PHD Thesis Bernardo R C Costa Lima Final Submission
PHD Thesis Bernardo R C Costa Lima Final Submission
PHD Thesis Bernardo R C Costa Lima Final Submission
MACROECONOMICS
McMaster University
c
Copyright
by Bernardo R. C. da Costa Lima, July 2013
ii
Abstract
The aim of this thesis is to provide mathematical tools for an alternative to the
mainstream study of macroeconomics with a focus on debt-driven dynamics.
We start with a survey of the literature on formalizations of Minskys Financial
Instability Hypothesis in the context of stock-flow consistent models.
We then study a family of macro-economical models that date back to the Goodwin
model. In particular, we propose a stochastic extension where noise is introduced in
the productivity. Besides proving existence and uniqueness of solutions, we show that
orbits must loop around a specific point indefinitely.
Subsequently, we analyze the Keen model, where private debt is introduced. We
demonstrate that there are two key equilibrium points, intuitively denoted good and
bad equilibria. Analytical stability analysis is followed by numerical study of the basin
of attraction of the good equilibrium.
Assuming low interest rate levels, we derive an approximate solution through perturbation techniques, which can be solved analytically. The zero order solution, in
particular, is shown to converge to a limit cycle. The first order solution, on the other
hand, is shown to explode, rendering its use dubious for long term assessments.
Alternatively, we propose an extension of the Keen model that addresses the immediate completion time of investment projects. Using distributed time delays, we
verify the existence of the key equilibrium points, good and bad, followed by their
stability analysis. Through bifurcation theory, we verify the existence of limit cycles
for certain mean completion times, which are absent in the original Keen model.
Finally, we examine the Keen model under government intervention, where we introduce a general form for the government policy. Besides performing stability analysis, we prove several results concerning the persistence of both profits and employment.
In economical terms, we demonstrate that when the government is responsive enough,
total economic meltdowns are avoidable.
iii
Acknowledgements
Success in finishing a PhD has been a long standing ambition of mine. During the
last five years, my life has been primarily dedicated towards this final and ultimate
goal. Borrowing the mathematical jargon for a bit, I must say that this feat would
almost surely have not been accomplished without the support and encouragement of
many. I would like to express my gratitude towards some of them below.
First, I must graciously thank Prof. Matheus Grasselli who afforded a nourishing
environment throughout my time here at McMaster University. More importantly, I
am utterly grateful for his inspirational support and ongoing collaboration throughout
these years. He has been a great mentor and role model for this young researcher,
providing pivotal guidance through my graduate studies.
I want to thank Prof. Tom Hurd for the enriching exposure during the time I
collaborated with his research in Systemic Risk in Financial Networks. Witnessing,
and also being part of his creative process has provided an in-depth perspective on
the essence of cutting edge academic research. I would like to also thank Prof. Traian
Pirvu, and Prof. Gail Wolkowicz for their many insights and encouraging discussions.
Additionally, I am grateful to Prof. Richard Deaves for his helpful criticism throughout my committee meetings. I will be eternally grateful to Prof. Jorge Zubelli, from
IMPA (Brazil), who has always inspired me, and introduced me to the world of Financial Mathematics. I must also thank Dr. Adrien Nguyen Huu, working with him
has been a great pleasure, and a remarkable lesson. His dedication and passion to
Mathematics are both astonishing and inspiring.
Finally, I have to thank those who are dear to me the most. My parents, Angela
and Nelson, and my siblings, Bruno and Fabrizzia, without whom I could not have
achieved this far, they have always encouraged me to aim high and strive. I am also
very thankful for the all the support my girlfriend, Veronica, has given me. She has
truly brought warmth to this otherwise stranded soul.
iv
To my mother,
Angela dos Reis Carneiro
and all the fond memories.
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
1 Introduction
2.1
2.2
12
2.2.1
12
2.2.2
16
3 Goodwin Model
20
3.1
Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . .
21
3.2
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
3.3
Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
3.4
26
29
4.1
. . . . . . . . . . . . . .
29
4.2
39
4.3
62
4.4
Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
4.5
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
vi
5 Keen Model
76
5.1
Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . .
77
5.2
79
5.3
82
5.4
Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
5.5
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
91
6.1
Zero-order solution . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
6.2
First-order solution . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
6.3
100
6.4
Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
106
6.5
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
107
109
7.1
Alternative formulation . . . . . . . . . . . . . . . . . . . . . . . . . .
110
7.2
Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
114
7.3
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
122
8 Government Intervention
8.1
124
Introducing government . . . . . . . . . . . . . . . . . . . . . . . . .
126
8.1.1
Finite-valued equilibria . . . . . . . . . . . . . . . . . . . . . .
130
8.1.2
Infinite-valued equilibria . . . . . . . . . . . . . . . . . . . . .
138
8.2
Persistence results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
144
8.3
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
157
8.4
Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . .
166
9 Conclusion
167
Appendices
170
vii
171
B XPPAUT Instructions
175
175
178
C Persistence Definitions
183
viii
List of Figures
3.1
26
3.2
Employment rate, wages, and output over time in the Goodwin model
27
4.1
Domain D and its covering by (Ri )i=1...8 . Since f (0) < 1 and y = 1 is
a vertical asymptote of , this corresponds to the general case, where
x = 0 is possible on R7 . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
40
4.3
72
Expected values of employment after one full loop T (left), and time
elapsed T (right). Computation performed in MATLAB, with 2000
simulations for every value single one of the 100 initial values taken
along the line = e. . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4
73
5.1
74
Phase portrait of employment and wages converging to a stable equilibrium with finite debt in the Keen model. . . . . . . . . . . . . . . .
ix
87
5.2
5.3
5.4
88
89
Basin of convergence for the Keen model for wages, employment and
private debt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
6.1
106
6.2
6.3
107
Examples of solutions of both (5.7), and (6.1) plus (6.32), for different values of the real interest rate. On the left column, we have the
evolution of versus t, while on the right column we have t. The
exact solution is drawn in a solid black line, whereas the approximate
solution is represented by the dashed blue line. . . . . . . . . . . . . .
7.1
108
7.2
115
7.3
116
7.4
117
Bifurcation diagrams on the left. 10% and 90% percentiles of the delay
distribution on the right. . . . . . . . . . . . . . . . . . . . . . . . . .
118
7.5
Brute Force bifurcation diagram for the model with discrete delay. . .
119
7.6
120
7.7
121
7.8
122
8.1
The part of the plane (s () v) where one can see the invariant
regions V and S. Every solution that enters V eventually makes it to
S and never leaves it. The region P is not invariant. Yet, solutions
that enter it must eventually leave it and enter the basin of attraction
of S, either directly, or after spending some time on (, v) [m, )R+ .154
8.2
159
8.3
160
8.4
160
8.5
Solution to the Keen model with and without a timid government for
good initial conditions. . . . . . . . . . . . . . . . . . . . . . . . . . .
8.6
Solution to the Keen model with and without a timid government with
bad initial conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.7
163
8.9
163
Solution to the Keen model with and without a timid government with
extremely bad initial conditions. . . . . . . . . . . . . . . . . . . . . .
8.8
162
164
Solution to the Keen model, starting close the good equilibrium point,
with positive (stimulus) and negative (austerity) government subsidies. 164
8.10 Solution to the Keen model, starting far from the good equilibrium
point, with positive (stimulus) and negative (austerity) government
subsidies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
165
List of Tables
2.1
Balance sheet, transactions and flow of funds for a general Minsky model. 11
2.2
Balance sheet, transactions and flow of funds for the standard Keen
model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3
13
Balance sheet, transactions and flow of funds for the Keen model with
government intervention. . . . . . . . . . . . . . . . . . . . . . . . . .
xii
17
Chapter 1
Introduction
Neoclassical economics has been largely criticized over the past years, specially since
the crisis that unfolded in 20072008. Be it for their unreasonable rational expectations assumption, or be it for the aggregate Law of Demand, technically refuted
by the Sonnenschein-Mantel-Debreu theorems [Son72], it is time to seek for an alternative. We will argue that even though stock flow consistent macroeconomic models
represent a radical substitute [God04], they might be the ideal one. Rather than
studying economies at the equilibrium, we suggest the very opposite, as economies
rarely find themselves in equilibrium, yet their time evolution matters. Stock flow
representation of the economy can be translated into a system of dynamical equations (either deterministic or stochastic) which will describe how the many different
variables fluctuate over time.
Perhaps a deeper concern regarding neoclassical economics is its disregard for the
financial system. As pointed out by Lavoie in [Lav08], the aforementioned crisis is a
reminder that macroeconomics cannot ignore financial relations, otherwise financial
crises cannot be explained. For this reason, a great deal of this thesis is focused on
a family of models that represent a mathematical formalization of Minskys Financial
Instability Hypothesis.
The structure of the thesis is as follows. We begin by introducing the concepts
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
behind stock flow consistent models in Chapter 2. In Sections 2.2.1 and 2.2.2, we
demonstrate that the Keen model both with (8.14) and without government intervention (5.7) conform to the stock-flow consistency requirements.
Next, Chapter 3 discusses the very basic foundation behind this entire work: the
Goodwin model [Goo67]. Goodwin elegantly proposes one of the most prominent
mathematical formulations of Marxs theory of class struggle. We show that the
solutions satisfy a constant Lyapunov equation, regardless of the choice of the Phillips
curve. In a later Section 6.3, we derive an analytical expression for the period of these
solutions, which depends exclusively on the initial value of the Lyapunov function.
A stochastic extension, where productivity is allowed to drift randomly according
to the level of employment, is then proposed in Chapter 4. In this chapter, based
on joint work with A. N. Huu and M. R. Grasselli [CGH13], we first obtain sufficient
conditions for existence and almost surely uniqueness of solutions. Next, we derive
a probabilistic estimate for the time it takes for solutions to deviate sufficiently far
from their initial orbit, in Lyapunov terms. More importantly, we show that solutions
almost surely orbit around a pivot point, indefinitely and in finite time. We end the
chapter by deriving a continuous extension of the proposed stochastic model that
approximates it when the volatility term is negligible. Such extension can be fully
solved analytically, as the sum of the solution to the Goodwin model plus a martingale
term.
The financial sector is taken into account in Chapter 5, where we introduce the
Keen model [Kee95] followed by a comprehensive analysis drawn from [GCL12], joint
work with M. R. Grasselli. After verifying the existence of two key equilibria, coined
good and bad, for they represent states of prosperity and collapse, respectively,
we perform local analysis. Necessary and sufficient conditions for their local stability
are obtained, showing the both fixed points are simultaneously stable under usual
conditions. Accordingly, solutions can converge to either equilibrium point, depending
on their initial position. As an illustration, we numerically integrate solutions starting
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
in a fine grid around the good equilibrium point, recording when they converge to this
fixed point, roughly obtaining the basin of attraction of the good equilibrium.
Chapter 6 presents an extension of this model where we assume regimes of extremely low interest rate. There, we use perturbation techniques similar to those
used in Section 4.3 to derive an approximate model which can be fully solved analytically. The corresponding zero-order solution resembles the Goodwin model (3.12),
except for its non-linear investment function. Regardless, we show that the zero-order
solution converges to a limit cycle, whose period is also analytically determined. The
first-order solution, on the other hand, is shown to grow quadratically with time, thus
spoiling the accuracy of this approximation for long-term investigations.
In Chapter 7, we propose a second extension of the Keen model (5.7), where we
consider non-immediate capital project completion times. Usually, to study such implications, one would have to deploy delay-differential equations machinery. Through
a clever mathematical device, commonly used in Mathematical Biology, we are able
to avoid these complications, and continue in the realm of ordinary differential equations, albeit with higher-dimension. The intuition behind this technique is that we
can approximate a discrete delay by the sum of exponentially distributed times, itself
Erlang distributed. As the number of intermediate stages increases to infinity, this
distribution converges to that of the Dirac delta, effectively representing the discrete
delay we started with. After verifying the existence of corresponding good and
bad equilibria, we show that the local stability of the first is only possible when the
mean completion time is shorter than a certain threshold. We perform bifurcation
analysis, which shows the existence of a super-critical Hopf bifurcation at that point.
In other words, for mean completion times higher than the mentioned threshold, we
have a limit cycle which attracts solutions which would otherwise converge to the
good equilibrium point. Unsurprisingly, the period of this limit cycle grows as the
mean completion time increases, until a second threshold is crossed, and the cycles
cease to exist. Beyond this point, solutions seem to converge exclusively to the bad
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
equilibrium point.
Chapter 8 is based on joint work with M. R. Grasselli, X-S. Wang, and J. Wu
[CGWW13]. We propose an original extension of the Keen model where government
intervention is introduced in a general setting, while respecting the stock flow consistent framework. First, we identify the good equilibrium, together with a variety
of bad equilibria characterized by zero employment and negative exploding profit
share, besides some extra finite equilibria associated to zero wage share, which are
either unattainable or unstable under usual conditions. Next, we obtain necessary
and sufficient conditions for their local stability, showing that all the bad equilibria can be successfully destabilized if the government subsidies are non-negative and
responsive enough in a vicinity of zero employment. Meanwhile, the same cannot be
said for a government operating under austerity. Moreover, and most importantly,
we show that through a variety of reasonable conditions associated with a responsive
government, we have that the model (8.14) is uniformly weakly persistent with respect
to both the exponential of profit share and employment ratio. Ultimately, this result
means that solutions can never remain trapped below arbitrarily low levels of profit
share or employment, regardless of the specific initial conditions.
Chapter 2
Stock-Flow Consistent Minsky
Models
There have been several attempts to model (Hyman) Minskys Financial Instability
Hypothesis (FIH henceforth), as surveyed by Lavoie in [Lav08], and Dos Santos in
[DS05]. A short tour through the key aspects of the most influential of these models
follows below.
One of the earliest attempts of formalizing the FIH was carried out by Taylor
and OConnell in their influential article [TO85], itself a variation of another paper
by Taylor [Tay85]. Taylor and OConnell [TO85] introduced innovative concepts, of
which three deserve further discussion. First, they used the notion of a portfolio
choice. Secondly, their investment function was modeled as a function of the spread
between expected profit rate of firms (actual profit rate plus a confidence indicator)
and the interest rate. Lastly, and perhaps most strikingly, they introduced cyclical
dynamics through a differential equation linking the confidence indicator and the
interest rate. Notwithstanding the remarkable impact of [TO85], their model has
been criticized for numerous flaws. As Lavoie [Lav08] and Dos Santos [DS05] both
point out, Taylor and OConnell [TO85] model is not stock-flow consistent (SFC
hereafter), as there are inconsistencies in how they treat the government debt. On
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
top of that, the banking sector and the firms leverage ratio do not play an explicit
role in the model, while the supply of money is not endogenous. Relatedly, Taylor and
OConnell do not model firms debt commitments, they can only finance themselves
through equity issuance.
Another attempt of modeling Minskys ideas was put forward by Franke and
Semmler in [FS91]. They modified Taylor and OConnell model incorporating the
banking system. In addition, they introduced the firms leverage ratio together with
the interest rate to the dynamical equation driving the confidence indicator. Unfortunately, Frank and Semmler model shares most of the criticism targeted towards Taylor
and OConnell [TO85]. For instance, the fact that stock price is not determined by
supply and demand, and the remaining problems with stock-flow consistency. In addition, Dos Santos [DS05] criticizes the (unrealistic) assumption that the whole stock of
high-powered money (i.e. cash) is kept by banks as reserves. Still, its most celebrated
contribution was to track the leverage ratio of the firms explicitly.
Yet another extension of Taylor and OConnell [TO85] was developed by Radke
in[Rad05]. He presented a rather complete model, which assumes that the confidence
indicator is driven by the spread between the profit rate and the interest rate, and the
firms debt ratio. Simultaneously, he allowed for credit rationing, by assuming that
banks can choose to grant a higher amount of loans to firms offering more collateral.
Delli Gatti and Gellagati published several papers in this topic by the 90s, for example [DGGG90] coauthored with Gardini,[DGGM98] coauthored with Minsky himself, and [DGG92]. The essence of their model is captured by the investment strategy,
which is modeled as a function of Tobins q ratio [BT68], retained earning and leverage over retained earning. The key insight is that when firms rely more heavily on
external finance, instability ensues, despite any interest rate movement. Nonetheless,
despite being faithful to Minskys ideas, their model has suffered extensive criticism.
As Lavoie [Lav08] observes, besides not being SFC, the model ignores the role interest payments from the firm debt have in the consumption function. Moreover, bank
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
reserves seem to have no counterpart. In addition, as Dos Santos [DS05] points out,
by assuming the supply of bank loans is a function of the interest earned alone, the
authors are (perhaps not intentionally) assuming as well that either the amount of
bank deposits or reserves do not fluctuate wildly, or that bank behaviour is not affected by such fluctuations. Lastly, the authors oversimplify the government sector
when, as Dos Santos [DS05] words it, they heroically assume that all taxes are zero.
Perhaps the earliest SFC formulations of the FIH was developed by Skott in
[Sko81]. The model, further enhanced in an article [Sko88] and a book [Sko89a],
introduces a firm that finances its investing through three sources: retained profits,
equity, and debt. At the same time, households choose to consume based on their
wealth, and decide their allocation on money or equities, while the government sector
is completely absent. As Lavoie [Lav08] remarks, the money supply in this model is
endogenous, while the stock price is determined by demand and supply, features not
found in Franke and Semmler [FS91], or Taylor and OConnell [TO85]. Lavoie continues to say that this model has had an unfortunate lack of impact, which can be
partially blamed on the fact that Skott leaves leverage ratios out of the picture. In a
more recent publication [Sko94], Skott introduces the leverage ratio as an upper bound
on the amount of loans supplied by banks, together with an investment function that
depends on financial variables coined fragility the sensitivity of investment to adverse shocks and tranquility the firms ability to meet their financial obligations
(as discussed in [DS05]).
None of the models mentioned up to this point consider household indebtedness.
As Isenberg concluded in her study on the FIH in the 1920s [Ise88], the production
sector, non-financial firms, which is at the center of the financial instability hypothesis,
did not exhibit a rising debt equity ratio. In a later article [Ise94], she explains
that even though firms did not experience a rise in their their leverage ratios up to
1929, households did suffer from higher debt ratios. With this in mind, we bring
our attention to Palleys work [Pal96], where he introduces two classes of households
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
that borrow/lend from one another. By linking the debt-to-income ratio of borrowers
positively with GDP, he is able to obtain Minskys paradox of tranquility (Lavoie
[Lav08]), along with an economic slowdown associated with higher levels of the interest
rate and/or debt-to-income ratio. Just like many of the other models discussed here,
this one is not SFC, as banks reserves share no counterpart (Lavoie [Lav08]).
More recently, Lavoie and Godley [LG01] have designed a simple SFC model with
only three sectors, households, banks, and firms, that has generated numerous extensions. For instance, Skott and Ryoo [SR08] made different behavioral assumptions,
changing the arguments of the consumption function, which led to rather different
results, indicating some sort of structural instability around the model specification.
On a different note, Zezza and Dos Santos [ZDS04] extend the model by adding the
central bank and a government sector, besides explicitly taking inflation in consideration. At a later stage, Zezza [Zez08] introduces two classes of households, the workers
and the rich. While the former would rent houses, or rely on mortgages when purchasing real estate, the latter rather purchase houses without the need of financing,
to collect rental income and/or capital gains (i.e. as an investment).
As an extreme example of how versatile this line of research can be, we turn to
the model developed by Eatwell, Mouakil and Taylor [EMT08]. Not only they include
a housing market, but they also split the financial sector in banks and their special
purpose vehicles (SPV), which issue the mortgages and pack them into mortgage-back
securities (MBS). Going further, repos are exchanged between the central bank and
every other bank to meet their reserve requirements (therefore relaxing the treasure
bills repurchase agreement mechanism). Furthermore, demand for houses is modeled
as decreasing with housing prices, yet increasing with their rate of change, and decreasing with the mortgage rate and the leverage ratio of both households and banks.
To end this quick survey, we mention the work developed by Steve Keen. In
[Kee95], he introduces a simple, yet beautiful extension of the popular, predator-prey
like, Goodwin model [Goo67]. Without attempting to suit SFC requirements (even
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
though such requirements can be met, as we will see), he directly derives the dynamics
of macroeconomical variables (wage share, employment, and capitalist debt) by introducing a nonlinear investment function that depends solely on the profit share of the
capitalists. Immediately after, he introduces an extension accounting for government
intervention, whose policy includes both spending, when employment shrinks, and
taxation, when profits soar. Its simplicity, and hence analytical tractability, allied
with the fact that it captures the essence of the FIH, invites further investigation,
turning it into the ideal candidate for a serious mathematical tour de force.
2.1
We have thus far discussed several implementations of the FIH, pointing out which
are SFC, and which are not, while leaving aside the description of this property. In
this section, we will explore the basic ideas and concepts involved with a SFC model.
Schematically, SFC modeling involves three steps:
1. double-entry accounting: build the balance sheets, together with the transactions table and the flow of funds;
2. determine the behavioral assumptions, e.g. investment, consumption, financing
flows;
3. perform comparative dynamics and/or prove desired analytical properties.
The output of the first two steps is a system of differential equations that seldom
can be solved analytically. A numerical approximation is thus useful when assessing
behaviour of a system under certain conditions. In addition, as explored throughout
this thesis, one can prove various properties of the system analytically, e.g. determine
its fixed points, together with their respective local/global stability, study bifurcations,
and establish persistence of key variables.
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
Borrowing a few words from Dos Santos [DS05], a fair depiction of Minskys Wall
Street paradigm requires an economy with households, firms, banks and a government
(including a central bank). Clearly, one can expand this list by including SPVs,
along with different classes of households, as previously discussed in some of the
surveyed models. These decisions are to be made by the researcher, having in mind
the trade-off between tractability and explanatory power. In praise of parsimony,
we shall stick with the first four sectors mentioned, plus the central bank. In terms
of balance sheet items, a decent starting set is composed by cash, deposits, bank
loans, government (treasury) bills, central bank advances and stock shares, besides,
naturally, the capital goods.
Without further delay, let us introduce a somewhat general balance sheet, transactions and flow of funds. The entries in Table 2.1 are in real terms and mimic closely
the framework developed by Godley and Lavoie [GL07] and Dos Santos [DS05].
A few assumptions go embedded in Table 2.1:
1. households can purchase stocks issued by both the firms and the banks (in line
with [GL07]);
2. the central bank has zero net worth (in line with Dos Santos [DS05]);
3. the firms current account distributes its profits to either the capital account,
or its shareholders (households). In other words, the resulting sum of all the
current account transactions is zero (in line with Godley and Lavoie [GL07] and
Godley [God04]);
4. banks distribute a portion Fb of their profits to their shareholders, which includes, but is not limited to, the households (shares can be traded amongst
other banks as well) here we depart slightly from Godley and Lavoie [GL07],
where they oddly assume that banks pay no dividends to the households;
5. banks do not pay taxes;
10
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
Table 2.1: Balance sheet, transactions and flow of funds for a general Minsky model.
Balance Sheet
Firms
Households
current
Cash
+Hh
Deposits
+Mh
+Hb
Loans
Bills
Equities
Banks
0
0
+L
Ef .pf
Central bank
advances
+Bb
+Bc
Eb .pb
A
Capital goods
0
+C
Vf
0
0
+A
+K
Vh
Sum
+Mf
+Bh
+Ef .pf + Eb .pb
capital
+K
Vb
Transactions
Consumption
Investment
+I
Government
expenditures
+G
Accounting memo
[GDP ]
[Y ]
Wages
+W
Interest on deposits
Dividends
Sum
0
G
0
GS
Th
Tf
+T
+rM .Mh
+rM .Mf
rM .M
rL .L
+rL .L
Interest on loans
Interest on bills
+GS
Government subsidies
Government taxes
+rB .Bh
+rB .Bb
+rB .Bc
rB .B
0
0
+Fd + Fb
+Fu
Fb
Fc
+Fc
Sh
Sf
Sb
Sg
+H b
Flow of Funds
Cash
+H h
Deposits
+M h
Loans
Bills
Equities
+L
+B h
+B b
+E f .pf + E b .pb
E f .pf
Central bank
advances
Sum
Sh
Sf
11
+M f
+B c
E b .pb
0
0
+A
Sb
0
Sg
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
6. the central bank sends its profits to the government, hence its zero net worth;
The final ingredient necessary is the set of behavioral assumptions. Naturally,
a crucial hypothesis is the investment function, which directly assigns the dynamics
of capital goods. Moreover, one needs to specify, among others, how households
consume, what the government policies are in terms of spending and taxation, and
how does the stock price evolve with time.
This crucial step is notably vulnerable to controversy and criticism. Once the
skeleton of the model is laid down, there are numerous ways to close the system,
each with its own advantages and disadvantages. Many of the models surveyed in
the last section shared similar accounting structure, yet possessed distinct behavioral
characteristics, rendering diverging results (e.g. Lavoie and Godley [LG01], and Skott
and Ryoo [SR08]).
By way of example, in the next section, we will show that both the Keen model
[Kee95], and its extension with government intervention, which will be explored later
in Chapters 5, and 8, satisfy the SFC requirements.
2.2
2.2.1
Our goal in this subsection is to derive the Keen model from the framework set in the
previous section. To this end, we need to simplify and eliminate most of the items
in Table 2.1, as the Keen model [Kee95] does not involve many of the sectors we just
described. To begin with, we remove both the government and the central bank. As
well, we get rid of stock shares, and cash (due to the absence of a central bank), while
assuming that rM = rL = r.
The simplified skeleton is described in Table 2.2, from which we obtain the
following
12
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
Table 2.2: Balance sheet, transactions and flow of funds for the standard Keen model.
Balance Sheet
Firms
Households
current
Banks Sum
capital
+Mf
Loans
+L
Capital goods
+K
Deposits
+Mh
Vh
+C
Vf
+K
0
Transactions
Consumption
Investment
+I
Accounting memo
[GDP ]
[Y ]
0
I
+W
+r.Mh
+r.Mf
r.M
Interest on loans
r.L
+r.L
Dividends
+Fu
Sf
Sb
+M f
+L
Sf
Sb
Wages
Interest on deposits
Sum
Sh
Flow of Funds
Deposits
+M h
Loans
Sum
Sh
13
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
M h = W C + rMh
M f L = Fu I = C W + rMf rL
L M = rL rM
(2.1)
(2.2)
(2.3)
with output
Y =C +I
(2.4)
Fu
F
=
Y
Y
= Y 1 (Y W r(L Mf ))
(2.5)
= 1 rd
where = W/Y , D = L Mf = Mh , and d = D/Y . Just like Keen [Kee95], assume
that investment is given by a general function of
I = ()Y
(2.6)
14
(2.7)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
D = L M f = W C + r(L Mf )
= Y (Y I) + rD
(2.8)
= Y ( 1 + () + rD)
= Y (() )
Following Goodwins [Goo67] steps, diligently explained in Chapter 3, we assume
that the capital to output ratio remains constant, K = Y , which yields the following
dynamics for Y
Y = Y
()
(2.9)
Next, we assume that productivity (in goods/workers) and the total labor force
both grow exponentially
a = a
(2.10)
N = N
(2.11)
Assuming full capital utilization, the employed labor force is then given by L =
Y /a, while the employment ratio is then = L/N . We now make another behavioral
assumption, that the bargaining equation for wages w = W/L follows from the Phillips
curve depending solely on the employment level,
w = ()w
As a result, we have the following three-dimensional model for , and d
15
(2.12)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
= [() ]
= ()
()
d = ()
(2.13)
and we have recovered the standard Keen model, just as defined in [Kee95], in full
STC form. Observe that consumption is fully determined by the investment strategy
of the capitalist sector,
C = Y I = Y [1 ()]
(2.14)
This shortcoming can be avoided by assuming households can decide their savings
policy, as done by Skott in [Sko89b], where the accommodating variable is the price
of goods.
2.2.2
In [Kee95], Keen immediately extends the model previously introduced by adding the
government sector. Table 2.3 shows the augmented balance sheet, transactions and
flow of funds representing all the relevant exchanges. Notice that we added a row
representing government subsidies, which add to the firms current account, yet do
not contribute to the economy output.
From the second and third columns of Table 2.3, we obtain
C + Ge + GS W T + r(Mf L) = Fu I = M f L
If we define the firms profit share as
16
(2.15)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
Table 2.3: Balance sheet, transactions and flow of funds for the Keen model with
government intervention.
Balance Sheet
Firms
Households
current
Deposits
+Mh
Loans
Bills
Government
+L
+Bb
+K
Vh
+C
Sum
+Mf
+Bh
Capital goods
Sum (net worth)
Banks
capital
Vf
0
+K
Transactions
Consumption
Investment
+I
Government
expenditures
+Ge
Accounting memo
[non-government GDP ]
[Y ]
Government
subsidies
Wages
Government taxes
Interest on deposits
0
Ge
GS
+W
Tf
+r.Mh
+r.Mf
r.M
r.L
+r.L
0
+T
+rg .Bh
Dividends
Sum
+GS
Interest on loans
Interest on bills
Sh
+rg .Bb
rg .B
+Fu
Sf
Sb
+M f
+L
0
Sg
Flow of Funds
Deposits
+M h
Loans
Bills
+B h
Sum
Sh
+B b
0
17
Sf
Sb
0
0
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
= 1 rd + g
(2.16)
Dk = L Mf
(2.17)
along with dk =
Dk
,
Y
g=
GS
,
Y
and =
Tf
.
Y
as
D k = L M f = W + Tf + r(L Mf ) C Ge GS
= W + T + rDk + I Y GS
(2.18)
(2.19)
where dg = B/Y , and ge = Ge /Y . In order to match the government debts dynamics of [Kee95], we need to assume that Ge = 0, that is, government expenditures
are zero. In Chapter 8 we will see another form of government expenditure that respects the non-negative consumption constraint. As before, consumption is implicitly
determined, this time by investment and government expenditure,
C = Y I Ge = Y [1 () ge ]
(2.20)
Once more, we follow Goodwins [Goo67] steps, and arrive at the following differential equations for , , and dk :
18
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
= [() ]
= ()
()
dk = () dk
(2.21)
We are left to specify the dynamics of g, and . One alternative is to follow Keen
[Kee95], who proposes non-linear functions that drive GS and T proportionally to
Y . In Chapter 8, we extend this idea by splitting GS and T into two variables each,
one with growth proportional to Y , and another that varies proportionally to itself.
Through this more general design, we are able to reproduce a plethora of behaviors,
including what was originally proposed by Keen [Kee95].
In any case, it is straightforward to see that the interaction between the government debt dg and the primary variables (, , dk , g, ) is unilateral: one does not
need dg to determine the value of the remaining variables. For this reason, even if
we consider that the government expenditure (excluding subsidies) is non-zero, the
dynamics of (, , dk , g, ) are not affected. On the other hand, determining dg now
requires the knowledge of ge , which is specified in a way that guarantees non-negative
consumption.
19
Chapter 3
Goodwin Model
Ever since its introduction in 1967, the model developed by Richard Goodwin in
[Goo67] has been heavily studied and extended. Its importance is due perhaps to
the fact that it pioneered the field of macroeconomics with endogenous economical
cycles, whereas most models had so far relied on exogenous shocks to produce the
same effect.
By adopting the Lotka-Volterra equations of population dynamics ([Lot25] and
[Vol27]), Goodwin proposed one of the first and most elegant formulations of Marxs
theory of class struggle. Indeed, the inspiration is evident: when profits are on the rise,
investments will follow, adding more jobs to the economy and, in consequence, giving
more bargaining power to the labor force. The workers will, in turn, demand higher
wages, depleting the capitalists surplus, which will lead to more frequent layoffs. The
labor force will have less bargaining power, becoming more susceptible to reducing
their wages, thus increasing profitability, and starting a new cycle.
In this chapter, we review the model under a minor modification, we introduce an
exploding Phillips curve in order to bound the employment rate from above by unity.
Moreover, we derive the solution in terms of a Lyapunov function, and illustrate the
general behaviour through examples.
20
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
3.1
Mathematical formulation
We start with a model for wages and employment proposed by Goodwin [Goo67].
Consider the following Leontief production function for two homogeneous factors
Y (t) = min
K(t)
, a(t)L(t) .
(3.1)
Here Y is the total yearly output, K is the stock of capital, is a constant capitalto
output ratio, L is the number of employed workers, and a is the labor productivity,
that is to say, the number of units of output per worker per year. All quantities are
assumed to be quoted in real rather than nominal terms, thereby already incorporating
the effects of inflation, and are net quantities, meaning that intermediate revenues and
expenditures are deducted from the final yearly output. Let the total labor force be
given by
N (t) = N0 et
(3.2)
L(t)
N (t)
(3.3)
a(t) = a0 et .
(3.4)
where and are constants. Finally, assume full capital utilization, so that
Y (t) =
K(t)
= a(t)L(t).
(3.5)
21
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
Specifically, denoting real wages per unit of labor by w, Goodwin assumes that
w = ()w
(3.6)
where () is an increasing function known as the Phillips curve. The second key
assumption is known as Says law and states that all wages are consumed and all
profits are reinvested, so that the change in capital is given by
K = (Y wL) K = (1 )Y K
(3.7)
where is a constant depreciation rate and is the wage share of the economy defined
by
(t) :=
w(t)L(t)
w(t)
=
.
a(t)L(t)
a(t)
(3.8)
It then follows from (3.5) and (3.7) that the growth rate for the economy in this model
is given by
Y
1
=
:= ().
Y
(3.9)
Using (3.4), (3.6) and (3.8), we conclude that the wage share evolves according to
w
a
= = () .
w a
(3.10)
Similarly, it follows from (3.2), (3.3), (3.4) and (3.9) that the dynamics for the employment rate is
Y
a
N
1
=
=
.
Y
a N
(3.11)
22
(3.12)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
3.2
Properties
In Goodwins original article [Goo67] the Phillips curve is taken to be the linear
relationship
() = 0 + 1 ,
(3.13)
for positive constants 0 , 1 , so that the system (3.12) reduces to the LotkaVolterra
equations describing the dynamics of a predator and a prey . Provided
1
> 0,
(3.14)
it is well known (see for example [HSD04]) that the trivial equilibrium (, ) = (0, 0)
is a saddle point, whereas the only non-trivial equilibrium
(, ) =
+ 0
1 ( + + ),
1
(3.15)
23
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
(3.16)
(0) <
(3.17)
lim () = .
(3.18)
It can then be verified again that (, ) = (0, 0) is a saddle point and that the non
trivial equilibrium
(, ) = 1 ( + + ), 1 ()
(3.19)
Z
1
(s)
log
( 0 ) = log
+
ds.
0
0
s
0
(3.20)
It follows that
Z
V (, ) =
x
dx +
x
(y) ()
dy
y
(3.21)
(3.22)
so that solutions starting at (0 , 0 ) (0, ) (0, 1) remain bounded and satisfy 0 <
< 1 because of condition (3.18). Moreover, conditions (3.16) and (3.17) guarantee
that the righthand side of (3.20) has exactly one critical point at = 1 () in
(0, 1), so that any line of the form = p intersects it twice at most, which shows
that the solution curves above do not spiral and are therefore closed bounded orbits
around the equilibrium (, ).
Unsurprisingly, the growth rate for the economy at the equilibrium point (, ) is
24
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
given by
() =
1
= + ,
(3.23)
3.3
Example
= 0.02,
= 0.01,
=3
(3.24)
1
0 ,
(1 )2
(3.25)
(0) = 0.04
(3.26)
so that = 0.96, and equations (3.16)(3.18) are satisfied. It is then easy to see that
the trajectories are the closed orbits given by
1
1
1
0
log (0 ) = (10) log 1 log
+1
0
0
1 0
(1 )(1 0 )
(3.27)
as shown in the phase portrait in Figure 3.1 for specific initial conditions (0 , 0 ).
The cyclical behaviour of the model can be seen in Figure 3.2, where we also plot the
25
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
total output Y as a function of time, showing a clear growth trend with rate
() = 0.045
(3.28)
0.98
0.96
0.94
0.92
0.9
0.88
0.7
0.75
0.8
0.85
0.9
0.95
Figure 3.1: Employment rate versus wage share in the Goodwin model
3.4
The Goodwin model has been extensively criticized for its structural instability, in
the sense that small perturbations of the vector field in (3.12) change the qualitative
properties of its solution. Most of the literature proposes some extension producing
a structurally stable limit cycle to address this issue, for example [Med79], [Sat85],
[DN88], [Chi90], [FK92], [Spo95], [FM98] and [MF01]. In particular, Desai [Des73]
26
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
9000
8000
0.95
7000
0.9
6000
5000
0.85
4000
0.8
3000
2000
Y
0.75
1000
0.7
10
20
30
40
50
time
60
70
80
90
100
Figure 3.2: Employment rate, wages, and output over time in the Goodwin model
shows that expressing wages in nominal rather than real terms (as is common in the
literature related to the Phillips curve) turns the non-hyperbolic equilibrium into a
stable sink with trajectories spiraling towards it, whereas relaxing the assumption
of a constant capital to output ratio leads to two non-trivial equilibrium points.
In a different direction, van der Ploeg [vdP85] shows that introducing some degree
of substitutability between labor and capital in the form of a more general constant
elasticity of substitution (CES) production function also leads to a locally stable equilibrium, whereas Goodwin himself [Goo91] showed that allowing labor productivity
to depend pro-cyclically on capital leads to unbounded oscillations, and the relative
strength of both effects were analyzed by AguiarConraria [AC08].
These and other extensions are reviewed by Veneziani and Mohun [VM06], where
it is argued that instead of being a shortcoming, the structural instability of the
Goodwin model can be used to analyze the factors that determine the fragility of
27
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
the basic mechanism. Not only they claim that structurally stable models do not
necessarily represent more satisfactory formalizations of Marxs theory of distributive
conflict, but also continue its structural instability is an extreme picture of the
fragility of the structure of the symbiotic mechanism regulating distributive conflict.
Under this interpretation, the proposed extensions also help to explain the poor
fitting of the original model to data as reported by Solow [Sol90] and Harvie [Har00],
with structural changes being responsible for the observed long run phase portraits
for (, ), which shows an overall tendency towards cycles, but nothing resembling
the closed trajectories implied by the Goodwin model.
Turning to the realm of stochastic dynamical systems, we introduce a stochastic
extension of the Goodwin model in the next chapter. By adding a random noise to
the productivity dynamics, we explore a variety of properties and concepts foreign to
deterministic systems, and not discussed thus far.
28
Chapter 4
Stochastic Goodwin Model
In this Chapter, we will introduce a stochastic extension of the Goodwin model discussed in the previous Chapter. Rather than the economic interpretation of the proposed extension, we believe that the tools developed here are the major contributions
to the field. Notions such as stochastic orbits and perturbation analysis will certainly
be ingredients of a more ambitious project where one considers stochasticity in a fully
developed Minsky model.
4.1
Unlike what we had in the Goodwin model, let us consider a growth rate of productivity which is not constant, but heterogeneous among the total labor force. We claim
that the effective growth rate of the productivity should depend negatively on the
level of employment. To understand why, imagine an economy close to full employment. The effective growth rate must be close to the average among the entire labor
force, as there can only be a few workers left out. On the other hand, consider the
opposite extreme: an economy where most of the labor force is unemployed. In such
situation, the turnover of the employed workers must be at its maximum, thus the set
of employed workers will compromise a different quality of workers at each instant of
29
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
time. We can then expect to witness a wild fluctuation of the effective growth rate of
productivity. In any situation in between, we should expect such variations to settle
down as the employment rises. For this reason, we propose the following dynamics
for productivity
dat := at dt = at (dt v(t )dWt )
(4.1)
where v() is the volatility term, which we will assume to be a non-negative and
decreasing function of satisfying v() = 0. The negative sign accompanying the
stochastic term was arbitrarily adopted simply for convenience. Here, Wt is a one
dimensional Brownian motion defined on a filtered probability space (, F, F, P) where
the filtration F = (Ft )t0 satisfies the usual conditions [Shr04].
Following the same definitions as in the Goodwin model, (3.1)(3.3), (3.5)(3.9),
and using Itos formula, we arrive at the following stochastic dynamical system
dt = t (t ) + v 2 (t ) dt + v(t )dWt
d = ( ) + v 2 ( ) dt + v( )dW
t
t
t
t
t
t
where is defined just like (3.9) as (x) =
1x
(4.2)
of the economy. Observe that the Dynkin operator associated to (4.2) applied to any
function f (t, , ) C 2,1 (D R+ ) is
f
f
f
+
() + v 2 () +
() + v 2 ()
t
2
2
2
1 2
f
+ v () 2 2 + 2 2 + 2
2
Lf =
(4.3)
30
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
t = t (t ) + v 2 (t )
= ( ) + v 2 ( )
t
t
t
t
(4.4)
e + v 2 ()
e = 0 and
e
given by ()
e := 1 ( + v 2 ()).
Remark 4.1. If we take the function as (3.25), and the function v() as
v() = (1 )2
(4.5)
(4.6)
(e
) (x)
dx +
x
Z
e
e
(y) ()
dy
y
(4.7)
1
1
e log 1 log
Ve2 () = 1 0 ()
+ 1
e
e
e
1 1
(4.8)
(4.9)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
to have a unique global solution given any initial value is to require that the functions
and are globally Lipschitz. As our choice of explodes when approaches 1, we
need an alternative. This is the purpose of the next assumption.
Assumption 2. There exists two constants k1 , k2 such that, for all (, ) D,
0 () () k1 Ve1 () + k2 and v 2 ()0 () k1 Ve2 () + k2
(4.10)
Remark 4.3. Notice that the inequality with is readily satisfied for the function
(3.9). The other bound concerns both and v, and ensures that t 1 almost surely.
We need one last assumption before we dive into the analytical results. It concerns
the magnitude of the volatility term.
e < 1 v 2 (0). This seemingly arbiAssumption 3. The volatility term is such that v 2 ()
2
trary assumption is easily achieved, and will be useful when determining the behaviour
of the system close to the origin.
Based on Lyapunov techniques, and flavors of Theorem 2.1 from [MMR02], we
prove existence and uniqueness of a regular solution in D. For such, we require conditions similar to those of Theorem 3.4 from [Kha12], which are recalled in Appendix
A for convenience.
Theorem 4.1. Provided Assumptions 1 and 2 hold, and assuming that (3.16)(3.18),
there exists a regular solution (t , t )t0 to system (4.2) starting at any point (0 , 0 )
D. Moreover, the solution is unique up to P-null sets, has the Markov property, and
remains in D with probability one.
Proof. The local Lipschitz growth and sub-linearity conditions of the coefficients of
the system on every compact subset included in D follow from the continuous differentiability of the function . We are left to check conditions (A.8) and (A.9) in
32
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
LVe (, ) =
ih
i
h
i
1h 2
e () ()
e 1 v 2 ()
e () ()
e + 1 v 2 ()0 ()
v () v 2 ()
2
2
2
1
e 1 v 2 ()0 ()
+ [(e
) ()] v 2 () v 2 ()
2
2
The first term is non-positive, while the second term is bounded from above by
h
i
1 2 e
e (0) and decreases to as goes to 1 . Condition (4.10) provides
v
(
)
(
)
2
the bound for third term since || < 1, and also the bound for the last line, as
the function v is bounded. Altogether, there exist constants k1 , k2 R+ such that
condition (A.8) holds for Ve . It is also clear from separation of variables in Ve that
inf
[0,+)
Ve (, ) = Ve2 () +
inf
[0,+)
Ve1 () = Ve2 ()
(4.11)
and tends to infinity when goes to 1 or 0. We also have inf (0,1) Ve (, ) going to
infinity as goes to 0 or +. Condition (A.9) is then satisfied, which allows to apply
Theorem A.2.
The next theorem states the divergence of system (4.2) in the path-wise sense. It
relies on another result of [Kha12], which is recalled in Appendix A as well.
Theorem 4.2. Let (t , t ) be a regular solution to (4.2). In addition to the requirements of Theorem 4.1, suppose that Assumption 3 holds as well. Then for any ,
(t , t )() cannot converge with time to any point (, ) in the closure of D, as t .
Proof. We will prove this result in three steps, for points in D first, then for the origin,
and last for points belonging to its boundary.
e in D, we have either [() + v 2 ()] 6= 0
1. For any point (, ) 6= (e
, )
or [() + v 2 ()] 6= 0. Then by the continuity of the functions and , there
33
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
(4.12)
for some small > 0. We can then find function W (x, y) defined in D (such as x,
K x, y or K y), yet non-negative in B, such that LW < in B.
We then apply Theorem A.3 of Appendix A and obtain that if (t , t ) = (, )
at some arbitrary time t 0, then it must almost surely leave the region B in finite
time.
e we have
Otherwise, at the point (e
, ),
h
i
e = 1 v 2 ()
e
e 0 ()
e
LVe (e
, )
e 0 (e
) > 0
2
(4.13)
Using the same argument, we get that the process exits a small region around
e in finite time almost surely.
(e
, )
2. Take now (, ) = (0, 0). We shall demonstrate that there exists a ball around
this point that must be exited in finite time almost surely. To see why, lets look at
the Dynkin operator applied to Ve at (0, 0)
h
i
1
2
2
e
e
e
) (0)
LV (0, 0) = v (0) v () (0) () + (e
2
(4.14)
34
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
(4.15)
RL = [0, d] [0, 1]
(4.16)
RU = [0, +) [1 d, 1]
(4.17)
(4.18)
RL = inf {t 0 : (t , t ) 6 RL }
(4.19)
RU = inf {t 0 : (t , t ) 6 RU }
(4.20)
We wish to prove that each of these regions is exited in finite time almost surely,
that is, the boundary of D cannot attract solutions indefinitely. Using the Markov
property of the solution, we will consider a starting point in each of these regions and
seek to prove that each corresponding stopping time is finite almost surely.
For the first region, we look at the Dynkin of 0
L = () + v 2 () M 0
(4.21)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
(4.22)
= 2
1 2
() v () 2 2 m 0
2
(4.23)
RB RL RU is exited in finite time almost surely. Instead, we show this property for
each of these regions individually. Conveniently, this will be sufficient for the analysis
that follows.
One could also be interested in the departure from the deterministic model (3.12).
The next result provides a probabilistic estimate of the time it takes for the process
V (t , t ) to exit a ball around V (0 , 0 ), where V is the Lyapunov function defined
in (3.21).
Theorem 4.3. Let (t , t )t0 be a solution of system (4.2) with initial condition
(0 , 0 ) D. For any and t 0, we define the quantity V0 := V (0 , 0 ) and
et () := V (t (), t ()) V0 , along with the stopping time
c () := inf{t > 0 : |et ()| c}
36
(4.24)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
I 2 (V0 , c)
1
2
(4.25)
with Bc, (t) := 21 R(V0 , c)t + t for two constants R and I depending only on V0 and
c.
Proof. Applying Itos formula to et , we find
1
det = v 2 () () () + () () 0 () + 0 () dt
2
+ v() () () + () () dWt
(4.26)
v(s ) () (s ) + (s ) () dW s for t > 0
(4.27)
(4.28)
0tc
Notice that for t and such that t < c (), we have that (, ) E(V0 , c) :=
{(, ) D : |V (, ) V0 | c}. With this in mind, we define the finite quantities
R(V0 , c) :=
I(V0 , c) :=
max
v 2 (y) () () 0 () + 0 () + () () (4.29)
(,)E(V0 ,c)
max
v() () () + () ()
(4.30)
(,)E(V0 ,c)
We can find the probability of the event A by first using conditional probability
and then following with Doobs martingale inequality for the absolute value of the
37
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
c`adl`ag modification1 of Mt , which is a sub-martingale that agrees with |Mt | for all
t > 0, namely
+
P A c = t fc (t) dt
0
#!
" Z
Z +
t
1
1
2
fc (t) dt
1 2E
v 2 (s ) (x) (s )(s ) () dsc = t
t 0
0
Z +
I 2 (V0 , c)
1
fc (t) dt
2
0
I 2 (V0 , c)
= 1
2
P [A ] =
(4.31)
where fc is the probability density function of the random variable c . At last, we
can integrate det to find the estimate, valid for any A and t < c ()
Z
1 t 2
v (s ) () (s ) s 0 (s ) + s 0 (s ) + (s ) () ds
|et ()|
2
Z t0
+ v(s ) () (s ) + (s ) () dWs
0
1
R(V0 , c)t + t = Bc, (t)
2
(4.32)
1
1
Finally, since Bc, (t) c for t Bc,
(c), we have that for A , c () > Bc,
(c),
which leads to
P c >
1
Bc,
(c)
1
P c > Bc,
(c)A P [A ]
I 2 (V0 , c)
1
2
(4.33)
In spite of the process |Mt | being discontinuous at t = 0, we can still verify through Chebyshev
inequality that
2
2
P lim+ |Mt | = P [|Z| /|h0 |] V ar [Z] (h0 /) = (|h0 |/) I 2 (V0 , c)/ 2
t0
38
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
4.2
Along the spirit of Theorem 4.2, we have no hopes of obtaining periodicity for the
the stochastic system (4.2). In this section, we will address this issue by proposing
a convenient and intuitive alternative, the notion that solutions must almost surely
transition amongst regions that cover the domain D indefinitely.
Definition 1. Let denote a random state, while (t0 , t0 ) represents a point on
e that is, t = e t while t (0, 1). As
the line L connecting the origin and (e
, ),
0
0
e 0
e to the vector {t ()
well, define t () as the angle from the vector {t0
e ; t0 }
e in clockwise direction. We define a stochastic orbit as the solution
e ; t () },
(t (), t ())t[t0 ,t0 +T ()] of the system (4.2), where T () = inf {t t0 : t () 2}.
The stopping time T () is called the period of the stochastic orbit.
Observe that the locus of the starting point was specifically chosen for a reason.
As we will see later in this section, solutions can only cross the line L in a specific
direction. The next theorem asserts the almost surely finiteness of the period.
Theorem 4.4. The period of any stochastic orbit is finite almost surely.
The rest of the section is devoted to prove Theorem 4.4. In order to achieve that,
we define regions (Ri )i of the domain D := (0, +) (0, 1), illustrated by Figure 4.1,
and prove that the system exits each of them in finite time, transitioning in some
appropriate direction.
Consider the concave decreasing function on R+
f () = 1 (() )
39
(4.34)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
R7
R8
R1
0.9
R6
0.8
0.7
0.6
R5
0.5
R3
0.4
R2
0.3
R4
0.2
0.1
0.5
1.5
Figure 4.1: Domain D and its covering by (Ri )i=1...8 . Since f (0) < 1 and y = 1 is a
vertical asymptote of , this corresponds to the general case, where x = 0 is possible
on R7 .
We also introduce the process := (t )t0 defined by t := t /t , which is a finite
variation, F-adapted process with dynamics
dt = t (() ()) dt = t ((f ()) ()) dt
40
(4.35)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
R1 := {(, ) D
R2 := {(, ) D
R3 := {(, ) D
8
R := {(, ) D
[
4
D=
Ri with
R5 := {(, ) D
i=1
R6 := {(, ) D
R7 := {(, ) D
R := {(, ) D
8
e and / e}
:
e
: f () }
: f () and
e}
:
e and / e}
(4.36)
e and / e}
:
e f ()}
:
: f () and
e}
:
e and / e}
e and
Ri = (e
, )
i=1
8
[
Ri = D .
(4.37)
i=1
e = f (e
e 6= 0, so that around the point
Moreover,
). We also emphasize that v()
e the system locally behaves like a diffusion on the plane, which implies, along
(e
, ),
Theorem 4.2, that
h
i
e for some t > 0 = 0 (0 , 0 ) D.
P (t , t ) = (e
, ),
(4.38)
Accordingly, we have that this particular point is almost surely never reached,
implying that, a priori, a solution can only leave a region Ri to one of its neighboring
regions Rj , where
j
{2, 8}
(4.39)
if i = 1
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
(4.40)
as the first-entry time to each region Ri . The dependence in s is implied in the above
formulation, yet we shall be exempt of any ambiguity in the proofs below: i (, )
is a short notation for i (0 , 0 ) with (0 , 0 ) = (, ). Our first proposition will be
extremely useful in the proofs of the proceding results.
Proposition 4.1. Define
%(, ) :=
e
t
,
t
e
(4.41)
and
Fc (x) = tan tan1 (x) + tan1 (c)
(4.42)
for any
1 !
M
0, e +
m
(4.43)
where
M :=
m :=
max
e [() ] 0
(
e ) [() ] ( )
(4.44)
min
e
v 2 () 0
(4.45)
e ]
(,)[
,e
][,
e ]
(,)[
,e
][,
(4.46)
is a super-martingale in Sc := D \ {e
%(, ) 1c }
e Our goal is to prove
Proof. Denote %t = %(t , t ), t := t
e , and t := t .
that the Dynkin operator applied to the function Fc %(, ) is non-positive in the
42
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
t t dt .
dt +
t (t )2
(t )2 t t
(4.47)
c
d%t +
d h%t i
1 %t c
(4.48)
(1 %c)2
3
2
LF
=
()
()
+
v
()
c
1 + c2
()()2 () + v 2 ()
2
c e
2
e
+ v () () e
+
e
1 %c
(4.49)
e ()
c
= 1
<
=
e %
e
1 %c
c %
/e
(4.50)
(1 %c)2
2
2
LF
()
()
+
v
()
()
()
+
v
()
c
1 + c2
e e
v 2 ()
= () [() ] () [() ]
e ]
0 in D \ [
,
e ] [,
(4.51)
e ].
We are left to
Which proves the desired result in the region Sc \ [
,
e ] [,
e ]
Sc . In order to accomplish that, let
show the same in the rectangle [
,
e ] [,
43
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
us rewrite (4.49) as
()2
(1 %c)2
LFc = () + v 2 () () + v 2 ()
2
1+c
#
"
2
e
(%
e
)
+ v 2 ()e
(% e) + 1
c %
= () + v 2 () () + v 2 ()
+ v 2 ()e
% e
( /c)
% c1
(4.52)
%e
%c1
1, thus
(1 %c)2
LFc [() ] [() ]
1 + c2
1
2
e+
e
+ v () +
e
c
= [() ] [() ]
1
2
e
v ()
e
c
1
e ]
M m
e 0 in [
,
e ] [,
c
(4.53)
44
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
Remark 4.6. Note that the whole proof works even if i = + for any i involved.
Notice also that Proposition 4.2 implies that if (0 , 0 ) R1 , then
2 (0 , 0 ) < 3 (0 , 0 ) < 4 (0 , 0 ) < 5 (0 , 0 ) P a.s.
(4.54)
e > 0, and
for which we have
2
e
1
3 2
v ()
L =
() + v ()
2
4
8
p
e 2 ()
e
v
e
< 0
e,
(4.55)
The theorem guarantees that (t , t ) leaves R1 in finite time, and this is only
possible via R2 .
Proposition 4.4. Take (0 , 0 ) R2 R3 . Then P [1 (0 , 0 ) 4 (0 , 0 ) < +] =
1.
Proof. Applying Itos formula to
, we find that
2
1
3 2
v ()
L =
() + v ()
2
4
8
e v 2 ()
e
< 0
e,
8
(4.56)
Resorting to Theorem A.3 once more, we obtain the desired result that the region
45
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
R2 R3 is exited in finite time P a.s., which translates into the statement of this
proposition.
The next couple of propositions deal with the opposite situation, concerning regions on the left of / = e.
Proposition 4.5. Take (0 , 0 ) R5 . Then P [6 (0 , 0 ) < +] = 1.
Proof. Define the sequence of regions {Bn }nN through
Bn = R5 { > k/n}
(4.57)
where k > 0 is small enough such that (0 , 0 ) B1 . Applying Itos formula to the
p
e gives us
function
q
1 2 2
1
2
e ) = p
L(
() + v () +
v ()
e
4
e
2
2
1
v 2 ()
3/2
e
8 ( )
(4.58)
1
k2
2 e
< 0 (, ) Bn
3/2 v ()
8
e
n n k
p
e ) 0 in R5 . DMCT implies that there exists the point-wise limit
while L(
q
e
lim
q
() =:
e
()
(4.59)
t6
for all . In addition, Theorem A.3 guarantees that every set Bn is exited in
finite time P a.s.. As consequence, if {5 = +}, we have that ( ) 0,
and consequently t () 0 as well. As Theorem 4.2 states, however, this is a
contradiction to the very existence of the solution.
Proposition 4.6. If (0 , 0 ) R6 R7 , then P [5 (0 , 0 ) 8 (0 , 0 ) < +] = 1.
46
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
(4.60)
e , we find that
1 2 2
1
2
() + v () +
e ) =
v ()
L(
4
e
2
e
1
2
v 2 ()
3/2
8 (e
)
1
k2
v 2 (1 k/n) < 0 (, ) Bn
8 n(ne
k)3/2
(4.61)
while L(
e ) 0 in R6 R7 . DMCT implies that there exists the point-wise
limit
lim
t
() =:
t5,8
()
(4.62)
47
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
[Kha12]. The key insight is that if the process alternates between regions R1 and R2
(or R5 and R7 ) indefinitely, then we arrive at a contradiction that defies existence.
Proposition 4.7. Take (0 , 0 ) R1 R2 . Then P [3 (0 , 0 ) < +] = 1.
Proof. 1. Let (vn )n0 be a sequence of stopping times defined by v0 = 0 and
e or (t , t ) R3 },
vn := inf{t vn1 : t =
n 1.
(4.63)
{3 (0 , 0 ) = +}
(4.64)
(4.65)
(4.66)
which measures the amount of time the process (t , t ) spends further than away
48
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
from the boundary R2 R3 . Observe that we can control the growth of t through
d log t () = ((t ()) (t ())) dt < 1{(t ())(t ())<} dt
(4.67)
Z
log t () < log(0 )
(4.68)
(4.69)
Z t
1
:= inf t 0 :
1{(s ())(s ())<} ds C ()
n
0
(4.70)
(4.71)
This immediately implies that limn (sn vkn )() = 0 for all E. Reminding
e in E, and invoking the continuity of the process (t )t0 , we
ourselves that vkn =
have
e
lim sn () = ,
n
E {3 (0 , 0 ) = +}
(4.72)
This contradicts (4.71), if we choose sufficiently small, proving the desired result
2
We refrain from using the nomenclature stopping time, as it is not applicable here. Indeed, these
random times are not F-adapted.
49
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
(4.73)
where k > 0 big enough such that (0 , 0 ) B1 . One can verify that the Dynkin
operator applied to the function / satisfies
L(/) = (/) [() ()]
1
2
n
(4.74)
(, ) Bn
while L(/) 0 in R5 R6 . DMCT implies that there exists the point-wise limit
lim(/)t7 () =: (/) ()
(4.75)
for every . Moreover, Theorem A.3 guarantees that every set Bn is exited
in finite time P a.s.. As consequence, if {7 (0 , 0 ) = +}, we have that
S
either (t , t )() converges to the set (R6 R7 ) {0} (0, 1 ((0) )), while the
ratio (t /t )() tends to (/) (). In other words, the solution (t , t )() converges
either to some point in R6 R7 (which contradicts Theorem 4.2), or to the segment
{0} (0, 1 ((0) )), in which case t () 0.
Alternative proof. Let (vn )n0 be a sequence of stopping times defined by v0 = 0 and
e or (t , t ) R7 },
vn := inf{t vn1 : t =
n 1.
(4.76)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
lim t () = +,
(4.77)
(4.78)
(4.79)
which measures the amount of time the process (t , t ) spends further than away
from the boundary R6 R7 . Observe that we can control the growth of t through
d log 1
t () = ((t ()) (t ())) dt < 1{(t ())(t ())>} dt
(4.80)
<
log(1
0 )
(4.81)
1
log(1
0 ) log( ())
< +
51
(4.82)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
,n := inf t 0 :
0
1
1{(s ())(s ())>} ds C ()
n
(4.83)
(4.84)
This immediately implies that limn (sn vkn )() = 0 for all E. Reminding
e in E, and invoking the continuity of the process (t )t0 , we
ourselves that vkn =
have
e
lim sn () = ,
n
E {7 (0 , 0 ) = +}
(4.85)
This contradicts (4.84), if we choose sufficiently small, proving the desired result
that E is a P-null set.
The next proposition addresses the transition from the boundary () = ()
to the regions R4 and R8 , and relies on a different proof for recurrence. The idea
in the following is to provide a lower bound for the probability to reach the desired
region R4 (or R8 ) given a specific locus for the starting point.
Proposition 4.9. Take (0 , 0 ) R2 R3 . Then P [4 (0 , 0 ) < +] = 1. Similarly,
if (0 , 0 ) R6 R7 , we can conclude that P [8 (0 , 0 ) < +] = 1.
Proof. 1. The proof for both statements are fairly similar. We will design the
arguments together, pointing out where the differences lie. Consider the process
Fc %(t , t ), as defined in (4.46), which, according to Proposition 4.1, is a superS
S
martingale in a set containing 3i=1 Ri 7i=5 Ri for some appropriate c.
52
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
(4.86)
Alternatively,
Fc (%(0 , 0 )) E Fc (%(5,8 , 5,8 ))
1
= cP [5 (0 , 0 ) < 8 (0 , 0 )] P [8 (0 , 0 ) < 5 (0 , 0 )]
c
(4.87)
c(c M )
> 0, (, ) R2 R3
c2 + 1
(4.88)
P [8 (0 , 0 ) < 5 (0 , 0 )]
c(c M )
> 0, (, ) R6 R7
c2 + 1
(4.89)
Rather,
53
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
through
v :=
inf{t un : (t , t ) R1 R4 R5 R8 }
n
u
n+1 := inf{t vn : (t , t ) (R2 R3 ) (R6 R7 ) or (t , t ) R4 R8 }
(4.90)
for n 1. By construction, we have that once vn =
e , then the process has reached
R4 R8 and um = vm = vn for all m > n. Similarly, if un =
e , then um = vm = un for
all m n. Accordingly, {4 (, ) = +} = n1 {vn 6=
e } for (, ) R1 R2 R3 .
e } for (, ) R5 R6 R7 . The
Alternatively, {8 (, ) = +} = n1 {vn 6=
sequence ({vn 6=
e })n1 is decreasing in the sense of inclusion (up to sets of measure
zero3 ), so that
P [4 (, ) = +] = lim P [vn 6=
e]
(, ) R2 R3 , or
e]
P [8 (, ) = +] = lim P [vn 6=
(, ) R6 R7
(4.91)
Using Bayes rule, allied with the fact that up to sets of measure zero, {un 6=
e } {vn1 6=
e }, we have
P [vn 6=
e ] = P [v0
n
n
Y
Y
6
=
e]
P vk 6=
e |vk1 6=
e
P [vk 6=
e |uk 6=
e]
k=1
k=1
(4.92)
3
54
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
P [vn 6=
e]
n
Y
k=1
n
Y
k=1
n
Y
c(c M )
1 2
c +1
(0 , 0 ) R2 R3 , or
(4.93)
k=1
n
Y
k=1
c(c M )
1 2
c +1
(0 , 0 ) R6 R7
proof.
The following couple of propositions finish the loop of transitions. As we unfortunately do not have a uniform bound on the partial transition probabilities to
work with, we develop a new technique that uses the weaker notion of convergence in
probability to achieve the desired result.
e Then P [5 (0 , 0 ) < +] = 1.
Proposition 4.10. Take 0 =
e and 0 < .
Proof. 1. We first claim that 2 (0 , 0 )5 (0 , 0 ) is finite almost surely for (0 , 0 )
(4.94)
DMCT implies that the stopped process h(t2 5 )() converges point-wise to
some h () for every .
Moreover, consider {2 (0 , 0 ) 5 (0 , 0 ) = +} and, for any small > 0,
e We can further bound the Dynkin operator
define the region R = (, +) (0, ].
55
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
1 2 e
v ()
8
(4.95)
Using Theorem A.3, we obtain that the region R is exited in finite time P a.s..
Since the process starting at (0 , 0 ) given by the random realization cannot exit to
neither regions R2 nor R5 in finite time, we have that there must exist some t < +
e .
for which t = and t < /e
We now claim that h = 0. To understand why, suppose by contradiction that
h > 0. We could then choose = h /2 and obtain that there will always exist
some future t > 0 for which t = < h , contradicting the fact that limt t =
h . As consequence, since h is a continuous bijection, and the solution must remain
confined to the region R3 R4 , we obtain that (t (), t ()) (0, 0). According to
Theorem 4.2, this is a contradiction, from which we obtain that 2 (0 , 0 )5 (0 , 0 ) <
+ P a.s..
2. Next, we show that if it is possible that solutions never make it to R5 , then we
can successfully establish the convergence of a certain transition probability.
If we take (0 , 0 ) R2 R3 , then Proposition 4.9 guarantees that the solution
reaches R4 in finite time P a.s.. On the other hand, if (0 , 0 ) R3 R4 , by step 1
we obtain that the process leaves R3 R4 in finite time P a.s.. With this in mind,
we define the following sequence of stopping times 0 = u0 v0 u1 v1 u2
v2 through
u
e or (t , t ) R5 }
n+1 := inf{t vn : t =
, n 0
v :=
inf{t un : (t , t ) R2 R5 }
n
(4.96)
By step 1 and Proposition 4.9, we have that P [un < +] = P [vn < +] = 1.
Moreover, we have the following chain of relations
{vn >
e } = {(vn , vn ) R2 } {un =
e } = {(vn1 , vn1 ) R2 }, n 1 (4.97)
56
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
Therefore, {5 (0 , 0 ) = +} =
n0 {vn
(4.98)
P [vn >
e ] = P [v0
n
n
Y
Y
e = P [v0 >
e]
>
e]
e |vk1 >
P [vk >
e |uk =
e]
P vk >
k=1
k=1
(4.99)
Bringing (4.98) and (4.99) together, P [5 (0 , 0 ) = +] > 0 implies that
e] = 1
e |un =
lim P [vn >
(4.100)
1 2 e
()t
1 2 e
v ()t
8
, obtaining
1 2
1
1
2
2
e v () dt +
v ()
() + v () dt + v()dWt
8
2
2
1
ekt v()dWt
2
e
<
(4.101)
(4.102)
(4.103)
57
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
or
E
vn ek(vn un ) |Fun un
(4.104)
Denote the sigma algebra generated by un by Gun Fun , so that iterated conditioning yields
E
vn ek(vn un ) |Gun = E E vn ek(vn un ) |Fun |Gun
E un |Gun = un
(4.105)
e
E ek[vn un ] vn (1{vn e} + 1{vn >e} )un =
e.
Since
(4.106)
(4.107)
e 1 P [vn >
e |un =
e]
ek[vn un ] 1 1{vn >e} un =
(4.108)
(4.109)
ek[vn un ] 1 1{vn >e} un =
e 0 as n +
(4.110)
58
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
ek[vn un ] 1 1{vn >e} > |un =
e 0 > 0
(4.111)
Bayes rule gives us that the LHS above is bigger than or equal to
P
ek[vn un ] 1 > |vn >
e , un =
e P [vn >
e |un =
e]
(4.112)
n1 {vn
(4.113)
>
e } {un =
e }. For {5 = +},
by the continuous mapping theorem, we must have that the solutions at consecutive
stopping times converge in probability, and that can only happen if the system overall
e which is a contradiction according to Theorem 4.2. Thus,
converges to the point (e
, ),
we must have that P [5 = +] = 0.
e Then P [1 (0 , 0 ) < +] = 1.
Proposition 4.11. Take 0 =
e and 0 > .
Proof. 1. First, Remark 4.7 implies that 1 (0 , 0 ) 6 (0 , 0 ) < + almost surely
for (0 , 0 ) R7 R8 . That being said, we can define a sequence of almost surely
finite stopping times 0 = u0 v0 u1 v1 u2 v2 through
v
n
u
n+1
:= inf{t un : (t , t ) R1 R6 }
:= inf{t vn : t =
e or (t , t ) R1 }
59
, n 1
(4.114)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
(4.116)
e ] = P [v0 <
e]
P [vn <
n
n
Y
Y
e |vk1 <
e = P [v0 <
P [vk <
e |uk =
e]
P vk <
e]
k=1
k=1
(4.117)
Bringing (4.116) and (4.117) together, P [1 (0 , 0 ) = +] > 0 implies that
lim P [vn <
e |un =
e] = 1
n
(4.118)
(4.119)
e zero only if = ,
e whereas v 2 () = 0 only if
Since () + v 2 () 0 for ,
e where
= 1, we have that L( 2 ) m 2 for
m := inf 2 (() ) + 3v 2 ()
e
[,1)
60
(4.120)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
(4.121)
(4.122)
E v2n em(vn un ) |Fun u2n
(4.123)
in other words
(4.124)
= u2n
that is, by choosing {un =
e }, we have
(4.125)
(4.126)
where we have used the result obtained in step 1. We can derive convergence in
probability from the convergence in mean, that is,
1 em(vn un ) 1{vn <e} > |un =
e 0 > 0
61
(4.127)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
Bayes rule gives us that the LHS above is bigger than or equal to
P
1 em(vn un ) > |vn <
e , un =
e P [vn <
e |un =
e]
(4.128)
n1 {vn
(4.129)
<
e } {un =
e }. For
{1 = +}, by the continuous mapping theorem, we must have that the solutions
at consecutive stopping times converge in probability, and that can only happen if
e which is a contradiction according to
the system overall converges to the point (e
, ),
Theorem 4.2. Thus, we must have that P [1 = +] = 0.
4.3
62
(4.130)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
(4.131)
= (0 ) + (0 ) + O( )
() = 0 + + O(2 )
(4.132)
= (0 ) + 0 (0 ) + O(2 )
while the function is affine, so that
() = (0 ) + 0 (0 ) + O(2 )
(4.133)
= (0 ) / + O( )
Using Itos formula in (4.130), we obtain the following equation for d
d(0 + ) = (0 + ) [((0 ) + 0 (0 ) ) dt + (0 )dWt ] + O(2 )
= 0 [(0 ) ] dt + {[ ((0 ) ) + 0 0 (0 )] dt + 0 (0 )dWt } + O(2 )
(4.134)
and d
d(0 + ) = (0 + ) [((0 ) + 0 (0 )) dt + (0 )dWt ] + O(2 )
= 0 [(0 ) ] dt + {[ ((0 ) ) + 0 0 (0 )] dt
+0 (0 )dWt } + O(2 )
(4.135)
The fundamental theorem of perturbation theory [SMJ98] allows us to group the
terms accompanying each power of from both sides and match them. Accordingly,
one finds that 0 and 0 solve the deterministic Goodwin model (3.12), while ,
63
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
d
= [ ((0 ) ) + 0 0 (0 )] dt + 0 (0 )dWt
d = [ (( ) ) + 0 ( )] dt + ( )dW
0
0
0
0
0
t
(4.136)
which is a linear stochastic dynamical system with periodic coefficients, through the
function 0 (t), 0 (t). Define ~v := [ , ]| , we can rewrite (4.136) in vector form as
d~v (t) = A(t)~v (t)dt + b(t)dW (t)
(4.137)
where
A(t) :=
p(t)
0 (t) (0 (t))
and
b(t) :=
0 (t)(0 (t))
0 (t)(0 (t))
(4.138)
In a later section, we compute the period of a generalized version of the LotkaVolterra model. Such result, stated as Theorem 6.1, can be used to obtain, in particular, the period of the functions 0 (t) and 0 (t), which we shall denote by T .
Consequently, we have that A(t) and b(t) are T -periodic as well. The initial conditions are given by (0 (0), 0 (0)) = (0 , 0 ), while ( (0), (0)) = (0, 0). We will also
e for the rest of this section.
assume that (0 , 0 ) 6= (e
, )
The next result provides a closed-form solution to (4.136).
Proposition 4.12. Let
p(t) := (0 (t))
(4.139)
q(t) := (0 (t))
0 (t)
0
G(t) :=
0
0 (t)
(4.140)
(4.141)
where the pair (0 (t), 0 (t)) solves the system (3.12) with initial condition (0 (0), 0 (0)) 6=
64
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
(4.142)
with 1 = [1, 1]| and I2 the identity matrix of M22 (R), and
(t) :=
p(t)
p(0)
q(0)x (t)
p(0)x (t)
q(t)
q(0)
q(0)y (t)
p(0)y (t)
0 (s)0 (0 (s))
ds
p(s)2
0
Z t
0 (s)0 (0 (s))
ds
y (t)) := q(t)
q(s)2
0
Z
(4.143)
x (t) := p(t)
(4.144)
Proof. For (0 (0), 0 (0)) 6= (, ) in the domain D, we can define the process ~z(t) :=
[x(t), y(t)]| := [ (t)/0 (t), (t)/0 (t)]| . Through Itos formula, we obtain that
d~z(t) = Az (t)~z(t)dt + bz (t)dW (t)
(4.145)
with
Az (t) :=
0 (t) (0 (t))
0 (t) (0 (t))
and
bz (t) := (0 (t))1
(4.146)
As the above elements are all functions of 0 and 0 , we have that Az and bz are
both T -periodic. First we claim that we can solve the homogeneous deterministic
problem
d~zH (t) = Az (t)~zH (t)dt
(4.147)
(4.148)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
To show this, we will build the state-density matrix and show that it can be
written as (4.143) . One can easily verify that ~z1 (t) := [p(t), q(t)]| is a solution of
(4.147). Denote by ~z2 := [x2 (t), y2 (t)]| another solution, linearly independent from
~z1 . A fundamental matrix can be constructed with these two linearly independent
solutions. Abels formula, together with the fact that tr(A0 ) = 0, implies that for any
t0
c := y2 (0)p(0) x2 (0)q(0) = y2 (t)p(t) x2 (t)q(t)
(4.149)
From which we see that if 0 (t) 6= then x2 (t) = c [(0 (t) ]1 , which is
well defined4 . Conversely, if 6= , then y2 (t) = c [(0 (0) ]1 , also well defined.
Otherwise, if neither 0 (t) = nor 0 (t) = , we can isolate either x2 (t) or y2 (t) and
substitute in (4.147) to obtain uncoupled one-dimensional ODEs
0 (t)0 (0 (t))
dx2 (t)
=
(c + x2 (t)q(q))
dt
p(t)
dy2 (t)
0 (t)0 (0 (t))
=
(c + y2 (t)p(t))
dt
q(t)
(4.150)
(4.151)
Solving these ODEs is trivial, and one immediately finds the solution to (4.147)
as
p(t)
x2 (0) + cx (t)
p(0)
q(t)
y2 (t) =
y2 (0) + cy (t)
q(0)
x2 (t) =
(4.152)
4
From Chapter 3, we know that if (0 (0), 0 (0)) 6= (, ), then (0 (t), 0 (t)) will belong to closed
orbit that does not contain (, )
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
Replacing c by its value in terms of x2 (0) and y2 (0) provides x2 (t) and y2 (t) in
terms of x2 (0) and y2 (0), which leads to the state-density matrix (4.143). Since (t)
is also a solution to (4.147), we have
d
d
((t))1 = 1 (t) ((t)) 1 (t) = 1 (t)Az (t)
dt
dt
(4.153)
(4.154)
= bz dWt
Taking into account that (0) = I2 , the identity matrix, we have the following
solution for ~z(t)
Z t
1
~z(t) = (t) ~z(0) +
(s)bz (s) dWs
(4.155)
t
1
(4.156)
q(t)p(t)
1
q(0)p(0)
(4.157)
which holds true whenever p(t) 6= 0 or q(t) 6= 0. As well, by the periodicity of the
integrands we see in (6.34), we find
x (nT + t) = x (t) + n
p(t)
q(t)
x (T ) and y (nT + t) = y (t) + n
y (T ) . (4.158)
p(0)
q(0)
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
It is then easy to see that det (t) = 1, and thus 1 (t) can be determined to be
1 (t) =
q(t)
q(0)
p(0)y (t)
p(0)x (t)
p(t)
p(0)
q(0)y (t)
q(0)x (t)
(4.159)
q(0)x (T ) = p(0)y (T ) =:
(4.160)
(T ) =
with inverse
p(0)
q(0)
q(0)
p(0)
1+
1 (T ) =
p(0)
1+
q(0)
q(0)
p(0)
(4.161)
(4.162)
~v1 =
p(0)
q(0)
(4.163)
which means that 1 is an eigenvalue with algebraic multiplicity equal to 2, but geometric multiplicity equal to 1. We should then be able to find a generalized eigenvector
of grade 2 associated to this eigenvalue. For that, we need to find a vector ~v2 such
that ((T ) I2 )2 ~v2 = 0, but ((T ) I2 ) ~v2 6= 0. Since ((T ) I2 )2 = 0I2 , our task
reduces to finding any vector that is not a multiple of ~v1 . For simplicity, we can use
~v2 =
p(0)
0
(4.164)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
(4.165)
v1 |~v2
Q := ~
1 1
J :=
0 1
(4.166)
where
(4.167)
To conclude this remark, we resort to Floquet theory to say that there exists a
T -periodic matrix P (t) and a constant matrix R such that
(t) = P (t)eRt
(4.168)
The next result provides the variance of the first-order solution at multiples of the
period.
Corollary 4.1. Define
1
Z
=Q
Ru
(u)11 P
(0 (u)) du Q|
R| u 2
(u)e
(4.170)
We then have that the variance of the solution (~v (t))t0 of the system (4.137)
69
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
satisfies
n
X
1 0 |
1 m
Q G(0)
m 1
0 1
m=1
(4.171)
Proof. As mentioned in Proposition 4.12, once we apply the appropriate initial condition to the solution (4.142), ~v (0) = [0, 0]| , we find the martingale
Z
~v (t) = G(t)(t)
(4.172)
(4.173)
where
M (t) =
1 1
1 1
2 (0 (t))
(4.174)
is a T -periodic matrix. We can then study how the variance behaves after multiples
of the period have elapsed using the results obtained in Remark 4.8
Z
nT
= G(0)
kT
k=0
n
X
mRT
Z
m=1
T
Ru
(u)M (u)P
(u)e
R| u
mR| T
du e
G(0)
n
X
1 m
1 0 |
Q G(0)
= G(0)Q
0
1
m
1
m=1
(4.175)
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
Corollary 4.1 shows that the variance of the first-order solution grows cubically
with respect to the cycles of the Goodwin model, suggesting that the first-order
approximation can only be accurate for short time horizons.
4.4
Example
To illustrate the results obtained in this chapter, we choose the functions , and v
as in (3.9), (3.25) and (4.5), with constants according to (3.24) and
1 () = 0.80
(0) = 0.04
(4.176)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
Figure 4.2: Examples of solutions of (4.2) for different values of volatility. On the
left column, we have a phase diagram , where the green star denotes the initial
point, while the red start represents the last point. On the right column, we have the
evolution of output Y over time t.
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
Figure 4.3: Expected values of employment after one full loop T (left), and time
elapsed T (right). Computation performed in MATLAB, with 2000 simulations for
every value single one of the 100 initial values taken along the line = e.
4.5
Conclusion
This Chapter accomplishes two goals. First, it extends the Goodwin model (3.12)
with a random productivity function. Secondly, and perhaps, more importantly, it
73
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
Figure 4.4: Examples of solutions of both (4.2) and (4.130) for different values of
volatility. On the left column, we have the evolution of versus t, while on the right
column we have t. The exact solution is drawn in a solid black line, whereas the
approximate solution is represented by the dashed blue line.
contains a variety of tools that, we believe, will ultimately be employed when dealing
with stochasticity in Minsky models. That being said, we do not exempt ourselves
from responsibly proposing the macroeconomical insights that will define the extended
model, hence the careful introduction of stochasticity in an intuitive, and plausible
manner.
The stochastic extension proposed proves to be a valid one: not only it conserves
the desired cyclical behaviour of its predecessor, it enriches it. By allowing the productivity to fluctuate randomly, where the noise decreases with employment, we have
created a stochastic model that continuously extends the Goodwin model, yet also introduces some stability in expectation, as verified numerically. More importantly, we
74
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
have produced several analytical properties, including: a probabilistic bound for the
time it takes before the stochastic solution deviates away from the Goodwin models
solution, the almost surely finiteness of the period of stochastic orbits, and a closedform solution to an approximation valid when the parameter is small.
Granted, the macroeconomical knowledge developed in this chapter might not be
revolutionary, yet the tools and ideas utilized should be easily extendable to a family
of more intriguing models. One immediate extension is to include the banking sector,
and study a stochastic version of the Keen model [Kee95], where the stochasticity
might arise from, for example, stochastic leverage ratio, credit worthiness, or risk
appetite.
75
Chapter 5
Keen Model
Minskys Financial Instability Hypothesis, described in numerous essays [Min82], links
the expansion of credit with the inherent fragility of the financial system. Minsky provided a verbal prognosis of such hypothesis that follow a sequence of events. At a time
when firms and banks are acting conservatively, perhaps due to the fresh memory of a
recent crisis, defaults are rare, and moderate levels of debt are perfectly sustainable.
Profit thirsty agents realize that they can, and ought to, borrow more to increase their
revenue. Lenders and borrowers all around start feeling compelled to expand their
balance sheet through credit to meet higher market expectations euphoria ensues.
Eventually, leverage reaches such a high level that any small downturn of the economy will be magnified beyond repair. To deleverage, some investors must sell assets,
adding negative pressure to the prices. Suddenly, liquidity has dried out, assets are
oversupplied, and euphoria becomes panic.
Despite his use of a persuasive verbal style aided by convincing diagrams and
incisive exploration of data, Minsky refrained from presenting his ideas in a formal
mathematical setting. This task was taken up by, among others, Keen [Kee95], where
a system of differential equations is proposed as a simplified model incorporating the
basic features of Minskys hypothesis.
Being itself an extension of the Goodwin model [Goo67], discussed in Chapter 3,
76
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
it benefits from the simplicity, while at the same time avoiding the much criticized
structural instability. Instead of centers, we verify the existence of two key fixed
points, which are locally stable under usual conditions. Notably, Keens ideas have
produced a much richer model, capable of exhibiting complex phenomena, resembling,
for instance, what happened in the Great Moderation, followed by the destructive
downward spiral initiated in 2008.
In this chapter, we introduce the Keen model, study its equilibria, determining
their local stability, and then investigate global stability through numerical representation of the basin of attraction.
5.1
Mathematical formulation
The extension of the basic Goodwin model proposed by Keen [Kee95] consists of
introducing a banking sector to finance new investments. By relaxing the assumption
that capitalists invest the totality of their profits, and thus by introducing the variable
D, the amount of debt in real terms, the net profit after paying wages and interest
on debt is
(1 rd)Y
(5.1)
where r is a constant real interest rate and d = D/Y is the debt ratio in the economy.
If capitalists reinvested all this net profit and nothing more, debt levels would remain
constant over time. The key insight provided by Minsky [Min82] is that current cashflows validate past liabilities and form the basis for future ones. In other words, high
net profits lead to more borrowing whereas low net profits (possibly negative) lead
to a deleveraging of the economy. Keen [Kee95] formalizes this insight by taking the
change in capital stock to be
K = (1 rd)Y K
77
(5.2)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
where the rate of new investment is a nonlinear increasing function of the net profit
share = (1 rd) and is a constant depreciation rate as before. Accordingly,
total output evolves as
(1 rd)
Y
=
:= (1 rd)
Y
(5.3)
(1 rd)
=
,
(5.4)
whereas the time evolution for the wage share remains (3.8).
The new dynamic variable in this model is the amount of debt, which changes
based on the difference between new investment and net profits. In other words, we
have that
D = (1 rd)Y (1 rd)Y
(5.5)
+ .
d
D Y
d
(5.6)
rd)
=
(1 rd)
+ + (1 rd) (1 )
d=d r
(5.7)
For the analysis that follows, we assume henceforth that the rate of new investment
78
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
(5.8)
lim 2 0 () <
(5.9)
(5.10)
Recalling from Section 2.2.1 that consumption in the Keen model is given by
C = Y [1 ()], condition (5.9) ensures that the consumption ratio will belong to
the [0, 1] interval.
5.2
We see that
( 0 , 0 , d0 ) = (0, 0, d0 ),
(5.11)
(1 rd)
d r
+ + (1 rd) 1 = 0,
(5.12)
is an equilibrium point for (5.7). Equilibria of the form (5.11) are economically meaningless, and we expect them to be unstable in the same way that ( 0 , 0 ) = (0, 0) is
saddle point in the original Goodwin model.
For a more meaningful equilibrium, observe that it follows from (5.9) that ( +
+ ) is in the image of so that we can define
1 = 1 (( + + ))
79
(5.13)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
( + + ) 1
+
1 = 1 ()
d1 =
(5.14)
( + + ) 1
+
(5.15)
and is an equilibrium for (5.7). This equilibrium corresponds to a finite level of debt
and strictly positive employment rate and is therefore economically desirable, so in
the next section we shall investigate conditions guaranteeing that it is locally stable.
As with the Goodwin model, it is interesting to note that the growth rate of the
economy at this equilibrium point is given by
( 1 ) =
(1 1 rd1 )
= + .
(5.16)
(5.17)
( + + ) 1 (( + + ))
= 1 (( + + )).
+
(5.18)
Provided (5.18) holds, we have that points on the line (0, , d1 ) are equilibria for (5.7)
for any value 0 < < 1. We see that equilibria of this form are not only economically
meaningless, but are also structurally unstable, since a small change in the model
80
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
(1 r/u)
r u2 [(1 r/u) (1 )] .
u = u
(5.19)
(5.20)
for the original system. This equilibrium for (5.19) corresponds to the economically
undesirable but nevertheless important situation of a collapse in wages and employment when the economy as a whole becomes overwhelmed by debt, rendering of
paramount importance to investigate its local stability. Observe that condition (5.9)
guarantees that
(1 r/u) ()
(5.21)
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
5.3
J(, , d) =
()
0
()
0
0
()
()(++)
r ()
(d)0 ()+
(r+)()+r(d)0 ()
(5.22)
At the equilibrium point (0, 0, d0 ) this reduces to the lower triangular matrix
J(0, 0, d0 ) =
(0)
( 0 )(++)
(d0 )0 ( 0 )+
(r+)( 0 )+r(d0 )0 ( 0 )
(5.23)
where 0 = 1 rd0 . Its real eigenvalues are given by the diagonal entries, and it
is hard to determine their sign a priori since d0 is given as the solution of equation
(5.12). Although they can be readily determined once specific parameters are chosen,
we observe that these equilibrium points are likely to be unstable, since a sufficiently
large value of 0 makes the second diagonal term above positive, whereas a sufficiently
small value of 0 (and correspondingly large value of d0 ) makes the third diagonal
term above positive.
At the equilibrium ( 1 , 1 , d1 ) the Jacobian takes the interesting form
0
K0
0
J( 1 , 1 , d1 ) = K1 0
rK1
K2
0 rK2 ( + )
82
(5.24)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
where
K0 = 1 0 (1 ) > 0
1 0 ( 1 )
>0
(d1 )0 ( 1 ) +
K2 =
K1 =
(5.25)
(5.26)
According to the Routh-Hurwitz criterion, a necessary and sufficient condition for all
the roots of a cubic polynomial of the form
p(y) = a3 y 3 + a2 y 2 + a1 y + a0
(5.27)
(5.28)
Our characteristic polynomial already has three of its coefficients positive; therefore all we need is
( + ) > rK2
(5.29)
(5.30)
and
Since we are already assuming that > 0 and > 0, we see that the equilibrium
( 1 , 1 , d1 ) is stable if and only if rK2 < 0, which is equivalent to
0 ( 1 )
r
1 ( + ) > 0.
83
(5.31)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
Because the real interest rate r can have any sign, condition (5.31) needs to be checked
in each implementation of the model. Observe, however, that once the sign of r is
chosen, the remaining terms are all independent of the magnitude of the interest
rate. For the most common situation of r > 0, condition (5.31) imposes interesting
constraints on the investment function . Condition (5.31) states that for 1 to
correspond to a stable equilibrium we must have 1 > , suggesting a lower bound
for equilibrium capitalists profits, while at the same time 0 ( 1 ) needs to be sufficiently
large, leading to a rapid ramp-up of investment for net profits beyond 1 .
As we mentioned in the previous section, equilibria of the form (0, , d1 ) depend
on a very specific choice of parameters satisfying (5.18), making them structurally
unstable. Therefore we are not going to discuss them any further, except by verifying
that they do not arise for the parameters used in the numerical example implemented
later.
Finally, regarding the point ( 2 , 2 , d2 ) = (0, 0, +), observe that the Jacobian
for the modified system (5.19) is
J(, , u) =
()
()
0
0
()
()(++)
()
r
u2
()(12u)+r0 ()(1/u1)+2u(1)(r+)
J(0, 0, 0) =
(0)
()(++)
()(r+)
(5.32)
from which all real eigenvalues can be readily obtained. Observe that the first two
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
eigenvalues are negative by virtue of conditions (3.16) and (5.8) on the functions
and , so this equilibrium is stable if and only if
() =
()
< r.
(5.33)
Recalling expression (5.3), we see that this equilibrium is stable if and only if the real
interest rate exceeds the growth rate of the economy at infinite levels of debt and zero
wages.
It is interesting to note that assumptions (3.16) and (5.8) were made in order
to guarantee the existence of the economically desirable equilibrium ( 1 , 1 , d1 ), but
perversely contribute to the stability of the undesirable point ( 2 , 2 , d2 ) = (0, 0, +).
Moreover, in view of (5.8), we see that a sufficient condition for (5.33) to hold is
+ < r.
(5.34)
5.4
Example
Choosing the fundamental economic constants to be the same as in (3.24) with the
addition of
r = 0.03,
(5.35)
taking the Phillips curve as in (3.25) and (3.26), and defining the investment function
as
(x) = 0 + 1 tan1 (2 x + 3 )
85
(5.36)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
(+) = 1,
(5.37)
1 = 0.16,
( 1 ) = 5
d0 =
0.021
(5.38)
32.503
The eigenvalues for J(0, 0, d0 ) for these two points are
(0.2915, 0.0650, 0.2763)
(0.0849, 0.0650, 0.0448),
confirming that the equilibrium (0, 0, d0 ) is unstable in either case, as expected.
Moving to the economically meaningful equilibrium, we obtain the equilibrium
values
( 1 , 1 , d1 ) = (0.8367, 0.9600, 0.1111).
(5.39)
(5.40)
(5.41)
so that (5.31) is satisfied and this equilibrium is locally stable. When the initial conditions are chosen sufficiently close to the equilibrium values, we observe the convergent
86
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
behaviour shown in the phase portrait for employment and wages in Figure 5.1. The
oscillatory behaviour of all variables can be seen in Figure 5.2, where we also show
the growing output Y as a function of time.
0 = 0.9, 0 = 0.9, d0 = 0.1, Y0 = 100
1
0.98
0.96
0.94
0.92
0.9
0.88
0.86
0.78
0.8
0.82
0.84
0.86
0.88
0.9
0.92
Figure 5.1: Phase portrait of employment and wages converging to a stable equilibrium
with finite debt in the Keen model.
We notice that condition (5.18) is violated by our model parameters, so we do not
need to consider the structurally unstable equilibria of the form (0, , d1 ). Moving on
to the equilibrium with infinite debt, we observe that
()
r = 0.04 < 0,
(5.42)
so that (5.33) is satisfied and ( 2 , 2 , d2 ) = (0, 0, +) corresponds to a stable equilibrium of (5.19). Therefore we expect to observe ever increasing debt levels when the
initial conditions are sufficiently far from the equilibrium values ( 1 , 1 , d1 ). This is
87
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
0.92
Y
d
9000
0.9
0.5
0.45
0.98
8000
0.4
0.88
7000
0.96
0.35
0.94
0.84
0.92
0.82
0.9
0.3
d
5000
0.86
6000
0.25
4000
0.2
3000
0.15
2000
0.8
0.88
0.1
1000
0.78
10
20
30
40
50
time
60
70
80
90
100
0.86
0.05
Figure 5.2: Employment, wages, debt and output as functions of time converging to
a stable equilibrium with finite debt in the Keen model.
depicted in Figure 5.3, where we can see both wages and employment collapsing to
zero while debt explodes to infinity. We also show the output Y which increases to
very high levels propelled by the increasing debt before starting an inexorable descent.
While it is difficult to determine the basin of convergence for the equilibrium
( 1 , 1 , d1 ) analytically, we plot in Figure 5.4 the set of initial conditions for which
we observed convergence to this equilibrium numerically. As expected, the set of
initial values for wages and employment leading to convergence becomes smaller as
the initial value for debt increases.
88
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
30
600
Y
d
25
4500
0.9
4000
0.8
3500
0.7
500
20
3000
0.6
0.5
15
2500
400
2000
0.4
300
10
1500
0.3
0.2
200
100
1000
10
20
30
40
50
time
60
70
80
90
100
0.1
500
Figure 5.3: Employment, wages, debt and output as functions of time converging to
a stable equilibrium with infinite debt in the Keen model.
5.5
Conclusion
89
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
Figure 5.4: Basin of convergence for the Keen model for wages, employment and
private debt.
investment projects are not immediately developed.
90
Chapter 6
Low Interest Rate Regimes
The purpose of this chapter is to further develop to analytical analysis of the Keen
model when the real interest rate is assumed to be close to zero. To begin with,
observe that if r = 0, the first two equations in (5.7) decouple, and we have the
following system for the variables, which we will write in terms of (0 , 0 ) for wage
share and employment, respectively
0 = 0 [(0 ) ]
(1
)
0
0 = 0
( + + )
(6.1)
with the capitalists debt, d0 , solving its own (non-autonomous, but one-dimensional)
ODE
(1 0 )
d0 = (1 0 ) (1 0 ) d0
(6.2)
0 = 1 (),
91
d0 =
0 (1 ( + + ))
(6.3)
+
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
In result, 0 (t), 0 (t) must follow cycles given by the energy function
Z
0 (t)
V (0 (t), 0 (t)) =
0
(1 0 ) (1 x)
dx +
x
0 (t)
(y) (0 )
dy
y
(6.4)
= V (0 (0), 0 (0)),
which can be verified from the fact that V = 0 everywhere. Theorem 6.1 gives the
period of such cycles, which depends only on the initial energy V (0 (0), 0 (0)).
The assumption of low interest rates will be translated into assuming that r equals
some arbitrarily small. Suppose next that we can expand the solution to the Keen
model (5.7) linearly as follows
(t) = 0 (t) + (t) + O(2 )
(t) = 0 (t) + (t) + O(2 )
(6.5)
(6.6)
= (1 0 ) ( + d0 )0 (1 0 ) + O(2 )
Differentiating (6.5) with respect to time, we have
0 + + O(2 ) = 0 + + O(2 )
(0 ) + 0 (0 ) + O(2 )
0
(6.7)
2
= 0 [(0 ) ] + [ [(0 ) ] + 0 (0 )] + O( )
92
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
"
(1 0 )
0 + + O(2 ) = 0 + + O(2 )
#
0 (1 0 )
+ O(2 )
( + d0 )
(1 0 )
= 0
( + + ) + O(2 )
(1 0 )
0 (1 0 )
+
( + + ) 0 ( + d0 )
(6.8)
d0 + d + O(2 ) = (1 0 ) ( + d0 )0 (1 0 ) + O(2 ) (1 0 ( + d0 ))
#
"
0
(1
)
(1
)
0
0
( + d0 )
+ O(2 )
(d0 + d + O(2 ))
(1 0 )
= (1 0 ) (1 0 ) d0
+ O(2 )
(1 0 )
d0
0
1 + + d0 d
+ ( + d0 ) (1 0 )
(6.9)
The fundamental theorem of perturbation theory [SMJ98] allows us to say that
the terms accompanying the powers of must be the same on both sides of the above
equations. Accordingly, one finds that 0 and 0 do indeed solve (6.4), while the
remaining variables solve
(1 0 )
d0 = (1 0 ) (1 0 ) d0
= [(0 ) ] + 0 0 (0 )
(1
)
0 (1 0 )
0
( + + ) 0
( + d0 )
=
d
(1
)
0
0
0
d = ( + d0 ) (1 0 )
1 + + d0 d
(6.10)
(6.11)
(6.12)
(6.13)
(6.14)
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
6.1
Zero-order solution
So far, we know that wage share 0 and employment 0 must oscillate in a cycle,
while d0 is given by
fd (u) hd (u) du
1
(6.15)
where
Z t
(1 0 (s))
0 (0) (+)t
fd (u) = exp
ds =
e
0 (t)
0
(6.16)
(6.17)
and
The following analysis follows closely the procedure developed in [ST11a]. The
homogeneous term is
dH (t) = fd (t)d0 (0) =
(6.18)
Denoting the period of the oscillations of 0 and 0 as T , one can easily see that
fd (T ) reduces to
Z
fd (T ) = exp
0
(1 0 (s))
ds = e(+)T < 1
(6.19)
n
0 (0) (+)nT
e
= e(+)T = fd (T )n
0 (nT )
(6.20)
dH (nT ) = d0 (0)fd (T )n
(6.21)
It can also be shown that the full solution for d0 at multiples of the period T
94
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
satisfies
1 fd (T )n
d0 (nT ) = fd (T ) d0 (0) + fd (T )
1 fd (T )
n
fd (u)1 hd (u) du
(6.22)
fd (u)1 hd (u) du
(6.23)
since fd (T ) < 1. In fact, we can show that d0 converges to a steady state solution.
For that purpose, let t (0, T ), n = 1, 2, , and observe that the particular solution
dp can be expressed as
Z
t+nT
fd (u)1 hd (u) du
0
Z T Z 2T
Z nT
Z nT +t
n
1
= fd (t)fd (T )
+
+ +
+
fd (u) hd (u) du
0
T
(n1)T
nT
("
Z
T
n
1
(n1)
= fd (t)fd (T )
1 + fd (T ) + + fd (T )
dp (t + nT ) = fd (t + nT )
+ fd (T )n
Z t#
)
fd (u)1 hd (u) du
"
2
fd (T ) + fd (T ) + fd (T )
= fd (t)
fd (u)1 hd (u) du
Z
+
t
1
fd (u) hd (u) du
0
1 fd (T )n
= fd (t) fd (T )
1 fd (T )
fd (u) h(u) du +
0
fd (u) hd (u) du
1
(6.24)
which converges, as n +, to the steady state solution dss as follows
fd (T )
dss (t + nT ) = fd (t)
1 fd (T )
T
1
fd (u) hd (u) du +
0
fd (u) hd (u) du = RHS(t)
1
(6.25)
95
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
Z T
Z t
fd (T )
1
1
1
= fd (t)
fd (u) hd (u) du + fd (T ) fd (u) fd (T ) hd (u) du
1 fd (T ) 0
0
Z T
Z t
fd (T )
1
1
fd (u) hd (u) du +
fd (u) hd (u) du
= X(t)
1 fd (T ) 0
0
= RHS(t)
(6.26)
Recalling the fact that the homogeneous term decays to 0 as t + (exponentially fast, with rate ( + )), we can conclude that d0 (t) converges to dss (t) as
t +.
6.2
First-order solution
(0 )
0 (0 )
0
0
(10 )
0)
=
0 (1
( + + )
0
0
(10 )
(10 )
1+
(d0 )
0
0 (10 )
+
d0 0
0
(10 )
d0 (d0 ) + d0
(6.27)
96
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
Likewise the perturbation model proposed for the stochastic Goodwin model in
Section 4.3, we can express the solution to (6.2) in closed-form. This is the objective
of the next proposition.
Proposition 6.1. Let
p(t) := (0 (t))
(6.28)
q(t) := (1 0 (t))/
0 (t)
0
G(t) :=
0
0 (t)
(6.29)
(6.30)
(6.31)
where the pair (0 (t), 0 (t)) solve the system (6.1) with initial condition (0 (0), 0 (0)) 6=
(, ), while d0 (t) is given by (6.15). Denote ~v (t) = [ (t), (t)]| . The solution of
the system (6.27) is then
Z t
1
1
~
(s)h (s) ds
~v (t) = G(t)(t) G (0)~v (0) +
0
Z t
= G(t)(t) 1 (s)~h (s) ds for ~v (0) = [0, 0]|
(6.32)
where
(t) :=
p(t)
p(0)
q(0)x (t)
q(0)y (t)
p(0)x (t)
q(t)
q(0)
p(0)y (t)
(6.33)
with
t
0 (s)0 (0 (s))
ds
p(s)2
0
Z t
0 (s)0 (0 (s))
y (t)) := q(t)
ds
q(s)2
0
Z
x (t) := p(t)
97
(6.34)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
Z
d (t) =
0
0 (1 0 (s))
0 (s) (+)(ts)
e
[ (s) + d0 (s)] 1 +
(d0 (s) ) ds (6.35)
0 (t)
A(t)
x
y
}|
0 0 (0 )
0
0
0 (10 )
x
y
}|
0
0
0)
d0 (1
(6.36)
x H
y H
0
0
0 (0 )
0 (10 )
xH
yH
(6.37)
With the exact same procedure developed in Proposition 4.12, we find that (t)
from (6.33) is the state-density matrix that solves (6.37), and thus ~z(t) = [x(t), y(t)]|
satisfies
Z t
~
~z(t) = (t) ~z(0) +
(s)h (s) ds
1
(6.38)
In other words,
~v (t) = G(t)~z(t)
Z t
1
1
~
= G(t)(t) G (0)~v (0) +
(s)h (s) ds
(6.39)
Taking into consideration that ~v (0) = [0, 0]| , we obtain (6.32). The solution for
d (t) follows immediately from the observation that
d
0 (1 0 (t))
0 (t)e(+)t d (t) = ( (t) + d0 (t))
(d0 (t) )
dt
98
(6.40)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
= (T )n
n Z
X
= (T )
(s)~h (s) ds
1
n
X
(u)~h (u + (k 1)T ) du
1
nk+1
((k 1)T )
k=1
n
X
(k1)T
k=1
n
kT
(u)~h (u + (k 1)T ) du
1
(T )
k=1
(6.41)
Observe that since
Z
(u)~h (u + (k 1)T ) du
1
1 (u)~hss (u) du
as k +
(6.42)
~hss (t) :=
0
0
0 (t))
dss (t) (1
(6.43)
nk+1 (T ) = V J nk+1 V 1 = V
1 nk+1
0
V 1 ,
(6.44)
we can conclude that both (nT ) and (nT ) diverge as n , indicating that the
approximate solution is only valid for finite time horizons.
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
6.3
The work presented here was motivated by [Hsu83]. In this section, we make the
contribution of deriving the period of a general non-linear Lotka-Volterra model1
([Lot25] and [Vol27]). Here, we consider general cross-dependence between x and y
through the functions f and g which are only assumed to be increasing, with a and b
in the interior of their respective images
x = x [a f (y)]
(6.45)
y = y [g(x) b]
The next Theorem provides a closed-form expression for the period of (6.45).
Theorem 6.1. If f (x) satisfies
f 1 (a)
Z
0
a f (y)
dy =
y
f (y) a
dy = +
y
f 1 (a)
(6.46)
log(xmax )
T =
log(xmin )
1
dz +
1
F1 (G(z))
log(xmin )
1
log(xmax ) F2
1
dz
(G(z))
(6.47)
g() b
d = V0 =
g 1 (b)
x(0)
g() b
d +
g 1 (b)
also
(6.48)
ez
b g(x)
dx
x
g 1 (b)
Z
G(z) = V0 +
y(0)
f () a
d
f 1 (a)
100
(6.49)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
f 1 (aw)
F (w) =
f 1 (a)
f (y) a
dy
y
(6.50)
to 0, a f (0) and (, 0], respectively.
Proof. Solutions of the dynamical system (6.45) are cycles centered at
x = g 1 (b)
(6.51)
y = f 1 (a)
(6.52)
Z
V (x, y) =
x
g() b
d +
f () a
d
(6.53)
that must remain constant on the trajectories, that is, for a given starting point
(x(0), y(0)), the solution (x(t), y(t)) must solve
V (x(t), y(t)) = V (x(0), y(0)) =: V0
(6.54)
The variable x(t) oscillates between xmin and xmax , where xmin < xmax are the
roots of
x
Z
x
g() b
d = V0
(6.55)
f ()
d = V0
(6.56)
x
a
x
101
(6.57)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
and thus
x
x
z }|
{
x = x [a f (y)] xf 0 (y)y [g(x) b]
x
x
(x)
2
0
1
1
a
a
=
f of
f
[g(x) b]
x
x
x
(x)
2
x
=
h a
[g(x) b]
x
x
(6.58)
(6.59)
We can write this second-order ODE as a system of first order differential equations
z = w
(6.60)
w = [b g (ez )] h(a w)
(6.61)
x
x
= a f (y).
w
dw
h(a w)
(6.62)
Suppose that at t = 0, x = xmin and y = y. We want to find out how long it takes
for x to reach xmax (at which point, y will return to y). Suppose this happens at
t = T1 . We know that for 0 t T1 , xmin x xmax and 0 < y y. Furthermore,
log(xmin ) z log(xmax ) and 0 w < a f (0). Hence,
Z
F (w) :=
0
d =
h(a )
log(xmin )
102
b g (e ) d =: G(z)
(6.63)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
Notice that
Z
F (w) =
0
f0
(f 1
f 1 (aw)
=
y
d
(a )) f 1 (a )
f (y) a
dy
y
(6.64)
exists for w (a Im(f )). If f is bounded from above, then a max{f } < w <
af (0), otherwise, < w < af (0). In any case, we will say winf < w < af (0),
with winf equal to either a max{f } or . Given that a is in the interior of Im(f ),
we obtain that
y
Z
0
a f (y)
dy =
y
y/2
+
0
y/2
a f (y)
y
dy > lim+ (a f (y/2)) log
= +
y0
y
2y
(6.65)
and
Z
y
f (y) a
dy =
y
Z
+
+
y
y
f (y) a
dy > lim (y y ) log = +
y+
y
y
(6.66)
for some y (y, +). It follows that F (w) must satisfy the following properties
F (0) = 0
(6.67)
(6.68)
(6.69)
lim F (w) =
wwinf
lim
F (w) = +
(6.70)
waf (0)
}|0
{ Z z
e
b g(x)
b g(x)
G(z) =
dx +
dx
x
x
xmin
x
z
Z
103
(6.71)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
Since
d
dx
Z
x
b g()
d
=
b g(x) > 0
x
< 0
if x < x
(6.72)
if x > x,
(6.73)
1
dz
(G(z))
(6.74)
F11 (G(z)) = w =
which implies that
log(xmax )
Z
T1 =
1
log(xmin ) F1
For the rest of the cycle, suppose now that at t = 0, x = xmax and y = y. We
wish to find how long it takes for x to reach xmin (when also y = y). Denote this time
length T2 . We have that z(0) = log(xmax ), z(T2 ) = log(xmin ), w(0) = 0, w(T2 ) = 0
and w(t) 0 for 0 t T2 . From (6.62), we have
V
}|0
{ Z z
w
xmax
z
e
g(x)
b
b g(x)
F (w) =
b g (e ) d =
d =
dx +
dx
x
x
0 h(a )
log(xmax )
x
x
Z
z
Z
= G(z)
(6.75)
Similarly, we can define F2 (w) to be the restriction of F (w) to (, 0] and conclude that
Z
T2 =
Remark 6.2.
log(xmin )
1
log(xmax ) F2
1
dz
(G(z))
(6.76)
104
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
(1 y)
a = ( + + )
By Theorem 6.1, 0 cycles between min and max , roots of
V0 =
()
d
(6.78)
11 ((w+++))
F (w) =
0
( + + ) (1 y)
dy
y
(6.79)
ez
G(z) = V0
0
(x)
dx
x
(6.80)
where 0 = 1 ().
Around the equilibrium point ( 0 , 0 ), linearization shows that the period converges to
2
q
0 0 0 (0 )0 (1
105
(6.81)
0 )/
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
6.4
Example
In this section, we take the parameters as (3.24), besides using the functions and
as defined in (3.25), and (5.36), calibrated according to (3.26) and (5.37).
First, to illustrate Remark 6.2, we refer to Figure 6.1, where the period of the
zero-order model (6.1) for different starting values of (0 , 0 ) is illustrated. There
seems to be a linear relationship between V0 and T , with vertical intercept at 3.2.
This value agrees with (6.81), which gives us a base period of 3.0039.
20
y = 3.6e+002*x + 3.2
15
10
Exact value
Linear Approximation
0.005
0.01
0.015
0.02
V0
0.025
0.03
0.035
0.04
0.99
0.98
0.97
6
0.96
0.95
0.94
2
0.93
0.92
0.78
0.8
0.82
0.84
0.86
0.88
0.9
Moreover, the limit cycle described in (6.25) can be observed in Figure 6.2, while
the quality of the overall approximation developed in this chapter can be assessed
through Figure 6.3.
As expected, due to the fact that the monodromy matrix (4.165) cannot be di106
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
0.8
0.6
d0
0.4
0.2
0
0.86
0.84
0.2
0.81
0.82
0.82
0.8
0.83
0.84
0.78
0.85
0.86
0.76
0.87
0.88
0.74
0.89
0.9
0.72
Figure 6.2: Example of the zero-order solution (0 , 0 , d0 ), depicting the limit cycle
in d0 as described in (6.25).
agonalized, the first-order solution grows unbounded with time, spoiling the approximation when t is large. In our numerical example, with a value of the interest rate
of order 104 , the approximation seems to be almost indistinguishable from the true
solution until at least t = 100, whereas once we raise the interest rate to 103 , at
t = 80, we can already see the growing trend of the error taking place.
6.5
Conclusion
In this chapter, we have studied the Keen model under regimes of extremely low levels
of the real rate of interest. Through perturbation methods, we are able to derive
simpler dynamical systems that can be fully solved analytically. More specifically,
we find that the zero order variables 0 , 0 belong to a cycle specified by an energy
functional, with period that can also be analytically determined. The zero order debt
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
Figure 6.3: Examples of solutions of both (5.7), and (6.1) plus (6.32), for different
values of the real interest rate. On the left column, we have the evolution of versus
t, while on the right column we have t. The exact solution is drawn in a solid
black line, whereas the approximate solution is represented by the dashed blue line.
d0 , on the other hand, converges a steady state periodic function of time, meaning
that the zero order solution, as a whole, converges to a limit cycle.
Unfortunately, as the monodromy matrix associated to the problem (6.37) cannot be diagonalized, the first-order solution inevitable grows quadratically with time,
compromising the accuracy of the approximation for large values of t. Still, as numerical examples show, when the interest rate is of order 104 , the error is virtually
inexistent for at least a century.
108
Chapter 7
Distributed Time Delay
Behind the capital goods dynamical equation (5.2) lies the assumption that capital
goods are added to the system as soon as the cash is invested. In reality, capital goods
expansion cannot be instantly developed. Every time capitalists decide to invest in a
new enterprise, the money spent at time t will only generate revenues at a later time
t + . In the context of the Keen model, this means that capital should not respond
to new investment immediately. We can attempt to capture this effect by adding a
delay in the the capital dynamics, representing the amount of time between when the
profits are invested and when new capital is effectively added to the economy.
Previously, in the Keen model, capital would be driven by
K = ()Y K
(7.1)
K(t)
= (t ) Y (t ) K(t)
(7.2)
When analyzing the dynamical equation (7.2), one needs to deploy the arsenal of
delay differential equations, notably quite involving. We can alternatively emulate
the time delay through a series of exponentially distributed times, as we will see next.
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
7.1
Alternative formulation
The technique described in this section is well documented in the Mathematical Biology literature (see, for example, [AD80]). Ultimately, the latent time for capital
expansion will follow the Erlang distribution. The main idea behind this trick is
that the sum of independent exponentially distributed random variables is Gamma
distributed1 . If we break the project implementation stage in n sub-stages, each exponentially distributed with mean
,
n
Erlang distribution with shape parameter n and scale parameter n . In the limit case
where n , this converges to the delay differential equation described by (7.2).
We will introduce a new variable, , to represent the amount of money invested
in projects yet to be completed. In the simplest case, where n = 1, we have
= ()Y 1
(7.3)
Equation (7.3) contains the inflow of capital goods, () and the outflow representing projects being completed at an exponential rate. The capital dynamics, in
this case, is simply
1
K = K
(7.4)
Capital assets are now completed according to the inflow represented by and
depreciate at an exponential rate given by . The debt dynamics remain unchanged,
as it is driven by the amount of money being currently invested.
In the general case, we have n intermediate investment stages.
1
If X n, n for n N, then X Erlang n, n . Also, E[X] = .
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
1 = ()Y n 1
n
2 = (1 2 )
..
.
n = n (n1 n )
n
K = n K
D = (() ) Y
(7.5)
k
,k
Y
= (() )
= n n ( + + )
n
d = () d
n
1 = () 1 n 1 + 1 n
n
n
2 = (1 2 ) 2
n
..
.
k = n (k1 k ) k n n
..
.
n = n (n1 n ) n n n
(7.6)
The added terms on the k equations are due to the evolution of Y , the economy
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
output (for more details, refer to the derivation of the Keen model in Chapter 5).
This model has an equilibrium, henceforward denoted as the good equilibrium,
determined by
1 = 1 ()
n,1 =
( + + )
n
..
.
ik
h
nk,1 = n,1 ( + + n/ )
n
..
.
h
in1
1,1 = n,1 ( + + n/ )
hn
i
1 = 1 1,1 ( + + n/ )
(7.7)
(
1 )
1
d1 =
+
1 = 1
1 rd1
If we linearize the system (7.6) around this point, we arrive at the following Jacobian matrix:
112
113
1 )
1 0 (
1 )
0 (
..
1)
1 0 (
...
...
n
1
r (1 0 (
1 )) ( + )
...
n d1
r0 (
1 )
...
n 1,1
...
n 2,1
..
.
..
.
...
...
++
++
..
..
.
++
n n1,1
+ 2 + 2 +
(7.8)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
There is also a second equilibrium (from now referred to as the bad equilibrium),
characterized by
2 = 0
2 = 0
1,2 = 0
..
.
(7.9)
k,2 = 0
..
.
n,2 = 0
d2 +
which, as one would expect, expresses the collapsed state of the economy.
7.2
Bifurcations
For the numerical experiments conducted in this chapter, we adopted the fundamental
constants according to (3.24). The function is the same as (3.25), with parameters
given by (3.26). In addition, the function is the one defined in (5.36), but calibrated
according to
() = 0,
(+) = 1,
(7.10)
1 = 0.16,
( 1 ) = 500
Armed with the Jacobian matrix, we can investigate what pairs of and n produce
a locally stable good equilibrium (that is, for what values of and n, matrix (7.8)
contains only eigenvalues with negative real part). Figure 7.1 depicts the threshold
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
0.065
0.06
0.055
0.05
0.045
0.04
10
15
20
25
n
30
35
40
45
50
Figure 7.1: Threshold values of where local stability of the good equilibrium disappears (for the model with project development delay). For each n, the good
equilibrium is stable for values of below this curve.
We can see that as the number of intermediate investment stages increases, the
critical value for the average completion time decreases. There seems to be an asymptote around t = 0.030, about ten days. This value is arbitrarily determined in terms
of the parameters chosen for this section.
Fixing the number of intermediate investment stages to n = 10, we can study
bifurcations with respect to the parameter . Figure 7.2 shows bifurcation diagram
containing the amplitude of for values of ranging from zero to one year(s). There
seems to be a Hopf bifurcation for somewhere between 0.035 and 0.040. The period
of the cycles is shown in Figure 7.3.
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
OMEGA
0.844
0.842
0.84
0.838
0.836
0.834
0.832
0.83
0.03
0.035
0.04
0.045
tau
0.05
0.055
0.06
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
Period
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.03
0.035
0.04
0.045
tau
0.05
0.055
0.06
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
case when the we use infinite intermediate stages of investment? To shed some light
on these concerns, we repeated the previous analysis for different values of n. Figure
7.4 shows the bifurcation diagram, together with the 10% and 90% percentiles of the
time delay for n in {2, 5, 10, 20, 50, 75}. The limiting case when n = +, that is,
when the time delay is discrete, was analyzed with brute force. By simulating the
model with capital dynamics given by (7.2) for a long period of time, with varying
from zero to one year, and recording the long-term behaviour of , we can observe
the stable attractors, be it a fixed point or a limit cycle (see Figure 7.5).
Figure 7.4: Bifurcation diagrams on the left. 10% and 90% percentiles of the delay
distribution on the right.
118
0.830
0.835
0.840
0.845
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
0.030
0.032
0.034
0.036
0.038
0.040
0.042
Figure 7.5: Brute Force bifurcation diagram for the model with discrete delay.
Altogether, we can study the dual effect of varying the number of investment
stages and the average time for completion on the stability of the system in a twoparameter bifurcation diagram, as depicted in Figure 7.6. The threshold value min
corresponds to the point at which the good equilibrium, previously locally stable,
becomes unstable. On the other hand, max denotes the maximum value of for
which there exists the stable limit cycle. The unstable limit cycle is not particularly
interesting as it does not attract solutions. Besides, it seems to exist for values of
that already generate the stable limit cycle. As shown in Figure 7.6, both threshold
values min and max drop quickly as soon as n grows from zero, settling thereafter
around 0.03534 and 0.04269, respectively.
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
0.9
0.5
0.88
0.86
1
0.8
0.5
0.84
0.82
0.9
0.5
0.88
0.86
0.6
0.4
n=1
n=5
n=20
0.2
10
0.5
0.84
0.82
0.6
0.4
=0.01
=0.02
=0.03
0
0.2
10
Figure 7.7: Solutions converging to the good equilibrium, for different values of n
and .
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
0.97
0.8405
0.84
0.12
0.965
0.8395
0.11
0.839
0.96
0.1
0.8385
0.838
0.09
0.955
wage share
employment
capitalists debt
0.8375
0.837
0.95
0.08
0.8365
0.836
10
12
14
16
18
20
0.945
7.3
Conclusion
In this chapter, we studied a modified model where projects are not immediately
completed, but take an Erlang distributed time to be developed. The resulting model
exhibits the same stability properties as the Keen model, namely two stable equilibria,
when the mean time for completion, , is small enough.
Beyond a threshold, which depends on the number of intermediate stages one
chooses to model, the good equilibrium becomes locally unstable, giving rise to a
stable limit cycle. As this parameter increases, we verify a complete instability around
the good equilibrium, where the limit cycle vanishes, leaving the bad equilibrium as
the single stable attractor.
Put differently, when the gap between the time investment decisions are made and
when projects start generating revenue becomes too large, the economy ceases to see
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
positive effects from the investments, which inevitably leads to the collapsed state.
123
Chapter 8
Government Intervention
Ever since the financial crisis that brought the worlds economies to their knees in 2008,
Minskys take on financial fragility has been a growing trend amongst economists.
Accordingly, mathematical formulations of his Financial Instability Hypothesis have
been gaining increasing popularity in the economics literature. Surprisingly, most
of these models have not given enough substance to the government role, confining
it to the task of regulating and/or issuing bonds to be purchased by more active
players such as firms and households. For instance, as Santos [DS05] points out,
Taylor and OConnell in their early influential article [TO85] fall short of completely
specifying the government policy, thus leaving room for hidden assumptions, that
is, model ingredients that were not acknowledged by the authors, let alone analyzed.
On the contrary, we model the government intervention explicitly, carefully analyzing
its impact on the overall economy.
As Minsky himself had already pointed out throughout [Min82], the debt-deflation
spiral can be interrupted with an appropriate intervention by the government, as government spending enter the Kalecki equation increasing firm profit. We model this
behaviour by introducing government expenditure, subsidies and taxation into the
Keen model in Section 8.1. We then perform local stability analysis of the many
equilibria available in Sections 8.1.1 and 8.1.2. As before, we identify a good equi-
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
librium, which proves to be stable under regular conditions, as expected. On the other
hand, there are many undesirable equilibria, both with finite and infinite (negative)
levels of profit. Fortunately, the finite undesirable equilibria are all either unstable or
unachievable under usual assumptions. Moreover, we prove in Proposition 8.1 that
all the equilibria with exploding negative profit are either unstable or unachievable
provided the size of government subsidies is large enough around zero employment
rate. In other words, even when the bad equilibrium is locally stable in the model
without government, we are able to design the government intervention in a specific
way that it destabilizes these unwanted fixed points associated to economic crises.
Moreover, we present in Section 8.2 our main result. As opposed to local stability
analysis, persistence theory [ST11b] studies the behaviour of the solutions around a
specific value for a single variable alone. Instead of answering questions about the
convergence to a point in the space, we are now concerned about the convergence
to a hyper-plane. Widely used in mathematical biology, where mathematicians are
interested in the survival of a specific species over the long term, or whether a certain disease-control protocol will be able to eradicate the pathogen, we bring the
same notion to macroeconomics. In our context, we are intrigued about the longterm survival of key economic variables, for instance profits, or employment. After
proving preliminary results addressing positive exploding profits in Proposition 8.2,
we show in Propositions 8.3 and 8.4 that under a collection of alternative reasonable
conditions on government policy, we obtain uniform weak persistence for both the
employment rate and the capitalists profit. For a precise definition of persistence,
we refer to Appendix C, though put simply, the main result proved in this chapter is
that if the government is willing to be responsive enough in times of crises, both the
employment rate and firm profit are guaranteed to not remain trapped at arbitrarily
small values. Like any persistence result, these statements are global, the initial state
of the economy is not addressed at all in the hypothesis. These represent a sharp
improvement from the model without government intervention, where employment
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
and profits were guaranteed to converge to zero and negative infinity, respectively,
and stay there forever, if the initial conditions were sufficiently bad.
8.1
Introducing government
(8.1)
(8.2)
where
G b = b ()Y,
G s = s ()Gs ,
(8.3)
Tb = b ()Y,
Ts = s ()Ts .
(8.4)
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
(8.5)
(8.6)
b () = lim b ()
(8.7)
()
()
=
lim 2 s0 () <
(8.8)
(8.9)
(8.10)
(8.11)
To simplify the notation, let the rate of growth of the economy be denoted by
() :=
()
.
127
(8.12)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
= [() ]
= [() ]
dk = () dk ()
g b = b () gb ()
(8.13)
b = b () b ()
g s = gs [s () ()]
s = s [s () ()]
dg = dg (r ()) + ge + gb + gs b s
Notice that the capitalist profit share in (8.10) does not depend on the government debt ratio dg , which implies that the last equation in (8.13) can be solved
separately from the rest of the system. Observe further that we can write
= rdk + g s + g b s b
= (() ) r(() ) + b () + gs s () s () s s ()
+ (rdk gs gb + s + b )()
= (() ) r(() ) + (1 )() + b () + gs s () s () s s (),
so that the model reduces to the following fivedimensional system:
= [() ]
= [() ]
g s =gs [s () ()]
(8.14)
s =s [s () ()]
= (() ) r(() ) + (1 )()
+ b () + gs s () b () s s ()
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
We shall base our analytic results on the reduced system (8.14), since this will
be enough to characterize the equilibria in which the economy either prospers or
collapses. Observe that when working with the reduced system (8.14), we cannot
recover dk , gb and separately, but rather the combination
rdk gb + b = 1 + gs s
(8.15)
For numerical simulations, however, we compute the trajectories for the full system
(8.13), so that the evolution of each individual variable can be followed separately.
For completeness, we can express the dynamics of total debt d = dk + dg as
d = dk + dg = () dk () + rdg dg () + ge + gb + gs b s
(8.16)
= () (1 rd) + ge
The hyperplanes gs = 0 and s = 0 are invariant manifolds, indicating that if the
initial value for either gs or s is positive (or negative), the corresponding solution
is also positive (or negative). In general, s 0, as we understand that tax cuts
will be captured by the decreasing function s as opposed to negative taxes. As well,
we will typically have gs > 0, as we want to represent a government attempting to
stimulate the economy with subsidies, although one could also have gs 0 in the case
of austerity measures intended to reduce the government deficit (as a naive attempt
to decrease the debt) when the economy performs badly.
We now return to the specification of government expenditures Ge . Observe that,
since Ge does not affect the profit share in (8.10), its dynamics can be freely chosen
without altering the solution of either the reduced system (8.14) or the full system
(8.13). In fact, the only other variable affected by Ge is government debt, which is
driven by (8.11).
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
(8.17)
we obtain
g e =
(, , , gs , s , Ge , Y )
ge ()
Y
(8.18)
In other words, as long as the dynamics for government expenditures does not
depend explicitly on the level of government debt, equation (8.18) can be solved
separately first and then used to solve the dynamics of dg . Equivalently, we can model
the government expenditure ratio directly as a function ge = ge (t, , , , gs , s ).
8.1.1
Finite-valued equilibria
It is straightforward to see that the only possible finite-valued equilibria for the system
(8.14) are given by the following six cases:
1. Define
1 = 1 ()
(8.19)
1 = ( + )
so that = = 0. Discarding the structural coincidences s (1 ) = + or
s ( 1 ) = + , the only way to obtain g s = s = 0 is to set g s1 = s1 = 0. This
leads us to
1 = 1 1
r(( + + ) 1 ) b (1 ) b ( 1 )
+
+
+
(8.20)
as the only way to obtain = 0. This defines what we call the good equilibrium for (8.14), that is, an equilibrium characterized by finite values for all
variables and non-zero wage share. As we will see next, all remaining cases have
the equilibrium wage share equal to zero.
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(8.21)
b ( 1 ) b (2 ) r(( 1 ) + 1 )
+
(1 1 )
+
+
(8.22)
so that g s = 0 and
g s2 =
so that = 0.
3. Take 3 = s3 = 0 and 3 = 1 and so that = = s = 0 as before. In
addition take g s3 = 0 so that g s = 0. To obtain = 0 define
3 = 1
b (r (( 1 ) + 1 ) (1 1 ) ( + ) + b ( 1 )) .
(8.23)
(8.24)
(8.25)
r(( 5 ) + 5 ) + (1 5 )s ( 5 ) + b (0) b ( 5 )
s ( 5 )
(8.26)
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
(8.27)
(8.28)
so that = 0.
To summarize, discarding equilibria whose existence depend on structurally unstable coincidences in the choice of parameter values, the finite-valued equilibria for
system (8.14) are given by
(, , gs , s , ) =
( 1 , 1 , 0, 0, 1 )
(0, 2 , g s2 , 0, 1 )
(0, 3 , 0, 0, 1 )
(8.29)
(0, 0, 0, 0, 4 )
(0, 0, 0, s5 , 5 )
(0, 0, g s6 , 0, 6 )
Once the system (8.14) converges to an equilibrium (, , g s , s , ), the dependent
variables gb , b , dg must solve
g b = b () gb ()
(8.30)
b = b () gT1 ()
(8.31)
dg = dg [r ()] + gs + gb + ge s b
(8.32)
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
(8.33)
Similarly, if the government expenditure ratio reaches an equilibrium value g e compatible with the equilibrium values for the remaining variables, then the government
debt ratio converges to
gs + ge + gb s b
() r
dg = +
if r < ()
if r > (), or r = () and g s + g e + g b s b > 0
if r = () and g s + g e + g b S b < 0
(8.34)
Local Stability
We begin our local stability analysis by determining the Jacobian matrix for the
system (8.14):
()
() ()
0 ()
()
gs 0s ()
s () ()
s () ()
s ()
s ()
0 () + 0b ()
+gs 0s ()
133
0 ()
gs 0 ()
s 0 ()
r ()
0
+ ()(1 r)
(0b () + s 0s ())
(8.35)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
Returning to the equilibria defined in (8.29), we have the following six cases:
1. Defining the constant
K = r + 0 ( 1 ) (1 1 r) ( + ) 0b ( 1 )
(8.36)
the characteristic polynomial for the Jacobian matrix (8.35) at the good equilibrium ( 1 , 1 , 0, 0, 1 ) can be written as
p1 (y) = y 3 + y 2 (K 1 0 ( 1 )) + y1 0 ( 1 ) 0b (1 ) 1 0 (1 )
0
0
( + )1 ( 1 ) 1 (1 )
s (1 ) ( + ) y s ( 1 ) ( + ) y
(8.37)
This equilibrium will be locally stable if and only if the polynomial (8.37) has
only roots with negative real part. We can identify two of the real roots to be
s (1 ) ( + ) and s ( 1 ) ( + ). The Routh-Hurwitz criterion gives us
the remaining necessary and sufficient conditions for stability:
s (1 ) < +
(8.38)
s ( 1 ) < +
(8.39)
1 > 0
(8.40)
1 0 ( 1 ) K
1 0 (1 )
>
+
1 0 (1 ) 0b (1 )
(8.41)
s ( 1 ) ( + ) y
y 3 + Ky 2
0
0
0
0
+ ( 1 ) 2 b (2 ) + g s2 s (2 ) g s2 ( + ) y + ( + ) 2 g s2 ( 1 )s (2 )
0
(8.42)
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
It follows that this equilibrium is locally stable if and only if the following
conditions are satisfied:
(2 ) <
(8.43)
s ( 1 ) < +
(8.44)
K<0
(8.45)
g s2 ( + ) 2 0s (2 ) 2 0b (2 ) > ( + )2 g s2 0s (2 )/K
(8.46)
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
are satisfied:
(3 ) <
(8.48)
s ( 1 ) < +
(8.49)
s (3 ) < +
(8.50)
K<0
(8.51)
(
+
)
(
> 1 ,
1
1
b
b
we have that
(3 ) > (1 ) = ,
which shows that this equilibrium is locally unstable whenever the good equilibrium is stable.
4. The Jacobian matrix at the equilibrium (0, 0, 0, 0, 4 ) is diagonal, hence we
can identify the eigenvalues at the diagonal and conclude that local stability is
equivalent to the following conditions:
( 4 ) < +
(8.52)
s (0) < ( 4 )
(8.53)
s ( 4 ) < ( 4 )
(8.54)
r + 0 ( 4 )(1 4 r) < ( 4 ) + 0b ( 4 )
136
(8.55)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
(8.57)
(8.58)
(8.59)
(8.60)
Since s (0) 0, this equilibrium can only be attained if s5 > 0. In that case,
we need s ( 5 ) < 0, for it to be locally stable, which would then force s (0) to
be negative. Since this is not economically meaningful, we can conclude that
this equilibrium will always be locally unstable to all effects and purposes.
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
are satisfied:
s (0) < +
(8.62)
s ( 6 ) < s (0)
(8.63)
8.1.2
(8.64)
(8.65)
Infinite-valued equilibria
Our original motivation to introduce a government sector was to prevent the economy from reaching the bad equilibrium (5.20) in the Keen model without government.
Because this equilibrium is characterized by infinitely negative profits caused by explosive private debt, we focus on the cases where .
Making the change of variable u = e , we obtain the system
=[() ]
=[(log u) ]
g s =gs [s () (log u)]
(8.66)
s =s [s (log u) (log u)]
u =u () r (log u) + log u + (1 log u)(log u)
+ b () b (log u) + gs s () s s (log u)
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
()
0 ()
(log u)
gs 0s ()
J5,1 (, u)
J5,2 (, gs , u)
0
0
0 (log u)/u
s () (log u)
0
gs 0 (log u)/u
,
0
s (log u) (log u) s [0s (log u) 0 (log u)]/u
us ()
us (log u)
J5,5 (, , gs , s , u)
where
J5,1 (u, ) = u(() + (log u))
J5,2 (u, , gs ) = u(gs 0s () + 0b () 0 ())
J5,5 (, , gs , s , u) = log u[(log u) r + 0 (log u)] + b () b (log u)
+ r [1 (log u) 0 (log u)] + gs s () s s (log u)
[() + (log u) + 0 (log u)] + 0 (log u)
0b (log u) s 0s (log u)
We can see that (, , gS2 , gT2 , u) = (0, 0, 0, 0, 0) is an equilibrium point for (8.66),
since all terms inside square brackets in the right-hand side of (8.66) approach constants as u 0+ , with the exception of log u, for which we have that u log u 0.
Assuming further that
b , s C 1 [0, 1]
139
(8.67)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
(0)
()
s (0) ()
s () ()
(() r)
where denotes any value, and we have replaced limu0+ [ log(u)] by . Recall that
we assumed in (5.9), (8.7), and (8.8) that the functions , b and s have horizontal
asymptotes satisfying s () < (). Local stability is then guaranteed if, in
addition to the standard requirements (3.17), (5.9), (5.33), and the new condition
(8.8), we impose that
s (0) < ()
(8.68)
140
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
=
vx
1
v =v
s ()
vx
1
1
s =s s
vx
vx
1
x =x s ()(1 x) r + vx (() ) + r
+ r
vx
1
1
1
+ b
+ s s
b () .
(1 )
vx
vx
vx
(8.69)
We then see that (, , v, s , x) = (0, 0, 0 , 0, 0 ) are equilibria for (8.69) since all
terms in the square brackets on the right-hand side of (8.69) approach constant values
as v 0 and x 0 . Recalling (5.10) and (8.9), the associated Jacobian matrix
for these fixed points is
J =
(0)
()
() s (0)
s () ()
s (0) r
(8.70)
Therefore, local stability for these equilibria is guaranteed by (3.17), (5.9), (8.8)
and the new condition
() < s (0) < r.
(8.71)
If we assume that s (0) 6= 0, two other possible equilibria for (8.69) are given by
s (0) r
(, , v, s , x) = 0, 0, 0 , 0,
,
s (0)
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
which are achievable provided either (i) gs (0) > 0 and s (0) < r (so that v 0+ and
), or (ii) gs (0) < 0 and s (0) > r (so that v 0 and ). The
associated Jacobian matrix for these fixed points is
(0)
J =
()
() s (0)
s () ()
r s (0)
(8.72)
Therefore, local stability for these equilibria is guaranteed by (3.17), (5.9), (5.33),
(8.8) and s (0) > r.
The case s (0) = 0 allows for the possible equilibrium (, , v, s , x) = (0, 0, 0 , 0, 0),
depending on the sign of initial condition gs (0), with associated Jacobian matrix
J =
(0)
()
() s (0)
0
0 ,
0
s () () 0
0
0
(8.73)
0
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
without government:
(, , gs , s , ) = (0, 0, 0, 0, )
(8.74)
(, , gs , s , ) = (0, 0, +, 0, )
(8.75)
(, , gs , s , ) = (0, 0, , 0, )
(8.76)
Assuming additionally that (5.33) is satisfied, the stability of these equilibria depend
on stimulus government subsidies gs as follows:
(a) When gs (0) > 0 (stimulus):
(i) if s (0) < (), then equilibrium (8.74) is locally stable, equilibrium
(8.75) is achievable but unstable, and equilibrium (8.76) is unachievable.
(ii) if () < s (0) < r, then equilibrium (8.74) is unstable, equilibrium
(8.75) is achievable and locally stable, and equilibrium (8.76) is unachievable.
(iii) if r < s (0), then equilibrium (8.74) is unstable, equilibrium (8.75) is achievable and unstable, and equilibrium (8.76) is unachievable.
(b) When gs (0) < 0 (austerity):
(i) if s (0) < (), then equilibrium (8.74) is locally stable, equilibrium
(8.75) is unachievable, and equilibrium (8.76) is achievable but unstable.
(ii) if () < s (0), then equilibrium (8.74) is unstable, equilibrium (8.75) is
unachievable, and equilibrium (8.76) is achievable and locally stable.
In other words, under a stimulus regime (gs (0) > 0), any achievable equilibria
with becomes unstable provided s (0) > r. On the other hand, under an
austerity regime (gs (0) < 0), there is no value of s (0) that eliminates the possibility
of local stability from all achievable equilibria with .
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
8.2
Persistence results
= ()
144
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
(8.78)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
Our core results are presented in the next two propositions. We first show that
government intervention can achieve uniformly weak persistence of the functional e
even when the bad equilibrium for the model without government is locally stable.
Proposition 8.3. Suppose that the structural conditions (3.16)(3.18), (5.8)(5.10)
and (8.5)(8.9) are satisfied, along with the local stability condition (5.33) for the
bad equilibrium of the Keen model (5.7) without government. Assume further that
gs (0) > 0. Then the model with government (8.14) is e -UWP if either of the following
conditions is satisfied:
(1) s (0) > r, or
(2) b () is bounded below as 0.
Proof. We prove it by contradiction. If lim supt (t) m for any given large
m > 0, there exists t0 0 such that (t) m for t > t0 . From the equation for ,
it follows that
(t) (t0 )e(tt0 )((m)) ,
for t > t0 . Choosing m > 0 large enough so that (m) < + (recall condition
(5.9)), we get that for any small > 0, there exists t1 > t0 such that (t) < for
t > t1 . From the equation for ,
this readily implies that
(t) < (t1 )e(tt1 )(()) ,
for t > t1 . Again, we may choose > 0 sufficiently small that () < (recall
conditions (3.16) and (3.17)). Hence, there exists t2 > t1 > t0 such that (t) < for
t > t2 . Finally, condition (8.8) guarantees that we can choose m large enough such
that
s () () < 0,
146
m.
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
It then follows from the equation for s that there exists t3 > t2 > t1 > t0 such that
s (t) < for t > t3 . In other words, we can bound , and s by for t large enough.
At this point, we need to consider the hypothesis s (0) > r and b () bounded
separately. Assume first that s (0) > r. Since is a decreasing function, we can
immediately see from the equation for g s that
g s
= s () () > s () (),
gs
for t > t1 . Moreover, since s (0) > r > () (see condition (5.33)), we can choose
small enough and/or m big enough such that s () > (m). Accordingly, for any
t > s > t1 , we have that
gs (t) > gs (s)e(ts)[s ()(m)] .
Using the equation for we have:
= [() ] r[() + ] + (1 )() + b () + gs s () b () s s ()
= [() ] r() + (r ()) + (1 )() + b () + gs s () b () s s ()
> r max{|()|, |(m)|} + (r ()) max{|()|, |(m)|} + b ()
+ s ()gs (t3 )e(tt3 )[s ()(m)] b (m) max{|s ()|, |s (m)|}
= C + A + DeEt
(8.79)
where C is finite and does not depend on t, D = s ()gs (t3 )et3 (s ()(m)) > 0 ,
A = r () > 0, and E = s () (m) > 0. Consequently, for t > t3 , we have
that (t) > y(t), where y(t) is the solution of
y = C + Ay + DeEt ,
147
y(t3 ) = (t3 )
(8.80)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
that is,
y(t) = (t3 )eA(tt3 ) +
C A(tt3 )
D
e
1 +
eEt3 eE(tt3 ) eA(tt3 ) .
A
EA
(8.81)
At last, since s (0) > r, we can choose sufficiently small and m sufficiently large
such that
E A = s () r + () (m) > 0,
which leads us to conclude that eEt dominates the solution y(t) when t , that is,
lim y(t) =
D
eEt = +.
EA
t
Yet, since (t) > y(t) for t > t3 , we must have also (t) +, which contradicts
the fact that (t) m for t > t0 .
Alternatively, assume now that b () is bounded from below as 0. We can
still bound , and s by for t large enough as before. Moreover, since b () > L
for some positive L as 0, we now have that b () > b ()/ > L/. From the
equation for we then have
= [() ] r[() + ] + (1 )() + b () + gs s () b () s s ()
= [() ] r() + (r ()) + (1 )() + b () + gs s () b () s s ()
> r max{|()|, |(m)|} + (r ()) max{|()|, |(m)|} + L/
b (m) max{|s ()|, |s (m)|}
=C()
+ Ay
(8.82)
where C can be made arbitrarily large by choosing sufficiently small, while A =
r () > 0. Therefore, for t > t3 , we have that (t) y(t), where y(t) is now
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
the solution of
y(t)
= C + Ay,
y(t3 ) = (t3 )
that is,
3)
3 ) eA(tt
C()
+ Ay(t
C
y(t) =
+. But this implies that (t) +, which again contradicts the fact that
(t) m for t > t0 .
Although profits play a key role in the model, from the point of view of economic
policy, arguably the most important variable in (8.14) is the rate of employment. Our
next and final result shows that under slightly stronger conditions we can still obtain
uniformly weak persistence with respect to the functional itself. Before stating it,
define the function
h(x) = r[(x) + x] + (1 x)(x) + b (0) b (x),
(8.83)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
exp
(s )ds <
t1
(+)(tt0 )
e
(t1 )
t > t1 ,
t > t1
In other words, given any large L > 0, provided we choose sufficiently small so that
s () > + , there exists t2 > t1 such that gs (t) > L for t > t2 . Alternatively, if
b () is bounded below as 0, given any large L > 0, we can choose sufficiently
small so that b () > L for < (since b () > L0 / > L0 /1 for some L0 > 0).
In either case, we can find > 0 small enough and/or t2 > t1 such that
[()]r[()+]+(1)()+b ()+gs s ()b () > (8.84)
for all [0, ], [0, ], [m, m] and t > t2 . Since lim sup > m and
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
lim inf < m, we can find t3 > t2 such that (t3 ) (m, m), from which it follows
from (8.84) and the equation for that (t
3 ) > 0. Furthermore, (t)
s
gs
v
= s () s ().
v
We can write in terms of v and h (defined in (8.83)) as
= [() ] () + h() + b () b (0) + gs [s () vs ()]
(8.85)
Let us now choose small enough such that () < , s () > + and
s ()
s () 2
> s ( 1 ),
s (0) + 2
151
(8.86)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
t > t0 ,
(8.87)
which grows exponentially since s () > + . Accordingly, we can find t1 > t0 such
that:
(i) gs (t) > [ (0) ()] + maxR [h()] and
s (0)2
40s (m)
and
(8.88)
(8.89)
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
for all t > t1 . In addition, we have that can be locally bounded from below by
> (m) max |h()| + b () b (0) + gs [s () vs ()]
(8.90)
[m,m]
> gs [ + s () vs ()]
(8.91)
for all t > t1 such that (t) [m, m]. We can therefore conclude that, for t > t1 and
(t) [m, m], if vs () s (0) + 2, then gs and if vs () s () 2,
then gs .
Moreover, we can globally bound v from both sides as
v
< s () s (),
v
s () s (0) <
(8.92)
so that < 1 (s ()) implies v < 0, whereas > 1 (s (0)) implies v > 0.
1
Observe further that lim inf 1
s (0), because when [m, s (0)] the lower
bound for becomes strictly positive for t > t1 , forcing to grow higher than 1
s (0).
We can therefore assume, without loss of generality, that 0 s ( 1 ) s (m), since
otherwise we would be done ( 1 = 1 ( + ) would be smaller than the lower bound
of the lim inf and could not go to zero).
We shall now define the following regions, contained in [m, m] R+
n
h
o
s (0)+2
V := (, v) [1
(0),
m]
,
+
:
()
v
()
(0)
+
2
;
s
s
s
s
s (m)
n
h
i
o
s ()2 s (0)+2
S := (, v) [1
,
:
()
v
()
(0)
+
2
;
(0),
m]
2
s
s
s
s (0)
2 ()
P :=
(1
s
(s (0)) , m] 0,
2 (0)+2
s (m)
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
Figure 8.1: The part of the plane (s () v) where one can see the invariant regions
V and S. Every solution that enters V eventually makes it to S and never leaves it.
The region P is not invariant. Yet, solutions that enter it must eventually leave it
and enter the basin of attraction of S, either directly, or after spending some time on
(, v) [m, ) R+ .
With the bounds on and v obtained above, one can observe the following (valid
for t > t1 ):
(i) The flow through v =
s (0)+2
s ()
[ (0) + 2] 0 ()
s
s
~nu :=
2
()
s
and notice that the flow going through the curve obeys
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
~nu
= [s (0) + 2] 0s () + 2s ()v [s () s ()]
v
= [s (0) + 2] 0s () + 2s ()
s (0) + 2
[s () s ()]
s ()
s ()2
s ()
[ () 2] 0 ()
s
s
~nl :=
2
()
s
which yields
~nl
= [s () 2] {0s () + s () [s () s ()]}
v
[s () 2] {0s ()gs + s () [s () s (0)]}
2s (0)
2s (0)
0
< [s () 2] s () 0
<0
4s (m)
4
(8.94)
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
(iii) the flow through the top side of P goes up. This is simply due to the fact that
if (, v) P , then > 1
s (s (0)), which implies that v > 0.
(iv) the flow through the left side of P goes inside P . To see this, notice that for =
1
s (s (0)) and v <
s (0)+2
,
s (m)
s (0)+2
s (0)
s (m)
< s () 2,
hence > 0.
(v) once (, v) V , there exists some t2 > t1 for which (, v) S. One can be
convinced of this from the fact that if (, v) V \ S, then it must be either that
1
< 1
s (s ()), in which case v < 0, or that > s (s (0)), and hence v > 0.
Finally, the last argument goes as follows. Once enters [m, m] (at time, say,
t), there exists some t2 > t1 for which (t) >
1 for all t > t2 . To see this, observe
that if ((t), v(t)) is above the curve v =
s (0)+2
,
s ()
region V , which then drives it to S at some future moment. If, however, ((t), v(t))
starts below the curve v =
s ()2
,
s ()
we are done, as we know that it will eventually enter S and stay away from
1 . If,
however, it enters either P , we are done as well, since from that region the solution
can either:
(i) leave P through its top side, entering the region of attraction of S, or
(ii) leave P through its right side, so becomes bigger than m, while v continues
growing. From there, the solution must return to [m, m] at some later time,
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
s (0)+2
s ()
their way to the region S. Hence, the importance of having s () < 0. If this
was not the case, we would not be able to eliminate cyclic solutions starting from the
region P , exiting to (m, +) R+ , returning to [m, m] above the band, completely
avoiding (v >
s (0)+2
s ()
[m, m] under the band and then return to P , which would not contradict the fact
that 0.
8.3
Examples
In this section, we compare the behaviour of the solutions to the Keen model without
government (5.7) to the model with government (8.14) studied in this chapter. We
fixed the basic parameters according to (3.24), and chose the functions and as in
(3.25) and (5.36), with parameters according to the following constraints
1 = 0.96,
(0) = 0.04,
(8.95)
1 = 0.16,
0 (
1 ) = 5,
(8.96)
(+) = 1
(8.97)
() = 0,
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
solution starts.
For the model with government (8.13), we use the functions
b () = 0 (1 )
(8.98)
s () = 1 2 3
(8.99)
b () = 0 + 1 e2
(8.100)
s () = 3 + 4 e5
(8.101)
ge (, ) = (1 ())(1 )4
(8.102)
(8.103)
g b1 = 0.004
b1 = 0.08
0.02
s (0) =
0.20
s (1 ) =
(8.104)
(8.105)
for a timid government,
(8.106)
for a responsive government
1
min { + , s (0)}
2
0s (1 ) = 0.5
(8.107)
(8.108)
lim b () = 0.20
(8.109)
b (0) = 0
(8.110)
1
s (
1 ) = ( + )
2
(8.111)
(8.112)
s (0) = 0
(8.113)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
8.4. We chose the value of b1 slightly higher than the historical average of government
taxation as we believe that the dataset available illustrates a period of extremely low
taxation.
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
On top of that, conditions (8.38)(8.41) hold as well, so that the good equilibrium
(, , gs , s , ) = ( 1 , 1 , 0, 0, 1 ) = (0.76067, 0.96, 0, 0, 0.16)
is locally stable. Moreover, we can verify that the conditions for stability of the other
finite-valued equilibria in (8.29) are easily violated for our choice of parameters, so
that none of them are locally stable.
As we have seen in Proposition 8.1, the stability of the infinite-valued equilibria in
the presence of government intervention depends crucially on the parameter s (0) =
1 corresponding to the maximum value of the stimulus subsidy function above.
It then follows from item (a) of Proposition 8.1 that in a stimulus regime, namely
for initial conditions with gs (0) > 0, equilibrium (8.74) is unstable in either case,
whereas equilibrium (8.75) is stable in the case of a timid government but unstable
in the case of a responsive government. On the other hand, it follows from item (b)
that in an austerity regime, that is for initial conditions with gs (0) < 0, equilibrium
(8.76) is locally stable in either case.
Moving to the persistence results in Section 8.2, observe that condition (1) of
Proposition 8.3 is satisfied in the case of a responsive government, but that neither
conditions in this proposition are satisfied in the case of a timid government. As a
result, provided gs (0) > 0, the responsive government above ensures uniformly weakly
persistence with respect to e , but the timid government does not.
Similarly, we can verify that condition (4) of Proposition 8.4 is satisfied by our
responsive government even when s (0) > 0, but none of the conditions in this proposition are satisfied by the timid government. Consequently, provided gs (0) > 0, the
responsive government above ensures uniformly weakly persistence with respect to ,
whereas the timid government does not.
We illustrate these results in the next six figures. Choosing benign initial conditions, that is to say, high wage share (90% of GDP), high employment rate (90%),
and low private debt (10% of GDP), we see in Figure 8.5 that the economy even161
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
0.4
0.8
0.3
0.6
0.2
0.4
0.2
0.3
0.15
20
40
60
80
100
time
120
140
160
180
0.12
0.1
0.09
0.05
0.08
20
40
60
80
100
time
120
140
160
180
200
0.11
0.1
0.1
200
g +g
gE
0.2
0.2
dg
0.1
0.4
0.4
0.2
0.6
Keen Model
+ Government
dK
0.8
Figure 8.5: Solution to the Keen model with and without a timid government for
good initial conditions.
As we move to worse initial conditions, that is lower wage share (75% of GDP),
lower employment rate (80%), and higher private debt (50% of GDP), we see in
Figure 8.6 that the free economy represented by the model without government
eventually collapses to the bad equilibrium of zero wage share, zero employment and
infinite private debt, whereas the model with a timid government is more robust and
eventually converges to the good equilibrium.
A timid government, however, is not capable of saving the economy from a crash
if the initial conditions are too extreme, for example a low wage share (75% of GDP),
low employment rate (75%) and extremely high level of private debt (500% of GDP),
162
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
1.5
1.5
1
0.8
Keen Model
+ Government
0.6
dK
(0) = 75%, (0) = 80%, dk(0) = 50%, gb(0) = 5%, gs(0) = 5%, b(0) = 5%, s(0) = 5%, dg(0) = 0%, r = 3%, rg = 0.5%, s(0) = 2%
2.5
0.4
0.8
0.1
40
60
80
100
time
120
140
160
180
0.1
0.08
0.06
0.04
0.02
200
0.06
g +g
gE
0.2
20
0.08
0.6
0.4
20
40
60
80
100
time
120
140
160
180
200
0.04
0.02
dg
0.5
0.2
0.5
Figure 8.6: Solution to the Keen model with and without a timid government with
bad initial conditions.
(0) = 75%, (0) = 75%, dk(0) = 500%, gb(0) = 5%, gs(0) = 5%, b(0) = 5%, s(0) = 5%, dg(0) = 0%, r = 3%, rg = 0.5%, s(0) = 2%
800
5
4
0.8
0.6
0.4
0.2
200
dK
600
400
1
Keen Model
+ Government
0.4
0.8
20
40
60
80
100
time
120
140
160
180
200
800
0.3
600
0.2
400
dg
B+S
gE
0.4
gB+gS
0
0.6
5
0.2
0
0.1
200
20
40
60
80
100
time
120
140
160
180
200
10
Figure 8.7: Solution to the Keen model with and without a timid government with
extremely bad initial conditions.
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
(0) = 75%, (0) = 75%, dk(0) = 500%, gb(0) = 5%, gs(0) = 5%, b(0) = 5%, s(0) = 5%, dg(0) = 0%, r = 3%, rg = 0.5%, s(0) = 20%
6
0.8
1.5
Keen Model
+ Government
0.6
dK
0.4
2
0.5
0.2
1
0
0.8
0.2
0.2
0.6
0.15
0.15
0.1
0.1
20
40
60
80
100
time
120
140
160
180
200
8
6
0.2
0.05
dg
0.05
20
40
60
80
100
time
120
140
160
180
200
10
0.4
g +g
gE
Figure 8.8: Solution to the Keen model with and without a responsive government
with extremely bad initial conditions.
(0) = 95%, (0) = 95%, dk(0) = 50%, gb(0) = 1%, gs(0) = 1%, b(0) = 1%, s(0) = 1%, dg(0) = 0%, r = 3%, rg = 0.5%, s(0) = 2%
1
1
1
0.8
0.8
0.8
0.6
gS(0)>0
0.4
gS(0)<0
0.2
0.4
0.03
0.3
0.02
0.2
50
100
150
time
200
250
300
0.08
0.06
0.04
0.1
0.01
dg
0.01
0.2
0.4
0.2
gB+gS
gE
0.4
0.6
dK
0.6
50
100
150
time
200
250
300
0.02
Figure 8.9: Solution to the Keen model, starting close the good equilibrium point,
with positive (stimulus) and negative (austerity) government subsidies.
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
(0) = 80%, (0) = 80%, dk(0) = 50%, gb(0) = 1%, gs(0) = 1%, b(0) = 1%, s(0) = 1%, dg(0) = 0%, r = 3%, rg = 0.5%, s(0) = 2%
1
1.5
8
gS(0)<0
0.6
dK
0.8
gS(0)>0
6
4
0.4
0.5
0.2
0.5
50
100
150
200
time
250
300
350
400
20
0.08
0.2
0
15
S
0.06
0.04
1
1.5
10
0.5
0.4
gB+gS
gE
0.6
50
100
150
200
time
250
300
350
400
dg
0.8
0.02
Figure 8.10: Solution to the Keen model, starting far from the good equilibrium point,
with positive (stimulus) and negative (austerity) government subsidies.
as shown in Figure 8.7. On the other hand, a responsive government effectively brings
the economy from the severe crisis induced by these extremely bad initial conditions,
as shown in Figure 8.8.
Additionally, the effects of austerity measures are exemplified in Figures 8.9 and
8.10. For a healthy initial state, we see that the transient period suffers from the negative spending, compared to a positive stimulus, without any long term consequences.
Once we push the initial state further away from the good equilibrium, we can immediately verify the disastrous consequences of austerity: the government focuses so
much on reducing public debt that it throws the economy into a path of eventual
collapse.
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
8.4
Concluding remarks
166
Chapter 9
Conclusion
In this thesis, we have explored several corners of macroeconomical modeling using dynamical systems. After reviewing the notion of stock flow consistency among
Minsky models in Chapter 2, we familiarized the reader with the double entry book
keeping framework that is necessary to guide any serious attempt of macroeconomical
modeling.
In Chapter 3 we carefully discuss the influential Goodwin model, where we make
our first contribution by proposing a non-linear Phillips curve which bounds the employment variable to the (0, 1) domain, providing the solution in terms of a Lyapunov
function. Chapter 4 discusses a novel extension with stochastic flavor. By introducing
random fluctuations in the dynamics of productivity, we obtain a continuous extension of the Goodwin model for which we examine several interesting properties. After
proving existence and almost surely uniqueness of solutions, we derive probabilistic
estimates based on the Lyapunov function derived for the Goodwin model. More
importantly, we show that every trajectory must indefinitely loop around a center
without ever converging to any point in the closure of the domain. We end the chapter showing that when the volatility of the random noise is negligible, the solution is
approximately that of the Goodwin system plus a martingale term.
The mathematical formalization of Minskys Financial Instability Hypothesis is
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
to both profits and employment. In other words, when the government is willing
to act responsively, capitalists profits and employment are guaranteed to never be
trapped under arbitrarily low levels, regardless of the initial state of the system.
We believe that this thesis paves the road for a variety of extensions. Among the
many future projects one could be interested in, we suggest a few. For instance, one
can model prices through a dynamical equation designed to converge to an equilibrium price, which itself could be obtained through supply-demand arguments. Such
a model should be able to answer questions regarding the effect of inflation in an
economy, reproducing stylized scenarios such as stagflation, or even hyperinflation.
Alternatively, stock prices could be introduced by allowing firms and/or banks to
capitalize themselves by issuing stocks. These should then take part in the portfolio choice of households, who should take into consideration the risk/return profile
of all the instruments available to them. Further down the road, one could enlarge
the model by adding securitization, issued by special purpose vehicles, and study the
effect of structured products, such as the stigmatized mortgage-backed securities, in
the whole economy. This thesis provides a useful toolbox for a thorough analysis of
any of the extensions suggested above. Even though one could argue that none of
the ideas just proposed are intrinsically novel, the depth and precision that one could
reach with the apparatus presented here would certainly be so.
169
Appendices
170
Appendix A
Stochastic Dynamical Systems
Toolbox
Consider the n-dimensional stochastic process ~xt , with dynamics
d~xt = ~b(t, ~xt )dt +
k
X
~ r (t)
r (t, ~xt ) dW
(A.1)
r=1
where W1 (t), W2 (t), , Wr (t) are mutually independent Brownian motions. The next
couple of results are Theorem 3.5, and Corollary 3.1 from [Kha12].
Theorem A.1. Assume that
k
X
||b(s, ~x) b(s, ~y )|| +
||r (s, ~x) r (s, ~y || B ||~x ~y ||
||b(s, ~x)|| +
r=1
k
X
(A.2)
r=1
hold in every cylinder R+ {||~x|| < R}, for any R > 0. Moreover, suppose that there
exists a nonnegative function V C 2 on the domain D such that for some constant
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
c>0
LV cV,
(A.3)
(A.4)
Then:
1. For every random variable ~x(t0 ) independent of the process Wr (t) Wt (t0 ) there
exists a solution ~x(t) of (A.1) which is an almost surely continuous stochastic
process and is unique up to equivalence;
s,~
xs
~x
Z
(t) = ~xs +
k Z
X
(A.5)
r=1 s
3. If the functions ~b(t, ~x) and ~r (t, ~x) are independent of t, then the transition
probability function of the corresponding Markov process is time-homogeneous;
and if the coefficients are T -periodic in t, then the transition probability is T periodic.
Corollary A.1. Consider an increasing sequence of open sets (Dn ) whose closure are
S
contained in D and such that n Dn = D. Suppose that in every cylinder R+ Dn the
coefficients ~b and ~r satisfy conditions (A.2) and there exists a function V (t, ~x), twice
continuously differentiable in ~x and continuously differentiable in t in the domain
R+ D, which satisfies condition (A.3) and the condition
inf
t>0,~
xD\Dn
V (t, ~x) as n
172
(A.6)
Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
then the conclusion of Theorem A.1 holds provided that also P [~x(t0 ) D] = 1. Moreover, the solution satisfies the relation
P [~x(t) D] = 1 for all t t0
(A.7)
The next Theorem is a slightly more general version of the Corollary above, though
applied to the stochastic dynamical system 4.2.
Theorem A.2. Consider an increasing sequence of open sets (Dn ) whose closure are
S
contained in D and such that n Dn = D. Assume that in each set Dn , the drift and
volatility coefficients of system (4.2) are Lipschitz and sub-linear functions. Assume
that there exists a function V(, , t) C 2,1 (D R+ ) such that
LV(, , t) k1 V(, , t) + k2
on D R+
(A.8)
(A.9)
D\Dn
then the system (4.2) possesses a unique almost surely continuous regular solution for
(0 , 0 ) L0 (D, F0 ) satisfying P [(t , t ) D] = 1 for all t 0.
Proof. The proof follows from applying Corollary A.1 with the function V = V +k2 /k1 ,
for which we have
LV = L (V + k2 /k1 ) = LV k1 V + k2 k1 (V + k2 /k1 )
(A.10)
= k1 V
Observe that condition (A.2) holds from the local Lipschitz and sub-linearity assumptions on the drift and the volatility, while V (, , t) satisfies conditions (A.3)
and (A.6), immediately giving us the desired result.
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
The next result is Theorem 3.9 from [Kha12], simply adapted to the notation of
Chapter 4.
Theorem A.3. Let (t , t )t0 be a regular process in D, with (0 , 0 ) U , for some
U D. Assume that there exists a function V (t, , ) C 1,2,2 (R+ U ) verifying
LV (s, , ) f (s)
where f (s) 0 and limt
Rt
0
(A.11)
time P a.s..
174
Appendix B
XPPAUT Instructions
B.1
#####################################
## keen_erlang.ode
#####################################
## EQUATIONS:
omega=omega*(Phi(lambda)-alpha)
tan_lambda=(1+tan_lambda^2)*PI*lambda*(g_Y-alpha-beta)
d=Kappa(pi_n)-pi_n-d*g_Y
theta_[75..2]=(n/tau)*(theta_[j-1]-theta_[j])-theta_[j]*g_Y
theta_1=Kappa(pi_n)-theta_1*(n/tau+g_Y)
## ALIAS
lambda=atan(tan_lambda)/PI+0.5
pi_n=1-omega-r*d
theta_n=theta_75
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
g_Y=n/tau/nu*theta_n-delta
phi0=(alpha*(1-lambda_eq)^2-Phi_min)/(1-(1-lambda_eq)^2)
phi1=phi0+Phi_min
kappa_eq=nu*(alpha+beta+delta)
kappa1=(kappa_U-kappa_L)/pi
kappa0=kappa_U-kappa1*pi/2
kappa2=kappa_pri*(1+tan((kappa_eq-kappa0)/kappa1)^2)/kappa1
kappa3=tan((kappa_eq-kappa0)/kappa1)-kappa2*pi_eq
aux auxLambda=atan(tan_lambda)/PI+0.5
aux auxTau=tau
## FUNCTIONS
Phi(u)=phi1/(1-u)^2-phi0
Kappa(u)=kappa0+kappa1*atan(kappa2*u+kappa3)
## PARAMETERS:
par tau=0.01,r=0.03,nu=3,alpha=0.025,beta=0.02,delta=0.01
par lambda_eq=0.96,Phi_min=-0.04
par pi_eq=0.16,kappa_pri=500,kappa_L=0,kappa_U=1
par n=75
## INITIAL CONDITIONS:
init omega=0.8366
init tan_lambda=7.916
init d=0.1137
init theta_[1..75]=0
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
## XPP SETUP
@ meth=gear
@ total=200,dt=0.01,bounds=1e10
@ maxstor=2000000,back=white
done
The sequence of commands below produce the bifurcation diagram shown in Figures 7.2 7.4:
1. open the file keen erlang.ode;
3. integrate it a couple more times, to make sure it converges and stabilizes at the
equilibrium: (I)nitialconds, (L)ast, twice;
5. change the axis to properly accommodate the diagram that will follow: (A)xes,
h(I)-lo: Xmin=0.03, Ymin=0.83, Xmax=0.06, Ymax=0.845;
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
B.2
## ALIAS
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
lambda=atan(tan_lambda)/PI+0.5
pi_n=1-omega-r*d
pi_n_d=1-delay(omega,tau)-r*delay(d,tau)
pi_n_2d=1-delay(omega,2*tau)-r*delay(d,2*tau)
g_Y=kappa(pi_n_d)/nu*z-delta
g_Y_d=kappa(pi_n_2d)/nu*delay(z,tau)-delta
phi0 = (alpha*(1-lambda_eq)^2-Phi_min)/(1-(1-lambda_eq)^2)
phi1 = phi0+Phi_min
kappa_eq=nu*(alpha+beta+delta)
kappa1 = (kappa_U-kappa_L)/PI
kappa0 = kappa_U-kappa1*PI/2
kappa2 = kappa_pri*(1+tan((kappa_eq-kappa0)/kappa1)^2)/kappa1
kappa3 = tan((kappa_eq-kappa0)/kappa1)-kappa2*pi_eq
aux auxLambda=atan(tan_lambda)/PI+0.5
aux auxTau=tau
#aux auxG_Y=g_Y
#aux auxPi=pi_n
#aux auxPi_d=pi_n_d
#aux omega_d=delay(omega,tau)
#aux rd_d=r*delay(d,tau)
## FUNCTIONS
Phi(u)=phi1/(1-u)^2-phi0
Kappa(u)=kappa0+kappa1*atan(kappa2*u+kappa3)
## PARAMETERS:
par tau=0.04,r=0.03,,nu=3,alpha=0.025,beta=0.02,delta=0.01
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
par lambda_eq=0.96,Phi_min=-0.04
par pi_eq=0.16,kappa_pri=500,kappa_L=0,kappa_U=1
## INITIAL CONDITIONS:
omega(0)=0.8366171
tan_lambda(0)=7.915815
d(0)=0.1127582
z(0)=0.9995501
init omega=0.836171
init tan_lambda=7.915815
init d=0.1127582
init z=0.9995501
## XPP SETUP
@ meth=qualrk4
@ total=150,delay=1,dt=1e-3,bounds=1e10,tol=1e-10
@ trans=100
@ maxstor=2000000,back=white
## STORAGE
@ output=bruteforce_KeenDelay_tau.dat
done
The above code was executed in silent mode, that is, through the command
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
## define variables
time <- rawdata[,1]
omega <- rawdata[,2]
tau <-rawdata[,7]
## plot set up
pdf("keen_delay_bruteforce_tau_n_inf.pdf",width=10, height=10/1.62)
par(mar=c(4,4,2,0.5))
plot(tau,omega, type="p",
xlab=expression(paste(tau)),
ylab=expression(paste(omega)),
col="blue", pch=".", cex=1, ylim=c(0.83,0.845))
dev.off()
with the following console command
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
182
Appendix C
Persistence Definitions
Let (t, x) : R+ X X be the semiflow generated by a differential system with
initial values x X. For a nonnegative functional from X to R+ , we say
is uniformly strongly persistent (USP) if there exists an > 0 such that
lim inf t ((t, x)) > for any x X with (x) > 0.
is uniformly weakly persistent (UWP) if there exists an > 0 such that
lim supt ((t, x)) > for any x X with (x) > 0.
is strongly persistent (SP) if lim inf t ((t, x)) > 0 for any x X with
(x) > 0.
is weakly persistent (WP) if lim supt ((t, x)) > 0 for any x X with
(x) > 0.
As an example, consider the Goodwin model (3.12). From Chapter 3, we know
that the solution passing through the initial condition (0 , 0 ) satisfies the equation
(3.20).
The closed periodic orbits implied by this equation are shown in Figure 3.1. Recalling that = 1 for this model, observe that remains bounded on each orbit,
so that lim inf t exp(1 ) > 0 and the system is e strongly persistent. However,
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Bernardo R. C. da Costa Lima PhD Thesis McMaster University Dept. of Math and Stats
since the bound on can be made arbitrarily large by changing the initial conditions,
we see that the system is not e uniformly strongly persistent. Finally, we see in
Figure 3.1 that becomes smaller than the equilibrium value infinitely often, regardless of the initial conditions. Therefore, taking < exp(1 ) shows that the
system is e uniformly weakly persistent. The exact same arguments show that the
Goodwin model (3.12) is SP, UWP, but not USP.
For the Keen model without government intervention defined in (5.7) the situation
is less satisfactory. Whenever the conditions for local stability of the bad equilibrium
(5.20) are satisfied, we cannot have either or e persistence of any form, since initial
conditions sufficiently close to the bad equilibrium will necessarily lead to = e = 0
asymptotically.
184
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