Microeconomics of Market Failures PDF

The Microeconomics of
Market Failures
Market Failures
Bernard Salanié
The MIT Press

Cambridge, Massachusetts
London, England
English edition © 2000 Massachusetts Institute of Technology. Originally published
in French under the title Microéconomie: Les défaillances du marché. © 1998 by Economica,
Paris.
All rights reserved. No part of this book may be reproduced in any form by any electronic
or mechanical means (including photocopying, recording, or information storage and
retrieval) without permission in writing from the publisher.
This book was set in Palatino by Best-set Typesetter Ltd., Hong Kong.
Printed and bound in the United States of America.
Library of Congress Cataloging-in-Publication Data
Salanié, Bernard.
[Microéconomie. English]
The microeconomics of market failures / Bernard Salanié.
p. cm.
Includes bibliographical references and index.
ISBN 0-262-19443-0 (hc. : alk. paper)

1. Equilibrium (Economics). 2. Efficient market theory. 3. Welfare economics.
4. Industrial organization (Economic theory). I. Title.
HB145 .S255 2000
338.5—dc21 00-038670
Contents
1 Introduction 1
1.1 The Fundamental Theorems 1
1.1.1 The Pareto Optimum 2
1.1.2 General Equilibrium 2
1.1.3 The Two Fundamental Welfare Theorems 3
1.2 Return to the Hypotheses 6
1.3 The Government’s Role 8
Bibliography 10
I Collective Choice 11
2 The Aggregation of Preferences 13

2.1 Arrow’s Theorem 17
2.2 Noncomparable Cardinal Preferences 22
2.3 Comparable Ordinal Preferences 23
2.4 Comparable Cardinal Preferences 24
2.5 Conclusion 25
2.6 Appendix A: Proof of Arrow’s Theorem 26
2.7 Appendix B: Theories of Justice 27
2.7.1 Utilitarianism 28
2.7.2 Rawls’s Difference Principle 29
2.7.3 Recent Developments 31
2.7.4 Nozick’s Historical Approach 33
2.7.5 Conclusion 34
Bibliography 35
3 Cost-Benefit Analysis 37
3.1 Measures of Welfare 37
3.2 First-Best 42
vi Contents
3.3 Second-Best 44
3.3.1 Shadow Prices 44
3.3.2 Nonmarket Goods 45
3.3.3 Incomplete Markets 46
Bibliography 46
4 Implementation 49
4.1 Dominant Strategy Equilibrium 50
4.2 Nash Equilibrium 52
4.3 Refinements of the Nash Equilibrium 57
4.4 Bayesian Equilibrium 58
4.5 Appendix A: Proof of the Gibbard-Satterthwaite
Theorem 59
4.6 Appendix B: Proof of Maskin’s Theorem 63
Bibliography 65
II Public Economics 67
5 Public Goods 69
5.1 The Optimality Condition 70
5.2 Implementing the Optimum 72
5.2.1 The Subscription Equilibrium 73
5.2.2 Voting Equilibrium 73
5.2.3 The Lindahl Equilibrium 74
5.2.4 Personalized Taxation 75
5.2.5 A Planning Procedure 76
5.2.6 The Pivot Mechanism 78
5.3 The Property of Public Goods 83
5.4 The Importance of the Free-Rider Problem 85
5.5 Local Public Goods 86
5.6 Appendix: Characterization of VCG Mechanisms 86
Bibliography 88
6 External Effects 89
6.1 The Pareto Optimum 90
6.2 Implementing the Optimum 92
6.2.1 The Competitive Equilibrium 92
6.2.2 Quotas 94
6.2.3 Subsidies for Depollution 94
6.2.4 The Rights to Pollute 95
Contents vii
6.2.5 Taxation 97
6.2.6 The Integration of Firms 97
6.2.7 A Compensation Mechanism 98
6.3 Must Prices or Quantities Be Regulated? 100
6.4 Coase’s Theorem 102
Bibliography 104
7 Nonconvexities 107
7.1 Consequences of Nonconvexities 107
7.1.1 Nonconvex Preferences 107
7.1.2 Nonconvex Sets of Production 109
7.2 Convexification by Numbers 112
7.3 Regulation of Natural Monopolies 113
7.3.1 Marginal Cost Pricing 116
7.3.2 Second-Best Pricing of Regulated Firms 117
7.4 Deregulation 124
Bibliography 125
III Industrial Organization 127
8 General Equilibrium of Imperfect Competition 131

8.1 Three Difficulties 131
8.1.1 The Firms’ Objectives 132
8.1.2 Price Normalization 133
8.1.3 The Quasi-concavity of Profit 133
8.2 Subjective Demand Equilibrium 134
8.3 Objective Demand Equilibrium 135
8.3.1 Equilibrium in Quantities 135
8.3.2 Equilibrium in Prices 136
8.4 Conclusion 137
Bibliography 138
9 Prices and Quantities 141

9.1 Monopoly 141
9.1.1 Social Distortion 143
9.1.2 How to Avoid Distortions 144
9.1.3 The Case of Durable Goods 146
9.2 Price Discrimination 148
9.2.1 First Degree 148
9.2.2 Second Degree 148
9.2.3 Third Degree 150
viii Contents
9.3 Oligopoly 150

9.3.1 Cournot’s Oligopoly 151
9.3.2 The Bertrand Equilibrium 152
9.3.3 Sketches of Resolutions of the Paradox 153
9.3.4 Strategic Substitutes and Complements 157
Bibliography 158
10 Product Choice 161

10.1 Definitions 161
10.1.1 A Model of Vertical Differentiation 162
10.1.2 A Model of Horizontal Differentiation 162
10.2 Differentiation and Monopoly 163
10.2.1 Optimal Quality Choice 163
10.2.2 Nonobservable Quality 165
10.2.3 Choice of Number of Products to Introduce 166
10.3 Differentiation and Oligopoly 167
10.3.1 The Maximal Differentiation Principle 167
10.3.2 Entry and Number of Products 170
Bibliography 177
11 Long-Term Entry and Competition 179

11.1 Sustainability and Contestability 179
11.2 Preemption 183
11.3 Limit Price and Predation 186
11.3.1 Limit Price 186
11.3.2 Predation 187
11.4 Research and Development 188
Bibliography 191
12 Vertical Relations 193

12.1 Double Marginalization 194
12.2 Justifications of Vertical Constraints 197
12.2.1 Retailer Effort Incentive 197
12.2.2 Price Discrimination 198
12.2.3 Tied Sales 198
12.2.4 Reduction of Price Competition 199
12.3 Comparison of Different Practices 199
12.4 Elements of Law 201
Bibliography 202
Contents ix
IV Incomplete Markets 203
13 Elements of the Theory of Incomplete Markets 205

13.1 The General Framework 205
13.2 Existence of Equilibrium 210
13.3 Inefficiency of Equilibrium 211
13.4 Equilibria with Production 213
13.5 Application to International Trade 215
13.6 Conclusion 216
13.7 Appendix: Nominal Assets 216
Bibliography 218
Index 221
Preface
As the title indicates, this book is organized around the unifying theme
of market failures in microeconomic theory. As such, I hope that it may
be useful to both advanced undergraduate and graduate students. The
book originates in a course I taught at the Ecole Nationale de la Statis-
tique et de l’Administration Economique in Paris and which was pub-
lished by Economica in 1998 as Microéconomie: Les défaillances du marché.
This course owed much to my predecessors, and in particular (in
chronological order) to Jean-Jacques Laffont, Jean Tirole, Roger Gues-
nerie, Pierre-André Chiappori, Jean-Charles Rochet, and Patrick Rey. I
thank Isabelle Braun-Lemaire, Elyes Jouini, Pierre-François Koehl, Jean
Lainé, Guy Laroque, Laurent Linnemer, Tom Palfrey, Anne Perrot,
Jérôme Philippe, and Patrick Rey for their comments on several ver-
sions of the original French manuscript. I stand, of course, responsible
for all errors and imperfections.
Market Failures
1 Introduction
This book has four parts. It treats, respectively, collective choice,

welfare economics, industrial organization, and incomplete markets.
As the title indicates, the unifying theme, which justifies the grouping
of these four parts, is constituted by market failures. These are cir-
cumstances where market equilibrium is not optimal. The first funda-
mental welfare theorem tells us that such situations can only be
produced if at least one of the standard hypotheses of the theory of
general equilibrium is violated. Therefore throughout this book the
objectives will be to examine diverse deviations from these hypothe-
ses, learning of their consequences, especially where optimum condi-
tions of equilibrium are concerned, and studying what economic policy
measures can help remedy them.
1.1 The Fundamental Theorems
The object of this section is to briefly remind the reader of the two
fundamental theorems of welfare economics. To this end, consider the
primitives of an economy:
• L goods indexed by l = 1, . . . , L and an initial dotation of the economy
represented by a vector w Œ IRL
• I consumers indexed by i = 1, . . . , I; consumer i is endowed with a
convex consumption set Xi included in IRL and a preorder of prefer-
ences1 i defined on Xi (or, under very general hypotheses, with a
utility function Ui)
1. Throughout this book, transitive and complete binary relations will be called
“pre-orders.”
2 Introduction
• J producers indexed by j = 1, . . . , J; producer j is represented by a

production set Yj Ã IRL or, under very general hypotheses, a produc-
tion function Fj such that
y j Œ Yj ¤ Fj (y j ) £ 0
1.1.1 The Pareto Optimum
By definition, a Pareto optimum of such an economy is an allocation.

That is, the Pareto optimum is a set of consumption and production
plans ((xi)i=1,...,I, (yj)j=1,...,J)),
• that is feasible—such that every consumer chooses from within his
or her own consumption set, every producer respects his or her pro-
duction set, and the sum of the consumptions does not exceed the
resources of the postproduction economy
Ï"i = 1, ... , I , xi Œ X i
Ô
Ì"j = 1, ... , J , y j Œ Yi
Ô I J
ÓÂi =1 xi £ w + Â j =1 y j
• that is efficient—in the sense that there is no other feasible allocation
((x ¢)
i i =1, ... , I , (yi¢) j =1, ... , J )
such that for every consumer, Ui(xi¢) ≥ Ui(xi), and that at least one of
these inequalities be strict.2
1.1.2 General Equilibrium
Now assume that the initial resources w and firms’ property rights are
spread among the consumers, every consumer i receiving a dotation wi
and shares (qij)j=1,...,J of the J firms, with
ÏÂiI=1 w i = w
Ì
Ó"j = 1, ... , J , ÂiI=1q ij = 1
In such a private property economy, the hypothesis of pure and perfect
competition is represented by the following behavior when met with a
price p:
2. It should be noted that if the Ui are strictly increasing, efficiency will again prohibit
that for the consumer, Ui(xi¢) > Ui(xi).
Introduction 3
• The producer j maximizes his profit and chooses
y j (p) Œ arg max p ◊ y j

y j ŒYj
• The consumer maximizes his utility under budgetary constraints and
chooses
Ïarg max U i (xi )

Ô
x i ( p) Œ Ìx i Œ X i
Ô J
Óp ◊ xi £ p ◊w i + Â j =1q ij p ◊ yi (p)
These rival supplies and demands so defined, a competitive general equi-
librium of private property is composed of a price system p* and of an
allocation ((x*i )i=1,...,I, (y*j )j=1,...,J) such that
Ï"i , x*i = xi (p* )
Ô
Ì"j , y j* = y j (p* )
Ô I J
ÓÂ i =1 xi* £ Âi =1 w i + Â j =1 y *j
I
1.1.3 The Two Fundamental Welfare Theorems
Armed with these definitions, we can now recall the two fundamental
welfare theorems:
theorem 1.1 (first fundamental welfare theorem) If the utility

functions Ui are strictly increasing, any equilibrium is efficient: if
(p*, (x*i )i=1,...,I, (y*j )j=1,...,J)
is a competitive equilibrium, then the allocation ((x*i )i=1,...,I, (y*j )j=1,...,J) is a

Pareto optimum.
Proof The proof is very simple and proceeds ad absurdum. If (x*, y*)
is not a Pareto optimum, then there is a feasible allocation (x, y) such
that
"i = 1, ..., I , U i ( xi ) > U i ( xi* )

But since x*i was the preferred basket of consumption of i in its budget
set for the prices p*, it can be deduced that xi cannot belong to this set,
by which
"i = 1, ..., I , p * ◊w i > p * ◊ w i + Â q ij p * ◊y*j

j
4 Introduction
So adding all the consumers yields

Ê ˆ
p * ◊ Â xi > p * ◊ Á w + Â y j* ˜
i Ë j ¯
But for every producer j, y*j maximizes the profit at prices p*, and we
then have
p* · y*j ≥ p* · yj
It can be deduced from these two inequalities

Ê ˆ
p * ◊ Â xi > p * ◊ Á w + Â y j˜
i Ë j ¯
which, since the prices are positive, contradicts the hypothesis that
(x, y) is feasible.
theorem 1.2 (second fundamental welfare theorem) If

• utility functions Ui are continuous, increasing, and concave (i.e., pre-
orders i are convex)
• consumption sets Xi are closed and convex
• production sets Yj are closed and convex
then every interior optimum can be decentralized in equilibrium: if the

allocation
(( x*)
i i= 1,...,I , ( y j*) j= 1,...,J )
is a Pareto optimum such that x*i is interior to Xi for every i, there is a
distribution of the economy’s initial resources
((w ) i i =1, ... , I , (q ij ) i =1, ... , I

j = 1 , ... , J
)
and a price system p* such that the price-allocation pair
(p*, ( x *i ) i =1, ... ,I , ( y*j ) j =1, ... , J )
is a competitive equilibrium for the economy of private property so
defined.
Proof Define the sets
Ci = {x ¢i U i ( xi¢) > U i ( xi¢)}

I
along with Z = Si=1 Ci and W = w + SIj=1 Yi. Z is in fact the set of aggre-
gate demands born of preferred individual consumptions via the
Pareto optimum, and W is the set of available postproduction
Introduction 5
resources. Under the hypotheses of the theorem, Z and W are convex.

Moreover they cannot have any point of intersection; otherwise, ((x*i ),
(y*j )) would not be a Pareto optimum. By the convex separation
theorem, there exists a vector p such that p · z > p · w for every z Œ Z and
w Œ W.
What this amounts to is that for every allocation ((xi), (yj)) such that
Ui(xi) > Ui(x*i ) for every i and yj Œ Yj for every j, we have
p ◊ Â xi > p ◊w + p ◊ Â y j
i j
or even, since Six*i = w + Sjy*j ,

p ◊ Â ( xi - xi* ) > p ◊ Â ( y j - y *j )
i j
Take all the xi close to x*i . Then in the limit we get
p◊ Â (y j - y j*) £ 0
j
that is, the production plans y*j maximize the profit to prices p. Now
let us equate all the yj to the y*j and make all the xk close to x*k for k π i.
Then we get
U i (xi ) > U i ( x*i ) fi p ◊ xi ≥ p ◊ xi*
Suppose that Ui(xi) > Ui(x*i ) and that p · xi = p · x*i . Choose xi¢ such that
p · xi¢ < p · xi. Then for every strict convex combination xi≤ of xi¢ and xi, we
have p · xi≤ < p · x*i. But one combination can be selected such that Ui(xi≤)
> Ui(xi*), and this is contradictory. We can conclude then that
U i (xi ) > U i ( x*i ) fi p ◊ xi > p ◊ xi*

which implies that x*i maximizes consumer utility i under the budgetary
constraint p · xi = p · x*i.
The convexity hypotheses are crucial here. They allow one to move
easily from differential conditions of optimality to global optimality. To
better understand this, consider the example of an exchange economy
(J = 0) with two consumers and two goods (I = L = 2). One could use an
Edgeworth box. Since preferences are convex, the two sets {U1(x1) ≥
U1(x*1 )} and {U2(x2) ≥ U2(x*2 )} are convex and have the point (x*1, x*2 ) in
common. One could separate them by passing a line through that point.
The slope of this line gives us the ratio of prices p*. If one chooses any
distribution of initial resources (w1, w2) on that line, then (x*1, x*2 ) is indeed
an equilibrium sustained by the price vector p*, as shown in figure 1.1.
6 Introduction
U2
p*
U1
x*
Figure 1.1
Second welfare theorem
1.2 Return to the Hypotheses
The “philosophical” importance of these two theorems cannot be

exaggerated. First they suggest that the market is an efficient organiza-
tion (first theorem). This was a perception of Adam Smith which he
stated in his famous metaphor of the invisible hand. The entire passage
from The Wealth of Nations (bk. IV, ch. 2) deserves repeating here (the
italics are mine):
As every individual, therefore, endeavors as much as he can both to employ

his capital in the support of domestic industry, and so to direct that industry
that its produce may be of the greatest value, every individual labors to make
the annual revenue of the society as great as he can. He generally indeed,
neither intends to promote the public interest, nor knows how much he is pro-
moting it. By preferring the support of domestic industry to that of foreign
industry, he intends only his own security; and by directing that industry in
such a manner as its produce may be of the greatest value, he intends only his
own gain, and he is in this, as in many other cases, led by an invisible hand to promote
an end which was no part of his intention. Nor is it always the worse for the society
that it was no part of it. By pursuing his own interest he frequently promotes
that of the society more effectively than when he really intends to promote it.
I have never known much good done by those who affected to trade for the
public good. It is an affectation, indeed, not very common among merchants,
and very few words need be employed in dissuading them from it.
What is more surprising, the results also show that the market permits
the attainment of any efficient allocation (second theorem). So even
Introduction 7
those among us who find revenue distribution to be uneven can turn

to the market to realize an optimum once the desired redistribution has
been carried out. In principle, this is the position adopted by the pro-
ponents of “market socialism,” and among them Léon Walras himself
can be counted.
Clearly, the hypotheses behind these two theorems merit examina-
tion. First, let us note that they assume the existence of a complete
set of markets. This hypothesis is particularly strong when we take
into account intertemporality and uncertainty: it then seems unlikely
that we can have a complete set of futures and contingent markets,
and in particular, in the presence of asymmetries of information or
transaction costs. However, the consequences of market incomplete-
ness are presented in the fourth part of the course. For now, we simply
note that the first theorem no longer applies (generically) in such an
economy.
Second, three failures concern what is called welfare economics:3
• In certain cases the use of goods by one consumer does not preclude
their being consumed by others. One speaks then of a public good (see
chapter 5) and aforementioned expressions, such as the sum Sixi, no
longer have meaning.
• Up to this point we have implicitly assumed that the utility of the
consumer i depends only on his or her own consumption xi, and that
the profit of the producer j depended only on a production plan yj. Such
is no longer the case in the face of external effects like pollution. This
will be the object of chapter 6.
• The hypothesis of the convexity of production sets Yj which subtends
the second theorem is particularly strong because it prohibits all form
of increasing yields in production. The convexity of preferences can
equally pose problems in certain cases. We will see the consequences
of nonconvexities in chapter 7.
Finally, third, we focus on the hypothesis of pure and perfect com-

petition which permits the definition of rival supplies and demands:
here lies the domain of industrial organization.
The phenomenon of macroeconomic fluctuations, accompanied by
unemployment and inflation, can be considered a market failure, but
macroeconomic aspects of markets will not be discussed in this book.
We will limit ourselves to the microeconomics. Nor will informational
3. The term public economics is reserved for the economics of the public sector (fiscal
policy, public firms’ pricing, etc.).
8 Introduction
aspects be covered (on this I take the liberty of referring the reader to
Salanié 1997). Nevertheless, in some form these issues will appear at
various junctures in the book. Finally, the book will not treat models
where there is an infinite number of goods or agents, for example,
models with overlapping generations.4
1.3 The Government’s Role
We will see again and again in this book that market failures can, in
theory, be moderated by adequate governmental intervention. Some
students may be quick to deduce from this that in situations of market
failure the government should intervene. This conclusion is a bit hasty,
as the classical economists had already found out:
It does not follow that whenever laissez faire falls short, government inter-
vention is expedient; since the inevitable drawbacks of the latter may, in any
particular case, be worse than the short-comings of private enterprise.5
The government has an undeniable advantage on the market: it has at

its disposal the monopoly of legitimate constraint and, in particular,
the power of taxation. It does not follow that the government possesses
unlimited power. The government is especially obligated to take into
account private agents’ responses to its decisions. Often cited in this
regard are the taxes on doors and windows that existed in various
European countries a few centuries ago. The unexpected effect was
considerably darkened houses. Economic history (even more recent) is
full of examples where the government has very badly foreseen the
reactions of private agents. The limited information in its possession
is largely responsible. One consequence of this is the government’s in-
ability to enact the lump sum transfers necessary to decentralize the
optimum in the sense of the second theorem.6 Therefore the govern-
ment will have recourse only to direct or indirect taxes which modify
the prices perceived by agents and which then create distortions in the
4. In overlapping generations models the value of the total resources can be infinite, so
it is easy to see that the proof of the first fundamental welfare theorem is no longer valid.
I refer the reader to the fairly technical survey by Geanakoplos-Polemarchakis (1991).
5. Henry Sidgwick, Principles of Political Economy (1887).
6. The characteristic of a lump-sum transfer is that the agent concerned cannot change
its value by altering his behavior. In the course of history there have been several
attempts to make use of such transfers. The old poll tax which affected men on the basis
of their social level, corresponded to this description in that that social class was deter-
mined initially by birth. More recently one can cite the aborted introduction of the poll
tax in the United Kingdom.
Introduction 9
economy. These distortions are at times desirable in order to correct

others (as in the case of externalities); at other times, for instance, to
finance a public good, they are a necessary evil.
As is generally the case in microeconomics, this book will content
itself with describing the government as a “benevolent planner” that
takes into account the preferences of agents for determining the social
optimum and then attempts to make use of it. The reader must realize
the narrowness of this point of view. It neglects all the complexity of
the political process, in which the agents act through their representa-
tives and with the aid of various lobby groups, and where the admin-
istrations themselves have a certain interpretive latitude in acting on
completed decisions. Ideally an analysis of measures that the govern-
ment can take to correct the effect of a market failure should take these
factors into account.7
Bibliographical Note
I will cite useful works and articles within the chapters themselves. A
general reference for welfare economics is Laffont (1988). The reader
may also profit from consulting Stiglitz (1988), which is less advanced
but contains numerous interesting discussions, and Henry (1989). In
regard to the industrial organization course, the reference work is
Tirole (1988), for the more ambitious student. I am greatly indebted
to the works of Laffont and Tirole, as the reader will perceive as the
chapters progress.
This book contains but elementary mathematics and requires only
the basics usually taught in introductory microeconomics courses (e.g.,
Varian 1992, Kreps 1990, or the first part of Mas Colell–Whinston–
Green 1995).
The Ecole Nationale de la Statistique et de l’Administration
Economique course which engendered this book calls for only eight
class sessions of two hours each. These are several parts of the book
that I do not teach in my own course for lack of time, in particular,
chapters 3, 8, and 13. Nevertheless, I think that a good student should
read the entire text.
Beginning on the principle that we can learn much from the
errors of those who have gone before, I have strived to give some his-
torical information on the theory’s development. Thus I hope to make
7. Readers interested in these aspects should consult Wilson (1989) and Dixit (1996).
10 Introduction
it better understood to the reader that economics is not a finished

science for which this text will reveal the Commandments, but that
it is in constant evolution, which makes studying it all the more
interesting.
Since this book is a manual, I did not seek to give an exhaustive
bibliography. Certain chapters have very short bibliographies because
the principles discussed have long been known or because one of the
references gives a longer bibliography of its own; this is particularly
the case in part III. Conversely, at times I give many more references,
for example, in part I where some of the results presented are fairly
recent.
Bibliography
Dixit, A. 1996. The Making of Economic Policy. Cambridge: MIT Press.
Geanakoplos, J., and H. Polemarchakis. 1991. Overlapping generations. In W.

Hildenbrand and H. Sonnenschein, eds., Handbook of Mathematical Economics, vol. 4.
Amsterdam: North-Holland.
Henry, C. 1989. Microeconomics for Public Policy: Helping the Invisible Hand. Oxford:
Clarendon Press.
Kreps, D. 1990. A Course in Microeconomic Theory. Oxford: Harvester Wheatsheaf.
Laffont, J.-J. 1988. Fundamentals of Public Economics Cambridge: MIT Press.
Mas Colell, A., M. Whinston, and J. Green. 1995. Microeconomic Theory. Oxford: Oxford
University Press.
Salanié, B. 1997. The Economics of Contracts: A Primer. Cambridge: MIT Press.
Stiglitz, J. 1988. Economics of the Public Sector. New York: Norton.
Tirole, J. 1988. The Theory of Industrial Organization. Cambridge: MIT Press.

Varian, H. 1995. Microeconomic Analysis. New York: Norton.
Wilson, J. 1989. Bureaucracy. New York: Basic Books.

I Collective Choice
The existence of market failures clears the way for government inter-
ventions that are destined to correct it. But according to what objectives
does the government act? Each consumer-voter has preferences, and it
is rare that a government decision has societal unanimity. I present the
problem of the aggregation of preferences in its most general form in
chapter 2. Chapter 3 is dedicated to cost-benefit analysis, a more
applied domain where the object is to evaluate the social significance
of a given government decision. Finally, in chapter 4 I analyze the con-
sequences stemming from the fact that the information the government
has at its disposal is limited.
2 The Aggregation of
Preferences
This chapter attempts to answer the question: Given a limited set of

possibilities, how does society make the “right” decision? Every econ-
omist brought up in the neoclassical tradition will think first of a Pareto
optimum. Indeed the criterion of Paretian efficiency has the great
advantage of being practically irrefutable. Every Pareto-dominated
allocation entails a waste of resources, and society should distance itself
from them as a whole. The Pareto principle consists precisely in pre-
ferring allocation A to allocation B if A dominates B in the Pareto sense.
Unfortunately, this criterion constitutes but a very partial order: when
there are n consumers in the economy, the set of Pareto optima is
(generically) a manifold of dimension n - 1. A position considered
orthodox in economics (and developed in particular by Robbins 1932)
has long been to remain at the Pareto optimum, since any other choice
criteria risks the intervention of comparisons between persons of
fragile status. As Robbins (1938) wrote,
Every mind is inscrutable to every other mind and no common denominator
of feelings is possible.
Such a rigid methodological choice made economists’ prescriptions

unconvincing. In the words of Harrod (1938),
If the incomparability of utility to different individuals is strictly pressed, not

only are the prescriptions of the welfare school ruled out, but all prescriptions
whatever. The economist as an adviser is completely stultified, and unless his
speculations be regarded as of paramount aesthetic nature, he had better be
suppressed completely.
At the end of the 1930s, the Pareto principle was therefore often com-
pleted by the “compensation principle.” This was also called the Hicks-
Kaldor criterion, and it led to what was called the new welfare economics.
14 Collective Choice
The compensation principle states that A must be preferred to B if in

leaving A and in effecting lump-sum transfers, one can attain an allo-
cation C that dominates B in the Pareto sense. The compensation prin-
ciple still defines only a very partial order,1 and it is far more debatable
than the Pareto principle. In effect, we saw in chapter 1, lump-sum
transfers are not easily put into practice. Consider, for example, the
opening of borders in a small economy. It is possible that it be harm-
ful to a certain group of producers, but in general, liberalization of
exchanges is more profitable for consumers than it is detrimental to
producers, so it is recommended by the compensation principle. This
is but meager consolation for the producers if the corresponding
transfers are never enacted, and so on.2
On the other hand, neither the compensation principle nor Pareto
optimality comprises the notion of equity. For example, an allocation
where one consumer assumes all of the economy’s resources are always
a Pareto optimum. One solution suggested by Foley (1967) for an
exchange economy entails allocations (x1, . . . , xn) that verify
"i , j , U i (xi ) ≥ U i (x j )
so that no consumer covets another’s allocation. Such an allocation is
said to be a no-envy allocation. This concept incorporates a certain idea
of justice but does not in fact imply optimality. For that it is necessary
to limit oneself to “fair” allocations, that are at once no-envy and
Pareto-optimal (Varian 1974).
It is easy to see that Walras equilibria obtained from the egalitarian
allocation3 are fair allocations. We let (p, x1, . . . , xn) be such an equilib-
rium and consider two consumers i and j. If the preferences are not sati-
ated, we have
p ◊ xi = p ◊ x j = p ◊
Â in=1w i
n
so xj is a possible choice for I. We must therefore have Ui(xi) ≥ Ui(xj),
and the equilibrium is no-envy. The first welfare theorem implies that
it is equally Pareto-optimal and therefore fair.
1. When utilities are nontransferable, that is, when there is no good whose marginal
utility is constant, the compensation principle does not even define an order: it is quite
possible that A be preferred to B and B preferred to A.
2. The history of new welfare economics and of its relative failure is reported by Chipman-
Moore (1978).
3. Where the initial allowance of each consumer is Sni=1 wi/n.
The Aggregation of Preferences 15
Do other fair allocations exist? Champsaur-Laroque (1981) showed

that the answer to that question is disappointing: if there is a contin-
uum of consumers, the only fair allocations are the Walras equilibria
obtained from the egalitarian allocation. Moreover the idea of no-envy
allocation does not allow for definition of an order (when does one allo-
cation generate “less envy” than another?), and it does not lend itself
well to generalizations in the case of production economies. We do not
have here simply a matter of technical difficulty. Suppose that John
enjoys greater productivity than Peter. Is it unfair that John have an
allocation that Peter finds preferable to his own? If it is not, then
the no-envy criterion becomes less attractive. However, most moral
philosophers argue that innate differences in productivities are morally
arbitrary and should not determine ethical judgments. If we think that
most differences in productivities are innate, then no-envy remains an
appealing property.4
Can we go further? This question actually can be broken down into
two subproblems:
1. Define a collective preference. This is the problem of the aggregation

of preferences.
2. Collect the information necessary in order to put the optimum
of this collective preference to use. This is the problem of
implementation.
To illustrate, let us take two examples. The first applies to distinguish-

ing a particular Pareto optimum from among all Pareto optima of an
economy. One way to proceed consists in defining a functional W(U1,
. . . , Un), called the Bergson-Samuelson functional, and maximizing it
over the set of possibilities. This supposes first of all that one knows
how to define the functional W (aggregation of preferences),5 and also
that one has obtained the necessary information—the utility functions,
which are private a priori (implementation).
The second example is that of an election where m candidates
compete for an office. If m = 2, it is easy to see that the simple major-
ity vote, which consists of electing a over b if the number of voters who
4. Production economies present a more technical difficulty: Pazner-Schmeidler (1974)

showed that contrary to the case of exchange economies, fair allocations may fail to exist.
However, Piketty (1994) showed that if more productive agents also face a lower
disutility of labour, then fair allocations do exist.
5. We will see in this chapter that very few of the Bergson-Samuelson functionals satisfy
minimal conditions.
prefer a to b is higher than the number of those who prefer b to a, is a

process that has good characteristics: it leads no voter to manipulate
his vote. One can hardly do better if (as is always the case in an elec-
tion) one cannot make use of more precise information on voters’ pref-
erences. Things become complicated when m > 2. One might think of
extending the simple majority vote in categorizing all the candidates
according to their performances in tournaments where they compete
in pairs. Then a would be ranked above b if the number of voters who
prefer a to b is higher than the number of those who prefer b to a. Unfor-
tunately, this process often ends in the famous Condorcet paradox
(1785): as the reader will easily see, it is very possible for this process
to lead to a situation where a is preferred to b, who is preferred to c,
while c is himself preferred to a. The resulting order is therefore not an
order, since it is intransitive, which makes finding the socially optimal
choice impossible. This phenomenon is not at all pathological. Sup-
pose, for example, that there are 3 candidates and 21 voters. Eight
voters prefer, in order, a > b > c, seven prefer b > c > a, and six c > a >
b. The reader will verify that these preferences result in Condorcet’s
paradox.6
The simple majority vote is not the only procedure that entails dif-
ficulties when there are more than two candidates. Consider, for
example, the classification procedure suggested by Borda (1781).
According to Borda, each voter can assign one point to his favorite can-
didate, two to his second favorite, and so on, and then the candidates
are ranked in order from the most points collected to the least. Note
that this is the exact method practiced in numerous sports competi-
tions, for example, in Formula One racing.7 The disadvantage to the
Borda method is the possibility that the relative ranking of a and of b
will depend on the ranking of c in relation to them. Thus, if Schumacher
has three points more than Hill before the last Grand Prix of the season,
their final order will depend on whether or not Berger or Alesi suc-
ceeds in defeating one of them. In regard to elections, any third candi-
date that pulls out of the election can affect the social classification of
6. It can be shown that if there were a perfect shuffle of the voters’ candidate preferences
then the probability of an electoral paradox is approximately 0.09 when there are three
candidates and a very large number of voters. This probability increases with the number
of candidates.
7. One trial (a Formula One Grand Prix) results in a ranking just like a voting poll.
So formally a sports event could correspond to an election, and the competitors to
candidates.
a and of b. Technically Borda’s method violates an axiom which we will

later call independence of irrelevant alternatives. All other imaginable
methods (organization of primaries, polls, two-round systems, etc.)
possess their own inconveniences. This is a consequence of Arrow’s
famous theorem, which is at the heart of the next section.
2.1 Arrow’s Theorem
Consider the following very general problem. The set of possible

choices is noted A. There are n agents i = 1, . . . , n. Using traditional
notation (with more common notation in parenthesis), we can repre-
sent individual preferences as
• aPib if i strictly prefers a to b [Ui(a) > Ui(b)]
• aIib if i is indifferent between a and b [Ui(a) = Ui(b)]
• aRib if i likes a at least as much as b [Ui(a) ≥ Ui(b)]
We can assume that relations Ri are preorders (transitive and complete).

Relations Pi and Ii are then obtained from Ri by
• xPiy iff (xRiy and not (yRix))
• xIiy iff (xRiy and yRix)
The aggregation of preferences problem consists in going from the

vector (profile) of individual preferences R = (R1, . . . , Rn) to a collective
preference R̂ on A. The social welfare function (SWF) will be the func-
tional R̂ = f(R). Several examples of SWFs were given in this chapter’s
introduction.
Note the considerable generality of this modelization. The descrip-
tion of the possible choices (also called social states) may contain the
set of all individual allocations, the production of public goods, gov-
ernmental orientations, and so on. As for preferences on social states,
they can be perfectly selfish or take into account equity concerns. As
Arrow (1950, p. 333) states, “It need not be assumed here that
an individual’s attitude toward different social states is determined
exclusively by the commodity bundles that accrue to his lot under
each.”
First let us examine the most natural case in the development of
the neoclassical methods, which was the object of Arrow’s founding
studies (1950, 1951): that of individual preferences being ordinal and
non-comparable among individuals. Where individual preferences are
ordinal, it is impossible to determine if i prefers a to b “more” than he

prefers b to c. Where they are noncomparable, it cannot be said whether
or not the preference of i for a over b balances the preferences of j for
b over a.
Arrow imposed on each SWF a series of four conditions which he
thought natural:8
1. Transitive criterion: for every R, f(R) must be preorder.

2. Pareto (or unanimity) criterion: if "i, aRib, then aRb.
3. UD (universal domain): f(R) must be defined for every imaginable
profile R.
4. IIA (independence of irrelevant alternatives). The social choice
between a and b must depend only on the individual preferences
between a and b. More formally, let R and R¢ be two profiles such that
af(R)b and {i|aRib} = {i|aRi¢b}. Then one must get af(R¢)b.
Condition 1 seems quite natural; still, we will see how it can be weak-
ened. Neither does the Pareto criterion of condition 2 seem debatable:
if all the individuals prefer a to b, it is unclear how the collectivity could
be of the opposite opinion. Condition 3 reflects the desire to impose
great generality on the model: if individuals have no idea a priori of
the nature of their preferences, then the SWF should furnish a univer-
sally applicable solution. The independence axiom 4 is the most com-
plicated to express. My discussion above of the Borda method shows
why such a criterion is necessary: without it, unforeseen and undesir-
able phenomena can result. Note that this axiom implies that the
sought-after SWF cannot help but take the form of the “generalized
vote,” where the rank of x in relation to y depends only on the list of i
who prefer x to y. It particularly prohibits taking into account the
“intensity of preferences” which specifies that agent i can prefer x to y
“more” than another agent j prefers y to x.
Finally an SWF will be called dictatorial if it systematically repro-
duces an individual’s preferences, whatever the individual preference
profile:
$i , "R , af (R) ¤ aRi b
We can now set forth Arrow’s theorem (often called general possibil-
ity theorem):
8. The list resumes pedagogical tradition by which the statements retained differ slightly
from Arrow’s original conditions.
theorem 2.1 (arrow) If A has at least three elements, every SWF

that verifies conditions 1 through 4 is dictatorial.
Proof See appendix A.
The conclusion of the theorem is of course quite disappointing

(Arrow goes so far as to say it is “the height of bad luck”). The ques-
tion that will concern us now is the possibility of weakening the
hypotheses of the theorem so as to invalidate its conclusion.
First, we note that if A has but two elements, then the simple major-
ity vote satisfies all of Arrow’s conditions (other procedures verify
these conditions, but a simple majority vote is the only way to satisfy
other natural conditions, as formulated by May 1952).
It does not seem very desirable to weaken the Pareto condition:
an SWF that would reverse unanimous choices would be quite
unsatisfactory.
Conversely, we could imagine weakening condition 1 by no longer
insisting that f(R) be an order but that it simply allow maximal elements
to be found. It suffices then that in every subset of A, there be one or
several elements preferred by the social choice. For this, we can show
that it is enough for the social preference to be acyclical, in other words,
that it verify
a1Pa2P . . . Pan fi a1Ran

which is considerably weaker than the transitivity implied by condi-
tion 1. Actually this weakening does not eliminate Condorcet’s paradox
within the bounds of a simple majority vote. Still there are SWFs that
verify conditions 2 through 4 and acyclicity, as shown in the following
example:
aRb ¤ a is not Pareto-dominated by b
The inconvenience of this SWF9 is that its curves of indifference are

dense. If neither a nor b Pareto-dominates the other, then a and b are
socially indifferent. If, on the other hand, each agent has veto rights,
aPib, then necessarily aRb. Indeed, it can be shown that every SWF
that verifies conditions 2 through 4 and is quasi-transitive10 obligato-
rily gives veto rights to a group of agents that can then be called an
9. Aside from its not being transitive.

10. Quasi-transitivity stipulates that only the strict order P is transitive; therefore it is a
slightly stronger condition than acyclicity. One can show (see Moulin 1988, ch. 11) that
fairly similar results are obtained for acyclical SWFs.
oligarchy. Therefore the weakening of transitivity is not a very promis-

ing solution.
The IIA axiom has been greatly disputed. Unfortunately, attempts
made to find a weaker version of it are often very technical and have
not led far. Its complete suppression would reestablish the existence of
nondictatorial SWFs, but they would suffer the aforementioned draw-
backs. Conversely, the IIA axiom could be replaced by a fairly natural
one of monotonicity (which says that if the ranking of x improves in
all Ri, then it cannot worsen in f(R)) without modifying the conclusion
of Arrow’s theorem.11 A better way to consider preference intensity is
to suppose that preferences are comparable, as we will see later.
The weakening of the hypothesis of universal domain is more
rewarding. It is very possible that in certain situations, one is privy to
information on agents’ individual preferences a priori. It is fitting then
to exploit that information.12 The most famous example, which dates
back to Black (1948), concerns single-peaked or unimodal preferences.
Suppose that all individuals agree to arrange the possible choices on
one axis and that each individual has single-peaked preferences on that
axis, as in figure 2.1.
We could think of political preferences where candidates are natu-
rally categorized as left or right. Under these conditions, it can be
illustrated that the simple majority vote leads to a social order that is
transitive (the Condorcet paradox cannot then appear with such pref-
erences13), and that this order is identified by the preferences of the
median voter—whose favorite candidate is in the middle of the axis
among all agents’ favorites.14 Note that this SWF is nondictatorial: the
median voter varies according to individual preferences, and therefore
there is no dictator whose choices are always followed.
This positive result is encouraging, but it allows (at least) two prob-
lems to remain. The first is that the rule of simple majority vote is far
less appealing in redistribution problems than in electoral contexts. For
example, consider the division of a cake among three agents i, j, and k,
11. We will find the axiom of monotonicity again in chapter 4.

12. In the framework of an economy, for example, it is possible to limit attention to
increasing or convex preferences; unfortunately, it can be shown that this does not
invalidate the conclusion of Arrow’s theorem.
13. Caplin-Nalebuff (1991) show that this result extends to certain classes of multi-
dimensional preferences, as long as the agents’ preferences do not differ too much and
the vote requires a majority higher than 64 percent.
14. Strictly speaking, it is necessary that the number of agents be odd in order to enable
the definition of a single median agent.
i's preferences
j's preferences
left right
Figure 2.1
Single-peaked preferences
each of whom is interested only in his own share. In the social state a,
i and j each have 40 percent of the cake and k has 20 percent. In state
b, 10 percent of k’s cake has been taken away and divided between
i and j. State b is clearly preferred to a by a simple majority vote.
Nevertheless, this conclusion does not seem fair: k, who was already
worst off under a, is even more impoverished under b! It is intuitively
clear that the simple majority vote is not a good solution in this context,
even with trivial preferences.
Now, it must be remembered that Arrow’s conditions constitute only
a very minimal set of demands. One could, for example, wish to pre-
serve individual rights in giving each individual final choice in circum-
stances pertaining to his reserved domain. For example, each individual
(at least each unmarried individual!) should be able to decide for
himself whether to sleep on his back or his stomach. Sen (1970) for-
malizes this notion of “minimal liberty” or of “liberalism” insisting that
each individual can impose his choices on at least a pair of decisions:
"i , $(a, b), af (R)b ¤ aRi b

Thus, if a and b are two decisions that differ only in that i sleeps on his
stomach in a and on his back in b, i must be able to impose his choice
on society. It is said then that i is decisive on (a, b).
Unfortunately, there is no Paretian liberal: no SWF (even if only
acyclical) satisfies altogether the condition of universal domain, the
Pareto principle, and the new condition of liberalism. Sen gives an
example of this based on Lady Chatterley’s Lover (henceforth LCL).15 Let
us concentrate on the two agents A, who is prude, and B who is a
libertine, and on three possible situations:
15. The book by D. H. Lawrence was published in 1928 and scandalized respectable
English society with its eroticism.
• x: only A reads LCL

• y: only B reads LCL
• z: neither A nor B reads LCL
A, being prudish, prefers that no one read LCL, but if someone must
read it, he prefers that it be himself rather than B, whom he judges to
be much too impressionable. B would like to read LCL, but he would
like it even more that the prude A read it and be horrified. Therefore
the preferences are
• for A: z > x > y
• for B: x > y > z
Conversely, liberalism suggests that A cannot be forced to read LCL,

nor B not to read it: A is decisive on (z, x) and B is decisive on ( y, z).
In this example the Pareto principle imposes that x > y on a social
level, while liberalism imposes z > x and y > z; the social preferences
are therefore cyclical. Yet it will be observed that this paradox rests on
the existence of nosy preferences: A and B have simultaneous preferences
that dictate not only what they prefer for themselves but also what they
prefer for other members of society. It is possible to show that only in
such cases does liberalism contradict the Pareto principle.
In fact it seems that in economic situations that interest us the most
directly, it is difficult to achieve positive results when individual pref-
erences are ordinal and noncomparable. The path we will take now
involves progressively refining the information available on individual
preferences.
2.2 Noncomparable Cardinal Preferences
It can be shown that switching to noncomparable cardinal preferences

does not improve the situation (d’Aspremont-Gevers 1977). I will just
give a sketch of this result.
To simplify, we will return to the notation in terms of utility func-
tions. Note that U = (U1, . . . , Un) is a profile of individual utilities and
that Û = f(U) is the corresponding social utility.16 The fact that the pref-
erences are noncomparable cardinals is translated by the requirement
16. Keep in mind that we are working from a welfarist perspective, where only the util-
ities are important; under fairly weak hypotheses, Û can be represented as a Bergson-
Samuelson functional Û = W(U1, . . . , Un). Therefore the problem lies in studying the
restrictions that W must satisfy.
that if all the Ui are submitted to possibly different affine transforma-

tions according to
U i¢ = aiU i + bi
then the new social utility Û¢ leads to preferences identical to those that
subtend the former Û:
Uˆ (x) ≥ Uˆ (y) ¤ Uˆ ¢(x) ≥ U

ˆ ¢( y )
Now let fi be any increasing transformation from IR to IR, and fix two
choices x and y. It is possible to find coefficients (ai) and (bi) such that
one gets for each i,
Ïf i [U i (x)] = aiU i (x) + bi

Ì
Óf i [U i (y)] = aiU i (y) + bi
since this is a regular system of 2n equations with 2n unknowns.
But as the preferences are noncomparable cardinals, in applying IIA
we can deduce that the preferences fi Ui lead to the same social
preferences as Ui, and we return to the case of ordinal noncomparable
preferences.
2.3 Comparable Ordinal Preferences
The case of preferences that are comparable between individuals is

more promising: It is clear a priori that the comparison between utili-
ties achieved by different agents produces useful information when
one examines a redistribution problem. Even in the seemingly simple
framework of Formula One racing, we saw that the adopted ranking
doesn’t satisfy IIA; in turn, the introduction of comparisons (here, the
racers’ times in the various stage races) would allow for a more satis-
fying ranking. By the same token, the introduction of a point system in
decathlon events allows for an escape from Arrow’s theorem. Now let
us study the impact of the comparability of individual preferences.
The fact that the preferences are comparable ordinals is illustrated
by the requirement that if all the Ui are submitted to the same increas-
ing transformation, then Û is transformed the same way.
In this setting, several anonymous SWFs17 exist that verify all of
Arrow’s conditions. They are all characterized by the existence of a
17. An SWF is anonymous when it is indifferent to the identity of individuals, that is,
when the order it engenders is unvarying regardless of permutations of individuals.
“positional dictator” defined by its place in the ordained utility vector,

which can then be the median agent, or the most advantaged, or the
most disadvantaged, or even the 13th most disadvantaged, and so on.
One might add one or several conditions in order to arrive at a single
characterization. It is in this way that Hammond (1976) adds one con-
dition that he calls equity:
U j (b) < U j (a) < U k (a) < U k (b)¸

˝ fi U (b) £ U (a)
"i π j , k , U i (a) = U i (b) ˛
This condition states that if the preferences of j and k between a and b
are opposed, although all the other agents are indifferent, and if j is
more disadvantaged anyways than k, then the social preference cannot
work against the wishes of j. The opinion followed is always that of
the most unfortunate—if this individual exists.
Hammond shows that if one adds anonymity and the equity axiom
to Arrow’s conditions, one obtains a leximin criterion. Without going
into too much detail, this preference is a lexicographical generaliza-
tion18 of the Rawls criterion, so U = mini=1,...,n Ui. Thus a state a is pre-
ferred to a state b if the most disadvantaged individual in state a is
happier there than the most disadvantaged individual in state b (who
of course is not necessarily the same individual). Hammond therefore
justifies applying the “difference principle”19 of Rawls (1971), whose
book has had and continues to have considerable influence on politi-
cal philosophy (see appendix B).
2.4 Comparable Cardinal Preferences
When preferences are cardinal and comparable, a like affine increasing

transformation of Ui must also be applied to U. There then exists only
one family of SWFs that verifies Arrow’s conditions: the utilitarist
family, which is given by
n
U = Â p iU i
i =1
where the pi possess the properties of a probability vector. If one further

insists that the chosen SWF be anonymous, then the only solution is
18. It takes into account possible indifferences.

19. The difference principle consists of not admitting inequalities except to the extent
that (especially for reasons of incentives) they benefit the individuals who are the least
well off.
U = –1n Sni=1Ui. In this way one rediscovers the result of Harsanyi (1955),
which justified it by invoking a social contract passed under a “veil of
ignorance,” before the agents know their identity. It is then reasonable
for the agents to be interested in the expectation of their utility. In this
sense the criterion of Rawls (whose theory is also tied to the social
contract school) corresponds to a case where the agents are infinitely
adverse to risk.
Finally, we note that in the extreme case where individual prefer-
ences are completely fixed (so that even affine transformations are
inadmissible), one again finds the Bergson-Samuelson approach men-
tioned at the beginning of this chapter: a very large number of com-
patible SWFs exist under Arrow’s conditions. Such a hypothesis is
obviously very remote from the framework of neoclassical analysis.
However, it can be useful if one is interested only in a unidimensional
indicator like income. Thus Atkinson (1970) and Dasgupta-Starrett
(1973) show that a distribution of income is preferred to another dis-
tribution of equal mean for all strictly quasi-concave and symmetrical
SWFs if and only if the corresponding Lorenz curve is nearer the diag-
onal, which is a customary definition of a more egalitarian distribution
of income.
2.5 Conclusion
We have seen that the best possible response to the problem posed by
Arrow’s theorem consists in enriching the available information on
individual preferences, in weakening the universal domain hypothe-
sis, or in resorting to interpersonal utility comparisons.
One can go even further, either by enriching the information tied to
the utility, as we did in the last two subsections, or by calling on inde-
pendent utility criteria. The last approach was adopted by Sen in his
recent work (e.g., see Sen 1992). This is a fairly vast body of writing
that also examines the possibility of introducing equity criteria into the
social preferences. Last we saw that the theory of optimal electoral pro-
cedures (e.g., see Moulin 1988, ch. 9) also plays a role in the aggrega-
tion of preferences.
To conclude, I would like to emphasize that it is impossible to be only
interested in distribution questions from the economist’s point of view.
There are more fundamental philosophical inquiries. The reader can
get a brief glimpse of some introductory elements of philosophical-
economics writings on social justice in appendix B.
2.6 Appendix A: Proof of Arrow’s Theorem
The idea for the following proof dates back to Vickrey (1960); it com-
prises three lemmas.
Denote I = {1, . . . , n}.
lemma 1 (neutrality) Given a division I = M » N and (a, b, x, y) Œ

A4 such that
• "i Œ M, xPiy and aP¢i b
• "i Œ N, yPix and bP¢i a
then
ˆ ¤ aPˆ ¢b
xPy (2A.1)
ˆ ¤ aIˆ ¢b
xIy (2A.2)
Proof The interpretation of lemma 1 is simple: If x and y are ordered

by each individual in P as a and b in P¢, then this must also be true of
social preference; that is, x and y must be ordered by P̂ as a and b by
P̂¢. If this weren’t the case, then the procedure of aggregation would
treat the pairs (a, b) and (x, y) in a nonneutral manner, hence the name
of the lemma.
Let us first prove (2A.1). Suppose that xP̂y and that (a, b, x, y) are dis-
tinct two by two. Let preferences P≤ be such that
• "i Œ M, aP≤i xP≤i yP≤i b
• "i Œ N, yP≤i bP≤i aP≤i x
(such preferences exist because the domain is universal). By IIA, since

x and y are ordered individually in P≤ as in P, we have xP̂≤ y. By the
Pareto principle, aP̂≤ x and yP̂≤ b. By transitivity, we get aP̂≤ b. Finally,
in reapplying IIA, we find that aP̂¢b. The cases where a or b coincide
with x or y are treated similarly.
Part (2A.2) is obtained directly. If xÎy but, for example, aP̂¢b, we
find the contradiction xP̂y by (2A.1) in reversing the roles of (a, b) and
(x, y).
Before stating Lemma 2, two terms must be defined. We say that a

set of agents M is almost decisive on (x, y) if for every P̂,
ˆ
(" i ŒM , xPi y and " i œM , yPi x) fi xPy
We say that M is decisive on (x, y) if for every P̂,

ˆ
(" i ŒM , xPi y) fi xPy
lemma 2 If M is almost decisive on (x, y), then it is decisive on (x, y).
Proof Suppose that "i Œ M, xPiy. By IIA, only individual preferences

count on (x, y); the others can be changed. Assume therefore, with no
loss of generality, that z exists such that xPizPiy if i Œ M and z is pre-
ferred to x and y by the other agents. Neutrality imposes that xP̂z
because the individual preferences are oriented as on (x, y) and M is
almost decisive on (x, y). Finally, the Pareto principle implies that zP̂y
and transitivity implies the conclusion that xP̂y.
Note that neutrality implies that if M is decisive on (x, y), it is deci-

sive on every other pair. We will therefore just say that M is decisive.
lemma 3 If M is decisive and contains at least two agents, then a strict

subset of M exists that is decisive.
Proof Divide M = M1 » M2, and choose a P̂ such that

• on M1, xPiyPiz
• on M2, yPizPix
• outside M, zPixPiy
As M is decisive, we have yP̂z.

One of two things results:
• Either yP̂x, and (since z doesn’t count by IIA) M2 is almost decisive
on (x, y), and therefore decisive by lemma 2
• Or xR̂y, and by transitivity xP̂z; then (y doesn’t count by IIA) M1 is
almost decisive on (x, y), and therefore is decisive by lemma 2
The proof is concluded by noting that by the Pareto principle, I as a

whole is decisive. In applying lemma 3, the result is an individual i that
is decisive, and this is therefore the sought-after dictator. The attentive
reader will have noted that lemma 1 remains valid when there are only
two alternatives, contrary to lemmas 2 and 3.
2.7 Appendix B: Theories of Justice
For our purposes it will suffice to take a brief historical glimpse at some
theories of justice. For readers who are interested in the original or
more subtle enlightenments, I encourage them to consult Roemer

(1996).
2.7.1 Utilitarianism
Modern theories of justice often define themselves in reference (or by

opposition) to utilitarianism, which dates back to the writings of
Jeremy Bentham at the end of the eighteenth century. Utilitarianism
was the dominant doctrine of classical economists at least up to John
Stuart Mill. Its hypothesis is based on the notion of “welfarism.” It
states that for each person and each state of the world, a single index
exists to measure welfare, and it is called “utility.”
The utilitarianists assumed that these utility indexes were cardinal
and comparable among persons, so that one could define their sum.
The task of the government then was simply to maximize that sum.
Bentham termed this task “the arithmetics of pleasures and pains.” The
objective was just to maximize (with obvious notation).
n
Â U i (x i )
i =1
under the scarcity constraints.

Utilitarianism became subjected to much criticism, however. First it
became apparent that the poor and the rich could not be treated in a
symmetrical manner, since that meant abandoning all redistributive
views. However, Edgeworth showed that this critique is not entirely
accurate. Thus suppose that individuals draw no utility except from
their income Ri, that total income is fixed at R, and that the utility
indexes are concave. Now utilitarianism calls for equality of marginal
utilities U¢i (Ri). If in addition the utility indexes of all individuals coin-
cide, then incomes must all be equalized, and this idea of course cor-
responds to the most progressive taxation possible.20
A second problem of utilitarianism is that it priviledges individuals
who easily transform their increments from income into utility. To see
this, suppose that the utility Ui(Ri) = U(Ri, ai) does not depend on i
directly but only through a parameter ai, that it is concave, and that
the marginal utility of income is increasing in a:
20. This argument is defective in that total income is independent of the redistributive
scheme which neglects all the disincentive aspects of taxes; we will see later in this section
that utilitarianism does lead to taxes that are not very progressive when the essential dis-
incentive effects of taxes are reintroduced into the analysis.
∂ 2U ∂ 2U
< 0 and >0
∂R2 ∂R∂a
clearly, a higher ai corresponds to a greater capacity to transform
income into utility. So it is easy to see that equalization of marginal
utilities of income leads to giving each individual i an income that
increases in his parameter ai. This doesn’t seem fair; consider, for
example, someone who is disabled or someone who has difficulty
transforming a dollar of extra income into utility because, for example,
his capacities for leisure consumption are reduced. Then his ai will be
low, and the utilitarist will assign him a lower income.
More generally, all the criticisms of welfarism also apply to utili-
tarism: The hypothesis of a sole utility index is reductive, it prohibits
the taking into account of liberties and human rights,21 and sundry
other essentials.
But the most important criticism of utilitarism is philosophical. In
adding up the utilities of different individuals, there is implicitly sup-
posed that the happiness of one human being can compensate for
the unhappiness of another. This idea violates the Kantian principle
that “one cannot treat people as means to an end. This is in fact the
main reason that utilitarianism was abandoned by twentieth-century
philosophers—though not by economists.
2.7.2 Rawls’s Difference Principle
John Rawls’s 1971 book became the cornerstone of all recent debates
on social justice. In Nozick’s words, “Political philosophers now must
either work within Rawls’s theory or explain why not” (1974, p. 183).
Rawls chose a framework, which he called the “original situation,” that
more or less corresponds to Rousseau’s pure state of nature. The orig-
inal situation exists before individuals have concluded a social contract.
However, the contribution that earned Rawls his fame is his introduc-
tion of the “veil of ignorance”: In the original situation, each indi-
vidual is unaware of who he is and what his place will be in society;
he does not even know what wealth and talents he will inherit. Under
such circumstances Rawls affirmed that each individual will first of all
want to be guaranteed elementary rights and liberties. Hence this is the
21. Liberties could be counted as arguments for the utility index, but this runs the risk
of bringing back into question the usual properties of the index, and thereby making the
result fairly unusable.
first principle, according to which each individual must have access to

the most extensive system of liberties that is compatible with an iden-
tical system for other individuals.22
Another idea that Rawls put forth is that there exist “primary goods”
(income surely, but also higher level variables like access to places of
responsibility) upon which one can define a utility index that is ordinal
and also comparable among individuals. Since individuals are all iden-
tical in the original situation, Rawls first affirmed that they will want
to equalize these utilities. However, total equalization of primary util-
ities of goods risks having disincentive effects that will distance the
society from a Pareto optimum, so it may be preferable to tolerate
certain inequalities if they benefit the most disadvantaged. This
brought Rawls to state his famous second principle, often called the
“difference principle”:23 “All social primary goods—liberty and oppor-
tunity, income and wealth, and the bases of self-respect—are to be dis-
tributed equally unless an unequal distribution of any or all of these
goods is to the advantage of the least favored” (1971, sec.46).
In mathematical terms what Rawls justifies is the “maximin crite-
rion,” according to which society’s utility function is given by
U (x) = min U i (xi )

i =1, ... , n
By way of illustration, I adopt an optimal taxation problem from

Roemer (1996). Assume that individuals do not differ except in their
marginal productivity w, which coincides with their salary in perfect
competition and which is distributed among the population according
to a uniform on [0, 1]. Their utility is (y - l2/2), where l is their job
and y their income, and y = wl. The government enacts a purely redis-
tributive affine tax T(y) = cy + d. How should parameters c and d be
chosen?
First, note that the individual w facing this tax solves
È l2 ˘
max Íwl - (cwl + d) - ˙
l Î 2˚
which gives l(w) = w(1 - c). The individual w therefore pays a tax
t(w) = cw 2 (1 - c) + d
22. This way the allocation of elementary rights eludes welfarism.

23. According to Rawls, this principle has inferior priority to the first principle. The
application of the difference principle must not, under any circumstances, lead to a vio-
lation of the liberties defined by the first principle.
In order for the tax to be purely redistributive, its balance must be zero.
That is, one must have
1
Ú0 t(w)dw = 0
from which one gets d = -c(1 - c)/3. The indirect utility of the indi-
vidual w becomes
2
l( w 2 ) (1 - c) c(1 - c)
v(w) = wl(w) - [cwl(w) + d] - = w2 +
2 2 3
A Rawlsian government would choose c so as to maximize the indirect

utility of the least advantaged:
max min v(w)

c w
This gives c = 1/2, and not c = 1 as one might imagine a priori. (An
overly confiscatory tax would dissuade the most productive indi-
viduals from working and would therefore reduce the mass of income
to be redistributed.) On the other hand, a utilitarian government would
seek to maximize the expectation of indirect utility:
1
max Ú0 v(w)dw
c
which gives c = 0, so the government revokes enactment of the tax.

This last result is dependent on the hypotheses, and could be modified
if redistributive objectives are introduced.24 Still the conclusion that
Rawls’s approach leads to a more progressive taxation remains true
–
in general: The poorest individuals (up to w = ÷2/3 here) prefer the
Rawlsian tax and the richest prefer the utilitarian tax.
2.7.3 Recent Developments
One could summarize Rawls’s argument on the allocation of primary

goods as follows:
• The inheritance (of wealth and talents) is morally arbitrary
• The ensuing social inequalities must therefore be abolished
• Complete equality must be replaced by the maximin criteria in order
to preserve Pareto-optimality
1
24. The reader can easily verify that if the government maximizes Ú0 m(w)v(w)dw, setting
m(w) = 1 - w to give a higher weight to the utility of the least productive individuals,
then it should set c = 1/3.
Several authors have been inclined to criticize this reasoning. Sen and
Harsanyi, for example, attacked some of its consequences. The maximin
criterion risks sacrificing a considerable welfare increase due five billion
individuals if it ever so slightly damages the most disadvantaged. Obvi-
ously this seems a bit extreme. Moreover there is the opposite argument
that has the individuals, in the original situation, not able to tolerate
any inequality, so they automatically imagine themselves as among the
most disadvantaged and promote behavior that is infinitely adverse to
risk. This too does not seem to correspond to reality.
Actually, Rawls’s approach does not seem to leave much room for
free will. His conclusion proceeds rather directly from the idea that
individuals are not morally responsible for their place in society.
Certain authors (see Roemer 1996, chs. 7 and 8) have reinterpreted indi-
vidual responsibility using the general framework of Rawls, but they
distinguish morally arbitrary circumstances, among which they cate-
gorize the inheritance of wealth and talents, and the free choices of
individuals, which include their efforts, and even their tastes insofar as
those tastes depend at least in part upon a choice. The maximin crite-
rion then must regard morally arbitrary circumstances so that it can
tolerate inequalities that affect an individual’s free choice.
This new approach is of course more difficult to model than a
straightforward utilitarianism or the maximin criterion. However, it is
confirmed to lead to an optimal taxation that is more redistributive
than the utilitarist criterion but less so than Rawls’s approach.
In this regard it is fitting also to cite the work of Sen (e.g., see Sen
1992). Sen completely rejects welfarism by refusing to accept Rawls’s
idea on the intercomparable utility of primary goods. Sen criticizes the
tendency of earlier theories to focus on a restricted group of variables.
He is inspired by the literature on positive rights to define a set of func-
tionings: to be well fed, well educated, in good health, and so on. The
set of functionings available to a human being constitutes his capability
set, and it is therefore multidimensional. In Sen’s view, after taking effi-
ciency into account, it is upon equalization of capability sets that social
justice theory must be founded. In practice, this means that we should
define an index that takes into account both the functionings (which
describe an individual’s welfare) and the capabilities (insofar as freedom
of choice has an intrinsic value25). Obtaining such an index may seem
25. By this Sen means that an individual whose preference is a benefits from the possi-
bility of choosing b, even if he would never choose it. For example, most people prefer
that there be an election rather than the establishment of a dictatorship of their favorite
candidate.
to be beyond reach. Nevertheless, Sen’s theory has the advantage of

considering the diverse talents of individuals. Certain individuals
have more difficulty in transforming an allocation of primary goods
into welfare, particularly if, for example, they are handicapped or have
an inferior social status. Thus the logic is to give more income to dis-
abled individuals, since their other functionings are reduced by their
handicap.
Sen’s approach has considerable real-world consequences. For
example, it suggests that the usual measures of the incidence of poverty
far underestimate it, since they do not account for the inadequacy
of incomes or for other handicaps (e.g., health-related) as being
much more widespread among the poor than among the rich. Inci-
dentally, the United Nations was largely influenced by Sen’s work in
defining their human development indicator, which weighs the statis-
tics of wealth, but also literacy and public health, to rank countries in
a way that differs appreciably from the usual ranking of GDP per
capita.
2.7.4 Nozick’s Historical Approach
The theories of Bentham, Rawls, and Sen adopt nonhistorical

approaches: an allocation is judged only by its characteristics at the
present time. Nozick (1974), inspired by Locke, took the opposite view.
He began by justifying the existence of a minimal state, burdened only
with defense, police, and justice departments, and showed that in the
absence of these services, citizens will refuse to contribute voluntarily
to institutions of mutual protection.26 Next he affirms that any exten-
sion of the state’s powers beyond the minimal state would be unjust.
To demonstrate this, Nozick proposed that an allocation is just if and
only if
• the original appropriation of goods respected justice (“justice in
appropriation”)
• subsequent transfers are likewise conducted according to justice
(“justice in transfers”)
Nozick’s theory obviously has practical value only if one precisely

defines the principles of the said justice of appropriation and of trans-
fers. Clearly, in Nozick’s mind, the free working of a market economy
26. This is the problem of free riding, which we will see again in discussing public
goods.
fills the requirement of the justice of transfers. The question of justice

in appropriation is more touchy. Since most natural resources are
rare, any appropriation of a part of those resources by an individual
encroaches on all others by definition and is therefore probably
“unjust.” Nevertheless, Nozick considers here, too, that a market
economy furnishes a good approximation of appropriation justice,
insofar as the appropriation of an unused resource by an individual
indirectly benefits others by increasing production possibilities.
The result is the libertarian philosophy according to which the
state must, above all, abstain from intervening in the economy,
especially through taxation of individuals (e.g., for redistribution
motives) beyond that which is demanded by the functioning of the
minimal state.
As brilliant as it may be, Nozick’s approach did not convince
philosophers, or even most economists. It was difficult to support the
statement that in the world in which we live, the principle of justice in
appropriation has always been respected; this simple remark consid-
erably reduced the scope of Nozick’s practical conclusions. However,
we should note that Nozick has widely renounced his former position
in recent writings. He now acknowledges a role for government in the
economy as a “solemn expression of the values of human solidarity”
(see Nozick 1989, ch. 25), though such concerns came under private
charity in his earlier book.
2.7.5 Conclusion
The recent philosophical debates on the theory of justice have only

barely touched most economists, many of whom still hold a very clas-
sical conception of utilitarism. Still, Rawls’s thinking offers an inter-
esting paradigm from which there can be extracted conclusions on
economic policies. This is not an easy task, since there are obvious dif-
ficulties in tracing a common boundary between morally arbitrary
circumstances and the agents’ free will. At the beginning of the third
millennium, we can recall from the Bible the famous parable of talents
(Matt. 25). The master, leaving on a voyage, entrusts three of his ser-
vants with money in direct proportion to their abilities (a none too
Rawlsian act, since such an allocation would tend to accentuate innate
advantages). Upon his return, he rewards those who show a profit from
the money left them but has only these words for the servant who saw
fit to bury his portion:
And cast the worthless servant into the outer darkness; there men will weep
and gnash their teeth. (Matt. 25:30)
Is this conclusion fair? If the “bad servant” did not make a profit from
the money entrusted to him, it is perhaps because he prioritized en-
joyment over altruism. On the other hand, it could be that he was
born lazy (for which he is not responsible) or that his parents did not
instill in him the spirit of initiative or—as the Gospel seems to
suggest—the taste for risk (which is not his fault either). To choose one
of these explanations, we have to clarify the question of free will, and
this has confounded philosophers and theologians alike for time
immemorial.
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3 Cost-Benefit Analysis
How does one evaluate the social value of a public project? Is it

necessary to build a new railroad line in Florida? Must one prohibit
the use of asbestos in construction projects? For an economist, all of
these questions relate to cost-benefit analysis. For that matter, jurispru-
dence emphasizes that every public decision must be justified by
arguments that show the benefits to exceed the costs. In particular, this
has been the case in the United States since the Reagan administra-
tion, even if the courts also base their decisions on extra-economic
arguments.
I will be content here in summarizing the problems raised by cost-
benefit analysis. I refer interested readers to Layard and Glaister (1994)
and to their references.
3.1 Measures of Welfare
In economics, and especially in this book, there is often a need for cal-
culating welfare variations among agents when prices are changed. For
firms this poses no real difficulty: If p(p) is a company’s profit when
the prices are p, then
p (p) = max p ◊ y
y ŒY
So it could be said, without ambiguity,1 that the welfare goes from p(p)
to p(p¢) when the prices vary from p to p¢.
Things are not so simple for consumers. It might seem natural to say
that the welfare of a consumer is given by his indirect utility, as
1. This is the case even if, strictly speaking, only the company’s shareholders have a
welfare, and so on.
Ïmax U (x)
Ô
V (p , R) = Ìx Œ X
Ôp ◊ x £ R
Ó
However, such a gauge of welfare depends on the choice of the utility
function U and so is not satisfactory. For our purposes a more stable
measure must be found that can be expressed, for example, in terms of
equivalent income. To this end, in the 1940s, Hicks introduced two
measures:2
• The equivalent variation of income E, which is the sum that must be
given to the consumer in the initial state in order for him or her to have
the same utility in the final state; that is,
V (p ¢ , R) = V (p , R + E)
• The compensating variation of income C, which is the sum that must
be deducted from the consumer in the final state in order for him or
her to have the same utility as in the initial state; that is,
V (p ¢ , R - C) = V (p , R)
First we note that these two measures are quite independent of the
choice of the utility function U. If the latter submits to an increasing
transformation, the indirect utility function V will be transformed in
the same way, since
V (p , R) = U [x(p , R)]
where x(p, R) is the Marshallian demand function. The variations E and

C therefore will be unaffected.
We note also that C and E are expressed very easily in terms of the
expenditure function
Ïmin p ◊ x
Ô
e(p , u) = Ìx Œ X
ÔU (x) ≥ u
Ó
since the definitions become
2. The reader should know that certain authors use different definitions from those that
I use. Sometimes E is defined as -E and C as -C, and I use Hicks’s notation.
Cost-Benefit Analysis 39
ÏC = R - e(p ¢ , V (p , R))
Ì
ÓE = e(p , V (p ¢ , R)) - R
Now, suppose that p¢ is very close to p so that p¢ = p + dp. Then E will
also be infinitesimal and to the first-order,
∂V ∂V
Â ∂ p dpi = ∂ R E
i i
Or, using Roy’s identity,
E = -Â xi dpi
i
The reader will verify that we get exactly the same formula for C.
Compensatory and equivalent variations coincide then to the first-
order, and can then be calculated on the basis of demand and price
variations.
This tidy property is of course no longer true for finite price varia-
tions. Yet there is a case where E and C coincide. Suppose there exists
a good (good 0) such that the marginal rate of substitution between any
two goods i and j does not depend on the consumption of this good.
Then there is a representation of the utility function such that the mar-
ginal utility of the good 0 is constant, and after normalization we can
write
U (x0 , x1 , ... , x L ) = v(x1 , ... , x L ) + x0
Let X be the vector (x1, . . . , xL), make p the corresponding price vector,
and normalize the price of the good 0 to 1.3 The maximization of utility
under budgetary constraints then amounts to
max[v(X ) + R - p◊X ]
X
from which v¢[X(p)] = p. Demands in goods 1, . . . , L are therefore

independent of consumer wealth. There is no wealth effect, and the
indirect utility is
V (p , R) = v[X (p)] + R - p ◊ X (p)
Because the utility is linear in R, the calculation of variations C and E

is extremely simple. Let us set
3. The good 0 is often called a numéraire, but one must avoid identifying it as money,
which in principle does not enter into the utility function.
S(p) = v[X (p)] - p ◊ X (p)
Then, by the definition of demands X(p), we have

S ¢ ( p) = - X ( p)
Immediately this obtains

p
C = E = S(p ¢) - S(p) = Úp¢ X (t) ◊ dt
So, in the presence of constant marginal utility, variations C and E still

coincide. Moreover C and E are easily calculated based on demand
functions.4
The quantity S(p) is called a Marshallian consumer surplus, even
though it was actually introduced by Jules Dupuit as early as 1844.
It has a very simple graphical representation when L = 1. One then
has to the nearest constant (making a change of variables and ignoring
integral convergence problems)
• X ( p)
S(p) = Úp X (t)dt = Ú0 P(x)dx - pX (p)
where P(x) = v¢(x) is the inverse demand function, so that S(p) is the
hachured area in the usual graph of demand (figure 3.1).
The hypothesis of the absence of wealth effect is of course very
strong: There are very few goods for which the demand is not sensi-
tive to income. What can be said in general? Suppose that p¢ differs
from p only by a decrease in price pi, which becomes p¢i < pi. Then a
Marshallian surplus variation can still be calculated:
pi
DS = Úp¢ xi (t , R)dt
i
It can be shown that if good i is normal (so that demand increases with
income), then we have the inequality
C £ DS £ E
4. The mathematician will have noticed that none of the above is defined independently
of the path of integration unless X(t) · dt is an exact differential form. For that, it is
necessary and sufficient that for every i and j,
∂ Xi ∂ X j
=
∂pj ∂ pi
However, this situation obtains in the absence of the wealth effect, since these deriva-
tives then form the Slutsky matrix, which is symmetrical.
S(p)
P(x)
X(p) x
Figure 3.1
Consumer suplus
To see this, recall that the derivative of the expenditure function e(p, u)
with respect to prices is the compensated demand Xc(p, u). Since R =
e(p, V(p, R)) = e(p¢, V(p¢, R)), one can then write
Ï C = Ú pi xic [t , V (p , R)]dt
Ô pi¢
Ì pi
ÔÓE = Úp¢ xic [t , V (p ¢ , R)]dt
i
Because good i is assumed to be normal, xi(p, R) increases in R. By the

identity x ci(p, u) = xi(p, e(p, u)), compensated demand therefore increases
in u. Now, since p¢i < pi and V decreases in p, one has
xic [t , V (p , R)] £ xi (t , R) = xic [t , V (t , R)] £ xic [t , V (p ¢ , R)]

for every p¢i £ t £ pi. This indeed produces (by integration) the sought-
after inequality.
The compensatory and equivalent variations thus bracket the
surplus variation. Moreover E and C converge toward DS when the
wealth effects become negligible.5
5. More precise inequalities can be obtained in terms of the income elasticity of demand;
see Willig (1976).
The use of consumer surplus therefore rests on the hope that wealth
effects can be considered small. A simple argument (which can be made
more rigorous) in fact shows that this is the case if the good in ques-
tion represents only a small portion of consumer spending.6 To see this,
we recall Slutsky’s equation
∂ xi ∂ xi ∂ xi
= - xj
∂pj ∂pj U
∂R
Moving on to elasticities, we get
∂ log xi ∂ log xi p j x j ∂ log xi

= -
∂ log p j ∂ log p j U
R ∂ log R
which well illustrates that the wealth effect’s contribution to demand

elasticity becomes negligible if the good whose price varies possesses
a small budgetary share (pjxj R).
In most of our applications, this condition will be approximately
verified. Therefore we will be able to use consumer surplus. To obtain
the social surplus, the common practice consists in adding the sur-
pluses of different consumers and of firms, possibly weighted to take
redistributive objectives into account. Chapter 2, however, shows
that this assumes certain ethical judgments that are not universally
accepted.
3.2 First-Best
Throughout this chapter we suppose that the problem of aggregation

of preferences has been settled: There exists a Bergson-Samuelson func-
tional W(u1, . . . , un) whose maximization gives the social optima.7
Furthermore in this section we make two assumptions:
•Markets are complete. There is one market for every good, on every
possible day, and in every possible state of the world.
• There is no market distortion or failure. No public good, no external
effect, no taxes on either income or transactions, all redistribution pro-
ceeding from lump-sum transfers.
6. Also see the discussion of surplus in the introduction of Tirole’s IO book.

7. Otherwise, it would of course be impossible to carry out a cost-benefit analysis, except
in the extremely rare cases where all agents agree on a public decision.
These obviously unrealistic hypotheses define a first-best problem.

We will show that under these conditions, cost-benefit analysis is an
extremely simple principle, for it is based on market prices.
To begin, consider a public project that amounts to modifying
resources w of the economy by a small vector dw. Certain components
of dw are positive, and others negative, because some resources are
used to produce new ones.
Should this project be carried out? In answer to this question, con-
sider the program defining social optima:
Ïmax x1 , ... , xn W [U 1 (x 1 ), ... , U n (x n )]

Ô n
Ì
ÔÓ Â x i £w
i =1
Let l be the vector of multipliers associated with the scarcity con-

straints. The first-order conditions give
∂W ∂U i
=l
∂U i ∂ x i
But under our hypotheses, the fundamental welfare theorems apply:
The optimum is an equilibrium, and the vector l is then proportional
to the vector of equilibrium prices p. It follows that we get, to the first-
order and up to a multiplying constant,
dW = p · dw
The public project therefore must be evaluated by using market prices

to weigh the transformation of resources that it entails, or rather,
by simply calculating the increase in national income8 due to the
project.
For noninfinitesimal projects, the calculation is more complicated
because market prices depend on economic resources, which are in
turn affected by the project. Still, it is simple to show that a project that
lowers national income evaluated at initial market prices cannot be
desirable in the sense of the Hicks-Kaldor compensation principle (see
chapter 2).
Suppose that the initial situation is characterized by resources w1,
prices p1, and equilibrium consumptions x1i (with Sni=1x1i = w1) and that
8. This “national income” measure must then (contrary to GDP) take into account the
destruction of natural resources, for instance.
the final resources are w2, with p1 · w2 < p1 · w1. Then let (x2i ) be any final
resource allocation. We get
n n
p1 ◊ Â x 2i < p1 ◊ Â x1i
i =1 i =1
Therefore a consumer i exists such that p1 · x2i < p1 · x1i . Since x1i was the
preferred consumption of i at prices p1, we can deduce that Ui(x2i ) <
Ui(x1i ). For every allocation of w2, there is at least one consumer whose
utility was reduced by the public project. So the Hicks-Kaldor criterion
tells us that the project should not be carried out.
3.3 Second-Best
Let us consider a more realistic situation. Now the economy is affected

by distortions. The fiscal system affects agents’ choices, markets are
incomplete, and so on. This situation is said to be second-best.
3.3.1 Shadow Prices
The immediate consequence of this change in perspective is that we

can no longer identify l and p as we did in the preceding section. If
there are distortions, the market price no longer coincides with the
scarcity constraint multiplier. If the markets are incomplete, the situa-
tion becomes even worse. By definition, for certain goods there is no
longer a market price. So we must calculate the shadow prices, l and
evaluate l · dw in order to assess the social value of an infinitesimal
project.
At times it is possible to have an idea of the direction in which
l departs from p. For example, how can the social cost of a wage-
earner’s employment in a project be evaluated? Salaries paid an
employee simply correspond to transfers between him and taxpayers,
so salaries do not then enter social welfare (expect for redistributive
considerations). For this reason the social cost of the employment
is simply the value of the leisure sacrificed by the employee in order
to work. In the first-best, this value coincides with his salary. In the
second-best, such as if the economy is in involuntary unemployment,
the value of leisure becomes inferior to that of the salary. The shadow
price of labor to be used in case of unemployment is then inferior to
the market salary.
3.3.2 Nonmarket Goods
Certain goods are difficult to assess for lack of a specific market. This
is the case, for example, with human life, transportation time, and
numerous natural resources (rare animal species, air quality, etc.).
Valuing transportation time is crucial to determining whether, for
instance, a new railroad line should be built. Under fairly restrictive
hypotheses (if labor supply is completely unconstrained), one hour of
leisure is worth one hour of salary. Transportation time, at least under
these conditions, can then be evaluated by “unearned” salaries.
Noneconomists are often horrified at the explanation that the
value of life is not infinite.9 They generally accept the cost-effectiveness
studies that calculate by lives saved the cost of diverse public health
regulations or measures and that show that some measures are
much more effective than others, but they refuse to go so far as to
attribute a price to human life. Still, each of us takes risks every day,
just by crossing the street. Would we do so if there were an infinite price
assigned to our life? Moreover it is readily observed that certain cate-
gories of wage-earners choose between high-paying but dangerous
professions and those that incur less risk and are also less well-paying.
This observation seems to suggest that we do indeed attach a finite
price to life.
How can this price be measured? Two main methods are used.
• The method of implicit valuations. This method is founded on the
theory of equalizing differences, whose principle dates back to Adam
Smith. It consists in observing trade-offs on the labor market in terms
of the salaries and risks incurred, and in deducing the cost assigned to
various risks.
• The method of contingent valuations. This method is very much
used, in particular, in the context of renewable resources. It relies on a
questionnaire whose subjects are asked to indicate the price they would
be willing to pay to have one of the risks that affects them decrease by
a given percentage.
9. It must be said in their defense that certain economists, and major ones at that, do not
go in for half measures. A dozen years ago, Larry Summers had signed a World Bank
memo that was destined to remain confidential but became famous. In it was argued that
sending waste from industrialized countries to underdeveloped ones was desirable
because these countries are much poorer, people’s life spans are shorter, and therefore
life there has a lesser value. This reasoning is of course quite disputable (e.g., see the
analysis of Hausman-McPherson 1996, ch. 2).
One remarkable result of these empirical studies is that these two

methods give comparable orders of magnitude: an average subject
(neither rich nor poor, neither young nor old) appears to assign to his
life a value on the order of two million dollars. This figure is obviously
quite fragile. Nonetheless, it suggests that a regulation imposing a cost
of 50,000 dollars per life saved should be adopted, while it is advisable
to hesitate before a regulation that saves lives at a uniform cost of 100
million dollars. Given the enormous dispersion recorded in costs per
life saved (e.g., see tables 1 and 2 in Viscusi 1996), this sort of indica-
tion is not as useless as one might, at first, think.
3.3.3 Incomplete Markets
In the real world many markets do not exist. This is especially the case
with markets for future goods or for state-contingent goods. Insofar as
most public projects have uncertain returns that lie in the future, the
corresponding shadow prices must be determined.
The interest rate for long-term government bonds is often used to
discount future returns on a public project, the rationale being that this
interest rate measures the cost of funds for the state. But the ideal rate
should reflect the individuals’ rates of preference for the present; then
the rate would depend on the nature of fiscal distortions or of capital
markets distortions. Unfortunately, as is often the case in second-best
theory, a general rule cannot be applied.
The situation is slightly more satisfying in regard to uncertainty. The
presence of uncertainty should add to the discount rate a term that
measures the risk premium. In fact Arrow-Lind (1970) showed that this
risk premium can be neglected when the public project concerns a very
large number of agents.
Bibliography
Arrow, K., and R. Lind. 1970. Uncertainty and the evaluation of public investment
decisions. American Economic Review 60: 364–78.
Dupuit, J. 1844. De la measure de l’utilité des travaux publics. Annales des Ponts et
Chaussées 8: 332–75. Published in English in P. Jackson, ed. 1996. The Foundations of Public
Finance. Cheltenham, England: Elgar.
Hausman, D., and M. McPherson. 1996. Economic Analysis and Moral Philosophy.
Cambridge: Cambridge University Press.
Layard, R., and S. Glaister, eds. 1994. Cost-Benefit Analysis. Cambridge: Cambridge
University Press.
Viscusi, W. K. 1996. Economic foundations of the current regulatory effort. Journal of

Economic Perspectives 10: 119–34.
Willig, R. 1976. Consumer’s surplus without apology. American Economic Review 66:
589–97.
4 Implementation
The results obtained in chapter 2, though negative, still allow for the
possibility that at least in certain cases, one can effectively aggregate
preferences in a satisfactory manner. As Arrow (1951) wrote,1 it none-
theless remains to collect the necessary information:
Even when it is possible to aggregate individual preferences into a coherent

model of collective ones, one still must define the rules such that individuals
actually express their preferences, even when they are reacting rationally.
So it is to the problem of implementation that we now turn. Our basic

hypothesis, as before, is that agents only act in their own interest. They
do not reveal information unless it benefits them to do so. For that they
must be offered proper incentives.2
We return to the general framework where the set of possible choices
is A and each agent i = 1, . . . , n has a utility function Ui which is known
a priori to belong to a set Ui. We will denote U = Pi=1 n
Ui. We are inter-
ested in the implementation of a given social choice function (SCF)3 that
associates to any profile of utility functions U = (U1, . . . , Un) a choice a
Œ A according to a = f(U). Suppose that there exists a “Center” charged
with putting a certain social choice function into practice. In the
absence of information on the agents’ preferences a priori, the Center
can only ask them various questions after having announced to them
how the information collected will be used. The most general approach
consists therefore of searching for a mechanism, which comprises spaces
1. Even though Arrow had posed the problem in the 1950s, it was not until the 1970s
that it received any detailed attention. Hurwicz (1986) provides the many references.
2. We take a view opposite from that of Borda. When someone showed Borda that the
method bearing his name could be manipulated, he was offended this response was: “Sir,
my method addresses only men of honor!”
3. The SCF could consist in a selection of a set of Pareto optima.
of messages M1, . . . , Mn and a rule of the game g which is a function

n
of M = Pi=1 Mi in A. Each agent i must select a message mi Œ Mi. It is
understood that if the agents choose m = (m1, . . . , mn), then the choice
put to use will be g(m) Œ A. This defines the game whose set of equi-
libria will be denoted h(U) Œ M (h is a correspondence a priori). Since
our goal is to implement the SCF f, we will define the following:
• Complete implementation. When g h = f, then all equilibria of the
game lead to the SCF.
• Weak implementation. When g h … f, there are then only certain
equilibria of the game that lead to the desired SCF.
I have voluntarily remained vague on the concept of equilibrium.

Equilibrium is retained for the mechanism-defined game where, as
we will see, the results depend heavily on the chosen equilibrium
concept.
4.1 Dominant Strategy Equilibrium
The strongest concept of equilibrium demands that the strategy (or

rather, the message) chosen by agent i be optimal whatever the other
agents do. Thus an equilibrium in dominant strategies is a n-tuple of
messages m* such that
[( )]
"i = 1, ... , "m Œ M , U i g mi* , m- i ≥ U i [ g(m)]
where I have used the usual notation

m- i = (m1 , ... , mi -1 , mi +1 , ... , mn )
If such an equilibrium exists, it is of course quite satisfactory, since then

every agent can judge it optimal to conform to his/her equilibrium
strategy without even having to wonder which strategies the other will
choose. Moreover, the revelation principle (see Myerson 1979), which
we will discuss next, shows that it is useless to seek complicated mech-
anisms; it is better to limit oneself to direct revealing mechanisms. A
direct mechanism is such that every agent directly announces a utility
function:4 "i = 1, . . . , n, Mi = Ui. It is revealing if stating the truth is
optimal for all agents.
4. Utility functions are generally supposed to be known up to a parameter qi; a direct

mechanism then corresponds to each agent announcing his qi.
Implementation 51
theorem 3.1 (revelation principle) If a mechanism (M, g) weakly

implements the SCF f in dominant strategies, then the direct revealing
mechanism (U, f ) also implements f weakly in dominant strategies.
Proof Let hS be a selection of the set of equilibria of the mechanism

(M, g) such that g hS = f (hS exists since (M, g) weakly implements f ).
In dominant strategies, the equilibrium strategies that implement f
weakly are therefore given by m*i = hSi (Ui), since the strategy of agent i
cannot depend on that of the other agents. We can proceed ad absur-
dum. If (U, f ) does not weakly implement f, then there exists a utility
profile U¢, an agent i and his utility function Ui such that
U i [ f (U i¢, U -¢ i )] > U i [ f (U i , U -¢ i )]
But since f = g hS, we immediately get
U i {g[ hSi (U i¢), hS-i (U -¢ i )]} > U i {g[ hSi (U i ), hS-i (U -¢ i )]}
which contradicts the hypothesis that (M, g) weakly implements f,

since when the true profile is (Ui, U¢-i), it is in the agent’s i interest to
lie by announcing hSi (U¢i ) rather than hSi (Ui).
The revelation principle in fact demonstrates that it is useless to ask
agents for more than their own information. Unfortunately, we will
expose a very negative result, the Gibbard-Satterthwaite theorem. This
theorem shows that the SCFs that are implementable in dominant
strategies are too rare to really be useful when the possible utility func-
tions are not restricted.
theorem 3.2 (gibbard 1973; satterthwaite 1975) if

• the SCF f is weakly implementable in dominant strategies
• f(U) has at least three elements
• U is of universal domain
then f is dictatorial.
Proof See appendix A.
As the formulation of theorem 3.2 suggests, the result is very close in

spirit to Arrow’s theorem (which the proof given in appendix A actu-
ally uses). It would be possible then to solve the paradox using the path
which we saw in chapter 1 was the most promising, and indeed weak-
ening the universal domain hypothesis. Unfortunately, this is not the
way to succeed in implementing the correspondence of Walrasian equi-

libria in an exchange economy, even if it is supposed that preferences
are convex. But there are two results that are more positive. The first
concerns situations where the choices can be arranged on an axis and
where the preferences are unimodal. Then the SCF that selects the pre-
ferred choice of the median voter, for example, is implementable. The
second positive result refers to quasi-linear preferences. These are
written Ui(x, t) = u(x, qi) + t, where x is an allocation, qi a known para-
meter of the agent i alone, and t a monetary transfer. If the preferences
are of this type, then optimal decisions can be implemented by a mech-
anism known as Vickrey-Clarke-Groves, which will be explained in
chapter 5.
The second way to avoid negative conclusions to the Gibbard-
Satterthwaite theorem is to employ a less demanding concept of
equilibrium; that is what we will do now.
4.2 Nash Equilibrium
Recall that m* is a Nash equilibrium if and only if every agent’s equi-

librium strategy is his best response to the other agents’ equilibrium
strategies:
"i = 1, ... , n, "mi Œ Mi , U i [ g(m *)] ≥ U i [ g(mi , m-*i )]
The Nash equilibrium is the basic concept of the theory of nonco-

operative games. It is habitually justified in noting that no agent is
interested in deviating from equilibrium if he anticipates that the other
agents are conserving their equilibrium strategies. This justification
is not convincing here unless we assume that every agent knows the
preferences of the other agents. One could imagine an iterative process
where every agent repeatedly takes the strategies of the others at iter-
ation t as fixed when he decides his strategy at iteration (t + 1), but this
argument merits a more rigorous modelization. It is also flawed in
its assumption of myopic behavior on the part of the other agents. The
implementation of an SCF through Nash equilibrium must therefore be
handled carefully. Here we will suppose that our use of the concept is
justified by context.
It is easy to show that if one is limited to direct mechanisms, nothing
is gained with respect to dominant strategy equilibrium:
Implementation 53
theorem 3.3 If f is implementable in Nash equilibrium by a direct

mechanism, then it is equally implementable in dominant strategy
equilibrium.
Proof It is enough to write the definitions for each utility profile. Weak
implementation in Nash equilibrium by a direct mechanism requires
that for every profile U, every agent i, and every U¢i,
U i [ f (U i , U - i )] ≥ U i [ f (U i¢, U - i )]
But, because this inequality holds for every U-i, it implies that f
is weakly implementable by a direct mechanism in dominant
strategies.
To obtain more satisfactory results than with dominant strategy equi-

librium, it is necessary for us to resort to spaces of messages Mi “larger”
than Ui. In fact one can demonstrate a revelation principle that stipu-
lates that if f is weakly implementable in Nash equilibrium, then it is
also implementable by a mechanism where Mi = U. The proof is iden-
tical to that of the principle of revelation in dominant strategy equilib-
riums. In both cases it is sufficient for each agent to give out all of his
information—which here means the whole utility profile.
Unfortunately, weak implementation is not a very useful concept in
Nash equilibrium. To see this, consider any SCF f and the following
mechanism:
• "i = 1, . . . , n, Mi = U
• g(m) = f(U) if "i = 1, . . . , n, mi = U
• Or else, “everybody’s dead”5
The idea of this mechanism is that if the agents succeed in coordinat-

ing themselves on a statement of the profile of the utilities, the corre-
sponding social choice is put to use. In the opposite situation, all of the
agents are severely punished. It is easy to see that the statement of the
true utility profile is a Nash equilibrium of this game, so this mecha-
nism weakly implements f. But many other Nash equilibria exist: all
those where the agents are coordinated on the same lie. Clearly, it is a
very weak form implementation, where g h contains f but also a whole
n-dimensional continuum of other equilibria.
5. A dissuasive fine is inflicted on all agents.

Therefore we had better study complete implementation. As in all of

this literature, we will henceforth assume that f is a correspondence. In
this regard we must slightly extend the revelation principle, which
stipulates (henceforth) that it suffices to limit oneself to mechanisms
where every Mi is U ¥ A. Maskin (1977, only published in 1999) used
this type of mechanism to completely characterize the implementable
SCF’s in Nash equilibrium.6 We begin by introducing two conditions:
• f is monotone if and only if
if there is a profile U and a Œ f(U), and a new profile U¢ such that
"i = 1, ... , n, "b Œ A , [U i (a) ≥ U i (b) fi U i¢(a) ≥ U i¢(b)]
then a Œ f(U¢).
• f verifies NVP (no veto power) if and only if
"a Œ A , "i = 1, ... , n, ["j π i , "b Œ A, U j (a) ≥ U j (b) fi a Œ f (U )]

These conditions are less complicated than they may appear. On the
whole the condition of monotonicity states that if one passes from one
utility profile under which a was socially chosen to another profile
where a has not gone down in the preferences of any agent, then a will
continue to be chosen. This is easily seen by defining the set of choices
to which i prefers a by
Li (a, U i ) = {b ŒA U i (a) ≥ U i (b)}

Then the premise of the condition of monotonicity expresses that for
every i, Li(a, U¢i ) contains Li(a, Ui). As for the NVP condition, it simply
states that if a is the preferred choice of all the agents except one, then
a will be chosen.
Maskin’s theorem gives a necessary condition and a set of sufficient
conditions for complete implementation in Nash equilibrium:
theorem 3.4 (maskin)

• If f is completely implementable, then f is monotone
• If f is monotone, verifies NVP, and if n ≥ 3, then f is completely
implementable
6. For technical reasons, often mechanisms are used that also comprise the statement of
an integer (see appendix B).
Implementation 55
Demonstration It is easy to see that the monotonicity of f is a neces-

sary condition. First, assume that f is not monotone; then there exist
two utility profiles U and U¢ and a choice a such that
• a Œ f(U)
• "i = 1, . . . , n, Li(a, U¢i ) … Li(a, Ui)
• a œ f(U¢)
If f is completely implementable, there exist a mechanism (M, g) and a

Nash equilibrium m* of that mechanism for the profile U such that
g(m*) = a. One deduces from this that for every i and m¢i,
[(
U i (a) ≥ U i g mi¢ , m*- i )]
But since Li(a, U¢i ) … Li(a, Ui), one gets for every i and m¢i,
[(
U i¢(a) ≥ U i¢ g mi¢ , m*- i )]
which shows that a = g(m*) remains a Nash equilibrium for U¢, and
therefore contradicts the hypothesis that a œ f(U¢).
The construction of the mechanism that implements f when the latter

verifies the sufficient conditions is a bit more complex; it is given in
appendix B.
Note that when there are two agents, the implementation is more dif-
ficult. The mechanism found by Maskin is based on the detection of an
agent who makes a false statement upon the confrontation of all state-
ments. When there are conflicting statements from only two agents,
it is obviously considerably more difficult to identify the agent who
cheats.7 In other respects the NVP condition is very strong with two
agents. It is easy to see that when there are three or more agents, the
NVP condition is rather weak, quite simply because it is rare that two
agents’ preferred choices coincide.8
The opposite can be observed in the condition of monotonicity. Take
the example of the construction and financing of a bridge. The utility
profile is summed up in a n-tuple of provisions to pay for the bridge.
We start from a profile such that the bridge is constructed and financed
according to some nonrandom apportioning of costs, and suppose that
7. Moore and Repullo (1990) study the case of n = 2; they also give a necessary and suf-
ficient (albeit a bit complicated) condition when n ≥ 3.
8. If, as in the example of an exchange economy of private goods, each agent covets the
economy’s resources, then the NVP condition is satisfied rather trivially.
the propensity to pay of agent i increases. Then monotonicity imposes

that the bridge construction continue (fortunately!) but above all that
the division of costs remain the same—which doesn’t seem very rea-
sonable. This difficulty is present in all problems involving a redis-
tributive element.
Moore’s (1992) excellent review provides another example.9 It is the
biblical case where monotonicity is violated by Solomon’s famous
judgment (I Kings 3:16–28). Solomon’s solution falls under Nash equi-
librium implementation, since each woman knows who is the real
mother. His solution, which consisted in threatening to cut the baby in
two, is not entirely foolproof: What would he have done if the impos-
tor had had the presence of mind to scream like a real mother? We will
attempt to reexamine Solomon’s dilemma via our general framework.
Two women quarrel over the baby: Anne and Beatrice. There are two
states: state a, where Anne is the real mother, and state b, where it is
Beatrice. Solomon has three possibilities: give the baby to Anne (a), give
it to Beatrice (b), or kill it (t). He would of course like to implement the
SCF f which gives the baby to its real mother: f(a) = a and f(b) = b. The
following are the individual preferences that correspond to the biblical
parable:
• For Anne,
in state a: a b t
in state b: a t b
• For Beatrice,
in state a: b t a
in state b: b a t
Each woman wishes, above all, to get the baby. If she is the mother, she
prefers to give the baby away rather than see it die. In the opposite
case, she would prefer the baby’s death to its being given to her rival.
Note that f(a) = a and that a is better placed for Anne and Beatrice
under b than under a. According to the condition of monotonicity,
one should therefore get f(b) = a, which is obviously not what we want.
The SCF then is not monotone, and by the necessary condition of
the Maskin theorem, it is not implementable in Nash equilibrium.
Solomon’s judgment is a more difficult problem than it appears in
reading the Bible.
9. Adapted from Glazer and Ma (1989).

Implementation 57
If the reader finds this example to be a bit far-fetched, here is another,

borrowed from Moulin (1988, ch. 9). Consider the method of the two-
round election (which well defines an SCF f when the agents vote sin-
cerely), such as is used for presidential elections in France, for example.
In such a vote, the two best-placed candidates after the first ballot face
each other in a second round. Suppose that there are three candidates
a, b, and c and 17 electors. The preferences U are
• a > b > c for 6 electors
• c > a > b for 5 electors
• b > c > a for 4 electors
• b > a > c for 2 electors
With these preferences, a, b, and c receive, respectively, 6, 6, and 5 votes

in the first round, which eliminates c. In the second round, a is elected
by 11 votes against 6 for b. In our notation then we have a Œ f(U). Now
change the preferences of the last group of two electors to a > b > c. This
causes a to come back up in the new preferences U¢, and a should then
always be elected if f is monotone. However, this is not the case; now
a wins 8 votes, b 4 votes, and c 5 votes in the first round, which elimi-
nates b; and c is elected over a by 9 votes against 8 in the second round,
even though the relative majority of a in the first round increased. Inci-
dentally, this enables us to illustrate the possibilities of manipulation
in such an election method: If the preferences are given by U and if the
voters of the first two groups and of the last group vote sincerely, the
four electors of the third group (whose preferred candidate b does not
survive the first ballot if they vote sincerely) have every interest in
voting for c in the first round, which permits them to avoid the elec-
tion of a, to whom they prefer c.
A last example is: Can the Walrasian equilibrium be implemented in
Nash equilibrium? It can be shown that in an exchange economy with
convex preferences, every completely implementable SCF must contain
the correspondence of Walrasian equilibria. Can this be taken further,
in other words, to implement this correspondence and nothing but?
Without going into details, the response is globally positive.
4.3 Refinements of the Nash Equilibrium
If the situation is therefore more encouraging in Nash equilibrium than

in dominant strategy equilibrium, some negative results remain. To
overcome them, recent research (see Moore 1992 and Palfrey 1998) has
shown interest in refining the Nash equilibrium concept, in particular,
using subgame perfect equilibrium.10 Moore and Repullo (1988) show
that the condition of monotonicity necessary (and nearly sufficient) for
subgame perfect implementation is much weaker than for implemen-
tation in Nash equilibrium, to the point that in economic environments,
“almost anything can be implemented”.11 One gets a comparable result
by using an equilibrium concept that eliminates weakly dominated
strategies (Palfrey and Srivastava 1991).12
One problem with the mechanisms used is that they are sometimes
based on methods that do not seem very realistic (like stating an
integer). Still one can often exploit the particular characteristics of an
implementation problem to construct simpler mechanisms (e.g., see
Jackson, Palfrey, and Srivastava 1994). Virtual implementation, which
is content to implement approximately a social choice function, also
seems to be a path of fruitful research (see Abreu and Sen 1991. The
underlying idea is that if f is not monotone, quite often there exists a
lottery f˜ “close” to f which is monotone.13
Another drawback of certain mechanisms is that they do not resist
renegotiation. During implementation of the mechanism, it is possible
that the agents find themselves in a situation where they would prefer,
by mutual agreement, to modify the mechanism so as to implement
an allocation preferred by all. In such situations it is difficult to see
what can keep agents from renegotiating. If they do renegotiate, the
incentive properties of the mechanism will be modified. Maskin and
Moore (1999) characterize the SCFs that can be implemented by a
renegotiation-proof mechanism.
4.4 Bayesian Equilibrium
The most simple justification in considering implementation in Nash

equilibrium concerns situations where every agent knows the identity
of all other agents. Without going that far, it is reasonable to assume
10. In a game with several stages, a Nash equilibrium is subgame perfect if it induces a
Nash equilibrium in each subgame, even out of the equilibrium path. This equilibrium
concept permits us to eliminate noncredible threats and therefore reduce the multi-
plicity of equilibria.
11. This still does not solve the problem of Solomon’s judgment.
12. Now the judgment of Solomon is implementable.
13. It is nevertheless possible that f˜ puts weight on suboptimal choices.
Implementation 59
that each agent possesses probabilist “beliefs” on the utilities of the

others. To simplify the notation, we assume that the utility function of
an agent is known up to one parameter: The utility of agent i is thus
Ui(a, qi). All agents (and the government) know that the vector of the
“types,” q = (q1, . . . , qn), is distributed according to a q(q) a priori on a
set Q, which I will assume to be finite. Every agent knows his type qi
and can therefore compute the conditional distribution of the types of
the other agents:
q(q i , q - i )
q(q - i q i ) =
Â q-¢ i q(q i , q -¢ i )
As is usual, a mechanism is a pair (M, g). An equilibrium in Bayesian
strategies will be a n-tuple of strategies s*i (qi) such that for all i, qi,
and mi,
Â q(q -i qi )U i {g[s i*(qi ), s -*i (q -i )], qi } ≥ Â q(q -i qi )U i {g[mi , s -*i (q -i )], qi }

q-i q-i
It is therefore, so to speak, a Nash equilibrium “in expectation.”

It is easily shown that the principle of revelation applies here
again: Anything that is implementable is implementable in revealing
direct mechanisms, where each agent truthfully announces his type in
equilibrium. (Interested readers can refer to Palfrey 1992 or Palfrey-
Srivastava 1993 for a study of Bayesian implementation.) For imple-
mentation in Nash equilibrium, the central difficulty is to eliminate
parasite equilibria. Here again, a property of monotonicity is necessary
(and nearly sufficient) for complete implementation.
4.5 Appendix A: Proof of the Gibbard-Satterthwaite Theorem
Several proofs of the Gibbard-Satterthwaite theorem exist. The proof

presented here is due to Schmeidler-Sonnenschein (1978). It has
the advantage of bringing to light a close relationship to Arrow’s
theorem.
To simplify the exposition, we will return to the notation used in pre-
orders of preferences. Thus aPib means that i strictly prefers a to b. We
will also assume that the individual preferences are all strict, in other
words, that there is never a situation of indifference. For all i, a and b,
one gets perforce aPib or bPia (this hypothesis is hardly restrictive if A
is finite).
Recall that the revelation principle allows us to limit ourselves to

direct mechanisms (P, f). We want to prove that if such a mechanism
is revealing, then f is dictatorial. We say that f is manipulable by i in P if
and only if P¢i exists such that
f (Pi¢, P- i )Pi f (P)
put differently, it behooves i to lie. A manipulable mechanism of course

cannot be revealing.
The demonstration comprises two lemmas; it consists of starting
from an implementable SCF f (which is implementable by a direct
mechanism, therefore, and is not manipulable) and of constructing
from it an SWF F that verifies Arrow’s conditions. One concludes from
this that F is dictatorial, which implies that f also is.
lemma 1 Suppose that f(P) = a1 and f(P¢i, P-i) = a2, where a2 π a1. Then
1. f is manipulable by i in (P¢i, P-i) if a1P¢i a2

2. f is manipulable by i in P if a2Pi a1
Proof In both cases it is sufficient to write the definition of

manipulability.
We will need the notation Pij, which, for a profile P and given agents
i < j, will represent the vector (Pi, . . . , Pj).
lemma 2 Let B be a subset of the image of f and P a profile such that
"a1 Œ B, "a2 œ B, "i = 1, . . . , n, a1Pi a2
Then f(P) Œ B.
Proof This can be shown by contradiction. Let a2 = f(P), and suppose

that a2 œ B. Let P¢ be a profile such that f(P¢) = a1 Œ B (such a profile
does exist, since B is included in the image of f and given the univer-
sal domain hypothesis). Now construct a sequence (a3i )i=0,...,n by
• a03 = a2 œ B
• for i = 1, . . . , n - 1, a3i = f(P¢1i, Pni+1)
• an3 = a1 Œ B
Let j be the first integer such that a3j Œ B. We then get

• f(P¢1j, Pnj+1) = a3j Œ B
• f(P¢j-1 n j-1
1 , Pj ) = a3 œ B
Implementation 61
and by the hypothesis of the lemma, a3jPja j-1

3 . Lemma 1 then implies that
f is manipulable.
Now construct an SWF F. Let P be any profile and a1, a2 two choices
in A. Define a new profile (using UD) P̃such that for each i,
• P̃i coincides with Pi on {a1, a2}
• P̃i coincides with Pi on A - {a1, a2}
• {a1, a2} is placed at the top of the preferences P̃i
(Strictly speaking, P̃ of course depends on a1 and a2, and the notation

should reflect this.)
Lemma 2 implies that f(P̃) Œ {a1, a2} (taking B = {a1, a2} and replacing
P by P̃ in the statement of the lemma). F can therefore be defined by
a1 F(P)a2 ¤ f (P˜ ) = a1
Now we can verify Arrow’s conditions:

• There are surely at least three choices.
• F is, by construction, of universal domain.
• F satisfies the Pareto principle: if for every i, a1Pia2, then a1 is at the
top of all preferences P̃i. By taking B = {a1, a2} in the statement of lemma
2, we indeed get f(P̃) = a1.
• F satisfies IIA: if this were not the case, there would exist P, P¢, a1 and
a2 such that
for every i, a1Pia2 ⇔ a1P¢ia2

a1 F(P)a2 and a2 F(P)a1
Now define a sequence (a3i )i=0,...,n by
a03 = a1
for i = 1, ... , n - 1, a3i = f (P˜1¢ i , P˜ in+1 )

an3 = a2
Lemma 2 implies that a i3 Œ {a1, a2} for every i. Therefore let j be the
first integer such that a3j = a2. This gives f(P̃¢1j, P̃nj+1) = a2 and f(P̃¢1j -1, P̃jn) =
a1. Now one of two things can result:
a1Pj a2
This implies a1Pj¢a2 and therefore a1P̃j¢a2, so lemma 1 implies that f is

manipulable.
a2Pja1
This implies a2P̃ja1, so lemma 1 again implies that f is manipulable.

But there is contradiction in both results: for every P, F(P) is clearly
a complete and asymmetrical binary relation. What remains for us is
to verify that it is transitive.
Take the opposite case so that we have a cycle on a triplet {a1, a2, a3}.
For every i, let P¢i which coincides with Pi on {a1, a2, a3} and on A -
{a1, a2, a3} be such that {a1, a2, a3} is at the top of P¢i (using UD). Lemma
2 implies that f(P¢) Œ {a1, a2, a3}; without any loss of generality, we
can assume that f(P¢) = a1. Since F(P) has a cycle on {a1, a2, a3}, we
necessarily get a2F(P)a1 or a3F(P)a1. Here again, without loss of gener-
ality, we can assume that a3F(P)a1. Now modify P¢ in P≤ by making
a2 move into third place in each individual preference (P≤ is admissi-
ble by UD). Note that a3Pia1 if and only if a3P≤i a1; in applying IIA (which
we have just shown is satisfied), we get a3F(P≤)a1, which again implies
a3 = f(P≤).
At the risk of seeming redundant, we now define a sequence (a4i )i=0,...,n
by
a04 = a1
for i = 1, ... , n - 1, a3i = f (P˜1¢¢ i , P˜ i¢+n1 )

an4 = a3
Lemma 2 implies that a4i Œ {a1, a2, a3} for every i. Therefore let j be the
first integer such that a4j π a1. One of two things results:
a4j = a2
but a1P≤j a2, since a2 is only in third position in P≤j . Therefore
f(P̃≤1 j-1, P̃¢j n)P≤j f(P̃≤1 j, P̃¢j+1
n
), so f is manipulable.
a4j = a3
Now, if a1P¢j a3, we also have a1P≤j a3. Therefore f(P̃≤1 j-1, P̃¢j n)P≤j f(P̃≤1 j, P̃¢j+1
n
), and
j n j-1 n
f is manipulable. If a3P¢j a1, we directly get f(P̃≤1 , P̃¢j+1)P¢j f(P̃≤1 , P̃¢j ), and f
is still manipulable. We are led therefore to a contradiction in every
case, which shows that F(P) is transitive.
Implementation 63
Since F verifies all of Arrow’s conditions, F must be dictatorial; let i

be the dictator. Let P be any profile and arrange the choices in such a
way that a1Pia2Pi. . . . Since i is the dictator, more precisely we have
a1F(P)a2 and therefore f(P̃) = a1. But, by construction, P̃ coincides with P
and f(P) is therefore a1, the preferred choice of i, which concludes the
proof showing that i is also a dictator for f.
4.6 Appendix B: Proof of Maskin’s Theorem
Recall from the text the necessity of monotonicity. To see the (near)
reciprocal, we can depend on the simplest construction of Maskin’s
mechanism, which is due to Repullo (1987).
In Repullo’s mechanism the message mi sent by each agent
consists of
• a statement Ui of the utility profile
• a choice ai
• an integer ki Œ IN
The allocation procedure g consists of two rules:
1. If there is a i such that for each j π i, mj = (U, a, k), then g(m) is ai if

ai Œ Li(a, Ui) and a if not.
2. Otherwise, g(m) = ai, where i is the smallest integer such that ki =
maxj=1,...,nkj.
This mechanism deserves some explanations. First note that it does

implement f weakly: U is the true profile, a Œ f(U), and k is any integer.
The statement (U, a, k) by all agents ends in g(m) = a, and it is clearly
a Nash equilibrium: if an agent i deviates, the first rule of g applies,
and the implemented choice is either a or the statement i if the latter is
less favorable. It is in no one’s interest to lie; any unilateral deviation
is sanctioned.
The second rule of g is more subtle; it comes into play if there are at
least three different statements. The effect is called an integer game. Each
agent, in order to win, must announce the largest possible integer,
which is of course an impossible task.
Now let us prove that every Nash equilibrium implements f. Let U
be the true utility profile and m a corresponding Nash equilibrium. We
will use a familiar lemma:
lemma If m is a Nash equilibrium for U and g(m) = a, let m¢i be a

deviation of i such that g(m¢i, m-i) = b. Then b Œ Li(a, Ui).
Proof Since m is a Nash equilibrium for U, we have, in particular,
U i [ g(m)] ≥ U i [ g(mi¢ , m- i )]
which is the definition of b Œ Li(a, Ui).
Three cases may present themselves:

• All agents announce the same mi = (U¢, a, k) with a Œ f(U¢). By rule 1,
we have g(m) = a, so we want to prove that a Œ f(U). Let b Œ Li(a, U¢i ),
so the new message for i is m¢i = (U¢, b, k). Then g(m¢i, m-i) = b by rule 1.
Now, since m is a Nash equilibrium for U, the lemma implies that b Œ
Li(a, Ui). From this we can conclude that Li(a, Ui) … Li(a, U¢i ), so the
monotonicity of f indeed implies that a Œ f(U).
• All agents announce a same mi = (U¢, a, k), but a œ f(U¢). Let b be
any choice of an agent i whose deviation is m¢i = (U¢, b, k¢), where
k¢ > minjπikj. Rule 2 applies, and g(m¢i, m-i) = b. By the lemma
above, we get b Œ Li(a, Ui). Since b and i are unconstrained, a is
the preferred choice of all the agents under U, and NVP implies that a
Œ f(U).
• In all other cases, two agents may send different messages. We can
call them 1 and 2, without loss of generality. Let i > 2, b Œ A, and the
deviation m¢i = (U, b, k¢), where k¢ > minjπikj. Rule 2 applies, and g(m¢i,
m-i) = b. The lemma implies that b Œ Li(g(m), Ui), and g(m) is therefore
the preferred choice of all the agents i π 1, 2 under U. In order to apply
NVP, we must prove that g(m) is also the preferred choice of either 1
or 2. To do this, let i π 1, 2; we inevitably get mi π m1 or mi π m2. Without
loss of generality, we can assume that mi π m1. Now consider a devia-
tion m¢2 = (U, b, k¢) for 2, where b is any choice and k¢ > minjπ2kj. The
application of rule 2 gives g(m¢2, m-2) = b, and the lemma gives b Œ
L2(g(m), U2). NVP can therefore be used to show that g(m) Œ f(U).
To conclude, we have proved that if m is a Nash equilibrium for U, then

we inevitably get g(m) Œ f(U). We can deduce from this fact that the
mechanism constructed above completely implements f.
Implementation 65
Bibliography
Abreu, D., and A. Sen. 1991. Virtual implementation in Nash equilibrium. Econometrica
59: 997–1021.
Arrow, K. 1951. Social Choice and Individual Values. New York: Wiley.
Gibbard, A. 1973. Manipulation of voting schemes: A general result. Econometrica 41:

587–601.
Glazer, J., and C.-T. Ma. 1989. Efficient allocation of a prize—King Solomon’s dilemma.
Games and Economic Behaviour 1: 223–33.
Hurwicz, L. 1986. Incentive aspects of decentralization. In K. Arrow and M. D. Intriliga-

tor, eds., Handbook of Mathematical Economics, vol. 3. North-Holland, Amsterdam.
Jackson, M., T. Palfrey, and S. Srivastava. 1994. Undominated Nash implementation in

bounded mechanisms. Games and Economic Behavior 6: 474–501.
Maskin, E. 1977. Nash equilibrium and welfare optimality. Published in the Review of Eco-
nomic Studies (1999), 66: 23–38.
Maskin, E., and J. Moore. 1999. Implementation and renegotiation. Review of Economic
Studies 66: 39–56.
Moore, J. 1992. Implementation, contracts, and renegotiation in environments with com-

plete information. In J.-J. Laffont, ed., Advances in Economic Theory, vol. 1. Cambridge:
Cambridge University Press.
Moore, J., and R. Repullo. 1990. Nash implementation: A full characterization. Econo-
metrica 58: 1083–99.
Moore, J., and R. Repullo. 1988. Subgame perfect implementation. Econometrica 56:
1191–220.
Moulin, H. 1988. Axioms of Cooperative Decision Making. Cambridge: Cambridge

University Press.
Myerson, R. 1979. Incentive compatibility and the bargaining problem. Econometrica 47:
61–73.
Palfrey, T. 1992. Implementation in Bayesian equilibria: The multiple equilibrium

problem in mechanism design. In J.-J. Laffont, ed., Advances in Economic Theory, vol. 1.
Cambridge: Cambridge University Press.
Palfrey, T. 1998. Implementation theory. In R. Aumann and S. Hart, eds., The Handbook of
Game Theory, vol. 3. Amsterdam: North-Holland.
Palfrey, T., and S. Srivastava. 1991. Nash implementation using undominated strategies.
Palfrey, T., and S. Srivastava. 1993. Bayesian Implementation. New York: Harwood
Academic.
Repullo, R. 1987. A simple proof of Maskin’s theorem on Nash implementation. Social

Choice and Welfare 4: 39– 41.
Satterthwaite, M. 1975. Strategy-proofness and Arrow’s conditions: Existence and

correspondence theorems for voting procedures and social welfare functions. Journal of
Schmeidler, D., and H. Sonnenschein. 1978. Two proofs of the Gibbard-Satterthwaite

theorem on the possibility of a strategy-proof social choice function. In H. Gottinger and
W. Ensler, eds., Proceedings of a Conference on Decision Theory and Social Ethics at Schloss
Reisenberg. Dordrecht: Reidel.
II Public Economics
The next three chapters highlight public economics, but they do

not exhaust this domain. Traditionally public economics is defined as
“the positive and normative study of government action over the
economy.” This definition is probably too wide, since it seems to
include, for example, the analysis of macroeconomic policies. My objec-
tive is much more limited. For one thing, I will confine myself to micro-
economic aspects of the subject; for another, I will discuss taxation and
the effects of public spending only as potential remedies for market
failures. Therefore my approach will treat rather what is called welfare
economics.
I will give little room to taxation questions, which really deserve
an entire book. For these issues I could do no better than advise the
interested reader to refer to the works of Atkinson-Stiglitz (1980) or
Myles (1995), or to that of Stiglitz (1988) for a less formal approach.
Bibliography
Atkinson, A., and J. Stiglitz. 1980. Lectures on Public Economics. New York: McGraw-Hill.
Myles, G. 1995. Public Economics. Cambridge: Cambridge University Press.
Stiglitz, J. 1988. Economics of the Public Sector. New York: Norton.

5 Public Goods
The private goods that so far have been the theme of this book possess
two properties that distinguish them. The first is that these goods are
rivals: consumption by an agent reduces the possibilities of consump-
tion by other agents (usually to nothing). For example, if I eat an apple,
no other agent can consume it after me. The second is economic. Private
goods are subject to exclusion, in other words, it is necessary to pay to
consume them.
There are other types of goods that do not necessarily possess these
properties. A nonrival good is called a public good. Pure public goods,
which are at the heart of this chapter, are therefore nonrival by defini-
tion; moreover they are not subject to exclusion.1 The standard example
concerns the services of national defense, the police, or emission stan-
dards for air quality.
Of course there are numerous intermediary cases. Thus research pro-
tected by a patent is subject to exclusion, since a royalty must be paid
to access it, but it is nonrival, since several agents can buy the rights.
A similar situation exists with coded or cable television or with a toll
road.2 The opposite is the case of a free parking space which is a good
without exclusion (because it is free) but a rival good (because two cars
cannot occupy it simultaneously). On the other hand, many public
goods are submitted to external effects (see chapter 6). Take highways
congested by heavy traffic, for example; clearly, their use value dimin-
ishes. We will dismiss these complications in this chapter; our interest
in pure public goods suffices to relieve us of the particularities of public
goods in general.
1. It is sometimes also assumed that their use is obligatory: no agent can choose not to
consume them.
2. This type of good is often called a club good.
70 Public Economics
The reader should not confuse public goods and publicly provided
private goods. To explain the latter case, in many countries (a large part
of) education and health care are provided by the public sector. Edu-
cation is not a public good: one can make access to it exclusive by
asking that a tuition be paid, and it is a rival good insofar as the cost
of educating an extra child is not negligible. The same can be said about
health care. This does not mean that there is no market failure in edu-
cation or health care but rather that such failures are due to external
effects. For example, contagion for health and the positive social effect
of a well-educated population.
The most simple test for determining whether a good is a public
good consists of asking
• whether its use can be rationed (i.e., whether it can be denied a given
agent)
• whether it is desirable to ration it (whether, on the contrary, the mar-
ginal cost of an additional consumer is zero; then, as we will see in
chapter 7, the price of the good should be zero at the optimum)
The good is public if the response to both questions is negative. This

test is not perfectly foolproof, but it permits us to forge a reasonable
idea of what is a public good. The reader will verify that education and
health do not pass the test.
5.1 The Optimality Condition
This section studies Pareto optima in an economy that comprises

public goods. To simplify things, we will consider an economy where
only two goods exist: a private good x (e.g., which may aggregate all
private goods) and a public good z. The consumer i = 1, . . . , n has a
utility function Ui(xi, zi) that is assumed to be increasing in its two argu-
ments. The public good is produced from the private good according
to a technology given by z = f(x), where f is increasing and concave. The
initial resources of the economy boil down to X units of the private
good.
The big difference between public and private goods resides in the
scarcity constraints. Assume that a quantity x of private good is set
aside to produce a quantity z of public good. Then, as usual, the scarcity
constraint for the private good expresses that the sum of consumptions
must not exceed what remains of the private good, or
Public Goods 71
n
Â xi £ X - x
i =1
On the other hand, the consumption of i in public goods is limited only

by the total disposable quantity since the public good is by definition
nonrival. One therefore gets
"i = 1, . . . , n, zi £ z
One obtains, as usual, the Pareto optima by fixing the utility of the
last n - 1 consumers and by maximizing the utility of i = 1 under the
feasibility constraints, or
Ïmax x1, ... , xn ,x U 1 ( x1, z1 )

Ô z1 , ... , zn , z
Ô "i = 2, ... , n, U i ( xi , zi ) ≥ U i
Ô n
Ì Âi=1 xi £ X - x
Ô
Ô "i = 1, ... , n, zi £ z
ÔÓ z £ f ( x)
Recall that in this program x represents the quantity of the private good
set aside for the production of the public good and z the quantity of
the public good produced. Since utility functions are increasing, one
gets "i = 1, . . . , n, zi = z = f(x). Hereafter we will denote g the cost in
the private good for the production of the public good:
z = f (x) ¤ x = g(z)
Obviously g is an increasing and convex function. The program is then

simplified to
Ïmax x1 ,... , xn ,z U 1 ( x1 , z1 )
Ô
Ì "i = 2, ... , n, U i ( xi , zi ) ≥ U i (li )
Ô n
Ó Âi=1 xi £ X - g( z) (m)
where the li and m are multipliers attached to different constraints. The

Lagrangean can be written
n n
Ê ˆ
L = U 1 ( x1 , z) + Â l i (U i ( xi , z) - U i ) + m X - g( z) - Â xi
i =2
Ë i =1
¯
Normalizing l1 = 1 to make the formulas symmetrical, the first-order

conditions are
72 Public Economics
Ï n ∂U i
ÔÔÂi =1 l i ∂ z = g ¢(z)
Ì
Ôl i ∂U i = m , "i = 1, ... , n
ÔÓ ∂ xi
From the second group of conditions we can derive
m
li =
∂U i ∂ xi
whence, by substituting in the first condition the Pareto-optimality con-
dition which sets the level of public good production,
n
∂U ∂ z 1
Â ∂U i ∂ x = g ¢(z) =
f ¢( x )
(BLS)
i =1 i i
The expression above is called the Bowen-Lindahl-Samuelson condition,

which we will denote (BLS) (see Samuelson 1954).
Notice that
∂U i ∂ z dxi
=-
∂U i ∂ xi dz Ui
is simply the marginal rate of substitution of consumer i, that is, his

propensity for sacrificing his private good consumption to instigate
growth in the level of his public good consumption. But as an increase
of public good production by definition benefits all consumers, its mar-
ginal cost g¢(z) must be compared to the sum of all propensities for
paying, not to that of a sole consumer as the case would be for a private
good. One must therefore equalize marginal cost and the sum of
propensities to pay, which is the aim of the (BLS) condition.3
The rest of this chapter will be devoted to examining different modes
of economic organization which are liable to lead to a Pareto-optimal
allocation a priori.
5.2 Implementing the Optimum
To simplify notation in what follows, we will take the private good as

the numéraire; that is, its price will be normalized to one.
3. The careful reader will note that we have only used nonrivalry to achieve the (BLS)
condition.
Public Goods 73
5.2.1 The Subscription Equilibrium
The first solution to consider consists of asking consumers to subscribe

part of their wealth to contribute to public good production. Assume
that the wealth of consumer i is Ri. He can then subscribe si to public
good production, thus consuming xi = Ri - si private good units. The
total quantity of public good produced will be simply f(Sni=1si).
The choice of i of the quantity si is made according to the principles
of the Nash equilibrium: i will take the quantity subscribed by other
consumers s-i as given and will resolve the program
Ïmax xi ,si ,z U i (xi , z)
Ô
Ì xi + si = Ri
Ô
Ó ( n
z = f Âi =1 s j )
or again, after obvious simplifications,
È Ê ˆ˘
max U i ÍRi - si , f Á si + Â s j ˜ ˙
si Î Ë jπi ¯˚
This leads to
∂U i ∂ z 1
=
∂U i ∂ xi f ¢ (Â n
s
j =1 j )
which clearly does not coincide with the optimality condition (BLS). In
effect, when a consumer decides to subscribe to the public good, he
takes into account only the increase of his own consumption of public
good. In his calculations he neglects the subsequent growth of the
utility of all other consumers, so the equilibrium cannot be optimal.
Under reasonable conditions, subscription equilibrium leads to a sub-
production of public good which is all the larger when consumers are
more numerous.
5.2.2 Voting Equilibrium
A variant of the preceding procedure consists in asking the agents to

vote for their preferred level of public good.4 To simplify things,
4. Historically this procedure was the first proposed by economists having studied
public goods; in particular, it was studied by Bowen.
74 Public Economics
suppose that the public good is produced at a constant marginal cost,

that is, g¢(z) = c for every z. It is conceivable to have a mechanism by
which each consumer announces a public good level zi knowing that
public good production is Z(z1, . . . , zn) and that its financing will be
distributed following a scheme ti(z1, . . . , zn). Such a mechanism would
give rise to all of the difficulties set out in chapter 4. We will limit our-
selves here to a simpler approach. It will be supposed, though it may
be irrational, that each consumer believes it is his statement that will
determine the level of public good production, and that the production
cost will then be distributed equally among all the consumers. There-
fore each consumer chooses zi in such a way as to maximize
Ê czi ˆ
Fi (zi ) = U i Ri - , zi
Ë n ¯
It is easy to verify that if U is concave in its two arguments, Fi is equally

concave and thus unimodal. The analyses in the first part of this book
show that the voting equilibrium consists, for planning purposes, of
choosing the median agent’s preferred level of production. With m as
the median agent, the retained level of production is therefore zm such
that F¢m (zm) = 0. So after these immediate substitutions we have
∂U m ∂ z Ê czm ˆ c
Rm - , zm =
∂U m ∂ x m Ë n ¯ n
This result of course does not coincide with the BLS condition, except
in the miraculous case where the marginal rate of substitution of the
median agent is equal to the average of the marginal rates of substi-
tution of all consumers. Still, note that contrary to the subscription
equilibrium, the voting equilibrium does not necessarily lead to a sub-
production of the public good: the direction of the comparison depends
on fine characteristics of the distribution of the marginal rates of sub-
stitution.
5.2.3 The Lindahl Equilibrium
Assume that personalized prices for the public good can be established.
Every consumer i must pay pi per unit of public good that he consumes.
The producer of the public good would then perceive a price p = Sni=1pi
and produce up to the level where his marginal cost equals p:
g ¢(z) = p
Public Goods 75
Every consumer chooses to equate his marginal substitution rate to his

personalized price:
∂U i ∂ z
= pi
∂U i ∂ xi
At equilibrium the amount of public good in demand by each con-

sumer must equal the amount produced, or
( ) ( )
"i = 1, ... , n zi pi* = z p *
From this result it is then deduced that

n n
∂U ∂ z
Â ∂U i ∂ x = Â pi = p = g ¢(z)
i =1 i i i =1
so the BLS condition is verified this time. Now the Lindahl equilibrium
(so named after the Swedish economist who came up with the idea in
1919) leads to a Pareto optimum.5
The disadvantage to this process is that it assumes as a matter of fact
the existence of n “micromarkets” upon which a sole consumer buys
the public good at his personalized price. In such circumstances it is
difficult to maintain the competitive hypothesis unless one can assume
that the consumers are divided in homogeneous groups from the point
of view of their propensity to pay for the public good—one market (and
one price) per group would then suffice. In the opposite case, it is in
every consumer’s interest to underestimate his demand, hoping that
the others will be more honest and that the level of produced public
good will then be high enough to meet his needs; this is the famous
free-rider problem which was evidenced for the first time by Wicksell
in 1896.
5.2.4 Personalized Taxation
Now suppose that every consumer i is taxed by the state in terms of

his consumption zi of public good: the budgetary constraint of the con-
sumer i then becomes
xi + ti (zi ) = Ri
5. It can be shown that the Lindahl equilibrium exists under the usual conditions and
that each Pareto optimum is decentralizable in a Lindahl equilibrium.
76 Public Economics
The consumer then will equate his marginal rate of substitution with
the private marginal cost of the public good, whence
∂U i ∂ z
= ti¢(zi )
∂U i ∂ xi
If the state chooses taxes of the form ti(zi) = p*i zi, where p*i is the per-
sonalized price of i in the Lindahl equilibrium, it is clear that the con-
dition of optimality will hold. Unfortunately, this operation assumes
that the state is privy to very detailed information about the tastes of
all consumers, which in general is not realistic.6 In practice, the financ-
ing of a public good is accomplished by fiscal resources which the state
levies on agents (taxes on income, consumption, etc.). Insofar as these
taxes bring on economic distortions and affect agents’ decisions, the
BLS condition must be modified. Under reasonable hypotheses, after
taking into account the fiscal distortions, this second-best problem
reduces the optimum production level of the public good.
5.2.5 A Planning Procedure
Malinvaud (1971) and Drèze and de la Vallée Poussin (1971) devised

the MDP method by which a planning office could, without using
decentralized information on consumers or even on the function of pro-
duction of the public good, implement the Pareto optimum. Their pro-
cedure takes place in continuous time t Œ [0, +•]. At each instant t, for
a given allocation (x1(t), . . . , xn(t), z(t)),
• every consumer i evaluates his marginal rate of substitution
∂U i ∂ z
Si (t) = (xi (t), z(t))
∂U i ∂ xi
and announces it to the planning office
6. In regard to implementation, it can still be shown that the Lindahl equilibrium (or
personalized taxation) possesses properties comparable to those of the Walrasian equi-
librium: it is not implementable in dominant strategy equilibria, but it is implementable
in Nash equilibrium. However, the core of an economy comprising public goods is not
reduced to the set of Lindahl equilibria when the number of consumers tends toward
infinity. Since any coalition that decides to pull out of the game in order to constitute its
own subeconomy sacrifices the contribution of other consumers to the financing of the
public good, the core must be quite large.
Public Goods 77
• the company announces its marginal cost g¢(z(t))

• the planning office readjusts the current allocation according to the
differential equations
Ïz ¢(t) = Âin=1 Si (t) - g ¢(z(t))
Ì 2
Óxi¢(t) = -Si (t)z ¢(t) + q i z ¢(t) "i = 1, ... , n
where the qi are positive constants whose sum equals 1

Let us try to interpret these differential equations. The right-hand side
of the first equation is zero at the optimum, by the BLS condition. This
equation describes therefore a process of trial and error by which the
level of public good continually draws nearer to the optimum. As for
the second group of equations, note that
dU i dt
xi¢(t) + Si (t)z ¢(t) =
∂U i ∂ xi
So the differential system can be rewritten as
dU i ∂U i 2
= q i z ¢(t)
dt ∂ xi
which shows that the utility of each consumer is an increasing func-
tion of time, and that the increase in social surplus is divided accord-
ing to the choice of the qi.
Let us show that the MDP method converges toward a Pareto
optimum. For this we note first that every stationary point of the
process (for which z¢ = x¢1 = . . . = x¢n = 0) is necessarily a Pareto optimum,
since the BLS condition applies. Let the function
n
F(t) = Â U i (t)
i =1
The preceding expression shows that F is an increasing function.

However, F cannot rise above a certain point (the economy’s resources
are finite). Finally F¢ is only zero under BLS, that is when the system
reaches a Pareto optimum. We deduce from this that F is a Lyapounov
function for the process, which therefore converges toward a Pareto
optimum.
We can also prove that the MDP procedure is neutral in the sense that
it favors no Pareto optimum: for every Pareto optimum, there exists a
choice of constants (qi) that leads to that optimum.
78 Public Economics
This method assumes that the agents announce their characteristics

without cheating. In fact it can be shown that even if the agents manip-
ulate their statements, the statement of truth is a maximin strategy for
each. In other words, to announce the truth is the best solution for an
agent who is infinitely adverse to risk and who thus fears that the other
agents will choose statements that are the most unfavorable for him.
This property of course is not perfectly satisfying. We must conclude
that in general, agents may well lie, and thus we put back into ques-
tion the optimality of the process.
5.2.6 The Pivot Mechanism
We encountered in chapter 3 the Vickrey-Clarke-Groves (VCG) mech-

anism which permits implementation of an optimal social decision in
a dominant strategy equilibrium when the utility functions are quasi-
linear. Consider an indivisible public good of which 0 or 1 unit can be
consumed (e.g., the decision to build a bridge) and for which the unit
costs C. The utilities of the agents are assumed to be quasi-linear: if xi
represents the consumption of the sole private good and z that of the
public good, we get
U i (xi , z) = xi + ui z
where ui is a parameter of propensity to pay for the public good which
is known only by consumer i, who has initial resources Ri in private
good at his disposal.
The decision to build a bridge brings a benefit of Sni=1 ui and a cost of
C. In the Pareto optimum the bridge should be built if and only if the
sum of propensities to pay exceeds the bridge’s cost: Sni=1 ui ≥ C.
A first possible mechanism consists of asking consumers to vote on
the opportunity of building the bridge, knowing (for example) that
each will contribute equally to its financing. Then the consumer i will
vote for the construction if and only if ui ≥ C/n. Let F be the cumula-
tive distribution function of the characteristics ui in the population. The
bridge therefore will be constructed if and only if F(C/n) ≥ 1/2, that is
if C/n does not exceed the median of F. The comparison with the Pareto-
optimal decision rule immediately shows that this mechanism is only
optimal if the median of F coincides with its average, which has no
particular reason to be true.7
7. If, for example, the ui are correlated with wealth Ri, they risk having an asymmetrical
distribution, since it is known that the distribution of wealth has a median distinctly
Public Goods 79
Now I will present a direct revealing mechanism that implements

this optimal decision rule. The intuition of this mechanism can be
understood by referring to the theory of auctions. Assume an indivis-
ible object is proposed to buyers i = 1, . . . , n whose propensities to pay
ui are known only to themselves. First, consider a first price auction,
where the winner (the one who bid the highest) pays the price he indi-
cated. If any consumer i announces a price vi, he will have a utility ui
- vi if he wins and zero if not. He can then guarantee himself a posi-
tive utility in expectation by announcing a price vi inferior to his true
disposition to pay ui, for he can win the bid if the other consumers are
not very tempted by the object.
Following the terminology of chapter 4, the first price auction is a
direct but nonrevealing mechanism. Vickrey (1961) showed that on the
contrary, the second price auction is revealing. The second price auction
consists of making the winner (who is still the one who indicated the
highest price) pay the price indicated by the person immediately after
him. Consider still consumer i, and let v̄i be the highest price announced
by the other consumers. If I announces vi > v̄i, he will win the bid and
will have a utility ui - v̄i; if he bids vi < v̄i, he will lose and will have a
zero utility. Thus he with choose to bid some vi > v̄i if ui > v̄i and some
vi < v̄i if ui < v̄i. In both cases his utility is independent of his bid, and
it is therefore not in his interest to lie.
In order to interpret this result, consider the social surplus
created when i buys the object. The price paid is simply a transfer
that does not intervene in the social surplus. If P is the price at
which the seller values the object, the social surplus is therefore
ui - P. But if the other consumers tell the truth, the increase of
utility of i in the second price auction when he raises his bid in
order to win is ui - ūi, which coincides with the increase in the
social surplus. The second price auction is a revealing mechanism
because it leads consumers’ objectives to align themselves on the social
objective.
Now let us return to the bridge’s construction. The VCG mechanism
does nothing but generalize the idea of the second price bid. The the-
oretical importance of this mechanism is such that Green-Laffont (1979)
devoted an essential part of their book to it. We will be content here to
look at some of its properties. To simplify the notation, I will subtract
from the ui the per capita cost C/n, which lets us rid ourselves of the
lower than its average. A vote would then sometimes lead to not constructing the bridge
even when it would be socially optimal.
80 Public Economics
cost C of the problem;8 the new ui can therefore be either positive

or negative, and the Pareto-optimal decision rule amounts to con-
structing the bridge when Sni=1ui ≥ 0. Let d(u) be the indicator of that
event.
Social surplus, with which the utility of each consumer must be iden-
tified, is Sni=1 uid(u). Let v = (v1, . . . , vn) be the statements of the agents.
In equilibrium, all consumers must tell the truth; the consumer i there-
fore evaluates the social surplus at
Ê ˆ
Á Â v j + ui ˜ d(v)
Ë j πi ¯
If a transfer ti(v) is deducted from him, we then have (up to a constant)
Ê ˆ
Ri - ti + ui d(v) = Á Â v j + ui ˜ d(v)
Ë j πi ¯
which implies (still up to a constant)
t i ( v) = - Â v j d( v)
j πi
In fact, we can even add to the transfers of every agent any quantity
independent of his statement. The category of VCG mechanisms is
therefore characterized by
• d(v) = 1, that is, the bridge is constructed if and only if
n
Â vi ≥ 0
i =1
• the consumer i pays a transfer

ti (v) = -Â v j d(v) + hi (v - i )
j πi
where hi(v-i) is any sum depending only on statements of the

other consumers
Groves-Loeb (1975) proved that all of these mechanisms are revealing

in dominant strategies. The proof is very simple. The utility of i under
this mechanism is written
8. This transformation can be interpreted by assuming that if the bridge is built, each
consumer contributes C/n even before the transfers linked to the VCG mechanism are
put into place.
Public Goods 81
Ê ˆ
Ri + Á Â v j + ui ˜ d(v) - hi (vi )
Ë j πi ¯
which depends on the statement vi of i only through d(v). The agent i

must then choose his statement in such a way as to maximize the
second term, which gives
d(v) = 1 iff Â v j + ui ≥ 0
j πi
But by definition,
n
d(v) = 1 iff Â vj ≥ 0
j =1
and vi = ui is therefore one of the solutions of the program of agent i,

whatever the statements of the other agents may be.
Clarke (1971) proposed a particularly interesting VCG mechanism.
Choose
Ê ˆ
hi (v - i ) = maxÁ Â v j , 0˜
Ë jπi ¯
Then there are four cases to consider:
1. If Sjπi vj > 0 and Snj=1 vj > 0, ti(v) = 0

2. If Sjπi vj > 0 and Snj=1 vj < 0, ti(v) = Sjπi vj > 0
3. If Sjπi vj < 0 and Snj=1 vj > 0, ti(v) = -Sjπi vj > 0
4. If Sjπi vj < 0 and Snj=1 vj < 0, ti(v) = 0
In cases 1 and 4, the statement of agent i does not change the decision
to build the bridge, and he pays a zero transfer. On the contrary, in
cases 2 and 3, agent i modifies the decision with his statement, and he
then pays a positive transfer; he is called a “pivot” agent. This prop-
erty gave its name to Clarke’s mechanism.
It is immediately ascertained that the pivot mechanism is not
“balanced”: no transfer is negative, and their sum is strictly positive
if there is at least one pivot agent. This property is unfortunately
common to all VCG mechanisms. To see this, consider the case of
two agents (n = 2) and assume the existence of a balanced VCG mech-
anism, in other words a choice of functions h1(v2) and h2(v1) such that
for every v = (v1, v2),
82 Public Economics
t1 (v) + t2 (v) = 0
Assume three statements v¢1, v1≤, and v2 such that
v¢1 + v2 > 0 and v1≤ + v2 < 0
The result clearly is d(v¢1, v2) = 1 and d(v1≤, v2) = 0, whence the two con-
ditions of the budget balance
Ïh1 (v2 ) + h2 (v1¢ ) = v2 + v1¢

Ì
Óh1 (v2 ) + h2 (v1¢¢) = 0
By subtracting the second condition from the first, we get

h2 (v1¢ ) - v1¢ = v2
which cannot be satisfied systematically, since the left-hand side, but

not the right-hand side, is independent of v2.
How are we to interpret this budgetary imbalance? It cannot be redis-
tributed to the agents, since the incentive properties of the mechanism
would be modified. The VCG mechanisms therefore do not allow a true
Pareto optimum to be attained, since such an optimum requires budget
balance as well as efficient decision making. This conclusion is some-
times summed up by saying the VCG mechanisms are only “satisfac-
tory.” Unfortunately, we can prove (see this chapter’s appendix) that
VCG mechanisms are the only mechanisms that are satisfactory when
the ui can assume all real values. Therefore it seems that we have
arrived at an impasse. In fact the situation is salvageable. On the
one hand, the pivot mechanism can be shown to be asymptotically
balanced: when the number of agents tends toward infinity, budget
imbalances become negligible9 and if they are redistributed to the
agents, stating the truth is nearly optimal for them. On the other hand,
we can imagine only proposing the mechanism to a representative
sample of agents, and redistributing the eventual surpluses to the non-
participatory agents. Finally d’Aspremont-Gérard-Varet (1979) showed
that if we only ask for implementation in Bayesian equilibriums (and
no more in dominant strategies), a fully Pareto-optimal mechanism
exists, which then simultaneously ensures efficient decision making
and budget balance.
9. This result is easy to understand: when there are many agents, the number of pivot
agents becomes quite small, since it is improbable that an agent can change the decision
on his own.
Public Goods 83
It is fitting to note that even if these mechanisms have abstract sat-

isfactory properties, at times they can give counterintuitive taxation
schemes. Such schemes tend to be very advantageous for the agents
with high propensities to pay.10
To conclude, return to the hypotheses. The fact that the good is indi-
visible simplifies the exposition, but actually matters little. If the quan-
tity of public good z could assume values in a set Z, it would be
necessary to write the agents’ utilities as
U i (xi , z) = xi + ui (z)
The optimal decision rule would then be given by
n
z * (u) = arg max Â ui (z)
z ŒZ i =1
and we can easily see that the pivot mechanism, for example, would
be associated with transfers
ti (v) = Â v j [z * (v - i )] - Â v j [z * (v)]
jπi jπi
On the other hand, the quasi-linearity of the agents’ utility functions is

absolutely crucial. It behooves us then to inquire in each particular case
whether this quasi-linearity indeed constitutes a good approximation
of reality.
5.3 The Property of Public Goods
If is often asserted that public goods should be provided by the public

sector, as is effectively the case for defense, police, or the justice system.
The underlying argument is that if the public good was provided by a
market mechanism, a consumer who would buy some (and would
then stimulate its production) would not take into account the benefit
that he would unwittingly give to other consumers. This is what is
called a positive externality in chapter 6; this phenomenon of course
leads to a suboptimal production of the public good. Coase (1974) has
10. Henry (1989) provides a good example of this. Consider the pivot mechanism and
suppose that the first n/2 agents are ready to pay 1, 9C/n and the last n/2 are ready to
pay only 0, 19C/n. Then the bridge will still be built, but we can easily see that if n ≥ 20,
no agent either pays or receives any transfer (on top of C/n)—this is obviously advanta-
geous for the first group.
84 Public Economics
nevertheless brought this reasoning into question. Consider the

example of a lighthouse showing the way for ships. The service ren-
dered is clearly a public good, since it is nonrival: the fact that a ship
sees the lighthouse beam of course does not keep any other ship from
seeing it also. For the same reason it is seemingly very difficult to
exclude ships from the service rendered by the lighthouse, which
makes their profitability quite doubtful for a private firm. Adam Smith
already thought that in such a situation, the state itself must provide
the service in question (The Wealth of Nations, bk. V, ch. 1):
The third and last duty of the sovereign or commonwealth is that of erecting
and maintaining those publick institutions and those publick works, which,
though they may be in the highest degree advantageous to a great society are,
however, of such a nature, that the profit could never repay the expence to any
individual or small number of individuals, and which it, therefore, cannot be
expected that any individual or small number of individuals should erect or
maintain.
After Smith, numerous classical authors, from John Stuart Mill to Paul
Samuelson, illustrated this principle through the lighthouse example.
Coase remarks, however, that through the vicissitudes of history,
British lighthouses were generally the responsibility of private national
organizations, and they perceived a fixed right which was compulso-
rily discharged by any ship landing in a British port. That arrangement
does not seem to have had tragic consequences for British naval com-
merce. Coase considers that private provision gives better incentives to
lighthouse keepers: since shipowners are more conscious of paying for
this service than if they financed it (like all other citizens, only through
general taxation), they will take to heart the need to verify that the
service is indeed rendered.
The fact remains that private provision of the service rendered by
lighthouses violates the condition of equality of price and marginal cost
which is necessary for the optimum (see chapter 7). Insofar as the mar-
ginal cost of lighting the way for an additional ship is zero, the price
of the service should also be zero. Making the ship pay may dissuade
it from visiting British ports, which would be inefficient.
In the specific example of lighthouses, we wonder whether this the-
oretical argument actually carries weight in the real world. This dis-
cussion nonetheless illustrates the difficulty of clearly expressing
general principles in public economics: each situation must be studied
in terms of its own characteristics.
Public Goods 85
5.4 The Importance of the Free-Rider Problem
The problem of the free rider is at the heart of the analysis of public
goods. It is then crucial to thoroughly determine its theoretical and
empirical relevance. First on the theoretical plane, note that in an
economy lacking public goods, the consumers—if they are very numer-
ous—gain nothing by trying to manipulate the Walrasian mechanism
(Roberts-Postlewaite 1976). In effect no consumer is “big” enough to
influence prices appreciably on his own. The situation is quite differ-
ent in an economy comprising public goods (Roberts 1976). This time,
a free rider who would announce a very small demand of public goods
would participate very little in the financing but would hardly suffer
from it, since the production level of the public good is practically inde-
pendent of his statement. One can then expect the free-rider problem
to be more important when the number of agents affected by the public
good is higher.
On the empirical plane now, there are good reasons to doubt the
importance of the free-rider problem. The first is that honesty is a social
norm that molds the behavior of individuals. The second is that at least
in a small group, where each consumer has a notable influence on the
level of public good, it is difficult for each agent to calculate the best
way to underevaluate its demand. Moreover the majority of decisions
of public goods production are made by elected representatives who
may have less tendency to announce levels that are very low.
An oft-quoted study of Bohm (1972) seemed to confirm this skepti-
cism. Bohm put into operation several different financing schemes for
a television program in Sweden. He then showed that the propensities
to pay announced by groups to whom he had submitted these differ-
ent schemes varied only little across schemes, which would imply that
the agents have a tendency to announce their true preferences.
Bohm’s experience has been criticized by numerous authors, who set
up their own experiments. Ledyard (1995) surveys this literature, from
which he gleans conclusions. The most recent experiments mitigate
Bohm’s conclusions quite a bit. In the current state of our under-
standing, it seems that
• the majority of subjects announce a propensity to pay that is inter-
mediary between the Nash equilibrium (behavior of the free rider) and
the Pareto optimum
• they contribute less if the game is repeated
86 Public Economics
• they contribute more if they are allowed to communicate with each

other before making their decision
These results are consistent with the theory, even if the theory certainly
exaggerates the extent of the free-rider problem in predicting negligi-
ble voluntary contributions. This problem is more acute when the
agents accumulate experience and can thus realize their capacity to
manipulate the financing plan; it also grows in importance when the
number of agents concerned is higher, since it then becomes more
difficult for them to communicate.
5.5 Local Public Goods
Up to this point we have assumed that public goods concern the com-
munity as a whole. In fact, quite a few public goods apply only to the
inhabitants of a given geographical area. For example, this is the case
with water (whose distribution usually is under communal supervi-
sion), garbage collection, or urban transports in some countries. Such
goods are called local public goods. Tiebout (1956) was the first to study
this theory. A fundamental feature of local public goods is that con-
sumers can decide in which local community they will establish them-
selves. If the production level of local public goods or the conditions
of their financing are not satisfying, they can “vote with their feet” by
moving to another community. Tiebout showed that under certain
hypotheses, this process has an equilibrium that is efficient. The most
recent literature (e.g., see Rubinfeld 1987) comes back to Tiebout’s
hypotheses: perfect information and perfect mobility of the consumers,
existence of a large number of communities, absence of land tax, a very
gross modeling of the supply of local public goods, and so on. It shows
that once these hypotheses are relaxed, the situation becomes much
more complex, owing particularly to the emergence of nonconvexities:
it is possible for equilibrium not to exist, and it is possible for it to be
inefficient when it does exist.
5.6 Appendix: Characterization of VCG Mechanisms
Consider a direct revealing satisfactory mechanism {d(v), [t1(v), . . . ,

tn(v)]}, which therefore allows effective decision making: d(v) = 1 if and
only if Sni=1 vi ≥ 0. The proof that this mechanism is necessarily a VCG
mechanism relies on two lemmas.
Public Goods 87
lemma 1 (independence of transfers) If d(v-i, v¢i ) = d(v-i, vi), then

ti(v-i, v¢i) = ti(v-i, vi). The transfer paid by i cannot therefore change except
when its statement modifies the decision.
Proof If, for example, ti(v-i, v¢i ) > ti(v-i, vi), then assume that i’s true
propensity to pay is ui = v¢i . We get
ui d(v - i , ui ) - ti (v - i , ui ) < ui d(v - i , vi ) - ti (v - i , vi )
so that the mechanism cannot be revealing.
lemma 2 (principle of compensation)
Ê ˆ
ti (v - i , vi¢) - ti (v - i , vi ) = Á Â v j ˜ [d(v - i , vi ) - d(v - i , vi¢)]
Ë jπi ¯
Proof Assume, without loss of generality, that
Ê ˆ
ti (v - i , vi¢) - ti (v - i , vi ) = Á Â v j ˜ [d(v - i , vi ) - d(v - i , vi¢)] + e
Ë jπi ¯
where e is a positive real. Lemma 1 implies that we inevitably get

d(v-i, vi) π d(v-i, v¢i ) (otherwise the transfers would be equal). Still,
without loss of generality, assume that
d(v - i , vi ) = 1 and d(v - i , vi¢) = 0
Define a statement v≤i = -Sjπi vj - e/2. We get Sjπi vj + v≤i < 0 and there-
fore d(v-i, v≤)
i = 0 = d(v-i, v¢i ), and lemma 1 gives ti(v-i, v≤)
i = ti(v-i, v¢i ). We
deduce from this that
e
ti (v - i , vi¢¢) - ti (v - i , vi ) = Â v j + e = - vi¢¢+
jπi 2
Now, if the true propensity to pay of i is ui = u≤,

i we get
e
ui d(v - i , vi ) - ti (v - i , vi ) = ui d(v - i , ui ) - ti (v - i , ui ) +
2
So the mechanism is not revealing.
The proof concludes simply by “integrating” lemma 2 (i.e., by fixing

v¢i ), whence
ti (v - i , vi ) = -Â v j d(v - i , vi ) + hi (v - i )
jπi
which is precisely the definition of VCG mechanisms.

88 Public Economics
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6 External Effects
We say that there is an external effect, or an externality, when an agent’s

actions directly influence the choice possibilities (i.e., the production set
or consumption set) of another agent. The word “directly” is very
important in this definition. We must not confuse genuine externalities
and what are at times improperly called “pecuniary externalities,”
which pass through the intermediary of the market and fit perfectly
within the framework of the most basic general equilibrium model.1
Even if this distinction dates back to Scitovsky (1954), it is at times
neglected in the applied literature.
Examples of externalities are many. The most famous one in the
literature is that of Meade, who considers a beekeeper and a nearby
located orchard. The beekeeper’s bees contribute to the productivity of
the orchard in fertilizing the flowers of the trees; in return, the trees
provide the bees with pollen which enters into the production of the
beekeeper’s honey. In each case one agent’s production function moves
upward because of another agent’s actions. It is therefore a matter of
positive crossexternalities of production.
All externalities are not so favorable. Pollution constitutes a typical
example of negative production (and also consumption) externality;
noise or cigarette smoke are negative consumption externalities.2 To
make up for this, the network effects tied to the extension of a tele-
phone network are positive consumption externalities. The arrival of a
new subscriber on the network allows all existing subscribers to call an
1. Pecuniary externalities so far create no inefficiency in the public economics framework

of this book. Such is no longer the case when competition is imperfect, as we will see in
chapter 10.
2. To take another example, if I derive a particular pleasure from seeing that my con-
sumption is superior to that of my neighbors, as in the (heterodox) theories of Veblen,
then their consumption inflicts on me a negative consumption externality.
90 Public Economics
firm 1 consumer
(on river A) (on river B)
firm 2
Figure 6.1
River pollution
extra correspondent and therefore contributes positively to their

welfare.3 Scientific research also constitutes a positive externality, from
the moment it falls into the public domain.
At times it is difficult to separate externalities from other market
failures. An agent who contributes to the financing of a public good
exerts a positive consumption externality over all other agents who can
benefit from an increased production of public good. Even the incom-
pleteness of markets refers to external effects (see chapter 13).
6.1 The Pareto Optimum
As in the preceding chapter, our first task consists in characterizing the

Pareto optimum in the presence of externalities. Consider here again a
simplified example that comprises two goods (1 and 2), two firms,
and one consumer. Good 1 is supposed to be polluting and good 2
nonpolluting.
As shown in figure 6.1, firm 1 is situated on river A and produces
good 1 from good 2 according to y1 = f(x2). The consumer is on river B
and has a utility function U(x1c, x 2c ). Now firm 2, situated beyond the
confluence of the two rivers, produces good 2 from good 1. Since it is
downstream from firm 1 (which ejects its wastes into river A) and from
3. The literature on the economics of networks has developed dramatically in recent

years, in connection with the explosion of telecommunications and the Internet. The
reader can find its principal themes in the symposium published in the Journal of Eco-
nomic Perspectives (spring 1994) or in the survey of Economides (1996).
External Effects 91
consumer 1 (who pollutes river B), its production suffers from two
negative externalities, so its production function
y 2 = g(x1 , y1 , x1c )
is increasing and concave in its first argument but decreases in its last
two arguments. Denote (w 1, w 2) the initial resources of the economy.
The Pareto optima of this economy are given by the following
program:
Ïmax x1 ,x2 , y1 , y2 ,x1c ,x2c U (x1c , x 2c )

Ô
Ô x1c + x1 £ w 1 + y1 (l1 )
Ô
Ì x 2c + x 2 £ w 2 + y 2 (l 2 )
Ô
Ô y1 £ f (x 2 ) (m 1 )
Ô y 2 £ g(x1 , y1 , x1c ) (m 2 )
Ó
where l1, l2, m1, and m2 are four multipliers. The first-order conditions
are
Ï∂ U ∂g
Ô ∂ x - l1 + ∂ x c m 2 = 0
Ô 1 1
Ô∂U
Ô∂ x = l2
Ô 2
Ô ∂g
Ôl1 = m2
Ì ∂ x1
Ô ∂f
Ôl 2 = m1
Ô ∂ x2
Ô ∂g
Ôl1 = m1 - m2
Ô ∂ y1
Ô
Ól 2 = m2
By eliminating the multipliers, we obtain
(∂ U ∂ x1 ) + (∂ U ∂ x2 )(∂ g ∂ x1c ) ∂g 1 ∂g
= = -
(∂ U ∂ x2 ) ∂ x1 ∂ f ∂ x 2 ∂ y 1
Although this condition of optimality may seem complicated, it is inter-

preted quite easily. The left-hand member is the marginal rate of sub-
stitution of the consumer, corrected by the fact that his consumption of
good 1 entails pollution downstream. This is called the social marginal
rate of substitution of the consumer, and it takes into account all of the
92 Public Economics
consequences of his consumption on social welfare. In the same way

the right-hand member is the marginal rate of transformation of firm
1, corrected by the effect of its pollution on firm 2. As for the central
term, this is the usual marginal rate of transformation of firm 2. In
effect the firm 2 does not pollute, and its social marginal rate of
transformation coincides therefore with its private marginal rate of
transformation.
The principle to remember is therefore the following: in the presence
of external effects, the usual optimality condition of equality between
the marginal rates of substitution of consumers and the marginal
rates of transformation of firms bears on the social values of these quan-
tities. The social values take into account the external effects of each
agent’s decisions on the rest of the economy. This principle is due to
Arthur Pigou in the 1920s. Its application permits us to justify, for
example, the establishment of toll booths on highways in the face of
excess traffic (when the arrival of one more vehicle reduces traffic
speed): the social marginal cost of this new vehicle is positive even if
its private marginal cost is practically nothing. The toll right then rep-
resents the marginal cost that the vehicle entering the highway imposes
on other vehicles.4
In our simplified economy, the consequences of externalities are
clear. At the optimum, firm 1 must produce less, and the consumer
must consume less of good 1 than in the absence of externalities. This
is perfectly intuitive: if a firm pollutes, it is advisable that its level of
production be reduced.
6.2 Implementing the Optimum
Now let us see what are the means at our disposal for implementing
the optimum; this is often called internalizing externalities.
6.2.1 The Competitive Equilibrium
Assume that firm 2 takes the pollution factors y1 and x1c as given. Then,
if the prices of goods are p1 and p2, the agents’ choices will lead to
(∂ U ∂ x1 ) p1 ∂ g 1
= = =
(∂ U ∂ x2 ) p2 ∂ x1 ∂ f ∂ x2
4. Logically it should depend on the degree of highway congestion at the location and
hour in question.
External Effects 93
which is inefficient. In competitive equilibrium the agents only take

into account the consequences of their choices on their own welfare, in
equating the private marginal rates of substitution and of transforma-
tion. Under usual conditions, one can show that there is too much of
good 1 being produced and consumed.
This analysis permits us to explain the phenomenon of overfishing,
that is, the overexploitation of fisheries resources by fishermen. The
quantity of fish in the oceans is the archetypal example of a “common
resource” that belongs to everyone and to no one.5 Every fisherman, in
pulling a fish from the water, reduces the capacity of the species’ repro-
duction and therefore future stocks; in so doing, he exerts a negative
production externality on other fishermen. The preceding results lead
us to think that the competitive equilibrium will indeed exhibit a phe-
nomenon of overfishing. It can easily be verified in a single-period
model, even if a dynamic framework would obviously be more appro-
priate. Normalize the price of a fish at 1. We will model the negative
externality by assuming that the cost of extracting Pi fish by the fish-
erman i when the set of fishermen takes S = Sni=1 Si fish from the ocean
is cSiS. As more fish are caught, they become more rare, so it becomes
more costly to find them. The private marginal cost of extracting a fish
is the derivative of cSi (Si + SjπiSj) in Si, that is, c(Si + S). On the other
hand, the social marginal cost must take into account the derivatives
in Si of the costs of the other fishermen j π i, which is cSj. When all fish-
ermen are identical, we get S = nSi. The competitive equilibrium E is
therefore given by the equality of the price and of the marginal private
cost so that
Ê SE ˆ
1 = cÁ + SE ˜
Ë n ¯
Now at the social optimum O price equals social marginal cost,
Ê ˆ
1 = cÁ SiO + SO + Â SOj ˜ = 2cSO
Ë jπi ¯
So we get
1 n
SO = < SE =
2c c(n + 1)
5. This is also the case with air quality or the climate (i.e., global warming).
94 Public Economics
and the equilibrium clearly leads to overfishing, as compared to the

social optimum.
6.2.2 Quotas
The simplest way to arrive at a Pareto optimum is of course for the

government to set quotas specifying that the externality-inducing
activities should be set at their optimal level. This is certainly an author-
itarian solution and one that assumes a very fine knowledge of
the characteristics of the economy. In the example studied here, it
would amount to calculating the Pareto-optimal levels of y1 and x1c ,
which we denote y*1 and xc1*, and to forbid firm 1 from producing more
than y*1 and the consumer from consuming more than xc1* of good 1. It
is nevertheless an oft-adopted solution, under a slightly less brutal
form, that consists of limiting the quantity of a certain type of pollu-
tants emitted by firms, or even by consumers (as in carbon emissions
of automobiles).
6.2.3 Subsidies for Depollution
It is sometimes possible to install dispositions to reduce pollution.

Suppose that firm 1 can invest in depolluting a quantity a of good 2
which, without affecting its production, reduces its pollution as if y1
were y1 - d(a).
First, let us look for the Pareto optimum. The scarcity constraint for
good 2 becomes
x 2C + x 2 + a £ w 2 + y 2
while the production constraint of firm 2 becomes
y 2 £ g[x1 , y1 - d(a), x1c ]

The quantity a also becomes a maximand of the program. We easily see
that the optimality conditions obtained above are unchanged, but that
we must add to them a condition to determine the optimal depollution
level a*:
1
d ¢( a *) = -
∂ g ∂ y1
What happens here with the competitive equilibrium? If firm 1 is not
prompted to depollute, it will of course choose not to do so. It still
External Effects 95
seems reasonable for the government to subsidize such an expense by

paying firm 1 a sum s(a). The profit maximization program of firm 1
then becomes
max[p1 f (x 2 ) - p2 x 2 - p2 a + s(a)]
x2 , a
while firm 2 maximizes in x1
p2 g[x1 , y1 - d(a), x1c ] - p1x1

The condition of profit maximization of firm 1 implies that
s¢(a) = 1
So choosing subsidy s(.) such that s¢(a*) = 1 induces the firm to realize
the socially optimal expenditures of depollution. Unfortunately, the
first-order conditions also entail
∂g 1
=
∂ x1 ∂ f ∂ x2
So the subsidized equilibrium is still not Pareto optimal.
6.2.4 The Rights to Pollute
Meade (1952) suggested a solution to the problem of external effects

which has generally found favor with economists (and more rarely
with decision makers). It rests on the finding that externalities con-
tribute to inefficiency only because no other market exists upon which
they may be exchanged. Therefore assume that the state (or any other
institution) creates a “rights to pollute” market: a right to pollute gives
the right to a certain number of pollution units. The polluters (firm 1
and the consumer) can then pollute with the proviso that they buy the
corresponding rights to pollute. Thus
• firm 1 pays q to firm 2 for each unit of y1
• the consumer pays r to firm 2 for each unit of x1c
The consumer’s program gives
∂U ∂x1 p1 + r
=
∂U ∂x 2 p2
while the profit maximization of firm 1 implies that
96 Public Economics
p1 - q 1
=
p2 ∂ f ∂x2
As for firm 2, it must take into account payments it receives for deter-
mining the number of pollution rights it is ready to sell; it solves
therefore
max [p2 g(x1 , y1 , x1c ) - p1x1 + rx1c + qy1 ]
x1 , y1 , x1c
whence
Ï p1 ∂g
Ôp = ∂ x
Ô 2 1
Ôq ∂g
Ì =-
p
Ô 2 ∂ y1
Ôr ∂g
Ô =- c
Ó p2 ∂ x1
All of these equalities amount to

( ∂ U ∂ x1 ) + ( ∂ U ∂ x2 )( ∂ g ∂ x1c ) ∂g 1 ∂g
= = -
∂ U ∂ x2 ∂ x1 ∂ f ∂ x 2 ∂ y 1
which is the condition of efficiency. The creation of markets for

rights to pollute thus implements a Pareto optimum.6 In other respects,
it is a system far less demanding than the imposition of pollution
quotas, since the calculation of such quotas implies that the state
knows the preferences and technologies of all agents. It is enough here
for the state to open pollution rights markets and let equilibrium
establish itself.7
This result merits some comments. First, note that the consumer and
firm 1 do not necessarily pay the same price to pollute at equilibrium. We
6. Neglected here are second-order conditions, which Starrett (1972) has shown are prob-
lematic. In effect, function g cannot be concave in y1 or x1c on all of IR+, since it is decreas-
ing and positive. The program of firm 2 is therefore nonconvex, and this can cause
difficulties for equilibrium, as we will see in chapter 7.
7. We assume that the pollutees are authorized to sell pollution rights, which results in
an optimal pollution level. In practice, these markets are often reserved to the polluters.
The pollution level attained depends then on the number of rights put into circulation,
which poses the problem of the government’s capacity to calculate the optimal pollution
level, to issue the correct number of rights to pollute, and to resist the pressure of agents
who would like to see that number modified.
External Effects 97
would get q = r only if the pollution was impersonal in the sense that g(x1,
y1, x 1c ) = G(x1, y1 + x1c ). Second, there is but one sole applicant and one sole
supplier on each open pollution rights market in this example, which
raises the problem of strategic behaviors. This solution is therefore better
adapted to situations where the pollution is of collective origin—we can
imagine similarly disposed polluters around the same lake.8
Finally, we have implicitly adopted the polluter-pays principle made
popular by the OECD in a 1972 report. In fact optimality does not at
all require that polluters make amends to the pollutees. One could
easily take a situation of serious pollution as a reference point and
impose “depollution rights” whereby the pollutees buy from the
polluters to reestablish optimality.9 Of course the distribution of utility
at equilibrium would not be the same.
6.2.5 Taxation
It is conceivable to tax the production of the good 1 at the rate t and

its consumption at the rate t. It is easy to see that if one chooses t = q
and t = r, where q and r are the equilibrium prices on virtual markets
of pollution rights, we again find a Pareto optimum. These tax levels
are often called Pigovian taxes, in honor of Pigou (1928). The disad-
vantage to this remedy is that like the imposition of pollution quotas,
one needs extraordinarily detailed information on the primitive data
of the economy, since the government must be able to calculate the
equilibrium prices on pollution rights markets.
6.2.6 The Integration of Firms
For simplicity’s sake assume that the consumer does not pollute, so
∂g/∂x1c = 0. In this case one can envision the two firms as merging (in
economic terms, we speak of “integrating”). The new firm thus formed
will maximize the joint profit as
8. One of the most spectacular applications of the pollution rights market functions in
the San Francisco bay area (see Henry 1989). The regulation of thermal power stations
and of sulfur dioxide emissions in the United States offers other examples. More recently
the summit on global warming held in Kyoto in 1997 decided to study the use of rights
markets at the world level.
9. A famous example is that of a city-dweller who retires to the country, settling next to
a farm. The question is, Does he have ground to ask for compensation if the farmer’s
rooster wakes him at the crack of dawn? A French court has decided that he is entitled
to compensation, but this judgment is much debated.
98 Public Economics
Ïmax x1 ,x2 , y1 [p1 f (x 2 ) + p2 g(x1 , y1 ) - p2 x 2 - p1x1 ]

Ì
Ó y1 £ f (x 2 )
which gives
p1 ∂g 1 ∂g
= = -
p 2 ∂ x1 ∂ f ∂ x 2 ∂ y 1
Again we have the Pareto optimum. The solution is obviously radical,

and it shows little regard for property rights. We often see in industrial
economics that the market power confered upon mastodons is not
without inconveniences. Nevertheless, the integration of firms is not to
be discarded entirely.
6.2.7 A Compensation Mechanism
Varian (1994) proposed a mechanism that implements the optimum

when every agent (but not the regulator) is informed of the parameters
of the whole economy. Suppose again that the consumer does
not pollute. If the regulator knows the production functions, he could
still calculate the Pareto optimum (x*1, x*2 ) and impose a Pigovian tax
on firm 1:
∂g
t* = -p2 [ x*1 , f ( x *2 )]
∂ y1
Firm 1 would then choose the socially optimal level of pollution y*1 =
f(x*2 ).
Now suppose that each firm knows the two production functions but
that the regulator does not have this information at his disposal. The
compensation mechanism has two stages:
1. Firm 1 announces a tax level t1 and firm 2 a tax level t2.

2. The regulator imposes transfers on both firms such that their profit
functions become
Ïp 1 = p1 f (x 2 ) - p2 x 2 - t2 y1 - a (t1 - t2 ) 2
Ì
Óp 2 = p2 g(x1 , y1 ) - p1x1 + t1 y1
where a is any positive parameter. The two firms make their produc-
tion decisions based on these modified profit functions.
External Effects 99
There exist multiple Nash equilibria in this game. In every equilibrium,

t1 and t2 are equal, but they may differ from the Pigovian tax t*, so the
optimal allocation is not implemented. On the other hand, we can show
that there exists a unique subgame-perfect equilibrium and that it
implements the Pareto optimum.
A subgame-perfect equilibrium is, by definition, a Nash equilibrium
in every subgame. Subgame-perfect equilibria are obtained by back-
ward induction, that is in starting from the end of the game, here at the
second stage. Therefore suppose that as t1 and t2 are announced, firm
1 maximizes its profit by choosing x2 such that
(p1 - t2 ) f ¢(x2 ) = p2 (1)
The solution to this equation is a decreasing function of t2, which we

will denote x2(t2). As for firm 2, it chooses x1 such that
∂g p1 (2)
=
∂ x1 p 2
In the first stage of the game, certainly it is optimal for firm 1 to choose
t1 = t2, considering the quadratic penalty. Company 2 must choose t2
in order to maximize its profit, which indirectly depends on t2 by the
intermediary of y1 and therefore on the function x2(t2). By differentiat-
ing, we get
∂p 2 Ê ∂g ˆ
= Á p2 + t1 ˜ f ¢(x 2 )x 2¢ (t2 )
∂ t2 Ë ∂ y1 ¯
Since f is increasing and x2 decreasing, t2 must be chosen at the level

where
∂g
p2 {x1 , f [x2 (t2 )]} + t1 = 0 (3)
∂ y1
By combining (1), (2), (3), and the equality t1 = t2, we again easily find
the optimality conditions
p1 ∂g 1 ∂g
= = -
p 2 ∂ x1 ∂ f ∂ x 2 ∂ y 1
and the equality of t1 and t2 with the Pigovian tax t*. This mechanism
effectively allows for the solution of the problem caused by the exter-
nality. The fact that it rests on the agents’ perfect knowledge of their
technologies can be a bit bothersome. Doubtless this situation is more
100 Public Economics
probable in cases of local pollution, where the parameters of the pol-

luter and the pollutee can be well known, even though the regulator
has difficulty obtaining unbiased information.
6.3 Must Prices or Quantities Be Regulated?
We have seen that regulation by quantities (e.g., in the form of emis-

sion quotas) and regulation by prices (e.g., by taxation) both permit the
restoration of Pareto-optimality of the equilibrium and are therefore
equivalent. Let us follow Weitzman (1974) and consider a pollutant
production q. The firm has costs C(q) and its profit is then (pq - C(q)) if
the price is p. The production entails “benefits” B(q) and a consumer
surplus B(q) - pq.
This modeling suggests two comments. The first is of a semantic
nature: many times students are amazed that pollutant production
can be beneficial. In reality there is often joint production of a useful
good and of pollution; what we call “benefits” of the pollutant pro-
duction is the value of useful production minus the cost inflicted by
pollution. The difference between benefits and costs is generally
maximal for a positive level of pollution; as was judged by a medieval
English court in the case of a candlemaker accused of smoking out his
neighborhood:
Le utility del chose excusera le noisomeness del stink. (The item’s usefulness excuses
the annoyance of the odor.)
The reader will note that in this interpretation,10 the social production
cost at once comprises the production cost C(q) and the pollution cost,
which was deducted from the value of production to get B(q).
The second comment is more technical. In the example we have
studied so far, q would be y1, the production of firm 1. This production
permits the consumer to raise his consumption of good 1, but it reduces
the production possibilities of firm 2. The sum of these two effects con-
stitutes the “benefits” of y1, which has moreover a cost given by the
technology of firm 1. Unfortunately, it is not possible to describe the
result in the form of a benefit minus a cost. Such a breakdown is in fact
more reasonable when the pollution injures consumers. In the interest
of not straying too far from our needs, we will disregard this con-
10. We could resort to a dual interpretation where q represents the nonpolluted good,
like air quality.
External Effects 101
sideration here. We will also make the usual hypotheses that B(q) is
increasing and concave and C(q) increasing and convex.
When information is perfect, the optimal emission quota is calcu-
lated by maximizing the social surplus (B(q) - C(q)). We then get
B ¢ (q * ) = C ¢ (q * )
As for the optimal tax rate, it is fixed in such a way that the prices verify
p * = B ¢ (q * ) = C ¢ (q * )
Then the firm effectively produces the optimal pollution level q*.
At this stage the two modes of regulation are perfectly equivalent.
Still, opinions on their respective advantages are generally quite clear-
cut. As noted by Weitzman (1974, p. 477):
I think it is a fair generalization to say that the average economist in the Western
marginal tradition has at least a vague preference toward indirect control by
prices, just as the typical noneconomist leans toward the direct regulation of
quantities.
The introduction of an imperfection of government information on the

costs and benefits will let us compare the two modes of regulation.
Thus suppose that either because the agents benefit from private infor-
mation or because the future is uncertain, the cost and benefit functions
are affected by independent shocks q and h so that they become
C(q, q) and B(q, h).
Ideally price or quantity would be fixed conditionally to realizations
of shocks in such a way as to verify
∂B ∂C
p * (q , h) = [q* (q , h), h ] = [q* (q , h), q ]
∂q ∂q
and regulations by prices and by quantities would remain strictly

equivalent.
In practice, this first-best solution is beyond reach, since the govern-
ment must make decisions knowing only the distributions of h and q and
not their realizations. In this second-best situation, the government choo-
ses the emissions quota q̂ so as to maximize the expected social surplus
E(B(q, h) - C(q, q ))
Things are only slightly more complicated for regulation by price.
If the government fixes a price p, the firm will choose a production
Q(p, q) such that
∂C
p= [Q(p , q ), q ]
∂q
The government must then fix the price at the level p̂ that maximizes
E{B[Q(p , q ), h] - C[Q(p , q ), q ]}
This time there is no more reason that the two modes of regulation be
equivalent. In fact it is easily checked (using second-order calculations
for small uncertainties) that the advantage (in terms of expected social
surplus) of regulation by prices on regulation by quantities is
s2 2
Ê∂ B ∂Cˆ
2
D Á 2 + ˜
2(C ¢¢) 2 Ë ∂q ∂ q2 ¯
where s2 is the variance of the marginal cost ∂C/∂q. Since B is concave
and C convex, the sign of D is ambiguous a priori. Note that if marginal
costs are almost constant, then regulation by quantities will dominate
regulation by prices. In effect a small error in price setting can lead to
a large error in the level of pollution attained.
It is clear that this purely normative perspective on the regulation of
polluters is insufficient. It must be completed by a more descriptive
analysis of the game that brings together polluters, consumer organi-
zations, and the administration. Finkelshtain-Kislev (1997) show that
in such a game, the two modes of regulation are no longer equivalent,
even with perfect information.
6.4 Coase’s Theorem
In a famous article Coase (1960) doubted the necessity of any govern-

mental intervention in the presence of externalities. His reasoning is
very simple: let b(q) be the benefit that the polluter draws from a level
of pollutant production q and c(q) the cost thus imposed on the pollu-
tee.11 When b is concave and c increasing and convex, the optimal
pollution level is given by
b ¢ (q * ) = c ¢ (q * )
Suppose that the status quo q0 corresponds to a situation where b¢(q0)
< c¢(q0), and thus the pollution level is too high. Then the polluter and
11. Be careful not to identify this notation with that of the preceding section.
the pollutee have an interest in negotiating. Let e be a small positive

number, and assume that the polluter proposes to lower the pollution
level to (q0 - e) against a payment of te, where t is comprised between
b¢(q0) and c¢(q0). Since t > c¢(q0), this offer raises the polluter’s profits;
and it is equally beneficial for the pollutee, since t < b¢(q0). Therefore
the two parties will agree to move to a slightly lower pollution level.
The reasoning does not stop here: so long as b¢ < c¢, it is possible to
lower the pollution level against a well-chosen transfer from pollutee
to polluter. The end result is the optimal pollution level. A very similar
argument applies in the case where b¢(q0) > c¢(q0). The “Coase theorem”
can thus be set forth as follows:
If property rights are clearly defined and transaction costs are zero, the parties
affected by an externality succeed in eliminating any inefficiency through the
simple recourse of negotiation.
Stated in this way, the theorem becomes more a tautology: if nothing

keeps the parties from negotiating in an optimal manner, they will
arrive at a Pareto optimum. In fact Coase (1988) explained in a collec-
tion of his articles that above all he wanted at the time to bring atten-
tion to the importance of property rights and of transaction costs.
Unfortunately, it is more than anything his “theorem” that has passed
into posterity.
What happens if the hypotheses of the theorem do not hold? First of
all note that in many industrial pollution cases, property rights are
not defined. For a common resource like ocean fish, for example, it
is impossible to identify the polluters and the pollutees and then
put a negotiation into place; Coase’s theorem is therefore of no
great help to us in attacking overfishing. Even if property rights are
clearly defined, transaction costs are rarely negligible. For example,
these costs include expenses incurred during negotiation (lost time,
necessary recourse to lawyers, etc.). In the argument above, an ele-
mentary negotiation improves the social surplus by (c¢(q0) - b¢(q0))e. If
the salary of the retained lawyer is higher, the parties will renounce
this stage of the negotiation and will stop before having attained the
optimal level.
Recent literature has above all insisted on the transaction costs due
to asymmetrical information.12 If the polluter has private information
on b¢(q0) and the pollutee has private information on c¢(q0), each will try
12. Farrell (1987) offers a good discussion of this topic.

to “hog the blanket,” thereby manipulating the transfer t. Myerson-

Satterthwaite (1983) show that under these conditions the parties will
not be able to achieve a Pareto optimum. When the concerned parties
become more numerous, it is of course much more difficult for each to
them to manipulate the “prices” t. In fact the negotiation becomes again
asymptotically efficient when the number of agents tends toward infin-
ity (Gresik-Satterthwaite 1989). One can still wonder about the capac-
ity of a large number of polluters and pollutees to negotiate together,
since other transaction costs then risk coming out.
These critiques do not reduce the interest in the Coase theorem
to nothing. In fact Cheung (1973) shows that in the case of the
beekeeper-orchard crossexternality made popular by Meade, there
exist in the United States contracts between two parties that seek to
internalize the externality. It is therefore reasonable to think that under
certain conditions the private agents, left to themselves, can effectively
negotiate to arrive at a level of externality that, if not optimal, is at
least more satisfying. The imperfections of such a solution to the
externality problem must in any case be compared to those of the
Pigovian solutions in a world where governmental information is quite
imperfect.
Bibliography
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7 Nonconvexities
Certain results that I recalled in chapter 1 strongly rest on the absence

of nonconvexities in the economy. This is particularly the case for the
theorem on the existence of equilibrium and the theorem on the decen-
tralization of Pareto optima (the second fundamental welfare theorem),
for which it must be assumed that preferences and production sets are
convex.
In the first part of this chapter (sections 7.1 and 7.2), I present various
theoretical elements on the treatment of nonconvexities. In the second
part (section 7.3) I focus on natural monopolies, their pricing and their
regulation.
7.1 Consequences of Nonconvexities
7.1.1 Nonconvex Preferences
We will begin by looking at the effect that nonconvexities have on

the existence of equilibrium.1 The hypothesis that preferences are
convex has a long history. Originally the marginalists expressed utility
l
under an additively separable form: U(x) = S k=1 uk(xk). The convexity
of preferences then referred to the concavity of each utility index uk.
The latter could be justified by appealing to introspection or to ex-
perimental psychology (and particularly to the Fechner-Weber law,
which states that the response to a stimulus grows less and less
quickly when the intensity of the stimulus increases). In the more
modern framework where utility is not separable, the convexity of
1. Nonconvexities of course not only affect the existence of the equilibrium; the second
fundamental welfare theorem rests in a crucial way on a theorem of separation of con-
vexes, which is no longer applicable a priori when the preferences or the production sets
are not all convex.
preferences has been presented by arguments that bear on the monot-

onicity of marginal rates of substitution: if, for example, there are only
two goods, bread and water, the quantity of bread that the consumer
will be ready to sacrifice for a glass of water is smaller when his thirst
is already slaked.
Even if such reasonings seem intuitive in a simple world, they are not
necessarily so in the real world where very many goods coexist. The
convexity of preferences implies that if the consumer is indifferent to
two baskets of goods x and x¢, he will like any convex linear combina-
tion (lx + (1 - l)x¢) of those two baskets at least as much. The reader will
have no difficulty imagining counterexamples. Now, if the preference
convexities cannot be affirmed, the indifference curves can be shaped in
such a way that demand is discontinuous, as shown in figure 7.1. In a
two-good economy a small change in relative prices (symbolized by
budget lines in the figure) makes the tangency point pass from one side
of the nonconvexity to the other, thereby creating a discontinuity of
demand, which becomes a correspondence with nonconvex values. The
discontinuity of demand can directly result in the nonexistence of equi-
librium if the supply function intersects the segment.
x x
2 1
demand
indifference curve
x p /p
1 1 2
Fiugre 7.1
Nonconvex preferences
Nonconvexities 109
decreasing
returns average cost
increasing returns
marginal cost
x y* y
Figure 7.2
Small nonconvexity in production
7.1.2 Nonconvex Sets of Production
Now consider production. Here the convexity hypothesis particularly

implies that returns to scale are decreasing. This is a strong hypothesis. At
least two different cases may present themselves. In the first, which we
will call the “small nonconvexity”2 case, returns are increasing (possibly
with a fixed cost) for small-scale production and then become decreasing.
The production function has the aspect given in figure 7.2. This is the
classic textbook case since Marshall where the average cost curve is U-
shaped. Recall that there then exists a finite “efficient scale,” which is the
scale of firms in the long-term (free-entry) equilibrium of the industry:
this scale y* is given by the minimum of the average cost in figure 7.2—
and it is also the point where marginal and average costs are equal.
The second case is that of “large nonconvexities,” where returns to
scale are constantly increasing.3 The production functions in both
2. The reader should be forewarned that in this chapter I adopt my own preferred
terminology.
3. The efficient scale of the firm then is infinite; in fact it would be sufficient for it simply
to be larger than the size of the market.
x
Figure 7.3
Large nonconvexities of production, I
figures 7.3 and 7.4 illustrate this case. In the first figure there are fixed
costs, and then a constant marginal cost. In the second figure, marginal
costs are stubbornly decreasing. The first case obviously can only be an
approximation, but this approximation seems to be realistic in numer-
ous cases,4 which confers upon it a central role in models of regulation
(see section 7.3) and of industrial organization.
The consequences of these nonconvexities on supply functions are
very different. In the case of small nonconvexities, the supply curve is
discontinuous. In effect, the necessary first-order condition, which
equates price and marginal cost, only defines the firm’s supply if it does
not bring on losses. The supply curve therefore coincides with the mar-
ginal cost curve only for prices high enough to make production prof-
itable, as presented in figure 7.5. There is a threshold price for which
supply is either nil or equal to the efficient scale.
When there are large nonconvexities, the situation is more cata-
strophic: depending the respective values of the price and of the mar-
ginal cost at infinity, supply can only be zero or infinite, and the supply
curve therefore degenerates completely.
In any case, the nonconvexity of the production set always implies
that the supply curve is discontinuous, and that equilibrium is
4. For a software producing firm, for example, the costs are essentially due to research and
development, so they are fixed. The marginal cost of manufacturing a copy of software can
be considered as a constant. The pharmaceutical industry has similar characteristics.
Nonconvexities 111
x
Figure 7.4
Large nonconvexities of production, II
p=dC(q)/dq
q=0
Figure 7.5
Supply in the presence of a small nonconvexity
therefore compromised. In the case of large nonconvexities, there

cannot be an equilibrium; for small nonconvexities and a fixed number
of firms, equilibrium only exists if the efficient scale is small enough in
relation to the market scale.
7.2 Convexification by Numbers
In this section we will show that the impact of certain nonconvexities

becomes negligible when the number of agents rises to infinity. The
suitable mathematical tool is the Shapley-Folkman theorem, which
inserts itself into a gamut of remarkable results on convex sets. We will
note co D the convex envelope of D, which is the smallest convex set
containing D.
theorem 7.1 (shapley-folkman) Let D1, . . . , Dn be any nonempty

sets of IRl and a point z Œ co Sni=1 Di. Then
Ï" i = 1,..., n, xi Œ coDi

Ô n
$x1 , ... , xn , ÌÂ i =1 xi = z
Ô{i = 1, ... , n x1 œ Di } has at most l elements
Ó
This theorem tells us that in order to attain a point of the convex enve-
lope of the sum of n sets, it suffices to add n well-chosen points of the
convex envelopes of these sets, of which at most l do not belong to the
sets themselves. The interest of this result stems from the case where,
l being fixed, n tends toward the infinite. Consider now the case where
aggregate excess demand z(p) is discontinuous, so equilibrium does not
exist. It is easy to show that in compensation, a pseudoequilibrium p*
exists which verifies 0 Œ co z(p*), as we see in figure 7.6.
But, if there are l goods and n agents, application of the Shapley-
Folkman theorem shows that
Ï" i = 1, ... , n, zi Œco z i (p *)

Ô n
$x1 , ... , zn , ÌÂ i =1 zi = 0
Ô{i = 1, ... , n z1 œ zi (p *)} has at most l elements
Ó
Therefore an approximate equilibrium is obtained where only l of at

the most n agents are not situated on their excess demand curve. When
n becomes very large, the proportion of these agents becomes negli-
gible in the economy, since the number of goods l is fixed. It is in this
Nonconvexities 113
co z(p*)
p* p
Figure 7.6
Pseudoequilibrium
sense that the multiplication of the number of agents makes the

economy convex.
Let us look at the two examples more closely. Figure 7.7 corresponds
to nonconvex preferences. In figure 7.8 we see how the problem posed
by small nonconvexities in production is (approximately) solved. Note
the interpretation of this second example: to obtain an approximate
equilibrium, it is sufficient to have a well-chosen number of firms
produce at their efficient scale and allow the others to close. This will
be indifferent to these firms because they would all have zero profit
anyway.
7.3 Regulation of Natural Monopolies
Convexification by number solves nothing in the presence of large non-

convexities: the supply function is not clearly defined, so there does
not even exist pseudoequilibrium. Since equilibrium concepts are
no help to us, we must resort to optimality concepts. Now, what is the
best way to organize production? Assume an industry where a sole
D S
Figure 7.7
Convexification of preferences
Figure 7.8
Convexification in production
Nonconvexities 115
technology exists that is characterized by a cost function C(q). If we

determine that the optimum requires a production Q, productive effi-
ciency supposes that the total production cost is minimized. Therefore
we must solve the program
Ïmin n , q1, ... ,qn Â in=1 C(qi )

Ì
Ó Â in=1 qi ≥ Q
which jointly determines the number of operating firms and the dis-
tribution of production among them.
When marginal costs are increasing, the efficient scale q* is zero, and
it would be fitting therefore to multiply small production units. In the
presence of small nonconvexities, it is easy to see that the solution is n
= Q/q* (to the nearest integer): a certain number of firms are closed and
the others are made to produce at the efficient scale. The case that inter-
ests us here is that of large nonconvexities. Then the cost function is
subadditive; that is, we get
Ê n ˆ n
ËÂ Â C ( qi )
" n, " i = 1, ... , n, C qi £
i =1
¯ i =1
and n = 1 at the optimum. Clearly, it would be better if a single firm

produced, rather than production being shared among several firms.5
It is then said that the sector in question is a natural monopoly. For purely
technological reasons, it is fitting to grant a monopoly to one of the
firms. However, the firm may then behave as a monopoly, which is
socially harmful as we will see in chapter 9. Even economists the least
prone to intervention have traditionally thought that in such a situa-
tion, adequate regulation is called for.6 Habitually cited are trans-
portation, telecommunications, and what are called utilities in the
United States: water, electricity, and so on. All these sectors operate
with high fixed costs (or heavy infrastructure). For example, it seems
that even if competition is desirable, it would be absurdly costly to
double the number of railroad lines.
The problem that interests us in this section therefore is how this type
of sector must be regulated. Both economists’ opinions and adopted
5. This is fairly obvious when there are large nonconvexities of the first type (C(q) =
F + cq), since then adding firms reverts to the multiplication of fixed costs without
gaining in efficiency in other respects.
6. The maxim followed is: “Competition where possible, regulation where necessary.”
policy measures have far evolved in the course of recent years, and we
will try to understand why.
7.3.1 Marginal Cost Pricing
As we well know, in any interior Pareto optimum, the marginal rates

of substitution and the marginal rates of transformation must be equal.
As an example, consider an economy with two goods x and y, with y
produced from x with a production function y = f(x). If the consumer
has a utility function U(x, y), this amounts to
∂U ∂ x
= f ¢( x )
∂U ∂ y
But at the consumer’s optimum, it takes the form
∂U ∂ x px
=
∂U ∂ y p y
Now the cost function is C(y) = pxf -1(y), which gives a marginal cost
C¢(y) = px/f¢(x). From this we can deduce the rule according to which
any good produced must be priced at the marginal cost, regardless of
the conditions of its production:
p y = C ¢( y )
This rule is due in its general form to Hotelling (1938),7 who expressed
it like this:
The optimum of the general welfare corresponds to the sale of everything at

marginal cost.
Note that if Hotelling’s rule is applied, a firm subject to large noncon-

vexities will necessarily make losses since its average cost is always
higher than its marginal cost (e.g., a firm with cost function C(y) = F +
cy will lose its fixed costs). The firm must therefore be subsidized,
preferably with lump-sum transfers in order not to distort the agents’
choices—which would take the economy away from the Pareto
optimum.
Marginal cost pricing was the object of a long debate until the end
of the 1950s. Certain authors judged that it was an unattainable
7. Even if Dupuit (1844) had already stated it in the case of pricing bridge use.
Nonconvexities 117
optimum, particularly because the lump-sum transfers destined to

finance firm losses are difficult to implement. They suggested that one
be content with average cost pricing, the firm then making zero profit
by construction. Maurice Allais’s famous allegory of the mine and the
forest illustrates the inefficiency of this solution, even in the absence of
nonconvexities. Suppose that the inhabitants of a town can warm them-
selves one of two ways: by extracting coal from a mine or by chopping
down trees in a forest on a neighboring hill. The coal is mined at a con-
stant marginal cost, while the woodcutting has an increasing marginal
cost—for example, because it is necessary to climb higher and higher
to find slopes that have not yet been cleared of trees. Of course the pro-
duction optimum consists of cutting trees until the marginal cost of this
activity coincides with that of the coal mining. What happens if one
then prices at the average cost?
• The average cost of coal mining coincides with its marginal cost
• The average cost of chopping down trees is lower than its marginal
cost
For a given price of energy (independent of its origin), average cost

pricing implies therefore that the trees are cut beyond the point where
their marginal cost meets that of the coal. This leads then to a waste of
labor and distances the economy from the production optimum.8
It is clear that average cost pricing is almost never optimal, but that
marginal cost pricing loses its character of optimality in the presence
of tax distortions. The choice of pricing then becomes a second-best
problem, which one can only study by modeling the sources of distor-
tion. This is what we will do below.
7.3.2 Second-Best Pricing of Regulated Firms
Many natural monopolies are state-owned firms (or are state-regulated

in some way). We will study two models that emphasize two aspects
of regulation. The Ramsey-Boiteux model9 studies optimal pricing of a
part of the regulated sector submitted to budgetary equilibrium when
only unit taxes are possible. Baron-Myerson (1982) analyze the problem
8. Another possible example is that a bridge should be constructed because it engenders

a positive net surplus, but no toll rights level exists that could finance its construction.
9. Ramsey (1927) was the first to solve the optimal taxation problem; Boiteux (1956)
was more interested in the problem posed in the text. Their two models are formally
equivalent, as is evident by the fact that the two authors are joined in its name.
of the regulator when the regulated firm possess private information

on its costs.
The Ramsey-Boiteux Formula

Here we take a set of regulated firms as one big firm. This firm pro-
duces goods (x1, . . . , xl) at a cost (C + c1x1 + . . . + clxl). The product k is
taxed at a rate tk, so its consumption price is pk = ck + tk. The firm faces
a budgetary constraint10
l
C = Â tk x k (p1 , ... , pl )
k =1
where the xk are demand functions.

In the economy there is a numéraire good y and a representative
consumer endowed with an income R whose utility function is quasi-
linear (absence of wealth effect) and additively separable: U = Slk=1 uk(xk)
+ y. With these specifications, the demand functions are simply
given by
uk¢ (x k (p k )) = p k
The regulator problem consists in maximizing the utility of the con-

sumer under the budgetary constraint:
Ïmax (p1 , ..., p1) ( Â lk= 1uk ( xk (p k )) + R - Â lk= 1pk xk ( pk ))

Ì
Ó C = Â lk=1 (p k - ck ) xk ( pk ) (l )
where l is the multiplier attached to the constraint.

The first-order conditions give
"k = 1, ... , l , uk¢ x k¢ - p k x k¢ - x k + lx k + l (p k - c k )x k¢ = 0
Put differently, after utilization of u¢k = pk and rearrangement of the

terms,
p k x k¢ p k - c k 1 - l
"k = 1, ... , l , =
xk pk l
If we denote ek = -pkx¢k/xk, the demand elasticity for good k and tk =
(pk - ck)/pk, the tax rate on this good, finally we get the Ramsey-
Boiteux formula:
10. The results will be qualitatively the same if one assigns only to the transfers given
to the firm a cost that takes into account the distortions entailed by their collection.
Nonconvexities 119
l -1
" k = 1, ... , l , e kt k =
l
The formula shows that the tax rate on a good must be inversely
proportional to the elasticity of demand for that good. This result is
easily understood: it is advisable to limit the distortions linked to
non-lump-sum taxation and therefore to tax more vigorously goods
that are the least sensitive to price variations. For that matter, the
Ramsey-Boiteux formula implies that at first-order, consumptions of
all goods are discouraged in the same way: simple calculations show
that if the fixed costs (and therefore the tax rates necessary to finance
them) are very small, the relative drop in the consumption of good k is
given by
dx k x k¢ tk
- =-
xk xk
x k¢ p kt k l - 1
=-
xk l
l -1
=
l
which is independent of the good k. Therefore an effective second-best
taxation must not equalize the tax rates but rather equally discourage
consumptions of different goods.11 Unfortunately, goods with high
price elasticities are often those with high income elasticity, that is,
luxury goods. Contrary to what common sense suggests, luxury goods
must therefore be taxed less heavily than necessities. Insofar as this
leads to taxing the consumption of the poor more heavily than that of
the rich, it is advisable to take into account the distributive concerns in
this model. Then the objective to maximize is no longer the utility of a
representative consumer but a weighted sum of the consumers’ utili-
ties: Sni=1 aiUi, where ai is the weight of the consumer i in the social utility
(and Sni=1 ai = 1). Very similar calculations to those above show that we
get at the second-best optimum
l - rk
t ke k =
l
where ek is the price elasticity of aggregate demand Xk for good k and

rk is the “distributive factor” of that good:
11. This result remains valid in the presence of income effects and cross-elasticities of
demand.
rk =
Â in=1a i xki
Â in=1xki
We note that rk is one in the absence of redistributive objectives (all ai
being equal); it is more than one if the consumers whose weight in
social utility is the highest (which are generally the “poor”) consume
more of good k than the others. The modified Ramsey-Boiteux formula
says therefore that for given elasticities, the tax rate must be smaller
on goods that are consumed mostly by the poor. This could also be
interpreted in noting that consumptions of the poor must be less
discouraged than those of the rich.12
It should be noted that the Ramsey-Boiteux formula is quite remote
from the situation of public transportation which operates on the
basis that each mode must be entirely financed by its users. Nothing in
the Ramsey-Boiteux model suggests a budgetary equilibrium on each
good is desirable. By comparison, a French report cited by Henry
(1989) calculated that given the estimated elasticities, three-quarters
of the fixed costs of rail transport should be financed by taxes on
highway traffic.
As suggested by the public transportation example above, Ramsey-
Boiteux pricing has not had much impact on decision makers, perhaps
because it gives no simple rule. Scott (1986) studies the example of the
United States Postal Service to show how real-world decision makers
try to draw inspiration from the Ramsey-Boiteux formula while taking
into account other extra-economic constraints.
Asymmetrical Information Regulation

Recent literature on the regulation of firms has insisted on the strate-
gic importance of private information at the disposal of regulated firms.
For example, it is clear that it behooves such a firm to overestimate its
costs to obtain a more favorable treatment. The regulator must, in order
to avoid this snag, construct a revealing mechanism which delegates
as much of the decision-making to the firm as possible. Formally, this
is a Principal-Agent problem where the regulator is the Principal and
the firm is the Agent.
Consider then a natural monopoly of cost function C(q, q) = K + qq.
The firm is brought to budgetary equilibrium by a subsidy t levied on
12. The numerical applications conducted on developing countries show that the
consumption of certain basic foods should in fact be subsidized.
Nonconvexities 121
the consumers, the social unit cost of which is l: when the government
collects a tax t, the utility of the consumers decreases by (1 + l)t because
of tax management costs and the economic distortions which accom-
pany taxes.13 The firm’s profit is therefore
t - C(q,q )
while the consumer surplus is

q
S(q) - (1 + l )t = Ú0 P(x)dx - (1 + l )t
where P is the inverse demand function. We will assume that the

regulator’s objective is social welfare, or the nonweighted sum of the
consumer surplus and of firm profit:
S(q) - C(q,q ) - lt
First-Best
Suppose that the regulator observes q. Then it would suffice him to
solve
Ïmax ( q ,t ) [S(q) - C(q, q ) - lt]

Ì
Ó t - C (q , q ) ≥ 0
whose solution is of course t = C(q, q) and S¢(q) = P(q) = (1 + l)q, that

is we see again the marginal cost pricing rule, corrected here for the
cost of public funds l.14
Second-Best
In fact it is more realistic to suppose that information is asymmetric:
the firm obviously knows q, but the regulator simply has a Bayesian a
priori such that q follows a cumulative distribution function F with
–
density f on an interval [q– , q ]. The revelation principle introduced in
chapter 4 applies here: the regulator must find a direct mechanism (q(q),
t(q)) that maximizes the objective
q
Úq {S[q(q )] - C[q(q ),q ] - lt(q )} dF(q )
13. Empirical estimates suggest that the value of l can be on the order of 0.2.
14. It is therefore the social marginal cost (1 + l)q that we must consider here.
under the incentive constraints (IC) and under the participation con-
straints (IR). The incentive constraints state that the firm q chooses the
pair (q(q), t(q)) destined to it and are written as
" q , q ¢ , t(q ) - C(q(q ), q ) ≥ t(q ¢) - C(q(q ¢), q ) (IC)
The participation constraints guarantee a nonnegative profit to the
firm:
" q , t(q ) - C[q(q ), q ] ≥ 0 (IR)
Denote by r(q) = t(q) - C(q(q), q) = t(q) - qq(q) - F the informational
rent of the firm q. This is the profit that it succeeds in obtaining
thanks to its private information. The incentive constraints can be
written
" q , q ¢ , r(q ) ≥ r(q ¢) - (q - q ¢) q(q ¢) (IC)

By exchanging the roles of q and q¢, we get a second inequality that can
be combined with the first. It is written as
" q , q ¢ , (q - q ¢) q(q ) £ r(q ¢) - r(q ) £ (q - q ¢) q(q ¢)
which implies, in particular, that (q - q¢)(q(q¢) - q(q)) ≥ 0 and there-

fore that q is decreasing. Moreover, by assuming that q £ q¢ and by
dividing by (q¢ - q), we get
r(q ¢) - r(q )
- q(q ) £ £ - q(q ¢)
q¢ -q
So at every point where q is continuous,15 r may be differentiated and
r¢ = -q. We can then deduce that r is convex since q is decreasing.
Since q is positive, r decreases, and the constraints of participation
–
amount to just r(q ) = 0. Therefore we get
q
r(q ) = Úq q(t) dt
which, incidentally, gives transfers t in terms of q:

q
t(q ) = C(q(q ), q ) + Úq q(t) dt
Having eliminated the transfers, it only remains for us to maximize
15. It is easily shown that q, being a strictly decreasing function, can only have a count-
able set of points of discontinuity.
Nonconvexities 123
q
( q
Úq S[q(q )] - (1 + l )qq(q ) - l Úq q(t) dt ) dF(q )
under the sole constraint q¢(q) £ 0.
To solve this program, we should first neglect the constraint that q
be decreasing. The optimum is then found by integrating the objective
by parts, whence
q Ê F(q ) ˆ
Úq ÁË S[q(q )] - (1 + l )qq(q ) - l f (q ) q(q )˜¯ dF(q )
This result is simply maximized point by point, whence finally
F(q )
S ¢[q(q )] = P[q(q )] = (1 + l )q + l
f (q )
If (q + F(q)/f(q)) is an increasing function (which is a hypothesis
verified by the majority of usual distributions), then the right-hand
member is increasing,16 and since P is decreasing, the solution q(q)
verifies the constraint q¢(q) £ 0.
Therefore the price this time is higher than the social marginal cost;
the supplementary term is used to compel the firm to reveal its type.
It leads to underproduction: q is smaller than in the first-best. This
underproduction stems from the fact that if a regulator wants to
augment production of a q while maintaining the incentives to reveal
their type for all the other types, what also must be augmented is the
informational rent of all types more efficient than q (the q ¢ < q). In more
intuitive terms, the regulator must keep the q-type firm from posing as
a less efficient firm q¢ > q and thus from pocketing a higher subsidy.
The most economical way to do this is to reduce the production of
inefficient firms, so that they get smaller subsidies, and more efficient
firms are not tempted to lie.
The reader should note here that the hypothesis that costs are unob-
servable is not always the most realistic. Laffont-Tirole (1986) intro-
duced a slightly different model where the costs C of the firm depend
simultaneously on an efficiency parameter and on a level of effort, both
of which are unobserved though the costs are (imperfectly) observable.
They show that the optimal incentive mechanism then consists in
offering a menu of linear prices t = a + bC where there are as many
(well-chosen) pairs (a, b) as possible parameters of efficiency. The most
16. For the study of the general case, I refer the reader to Salanié (1997, ch. 2).
efficient firm chooses a fixed-price contract (a tariff with b = 0, which

assures the firm a transfer independent of its costs), and the less
efficient firms choose a tariff corresponding to a higher b.17
7.4 Deregulation
The regulation of natural monopolies has greatly evolved over the last
few years as numerous firms of these industries were privatized and
often faced fierce competition. The designation of this movement as
“deregulation” is somewhat inaccurate, since one form of regulation
was being displaced by another.18 This has proved to be a fertile field
of application for the models of section 7.3.2.
Deregulation is based on a triad of beliefs:
• Technological innovations make natural monopolies more expend-
able: even where the fixed costs remain high at the infrastructure level,
infrastructures can be separated from the actual service activity, where
returns to scale are rarely increasing.19
• The diversified product range of these sectors (one need only think
of the new telecommunication products), makes detailed regulation

problematic.
• Competition in itself is an effective regulator.
The first two statements do not call for any further comment.
Concerning the third belief, there is the idea proposed a number of
decades ago by Demsetz (1968). Demsetz argued that rather than
regulate the distribution of utilities like electricity, the right to distrib-
ute electricity could be put up for auction and then granted to the
highest bidder. In this way the monopoly rents would transfer to the
taxpayer who could select the most efficient operator. The most recent
literature insists on the use of competition to reduce informational
rents: the underlying idea is that if the industry is left to several firms
whose costs are correlated, the observation of their behavior in the
17. A similar contract menu was effectively used in the United States for regulating the
“Baby Bells,” the seven local telephone firms created by the breakup of AT&T.
18. Kay and Vickers (1988) provide a good explanation of how deregulation has replaced
structural regulation (which defines the marketplace and the number of firms that can
operate there) by regulation of the behavior of firms.
19. This distinction particularly applies to network industries like transportation,
telecommunications, and energy.
Nonconvexities 125
framework of the Baron-Myerson model permits their costs to be better

understood.
Clearly, it is incorrect to say that deregulation is a uniform and uni-
versal movement. Still we can point to certain common themes (aside
from privatization, which would merit special treatment):
• Infrastructures and the services used by these infrastructures are sep-
arated. This permits the activity with increasing returns to be isolated.
The infrastructures have sometimes been regrouped in a firm created
for that purpose; in other cases (e.g., telecommunications) one firm has
conserved all the property or infrastructure but has allowed its com-
petitors access, subject to the payment of a fee.
• Increasing returns are a negligible phenomenon in the opening of
services to competition in the market.
• Price-cap type of regulation formula is adopted. The most popular
was recommended by the 1983 Littlechild Report in the United
Kingdom and is called RPI - X; it consists of limiting the growth rate
of a weighted average of the firm’s prices to the increase of the retail
price index after deduction of an expected productivity growth rate.
Bibliography
Baron, D., and R. Myerson. 1982. Regulating a monopoly with unknown costs.
Boiteux, M. 1956. Sur la gestion des monopoles publics astreints à l’équilibre budgétaire.
Econometrica 24: 22– 40.
Demsetz, H. 1968. Why regulate utilities? Journal of Law and Economics 9: 55–65.
Dupuit, J. 1844. De la mesure de l’utilité des travaux publics. Annales des Ponts et Chaussées
8: 332–75. Published in English in P. Jackson, ed. The Foundations of Public Finance. Elgar,
Cheltenham, England, 1996.
Henry, C. 1989. Microeconomics for Public Policy: Helping the Invisible Hand. Clarendon
Press, Oxford.
Hotelling, H. 1938. The general welfare in relation to problems of taxation and of railway
and utility rates. Econometrica 6: 242– 69.
Kay, J., and J. Vickers. 1988. Regulatory reform in Britain. Economic Policy 7: 286–351.
Laffont, J.-J., and J. Tirole. 1986. Using cost observation to regulate firms. Journal of
Political Economy 94: 614–41.
Ramsey, F. 1927. A contribution to the theory of taxation. Economic Journal 37: 47–61.
Salanié, B. 1997. The Economics of Contracts: A Primer. MIT Press, Cambridge.

III Industrial Organization
The next five chapters are devoted to industrial organization. Indus-

trial organization theory, or simply IO, applies to the operation of
isolated markets as opposed to general equilibrium theory, which is
concerned with the global operation of an economy. IO theory rests on
a fairly detailed description of strategic interactions among agents.
The terms “imperfect competition” and “industrial organization” are
nearly synonymous. They can be defined by antithesis in recalling the
four classical conditions for perfect competition:
• The number of each market’s buyers and sellers is very high
• There are no barriers to entering the market
• The exchange concerns a homogeneous good
• The information on prices is perfect
One enters the domain of industrial organization as soon as one of

these conditions ceases to be verified.
The classical economists were interested, above all, in perfect com-
petition. Their aversion to what they called “monopolies” had to do
more with the existence of barriers to entry than with the modern
concept of a monopoly.1 The theory of imperfect competition came
from Cournot (1838)2 and its criticism from Bertrand (1883). Unfortu-
nately, both contributions did not receive the welcome they deserved,
1. Thus Adam Smith under the name “monopoly” deplored the existence of professional
corporations that regulated the behavior of participants on the market.
2. I can’t praise highly enough Cournot’s little book. In it one finds a strong plea for the
use of mathematics in economics (introduction), the discovery of demand curves and
(with very few exceptions) of elasticity (chapter IV), the resolution of the monopoly
problem (chapter V), a theory of oligopoly (chapter VII) and its competitive limit (chapter
VIII). All this appeared at least fifty years ahead of its time, and in a form that remains
quite modern.
128 Industrial Organization
so Alfred Marshall’s (1890) famous manual in this domain really redis-

covered Cournot’s theory of monopoly.
Three important contributions must be noted between the two World
Wars. Hotelling (1929) invented the horizontal differentiation model
which remains at the foundation of numerous developments in indus-
trial and spatial organization. Chamberlin (1933) and Robinson (1933)
had both ambitiously planned to re-establish microeconomics around
a theory of imperfect competition: monopolistic competition for
Chamberlin, and monopoly for Robinson.
Industrial organization developed more fully in the 1950s. It was
first dominated by Harvard, where the central figure was Joe Bain.3
The work of the Harvard economists was hinged by the triptych
structure–conduct–performance: it went from the organization of a sector
of the economy, to its consequences on the behavior of firms, and then
to their profits. A typical application of this type of analysis was a
regression of the profit rates of different sectors on a measure of
the sectors’ concentration. In the 1960s Chicago economists, grouped
around George Stigler, offered a more Popperian approach of indus-
trial organization, which consisted of proposing different theories and
then attempting to choose from among them on the basis of empirical
tests.4 In the 1970s the torch was taken up by economists who system-
atically applied to industrial organization tools of game theory by
constructing more elaborate theoretical models which often contained
an explicit dynamic component. This part of the book is situated in the
theoretical models of this last school. The reader should not presume
to find an exhaustive treatise on industrial organization. Only the
basics are given with illustrations using concrete examples. Readers not
satiated by the contents of the next few chapters can refer to Tirole’s
manual cited in the introduction.
The reader will notice a certain analytical rupture between parts II
and III. In part III we effectively exit the framework of general equi-
librium and interest ourselves in partial equilibrium models. Chapter
3. We should note also the work of Schumpeter (1942), which led among other
things to the rehabilitation of large firms as essential vectors of “creative destruction.”
Schumpeter argued that innovative firms constantly replaced those that had exhausted
their innovation potential. Unfortunately, we do not yet have sufficiently sophisticated
dynamic models to test Schumpeter’s intuitions, but we will but touch upon them in
chapter 11.
4. The Chicago school is also associated with an optimistic vision of the principle of
laissez-faire and with deep skepticism about the government’s capacity for worthwhile
intervention.
Industrial Organization 129
8 attempts to explain why it is difficult to construct a theory of general

equilibrium of imperfect competition.
Bibliography
Bertrand, J. 1883. Théorie des richesses. Journal des Savants, 499–508. Published in English
in A. Daughety, ed. Cournot Oligopoly: Characterization and Applications. Cambridge: Cam-
bridge University Press (1988).
Chamberlin, E. 1933. The Theory of Monopolistic Competition. Cambridge: Harvard

University Press.
Cournot, A. 1838. Recherches sur les principes mathématiques de la théorie des richesses. Pub-
lished in English as Mathematical Principles of the Theory of Wealth, James and Gordon, San
Diego, CA 1995.
Hotelling, H. 1929. Stability in competition. Economic Journal 39: 41–57.
Marshall, A. 1890. Principles of Economics. London: Macmillan.
Robinson, J. 1933. The Economics of Imperfect Competition. London: Macmillan.
Schumpeter, J. 1942. Capitalism, Socialism and Democracy. New York: Harper and
Brothers.
8 General Equilibrium of
Imperfect Competition
While models of welfare economics are set in a general equilibrium

framework, those of industrial organization are partial equilibrium
models. The study of the complexity of strategic interactions is a suffi-
ciently difficult problem without our adding in general equilibrium
considerations. Moreover, if we limit ourselves to the most simple
strategic concepts—what is called Cournot or Bertrand competition—
their immersion in a framework of general equilibrium poses delicate
problems for which no satisfactory solution has yet been found. This
chapter is dedicated to this second point. Without sinking into overly
technical digressions, we will get an idea of the difficulties entailed
when attempts are made to construct models of general equilibrium in
imperfect competition. It will be worthwhile for those interested in
these questions to refer to Bénassy (1991), Bonanno (1990), or Hart
(1985).1
8.1 Three Difficulties
Each model of general equilibrium in imperfect competition harbors its

own difficulties. The objective of this section is to discuss three
problems that intervene in all of these models: the ambiguity of the
objective of noncompetitive firms, the noninvariance of equilibrium
in relation to the price normalization rule, and the impossibility
of finding conditions that ensure that profit functions have good
properties.
1. The reader should be alerted to the fact that these writings use the term “mono-
polistic competition” in a much broader sense than the one I define in chapter 10.
8.1.1 The Firms’ Objectives
In competitive equilibrium a firm’s shareholders are unanimous in

asking for profit maximization. Assume in effect a consumer i with
initial resources wi who possesses a fraction qi of the firm. If he is faced
with prices p, his indirect utility will be
Vi (p , p ◊w i + q ip )
when the firm makes a profit p. This expression is of course increasing

in p; in other respects, the firm, like its stockholders, takes prices as
givens by definition. It is therefore clear that all the stockholders desire
the firm to maximize its profits.
Things are more complicated when the firm has market power. Then
its profit p(p) is a function of a variable that it can choose (the price that
it tariffs, e.g., p1). Say, the firm takes the prices of other goods as givens.
It will then be necessary for it to maximize in p1 in order to please con-
sumer i,
Vi [p , p ◊w i + q ip (p)]
So clearly this problem is no longer reduced to profit maximization.

Rather the maximum is characterized by
∂Vi ∂Vi Ê ∂p ˆ
+ Á w i1 + q i ˜ =0
∂ pi ∂ R Ë ∂ p1 ¯
Using Roy’s identity and denoting the excess demand for good 1 zi1 =
xi1 - wi1, the result is
∂p zi 1
=
∂ p1 q i
If p is concave in p1 and zi1 is positive, that is, if the consumer i con-
sumes a lot of the good 1 that the firm produces, the latter will have to
reduce the price p1 below the level that maximizes its profits. Moreover
there is no reason for the firm’s shareholders to agree on the price that
must be fixed. It is therefore impossible to assign the firm a simple
objective.
This theoretical difficulty is not final. Most consumers actually buy
a very small quantity of the products of firms in which they are stock-
holders. It is reasonable, at first approximation, then to think that profit
General Equilibrium of Imperfect Competition 133
maximization is an acceptable objective. In any case, this chapter rests

on this hypothesis.2
8.1.2 Price Normalization
In competitive equilibrium only the relative prices are important. The

price vector can be normalized by fixing any price at 1, for example,
without that choice having the least influence on the relative prices or
on the allocations of equilibrium. Such is no longer the case in imper-
fect competition. To see this, it is enough to consider a monopoly that
chooses its prices for profit maximization: if its price is fixed at 1, its
maximization program does not make any sense.
This difficulty is inherent in all models where agents perceive a rela-
tion between prices and quantities: equilibrium depends on the rule of
normalization.3 Subsequently we will implicitly assume the existence
of a good (the numéraire) produced and consumed in competitive con-
ditions, and we will normalize its price at 1. However, this solution is
not really satisfactory. In fact the definition of the firms’ objective and
the choice of a price normalization rule are inextricably bound together.
Dierker and Grodal (1996) propose solving this double problem by
assuming that firms maximize what they call the “real stockholder
wealth,” but this proposition raises other difficulties.
8.1.3 The Quasi-concavity of Profit
The last general difficulty we will address is important, albeit a bit tech-
nical. Our objective is to demonstrate the existence of general equilib-
rium based on Kakutani’s fixed-point theorem. The theorem applies to
semi-continuous correspondences with convex values. Whatever is the
firm’s desired strategic variable (its price or its quantity) then that is
its best choice correspondence be of convex values. The only simple
hypothesis that assumes this property deals with the quasi-concavity
of the profit function with respect to the strategic variable.4
In partial equilibrium, it is easy to find hypotheses that ensure
that the profit function is quasi-concave—concavity of demand, for
2. We will see a similar, though more pronounced, difficulty in chapter 13.

3. This discussion is a bit abstract; it will be illustrated by an example in section 8.3.1.
4. Recall that a function f of IRn in IR is quasi-concave if and only if, for every y of IR,
the set of x such that f(x) ≥ y is convex. The standard example of a function that is not
quasi-concave is a multimodal function.
example. Unfortunately, in general equilibrium demand functions

themselves proceed from consumer optimization, and they cannot be
assumed to be concave a priori. In fact, as Roberts and Sonnenschein
(1977) showed, there are instances where the basic economic data
satisfy all the usual hypotheses and yet the profit function is not quasi-
concave, which brings about the nonexistence of equilibrium.
At the present time there does not exist a satisfactory solution to this
problem. All the literature therefore just assumes that profit functions
are quasi-concave, as we will do in this chapter.
8.2 Subjective Demand Equilibrium
The first contribution of import in this domain is due to Negishi (1961).

Negishi’s analysis centers on subjective inverse demand functions, in
other words, price variations that noncompetitive firms foresee when
they modify their production.
Let s = (p, x, y) be an economic state where p represents the price
vector, x the consumptions, and y productions. Negishi supposes that
the firm j is endowed with a “conjecture” or subjective demand
Pj (y ¢j , s)
which represents the price that the firm anticipates for goods for which
it has market power when, leaving the state s, it modifies its produc-
tion yj to y¢j . Negishi dictates only that this conjecture be compatible
with the state of departure in the sense that
Pj (y j , s) = p j
so that the firm, when it does not deviate from the original state, is not
mistaken about the price.
A subjective demand equilibrium is an economic state s* = (p*, x*, y*)
which balances all markets, such that consumers maximize their utility
under budgetary constraint and that for each noncompetitive firm j of
the production set Yj, y*j maximizes its perceived profit
Pj (y j , s *) ◊ y j
in yj Œ Yj.
We can show that equilibrium exists under the usual hypotheses
when perceived profit functions are quasi-concave.5
5. This is for example, the case where conjectures are linear, as Negishi supposed.
8.3 Objective Demand Equilibrium
The disadvantage to subjective demand equilibrium is that it intro-

duces an unknown element (conjectures) into the determination of the
equilibrium. There is a risk therefore of the existence of very numer-
ous equilibria when the conjectures vary. Thus competitive equilibrium
is obtained when for each noncompetitive firm j and each production
plan yj,
Pj (y j , s *) = p*j
Moreover certain of these equilibria can rest upon conjectures that are
entirely wrong. The polar hypothesis consists of assuming that non-
competitive firms perceive “objective demands,” that is, true demand
functions. We will see two examples of this, depending on whether
firms compete in quantity or in price.
8.3.1 Equilibrium in Quantities
Gabszewicz and Vial (1972) integrated Cournot competition in a model

of general equilibrium. Suppose that all firms j = 1, . . . , m are non-
competitive. If they choose production plans y = (y1, . . . , ym), define the
“modified resources” of consumers as
m
w i¢(y) = w i + Â q ij y j
j =1
which take into account the fact that each consumer is owner, through
his shares, of a portion of the firms’ production.
Now consider the “modified economy” where there is no firm and
where consumers’ resources are the modified resources w¢i (y). If all goes
well (we return later to this point), this economy has a unique equilib-
rium price vector P(y) that continuously depends on y. We can conse-
quently define a “Cournot-Walras equilibrium” by a price vector p*,
consumptions x* which maximize consumers’ utility under budgetary
constraint, and productions y* such that
• p* = P(y*)
• the equilibrium is a Cournot equilibrium, in the sense that for every
firm j, y*j maximizes P(yj, y*-j ) · yj in yj Œ Yj

Here again, an equilibrium exists under the usual hypotheses if the

profit functions are quasi-concave (recall that this is not in any way
guaranteed). Nevertheless, it is not at all obvious that the function P is
well defined. It is perfectly possible that w¢i (y) is no longer within the
consumption set Xi (e.g., because the modified resources have negative
components), in which case the modified economy has no equilibrium.
If, on the contrary, the economy has several equilibria, it may further
be impossible to continuously select only one. We are therefore reduced
to assuming that the modified economy has a sole equilibrium, and this
is hardly satisfactory.
It is easy to see by this definition why equilibrium depends on
the chosen rule for normalizing prices. For example, define the
q-norm by
1q
p q = ÊË Â pl ˆ¯
q
If one decides to normalize the prices in a modified economy by divid-

ing them by their q-norm, then in Cournot equilibrium firm j must
maximize in yj the expression
P( y j , y *- j )
◊yj
P( y j , y *- j ) q
But, since the denominator depends on yj, different choices of q—which

have no effect in perfect competition—will lead to different Cournot
equilibria.
Note also that this model is meaningless unless the only inputs are
possessed by the consumers. If firm j uses as input a good produced
by firm k, then when j changes its production, it modifies its demand
to k and then cannot reasonably expect the production of k to remain
constant, which is the definition of Cournot’s equilibrium. Therefore
this concept cannot take into account the existence of pure intermediate
goods, for which the final consumer demand is nil.
8.3.2 Equilibrium in Prices
We can equally integrate price competition in a general equilibrium

model. No concept truly prevails here. We will briefly review that of
Bénassy (1988), which depends on fixed price equilibrium. By analogy
with the Cournot-Walras equilibrium, suppose that the firms have

chosen prices p = (p1, . . . , pm). An initial difficulty is that at these prices,
there may not be feasible production plans that equalize supply and
demand. It may even be optimal for certain firms not to serve all the
demand applying to them. For these two reasons Bénassy is interested
in fixed price equilibrium when the prices are p. Let y*j (p) be the pro-
duction plane of firm j at fixed price equilibrium (assuming here again
that it is unique). Then firm j will choose a price pj that maximizes its
profit, given other firms’ prices so that
max p j ◊ y *j (p j , p - j )
pj
If the profit is quasi-concave, this defines a correspondence pj Œ

yj (p- j), which has good properties. The general equilibrium of imper-
fect competition in prices is then the Nash equilibrium in prices p*
given by
"j = 1, ... , m, p*j Œy j ( p *- j )
The drawback to this concept is of course that it rests on the theory

of fixed price equilibrium and is therefore subject to the same
criticisms.
8.4 Conclusion
What lessons can we extract from this brief overview of general equi-
librium in imperfect competition? The one objection we can make is
that insofar as we know nothing of firms’ conjectures, the equilibrium
remains largely indeterminate. We can even show that very many pro-
duction plans make up subjective demand equilibria when conjectures
are varied. The disadvantage to objective demand equilibria is different:
each firm is assumed to be capable of calculating the equilibrium of the
economy as a whole and to take into account all the effects of general
equilibrium in modifying its decisions. This is clearly not realistic.
One way to handle the problem consists in making objective demand
equilibria less demanding in terms of information by authorizing firms
to neglect certain effects. Thus Laffont-Laroque (1976) assume that the
firm neglects its indirect influence on markets where it does not directly
intervene. Hart (1985) supposes that the firm neglects the “Ford effect,”
that is, the impact of its profits on demand aimed at the firm by con-
sumers who may also be its stockholders.
Another solution consists in imposing more restrictions on conjec-

tures in a subjective demand equilibrium. Gary-Bobo (1989) thus shows
that if firms cannot be mistaken about the local demand elasticity that
applies to them, then the equilibrium coincides with the Cournot-
Walras equilibrium. This contribution reconciles to some extent sub-
jective and objective demand equilibria by showing that they can end
in the same allocation.
At this stage of research it is very evident that the transposition of
the Arrow-Debreu general equilibrium model to an economy where
competition is imperfect poses too many problems that have not yet
been solved.6 The literature concentrates, as a result, on the analysis of
partial equilibrium situations by using noncooperative game theory;
this is the approach we will take in the next five chapters.
Bibliography
Bénassy, J.-P. 1988. The objective demand curve in general equilibrium with price makers.
Economic Journal 98: S37–S49.
Bénassy, J.-P. 1991. Monopolistic competition. In K. Arrow and M. Intriligator, eds., Hand-
book of Mathematical Economics, vol. 4. Amsterdam: North-Holland.
Bonanno, G. 1990. General equilibrium theory with imperfect competition. Journal of

Economic Surveys 4: 297–328.
Dierker, E., and B. Grodal. 1996. The price normalization problem in imperfect competi-
tion and the objective of the firm. Working paper 9616. University of Vienna.
Gabszewicz, J.-J., and J.-P. Vial. 1972. Oligopoly “à la Cournot” in General Equilibrium
Analysis. Journal of Economic Theory 4: 381–400.
Gary-Bobo, R. 1989. Cournot-Walras and locally consistent equilibria. Journal of Economic

Theory 49: 10–32.
Hart, O. 1982. A model of imperfect competition with Keynesian features. Quarterly
Journal of Economics 97: 109–38.
Hart, O. 1985. Imperfect competition in general equilibrium: An overview of recent

work. In K. Arrow and S. Honkapohja, eds., Frontiers of Economics. Oxford: Basil
Blackwell.
Laffont, J.-J., and G. Laroque. 1976. Existence d’un équilibre général de concurrence
imparfaite: Une introduction. Econometrica 44: 283–94.
6. These underlying ideas have nevertheless found a fertile field of application in

macroeconomics. Following Hart (1982), numerous authors demonstrated that the
introduction of imperfect competition can engender Keynesian effects of economic
policy measures in a general equilibrium model. Their work is discussed by Silvestre
(1993).
Negishi, T. 1961. Monopolistic competition and general equilibrium. Review of Economic

Studies 28: 196–201.
Roberts, J., and H. Sonnnenschein. 1977. On the foundations of the theory of monopolistic
competition. Econometrica 45: 101–13.
Silvestre, J. 1993. The market-power foundations of macroeconomic policy. Journal of

Economic Literature 31: 105–41.
9 Prices and Quantities
In this chapter we review the bases of imperfect competition: the theory

of monopoly and the theory of oligopoly in prices and in quantities.
These theories are often covered in introductory courses, so readers
having mastered them can pass over this chapter without considerable
loss of continuity.
9.1 Monopoly
The usual definition of a monopoly corresponds to the case of a single

firm in the market place. If we accept this definition, there exists no
economic sector that can be said to have a monopoly. Even firms that
manage to create temporary monopolies for themselves, thanks to an
invention (e.g., Sony’s Walkman), are subject to competition from firms
that produce substitutes (e.g., transistor radios). Yet certain firms, for
reasons that we will explore below (see chapter 11), effectively succeed
in ensuring themselves a monopoly on the market of a good that has
no very close substitute. So at first glance, their situation has the
semblance of a monopoly.
The distinction between a competitive firm and a monopoly is clear:
• A competitive firm with decreasing returns takes the price p as a
given and produces a quantity q such that price and marginal cost are
equal: p = C¢(q)
• A monopoly chooses its price p while considering the effect on the
demand q = D(p)
A monopoly with cost function C(q) (with marginal costs assumed to

be increasing) chooses the price p that maximizes the profit (pD(p) -
C(D(p))), or symmetrically, the quantity q that maximizes the profit
(P(q)q - C(q)). The first-order condition of this second program is

written
C ¢(q m ) = P(q m ) + qP ¢(q m )

The right-hand member of this equation is, by definition, the marginal
revenue, MR(qm), which is lower than the price because P¢(q) is negative:
the monopoly takes into account the fact that it cannot sell off increased
production except by accepting a price cut. We can already deduce
from this that a monopoly tariffs above its marginal cost. As for the first
program, after some elementary calculations, it gives the formula
p m - C ¢ (q m ) 1
=
p m
e (p m )
where e(p) = -dlog D(p)/dlog p is the elasticity of demand. The left-hand
member of this formula is called the Lerner index or markup. It is a
measure of a firm’s capacity to exploit its market power in order to
tariff above its marginal cost.
We note also that since pc = C¢(D(pc)) while pm > C¢(D(pm)), and since
the function (p - C¢(D(p))) is increasing,
pm > pc and q m < q c
The monopoly therefore reduces quantities and raises prices. This is

sometimes called a Malthusian behavior.
Let us point out one last property: it is easy to show that the monop-
oly price pm is an increasing function of the costs, as intuition suggests.
To see this, let C1 and C2 be two cost functions, and p1m and pm2 be the
associated monopoly prices. Since, by hypothesis, pmi maximizes the
monopoly profit for the costs Ci, we get
Ï p1 D ( p1 ) - C1[D ( p1 )] ≥ p2 D ( p2 ) - C1[D ( p2 )]
m m m m m m
Ì m
Ó p2 D ( p2m ) - C2 [D ( p2m )] ≥ p1m D ( p1m ) - C2 [D ( p1m )]
whence, by adding these two inequalities,
C1[D( p2m )] - C1[D( p1m )] ≥ C2 D( p2m ) - C2 [D( p1m )]

The last result is rewritten as
( )
D p2m
ÚD( p ) [C1¢ (q) - C2¢ (q)] dq ≥ 0
m
1
If, for example, C¢1(q) > C¢2(q) for every q, then we must get pm1 ≥ pm2 .
Prices and Quantities 143
In this regard note that even with constant returns, a cost increase
can be amplified or, conversely, absorbed in the price, unlike the com-
petitive case where the cost increase is entirely transmitted in the price.
Three technical observations can be made about monopolistic
behavior:
• The preceding formulations are only valid if the elasticity e is greater
than 1; in the opposite case, the monopoly sets an infinite price. Then,
as quantities become arbitrarily small, the monopoly’s returns will tend
to infinity.
• The second-order condition of the monopoly program implies P≤,
about which we can deduce very little. In practice, it is always assumed

that P≤ is such that the second-order condition is verified. Moreover it
is necessary to suppose that the fixed costs are small enough for the
monopoly not to take any losses.
• Monopoly production can be inferior or superior to the efficient scale;
everything depends on the demand.
9.1.1 Social Distortion
We saw that a monopoly tariffs above the marginal cost. Considering

the results of chapter 7, there is a distortion in relation to the social
optimum. Figure 9.1 shows the results of the analysis of surplus in the
competitive case (on the left) and in the monopolistic case (on the
right). S represents here the consumer surplus and p that of the pro-
ducers (the profit).
Social loss corresponds to the hachured triangle or Harberger triangle.
Empirical estimates show that it is fairly small.1 Nevertheless, as stated
by John Hicks, “the best of all monopoly profits is a quiet life”: a monop-
oly can be permitted to keep high costs, which entails a social loss often
called X-inefficiency and not quantified in figure 9.1. In fact it was the
principal objection of classical authors to the monopoly, as shown in this
quotation from Adam Smith (The Wealth of Nations, bk. I, ch. 11):
Monopoly, besides, is a great enemy to good management, which can never

be universally established but in consequence of that free and universal
competition which forces every body to have recourse to it for the sake of
self-defence.
1. Harberger (1954) estimates it at a fraction of a percent of the gross domestic product;

more recent estimates converge at a slightly higher scale order.
p p
C'(q) C'(q)
m
S
Sc social loss
pm
c
p
πc
P(q) πm P(q)
MR(q)
qc q qm q
Figure 9.1
Competitive and monopolistic surplus
9.1.2 How to Avoid Distortions
Now let us see how it is possible to eliminate the social distortion inher-
ent in the monopoly’s tarification.
Two-Part Tariffs
Assume that in addition to the unit price p, the monopoly can demand
payment by consumers of a franchise of access A. The two-part tariff
practiced is thus
ÏA + pq if q > 0
Ì
Ó0 if q = 0
In the absence of income effect in the utility function of consumers, the
franchise does not modify their demand function. The monopoly can
therefore choose to tariff p = pc and
+•
A = Sc = Úp c D(p) dp
as in figure 9.2.
p
C'(q)
c
p
P(q)
q
Figure 9.2
Two-part tariff and monopoly
All social surplus is then appropriated to the monopoly, which real-

izes perforce the maximum of its profit, but the social distortion dis-
appears. The obvious disadvantage of such a “solution” is that it entails
a transfer of consumers to monopoly stockholders, which may not be
very redistributive. Moreover, we have implicitly assumed here that all
the consumers are identical. Where this is not the case, the monopoly
could demand a personalized franchise from each consumer, though
the law generally forbids this practice. Such a tarification also assumes
that the monopoly has very detailed consumer preference information
at its disposal.
Taxation
We could imagine taxing the monopoly’s profit, the product price
remaining p at production but becoming (p + t) at consumption. The
monopoly then seeks to maximize its profit
pD(p + t) - C[D(p + t)]

whence
{p - C ¢[D(p + t)]}D ¢(p + t) + D(p + t) = 0

Social optimality will be re-established if p + t = pc = C¢(D(p + t)).

However, this brings
D(p + t) D(p c )
t= =
D ¢ ( p + t) D ¢ ( p c )
which is negative. This says that it is necessary to subsidize the monop-
oly. This result is in fact logical: since the monopoly tends to restrain
production, it is fitting to encourage more production. Here again,
the solution is not particularly redistributive, so it is not likely to be
popular. Moreover it implies a demand curve and cost information that
the government may not have.
Regulation
Most governments prefer to regulate monopolies in some way.
Researching the optimal regulation mode depends on the methods
presented in section 7.3.2.
9.1.3 The Case of Durable Goods
The introduction of time in the preceding analysis means that certain

themes must be ajusted. Take the monopoly that sells a durable good.
It can be thought to compete with itself, since consumers may buy the
good at the moment when they think it the least expensive.
To illustrate this phenomenon more fully, consider a model with two
periods, t = 1, 2, where the demand for services provided by the durable
good is given by D(p) = 1 - p in each period. The good is produced
at no cost, and the discount rate common to the monopoly and to the
consumers is d.
If the monopoly rents the durable good, at each period it will maxi-
mize the profit p(1 - p), whence p1 = p2 = 1/2 and a discounted profit
(1 + d)/4. When the monopoly decides to sell the durable good, things
become a bit more complicated, since the consumers must choose the
moment of purchase (if there is a purchase). It is therefore necessary to
apply backward induction, that is, to reason starting from the second
period.
Suppose that the quantity q1 had been sold in the first period. We
now find ourselves at the beginning of the second period. The residual
demand is (1 - q1 - p). The monopoly must then choose the quantity it
sells, q2, so as to maximize q2(1 - q1 - q2), whence p2 = q2 = (1 - q1)/2 and

a second period profit p2 = (1 - q1)2/4.
Let us go back to the beginning of the first period. Since the buyer
knows that he can resell the good in the second period at the price p2,
he is ready to pay dp2 more than the normal price (1 - q1). We get
Ê dˆ
p1 = 1 - q1 + dp2 = (1 - q1 ) 1 +
Ë 2¯
The discounted profit is therefore
2
Ê dˆ (1 - q1 )
q1 (1 - q1 ) 1 + +d
Ë 2¯ 4
We immediately get q1 = 2/(4 + d) and a discounted profit
2
(2 + d ) 1 + d
p= <
4(4 + d ) 4
which shows that the monopoly’s profit is smaller when the good is
sold than when it is rented. In effect a monopoly that sells the durable
good cannot commit itself not to lower the price of that good later; this
encourages consumers to wait. As a matter of fact, we can show in our
model that p1 > p2 so that the monopoly effectively lowers its prices in
the course of time.
One intuitively senses that the monopoly’s profit is smaller because
it has more difficulty committing to a price policy. For that matter,
Coase has conjectured that if the periods were cut into shorter and
shorter subperiods and if correspondingly d tends toward 1, then the
discounted profit of the monopoly will tend towards 0. In the limit the
monopoly’s power disappears completely. The validity of this conjec-
ture was demonstrated at the beginning of the 1980s.
To conclude, note that the policy of decreasing prices practiced by
the monopoly in our model does correspond to reality. For example,
we can cite publishing houses that issue expensive hardbound books
and sell them to their strongest customers before issuing more afford-
able paperbound editions, without the production costs being affected.
This is what is meant when one speaks of skimming the demand.
For this reason we sometimes see a firm commit to compensating its
present customers if it ever lowers its prices. This practice does not
amount to a gift for consumers; rather, it is simply an incentive for not
lowering prices, allowing the firm to protect its profits.
9.2 Price Discrimination
We speak of price discrimination when two units of the same good

are sold at different prices without this difference being justified by
differences in cost. We are interested here in price discrimination by a
monopoly.
In 1920 Pigou distinguished three degrees of discrimination:
• First degree: perfect discrimination. The monopoly observes the char-
acteristics of all consumers and appropriates all of their surplus
• Second degree: discrimination. The imperfect observation of con-
sumer characteristics depends on self-selection (as in section 7.3.2) or on

nonlinear tariffs, whereby consumers choose different quantities at dif-
ferent prices
• Third degree: discrimination founded on observable signals.
Gender, age, and other distinguishing consumer characteristics come
into play
Note first that discrimination by prices is limited by phenomenons of

arbitrage: if the elderly cannot be kept from letting younger people
travel on their airline ticket, establishing reductions that target the
elderly serves little purpose.
9.2.1 First Degree
We have already seen in section 9.1.2 how the monopoly can appro-
priate consumer surplus as a whole: to this end it is enough for the
monopoly to practice a two-part tariff such that the unitary price is the
competitive one and franchise Ai applied to consumer i is consumer
surplus. Once again, the use of such a tariff of course assumes that
the monopoly perfectly observes the demand functions of all the
consumers. Nor is there much regard for arbitrage problems: it is
often difficult to keep consumers from sending a delegate to buy what
they want.
9.2.2 Second Degree
Assume that there are two types of consumers, i = 1, 2, and that con-
sumer i has as his utility qiqi - pi, where qi is the quantity he consumes,
pi the sum he pays to the monopoly, and qi a taste parameter. We will
suppose that q2 > q1 so that consumer 2 is more eager for the good than
consumer 1.
If the monopoly could observe each consumer’s type, it would solve
for each type
Ïmax qi ,pi [pi - C(qi )]

Ì
Ó q i qi - p i ≥ 0
This gives the first-best solution (q*i , p*i ) such that
Ïq i = C ¢ ( q i*)
Ì
Óp*i = q i qi*
In the cases that interest us, each consumer of course knows his type
qi. In contrast, the monopoly cannot observe a consumer’s type but
knows only that the proportion of the types 2 in the population is m.
According to chapter 4, the monopoly must therefore offer a direct
revealing mechanism: two contracts (q1, p1) and (q2, p2) such that the
type 1 consumer chooses the first and type 2 the second. If the monop-
oly has production costs C(q), the program it must solve is then
Ïmax q1 ,p1 ,q2 ,p2 {m[p2 - C(q2 )] + (1 - m )[p1 - C(q1 )]}

Ô
Ô q 1q1 - p1 ≥ q 1q2 - p2 (IC1 )
Ô
Ì q 2 q2 - p2 ≥ q 2 q1 - p1 (IC 2 )
Ô (IR 1 )
Ô q 1q1 - p1 ≥ 0
ÔÓ q 2 q2 - p 2 ≥ 0 (IR 2 )
where the (ICi) are incentive constraints that express that each type
must voluntarily choose the contract meant for him and the (IRi) are
participation constraints which guarantee to each type a nonnegative
utility.
This program is analyzed in detail in Salanié (1997, ch. 2). A pre-
liminary study of the program shows that the only active constraints
are (IR1) and (IC2). It remains therefore to solve
Ïmax q1 ,p1 ,q2 , p2 {m[p2 - C(q2 )] + (1 - m )[p1 - C(q1 )]}

Ô
Ì p2 - p1 = q 2 (q2 - q1 )
Ô
Ó q 1q1 = p1
After replacing p1 and p2 by their values, we have
max{m[q 1q1 + q 2 (q2 - q1 ) - C(q2 )] + (1 - m )[q 1q1 - C(q1 )]}
q1 , q2
Now we easily get
ÏC ¢(q2 ) = q 2
Ô
Ì m
ÔÓC ¢(q1 ) = q 1 - 1 - m (q 2 - q 1 )
This result shows that if q2 is equal to the first-best solution q*2 , q1 is

less than q*1. There is then a loss of efficiency owing to the possibility
that consumer 2 understate his type, so the informational rent is
measured by q2q2 - p2. The intuition of this result is very similar to
that noted in section 7.3.2: the monopoly must prevent type 2 from
masquerading as type 1 in order to pay the smaller price consigned
to type 1’s. To succeed in this, the monopoly will sell a smaller
quantity to type 1, which makes the lower price less interesting to
type 2.
9.2.3 Third Degree
Assume that there are m groups of consumers, i = 1, . . . , m, distin-

guishable by observable characteristics: the demand of group i is
a function Di(p) known to the monopoly, and it can therefore tariff
distinct prices p1, . . . , pm that maximize the total profit
m m
Â pi Di (pi ) - CÊË Â Di (pi )ˆ¯
i =1 i =1
We immediately get the classical formula, but this time good by good:
pi - C ¢ 1
" i = 1, ... , m, =
pi ei
where ei is the demand elasticity of group i. As in the Ramsey-Boiteux
formula, it is necessary to make groups with the least elastic demand
pay more. These groups of course prefer a tariff that does not dis-
tinguish between groups, so the conclusions are ambiguous where
surplus is concerned.
9.3 Oligopoly
It is said that a market can be called an oligopoly when only a small

number of firms can compete in it. This is in fact the case that best rep-
resents the majority of economic sectors. In the nineteenth century the
oligopoly was the focus of two penetrating analyses whose premises

and conclusions were radically opposed.
9.3.1 Cournot’s Oligopoly
Cournot (1838) studied oligopolistic competition when firms utilize

produced quantities as strategic variables. Suppose that there are
n firms and that firm i has a cost function Ci(qi). Let P (Sni=1 qi) be
the inverse demand function on the market. We look for the Nash
equilibrium in quantities, that is, a vector of quantities (q1, . . . , qn) such
that producing qi is optimal for firm i when it takes q-i as given.2 In
solving
È Ê n ˆ ˘
max Íqi PÁ Â q j ˜ - Ci (qi )˙
qi Î Ë j =1 ¯ ˚
we easily find the formula that gives the Lerner index for firm i:
P - Ci¢ si
"i = 1, ... , n, =
P e
where si is the market share of firm i (which is of course endogenous),
qi
si =
Â nj =1q j
As in the case of the monopoly, we find a price that is superior to
the marginal cost (even if the distortion is less, since si £ 1): each
firm realizes a positive profit, even if it produces at constant returns,
Ci(qi) = ciqi.
We will note that the average markup of the firms of the sector,
weighted by their market shares, is given by
n
P - Ci¢ H
Â si P
=
10, 000e
i =1
where H = Sni=1 (100si)2 is the Herfindahl index. This index measures

the concentration in the studied sector; it is worth 10,000 in the case of
a monopoly and 10,000/n for an oligopoly of n identical firms. As
appears from the formula above, this index is directly linked to the
2. We can show that equilibrium exists if qP¢(q) is a decreasing function, and that it is
then unique if the marginal costs are constant.
average markup; it is for this reason that the index serves as the basis
of the mergers and acquisitions control policy in the United States.
Note that if all the firms are identical, then at equilibrium one will
have "i = 1, . . . , n, si = 1/n. It is easily seen that in this case there is a
convergence toward equality between price and marginal cost when n
tends to infinity. Therefore we again find the competitive equilibrium
when the firms are very numerous.
9.3.2 The Bertrand Equilibrium
The underlying game in the Cournot equilibrium is a bit surprising.

Competitors choose their quantities and wait with no further action
while the equilibrium price establishes itself on the market, though it
would be quite easy for them to modify their prices, which are gener-
ally much easier to adjust than quantities. In his famous criticism of
Cournot’s book, the French mathematician Bertrand (1883) proposed
rather that prices be considered the strategic variables. The strength of
Bertrand’s critique is best shown in the example of two identical firms,
i = 1, 2, producing at constant returns, Ci(qi) = cqi. To compute the Nash
price equilibrium, we use reductio ad absurdum:
• p < c or p < c is impossible, since the firm in question would take
1 2
losses
• c < p1 < p2 is impossible, since 2 sells nothing and gains by reducing
its price to just under p1.
• p1 = c < p2 is impossible, since 1 would gain in slightly raising its price,
resulting in p1 = p2 ≥ c
• p1 = p2 > c is not an equilibrium, since each firm then has only a part
of the market and would gain then by slightly lowering its price,
thereby taking control of the whole market
In what must be one of the most concise texts in the history of

economic thought, Bertrand concludes (p. 503):
Whatever the common adopted price, if only one of the competitors lowers his
price, he attracts, neglecting trivial exceptions, the totality of the sale, and he
will double his revenue if his competitor lets him. If Cournot’s formulas mask
this obvious result it is because by a singular mistake, he introduces with names
D and D¢ the quantities sold by the two competitors and, treating them as inde-
pendent variables, he assumes that if one happens to change through the will
of one of its owners the other can remain constant. In fact, the contrary is
obvious.
The conclusion is simple: the only Nash price equilibrium is p1 = p2 =

c, that is, competitive equilibrium, even though there are only two firms
on the market. This is called the Bertrand paradox. Current observation
shows us that firms compete in price (as observed Bertrand) rather than
in quantities (Cournot); still the results obtained in a Cournot oligop-
oly are much more satisfying intuitively than those of the Bertrand
equilibrium. The following chapters in large part will be devoted to
solving this paradox.
9.3.3 Sketches of Resolutions of the Paradox
The overarching question is: How can firms make positive or supra-
normal profits3 (confirmed by empirical studies) when they are
engaged in a price competition (which better corresponds to the work-
ings of the marketplace)?
We can say, first of all, that if the two firms have constant but dif-
ferent marginal costs, c1 < c2, then we can distinguish two cases depend-
ing on whether the monopoly price pm1 corresponding to the constant
marginal costs c1 is less or more than c2:
• If c2 £ pm1 , then at equilibrium firm 1 tariffs at c2 and 2 leaves the market
(if it resisted, 1 could slightly lower its price to chase it out)
• If c2 > pm1 , then at equilibrium firm 2 leaves the market and 1 tariffs
at the monopoly price
Firm 1 succeeds in making profits by exploiting its technological

superiority. This obvious conclusion does not, however, solve the
paradox satisfactorily, since the paradox remains infact when costs are
identical.
Capacity Constraints
Edgeworth noted in 1897 that in the presence of capacity constraints,
firms can avoid throwing themselves into a price war. Assume,
for example, that the capacity of firm 1 is K1 = D(p) with p > c. If
both firms tariff p1 = p2 = p, firm 1 will have no interest in unilaterally
lowering its price, for it will not be able to serve the whole market
anyway.
3. I define supranormal profits as profits greater than what can be obtained in a perfectly
competitive market.
Clearly, the choice of capacities is endogenous. A firm compares the

discounted benefits of an increased capacity and the corresponding
investment cost. Here it is important to specify well the rationing
scheme: if p1 < p2 and D(p1) > K1, certain consumers will not be able to
buy at the price p1; some of them will pull out of the market and others
will apply to firm 2. The rationing scheme defines the demand aimed
at firm 2 in these conditions. Kreps-Scheinkman (1983) adopted the so-
called efficient rationing scheme4 where D2 = max[D(p2) - K1, 0]. They
analyzed a game in two stages where firms chose their capacities (by
incurring a cost proportional to the installed capacity) before fixing
their prices. Their result was remarkable in that the subgame-perfect
equilibrium was a Cournot equilibrium.5 Therefore they justified
Cournot’s analysis in sectors where investments rigidly determine the
production capacities.6 Still this result does not solve the Bertrand
paradox if the firms can easily raise their capacities in response to
demand.
Product Differentiation
Thus far we have assumed that products were homogeneous. In reality
two firms rarely produce the same product, though two products
can more or less replace each other. It is intuitively clear that this
gives the firms then a certain market power. Demand functions con-
tinuously depend on two prices, contrary to the case of a homogeneous
product. If firm 2 lowers its price below that of firm 1, the latter will
lose only a part of its market. If, for example, the demand functions are
given by
ÏD1 = 1 - p1 + q p2
Ì
ÓD2 = 1 - p2 + q p1
where q is a positive parameter, then the Bertrand equilibrium with
constant marginal costs c becomes
4. This rationing scheme is called efficient because, while the consumers’ propensities to
pay differ, it maximizes the total surplus of consumers by having the firm that tariffs the
lowest prices serve the consumers who are the most greedy for the good.
5. As shown by Davidson-Deneckere (1986), this result narrowly depends on the choice
of the efficient rationing scheme.
6. More generally, it can be shown that competitive equilibrium is no longer a Nash price
equilibrium if the cost functions are strictly convex. The analysis of this case is unfortu-
nately fairly technical.
1 - (1 - q )c
p1 = p2 = c +
2 -q
Note that 1 - (1 - q)c is the common value of demands when p1 = p2 =
c; clearly, it must be positive, so the Bertrand equilibrium prices are
superior to the marginal cost.
It remains to give a microeconomic foundation to such demand func-
tions; we will see how to do so in chapter 10.
Accounting for Dynamic Aspects

In Bertrand’s reasoning, an equilibrium where firms tariff above the
marginal cost is not stable because it is in each firm’s interest to lower
its prices to monopolize the market. However, this supposes that the
deviant firm is not “punished” by its competitor. This hypothesis is less
solid when the interaction is repeated, since then the two firms run the
risk of being thrown into a price war which would be harmful to both.
As Fisher (1898) wrote early on:
No business man assumes that either his rival’s output or price will remain
constant . . . On the contrary, his whole thought is to forecast what move the
rival will make in response to his own.
If the game is repeated infinitely, the monopoly price can become an

equilibrium. Assume, in effect, that there is an infinite number of
periods t = 0, . . . , +•, that the discount rate is d, and that J identical
firms of constant marginal cost c interact. Consider the following
strategies, where pm is the monopoly price:
Ïp t = p m if no one has yet tariffed below p m

Ì
Óp t = c in the contrary case
Let p m be the monopoly profit; when the J firms tariff pm, each makes
a profit p m/J. If one of the J slightly lowers its price, it will make a profit
close to p m, and the others will have a zero profit. Therefore, if a firm
“deviates” on date t in fixing a price that is slightly lower than pm, it
will make a profit close to p m on date t and nothing afterward, since
the price will then establish itself at the level of marginal cost. The firm
will obtain a discounted profit p m as a result. If, on the other hand, the
J firms still fix pm, each makes a profit
pm pm
(1 + d + d 2 + ...) =
J J (1 - d )
But, if d ≥ (J - 1)/J (so that firms are fairly patient),
pm
≥pm
J (1 - d )
and the monopoly price is a Nash equilibrium of the game.
Two observations demand attention. On the one hand, this reason-
ing applies just as well when replacing pm by any price superior to the
marginal cost. There exist then many other equilibria where p1 = . . . =
pJ > c if d is close enough to 1, so this model is not entirely satisfactory.
On the other hand, deviations are not easily observable perforce;
each firm observes above all the demand which concerns it, which
depends on the price practiced by its competitors but also on numer-
ous other factors. It is not always easy then to determine whether a
firm has deviated from the equilibrium. Green and Porter (1984) study
a model where the decrease in the demand aimed at a firm can be
due to a decrease of the price of a competitor and also to a shock
to global demand (a recession). They show that the monopoly price
can no longer be sustained with certainty: recessions lead to price
wars.
Illegal Agreements
The preceding discussion dealt with what is called tacit collusion
between firms. At times collusion is more explicit. In this regard we
should not neglect Adam Smith’s cynical remark (The Wealth of Nations,
I, ch. 10):
People of the same trade seldom meet together, even for merriment and diver-
sion, but the conversation ends in a conspiracy against the publick, or in some
contrivance to raise prices.
Recent history is spotted by numerous instances of such illegal agree-

ments, called cartels. One of the most spectacular concerned electrical
appliances makers in the United States in the 1950s. Their cartel system
rested on three pillars:
• an agreement on prices for standard appliances
• for large appliances, price books comprising several hundred
pages
• a rolling system based on lunar phases that aimed to give the
appearance of cut-throat competition and answer supply procurement
calls
On the European side, in the 1990s firms of the cement sector found
themselves subject to very heavy fines by the European Commission
because of their noncompetitive pricing practices.
Certain authors, following Stigler (1950), consider cartels to be intrin-
sically unstable. The reasoning is simple. A cartel comprises k firms that
act as a single firm. In Cournot competition the entire cartel obtains the
same profits as any firm situated outside the cartel. A cartel member
therefore multiplies its profit by k in leaving it. This reasoning, how-
ever, only applies if the competition is strictly that of Cournot, without
the cartel having a strategic advantage. Such a hypothesis is in all like-
lihood not realistic. Stigler’s remark stimulated a vast literature on the
stability of cartels; as often happens in industrial economics, exact con-
clusions depend on the type of competition that prevails in the sector
under consideration. We will simply note that the concrete examples
cited above show fairly well that some sectors can succeed in creating
durable cartels.
9.3.4 Strategic Substitutes and Complements
The Bertrand paradox finds its source in the very different structures
of the two games, of quantities and of prices. First, consider a Cournot
duopoly of inverse demand function P. If the firm 1 has a cost function
C1, its profit is
p 1 = q1 P(q1 + q2 ) - C1 (q1 )
Fix q2; the best response function of 1 is given by
∂p 1 ∂ 2p 1
[R1 (q2 ), q2 ] = 0 and [R1 (q2 ), q2 ] £ 0
∂ q1 ∂ q12
By the implicit function theorem, its derivative is therefore
∂ 2p 1 ∂ q1 ∂ q2
R1¢ (q2 ) =
∂ 2p 1 ∂ q12
and has the sign of the crossed derivative
∂ 2p 1
= q1 P ¢¢(q1 + q2 ) + P ¢(q1 + q2 )
∂ q1 ∂ q2
If P≤ is not too positive, then the crossed derivative is negative, and the
best response function R1 is decreasing: in Cournot competition a firm
tends to produce less when its competitors produce more. It is then

said that quantities are strategic substitutes (Bulow, Geanakoplos, and
Klemperer 1985).
What about price competition? In the case examined by Bertrand,
profit functions are discontinuous, which makes the application of
these notions difficult. If, however, there is product differentiation
as in the example of section 9.3.3, then we easily see that the crossed
derivative
∂ 2p 1
=q
∂ p1 ∂ p2
is positive. The prices are therefore strategic complements. A firm reacts

to its competitors’ price decrease by lowering its own, which is quite
natural.
The strategic complements and substitutes games are very different.
It may be obvious that competitors’ attitudes are more aggressive in a
game of strategic complements and more peaceful in a strategic sub-
stitute game. This helps explain why profits are higher in competition
by quantities than they are in competition by prices.
Bibliography
Bertrand, J. 1883. Théorie des richesses. Journal des Savants, pp. 499–508. Published
in English in A. Daughety, ed. Cournot Oligopoly: Characterization and Applications.
Cambridge: Cambridge University Press (1988).
Bulow, J., J. Geanakoplos, and P. Klemperer. 1985. Multimarket oligopoly: Strategic

substitutes and complements. Journal of Political Economy 93: 488–511.

University Press.
Cournot, A. 1838. Recherches sur les principes mathématiques de la théorie des richesses.
Published in English as Mathematical Principles of the Theory of Wealth. San Diego, CA:
James and Gordon (1995).
Davidson, C., and R. Deneckere. 1986. Long-run competition in capacity, short-run

competition in price, and the Cournot model. Rand Journal of Economics 17: 404–15.
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Green, E., and R. Porter. 1984. Noncooperative collusion under imperfect price
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10 Product Choice
As we saw in the previous chapter, product differentiation is one way

to escape the Bertrand paradox. In perfect competition, firms price at
marginal cost whether the products are differentiated or not. In this
chapter we will see that when competition is imperfect, an oligopoly
engaged in price competition will obtain supranormal profits (superior
to competitive profits). That is to say, if products are no longer homo-
geneous, each competitor can raise its prices above the marginal cost
without abruptly losing all of its clients.
10.1 Definitions
Classically two distinctions are made regarding product differentiation:

• Vertical differentiation. All consumers agree to regard products in the
same way (e.g., because different products correspond to different

qualities of the same product).
• Horizontal differentiation. Consumers show differences in subjective
tastes.
These two consumer choice models are clearly opposed if all differen-
tiated products are sold at the same price. Then in the model of verti-
cal differentiation, we can assume that there is a blitz of demand upon
a single product, whereas in the model of horizontal differentiation
demand is spread over several products.
Here are two classic examples we can look at right away.
10.1.1 A Model of Vertical Differentiation
Consider a good for which there exist two qualities indexed by a para-
meter s: s1 and s2, with s1 < s2. The price of the best quality is of course
higher, p1 < p2. The consumers can have at most one unit of the product;
their utility function is U = qs - p, where q is a quality preference para-
meter. The consumers must therefore compare utilities qs1 - p1 (pur-
chase of quality 1), qs2 - p2 (purchase of quality 2), and 0 (no purchase).
Accordingly they are classed in terms of how much they appreciate
quality. Those who value quality most will of course buy the highest
quality.
10.1.2 A Model of Horizontal Differentiation
Hotelling (1929) proposed the first horizontal differentiation model. It

is still the most used model today.
Imagine a linear city as segment [0, 1]. There are two stores each
located at an extremity of the city; their tariff prices are p0 and p1. The
consumers are uniformly spread throughout the city. Consumers face
a transport cost t per unit of distance covered. There are two possible
models:
• of travel costs incurred by consumers in order to reach each store
• of transport costs incurred by the stores in delivering products to
consumers.
The second situation is appropriate for certain industrial sectors. The

first relates more to retail shoppers: think of two supermarkets. In an
abstract interpretation of the model that is less geographical, the
product could present itself in numerous varieties indexed by a para-
meter of [0, 1]. Company 0 chooses to sell variety 0 and firm 1 variety
1. The consumer situated in x prefers variety x and must discharge a
cost (in utility) t|x - x¢| if he buys variety x¢.
The consumer located in x perceives therefore a generalized price,
p0 + tx, for the store situated in 0, and thus p1 + t(1 - x) for the store
situated in 1. If P is the subjective value of the good for each consumer,
the demand aimed at store 0 is given by the smallest x such that
Ï p0 + tx £ p1 + t(1 - x)
Ì
Ó p0 + tx £ P
Product Choice 163
P P
p1
p1 p0
p0
0 x 1 0 x 1
Figure 10.1
Generalized prices in the Hotelling model
It is then
Ï p1 + t - p0 p1 + t + p0
Ô if £P
2t 2
ÔÔ
Ì P - p0 p1 + t + p0
Ô t if < P and p0 £ P
2
Ô
ÔÓ 0 if p0 > P
In figure 10.1 the panel on the left represents the first case and the panel
on the right the second case.1
10.2 Differentiation and Monopoly
Before studying differentiation in oligopolistic competition, we need to

examine some novel distortions that can intervene when a monopoly
sells differentiated products.
10.2.1 Optimal Quality Choice
Take now a vertically differentiated good of quality index s. Suppose

that the inverse demand function of the consumers is P(q, s), which
increases in s since the price that they are prepared to pay increases
1. The market is covered in the first case only.

with quality. The production cost of the good C(q, s) also rises with
the quality. The monopoly chooses the quantity and the quality that
maximize its profit:
max[qP(q, s) - C(q, s)]

q ,s
The first-order condition in s is written
∂P ∂C
q (q, s) = (q, s)
∂s ∂s
Therefore the monopoly decides on its quality in considering the
marginal consumer: the consumer who buys the last unit of the good.
What should a planner do in maximizing the social surplus? First, the
planner computes
q
Ú0 P(x , s)dx - C(q, s)
whence the first-order condition in s is
q ∂P ∂C
Ú0 (x , s)dx = (q, s)
∂s ∂s
The planner has to consider next the average consumer. Clearly, at a
given quantity, the monopoly does not generally choose the socially
optimal quality.
Take another example where the consumers have a utility U =
qs - p and q is uniformly distributed on [0, 1]. The demand is
D(p, s) = 1 - p/s, and its inverse is P(q, s) = s(1 - q). We deduce from
this that the cross-derivative ∂ 2P/∂q∂s is negative. So
q ∂P ∂P
Ú0 (x , s)dx > q (q, s)
∂s ∂s
In the reasonable case where C is convex in s, it is easily seen that the
monopoly will furnish suboptimal quality for any given quantity.
This result of suboptimality is not general, however. One must not
forget that the monopoly can reduce the quantity sold (in relation to
the social optimum). Barring miracles, it remains that a monopoly nor-
mally does not choose the socially optimal quality.
Product Choice 165
10.2.2 Nonobservable Quality
Now let us look more realistically at a situation where only the seller
knows the quality s of his product. The buyer knows only that s is
uniformly distributed on [0, 1] a priori. The seller’s utility is p - q1s,
and that of the single buyer is q2s - p, where q1 and q2 are two publicly
known parameters such that q1 < q2; hence it is socially optimal for the
buyer to acquire any quality from the seller. Still we will see that the
unobservability of quality can cause a market breakdown.
First, we must keep in mind that when a good’s quality is unob-
servable, all qualities can be sold at the same price. If buyers anticipate
an increasing relationship between price and quality, then it is always
in the sellers’ interest to tariff the highest price. Clearly, this fact con-
tradicts the buyers’ expectations. When a seller proposes a price p, the
buyer could reason as follows:
•If the seller can sell at the price p, it is that p ≥ q1s. There are two
possibilities:
1. p ≥ q1. In this case I learn nothing of the quality by observing the
price. So I must evaluate the average quality of goods for sale at 1/2,
and the average utility that I can have in purchasing is therefore
q2/2 - p. If q2 < 2q1, this utility is then negative, so I will not buy.
2. p < q1. The average quality that I can expect is p/2q1, which gives
me an average utility pq2/2q1 - p. This utility is still negative if
q2 < 2q1.
• Whatever the price proposed by the seller, it is not in my best inter-
est to buy if q2 < 2q1.
The conclusion is immediate: if q1 < q2 < 2q1, no quality of the good
can be sold, even though a sale would be socially optimal. This situa-
tion is often called the lemon problem after a famous example of Akerlof
(1970).2
The lemon problem is fortunately not irremediable; it can be solved
by introducing a signal of quality for a good. For example, if the
product’s quality coincides with its reliability, introducing guarantees
can establish a more satisfying (albeit second-best) equilibrium: only
the sellers of good cars will offer a guarantee, since it would be costly
2. In the United States, a new car that has continual mechanical problems is called a
lemon.
for sellers of bad cars to include guarantees. The offer of guarantee then
signals a good car and allows pertinent information to be revealed in
equilibrium.
10.2.3 Choice of Number of Products to Introduce
Now, in a horizontal differentiation model, would a monopoly intro-

duce too many or too few products with relation to the social optimum?
There are two opposing arguments to begin with:
• A monopoly can restrain the variety of its products. In the absence
of first-degree discrimination, a monopoly’s profit p is less than the
social surplus S, which also comprises consumer surplus. A monopoly
will not introduce a new product if its fixed cost f is between p and S,
though the product is socially desirable.3
• Monopoly prices above the marginal cost on a product can make
other varieties of the product more interesting for consumers. The
monopoly can therefore gain by introducing new varieties even if not
socially optimal.
Since the second argument is less direct than the first, let us examine
it more closely using Hotelling’s linear city example. In this model the
social surplus maximization reverts to minimizing the sum of the trans-
port costs and firms’ production costs (the prices paid are merely
transfers between consumers and firms). Suppose that the marginal
costs are zero but that opening a store costs f. If there is only one store,
situated at 0, the total transport costs are given by Ú10txdx = t/2; if
the monopoly opens a new store at 1, the transport costs become
2Ú1/2
0 txdx = t/4. If the cost of creating a new store f is superior to t/4, it
is therefore socially preferable to open only one store.
What about a monopoly? Suppose that the subjective value of the
good P is high so that the whole market is covered. Then the monop-
oly will tariff so as to leave a zero surplus to the most distant consumer:
p = P - t if there is only one store, and p = P - t/2 if there are two. The
opening of a new store reduces the transport costs of consumers and
thereby allows the monopoly to raise its prices. Since the monopoly’s
3. This is a common argument, but it depends as much on nonconvexities (the cost f

of creating a new product) as in imperfect competition: when f is not zero, it is equally
possible that a competitive sector will not create a new product, though it may be socially
optimal.
Product Choice 167
profit (outside of opening costs) is simply p = p, it will prefer opening

two stores provided that f < t/2.
Where t/4 < f < t/2, the monopoly will indeed open one more store
than a social planner.
10.3 Differentiation and Oligopoly
When there are several firms, one must take into account their differ-
entiation strategies. This is what we will do within the framework of
Hotelling’s model. We can then study the optimality of the entry
process of the firms, which permits us to present the monopolistic com-
petition model of Chamberlin and his more recent descendants.
10.3.1 The Maximal Differentiation Principle
Let us return to the horizontal differentiation model of Hotelling for

two firms, by way of three new hypotheses:
• P is very high, and consumers then always buy.
• The marginal production costs are constant and equal to c.
• The transport costs are quadratic.
The last hypothesis is a technical one that enables us to avoid problems

owing to the discontinuity of demand functions when transport costs
are linear. Consider in effect that situation represented in figure 10.2,
where the firms are localized in a and 1 - b and the lines represent the
generalized prices when the transport costs are linear. Then firm a’s
market is (approximately) [0, 1 - b] and that of firm 1 - b is [1 - b, 1].
Now, if firm a lowers its price a little, it will monopolize the entire
segment since its generalized price will be everywhere inferior to that
of (1 - b).
It is easy to see that if the prices are fixed in the quadratic costs
model, the firms will gain by drawing nearer the center of the segment:
it is the minimal differentiation principle stated by Hotelling (1929):
• Buyers are confronted everywhere with an excessive sameness.
To see this, assume that the prices practiced by the two firms are equal
(to p), that one firm is localized in a, and the other firm in (1 - b), with
a £ 1/2 £ 1 - b. So the limit of markets is given by
0 a 1-b 1
Figure 10.2
Discontinuity with linear transport costs
2 2
(x - a) = (1 - b - x)
being x = (a + 1 - b)/2. The first firm’s profit is px, and the profit
increases in a. So its dominant strategy is to be established in 1/2. The
same reasoning also brings the second firm to set up in 1/2. We have
indeed a matter of “excessive sameness,” since the social optimum is
to minimize the total transport costs4
3
( a +1- b ) 2 2 1 2 (1 - b - a) a3 b3
Ú0 (x - a) dx + Ú( a+1-b ) 2 (1 - b - x) dx = + +
3 3 3
From the expression above we easily see that the costs are minima for
a = b = 1/4.
We will find, however, that this principle must be inverted when the
prices are freely chosen by the firms (d’Aspremont, Gabszewicz, and
Thisse 1979). In a first stage the two firms are free to localize themselves
wherever they wish on the segment [0, 1]. So we can consider this as a
two-stage game:
4. The production costs do not come into play since the market is supposedly covered.
Product Choice 169
1. The firms choose their localizations a and (1 - b).

2. They subsequently choose their prices p0 and p1.
We want to characterize the perfect equilibrium of this game and to

demonstrate that a = b = 0 at equilibrium, that is, that the firms seek to
differentiate themselves to the maximum.
As usual, we start at the game’s end, after the firms have settled
in their localizations. If they tariff p0 and p1, the coordinate x that
delimits their territories is given by
2 2
p0 + t(x - a) = p1 + t(1 - b - x)
The demand applying to 0 is therefore
1- b - a p1 - p 0
D0 ( p0 , p1 , a, b) = a + +
2 2t(1 - b - a)
It comprises three terms. The first is the territory situated to the left of
firm 0, the second corresponds to the equal distribution of the territory
situated between the two firms, and the last is the effect due to the dif-
ference in price. Naturally the demand aimed at 1 is simply
D1 (p0 , p1 , a, b) = 1 - D0 (p0 , p1 , a, b)
If each firm has a constant marginal cost c, the Nash price equilib-
rium is given by the maximization of (p0 - c)D0 in p0 and of (p1 - c)D1
in p1. Simple calculations give
Ê a - bˆ
p0 (a, b) = c + t(1 - b - a) 1 +
Ë 3 ¯
and a similar formula for p1(a, b).
Now we can go to the first stage. The firm 0 will choose a to
maximize
p 0 = (p0 (a, b) - c)D0 (p0 (a, b), p1 (a, b), a, b)

In differentiating, it gets
∂p 0 ∂p 0 ∂ p0 ∂D0 ∂ p1 ∂D0
= + (p0 (a, b) - c) + (p0 (a, b) - c)
∂a ∂ p0 ∂ a ∂ p1 ∂ a ∂a
The first term is zero by the envelope theorem, since p0 makes p0
maximal. The second term is the strategic effect: as ∂D0/∂p1 > 0 and
∂p1/∂a < 0, to reduce a (to distance itself from 1) permits 0 to incite 1 to
raise its price, and therefore 0 acquires more clients. The last term
simply expresses (as ∂D0/∂a > 0) that being near downtown permits 0
to take clients from 1.
After some calculations of little interest, we can show that the strate-
gic effect carries it away and that one therefore gets ∂p0/∂a < 0, so at
equilibrium, 0 is positioned in a = 0. Identical reasoning bearing on 1
would show that one gets b = 0, so the two firms choose to position
themselves at the two extremities of the segment. This is the maximal
differentiation principle. It calls for several remarks:
• This principle is not very general. For example, it is false if the
transport costs are in x3/2, which is no less reasonable than quadratic
costs.
• Various considerations are missing from the model. For example,
consumers’ prospecting costs would militate rather in favor of a
minimal differentiation principle, which is effectively observed at work
in many of our cities.
• The maximal differentiation principle induces too much differentia-
tion. We saw earlier than a = b = 1/4 at the social optimum.
10.3.2 Entry and Number of Products
The previous model did not include fixed costs of entering the market
and did not therefore permit us to examine entry decisions. To this end,
consider another model by Salop (1979). This model’s theme is that
there are too many products at equilibrium in relation to the social
optimum. A firm that enters the market does not take into account the
decrease of profits of already installed firms induced by its own entry:
this is business stealing. What is concerned therefore is a pecuniary
externality, which can create an inefficiency here only because the com-
petition is imperfect.
Consider then a circular city upon which consumers and n firms are
uniformly arranged.5 The transport costs are supposed linear (in tx).
Here again the value of the good for each consumer is assumed to be
very high, so the whole market will be covered at equilibrium. Each
firm has a fixed cost f and constant marginal costs c.
5. Readers desiring another concrete illustration can think of the positioning of

London–New York flight schedules on a clock.
Product Choice 171
p'
p p
p p
p
p
Figure 10.3
The Salop model
At symmetrical equilibrium, all firms tariff the same price p. A firm

that chooses a different price p¢, as in figure 10.3, can ensure itself a
market of size x to both sides of itself, where x is given by
Ê1 ˆ
p ¢ + tx = p + t -x
Ën ¯
The firm would then get a demand (1/n + (p - p¢)/t) and a profit
Ê 1 p - p¢ ˆ
p = (p ¢ - c)Á + ˜-f
Ën t ¯
which is maximal in p¢ = (p + t/n + c)/2. The condition p¢ = p gives the
equilibrium
t t
p =c+ , p = 2 -f
n n
At free-entry equilibrium, this_profit is zero and there are therefore (to
the nearest round number) ÷t/f firms.
The social optimum minimizes the sum of fixed costs nf and _ of trans-
port costs 2nÚ01/2ntxdx = t/4n, which this time gives n = 1/2÷t/f. Clearly,
there are too many firms at free-entry equilibrium, because of business
stealing: a firm that decides to enter the market takes into account only
its own profit and not the decrease of profits of firms already there.
The conclusion of this model recalls that of the classical analysis of

monopolistic competition of Chamberlin (1933). Chamberlin considers
a sector producing differentiated products where
1. the demand that applies to every firm decreases when the firm raises
its price
2. every firm makes zero profits
3. when a firm changes its price, the effect on other firms’ demands is
negligible
The third hypothesis is of course crucial: since no firm has an impact

on the others, each one can neglect reactions to its price policy and have
no further reason to carry a strategic policy. In fact all strategic con-
siderations are abandoned, and this greatly simplifies the analysis.
So one can characterize equilibrium quite simply. Let D(pi|p-i ) be the
demand function6 of the firm i. By hypothesis, it decreases in pi (and
increases by p-i). Since the firm can neglect its competitors’ reactions, it
maximizes
pi D(pi p - i ) - Ci [D(pi p - i )]
by taking pi as given. Assume (which is the only interesting case) that
the average cost curve engendered by Ci is U-shaped. Since the firm
must make zero profit, it must tariff on a point where the demand curve
cuts the average cost curve; the demand curve must even be tangen-
tial to the average cost curve from below: failing this, the firm could
make positive profits by modifying its price. We then get the “tangency
solution” represented in figure 10.4. Note that at such a point, each
firm’s production is inferior to the efficient scale q*. It is generally said
that the sector has an excess capacity.7
The solution Chamberlin proposed has since lost much of its attrac-
tion. It was quickly realized that there were very few industries to
which hypotheses 1 to 3 could be applied. It is generally hypothesis 3
that poses a problem; in retail trade, for example, it is without a doubt
true that the price policy of a small retailer little affects the demand of
a sufficiently distant competitor, but it is from current experience that
a supermarket’s installation has important effects on neighborhood
6. Note that the demand function depends on prices tariffed by the firm i and by its com-
petitors, and therefore implicitly on the number of firms in the industry.
7. There could be several points of tangency, but each point would correspond to an
excess capacity.
Product Choice 173
C(q )/q
i i
p
i
D(p | p )
i -i
qi q*
Figure 10.4
The tangency solution
retailers. Then, as in many comparable cases, the situation is best

modeled as an oligopoly (here, local).
What interests us in Chamberlin’s and Salop’s analyses converges
upon an important point: at equilibrium firms price too high, sell too
little, and are too numerous. This analogy is actually less strong than
it appears. On the one hand, Salop’s model is not a true monopolistic
competition model, since it violates hypothesis 3: a firm’s price policy
affects its neighbors in a non-negligible way. On the other hand, Cham-
berlin’s approach completely neglects the positive effect due to the
entry of a new firm, which increases the variety of available products
(this is manifested by decrease of transport costs in Salop’s model). It
is therefore not obvious that there are always too many firms at equi-
librium. In effect, if the business-stealing phenomenon presented by
Salop leads to too many firms, the fact that the entering firms cannot
appropriate all of the social surplus implies that there may be too few
at equilibrium.
Dixit-Stiglitz (1977) proposed a monopolistic competition model that
verifies hypotheses 1 to 3 and allows for the study of trade-offs between
these two pecuniary externalities. It is assumed that there exists in the
economy a nondiversified good x0 and diversified goods (xi)i=1,...,•. Each

diversified good is produced by a firm of fixed cost f and of constant
marginal cost c.
The economy’s demand side can be summarized by a representative
consumer with a utility function
g
U = x0- ÈÍÂ v(xi )˘˙
1 g
Îi ˚
where 0 < g < 1 and v is a concave increasing function. All things being
equal in other respects, the introduction of a new variety therefore
raises the consumer’s utility: it is said that the consumer has a taste for
variety.
The consumer’s budgetary constraint is written (the good x0 being
the numéraire)8
x 0 + Â pi x i £ R
i
The consumer’s demands are obtained by maximizing

1-g g
ÊR - p x ˆ È v(x )˘
Ë Â i i¯ Â i ˚˙
ÎÍ i
i
whence
(1 - g )pi g v ¢( x i )
=
R - Âi pi xi Âi v(xi )
Now suppose that there are very many firms (and therefore products).
Then the terms Sipixi and Siv(xi) are practically independent of pi, so
hypothesis 3 is valid asymptotically: firm i approximately perceives a
demand given by pi = Kv¢(xi), whose elasticity is
v ¢( x i )
ei = -
xi v ¢¢(xi )
the symmetrical equilibrium is characterized by a price pe and a pro-
duction xe for every firm and a number of firms ne. Every firm tariffs
like a monopoly, so
8. The consumer is assumed to have at his disposal a quantity R of cash; since the profits
will be zero at equilibrium, the problem of their distribution does not arise.
Product Choice 175
c c
pe = = (1)
1 - (1 e e ) 1 + [v ¢( x e ) x e v ¢¢( x e )]
Since the profits are zero, we get
( p e - c) x e = f
so
cx e + f v ¢( x e )
=- (2)
f x e v ¢¢( x e )
We lack only one equation. We know that at equilibrium the consumer
determines xe by maximizing in x
1-g g
(R - ne pe x) [ne v(x)]
We now define two auxiliary functions
xv ¢( x) gr( x)
r ( x) = and w( x) =
v( x ) gr( x) + 1 - g
The function r(x) plays an important role in what follows. As p and
v¢(x) are proportional, r(x) is proportional to the ratio of px (the firm’s
revenue) to v(x), the gross social surplus brought in by its production.
The firm’s capacity to appropriate surplus engendered by its produc-
tion x is therefore measured. Elementary calculations show that the
consumer demand is given by
Rw (x e )
xe =
ne p e
so, utilizing the zero profit condition, we obtain
w (x e ) (3)
ne =
f + cx e
Under reasonable hypotheses, equilibrium is well defined by these
three equations: (2) gives xe, then (1) gives pe, and (3) gives ne.
What about the social optimum? In the face of the facts, the planner
must make firms tariff at the marginal cost p = c and subsidize their
fixed costs by levying nf of the consumer. The consumer then deter-
mines his demand, from which
1-g g
V (n) = max(R - npx) [nv(x)]
x
and the planner permits the entry of a number of firms equal to n* =

arg maxn V(n). Simple calculations show that we get
cx * g
r (x * ) = and n* =
f + cx * f + cx *
Dixit and Stiglitz demonstrate the following theorem:
theorem 10.1
1. xe > x* ⇔ r(x) increases in x.

2. If r(x) increases in x, then ne < n*.
To better see this, return to the zero-profit condition

cx e + f v ¢( x e )
=-
f x e v ¢¢(x e )
Noting that
xv ¢¢(x) xr ¢(x)
1+ = r ( x) +
v ¢( x ) p( x )
this condition can be rewritten in the form
cx e x e r ¢( x e )
= r (x e ) +
cx e + f r (x e )
Comparing it with the social optimum condition
cx *
r (x * ) =
f + cx *
we easily get conclusion 1 of the theorem. Conclusion 2 is demon-

strated by noting that the function w(x) is always smaller than g,
whence
g
ne <
a + cx e
But, if r(x) increases by x, then xe > x*, and therefore
g g
ne < < = n*
a + cx e a + cx *
There is then excess capacity at equilibrium (as with Chamberlin’s
model) if and only if a firm that raises its production sees its portion
Product Choice 177
of the social surplus diminish. In the opposite case, there will be too
few firms and they will be too large. It therefore seems that many con-
figurations are possible a priori. Dixit and Stiglitz cite a simple example
where v(x) = xr, with 0 < r < 1. Then r(x) is constant and the scale of
the firms at equilibrium is socially optimal, contrary to the analyses of
Chamberlin and Salop9. This particular case of the Dixit-Stiglitz model
is often used today in international trade theory and in spatial
economics.
Bibliography
Akerlof, G. 1970. The market for “lemons”: Qualitative uncertainty and the market mech-
anism. Quarterly Journal of Economics 84: 488–500.
d’Aspremont, C., J.-J. Gabszewicz, and J.-F. Thisse. 1979. On Hotelling’s “stability in com-
petition.” Econometrica 47:1145–50.

University Press.
Dixit, A., and J. Stiglitz. 1977. Monopolistic competition and optimum product diversity.
American Economic Review, 67: 297–308.
Hart, O. 1985a. Monopolistic competition in the spirit of Chamberlin: A general model.

Review of Economic Studies 52: 529–46.
Hart, O. 1985b. Monopolistic competition in the spirit of Chamberlin: Special results. Eco-
nomic Journal 95: 889–908.
Salop, S. 1979. Monopolistic competition with outside goods. Bell Journal of Economics 10:
141–56.
9. Hart (1985a, b) criticized the behavior of the utility function used by Dixit and Stiglitz
in the case where n becomes very large. In a more vigorous model, very similar results
to those of Dixit-Stiglitz are obtained, albeit less easily interpretable ones.
11 Long-Term Entry and
Competition
Among the factors that can explain how firms that compete against
each other by price can sustain supranormal profits, barriers to entry
hold an important place. Several different types of barriers to entry can
exist:
• Legal barriers. In certain countries the law restricts the number
of operators in certain professions. Other industries are ruled by

permit systems or by claim to financial guarantees1 (e.g., financial
institutions).
• Scale economies. The presence of fixed costs or of subadditive cost
functions can constitute effective barriers to entry.
• Strategic barriers. We will see that these can also exist.
• Technological barriers. A firm with technological advantages at its
disposal (e.g., an accumulation of research and development) can make

entry difficult for competitors.
11.1 Sustainability and Contestability
Baumol, Panzar, and Willig (1982) (hereafter BPW) showed that the
force of potential competition (the existence of potential entrants)
could be sufficient to compel even a natural monopoly to behave in
a (nearly) socially optimal manner. To this end they considered perfectly
contestable markets where, by definition, entry and exit are executed
at no cost. This definition particularly excludes unrecoverable expenses
(sunk costs) that a firm can be induced to incur upon entrance to the
market.
1. Moreover some countries practice, or have practiced, a policy of promoting “national

champions” which favors the existence of very large conglomerates.
Consider therefore a market where all firms are identical, with

costs C(q). Let D(p) be demand and P(q) its inverse function. BPW
propose focusing on sustainable configurations, defined as follows:
A configuration of m firms (p, q1, . . . , qm) is sustainable if and
only if
• supply and demand are equal,
m
Â qi = D(p)
i =1
• none of the m firms takes any losses,
"i = 1, ... , m, C(qi ) £ pqi

• no entrant firm can propose a lower price and still make positive
profits, /
$ (p¢, q¢) such that
Ïp ¢ < p
Ô
Ìq ¢ £ D(p ¢)
ÔC(q ¢) < p ¢q ¢
Ó
Note first of all that in a sustainable configuration, all firms make
zero profits. Suppose otherwise, and let i be a firm whose profits are
strictly positive:
pqi > C(qi )
Then an entrant firm can tariff p - e, sell qi, and obtain a profit that is
smaller but always strictly positive.
Moreover no firm can tariff below its marginal cost. If one had p <
C¢(qi), then an entrant could tariff at the same price p, sell a quantity
(qi - e), and make a profit:
p (qi - e ) - C (qi - e ) [ pqi - C (qi )] + e [C ¢ (qi ) - p] > 0
Now assume that there exists a sustainable configuration of several

firms (p, q1, . . . , qn) where firm 1 prices above marginal cost: p > C¢(q1).
If an entrant tariffs p - e 2, it will be able to sell all or part of D(p - e 2).
But this new demand is larger than
D(p) = q1 + ... + qn
The entrant firm can then choose to sell q1 + e if e is fairly small. It will
thus obtain a profit
Long-Term Entry and Competition 181
p
P(q)
A AC(q)=c+f/q
Figure 11.1
Sustainable configuration
(p - e 2 )(q1 + e ) - C(q1 + e ) [pq1 - C(q1 )] + [p - C ¢(q1 )]e > 0

Such a sustainable configuration therefore cannot exist.
We know then that in a sustainable configuration of several firms,
the price must be at once equal to both average and marginal costs; that
is, each firm must produce at its efficient scale. But this is only possible
if the demand at minimum average cost is an exact multiple of the effi-
cient scale, which could only result from a miraculous coincidence. The
theory of contestable markets therefore faces serious difficulties in
accounting for an equilibrium where several firms coexist.
BPW then apply their demonstration to the following example:2
Suppose that C(q) = f + cq so that the market is in a state of natural
monopoly. Then, by the preceding discussion, there can only exist one
sustainable configuration: it is point A in figure 11.1, situated at the
intersection of the inverse demand and average cost curves. It is easy
to verify that this configuration is indeed sustainable.
This configuration is remarkable for several reasons. First of all, a
single firm produces, which is indeed desirable since the sector is a
2. Contestable market theory is more complex than it appears in my treatment here. The
question of multi-product firms, in particular, holds an important position in the theory.
I will be content to refer the reader to the book by BPW or to Baumol (1982).
natural monopoly. Moreover the monopoly has zero profits, which

seems to contradict all the normal intuitions: the danger of entry alone
is enough to force it to lower its price. Finally, this configuration is a
constrained social optimum (in the absence of compensatory transfers,
which would allow the monopoly to price at marginal cost).
Unfortunately, or perhaps fortunately for our intuition, the attention
given to sustainable configurations by BPW is excessive. First of all, no
sustainable configuration can exist, as shown in figure 11.2, where the
average cost curve is U-shaped and intersects the inverse demand
curve to the right of the minimum of average cost. Only point B in the
figure could be sustainable, but it is destabilized by a point such as
point C, where the entrant rations demand.
BPW make strong claims for their theory, emphasizing that like
Chamberlin’s monopolistic competition theory, it allows very strong
results to be obtained without burdening oneself with strategic con-
siderations. Still we would like to have at our disposal a description of
the firm’s strategic behavior which could justify the sustainable con-
figurations. In fact such a game does exist, and it has two stages. In the
first, firms choose their prices; then they decide whether or not to
enter, and they fix their production level. Such a game is open to
p
AC(q)
P(q)
B
C
Figure 11.2
Absence of Sustainable Configuration
criticism, for price decisions are habitually more flexible than those
of entry and production. The behavior credited to firms by BPW
particularly assumes that established firms do not modify their
prices when an entrant presents itself, which is somewhat hazardous
practice.
Finally, no doubt the hypothesis of the absence of unrecoverable
expenses is very strong, and it can describe realistically only a limited
number of markets. The case of airline companies is interesting insofar
as their deregulation in the United States in the 1980s was considered
a propaganda success of the contestable market theory.3 It seems a
priori that the air transportation market is close to contestability, inas-
much as airplanes can be rented or resold fairly easily. In a first period
one effectively observed a massive entry in this industry, where the
number of firms tripled. But in a second phase, a consolidation of
supply was witnessed: the number of airline companies is even smaller
than before regulation, and the Herfindahl index of the industry has
fallen to its original level. The barriers to entry were probably under-
estimated: the scarcity of airport slots, the role of electronic reservation
systems, and so on. More fundamentally, potential competition seems
to be an imperfect substitute for the real thing, and the reactions of
established firms have certainly been underestimated.
In conclusion, it does not seem that the concepts elaborated by BPW
overturn classical analysis. They succeed nonetheless in attracting out
attention to the influence of the existence of potential entrant firms on
the behavior of those already established on a market.
11.2 Preemption
Under the rubric of preemption are regrouped strategies that consist,

for a firm established on a market (an incumbent), in accumulating
capital to dissuade entry. Here the term “capital” has acceptance in a
much larger sense than that of physical or production means; it can also
refer to experience accumulated in a learning situation (learning-by-
doing), to goodwill acquired in the course of time, or again to a stock
of patents resulting from previous R&D.
The intuition of the fact that accumulating capital can dissuade
entry dates back to Stackelberg in 1934, and it was later developed
by Spence (1977) and Dixit (1980). Consider therefore an established
3. I refer the reader to the article of McGowan and Seabright (1989) for details.
firm and a potential entrant. We assume that the timing of the game is
as follows:
• The established firm decides to carry its capital to K1.
• Then the potential entrant carries its capital to K2 (one can have
K2 = 0, in which case there is no entry).
• Finally, the two firms compete in prices.
To simplify, we do not specify the conditions of the price competition.

We simply suppose that at the given capitals K1 and K2, it leads to
reduced profit functions p1(K1, K2) and p 2(K1, K2) which verify the fol-
lowing properties:
• p1 is concave in K1.
• ∂p1/∂K1 is a decreasing function of K2.
• p 2 possess two symmetrical properties.
Numerous forms of price competition in fact lead to reduced profit

functions which verify these properties. To fix ideas, we will give the
following analytical form to reduced profit functions:
"i = 1, 2, p i (K1 , K 2 ) = K i (1 - K1 - K 2 )
If the capacities game was simultaneous (the established firm and the
potential entrant firm choose their Ki at the same time), then the equi-
librium would of course be symmetrical, and it is easily calculated that
K1 = K2 = 1/3 and p1 = p 2 = 1/9. This equilibrium of the simultaneous
game will serve us as a reference.
The game that interests us is a Stackelberg game, with a leader that
is the established firm and one follower that is the potential entrant. It
is easily seen that the reaction function of the entrant (i.e., the K2 which
maximizes p 2 to given K1) is
1 - K1
K 2 = R2 ( K 1 ) =
2
As for the established firm, it will choose K1 to maximize p1 [K1, R2(K1)],
which at equilibrium finally gives
1 1 1 1
K1 = , K2 = , p 1 = , p 2 =
2 8 8 16
The established firm therefore overinvests and thus succeeds in limit-

ing entry (since K2 is smaller than in the simultaneous game). It guar-
antees itself a higher profit than that of the entrant firm.
Note that the hypothesis that the established firm can commit on the
value of K1 is very important: if it updates the value of K1 after having
observed K2, it will choose K1 = R1(K2) = 3/8 and not K1 = 1/2. There-
fore the investment must have an irreversible character.
Thus far we have assumed that the entry costs are zero, which is
unrealistic. Now suppose that every entrant must discharge a fixed cost
f that is still inferior to the equilibrium profit p 2 = 1/16, so entry remains
viable a priori. The reaction curve of the entrant then becomes discon-
tinuous. In effect we let the capital level be K1b which verifies
max[K 2 (1 - K1b - K 2 ) - f ] = 0
K2
If K1 > K1b, the potential entrant has no interest in entering, since its
profits do not allow for the recovery of the entry cost. We therefore
get R2 (K1) = 0. If K1 < K1b, we again see the reaction function R2(K1) =
(1 - K1)/2. Figure 11.3 shows the reaction function R2.
The established firm must always maximize p1[K1, R2(K1)], which is
now discontinuous. Two cases must therefore be distinguished:
K
2
Kb K
1
1
Figure 11.3
Reaction function in the Dixit-Spence model
• If K1 < K1b , the entrant firm participates in the market; since f < 1/16,
we find that K1b > 1/2. The optimum is therefore always K1 = 1/2, which
gives a profit p1 = 1/8
• If K1 ≥ K1b , the potential entrant remains outside the market. The
maximum profit is reached when K1 = K1b (so that the entrant is just dis-
– –
suaded from entering), and it is worth p1 = 2÷f (1 - 2÷f ).
– –
Take f close to 1/16. Then 2÷f (1 - 2÷f ) is close to 1/4, and therefore the
established firm’s optimum is to invest up to K1b to completely dissuade
entry, which permits the firm to double its profits in relation to the
Stackelberg equilibrium without entry costs.
11.3 Limit Price and Predation
In this section we consider the potential reasons for an established firm

to practice an aggressive low-price policy.
11.3.1 Limit Price
We speak of limit price when the incumbent firm practices a price that
is less than the monopoly price to make entry less profitable. The
underlying idea is that the entrant knows it will have to practice even
lower prices in order to win a market share, so the game is not worth
the candle.
This classical intuition, however, is not very satisfactory: it assumes
a none too credible engagement in prices, since they are an easily
changeable strategic variable. If the potential rival firm effectively
enters the market, then the best response of the established firm is to
maximize its profits in this new situation, and not to maintain an
artificially low price. Moreover it is essential for the established firm to
persuade the entrant of its readiness to fight. Sacrificing its present
profits to this end may not be optimal.
Milgrom and Roberts (1982) propose a more modern analysis of limit
pricing that rests on the asymmetry of information between the entrant
firm and the established firm. Actually it is reasonable to suppose that
the established firm is better informed of its costs than is the entrant.
In fixing a low price, it signals to the entrants that its costs are low and
that it can therefore easily sustain a price war. Such a game is one of
signals and, as such, comprises large numbers of perfect Bayesian equi-
libria. Nonetheless, the most reasonable among them (the only one that
is stable in a certain sense) indeed possesses the property that the firms
having a low cost tariff below the monopoly price.
Analysis of the limit price in terms of social surplus is ambiguous.
The established firm succeeds in dissuading entry when its costs are
low, which tends to reduce social surplus, but it tariffs below the
monopoly price, which in compensation is desirable. The regulation
policy that the state must adopt in the situation must then be exam-
ined in terms of a specific situation.
11.3.2 Predation
Predation, For an established firm that has just seen an entrant firm
take a share of its market, consists in launching into a price war to
oblige the entrant to exit the market.4 This behavior is fairly common.
It can reach a high degree of savagery, as in the case of the Cola wars
between Coca-Cola and Pepsi-Cola in the 1970s: at times the prices
for these beverages were barely sufficient to cover the cost of the alu-
minum cans containing them.
Any explanation of predation assumes that the established firm has
an advantage over the entrant, which permits it to win a price war. This
advantage can of course be due to a lower production cost. Cabral and
Riordan (1994) invoke learning effects that result in the established
firm’s seeing its costs decrease according to past production. Things
can become more complex if the two firms are identical. Then one can
appeal to imperfections of the capital markets. A common explanation
is that the entrant firm’s bank fears that a price war will raise its prob-
ability of failure and thus refuse to continue to finance the firm if it
jumps into an aggressive policy of low prices.
The asymmetry of information is another possible explanation. It
is based on “reputation effects”: if the entrant firm thinks there is
more than a slight possibility that the established firm will fight
entry even when it is not in the latter’s interest to do so, and the game
can be prolonged for a fairly long period of time, then the potential
entrant will not enter the market. It is therefore desirable for the
established firm to create a reputation of aggressiveness in order to
dissuade entry.
4. Strictly speaking, there is predation only when the price is so low that both firms take
losses.
11.4 Research and Development
As shown by the growth accounting literature, research and develop-

ment is responsible for a fair portion of economic growth. This alone
suffices in justifying that industrial organization is interested in the
impact of market organization on R&D. Its place in this chapter is pri-
marily due to the advantage that a firm can draw from its stock of R&D,
thanks to the patent system which confers upon it de facto a tempo-
rary monopoly.
As early as 1942 Schumpeter5 expressed the idea that monopolies are
a necessary evil for the development of R&D inasmuch as
1. being larger, it is easier for monopolies to invest and benefit from

increasing returns tied to R&D activity
2. being more diversified, monopolies are better prepared to absorb the
inherent risks of R&D
3. having monopoly power makes innovation more profitable for
them.
We will interest ourselves, above all, with the last reason. We will
see that we must slightly temper Schumpeter’s enthusiasm for
monopolies.
We should first note that effectively the patent system is a compro-
mise. After an innovation occurs, the socially optimal decision would
be to make it public. However, this would leave no incentive for firms
to innovate, and for this reason all countries protect innovations for a
limited duration. For our purposes, we will nevertheless take the
patent system as given.
We will depend on Arrow’s (1962) article, where he considers a
process innovation6 that reduces the (constant) marginal production
cost of a good from c̄ to c < c̄. This innovation is protected by a patent
¯
that, to simplify, we assume has an unlimited life span. Let r be the
interest rate, so that the discounted value (in continuous time) of a
constant flux of 1 is Ú+0• e-rtdt = 1/r.
5. See Schumpeter (1942). This book is reputed to have predicted an inescapable con-
vergence of capitalism and socialism, a vision that seemed reasonable until the 1980s but
has fallen out of fashion since.
6. A process innovation reduces the production cost of an existing good, in opposition
to a product innovation, which creates a new good.
First, we compute the social value of this innovation. If the demand

function for the good is D(p), the discounted social surplus when the
production cost is c is
1 +•
D(x)dx
r Úc
since at the social optimum firms must price at marginal cost. The social
value of innovation is therefore
1 +• 1 +• 1 c
Vs = D(c)dc - Úc D(c)dc = Úc D(c)dc
r Úc r r
What about different market organizations? Suppose, first of all, that
innovation is realized by a monopoly. Let pm(c) be the monopoly price
when the cost is c and p m(c) the corresponding profit:
p m (c) = max(p - c)D(p)
p
By the envelope theorem we get
∂p m
= -D[p m (c)]
∂c
Therefore the value of innovation for the monopoly is due to the

increase of its profits when its costs go down:
1 c
V m = p m (c) - p m (c ) = D[p m (c)] dc
r Úc
Right away we note that since the monopoly prices above marginal
cost, we get D(pm(c)) < D(c) and therefore Vm < Vs: the value of innova-
tion for the monopoly is less than its social value. This was foreseeable,
insofar as cost reduction operates on a smaller production because the
monopoly restricts its production.
Now assume that the starting situation is competitive, with several
firms all pricing at the marginal cost c̄. If the innovator continues to
conduct itself in a competitive manner, its profits will remain zero, and
the encouragement to innovate is therefore nonexistent. Yet this
hypothesis is obviously unrealistic. So we must distinguish two cases,
depending on whether the new monopoly price pm(c ) is superior or
¯
inferior to c̄:
• If pm(c ) ≥ c̄ (nondrastic innovation), the innovator is interested in

¯
tariffing a slightly inferior price to c̄ in order to monopolize the market.
Then it realizes a discounted profit
1
V c = (c - c)D(c)
r
which is easily shown to lie between Vm and Vs
• If pm(c ) < c̄ (drastic innovation), the innovator is interested in tariffing
¯
the monopoly price pm(c ). In this case it will still monopolize the market
¯
and realize a discounted profit
1 m
Vc = (p (c) - c)D(p m (c))
r
Here again, it is easily shown that Vc lies between Vm and Vs.
Therefore in all cases we get the conclusion Vm < Vc < Vs. Schumpeter’s
perception is likely wrong: a monopoly gains less than a competitive
firm by innovating (and is consequently less encouraged to do so).7
This is the replacement effect: through innovation the monopoly replaces
itself, and if it earns new profits, it loses old ones.
Still it would not be right to conclude from this analysis that an inno-
vation always has more value for a potential entrant than for an estab-
lished monopoly. It all depends on the conditions of innovation. If
innovation has been achieved by a firm outside the market looking to
sell to the monopoly or to the entrant, we can show that under rea-
sonable conditions, the monopoly will be prepared to pay more than
the entrant for innovation. This is due to the monopoly’s fear of seeing
itself in competition with the entrant and to the efficiency effect, which
implies that competition reduces total industry profits. This result
explains, for example, why Xerox remained for a long time a monop-
oly on the photocopier market; it got in the habit of buying back all the
patents registered in this domain and of leaving them “dormant,” at
least until forbidden this practice by antitrust authorities.
In this chapter we sidelined many interesting subjects. There are
more dynamic aspects of R&D, in particular, the “patent races” that
dissipate at least a part of the monopoly rents potentially brought in
ex ante by an innovation.
7. Neither type of firm receives all of the social advantages of innovation, since the con-
sumers benefit as well.
Bibliography
Arrow, K. 1962. Economic welfare and the allocation of resources for inventions. In
R. Nelson ed., The Rate and Direction of Inventive Activity. Princeton: Princeton University
Press.
Baumol, W. 1982. Contestable markets: An uprising in the theory of industry structure.

American Economic Review 72: 1–15.
Baumol, W., J. Panzar, and R. Willig. 1982. Contestable Markets and the Theory of Industry
Structure. New York: Harcourt Brace Jovanovic.
Cabral, L., and M. Riordan. 1994. The learning curve, market dominance, and predatory
pricing. Econometrica 62: 1115–40.
Dixit, A. 1980. The role of investment in entry deterrence. Economic Journal 90: 95–106.
McGowan, F., and P. Seabright. 1989. Deregulating European airlines. Economic Policy 9:
284–344.
Milgrom, P., and J. Roberts. 1982. Limit pricing and entry under incomplete information:
An equilibrium analysis. Econometrica 50: 443–60.
Schumpeter, J. 1942. Capitalism, Socialism and Democracy. New York: Harper and Brothers.
Spence, M. 1977. Entry, capacity, investment and oligopolistic pricing. Bell Journal of
12 Vertical Relations
Until now we have considered producers who were selling their prod-
ucts directly to consumers. In reality such a configuration is very rare.
There are generally one or several intermediary participants. In this
chapter we will concern ourselves with what are called vertical relations,
which link a producer (often a monopoly in this chapter) and its dis-
tributors. The central question deals with the control that the producer
would like to exercise over the distributors’ activities. The producer
could have recourse to a contract that specifies a nonlinear tariff.
However, in practice, the producer usually cannot observe the sales of
each of its distributors, so a nonlinear tariff is diverted from its objec-
tives by arbitrage between distributors.1 The producer is then only able
to utilize linear prices, possibly with a franchise, if he can observe
whether the retailer actually sells his product. Historically these
instruments of control have seemed insufficient to producers, who
have compensated with a whole panoply of vertical restrictions (or
constraints):
• Exclusive territories. In this system the producer grants a local
monopoly on his product to each distributor. By way of example, we

can cite franchise stores or insurance agents. Note that the term “local
monopoly” is not necessarily a purely geographical one. The producer
could, for instance, grant to each distributor a monopoly on a variety
of his product.
• Exclusive dealing. The producer prohibits the distributor from selling
a brand that rivals his product. Examples are European automobile
dealers and, again, insurance agents.
1. At least this is the point of view we will take in this chapter. Note, however, that in
the retail sector, certain participants practice a complex system of rebates that actually
has much in common with a nonlinear tariff.
• Tied sales. The producer assumes that the distributor will transform
the product by combining it with products of others. Then the producer
could insist that the distributor buy all of his products from him.
• Resale price maintenance. The producer directly fixes the retailer’s
selling price (or at least a minimal price). This practice is common in
the European book business (where several EU states fix minimum
prices) and perfumery.
•Buying quota. The producer imposes on the retailer a minimal
amount of sales.
These restrictions are the subject of still quite animated legal and eco-
nomic debates that center upon the comparison of the economic effi-
ciency and the anticompetitive distortions that restrictions can bring
on. The Chicago school, which was very influential in the United States
in the 1980s, has insisted upon the efficiency contribution of vertical
relations and has praised laissez-faire in this regard. American jurispru-
dence therefore fluctuated over that period of time. In Europe the
dominant attitude is much more skeptical; the accepted wisdom is
more restrictive legislation.
A vertical relation could be analyzed, from a technical point of view,
as a hierarchy, that is, as a Principal-Agent problem where the Princi-
pal is generally the upstream firm and the Agent the downstream one.
However, the difficulty is dealing with the fact that in the most inter-
esting cases there are actually several Principals in competition and
several Agents also in competition. This type of model is at the limit of
what economists currently know how to solve, and we only skim the
surface of its complexity.
12.1 Double Marginalization
We will start by examining a classic model due to Spengler (1950). This

model gave rise to the dictum that “the only thing worse than a monop-
oly is a chain monopoly.”
Consider first the producer in this monopoly situation. He produces
with a constant marginal cost c; he sells the good at a price w to a
retailer, himself in a situation of monopoly, who distributes the good
at a price p without incurring costs. The final demand is given by D(p)
= d - p, where d is supposed to be larger than c (in order for produc-
tion to be profitable). This situation is summarized by figure 12.1.
Now consider the retailer’s program. For a given upstream price w,
he must choose the downstream price p to maximize his profit
Vertical Relations 195
M cost c<d
M cost 0
D(p)=d-p
Figure 12.1
Chain monopoly
p d = (d - p)(p - w)
We can deduce from this that
2
w+d ( d - w) d-w
p= , pd = , x=
2 4 2
where x is the quantity sold. As for the producer, he must choose
w to maximize his profit which, since x = (d - w)/2, is written p =
(w - c)(d - w)/2. So we get
2
c+d ( d - c)
w= , p=
2 8
where finally for prices
c+d c + 3d
c<w= <p=
2 4
This double inequality is called “double marginalization”; it refers to
the fact that each of the links of the chain, being a monopoly, prices
above its marginal cost.
For the profits of the producer, the distributor, and the vertical struc-
ture, we find that
2 2 2
( d - c) ( d - c) 3(d - c)
p= , pd = , pt =
8 16 16
Interestingly, in comparing these results with those of the “integrated

structure,” we see that if the producer and the retailer had merged into
a single firm, this structure would tariff a price q that maximizes pi =
(d - q)(q - c), whence
2
c+d ( d - c)
q= , pi =
2 4
The profit of the integrated structure is therefore higher than the total
profit of the nonintegrated chain.2 Moreover the downstream price q is
inferior to the price p, and therefore the consumers’ surplus increases.
From this result we can deduce that vertical integration of a chain
monopoly leads to that fairly rare thing in economics, an improvement
in the Pareto sense. In effect there exists in this vertical relation an exter-
nality exerted by the retailer upon the producer: when the retailer
lowers p, x increases, and as w > c, the producer’s profit increases. The
inefficiency of the chain monopoly is in part due to the fact that the
retailer does not take this effect into account; vertical integration inter-
nalizes the externality.
Could we arrive at the same result without making the simple but
radical decision to integrate the two firms? With perfect information,
the response is yes. The producer could in effect adopt three vertical
restrictions that are perfectly equivalent:
• Price at marginal cost w = c. The retailer would then tariff p = q, and
the producer could recuperate his profits by making him pay into a
franchise A = pi.
• Impose onto the retailer the selling price p = q (or simply impose
p £ q). The optimal downstream price is then w = q.
• Impose onto the retailer a sales quota x ≥ D(q).
With asymmetric information, things become more complicated.

For example, if the retailer is better informed on the state of demand
than is the producer, it will be necessary to consider optimal
2. To see this more directly, note that by definition,
p i = max( d - q)( q - c) > ( d - p)( p - w) + ( d - p)( w - c) = p d + p = p t

q
risk-sharing and a good decentralization of decisions, as we will see

section 12.3.
12.2 Justifications of Vertical Constraints
Let us start by examining the arguments that could justify (from either
a private or a social optimum point of view) the use of a certain verti-
cal constraint.
12.2.1 Retailer Effort Incentive
We can designate as retailer effort all services that permit the retailer
to increase sales. Grouped under retailer effort are advice to clients,
local publicity, after-sales service, among other such business promot-
ing sales. The retailer effort has two external effects:
• Toward other retailers. Certain consumers seek advice from one
retailer and end up buying from another retailer the same product
at a lower price. Thus a discount audio store can benefit from its
proximity to a more traditional store by “diverting” its clients. This
type of external effect leads to a suboptimal effort
• Toward the producer. Anything that increases the sales of the retailer
likewise increases the sales of the producer. Here again, the retailer’s
effort risks being inferior to that of an integrated structure.
Clearly, it is advisable for the producer to encourage the retailer to

make the effort to provide sales enhancement services. One of the tools
used most often to this end is resale price maintenance. By assuring a
margin to the retailer, such effort is validated, and he is encourage to
continue his services in behalf of the producer. The retailer incentive
argument dates back to Telser’s (1960) classical article. Moreover, if this
practice is imposed on all distributors, client diversion to discount
stores will be avoided,3 allowing retailers to fully harvest the fruits of
their labor. Laws on book pricing have been motivated in various
European countries by such consideration.4
3. Discount stores actually practice free-riding on the public good which comes from the
effort of other retailers.
4. The more commonly accepted justification of resale price maintenance has had more
to do with the influence of the Fair Trade Movement in the United States which sought to
preserve small traditional stores. This movement was quite powerful between the 1930s
and the first oil crisis in 1973.
12.2.2 Price Discrimination
Assume the existence of several submarkets which have different

demand elasticities. Then the producer will want to appropriate as
much as possible the consumer surplus by causing more to be paid on
submarkets with less elasticity. For example, a publisher will sell hard-
bound books to libraries (whose demand is generally inelastic) and
paperback books to individual consumers.
The producer will be able to achieve his ends by granting exclusive
territories to his retailers, each “territory” here being a submarket. He
may then resort to a two-part tariff to recover the maximum profit.
12.2.3 Tied Sales
Consider the situation represented on figure 12.2. The retailer buys an

input x at the price w from a monopoly, producing it at constant
marginal cost c, and another input x¢ from a competitive sector,
which prices x¢ at the marginal cost w¢ = c¢. Under these conditions w/w¢
> c/c¢, since w > c. So the input x will be underused by the retailer,
which will not do for the monopoly concerned. The monopoly could
solve this problem by dictating that the retailer also procure the input
x¢ from them, and by pricing it at a price w¢ such that w/w¢ = c/c¢: this
is the practice of tied sales. Note that in this very simple case, tied sales
have a social utility, for they bring the retailer to the production
optimum.
M c M c'
w x w' x'
Figure 12.2
Tied Sales
12.2.4 Reduction of Price Competition
The effect of certain vertical constraints can be to reduce the down-

stream competition. This is the case with exclusive territories, since
each retailer becomes a local monopoly. Rey and Stiglitz (1995) show
that when producers of differentiated products sell to consumers
via retailers, the establishment of exclusive territories permits them
to reduce price competition not only downstream but also up-
stream and therefore to come closer to a cooperative optimum; pro-
ducers can then recuperate retailers’ excess profits by a franchise.
Naturally, such a strategy is detrimental to consumers and to social
welfare.
12.3 Comparison of Different Practices
In most cases the retailer is better informed of the state of demand than
is the producer. Any comparison of different vertical restrictions must
take this factor into account. Following Rey and Tirole (1986), we
assume in this section that the final demand D(p) = d - p contains a
parameter d that is known to the distributor but not to the producer. It
is a matter then of the producer’s solving a self-selection problem, with
two new imperatives:
• assure a good decentralization of decision making. This implies
that the downstream price p must vary with the intensity of the
demand d.
• Achieve good risk-sharing with the retailer. The producer presum-
ably intervenes in numerous independent markets and can therefore

by supposed to be risk-neutral; the retailer presumably finds it more
difficult to diversify away his risk. The producer must therefore par-
tially insure the retailer.
We will compare three possible strategies for the producer from the
perspective of these two new imperatives introduced by the asymme-
try of information:
• Exclusive territories. Each retailer is a local monopoly.
• Resale price maintenance. The imposed price is supposed to be the
same for each retailer.
• Competition among retailers. The producer can simply let competi-
tion take place among retailers.
First of all, where the decentralization of decisions is concerned,

•in resale price maintenance, the downstream price is set by the pro-
ducer, who does not know d; the price cannot then vary with d
•the retailer who discharges an exclusive territory is a local
monopoly and will therefore make p vary with d
•in perfect competition between retailers, the retailers price at (sup-
posedly constant) marginal cost, which does not depend on d, so the
downstream price is independent of d
In view of this criterion, exclusive territories appear to be preferable to

competition and to resale price maintenance.
Let us move on to the second criterion: insurance of retailers against
profit variations.
• with resale price maintenance, the retailer’s profit is an increasing
function of the risky parameter d

• this is also the case with exclusive territories
• by contrast, in competition, the retailers’ profit is zero and therefore
independent of d
This time competition clearly dominates exclusive territories and resale

price maintenance.
From this analysis we can conclude that resale price maintenance is
dominated according to our two criteria.5 The comparison between
exclusive territories and competition depends on the retailer’s aversion
to risk. If the retailer is very adverse to risk, the imperative of insur-
ance will prevail and competition will be the best of systems; if his
aversion to risk is weaker, exclusive territories will be preferable to
competition.
The comparison that we just executed used the perspective of the
economic efficiency of the vertical relationship. If we place ourselves
now at the social point of view, we must further take into account
consumer surplus, which is a decreasing function of the downstream
price p. Competition, which assures the lowest downstream price, is
of course the consumers’ preferred system. It can even be shown that
whatever the retailer’s aversion to risk, competition maximizes the
5. Deneckere, Marvel, and Peck (1997) show, however, that if retailers supply themselves
from the producers before observing demand, then resale price maintenance can raise
the profits of the producer and even consumer surplus.
social surplus, which is hardly surprising. This conclusion could

change if we consider the necessity of providing effort incentive to
retailers or the existence of specific investments, or even if we consider
the impact of vertical restrictions on interbrand competition.
12.4 Elements of Law
To conclude this chapter, I would like to provide the reader with some
notion of the legislation and jurisprudence in the domain of vertical
restrictions. At issue is one of the essential elements of what the Amer-
icans call antitrust policy and the Europeans prefer to call competition
policy. In the United States, antitrust policy has its origins in the
Sherman Act of 1890 and the Clayton Act of 1914. But these texts are
fairly vague, and the jurisprudence has been altered in the course of
time (see Scherer and Ross 1990). The European laws are more recent.
The 1957 Treaty of Rome, which created the Common Market, played
a driving role,6 particularly through Article 85 which forbids agree-
ments that thwart competition without improving productive effi-
ciency (see Korah 1994). It has been completed since that time by
several “common regulations.”
There have long been debates in the law and economics literature
with the objective of finding out if one should decree the systematic
legality or illegality per se of different vertical constraints, or if their
effect on social welfare should be examined on a case by case basis (by
applying what is called the rule of reason). In the United States, the
Chicago school thus lobbied for the legalization, as such, of most ver-
tical constraints. In practice, jurisprudence generally depends on the
rule of reason rather than on per se decisions. For example, it evalu-
ates the degree of interbrand competition in order to justify its deci-
sions: vertical restrictions are judged less severely where interbrand
competition is brisk. Still there are major exceptions. For instance, the
practice of resale price maintenance is illegal per se in Europe.7 It has
also been illegal in the United States since the oil crisis, after a much
more permissive period which, in numerous states, even imposed upon
retailers to respect the minimum prices decreed by producers. The
practice of tied sales, as such, is also considered illegal. On the other
6. Like all international treaties, the Treaty of Rome has a higher value than the national
laws of its signing countries.
7. Even if minimum prices for books are authorized, for reasons that are more political
than economic.
hand, exclusive territories and exclusive dealing are usually judged

according to the rule of reason. European authorities even issued an
exemption rule which authorized the automobile retail sector to con-
serve its own particular practices.
Bibliography
Deneckere, R., H. Marvel, and J. Peck. 1997. Demand uncertainty and price maintenance:
Markdowns as destructive competition. American Economic Review 87: 619–41.
Korah, V. 1994. EC Competition Law and Practice. London: Sweet and Maxwell.
Rey, P., and J. Stiglitz. 1995. The role of exclusive territories in producers’ competition.
Rand Journal of Economics 26: 431–51.
Rey, P., and J. Tirole. 1986. The Logic of Vertical Restraints. American Economic Review 76:
921–39.
Scherer, F., and Ross, D. 1990. Industrial Market Structure and Economic Performance.
Boston: Houghton Mifflin.
Spengler, J. 1950. Vertical integration and antitrust policy. Journal of Political Economy 58:
347–52.
Telser, L. 1960. Why should manufacturers want fair trade? Journal of Law and Economics
3: 86–105.
IV Incomplete Markets
The theory of general equilibrium proceeded in the 1960s on the

hypothesis that all markets necessary to the successful working of the
economy could be opened. It escaped no one that this hypothesis was
a bit heroic, that its negation would generally destroy the optimality of
the equilibrium, but the technical tools that would permit one to treat
the incompleteness of markets did not yet exist. The technical tools
did not become available until the beginning of the 1970s. The theory
of incomplete markets has rapidly developed since then, to the point
where it seems possible to me today to give a summary without
becoming mired in overly technical considerations. This is what I will
endeavor to do in this last chapter. The general theme will be that
when markets are incomplete, equilibrium can be efficient only in very
exceptional cases, which clears the way (in theory at least) for a new
genre of governmental interventions.
13 Elements of the Theory of
Incomplete Markets
The first section in this chapter shows how the theory of general equi-
librium in incomplete markets naturally ensues from the generaliza-
tion of the model of general equilibrium to situations of uncertainty.
The next two sections study the question of the existence of equilib-
rium, which poses delicate problems, and that of its inefficiency. These
three sections are located within the framework of an exchange
economy; I will subsequently expose the difficulties presented by the
introduction of production.
These four sections are fairly theoretical, as is, unfortunately, the bulk
of the literature on the subject. In order to refute the impression that
incomplete market theory is but a gadget for specialists, I devote the
last section of this chapter to one of its applications.
The experienced reader will note that I am only interested here in
real assets. Indeed, I consider that this is the most natural framework
in which to present market incompleteness. Nevertheless, in the appen-
dix, I give some elements of the theory’s development when the avail-
able assets are nominal.
13.1 General Framework
As the reader knows, the model of competitive general equilibrium was

extended by Arrow (1953) to account for uncertainty. This extension
consists simply of assigning to each good at each date and in every state
of the world a market and a price. To simplify the account, for the bulk
of this chapter we will place ourselves in a two-period exchange
economy where uncertainty is symmetrical and bears on the state of
the world at the second date.1 We will assume that there exist L goods,
1. This choice of modelization is not necessary for treating the case of complete markets,
but it will become very useful when we approach the question of incomplete markets.
206 Incomplete Markets
l = 1, . . . , L, available on each date and in every state of the world and

n consumers, i = 1, . . . , n. At date 0, the consumers do not know what
the state of the world will be at date 1; they simply know that it can be
s = 1, . . . , S. A consumption plan for a consumer i is therefore a vector
(x0, x1, . . . , xS), where x0 Œ IRL is his consumption vector on date 0, and
for s ≥ 1, xs Œ IRL is his consumption vector at date 1 in state s. His utility
is written then Ui(x0, x1, . . . , xS). We will denote (w 0i , w 1i , . . . , wSi ) Œ IRL(S+1)
the resources of the consumer i on date 0 and in every state of the world
on date 1.
Arrow’s original idea consists of introducing contracts of contingent
delivery, one for every good in every state of the world. It is then said
that the markets are complete. Thus a consumer, for example, could buy
at date 0 three units of good l which are deliverable only if the state
s is realized and at a unit price which we will denote pls. A consumer
who wants to have a delivery of good l whatever the realized state then
will have to pay SSs=1 pls for each unit. If p0 is the price vector for avail-
able goods at date 0 and ps the price vector for available goods at date
1 in state s, the consumer’s budgetary constraint i is then written
S S
p0 ◊ x0i + Â p s ◊ x si £ p0 ◊ w 0i + Â p s ◊ w si
s =1 s =1
The conditions of equilibrium are

n n
Â x0i = Â w 0i
i =1 i =1
at date 0 and
n n
Â xsi = Â w si
i =1 i =1
in state s at date 1.
Under the usual hypotheses (in particular, that of the convexity of
preferences2), equilibrium exists and the two fundamental theorems
of welfare apply: the equilibrium is Pareto-optimal and any Pareto
optimum can be decentralized. We will denote p*0 , (p*s )s=1, . . . , S, x*0 and
(x*s )s=1, . . . , S the prices and allocations of equilibrium.
The introduction of complete contingent markets therefore appar-
ently permits the accounting of uncertainty with a large economy of
2. Recall that in the expected utility framework, preferences are convex if and only if the
agents are averse to risk.
Elements of the Theory of Incomplete Markets 207
means.3 This extension of the general equilibrium model is nonetheless

open to criticism in at least two ways:
• Even in a simple two-date model, a very static vision of the world is
proposed: all markets are open on date 0, and there is no need to reopen
them on date 1, for all contingent transactions have been already
decided.
• A very high number of markets is required: L(S + 1).
We will see below that incomplete market theory partially responds to

the first criticism with a more sequential vision of exchange. Arrow also
proposed a sequential implementation of the equilibrium that requires
only (2L + S) markets, as we will now see.
Suppose that the consumers can exchange assets so as to transfer
buying power between date 0 and the different states of the world
on date 1. For s = 1, . . . , S, asset as entitles to a unit of the good 1 (taken
arbitrarily as numéraire so that p1s = 1 for all s) in state s and to nothing
at all in the other states: such an asset is called an Arrow-Debreu
asset. The total supply of each asset as is zero. At date 0, consumers
exchange goods consumed on that date and assets: (L + S) markets
are therefore open.4 If state s is realized on date 1, consumers use their
goods endowments and their asset stocks as to finance their consump-
tions. Let qs be the asset price s on date 0 and q si the quantity of assets
bought by consumer i; the budgetary constraints of this consumer are
written
S
p0 ◊ x0i + Â q si qs £ p0 ◊ w 0i
s =1
at date 0 and
p s ◊ x si £ q si + p s ◊ w si
in state s at date 1.
In this chapter (as in all these writings) we will use the rational expec-
tations hypothesis: each consumer anticipates without error at date
0 the prices ps that will prevail on each “spot” market at date 1. We
multiply then the second constraint by qs, sum on s, and add the
result to the first constraint, so we get
3. This explains the subject’s occupying only four pages in Debreu (1959, ch. 7).
4. It will be noted that consumers cannot gauge their asset purchases by future resources.
S S
p0 ◊ x0i + Â qs p s ◊ x si £ p0 ◊ w0i + Â qs p s ◊ w si
s =1 s =1
The expression above is exactly equivalent to the consumer’s bud-

getary constraint in the complete contingent markets equilibrium, as
we see by posing p0 = p*0 and p*s = qsps. The equilibrium of this new model
is therefore identical to that of the first. Agent i buys q*s i = p*s · (x*s i - w si )
units of asset as to finance his consumptions in state s, so there is equi-
librium on each asset market:5
n
Ê n *i n
ˆ
Âq *s i = p s* ◊ Â
Ë i =1
x s - Â w si = 0
¯
i =1 i =1
However spectacular it may seem, the fall in the number of markets

necessary for implementing an equilibrium has a slightly abstract char-
acter: in a real world, S is an enormous number (perhaps infinity), and
it is improbable that markets exist for all Arrow assets. Nevertheless,
the Arrow construction is important because it introduces the basic
concepts of incomplete market theory, such as was created by Radner
(1972).
Radner generalized Arrow’s idea in two directions:
• The asset market includes only J £ S assets aj, j = 1, . . . , J.
• Each asset entitles the consumer to a basket of different goods in
every state of the world. Thus aj entitles to alsj units of good l in state s.
In this way each asset is represented by a matrix (L, S).
We will maintain the hypothesis that the total supply of each asset aj is
zero, so that we have S ni= 1 q ji = 0 in equilibrium,6 where q ji is the demand
for asset j of consumer i. The budgetary constraints of the consumer i
are now written
J
p0 ◊ x0i + Â q ji q j £ p0 ◊ w 0i
j =1
at date 0 and
J
p s ◊ x si £ p s ◊ Â q ji a sj + p s ◊ w si
j =1
5. We will note that Arrow’s asset prices are indeterminate at equilibrium: we can change
qs to lsqs on the sole condition of modifying ps to ps/ls so as to maintain the equality
p*s = qsps. Such a transformation affects asset purchases but not consumptions.
6. If the asset aj is an action of firm j in a model with production, we would actually have
Sni = 1 q ij = 1. This would not fundamentally change the analysis.
in state s at date 1, where ajs Œ IRL denotes the basket of goods to which
asset j entitles in state s and qj is the price of asset j at date 0.
It is necessary to introduce some notation. If s is a state of the world
and j an asset, we can denote by ps · asj the value in state s of the basket
of goods to which aj entitles the consumer:
J
p s ◊ a sj = Â pls a lsj
l =1
Let p aj be the vector of IRS that represents the value of the asset aj in
different states of the world:
Ê p1 ◊ a1j ˆ
p aj = Á M ˜
ÁÁ ˜˜
Ë ps ◊ aSj ¯
Finally, let p a be the matrix (S, J) whose column j is p aj. With this
notation, the budgetary constraints on date 1 can be written in a con-
densed format:
p (x i - w i ) £ q i ◊ (p a)
If the utility functions are strictly increasing so that the budgetary
constraints are active at equilibrium, this constraint is at equality
and therefore implies that
p (x i - w i ) Œ sp (p a)
where sp(M) is the image (span) of matrix M, that is, the subspace of
IRS engendered by the columns of M. Note that the weights of the dif-
ferent columns of p a in p (xi - wi) are simply the portfolios q ji .
The vector p (xi - wi) of IRS formed by the values of excess demand
in different states is therefore constrained to evolve within the span
of p a. All of this will seem less abstract if we return to the Arrow-
Debreu assets. Then J = S, and the matrix p a is simply the identity
matrix (S, S), since (good 1 being the numéraire) ps · aj = dsj where d is
the Kronecker symbol.7 The image of p a is therefore IRS, which trans-
lates as the fact that when markets are complete, the consumer can
transfer buying power among states as he wishes.
The interesting case is of course that where sp(p a) π IRS, which is
necessarily true if J < S. Now the consumer’s capacity for transferring
buying power among states is limited by the lack of assets: markets are
7. Recall that dsj is worth 1 when s = j and 0 if not.

said to be incomplete. Let V is be the value of excess demand of con-

sumer i in state s at Arrow-Debreu equilibrium:
Vsi = p s* ◊ (x *s i - w si )
If we wanted to implement this equilibrium with assets a1, . . . , aJ, it
would be necessary to solve in q 1i , . . . , q Ji the S equations
J
Vsi = Âq ji ps* ◊ a sj
j =1
But there are S equations in J unknowns, which is clearly impossible

to solve once J < S.8 For example, if there is only one asset and it enti-
tles one unit of good 1 in every state of the world, then the consumer
can only equalize the value of his demand excess in every state of the
world, which represents an overly strong constraint on consumption
plans.
13.2 Existence of Equilibrium
One might think that the existence of equilibrium in incomplete

markets naturally ensues from that of complete market equilibrium.
Unfortunately, the problem is much more complex. This is a fairly tech-
nical subject, but I will nonetheless devote some paragraphs to it in
order to give the reader a sense of the difficulties.
For example, assume that there are two goods (of which the first is
always taken as numéraire), two states and two assets. The first asset
brings a unit of good 1 into every state of the world and the second a
unit of good 2 into every state. So the matrix p a is simply
Ê 1 p21 ˆ
pa =Á ˜
Ë 1 p22 ¯
If p21 π p22, this matrix is of full rank: sp(p a) = IR2. We see that the
demand functions for the goods coincide with those of the Arrow-
Debreu model. If, on the other hand, p21 = p22, then sp(p a) is unidi-
mensional, and the consumer is constrained in his capacity for
transferring purchasing power between the two states because the two
assets have become linearly dependent. This phenomenon, called drop
in rank and evidenced by Hart (1975), introduces a discontinuity in the
8. It is an equally impossible computation when J = S, when one of the assets is a linear

combination of the others.
budgetary constraints, and therefore a fortiori in demand functions. It

is easy to see that when p22 tends toward p21, the asset demands tend
toward infinity. Radner (1972) avoided this difficulty by imposing a
constraint of the form q ji ≥ -C on asset short sales, but this “solution”
is artificial, since the equilibrium can then depend on the arbitrary
constant C.
Still we have hope that the nonexistence of equilibrium caused by a
demand function discontinuity accompanying a drop in rank is an
exceptional phenomenon. This is what Duffie and Shafer (1985) effec-
tively showed. Since the difficulty stems from dimensional changes of
sp(p a), Duffie and Shafer defined a pseudoequilibrium by writing
the budgetary constraint
p (x i - w i ) Œ L
where L is a given subspace of IRS. From this it was easy to see that
a pseudoequilibrium could exist under usual conditions. Duffie and
Shafer proceeded to show, using fairly sophisticated tools of differen-
tial topology, that the demand functions are almost always continuous
in pseudoequilibrium. But then the space sp(p a) is locally indepen-
dent of p, and as a result it can be substituted for L in the definition of
pseudoequilibrium. The latter now becomes a true equilibrium. Duffie
and Shafer actually obtained a result of generic existence: the set of
specifications of utility functions, of initial resources, and of assets such
that equilibrium does not exist is of (Lebesgue) measure zero in the
set of possible specifications. In other words, if we take “at random”
one specification of economy among all the possibilities, equilibrium
exists with a probability of one.
13.3 Inefficiency of Equilibrium
When it exists, equilibrium is of course not Pareto optimal when J < S.

Indeed, we saw in section 13.1 that the Arrow-Debreu equilibrium
cannot generally be implemented in incomplete markets. But since all
Pareto optima are decentralizable by an Arrow-Debreu equilibrium, it
follows that incomplete market equilibrium is suboptimal.
This result is not all that interesting. If the government wanted to
intervene to restore the optimality of equilibrium, it would be neces-
sary for it to proceed with transfers of purchasing power inaccessible
to agents, that is, in order to complete the markets. This is a lot to ask
of government: if there are profound reasons (as we will see in the
conclusion of this chapter) for markets’ being incomplete, these reasons

are generally imposed both on the government and on consumers.9
We must therefore define a less demanding concept of optimality
that takes market structure as a given. For this we turn to constrained
Pareto optimality as defined by Diamond (1967).10 We assume that the
only possible reallocations must take place on the market for goods and
that for assets at date 0. A planner can only intervene in redistributing
the assets and consumptions on that date; the planner must subse-
quently let the equilibrium be established on date 1. A constrained
Pareto-optimal allocation is therefore such that there exists no real-
location of goods and of assets at date 0 that raises the utility of all
consumers.
Diamond demonstrated that when there is only one good (L = 1),
equilibrium is always constrained Pareto optimal. Hart (1975) exhib-
ited an example in a two-good model where one equilibrium domi-
nates another, which shows that Diamond’s result cannot be extended
to L ≥ 2. Stiglitz (1982) showed in a fairly general model (with pro-
duction) that constrained optimality was only obtained in very specific
cases. Finally Geanakoplos-Polemarchakis (1986) confirmed this result
by demonstrating that if there are at least two goods and two agents,
a planner who can redistribute asset portfolios and consumptions on
date 0 and possibly proceed to interagent transfers can improve the
equilibrium in the sense of constrained optimality, and at that for a
generic set of economies. It will be noted that most of the incomplete
market models used in macroeconomics rely on the representative
agent hypothesis (n = 1), which guarantees constrained optimality of
equilibrium.
The proof of these inefficiency results is not simple, but we will
attempt at some intuition. When the government redistributes at date
0, it changes consumer wealth in all states (direct effect). This results
further in modification of the equilibrium prices on date 1 (indirect
effect). While the government can at once take into account the direct
9. There are important exceptions. The equilibrium’s inefficiency in the presence of exter-
nalities can be analyzed as a phenomenon of market incompleteness, since, for example,
the market for the pollution imposed on consumers by a firm does not exist. The
creation by the government of a market for pollution rights reverts then to completing
the structure of markets.
10. Grossman (1977) invoked a concept of “Nash social optimality” that permits the two
fundamental welfare theorems to be obtained: any equilibrium is a Nash social optimum
and any Nash social optimum is decentralizable in equilibrium. Still this optimality
concept is very weak, as shown in Hart’s example cited below.
and indirect effects, the consumers only integrate the direct effect in
their calculations, since their price expectations are independent of the
portfolios they hold. It is this superiority of government that leads to
constrained inefficiency of equilibrium.
At always in such cases, we must ask ourselves the question of how
much information is necessary for the government in order to proceed
to such interventions. It can be shown that the government must in fact
know demand functions of goods and assets in terms of current and
expected prices, which sounds of course fairly presumptuous. More-
over it would be dangerous to believe that opening a new market when
several markets are missing necessarily improves welfare; Hart (1975)
gave a famous counterexample.
13.4 Equilibria with Production
The introduction of production in the model of incomplete markets

comes up against serious obstacles that stem from the difficulty in
assigning to firms an objective on which their shareholders are unani-
mous. When markets are complete, all shareholders agree that the firm
should maximize its market value. Indeed, imagine a firm whose input
on date 0 is a vector y0 Œ IRL and output on date 1 is a vector ys Œ IRL
in state s = 1, . . . , S. In the Arrow-Debreu model, a consumer who
possesses a nonzero share q i of this firm will have as budgetary
constraint
S S
Ê S ˆ
p0 ◊ x0i + Â p s ◊ x si £ p0 ◊ w 0i + Â p s ◊ w si + q i
ËÂ
p s ◊ y s - p0 ◊ y 0
s =1 s =1 s =1
¯
Clearly, in order to relax this constraint as much as possible, each share-

holder will want the firm to maximize its profit
S
Â p s ◊ y s - p0 ◊ y 0
s =1
Things are not so simple when markets are incomplete. Considering

the obstacles that keep consumers from transferring their buying
power between states ad libitum, the production vector (p1 · y1, . . . ,
ps · ys) can no longer be evaluated in a way that enables unanimity
between shareholders. To give an example, a shareholder who
fears finding himself unemployed in state s and who has difficulty
transferring a sizable purchasing power there because of market
incompleteness will want the firm to emphasize the term ps · ys. The
other shareholders, however, will not agree if the same state is not unfa-
vorable to them.
More precisely, let qis be the marginal rate of substitution of consumer
i between the numéraire at date 0 and the numéraire at date 1 in state
s. It is easy to see that if ps is the firm’s profit in state s, the consumer i
would want the firm to maximize SSs=1qsi ps. But since this expression
depends on i, the consumers will not generally agree between them.11
As noted by Drèze (1974), the firm’s profit takes on the character of a
public good for which consumer evaluations are divergent, and we
know from chapter 5 all the difficulties created by this situation.12
A natural solution consists of constructing an objective for the firm
by weighting that which the shareholder i would like to assign the firm
by his share q i. The firm should then maximize.
n S
Âq i Â qsip s .
i =1 s =1
This solution, due to Drèze and to Grossman and Hart (1979), possesses
certain properties of optimality (see Laffont 1989). Still it is not uni-
versally accepted. The reason is that the “state prices” qsi are unob-
served; they depend on subjective probabilities of states for consumers,
on their resources, and on their utility functions. It is therefore difficult
to imagine that a firm’s director can easily maximize such an objective.
Moreover any change of stockholder composition should, according to
this formula, modify the firm’s objectives (so that a long-term decision
becomes inverted after some operations on the stock exchange!).
Finally there is the risk that managers will follow the majority share-
holders’ directives and neglect the desires of minority shareholders. It
is clear that to this day there does not exist a fully satisfactory way to
integrate production into the incomplete markets model, except in very
special cases.
To end this section, I would like to point out that if the Drèze–
Grossman-Hart solution to the problem of shareholders’ nonunanim-
11. This difficulty does not arise in complete markets, since then the consumers can
equalize the marginal rate of substitution rate and the price. Therefore we get qis = p*1s for
each consumer i, where p*1s is the Arrow-Debreu numéraire price in state s. Since this
price obviously does not depend on i, there will be unanimity on the firm’s objective.
12. The problem of defining the firm’s objectives recalls one that we saw in chapter 8.
But here there is no good reason to consider it negligible.
ity is adopted so that firms have well-defined objectives, then con-

strained Pareto optimality could be examined for the production equi-
librium. The reader will not be surprised to learn that to exchange
inefficiencies are added those of production, so equilibrium is
even less often optimal under this constraint than in an exchange
economy.
13.5 Application to International Trade
Accounting for market incompleteness may transform certain results

that seem to be the best established. Thus the majority of economists
consider that under certain reasonable hypotheses, free trade improves
social welfare without ambiguity. Newbery and Stiglitz (1984) show,
however, that the optimality of free trade may not hold when markets
are incomplete.
To see this, suppose that American and Australian farmers can
produce wine and wheat at zero cost. Wheat production is indepen-
dent of climactic hazards, while wine production is very strongly
affected. It is not possible to insure farmers against these risks (it is here
that market incompleteness comes into play). Australia is very far from
the United States. As is well known the weather down under is per-
fectly anticorrelated with American weather: when it is sunny in the
United States, it rains in Australia, and vice versa. In addition con-
sumers’ demand functions have a unitary price elasticity, so agricul-
tural profit does not depend on the price of wine. In autarky, price
variations therefore perfectly insure farmers against climactic hazards.
Now, if we let open the borders, the price of wine becomes perfectly
stable, and the agricultural profit from wine in each country becomes
variable. This then raises the perceived risk of each country’s farmers
and reduces their welfare.
What about the consumers? In autarky, they withstand the entire
risk. The opening of the borders has a beneficial effect for them in
this sense. On the other hand, profit variability introduced by free
trade pushes the farmers to produce less wine, and the price increases,
thereby reducing consumer surplus. It is conceivable that by choosing
the parameters of the model well, this second effect can dominate
the first. Free trade therefore hurts both agricultural producers
and consumers, contrary to the gospel taught by generations of
economists.
13.6 Conclusion
The theory of incomplete markets promises to furnish new bases for

finance, for the study of the effect of bankruptcies, but also for macro-
economics (especially by providing more solid arguments for the
theory of money). It is also a means by which to analyze precisely the
reasons why markets are incomplete, and to characterize in an end-
ogenous way the market structures that are the most likely to arise.
Until recent years, justifications of market incompleteness relied
above all on the existence of transaction costs (a market will only be
open if the surplus to which it gives access is superior to private costs
incurred in creating it) and asymmetries of information (which make
markets less liquid, particularly in the domain of credit and insur-
ance).13 One approach can be to fix the number of assets and to find out
which combination of assets is the most efficient (Demange and
Laroque 1995).
The reader who wants to explore the theory of general equilibrium
in incomplete markets, more than I have completed here, can refer to
the reviews of Duffie (1992), who is particularly interested in financial
markets, Geanakoplos (1990), and Magill and Shafer (1991) or to the
book by Magill and Quinzii (1996) if he is feeling particularly
ambitious. I must, however, point out that these references are fairly
technical.
13.7 Appendix: Nominal Assets
The assets with which we were interested in this chapter were real; that
is, they procure a basket of goods in each state of the world. Their return
was therefore linear in relation to prices. Indeed, this hypothesis corre-
sponds to future contracts on goods or, as we have seen, to firms’
actions. It could also take into account derivative assets, like options,
provided that we accept returns that are nonlinear in relation to prices.
Cass (1984) and Werner (1985) proposed considering nominal assets,14
whose return does not depend on prices. To see this, suppose that con-
13. There are different reasons for market incompleteness, even in the absence of uncer-
tainty. In the presence of externalities, for example, the reasons relate to the inexistence
of markets for personalized goods, like “consumption by consumer i of a pollutant pro-
duced by firm j”.
14. These are also often called financial assets, which can create confusion, real assets
having just as much right to that name.
trary to the rest of this chapter, there exists in the economy a good that
does not enter into agents’ utility function. This good is therefore only
used for transferring purchasing power between dates. Since a price
can be normalized to one on all dates and in all states of the world, we
choose to do so with this particular good. It is then an accounting unit
sometimes called “money,” even though, as we will see, it does not
possess all the properties of money. A nominal asset is then a asset
whose return is expressed in money.
To take a more precise example, a one-dollar bond that bears an inter-
est rate r can be represented by a nominal asset that brings 1 + r in each
state of the world. Money itself can be represented by a bond with zero
interest rate. More generally, a structure of J nominal assets will be a
matrix (J, S) of which the element ( j, s) represents the return of asset j
in state s, measured in money.
The primary advantage of considering nominal assets is that the exis-
tence of equilibrium holds under the usual hypotheses: the analogue
of “drop in rank” cannot happen here because the returns no longer
depend on prices. This was actually the primary motivation of the
introduction of nominal assets.
The indetermination of equilibrium is also a property that differen-
tiates nominal assets from real ones. In the real asset model, it can be
shown that equilibrium is usually locally unique: for nearly all possi-
ble specifications, there is a finite number of equilibria, as in the Arrow-
Debreu model. Geanakoplos and Mas Colell (1989), on the contrary,
showed that if markets are incomplete (J < S) in the nominal asset
model and if there are at least ( J + 1) consumers,15 the set of equilib-
rium allocations is generically (S - 1)-dimensional. It is in fact easy to
see that in this model, inflation rate variations between different states
are indeterminate; this is what engenders the (S - 1) dimensions of the
space of equilibrium allocations. We may note moreover that the result
of equilibrium indetermination implies a fortiori that equilibrium is
generally not constrained Pareto optimal.
In my opinion, the nominal assets model suffers several lacunae.
The first is that assets’ returns are postulated as exogenous variables,
even though it would be desirable that they be determined in an
endogenous manner: thus the interest rate of a bond is in principle
an equilibrium price. The second is how “money” is introduced in this
model. In effect it can be shown that money can only have a positive
15. Here again, this assumption excludes the representative agent models.
price if its stock is zero, so certain agents can detain a negative quan-
tity of money and stand counterpart for those who detain a positive
quantity. The result is what is called inside money, by opposition to
outside money, which is created by the Central Bank and whose stock
is positive. Moreover, while this so-called money fulfills its functions
as a unit of account and a means for liquidity in this model, its essen-
tial role for facilitating transactions is left aside. Finally the indetermi-
nacy result leaves the impression that the model is not completely
specified.
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Debreu, G. 1959. Theory of Value. New York: Wiley.
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Index
Abreu, D., 58 Business stealing, 171

Acyclicity, 19 Buying quota, 194
Akerlof, G., 165
Allais, M., 117 Cabral, L., 187
Antitrust law, 201–202 Capabilities, 33
Arrow, K., 17, 46, 49, 188, 205 Caplin, A., 20n
Arrow conditions, 18 Cartel, 156
Arrow theorem, 19 Cass, D., 216
proof, 26–27 Chain monopoly, 194
d’Aspremont, C., 22, 82, 168 Chamberlin, E., 128, 172
Assets Champsaur, P., 15
Arrow-Debreu, 207 Cheung, S., 104
nominal, 216 Chipman, J., 14n
real, 208 Circular city, 170
Atkinson, A., 25, 67 Clarke, E., 81
Auctions, 79 Coase, R., 83–84, 102–104, 147
Coase conjecture, 147
Bain, J., 128 Coase theorem, 102
Baron, D., 117 Compensating variation, 38
Baron-Myerson model, 120 Compensation principle, 13
Barriers to entry, 179 Competition policy, 201–202
Baumol, W., 179, 181n Condorcet, M. de, 16
Bénassy, J.-P., 131, 136 Condorcet paradox, 16
Bentham, J., 28 Consumer surplus, 37–42
Bergson-Samuelson functional, 15, 42 Contestable markets, 179
Bertrand, J., 127, 152 Contingent valuations, 45
Bertrand paradox, 152 Convexification, 112
Black, D., 20 Cournot, A., 127, 151
Bohm, P., 85, 151 Cournot oligopoly, 151
Boiteux, M., 117n, 204 Cournot-Walras equilibrium, 135
Bonanno, G., 131
de Borda, J.-C., 16 Dasgupta, P., 25
Borda method, 16 Davidson, C., 154n
Bowen, H., 72 Debreu, G., 207n
Bowen-Lindahl-Samuelson condition, Demange, G., 216
72 Demsetz, H., 124
Bulow, J., 158 Deneckere, R., 154n, 200n
222 Index
Depollution, 94 Gérard-Varet, L.-A., 82

Deregulation, 124 Gevers, L., 22
Diamond, P., 212 Gibbard, A., 51
Dictatorial Gibbard-Satterthwaite theorem, 51
social choice function, 51 proof, 59
social welfare function, 18 Glaister, S., 37
Dierker, E., 133 Glazer, J., 56n
Difference principle, 30 Government, role and limits, 8
Discrimination, 148, 198 Green, E., 156
Distributive factor, 119 Green, J., 9, 79
Dixit, A., 9n, 173, 183 Gresik, T., 104
Dixit-Stiglitz model, 173 Grodal, B., 133
Double marginalization, 194 Grossman, S., 212n, 214
Drop in rank, 210 Groves, T., 79
Drèze, J., 76, 214
Duffie, D., 212, 216 Hammond, P., 24
Dupuit, J., 40, 116n Harberger, A., 143n
Harberger triangle, 143
Economides, N., 90n Harrod, R., 13
Edgeworth, F.-Y., 28, 153 Harsanyi, J., 25
Efficiency effect, 190 Hart, O., 131, 137, 138n, 177n, 210, 212–14
Elections, 15, 57 Hausman, D., 45n
Envy, 14 Henry, C., 9, 83n, 97n, 120
Equity, 24 Herfindahl index, 151
Equivalent variation, 38 Hicks, J., 38, 143
Exclusion, 69 Hicks-Kaldor criterion, 13
Exclusive dealing, 193 Horizontal differentiation, 162
Exclusive territories, 193, 199, 202 Hotelling model, 162
Externality Hotelling, H., 116, 128, 162, 167
examples, 89 Hotelling’s rule, 116
pecuniary, 89 Hurwicz, L., 49n
Fair allocation, 14 Implicit valuations, 45

Farrell, J., 103n Incumbent, 183
Finkelshtain, I., 102 Independence of irrelevant alternatives,
Firms’ objectives 18
imperfect competition, 132 Informational rent, 122, 150
incomplete markets, 213 Integrated structure, 116
First best, 42
Fisher, I., 155 Jackson, M., 58
Foley, D., 14
Franchise, 144 Kaldor, N., 36, 84
Free rider, 75 Kay, J., 124n
Functionings, 33 Kislev, Y., 102
Fundamental welfare theorems Klemperer, P., 158
discussion, 6–7 Korah, V., 201
first theorem, 3 Kreps, D., 9, 154
second theorem, 4
Laffont, J.-J., 9, 79, 123, 137, 214
Gabszewicz, J.-J., 135, 168 Laroque, G., 15, 137, 216
Gary-Bobo, R., 138 Layard, R., 37
Geanakoplos, J., 8n, 158, 212, 216, 217 Ledyard, J., 85
Generalized price, 162 Lemons problem, 165
Index 223
Lerner index, 142 Negishi equilibrium, 134

Leximin criterion, 24 Newbery, D., 215
Liberalism, 21 New welfare economics, 13
Limit pricing, 186 No veto power, 54
Lind, E., 46 Nonconvexities in production
Lindahl equilibrium, 74 large, 109
Lindahl, E., 74–75 small, 109
Linear city, 162 Nozick, R., 29, 33–34
Local public goods, 86
Loeb, M., 79 Oligopoly, 150
Lorenz curve, 25 Original situation, 29
Lump sum transfers, 8 Overfishing, 93
Ma, C.-T., 56n Palfrey, T., 58–59

Magill, M., 216 Panzar, J., 179
Malinvaud, E., 76 Pareto principle, 13
Marginal cost pricing, 116 Pazner, E., 15
Marginal revenue, 142 Peck, J., 200n
Markup, 142 Pigou, A., 92, 148
Marshall, A., 128 Pigovian taxes, 97
Marvel, H., 200n Planning, 76
Mas Colell, A., 9, 217 Polemarchakis, H., 8n, 212
Maskin, E., 54, 58 Porter, R., 156
Maskin theorem, 54 Postlewaite, A., 85
proof, 63 Predation, 187
Maximal differentiation principle, 170 Preemption, 183
Maximin criterion, 30 Price war, 155
May, K., 19 Primary goods, 30
McGowan, F., 183n Process innovation, 188
McPherson, M., 45n Public property, 83
MDP procedure, 76
Mechanism, 49 Quinzii, M., 216
direct revealing, 50 Quotas, 94
pivot, 81
Vickrey-Clarke-Groves, 78 Radner, R., 207, 211
Meade, J., 89, 95 Ramsey, F., 117n, 204
Median voter, 20 Ramsey-Boiteux formula, 118
Milgrom, P., 186 Rationing scheme, 154
Mill, J. S., 28 Rawls, J., 24, 29–30
Minimal differentiation principle, 167 Regulation, asymmetric information,
Minimal state, 33 120
Monopolistic competition, 172 Relative discouragement, 119
Monopoly, 141 Replacement effect, 190
Monotonicity, 20, 54 Repullo, R., 54n, 58, 63
Moore, J., 55n, 56, 58 Resale price maintenance, 194, 197, 199
Moore, J. C., 14n Research and development, 188
Moulin, H., 19n, 25, 57 Retailer effort, 197
Myerson, R., 50, 104, 117 Revelation principle, 50
Myles, G., 67 Rey, P., 199
Rights to pollute, 95
Nalebuff, B., 20n Riordan, M., 187
Natural monopoly, 115 Rivalry, 69
Negishi, T., 134 Robbins, L., 13
224 Index
Roberts, J., 85, 134, 186 Two-part tariff, 144

Robinson, J., 128 Two-round election, 57
Roemer, J., 28, 30, 32
Ross, D., 201 Unimodal preferences, 20, 52
Rubinfeld, D., 86 Universal domain, 18
Rule of reason, 201 Utilitarianism, 28
Salanié, B., 8, 123n, 149 Vallée Poussin, D. de la, 76

Salop, S., 170 Value of life, 45
Salop model, 170 Varian, H., 9, 14, 98
Samuelson, P., 72 Veblen, T., 89n
Satterthwaite, M., 51, 104 Veil of ignorance, 29
Scheinkman, J., 154 Vertical differentiation, 162
Schmeidler, D., 15, 59 Vertical relations, 193
Schumpeter, J., 128n, 188 Vertical restraints, 193
Scitovsky, T., 89 Vial, J.-P., 135
Scott, F., 120 Vickers, J., 124n
Seabright, P., 183n Vickrey, W., 27, 79
Second best, 44 Viscusi, W., 46
Sen, A., 21, 25, 32–33, 58 Voting equilibrium, 73
Shadow prices, 44
Shafer, W., 212, 216 Weitzman, M., 100
Shapley-Folkman theorem, 112 Welfarism, 28
Sidgwick, H., 8n Werner, J., 216
Silvestre, J., 138n Whinston, M., 9
Smith, A., 6, 84, 143, 156 Wicksell, K., 75
Social choice function, 49 Willig, R., 41n, 179
Social loss, 143 Wilson, J., 9n
Social states, 17
Social welfare function, 17
Solomon’s judgment, 56
Sonnenschein, H., 59, 134
Spence, M., 183
Srivastava, S., 58–59
Starrett, D., 25, 96n
Stigler, G., 128, 157
Stiglitz, J., 9, 67, 173, 199, 212, 215
Strategic complements, 158
Strategic substitutes, 158
Subscription equilibrium, 73
Summers, L., 45n
Supranormal profits, 153
Sustainable configurations, 180
Tacit collusion, 155

Tangency solution, 172
Telser, L., 197
Thisse, J.-F., 168
Tiebout, C., 86
Tiebout equilibrium, 86
Tied sales, 198
Tirole, J., 9, 123, 199

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The Microeconomics of

The MIT Press

Library of Congress Cataloging-in-Publication Data

ISBN 0-262-19443-0 (hc. : alk. paper)

2 The Aggregation of Preferences 13

III Industrial Organization 127

8 General Equilibrium of Imperfect Competition 131

9 Prices and Quantities 141

9.3 Oligopoly 150

10 Product Choice 161

11 Long-Term Entry and Competition 179

12 Vertical Relations 193

IV Incomplete Markets 203

13 Elements of the Theory of Incomplete Markets 205

This book has four parts. It treats, respectively, collective choice,

1.1 The Fundamental Theorems

• J producers indexed by j = 1, . . . , J; producer j is represented by a

1.1.1 The Pareto Optimum

By definition, a Pareto optimum of such an economy is an allocation.

1.1.2 General Equilibrium

• The producer j maximizes his profit and chooses

y j (p) Œ arg max p ◊ y j

• The consumer maximizes his utility under budgetary constraints and

Ïarg max U i (xi )

1.1.3 The Two Fundamental Welfare Theorems

theorem 1.1 (first fundamental welfare theorem) If the utility

is a competitive equilibrium, then the allocation ((x*i )i=1,...,I, (y*j )j=1,...,J) is a

"i = 1, ..., I , U i ( xi ) > U i ( xi* )

"i = 1, ..., I , p * ◊w i > p * ◊ w i + Â q ij p * ◊y*j

So adding all the consumers yields

It can be deduced from these two inequalities

theorem 1.2 (second fundamental welfare theorem) If

then every interior optimum can be decentralized in equilibrium: if the

((w ) i i =1, ... , I , (q ij ) i =1, ... , I

Proof Define the sets

Ci = {x ¢i U i ( xi¢) > U i ( xi¢)}

resources. Under the hypotheses of the theorem, Z and W are convex.

or even, since Six*i = w + Sjy*j ,

Take all the xi close to x*i . Then in the limit we get

U i (xi ) > U i ( x*i ) ﬁ p ◊ xi > p ◊ xi*

1.2 Return to the Hypotheses

The “philosophical” importance of these two theorems cannot be

As every individual, therefore, endeavors as much as he can both to employ

those among us who find revenue distribution to be uneven can turn

Finally, third, we focus on the hypothesis of pure and perfect com-

1.3 The Government’s Role

The government has an undeniable advantage on the market: it has at

economy. These distortions are at times desirable in order to correct

it better understood to the reader that economics is not a finished

Dixit, A. 1996. The Making of Economic Policy. Cambridge: MIT Press.

Geanakoplos, J., and H. Polemarchakis. 1991. Overlapping generations. In W.

Kreps, D. 1990. A Course in Microeconomic Theory. Oxford: Harvester Wheatsheaf.

Laffont, J.-J. 1988. Fundamentals of Public Economics Cambridge: MIT Press.

Salanié, B. 1997. The Economics of Contracts: A Primer. Cambridge: MIT Press.

Stiglitz, J. 1988. Economics of the Public Sector. New York: Norton.

Tirole, J. 1988. The Theory of Industrial Organization. Cambridge: MIT Press.

Wilson, J. 1989. Bureaucracy. New York: Basic Books.

This chapter attempts to answer the question: Given a limited set of

Such a rigid methodological choice made economists’ prescriptions

is a competitive equilibrium, then the allocation ((xi )i=1,...,I, (yj )j=1,...,J) is a

or even, since Sixi = w + Sjyj ,

U i (xi ) > U i ( xi ) ﬁ p ◊ xi > p ◊ xi