growth rate intact. Bruno and Sachs (1985) emphasize the
role of demand factors in their empirical analysis of worldwide stagflation in the
1970s, but analytically demand and supply sides are not fully integrated. Their
analytical framework focuses on the supply side. Yoshikawa (1995; Chapter 4)
shows a way to integrate both demand and supply constraints within a unified
framework, and also demonstrates that the demand constraints were indeed very
important in the case of the Japanese economy in the 1970s.
Another approach which focuses on the supply side of the economy to explain
deceleration of growth is Jorgenson (1988). Jorgenson emphasizes the bias of
technical change. A ‘bias’ of technical change indicates how the technical change
affects factor shares. Kuroda, Yoshioka and Jorgenson (1984) show that in the
post-war Japanese economy (1969–79), the energy using technical progress was
dominant in almost all the industries. The energy using technical progress means
that an increase in the real energy price makes the rate of technical change deceler-
ate. This is the essence of the Jorgenson argument. The conclusion of his analysis
is as follows:
The most important single factor in the Japanese slowdown is the sharp decline
in the rate of technical change. I have now succeeded in linking that decline
directly to energy prices through the energy using bias of technical change in
Japan.
Jorgenson’s analysis rests entirely on the traditional growth accounting. His result
actually attributes nearly a half of a fall in the rate of economic growth in the 1970s
to capital input. And yet, noting that ‘after the energy crisis as well as before, the
growth rate of capital input was higher than that of output’, Jorgenson argues that
‘rather than causing the slowdown, the growth of capital after the energy crisis
contributed to the continued growth of output at unsustainable levels’. Thus he
concludes that ‘the decline in the growth rate of capital is not the cause of the
slowdown in Japanese economic growth’.
Capital accumulation is certainly not exogenous, however. And what brought
about a fall in the rate of economic growth and thereby caused the decline in the
growth rate of capital in the first place? Jorgenson’s answer is, of course, a fall
in the rate of technical change caused by the oil shock. In the growth accounting,
technical progress is identified as the part of the blossom of economic growth
which cannot be explained by contributions of production factors, the so-called
‘residual’. I argue, however, that whether the residual measured by the growth
accounting method really captures technical progress or not is an open issue.
Two approaches, Bruno and Sachs (1985) and Jorgenson (1988), both attribute
a fall in the rate of economic growth in the 1970s ultimately to the oil shock in
1973. These approaches cannot explain, however, why the second oil shock which
occurred in 1979 did not bring about a similar fall in the rate of economic growth;
the average growth rates of real GNP for 1973–80 and 1981–90 are 4.1 per cent
and 4.2 per cent , respectively. During the first oil crisis, the oil price quadrupled
in 1973–74 while in the second oil crisis it only doubled in 1979–80. It might be
argued that the first oil crisis hit the Japanese economy harder than the second oil
crisis. But transfer payments to OPEC necessitated by an increase in the oil price,
when seen as relative to GNP, were actually comparable during the two oil crises,
at 3.8 per cent and 4.1 per cent , respectively. The supply side analyses, such as
Bruno and Sachs (1985) and Jorgenson (1988), which attribute a fall in the rate of
economic growth to the first oil crisis are, therefore, inconsistent with the fact that
the second oil crisis did not entail a similar fall in the growth rate. Nor can they
explain why the growth rate of the oil-importing Korean economy fell so sharply
at the second oil crisis while the effect of the first oil crisis was relatively small,
which is converse to the Japanese case. I maintain that demand is an indispensable
part of any reasonable explanation of the 1970s.
However, I do not mean to argue that the oil crisis did not affect the supply
side of the Japanese economy, but I do argue that a permanent fall in the rate of
economic growth beginning in the early 1970s was caused by a domestic structural
change as explained above rather than the first oil crisis. In this respect, I concur
with Maddison (1987). By his careful growth accounting, Maddison finds that the
growth rate of real GDP in Japan would have been 3.8 per cent during 1973–84
as against its actual value 3.6 per cent , ‘if it had been possible to maintain the
relation between energy growth and GDP growth in the previous period’ (his Table
15b). In his view, the effect of the oil shocks on growth rate is plainly minor.
2.3 The post high growth period: 1970–90
In any case, the Japanese economy saw a major structural change around 1970.
Table 2.1 shows that the rate of growth of real GNP was halved from 10 per cent to
4 per cent , but at the same time the Japanese economy became much more stable
than in the high growth period. I address this issue of stability first.
2.3.1 Investment and the stability of the Japanese economy

In the 1970s and 1980s the Japanese economy became much more stable than
in the high growth period. Taylor (1989), for example, comparing the Japanese
and American business cycles notes that ‘real GNP fluctuations in Japan are so
small compared to those in the United States, especially in the last 12 years that
actual GNP in Japan behaves much like the smooth potential GNP trend for the
United States’. Taylor explains the difference between the two economies by taking
money as the major shocks to the economy. Nominal wages in Japan are much
more flexible than those in the US because in Japan a standard length of wage
contract is one year as against three years in the US, and also wage negotiations are
synchronized by the nation-wide Shunto, or the Spring Offence. Thus, monetary
shocks exert less impact on the real economy in Japan than in the US, so goes
Taylor’s explanation. In the high growth period, however, real GNP fluctuations
in Japan were actually greater than those in the US. Shunto had been in place
throughout the period. Taylor’s explanation is, therefore, not satisfactory. Then,
what brought about the stability of real GNP in Japan beginning in the 1970s?
West (1992), based on a simple macro model of the Japanese economy, de-
composes movements of the real GNP during 1975 to 1987 to cost shocks and
demand shocks. His conclusion is that demand shocks are estimated to account
for nearly nine-tenths of the movement of GNP. This result and my own analysis
in Section 2.2 naturally lead us to analyze the behaviour of demand components
for 1956–73, and 1974–89. Table 2.5 shows the means and standard deviations of
the growth rates of demand components. The stability of consumption measured
by standard deviation of growth rate remains the same for the two periods. Ex-
ports and residential construction are actually more unstable in 1974–89 than in
1956–73. The stability of real GNP in 1974–89 was clearly brought about by the
behaviour of fixed and inventory investment. Here I focus on fixed investment.
There are three closely related causes to explain why investment became much
more stable beginning in the 1970s than in the high growth period. First is the
effect of aggregation. Investment in the economy as a whole, I is, of course, the
sum of investment in many sectors,Ij . Therefore the variance of the growth rate of
I , Var (İ /I ) is the sum of variances of İj /Ij and their covariances. One is apt to
forget the effect of changes in covariances.
In the high growth period, economy-wide factors spurred investment and as a
result, strong parallels in investment across various industries are observed. This
situation changed dramatically around 1970. Yoshikawa (1995) shows that even if
the variances of investment in individual industries had not declined, the variance
of investment in the economy as a whole would still have declined from 15.6
Table 2.5 Stability of demand components, 1956–89 (%)
1956–73 Mean s. d. 1974–89 Mean s. d.

Real GNP 9.2 2.4 3.9 1.6
Consumption 8.7 1.7 3.6 1.7
Private fixed investment 17.3 14.0 5.4 6.7
Residential construction 15.3 6.3 1.7 8.4
Inventory investment 93.6 271.5 46.9 131.0
Government expenditure 7.1 4.0 2.5 3.7
Exports 13.2 6.7 8.6 7.6
Imports 15.0 10.5 5.2 9.1
Source: EPA, National Income and Product Accounts.
to 10.5. At the beginning of the 1970s, investment seems to have been governed
by micro-specific rather than economy- wide factors. By way of enlarging the
diversity of timing of investment among industries, it enhanced the stability of
investment in the economy as a whole. The calculation above suggests that this
effect alone explains more than half of a decline in the variance of the growth rate
of investment in the second period.
Beyond that, there are two other factors to have enhanced stability of invest-
ment. One is an increase in the share of the non-manufacturing sector. Investment
in the non-manufacturing sector, compared to that in the manufacturing sector is
stable: standard deviations are 14.1 and 8.8 (1968 Q2 – 90 Q4), respectively. An
increase in the share of the non-manufacturing sector from 54.1 per cent (1968–72)
to 63.8 per cent (1973– 90), therefore, necessarily contributed to the stability of
total investment beginning in the 1970s.
The motives for investment also saw a major change. The Development Bank
of Japan has surveyed the investment of large corporations by motives. According
to this survey, 67.4 per cent of investment was done for the purpose of augmenting
capacity in 1969, but its share declined to 28.7 per cent in 1980. Now measured
by standard deviation of the growth rate, the volatility of investment motivated
by capacity augmentation is much greater than that of other kinds of investment
such as labour/energy saving and R and D (research and development): 24.8 per
cent and 14.0 per cent , respectively (1977–90). The change in the motives for
investment, therefore, also contributed to the stability of total investment.
I emphasize that the three factors mentioned above are all real rather than
monetary. Some economists argue or used to argue that stability of monetary
growth is responsible for the stability of real GNP. Suzuki (1985), for example,
argues that
Since 1975 the money growth rate in Japan has become more stable and the
uncertainty associated with it has also decreased due to the announcements
of targets. And since 1976 there has been an accompanying decrease in the
variability of the inflation and real growth rates.
However, the major influence which brought about the stability of real GNP be-
ginning in the 1970s was investment, and the stability of investment in turn was
60 Equipment
50
Percent
40
Construction
30
1960 1965 1970 1975 1980 1984
Figure 2.5 Equipment and investment in plant, 1960–84.
caused by three real factors.1 The stability of monetary growth is basically nothing
but a mirror image of the stable real economy.
2.3.2 Equipments and structures

Before leaving investment, it is worth pointing out another interesting fact. Figure
2.5 shows the respective shares of equipment investment and investment in struc-
tures. In post-war Japan, during the high growth period the share of structures was
higher than that of equipment investment. After the average growth rate fell from
10 per cent to 4 per cent , the share reversed and by the mid 1980s, the share of
equipment investment had risen to about two-thirds.
De Long and Summers (1991), on the other hand, using the UN and the Penn
World cross-country data, document a strong correlation between the growth rate
of real GDP per worker and the equipment investment/GDP ratio. According to
their estimates, over 1960–85, a one percentage point increase in the equipment
investment/GDP ratio is associated with an increase of about 0.3 percentage points
in the growth rate of per capita GDP. They interpret this correlation as causal
and argue that it is not just investment but equipment investment that promotes
economic growth. The average investment/GDP ratio in Japan is about the same
in real terms for both the high growth period (1955–70) and the subsequent low
growth period (1970–90). Figure 2.5, therefore, shows that the post-war Japanese
experience contradicts the De Long/Summers thesis that equipment investment
promotes economic growth more than investment in structures.
De Long and Summers seem to hold that most new technologies are embodied
in equipment rather than in structures. However, equipment which embodies new
technology may be introduced simply to substitute other inputs such as labour
and energy for given output. The introduction of new equipment is, therefore,
not straightforwardly linked to growth. On the other hand, high economic growth
sooner or later, necessitates the augmentation of capacity, and thereby investments
in structures. Besides, within a reasonable range, new technology often, if not
necessarily, enlarges the optimal size of plant. For Japan, this was indeed the
case during the 1950s and 60s. Gigantic steel mills and chemical plants in coastal
industrial regions were surely the symbol of the high growth era. They all involved
high investments in structures.
All this is, of course, not to say that it is investment in structures that promotes
economic growth. Rather the role played by equipment investment and investment
in structures in the process of economic growth is much more complex than De
Long and Summers assert. Saying that it is equipment investment that promotes
growth is too simplistic. In any case, I have already explained how high economic
growth was generated during the 1950s and 1960s in Japan. Investment in structures
naturally played a major role during the period. On the other hand, in the 1970s
and 1980s when the growth rate was much lower than in the previous period,
equipment investment was done mainly for the sake of substituting for labour and
energy, particularly in the machinery industries and the non-manufacturing sector.
De Long and Summers take the negative correlation between the relative price
of equipment and the equipment investment share as the major evidence for their
argument that high rates of equipment investment were driven by rightward shifts
of the supply curve for equipment, rather than by rightward shifts of the demand
curve for equipment. However, this argument is not convincing. Since most equip-
ment is tradable, its prices are equalized across countries. When high productivity
growth occurs mainly in the tradable sector or the manufacturing industry, the rel-
ative price of tradables necessarily declines in those countries where productivity
grew: Balassa (1964). This was indeed the case for Japan. The negative correla-
tion between the relative price of equipment and quantities, therefore, does not
establish the supply-side causation.
2.3.3 The dominant role of exports

High growth in the 1950s and 1960s was led by domestic demand. In contrast,
growth and cycles of the Japanese economy beginning in the early 1970s and up
to the mid-1980s were dominated by exports. Vigorous investment in the 1960s
had gradually endowed the Japanese manufactured goods with international com-
petitiveness. At the same time, high growth itself also contributed to the compet-
itiveness by allowing firms to take advantage of increasing returns and learning
by doing. Given this background, the structural change around 1970 made exports
the engine of growth of the Japanese economy in place of domestic demand. This
change is clearly visible for the automobile industry, the most important export
industry. The growth of the Japanese automobile industry was led by domestic
demand in the 1960s but by exports thereafter.
Meanwhile the two oil shocks in the 1970s provided a strong stimulus to the
Japanese manufacturing industry. Innovations were particularly vigorous in the
machinery industries and by the end of the 1970s the machinery sector came to
dominate growth and cycles of the Japanese economy. The first oil shock, though
it was an adverse supply shock to the economy as a whole, in fact had some
favourable effects on machinery industries. In the first place, an increase in costs
due to the oil embargo was very slight in machinery industries as compared to
other industries such as chemical, paper, and metal; the energy coefficients in the
1973 Input–Output Table were 9.9 (chemical), 7.2 (iron and steel), 4.7 (paper and
pulp) and 1.4 (machinery), respectively. Therefore when a sharp increase in the
real price of oil, equivalent to terms of trade deterioration in Japan, brought about
the 10 per cent depreciation of the yen, machinery industries benefited by gain-
ing international competitiveness. During the period 1973–76, WPI in machinery
industries increased only by 25.2 per cent in Japan as against the 40.3 per cent
increase in the US. And yet during the same period, the yen depreciated against
the dollar by nearly 10 per cent . This means that the price competitiveness of the
Japanese machinery industries improved by 25 per cent .
Beyond that, the oil embargo created a huge transfer of money from the oil im-
porting countries to OPEC; about 16 billion dollars in the case of Japan. This newly
created oil money was then eventually loaned to developing countries through inter-
national financial intermediaries. To sustain domestic investment, growth- oriented
developing countries needed to import machinery. In this way, machinery became
the product which enjoyed exceptionally high demand, particularly by developing
countries, in the generally depressed post first oil-shock world economy. Helped
by price competitiveness, the Japanese machinery industries emerged as a chief
supplier to meet this worldwide demand.
After the second oil shock in 1979, amid debt crisis many developing countries
had to curtail their imports. But in place of them, the US, backed by the overvalued
dollar, became the major importer of machineries. In 1970, the share of machineries
in Japan’s total exports was still less than half, but by the mid-1980s it had risen
to three-quarters.
The emergence of the machinery sector was brought about by vigorous invest-
ment in this sector. The machinery block consisting of general machinery, electric
machinery, and transportation equipment shares a common catalyst, the integrated
circuit (IC). ICs enormously enlarged the potential areas of innovations.
Exports consisting mainly of machinery have indeed become a major force
generating growth and cycles of the economy as a whole in the 1970s and the
1980s. For example, the contribution of exports to the growth rate of real GNP
was on average –0.2 per cent for the period 1956–70 but it rose to 0.6 per cent for
the period 1976–85. For 1980–84, the relative (percentage) contribution of exports
to growth reached the astonishing level of 40 per cent!
Likewise the simple correlation between industrial output and exports is −0. 41
for the period 1960 Q3 to 1971 Q4, but it is 0.56 for the period 1972 Q1 to 1985
Q4. The change in the sign of correlation is consistent with the view that in the
1960s changes in output were not export-led, rather exports were a ‘vent for surplus
production’; In contrast, in the 1970s and the early 1980s exports have become a
major force generating business cycles in Japan.
Table 2.6 Variance decompositions of investment, export and gross domestic demand
SE Investment Exports GDD

1966(IV)–1973(I) (10 quarters ahead)
Investment 0.068 72.93 14.48 12.58
Export 0.060 29.35 39.91 30.74
GDD 0.034 51.96 20.18 27.86
1973(II)–1984(IV) (10 quarters ahead)

Investment 0.049 43.28 51.53 5.18
Export 0.091 25.92 71.51 2.57
GDD 0.011 32.50 27.80 39.70
Note: Based on VAR with 4-period lags.
I also estimated a vector autoregression (VAR) model containing quarterly

investment, exports, and gross domestic demand to obtain the variance decompo-
sition of each variable (Table 6). Since the variance decompositions stabilize after
10 quarters, the results in Table 2.6 are for 10 quarters ahead. They suggest that
investment was very autonomous in the period 1966 Q4 to 1973 Q1 while exports
were very autonomous in the period 1973 Q2 to 1984 Q4.
In passing, these results imply first that the assertion that the flexible exchange
rate regime is responsible for a fall in the rate of economic growth presumably
because it makes exchange rate more unpredictable, is simply inconsistent with the
facts, at least in the case of the Japanese economy, since the relative contribution
of exports to growth increased in the 1970s and 1980s. On the other hand, they are
consistent with the fact that the correlation between the Japanese and American
business cycles is higher in the 1970s and the 1980s than in the previous period.
2.4 The 1990s

Export-led growth created a surge of current account surplus and trade frictions
among Japan, US and Europe in the 1980s. At the same time, the yen appreciated
from Y=240 per dollar to Y=125 during the short period 1985–87. Yoshikawa (1990)
demonstrated that the sharp appreciation of the yen during this period broadly
reflected increases in labour productivity in the Japanese export (machinery) in-
dustries.
After a brief recession in 1986, the Japanese economy enjoyed an average five
per cent growth for four years beginning in 1987. The growth during this period
was basically domestic demand-led; It was in sharp contrast to the growth during
1975–85. The biggest problem of this period was that the asset price ‘bubbles’
accompanied sizable bad investment. When the economy entered recession in
1991, it was left with gigantic bad loans. As of 1997, the amount of bad loans the
banks carry is estimated to be 80 trillion yen or 16 per cent of GDP. This is not
the place to make premature comments on the current issues, but I suggest that
the fragility of the financial sector is not the only and the most important cause of
the extremely poor performance of the Japanese economy during the 1990s.2 The
average growth during 1991–97 is below 2 per cent .
An important fact is a sharp contrast between manufacturing and non-
manufacturing sectors. The Japanese manufacturing industry such as automobile
and electric machinery still keeps high competitiveness today whereas the appreci-
ation of the yen made a significant portion of non-manufacturing sectors non com-
petitive. Investment in these declining industries (mostly in non-manufacturing,
and particularly in small firms) understandably stagnates, and an extraordinarily
depressed investment in these industries is one of the major factors underlying the
deep recession in the 1990s.
2.5 Concluding remarks

Several important conclusions can be drawn from the analyses of growth and busi-
ness cycles in post-war Japan. First, the best model to explain high growth of the
Japanese economy during the 1950s and 1960s is the Lewis model supplemented
by demand factors. Technical progress is certainly a very important factor but the
extent to which technical change bears fruit in economic growth depends crucially
on how it affects growth of demand; see Aoki and Yoshikawa (2001). Technical
change, for example, by lowering the prices of consumer durables, certainly con-
tributed to high growth of domestic demand in the 1950s and 1960s. It also made
the Japanese manufacturers a world major exporter of machinery in the 1970s and
1980s. In both cases, technical change raised the growth rate in a roundabout way
through its subsequent effects on demand, not through a direct shift of production
function. The causality could also run the other way; there is a significant effect
of demand on technical progress. The effect of the emergence of the Japanese IC
industry on the IC equipment manufacturers is a good example.
Second, exports and the exchange rate are the key variables to understanding
the Japanese economy from the mid-1970s up to the present time. Japan imports
virtually all its raw materials from abroad, and, therefore, to finance imports,
exports have always played an essential role. However, the high growth during
the 1950s and 1960s was basically led by domestic demand. In contrast, growth
during the period 1975–85 was export led. The appreciation of the yen beginning
in 1985 stopped the export-led growth, and, at the same time induced reforms of
the declining industries. The sectoral adjustment, however, has not been smooth,
and the economy has suffered from a long stagnation during the 1990s.
Acknowledgement
Section 2.2 draws upon Yoshikawa (1995).
Notes
1 However, the real business cycle theory cannot adequately explain business cycles in
Japan. See chapter 2 of Yoshikawa (1995) for details.
2 Beginning in 1997, the credit crunch exerted substantially negative effects on the real
economy, however. See Motonishi and Yoshikawa (1999).
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nomic Reforms, London: Macmillan.
3 Productivity–investment
fluctuations and structural
change
Bernhard Böhm and Lionello F. Punzo
3.1 Introduction
In this chapter, we review the growth experience in the last two decades of three
European countries and compare it with that of Japan and the US. The stylized
facts of these economies (Sections 3.5–3.6) exhibit dynamic discontinuities in
whose interpretation we employ concepts imported from dynamical systems and
bifurcation theories. Relying upon a framework of cross-regime dynamics (intro-
duced in Sections 3.2–3.6) where regime switches represent structural changes,
our approach departs from traditional and modern growth theories also in two
other respects. On one side, we use a sectorally disaggregated description of the
economies. On the other, we focus on observed sectoral paths which are typically
fluctuating, instead of looking for long run, steady states homogeneous within the
dynamic structure of a given economy. In fact, dynamic heterogeneity turns out to
be so pervasive that it has surfaced in recent growth literature.1
From a methodological point of view, the present exercise goes in the direction
of developing qualitative econometrics as suggested by R. H. Day (e.g. in (1992),
(1993), (1994)). This approach to modelling stresses emulation of qualitative prop-
erties of given historic data sets rather than quantitative estimation and prediction.
We make an attempt at marrying it with an inductive framework which focuses
upon the interdependence between structure and dynamics. Sections 3.10–3.14
experiment with recent techniques for the econometric modelling of structural
change, while the last section reviews some of the open issues in the research
agenda on multi-regime dynamics for large scale model economies.
3.2 Dynamic approaches

The analytical framework that we employ is constructed as follows. An economy
is described as a set of ‘sectors’, just like in certain static models of the input–
output family. The relationship with the latter ends there, though. We focus on an
economy’s time evolution by trying to reconstruct, from stylized historical facts, the
dynamic relationship between capital accumulation and productivity. Employment
and its dynamics, which form an increasingly important policy concern in particular
48 Bernhard Böhm and Lionello F. Punzo
for European economies, is left to lie behind the scene, so to say, as a shortcoming
of our choice of isolating certain aspects of dynamics in relation to technological
change. Thus, our framework and empirical analysis can only give a partial account
of the historical evidence. This is certainly a high price to pay for making the first
steps in the direction of structural dynamics, the dynamics of interconnected sectors
within an economy.
One way to present the heuristic model implicit in our approach is by compar-
ison with the available alternatives which try to explain dynamics through tech-
nology and generally conditions of production.2 For the sake of our argument, one
may lump them into two ‘traditions’ . In the one associated with neo- (and new-)
classical theories of growth, modelling style requires that production functions be
assumed, factors be measured by their physical quantities, and capital (defined
by a variety of measures, as fixed as well as human capital) appears as the stock
available at a given date. In other words, stocks are functionally related to the flows
they are assumed to generate, and the posited production functions relate, at each
point of time, productivity, factors endowments and technology. Growth reflects
their properties.
We will experiment, instead, with an analytical framework which does not
contain production functions explicitly. We maintain that it does not require them,
though assuming such functions cannot cause great harm. It may cause some con-
fusion, but it may also help to follow the argument. We try this out for several
reasons. First, we share the view that the notion of capital stock becomes rather
shaky when we come to make it operational, as it is an estimate whose value
depends crucially on a number of assumptions.3 Second, our empirical analysis
is sectorally disaggregated and requires several (sector-specific) production func-
tions. This would amplify statistical and conceptual problems that practitioners, of
for example input–output analysis, know very well. Capital stock as an argument in
a production function seems to perform better, if at all, in the aggregative parables
of growth or long run dynamics, while here, we look for an account or explanation
of shorter run sectoral dynamics. The latter exhibits time variability that is hardly
compatible with the stability of some underlying technical or production relations.
This is all the more important as we allow for the possibility of innovations which,
by definition, reflect such instability.
The alternative tradition4 still offers two possibilities. The ‘growth view’ sees
technical progress as a fundamentally embodied phenomenon and, thus, maintains
that investment is its (privileged) vehicle. Productivity growth is faster, the faster
the process of capital accumulation. This view claims to be dynamic as it is best
formulated in terms of growth rates, rather than levels, of the chosen variables
(productivity and investment). At least in one version, it ‘regresses’ the growth
rate of aggregate net output on the growth rate of aggregate investment. The idea
of explaining the dynamics of a flow (productivity) with the dynamics of another
flow (investment as capital formation, instead of capital stock), thus without using
the production function,5 has received a variety of formulations. Still, the classic
one that deploys the ‘technical progress function’, was first proposed by Kaldor
(1957) and later cleared up, analytically, in Kaldor and Mirrlees (1961).6
Productivity–investment fluctuations and structural change 49
The lack of consideration for stocks and related production functions is a
feature typical also of the neo-Schumpeterian view. One can even say that the
latter questions the very existence, not merely the relevance, of a functional or
causal association between capital stock and investment with productivity. The
dynamics of the latter is explained by the pace of innovations, e.g. new forms of
organization of production and distribution and/or new products which increase
the gap between material input costs and sale prices (i.e. value added). Investment,
hence, could really ‘explain’ neither the time profile of economic performance nor
that of technological advance. Models in the neo-Schumpeterian style are thus
best formulated as pure flow relations, in terms of variables like employment and
productivity measured by value added (though more recent versions do allow for
some role for capital). This is reflected in the fact that, in contrast to the two
versions of the growth view above, full employment is neither an assumption nor
a result. The possibility arises of macroeconomic and sectoral fluctuations which
involve the dynamics of labour employment. In some cases, they may be so severe
as to lead to dynamic discontinuities, or structural changes.
In synthesis, the analytical framework we are going to present, is based upon
only flow variables, which do appear in the latter tradition but are treated differently
here, in a context where growth, oscillations and structural change are facets of a
unique dynamics. Here too, variables are manipulated so as to obtain growth rates
from levels but they are also sectorally disaggregated. The parallel dynamics of the
sectors in an economy is thus represented as a set of fluctuations in growth rates,
an idea that belongs to the Schumpeterian tradition and was fully articulated by
R. M. Goodwin for the dynamics of income distribution in a one-sector, two-class
economy.7 We use growth rates to identify and to characterize structural cycles:
i.e. a form of basically irregular oscillations across growth paths.
The typical output is illustrated briefly in Sections 3.5–3.6, with reference to
the histories of the European economies, Japan and the US.
3.3 The Framework Space

We are thinking of an economy as the analogue of a complex dynamical system. It
has therefore an architecture given for example by the economy’s sectoral wiring.
This notion can be naturally extended to a set of interconnected economies. It
can also be simplified as when we think of an economy as a single aggregate
sector. Hereafter, economies are blocks or systems of interdependent sectors but
the logical construction is indifferent to the level of aggregation.8
As anticipated, the construction9 starts with time series of value added (VA),
gross physical capital formation (I ), both taken in real term, and employment
(E). Dividing VA and I by E, time series data are converted into intensive form
ν and i respectively. In the following, we will consider the growth rates of value
added per person employed and of physical gross capital formation per person
employed, as the state variables of the systems investigated. Data sets are taken
in a disaggregation into economic sectors, and whenever available the OECD
classification of productive sectors is followed.10 We examine such time series for
three European countries (Italy, Germany, France), using Japan and the US as a
benchmark.
The growth rate of value added per person employed in sector j at a given date
(which is omitted here), is defined11 as
gv,j = d(VAj /Ej )/dt∗(Ej /VAj ) (3.1)
while the growth rate of gross, physical capital formation per person employed in
sector j is
gi,j = d(Ij /Ej )/ dt∗(Ej /Ij ) (3.2)
An economy is a set of, say, k distinct sectoral sub-systems whose time behaviour
is represented by paired time series of growth rates. There will be therefore 2 by
k pairs at each date in the history of the given economy (from 2 by k time series).
A pair of values for (3.1)–(3.2) gives the co-ordinates of the dynamic path of the
assigned sector in a plane. The co-ordinate space will be called the Framework
Space (FS) after a certain logical operation has been performed.
The plane is endowed with the Innovation and Accumulation axes, taken in
the conventional order. The former is associated with the growth rate of value
added per person employed, the index of productivity growth chosen here. The
other axis records the pace of investment plotting values of the variable (3.2),
indexed again by both the sector index and the corresponding date. The abscissa
axis is the Innovation axis as in the neo-Schumpeterian interpretation produc-
tivity dynamics is functionally independent of capital accumulation, at least in
the extremistic representation proposed before. Conversely, the Accumulation axis
monitoring the process of change in investment intensity, would be the focus of
conventional aggregate theories of growth and technological progress. One novelty
in our framework is that the two axes are plotted one against the other.
A sectoral path can now be traced as a sequence of dated states, or pairs of
co-ordinate values in FS. States are dated according to a ‘clock’, a device defin-
ing conventionally the relevant time horizon. One can consider the time intervals
used to construct the original data set as the ‘natural’ periodisation. This does not
necessarily provide a clock for all investigative purposes.12 A sort of temporal ag-
gregation, imposing an ‘artificial clock’, can often be useful as a simple smoothing
device.
The criteria to carry this out may vary with the phenomena under investigation
and one’s own viewpoint. One may, thus, orientate the determination of time in-
tervals on for example the macroeconomic business cycle, measured by the time
span between peak or trough values of the GDP growth rate. Alternatively, external
or supplementary information can be used to determine a time breakdown based
upon the rhythms of some exogenous and/or domestic shocks.13 These two can
be interpreted as macroeconomic clocks and be employed to generate a uniform
periodisation across sectoral dynamics. However, a general cycle period may differ
sensibly from sectoral cycles and, for certain investigations, the properties of sec-
toral fluctuations must be retained. A clock and the implied periodisation should,
therefore, be considered as experimental devices whose worth depends upon what
they allow us to see.
Once the clock has been chosen, a state represents the growth path of a sector
at a given date. Thus, in comparison to more conventional approaches, the second
novelty introduced here is that a path is observed through two state variables,
instead of one. Moreover, overall evolution is reconstructed as a set of growth
paths akin to a segmented trend14 , exhibiting in general a variety of oscillatory
patterns. Of course, segmentation collapses into a single (smooth) trend if we
choose the long run periodisation, i.e. a ‘date’ as long as the total span of the time
series available.15
The geometric representation in FS borrows concepts from the mathematical
theory of dynamical systems. Any given sequence of dated states or trajectory
is, thus, a single instance or realization of the phase portrait in the sector’s own
phase diagram. As any of the empirically given pairs of growth rates typically
jumps dramatically around its state space, sectors are similar to generic dynamical
systems in the two-dimensional (gv , gi ) plane with some complicated dynamics.
That the actual history of a sector gives us a single time series of states (single
trajectory) implies that it is difficult (if not impossible) to recover the mathematical
model, or dynamic equation(s) with appropriate restrictions, which generate such
trajectory. However, one such system does exist conceptually and it may take a
formulation in continuous or discrete time.
We pick up the latter as it is standard in the econometric and time series
approaches. Some of the econometric issues involved in practical model selection
are discussed in Sections 3.10–3.14, where also the issue of linear versus non-linear
specifications is reviewed. Here, for the sake of the argument, we may simply posit
a two-dimensional system of first order, generally non-linear difference equations
g j,t+1 = Φ(g j,t ) (3.3)

where vector g j = [gv , gi ] is defined on the whole of the FS.16 Although (3.3)
may appear structurally simple, it can still be a monster to analyze, depending on
the properties of the Φ maps (recall that Φ is a 2-dimensional vector). The latter
can be simply assumed positing models and family of models parameterized by
their associated Φs. This would be the instinctive attitude of a well trained model
builder. If, instead, we refrain from of this procedure and try to recover them from
actual data, the issue looks altogether different. It is reasonable to expect that the
system be implicitly defined only locally, around an observed path, and then a
linear representation might prove good enough, or be all we can get. But then, if
we are lucky, we get more than one model (3.3) for a sector. Let us look at the
architecture of the economy.
Treating an economy as a system of interdependent sectors, we have two equa-
tions in discrete (continuous) time for each of its k component
g j,t+1 = Φj (g 1,t , g 2,t , . . . , gk,t ) (3.4)

Equation (3.3) is then the special or degenerate case of (3.4) whereby sectoral
paths are independent of each other or dynamically de-coupled17 , hence, only
‘own’ state variables appear. It is suited to represent either a closed aggregate
economy or a sector whose dynamics has no functional relation to the dynamics of
any other sector. A system (3.4) could be built by simply assembling k such pairs
of equations and therefore it would be appropriate to represent a set of economies
considered to be close to each other, and/or many parallel but de-coupled sectors.
The former is typical of frameworks for cross country studies: if we reproduced the
performance of a set of economies in one and the same FS, we would be using this
formulation implicitly or explicitly.Although such exercises can be performed (and
they have been presented elsewhere18 ), in this chapter we focus on the interplay
between sectoral paths within an economy, and therefore cannot expect a standard
version of (3.4) to have such an orderly block diagonal structure.
On the other hand, we may also simplify a system (3.3) by de-coupling the
state variables of the sector. In this case one of the variables would drive the other
without feedback from the latter’s dynamics. We have a triangular system which
can be treated as a one-equation model once we know (or assume to know) the
dynamics of the driving variable. Compared to (3.4) this yields a doubly simplified
system that can be used for example to illustrate how to begin with an econometric
approach (it is used in Section 3.12).
All in all, it is justified to resort to simplified systems either when dynamic
independence is an acceptable assumption, or else as a step in the process of
learning how to build a model of full interdependence between sectoral paths.
Equations like (3.3), for an individual sector are best treated as proper subsystems
of (3.4).
3.4 Regimes
It was pointed out how the one-sector model (3.3) reveals a twofold difficulty. One
relates to mathematical formulation for, if we choose discrete time dynamics as
might seem more appropriate, we are already in a realm of chaotic dynamics, given
that in principle the system (3.3) is non-linear.19 The other difficulty is related to
the insufficient statistics to identify the explicit form of the system, i.e. the pair of
Φ1 , Φ2 functions. Both problems are obviously greatly amplified once we allow
for a feature of reality, that is the dynamic interdependence among sectors as when
positing system (3.4). In fact we are dealing with k parallel but coupled two-
dimensional dynamical systems (the k sectors). Even if we assume each of them
to be relatively simple, the overall or global dynamics can be anything one may
imagine. An economy has a complex dynamical scheme.
One may hope that, in real economies, interdependences take up some simpler
form, with asymmetries introducing hierarchical orders, hence decomposing (3.4)
into smaller, in principle computationally simpler, block systems of non-linear
equations. At any rate, modelling (3.4) explicitly requires a strategy to choose
the Φs.20 On one hand, one can introduce a set of hypotheses into an estimated
version of (3.4), to the effect of generating dynamic behaviours compatible with
actual, observed dynamics. The latter approach is based on an understanding of
the dynamical system (3.4) as a reduced form of some, yet unspecified, model in
structural form. Alternatively, in a more standard approach we start with assump-
tions defining one such structural form, then the corresponding reduced form is
derived, and finally simulated dynamics is analyzed in comparison with the actual
one. Along this line, it would be natural to begin with production functions, etc. as
is typical of production-oriented modelling, and to choose them with the desired
properties. The two strategies are, of course, compatible with one another, the dif-
ference being essentially methodological. (Section 3.11 illustrates an application
of the former and reviews some of the estimation problems involved.21 )
Here we proceed heuristically as follows. If (3.4) above renders the global
model of a sector’s behaviour, one can segment it into a set of local models giving
dynamics under particular conditions, i.e. for certain values of the two co-ordinates.
The intuition is that, locally, one can almost always represent dynamic behaviour
with a description that in principle is simpler (often, linear) than the overall dynamic
model. Such a local model is meant to explain the dynamics from one path to a
‘nearby’ path of the same family or dynamical class. A regime is a family of growth
paths that are all generated by one and the same standard model, a canonical model,
for restricted sets of values of its parameters. Thus, in the FS we are going to
distinguish six regimes plus one special regime, the Harrodian generalized set. To
start from the latter, all paths exhibiting the typically steady state property of time-
constant ratios of investment to value added belong to the Harrodian generalized
set. The latter, of course, includes as special cases those steady state paths of
constant levels of capital and output discussed in conventional growth theories (as
shown in Section 3.7). The set of paths with investment and value added growing at
the same rate (i.e. the 45◦ line) represents the natural extension. When it comes to
empirical analyses, it will be convenient to speak of a Harrodian corridor around
the line, where the two rates are almost equal to each other, to allow for small
deviations, errors and the like, in statistical data. As the problem is general,22 it is
better to keep calling it a set, and recall that it has the typical property of Harrodian
paths, being a knife-edge.23
The ratios between growth rates in the FS yield one of the parameters of the
canonical model. They can be either larger or smaller than one, and a ratio of
exactly one can be treated as a bifurcation value. It corresponds to all paths be-
longing to the Harrodian set, and it can be used to characterize that set compared
to all others. On the other hand, the four semi-axes can be used to yield the second
parameter. All pairs of values of growth rates in the first and third quadrants pre-
serve the same signs (positive value is associated with positive, and vice versa),
while for paths in the 2 and 4 quadrants signs are interchanged. This reflects the
fact that the underlying relationship between levels of variables, i.e. ν and i, is
increasing or decreasing.24 This can be represented, simplistically, by a second
parameter ranging on the real line: for positive (or negative values) we get thus
either relationship.25 The Harrodian set together with the other 4 semi-axes in
a FS 26 can now be used to induce a particular partition into dynamical regimes.
Each regime corresponds to a family of realizations of the canonical model for
values of the two parameters in partitions of the parameter space induced by their
‘bifurcation values’, 1 and 0, respectively.
In the Innovation regime (regime I) corresponding to the area of the first quad-
rant below the Harrodian set, all paths show positive productivity growth rates
exceeding positive investment growth rates. The name was justified in Section
3.2, as in this area the functional association between productivity growth and
gross capital formation is nil or weak. Likewise, the area above the set, where
productivity falls behind investment growth, is the regime that can be associated,
though not uniquely, with conventional growth theories which conceive growth as
a capital-driven path (it is regime VI). With the quadrants numbered clockwise,
beginning with the innovation regime – and observing that the positive and the
negative quadrants are further subdivided by the Harrodian set – a classification is
obtained: with number II being associated with ‘restructuring’ and showing nega-
tive investment growth but positive productivity growth, while the remaining three
are mirror images of those just described.
It is only when the (gv , gi ) co-ordinate plane is endowed with this theory-
induced partition, that it makes sense to call it framework space. Traditionally,
theories see only regimes I and VI (and their polar cases, III and IV respectively)27 .
The introduction of regimes II and V presents us with the possibility of analyzing
oscillations that fall outside standard economic dynamics. Oscillations are now
fundamentally across growth paths and these cannot be treated as (sometimes,
purely virtual) long run equilibria or steady states.
Dynamics that takes across regimes can be associated with structural change,
for it is the ‘model of growth’ that is changing, then, not just its quantitative
properties.
3.5 Of country movies

In general, an economy may refer to a district, a group of firms or an economic
and/or administrative region or finally a macro-region like the Italian North-Centre
and the Mezzogiorno.28 Here, it stands for whole countries: Italy, France, Germany,
Japan, the US. Sectors located in a given economy are treated as component sys-
tems, location in different economies making them different. The ultimate, binding
constraint to the detailed treatment of economies is in the available data, the way
in which it is collected and the level of aggregation in which it is finally released.
Ideally, systems should be as close as possible to microeconomic decision units.
The foregoing results span a set of stylized facts: against them the empirical rel-
evance of certain theoretically defined dynamic behaviours represented in the FS
may be tested, and assessed also against other theoretical scenarios.
There is a distinct FS for each economy. Visualizing its history requires to
take as many pictures as there are ‘dates’, their sequence becomes a movie.29 In
each picture, the simultaneous states of several sectors are super-imposed giving
a cloud of growth paths. For one sector at a time, a single picture can represent its
evolution giving the cloud of paths followed by the sector over different dates.
As changes of regime are structural changes, the dynamics of sectors and/or
economies that take them across regimes, is a structural dynamics, represented
with the level of precision allowed by a low dimensional dynamical model.30 Our
stylized facts refer to such structural dynamics, and do not retain the fine description
of the business cycle approach.
We reproduce the movies of five countries (See Appendix 3.2). Any one of
the graphs, with all dots erased, is an example of FS, showing the partition in
six regimes and the Harrodian set. Each movie is a set of five sequential shots or
pictures in the respective FS, plus one picture that represents the long run average
path in the two growth rates. The latter is reproduced for comparison with the
viewpoint of the conventional growth approach: it looks however at only one of
the two rates, the one defined only for the aggregate economy. A uniform clock
marks the time for all three European countries; it is derived from the timing of
the business cycles in the countries GDP growth rates (peak to peak). This unique
clock cannot capture some important features of the observed fluctuations, like
the different phasing of Europe with respect to the US and Japan (for the latter,
a different periodisation has been used), but this is not the focus of our analysis.
In any case, a date is a full cycle; hence, in each picture, the growth dynamics
in terms of net rates in the two chosen variables is averaged over the set of years
spanning the corresponding business cycle. Likewise, for ease of comparison, a
fixed focus has been adopted: in other words, all graphs have the same maximum
and minimum values, so that clouds of sectoral paths can be directly compared
across time and/or across countries.
We see at once that equilibrium behaviours, i.e. growth paths that persist for
longer than one date, and paths along the 45◦ line are generally ‘rare’. Moreover,
even when they represent the behaviour of a sector over some date, they are soon
abandoned. (This cannot be said to be entirely true of Japan and for this reason
it would be better to separate the comments on this country’s dynamics from the
others.)31
Over the time span of our graphs, Italy, France and Germany share the many
difficulties of devising and shaping their participation in the process of European
unification. Japan, instead, is exiting the era of fast growth and has to cope with
the problems of building up and then adjusting to a new role as an internationally
integrated economy. That countries are in quite different initial conditions, is an
interesting feature of the exercise.
One can see that, between 1970 and the early 1990s, there is not a single sector
in the three European countries selected, which does not cross from path to path.
In other words, ‘traverse dynamics’ showing the instability of individual growth
paths is a generic property both across sectors and across countries. The actual
histories of the sectors, moreover, appear to be made up of sets of traverses taking
them from path to path but also from regime to regime. In this light, for the three
European countries under investigation, the 1970s and 1980s are years of high
instability. Its intensity is not the same, though. Italy seems to be the least stable;
France is somewhere in between, at the other extreme lies Germany.
A marked tendency to instability seems to be characteristic of the history of
Italy in particular. Although this may also reflect some distortion in the data, we
are inclined to believe that it is evidence of a drive towards increasing flexibility
in response to increased uncertainty in the economic environment. The Italian
literature has seen this in the strategies of the industrial sectors. On the other hand,
alternating low and high levels of investment, perhaps, are due to the incapability
of facing structural problems (linked with technological lags, geographic dualism,
and other issues), and the general inadequacy of the various economic policies
implemented.
Italy, with France, had a fast accumulation-driven development in the 1960s
and shared the experience of great social unrest at the end of that decade signalling
change of political and economic atmosphere. The 1960s were a time of rapid
output expansion at low labour cost, and often obsolete technologies, fuelling
rapid accumulation. This path came to an end due to causes internal to the countries
and to the accumulated effects of the past history, thus totally endogenously. For
both countries, a sequence of labour union strikes marked the end of the post-war
reconstruction period and the need to find a new development model.
One notices that in the 1970–73 period, France’s sectoral behaviours seem to be
more dispersed, and along a different direction( the gi -, instead of gv -axis). Italian
sectors reacted to the new relative factor prices with a comparatively higher, and
sectorally more homogeneous, recourse to innovative behaviours. If development
at the very end of the 1960s has seen the restructuring of traditional sectors in the
industrial core of the country, with fast productivity gains, this process seems not
to have spent all of its impetus yet.
The dynamics of the French economy, on the contrary, appears to be markedly
diversified across sectors, though all sectoral paths are in the first quadrant (with
a majority of them actually in regime VI of fast accumulation). Such behaviour
can perhaps be explained (in accordance with the existing literature) recalling
the devaluation of the French franc at the end of 1969, which created sheltered
conditions for French firms. Such sheltering effects persist in France while Italy
begins to face the need for re-structuring already in the following period, 1973–
79. For the French it is the time to invest, with the bonus of generous investment
subsidies, in certain industries at least; Italian sectors, making a bare living under
the shelter of a French-style exchange rate policy, begin to feel the shortcomings
of the relatively high investment levels realized in the earlier period. This induces
paths to fall into regime I and slowly move towards regime II, with not a few sectors
landing there quite soon. Accommodating macro-policies have the sole effect in
both countries of postponing or slowing down a much-needed re-adjustment.
Adjustment comes in the first half of the 1980s. Both countries have to restruc-
ture, there is however a process of diversification in the reactions to new environ-
mental conditions between traditional and modern sectors. In Italy, restructuring
affects all key sectors of the economy, with dramatic labour shedding (partially
absorbed by an expanding tertiary or service sector) and little investment. Here,
all evidence indicates that data disaggregated by firm size would be much more
informative. Different strategies by big and small firms vis-à-vis investment and
innovation lie behind the drive for sectoral specialization. The Italian traditional
sectors (the stronghold of its SMEs) had shown from the end of the 1960s great
capacity to re-organize and to do well in an internationally competitive market.
Firm-level evidence points out that this is again the case in these years. With their
inventive rearrangements (in industrial districts, network firms, etc.) SMEs are
the candidate to play the role of backbone of Italian economy and its rescuer of
last resort. In France, the slowing down of the investment pace is somewhat more
diversified across sectors.
The years between 1984 and 1988 see in both countries an apparent return
to a more or less dominant accumulation-driven growth model. A new phase of
accumulation seems to be under way. During the period 1988–92 (or 1993), both
economies re-enter a phase of restructuring, more pronounced in France, possibly
related with the creation of the common European market.
While Italy, on average, during the 1970s is going through a process of acceler-
ating investment, restructuring and innovation introduction with quickly harvested
productivity gains, Germany begins with a process of restructuring especially in
manufacturing. This is triggered by labour market rigidities and income distribu-
tion dynamics, that is favouring labour share to the detriment of profits. The first
oil shock is accommodated better than in the nearby countries (or Japan and the
US for that matter) by refuelling the economy with public deficit that sustains the
pace of accumulation in certain sectors. It is the next oil shock (in contrast to what
happens to the other European countries) that brings about a dramatic change in the
dynamical scene, causing a remarkable and widespread shrink in both productivity
and the accumulation paces across the sectors (a shrinking of the sectoral cloud that
proves to be much more serious than in the other countries). With the beginning
of the process of labour shedding, capital formation comes to almost a halt, with
some investment being diverted to R and D (at least according to external infor-
mation available in the literature). This re-allocation of investment outlays from
traditional capital goods to R and D and other likewise intangible assets, seems to
be a common trend that begins to manifest itself in all European countries, except
the UK, and perhaps including Japan, and it is one of the ways (perhaps a diffused
one) of realizing industrial restructuring32 . In the next period the overall sectoral
behaviour shows a resumed path of fast accumulation. The predominant dynamics
in regime VI last well into the final period of our investigation, i.e. 1988 to 1992,
a period in which, on the contrary, the other two European countries already fell
into the incumbent crisis of the early 1990s. This phenomenon is largely attributed
to the positive external shock of the German re-unification.
The relative predominance of dynamics in the regime of accumulation driven
growth which is characteristic of Germany vis-à-vis France and Italy, emerges also
from the inspection of the graphs for the US. In other words, the pattern of structural
fluctuations shows more similarities between Germany and the US, than between
the former and the other two European countries. To compare, the US begins the
1970s still on a sectorally widespread path of fast accumulation that continues,
though at a lower average speed and at the cost of productivity slow down, almost
through to the end of the 1970s. The period across the two decades, comprising
the second oil shock, sees a set of measures that, in relative terms, in comparison
to the other countries, managed to cushion investment preventing a dramatic and
widespread fall. The sectoral cloud spreads out over various regimes in the next and
the final period, reflecting indirectly the sectoral reallocation of resources linked
with the restructuring of traditionally important production sectors.
3.6 Some stylized facts of structural dynamics
From this slice of the European growth experience, one may conclude that the
relative weight of innovative behaviours has been on the whole moderate and
certainly only cyclical. Search for greater flexibility, which has been said to have
generated ‘development without accumulation’, does not seem to have been the
driving force behind adjustments to inflation in the 1970s or to the slumps in activity
levels at the beginning of the 1980s. These resulted basically in changes in regime
dynamics. The US exhibit a different picture, more centred on accumulation-driven
behaviours.
These bold generalizations need certainly to be qualified when it comes to
Japan. Its history could be better narrated by what it does not show (as compared
to the others), than by what it does show. Japan has a system of seasonal concerted
wage bargaining (shunto); a tradition of structural or real, instead of simply fiscal
or monetary, policies which is comparable with the experience of Italy and France.
It has a tradition of economic dirigism which is also typically French. Finally, it
also has the large economic presence of the state. However, in its graphs there
are a sufficient number of sectors showing a tendency to equilibrium or persistent
dynamical behaviours; some of them, and some of the internationally competitive
sectors, stay in the Harrodian corridor (as close as they could be) for long stretches
of their histories.
These key stylized facts33 are reproduced also in the sectoral behaviours. After
the high growth period of the 1960s there is a marked and generalized slow down
and a decreased correlation of investment paths across sectors. There is a fall in
growth rates in the 1980s compared to the exceptional performance of the 1960s.
This confirms the thesis that the country entered the 1970s already in the middle of
a major structural adjustment, but it managed to surf over the troubled years of the
1970s and 1980s. Therefore, Japan’s cross-sectorally dominant behaviours lie in
the first quadrant, between the innovation and the accumulation regimes, actually
with a relative predominance of the latter. There is a very limited area and a short
time for re-structuring via regime II, hence by labour shedding (which is dominant
in the European countries, instead). This is experienced only by some sectors, not
the prominent nor in particular those classified as traditional ones (an interesting
fact when compared to the experience of Italy). On the other hand, the new role
of driving forces for the country expansion played by sectors such as general
machinery, automobile, and electronics, is clearly prepared by their innovative
behaviours in the 1970s. The re-shaping of the structure of the Japanese economy
taking place then, is evident from the picture of 1969–72, preceding the first oil
shock.
In all countries, sectors exhibit structural cycles but, compared with their Euro-
pean counterparts, for a typical Japanese sector the sequence, the number of regime
switches and their timing are generally different. Japan’s restructuring takes place
in the 1970s (in particular, it is accomplished after the first oil shock), compared to
the later adjustments in Europe. Here, the intuition is that Japan has been forced to
adjust immediately to the new relative prices for raw materials, searching for new
technologies (hence restructuring in the first regime) and realizing substitution in-
vestment in the 1980s (at a lower level of macroeconomic performance). Reaction
in the European countries has, at first, taken the form of an assorted variety of short
term adjustments induced by macro-policies (perhaps in the belief that the shock
was only temporary and isolated). The second oil shock forced them to realize
that something had changed at a fundamental level. The capability to respond with
structural adjustments, was far greater in Japan.
This shows up in a stable structure with a dynamics founded upon investment
expansion under balanced conditions shared by all sectors. For this, no unique
explanation exists, a mixture of policy considerations, cultural values strongly
shared by the country and economic factors all had a role to play.
This sketchy illustration of the empirical use of the FS is meant to isolate some
stylzsed facts of structural dynamics.34
• Structural change has been the all-pervasive phenomenon throughout the

period under observation, seeing repeated regime shifts in a large proportions
of sectors and of economies investigated.
• Structural change phasing was different across countries, a facet which can
be partially attributed to political business cycles and other related phenom-
ena (e.g. implementation of specific economic and industrial policies).
• Sectoral structural fluctuations seem to follow different patterns in Europe

and the US compared to Japan.
• Innovation and capital accumulation are not alternative growth behaviours:

they belong to regimes alternating in irregular sectoral fluctuations.
• Finally, Harrodian paths are rarely observed and generally they are short
lived.
The latter observation is quite important for the ensuing discussion: the Solow-path
belongs to the same set. But, if we compare our representations of the long-run
(the last picture in each movie), they tend to look similar, which seems to support
the hypothesis of a steady state prevailing in the long-run. Against our structural
dynamics that is working in the medium run, this observation suggests also that long
run (equilibrium) theories basically disregard important segments of the histories
of actual economies. They rely on the optical deformation created by comparing
pictures that are taken at some time distance and leave out the history to which
they belong.
It is this history that reveals the interesting feature, it exhibits the fluctuating
character of economic dynamics and the changing structures that support it. It can
be directly compared with some of the findings of the growth empirics literature.
(See, for instance, van Ark and Toniolo (1996).)
3.7 Growth theories and the fluctuating dynamics in the FS
This was a brief review of what can be seen by applying our analytical framework
to data that are normally analyzed by a growth approach. The fact that growth rates
are here allowed to fluctuate, shows that the FS dynamics tries to embed growth
into the more general phenomenon of economic oscillations. With the possibility
of moving across regimes (travelling along special ‘traverses’35 ) such structural
dynamics incorporates business oscillations and growth as special cases.36
For a comparison with growth analyses, we shall briefly consider the neoclas-
sical aggregate theory of growth and the disaggregated one associated with the
multisectoral model. Both versions employ, explicitly or implicitly, some notion
of production function. We have argued that our approach does not (necessarily)
depend on production functions. This is in a sense trivially true as, technically
speaking, we have no ‘formal model’. Or, at least, not a unique model: for formal
models we depend on others, and this is what we want to show in this and partly
the next sections.
Aggregating over our sectors, we may obtain the growth rates of aggregate
productivity and investment intensity. If then, we average them over the whole
time period (collapsing the sets of dates into a single one), we get something
resembling long run values. Finally, projection onto the horizontal axis of the
history of a given economy thus condensed by a single long-run path, gets us the
key ingredient in the aggregate growth description. Output is GDP, and hereafter
will be denoted as usual by Y . It can readily be calculated as the sum of VAs. Theory
starts with level values of aggregative variables, Y , K for total capital stock, E and
N for total employment and labour force, and so on so forth. From this first level
are derived ratios like y ≡ Y /E, k ≡ K/E, and the like. Finally, we obtain growth
rates to describe dynamic properties of growth paths. In the FS also, we follow a
similar three-layer procedure, keeping the hierarchy between the lowest layer of
raw data and the subsequent layers which are obtained via simple manipulation.
In other words, just like in the standard growth approach, from the lower we also
derive a unique ‘higher level’ description. In the opposite direction the procedure,
of course, does not lead to unique identification, at most it leads to a whole class.
That is, a path in the FS is associated with a whole set of pairs of time series in
the level values of the variables. (This is an implication of the parameterization
introduced above to define regimes.)
Now it can be argued that a notion of regime(s) can be traced in growth theory
and is implied by much of recent growth empirics, though there it is interpreted
as an equilibrium behaviour. In fact, neo-classical growth theory is said to imply
a ‘prediction’ on the long-run dynamic behaviour of an economy. This says that,
under certain conditions (basically reflecting properties of the production function),
(i) there will be a unique steady state value of y, the output per capita, equal under
full employment, labour productivity; and that (ii) the long run growth rate of output
per capita obeys the equation: g − n = λ, where n ≡ dN /dt N −1 , the natural rate
of growth, and λ is the rate of growth of technological progress. The long-run rate
of growth of per capita output would then be controlled entirely by exogenous
technological progress. In the absence of the latter, it would be equal to zero. One
can express this result by saying that the endogenous growth rate of per capita
output gy = (g − n), and of labour productivity (if e = n), are both equal to zero.37
One can write this growth rate as a co-ordinate value on a real line and call
the latter the growth line of per capita output or productivity. Each gy value is
associated with a growth path. Hence, in the absence of technological progress of
exogenous type, the neo-classical theory says that the long-run steady state value
of gy lies at the origin of this line. If it is not there, this is due to λ not being zero.38
Thus, the productivity growth line decomposes into the union of an equilibrium set,
made up of a unique point, and the set of all other paths.39 While the equilibrium
path is sustainable in the long run, all other growth rates can only be associated
with short run dynamics. At any rate, stability in the large implies that, eventually,
the observed path is the one associated with the origin (or with the exogenously
fixed value gy = λ). Any other path describes transient dynamics.
Notice that, due to this twofold property, uniqueness of equilibrium and sys-
tem global stability, this is a non-linear dynamical systems with properties in a
sense typical of a linear one.40 On the other hand, notice also that many dynamic
paths that are associated with different time series in levels (e.g. in the values of
Y, K, E and N) are zoomed into a single equilibrium path on the gy -axis. This
shows that the latter is like a one-dimensional version of our Framework Space.
Hence, in principle, we can introduce a two-regime classification here too: the
systematic equilibrium behaviour, and the transient dynamics. An analogous idea
can be found in, for example, the endogenous growth literature which, re-phrased
in our jargon, distinguishes between a Solow-type regime (with diminishing re-
turns to capital) and a ‘non-Solow’ regime. Any growth rate greater than the Solow
rate is sustainable in the long run if constant returns to capital are at work, hence
parameterization of the implied model is made with reference to the parameters
of the underlying production functions. The introduction of two distinct regimes
on the gy -line rationalizes the possibility of settling into different steady states.41
The related partition can be used to explain as an equilibrium outcome, the per-
sistent difference of growth rates (increasing divergence of growth paths) across
countries, a feature of recent growth empirics that has found large support in the
evidence.
We can re-set the above argument to show some other of its implications and
ramifications. The Neo-classical theory ‘exhausts’ itself on the gy -axis, for the
steady state goes along with full employment and the long run rate is exogenously
fixed by demographic forces as well as by technological change. There is no need
for supplementary endogenous explanatory variables, except for the adjustment
or transitional dynamics. It is really this that, together with the full employment
hypothesis, implies uniqueness of the equilibrium solution.42 This is also the con-
ceptual basis for cross country studies. With each economy one can associate the
observed growth path as a pair of values for y and the productivity growth rate.
All such paths however, monitored through growth rates, are expected to eventu-
ally converge to one and the same value on the growth line. It is the simplicity
of the neo-classical model, that the properties of one dynamics mirrors to a large
extent the properties of the other. Still, we have a dual dynamics between levels
and growth rates that is put together via the various notions of convergence (but,
in particular, with β-convergence).43
The key idea that has emerged in the literature, is that there may be more than
one long-run growth rate, so that countries need not converge to a unique long-
run value of y (nor to a small interval around it). It is to account for the apparent
diversity of the growth phenomenon across countries, if not yet to ‘explain it’,
that the single axis of productivity growth is not enough. Kaldorian and (all) other
endogenous growth theories do exactly this: they introduce extra axes to explain
the long run. They are at variance for their choices of such axes.
Thus, some of the debates raging in the recent, theoretical and empirical, growth
literature can be re-interpreted as statements about long-run dynamics in a one-
dimensional or a two- (or larger) dimensional version of our FS. In the latter,
one can enrich the picture on the gy -axis in different ways, e.g. admitting the
possibility of multiple steady states. This idea extends naturally to the analysis of
many countries, belonging to different clubs and then settling in the neighborhood
of a distinct long run attractor. Or else one can do country by sectors studies, like
those in our ‘movies’ of Appendix 3.2. In all cases, stability is an open issue and
a sensitivity to initial conditions exhibits properties typical of some non-linear
chaotic systems.
Therefore, in our two-dimensional FS there is room for all long run theories:
at its origin, if they are of the neo-classical sort; at its right, if they are projections
of paths in two (or more) variables, one variable being used to explain the other.
Hence, exogenous and endogenous explanations of growth can be taken to rep-
resent two classes of ‘models of long run dynamics’ that need not be excluding
each other. They may replace one other at different times as well as in different
economies or sectors. This depends on the empirical evidence.
However, to capture the stylized facts of the experience of growth, we need to
do more and account for the whole FS. The common long run view of the growth
theories confines itself to the first quadrant and to the equilibrium phenomena that
may appear there. To charter the rest of the plane, we need to look at dynamics
in the ‘shorter run’, shorter than the time span preferred by the growth views, and
allow for out-of-equilibrium dynamics. This is the dynamics taking across paths.
Treatment of investment as the second variable brings in the instability due to
its typically volatile time profile. Growth theory can speak of stable monotonic
dynamics as long as investment is kept out of the picture (e.g. by confining it to
explain the short run adjustment at business cycle frequencies). Once a shorter time
horizon is chosen, investment behaviour re-appears. No surprise, then, in the FS
the typical mode becomes oscillatory dynamics in growth rates. In those instances
where it is fairly regular, it resembles a growth cycle. The empirical evidence shows
that in general it is not of such a simple dynamic variety.44 In the appropriate time
horizon, in fact, the interplay between the chosen variables may take up different
forms: multiple schemes, time varying schemes. To accommodate this feature, the
model implicit in our FS has to be a non-linear system of simultaneous equations
with (at least) two state variables for each sector.
In this light, the Framework Space is a heuristic tool to classify certain em-
pirically observed ‘growth phenomena’, when they can be actually observed, and
inductively produce a usable theory of actual facts.45 And it is proposed as a first
step for an approach that recognizes the possibility of different theories being
consistent to each other.
3.8 Some related issues

The FS has been applied to economies in a multi-sectoral breakdown. A few words
on the tradition of multi-sectoral dynamics may be appropriate. It is useful perhaps
to recall something of this approach, that, in fashion not so many years ago, was
progressively marginalized from macroeconomics by the advent of models pop-
ulated by representative agents and likewise representative firms. Multi-sectoral
theories are in a sort of nobody’s land, for they deploy the notion of a ‘sector’which
is normally neither the macroeconomy nor the individual firm, though it has some
statistical advantages in the way some databases are collected. There, we basically
encounter growth theory but virtually no account of oscillations. These models talk
of long run dynamics in terms of flows, and focus upon equilibrium paths, steady
states in a generalized form. Along a balanced growth path all sectors can grow
if they grow at the same rate, and if they do so, the path of the economy enjoys
certain optimality properties. This balanced growth path, in fact, is the fastest path
an economy can afford in the long run and it is expansionary if (or as long as)
there are no stock or resource constraints. Moreover, what is important for our
argument is that it is endogenously determined in the sense that it is the result of
purely technological properties. The optimal and cross-sector sustainable rate is,
in fact, a purely technological index. As a consequence, differences in balanced
growth rates across economies (like their changes over time) reflect ‘technological
change’ and possibly labour supply (or other resource) constraints.
Multisectoral modelling focuses on an analysis of growth as a cloud of (sec-
toral) dynamical paths, with sectoral employment practically undetermined. We
borrow both notions into our framework, but we enrich them with a description of
‘shorter run dynamics’. This may also account for ‘why growth rates differ’ across
sectors within the same economy or for the same group of sectors across differ-
ent economies. We also do away with the full employment assumption typical of
aggregate growth theories.
Let us now turn to the notion of regime as used here in comparison to its use
in the literature. As already pointed out, a notion of regime has found its way into
the growth field, though only in connection with the cross country convergence
issue. A regime in the sense for example of Durlauf and Johnson (1995) is, intu-
itively speaking, a steady state value of the relevant variable that represents a local
(global) attractor with respect to a (sub)set of countries. Multiple attractors can
then be simultaneously present, implying a globally non-linear growth model. Our
definition differs for two basic reasons: our regimes are defined with respect to
(two) rates of growth and not (the single level of) output per capita. More impor-
tantly, they need not be attractors, though obviously this may not be excluded. In
our interpretation, a regime is a given class of realizations of what might be called
a canonical model, a class corresponding to an open subset of the two parameter
space. Even though within a class there may be one or more point attractors, dy-
namics becomes really interesting when it is not so. This turns out to be almost
always the case in our graphs. Thus, the local model for a given class may or
may not be linear46 . What is clearly non-linear is the dynamics that takes from
one regime to a distinct one. Such shifting from one dynamical model (regime)
to another is the two-dimensional analogue of what has been called ‘multi-phase
dynamics’ by R.H. Day (1994). It is related to the Hicksian ‘traverse’.
Finally, our definition of structural change is clearly context-defined. Having
opted for a description of the economy as a set of ‘sectors’, any change in their
mutual relationships has to be taken to imply a structural change. There is a large
literature on this, basically of input/output affiliation, which identifies structure
with the set of functional (essentially complementarity) relations among producing
sectors (the inter-industry part), with the Final Demand (columns) and Payment
(along the rows) Sectors. If this is the notion of structure entertained, any changes
in coupling parameters and/or in functional forms linking sectors, is naturally
synonymous with structural change.
We have altered in an essential way this static picture of structure. Hence,
sectors are not represented simply by flows of inputs and outputs per unit of
time; they are rendered in their dynamics through the relative rates of changes
of both output (in labour intensive form, hence taking into account dynamics
within the Payment Sector) and of investment (again in intensive form, taking
thus into account, after the necessary statistical modifications, dynamics within
the Final Demand Sector). Therefore, an economy is described by a moving cloud
of paths in two variables. Its implicit system law describes a ‘dynamical structure’,
represented by a model of dynamic complementarities, a generalized scheme of
dynamical couplings in the sense of R.M. Goodwin.47 Any change in the qualitative
properties of the cloud reveals a change in the underlying dynamical structure, a
structural change in this special sense.
The notions of regime and of regime change are introduced to make progress
towards identifying such structural changes through a more adequate definition of
qualitatively ‘similar-and-different’ growth paths. Analysis of empirical evidence
in the FS is preliminary to the inductive identification of such breaks and their
causes.
3.9 Beyond stylized fact

It is clear that all exercises of the above type identify an agenda for further research.
Collecting stylized facts may be a nice exercise, and a large slice of modern growth
analysis is concerned with collecting such facts, explaining them by finding reg-
ularities, etc. The approach above provides one sectorally disaggregated version
of the same sort of growth exercise, and this might be, by itself, a justification
good enough for experimenting with it. From this point of view, it shows its close
relation with a trend towards more disaggregated analysis surfacing in the recent
literature.
There are other uses of our approach beyond reproducing Growth and Business
Cycle theories and empirics. One can address issues like ‘Is there convergence of
the European countries to a common dynamic path, interpreted as a structural
oscillation?’ and then proceed to discuss the policy implications of the answer
to the above question. (This exercise has been proposed in another paper, Punzo
(1997).) Or, one may inquire about the relationship between structural dynamics
at the sectoral levels and macroeconomic performance. However, beyond facts
collection and inductive theorizing, one should make some attempts at explicit
model building and treat the FS as something more than a descriptive device.
One can try several more or less conventional approaches to model building,
that can be best collected into three lines of attack. The first one, naturally, begins
postulating a theoretical model. In our case, this should have all the standard ingre-
dients of a growth model which (more than other models in economic dynamics)
is typically supply-oriented, and this is just like our framework above. What we
would need to do, is to lay down production functions with specific assumptions
(as to their arguments and properties) and to opt for exogenous or endogenous
growth theories, or even a mix of them. Alternatively, we can employ surrogates of
production functions, postulating functional relations between our key variables.
In other words, we can adopt one theory and do modelling in the corresponding
style, either axiomatically or incorporating hypotheses suggested by previously
sifted empirical evidence. From one such model ‘in structural form’, one derives
a reduced form expressing the observable implications of the hypotheses and then
proceeds to test them against evidence. This is one version of what we may call
the econometric approach. This leads to either accepting or rejecting one specific
theory or model, hence in principle rejecting the possibility of having a variety of
locally valid models. But, it is on this possibility that the FS was built.
An alternative direction of attack tries to model mathematically time series data
directly and, by comparison with the former, may thus be called phenomenological.
There is a number of formulations along this line, fromVAR econometrics to Neural
Nets and the like. It is model building along this line that is hinted at in Sections
3.3–3.4, where we introduced some standard dynamic equations that might be able
to emulate actual time series (or, at least, some of their properties). This is in fact
the development that will be sketched out in Sections 3.10–3.14, where some of
those properties are shown to be implied by the dynamic equations, and structural
change is interpreted as parameter change.
The final section goes cursorily over the last of these three lines of attack, the
one which is based upon the typically non-linear notion of Multiphase Dynamics.
There, it is the model that is allowed to change (and to change discontinuously) in
the dynamics depicted in the FS. Such discontinuous change can be represented
with a parameterized model whose parameters exhibit critical values for the asso-
ciated dynamics. This is the interpretation suggested in Section 3.4 to explain the
notion of regime and regime switches.
3.10 Econometric approaches to structural change
Structural change in the context of economic growth is here associated with a dis-
continuous movement between growth regimes. A relationship that is considered to
imply a particular growth regime, must therefore change in a specific way in order
to be associated with a regime shift. In a descriptive framework one may be able
to observe such shifts if growth rates, taken as representative descriptions of the
growth phenomenon under consideration, move across certain boundaries which
partition the space in association with clearly defined rules. Such rules could relate
to specific values or ranges of growth rates or combinations thereof. The latter we
chose in our framework, as indicated in Section 3.4.
It must be emphasized that in our framework we depict a system working its
way through historical time. What we can hope to construct by using econometric
methods is, on the other hand, a model derived from theoretical insights or from
data-mining which comes close to capturing the variability of the data and at the
same time the implications of economic theory. Several approaches can therefore
usefully be applied. In the following overview we intend to show some of the more
recent developments in a somewhat simplified manner. It should be clear, how-
ever, that even the ‘smallest’ theoretical model underlying the previous analysis
(i.e. the model for an aggregate economy) must be represented by at least two of
these econometric relationships – accounting for the symmetric dynamic interplay
between investment and productivity – which could be considered as constituting
some sort of structural form (in the econometric sense). Nevertheless, we shall dis-
cuss econometric issues of structural changes by analyzing single equation models.
Generalizations to systems of equations have to deal with the complications aris-
ing from simultaneity, co-integration (Johanssen (1988), MacKinnon (1991)) and
identification. This can be achieved along the approach adopted e.g. by Hendry,
Neale and Srba (1988) and implemented e.g. in PcFiml by Doornik and Hendry
(1994).
3.11 Traditional econometric models

Traditional modelling may follow the lines of the ‘general to specific’ strategy
of the LSE school (e.g. Hendry, Pagan, Sargan (1983), Hendry (1987)) which
concentrates on linear dynamic equations of the form
A(L)yt = B(L)xt + u(t) (3.5)

with A(L) and B(L) denoting lag-polynomials of a specific order, y and x are
measured variables of interest and u(t) a white noise error process. In terms of
the variables employed in the present analysis, the chosen variables are levels of
investment per employed and of labour productivity. The above specification is
derived from a general data generation process (DGP) which represents the joint
probability distribution of all involved variables.
Let zt be a vector of observations on all variables in period t and Zt =
(zt−1 , . . . , z1 ) . The joint probability of the sample zt i.e. the DGP is

t
D(zτ |Zτ −1 ; Θ) (3.6)
τ =1
with Θ the vector of unknown parameters. Simplifying this general formulation

by adding restrictions will yield an estimable model: The assumptions involved
are
• Marginalizing, i.e. dividing the set of variables into a subset of variables of

interest and those not of interest (Wt ) for the current problem.
• Conditioning by selecting that subset of variables of interest considered to
be endogenous (Yt ) which is determined (conditioned) by the remaining
variables of interest (Xt ). The latter should be at least weakly exogenous i.e.
Xt independent of Yt .
• Selecting a functional form.
• Replacing the unknown parameters by estimated numerical values.
Then the distribution can be written
D(zt |Zt−1 ; Θ) = F(Wt |Zt ; a)G(Yt |Yt−1 , Xt ; b)H (Xt |Yt−1 , Xt−1 ; g). (3.7)
One can recall some standard definitions, in the present context. Thus, strong exo-
geneity would require (Xt |Xt−1 ; g), lagged endogenous variables may not influence
the exogenous ones. Super exogeneity would in addition require parameter vectors
b and g to be independent. Then a change in b would not influence g, which makes
the specification immune to the Lucas-critique.
The partial log-likelihood function of the model can be written

log L(Θ) = L(Θ; yt |xt , yt−1 ) (3.8)
t
and forms the basis for estimation. As the chances of arriving at a correct model
specification according to the true model and the restrictions found by conditioning
and marginalizing are very small, the aim is to find an ‘adequate’ model, i.e.
statistically acceptable and not outperformed by rival models.
Assuming that this process has been successful (and in particular, the condition-
ing step has identified exogenous and endogenous variables), a general dynamic
specification of the form (3.5) allows a derivation of long run properties of the rela-
tionship between the two variables on the basis of the short run information on the
data over the period of observation. Now, to recall our variables, let us interpret y
for the level of labour productivity, and re-denominate x with i = I /E, the intensity
of gross capital accumulation (or gross capital accumulation per employed), and
to simplify, let us use national aggregate data. Moreover, for analytical reasons,
growth rates in the FS will be approximated by logarithmic first differences.
Taking only first order lags in the general equation (3.5), one has
log y = a + b log i + c log i−1 + d log y−1 (3.9a)
which has a static steady state solution for growth rates: D log y = D log i = 0 given
by
log y = a/(1 − d) + [(b + c)/(1 − d)] log i. (3.10)

If [(b+c)/(1−d)] = 1, i.e. with the restriction that b+c +d = 1, we obtain log y =
a +log i corresponding to direct proportionality of y and i. If the restriction does not
hold we obtain the more general relation log y = a + b log i, where a = a/(1 − d)
and b = (b + c)/(1 − d). Thus, the set of solutions of the family of equations
parameterized by the unrestricted vector (a, b, c, d), is associated with the origin
of the framework space, which is in the ‘Harrodian set’ because of the specific
relationship between the two growth variables.
A straightforward generalization to growth paths other than those in the origin
of the FS will yield a dynamic long run solution. This is achieved by setting
log it−1 = log it − gi , etc., where gi = D log i is a given (and not necessarily zero)
long-run growth rate. Then, for any pair of given values of the long run growth
rates, we get by substitution the generalized equation
log y = a + b log i + c(log i − gi ) + d(log y − gy ) (3.9b)

and finally
log y = a/(1 − d) + [(b + c)/(1 − d)] log i − c/(1 − d)gi − d/(1 − d)gy (3.11)
showing proportionality of levels as depending on their given long-run growth

dynamic performance. We may also interpret this result as a long-run growth rate
relationship
gy = (−c/d)gi + [((b + c)/d) log i − ((1 − d)/d) log y + a/d] (3.12)
which is seen to depend on the levels of the variables involved and shows, therefore,
time dependence.
Taking differences of (3.11), assuming the restriction b + c + d = 1 to hold
and the long-run growth paths constant (but different from zero), yields D log y =
D log i, a result compatible with points on the 45◦ line of the diagram. This is
a generalized Harrodian set as it is the linear (one-dimensional) subspace of FS
spanned by all paths with growth rates for the two variables, equal to each other.
A narrow band around it was previously called the ‘Harrodian corridor’ in view
of measurement inaccuracy or, as in the present case, the stochastic nature of the
approach. If, on the other hand, the restriction b + c + d = 1 does not hold, we
have a long run steady state of proportional growth off the Harrodian subspace48 .
The value of [(b + c)/(1 − d)] will indicate the nature of such a long run state
and may classify the system under investigation to which regime the path would
belong. (This justifies the treatment of the model as parameterized by that value, as
carried out in Section 3.4). If, finally, actual growth paths were variable, shifts of
the linear relationships of growth rates would be implied, that depend on second
order dynamic movements, the changes of long run growth rates or (long run)
growth acceleration. In fact it is easily seen that taking differences of (3.11) is
equivalent to assuming a dynamic (first order) relationship in growth rates apart
from a constant, i.e.
gy = bgi + cgi−1 + dgy−1 (3.13)

Its ‘static’ long run solution (i.e. the one with constant growth paths) is gy = [(b +
c)/(1 − d)]gi = b gi as above, while its dynamic long run solution is equivalent
to the first difference of equation (3.11). By comparison to this which is the focus
of much of growth empirics, paths characterized by variability of growth rates, or
‘traverse paths’, belong in a sense to a second order dynamics. They seem to be
typical of the dynamics in the FS associated with any actual economy.
We notice that the short run parameters are compatible with different long run
theories expressed by proportionality in levels, growth rates or accelerations49 . In
fact, the rather general, but linear, dynamic model specification introduced above
may in principle generate different particular solutions or steady states, depending
on whether we assume the given long-run growth paths to be zero, or non-zero
constants, or else variable. In order to analyze changes in the structure implying
these properties we need to investigate the time stability of its parameters.
3.12 Testing for constant parameters

If we remain in the linear framework, testing the dynamic specification proceeds
along the conventional econometric arsenal. In view of the discussion about non-
linear specification below we may mention specifically the use of the RESET test by
Ramsey (1974) which is employed by constructing a more general model including
a higher order polynomial to approximate a different functional form. Taking the
difference between the general and the (linearly) estimated model one should not
find much explanatory power if the null-hypothesis of linearity is correct. So, this
test checks the significance of the parameters in the difference of the two models
by computing the value of the appropriate T.R2 which is distributed χ2n under the
null.
Given the focus of the linear dynamic model on representing steady states it
is necessary to devote our attention to the question of stability of its parameters.
As the resulting states depend on the values of the parameters, any change in them
would also lead to a change in the corresponding steady state. This is especially
important for the question of unit elasticity, i.e. the property of belonging to the
steady state corridor.
The relationship of the phenomenon of structural change as understood in the
Böhm/Punzo approach to the econometric one can be best explained by using a
scatter diagram of the two relevant growth rates gy and gi . The use of prior in-
formation about the time periods over which to compute averages of growth rates
simplifies the scatter diagram. The points observed now show less fluctuation and
usually, due to the selection principle, will show a special pattern that forms the ba-
sis for the interpretation of (sectoral and/or regional) histories and their attribution
to growth regimes. Discontinuously moving across regimes – i.e. travelling along
‘traverse’ paths – is associated with structural change and it is by visual inspection
that changes in regimes are ascertained. The econometric approach typically uses
all data of the scatter diagram and fits a curve under the assumption of constant pa-
rameters taking account of short run dynamic effects. The implied long-run steady
states are those relationships that obtain when short run dynamics has ceased to
have an effect. If constant parameters cannot be rejected by parameter instabil-
ity tests, the model represents one single steady state behaviour derived from the
whole period of observation.
Linear dynamic econometric models need, therefore, to be tested for parameter
constancy (usually the null-hypothesis) against the alternative of a structural break.
The Chow test can be used for this purpose if the time period of the structural
break is known in advance, if not then Hansen’s test (Hansen (1992)) informs
about potential parameter instability. If the sub-period (for which a different set
of parameter values looks likely to prevail) is smaller or equal to the number of
parameters one can use the one-step-ahead forecast errors in Hendry’s forecast test
which is χ2 – distributed with degrees of freedom according to the length of the
sub-period tested.
An alternative way to temporally locate structural breaks is via recursive es-
timation which provides a more general framework and does not require prior
knowledge of possible breaks. This technique estimates the parameters by least
squares over successively increasing periods and, thus, generates a time series of
the estimated parameter vector, say b(t) in the model yi = bt xi + ui for i = 1, . . . , t
and t = k, . . . , T with k the numbers of explanatory variables and T the to-
tal number of observations. The important assumption, however, is that the true
parameter vector is constant. We simply derive varying estimates from differ-
ent data sets. A sudden change in the estimates indicates a structural break, i.e.
the relationship has changed its character. Smooth changes are usually indica-
tive of misspecification. Once some type of instability of parameter estimates is
discovered the above mentioned diagnostic checks can be applied. Furthermore,
the forecast errors from one step ahead predictions can be usefully employed in
the construction of stability tests: vt = yt − bt−1 xt (t = k + 1, . . . , T ). Under the
null hypothesis of constant b and assuming u ∼ N (0, σ 2 ), the v are distributed
N (0, σ 2 ft ) with ft = 1 + xt (Xt−1

Xt−1 )−1 xt where Xt−1 = (x1 , . . . , xt−1 ) . They are
also uncorrelated. The standardized prediction errors are the recursive residuals
wt = [vt /(ft )1/2 ] which are again distributed N (0, σ 2 ). Since they contribute in the
updating formula for the residual sum of squares RSSt = RSSt−1 + wt2 , sequential
Chow-tests can be constructed, e.g. one step or n-step Chow-tests, testing a struc-
tural break occurring one or n steps later. They also appear in the CUSUM-test
which sums the recursive residuals normalized by the sample standard deviation,
and the CUSUMSQ test which basically compares RSSt with that of the whole
sample.
Examples of an empirical investigation using the aforementioned approaches
can be found in Böhm (1996). It was shown how the application of careful spec-
ification searches and tests leads to identification of regime changes, implying
different long-run growth paths.
3.13 Discrete switching models

While for the specification strategy mentioned detection of a change in parameters
is rather the exception than the rule, because the objective is to find a constant
parameter specification satisfying the assumption of linearity inherent in the fun-
damental model, classes of models specified as non-linear from the outset are able
to capture presumed changes or shifts in parameters.
The general class of switching regression models assumes a finite number
of regimes, incorporate (observable or non-observable) switching variables and
may assume the switch point from one to the other regime as known or unknown
(Goldfeld and Quandt, 1972). One can further distinguish cases where the shifts
from one to the other regime are discrete or continuous.
As an example of a discrete switching regression model of two variables yt and
xt take
yt = α1 [1 − D(zt )] + α2 D(zt ) + {β1 [1 − D(zt )] + β2 D(zt )}xt

+[1 − D(zt )]u1t + D(zt )u2t
where the errors of the two regimes (uit ) are identically normal distributed with
mean zero and constant variance and zt is a transition variable. D(zt ) is a heavyside
function taking the value of one if zt ≥ c and zero if zt < c, a constant. Since esti-
mation of this discrete model is complicated Goldfeld and Quandt have suggested
a continuous approximation to the transition function using the cumulative normal
(c, σ 2 ) distribution function with the errors of the regimes assumed to have equal
constant variances σ 2 . Using this transition function in the regression will define
a smooth transition regression model. This idea has also been followed up in the
time series literature (cf. Tong (1990)).
Markov switching times series models may be suitable for time series with
jumps or oscillations (i.e. sequences of short jumps). One can proceed along the
following steps. First test hypotheses on the number of states present in a particular
time series. This determines the number of regimes. Then define the appropriate
switching model and its likelihood function. Thereafter estimate the parameters
and calculate the ML estimates of transition probabilities.
A simple example of the application of a discrete time series switching model
to a two-dimensional problem like productivity and investment intensity growth is
the following. Assume that each of the variables may be observed in two regimes,
one defined by positive growth rates, the other one by negative ones. A combination
of the two variables involved will therefore produce four regimes in which each
pair of growth rates can be found. The sequence of regimes obtained can then be
read off the simplified time series graph which contains just the averages of values
of the whole period in which one regime obtains. Representing this in the phase
space gives just four points (i.e. average growth rates), each in one regime, each
of which is characterized by one of the four quadrants.
3.14 Smooth transition models

A smooth transition between two extreme regimes is sometimes more useful than
just assuming the discrete transition between them. As a further alternative to
modelling the smooth transition function Maddala (1977) has proposed the logistic
function D(zt ) = (1 + exp(β1 + β2 zt ))−1 . Indeed a number of further alternatives
have been proposed and have given rise to a detailed analysis of smooth transition
regression (STR) models and have emphasized their attractiveness as compared to
conventional switching models.
STR models can easily accommodate more than two regimes, which is more
realistic to assume in many cases (e.g. in business cycles, or when considering
reactions of economic agents). The two regime model is thus a special case of
the STR model. A further advantage of such models is that they can serve as an
alternative against which to test parameter constancy in a linear model. Here the
alternative to constant parameters is a continuous change in parameters which may
be more convenient to handle statistically than a discrete change (as has already
been stressed by Goldfeld and Quandt (1972)). Such a STR model can be written
by using time as the transition variable: yt = (α + βD(t))xt + ut . xt represents
a vector of explanatory variables,α and β are parameter vectors. If parameter
constancy (implying D(t) identically zero) is rejected then β can be estimated and
the influence of the explanatory variables changes over time according to their
marginal effects in (α + βD(t)). More generally, the transition variable can be any
observable economic variable, or even an unobservable one. A specific practical
problem is therefore the specification of the transition function.
In a survey of STR models Teräsvirta (1996) presents a selection of popular
definitions for the transition function, denoted G(. ). Typical are the exponential
and the logistic variants. The logistic STR model uses
G(γ, c, St ) = (1 + exp (−γ(St − c)))−1 .

G is monotonically increasing in the transition variable St . γ, a positive constant
(positivity assumed as identifying restriction) indicates how fast the transition from
zero to unity is a function of St . The constant parameter c determines where the
transition occurs. The larger the value for γ the faster the transition tending to a two
regime switching model in the limit. There also exist non-monotonic alternatives
to the above mentioned transition function which are useful when reswitching
occurs.
Another example is given by the exponential STR model
Figure 3.1 Scatter plot of growth rates of investment and value added per employment
G(γ, c, St ) = 1 − exp (−γ(St − c)2 ) with γ > 0.

For very large values of γ, G(. ) will tend to zero and therefore the transitory
influence on parameter estimates will tend to vanish. All these transition functions
can also employ time as a transition variable and may with suitable changes in
their parameterisations also generate nonmonotonic transitions. Examples can be
found in Teräsvirta (1996).
The STR models then have the form yt = (α +βG(γ, c, St ))xt +ut . For specific
cases one may choose St = t. The major advantage of using STR models is that
any rejection of the constancy of parameters of the linear relationship is a rejec-
tion against a specific parametric alternative. If a rejection of constant parameters
occurs, e.g. as typically the result of Chow tests etc., then the parameters of the
alternative can be estimated. This helps in determining where and how parameter
constancy of the linear model breaks down.
On statistical inference and the model building procedure for STR models
consult Teräsvirta (1996). Future research will focus on the applicability of such
modelling approaches. A major problem to be expected will be the availability
of relatively short annual time series observations which will severely limit the
specification search process.
The following demonstration (Figure 3.1) for the paper-production sector of
Austria should just serve as example for the potential complexity of the transition
function when applied to the investment – productivity relationship. The scatter
plot of the growth rates of investment and value added per employment for the
sample period shows a dominating variation of the investment rates. Some of them
are exceptionally high, especially during those periods when due to environmental
legislation most paper producing firms had to engage in investing in new produc-
tion techniques. In contrast, the productivity growth rates as measured by value
added per employment remain with one exception within a narrow band of approx-
imately ten percent. The demonstration thus focuses only on the use of investment
as a potentially useful transition variable in the explanation of productivity devel-
opment.
The dependent variable is the logarithm of value added of the paper sector. It
is modelled as a second order autoregressive process with investment and its lag
as transition variables in an exponential STR model.
Estimation proceeded by NL least squares. The results of the model
log(v34) = a0 + log(v34−1 )(a1 (1 − exp(−a2 (log(i34) − a3 )2 ))) +

+ log(v34−2 )(a4 (1 − exp(−a5 (log(i34−1 ) − a6 )2 )))
are contained in Table 3.1 (using an annual sample 1972 to 1991).

The two estimated transition functions can be inspected in Figures 3.2 and 3.3.
Although the estimation result may still be improved by further specification
search, it is easily recognizable that the interaction between the two variables is
seemingly complex. Now we need to remember that the dynamic system we are
interested in is, in fact, of higher dimensions. Requirements concerning the length
of observation periods as well as the task of identifying systems relationships make
an extension of these econometric techniques into dynamical systems not an easy
task.
A simple simulation study of a two dimensional system with an exogenous
transition function may serve as an instructive example. We can show the nature
of the simulated paths represented in the framework space to be quite similar to
those that have been observed from actual economies.
Let the two endogenous variables be given by
y1(t) = (a + b. G1(γ, c, St ))x(t) + ut

y2(t) = (f + g. G2(γ, c∗ , St ))x(t) + vt
where G1 and G2 are two transition functions, St = t the transition variable, x(t)
an exogenous variable (here a noisy trend), and u and v normally distributed error
terms. The transition functions have been chosen as to produce structural shifts
during the first third of the thirty observations generated. They are given by:
G1(γ, c, St ) = 1 − exp(−γ(S(t) − c)2 ) with γ = 3 and c = 10,

G2(γ, c∗ , St ) = (1 + exp(−γ(S(t) − c1 )(S(t) − c2 )(S(t) − c3 )))−1
with c∗ = (c1 , c2 , c3 ) = (2. 5, 5, 10) and γ = 3.
Thus, changes should occur during the second and third and around the fifth and
the tenth period. We can graph these generated series y1 and y2 in diagram (Figure
3.4) and draw a scatter plot (Figure 3.5):
One can observe that there are structural changes in both series around period
10. Whereas series y1 exhibits the break only during one period the regime shift
in series y2 lasts longer (between periods 2 and 3, and 5, and 10). Turning to their
growth rates and the framework space as given by the following graphs in Figures
3.6 and 3.7, one can readily recognize the regime shifts despite the added noise to
the two equations in levels.
Table 3.1 Estimation of NL least squares
Variable Coefficient Std.Error t-value t-prob PartR2

a0 0.41354 0.21293 1.942 0.0741 0.2249
a1 0.88103 0.067007 13.148 0.0000 0.9301
a2 20.124 59.592 0.338 0.7410 0.0087
a3 1.0855 0.24027 4.518 0.0006 0.6109
a4 0.020500 0.010746 1.908 0.0788 0.2187
a5 3.1438 5.1151 0.615 0.5494 0.0282
a6 1.2750 0.38476 3.314 0.0056 0.4579
R2 = 0.982549 F(6, 13) = 121.99 [0.0000] I = 0.0368252 DW = 2.39
RSS = 0.01762927324 for 7 variables and 20 observations.
Figure 3.2 Transition function 1
Figure 3.3 Transition function 2

100
y1, y2
50
0
0 10 20 30
Time
y1(t)
y2(t)
Figure 3.4 Time series graphs of simulated series
100
y1
50
0 10 20 30 40
y2
(y2, y1)
Figure 3.5 Scatter plot of series in levels
The framework space makes these shifts even more obvious. We find four pairs
of growth rates to be located outside the area around the zero point, corresponding
to regime shifts according to the construction of the transition function.
The cluster around zero follows from the relationship of both endogenous
variables with the exogenous variable, a stochastic trend, which enters linearly
whenever the effect of the transition function is zero. Transformed into growth
rates both variables have to fluctuate around zero due to the noise component.
We may conclude this excursion into some of the more promising econometric
approaches which might shed some light on the possibility to identify relevant
economic processes with a quotation by T. Haavelmo in his Nobel lecture (1989):
I believe that econometrics can be useful. . . . the possibility of extracting

information from observations of the world we live in, depends on good eco-
nomic theory. Econometrics has to be founded on theories that describe in a
reasonably accurate way the fashion in which the observed world has operated
in the past.
4
2
Growth rates
2
0 5 10 15 20 25 30
Time
dy1
dy2
Figure 3.6 Time series of growth rates of y1 and y2
2
dy1
2 1 0 1 2
dy2
(dy2,dy1)
45°-Line
Figure 3.7 The Framework Space
We have seen that our framework space can accommodate several theories. How-
ever, it still remains a formidable task for theories to provide a convincing expla-
nation of economic change and for the econometrician to account for the changing
relevance of those theoretical explanations.
3.15 Multiregime dynamics in large model economies

As a complement to the econometric approach, rather than in contrast to it, we
can do some more mining of stylized facts, and then try out different modelling
techniques. Before outlining these further developments, we have to re-consider
the dynamics we have been talking about.
So far, we have developed the Framework Space (FS) with its structure. This
is a partition which is based upon the notion of regime as a collection of states
(-paths) qualitatively similar though quantitatively different. Our chosen partition
depends upon a classification criterion which is derived from existing theories of
growth and cycle.
That each state in our FS is a growth path monitored on the basis of two co-
ordinates makes for the first of such differences. Furthermore, a particular state can
be either a long-run trend or steady state, or else a kind of ‘short-run trend’ (short
as the length of the averaging period of time) when, for example, it belongs to a
sequence of short run paths. Thus, in our dynamics there is no role for the distinc-
tion between growth and oscillations. Actually, our choice of growth rates as state
variables instead of levels, is intended to unify the treatment of the two, so that
we start with growth cycles (or cycles in growth rates) as the simplest dynamical
phenomenon to expect50 . A sequence of short run paths may turn out to be regu-
lar or irregular to various degrees, both in theory and in empirical evidence. This
introduces the possibility (and the likelihood) of encountering chaotic and gen-
erally complex dynamical behaviour in our framework. Those regular behaviours
studied by a conventional approach51 are naturally viewed as simple and limiting
cases of an otherwise, generically, irregular dynamics of sectorally disaggregated
economies.52
All this can be expressed by saying that there are three dynamics going on in the
FS: the first is implicit in the definition of a state as a path; the second takes across
paths, the last one is in principle across regimes (that one stays for sometime within
a given regime cannot be excluded on first principles). Each kind of dynamics is
in a sense typical. The first dynamical layer implies a relation between growth
rates and behind it a parameterized relationship between levels (VA, E, I). It can
be associated with equilibrium growth theories, which look, among all paths in
growth rates, for the one that is a long run attractor (‘the data implied steady state’).
Recent growth literature on endogenous growth and on non-ergodic growth has
expanded the scope of such an approach by allowing for multiple steady states
equilibria. At any rate, the growth approach can be said to imply a partition of the
FS between the subset of attractors and all other dynamical paths that are short
run or disturbed paths or ‘disequilibria’. The second layer of dynamics is best
interpreted in terms of Hicksian traverses: each path is in principle unstable, and
the system ‘traverses’ from one to the other in disequilibrium. What is new is the
third one, i.e. the dynamics of structural change according to our definition, where
disequilibrium may be systematic.
This refers fundamentally to regime behaviour. Of this it emphasizes the ‘qual-
itative part’, in the sense that the dynamical behaviour of the selected system is
now looked at through the sequence of the regimes it goes through (or ‘stays in’),
rather than through the sequence of states. We loose the (quantitative) precision of
the state representation, we no longer retain information on the history of where
the system was, nor can we try to ‘predict’ ‘where’ exactly it will go; these are the
effects of the particular texture of the state space when co-ordinates are reals. But
we may hope to find out the region where it was/is/or it will go. The information
conveyed by the regime description can be better characterized by saying that it
tells us where the system does not lie. With this spatial notion, we are practically
very close to the point of view of recent chaotic dynamics literature, which is con-
cerned with such problems as: is there a ‘region attractor’; what is the likelihood
for a given system to visit a particular region of the state space; what is the likely
length of time spent in a particular region, and the like.
There is a relationship between these three types of dynamics but one goes
from one to the other, in a sense, by a process of reducing information retained.
Therefore, there are costs in choosing one or the other, and one must be aware of
this. In the following, it is assumed that type-one and type-two dynamics are well
known, and we consider only the implications and the possibilities linked to the
study of type-three dynamics.
Using the notion of regime implies a coarse partition of our two-dimensional
state space which is ‘spatially’ discretized. Thus, we do not need to use pairs of real
numbers as co-ordinates, it is sufficient to use single naturals (and a finite number
of them) or else a finite set of symbols in a given alphabet. Any other trick will
do. A colouring scheme, like the one often used by Ralph Abraham, could also be
used, and would be better for purposes of visualization and animation. In our case,
a partition into six regions-regimes is used, but the principle is the same, whenever
we can use the notion of regime.
In more abstract terms, we may speak of k regimes. Hence we need k symbols
only (and not two real lines), say letters from A to F. The history of a chosen system
can now be encoded through this alphabet, and will result in a string of letters or
‘word’. With such dynamic representation, one can address questions like: is there
regular regime behaviour? is ‘irregular’ behaviour dominant throughout the history
of the chosen system (or across the histories of a set of systems)? But obviously
the terms themselves will need a new definition. To extract such information, if
it is there to be found, one can imitate standard econometric techniques, looking
across data: (i) within the individual strings that record histories, one string at a
time, looking for regular/irregular behaviours; (ii) across the screen of all strings of
the sectors within a given economy, to find cross-strings regularities; (iii) finally,
country screens can be compared. To characterize the dynamic properties in a
string one can resort to ideas that belong to information theory, and in particular
to the theory of information complexity.
The presence of a ‘fairly’ regular ‘string’ would be the basis of the qualitative
approach to econometrics: it tells us of the presence of a stable sequence of regimes
that the system may go through. The idea is not entirely new, as something similar
was originally formulated in the theory of stages by development theorists. This
shows, once again, that in the dynamics of regimes there is no longer a distinction
between ‘growth’ and ‘development’, the former implying structural stability, the
latter allowing for structural change. It is a means to unify them into a unique
dynamical theory. This new formalism may also provide the way (or one of the
ways) of treating growth and oscillations as disequilibria.53
Turning now to the other line of development. Its roots lie in distributional
dynamics as implemented by Quah (1993). Quah who is working in the area of
‘growth empirics’ questions the relevance (and/or interest) of representing the
time behaviour of cross-country performance by synthetic properties, like average
growth rates, and/or levels of per capita productivity or income. Distributional
dynamics takes the original cross country distribution at say time 0 and follows its
evolution over time, trying to figure out, from stylized facts, guessing or theoretical
construction, asymptotic properties of the distribution itself while keeping track of
the intra-distribution dynamics (poor becoming richer and the like). In a discretized
version of this (where an income indicator is discretized in intervals of values and
countries are classified accordingly), we obtain a dynamics that is easily converted
into our coding approach.54
At each date, the economy is represented by a state vector with k entries, each
entry being the relative proportion of sectors in a given economy that is lying in
the corresponding j-th regime at that date (that, as usual, can be a whole time
interval). The dynamics is therefore represented by a sequence of dated vectors
[xτ ] telling us the time evolution of the relative frequencies in which each regime
is being visited. The sample can be taken to be the sectors of a single country, one
sector across a pool of countries, or finally sectors in all countries for which data is
available. Between two subsequent vectors one can compute transition probability
matrices, and then see if certain tendencies can be identified.
The candidate model in any case must account for a dynamics that takes a
system across regimes. This model will have to take regimes seriously, but this is
ongoing research.
Acknowledgement
The author acknowledges the financial support of the Italian CNR, contract n°
98.03803.CT10, and of Siena University through its PAR 1999. Part of the work
reported in this chapter is also the result of a close collaboration with other re-
searchers in the IDEE Project, financed by the EU within the TSER action of the
5th S & T Framework Programme.
Notes
1 See for instance Bernard and Jones (1996), and the debate on convergence reviewed
e.g. in Durlauf and Quah (1998).
2 Thus, we leave aside models explaining dynamics on the basis of demand behaviour
(e.g. Keynesian models) or which conjugate demand and supply conditions in a theory
of structural dynamics (e.g. see the work of Morishima , Pasinetti, for instance). We
also leave out the neo-Austrian approach, see Amendola and Gaffard (1988), (1998).
Our framework is compared with the latter in Amendola, Gaffard and Punzo (1999).
3 For a discussion, see Scott (1989).
4 We mean alternative to the neo-classical one.
5 Again, that there is no production function does not mean that one cannot insert one
compatible with the Kaldorian model.
6 There is, of course, a distinct formulation of the model of growth as reflecting embodied
technological progress, the one proposed by Arrow with capital stocks of different
vintages, but this version is not the one that interests us here.
7 Goodwin (1967); see also Goodwin and Punzo (1987); Goodwin (1990). Punzo (1995)
discusses the relation between the present empirical framework and those works.
8 This construction was introduced in Böhm and Punzo (1994) and discussed in Punzo
(1995).
9 In fact, one can build an analogue framework with different sets of variables and their
growth rates (if this is suggested by the problem to investigate). Notice that gross
capital formation could have been re-defined so as to measure capital in a broader
sense. Here, we take physical capital in the Kaldorian and Solow type of tradition.
10 See Appendix 2 on statistical data bases.
11 In the econometric sections below, it will be useful to resort to a log approximation of
(3.1) and (3.2).
12 Clearly, there is nothing natural about this choice that is dictated by statistical difficul-
ties rather than the timing of the phenomenon under observation.
13 The clock chosen in Böhm and Punzo (1994) reflects the hitting of exogenous shocks,
and therefore reports external information. It was used to check, to a certain extent at
least, the relevance of aggregate shocks in explaining the structural dynamics of the
economies involved.
14 There is some relation with the econometric literature on broken trends here. The idea
of a variety of patterns of growth, on the other hand, has recently become popular even
in the tradition of endogenous and exogenous aggregative theories. There, however, it
is attributed to cross country, rather than to cross sectoral dynamics.
15 This is the choice of growth theory as is done in the conventional mode. Thus, the
latter too has a place in the FS, justifying the latter’s name.
16 The system (3.3) is taken to have no forcing term, an analogue of a VAR formulation.
The forcing term may be used to introduce consideration of the influence on system
dynamics of an exogenous force, that can have stochastic or deterministic nature or can
sum up the two. The forcing term will therefore determine, together with endogenous
properties, the system’s long run dynamics. For such formulation see again Section
3.11.
17 A notion introduced by R. M. Goodwin in his (1947) contribution, and later investigated
in Goodwin and Punzo (1987).
18 Again, see Böhm and Punzo (1994, 1995). The literature on Growth Empirics is full
of exercises of the first type.
19 The alternative is obviously to assume that the model be linear, as done in much
econometric practice, again see Sections 3.11–3.12, but this has a cost that may be
quite high.
20 The dynamic specification of (3.4) as a first order autoregressive process is only used
for simplicity. The appropriate order of these processes needs to be derived from
econometric testing.
21 Of course, estimation problems enter the latter modelling strategy as well, though at a
different stage.
22 It will be the case whenever we will deal with borderlines between regimes in the
partition of the state space below.
23 It is a linear one-dimensional subspace, whence the knife edge property descends.
24 Compare with the discussion in Section 3.11 below, in particular equation(s) 3.10,
which contains two parameters.
25 And of course, its zero would be again a bifurcation value and correspond to the origin.
In this light, the origin corresponds to a bifurcation value of a family of maps with two
parameters.
26 We are using orthogonal co-ordinates.
27 Actually the former, i.e. the first quadrant , can be called the growth and the latter, the
third quadrant, the contraction quadrants.
28 As in Böhm and Punzo (1995).
29 Movie refers also to the possibility of animating the graphs, a technique amply demon-
strated in the work of R. Abraham and C.D. Shaw (1989).
30 It is to be understood that the decision of defining the FS in terms of two coordinate
variables (the growth rates) already makes the model more complex than a standard
growth model. The latter can be argued to imply a one-variable version of our FS.
Two however is surely less than the desired dimension for describing an economy;
however, it already shows all the complexity that a higher dimensional model would
have to deal with. The choice is made, among other reasons, to keep a multisectoral
model manageable.
31 This and the following sections are based upon Punzo (1997).
32 See Industrial Policies in OECD countries, Annual Review 1990, OECD Paris, 1990.
33 See for example Komiya, Okuno and Suzumura (1998), Moriguchi (1991), (1995),
Yoshikawa (1995).
34 For more details, the interested reader is invited to look at the publications from the
ongoing IDEE project, as well as previous publications by the authors and other as-
sociates to the project. See also Amendola and Gaffard (1988), (1998); Amendola,
Gaffard and Punzo (1999); Böhm (1996).
35 ‘Traverse’ in the sense of Hicks (1973).
36 This belongs to the tradition of Schumpeter and Goodwin.
37 Let d Log E/dt ≡ e , E being aggregate employment. It is typical of neo-classical
and endogenous growth models (including the Kaldorian and Cambridge tradition) to
introduce as an assumption a full employment equilibrium path. This is not accepted
into our framework.
38 Given a known value of λ, we can always measure the economy’s growth rate in
deviation. Thence, the notion of ‘endogenous growth rate’ is in this case the residual.
39 The latter is the union of two open intervals, the closure being the equilibrium set (a
point or one-dimensional manifold).
40 In particular, the long-run state, an attractor, is a monotonic equilibrium path, the
simplest of all dynamic morphologies.
41 An idea ingenuously exploited by, for example, Durlauf and Johnson (1995), though
in a different analytical setting.
42 If employment levels were let free, in other words, we would have an interval of long-
run values. Typically, in growth theories this possibility is associated with short-run
dynamics, as they all descend one way or another from Harrodian dynamics.
43 β-convergence is an exercise whereby a cross country regression is run of growth rate
on initial level of output per capita. Hence, a one dimensional FS space with a single
gy -axis is completed with a y-axis. We talk of β-convergence when the regression
coefficient is negative, implying that the poor countries run faster towards the long run
equilibrium path. See the vast literature, e.g. Baumol, Nelson and Wolff (1994), and
Durlauf and Quah (1998). The resulting framework is not a two-dimensional FS in our
sense, for one variable is interpreted as the explanation of the other. Econometrically
it is a one equation model, rather than the two equation system implicit in our FS and
described by (3.3) or (3.4) above.
44 See for instance Goodwin (1991), or two collections, Benhabib (1992) and Day and
Ping Chen (1993) for a sample of a burgeoning theoretical literature.
45 Changing the variables on the y-axis then can lead to different frameworks where the
classification between exogenous and endogenous changes too, as they are relative to
the explanatory variables on such axis. In our framework where the system is generally
simultaneous, the distinction in principle does not make sense.
46 See for instance some of the models used later to discuss the econometrics of structural
change.
47 See Goodwin (1990).
48 See the treatment of the model as parameterized by b = (b + c)/(1 − d), in Section
3.4.
49 This result obtained from using a first order lag-polynomial is easily generalized
to higher orders. The relevant restriction for long run unit elasticities will then be
B(L)[A(L)]−1 = 1 in the notation of eqn. (3.5).
50 Thus, the emergence of a long run steady state in growth rates is a very special case
of a growth cycle collapsing to a single state as a global attractor.
51 We are referring to non-stochastic versions of Growth and BC theories with which
our approach can be directly compared. In fact, the latter can be best understood
if formulated with a deterministic, i.e. non stochastic, model, where irregularity is
endogenously generated rather than shock induced. This does not imply that stochastic
phenomena cannot be added to it. Most of modern economic dynamics is formulated
in a stochastic environment, to derive some statistically average long-run properties.
52 Actually there is a growing literature on this, (see for instance some of the papers
in the collection edited by Benhabib (1992)), though such literature focuses upon the
theoretical possibility of observing highly irregular dynamics associated with relatively
simple multisectoral models.
53 See Brida, Puchet and Punzo (2000).
54 See Lavezzi (2000).
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Appendix 1
The data base
Data for Germany, France, and the USA are taken from the OECD publication,
National Accounts, Vol II (various issues).
Value added by kind of activity (table 12), gross fixed capital formation by kind
of activity of owner (table 3), employment by kind of activity (table 15).
For Germany all value added and investment data at constant prices have been
converted into 1991 prices, for USA they have been converted into 1985 prices,
and for France into 1980 prices.
Data for Italy have been constructed from two sources: for the period 1970
to 1980 they are from the ISTAT publication Annuario di Contabilità Nazionale,
Tomo 2, tav1.7 etc. and converted into constant prices 1980. For data from 1980
until 1992 they are from the ISTAT data bank as published. Tests for the period
1980 to 1983 for both data sets have shown that the calculated growth rates are not
affected by the systemic change in Italian national accounts.
The data for Japan come from regular publications by EPA, Economic Planning
Agency.
Table 3.A1 Coding of sectors
Sector OECD ISTAT EPA

(Germany, (Italy) (Japan)
France, USA)
Agriculture, hunting, forestry and fishing 1 01 02
Mining and quarrying 2 03
Manufacturing 3 IND
Food, beverages and tobacco 31 36 04
Textiles, wearing apparel and leather industries 32 42 TEX
Wood, and wood products, including furniture 33
Paper, and paper products, printing and publishing 34 47 06
Chemicals and chemical petroleum, coal, rubber 35 17 07
and plastic products
Non-metallic mineral products except products of 36 15 09
petroleum and coal
Basic metal industries 37 13 10
Fabricated metal products, machinery and 38 24, 28 MACH
equipment
Metal products, except machinery and transport 11
equipment
Office and data processing machines, precision PREC
and optical instruments
Electrical goods ELTR
Transport equipment 28 AUT
Other manufacturing industries 39 50 16
Electricity, gas and water 4 06 08,18
Construction 5 53 17
Wholesale and retail trade, restaurants and hotels 6 58 19
Transport, storage and communication 7 60 22
Finance, insurance, real estate and business 8 69 20
services
Community, social and personal services 9 74 21
Producers of Government services 9 86 23
Appendix 2
Italy 1970-73
Italy 1973-79
25 25
13
20 20
15 42 15
10 60 10
58
01 28
5 53 36 5
01
gi
gi
15 24 53 5036
0 060 28 47 24
47 60 15
74 06 17 58 42
-20 -15 -10 -5 -5 0 86 5 1050 15
69 20 25 -20 -15 -10 -5 -5 0 86 5
74 10 15 20 25
69
-10 17
-10
-15 -15 13
-20 -20
gv gv
Italy 1979-84 Italy 1984-88
25 25
20 20
15 15
10 47
10 17 28
47 58 50 24
15 28 1
5 36 586 69 156036
6 6042 13
124 17 6 42
58 13 0
0 53 53
69 86
-20 -15 -10 -5 0 550 10 15 20 25 -20 -15 -10 -5 0 5 10 15 20 25
-5 74
-5
74
-10 -10
-15 -15
-20 -20
gv gv
Italy 1988-92 Italy 1970-92
25 25
20 20
15 13 15
6
10 10
28 28
5 60 5 6013
58 36 6 15 47
gi
4236
gi
53 15 58
1 24
0 69 17 50 0 53 50
74 24 60
86 86 5
-20 -15 -10 -5 0 5 10 15 20 25 -20 -15 -10 -5 74 0 17
10 15 20 25
-5 47 42 1 -5
-10 -10
-15 -15
-20 -20
gv gv
Figure 3.A2.1 Italy

France 1970-73 France 1973-79
25 25
20 1 20
37 36
33
15 15
7
10 9 10
32 4
6
5 5 2
5 38 34
gi
gi
8 5
2 31 4 7 1 31
0 0 35 39
34 638
39
-20 -15 -10 -5 0 5 10
36 15 20 25 -20 -15 -10 -5 0 9 335 10 15 20 25
-5 35 -5 8
32
37
-10 -10
-15 -15
-20 -20
gv gv
25 25
34
20 20
15 15 36
2 32 38
10 10 9
35 33
32
6
5 9 5 31 5 7 37
3339 1
gi
gi
37
31 1 390 8
0 6 3836
34 35 4
-20 -15 -10 -5 0 8 55 10 15 20 25 -20 -15 -10 -5 -5 0 54 10 15 20 25
-5 7
2
-10 -10
-15 -15
-20 -20
gv gv
25 25
20 20
15 15
10 10
36
5 5 39 5 34 7 1
9 38
gi
7
gi
35 5 3233
1
0 9 0 63139
638 31 3735 4
8 36 4 -20 -15 -10 -5 0 8
2
5 10 15 20 25
-20 -15 -10 -5 0 5 10 15 20 25
-5 33 32 -5
34 37
-10 -10
-15 2
-15
-20 -20
gv gv
Figure 3.A2.2 France

Germ any 1970-73
Germany 1973-79
25 25
20 20
15 15
2 9
4
9
10 10
2 39
33 1
34
5 8 1 5 5
38
6 36 32
gi
gi
7 39 36
0 0 333735
315 317 4
37 6
-20 -15 -10 -5 0 5 10 15 20 25 -20 -15 -10 -5 0 8 5 10 15 20 25
-5 32 -5
38
-10 -10
34 35
-15 -15
-20 -20
gv gv
Germany 1979-84 Germ any 1984-88
25 25
20 20
34
15 15
10 10 39
6
33
35 383632
5 25 37
2 4 32 7
gi
31 5
gi
1
37 9 7
0 39 838 0
36 31 9 1
-20 -15 -10 -5 6 35 345
0 10 15 20 25 -20 -15 -10 -5 0 4
-5 33 -5 85 10 15 20 25
5
-10 -10
-15 -15
-20 -20
gv gv
Germany 1988-92 Germany 1970-92
25 25
20 20
15 15
5
10 33 10
31 36
6 37 1 2 7
5 8 39 5 39 1
3336 338
4
gi
gi
9
35 38 6 4
32 37
531732
0 4 7
2
0 35
34 8
-20 -15 -10 -5 0 5 10 15 20 25 -20 -15 -10 -5 0 5 10 15 20 25
-5 -5
-10 -10
-15 -15
-20 -20
gv gv
Figure 3.A2.3 Germany

USA 1970-73 USA 1973-78
25 25
20 20
15 15
3731
10 734 10
21 3836 37
3339
8 4 36 32 34
5 69 2 5 31335
9
38 33
7
gi
gi
5 136
0 0 5
6 32
-20 -15 -10 -5 0 5 10 15 20 25 -20 -15 -10 -5 0 8 94 5 10 15 20 25
-5 -5
-10 -10
-15 -15
-20 -20
gv gv
USA 1978-84 USA 1984-88
25 25
20 20
15 15
37
10 6 10 39
38 34 4
5 9 5 9
gi
31
gi
32 36 35
8 234 31 5
0 0 7 32 38
435
7 39 33
36 8
-20 -15 -10 -5 -5 0 37 5 101 15 20 25 -20 -15 -10 -5 -5 0 6 15 10 15 20 25
5 33 2
-10 -10
-15 -15
-20 -20
gv gv
USA 1988-91 USA 1970-91
25 25
20 20
35
15 15
31
10 10
38 37 3731
5 36 1 5 34 35
38
34 39
gi
6 39
gi
2 36 32
0 6 08 94 6 1
2 4 32 5
33
-20 -15 -10 -5 -5 0 57 10 15 20 25 -20 -15 -10 -5 -5 0 5 10 15 20 25
9 33
5 8
-10 -10
-15 -15
-20 -20
gv gv
Figure 3.A2.4 United States

Japan 1969-72 Japan 1972-79
25 25
20 20
15 15
23 4 ELT R
18
T EX
10 17 10
23 11
22 3 PR EC
2 21 19
AUT
5 6 5 918 AUT 7
16IND 2 19 6
gi
gi
17 MACH
0 8 PR EC 9 3
0 IND T EX
11 20 22 4 10
-20 -15 -10 -5 0 5 10 15 20 7 25 -20 -15 -10 -5 0 20 5 10 15 20 25
10
-5 -5 21
8
16
-10 MACH -10
ELT R
-15 -15
-20 -20
gv gv
Japan 1979-85 Japan 1985-88
25 25
20 20 20
PR EC T EX
22 16
15 ELT R 15 17
AUT 4 19
21
23 MACH
10 22 7
10
6 6 IND
16 11 23
T EX 9
5 9 8 8 5 2 7
21 11 10
gi
gi
3IND 20 P REC
18 3 AUT
4 10 18 EL T R
0 0
17 2 MACH
-20 -15 -10 -5 0 19 5 10 15 20 25 -20 -15 -10 -5 0 5 10 15 20 25
-5 -5
-10 -10
-15 -15
-20 -20
gv gv
Japan 1988-92 Japan 1969-92

25 8 25
20 20
10
15 15
AUT
9 11
10 17AUT 7 10 23
P REC
MACH 3 11
20 22 18 17 22 9 ELT R
4 T EX
5 16 5 2
23 IND 81820 7
gi
4 19
gi
2
21 19 EL T R 10 3
21
0 P REC 0 IND 16 6
MACH
-20 -15 -10 -5 T EX0 5 10 15 20 25 -20 -15 -10 -5 0 5 10 15 20 25
-5 6 -5
-10 -10
-15 -15
-20 -20
gv gv
Figure 3.A2.5 Japan

Part II
The macroeconomy and its

dynamics
4 Qualitative dynamics and
macroeconomic evolution
in the very long run
Richard H. Day and Oleg V. Pavlov
When it is impossible to incorporate all the complex components involved, and it is

therefore hopeless to represent behavior in accurate quantitative terms, it may still
be possible to derive a compelling qualitative explanation of behavior. This possi-
bility is illustrated in this chapter with the problem of explaining macroeconomic
evolution. The resulting nonlinear, multiple–phase model exhibits trajectories that
are qualitatively like those in the historical and archaeological records. Implications
for understanding current transitions in the global economy and for development
policy are suggested.
Dynamic macroeconomics is concerned with explaining growth, fluctuation,
and structural change at aggregative levels. In this discussion our interest is with pe-
riods of time long enough to characterize fundamental changes in technology and
economic organization that arise endogenously through the interaction of popula-
tion, welfare, learning, and the internal and external economies and diseconomies
of production.
The evolutionary character of the economic process is most easily seen at the
microeconomic level when one kind of production technology displaces another.
For example, exhibits in the old town museum of Århus describe in graphic de-
tail the history of passenger transport in Denmark. Stage coach travel, which was
introduced early in the nineteenth Century, was estimated to have produced some
8,300 passenger trips in 1833; 65,000 in 1846; and by 1859, 122,000. In the mean-
time, around mid-century the railroad had been introduced which, by the century’s
end, had all but eliminated the stage coach and was accounting for 15,000,000
passenger trips a year! Underlying this spectacular growth was a profound struc-
tural change in the nature of conveyance and human activity. Examples of similar
growth waves abound throughout the industrial era.
Growth theory, as usually constituted, does not mimic the endogenous insta-
bilities that drive an economy away from where it has been operating. In what
follows a model of evolving macroeconomic structure is described that addresses
these two aspects of very long run economic development, that is, (i) qualitative
differences in economic structure and (ii) inherent instabilities.1
There is already a small literature on economic development in the very long
run. Papers by Lee (1988), Kremer (1993), and Jones (1999) focus on the greater-
96 Richard H. Day and Oleg V. Pavlov
than-exponential increase in human numbers, emphasizing demographic and pro-
ductivity aspects of the story, but do not take into account structural changes in
the process. Day and Walter (1989) introduce the role played by fundamentally
different production technologies and social organizations based on discrete infras-
tructural requirements, the diseconomies that emerge as population grows within
a given system, and the economies that can be achieved either by replicating
economic units or by switching to a more ‘advanced’ system. Their approach em-
phasizes endogenous instabilities that drive the switch from one system to another.
In a subsequent discussion of the model, David (1994) suggested incorporating
structural changes induced by shocks that perturb a trajectory exogenously from
a given basin into an alternative basin of attraction.
This study is a further development of the Day and Walter model that incorpo-
rates continuous learning by doing within given technological regimes. The model
has been re-calibrated for European conditions with an emphasis on obtaining a
better characterization of the long run qualitative trajectory of population and the
structural episodes that underlie it.
Our approach requires a particular way of thinking about the facts which we
call Qualitative Dynamics. It takes up as a general methodological issue in Section
4.2. In Section 4.3 the model components are described in nontechnical terms;
Section 4.4 pulls all the parts together and the theoretical properties are summa-
rized in Section 4.5. Section 4.6 is concerned with fitting the qualitative record
from the hunting and food collecting stage of development to the present global
information economy. A simulation example is presented that mimics features of
development that stand out in the historical and archaeological records. Section
4.7 summarizes the contributions the qualitative econometric, multiple phase dy-
namics seems to offer. In the reference cited above, David rightly emphasize the
fact that mathematical models such as this can only encompass a few of the factors
necessary to explain the details of economic history. What they can do is capture
in a stylized way some of the most obvious and important aspects of the historical
process. In our opinion the present exercise amply demonstrates this potential.
4.1 Facts and qualitative econometrics

Economic science involves two distinctly different categories of fact: (i) facts about
causality or how things work and (ii) facts about how states of the world change
over time. The first set of facts provides the basis for a theory. A theory is then
judged – from the scientific viewpoint – on how well it explains and predicts facts
of the second kind. To put it slightly differently: (i) How well does a theory reflect
facts of the first kind? (ii) How well does it ‘explain’ and ‘predict’ facts of the
second kind?
When the focus is on microeconomic data, one can answer these questions in
terms of facts garnered through direct observation. How and why things are done
the way they are in particular industries or farming regions can form the basis for
identifying decision criteria and causal structure. The resulting models can then be
compared to data series at very low levels of aggregation obtained from individual
Qualitative dynamics and macroeconomic evolution 97
firm records. Such was the case in the recursive programming models developed
to simulate multi-product output, investment in alternative technologies and the
changing patterns of resource utilization and productivity in various economic
sectors.2
Micro detail is virtually impossible when the focus is on an entire national
or world economy. Index numbers, aggregate variables or averages must be used
to represent state variables. These are not observational variables but statistical
artifacts that merely reflect economic activity as a whole. Indeed, if the concern is
with periods of time long enough to reveal episodic structural changes, adequate
numerical data do not exist at all; the quantitative record can – at best – only be
pieced together from scattered fragments of information. It is here that recourse
to qualitative as opposed to quantitative explanation is necessary: growth – not
specific growth rates; fluctuation – not specific periodicity or statistical spectrum;
stylized regimes – not specific technologies or institutions. Arranged in a time line,
these qualitative facts become specific scenarios of development whose theoretical
explanation can be contemplated.
Likewise, the specific attributes of the various technologies and institutions
must be subsumed within coarse and abstract characterizations. We end up, then,
not fitting a model to data or testing specific numerical hypotheses. Rather, we
calibrate our model so that it can generate the qualitative development scenarios
of interest. If this can be done on the basis of components that reflect what we
know about crucial economic structure, then we have accomplished the task of
qualitative econometrics. The work described here is undertaken in accordance
with this methodological stance.
The qualitative facts of economic history writ large, so to speak, and that
demand explanation include: (i) long run growth in population and production;
(ii) intergenerational fluctuations in labor and welfare; (iii) secular increases in
per capita productivity; (iv) periods of stagnation or declining productivity; (v)
episodic regime switching involving fundamental differences in both socioeco-
nomic organization and production technology; (vi) the replication and merging
of similar economies, and (vii) the integration of economies to achieve a more
advanced stage and the disintegration of an economy into a number of smaller
economies at a less advanced stage of development. These various qualitative
changes have occurred throughout history, as shall be summarized later.
4.2 Modeling macroeconomic evolution

4.2.1 The family function
Time is reckoned in the classical unit of a human generation or a quarter century,
each period is represented by a population of adults and their children who inherit
the adult world in the next generation. Each generation must provide its own capital
goods which only last the period. In what follows we measure population in terms
of the number of individuals.
The number of children who survive to adulthood, b, depends on the aver-
age standard of living, y, on preferences (which may be presumed to depend on
the broader aspects of culture) and on the survival characteristics of the social–
technical regime. This relationship is called the family function,
⎧
⎨ 0 , 0≤y≤η
b = g(y) := (α/q)[y − η] , η ≤ y < ζ (4.1)
⎩
1+n , ζ ≤y
where n is the maximal increase in population, η is the birth/welfare threshold,
and ζ is the threshold above which no further children are desired, or can survive,
where ζ = η + (1 + n)q/α.
The threshold standard of living below which no children survive to adulthood,
exists for physiological reasons alone. Its empirical relevance is evident whenever
famine or social conflict become severe, as has been the case recently, for example,
in Africa and North Korea. A family function with such a threshold can also be
derived from a reasonable specification of family preferences and constraints.3
4.2.2 Economies
To capture the character of economic development in the very long run, a number
of factors, some of which have not been incorporated into macroeconomic growth
models so far, have to be taken into account. We take these up now.
The organizational structure of a modern society includes households, pro-
ducers, marketing firms, financial enterprises, and public institutions of various
kinds. It provides the coherent framework of rules and procedures within which
work can occur. It must be supported by human effort. The humans devoted to
this effort form the infrastructure for a given socioeconomic system that mediates
the human energy devoted to coordinating production and exchange, to providing
social cohesion for effective cooperation, for training and enculturating the work
force, and for producing the public goods such as waste disposal and public safety
required for the well being of the work force. The knowledge that makes this hu-
man infrastructure effective is the administrative technology, a term due to Ester
Boserup (1996). It must augment the production technology. As there are many
institutions that are involved in the various infrastructural functions, the broader
term social technology might be preferred.4
The adult, x, population is divided between the labor force engaged directly
in production, L, and the administrative or social workforce that manages the
infrastructure, M . In a decentralized economy much of this will be part of the
private sector. A large part will also be part of the public sector. Both are necessary
for a productive labor force. Given this, the number of adults in the infrastructure
outside the family, M , and the number of adults in the labor force, L = x − M .
Diseconomies are, of course, implied by resource scarcity. They also accrue
because of the increasing complexity of planning, communicating, and coordi-
nating production as the economy grows. The ability to overcome them depends
on the administrative technology and on the social space which this technology
‘produces’. The social space defines the maximum number of individuals, say N ,
compatible with an effective socioeconomic order and with the feasible operation
of the society’s production process. Social slack is the difference between the so-
cial space and the current number of people, S = N − x. If there is positive social
slack, then more people can be accommodated within the economy. As social space
is ‘used up’, cooperation becomes increasingly difficult, social conflict increases,
and productivity declines. When S ≤ 0, society cannot function. Only when the
social slack is positive can the society function. These internal diseconomies can
yield absolutely diminishing returns to population within an economy.
Assume that the technological production function satisfies the usual assump-
tions:
Y = BG(L, S), L ≥ 0, S ≥ 0, (4.2)

it is a continuous, strictly concave, homogeneous function of social slack and labor
satisfying
G(0, S) = G(L, 0) = 0 for all L, S (a)

(4.3)
limL→0 ∂G
∂L = limS→0 ∂G
∂S = ∞. (b)
These assumptions imply that both labor and social slack are necessary for positive
production and that both labor and social slack contribute positive but declining
marginal productivities. The parameter B is, as usual, the total factor productivity
level. Substituting for L and S, the production function can be written

0 , x ∈ \(M , N )
H (x) := (4.4)
G(x − M , N − x) , x ∈ (M , N ).
In words, output depends on the technology level, labor effect and slack effect.5
A given economy is characterized by (i) its (representative) family function
(4.1), and (ii) its aggregate production function (4.4).6
4.2.3 Cultures
The key characteristic of the above formulation is that an economy based on a given
social technology is bounded by its social space. At some point in the expansion
of human numbers within an economy, the population may reach a level at which
a new economy with the same system but with a newly constituted infrastructure
can be split off in such a way as to increase welfare, in this way overcoming the
internal diseconomies of population size. In effect, the social space is increased by
increasing the number of similar economies. Contrastingly, if productivity were to
fall sharply enough as the potential limit is approached, separate economies could
merge to form a smaller number of economies with the same type of system, in
effect economizing on infrastructure.
Here we assume these possibilities can be represented by fission and fusion, the
former being the splitting of a given economy into two; the latter being the fusion
of two or more economies to form a single one using the same basic system as
before.7 In addition to the internal diseconomies implied by resource scarcity and
social space, external diseconomies should be recognized that derive from the total
population of all the economies together. These diseconomies are, for example,
caused by the environment’s diminishing waste absorbing capacity as population
expands and by the increase of the cost of extracting and refining resources as stocks
decline. We assume that the environmental capacity depends on the production and
administrative technology of a given system and can be expressed in terms of a
maximum population density. Diminishing absolute returns to the work force can
eventually come to pass as the total population becomes large.
For a given culture, the internal diseconomies of population can be overcome by
replication; the external ones cannot. The aggregate effects of these diseconomies
on production can be represented by a continuous environmental damage function
d = D(x̄ − x) (4.5)

where D(x̄) = 1, D(0) = 0 and D (x̄ − x) ≤ 0. The damage function reduces pro-
ductivity as environmental capacity becomes progressively exhausted. We refer
to x̄ as the environmental capacity and the term x̄ − x as the environmental slack.
Once the world is full in the sense that external diseconomies become important,
the replication of economies with the same basic structures must eventually come
to an end.8
Out of all the conceivable numbers of economies that could exist by means of
the fission/fusion process for a given system, we choose the one that is locally effi-
cient for the population of a given size. We call a production function that optimizes
the number of economies in the system the cultural production function. Given
the environmental damage function, the locally efficient number of economies is
j = J (x) := arg max 2l BH (x/2l )D(x̄ − x). (4.6)

| +
l∈N
Then the environmentally constrained production function is
KJ (x) (x) := 2J (x) BH (x/2J (x) )D(x̄ − x). (4.7)

If, when the environmental space is approached, a collapse occurs due to a
very powerful drop in productivity, the population may reorganize itself by fusing
groups into a smaller number of economies with the same technology as before.
Then the stage is set for a new growth process through internal growth of the
individual economies and through a resumption of replication in their numbers.
Fluctuations in the numbers of economies, as well as in total population, could
ensue, perhaps in a highly irregular way for a very long time.
Putting all this together leads to the concept of a culture: a society divided into
a collection of similar economies, each based on the same socioeconomic system
and whose total population is constrained by a common damage function. A small
number of such economies can expand internally, then bifurcate, forming a new
set of economies. This growth through fission can continue until the environmental
slack is so diminished that productivity begins to fall in all the economies. Vari-
ous possibilities follow, including population fluctuations and, if the productivity
decline is sharp enough, fluctuations in the number of economies as well when
existing groups merge to form a smaller number of economies, thus economizing
on infrastructure and by this means re–establishing the possibility for resuming
growth.
4.2.4 Cultural evolution

Now consider the existence of several cultures, each characterized by the ingredi-
ents just described but each with distinct parameters. These alternatives represent
different ‘family values’, ‘ways of life’, ‘development blocks’, distinct administra-
tive and production technologies, and different environmental damage functions.
We assume that the technologies have a natural order: each successive system in
the order requires a greater overhead of human capital, possesses enhanced social
and environmental spaces, and higher attainable production and population levels
than its predecessors. We can therefore identify each distinct system by an index,
i ∈ T := {1, . . . , τ }.
Each culture is defined by the functions and parameters involved in its structure
and by the set of potential alternative numbers of economies that can be replicated
within its environmental constraints. A collection of potential cultures is a cultural
menu.
In the face of limited environmental space for a given culture, a process of
growth through replication of a given system cannot continue indefinitely. How-
ever, the integration and reorganization of infrastructure of existing economies
could permit a jump to a more advanced system with a more demanding infras-
tructure if by doing so the environmental space of the new system is greater. Such
a change in regime occurs in the present theory if average productivity is enhanced
by doing so. This does not mean that each successive technology is uniformly more
productive than its predecessor, but only that at a given current total population
the switch to a new regime will enhance total factor productivity and, hence, the
standard of living at that population level.
If in the process of growth, productivity falls enough as the regime’s environ-
mental capacity is exhausted and a higher regime is uneconomic or unavailable,
existing economies could be forced to disintegrate into a larger number of smaller
economies that require less elaborate infrastructures, in effect economizing on hu-
man capital by reverting to less infrastructure intensive regimes in a process of
cultural reversion. Uniform progress through the natural order of cultures cannot
be presumed.
Local efficiency must now be determined with respect to both the culture and
the number of economies adopting it. Define
(i, j) = IJ (x, B) := arg max max 2l Bp Hp (x/2l )Dp (x̄p − x). (4.8)
p∈T l∈N
++
where B = (B1 , . . . , Bτ ). Then the locally efficient culture is given by the culture,
i, with 2j similar economies. The production function that is locally efficient with
respect to the selection of a culture and the number of economies using it, is given
by
Y = K(x, B) := KIJ (x,B) (x, B) :=

(4.9)
= 2l Bi Hi (x/2l )Di (x̄i − x), x ∈ X ij
where
X ij = {x | (i, j) = IJ (x, B)}.
The set X ij is the set of population sizes for which Kij (x, B) is the cultural production
function with the number of economies yielding the highest total (and average)
output.
4.2.5 Learning by doing

Output per unit of labor is assumed to increase as experience within a given regime
accumulates, but continuing productivity advance based on experience or learning
by doing can only occur within an ‘active’ system, that is, the one with the cur-
rently adopted basic technology (for only then can practical knowledge based on
experience accumulate).
If i is the system index identifying the active regime, then
i
Bt+1 = (1 + ρi )Bti − ρi (Bti )2 /B̃i (a)
j (4.10)
Bt+1 = Btj , j = i. (b)
Recall B to be the vector of system technology levels, B := (B1 , . . . , Bτ ). Given
definition of the locally efficient culture (4.9), an operator,
T : (xt , Bt ) −→ Bt+1 , (4.11)
is defined by the τ equations (4.10). The ith component of the vector Bt+1 is
generated by equation (4.10a) and the remaining coefficients by equation (4.10b).
The process is asymptotically monotonically stable if 0 < ρi < 1. We can think of
B̃i as the technology potential for a given system.
The value of the technology parameter when a given regime is entered for the
first time will be called the innovating technology level, denoted by the parameter
B0 . Assuming that 0 < B0 < B̃, productivity will grow. The larger the potential and
the smaller the innovating level, the larger is the technology gap; and the larger
this gap, the more rapid the initial rate of productivity enhancement. As the stock
of practical knowledge accumulates, the rate of accumulation eventually declines,
and the technology level approaches its potential asymptotically. This process does
not expand the ultimate limits on population allowed by the associated social and
environmental spaces. That can be done in this model only by switching to a more
advanced system.
4.3 GEM: A ‘general’ evolutionary model
4.3.1 The complete model
Putting all the above together, a ‘general’ evolutionary model, which we call
‘GEM’, emerges in which a society is portrayed as evolving through a sequence
of alternative numbers of economies of a given type and switching among alter-
native socioeconomic systems in response to the standard of living of the current
generation and to the potential standard of living that can be ‘selected’ through
‘self–reorganization’.
Let us summarize the model as a whole. The state variables of the system consist
of the population, xt , and the vector of productivity levels, Bt = (Bt1 , . . . , Btτ ) with
one element for each member in the cultural menu. The locally efficient culture
and number of economies is given by
(i, j)t = IJ (xt , Bt ), (4.12)

defined by equation (4.8). Output is given by the cultural production function,
defined by equation (4.9),
Yt = K(xt , Bt ). (4.13)
Average per capita standard of living is given by
yt := ω(xt , Bt ) = Yt /xt (4.14)

and the succeeding population of adults by

xt+1 = g ω(xt , Bt ) · xt (4.15)
where g(yt ) is defined in equation (4.1). With (4.11) we have, also,

Bt+1 = T xt , Bt . (4.16)
Given initial conditions x0 = x, B0 = (B01 , . . . , B0τ ), the τ + 1 equations of (4.15)–

(4.16) generate the trajectories for the state variables (xt , Bt ).
4.3.2 The multiple phase characterization of trajectories

Defining the structurally dependent average standard of living, ωIJ (x,B) :=
KIJ (x,B) (x, B)/x, we obtain the phase zones
D0(i,j)t := {x ∈ X ij | 0 < ωij(x,B) ≤ η i }

Ds(i,j)t := {x ∈ X ij | η i ≤ ωij(x,B) ≤ ζ i } (4.17)
Dn(i,j)t := {x ∈ X ij | ωij(x,B) ≥ ζ i }
Then at each time, t + 1,
⎧
⎪
⎨ 0 , xt ∈ D0(i,j)t
xt+1 = (α/q) 2it Btit (xt − M it )βit (N it − xt )1−βit − ηxt , xt ∈ Ds(i,j)t
⎪
⎩
(1 + n)xt , xt ∈ Dn(i,j)t .
(4.18)
Whenever xt enters a different phase zone, the equations governing (xt , Bt ) change.
This happens if the culture does not change (it+1 = it ) but the number of economies
increases or decreases due to fission or fusion does change, i.e. jt+1 = jt . Or, it
happens if a different culture is adopted so that it+1 = it . In this case, the number
of economies making up the new system will also change through integration or
disintegration, as the case may be.
When xt enters a new phase zone, we denote the entry time ‘s’. Then s0 = 0 and
sk is the entry time for the k th regime switch. With this definition we can define
an episode as the consecutive length of time periods governed by a given regime.
With this definition (i, j)t is the same for t ∈ {sk , sk + 1, . . . , sk+1 }. A sequence,
(i, j)sk , k = 0, 1, 2, 3, . . . describes a trajectory in terms of the sequence of phases
and phase zones through which it passes. Such a sequence is called a scenario.
Given this interpretation, the following additional endogenous variables are
determined:
aggregate production Yt
average welfare per capita yt
the time of entry into each episode sk
the duration of each episode sk+1 − sk
the governing regime in each episode (i, j)t = IJ (xt , Bt )
the governing number of economies
using the dominant system 2jt
the size of the aggregate infrastructure 2 M it
jt
the size of the labor force xt − 2jt M it

the social slack st = 2jt N it − xt
the environmental slack x̄it − xt
4.4 Theoretical properties

The mathematical analysis of the system is intriguing but requires a considerable
amount of nomenclature to develop rigorously. A complete global analysis is given
in Day (2000, Chapters 21–24). The upshot of it is that the conditions for various
qualitative evolutionary scenarios to occur can be derived from an analysis of the
structural properties of the model. To summarize:
(1) Evolution in this theory is driven by an unstable, deterministic process. (Of

course, random shocks could be introduced but our purpose here is to isolate
the intrinsic dynamics of the development process.)
(2) The probabilities of various possible historical scenarios can be derived
in terms of sequences of qualitative events due to replication/merging and
integration/disintegration.
(3) If the number of systems is finite and if the technology potentials of each
system are not too high, then model histories must involve endless fluctu-
ations, eventually sticking with a given regime or cycling in a nonperiodic
fashion through an endless sequence of regimes with replication, merging,
integration, and disintegration with jumping and reverting among systems.
(4) If there were a reachable regime with an asymptotically stable stationary
state when its technological potential is reached, then the model’s histories
would very likely converge (with ‘positive measure’) possibly after many
periods of local chaos to a classical equilibrium.
(5) Under some conditions trajectories can escape the zone of definition with
positive measure. This corresponds to the demise of a society.
(6) If there is an upper bound on the environmental capacity for all systems,
then the advancing regimes must become progressively ‘squashed’ against
this ultimate bound.
(7) With large enough potential technology levels, the processes of output
growth, replication, integration, and regime switching are accelerated as pro-
ductivity is enhanced. Over the long run the system becomes more unstable.
Development is less likely to get ‘stuck’ within a given regime. Economies
are more likely to replicate or integrate and jump. Moreover, asymptotic con-
vergence cannot then occur with positive measure. The chance of demise
increases and the pressing of population against the environmental capacity
occurs with ever greater speed, with an ever greater likelihood of collapse.
4.5 Fitting the evolutionary record

We now show that GEM can be calibrated so as to capture salient qualitative
features of economic development listed at the end of Section 4.2.
4.5.1 The qualitative record

Historians of the nineteenth century noticed that prior to the industrial take-off,
economies had passed through distinct stages of development characterized by
differences in production technology and in the organization of exchange and
governance. Archaeologists, aided by modern methods of dating materials, began
extending this picture backwards in time. By now they have constructed an ap-
proximate but coherent chronology of major developments on a worldwide basis
that stretches back to the earliest evidence of a ‘modern’ human presence.
The great variety of human societies can be roughly grouped into a relatively
small number of stages based on production technology and social infrastructure.
Table 4.1 A time-line for the major systems
Index Description Permanent Duration: Duration:

Entry: Generations Years
Generation
1 The hunting band 1 3702 100,700–8,125 BC

2 Village agriculture 3703 181 8,126–3,600 BC
3 The city-state 3884 126 3,601–450 BC
4 Trading empires 4010 89 451 BC–1775 AD
5 Industrial societies 4099 8 1776–1975 AD
6 Global information economy 4107 2 1976–Present
To describe the major developments throughout the entire span of Homo Sapi-
ens Sapiens and to take advantage of the known archaeological information, a
reasonable minimal specification would be:
I. Hunting and gathering

II. Settled (village) agriculture
II. Complex societies and the city state (civilization)

IV. Trading empires
V. Industrial economies and the nation state

VI. Global information economies
In reality, various geographical areas traversed these stages at very different times
and the advance through them did not increase uniformly from lower to higher
index. Rather, progress from one to another, especially in earlier times, was in-
terrupted by reversions to lower level stages. Moreover, fluctuations in income,
population and capital have been typical. The overall picture is one of growth at
fluctuating rates with sometimes smooth, sometimes turbulent transitions when
jumps and reversions occurred until a ‘higher’ stage became firmly established. A
summary of the archaeological and historical evidence concerning the transition
through these stages and the various regime switching events is presented in Day
(2000, Chapter 23).
A rough time line for the permanent transitions to the several stages is given
in Table 4.1.
4.6 A simulation exercise

Given the qualitative patterns outlined above, we set for ourselves the task of cali-
brating the model so as to produce output, productivity and population trajectories
through the six stylized systems according to the following scenario: population
grows with intermittent fluctuations; numbers of economies within various systems
increase (and perhaps decrease) through fission and fusion; economies integrate
Table 4.2 System parameters for the mathematical history

Index i η α q n B B̃ ρ β M SS( ≡ N ) δ x̄
1 0.5 0.6 1 0.012 2.97 7 0.003 0.9 5 30 0.1 1.60E+06

2 0.5 0.6 0.95 0.015 2.055 10 0.02 0.6 250 2000 0.1 6.00E+06
3 0.45 1 1.5 0.022 2.2 12 0.12 0.6 2.50E+05 1.00E+06 0.1 1.90E+07
4 0.63 1 2 0.02 2.1 28 0.41 0.6 2.50E+06 1.00E+07 0.1 1.11E+08
5 0.7 1 2.2 0.19 2.1 60 0.4 0.6 2.00E+07 3.50E+08 0.1 4.00E+08
6 0.8 1 3 0.1 300 3000 0.45 0.4 3.98E+08 9.00E+08 0.1 9.00E+08
within a given culture to form smaller numbers of more advanced economies;

economies disintegrate and revert of to a ‘lower’ stage. Parameter values, given
in Table 4.2, were chosen by grid search so as to capture stylistic historical facts
from Europe.
A simulation was begun with an initial population of 100 individuals (x0 = 100)
and was continued for 4,168 periods or generations, a span of 102,700 years. Fig-
ure 4.1 shows the graph of population for this run – virtually a vertical line over
the present, caused by explosive growth after a take-off a few centuries ago. In
terms of sheer numbers, human population – relative to the present–appears ut-
terly insignificant until the most recent centuries. This is the very long run trend
explained in related ways by Lee, Kremer, and Jones in the studies referred to
in Section 4.1. But the present model adds to this by identifying the action go-
ing on during the long duration when human numbers appear insignificant from
the current perspective. This can be seen by transforming population numbers to
logarithms as in Figure 4.2. In effect this change gives a heavy weight to small pop-
ulations. It reveals some prominent features that are disguised within the thickness
of the horizontal axis in Figure 4.1. Thus, at the beginning of the run population is
seen to fall, then increase slowly for a long time. Next, it undergoes a considerable
span of pronounced growth, reaches a plateau, then enters an era of very rapid
1.00e+009
5.00e+008
1 1 1 1
0
1.00 1025.50 2054.00 3080.50 4107.00
Figure 4.1 A simulated population history (number of families). Time is measured in

‘generations’ of 25 years; population in number of individuals.
growth interspersed with fluctuations. Finally, the explosive growth of the recent
past emerges.
When the data are plotted for shorter time spans, still more detail in the behav-
ior of population emerges. This is shown in Figure 4.3. Panel (a) plots population
for a span of some 3720 generations or 93,000 years. On this scale, irregular fluc-
tuations of increasing magnitude appear. Panel (b) shows the next 388 generations
or 9,700 years. There, too, periods of growth within the hunting and food collect-
ing system are generated, interrupted by population decreases. Note the explosive
trend of the most recent years.9 Turning to the underlying economic forces, we
find much more going on in terms of structural evolution. This can be seen in Fig-
ure 4.4, which displays the dominant system index, it , at each generation. These
pictures reveal a model-generated history in which much of significance was hap-
pening economically, even though population was extremely small compared to
the modern era.
The initial population in our simulation adopted the first system and remained
with it for 3,702 generations. Growth occurred by means of the fission process,
continuing for 10 millennia or so until the number of economies reached a level
that persisted for many thousands of years thereafter. This long history of growth
through replication underlies the population growth shown in Figure 4.3a.
The different time scale of Figure 4.5 reveals more detail. A system jump first
occurs in period 3,630, some 11,975 years ago or about 9975 BC. It involves the
integration of the very large number of hunting bands into a considerably smaller
number of agricultural economies. They disintegrate within a generation, however,
back into the original number of system 1 groups. Then structural fluctuation
occurs, involving successive integrations and disintegrations, until the society locks
into system 2. Growth then continues within this village agriculture system for 181
generations or nearly 4,525 years.
A similar round of structural fluctuations occurs between systems 2 and 3 (city-
states) and systems 3 and 4 (trading empires), with corresponding fluctuations in
the number of economies as the processes of integrations and disintegrations bring
about system jumps and reversions. These outcomes reflect similar changes that
are known to have occurred in reality, as briefly described above.
Once system 4 is locked in, reversions brought about by disintegration cease.
After a few centuries, a jump to system 5 (the industrial revolution) occurs. The run
terminates with a jump to system 6 (the global information economy), which in this
simulation is interpreted as occurring in the last quarter of the twentieth century
and takes place through an integration of industrial economies. The structural
developments just described are shown in more detail by plotting the generated
scenario for the most recent four millennia.
By the time the industrial system emerges, the number of autonomous
economies has been drastically reduced. Nonetheless, the number of such states
increases about the mid-nineteenth century and again about the mid-twentieth cen-
tury. These are treated in the model as autonomous, but should be thought of not
as isolated units but as interrelated yet identifiably distinct political entities.
9.00
5.00
1.00
1.00 1025.50 2054.00 3080.50 4107.00
Figure 4.2 Logarithm of simulated population
2500000.00
1250000.00
a)
0
1.00 930.75 1860.50 2790.25 3720.00
5.50e+008
276e+008
b)
2000000.00
3721.00 3818.00 3915.00 4012.00 4109.00
Figure 4.3 Details of the population dynamics (number of families). Note the changing
time scale from (a) to (b)
6.00
3.50
1.00
1.00 1027.00 2053.00 3079.00 4105.00
Figure 4.4 Simulated history of structural change
5.00
3.00
1.00
3600.00 3727.25 3854.50 3981.75 4109.00
Figure 4.5 Simulated history of structural change in greater detail
4.6.1 Conditional future scenarios

The above simulation covers a span of history that brings the model society to the
close of the twentieth century. Without adding a new breakthrough that would jus-
tify incorporating still another macro system but retaining exactly the same model
structure as above, the simulation has been continued using as initial conditions
the variable levels reached at the end of the first simulation, i.e. generation 4108
or year ‘2000’.
In this extended story, population divides into several ‘global systems’, which
allow the population explosion shown to continue until the environmental space
is so crowded that productivity is drastically lowered and the model economy
experiences a disastrous fall in population. Using our modeling language, the
society ‘self-destructs’ in some nine generations, which corresponds in real time
to 2175 AD.
9.83e+008
1 2
4.91e+008
1 2 1 2
20.75 1 2
4070.00 4090.00 4110.00 4130.00 4150.00
Figure 4.6 Two extended simulations compared. 1 and 2 indicate number of families
This extended simulation is not a forecast. Rather, it is a conjecture about what

could happen if no fundamental changes in technology or social parameters take
place. Of course, we expect that such fundamental changes will take place.
In a second simulation all the parameters are retained except the natural growth
rate ‘n’ which is reduced to one half of the initial level. Without any advances in
the technological culture, a mere reduction in the population growth extends the
life of the model civilization to 17 generations after the first switch to the ‘global
economy’. The new ‘end of the world’ comes in 2400 AD. The situations with the
two natural growth rates compared is shown in Figure 4.6. Obviously, attainment,
of course, as long as the environmental capacity of the world is maintained, a
‘zero population growth’ rate, n = 0, would (in this model) extend the life of the
civilization indefinitely.
4.7 Conclusion
Growth theory in the hands of Tinbergen, Solow, and Swan was designed to explain
growth in the industrial countries during the first half of the twentieth century. Their
models performed remarkably well and revealed the important roles of capital
accumulation and productivity improvement. The elaboration of the theory by
Lucas, Romer, and their followers incorporates the allocation of capital and human
capital to productivity enhancement, in this way endogenizing the explanation of
productivity advance in the long run, that is, over a century or more.
The approach described here emphasizes another fundamental aspect of eco-
nomic growth that arises when the process is viewed over the very long run, that
is, over millennia. That aspect is the discrete change in the production technology
and social organization that has occurred along with the proliferation of economies
with a given fundamental system, the fluctuation in their numbers, their unification
in the transition to a system with a more elaborate social infrastructure, and their
occasional disintegration into a larger number of economies with less elaborate
infrastructures. The present theory characterizes this process and shows that pa-
rameter values can be chosen that fit the historical record in a qualitative sense,
that is, that lead to model generated regime changes and fluctuations among the
number of economies that are known to have occurred, according to a time line
roughly in accord with the evidence.
This would seem to be the most that can be expected of such an exercise.
Nonetheless, several important insights are suggested. First, population growth
within a given fixed social system seems to have been limited, even given the
presence of improving productivity. To overcome the limits, something more than
doing the same thing better seems to be indicated. Overcoming the diseconomies
that eventually emerge would seem to require basic changes in the way things
are done, both in production and social organization. Endless growth, therefore,
cannot be taken for granted, and collapses worse than a great depression can befall
any culture that allows population growth to go unchecked while failing to re-
organize itself to provide the means for maintaining coherence and symbiosis in
its increasing numbers.
Our representation of cultural selection and phase switching does not explain
the process by which such transitions are actually brought about. It only ex-
plains conditions sufficient to force such changes to occur. Both historian Quigley
(1979) and archaeologist Flannery (1999) have discussed these processes in socio–
political terms. Models like ours are very limited in what they can contribute to
an understanding of the grand process of macroeconomic evolution. But they can
provide a rigorous explanation of the demoeconomic forces that are involved.
Acknowledgement
This chapter was presented as a paper at the Summer School on ‘Economic Fluc-
tuations and Structural Change’ held in Siena, June 27–July 7, 1998. It draws on
an earlier lecture presented at the Columbia University Conference, ‘Managing
Plant Earth’, April 1997.
Notes
1 Growth economists have, until recently, thought of the ‘long run’ in terms of centuries.
Here we mean by ‘very long run’ growth over millennia.
2 See Day and Cigno (1978) and Day (1994). In the meantime, many alternative ap-
proaches have been and are being explored for studying microeconomic evolution.
3 The basis in household preferences for this Malthusian form is derived and the empirical
basis for it reviewed in Day, Kim and Macunovich (1989). A more general version that
incorporates a declining birth rate at high income levels is also introduced there. This
feature, however, is not incorporated in the present model.
4 It must be emphasized that infrastructural functions are carried out in both public
and private domains. The importance of the latter is sometimes overlooked. Large
scale corporations allocate roughly half their expenditures on educational, research,
managerial and administrative functions and roughly half on the production of goods
and services. Although some economists would include such things in the category
of intermediate goods used in the production process, it is worth distinguishing them
because their individual productivity cannot be measured in the usual ways (output
per hour expended). Their productivity, like that of the elements of public infrastruc-
ture, is only reflected in the productivity of the entire organization. The contribution
to the organization’s success of individual scientists, teachers, managers, accountants
is impossible to measure except by profit comparisons among similar organizations.
Likewise, a productive public infrastructure will be reflected in some measure of ag-
gregate accomplishment such as political, military or economic dominance, and/or a
high level of culture and wide distribution of welfare.
Infrastructure has recently been receiving increasing attention. See North (1981) for
very broad aspects and the World Bank (1994) for numerous details. For a suggestive
attempt to quantify infrastructural effects on productivity, see the working paper by
Charles I. Jenes and Robert Hall, ‘Measuring the Effects of Infrastructure on Economic
Growth’, Stanford University.
5 The conditions expressed in (4.3b) imply that lim H (x) = ∞, that lim H (x) = −∞,
x→M x→N
and that for all x ∈ (M , N ), H (x) < 0, so H (·) is strictly concave on [M , N ].
6 Day and Min (1996) show that such an economy can display all the simple and complex
possibilities: convergent growth, cycles, erratic fluctuations, and collapse. If contin-
uous (exponential) productivity improvement is incorporated, then growth or fluctu-
ations around a rising trend are possible or, as before, growth – possibly expanding
fluctuations around a rising trend – followed by a collapse.
7 An alternative which allows for emigration and immigration is the process of ‘shedding
and assimilation’ introduced in Day (2000).
8 For a discussion of the existence of an upperbound on population, see Cohen (1995a,b).
9 Necessarily, a different scale is used in each graph.
References
Boserup, Ester (1996) ‘Development Theory: An Analytical Framework and Selected Ap-
plication’, Population and Development Review, 22, 505–15.
Cohen, Joel (1995a) ‘Population Growth and Earth’s Human Carrying Capacity’, Science,
269, 34–6.
Cohen, Joel (1995b) How Many People Can the Earth Support?, Norton and Co., London,
UK.
David, Paul (1994) ‘Do Economies Diverge? Comment on Day’, in G. Silverberg and
L. Loete (eds), The Economics of Growth and Technological Change: Technologies,
Nations, Agents, Edward Elgar Publishers, Hants, UK, pp. 69–71.
Day, Richard H. and Alessandro Cigno (eds) (1978) Modeling Economic Change, Elsevier
North–Holland, Amsterdam.
Day, Richard H., Kyoo–Hong Kim and Diane Macunovich (1989) ‘Complex Demoeco-
nomic Dynamics’, Journal of Population Economics, 2, 139–59.
Day, Richard H. and Jean–Luc Walter (1989) ‘Economic Growth in the Very Long Run: On
the Multiple–Phase Interaction of Population, Technology, and Social Infrastructure’,
Chapter 11 in W. Barnett, J. Geweke, K. Shell (eds), Economic Complexity: Chaos,
Sunspots, Bubbles and Nonlinearity, Cambridge University Press, Cambridge.
Day, Richard H. and Min Zhang (1996) ‘Classical Economic Growth Theory: A Global
Bifurcation Analysis’, in T. Puu (ed.), Chaos, Soilitons, and Fractals, 7, 12, 1969–88.
Day, Richard H. (2000) Complex Economic Dynamics, Volume II, An Introduction to Dy-
namic Macroeconomics, The MIT Press, Cambridge, MA.
Easterlin, Richard A. (1978) ‘The Economics and Sociology of Fertility: A Synthesis’,
Chapter 2 in Charles Tilly (ed.), Historical Studies of Changing Fertility, Princeton
University Press, Princeton, NJ.
Flannery, Kent V. (1999) ‘Process and Agency in Early State Formation’, Cambridge Ar-
chaeological Journal, 9, 3–21.
Jenes, Charles I. and Robert Hall (1999) ‘Measuring the Effects of Infrastructure on Eco-
nomic Growth’, Working Paper, Stanford University.
Jones, Charles I. (1999) ‘Was an Industrial Revolution Inevitable? Economic Growth Over
the Very Long Run’, Mimeo, Department of Economics, Stanford University.
Kremer, Michael (1993) ‘Population Growth and Technological Change: One Million B.C.
to 1990’, Quarterly Journal of Economics, 108,4, 681–716.
Lee, Ronald Demos (1988) ‘Induced Population Growth and Induced Technological
Progress: Their Interactions in the Accelerating Stage’, Mathematical Population Stud-
ies, 1, 265–88.
North, Douglass (1981) Structure and Change in Economic History, W.W. Norton & Co.,
New York.
Quigley, Carroll (1979) The Evolution of Civilization, Liverty Press, Indianapolis, IN.
World Bank (1994) World Development Report 1994, Infrastructure for Development, Ox-
ford University Press, Oxford.
5 Out-of-equilibrium
dynamics
Mario Amendola and Jean-Luc Gaffard
5.1 Equilibrium dynamics

An instructive way to understand what we mean by out-of-equilibrium dynamics,
and what kind of modeling it implies, is to compare it with the interpretation of
economic theory as made explicit, e.g. in the real business cycle analysis. Accord-
ing to this view, the theory is reckoned to provide: ‘an explicit set of instructions
for building . . . a mechanical imitation system..to answer a question’ (Kydland
and Prescott 1996, p. 72).
The set of instructions consists mainly in behavioural axioms focusing on
rational behaviour and concerning intertemporal optimization. Among them the
axiom of technical efficiency, that is, to be always instantaneously on the optimal
production function1 .
The system to be built by means of the above instructions is aimed at imitating
a reality interpreted as an equilibrium state (on the assumption that the economy
always really works as if it were in equilibrium), except for the case of stochastic
shocks. The theory must then identify the stochastic properties of statistical series
and reproduce them by means of a model. Modeling means describing scenarios
which reproduce the ‘reality’ as defined above.
The shocks may affect production functions, thus bringing about exogenous
cycles of a stochastic nature (Lucas 1980, Kydland and Prescott 1982), or they
may affect the properties of production (and/or utility) functions, thus bringing
about endogenous cycles of a deterministic nature (Grandmont 1985, Benhabib
and Nishimura 1985). In both cases the ‘fundamentals’of the economy are affected,
and determine what happens to the economy itself.
5.2 Out-of-equilibrium dynamics

We do not consider axioms but formulate hypotheses such as to let dynamic eco-
nomic phenomena stand out. In particular we believe that dynamics has to do with
phenomena which are in the nature of change. Not ‘quantitative’ change, though –
that is, a simple modification of the intensity of a given functioning of the economy
– but ‘qualitative’ change, a change in the very way the economy actually operates.
116 Mario Amendola and Jean-Luc Gaffard
Qualitative change is a structural phenomenon that can only take place through an
adjustment. This implies a disruption of the productive structure on the operation
of which the behaviour of the economy as a whole depends, and the shaping out
of a different productive capacity. The adjustment thus comes down to an out-of-
equilibrium process, characterized by the appearance of problems of co-ordination
of economic activity which originate in the production side of the economy, since
they actually reflect a breaking of the intertemporal complementarity of the pro-
duction process as the result of the attempt to bring about a qualitative change (this
will be shown more clearly in Section 5.4).
All this is canceled in the equilibrium analytical approach. Intertemporal op-
timization implies in fact intertemporal co-ordination. Equilibrium – as we have
seen in the preceding section – is then no longer a position of the economy, but
its very way of being (even when this takes the form of fluctuations which be-
come themselves an expression of equilibrium). It becomes in a way the language
through which economic theory expresses itself. We can thus no longer make the
distinction between being in equilibrium or being out of equilibrium, a distinction
which characterizes the analysis of the economists of the beginning of the century,
like Wicksell or Marshall, and which still exists in Solow’s model.
We look instead at equilibrium as at a state of the economy, defined with refer-
ence to given facts: namely, that the intertemporal complementarity of production,
and hence the intertemporal co-ordination of economic activity, are assured. A
breaking of this state implies the appearance of problems of complementarity and
co-ordination, which throw the economy out of equilibrium
The co-ordination problems we are referring to are therefore different from
the co-ordination problems of the standard approach, which are compatible with
equilibrium. In this approach different co-ordination modes exist in relation to
different informational contexts; co-ordination failures result then in the existence
of multiple sub-optimal equilibria, not in the breaking of equilibrium.
This different perspective implies a different view of the very role of economic
theory.
We no longer aim at describing scenarios, at reproducing ‘realities’ interpreted
as the different facets of a given way of being of the economy. We focus on a
particular state of the economy, its being out of equilibrium, which is in the nature
of a process. Our aim is then to interpret this process, to analyze the salient moments
and links which make it up.
5.3 The analytical implications of a change of focus

The above change of focus has momentous analytical implications. In the first
place the usual distinction between a long term, where equilibrium obtains, and a
disequilibrium short term disappears. A process is neither a short nor a long term: it
is a sequence of disequilibria which link on and shape the evolution of the process
itself.
The analysis of this process, on the other hand, does not call for a traditional
type of model, that is, a model capable of generating a ‘solution’ in the sense of
Out-of-equilibrium dynamics 117
a type of behaviour of the economy (the attainment of a point or the following
of a path) characterized by certain specific features (efficiency, optimality, and
so forth). What we are after, instead, is to follow the evolution of the economy,
traced out step by step by the above mentioned sequence of disequilibria, in order
to investigate its viability. The essence of a thorough process, in fact – when the
focus is on the process itself, as it should, not on its point of arrival which can’t
even be defined abstracting from how the process builds up along the way – is in
its going on, that is, in its being viable. The concept of ‘solution’ interpreted as
a given configuration of the economy, therefore, has no meaning when we refer
to an out-of-equilibrium process, unless by this we mean the sorting out of the
conditions which, step by step, make it viable. This calls for a monitoring of the
process itself to bring to light its salient moments: which can only be achieved
by means of numerical experiments, that is, by simulations that, under certain
conditions (chosen so as to stress aspects relevant to the analysis) allow to unveil
what happens ‘along the way’.
In this light also the usual distinction between the terms ‘exogenous’ and ‘en-
dogenous’ must be interpreted in a different way. In a model there are variables and
there are parameters which reflect the existing constraints. In the standard analysis
the constraints, which exist outside and above the economy and which determine
its behaviour, are taken to be exogenous.
But once we recognize that the time over which change takes place is a con-
tinuing and irreversible process which shapes the change itself, as we have
to do when we consider a qualitative change, ‘it is impossible to assume the
constancy of anything over time . . . The only truly exogenous factor is what-
ever exists at a given moment of time, as a heritage of the past’. (Kaldor 1985,
p.61) In the analysis of an out-of-equilibrium process . . . we thus have to
consider as a parameter, and hence as exogenous, not some given element
chosen beforehand by reason of its nature or characteristics, but whatever, at a
given moment of time, is inherited from the past. What appears as a parameter
at a given moment of time is therefore itself the result of processes which
have taken place within the economy: processes during which everything –
including resources and the environment, as well as technology – undergoes
a transformation and hence is made endogenous to the change undergone by
the economy. Thus, while the standard approach focuses on the right place to
draw the line between what should be taken as exogenous and what should be
considered instead as endogenous in economic modeling – a line that moves
according to what we want to be explained by the model – out of equilibrium
. . . the question is no longer that of drawing a line here or there but rather
one of the time perspective adopted. Everything can be considered as given at
a certain moment of time, while everything becomes endogenous over time.
(Amendola and Gaffard 1998, pp.32–3)
Finally, it must be stressed that the ‘fundamentals’, which determine the equilib-
rium values of the relevant magnitudes of the economy, no longer play the same
role out of equilibrium, when the focus is on a process rather then on a given
configuration of the economy. Different evolution paths can be associated in fact
to given fundamentals, according to how the out-of-equilibrium process actually
evolves, and the fundamentals themselves undergo a change during this process,
given the very definition of qualitative change. The ‘fundamentals’, in other words,
are no longer fundamental.
5.4 Intertemporal complementarity and co-ordination

A viability problem arises, out of equilibrium, due to the appearance of co-
ordination problems. There is no question of viability when co-ordination is as-
sured, as it is the case in equilibrium. Co-ordination problems arise, during the
adjustment process stimulated by the attempt to bring about a qualitative change,
as the result of what happens in this case to productive capacity – namely, what
happens sequentially in time to productive capacity.
Consider a shock which affects the regular behaviour that defines an equilib-
rium state of the economy. The immediate effect of this shock is to throw the
economy itself out of equilibrium. This depends on a modification in the structure
and the functioning of the underlying productive capacity which actually deter-
mines the behaviour of the economy. The focus therefore must be put in the first
place on the production process.
This can be illustrated by considering a neo-Austrian representation of the
process of production (Hicks 1973, Amendola and Gaffard 1988, 1998), that is,
by portraying it as a fully vertically integrated process (where labour is the only
primary input contemplated) taking place through a sequence of periods which
make up a phase of construction and, following it, a phase of utilization of pro-
ductive capacity. Although in equilibrium this representation comes down to a
representation in terms of standard production functions, it also allows us to show
what happens out of equilibrium, when the time dimension of production comes to
the fore, with all its analytical implications. In neo-Austrian terms an equilibrium
structure of productive capacity is represented by an array of production processes
in the (different periods of the) phase of construction and of the phase of utilization
which are consistent with each other, in the sense of supporting a steady-state of
the economy. This age structure of productive capacity implies not only a hori-
zontal dimension, the number and age structure of production processes at each
given moment of time, but also a vertical dimension, the time pattern of production
consistent with the former. When this is so, not only construction and utilization,
but also the economic activities behind these phases, investment and consumption
and supply and demand of final output, are consistent with each other, at each
moment of time and over time. The complementarity over time of the production
process (that is, the complementarity between construction and utilization) implies
the co-ordination over time of the decision (and allocation) process. Production,
as to its effects, is synchronized.
As we have mentioned, the attempt to bring about a qualitative change, that is,
to modify the existing behaviour of the economy, throws the economy itself out of
equilibrium in that it results in a breaking of the intertemporal complementarity
of the ongoing processes of production. This, we have also seen, implies the ap-
pearance of problems of co-ordination, as saving, investment and consumption, as
well as supply and demand of final output, go out of balance in correspondence to
the fact that the phases of construction and utilization of productive capacity are
no longer consistent with each other. The time dimension of production becomes
relevant.
Reaction to these disequilibria, and the adjustments of productive capacity
aimed at re-establishing the consistency over time of construction and utilization
disturbed by the original shock, stimulate an out-of-equilibrium process that prop-
agates the initial distortion over time without needing any further shock. What
happens then to the economy must be looked at as a process sketched out step by
step by sequentially interacting disequilibria which engender a complex dynam-
ics. The backbone of this process is the accumulation through which adjustments,
which necessarily imply a restructuring of productive capacity, take place in time.
A modeling of this process, besides the above mentioned neo-Austrian rep-
resentation of the production process, requires the hypothesis of an adaptive be-
haviour of agents, so as to stress the sequential character of the decision process
and its interaction over time with the production process. A sequence ‘constraints–
decisions–constraints’, fed by complementarity and co-ordination problems inter-
acting over time, sketches out the evolution path of the adjustment process un-
dergone by the economy (see the model expounded in Section 5.6 which gives
analytical structure to this argument).
5.5 Analysis and policy conclusions

Out-of-equilibrium dynamics, we maintain, consists in the analysis of the above
defined adjustment processes to shocks implying structural modifications, that is,
altogether different behaviours of the economy.
We shall now consider the processes associated with specific shocks, selected
so as to show not only how the analysis itself is carried out but how it may lead
to policy conclusions which are often just the opposite to those resulting from
standard equilibrium analysis.
Under so called rational expectations, an economy originally in a steady state
which faces a forward biased technological change (i.e. with reference to a neo-
Austrian framework, a technological change implying higher construction costs
more than compensated for by lower utilisation costs) converges towards a new
steady state characterized by a higher level of productivity and higher real wages.
The unemployment and the productivity slowdown which appear as a consequence
of the emergence of a human resource constraint are transitory (Amendola and Gaf-
fard 1998, pp. 157–8). As a matter of fact, the hypothesis of rational expectations
makes it possible to maintain the consistency over time of construction and utiliza-
tion and hence hampers the problems linked to the intertemporal complementary
of production processes from appearing. With more realistic assumptions about
firms’ behaviours, the evolution of the economy is more complex and depends on
the way co-ordination issues are dealt with. In what follows, on the one hand we
shall assume that firms also have ‘rational expectations’ in the sense that they try
to maintain an investment behaviour aimed at preventing the distortions of pro-
ductive capacity from occurring. Investment thus will only be constrained by the
availability of financial/or human resources. But on the other hand we shall assume
that firms have an adaptive behaviour in the sense that they determine their current
supply in each period in reaction to market disequilibria perceived in the previous
periods.
Three different scenarios of evolution will be analysed hereafter, each one
corresponding to a different monetary policy.The first one is characterized by a
growth rate of money supply which is maintained equal to the current growth
rate of the economy (a Friedman rule). The second one is characterized by a
monetary policy aimed at keeping the price level stable (the rule that could be
applied by an independent central banker). The third one is characterized by a
discretionary monetary policy implying a temporary increase in the growth rate of
money supply. In each case 900 simulations have been performed corresponding
to different values of the prices and wages reaction coefficients randomly chosen
in the interval [0–1.5].
In the first scenario (Figures 5.1), 99 per cent of viable paths end with lower
real wages, lower productivity and persistent unemployment. In other words the
co-ordination conditions prevent the economy from capturing the benefits of the
new technology. Things get worse in the second scenario (Figures 5.2): 100 per
cent of viable paths end with lower real wages, lower productivity and persistent
unemployment; moreover the final level of these variables is lower than in the
previous scenario. On the contrary, in the third scenario (Figures 5.3), 95 per cent
of viable paths end with higher real wages and higher productivity, and 98 per
cent end with full employment. It is worth mentioning that an increase in the
saving rate (a decrease in the take out, as defined in Section 5.6) in scenarios of
the first category does not improve the results (Figures 5.4): only 80 per cent of
the simulated paths are viables; 83 per cent of the viable paths end with lower
real wages, 91per cent with lower productivity, and 91 per cent with persistent
unemployment.
These numerical experiments help to solve some puzzles in modern growth
economics. First, they provide an original and robust explanation of the so called
productivity paradox. As a matter of fact we have been able to show that the pro-
ductivity slowdown which, at the economy’s level, can result from the introduction
of a new and superior technology must be attributed to the co-ordination failures
arising during the out-of-equilibrium process of adjustment to the technology it-
self. These experiments also throw a different light on the role played by monetary
policy in a growth process. In a context where this process is properly seen as
an out-of-equilibrium adjustment whose main requirement is viability, monetary
policy cannot be neutral. It must be aimed at keeping the evolution of the economy
within a stability corridor which assures viability. Finally, these experiments con-
firm that the saving rate does not really matter. More precisely, an increase in the
saving rate does not make sure that potential productivity gains will be realized.
7
5
F (w/p-w0/p0)
4
a)
3
0
–0.30 –0.25 –0.20 –0.15 –0.10 –0.05 0 0.05
w/p-w0/p0
7
5
F (P-P0)
4
b)
3
0
–1.00 –0.80 –0.60 –0.40 –0.20 0 0.20
P-P0
8
5
F (U-U0)
c) 4
0
0 0.05 0.10 0.15 0.20 0.25
U-U0
Figure 5.1 Scenario 1: a) distribution of real wages, b) distribution of productivity, c) dis-

tribution of final employment.
8
F (w/p-w0/p0) 6
a) 4
0
–0.25 –0.20 –0.15 –0.10 –0.05 0
w/p-w0/p0
8
5
F (P-P0)
b) 4
0
–0.9 –0.8 –0.7 –0.6 –0.5 –0.4 –0.3 –0.2 –0.1 0 0.1
P-P0
35
30
25
F (U-U0)
20
c)
15
10
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
U-U0

16
14
12
F (w/p-w0/p0)
10
a) 8
0
–0.12 –0.10 –0.08 –0.06 –0.04 –0.02 0 0.02 0.04
w/p-w0/p0
70
60
50
F (P-P0)
40
b)
30
20
10
0
–0.40 –0.30 –0.20 –0.10 0 0.10 0.20
P-P0
100
90
80
70
60
F (U-U0)
c) 50
40
30
20
10
0
0 0.01 0.02 0.03 0.04 0.05 0.06
U-U0

12
10
8
F (w/p-w0/p0)
a) 6
0
–0.4 –0.3 –0.2 –0.1 0 0.1 0.2
w/p-w0/p0
14
12
10
F (P-P0)
8
b)
6
0
–1.4 –1.2 –1.0 –0.8 –0.6 –0.4 –0.2 0 0.2
P-P0
18
16
14
12
F (U-U0)
10
c)
8
0
0 0.05 0.10 0.15 0.20 0.25 0.30
U-U0
Figure 5.4 Scenario 1 with increase in saving rate: a) distribution of real wages, b) distri-
bution of productivity, c) distribution of final employment.
On the contrary, it brings about intertemporal complementarity problems that may
result in cumulative imbalances and erratic fluctuations in the growth rate of the
economy which hamper its viability.
5.6 Appendix
The following model which has been used for the simulation analysis whose re-
sults have been summed up in the figures in the text, is based on a neo-Austrian
representation of the production process, where
a = [a1 , . . . , anc , anc +1 , . . . , anc +nu ]
is the vector of labour input coefficients, and
b = [bnc +1, . . . , bnc +nu ]
is the vector of final output coefficients, in the different periods of the phase of
construction (1, . . . , nc ) and of the phase of utilization (nc + 1, . . . , nc + nu ).
Productive capacity at time (t) can be written
xc (t) = [x1 (t), x2 (t), . . . , xnc (t)]
xu (t) = [xnc +1 (t), xnc +2 (t), . . . , xnc +nu (t)]
where xc (t) and are xu (t) the vectors of the production processes in the phase of
construction and the phase of utilisation, respectively.
In a steady state:
xj (t) = xj−1 (t − 1) = xj+1 (t)G
xj (t) = xN (t)G N −j ; N = nc + nu ; j = 1, . . . , N
where G=1+g is the growth factor.
The labour supply is equal to:
LS (t) = LS (0) [1 + g]t
where g is the natural growth rate.
The resources required to carry out production and to sustain consumption are
financial resources. There are ‘external’and ‘internal’financial resources.
m(t) = min [p(t)s(t), p(t)d(t)]are the money proceeds from sales (internal fi-
nancial resources), where s and d are the supply and demand of final output and p
its price. f (t) is money supply (external financial resources)
In a neo-Austrian model, the wage fund represents the resources which sustain
the process of capital accumulation. It is determined by the minimum between the
available financial resources F(t) and the wage fund constrained by the available
human resources:
ω(t) = min [F(t), w(t)Ls (t)]
with
F(t) = m(t − 1) + hf (t − 1) + f (t) − c(t)
where c(t) is the take out (the resources withheld from financing production pro-
cesses) and hf (t) the monetary idle balances of finance which pile up when the
human resource constraint is more stringent than the financial constraint:
hf (t) = F(t) − w(t)Ls (t)
The decisions are taken as follows:
Production decisions The current production is the difference between the cur-
rent supply and the stocks actually put back on the market. It cannot be greater
than the existing output capacity inherited from the past.
q t) = s(t) − o(t − 1) ≤ b
x(t − 1)
where o(t) = s(t) − d(t). It determines the vector of production processes in the
phase of utilisation xu (t)
The money value of current supply is determined on the basis of expected
money proceeds
p(t)s(t) = Em(t)
Consumption decisions The money value of current households’ final demand

is determined by their financial constraint
y(t) = p(t)d(t) = ω(t) + c(t) + hh (t − 1)
where hh (t) = max [p(t) (d(t) − s(t)) , 0] are the monetary idle balances of house-
holds which pile up when the value of final demand exceeds the value of current
supply.
Investment decisions All the available financial resources can be invested:

i(t) = ω(t) − ω u (t)
where ω(t) = w(t)au xu (t).
Alternatively the investment can be determined as follows:
i(t) = min [ω(t) − ω u (t), w(t)ac (t)xc (t)]
where x1 (t) = xnc (t) [1 + g ∗ (t)]n +1 and g is the growth rate which makes it possible
c
to prevent distortions in the age structure of productive capacity.
Price and wage decisions Price and wage change from one period to the next in
reaction to market disequilibria
gp (t) = κΦ(t − 1)
gw (t) = νΨ(t − 1)
where gp (t) is the rate of variations of price, Φ(t − 1) the rate of excess demand
for the final ouput, gw (t) the rate of variation of wage,Ψ(t − 1) the rate of excess
demand for labour.
Control variables The money supply is alternatively determined on the basis of

the current growth rate
f (t) = f (t − 1) [1 + gm (t − 1]
or in such a way to keep the price level stable
f (t) = f (t − 1) [1 + gf (t − 1) − ςgp (t)] ; ς > 0
or on the basis of a targetted growth rate
f (t) = f (t − 1) [1 + g (t − 1]
The take out is alternatively determined on the basis of the current growth rate
c(t) = c(t − 1) [1 + gm (t − 1]
or in such a way as to gradually increase the amount of resources devoted to finance
investment
c(t) = c(t − 1) [1 + g(t − 1)t−tc ]
where tc is the date at which the rate of take out starts decreasing.
Notes
1 It may be noted that this does not allow the typical problems connected with the
phenomenon of production (its time dimension, the dissociation of inputs and output
and hence of costs and proceeds over time, etc.) to stand out at the analytical level. This,
we shall see in what follows, represents the real watershed between an equilibrium
and an out-of-equilibrium analytical context.
References
Amendola, M. and J-L. Gaffard (1988) The Innovative Choice Oxford: Basil Blackwell.
Amendola, M. and J-L. Gaffard (1998) Out of Equilibrium, Oxford: Clarendon Press.
Benhabib, J. and K. Nishimura (1985) ‘Competitive Equilibrium Cycles’, Journal of Eco-
nomic Theory, 35, 284–306.
Grandmont, J.M. (1985) ‘On Endogenous Competitive Business Cycles’, Econometrica,
53, 995–1046.
Hicks, J.R. (1973) Capital and Time, Oxford, Clarendon Press.
Kaldor, N. (1985) Economics without Equilibrium, Cardiff University College Press.
Kydland, F.E. and E.C. Prescott (1982) ‘Time to Build and Aggregate Fluctuations’, Econo-
metrica, 50, 1345–70.
Kydland, F.E. and E.C. Prescott (1996) ‘The Computational Experiment: an Econometric
Tool’, Journal of Economic Perspectives, 10, 69–85.
Lucas, R.E. (1980) ‘Methods and Problems in Business Cycles Theory’, Journal of Money,
Credit and Banking, 12, 696–715.
6 Disequilibrium growth in
monetary economies
Basic components and the KMG
working model
Peter Flaschel
6.1 Introduction
This chapter surveys, also for the general reader interested in non-market
clearing models of growth and fluctuations, the foundations – and the core
K(eynes)M(etzler)G(oodwin) model built on them – of the general framework un-
derlying joint past and present work with Carl Chiarella and others on integrated
disequilibrium models of monetary growth. The core KMG model of disequilib-
rium growth and its analysis is founded in this chapter on specifically reformulated
and extended partial dynamic models of the literature, the PC-AC approach (of
Goodwin and Rose) and the IS–LM–PC approach (of the textbook literature). We
also briefly indicate at the end how the fundamental KMG model thus obtained
(with its six basic laws of motion) can be extended into the direction of fairly
detailed, high-dimensional macrotheoretic disequilibrium growth models of mon-
etary economies, with strong relationships to a variety of models currently used
for structural macroeconometric model-buildings and their applications.
In order to indicate the scope and perspective of the macrodynamics to be
considered in the following let us briefly discuss here the following graphical
representation of the essential components of an integrated Keynesian disequilib-
rium growth theory. Figure 6.1 shows in the middle what might be considered
the backbone of Keynes’s (1936) General Theory, the basic causal nexus, that
makes goods markets behavior (via the investment decisions of firms) dependent
on what is achieved on financial markets, and labor markets in turn on the outcome
on the goods markets with their determination of output through expected sales
(and intended inventory changes), which in turn depend on actual aggregate goods
demand (and actual inventory changes).
In the center of interest in the macroeconomic debate of the last two decades
has been, however, quite a different module of the macroeconomy, representing
so-called supply side features or, as we prefer to characterize it, the dynamic wage–
price block of a fully-specified approach to macrodynamics. This block is shown
as wage–price spiral bottom left in Figure 6.1 and it is surely no exaggeration
to state that it has dominated the development of macrostatic and macrodynamic
mainstream thinking in the last decades, even to the extent that it was claimed that
Disequilibrium growth in monetary economies 129
Money supply rule Dombusch exchange
rate dynamics
Taylor interest rate rule
Asset
Keynes effect Blanchard equity and
markets
r, rI ,… bond dynamics
Short- and medium-term

profit rates Investment
Fisher debt and

Pigou effect
Goods Metzlerian sales
Price markets inventory
inflation Saving, investment adjustments Capacity
propensities effect on I
Expected medium-run
inflation (Mundell effect)
Real wage Rose effects Production

dynamics function Capacity
effect of I
Wage Labour
Wage price spiral
inflation markets Fiscal policy rules
Figure 6.1 The scope of traditional Keynesian theory
the Keynesian IS–LM model when integrated with supply side effects and rational
expectations degenerates to a supply side dynamics without effective demand con-
straint, see Sargent (1987, Ch.5), and for a critique of such results Flaschel (1993)
and Flaschel, Franke and Semmler (1997).
The consequences of such a concentration on supply side issues has been
that neither the relevance of the above considered causal nexus, nor the feedback
structure (or repercussions) we have added to it in Figure 6.1 (the arrows leading
back into the middle area) have received much interest. It may indeed be claimed,
that interest in the many stabilizing or destabilizing feedback chains known from
the literature on Keynesian macrostatics or macrodynamics (of the 1960s and the
1970s in particular), was nearly absent in more recent mainstream macrodynamic
model building with their wage–price interactions, up to the use of stabilizing
Keynes- and Pigou-effects, which often simply served the purpose of providing
for the deterministic part of the models a shock-absorber scenario on the basis of
which impulse-propagation mechanisms could be erected and investigated. Even
if the Keynesian demand constraints were given some attention, there was thus
generally the scenario of rapid convergence to full employment steady states unless
shocks continued to hit the economy.
Yet, there are (locally) destabilizing effects of rising inflation and expected
inflation (Mundell-effects), there are destabilizing Metzlerian inventory acceler-
130 Peter Flaschel
ator mechanisms, there is the implication (when aggregate demand depends on
income distribution and the real wage) that either price of wage flexibility must be
destabilizing (normal or adverse Rose-effects), there are destabilizing Fisher debt
deflation effects with respect to investment or consumption behavior (in particular
if there is high debt of firms or certain types of households). There are cumulative
processes in exchange rate and asset market dynamics (where expected and actual
rates of appreciation or depreciation do exercise a positive feedback on each other).
There are therefore a variety of reasons on the basis of which one might form the
expectation that steady states of integrated Keynesian or disequilibrium growth
models are more likely surrounded by centrifugal forces than by centripetal ones,
implying the necessity to introduce additional nonlinearities should the dynamics
depart too much from the steady state due to these forces.
The study of growth with endogenously generated fluctuations, based on the
disequilibrium adjustment processes and resulting feedback chains of the just char-
acterized type and coupled with additional nonlinearities they may give rise to far
off the steady state, thus should remain on the agenda of macrodynamic theoriz-
ing, if only as a (not yet) well-established and well-known alternative scenario
with which achievements of other integrated approaches to monetary growth can
be compared. There is of course always the still fashionable possibility, to model
economic dynamics such that the jump variable technique can be applied, which
by definition removes from sight all instabilities initially present in the dynamics,
so that there remains not much to be compared. Models employing the jump vari-
able technique have however stressed the importance of a treatment of anticipated
future events, a topic that should also be considered and solved by the integrated
disequilibrium growth dynamics we have in mind.
In Figure 6.1 we finally present (top-left and bottom-right) the addition of
policy feedback rules to the considered interaction of a structure of market domi-
nance with the wage–price spiral and the shown feedback structures of the private
sector, yet not necessarily with the understanding that policy can just manipulate
this scenario from the outside, but that there may be interactions with the behav-
ior assumed for the private sector and the type of policy considered to a more or
less significant degree. Policy issues are however not yet well developed in the
approaches we shall review in this chapter and thus only supplement here our
summary of traditional Keynesian model building, or better what it should have
been, but has not been yet, at least from the dynamically fully integrated point of
view.
Our perspective in the following is thus integrated Keynesian disequilibrium
growth analysis as it is obtained, on the one hand, from prototypic models of
fluctuating growth and, on the other hand, of inflation and stagflation. This implies
in a natural way the inclusion of four of the feedback structures just discussed,
with particular stress on the dynamics of wages and prices, but also of quantities
and thus on goods and labor market reaction patterns. Due to space limitations we
arrive here however only at a fundamental or working model type of integrated
disequilibrium growth theory, while further extensions (open economies, financial
markets, policy rules and more) remain for future research.1
In Section 6.2 we will consider supply side dynamics in isolation, yet not one
of the AS variety, but one that extends the basic approaches of Goodwin (1967)
and Rose (1967), here called AC-PC analysis, towards more refined treatments of
wage–price dynamics, and also towards an inclusion of adverse adjustments of
real wages and real debt (in the case of deflation in particular). Section 6.3 will
then consider the textbook IS–LM model augmented by a certain type of wage–
price dynamics (based on Friedman’s views on full employment and the role of
inflationary expectations). We will show there that the resulting dynamical system
is far from being well understood, and that it will give rise more likely to persistent
fluctuations in employment and inflation rates rather than to the shock absorber
behavior that is generally believed to be the outcome here. We also show that the
labor market NAIRU may be a goods market NAIRU in fact.
Section 6.4 considers modern discussions of wage or price Phillips curves and
tries to offer a unifying framework for such discussions (to be integrated into our
systematic evolution of integrated models of disequilibrium growth later on). This
section therefore demonstrates that much remains to done when AC-PC analysis is
extended to include Keynesian goods market dynamics. We also show that a special
case drawn from this general framework of wage–price dynamics may give rise
to an endogenous explanation of the NAIRU which is quite different from the one
that rules the roost in the literature on this rate of (un)employment.
In Section 6.5 we then present and investigate our basic modeling of integrated
disequilibrium growth theory, the core or working KMG model of disequilibrium
growth obtained as endpoint of a systematic discussion of models of monetary
growth of orthodox type in Chiarella and Flaschel (2000b), namely the Tobin type
models, the Keynes–Wicksell model types, the IS–LM growth model (without and
with smooth factor substitution) and IS–LM growth models based on the (incom-
plete) dynamic multiplier story. This section briefly presents theoretical results
on the resulting six-dimensional dynamics (non-existing so far in the literature
on this level of generality). We here study Rose-, Mundell- and Metzler-effects
in their interaction and thus obtain, on the one hand, an already fairly advanced
feedback structure between goods and labor markets (including the conventional
Keynes-effect as short-cut to the interaction with asset markets). In comparison
to Figure 6.1, on the other hand, the working KMG model still provides only a
limited picture of the working of actual economies (on the macrolevel). The reader
is referred to Chiarella and Flaschel (2000h) for the consideration of extensions of
this fundamental integrated disequilibrium growth model towards the other top-
ics included in figure 6.1 and also towards a theoretical penetration of modern
structural macroeconometric model building of the type shown in the figure.
6.2 AC-PC analysis

In static or dynamic models of the supply side of Keynesian and other types, the
marginal productivity hypothesis, the equality of real wages with the marginal
product of labor, or of the price level with marginal nominal wage costs, appears
in many respects as the central element on which analyses of supply side processes
132 Peter Flaschel
are to be founded, by way of a conventional Aggregate Supply (AS-) curve as the
competitive theory of the price level in a Keynesian setup (where firms are to be
treated as quantity-takers and price-setters) or in a Walrasian setup as theory of
labor demand, giving rise there to a Lucas type supply function with price-taking
firms. Yet, at least in a Keynesian environment with price setting firms, supply side
processes should have an explicit and detailed representation even in the case of
fixed proportions in production, where the marginal productivity theory is no longer
applicable, since this type of modeling of macroeconomic interactions should not
depend on the assumption of so-called neoclassical production functions (though
its implications may vary to some extent with the assumptions that are made with
respect to available production technologies).
We shall show in this chapter that attempts to include the dynamics of supply in
models with a Keynesian short-run give rise to an analysis of wage–price dynamics
that can be usefully compared with the implications of the supply side dynamics
of Goodwin’s (1967) model of ‘the growth cycle’, and its extensions, which we
will discuss in this section in simple graphical terms. In further sections we will
exemplify that this claim indeed holds with respect to integrated Keynesian models
of traditional type. In order to provide a lively idea of the dynamics we have in
mind when reconsidering integrated models of disequilibrium growth later on, we
will present the Goodwin model here in even simpler terms than were used in the
original work of Goodwin, the fundamental and prototypic nature of which has
been stressed by Solow (1990) in particular.
After our brief representation of Goodwin’s (1967) seminal contribution we
will indicate what has to be added to it when integrated into a Keynesian framework
as far as goods and asset markets are concerned. We then provide two extensions
of the Goodwin model (and their synthesis) which serve to indicate its potential
for more advanced types of analyses. The first of these extensions is related to
the goods-market analysis of Rose (1967), another early contribution to cyclical
growth and employment fluctuations which at least in some respects is closely
related to the Goodwin approach. The second integrates stocks and financial assets
in a very fundamental way in order to add the topic of debt deflation to the price
dynamics of the Rose approach.
6.2.1 AC-PC model building: the basic structure

Figure 6.2 provides the basic elements needed to derive the Goodwin (1967) over-
shooting growth cycle mechanism in an environment where we are still abstract
from technical change. We are implicitly assuming a fixed proportions technology.
We have top left a real wage Phillips Curve (PC), relating the rate of growth ω̂ of
real wages ω with the state of the labor market, expressed by the rate of employ-
ment V . This curve has been drawn as strictly convex, but it needs in the minimum
only fulfill the following three conditions in order to obtain the conclusions of the
Goodwin (1967) model:2
(A) Phillips curve (B) Accumulation curve
. (PC) . (AC)
ω/ω V/V
NAIRU V NAGRW ω
V ω
ω V
V ω
V ω
V V
V V
ω ω
ω
Figure 6.2 The Goodwin (1967) growth cycle model
• there is a uniquely determined Non-Accelerating-Inflation Rate of Utilization V̄ of

the labor force,3
• the PC curve exhibits negative values below this rate, and thus implies falling real
wages in this domain,
• the PC shows positive values to the right of the NAIRU, implying rising real wages
on this side.
We thus do not need in the following that the PC, in terms of employment, is
monotonically increasing as is generally assumed in the literature.4
Corresponding to this real wage PC we assume as second building block of
our model an Accumulation Curve (AC) which postulates that the rate of change
134 Peter Flaschel
V̂ of the rate of employment is a function of the level of real wages, again with
a benchmark value, a Non-Accelerating-Growth Rate of Wages or NAGRW ω̄
of the real wage, which separates rising from falling rates of employment. As
before the conclusions of the Goodwin growth cycle analysis do not demand the
monotonicity of the curve shown top right in Figure 6.2.
On the basis of the AC and PC curves shown in Figure 6.2 one gets the ad-
justments of the rate of employment and of real wages as shown in the middle of
this figure. In order to get from that the dynamic consequences for the interaction
of real wages with the rate of employment (bottom right in Figure 6.2) one has
to mirror the implications of the PC part of the model along the 45◦ degree line
(bottom left). The phase space bottom right then simply integrates the neighboring
situations as shown by the arrows in this space. The further implication of this
model, that all curves (in the positive part of the phase space shown) must be
closed orbits, can of course not be proven in this way. In order to get this result
in an intuitively understandable way, one has to consider the following type of
function:
V ω
L(ω, V ) = PC(Ṽ )/Ṽ d Ṽ − AC(ω̃)/ω̃ d ω̃.
V̄ ω̄
The graph of this (Liapunov) function has the form of a global sink (under the
assumptions made) with its minimum at ω̄, V̄ and with all level curves (where the
function assumes a given value) closed. Projected into the ω, V phase space these
closed curves are just the orbits of the considered dynamics, since it is easily shown
that L is constant along the trajectories of the investigated dynamical system (Figure
6.2, bottom right). We add that this proof applies, on the one hand, to very general
situations as far as functional shapes of the PC and AC curves are concerned, but
that it, on the other hand, has to be checked carefully for (and will often not be
applicable to) systems which do not rely on the simple cross-dual nature of the AC-
PC interaction shown in Figure 6.2.5 We thus have that all trajectories generated
by the interaction of PC and AC dynamics represent periodic motions of the real
wage and the rate of employment as the one shown in Figure 6.2, bottom right. We
do not describe this overshooting dynamic in its details here, since this has been
done many times already, including the original article of Goodwin (1967).
6.2.2 AC-PC model building: Extensions

Desai (1973) has extended the Goodwin (1967) model by an explicit treatment of
price level dynamics, through a delayed type of markup pricing.6 Using in addition
a money wage Phillips curve where price inflation enters additively with a factor
η that can be less, equal or larger than one, one can show by means of the above
Liapunov function L that the steady state ω̄, V̄ of these extended dynamics becomes
a global sink (a global source) if η < 1 (η > 1) holds true (while η = 1 gives the
original Goodwin growth cycle). This is one among a variety of examples which
shows that this growth cycle represents a border case between asymptotic stability
and instability, of which one can thus say that its closed orbit structure is exceptional
(structural unstable). Nevertheless, its message of overshooting conflicts about
income distribution can be found in many advanced models that rely on a dynamic
interaction of wages, prices and factor utilization rates.
It is thus one aim of the present chapter to show the power of this approach for
disequilibrium growth theories of an integrated nature where also goods and asset
market behavior is taken into account. Figure 6.3 indicates already what complex-
ities this might add to the growth cycle just considered. Taking disequilibrium on
the market for goods seriously in our view means that measures that represent this
disequilibrium must be introduced into the analysis, which we shall do here by re-
ferring, on the one hand, to the rate of capacity utilization Uc = Y /Y p of firms and,
on the other hand, to their rate of inventory disequilibrium Un = N /N d . Here, Y
denotes actual output and Y p potential output (for a fixed proportions technology),
while N denotes actual inventories and N d desired ones. Furthermore, we shall use
in the following Figure 6.3 also the measures V w , the (inside) employment rate
of the employed workforce (based on over- or under-time work of the employed),
and ρl , rl − π e the rate of profit and the real rate of interest, π e the expected rate of
inflation (everything here conceived as average over the longer run). Allowing for
varying rates of capacity utilization within firms implies in addition that we now
have to distinguish between the actual rate of employment, V , and the potential
one, V p , based on fully utilized capital stock and on a normal working-day for all
members of the workforce.
Obviously, the AC dynamics now concerns the relationship between real wages
and the growth rate of potential output, via the resulting profitability of firms and
the investment decision based on them. Furthermore, rates of capacity utilization
Uc and the real financing costs of firms as measured by rl − π e may also influence
their investment decision and thus the growth path of potential employment and
must therefore be added as ‘shift’ terms to the AC dynamics as indicated in Figure
6.3, top right. Note that we do not add the inventory measure here since we believe
that this measure is related to short-run (pricing) decisions of firms solely. The
PC dynamics top left, on the other hand, will be positively influenced by the
internal employment rate of firms, and negatively (through positive changes in
the price level) by the two rates that characterize the disequilibrium experienced
by firms on the market for goods. We thus get that the two curves underlying the
Goodwin growth cycle mechanism are neither fixed nor do they give rise to a
unique phase space diagram, bottom right, as was the case in Figure 6.2. Instead,
actual and potential employment will in general differ in a model that includes
the interaction of goods and asset markets (the real–financial interaction), so that
higher dimensional representations may become unavoidable when the analysis of
these additions is approached.
The phase diagram bottom right in Figure 6.3 indicates in simple terms that the
outcome of an integrated treatment of income distribution, goods market and asset
market behavior may be very uncertain at the present stage of the investigation and
may or may not lead to the result that Figure 6.2 will continue to play a prominent
role in such extended dynamics. We will return to this topic in Section 6.5.
136 Peter Flaschel
(A) Phillips curve (B) Accumulation curve
. (extended) . (extended)
ω/ω p
V /V p
Changes with: Changes with:

V w Uc Un ρl Uc rl – πe
NAIRU V ≤ Vp NAGRW ω
V ω
ω Vp
V ω
V ω
V V,V p ?
V V
?
ω ω
ω
Figure 6.3 The Goodwin growth cycle model in perspective (with influences from goods
and asset markets)
We now investigate two fundamental extensions of the Goodwin growth cycle

model that integrate, on the one hand, financial assets and liabilities, and, on the
other hand, aspects of fluctuating goods market behavior in a still very basic way, in
order to show how the dynamics are changed by such basic additions. We consider
first the case where firms do not only use retained profits to finance their investment,
but also loans (from asset holding households), while worker households still spend
what they get. The following model is based on Keen (1999), investigated in detail
in Chiarella, Flaschel and Semmler (2000), and is only briefly considered in the
present chapter. For further details the reader is therefore referred to these other
works.
Let us consider the budget equation of firms first. This equation simply states
that the excess of nominal investment expenditures pI over pure profits of firms
ρpK is to be financed by new loans as shown in the following equation (D the
stock of loans of firms):
Ḋ = pI − ρpK.
This is here to be combined with a money wage PC of the type ŵ = βw (V − V̄ ),
with w the money wage, and an investment equation of the type I /K = α(ρ − ρmin )
with α > 1 and the definition of the pure rate of profits ρ (net of interest payments
rD on loans) given by: pY p − rD − wly Y p /pK. The price level p and the rate of
interest r are considered as given for the time being, as well as the benchmark
rate of profit ρmin . The parameter ly denotes the labor coefficient of the assumed
fixed proportions technology (the other coefficient being given by the potential
output–capital ratio yp = Y p /K). Since this is still a supply side dynamics, actual
output will always equal potential output due to Say’s Law assumed to prevail in
this type of model.
These structural equations of the model give rise to:
û = βw (V − V̄ ), the reformulated PC dynamics (6.1)

V̂ = α(ρ − ρmin ) − n, the reformulated AC dynamics (6.2)
ḋ = α(ρ − ρmin )(1 − d) − ρ, the reformulated budget equation of firms (6.3)
where u = w/p · ly denotes the share of wages, where n is the rate of natural growth,
d the debt to capital ratio D/(pK) of firms, and ρ = yp (1 − u) − rd the pure rate
of profit in this supply driven approach to economic growth. The dynamically
endogenous variables are u, V as in the original Goodwin model and a new one,
the ratio d.
Proposition7
1. Assume 0 < r < n. Then: The steady state
Vo = V̄ + n/βw (6.4)
uo = (yp − ρo − rdo )/yp , ρo = ρmin + n/α (6.5)
do = 1 − ρo /n = (α − 1)/α − ρmin /n (6.6)
of the dynamics (6.1) – (6.3) is locally asymptotically stable.
2. This steady state is not globally asymptotically stable with respect to shocks
of the debt capital ratio d which, when sufficiently large, can lead to an
explosive development of the debt ratio d.
The details of this proposition and its proof are provided in Chiarella, Flaschel
and Semmler (2000) and will not be repeated here. We simply conclude here that
debt financed investment makes the Goodwin growth cycle convergent for small
138 Peter Flaschel
r and for small shocks in its state variable d, but that sufficiently high debt per
unit of capital can make these dynamics divergent ones, implying a situation of
corridor stability in the place of the closed orbit structure of the original Goodwin
(1967) approach.
6.2.3 Debt deflation and adverse real wage adjustments

It is of course necessary to add a theory of the price level to the above model of
debt accumulation, due to its long-run nature. We thus now integrate goods market
disequilibrium and fluctuating rates of capacity utilization. This will lead us to
a framework that now exhibits two interacting processes of income distribution,
between profits and wages on the one hand and between profits and interest on
the other hand. We thus now extend the model (6.1) – (6.3) to include in it the
possibility for price level deflation and thus the possibility for the occurrence of
debt deflation (high levels of debt combined with declining profitability due to
falling output prices) as well as the possibility of an adverse adjustment of real
wages, both leading to instability of the steady state of the model. This gives rise to
the following nominal dynamics for wages w, prices p and the debt ratio d coupled
with an investment driven growth and employment path, the details of which are
explained below:
1
ŵ = [βw (V − V̄ ) + κw βp (Uc − Ūc )](+π e ) (6.7)
1 − κw κ p
1
p̂ = [κp βw (V − V̄ ) + βp (Uc − Ūc )](+π e ) (6.8)
1 − κw κ p
V̂ p = α(ρ − ρmin ) − n (6.9)
ḋ = α(ρ − ρmin )(1 − d) − ρ − p̂d (6.10)
These dynamics are based on the following static (and linearized) relationship
representing Keynesian goods market equilibrium, here directly expressed in terms
of the rate of capacity utilization Uc of firms:
w w
Uc = Ū + d1 ( − ( )o ) + d2 (d − do ), d1 , d2 ≤ 0
p p
This equation is used here as a shortcut for the delayed feedback chain on the
market for goods to be introduced in Section 6.5 on the KMG model type (and its
richer concept of aggregate demand). We have assumed in this equation that output
and capacity utilization depend negatively on the real wage, based on the particular
view that the negative real wage effect on investment dominates the positive one
on consumption (the orthodox point of view), and have also assumed that output
and capacity utilization depend negatively on the debt ratio d, again because in-
vestment depends negatively on it. The above goods market representation allows
for Rose (1967) type real wage effects of traditional type (where price flexibility
will be destabilizing) and for Fisher debt effects (where price flexibility will also
be destabilizing), but it still excludes Mundell-effects, for example based on the
inflationary expectations mechanism considered in Section 6.3. Finally we have
ρ = y − wp ly y − rd = yp Uc (1 − wp ly ) − rd for the rate of pure profits ρ.
The first two laws of motion for wages w and prices p can be easily derived,
under one additional assumption stated below, from the following wage–price
adjustment equations:
ŵ = βw (V − V̄ ) + κw p̂ + (1 − κw )π e , 0 < κw < 1 (6.11)

p̂ = βp (Uc − Ūc ) + κp ŵ + (1 − κp )π e , 0 < κp < 1 (6.12)
These equations represent two symmetrically formulated Phillips curves in the

place of the hybrid single one that is usually considered in the literature, see
Solow and Stiglitz (1968) for an early formulation of this type, Rose (1990) for a
recent and more advanced one, and Fair (1997a,b) for an application of such an
approach. These two equations state that wage as well as price inflation depend
positively on the demand pressure in the market for labor or goods, respectively,
and on a weighted average of the relevant cost-push expression for each of these
PCs, actual price inflation in the first and actual wage inflation in the second case
both combined with an average rate of inflation π e that is expected to hold over
the medium run. This rate is set equal to zero in the present section for reasons
of simplicity. Investigation of the role of expected inflation will be started with
Section 6.3 and will be fully present in Section 6.5 on the KMG model.
The equations (6.7), (6.8) are easily interpreted and they state that wage as well
as price dynamics can be expressed in terms of both demand pressure variables
solely, by appropriate elimination of the cost-push expressions originally contained
in them. Increasing demand pressure in one of these markets is therefore already
sufficient to raise both wage and price inflation rates. Note in this regard also that
there is a second Non-Accelerating-Inflation Rate of Utilization Ūc now present
in the model, for the goods market and the rate of capacity utilization of the capital
stock, which plays a similar benchmark role for price inflation as the rate V̄ did
for wage inflation. Demand pressure is therefore always measured relative to such
benchmark rates, both assumed to be less than 1.
The other two equations (6.9), (6.10) are the same as before, with the exception
that p̂d has now to be added to (6.10), due to the definition d = D/(pK) of the
ratio d, since the price level p is now a variable of the model. For simplicity we
assume in the following that the minimum rate of profit ρmin of investors is equal
to r, the given rate of interest of the model.8
Note finally that the equation for the actual rate of employment V is related to
the rate of capacity utilization Uc in the following way:
V = ly Y /L = ly (Y /Y p )(Y p /K)/(L/K) = ly Uc yp /l, L the supply of labor, L̂ = n
where ly , yp are again the labor coefficient and the potential output–capital ratio
of the fixed proportions technology and where l = L/K denotes the factor endow-
ment ratio of the economy. Note also that the third equation (6.9) now concerns the
evolution of the potential rate of employment V p = ly Y p /L as discussed in connec-
tion with Figure 6.3. The dynamical system (6.7) – (6.10) therefore needs further
140 Peter Flaschel
reformulation in order to make it an autonomous system of dimension 4, since V
now depends on Uc , l and since (6.9) can no longer be used to describe the actual
evolution of the rate of employment as in the earlier treatments of the Goodwin
approach to cyclical growth (where there was always full capacity growth).
A simple reformulation of the dynamics (6.7) – (6.10) is in this regard provided
by making use of the following relationship between the ratios V , V p , Uc :
V = ly Y /L = ly Y p /L · Y /Y p = V p · Uc
by which the rate of employment can be removed from the above 4D dynamics
which are then based on the state variables w, p, V p , d, since Uc has been assumed
to be a function of w, p, d.
The interior steady state of these dynamics in the state variables w, p, V p , d is
characterized by:9
do = 1 − r/n (6.13)
Uco = Ūc (6.14)
Vop = Uco /V̄ (6.15)
ρo = yp Uc (1 − ωo ly ) − rdo = r, ω = w/p (6.16)
1 − (ρo + rdo )/(Uc yp )
ωo = (6.17)
ly
po = determined by initial conditions (6.18)
wo = po ωo (6.19)
It is therefore in fact not uniquely determined as far as nominal magnitudes are

concerned, since these dynamics can be further reduced to an autonomous system
in the real variables ω, V p , d due to the fact that equations (6.7), (6.8) can be
transformed into the single law of motion for the real wage:
1
ω̂ = [(1 − κp )βw (V − V̄ ) − (1 − κw )βp (Uc − Ūc )] (6.20)
1 − κw κ p
All dynamical equations (6.7) – (6.10) therefore only depend on ω, V p , d which

means that the 4D system has a singular Jacobian at the steady state and thus
exhibits zero root hysteresis with respect to the nominal variables of the model
which are thus determined in their long-run behavior by historical conditions. The
law of motion for nominal prices (and wages) thus can be treated as appended to
the real dynamics.
Proposition
1. Assume 0 < r < n and that βp , κp , d2 are all sufficiently small. Assume fur-
thermore that the investment parameter α is such that αr − n > 0 holds
true. Then: The steady state (6.13) – (6.19) of the dynamics (6.7) – (6.10) is
locally asymptotically stable.
2. The steady state (6.13) – (6.19) of the dynamics (6.7) – (6.10) is not locally
asymptotically stable for all price adjustment speeds βp chosen sufficiently
large.
3. Assume that nominal wages are completely fixed (βw , κw = 0). Then: The
dynamics (6.7) – (6.10) is monotonically explosive, implying higher and
higher real wages and debt to capital ratios, for all initial debt capital ratios
sufficiently high and all real wage levels above their steady state value.
Proof: See Chiarella, Flaschel and Semmler (2000).

Sufficiently sluggish price level adjustments are thus favorable for local asymp-
totic stability, while sufficiently flexible price levels will definitely destroy it. This
is due to the joint working of destabilizing Rose (1967) real wage and Fisher (1933)
debt deflation mechanisms. On the one hand, if prices are more flexible than wages
we see that depressions will increase the real wage, since prices fall faster than
nominal wages in such a situation, deepening the depression already under way,
an adverse Rose-effect as in Rose (1967), but here no longer in a framework of
Keynes–Wicksell type. On the other hand, if prices are sufficiently flexible, down-
ward in the case of a deflationary situation, they will raise real debt d significantly,
see (6.10). This depresses economic activity further and thus leads to a deflationary
spiral as was happening to some extent during the Great Depression of the 1930s
and as has again to some extent been the fear of policy makers in the years 1998/9
and at present.
We thus end up with a model type and its implications that has extended AC-PC
analysis considerably, with regard to the range of PC-dynamics to be used and with
respect to Keynesian demand pressure appearing in PC as well as AC-dynamics
now. Nevertheless, the PC discussion as well as the treatment of Keynesian demand
problems must be further improved in the light of what has been shown, which
will be done in Sections 6.4 and 6.5, thereby continuing the discussion of the
question mark in Figure 6.3 of this section. In closing this section we remark that
the Goodwin (1967) growth cycle model has of course been extended in numerous
other ways after its publication which cannot be surveyed here due to lack of
space. In this regard the reader is referred to Flaschel, Franke and Semmler (1997),
Chiarella and Flaschel (2000b), where also the question of global boundedness of
locally diverging dynamics is pursued, see also Section 6.5 of the present chapter.
6.3 IS–LM–PC analysis

We have started the analysis of disequilibrium growth in the preceding section
from the supply side and from a prototype growth cycle model whose relevance for
integrated disequilibrium growth still remains to be investigated. However, we have
augmented these dynamics by a rudimentary theory of aggregate demand already,
and have seen that this introduces significant destabilizing feedback mechanisms
into these supply side dynamics, the real wage Rose-effect and the Fisher debt-
effect, when the price level is assumed as sufficiently flexible. In this section we
142 Peter Flaschel
will now start the analysis from the opposite end, from the demand side, by means
of the conventional IS–LM model, augmented by PC dynamics which includes an
adaptive formation of expectations, as has often been discussed in the literature.
We will argue in this section that this well-known dynamic model should be well-
understood meanwhile, but that indeed just the opposite will turn out to be true.
This section will therefore reveal, on the one hand, the true power of the IS–
LM–PC approach (giving rise to persistent fluctuations in place of the generally
assumed global asymptotic stability of the steady state of the model). It will,
on the other hand, show that this approach is at least as limited with respect to
a full understanding of the fundamental feedback mechanisms of macrodynamic
systems, discussed in the introduction to this chapter, as the basic AC-PC dynamics
considered in Section 6.2. Both approaches are in fact complementary to each other,
since IS–LM–PC analysis introduces interest rate flexibility (and the Keynes-effect
that is based on it), inflationary expectations (and the Mundell-effect that derives
from them). We will study their interaction with respect to the stability question
once again. Sections 6.2 and 6.3 will be integrated with each other in Section 6.5
of this chapter which will move us closer to the perspective introduced in Section
6.1.
6.3.1 Medium-run IS–LM analysis?

Macroeconomic textbooks (also on the advanced level) usually include sections
which extend the IS–LM model to the medium run, where wage and price adjust-
ment occur depending on the state of the labor market, where then expectations
and NAIRU augmented PC dynamics are employed in order to show or indicate the
stability of the ‘full employment’ or NAIRU-equilibrium. This is in particular true
for the prominent textbook by Dornbusch and Fischer (1996) who make use of the
following simple IS–LM–PC dynamics10 in order to discuss on this basis supply
side and demand side shocks and the subsequent readjustments to the NAIRU rate
of employment:11
Ẏ = a1 (µ̄ − π) + a0 f¯, M̂ = µ̄ = const

π = p̂ = ŵ = βw (Y − Ȳ ) + π e
π̇ e = βπe (
p − πe )
This model is based on a dynamic theory of effective demand whereby the time
rate of change Ẏ of IS–LM equilibrium output Y is postulated to depend positively
on the rate of change of real balances M /p:

M /p = M̂ − p̂ = µ̄ − π
(due to the conventional Keynes-effect of static IS–LM theory) and on an exoge-

nously given dynamic fiscal policy parameter f¯. The next equation then adds a
linear, expectations augmented, natural rate (money wage and price level) PC here
based on output levels in the place of rates of unemployment. Since this model
is based on fixed proportions in production, on a constant labor supply and on
a constant markup on average wage costs, this equation can however easily be
translated back into one that shows rates of unemployment (or employment) in the
place of Y , see below. Furthermore, the assumption on markup-pricing immedi-
ately implies that wage and price inflation can be identified and represented by a
unique magnitude π.
The third equation finally is the conventional adaptive expectations mechanism
of elementary inertia theories of inflation and stagflation (π e the expected rate of
inflation). The above model can be reduced to the following form:
Ẏ = a1 (µ̄ − π) + a0 f¯
π̇ = βw a1 (µ̄ − π) + βπe βw (Y − Ȳ ) + βw a0 f¯
which is a linear autonomous differential equations system of dimension 2 in the

variables output Y and inflation π.
These dynamics imply everything one would like to find in a basic model of
monetarist wage–price dynamics with adaptive expectations, here however in the
context of a system that is apparently of IS–LM–PC type. There is a unique and
economically meaningful steady state Yo = Ȳ , πo = µ̄ + a0 f¯/a1 which reduces to
Yo = Ȳ , πo = µ̄ if fiscal policy is stationary. This steady state is globally asymptoti-
cally stable in the whole phase plane for all possible parameter values of the model
and is surrounded by cyclical forces when adjustment of inflationary expectations
is fast. There hold the monetarist propositions on monetary policy, accelerating
inflation, periods of inflation and stagflation, long-run neutrality, changing expec-
tations mechanisms in this framework of medium run IS–LM dynamics hold. A
detailed discussion of all this – which due to the linearity of the model is straight-
forward – is provided in Dornbusch and Fischer (1996) and Flaschel and Groh
(1996, Ch.4, 1998) and will not be repeated here, since we shall argue now that
this model of Dornbusch and Fischer (1996, Ch.16) is not a valid extension of their
linear IS–LM analysis12 (which we do not question) towards an inclusion of the
dynamics of wages, prices and inflationary expectations.13
Two simple observations must here suffice to justify the claim that this type of
analysis is an invalid one even when based on the assumptions made in Dornbusch
and Fischer (1996). The first observation is that investment depends on the real rate
of interest in their book (rising inflationary expectations π e will move the IS-curve
to the right and thus must enter the original Ẏ equation by mathematical necessity).
The second observation is that the Phillips curve is a nonlinear dynamic equation,
since it is based on the growth rate and not the time derivative of money wages.
However transformed, the dynamics to be analyzed is thus always nonlinear and
thus cannot be represented in the large by the above linear dynamics. The above
dynamics therefore do not represent a correct formalization of the Dornbusch
and Fischer (1996) assumptions about the Keynesian short- and the monetarist
medium-run and thus should be dismissed from this book for these and other
reasons.
144 Peter Flaschel
6.3.2 IS–LM–PC analysis proper
The first correction of the dynamics on the basis of the assumptions of Dornbusch
and Fischer’s (1996) book, is that the outcome of their linear IS–LM model, re-
formulated in terms of the employment rate V = (Y /x)/L, should be represented
as follows
V = a0 + a1 m + a2 π e , a1 , a2 > 0 (6.21)
An increase in real balances m = M /p will increase IS–LM equilibrium output

(the Keynes-effect) and an increase in inflationary expectations π e will do the
same (the so-called Mundell effect), due to the rightward shift of the IS curve that
results from this parametric change. The PC dynamics, easily transformed to rates
of change m̂ of real balances m for a given growth rate µ̄ of the money supply, then
read (when output data are transformed to rate of employment expressions):

m = µ̄ − βw (V − V̄ ) − π e (6.22)
π̇ e
= βπe (π − π ) = βπe βw (V − V̄ )
e
(6.23)
There is of course again the assumption of adaptive expectations in order to make

the model determinate. This is the complete model of the Dornbusch and Fis-
cher (1996) approach to medium run wage–price dynamics. We shall see that this
proper IS–LM–PC dynamics has little in common with the dynamics considered
in the preceding subsection, which at least disqualifies some of the monetarist
conclusions there stated.l̇ongpage
Proposition
There are always two steady states of the dynamics (6.22) – (6.23), one that is
interior to the right half of the phase plane and thus economically meaningful
and one that lies on its boundary:
m0 = (V̄ − a0 − a2 µ̄)/a1 > 0, π0e = µ̄ and m0 = 0, π0e = (V̄ − a0 )/a2 > 0.
The dynamics around the border steady state are always of saddlepath type
(det J < 0), while the dynamics around the interior steady state will be rep-
resented by a stable node, a stable focus, an unstable focus and an unstable
node as the parameter βπe , the adjustment speed of inflationary expectations,
is increased from close to zero to close to infinity.
The proof of these results is simple and is provided in Flaschel and Groh (1998).
These results in particular state that the dynamics are never globally asymptotically
stable and are also not locally asymptotically stable if inflationary expectations
are adjusted with sufficient speed. The monetarist belief in the overall asymptotic
stability of the private sector is therefore not justified in an IS–LM–PC framework,
which at best allows for corridor stability (when the Keynes-effect is sufficiently
strong relative to the Mundell-effect), but not for more, see also Groth (1993) on
this matter. If the steady state is locally explosive it will be globally explosive. The
dynamics therefore are not a viable one in this case and the question must be posed
as to what can make them bounded in such an explosive situation.
Keynes (1936) in fact did provide the basic answer to this question when
stating:
Thus it is fortunate that workers, though unconsciously, are instinctively more
reasonable economists than the classical school, inasmuch as they resist re-
ductions of money-wages, which are seldom or never of an all-round character
. . . (p.14)
The chief result of this policy (of flexible wages, P.F.) would be to cause a
great instability of prices, so violent perhaps as to make business calculations
futile . . . (p.269)
We use the following stylized modification of the PC used so far in order to provide
a mathematical expression for the institutional fact just quoted:
= max{βw (V − V̄ ) + π e , 0}.
w
This Phillips curve says that money wages behave as in the preceding subsec-
tion if their growth rate is positive, but stay constant if they would be falling in the
previous situations. There is thus no wage deflation possible now. This assumed
kink in the money wage PC could be smoothed or some wage deflation could be
allowed for, but this will not alter the conclusions significantly. We consider this
kinked Phillips curve as a much better description of reality than the one that is
linear throughout.14
The immediate consequence of this new form of the Phillips curve is that
system (6.22) – (6.23) now only applies when βw (V − V̄ ) + π e ≥ 0 holds while it
must be replaced by
= µ̄
m (6.24)
π̇ e = −βπe π e (6.25)
in the case βw (V − V̄ ) + π e < 0.15 We thus get a system of differential equations
which is only continuous now, but which can be made a smooth system in an
obvious way. We call this system the patched system while we refer to the earlier
dynamics as the unpatched one.
There are a variety of propositions that can be formulated in the context of such
a patched dynamics, see Flaschel and Groh (1998), but due to space limitations we
will consider here only one of them which describes the outcome of the explosive
case of the Dornbusch and Fischer (1996) model in the patched situation when
there is steady state inflation (µ̄ > 0). This proposition refers to Figure 6.4 which
represents the considered dynamics in the m, π e state space.
Proposition (Viability Theorem):

1. There exist exactly three steady states for the patched dynamics: So , S1 , W
if µ̄ > 0 holds. These steady states are connected by the π̇ e = 0 isocline.
146 Peter Flaschel
πe
S0
.
πe = 0
D0
S
E'
. D1
w=0 W
.
m=0
S1 .
πe = 0 E m
Figure 6.4 Implications of the kinked Phillips curve in the case of steady state inflation
2. Assume that the interior steady state W is locally repelling (for values of
βπe chosen sufficiently large). Then: every trajectory in Do converges to a
persistent cycle around W (and in D1 ).
As this figure shows the domain below the separatrix S of the saddlepoint
S0 in the nonnegative orthant is now an invariant domain Do , i.e. no trajectory
which starts in it can leave it. Note also that the domain below the ẇ = 0 isocline
is governed by the above revised dynamics in place of the one of the preceding
section, which however only alters the direction of the dynamics on the horizontal
axis. This axis is now also an isocline (π̇ e = 0) of the patched dynamics (up to point
E). Note furthermore that the trajectory which starts in E, followed up to point E
and then vertically continued up to the m-axis, also defines an invariant domain
D1 of the patched dynamics which moreover is attracting for all trajectories in the
interior of Do . We thus have that all orbits in Do (with the exception of the ones
on the vertical axis) are either inside of D1 or are entering the domain D1 (from its
left) at some point in time.
We do not go into the proof of this proposition or a deeper explanation of the
phase diagram shown, but refer the reader to the quoted work in this regard. We
simply close this subsection here by stating that the situation in Figure 6.4 is far
away from anything that can be found on IS–LM–PC dynamics in the literature,
and this is simply due to the fact that there is a Mundell effect (in IS–LM), a
simple growth law (of wages) and a fundamental institutional asymmetry in the
PC analysis to be used, which makes the overall dynamics viable up to shocks in
m or π e that go beyond the shown separatrix S.
6.3.3 Reinterpreting the IS–LM–PC dynamics
We now proceed to a reformulation and reinterpretation of the structural equations
underlying the dynamical system of the preceding subsection which do not alter its
mathematical formulation and stability features, but which give the goods market
in the place of the labor market the decisive role in the explanation of the stability
and instability scenarios just discussed. In order to introduce this reformulation of
IS–LM–PC analysis, we start from a stylized representation of empirical results
(testing the conventional NAIRU model) provided by Fair (1997a), and extended
in Fair (1997b). Fair’s (1997a) reconsideration of the structural price and wage
equation leads him to the (here simplified) result that it is in fact the price Phillips
curve which determines the shape of the integrated Phillips curve of the literature,
while wage inflation is following price inflation more or less passively. This implies
that the integrated Phillips curve now refers to demand pressure on the goods
market and not as is customarily assumed on the labor market. In terms of the two
wage and price Phillips curves considered in the preceding section this leads to
their following special reformulation
ŵ = κw p̂ + (1 − κw )π e (6.26)
p̂ = βp (Uc − Ūc ) + κp ŵ + (1 − κp )π e (6.27)
or, if solved as in the preceding section, but now with expected medium-run infla-
tion shown explicitly (no longer equal to zero):
κp
p̂ = βp (Uc − Ūc ) + π e . (6.28)
1 − κw κp
This is the same type of Phillips curve as used in the preceding subsection with
the only (economically seen very important) difference that the rate of capacity
utilization is now used in place of the rate of employment as the measure of demand
pressure that drives price inflation. Formally seen, this Phillips curve can even be
represented exactly as in the preceding subsection (in the situation where labor
supply and capital stock growth are still excluded due to the medium run nature
of the performed analysis), if one makes use of the following implications of our
assumption of a fixed proportions technology:
Y 1 Ld 1 L Ld 1 L
Uc = p
= p = p = p V
Y ly y K ly y K L ly y K
The price Phillips curve thereby becomes of the form:
p̂ = β̃p (c1 · V − c0 ) + π e . (6.29)
without implying that the demand pressure driving inflation is a labor market
phenomenon and also not that the NAIRU rate c0 /c1 implied by it is related to
labor market issues. In fact the above shows that it is the NAIRU rate of firms’
capacity utilization while there is in fact no labor market NAIRU at work in the
148 Peter Flaschel
present model. This shows that there is the possibility that the literature on the
conventional type of integrated PCs has completely misinterpreted the NAIRU
phenomenon.
On the basis of this reformulation of the (across markets) integrated Phillips
curve the analysis of the preceding subsection can be repeated word by word, with
the interpretational differences just stressed, namely that Ūc determines the steady
state value of the rate of employment and that the destabilizing Mundell-effect is
now basically due to the behavior of firms. This implies that price flexibility is bad
for economic stability in a third way (adding to the Rose adverse real wage effect
and the Fisher debt effect considered in the preceding section), namely through
the so-called Mundell effect, which says that there can be a positive (destabilizing)
feedback mechanism leading from rising inflation to rising expected inflation and
then, via the real rate of interest, to rising aggregate demand and thus to further
inflationary impulses. This holds if the negative Keynes effect on aggregate demand
based on a positive correlation of the nominal price level and the nominal interest
rate is not strong enough to overthrow this cumulative tendency in the interaction
of expected and actual rates of price inflation. Price flexibility must therefore be
regarded with suspicion from at least three different angles, in particular when
it occurs under deflationary pressure, since floors to economic activity are not
so easily established as ceilings (the latter are built into the system via supply
bottlenecks or via monetary and fiscal policy, which may stop accelerating growth,
but which cannot so easily revive declining economic activity).16
A recent discussion concerning the core of practical macroeconomics, in the
papers and proceedings issue of the American Economic Review 1997, provides
numerous statements for and against the scope and relevance for traditional Key-
nesian dynamics, in particular in the applied area, with those against generally
referring to a lack of microfoundations of Keynesian analyses, contrasted to the
progress of the macroeconomic theory of the last two decades. Nevertheless, there
continue to exist observations of the kind:
Right or wrong, the IS–LM model, and its intellectual cousins, the Mundell-
Fleming model and the various incarnations of aggregate supply – aggregate
demand models, have proved incredibly useful at analyzing fluctuations and
the effects of policy.
(Blanchard, 1997, p.245)
Yet, in view of the analysis presented in this subsection and in the preceding one,
it can be claimed that traditional Keynesian analysis is in fact still poorly understood
(or at least represented) even on the textbook level, but also in more advanced types
of analysis, if dynamic issues are addressed. There is no thorough discussion of
the many scenarios even the simple IS–LM–PC model can give rise to, there is no
investigation of the additional instabilities arising from Rose real wage and Fisher
debt effects considered in Section 6.2, nor is there any far-reaching analysis of the
full picture of traditional Keynesian dynamics as sketched in the introduction to
this chapter and as filled with more details in the sections that will follow. There is
no well-documented general analysis of the wage–price spiral, based on demand
pressure and cost-push terms, as we shall present it in the following section, no
detailed study of an integrated theoretical model with sluggish price as well as
quantity adjustments and varying rates of capacity utilization for both labor and
capital as we shall present in Section 6.5 and no such model where endogenous
growth, financial markets, policy feedback rules, and more, are systematically
investigated as to their contribution to the overall behavior of the macroeconomy.
There are however integrated models which address many of these issues,
but not all of them, from an applied perspective. Yet, these empirically motivated
structural macroeconometric models have until recently17 never been analyzed in
detail from the theoretical perspective, since these models are filled with a lot of
empirical details, often not complete with respect to long run aspects, and not
represented in the form of a theoretical reference model in the literature which
would have allowed the analysis of their steady states, their stability and of the
mechanisms that would ensure global boundedness should the steady state be
surrounded by centrifugal forces. Recent structural macroeconometric models tend
to include long run considerations now, but they still continue to believe that
the deterministic part of the dynamics they consider is behaving like a shock
absorber, a dynamic scenario with a long tradition in dynamic economic theory,
but nevertheless only one possibility of many others, in particular in the high order
dynamical systems any integrated macrodynamics will necessarily lead us to.
It can therefore not really be claimed that we do not have models at our disposal
which enrich the early Keynesian quantity dynamics by detailed price dynamics,
asset market behavior, questions of open economies and more, but it is surely true
that these models or better their common theoretical core (removing lags, special
features and pure replication by disaggregation) are not presented and investigated
to any satisfying degree. In view of the above quotation from Blanchard (1997) it
must therefore be added that the usefulness of dynamic IS–LM analysis is to be
regarded as very limited and fragmented in its present state where there are only
more or less isolated examples for such an analysis available, see Turnovsky (1977,
1995) for some left over ruins of this type. Furthermore, there is still the confusion,
see again the above quotation from Blanchard (1997), that AS–AD dynamics, as
for example presented in Sargent (1987, Ch.5), is the model of Keynesian medium
and long run analysis. This, however, cannot be true simply due to the fact that
capital is always fully employed in these approaches, while Keynesian dynamics
should in principle study the reasons for the possible under-employment (or over-
employment) of all factors of production.
We thus conclude that even those authors who show some sympathy for the
traditional Keynesian way of analyzing the macroeconomy do not really describe
what such analyses have been capable of solving and where they are still in their
state of infancy (as far as systematic explorations of the fluctuating growth pat-
terns they can give rise to are concerned). A step closer to the true alternative in
macrodynamic analysis is Barro (1994) when he states:
We have available, at this time, two types of internally-consistent models that
allow for cyclical interactions between monetary and real variables. The con-
150 Peter Flaschel
ventional IS/LM model achieves this interaction by assuming that the price
level and the nominal wage rate are typically too high and adjust only gradually
toward their market-clearing values. The market-clearing models with incom-
plete information get this interaction by assuming that people have imperfect
knowledge about the general price level.
(Barro, 1994, p.4)
It is clearly stated here that both wage and price rigidity and imbalance in the
labor and the goods market are the basic building blocks of the disequilibrium
approach to macrodynamics, which cannot therefore be of the AS–AD variety. Yet
it is not made clear that the analysis of the ‘cyclical interactions between monetary
and real variables’ is not yet very far developed if the scenario presented in the
introduction of this chapter (and more) is really taken seriously. However, it is
admitted by Barro that disequilibrium approaches can be internally consistent.
We conclude this section with the observation that much remains to be done
even on the level of traditional Keynesian IS–LM growth dynamics in order to
obtain a well-understood reference situation against which the achievements of
more recent studies of the dynamic implications of market imperfections, supply
side bottlenecks (and also of perfect market clearing approaches) can be evaluated
and put into perspective. To demonstrate this in more detail will be the topic of the
remaining sections of this chapter, see also Chiarella, Flaschel, Groh and Semmler
(2000).
In Section 6.4 we shall start this discussion with a critical evaluation of the
generality of applied Phillips curve analyses where, as we shall see, numerous
alternatives have been proposed and claimed to be the crucial ones, but where no
unifying approach so far exists which conceives all these studies as special cases
of a general formulation of wage price dynamics based on non-market clearing
in both labor and goods markets. We shall supply in the section such a general
approach as an extension of what we formulated in Section 6.2 and as a detailed
representation of the supply side features of traditional Keynesian theory as sum-
marized in Figure 6.1 of Section 6.1. On the basis of this general framework for
wage price interactions we shall then add to the discussion of Section 6.3 a model
(and theoretical interpretation) of the presented IS–LM–PC analysis of this section
which further contributes to the insight gained, which are, that there is in fact not
a uniquely determined understanding of medium-run IS–LM analysis. The topics
to be solved by the IS–LM–PC approach are thus far from being settled and un-
derstood, in contrast to what is generally declared to be the case in the literature,
where these issues are generally considered as sufficiently treated and on the basis
of this understanding as outdated.
6.4 The two PCs approach: extensions and applications

We have already stressed the importance of the usage of separate Phillips curves
for wage and price dynamics at several places in this chapter, in contrast to many
statements on ‘the Phillips curve’ that can be found in the literature on macroe-
conomic theory. Nevertheless, there exists a long (basically off-stream) tradition
to make use of two such curves in the economic theorizing, in particular in the
growth cycle literature. We have already pointed in Section 6.2 to the work of Rose
(1967), who uses two PCs with demand pressure terms solely and to Desai (1973)
where an augmented wage PC is coupled with a delayed price PC of the cost-push
variety. There is the early article of Solow and Stiglitz (1968) where symmetrically
formulated wage and price PCs are used (both with demand pressure and cost-push
terms) to investigate medium run dynamics where regime switching occurs, and
there is the related macroeconomic literature of non-Walrasian type (Malinvaud
(1980), Benassy, Picard, Hénin, Michel and others) where such PCs have often
been used in conjunction with both labor and goods market disequilibrium, see
Malinvaud (1980) for a typical example. Rowthorn (1980) makes use of a dynamic
price PC coupled with a static wage PC in order to show how the conflict over
income distribution allows for an endogenous determination of the NAIRU rate
of capacity utilization of both labor and capital, an approach that will be briefly
reconsidered in this section. There is finally recent work by Rose (1990) where PCs
of the type (6.11), (6.12) were introduced and discussed. With respect to applied
work on Phillips curves, we have already considered in simplified form the ap-
proach of Fair (1997a,b) in Section 6.3. Fair’s (1997a) approach exhibits of course
further arguments in the structural wage and price equations he proposes, see his
page 6, such as labor productivity growth and import prices.
Specifying wage and price dynamics as two separate equations is highly desir-
able in theoretical as well as applied macroeconomic analyses since it will make
explicit the reasons that may lead us to a single integrated Phillips curve later on,
and since demand pressure variables should be specific to the price variable to
considered and only be substituted by measures referring to other markets if there
is good reason to do so. The scene is thereby set to further consider the two PC
approach with demand as well as cost pressure terms from the theoretical as well as
the applied perspective and to investigate its true degree of generality before special
restrictions are added as far as the perspective of theory based macroeconometric
model building is concerned.
As already stressed, Rose (1990) has revived the theoretical consideration of
the two PCs approach, an approach taken up in Chiarella and Flaschel (2000b) in
their formulation of the wage–price module of a hierarchically structured series
of integrated models of monetary growth. In this work, however, the degree of
generality chosen basically remained limited to the PCs (6.11), (6.12) as presented
in Section 6.2 of this chapter.Yet, these two equations are but the beginning of truly
general PC formulations and investigations (leaving aside here that PC findings
are not limited to the macro-markets for goods and labor, but may also apply to
sectoral macro-considerations concerning housing, agriculture and more). In order
to show this at least partially, the return to an even older article of Phillips than the
one from which the discussion of PCs started is of great help.
Phillips (1954) investigated three possible types of fiscal policies, proportional,
derivative and integral feedback policy rules (or controls) which change for ex-
ample government expenditures, broadly speaking, in proportion to output gaps,
in proportion to their time rate of change and in proportion to the accumulated
152 Peter Flaschel
differences of such gaps, of course with a negative feedback sign in order to coun-
teract less than normal situations in particular. Similarly, inflation rates may be
driven by factor utilization gaps or in the case of wage inflation specifically by
deviations of the rate of employment from its NAIRU level, by the rate of change
of the employment rate or also by the accumulated differences (where positive and
negative signs may occur) of the deviation of unemployment rates from normal
levels. Though not framed in this type of language, all three possibilities are in
fact taken into account in early or recent investigations of the PC approach, as we
shall see in more detail below, the proportional control by the original approach of
Phillips, the derivative control in form of the so-called Phillips loops discussion
and the paper by Kuh (1967) where the level of wages (and not its rate of change)
was related to the rate of unemployment, and the integral control when it was
claimed that the rate of unemployment is not in fact determining the rate of infla-
tion itself, but rather its time rate of change. Marrying Phillips (1954) with Phillips
(1958) with respect to a treatment of wage and price inflation thus provides a fairly
general framework on the basis of which the various findings in the literature on
‘the’ Phillips curve can be evaluated and investigated in a unified way.
Let us first extend our formulation of the wage and price PCs (6.11), (6.12) of
Section 6.2 by these additional measures of demand pressure and their influence
on wage and price inflation, leaving aside here the same issue for the cost-pressure
terms. The wage and price PCs then read:

ŵ = βw1 (V − V̄ ) + βw2 V̂ + βw3 (V − V̄ )dt + κw p̂ + (1 − κw )π e (6.30)
o

p̂ = βp1 (Uc − Ūc ) + βp2 Ûc + βp3 (Uc − Ūc )dt + κp ŵ + (1 − κp )π e (6.31)
o
Note here that dimensional homogeneity demands that we should express deriva-
tive control in terms of growth rates and not as time rates of change and that it may
be preferable to use V /V̄ − 1, Uc /Ūc − 1 as measures of demand pressure (in the
place of the simple differences shown above) which at present however only leads
to proportional changes in the sizes of the parameters employed in these Phillips
curves.
Next there exists another important extension of the considered PCs which
takes note of the fact the labor and goods market disequilibrium is in fact reflected
in at least two qualitatively different measures of such disequilibrium. In the case
of the labor market this is through the external rate of employment of the labor
force and the internal rate of employment of the employed, and in the case of the
goods market through the rate of utilization of the capital stock and the rate of
utilization of the stock on inventories. This leads us to the following alternative
extension of the two PCs approach:
ŵ = βw1 (V − V̄ ) + βw2 (V w − 1) + κw p̂ + (1 − κw )π e (6.32)
p̂ = βp1 (Uc − Ūc ) + βp2 (Un − 1) + κp ŵ + (1 − κp )π e (6.33)
Here V = Lw /L, V w = Ld /Lw denotes the external and the internal rate of em-
ployment and Uc = Y /Y p , Un = N /N d the rate of capacity utilization and the
inventory / desired inventory ratio (Ld actual employment in ‘hours’, Lw the em-
ployed part of the workforce, also representing the normal working day, and Y , Y p
actual and potential output, N , N d actual and desired inventory levels). Note here
that normal working hours may be diminished by some average rate of ‘absen-
teeism’ which would imply that the rate of employment of the employed is to be
compared with a number smaller than one (in the place of the one used above).
Note also that equations (6.30), (6.31) as well as equations (6.32), (6.33) are
of the general form
ŵ = βw (·) + κw p̂ + (1 − κw )π e
p̂ = βp (·) + κp ŵ + (1 − κp )π e
and thus represent (when appropriately reordered) two linear equations in the
unknowns ŵ − π e , p̂ − π e that can be uniquely solved for ŵ − π e , p̂ − π e when
κw , κp ∈ [0, 1] fulfill κw κp < 1, giving rise then to:
1
ŵ − π e = [βw (·) + κw βp (·)]
1 − κw κ p
1
p̂ − π e = [βp (·) + κp βw (·)].
1 − κw κ p
Integrating across markets for example the two PCs approach (6.32), (6.33) in
this way, would thus imply that four qualitatively different measures for demand
pressure in the markets for labor as well as for goods have to be used both for money
wage and price level inflation for describing their deviation from expected inflation
in the usual way by an expectations augmented PC, see Laxton et al. (1998) for
a typical example, where, as is customary, only one measure of demand pressure
(on the labor market) is considered. Making furthermore use of Phillips’ (1954)
three types of control, the obtained integrated PCs will be further differentiated,
leading to twelve types of expressions for demand pressure that may appear in the
integrated (across markets) price level PC that rules the roost in the mainstream
literature. Furthermore, as in Section 6.2, two different types of NAIRUs will
then be present in the integrated (wage and) price PC which in general cannot be
identified with each other. Finally, as already mentioned, further differentiation
may concern the cost pressure terms of the PCs shown above, but will not be
considered here in its details.
The stage of wage and price Phillips curves considerations now reached thus
exhibits in each case six different measures of demand pressure in the correspond-
ing PC, which, when transformed into integrated PCs, spanning across markets,
in the way just shown, leads us to the following fairly complex expressions for
expectations augmented PCs:18

1
ŵ = π e + [βw1 (V − V̄ ) + βw2 V̂ + βw3 (V − V̄ )dt
1 − κw κp 0

+ βw4 (V w − 1) + βw5 V̂ w + βw6 (V w − 1)dt
0
154 Peter Flaschel

+ κw (βp1 (Uc − Ūc ) + βp2 Ûc + βp3 (Uc − Ūc )dt
0
+ βp4 (Un − 1) + βp5 Ûn + βp6 (Un − 1)dt)] (6.34)

0

1
p̂ = π e + [κp (βw1 (V − V̄ ) + βw2 V̂ + βw3 (V − V̄ )dt
1 − κw κp 0

+ βw4 (V w − 1) + βw5 V̂ w + βw6 (V w − 1)dt)
0
+ βp1 (Uc − Ūc ) + βp2 Ûc + βp3 (Uc − Ūc )dt
0
+ βp4 (Un − 1) + βp5 Ûn + βp6 (Un − 1)dt] (6.35)
0
As should be obvious now, the second of these equations represents ‘the’ inte-
grated price Phillips curve of this extended approach to wage and price inflation
and its various measures of demand pressure (where the actual wage and price in-
flation cost-push cross reference has been removed by mathematical substitution).
Obviously, this equation is much more complicated than the simple expectations
augmented price Phillips curve of the theoretical literature (or its Walrasian rein-
terpretation as a Lucas supply curve).
Let us briefly consider various applied approaches to PC measurements on the
basis of the equations (6.34), (6.35). Fair (1997a,b), as already shown, provides
one of the rare studies (disregarding structural macroeconometric model building
for the moment) which start from two PCs, though he makes use of βp1 = 0 solely
as far as demand pressure variables are concerned. In his view the price Phillips
curve is therefore the important one.
Concerning, modern macroeconometric model building, we find in Powell and
Murphy (1997) a money wage Phillips curve with βw1 , βw2 = 0 and a price Phillips
curve that appears to be based on cost-push terms solely, but which (when appro-
priately reformulated, see Chiarella, Flaschel, Groh, Köper and Semmler (2000a))
in fact also makes use of βp1 = 1 implicitly. Furthermore, the parameter βw2 is
in their study about eight times larger than βw1 when the nonlinear wage Phillips
curve measured in this work is linearized at the steady state, which supports Kuh’s
(1967) assertion that the wage Phillips curve is a level relationship rather than
one concerning rates of inflation (and which at the same time stresses the impor-
tance of Phillips loops as already observed by Phillips (1958) himself). Indeed,
if ŵ = βw2 V̂ represents the dominant part of the money wage Phillips curve, we
get by integration w = constV βw2 and thus a wage curve as considered on the mi-
crolevel by Blanchflower and Oswald (1994) for example. In this view the wage
Phillips curve, with derivative control solely, is therefore the important one.
Laxton et al. (1998) use for the Multimod mark III model of the IMF an inte-
grated (or hybrid) PC of the type (6.35) with only βw1 = 0, and thus the most basic
type of PC approach, but stress instead the strict convexity of this curve and the
dynamic NAIRU considerations this may give rise to. In their view, therefore, the
wage Phillips curve with proportional term only is the important one. Stock and
Watson (1997) find evidence for a Phillips curve of the type π̇ = βw3 (V − V̄ ), π = p̂,
which shows that this view is in fact based on an integral control in the money
wage Phillips curve (solely) and possibly also on an implicit treatment of inflation-
ary expectations in addition. Roberts (1997) derives a conventional expectations
augmented price Phillips curve from regional wage curves as in Blanchflower and
Oswald (1994) and thus argues that proportional control is relevant in the aggregate
even if derivative control applies to the regional level.
We thus find in this brief discussion of applied approaches a fairly varied set of
opinions, which is, however, not so varied as to pay attention to inside employment
rates and inventory utilization rates and which only in the case of Fair (1997a,b)
takes account of the possibility that demand pressure on the goods market may
be qualitatively and quantitatively different from demand pressure on the labor
market with respect to extent and implications. Otherwise, however, at least the
possibility for proportional, derivative and integral control is taken into account by
this literature (though not reflected and compared in these terms). It must therefore
be noted that the discussion on Phillips curves is at present again a lively one, a still
unsettled one, but also one with still a very limited horizon. Of course, not all of
the expressions shown in (6.35) must be relevant from the empirical point of view,
at all times and in all locations. But this should be the outcome of a systematic
investigation and not the result of more or less isolated views and investigations of
already very specialized types of PCs. Despite the new approaches to PC analysis
it therefore appears as if the analysis and investigation of these curves should start
anew from the extended perspective we have tried to describe above.
Let us close this section by considering a theoretical approach by Rowthorn
(1980) which makes use of a price Phillips curve with proportional control and
a wage Phillips curve with derivative control in order to provide an IS–LM–PC
model, in his case in fact a monetarist model of inflation and stagflation, which is
formally of the same type as the ones considered in the preceding section, but which
allows for an endogenous determination of the NAIRU rates V̄ , Ūc (based on the
conflict over income distribution). This is an interesting extension of the IS–LM–
PC dynamics considered in Section 6.3 and it furthermore provides a theoretical
example on how the use of various special types of Phillips curves (appropriately
combined) can lead to quite different views of the interaction of unemployment
and inflation as compared to the conventional one.19
The fundamental features and building blocks of Rowthorn’s reformulation of
this interaction (here augmented by IS–LM analysis in the place of his simpler
quantity theoretic approach) are the following ones:
p̂ = βp (Π∗ − Π) + π e (6.36)
∗
Π = Π∗ (Uc ) (6.37)
Π = 1 − u, u = (w/p)ly the share of wages (6.38)
ŵ = βw V̂ + p̂ (6.39)
156 Peter Flaschel
We have a price Phillips curve of the proportional kind (based on a kind of
self-reference to price inflation expected to hold over the medium run) and a wage
Phillips curve of the derivative type with myopic perfect foresight as far as price
inflation is concerned. Price inflation is driven by the gap between the desired
profit share Π∗ and the actual one, Π, with the desired profit share being a positive
function of the rate of capacity utilization Uc of firms. In the background of this
model we have our fixed proportions technology with given labor productivity 1/ly
and thus get from this a strict proportionality between the rate of capacity utilization
and the rate of employment, as a very simple form of Okun’s law, Uc = const · V ,
as shown in Section 6.3. Furthermore, the money wage Phillips curve gives rise to
(by its integration): w/p = const V βw , a functional form that then also applies to
the wage share u in the place of the real wage w/p. Inserting all these expressions
into the price Phillips curve p̂ = βp (Π∗ − Π) + π e gives rise to
p̂ = βp (Π∗ (V ) − (1 − u(V ))) + π e = βp (Π∗ (V ) + u(V ) − 1) + π e
with both Π∗ , u being strictly increasing functions of the rate of employment V .
On the surface this is just an ordinary PC of the monetarist type (as we have
employed in Section 6.3), though now possibly a nonlinear one. Disregarding
this latter possibility and assuming that parameters are such that there is a solution
V̄ ∈ (0, 1) where Π∗ (V̄ ) + u(V̄ ) = 1 holds (which is then uniquely determined), we
then get from these alternative underpinnings of the IS–LM–PC model analyzed in
Section 6.3 an endogenous explanation of the NAIRU rate of employment, which
was there given as a parameter. This NAIRU rate, among others, now depends on
the relationship Π∗ (Uc ), and thus negatively on the steepness of this curve (which
characterizes the strength with which capital owners defend their income shares)
and also negatively on the parameter βw which measures the strength with which
labor defends its income share. Therefore, the stronger the conflict over income
distribution, the lower is the NAIRU rate of employment at which the income
shares demanded, Π∗ (V ) + u(V ), become compatible with what is available for
distribution, thereby allowing for a steady behavior of wage and price inflation. This
provides in simple terms a simultaneous interpretation of both the NAIRU rate of
capacity utilization and the NAIRU rate of employment, surrounded by dynamics
that are of the same type as the one investigated in the preceding section. We stress
again that this has become possible through a simple specialization of the very
general type of PCs we have investigated in the present section. Note however that
Okun’s law, which is based on a positive correlation between the rate of capacity
utilization and the rate of employment, has been used here in order to derive this
specific view on the explanation of steady state rates of factor utilization.
We conclude this section with the observation that much remains to be done in
the theoretical discussion of the form and the implications of PC approaches, where
many more outcomes may be obtained than is generally believed. The same holds
true for empirical studies of Phillips curves, where there is a lack of systematic
investigation of the wealth of possibilities our extended presentation of the wage
price module of integrated dynamical models of Keynesian or other variety can give
rise to. Furthermore, what has been discussed here for PCs can also be applied to the
ACs considered in Section 6.2, see Figure 6.3, where derivative control terms for
the impact of capacity utilization on capital stock and employment growth would
introduce Harrodian accelerator aspects into the growth cycle there considered and
where integral terms (for profitability) would represent one possibility to introduce
medium run aspects into such AC analysis.
6.5 The basic integrated KMG model of fluctuating growth

Let us now put together what we have learned about AC-PC analysis and IS–LM–
PC analysis with their common element, the wage price adjustment equations. We
shall use these wage price equations in the form (6.11), (6.12) used in Section 6.2
and will thus considerably generalize the conventional representation of the expec-
tations augmented integrated price Phillips curve of Section 6.3, but will not yet
take the generalizations of this approach proposed in Section 6.4 into account here.
Instead, our presentation of an integrated IS–LM–PC-AC or K(eynes)–M(etzler)–
G(oodwin) model will add to Section 6.2 an investment function that now includes,
besides a measure of the profitability of firms, the real rate of interest as in IS–LM–
PC analysis and the rate of capacity utilization as in Malinvaud (1980), but not yet
in derivative form as in Harrodian knife edge growth analysis. Furthermore, we will
have two types of households in the following, workers who consume what they
get and asset holders who consume and save (the latter in the form of money, gov-
ernment bonds and equities). Finally, we will reformulate and extend this synthesis
of AC-PC and IS–LM–PC analysis by means of a Metzlerian inventory adjustment
mechanism and thus allow, in addition to sluggish wage and price adjustment, also
for sluggish quantity adjustment. In the place of IS-equilibrium we thus will have
the interaction of sales expectations, output (including desired inventory changes)
and actual aggregate demand and on this basis unintended inventory changes and
an inventory adjustment mechanism. Since this model wants to be complete with
respect to the interaction of all these elements we have, of course, to consider the
capacity effects of investment in addition to the income effects of investment and
thus to include growth which we here supplement by natural growth of labor force
in the usual way (and implicitly also by Harrod neutral technical change, which
does not alter the presentation of the model, but demands only that all labor and
wage expression have to reinterpreted in terms of efficiency units).
The model we obtain in this way is the working model of Chiarella and Flaschel
(2000b) which is derived in their book as a systematic extension of the earlier
traditional models of monetary growth of Tobin, Keynes–Wicksell and IS–LM
type. This model type is easily extended to include, besides the neutral form of
technical change just mentioned, a more refined government sector, in particular
with respect to income taxation, smooth factor substitution in the place of the fixed
proportions technology of this chapter, more advanced modeling of expectation
formation than are considered in this section as a generalization of Section 6.3’s
adaptive expectations mechanism, and more. These generalization are considered
in their relevance and with respect to their implications in Chiarella and Flaschel
(2000a) amd Chiarella, Flaschel, Franke and Lux (2001) and will here be left
158 Peter Flaschel
aside for reasons of simplicity, since they do not alter the conclusions drawn in
this section in a significant way.
The working model of Chiarella and Flaschel (2000b) that we reconsider be-
low, here called for simplicity the KMG model due to its synthesis of IS–LM
analysis (Keynes, 1936, Hicks, 1937), delayed goods market adjustment processes
(Metzler, 1941) and the classical growth cycle mechanism (Goodwin, 1967), rep-
resents in our view the basic format for traditional Keynesian monetary growth
analysis with which all extensions, modifications, reformulations or completely
new formulations of monetary growth models not based on Say’s Law should be
compared and evaluated. We are of course aware that the Hicksian representation
of the Keynes-component of this model type will not be considered as a proper
Keynes-representation by a variety of macroeconomists. Nevertheless it is impor-
tant to have such a traditional reference case, the KMG model, at our disposal in
order to allow for a precise presentation of where one should depart from it in order
to get a better theory of fluctuating growth of deterministic or even of stochastic
type. The problem in the literature on monetary growth, see Orphanides and Solow
(1990) for example, was, that such a traditional integrated Keynesian prototype
dynamics was completely missing as an explicitly spelled out model type, not to
speak of the analysis of the six dimensional dynamics this model type will give rise
to (which in fact is the minimum dimension of a theory of monetary growth with
sluggish adjustments in wages, prices, expectations, quantities and inventories).
We start with an overview on the intensive form representation of these 6D
KMG-dynamics. The extensive form of this KMG model is presented in its details
in Chiarella and Flaschel (2000b), where it is shown that the intensive form results
from a model type that is coherently formulated with respect to behavior and budget
restrictions and thus does not allow for demand of agents that is not backed up by
the supply of funds (or loans) that finance these expenditures.
The first dynamical law concerns the real wage ω = w/p and is in fact a
direct consequence of the integrated two PCs approach (6.7), (6.8), as it has been
derived from the wage–price dynamics (6.11), (6.12), now including inflationary
expectations π e = 0 which however cancel in the calculation of the real wage.
Subtracting (6.8) from (6.7) gives immediate rise to
ω̂ = ŵ − p̂
1
= [(1 − κp )βw (V − V̄ ) − (1 − κw )βp (Uc − Ūc )] (6.40)
1 − κw κp
which simply states that the adjustment of real wages depends positively on the
demand pressure on the market for labor and negatively on that in the market for
goods. We immediately realize that either wage or price flexibility should bring
instability to the dynamics of this section (normal or abnormal Rose effects),
depending on whether real wage increases increase or decrease economic activity.
The next law of motion concerns the dynamics of the factor endowment ratio
l = L/K, where it is assumed that labor supply L grows with the given natural rate
n and the capital stock (in our Keynesian approach) with the rate of net investment
K̂ = I /K. The essential element in this law of motion is therefore given by the
investment function which is specified as follows:
I /K = i1 (ρe − (r − π e )) + i2 (Uc − Ūc ) + n.
This function (as all other behavioral equations) is assumed as linear (just as our
simple production function Ld = ly Y , Y p = yp K, ly , yp is given magnitudes) in order
to keep the model first as linear as is possible which allows us to concentrate on
its intrinsic or unavoidable nonlinearities in the beginning of the analysis of KMG
growth dynamics. Note also that the trend term in this investment equation is given
by the natural rate of growth.
Investment per unit of capital thus depends on the expected rate of profit ρe ,
the real rate of interest r − π e and the rate of capacity utilization Uc , representing
Tobin’s q and the capacity effect considered by Malinvaud (1980) and others. All
these magnitudes are actual (short run) values and should be replaced by medium
run averages in applications of this model type, which increases the dimension of
the dynamics without adding too much new structure to the model, see Flaschel,
Gong, and Semmler (1999) for such extensions and applications of the model.
Here, however, only the above simple formulation of investment behavior will be
used and it gives rise to the following law of motion for the labor intensity l = L/K :
l = −i1 (ρe − (r − π e )) − i2 (Uc − Ūc ). (6.41)
The next dynamical law is basically equation (6.12) of Section 6.2, since it is
based on the definitional equation for real balances (per unit of capital now):
m = M /(pK), implying m̂ = µ̄ + n − l̂ − p̂, where it is again assumed that the
money supply M grows with a given rate µ̄ (since policy questions are not of
interest here). Making use of the equation (6.12) for p̂ this gives:
1

m = µ̄ − n + l̂ − π e − [βp (Uc − Ūc ) + κp βw (V − V̄ )] (6.42)
1 − κw κp
We see that increased capacity utilization on both the labor and the goods market
will speed up inflation and thus reduce the growth rate of real balances, leading to
corresponding nominal interest rate changes due to the Keynes-effect as it derives
from the simple LM-curve still present in this model type.
Next we have the law for inflationary expectations which is a simple extension
of the one used in Section 6.3, since we now determine these expectations as an
average of backward and forward looking behavior (time series methods and fore-
casts by means of small theoretical models). Time series methods can in principle
be as complicated and refined as possible, when only numerical simulations of the
model are intended. From the viewpoint of theory they should at first be chosen to
be as simple as possible in order to allow for an analytical treatment of stability
issues as we shall provide below, i.e. they will be of the simple adaptive type made
use of already in Section 6.3 (they can be made a humped shaped average of past
160 Peter Flaschel
observations of inflation by means of nested adaptive expectation schemes in a
next step for example). Forward looking expectations can be based on the p-star
concept of the FED and the German Bundesbank for example, which says that
inflation rates will converge to the difference of µ̄, the growth rate of the money
supply, and Ŷ p , the growth rate of potential output (as long as the velocity of money
can be considered a given magnitude). Made as simple as possible again this shows
that inflationary expectations of this type assume that there is convergence of these
actual inflation rates back to the steady rate µ̄ − n giving rise to:
π̇ e = βπ [α(p̂ − π e ) + (1 − α)(µ̄ − n − π e )] (6.43)

1
= βπ [α [βp (Uc − Ūc ) + κp βw (V − V̄ )]
1 − κw κ p
+ (1 − α)(µ̄ − n − π e )]
where α denotes the weight attached to the backward looking type of expectations
and 1 − α the one for the forward looking type. The destabilizing role of the
Mundell effect is clearly visible in this extended equation, since economic activity
depends positively on expected inflation (due to the assumed investment behavior)
and since an increase in economic activity, here measured by two rates of factor
capacity utilization, speeds up the increase in inflationary expectations as shown
by this equation.
There remains the quantity adjustment process on the market for goods which is
driven by the adjustment of sales expectations and the changes in actual inventories,
ye = Y e /K and ν = N /K, both already in per unit of capital form here, see Metzler
(1941) for the original approach. The two laws of motion for these variables read:
ẏe = βye (yd − ye ) + l̂ye (6.44)

ν̇ = y − y + (l̂ − n)ν
d
(6.45)
where the terms involving l are simply due to the fact that everything is expressed
in per unit of capital terms. Sales expectations of firms, ye , are here assumed to
change in an adaptive fashion, following actual demand yd with some time delay,
while actual inventories changes are given by definition through the difference
between actual output y and actual demand yd , again corrected by a term that takes
account of the intensive form under consideration.
This closes the description of the laws of motion of the state variables of our
basic KMG dynamics which concern income distribution, relative factor growth,
inflation as measured by the change of real balances, inflationary expectations,
sales expectations and actual inventory changes. These dynamical laws do not yet
form a complete system, but must be supplemented by some algebraic equations
which define the statically endogenous magnitudes we used in the above differen-
tial equations. They are given by:
y = βn (βnd ye − ν) + (1 + nβnd )ye

y d
= ωly y + (1 − sc )(ρe − t n ) + i1 (ρe − (r − π e )) + i2 (Uc − Ūc ) + n + δ + g
V = l d /l, Uc = y/yp , l d = Ld /K = ly y,
ρe = ye − δ − ωly y, r = r0 + (h1 y − m)/h2 .
These equations describe output y equal to expected sales and voluntary inventory
changes, which follow βnd ye − ν, the difference between desired inventories and
the actual ones, with speed βn , aggregate demand yd which is composed of the
real wage sum per unit of capital, consumption of asset holders based on expected
profits after taxes ρe − t n , where taxes net of interest t n are treated as a parameter
of the model as in Sargent (1987, Ch.5), gross investment (δ the depreciation rate)
and government expenditures g, again assumed a parameter of the model. The
remaining equations then define the rate of employment V , the rate of capacity
utilization Uc , the expected rate of profit, ρe , based on the sales expectations of
firms, and the nominal rate of interest as defined by a linear form of money market
equilibrium. These explanations must suffice here as a presentation of the static
part of the working KMG model of Chiarella and Flaschel (2000b). For additional
explanations and a detailed analysis of this model type the reader is referred to this
earlier work.
Note with respect to the full structure of traditional Keynesian monetary growth
analysis presented in Section 6.1, that we now employ an advanced description of
the wage–price module (but not as advanced as the one arrived at in Section 6.4),
that we still consider a closed economy (no Dornbusch exchange rate dynamics),
that internal asset markets (up to a static LM determination of the nominal rate
of interest) are not considered explicitly (as in Sargent (1987, Ch.1–5), i.e. we do
not yet consider Blanchard (1981) type asset market dynamics, we do not yet have
Fisher debt effects, as considered in Section 6.2 and have also abstracted from
wealth (Pigou) effects in consumption as well as money demand. The above dise-
quilibrium growth model therefore just represents the beginning of the analysis of
integrated disequilibrium growth of a traditional Keynesian type, with particular
stress on sluggish wage, price and quantity dynamics and resulting capacity uti-
lization problems for labor as well as capital, but not yet a real-financial interaction
in the proper sense of this phrase.
Despite the linearity assumed for all behavioral equations the model is intrin-
sically nonlinear, due to its use of growth laws of motion and the unavoidable
appearance of products and quotients of state variables in various places. The con-
sidered dynamics therefore should be capable of generating limit cycles and also
more complex attractors for its trajectories even on this basic level of its formula-
tion.
Proposition
1. Assume sufficiently sluggish adjustment for wages, prices, and inflationary
expectations, and a strong Keynes-Effect (h2 small). Then:
The interior steady state of the 6D dynamics (6.40) – (6.45), which is easily
calculated and uniquely determined, is locally asymptotically stable for all
adjustment speeds of sales expectations βye chosen sufficiently large and
speeds of adjustments of inventories, βn , chosen sufficiently low.
162 Peter Flaschel
2. The 6D determinant of the Jacobian of the dynamics at the steady state is
always positive.
3. If βπe , βn , h2 are chosen sufficiently large, the steady state equilibrium be-
comes locally repelling. The system therefore undergoes a (generally non
degenerate) Hopf bifurcation at intermediate value of these (and other) pa-
rameters, which generates persistent fluctuations, that are attractors in the
supercritical case and repellers in the subcritical case.
The periodic fluctuations obtained in this way integrate the growth cycle anal-
ysis of Section 6.2 with the inflationary dynamics of Section 6.3, coupled with
Metzlerian quantity adjustment in the market for goods and they are generated in-
dependently of any kink in the money wage PC. Further details on this proposition
are provided in Chiarella and Flaschel (2000b). We do not go into a proof of this
proposition here, but simply add some explanations to the assertions made. The
steady state of the system is asserted to be locally attracting for all price and quan-
tity adjustment speeds (including the Metzlerian inventory accelerator) sufficiently
low (up to sales expectations which mirror the stable Keynesian multiplier dynam-
ics which improve stability if chosen sufficiently large). In addition we should
have a fairly interest-inelastic money demand function in order to produce large
positively correlated swings of the nominal rate of interest when the price level
rises or falls. Partial insights on the stability of Keynesian dynamics (augmented
with what we know for the Rose effects considered in Section 6.2) or even static
conclusions of Keynesian theory, appropriately combined, thus allow here for a
stability assertion for the full 6D dynamics of the integrated KMG growth model.
Furthermore, since the determinant does not change sign when the parameters of
the model are changed, we know that loss of stability can only occur in a cyclical
fashion since eigenvalues must then cross the imaginary axis excluding 0 (and will
generally do so with positive speed). The resulting situations of Hopf bifurcations
then generally imply that this change in stability of the system is accompanied by
either the ‘birth’ of a stable limit cycle (with increasing amplitude) to the right
of the critical bifurcation value (where a pair of eigenvalues has become purely
imaginary) or the death of an unstable limit cycle (via its shrinking amplitude and
the disappearance of a ‘stable corridor’) to the left of this critical point when this
point is approached.
The above brief considerations of KMG growth dynamics must here suffice to
indicate what results might be expected from an integrated AC-PC analysis with
Keynesian supply rationing and with also sluggish quantity adjustment processes
of Metzlerian type. These topics are further pursued in particular in Chiarella and
Flaschel (2000b) and Chiarella, Flaschel, Groh and Semmler (2000).
6.6 Conclusions and outlook

We have started in this chapter from a simple classical growth cycle model, based
on AC-PC analysis, and from the textbook understanding of IS–LM analysis aug-
mented by inflation and expected inflation in the conventional way, here called
IS–LM–PC analysis. We could show for this latter model type that textbook presen-
tations of it are generally misleading and incomplete and give rise, when corrected
and completed, to persistent fluctuations of employment and inflation around the
steady state position when adjustment of adaptively formed expectations is suffi-
ciently fast. These two model types therefore provide us with a theory of medium
run as well as long run employment cycles which, due to their common PC element
suggests, that it should be possible to integrate them into larger dynamics with at
least two cycle generating mechanisms. We have supplemented this discussion of
growth and inflation with a reconsideration of theoretical and applied PC discus-
sions and have found, that the former is still fairly underdeveloped with respect
to a general treatment of wage–price dynamics and the demand pressure elements
that may be involved in this interaction, that applied approaches have generally
been broader here, but did not treat their empirical investigations from the unified
point of view that we could present in Section 6.4.
In Section 6.5 we then presented a coherent core model of short run Keynesian
quantity dynamics, medium run wage–price dynamics and long run growth cycle
dynamics, based on fully specified budget equations for all sectors considered, see
Chiarella and Flaschel (2000b), and on a unique economic steady state reference
path, which integrated the Goodwin (1967) AC-PC analysis and the Rose (1967)
real wage effects considered in Section 6.2 with the IS–LM–PC analysis (based
on Keynes- and Mundell-effects) of Section 6.3, by making use in addition of
Metzlerian quantity adjustment processes on the market for goods in the place of
an infinitely fast adjustment to Keynesian goods market equilibrium. This model
type, the working model of Chiarella and Flaschel (2000b), there obtained from
a hierarchical structured sequence of disequilibrium models of monetary growth,
should long since have been the basic reference case for an integrated treatment
of Keynesian disequilibrium growth theory, but has indeed been completely ne-
glected in the literature on monetary growth, see Orphanides and Solow (1990) for
example, possible due to the fact that AS–AD growth, as treated in Sargent (1987,
Ch. 5) was generally considered as the Keynesian theory of monetary growth and
due to the fact that the analysis of nonlinear economic dynamics seemed to be
restricted to dimension 3 or less, while our integrated model of Keynesian mon-
etary growth, with sluggish price and quantity adjustment (including expectation
formation on aggregate demand as well as inflation), and under- or over-utilized
labor as well as capital is in the minimum of dimension 6.
The scene is thereby set for a fundamental and at the same time already very
general labor and goods market disequilibrium growth model (with a simple LM
treatment of asset markets still) which can be analyzed from the theoretical as
well as from the numerical and empirical perspective, see Chiarella and Flaschel
(2000b) and Flaschel, Gong and Semmler in particular, and which should serve
at the least as a reference case for judging the progress made by other types of
approaches in the recent past, by those who believe that Keynesian monetary
growth theory (which was never fully presented and analyzed as we have shown)
is nowadays dated or even outdated. In our view such a judgment is not based on
knowledge of the true potential of integrated Keynesian theory of traditional type,
164 Peter Flaschel
as we have attempted to show here and in more detail in Chiarella and Flaschel
(2000b). Chapters 3 to 5 in Chiarella, Flaschel, Groh and Semmler (2001) show in
this respect furthermore, that recent non-Walrasian macro-approaches with their
regime switching methodology can easily be used to make the traditional KMG
approach even more coherent, by adding to it supply side constraints of a fairly
secondary nature, and that also the partial new Keynesian treatment of market
imperfections can be integrated into this KMG approach or used to overhaul certain
of the modules of this approach from these partial perspectives.
There are further topics which are relevant for the discussion presented in
this chapter but which are not considered here due to space limitations. Opening
the KMG economy is the subject of Chiarella and Flaschel (1999, 2000c) and
Chiarella, Flaschel, Franke and Lux (2001), developing the model further towards
applied integrated macrodynamics is the subject of a series of papers, Chiarella
and Flaschel (2000d–f), Chiarella, Flaschel and Zhu (2000a-c), Chiarella, Flaschel,
Groh, Köper and Semmler (2000a,b) and debt deflation is further analyzed in a very
general model type in Chiarella and Flaschel (2000d,g ), Chiarella, Flaschel and
Semmler (2000, 2001). Taking this work together I hope will lead to the formulation
of an applicable theoretical model of integrated disequilibrium growth with all the
features shown in Figure 6.1 and thus of a fully specified traditional Keynesian
model type that is consistently formulated with respect to budget equations and
steady state calculations and that allows for a detailed analysis of the various
feedback hypotheses that can be associated with a structure as shown in Figure
6.1. This figure thus represents the perspective of this chapter which I believe has
shown that such an aim can be pursued by a systematic integration and extension
of classical AC-PC and Keynesian IS–LM–PC analysis.
Notes
1 See Chiarella and Flaschel (2000h) for the full details of what is sketched in this chapter
and in the concluding section in particular.
2 continuity of this curve is of course assumed in addition.
3 Note that we reinterpret the NAIRU of the literature here in terms of the rate of
employment (or utilization) V of the labor force, not in terms of unemployment.
4 Note that a real wage PC is obtained from the conventional money wage PC (augmented
by inflationary expectations of course) by assuming myopic perfect foresight with
regard to the expected rate of inflation.
5 See Flaschel, Franke and Semmler (1997) for the consideration of cross-dual macro-
dynamics on various levels of generality.
6 See also Flaschel (1984) in this regard.
7 We assume that the parameters of the model are such that both uo and do are positive.
8 See Chiarella, Flaschel amd Semmler (2000) for a much more general treatment of
such growth dynamics.
9 We assume again that parameters are chosen such that all steady state values are
meaningful.
10 These authors employ a special discrete time version of the following model, a differ-
ence which however is not essential for our following discussion of their model.
11 Note that the real wage is constant in this approach which is therefore clearly comple-
mentary to one in the preceding section.
12 which in principle should be well-known from Tobin (1975) and subsequent work, but
which is still unfamiliar, see Groth (1993).
13 Note that – though globally asymptotically stable – the model is still incomplete since
the right half of the phase plane is not an invariant set of this dynamics, i.e. output
can be become negative along trajectories that start in an economically meaningful
domain.
14 See Laxton, Rose and Tambakis (1997) for an empirical discussion of the kind of
nonlinearity that may be involved in the integrated price level – rate of unemployment
Phillips curve.
15 The two systems are identical at the border line ŵ = βw (V − V̄ ) + π e = 0.
16 See here also Flaschel (1994) and Flaschel and Franke (1996,98).
17 See for example Barnett and He (1998) for an exception.
18 Note that π e was set equal to zero in the models considered in Section 6.2.
19 See also Flaschel (1993) and Flaschel and Groh (1996) in this regard.
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7 Schumpeterian dynamics
A disequilibrium theory of
long run profits
Katsuhito Iwai
7.1 Introduction
The subtitle of this chapter may sound a contradiction in terms. In the traditional
economic theory, by which I include both classical and neoclassical economics, the
long-run state of an economy is an equilibrium state and the long-run profits (if they
ever exist) are equilibrium phenomena. Figure 7.1 illustrates this by drawing two
supply curves that can be found in any textbook of economics. In the upper panel
is an upward-sloping supply curve which aggregates diverse cost conditions of
the existing firms in an industry. Its intersection with a downward-sloping demand
curve determines an equilibrium price, which in turn determines the amount of
profits (represented by the shaded triangle) accruing to the industry as a whole. As
long as the supply curve is upward-sloping, an industry is able to generate positive
profits.
In traditional theory, however, this is merely a description of the ‘short-run’state
of an industry. Whenever there are positive profits, existing firms are encouraged
to expand their productive capacities and potential firms are induced to enter the
industry, both making the supply curve flatter and flatter. This process will continue
until the industry supply curve becomes totally horizontal, thereby wiping out any
opportunity for positive profits. The lower panel of Figure 7.1 describes this ‘long-
run’ state of the industry.
This implies that if there are any profits in the long run, it must be the ‘normal’
profits which have already been incorporated into cost calculations. In fact, it is how
to explain the fundamental determinants of these normal profits which divides the
traditional economic theory into classical and neoclassical approaches. Classical
economics (as well as Marxian economics) has highlighted an inverse relationship
between the normal profit rate and the real wage rate, and reduced the problem
of determining the former to that of determining the latter and ultimately to that
of distributional conflicts between classes. Neoclassical economics has identified
the normal profit rate with the interest rate plus a risk premium and reduced the
problem of its determination to that of characterizing equilibrium conditions for
intertemporal resource allocation under uncertainty. But no matter how opposed
their views might appear over the ultimate determinants of normal profits, they
170 Katsuhito Iwai
Price
Quantity
0
Price
normal profit
Quantity
0
Figure 7.1 Industry supply curve in the short run and in the long run
share the same ‘equilibrium’ perspective on long-run profits – any profits in excess
of the normal rate are ‘disequilibrium’ phenomena which are bound to disappear
in the long run.
It is Joseph Schumpeter who gave us a powerful alternative to this deep-rooted
‘equilibrium’ tradition in the theory of long-run profits. According to Schumpeter,
it is through an ‘innovation’or ‘doing things differently’that positive profits emerge
in the capitalist economy. ‘The introduction of new commodities . . . , the techno-
logical change in the production of commodities already in use, the opening-up of
new markets or of new sources of supply, Taylorization of work, improved handling
of material, the setting-up of new business organizations.’ (Schumpeter (1939), p.
84) etc. allow the innovators to charge prices much higher than costs of production.
Profits are thus the premium put upon innovation. Of course, the innovator’s cost
advantage does not last long. Once an innovation is successfully introduced into
the economy, ‘it becomes much easier for other people to do the same thing’.1 A
subsequent wave of imitations soon renders the original innovation obsolete and
Schumpeterian dynamics 171
gradually wears out the innovator’s profit rate. In the long run, there is therefore
an inevitable tendency towards classical or neoclassical equilibrium which does
not allow any positive profits in excess of the normal rate. And yet Schumpeter
argued that positive profits will never disappear from the economy because capi-
talism is ‘not only never but never can be stationary’. It is an ‘evolutionary process’
that ‘incessantly revolutionalizes the economic structure from within, incessantly
destroying an old one, incessantly creating a new one’.2 Indeed, it is to destroy
the tendency towards classical or neoclassical equilibrium and to create a new in-
dustrial disequilibrium that is the function the capitalist economy has assigned to
those who carry out innovations. ‘Surplus values [i.e. profits in excess of normal
rate] may be impossible in perfect equilibrium, but can be ever present because that
equilibrium is never allowed to establish itself. They may always tend to vanish
and yet be always there because they are incessantly recreated’.3
It is the first objective of this chapter to formalize this grand vision of Joseph
Schumpeter from the perspective of evolutionary economics.4 It makes use of a
simple evolutionary model of Iwai (1984a, b) and demonstrates the Schumpeterian
thesis that profits in excess of normal rate will never disappear from the economy
no matter how long it is run. Indeed, it will be shown that what the economy will
approach over a long passage of time is not a classical or neoclassical equilibrium
of uniform technology but (at best) a statistical equilibrium of technological dis-
equilibria which reproduces a relative dispersion of efficiencies among firms in
a statistically balanced form. Although positive profits are impossible in perfect
equilibrium, they can be ever present because that equilibrium is never allowed to
establish itself.
This chapter is organized as follows. After having set up the static structure
of an industry in Section 7.2, the following three sections will develop an evolu-
tionary model of industrial dynamics and examine how the firms’ capacity growth,
technological imitations and technological innovations, respectively, move the in-
dustry’s state of technology over time. It will be argued that while both the dif-
ferential growth rates among different efficiency firms and the diffusion of better
technologies through imitations push the state of technology towards uniformity,
the punctuated appearance of technological innovations disrupts this equilibriating
tendency. Section 7.6 will then turn to the long-run description of the industry’s
state of technology. It will indeed be shown that over a long passage of time these
conflicting microscopic forces will balance each other in a statistical sense and
give rise to a long-run distribution of relative efficiencies across firms. This long-
run distribution will in turn allow us to deduce an upward-sloping long-run supply
curve in Section 7.7. The industry is thus capable of generating positive profits
even in the long run! Hence, the subtitle of this chapter – ‘a disequilibrium the-
ory of long-run profits’. Section 7.8 will then examine the factors determining the
long-run profit rate of the industry.
The present chapter will adopt the ‘satisficing’ principle for the description
of firms’ behaviors – firms do not optimize a well-defined objective function but
simply follow organizational routines in deciding their growth, imitation and in-
novation policies.5 Indeed, the purpose of the penultimate Section 7.9 is to show
172 Katsuhito Iwai
that our evolutionary model is able to ‘calibrate’ all the macroscopic characteris-
tics of the neoclassical growth model without having recourse to the neoclassical
assumption of fully optimizing economic agents. If we look only at the aggregative
performance of our evolutionary economy, it is as if aggregate labor and aggregate
capital together produce aggregate output in accordance with a well-defined ag-
gregate production function with Harrod-neutral technological progress. Yet, this
macroscopic picture is a mere statistical illusion. If we zoomed into the microscopic
level of the economy, what we would find is the complex and dynamic interactions
among many a firm’s capital growth, technological imitations and technological
innovations. It is simply impossible to group these microscopic forces into a move-
ment along an aggregate production function and a shift of that function itself. The
neoclassical growth accounting may have no empirical content at all.
Section 7.10 concludes the chapter.
7.2 Construction of the industry supply curve

The starting point of our evolutionary model is an observation that knowledge is
not a public good freely available among firms and that technologies with a wide
range of efficiency coexist even in the same industry. And one of the end points of
our evolutionary model is to demonstrate that technologies with a wide range of
efficiency will indeed coexist even in the long-run.
Consider an industry which consists of many firms producing the same
product.6 Let us denote by n the total number of technologies coexisting in the
industry and assume that each of these technologies is of Leontief-type fixed-
proportion technology with labor service as the sole variable input and capital
stock as the sole fixed input. If we further assume that only labor productivity
varies across technologies, we can express the ith technology as:

l k
q = min , , (7.1)
ci b
where q, l and k represent final output, labor input and capital stock, and ci and
b are labor and capital coefficients. Let us choose money wage as the numeraire.
Then, the labor coefficient ci determines the unit cost of each technology up to a
productive capacity k/b. I will slightly abuse the term and call ci the ‘unit cost’ of
technology i. It is then possible to rearrange the indices of technology and let cn
stand for the lowest and c1 the highest unit cost of the industry without an loss of
generality, or:
cn < cn−1 < . . . < ci < . . . < c1 . (7.2)

I now have to introduce several notations in order to construct the supply curve of
the industry in question. Let kt (ci ) represent the sum of the capital stocks of all the
firms whose unit cost is ci at time t, and let Kt (ci ) ≡ kt (cn )+ . . . +kt (ci ) represent
the cumulative sum of all the capital stocks of the firms whose unit costs are ci or
lower at time t. The industry’s total capital stock at time t can then be represented
St (c )
1
c
0 cn cn–1 …… c2 c1
Figure 7.2 Cumulative distribution of capacity shares
by Kt (c1 ), but will be denoted simply as Kt in the following discussion. Next, let
st (ci ) and St (ci ) represent the ‘capital share’ and the ‘cumulative capital share’ of a
unit cost ci at time t. Of course, we have st (ci ) ≡ kt (ci )/Kt and St (ci ) ≡ Kt (ci )/Kt .
As a convention, we set St (c) = St (ci ) for ci ≤ c < ci−1 . Figure 7.2 exhibits a typical
distribution of cumulative capital shares in the industry. It illustrates the ‘state of
technology’ at a point in time by showing us how technologies with diverse unit
costs are distributed among capital stocks of an industry.
The state of technology thus introduced, however, represents merely the pro-
duction ‘possibility’of an industry. How this possibility is actualized depends upon
the price each firm is able to obtain in exchange for its product. Let us assume
that the industry in question is a competitive industry in which a large number of
firms are producing the same homogeneous product and charge the same price for
it.7 Let us denote by Pt the product price (measured in terms of money wage) at
time t. Then, under the assumptions of homogeneous product and fixed proportion
technology, firms with unit costs strictly smaller than Pt decide to produce up to
their productive capacity k/b, and firms whose unit costs are strictly higher than Pt
decide to quit all production. Firms with the unit cost equal to Pt are indifferent to
their production level, as long as it does not exceed their productive capacity. (We
ignore here the cost of shutting-down of a factory as well as the cost of setting-up
of a new production line.)
It follows that when ci−1 > Pt > ci the total supply of the industry product
becomes equal to Kt (ci )/b and that when Pt = ci it takes any value from Kt (ci+1 )/b
to Kt (ci )/b. Hence, if we denote by Yt (P) the industry’s ‘short-run supply curve’
174 Katsuhito Iwai
Pt
c1
cn
yt
0 St (cn ) St (cn –1 ) …… St (c2 ) St (c1 ) = 1
Figure 7.3 Relative form of industry supply curve
(or short-run supply correspondence, to be precise ) at time t, it can be written as
Yt (P) = Kt (ci )/b if ci < P < ci−1

(7.3)
ε[Kt (ci+1 )/b, Kt (ci )/b] if P = ci .
Dividing this by the total productive capacity Kt /b, we can also express it as:
yt (P) ≡ Yt (P)b/Kt = St (ci ) if ci < P < ci−1
(7.4)
ε[St (ci+1 ), St (ci )] if P = ci .
(7.4) is nothing but the ‘relative’ form of industry supply curve at time t, which
has neutralized the scale effect of changes in the total capital stock of the industry.
Since the forces governing the motion of St (c) are in general of different nature
from those governing the motion of Kt , I will be concerned mostly with this relative
form of industry supply curve in what follows.8
Figure7.3 depicts the relative form of industry supply curve, yt (P) ≡
Yt (P)b/Kt , in a Marshallian diagram with prices and costs (both in terms of money
wage) measured along the vertical axis and quantities per unit of total productive
capacity measured along the horizontal axis. Indeed, it merely turns Figure 7.2
around the 45 degree line. It is an upward-sloping curve as long as different unit
costs coexist within the same industry.
7.3 Darwinian dynamics of the state of technology

Any freshman knows that the industry supply curve is a horizontal sum of all the
individual supply curves existing in the industry. But the problem we now have
to tackle is to ascertain how the dynamic competition among firms will mold the
evolutionary pattern of the supply curve and govern the fate of the industry. This is
not the problem for freshmen. Since there is a one-to-one correspondence between
the relative form of industry supply curve and the cumulative distribution of capital
shares, the analysis of the dynamic evolution of the former can be reduced to that
of the latter.
Now, the state of technology in our Schumpeterian industry is moved by com-
plex interactions among the dynamic forces working at the microscopic level of
individual firms – successes and failures of technological innovations and imita-
tions and the resulting differential growth rates among competing firms. Let us
examine the effect of differential growth rates first.
‘Without development there is no profit, without profit no development’, so
said our Schumpeter.9 The following hypothesis relates the growth rate of capital
stock to the rate of profit:
Hypothesis (CG): The capital growth rate of a firm with unit cost ci is linearly
increasing in its current rate of profit rt (ci ), or it is equal to:
γrt (ci ) − γ0 ; (7.5)

where γ > 0 and γ0 > 0 are given constants.
This hypothesis needs little explanation. It merely says that a higher profit rate
on the existing capital stocks stimulates capital accumulation, either by influencing
the expected profitability of new investment projects or by directly providing an
internal fund for the projects. The parameter γ (or, more precisely, γ/b) represents
the sensitivity of the firm’s growth rate to the current profit rate, and the parameter
γ0 represents the rate of capital depreciation of the break-even firm. As I have
already indicated in Section 7.1, the present chapter follows the strict evolutionary
perspective in supposing that firms do not optimize but only ‘satisfice’ in the sense
that they simply follow organizational routines in deciding their growth, imitation
and innovation policies. Indeed, one of the purposes of this chapter is to see how
far we can go in our description of the economy’s dynamic performance without
relying on the assumptions of individual optimality. I will therefore assume that
the values of γ and γ0 are both exogenously given.10
We have already assumed that every firm in the industry produces the same
homogeneous product and faces the same price Pt . If we further assume that
the price of capital equipment is proportional to Pt , we can calculate the profit
rate r(ci ) ≡ (Pt yt − ci lt )/Pt kt as b(Pt − ci )/Pt , which we will approximate as
b(log Pt − log ci ) for analytical convenience. Then, by simply differentiating the
cumulative capacity share St (ci ) with respect to time, Hypothesis (CG) allows us to
deduce the following set of differential equations for the dynamics of cumulative
capital shares11
Ṡt (ci ) = γδt (ci )St (ci )(1 − St (ci ))(i = n, n − 1, . . . , 1). (7.6)
176 Katsuhito Iwai
In the above equations, δt (ci ) represents the difference between the logarithmic
average of a set of unit costs higher than ci and the logarithmic average of a set of
unit costs not higher than ci , or:

i−1
(log cj )st (cj )
n
(log cj )st (cj )
δt (ci ) ≡ − > 0. (7.7)
j=1
1 − St (ci ) j=i
St (ci )
Its value in general depends on t and the whole distribution of ci . I will, however,
proceed with the following analysis as if it were an exogenously given constant δ,
uniform both across technologies and over time. This will simplify the exposition
of our evolutionary model immensely without losing any of its qualitative nature.12
Then, we can rewrite (7.6) as:
Ṡt (ci ) ∼
= γδSt (ci )(1 − St (ci )) (i = n, n − 1, . . . , 1). (7.8)
Each of the above equations is a well-known ‘logistic differential equation’ with
a logistic parameter µ, and can be solved explicitly to yield:
1
St (ci ) = (i = n, n − 1, . . . , 1), (7.9)
1 + (1/ST (ci ) − 1)e−γδ(t−T )
where e stands for the exponential and T (≤ t) a given initial time.13
Differential growth rates among firms with different cost conditions never leave
the industry’s state of technology static. As the firms with relative cost advantage
grow faster than the firms with relative cost disadvantage, the distribution of capital
shares gradually shifts in favor of the lower unit costs, thereby reducing the average
unit cost of the industry as a whole. This process then eliminates the relative cost
advantage of the existing technologies one by one until the capital share of the
least unit cost completely overwhelm those of the higher ones. Only the fittest
will survive in the long run through their higher growth rates, and this of course
is an economic analogue of the ‘Darwinian’ natural selection mechanism. The set
of logistic equations (7.9) describes this ‘economic selection’ mechanism in the
simplest possible mathematical form, and its evolutionary dynamics is illustrated
by Figure 7.4. In particular, the equation for i = n shows that the cumulative
capital share of the lowest unit cost St (cn ) moves along an S-shaped growth path.
It grows almost exponentially when it occupies a negligible portion of the industry,
gradually loses its growth momentum as its expansion narrows its own relative cost
advantage, but never stops growing until it swallows the whole industry.
7.4 Lamarkian dynamics of the state of technology

Next, let us introduce the process of technological imitation and see how it molds
the dynamics of the state of technology. In this chapter I will suppose that technol-
ogy is not embodied in capital stocks and hypothesize the process of imitations as
follows:14
1
St (cn –2 )
St (cn –1 )
St (cn )
0
T t
Figure 7.4 Evolution of the state of technology under the pressure of either economic
selection or technological diffusion
Hypothesis (IM ): The probability that a firm with unit cost ci succeeds in imitating
a technology with unit cost cj is equal to:
µst (cj )dt if cj < ci and 0 if cj ≥ ci , (7.10)

for a small time interval dt; where µ(> 0) is assumed to be a constant uniform
across firms.
One of the characteristic features of technology is its non-excludability. It may

be legally possible to assign property rights to the owners of technology. But, as
Arrow has remarked in his classic paper (1962), ‘no amount of legal protection
can make a thoroughly appropriable commodity of something so intangible as
information’, because ‘the very use of the information in any productive way is
bound to reveal it, at least in part’.15 The above hypothesis mathematically captures
such spill-over effects of technology in the simplest possible manner. It says that
it is much easier for a firm to imitate a technology with high visibility (i.e., a large
capital share), than to imitate a technology with low visibility (i.e. a small capital
share). Needless to say, the firm never imitates the technology whose unit cost is
not smaller than its current one. The imitation coefficient µ in the above hypothesis
represents the effectiveness of each firm’s imitative activity. There is a huge body
of literature, both theoretical and empirical, which identifies factors influencing
the effectiveness of firms’ imitation activities.16 The main concern of the present
chapter is, however, to work out the dynamic mechanism through which a given
imitation policy of firms structures the evolutionary pattern of the industry’s state
of technology. In what follows I will simply assume that µ is an exogenously given
parameter, uniform across firms and constant over time.17
178 Katsuhito Iwai
In order to place the effect of technological diffusion in full relief, let us ig-
nore the effect of economic selection for the time being. Then, the hypothesis
(IM’) allows us to deduce the following set of logistic differential equations as a
description of the evolution of the state of technology under the sole pressure of
technological imitations:18
Ṡt (ci ) = µSt (ci )(1 − St (ci )) (i = n, n − 1, . . . , 1). (7.11)

We have again encountered logistic differential equations, which can then be solved
to yield the second set of logistic equations in this chapter!
1
St (ci ) = (i = n, n − 1, . . . , 1), (7.12)
1 + (1/ST (ci ) − 1)e−µ(t−T )
where T (≤ t) is a given initial time.
Since the second set of logistic equations (7.12) is mathematically equivalent to
the first set of logistic equations (7.9), Figure 7.4 in the preceding section can again
serve to illustrate the dynamic evolution of the cumulative capacity shares under
the sole pressure of technological diffusion. And yet, the logic behind these second
logistic equations is entirely different from that of the first. ‘If one or a few have
advanced with success many of the difficulties disappear’, so wrote Schumpeter,
‘others can then follow these pioneers, as they will clearly do under the stimulus
of the success now attainable. Their success again makes it easier, through the
increasingly complete removal of the obstacles . . . , for more people follow suit,
until finally the innovation becomes familiar and the acceptance of it a matter of
free choice.’(Schumpeter (1961), p. 228) The logistic equations (7.12) describe
this swarm-like appearance of technological imitations in the simplest possible
form. In particular, the equation for i = n shows that the cumulative capital share
of the lowest unit cost moves along a S-shaped growth path, initially growing
at an exponential rate but gradually decelerating its growth rate to approach unity
asymptotically. In the long run, therefore, the lowest cost technology will dominate
the whole industry, simply because it will eventually be diffused to all the firms
in it. This technological diffusion process is nothing but an economic analogue of
the ‘Lamarkian’ model of biological evolution – the achievement of one individual
are passed directly to the other individuals.
Let us then bring back the Darwinian process of economic selection into our
industry and add (7.6) to (7.11). The result is the third set of logistic differential
equations in the present chapter:
Ṡt (ci ) = (γδ + µ)St (ci )(1 − St (ci )) (i = n, n − 1, . . . , 1), (7.13)
which can again be solved explicitly as:
1
St (ci ) = (i = n, n − 1, . . . , 1), (7.14)
1 + (1/ST (ci ) − 1)e−(γδ+µ)(t−T )
for t ≥ T . (We refrain from drawing a diagram for the third set of logistic equations
(7.14) which is qualitatively the same as Figure 7.4.)
We have thus shown how the mechanism of economic selection and the pro-
cess of technological diffusion jointly contribute to the logistic growth process of
cumulative capital shares – the former by amassing the industry’s capacities in
the hands of the lowest cost firms and the latter by diffusing the advantage of the
lowest cost technology among imitating firms. While the former is Darwinian, the
latter is Lamarkian. But, no matter how opposed the underlying logic might be,
their effects upon the industry’s state of technology are the same – the lowest cost
technology will eventually dominate the whole capital stocks of the industry.
7.5 Punctuated dynamics of the state of technology

Does this mean that the industry’s long-run state is no more than the paradigm of
classical and neoclassical economics in which every market participant is supposed
to have a complete access to the most efficient technology of the economy?
The answer is, however, ‘No’. And the key to this negative answer lies, of
course, in the phenomenon of innovation – the carrying out of what Schumpeter
called a ‘new combination’. Indeed, the functional role of innovative firms is
precisely to destroy this tendency towards static equilibrium and to create a new
industrial disequilibrium.
Suppose that at some point in time one of the firms succeeds in introducing
a new technology with unit cost cn+1 smaller than cn . Let us denote this time by
T (cn+1 ) and call it the ‘innovation time’ for cn+1 . Then, a new cumulative capital
share St (cn+1 ) emerges out of nothing at T (cn+1 ). Because of the disembodied
nature of technology, ST (cn+ 1 ) (cn+1 ) is identical with the capital share of the inno-
vator of cn+1 . Moreover, if the innovator’s unit cost was, say, ci before innovation,
all the cumulative capital shares from St (ci+1 ) to St (cn ) also experience a jump
of the same magnitude at time T (cn+1 ). In no time the innovator starts to expand
its capital stocks rapidly, which then induces all the other firms to seek opportu-
nities to imitate its technology. Through such selection mechanism and diffusion
process, the newly created cumulative capital share begins to follow a S-shaped
growth curve described by (7.14).
Innovation is not a single-shot phenomenon, however. No sooner than an inno-
vation occurs, a new round of competition for a better technology begins. And no
sooner than a new winner of this game is named, another round of technological
competition is set out. The process repeats itself forever, and technologies with ever
lower unit costs, cn+2 > cn+3 > . . . > cN > . . . will be introduced into an industry
one by one at their respective innovation times T (cn+2 ), T (cn+3 ), . . . , T (cN ), . . . .
Figure7.5 shows how the industry’s state of technology evolves over time,
now as an outcome of the interplay among three dynamical forces working in
the industry – economic selection mechanism, technological diffusion through
imitations and creative destruction of innovations. In fact, while the former two
work as equilibriating forces which tend the state of technology towards uniformity,
the third works as a disequilibriating force which destroys this leveling tendency.
180 Katsuhito Iwai
1
St (cn –2 )
St (cn –1 )
St (cn )
St (cn +1 )
St (cn +2 )
St (cn +3 )
0
T (cn ) T (cn +1 ) T (cn +2 ) T (cn +3 ) t
Figure 7.5 Evolution of the state of technology under the joint pressure of economic
selection, technological diffusion and recurrent innovations
A new question then arises: is it possible to derive any law-like properties out of
this seemingly erratic movement of the industry state of technology?
In order to give an answer to this question, it is necessary to introduce two
more hypotheses – one pertaining to invention and the other to innovation. The
conceptual distinction between invention and innovation was very much empha-
sized by Schumpeter. Invention is a discovery of new technological possibility
which is potentially applicable to the production processes of the economy. But,
‘as long as they are not carried into practice’, so says Schumpeter, ‘inventions
are economically irrelevant’, and ‘to carry any improvement into effect is a task
entirely different from the inventing of it’.19
Denote then by C(t) the unit cost of potentially the best possible technology
at time t and call it ‘the potential unit cost’. The following is our hypothesis about
the process of inventions:20
Hypothesis (PC): The potential unit cost is declining at a positive constant rate
λ over time.
C(t) = e−λt , (7.15)

where the scale of C(0) is chosen to be unity.
The declining rate of potential unit cost λ reflects the speed at which the stock
of technological knowledge is being accumulated by academic institutions, private
firms, government agencies and amateur inventors throughout the entire economy.
In the present chapter which follows an evolutionary perspective, however, it is
assumed to be given exogenously to the industry.
We are then able to characterize the notion of ‘innovation’ formally as an event
in which the potential unit cost is put into actual use by one of the firms in the
industry. This is tantamount to saying that when an innovation takes place at time
t, it brings in a technology of unit cost C(t) for the first time into an industry.
This also implies that if a technology with unit cost c is presently in use, it must
have been introduced at time t = T (c) where T (c) is the inverse function of C(t)
defined by:
T (C(t)) ≡ t or C(T (c)) ≡ c. (7.16)

The function T (c) thus defined is nothing but the ‘innovation time’ for unit cost c
we have already defined at the beginning of this section. Under the specification
of the dynamics of potential unit cost in (7.15), we have T (c) = −(log c)/λ.
Next, let us introduce the hypothesis about the process of innovations:
Hypothesis (IN − a): The probability that a firm succeeds in an innovation is equal
to:
vdt, (7.17)
during any small time interval dt, where ν is a small positive constant.
The parameter ν represents the effectiveness of each firm’s innovative activity in

the industry. Its value should in general reflect a particular innovation policy the
firm has come to adopt in its long-run pursuit for technological superiority, and
there is a huge body of literature identifying the factors which influence the firm’s
innovation policy.21 Our main concern in the present article, however, is rather to
examine how a given innovation policy will mold the evolutionary pattern of the
industry’s state of technology in the long run. In what follows I will simply assume
that ν is an exogenously given parameter, uniform across firms and constant over
time.22 I will, however, suppose that the value of ν is much smaller than either that
of µ or of γδ, for the innovations are by their nature much more difficult activities
than imitations or growth.
Implicit in the above hypothesis is the supposition that an innovation can be
introduced at any time and by any firm, irrespective of at what time and by which
firm the last innovation was introduced.23 Indeed, if we let M denote the total
number of firms in the industry, the probability that there is an innovation during
a small time interval dt is equal to (νdt)M = νM dt. Hence, the process of techno-
logical innovations in the industry as a whole constitute a Poisson process, which
is sometimes called the law of rare events. As time goes by, however, innovations
take place over and over again, and out of such repetitive occurrence of rare events
a certain statistical regularity is expected to emerge.
7.6 The state of technology in the long run

Indeed, not only the process of innovations but also the entire evolutionary process
of the state of technology is expected to exhibit a statistical regularity over a long
182 Katsuhito Iwai
passage of time. To see this, let Ŝt (c) denote the expected value of the cumulative
capital share of c at time t. For the purpose of describing the long-run pattern
of the industry’s state of technology, all we need to do is to follow the path of
Ŝt (c). Indeed, it is not hard to deduce from Hypothesis (IN-a) the following set of
differential equations for Ŝt (c):24 :
Ŝt (c) = (γδ + µ)Ŝt (c)(1 − Ŝt (c)) + ν(1 − Ŝt (c)), (7.18)
for t ≥ T (c). It turns out that this is the fourth set of logistic differential equations
of this chapter, for each of which can be rewritten as ẋ = (γδ + µ + ν)x(1 − x)
with x ≡ (Ŝt (c) + ν/(γδ + µ))/(1 + ν/(γδ + µ)). It can thus be solved to yield:
1+ ν ν
Ŝt (c) = γδ+µ − , (7.19)
1+ γδ+µ
e−(γδ+µ+ν)(t−T (c)) γδ +µ
ν
for t ≥ T (c).25
Of course, we cannot hope to detect any regularity just by looking at the motion
of expected cumulative shares Ŝt (c) given above, for they are constantly pushed to
the lower cost direction by recurrent innovations. If, however, we neutralize such
declining tendency by measuring all unit costs c relative to the potential unit cost
C(t) and observe the relative pattern of the cumulative capital shares, a certain
regularity is going to emerge out of the seemingly unpredictable vicissitude of
the industry’s state of technology. Let us thus denote by z the proportional gap
between a given unit cost c and the current potential unit cost C(t), or
z ≡ log c − log C(t). (7.20)

We call this variable the ‘cost gap’ of a given technology at time t. Since the inverse
relationship between innovation time T (c) = −(log c)/λ and the potential unit cost
C(t) = e−λt implies z = λ(t − T (c)), it is possible to rewrite (7.19) in terms of z
as follows:
1+α
Ŝt (c) = S̃(z) ≡ 1+ α − α, (7.21)
1 + (1/α)e− αβ z
where α and β are composite parameters respectively defined by:

ν λ
α≡ and β≡ . (7.22)
γδ + µ ν
This is the fifth time we have encountered a logistic curve. This time, it represents
the ‘long-run cumulative distribution’ of cost gap z, towards which the relative
form of the industry’s state of technology has a tendency to approach in the long
run. This distribution is a function only of the cost gap z and is totally independent
of calendar time t. Figure 7.6 illustrates this distribution.
As is seen from (7.21), the shape of S̃(z) is determined completely by two
composite parameters α and β, whose values are in turn determined through
1
αβ
0 z
Figure 7.6 Long-run cumulative distribution of capacity shares
(7.22) by such basic parameters as γδ, µ, ν and λ. The first composite parame-
ter α ≡ ν/(γδ + µ) represents the relative strength between the disequilibriating
force of creative-cum-destructive innovations and the joint equilibriating force
of economic selection mechanism and diffusion process through imitations. The
second composite parameter β ≡ λ/ν, on the other hand, represents the relative
strength between the force of inventions and that of innovations. Since the expected
rate of innovation per unit of time is 1/ν and the reduction rate of the potential unit
cost per unit of time is λ, β can also be interpreted as the expected cost reduction
rate of each innovation.26 It is not difficult to show that:27
∂ S̃(z) ∂ S̃(z)
<0 and < 0. (7.23)
∂α ∂β
As is illustrated in Figure 7.6, an increase in both α and β thus shifts S̃(z) clock-
wise, thus rendering the distribution of efficiencies across firms more disperse than
before.
The long-run cumulative distribution S̃(z) thus deduced is a statistical summary
of the way in which a multitude of technologies with diverse cost conditions are
dispersed among all the existing capital stocks of the industry. It shows that, while
the on-going inventive activities are constantly reducing the potential unit cost,
the unit costs of a majority of production methods actually in use lag far behind
this potential one. The state of technology therefore has no tendency to approach
a classical or neoclassical equilibrium of uniform technology even in the long run.
What it approaches over a long period of time is merely a ‘statistical equilibrium
of technological disequilibria’.
184 Katsuhito Iwai
7.7 The industry supply curve in the long run
Now, the fact that the state of technology retains the features of disequilibrium
even in the long run does have an important implication for the nature of the
industry’s long-run supply curve. For, as is seen by (7.4), the relative form of
industry supply curve yt = Yt (Pt )b/Kt traces the shape of St (c), except for the
portions of discontinuous jumps. Hence, if the expectation of St (c) tends to exhibit
a statistical regularity in the form of S̃(z), the expectation of the relative form of
the industry supply curve should also exhibit a statistical regularity in the same
long-run form of S̃(z). Let us denote by pt the relative gap between a given product
price Pt and the potential unit cost C(t), or
pt ≡ log Pt − log C(t), (7.24)

and call it the ‘price gap’ at time t. Then, we can obtain the following proposition
without paying any extra cost.
Proposition (SC): Under Hypotheses (CG), (IM’), (PC) and (IN-a), the expected
value of the relative supply curve of the industry yt = Qt (Pt )b/Kt will in the long
run approach a functional form of
1+α
S̃(pt ) ≡ 1+ α − α. (7.25)
1 + (1/α)e− αβ pt
Figure 7.7 exhibits the relative form of the industry’s long-run supply curve as
a function of price gap p. As a matter of fact, it has been drawn simply by turn-
ing Figure 7.6 around the 45 degree line. It therefore moves counter-clockwise as
either of the composite parameters α and β increases. This implies that the long-
run supply curve becomes more upward-sloping, as the disequilibriating force of
creative-cum-destructive innovations becomes stronger than the joint equilibriat-
ing force of economic selection and technological diffusion or as the average rate
of cost reduction of each innovation becomes larger.
What is most striking about this long-run supply curve, however, is not that
it is the ‘sixth’ logistic curve we have encountered in this chapter but that it is an
upward-sloping supply curve!
Let us recall the lower panel of Figure 7.1 of the introductory section. It re-
produced a typical shape of the long-run supply curve which can be found in any
textbook of economics. This horizontal curve was supposed to describe the long-
run state of the industry in which the least cost technology is available to every
firm in the industry and all the opportunities for positive profits are completely
wiped out. However, the relative form of the long-run supply curve we have drawn
in Figure 7.7 has nothing to do with such traditional picture. There will always be
a multitude of diverse technologies with different cost conditions, and the industry
supply curve will never lose an upward-sloping tendency, just as in the case of
the ‘short-run’ supply curve of the upper panel of Figure 7.1. There are, therefore,
always some firms which are capable of earning positive profits, no matter how
competitive the industry is and no matter how long it is run.
p
αβ
y
0 1
Figure 7.7 Long-run industry supply curve
We can thus conclude that positive profits are not only the short-run phe-
nomenon but also the long-run phenomenon of our Schumpeterian industry. It is
true that the positivity of profits is a symptom of disequilibrium. But, if the industry
will approach only a statistical equilibrium of technological disequilibria, it will
never stop generating positive profits from within even in the never-never-land of
long-run.
7.8 The determination of the long-run profit rate

It is one thing to demonstrate the existence of positive profits in the long run. It is,
however, another to analyze the factors which determine the long-run profit rate.
Let us then look at Figure 7.8 which superimposes a demand curve on Figure
7.7. If we suppose that this demand curve is shifting to the right at the same rate
as that of the industry’s total capital stock and shifting to the bottom at the same
rate as that of the potential unit cost, its relative form will become invariant over
time. The intersection e∗ of this relative demand curve with the long-run relative
supply curve then determines the long-run equilibrium price gap p∗ and the long-
run equilibrium output-capacity ratio y∗ = S̃(p∗ ). Since we have approximated
the profit rate (Pq − cl)/Pk of each technology by b(log P − log c), we can also
express it as b((log P − C(t)) − (log c − C(t))) = b(p − z). This is nothing but the
vertical distance between a given price gap and a point on the upward-sloping
supply curve. Summing these individual profit rates from z = 0 to z = p∗ with
capital shares s̃(z) ≡ S̃ (z) as relative weights, we can calculate the long-run profit
186 Katsuhito Iwai
p
e*
p*
r*
y
0 y* 1
Figure 7.8 Determination of long-run profit rate
rate r ∗ of the industry as a whole. Graphically, it can be represented by the shaded

area 0e∗ y∗ in Figure 7.8. Algebraically, it can be expressed as:28
p∗ p∗
r∗ = b(p∗ − z)s̃(z)dz = b S̃(z)dz (7.26)
0 0

αβb ∗ y∗
= − log(1 − y ) − α log 1 + > 0.
1+α α
We have thus succeeded in giving a complete characterization of the long-run profit

rate of our Schumpeterian industry. It is positive, indeed.
It is now time to do some comparative dynamics. First, demand effects. It does
not require any graphical explanation to see that an upward shift of the relative
demand curve works to increase the long-run profit rate of the industry r ∗ . In fact,
a differentiation of (7.26) with respect to y∗ leads to:

∂r ∗ αβb 1 α
= − > 0. (7.27)
∂y∗ 1 + alpha 1 − y∗ α + y∗
A further differentiation of (7.27) leads to:

∂2r∗ αβb 1 α2
= + > 0. (7.28)
∂y∗2 1 + α (1 − y∗ )2 (α + y∗ )2
The industry’s long-run profit rate r ∗ is thus seen to be an increasing and convex
function of the equilibrium output-capacity ratio y∗ .
This convex relationship between long-run profit rate and output–capital ratio
would have a particularly important implication for the dynamic stability, or more
appropriately, dynamic instability of our Schumpeterian economy. For Hypothesis
(CG) immediately implies that the growth rate of fixed investment also becomes
on average an increasing and convex function of output–capacity ratio, which is
very likely to violate the stability condition for investment-saving equilibrium of
the economy as a whole. In the present chapter, however, we can only mention this
possibility in passing and must resume our comparative dynamics.
Next, let us examine the effects of a shift in the long-run supply curve on
the industry’s long-run profit rate. This, however, turns out to be a much more
involved exercise than that on the demand effects. I will therefore relegate the
detailed discussions to Appendix and only summarize the results obtained therein.
When the relative demand curve is perfectly elastic with respect to price change,
we have:

∂r ∗ ∂r ∗
< 0 and < 0. (7.29)
∂α p∗ = const. ∂β p∗ = const.
In this case, both an intensification of the force of innovations relative to the force
of growth and imitations and an intensification of the force of invention relative
to that of innovations reduce the profit rate of the industry in the long run. But the
assumption of perfectly elastic industry demand curve is empirically of limited
relevancy (except for the case of price regulation), and we had better proceed to
another special case.
When the relative demand curve is absolutely inelastic with respect to price
change, we have:

∂r ∗ ∂r ∗
> 0, and > 0. (7.30)
∂α y∗ = const. ∂β y∗ = const.
In this case, as the disequilibriating force of creative-cum-destructive innovations
becomes stronger than the equilibriating force of economic selection and swarm-
like imitations, or as the average rate of cost reduction of each innovation becomes
greater, the industry is expected to generate a higher profit rate in the long-run.
Innovation is not only the source of short-run profits but also the source of long-run
profits in an industry with inelastic demand.
Finally, when the relative demand is neither perfectly elastic nor absolutely
inelastic, we have:
∂r ∗ ∂r ∗
> (<)0 and > (<)0, (7.31)
∂α ∂β
when demand curve is inelastic (elastic).
In this general case, as the disequilibriating force of creative-cum-destructive
innovations becomes stronger than the equilibriating forces of economic selection
and swarm-like imitations, or as the average rate of cost reduction of each inno-
vation becomes greater, the industry is expected to generate a higher profit rate in
188 Katsuhito Iwai
the long run, as long as the price-elasticity of the demand curve is not so large.
However, this tendency will be reversed when the industry demand curve becomes
sufficiently elastic with respect to price change.
7.9 Pseudo-aggregate production functions

Since the pioneering works of Solow (1956, 1957), it has become the standard
method of neoclassical economics to use the concept of an ‘aggregate production
function’ in accounting the sources of economic growth. It allows economists to
decompose variations in GNP into those due to movements along the aggregate
production function and those due to shifts of the aggregate production function
itself. The former can be attributed to changes in measurable inputs, usually cap-
ital and labor, and the latter to changes in technology, an unobservable variable
usually inferred from the data as a residual. Early empirical studies of the long-
term macroeconomic growth in advanced capitalist economies found that only a
very small portion of GNP growth can be accounted for by increases in capital
and labor and that most of the growth being explained by technological progress
– an increase in the residual factor. More recent efforts by Maddison (1987) and
others, however, have succeeded in reducing the size of the residual factor sub-
stantially by incorporating variations in the qualities of capital and labor and other
supplementary effects.
The ‘success’ of the neoclassical growth accounting exercises is quite impres-
sive. The challenge to any theory claiming to challenge the neoclassical ortho-
doxy is therefore to match its power of tracking down the empirical patterns of
the macroscopic growth processes of advanced capitalist economies. The most
straightforward way to do this is, of course, to set up an empirical study of our
own. But in order not to lengthen this already lengthy chapter, I choose a short-cut.
The purpose of this section is to demonstrate that our evolutionary model is capable
of ‘calibrating’ all the characteristics of a neoclassical aggregate production func-
tion both in the short run and in the long run.29 If the neoclassical growth model is
capable of accounting the actual macroeconomic growth paths of advanced cap-
italist economies, then our evolutionary model is equally capable of performing
the same task. There is no way to differentiate these two models empirically at
the macroscopic level. Moreover, our evolutionary model has a decided advantage
over the neoclassical model in its ability to integrate microeconomic processes with
macroeconomic phenomena. While the neoclassical growth theory simply ignores
the complexity of the growth processes we daily observe at the microscopic level,
its recognition is the very starting point of our evolutionary model.
Let me begin this ‘calibration’ exercise by computing the amount of labor
employment for each level of product demand. When product demand is small so
that price Pt just covers the minimum unit cost cn , only the first-best technology
firms can engage in production and the level of product demand determines that
of output Yt . Because of the fixed proportion technology (7.1), the level of total
employment Lt associated with this output is cn Yt . When the demand reaches the
total capacity of the best technology kt (cn )/b = st (cn )Kt /b, a further increase
in demand is absorbed solely by an increase in Pt , while output is kept at the
capacity level. But when Pt reaches cn−1 , the second-best technology firms start
to produce and all the increase in demand is absorbed by a corresponding increase
in output. The relation between output and employment can then be given by
Lt = cn st (cn )Kt /b + cn−1 (Yt − st (cn )Kt /b) until Yt reaches the total productive
capacity of the first- and second-best technology firms (st (cn ) + st (cn−1 ))Kt /b. In
general, the relation between Yt and Lt can be given by

j=i

j=i
Lt = cj st (cj )Kt /b + ci−1 Yt − st (cj )Kt /b
j=n j=n
ci
≡ cdSt (c)Kt /b + ci−1 (Yt − St (ci )Kt /b)
0
whenever S(ci )Kt /b ≤ Yt < S(ci−1 )Kt /b. If we divide this relation by Kt /b and take
its inverse, we can construct a functional relation between the industry-wide labor–
capacity ratio lt ≡ Lt b/Kt and the industry-wide output–capacity ratio yt ≡ Yt b/Kt
as:
yt = ft (lt ), (7.32)
ci
where l ≡ 0 cdSt (c) + ci−1 (ft (l) − St (ci )) whenever St (ci ) ≤ ft (l) < St (ci−1 ).
Figure 7.9 depicts this functional relation in a Cartesian diagram which measures
labor–capacity ratio l along horizontal axis and output-capacity ratio y along a
vertical axis. It is evident that this relation satisfies all the properties a neoclassical
production function is supposed to satisfy.30 Y is linearly homogeneous in L and
K, because y ≡ Yb/K is a function only of l ≡ Lb/K. Though not smooth, this
relation also allows a substitution between Kt and Lt and satisfies the marginal
productivity principle: ∂ȳt /∂lt ≤ 1/Pt ≤ ∂ + yt /∂lt . (Here, 1/Pt represents a real
wage rate because of our choice of money wage rate as the numeraire, and ∂ȳ/∂l
and ∂ + y/∂l represent left- and right-partial differential, respectively.) Yet, the
important point is that this is not a production function in the proper sense of
the word. It is a mere theoretical construct that has little to do with the actual
technological conditions of the individual firms working in the industry. It is in
this sense that we call the relation (7.32) a ‘short-run pseudo-aggregate production
function’, with an emphasis on the adjective: ‘pseudo’.
The shape of the short-run pseudo-production function y = ft (l) is determined
by a distribution of capital shares {St (ci )} across technologies. Hence, as this
distribution changes, the shape of this short-run function also changes. And in our
Schumpeterian industry, the distribution of capital shares is incessantly changing
over time as the result of dynamic interplay among capital growth, technological
innovation and technological imitation. The most conspicuous feature of the short-
run pseudo-production function is, therefore, its instability.
In the long run, however, we know we can detect a certain statistical regularity
in the distribution of capital shares {St (ci )} out of its seemingly unpredictable
190 Katsuhito Iwai
yt
1
xt
0
Figure 7.9 Short-run pseudo-aggregate production function
movement. We can thus expect to detect a certain statistical regularity in the pseudo-
production function as well out of its seemingly unpredictable movement. Let l̂ and
ŷ denote the expectation of labor–capacity ratio l ≡ Lb/K and of output–capacity
ratio y ≡ Yb/K, respectively. Then, we indeed arrive at:31
Proposition (PF): Under Hypotheses (CG), (IM’), (PC) and (IN-a), the func-
tional relationship between the expected labor–capacity ratio and the expected
output–capacity ratio will in the long run take the form of :

ŷ = f˜ l̂eλt , (7.33)
where the function f˜(·) is defined implicitly by the following identity:

˜f (leλt ) 1+βαα
α+y
leλt ≡ dy. (7.34)
0 α(1 − y)
At the seventh time, we have finally graduated from the tyranny of logistic equa-
tions! What we have obtained in Figure 7.10 is a well-behaved function which
satisfies all the characteristics a neoclassical production function should have. In-
deed, it is not hard to show that f˜(0) = 0, f˜ (·) > 0, f˜ (·) < 0.32 It is as if total labor
force L and total capital stock K produce the total output Y in accordance with
an aggregate neoclassical production function f˜(·) with pure labor augmenting
(or Harrod-neutral) technological progress eλt . It is, in other words, as if we had
entered the Solovian world of neoclassical economic growth where the economy’s
macroscopic growth process could be decomposed into the capital–labor substitu-
tion along an aggregate neoclassical production function and the constant outward
shift of the aggregate neoclassical production function itself. This is, however, a
y
1
xe λt
0
Figure 7.10 Long-run pseudo-aggregate production function
mere statistical illusion! If we zoomed into the microscopic level of the economy,
what we would find is the complex and dynamic interactions among many a firm’s
capital growth, technological imitation and technological innovation. In fact, as
is seen from (7.34), the functional form of f˜(·) is a complex amalgam of such
basic parameters of our Schumpeterian model as α ≡ ν/(γδ + µ) and β ≡ λ/ν. It
is just impossible to disentangle various microscopic forces represented by these
parameters and decompose the overall growth process into a movement along a
well-defined aggregate production function and an outward shift of the function
itself.33 Indeed, it is not hard to show that both an increase in α and in β shift the
function f˜(·) in the downward direction,34 or
∂ f˜(·) ∂ f˜(·)
< 0 and < 0. (7.35)
∂α ∂β
We are after all living in a Schumpeterian world where the incessant reproduction
of technological disequilibria prevents the aggregate relation between capital and
labor from collapsing into the fixed proportion technology of individual firms. It
is, in other words, its non-neoclassical features that give rise to the macroscopic
illusion that the industry is behaving like a neoclassical growth model. It is for this
reason we will call the relation (7.33) the ‘long-run pseudo aggregate production
function’.

In the traditional economic theory, whether classical or neoclassical, the long-run
state of the economy is an equilibrium state and the long-run profits are equilibrium
192 Katsuhito Iwai
phenomena. If there is a theory of long-run profits, it must be a theory about the
determination of the normal rate of profit.
This chapter has challenged this long-held tradition in economics. It has intro-
duced a simple evolutionary model which is capable of analyzing the development
of the industry’s state of technology as a dynamic interplay among many a firm’s
growth, imitation and innovation activities. And it has demonstrated that what the
industry will approach over a long passage of time is not a classical or neoclassical
equilibrium of uniform technology but a statistical equilibrium of technological
disequilibria which maintains a relative dispersion of efficiencies in a statistically
balanced form. Positive profits will never disappear from the economy no matter
how long it is run. ‘Disequilibrium’ theory of ‘long-run profits’ is by no means a
contradiction in terms.
Not only is a disequilibrium theory of long-run profits possible, but it is also
‘operational’. Indeed, our evolutionary model has allowed us to calculate (only with
pencils and paper) the economy’s long-run profit rate as an explicit function of the
model’s basic parameters representing the microscopic forces of capital growth,
technological imitations, recurrent innovations and steady inventions. ‘Without
development there is no profit, without profit no development’, to quote Joseph
Schumpeter once more.35 The model we have presented in this chapter can thus
serve as a foundation, or at least as a building block, of the theory of ‘long-run
development through short-run fluctuations’ or ‘growth through cycles’. To work
out such a theory in more detail is of course an agenda for the future research.
Appendix: Comparative dynamics of supply-side determinants of

long-run profit rate
The purpose of this Appendix is to deduce (7.29), (7.30) and (7.31).
Consider first the case of perfectly elastic demand curve. Although the eco-
nomic relevancy of this special case is of limited nature, it serves as a useful
benchmark for the other cases. Figure 7.A1 juxtaposes a horizontal demand curve
on the relative form of a long-run supply curve. We already know from Section
7.8 that an increase in either α or β moves the inverted logistic shape of the supply
curve counter-clockwise. As is seen from Figure 7.A1, such a supply curve shift
transfers the equilibrium point from e∗ to e∗∗ along the horizontal demand curve
and squeezes the long-run profit rate by the magnitude equal to A ≡ 0e∗ e∗∗ . We
can easily confirm this graphical exposition by differentiating (7.26) with respect
to α and β, keeping p∗ constant.

∂r ∗
p∗ ∂ S̃(z)
∂α p∗ = const. =b ∂α dz ≡ −Aα
0
βb
1
= − α(1+α) e−( 1+ y∗ ) / ( αβ) u−1−log
(1+u)2 du < 0;
u
p∗ ∂ S̃(z)
∂r ∗ (7.A1)
∂β p∗ = = b 0 ∂β dz ≡ −Aβ
const. 1
= − (1+β)b
αβ e−( 1+ y∗ ) / ( αβ) (1+u)2 du < 0.
u−1−log u
This is nothing but (7.29) of the main text. Note also that since ∂ S̃(z)/∂ν >
∗
0, ∂r ∗
∂ν p = const. > 0.
Next, consider the case of an absolutely inelastic demand curve. As is shown
in Figure 7.A2 which juxtaposes a vertical demand curve on an inverted logistic
shape of the long-run supply curve, an increase in either α or β moves the latter
counter-clockwise and transfers the equilibrium point from e∗ to e∗∗ along the
vertical demand curve. This raises p∗ to p∗∗ , while keeping y∗ the same as before.
The long-run profit rate thus changes from 0e∗ p∗ to 0e∗∗ p∗∗ . We have to examine
whether this amounts to an increase or decrease of r ∗ . To see this, Figure 7.A2
decomposes this change of profit rate into two components – A ≡ 0e∗ e∗∗ and
B ≡ p∗ e∗ e∗∗ p∗∗ . The first component A represents the ‘loss’ of profit rate due to
a universal increase of cost gaps, which corresponds to the profit loss A of the
previous case. In the present case of absolutely inelastic demand curve, however,
an increase in the long-run equilibrium price gap gives rise to a ‘gain’ of profit rate,
as is represented by the second component B. Whether r ∗ increases or decreases
thus depends on whether A is smaller or larger than B. This can be checked by
differentiating (7.26) with respect to α and β, keeping y∗ constant. We have:
S̃−1(y∗ )
∂r ∗ ∂ S̃(z) ∂ S̃ −1 (y∗ )
= b dz + by∗
∂α y= const. 0 ∂α ∂α
≡ −Aα + Bα .

βb ∗ y∗
= − log(1 − y ) − α log 1 +
(1 + α)2 α

y∗ y∗
−(1 + α)α log 1 + − ; (7.A2)
α α + y∗
S̃ −1 (y∗ )
∂r ∗ ∂ S̃(z) ∂ S̃ −1 (y∗ )
= b dz + by∗ ≡ −Aβ + Bβ
∂β y∗ = const. 0 ∂β ∂β

αb y∗
= − log(1 − y∗ ) + α log 1 + > 0.
1+α α
∗ ∗ ∗
Although both − log(1 − y∗ ) − α log(1 + yα ) and (1 + α)α(log(1 + yα ) − α+y y
∗ ) are
positive in the first expression, the former dominates the latter if we let α → 0. Since
α ≡ ν/(γδ + µ) is assumed to be small, it does not seem unreasonable to suppose
the first expression to be positive. The second expression is always positive. Hence,
∗
(7.30) of the main text. Note that we can also calculate ∂r ∗
∂ν y =const. as

−α2 βb αy∗ y∗ y∗
− log(1 − y∗ ) − − log 1 + − < 0.
(1 + α)2 α + y∗ α α + y∗
Finally, let us consider the general case where industry demand curve is neither
perfectly elastic nor absolutely inelastic. As is seen from Figure 7.A3, an increase
in either α or β transfers the equilibrium point upward from e∗ to e∗∗ along this
downward-sloping demand curve. This raises p∗ to p∗∗ but lowers y∗ to y∗∗ , thereby
changing r ∗ from 0e∗ y∗ to 0e∗∗ y∗∗ . We can then decompose this change again into
A ≡ 0e∗ e∗∗ and B ≡ p∗ e∗ e∗∗ p∗∗ . A represents the ‘loss’ of r ∗ due to a universal
194 Katsuhito Iwai
αβ
e ** e*
p*
A
y
0 y ** y*
Figure 7.A1 The case of a perfectly elastic demand curve
e **
p **
αβ
B
e*
p*
A
y
0 y*
Figure 7.A2 The case of an absolutely inelastic demand curve
p
αβ
e **
p **
B'
e*
p*
A
y
0 y ** y*
Figure 7.A3 The general case
increase of cost gaps, and B represents the ‘gain’ due to an increase in the long-run
equilibrium price gap. However, B in Figure 7.A3 is not as large as B in Figure
7.A2, for the price elasticity of the demand allows the effect of cost increases to
be absorbed not only by price hike but also by quantity reduction. This means
that when the demand curve is steeply sloped, the gain component B is likely
to outweight the loss component A. But, when the demand curve becomes more
elastic, B becomes smaller, and in the limiting case of perfectly elastic demand
curve it shrinks to zero.
This graphical explanation can be formalized as follows. First write down the
relative form of industry demand function as yt = D̃(pt ). Then, p∗ is determined
by the supply-demand equation: S̃(p∗ ) = D̃(p∗ ). Differentiating this with respect
to α and β and rearranging terms, we have: ∂p∗ /∂α = (−p∗ ∂ S̃(p∗ )/∂α)/(ε +
η) and ∂p∗ /∂β = (−p∗ ∂ S̃(p∗ )/∂β)/(ε + η), where ε and η are the price-
elasticity of the supply curve and of the demand curve, respectively defined by
(∂ S̃(p)/∂p)/(S̃(p)/p) and (∂ D̃(p)/∂p)/(D̃(p)/p). Keeping this in mind and dif-
ferentiating (24), we obtain:

∂r ∗
S̃ −1 (y∗ ) ∂ S̃(z) ∗
∗ −∂ S̃(p ) 1
∂α S̃(p∗ )=D̃(p∗ ) =b 0 ∂α dz + bp ∂α η+ε ≡ −Aα + Bα . ;
∂r ∗
S̃ −1 (y∗ ) ∂ S̃(z) ∗
∗ −∂ S̃(p ) 1
(7.A3)
∂β S̃(p∗ )=D̃(p∗ ) = b 0 ∂β dz + bp ∂β η+ε ≡ −Aβ + Bβ .
Note that the component B in either expression is decreasing in η. In particular,

when η = ∞, B becomes equal to 0; when η = 0, B becomes equal to B in (7.A2).
Hence, we have obtained (7.31) of the main text.
196 Katsuhito Iwai
Acknowledgement
This is a simplified version of the paper I presented at ISER XI Workshop at the
Certosa di Pontignano, Siena, on July 1, 1998. Since the workshop paper developed
a series of new evolutionary models, it was very long and mathematically compli-
cated. The present work has used the simpler evolutionary model of Iwai (1984b)
so that I can present the same thesis much more economically. A revised version
of the workshop paper is to be published as ‘A Contribution to the Evolutionary
Theory of Innovation, Imitation and Growth’ in Journal of Economic Behavior
and Organization. I am grateful to the participants of the Siena Workshop for their
suggestions as well as to the members of Macro Workshop at University of Tokyo
for their comments. The remaining errors are exclusively mine.
Notes
1 Schumpeter (1939), p.100.
2 Schumpeter (1950), p. 83.
3 Schumpeter (1950), p. 28.
4 See, for instance, Nelson and Winter (1982), Dossi, Freeman, Nelson, Silverberg and
Soete (1988), Metcalfe and Saviotti (1991), and Anderson (1994) for the comprehen-
sive expositions of the ‘evolutionary perspective’ in economics.
5 The term ‘satisficing’ was first coined by Simon (1957) to designate the behavior of a
decision maker who does not care to optimize but simply wants to obtain a satisfactory
utility or return. The notion of ‘organizational routines’ owes to Nelson and Winter
(1982). Organizations ‘know’ how to do things. In Iwai (1999) I have provided a legal-
economic-sociological framework for understanding the nature and sources of such
organizational capabilities.
6 Or we can think of this as a one-commodity economy with many competing firms.
7 Our evolutionary model can also accommodate a wide variety of industry structures.
See Appendix A of Iwai (1984b) for the way to deal with the case of monopolistically
competitive industry.
8 It is easy to show from (7.8) below that: K̇t /Kt = γ(log Pt − i (log ci )st (ci )) − γ0 ,
so that the growth rate of the industry’s total capital stock is linearly dependent on
the proportional gap between the price-wage ratio Pt and the industry-wide average
unit cost. If K̇t /Kt is pre-determined (probably by the growth rate of the demand for
this industry’s products), this equation can be used to determine Pt . If, on the other
hand, Pt is pre-determined (probably by the labor market conditions in the economy
as a whole), this equation can be used to determine K̇t /Kt . In either case, the forces
governing the motion of Kt are in general of the different nature from those governing
the evolution of {St (c)}.
9 Schumpeter (1961), p. 154.
10 It is, however, not so difficult to deduce an investment function of this form by explicitly
setting up an intertemporal optimization problem with adjustment costs, as in Uzawa
(1969).
11 The actual derivation is as follows.

n
Ṡt (ci ) ≡ ṡt (cj )
j= i

n
= (k̇t (cj )/kt (cj ) − Kt /Kt )st (cj )
j= i

n
n
= (γ(log pt − log cj ) − γ0 ) − (γ(log pt − log ch ) − γ0 )st (ch ) st (cj ) by (7.5)
j= i h= 1

n
n
= γ (log ch )st (ch ) − log cj st (cj )
j= i h= 1
i−1

n
= γ (log ch )st (ch )St (ci ) − (log ch )st (ch ) (1 − St (ci ))
h= 1 h= i
= γδt (ci )St (ci )(1 − St (ci )).
12 This is the simplification I also adopted in Iwai (1984b). However, in a recent article
Franke (1998) indicated that the value of δt (c) may actually vary considerably as the
parameter values of γ, ν and λ as well as the value of c vary. A caution is thus needed
to use this approximation for purposes other than heuristic device.
13 A logistic differential equation: x = ax(1 − x) can be solved as follows. Rewrite it as:
x /x − (1 − x) /(1 − x) and integrate it with respect to t, we obtain: log(x) − log(1 −
x) = log(x0 ) − log(1 − x0 ) + at, or x/(1 − x) = eat x0 /(1 − x0 ). This can be rewritten
as: x = 1/(1 + (1/x0 − 1)eat ), which is nothing but a logistic equation given by (7.9).
14 The reason I have designated this Hypothesis by (IM ) is to differentiate it from a
slightly different hypothesis adopted in Iwai (1984a). Its Hypothesis (IM) assumes
that the probability of imitating a better technology is proportional to the frequency
(rather than their capital share) of the firms using it. On the other hand, Iwai (2000) has
adopted yet another hypothesis which assumes that firms imitate only the best practice
technology and the probability of its success is proportional to the frequency of the
firms using it.
15 p. 615.
16 See, for instance, Mansfield, Schwartz and Wagner (1981), Gorts and Klepper (1982)
and Metcalfe (1988).
17 It is, however, possible to incorporate a trade-off between the resources devoted to
capital growth and the resources devoted to imitative activities into our model. For
instance, the growth parameters γ and/or −γ0 in (7.5) can be made a decreasing function
of the imitation coefficient µ.
18 The actual derivation is as follows. The value of St (ci ) increases whenever one of the
firms with unit costs higher than ci succeeds in imitating one of the technologies with
unit costs ci or lower. Indeed, because of the assumption of the disembodied nature of
technology, it increases by the magnitude equal to the imitator’s capacity share. Note
that St (ci ) is not affected by the imitation of any of the firms with unit costs ci or less,
for it only effects an infra-marginal transfer of capacity share. Let Mt (ci ) denote the
number of firms with unit costs ci or lower. Since the average capacity share of the
firms with unit costs higher than ci is (1 − St (ci ))/(M − Mt (ci )) and the probability
of a successful imitation for each of those M − My (ci ) firms is µSt (ci )dt during a
small time interval dt, we can calculate the expected increase in St (ci ) during dt as
((1 − St (ci ))/(M − M (ci ))(µSt (ci )dt)(M − Mt (ci )) = (µSt (ci )dt)(1 − St (ci ))). If the
number of firms is sufficiently large, the law of large numbers allows us to use this
expression as a good approximation of the actual rate of change in St (ci ). Dividing
this by dt and letting dt → 0, we obtain (7.11).
19 Schumpeter (1961), p. 88.
20 Iwai (2000), however, presents an evolutionary model which does not separate inno-
vators from inventors and assume that each innovation raises the productivity of the
industry’s best technology by a fixed proportion.
21 See, for instance, Kamien and Schwarts (1982), Grilliches (1984) and Scherer and
Ross (1990).
198 Katsuhito Iwai
22 It is, however, possible to incorporate a trade-off between the resources devoted to
capital growth and the resources devoted to innovative activities into our model. For
instance, the growth parameters γ and/or −γ0 in (7.5) can be made a decreasing function
of the innovation coefficient ν.
23 Iwai (1984a, 2000) also develops versions of evolutionary models which assume that
only the firms currently using the best technology can strike the next innovation. In
this case, the process of technological innovations is no longer a Poisson process, so
that it is necessary to invoke the so-called ‘renewal theory’ in mathematical probability
theory to analyze the long-run performance of the state of technology.
24 The derivation is as follows. Whenever one of the firms with unit costs higher than
c succeeds in innovation, the value of St (c) increases by the magnitude equal to the
innovator’s capacity share. (St (c) is, however, not affected by the innovation of any of
the firms with unit costs c or less, because it only effects an infra-marginal transfer of
the capacity share.) As in note 11, let M − Mt (ci ) denote the total number of firms with
unit costs higher than ci . The average capacity share of the firms with unit costs higher
than ci is (1 − St (ci ))/(M − Mt (ci )) and the probability of a successful innovation for
each of those M − Mt (ci ) firms is νdt during a small time interval dt. We can then
calculate the expected increase in St (ci ) due to an innovation as ((1 − St (ci ))/(M −
M (ci )))(νdt)(M − Mt (ci )) = (νdt)(1 − St (ci )). If we divide this by dt and add to it the
effects of economic selection and technological imitations given by (7.13), we obtain
(7.18).
25 In deducing (7.19), we have employed a boundary conditionŜT (c) (c) = ν or ŜT (c) (c) =
0.
26 In our companion paper [1998] which assumes the step-by-step nature of innovations,
it is β that is assumed to be exogenously given.
27 The derivation is as follows.
∂ S̃(z)/∂α = (1 − (1 + α)z/(αβ) − e−(1+ α)z/ (αβ) )α−2 e−(1+ α)z/ αβ

(1 + e−(1+ α)z/ αβ /α)−2 < 0
and
∂ S̃(z)/∂β = −(1 + α)z/(αβ)−2 e−(1+ α)z/ αβ (1 + e−(1+ α)z/ αβ /α)−2 < 0.
Note that ν appears both in α and in β. But its impact on S̃(z) can be calculated as
∂ S̃(z)/∂ν = α∂ S̃(z)/∂α − β∂ S̃(z)/∂β

= 1/α−1 (1 − e−(1+ α)z/ αβ )e−(1+ α)z/ αβ (1 + e−(1+ α)z/ αβ /α)−2 > 0.
28 The derivation is as follows.

p∗
r∗ = b S̃(z)dz
0
S̃(p∗ )
= b S̃(dz/dS̃)dS̃
0
S̃(p∗ )
= b S̃(αβ/((1 − S̃)(α + S̃)))dS̃
0
S̃(p∗ )
= (bαβ/(1 + α)) (1/(1 − S̃) − α/(α + S̃))dS̃
0
Noting that y∗ = S̃(p∗ ), an explicit integration leads to (7.26).

29 In this sense, this section follows up the simulation exercises of Nelson and Winter in
(1974) and chapter 9 of (1982).
30 See Sato (1975) for the general discussions on the aggregation of micro production
functions.
31 The derivation of this Proposition is as follows. Since the short-run ‘pseudo’production
P P
function (7.27) implies that l = 0 cdSt (c) whenever y = St (P), we have l̂ = 0 cdŜt (c)
whenever ŷ = Ŝt (p). But from (7.21) we then have
P
l̂ = cdS̃(z)
0 p
= ez+ lo g C(t) dS̃(z)
0
p
= e−λt ez dS̃(z)
0
S̃(p)
−λt
= e ez( S̃) dS̃
0
S̃(p)
= e−λt ((α + s̃)/(α(1 − S̃)))αβ/ (1+ a) dS̃
0
and ŷ = Ŝ(p). Putting these two relations together, we obtain (7.29).

32 More precisely, we have dy/d(leλt ) = ((α + y)/(α(1 − y)))−αβ/ (1+ α) > 0 and
d2 y/d(leλt )2 = −β((α + y)/(α(1 − y)))αβ/ (1+ α)−1 /(1 − y)2 < 0.
33 It is true that in the present model the rate of pure labor augmenting technical progress
in the pseudo-aggregate production function is a given constant λ which is determined
exogenously by inventive activities outside of the industry. However, in some of the
models presented in a companion paper (Iwai [2000]) this rate becomes also an amal-
gam of the parameters representing the forces of economic selection, technological
diffusion and recurrent innovations.
34 This can be shown as follows. Let us differentiate (7.34) (or an equivalent ex-
pression given in note 33) with respect to α. We then have: 0 = ez( S̃) ∂ ˜f /∂α +
s̃ z( S̃)
0
e (∂z(S̃)/∂α)dS̃. Since z(S̃) is an inverse function of S̃(z) and ∂ S̃(z)/∂α < 0
by (7.23), we have ∂z(S̃)/∂α > 0. Hence we have ∂ ˜f /∂α < 0 as in (7.35). We can
also show that ∂ ˜f /∂β < 0 in exactly the same manner.
35 Schumpeter (1961), p. 154.
References
Anderson, E.S. (1994) Evolutionary Economics: Post-Schumpeterian Contributions, Pinter:
London.
Arrow, K. (1962) ‘Economic welfare and the allocation of resources for inventions’, in R. R.
Nelson (ed.), The Role and Direction of Inventive Activity, Princeton University Press:
Princeton.
Dossi, G., G. Freeman, R. Nelson, G. Silverberg and L. Soete (1988) (eds), Technical
Change and Economic Theory, Pinter: London.
Franke, R. (1998) ‘Wave trains and long waves: a reconsideration of Professor Iwai’s Schum-
peterian dynamics’, forthcoming in D. Dell Gatti, M. Gallegati and A. Kirman (eds),
Market Structure, Aggregation and Heterogeneity. Cambridge University Press: Cam-
bridge.
200 Katsuhito Iwai
Gorts, M. and S. Klepper (1982) ‘Time-paths in the diffusion of product innovations’,
Economic Journal, 92, 630–53.
Grilliches, Z. (1984) (ed.) R and D, Patents and Productivity, Chicago University Press:
Chicago.
Iwai, K. (1984a) ‘Schumpeterian dynamics: an evolutionary model of innovation and imi-
tation’, Journal of Economic Behavior and Organization, 5, (2) 159–90.
Iwai, K. (1984b) ‘Schumpeterian dynamics, Part II: technological progress, firm growth and
“economic selection”,’ Journal of Economic Behavior and Organization, 5, (3) 321–51.
Iwai, K. (1999) ‘Persons, things and corporations: Corporate personality controversy and
comparative corporate governance’, American Journal of Comparative Law, 45, 101–50.
Iwai, K. (2000) ‘A contribution to the evolutionary theory of innovation, imitation and
growth’, Journal of Economic Behavior and Organization, 43, 167–98.
Kamien, M. and N. Schwarts (1982) Market Structure and Innovation, Cambridge Univer-
sity Press: Cambridge.
Maddison, A. (1987) ‘Growth and slowdown in advanced capitalist economies: Techniques
of quantitative assessment’, Journal of Economic Literature, 25 (June 1987), 649–98.
Mansfield, E., M. Schwartz and S. Wagner (1981), ‘Imitation costs and patents: an empirical
study’, Economic Journal, 91, (364), 907–18.
Metcalfe, J.S. ‘The diffusion of innovation: an interpretive survey’, Chapter 25 of G. Dossi
et al. (1988).
Metcalfe, J.S. and P. Saviotti (1991) (eds) Evolutionary Theories of Economics and Tech-
nological Changes, Harwood Publishers: Reading. PA.
Nelson, R. and S. Winter (1974) ‘Neoclassical vs. evolutionary theories of economic growth:
critique and perspective’, 84, 886–905.
Nelson, R. and S. Winter (1982) An Evolutionary Theory of Economic Change, Harvard
University Press: Cambridge, MA.
Sato, K. (1975) Production Functions and Aggregation, North-Holland: Amsterdam.
Scherer, F.M. and D. Ross (1990) Industrial Market Structure and Economic Performance,
Houghton and Mifflin: Boston.
Schumpeter, J.A. (1939) Business Cycles, McGraw-Hill: New York.
Schumpeter, J.A. (1950) Capitalism, Socialism and Democracy, 3rd. ed., Rand McNally:
New York.
Schumpeter, J.A. (1961) The Theory of Economic Development, Oxford University Press:
Oxford.
Simon, H.A. (1957) Models of Man, Wiley: New York.
Solow, R.M. (1956) ‘A contribution of the theory of economic growth’, Quarterly Journal
of Economics, 70, (1), 65–94.
Solow, R.M. (1957) ‘Technical change and the aggregate production function’,Review of
Economics and Statistics. 39, 312–20.
Uzawa, H. (1969) ‘Time Preference and the Penrose Effect in a Two-Class Model of Eco-
nomic Growth’. Journal of Political Economy, 77 (July/Aug.), 628–92.
Part III
Dynamics by interaction
8 Asymmetrical cycles and
equilibrium selection in
finitary evolutionary
economic models
Masanao Aoki
8.1 Introduction
This chapter takes a fresh look at economic phenomena with multiple equilibria,
and some of the issues associated with modeling asymmetrical business cycles. It
summarizes an approach proposed in Aoki (1996, 1998), which is based on using
continous time Markov chains and illustrates the advantages of using models with
at most countable state spaces to answer these questions. The main advantage of
employing models with a finite number of agents is simplicity in clarifying the issue
of asymmetrical oscillation and naturalness of the criterion proposed for equilib-
rium selection. Two examples are presented to illustrate the proposed method. One
is a binary choice model in which agents choose one of two alternative decisions
or behavioral rules. Agents’ decisions are interdependent due to externalities, such
as congestion, bandwagon, group sentiments, network externalites and the like.
The other example is a finite agent version of Diamond’s search model, Diamond
(1982).
8.2 Structure of transition rates

Dynamic behavior of a collection of (micro)economic agents is modeled by a
continous time Markov chain, i.e., Markov process with at most countable states,
called a jump Markov process. It describes time evolution of the probabilities of
states of Markov chains by accounting for probability flows into and flows out of
(sets of) states. The equations that do this accounting of probabilities are called
the Chapman–Kolmogorov equations in the stochastic processes literature, see
Whittle (1976, 156, 175), or Karlin and Taylor (1981, 286), for instance. We use a
version of them which is easier to interpret. They are called the master equations in
the physics and mathematical sociology literatures and we will use the same name
in this chapter.1 The master equations describe time evolution of probabilities of
states of dynamic processes in terms of the probability transition rates and state
occupancy probabilities. See Kelly (1979, 3) for example.
We must specify transition rates from a microeconomic consideration to de-
scribe the dynamics, and to draw macroeconomic implications of the Markov
204 Masanao Aoki
processes thus specified. With countable state spaces, transition rates specify well-
behaved jump Markov processes, that is, processes do not execute an infinite
number of jumps instantaneously. See Kendall (1975), Kelly (1976) or Breiman
(1968).
In modeling a collection of interacting microeconomic units, henceforth called
agents, the variables which are most important in influencing macroeconomic
dynamics of the model are often not the absolute numbers of agents who occupy
particular sets of states of the model, but rather the proportions or fractions of
agents in these sets of states,2 and possibly some function of the total number of
agents, N , as an indicator of the scale or size of the model.
With state variable X (t), the probability distribution is governed by the master
equation

dP(X , t)
= {wN (X |X )P(X , t) − wN (X |X )P(X , t)}dX
dt
where we indicate N explicitly as subscript to the transition rates from state X to
X as wN (X |X ). We give a heuristic derivation of this relation for discrete state
variables.
When X is discrete, the integral is replaced by summation. Let X (t) be a state
vector in a finite set S. Using the backward Chapman-Kolmogorov formulation
we derive

dP(X , t)/dt = P(X , t)wN (X |X , t),
X
which is known as the master equation. See Aoki (1996, 116) for a heuristic
derivation. See Karlin and Taylor (1981, Chapter 14 ), for example, for the proof
of the existence of the time derivative.
Write the sum of the right hand side separately for state X and the rest as

P(X , t)wN (X |X , t) = P(X , t)wN (X |X , t) + P(X , t)wN (X |X , t),
X X =X

and substitute − X =X wN (X |X , t) in w(X |X , t) to rewrite the master equation
as

dP(X , t)/dt = P(X , t)wN (X |X , t) − P(X , t)wN (X |X , t).
X =X X =X
This is the usual form in which the master equation is stated. The first term is
the sum of the probability flows into state X , and the second the probability flow
out of state X .
We assume that x(t) = X (t)/N behaves as an intensive variable, and that the
transition rates depend on X and N through x, except possibly for the scale factor
which may depend on N .3
We make one additional key assumption that the change in the state variable, i.e.
X −X , remains the same for different values of N . This assumption is certainly met
Asymmetrical cycles and equilibrium selection 205
in birth-and-death processes, or birth-and-death with immigration processes, since
jumps are restricted to be ±1 or some fixed integers from any state regardless of the
total number N . This assumption actually is concerned with the scaling properties
or homogeneity properties of the transition rates. Loosely put, it means that each
of N agents contribute approximately equally to the transition events. To make this
explicit, express the transition rate as a function of the starting state, X , and the
jump (vector), r = X − X as
wN (X |X ) = wN (X ; X − X ) = wN (X ; r),
and assume that
wN (X ; r) = wN {N (X /N ); r} = NwN (X /N ; r) = N Φ(x ; r),
for some function Φ, and where x = X /N . Using the same function we can express
the transition rate in the opposite direction as
wN (X |X ) = N Φ(x; −r).
A scaling property which is seemingly more general is
wN (X ; r) = f (N )Φ(x ; r),
for some positive function f (N ). Actually, this factor f (N ) can be arbitrary, since
it can always be absorbed into the choice of time unit.
More generally, the transition rates may take the form
wN (X |X ) = f (N ){Φ0 (x ; r) + N −1 Φ1 (x ; r) + N −2 Φ2 (x ; r) + · · ·}, (8.1)
where higher order terms in N −1 may represent higher order interactions among
microeconomic units beyond those which are captured by the leading term. In
terms of these transition rates the master equation may be rewritten as

dP(X , t) X −r X −r
= f (N ) {Φ0 ( ; r) + N −1 Φ1 ( ; r) + · · ·)}P(X − r, t)dr
dt N N

X X
− f (N ) {Φ0 ( ; r) + N −1 Φ1 ( ; r) + · · ·}P(X , t)dr. (8.2)
N N
Series expansions in terms of some fractional powers of N , such as N −1/2 may
also be used in some problems.
Example 1: A binary choice model Suppose that the total number of agents,
N , is constant and that each of N agents face a binary choice, c1 , and c2 . Merits
or benefits of choices depend on the fraction of agents with particular choices.
Let n be the number of agents with c1 at a point in time. This, or its fractional
form n/N serves as state variable. We suppress time argument for simplicity. If an
agent can make his choice independent of others, then the transition rate from n to
206 Masanao Aoki
n + 1 occurs because one of the agents with choice 2 has changed its mind, while
n becomes n − 1 if one of the agents with choice 1 changes its mind. We assume
no entry and no exit to keep N fixed in this example, for simplicity. To account for
the externality of decisions we posit
n
rn := wN (n + 1|n) = a(N − n)η = aN (1 − x)η(x),
N
and n
ln := wN (n − 1|n) = bn 1 − η = bNx[1 − η(x)].
N
Without the factor η, each of N − n agents changes its mind at the rate a∆t over a
small time interval ∆t, and each of n agents does likewise at the rate b∆. Here a
and b are some constant. To simplify presentation we let them be the same.
The master equation is given, in terms of n(t)
dP(n(t))/dt = rn−1 P(n(t) − 1) + ln+1 P(n(t) + 1) − [rn + ln ]P(n(t)),
together with boundary conditions at 0 and N , where n(t) is the number of agents
with the first choice. Equivalently it can be written in terms of x(t) = n(t)/N .
Example 2: A finitary search model To reformulate the search model of Diamond

as a finitary evolutionary model, let n be the number of employed. This, or its
fractional form e(t) = n(t)/N serves as state variable. The number increases from
n to n + 1 when one of the unemployed encounter the production opportunity with
production cost less than his reservation cost. The probability of this implies that
n
rn := wN (n + 1|n) = (N − n)aG(c∗ = N (1 − e)aG(c∗ (e)),
N
where e = n/N is the fraction of employed, using Diamond’s notation, and c∗ (e)
is the reservation cost when the fraction is e. See Diamond (1982) or Aoki (1999).
The number n decreases by two when two employed randomly encounter to
exchange their output. This probability involves a chosen agent finding his trading
partner as well as a possibility of two of the other employed forming a pair without
his finding his partner. We define b(e) so that
N
ln := wN (n − 2|n) = eb(e).
2
The master equation is
dP(n(t))/dt = rn−1 P(n(t) − 1) + ln+2 P(n(t) + 2) − (rn + ln )P(n(t)),
together with some boundary conditions. See Aoki and Shirai (2000) for exact
expressions.
8.3 Two types of state variables
In these two examples the number of agents of some type ( with choice c1 or
employed) serves as state variable. In situations with K types of agents, K ≥ 3, we
need a vector n = (n1 , . . . , nK ) as state vector in general, where ni is the number
of agents of type i or choice i. More correctly, the vector with fractions ni /N as
the ith component, i = 1, . . . , K is a state vector. When the total number of agents
is fixed, then a K − 1 dimensional vector may be used.
Although this choice of state variables seems quite natural, many situations can
be described better using an alternative choice of state variables. This alternative
choice is in line with the occupancy problems in statistics or in physics. See Feller
(1970) for example. In this definition, think of types as boxes and agents as indis-
tinguishable balls. That is, agents are treated as exchangeable, the labels attached
to agents being non-intrinsic and for mere convenience. The state vector is defined
by a vector a with components ai being the number of types (boxes) with exactly
i agents
in each of them, i = 1, . . . , K. By definition, then we have i iai = N ,
and i ai ≤ K. This type of state variables are useful in dealing with distributions
of firms by size, for example. Here interpret size broadly such as the number of
employees, amount of capital or output in some convenient units, and so on. We
do not have space to discuss models of this kind of state variable description. See
Aoki (1999, 2000) for examples.
8.4 Dynamics
We use dynamic equations for the probability distributions of the states as the
basic dynamic description of economic models with many agents of several types
or agents with discrete choices. We do not use ordinary differential equations for the
states themselves. This distinction is important, if subtle. That is, we do not derive
differential equations for X (t)/N , such as the fraction of the employed in Diamond
(1982), for example, but rather the differential equation for the probability of the
fraction of the employed. This is what we call the master equation. In Weidlich and
Haag (1983) we find examples of a birth-and-death stochastic process, which has a
simple master equation, being adapted to model diffusion of opinion or information
such as brand choices of consumer goods or voting for political candidates among
a population. Recent examples in which master equations are used in economic
models are Kirman (1993), Weidlich (1994), and Aoki (1995, 1998, and 1999).
8.4.1 Equilibrium distribution

We first describe models with scalar state variables. By setting the left hand side
of the master equation to zero, we obtain the condition for a stationary solution.
Let Pe (x) denote a stationary probability distribution. In stationary state or in
equilibrium the probability in- and out-flows balance at every state y, the relation

w(y|x)Pe (x) = w(x|y)Pe (y)
x=y x=y
208 Masanao Aoki
holds for all y. This is the balance condition of probability flows, called the full
balance equation, Kelly (1979, 5).
If the probability flows balance for every pair of states, then the equation
w(y|x)Pe (x) = w(x|y)Pe (y)
holds for all x and y. This is called the detailed balance conditions. See Kelly (1979,
Sec.1.5) for Kolmogorov criteria for stationary Markov chains and processes to
satisfy the detailed balance condition.
Given an irreducible Markov chain, for any state xi there is a finite sequence
of states which reaches it from some initial state, x0 , x1 , . . . , xi . If the detailed
balance condition holds, we have

i−1
Pe (xi ) = Pe (x0 ) [w(xk+1 |xk )/w(xk |xk+1 )].
k=0
This probability distribution is a Gibbs distribution since we can express this

as an exponential distribution
Pe (x) = constant exp {−βNU (x)},
with

w(xk+1 |xk )
−βN [U (xi ) − U (x0 )] = ln , (8.3)
w(xk |xk+1 )
where β is a parameter introduced to embody uncertainty or imprecision involved in
making decisions, seeAoki (1996, 138), for example. Note that (8.3) is independent
of paths from x0 to xi , i.e. U (x) is a potential. See the Kolmogorov criterion in
Kelly (1979, 21).
A binary choice model continued Example 1 has the equilibrium distribution

which can be put as
Pe (x) = B exp[−βNU (x)],
where
n
n
k
−βNU = const + βg + N ln CN ,n + O(1/N ),
N N
k=1
with k−1
k η
g := ln N
k + O(1/N ).
N 1−η N
Parameter β is introduced to indicate the degree of uncertainty about the relative
merits of alternative choices. See Aoki (1996, 138) or Aoki (1998, 2000) for further
discussion on this parameter.
We can show that the local minima of the potential corresponds to locally stable
equilibria of the aggregate dynamic equation which is derived next.
In models with several types of agents or several choices, suppose we use
x(t) = (x1 (t), . . . , xK (t)) as
the vector. The equilibrium distribution Pe (x) in
some cases has product form k πk (xk ). See Kelly (1979) and Pollett (1986) for
several examples of this representation. Even when this product form is not exact,
it may serve as an approximation in some cases. In terms of the alternative state
vector a mentioned at the end of the previous section, there is multi-variate Ewens
distribution which seems to hold much promise in economic modeling.Aoki (1996,
238) has some preliminary account of this distribution. In problems with agents
of many types in which random partitions of agents into types or choices are
important, the Ewens distributions invariably arise. Kingman (1978) defines the
notion of random partitions. See Aoki (1999, 2000) for economic examples.
8.5 Approximate solutions of master equations

Only a special class of master equations admits closed form analytic solutions.
Notable among them are generalizations of birth-and-death processes which are
discrete state Markov chains in which states are integer-valued and jumps are
restricted to be ±1.
When we cannot solve the master equation explicitly, we may approximately
solve it by expanding the solution in some parameter such as the size of the model,
N , in the transition rate expressions. The parameter should be such that it governs or
influences the size of fluctuations of the probabilities by affecting the jumps. As the
parameter value approaches a limit the size of fluctuations should approach zero,
so that this solution method produces a macroeconomic (aggregative) equation
which is appropriate for a model with a large number of agents.
We mention two methods for approximately solving the master equation, and
use one to derive the macroeconomic (aggregate) dynamic equations. This method
is in the time domain, utilizing the Taylor series expansion mentioned above. The
other is in the transform domain, and uses probability generating functions and
possibly their approximate solutions to derive differential equations for the first
few moments, such as mean and variance via the cumulant generating functions.
See Cox and Miller (1967, 158). We do not discuss this method here.
8.5.1 Power series expansion

We base this subsection on van Kampen ( 1992). The idea is to expand the master
equations in N −1/2 retaining terms only up to the order O(N −1 ).4
When we anticipate that the probability of the state√will show a well-defined
peak at some X of the order N and spread of the order N , if the initial condition
is
P(X , 0) = δ(X − X0 ).
In such cases we change variable by introducing two variables φ and ξ, both
of order one, and set (recall that x(t) = X (t)/N )
x(t) = φ(t) + N −1/2 ξ(t).

210 Masanao Aoki
If this change of variable decouples the equation for φ and that for ξ, and if the
coefficients in the dynamics for ξ involve only φ, then we are in a position to derive
a deterministic aggregate dynamic equation and a stochastic dynamic equation for
the fluctuations around the aggregate equations. Later we show that φ is the mean
of the distribution when this change of variable is applicable i.e. φ(t) keeps track
of the mean of x(t).
We next show that the scaling by the square root of N introduced above is
the right one because terms generated in the power series expansion of the master
equation separate into two parts. The first part, which is the largest in magnitude, is
an ordinary differential equation for φ. This is interpreted to be the macroeconomic
or aggregate equation. The remaining part is a partial differential equation for ξ
with coefficients which are functions of φ, the first term of which is known as the
Fokker–Planck equation.
To obtain the solution of the master equation, we may set the initial condition
by φ(0) = X0 /N .5
We rewrite the probability density for ξ as
Π{ξ(t), t} = P{X (t), t},
by substituting N φ + N ξ into X .6 In rewriting the master equation for Π we
1/2
must take the partial derivative with respect to time by keeping x(t) fixed, i.e. we
must impose the relation
dξ dφ
= −N 1/2 ,
dt dt
and we obtain
dP dΠ ∂Π dξ ∂Π dφ ∂Π
= = = − N 1/2 .
dt dt ∂ξ dt ∂t dt ∂ξ
We also note that we need to rescale time by
τ = N −1 f (N )t.
Otherwise, the random variable ξ will not be of the order O(N 0 ) contrary to our
assumption, and the power series expansion will not be valid. But, f (N ) = N in
this section. In general τ = t. We use τ from now on to accommodate this more
general scaling function.
The master equation in the new notation is given by
∂Π(ξ, τ ) dφ ∂Π ∂ 1 ∂2
− N 1/2 = −N 1/2 {α1,0 (x) · Π} + {α2,0 (x) · Π}
∂τ dτ ∂ξ ∂ξ 2 ∂ξ 2

1 −1/2 ∂ 3 −1/2 ∂
− N α3,0 (x) · Π − N α1,1 (x) · Π
3! ∂ξ 3 ∂ξ
+ O(N −1 ),
where x = φ(τ ) + N −1/2 ξ, and where we define the moments of the transition rates
by
αµ,ν = r µ Φν (x; r)dr. (8.4)
See van Kampen (1992, 253) for the terms not shown here.7
In this expression we note that the dominant term O(N 1/2 ) on the both sides are
equated. In the next section, the first moment of the leading term of the transition
rate wN (X |X ), α10 (φ), will be shown to determine the macroeconomic equation.
8.6 Macroeconomic equation

In the power series expansion we collect and equate the largest terms on both sides.
This produces
dφ
= g(φ), (8.5)
dτ
where we rename α1,0 (φ) as g(φ) for short, where α1,0 , defined in (8.4), is the
first moment of the function Φ0 , which appears in the transition rate expression
wN (X |X ) = f (N )Φ0 (x ; r), with respect to r. This is a deterministic aggregate
equation for the average, X /N , which is the limiting dynamics as the number of
agents goes to infinity.
The zeros of the right-hand side of this function are the equilibria of the macroe-
conomic model.
A binary choice model continued With the transition rates of Example 1, the
macroeconomic equation becomes
dφ
= (1 − φ)η(φ) − φ[1 − η(φ)]. (8.6)
dt
In terms of g(·) introduced in Example 1, the critical points of the macroeconomic
equation is
φ
exp[βg(φ)] = .
1−φ
A finitary search model continued In Example 2, the macroeconomic equation

becomes
dφ
= (1 − φ)aG(c∗ (φ)) − φb(φ). (8.7)
dt
This is the same equation as that in Diamond (1982), because this term represent
the first term in the Taylor series expansion. The difference lies in the equation for
the fluctuation which is absent in Diamond since he assumes N to be infinite from
the beginning.
One should realize that the method presented above applies to models with
discrete state space. See Aoki (1995, 1998, and 1999) for examples.
8.6.1 Multiple equilibria

The zeros of the right-hand side of the aggregate equation (8.5) are critical points
and potential candidates for the equilibria. If the sign of the first derivative g (·) is
212 Masanao Aoki
negative at a critical point, then that critical point is a locally stable equilibrium. If
g(·) is continous, then multiple zeros are alternatingly locally stable and unstable.
In the example in Aoki and Shirai (2000), g(·) is discontinous, and g is negative
except at the point of discontinuity. This example is the case of two locally stable
equilibria. That is, the dynamics have two basins of attractions, and the centers are
locally stable. We return to this example later in connection with the problem of
equilibrium selection.
After the equation for φ is determined, the remainder of the master equation
governs the density of ξ. This equation is called the Fokker–Planck equation when
terms of the order O(N −1/2 ) and smaller are neglected.
Later we return to this equation and calculate, among other things, the mean
of ξ and show that it remains at zero under certain conditions, i.e., that the mean of
x(t) is given by φ(t), and ξ describes the spread about the mean, as we have claimed
earlier. The variable φ may be thought of as the peak (the maximum likelihood
estimate) of the distribution for x in a single peaked distribution, while ξ keeps
track of the ‘spread’ of the distribution about the peak.
In some models α1,0 (φ) or c1 (φ) is identically zero, then φ(τ ) = φ(0) for all
non-negative τ , and a small deviation in φ(0) does not decay to zero. In this case
we need to expand the master equation in X /N , and redefine τ by N −2 t. Then we
are lead to what is known as the diffusion approximation. We do not discuss this
case here. See Aoki (1996, Chap.5 ), for example.
8.6.2 Macroeconomic relations: case of infinite number of agents

When N becomes very large only the first terms in the Taylor series expansions in
the right-hand side of (8.1) and (8.2) remain. Traditional macroeconomic relations
are recovered in the limit of N going to infinity. For example, (8.6) and (8.7) are the
macroeconomic dynamic equations in terms of fractions of agents with infinity of
agents assumed. For example, (8.7) is exactly the equation derived by Diamond.
8.7 Dynamics for fluctuations

After the terms which determine the macroeconomic equation are removed, and
when we retain terms up to O(N −1/2 ), then we are left with
∂Π(ξ, t) ∂(ξΠ) 1 ∂2Π

= −α10 (φ) + α2,0 2 .
∂t ∂ξ 2 ∂ξ
This equation is linear in ξ, and is called a linear Fokker-Planck equation, Aoki
(1996, 137, 158). This equation can be used to obtain the dynamics for the mean
and variance of ξ, or as in the case of Example 1, the distribution function for ξ
can sometimes be determined.
A finitary search model continued The dynamic equation is given by
∂Π/∂t = A(φ)∂(ξΠ) + C(φ)∂ 2 Π/∂ξ 2

up to O(N −1/2 ), with A(φ) = −Φ (φ), where Φ(φ) = (1 − φ)aG(c∗ ) − φb(φ), and
C(φ) = (1/2)(1 − φ)aG(c∗ ) + φb(φ). Near locally stable equilibrium the deriva-
tive Φ (φ) is negative, that is A(φ) is positive, and the stationary distribution of
this equation turns out to be normal with zero mean and variance C(φ)/A(φ).
8.8 Equilibrium selection

There is a large body of literature on the issue of how to select an equilibrium in
models with multiple equilibria. With an infinite number of agents, the dynamics
are deterministic. Hence the basins of attractions are fixed and once a state is in one
of the basins, the state does not leave that basin, unless there are some exogenous
devices such as sudden changes in expectations which force the state to jump from
one basin to another.
With a finite number of agents in stochastic models such as those in this chapter,
the state oscillate persistently among different basins of attractions and generate
generally asymmetrical oscilaltions. See appendix in Aoki (1998) for a simple two
state example to illustrate this point. See also Aoki and Shinai (2000).
In the framework employed in this chapter, the probability that the model stays
in different basins can be used to show which basin of the model the state is likely
to settle. Two examples will illustrate the method.
8.8.1 Case of two locally stable equilibria

We approximately evaluate the mean transition times from one basin of attraction
to the other, and calculate the equilibrium probabilities that the employed fraction
of population stays in each of the two basins of attraction. Moreover, we show that
our analysis provides the basis for equilibrium selection in the deterministic version
of the model. Since the states off the equilibria approach them exponentially fast,
only the events of disturbances which are large enough to bring the states from one
basin of attraction to another will force the model to move from one equilibrium
to another.
Approximate analysis
First we recognize that we need to calculate only the event from one of the equilib-
rium state to the boundary between two basins of attraction, ψ, which is introduced
in the example above. The reason is the same one used by van Kampen (1992) as
quoted in Aoki (1996, 151). The time needed for φ to reach its equilibrium value,
φ1 , or φ2 depending on the initial value, is much shorter than the time needed to
go from one basin of attraction to the other.
A quick way to see this is to solve the deviational equation for φ. To be definite,
suppose that φ is in the domain of attraction to φ1 and let x := φ − φ1 . Then, it is
governed by
dx/dτ = Φ1 (φ1 )x = −A(φ1 )x,
with the initial condition x(0) = φ(0) − φ1 .
214 Masanao Aoki
The solution is x(τ ) = x(0) exp[−A(φ1 )τ ]. Recalling that exp(−4. 5) = 0. 01,
it takes about τ = 4. 5/A(φ1 ) to reduce the distance from φ1 to about 0.01 of the
original value. In the case where a = 1 and p = 0. 2, we have φ1 = 0. 358 and
A(φ1 ) = 0. 92. Thus, it takes about 4.9 or 5 time units to reach the equilibrium
point. As we show later in example 2, the mean first passage time for this example
is of the order of 103 when N = 100. Therefore we are justified in assuming that
φ is initially at one of the equilibrium points in calculating the mean first passage
time. The procedure is as outlined in Appendix in Aoki (1998). We set up a two
state Markov chain, because there are two locally stable equilibria in the example.
Let π1 and π2 = 1 − π1 be the probability that the employed fraction are in basins
of attraction for φ1 and φ2 , respectively. These probabilities evolve according to
the differential equation,
dπ1 /dτ = W2,1 π2 − W1,2 π1 ,
where W1,2 is the transition rate from φ1 to the boundary of the two basins of
attractions, that is, ψ, and W2,1 is that from φ2 to ψ. In the stationary state dπ1 /dτ =
0, we have
1
π1 = .
W1,2 /W2,1 + 1
The mean first passage time is given by
1
τ1,2 = .
W1,2
See Aoki (1996, 152).

To calculate W1,2 we use the probability that
W1,2 = Pr[ξ ≥ ξc ],
with
ξc
φ1 + √ = ψ.
N
√
Analogously W2,1 is approximated by√the probability that ξ is smaller than N (ψ −
φ2 ), or equivalently it is larger than N (φ2 − ψ).
8.8.2 Limiting case of N infinity

As we increase the number of agents in the economy to infinity, our model con-
verges to that of Diamond. This can be seen by the fact that variance of employed
fraction of agents converges to zero as N taken to infinity as it is indicated by
the density function . This suggests that the stationary (invariant) distribution over
fraction of employed agents in the economy become spikes with probability masses
of π1 and π2 assigned for employed fractions e = φ1 and e = φ2 , respectively.
These probability masses for each locally stable critical points provide the
criteria for equilibrium selection for the model of multiple equilibria with infinite
number of agents. One can easily check that our special case given in section
8.6 yields exactly the same stationary fractions of employed agents φ1 and φ2 in
Diamond’s model if we set the same matching function b and cost distribution
function G.
What we are left to do is to calculate π1 when N is taken to infinity. As suggested
above, we have
√
W1,2 = Pr ξ > N (ψ − φ1 )
∞
1 ξ2
= √ exp − dξ,
N (ψ−φ1 ) 2πσ 2 (φ1 ) 2σ 2 (φ1 )
and
√
W2,1 = Pr ξ < N (ψ − φ2 )
∞
1 ξ2
= √ exp − dξ.
N (φ2 −ψ) 2πσ 2 (φ2 ) 2σ 2 (φ2 )
It is easy to see that both W1,2 and W2,1 approach zero as N is brought to in-
finity. Hence, we can approximate limN →∞ W1,2 /W2,1 by limN →∞ (dW1,2 /dN )/
(dW2,1 /N ). This is given by,
W1,2 dW1,2 /dN

lim = lim
N →∞ W2,1 dW2,1 /dN
N →∞

(φ2 − ψ)2 (ψ − φ1 )2 (ψ − φ1 )σ(φ2 )
= lim exp − N .
N →∞ 2
2σ (φ2 ) 2
2σ (φ1 ) (φ2 − ψ)σ(φ1 )
From this, it is straightforward to see that as N approaches infinity, W1,2 /W2,1

approaches 0 if and only if (ψ − φ1 )/σ(φ1 ) > (φ2 − ψ)/σ(φ2 ); infinity if and
only if (ψ − φ1 )/σ(φ1 ) < (φ2 − ψ)/σ(φ2 ); and 1 if and only if (ψ − φ1 )/σ(φ1 ) =
(φ2 − ψ)/σ(φ2 ).
The result is summarized as follows;
⎧
⎨ 1 if (ψ − φ1 )/σ(φ1 ) > (φ2 − ψ)/σ(φ2 ),
lim π1 = 1/2 if (ψ − φ1 )/σ(φ1 ) = (φ2 − ψ)/σ(φ2 ), and
N →∞ ⎩
0 if (ψ − φ1 )/σ(φ1 ) < (φ2 − ψ)/σ(φ2 ).
The larger the distance between the critical point and the boundary of basins
of attraction, or the smaller the variance of fluctuation around the critical point, it
is likely that this critical point is selected as an equilibrium in a model with infinite
number of agents.
8.8.3 Case of two locally stable and one unstable equilibria

We discuss the first passage between two locally stable equilibria when they are
on both sides of a locally unstable equilibrium. The method for calculating first
216 Masanao Aoki
passage probabilities or mean first passage time is well known, and is discussed in
Cox and Miller (1965, Section 3.4), Grimmett and Stirzaker (1992, Section 6.2),
or van Kampen (1992, Chapter XII) to mention a few textbooks.
Basically, in going from the basin containing one locally stable equilibrium to
the other basin, the model must overcome or go over the potential barrier Ub − Uc
or Ub − Ua , where we use the short-hand notation to denote U (φa ) by Ua , and so
on, where the three critical points are states,φa < φb < φc , and φb is the unstable
equilibrium point. That is, the model must go over the potential of height U (b)
in going from one basin to another. The model starts from potential height of
U (a) in one direction and U (c) in another. Suppose that U (a) < U (c). Since the
equilibrium probability of going from φa to φc is proportional to exp[−β(Ub −
Ua )], and the reverse direction has the probability proportional to exp[−β(Ub −
Uc )], the model is more likely to stay in the basin containing state φa than that
containing state φb . Put differently, the transition from state a to state c takes longer
than the reverse direction.
In the limit of N going to infinity, the model will stay longer and longer in the
basin containing state φa , and in the limit this is the equilibrium selected.
To make this heuristic argument rigorous, we need to calculate the first passage
times from one basin to another and its reverse and calcualte the probability that
the model stays in each of the two basins. See Aoki (1996, Sec. 5.11)

We have proposed a probabilitic way for dealing with multiple equilibria and the
associated asymmetrical cycles by formulating the dynamics for jump Markov pro-
cesses as the master equation. Out of the master equation we derive deterministic
aggregate or macroeconomic dynamics which determine locally stable equilibria
as the centers of basins of attractions. The ratios of the probabilities of individual
basins of attractions determine which of the equilibria is selected in the limit of
the size of the model (the number of agents in the model ) goes to infinity.
In the limit of N going to infinity, the aggregate dynamics become the same as
those derived by assuming at the outset that there are infinity numbers of agents
and working with appropriate fractions. You might ask, therefore, what is the
advantage of the proposed procedure? The advantage lies in the way the model
naturally introduces asymmetrical cycles, and selects one basin out of several as
the one with most probability mass as N is increased.
Notes
1 There are many references on master equations. Van Kampen (1992, 97) tells us the
origin of the word ‘master’. See also Weidlich and Haag (1983), or Kubo (1975).
2 This fraction is an example of variables called intensive variables. This is often implicit
in the search literature, where it is routinely assumed that there are an infinite number
of agents, and fractions of one kind or another are posited from the beginning, such as
the fraction of employed, or fractions of agents which hold some specified assets, and
so on. See Diamond (1982) or Kiyotaki and Wright (1993) for example.
3 When the size N is fixed, this assumption is innocuous because it can be absorbed into
time units. If N is a random variable as in models open to entry and exit by agents, it
must be explicitly incorporated into the transition rates. Kelly (1979) has an example
in which N is a random variable.
4 When these terms are zero, we may want to retain terms of the order N −2 . Then we have
diffusion equation approximations to the master equation. Diffusion approximations
are not discussed here.
5 We need not be precise about the initial condition since an expression of the order
O(N −1 ) or O(N −1/ 2 ) can be shifted between the two terms without any consequence.
Put differently, the location of the peak of the distribution can’t be defined more
precisely than the width of the distribution which is of the order N 1/ 2 .
6 Following the common convention that the parameters of the density are not carried as
arguments in the density expression, we do not explicitly show φ when the substitution
is made.
7 Note that
∂ ∂Π ∂
{α10 (x)Π} = α10 (φ) + N −1/ 2 α10

(φ) (ξΠ) + O(N −1 ).
∂ξ ∂ξ ∂ξ
References
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Externalities’, Japanese Economic Review, 46, 148–65.
Aoki, M. (1996) New Approaches to Macroeconomic Modeling: Evolutionary Stochastic
Dynamics, Multiple Equilibria, and Externalities as Field Effects , Cambridge University
Press, New York.
Aoki, M. (1998), ‘Simple Model of Asymmetrical Business Cycles: Interactive Dynamics
of a Large Number of Agents with Discrete Choices’, Macroeconomic Dynamics, 2,
427–42.
Aoki, M. (1999), ‘Open Models of Share Markets with Several Types of Participants’,
presented at Workshop on Heterogenous Interacting Agents, Univ. Genoa, June.
aoki, M. (2000) ‘Cluster Size Distributions of Economic Agents of Many Types in a Market’,
Journal of Mathematical analysis and Applications, 249, 32–52.
Aoki, M. (2001) Modeling Agregate Behavior and Fluctuations in Economics: Stochastic
Views of Interacting Agents, Cambridge University Press, Cambridge.
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Search Equilibrium’, Macroeconomic Dynamics, 4, 487–505.
Breiman, L. (1968) Probability Theory, Addison-Wesley, Reading, MA.
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London.
Diamond, P. D. (1982) ‘Aggregate Demand Management in Search Equilibrium’. Journal
of Political Economy, 90, 881–94.
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& Sons, New York.
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Kelly, F. (1979) Reversibility and Stochastic Networks, Wiley & Sons, New York, 1979
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218 Masanao Aoki
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spective in Probability and Statistics, Academic Press, New York.
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880–900.
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9 The instability of markets
Tad Hogg, Bernardo A. Huberman and
Michael Youssefmir
9.1 Introduction
The explosive growth of computer networks and of new forms of financial products,
such as derivatives, is leading to increased couplings among previously dispersed
markets. This increased fluidity raises questions about the stability and efficiency
of the international financial system. On the one hand, it is quite apparent that
increased overall connectivity among markets allows transactions that were not
previously possible, increasing the net wealth of people and leading to more effi-
cient markets. But at the same time, this implies that a transition is taking place,
from a more static scenario in which isolated markets can be considered in equi-
librium on their own, to a global economy that knows of no geographical borders.
One may wonder about the nature of this transition, i.e. is it gradual in the sense of
a smooth change in prices, or punctuated by abrupt changes in the value of certain
commodities, cascading bankruptcies and market crashes?
Underlying these questions is the old problem of the existence and stability
of equilibria in markets, which general equilibrium theory has addressed under
various conditions (Arrow and Hahn 1971, Arrow 1988) . In this chapter, we will
focus on the stability of markets by treating the couplings among them as dynamical
entities in the spirit of other evolutionary approaches (Day 1975, Nelson and Winter
1982, Brock 1988) .
The standard approach to characterizing the stability of markets postulates that
agents opportunistically optimize their portfolios in such a way so as to minimize
their risk while maintaining a given return. But in real life agents are not always
able to perfectly process information about the system in which they are embed-
ded. Under such conditions, adaptive agents continuously switch between different
behavioral modes in response to a constantly changing environment, favoring be-
haviors that can temporarily lead to increased rewards. It follows that while agents
are continuously learning about the relative merits of different commodities and
markets, the couplings between such commodities evolve in rather complicated
ways.
Other examples of such evolving couplings are provided by agents linking
limited baskets of commodities efficiently while ignoring other commodities in the
220 Tad Hogg, Bernardo A. Huberman and Michael Youssefmir
process. Of more recent interest are the couplings that agents introduce through
complicated derivatives created in an effort to optimize and hedge away risks.
Due to their complex nature such derivatives can be poorly understood by the
people who use them, once again leading to extraneous couplings. Finally, highly
leveraged hedge funds can also introduce couplings into the market by being forced
to cover certain leveraged positions with other unrelated positions.
In this chapter, we assume the existence of an equilibrium within a network of
markets and examine its dynamic stability through the use of a general model of
the adjustment processes within the markets. We focus on cases where agents are
not perfectly rational or identical and therefore can introduce couplings that may
not always represent the best possible allocation of their resources. We then discuss
how the stability of such a system scales as different markets become increasingly
coupled. We show that the class of systems that are stable becomes smaller and
smaller as the number of coupled markets scales up. These instabilities are shown
to exist even when couplings are so weak that the markets are near decomposable.
Such instabilities in turn require a heightened degree of learning on the part of
market participants, a process in which the learning itself is marked by periods of
instability until equilibrium is reestablished. These results, which run counter to
prevailing notions of stability in large coupled markets, offer a cautionary note on
treating emerging markets with the tools of equilibrium economics.
9.2 Dynamics of coupled markets

In what follows we will consider the case of a number of markets, each of which
is in stable equilibrium when decoupled from the others. These markets contain
arbitrarily large number of agents that buy and sell commodities, using a diverse set
of strategies. Heterogeneous agents will provide diverse couplings between these
markets through mechanisms such as physical substitutions, financial derivatives,
arbitraging between geographically disperse markets, and expectations (be they
rational or irrational) that movements in one market will lead to changes in others.
Given a single commodity, a reasonable dynamic model of price adjustments
postulates that the price rises when demand exceeds supply and that they fall when
supply exceeds demand. This can be described by a differential equation of the
form
dp
= f (p) (9.1)
dt
where p is the price of the commodity and the function f represents the excess
demand at the given price. This equation can be thought of as a description of
market adjustments via the tatonnement process by which an auctioneer calls
out adjustments in prices in order to satisfy excess demand. The function f then
defines the equilibrium p(0) where f (p(0) ) = 0. The value f (p(0) ) represents a
linearization of the adjustment process near the equilibrium. If this quantity is
negative the equilibrium at p(0) is stable against small perturbations, which relax
−1
on a time scale given by τ = f (p(0) ) .
The time scale τ for adjustment is reduced when transaction costs are negligible
or agents are confident about the value of the commodity. Any movement of p
The instability of markets 221
away from p(0) will be immediately corrected for by agents seeking to make a
profit on the differential p − p(0) . On the other hand, when agents are uncertain or
have diversity of expectaions for the fair price for the commodity, the adjustment
process may depend on agent interactions, communication, and analysis. In this
case, the time scale for adjustment τ will tend to be greater than before.
These remarks are readily generalized to the adjustment processes where there
is more than one commodity and multiple markets. We will study the dynamics
that result from the couplings of these markets, and in particular how the couplings
affect the stability of the system as whole. To do this we interpret p as a price vector,
whose entries correspond to prices in the separate markets, and f as a vector of
excess demands that characterizes both the dynamics of the isolated markets and
their couplings. The dynamics is then given by an equation of the form (9.1), which
relates the evolution of prices of all commodities in all markets as a function of
each other and the degree to which they are coupled (Arrow and Hurwicz 1958,
Samuelson 1941, Metzler 1945).
The excess demand in the case of each commodity is controlled by the individ-
ual agents within the respective markets. In stable equilibrium the individual excess
demands of the agents are such that in totality prices are held at the equilibrium
values. We can then study stability around an equilibrium peq where f (peq ) = 0, by
assuming the existence of a temporal departure from this equilibrium by a small
amount δ. In order to see if this disturbance decays in time or grows, we perform
a Taylor series expansion around the fixed point to get an equation of the form
dδ
= M (peq )δ (9.2)
dt
where the n by n matrix M is the Jacobian of f evaluated at the fixed point peq and
n is the number of commodities.
The components of this Jacobian matrix describe how a small increase in the
price of one item in one market changes that of another item. Thus, the diagonal
elements of M show the direct effect of a small change in excess demand on the
price of a given commodity. We assume that each market by itself is stable so as
to counteract the original change, which implies that the diagonal elements will
be negative, with average value that we denote by −D, and which quantifies the
speed of adjustments within that market.
On the other hand an off-diagonal entry describes the direct effect on an item
from a change in the price of another, which can be of either sign. Notice that if the
couplings between commodities are weak, it would translate into a matrix whose
largest entries are on the diagonal and the off-diagonal terms would be small. As
discussed above, these off-diagonal couplings are the result of individual agents
whose expectations, whether informed or uninformed, link the two commodities
such that an increase in one commodity’s price affects that of the other.
The effect of slight disturbances around the fixed point perturbation is deter-
mined by the eigenvalue of M with the largest real part, which we denote by E.
Specifically, the long time behavior of the perturbation is given by δ ∝ eEt , imply-
ing that if E is negative, the disturbance will die away and the system will return
to its original equilibrium. If, on the other hand, E > 0, the smallest perturbation
will grow rapidly in time, leading to instability.
We note that this model of dynamic adjustment and its stability has also been
used in the economic literature focusing on qualitatively specified matrices (Quirk
1981). In that literature, the focus is on specifying the stability properties of matri-
ces given only the qualitative nature of the signs of the entries (positive, negative,
or zero). In that case, for example, commodities that are gross substitutes would
be coupled via positive nondiagonal matrix elements while gross compliments
would be coupled by negative coefficients. Such an approach, however, suffers
from the disadvantage that the cases for which stability criterion are specified are
quite restricted.
9.3 Market instabilities

The approach taken in this chapter looks at the average behavior of a market given
an ensemble specifying uncertain knowledge of parameters within the stability
matrix. For the sake of treating a very general case, we will assume as little knowl-
edge about these mechanisms as it is possible. This implies that all matrices that
are possible Jacobians can be considered, and that there is no particular basis for
choosing one over the other. This is the class of the so-called random matrices,
in which all such matrices are equally likely to occur. Matrices in this class are
such that each entry is obtained from a random distribution with a specified mean
and variance. By taking this approach, we can make general statements about the
stability of these systems as the number of markets and diversity of agent behaviors
grows. We show quite generally that as the number of coupled markets and the
diversity among the agents grows, it is more and more likely that the system as a
whole will be unstable.
The precise value of the largest eigenvalue E depends on the particular choice
of the Jacobian matrix. Methodologically, the study of the general behavior of
these matrices is performed by examining the average properties of the class that
satisfies all the known information about them. A class of plausible stability ma-
trices is determined by the amount of information one has about particular market
mechanisms and their couplings. This information, which is far from perfect, de-
pends on the nature of expectations that agents have about future values, which are
based on limited knowledge, and on the extent to which individuals understand
the nature of the financial instruments that they invest in.
In spite of their random nature, these matrices possess a number of well defined
properties, among them the behavior of their eigenvalues (Wigner 1958, Mehta
1967, Cohen and Newman 1984, Furedi and Komlose 1981, Juhasz 1982, Edelman
1989). This means that we can use these properties to ascertain the stability of the
markets against perturbations in the excess demand. In what follows we will show
that in the general case, as these matrices grow in size, or the variability of their
entries, their largest eigenvalue becomes positive, thus leading to market instability.
This is a result that applies not only to markets but also to complex ecosystems
(May 1972, McMurtie 1975, Hogg, Huberman and McGlade 1989).
The simplest case, albeit not very realistic, would correspond to a situation
where commodities are equally likely to be substitutes or complements of each
other. This means that on average the nondiagonal elements would be of zero mean.
We will also assume that the Jacobian random matrix will have symmetric entries
and be bounded in magnitude. For this case, the distribution of eigenvalues as a
function of the size of the Jacobian is given by Wigner’s law (Wigner 1958), i.e.,
√
E = 2σ n − D (9.3)
This implies that as the matrix gets large enough, its largest eigenvalue will become
positive.
A more realistic case relaxes the requirement that the changes in one price are
on average balanced by changes in others. This corresponds to having a non-zero
mean. In this situation a theorem of Furedi and Komlos (1981) states that, on
average, the largest eigenvalue is given by
E = (n − 1)µ + σ 2 /µ − D (9.4)
where µ is the average value of the couplings, which we assume to be positive.

Moreover, the actual values (as opposed to the average) of E are normally dis-
tributed around this value with variance 2σ 2 . This implies, that as the size of
the system grows, most such matrices will have positive largest eigenvalue, thus
making the equilibrium point unstable.
Consider next the case of a non-symmetric matrix, with nondiagonal terms with
positive mean, µ, standard deviation, σ, and whose diagonal terms have mean, −D.
In this case the largest eigenvalue grows with the size of the matrix as (Furedi and
Lomos 1981, Juhasz 1982):
E ∼ µ(n − 1) − D (9.5)
Since µ is positive these results imply that even if markets are stable when small,
they will become unstable as their size becomes large enough for E to change sign.
One argument that could be given for the stability of coupled markets in spite
of their size is that not all commodities happen to be coupled to each other. In
terms of our theory, this amounts to a near decomposability of the Jacobian matrix
whereby the entries are such that the further they are from the diagonal the smaller
they become (Simon and Ando 1961). Such systems, sometimes called loosely
coupled, are very relevant to situations when markets are initially weakly coupled
to each other. But as we will now show, even in this case, as the size of the markets
increases the equilibrium will be rendered unstable. In terms of the interactions, this
situation can describe either the fact that a given item’s price is strongly influenced
by a few others and weakly by the rest, or a more structured clustering, where
the commodities appear in groups (e.g. technology stocks, foreign currencies)
wherein their members strongly interact but members of different groups have
weak interactions with each other.
Another way of considering this scenario is to imagine initially isolated mar-
kets that eventually get coupled through the interaction of mediating interactions,
such as roads and communications. In this case the coupling between the initially
isolated markets grows in time.
The corresponding matrices for the first case are constructed by selecting off-
diagonal entries at random with large magnitude 1 with probability p and small
value < 1 otherwise. In this case µ = p + (1 − p) and σ 2 = p(1 − p)(1 − )2 .
As shown in (9.5), the largest eigenvalue will become positive when the system
becomes large enough.
In the second case, the commodities are grouped into a hierarchical structure
which we assume to be of depth d and average branching ratio, b. In this representa-
tion, the strength of the interaction between two commodities will decrease based
on the number of levels in the hierarchy that one has to climb to reach a large
enough common group to which they both belong. Specifically, the interaction
strength will be taken to be Rh , with h the number of hierarchy levels that separate
the two commodities, and R characterizes the reduction in interaction strength that
two commodities undergo when they are separated by one further level. The aver-
age size of the matrix is given by n = bd and the mean of the non-diagonal terms
can be computed to be
d
h=1 b
h−1
(b − 1)Rh (Rb)d − 1 R(b − 1)
µ= = (9.6)
d
h=1 b
h−1 (b − 1) bd − 1 Rb − 1
In order to represent a reasonable clustering, we need to specify whether the total

interaction is dominated by either those neighboring commodities in the hierarchy
we choose, or distant ones. In the first case, this amounts to restricting R to be
R < 1/b. Notice that this choice makes the decreasing interaction strength be-
tween commodities overwhelm the increase in their number as higher levels in the
hierarchy are considered.
In this situation when n is large Eq. 9.6 becomes
R(b − 1)
µ=
(1 − Rb)n
which implies that as n grows the mean goes to zero, leading to a stable system
because of (9.5). However, if fluctuations in the coupling strength were taken into
account some of the matrices could still become unstable, as was shown above.
In the second case, R > 1/b. This implies that in the large n limit the average
becomes
R(b − 1) R(b − 1) l n R
µ = Rd = n ln b (9.7)
Rb − 1 Rb − 1
Note that since −1 < ln ln b < 0, µ goes to zero as the size of the matrix grows,
R
but in slower fashion than the case above. Nevertheless, this subtle difference
in convergence to zero amounts to a qualitative difference in the stability of the
system. To see this, notice that (9.5) implies that the largest eigenvalue of a random
matrix grows as µn, which increases with n for this value of µ, thus leading to
instability when the system gets large enough. The growth in largest eigenvalue
with the size of the system is exhibited in Figure 9.1 for a particular choice of
5
3
largest eigenvalue
3
0 1 2 3 4 5 6 7 8 9 10
d
Figure 9.1 Plot showing the growth of the largest eigenvalue of a hierarchical matrix with
branching ratio b = 2 and R = 0. 55 as a function of d = log2 n. The diagonal
elements were chosen to be D = 3.The points are the theoretically predicted
values of Eq. 9.6 and lie very close to the computed eigenvalues shown by the
solid line.
parameters. Notice that the system becomes unstable for d ≥ 5. Given these results
we see that the size of the matrix for which this instability takes place is much
larger than the one in the absence of a hierarchy of interactions.
A final possibility is for the commodities to include aggregate structures, such
as stock indices. In this case the couplings will correspond to situations where each
commodity interacts with itself, the other components of its aggregate, or its higher
order aggregate. The ensuing Jacobian will have blocks of nonzero elements and
zero entries elsewhere. For this case, we have shown that an even slower growth
of the largest eigenvalue with size is obtained (Hogg,√Huberman and McGlade
1989). Specifically, the eigenvalues grow no faster than ln n, implying that much
larger coupled markets can be stable when they are structured in a highly clustered
fashion. Fluctuation corrections make the eigenvalues grow like ln n, still implying
a higher degree of stability than in the previous cases.1
9.4 Discussion
Recent dramatic fluctuations and losses in the world financial markets have raised
concerns about the inherent stability of the global financial system. These concerns
have been prompted by the heightened fluidity of global currency flows and the
emergence of complicated derivatives, which now allow market players to make
financial bets as never before. For example, the derivatives debacle in Orange
County, California points to the fact that some market players do not understand
the full risks that are being taken. At the same time, the result of the current trends
in global finance is that markets are now more and more coupled and individual
governments have less and less power to control these perhaps destabilizing cou-
plings. The recent economic crisis in Mexico, the ‘tequila effect’, resulted in added
volatility and ‘corrections’ in many emerging equity markets all over the world.
In this chapter, we modeled this system as a web of interacting markets much
like a biological ecosystem (Rothschild 1990). By doing so we obtained a gen-
eral result showing that as couplings between previously stable markets grow, the
likelihood of instabilities is increased.
These results allow us to understand phenomena that are likely to arise as a
system grows in diversity, strength of couplings and in the size of the overall number
of the coupled markets. In a sense these results appear to be counterintuitive, for
one expects that as a system gets larger, disturbances in a particular part would
exhibit a kind of decoherence as they propagate through the system, making it very
unlikely that after a given time they would once again concentrate on a particular
node and amplify it. But the properties of random matrices make it probable for
this conspiracy of perturbations to concentrate on given parts of the market. As the
size and diversity of couplings in such matrices grow, the complicated effects of
the couplings are more and more likely to result in instability that leads to motion
away from equilibrium.
One may ask about the fate of the lost equilibrium and the ensuing evolution of
an unstable market. We speculate that once the instability sets in adaptive agents
will modify the respective couplings in such a way as to stabilize the entire system
once more at another equilibrium. If this is indeed the case, the volatility brought
about by instabilities in such large systems is the natural mode by which couplings
are modified to achieve a more efficient market. Due to the lack of information
needed for appropriate centralized control, it is also by no means clear that regu-
latory approaches to controlling the market structures will be effective. Improper
controls could introduce additional couplings in such an uncertain way that the
system may be further destabilized. Accepting the instabilities of these larger sys-
tems as the best way for market participants to learn the correct couplings may,
therefore, be the most reasonable course of action.
Last but not least, these results cast light on the related problem of the dynamics
of multiagent systems in distributed computer networks, which have been shown
to behave like economic systems (Huberman and Glance 1995). In the case of
only two resources their equilibrium is punctuated by bursts of clustered volatility
(Youssefmir and Huberman 1997), a fact which renders the notion of equilibrium
suspect. This paper shows that as distributed computing systems get large and more
coupled, they will also exhibit a loss of stability, on their way to finding a new and
more efficient equilibrium.
Notes
1 A further interesting possibility (Trefethen et al 1993) is that the linearized system can
produce an initially growing transient even when E < 0 so the perturbations eventually
decay. During this transient growth, the perturbations may become large enough to be
sustained by nonlinear corrections, thus giving another possible source of instability.
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10 Heterogeneity, aggregation
and capital market
imperfection
Domenico Delli Gatti and
Mauro Gallegati
‘there can be many representative firms’

Alfred Marshall, Correspondence: III, 377
10.1 Introduction
The representative agent framework has a long tradition (Marshall, 1920) and has
been one of the most successful tools in economics (Hartley, 1997). 1 It is still
the cornerstone of microfoundations in macroeconomics and of aggregation in the
classical literature (see e.g. Lucas, 1975; Kydland and Prescott, 1982; Long and
Plosser, 1983) because the aggregation process allows any difference between the
behaviour of individually optimizing agents and that of aggregate variables to be
ruled out.
Despite its success, economists are growing more and more dissatisfied with
the representative agent framework2 for a number of different reasons.
First of all, the set of assumptions necessary to reach exact aggregation in a
logically consistent way is impressive.3 Martel (1996: 128) lists the following:
• homothetic preferences (i.e. linear Engel’s curve);
• linearly homogeneous production functions, identical for all firms;
• homogeneous and infinitely divisible commodities and factors of produc-
tion;
• a common set of prices with constant relative ratios;
• constant distribution of income and endowments over time.
Of course, none of the requirements above matches real world features. Empiri-
cal analysis at the disaggregated level has shown that heterogeneity matters and
aggregation of heterogeneous agents is of central relevance, since there is sys-
tematic evidence of individual differences in economic behaviour (Stoker, 1993).
In econometrics, several contributions show that aggregate dynamics and indi-
vidual heterogeneity are intertwined. On the one hand, neglecting heterogeneity
230 Domenico Delli Gatti and Mauro Gallegati
in aggregate equations generates spurious evidence of dynamic structure. On the
other hand, aggregation of very simple individual behaviour may lead to aggregate
complex dynamics.4
Apart from the obvious objections to the Representative Agent Hypothesis
(RAH) emerging from the empirical evidence, there are theoretical reasons to
reject the use and misuse of the representative agent.
First of all, Arrow, 1951 (and later Sonnenschein, 1972, and Mantel,1976)
have shown that the proposition according to which the properties of an aggregate
function will reflect those of the individual function has no theoretical foundations.
Second, ignoring heterogeneity may lead to a fallacy of composition (see Weiss,
1967) which may be relevant when agents’decisions are not perfectly synchronized
– so that composition effects arise (Caballero, 1992) – and misleading for empirical
analysis.5
Third, the representative agent framework can be unsatisfactory for a wide
range of economists, even belonging to opposite sides of the academic spectrum.
The very idea of asymmetric information of the new keynesian economics (Mankiw
and Romer, 1991) is inconsistent with the representative agent hypothesis (Stiglitz,
1992). In fact, the asymmetric information context is based on heterogeneity; only
if agents are different from one another may phenomena such as adverse selection
or moral hazard exist.
From a different theoretical perspective, the general equilibrium theorist may
not feel at ease with the representative agent assumption because some of the
building blocks of general equilibrium theory do not hold in the presence of the
representative agent (e.g. the ‘Weak Axiom of Revealed Preferences’ or Arrow’s
‘Impossibility Theorem’, Kirman, 1992: 122).
Therefore there are theoretical as well as empirical reasons to question the reli-
ability of the representative agent as an economic tool. This chapter reviews some
macroeconomic models in which agents’ heterogeneity is the main ingredient.
The literature on heterogeneous agents is burgeoning. We can distinguish a
number of different strands of this literature: models of distribution and growth,6
employment and aggregate supply, 7 aggregate demand, 8 and macrodynamics,9
capital market imperfections and business fluctuations.10 Some authors identify
a law of motion of the average level of the state variable, keeping constant the
distribution; others analyze the dynamics of distribution alone, making inferences
on aggregate dynamics.
The assumption of a constant distribution of income, wealth or endowments
over time is quite implausible when coping with dynamics. In a sense, agents’
heterogeneity is the logical requirement for dynamic analysis. According to the
impulse-propagation approach small shocks may lead to large fluctuations once
a sufficiently large amplification mechanism is activated. If agents are heteroge-
neous, the propagation mechanism is affected by changes in the distribution of
agents. In such a case, business cycles are not ‘all alike’ (Lucas, 1977). Moreover,
if we give up the RAH, fluctuations and cycles may emerge as the result of changes
in the distribution of agents rather than the consequence of aggregate (as well as
idiosyncratic) exogenous shocks to which a ‘representative’ individual reacts, as
Heterogeneity, aggregation and capital market imperfection 231
in the standard literature on the business cycle. In other words, if the RAH is
abandoned, the change in distribution may produce endogenous fluctuations. The
literature we review in the next sections emphasizes these aspects.
In the following we survey extensively a class of models in which heterogene-
ity and aggregation play a crucial role. We deal with macroeconomic models based
on capital market imperfections due to asymmetric information. Greenwald and
Stiglitz, 1988, 1990, 1993 (section 2), Bernanke and Gertler, 1989, 1990 (section
3), Kiyotaki and Moore, 1997 (section 4), have put forward an approach which
emphasizes the role of financial fragility through the interaction of heterogenous
agents. These authors, however, don’t push the analysis to the point of identify-
ing the law governing the distribution of agents. On the other hand, models by
Galor and Zeira, 1993, and Aghion and Bolton, 1997 (section 5) emphasize how
the distribution of income and wealth across agents affects growth, i.e. the dy-
namic behaviour of the system. Section 10.6 discusses an approach which is at the
crossroad of the existing literature: while providing an aggregation mechanism, it
identifies the law of motion of agents’ distribution and shows how aggregation of
heterogenous individuals affects the dynamics. Section 10.7 concludes.
10.2 Bankruptcy costs, net worth and employment

In a series of papers, Greenwald and Stiglitz (GS hereafter) have put forward a the-
ory of supply decisions in which the financial structure of the firm plays a relevant
role (Greenwald and Stiglitz, 1988, 1990, 1993). Their theoretical framework is
based on the assumption that firms are unable to raise external finance by issuing
new equities because of asymmetric information between managers and potential
shareholders on the value of the firm (equity rationing). Due to this financing con-
straint firms rely first and foremost on internal funds in order to finance production
and resort to bank credit in order to raise external finance if internal funds are
insufficient.
As a consequence, the firm runs the risk of bankruptcy, which occurs when
the realization of the random individual price happens to be lower than average
operating cost, which in this framework coincides with debt commitments. The
probability of bankruptcy is a decreasing function of firms’ net worth (or equity
base).
In this framework, the scale of production of each firm depends upon its net
worth. In fact, the higher is net worth, the lower is the probability of bankruptcy
and the higher employment and production. As a consequence, the risk-neutral
firm which takes explicitly into account the cost of bankruptcy behaves as if it
were risk averse. Production in the presence of bankruptcy risk, in fact, is smaller
than in the case in which the firm does not face financing constraints, i.e. capital
markets are perfect and the Modigliani–Miller theorem holds true.
It is worth noting that the representative agent assumption is perfectly con-
sistent with the Modigliani–Miller perfect capital markets world which delivers
the first best solution. In this case, in fact, individual financial conditions – albeit
different from one agent to another – are irrelevant for employment and produc-
tion decisions. When agents are different, however, the dispersion of financial
conditions may generate composition effects and affects the dynamics.
10.2.1 Background assumptions

GS consider an economy populated by firms, households and banks. The produc-
tion function of the i-th firm in period t is Yti = F(Nti ) where Yti is output, Nti
employment, and F(. ) is a well behaved production function 11 uniform across
firms. Therefore Nti = Φ(Yti ) is the labour requirement function: Φ = F −1 .
GS assume that ‘production takes time’. Therefore there will be a one-period
time lag between production and sale of the good. Because of the production lag,
i
the price Pt+1 at which the firm will sell in t+1 output produced in t is uncertain.
i
Uncertainty is captured by assuming that Pt+1 is a random variable with expected
value equal to the general (average) price level Pt+1 . As a consequence the relative
i
price ut+1 i
= Pt+1 /Pt+1 is also a random variable, with c.d.f. G(. ) and expected
i
value E(ut+1 ) = 1.
Firms finance production costs, i.e. the wage bill, at least partially by means of
internally generated funds, which will be referred to as net worth. A financing gap
occurs when net worth is not sufficient to pay the wage bill. In this case the firm has
to resort to external finance. In principle there are two sources of external finance:
bank loans and the issue of new equities. GS assume that the issue of new equities
is not feasible because of equity rationing (Myers and Majluf, 1984; Greenwald et
al., 1984), which is due to asymmetric information between potential shareholders
and managers of the equity-issuing firm.
Therefore, the only source of external finance available to the firm is credit.12
The demand for loans in t is Bti = Wt Nti − Eti where Wt is the nominal wage, and Eti
is equity base or net worth in nominal terms. The demand for loans in real terms is
i
Eti
bit = wt Nti − Ait where bit ≡ BPtt , wt ≡ W
Pt is the real wage and at ≡ Pt is the equity
t i
base in real terms.

Banks extend credit on demand at the interest rate iL . GS assume that debt is
repaid completely in one period (there is no accumulation of debt). In this case,
nominal debt commitments for the firm in t+1 are (1 + iL ) Bti .
The firm’s profit in nominal terms in t+1 is
Πt+1 = Pt+1 Yt − (1 + iL ) Bt
i i i i
Dividing by Pt+1 we get profit in real terms:
πt+1 = ut+1 Yt − Rbt

i i i i
where R ≡ (1+r) = (1 + iL ) PPt+t 1 is the gross real interest rate, which will turn out to
be equal to the rate of time preference and exogenous. Recalling that bit = wt Nti −Ait
and that Nti = Φ(Yti ) the expression above can be written as:

πt+1 = ut+1 Yt − R wt Φ(Yt ) − At
i i i i i
The relative price being stochastic, also profit is a random variable with expected
value:
E(πt+1 ) = Yt − R[wt Φ(Yt ) − At ]
i i i i
The firm goes bankrupt if πt+1

i
< 0, i.e. if
ut+1 < R[wt Φ(Yt ) − At ]/Yt ≡ ūt+1

i
i i i i
In other words the firm goes bankrupt if the relative price in t+1 happens to be
i
lower than a critical threshold ūt+1 , which coincides with debt service per unit of
output. The probability of bankruptcy is13

R[wt Φ(Yti ) − Ait ]
Pr(ut+1 < ūt+1 ) = G(ūt+1 ) = G , G > 0.
i i i
Yti
In other words, the probability of bankruptcy is increasing with the interest rate
and decreasing with the equity base. It is increasing also in the level of output if
the elasticity of the input requirement to the level of output is greater than one.
Finally, GS assume there are bankruptcy costs which are increasing with the
scale of production:14 CBti = c(Yti ), c > 0.
10.2.2 Perfect capital markets: the first best case

Let’s assume, for the sake of discussion, that firms don’t incur bankruptcy costs.
In this case, the problem of the firm is:
E(πt+1 ) = Yt − R[wt Φ(Yt ) − At ]

i i i i
max
i
Yt
1
Solving the problem yields: Yti = Φ−1 Rw .Therefore Nti = F −1 (Rwt ) . It is easy
to see that for each firm output supply (and labour demand) is a decreasing function
of the real interest rate and the real wage, which are uniform across firms. In
other words employment and production decisions react only to changes of the
real interest rate and the real wage. Heterogeneity of financial conditions, albeit
present, is irrelevant.
Notice that there are two sources of heterogeneity in this framework. The
first one is the random relative price, which can be thought of as an idiosyncratic
shock. This source of heterogeneity is irrelevant in this context when GS consider
the profit maximization problem of the firm. By assumption, in fact, each firm is
risk neutral and its expected relative price is one (expectations are homogeneous
and rational). The second source of heterogeneity is the equity base. Firms can
differ, in fact, according to their degree of financial robustness as measured by
the equity base. In the first best case, however, also this source of heterogeneity is
irrelevant because there are no bankruptcy costs.
In this case, individual agents, albeit heterogeneous, may be modeled as iden-
tical with no loss of generality. Aggregation would yield an aggregate supply and
labour demand functions which keep the basic features of the individual functions.
Let’s now turn briefly to the household sector. The representative infinitely
lived household maximizes the discounted sum of utilities – which are assumed
to be linear in consumption and labour supplied – over an infinite horizon subject
to a budget constraint according to which total wealth consists of labour income
and the return on assets. According to the solution of this dynamic programming
problem:
• in each period, consumption is equal to income (output) of the previous
period;15
• the real interest rate is equal to the (given and constant) rate of time prefer-
ence;
• the supply of labour is an increasing function of the real wage only.
fe fe fe fe
Equilibrium on the labour market yields wt = w (R) and Nt = N (R) where the
fe stands for full employment. As a consequence

fe fe fe
Yt = F N (R) = Y (R)
is full employment output. It is easy to see that full employment output is a de-
creasing function of the real interest rate, provided certain plausible conditions on
the configuration of parameters are met.
The law of motion of the equity base of the i-th firm in real terms can be written
as follows:
At+1 = ut+1 Yt − R wt Φ(Yt ) − At − Dt+1
i i i i i i
(10.1)
where Ait+1 , the equity base in period t+1, is equal to retained profit, which in
i
turn is equal to total profit less dividends Dt+1 . GS assume that dividends are an
increasing function of the equity base Dt+1 = d(Ait ), d > 0.
i
The law of motion of the aggregate equity base can be derived from (10.1)
through aggregation. Assuming that the labour market is in equilibrium, the law
of motion is:

At+1 = Y fe (R) − R wfe (R) N fe (R) − At − d(At ) (10.2)
(10.2) is a non linear difference equation in the state variable At , the aggregate
(average) equity base, parametrized to the real interest rate. The steady state solu-
tion(s) will be generically written as follows: A∗ = A∗ (R). In the steady state the
equity base on average depends negatively upon the real interest rate. Under quite
general assumptions, the steady state is stable.
10.2.3 Imperfect capital markets

If firms run the risk of bankruptcy and bankruptcy is costly, the problem of the
firm must be rewritten as follows:
E(πt+1 ) − CBt Pr(ut+1 < ūt+1 )
i i i i
max
i
Yt
After substitution one gets

R[wt Φ(Yti ) − Ait ]
Yt − R[wt Φ(Yt ) − At ] − c(Yt )G
i i i i
max
i
Yt Yti
Solving the problem yields:
Yt = Y i (Rwt , At ).
i i
Therefore
Nt = Φ Y i (Rwt , At ) = N i (Rwt , At )
i i i
In this case, for each firm output and labour demand are increasing functions of
the equity base, given the real interest rate and the real wage, which are uniform
across firms.
Notice that firms can differ in their production and employment decisions due
to the different degrees of financial robustness as measured by the level of the equity
base. This is the only source of heterogeneity which persists once the problem of
the firm is solved as it is clear from the formulae above. In the first best case, on the
contrary, production and employment decisions were independent from the equity
base.
According to GS, aggregation would yield an aggregate supply function and
a labour demand function which keep the basic features of the corresponding
individual functions. Therefore aggregate (average) output and labour demand are
increasing functions of the aggregate (average) equity base, given the real interest
rate and the real wage, which are uniform across firms. GS would be perfectly
right if the equity base were the same for each and every firm (i.e. if they had
assumed the representative agent framework from the beginning) – in which case
there would not be any difference between the individual and the average relations
– and/or if the individual relations were linear. If financial robustness is different
from one firm to the other, however, and if the Y i (. ) function is non-linear, also
the higher moments of the distribution of the equity base across agents should be
taken into account in the aggregation procedure. In particular, the variance of the
equity base across agents should show up in the aggregate relation.
If we take GS’ aggregation procedure at face value and ignore the higher
moments of the distribution, equilibrium on the labour market, yields wtfe =
wfe (R, At ) , Ntfe = N fe (R, At ). As a consequenceYtfe = Y fe (R, At ) is full employment
output. If we amend GS’ aggregation procedure to take heterogeneity seriously,
however, the variance of the equity base would be an important part of the story, it
should show up in the full employment formulae for real wage, employment and
output and composition effects would be crucial.
The law of motion of the equity base of the i-th firm in real terms has been
derived above (see eq.(10.1)). The law of motion of the aggregate equity base can
be derived from (10.1) through aggregation. Summation and averaging over firms
yields :

At+1 = Y fe (R, At ) − R wfe (R, At ) N fe (R, At ) − At − d(At ) (10.3)
where, once again, we have borrowed the aggregate formulae from GS, ignoring
the variance. (10.3) is a non linear difference equation in the state variable At ,
parameterized to the real interest rate, which can yield different types of dynamics.
The steady state solution(s) will be generically written as follows: A∗ = A∗ (R).In
the steady state the equity base on average depends upon the real interest rate. As
a consequence in the steady state also output and employment depend on the same
variables: N ∗ = N fe (R, A∗ (R)) = N ∗ (R), Y ∗ = Y fe (R, A∗ (R)) = Y ∗ (R).
A monetary shock affects the steady state only if unexpected (GS policy effec-
tiveness proposition).16
If we amend GS’ aggregation procedure to take heterogeneity seriously, how-
ever, the variance of the equity base would be an important part of the dynamic
story too. In fact, (10.3) should revritten as follows:

At+1 = Y fe (R, At , σA2 ) − R wfe R, At , σA2 N fe R, At , σA2 − At − d(At )
where σA2 is the variance of the distribution of the equity base in t.
10.3 Monitoring costs, entrepreneurs’ savings and capital

accumulation
Bernanke and Gertler (1989,1990) (BG hereafter) have put forward a different the-
oretical framework which yields basically the same results as in GS. BG assume
that in a principal-agent relationship between borrowers and lenders, character-
ized by asymmetric information and moral hazard, the latter can assess the return
on investment only incurring monitoring costs. As a consequence, also in their
framework the scale of production depends upon a proxy of borrowers’ net worth,
namely entrepreneurial savings. In fact, the higher is net worth, the lower is the
cost of monitoring firms on the part of banks and the higher the volume of credit
extended, investment and production. The scale of production in the presence of
costly state verification is smaller than in the perfect capital markets case.

BG assume that there are overlapping generations of agents who live for two peri-
ods. In each generation (and in total population, constant by assumption) there is a
constant proportion η of entrepreneurs. An entrepreneur is an agent endowed with
an investment project. Non-entrepreneurs (1 − η of each generation), on the con-
trary, cannot access the investment technology. The distribution of agents according
to their nature (entrepreneurs and non-entrepreneurs) is exogenous (unexplained)
and constant.
There are two types of goods, output and capital. Output can be consumed,
invested – i.e. used up as an input of the investment project – or ‘stored’. If stored,
each unit of output yields a constant return r. In this framework, storage is a
synonym for investment in a risk-free (safe) asset whose return is r. Alternatively,
we can think of stored output as the volume of saving which is lent (loanable funds
supplied) at the exogenous interest rate r.
Output is produced by means of a constant returns to scale technology which
uses capital and labour. The production function in per capita terms is: yt = θf (kt )
where yt is per capita output, θ is a technological shock, f (. ) is a well behaved
production function and kt is per capita capital. Assuming that capital depreciates
completely in one period we can claim that capital ‘tomorrow’ is equal to the flow
of investment carried out ‘today’.
The i-th entrepreneur in t is characterized by a degree of inefficiency ωti , a
random variable distributed uniformly with support [0, 1]. The input requirement
for each investment project in t (xti ) is an increasing function of the degree of
inefficiency: xti = x(ωti ), x > 0, x(0) > 0, x(1) = xmax . The return on investment
– i.e. the quantity of capital generated by each investment project – is a discrete
random variable k̃, with expected value

E k̃ = πb kb + (1 − πb )kg , kb < kg , πb > 1/2.
πb is the probability of the lower outcome of the investment project, i.e. the prob-
ability of the ‘bad’ state of the world. Alternatively, we can think of πb as the
proportion of investment projects undertaken by entrepreneurs which yield the
lower outcome.
The return on investment is known to the entrepreneur. Non-entrepreneurs can
observe the return on investment only incurring monitoring costs equal to γ units
of capital. Therefore, capital in period t+1 is:

kt+1 = E k̃ − hγ It (10.4)
where I is the number of investment projects per capita undertaken, while h is the
proportion of projects monitored.
Each agent is endowed with 1 unit of labour which is supplied inelastically
(there is no disutility of labour). In equilibrium the real wage will be equal to the
marginal productivity of labour: wt = θ [f (kt ) − kt f (kt )].
Preferences are such that entrepreneurs save all their labour income when
young and consume all the return they receive on their project when old. Therefore
the entrepreneur’s saving (when young) is Ste = wt η.

If information were perfect, there would not be monitoring costs: γ = 0. An en-
trepreneur would carry out his investment project in t if the return on investment,
i.e. the product of the price of
capital
in t+1 times the expected real return of
the investment project, qt+1 E k̃ , were greater than or equal to the opportunity
cost of investing, which in turn is equal to the return on storage times the input
requirement of the investment project rxti = rx(ωti ):

qt+1 E k̃ ≥ rx(ωti ) (10.5)
Therefore, we can detect a critical degree of inefficiency:
⎛ ⎞
qt+1 E k̃
ω̄t = x−1 ⎝ ⎠ (10.6)
r
such that all the entrepreneurs with inefficiency lower (higher) than ω̄t will invest
(not invest).
Thanks to the assumption according to which ωti is a random variable distributed
uniformly with support [0, 1], the critical degree of inefficiency ω̄t represents also
the proportion of entrepreneurs who invest. Therefore the distribution of agents
between entrepreneurs and non-entrepreneurs is exogenous while the distribution
of entrepreneurs between investors and non-investors is endogenous. The higher
the return on investment (the lower the return on storage), the greater the share of
entrepreneurs who invest in the population of entrepreneurs.
At this point of the analysis agents can be classified into three groups:
• entrepreneurs who invest, whose share in total population is η ω̄t ,
• entrepreneurs who don’t invest η(1 − ω̄t ),
• non-entrepreneurs (1 − η).
The last two groups do not invest and therefore employ their saving in the storage
technology. In other words they lend their savings (supply loanable funds) to the
entrepreneurs. The distinction between borrowers (entrepreneurs who invest) and
lenders (entrepreneurs who don’t invest and non-entrepreneurs) is at least partially
endogenous.17 In a sense, in this framework, entrepreneurs who are ‘too inefficient’
– i.e. entrepreneurs whose degree of inefficiency is ωti > ω̄t – give up investment
and join the ranks of non-entrepreneurs in lending their funds to entrepreneurs who
are ‘enough efficient’ to invest – i.e. entrepreneurs whose degree of inefficiency is
ω i ≤ ω̄.
The supply of capital is obtained from (10.4) substituting It = η ω̄t and γ = 0.
We obtain:
kt+1 = E k̃ η ω̄t (10.7)
Substituting the definition of ω̄t into (10.7) and solving for qt+1 we get:

rx kt+1 /E k̃ η
qt+1 = (10.8)
E k̃
(10.8) is the

equation of the supply of capital. Assuming, for the sake of simplicity
that E k̃ = 1, (10.8) specializes to:
qt+1 = rx (kt+1 /η) (10.9)

The demand for capital is obtained from the usual condition according to
which the marginal productivity of capital must be equal to the real remuneration
of capital:
qt+1 = θf (kt+1 ) (10.10)
∗ ∗
Equilibrium in the market for capital yields kt+1 = k(θ); qt+1 = q(θ). Per capita
∗
output is y = θf (k(θ)) .
There are no true dynamics in the first best case. Both the price and the quantity
of capital (and therefore output produced) depend in each period only on the
technological parameter.18 For instance an exogenous technological innovation
would yield a once and for all increase in the price and the quantity of capital.
Moreover, the volume of internal finance (entrepreneurial savings) has no role
to play in capital accumulation and production decision. Financial conditions are
irrelevant.
10.3.3 Imperfect capital markets

In the presence of asymmetric information between borrower and lender, the former
has an incentive to lie to the latter, declaring the bad outcome when the good
one has occurred. The lender can ascertain the true return on investment only by
incurring monitoring costs. BG derive the optimal financial contract and show that
auditing/monitoring occurs only if the bad state of the world is declared. They
distinguish between the case of full collateralization and the case of incomplete
collateralization.
For the i-th entrepreneur, full collateralization occurs if:

qt+1 kb ≥ r x(ωti ) − Ste (10.11)
i.e. if the return on investment in the bad state of the world is high enough
to allow the entrepreneur to reimburse its debt. Manipulating (10.11) one realises
that full collateralization occurs if the volume of internal finance is higher than a
threshold which is increasing with the degree of inefficiency:
qt+1 kb
Ste ≥ x(ωti ) − = S(ωti )
r
In this case the probability of auditing is equal to zero: pa = 0.
Incomplete collateralization occurs if:

qt+1 kb < r x(ωti ) − Ste (10.12)
In this case the probability of auditing is positive.

The return on investment in this case is qt+1 E k̃ − πb γpa . An entrepreneur
with inefficiency ωti will invest if

qt+1 E k̃ − πb γpa ≥ rx(ωti ) (10.13)
(10.13) replaces (10.5) as a condition for investment to be carried out.
Entrepreneurs with inefficiency higher than ω̄t as defined in (10.6) will not
invest in the perfect information case. Therefore, they will not invest even in the
imperfect information case, whatever their level of internal finance. They will be
labelled poor entrepreneurs for short. Poor entrepreneurs are lenders.
If the probability of auditing is one, the condition for investment is

qt+1 E k̃ − πb γ ≥ rx(ωti ) (10.14)
Therefore, we can define a critical degree of inefficiency:

⎛ ⎞
qt+1 E k̃ − πb γ
ω t = x−1 ⎝ ⎠ (10.15)
r
Entrepreneurs with inefficiency lower than ω t (good entrepreneurs for short) will
invest even if the probability of auditing is one, whatever their level of internal
finance. Fully collateralized good entrepreneurs will never be monitored. As far as
incompletely collateralized good entrepreneurs are concerned, BG show that they
are monitored with probability

r x(ωti ) − Ste − qt+1 kb
pa = = p(ωti , qt+1 , Ste ); ωti < ω t (10.16)
qt+1 [(1 − πb )(kg − kb ) − πb γ]
The probability of auditing is increasing (decreasing) in the degree of inefficiency

(internal finance).
Entrepreneurs with inefficiency higher than ω t and lower than ω̄t will be la-
belled fair entrepreneurs for short. By assumption a fair entrepreneur will invest
only if he is fully collateralized, that is if his internal finance is higher than the
threshold level of internal finance:
Ste > S(ωti ); ωt < ωti < ω̄t
−
or:
rSte
s(ωti , qt+1 , Ste ) = >1 (10.17)
rx(ωt ) − qt+1 kb
i
At this point of the analysis agents can be classified into six groups:
• fully-collateralized good entrepreneurs: they invest, whatever their level of

internal finance, and will never be monitored;
• incompletely-collateralized good-entrepreneurs: they invest, whatever their
level of internal finance and will be monitored with probability as in (10.16);
• fully-collateralized fair-entrepreneurs: they invest and will never be moni-
tored;
• incompletely-collateralized fair-entrepreneurs: they will not invest;
• poor entrepreneurs;
• non-entrepreneurs.
The taxonomy of agents in the imperfect capital markets case is much richer than
in the perfect capital markets case. The first three groups consists of entrepreneurs
who invest. Good entrepreneurs, whose share in total entrepreneurial population
is ω t , will always invest but the return on investment should be diminished to take
into account monitoring costs if the good entrepreneur is not fully collateralized.
The probability that a good entrepreneur is not fully collateralized is
ωt
−
p(ωti , qt+1 , Ste )dωti .
0
As to fair entrepreneurs, by assumption only fully collateralized entrepreneurs

invest. The share of fully collateralized fair entrepreneurs in total entrepreneurial
population is
ω̄t
s(ωti , qt+1 , Ste )dωti .
ωt
The last three groups do not invest and therefore employ their saving supplying
loanable funds to the entrepreneurs.
Therefore, the supply of capital will be:
⎡ ⎤
ωt
kt+1 = E k̃ η ⎣ω t − πb γ p(ωti , qt+1 , Ste )dωti ⎦ +
0
⎡ ⎤
ω̄t
⎢ ⎥
+E k̃ η ⎣ s(ωti , qt+1 , Ste )dωti ⎦ (10.18)
ωt

Imposing E k̃ = 1 and rearranging we get:
⎡ ⎤
ωt ω̄t
⎢ ⎥
kt+1 = ω̄t − ⎣πb γ p(ωti , qt+1 , Ste )dωti + [1 − s(ωti , qt+1 , Ste )]dωti ⎦ η (10.19)
0 ωt
The supply of capital in the imperfect capital markets case depends not only on the
price of capital as in the first best case but also on the volume of entrepreneurial
savings.
The demand for capital is obtained from the usual condition according to
which the marginal productivity of capital must be equal to the real remuneration
of capital (see (10.10) above).
Equilibrium in the market for capital yields19 :
∗ ∗
kt+1 = k(θ, Ste ); qt+1 = q(θ, Ste )
In the imperfect capital markets case, therefore, the price and the quantity of capital
are influenced not only by the state of technology as in the first best case but also
by the financial conditions of investing entrepreneurs captured by their savings.
For instance an increase in entrepreneurial savings would produce an increase in
the quantity of capital and a decrease of its price.
We recall that entrepreneurial savings in period t are equal to Ste = wt η and that
wt = θ [f (kt ) − kt f (kt )]. Therefore, the equilibrium stock of capital in period t+1
will be:
kt+1 = k (θ, ηθ [f (kt ) − kt f (kt )]) (10.20)
(10.20) is a generally non-linear first order difference equation in the stock of
capital. In this case, there are true dynamics and the volume of internal finance
(entrepreneurial savings, i.e. the wage bill of entrepreneurs who invest) has an
important role to play. The steady state of (10.20) will be k ∗ = k ∗ (θ).
10.4 Moral hazard, financing constraints and collateralizable wealth

Kiyotaki and Moore (1997) (KM hereafter) assume that in a principal-agent re-
lationship between borrowers and lenders, characterized by asymmetric informa-
tion and moral hazard, borrowers face a financing constraint: the loans they get
are smaller or equal to the value of their collateralizable assets, which plays, in
this framework a role analogous to that of net worth or the equity base in GS and
entrepreneurs’ savings (internal finance) in BG. As a consequence, also in their
framework production depends upon ‘net worth’. In fact, the higher is net worth,
the higher is the volume of credit extended, investment and production. Production
in the presence of this financing constraint is smaller than in the case in which the
firm does not face financing constraints, i.e. the perfect capital markets case.

KM assume that there are infinitely lived agents. In total population, there is a con-
stant proportion of financially constrained agents (‘farmers’). Farmers are agents
endowed with inalienable human capital. Therefore, they can get from lenders
no more than the value of their collateralizable assets. This is the reason of the
financing constraints.20
Notice that the financing constraint here is binding. Non-farmers (‘gatherers’),
on the contrary, are not endowed with inalienable human capital. Therefore they
do not face financing constraints. It will turn out that all the farmers (gatherers)
will be borrowers (lenders). The distribution of agents according to their nature
(financially constrained farmers/borrowers and unconstrained gatherers/lenders)
therefore, is exogenous and constant.
There are two types of goods, output (‘fruit’) and a collateralizable, durable,
non-reproducible asset (‘land’) whose total supply is fixed (K̄). Output can be
consumed or lent. If lent, each unit of output yields a constant return R = 1 + r.
Output is produced by means of a technology which uses land and labour.
By assumption farmers and gatherers have access to different technologies.
The production function of each financially constrained agent (farmer) is:
ytF = (α + c̄)kt−1
F
where ytF is output of the farmer in t, α,c̄ are positive tech-
F F
nological parameters and kt−1 is land of the farmer in t-1. c̄ kt−1 is the output
which deteriorates (‘bruised fruit’) and is therefore non-tradable.
Each farmer’s technology is idiosyncratic in the sense that once production
has started only the farmer has the skills to successfully complete the production
process, i.e. to make land bear fruit. If the farmers withdrew their labour, production
would not be carried out, i.e. land would bear no fruit. In the words of Hart and
Moore (1994), the farmer’s human capital is inalienable. As a consequence, if the
farmers are indebted, they may have an incentive to threaten their creditors to
withdraw their labour and repudiate her debt. Creditors protect themselves against
this threat by collateralizing the farmer’s land. This is the reason why farmers face
a financing constraint:
qt+1 ktF
bFt = (10.21)
R
According to (10.21), the maximum amount of debt a farmer succeeds to get ‘today’
bFt is such that the sum of principal and interest RbFt is equal to the value of the
farmer’s land when the debt is due, i.e. qt+1 ktF where qt+1 is the price of land at
time t+1.
Farmers face also a flow-of-funds constraint:
ytF + bFt = qt (ktF − kt−1

F
) + RbFt−1 + ctF (10.22)
where ctF is the farmer’s consumption. Substituting (10.21) into (10.22) we get the
budget constraint:
ctF = (α + c̄)kt−1
F
− µt ktF (10.23)
where µt = qt − qt+R 1 .
Preferences are such that farmers consume only non-tradable output, i.e. ctF =
c̄ kt−1 . The farmer’s demand for land, therefore, is:
F
1 α F
ktF = F
(α + qt )kt−1 − RbFt−1 = kt−1 (10.24)
µt µt
Substituting (10.24) into (10.21), we obtain:
qt+1 1
bFt = F
(α + qt )kt−1 − RbFt−1 (10.25)
R µt
G
The production function of each gatherer is: ytG = f (kt−1 ) where ytG is output of
G
the gatherer in t, f (. ) is a well behaved production function and kt−1 is land of
the gatherer in t-1. The gatherers’ human capital is not inalienable. Therefore,
gatherers face only a flow-of-funds constraint:
t−1 = qt (kt − kt−1 ) + bt + ct

ytG + RbG G G G G
(10.26)
Substituting the production function of gatherers and the financing constraint

of farmers into (10.26) and assuming, for the sake of simplicity and without loss
of generality, that population consists only of one farmer and one gatherer so that
ktF = K̄ − ktG we get the budget constraint:

ctG = f (kt−1
G
) + µt K̄ − ktG (10.27)
Preferences of the gatherer are such that Rµt = f (ktG ). The demand for land,
therefore, is:
−1
ktG = f (Rµt ) (10.28)

In the absence of asymmetric information, i.e. in the perfect capital markets case,
there would not be any financing constraints. In equilibrium, the marginal produc-
tivity of land for the farmer should be equal to the marginal productivity of land
for the gatherer, i.e.
f (ktG ) = α + c̄ (10.29)
From (10.29) it follows that the equilibrium quantity of land for the farmer and
the gatherer are:
ktG = f −1 (α + c̄)

ktF = K̄ − f −1 (α + c̄)
Moreover, from (10.28) it follows that

f f −1 (α + c̄)
µt =
R
There are no true dynamics in the first best case. The quantity of land allocated to
(and therefore output produced by) the farmer and the gatherer respectively depend
in each period only on the parameters which characterize the production functions
of the two types of agents.
Moreover, the value of collateralizable wealth has no role to play in the process
of allocation of land and in production decisions. Once again, in the perfect capital
markets case, financial conditions are irrelevant.
10.4.3 Imperfect capital markets and dynamics

From (10.28) we know that the following must be true:
−1
K̄ − ktF = f (Rµt ) (10.30)
Substituting this expression into (10.24) and rearranging we end up with the fol-
lowing:
Rα
ktF = kF (10.31)
f (K̄ − ktF ) t−1
(10.31) is a non-linear difference equation in the state variable ktF .
In the steady state ktF = kt−1
F
= k ∗ Therefore µ∗ = a. Moreover qt = qt+1 = q∗
k∗
so that q = α R−1 . Finally bt = bFt−1 = b∗ so that b∗ = α R−1
∗ R F
. From the steady
−1
∗ ∗
state condition kt = kt−1 = K̄ − k follows K̄ − k = f (Rα). As a consequence:
G G
−1
k ∗ = K̄ − f (Rα) .
In this setting KM show that small shocks – for instance to technology – can
produce large and persistent fluctuations in output and asset prices. In their model,
in fact, the durable, non-reproducible asset (land) plays the dual role of a factor of
production for both constrained and unconstrained agents and of collateralizable
wealth for financially constrained agents. Therefore the price of assets affects the
borrowers’ financing constraint. At the same time, the size of the borrowers’ credit
limits feeds back on asset prices. ‘The dynamic interaction between credit limits
and asset prices turns out to be a powerful transmission mechanism by which the
effects of shocks persist, amplify and spread out’ (Kiyotaki and Moore, 1997:212).
10.5 Capital market imperfections, distribution and growth

In this section we briefly present and discuss the papers by Galor and Zeira (1993)
and Aghion and Bolton (1997)21 in which the authors describe the evolution over
time of the distribution of wealth among individuals in the presence of capital mar-
ket imperfections. Changes in the distribution of wealth, in turn, affect production
decisions and economic growth.
Galor and Zeira (1993) (GZ hereafter) assume there are overlapping genera-
tions of agents who live for two periods. In each generation (and in total population,
constant by assumption) there is a proportion of agents who are skilled (unskilled).
An unskilled agent is endowed with a constant returns to scale technology to pro-
duce output (Y U ) by means of labour (N U )in both periods of life. The production
function of the unskilled is Y U = wu N U where wu is the real wage of the un-
skilled worker. A skilled agent invests in human capital (and does not work) when
young to acquire a higher degree of efficiency at work when old. The amount of
investment in human capital is fixed (H ). The production function of the skilled
is Y S = f (N S , K) where K is the capital stock, f (. ) is a well behaved production
function, ws is the real wage of the skilled worker, ws > wu . The distribution of
agents according to their nature (skilled and unskilled) is endogenous, as we will
see.
Each agent has one parent, whom she receives a bequest from, and one child,
whom she leaves a bequest to. Preferences are defined over consumption in the
second period of life (for the sake of simplicity, young agents do not consume)
and bequest to the child. Each agent’s wealth is the sum of labour income in the
first period of life and initial wealth, i.e. bequest received from the parent.
Capital markets are imperfect. Borrowers can evade debt payments by ‘mov-
ing to other places’ and lenders can avoid this occurrence by ‘keeping track’ of
borrowers. This is a moral hazard problem in a principal-agent relationship that
the principal can deal with by incurring monitoring costs. In other words the basic
setting is reminiscent of BG’s. The actual interest rate on loans (ρ) is a mark up
on the exogenous ‘basic’ interest rate (r) which GZ think of as the ‘world rate of
interest’.
There are three types of individuals in this economy:
• unskilled workers who are lenders;
• skilled workers who inherit less than the value of the investment in human
capital. They are borrowers;
• skilled workers who inherit more than the investment in human capital. They
are lenders.
GZ assume that preferences are homogeneous across agents. Whatever her type,
each agent has a linear logarithmic utility function defined over consumption (when
old) and bequest.
From utility maximization follows that the (optimal) bequest each agent leaves
to his child at time t – which will be the initial wealth of the newborn in t+1 (say
Zt+1 ) – is a fraction (1 − ζ) of her wealth, which in turn is the sum of labour income
in the first period of life and initial wealth.
The definition of agents’s wealth however is not uniform across agents. There-
fore the bequest received from her parent by each child can be described by the
following laws of motion:
Zt+1 = (1 − ζ) [(1 + r) (Zt + wu ) + wu ] (10.32)
Zt+1 = (1 − ζ) [−(1 + ρ) (H − Zt ) + ws ] ; Zt < H (10.33)
Zt+1 = (1 − ζ) [(1 + r) (Zt − H ) + ws ] ; Zt > H (10.34)

(10. 32) is the bequest received by the child of an unskilled worker. The unskilled
worker’s wealth is the sum of labour income when old and the gross return on her
saving when young, which in turn is the sum of initial wealth and labour income
when young augmented by the flow of interest on loans. (10.33) is the bequest
received by the child of a skilled worker who has to borrow in order to carry
out her investment in human capital. (10.34) is the bequest received by the child
of a skilled worker who lends the difference between her initial wealth and her
investment in human capital.
The system (10.32), (10.33), (10.34) generates a piecewise linear phase dia-
gram with two steady states, Zu∗ and Zs∗ , Zu∗ < Zs∗ , which are both stable. Let’s
assume that there is an exogenously given initial distribution of wealth across
individuals. For the sake of simplicity, we can assume that individual wealth is
distributed uniformly with support [0, Zmax ] .If an individual has initial wealth
smaller (greater) than a critical degree of wealth Z̄, she leaves a bequest to her
child smaller (greater) than her initial wealth and her child would do the same, so
that with the passing of time the individual wealth of her descendants will converge
to Zu∗ ( Zs∗ ). The distribution of wealth, in other words, will converge to a stationary
distribution in which a portion
population(Z̄/Zmax ) has wealth equal to
of total
Zu while the remaining 1 − Zm a x has wealth equal to Zs∗ . Per capita wealth is
∗ Z̄
therefore:
Z̄
z = Zs∗ − (Z ∗ − Zu∗ )
Zmax s
This is a simple theory of the persistence of heterogeneity. If the dynamics of
the distribution were such as to converge to a stationary distribution in which only
one type of agents survive, we would have had the restoration of the representative
agent. This is what would happen in case of perfect capital markets.
Aghion and Bolton (1997) (AB hereafter) assume that agents live for one period
during which they work and invest. Income earned by working and investing is
divided between consumption and bequests. Each agent has one parent, whom she
receives a bequest from, and one child, whom she leaves a bequest to. Each agent’s
wealth is equal to the bequest received from the parent. The distribution of wealth
endowments will be represented by G(Z).
Each agent is endowed with one unit of labour which she supplies inelastically
(there is no disutility of labour). She can use her unit of labour to work on a
‘backyard activity’ which yields r̄ with certainty or invest in an ‘entrepreneurial
activity’ which yields r with probability pr or 0 with probability (1 − pr ).
In order to invest, the agent must commit one unit of wealth. If the agent doesn’t
invest, she can employ her wealth in an economy-wide ‘mutual fund’.
Capital markets are imperfect. The borrower–lender relationship is character-
ized by a moral hazard problem with limited wealth constraints. In this setting
AB show that, due to the features of the optimal lending contract, the probability
of success pr and the interest rate charged on loans ρ are functions of individual
wealth, with ∂p ∂ρ
∂Z > 0, ∂Z < 0.
r
In equilibrium the expected return on loans must be uniform across borrowers

and equal to the interest rate of the mutual fund rm , i.e. rm = pr (Z) ρ (Z)
An agent endowed with less than one unit of wealth will borrow the amount
(1 − Z) and invest (alternatively: carry on her backyard activity and lend) if the
expected net return on investment is greater (smaller) than the opportunity cost of
investing, which in turn is equal to the sum of the return on the mutual fund and the
return on the backyard activity. This condition entails a critical level of individual
wealth Z̃, which is an increasing function of the return on the mutual fund, such that
those individuals with initial wealth Z > Z̃ (rm ) (alternatively: Z < Z̃ (rm )) prefer
to borrow and invest (work in the backyard and lend). The demand for loans,
1
therefore, will be Bd := Z̃(rm ) (1 − Z)dG(Z) = Bd (rm ) . The supply of loans, on
the other hand, is the sum of resources made available by poor individuals who
prefer to lend and by rich individuals who hold wealth in excess of the input
requirement of investment:
Z̃(rm ) ∞
s
B := Z dG(Z) + (Z − 1) dG(Z) = Bs (rm )
0 1
Equilibrium on the credit market, whenever it exists, yields the equilibrium

level of the interest rate, say rm∗ ,which depends only on the features of the distri-
bution function, and the equilibrium distribution of agents into three groups:22
• fully-collateralized (rich) entrepreneurs who invest and lend the excess of

their wealth over the input requirement of investment;
• incompletely-collateralized (poor) entrepreneurs who invest and borrow the
excess of the input requirement of investment over their wealth;
• poor non-entrepreneurs who work in the backyard and lend.
The nature of borrowers and lenders, in other words, is endogenously determined

as in BG and GZ.
From utility maximization follows that the (optimal) bequest each agent leaves
to her child at time t – which will be the initial wealth of the newborn in t+1 (say
Zt+1 ) – is a fraction (1 − ζ) of her wealth, which in turn depends on the type of
activity she has carried out and her initial wealth.
The bequest received by each child from her parent when the latter is a poor
non-entrepreneur (Z < Z̃ (rm ))can be described by the following law of motion:
Zt+1 = (1 − ζ) (rm∗ Zt + r̄)
When the parent is a rich entrepreneur, the law of motion is:
Zt+1 = (1 − ζ) [r + rm∗ (Zt − 1)] (10.35)

if the investment project yields the high return r, with probability pr = pmax where
pmax is the maximum probability of success.
On the other hand, if the investment project yields the low return 0, with
probability 1 − pmax , the law of motion is:
Zt+1 = (1 − ζ) [rm∗ (Zt − 1)] (10.36)

Finally, when the parent is a poor entrepreneur, the law of motion is:
Zt+1 = (1 − ζ) [r − ρ (Zt ) (1 − Zt )] (10.37)

if the investment project yields the high return r, with probability pr = pr (Zt ).
On the other hand, if the investment project yields the low return 0, with
probability 1 − pr (Zt ) , the law of motion is:
Zt+1 = 0 (10.38)
According to AB the economy will grow until all investment opportunities are
exploited. In this case the equilibrium interest rate on the mutual fund is rm∗ = 1.
Applying the results on convergence for monotonic Markov processes AB show
that the distribution of wealth converges to a stationary (long run) distribution.
As in GZ, AB have put forward a simple theory of the persistence of hetero-
geneity in wealth endowments in an imperfect capital markets framework.
10.6 The evolution and the persistence of heterogeneity: a proposal

In the search for a satisfactory way to model the evolution and persistence of
heterogeneity over time and its role in the transmission and amplification of shocks,
we have put forward a simple model (Delli Gatti, Gallegati and Palestrini, 2000)
with imperfect capital markets along the lines of Greenwald and Stiglitz.
Our economy is characterized by a large number (say z) of firms. Each firm
produces a homogenous good by means of a constant returns to scale technology
in which capital is the only input:
Yti = νKti (10.39)
where Yti and Kti are output and capital of the i-th firm in period t, ν is the output-
capital ratio, uniform across firms.
Firms sell their output at an uncertain price because of their limited knowledge
of market conditions. In order to capture uncertainty, we model the individual
selling price Pti as: Pti = uti Pt where uti is a random idiosyncratic shock iand
Pt
is the average market
price, uniform across firms. We assume that E u t = 1.
Therefore E Pti = Pt .We can also interpret the random shock as the relative
price: uti = Pti /Pt .
Moreover we assume that firms cannot raise funds on the Stock market because
of equity rationing but they have unlimited access to credit. This means that firms
do not issue new equities but can obtain from banks all the credit they need to
finance production at the (exogenous) rate of interest, r, uniform across firms and
time invariant.
Firms differ according to their financial conditions. The financial robustness
of a firm is proxied by the equity ratio, i.e. the ratio of its equity base to capital
ait = Ait /Kti .
Each firm incurs financing costs CFti equal to debt commitments:
CFti = r(qKti − Ait−1 ) (10.40)
where q is the real price of capital and Ait−1 is the net worth or equity base in real
terms inherited from the past.
Assuming that there is no depreciation, capital accumulates according to the
investment equation Iti = Kti − Kt−1 i
.
We assume that the firm incurs capital adjustment costs, CAit which are increas-
ing in the ratio Iti /Kt of its investment Iti to the average capital stock Kt . We make
the additional technical assumption that the adjustment costs function is quadratic:
2 2
γ Iti γ Kti − Kt−1
i
CAit = = (10.41)
2 Kt 2 Kt
Quadratic adjustment costs are well known in the literature on investment. The
novelty of the expression above consists in assuming that adjustment costs are
decreasing with the average capital stock: the higher is the average capital stock
– i.e. investment activity on the part of other firms – the lower is the level of
adjustment costs for the individual firm. This formulation captures an externality
in investment activity.
Real profit is the difference between real revenue and real cost, which in turn
is the sum of financing and adjustment costs:
πti = uti Yti − CFti − CAit =
2
γ Kti − Kt−1
i
= uti Yti − r(qKti − Ait−1 ) −
2 Kt
In this framework, bankruptcy occurs if net worth becomes negative. Net worth
‘today’ is equal to net worth ‘yesterday’ plus retained profit, which in turn is equal
to profit less the flow of dividends (Dti ). The bankruptcy condition therefore is:
Ait = Ait−1 + πti − Dti = Ait−1 + ui Yti − CFti − CAit − Dti < 0 (10.42)
The inequality (10.42) is verified if the sum of financing costs, adjustment costs and
dividends is higher than revenues – thereby generating a loss – and the associated
loss is higher than the equity base inherited from the past. In order to simplify the
argument, in the following we will assume that the flow of dividends is proportional
to net worth inherited from the past:
Dti = rAit−1 (10.43)
In a sense this is tantamount to assuming the net worth is remunerated at the same
rate as bank loans.
Substituting(10.40)(10.41)(10.43) into (10.42) we get:
2
γ Kti − Kt−1
i
At−1 + ut Yt − r(qKt − At−1 ) −
i i i i i
− rAit−1 < 0
2 Kt
Using (10.39) and rearranging, we can write the bankruptcy condition as follows:
2
q Ait−1 γ Kti − Kt−1 i
ut < r −
i
+ (10.44)
ν νKti 2 νKti Kt
In the following, for the sake of analytical convenience, we will adopt a sim-
plified version of the bankruptcy condition, namely:
q Ai
ui < r − t−1i ≡ ūti (10.45)
ν νKt
The condition (10.45) essentially ignores capital adjustment costs. This assumption
makes computations much less cumbersome without loss of generality. According
to the bankruptcy condition, bankruptcy occurs if the realization of the random
relative price uti falls below a critical threshold ūti which in turn is a function, among
other variables, of the capital stock and of the equity base lagged one period.
Let’s assume, for the sake of simplicity, that uti is a uniform random variable,
with support (0,2). In this case, the probability of bankruptcy is:

i ūti 1 q Ait−1
Pr ut < ūt = =
i
r −
2 2 ν νKti
Following Greenwald and Stiglitz, moreover, we assume that bankruptcy is

costly (Gordon and Malkiel, 1981; Altman, 1984; Gilson, 1990; Kaplan and
Reishus, 1990). In particular, bankruptcy costs, CBti are a decreasing function
of the degree of financial robustness, proxied by the equity ratio of the previous
period and an increasing function of output:

CBti = α1 − α2 ait−1 Yti (10.46)
where α1 and α2 are positive parameters.

In this setting, the problem of the firm is:
max E(πti ) − CBti Pr(uti < ūti )
After substitution, we can reformulate the firm’s problem as:

2
γ Kti − Kt−1
i
max νKti − r i Kti − +T
i
Kt 2 Kt
where α1 α2 i
r i = rq 1 + − at−1
2 2
α1 α2 i
i
T = At−1 r + − at−1
2 2
r i is the bankruptcy cost augmented interest rate. It is determined as a mark-up
over the interest rate charged by banks on loans. This mark-up is a decreasing
function of the equity ratio: the higher the equity ratio, i.e. the financial robustness
of the firm, the lower the bankruptcy cost augmented interest rate. In a sense, this
mark-up captures the idea of the risk of the borrower. Solving the problem yields:
Iti K i − Kt−1
i
ν − ri
= t = = T0 + T1 ait−1
Kt Kt γ
q α1
where
ν
T0 ≡ −r 1+
γ γ 2
rqα2
T1 ≡
γ2
Since the issue of new equities is ruled out by assumption, in this framework each
firm can increase its net worth inasmuch as it accumulates internal funds. The
change of the equity base, therefore, coincides with retained profits.
The law of motion of the equity base of the i-th firm is:
2
γ Kti − Kt−1
i
Ait = Ait−1 + uti Yti − rqKti − (10.47)
2 Kt
Dividing by the individual capital stock, we can derive the law of motion of the
equity ratio:
2
i
Kt−1 γ Kti − Kt−1i
at = at−1 i + ut ν − rq −
i i i
Kt 2 Kt Kti
i
The expression Kt−1 /Kti can be written as:
i
Kt−1 I i Kt
i = 1− t i
Kt Kt K t
In order to make the analysis manageable, we assume that the ratio Kti ∼
= 1. Thanks
Kt
to this simplifying shortcut we get:
i
Kt−1 Iti
= 1 − = 1 − T0 + T1 ait−1
Kti Kt
Moreover, thanks to the assumption Kti ∼

= 1 the expression
Kt
2 i 2
Kti − Kt−1
i
It
=
Kt Kti Kt Kti
boils down to: 2

Iti 2
= T0 + T1 ait−1
Kt
As a consequence, after substitution, the law of motion of the individual equity
ratio becomes:
γ 2
ait = ait−1 1 − T0 + T1 ait−1 + uti ν − rq − T0 + T1 ait−1
2
After some algebraic manipulation, we get:
at = Γ1 at−1 − Γ2 a2t−1 + Γi0 (10.48)
where
Γ1 = 1 − T0 (1 + γT1 )
γ
Γ 2 = T1 1 + T 1
2
γ
Γi0 = uti ν − rq − (T0 )2
2
From (10.48) through aggregation, assuming that the law of large numbers holds
true, we obtain the law of motion of the average equity ratio:
at = Γ1 at−1 − Γ2 a2t−1 + Γ0 − Γ2 Vt−1 (10.49)
where Vt−1 is the variance of the distribution of the equity ratio in t-1. From the
definition of the variance at time t, we derive the following:
Vt = Γ22 (βt−1 − 1)Vt−1

2
+ 2Γ1 Γ0 µ3t−1 + 4Γ22 a2t−1 Vt−1
−4Γ1 Γ2 at−1 Vt−1 + Γ21 Vt−1 + 4Γ22 µ3t−1 at−1
where βt−1 is a parameter capturing the kurtosis of the distribution, while µ3t−1 is
the third moment from the mean.
(10.49) and (10.50) is a system of two non-linear difference equations in the
state variables at and Vt which describes the evolution over time of the first two mo-
ments of the distribution of the equity ratio. This dynamical system is described by
a two-dimensional non-linear map which yields multiple equilibria (steady states),
with different dynamical properties depending upon the chosen configuration of
parameters.23
The map generates a wide range of dynamic patterns: convergence to a sta-
tionary distribution, periodic or aperiodic cycles, chaotic dynamics, divergence. To
each dynamic pattern of the distribution corresponds a dynamic pattern of aggre-
gate production and investment (capital accumulation). Therefore, the framework
can generate convergence to a steady growth path, endogenous regular or irreg-
ular business cycles and growth – which we label fluctuating growth for short –
divergent trajectories, which we label ‘ financial crises’ .24
A fluctuation can be generated also by an exogenous stochastic shock forced
upon the system when it is in a steady state position. In particular, elsewhere we
have shown that the higher the degree of heterogeneity, the larger the effects of a
shock and the longer its persistence.
Heterogeneity is very important in business cycle analysis for empirical as well
as theoretical reasons (Kirman, 1999). When firms are heterogeneous, knowledge
about the distribution of firms is crucial in order to understand the response of the
system to aggregate and idiosyncratic shocks.
10.7 Conclusion
Heterogeneity is a catch word which can mean different things to different people
in different contexts. We have to refine the notion in order to draw meaningful
conclusions on its importance in macroeconomics. In this section we will try such
a refinement with the help of the following table, which summarizes the main
features of the different theoretical frameworks examined so far. The columns are
Greenwald Bernanke Kiyotaki Galor Aghion

Stiglitz Gertler Moore Zeira Bolton
Borrowers Entrepreneurs Financially Skilled non- Middle class

Firms who invest constrained wealthy investors
agents agents
(farmers)
Agents
Non- Non-
Households entrepreneurs constrained Skilled
Banks Entrepreneurs agents wealthy agents Poor workers
Lenders who donít invest (gatherers) Unskilled agents Rich investors
(Consumption) (Consumption)
(Consumption) (Consumption) (Consumption)
goods goods
goods goods goods
Markets Labour
Capital Non-reproducible
Labour Labour
Labour assets (land)
Credit Credit Credit
Credit Credit
Asymmetric Asymmetric
Asymmetric information on information on Asymmetric
Asymmetric information on effort effort information on
Capital market
information on investment return ¯ ¯ effort
imperfections equity market ¯ Moral hazard Moral hazard ¯
Moral Hazard (inalienable human (take money and Moral hazard
capital) run)
¯ ¯ ¯
Implication of Equity rationing Monitoring costs Financing
Financing (costly state constraint: External financial Possibility of credit
capital market hierarchy verification) credit= collateral premium rationing
imperfections Bankruptcy risk if auditing occurs wealth
First best No true dynamics No true dynamics No true dynamics

Convergence
Dynamics
Convergence Convergence
Second best Wide range of Wide range of to stationary to stationary
Complex dynamics possible dynamics possible dynamics wealth distribution wealth distribution
Endogenous Investors Skilled workers Investors
Distribution of Non-investors Unskilled Non-investors
net worth workers
Distri-
Heterogeneity bution
of firms,
households,
banks Entrepreneurs Farmers
Exogenous Non-entrepreneurs Gatherers
Figure 10.1 Main features of theoretical frameworks examined
entitled to the models presented, while the rows are devoted to a classification of
agents (grouped in the two broad categories of borrowers and lenders), markets,
type and implications of capital market imperfections, type of dynamics, nature of
heterogeneity.
First of all, we can draw a distinction between weak and strong heterogeneity
(Rios-Rull, 1995; Gaffeo, 1999). Heterogeneity is strong if agents, differentiated
according to some economically relevant characteristics, interact strategically. In
this case the benchmark theoretical framework is game theory. Heterogeneity is
weak if agents, albeit different, are not aware of – or simply do not take into
account – the fact that the decision taken by each one of them affects the payoff of
the others. The models surveyed so far explore the macroeconomic implications
of weak forms of heterogeneity. In other words none of the models reviewed in
this chapter is embedded in a game theoretical framework.
A second distinction can be drawn between exogenous and endogenous het-
erogeneity. Heterogeneity is exogenous if the distribution of agents according to
some economically relevant characteristics is postulated by the modeler, it is en-
dogenous if the distribution if determined in the model. An example can help in
clarifying this distinction.
A fundamental differentiation of agents in models with imperfect capital mar-
kets is the distinction between borrowers and lenders. GS simply assume that
distinction from the beginning of the analysis. They start, in fact, from the fact that
some agents specialize in lending funds to other agents who specialize in borrow-
ing; they don’t bother to assess which characteristics of the agents brought about
that specialization. In a sense, the same can be said of KM. In their framework,
the distribution between financially constrained and unconstrained agents is ex-
ogenous. It turns out that the former are borrowers and the latter are lenders. In
this sense also the distribution between borrowers and lenders is exogenous.
On the contrary BG exogenously assume the distribution of agents between
entrepreneurs and non-entrepreneurs but are able to derive the distribution of
entrepreneurs in investors and non-investors endogenously. It turns out that en-
trepreneurs characterized by a degree of inefficiency lower than a certain – en-
dogenously determined – threshold level of inefficiency carry out their investment
projects and therefore are seeking funds to finance investment activity, i.e. they
specialize in investing and borrowing, while entrepreneurs with a degree of ineffi-
ciency higher than the above mentioned threshold give up investment and ‘store’.
In other words, inefficient entrepreneurs and non-entrepreneurs specialize in lend-
ing. In a sense, the distinction between borrowers and lenders is partly endogenous
due to the fact that the distinction between efficient and inefficient entrepreneurs
is endogenous. Similarly, AB endogenously determine the specialization of agents
as borrowers or lenders. GZ endogenously determine the (long run or stationary)
distribution between skilled and unskilled agents.
There is a strong correlation between the degree of capital market imperfec-
tions, the relevance of heterogeneity for macroeconomic performance, the relative
complexity of the dynamics of the state variables involved in the analysis and the
response of individuals and the aggregate to impulses.
Generally speaking, in the models surveyed so far, we can distinguish between
the general case of imperfect capital markets – imperfections being generally
informational in nature – and the special, simpler case of perfect capital markets.
For instance, if we relax the assumption of equity rationing and bankruptcy costs
in the GS model we go back to a perfect capital markets world in which the
financial conditions of the individual firm do not matter as far as employment
and production decisions are concerned. If we relax the assumption of monitoring
cost in the BG model we go back to a perfect capital markets world in which the
financial conditions of the individual entrepreneurs do not matter as far as capital
accumulation is concerned.
In all these cases, the representative agent assumption is a legitimate working
hypothesis. If employment and production decisions in the GS world or capital
accumulation decisions in the BG world do not depend on individual financial
conditions, but only on characteristics of the economy (for instance: technology)
which are uniform across firms, heterogeneity is irrelevant and the Modigliani–
Miller rule is reinstated. This is the first best case. It turns out that in this case the
dynamics of the state variables under examination are relatively simple to analyze
and determine a stable steady state (this is the GS’ first best case) or the dynamics
are trivial because . . . there are no dynamics, i.e. no feedback of the value of the
state variable in one period to the values the variable will assume in the future (this
is BGs first best case, for instance).
A slightly different story holds true for the KM framework, in which farmers
and gatherers are differentiated not only because of the presence or absence of a
financial constraint, but also because different agents can access different tech-
nologies. In particular farmers produce by means of a constant marginal returns
technology while gatherers are characterized by a well behaved production func-
tion with decreasing marginal returns. This second distinction is inessential as to
the deep characterization of the two groups of agents. In other words, what is re-
ally important in the model is the financial distinction and not the technological
one. There is no reason, moreover, to link the nature of financially constrained
(unconstrained) agent to the type of technology available.
The technological distinction, however, enables us to define in a peculiar and
interesting way the perfect capital markets case. The special case of perfect capital
markets in KM framework is obtained removing the crucial assumption of finan-
cial constraint but not the technological one. The first best is achieved when the
marginal product of the durable non-reproducible asset (land) for the farmer, which
is a given constant thanks to the constant marginal returns assumption, is equal to
the marginal product of land for the gatherer, which is decreasing with the amount
of land used in production by the gatherer thanks to the decreasing marginal returns
assumption. This is tantamount to assuming a no-arbitrage condition.
Thanks to the technological differentiation, from the no-arbitrage condition
KM can derive the amount of land allocated to each group of agents and the
equilibrium values of the other variables without recurring to any laws of mo-
tion. In other words, in KMs framework, the first best case is characterized by
(technological) heterogeneity but there are no dynamics. As we said before, the
technological heterogeneity which persists in KM even when the financial con-
straints are removed is inessential as far as the macroeconomic implications of
capital market imperfections are considered but they are necessary to derive uni-
vocally the allocation of the durable assets. If technology were uniform across
agents the no-arbitrage condition would have left the allocation of land to farmers
and gatherers undetermined.
Summing up, the perfect capital markets – Modigliani–Miller – first best is
a special case characterized by the absence or irrelevance of heterogeneity (in
our case, heterogeneity of financial conditions) which makes the representative
agent assumption legitimate and the dynamics simple or even trivial. In this case
shocks typically have symmetric effects. This is the consequence of homogeneity
of employment, production and/or capital accumulation responses to exogenous
shocks.
Symmetrically, in the imperfect capital markets case heterogeneity (in our
case, heterogeneity of financial conditions) is essential, it invalidates the adoption
of the representative agent assumption and generally determines a more interesting,
sometimes complex dynamics. In this case shocks typically have asymmetric ef-
fects. This is the consequence of heterogeneity of employment, production and/or
capital accumulation responses to exogenous shocks.
We now turn to aggregation. We can distinguish two procedures in the models
discussed so far. In GS there is an initial continuous distribution of agents according
to their level of equity base or net worth. This distribution can be characterized
by its moments and in particular by the mean – the average equity base or the
equity base of the average agent – and the variance. GS derive the law of motion
of the average equity base, neglecting the role of the variance and its evolution
over time. In the extension considered in Section 10.7, on the other hand, we show
how to describe simultaneously the laws of motion of the average equity ratio and
its variance. The initial distribution of the equity ratio may converge or not to a
stationary distribution. As a matter of fact a wide range of dynamical behaviors
are possible, within this framework depending upon the values assumed by the
parameters.
The rest of the papers considered follows a different procedure. BG start from
a continuous initial distribution of degrees of inefficiency and boil it down to a
polarized distribution of efficient and inefficient entrepreneurs by means of a simple
criterion of choice. GZ start from a continuous initial distribution of wealth and boil
it down to a polarized distribution of skilled/rich individuals and unskilled/poor
individuals by means of a piecewise linear phase diagram. Aggregate outcomes,
therefore, emerge as weighted averages of polarized outcomes. In other words it
is as if the modeler could compute the expected value of a discrete distribution
consisting of only two characters. AB follow a similar line of inquiry in a more
sophisticated setting. In fact, they treat the evolution over time of the distribution
of wealth by means of the theory of convergence for monotonic Markov processes.
Notes
1 Stoker (1993: 1829) notes that representative agent modelling ‘has proved a tremen-
dous engine for the development of rational choice models over the last two decades,
and their empirical application has developed into an ideology for judging aggregate
data models’.
2 Kirman, 1992, is the locus classicus of this literature.
3 See, among others, Leontief, 1947; Gorman, 1953; Theil, 1954; Eisenberg, 1961;
Fisher, 1969, 1982; Muellbauer, 1975, 1976; Lau, 1977, 1982. On aggregation see
Daal and Merkies, 1984.
4 Lippi, 1988; Lippi and Forni, 1996; Lewbel, 1992.
5 Stoker, 1986,1993; Caballero et al., 1997; Geweke, 1985, demonstrate the pitfalls of
using the representative agent framework to explain aggregate behaviour. Grunfeld and
Griliches, 1960, show that macromodels with large aggregation bias can produce better
prediction than their disaggregated counterparts. Empirical evidence on distribution
of income and wealth in the growth process can be found in Kuznets, 1955, and Levy
and Murnane, 1992.
6 Banerjee and Newman, 1988; Acemoglu, 1997; Benabou, 1993, 1996; Bertola, 1993;
Hopenhayn, 1992; Solon, 1992; Persson and Tabellini, 1994; Zarembka, 1975; Chiu,
1998.
7 Roller and Sinclair-Desgagne, 1996; Bertola and Caballero, 1994; Keane et al., 1988;
Heckman and Sedlacek, 1985.
8 Keane, 1997; Amable and Chatelain, 1997; Guariglia and Schiantarelli, 1998; Zietz,
1996; Fortin, 1995; Caballero, 1995; Gordon, 1992; Perotti, 1993;Attanasio and Weber,
1995.
9 Cho, 1995; Aoki, 1994; Chiaromonte and Dosi, 1993; Caballero and Engel, 1993; Das,
1993; Zeira, 1994; Caplin and Leahy, 1991, 1994; Kydland, 1984.
10 Aghion and Bolton, 1997; Banerjee and Newman, 1988, 1993; Galor and Zeira, 1993;
Piketty, 1997; Bernanke and Gertler, 1989, 1990; Greenwald and Stiglitz, 1988, 1990,
1993; Kiyotaki and Moore, 1997.
11 In other words, there are marginal decreasing returns. The analysis could be carried
out, however, also in the presence of constant marginal returns, i.e. in the case in which
F is linear.
12 When capital markets are affected by informational imperfections such as asymmetric
information, a financing hierarchy (pecking order) can be envisaged. Internal finance is
the most preferred source of finance. As to external sources, credit has a cost advantage
over the issue of new equities (Fazzari, Hubbard and Petersen, 1988). A different
wording is used in the literature on the so-called credit view: bank loans are imperfect
substitutes of the issue of new equities (credit is ‘special’). See, for instance, Bernanke
and Blinder (1988).
13 As a matter of fact, the procedure followed by GS to derive an equation for the prob-
ability of bankruptcy is much more complicated. In this survey we can simplify the
argument without loss of generality.
14 On bankruptcy costs, see Gordon and Malkiel, 1981; Altman,1984; Gilson, 1990;
Kaplan and Reishus, 1990.
15 There is only demand for consumption goods in this framework. Investment is ruled out
by assumption, since production takes place using only labour as an input. Financial
factors do not play any role in the determination of aggregate demand, due to the
peculiar way in which consumer’s preferences and the budget constraint are modelled.
It is clear, however, that if firms use capital together with labour to carry on production,
financial factors can influence aggregate demand through their impact on investment
activity. Even in the absence of investment activity, however, demand can be influenced
by financial factors. If firms pay-out dividends based on their net worth, in fact, and
dividends are part of consumers’income, consumption expenditure is affected by firms’
financial conditions.
A different and somehow more indirect way to explore the impact of financial factors
on aggregate demand is adopted in Delli Gatti and Gallegati (1997).
16 This point is discussed at length in Delli Gatti and Gallegati (1997).
17 BG do not impose equilibrium on the market for loanable funds. They simply assume
that the volume of loans available is sufficient to fill the financing gap of borrowers. The
rate of return on storage therefore is not determined in equilibrium but is exogenous.
18 For the sake of simplicity, we are ignoring the effects on the price and the quantity of
capital of changes of other exogenous variables, such as the interest rate and the share
of entrepreneurs in total population.
19 For the sake of simplicity, we are ignoring the effects on the price and the quantity of
capital of changes of other exogenous variables, such as the interest rate, the share of
entrepreneurs in total population, the cost of monitoring etc.
20 On this issue see Hart and Moore (1994, 1998).
21 As a matter of fact, the paper by Aghion and Bolton has been circulating for some
years as a LSE working paper.
22 As a matter of fact, AB show that under appropriate conditions there can be credit
rationing. We will not dig deeper into this issue.
23 If agents were identical – i.e. if the Representative Agent Hypothesis held true – the
law of motion of the equity ratio of the representative firm would be:
at = Γ1 at−1 − Γ2 a2t−1 + Γ0
which is a quadratic map topologically conjugated to the logistic map.

The dynamical properties of the logistic map are well known (for a comprehensive
survey see Day, 1994). It is worth mentioning, however, that there are configurations
of the parameters such that the dynamics are chaotic, i.e. the equity ratio oscillates
apparently at random around the steady state. In this case the economy follows a path
of endogenously determined fluctuating growth.
24 See Agliari et al. (2000) for a thorough analysis of the dynamic properties of this type
of system.
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11 Toward the
microeconomics of
innovation
Growth engine of market
economies
William J. Baumol
The Bourgeoisie (i.e. capitalism cannot exist without constantly revolution-

izing the instruments of production. . . . Conservation of the old modes of
production in unaltered form was, on the contrary, the first condition of ex-
istence for all earlier industrial classes. . . . The bourgeoisie, during its rule
of scarce one hundred years has created more massive and more colossal
productive forces than have all preceding generations together. . . . It has
accomplished wonders far surpassing Egyptian pyramids, Roman aqueducts
and Gothic cathedrals. . . .
(Marx and Engels, The Communist Manifesto, 1847)
11.1 Introduction: the big puzzle: why do all rival systems trail so
far behind free market growth rates?
Undoubtedly, the spectacular and unmatched growth rate of the industrialized
free-market economies is what most distinguishes them from all other economic
systems. In no other system, current or in the past, has the average income of the
general public risen anywhere nearly as much or as quickly as it has in North
America, Western Europe and Japan. Though the former Soviet Union planned its
economy and forced its population to invest heavily in factories and hydroelectric
dams, its failure to produce enough to raise the standard of living of its population to
that of the free-market economies undoubtedly played a major role in its downfall.
There have been great civilizations with extraordinary records of invention and
engineering – medieval China and ancient Rome are clear examples. But none
has approached the growth record of modern free-market economies. What is the
secret of their extraordinary success? That is the economic puzzle that is absolutely
critical to the degree of prosperity our future is able to achieve. Its solution is what
the world’s poorer countries are anxious to learn.
Toward the microeconomics of innovation 267
11.2 The free market’s growth record
The growth of per-capita income and productivity in the free-market economies
is so enormous that it is virtually impossible to comprehend. In contrast, average
growth rates of per-capita incomes were probably approximately zero for about
1,500 years before the Industrial Revolution at the time of George Washington. In
1776, even the wealthiest consumers in England, then the world’s richest country,
had only a half-dozen consumer goods that had not been present in ancient Rome.
These new products included (highly inaccurate) hunting guns, (highly inaccurate)
watches, paper, window glass and very little else. And, remarkably enough, Roman
citizens enjoyed a number of amenities, such as hot baths and good roads, that had
long disappeared at the time of the American Revolution.
In contrast, in the past century and a half, per-capita incomes in the typical
capitalist economy have risen by amounts ranging from several hundred to several
thousand percent. Recent decades have yielded an unmatched outpouring of new
products and services: color television, the computer, jet aircraft, the VCR, the
microwave oven, the hand-held calculator, the cellular telephone, and so on and
on. And the flood of new products continues. When, a few years ago, many of the
world’s communist regimes collapsed and when even the masters of China turned
toward capitalist enterprise, surely part of the reason was the public’s desire to
participate in the growth miracle of the capitalist economies that Marx and Engels
– those high priests of anticapitalist movements – were among the first economists
to discern (as the opening quotation demonstrates).
The explanation of this miracle must be sought in the activities of industries
and the business firms of which they are constituted, for they are the producers of
the increasing outpouring of goods and services that constitutes the growth record
of capitalism. It must be something about business firms and the decisions they
make that plays a vital part in this prime attribute of our economy. Yet the standard
core microeconomic theory of firms and industries, while it has included some
outstanding contributions on the theory of innovation, has not provided anything
suggesting what features of business behavior and decision-making can account for
all this growth. Indeed, as we know, mainstream microeconomics offers reasons to
expect that the capitalist economy will be characterized by a growth performance
that is far from optimal.
Here, I will describe some features of competitive markets to which the growth
performance of business firms can be attributed, features that literally force busi-
nesses to do all they can to contribute to the growth miracle. I will then provide
some hints of a microeconomic model, using the most elementary of microeco-
nomic tools to analyze this process.
11.3 Innovation and the growth process

Without a doubt, a primary source of the growth miracle of the past two centuries
is ‘the wave of new gadgets’1 – the surge of innovation that probably first reached a
substantial pace in the first third of the nineteenth century. Though it is difficult to
268 William J. Baumol
prove statistically (because improved education and the construction of factories,
roads and other influences certainly made substantial contributions), a very large
proportion of all of the economic growth that has occurred since the eighteenth
century probably is ultimately attributable to innovation. Indeed, the incredible
poverty of earlier centuries that the inventions of the Industrial Revolution helped
to bring to an end meant that previously society simply could not afford to spend
much on education or on the construction of plant and machinery. For us, the
magnitude of that poverty is difficult to grasp. The following passage from the
writings of a noted historian suggests how serious the problem was only a few
centuries ago:
The poor in the towns and countryside lived in a state of almost complete
deprivation. Their furniture consisted of next to nothing. . . . Inventories
made after death . . . testify almost invariably to the general destitution . . . a
few old clothes, a stool, a table, a bench, the planks of a bed, sacks filled with
straw. Official reports for Burgundy between the sixteenth and the eighteenth
centuries are full of references to people (sleeping) on straw . . . with no
bed or furniture, who were only separated from the pigs by a screen. . . .
Paradoxically the countryside sometimes experienced far greater suffering
(from famines than the townspeople). The peasants . . . had scarcely any
reserves of their own. They had no solution in case of famine except to turn to
the town where they crowded together, begging in the streets and often dying
in public squares. . . .
(Fernand Braudel, The Structures of Everyday Life, Vol. I, New York: Harper
and Row, 1979, pp. 73–75 and 284–286).
Only the growing outputs that innovation, first in agriculture and mining and then in
manufacturing and transportation, made feasible produced the enormous increases
in productive plant and equipment and in education (and other forms of investment
in human capital) that are widely judged to have contributed greatly to economic
growth. Thus, it can be argued not only that innovation has facilitated the growth
process, but that without it the process would have been reduced to insignificance.
Two of the leading analysts of economic growth conclude:
As yet, no empirical study proves that technology has been the engine of
modern-day growth. Still, we ask the reader to ponder the following: What
would the century’s growth performance have been like without the invention
and refinement of methods for generating electricity and using radio waves to
transmit sound, without Bessemer’s discovery of a new technique for refining
iron, and without the design and development of products like the automobile,
the airplane, the transistor, the integrated circuit, and the computer?
(Gene M. Grossman and Elthanan Helpman, ‘Endogenous Innovation in the
Theory of Growth,’ Journal of Economic Perspectives, Vol. 8, Winter, 1994,
p. 32).
Table 11.1 Some leading growth macromodels

A. Autonomous innovation:
1) Yt = A(t)f (Lt ) Ricardo (1817)
2) Y = A(t)K a L(1−a) Solow (1956)
B. Endogenous innovation
(1−a)
3) Yj = A(K)Kja Lj Arrow (1962) (spillovers, learning by doing)
4) Yj = A(R)F(Rj , Kj , Lj ) Romer (1986) (R = Knowledge)
5) Yj = A(H )F(Kj , Hj ) Lucas (1988) (H = Human capital)
6) Y = A(H )K a H b L(1−a−b) Mankiw, D. Romer and Weil (1992)
11.4 Macroeconomic endogenous growth models: where is the Prince

of Denmark?
The model building that is introduced here relies heavily on the current macroeco-
nomic growth literature. None of what follows, consequently, is to be interpreted
as criticism, much less denigration, of the earlier writings. However, since it is my
hope to carry the study of the subject a step beyond what that work has been able to
achieve, I must begin by indicating what this literature has not yet succeeded in do-
ing. In my view left by that literature is its failure to grapple with the extraordinary
growth record of the capitalist economies. Indeed, as is well known, and as my
brief review of this literature will note, the earlier contributions took innovation to
be an autonomous contribution of the passage of time. It was, in effect, described
as a sort of manna dropped in a steady stream from some unspecified source, and
that could just as well emerge from a capitalistic economy or from any other. Later
model builders recognized that this formulation was inadequate, and that there
were features inherent in the economic processes that account for innovation and
growth. This led to the valuable line of analysis referred to as ‘endogenous growth
theory.’Yet the features cited in this literature as sources of innovation, notably the
externalities of innovation, and the acquisition of human capital, in part through
learning by doing, apply to many forms of economic organization, and not only to
the free-market economies.
In my view, these very valuable contributions are like performances of Hamlet
that include the King, Ophelia, Gertrude, and many other of the crucial characters,
but omit the Prince of Denmark. They tell us much about innovation and growth,
but they fail to account for the most salient and extraordinary feature of the growth
record, the entirely unparalleled success of the free-market economies. I will sug-
gest that they fail to do so because they are macromodels, something patently
unobjectionable in itself, but that is a major handicap for study of the issue before
us, which, I believe, is explainable primarily in terms of microeconomic behavior.
Recent growth analysis had its beginnings in the 1950s with the work of Solow
and Swan – work which deservedly elicited renewed interest in models of growth
and in approaches compatible with statistical estimation. The models themselves
represented no break with the past, and clearly have their roots in the work of
the classical economists, notably that of David Ricardo. Table 11.1 is a vastly
oversimplified summation of some of the leading models in the more recent group,
including also the Ricardian model for contrast.
The Ricardian model is sufficiently familiar and needs little review here. In
short, it postulates linear relationships throughout, with the exception of dimin-
ishing returns to labor and capital invested on a given stock of land. In an early
stage of the economy, with highly productive land abundant, output, in the model,
is more than sufficient to provide subsistence to agricultural workers. This ini-
tially yields high profits, induces increased innovation and expands the demand
for labor. Higher wages stimulate an expanded population, and diminishing re-
turns mean that a second round in this parable yields a second set of increases
in profits, wages, population and output, but all of them smaller than those in the
previous round. This process continues until, finally, diminishing returns cut out-
put to a level only capable of providing subsistence wages to workers, and there
the process would end in the stationary state, were it not for innovation. But Ri-
cardo and other classical economists recognized that innovation does occur. This
results in a shifting of the production function and postponement of the stationary
state, something that can occur repeatedly and can keep the economy expanding
indefinitely. What is missing in the Ricardian story is any explanation of the in-
novation process, and certainly of any endogenous innovation model. That is why
the innovation process is represented simply as A(t), as a function of time and
nothing else, and with no distinguishing features that differentiate the process in
a capitalist economy from that in any other form of economic organization. Thus,
Ricardo’s story emphatically contains no role for the Prince of Denmark.
The original Solow model, the prototype neoclassical model, contains a rep-
resentation of innovation not much different from Ricardo’s, with innovation also
autonomous, and undifferentiated as between free-market economies and other
economic forms. The model also assumes that there are diminishing returns to
capital, an attribute that can be used as a hypothesis that predicts convergence
of productivities and per-capita incomes in different economies, because wealth-
ier economies have relatively large capital stocks whose productivities, relative
to those of poorer countries, are consequently reduced severely by diminishing
returns.
Two observations led Romer to argue that the neoclassical model required
some modification. First, he observed that the universal convergence apparently
predicted by the theory is not sustained by the facts. Indeed, the many statistical
studies of the convergence hypothesis generally conclude that while the wealthi-
est economies have, indeed, been converging toward one another’s productivities
and per-capita incomes, most of the impecunious nations are falling further be-
hind. Second, he noted (as students of the subject such as Schmookler had long
observed) that the innovation process is neither largely autonomous nor largely
fortuitous. The amount of activity devoted to innovation, and the output of that
activity, is influenced substantially by what is going on in the economy. This argu-
ment served to reorient research toward the endogenous growth models. Some of
these, including several that preceded Romer’s writings, are also characterized in
Table 11.1. For example, the Lucas model of 1988 can be described as taking the
innovation function as A(H ), where H is the investment in human capital of the
entire society, as distinguished from Hj , the corresponding investment by agent
J .2 Similarly, the Arrow model of 1962 uses an innovation function such as A(K),
where K is society’s investment in physical capital, as distinguished from that of
agent J . The other entries in Table 11.1 can easily be interpreted analogously by
the reader.
Again, what is to be noted is that none of these formulations distinguish the
free-market economy from other economic forms. Thus, whatever their virtues,
none of them assigns a part in the scenario to the Prince of Denmark. Nor should
this be surprising. Other economies, both historical and modern, have stressed
education, have innovated, have experienced spillovers from innovation, in short,
they have exhibited all the endogenous innovation features of the newer models.
None of these new models seems to have stressed any special features of the
capitalist economy. To do so, I believe, it is necessary to turn to microeconomics.
11.5 What is different about free-market economies?

Invention alone is not the complete answer to the great puzzle – the explanation
of the free market’s unmatched growth performance. Earlier societies have had a
spectacular invention record. The Chinese are the outstanding example. Centuries
before Columbus they had invented printing, the compass, complex clockwork,
gunpowder, spinning machinery, a cotton gin, porcelain, matches, toothbrushes,
playing cards and much more. There have been other countries in history with a
considerable record of new products and new technology. Moreover, education
was highly valued in the Chinese culture and others, though, it must be admitted,
much of the population was uneducated. Yet, economically, these inventions and
this education never produced economic growth anything like that in the modern
market economies.3
It should be added that markets of substantial importance exist in virtually every
economy of the world and have existed throughout recorded history. What, then,
is different about modern markets that not only gives them the capacity to produce
growth miracles but seems to get those miracles to happen very frequently? There
can be no simple answer, indeed, any proposed answer is bound to leave out key
features, ranging from political changes, evolution of religious beliefs and even
historical accident. However, here it will be argued that two features of our econ-
omy have played a crucial role. The first such feature is free competition, that is,
competition not handicapped by tight government regulations or closely enforced
customary rules, like those of the medieval guilds, which prevented gloves-off
combat among rival firms. The second crucial development is the fact that in to-
day’s economy many rival firms use innovation as the main battle weapon with
which they protect themselves from competitors and with which they seek to beat
those competitors out. The result is like the case of two countries, each of which
fears that the other will attack it militarily and therefore feels it necessary always
to match the other country’s military spending. Similarly, either of two competing
firms will feel it to be foolhardy to let its competitor outspend it on the development
and acquisition of battle weapons. Each is driven to feel that at least matching effort
and spending on the innovation process is a matter of life and death. Naturally, in
an economy in which this is so, a constant stream of innovations can be expected
to appear, because firms do not dare to relax their innovation activities.
11.6 Innovation versus price as the competitor’s prime weapon

There are substantial sectors of the economy in which it seems quite clear that
the weapon of choice for competitive battle is innovation. Price does, of course,
matter, but it is improvement in processes and products that capture the attention
of management. In product lines as diverse as computers and computer software,
automobiles, cameras and productive equipment, models are constantly improved,
and the improvements are instantly and widely advertised. The firm that can come
up with a model better than those of its rivals will have an advantage that cannot
be matched by the latter as quickly and easily as a price cut, and certainly the
advantage of a dramatic new product is likely to be more substantial. In all of
the economy’s ‘high-tech’ industries this appears to be true, and the relationship
probably plays a role in many others.
Here, one must not exaggerate. There is strong evidence that most innovation
is contributed by a very few industries in a very small number of countries. It
is reported that about 80 percent of industrial outlays on new product research
and development comes from the chemical and machinery and equipment sectors
of manufacturing. Yet, even in many of the industries where product and process
development may not be the leading instruments of competition, management
cannot afford to neglect them or to leave them to chance. For if one firm fails
to do enough in this arena – if it delays in adopting the latest technology, even
technology created by others, it becomes far easier for rivals to get ahead of it,
very possibly with disastrous consequences for its sales.
As a result, at least in the high-tech sectors of the economy firms do not dare to
leave their innovation to chance. Rather, the pressures of the competitive market
force them to systematize the innovation process and to seek so far as possible to
remove risk from the undertaking. Nowadays this drives business firms systemat-
ically and routinely to determine the amounts they will invest in the research and
development (R and D) process, systematically decide on the ways in which they
will promote and price their innovation and even systematically determine what it
is that the company’s laboratories should invent.
This kind of business firm innovation activity is far easier to analyze using
the tools of the microeconomics of the firm and the industry than it is to analyze
what can be described as the ‘Eureka!’ process, in which a lone inventor working
in a basement or garage happens to come up with a brilliant invention. Business
innovation is easier to analyze because the decision process related to this activity
has become routine, carried out in a manner with much in common with other
decisions of the firm, such as how much of one of its output commodities to
produce, how much to spend on advertising, and so forth.
11.7 How much will the profit maximizing firm spend on innovation?
The basic story I am telling is that competition forces many firms in the economy
to keep up their expenditure on research and development, the activity that creates
the company’s inventions and prepares them for market. While some of the money
spent on R and D will be a failure, other such spending will be spectacularly
successful, and most R and D outlays will yield modest advances. Taken as a
whole, then, one can expect that the more firms spend on R and D, the greater
will be the number of innovations that contribute to the economy’s GDP. The key
questions, then, are how much can we expect firms to spend on R and D, and how
will competition affect that amount?
The obvious answer that the firm will act to maximize its profit, spending
to the point at which expected marginal revenue equals expected marginal cost,
simultaneously tells us everything and nothing about the R and D decision. It
tells us nothing about the shapes of the relevant functions or, in particular, how
the relationships are affected by competition which, I have suggested, plays an
important role in the R and D decision. Nor has the discussion told us what insights
we get from the analysis about business behavior and its contribution to economic
growth.
11.8 The profits of innovation

Many discussions of innovation start off with the assumption that innovators expect
to earn very high profits. And this is obviously true of those innovators who create
unusually successful innovations. We have all heard of innovators like Thomas
Edison, Alexander Graham Bell and, more recently, Bill Gates and others in the
computer industry, who have acquired great riches from their ability to invent,
bring the innovations to market and sell their products. But for every successful
innovator there are many others who have plowed the family savings into their
new gadgets, and lost all they have spent. It is possible that on average, inventors
have earned zero profits or even less.
Now, entry into innovation is not perfectly easy, but it is much easier than
entry into many other economic activities. That means we cannot be certain that
economic profits to invention will tend exactly toward the zero level, but we can
expect them to be very low on average. In other words, while invention activity
will sometimes pay off enormously well, there will also be big failures, so that
the average comes out close to zero. This is particularly likely to be true of a
large firm that has a big R and D division that simultaneously works on many
possible innovations. The law of large numbers makes it very likely that some of
these efforts will fail and that some will succeed. Thus, zero economic profit from
innovation activity of the firm, that is, profit no greater than what is currently usual
in the competitive industries of the economy, is to be expected to be a frequent
occurrence in industries with a great deal of innovative activity.
Does this conclusion fit in with the facts? We have no systematic study for all
inventive activities. But the high-tech industries do provide a useful case study,
and this is particularly true of computers, because it is an industry in which many
fortunes have been made and have received much publicity. Here is one report:
The computer industry hasn’t made a dime . . . Intel and Microsoft make
money, but look at all the people who were losing money all the world over.
It is doubtful the industry has yet broken even,’ said Peter Drucker in a recent
interview . . . but is it true? Paul Gompers of the Harvard Business School and
Alon Brav of the University of Chicago . . . looked at companies that went
public from 1975 to 1992, most of which were high-tech firms, and found
their rate of return to be about average [i.e., zero economic profit], once they
adjusted for risk and company size (‘The Rewards of Investing in High Tech,’
Federal Reserve Bank of Boston, Regional Review, Vol.6, Fall 1996, p. 14).
11.9 Risk reduction through technology sharing

We have just shown that low average profits from innovation are in good part
attributable to the riskiness of the activity – the high likelihood that effort devoted
to an apparently promising innovation will not pay off, and the investment will
go down the drain. Firms can be expected to try to minimize the risks, and they
do. For example, management may try to avoid providing money to their research
laboratories to finance the development of those inventions that are judged to be
impractical. There are, however, many examples where management’s foresight
has not proved to be brilliant – as, for instance, when the Western Union Telegraph
Company turned down the newly invented telephone.
Firms also often try to protect themselves by going into partnerships, research
joint ventures, that enable them to share the risks, with each of, say, five partner-
firms supplying only one fifth of the funds needed to bring an innovation to com-
pletion. What is less well known is that many firms try to reduce their risks by
systematic technology trading. It is widely believed by those who have not studied
the matter that when a firm succeeds in producing a promising new invention it
will generally try to keep its competitors from getting hold of it, in order to retain a
competitive advantage over its rivals. But that is often not true. Fearing that its own
laboratories may conceivably come up with only failures in the year 2003, while
its competitor may possibly have better luck that year, a firm will often choose to
sign an agreement with that competitor in which each shares with the other all of
its successful future innovations, say, for the next five years. This helps to cut down
other risks for the two technology-sharing firms. In photography, for example, one
camera manufacturer may introduce an improved automatic focus device, another
an automatic light adjustment, and a third may invent a way to make the camera
lighter and more compact. Each of these three firms has the choice of keeping its
invention to itself. But if two of them get together and agree to produce cameras
combining the features each of them has contributed, they will be able to market a
product that is clearly superior to what each could have produced alone. They are
then likely to be in a far better position to meet the competition of the third camera
manufacturer.
There are many firms and industries that engage in this practice, industries
ranging from steel production to computer manufacturing. The exchanges may be
entirely informal, or they may be based on detailed contracts, even requiring each
firm to train technical experts from the other in the use of the new technology, and
even specifying whether the company requesting such training will pay the travel
and living expenses of these experts.
Indeed, the activity of business firms in providing their technology to others,
for a profit, has become so commonplace that the Massachusetts Institute of Tech-
nology runs a seminar for business firms, teaching how they can be more effective
in the technology provision business.4
11.10 A kinked revenue curve model of spending on innovation

The discussion so far leaves a basic question unanswered. If innovation takes
much effort and money, if it is so risky, and if the economic profits to be expected
from innovation activity are near zero, why do firms do it? Why does not every
firm refuse to join this unattractive game? The answer, at least in part, is that
the competitive market mechanism gives them no choice. If they do not keep up
with their competitors in terms of attractiveness of their products and efficiency
improvements that permit them to keep their costs low, they will lose out to their
rivals, and end up losing market share and losing money. Zero economic profits,
that is, profits that yield normal competitive returns to investors, surely are better
than negative profits.
The result is like an arms race between two countries, each of which fears inva-
sion by the other. Each is driven to keep up with the other’s military expenditure.
Raising its armaments expenditure will probably get it nowhere, because it can
expect the other nation to match any such increase, raising expenditure without
improving the nation’s military security. But, at the same time, neither nation will
dare to cut its arms spending unilaterally, since that will simply invite invasion by
the other.
This story can be made more explicit with the help of a microeconomic model
very similar to the well-known kinked demand curve model of monopoly pricing
that has been proposed to explain why in oligopoly markets prices tend to be
‘sticky’ because no firm dares either to raise or to cut its price. It will be recalled
that the underlying mechanism is an asymmetry in the firm’s expectations about
the behavior of its competitors. The firm fears either to lower its price below that
of its rivals, or to raise it above theirs. It is afraid that if it lowers its product price
its rivals will match the price cut, so that our firm will end up with hardly any more
customers. On the other hand, if it increases its price it fears that the others will
not follow, so that it will be left all by itself as the seller of an overpriced product
and will lose all its customers. The result is that normally such a firm will set its
price at the industry level, no more and no less, and leave it there unless there is a
major change in cost or demand or some other extreme change.
The innovation story is similar. Consider an industry with, say, five firms of
roughly equal size, and that the firm with whose decision we will be concerned,
Company X, sees that each of the other firms spends about $20 million a year
on R and D. X will not dare to spend much less than $20 million on R and D
itself, because if it does so its next year’s product model will probably not be
nearly as good as those of some or all of its rivals. On the other hand, it sees little
point in raising the ante, say, to $30 million, because it knows the others will feel
themselves forced to raise their R and D budgets correspondingly.
The story can described graphically in a figure that shows the MC curve and a
kinked MR curve as functions of the firm’s total R and D expenditure. The shape of
the MC curve does not matter in this story. The MR curve, however, has a vertical
break at the, say, $20 million level of R and D investment that currently is the norm
in the industry. The story behind this curve is simple. Consider any lower level of
R and D spending by Company X , say, $5 million per year. Then its products will
grow exceedingly inferior to those of its rivals, but it may sell a few because they
meet the special needs of a few customers at highly discounted prices. More R and
D spending will bring in a small additional revenue and still more increases in R
and D will add more to revenue. The peak will occur at the point where Company X
meets the industry standard and its product becomes really competitive with those
of its rivals. However, further spending is not very revenue producing because if
Company X inflates its R and D budget further its rivals will feel threatened, and
match the increase. Thus, further increases in R and D spending by Company X
yield a very low MR, as shown by a low segment of the MR curve to the right
of the $20 million spending level. The MC and MR curves meet at a point in the
vertical part of the MR curve, so it will pay Company X to follow industry practice,
investing $20 million a year in R and D, at which MC = MR. It may go on doing
so, year after year.
But that is not the end of the story. All five firms in the industry will continue to
invest the same amount, until some year one of them has a research breakthrough
and comes up with a wonderful new product (as happens in most high-tech indus-
tries from time to time). Then, for that firm it will pay to expand its investment in
the breakthrough product, because that will pay off even if the other firms in the
industry match the increase. The MR curve for that breakthrough firm will rise,
its MC = MR point will move to the right, say to $25 million. Other companies in
the industry will feel forced to follow, and now the industry norm will no longer
be a $20 million investment per year, but will instead be $25 million per firm.
The story, then, is that competition forces firms in the industry to keep up with
one another in their R and D investment. But once they have caught up, the invest-
ment level remains fairly level until, from time to time, something induces one firm
to break ranks and increase its spending, with all the other firms following behind.
Such an arrangement is described as a ‘ratchet’ (in analogy with the mechanical
device that prevents a spring that is being wound up from suddenly unwinding).
It is an arrangement that holds matters steady, permits them under certain circum-
stances to move forward, but generally does not allow them to retreat. R and D
spending can then be expected to expand from time to time, but once the new level
is reached, the ratchet – the competitive market – prevents a retreat to the previous
lower level.
This, in my view (based on considerable but unsystematic observation), is a
critical part of the mechanism that accounts for the extraordinary growth record of
free-enterprise economies and differentiates them from all other known economic
arrangements. It is the competitive pressure that forces firms to run as fast as they
can in the innovation race just in order to keep up with the others.
11.11 Three growth-creating properties of innovation

We have just seen reasons to expect that the market mechanism will force firms
to devote at least a steady stream of resources to innovative activities, notably to
R and D. With luck, such a level R and D effort will yield a fairly level flow of
innovations. But a level flow of innovation does not mean that GDP will remain
level. Rather, a level flow of innovation can be expected to result in steady growth of
the economy’s output. Here we must take note of three critical features of innovation
that can, so to speak, magnify the contribution of technical change to the economy’s
GDP. These three features are (i) the cumulative character of many innovations,
meaning that many innovations do not merely replace older technology and make
that technology obsolete; rather, they add to what was previously available, thus
constituting a net increase in the economy’s inventory of technical knowledge. (ii)
In addition, an innovation, once created, need not contribute only to the output of
the firm that made the breakthrough. At little or no additional cost it can also add
to the outputs of other enterprises. This is, of course, the public good property of
technical knowledge. (iii) Finally, there is what can be called an accelerator feature
of innovation – a level stream of innovation usually means that output will not be
level, but growing. It is like stepping on the accelerator, where a steady unchanging
pressure makes an automobile move, that is, change position, more rapidly. It is
only the last of these attributes that may be unfamiliar to readers, and so a few
further words on the subject may be appropriate.
11.12 The accelerator property of innovation

It should be clear that each successful innovation adds to the nation’s GDP by per-
mitting more products to be created with a given quantity of resources (a ‘process
innovation’) or by making new and more valuable products available (‘product in-
novation’). Thus, an economy whose R and D produces a steady and unincreasing
output of one innovation per month will obtain a GDP that is higher each month
than it was in the previous month. That is to say, the economy’s output will grow
without letup even though the flow of innovations that fuels that output growth
remains steady at one invention per month. This acceleration relationship applies
to innovation generally, so that if the competitive market mechanism were only
to lead firms to supply a level quantity of resources to R and D, we would expect
continued growth of GDP to result. Of course, as noted, the ratchet principle tends
to increase the expenditure of resources on innovation, and not just to leave them
level. The acceleration principle tells us about the effects of this too. It tells us that
if the level of R and D spending were to increase just once, for example, and stay
at that new higher level forever, then the growth rate of GDP would also move to
a higher rate, and GDP would increase at this new faster rate forever.
All of this reinforces the role of the competitive market mechanism and its stim-
ulation of innovation as a contributor to the extraordinary growth that characterizes
the world’s free-enterprise economies.
11.13 Free-enterprise growth: routine versus independent

endogenous innovation
The analysis of this chapter has focused on the innovation activities of large busi-
ness firms, their routinized character and the influence of the competitive market
forces on the magnitude of such routine innovative activities of the firm. The size
of such activity is substantial, and its funding can be estimated to amount to some
70 percent of R and D expenditure in the United States.
Still, a good deal of important innovative activity takes place outside the large
corporation. Indeed, there is evidence that a very considerable portion of the in-
novative breakthroughs of the twentieth century are to be attributed to such inde-
pendent innovation activities. Nothing in my discussion is meant to minimize their
importance. They have not been emphasized here only because they are not so
directly influenced by the market mechanism, and therefore probably play less of
a role than routine innovation activity in explaining the growth record of the free
market – which is my main focus here. Of course, even independent innovation
is subject to endogenous influences. For one thing, the success of one innovator
is likely to encourage the activity of another and to make it easier for the latter
to obtain funding. Nevertheless, independent invention activities have occurred in
abundance in some non-capitalist economies, as we know. They are part of my
story, but important though they may be for society, they do not seem to be at the
heart of the explanation of the growth record of free-market economies.
11.14 On the efficiency of innovation activity in free markets

As already noted, standard microtheory suggests that the economy’s innovative
activity is apt to be extremely far from optimal. The prime reason for this is the
spillovers of innovation – their presumably beneficial externalities, which on stan-
dard grounds lead us to expect that investment in innovation will be less than
optimal. The second apparent shortcoming of the process is the proprietary behav-
ior of innovating firms that leads them to resist sharing their knowledge with others,
thereby condemning rivals (and others) to use obsolete methods and to provide ob-
solete products. Here, to keep my discussion from growing too long, I will only
hint at my analysis of these issues. With regard to the second issue – unwillingness
to share technology – I have already indicated that market behavior is often very
different. Rather than struggling to prevent others from obtaining their proprietary
information, firms often trade it for access to the proprietary information of other
firms, including competitors. They are often forced to do this by market pressures
because firms that share information are likely to end up with better products, ben-
efitting from the combined innovations of all the participants, than the products
of firms that rely on just their own innovative resources. The amount of sharing
that results may or may not approximate optimality, but it is surely better than a
world where secrecy, patents and other influences systematically prevent much of
economic activity from use of the latest technology.
Many economists also, apparently, believe that the market does not induce
private firms to invest the socially optimal amount in innovation. They believe
that many innovations whose benefits would exceed their costs are never carried
out by industry, because the firm that spent the money to produce the innovation
would get only part of the benefit, a part insufficient to cover the cost. This is why
a good deal of innovation and research activities are financed by governments and
carried out by research institutions such as universities. This is particularly true
of basic research. However, it does not follow that the market’s performance on
quantity of resources devoted to innovation is as far below the optimum as might
have been believed. The reason is that there is a tradeoff between increased flow
of invention and the distribution of benefits because of which zero externalities
cannot be expected to be optimal. Moreover, there is no one level of expenditure
that is unambiguously optimal. Instead, there is a range of values of what I call the
spillover ratio, that is, the share of the benefits of innovation that goes to persons
other than the investors, such that all values of the ratio within this range are
Pareto-optimal. Consequently, there is no way in which economic analysis alone
can choose among them. Rather, value judgments must be employed in making
that selection. Indeed, it can even be suspected that the high spillover ratio found
in reality falls within the range of Pareto optimality.5 The reason for these results
is that, in contrast with most of the literature (except that on optimal duration of
patents), here spillovers are considered to be capable of offering social benefits,
and do not always simply impede or prevent the attainment of optimality. The
point is that there is an inevitable tradeoff between the number of innovations
actually produced and the standard of living of the majority of the population. In
this scenario, as overall GDP is raised, any increase in workers’ standards of living
constitutes a rise in the spillovers from innovation that depresses the flow of further
innovation. Thus, the more the general public benefits from such growth in GDP,
the slower that growth must be.
This is more than just an embellishment of the old story of the tradeoff between
output and distributive equality. The mechanism under discussion here is very
different, and does not involve the disincentive to work that results from a reduction
of the marginal return to worker effort. Rather, we are concerned here with the
heart of the capitalist growth process: the payoff to innovation and the speed with
which new technology and new products become available.
Our scenario is by far the more dramatic. Romer notes in passing that, if
the innovator were totally immune to the disincentives of spillovers, then none
of the benefits would have gone to others. But, if that were so, then real wages
would hardly have risen from their levels before the Industrial Revolution!6 It is
almost impossible to imagine how great a difference that would have entailed.
If we assume the most extreme case – that the spillovers from innovation are
reduced to (anywhere near) zero – the living standards of the vast majority of the
citizens of today’s rich countries would have stalled at pre-Industrial Revolution
levels. One can hardly accept the notion that it would be socially preferable to
achieve a total GDP that is far higher than today’s through enhanced incentives
for innovation, while the bulk of the population is condemned to near-medieval
living standards, but that is where such a premise leads us. Even the fortunate
few innovators who might amass unimaginable wealth in such a zero-spillover
world would undoubtedly prefer somewhat better conditions for their impoverished
compatriots, and would themselves probably be better off not only in terms of the
social environment, including reduced violence and disease. In addition, innovators
would probably have higher absolute incomes because more can be produced by
a labor force that is better fed, healthier and better educated. But any such gain to
labor is unavoidably a rise in the percent of the return to innovation that goes to
others than the innovator – it is necessarily a rise in the share of spillovers.
Once again, my conclusion is not that the market’s performance in response
to the externalities of innovation is optimal. I suggest only that it is far better than
standard theory might lead us to expect, and comes closer to consistency with the
observed and unprecedented growth record of the free-enterprise economies.
Acknowledgement
I am grateful to the Russell Sage Foundation and the C. V. Starr Center at New
York University for their support of this work.
Notes
1 According to the late British economist, T.S. Ashton, in his classic book, The Industrial
Revolution, 1760–1830, London: Oxford University Press, 1948, that phrase is how
one schoolboy (quite appropriately) described the Industrial Revolution.
2 See R.E. Lucas, Jr (1988) ‘On the Mechanics of Economic Development’ Journal of
Monetary Economics, 22, 3–42
3 For a discussion of possible reasons for failure of the economies of Ancient Rome
and Medieval China to achieve outstanding growth records, see my Entrepreneur-
ship, Management and the Structure of Payoffs, Cambridge, Mass.: MIT Press, 1993,
Chapter 2.
4 For further materials on technology sharing in practice, with a number of concrete
examples, see Chapter 10 of my recent book (op.cit., 1993).
5 My own value judgment is summed up in George Bernard Shaw’s dictum that there
is no crime greater than poverty. Consequently, I am inclined to prefer a fairly high
spillover ratio, perhaps not far from its current value.
6 ‘This pattern of industrialization without wage gains is what it would take to ensure that
the industrialist captures all of the benefits he creates when he introduces machinery.
. . . [this] cannot be a historically accurate description of the process of development in
industrial countries, for if it were, unskilled labor would still earn what it earned prior
to the industrial revolution’ (Paul Romer, ‘New Goods, Old Theory and the Welfare
Costs of Trade Restrictions,’ Journal of Development Economics, Vol. 43, 1994, p. 29).
Part IV
Challenges for quantitative

methodologies
12 Business cycle research
Methods and problems
Edward C. Prescott
12.1 Methods in business cycle theory

Basketball and soccer are different games. The objective is the same, namely to
put the ball in the net. In both games, a ball is dribbled and passed. However, what
is good dribbling and passing practice when playing one game is bad practice
when playing the other. A soccer player who picks up the ball and bounces it as he
runs down the field is not following good practice. This activity, however, is good
practice when playing basketball.
In scientific inference there are two fundamentally different and complimentary
games. The objective of both is the same, namely to draw scientific inference.
One game is drawing deductive inference. The other game is drawing inductive
inference. Both require the selection of a model. In this way they are similar.
However, the rules for selecting the model are fundamentally different in these
games.
This distinction between inductive and deductive scientific inference is impor-
tant in business cycle research. Business cycle theory uses deductive or quantitative
theoretic inference. Often practitioners do it poorly. They do it poorly because they
use practices that are good for inductive inference but bad for deductive inference.
Their behavior is analogous to the soccer player who picks up the ball and bounces
it as he runs down the field.
I first clarify the differences between these forms of scientific inference. I
do this by reviewing the role played by each in the development of the natural
sciences. With inductive or empirical inference the product is the model or the
law. In physics an example of inductive inference is Kepler’s laws of planetary
motion. These laws are:
• Law 1 – Planets follow an elliptical orbit around the sun,
• Law 2 – The sun is at one of the foci of each ellipse,

• Law 3 – An equal area of a planet’s ellipse is swept out in a given time
interval by the planet.
284 Edward C. Prescott
Another example of successful inductive inferences in the natural sciences is
Galileo’s discovery of the law of motion or model of a ball rolling down an incline
plane. This law is:
D = 1/2g sin (θ)t 2 ,

where D is distance, t is time, θ is the angle of the inclined plane, and g is a constant.
An important feature of the formula is that the weight of the ball does not appear.
I do not use the term mass as that concept did not exist when Galileo discovered
his law. The concept of mass had meaning only subsequent to the development of
Newton’s theory of mechanics.
Inductive or empirical inference was productive in the natural sciences. Good
inductive practices involved model estimation and testing. This raises the question
of why inductive or empirical inference proved sterile in business cycle research.
This sterility was not due to the incompetence of the researchers who pursued the
inductive approach. The group who pursued this research program included a dis-
proportionate number of the best minds in economics. The reason these inductive
attempts failed, I think, is that the existence of policy invariant laws governing
the evolution of an economic system is inconsistent with dynamic economic the-
ory. This point is made forcefully in Lucas’ famous critique of econometric policy
evaluation.
With deductive inference a model is a tool or measurement instrument used
to deduce the implication of theory. This statement requires a discussion of what
theory is. The definition that I will use is an implicit set of instructions for con-
structing a model economy for the purpose of answering a question. Two examples
of theory drawn from physics are: (i) Newtonian Mechanics – force equals mass
times acceleration – and (ii) Newton’s Universal Law of Gravitation – the gravi-
tational force operating on two bodies is proportional to the product of the masses
and inversely proportional to the square of the distance. These theories provide a
theoretical foundation for Kepler’s laws of planetary motion and Galileo’s law of
a ball rolling down an inclined plane. Newtonian mechanics has proven useful.
This theory is used to construct models for all kinds of purposes in the engineering
sciences. It is used to predict the path of rocket ships and to control their paths. It
is used to design machinery in factories.
In economics there is Walrasian general equilibrium theory, which Schumpeter
(1954) judged to be ‘the only work by an economist that will stand comparison
with the achievements of theoretical physics’. I do not agree with the ‘only’ part of
Schumpeter’s statement, as I would add Arrow–Debreu general equilibrium theory
to this list, and possibly game theory and the closely related mechanism design
theory.
General equilibrium theory is theory in the language sense but not in the sense
that I am using the word theory. The reason is that without restrictions on pref-
erences and technology general equilibrium theory is virtually vacuous. Conse-
quently it is not a set of instructions for constructing an instrument to measure
something or predict the consequences of some policy. Growth theory with mea-
Business cycle research 285
sures of the elasticity of substitutions and transformations and share parameters
is theory in the sense I am using it here. Growth theory provides instructions for
constructing a model economy to address some question of interest. The quan-
titative answer to the question is deduced for the model economy. In business
cycle studies, growth theory is the theory used. Indeed, business cycle research
is largely drawing inference from growth theory for business cycle fluctuations.
Growth theory is also heavily used to construct models to estimate the welfare and
other quantitative effects of tax policies and social security systems.
In this review I will explain why using estimation theory to select a model used
in drawing deductive inference is bad practice. Unfortunately, or maybe fortunately
for economists, there is no set of mechanical rules for selecting a good model for
deducing some inference of a theory. I say it may be fortunate because, if it were
mechanical, computers could replace economists. I now illustrate this point by
reviewing the development of modern business cycle theory.
12.1.1 History and overview of business cycle theory

Burns and Mitchell (1946) developed a statistical definition of a business cycle.
This definition did not prove useful. Kuznets (1946a, 1946b) and his students were
more successful. They systematically reported economic events using a system of
national income and product accounts. They also reported aggregate factor inputs
of capital and labor. A set of growth facts emerged from this reporting. These facts
guided researchers in the development of growth theory. Some of these facts are as
follows. Labor share of product, using Kravis’ (1959) economy-wide assumption,
is more or less constant over time even though the real wage increased dramatically
relative to the rental price of capital. Subsequently Gollin (forthcoming) found that
this regularity held across countries with deviations being unrelated to the level
of development. Another fact is that the investment share of product is more or
less constant. This led to the Solow growth model with a Cobb–Douglas aggregate
production function. The Solow model with its aggregate production function and
factors being paid their marginal product is a theory of the income side of the
national income and product accounts.
The Solow growth model, with its exogenously determined savings rate, led the
economic theorists Cass (1965), Koopmans (1965) and Diamond (1965) to develop
a theory of the allocation of product between consumption and investment. Brock
and Mirman (1972) extended this theory to stochastic environments.
Lucas (1977) defined business cycles to be recurrent fluctuations of output
and employment about trend. He wrote that the key business facts were the co-
movements of the economic time series. Hodrick and I (1980) developed a statis-
tical definition of the business cycle component of an economic time series. The
regularities that appeared are all tied to the variables in growth theory. They are: (i)
consumption is strongly procyclical and fluctuates about a third as much as output
in percentage terms; (ii) investment is strongly procyclical and fluctuates about
three times as much as output; (iii) two-thirds of output fluctuations are accounted
for by variations in the labor input, one-third by variations in TFP and essentially
zero by variations in the capital input; (iv) the only important lead–lag relation is
that the capital stock lags the cycle with the lag being greater the more durable
the capital good; (v) the deviations of output from trend, that is the business cycle
component, displays a moderately high degree of persistence; (vi) the real wage
is procyclical but is roughly orthogonal to the labor input.
These facts were bothersome for theorists. Why should leisure be low when
consumption is high? After all, consumption and leisure are normal goods and
leisure is not high when the real wage is high. Another question is why labor
productivity is high when labor input is high. This violates the law of diminishing
returns.
Many have argued that Hodrick and my facts are not interesting because we
did not correctly measure the business cycle. This criticism is spurious. An opera-
tional definition can be neither right nor wrong and our definition is an operational
definition. Economics is not the only science where operational definitions have
proven useful. In the natural sciences prior to the development of the theory of an
ideal gas the definition of temperature was an operational one being neither right
nor wrong. An environment was 50 degrees Celsius if the thermometer registers
half way between what it registers when it is immersed in ice water and what it
registers when immersed in boiling water.
In retrospect, Hodrick and my representation of time series as the sum of
two components, one we called the growth or trend component and the other
the business cycle or deviation component, turned out to be a useful one. Our
representation revealed some behavior that was in apparent contradiction with
theory and fostered the development of some good theory.
I emphasize that these facts were in apparent contradiction with theory. Until
the dynamic applied general equilibrium tools were developed to derive the im-
plication of growth theory for business cycle fluctuations, economists had to rely
on their intuition derived from price theory. This price theory intuition proved to
be wrong.
Exploiting Arrow–Debreu language, recursive methods, and computational
methods, Kydland and I (1982) derived the implications of growth theory for
business cycle fluctuations. To our surprise we found that, if total factor productivity
(TFP) shocks are persistent and of the right magnitude, business cycle fluctuations
are what growth theory predicts. Subsequently I (1986) found that these TFP
shocks are highly persistent and of a magnitude that implies that they are the
major contributor to business cycle fluctuations. This success of growth theory led
me to have greater confidence in public finance findings that use growth theory to
evaluate tax policies.
Kydland and Prescott (1982) examine the consequence of people valuing
leisure more if they have worked more in the past. The introduction of this feature
into growth theory preserved the consistency of the theory with the growth facts.
Its introduction increases the inter-temporal elasticity of substitution for leisure
and results in the prediction of the model being more in conformity with aggre-
gate observations. However, this feature should not be part of the measurement
instrument used to answer the question of how volatile the US economy would
have been if TFP shocks were the only shocks until other evidence is provided
that it indeed quantitatively describes people’s preferences. Such evidence never
materialized. This leads to the following two principles for selecting the model
used in business cycle research.
Principle 1: When modifying the standard model of growth theory to address a

business cycle question, the modification should continue to display the growth
facts.
Principle 2: The model economy being used to measure something should not
have a feature which is not supported by other evidence even if its introduction
results in the model economy better mimicking reality.
When modifying the standard growth model to address a business cycle question,
Hansen (1985) introduced another feature, a labor indivisibility, and permitted
people to enter into mutually beneficial insurance contracts as in Arrow–Debreu
theory. This increases the magnitude of the response to TFP shocks. There is empir-
ical evidence that justifies incorporating this feature. First, there is unemployment
insurance and people have a close substitute for insurance, namely the holding of
liquid assets. Second, and most important, most of the cyclical variation in the
labor input is the result of variation in the number of people that work in a given
week and not in the average workweek length. Like Kydland and my non-time-
separable utility function, the introduction of this feature results in leisure having
a high intertemporal elasticity of substitution.
Hansen finds that for his model economy, cyclical fluctuations induced by
technology shocks are as large as the observed fluctuations. Given the strong
empirical support for labor indivisibility, this mechanism should be part of the
model or measurement instrument used to answer the question of the importance
of technology shocks for business cycle fluctuations. A problem, however, is that
not all fluctuations in hours are the result of the variations in the number of people
working. An important fraction is the result of variation in the workweek length
as in Kydland and my model economy. This number is somewhere between the
Hansen and the original Kydland and Prescott estimates.
What was needed was better theory. The reason is the following. Why is the
workweek fixed? If it is permitted to vary and aggregate hours is the labor input to
production, then all work and all variation is in the workweek length. There was a
major inconsistency between observation and theory. Better theory was developed
that reduced this inconsistency. I begin with the aggregate production function and
its inadequacy for the purpose of understanding the determination of workweek
length.
12.1.2 The aggregate production function

The aggregate production function with labor and capital as its inputs, which has
proven so useful in public finance, fails to capture the intensity with which people
and machines work. These intensities vary over the cycle. Before discussing how to
extend the theory of the production function utilizing results of Mas-Colell (1975),
Hart (1979), and Jones (1984) on differentiated commodities, the aggregation
theory underlying the aggregate production function will be reviewed.
The technologies underlying the aggregate production function:
(i) There are n factor inputs and a composite output.
(ii) The vector of inputs is x ∈ n+ and the output good is y.

(iii) A plant technology is indexed by x ∈ T with f (x) being plant type x output.
(iv) X ∈ n+ is the vector of aggregate inputs and Y aggregate output.
Definition: An aggregate production function F(X ) is the maximum output that

can be produced given the input vector X .
Assumption 1: Any number of technologies of type x ∈ T can be operated.

Assumption 2: For all x ∈ T , x is infinitesimal relative to X .
Assumption 3: T ⊂ n++ and T is compact.
Assumption 4: f : T → is continuous.
The aggregate production function is the solution to the following program, where
M+ (T ) is the set of measures on the Borel sigma algebra of T ,

F(X ) = max f (x)z(dx)
z∈M+ (T )
subject to
xi z(dx) ≤ Xi i = 1, 2, . . . , n.
T
Given the assumptions, the constraint set is compact and non-empty and the ob-
jective function is continuous in the weak star topology. Therefore the program
has a solution. Two well-known results are the following.
Proposition: F(X ) is weakly increasing, continuous, weakly concave, and ho-

mogenous of degree one.
Proposition: If there is free entry, profit maximization results in output being
maximized.
12.1.3 An example
The Cobb–Douglas production function has come to dominate in aggregate applied
general equilibrium analysis. The reason is that both over time and across countries,
labor’s share of product is surprisingly constant at about 70 per cent.1 The Cobb–
Douglas production function, with its unit elasticity of substitution, is about the
only aggregate production function with the property that factor income shares are
the same even though relative factor prices are very different.
An example of an underlying set of plant technologies for the Cobb–Douglas
production function is the following one. Suppose that the factor inputs are capital
k and labor n and that the plant technologies are g(n)k θ . In addition, the function
g is such that the function g(n)nθ−1 has a maximum. This maximum is denoted
by A and a maximizing n by n∗ .
Proposition: For this example, the aggregate production function is F(K, N ) =

AK θ N 1−θ .
Outline of proof: Given there are two constraints and an optimum exists, there
is an optimum with at most two types of plants operated. Consider one such
optimum. Let (Ki , Ni ) be the aggregate factor inputs used to operate type i plants
for i = 1, 2. Allocate (Ki , Ni ) equally to m plants of type (Ki /m, Ni /m). The m for
which Ni = mn∗ is an optimum. Thus, all operated plants for this optimum have
n = n∗ . Output maximization requires marginal products of capital be equated
across operated plants. Thus, operating N /n∗ plants each with n∗ workers and
K/(N /n∗ ) units of capital is optimal. Using this result,

N ∗ K
F(K, N ) = ∗ g(n ) = AN (1−θ) K θ
n Nn∗
12.1.4 Workweek of capital

The workweek of capital and labor varies cyclically. The aggregate production
function does not capture this. The following technologically does:
y ≤ Ahk θ ,
where 0 < θ < 1 and where y is the output produced by an individual, k is the
capital that that individual uses and h is the length of that individual’s workweek.
Element h belongs to the set H ⊂ (0, 1].
Question: What is the aggregate production function for this technology?

A workweek of different length is a different factor input. Following the pro-
cedures above, there is an aggregate production function. The inputs are capital
K and the measure N ∈ M (H ). N (B) is the measure of people working a work-
week belonging to Borel measurable set B ⊆ H . An aggregate production function
F(K, N ) exists.
To simplify the exposition I deal with the case that there are only a finite
number of possible workweek lengths. Then Nh is the measure of workweeks of
length h ∈ H and N is a finite dimensional vector. For each workweek length the
aggregate production function is
Fh (Kh , Nh ) = hAKhθ Nh1−θ

The aggregate production function F(K, N ) is obtained by equating marginal prod-
ucts of capital across these aggregated technologies and summing over h.
Finn Kydland and I (1991) introduced this feature into the growth model. In
our model economy households’ preferences are ordered by the expected value of
∞ 1−Ψ 1−σ
ct (1 − ht )Ψ
β t
.
t=0
1−σ
The technology parameter {At } is governed by a first order auto-regressive pro-

cess with high persistence. Capital depreciates exponentially. The parameters are
selected so that the model economy displays the growth facts including the rough
constancy of the fraction of time allocated to the market, including commuting
time.
Kydland and I were surprised that cyclically only the fraction of the population
working varied and not hours per employed worker. Hornstein and I (1993) find
that this is precisely what theory predicts. The fact that hours per worker do vary,
however, is not bothersome for theory. Kydland and I (1991) find tiny costs of
moving between the market and the household sector results in h varying cyclically,
as it does.
Introducing this option to vary the length of the workweek along with some
moving costs results in observations being in better conformity with theory. The
costs are selected so that the relative variability of employment and workweek
length match observations. The resulting model, or measuring instrument, is a
better one than either Kydland and my original model or the Hansen model. Using
this model Kydland and I estimated that the US post-war economy would have been
70 per cent as volatile if total factor productivity shocks were the only disturbance.
Here the volatility measure is the variance of the business cycle component.
With this 70 per cent number, an implication of theory is that labor produc-
tivity should be orthogonal to the labor input, as it is in the data. If the estimate
contribution had been near 100 per cent, an implication of theory is that labor pro-
ductivity and the labor input should be highly correlated, as they are in the model
economy. This leads to an important methodological point. Given this 70 per cent
estimate, if both the model and the actual economy had high correlation between
labor productivity and the labor input, it would have been a basis for rejecting the
model as a good instrument for measuring the importance of TFP shocks.
Principle 3: A model that better fits the data may be a worse measurement
instrument. Indeed, a model matching the data on certain dimensions can be the
basis for rejecting that model economy as being a useful instrument for estimating
the quantity of interest.
Principle 4: A corollary of Principle 3 is that using statistical estimation theory

to estimate models used to deduct scientific inference is bad practice. Estimating
the magnitude of a measurement instrument, whether it is a thermometer or a
model economy makes no sense.
There are legitimate challenges to this 70 per cent number. Challenging the result
because the model used is not realistic is not one. As stated previously all models
are abstractions and therefore unrealistic. A legitimate challenge is to introduce
some feature into the model economy in a quantitatively reasonable way and show
that the answer to the question changes. There have been many such challenges
to Kydland and my finding, but the result has been found to be robust. Before
proceeding with a review of some of these studies, I will state the fifth and last
principle.
Principle 5: A legitimate challenge to a finding is to introduce a feature into the

model economy that is serving as the measurement instrument in a quantitatively
reasonable way and show the answer to the question changes.
12.1.5 Robustness of results to increasing returns and monopolistic

competition
Hornstein (1993) extends the neoclassical growth model to incorporate monopo-
listic competition and increasing firm-specific returns to scale. In his model world
there is a large and fixed number of firms where each firm has market power for
its own product. Increasing returns to scale at the firm level is introduced through
a fixed cost to production. This assumption allows for an equilibrium where firms
make profits but, on average over time, profits are zero.
To account for the role of productivity shocks as sources of output fluctua-
tions Hornstein focuses on two effects. First, he studies to what extent the Solow
residual overestimates actual productivity changes. Second, he demonstrates that,
compared with the basic neoclassical growth model, a productivity shock gener-
ates a stronger output response and a weaker employment response. The change
in output and employment responses can be attributed to the increasing returns
to scale that also generates stronger wealth effects and thereby dampens employ-
ment fluctuations. Finally, he suggests that the net effect of increasing returns to
scale and monopolistic competition is to lower the contribution of the productivity
shocks to output fluctuations somewhat, but that this effect is limited when mark-
ups and returns to scale are not unreasonably large. Cooley, Hansen and Prescott
(1995) find similar results for a model economy with idle capital.
Devereux, Head and Lapham (1996) incorporate technology shocks into a real
business cycle model with monopolistic competition and increasing returns to both
specialization and scale. They find that market power and increasing returns due to
fixed costs have no effect on the responses of aggregate variables to a technology
shock when compared to those exhibited by a standard, perfectly competitive real
business cycle model. The responses of aggregate variables to technology shocks
are actually increased by returns to specialization and reduced by returns to scale
in variable factors. They find that returns to specialization and scale also affect the
measurement of technology shocks. The variance of the Solow residual understates
the variance of the technology shock due to increasing returns to scale. Returns to
specialization result in the opposite bias. When both types of increasing returns are
present, the authors find the variance of output is increased relative to a standard
competitive model despite a significant reduction of the variance of technology
shocks.
Fagnart, Licandro and Portier (1999) investigate the phenomenon of under
utilization of productive equipment and its implications for business cycles. The
authors introduce the concept of capacity utilization (as opposed to capital uti-
lization) into a stochastic dynamic general equilibrium model. Monopolistically
competitive firms use a ‘putty–clay’ technology and decide on their productive ca-
pacity and technology under idiosyncratic (demand) uncertainty. It is shown that
the proportion of firms with excess capacities plays an important role in magnify-
ing and propagating aggregate (technological) shocks. Furthermore, they find that
idiosyncratic uncertainty about the exact position of the demand curve faced by
each firm explains why some productive capacities may remain idle in the sequel
and why individual capacity utilization rates differ across firms. Finally, the vari-
ability of capacity utilization allows for a good description of some of the main
stylized facts of the business cycle and generates endogenous persistence.
12.1.6 Consequence of capital and capacity utilization variation for

the estimate
Greenwood, Hercowitz and Huffman (1988) make the assumption that the intensity
with which capital is used, say x, is a choice variable. The amount of capital
services is the product of x and K. The cost of using capital more intensely is that
depreciation is greater.
Kt+1 = (1 − δ(x))Kt + It .
The function δ(x) is increasing in x.
Introducing this feature results in aggregate observations being in greater con-
formity with theory. However, there is a problem. The problem is the lack of
micro-observations to back up the depreciation assumption. Does capital depreci-
ate more in boom periods? Until other evidence is provided for this depreciation
assumption, this feature is best not incorporated into the measurement instrument.
12.1.7 Labor hoarding over the business cycle

Burnside, Eichenbaum, and Rebelo (1993) introduce the possibility of labor hoard-
ing. In their model world, people must commit to how many hours they will work
over the next three months. During this three month period only the worker’s work
intensity can vary. In fact hours of employment are not fixed for such long periods
for much of the workforce. There are weekly layoffs and variations in the length
of the workweek. Their analysis is important for it nicely puts to rest the labor
hoarding story that has confused the profession for so many years. The paper ac-
complishes this by showing how extreme and implausible assumptions must be for
labor hoarding to be a factor in understanding business cycle fluctuations. Labor
hoarding is only important when shocks are temporary, not when they are highly
persistent, and business cycles are responses to highly persistent shocks.
12.2 Problems in business cycle theory

There is no shortage of important open problems in business cycle theory. What is
in short supply are problems that are both important and analyzable using existing
tools. My view is that whenever new tools are developed, it is a good time to search
the set of important open problems for one that can be analyzed using these new
tools. With this in mind I focus only on problems for which the needed tools have
been recently developed or are being developed.
12.2.1 The role of organizations in business cycles

Fitzgerald (1996, 1998) develops and uses a general equilibrium framework in
which the number of hours and the employment levels of heterogeneous workers
is endogenously determined. He does this in an environment where production
requires coordinating work schedules of different worker types, a characteristic
that he refers to as team production. In particular, he assumes that all workers in a
production team must work the same hours. Output is produced by a large number
of teams, where team composition and the hours a team works can differ across
teams.
He has two types of workers, the skilled and the unskilled. Skilled workers lose
human capital if they are not employed. He finds that constraints on workweek
length increase the welfare of the high income skilled people and reduce the welfare
of the low income unskilled workers. Furthermore, he finds there is an increase
in the employment rate of the unskilled workers. He also finds that introducing
this realistic feature does not change the estimate of the importance of total factor
productivity shocks for business cycle fluctuations.
In the model he uses preferences that are additively separable in consumption
and leisure, namely
*∞ +

E β t [log(ct (s)) + ν(1 − ht (s))] .
t=0
A suitable commodity vector is {ct (s), it (s), kt (s), n1t (s), n2t (s)}s∈S,t∈{0,1,2, . . . } .
Here s is the state, c and k are scalars, and the ni are signed measures on the
Borel sigma algebra of [0,1]. Restrictions on a type i consumption set are that ni
be a probability measure and nj=i be the null measure.
The plant production function is
hf (k, x1 , x2 )
where h is hours the plant is operated, k is the capital employed and the xi are
the number of type i workers. Suppressing the t and s index, a period aggregate
production possibility set is
Y = {(C, I , K, N1 , N2 , ) ≥ 0|∃ measure z(dh × d k × d x1 × d x2 ) for which

C + I ≤ f (k, x1 , x2 , )d z
k dz ≤ K
d z ≤ Ni (B) all measurable B all i}
{(h,k,x1 x2 )|xi ∈B}
The set is a convex cone so there are constant returns to scale.
12.2.2 The role of money in business cycles

A question that has received a great deal of attention is whether monetary factors are
a major contributing factor to business cycle fluctuations. Friedman and Schwartz
(1963) argued in a comprehensive empirical study that monetary shocks are the
major cause of business cycle fluctuations. They observed that sharp declines in
the money stock occurred prior to severe economic downturns.
The apparent inconsistencies of economic fluctuations with economic theory
that abstracted from money led to widespread acceptance of the Friedman–Schwarz
view even though a theoretical foundation was lacking. Real business cycle theory
finds that a major fraction of US postwar business cycle fluctuations is accounted
for by persistent shock to total factor productivity (see Kydland and Prescott (1991).
If money is not a major contributing factor to business cycle fluctuations, why
is money highly correlated with output? Freeman and Kydland (1998) provide a
possible answer using a transaction based theory of money. They introduce a costly
transaction technology into the standard growth model and find that monetary
aggregates and output are positively correlated even though there is no causal
relationship between money and output.
An issue is how to model money. Currently the dominant view is that money is
valued because it facilitates existing trade or permits new trade. Lucas (1980) and
Lucas and Stokey (1987) develop cash-credit goods and cash-in-advance models.
Ireland (1995), Schreft (1992), Freeman and Kydland (2000) develop transac-
tion based models. Saving (1971) and McCallum and Goodfriend (1987) develop
shopping time models. Christiano and Eichenbaum (1995), Fuerst (1992) and
Scheinkman and Weiss (1986) develop and explore limited participation models.
A question is how do these transaction-based theories stand up to the facts. At
the very low frequencies their performance is impressive (see Lucas (1988)). At the
very high frequencies there are no problems because, with Lucas surprises, these
theories place weak restrictions on observations. At the intermediate frequencies,
these theories with their empirical demand for money relations fail and fail spec-
tacularly. These theories predict much larger variations in the demand for money
than observed given the variation in the nominal interest rate.2
Whether this failure is a serious one for evaluating the importance of monetary
shocks for business cycle fluctuations is an open issue. This question, however, is
likely to be central for evaluation and design of monetary stabilization policies,
which is the principal reason for developing a theory of money. Perhaps the em-
piricists are right that adopting a constant money growth rule will result in less
fluctuations in output and employment. Perhaps they are wrong. Until there is
some strong monetary theory the answer is unknown. Once there is a good theory
for evaluating monetary policy, the welfare benefits and costs of alternative policy
rules can be deduced.
Researchers using the RBC methodology and a transaction based theory of
money have found that money contributes little to business cycle fluctuations.
I, however, see these exercises as being far from conclusive in establishing that
monetary policy cannot be used to stabilize the economy. The reason is the failure of
these theories at the intermediate frequencies. Diaz-Gimenez, Prescott, Fitzgerald
and Alvarez (1992) have taken an alternative approach that may not suffer from
this deficiency. Key in their model is the government issuance of nominal debt
that is held directly or indirectly by households. In the United States in 1986 they
report that the outstanding stock of nominal debt is nearly one annual GNP. There
is also a large amount of lending by old to young in nominal terms to finance home
purchases. Consequently there is a lot of risk associated with unexpected inflation
that results in redistribution between young borrower and old lenders. A question
is why markets do not develop to eliminate this risk. One answer is that there is no
way for the unborn to contract with people alive to avoid the risk associated with
this redistribution.
Another possibly important component of a successful theory for evaluating
monetary policy rules is intermediation costs. These costs are large. In rich coun-
tries the spread between the average borrowing and average lending rates is about
5 per cent. The total product of the financial sector was 9 per cent in the United
States in 1986. Still another possibly important component of a successful the-
ory for evaluating monetary stabilization policy is moral hazard associated with
allocating idiosyncratic risk. Empirically much of this idiosyncratic risk is not
eliminated, which would be the case if financial markets were frictionless and
there was a complete set of state contingent commodities.
12.2.3 The role of policy in determining labor-leisure time allocation

The Great Depression in the United States is an example of a large deviation from
the neoclassical growth theory that is not accounted for by variations in TFP. In
1939 hours worked per adult were still 23 per cent below what they were in 1929,
the year prior to the start of the Depression. During this ten year period output
per hour increased by about 10 per cent, which is only a little below the historical
average. The question is why employment did not return to its 1929 level. The
only candidate for an answer is policy that changed the nature of the game being
played by the economic actors.
12.2.4 International business cycles
Backus, Kehoe and Kydland (1992) use theory to construct a model economy to
derive the implications of growth theory for international business cycles. They as-
sume, among other things, that there is a full set of state-contingent commodities.
They find that observations deviate from the predictions of this theory in impor-
tant respects. First, consumption is less correlated between countries than theory
predicts. Second, investment and labor supply are too negatively correlated with
output.
Their findings led Baxter and Crucini (1995) and Kollmann (1996) to restrict
international asset markets to just borrowing and lending. With this restriction, the
model economy better mimics reality. But, the question of why there is this re-
striction is left unanswered. This led Kehoe and Perri (forthcoming) to endogenize
these debt contracts. They consider a model economy with limited enforcement of
international contractual constraints.
12.2.5 Introducing contractual constraints using modern contract

theory
Cooley and Quadrini (1998) find that if firms face debt constraints, monetary policy
has quantitatively important welfare effects. The economy mimics reality on a
number of dimensions suggesting that indeed firms are debt constrained. Here, I
think, there is a need to use modern contract theory to endogenize these constraints.
The issue of what is a firm, which in Arrow–Debreu theory is just a technology set,
must be addressed. Here I conjecture that modeling a firm as a coalition or club is
the way to proceed. Useful tools include the recursive dynamic contract theory of
Thomas and Worrall (1990), Marcet and Marimon (1992), and Atkeson and Lucas
(1992,1995). Other useful tools might be general equilibrium theory with clubs as
in Cole and Prescott (1997) and financial intermediary coalitions as in Boyd and
Prescott (1986).
Carlstrom and Fuerst (1997), in an interesting paper, deduce some quantita-
tive implications of these informational constraints on contracting. Alvarez and
Jermann (2000) quantitatively explore the implications of limited enforcement for
asset prices in an applied general equilibrium study of asset pricing.
12.2.6 Introducing plant irreversible investment

Marcelo Veracierto (1998) develops a method for introducing irreversible invest-
ment at the plant level into equilibrium business cycle models. The author’s compu-
tational strategy makes the analysis of the class of (S, s) economies fully tractable.
He finds that introducing this feature has no effects on aggregate business cycle
dynamics.
12.2.7 Computing equilibrium when a distribution is part of the state
variable
There are many issues that require a model with a distribution as a state variable.
Existing tools for analyzing such model economies are limited although tools are
improving. The problem is the well know curse of dimensionality in recursive
analyses. Veracierto (1998) finesses this problem with his approach. Krusell and
Smith (1998) have developed methods to find an approximate solution that turns
out to be a remarkably good approximation in some applications.
12.2.8 Role of financial costly financial intermediation in business

cycle
Large amounts of resources are used in financial intermediation. In the United
States resources used are approximately 9 per cent of GNP. The average spread
between average borrowing and average lending rates is about 5 per cent. Further
there is a lot of borrowing and lending between households that is intermediated
through the business sector. In the United States, governments have nominal lia-
bilities equal to about one annual GNP.
12.2.9 The role of varying number of shifts that plants are operated
A margin of adjustment that has not been introduced into any applied general
equilibrium analysis of business cycles is the option to vary the number of shifts
a plant is operated. This is an important margin of adjustment in the automobile
industry.
Acknowledgement
Prepared for the International School of Economic Research’s XI Workshop, ‘Cy-
cles, Growth and Technological Change.’ 29 June 1998, to 7 July 1998. Certosa di
Pontignano-Siena (Italy). I thank Dirk Krueger and Jessica Tjornhom for useful
comments and assistance in preparing the list of references. The views expressed
herein are those of the author and not necessarily those of the Federal Reserve
Bank of Minneapolis or the Federal Reserve System.
Notes
1 See Gollin (1997) for the cross-country numbers. He uses the Kravis’ (1959) economy-
wide assumption for assigning proprietor’s income and indirect business taxes to capital
and labor.
2 See Chari, Kehoe and McGrattan (2000) for a critical evaluations of mechanisms to
propagate the effects of monetary shocks.
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13 Complexity-based methods
in cycles and growth
Any potential value-added?
William A. Brock
13.1 Introduction
This chapter is written in the style of both a brief survey and a research proposal in
order to present a set of analytical and statistical tools, some of which are related
to what might be called ‘complexity theory’, and to explore whether these tools
have anything to contribute to the study of cycles, growth, and structural change.
The reader is warned at the outset that much of what is said here is extremely
speculative and may end up being a research dead end. Furthermore, this article
will keep mathematics and technical details to a minimum in order to attempt to
reach a wide audience.
I shall give a lot of references to the literature for details. I apologize in ad-
vance for the literary license with subtle ideas that I take here. Defining whether
an analytical, theoretical, or statistical tool is ‘complexity-based’ or a research
methodology is ‘complexity-based’ is hard to do within the scope of a short arti-
cle. So we shall just ask the reader to take a look at ‘complexity-based’ references
to see what the word ‘complexity’ refers to. Many ‘complexity-based’ research
groups have been operating for a long time.
There are, for example, the Brussels School (cf. Day and Chen (1993) for a
sampling from the Brussels/Austin school as well as others), Day’s two volume
work (Day (1997) which covers not only the work of himself and his co-workers
but also work of many others), Rosser’s book ((1991), which gives an exceptionally
broad based coverage of the work of different research groups in this area), and
the Stuttgart School (cf. Weidlich (1991)).
The current article will draw on my association with the Santa Fe Institute (SFI).
The books, Anderson, Arrow, and Pines (1988), (SFI (I)), and Arthur, Durlauf, and
Lane (1997), (SFI (II)) give nice descriptions of ‘complexity theoretic’ approaches
to economics from the ‘SFI’ point of view. For example, let us quote from the
introduction to SFI (II):
But just what is the complexity perspective in economics? That is not an easy
question to answer. Its meaning is still very much under construction. . . . To
describe the complexity approach, we begin by pointing out six features of
302 William A. Brock
the economy that together present difficulties for the traditional mathematics
used in economics.
We list them as follows: (i) dispersed interaction; (ii) no global controller; (iii)
cross-cutting hierarchical organization; (iv) continual adaptation; (v) perpetual
novelty; (vi) out-of-equilibrium dynamics. Notice that all six being present together
is what prevents one from treating them by simply re-interpreting the usual Arrow–
Debreu–McKenzie apparatus of general equilibrium theory with enlarged state
spaces of date-event contingent goods and the like with disequilibrium dynamics
grafted onto this apparatus. For example, Norman Packard argued in his piece
in Anderson, Arrow, and Pines (1988) that the state space’s dimensionality is
constantly changing so we cannot use conventional dynamical systems theory to
analyze the dynamics generated by his complex system.
Readers can be quite assured that the piece they are reading is ‘complexity-
based’ if they encounter the following terms and ideas: (a) path dependence; (b)
self-organized criticality; (c) edge of chaos; (d) power law scaling; (e) renormal-
ization group; (f) fractals and other types of self similarity; (g) genetic algorithms,
emergent computation, (h) adaptive neural networks, complex adaptive systems;
(i) chaos theory, embedding theorems, correlation integrals; (j) interacting particle
systems, statistical mechanics, mean field theory, non-ergodic systems, breakdown
of the law of large numbers.
My paper, (1999), compares and contrasts, for applications to economics, the
SFI (I,II) version of complexity approaches to economics with another closely
related approach which emphasizes a hierarchy of dynamical systems at temporal
and ‘spatial’ scales and uses bifurcation theory from dynamical systems theory.
The latter research style is common in ecology. Rather than repeat what has
been stated already in SFI (I,II) and Brock (2000), we shall just sketch the needed
highlights here and refer to these sources for details. The econometric and statistical
tools reviewed by Brock and Durlauf (1999) will be used to propose a different
twist to the problem of detecting and measuring the ‘endogeneity’ in ‘endogenous’
growth theories. I shall also outline some proposed research strategies to help
narrow the dispute between those who prefer rational expectations models of cycles
and growth and those who prefer ‘boundedly rational’ approaches.
13.2 Total factor productivity viewed with ‘complexity-based’ tools

Before we are in a position to discuss whether there is any potential value-added
of complexity-based tools to students of economic growth and cycles, we must
spend some time reviewing ‘conventional’ work. I shall use optimal timing theory
to add a tiny bit of novelty, perhaps, during this part of the review.
Let us use a discussion of Total Factor Productivity (TFP) dynamics as the
central organizing idea of this part of the chapter. A related ‘complexity-based’
review of the dynamics of technical change, i.e. TFP dynamics, is Dosi (2000).
My objective will be to isolate the value-added of complexity-based approaches
to econometrics of TFP dynamics. For example, the notion of path dependence
Complexity-based methods in cycles and growth 303
raises the issue of testing for ‘true state dependence’ versus ‘spurious state depen-
dence’ as in Heckman and Singer (1985) and measuring the relative strengths of
each using actual data. The notion of ‘lock-in’ raises the issue of alternative stable
states, especially Pareto-ranked alternative stable states, which, in turn, raises the
econometric issue of testing and measuring such effects. It also raises the issue
of isolating the obstruction to economic agents or coalitions of economic agents
to picking up ‘big bills on the sidewalk’. (Olson (1996)). This raises the econo-
metric issue of isolating and gathering the kinds of data we need to identify these
obstructions and to measure the size of these obstructions. Indeed one might say
it raises the issue of how one might test whether the current state of an economy
is Pareto Optimal or not. Turn now to TFP.
It seems fair to say that one of the key issues discussed by several papers
presented at the Siena School in July, 1998 was finding good explanations for the
tremendous differences in Total Factor Productivity (TFP) growth across countries
as well as standards of living measured by, for example, Gross Domestic Product
(GDP) per capita.
Baumol’s paper (Baumol (2001, this volume)) discussed relative TFP growth
over long time scales of centuries and stressed institutions which greased the com-
petitive process which generated pressure for each business to innovate before
some other business out innovates it and puts it out of business. He also stressed
large spillovers that were not captured by the innovators. He used the illustration
problem of the optimal time to buy a computer when the cost was falling and the
usefulness was rising in order to focus attention on economic forces that speed up
adoption of innovations. In the spirit of Baumol, I shall use optimal timing theory to
shed light on some economic forces that influence TFP growth. This will motivate
a later discussion of the econometrics of duration models with social interactions
following Brock and Durlauf (1999) and Heckman and Singer (1985). Surveys
of growth theory, both empirical and theoretical, such as Barro and Sala-i-Martin
(1995) attack the TFP question at several scales of time ranging from centuries to
decades, especially post-World War II performance. Barro’s recent book (Barro
(1997)) surveys his well known work. The paper of Durlauf and Quah (1999)
gives us a tour of the whole field as well as a critical review which stresses basic
econometric identification issues. A major theme that emerges from this aggrega-
tive literature is the difficulty of empirically differentiating between modifications
of ‘Solovian’ type theories and the ‘new endogenous growth’ theories, by condi-
tioning on regressors that attempt to capture the quality of infrastructure, human
capital, and incentive structures (embodied, for example, in the quality of institu-
tions) Indeed Durlauf and Quah (1999) show how difficult it is to use available
data to even distinguish between a broadly interpreted Solow type model with
‘mechanical’ savings behavior such as Mankiw, Romer and Weil with two types
of capital from an optimizing model of such savings behavior. I.e. it appears to be
difficult to use available data to distinguish between growth models that are ‘Solo-
vian modifications’ with ‘boundedly rational expectations’ from models which are
modifications of the rational expectations ‘forward looking’ neoclassical models.
Turnovsky’s book (1995) has reviewed generalizations of the basic intertempo-
ral general equilibrium growth models to include policy induced distortions such
as taxes at both the corporate level and the personal level, inflationary finance,
debt finance, and others. This raises the issue of how to use these recently devel-
oped growth models to design econometric methods that help detect the ‘jump
variable’ effects associated with anticipated (via forward looking rational expec-
tations type behavior) policy changes (cf. Solomou (1998) for a critical review in
applications to cycles and growth analysis). A major issue is how to empirically
distinguish lagged effects such as adjustment cost effects in rational expectations
models from lagged effects due to backwards looking expectations (e.g. boundedly
rational expectations as in Sargent (1993)).
The usual approaches in rational expectations econometrics include, for ex-
ample: (i) formulate and test cross equation restrictions (cf. Hansen and Sargent
(1991)); (ii) formulate and test for the small real impact of, for example, ‘antici-
pated money’ in contrast to the predicted larger real effect due to ‘un-anticipated
money’ under rational expectations in contrast to some types of boundedly rational
expectations. See Solomou (1998) for a review and critique of attempts to use (ii)
in cycle analysis.
A ‘nested’ approach applied by Baak (1999) might be adaptable to the problem
of distinguishing boundedly rational expectations from rational expectations in
cycles and growth models. Here one writes down a general model where rational
expectations are costly and compete against other types of expectations which are
more boundedly rational and also ‘cheaper to purchase’. One then estimates this
model and tests the null hypothesis whether the ‘extra parameters’ that appear
in the general equilibrium equations are ‘significant.’ This is one way to assess
whether the extra free parameters brought in by boundedly rational theorists cover
their ‘cost’ in terms of extra predictive and explanatory power. More will be said
about this kind of procedure later.
Empirical problems loom even larger when one attempts to econometrically
identify and estimate the ‘spillover’ effects (which have ‘social multipliers’) which
were stressed in the early Romer models and in the Lucas/Uzawa models reviewed
in Barro and Sala-i-Martin (1995). Recall that the early Romer models stressed
noninternalized spillovers from production activities and the Lucas/Uzawa type
models stressed noninternalized spillovers from human capital. We shall discuss
this last econometric identification problem using econometric work on identifying
‘true social interactions’ from ‘spurious social interactions’ using tools reviewed
by Brock and Durlauf (1999) and Manski’s paper in SFI (II).
Prescott’s Klein Lecture (Prescott (1998)) attacks the same issue of explaining
the huge disparity of living standards across countries and argues that we need
a theory of TFP. He outlines some basic historical facts about TFP performance
across countries and within industries and poses strong challenges to the literature
to date to explain these facts. Hahn’s paper (2001, this volume) argues that so
called ‘endogenous growth theories’ are ‘exogenous’ after all. I.e. much is left
unexplained rather than explained. It may be useful to get started by thinking
about cycles, growth, and structural change in terms of a hierarchy of time scales
because different tools, methods of analysis and explanations may be needed at
different time scales. Day and Pavlov’s (2001, this volume), Baumol’s (2001, this
volume), and Barro’s (1997) work stressed the determinants of economic progress
on the time scale of millenia, centuries, and decades respectively. Business cycle
workers concentrate on yearly frequencies because business cycles have a modal
length of 3–5 years. Call these time scales, very slow, slow, medium, and fast.
Useful analytical hierarchies of time scales are common in economics. For
example, in financial analysis the typical hierarchy of time scales ranges from
business cycle scale as slowest to minute by minute scale (tic by tic) as fastest.
Furthermore in subjects where the phenomena under scrutiny have a ‘spatial’ re-
lationship (where ‘space’ is broadly interpeted) as well as a temporal relationship
it is useful to introduce a spatial ‘hierarchy’ of scales from largest to smallest.
This theme of a hierarchy of time scales is used in Böhm and Punzo (2001, this
volume) where they stress that variables at a slower scale serve as approximate
comparative statics parameters for variables moving at the next faster time scale
in the hierarchy. Indeed the slower variables may pass through a bifurcation point
in the Böhm and Punzo work.
The work of the Resilience Network, hereafter, ‘Rnet’, discussed in Brock
(2000) stresses the use of a hierarchy of time scales as a useful device in organizing
an analysis of dynamic phenomena in general. See Gunderson, Holling, and Light
(1995) for a sample of Rnet approaches to ecosystem management. Rnet models
stress the potential presence of alternative stable states and adducing empirical
evidence for or against the alternative stable state hypothesis. This emphasis is
similar to the nonconvex models of economic growth of Azariadis and Drazen,
Galor and Ziera, Murphy, Shleifer, and Vishny which are discussed by Durlauf
and Quah (1999) (DQ) from the perspective of adducing empirical evidence for
clustering and divergence in growth performance across countries.
Rnet models, however, stress the existence of slow moving background dynam-
ics which are difficult for observers to detect and measure but which nevertheless
impact the ability of the ‘primary dynamics’ to absorb shocks and revert to steady
state after such shocks. In DQ (1999) continuous time dynamical language, the
stable eigenvalue (a pair of eigenvalues with negative real parts if complex) which
measures how fast the system reverts back to its steady state path after a shock is
impacted by a state variable driven by a slow moving dynamics (which is fed by
the primary dynamics).
For example the stable eigenvalue may be gradually moved closer to zero by
the slow moving dynamics, but this is not apparent to observers, (both to scientists
outside the system and to actors inside the system) until a really large shock hits
the system which makes it apparent to both outside and inside observers that the
system’s capacity to absorb shocks has been compromised. For an example of this
type of modelling of ecosystems that might be adapted to growth modeling see
Brock (200) and, especially, Gunderson et al. (1995).
In some phenomena such as climate dynamics there will be a rough positive
relationship between the time scale and spatial scale of an activity. A quantitative
depiction of this relationship, which plots some measure of relative strength or
relative frequency of the activity in that portion of space/time relative to other
portions of space/time such as spectral power on a temporal axis and on a spatial
axis in an attempt to isolate ‘spectral mountains’, called a Stommel diagram, is
sometimes used in other sciences but I have not seen it used in economics. See
Gunderson et al. (1995) for some examples of Stommel diagrams. More will be
said about this later when we discuss ‘spatial’growth models (e.g. Durlauf’s lattice-
growth model and its relatives discussed in Durlauf’s piece in Anderson, Durlauf
and Lane (1997)), and ‘sandpile’ models (See, e.g. Scheinkman and Woodford
(1994), Krugman (1996), and references to Bak and others).
For example, on the slow time scale of centuries, Baumol’s (2001, this volume)
work stressed the determinants of economic progress over centuries and asked the
question why, at this time scale, have the Western nations progressed so far in
improving the standards of living of their peoples relative to many others. He
stressed the institutional environments in these countries which puts such strong
pressure on each business to innovate under the constant threat of being put out of
business by one who does. He stresses the construction of these institutions and
the degree of spillovers that accrue to the population at large.
This theory is related to the ‘Schumpeterian’ family of theories, which appear,
for the most part, to be designed to explain economic progress at the medium scale.
See, for example, writers such as Aghion and Howitt, and Jovanovic in Kreps and
Wallis (1997), Jovanovic and Lach (1997), as well as Iwai (2001, this volume) and
Aghion and Howitt’s recent book (1998). I shall stress incentives to adopt existing
technology in the style of Jovanovic’s piece in Kreps and Wallis, Harberger (1998),
Prescott (1998), rather than the grand incentive to innovate new ideas. The grand
incentives and Schumpterian waves of innovations can be thought of as forces on a
slower time scale than we wish to discuss here. See, for example, Solomou (1998)
for a general empirical and historical analysis on these slower scales in comparison
to business cycle analysis on faster time scales. I adopt a perspective much like
Solomou here. For example, he writes (1998, p. 119)),
The many changes in cyclical features reflect, among other things, changes
to economic structures, policy frameworks and behavioural patterns. Given
that these features are in a state of flux we are unlikely to observe a universal
business cycle structure that can be understood by explaining the features of
particular periods. . . . We cannot hope to predict business cycles, but we can
hope to understand the foundations for making conditional predictions.
The analysis, to be conducted below, of the impact of five industrial organizational
and regulatory arrangements, I1–I5, on the rate of adoption of TFP enhancing
activities is in the spirit of attempting to understand some ‘foundations for making
conditional predictions’about the impacts of different ‘economic structures, policy
frameworks, and behavioural patterns’. One might also imagine that innovation
activity at the Schumpetarian level of ‘high imagination’ would be much more
difficult to do econometrics on (how do you predict when the next Crick/Watson
type of finding will set off a biotech type wave?) than predicting what will happen
if one restructures a state-owned enterprise so that top executives are paid a bonus
based upon a measure of TFP growth rather than flat government salaries. We
doubt, anyway, that the relative performance of North Korea relative to South
Korea is due to a lack of Crick and Watsons in North Korea relative to South
Korea. A major challenge posed by writers like Olson (1996) and Prescott (1998)
is this. Why doesn’t the set of nonadvanced nations, especially the poor ones, of
the world simply copy and adopt the huge existing stock of technology built up by
the advanced nations and give their peoples a ‘free’ one-shot jump in their standard
of living up to the level of the advanced nations? Here we might think of the ‘stock
of technology’ broadly enough to include the stock of organizational ‘capital’ and
the stock of institutional ‘rules-of-the-game’ experience that is embodied in the
collective experience and history of the advanced nations. What explanation for
this ‘adoption gap’ can economists give that advances the discussion beyond that
given by a perceptive tourist? Let us peer inside the black box of TFP growth with
a slightly different emphasis than the recent literature by using some tools from
optimal timing theory including the Feynman–Kac formula. The approach will
draw attention to variability of the stock of ‘latent options’ on both the quality
improvement side and finding new markets side of growth as well as the cost
reduction side of growth. For example, the idea of latent options is to shed some
potentially slightly different light on the economic incentives to pay a cost in
order to obtain what Harberger (1998) calls a Real Cost Reduction (RCR). Our
use of optimal timing theory to expose economic forces that drive TFP dynamics
motivates the proposed use of the econometrics of hazard function estimation to
be discussed below.
The kind of cost we have in mind is more related to that paid by a business
executive in dealing with a recalcitrant middle management and labor force to
change their past set of habits and work routines in order to adopt the new way of
production. Or, at a more personal level, think of the problem of finding the optimal
time to switching to a new word processing program as the hassle and learning
cost of switching falls at some rate (perhaps because newer, more user-friendly
versions are appearing and more people in your office are switching so you can
learn from them) while the benefit to switching rises at some rate. In any event, as
Harberger stresses, this kind of RCR takes place in 1001 ways.
To put it more bluntly we attempt to shed light on forces that remove incentives
to indulge in gross inefficiencies such as over-manning an enterprise, over-use of
capital, ignoring obvious markets, catering to factions such as labor and manage-
ment by using the enterprise as a comfort and security vehicle for these factors
instead of serving the customers for whom the enterprise was set up to serve. These
types of obviously inefficient behaviors are typically associated with regulated in-
dustries, government departments, protected industries, and the like. These kinds
of perceived gross inefficiencies gave rise to reforms such as the ‘corporatization
reforms’ in New Zealand as well as ‘privatization’ reforms world wide. See Evans
et al. (1996) for a careful review of the restructuring experience in New Zealand.
See Olson (1996) for a dramatic argument that ‘economic performance is deter-
mined mostly by the structure of incentives – and that it is mainly national borders
that mark the boundaries of different structures of incentives’ (Olson, 1996, p. 22).
For a dramatic review of problems in transition economies see the Symposium on
Transition from Socialism in the same issue of JEP as Olson’s lecture. See also
McMillan and Dewatripont and Roland’s pieces in Kreps and Wallis (1997) for
what might be called the ‘1001 complementarities’ of institutional and experien-
tial infrastructure that are needed to align the structure of incentives of the agents
within a transition economy in the direction of the public interest.
Notice that both Prescott (1998) and Jovanovic’s piece in Kreps and Wallis
(1997) stress factors closely related to the structure of incentives when they stress
how successful resistence to adopting pieces of the huge already existing stock
of usuable technology (not only technical but also administrative) might explain
a lot of the current gap in standards of living across countries. We do not wish
to focus on the costs of developing new technologies like biotech because that is
well covered in the literature, although the idea of latent options stressed in this
article applies here too. See especially Romer’s website (Romer (1998)) for a vivid
discussion of the power packed by potential innovations for economic progress.
We also abstract away from negative externalities such as environmental degra-
dation due to adoption of the RCR. I.e. we are assuming that incentives are aligned
to the social accounting system which ensures that all social costs are borne on
private account. I.e. Social RCRs are identical to Private RCRs. We are focusing on
the general problem of harnessing the Schumpeterian engine of economic progress
for the social interest. The very important problem of incentive design to redirect
the Engine from finding private RCRs that advance the private interest and that
off-load costs onto the rest of society towards finding social RCRs that advance
the social interest where all cost-causers bear their own costs is left for another
time. The approach taken here will be to write out a traditional structure–conduct–
performance outline of the impact on the rate of TFP growth due to the speed of
adoption of RCRs using existing technology. I hasten to add that, except possibly
for a slightly different approach to the idea of latent options, I am adding nothing
new to a very well-worked literature. However some of the ‘tool scenery’ may be
interesting and may be useful to readers in their own research. After, hopefully
shedding some light on quantifying the strengths of carrot and stick forces to in-
novate under different industrial structures and different institutions and, perhaps,
adding a little bit to the well developed literature in this area, I wish to turn to the
problem of identification of spillovers that catalyze growth and positive feedback
forces in the ‘growth regressions’ literature reviewed by Durlauf and Quah (1999).
This discussion will use some econometric tools developed in Brock and Durlauf
(1999) and some theoretical ‘complexity’ tools developed in Brock (1993) and
Brock and Durlauf (1995).
The treatment of the well-worked area of innovation and adoption theory will be
brief. Optimal stopping theory and the Feynman–Kac formula (e.g. Duffie (1988))
will be used to add a little material on the interaction of variance and industrial
structure that might shed a bit of light on forces behind TFP growth that the existing
literature has not quite cleaned up. Before we begin this part let us consider growth
regression analysis and state the basic equation used in such analysis from Durlauf
and Quah (1998), which we denote by (DQ.19),
log y(t) = log(b) + log(A(0)) + gt + [log(y(0)/b)

+ log(A(0))] exp(lt), (DQ.19)
where y(t) denotes observable per capita income, A(t) denotes the ‘level of tech-
nology’ at date t, d log(A(t))/dt = g, b and l are parameters that depend upon other
economic parameters such as the rate of population growth, the rate of depreciation
on capital stock, savings rate, and the like.
DQ show how (DQ.19) emerges from the main received growth theories and
they use it to organize their discussion of forces that shape the ‘convergence com-
ponent’ (the term involving exp(lt)) and the ‘levels component (the rest of the
right-hand side).’ They point out that since log(b), log(A(0)) are unobserved just
about any pattern of ‘cross-country growth and convergence is consistent with the
model.’
Thus we need to model the forces that shape dlog(A(t))/ dt = g, A(0), b, and
l. Hence, while reading the part below, readers might view it as a discussion of
some economic forces that shape g = d log(A(t))/ dt. Readers should imagine that
they have data on a cross section of countries where they are estimating a system
like (DQ.19) with one equation like (DQ.19) for each country. Theory like that
presented below will be used to suggest regressors to be inserted into the RHS of
(DQ.19). Economic forces that lead to diffusion of technologies across different
countries such as international trade lead to cross equation dependence for systems
like (DQ.19). More will be said about estimation of such systems later.
13.2.1 Optimal stopping tools and exploitation of options

I shall draw on optimal stopping theory to compute the optimal time of intro-
duction of an innovation or project which costs F at time of introduction and
generates a flow S(t) of net benefits from that time on. The introduction cost F
will be widely interpreted so that it can stand for costs of overcoming employee
resistence to a new technique following Prescott’s discussion (1998). A complete
benefit/cost welfare analysis might include the discomfort costs of adapting to the
new innovation that are borne by the businesses executives, middle management,
and workers. More will be said about this below. The results will be organized
in a structure–conduct–performance framework so familiar from theoretical in-
dustrial organization analysis, but carried out from the vantage point of optimal
stopping theory with the goal of contributing to understanding of the forces that
shape the rate of growth of TFP. This analysis exploits elements of the work of
Brock, Rothschild and Stiglitz (1989) and their references to the works of Brock,
Miller, Scheinkman, and Ye.
I will use these qualitative results to discuss probable impacts on the time
evolution of the TFP quantity, log(A(t)), in the Durlauf and Quah (1999) survey of
the empirical growth literature. First I shall analyze the simplest possible problem
under five industrial organizations.
• I1 denotes the social interest.
• I2 denotes a monopolist who cannot price discriminate and, hence, cannot
extract out the area under the demand curve at each point in time.
• I3 denotes competition between two firms and the first to build will have a
monopoly from that point on.
• I4 is same as I3 but the monopolist will have the ability to price discriminate.
• I5 Is the general problem where the incentive system of the management
puts different weights on the consumer benefit side and the adoption side of
the problem. This may be due to distortions induced by regulatory systems,
the tax system, whether the enterprise is a government department, a partial
public-private enterprise, etc.
Although we placed I5 at the end, it may be the most important of all. Many
economic entities operate under incentive systems placed upon them that ‘tax’
heavily the gains to potential productivity enhancing changes in operations but
do not give full ‘tax-offset,’ i.e. ‘tax deduction,’ for the costs, both personal and
corporate of carrying out those changes. For example the management may realize
that their enterprise is inefficient but it does not want to bear the personal cost of
having to fire people. If the tax system captures 50 per cent of the profit, the
management’s contract gives a bonus of 1 per cent of the after tax profit, and the
personal income tax takes half of this, while there is no offset for the personal cost
of ‘taking the heat’ of firing the excess labor, it is not surprising that the enterprise
will be over-manned. See Gibbons’ piece in Kreps and Wallis (1997) for a review
of work on incentive design in organizations. Turn now to the simplest problem
for analysis. Let’s start with the simplest problem which I shall call the DATRAN
problem after one of my articles in Evans (1983, Chapter 8)). This is the problem
of identifying a new market whose profit flow is growing at some fixed positive
rate and building a facility that costs F to capture it. Properly measured TFP will
stress the development of new goods and new markets as emphasized by DeLong
where he reviews the proper measurement of economic progress at his website
(1998).
Later we shall discuss the cost side where there is technology available which
is reducing unit cost at some rate ‘c’ but it costs F to adopt it. The analysis of both
problems is similar. Here is a prototype example problem. When should society
build a digital data transmission network which will cost a fixed setup cost F, will
serve the entire demand, and where the social value of the net benefit per period
generated by the network S(t) is growing at constant rate a? To be more precise
let demand at date t be given by D(q) exp(at), gross benefit be the area under
D(·) exp(at) up to q, call this B(q) exp(at), let unit cost be constant in q at all
dates t and be given by c exp(at). Letting gross benefit and cost grow at the same
rate makes the mathematics simple. This problem is the problem of when to build
a facility to capture a new market. Alternatively it may be viewed as the problem of
selecting the optimal time to adopt an RCR which yields a gain which is growing
at rate ‘a’ but the cost to adopting is F. The RCR adoption problem where cost F is
falling at some rate has a very similar structure to this problem, as we shall see later.
Both problems concern finding the optimal time to take advantage of a latent oppor-
tunity. The net benefit, S(t) = exp(at)[max{B(q) − cq}] = exp(at)S(0), includes
operating costs, depreciation, additions to accomodate growing demand, etc. To
repeat, under the constant growth rate assumption we have S(t) = S(0) exp(at).
Let the interest rate be the constant r.
I1: The social interest

We take society’s problem to be to choose t to
maximize PV (t) − F exp(−rt), (13.1)
where PV (t) denotes the present value of the stream of net benefits from t on. This
quantity is computed by integrating the present value flow, exp(−rt)S(t) from t
to infinity. One obtains
PV (t) = exp[(−r + a)t]S(0)/(r − a). (13.2)

Note that r > a for this to be finite. This makes sense. If net benefits grow faster than
the rate of interest then the value of the project grows without bound. Differentiate
(13.1) to obtain the first order condition for a maximum, and solution T ,
S(0) exp(aT ) = rF, T = (1/a) ln(rF/S(0)). (13.3)

It is easy to check that T is the only solution to the first order condition and
it satisfies the second order necessary condition for an interior maximum. This
completes the analysis of the deterministic case for the social interest. Note that
larger S(0) and smaller rF gives earlier building time T . If T is negative in (13.3)
then the ‘corner’ solution, T = 0, is optimal. We shall assume interior solutions in
the following.
I2: Monopoly
For monopoly, profit, P(t) = exp(at)[max{D(q)q − cq} = P(0) exp(at). Copy
the procedure above to obtain
P(0) exp(aT ) = rF, T = (1/a) ln(rF/P(0)). (13.4)

Since P(0) < S(0) we have building at a later date than socially optimal. However,
there is spillover at each date beyond the building date. The spillover not captured
by the monopolist is the area S(q∗ ) − D(q∗ )q∗ where q∗ is the monopoly level of
output. This spillover grows at rate a after building the facility to produce for the
market.
I3: Competition to capture a monopoly position
Consider now a form of competition where there is a race and the first one to enter
by paying F at date T , captures a monopoly like the above from T on. This forces
PV (T ) = exp[aT ]P(0)/(r − a) = F, T = (1/a) ln((r − a)F/P(0)). (13.5)
Here, as one would expect, building takes place earlier. The ‘winning’ monopolist
builds earlier and captures no PV profit for herself. She does this to avoid being
displaced in the market by an earlier builder if there were still positive net PV
to be had. Spillovers of the amount exp(at)[S(q∗ ) − D(q∗ )q∗ ] are captured by the
public from the building date on.
I4: Competition to capture a perfectly discriminating monopoly position

Consider a form of competition where the first one to enter by paying F at date
T captures a perfect price discriminating monopoly where the operator can use
nonlinear prices to sweep out the entire area under the demand curve and capture
the entire net benefit at all dates after entry. This forces
PV (T ) = exp[aT ]S(0)/(r − a) = F, T = (1/a) ln((r − a)F/S(0)). (13.6)
Here we have an earlier date of building the facility than competing monopolists
for the market in (13.5). Furthermore if F measures the true cost to society as a
whole of undergoing the cost of building, the entire capitalized net benefit from
building is absorbed by the cost. This is truly a society that is enslaved by a rat
race of building too early to service emerging demands. The discussion in Evans
(1983, p. 219) shows how the welfare cost of this kind of rat race can be very large
to the society as a whole. The ‘dual’ case of a rat race of adopting RCRs too early
from a social welfare benchmark will be examined below.
I5: ‘Actual’ enterprise management in the ‘real world’

Given the above analysis of the social interest I1, the analysis of I5 is trivial. We
simply replace (13.1) with (13.1 ) maximize (w1)PV (t) − (w2)F exp(−rt), where
w1 is the fraction of PV (t) captured by the managers and w2 is the fraction of
costs borne by the managers as determined by their incentive contracts. Much of the
discussion on the New Zealand restructuring reforms reviewed by Lewis Evans
et al. (1996) involved the design of appropriate ways of designing managerial
contracts to provide an incentive to the managers to capture potential gains to the
enterprise as a whole on the public interest standard rather than ‘rent-seeking’ the
enterprise for themselves. Writers on this subject (e.g. Evans et al. (1996), Gibbons
in Kreps and Wallis (1997) stress how the careful design of an accounting system
upon which to write incentive contracts for management and workers is critical.
Equation (1 ) is a crude attempt to capture this idea in a tractable way within the
scope of this review.
Naturally if w1 is close to zero and the management has to bear the costs of
unpleasant relationships with their workforce in gearing up to enter a new market or
adopting different work practices to capture an RCR then obviously management is
not going to feel motivated to do it. Indeed when one looks at the implied incentive
contracts in some enterprises (not just the public ones) it is amazing that anything
gets done at all. We encourage the reader to look at Lewis Evans et al (1996) for
a thorough and vivid discussion of the New Zealand case.
To sum up, this deterministic analysis of building to service emerging markets,
and bearing costs now to adopt RCRs which promise to deliver future gains, while
trivial, does help us focus attention on cases where the rate of building may be
too fast or too slow for the social interest, depending upon the industrial structure.
This kind of analysis is easy to extend to include imperfect property rights, rule of
law and theft, taxation, possible government confiscation without compensation,
political instability, infrastructure (which impacts profit flow through cost of pro-
duction), human capital (which reduces adoption cost F), subsidies to reduce F,
etc. These factors impact adoption time and would feed into the forces emphasized
by Barro (1997). Turn now to uncertainty in the growth of value produced by a
facility once it is built.
The general theme will be that an increase in the instantaneous variance of
uncertainty creates an ‘option’ whose ‘option value’ will lead to an increase in the
expected present value of the problem and a shortening of the building time due to
exercise of this ‘option.’ This theme has been explored by Brock and Rothschild
(1986), Brock, Rothschild, Stiglitz (1989), hereafter ‘BRS’, Dixit and Pindyck
(1994), Malliaris (1982), and others.
In order to get started, consider the problem
Maximize E{exp(−rT )(PV (T ) − F)}, (13.7)

subject to the Ito stochastic differential equation,
dS(t)/S(t) = a(S(t))dt + b(S(t))dZ, S(0) = S0 given. (13.8)

where E{·} denotes mathematical expectation conditioned on S(0). Here one
chooses the ‘stopping time’ T as a function of past information to solve (13.7). To
do this, one can first compute the quantity PV (T ) in (13.7), for each given S(T )
by using the Feynman–Kac formula as in Duffie (1988, p. 307). I.e. E{PV (T )}
such that (13.8) starting at initial condition S(T ) solves an ordinary second order
differential equation with two boundary conditions. Doing this, one can write the
quantity in the form R(S(T )). Then one may follow BRS (1989) and solve
Maximize E{exp(−rT )(R(S(T )) − F)}, (13.9)

subject to (13.8), by using ‘barrier strategies’. Do this by restricting oneself to the
set of stopping times T that stop the first time a barrier S > S(0) is hit by (13.8)
starting from S(0). Hence the problem reduces to the scalar problem
Maximize L(S; S(0))[R(S) − F], (13.10)

where L(S; S(0)) is the Laplace transform of first passage of (13.8) from S(0)
to S. The Laplace transform solves an ordinary differential equation of second
order with two boundary conditions of particularly simple form. There is a subtle
technical issue of locating sufficient conditions for the restricted class of barrier
solutions to be optimal in the class of all stopping times adapted to the filtration
generated by (13.8) but we refer the reader to BRS and references for that. Barrier
solutions will be adequate for the illustration to be given here. Let us do the simplest
case where a(S) and b(S) are constant.
Put v equal to one half the square of b. The solution of (13.8) is lognornal,
i.e. ln(S(t)) is normally distributed with mean m(t) = ln(S(0)) + (a − v)t and
variance (2v)t. It is convenient to transform variables by putting X = ln(S) in
(13.8) to obtain
dX = a dt + bdZ, a = a − v, X (0) = X0 , (13.11)

which, as we said above, has solution X (t) normally distributed with mean X 0+a t,
variance (2v)t. Compute the expectation conditional at date t on X (t) of the integral
from t to infinity of exp(−rs + X (s)) by the moment generating function formula
for the normal (E exp(bZ(t)) = exp[(2v)t]) to obtain
R(S(t)) = S(t)/(r − a) − F. (13.12)

Recall that we have changed units by putting X = ln(S) to get a linear stochastic
differential equation for the dynamics of the state variable. Suppress X in the
notation for L(X ; X (0)), and write L(X ; X (0)) = L(X (0)). It is easy to show (See
BRS (1989) or Malliaris (1982), for example) that L satisfies the second order
ordinary differential equation,
rL − a L − vL = 0 (13.13)
The roots of the characteristic equation for (13.13) are both real with one positive
and one negative. Since a Laplace transform is always positive and bounded above
by unity only the positive root is sensible. Furthermore if S(0) is already at the
target barrier S, the Laplace transform is unity. Hence the solution of (13.13) is
given by
L(X ; X (0)) = exp(P(X (0) − X ), (13.14)

where P denotes the positive root of the characteristic equation for (13.13),
P = (1/2){−a /v + [(a /v)2 + 4(r/v)] 2 }.

1
(13.15)
We may now write the value of a problem starting at X (0) with barrier target X ,
W (X (0); X ) = exp(P(X (0) − X ))[exp(X ))/(r − a) − F]. (13.16)

The procedure above has reduced a hard problem to a simple problem. For optimum
problems one can now apply simple calculus to differentiate (13.16) with respect
to X and use the first and second order conditions for a maximum to locate optimal
X . For entry competition problems, one finds X that sets (13.16) to zero. One can
now analyse the impact of parameter changes such as increases in variance ‘v’
upon the entry times under the structures I1–I5. Here is a very brief sketch of how
to do this.
First, assuming the economically sensible case r/a > 1, it is easy to show
that P(v) decreases from P(0) = r/a to zero as v increases from zero to infinity.
Since P(·) acts like a discount rate, a decrease in the discount rate tends to make
the problem worth more. An increase in variance ‘v’ tends to make the optimal
stopping size X bigger. Increases in stopping size X tend to make building occur
slower, i.e. at later times, in order to capture the option value contained in the
variance increase. Notice that one must restrict P to be between unity and r/a in
order for the optimum problem to be well posed.
The mathematics sketched above now makes it possible to analyze the impact
on problem values and X-sizes of parameter changes under different structures
like I1–I5. We refer the reader to BRS (1989) and Dixit and Pindyck (1994) for
details on how to solve these kinds of problems. BRS (1989) shows how to do local
comparative statics such as local increases in variance for a wide class of these
problems as well as do the mathematics of ‘smooth pasting’ and ‘free boundary’
problems in general. Malliaris (1982) has an especially straightforward treatment
at an elementary level. Second, I1 gives a benchmark, the social optimum, against
which the performance of other arrangements like I2–I5 may be measured. Turn
now to the ‘dual case’ of RCR theory.
Let F(t) be given by the Ito stochastic differential equation,
dF/F = −cdt + bdZ, (13.17)

and let the instantaneous standard deviation be zero on the growth of demand in the
above problem in order to avoid dealing with multidimensional optimal stopping
problems as in Brock and Rothschild (1986). Think now of F as a cost to be paid
which lowers unit cost of production by a fixed amount and this translates into
an increase in profits (or social benefit) of D to be computed below. Before going
further we must make a digression on expectations, general equilibrium effects,
and game theory.
We ignored general equilibrium effects and game theoretic modelling issues in
setting up the five comparison ‘institutions’ I1–I5 on the ‘primary side’ treatment
above. On the RCR side of the analysis we must confront head on the issue of how
long the adoptor of an RCR expects to keep her new advantage before another
competitor adopts an RCR and uses that advantage to undercut her.
We can only speculate about what a completely specified game theoretic ‘com-
mon knowledge’ model of adoption ‘equilibria’ would produce. One might, how-
ever, confidently predict that there will be ‘surprises’ like parallels to ‘no trade’
theorems for example. See Dekel and Gul’s survey in Kreps and Wallis (1997)
for a discussion of common knowledge frameworks in game theory and results
like no-trade theorems. We wish to warn the reader about the complexities that a
proper game theoretic treatment of adoption equilibria would raise. See Aghion
and Howitt’s (especially their references to patent race games and the like) piece
in Kreps and Wallis (1997) for subtleties that should be dealt with, that we are
ignoring here.
This complexity is simply avoided here by positing a random variable K = T −t
with density Pr{K = k} that determines how long the RCR adoptor will be allowed
to keep her advantage before she is displaced by someone else if she adopts an
RCR at date t and her advantage is snatched away at T > t. I.e. think of K = T − t
as the keeping time of the advantage. Let the expected value conditional on t of
this advantage be denoted by g (g for ‘gain’) which is assumed, for simplicity, to
be independent of t. Then her problem is to choose a stopping time t (adapted to
the process (13.17) for F(t)) to solve,
maximize E{exp(−rt)[g − F(t)]} (13.18)

subject to (13.17) above. Notice that this problem would be analytically similar
to that already treated above if ‘g(t)’ was governed by the dynamics (13.8) and
F(t) were constant in t. I.e. R(g(T )) in (13.9) would be interpreted as the expected
present value of incremental profits from the RCR which costs F to implement.
Thus the reader may use (13.8) and (13.9) as the solution procedure for an appro-
priate class of RCR timing problems as well as the market timing problems treated
above. In view of this similarity we shall be very brief in the following exposition
of an RCR problem that cannot be completely mapped into the treatment of (13.9)
above.
Let X (t) = ln(F(t)) and consider the Laplace transform of first passage from
initial cost X (0) down to X < X (0),
L(X ; X (0)) = exp(P(X (0) − X )), (13.19)

where P is now the negative root of the equation (13.13) with a = −c − v. As in
the above analysis, we may reduce problem (13.18) to
maximize {exp(P(X (0) − X ))[g − exp(X )]}. (13.20)

Analysis of this problem is now a straightforward adaptation of the above analysis.
13.2.2 A summing up
We have seen enough variations on the ‘adoption problem’ to sum up what we have
learned. First, the structure I5 stresses the importance of designing an accounting
system upon which award to bonuses to the management of the enterprise for the
unpleasantness of bearing cost F in order to get the gain ‘g’ from an RCR or
identifying and entering a new market. Much of the wave of privatizations and
‘corporatizations’ during the 1980s and 1990s was driven by the recognition that
incentives to innovate were severely distorted by the regulatory framework in many
countries. See, for example, Lewis Evans et al. (1996) for a review of the New
Zealand ‘experiments’ in restructuring to induce efficiency enhancing incentives.
Second, the instantaneous variance in the rate of gain to new market entry or
gains to RCR as well as the instantaneous variance in the rate of fall of adoption
cost is a modelling way of capturing the presence of latent options the size of which
increase with these instantaneous variances. The performance of the four different
structures I2–I5 in capturing these opportunities (measured relative to the public
interest standard I1) can vary widely. Perhaps the worst structure from the social
welfare perspective is a variation of I5 where the management gains a trivial part
of ‘g’ but bears a nontrivial part of F if they adopt.
Third, the list of parameters in the structures I1–I5 can be given more concrete-
ness by looking at a particular industry such as agriculture where cost and benefit
parameters to an individual farm of adopting a technology are influenced by the
regulatory and tax framework; reference group adoptions; activities by government
such as agricultural experiment stations and extension offices; relative input price
shifts which induce innovations in the direction of economizing on relatively more
expensive inputs as in the Ruttan theory of induced innovations; volatility in the
ratio of benefits/costs to adoption of RCRs; and more. See Carlson et al. (1993)
for an excellent review of agricultural and resource economics issues.
13.2.3 TFP growth, aggregate production functions, and growth

regressions
The reader may ask at this point: What does the analysis of RCRs and market seizing
have to do with the dynamics of technical change that appear in the endogenous
growth theories reviewed by Baumol (2001, this volume), Hahn (2001, this volume)
and others at the Siena School, July, 1998? For example, let X be a vector of inputs,
let F(X ) be a constant returns production function, let d log(A(t))/dt measure TFP
growth. Then given a set of input prices, assuming they are constant though time,
suppressing them in the notation, we can write the unit cost function as C(0)/A(t)
so that the rate of RCR is measured by |d log(C(0)/A(t))/dt| = d log(A(t))/dt.
Consider the parameter changes and changes in industrial structure that make
adoption times earlier in the analysis above. It is plausible to conjecture that the
same changes are positively related to changes in at least the level of A(t) (and
perhaps the rate of growth, dA/Adt, over an appropriate interval of time). While the
above analysis has to be aggregated up to the economy-wide level to be connected
to the determinants of A(·) as a complete theory like Aghion and Howitt’s work
in Kreps and Wallis (1997) and their book (1998) has done, it is nevertheless
suggestive of regressors that one might want to insert into growth regressions. The
analysis is used to organize a list.
Think of the analysis of the time to build as not only the time to build to service
more of an existing market, but also the time to build to service a latent or emerging
market. One could think of the actual building to service a latent or emerging market
as the production of a ‘new’ good. On the ‘dual or cost’ side think of the time to
adopt an RCR that will increase profits by an amount D which grows at some rate
after adoption. Consider the key parameters in the above analysis: (Primary or
Revenue Side) (i) the rate of growth ‘a’ of potential profits in an emerging market;
(ii) the instantaneous standard deviation, ‘v’ of those potential profits; (iii) the cost
F of building a facility to service the emerging or latent market; (Dual or Cost Side)
(iv) the rate of fall of adoption cost ‘c’; (v) the instantaneous standard deviation
‘v’ of the rate of fall of adoption cost; (vi) the rate of growth of profit before and
after adoption of the RCR.
The parameters and their effects help focus concentration on key determinants
of growth on the revenue side and the cost side. First, the impact of infrastructure
such as reliable transportation, communications, and power networks is hidden in
the notation. Obviously it takes a modicum of infrastructure to even build a facility,
much less use it to service a market and extract profit from this activity. Hence if
this kind of public infrastructure is complementary with production and decreases
in its quality and quantity take place, we can expect building to be delayed and
RCR adoption to be delayed.
‘Rules-of-the-game’ embodied in good government and societal norms
(stressed by ‘social capital’ in sociology (Coleman (1990)), Putnam (1993) in po-
litical science and, Alesina (1997) and North in Arthur, Durlauf and Lane (1997)
in economics) are key to good growth performance. Crafts in Kreps and Wallis
(1997) gives a broad ranging survey of research into the forces that drive disparities
in growth performance.
Putnam’s (1993) study of the regional governments of Italy comes close to a
‘natural experiment’ to study the differential impacts on economic performance of
differential quality measures of regional governments. Putnam argues that cross
regional differences in governments in Italy have dramatic impacts upon regional
performance, not only in economic growth, but also in satisfaction of the residents
with their lives. We shall list below a collection of things that governments can do
that stifle growth. Doing the reverse would, of course, stimulate growth. To put it
bluntly, government can be a macroparasite or a macrocatalyst when it comes to
growth. For example if government is allowed to do the equivalent of randomly
take the facility’s profit income after it is built, this factor increases the ‘effective’
interest rate ‘r’ and building is delayed. If government taxes profits at a constant
rate, this factor multiplies the present value of after-tax profits, building is delayed
and RCR adoption is delayed (if, as is realistic, the full cost, both material and
mental, of RCR adoption is not off-set by tax deductions). If taxation of current
profits is progressive, then the value of latent options on the revenue side may be
lessened and building delays may result. If criminal activity by both citizens and
government officials cuts profits or increases building costs then building will be
delayed.
If government installs a set of rules and regulations that place burdens on
RCR adoption by increasing the cost, F, then RCRs will be delayed. For example,
government can impose rules that help workers and management resist adoption
of RCRs and, hence, increase F. Government can impose burdens on foreign
competition by closing off international trade via protectionism. But notice that
while this closing off will reduce adoption time, it could also reduce adoption rat
races so the social welfare calculation becomes subtle. By this point, the reader can,
no doubt, list many ways, perhaps 1001 ways, that the incentives to adopt RCRs
and to build facilities to enter and to service markets can be cut. Hence an economy
may die (or grow) due to 1001 cuts (enhancements) in incentives. Putnam (1993)
puts forth 12 measures of institutional performance of regional governments that
enhance economic growth and satisfaction of its people. One can imagine building
an index of quality of government (much in the spirit of Barro (1997)) following
Putnam and inserting it as a regressor in growth regressions. Opening the mind
even further towards finding the avenues of death of an economy from 1001 cuts in
incentives shows up in Harberger’s article (1998) by looking at more disaggregated
levels than is usual in growth studies.
Harberger (1998) makes a strong plea to analyze the determinants of TFP
growth (both positive and negative) at a much more disaggregated level than the
national level, i.e. the industry level is better and the individual firm level is better
yet. In our notation Harberger, imagines each individual firm continually solving
little optimal adoption problems by choosing adoption times of RCRs in 1001
ways and each firm is embedded in an industry of firms doing the same thing.
13.2.4 Welfare, spillovers, adoption avalanches

The five structures I1–I5 were introduced, not only to illustrate how the industrial
structure impacts entry and adoption times but also to expose subtlety in welfare
analysis. Notice that structure I4 illustrates a rat race scenario where building to
service new markets and adoption of RCR’s takes place too fast relative to the
social interest. It should be emphasized that all of the analytics here are partial
equilibrium analytics so one can not do a serious welfare analysis. Obviously the
analysis should be carried out in a general equilibrium framework more like that
in the surveys of Aghion and Howitt, Crafts, Jovanovic in Kreps and Wallis (1997)
and their references to writers such as Grossman and Helpman, Lucas, Romer,
Krugman and others working the general equilibrium approach, as well as Iwai
(2001, this volume). A general equilibrium approach with heterogeneous agents
along these lines that might be extended with optimal timing components under
different industrial structures is that of Horvath (1998) who has an entry/exit com-
ponent in his model for firms. One could imagine building a general equilibrium
model by using building blocks like the asset pricing models studied by Altug and
Labadie (1994) and Akdeniz and Dechert (1996) where there is a representative
consumer who generates the stochastic discount factors which are taken as para-
metric to value maximizing firms and where the value maximizing firms are solving
intertemporal allocation problems and optimal timing problems like those treated
above. The representative consumer drives the asset trading market in claims to the
earnings of these firms which, in turn, determines the stochastic discount factors
taken as parametric to these firms. This structure is general equilibrium ‘enough’
to do welfare economics and general equilibrium analysis. I suspect that some of
the insights gained from the optimal timing analysis above will still hold in this
general equilibrium setting. Of course the simple analytics of the above treatment
will no longer survive and quite possibly numerical methods as in Judd’s book
(1998) as used by Akdeniz and Dechert (1997) will have to be adapted to carry out
this kind of analysis. But, at least the numerics are now within our reach.
Another future research project, more along the lines of complexity theory as
described by the introduction to Arthur, Durlauf and Lane (1997) is to trace the
linkages through the system of an adoption of an RCR by one firm. For example,
it seems sensible that adoption of an RCR by one firm is going to put pressure
on the other firms in the industry. If they fail to adopt the RCR, the adoptor can
shave her price and capture all of the market (assuming perfect substitutes are
being produced) up to her production capacity. If she builds additional plant up
to market capacity at her shaved price, she could capture the whole market. The
analysis above can be adapted to include pressure on others to adopt after the first
adoptor uses a limit pricing strategy. Prescott (1998) stresses how relative price
changes on competing goods can put pressure on firms within an industry to adopt
RCR’s rather like the pressure applied by a limit pricer with lower costs. Barro and
Sala-i-Martin (1995) treat diffusion models of technical adoption as an impetus in
TFP dynamics and discuss empirical issues raised in the estimation and testing of
such models. More will be said about diffusion models later.
Complexity-based models of impulses propagated by locally interacting agents
along the lines of Durlauf (1993) and Scheinkman and Woodford (1994) may be
useful in thinking through the impact of an RCR adopted by one firm in a ‘graph
network’ of imperfectly competitive firms where the strength of the graph link
between firm i and firm j increases with the size of the cross elasticity of demand
between i and j. This linkage through a network of cross-elasticities is a factor
determining the size of the ‘avalanche’ of RCR adoption across this network of
firms set off by an RCR adopted by one firm who lowers her price. If enough
imperfectly competing producers cut their prices, that could force a ‘distant’ firm
(‘distance’ measured via the size of cross-elasticity of demand) to adopt an RCR
earlier which, in turn puts pressure on its nearest competitor to adopt an RCR. One
could then study the size distributions of such ‘waves’ of RCR’s and see if power
law scaling and thick tailed distributions appear as discussed by Scheinkman and
Woodford (1994).
Here is another way to illustrate TFP dynamics in an instructive way. Harberger
(1998) presents a construct which he calls a ‘sunrise/sunset’diagram where he plots
percentage of value added on the horizontal axis and percentage of total RCR on
the vertical axis. He shows for four 5-year time spans, for US manufacturing, that
the percentage of value added by the top ranked RCR achievers to achieve 100 per
cent of the total RCR varies from around 12 per cent to almost 50 per cent and
the industries that achieve this vary highly across the four 5-year time spans. He
also shows that a lot of industries achieve negative TFP (i.e. their costs go up, not
down) during his four 5-year periods. He reports on studies that show that similar
patterns appear at the firm level as well as the industry level. Curiously this suggests
a ‘fractal’ type of self similarity structure in TFP dynamics. That is to say that the
relationship of TFP at the firm level to TFP at the industry level is rather like the
relationship of TFP at the industry level to TFP at the level of the whole economy.
This ‘self’ similarity of structure of sunrise/sunset diagrams at different levels of
aggregation is useful discipline for theory. For example, Harberger states that the
‘mushroom’ pattern revealed in his sunrise/sunset diagrams suggests that the kind
of external spillover (if any) in propagating TFP dynamics is much more likely to be
a localised spillover rather than a global spillover. This insight already sheds light
and discipline on how one should enter spillovers in theoretical growth models in
order to generate simulated sunrise/sunset diagrams consistent with Harberger’s
evidence. Returning to the issue of useful tools to detect evidence of spillovers,
one might use Harberger’s construct to see if there is any difference in the TFP
dynamical pattern found as revealed in sunrise/sunset diagrams across different
historical periods, in particular, the current period, which many commentators
think is ‘different’ due to the huge potential spillovers from the nonrivalry and
nonexcludability of the ‘information goods’ being produced in todays leading
sectors. Recall that Baumol (2001, this volume) stresses the apparently large size
of spillovers of benefits from inventors and creators to the population of past
technological revolutions as well as the present, even though the earlier innovators
did not have to deal with problems of nonrivalry and nonexcludability as much as,
perhaps, in current times.
For example, DeLong (1998) reviews, from a long term historical perspective,
whether there is any difference between today’s ‘leading sectors’ of growth such
as the computer industry centered in Silicon Valley and the biotech industries cen-
tered near university complexes. He argues that there is not an order-of-magnitude
difference in TFP growth between these current media-hyped leading sector in-
dustries and leading sector industries of the past when TFP is properly measured
as the number of hours a median worker must work to purchase a unit of service
flow. The prosaic service of casting light into a dark room is a useful example to
gain perspective. Think of the dramatic drop in number of median worker hours
needed to purchase a unit of this service during the ‘Edison’ period. However,
DeLong (1998) does say that there is a major difference in the degree of (i) rivalry,
(ii) excludability, and (iii) transparency in the goods being produced by today’s
leading sectors in contrast to leading sectors in the past.
This difference in degree of three essential components of private property,
stressed by DeLong, may translate into an increase in the degree of spillovers in the
adoption models sketched above. This concern with spillovers due to an increase
in the permeability of property rights brings us to the use of random field models,
such as Ising models (Durlauf (1993)) in cycle and growth modeling and related
empirical work using such models. Brock (1993) shows how useful approximations
to the complicated probability structures of random field models, called ‘mean
field approximations,’ may be introduced in such a way that one can borrow from
the large literature in discrete choice econometrics in order to do empirical work
with interacting systems models. See also Durlauf’s and Ioannides’s papers in SFI
(II) for more on this. A central message of random fields models and their mean
field approximations is this: as the degree of complementarity increases in these
frameworks, alternative stable states appear and nonergodic behavior appears at a
critical level of the degree of complementarity.
Hence, we can imagine building a model where there is a ‘lattice’ of ‘sites’ at
which are located economic entities facing adoption decisions as in Durlauf (1993)
or there is a graph structure as in some of the papers in SFI (II) that describes the
network of potential spillovers amongst adoptors. I.e. let there be an arrow from
agent i to agent j if i’s adoption lowers the cost to j’s adoption and let the ‘size’
of this arrow denote the strength of the cost reduction spilled over from i to j. Let
each adoptor solve optimal adoption timing problems like the above but ignore her
effects on the others connected to her. Brock (1993) and Durlauf (1993) discuss
mathematics of these kinds of models in the simpler setting where the decision is
binary: Adopt or Do not Adopt. The treatment of optimal timing to adoption in
such interacting settings will surely be complicated.
However, one could imagine building general models that contain modules
from the received innovations literature such as Aghion and Howitt, Jovanovic in
Kreps and Wallis (1997), Brock’s (1993) work using Curie–Weiss models and mean
field approximations to general interacting particle systems models, Durlauf’s
(1993) work using random fields models that might possibly isolate a critical
degree of spillovers due to changes in the degrees of rivalry and excludability
in leading sector industries as they impact the rest of the economy. This kind of
modeling might shed light on the consequences for general TFP performance for
the economy as a whole of changes in the degree of rivalry and excludability in
particular industries, especially leading sector industries. At the very minimum
just thinking about carrying out such a modeling exercise motivates work on the
econometrics of spillovers to be discussed below.
13.3 Complexity theory: any help?

In Section 13.2 we set out a context for evaluation of what complexity theory
has to offer the study of cycles, growth, and structural change. We organized the
discussion in terms of optimal timing for adoption of an RCR and briefly mentioned
the importance of thinking in terms of a hierarchy of time scales and ‘spatial’ scales
from the slow to the fast in time and from the large to the small in ‘space’. The
study of dynamics on smaller scales tends to require more disaggregated data.
Recall that Harberger (1998) stressed a disaggregated approach to understand-
ing dynamics of TFPs and RCRs and used the expository device of ‘Yeast versus
Mushrooms’ to capture the idea that TFP/RCR progress in any given decade is
concentrated in a handful (mushrooms) of industries rather than being spread out
more evenly across the whole economy. At the risk of repeating what was said in
Section 13.2, one could imagine constructing Stommel type diagrams where the
‘spatial axis’ is not ‘geographic space’ but ‘space’ measured as an index of similar-
ity across industries. A spectral version of a Stommel diagram does not strike me
as useful in this context as a version where the temporal scale measured the size of
the TFP movement at the industry index, where the index would be measured by
something like an SIC code but quantised to stress product similarity. The pictoral
scaling of TFP movement on Stommel plots may help locate patterns that would
not be easy to see otherwise. See Clark (1985) and his references for the very
real technical problems of constructing Stommel diagrams and see Gunderson et
al. (1995) for some samples of the helpfulness of Stommel diagrams in pattern
detection in applications. Although we have been purposely vague in exact details
describing how to construct a Stommel diagram one can imagine creating such
constructs after looking at the examples in Gunderson et al. (1995) and using them
on ‘spatial/temporal’ (‘space’ being interpreted broadly) data sets to test the hy-
pothesis as to whether the data is generated by a ‘sandpile’ type threshold-cellular
automata type model used by Scheinkman and Woodford (1994), for example.
If one plotted the time to relaxation and the ‘size’ of ‘avalanches’ generated by
such models on a Stommel diagram, one might see ‘blobs’ at all size scales on the
diagram.
In contrast ‘Rnet’ models stress two or three critical scales of time at which
the bulk of activity ‘clumps’ with corresponding ‘clumping’ on spatial scales. See
Gunderson et al. (1995) for explanation of this reasoning, case studies where it
appears to fit, and Stommel diagrams that illustrate it. Notice that ‘Rnet’ type
models are essentially stochastic differential equation models whose underlying
deterministic differential equation (called the ‘skeleton’of the model) with a two or
three level hierarchy of time scales where the slower time scale movements serve
as bifurcation parameters for the faster scale below it in this hierarchy. ‘Spatial’
dynamics are linked by dynamical spillovers in these models. We have said enough
about the ‘theory’ of pattern formation. Turn now to the ‘econometrics’ of pattern
recognition.
13.4 Pattern recognition

In the first SFI Volume on the Economy, SFI (I), there was some attention focused
on the problem of pattern recognition including the problem of testing for patterns
in noisy data, especially left-out structure in forecast errors of fitted models. Let
me give a quick sketch of this development up to the present following Brock’s
papers in SFI (I), SFI (II) and add a bit on newer work.
Suppose a model is fitted to data and the forecast errors {e(t)} are saved for
testing. For example, consider growth regressions as reported in Barro (1997) and
Durlauf and Quah (1999). Under the null hypothesis that the model being fitted is
the true model up to Independently and Identically Distributed (IID) errors, then
one can use a test for independence on these errors to specification-test the null
model. However problems arise because the true errors are not known, only the
estimated errors are available for test.
Brock, Dechert, Scheinkman, and LeBaron (1996), hereafter, BDSL, devel-
oped a test for IID errors which has the same first order asympotic distribution on
estimated errors as on the true errors (under regularity conditions typically satisfied
in applications). Their theorem takes care of the problem, that only fitted model er-
rors are available to test, but there is another problem. In most applications forecast
errors of fitted models (and perhaps the ‘true’ models) tend to display persistence
in variance (called autoregressive heteroscedasticity). So applicators of BDSL’s
work tend to estimate a model of this error variance persistence and ‘standardize’
the errors so that if the model is ‘true’ these estimated standardized errors should
be approximately IID.
Here is an example directly relevant to cycles and growth issues being discussed
here. Altug et al. (1999) tested for linearity of detrended real per capita US GNP,
call this data {x(t)}, and adduced evidence that {x(t)} was not generated by a
linear stochastic process. I.e. {x(t)} is not a sample from a stochastic process that
has a Wold representation with IID ‘drivers’ (‘innovations’ in technical language).
They used several tests for nonlinearity including BDSL (1996). The pattern of
rejections of linearity suggested the rejection was not just due to persistent volatility
of the fitted model errors.
They computed ‘Solow Residuals’ (a conventional measure of TFP), call these
{s(t)}, tested the hypothesis that {s(t)} came from a linear stochastic process, and
failed to reject. This lead them to examine the aggregate labor market where they
adduced evidence that the dynamics appeared to be asymmetric. This lead them
into an exploration of ‘labor hoarding’ type models, where there is an asymmetry
in cost between hiring and firing a unit of labor input, a review of the literature on
such models, and simulation of such models to see if such models could produce
simulated data consistent with their findings. While the jury is still out on whether
linear TFP dynamics coupled with asymmetric dynamics in aggregate employment
dynamics is consistent with the aggregate data used in their study, their study
illustrates potential usefulness of the kind of tools we are discussing here. See
Granger and Terasvirta (1993), Pesaran and Potter (1992), and Potter (1995) for
methods in nonlinear time series econometrics that are very useful for the study
of cycles and growth.
See Dechert (1997) for other examples of applications of nonlinearity tests,
including some applications to cycle and growth empirics. For practical guides
to nonlinearity testing see Dechert’s (1998) and LeBaron’s (1998) websites for
details, references, practical hints for applications, software discussions, research
findings, etc.
Testing procedures like the above tests for ‘left out structure’ of fitted model
residuals generated much evidence for ‘departures’ from ‘conventional’ models
but did not give good guidance towards the causes of these ‘departures.’ So this
evidence generated a flurry of activity towards designing sophisticated procedures
which used computational inferential procedures such as variations on bootstrap-
ping to produce null model distributions of statistics generated by the purposive
economic behavior being modeled, such as profits from trading strategies and
means and variances from returns to trading strategies.
The bootstrap could be used to bypass analytical impossibilities of calculating
null model distributions for such complicated econmically motivated statistical
objects such as trading profits, that are much more germane to the purposive
economic behavior that the null model purports to be describing. This procedure
has been pursued by Brock, Lakonishok, and LeBaron, (1992). See LeBaron’s
website (1998) for much more on this. The bottom line is that many conventional
models overpredicted volatility of returns and under predicted returns following
buy signals. If one tested the Efficient Markets Hypothesis (EMH) by testing
whether past returns help predict future returns by averaging the mean squared error
of prediction over all periods, one would miss these ‘pockets of predictability’.
However, it is not clear that this kind of evidence is a rejection of the EMH or is
simply evidence of mismeasured risk. Indeed it is not even clear that this evidence
(based on 100 years of daily data up to 1986) holds up on post-1986 data (cf.
Sullivan et al. (1999)). See Acar and Satchell (1998) for papers carrying out related
work (including one by LeBaron).
Authors doing this kind of pattern recognition work were concerned about ‘data
snooping’ biases. Data snooping biases arise from researchers and the research
community at large experimenting with different candidate models or trading rules
over the huge space of potential models or trading rules. To put it another way, the
research community, sharing a data base, can not avoid pre-test bias that comes
from fitting different models using that data base and spreading this knowledge to
a particular researcher who does a ‘specification search’ and selects a particular
model to be fitted to that data. The point is that the conventional test statistics used
to assess the quality of ‘fit’ of that model have contaminated null distributions
because of the communal and individual specification searches.
A major way to deal with this problem is out-of-sample testing provided that
the data used for the ‘out-of-sample’ testing is ‘truly’ out-of-sample and is not just
a ‘hold-out’ sample saved for ‘out-of-sample’ testing. Even hold-out samples may
have been inadvertantly snooped during the model selection phase. Recently work
by Sullivan, Timmerman, and White (1999) has presented methods that build on
recent advances in statistics that deal with this data snooping problem. It would be
desirable to apply the set of tools reviewed above to empirical work in cycles and
growth, especially the work on testing for ‘left out structure’ of received growth
regressions. Even more important would be to adapt the work of Sullivan et al.
(1999) on correcting for data snooping biases in received growth regressions.
However, there is a serious problem in using such data hungry procedures in
a field where data is scarce. Financial markets generate data at the ‘tic by tic’
frequency but cycle and growth empiricists must make do with quarterly and
yearly data with a few advanced countries producing monthly data. Nevertheless,
even a modest attempt to correct for data snooping biases using the above tools on
growth regression exercises like those reviewed by Durlauf and Quah (1999) may
produce useful information on reliability of the findings of that literature.
Indeed Durlauf and Quah (1999) are quite critical of received work on growth
regressions partly because of the rather large number of regressors (relative to the
size of the underlying data set) that have appeared in growth studies. One could
do, in principle, a simulation exercise inspired by Sullivan et al. (1999) to correct
‘significance tests’ for ‘regression fishing’ of the type that concerns Durlauf and
Quah (1999).
Another part of their critique concerns inadequate attention being paid by stu-
dents of growth in addressing plausible alternative hypotheses such as threshold
nonlinear models where the poor do not eventually catch up with the rich in con-
trast to conditional convergence models where they do eventually catch up (after
conditioning on the same regressor set in both types of models). Policy prescrip-
tions differ drastically across these two sets of models. Indeed much in the spirit of
‘lump analysis’ in ecology (cf. Allen et al. (1998)), Durlauf and Quah apply ‘lump
detection’ methods from statistics to adduce evidence of multimodality (cf. their
Figures 1, 7, 11a,b and surrounding text) to adduce evidence in support of lack
of convergence even after conditioning on conventional regressors that indicate
conditional convergence in ‘linear’ studies. It would be interesting to compare the
lump detection methods of Allen et al. (1998) in ecology with those of Durlauf
and Quah (1999) to detect evidence of possible multimodality in economic data,
not only in the raw data, but especially after conditioning on received regressors
that indicate ‘conditional convergence clubs’.
13.5 Complex econometrics

We have given the reader a foreshadowing of the econometric issues raised by
the use of complexity-based methods in economics and finance. Brock (1993)
and my paper in SFI (II) gives a discussion that relates tools such as statistical
mechanics, interacting particle systems theories, mean field theory, non-ergodic
models, and ensemble analysis to received discrete choice econometric methods.
Pure theory of social interactions and the complementary econometric theory is
reviewed Brock and Durlauf (1995), (1999). We give some speculations here on
possible use of these tools for growth empirics. The ‘new’ growth theory makes
much out of spillovers whether due to production and technology as in Romer’s
work or to human capital as in Lucas’s work (cf. discussions in Barro and Sala-i-
Martin (1995), Durlauf and Quah (1999), Hahn (2001, this volume), Hall (1996)).
These kinds of spillovers have a ‘social multiplier’ in the sense that benefits to
a spillover originator multiply out across to others in a type of chain reaction or
percolation throughout the whole system. The potential presence of such spillovers
raises the econometric issue of identifying such spillovers and separating them from
other effects that are essentially observationally equivalent to spillovers but have
no social multiplier. See Manski’s paper in SFI(II) for a review of his pioneering
work on this problem.
In view of the central role of spillovers and the resulting lack of Pareto Op-
timality in the ‘new’ growth theory, it is surprising to find so little written on the
practical econometric problem of identifying and measuring such spillovers using
econometric methods. See Barro and Sala-i-Martin (1995, Chapter 8, especially
their discussion of how spillover effects can lead to a dependency of country i’s
growth rate on country j’s variables, p. 275) for work on estimating diffusion type
models. Baumol’s work (2001, this volume) makes such a good case for the ex-
istence and size of such spillovers that, perhaps, formal econometric work is not
necessary to make a persuasive case for their existence, but it will still be useful
for measuring their size and importance.
In any event we sketch how tools reviewed in Brock and Durlauf (1999) might
be used to help identify and measure spillover effects in growth regressions. Con-
sider Barro and Sala-i-Martin’s (1995) and Barro’s (1997) growth regressions and
the framework laid out in DQ (1999) especially (DQ.19) in Section 13.2.
Following Manski in SFI (II) we say an ‘endogenous’ social interaction or ‘true
spillover’ appears in a set of growth regressions in a cross section of economies
(e.g. US counties (Wheeler (1998)), Japanese prefectures, regions in Europe, US
states, and countries (Barro and Sala-i-Martin (1995), regions of Italy (Putnam
(1993)) if an average over some set of economic entities, call it S, of an outcome
variable appears in the right-hand-side of some of the regressions. Hence, if we
averaged the left-hand-side of the outcome equations of these economic entities in
S over S we have a set of simultaneous equations to solve for the outcome average
over S.
This is a type of simultaneous equations problem which can lead to a lack of
identification of parameters when estimation is done. Manski’s paper in SFI(II)
stresses that, in other applications, this problem raises fundamental issues of econo-
metric identification of ‘true endogenous’ effects (those which have a social multi-
plier potentially exploitable for the public good by appropriate policy intervention)
and ‘spurious’ effects (those which have no such multiplier and where beneficial
policy intervention that exploits autocatalytic effects and positive feedback loops is
not possible). Examples of ‘true endogenous’ effects include production spillovers
via learning-by-doing, human capital spillovers due to average economy-wide hu-
man capital appearing as an argument in each firm’s production function, and
learning spillovers from highly skilled to new learners as in Glaeser’s (1997a,b)
city models.
Examples of ‘spurious’ effects that might generate false evidence of true en-
dogenous effects if omitted in regressions include unobservables that are correlated
across an economy that increase productivity to individual firms in that economy.
Furthermore productive variables at the individual firm level whose economy wide
averages also improve productivity at the firm level, but yet have no social multi-
plier also can cause identification problems as pointed out by Manski.
Here is the simplest example I can think of to expose the problem quickly.
Following Manski’s paper in SFI (II), suppose y is the outcome variable of an
economic entity which is of interest, E(y|g) is the average of y in reference group
g, x is an input variable that effects y, E(x|g) is the average of x in reference group
g which may also impact y. Consider the regression in levels, where E(e|x, g) = 0,
y = a + bE(y|g) + cx + dE(x|g) + e. (13.21)

To obtain a social ‘rational expectations’ equilibrium reduced form, take the aver-
age of both sides with respect to reference group g in (13.21) to find E(y|g) and
insert that back into (13.21) to obtain the reduced form
y = A + cx + DE(x|g) + e, (13.22)
where A = a/(1 − b), D = (bc + d)/(1 − b)
Notice that a social multiplier, [1/(1−b)−1] = b/(1−b), exists for the regres-
sor E(y|g). I.e. if, for example, the constant term ‘a’ increases by an amount f for
one member of a group g of size |g|, the group mean, E(y|g) increases by a factor
f /[(1 − b)|g|] for a net gain of bf /[(1 − b)|g|] over the case b = 0 of no multiplier.
This is a measure of the size of spillover. We see the source of the econometric
problem raised by (13.21) and (13.22) right away. There are only three regressors
in the reduced form (13.22) but there are four parameters to estimate in the primary
form (13.21). The coefficient of interest, ‘b’, cannot be separated from d. Taking
first differences does not solve the problem because the constant is lost. We still
have one more parameter relative to regressors in the reduced form.
It is easy to see how this kind of situation can arise in a growth regression
exercise inspired by the spillover models of Romer (spillovers to our economic
entity’s output, y, across outputs in reference group g due to learning by doing
spillovers) and Lucas (spillovers from average human capital in reference group
g) (discussed in Barro and Sala-i-Martin (1995), Barro (1997), and Durlauf and
Quah (1999)).
Following Brock and Durlauf (1999), we consider four avenues of potential
escape from this problem. First, one might be able to use economic theory to argue
that a term like cx + dE(x|g) should not appear in the regression equation and that
it should be replaced by a term like cx + dE(w|g) where w may be correlated with
x but is not identical to x. This escape route leads to a reduced form with the same
number of regressors as parameters but the ‘X X ’ matrix of the regression may be
near singular if the correlation between x, and w is strong enough.
A second avenue is to attempt to use economic theory to argue that the linear
combination of x, E(x|g), should be replaced by, for example, a linear combi-
nation of nonlinear functions of x, E(x|g). In this case the ‘X X ’ matrix would
be nonsingular except for ‘hairline’ cases. However the X X matrix may be near
singular if the nonlinearity was weak or the support of x, E(x|g) was narrow so
that there was not enough variation to let the nonlinearity speak loudly enough to
the nonsingularity of X X . A third route of escape is to replace the current values
of E(y|g), E(x|g) in the regression (13.21) above and replace them by their lagged
values or, perhaps, some function of their lagged values. See Manski’s paper in
SFI (II) for a critique of this route of escape as well as some other routes in other
applications than cycles and growth applications. His critique may, however, be
of concern to researchers in cycles and growth. A fourth route of escape is to con-
sider other ways of adducing evidence for spillovers than growth regressions. For
example if one were studying TFP dynamics at the individual firm or individual
industry level, then one could define a relevant comparison group g and attempt
to estimate spillovers from the average adoption time of a particular innovation of
the reference group g upon the time of adoption of the individual firm. Turn now
to one possible way to implement this strategy.
13.5.1 Hazard model econometrics and measuring spillovers in

adoption times
A fundamental concern in measuring whether there are spillovers across economic
units in TFP dynamics is the problem of estimating the impact of spillovers (e.g.
the impact of the average time to adoption in a relevant ‘peer group’ sector to
individual firms in that sector). We sketched above how optimal timing theory
suggests forces that shorten the time to adoption of an RCR to an individual
adoptor. If there are ‘true spillovers’, for example a decrease in the average time
to adoption of a ‘reference group’ of firms decreases the cost F to adoption for
an individual member of that reference group, then we expect from the theory
sketched above that our firm will adopt earlier.
A social adoption equilibrium can be defined in terms of hazard functions as
follows. Let T be a non negative random variable giving the adoption time and let
T have absolutely continuous cumulative distribution function F(t) = Pr{T < t}
and density function f (t) = dF/ dt. Then the hazard function, h(t), is defined by
h(t) = f (t)/[1 − F(t)]. (13.23)

Since the survivor function Pr{T > t} is the exponential of the negative of the
integral of the hazard up to t, therefore once the hazard function is specified, we may
write F(t) and f (t) in terms of the hazard. Consider an example of individual firms
in an industry potentially adopting a piece of cost-reduction technology. Imagine
one has a data set on firms and industries that contains individual characteristics,
x, of the firms, industry characteristics, w, (some of which are just averages of
firm characteristics within each industry), and adoption times for different cost-
reducing practices. Suppose one wishes to adduce evidence for or against existence
of ‘true spillovers’ of adoption times of each firm within an industry from the rest
of the firms in that industry using this data set. Here is a possible strategy. Write
down a hazard function h(t, x, w; E(t|g)) for each individual firm as a function of
individual characteric vector x, its industry characteristic vector w, and the firm’s
expectation E(t|g) on the average adoption time of firms in that industry. Here
‘g’ denotes the industry of the firm. The actual average time to adoption in that
industry is the average over the set of firms in that industry of the individual firm
adoption times (assume all firms eventually adopt). If the term, E(t|g) did not
appear in the individual firm’s hazard, we could simply apply standard methods
of hazard function estimation (cf. Heckman and Singer (1985)) and estimate the
hazard function. We illustrate how to modify the standard methods to include social
interactions in the hazard.
Consider the simple example of exponential hazard constant in t. Then Et is
just the reciprocal of the hazard. Now specify a functional form, h(x, w; E(t|g))
for this hazard in ‘regressors’ x,w,E(t|g). We have by the properties of exponential
hazards
E(t|x, w; E(t|g)) = 1/h(x, w; E(t|g)) (13.24)

Impose ‘rational expectations’ for each reference group g,
E(t|g) = sum {1/h(x, w; E(t|g))} over members of g. (13.25)

Equation (13.25) defines a rational expectations equilibrium. Brock and Durlauf
(1999) show that if the hazard is known up to a finite dimensional parameter vector,
then maximum likelihood may be used to estimate (13.24) subject to (13.25). For
a simple example consider,
h(x, w; E(t|g)) = exp[−(a + bx + cw + dE(t|g))]. (13.26)

One may locate sufficient conditions for existence of a rational expectations equi-
librium (i.e. a solution to (13.25)). When one checks the identification condition
for (a, b, c, d) to be identified in (13.26) one ends up looking at the matrix of sec-
ond order partial derivatives of the likelihood with respect to the parameter vector.
This gives one a matrix that looks rather like a regression matrix X X . A suffi-
cient condition for identification requires that this matrix be nonsingular subject
to solutions of (25).
If one has a data set on individual firms and industries where there is variation
in adoption times across individual firms and there is variation in mean adoption
times across industries then the identification condition will be typically be satisfied
even if ‘regressor pairs’ like (x, E(x|g)) appear in the hazard. Notice that regressor
pairs (x, E(x|g)) destroy identification of ‘true’spillovers in the linear case (13.21),
(13.22). The reason for this happy identification result in the hazard case is the
nonlinearity of the hazard and the rational expectations equilibrium condition keeps
the identification matrix from becoming singular on the set of solutions to (13.25),
unlike the linear case (13.21), (13.22). See Brock and Durlauf (1999) for the details.
We wish to warn the reader at the outset that carrying out this strategy of adducing
evidence for ‘true spillovers’ in adopting RCRs may be difficult to carry out in
practice. The main point we wish to make here is this. The application of optimal
timing theory above and thinking about interactive relationships amongst firms
solving optimal timing problems suggests pathways for ‘true spillovers’ to operate.
This, in turn suggests econometric exercises using estimation techniques from
duration analysis which are discussed in Heckman and Singer (1985). Our sketch
of extending these techniques to interactions models is intended as an illustration
of what future research using these tools might look like. Turn now to another area
of potential application of tools discussed here.
13.5.2 Expectations: rational or other?

The debate on whether it is more useful in cycles and growth research to use rational
expectations modeling or some type of adaptive or boundedly rational modeling
was quite intense during the Siena School. SeeAmendola and Gaffard (1998), Hahn
and Solow (1995), and Flaschel (2001, this volume), for alternatives to rational
expectations modeling. In this section of the chapter, I discuss whether some
recent work might contribute to a narrowing of the disagreement among scholars.
There are several potentially useful approaches to use data and ‘complexity-based
evolutive’ type theory to help narrow disagreements in this area. First, consider the
use of asset pricing data in the countries that have well-developed asset markets.
One can insert an asset pricing module into their favorite macro model that departs
from rational expectations and crank out the implied time path of stock prices
and trading volume and ask whether these look like aggregative stock price data
and aggregate trading volume data. This strategy allows one to check whether
their favorite macroeconomic model generates implied asset prices consistent with
empirical findings from the large event studies literature in finance. This strategy
imposes useful discipline on model building. Some examples are given in my paper
in SFI (II). Here are some more.
For example, rational expectations type asset pricing models like those with
a production side with tax distortions reviewed in Altug and Labadie’s recent
text (1994) generate the following testable (in principle) prediction. If one has a
sufficient state space dynamical representation for the dynamics of the ‘real side’
of the model then here is a sharp empirical implication. Past asset prices do not help
incrementally predict future asset prices given past and current values of the state
variables of the ‘real side’. Hence one can do a nonparametric regression of future
prices upon past prices and past state variables and set up a test of ‘significance’
of past prices in this regression given past state variables. If one fails to reject the
null hypothesis that past prices do not enter this regression, then one has adduced
evidence consistent with the hypothesis of rational expectations in this context.
This procedure is sometimes called ‘Granger causality testing’ in the time series
literature. Parenthetically, we wish to emphasize that this strategy is not nearly as
easy to carry out in practice as we are making it sound in this brief exposition. We
are burying a lot of problems in order to expose the basic structure of the strategy.
One major problem is unmeasured components of the state space representation
that impact future earnings which the stock traders see but the scientist does not.
These unmeasured components could be correlated with the measured components
and this correlation induces a pathway through which past prices might help predict
future prices. Hence a researcher might erroneously reject the rational expectations
hypothesis when it is true. In any event let us continue with our discussion.
Second, we may be able to adapt a strategy used in the debate on the causes
of mean reversion (if any) in asset prices. Jog and Schaller (1994) argued that
mean reversion is likely to be due to liquidity constraints and differential access
to raising funds in the financial markets that especially bite on the smaller firms
relative to the larger firms. This hypothesis had distinct implications in a production
based rational expectations asset pricing model where some firms were liquidity
constrained and others were not. For example this kind of economic force would
lead one to expect mean reversion to loom larger in smaller firms and to loom larger
during periods of economy wide liquidity stress such as deep recessions. Indeed
the evidence for mean reversion appears weak for large firms post-World War II
in contrast to small firms during deep recessions. See Jog and Schaller (1994) for
the details.
Third, one might ‘back-off’ from rational expectations in the direction of evo-
lutionary dynamical models where ideas from evolutionary computation theory
are used to set up models where expectational schemes co-evolve while compet-
ing with each other in a ‘Digital Darwinistic’ struggle for existence. See Arthur
et al., and Darley and Kauffman in SFI (II), LeBaron (2000), and Sargent (1993)
for work of this type. See, especially LeBaron’s review (2000) for an excellent
picture of this area. One can contrast the prediction of no ‘Granger causality’ of
past prices for future prices in rational expectations models above with simulated
data from evolutionary computational models to see if one gets a Granger causal
pattern that mimics that found in actual asset market data.
Fourth, one can estimate a class of models containing a mixture of rational
expectations agents and boundedly rational agents as did Baak (1999) for the case
of cattle ranchers and did Chavas (1999) for the case of pork producers, where
the data are free to speak to the presence of boundedly rational agents. Baak, for
example, was able to test, using data on the cattle industry, whether the extra ‘free
parameters’ brought by two different types of bounded rationality, naive simple
backwards looking predictors, and Nerlove’s Quasi-Rational predictors were ‘sig-
nificant.’ His evidence suggests some support for Nerlove’s level of rationality in
contrast to fully rational expectations.
Nerlove’s quasi-rational expectations impose some level of rationality for in-
stance being consistent with historical features of the data such as the autocorrela-
tion structure of past prices. This idea is related to Kurz’s Rational Beliefs (1997)
which impose rationality to the level of consistency with the data but do not require
agents to possess full knowledge of the structure of the world that they live in. Kurz
adduces evidence to support his type of departures from full rational expectations.
Baak’s evidence suggests that boundedly rational ranchers left a detectable trace
in the data. However, his rejection of the null hypothesis of purely rational expec-
tations ranchers may have been due to specification of a false rational expectations
model on a truly rational expectations sector even though he used the well-known
Rosen, Murphy, Scheinkman model for his null model. His rejection could also
be due to unintended data snooping bias of the type discussed by Sullivan et al.
(1999). But, it is encouraging that Chavas (1999), using different methods than
Baak, obtained rather similar results using data on the pork industry.
Fifth, Brock and Hommes (1997), (1998) develop a theory of ‘Adaptively
Rational Equilibrium Dynamics’ (ARED) where economic agents hold a set of
different belief systems and models of the economy and use a discrete choice
model to select their own belief out of this set based upon performance indices.
The performance index for a particular belief system or predictive model is built
out of past profits gained from using that system, for example, a distributed lag of
past profits. Rational expectations are also available (for a fee). Costs of obtaining
information to implement a belief system (e.g. rational expectations) are subtracted
from the profits gained by using it. The system of beliefs co-evolve in ARED theory
in an evolutionarily dynamic equilibrium.
The ARED theory is rather like the computationally-based work of Arthur
et al., Darley and Kauffman in SFI (II), and LeBaron (2000), but is framed in
a setting where analytical results can be obtained and econometric work can be
done. In this way the massive literature on discrete choice econometrics and rational
expectations econometrics can be adapted to do econometric work in testing for
the presence of expectational schemes being used by economic agents other than
rational expectations. The work of Baak and Chavas discussed above is an example
of recent econometric work that allows data to speak to the presence of boundedly
rational agents.
In ARED theory one introduces evolutionary competition of different belief
systems in generating trading profits net of costs of obtaining the information
needed to implement those belief systems. Rational expectations will be acquired
by economic agents only when there appears (based upon the recent past) to be
enough value to the extra predictive accuracy to cover the cost of obtaining ra-
tional expectations. So the theory ‘backs off’ from fully rational expectations but
does not abandon the intelligence of agents. Hence there will be ‘phases’ where
cheaper boundedly rational expectations are used by the bulk of agents. In these
‘phases’ past prices (in most formulations of such mixed expectational models)
will incrementally help predict future prices given past and current values of the
state variables of the ‘real side.’ I.e. past prices will Granger cause future prices
during these temporal phases in this setting. This implication may allow one to
use Granger causality testing methods in time series econometrics to test for the
presence of boundedly rational traders.
Whether any evidence of departure from rational expectations is detected by
empirical testing will, of course, depend upon the size of the phases where rational
expectations are abandoned by the bulk of agents because they are not worth the
cost of obtaining. Notice that if empirical evidence of departures from rational
expectations is easy for econometricians to find in this kind of model the very
logic of the model says that rational expectations are likely to be used by most of
the agents (unless, for some reason, the costs of obtaining rational expectations are
very large). In turn the size of such phases will depend upon the tradeoff between
the gains to the extra ‘look ahead’ value of rational expectations in improving
profits to the costs of obtaining such sophisticated expectations.
During stable periods where the state of the system is not changing very much,
one might expect the gains to exploiting rational expectations not to cover their
cost in which case the agents revert to naive schemes. Nevertheless, even a modest
amount of predictability out-of-sample of future returns using past returns is incon-
sistent with a lot of empirical evidence in asset markets, especially heavily traded
assets in well developed markets. Yet, common knowledge rational expectations
frameworks tend to lead to small trading volume whereas trading volume in reality
is very large. This lead to development of a related type of modeling strategy that is
still evolutive. See LeBaron (2000) for an excellent review of the latest compexity-
based approaches to this problem. I describe very recent theoretical work of de
Fontnouvelle here. de Fontnouvelle (2000) builds on Brock and Hommes (1997)
to build an evolutionarily dynamic version of the ‘Grossman-Stiglitz information
paradox’ in the context of noisy rational expectations models where agents can
expend resources to obtain more accurate ‘signals’ of future prices, but the market
‘leaks’ these efforts, allowing other agents who do not pay for better signals to free
ride on the efforts of those who do. This tension between information gatherers
and information parasites creates a layer of dynamics across signal types (more
precise signal types must pay more for their signals but earn higher gross profits
from trading on those signals) that generates patterns of returns and trading vol-
ume that looks rather like real data on returns and trading volume. I.e. the EMH is
accepted by simulated returns data (as it tends to be in real markets) from his model
and trading volume not only fails to ‘dry up’ (as in Sargent’s (1993) discussion of
‘no-trade’ theorems under rational expectations and common knowledge) but is
highly persistent (as it is in the data from real markets).
Sixth, while the complexity-based approaches to ‘backing off’ from fully ra-
tional expectations are exciting there is a problem in taming the number of free
parameters that the plethora of competing expectational schemes (also constantly
evolving in the evolutionary computational approaches) brought to the analysis.
As Sargent (1993) argues, economists are in the market for theories that reduce the
number of free parameters not theories that increase them. Brock’s paper in SFI
(II) reviews recent work of Brock, Hommes, and de Fontnouvelle that adapts meth-
ods inspired by large system limits in statistical mechanics to reduce the number
of parameters by recognizing that equilibrium conditions in asset markets contain
terms from the heterogeneity of agents that resemble sample moments. If one takes
the number of agents to infinity these ‘sample moments’ converge to population
moments. If the ‘building blocks’ of the agent types come from parsimoniously
parameterized ‘building block’ distributions then the number of free parameters is
reduced from essentially the number of agents (huge) to the number of parameters
in the building block distributions from which agents are built out of evolutionarily
adaptive characteristics. This theory, which is still very much under construction
looks rather like large economy limit theory in general equilibrium theory (cf.
writers such as Aumann, Debreu, Hildenbrand, Kirman, et al.).
Seventh, the evolutive signal access models of de Fontnouvelle (2000) might
be generalized to get ‘use it and lose it’ type results where better signals are
purchased from ‘policy insiders’ for a fee, are traded on and generate superior
profits as policy unfolds which impacts the potency of policy if its potency depends
upon the policy makers being able to move faster than the agents or having better
information than the agents. This might allow a better understanding of where
policy invariance results are likely to be a problem in practice. It also may help
identify channels where policy benefits are likely to be greater than its costs,
especially when correction for practical problems such as political intervention
and rent seeking is done.
13.6 Summary
This chapter has attempted to bring to the reader’s attention some ‘complexity-
based’ tools and has attempted to assess whether they might add some value to
discussion of issues raised at the Siena School of Summer 1998. These issues
included (i) explaining the wide differences in growth performance across regions,
countries, and periods; (ii) identification of ‘spillover’ effects in growth contexts;
(iii) and the desirability of ‘backing off’ from fully rational expectations modeling.
We addressed these issues by first setting out a little bit of optimal timing theory
and applying it to uncovering forces that lead to faster adoption of RCR’s and faster
development of new markets. We then used this framework to set out five sample
industrial and regulatory settings to stress how to go point-by-point through how
the structure of incentives (but not all of Harberger’s 1001 ways) impact optimal
adoption times. Much was said about the many things that government can do
to help or to hurt. After spending a fair amount of time setting out this context,
we turned to a discussion of channels for potential spillovers. Then we turned to
discussion of the basic econometric problem, following Manski’s paper in SFI (II),
of identifying spillovers in growth regressions. We suggested some ways based on
Brock and Durlauf (1999) to deal with this problem. We then turned to a sketch
of econometrics of hazard function estimation with spillover effects following
Brock and Durlauf (1999) which is close to the spirit of the optimal timing theory
developed above.
The bottom line is this. Complexity-based methods offer stimulating sugges-
tions of potentially useful research strategies for work in the area of cycles and
growth.
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14 Information, complexity
and the MDL principle
Jorma Rissanen
14.1 Introduction
In intuitive terms, the objective of statistical modeling is to separate a given data
sequence into useful learnable information and the rest, which may be viewed
just as noninformative noise. The difficulty is in the formalization of the two
constituents: the ‘useful information’ and the ‘noise’. Traditionally, modeling is
done by envoking a metaphysical ‘true’ data generating distribution, which is to
be estimated from the data by minimization of an appropriate mean performance
criterion, which itself is to be estimated from the data. Since the basic issue of
how to formalize the useful information and the noise is not addressed such an
approach cannot provide a rational explanation of why the best approximation of
the ‘truth’ is not the most complex model fitted to the data. To avoid this disastrous
conclusion one has to add an ad hoc term to the criterion to penalize the model
complexity. But because the added term lacks any deeper meaning it does not
reflect adequately the model complexity nor its effect to models’ performance,
and such a metaphysical assumption does not provide a sound basis for a fruitful
theory of modeling.
There are alternative purely empirical approaches to model building, which are
based on reuse of the available data, such as in cross-validation and bootstrapping.
In these a desired data dependent performance criterion is minimized in a portion
of the data and the result tested either on the rest or on a reshuffled version of the
data. Although computationally demanding such approaches may work reasonably
well in individual cases, but, being empirical processes, they cannot serve as a basis
for a theory of modeling.
It seems to us that no solid theory of modeling can be achieved without a formal
definition of ‘information’ in a data sequence, which can be done by a generaliza-
tion of Shannon’s information measure along the lines of the algorithmic theory of
complexity (Hansen and Yu 1998), Kolmogorov 1965). The basic concepts have
undergone a somewhat tortuous evolution, (Rissanen 1986, 1987, 1996), and in this
chapter we outline some of the most recent developments (Barron et al 1998, Bala-
subramanian 1996, Grünwald 1998, Rissanen 2000a, 2000b). Briefly, the stochas-
tic complexity of a data sequence xn = x1 , . . . , xn (or (yn , xn ) = (y1 , x1 ), . . . , (yn , xn )
340 Jorma Rissanen
as in regression problems), relative to a class of parametric probability models
M = {P(xn ; θ)}, is defined to be the shortest code length with which the data xn
can be encoded, when advantage is taken of the models. Such a code length may
be identified with the negative logarithm of a probability distribution P(xn ; M),
which factors as follows
P(xn ; M) = H (xn |θ̂(xn ))Q(θ̂(xn )), (14.1)
where the first factor has no useful information about the data while the second
factor, describing the maximum likelihood estimate θ̂(xn ), has it all. To such a
factorization there often corresponds a decomposition of the data of the kind xn =
x̂n + en , where en is the ‘noise’ part that cannot be compressed with the given
models, and x̂n = F(xn ; θ̂(xn )) is the purely predictable part. We define the code
length for encoding x̂n the amount of information in the data xn that can be retrieved
with the model class M. A similar factorization and decomposition exist also for
data of type (yn , xn ) = (y1 , x1 ), . . . , (yn , xn ), where yn , given another data sequence
xn , is to be modeled.
We may then state that the objective in building models for data is to ob-
tain a decomposition of the kind described, which so far has been done only for
probabilistic models. Hence the goodness of any of the numerous model selec-
tion criteria proposed in the literature can be judged on how well they accomplish
such a decomposition. Also, principles like Occam’s razor and its variants express
only the obvious: redundancy cannot add to the useful information, and since
such a model cannot achieve the stochastic complexity it should be removed. The
reader will recognize the factorization (14.1) as a generalization of the ordinary
sufficient statistics factorization, which includes parameters, to a universal suffi-
cient statistics factorization, which has no parameters. This, in turn, is an analog
of Kolmogorov’s sufficient statistics decomposition in the algorithmic theory of
complexity (Cover and Thomas 1991) . A perfect separation of the noise from the
useful information is possible only for special model classes, but asymptotically
it can be done for all the usual model types. A perfect nonasymptotic separation
exists in the linear quadratic regression case, (Rissanen 2000a), which we describe
below.
14.2 Models
We begin with a brief discussion of the formal definition of a model. We consider
sets X and Y and their cartesian product X × Y together with the extension X n × Y n
to sets of strings of length n. Perhaps the most common type of model is defined
by a function F : X n → Y n , together with an error function δ(yn , F(xn )), for which
we take the logarithmic one δ(yn , F(xn )) = − log f (yn |x̂n ), where x̂n = F(xn ) and
f (yn |x̂n ) is a conditional density function. Many of the usual error functions, above
all the quadratic one, define conditional density functions. Of particular interest to
us are the parameteric models, Mγ = {f (yn |xn ; γ, θ)}, where γ is a structure index,
such as the pair of orders p, q in ARMA models, and θ = θ1 , . . . , θk ranges over
some, usually compact, subset of the k-dimensional euclidean space, k depending
Information, complexity and the MDL principle 341
,
on γ, such as k = p + q in the ARMA models. Put M = γ Mγ , an example
of which is the set of all ARMA models. The parameters θ often include both
parameters in the function F and others in the conditional probability or density
function, such as the variance. Finally, an important special case is the one where
the data sequence xn is absent, in which case we write xn rather than yn for the
single data sequence, and f (xn ; γ, θ) for the models. The theory is similar for both
types of data.
14.3 Two minmax problems

As indicated in the Introduction the key to the desired factorization and decom-
position is the shortest (ideal) code length with which the data sequence xn can
be encoded, when the codes are , somehow designed with the models in the classes
Mγ = {f (xn ; γ, θ)} and M = γ Mγ , respectively. The word ‘ideal’ refers to
the convenient habit of dropping the requirement that a real code length must be
integer-valued, and hence the negative logarithm of any probability or density is
regarded as an ideal code length. By a harmless abuse of notation even the word
‘ideal’ is often dropped. For those unfamiliar with coding we add that all these
conventions are justified, because we can design codes for a large set of objects
such that the real integer-length code lengths differ from the ideals by a negligi-
ble amount. And since our objects are typically the sequences xn and real-valued
parameters, their sets are certainly ‘large’. Whenever we need to consider code
lengths for small sets of objects, such as structure indices, we use the real code
lengths or their accurate estimates. To summarize, a probability distribution, a
model, and a code can be identified.
With these agreements the shortest code length, relative to the class Mγ , we are
searching for will be of the form − log f (xn ; γ), which means that we are looking
for a model that is universal for the class Mγ in question. The very best code length
we could hope to get would, of course, be minθ log 1/f (xn ; θ, γ), obtainable with
the ML (maximum likelihood) estimate θ̂(xn ), but f (xn ; θ̂(xn ), γ) is not a valid
model, because its integral exceeds unity. This suggests the following minmax
problem (Barron et al. 1998),
f (X n ; θ̂(X n ), γ)
min max Eθ log , (14.2)
q θ q(X n )
where q(xn ) is any density function and the expectation is with respect to f (xn ; θ, γ).
It is clear that the models in the class Mγ cannot express all the statistical
properties in real world data sequences xn , no matter how large n is. For instance,
if we generate the data with a density function g(xn ) which is outside the class,
the data will have statistical properties different from those expressible with the
models in the class. Notice, that we do not want to make the claim that the data are
a sample from any distribution. Rather, we are simply using density functions to
describe statistical properties in the data. This suggests that we should generalize
the minmax problem (14.2) as follows
342 Jorma Rissanen
f (X n ; θ̂(X n ), γ)
min max Eg log , (14.3)
q g∈G q(X n )
where G is a class larger than Mγ . In fact, we can let G consist of all distributions
such that G = {g : Eg log(g(X n )/f (X n ; θ̂(X n ), γ) < ∞}. This excludes the singular
distributions, which clearly do not restrict the data in any manner and hence do not
specify any properties in them. Also, both the minimum and the maximum will be
reached.
Theorem 14.1 If Ω is such that the integral

Cn (γ) = f (yn ; θ̂(yn ), γ)dyn (14.4)
θ̂(yn )∈Ω
is finite, the solution to the minmax problem (14.3) is the universal NML (normal-
ized maximum likelihood) model
f (xn ; θ̂(xn ), γ)
fˆ(xn ; γ) = . (14.5)
Cn (γ)
Clearly,
f (X n ; θ̂(X n ), γ)
Eg log = log Cn (γ) (14.6)
fˆ(X n ; γ)
for all g.
The proof is given in Rissanen 2000b.
Interestingly, for discrete data the solution P̂(xn ; γ) also solves Shtarkov’s
minmax problem (Shtarkov 1987),
P(xn ; θ̂(xn ), γ)
min max log = log Cn (γ). (14.7)
Q n x Q(xn )
Notice the important fact that the best model fˆ involves only the models in the class
Mγ . Under certain conditions, satisfied for the classes of exponential distributions
(Rissanen 1996),

k n
log Cn (γ) = log + log |I (θ)|dθ + o(1), (14.8)
2 2π Ω
where one of the assumptions requires the convergence

∂ 2 ln f (X n |θ)
−n−1 {E } → I (θ),
∂θi ∂θj
to the Fisher information matrix
∂ 2 log f (X |θ)
I (θ) = {−Eθ }.
∂θi ∂θj
There is another related minmax problem, originally defined for universal cod-
ing
f (X n ; θ, γ)
min max Eθ log , (14.9)
q θ q(X n )
where q(xn ) is any density function and the expectation is with respect to f (xn ; θ, γ).
The minimizing model turns out to be given by a mixture

fw̄ (x ; γ) = w̄(θ)f (xn ; θ, γ)dθ,
n
(14.10)
where w̄ is for many model classes approximately given by Jeffreys’ prior

|I (θ)|
w̄(θ) = . (14.11)
Ω |I (η)|dη
Moreover, the minmax value is the so-called capacity of the channel Θ → X n ,

(Merhav and Feder 1995, Clarke and Barron 1990) which for iid models satisfies
Kn (γ) = Cn (γ)e−k/2+o(1) . (14.12)
One can show that − log fˆ(xn ; γ) and − log fw̄ (xn ; γ) behave similarly for large n.
14.4 Complexity and information

If we restrict the data generating model class to the same on which the codes are
designed, G = Mγ , then for all codes the bound log Cn (γ) in (14.6) is in essence a
lower bound for all models g = f (xn ; θ, γ) (and not only for the worst case model)
except when θ ranges over a set whose volume goes to zero as n grows to infinity
(Rissanen 1986). A generalization of this is given in Rissanen 2000b. Accordingly
we are justified to define the ideal code length
− log fˆ(xn ; γ) = − log f (xn ; θ̂(xn ), γ) + log Cn (γ) (14.13)
to be the stochastic complexity of the data string xn , relative to the model class
Mγ . We can rewrite it as
− log fˆ(xn ; γ) = − log f (xn |θ̂(xn ), γ) − log π(θ̂(xn )) (14.14)

f (x ; θ̂(x ), γ)
n n
f (xn |θ̂(xn ), γ) = (14.15)
g(θ̂(xn ))
g(θ̂(xn ))
π(θ̂(xn )) = , (14.16)
Cn (γ)
where g(θ̂(xn )) is a density function induced by f (xn ; θ̂(xn ), γ).

For sequences yn such that θ̂(yn ) = θ̂(xn ) the code lengths − log f (yn ; θ̂(yn ), γ),
divided by n are virtually equal for large n, which means that we cannot compress
344 Jorma Rissanen
the first term in (14.14) further with the given model class; it is the length of a code
defining just incompressible noise having virtually no information to add to the
regular learnable features that are in the optimal model f (yn ; θ̂(xn ), γ) as defined
by the second term. We call it the amount of information in the data sequence xn
that can be extracted with the given model class. It has also been called the ‘model
cost’.
It was shown by Balasubramanian (1996) using quite elegant arguments of
differential geometry without any appeal to the code length that one can derive
a continuum of distinguishable distributions such that their equal distance lattice
in the natural metric defined by the quadratic form θ I (θ)θ for data sequences
of length n has the number of elements given by Cn (γ), (14.8). Its logarithm is
called there the geometric complexity. Also the NML density function itself fˆ(xn ; γ)
has an interpretation in terms of differential geometry. Myung et al. (2000) also
has illuminating examples showing that the MDL criterion (discussed in the next
section) resulting from the minimization of the stochastic complexity (14.13) and
having the important structure dependent terms gives results, especially for small
amounts of data, which are superior to the common criteria, where the model
complexity is penalized only through the number of parameters.
The ML estimate θ̂(xn ) and the optimal model it defines have further properties
to justify their fundamental importance. Let θg define the model that is nearest to
a density function g in or outside the model class Mγ in the KL distance:
g(X n )
min Eg log = D(g(X n )||f (X n ; θg , γ)). (14.17)
θ∈Ω f (X n ; θ, γ)
Then at least for iid models such that the parameters range over a compact set Ω the
ML estimates θ̂(xn , γ) and −(1/n) log f (xn ; θ̂(xn ), γ) can be shown to converge to
θg and −Eg log f (X ; θg , γ), respectively, with g-probability one (Grünwald 1998,
White 1994).
14.5 The MDL principle

The fundamental properties of the ML-estimates and the associated measures of
complexity and useful information in a data sequence are valid for each model class
Mγ . The model selection problem refers to finding the best structure index γ. With
the same arguments as in the preceding section this amounts to finding the,shortest
code length for the data sequence xn , relative to the model class M = γ Mγ ,
which again leads to a decomposition of the data into the useful information; i.e.
the optimal model, and the incompressible noise, relative to the class M. The
criterion to find the shortest code length is the MDL principle, which seeks to find
the index γ̂(xn ) that minimizes the stochastic complexity, or
min{− log f (xn ; θ̂(xn ), γ) + log Cn (γ)}. (14.18)

γ
The justification of this principle, as we discussed above, is the fact that it

achieves the desired decomposition of the data into the information bearing part
and the noise. We’ll illustrate that in the concrete example of linear quadratic
regression problem in the next section. Our thesis then is that the MDL principle
and the NML universal model capture all the regular features in the data sequence
xn that can be described by the model classes in question.
We conlude this section with a discussion of an inherent problem with Akaike’s
AIC criterion and how to replace it with another that has no such difficulty. In
the early 1970s, Akaike (1974) set out to obtain a model selection criterion by
asking for the model in a parametric class which is closest to a data generating
model in KL-distance lying outside of the parametric class. This then, by less than
convincing arguments aimed at estimating the distance, led into his criterion AIC.
That there is a problem in the arguments is best illustrated by the fact that if we
consider a class of nested subclasses, such as the set of polynomials of all orders k,
where k > m implies that f (xn ; θ̂k ) ≥ f (xn ; θ̂m ), and let the data generating model
be a nonparametric one, then the nearest model is either the one with the maximum
number of parameters or it does not exist at all. Also AIC fails to recover the data
generating model even in the case where it lies within the model class.
The difficulty can be avoided if we consider the following problem
fˆ(X n ; γ̂(X n ))
min max Eg log , (14.19)
q g q(X n )
which gives the solution

fˆ(xn ; γ̂(xn ))
fˆ(xn ) = ,
Cn
where Cn is a normalizing constant not depending on the MDL estimate γ̂(xn ) of the
structure parameter. Hence there is no need to evaluate the normalizing constant
nor to add its code length to the MDL criterion (14.18).
14.6 Linear quadratic regression

Because of the term o(1) in (14.8) we cannot calculate the normalizing constant and
hence the stochastic complexity (14.13) to any desired precision for general model
classes. However, the important normal family is an exception, and for the Gaussian
linear regression problem the NML universal density function can be calculated
to any precision even for small samples, and we get a complete decomposition of
the data into the optimal model and an incompressible noise. This also delivers
the information bearing signal sought for in the so-called denoising problem. We
outline the recent findings on these developments (Rissanen 2000b) .
In the linear regression problem the data consist of pairs (yt , x1t , x2t , . . . , xmt ) =
(yt , xt ) for t = 1, 2, . . . , n, which we also write as (yn , xn ), and we wish to learn how
the values xit , i = 1, 2, . . . , m, of the regressor variables influence the corresponding
values yt of the regression variable. We fit a linear model of type

yt = β xt + t = βi xit + t , (14.20)
i∈γ
346 Jorma Rissanen
where γ = {i1 , . . . , ik } denotes a subset of the indices of the regressor variables;
the prime denotes transposition, and for the computation of the required code
lengths the deviations t are modeled as samples from an iid Gaussian process of
zero mean and variance τ = σ 2 , also as a parameter. In such a model the response
data yn = y1 , . . . , yn are also normally distributed with the density function
1
− 2τ1 t (yt −β x t ) ,
2
f (yn |xn ; γ, β, τ ) = e (14.21)
(2πτ )n/2
where the indices in γ pick out the components of the regressor variables and define
the k × n matrix Xγ = {xit : i ∈ γ}. Write Zγ = Xγ Xγ = nΣγ , which is taken to be
positive definite. The development for a while will be for a fixed γ, and we drop
the subindex γ in the matrices above as well as in the parameters. The maximum
likelihood solution of the parameters is given by
β̂(yn ) = Z −1 X yn (14.22)
1
τ̂ (yn ) = (yt − xt β̂ (yn ))2 . (14.23)
n t
We next consider the NML density function
f (yn |xn ; γ, β̂(yn ), τ̂ (yn ))

fˆ(yn |xn ; γ) = , (14.24)
Y (τ0 ,R) f (z n |xn ; γ, β̂(z n ), τ̂ (z n ))dz n
where yn is restricted to the set
Y (τ0 , R) = {z n : τ̂ (z n ) ≥ τ0 , β̂ (z n )Σβ̂(z n ) ≤ R}, (14.25)
which is to include yn . In order to simplify the notations we suppress the super

index n in all the variables.
The numerator in Equation (14.24) has the simple form
f (y|x; γ, β̂(y), τ̂ (y)) = 1/(2πeτ̂ (y))n/2 , (14.26)
and the problem is to evaluate the integral in the denominator. We can do it by

using the facts that β̂ and τ̂ are sufficient statistics for the family of normal models
given, and that they are independent by Fisher’s lemma. With the notation θ = (β, τ )
rewrite f (y|x; γ, β, τ ) = f (y|x; γ, θ), and after some computations; for details see
Dom 1996, where such calculations were done first without use of Fisher’s lemma),
we get

C(τ0 , R) = f (y|x; γ, θ̂(y))dy (14.27)
Y (τ0 ,R)
k/2
2 R
= An,k Vk , (14.28)
k τ0
where
2π k/2 (nR)k/2
Vk Rk/2 = |Σ|−1/2 , (14.29)
kΓ(k/2)
is the volume of the ellipsoid BR = {β : β Σβ ≤ R} and
n−k
|Σ|1/2 ( n−k
2e )
2
An,k = k/2 n−k
. (14.30)
(2π) Γ( 2 )
We then have the NML density function itself for 0 < k < m
n k R n−k k 4 n
− log fˆ(y|x; γ, τ0 , R) = ln τ̂ (y)+ ln −ln Γ( )−ln Γ( )+ln 2 + ln(nπ).
2 2 τ0 2 2 k 2
(14.31)
In order to get rid of the two parameters R and τ0 , which clearly affect the
criterion in an essential manner, set them to the values that minimize (14.31): R =
R̂(y) = n−1 β̂ (y)Σβ̂(y) and τ0 = τ̂ (y). However, the resulting fˆ(y|x; γ, τ̂ (y), R̂(y))
is not a density function. We rectify this by the same normalization process as
above:
fˆ(y|x; γ, τ̂ (y), R̂(y))
fˆ(y|x; γ) = , (14.32)
Y f (z|x; γ, τ̂ (z), R̂(z))dz
ˆ
where the range
Y = {z : τ1 ≤ τ̂ (z) ≤ τ2 , R1 ≤ R̂(z) ≤ R2 }
will be defined by four new parameters. Again the integration can be performed,
(Rissanen 2000a), and the negative logarithm of fˆ(y|x; γ) is given by
n−k k n−k k
− ln fˆ(y|x; γ) = ln τ̂ (y) + ln R̂(y) − ln Γ( ) − ln Γ( )(14.33)
2 2 2 2
2 n τ2 R2
+ ln + ln(nπ) + ln ln . (14.34)
k 2 τ1 R1
This time the last term involving the new parameters does not depend on γ nor
k, and we do not indicate the dependence of fˆ(y|x; γ) on them. Almost the same
criterion was obtained in Hanson and Yu 1998 by evaluation of a mixture density
for Zelner’s prior.
As a final step we wish to extend the density function fˆ(y|x; γ) to the larger
class of models, defined as the union over all the index sets γ, and to obtain a
criterion for finding the optimal index set and the associated optimal model. We
begin with the MDL estimator γ̂(·), obtained by minimization of the ideal code
length for the data − ln fˆ(y|x; γ) with respect to γ. Although the result fˆ(y|x; γ̂(y))
is not a density function we get one by the normalization process
fˆ(y|x; γ̂(y))
fˆ(y|x; Ω) = , (14.35)
γ̂(z)∈Ωfˆ(z|x; γ̂(z))dz
348 Jorma Rissanen
where Ω is a set of indices such that it includes γ̂(y). The denominator, call it C,
is given by
C= P̂n (γ), (14.36)
γ∈Ω
where
P̂n (γ) = fˆ(z|x; γ̂(z))dz. (14.37)
{z:γ̂(z)=γ}
In analogy with fˆ(y|x; γ) we call fˆ(y|x; Ω) the NML density function for the
model class with the index sets in Ω, and we get the final decomposition
− ln fˆ(yn |xn ; Ω) = n−2 k̂ ln τ̂ (yn ) + k̂2 ln R̂(yn ) − ln Γ( n−2 k̂ )

(14.38)
− ln Γ( k̂2 ) + ln k̂1 + Const ,
where we include in Const all the terms that do not depend on the optimal index
set γ̂ of size k̂. The terms other than the first define the length of a code from
which the optimal normal model, defined by the ML parameters, can be decoded,
while the first term represents the code length of the part of the data that adds no
further information about the optimal model. It may be viewed as noise. Hence
this decomposition is similar to Kolmogorov’s sufficient statistics decomposition
in the algorithmic theory of information, and it is also seen to extend the ordinary
sufficient statistics, as defined for certain parametric families, to parameter free
universal sufficient statistics.
By applying Stirling’s approximation to the Γ-functions we get the NML cri-
terion for 0 < k ≤ m
n
min{(n − k) ln τ̂ (yn ) + k ln(nR̂(yn )) + (n − k − 1) ln − (k + 1) ln k},
γ∈Ω n−k
(14.39)
where k denotes the number of elements in γ.
It seems that in order to find the minimizing index set γ we must search through
all the subsets of the rows of the m × n regressor matrix X . However, this can be
avoided if we make a linear transformation AX of the regressor matrix such that
AX XA /n = I . Let the new ML parameters for the maximal number of rows k = m
be α̂ = α̂1 , α̂2 , . . . , α̂m and σ̂ 2 . Further, let α̂(1)
2
≥ α̂(2)
2
≥ . . . ≥ α̂(m)
2
so that (i) is
the index of the i’th largest parameter in absolute value. Then the criterion (14.39)
is equivalent with
n
min{(n − k) ln(α̂ α̂ − R̂) + k ln(nR̂) + (n − k − 1) ln − (k + 1) ln k},
k n−k
(14.40)
where R̂ is the sum of either the k largest or the k smallest squares α̂i2 (Rissanen
2000a). Hence, the optimum γ can be found with no more than m evaluations of
the criterion.
For the so-called denoising problem an orthonormal regressor matrix is easily
obtained with wavelets, and the criterion (14.40) provides a natural separation of
noise from the data as its incompressible part, which appears to be superior to any
ad hoc criterion; for numerical examples see Rissanen 2000a.
14.7 Conclusions
We have described a formal definition of complexity and (useful) information in a
data sequence, as a foundation for a theory of model building. These notions can
be obtained to a good approximation from a universal NML (Normalized Maxi-
mum Likelihood) model for parametric model classes as a solution to a minmax
problem. Theorems exist which demonstrate that such a universal model provides
an extension of the usual sufficient statistics decomposition to a parameter free
universal sufficient statistics decomposition, and accomplish the desired decom-
position of the data. For a collection of model classes the best can be found with
the MDL principle. We illustrate such a decomposition and the resulting criterion
for the Gaussian family in the basic linear regression problem, for which they can
be computed exactly even for small amounts of data.
References
Akaike, H. (1974), ‘Information Theory and an Extension of the Maximum Likelihood
Principle,’ Second international symposium on information theory, ed. B.N. Petrov and
F. Csaki, Akademia Kiedo, Budapest, 267–81
Balasubramanian, V. (1996)‘Statistical Inference, Occam’s Razor and Statistical Mechanics
on the Space of Probability Distributions’, Neural Computation, 9, (2), 349–268, 1997
http://arxiv.org/list/nlin/9601
Barron, A.R., Rissanen, J. and Yu, B. (1998) ‘The MDL Principle in Modeling and Cod-
ing’, special issue of IEEE Trans. Information Theory to commemorate 50 years of
information theory, IT-44, (6), October 1998, pp 2743–60
Clarke, B. S. and Barron, A. R. (1990) ‘Information-Theoretic Asymptotics of Bayes Meth-
ods’, IEEE Trans. Information Theory, IT-36, (3), 453–471, May 1990.
Cover, T. M. and Thomas, J. A. (1991) Elements of Information Theory, Wiley, New York,.
Dom, B. (1996) ‘MDL Estimation for Small Sample Sizes and Its Application to Linear
Regression’, IBM Research Report RJ 10030, June 13, 1996.
Grünwald, P.D. (1998) The Minimum Description Length Principle and reasoning under
Uncertainty, PhD thesis, Institute for Logic, Language and Computation, Universiteit
van Amsterdam.
Hanson, M. and Yu, B. (1998) ‘Model Selection and the Minimum Description Length
Principle’, to appear in Journal of American Statistical Association, June, 2001; see also
http://cm.bell-labs.com/stat/doc/mdl.pdf
Kolmogorov, A.N. (1965) ‘Three Approaches to the Quantitative Definition of Information,’
Problems of Information Transmission 1, 4–7.
Li, M. and Vitanyi, P.M.P. (1997) An Introduction to Kolmogorov Complexity and Its Ap-
plications, second edition, Springer-Verlag, New York.
Merhav, N. and Feder, M. (1995) ‘A Strong Version of the Redundancy-Capacity Theorem
of Universal Coding’, IEEE Trans. Information Theory, IT-41, (3), 714–22.
Myung, I., Balasubramanian, V. and Pitt, M. (2000) ‘Counting Probability Distributions:
Differential Geometry and Model Selection’ Proceedings National Academy of Science,
97, 21 (October 2000), 11170–5.
350 Jorma Rissanen
Rissanen, J. (1986) ‘Stochastic Complexity and Modeling’, Annals of Statistics, 14, 1080–
100
Rissanen, J. (1987) ‘Stochastic Complexity’, J. Royal Statistical Society, Series B, 49, (3)
(with discussions) 223–65
Rissanen, J. (1996) ‘Fisher Information and Stochastic Complexity’, IEEE Trans. Informa-
tion Theory, IT-42, (1), pp 40–7
Rissanen, J. (2000a) ‘MDL Denoising’, IEEE Trans. on Information Theory, IT-46, (7),
November 2000. Also http://www.cs.tut.fi/ rissanen/.
Rissanen, J. (2000b) ‘Strong Optimality of Normalized ML models as Universal Codes and
Information in Data’, http://www.cs.tut.fi/ rissanen/ (to appear in IEEE Trans. Informa-
tion Theory, 2001
Shtarkov,Yu. M. (1987) ‘Universal Sequential Coding of Single Messages’, Translated from
Problems of Information Transmission, 23, (3), 3–17.
White, H. (1994) Estimation, inference, and specification analysis, Cambridge University
Press, Cambridge, UK.
15 The ‘exogenous’ in
‘endogenous’ growth
theory
Frank Hahn
15.1 Introduction
In this chapter I examine the claim of the new growth theories that they provide
an ‘endogenous account’ of economic growth. The first question is: what would
count as an endogenous account? Or perhaps better: what would not count? For it
is easier to answer this.
The Solow model [1956] is perhaps the paradigm of an exogenous growth
model. In that model the rate of growth of Harrod-neutral technical progress is
taken as given. Of course the type of technical improvement is also fixed. On
the other hand economic theory is used to show that every equilibrium path of
the economy seeks the steady state. Many if not most of current theories do not
even ask or answer the question of convergence. Yet again Solow does not explain
why we need only consider equilibrium paths (Harrod had distinguished between
the ‘warranted’ and the ‘ actual’ growth rate). But the same is true of current
‘endogenous’ theories. One can continue . . . for instance the saving rate is given
in Solow while it is deduced from the maximisation of a Ramsey integral of a given
intertemporal utility function in many ‘endogenous’ theories.
So it seems clear that while it is easy to distinguish between endogenous and
exogenous variables it is uncertain how to distinguish theories by the same criteria.
On reflection one reaches the conclusion that all theories in all subjects must base
themselves on some exogenously given elements. But in that case what is it about
recent growth theories that entitles them to be called ‘endogenous’? It seems that
the answer is simply that the exogenously given rate of technical progress has
been abandoned in favour of a stochastic profit maximising use of resources to
search for technical improvements. This can be combined with an intertemporal
utility maximising choice of training. Since both can involve increasing returns
and externalities some fairly drastic changes in received theory may result: e.g.
multiple steady states and non-perfect competition. But all arguments are derived
from equilibrium conditions and functional forms are chosen to ensure that some
steady state equilibrium exists.
Clearly the scope of the theory has been enlarged. Has this been sufficient to
earn the character of an ‘endogenous’ theory? It is time to ask what we mean by the
352 Frank Hahn
question. In an endogenous theory all the elements which explain and determine
growth are explained as an outcome of the rational decisions of agents. So one
may be willing to accept rationality of agents as exogenously given. Or more
ambitiously it may be deduced from less immediate postulates, e.g. a Darwinian
struggle for survival. How the theory proceeds on this has important consequences
for its scope; to how many societies can it be applied?
Let us stick to a less ambitious interpretation to see what is needed. Rationality
of decisions (or of beliefs) by itself will not get us very far. We shall surely need
to be given the preferences of agents, their distribution, their endowments and
their distribution. Of course we shall need to know what information is available.
In preferences one includes attitudes to uncertainty and beliefs. One also must
find a way to describe the educational endowments and the production process of
education. The structure of production i.e. the mix of industry and agriculture as
well as the mix of competitive and imperfectly competitive activities all need to
be specified or explained. So will the endowment of public goods.
The list is large but I have included in it only what seems most relevant for
growth and have as yet not even mentioned an account of the manner of technical
progress.1 This will be discussed below. At this stage I hope that it will be clear
why I regard the somewhat hopeless task of a general definition of ‘endogenous’ to
be unnecessary. It is clear to the naked eye that functions describing the outcome
of the choice of education, or the acquisition of new technological knowledge by
a choice of R and D will not suffice for a model of growth in which the most
important elements are themselves explained, by the model. That of course does
not mean that these partially endogenous models are not valuable. But it does
suggest that it would be useful for consumers of these theories to be warned that
they are not being offered a theory of economic history. This chapter constitutes
such a warning.
15.2 Schumpeterian models

In my view the clearest exposition of a possible modern Schumpeterian approach
is that of Aghion and Howitt (1998). It also allows me to make some of the above
points more precisely. I shall only discuss some of their simpler models but I will
note the results of variation.
The idea is that output y is produced according to a monotone concave pro-
duction function with intermediate goods as inputs:
y = AF(x)
Innovations replace some intermediate goods by others and raise A. They assume
that A is raised by a constant factor γ > 1. Labour in turn can be allocated to
research or to the production of intermediate goods. Innovations arrive at a Poisson
rate λ. φ(n) where λ indicates the productivity of the research technology and n
workers are allocated to research. This amount (n) is given by the equilibrium
condition
wt = λφ (nt )Vt+1
The ‘exogenous’ in ‘endogenous’ growth theory 353
where t is the number of the innovations which have taken place and Vt+1 is the
expected pay-off to the (t + 1)th innovator:
Vt+1 = πt+1 /(r + λφ(nt+1 ))
where nt+1 is the amount of labour in R and D after the (t + 1)th innovation and
πt+1 is the flow profit of the (t + 1)th intermediate good monopolist. They interpret
(r + λφ(nt+1 )) as the ‘obsolescence adjusted interest rate’ and define
πt+1 = At+1 π̂(wt+1 /At+1 ) with π < 0.
There must be no profit that can be made (in equilibrium) by reallocating labour
between the production of intermediate goods and research. Hence
wt = λφ (nt )γπ(wt+1 )/{r + λφ(nt+1 )}
Since labour can be indifferently used in research or production we obtain a second

equilibrium equation for labour market clearing (0 = labour allocated to the
production of x)
L = nt + 0 (wt )
In steady state one drops the ‘t’ and has
w = λφ (n)φπ(w)/{r + λφ(n)}
n + 0 (w) = L
This can easily be shown to yield a unique steady state growth rate which will be
an increasing function of λ and γ and L and a declining one in r.
Schumpeter believed that competition increased the rate of innovations. If we
write F(x) = xα (0) the parameter α is a measure of the degree of competition and
the derived demand curve faced by the monopolist has an elasticity 1/(1 − α). So
now
π = wx(1 − α)/α ≡ w(L − n)(1 − α)/α
So A becomes
1 = λφ (h∗ )γ(L − n∗ )(1 − α)/α//{r + λφ(n∗ )}
So n∗ is declining in α and competition is bad for growth. (It is not clear that this
is a faithful interpretation of Schumpeter.)
The above are the bare bones of A. H and the simplest. By introducing many
sectors, capital into R and D and a sequential adoption rate they can attain an-
swers which make competition good or bad for growth depending on parameter
values. They are to be praised for isolating some of the key issues which include
obsolescence and the foresight of it.
It is not the lack of realism of the construction which is of interest here, but
rather the question: is it an endogenous model of growth. It is clear that some
of the more important parts are exogenous. Not only λ, γ and α which play a
354 Frank Hahn
considerable part but the functional forms, the postulate of universal risk neutrality,
the homogeneity of the ability composition of labour, the perfect information of
agents and the postulate that there is either a steady state or that markets clear at
all dates. Of course even pretty closed theories depend on parameter values: for
instance in particle physics, the spin of the electron. But physicists can measure
these parameters, to put it no more strongly, better than we can the ones relevant to
us. It is precisely our relative inability to measure that leads to exogenous variables.
That and the immense complexity of an interdependent group of variables.
I hope that it will not be regarded as presumptuous if I sketch a model I
published in 1990 (for a 1988 conference) to illustrate the latitude plausibility
gives us in functional specification of R and D. Nowadays this approach is rather
commonplace.
Unlike AH I took the production function of firm as Cobb–Douglas:
yt = at lt kt
where kt is the capital to efficiency units of labour. I took a deterministic formulation

because I did not know how to deal with risk aversion of firms.2 Also I worked
in continuous time, so that if rt is the amount of research per efficiency unit one
obtained
ȧ = h(rt )at
where h(r) = α + λ(r) and λ(r) is logistic. I then postulated that the difficulties
of innovation first decline and then increase with the amount of research already
done. (Compare this with Romer [1986].)
To capture externalities I let the cost of research per efficiency unit c(·) be
c(r, R) with R = average amount of research in the economy,
cr (r, R) > 0 all (r, R), crr (r, R) > 0 when r = R, crr + crR = 0 if r = R.
One can find many plausible arguments for those kinds of externalities. These
include those given by AH but also the likelihood that a higher R will lead to a larger
community of research workers and a greater familiarity with research methods.
For the rest I do not at all improve on A H. R and D is carried out inside the
producing firm and there are no specialised research firms. I then use the calculus
of variations to find the firm’s present value profit maximising plans. To this I add
the equilibrium condition of self-finance which is clearly implausible. Moreover I
take an exogenous Solowian saving ratio. That too is limiting, but the model can be
rectified in the Chicago direction rather easily but at the Chicago cost that savers
have perfect foresight. (Actually in a deterministic setting such as here this would
not be as silly as usual.) There is only one good produced.
It turns out that there are three steady states: one with no research at all so that
α is the Solowian rate of technical progress. In all steady states r = R but there is
more research in one than the other. Of course in steady state all firms take R as
given when they decide on r, (and only two can be stable).
I think that I prefer the AH model with its spirit of struggle for existence and
its explicit attention to obsolescence.3 On the other hand in my version the role
of R is also plausible. It indicates whether one lives in a world where research is
a habit or not, and my formulation suggests that it may or may not be the case.
This gives just a little greater flavour of endogeneity. On the other hand again I,
no more than others, distinguish labour by its ability to research and produce. But
my point is not realism. It was to underline how much that is taken as endogenous
here really is not. This brings me to my next point.
15.3 Equilibrium
When we say that a variable z is endogenous to a model we mostly mean that it is
determinable by equilibrium conditions. In growth theory this involves expecta-
tions in an essential way. In my view they also involve risk attitudes in an essential
way. For instance it seems that in nineteenth century England risk aversion was less
than in Germany. One also needs to make postulates concerning the information
flows and the ability to use them. So when one writes down equilibrium equations
one is bringing into the story many elements not explained by any theory.
If we pay explicit attention to these then we may find our way to a theory which
can indeed explain our observation – say economy A grows slowly because it has
many risk-averse producers – say farmers. But it is not clear whether economists
can say why this should be so.
My point as usual here is not the usual grouse concerning lack of realism. It
is that when we use the canonical paradigm of economic theory we are rarely in a
position to attain purely endogenous results. Or rather what seems like those are
not genuinely so.
As a good example take the influential Chicago method. It needs to postulate
that expectations are rational and look at rational expectations equilibria. The time
horizon for these expectations is infinity. There are possible learning theories –
which lead to rational expectations over finite time but, in the nature of the case,
such expectations over the infinite future could not be proved. But equally seriously,
almost all of the authors make no allowance for risk attitudes or, rather, postulate
risk-neutrality. In our theories of course this means that utility functions are taken
to be exogenous. This in my view is the right attitude to take by economists.
But there is almost surely a distribution of such functions so that the more subtle
hypothesis that tastes etc. are stable through infinite time needs to be invoked. This
is certainly an exogenous piece of theory. For long run growth theories it does not
seem persuasive. In addition Chicago always postulates perfect competition and
production possibilities which allow this. All improvements in technique are either
in intermediate goods or due to improved education but, as in Marshall, external
to the competitive firm. This feature is not explained by the theory and hence an
exogenous element. I do not know whether it is claimed that the theory fits the
facts (so far), but I doubt it, since it is too aggregated to be confronted with some of
the most important episodes in growth, e.g. the introduction of the steam engine,
the internal combustion engine, electricity, computers, etc.
This discussion mirrors one in evolution. Gould maintains if exogenously given
stochastic geological events had been different than they have been, present day
356 Frank Hahn
creatures would not only be different but intelligence might never have arisen. The
Cambridge professor of evolution believes that whatever the external geological
etc. events intelligence would have evolved – I assume because of the advantage it
confers. In our context this is something of a pro-Chicago argument, since it argues
that competition and intelligence are almost bound to lead to growth whatever
particular form it takes. But that is a good deal more modest claim. In particular
it could be made without the full equilibrium paraphernalia since in this context it
is bound to be ill-defined without imports of exogenous factors.
As a matter of fact there are theories which give a fully endogenous account of
why a (Nash) equilibrium should occur. But they rarely include technical progress
and, what is worse, the equilibrium is not unique. This then requires initial condi-
tions and a more or less exogenously specified process of the evolution of strate-
gies to get us where we want to be. A variety of these have been proposed but
the outcome is more in the spirit of Gould since some processes converge to the
risk-dominant rather than the Nash equilibrium.
15.4 Expectations again

Chicago has rational price expectations which seems somewhat odd in a world in
which future technology has to be anticipated albeit stochastically. Suppose for
instance that a high probability is given to a future viable electric car. The first
question then is – say for oil companies – its probable date. The second is how
would it affect all the prices relevant to an agent’s decision. This is an example
which is not in itself hostile to endogeneity – for instance the pollution etc. of
present cars is what sets off the search for electric ones. But it is, I believe, hostile
to endogeneity as perpetual equilibrium.
AH – good Schumpeterians – pay special attention to anticipated obsolescence
and to Schumpeterian temporary monopoly for the innovator. But I think that there
is at any time a rather large array of vintages so that at best one could calculate the
expected average length of anyone of them. (Like everything else, this is discussed
by AH). I am sure that one can construct some kind of theoretical picture of this
– indeed AH do so. But this depends on information available, (or obtainable),
and on good telescopic faculties which we recall was one of the characteristics of
the Schumpeterian entrepreneur. Relatively small risk-aversion was another. The
proportion of these in a population at present is exogenous. (See below).
Moreover as a matter of plain fact the economies we know have not been
and are not perfectly competitive. Firms then must forecast expected demand
functions – their own prices they will set in the light of these forecasts. Except for
the simplest cases neither I nor, I believe, any one else knows how to construct
an equilibrium model consisting of these kind of firms. But even if we did it is
obvious that there would be a very large exogenous – unexplained – component.
In most endogenous theories new products are continually appearing and it is not
clear how past experience leads one to rationally expected demand curves. One
could in the usual way think of agents engaged in a game. But the game will be
changing as strategy sets and payoffs are changed by ‘progress’. Again it will be
possible to cobble together some hypotheses but, on the whole, it seems to me that
this will replace what might have been taken as a simple exogenous process by a
more fancy one which must cast doubt on, for instance, agents at all time being in
a Nash equilibrium.
Returning to some remarks of AH it must be doubtful that the elasticity of
demand should be treated as exogenous in an endogenous theory or that it is a
good measure of monopoly. For one thing most firms produce multiple products,
for another one would have thought the position of the demand curve – share of
market – was also a monopoly indicator. Of course there is something to the Kalecki
measure but not enough, it seems to me, to measure the extent of competition. For
instance legal restrictions on market share or on getting together on pricing also
matter. There probably exist, or could exist, ‘endogenous’ theories of how such
laws arise, but we must take this as exogenous. (Recall that not so long ago kings
licensed monopolies.)
15.5 Information
Risk-attitude and competence characterised the Schumpeterian entrepreneur. Their
number Schumpeter believed to depend on the culture including the religion of the
economy. Interestingly enough he thought that the ‘routinisation’of R and D would
not only lead to the entrepreneur’s obsolescence but to a slow down in technical
progress. But in any case R and D does depend essentially on people capable of
generating new knowledge not only inside the R and D outfit but outside it.
The DNA revolution has transformed R and D in pharmaceutical industries
and agro-businesses. But this revolution in knowledge was brought about by ex-
ceptional people the fraction of which in the population is surely at present exoge-
nously given. It also, of course, depended on these people having the opportunity to
do their revolutionising, on the transmission of this knowledge and on sufficiently
educated people to recognise the importance of what has been accomplished, (see
Arrow [1974]). Clearly this brings education into the picture (see below) and the
whole is permeated by externalities.
There are many examples of revolutionary inventions which for a long time
were ignored by those who could have profited from their use. The jet engine is
one, as is the fluorescent bulb. I do not know the facts sufficiently well to make
the point completely convincing: to recognise the benefit of an advance is almost
as important as the advance itself. For instance it is reported (Nelson and Wright
[1992] in their excellent survey) that when electronic computers first emerged to
satisfy the US military, the general view, including at IBM, was that it would have
few civilian uses and the invention languished for quite some time. This seems to
me more than a case of mistaken expectations but rather a lack of imagination.
People with that kind of imagination are rare and certainly their representation in
the economy is a matter of culture and largely exogenous for the economist. It is
not clear whether one could regard it as an output of education. (See below.)
But the basic point is (i) finding new information, (ii) whether and how it is
transmitted and (iii) how widely it can be used. I doubt that Crick and Watson
358 Frank Hahn
needed the incentive of monetary rewards to generate DNA information. How-
ever, there were other rather obvious incentives (which I have not seen discussed
in the literature). Certainly adoption by commercial firms or even feeding new
information into R and D depends on their calculation of probable rewards. These
in turn, as Schumpeter argued, depend on the estimated length of money rents (and
on patent law). Clearly that is part of an ‘endogenous’ argument, but only a part.
Many of the relevant elements seem clearly exogenous.
Of course much information is available by publication and word of mouth.
But because information is a public good its transmission, if it is valuable, will
be imperfect. Some information when it is directly or indirectly revealed will
be a surprise to agents who cannot be assumed to have already adapted to it in
their plans. Except, that is, for one thing: the positive probability given to the
appearance of new innovations. This aspect, ‘creative destruction’ is well treated
by AH and of course enters growth as a negative element. But the probability of
such obsolescence will vary from case to case, e.g. the biro does not, and computers
do, seem to run this risk with low and high probability. Once again exogenously
given risk attitudes play as large a part as probabilities. These in turn depend on
the time horizon (effective discount rate) of the agents concerned.
Nelson and Wright [1992] report the predominant view of historians that the
early US productivity superiority (from about 1840) to 1960 had a great deal to
do with superior organisation, abundant raw material and a large domestic market.
(The US for long was a high tariff economy.) None of this seems to be captured
by existing models of endogenous growth.4 Organising and managing etc. is not
a matter of R and D but rather the fruit of a culture which led to the emergence
of a ‘correct’ way to organise and manage. They emphasise that the education
system did not produce many outstanding scientists who had to go to Europe to
study. It is true that high real wages made general education advantageous and that
a generally educated workforce helped to sustain the US system of production.
But later, immigration of ill-educated workers does not seem to have led to a slow
down. Certainly information was of almost a collective sort – one knew how others
organised and so the usual caveats do not apply.
However after the second world war and the rise of ‘science based’ industries,
this picture changes. In particular there was a great growth in resources devoted
to R and D in the US and secrecy of certain aspects of a firm’s activities became
important. But it was never watertight. The rise of science based industries itself
depended on the progress of science which, in turn, depended on the evaluation of
the benefits it would confer. It is interesting to note that the government played a
significant role in this (partly through universities, partly through subsidies).
As a description these (and other aspects in particular education) matters are
very clear. How to convert this into an endogenous theory of growth is not. The
very fast growth in scientific knowledge seems like an exogenous cumulative
process. Once the genetic code was discovered for instance, many researchers
were attracted to this field and new knowledge rapidly accumulated. This sort of
thing is hard to embody persuasively in a functional form and even if it could be,
it would be quite unclear whether it also embodies other scientific advances, e.g.
in superconductivity. Least clear is whether these various theoretical descriptions
can be made to yield a steady state with exponential growth.
But of course post-war scientific advances, for a time, led to greater encour-
agement of fundamental science and the hiring of scientists by firms. So of course
there is something ‘endogenous’ to economic models of growth. My contention
is that it is only ‘something’. For instance, many fundamental sciences by now
receive greatly reduced subsidies from government and firms often use scientists
as managers. There are still considerable risks in new science based economic ven-
tures, for instance in bio-technology. Everyone sees ever more scientific knowledge
emerging, but there is not only uncertainty as to timing but as to kind. This brings
me back to risk attitudes.
15.6 Risk attitudes

I have proceeded on the hypothesis that these are exogenously given. This is only
correct in a somewhat subtle way.
Prevailing risk attitudes of others have an effect on my own when I manage
a firm. When I see others taking considerable risks with some success I shall be
more inclined to do so also. Moreover the high risks undertaken by others may spell
danger for me (obsolescence, etc.). This is only partly an effect on risk attitudes but
on the risks I run. Here is what AH mean by the effects of competition or a higher
arrival rate of new knowledge. But attitudes to risk are also likely to be affected. If
the prevailing social norm is not to flinch from risk, my attitude will be different
from what it would be for a more conservative social norm. If one thinks of these
matters game-theoretically, there may be many outcomes (e.g. Nash equilibria)
and it will be a path-dependent and so partly an exogenously given one.
I believe that it would not be too hard to put all of this formally. But I am not
attempting a new theory here. My objection to most endogenous growth models is
that they ignore risk attitudes. It seems to the highest degree unlikely that managers
do not consider the variance (possibly also higher moments) of their expectations
of future new knowledge and of their chances of producing it, say by R and D.
In production, risk attitudes of firms is related to the way they are governed. A
single powerful manager is not all that common and much is decided by commit-
tees, etc., and I do not know of a good theory of the risk attitudes of such bodies.
But, as I say, there are cases where the attitude of a dominant manager is determin-
ing. In the UK Lord Weinstock at GEC was such a person and he was extremely
risk averse. At some time the company had two billion pounds of liquid assets and
Britain never developed its own viable electronics industry. An economy where
powerful magnates are the norm is not one in which attitudes to risk can be ignored
and they, I believe, must to some extent at least be taken as exogenous.
But there is hope for serious endogenous theorising in the global economy.
The reasons are clear – peculiarities, whether national or personal have a harder
time of surviving or being important. Although there seems to be a furious debate
on ‘convergence’, this I think cannot be denied and differences will turn more
on exogenous physical peculiarities such as infra-structure etc. That is, ultimate
360 Frank Hahn
differences will depend on these. On the way there are inherited education systems
etc., all of which are exogenous.
Lucas who believes that a good way of modelling growth is by a Ramsey model
and who is willing to invoke externalities (here in education), simply appeals to
different steady state equilibria. But I find the analysis very unconvincing, if for
no other reason than that one needs to invoke all sorts of institutional elements to
argue that, say, Mozambique is on an optimum growth path. Nonetheless, as usual,
I admit that the analysis evidently reveals some real features – the importance and
externalities of education.
Now under the present theory of uncertainty all this may sound as if I am
objecting to a particular parametisation of utility functions which we all take as
exogenous. But a social historian would not do so. Exogenously given utility
functions are acceptable when we are studying situations over a stretch of time
which is relatively brief. But as t goes to infinity that seems far-fetched. At a
guess a medieval farmer was much more risk averse than a modern one. If nothing
else we are looking at the curvature of U at higher income levels, and it simply
is implausible that it is linear over the whole set of possible outcomes. For an
endogenous theory of growth we would like an endogenous theory of the evolution
of risk attitudes. As far as I know we do not have one.
15.7 Labour and education

There have been many attempts at endogenous population theories. They are com-
plex and contentious and most endogenous growth models (but see Jorgenson
[1961]) take them as exogenous. Notoriously, of course, not Malthus and other
classical economists.
Even so, if labour is measured in efficiency units some possibilities of endo-
geneity arise even for constant population. Indeed since economists universally
regard labour as a source of disutility, the vast increase in labour productivity
which has taken place must lead us to suppose that the supply of raw labour has
diminished over the century – the amount of time taken in leisure has increased.
Curiously enough this rather obvious manifestation of endogeneity in growth the-
ory is rarely analysed. Here too there will be important externalities as to what
counts as ‘long hours’ or even ‘hard work’ changes. In a study of exponential
steady state growth this matter deserves attention since the limit of the growth of
efficiency and leisure choices need to be given. (See Hahn [1990]).
But I do not propose to delve deeply into this. What has become important
in endogenous theorising (ever since Lucas’s [1988] paper) has been the role of
education. The choice of education is discussed in the usual marginal expected
benefits and costs terms. Education itself adds to the number of efficiency units of
labour directly, quite apart from its contribution to the process of innovation. But
the ‘education industry’ if one may so call it is left largely unmodelled. There have
been large innovations in that ‘industry’ from a concentration on the classics and
theology, to mathematics and science. No doubt these changes can also be partially
accounted for by the greater use business could make of numerate workers. But a
good deal of these changes were due to governments and one needs a theoretical
account of how these respond to the needs of business. There is also an egg and
chicken problem. Until there are sufficiently trained numerate people, others cannot
choose to be trained in these skills. The Lucas procedure of letting the education
of labour be entirely determined by the trade-off between benefits and costs is only
partly convincing.
Of course within this class of problems there is the question of what mix of
education is offered and chosen. Nelson and Wright report a marked decline in the
US recently in the support for fundamental science. The same is true in the UK.
Practically oriented people have never valued fundamentals – Ford pooh-poohed
their pursuit and Mrs. Thatcher lamented that the U.K. produced too many Nobel
Prizes. There are a number of conjectures of why this should be so, but there are
also a number of R and D models in which variously trained labour appears in
the functions, which argue what choices a growth maximising government should
make. (AH and, for instance, Phelps [1966]).
It seems widely agreed that there are increasing returns in the acquisition of
human capital (Lucas [1988]) and this allows for education to have an effect on
growth5 . Education is also meant to increase mobility of labour with the same
consequence. But here other exogenously given (for a time) factors may work
in the opposite direction as the distribution of the housing stock and preferences
for locality. But one thing is clear: education is required for R and D, that is,
it is now a necessary condition for growth. That was not true in the nineteenth
century when R and D hardly existed, and yet there was much growth by learning
by doing and shrewd management. So it depended on the (for us) exogenously
given progress of science and engineering whether R and D is the main vehicle for
growth by innovation. (See below.) Clearly there will be different mixes possible
and to some extent the market will determine these if the educational manpower
makes it feasible.
A much debated question is whether there are externalities to education. It is
usual to suppose that more skill etc. is reflected in higher wages. Looking at the
wages of engineers in the UK that is by no means obvious. (It clearly also depends
on the number undertaking the particular training.) But if education contributes to
the productivity of R and D, it is unlikely that its benefits will be fully internalised.
Moreover Spence’s [1973] famous work suggests that the amount and kind of
education is unlikely to be optimal. There are also other simple externalities when
an extra skilled worker raises the productivity of those already in place.
AH in their chapter on education use many common-sense ideas to put into
their functional dress. They do not provide a precise theory of the development of
the educational sector and indeed often discuss policy measures to improve this,
which suggests that the endogenous market theory does not suffice. But it would
be wrong to deny that the market has considerable influence. However when the
amount of education in an economy changes it is likely that the decision to pursue
education will also change independently of the pecuniary benefits. For instance,
for many people higher education is a vehicle for social mobility and is also driven
by a herd effect. These are all matters which make full endogenisation harder.
362 Frank Hahn
I have already mentioned Nelson and Wright’s emphasis on organisation.
Recognition of this led to business schools in the US and then in almost all Eu-
ropean countries. Whether this can be explained as an equilibrium outcome I do
not know. Once again there are surely herd effects. Moreover many firms prefer
to do their own management training and rely on learning by doing. On the other
hand most business schools are heavily subsidised by business, which argues in
another direction. I find it hard to see how this development is to be fitted into an
R and D theory, and certainly think imitation a plausible model – imitation of a
spontaneous bright idea.
15.8 R and D again

I want to include a somewhat more formal problem which is too little discussed. It
is my belief that just stating it will contribute to an understanding of endogenous
growth theory.
Consider a firm i and the orthodox theory of its production decision. We endow
it in the first instance with a production set Yi , a subset of commodity space. How
do we interpret Yi ? Is it the set of activities known to i? Or should we straight away
write Yi (t), the set known to i at t? Clearly here we had better opt for the latter.
But the commodity space of which Yi (t) is a subset should also be indexed by t
for obvious reasons. Note that by commodities I include labour of different types
and efficiency. Or again we may write Ŷi as its ‘divine’ set by which I mean the set
of all activities that will ever be known to i, (embedded in a ‘divine’ commodity
set). Then Yi (t) ⊂ Ŷi .
Whichever more or less equivalent approach we adopt there remains a problem:
how do we represent organisation (or management) choice. (See Arrow [1974] and
Radner [1993].) We have learned a great deal in recent years of the reasons why
production cannot be completely decentralised – bargaining costs, information and
other transaction costs are cited. So we ought to include organisation in Y (and Ŷ .
If that involves costs, i.e. inputs, for instance of managers, then this will have to be
taken into account. There are also intangibles, (e.g. organisation charts) quite apart
from possible special managerial input. It is seen that the sheer act of description
is not a simple one nor will the usual properties of production sets remain.
Now R and D is to provide information to i of the divine set. (It may also be
used strategically but matters are hard enough already.) The amount of R and D
undertaken by i will depend on expected costs and benefits. (One uses an arbitrary
equation of AH.) I have already argued that these depend on the amount of related
research done by others in the economy, as well as on what, for a better expression,
one might call the ‘research atmosphere’ in the economy. Of course it also depends
on more conventional inputs of specialised and ordinary labour, and on equipment.
There are good arguments to suggest that R and D has an increasing returns segment
in all of its inputs. But these increasing returns – or the way we look at returns –
may refer only or largely to the probability of success.
But success in what? The endogenous macro-economic growth modellers lump
it all into productivity. That however does not tell a firm in which direction to point
its research. It seems to me that the past here has some explanatory power. Once
there are computers, smaller and faster ones are natural projects. Before that it
was not at all clear that research here was worthwhile. Once the genetic code was
discovered the direction of further research and development was fairly obvious,
at least in broad outline. There will also be many past failures to point to which
one now seeks to avoid. As far as decision makers are concerned the past is given,
as far as economists are concerned it can be painfully explained but it is a complex
task. I do not think that enough attention has been given to the stochastic which
we all agree is needed. We do not really know what is the process from which
realisation of say fundamental new knowledge or clever new production ideas are
drawn. As far as I can see we claim to have an endogenous (macro) theory where
peculiarities ‘wash out’ and where the machine is driven by a search for profit.
That is better than nothing, but since we have no idea what the functional forms
are, it leaves a great deal to ‘it fits the data so far’.
As far as benefit calculations are concerned one needs to specify the organ-
isation of the firm and its risk attitudes. One also must hope that benefits are
recognised and properly evaluated. At the moment R and D expenditure in the UK
is falling and has fallen rather a lot already. Why? I suspect it has a good deal to
do with the type of British manager, but there are almost surely more profound
explanations. As far as I can see competition from abroad has increased because
of the high pound and it may be that foreigners are regarded as so much more
‘dynamic’ that the obsolescence risk is put very high.
One of the difficulties with present theorising is that it makes it very difficult
to explain why some executives are highly paid and others less so. They all form
policy in the same way and indeed the latter could be put on a computer and all
a firm then needs is to recognise the ‘conditional’ of a policy and press the right
button. There is in other words far too little said about different abilities to evaluate
probabilities of different people (and societies) and again different risk attitudes
disappear. There is no Schumpeterian entrepreneur and the analysis is all routine
maximisation. Of course I am not arguing that orthodox maximisation approach
yields no useful insights – only that we need more before we claim to have an
‘endogenous’ theory.
Innovations, as Schumpeter argued, disrupt routine. Many people dislike such
disruption. Comparing the US with the UK it seems clear that this dislike is less
pronounced in the former. Why? Do we build that into our equations?
R and D activity maps into the probability space of the divine set. What that
map is I do not think we know, even at the macro-level, and we choose maps which
can either yield steady states or cycles or both. Since both have been observed this
is not to be despised but it seems to me rather mad to do this over infinite time and
with an unchanging stochastic process, let alone under perfect competition.
My conclusion is this: given a history and a market economy we can single out
elements which are important in producing growth and that in a fairly orthodox
fashion. To do more is fiction.
364 Frank Hahn
15.9 The global economy (an addendum)
Up to now I have had the traditional ‘economy’ of our textbooks in mind. Even
in that case – supposing it to be closed to foreign trade, it will receive relevant
technological signals from the outside world. But that is well understood. However
if we are concerned with large geographical areas where trade is free and capital
(and possibly labour) movements are free, then one needs to think again. Like
other areas, the literature on the subject is well summarised by AH.
I am only concerned with what becomes of the exogenous–endogenous distinc-
tion. It would seem that the scope of the endogenous is enlarged simply because
there will be a ‘washing out’ of the special occurrences and in particular it may be
that the importance of risk attitudes is diminished. I do not really know. But the
scope for cartelisation is also enlarged, which points in the opposite direction.
Clearly one is now discussing convergence. But quite often, because of in-
creasing returns and externalities, one would expect divergence. Southern Italy is
a good example and I fear that parts of Europe will have a similar fate relative to,
say, Germany. But of course we shall no longer sensibly speak of countries pro-
vided labour mobility and capital mobility is large. Much here depends on local
authorities and exogenously given elements like language.
I would hesitate to predict that the global economy can be modelled in a similar
way to local economies, and I would find it hard to believe that the plausibility of
functional forms, e.g. for education etc., will remain unaffected. On the other hand
national characteristics, utility functions and organisation, may indeed converge
and so, as I have said already, make an endogenous theory more possible. I have
not seen any very formal discussion of this, but it is often mentioned. Note that
this suggests for instance an endogenous theory of tastes and aspiration.
So I conclude by saying that for a global economy the ‘exogenous’, beyond
obvious elements like climate, may be less important. However for a long time
to come functional forms needed by an endogenous theory will have little but
plausibility to support them.
Acknowledgement
I am grateful to Kenneth Arrow and Robert Solow for comments.
Notes
1 Kenneth Arrow (in conversation) interprets ‘endogenous’ in this way, that is, to get a
growth equation on the basis of rational choice. He takes Solow’s given rate of growth
as the point of departure. There is no monopoly in terminology and if ‘endogenous’ is
taken to have the Arrow meaning this chapter has the wrong title. I however regard the
word to apply to a theory which does not treat as exogenous what economic theory is
in principle equipped to explain. Moreover one is likely to be led into error when the
full set of factors taken to be exogenous is not enumerated.
2 If one postulates risk neutrality, stochastic elements present no technical problems.
3 Solow has pointed out to me that some innovations may be complementary to existing
technology.
4 What is sometimes discussed is the relative efficiency of various organisation forms
(largely connected with the absorption of information) but it is not integrated into what
we are used to call the production set. Of course these findings suggest that there were
important increasing returns to scale. But it may also be that a large market reduces
risk.
5 This may be taken to mean that the productivity of educational resources increases with
the general educational level of society or that the cost of acquiring extra education by
an individual is a decreasing function of his educational level.
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Nelson and Wright, G. (1992) ‘The Rise and Fall of American Technological Leadership’,
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Phelps, E. (1966) ‘Investment in Humans, Technological Diffusion and Economic Growth’,
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Radner, R. (1993) ‘The Organisation of Decentralised Information Processing’, Economet-
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366 Index
Index 367
Index
AC-PC (Accumulation Curve-Phillips Aghion, P. 3, 231, 245, 247–9, 254, 255,

Curve) analysis 131–8, 141, 162, 163, 257, 316, 317, 319, 352, 356, 357,
164 358, 359, 361, 364
Acar, E. 325 agricultural sector 11–12, 29, 30
accelerator property, of innovations AH see Aghion, P.
277–8 AIC, Akaike’s criterion 345
accounting method, of growth 37, 172, 188 Akaike, H. 345
accounting systems, of incentives 310, Akdeniz, L. 319, 320
316 Allen, C. 326
Adaptively Rational Equilibrium alternative stable states 303, 305
Dynamics (ARED), theory of 332, 333 Altug, S. 319, 324, 331
adjustment costs, of capital 249–50, 251 Alvarez, F. 295, 296
adjustment mechanisms 157, 160; of gold Amendola, M. xxi–xxii, 117
standard period 10, 15, 17, 22; to American Economic Review 148
shocks 15–16, 17, 22, 116, 118, 119; analysis: AC-PC (Accumulation Curve-
within markets 220–5, 227 n1 Phillips Curve) 131–8, 141, 162, 163,
administrative technology 98 164; of business cycles 253, 306;
adoption: models 320, 321–2, 328; of disaggregated 22, 64, 322; duration
Real Cost Reduction (RCR) 311, 330; economic 116; empirical 229,
316–22, 329–30, 334–5 230; equilibrium 288, 297, 319; of
agents 204, 216 n2, 303; behavioural growth 64, 120–5, 130, 269, 308–9,
modes of 219, 254; belief systems of 317–19, 323, 325; Keynesian 148–9,
332; classifications of 238, 240–1, 162; multi-sectoral 60, 63, 86–91;
246, 248, 254, 257; distribution of optimal timing/stopping 319–20;
230–1, 242, 245, 257; financially Phillips curves (PC) 150, 163; welfare
constrained 242, 243; generations of 319
236, 245; heterogeneous 230, 231, Anderson, P. 301, 321, 323
233, 319; infinity of 212, 213, 214–15, Aoki, M. xxii, 44, 203, 204, 206, 207,
216; number of 334; rationality of 208, 209, 211, 212, 213, 214, 216
332, 333, 352; skilled 245, 246; appreciation, of yen 43, 44
unskilled 245, 246, 293; wealth of archaeologists 105
245, 246, 247 ARED see Adaptively Rational
aggregate dynamics 229–30, 235–6, 253 Equilibrium Dynamics (ARED)
aggregate equity, law of motion of 234, Arrow, K.J. 177, 230, 269, 271, 301, 321,
235 323, 364 n1
aggregate functions 230, 233, 235–6; Arthur, W. 301–2, 320, 322, 326, 327,
production 285, 287–91 328, 331, 332, 334, 335
aggregation 229–30, 235, 257 n5; theory assets: collateralizable 242; prices of 245,
288, 303 330–2, 333
368 Index
asymmetric information 230, 231, 232, building cycle theory 33–4
239, 258 n12 Burns, A.F. 285
asymmetrical business cycles 203, 216 Burnside, C. 292
attraction, basins of 213, 216 business cycles: analysis 253, 306;
auditing, probability of 239, 240 asymmetrical 203, 216; definition of
automobile industry, in Japan 41 285; duration of 4, 19, 305;
fluctuations of xviii, 22, 230–1, 285,
Baak, S. 304, 332 286, 287, 294, 296; and growth
Backus, D.K. 10, 17, 296 theories xviii, 286, 296; historical
Balassa, B. 41 perspective needed 3, 22–3;
Balasubramanian, V. 344 international 14, 17, 19–21; models of
Balke, N.S. 19 203, 287–97; and policy frameworks
bank credit 232, 258 n12 3, 4, 15, 18; research 284, 285; shocks
bankruptcy: costs 233, 251; probability of 15, 16; stylised facts approach 3–4,
231, 233, 250, 251 21–2; theories xviii, xx, xxi, 4, 21,
barrier solutions 313, 314 283, 285–7, 293, 294; volatility of 16,
Barro, R. 149–50, 303, 304, 305, 313, 18–19; and weather shocks 10–14
320, 323, 326, 327 business schools 362
basins of attraction 213, 216
Baumol, W.J. xxii, 303, 305, 306, 317, capacity utilization 292
321, 326 capital: adjustment costs 249–50, 251;
Baxter, M. 296 demand for 241, 247; depreciation of
Bayoumi, T. 6 292; formation 49, 81 n9; human 36,
BDSL see Brock, W.A. 242, 243, 244, 245, 246, 293, 365 n5;
belief systems 332, 333 stock 250, 286; supply of 241, 247–8;
bequests 245, 246, 247, 248 utilization 292
Bernanke, B.S. 15, 231, 236–42, 254, capital markets, models of 232–53, 254,
255–6, 257, 258 n17 255–7, 258 n18, n19
BG see Bernanke, B.S. Carlson, G. 317
biases, data snooping 325, 332 Carlstrom, C. 296
bilateral cyclical linkages 19–20 cartelisation 364
binary choice, models of 205–6, 208–9, Cass, D. 285
211 Catao, L.A.V. 9
Blanchard, O.J. 148, 149 change: cultural 104; qualitative 115–16,
Blanchflower, D.G. 154, 155 117, 118–19; structural xix, xxi, xxiii,
Böhm, B. xxi, 71, 305 3, 12, 18, 26, 35, 38, 47, 54–5, 59, 64,
Boltho, A. 33 66, 70, 79, 95, 108, 110; technological
Bolton, P. 231, 245, 247–9, 254, 255, 257 36–7, 41, 44, 63, 95, 119, 120, 170,
bootstrapping 339 317, 352
borrowers 255, 297 chaotic dynamics 52, 79, 259 n23
Boserup, E. 98 Chapman–Kolmogorov equations 203,
bounded rationality 304, 332, 333 204
Braudel, F. 268 Chavas, J. 332
Breiman, L. 204 Chenery, H.B. 33
Bretton Woods exchange system 17–18 Chiarella, C. 128, 131, 136, 137, 141,
Britain see United Kingdom 150, 151, 157–8, 161, 162, 163, 164
Brock, W.A. xxiv, 285, 302, 303, 304, Chicago method 355, 356
305, 308, 309, 313, 314, 315, 321, choice: binary 205–6, 208–9, 211; discrete
322, 323, 324, 326, 328, 329–30, 331, 321, 332; of state variables 207
332, 333, 334, 335 Chow test 70
BRS see Brock, W.A. Christiano, L. 294
Bruno, M. 36, 37 Clark, W. 323
Brussels School 301 classical economic theory 169, 270
budget constraints 243, 244 co-ordination problems 116, 118, 119, 120
Index 369
Cobb–Douglas production function cumulative capital shares 173, 175, 176,
288–9, 354 178, 179, 182
code length of data 340, 341, 343–4, 348 CUSUM-test 70–1
collateralizable assets 242 cycles: asymmetrical 203, 216; growth
collateralization 239, 240, 241, 243, 248 78, 83 n50, 132–8, 141, 162, 330;
collection of data 96, 97 structural 49
common knowledge frameworks, in game cyclical linkages 19–21
theory 35–6
competition: of belief systems 333; and Danthine, J.P. 20
growth 353, 356; on innovation Darwinian evolution 176, 178–9
activities 271–3, 275–7, 303, 353, data: code length of 340, 341, 343–4,
359; monopolistic 291–2, 357; perfect 348; collection 96, 97; discrete 342;
320, 355, 356 sequences 339–43, 344; statistical
complex dynamics xx, xxiv, 230, 257, properties of 341
302 data generating models 345
complexity 349; approaches to economics data generation process (DGP) 66–7
301–2, 320, 322, 330, 334; geometric data snooping biases 325, 332
344; models 320; research 301, 321; David, P. 96
stochastic xxiii, 339–40, 343, 344, Day, R.H. xxi, 47, 64, 96, 104, 106, 113
345; tools 302, 326, 334, 335 n6, 301, 305
computer industry, profitability of 274 debts 137–8, 232, 296
computer networks, stability of 226 Dechert, W. 319, 320, 323, 324
construction sector 11, 33 decomposition 340, 341, 349; of data
consumer durables, diffusion in Japan of 344–5, 348
32 deductive inferences 283, 284
consumption 234, 246, 247, 258 n15; definitions: of business cycles 285; of
contribution to GDP 34; and leisure episodes 104; of firms 296; of
293; volatility of 22, 285 information 339, 349; of models 340;
contract theory, modern 296 operational 286; of scenarios 104; of
convergence: debate on xv, 351, 359, 364; theories 284
of dynamic paths 65; and growth xvi, deflationary situations 141, 148
309, 326; hypothesis 270; notions of degree: of efficiency 245, 255, 257; of
62, 82 n43 inefficiency 237, 238, 239, 240, 255,
Cooley, T. 291, 296 257
corporatization reforms 307, 312–13, 317 Delli Gatti, D. xxii, xxiii
costs: of bankruptcy 233, 251; of capital DeLong, B. 40–1, 310, 321
adjustment 249–50, 251; financing demand: for capital 241, 247; curves 185,
249; of introducing innovations 309; 186, 187, 192–5, 292, 353, 356;
of monitoring 236, 237, 239, 241, 246 functions 356; for land 243, 244;
country by sectors studies 62, 86–91 pressures 145, 147, 152, 153, 155;
coupled markets 220, 221, 223–5, 226 shocks 38
Cox, D.R. 209, 216 demand-led growth 30, 32, 33–4, 44
Crafts, N.F.R. 318, 319 demise, of a society 105, 110–11
creative destruction 358 density functions 341, 342, 344;
credit 232, 245, 249, 258 n12 normalized maximum likelihood
creditors 243 (NML) 345, 346, 347–8
crises: economic 226; financial 15, 253 depreciation, of capital 292
cross-elasticities, network of 320 derivatives 220, 226
cross-validation 339 Desai, M. 134, 151
Crucini, M. 296 destabilizing effects 129–30
cultural change 104 development: economic 105, 106–8, 175,
cultural production functions 100, 101–2, 192; and growth 79; of theories 287
103 Development Bank of Japan 39
cultures 100, 101–2, 103 Devereux, M. 291
370 Index
DGP (data generation process) 66–7 144–50, 155–7, 163, 164; macro
Diamond, P.D. 203, 206, 207, 211, 212, xv–xvi, xvii, xx, 128, 129–30, 148–50,
214, 285 164, 216; multi-phase xxi, 64, 65; of
Diaz-Gimenez, J. 295 oscillations xvii–xviii; out-of-
diffusion: of consumer durables 32; equilibrium xxi, 62, 115–19;
models 320, 326; technological 178, qualitative 96; regime xxi, 53–4, 79;
179, 309 slow moving 305; structural xviii–xix,
disaggregated analysis 22, 64, 322 48, 49, 54–5, 59, 60, 80 n2; supply
discrete: choice econometrics 321, 332; side 129, 131, 132; total factor
data 342 productivity (TFP) 302–3, 320–1,
diseconomies 98–100 322–3, 328; wage-price 128, 129, 131,
disequilibria xxii, 79, 116, 119, 152, 170; 132, 139, 141, 143–5, 147, 151, 152,
dynamics 302; technological 171, 183, 158, 163
191, 192
disequilibrium growth: models 128–30, econometrics 65, 77, 229; discrete choice
131, 135, 137–41, 157–62, 163, 164; 321, 332; models xvii, 66–9, 70, 71–7,
theories 128–30, 135, 150, 163, 192 83 n49; of pattern recognition 323–6;
distributional dynamics 79–80 qualitative xxi, 47, 97; rational
distributions 342, 344; of agents 230–1, expectations 304, 330, 331, 332–4; of
242, 245, 257; of equilibria 207–9; spillovers 322, 326–30
functions 248; Gibbs 208; probability economic behaviour 229
340; stationary 247, 249, 257; of economic crises, in Mexico 226
wealth 245, 246–9, 257 economic development 105, 106–8, 175,
divergence, expectations of 364 192
Dixit, A. 313, 315 economic instabilities xxiii, 55–7, 88, 89
DNA revolution 357 economic restructuring 56, 57, 58
Dom, B. 346 economic science 96
Donaldson, J.B. 20 economic selection 176, 178–9
Dornbusch, R. 142, 143–4, 145 economic structures xix
Dosi, G. 302 economic theory 328, 351, 355, 364 n1;
DQ see Durlauf, S.N. classical 169, 270; dynamic 284;
dual economies 29, 30 models in 283–5
Duffie, D. 313 economics: complexity theoretic
duration: analysis 330; of business cycles approaches 301–2, 320, 322, 330, 334;
1, 4, 305 evolutionary 171; see also
Durlauf, S.N. 63, 301–2, 303, 304, 305, macroeconomics; microeconomics
308–9, 320, 321, 322, 323, 325–6, economies: conceptions of xix–xx, 47–8,
327, 328, 49, 54, 64, 99; death of 319; dual 29,
329–30, 331, 332, 334, 335 30; export-led 32–3, 42–3; global 219,
dynamic economic theory 284 364; industrialized free-market 266–7,
dynamic heterogeneity 47, 230, 249–53, 269, 280; number of 100–1, 103, 104,
256–7 108; steady states of 119; transition
dynamic macroeconomics 95, 216 308
dynamic paths, convergence of 65 ecosystems, models of 305
dynamical systems 61, 82 n40; theories of education 358, 360–1
51, 302 efficiency: degree of 245, 255, 257; local
dynamics 78–9, 230, 254; aggregate 229– 101; see also inefficiency
30, 235–6, 253; chaotic 52, 79, 259 Efficient Markets Hypothesis (EMH) 325
n23; complex xx, xxiv, 230, 257, 302; Eichenbaum, M. 292, 294
of coupled markets 220–2; Darwinian Eichengreen, B. 6
176; disequilibria 302; distributional EMH see Efficient Markets Hypothesis
79–80; equilibrium 115, 170–1, 332; (EMH)
evolutionary 176, 177, 178, 207, 333; empirical inferences 283–4
industrial 171, 172–4; IS-LM-PC 142, empirical testing 333, 339
Index 371
employment 29, 235; full 61, 63, 82 n42, evolutionary competition, of belief
149, 234, 235; rates of 133–4, 139, systems 333
143, 144, 152, 156, 293, 295 evolutionary computation theory 331
endogenous effects, identification of 327 evolutionary dynamics 176, 177, 178,
endogenous growth: models 269–71, 207, 333
352–5, 358, 359, 362; theories xvii, evolutionary economics 171
xviii, xix, 61, 62, 130, 269, 302, 304, evolutionary models xviii, 102, 113 n6,
351–2, 355, 358, 360, 362, 363, 364 117, 171, 207–8, 331–2; finitary 206;
endogenous instabilities 96 general (GEM) 103–12; long-run
endogenous population theories 360 172–91, 192–5, 196 n7, n8, n11, 197
endogenous variables 104, 117, 351, 355, n12, 198 n23–n28
364 n1 Ewens distribution 209
Engels, F. 266, 267 exchange rate systems 9, 21, 23 n1;
entrepreneurs 236–8, 239–41, 248, 255; Bretton Woods 17–18; fixed 15;
Schumpeterian 356, 357, 363 flexible 16, 43
environmental capacity 100, 105 exchange rates, in Japan 43, 44
environmental damage functions 100 exogenous, variables 117, 351, 354, 355,
environmental space 100, 101, 110 356, 357–8, 359–60, 364
episodes, definition of 104 exogenous growth: models 351; theories
equations: Chapman–Kolmogorov 203, xvii, 78, 351, 355
204; differential 207, 220, 313–14; expectations: boundedly rational 303,
Fokker–Planck 210, 212; master 203, 304; of divergence 364; rational 304,
204, 205, 206, 207, 209–11, 212, 216, 330, 331, 332–4, 355, 356
217 n4–n6; simultaneous 327 export-led economies 32–3, 42–3
equilibria 61, 116, 220, 226, 351; in exports, Japan 41–3
capital markets 242, 244, 247, 248, external diseconomies 100
253; conditions 352, 355, 356; external finances 232, 258 n12
distribution 207–9; long-run xxii, 169, external spillovers 321
171, 185, 191–2; multiple xxii, 116, externalities 279–80, 308, 354, 360, 361
203, 209–16, 253; rational
expectations 355; selection of 213–16; factorization 340, 341
social adoption 329; stability of 212, Fagnart, J. 292
214, 215–16, 219, 221; see also Fair, R. 139, 147, 151, 154, 155
stability; stable states; steady states family functions 97–8
equilibrium analysis 288, 297, 319 farmers 242, 243, 256
equilibrium dynamics 115, 170–1, 332 feedback structures, in growth models
equilibrium growth: models 115, 304, 129–30, 131, 141
319, 356; theories xvii, 78, 115, 170, Feller, W. 207
230, 284–5, 302, 334 Feynman–Kac formula 313
equilibrium paths 351 finances 232, 239, 258 n12
equipment, investments in 40–1 financial conditions 233, 255
equity: aggregate 234, 235; base 233, financial crises 15, 253
234, 235–6, 242, 250, 252, 257; ratio financial fragility 231
249, 251, 252, 253, 257, 259 n23; financial markets xxiii, 219, 225–6
rationing 231, 232, 249 financial robustness, of firms 235, 249,
errors: functions 340; Independently and 251
Identically Distributed (IID) 323–4 financial sector, in Japan 43–4, 45 n2
estimation techniques 330 financially constrained agents 242, 243
Evans, L. 307, 310, 312–13, 317 financing constraints 231, 242, 243, 244,
evolution 104, 355–6; of cultures 101–2; 256
Darwinian 176, 178–9; in distribution finitary search models 206, 211, 212–13
of wealth 245; Lamarkian 178, 179; firms: adoption of Real Cost Reduction
paths 118, 120 (RCR) 320, 329–30; behaviour of 120,
372 Index
171, 173, 175, 231, 232, 233, 235, 232, 237, 243–4, 245, 285, 287–91,
267, 292, 356; competing on 293, 352, 354; supply 132;
innovation activities 271–3, 275–7, technological production 99; transition
303; debt constraints of 232, 296; 72, 73–5; utility 246, 355, 360
definition of 296; financial constraints Furedi, Z. 223
of 231, 232–3, 258 n12; financial fusion 99, 104
robustness of 235, 249, 251; financing
costs of 249; growth rates of 176; Gaffard, J.L. xxi–xxii, 117
imitative activities of 177, 197 n17; Galileo’s law of motion 284
innovative activities of 179, 181, 198 Gallegati, M. xxii, xxiii
n23, 362–3; management of 359; Galor, O. 231, 245–6, 254, 255, 257
returns to scale 291–2; seeking to game theory 35–6, 254, 284, 356, 359
maximize profits 273; see also gatherers 242, 243–4, 256
entrepreneurs GDP: cycles 7, 8, 10, 18–19; growth rates
fiscal policies 151–2 27–8, 188; sectoral shares in 11, 13
Fischer, S. 142, 143–4, 145 GEM see Generalised Evolutionary
Fisher, I. 141 Model (GEM)
fission 99, 100, 104, 108 general equilibrium theory 284, 302, 319,
Fitzgerald, T. 293, 295 334
Flannery, K.V. 112 Generalised Evolutionary Model (GEM)
Flaschel, P. xxii, 16, 129, 131, 136, 137, xxi, 103–12
141, 144, 145, 150, 151, 157–8, 159, geometric complexity 344
161, 162, 163 Germany: economic instability 55, 89;
flow-of-funds constraints 243, 244 economic restructuring 57; GDP
flows: information 355; population 30, cycles 8; impact of oil shocks 57;
32, 33; probability 208 weather-sensitive sectors 11
fluctuations 6, 230–1, 285–6; of business Gertler, M. 231, 236–42, 254, 255–6,
cycles 22, 230–1, 285, 286, 287, 294; 257, 258 n17
caused by shocks 245, 253, 256, 294; Gibbons, R. 310
and growth xvi, xviii, xx, xxi, xxii, Gibbs distribution 208
xxiii, 253 Glaeser, E. 327
Fokker–Planck equations 210, 212 global economies 219, 364
de Fontnouvelle, P. 333–4 gold exchange standard 15
framework space (FS) 49–50 gold standard period (pre-1913 era) 4, 6,
France: economic instability 55, 56, 57, 10, 12, 14, 21, 22; adjustment
88; economic restructuring 57; GDP mechanisms of 10, 15; policy
cycles 7; weather-sensitive sectors 11 framework of 6, 9
Franke, R. 129, 141, 157, 164, 197 n12 Goldfeld, S.M. 71
Freeman, S. 294 Gollin, D. 285
Friedman, M. 294 Gong, G. 159, 163
Frisch, R. xx Goodfriend, M. 294
Fuerst, T.S. 294, 296 Goodwin growth cycle model 132–4,
full collateralization 239 135–6, 141, 164 n3, n4
functions: adjustment costs 250; Goodwin, R.M. xv, xvii, xx, xxiii, 49, 64,
aggregate 230, 233, 235–6, 285, 131, 132, 134, 163
287–91; cultural production 100, Gordon, R.A. 19, 34
101–2, 103; demand 356; density 341, Gould, S.J. 355–6
342, 344, 345, 346, 347–8; distribution government sector 18
248; environmental damage 100; error governments: financing Research and
340; family 97–8; hazard Development (R and D) 279;
329–30, 335; innovation 270–1; stimulating growth 318–19, 335, 361
investment 157, 158, 196 n10; Granger causality testing 331, 332
Liapunov 134; production 48, 60, 65, Greenwald, B. 231, 232–3, 235–6, 242,
115, 132, 172, 188–91, 199 n31–n34, 249, 251, 254, 255–6, 257
Index 373
Greenwood, J. 292 growth rates 60–2, 76, 77, 78, 175, 351;
Grimmett, G.R. 216 differences in 318, 334; of firms 176;
Groh, G. 144, 145, 150, 162, 164 of industrialized free-market
Grossman, G.M. 268 economies 266–7, 269, 280; long-run
Grossman-Stiglitz information paradox 62, 63, 69, 71; of money supply 120
333 growth regimes 66, 72
Groth, C. 145 growth regressions 309, 317–19, 326–8,
growth: and competition 353, 356; and 335
convergence xvi, 309, 326; demand- growth theories xv, 48–9, 62–3, 65, 80 n5,
led 30, 32, 33–4, 44; and development n6, 81 n14, n15, 82 n42, n45, 83 n51,
79; and education 361; export-led 95, 111, 351; and business cycle
41–3; and fluctuations xvi, xviii, xx, fluctuations xviii, 286, 296;
xxi, xxii, xxiii, 253; government disequilibrium 128–30, 135, 150,
stimulated 318–19, 335, 361; and 163, 192; endogenous xvii, xviii, xix,
household formation 32, 33; and 61, 62, 130, 269, 302, 304, 351–2,
innovations xvii, 179, 267–9, 277–8, 355, 358, 360, 362, 363, 364;
279, 306; and investments 40–1, 44, equilibrium xvii, 78, 115, 170, 230,
48, 49, 62, 120; long-run xxii, 62, 69, 284–5, 302, 334; exogenous xvii, 78,
95, 112 n4, 306; and monetary 351, 355; monetary 163; multi-sectoral
policies 120; and oil shocks 36–7, 42, model 60; neo-classical xvii, 48, 60–2,
57, 59; and oscillations xvii, xviii, 54; 82 n37, n38, 169, 171, 172, 188, 295;
and population flows 30, 32, 33; and neo-Schumpeterian 49; RBC xviii; and
profits 175; and structural change 66, spillovers 326; surveys of 303, 309
79; and technological change 41, 44, GS see Greenwald, B.
48, 120, 170, 317, 351, 352; in total Gunderson, L. 305, 306, 323
factor productivity (TFP) 303, 307, GZ see Galor, O.
308, 309, 319, 321; very-long-run
95–6, 98, 112 n1; and weather shocks Haag, G. 207
10–14 Haavelmo, T. 77
growth accounting method 37, 172, 188 Hahn, F.H. xix, xxiv, 304, 317, 354
growth analysis 64, 120–5, 130, 269, Hansen, G.D. 287, 291
317–19; regression 308–9, 317–19, Hansen’s test 70
323, 325, 326–8, 335 Hanson, M. 347
growth cycles 78, 83 n50; models of Harberger, A. 306, 307, 319, 320–1,
132–8, 141, 162, 330 322
growth empirics xv, 59, 60, 61, 69, 80, 81 Harrodian growth paths 53, 59, 68
n18, 326 Hart, O. 243, 288
growth facts 285, 287 hazard functions 329–30, 335
growth models 65, 101–2, 117, 330; Head, A. 291
disequilibrium 128–30, 131; Heckman, J. 303, 330
endogenous 269–71, 352–5, 358, 359, hedge funds 220
362; equilibrium 115, 304, 319, 356; Helpman, E. 268
evolutionary xviii, 103–12, 113 n6; Hendry’s forecast test 70
exogenous 351; Goodwin 132–4, 135– Hercowitz, Z. 292
6, 141, 164 n3, n4; neo-Austrian xviii, herd effects 361, 362
125–6; neo-classical 188, 270, 291; heterogeneity: and business cycle analysis
neo-Schumpeterian xviii, 49–54, 81 253; dynamic 47, 230, 249–53, 256–7;
n16, n19–26, 82 n30, 352–5; non- of financial conditions 233; individual
convex 305; Solow’s xvii, xviii, 61, 229–30, 231, 233, 254; models of
270, 285, 291, 303, 351; spatial 306; 249–53, 255, 259 n2; nature of 254–5;
spillovers in 321; very-long-run 96, notion of 253; technological 256;
99, 103–12 theory of persistence of 247
growth paths 53–4, 55, 56–8, 59–61, 63, Hicks, J.R. 5, 6
68–9, 78 Hicks neutral technical regress 36
374 Index
Hicks-Goodwin theory of self-sustained innovations 181, 239, 352–4, 363, 364 n3;
oscillations xvii accelerator property of 277–8;
hierarchies, of time scales 304–5, 322 competition on 271–3, 275–7, 303,
Hillinger, C. 19 353, 359; in education 360–1;
historical perspective, need for 3 externalities of 279–80; functions
Hodrick, R. 285–6 270–1; and growth xvii, 179, 267–9,
Hodrick-Prescott (H-P) filter 10, 17 277–8, 279, 306; independent 278;
Hogg, T. xxii introduction cost of 309; and
Holling, C. 305 inventions 180, 197 n20, 271, 357–8;
Hommes, C. 332, 333, 334 in macroeconomic models 269–71;
Hopf bifurcations 162 and profits 170–1, 187–8, 273–4; rate
Hornstein, A. 290, 291 of 183–4, 198 n22; and research and
Horvath, M. 319 development (R and D) 354–5, 357,
household formation, and growth 32, 33 361, 363; spillover ratio of 279, 280 n5
Howitt, P. 36, 316, 317, 319, 352, 354, instabilities 62, 95; of coupled markets
356, 357, 358, 359, 361 220, 226; economic xxiii, 55–7, 88,
Huberman, B. xxii 89; endogenous 96
Huffman, G.W. 292 integration: causing instability xxiii; of
human capital 293; inalienable 242, 243, international economy 10
244; investments in 36, 245, 246, 365 inter-war period 6, 14, 15–17, 21
n5 interest: rates 232, 233, 234, 236, 246,
human generations, as time units 97 247, 248, 251; social 308, 310, 311
internal diseconomies 99–100
IDEE (Industrial Dynamics and European internal finances 239
Employment) project 80 international business cycles 14, 17,
IID see Independently and Identically 19–21
Distributed (IID) errors international integration, of economy 10
incentives: contracts 312–13; structures of international migration 6
31, 307, 308, 310, 316, 334–5 international trade 9
incomplete collateralization 239 intra-bloc cyclical linkages 20–1
Independently and Identically Distributed intra-regional trade 18
(IID) errors 323–4 inventions: and innovations 180, 197 n20,
individual heterogeneity 229–30, 231, 271, 357–8; revolutionary 357
233, 254 investments 143; by agents 238; debt
inductive inferences 283–4 financed 137–8; decisions 236–40,
industrial dynamics, models of 171, 241, 247–8; in equipment 40–1;
172–4 function 157, 159, 196 n10; and
industrial production, in United Kingdom growth 40–1, 44, 48, 49, 62, 120; in
5 human capital 36, 245, 246, 365 n5;
industrial structure 309–10, 319 overseas 16–17, 23 n3; projects 236,
industrialized free-market economies, 237, 248, 255; return on 237, 239;
growth rates of 266–7, 269, 280 stability of 38–9; volatility of 22, 285
industries, science based 358 Ioannides, Y.H. 321
inefficiency 307; degree of 237, 238, 239, Ireland, P. 294
240, 255, 257; see also efficiency irregular fluctuations 6
inflation, rates of 139, 148, 152, 159–60 IS-LM model 128, 129, 131, 142–3, 148,
information: asymmetric 230, 231, 232, 149–50, 162–3, 165 n13
239, 258 n12; definition of 339, 349; IS-LM-PC dynamics 142,144–50, 164;
flows 355; theories xxiii; transmission models of 150, 155–7, 163
of 358 Ising models 321
infrastructural functions 98, 112 n4 Italy: economic instability 55–7, 87;
infrastructure 99, 318 economic restructuring 56, 57;
innovating technology level 102 regional differences in 318
innovation regime 54 Iwai, K. xxii, 171, 319
Index 375
James, H. 15 KMG (Keynes Metzler Goodwin) model
Japan: agricultural sector in 29; 128, 131, 138, 157–62, 164
automobile industry 41; construction Kollmann, R. 296
sector 33; diffusion of consumer Kolmogorov, A.N. xxiii, 348
durables 32; domestic demand in 30, Kolmogorov criterion 208
32, 33–5, 38–9; dual economy 29, Komlos, K. 223
30; economic growth in xx–xxi, 27– Kondratieff wave 5–6
8, 29, 36–7, 41–4, 58, 91; economic Koopmans, T. 285
restructuring 58–9; economic Köper, C. 164
stability 38, 39–40; exchange rates Kravis, I. 285
43, 44; exports 41–3; financial sector Kremer, M. 95–6, 107
43–4, 45 n2; household formation in Kreps, D. 306, 308, 310, 312, 315, 316,
30–2; impact of demand shocks 38; 317, 318, 319
impact of oil shocks 36, 37, 42, 59; Krusell, P. 297
imports 34, 44; investments 34–5, Kuh, E. 152, 154
38–40, 41, 44; machinery sector 41; Kuroda, M. 36
manufacturing sector in 29–30, 44; Kurz, M. 332
non-manufacturing sector 39; Kuznets, S. 18, 285
petrochemical industry 35; Kuznets swings 5, 14, 16, 17, 22–3
population flows 30–2; structural Kydland, F. 286, 287, 290–1, 294, 296
changes in
xx–xxi, 28, 35, 38; technical change Labadie, P. 319, 331
in 36–7; wage structure in 30, 32, 38 labour: amount in research and
Jermann, U. 296 development 352–3; force 98;
Jevons, W.S. 11 hoarding 292–3, 324; indivisibility
Jog, V. 331 287; market 234, 235, 324;
Johnson, P.A. 63 productivity 290, 360
Jones, C.I. 95–6, 107 Lakonishok, J. 324
Jones, L. 288 Lamarkian evolution 178, 179
Jorgenson, D. 36–7 land, demand for 243, 244
Jovanovich, B. 306, 308, 319 Lane, D. 301–2, 320, 322, 326, 327, 328,
Judd, K. 320 331, 332, 334, 335
Juglar trade cycle 5 Lapham, B. 291
Laplace transform 314, 316
Kaldor, N. 48, 62, 117 latent options 307, 308, 317
Kalman filter 6, 7–9 laws: Galileo’s of motion 284; Kepler’s of
Karlin, S. 203, 204 planetary motion 283, 284; of motion
Keen, S. 136 246, 248, 252, 253, 256, 257, 259 n23;
Kehoe, P.J. 10, 17, 296 Okun’s 156; Wigner’s 223
Kelly, F. 203, 204, 208, 209 Laxton, D. 153, 154–5
Kendall, D.G. 204 learning by doing 102
Kepler’s laws of planetary motion 283, LeBaron, B. 323, 324, 331–2, 333
284 Lee, R.D. 95–6, 107
Keynes, J.M. 128, 145 leisure 286, 293
Keynes Metzler Goodwin (KMG) model lenders 240, 242, 255, 297
see KMG model Lewis, W.A. 5, 30, 32, 44
Keynesian analysis 148–9, 162 Liapunov functions 134
Keynesian disequilibrium growth theory Licandro, O. 292
128–30 Light, S. 305
Kindleberger, C.P. 16, 33 Lincoln, E. 29
Kingman, J.F.K. 209 linear quadratic regressions 345–9
Kirman, A. 207 linkages, cyclical 19–21
Kiyotaki, N. 231, 242–5, 254, 255, 256 living standards, variations in 304
KM see Kiyotaki, N. local efficiency 101
376 Index
lock-in, notion of 303 MC curves 276
long-run: equilibria xxii, 169, 171, 185, MDL (minimum description length):
191–2; growth xxii, 62, 69, 95, 112 criterion 344; estimator 347; principle
n1, 306; profits 171, 185–6, 192; 344–5, 349
supply curves 171, 184, 187, 192; mean reversion, in asset prices 331
theories 62, 97–8, 355; see also very- methodological issues xxiii, xxiv, 326
long-run Metzler, L.A. 158, 160
Lucas, R.E. xix, 21, 269, 270, 284, 285, Mexico, economic crisis in 226
294, 304, 326, 328, 360 microeconomics 96–7, 267, 275–6
lump detection methods 326 migration, international 6
Lux, T. 157, 164 Miller, H.D. 209, 216, 309
Min, Z. 113 n6
McCallum, B. 294 Minami, R. 30
machinery sector, in Japan 41 minmax problems 341–3, 349
macrodynamics xv–xvi, xvii, xx, 128, Mirman, L. 285
129–30, 148–50, 164, 216 Mirrlees, J. 48
macroeconometric models 148–50, 154 Mitchell, W.C. 285
macroeconomics 148; data collection 97; ML see maximum likelihood (ML)
dynamic 95, 216; models of 2, 97–8, models: of adjustment processes within
112 n3, 204, 209–16, 217 n3, 230, markets 220–5, 227 n1; of adoption
231, 269–71, 331 320, 321–3, 328; of binary choice
Maddala, D.S. 72 205–6, 208–9, 211; building of xix,
Maddison, A. 37, 188 65, 97, 117, 129, 130, 131, 154, 319,
Malinvaud, E. 151, 157, 159 322, 331, 339–40; of business cycles
Malliaris, A. 313, 314 203, 287–97; of capital markets
Malthus, T.R. 360 232–53, 254, 255–7, 258 n18, n19;
management, of firms 359 complexity-based 320; cost 344; data
Mankiw, N. 269, 303 generating 345; of debt accumulation
Manski, C.F. 304, 326, 327, 328, 335 137–8; definition of 340; diffusion
manufacturing sector: in Japan 29–30, 44; 320, 326; of disequilibrium growth
population flows from agricultural 128–30, 131, 135, 137–41, 157–62,
sector 30 163, 164; econometric xvii, 66–9, 70,
marginal productivity 132, 244 71–7, 83 n49; in economic theory
markets: capital 232–53, 254, 255–7, 258 283–5; of ecosystems 305; of
n18, n19; coupled 220, 221, 223–5, endogenous growth 269–71, 352–5,
226; financial xxiii, 219, 225–6; 358, 359, 362; of equilibrium growth
labour 234, 235, 324; stability of 219, 115, 304, 319, 356; errors 323–4;
222 evolutionary 102–5, 117, 171, 172–91,
Markov chains 203, 208, 209, 214 192–5, 196 n7, n8, n11, 197 n12, 198
Markov processes 203–4, 216 n23–n28, 206–8, 331–2; of exogenous
Marshall, A. 229 growth 351; fitted to data 323;
Martel, R. 229 Goodwin growth cycle 132–4, 135–6,
Marx, K. 266, 267 141, 164 n3, n4; of growth 65, 101–2,
Mas-Colell, A. 288 117, 330; of growth cycles 132–8,
Massachusetts Institute of Technology 141, 162, 330; of growth regressions
(MIT) 275 309, 317–19; of heterogeneity 249–53,
master equations 203, 204, 205, 206, 212, 255, 259 n23; of industrial dynamics
216; in economic models 207; solving 171, 172–4; IS-LM 128, 129, 131,
of 209–11, 217 n4–n6 142–3, 148, 149–50, 162–3, 165 n13;
mathematical models 96 of IS-LM-PC dynamics 150, 155–7,
mathematical theories 51 163; Ising 321; KMG (Keynes Metzler
matrices, stability properties of 222, 225 Goodwin) 128, 131, 138, 157–62, 164;
maximum likelihood (ML) 341; estimates macroeconometric 148–50, 154;
344; parameters 348; solution 346 macroeconomic 2, 97–8, 112 n3, 204,
Index 377
209–16, 217 n3, 231, 269–71, 331; and
master equations 207; mathematical NAGRW (Non-Accelerating-Growth Rate
96; microeconomic 275–6; of of Wages) 133–4
monetary growth 157–8, 163, 294, NAIRU (Non-Accelerating-Inflation Rate
295; multiple cycle 6, 7–9; of multiple of Utilization) 131, 133, 139, 142,
equilibria 209–16; neo-Schumpeterian 147–8, 153, 164 n3; rates 155–6
xviii, 49–54, 81 n16, n19–26, 82 n30, Nakamura, T. 27
352–5; of optimal timing/stopping natural sciences, inductive inferences in
310–16, 322; out-of-equilibrium 119, 283–4
125–6; parameter constancy of 70–1, Nelson, R.W. 358, 361, 362
72, 73; parameteric 340–1; of price neo-Austrian approach xviii, xxi–xxii,
adjustments 220; random field 321–2; 118, 125–6
rational expectations 332–4; recursive neo-classical: growth models 188, 270,
programming 97; of Resilience 291; theories of growth xvii, 48, 60–2,
Networks (Rnet) 323; Ricardian 270; 82 n37, n38, 169, 171, 172, 188, 295
search 206, 211, 212–13; selection of neo-Schumpeterian models xviii, 49–54,
285, 287, 290–1, 340–9; simulation of 81 n16, n19–26, 82 n30, 352–5
73–7, 106–11, 120–5, 324; smooth Nerlove, M. 332
transition regression (STR) 72–7; New Zealand, corporatization reforms
Solow growth xvii, xviii, 61, 270, 285, 307, 312–13, 317
291, 303, 351; testing procedures of Newtonian mechanics 284
323–5; theory of 339; time series NML see normalized maximum
switching (Markov) 71–2; of very- likelihood (NML)
long-run growth 96, 99, 103–12 no-arbitrage conditions, assumption of
Modigliani-Miller theorem 231, 256 256
monetarists xvii, 144 non-entrepreneurs 238, 241, 248
monetary growth models 157–8, 163, non-farmers 242, 243
294, 295 non-manufacturing sector, in Japan 39
monetary policies 17, 21, 120, 143; nonconvex models of growth 305
evaluation of 295; and growth 120 nonlinearity tests 324
monetary shocks 16, 38, 236; and normalization processes 347
business cycle fluctuations 294 normalized maximum likelihood (NML)
monetary theories 163, 294–5 342, 344, 348; density functions 345,
monitoring costs 236, 237, 239, 241, 246 346, 347–8; universal model 345, 349
monopolistic competition 291–2, 357
monopolists 310, 311–12, 353 oil shocks 28, 35–6; impact on growth
monotone concave production functions 36–7, 42, 57, 59
352 Okazaki, T. 28
Moore, J. 231, 242–5, 254, 255, 256 Okun’s law 156
moral hazard problem 246 Olsen, M. 307
motion, laws of 234, 235, 246, 252, 253, optimal timing/stopping: models 310–16,
259 n23, 283, 284 322; theory 303, 307, 309–10, 319–20,
MR curves 276 329–30, 334, 335
multi-modality 326 options, latent 307, 308, 317
multi-phase dynamics xxi, 64, 65 organization forms 358, 362, 365 n4
multi-sectoral analysis 60, 63, 86–91 organizational routines 171, 196 n5
multiple cycles 7–9, 10, 11, 236 Orphanides, A. 158, 163
multiple equilibria xxii, 116, 203, oscillations: dynamics of xvii–xviii; and
209–16, 253 growth xvii, xviii, 54; see also
multiple steady states 62, 354 fluctuations
multiplier, social 326 Oswald, A.J. 154, 155
Mundell effects 146, 148, 160 out-of-equilibrium: approach xxi–xxii;
Murphy, C. 154 dynamics xxi, 62, 115–19; models
Myung, I. 344 119, 125–6
378 Index
out-of-sample testing 325 prices, of assets 245, 330–2, 333
output 20, 21 privatization reforms 307, 312–13, 316
overseas investments 16–17, 23 n3 probability: of auditing 239, 240; of
bankruptcy 231, 233, 250, 251;
Packard, N. 302 distributions 340; flows 208; of
parallel view of the economic system xix, success in research and development
xx (R and D) 362
parameteric models 340–1 production decisions 231–2, 235, 242
parameters 117; configuration of 253, 259 production functions: aggregate 285,
n23; constancy of 70–1, 72, 73; 287–91; Cobb-Douglas 288–9, 354;
maximum likelihood (ML) 348; cultural 100, 101–2, 103; in economic
number of 334; selection of 290; models 48, 60, 65, 115, 132, 172,
technological 239 188–91, 199 n31–n34, 232, 237, 243–
paths: dependence 302–3; dynamic 65; 4, 245, 293; monotone concave 352;
equilibrium 351; evolution 118, 120; technological 99
of growth 53–4, 55, 56–8, 59–61, 63, production processes 118, 119
68–9, 78 productivity 61, 82 n39, 100, 102, 113 n4,
pattern recognition 323–6 362; labour 290, 360; marginal 132,
Pavlov, O.V. xxi, 305 244; paradox 120; shocks 291
PC see Phillips curves profits 232–3, 250; and growth 175; and
perfect competition 320, 355, 356 innovations 170–1, 187–8, 273–4;
performance 332, 339 long-run 171, 185–6, 192; normal
Perri, F. 296 169–71; surplus 171
petrochemical industry, in Japan 35 progress 101, 310
phase zones 103–4 Punzo, L.F. xxi, 81 n7, 305
Phillips, A.W. 151–2, 153 Putnam, R. 318, 319
Phillips curves (PC) 131, 132–3, 139,
143, 145, 147–8, 164 n4; analysis 150, quadratic adjustment costs 250
163; approach 150–7, 158 Quadrini, V. 296
Pindyck, R. 313, 315 Quah, D. 35, 79–80, 303, 308–9, 323,
Pines, D. 301, 321, 323 325–6
Poisson processes 181 qualitative change 115–16, 117, 118–19
policies: feedback rules 130; fiscal 151–2; qualitative dynamics 96
innovation 181; monetary 17, 21, 120, qualitative econometrics xxi, 47, 97
143, 295; role of 295 qualitative explanations of behaviour 95
policy frameworks: and business cycles 3, Quandt, R.E. 71
4, 15, 18; of gold standard period 6, 9; quantitative, scientific inference 283
in inter-war period 15; of post-war quasi-rational expectations 332
period 17–18 Quigley, C. 112
Pollett, P.K. 209
population: endogenous theories of 360; R and D see research and development (R
flows and growth 30, 32, 33; growth and D)
30, 96, 99, 107–8, 109, 110–11, 112; RAH see Representative Agent
internal diseconomies of 100 Hypothesis (RAH)
Portier, F. 292 Ramsey, J.B. 69
post-war period 17–21 random field models 321–2
poverty 268 ratchets 276, 277
Powell, A. 154 rational expectations: econometrics 304,
Prescott, E.C. xxi, xxiii, 285–6, 287, 330, 331, 332–4; equilibria 355; of
290–1, 295, 304, 306–8, 309, 320 price movements 356
price movements 21, 120, 134, 220; rationality 332, 352; bounded 304, 332,
theories of 138, 233, 356; and wages 333
128, 129, 131, 132, 139, 141, 143–5, RBC theory (Real Business Cycle) xviii,
147, 151, 152, 158, 163 xxiii, 295
Index 379
RCR see Real Cost Reduction (RCR) Schaller, H. 331
Real Cost Reduction (RCR) 307, 308; Scheinkman, J. 294, 309, 320, 323, 324
adoption of 311, 316–22, 329–30, 334 Schreft, S. 294
Rebelo, S. 292 Schumpeter, J. 170–1, 175, 178, 179, 180,
recursive estimation 70 192, 284, 353, 357, 363
recursive programming models 97 Schumpeterian entrepreneurs 356, 357,
reforms, privatization 307, 312–13, 316 363
regimes: dynamics of xxi, 53–4, 79; of Schumpeterian vision of market economy
growth 66, 72; innovation 54; notion xvi–xvii, 306, see also neo-
of 63–4, 78, 79; switches of 47, 54–5, Schumpeterian models
64, 66, 70, 71–7, 101, 105; Schwartz, A.J. 294
technological 96, 101, 105 science: based industries 358; economic
regression growth analysis 308–9, 317– 96; fundamental 361; natural 283–4
19, 323, 325, 326–8, 335 scientific advances 358–9
regressions, linear quadratic 345–9 scientific inferences 283–4, 290
regressor variables 346 search models 206, 211, 212–13
representative agent framework, sectoral shares in GDP 11, 13
aggregation within 229–30, 257 n5 sectors: coding of 86; leading 321; notion
Representative Agent Hypothesis (RAH) of 63; weather-sensitive 11
230–1, 255, 259 n23 selection: economic 176, 178–9; of
research and development (R and D): equilibria 213–16; of models 285, 287,
amount of labour in 352–3; budgets 290–1, 340–9; of parameters 290
for 273, 276; government financed Semmler, W. 129, 136, 137, 141, 150,
279; and innovations 354–5, 357, 361, 159, 162, 164
363; probability of success 362 serial view of the economic system xix
RESET test (Ramsey) 69 Shannon, C.E. xxiii
Resilience Network (Rnet) 305, 323 shares, cumulative capital 173, 175, 176
restructuring, economic 56, 57, 58 Shirai, Y. 206, 212, 213
returns: to scale 291–2, 294; to Shishido, S. 33
specialization 291–2 shocks 115, 230; business cycle 15, 16;
revolutionary inventions 357 causing fluctuations 245, 253, 256,
Ricardian model 270 294; demand 38; monetary 16, 38,
Ricardo, D. 269 236, 294; oil 28, 35–7, 42, 57, 59;
risk attitudes 355, 356, 358, 359, 360, 364 productivity 291; system’s capacity to
Rissanen, J. xxiii, 342, 343, 349 absorb 305; technology 287, 291–2;
Rnet see Resilience Network (Rnet) total factor productivity (TFP) 286–7,
Roberts, J.M. 155 290, 294, 295; weather 10–14
Romer, C. 18–19 Siena School xv, 303, 317, 330, 334
Romer, P. 269, 270, 279, 280 n6, 303, Sims, C.A. xxiii
304, 308, 326, 328 simulation, of models 73–7, 106–11,
Rose, H. 131, 132, 138, 139, 141, 151, 120–5, 324
163 simultaneous equations problem 327
Rosser, J. 301 Singer, B. 303, 330
Rothschild, M. 309, 313, 314, 315 skilled workers 245, 246, 293
Rowthorn, B. 151, 155 slow moving dynamics 305
Smith, A. 297
Sachs, J. 36, 37 smooth transition regression (STR)
Sala-i-Martin, X. 303, 304, 320, 326, 327 models 72–7
Santa Fe Institute (SFI) 301 social adoption equilibrium 329
Sargent, T. 13, 129, 149, 161, 334 social interactions 304
Satchell, S. 325 social interest 308, 310, 311
Saving, T. 294 social multiplier 326
savings 120, 241, 242, 351 social slack 99
scenarios, definition of 104 social space 98–9
380 Index
social technology 98, 99 Japan xx–xxi, 28, 35, 38; in post-war
societies: demise of 105, 110–11; period 18; as regime switches 47,
development stages of 105–6, 108 54–5, 59, 64, 70
Solomou, S. xx, 6, 9, 11, 304, 306 structural cycles 49
Solow, R. xvii, 132, 139, 151, 158, 163, structural dynamics xviii–xix, 48, 49,
188, 269, 364 n3 54–5, 59, 60, 80 n2
Solow’s growth model xvii, xviii, 61, 270, structure indices 344
285, 291, 303, 351 structures, of incentives 307, 308
spatial data sets 323 Stuttgart School 301
spatial scales, and time scales 305–6, stylised facts approach, to business cycles
322–3 3–4, 21–2
Spence, M. 361 Sullivan, R. 325, 332
spillovers: econometrics of 322, 326–30; Summers, L. 40–1
effects 304, 311–12, 334; external supply: of capital 241, 247–8; curves 169,
321; in growth models 321; and 170, 172–5, 184, 187, 192; functions
growth theories 326; identification of 132
308; potential 322, 335; ratio of supply side dynamics 129, 131, 132
innovations 279, 280 n5 surplus profits 171
spurious effects 327 Suzuki, Y. 39
spurious social interactions 304 Swan, T. 269
stability xxiii, 61, 62, 69, 162, 187; of
computer networks 226; of equilibria tatonnement processes 220
212, 214, 215–16, 219, 221; of Taylor, H. 203, 204
investments in Japan 38–9; of markets Taylor, J.B. 38
219, 222; see also equilibria; stable technological change 36–7, 63, 95, 119; and
states; steady states growth 41, 44, 48, 120, 170, 317, 351,
stable states: alternative 303, 305; steady 352
256; see also equilibria; stability; technological diffusion 178, 179, 309
steady states technological disequilibria 171, 183, 191,
standards of living 97–8, 103, 308 192
state variables 207 technological heterogeneity 256
stationary distribution 247, 249, 257 technological imitations 176–7, 178, 197
statistical properties, of data 341 n14
steady states 69, 70, 119, 353; of capital technological parameters 239
markets 234, 235, 246; of disequilib- technological production functions 99
rium growth models 130, 137, 140–1, technological regimes 96, 101, 105
161–2; of IS-LM model 143, 144, technology: administrative 98; gap 102;
145–6; multiple 62, 354; stable 256; ownership of 177; potential of a system
see also equilibria; stability; stable 102, 105; sharing 274–5,
states 278–9; shocks 287, 291–2; social 98, 99;
Stiglitz, J. 139, 151, 231, 232–3, 235–6, state of 172–3, 175, 176–7, 178, 179–80,
242, 249, 251, 254, 255–6, 257, 309, 181–2, 183–4, 192; stock of 307, 308
313, 314, 315 temporal data sets 323
Stirzaker, D.R. 216 Teräsvirta, T. 72, 73
stochastic complexity xxiii, 344, 345; of testing: empirical 333, 339; Granger
data sequences 339–40, 343 causality 331, 332; of models 323–5;
Stock, J.H. 155 out-of-sample 325
Stokey, N. 294 tests: Chow 70; CUSUM 70–1; Hansen’s
Stommel diagrams 306, 322–3 70; Hendry’s forecast 70; nonlinearity
storage 236 324; RESET (Ramsay) 69
STR (smooth transition regression) TFP see total factor productivity (TFP)
models 72–7 theories: definition of 284; development of
structural changes xix, xxi, xxiii, 3, 12, 287; of models 339; see also growth
64, 95, 108, 110; and growth 66, 79; in theories
Index 381
time scales: hierarchies of 304–5, 322; volatility: of business cycles 16, 18–19; of
and spatial scales 305–6, 322–3 consumption 22, 285; of inter-war
time series 65; economic 285; switching period 16; of investment 22; measure
models 71–2 of 290; of output 20; of post-war
Timmerman, A. 325 period 18–19
total factor productivity (TFP): dynamics
302–3, 320–1, 322–3, 328; growth wages: level of 152, 279–80; and price
303, 307, 308, 309, 319, 321; shocks dynamics 128, 129, 131, 132, 139,
286–7, 290, 294, 295; theory 304 141, 143–5, 147, 151, 152, 158, 163;
trade: international 9; intra-regional 18 real 133–4, 286; rigidity of 16;
transition economies, problems of 308 structure in Japan 30, 32, 38
transition functions 72, 73–5 Wallis, D. 306, 308, 310, 312, 315, 316,
transition rates 203–6, 210–11, 217 n3 317, 318, 319
transmission mechanisms 245 Walrasian general equilibrium theory
true social interactions 304 284
Turnovsky, S. 149, 304 Walter, J.L. 96
Watanabe, T. 33
uncertainty 313; idiosyncratic 292; theory Watson, M.W. 155
of 360 wealth: of agents 245, 246, 247;
United Kingdom: GDP cycles 7; distribution of 245, 246–9, 257
industrial production 5; overseas weather shocks, and business cycles
investments 16; research and 10–14
development expenditure in 363; Weidlich, W. 207
weather-sensitive sectors 11 Weil, D. 269, 303
United States: collapse of financial Weiss, L. 294
system 15; domestic demand in 34; welfare analysis 319
economic growth in 57, 90; economic West, K. 38
leadership of 16; economic volatility White, A. 325
of 290; education system 358; GDP Wigner’s law 223
cycles 8, 18–19; Great Depression Wittle, P. 203
295; immigration 33; productivity Woodford, M. 320, 323
superiority 358; research and workers 245, 246, 293
development in 358 workweek, length of 287, 289–91, 293
unskilled workers 245, 246, 293 Wright, G. 358, 361, 362
utility functions 246, 355, 360 Wu, W. 11
value added per person 49–50 Ye, J. 309

van Kampen, N.G. 209, 211, 213, 216 yen, appreciation of 43, 44
variables: aggregate 291; endogenous Yoshida, K. 36
104, 117, 351, 355, 364 n1; exogenous Yoshikawa, H. xx–xxi, 28, 36, 38–9, 43,
117, 351, 354, 355, 356, 357–8, 359– 44
60, 364; regressor 346; state 207 Yu, B. 347
Veracierto, M. 296, 297
very-long-run growth 95–6, 98, 112 n1; Zarnovitz, V. 20
models of 96, 99, 103–12 Zeira, J. 231, 245–6, 254, 255, 257
viability problems 118 Zhu, P. 164
382 Index
Index 383
384 Index
Index 385
386 Index
Index 387
388 Index
Index 389
390 Index

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Running head i

Cycles, Growth and Structural

Lionello F. Punzo is Professor of Economics at the University of Siena.

The Siena Summer School hosts lectures by distinguished scholars on topics

International Problems of Economic Interdependence

Ethics, Rationality and Economic Behaviour

The Politics and Economics of Power

The Evolution of Economic Diversity

Cycles, Growth and Structural Change

The Routledge Siena Studies in Political Economy Series gives a comprehensive

Cycles, Growth and

London and New York

First published 2001

All rights reserved. No part of this book may be reprinted or reproduced

British Library Cataloguing in Publication Data

Library of Congress Cataloging in Publication Data

HB3711 .P93 2001

ISBN 0-203-25909-2 (Adobe eReader Format)

1 Economic cycles since 1870 3

2 Growth and fluctuations: the post-war Japanese 27

3 Productivity–investment fluctuations and structural

4 Qualitative dynamics and macroeconomic evolution

5 Out-of-equilibrium dynamics 115

6 Disequilibrium growth in monetary economies:

8 Asymmetrical cycles and equilibrium selection in

9 The instability of markets 219

10 Heterogeneity, aggregation and capital market

11 Toward the microeconomics of innovation: growth

12 Business cycle research: methods and problems 283

13 Complexity-based methods in cycles and growth:

14 Information, complexity and the MDL principle 339

15 The ‘exogenous’ in ‘endogenous’ growth theory 351

1.1a Kalman filter decomposition of UK GDP cycles,

1.1 International output correlation: pre-1914 11

Mario Amendola is Professor of Economic Analysis at the University of Rome

Masanao Aoki is Professor Emeritus at UCLA, and Professor in the Department

William J. Baumol is Professor of Economics at New York University and

Bernhard Böhm is Professor for Economics and Applied Econometrics at the

William A. Brock is Vilas Research Professor of Economics at the University of

Richard H. Day is Professor of Economics at the University of Southern

Domenico Delli Gatti is Economics Professor, Catholic University, Milan, Italy.

Peter Flaschel is Professor of Economics, Bielefeld University and Visiting

Jean-Luc Gaffard is Professor of Economics at the University of Nice-Sophia

Mauro Gallegati is Professor of Economic Dynamics, University of Teramo,

Frank Hahn is Professor Emeritus at the University of Cambridge, Professore

Bernardo A. Huberman is the Scientific Director of the Hewlett Packard Sand

Katsuhito Iwai is Professor of Economics at the University of Tokyo. His works

Oleg V. Pavlov is a postdoctoral fellow at the Boston University of School of

Edward C. Prescott is Regents’ Professor at the University of Minnesota and an

Lionello F. Punzo is Professor of Economics at Siena University. His current

Jorma Rissanen is a member of the research staff at IBM Almaden Research

Hiroshi Yoshikawa is Professor of Economics at the University of Tokyo. His

Michael Youssefmir Michael Youssefmir is Director of Product Management for

Convergence: the new impetus to growth theory and growth

Cycles, growth and structural change

Dynamics from structural change

Parts III and IV

Facts and interpretations of

1.2 1870 – 1913

1.2.2 Rules-driven policy framework and business cycles