Discrete Dynamical Systems, Bifurcations and Chaos in Economics, Volume 204 (Mathematics in Science and Engineering)

Discrete Dynamical Systems, Bifurcations Bifurcations and Chaos in Economics

This is volume 204 in SCIENCE AND AND ENGINEERING ENGINEERING MATHEMATICS IN SCIENCE Edited Edited by C.K. Chui, Stanford University in this series series appears appears at at the the end end of of this this volume. volume. A list of recent titles in

Discrete Dynamical Systems, Bifurcations Bifurcations and Chaos Chaos in Economics

WEJ-BJN ZHJ4NG WEI-BIN ZHANG COLLEGE OF ASIA PASIFIC MANAGEMENT MANAGEMENT RITSUMEIKAN ASIA PASIFIC UNIVERSITY UNIVERSITY BEPPU-SHI, BEPPU-SHL OITA-KEN OITA-KEN JAPAN

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Preface Difference equations equations are are now now used used in modeling modeling motion motion and and change change in all areas Difference areas of science. In particular, applications of difference difference equations in economics economics have have recently been been accelerating mainly because because of rapid development recently accelerating mainly development of of nonlinear nonlinear theory and computer. computer. Application of difference difference equations equations to to economics economics is a vast and vibrant area. Application Concepts and theorems related to difference Concepts and theorems related difference equations equations appear appear everywhere everywhere in academic journals journals and textbooks in economics. economics. One can can hardly hardly approach, approach, not not to to mention of economic mention digest, digest, the literature of economic analysis analysis without without "sufficient" "sufficient" knowledge Nevertheless, the the subject knowledge of difference difference equations. Nevertheless, subject of applications applications of difference equations to to economics difference equations economics is not systematically systematically studied. studied. The subject subject is often mathematical economics. Due often treated as a subsidiary part of (textbooks of) mathematical to the the rapid rapid development development of of difference difference equations equations and wide wide applications applications of of the the theory there is a need theory to economics, economics, there need for for aasystematic systematic treatment treatment of of the the subject. subject. This This book book provides provides aa comprehensive comprehensive study study of of applications applications of of difference difference equations to economics. economics. This book is aa unique unique blend blend of of the the theory theory of ofdifference difference equations equations and and its its exciting applications to to economics. The book provides not only a comprehensive introduction applications of theory of of linear linear (and (and linearized) linearized) comprehensive introduction to to applications difference equations to economic economic analysis, analysis, but but also also studies studies nonlinear nonlinear dynamical dynamical difference systems which which have have been been widely widely applied applied to to economic economic analysis analysis in in recent recent years. years. ItIt provides a comprehensive introduction to most important important concepts concepts and theorems theorems in difference equations theory in a way that can be understood by anyone difference equations theory that can be understood by anyone who who has basic basic knowledge knowledge of calculus calculus and and linear linear algebra. algebra. In In addition addition to to traditional traditional applications of of the the theory applications theory to economic economic dynamics, dynamics, it also also contains contains many many recent recent developments different fields fields of of economics. developments inin different economics. We We emphasize emphasize "skills" "skills" for for application. Except conducting analysis of the economic application. Except conducting mathematical mathematical analysis economic models models v

vi

PREFACE

like like most most standard standard textbooks textbooks on on mathematical mathematical economics, economics, we we use use computer computer simulation to demonstrate motion of of economic systems. A A large simulation to demonstrate motion economic systems. large fraction fraction of examples examples in this book book are are simulated simulated with with Mathematica. Mathematica. Today, Today, more more and and more more researchers and educators tools to solve onceseemingly seemingly researchers and educators are using computer computer tools solve —once impossible to calculate even even three three decades decades ago ago—- complicated and tedious impossible to calculate and tedious problems. problems. I would would like like to to thank thank Editor Editor Andy Andy Deelen Deelen at at Elsevier Elsevier for for effective effective cooperation. operation. I completed completed this book book at at the the Ritsumeikan Ritsumeikan Asia Asia Pacific Pacific University, University, am gratefifi grateful to the the university's university's free freeacademic academic environment. environment. II take take great great Japan. I am pleasure in expressing my gratitude to wife, Gao Gao Xiao, Xiao, who who has has been been pleasure in expressing my gratitude to my my wife, supportive writing this Beppu City, City, Japan. Japan. She also supportive of of my my efforts efforts in in writing this book book in in Beppu She also to draw draw some some of ofthe thefigures figures in inthe thebook. book. helped me to Wei-Bin Zhang

Contents Preface

v

Contents

11

22

vii

Difference equations equationsin ineconomics economics Difference 1.1 Difference 1.1 Difference equations and economic analysis 1.2 Overview linear difference equations Scalar linear Linear first-order difference difference equations 2.2 Some concepts 2.2 2.3 Stabilities 2.3 2.4 Stabilities of nonhyperbolic equilibrium points 2.5 On dissipative maps 2.6 Linear difference equations of of higher order order 2.7 Equations with constant coefficients 2.8 Limiting behavior 2.1 2.1

dynamical economic economic systems 33 One-dimensional dynamical 3.1 A model of inflation and unemployment 3.1 3.2

The one-sector growth (OSG model) 3.3 The general OSG model 3.4 The overlapping-generations (OLG) model 3.5 Persistence of inequality and development 3.6 Growth with creative destruction 3.7 Economic evolution with human capital capital 3.8 Urbanization with human capital externalities 3.9 The OSG model with money 3.10 The TheOSG OSG model model with with labor labor supply

vii vn

11

22 77

13 13 15 21 21 27 42 49 54 60 67 79 80 80

83 83 88 93 93 97 100 106 106 112 112 118 118 126 126

CONTENTS

viii 44

Time-dependent solutions solutions of scalar scalar systems Periodic orbits 4.2 4.2 Period-doubling biflircations bifurcations 4.3 4.3 Aperiodic orbits 4.4 Some types of of biftircations bifurcations 4.5 Liapunov numbers 4.6 Chaos 4.1 4.1

Economicbifurcations bifurcations and andchaos chaos 5 Economic 5.1 5.1 5.2 5.3 5.3 5.4 5.5 5.5 5.6 5.7

Business cycles with knowledge spillovers A cobweb model with adaptive adjustment Inventory model with rational expectations Economic growth with pollution The Solow and Schumpeter growth oscillations Money, growth and and fluctuations fluctuations Population and economic growth

66 Higher Higher dimensional dimensional difference equations 6.1 6.1 Phase space analysis of planar linear systems systems 6.2 6.2 Autonomous linear linear difference equations 6.3 Nonautonomous linear linear difference equations equations 6.3

135 135 148 148 155 155 163 163 170 170 177 177 185 185 186 186 193 193 195 195 202 202 205 205 213 213 219 219

Stabilities Stabilities Liapunov's Liapunov's direct direct method method Linearization ofdifference of difference equations equations Conjugacy and and center center manifolds The Hénon Henon map map and and biflircations bifurcations The The Neimark-Sacker (Hopf) (Hopf) bithrcations bifurcations 6.10 The Liapunov numbers and chaos The Liapunov numbers and chaos

227 227 228 228 240 248 248 261 261 270 276 276 281 281 289 289 296 296 301 301

Higher Higherdimensional dimensional economic economic dynamics dynamics 7.1 7.1 An An exchange exchange rate rate model model 7.2 A A two-sector two-sector OLG OLG model model 7.3 Growth with government spending 7.3 7.4 Growth with fertility fertility and old age support 7.5 Growth with different different types of economies 7.5 7.6 Unemployment, inflation inflation and chaos 7.7 Business cycles with money and capital 7.7 7.8 The OSG OSG model with with heterogeneous heterogeneous households households 7.8 7.9 7.9 Path-dependent evolution with education Appendix A.7. A.7.11 Proving Proving proposition proposition 7.81 7.81 A.7.2 Proving Provingproposition proposition 7.9.1 7.9.1

305 305 306 306 310 310 315 315 320 320 327 327 331 331 335 335 341 341 358 358 374 374 374 374 379 379

6.4 6.5 6.5 6. 66 6.7 6.7 6.8 6.8 6.9 6.9

77

135 135

CONTENTS

ix

8 Epilogue

385 385

Appendix A.1 A.I Matrixtheory Matrix theory A.2 Systems of linear equations A.3 A.3 Metric spaces A.4 The Theimplicit implicitfUnction function theorem A.5 The TheTaylor Taylorexpansion expansionand andlinearization linearization A.6 Concave and quasiconcave Concave and quasiconcavefimctions functions A.7 Unconstrained Unconstrainedmaximization maximization A.8 A.8 Constrained maximization A.9 Dynamical Dynamicaloptimization optimization

391 391 391 397 397 398 398 401 401 408 410 415 415 418 418 422

Bibliography

427

Index

439 439

Chapter Chapter 11

Difference economics Difference equations in economics The necessity difference equations equations is evident evident ifif one one necessity of knowledge about theory of difference opens current journal subfield of theoretical theoretical as well as applied applied opens almost almost any any current journal in any subfield economics. Nevertheless, Nevertheless, there there is no book book which is concentrated concentrated on on applications applications of of contemporary contemporary theory theory of of difference difference equations equations to to economics. economics. The The purpose purpose of of this book to introduce introduce the the theory theory of of difference difference equations equations and and its its applications applications to book is to economics. A difference difference equation equation expresses expressesthe therate rate of of change change of of the the current current state state as as a function of the current state. A simple illustration of this type of dependence is function simple illustration this type of dependence is changes of the GDP (Gross Domestic Product) over years. Consider the GDP of the economy in year as the state variable variable in period denoted by by x(t). x(t). Let period t, which is denoted economy year t as change of the the GDP GDP isis constant. constant. Then, Then, the the motion motion us consider a case that the rate of change as of the GDP is described mathematically as

x(t+i)—x(t) =g. x(t) ~SAs the growth growth rate given for for each each year, year, the the GDP GDP in in period period t is given by As the rate g isis given solving the difference difference equation

x(t+1)=(1-i-g)x(t). x(0), then the GDP in year tt is is given by If we know a special special year's year's GDP, x(O), x(t)= x(OX1 + g)'.

1

2

IN ECONOMICS ECONOMICS 1. DIFFERENCE EQUATIONS IN

In fact, fact, if if we we know know any any special special year's year's GDP, GDP, then then the the equation equation predicts predicts the time. We We can can explicitly explicitly solve the above difference difference function function because gg GDP in any time. constant. ItIt isis reasonable reasonable to consider consider that the growth rate is affected affected by many is aa constant. factors, such the current current state state of of the the economic economic system, system, the knowledge knowledge of the the factors, such as the economy, international international environment. constant and is economy, environment. When When the the growth growth rate rate is not constant considered to affected by the the current current state state and and other other exogenous exogenous factors factors like like considered to be affected global economic economic conditions conditions (which global (which are are measured measured through through the the variable, variable, t), t), then economic growth is described by

x(t+1)—xQ)

= g(x(t), t).

XV)

general, it is is not not easy easy to toexplicitly explicitly solve solve the the above above function. function. There There are are In general, different methods of solving solving different different types types of ofdifference difference equations. equations. different established established methods This introduces concepts, concepts, theorems, theorems, and methods methods in in difference difference equations equations This book introduces theory are widely widely used used inincontemporary contemporary economic economic analysis analysis and and provides provides theory which are many traditional as well as contemporary applications of the theory to different many traditional well as contemporary applications of different fields in economics. economics.

1.1

Difference and economic economic analysis analysis Difference equations and

unique blend blend of of the the theory theory of of difference difference equations equations and and their their exciting exciting This book is aa unique economics. First, itit provides aa comprehensive comprehensive introduction introduction to most most applications to economics. important concepts and theorems in difference difference equations theory in a way way that that can can important by anyone anyone who who has has basic basic knowledge knowledge of of calculus calculus and and linear linear algebra. algebra. be understood by In addition addition to traditional traditional applications applications of economic dynamics, of the the theory theory to economic dynamics, itit also also contains developments in different different fields fields of economics. It is is mainly mainly contains many recent developments concerned difference equations solve and provide provide concerned with with how how difference equationscan can be be applied applied to to solve insights into economic dynamics. We emphasize "skills" for application. insights economic dynamics. emphasize for application. When applying the theory to economics, we outline the economic problem applying the economics, problem to be solved and then derive difference difference equation(s) equation(s) for this problem. These equations are then for problem. equations analyzed and/or simulated. We use computer simulation to demonstrate analyzed simulated. computer simulation to demonstrate motion motion of of economic systems. A large fraction of examples in this book are simulated with economic systems. examples this book are simulated with Mathematica.' and more more researchers researchers and and educators educators are are using using Mathematica. Today, Today, more more and computer tools such as Mathematica to solve once seemingly impossible computer tools such as Mathematica to solve — once seemingly impossible to calculate even even three three decades decades ago ago - complicated calculate complicated and and tedious tedious problems. problems. This comprehensive introduction introduction to This book provides not only a comprehensive to applications applications of of linear and linearized linearized difference difference equations equations theory theory to economic economic analysis, analysis, but also also linear studies nonlinear dynamical systems which have been widely applied to economic 1

Ennsand and McGuire McGuire(2001), (2001),Shone Shone(2002), (2002),and andAbell Abelland andBraselton Braselton(2004). (2004). Enns

1.1. DIFFERENCE EQUATIONS AND ECONOMIC ANALYSIS

33

analysis determines what analysis only only in recent years. Linearity Linearity means means that that the rule that determines what a piece of a system is going to do next is not influenced influenced by what it is doing now. The mathematics of linear systems exhibits a simple geometry. The simplicity allows us to capture the essence of the problem. problem. Nonlinear Nonlinear dynamics is concerned concerned with the equations are nonlinear. study of systems whose time evolution equations nonlinear. If If a parameter parameter that that qualitative nature of the behavior remains describes a linear system is changed, the qualitative the same. But for nonlinear nonlinear systems, systems, a small small change change in a parameter parameter can lead to sudden and dramatic changes in both the the quantitative quantitative and qualitative qualitative behavior of the system. system. the Nonlinear dynamical theory reveals how nonlinear interactions can bring about qualitatively new different from its qualitatively new structures structures and and how how the whole whole is related related to and different individual nonlinear dynamical dynamical theory been individual components. components.The The study study of of nonlinear theory has has been enhanced with developments developments in computer computer technology. technology. A modem modern computer computer can can enhanced explore explore a far far wider wider class class of phenomena phenomena than than itit could could have have been been imagined imagined even even aa few decades ago. ago. The The essential essential ideas ideas about about complexity complexity have have found found wide wide few decades applications range ofofscientific scientific disciplines, disciplines, including including physics, physics, applications among among aa wide range cognitive science, science, economics economics and and sociology. sociology. Many Many biology, ecology, psychology, cognitive complex found to share share complex systems systems constructed constructed in in those those scientific scientific areas areas have have been been found commonproperties. properties. The Thegreat greatvariety variety of ofapplied appliedfields fields manifests manifests aapossibly possibly many comnon unifying methodological methodological factor the sciences. sciences. Nonlinear Nonlinear theory theory is is bringing bringing factor in in the scientists they explore explore common common structures structures of of different different systems. systems. ItIt offers offers scientists closer closer as as they scientists tool for for exploring exploring and and modeling modeling the the complexity complexity of nature and and scientists aa new new tool of nature society. The new techniques and concepts provide powerful methods for modeling society. The new techniques and concepts provide powerfUl methods for modeling and simulating trajectories sudden and and irreversible irreversible change change in in social social and and natural natural and simulating trajectories of of sudden systems. systems. Modern nonlinear theory begins with Poincare Modem Poincaré who revolutionized revolutionized the the study study of of nonlinear differential equations by introducing the qualitative techniques of nonlinear differential equations by introducing the qualitative techniques of geometry topology rather strict analytic analytic methods methods to discuss discuss the the global global geometry and and topology rather than strict properties of solutions of these systems. He considered it more important to have a system than the global understanding understanding of of the the gross gross behavior behavior of of all solutions solutions of of the the system local behavior behavior of particular, particular, analytically analytically precise the local precise solutions. solutions.The The study study of of the dynamic systems systems was furthered furthered in the Soviet Soviet Union, Union, by by mathematicians mathematicians such such as as dynamic Pontryagin, Andronov, Liapunov, Pontryagin, Andronov, and others. Around Around 1960, the study study by Smale in United States, States, Peixoto Peixoto in Brazil Brazil and and Kolmogorov, Kolmogorov, Arnold Arnold and Sinai in in the the the United and Sinai Soviet significant influence Soviet gave gave aa significant influence on on the the development development of of nonlinear nonlinear theory. theory. Around Around 1975, 1975, many many scientists scientistsaround aroundthe theworld world were were suddenly suddenlyaware awarethat thatthere there isis aa new new kind of of motion motion - now chaos - in dynamic systems. systems. The motion is erratic, erratic, kind now called called chaos The new new motion is in dynamic but not not simply simply "quasiperiodic" "quasiperiodic" with with aa large largenumber number of ofperiods.2 periods.2 What What is is surprising surprising but is that that chaos chaos can can occur occur even even in in aa very very simple simple system. system. Scientists Scientists were interested in is were interested in complicated motion advent of computers, complicated motion of of dynamic dynamic systems. systems. But But only only with with the the advent of computers, 22

solar system, the motion traveled around the earth in month, the earth around the In the solar 11.867 years. Such systems with sun in about a year, and Jupiter around the sun in about 11.867 quasiperiodic. multiple incommensurable periods are known as quasiperiodic.

4

1. DIFFERENCE INECONOMICS ECONOMICS DIFFERENCE EQUATIONS [N

capable of of displaying displaying graphics, scientists been seethat that with screens capable graphics, have have scientists been able able to to see demonstrated in this dynamic systems systems have many nonlinear dynamic have chaotic chaotic solutions. As As demonstrated book, nonlinear dynamical dynamical theory theory has has found found wide applications applications in different different fields fieldsof of economics.3 The ascatastrophes, catastrophes, economics.3 The range range of of applications applications includes includes many many topics, such as bifurcations, trade cycles, economic chaos, urban pattern pattern formation, formation, sexual sexual division division biftircations, of labor and economic development, economic economic growth, growth, values valuesand andfamily family structure, structure, ofstochastic stochastic noise noise upon upon socio-economic socio-economic structures, structures, fast fast and andslow slow sociosociothe role role of the economic processes, and relationship between microscopic and macroscopic economic processes, and relationship between microscopic and macroscopic beeffectively effectively examined examined by by traditional traditional analytical analytical structures. cannot be structures. All All these these topics topics cannot methods which are concerned with linearity, stability and static equilibrium points. methods which are concerned with linearity, stability and static equilibrium points. Nonlinear dynamical theory has changed economists' views about evolution. Nonlinear dynamical theory has changed economists' views about evolution. For For instance, relations between laws and andconsequences consequences -instance, the the traditional traditional view view of of the the relations between laws between cause and and effect effect -- holds holds that that simple simple rules rules imply imply simple simple behavior, behavior, therefore therefore between cause complicated behavior must arise from complicated rules. This vision had been held complicated behavior must arise from complicated rules. This vision had been held by professional economists for a long time. But it has been recently challenged due by professional economists for a long time. But it has been recently challenged due the development of nonlinear theory. Nonlinear theory shows how complicated to to the development of nonlinear theory. Nonlinear theory shows how complicated behavior simple rules. illustrate this idea, we weconsider consider the behavior may may arise arise from from simple rules. To To illustrate this idea, the following model following model x(t + l) - x(t) . ix -^ x(t) fr—y-L=a-\-ax{t), a>O. a>0. x{t) is a linear function function of the x{t). We We may may rewrite rewrite the The growth rate is the state state variable variable x(t). the above equation as asfollows follows

x(t + l) = ca(t\l —- x(t)), x{t)), a>0. + i) a> 0. = ax(tXi This is is the thewell-studied well-studied logistical map. This seemingly seemingly simple simple map map exhibits exhibits very very For instance, instance, figure figure 1.1.1 1.1.1 depicts depicts complicated complicated behavior as as we will analyze later on. For chaotic behavior of of the difference difference equation with a given parameter value and initial condition. chaos implies implies that no one onecan canprecisely precisely know know what what will will existence of chaos The existence that no happen in society society in in the fttture, future, except in some bounded happen in except that that it will will be be changing in To illustrate illustrate why no one onecan canprecisely precisely foresee foresee the theconsequences consequences of of the the area. area. To why no intervention policy, find out outwhat what happen happen to tothe thechaotic chaotic system system intervention policy,let let us us try to fmd starts from from two two different different but very very near states. states. In In figure figure 1.1.2, 1.1.2, we when when it starts simulate the 5.75. Let us consider two twocases cases of ofx0 x0 = = 0.400 and us consider and simulate the case case of aa ==5.75. 3

For applications of of nonlinear nonlinear theory to toeconomics, economics, see seeDendrinos Dendrinos and andSonis Sonis(1990), (1990), For applications Rosser (1991), (1991), Zhang Zhang (1991, (1991, 2005a), 2005a), Lorenz Lorenz (1993), (1993), Azariadis Azariadis (1993), (1993), Puu Puu(1989), (1989), Rosser Ferguson and Lim(1998), Flasehel Flaschel et al(1997), (1997), Chiarella Chiarella and andFlaschel Flaschel (2000), (2000), and Ferguson and Lim et a! and Shone (2002).

1.1. DIFFERENCE EQUATIONS AND ECONOMIC ANALYSIS

55

over 100 years. It can be seen that the two behaviors behaviors are varied varied over over xx0 = 0.405 over 0 = time. time.

xt 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

20 20

40 40

60 60

80 80

100 100

120 120

140 140

t

Chaoswhen when a = 5.75 Figure 1.1.1: 1.1.1: Chaos 5.75 and x0 x0 == 0.4

xt [0.400] 1

xt [0.405] 1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2 20 20

40 40

60 60

= 00.400 .400 (a) x0 =

80 80

t

100 100 t

20 20

40 40

60 60

80 80

t 100 100 t

(b) x0 = = 00.405 .405

dynamics with with different different initial initialconditions, conditions, aa = 5.75 5.75 Figure 1.1.2: The dynamics

.405] between the path We calculate calculate the the difference difference x[t, 0.400] 0.400]— - x[t, x[t, 00.405] between the path We x0 = 0.405 over 100 years as in figure 1.1.3. 1.1.3. started x0 = 0.400 and the one at x0 started at x0

1. DIFFERENCE DIFFERENCE EQUATIONS EQUATIONS IN ECONOMICS

6

[t, - At, x[t , 00.400] .400] − x[t , 00.405] .405]

11

0.5 0.5

20

40

60

80

100 t

-0.5 -0.5

-1 Figure 1.1.3: 1.1.3: Small differences differences at the beginning signify much

shown when the parameter small,the thedifference difference equation equation has has aa It can be shown parameter a isissmall, unique parameter aa exceeds exceeds aacertain certain value, value, the the steady steady unique equilibrium equilibrium point. point. As the parameter state ceases approached monotonically, state ceases being being approached monotonically, and and an an oscillatory oscillatory approach approach occurs. occurs. IfIf a is increased further, the steady state becomes unstable and repels nearby points. a is increased thrther, the steady state becomes unstable and repels nearby points. As a increases, one can find a value of a where the system possesses a cycle As a increases, one can find a value of a where the system possesses a cycle of of 1.1.4). Also, Also, there there exists exists an an uncountable uncountable period for arbitrary arbitrary kk (see (seefigure figure 1.1.4). period kk for number of initial conditions from which emanate trajectories that fluctuate in a number of initial conditions from which emanate trajectories that fluctuate in a bounded and aperiodic fashion and are indistinguishable from a realization of some bounded and aperiodic fashion and are indistinguishable from a realization of some stochastic (chaotic) process. stochastic (chaotic) process. Nonlinear dynamical dynamical systems sufficient to determine the Nonlinear systems are are sufficient to determine the behavior behavior in the sense that solutions of the equations do exist, it is frequently difficult difficult to figure figure out be. ItIt is is often often impossible impossible to explicitly write down solutions solutions in in what behavior would be. algebraic expressions. Nonlinear algebraic expressions. Nonlinear economics economics based based on nonlinear dynamical theory attempts provide a new vision vision of economic economic dynamics: dynamics: aa vision vision toward toward the attempts to to provide the unpredictable, and multiple, the temporal, the unpredictable, and the complex. complex. There There is a tendency to replace simplicity with complexity and specialism with generality in economic economic replace simplicity with complexity and specialism with generality research. concepts such such as astotality, totality,nonlinearity, nonlinearity, self-organization, self-organization, structural structural research. The concepts changes, order and chaos have found found broad broad and and new new meanings meanings by by the the development development of this this new new science. science. According According to this this new new science, science, economic economic dynamics dynamics are are considered to resemble aa turbulent turbulent movement liquid in which which varied varied and and considered to resemble movement of of liquid relatively stable forms of of current current and and whirlpools whirlpools constantly constantly change change one one another. another. relatively stable These changes consist consist of dynamic dynamic processes processes of self-organization self-organization along These along with with the

DIFFERENCE EQUATIONS AND ECONOMIC ANALYSIS 1.1. DIFFERENCE

7

spontaneous formation formation of of increasingly subtle and complicated spontaneous increasingly subtle complicated structures. structures. The The accidental nature and and the the presence accidental nature presence of structural structural changes changes like catastrophes catastrophes and and biflircations, which are are characteristic of nonlinear bifurcations, which characteristic of nonlinear systems systems and whose whose further further trajectory is is determined by chance, chance, make make dynamics dynamics irreversible. irreversible.

x 1

0.8

0.6

0.4 0.4

0.2 0.2

0 2

2.5 2.5

3

3.5 3.5

4

4.5 4.5

5

5.5 5.5

a

The map mapof ofbifurcations biftircationsfor foraaec [2,5.75] Figure 1.1.4: The

1.2

Overview

This book presents the mathematical theory in linear mathematical theory linear and and nonlinear nonlinear difference difference equations and and its its applications applications to to many many fields fields of of economics. The book book is for equations economics. The for economists concerned to and economists and and scientists scientists of of other other disciplines disciplines who who are are concerned to model and understand the to understand the time time evolution evolution of of economic economicsystems. systems.ItItisis of of potential potential interest to advanced graduate students students in economics, economic economic in economics, advanced undergraduates undergraduates and and graduate professionals, professionals, applied applied mathematicians mathematicianswho who are are interested interested in in social social sciences, as well as researchers in social sciences. The book is organized as follows. follows. Chapter 2.1 deals with linear Chapter 22 is organized organized as follows. follows. Section Section 2.1 linear first-order first-order difference equations. We also examine dynamics of two economic models, a price difference equations. We also examine dynamics of two economic dynamic model amortization. In dynamic model with with adaptation adaptation and and aa model model of amortization. In section section 2.2, 2.2, we introduce dimensional introduce some some basic basic concepts concepts which which will will be be used used not not only only for one dimensional problems problems but but also also for for higher higher dimensional dimensional ones. ones. Section Section 2.3 2.3 introduces introduces concepts concepts of of stability and stabilities of one-dimensional one-dimensional difference difference stability and some basic theorems about stabilities with equations. We apply these results results to the cobweb model and a dynamic model with

88

1. 1. DIFFERENCE EQUATIONS EQUATIONS IN IN ECONOMICS ECONOMICS

inventory. Section 2.4 is is concerned concerned with with conditions conditions for for stabilities stabilities of of nonhyperbolic nonhyperbolic equilibrium points. Section 2.5 deals with dissipative maps. Section 2.6 introduces introduces equilibrium difference equations and provides provides general general solutions solutions to the the system. system. difference equations of higher order and Section 2.7 examines difference difference equations equations of higher orders orders with with constant constant Section 2.7 examines of higher coefficients. In section 2.8, are concerned concerned with with limiting limiting behavior behavior of of linear linear coefficients. In section 2.8, we are difference equations. We examine examine limiting limiting behavior behavior of of the the Samuelson Samuelson multipliermultiplierdifference interaction model. model. accelerator interaction Chapter 33 applies Chapter applies the concepts concepts and theorems theorems of the the previous previous chapters chapters to analyze different different models in economics. economics. Although the economic economic relations in these these models are are complicated, complicated, we dynamics of all all these these models models are are models we show show that that the dynamics determined one-dimensional difference determined by by one-dimensional difference equations. equations.Section Section3.1 3.1 examines examines aa interactions between between inflation inflation and and unemployment. unemployment. The The model model traditional model of interactions built on on the the expectations-augmented expectations-augmented version Phillips relation is built version of of the the Phillips relation and and the adaptive adaptive expectations expectationshypothesis. hypothesis.WeWesolve solvethe themodel modeland and show show that that the characteristic characteristic equation equation may may have have either: either: (1) (1) distinct distinct real real roots; roots; or or (2) repeated real (3) complex complex roots. roots. Section Section 3.2 3.2 introduces introduces the the one-sector one-sector growth growth real roots; or (3) model. different from growth models in the literature literature model. The model model is different from most of the growth in it treats saving as as an an endogenous endogenous variable variable through through introducing introducing wealth wealth into into in that that it treats saving utility function. We demonstrate that unique stable stable utility function. We demonstrate that the the OSG OSG model model has has aa unique equilibrium. section 3.2. 3.2. equilibrium. Section Section 3.3 3.3 generalizes generalizes the the OSG OSO model model proposed proposed in in section Section 3.4 3.4 deals deals with withthe theoverlapping-generation overlapping-generation (OLG) (OLG)model model—- one one of of the the Section most among economists. most popular popular models models among economists. The The model model is is essential essential for for the the reader reader to to approach some of the models in this book as well. Different from the OSG approach some of the models in this book as well. Different from the OSG models in in the the previous previous two two sections, sections, ininthe theOLG OLGanalytical analytical framework, framework, each each models person lives for only two periods. This is the main shortcoming of the model; person lives for only two periods. This is the main shortcoming of the model; nevertheless, its its popularity popularity is sustained partly because this this framework framework often often is sustained partly because nevertheless, simplifies complicated analytical issues. Section 3.5 introduces a growth model to simplifies complicated analytical issues. Section 3.5 introduces a growth model to demonstrate persistence of inequality. In this model, the evolution of income demonstrate persistence of inequality. In this model, the evolution of income within each each dynasty dynasty in in society society is is governed governed by by aa dynamical dynamical system system that generates within that generates aa poverty poverty trap trap equilibrium equilibrium point point along along with with aahigh-income high-income equilibrium. equilibrium. Poor Poor dynasties, those with the income at the threshold level, converge to a low dynasties, those with the income at the threshold level, converge to a low income steady state, whereas dynasties with income above the threshold income steady state, whereas dynasties with income above the threshold converge high income income steady. steady. Section Section 3.6 3.6 studies studies aa model model to to provide provide converge to to aa high insights into evolutionary processes of Schumpeterian creative destruction. insights into evolutionary processes of Schumpeterian creative destruction. Section 3.7 is concerned with interactions among human capital accumulation, Section 3.7 is concerned with interactions among human capital accumulation, economic growth, and inequality. The model exhibits three possible equilibrium economic growth, and inequality. The model exhibits three possible equilibrium points: a low-growth trap, a pair of equilibrium points in the intermediate and points: a low-growth trap, a pair of equilibrium points in the intermediate and advanced development phase. If these equilibrium points exist, it can be shown that advanced development phase. If these equilibrium points exist, it can be shown that the poverty trap is stable, while in development phase, the first equilibrium point the poverty trap is stable, while in development phase, the first equilibrium point will be unstable and the second one stable. Section 3.8 studies an urban dynamic will be unstable and the second one stable. Section 3.8 studies an urban dynamic model to highlight how the trade-off between optimal and equilibrium city sizes model to highlight how the trade-off between optimal and equilibrium city sizes behaves when human capital externalities are introduced into urban dynamics. behaves human capital externalities into urban The dynamics. Section when 3.9 introduces a growth model are of introduced monetary economy. model Section 3.9 introduces a growth model of monetary economy. The model addresses the Tobin effect and the existence of monetary economy. Section 3.10 addresses the Tobin effect and the existence of monetary economy. Section 3.10

1.2. OVERVIEW

99

introduces endogenous time distribution between leisure and work into the OSG model. Chapter 4 examines examines periodic, periodic, aperiodic, chaotic solutions solutions of scalar scalar systems. systems. Chapter defines concepts concepts such as periodic or or aperiodic aperiodic solutions solutions (orbits). This Section 4.1 defmes introduces some some techniques techniques to to fmd find periodic periodic solutions solutions and and provides provides section also introduces stability of periodic solutions. Section 4.2 is concerned conditions for judging stability concerned with period-doubling bifurcations. section introduces introduces concepts concepts such as as branch, branch, period-doubling bifurcations. This This section bifurcation values, period-doubling period-doubling bifurcation bifurcation bifurcation route route to to chaos, chaos, Myrberg's Myrberg's number and Fiegenbaum's Fiegenbaum's number. number. Section Section 4.3 4.3 deals deals with with aperiodic aperiodic orbits. orbits. This This section section introduces the Li-Yorke theorem and the Sharkovsky theorem, which are introduces the Li-Yorke theorem and the Sharkovsky theorem, which are important for proving existence of chaos in scalar systems. Section 4.4 studies important for proving existence of chaos scalar systems. Section 4.4 studies some typical types of of biflircations. bifurcations. They include supercritical supercritical fold, fold, subcritical subcritical fold, fold, supercritical pitchfork, subcritical pitchfork, transcritical bifurcations. Section pitchfork, pitchfork, transcritical biflircations. Section 4.5 introduces theory of Liapunov numbers. In this this section, section, we we also also examine examine behavior behavior introduces theory of Liapunov numbers. In of a model of labor market. In section 4.6, we study chaos. We simulate a demand of a model of labor market. In section 4.6, we study chaos. We simulate a demand and supply model to demonstrate chaotic behavior. and supply model to demonstrate chaotic behavior. applies concepts concepts and theorems of the the previous previous chapters chapters to analyze analyze Chapter 5 applies different economics. The models in this this chapter chapter exhibits exhibits periodic, periodic, different models models in in economics. The models aperiodic, or chaotic behavior. Section 5.1 studies studies aa model model of of endogenous endogenous business business cycles in the presence of of knowledge knowledge spillovers. spillovers. Many economic economic indicators, indicators, such such as as GDP, exhibit asymmetry asymmetry as ifif they they repeatedly repeatedly switch switch between between different different regimes. regimes. For instance, instance, it has has been been found found that that (i) (i) positive positive shocks shocks are are more more persistent persistent than than negative United States States and France; France; (ii) negative negative shocks shocks are more negative shocks shocks in the United persistent persistent than negative shocks in the United United Kingdom Kingdom and and Canada; Canada; and (iii) (iii) there is almost no asymmetry asymmetry in in persistence persistence in Italy, Japan, and (former) (former) Germany. The model in this section section provides some insights into well-observed asymmetric nature of business business cycles. cycles. Section Section 5.2 cobweb model model with normal normal 5.2 studies studies aa nonlinear nonlinear cobweb demand and supply, supply, naïve naive expectations expectations and and adaptive adaptive production production adjustment. adjustment. The The model exhibits a horseshoe. Section 5.3 examines an inventory model with model exhibits a horseshoe. Section 5.3 examines an inventory model with rational expectations. expectations. In In section section 5.4, 5.4, we we discuss discuss an an economic economic growth growth model model with with rational pollution. The model is an extension of the standard neoclassical growth model pollution. The model is an extension of the standard neoclassical growth model which has has aa unique unique stable stable equilibrium equilibrium point. point. Chaos Chaos exists exists in in the the model model because because which of the effects of pollution upon production. It is know that the neoclassical of the effects of pollution upon production. It is know that the neoclassical growth based on on the the Solow Solow growth growth model model focuses focuses accumulation accumulation as an growth theory theory based as an engine of growth, while the neo-Schumpeterian growth theory stresses engine of growth, while the neo-Schumpeterian growth theory stresses innovation. Section Section 5.5 5.5 studies studies aa model model to to capture capture these these two two mechanisms mechanisms within within innovation. the same framework. The model generates an unstable balanced growth path and the same framework. The model generates an unstable balanced growth path and the economy achieves sustainable growth cycles, moving back and forth the economy achieves sustainable growth cycles, moving back and forth between the the two twophases phases— - one one is is characterized characterized by by higher higher output output growth, growth, higher higher between investment, no innovation, and a competitive market structure; the other by investment, no innovation, and a competitive market structure; the other by lower output growth, lower investment, high innovation, and a more lower output growth, lower investment, high innovation, and a more monopolistic structure. structure. Section Section 5.6 5.6 identifies identifies economic economic fluctuations fluctuations in in aa monopolistic monetary economy within the OLG framework. Section 5.7 shows chaos in aa monetary economy within the OLG framework. Section 5.7 shows chaos in model of interaction of economic and population growth. model of interaction of economic and population growth.

10

DIFFERENCE EQUATIONS IN ECONOMICS 1. DIFFERENCE

is organized organized as as follows. follows. Section 6.1 studies phase space analysis of Chapter 6 is planar linear difference difference equations. equations. This planar This section section depicts depicts dynamic dynamic behavior behavior of the characteristic equation equation has two distinct distinct eigenvalues, or repeated repeated system when the characteristic eigenvalues, or complex complex conjugate conjugate eigenvalues. eigenvalues. Section 6.2 6.2 studies studies autonomous autonomous eigenvalues, difference equations. equations. This This section section provides finding general general linear difference provides a procedure of fmding solutions of the the system. system. Section Section 6.3 6.3 studies studiesnonautonomous nonautonomous linear linear difference difference solutions equations. This section section provides provides a procedure procedure of finding finding general general solutions solutions of of the the equations. system. We also examine a few models of economic dynamics. They include system. We examine few models economic dynamics. They include a dynamic input-output dynamic input-output model model with withtime time lag lag in in production, production,aacobweb cobweb model model in two interrelated markets, a duopoly model, a model of oligopoly with 3 interrelated markets, a duopoly model, a model of oligopoly with 3 firms, firms, and and aa model of international trade between two countries. This section also shows how model of international trade between two countries. This section also shows how the one-dimensional difference equation of higher order can be expressed in multithe one-dimensional difference equation of higher order can be expressed in multidimensional equations order. Section Section 6.4 stabilities and dimensional equations of of first first order. 6.4 defines defines concepts concepts of of stabilities and relations among these concepts. This section also provides conditions for stability stability relations among these concepts. This section also provides conditions for or instability of of difference difference equations. equations. Section Section 6.5 6.5 studies studies Liapunov's Liapunov's second second or instability method. The theory of Liapunov fianctions is a global approach toward determining method. The theory of Liapunov functions is a global approach toward detennining asymptotic asymptotic behavior behavior of of solutions. solutions. Section Section 6.6 6.6 studies studies the the theory theory of of linearization linearization of of difference equations. There are two possible ways to simplify dynamical systems: difference equations. There are two possible ways to simpli& dynamical systems: one is is to to transform transform one one complex complex system system to to another another one one which which is is much much easier easier to to one analyze; and the other is to reduce higher dimensional problems to lower ones. The analyze; and the other is to reduce higher dimensional problems to lower ones. The center manifold manifold theorem to reduce reduce dimensions dimensions of of dynamical dynamical problems. problems. center theorem helps helps us us to Section 6.7 defines the concept of conjugacy and shows how to apply the center Section 6.7 defines the concept of conjugacy and shows how to apply the center manifold theorem. Section 6.8 studies the Henon map, demonstrating bifurcations manifold theorem. Section 6.8 studies the Hénon map, demonstrating biflircations and chaos of the planar difference equations. Section 6.9 studies the Neimarkand chaos of the planar difference equations. Section 6.9 studies the NeimarkSacker (Hopf) bifurcation. This section identifies the Hopf bifurcation in the Sacker (Hopf) bifurcation. This section identifies the Hopf bifUrcation in the discrete Kaldor model. Section 6.10 introduces the Liapunov numbers and discrete Kaldor model. Section 6.10 introduces the Liapunov numbers and discusses chaos for planar dynamical systems.

discusses chaos for planar dynamical systems. Chapter 7 applies Chapter applies the concepts concepts and theorems theorems of the the previous previous chapter chapter to to examining different economic economic systems. systems. Section 7.1 7.1 studies examining behavior behavior of different Dornbush's show how aa monetary monetary expansion expansion will Dornbush's exchange exchange rate rate model. model. We We show result immediate depreciation depreciation of the currency and sustain sustain the inflation inflation as as result in an immediate the price level level gradually gradually adjusts adjusts upward. upward. Section Section 7.2 7.2 studies studies aa two-sector two-sector OLG OLG model with the the Leontief Leontief production production functions. functions. The economy produces produces two, two, model with The economy consumption and investment, goods; it has two, consumption and investment consumption and investment, goods; two, consumption and investment sectors. We provide conditions when the system is determinate or indeterminate. indeterminate. Section 7.3 7.3 introduces introduces aa one-sector one-sector real business cycle model with mild increasing increasing returns-to-scale with government spending. Section 7.4 introduces endogenous introduces endogenous returns-to-scale government spending. Section fertility and age support support into into the OLG model. model. Section Section 7.5 examines aa model model to to fertility and old old age the OLG 7.5 examines capture the historical evolution of population, technology, and output. The capture the historical evolution of population, technology, and output. The economy evolves evolves three three regimes regimes that that have havecharacterized characterized economic economic development: development: economy from a Malthusian regime (where technological progress is slow and population from a Malthusian regime (where technological progress is slow and population growth prevents any sustained rise in income per capita) into a post-Malthusian growth prevents any sustained rise in income per capita) into a post-Malthusian regime (where (where technological technological progress progress rises rises and and population population growth growth absorbs absorbs only only part part regime

1.2. OVERVIEW

11 11

modern growth growth regime regime (where (where population population growth growth is is reduced reduced of output growth) to a modem is sustained). The model is defined within the OLG framework and income growth is single good good and and ititexhibits exhibits the thestructural structural patterns patterns observed observed over over history. history. with a single of unemployment and and inflation. inflation. We We demonstrate demonstrate that that Section 7.6 examines a model of is built built on on the the well-accepted well-accepted assumptions assumptions may may behave behave chaotically. chaotically. the model which is Section 7.7 provides provides aa model model of oflong-run long-run competitive competitive two-periodic two-periodic OLG OLG model model Section and capital. capital. Section Section 7.8 introduces introduces heterogeneous heterogeneous groups with money and groups to the OSG model. Section Section 7.9 7.9 examines examines interdependence interdependence between between economic economic growth growth and and model. human capital accumulation in the OSG modeling framework. human capital accumulation in the OSG modeling framework. concluding remarks remarks to this this book, book, we we address address two two important important issues, issues, which which As concluding been rarely rarely studied studied in in depth depthinineconomic economicdynamical dynamicalanalysis, analysis, changeable changeable have been issues is is essential essential speeds and economic structures. The understanding of these two issues only for for appreciating appreciating validity validity and limitations limitations of different different economic models in not only literature, but also also for developing general economic theories. We also include a the literature, mathematical appendix. shows how to to solve solve mathematical appendix. A.I A. I introduces introduces matrix matrix theory. theory. A.2 A.2 shows equations, based some linear equations, based on matrix theory. A.3 introduces metric spaces and some concepts and theorems theorems related related to metric metric spaces. spaces. A.4 defines defines some some basic basic basic concepts concepts in the study study of of functions functions and and states states the the implicit implicit function function theorem. theorem. A.5 A.5 concepts gives a general general expression expression of the the Taylor Taylor Expansion. Expansion. A.6 isis concerned concerned with with gives convexity of of sets sets and and functions functions and and concavity concavity of of functions. functions. A.7 A.7 shows shows how to solve solve convexity how to unconstrained maximization we introduce introduce conditions conditions for for unconstrained maximizationproblems. problems.InIn A.8, A.8, we constrained maximization. A.9 introduces theory of dynamic optimization. constrained maximization. A.9 introduces theory of dynamic optimization.

Chapter 22 Chapter

linear difference difference equations Scalar linear In discrete discrete dynamics, discrete variable variable so that dynamics, time, time, denoted denoted by by /, t, is taken taken to be a discrete the variable variable t/ is is allowed allowed to take only integer values. For example, certain to only integer values. example, if a certain (t + l)st + l) population has discrete generations, the size of the (t + 1) 1) is a st generation generation x(t + function of the / th generation x(t). Different from the continuous-time dynamics fUnction of the t th generation x(t). Different from the continuous-time dynamics where pattern of change of variable xx isis embodied embodied in in its its derivatives derivatives with with where the the pattern of change of aa variable respect to the change of time t which is infinitesimal in magnitude, in the discrete respect to the change of time t which is infmitesimal in magnitude, in the discrete dynamics the pattern of change of variable xx isisdescribed described by by"differences", "differences", rather rather dynamics the pattern of change of variable than by derivatives of x. Hence, the system in discrete time is in the form than by derivatives of x. Hence, the system in discrete time is in the form of of difference equations, equations, rather differential equations. the value value of of variable variable difference rather than than differential equations. As As the x(t) will will change change only only when when the variable variable tt changes changes from one integer value to the the next, such such as as from from 2004 2004 toto2005 2005during during which which nothing nothing isis supposed supposed to to happen happen to to next, x(t), in difference equation theory the variable t is referred to as period, in the x(t), in difference equation theory the variable t is referred to as period, in the analytical Thetime timeinterval interval between between analytical sense, sense, not not necessarily necessarily in in the the calendar calendar sense. sense. The successive states is usually usually suggested suggested by process itself. For example, example, two successive by the real process itself For x(t l) could be separated x(/j by by one onehour, hour, one oneday, day, one one week, week, one one month, month, x(t + i) separated from x(t) etc. introduce the To describe the pattern pattern of of change change in in xx as a function describe the function of t,t, we introduce difference / At. As As t has difference quotient Ax Ax/At. has to take integer integer values, values, we choose choose At = 1. difference quotient quotient Ax is Hence, the difference AxI/AtAtisis simplified simplified to to the the expression expression Ax; Ar; this is called the first difference difference of of x, and and is is denoted by

Ax(t)_x(t +

i)— x(t),

13 13

14

2. SCALAR LINEAR DIFFERENCE EQUATIONS

where x(t) x(t) is the value of x in + i) is its value where value of in the the tt th th period period and and x(t x(t +1) value in the the period immediately following during two two period immediately followingthe thet tthth period. period. The The change change in in x during consecutive time affected by many factors. factors. We may may express express the the consecutive time periods periods may may be affected of x by, pattern of change of by,for for instance instance a — + l)=-ax Ax(t + (t).

Equations of of this equations.By By the the way, way, we this type type are are called called d(fference difference equations. we use use

difference equations equations and discrete discrete dynamical dynamical systems systems exchangeable.' exchangeable.' There There are are difference of difference, difference, which which are are equivalent equivalent to the above equation, equation, for for instance instance other forms of

x(t + i) — x(t) = — or or

x(t + i) = x(t) —

f(x(t)).

given by by the the sequence sequence The evolution evolution of of the the system system starting from from x0 x0 isis given The x0 (= x(O)), x(1) = *(i)=/(*„* x(2) == f(x(l)) ,(2) /(*(!)) == f(f(x0 f(f(xQ)), )), = f(x(2)) x(3) = f(ff((x0)))... 2 usuallywrite write ff2 (9, f3 We usually (x), / 3 (x),

ininplace placeofof

f(f(x)), f(ff((x))), Hence, we have

x(t + i) = f(x(t)) =

1

When When mathematicians mathematicians talk the talk about about difference difference equations, equations,they they usually usually refer refer to to the analytical analytical theory theory of the the subject, subject, and and when when they they talk talkabout aboutdiscrete discretedynamical dynamicalsystems, systems, they generally refer refer to to its its geometrical geometrical and and topological topological aspects. aspects.

2.1. LINEAR FIRST-ORDER DIFFERENCE EQUATIONS 2.1. EQUATIONS

15

f(x0) iterateof of xx0 under/f;; ftf'(x (x0) f(x is called called the Hie first iterate called the the tti/i th iterate iterate of of 0) is 0 under 0) isiscalled under f. xx0 / . The Theset setofofall all(positive) (positive) iterates iterates 0 under

{ft(x): t

o}

called the the (positive) (positive)orbit orbitof of xx0 and is is denoted denotedby by 0(x0). x0 is called O(x0). where f° (x0) (x0) = x0 0 and If the equation equation /f isisreplaced replaced by by aafunction function gg ofoftwo twovariables, variables, that that is, is,

g: R x+Z, where Z + is the set of nonnegative integersand andRR is is the set of real g:RxZ nonnegative integers numbers, then we have

= g(x(t), t). x(t + i)\)=g(x(t),t). This equation is called nonautonomous nonautonomousorortime-variant, time-variant,whereas whereas

x(t + i) =l)=f(x(t)), is time-invariant. is called autonomous or time-invariant.

This chapter is organized organized as follows. Section 2.1 deals with linear linear first-order first-order difference dynamics of two economic models, a price difference equations. We also examine dynamics dynamic model with with adaptation and a model In section dynamic model adaptation and model of amortization. amortization. In section 2.2, we introduce concepts, which not only only for for one-dimensional one-dimensional introduce some some basic basic concepts, which will will be used not problems but also for higher dimensional dimensional ones. problems ones. Section Section 2.3 2.3 introduces introduces concepts concepts of of stability basic theorems theorems about about stabilities stabilities of of one-dimensional one-dimensional difference difference stability and and some basic equations. We also apply these results results to the cobweb model model and a dynamic model with inventory. inventory. Section Section 2.4 2.4 is concerned concerned with with conditions conditions for stabilities of for stabilities nonhyperbolic equilibrium points. Section 2.5 deals with dissipative dissipative maps. maps. Section Section 2.6 introduces introduces difference difference equations of higher order order and and provides provides general general solutions solutions to the the system. system. Section Section 2.7 2.7 examines examines difference difference equations equations of higher higher orders orders with with constant coefficients. In section 2.8, we study limiting behavior of linear difference constant coefficients. In section 2.8, we study limiting behavior of linear difference equations. examine limiting limiting behavior of the the Samuelson Samuelson multiplier-accelerator multiplier-accelerator equations. We We examine behavior of interaction model. interaction model.

Linear first-order difference 2.1 Linear difference equations A typical linear homogenous first-order equation is is given by

x(t x(t0) x(t + i) l ) = a(t)x(t), a{t)x{t), x(t0) = = x0, x 0 , tt >t0 t o >0,0 ,

(2.1.1) (2.1.1)

2. 2.SCALAR LiNEAR LINEAR DIFFERENCE EQUATIONS EQUATIONS

16

where a(t) 0. One One may by aa simple where a{t) * 0. mayobtain obtain the thesolution solution of of equation equation (2.1.1) (2.1.1) by simple iteration x(t0 x{t 0 +

i)+l)=a{t = o)xo,

2) = + l)x{t 1)x(t0 x(t0 = afr0 + i) = a{t a(t00 + + 2) a{t0 + l) = + 1)a(t0)x0, l)a(to)xo, x(t 0 + 0 +

x(t0 2) = + 2)a(t0 x{t 3) = + 2)x(t0 2)x{t0 + + 2) = a(t0 a{t0 + 2)a(t0 + + i)a(t0)x0. l)a(to)xo. a(t00 + 0 + 3) = a{t

And inductively, inductively, itit is easy to see that

x(t) = x(10 o++t-t t —ot0) = [fiao)]xo. x{t)=x{t ) =

(2.1.2)

1=10

Example There There are are 2n as p(n), p(n), to 2n people. people. Find Find the thenumber number of of ways, denoted denoted as

group these these people into into pairs. To group 2n a 2«people peopleinto intopairs, pairs, we we first firstselect select aaperson person and and find find that that person person a partner. Since the partner can be taken to be any of the other 2n — 1 persons in the partner. Since the partner can be taken to be any of the other 2n — \ persons in the original 2«——1 1ways waystotoform formthis thisfirst firstgroup. group.We Weare areleft leftwith withthe the original group, group, there there are are2n problem persons into the number of ofgrouping grouping the theremaining remaining 2n 2«—- 22 persons into pairs, pairs, and and the number of problem of ways ways of of doing doing this this is isp(n p(n—- i). l). We have

p(n)=(2n—1)p(n—1). p(n) = {2n-\)p{n-\). Since two two people people can can be be paired only one way, 1. To Since way, we wehave have p(1) p(i) == 1. Toapply apply as formula (2.1.2), we rewrite the above formula as

p(n+1)=(2n+1)p(n). According According to to formula formula (2.1.2), (2.1.2), we we have have

p(n) = [f\(2i + l)jp(l) = (2» - 1)(2» - 3) P(n)[fl(2i+l)JP(1)(2n1)(2n3)."1 The nonhomogeneous nonhomogeneous first-order equation associated associated with with equation equation first-order linear equation (2.1.1) is isgiven by a{t)x{t) + + g(t), g{t\ 4x(t + li)) == a(t)xfr)

x(t0) 0.. x{t0) = = x0, x 0 , t > t to o > O

(2.13) (2.1.3)

LINEAR FIRST-ORDER DIFFERENCE EQUATIONS EQUATIONS 2.1. LiNEAR

17 17

equation (2.1.3) (2.1.3) is is found found as as follows follows The unique solution to equation a{to)x + g{t()), x{to+l) )x0 + i) == a(t0 o + x{t0 + x(t0

2) a{t0 ++ l)x(t0 l)x{t0 + l) g{t0 + + i) l) 2)== afr0 i)++ g(t0 l)a{to)x + a{t0 + l)g{to)++ g(t + 1)gfr0) + l). i). )x0 = a(t0 ++1)a(t0 g(t00 + o +

Inductively, it can be shown shown that the the solution solution is

x(t) =

[fiao)]xo 1=10

(2.1.4)

+ r=(0 r=10

\_i=r+\ i=r+1

Example Solve the equation

)\, x(0)=1, JC(0) = 1, t>O. t>0. x(t+1)=(t+1)x(t)+2'(t+l)!, (2.1.4), we have By formula (2.1.4),

= [11(1 +i)]x0 + 1=0

i=0 I /=*+! k=0 i=k+1

t! + + 1)12'(k + I)! =

= =2't\ k=0

independent of A special case of equation equation (2.1.3) (2.1.3)isisthat that a(t) a(t) is independent of t x{t + l ) = ax{t) + g{t), x(t0)=x0, x(t0) = x 0 , ttt0 >to>O. x(t+l)=ax(t)+g(t), According to to equation equation (2.1.4), (2.1.4),the the solution solution to to equation equation (2.1.5) (2.1.5) isis According

x(t) = a'x0 +

where we set t0 = = 0. 0. Example Find Findaasolution solution to to the the equation equation Example x(t + i) l) == 2x(t) ++ 3', x(l) = 0.5. x(t +

(2.1.5)

2. 2. SCALAR LINEAR DIFFERENCE EQUATIONS EQUATIONS

18

The solution is is given by by

x(t) == f - V 1++ f;2'-i"13*==31 3'—-5•5 -2'"1. x{t) k=i

Example @rice There are are two (pricedynamics dynamics with with adaptive adaptive expectations) expectations) There twofinancial financial assets available to investors; a riskless riskiess bank bank deposit deposit yielding yielding aa constant constantrate rate r in in perpetuity, and a common share, that that is, an equity perpetuity, common share, equity claim claim on on some firm, which pays out aa known per share, share, {^(s)}^. {d(s)}L. Let Let p(s) p(s) be known stream stream of dividends dividends per be the pays actual market price of a common common share share at the the beginning beginning of of period period s, before the market price before the dividend d(s)> 0 is paid. Suppose also that the future share prices are unknown d(s) > 0 is paid. Suppose also that the future share prices are unknown have the common common belief belief at at /t = ss that the but that all all investors investors have the price is is going to be pe pe(s ++1) l) at atthe the beginning beginning of ofthe following following period. We consider the following arbitrage condition

(i + r)p(t)=d{t)+p'(t r)p(t) = d(t) + pe(t + + l). 1). (l This equation means that that if a monetary monetary sum sum of p(t) p(t)dollars dollars were were invested invested in inthe the stock market at time t, it should yield at t + 1 an amount whose expected value at t, yield at + 1 an amount

d(t)+ pe(t + + l) i) d(t)+p'{t equals the The theprincipal principal plus interest interest on onan anequal equal sun suninvested invested in inbank bank deposits. deposits. The by adaptive expectation hypothesis is is described by

pe(t = ap(t) + (i — a)pe(t), p'(t ++ i)l)=ap(t)+(l-a)p'(t\

where the speed where the theparameter, parameter, aaee[o, [0,iJ, l], describes describes the speed of learning. learning. Using Using the arbitrage condition to eliminate eliminate expected expected prices the adaptive adaptive expectation expectation arbitrage condition prices from from the equation yields

p(t + i) = where

ApQ) +

2.1. 2.1. LiNEAR LINEAR FIRST-ORDER FIRST-ORDER DIFFERENCE DIFFERENCE EQUATIONS EQUATIONS

K

19

d(t+1)—(1—a)d(t)

1—a , 'l l+r—a + r-a

l1+r—a + r-a

We haveX2ee (o, i) if r >0 We have (0, l) > 0and andaea e(o,i). (0, l).The Thesolution solution of of

p(t+1)=Ap(t)+b(t) is given by

-

p(t) = 2'p(0) +

are constant. constant. Another special special case case ofequation of equation(2.1.3) (2.1.3)isisthat thatboth botha(t) a(t)and and g(t) are That is

x(t x{t ++ 1) \ ) ==a ax(t) x { t ) ++b,b , Xx(t0) ( / 0 ) = x0, JC0> t > t t0o > O0..

(2.1.6)

Using formula (2.1.4), (2.1.4), we we solve solve equation equation (2.1.6) (2.1.6) as as

x(t)= «(') =

a'x0+b a'xx o+b\o

a -\ 'a' — ififif a a*l, a-I) La—i)

[ I ) x0+b, x0 + b, if if a=1. a = 1.

(2.1.7)

Example Amortization Amortizationisisthe the process process by by which which aa loan loan isis repaid repaid by aa sequence sequence of periodic payments, each of which is is part part payment payment of of interest interest and and part part payment payment to to outstanding principal. principal. Let w(t) w(t) denote denote the theoutstanding outstanding principal principal after after reduce the outstanding

the t th payment payment m{t). m(t). Suppose that interest interest charges chargescompound compoundatatthe therate raterr per Suppose that payment period. The outstanding i) st payment is outstanding principal w(t w(t ++ i) l) after after the (t(t + l)st is equal to the the outstanding outstanding principal principal w(t) after the payment plus the the interest interest equal w(t) after the t th payment l)th period minus the that is is rw(t) incurred during the (t(t ++ 1)th the t th payment m(t), that w(t + w(t)++rw(t)— + l)= l) = w(t) rw(t) - m(t). m{t).

determinationofof w(t) w(t) is to solve Let w0 w0 stand for the initial initial debt. Then, Then, determination solve the following difference equation equation

2. SCALAR LINEAR DIFFERENCE DIFFERENCE EQUATIONS EQUATIONS

20

i) == (i(l + r)w(t) —- m(t), m(t), w(0) = w0. w(t + + l) w(0)=w o.

(2.1.8)

equation (2.15). (2.15). Applying the solution solution Equation (2.1.8) belongs to the type of equation (2.1.8), we get get of equation (2.1.5) to equation (2.1.8),

—

+

hi particular, then the the above above solution solution In particular, ifif the thepayment payment m{k) m(k) is constant, constant, say say M, then becomes

w(t)=(1+r)tw0 —((1+r)' loan is to be paid paid off w(t) = 00), ), the monthly monthly If the loan off in t payments (that is, w(t)

payment payment M isisgiven given by

1k!

=

(1 +

(1+r)' —1

rw0.

Example The TheLees Leesare are purchasing purchasing aa new new house house costing costing $200,000 $200,000 with with a down payment of $25,000 $25,000 and aa 30 30 -year -year mortgage. mortgage. Interest Interest on payment on the the unpaid unpaid balance balance of of mortgage is be compounded compounded at monthly rate rate of 1%, 1%, and and monthly monthly the mortgage is to be at the monthly payments paymentswill willbe be $1800. $1800. How How much muchwill will the the Lees Lees owe owe after after tt months of payments? this question, let x(t) x(t) denote in dollars that will will be To answer answer this question, let denote the balance balance in dollars that months payment. Then, as the previous example owed on the mortgage after t payment. the previous example owed mortgage after shows, we have 1.olx(t — i) — 1800, t > 11.. x(t) jc(/) = 1.01jc(f-l)-1800,

The The amount amount owed owed initially initially is the purchase purchase price minus minus the down down payment, payment, so so x(0) = 175,000. We thus solve 175,000.

i) = 180,000 -— sooo(i .01 .01' )i 75,000 x(t) = (i(l.Ol')l x(t) 75,000— - 180,000(1.01' 180,000(l.01' -— l)= 5OOo(l.Ol

2.1. 2.1. LINEAR FIRST-ORDER DIFFERENCE EQUATIONS

21 21

loan after 20 years (240 (240 months) months) of of payments payments For example, the balance of the loan is x(240) = 125,537. 125,537.

Exercise 2.1 2.1 1 Find Findthe thesolutions solutionsof ofthe thefollowing following difference difference equations: equations: x(t +l)—(t (a) *(/+ 1) - (/++l)x(t)=0, l)x(') = 0, x(0)=x0; x(0)=x0; 2l x{t) = 0, *(0) x0; (b) 1) — - ee2tx(t) (b) x{t xQ + i) x(0) = = x0; ((c) c)JC(/ +

l ) x(t+1)—

x

tx(t)=O, ( r ) 0 >

*(o) xo; x(0)=x0;

(d) x(t *(f + i) l) — - 0.5x(f) x(o) = ;to; 0.sx(t) = 2, x(0) (e)x(? ) ==e e', ' , x(0) x(0)=x (e) x(t + li)) —- ^x(t) = x0.0. 2 AA debt debtof of$12,000 $12,000isistotobe beamortized amortized by by equal equal payments payments of $380 at at the the end end of each month, plus aa final final partial each month, plus partial payment paymentone onemonth monthafter afterthe thelast last$380 $380 isis paid. paid. IfIf interest is at an annual rate of 12% compounded monthly, construct an interest is at an annual rate of 12% compounded monthly, construct an the required amortization schedule schedule to to show show the required payments. payments.

2.2 2.2

Some Concepts

Let us consider the one-dimensional difference equation

x{t f{x{t)) == f M { x 0 ) , i) = x(t + l) = f(x(t))

0,l,-, 1, ... t ==0,

(2.2.1)

When studying studying the the motion motion where ff::R^> R —* R is aa given given nonlinear nonlinear function function in x(t). When of difference difference equations, attempt to to determine equilibrium points periodic equations, we we attempt determine equilibrium points and periodic points, to analyze their stability and asymptotic stability, and to determine aperiodic points chaotic behavior. equation (2.2.1) scalar (or (or oneonepoints and and chaotic behavior. We We refer refer to to equation (2.2.1) as as a scalar dimensional) dynamical dynamical system. The function / isiscalled calledthe themap mapassociated associated with function f nat satisfies equation (2.2.1). A solution of equation equation (2.2.1) is a sequence sequence {(*,}7=o tthat equation (2.2.1). satisfies

given, the the equation for for all all itt = 0, 0,1, . If initial condition is given, the equation If an an initial condition x(o) x(0) = =x x00 is problem of solving equation (2.2.1) so that the solution satisfies the initial condition the initial initialvalue value problem. problem. The The general general solution to equation equation (2.2.1) (2.2.1) is is aa is called the sequence { $ } " „ that satisfies equation (2.2.1) for all t = 0 , 1 , a n d involves sequence that satisfies equation (2.2.1) for all t 0, 1, ... and involves a

22

EQUATIONS 2. SCALAR LINEAR DIFFERENCE EQUATIONS

thatcan canbe bedetermined determined once oncean aninitial initial value value isis prescribed. prescribed. A A particular particular constant C that 0, 1, . solution is a sequence {^r}^0 that satisfies equation (2.2.1) for all itt = = 0, 1, 2.2.1. The The sequence sequence Definition 2.2.1. {x0,

x,,

x1,

the system starting from is denoted is denoted by by O(x 0(x0) called the the orbit orbit or trajectory trajectory of the system starting 0) and is called Xs.

Definition 2.2.2. 2.2.2. A x*isiscalled called aa stationary point of of equation equation(2.2.1) (2.2.1) if if Definition A point point I stationarypoint x=f(x). I = f(x*).

(2.2.2)

Each x* can can be regarded regarded either either as thethedynamical Each x* as aa state stateofof dynamical system system

x{t

\)=f{x(t)\ =

x(t ++ 1)

satisfying equation (2.2.2) (2.2.2) or satisQying equation or as as aa solution solutionto tothe thesystem system of ofequation equation x=

*

=

f(x). /(*)

point of of f.f. We also afixed (or (or stationary stationary or or equilibrium) also call call xx ajIxed equilibrium) point Example Everystationary stationary state state of of the the system system Example Every

x(t+i)=axQ)(i—x(t)) x{t + l)=ax{t){l-x{t)) the equation equation must satisfy the

x = ax(l - x). x=ax(1—x). We see stationary state Another is a stationary state regardless regardless of ofthe the value value of of a. Another We seethat thatx*I == 00 is stationary point is given by

x=

a-\ a—i a

.

23

2.2. SOME CONCEPTS CONCEPTS For every wecan canvisualize visualize the the fixed points of every aa we

f{x) ax(l-x), f(x) ==ax(i —4, since they are given since given by bythe theintersection intersectionofofthe thegraph graph/ f

with the the line line yy == x. Figure

2.2.1 2.2.1 depicts depicts the the case caseof of aa = 3. 3. By the the way, way, we weintroduce introduce how howtotovisualize visualizesolutions solutionstotoone-dimensional one-dimensional difference equations. A frequently used plot plot isisthe theso-called so-called sta stair-step diagram or or difference fr-step diagram stafrcase diagram, or cobweb staircase cobweb diagram. The diagram diagram is a plot plot in in aarectangular rectangular coordinate system of: of: (1) (1) the the graph graph of the function f(x); (2) the identity line coordinate system function y == fix); y = x; and (3) a polygonal line that results from joining the points and polygonal (x *,) (x1, x1), (x1, x2), (x (x2,xx3), (x3, x3), x3), (x0, (x2, (x2, (x3, 0, x1), 2,xx2), 2), 3), Figure 2.2.2 shows that line segments segments of of the the polygonal polygonal line line create create the the Figure 2.2.2 shows that the line of stairs. stairs. impression of

f

1 0.8 0.8

0.6 0.6 0.4 0.4

0.2 0.2

x

00.2 .2

00.4 .4

00.6 .6

ax(l -— 4 x) Figure 2.2.1: 2.2.1: The The fixed fixedpoints pointsof of f(x) f(x) == ax(1

00.8 .8

1

2. SCALAR LiNEAR LINEAR DIFFERENCE EQUATIONS EQUATIONS

24 x (t + 1) 1

\

0.8 0.8 0.6 0.6

7 0.4 0.4 0.2 0.2

0.2 0.2

A 0.4A 0.4

0.6 0.6

x (t) 11 x(t)

0.8 0.8

Figure 2.2.2: Staircase Staircase diagram diagram of of O(0.4) 0(0.4) for f(x) f(x) == 4x(1 4x(l -— 9 x)

Mother plot used Another plot used for for visualizing visualizing solutions solutions totoone-dimensional one-dimensional difference difference equation is called time series. ItIt consists consists of of aarepresentation representationof ofthe thevariable variable x{t) x(t) as a function of t.t. See function See figures 2.2.3 and and 2.2.4. 2.2.4.

f 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 t 10 10

20 20

30 30

40 40

50 50

60 60

Figure 2.2.3: Time series series plot plot of of O(0A) 0(0.4) for f(x) f(x) == 4x(l 4*(l -— 9 x)

2.2. SOME CONCEPTS

25

x(t) x (t) 11 0.8 0.6

0.4 0.2 0

20

40

60

t

Figure 2.2.4: - x) x) Figure 2.2.4:Plot Plotofofpoints pointsofofO(0.4) 0(0.4)for for f(x) f(x) = = 4x(l 4x(1 —

Definition saidtotobebeanan eventuallyequilibrium equilibrium (or (or Definition 2.2.3. 2.2.3. AA point point x' x es RR isissaid eventually stationary) stationary)point point for for equation equation(2.2.1) (2.2.1)ororan aneventually eventuallyfaed fixedpoint pointforfor/ f ifif there there 2 andaafixed fixed point point x of f/ such suchthat2 that exists a positive positive integer integer r and

Example The logistical difference difference equation

ff=4x(1—x) = 4x{l-x) and 3/4. 3/4.Finding Findingeventually eventuallyfixed fixedpoints pointsisisto to solve solve has two fixed fixed points, points, 0 and

fr(x)

o,2-,

where r is aa positive positive integer integer greater greater than For instance, instance, with and / 4 and than 1. For with x = 33/4 r = 2, 2, we obtain the algebraic equation 22

sa 0 A sequence sequence {xQ))]°0 {*(0|"o is id tto have A is said have eventually eventuallysome someproperty propertyP,F, ifif there there exists exists an an

{x^)}^ has integer NN > kk such that integer that every term of {X(l)rN has this property.

EQUATIONS 2. SCALAR LINEAR DIFFERENCE EQUATIONS

26

16x(1 x)[1 4x(1 x)]=— 16x(l-;c)[l-4jt(l-*)] = -.

4

It is easy to check that the equation has the following eventually fixed fixed points

1 1 73 1

4'2 4' 2 + 44''

1 1 72 2' 22 + 4' 4 ' 1

because

1 4' A) 4' 3

3

4

4'

=

Exercise Exercise 2.2 2.2 1 Show: Show:

(a) f(x) cos x has for x ce [o, i]; (a) f{x) = = cosx has aaunique unique fixed point for [0, l]; (b) f(x) f(x) == xx33 - 2x ++ 11 has three fixed points. points. (b) equation 2 AApiecewise piecewise linear linear version version of of the the logistic logistic equation equation is is the tent tent equation 2x(t), 2x{t),

if ifx{t)<x(t)

2(1 — x(t) 2(l-x(t)\ifx(t)>^

2

I

>—.

This map may be written in the form

2

2.2. SOME CONCEPTS

27 27

equilibrium points; points; and and (b) (b) the the point point 11/2 aneventually eventually Show: (a) there are two equilibrium / 2 isis an point. fixed point. 3 AApopulation population of ofbirds birds isis modeled modeled by bythe thedifference difference equation equation

2x(4 , , J13.2x(t), /

v

;

if 0 xQ) 1,

[0.5x(/)+2.7, + 2.7, if xQ)> 1,

where x(t) x{t) is is the the number number of of birds birds in in year year t.t. Find Findthe theequilibrium equilibrium points and then where their stabilities. stabilities. determine their

2.3

Stabilities Stabilities

dynamical system system might have have unpredictability unpredictability property, property, which which means means that A dynamical orbits far apart apart at at some some orbits starting starting at at points points very very close close to each other can be quite far later orbits remain remain confmed confined in in aa bounded bounded region. region. Since, Since, later time, even though the orbits on experimental state of of an an orbit orbit isis never never known known accurately, accurately, experimental grounds, the initial state cannot "predict" "predict" where where the system system will be at at some some later later time. time. To To explain explain we cannot unpredictability, we definition of stable and and unstable unstable orbits. orbits. The The unpredictability, we introduce introduce the the definition of stable presence of of unstable unstable orbits orbits plays plays an an important important role role in in dynamical dynamical systems. systems. presence Let the one-dimensional one-dimensional difference difference equation equation Let us us consider consider the x{t \)=f{x{t)\ ,--. x(t + i) = f(x(t)), t? = 00,, l1,•••.

(2.3.1)

Definition fixed point equation(2.3.1) (2.3.1)isislocally locallystable stable ifif for for every every Definition 2.3.1. A fixed point xx*ofofequation s6>0 > 0there there exists exists S8>0 > 0 suchthat such that

Oxo

—

I< S 5 implies impliesthat that llx, -— x* < s& for 1. forall allft>i.3

In this book, the norm ||jc|| denotes defined by denotes the the Euclidean Euclidean norm normof of x, x, defined 1/2

lIxII

=

LINEAR DIFFERENCE DIFFERENCE EQUATIONS EQUATIONS 2. SCALAR LiNEAR

28

A fixed point that is is not stable stable is said to be unstable. unstable. An unstable equilibrium point repeller. is called a source, or a repeller. Stability Stability means means that once we have chosen chosen how how close we want to remain remain xx* in the future, future, we can find how close we must start at the beginning. beginning. Figures Figures 2.3.1 2.3.1 and and the two two concepts. concepts. 2.3.2 illustrate the

(t) xx(t)

x+e *

xo

x

*

*

x—E

x* -o t

Figure 2.3.1: Stable x0 x0

Example Consider Consider x{t + l) = 1 —x(t). - x(t). xQ+1)=1 —0.5 0.5 isisthe theonly onlyfixed fixedpoint pointof of f./ .For Forevery everyother otherinitial initialstate state x0 x0 we we The point x* = have X\ = 11

XQ , — x0,

X2 = x2

XQ . x0.

We thus have 0.511 = = Oxo I1k K - Ml Ik - 04 —

—

STABILITIES 2.3. STABILITIES

29

x(t)

+E

+5 xo

x

x —e

-5 t

2.3.2: Unstable x0 x0 Figure 2.3.2:

- e s in in the definition). for all all t/ > 1. Hence, the fixed point point is stable stable (by (by selecting selecting SS = Example Let as Example Let f:R-*Rbe f: R R be defined as

for x0.

f(x) = 0.5x, for x < 0, /(*) = for x>0. x > 0. 2x, for equilibrium x* == 0. 0. Every orbit The system has a unique equilibrium orbit starting starting to to the the left left of of the origin will converge converge to x*, while every orbit starting to the right of the the origin origin Thus, x* is an unstable fixed fixed point. will go to infinity. Thus, Definition 2.3.2. The The point point x* is said said to be attracting if there exists exists 7] such Definition ij >>00 such that

x(o)-x*
2. 2.SCALAR LINEAR DIFFERENCE EQUATIONS EQUATIONS

30

I isiscalled If 7]==cc, attracting. The Thepoint pointx* I isisan °°, x* called aaglobal global attractor attractor or globally globally attracting. an asymptotically stable stable point point or or aa sink s Mif ifititisisstable stable and andattracting. attracting. IfIf17 rj = = ac', <», xx* isis globally asymptotically asymptoticallystable. stable. said to be globally

Let f/ bebea a fUnction interval Q. Q of function defined defined on some interval ofRRwith withvalues values in inCI. Q.ItItisis known that under under certain certain conditions conditionsthe thestability stabilitytype typeofofananequilibrium equilibrium point of point x*Iof difference equation (2.3.1) is isthe same same as asthe the stability stability type type of of the the equilibrium equilibrium point point difference of the corresponding linearized equation =

y(t + +i)l) =ft(x*)y(t). f'{x')y(t). y(t

(2.3.2) (2.3.2)

A simple argument is is that that for for x close butnot notrigorous rigorous argument close to to the thevalue value of ofan an simple but equilibrium point x* we have HOT, f(x) = f(x)+ f(x')+ f'(x)(x f'{x){x —- x ) + HOT, f(x)=

I.

Taking into intoaccount accountthat thatx*I where HOT HOTdenotes denotesthe thehigher-order higher-order terms terms in inxx— - x*. Taking the anequilibrium equilibrium point, neglecting neglecting HOT, andreplacing replacing xx == u weobtain obtain the is an HOT, and u,,t, we approximate equation

I

= x+f(x u(t i) = + f(x )(u(t) — u(t++ l) )(«(/)-*').

Finally, setting

y(t) = y(t) = u(t) -— x, we obtain equation equation (2.3.2). (2.3.2).

Definition 2.3.3.Assume Assume/ f isiscontinuously differentiableininQ..Cl. Let be aa Definition 2.3.3. continuously differentiable Let x*I be steady state e Q. CI. If If

= 1, then x* is non-hyperbolic. If If

STABILITIES 2.3. STABILITIES

31 31

1,

/

x is is hyperbolic. hyperbolic. non-hyperbolic,itsitsstability stability type type cannot cannot be bedetermined determined by by its its first first If /x*is is non-hyperbolic, derivative, as as discussed later on. 2.3.1. (linearized (linearized stability) stability) Let Let x* be be aa hyperbolic hyperbolic steady steady state state ee Q. Theorem 2.3.1. Then 1, then then x* is asymptotically (i) If / ' ( / ) < <1, asymptoticallystable. stable. (ii) If 1, then i is unstable. (ii) If |/'(x*)| > 1, then x' is unstable. c 1, consider /1 such that the case of Proof. In the case of If'yx") < 1, consider /? such that

Proof.

V(x*)I

\r(x')\
(x* — By continuity continuity of we have have If'(x)l |/'(x)| < /3 on on some some interval interval [x* - e, e, xx + +4 e) < /3 By of f'{x), f'(x), we 0. For any any xx ininthis thisinterval, interval,applying applying the themean mean value value theorem theorem for with e >> 0.

derivatives yields

f(x)_f(x*)=(x_x*)f(x* +e(x_x*)), f{x)-f{x)={x-x)f{x+e(x-x\ where 00 << 0 + O(x 9\x — — x*) x jbelongs belongs(x[x* have < 1. where S < 1. Since x + ——e,£, x*+ +4,ej, wewehave

\f{x)-f(x)\
x(t ++ \) = f(x(t)) O=f(x(t)) x0 satisfies satisfies starting at x0

32 32

LINEAR DIFFERENCE EQUATIONS EQUATIONS 2. SCALAR LII'JEAR — - x'\ = f{x')\ {t + 1) 1)— = \f{x{t)) -

ii

- x

—

-— x xi.

We conclude that the sequence converges x0 e \x' + 4, e\ converges to to xx* for all x0 (x — e, e, x* + and x* is islocally locally stable. stable. This This also also implies implies that that the the whole whole sequence sequence remains remains in in the the and The same same proof proof applies applies to toaacorner corner steady steady state state with with neighborhood neighborhoodof ofx*. x. The x0 e (0, (o, e). e). The equilibrium isasymptotically asymptotically stable when / ' ( / ) < -> 00, °°, equilibrium point point is c 1. 1. As It —* we have 0.

x[t)-.

We conclude conclude asymptotical asymptotical stability. stability. In the case of /'(**) > 1, consider consider /3 J3such such that >1, > /3>1.

continuity of we have P on on some interval interval (x (x -— e, £, x* + s) By continuity of f'(x), f'(x), we > /3 have |/'(x)| If(x)I > 4 any xx ininthis thisinterval, interval, applying applying the the mean mean value value theorem theorem for with e For any for e>> 0. 0. For derivatives yields

f(x)_f(x*)=(x_x*)ft(x*+9(x_x*)), f{x)- f{x')= {x - x*)f'(x' + e{x -x')), withOcOci. with 0 < 6 < 1.

i)

belongs \x' - £, x* + + 4, £), we we have Since x* + O(x 8\x -— xj belongs (i — e, x — \f(x)-f{x)\>p\x-x. > — xi.

(x* x0 e [x* + £), the sequence For any x0 — £, e, x + 4, the

x(/ + l) /(*,) x(t i)==f(x1) x0 satisfies satisfies starting at x0

STABILITIES 2.3. STABILITIES

x{t + l ) - x \ = \f(x(t)) + 1)-

-- f(x* f{x )I)| >>

33

fi\x(t) -- x \ >> 0 ' \ x o -- x ,

which holds for the terms

+4. xt € [x* -_s,x* £, x" + e). does not not converge converge to to x* for all all We conclude that the sequence does [x* — e, x*++ 4, e), x0 € (x* — e, and x* unstable. x is locally unstable. 2.3.1, if the the equilibrium equilibrium point point x* of equation equation (2.3.1) (2.3.1) is From theorem 2.3.1, From theorem hyperbolic, then it must be either either asymptotically asymptotically stable or unstable, and the stability stability type is determined determinedfrom fromthe thesize sizeofoff'\x*). f'(x). Figure 2.3.3 depicts different different stability stability types. types. Example cobwebmodel modelofofdemand demandand andsupply4) supply4) Consider Consider the Example (a(acobweb the market market for for a Assume that that the the output output decision decision in inperiod period t is based on the then single commodity. Assume prevailing price and that that the the output output planned planned in period period t will not be available available prevailing price Pit) P(t) and s for the sale, Wethus thushave haveaalagged lagged supply supply fimction function sale, Q QS(t(t++ l), i), until period tt ++1.1.We

+ i)= s(P(t)). Equivalently QS(t) Q°(t) =s(P(t S(p(t-l)). — i)).

function interacts with a demand ftinction function of the When such a supply thnction the form Qd(t) Qd(t)=D( = P(t)),

the price dynamics will be be determined determined by by the the balance balance condition condition the

1

Thisisisfrom fromchapter chapter16 16ininChiang Chiang(1984). (1984). "This

34

2. SCALAR LINEAR DIFFERENCE EQUATIONS EQUATIONS

(a)

f(x*)=1

x(t)

x(t+1)

x(t)

(b) 0< ft(x*)< !

ft(x*)>1! (c) (c)

1

+i)

i)

(d) —1> ft(x*)<

0

x(t)

(e)—l
*

Figure 2.3.3: Stability types of scalar difference difference equations

x(t)

2.3. STABILITIES STABILITIES

Qd(t)

35

QS(t)

Specify the demand and supply supply fimctions functions as as follows follows Speci&

Qd(t)=ad Qd(t)=ad-bdP{t), —bdP(t), Qs(t) - i), l), ad, bd, bd, a,, bs >0. > 0. Qs(t) =-a — s+ + b,P(f — The price dynamics is given by

bd bd

=0. = 0.

bd

The solution is given by

P(t)=IP(o)—

~tbd)

b$+bd

Introducing = a, + +

P =

bd

solution as as where P* is the fixed point, we may write the solution

P(t) = (p(o) -

+

Because Z;s and bd bd are arepositive, positive,the thetime timepath path will will be be oscillatory. oscillatory. We We see see

* I -=

*

According to to the the values values of of bs and and bd, bd, we we have havethree threepossibilities, possibilities, (i)

> bd; bd; (ii) (ii) uniform uniform if bb5s = bd; bd; and (iii) damped damped if bb5 We explosive if bs > and (iii) explosive c bbd. s < d. We 2.3.4. depict (i) and (iii) as in figure 2.3.4.

36

2. SCALAR SCALAR LII'4EAR LINEAR DIFFERENCE EQUATIONS EQUATIONS

Q QS

QS

P

P (the supply supply curve curve isis steeper) steeper) b) b5 bs < bd(the (thesupply supplycurve curveisis flatter) flatter) a) b5 bs >> bbdd (the cbd Figure 2.3.4: The cobweb model

Example (a cobweb cobweb model model with with the thenormal-price normal-price expectation5) expectation5) Consider Consider the the market market for aa single single commodity. commodity. Assume that the the output output decision decision in in period period t is based on on the the then thenexpected expectedprice priceP'r (t). The supply function finction is is specified as Qs{t) = =— -asa5 +

+ bpe(t) bsPe{t), a5, a,, 15 bs > 0. 0.

The demand demand function function is is Qd(t) Qd(t)=a

-bd—P(t), = dad bdP(t),

> 0. ad,b k1 d>Q.

To form price expectations, we introduce a concept of "normal price", price", denoted denoted as P, asasthat thatprice pricewhich whichproducers producers would would think think sooner sooner or or later later to to obtain obtain in in the the If the the current current price price isis different different from the the normal normal price, price, they they think think the the former former market. If moving toward toward the the latter. latter. A A simple simple way way to toformalize formalize this thisisis to to specify specify will modify, moving

0cacl.

Pe(t) = P(t - l) + a(¥ -— P(t —- l)), 0 < a < 1. The balance condition of

' This This and and the thefollowing following example examplearc arefrom from Gandolfo Gandolfo(1996: (1996:38-43). 38-43).

STABILITIES 2.3. STABILITIES

37

Qd(t) =

is now expressed as - a5 as + bsP(t —- 1i)) + + ab,P —- ab5P(t absP(t — -\)i) == ad ad- — bdP(t). bdP(t). + b5P(t

—

The solution to this equation is

-

P(t) =

bd

+

)

the solution solution is is oscillatory oscillatory and the equilibrium equilibrium point P(t) P(t) ==1P is We see that the is stable if —

a) c

bd.

cobweb model model with adaptive adaptive expectations) expectations) Like the previous previous Example (a (a cobweb Like in the supply and and demand demandfUnctions functions are specified specified as as example, the supply d Qd(t) {t)=a ,bd>0, >0, Q d-b—dP{t), = ad bdP(t), ad,adbd e (t), as,bs>0, >0, Qs(t)=-a a5 + a5, sPb5pe(t), — s+b

where Pre(t) expected price. assume that expectations expectations are adapted adapted in in where (t) is is the expected price. We assume period on the basis basis of of the the discrepancy discrepancy between the observed observed value and the each period expected value, value,that that is6 is6 previous expected

pe(t)_pe(t_l)a(p(t_1)_pe(t.....1)), P'(t)Pe{t -1) = a(p(t - l) - P'(t -1)), o
Substituting Substituting

66

The was initially initially proposed proposed by by Nerlove Nerlove (1958). (1958). The adaptive expectation was

2. SCALAR SCALAR LINEAR LINEAR DIFFERENCE DIFFERENCE EQUATIONS EQUATIONS

38 38

+Q(t)

pe(t)

into the above equation equation yields

a, + Qs{t) - (1 - a)(a, + Qs{t —1))_—kceP(t—1). - l)) = By Qd(t) =

we get

bdPQ). Qs(t)=ad-b—dP(t\

Substituting QS(t) = ad — bdP(t)

into the previous previous equation equation yields yields

ha

The equilibrium point point isis given given by by

aS+ad bS+bd

The general solution of the problem is thus given by The

P(t)=AlP(t) = A[1

—

U \+P\ f + + li}r]+

The stability condition is is

(b

—i-1-+lial<1.

)I

2.3. 2.3. STABILITIES STABILITIES

39 39

Example (a(amarket marketmodel modelwith withinventory) inventory) We Wenow nowextend extend the the cobweb model Example model to aa can keep an an inventory of of the commodity. Assume that the the quantity case that sellers can and the quantity quantity currently currently produced, produced, Q5 Qs(t), unlagged demanded, Q" Qd(t), (t), and (t), are unlagged demanded, linear functions of price P{t). The adjustment of price is effected not through linear functions of price P(t). The adjustment of price is effected not through market clearance in every period, but through a process of price-setting by the market clearance in every period, but through a process of price-setting by the sellers: at the beginning of each period, sellers set a price for that period after taking sellers: at the beginning of each period, sellers set a price for that period after taking inventory situation. into consideration consideration the into the inventory situation. If If inventory inventory accumulated accumulatedas as aa result result of of the preceding-period price, the current-period price is set a lower level than before, the preceding-period price, the current-period price is set a lower level than before, and vice vice versa. versa. Moreover, Moreover, the the price price adjustment adjustment made made from from period period to to period period is is and inversely proportional observed change the inventory. inventory. With With these these inversely proportionaltoto the the observed change in in the conditions, we we can canwrite write the thefollowing following equations equations conditions,

Qd(t):=ad s Qs(t)=_a +b5P(t), Q (t) = -as+ bsP{t), ad,bd,aS,bS ad, bd, a,, bs >0, > 0,

P(t +

1)

= P(t)

—

where a denotes the the stock-induced-price-adjustment stock-induced-price-adjustment coefficient. coefficient. Substituting Substituting the first two equations into the last one yields the one Pit +1) - [l - a{bs +bd)}P(t)=c(aS + bd )]p(t) = a{as + ad). +ad). P(t+O—[l—a(b3

Let P* be be the the fixed fixed point. point. Then, Then, we wemay may write write the theabove above solution solution Let

P(t) == (p(o)-P')(l-a(b s+bd+))' — P)(l — P(t) bd)Y + + P', where p* -

+Qd +

bd

We see seethat that the thestability stability isisdetermined determinedby bythe theterm term1 1—- o(bs + + bd). bd). We (the Newton-Raphson method) The TheNewton-Raphson Newton-Raphson method is is aa wellwellExample (the forfinding finding the theroots roots of ofthe the equation equation method for known numerical method

g(x) = = 0.

2. SCALAR LINEAR DIFFERENCE EQUATIONS

40 40

finding a zero zero x* of of g(x) g(x) is thedifference difference Newton's algorithm for finding Newton's algorithm is given given by by the equation g(x(t)) x(t +1) = x(t) -

g (x(t))

theinitial initial guest guest of of the the root root x*. To determine whether whether Newton's Newton's where JC To determine x0 0 is the \x(t)\ that converges converges to to x* we calculate algorithm provides aasequence {x(t)} I

f

Wl. I

= 0,

0. We We conclude that where we use g(x*) g(x) == 0. = JC*, \imx(t) limx(t) =

if if xxc0 is close closeenough enoughtotox'x and and g'(x*)* g' (x*) 0. 0. Example Consider Consider Ax2(t), x(? 1) = = 11 -— Ax2(t), x(t + l) where XE [—1,1], 2€ (0,2]. xe[-l,l], Ae(0,2].

There are are two two equilibrium equilibrium points points

,

*

*

l2

- 1 + ^ 1 + 4/1 —1±41+42 2l " 22

It is straightforward to check that that x* unstable for 2]; is straightforward to check x1 is unstable for all 2A ee (o, 2J; asymptotically f O0< 33/4.

x*2 is

2.3. STABILITIES

41 41

Definition asymptotically stable stable fixed Definition 2.3.4. 2.3.4. Let Let x* I bebeananasymptotically fixedpoint pointof ofaamap map/f.. The of attraction attraction 8[x) of of xx* is is defined defined as that contains contains x" basin of as the the maximal maximal set set J that and is such that f'(x)-*x*

as / —» oo foreveiy for every XE x €J.J. as t—*oo

attraction of an attracting defined in The basin of attraction attracting periodic periodicpoint pointof ofperiod period pp isis defmed analogous fashion, replaced by the pp-th -th iterate. iterate. analogous fashion,with withthe themap map/f replaced Example The The map Example

/(x)=2x(l-4 f(X) = 2x(1 — has has

an attracting fixed x* = (0, i). l). fixed point X* = 1/2 1/2 with a basin basin of of attraction attraction K(l/2) R(1/2)==(o,

We will see seelater later that that basins basins ofofattraction attraction may mayhave havecomplicated complicated structures structures even for simple looking maps. Definition 2.3.5. is said a map / fififf(M) M, Definition 2.3.5. A A set setM M is saidtotobe beinvariant invariantunder under a map f(M) ac M, the elements elements of O(x) belong to M. M. that is, if for for every every xXee M, M, the of 0(X) Theorem 2.3.2.7 Let X x*be bean anattracting attracting fixed point Then the basin of pointof ofaamap map /f.. Then of attraction

N(X*J is an invariant open open interval. interval.

Exercise 2.3 2.3 1 Given Givendemand demandand andsupply supplyfunctions functions for forthe theCobweb Cobwebmodel modelas asfollows: follows: d Qd(t) Q (t)=l*-3P{t\ 18 —3P(t),

= —3 + 4PQ — d

1);

Qd(t) {t) = l9-6P{t), = 19 — 6P(t), r.^Q (n) Q>(t) = -5 ÷6P(t + 6P(—ii). t-l).

QSQ)5

1

See Elaydi Elaydi (2000). (2000). See

2. SCALAR LINEAR DIFFERENCE EQUATIONS EQUATIONS

42

Find the the intertemporal intertemporal equilibrium equilibrium prices, determine whether Find prices, and and determine whether or or not the are stable. equilibrium points are

2 Let

f(x)=x2. stableand andx* x == 1 is Show that is unstable. unstable. Show thatx*i == 00 is stable 1

theequilibrium equilibrium points pointsand anddetermine determinetheir theirstability stability for forthe theequation equation Findthe 3 Find - «6/ x(t+1)=5—

x(t + l) = 5

x{t) xv

4 Fielou Pielou 's logistic logistic equation equation is is defined as

x(t+l)=

av(t)

l+/lv(t)'

cx>!, /3>0.

Find the thepositive positive equilibrium equilibrium point; point; and and(b) (b)demonstrate, demonstrate, using using the thestair stair step step (a) Find diagram, that the positive positive equilibrium equilibrium point point is is stable, stable,taking taking aa == 2 and /3 1. /? == 1.

2.4

nonhyperbolic equilibrium points Stabilities of nonhyperbolic points

information about aboutthe thesign signof ofthe thefirst firstderivative derivativevalue valueofofthe themap map/ f at at an From the information equilibrium point, point, theorem properties of of the equilibrium theorem 2.3.1 2.3.1 determines determines the the stability properties equilibrium point point when it is equilibrium is hyperbolic. This section section addresses addresses the the nonhyperbolic nonhyperbolic

Further information case where information about about derivatives derivatives is needed needed to case where /'(**) = 11.. Further determine stability stability properties. properties.We We discuss discuss f'[x*)=l ft = 1 and f'[xj f (x') == -l —1 separately. determine separately. hasa acontinuous continuoussecond secondderivative. derivative. This section section assumes assumes that that /f has 2.4.1. An An equilibrium equilibrium point x* of Definition 2.4.1.

i)= f(x(t)), x(t + l)=f{x{t)),

(2.4.1)

2.4. STABILITIES OF 2.4. OF NONHYPERBOLIC EQUILIBRIUM EQUILIBRIUM POINTS POINTS

43

is semistable (from the the right) semistable (from right) if given given e£ >> 0 there there exists exists 88> > 0 such such that that if x(o) — x(O)> <8,S, then <e. Semistabilityfrom from the the left x(o) > xx,, x(o) - x < thenx(t) x(/)— - x* < £. Semistabiliry left is defined similarly. If in addition

limx(t)= x, limx(t) = x*, (—*00

whenever x(o) x(0) —- xx* < <77, stable from thethe T],then then x isissaid saidtotobebesemiasymptotically semiasymptotically stable from

right. We semiasymptoticalstability stabilityfrom fromthe theleft. left. We can can similarly similarlydefine definesemicisymptotical

The following two two theorems theoremsare arereferred referred totoElaydi.8 Elaydi.8

Theorem 2.4.1. 2.4.1. Suppose Suppose that that for for an anequilibrium equilibrium point point x* of equation equation (2.4.1), (2.4.1), 1. The following statements statements then then follow: follow: / ' (JC* ) = 1. ft

(i) If / * ( / ) * 0, then x" is unstable. unstable. md m x h e n x isis unstable. fm(x*)> (ii) f'(x) = 0° and f { ')> °> 0, tthen (ii) IfIf/"(**)= " unstable.

(iii) f (x )) = o0 and j (x )
then the curve curve is is either either concave concave upward upward if f[x*) > 0 or or concave concave downward downward if f"(x') 0, 0, then

f'(x) / ' ( * ) > M >>1, 1, for all x in interval II == (xe, for all in a small interval [x", x" + 4. e). Then, Then, we we have have jx(i) - x*I = f(x(0))

-

=

where where £ e I./.By Byinduction inductionwe weconclude conclude

81

Section Section 1.4 in in Elaydi Elaydi (1999). (1999).

- x1>

-4

- x*I,

44

2. SCALAR LINEAR DIFFERENCE EQUATIONS (t)-x* >M'[x(o)—x[ >M'x(0)-x'.

We conclude that xx* is unstable. We can discuss other cases similarly. It should

0), be noted if /'(**)> 0 o ((fN(x*) the be noted that that as asproved proved in inSandeftir,9 Sandefur,9 if / " ( / ) << 0 ), then then the equilibrium point x* is semistable from the left (right).

Consider Example Consider

x(t+O=x2(t)+x(t), x(t + l) = x2(t)+x(t), t = 0,1,

.

equilibrium point x* = The difference equation has aaunique equilibrium = 0. 0. As

ft(o)1, f'(0)=2>0, /"(0)=2>0, isunstable. unstable. To examine examine its its semistability, semistability, theorem 2.4.1 we we conclude conclude that that x* is by theorem first first assume x0 >> 0. 0. Then

x(1)= x2(0)+ x(0)> x(O). — l). {x(t)\ either converges to aa By induction induction we x{f) >x(t x{t— we get get x(t)> 1). The sequence {x(t)} fixed point or diverges to to infinity. infinity. Since Since the the only only fixed fixed point point isis zero, zero, {x(t)} {x(t)} diverges diverges if x e (-1 / 2, 0) to infinity. infinity. On the other hand, the otherhand, if x0Q e (—1/2, 2

) = x2(0)+ x (o)+x(o)>x{o), x(1)= x(0)> x(0), and x(1) x(l)ee (—i (—1/2, / 2, 0). 0). By induction, we conclude x(t) > x(t

—

i).

{x(t)} converges The sequence {x(t)} converges to to0. 0. Consider Example Consider

9

Sandefur (1990). (1990). Sandefur

2.4. STABILITIES STABILITIES OF OF NONHYPERBOLIC NONHYPERBOLIC EQUILIBRIUM EQUILIBRIUMPOINTS POINTS 45 45 2.4.

x(t+1)

x

x

x(t)

unstable, fff(x*)> /'(**)> 00 (a) unstable, left) (semistable from the left)

x(t)

unstable, / ' ( / ) < 00 (b) unstable, from the the right) right) (semistable from

x(t+1)

x

(c) unstable unstable

f'(x) = o, f"(x)> o /-(**)=o,r(**)>o

x(t)

x (d) asymptotically stable ft(x*) =0, )<

Figure 2.4.1: Stability (JC* ) = 11 Stabilitytypes typesininthe thecase caseofof/ f'' (x*)

x(t)

46

LINEAR DIFFERENCE DIFFERENCE EQUATIONS EQUATIONS 2. SCALAR LiNEAR x(t x(t), ft =0,1,.... JC(/ + 1) = JC 3 (0+*(/), = 0,l,-.

difference equation has a unique equilibrium point x* = 0. 0. As The difference

r(o)=o, f'(O)=l,, f'(O)=O,

/»=6>o,

by theorem 2.4.1 2.4.1 we conclude conclude that that xx* is is unstable. determine this unstable. We We can can determine this by by theorem induction. Assume that x0 x0 >> 0. Then 0.

x(l) = x3(0)+x(0)>x{0). x(1)= x(0)> x(O). By induction induction we get get x(t)> x{t) >xQ x(t—— i). l). The {x(t)} converges to to The sequence sequence {x(t)}either either converges a fixed fixed point point or or diverges diverges to to infinity. infinity. Since Since the the only onlyfixed fixed point point isiszero, zero,{x(t)} {x(t)} diverges to infinity. 0, we we have infmity. On Onthe theother otherhand, hand,ififxx0 <0, 0 <

x(l) =

x(O)< x(O).

By induction induction x(t) x(t)c<x(t x(t—- 11). ) . The sequence sequence {x(t)} {x(t)}diverges divergestoto—oo. -°°. Hence, Hence, the equilibrium point is a source. Example Consider Consider x{t + 0,l,-. i)= —x3(t)+ x(t), f t= =0,1,... x(t + l)=-x'{t)+x(t),

The difference 0. As difference equation equationhas hasaaunique uniqueequilibrium equilibriumpoint pointx*x = 0. / ' ( 0 ) = 0,f"(0)=—6
2.4. STABILITIES STABILITIES OF OFNONI-IYPERBOLIC NONHYPERBOLIC EQUILIBRIUM EQUILIBRIUM POINTS POINTS

47 47

f(f(p)) = p, f(f(p))=P, with from f\p). f(p). Then with pp different different from Then p isisaaperiodic periodic point point of ofperiod period two. two. This This of solutions solutions of equation (2.4.1) is called period-doubling change in global behavior of period-doubling bjfurcation. or flip bifurcation.

Example Consider Consider

x(t+1)=—x(t), t=o,i,... The unique point is zero. unique equilibrium equilibrium point zero. Every Every solution solution of of this this equation, equation, except except = 0, isisperiodic x* = periodicwith with period period two.

Example Consider Consider 0,l,-. x{t l) = -x(t)+x2(t), = x(t + i)=—xfr)+ x2(t), t =0,1,•••

The equilibrium pointsofof this this equation equationare are x* 4 = = 0 and The equilibrium points and xx*=2 =2.2.To Toexamine examine |x:(2&)}, we we note note behavior of the sequence {x(2k)}, 3 4 (2k). x(2k + x(2k + + l)+ i) + x2 (2k + i) = = x(2k) — x3 (2k) + x4 {2k). + \) x(2k)-x {2k)+x + 2) 2) ==—-x(2k x2(2k

Setting

y(k) = y{k) = x(2k), x{2k), we rewrite the above equation as y{k y(k)-/{k) / (*) ^ g(y(k)). g{y{k)). y(k + l) i) == y(k) — y3(k) + + y4(k)

(2.4.2)

Since if! \ .wi \ ,i \ g'(0) = l, g'(0) = 0, g g"{0)=-S,

theorem 2.4.1 sink. Likewise, Likewise, we we conclude conclude that by theorem 2.4.1 we we conclude conclude that that x[ 4 = 0 isis aasink. x*2 ==22isisa asource. source.

48

2. SCALAR LINEAR DIFFERENCE EQUATIONS EQUATIONS

Similarly, we conclude that the sequence {x(2k + i)} l)} satisfies equation (2.4.2). convergent to to the the zero zeroequilibrium. equilibrium. Consequently, Consequently, both both even-indexed even-indexed Hence it isis convergent terms and odd-indexed sink, while while odd-indexedterms termsare areconvergent convergenttotozero, zero,hence hencex\x = 0 isis aasink, x* source.Also Alsoapplying applyingthe theabove abovemethod, method, check check that that the the origin origin isisaa 4 2 ==22isisa asource. equation source of the following difference equation

x(t + !) l)== —x(t) *(/ -*(,)-**(,). — x3(t). Before stating stating stability introduce Before stability conditions conditions for for the the case case of of f'[x*)= f' (x) = —- 1, 1, we introduce the notion of the the Schwarzian Schwarzianderivative derivativeofofaafunction function f/

_r(x) sf(x) Sf(x)="

f'(x)

3r

irj'C]2

2 [f'(x)

F o r /f'(x)=—l, ' ( * * ) = - 1 , then then For

Sf(x)= Theorem 2.4.2. Suppose that for an an equilibrium equilibrium point point x* of equation equation (2.4.1), (2.4.1), 2.4.2. Suppose that for /f'' (x) [x* j==—1. - 1 .The Thefollowing followingstatements statementsthen thenfollow: follow: (i) stable. (i) IfIf Sf[x') Sf(x) << 0, then xx* is asymptotically stable. (ii)If 0, then then xx* is unstable. (ii) IfSf(x*)> Sf(x)> 0, unstable. Example Consider Consider x{t + + \) x2{t)+3x{t). x(t i) = = x2(t) + 3x(t). The equilibrium points and—2. - 2 . We We have have points are are 0 and f' = 2x + 3. f'=2x+3. Since f' /'(o) equilibrium point 00 isisunstable. unstable.As Asf' /(—2) ' ( - 2 )==—1, - 1 , theorem theorem Since (o) = 3, the equilibrium 2.4.2 applies. As

2.4. STABILITIES OF OFNONHYPERBOLIC EQUILIBRIUM EQUILIBRIUM POINTS POINTS 49 2.4. 49 S / ( - 2 ) == -—12 1 2
Exercise 2.4 1 Find Find the theequilibrium equilibrium points points and anddetermine determine their their stability stability of ofthe thefollowing following difference equation + l) = ax2(t) + bx(t) + c, a * 0. x(t x(t+i)=ax2(t)+bx(t)+c,

ft(x*)= Suppose that if f(x')=l, 2 Suppose

then

1,

/ " ( / ) * o. 0. Prove that x

is (i) is (i)

semiasymptotically stable and(ii) (ii)semiasymptotically semiasymptotically semiasymptotically stablefrom fromthe theright rightififf\x*) f'(x) c<0;0;and stable 0. Applying Applying this result, result, determine determine whether whether the stable from from the the left left if f"[x*) > 0. the point x* = of the thefollowing following equations equations isissemiasymptotically semiasymptotically stable stable equilibrium point = 00 of from from the left left or or from from the the right

(a) x(t + I) = x3(t)+ x2Q)+ x(t); 2 (b)x(t (t)+x(t). (b) xfr + x3(t)— x2(t)+ x(t). + l)=x'(t)-x

Find the theequilibrium equilibrium points points and anddiscuss discussthe thestability stability of ofthe thefollowing following difference difference 3 Find equations (a) + l ) = -x2{t) + x{t), t = 0,l, (a) x(t x(t+1)=_x2&)+x(t),

;

(b) x{t + 1) - 1 - | * 2 ( 4 , t=0,l,•• /= 0,l,-.

2.5 2.5

On On dissipative maps

the one-dimensional difference difference equation equation Consider the x(t + l)=f(x{t)), = 0,
(2.5.1)

[a, b] b] is is called called an anabsorbing absorbinginterval, interval,ororananattracting attractinginterval intervalif if An interval [a, ofequation equation (2.5.1) (2.5.1) eventually enters enters the the interval. interval. An Aninterval interval isiscalled called an every orbit of an

50

2. SCALAR LINEAR DIFFERENCE EQUATIONS EQUATIONS

invariant if every invariant interval interval if every orbit orbit that that enters enters the interval interval stays stays in itit forever. forever. A A difference difference equation (map) (map) with with an an absorbing absorbing interval interval is is called called dissipative dissipatived(fference difference equation equation(dissijpative {dissipativemap). map). Theorem bj be of real realnumbers numbersand andassume assumethat that/ f is Theorem2.5.1.10 2.5.1.10 Let [a, [a, b] be an an interval interval of is a continuous, nondecreasing thnction function such that

f([a, bD c [a, b], and equation (2.5.1) has has aa unique unique equilibrium equilibrium x'. Then every solution of equation

(2.5.1) with with initial initial point point in in [a, b] b] converges convergesto to x*. x. IfIfequation equation(2.5.1) (2.5.1) isis dissipative dissipative

then every solution of equation (2.5.1) converges to x*.

LI

Example Consider Consider x(t) *(' + l) = T ^ 7 - y , 't=O,1,•••. = 0,1,-. 1+x(t) 1 + x{t)

The map theorem 2.5.1 2.5.1 on the interval interval [o, i]. Thus, map satisfies satisfies all all conditions conditions of theorem [0, l]. every solution solution of this this equation equation that that starts starts in in the the invariant invariant [o, [0, 1]. l]. in In addition, addition, every every solution that starts in in [i, [l,cc) oo) in in one step enters the invariant interval and converges to the the zero zero equilibrium. equilibrium. ItIt should should be be noted noted that that the the situation situation for for negative negative initial initial conditions is more complicated as we we face conditions is complicated as face the problem problem of the the existence existence of the the solution.

Theorem 2.5.2. Let Let [a, [a,b\b]be be an an interval intervalofofreal realnumbers numbersand andletlet/ f be be a Theorem 2.5.2. continuous, b]) ac [a, [a, b] and and equation equation continuous, nonincreasing nonincreasingfunction fimctionsuch suchthat that /([a, f([a, bJ) (2.5.1) has no prime period- 2 solution. solution. Then Then every every solution solution of of equation equation (2.5.1) (2.5.1) that enters eventually enters eventually [a, [a, b] converges converges to x\ In addition, addition, the the subsequences subsequences {x(2t)} {x(2t)} and {x(2t {x(2t + + i)} l)} of every solution of equation (2.5.1) converge monotonically to x* by oscillating about the equilibrium equilibrium such such that + i)— (x(t x)(xQ)— x*)< 0, for nM== 0, (x(t + l)-x')(x(t)-x')<0, 0 , 1,1 , - .

Example Example The Thedifference difference equation equation 101

This This theorem theorem and andthe thenext nextone oneare arereferred referredtotoKulenovié Kulenovicand andLadas Ladas(2001). (2001).

2.5. ON DISSIPATIVE MAPS MAPS

51 51

1, x{t ++ i) \) == e - * { ' \ tf ==0, 0, 1,

has an invariant invariant interval. interval. The corresponding corresponding map is decreasing decreasing everywhere and itit does not possess a prime period- 2 solution. solution. Indeed, Indeed, ifif such such aa solution solution would would exist exist it would satisfy —x

e-" =x, = x,

for some some p x*. In fact, it is straightforward straightforward to show that the function function e~e = xx has a unique solution on the [0, l]. Thus, all all conditions conditions of theorem 2.5.2 on has the interval interval [o, i]. Thus, interval {o, [0, l] are satisfied satisfied and every every solution solution of the equation that starts in in the the the interval invariant interval interval [o, [o, l]1] converges unique equilibrium equilibrium point. point, In hi addition, addition, invariant convergestoto the the unique every solution that starts in R enters the invariant interval in at most two steps and every solution that starts in R enters the invariant interval in at most two steps and so so converges converges to to the the equilibrium equilibrium point. point. Example ExampleConsider Considerthe thedifference difference equation equation

(t xx(t

+

+ \)=\ i)=i + ^ +

A r

xiO(t)

,

t = 0,l,-

(2.5.2)

p, and p, and x(o) x{0) are positive positive numbers. numbers. This This equation equation has an an invariant invariant interval [i, [l, 1 + A] A] and the corresponding corresponding map is is decreasing decreasing for for all all positive positive values. values.

where A,

The has aa unique positiveequilibrium, equilibrium, wherex'I isis the The equation equation has unique positive x*,x,where theunique unique positive positive root of of the equation — x" -A — A = 0. xp + ' -x"

If pp 1, 1 isx' locally asymptotically < 1, is locally asymptoticallystable. stable.Similarly, Similarly, if p >> 1,1, xx* is locally locally asymptotically stable provided that *

x

p P p-\

p-i

One can show that this condition is equivalent to the condition

2. 2.SCALAR LINEAR DIFFERENCE DIFFERENCE EQUATIONS EQUATIONS

52 52

A<

(2.5.3)

(p-I)

When condition condition (2.5.3) (2.5.3) holds, the Schwarzian Schwarzian derivative derivative can can be beused used to toshow show asymptoticallystable. stable.Thus, Thus,ininboth bothcases, cases,(a) (a)pp <1,1,and that i is is locally locally asymptotically and(b)(b) that x* is locally locally and condition > 1 and condition (2.5.3), the unique unique equilibrium equilibrium point point x* is p> asymptotically stable.To To show show that that equation (2.5.2) has has no asymptotically stable. equation (2.5.2) noprime prime periodperiod- 2 1

solution we consider the second iterate

=1+

f2(x) =

Introduce 2(x) — H(x) H{x)=f{x)-x.

is a solution of the the equation equilibrium xx' is Clearly, the equilibrium

H(x)==0.0. H(x) We now that H(x) We nowshow show that that x* x* is is the theonly only solution solution by by checking checking that H{X) is isstrictly strictly As decreasing. As I

\2

where A 0

function H0 (x) has aa minimum at at Since the function H0(x) II \ \l/p x = x=((p-l)Af",

2.5. ON DISSIPATIVE MAPS

53 53

we have P

H0(x)H0(xj= H 0{x)*H0{x)

P-i p-\

Thus we have

(

\

I

\

_iIAIP_1,r pP

I

—NO,

H(x) has has a if condition (2.5.3) holds, where equality equality holds holds only only ifif JC x = 3c. Hence, H(x) unique zero and and this this isisthe theunique uniquepositive positiveequilibrium equilibrium point. point. Hence, Hence, equation equation unique (2.5.2) does not possess possess aa prime prime periodperiod-22 solution solution and by theorem theorem 2.5.2 every every (2.5.2) solution of this equation equation that that starts starts in in the the invariant invariant interval interval [i, [l, 1 + A] A] converges to equilibrium point, point, x*. In addition, the unique equilibrium addition, every everysolution solutionthat thatstarts startsinin[0, [o,oo) x) in at most two steps steps enters enters the the invariant invariant interval, interval, hence converges to the equilibrium. the invariant invariantinterval interval[1, [l,11++ A] A] is an absorbing absorbing interval, interval, and the difference difference Thus the is dissipative. dissipative. equation is The remaining case of

p>l, A<1

P

P

kp—1J

is qualitatively qualitatively different. shown that that equation equation (2.5.2) (2.5.2) has two periodic periodic different. ItIt can can be shown solutions, which are areasymptotically asymptotically stable stable and and global global attractors attractors solutions, each each if period 2, which with a basin of attraction (o, (o, x) and (x, \x\ ce) <») respectively.'1 respectively.11

Exercise 2.5 Showthat thatfor forthe thedifference difference equation equation 1 Show x 2(e)

x(t+1)= 1

+ x2(t)

,

the interval [o, [0,1] l] isisan anabsorbing absorbinginterval intervaland andzero zeroisisaaglobal global attractor. attractor. DeVault et e*aT a/. (1992). (1992). See DeVault

2. SCALAR LINEAR DIFFERENCE EQUATIONS

54

higher order order 2.6 Linear difference difference equations of higher The normal form form of of aakA:th-order th-ordernonhomogeneous nonhomogeneous linear linear difference difference equation equation isis given by x(t + + k)+P]p1(t)x(t — x(t (t)x{t + kk-\)+-

+ + ppk(t)x(t) k (t)x(t) = g(t),

(2.6.1)

where p. definedfor fort t >t0t0 and where pt (t) and g(t) g(t) are arereal-valued real-valued functions functions defined and

for all tt >t0. Here, the sequence g(t) g(t) isiscalled calledthe theforcing forcing term, term,the the external externalforce, force, for all t0. Here, the control, or the the input input of of the the system. system. If If g(t) g(t) isisidentically identically zero, zero, then then equation equation (2.6.1) is said to be a homogeneous equation. equation. We We can canevaluate evaluate x(k) x(k) with

x(k) (o)x(k -1) — 1) 4k) == gg(O) (o) - p, pj (oM*

(o)x(o). P* (oMo).

Similarly, we can evaluate evaluate x(k x(k ++ i), l),and andsosoon. on.AAsequence sequence{x(t)} {x(t)}™isissaid saidtoto be a solution solution of equation (2.6.1) if it satisfies satisfies the equation. If

x(to\---,x(to + k - l ) are specified, we have the initial initialvalue value problem. problem.

Example Consider Consider + 1)l)— = t, — x(t x{t + 2)+ tx(t + - 3x(t) =

x(t + 3) 3) —

where the initial conditions are

x(1)=O, x(2)=—l, x(l)=0, x{2) = -\, x(3)= x(3) =1.l. It is straightforward straightforward to evaluate

x(4)= 20.9. x(7) ==20.9. *(4)=f, x(5)= * ( 5 ) = - | , xx(6)= ( 6 ) =—1, - | , ,(7)

2.6. LINEAR DIFFERENCE EQUATIONS OF 2.6. OF HIGHER HIGHER ORDER ORDER

55 55

homogenous difference difference equation equation Consider the homogenous x(t k — x(t + k) k) + + Plp1(t)xQ (t)x(t ++ k-l)+-

•.. + pk(t)x(t) p t (t)x(t) = 0. 0.

(2.62) (2.6.2)

Definition 2.6.1. 2.6.1. TThe functionsf fx (t), Definition h e functions ( t \ f 2 (t),.•., {t\

ffrr {(t) t ) are said said to t o be b e linearly linearly

dependent Sax for t>t t 0t0 ifif there are are constants a1, ax, a2, a2, dependent

zero, such ar, nnot o t all zero, s u c h that that

/ ( / ) = 0,

t10. t>t0.

Definition 2.6.2. A set set of of kk linearly linearlyindependent independent solutions solutions of equation equation (2.6.2) is called afundamental a fundamental set setof of solutions. solutions. It is often often difficult difficult to check check the linear linear independence independence of of aa set set of of functions functions using using the defmition. method to check check linear linear independence independence of of definition. We now introduce a simple method solutions using the so-called so-called Casoratian. Casoratian.

Definition 2.6.3. The The Casoratian Casoratian w(t) w(t) of the the solutions solutions

x1Q),x2(t),..., xx(t),x2{t),-, xr(t) is given by'2 by12

W{t) =

x(t)

x(t) x2(t)

x1Q+i)

x2(t+1)

x(t) xrQ) ...

Xr(t+l)

x1fr+ r —i) x2(t xr(t + 2(t +r —i) ...

r—

Example It is straightforward to check that the the sequences sequences (....3)' 21, 1, (-3)', 2',

1,

are solutions of of the the difference difference equation equation 7x(t + i) x(t + 3) 3) —- lx(t l) + 6x(t) 6x(t) = 0. 0.

x(t

121

This is the discrete analogue of the Wronskian in differential equations.

This is the discrete analogue of the Wronskian in differential equations.

56

EQUATIONS 2. SCALAR LINEAR DIFFERENCE EQUATIONS

The following calculation verifies the linear independence independence of of the the solutions solutions 11

wQ)= i

2'2 '

(-3)' (—3)' f-3)'

+1

2 ' +1

=_20(2)t(_3)1.

i

following Abel formula formula gives gives an aneffective effective method methodfor forveriQ,'ing verifying linear linear The following independence of solutions. Lemma 2.6.1. 2.6.1. (Abel's (Abel's lemma)'3 lemma)13 Let Lemma x (t) x(t) x1(t), x2(t),

,

x (t) X/jt),

be solutions be their Casoratian. Then for solutions of of equation equation (2.6.2) (2.6.2) and andlet let W\t) w(t) be for t > tt00

w(t)=

(_lt(''0)[fiPk(O]w&O). 1=10 V='o

)

We see pk{f) * 00 for t0, then the Casoratian Casoratian W(t) w{f) ^ 00 if and for all t > to, see that that if if pk(t)

W(to)*0. only if w(t0) 0. 2.6.1.The Theset set of of solutions solutions Theorem2.6.1. Theorem

x1(t),x2(t),..., xk(t) + of is is a fundamental set ifset andif and only only if forif some t0 ee Zt0 , the of equation equation(2.6.2) (2.6.2) a fundamental for some

w(t0)* Casoratian Casoratian wQ0)

0. 0.

Example Solvethe thedifference difference equation equation Example Solve x(t ++ 3) + 2) l) —- I2x(t) = 0. 0. 3) + 3x(t 3x(t + 2) -— 4x(t + i) l2xQ) = We veriQy' verify that We thatthe the solutions solutions

13 13

The to Elaydi Elaydi (1999). (1999). The proof is referred to

2.6. 2.6. LINEAR DIFFERENCE EQUATIONS OF HIGHER HIGHER ORDER ORDER

57 57

2', (-2)', (—37, (-3)', form aa fimdamental fundamental set setof ofsolutions solutionsofofthe theequation. equation.This Thisisisproved provedby bycalculating calculating

(-2)' (-3)' 2< 2' (—2)' (—3)' + 1 +I + (3)1+1 21+1 w(t) = 2' (-2)' (-3)' w(t)= 2/+2 (_2)t+2 (3)1+2

1

1

>w(o)=

1 1 1

1

22

=- 2 0 . - —2 2 33 =—20. 4 44 9 9

Theorem 2.6.2. (the (thefundamental fundamental theorem) theorem) If Pk pk{t)^0 for all all t > to, then Theorem 2.6.2. (t) 0 for t0, t0. D equation (2.6.2) has has aa fundamental fimdamentalset setof ofsolutions solutionsfor fortt >t0. Principle.If If x2(t), Superposition Principle. x1 x(t), x(t\x2 (2.6.2), then

,

; (t) are solutions of equation

xk{t) are solutions of equation

x(t)= a, are areconstant. constant. is also a solution of of equation (2.6.2), where where a,

D

xx(t), x2 x2(t), Let x1

, xk(t) fundamental set set of ofsolutions solutions of ofequation equation (t) be aafundamental solution of ofequation equation (2.6.2) (2.6.2) is isgiven by (2.6.2). Then the general general solution by

x(t) =

Xi(t).

for at. Any Anysolution solution of of equation equation (2.6.2) may beobtained obtained from from for arbitrary arbitrary constants constants a•. may be the general solution by by a suitable choice of the the constants constants a,. a,. We now examine nonhomogeneous nonhomogeneous equation equation (2.6.1). (2.6.1). ItIt is iscustomary to torefer to to general solution solution of ofthe thehomogeneous homogeneous equation equation (2.6.2) (2.6.2) as asthe thecomplementary complementary the general to nonhomogeneous nonhomogeneous equation equation (2.6.1). (2.6.1). Denote the thecomplementary complementary solution solution solution to by xc\t). Then we have

2. SCALAR LINEAR DIFFERENCE EQUATIONS

58

;

where x1 (t), x2 (t), (t) isis aa fbndamental where x,(?), x2(t), fundamental set set of of solutions solutions of of equation equation (2.6.2) (2.6.2) , xk{t) are arbitrary arbitrary constants. constants. A A solution solution of ofnonhomogeneous nonhomogeneous equation equation (2.6.1) (2.6.1) isis and a a,t are solutionand and will will be be denoted denoted by x,, xp (t). (/). called a particular particular solution

Theorem Theorem2.6.3. 2.6.3.Any Anysolution solutionof ofequation equation (2.6.1) (2.6.1) may may be be written written as as + p(t). x(t) = x(t) = xc(t)+x

The general general solution solution of of equation equation (2.6.1) is is given by

x(t) = x{t) = xc(t)+xp(t).

As we studied (t), we now studied how how to to find find xc(t), now learn learn how how to fmd find xp(t). We only only (t). We introduce (t). In this introduce the the method method ofofundetermined undeterminedcoefficients coefficients to to compute xp{t). this method, we first first guest guest the the form form of of the the particular particular solution solution and and then then substituting substituting this this function into the difference effective mainly mainly when when g(x(t)) difference equation. This method is effective g(x(t)) forms is a linear combination or products of terms, each having one of the forms 1 aa', , sin(bt), sin(bt), cos(bt), cos(bt\

t".

t".

Example Example Solve Solvethe thedifference difference equation equation

xQ 12x(t) = t2'. x(t + 2) + x(t + i) l) —- I2x(t) O!. characteristic roots roots are are p p1x = = 33 and p2 p2 == —4. - 4. Hence, The characteristic

xc(t)==ai3< a13' ++a2(-4)'. Wetry We try = C] c12' c2t(27. xp(t) = 2' ++c 2t{2). Substituting Substituting

this into equation (2.6.3) (2.6.3) yields yields

(2.6.3) (2.6.3)

2.6. LINEAR DIFFERENCE DIFFERENCE EQUATIONS EQUATIONSOF OFJ-IIGHER HIGHER ORDER ORDER 21+2 —

+

(t + 2)(27+2 } +

l$2' +

+

(t +

59

}

= t2',

or 6a32t —6a2t2' = t2'. (10a2 — (l0a 2 - 6 ^ ) 2 ' -6a2t2' =t2'.

As this this is held for we should should have have As is held for any any 1, /, we

10a2—6a1=O, —6a2=1. 10a 2 -dax = 0, - 6 a 2 = 1 . Hence Hence 55

1 , a, = — . 18' 1 18 2 6 The general solution is a, =

1

2

x(t)=a,3 + aJ-4)' x(t)=a3' t+a

_It(2)t -—2'--t

Exercise Exercise 2.6 2.6 1 Find Findthe theCasoratian Casoratianof ofthe thefollowing following functions functions 35t+1 +1 (i) e'; (i) 5t, 5', 35' , e<;

t5', p251 (ii) 5', 5', t5', « 2 5'. 22

For the following following equations equations and and their their accompanied accompanied solutions, solutions, determine determine whether whether these solutions are are linearly linearly independent independent and and find, find, ififpossible, possible,the the general general solution: solution: 3)— i) —- x(t) == 0; (i) x(t + + 3) - 3x(t + 2) 2) ++ 3x(t 3x(t ++ l) 0; 1, t,t, t2; t2;

3)+ (ii) x(t + 3) + x(t x{t ++ 2)— 2) - 8x(t + i)— l) - l2x(t)= 12JC(/) = 0; 0; 3', 3 ' , (2)', (_2)t+3 (- 2)' +3 . (—2)', 3 Find Find the the general general solutions solutions of of the the following following difference difference equations equations

60

2. SCALAR LINEAR DIFFERENCE EQUATIONS EQUATIONS

(i) = 8(2)'cos(?;z72); (i) x(t + 2) + 4x(t) 4x(t) = t(2)'. (ii) + 2} + 8x(t 8x{t + l) lx{t) = t(2)'. (ii) x(t + 2) + i) + 7x(t)

2.7 Equations Equationswith with constant constantcoefficients coefficients We consider equation (2.6.2) when pi are when p. are constant x(t + k) p,x(t + k — - i)+ l) + •••+ + Pkx(t)=0, pkx(t) = 0, x(t k)++ p1x(t+

(2.7.1)

Suppose that solutions of p'. where pk * 0. 0. Suppose that solutions of equation equation (2.7.1) (2.7.1)are areinin the the form form p'. where J)k Substituting intoequation equation(2.7.1) (2.7.1) yields yields Substituting p' into + ^yO /? +

+ +

+ pk = 00.. +

{Z.I.2) (2.7.2)

This is called called the characteristic equation This equation of of equation equation (2.1.7) (2.1.7)and andits itsroots rootspp are called the characteristic roots. characteristic roots. Case a: a: IfIfthe thecharacteristics characteristics roots roots {p1, {px, p2, p2, shown that the set

pk}) are aredistinct, distinct, then then ititcan canbebe

is a fundamental fimdamental set of solutions. Hence, the general solution of equation (2.7.1) is

x(t)= Example Example (the Fibonacci sequence and the rabbit problem) The problem appeared in 1202 1202 in in aa book book about about the theabacus, abacus,written written by bythe thefamous famousItalian Italianmathematician mathematician Leonard di Pisa, known as Leonard as Fibonacci. Fibonacci. The The Fibonacci Fibonacci problem problem is as as follows: follows: how how many pairs of rabbits will there be after one year if starting with one pair of mature rabbits, if each pair of rabbits gives birth to a new pair pair each each month month starting when it reaches its maturity maturity age age of of two two months? months? If If x(t) xQ) is the number of pairs of rabbits rabbits at at the end of tt months, months,then thenthe therecurrence recurrencerelation relationthat thatrepresents represents this this model model is is given given by the second-order linear difference equation by the second-order linear difference equation

2.7. EQUATIONS WITH CONSTANT CONSTANT COEFFICIENTS 2.7. JC(2)=2, x{t + 2)=x(t + l)+x(t), x(o)=1, * ( 0 ) = l , x(2)=2, x(t+2)=x(t+1)+x(t),

61 61

0 < f <10.

sequence This equation is a special case of the the Fibonacci Fibonacci sequence

x(t+2)=xQ+1)+x(t), x(t + 2) = x(t + i)+x(t), x(O)=1, x ( 0 ) = l , Jx(2)=2, C ( 2 ) = 2 , tO. t>0. The characteristic equation is is p2 —p—1= - p - 1 = 0. 0. p2 are The characteristic roots are

P1,2

2

2

is The general solution is

fi(1 VsY+ (1 fi VsY 2)

= a, — + —

\2

2)

+ a,

\2

-!!2— .

2)

Substituting Substituting the the initial initial conditions conditions into into the the general general solution solution leads leadstoto aa12 = ± V5. x2 = I.?. Case b: Suppose Suppose that that the the distinct distinct characteristic characteristic roots roots are p1, /?,, p2, p2,---,

pPr with r with multiplicities in1, mx, in2, m2, •', in,.. mr. Then, the the general general solution solution of equation equation (2.7.1) (2.7.1) is is multiplicitics given by by

x(t) = Ep1'

+ a,1! + ai2t2

+

+ aim

Solve Example Solve x(t + + 3) - 3x(t 3x(t + 2)+ 2) + 16x(t 16x(? + l) - 12x(?) 0, x(t 3)— i)— l2xQ)== o, JC(0) = 0 , jc(l)=l, x(2)=l. x(1)=1, x(2)=l. x(0)=0, It is straightforward to check that the characteristic characteristic roots are are

62

2. SCALAR LINEAR LiNEAR DIFFERENCE DIFFERENCE EQUATIONS > p3=3. P-i = p1=p2=2,

A = Pi=

2

3

The general solution is 1 + ajl x(t)=a x(t)= a02' a1t2' + a a33'. o2' + 33'.

Substituting the initial conditions into the general solution, we solve

a0 3 , a1=2, a, = 2 , a3 a3 =—3. =-3. a0 ==3, The The solution solution of of the the initial initial value value problem problem isis 31+1• x(t) = = 3(2)' + + t2M — - 3M.

Example Kobayashi and Anders have agreed to bet one dollar on each flip flip of a fair fair continue playing until one of them wins all of the other's money. What coin and to continue is the probability that Kobayashi will win all of Anders' money if Kobayashi starts

dollarsand and Anders Anders with b dollars? dollars? with aa dollars To analyze this game, we denote x(t) the the probability probability that that Kobayashi Kobayashi will win all of has t dollars. of Anders' Anders' money money ifif Kobayashi Kobayashi currently currently has dollars. Let A A = aa ++ b be the total amount amount of money money available available to the the players. players. Note Note that that x(0) x(o) = = 0 because because total X(A) = = 11 because Kobayashi has all the money. Kobayashi has no money left, left, and x(A) Moreover, ifif 1 < t < A - 1, then Kobayashi has has a 0.5 probability of winning one dollar on the next flip (raising the amount of money he has has to to t + + 11 dollars) dollars) and 0.5 probability of losing one the amount of money he dollar flip (reducing (reducing the amount of dollar on the next flip the amount of money money he he has has to tt - 1 dollars. dollars. Hence 1

x{t) = 0.5x(t + l) + 0.5x(t - l), 1 < t < A - 1. Rearrange the above equation x(t + l) - 2x{t) + x(t - l) = 0, 1ltA—l. < t < A - 1. x(t+1)—2x(t)+x(t-.1)=o,

2.7. EQUATIONS WITH CONSTANT COEFFICIENTS

63 63

The characteristic equation equation

p2 -—2p+l=O, 2p + 1 = 0, p2 has a double root, root,

x(t) =

(a1

solution is p]2 == 1.1. Hence, the general solution p12

+ a2t)l'.

From x(o)=o =00 and From x(o) = 0 and and x(A)=l, X(A) = 1,we wedetermine determine a1 ax = and a1 ax =1/A. =1/ A. We We thus thus obtain t

a+b When Kobayashi has a dollars, Kobayashi has dollars, the the probability probability of Kobayashi's Kobayashi's winning the game game is is

x(a) =

a

-,

a+b a +b

and the probability probability of Anders' Anders' winning the game is

1—

a

a+b a +b

=

b

a+b a +b

Case Case C: c: Suppose that the characteristic characteristic equation has complex roots. For simplicity, simplicity, we we are concerned concerned with

x(t ++ 2) 2) ++ p1x(t pxx{t ++ i) l) + p2xQ) p2x{t) == o. 0. The The complex roots roots are are given given by by

p12 A.2 ==a±i/3. WIt can be shown that the general solution is given by

64

2. SCALAR LINEAR LiNEAR DIFFERENCE EQUATIONS Art cos(et x(t) x(t) == Ar' cos(fft -— a),

arbitraryconstants constantsand and where A v4 and and wa>areare arbitrary

r=Ja2+/32, We We have been concerned with with finding finding solutions solutions of of linear linear difference difference equations. equations. Sometimes nonlinear Sometimes nonlinear difference difference equations equations can can be be transformed transformed to to linear linear ones. We equations transformable transformable to to linear linear equations. equations. discuss some types of nonlinear equations

Example (equations (equations of of Riccati Riccati type) Consider Example

x(t ++ I)x(t) lMO + p(t)x(t P(f)x{t ++ i) l) + q(t)x(t) qif)x(t) = = 0. 0. Introduce = 1/x(t). Equation Introduce z(t) z(t)=\lx(t). Equation(2.7.3)becomes (2.7.3) becomes q(t)xQ + i) + p(t)z(t) + 11 == 0.0. For the nonhomogeneous equation

x(t ++ l)x{t) 1)x(t) + + pQ)xQ p(t)x(t ++ 1) 1)++ q(t)x(t) = g(t), g(t), we introduce introduce

x(t)=

z(t + i)

—p(t).

The above above equation equation becomes becomes The z(t ++ 2) + (q(t) {q(t) —- p{t ++ 1))z(t \))z{t ++ 1) l) —- (g(t) {g{t) ++ p(t)q(t))z(t) pif)q{t))z{f) == 0. 0. Pielou logistic logistic equation equation Consider the Pielou /

xv +

1)

=

av(t)

1+/lv(t)

(2.7.3) (2.7.3)

WITH CONSTANT CONSTANT COEFFICIENTS COEFFICIENTS 2.7. EQUATIONS WITH

65 65

x{t) == 1/ \l z(t), z{t), this equation becomes Under transformation transformation x(t) a z(t z(t + i) l) == z(t) ++/7. p. Example (equations of of general general Riccati Riccati type) Consider Example (equations

a1(x)x(t)+a2(x)

{2.1 274A)

a3{x)x{t)+a,{x) where where a3{x)*0, 0, aa1(x)a4(x)— l(x)aA{x)-a2{x)a2(x)*Q,

0,

t t>0. 0.

To solve this equation, let

y(t + I) + a4Q) = y(t)

Under this transformation, equation (2.7.4) becomes y(t ++ l) 0, i) + p,{t)y(t + l) i) + p2(t)y{t) == 0, where a3(t)a4(t + +$ i)++ PiiM a1(t)a3(t + + $ i) (,)\ s— _ OikM ait) a3V)

(a1(t)a4(t) — a2Q)a3(t))

a

(t + i) a3

Consider Consider I

\

2x(t)+3 3x(t) +

2

Using the transformation

2. SCALAR LINEAR DIFFERENCE EQUATIONS 2.

66

y(t+i)

3x(j)+2= 3x (,) +2 =i^J), y(t)

we see see that the the equation becomes + 2)

—

4y(t + i)

—

5y(t) =

0.

Example of the type Example Homogeneous Homogeneousdifference difference equation equation of

x(t+1)

4)

—o '

—

Use the the transformation transformation

z(t)=

x(t+1) x(t)

to convert such an an equation an equation to an equation in inz(t). z(t).

Exercise 2.7 Exercise 2.7 1 Introduce Introduce the the symbol symbol E"x(t) Ekx(t) == x(t x(t ++k). k). Find Find the thegeneral general solutions solutions of of the the following difference equations

(i) x(t 16x(t) = = 0;0; *(/ ++ 2) 2) —- \6x{t) 3)2

(E22 +4>(/)=0; (ii) (E — 2 (£ + 4)r(t) = 0; (ii)(£-3)

(iii) (E22 ++2fx{t)=0. 27 x(t) = 0. (iii)(£

2 Solve 2 Solvethe thefollowing following difference difference equations equations 2 (i) x2(t (2 + 2tr2(t) t)xfr + = 0; 0; x2(t ++ i) l) —- (2 + t)x(t + l)x(t) —- 2tx (t) =

(ii) xQ + l)x(t) —

(iii) x(t + i) =

x2

2

(t);

1

5

+ i) + —x(t) = —;

2.7. EQUATIONS WITH CONSTANT COEFFICIENTS

(iv)

67

+ 2) =

(v) xQ +

2.8

1)

=

1 —

x2(t)/2

1—x(t)

Limiting behavior

For simplicity, we first first examine examine limiting limiting behavior behavior of of two-dimensional two-dimensional difference difference equations. We now examine limiting limiting behavior of of solutions solutions of of the the following following secondsecondorder linear difference difference equation

x(t ++ 2) 0. 2)++ p1x(t pxx{t + i) l) + p2x(t) p2x(t) == 0.

(2.8.1)

Suppose and p2 p2 are are the the two twocharacteristic characteristic roots roots of of the the equation. equation. Suppose that that pp1 x and Following the previous section, we have the following three cases. Case and p2, p 2 , are aredistinct, distinct,then thenthe thegeneral general solution solution Case a: a:IfIfthe thecharacteristic characteristic roots, roots, p1 px and of equation (2.8.1) is (2.8.1) is

x(t)= a1p ++ a2p'2. x{t)=a,p{ Without loss loss of generality, assume \px >> \p2\. Then

x(t) = [ai +

Since jp2 I < 1, it follows that p2/p1\
— I -> 0 as t -^ oo. ¼¼P1)

Consequently

68

2. SCALAR LINEAR DIFFERENCE EQUATIONS EQUATIONS

limx(t)= limx(;) =lima1pjt. lima,/?,'. (-*00 different situations that may arise depending on the the value value of of /?,. p1. There are six different (1) If p1 >> 1,1, the sequence If/?, sequence {a1p; {a,/?,'}} diverges diverges to oo °° (unstable system).

(2) (2) If If p1 /?, =- 1,1, the sequence sequence {a1p; {a,/?/}} is is a constant sequence. } is monotonically decreasing to zero (3) If 00 << /?, < 1, the thesequence sequence{a1p: {a,/?,'} monotonically decreasing zero p1 <1, (stable system). system). (4) <0,0,the (4) If—i If -1 << p1 /?, < thesequence sequence{a1p1'} {a,/?,'} is is oscillating around zero (i.e., (i.e., altering altering in sign) and converging to zero zero (stable (stable system). system). (5) If p1 = —1, the sequence (5) If /?, = - 1 , sequence {a,/?,') is oscillating oscillating between between two values values a1 ax and and a2 .

(6) IfIfp1 } is oscillating \axp[} oscillating but increasing increasing in magnitude magnitude /?,<—1, < - 1 ,the thesequence sequence{a1p; (unstable system).

Case b: pp = pp1x == p2. p2. The general solution solution of of equation equation (2.8.1) (2.8.1) is is given given by Case b: x(t) = a2t)p'. x{t) = (a1{a+x+a 2t)p'.

|/?| > 1,1, the solution solution x(t) x(t) diverges diverges either either monotonically monotonically ifif p >1 1 or by by If Ipi

oscillating oscillating ifif pp <—1. - 1 .However, However, ifif |/?| c< 1,1, then thensolution solution converges converges to zero. Case Case c:c:Suppose Suppose that that the the characteristic characteristic equation equation has has complex complex roots, roots, denoted denoted by

p1,2 p l 2 =a± i/i. ifi. The general solution solution is given by x(t) x{t) == Ar' cos(9t cos(fft —- a), where A and where A and wa are arearbitrary arbitrary constants constants and ÷132

J§_ a

" [a

2.8. LIMITING LIMITING BEHAVIOR

69

(1) If x(t) isisoscillating oscillating but but increasing increasing in in magnitude magnitude (unstable (unstable If r >>1,1, the solution x(t) system). (2) If If r == 1,1, the solution x(t) x(t) isisoscillating oscillating but but constant constant in in magnitude. (3) If If r < .c 1, 1, the solution solution x(t) isis oscillating oscillating but but decreasing decreasing in in magnitude magnitude (stable (stable system). Example (gambler'sruin) ruin)AAgambler gamblerplays playsaasequence sequence games games against an adversary Example (gambler's in which the probability that the gambler gambler wins wins $1.00 in any given given game is a known value q, and HeHe value and the theprobability probability of ofhis hislosing losing$1.00 $1.00isis1 1— - q, q, where where 00
p(t) = qp(t +

1)

+

(1 —

q)pQ

—

i).

We have have 1

p{t + 2)--p{t + l)+ p(t+2)——p(t+1)+ qq

1—q X -^3-p{t) = 0 , tt=0,l,.••,N, = 0,l,-,N p(t)=0, qq

with p(0) p(o) = = 1I and and p(N) P(N) == 0. equation is is with 0. The characteristic equation

p2 - I 2

q

1—q qq

p +1^1

= 0,0 t = 0,l,-,N.

The two characteristic roots are A=

\-q 1—q -» qq

Pi = 1 -

1 / 2, the thegeneral generalsolution solutionisis In the case of q * 1/2,

_____ 70

LINEAR DIFFERENCE EQUATIONS EQUATIONS 2. SCALAR L[NEAR

p\t)

, ^ r »

1—q

i

q

q

\-q q

the two characteristic roots are equal. In this case,

In the case of q = 1 / 2, the two characteristic roots are equal. In this case, p(x) = a1 + a2t. p(x) = a, + a2t.

As p(0) = and p(N) = 1

0,

we have

As p(o) = 1 and p(./V) = 0, we have p{n)=l--. For instance, instance, ifif one starts with dollar and probability that one starts with $4 dollar and the probability that one one wins wins a 0.3, and andone onewill willquit quitififone oneruns runs out out of of money money or has a total of $10.With With dollar is 0.3, of$10. nn=4, = 4, q=O.3, q = 0.3, NN=10, = 10, the probability of being being ruined ruined isis p(4) == 0.994.

We now now examine examine nonhomogeneous nonhomogeneous difference input is difference equations equations in in which which input constant. We have x(t + l) x(t + + 2) + Plp1x(t ii) + pp2x(t) 2x(t) = b,

(2.8.2) (2.8.2)

where bb isisaanon-zero non-zeroconstant. constant. AAparticular particular equation equation (equilibrium (equilibrium in this case) x* is determined by x + pxx + + p2x = b. b. That is b

*

x

=

1 + A + Pi l+p1+p2 Consequently, the general solution of equation (2.8.2) is given by

2.8. LIMITING BEHAVIOR + x{t) = = xc(t)+x, x(t)

71 71 (2.8.3)

general solution solution of of the the corresponding corresponding homogeneous homogeneous equation equation where xc(t) (t) is the general (2.8.1). From formula formula (2.8.3), (2.8.3), we we conclude conclude that thatx(t) x(t)—* —» x*ififand andonly onlyifif xc{t) —> 0 x{t)oscillates oscillatesaround around x*ififand andonly onlyifif x(t) oscillates as t —> °o. Furthermore,x(t) c(t)oscillates 14 about zero.14 zero. Summarizing Summarizing the the above above discussions, discussions, we we obtain obtain the the following following theorem. Theorem 2.8.1.The Thefollowing following statements statements are are held. Theorem 2.8.1. (1) All All solutions solutions ofofequation equation (2.8.2) (2.8.2) oscillate oscillate about about x* if only only if none none of of the the equation (2.8.1) (2.8.1) is is aa positive real number. number. characteristic roots of equation —*>oc°° ififand andonly onlyifif Allsolutions solutions of of equation equation (2.8.2) (2.8.2) converge (2) All convergetoto x* x asasitt —

where px, pp22 are the characteristic roots of homogeneous equation (2.8.1). The following following lemma provides criteria for stability stability based based on the values of the 15 p2. coefficients p1 px and p2. coefficients Lemma 2.8.1. The conditions Lemma 2.8.1. + p2 >0, > 0,I—p1 1 - px ++ p2 p2 >0, > 0, i—p2 1 - p2 >0, > 0, 1l+p1 + pl +p2

(2.8.4) (2.8.4)

are necessary necessary and sufficient sufficient conditions conditions for solution of equations equations (2.8.1) (2.8.1) and for the solution (2.8.2) to be asymptotically stable (i.e., all all solutions solutions converge converge to to x*). Example (the (theSamuelson Samuelsonmultiplier-accelerator multiplier-accelerator interaction interaction model) model) Consider Consider the the national national income income model. model. The The national national income income Y(t) Y(t) at at year year tt is equal equal to to the the sum sum of consumer constant government government investment, l(t), 1(t), and constant consumer expenditure expenditure C(t), c(t), private investment, expenditure, That is expenditure, G. That Y(t) = c(t) + 1(t) + 6.

14 14

We x{t) oscillates oscillates about about xx' if x(t) x{t)—— x*alternates alternatessign, sign,i.e., i.e., ififx(t)> x(i)>x', then x, then We say x(t) x(t + l) I) < c x'.

15 15

is referred referred to to Elaydi Elaydi (1999: (1999: 82). 82). The proof is

2. SCALAR LINEAR DIFFERENCE EQUATIONS EQUATIONS

72

Assume that that consumption expenditure isis proportional to the national Assume consumption expenditure proportional to national income income Y(t — - 1i)) in the preceding year, that is

c(t) i), C(t)== aY(t -— l), whereaa ((00 < aa
i(t) = /3[C(t) — C{t-1)], c(t — i)], ji> 0. Substituting this equation and

C(t)=aY(t_1) C(t) = aY{t - l) into into

v(t) = c(t) + 1(t) + G yields

Y(t + 2)— a(1 + /3)Y(t 2) - «(l 0)Y(t + i)+ l) +cc'/Jv(t) a/3Y(t)==1.1. The equation has a unique unique equilibrium equilibrium state state Y* Y given by 1

1-a \-a Applying lemma 2.8.1 2.8.1 to to the the problem, problem,we weconclude concludethat thatififaa < c 1 and e43 Applying lemma afi c<1, 1, the the equilibrium point point is asymptotically equilibrium asymptotically stable. stable. The national national income income Y(t) Y(t) fluctuates fluctuates and only if around the equilibrium state Y* ififandonlyif

Ap ac (1+,o)2 As we can can explicitly explicitly solve solve the the problem, problem, we we can can examine examine its its stability stability properties. properties. Figure Figure 2.8.1 2.8.1 shows shows the the stability stability properties properties of the the Samuelson Samuelson multipliermultiplieraccelerator interaction model.

2.8. 2.8. LIMITING BEHAVIOR BEHAVIOR

a

73 73

a/3=I uniform stepped stepped|fluctuation fluctuation unstable; no cycle cycle

a

418

explosive stepped fluctuation steppe4 fluctuation damped stepped

1

1

2

33

2

Figure 2.8.1: Stability of the multiplier-accelerator multiplier-accelerator interaction interaction model model

th-order scalar scalar equation We now consider the kk th-order

x(t+/c)+ x(t + k) + p1x(t+k plx(t + k —1)+p2x(t+k - l) + p2x(t + k - 2) +

+ Pkx(t)_0, pkx(t) = 0,

(2.8.5) (2.8.5)

where p, where pt 's's are are real real numbers numbers and Pk pk * 0. The The characteristic characteristic equation equation of equation equation (2.8.5) is +

=

+

+

= 0. 0. + Pk =

(2.8.6)

As discussed discussed in section 2.7, if the distinct characteristic roots are As if the in,., then, the the general of /?,, P2' p 2, pr with multiplicitics multiplicities in1, mu in2, m2, m general solution solution of r, then, Pi' Pr

equation (2.7.1) is given by r

+ a,1t + a,2t2

x(t) =

+ ... +

a,m_itmi_1). ai

1=1

It can can be be shown shown that that the thezero zerosolution solution of ofequation equation (2.8.5) (2.8.5) isis asymptotically asymptotically stable if and and only only if if \p\ < ci 1for stable forevery everycharacteristic characteristic root p. p. Furthermore, Furthermore,the the zero zero solution (2.8.5) is is stable stable ifif and and only only if |/?| <1 1and solution of equation equation (2.8.5) andthose thosecharacteristic characteristic

1). On the other other hand, if roots with Ipl |p| = I1 are are simple simple (i.e., (i.e., not repeated, repeated, in. mi ==1). roots pp with

2. SCALAR LINEAR DIFFERENCE EQUATIONS

74 74

there is aamultiple multiple characteristic characteristic root with with |/?| == 1,1, then the zero zero solution solution of there is then the equation (2.8.5) important to whether the is important to judge judge whether equation (2.8.5) isis unstable. unstable. We We see see that that it is characteristic roots fall inside the unit cycle. The Schur-Cohn criterion an characteristic roots fall inside cycle. The Schur-Cohn criterion gives an answer. answer. Before introducing A= = [a,j Before introducingthe the criterion, criterion,we wedefine defmethe theinners irners of of a matrix A [a11 j.\. The inners inners of a matrix are the matrix itself and all the matrices obtained by by omitting omitting The successively the first first and last rows and the first first and last last columns. columns. For For instance, instance, the the successively inners [a,-,133 ] 3x3 consists [a;> j133 and the scalar a . For l«^J , they are inners ofof[a11 consists of of [a11 and the scalar a22. For [a11 j are 3x3 22

a11

a12

a13

a14

a15

a21

a22

a23

a24

a25

a22

a23

a24

a31 -*31

a32 a32

a33 a:33

a1134

a35

a32

a33

a34 ,

a41 "41

a42 "42

a43 "43

a44

a45

a42

a43

a44

a,, a51

a,, a52

a,, a53

a. a54 '54

a55

a33.

A matrix matrix A is is said said to to be bepositive positive innerwise innerwise if the determinants determinants of all of its its inners are positive. positive. Theorem 2.8.2. 2.8.2. (the (the Schur-Cohn Schur-Cohn criterion'6) criterion16) The zeros of of characteristic characteristic equation equation (2.8.6) lie inside and only only if if we we have: have: (i) p(1)> p(\) >0; 0; (ii) inside the the unit unit cycle cycle ifif and

(1)k(1)> (-if p(-1) > 0; and (iii) (iii) the the (k(k—-l)x(k-l) 1)x(k — i) matrices 0; and 11

00

0

0

00

p1 A

11

00

0

00

0

Pk Pi

_A Pk

Pk-X Pk-l

... o Pk Pk Pk-i •"PkPk-1

4-i = Pk-, Pk-3

P Pk-2 k- 2

Pk-3

A p1

1 1_

innerwise. are positive innerwise. Example For Forthe theequation equation Example x(t + 2) 2) + + p1x(t pYx(t + 1) l) + p2x(t) p2x(t) == x(t +

16 16

See Jury (1964).

0,

A p3A

A p2

2.8. LIMITING BEHAVIOR

75 75

the Schur-Cohn criterion is given by

+p2 >0, (-lfp(-l) = l-Pl+p2 +p2>0, >0, p(\) = \ + /

\2

I

P]+p2>0,

\

> 0. A? =l±p2 = 1 p2 >0. We thus conclude that the zero solution is if is asymptotically stable if if and only if |A|

< ! + Pi <2 -

Okuguchi and Irie show that the necessary and sufficient sufficient conditions for for a third order difference difference equation x(t + 3) + pxx{t + 2) + p2x(t + i) l) + + pp3x(t) = 0, 0, 3x(t) = 17 are given by by'7

1+ 1 + p1+ A +p2 Pi ++p3 Pi >>0,0, l-Pi+P2-Pi>0, 1—p2 1 - Pi + PiPi ~ p] >0. > 0.

(2.8.7) (2-8-7)

Example (inventory (inventory cycles) cycles) The The economy economy has has a single commodity commodity which which can be be used either for for consumption consumption and andinvestment. investment. The Thebasic basic equation equation of ofMetzler Metzler used either equation is'8 is18 Y(t) = u(t) + (O(t)

- Q(t - 1)) +

period t,t, u{i) u(t) is is output to to be where Y(t) Y(t) is is the output in in period becurrently currently sold sold according according

on sales, sales, O(t) Q(t)is isthe thedesired desired level level ofofinventories, inventories, to producers' producers' expectations expectations on is exogenous output Q(t —- l) is the inventory inventory level of ofthe previous period, and and / 0 is i) is of investment goods. Assume Assume that that producers wish to to maintain maintain aa constant constant ratio ratio of investment goods. producers wish between inventories inventories and and sales sales (the (the inventory inventory accelerator). accelerator). Producers Producers apply apply this ratio in forming the desired level of ofinventories as follows follows 17 17

See Okuguchi and Irie Irie (1990). (1990). Metzler (1941) proposed one of Metzler ofseminal seminal models models for forexplaining explaining inventory inventory cycles. cycles. The The model here is is referred to Gandolfo (1996: 95-98). 18

76

2. SCALAR LINEAR DIFFERENCE DIFFERENCE EQUATIONS EQUATIONS — = kU(t W{t-\), O(t)=

period t,t, the the actual actual inventory inventory level level Q{t) Q(t) is where Ic& is a constant. constant. In any period is equal to

the level level that producers producers had had planned planned for for that that period period Q(t), Q(t),minus minus the the unintended unintended variation in inventories inventories occurring occurring because the difference difference between between realized variation because of the realized and and expected sales, i.e. sales, cQ) C(t)—- U(t), U(t), i.e. Q(t) = ä(t) — (c(t) — u(t)). Q(t)=Q{t)-(c(t)-U{t)).

Producers form form their their expectations expectations on onsales sales as asfollows follows U(t) c(t -— i) U(t) == C(t \) ++

— 1)— c(t — 2)), a(c{t-1)-C{t-2%

where c(t) C(t)isisthe thecurrent currentconsumption consumption level. level. No Nolag lagisisassumed assumed to to exist exist between between and current income. The Theconsumption consumption function function is thus given consumption and current consumption by

c(t)=bv(t), O
v(t) k)Q + l]Y(t -—1i)) + + k)(1 + 2a)Y(t Y(t) —- b[(i b[(\ + + k)(\ + a) + + \}Y(t + b(l b{l + k){l + 2a)Y(t -— 2) —cth(1+k)Y(t—3)=10. A particular solution is by is given by

y*=Io 1—b \-b

are Applying conditions (3.8.7) to to this problem, we we see seethat the the stability conditions are given by

33—b(2k+3)>O, - b{2k + 3) > 0, 2 (1 + k)(2 k)eib2 — > 0. 0. (l Jfc)(2 ++ k)ab - (1(l + + k)(l jfc)(l ++ 2a)b 2a)b + + 1> 1

2.8. LIMITING BEHAVIOR

77

Exercise 2.8 2.8 1 Determine Determine the the equilibrium equilibrium points, points, their stability, stability, and oscillatory oscillatory behavior of the the following equations equations solutions of the following 2) - 2x{t ++ 1)+2x(t)= l) + 2x(t) = 1; (a) x(t + 2)—2x(t 1; Q.Sx{t)== - 5. x{t + + 2) + (b) x(t + x(t + l) 1) + o.sx(t) 2 2

Determine the limiting behavior of solutions of the following following equation x{t + 2) - aj3x{t + l) + a/3x{t) = ad, a,a,/3, 0, 6>0, 6 > 0, x(t+2)—crflv(t+l)+aflv(t)=aO,

if (i) a/I a/3 = = 1;1; (ii) (ii) a/I a/3 == 2; 2; and (i) if(i) (i) or/3 a/I == 0.5. Show that that the the zero zero solution solution of 3 Show

x(t + 3)+ 2)++ p2x(t x(t 3) + pp1x(t p2x(t ++i) l) ++ p3x(t)=0, p3x(t) = 0, xx{t + 2) is asymptotically stable if and only if |/7, +pi\
+ p2,

\p2 -

\

1

]

Chapter 3 Chapter

One-dimensional dynamical economic systems concepts and theorems of of the the previous previous chapters chapters to to analyze analyze This chapter applies the concepts different models economics. Although the economic economic relations in these these models models different models in economics. complicated, we show that the dynamics of all these models are determined by are complicated, one-dimensional difference one-dimensional difference equations. equations. Section Section 3.1 3.1 examines examines a traditional traditional model model of interactions between inflation and unemployment. The model is built interactions between inflation and unemployment. The model is built on on the expectations-augmented version version of of the the Phillips Phillips relation relation and and the the adaptive expectations hypothesis. the characteristic characteristic expectations hypothesis.We Wesolve solvethe the model model and and show show that that the equation may have either: (1) distinct real roots; or (2) repeated real roots; or or (3) (3) equation may have either: (1) distinct real roots; or (2) repeated real roots; complex roots. Section 3.2 introduces the one-sector growth model. The model complex roots. Section 3.2 introduces the one-sector growth model. The model is different different from from most most of of the the growth growth models models in in the the literature literature in in that that itit treats treats is saving as an endogenous variable through introducing wealth into utility saving as an endogenous variable through introducing wealth into utility function. stable equilibrium. equilibrium. frmnction.We Wedemonstrate demonstratethat thatthe theOSG OSGmodel modelhas has aa unique unique stable Section 3.3 generalizes the OSG model proposed in section 3.2. Section 3.4 3.4 Section 3.3 generalizes the OSO model proposed in section 3.2. Section deals with the overlapping-generation (OLG) model one of the most popular deals with the overlapping-generation (OLG) model — one of the most popular dynamic models to dynamic models among among economists. economists.The The model model isis essential essential for for the the reader reader to approach some of the models in this book as well. Different from the OSG approach some of the models in this book as well. Different from the OSG models in in the the previous previous two twosections, sections,ininthe theOLG OLGanalytical analyticalframework, framework, each each models person lives for only two periods. This is the main shortcoming of the model; person lives for only two periods. This is the main shortcoming of the model; nevertheless, popularity is is sustained sustained partly because this this framework framework often often nevertheless, its its popularity partly because simplifies complicated analytical issues. Section 3.5 introduces a growth model to simplifies complicated analytical issues. Section 3.5 introduces a growth model to demonstrate persistence model, the the evolution evolution of of income income demonstrate persistence of of inequality. inequality. In In this this model, within each dynasty in society is governed by a dynamical system that generates within each dynasty in society is governed by a dynamical system that generates a poverty trap equilibrium point along with a high-income equilibrium point. a poverty trap equilibrium point along with a high-income equilibrium point. Poor dynasties, those with the income at the threshold level, converge to a low Poor dynasties, those with the income at the threshold level, converge to a low income steady state, whereas dynasties with income above the threshold income steady state, whereas dynasties with income above the threshold converge to a high income level. Section 3.6 studies a model to provide insights converge to a high income level. Section 3.6 studies a model to provide insights 79

80

3. ONE-DIMENSIONAL SYSTEMS ONE-DIMENSIONAL DYNAMICAL ECONOMIC SYSTEMS

of Schumpeterian Schumpeterian creative creative destruction. destruction. Section Section 3.7 3.7 isis concerned concerned into processes of with interactions between between human human capital capital accumulation, accumulation, economic with interactions economic growth, growth, and and inequality. The model exhibits three possible equilibrium points: a low-growth trap, a pair of equilibrium points in the intermediate and advanced development phase. If these equilibrium points exist, it can be shown that the poverty trap is is stable, stable, while while development phase, the the fir first the second second in development St equilibrium equilibrium point point will will be unstable and the one stable. stable. Section Section 3.8 highlight how how the the one 3.8 studies studies an an urban urban dynamic dynamic model model to to highlight trade-off between optimal and equilibrium city sizes behaves when human trade-off between optimal and equilibrium city sizes behaves when human capital externalities are introduced into urban dynamics. Section 3.9 3.9 introduces a economy. The model addresses addresses the Tobin Tobin effect effect and and growth model of monetary economy. existence of of monetary monetary economy. economy. Section Section 3.10 3.10 introduces introduces endogenous endogenous time time the existence and work into into the the OSG OSG model. model. distribution between leisure and

3.1

A model model of of inflation inflation and and unemployment

1 inflation and unemployment. We now consider an interaction of inflation unemployment.' Denote the rate of inflation whichisisdefmed defined by inflation p(t), which

f\

PQ+1)—P(t) P(t) '

where P(t) P(t) is the the price. price. The Theexpectations-augmented expectations-augmented version the Phillips Phillips where version of of the relation assumes assumes the following following relationship relationship between the rate of inflation, the between the rate of inflation, the unemployment andthe the expected expected rate of inflation, unemploymentrate, rate, u(t), u(t), and inflation, m{t) ir(t)

p{t)=a-bU(t)+h7r(t), (0 < h < p(t)= a — bU(t)+ hir(t), (0< 1),1),

(3.1.1)

where and hhare arepositive positiveparameters. parameters.The Theadaptive adaptiveexpectations expectationshypothesis hypothesis where a, b, and establishes a rule of the expected rate of inflation as follows n{t + \)-7t{t)=j{p{t)-7z(t)), + 1)— 7r(t) = j(p(t) —

00 <<;j< l 1,,

(3.1.2)

which parameter. The equation states actual rate of of inflation inflation The equation states that that ifif the actual which jj isis a parameter. exceeds the expected expected rate of inflation, inflation, then expected rate inflation tends to exceeds then the expected rate of inflation rise. Denote the andits itsrate rate of of growth growth by the nominal nominal money money balance balanceby by M\t) M(t) and 1

model isis based based on onChiang Chiang(1984: (1984:59 591-596). The model 1-596).

3.1. 3.1. A MODEL OF INFLATION AND UNEMPLOYMENT

/

iX

m(t + 1) =

81 81

M(t + \)-M(t) M(t) + i)— i

'-pr

)J

-.

Assume that m is constant over over time. time. The The model model contains contains aa feedback feedback from from is constant inflation to unemployment U(t + + i) l)-u(t) = -k(m-p(t l)), kk>0. uQ — u(t) = —k(m — p(t + i)), > 0.

(3.1.3)

The model model consists consists of three three equations, equations, (3.1.1), (3.1.1), (3.1.2), (3.1.2), and and (3.1.3), (3.1.3), with with three three The variables, p, U,and and,r.n.We Wenow nowshow showthat thatthe thedynamics dynamics can canbe bedescribed described by aa variables, p, U, equation. second-order linear difference equation. From equation (3.1.1), we get P{t

+ i) I) -— Pp(t) {t) = + i) -1/(/)) + h{4 +1) + i) -— 4)) = -— b{u(t b(U(t +1) — u(t)) + = kb(m p{t + l)) + hj{p(t) 4% = kb(m + i)) + hj(p(t) -

{

use equations equations (3.1.2) (3.1.2) and and (3.1.3). (3.1.3). To eliminate eliminate lr(t) 7r(t) from from equation equation where we we use (3.1.4), we observe from equation (3.1.1)

h,r(t) == p(t) — aa + + bU(t). hx(t) pif) ~ bU{t). Substituting this into equation (3.1.4) yields (l kb)p{t + i) l) —- (l bjUit) = kbm + + a]. aj. (i + kb)p(t (i -— j(\ -— h))p(t) + bjU(t)

(3.1.5)

From equation (3.1.5), we get

bj(U(t + i) — u(t)) = - (1 kb)P(t ++2)2)++(2(2—- 1(1 Xl —-h)h) + kb)p{t ++I)l)— - (i(l -— j(\ —- h))p(t). h))p(t). + kb)p(t —. (i + kb)p(t Inserting equation (3.1.3) into this equation equation finally finally yields yields the the equation equation involving involving Inserting equation (3.1.3) only p only p p{t + 2) + a1p(t alP(t + i) l) + a2pQ) a2p{t) = c, c, where

(3.1.6)

82

SYSTEMS 3. ONE-DIMENSIONAL DYNAMICAL ECONOMIC SYSTEMS

=_[2_i(1_h)+(l_J)kbj

ai

1+kb j(1

a22

—

h)

Il ++ kb kb

_ kbjm Jthjm 1l + + kb kb'

C=

The fixed fixed point of of equation equation (3.1.6) (3.1.6) is given by The *

p =

C

=m. 1+a1 +a2 = m. 1+ a +a x

2

The equilibrium rate of inflation is exactly equal to the rate of monetary expansion. expansion. The characteristic equation is is + a p2 + + aa1p p2 a22 = 0. xp +

We have have We a\ >(c) (<)4a2 Aa2 if and only if (2 (2 —- j(1 j(l

-h)+{\h) + (i

—

j)kbf > (<)4(1 4(1 - .j(i j{\ —- h)X1 h)\\ ++ kb). j)kb)2 kb). (<)

straightforward to check check that that there there may may arise arise either: either: (1) (1)distinct distinct real real ItIt isis straightforward roots (e.g., = 0.2, j = 1/3 and and bk=5 bk = 5 => a1 a, =31/5 = 31/5 and and a2 a2 ==5); 5 ); or or (2) (2) roots (e.g.,hh=0.2,j=1/3 repeated repeated real real roots roots (e.g., (e.g., hh == 1,\, j == 1/3, and and kb kb == 33 => a\ == 4a2 4a 2 ); or or (3) (3) roots (e.g., (e.g., hh=1,j=1 = 1, j = 1 => a1=2 ax = 2 and and a2=1+bk>l). a2 = 1 + bk > 1).Figures Figures complex roots 3.1.1 and and 3.1.2 3.1.2 depict depict respectively and case case (3) ( 3 ) with with the thefollowing following 3.1.1 respectivelycase case(1) (I) and specified values values of of the the parameters parameters specified m = 0.06, 0.06, b=0.3, b = 0.3, a=0.05, a = 0.05, m =

on U -plane. The reader is encouraged on the pp-U-plane. encouraged to carry our stability stability analysis analysis of the the model model in in detail. detail.

3.1. A MODEL OF OF INFLATION AND UNEMPLOYMENT

83

U (t) 0.14 0.12

p (t) 0.057 0.057

0.058 0.058

0.059 0.059

0.061

0.062

0.063

0.08 0.06 0.04

Figure 3.1.1: Different Different real eigenvalues

U (t) 0.18 0.16 0.14 0.12

p (t) 0.0525

0.055 0.0575

0.0625 0.065 0.0675

0.08 0.08 0.06 0.06

Figure 3.1.2: Complex eigenvalues eigenvalues

3.2

The one-sector growth (OSG) model The

We now represent aa one-sector growth (OSG) model in in discrete time. The economy an infinite infinite ftiture. future. We We represent represent the passage in aa sequence sequence of of periods, periods, has an passage of time in and indexed by tt = =0, 0, 1, 1, 22,.... Time 0, being being referred referred to to the the numbered from zero and Time 0,

84

ONE-DIMENSIONAL DYNAMICAL ECONOMIC SYSTEMS 3. ONE-DIMENSIONAL

beginning of period 0, 0, represents represents the theinitial initial situation situation from which economy starts to grow. The end of period period t -— 11 coincides with the beginning of period t;t; it can also transactions are are made made in in each each period. period. be called time t.t. We assume that transactions population grows at The model assumes that each individual lives forever. forever. The population rate n; n; thus thus

N(t) n)N(t —- l). 1). N(t) = (1 (1 + ri)N(t Each individual one unit of labor period t individual supplies supplies one labor at each each time time t.t. Production in period uses inputs amount K(t) of capital and amount N(t) of labor services. It uses inputs amount K(t) of capital of labor services. supplies supplies amount YQ) Y{f) of goods. Here, production production is assumed assumed to be be continuous continuous during during the the amount period, but then use the same same capital that that existed existed at at the beginning of the period. The production function function is is

F(K(t),N(t))= AKa(t)NPQ), = \, I, a,a,j3>0. AKa{t)Nfi(t), a ++ /3 p= /3>0. The production function has constant returns to scale. scale. Markets are competitive; thus thus labor and capital earn their marginal products, and firms earn zero profits. Let us marginal products, firms earn zero profits. us assume that depreciation and denote the rate assume that depreciation is proportional proportional to capital capital and denote the rate of of depreciation by 8k• Sk. The The total amount of of depreciation depreciation is equal equal to SkK(t). SkK\t). The The real interest rate and the wage of labor are given by +

= aF(t)

K(t)

Q/3F(t) N(t) capital stock stock K0 that isis owned owned equally equally by all individuals individuals There is some initial capital Ko that at the initial period. We write the marginal conditions conditions in capital intensity

r(t) = r(f)++Sk=aAk->(t), w(t)= /JA/e(t),)

(3.2.1)

where where k(t) = K(t)IN(t). K.{f)lN(t). We now model model behavior of consumers. consumers. Consumers Consumers obtain income in period tt from payment, r(t)K{t), r(t)K(t), and from the interest interest payment, andthe the wage wage payment, w(t)N(t) w\t)N{t)

3.2. THE THEONE-SECTOR ONE-SECTOR GROWTH GROWTH (OSG) (OSG) MODEL MODEL

85 85

Y(t) = r(t)K(t) + w(t)N(t). Y(t)=r(t)K(t)+w(t)N(t).

We call Y(t) Y(t)the thecurrent currentincome. income.The Thetotal totalvalue valueof ofwealth wealth that that consumers consumers can cansell sell purchase goods goods and to save is equal equal to K(t). K(t). The Thegross gross disposable disposable income income is to purchase equal to = Y(t) + K(t). Y{t)=Y{t)+K{t).

The gross gross disposable disposable income income is used for for saving saving and and consumption. consumption. In period period t,t, consumers would distribute the total total available available budget budget among among savings, savings, S{t), s(t), and consumers would distribute the consumption of of goods, CQ). C\t).The Thebudget budgetconstraint constraint isisgiven given by

c(t) + s(t) = C(t)+s(t)=Y(t). We level, ll(t), u(t), that We assume assume that utility utility level, thatthe theconsumers consumersobtain obtainisisdependent dependent on on

the consumption level of of commodity, commodity,C(t), C(t), and andthe thenet netsaving, saving,S(t), s(t), in period t. consumption level period /. We use the Cobb-Douglas utility function to describe consumers' preference We use the Cobb-Douglas utility function to consumers' preference

uQ)

=

+2=

1,

2 > 0,

in in which which £ and and 2Xare arerespectively respectively the thepropensities propensities to toconsume consume goods goods and andto toown own wealth. wealth. Households Householdsmaximize maximize utility utility subject subject to tothe thebudget budget constraint. constraint. We Wesolve solvethe the optimal optimal choice choice of ofthe the consumers consumers as as

c(t) = ef(t), s(t) C(t)=t;Y(t), s(t) = 2Y(t). Amount Amount K(t K.(t++i)l) ininperiod period tt isisequal equal to to the thesavings savings made in in period t,t, i.e. i.e.

K(t+ K(t + l)=S(t). l)=S(t). Since force N(t) the initial value K0 Ko and andthe the labor force N(t) are areexogenously exogenously given, given, the Since the above above equation equation allow allow us us to to calculate calculate recursively recursively all the K(t). K(t). Capital Capital K(1) K{\) isis obtained obtained from from K0 Ko and N0; No; K(2) K(2) isisobtained obtained from from K(l) and and N(1),..., JV(l),...,etc. etc.Then Then we r{t),w(t), w(t),s(t), s(t),and, and,C(t) c(t)from fromthe therelated relatedequations. equations. we directly directly calculate calculate r(t),

86

ONE-DIMENSIONAL DYNAMICAL DYNAMICAL ECONOMIC SYSTEMS 3. ONE-DIMENSIONAL We now rewrite the the dynamics dynamics in in per per capita capitaterms. terms.With With S(t) S(t) = 2Y(t) AY(t) and a l) f(t) = AKa Y{t)=AK {f)N (t)+SK(t), (t)is,rfl (t) +

where 5k, the the capital accumulation, where S = 11 —- 5k'

K(t+1)=S(t), K{t + l)=S(t), given by by is given K(t ++ 0= l) = AAK ASK{t). AAra{t)NP ++ 28K(t).

Dividing the above equation equation by by N yields yields

KQ + i) = Mka(t) + N(t) Substituting Substituting

N(t)=

N(t + 1)

1+n \ +n

into the above equation, we have have

(i (l ++ n)k(t n)k(t + i) l) == AAk" XAka (t) if) ++ 28k(t). XSkit).

(3.2.2) (3.2.2)

This is is aa nonlinear nonlinear difference difference equation We may may rewrite rewrite this this equation equation This equation in in k(t). We as A k(t *(/ ++ 1) l) == w(k(t)) V{k(t)) s (Aka(t) (Aka{t) ++ &(t)) <Sfc(f))'\1+n +n

For For this this difference difference equation equation to to be be in in steady steady state, state, we we have have

k(t+ k(t + l)= l) = k(t)= k(t) = k. k'.

3.2. THE ONE-SECTOR GROWTH (OSG) MODEL

87

Substituting this condition into equation (3.2.2) yields = AAk*a. (i AAk'a. (l ++ n -— 28)k* AS)k* =

The equation has a unique positive solution solution

k*=I

\11/fl

n-XSy \\ + n—AS) It It is is straightforward straightforward to to show show

d'P(kQ)) - a +— < a + p = 1. dk(t) k(t)=k + n


1

We We conclude conclude that that the the unique unique equilibrium equilibrium point point isis stable. stable. Figure Figure 3.2.1 3.2.1 shows shows the the relation relation between k(t kit ++ i)l) and k(t), k{t), which whichwe we express express by by k(t + i) = The The slope of 'P(k(t)) ^{kifj) isis infmite infinite at k(t) k(t) == 00 and and diminishes diminishes toward toward aa constant constant

28<1+n. 18 < 1 + n. The The thnction function 'P(k(t)) ^(k^)) crosses crossesthe the45 45-degree -degreeline line at at the the steady-state steady-state value, k*. The The capital capital stock stock monotonically approaches its unique equilibrium point point as as time time passes. passes. The The equilibrium equilibrium point point is is stable stable because because the the curve curve w(k(t)) ¥(&(?)) isisalways always upward upward sloping, and it crosses the 45 -degree line from above. sloping, and it 45 -degree line from above. Summarizing Summarizing the the discussions discussions in in this this section, section, we obtain obtain the the following following theorem.

Theorem Theorem 3.2.1. 3.2.1.Given Giventhe theCobb-Douglas Cobb-Douglas production production and and utility utility functions, functions, and a constant constant rate rate of of population population growth, growth,the thecapital-labor capital-labor ratio ratioconverges convergesmonotonically monotonically to to aa unique unique positive positive equilibrium equilibrium point. point. The Theunique unique equilibrium equilibrium point point isisstable. stable. Moreover, Moreover, the aggregate aggregate capital capital stock stock and and real real aggregate aggregate output output converge converge to the the balanced exponential exponential growth growth paths pathsproportional proportional to to the thepopulation population growth. growth. balanced

ONE-DIMENSIONAL DYNAMICAL ECONOMIC SYSTEMS 3. ONE-DIMENSIONAL SYSTEMS

88

k(t+1) k{t + 4

k2

F

k (t)

Dynamics in inthe the OSO OSG model model Figure 3.2.1: Dynamics

3.3

The general OSG model

We now generalize generalize the model in discrete discrete time section 3.2 the OSG model time proposed proposed in in section 3.2 by replacing and utility functions with more replacing the Cobb-Douglas Cobb-Douglas production production and utility fiinctions more general general production and and utility utility functions. functions. population grows grows at atrate rate n; thus As in section 3.2, population

N(t) =

(1

+ n)Nfr

—

i).

Each individual supplies one unit of of labor labor at at each each time timeit.t. Production in period period itt uses inputs uses inputs amount amount K(t) K[t) of of capital capital and and amount amount N(t) N\t) of of labor labor services. services. The The production function is

F(t) = F(K(t),N(t)). The marginal conditions can be expressed by

3.3. THE THE GENERAL 3.3. GENERALOSG OSGMODEL MODEL

89 89

r(t) = w(t) = f(k(t)) k(t)f'(kj, At)=f{k(t))-k(t)f'(k,),

(3.3.1)

—

where

f(t)

k(t)

= F(k(t), 1).

The consumer is to to choose choose his most most preferred preferred bundle bundle (c(t), s(t)) s(t)) of consumer is consumption consumption and saving saving under under his hisbudget budget constraint. constraint. The Theutility utilitymaximizing maximizing problem by atany anytime time isisdefined defined by problem at

u(c(t),s(t))

Max o

s.t.: c(t)

f(k(t)) +

+ s(t) = ;(t)

We We require require U aaC2 C 2function, function, and andsatisQj satisfy Uc >>00 and and Us >>00 for any

(c(4 s(t))> s(t)) >0.0.Construct Constructthe theLagrangian Lagrangian

L(c(t), c(t) — s(t)). L(c(t), s(t), s(t), 1(t)) 1(0)== u(c(t), U(c(t),s(t)) ,(,))++ 1(t)(p(t) 1(0(^(0 -— c{t) - s(t)). The The first-order first-order condition condition for formaximization maximization is is = 1(t), Uc=U = s=A(t),

y(t) — c(t) — 5(t) = 0. j>(0-c(0-*(0=o.

(3.3.2)

The The bordered bordered Hessian Hessian for forthe the problem problem is is

01 0 1 = 11 Ucc I1

1

== 2£/cs —- [/„ — cc - U

£/„

The The second-order second-order condition condition tells tells that that given given aastationary stationary value value of ofthe thefirst-order first-order condition, positive \H\ is issufficient sufficient to to establish establish it it as as aa relative relative maximum of ofU. U. condition, aapositive Taking Taking derivatives derivatives of ofequations equations(3.3.2) (3.3.2)with withrespect respecttoto y', , yields yields

______

3. 3. ONE-DIMENSIONAL DYNAMICAL DYNAMICAL ECONOMIC ECONOMIC SYSTEMS SYSTEMS

90

dc

dc dy

dc

ds

+

+

ds

ds dy

We solve these functions functions ds _ Usc - Ucc 2 CS - Uss - Ucc dy 2U Ucc'

dc _

Usc -— USC

dy 6'

Uss Uss

2 CS — U55 2U - U - Ucc" ss —

We see see that . ds ds . . dc dc , o<—<1, o<—<1, 0< — < 1 , 0< — <1, d

dy dy in the case in case of Usc >0 0under underthethe second-ordercondition conditionofofmaximization. maximization. We We second-order denote an optimal solution as as function of of the disposable income income (c(t), s(t)) = (c(5'(t)), s(;(t))). (c(t],s(t)h(c(y(t)\s(y(t)))The capital accumulation equation is is given given by by

K(t ++ 1)l)=s{y{t))N{t). = s(5'(t))N(t) Dividing the two two sides sides of of this this equation equationby by ./V(f N(t ++ i) l) yields

k(t +

1)

=

1+n

mapping controls controls the motion motion of the the system. system. A A stationary stationary state state for for the the This mapping growth progress progress is is aacapital-labor capital-laborratio, ratio,k', k, that thatsatisfies satisfies

k= k = «?L. 1+n

(3.3.3)

3.3. THE GENERAL OSG MODEL

91 91

Define s(5') S y k0. (\ - 1, k>0. (i + + n)k n)k

(3.3.4)

When is also When k isisapproaching approaching zero, zero, 5'y (= (= /f(k) ( £ ) + &k) <5ifc) is also approaching approaching zero, zero, and and hence s(5)) As 0 < s'(y) << 11 and —>0 0 s(y) isis coming coming near zero. As and f'(k) f'(k) —> -> cc oo as as kk—* = s'(O)(f'(O)

s($')

+ s)

(1+n)

k_*O(1+n)Jç

(3.3.5)

1.

When k^ isisapproaching approachingpositive positive infinity, infinity, 5' y isis coming coming to to positive positive infinity. infinity. As As

0<s'(5')<1, 0 < s\y) < 1, f'(k)—÷Oask—÷+cc, f(k) - 0 as k -* + oo, we we have have

cc)(f'(+ cc) lim s(5') «» == s'(+ ^ « y ( + » ) +^s)) k—>+ao(1+n)k (1+n)

k^^(\

+ n)k

< L1.

(3.3.6) (3.3.6)

(l + n)

Taking (3.3.4) with with respect respecttoto kk yields Taking derivatives of equation (3.3.4) —

dk

[s'(5')k(f'(k) + s(5') s(y) [

sG) j(1+n)k2

We now show showthat d'J?Idk<0 prove this, weweuse dk < 0fork for >0. k >To 0. To prove this, use ds I dy <1 <1 We now that dQ>l and and the inequality inequality

f'(k) < f(k) k

which w>> 0. By (3.3.3) and and the the definition definitionofoff', which also guarantees guarantees w By equation equation (3.3.3) y, we we have have

+ s) (i + n)s(5')

s'(5')(f(k) + 8k)—_ (1 + n)s(LP)

s'(5')$' s'(y)y

(1 ( + n)s(5') >()

; t

92

ECONOMIC SYSTEMS SYSTEMS 3. ONE-DIMENSIONAL DYNAMICAL ECONOMIC

where we we use use s'(y) s'(j)) <1 where < 1 and and yl s(y) <1 1totoguarantee guaranteethe theright rightinequality. inequality.We Wethus thus conclude d&ldk << 00 for equation, for k >> 0. The equation,

=0, for for kk >> 00 has has aa unique unique solution solution because because ofofequations equations (3.3.5), (3.3.5), (3.3.6), (3.3.6), and and d®l' dkc<0.0.We Wenow nowdemonstrate demonstratethat thatthe theunique uniquestationary stationary state state is stable. dD/dk steady state to be stable, stable, the following conditions conditions must must prevail prevail For the steady

—1<

s'&)(f(k)+8)1 1+n

As the steady-state values, k* f'(k') are positive. positive. k* and and y* are are positive, positive,s'(y') s'(') and ft(k*) The left left inequality inequality always always holds. prove the right right inequality, inequality, we use use The holds. To To prove dsldy <1 and and the theinequality inequality ds! 4' <1

flk) f(k) >

(which also also guarantees guarantees ii', w, > > 0). 0). By equation y, we we equation (3.3.3) (3.3.3)and andthe thedefinition definitionofof;, have have + s)

l+n 1+ n

st(;*)(f(kt)/k* s) - st(;*)y/k* s'(y )(f{k')lk' ++S)_ s\y )y Ik' 1+n — l+n 1+ n 1+ n

- s?(y);* s\y)y

=

—

s{y*)

Since s'($')
Theorem 3.3.1. that is 'neoclassical' 3.3.1. Given Given aa production production function function that 'neoclassical' and and aa utility utility function that that is is aa C2 C2 function, function, and andsatisfies satisfies Uc >> 00 and and Us >> 00 for for any any function s,)> 0. 0. Let the bordered bordered Hessian be positive for any nonnegative. nonnegative. Then the (c,, s1)> capital-labor ratio converges monotonically unique positive positive steady steady state. state. The The capital-labor ratio converges monotonically to to aa unique unique stationary stationary state state isisstable. stable. unique

OVERLAPPING-GENERATIONS (OLG) MODEL 3.4. THE OVERLAPPING-GENERATIONS

93 93

The Theoverlapping-generations overlapping-generations(OLG) model

3.4

2 finite-horizon households. It This section introduces introduces the model model of finite-horizon households.2 It is is common common to to assume, the OLG OLGmodeling modeling framework, assume, in the framework, that that each each person person lives lives for for only only two two periods. People People work work in the first first period period and retire retire in in the second. second. If If one one thinks thinks of of reality, perhaps last last over over 30 years. years. As we of reality, one period period perhaps we neglect neglect any any possibility possibility of generations, people people consume transfers from government or from members of of other generations, in both periods; they pay pay for for consumption in in the the second period by by saving in in the thefirst first period. The time t is Thecohort that is born at time is referred to to as as generation t.t. Members of of this generation are young in in period itt and oldin inperiod period itt ++1. 1. At At each point in in time, time, and old members of of only only two twogenerations are are alive. Each person maximizes lifetime utility, on consumption in the which depends on the two two periods periods of life. life. It is is assumed assumed that that people people no assets assets and and don't care care about about events events after theft their death; they are are not not are born with no altruistic toward children, and therefore, therefore, do not not provide provide bequests bequests or or other other altruistic toward their their children, to members of of the next generations. Their lifetime transfers to lifetime utility is specified specified as as

u(t) =

— 1

1—0

+

1

+

i)

1—0

—

I

(3.4.1)

where 6>>0,0,p>p 0, andand c.(t) is consumption where 6 > 0, c^t) is consumptionofofgeneration generation 1,/', i = 1,1, 2. Each Each individual supplies supplies one unit of labor labor inelastically inelastically while and receives the individual unit of while young young and wage income w(t); stand for the amount saved w(t); he he does not not work when old. old.Let Let s, stand The budget constraint for for period t is in period t. The is c1(t) + s(t) = w(t). cXt)+s{t)=w(t).

(3.4.2) (3.4.2)

Let r(t 1) denote rate on periods tI r(t ++1) denote the interest interest rate on one-period one-period loans loans between between periods and

f ++1.1. In In period period itt ++1,1, the the individual individual consumes consumes the savings plus the theaccrued accrued the savings interest it

c2(t l ) == ((il ++ r(t r(( ++ 1))s(t). l))4 4 ++ i)

22

(3.4.3) (3.4.3)

The model model was wasinitially initially examined examined by by Samuelson Samuelson (1958) (1958) and and Diamond Diamond (1965). (1965). The The model here here is isbased based on onBlanchard Blanchard (1985) (1985) and andBarro Barro and andSala-i-Martin Sala-i-Martin (2004) (2004) in insection section 3.8. Some Some extensions extensions of ofthe model model are arereferred to to de de la la Croix and andMichel (2002).

94

3. ONE-DIMENSIONAL DYNAMICAL DYNAMICAL ECONOMIC ECONOMIC SYSTEMS SYSTEMS 3.

w(f) and +1) are exogenous; exogenous; he hechooses chooses c1(t) c^t) and each individual, individual, w(t) For each and r(t r(t + i) are and s(t) to equations (3.4.2) (3.4.2) and and (3.4.3). (3.4.3). Substitute Substitute equations equations (3.4.2) (3.4.2) 5(t) (and (and c2M)) subject to utility to todelete delete c1(t) cx(t)and andc2(t c2(t++i)l) and (3.4.3) into the the utility (w(t) —

1

1

(i + r(t +

1+p)

1—0 1-0

1

1—0

s(t) yields The first-order condition with with respect respect to to s(t) - s(t))°, s{t))~e, s~e(tXl ++ r(t r(t ++ l)J'0 = = (l (i ++ p){w(t) p)(w(t)— s°(tXi

(3.4.4)

(3.4.4)

which also implies, under equations (3.4.2) and (3.4.3)

c2(t+1)11+r(t+lfl c,(t) c1(t)

{

11+p +p

110

(3.4.5)

.

)

Solve equation (3.4.4) =

w(t)

s(t) s(t) = - r ^ j y

(3.4.6) (3-4-6)

,

where (

l+p 1/0

>1.

see We see =

s.

8s(t) as(t)

raQ+1)

_(\-0

=

Bs(t) = w

We have We

Uw(t)

i+p

s(t)

0 1


,

O<s <1. w

(3.4.7)

(OLG) MODEL 3.4. THE OVERLAPPING-GENERATIONS (OLG)

95 95

f>o 0 if e 9<1, | =0 = 0 i / (9 = 1, 9=1, [<0 8>1.

sr

Firms' behavior behavior isisthe the same same as asin inthe the OSG OSG model model in inthe previous section. They Firms' have the neoclassical production function yQ) =

where M

Y(t) Y(t)

y(t) (t) = N(t)

, M Kit) K(t) kQ) = — U jfc(f) N(t)

where N(t) N{t)is isthe thenumber number of ofyoung young people, people, Y(t) is the the total output, output, K(t) is the the where K(t) is are total capital. The marginal conditions are r(t) == f'(kQ)) f'(k(t))-S r(t) — 8 k, f(k(t))-k(t)fik( w(t) = = f(k(t)) — t)),

(3A8)

Sk is ofcapital. capital. where 8k is the depreciation rate of theclosed closed economy, economy, households' assets assets equal equal the thecapital capital stock. stock. Aggregate Aggregate In the net investment equals total income minus minus total total consumption consumption + i) l) -— Kit) w(t)N{t) ++ r(t)K(t) c,{t)N{t) —- c2Q)N(t c2{t)N{t —- i). l). K(t K(t + K(t) == w(t)N(t) r(t)K(t) -— c1Q)N(t)

(3.4.9)

(3.4.9), we get Substituting equation (3.4.8) into equation (3.4.9),

K(t ++ 1) l) -— Kit) F(Kit\ N(t)) Nit)) -— Cit), K(t) == F(K(t), c(t), where

C(t) cl{t)N(t)+c = c1(t)N(t) + c2(t)NQ — c(t) = 2{t)N(t-\). From equations (3.4.9), (3.4.2), and (3.4.3), we obtain

(3.4.10) (3.4.10)

96

3. ONE-DIMENSIONAL DYNAMICAL ECONOMIC SYSTEMS SYSTEMS

K(t — s,_1(t)N,_1(t)). s(t)N{t)++(i (1++r(t))(KQ) r{t))(K(t) s^N^t)). K{t + i) 1) == s(t)NQ) Assume Assume that that the the economy economy starts starts with with the the condition condition C2

(I)N(o) = (i + r(1))K(1),

which is equivalent equivalent to K(2) K(2) == s(1)N(l) S(I)N(I) with equations equations (3.4.9) and (3.4.2). (3.4.2). Hence, Hence, the above equation becomes

K(t = s(t)N(t), K{t ++ 1)l)=s{t)N{t), which means that the savings period's capital savings of the young equal the next period's capital stock. stock. We We can can write write this this equation equation in in per per capita capita terms terms

k(t+1)=

s(t)

1+n 1+ n

Substituting Substituting sQ) s(t) ininequation equation(3.4.6) (3.4.6)into intothe theabove aboveequation equation yields yields =

+

+ (i

(3A.11)

in i).l).Inhithethecase utility in which which we use the the definition definition of çQ
logarithmic logarithmic ((86 == 1), 1), itit can can be be shown shown that that equation equation (3.4.11) (3.4.11) becomes becomes k(t + i)

=

1

(1+nX2+p)

[f(kQ)) — k(t)f'(k(t))].

(3.4.12)

The The analysis analysis of of equation equation (3.4.12) (3.4.12) can be be similarly similarly conducted conducted as for for equation equation (3.2.2). (3.2.2). The The system system has hasaaunique uniqueequilibrium equilibrium point point and and ititisis stable stable (check!). (check!).

Exercise Exercise3.4.1 3.4.1 11

Discuss Discuss the the existence existence of ofequilibrium equilibrium point point and and find findstability stability conditions conditions when when

81.

3.5. PERSISTENCE OF iNEQUALITY INEQUALITY AND AND DEVELOPMENT

97

3.5 3.5 Persistence of inequality and and development development This This section section introduces introduces a growth growth model model proposed proposed by by Moav Moav to todemonstrate demonstrate persistence In this this model, model, the evolution inequality.3 In evolution of income income within each each persistence of inequality.3 dynasty in society society is is governed governed by by aadynamical dynamical system system that that generates generates aapoverty poverty trap high-income equilibrium equilibrium point. point. Poor Poor trap equilibrium equilibrium point point along along with with aahigh-income dynasties, those with dynasties, those with income income at at the the threshold threshold level, level, converge converge to to aa low lowincome income steady the threshold threshold converge converge to to aa steady state, state, whereas whereas dynasties dynasties with with income income above above the high income steady state. high income steady state. Consider small open open overlapping-generations overlapping-generations economy economy that that operates operates in in aa Consider aa small one-good world. In each period the economy produces a single homogeneous one-good world. In each period the economy produces a single homogeneous product can be beused usedeither either for forconsumption consumption or orfor forinvestment. investment. Production Production product that that can occurs within a period according to a concave, constant-returns-to-scale occurs within a period according to a concave, constant-returns-to-scale technology. The output output produced produced at at time time tt uses uses capital capital KQ), K(t), and andhuman humancapital capital technology. The efficiency efficiency units, H(t), H[t),as asfollows follows Y(t) n'(t)), Y{t)==F(K(t), F{K{t),H(t)),

where investment in physical and and human human capital capital isis made made one one period period in inadvance. advance. investment in Assume Assume that the world world capital capital rate of of return return remains remains constant, constant, denoted denoted by R. R. Unrestricted international capital capital movement yields FK(K(t), HQ)) == R. FK{K{t),H(t)) R.

(3.5.1) (3.5.1)

From From the the properties properties of of the the production production fimction, function, we know that the wage per per unit unit of human capital, w, is uniquely determined given the rate of return to capital, determined the rate of return to capital, of human capital, w, is R, and andisistherefore, therefore, constant constant over over time.4 time.4 Like section, individuals individuals R, Like in the previous section, live in in two twoperiods. periods. Individuals, Individuals, within within as as well well asasacross acrossgenerations, generations, are are live identical in in their their preferences preferences and and their their technology technology of ofhuman human capital capital formation. formation. identical They may may differ differ in intheir their initial initial wealth, wealth, inherited inherited from fromtheir theirparents. parents.Individuals Individuals They cannot in order order to to fmance finance investment investment in inhuman human capital. capital. Individual Individual born born cannot borrow borrow in in in period period tt allocates allocates her her second second life life period period income, income, I./,(? !), between between (t ++ 1),

3 This This section section isisbased basedon onMoav Moav(2002). (2002).See Seealso alsoother otherapproaches approachesby, by,for forinstance, instance,Galor Galor and and Zeira Zeira (1993), (1993), Abe Abe(1995), (1995), Benabou Benabou (1996), (1996), Durlauf Durlauf (1996), (1996), Piketty Piketty (1997), (1997), to to issues issues related related to to development development and and distribution. distribution. Aghion Aghion and and Bolton Bolton (1997), (1997), Aghion Aghion and andHowitt Howitt (1998) (1998) and and Matsuyama Matsuyama (2000) (2000) also also provided provided some some interesting interesting insights insights into into evolution evolution of of inequality inequality due dueto to"institutional "institutional frictions". frictions". " 4 As R = f'(k), i), we As R = f'{k), where where kk =K/H, KlH, ff e=F(k, F{k,\), we have have kk==g(R). g{R). Hence, Hence, kk isis constant. As w — Rk, w is constant. constant. As w = = ff - Rk, w is constant.

98

3. ONE-DIMENSIONAL DYNAMICAL ECONOMIC SYSTEMS

household consumption, i). The consumption, c1 c,(f(t + i), l), and a bequest to the offspring, offspring, kbt(t + l). budget constraint is given by

i.Q++ i)= 7,(f l) = cQ c,(/ + 1)+b,(t+1). l)+*,(f + l).

(3.5.2)

Preferences are defined by by the the utility utility function function /3)logc,(t I) + fllog{O + l)}, U,{t) = (i(1 —- P)\og + 1) + /Hogfc + k(t b,{t + 1)}, Ci{t +

where 1) and where /3 /? e (o, (0, l) and S 6 >> 0. 0. The optimal optimal transfer transfer of of individual individual /'i born in born in period t is given by

/ \ / / „ lo [0 i); == b(J(t i)) == \ , , b.(t + l) bll.lt+ A + V A + l))

"

v

v

if I(t+1)o, if/,(f + l)<0, I

\ Lfl(11(t+O—9) p { i { t + \ ) e ) \ sifJ1(t+1)>O, i { t +\ ) > e

(3.5.3) (3-5.3)

'

where 0 = 0(l We now discuss st period period of their lives discuss formation formation of of human human capital. capital. In In the the fir first in period t,t, individuals devote devote theft their entire time for for the the acquisition acquisition of of human human Individuals acquire one efficiency capital. Individuals efficiency unit unit of of labor-basic labor-basic skills. skills. The The level level of of human capital of of an an individual individual /,i, kht(t human capital {t ++1), l), is an increasing concave concave function function of real resources resources invested investedinineducation, education,e1et(t)

h(t + 1) = h(e(t)) =

Ii +

2e.(t)

if e.(t)< e,

if e(t)ë.

(3.5.4)

.c ë, It is is assumed assumed that that the themarginal marginal return return to tohuman human capital, capital,for fore•et(t) (?) < e, is is larger than the marginal return to physical capital, wy > R. We also require that marginal return to physical capital, wy> R. We also require income level below which which individuals individuals choose choose a zero bequest, bequest, 0, is larger larger the income than the wage rate, w, WeWe also assume w, 0> 6 w. > w. also assumethat thatthe thereturn returntotophysical physical capital, capital, R, sufficiently low, i.e., fiR fiR c<1.1.InInsummary, summary, we we assume assume R, isis sufficiently

0>w>—, flR<1. PR<\.

Ir

(3.5.5)

3.5. 3.5. PERSISTENCE OF INEQUALITY AND DEVELOPMENT

99

Under the above above constraint, constraint, the second second life life period period income, income, I.It(t(t + + i), l), is is Under determined by first life period bequest (t) uniquely determined bequest kbt(t)

jw(l+1tb,(t)) 1¼(

ii' if b,(t)ë, if b.(t)ë.

(356)

From equations equations (3.5.3) income within From (3.5.3) and and (3.5.6), (3.5.6), the the evolution evolution of of income within aa dynasty is is uniquely determined by

(w 11(t =M w(1 — faif)) = l + ,.fl(l,(t) tf(l,(t)/,(/ + i)1)== 9(1,(t))

if fl(I,(t)—9)
0)) w(i ++je)+ 2e')+ R(/3(11(t)— o)— e) if /3(I,(t)— o)> e, [w{\ R{fi{l,{t)-9)-e) p{lfc)-9)>e,

(3.5.7)

/ ( (o) is given. given. We w for for all allt.t.Under Underequation equation(3.5.5), (3.5.5),from from where 11(o) We have have /,(?) 11(t)> w equation (3.5.7) (3.5.7) we conclude that there exists exists a low low income, income, locally locally stable, stable, equation we conclude that there steady state poverty trap steady state 1 == w because

I(t I,{t ++ i)1)== ço(i,(t)) ?(/,(')) == w, w, for all I,(t) /,(*) Ee [w, [w,9]. e\ It is assumed assumed that that the the return return to to human human capital, capital, yvv, yw, and and its its potential potential

magnitude, e, are are sufficiently sufficiently large, such such that that an an individual individual who who receives receives aa magnitude, bequest bequest /',bt(t) = e will transfer her offspring aa higher bequest

b,{t + l)>bl(t)=e. k(t+1)>k(t)=e. This This assumption assumption isis expressed expressed as

fl[w(1 + je) — a]> ëi P[w{\ +je)-e]>e.

(3.5.8) (3.5.8)

This assumption assumption assures range of income, income, above above the poverty trap range of This assures aa range the poverty trap range

income, income, in in which which

i(t + i) >

100

ONE-DIMENSIONAL DYNAMICAL DYNAMICAL ECONOMIC SYSTEMS 3. ONE-DIMENSIONAL

From higher income income steady steady state state 1'/ From equation equation (3.5.5), (3.5.5), we we see see that that there there is aa higher (higher than poverty trap) which is determined by

1-/JR 1This steady state is is locally locally stable. stable. In the dynamic there exists exists an an income threshold, I,1, I1'
This steady steady is is locally locally unstable. unstable. It It can can be be seen seen that that dynasties dynasties with with income income below below the to the poverty the threshold threshold ( 10(t) I0(t)< cI I )) converge converge to poverty trap trap income income level, level, and and

dynasties with income income above abovethe the threshold threshold(I(10(t)> to the high converge to high dynasties with 0(t)>I) I) converge income steady state. state. (t)), depicted in figure In summary, summary, the the dynamical dynamical system, system, I.It(t(t + + i) l) = ^(//(?)), figure 3.5.1, generates generates three three steady steady states, states, a poverty poverty trap, a high income 3.5.1, income steady steady state, state, Dynasties with with initial initial income income below below the threshold threshold level and a threshold income. Dynasties converge to a low converge to low income income steady steady state; state; dynasties dynasties with with income income above above the the threshold level converge to a high income steady state.

3.6 Growth with creative destruction The year 1883 and the the history history of of 1883 is special for for the history history of the world world economy economy and economic analysis. analysis. The year saw the death of Karl Marx (1918-1883), the birth of Maynard Keynes Keynes (1883-1946), (1883-1946), and of Josephy Josephy Alois Alois Schumpeter Schumpeter John Maynard and the birth of (1883-1950). In Theory of of Economic Economic Development Development published 1911, (1883-1950). In the the Theory published in 1911, changes Schumpeter argued that development should be understood as only such changes economic life not forced forced upon itit from from without without but arise arise by by its its own own in economic life as are not initiative, from within. Schumpeter Schumpeter held successful carrying carrying out of of new new initiative, from within. held that successful services is is the the essence essence of of this this process. process.Schumpeter' Schumpeter's combinations of productive services s ideas development and creative destruction destruction have recently been been modeled. modeled. ideas about development represents such such aa model model by by Aghion Aghion and andHowitt.5 Howitt.5 This section represents

''Aghion Aghionand andHowitt Howitt(1992). (1992).

3.6. GROWTH WITH CREATIVE DESTRUCTION DESTRUCTION

101 101

450

—I

I =w I=w

ee iI 9+ë/fl e+eip

71

i(t)

3.5.1: Multiple steady steady states and persistent income inequality Figure 3.5.1:

Each innovation innovation consists consists of intermediate good can be used used to to Each of a new intermediate good that that can produce fmal final output output more more efficiently efficiently than than before. before. Research Research firms firms are are motivated motivated produce by the the process process by by the the prospect prospect of of monopoly monopoly rents that can be captured captured when a by successful innovation destroyed by successfiul innovationisis patented. patented.But Butthose those rents rents in in turn turn will will be destroyed the next next innovation, innovation, which which will will render render obsolete obsolete the the existing existing intermediate intermediate good. good. the is determined determined by aa forward forward difference difference equation, equation, according according to to which which Equilibrium is the amount amount of of research research in in any any period period depends depends upon upon the the expected expected amount amount of the research next next period. period. Here, Here, we we represent represent aa simplified simplified version version of of the the model model by by research of Solow.6 A Solow.6 A period is the time between two successive innovations. The length of each period period isis random random because because of of the stochastic stochastic nature of of the innovation innovation process, process, each but the the relationship relationship between between the the amount amount of of research research in in two two successive successive periods periods but can be be modeled modeled as asdeterministic. deterministic. The The amount amount of of research research this this period period depends depends can negatively upon upon the the expected expected amount amount next next period, period, through through two effects. The The first first negatively two effects. is that of creative destruction. The payoff from research in this period is the is that of creative destruction. The payoff from research in this period is the prospect of monopoly rents in the next period. Those rents will last only until prospect of monopoly rents in the next period. Those rents will last only until the next next innovation innovation occurs. occurs. The The expectation expectation of of more more research research next next period period will will the discourage research this period. The second effect is that of a general discourage research this period. The second effect is that of a general equilibrium in in the the labor labor market. market. Workers Workers can can be be used used either either in in research research or or in in equilibrium manufacturing. To maintain labor market equilibrium, the expectation of more manufacturing. To maintain labor market equilibrium, the expectation of more research next next period period must must correspond correspond to an expectation expectation of of higher higher demand demand for for research to an labor in research next period, which implies the expectation of a higher wage labor in research next period, which implies the expectation of a higher wage 66

Solow Solow (2000).

102

ONE-DIMENSIONAL DYNAMICAL ECONOMIC SYSTEMS 3. ONE-DIMENSIONAL

rate. Higher next period rate. Higher wages wages next period will will reduce reduce the monopoly monopoly rents rents that can can be be gained gained by exclusive knowledge knowledge of how to produce produce the the best best products. products. Thus Thus the the expectation of more research will discourage research this period. The model assumes assumes a constant constant employment. employment. There classes of The model There are are three three classes consumption good, and an intermediate intermediate good. There is tradable objects: labor, a consumption continuum ofofinfmitely-lived infinitely-lived individuals, individuals, with withidentical identicalintertemporally intertemporally a continuum additive preferences preferences defmed defined over lifetime lifetime consumption, and the constant consumption, and constant rate of time preference. The marginal utility of consumption is equal time preference. The marginal utility of consumption is equaltoto the the rate rate of of interest. There are three categories of labor: unskilled labor, which can be used interest. There are categories of unskilled labor, which only in in producing producing the consumption consumption good; (skilled) labor, used good; (skilled) labor, which which can can be used either in research or in the intermediate intermediate sector; and specialized specialized labor, which can be used used only in in research. research. Each Each individual individual is endowed endowed with aa unit unit flow flow of of labor. labor. final good good is is produced produced by the the fixed fixed quantity quantity of of unskilled unskilled labor labor and and Only one fmal skilled labor Theproduction production fimction function is is labor x. The

F(t) == Af(x), Af(x),

where where we we omit expressing unskilled labor and and f(x) f{x) is increasing (f'(x) (f"(x) c< 0). (/'(*) >> 0) 0) and and concave concave (/"(*) 0). The The variable variable A A is a technological technological variable. Final good is used as numeraire. numeraire. Some skilled labor is devoted to R&D. When successful, successful, the innovation is a new intermediate good that that allows allows a higher higher value value of of A and intermediate good and thus thus makes makes the the old old intermediate good good obsolete. obsolete. Let Let t stand intermediate stand for for the t th innovation innovation (not time). time). For For convenience of description, we we consider consider that that each eachsuccessful successful innovation innovation convenience of description, increases the final output output producible by aa multiplicative multiplicative factor factor 7,y, increases the final producible with with any any xx by i.e.

A(t+l) A(t) At)

=7

~'

Suppose Suppose n units units of oflabor labor are aredevoted devoted totoR&D R&Dand andinnovations innovations arrive arrive according to a Poisson according to Poisson process with arrival rate An. An. It should should be noted noted that that in in the original model the specialized labor affects affects the arrival arrival rate. Since the number of specialized labor is prefixed, we omit this type type of labor. specialized labor prefixed, we omit mentioning mentioning this labor. The The probability of an an innovation in aa given unit of time equals probability of innovation in given short short unit equals An, An, and the probability probability of no no innovation innovation isisequal equaltoto1 1—- An, An, and the probability probability of two or or innovations is equal to zero. The assumption assumption of the Poisson Poisson process process says says more innovations probability of making making an innovation innovation of depends only n, that the probability of given given size size depends only on n, independent of past history of innovation. In fact, innovation is hard to model independent past history innovation. fact, innovation model in probability sense. sense. even in

DESTRUCTION 3.6. GROWTH WITH CREATIVE DESTRUCTION

103 103

innovating firm acquires a monopoly monopoly on the the final final production production that is is The innovating firm acquires useful until the next innovation. Thus the t th innovation brings a negative useful until the next innovation. Thus the t innovation brings negative externality throughkilling killingthe the rents rents of of the the firm that externality through that produced produced the (t{t — -1)1) st innovation and aa positive positive externality externality through through making making the the (it (t ++1) 1) st innovation innovation innovation successful innovator innovator is a monopoly monopoly of the intermediate intermediate good possible. A successful good and is faced with a demand curve from the final-goods industry

Af'(x(t)) = Af'(x{t))=P{t), where where P(t) isisthe theprice price of of the the intennediate intermediate good. good.

We We introduce introduce v(t) V(t) and and 11(t) n(/) to respectively respectively stand stand for for the the expected expected discounted rents associated associated with successful innovation innovation and the constant constant discounted with the t th successful flow of rent expected by the / th innovator during the profitable life the flow rent expected the it innovator during the profitable life of the innovation. Let stand for for the the discount discount rate rate of of the the rent rent expected expected by by the the itt th th innovation. Let pp stand innovator. Then the the Fisher Fisher equation equationtells tellsthat thatthe the interest intereston on the the value value of of innovator. Then innovation equals current income income plus plus the the expected expected capital capital gain, gain, which which innovation equals the the current equals An(t)(—V(t))+ An(t))'o. An(t)(-V(t))+(1— (l-Mt)H That is

pV(t) — An(t)V(t). = fl(t) n{t)-An{t)v{t). PV{t) = The above above equation equation is is solved solved as as

v(t)=

n(t) p+An(t)

(3.6.1) (3.6.1)

equation says says that value of n(t) n(t) reduces reduces v(t). V{t). In hi other other words, words, The equation that a large value like capital capital investment, investment, is discouraged discouraged by the prospect of future R&D. research, like Free entry entry and and risk risk neutrality neutrality in in R&D R&D guarantees guarantees that that entry entry will will occur occur until until the the Free cost of of conducting conducting R&D R&D is is equal equal to the expected expected value value of of the innovation innovation cost An{t)v{t ++i)l)++(i(l-An{t))O w(t)n(t) == An(t)v(it — An(t)).o => w(t) = XV(t + l). Labor market market isiscleared clearedfor forevery everyitt Labor

(3.6.2) (3.6.2)

___ 3. ONE-DIMENSIONAL DYNAMICAL ECONOMIC SYSTEMS SYSTEMS

____

104 104

n(t) + x(t) == N, n(t)+x(t)

(3.6.3) (3.6.3)

where N isisthe where theconstant constant volume volume ofofemployment. employment. Solow Solow holds holds that that equation equation (3.6.3) contains contains a major major limitation limitation of this this model: model: "one "one of the true risks of R&D is that economic conditions should should be cyclically cyclically weak weak during during the the effective effective life life of of innovation, so turns out out to to be beunprofitable unprofitable because because sales sales of of fmal final an innovation, so that that it turns product are poor". poor". The The intermediate intermediate good good is produced using skilled labor alone. alone. The The produced using production way that that the the intermediate intermediate product product isis production function function is is specified specified in such a way skilled labor labor used in in the the intermediate intermediate sector. With the oneequal to the flow of skilled to-one technology for producing intermediate good, the monopolist maximizes P(t)x(t) = A(t)f'(x(t))x(t) P{()x{t) — - w(t)x(t) w{t)x{t) = A{t)f{x{t))x(t) — - w(t)x(t). w{t)x(t).

(3.6.4) (3.6.4)

Provided that marginal revenue optimal x(t) x(t) is is aa Provided that marginal revenue falls falls with with x(t), x(t), the optimal Tl(t)/ A(t) A(t) decreasing function function of of w(t)! w(t)l A(t), A(t), and andthe thebest bestachievable achievablevalue valueofof decreasing 111(t)! w{t)l A{t) rises. rises.Denote Denote this this function function by by falls as w(t)!A(t)

x(t) = From equation (3.6.3), n(t) function of of w(t)/A(t). equation (3.6.3), nif) isis an an increasing increasing function w{t)l A{t). We We have

n(t) = N

- x(t) = N

-

Solving the above above equation with wQ)! w(t)/A(t) as the the variable variable yields yields A(t) as

= çb(n(t)), Mt%

(3.6.5) (3-6.5)

where Ø(n(t)) in in n(t). Now equations (3.6.1) andand (3.6.2) imply
Afl(t + i)

w(t)= 2V(t + p

3.6. GROWTH WITH CREATIVE DESTRUCTION

105

Inserting equation equation (3.6.5) and A(t A(t + +1) into the the above above equation equation Inserting (3.6.5) and i) == yA{t) into yields w(t) = y((n(t + + i)) A(t) p+An(t÷1) A{t)

(3 V 66)

As 1)/A(t 1)/A(t 1) and As H(t Tl(t + \)l A{t + i) l) is is a decreasing decreasing function function of w(t w(t + l)/ A(t + + l) 1)/A(t i) w(f + l)/^(? + l) is an increasing function of n(t + l), n ( ; + \)l A{t + l) falls as w(t flmction of 1), n(t 1)/A(t i) n(t +1) rises. Consequently, the right-hand side side of of equation equation (3.6.6) (3.6.6) is is aa rises. Consequently, the right-hand l)). Equations Equations (3.6.5) and and decreasing function function of n{t + l), decreasing of n(t i), denoted denoted by y/{n(t + I)). (3.6.6) imply Ø(n(t)) = w(n@ +

1)).

y/(n(t ++ i)) l)) falls in the variable, This dynamic equation closes the model. As w(n(t + l) 0{n(t)) rises n(t + 1) and and Ø(n(t)) rises in in the the variable variable n{t), n(t), we we rewrite rewrite the the above above map map in the following form form

n(t i) = h(n(t)), c 0. n{t ++ l) h{n{t)\ h' h'<0.

(3.6.7) (3.6.7)

An equilibrium equilibrium point is a solution of

n == h(n). h(n). In general, as shown in figure 3.6.1, there will be a unique steady state. We know < 1 for for all nn and and will will converge converge locally locally if know that nQ) n(t) tends tends to nn if \h'(n)\ <1 the following following inequality inequality holds holds h(n)\<\.<1.

Once we determine the value value of n(t), we wedetermine determine all all the the other other variables variables in in the the system. We may may conduct conduct usual usual comparative comparative statics statics analysis analysis with with respect respect different different parameters in in the the system. system.

3. ONE-DIMENSIONAL DYNAMICAL ECONOMIC SYSTEMS SYSTEMS

106 106

450

h(n)

n

*

n

12

n

Figure3 .6.1: A A steady Figure3.6.1: steady state state in in the the Aghion-Howitt Aghion-Howitt model model

3.7

Economic evolution evolution with with human capital

This section section is is concerned concerned with with interactions interactions between between human human capital capital accumulation, accumulation, economic economic growth, growth, and andinequality.7 inequality.7 hi the the economy, economy, aa competitive competitive fmal final goods goods sector sector produces produces one one homogenous homogenous output output using using human human capital capital and and aa variety variety of ofintermediate intermediate inputs with the following production function function

D(t)

where thethenumber where 85 (0( 0c <S£c<1)l )isi sa aparameter, parameter,D(t) D(t)is is numberofofdifferent differentintermediate intermediate

goods goods used used in inproduction, production, n(e, «(/', t) t) isis the quantity quantity of of the the ith ithintermediate intermediate good good employed, and H H isisthe employed, and theskilled skilled adjusted adjusted stock stock of of labor labor in in the theeconomy. economy. The The intermediate goods sector is competitive, and each intermediate good requires one intermediate goods sector is competitive, and each intermediate good requires one unit unit of of capital, capital, k, k, tototransform transform aanew newtechnology technology into intoaanew newintermediate intermediategood. good. The The symmetric symmetric use use of of inputs inputs implies implies n(i, t) =n(t). n(i,t) n(t). 7

The The model model of ofthis this section section isisbased based on onEither Either and andGarcIa-Penalosa Garcia-Penalosa (2001). (2001).

3.7. ECONOMIC EVOLUTION WITH WITH HUMAN HUMAN CAPITAL CAPITAL

107

Hence, the production function becomes l s v(t) = - {t)Hs{t). Y(t)=D(t)n

There are two, skilled and unskilled, types of workers. The skilled-adjusted skilled-adjusted stock of labor is given by 1/a

+ H(t) == (D(t_l)ua Hit) ^fU~a + s;(t)J S-p°(t)\

1, ,, 00 > aa>>—- 1,

(3.7.1)

denotes skilled skilled labor where Sp(t) denotes labor employed employed in production production and U(t) u{t) unskilled unskilled l/(l ++ a), falls fallsininthe theinterval interval(1, (l,cc). oo). Here, the labor. The elasticity of substitution, 1/(1 rate of technological change is measured by D(t)/ D{t)lD(t D(t— - i). l). The above information information takes account of the the hypothesis hypothesis that the the greater greater the the speed speed of oftechnological technological change, change, the relatively more productive skilled labor becomes, compared to unskilled labor. (t), and wu(t) stand Let w(t), w(t\ ws(t), stand for, for, respectively, respectively, the relative relative wage of of skilled skilled to to of the the skilled skilled and and unskilled unskilled labor. labor. Perfect Perfect competition competition in in unskilled labor, the wage of markets yields yields labor markets

/ W

(t)=

-

—1

-

\1+a 1

D&) D(t

372

(3.7.2)

Agents work when when young and consume only when when old. old. Let Let Agents live two periods; work consume only

s(t) bebethe thetotal totalnumber numberofofskilled skilledlabor. labor.We Wenormalize normalizethe thetotal totalpopulation population of of each each generation

s(t)+ uQ)=i. S(t)+U{t) = l. Skilled workers can be employed (t), or in Sp(t), in research research Skilled workers employed either either in in production, production, 5,, Sr When net net technologies technologies are are introduced, introduced, agents agents must must learn learn to to work work with with these these Sr (t). When technologies technologies to become skilled skilled labor. Hence, at at the the start start of oftheft theirworking working lives, lives, agents agents have to decide decide whether to invest invest in in education education or or to toremain remain unskilled. unskilled. We We assume that agents assume that agents differ differ in their abilities abilities to learn, ,u, //, and and that that theft their abilities abilities are

distributed andthe the cost cost of distributed uniformly, uniformly, pjuee[o, [0,i]. l]. Education Education is instantaneous instantaneous and education is equal to education is to c(t)wu (t)i' (t)/ p. ju. ItItisisfhrther further assumed assumed that that the direct direct cost cost of

108

3. 3. ONE-DIMENSIONAL DYNAMICAL DYNAMICAL ECONOMIC ECONOMIC SYSTEMS SYSTEMS

education c(t)wu{t) (t) is of the is a decreasing decreasing ftmction function of thenumber number of ofagents agents being being educated, i.e. educated, i.e.

c(t)=c(SQ)),\ e< 0. c'<0. We specify specify

c(t)=pS-°(t). cQ)= Assume that an anincome income tax taxz-r is is imposed imposed only on onskilled skilled wages. wages. The Theincome income of a skilled worker born born at at tt is is

-

-

(3.7.3)

M Agents choose to toinvest invest in inskills skills ififthe the income income in inequation equation (3.7.3) (3.7.3) exceeds exceeds that of remaining unskilled

lç(t)= Yu(t)=wu(t). Equality between these two two expressions expressions gives the level of of ability of of the agent that is is indifferent between investing investing in ineducation education and andworking working as asunskilled unskilled

Substituting the theuniform ability ability distribution, distribution, i.e.

s(t)= 1— S(t)=l-fi(t) the above condition yields the into the the following inverse labor supply equation

1

(

11+

pS°(t) i—s(t)

.

(3.7.4)

3.7. ECONOMIC EVOLUTION WITH HUMAN CAPITAL

109

From equation (3.7.4), conclude that externalities in education, a, a, From equation (3.7.4), we we conclude that externalities in education, generate a U -shaped relationship between the relative wage and the supply generate U -shaped relationship between the relative wage and the supply of skilled labor. labor. The The greater greater the externality, the initial decline skilled the externality, the more more prolonged prolonged the the initial decline of of the relative wage. Increases differential Increases in inthe thedirect directcost costof ofeducation, education,p,p, or in the differential tax on skilled labor, r,T,require requirehigher higherwage wagefor foreach eachlevel level of of skilled skilled labor labor supply. supply. We take account of two ways of of technological technological change, learning-by-doing learning-by-doing and costly investment investment in in research research and and development development (R&D). (R&D). Learning-by-doing Learning-by-doing takes takes skilled workers workers serendipitously serendipitously discover new types of intermediate intermediate goods goods place as skilled during production process. This source of technological technological change is given given by by DQ)

D(t-1)

=l+yS(t—l).

Research which is undertaken undertaken only only by by the government government is financed financed by a public entity (t) researchers are entity that raises raises revenues revenues through through tax collection, collection, r. When When Sr Sr(t) employed, the economy produces technological blueprints according to

D(t-1)

= (1+ fls(t

-

- i).

As skilled wewe have skilled workers workersare arefully ffillyemployed, employed, have

= s(t). S{t). Sp(t) + + S,(t) = is The budget constraint of the government is = ws(t)Sr(t). zws{t)s{t)=w s{t)Sr(t)The above two conditions yield

Sr{t)=rS{t), rS(t), = (i

—

r)S(t).

We assume that technological technological change takes place either via LBD or R&D, but not both in in one one period. period. ItIt is is not not difficult difficult to to show show that that utilizing utilizing the the equations equations above, above, we can express express the inverse inverse relative relative labor demands under the "LBD regime" labor demands under the "LBD regime" and and "R&D regime" regime" as as follows follows (check!) (check!)

3. ONE-DIMENSIONAL DYNAMICAL DYNAMICAL ECONOMIC ECONOMIC SYSTEMS SYSTEMS

110

-s(t)) = (i + yS(t - 1))(1

1+a 3 7 5 (3.7.5)

(-->

s(t)J

+ ys(t -- l»S(f--o[' l/2_gp

=

1+a

j ,

(3.7.6)

(3.7.6)

where

rT

fir =

The general equilibrium equilibrium in in the the goods goods and and factor factor markets markets isisattained attainedby byequating equating the labor inequation equation (3.7.4), to tothe thelabor labor demand demand conditions conditions labor supply, supply, expressed expressed in (3.7.5) and (3.7.6). is (3.7.6). That is

=

Iil ++ys(t yS(t - i), l),

W

+

—

1))s(t

—

for LBD, for i), for R & D,

(3.7.7)

where

1

11+ pS

—c

1+a

s&)

It is is The evolution of the The evolution of theeconomy economy isisgoverned governed by byequation equation (3.7.7). (3.7.7). It straightforwardto to see see that that for for any any given given level level of of s(t\ s(t), all are straightforward allthe theother othervariables variables are uniquely determined. In Insteady steadystate state

s(t —1)=s(t)=s, S{t-l)=S{t)=S', the steady growth rate is given by for LBD, Y(t)-Y(t-l) _ D(t)— D(t)-D(t-\) \yS', for Y(t)— v(t — ii) — D(t — i) _ 57s, Y(t) — D(t) Y(t) ~ D(t) —~ lflr(1+iS*$1* \pT (l +fi*)s'—1, - 1, for forR&D. R&D. We now analyze ofequation equation (3.7.7). (3.7.7). To simplify simplify the the analysis, analysis, in inthe the analyze behavior of remainder of this section we impose

3.7. ECONOMIC EVOLUTION WITH WITH HUMAN HUMAN CAPITAL CAPITAL

111 111

1 + a > a, which is not not strong given the elasticity which means means that the education education externality externality is strong given elasticity of substitution between the the two two types types of of labor. labor. Under Under this this requirement, requirement, we we have have

db d dS(t) dS{t)

>0 >0

It can for s(t) S(t)eE [o, [0, 1]. l]. This This also also leads leads to to os(t)/ 8S(t)/dS(t l) >0. 0. It can be be shown shown that that for dS(t —- i)> = cc. Figure = 0 and 3.7.1
s=1 S =1

s(t) 1/? Figure 3.7.1: Three possible equilibrium equilibrium points

112

3. ONE-DIMENSIONAL ONE-DIMENSIONAL DYNAMICAL ECONOMIC SYSTEMS

3.8

Urbanization with human capital externalities externalities

introduces an trade-off This section introduces an urban urban dynamic dynamic model model to to highlight highlight how how the the trade-off between optimal and equilibrium city sizes behaves behaves when when human human capital capital between optimal and equilibrium city sizes 8 The economy consists N externalities are introduced into urban dynamics. externalities urban dynamics.8 The economy consists of N identical workers. All workers are members of single-worker single-worker firms who ex ante identical live supply labor inelastically, inelastically, and consume live for one period, supply consume what what they they produce. location and human capital In the beginning of the period workers make a joint location investment investment decision. decision. In In the the middle of the period period they work, and at the end of the consume. Workers period they consume. Workers may may choose choose to to reside reside either either in in urban urban areas areas or or in rural areas. areas. Workers Workers in rural location location have incentive to invest in human human rural in rural have no no incentive to invest levels capital. Workers in the rural areas receive the earning levels 1r(t) =

A(t) describes describes the economy's level level of technology technology at and Gr is an where A(t) at time time t and exogenous productivity rural exogenous productivityfactor, factor,representing, representing,for forinstance, instance,the the quality quality of of rural infrastructure. Gross earnings received by a worker in urban location are a GA(t)ha(t), w(t) = GA{t)h w{t) {t\ a
where is the the quality quality of of public public infrastructure infrastructure in the where G G is in urban urban areas areas and and h{t) h(t) is the amount human capital amount of human capital possessed possessed by by the the urban urban worker, worker, which which can can be obtained Assumethat that the the utility utility levels levels of workers in the through a per unit level of effort effort //. p. Assume urban location are population, due to urban location are negatively negatively affected affected by by the the size size of the local population, urban external diseconomies. diseconomies. The earnings net of educational educational urban external The urban urban residents' residents' earnings net of effort effort and congestion are

i(t) I(t) == GA(t)ha(t) GA(t)ha(t)—- ,ith(t) jMt) —- b{z{t)N)ys, 8>8>\, 1, where (0 < 1) the is the proportion populationthat thatlocates locatesininurban urban where zz (0 z z1)< is proportion of of thethe population areas, and b is a constant. Urban workers maximize l(t) by choosing h(t). The and is a constant. Urban maximize 1(t) h(t). The optimal solution is

!

This section section isis based based on on Bertinelli Bertinelli and andBlack Black (2004). (2004). This

URBANIZATION WITH HUMAN HUMAN CAPITAL CAPITAL EXTERNALITIES EXTERNALITIES113 113 3.8. URBANIZATION

h(t)= where e l/(l where /3 /? = a). With With costless migration, the proportion of ofthe population population i(i —- a). that chooses to locate in in the urban urban areas, areas, 2'(t), z(f), adjusts adjusts until until the thepoint point where that chooses to locate i.e. Ir(t), i.e. I(t)==t'r(t), 1

[GA(t)h*(t) -

- GrA(t)

GA{t)h'(t)-juh'(t)-GrA(t) z(t)=i :/ = bN us [

'

(3.8.1)

offi.ill full urbanization. urbanization. It is is straightforward straightforward to prove that that the until z(t) z(t) =-11 of or until to prove the following city size S0~z(t)N, where S

1+8 maximizes per capita capita net netoutput output 7(t). / (t).9 When When choosing choosing to locate locate in in the thecity, city, maximizes per individual workers do do not not take take account of the impact individual workers account of impact of oftheir their location location on onthe the costs of living for all other workers. The optimal city size is different from the of living for all other workers. The optimal size is different from the city size size given by by equation equation (3.8.1). (3.8.1). If Ifthe thesize sizeof ofthe thecity cityisisrestricted restrictedtoto801'(t)N, <50z(?)iV, larger than that in the rural area, area, making city dwellers earning in the city will be larger better off. off. Substituting

h*(t)=

into equation equation (3.8.1) (3.8.1) gives gives the the optimal optimal level level of of net net earnings earnings for for urban urban workers workers into J*(t) = ce0A'3(t) —

9

Aplanner planner chooses chooses z(t) z(t)to tomaximize maximize z(t)I(t) z(t)l(t) ++ (i(l—- z(t))I,(t). z(t))lr(t). A

optimal problem is isgiven given by by Soz;(t)N.

Thesolution solution of ofthe the The

3. ONE-DIMENSIONAL DYNAMICAL ECONOMIC SYSTEMS SYSTEMS

114 where

a"G (&G

citM The motion of the overall overall level level of technology technology A(t) A(t) is specified specified as A(t A(t++ i)l)== max[A(t), max[^), (z(t)h(t))°j. WWif J.

(3.8.2) (3.8.2)

There is no depreciation depreciation in A. A. Provided Provided that that A(t) c (z(t)h(t))°, A(t)<(z(t)h(t)T, see we see

^ ) = | h(t) M can be rewritten as

h(t) =

—

l)h(t

—

(3.8.3)

Urbanization Urbanization acts through two two channels channels on on technological technological change: change: directly directly via z(t) z(t) in inexpression expression (3.8.2) (3.8.2) and and indirectly indirectly through through the the impact impact of of z(t z(t—i) - l ) on on h(t) as as expressed expressed in in equation equation (3.8.3). (3.8.3).

For For the the dynamic dynamic path path ofoftechnology technology {A(t)}0, {v4(?)}jl0, three three cases cases may may be be considered, no-urbanization no-urbanization z(t) urbanization 00
z(t) =

1.

incentive for urbanization urbanization to occur. occur. When z(t) == 0 and l(t) < 'When and 1(t) c Ir(t), (t), there is no incentive satisfies This will be the case case whenever whenever the the initial initiallevel levelofoftechnology technologyA4o satisfies

URBANIZATION WITH WITH HUMAN HUMAN CAPITAL CAPITAL EXTERNALITIES EXTERNALITIES 115 3.8. URBANIZATION

A0<j_L The economy remains in in aa development development trap, experiencing no no growth. ItIt can canbe be seen that that leaving leaving aside aside the case case of z(t) z(t) = = 0, 0, the the dynamic dynamic path path ofof{A(t)}70 {^(f)]f=o is ^s seen given by G1 (A(t)) it'M0 z(t) < 1,

A(t), 0, (A(t))> o, (z(t)hQ))° G,WO)>0, (z(t)h(t)Y> > A(t),

A(t + i) = G(A(t)) = G G2(A(t)) z(t) ==1,h G2(A(tj)> 0, (z(t)hQ))° l)=G(A(t)) Xz{t) G2(A{t))>0, (z{t)h(t)f > A(t\ 2(A(t)) if A(t) otherwise,

where G2(A(t))

(A(t))

b0

(t)

—

r.

G^A) governs function G1(A) The function governsthe thedynamic dynamicevolution evolutionofofA(t) A(t)in in the the case case of of partial urbanization. urbanization. The Thefull full urbanization urbanizationdynamics dynamicsisischaracterized characterizedbyby G (A). Cl2 (A). 2 the point It is easy to prove that at the

z(t)=0, G2(A(t)), z(t) = 0, G1(A(t))= G1(A(t))=G 2(A(t)\ G2(A)isisstrictly strictlyless lessthan thanthat thatof ofG1 G^A). We now now illustrate different different the slope of 02(A) (A). We difficult and cases. The technical parts are are not difficult and left the the reader to check. Case 1: 1: The Theeconomy economy always always remains remains at its its initial initial levels levels of of urbanization urbanization and aggregate of technology. This case case occurs occurs whenever whenever aggregate output, for all initial levels of the GX(A) (A) function concave and remains remains below the the 45° 45° line. line. This This happens happens function is concave low and/or and/or when when the incentives incentives to invest invest in human human capital capital are when is too low whenAQ 4 is 10 too low. low.The Thesituation situation isisillustrated illustratedininfigure figure3.8.1.10 3.8.1.

10

The plot is ' The is for for concave concave Gt (A). (A). When When GGisisconvex, convex,similar similarconclusions conclusionsare areheld. held.

3. ONE-DIMENSIONAL DYNAMICAL ECONOMIC SYSTEMS SYSTEMS

116 116

Case 2: the 45° 45° line. 2: This This refers refers to to the thecase casewhen whenG1(A(t)) G^Aty)) intersects intersects the line. If the the G,(/4(?)) equilibrium points: (AQ)) is is concave, concave, there there are are two two equilibrium points: a lower lower unstable unstable one and upper stable stable equilibrium. equilibrium. For all levels levels of of technology technology above above the lower lower an upper For all intersection (A(t))function functionand andthe the 45° 45° line, the economy experiences intersection of the G1 Gx{A(t)) experiences concave, the the upper upper growth of technology technology and and urbanization. urbanization. If If G1(A(t)) Gx{A{t)) is growth is concave, and partly partly urbanized, urbanized, as as illustrated illustrated in in figure figure 3.8.2. 3.8.2. equilibrium point is stable and

A{t + \

-A(t) 4') Figure 3.8.1: No urbanization

If If

G1 (A(t))isisconvex, convex,the theeconomy economyexperiences experiences full full urbanization. urbanization. After full G,(^(?)) full urbanization, the economy economy isis characterized characterized by byG2 G2(A(t)). G2(A(t)) (A(t)). If G2 (A(t)) is is concave, the economy economy will steady state state level level of of technology technology as depicted depicted in will grow grow up up to a steady figure is convex, convex, the theeconomy economyexperiences experiences unbounded unbounded figure 3.8.3a. 3.8.3a. If If G2(A(t)) G2(A(t)) is growth with full urbanization as illustrated illustrated in in figure figure 3.8.3b. 3.8.3b.

3.8. URBANIZATION URBANIZATION WITH WITH HUMAN HUMAN CAPITAL CAPITAL EXTERNALITIES EXTERNALITIES 117

A(t)

points Figure3.8.2: Partial urbanization with two equilibrium points

A{t + \) A(t+1)

A(t + \ 0

G2(A

450

GX{A)

A{t) 4')

A(t) A(t)

a) with withconcave concaveG2 G2(A) (A) a)

b) convex convexG2 G2(A) (A) b)

low equilibrium equilibrium Figure3.8.3: Full urbanization with a stable low

118

3. ONE-DIMENSIONAL DYNAMICAL ECONOMIC SYSTEMS SYSTEMS

3.9

The OSG model with money1' money11

Modem analysis Modern analysis of the the long-term long-term interaction interaction of of inflation inflation and and capital capital formation formation begins with Tobin's seminal contribution in 1965. Tobin deals with an isolated begins seminal contribution 1965. isolated economy in which "outside money" competes with real capital in the portfolios of economy in which "outside money" competes with real capital in the portfolios of I2 agents Since then, then, many many growth growth agents within within the the framework framework of of the the Solow Solow model.'2 model. Since 13 This section models of monetary economies are built within the OLG framework.'3 models of monetary economies are built within the OLG framework. This section introduces money within the OSG framework proposed in section 3.2. We present introduces money within the OSG framework proposed in section 3.2. We present the model in discrete time, numbered from zero and indexed by t = 0, the model in discrete time, numbered from zero and indexed by t = 0,1, 2, . Time 0, being referred to the beginning of period 0, represents the initial situation Time 0, being referred to the beginning of period 0, represents the initial situation from starts to to grow. grow. The The end end of of period period tt — coincides with with the from which which economy economy starts - 11 coincides the period t; it can also be called time t. We assume that transactions are beginning of period made in each each period. period. The The model model assumes assumes that that each each individual individual lives lives forever. forever. The population, N, N, isisconstant. constant.Each Each individual individual supplies supplies one one unit unit of of labor labor at each time t. Production in period period tt uses amount capital and of labor labor amount K(t) K(t) of capital and amount amount NN of services as as inputs. inputs. It It supplies supplies amount amount Y(t) Y(t) of ofgoods. goods.Here, Here,production production isis assumed assumed services continuous during during the the period, period, but then then use use the the same same capital capital that that existed existed at at the the to be continuous beginning of the period. Production is made through a neoclassical constant return to scale technology. The The real interest interest rate rate and and the the wage wage of of labor labor are are given given as as before before by by

1, 2,.

r(t) = f'(k(t)), w(t) f(k(t)) — k(t)f'(k(t)). w{t)= = f{k{t))-k{t)fik{t)). r(t)+S + t=f{k{t)\

(3.9.1) (3.9.1)

For simplicity we normalize N == 1. For simplicity of expression, expression, we normalize N 1. There There is some some initial initial capital stock stock k0 that is owned equally by all individuals at the initial period. We assume that agents agents have perfect perfect foresight foresight with respect to all all fhture future events events and capital capital markets markets operate operate frictionless. frictionless. The government levies no taxes. Money is is introduced by assuming that a central bank distributes at no cost to the population introduced by assuming that a central bank distributes at no cost to the population aa per of fiat fiat money, money, M(t) M(t)>>0.0.The per capita capita amount amount of Thescheme scheme according according to to which which the the money stock evolves over time is deterministic and known to all agents. With p money stock evolves over time is deterministic and known to all agents. With ju being net growth rate, the M(t), evolves being its its constant constant net growth rate, the money money stock, stock, M(t\ evolves over over time time according to

M(t)=(l+p)M(t—1), M{t) = (l + [i)M(t - l), p>O. //>0. 11

This model isis referred referred to to Zhang Zhang (2005b: (2005b:chapter chapter 3). 3). See Tobin (1965). Outside money money isis the the part part of money stock which is issued (1965). Outside issued by by the the See also also Burmeister Burmeister and andDobeIl Dobell (1970) (1970) and and Zhang Zhang(1999). (1999). government. See 13 13 See See Tobin Tobin (1969). (1969). 12 12

3.9. 3.9.THE THEOSU OSGMODEL MODELWITH WITHMONEY MONEY

119 119

At of period At the the beginning beginning of period t/ the government government brings brings M(t) M(t)——M(t M(t ——i) l) additional units capita into into circulation circulation in order to to fmarice finance all all additional units of of money money per per capita in order work, government expenditures via seigniorage. For the seigniorage mechanism to work, injections of the additional units of of money take place before the the other other markets markets open. open. Let m(t) stand for the real value of money per capita measured in units of the Let stand real value of money per capita measured units the can rewrite rewrite the the above above equation equation as as output good. Then, we can

g(t) = M&)

M(t

—

=

m(t).

(3.9.2) ( 392)

In this model, money acts as a pure money acts pure store store of of value. value. The The demand demand for for money money relies exclusively exclusively on the speculative speculative conjecture conjecture that the the fUture future exchange value of money in terms of goods money goods will will be be positive positive because because people people will will express express aa positive positive demand for money in the uncertainty and with money not the fUture. future. In the absence of uncertainty being required for transaction purposes, purposes, this this "bubbly "bubbly view" view" implies that being explicitly explicitly required for transaction implies that in a monetary equilibrium point the return on money needs to be equal to the return in a monetary equilibrium point the return on money needs to be equal to the return on projects. In In period period t,/, on competing competing assets assets such such as as claims claims on on productive productive investment investment projects. each receiving the the per per capita money stock stock M each consumer consumer receiving capita nominal nominal money M believes believes that that will be be exchanged exchangedatatthe theexpected expectedfUture futureprice priceptm pm(t(t + l) money will i)>> 0. The price of m (t) ptm money p (t) is is in terms of goods or itit expresses the amount of goods that can be

purchased by one unity of money in each period 1. t. According According to to the the definitions, we have 1

p#i(()

A necessary condition for for a monetary monetary economy economy to to exist exist is that the price of money necessary condition must be positive. If money money has has aa positive positive value, value, people people have have an an opportunity opportunity to to save save According to to the definition in money. money. According definition of the price price of of money, money, the the deflation deflation rate rate P(t)/ P(t + l) i) coincides with the real P{t)l real return return on on money. money. In In aacompetitive competitive economy economy of arbitrage arbitrage opportunities opportunities entails entails equality equality of of return return on on assets assets the absence of P(t) -4 P(t+l) p(t As

120

3. ONE-DIMENSIONAL DYNAMICAL ECONOMIC SYSTEMS SYSTEMS

M(t +1) —_ m(t+l) m{t +1) P(t) —_ P(t) P(t) M'(t+l) P(t + i) l) — ~ P(t P{t ++ i) l) (i (l + p)M'(t) n)M{t) — ~ (i (l + .p(t the above condition becomes 1+

f'(k(t

+ i))

=

m(t +

i)

(393)

—

(1 +

The consumer obtains obtains income income in period period tt from the interest payment The consumer from the interest payment andthe thewage payment w(t) w(t) r(t)k(t) and (3.9.4) (3.9.4)

y{t)=r{t)k{t)+w{t), y(t) = r(t)k(t) + w(t),

calledthe thecurrent currentincome. income.We Wedefme define the the disposable disposable income income as the where y(t) isiscalled the + m(t) m\i) and sum of k(t) + and the current income, i.e. m{t). y(t) + k(t) + m(t).

(3.9.5)

is given by The budget constraint is c{t)+s{t) = y{t). c(t) + s(t) = For simplicity, take the theCobb-Douglas Cobb-Douglas utility utility fimction function to to describe simplicity, we we take consumers' preference preference

u(t) =

+ 2 =

1,

2>

0,

and 2Aare arerespectively respectively the thepropensities propensities to to consume consume goods and and to own in which £ and wealth. A typical household maximizes the utility subject subject to the the budget constraint. We solve the the optimal choice of the consumers as c(t) = = # ( / ) , *(') s(t) == #2;(t). *(/) (')

(3.9.6) (3-9-6)

to the the definitions, the per capita capital + l), According to capital in period period t ++1,1, k(t + 1), is to the the saving saving made made in inperiod period itt minus the real value equal to i.e. value of of money money in in period period t, i.e.

3.9. THE OSG MODEL WITH MONEY

k(t i) = s(t) — m(t). k(t ++ \)=s(t)-m(t).

121 121 (3.9.7)

The economy starts tooperate operate in init/ = 0. Each member endowed starts to = 0. member in period period 0 isisendowed with M(— M{-\) units of of money money and andowns owns k0 (= (= K(0)I K(0)/N(O)) units of ofphysical physical i) units N(0)) units The labor force N is exogenously given. We will show that the above capital. capital. The force N is exogenously given. We will show that the above equations allow allow us us to to calculate calculaterecursively recursivelyall allthe thek(f) k(t) and andm(t). m{t). The government spending is is given by equation (3.9.2). From equations (3.9.5) and (3.9.6), we obtain

s(t) + k(t) + m(t)). s(t) == 2(y(t) A(y(t)+k(t)+m{tj). Substituting equations (3.9.1) and (3.9.4) into the above equation yields

s(t) 2(f(k(t)) ++ S4(t) + m{t)\ m(t)), Sk{t) + s{t) == A(f(k(t))

(3.9.8) (3.9.8)

where 8k• Substituting equation (3.9.8) into equation (3.9.7), we obtain where 58 = 11 — - 8 k. Substituting equation (3.9.8) into equation (3.9.7), we obtain k(t + i) == A(f(k(t)) *(/ + l) JL{f(k{t)) ++ &(t) Sk{t)++m(t)) m{t)) —- m(t). m(t).

(3.9.9) (3.9.9)

From equation (3.9.3), (3.9.3), we obtain

f(k(t ++l))]m(t). m(t + + i) 1) = = (i(1 + + n)[S+ + f'(k(t l))]m(t). m{t

(3.9.10) (3.9.10)

Definition Definition 3.9.1. 3.9.1. Given Given the theinitial initial capital capital stock k0 k0 and andthe theinitial initial money money stock stock M(— given quantities M(— i), l),a acompetitive competitiveequilibrium equilibriumisis givenbybya asequence sequenceofof quantities {m(t), cfr), s(t), s(t), g(t), g(t), k(t {m(t), c(t), k(t + + l)} 1)} and aa sequence of of prices prices {r(t), 2,• : {r(t), w(t), w(t), P(t)} P(t)} such such that that for for all all periods periods tt = =0, 0 , 11,, 2, (i) competition to competition ensures ensures that factors factors get get paid their their marginal marginal products products according according to equations (3.9.1); (ii) given

M(t)=(1 ÷p)M(t

—

budget (3.9.2) of the government (iii) given given the government is satisfied; satisfied; (iii) theprice price budget constraint constraint (3.9.2) consequence, agents solve solve optimally optimallythe the decision decisionproblem; problem;(iv) (iv)the the evolution evolutionof of consequence, agents money balances balances satisfies satisfies equation equation (3.9.10); investments and savings are money (3.9.10); (v) investments and savings are

the the

122

3. ONE-DIMENSIONAL DYNAMICAL ECONOMIC SYSTEMS SYSTEMS

matched equation (3.9.7); (3.9.7); and (vi) all all the the markets markets clear clear with with the theequilibrium equilibrium matched by equation as following: following: conditions being as Labor market:

w(t) = f(k(t)) — k(t)f'(k(t)); Money market:

m(t) == M(t)/P(t); M(t)/P(t); m(t) market: Goods market: /(*(/)) k(t) ==g(t) g(t) ++Skk(t) Skk{t) + c(t) + k(t + 1). f(k(t)) + k(t) + c(t) i). Equilibrium as as inside andand outside money equilibrium: Equilibrium can canbe befurther furtherclassified classified inside outside money equilibrium: inside money money equilibrium equilibrium is associated associated with situation in in which which outside outside an inside with aa situation are zero; zero; an an outside outside money equilibrium equilibrium is is associated associated with positive outside balances are money balances. balances. We now now examine examine dynamic dynamic properties properties of equations equations (3.9.9) and (3.9.10). (3.9.10). At money equilibrium equilibriumwith with m{t) m(t) = 0, inside money 0, we we have k{t + + \) k(t i)== l{f{k(t))+Sk{t)). 2(f(k(t))+ &(t)).

(3.9.11) (3.9.11)

Government Governmentspending spendingisisreduced reducedtoto zero. zero. As As shown shown before, before, this this system systemhas has aa unique stable equilibrium point given by by

1

2 X

(3.9.12) (3912)

are mainly mainly interested interested in in monetary monetary economy economy which which isischaracterized characterized by bythe the We are three difference (3.9.2), (3.9.9) (3.9.9) and (3.9.10). (3.9.9) and three difference equations equations (3.9.2), (3.9.10). Equations Equations (3.9.9) (3.9.10) do In fact, fact, once m(t) m(t) isisdetermined, determined, we determine (3.9.10) do not not contain contain g(t). g(t). In determine g(t) by (3.9.2).14 The dynamics dynamics of the the monetary monetary economy economy are are described described by by equations equations by and (3.9.10). We may rewrite the (3.9.9) and We may the system as follows 14 14

Here, we we consider that the the conduct conduct of of monetary monetary policy takes takes precedence over all all fiscal fiscal Here, chapter 2) 2) provides provides aadiscussion discussion of ofhow howto toresolve resolvethe thecoordination coordination matters. Gale (1983: chapter monetary and and fiscal measures measures in in aa similar similar context. context. problem between monetary

3.9. THE OSG MODEL WITH MONEY 1) = + l) k(t + = 2(f(k(t)) A(f(k(t)) + — in(t + i) (i + ftp + + /"(*(' f'(kQ + 1))]m(t). m(t 1) = (1 l))]w(/)-

123

(3.9.13)

A steady state of the monetary economy needs to satis& satisfy

11 == (i(1++//X<5+ + f'(k)). /"(*)).

(3.9.14) (3.9.14)

From 11 = (l (i + + ju%S + /'(&)), f'(k)), we get —

1+jt

(3.9.15)

.

///(l + iii), //), equation equation (3.9.15) (3.9.15) From the we see see that that ifif 8k Sk >> 1ui(i From the properties propertiesofof /f,, we Fromthe thefirst firstequation equationin in(3.9.14), (3.9.14), we solve has a unique solution, solution, denoted denoted by by k*. k. From

k

m

-.

(3.9.16) (3.9.16)

A,

K

For m >> 0, For m 0, we should require

f(k)e5 F k"

'~AA+

k

'

Comparing equations (3.9.12) (3.9.12)and and (3.9.16), (3.9.16), we we see see that the Comparing equations the monetary monetary economy has a unique equilibrium point if equilibrium > ft). M>M.

k

(3.9.17) (3.9.17)

k

As As

d(f/k) d{flk) dk

=

f-— f/k <0 k

ONE-DIMENSIONAL DYNAMICAL ECONOMIC SYSTEMS 3. ONE-DIMENSIONAL

124

becauseofofstrict strictconcavity concavity decreases in in k. Hence, because of of / , f,f fIIkk strictly strictly decreases Hence, equation equation (3.9.17) implieskk>> k* if both both kk and exist. As As /f'' decreases (3.9.17) implies and k* exist. decreases in k, we we see see that if

then kk>>k*. then k*. In summary, we have the following proposition. proposition. Proposition 3.9.1. has a unique 3.9.1. Suppose Suppose that that the the inside inside competitive competitive equilibrium equilibrium has non-trivial steady state state k >> 0. Then, the monetary has a unique steady non-trivial steady 0. Then, monetary economy economy has steady state Fk*ififthe thefollowing followinginequality inequality holds: holds:

The The steady steady state state isis aa saddle saddle point. point.

Proof Proof. We Weonly onlyneed needtotoexamine examine the the stability stability of of the the steady steady state. state. The The Jacobian Jacobian at at steady state for equation (3.9.13) is given by = [au an

a2 a221 a22j

in which a11

= a21 a21 = = Ainf" Amf", a22 //)m/"£ a 2 2== 1l -—( (il ++ .u)mfht4t. a12

The The characteristic equation is is given by

H(p) == pp22 -— Trace{j)p Trace(J)p + + Det(J), Det(j\

3.9. THE OSG MODEL WITH MONEY

125 125

where

Trace(J) + 1 — (i + / / ) » / " £ > Trace{j) = aa11 a22 = 2(1' X{f + a) S)+l-(l > 0, 0, u + a22 Det(J) a)>> 0. a12a21 Det(j) = = aa,1a22 X{f + S) 0. na22 -— a l2a21 = 2(1' We have

H(l) Trace(J) + + Det(j) Det(J) == —1 (i + H{\) == 1 —- Trace{j) -1 + (l + p)mfflct //)«/"£ < 0. 0 1

We thus conclude that the the two two eigenvalues eigenvalues are arereal realand andsatisi3 satisfy

0
dp

1

(1+p)

0

the rate rate of of monetary monetary expansion expansion p ji isispermanently permanently raised, raised, then then the the per per capita capita If the capital capital stock stock in the new new monetary monetary steady steady state state is is higher higher than than before. before. That That is, is, the the 15 Tobin effect prevails.'5 This is illustrated as in figure 3.9.1. As /1 is shifted, the effect prevails. This is illustrated as figure ju is shifted, the equation accumulation equation

is not affected; the arbitrage relation

1=(1+PXS+f'(k)) is shifted shifted by lowering lowering the return return on on real real balances balances via viaincreased increased inflation inflation taxation. taxation. rebalance the arbitrage arbitrage relation, relation, the composition composition of the the portfolios portfolios of of agents agents To rebalance in favor favor of of capital. capital, In hi the the new new steady steady state, state, the the capital capital stock stock will will be be needs to readjust in higher than than before. before. higher 15

The substitution substitution of of capital capital for for fiat fiat money money in reaction reaction to an an increase increase in in anticipated anticipated inflation inflation is called called the Tobin effect, effect, as described described in Tobin Tobin (1965). (1965). See See also also Champ Champ and and Freeman (2001). (2001).

126

SYSTEMS 3. ONE-DIMENSIONAL DYNAMICAL ECONOMIC SYSTEMS

m m

2(f(k) + & + m)-m

as pn increases as mcreases = (i

k*

k*

+ p)(8

+

f'(k))

k

k

Figure 3.9.1: The Tobin Tobin effect effect

3.10 The OSG OSG model model with labor supply'6 supply16 3.10 The This section section extends extends the OSG OSG model model to to treat treat time timedistribution distribution among among various various This activities as endogenous variable. Most of the assumptions and variables activities endogenous variable. Most of assumptions and variables are are the stand for for the flow flow of of labor labor services services used same as in section section 3.2. 3.2. Let Let N(t) stand used at at time time t T(t) for production. production. The Thetotal totallabor laborforce forceN{t) N(t) is is given given by by N(t) N(t) =- T(t)N T(t)N0, 0, where T(t) thepopulation. population. Let Let the work work time time of ofaarepresentative representative household household and N0 No isis the is the Themarginal marginal conditions conditions are given by K(t)/ N(t). The k(t) = K{f)l

r(t) + 8k =f'(k(t)), = fik(t)\ r(t)+ök

—

k(t)f'(k(t)).

(3.10.1)

stand for for per per capita capita wealth. wealth. According According to to the definition definition Let k* k*(t)(= K(t)/NN0) Let (t) K(t)/ 0) stand k(t) and and k*(t), k*(t), we we have have k*(t) k'(t)== k(t)r(t). capita current current income of k(t) k(t)T(t). Per Per capita income from from the interest and the wage payment w(t)T(t) w(t)T(t) isis given given by interestpayment payment r{t)k'{t) r(t)kt(t) and

y(t) y{t) == r(t)k*(t) r(t)k'{t)+w(t)T(t). + w(t)T(t).

16 16

This section based on on Zhang Zhang (2005c). (2005c). section is based

3.10. THE OSU OSG MODEL WITH LABOR SUPPLY

127 127

The per capita disposable income is is given by disposable income y{t)=y(t)+k'{t). ;(t)= y(t)+ k(t). The disposable income is is used for forsaving and consumption. At Ateach point of of time, a consumer distribute the total total available among saving, saving, s(t), consumer would would distribute available budget budget among s(t), consumption c{t).The Thebudget budget constraint constraint is is given by consumption of of goods, goods, c(t). by = KO = = r(t)kQ) (0++ w(t)T(t) Mt)r(t) ++ k'{t). c(t) c(t) + + s(t) s(t) = k(t).

(3.10.2)

theleisure time at at time tt and andthe the(fixed) (fixed) available available time time for forwork Denote Th(t) the To. The time time constraint constraint is isexpressed by and leisure by T0. by T(t ) + Th (t )=

T0

this fUnction function into the budget constraint yields yields Substituting this

w(t)7 M{t)T c(t)++s(t) s(t)== y{t) =r(t)k(t)+ r{t)k'{t) + w{t)T+0 k*(t). + k'{t). w(t)7,(t)+ h{t) +c(t)

(3.10.3) (3.10.3)

model, atateach eachpoint pointofoftime, time,consumers consumers have havethree threevariables variables to todecide. decide. In our our model, utility level UQ) U{t)that that the theconsumers consumers obtain obtain is isdependent dependent on on the the We assume that utility time, Th, the thesaving, saving, s(t), s(t), leisure time, the consumption consumption level level of of commodity, commodity,c(t), c(t), and and the follows as follows U(t) =

calledthe thepropensity propensitytotouse useleisure leisure time, time, £, the propensity to to consume, where ao isiscalled X,the thepropensity propensity to toown ownwealth. wealth. For Forany any individual, individual, the w(t),and and and 2, the wage wage rate, rate, w(t), rate of of interest, interest, r(t), aregiven given in in markets markets and the level the rate r(t), are level of of wealth, wealth, k(t), k(t), is decision. predetermined before decision. U(t)subject subjecttotothe thebudget budgetconstraint constraint (3.10.3) (3.10.3) yields Maximizing U(t) w{t)Th{t)=dy{t), c(t) = &(t), s(t) s(t) == Xy{t). = 25(t).

(3.10.4)

128

3.ONE-DIMENSIONAL ONE-DIMENSIONALDYNAMICAL DYNAMICALECONOMIC ECONOMICSYSTEMS SYSTEMS 3.

According to tothe definitions, definitions, the household's wealth in in period tt + + I1 is is equal to to the savings made in period period t k*(t k'(t + I) l) = = s(t) = 2p(t). Ay(t).

(3.10.5) (3.10.5)

completed the model. We now analyze dynamic properties We have thus completed properties of of the system. system. We now examine fromthe the examine the theexistence existence of ofequilibrium equilibrium and andstability. stability. First, First, from definition of of.V(t) y(t) and and equations (3.10.1), we obtain y= = (f(k)-

—

8k)k' + + (f{k)-kf{k))T (f(k) — kf (k))T0 +k\ F. 0 +

(3.10.6)

Substituting this this equation equation into into w(t)Th(t) (t) = = oy(t) dy(t) yields yields

{S++ Xif{k) - kf'(k))T0 kf(k))T0==(f'(k) (f(k)+ + s)ok* + (/(*) - kf'{k))r, + (f(k) — kf'(k))T, AXf(k) — weuse useequation equation (3.10.1) (3.10.1) and andT(t) + Th(t) == To. Inserting Inserting k"{t) T(t)k(t) where we kQ) == T(t)k(t) into the above equation, we solve T(t)

—

-

(f(k) —

+ 2)T0

where

S = \-Sk,

f-kf>0 f—kf>0

for any positive k. We Wesee see that that for for any positive positive k. positive k

0
o
(3

107)

SUPPLY 3.10. THE OSG MODEL WITH LABOR SUPPLY

129

We see that that T(t) T(t) is uniquely uniquely determined function of k(t). From From We see determinedasas aa function of k(t). k*(t) = k*(t) - T(t)k{t) T(t)k(t) and equation (3.10.5), we obtain

T(t + + 1)k(t l)k(t + + 1) 1) = AyQ). 25(t). k*(t) = T(t)k(t) T(t)k(t) and equation (3.10.6) into this equation, we obtain Inserting k* (t) = l))k(t + i) 1) = ))(/(*(/)) -— k(t)f(k(t))), A0((f + 1))k(t k(t)f'(k(t))), = A(*(f A(k(tD(f(kQ))

(3.10.8)

where A0(*(f + l)) =

(f(k(t + 1)) - kQ k(t + 1)f'(kQ l)f(k(t ++ 1))X£ + X) 1))— (f(k(t + (f'(k(t + 1)) 1)) + S)ok(t + 1) + 1)) - k{t 1) + (f(kQ/ + 1))— 1)f'(k(t + + 1)))' kQ + + l)f(k(t 8)ok(t + + o)k(t) + (f(k(t)) — k(t)f'(k(t))) A(kQ)) = [ (f'(k(t)) — [(f'(k(t)) + S)ok(t) + (f(k(t)) — k(t)f'(k(t)))

> 0

We calculate 1))] d[k(t + i)Ajk(t l)A0(*(/ + 1))] dkQ + 1)

(f(k) — kf'(k)))A0(k) (.f(k) + S)ok 5)ok + + {/{k) (f(k) -— kf'(k)) kf'(k))

2 f(k) - kfjk))) ok f"A0(k)(if'(k) +8)k S)k + + (f(k) ck2f"A(k)((f'(k)+ >0 kfik))[{f{k) ++8)ak (f(k) -— kf'(k))[(f'(k) S)ck ++ (f(k) (f(k) -—kf'(k))} Içf(kj)]

according to to the the implicit implicit function function theorem, theorem, d{hok)l dk >> 00 for any positive As d(A0k)/ positive k, according from equation as follows follows equation (3.10.8) (3.10.8)we wecan canexpress express k(t k(t ++ l) i) as a function flinction of of k(t) as A'(k(t)), k(t + + l) i) == A*(k(t)),

(3.10.9) (3.10.9)

where same degree degree of smoothness smoothness of f"(k). The Thedifference difference where A*(&) K (k) has has the same of f"(k). equation equation involves involves aa single single variable variable k(t). k(t). With a positive initial condition, the onedimensional dimensional difference difference equation equationdetermines determines k{t) k(t) in any period of time.

130 130

3.ONE-DIMENSIONAL ONE-DIMENSIONALDYNAMICAL DYNAMICALECONOMIC ECONOMICSYSTEMS SYSTEMS 3.

Forany anypositive positive solution solution k(t) k(t)of ofequation equation (3.10.9), (3.10.9), all the other 3.10.1. For Lemma 3.10.1. all the variables are are uniquely determined by the the following T(t) by variables uniquely determined following procedure: procedure: T(t) by equation equation (3.10.7) -> Th(t)=T0 —- r(t) T(t)—> - > k*(t)= k'(t)= T(t)k(t) T(t)k(t) —+ - > r(t) r(t)and and w(t) by equations equations (3.10.7) —> w(t) by - > y(t) byequation equation (3.10.2) (3.10.2) —> - > j(t) y(t)bybyequation equation(3.10.3) (3.10.3)—÷ - » c(t) c(t)and and (3.10.1) (3.10.1) —3 5(t) by by equations equations (3.10.4) - > F(t) F(t)= N(t)f(k(t)). s(t) by (3.10.4)-^ —3f(k(t)) f(k(t)) —* = N(t)f(k(t)). determine the 3.10.1 guarantees guarantees that Lemma 3.10.1 that once once we we determine the dynamic dynamic properties propertiesof of equation candetermine determine the of all all the theother other variables variables in inthe the equation (3.10.9), (3.10.9), we we can the behavior of Hence, itit is issufficient for for us us to to be be concerned concerned with with equation equation (3.10.9). (3.10.9). system. Hence, By equation (3.10.8), (3.10.8), an anequilibrium equilibrium point point isisgiven given by by A0(k)k=A(kXf(k)-kf'(k)). = A(kxf(k) — kf'(k)). Inserting A0 (k) and A0(A;) andA(k) \(k) into intothe theabove aboveequation equation yields yields

(3.10.10)

The equation has aa unique positive solution if if

With lemma We lemma 3.10.1, 3.10.1, we wesee seethat that the thesystem system has hasaaunique uniquepositive positive solution. solution. We denote the equilibrium equilibrium value by k*. k*.To Tocheck check stability stability of of the theunique unique denote valueof of kk by equilibrium, we Taking derivatives derivatives of equilibrium, we calculate calculate dA dA1 / dkatatk'. F. Taking of the the two two sides sidesof of }/dk equation (3.10.8), (3.10.8), we wehave have equation d[A0((t dk(t ++ l) i) — d[A(k(t))(f(k(t)) l))k\ d[A0((t ++i))k(t — l)] ++ 1')] k— dk(t + 1) dk(t+1) dk(t) — dk(t) ' — dk(t) (3.10.11) (3 .10. 11)

Substituting Substituting d(kA0) d(kA0) dk dk

2

— ok f" A o ((/' ++ £)fc+ + ((f/ —- kf')) kf)) kt")A (f-kf')K (f'+S)ok+(f—icf') (f + S)ok + (f-kf) (f—kf')[(f'+S)ok+(f—kf)]' (f-kf')[(f'

3 .10.THE THEOSG OSGMODEL MODELWITH WITHLABOR LABORSUPPLY SUPPLY 3.10.

[+8

dA[

dk dk[(f'+S)k+ (f-kf')

(p+8)ck+(f-kp) (f-kf)

131 131

A,

into equation (3.10.11), we have

u

' dk dk

=F2(k),

in which F1(k)

f — kf'

—

± 2Xf+8)k

—

-

+

[+8

f-kf' /-¥' at Using /[>0, ' > 0,['<0 / " weseethat 0, we see thatF1F>0 and F2 >0. > 0. at k'. F. Using f—kf'>O, l > 0and It is straightforward F^>> F2. Hence, we conclude conclude straightforward to check check Fj dA1

dk

F =_L€(o,1). /Pj;

This implies that the equilibrium point is stable.

Theorem 3.10.1. 3.10.1.The Thedynamic dynamicsystem system has has aaunique unique stable stable equilibrium. equilibrium. Theorem We showed that the dynamic dynamic system system has aa unique stable stable equilibrium. We now of change in some parameters. First, we introduce introduce technological technological examine impact examine of change in some parameters. First, we change by specifying change by speciQving

f{k)=Ah(k), f(k) = Ah(k), where A A describes describes the thelevel levelofof technology. Taking Taking derivatives derivatives of ofequation equation where technology. A yields yields (3.10.10) with respect respect to to A

ONE-DIMENSIONAL DYNAMICAL ECONOMIC SYSTEMS 3. ONE-DIMENSIONAL SYSTEMS

132

dk _

kf kf >0, dA (.f—kf')A> ' dA~{f-kf')A

f

where Astechnology technology is isimproved, improved, the thecapital capital intensity, intensity, k, is where f/k fi k —- f > > 0. As is increased. From equation equation (3.10.7), (3.10.7), we we obtain obtain dT = dA dA

kf')f' + + kff"){f kff")(f + + bk)kTc ((/ - kf')f Sk)kTa ((f —

A(f—kft[(f'+S)ck+(f—kf')I

The sign sign of of dT/dA dT I dAisisthe thesame sameas as that that of (f —

kf')f' + kff".

As c 0, the If /f takes / - kf')f'> kf')f > 00 and and kff" kff" < theimpact impact is is ambiguous. ambiguous. If takes on onthe the As (f( — Cobb-Douglas form, form, i.e. i.e.

f = Aka, then fimction takes takes on the CES form then dT dT/ Ic/A dA == 0. 0. If the production function

A(akp ++ \)vp, f/ ==A(ak" where A are positive, where pp ci, < 1,a and a and A are positive,we wecalculate calculate

(/ —(f

kf')f' ++ kff"= ¥')f W" = apA2kp-\akp + l) 2/p " 2 .

We see that that ifif pp>0, > 0,then then dT/dA>0; dT/dA>0; if if p=O, p = 0, then then dT/dA=0; dT/dA = 0; and and ifif

ppcO, < 0,then then dT/dA<0. dTldA< 0.By By k k"=kT, = kT,wehave we have dk* dA

dT dk =k—÷T—. dA

dA

If the work time, then per the improvement improvement in intechnology technology increases increases work percapita capita wealth wealth increases; otherwise otherwise the impact impact is ambiguous. ambiguous. From From equation equation (3.10.1), (3.10.1), we we definitely increases; obtain

_______ 3.10. THE OSG MODEL WITH LABOR SUPPLY

dA dA

A

dA dA'

dw

w

dk

dA dA

A

dA dA

133

The wage rate is increased due to technological improvement; but the impact on the ambiguous. The The impact on on the the output output level level is is given by rate of interest is ambiguous.

dA dA

A

dA dA

We now examine the impact of change in in preference. As

a change change in in the the propensity propensity to to consume consume leisure leisure has to to be be associated associated with with some some changes in other other propensities. propensities. For simplicity, simplicity, we specify the preference preference change as follows

dc=—dA, dç'=0. Taking derivatives of equation equation(3.10.10) (3.10.10)with withrespect respecttotoaa yields derivatives of dk dk _

%k2


2< da (f—kf')22 ~da~~~{f-kf')A

As the the propensity propensity to use use leisure leisure increases increases and and the the propensity propensity to to save save declines, declines, the the capital intensity declines in in the the long longterm. term.ItItisisimportant importanttotonote notethat thatififda da = —- di; intensity declines

and d2 / da >> 0. dX == 0, 0, then we have dk dk/da 0. That is, ifif the the propensity propensity to use use leisure leisure time increases increases and and the the propensity propensity to to consume consume declines, declines, then the capital capital intensity intensity increases. Another Mother possible possiblepattern patternof ofpreference preference change change isis given given by by

da=—d(e+2),

2

In this case, case, we we have have dk/ dk / da da == 0. 0. As preference may change in different different ways, its impact impact on kk isisdependent dependent on onthe thespecified specified pattern. pattern. In In what what follows, follows, we we limit limit our our discussion to the case of

ONE-DIMENSIONAL DYNAMICAL ECONOMIC SYSTEMS 3. ONE-DIMENSIONAL

134

da = - dA, dg = 0. dc=—dA, equation (3.10.7), (3.10.7), we we obtain obtain From equation ]_dT_ _ ~ dk (p-l-SXf—/cf')c_(f-l-&)aJcf"l 1.41 Tdc[ [(fT+s)ak+(f kf)J(f kf')Jda Tda~ IT/' + S)ak + (/ - */')](/ - ~kT)da (f'+s)k

11

(£ + X)

(f'+s$+(f—kf') as the propensity to stay at home is increased, the time of staying We conclude that as at home increases. increases. It should should be be remarked remarked that this conclusion conclusion may not hold if we specify different patterns of preference change. By k* k* =- kT, we we have have

dp dp

dT dp

dk

=k—+T——cO. = k + T < 0 . dp dp

dp

propensity to to use use leisure leisure increases, increases, per capita capita wealth wealth declines declines in the long long As the propensity term. From equation (3.10.1), (3.10.1), we obtain

f

da

>

^

da

k cia

da

r

<

cia0

da

The The wage rate is is increased increased and and the rate of interest interest is is increased. increased. The impact on the

output level is is given given by

da

da da

Chapter 4 Chapter

Time-dependent solutions of scalar systems chapter examines systems. This chapter examines periodic, periodic, aperiodic, aperiodic, chaotic chaotic solutions solutions of scalar systems. Section 4.1 defmes defines concepts concepts such such as as periodic periodic or or aperiodic aperiodic solutions solutions (orbits). (orbits). Section This section section also also introduces introduces some some techniques techniques to find periodic periodic solutions solutions and This to fmd provides conditions for stability of periodic periodic solutions. solutions. Section Section 4.2 is provides conditions for judging judging stability 4.2 is concerned with with period-doubling period-doubling biftircations. bifurcations. This This section section introduces introduces concepts concepts such as branch, branch, biftircation bifurcation values, values, period-doubling period-doubling bifurcation bifurcation route route to chaos, number and and Fiegenbaum's Fiegenbaum's number. number. Section Section 4.3 4.3 deals deals with with aperiodic aperiodic Myrberg's number orbits. This section section introduces introduces the Li-Yorke Li-Yorke theorem theorem and and the the Sharkovsky Sharkovsky orbits. important for for proving proving existence existence of of chaos chaos in in scalar scalar systems. systems. theorem, which are important Section 4.4 studies studies some some typical typical types types of ofbifturcations. bifurcations. They include supercritical supercritical fold, subcritical fold, supercritical supercritical pitchfork, pitchfork, subcritical subcritical pitchfork, pitchfork, transcritical transcritical fold, subcritical bifurcations. Section 4.5 introduces theory of Liapunov biftircations. Liapunov numbers. In this section, behavior of of aa model model of of chaotic chaotic labor labor market. market. In In section section 4.6, 4.6, we we we also examine behavior study chaos. We simulate a demand demand and and supply supply model model to to demonstrate demonstrate economic economic chaos.

4.1 Periodic orbits 4.1 Definition sequence {x(t)} {x(t)} isis said simply to Definition 4.1.1. A sequence said to to oscillate oscillate about about zero zero or or simply oscillate the terms terms x(t) x{t) are areneither neither eventually eventually all all positive positive nor nor eventually eventually all all oscillate if the negative. sequence is called called nonoscillatory. nonoscillatory. A sequence sequence {x(t)} {x(t)} is negative. Otherwise Otherwise the sequence called 0, there there exist exist t1, /,, t2 t2 >t0t0such suchthat that called strictly strictlyoscillatory oscillatoryififfor forevery everytQt0> 0,

135 135

136

4. TIME-DEPENDENT SOLUTIONS OF SCALAR SYSTEMS

x(t1).x(t2)<

o.

— x*oscillates. oscillates. {x(t)} is said to oscillate about x* if the sequence sequence x(t) x(t)— A sequence {x(t)} — x* The sequence {x(t)} {x(t)} is called strictly oscillatory sequence x(t) x{t)— oscillatory about about xx* if the sequence is strictly oscillatory. Theorem 4.1.1. 4.1.1. (linearized (linearized oscillation oscillation theorem) / :RR—* -> R be aa continuous continuous theorem) Let Let f: function function such that

uf(u)>O, «/(«)> 0, for for u

0, 0,

and ft /'(o) 1. If there (o) == 1. there exists exists /r > 00 such that either

f(u)u, for [0, r] or or f(u)> /(M) >u w e [-r,0],0],then thenevery everysolution solution of for r e [o, forforr er [—r, x(t + l) i) = x(t) — pf(x(t)), Jc(/ + x(f)-/7/(x(0),

oscillates if and only only if every every solution solution of of the the corresponding corresponding linearized linearized equation equation y(t i) == y(t) — py(t), y{t + l) y{t)-py{t),

oscillates, oscillates,that thatis,is,ififand andonly onlyif ifpp>1.1. Definition 4.1.2. called aa periodic if Definition 4.1.2.AA point point pp ee R R isis called periodicpoint point ofofperiod periodkk if k e (p) = Pffk {p)~ called a periodic Thepoint point pp is called periodic point point of of minimal minimalperiod periodkk ox or prime prime p. Th period period k point if

fk(p)=

fk(p)=P, and and

k isis the the smallest smallest positive number for periodic point, point, for which which this thisholds. holds.IfIf p isisaaperiodic then 0(p) O(p) isiscalled called the the periodic periodic orbit. orbit. Orbits Orbits that are are not not periodic periodic are sad to be then aperiodic. aperiodic.

ORBITS 4.1. PERIODIC ORBITS

137 137

choice of "minimal" "minimal" is motivated motivated by the the fact fact that that the the orbit orbit cannot cannot be be The choice loops. A fixed fixed point can can be regarded as a periodic point of decomposed into smaller ioops. period Since period 1. Since x0

(x0) whenever pp is whenever is the the period periodofof O(x 0(x0) wealso alsohave havefm7(x0) 0) we

x0 for for every every x0

m
Every point

x(t), tt=0,1,•••,p—1, = 0,l,-,p-l, x(t), O(x0) of ofperiod period ppis periodicof ofthe thesame same period. period. Thus, Thus, 0(x0) O(x0) of the period orbit orbit 0(x0) is periodic distinct periodic periodic points of period p. Sometimes, Sometimes, we we use contains exactly exactly p distinct {x(0),

..., x(p

—

i)}

p. to denote a periodic orbit of period period p. The aregiven givenby bythose thoseintersections intersections of the The periodic periodic orbits orbitsof ofperiod period22 of of /f are graph of ( / ( x ) )with withthe theline line graph of /f(f(x))

y=x, which are Plotting all f(f(x)), which are not not on on the the graph graphofoff{x). f(x). Plotting allthree threefunctions functionsf(x), f(x), f(f(x)), solvesthis thisproblem. problem. Suppose Suppose and xx solves fix) 3.2x(\-x). f(x) == 3.2x(1 — x). Figure 4.1.1 hastwo twofixed fixedpoints, points,which whichare arealso also fixed fixed for for its its second second 4.1.1 shows shows that that /f has graph of of the the second second iterate iterate crosses crosses the the line line iterate. The two points where only the graph yy = xx are periodic points of period period 2. function The function /(x) = = 3.82x(l 3.82x(l-x), f(x) — x), stable periodic periodic orbit of of period period 3. 3. Figure Figure4.1 4.1.2a describes the has a stable .2a describes the orbit orbit with with the initial condition condition x0 x0 == 0.3. 0.3. The orbit appears appears chaotic, chaotic, but this this impression impression is is initial The orbit incorrect. The orbit is asymptotically periodic of period 3, as shown in figure incorrect. The orbit is asymptotically periodic of period 3, as shown figure 4.1.2b. 4.1.2b.

138

SOLUTIONS OF OF SCALAR SCALAR SYSTEMS SYSTEMS 4. TIME-DEPENDENT SOLUTIONS x

1

f (x )

0.8 0.8

0.6 0.6

f ( f (x ))

0.4 0.4

0.2 0.2

0.2 0.2

0.4 0.4

0.6 0.6

0.8 0.8

11

Figure 3.2x(l -— x) x) Figure 4.1.1: 4.1.1: The Thesolution solutionofofperiod period2 2totof(x) f(x) = 3.2x(I

11

1U

00.8 .8

00.8 .8

00.6 .6

00.6 .6

00.4 .4

00.4 .4

00.2 .2

00.2 .2 0 .2 0 .4 0.2 0.4 a) the orbit

0 .6 0.6

0 .8 0.8

11

0 .2 0 .4 0 .6 00.8 .8 11 0.2 0.4 0.6 b) asymptotically periodic periodic of of period period 33

Figure 4.1.2: 3.82*(1 — - x) with x0 x0 == 0.3 4.1.2:The Theorbit orbitofof /(*) f(x) = 3.82x(l

calledananeventually eventuallyperiodic periodicpoint pointofofminimal minimal Definition 4.1.3. A point point pp*e eRRis is called eventuallyperiodic periodicpoint pointfor forthe themap map period kk for fordifference differenceequation equation(2.2.1) (2.2.1)or oran aneventually thereexists existsaapositive positive integer integer rr and andaaperiodic periodic point point p of ofminimal minimal period period kk f/ ififthere such that r1 *\ r{p')=p, frl{p *\ r—1'

ORBITS 4.1. PERIODIC ORBITS

139

Example Thelogistical logistical difference difference equation equation Example The / = 4x(l - x), f=4x(1—x), has two two period-two period-two solutions, solutions, 5 / 8 ± solutions of

/ 5,, which

are the solutions, solutions, except except the the

x = 4x(l - x), x=4x(1—x), of the following equation x = /f(f(x)) ( / ( * ) )= =16x(1 16x(l -— x)[l -— 4x(l -— x)]. 4]. r p, where is a Finding eventually eventually periodic periodicpoints pointsleads leadstotothe theequation equationf f (x) (9 = where r is = p, positive with positive integer integergreater greaterthan than1.1.For For/ f = 4x(l -— x) 9 with

p=—±-—, r=2, we obtain the algebraic equation

16x(1

—

x)[1

—

4x(1

—9] =

±

2!@1

Example Consider the difference equation

x a n eeven v e n positive number if -1-x(t), W' ^ x(t) *(*)isi s an positive number I i\ = J ~ x(t + i) = v "*" v | 2 \3x(t) x(t) isisan anodd odd positive positive number, number, itt == 0, 0, 1, 3x(t) ++ 1,1, if x(t) 1,

,

positive integer. integer. The Thedifference difference equation equation has has no noequilibrium equilibrium points. points. x0 isisaapositive where x0 The equation has a period-three {l, 4, 2} 2} and and many many eventually eventually period-three period-three solution solution fi, solutions. For example, the solution that starts at 8 has hasthe the corresponding corresponding orbit orbit {{8, 8 ,4, 4 ,2, 2 ,1, 1 ,4, 4 ,2, 2 ,1, 1,-}. .

.

140

SOLUTIONS OF SCALAR SCALAR SYSTEMS SYSTEMS 4. TIME-DEPENDENT SOLUTIONS

Calculate O(l), etc. ItIt isisknow knowthat thatthere thereare areinfinitely infinitely many many eventually eventually periodic periodic Calculate 0(7), etc. 1 The Collatz conjecture, one of the most open difficult problems in the the solutions. solutions.' The Collatz conjecture, one of the most open difficult problems in theory of difference equations, establishes that every solution of the above equation solution {i, {l, 4, 4, 2}. 2}. is eventually periodic to the period-three solution Since of minimal minimal period period k isis an an equilibrium equilibrium point point of the Since aa periodic periodic point point pp of map ft", / * , the notion the stability stability of follows from from the the definition definition of an an map notion of of the of p follows equilibrium point, can to equilibrium point,and andthe the linearized linearizedstability stabilityresult result canbebeapplied appliedtoto/ *f to p. determine the stability stability type type of of p. Definition 4.1.4. 4.1.4. A Aperiodic periodicpoint pointpp of x{t + + 1)=l)=f(x(t)) x(t f(x(t)) minimal period period k isis said said totobebestable, stable,asymptotically asymptotically stable, stable, unstable, unstable, or aa of minimal global attractor attractor ifif ppis,is,respectively, respectively,stable, stable,asymptotically asymptotically stable, stable, an an unstable unstable equilibrium *. equilibrium point, point, or oraaglobal globalattractor attractorofof /f". 2 , we weobtain obtainthat thataatwotwoApplying the Applying the chain chain rule rule and and theorem theorem2.3.1 2.3.1totomap map ff2, periodic solution stable if periodic solution pp isis stable

<1, \/ip)fif(p))\
and unstable unstable if

>1. The number

2= f'(p)f'(f(p)) is called a multiplier of the orbit. Likewise, the multiplier multiplier of a periodic orbit of any canbe bedefmed defined by period kk can

1

See Bernstein and and Lagarias Lagarias (1996) (1996) and and Lagarias Lagarias (1985). (1985). See

4.1. PERIODIC ORBITS

141 141

ft&k1(p))

2= Example Consider

f-x for-l<x1, forx>l, x(t+1)= —2x+1

[-2x-l forx<-l. —2x—1 forx<—1. Notice ( l ) = -—11 and / ( - l i) ) = period orbit orbitfi,{l,—- li}} is is unstable. In Notice that that /f(1) In and f(— = 1. 1. The period fact, for x >>1,1, we have fact, /(/{*)) = 4 ( * - 0+1. l)+l. f(f(x))=4(xThus, the is easy easy to toshow show that that the theobit obitO(x(O)) O(x(o)) with with x(O)> x(o) >11 is is going to + oo• °°. Thus, the It is to + is an anunstable unstable fixed fixed point second iterate Hence,the theperiodic periodic state 11 is point for the second iterate of /f..Hence, - l }is unstable. unstable. {l,—i} orbit {i,

Theorem 4.1.2. If If Theorem

O(p)=

...

is orbit of of the kk -periodic -periodic point where f/ is is a continuouslydifferentiable differentiable is the the orbit point p, p, where a continuously then we we have: have: (i) (i)ppisisasymptotically asymptotically stable stable ififthe the multiplier multiplier of the function function at at p, then the > 1.1. orbit is less lessthan thanunit, unit,that thatis,is,X2 Example (a(agenotype genotypeselection selectionmodel) model)Consider Consider aalocust locust with with two two alleles: alleles: allele allele A and allele aawith withfrequency frequency A with frequency pp and

q=1—p, < ? = 1 - P, in a diploid population! population.2 The change in in gene gene frequency frequency between one generation generation /t and (/ + l) and the next generation generation (t i) is given by

22

See 1.9 in inElaydi Elaydi (2000) (2000) See Section 1.9

142

SOLUTIONS OF OF SCALAR SCALAR SYSTEMS SYSTEMS 4. TIME-DEPENDENT SOLUTIONS

p(t + i)

=

p(t)g(p(t)) p(t)g(p(t)) + — p(t)

(4.1.1)

1

where g{p(t)) isis the thefrequency-dependent frequency-dependent fitness fitness that where g(pQ)) that the the genotype genotype AA AA has. has. The difference equation equation has has three three equilibrium equilibrium points: points: p\ = 0, 0, p*2 = 1,1, p\ determined determined byby g[p* g(p)= 1. 3)=l. Now

\_ g(p)+p(1—p)g'(p) [pg(p)+l—p] 2

We have /'(o) = g(o) and and Wehave f'(O)=g(O) 1

f'(l)= f\=g(0). The and p\p = = 1, unstable ifif g(o) which 1, are The two two equilibrium equilibriumpoints, points,p*p = 00 and are unstable g(0)>> 1,1, which is true when

g(p)= P >> 0. We assume with /3

g(p) = A1 - 2p) henceforth. becomes henceforth. Under Underthis thisassumption assumption/f becomes /3(1-2p)

+ i—p Hence

4.1. PERIODIC ORBITS

143 143

/3

1

for />* is an an asymptotically asymptotically stable stable gene gene frequency.3 frequency.3 If for 0 < c //3? <<44,, p P 2 ) = —1. direct calculation calculation shows shows that that the the Schwarzian Schwarzian /3 = 4, 4, then then f'/ ' ((il //2) — 1. A direct derivative Hence, Hence,

sf\\\
p(t) , 1—p(t)

(4.1.3)

which transforms = e^1 transforms equation equation(4.1.1) (4.1.1)under under g(p) g(p) =

x(t + l) x(/)exp P1 ~ Xy, i) == x(t)exP[fl'

2p - 2p) '

= F(x(t)).

into

(4.1.4)

-cycle of of equation equation (4.1.4). (4.1.4). Then Then Let{x1, {x,,x2 x2}} be a 22-cycle Let x, exp( P-—^ xx2 =x1expi/? 2 = Xj x1

V

+ x 1+x1) L \ J

(

i—x2

= x 2 exp /9 P

,

"1+;

Multiplying the the two two equations equations yields yields

3

/? ^ 4, the equilibrium equilibrium point l / 2 is globally point p* p == 1/2 00 c< /9 asymptotically stable on (0, i). l). See Kocic and Ladas Ladas (1993). (1993). asymptotically on the the interval interval (o, proved that that for It is proved

4. TIME-DEPENDENT SOLUTIONS SOLUTIONS OF OF SCALAR SCALAR SYSTEMS SYSTEMS

144

(f ,8o l - X , |

x1x2 =

( /> 1 /3

x1x2 xxx2 = x,x 2 exp p——l+xQ 1 + x J exp' ^ p-1 + x2

1+;

Hence

(

1—x1x2 2

exp1218 e*p|2/?A

.

yf

J 1=1. = l-

(1+x1Xl+x2))

We obtain We XX == 11. x1x2

the corresponding From equation equation (4.1.3), (4.1.3), the corresponding 22 -cycle -cycle of of equation equation (4.1.2) is From {p, i1 ——p}. p}.To Tocheck checkthe thestability stabilityofofthis thiscycle, cycle,we wefirst firstcalculate calculate 2x/3

,1-x

F'(x)=[l_ 2xfl1[fl1_x) F'(x) = 1exp I+x 1+ x (1 + xf j (l+x) [ We have

F'(x1)F'(x2) =

F'(xi)F'[iJ =

—

2flx1

(i+x1) 2]

<1.

By theorem 4.1.2, it follows that the cycle cycle is is asymptotically stable. stable. Example Consider the map Example Consider

f(x) =

— 0.85,

2]. Find Find the 2 cycles interval [—2, [-2, 2J. cycles and anddetermine determine their theirstability. stability. defined on the interval -periodic points are obtained by solving The 22 -periodic

f2(x) = i.e.

4.1. PERIODIC ORBITS 2 x4 -1.7x 1 2 7 5 = 00.. — 1.7x2 —xx- 0 —. 0.1275

145 (4.1.5) (4.1.5)

These two two fixed fixed This equation equation has has four four roots, roots, two two of of which which are are fixed fixed points pointsof of /f.. These points are the roots of the equation x2 -— x -— 0.85 = 0. 0. fromequation equation(4.1.5), (4.1.5),we wedivide dividethe the leftleftTo eliminate these these fixed fixed points points of of / from hand side of equation (4.1.5) by the left-hand side of the above equation to obtain hand side of equation (4.1.5) by the left-hand side of the above equation to obtain the second-degree equation x2 + xx + + 0.15 = 0. o. The equation has two solutions solutions

x12 = 12

2

—

are the the 22-periodic -periodic points. points.To Tocheck checkthe thestability stabilityofofthe thecycle cycle{x1, {x,,x2x2}, }, we which are calculate

\f(x1)\\f(x2)\

= 0.6
stable. The cycle is asymptotically stable. We now introduce introduce a method method to to find find periodic periodic points points of of aa given given function. function. This This method is based on the following lemma. lemma. Lemma 4.1. 4.1. (Carvalho's (Carvalho's lemma4) lemma4) IfIf x(t) isis kk-periodic, -periodic, then then either Lemma

x(t) =

+

(2j'a\ , . (2fa + d1 sin[21'2r]], -!— + d, sin -^—

c, coss E[ci

if k > 1isoddandm=(k—1)12, is odd and m = (k - ])/ 2, or or ifk>l 4

SeeCarvalho(1998). See Carvaiho (1998).

(4.1.6) (4.1.6)

146 146

4. TIME-DEPENDENT SOLUTIONS SOLUTIONS OF OFSCALAR SCALAR SYSTEMS SYSTEMS 4.

x(t) = c0 ++ ( = c0

+

2?n E[cjcos[21?J

+

n

(4.1.7)

if k = = 2m 2m and and tf >1.1. jfk dynamics Example Consider aa population dynamics xQ + i) = x(t)exp(r(1 — x(t))).

(4.1.8)

to grow at at low low densities and andto to decrease at at high densities. The The The population tends to nontrivial As nontrivial fixed fixedpoint pointof ofthe theequation equationisisxx = = 1. 1. As

[(1) = I

—

r,

x* =1 = 1 is is asymptotically asymptotically stable if 0 if < rO
x(t)=a x(t)= a ++(-l)'b. into equation equation (4.18) (4.18) yields yields Plugging this into a — (—I)'b

(= (a + (_1)tb)exp(r(1 -— aa —- (_1)tb)).

The shift it/ F—> h-> t it++ 1 gives I gives

a+

= (a

—

(—lYb)exp(r(l — ~ aa ++ \(—1)'b)).

We have 2 2 = (a2 2 2 a a2 -b — ={a -b—)b2)exp(r(1 — a)). exP{r(l-a)).

either aa == b, b, which = 1.1. A 2 Thus either which gives gives the the trivial trivial solution solution zero, zero, or or a = periodic solution has has the the form

4.1.PERIODIC PERIODIC ORBITS 4.1.

147

x(t)= 1 + (—1)b. Plugging this equation again into into equation equation (4.1.8) (4.1.8) yields yields —

(—1)'b = (i + (_1)tb)exp((_1)t+lrb).

Introduce Introduce = Then

r=141+YJ=g(y). y

Y

1—Y \y-y

The g(y) has 2. For For r c<2, The function function g{y) has its its minimum minimum at at 0, 0, where where g(0) g(o) == 2. 2, g(y) == rr has each r > hasno nosolution, solution,and and we wehave have no no periodic periodic points. points. However, However, each >2 determines ±+ yr.

Exercise Exercise 4.1 4.1 1 Find a periodic orbit of period 2 of 1 Find a periodic orbit of period 2 of x(t + 1) = —2x2(t) + 1. x{t + \)=-2x2{t) + \. 2 Let 2 Let x(t + 1) = _ax2(t) + 1. x(t + l) = -ax 2 (/) + l. For what values values of of the the parameter parameteraa does does aa function function /f have For what havenone none or or one one periodic periodic

orbit of period period 22?? Are orbit of Are there there any any values values of aa for for which which there there is is more more than than one one periodic orbit of of period period 22??

148

SOLUTIONS OF SCALAR SCALAR SYSTEMS SYSTEMS 4. TIME-DEPENDENT SOLUTIONS

3 Assume Assume that x0 x0 isis aa periodic periodic point of point 22 for for aadynamical dynamical system system governed governed by aa differentiable differentiable function function f: /:/?—> R. Prove that the derivatives derivatives of the second second R —> R. and at at x(l) x(l) == f(x0) f(x0) are the same. iterate of x0 and iterate of /f atatx0

4 Assume Assume that x0 x0 isis aa periodic periodic point of point 33 for for aadynamical dynamical system system governed by aa differentiable differentiable function function /f:' : RR—, —>R. R. Prove Prove that the derivatives derivatives of the third third iterate ofof/ f at at x0, x0, x(1)= x(l) = f(x0), f(x0), x(2) x(2) == /(x(l)) same. iterate f(x(l)) are the same. Show the the Schwarzian Schwarzian derivative for the the map map 5 Show derivative <Sy(l/2) sjQ / 2)
f(x)

/to-=

6 6

2x)

+I

—

Let

[ l—xl xe(0,oo). (o, f(x)= xexplfl k P >4,4, XE [ I+Xj

4

Let Let{X1, {x,, X2 x2}} be

4.2 4.2

a 22 -cycle -cycle of f./ .Show Showthat thatthis this2 2-cycle -cycleisisasymptotically asymptotically stable. stable.

Period-doubling bifurcations bifurcations

Let and f(x, Let I/ be bean aninterval, interval, aaEE I, /, and f(x, a) a)be bea aone-parameter one-parameterfamily family of scalar scalar maps. Assume that that f/ isisdifferentiable differentiablewith withcontinuous continuousderivatives derivatives with with respect respect to to X beaa x and a. Let Let J c / beanintervalwithmorethanonepointandx:J—*R be an interval with more than one point and x: J -* R be continuous function and x(a) x(a)has hasthe theproperty property continuous functionsuch suchthat thatfor forevery everya ae eJi and

f(X(a), a)=x(a), /(x(a), a) = x{a), i.e., x(a) is a stationary stateofoff(x, f(X, a) a) for for every every oa ec /J.. Then i.e., x(a) stationary state Then the the graph graph of of the the map x(a), x(a), i.e., i.e., the the curve curve (a, (a, X(a)), x(a)), aase i Jis called is called a branch of fixed points of the a branch of fixed points of the one-parameter family f(x, a). For simplicity, x(a) sometimes called called aa branch. branch, one-parameter family f(x, a). For simplicity, x(a) isissometimes hi a similar manner, we define p. defme branches of periodic periodic points points of period period p.

BIFURCATIONS 4.2. PERIOD-DOUBLING BIFURCATTONS

149 149

Example Let

xe [o,l], [0, l], a f(x, a) = ax(\ -x), XE f(x, a)=ax(1—x), a>> 0.0. From f(x, a) = - x, x, we have From f(x, a) have

a—I = 0, x2(a)= x2(a) = . x,(a) x1(a)=O, aa The branch of of fixed fixed point point x1(a) of a; while of x^a) = 00 is is independent independent of while that of

xX2

/

=

a—l a-\ a

As is dependent dependenton on a. a. As

f'(X, a)= a(l

—

2x),

conclude that: is an an asymptotically asymptotically stable and we conclude that: (i) (i) 0 is stablefixed fixedpoint pointfor for00>1. I, we have /f'(O)= ' ( 0 ) = 1, l , f"(O) /"(0) = = —2. -2. mayconclude conclude that 00 isisunstable unstable ififwe weconsider consider negative negative as aswell well as aspositive positive We may initial points in the neighborhood of 0. Since negative initial points are not 0. Since negative initial are not in in the the initial points the neighborhood domain 00 is is semiasymptotically domain of f, / ,and and semiasymptoticallystable stablefrom fromthe theright, right,we weconclude conclude that that 0 isisasymptotically asymptotically stable stableininthe thedomain domain[o, [o,1]. l]. For X2 x2 (a) ee (o, (0,1], l], we require Applying require a a>>1.1.Applying For

f{x,a)=a{l-2x) f'(x, a) = a(l — 2x) to to a—l /i \ xAa) = a -I , x2 a

4. TIME-DEPENDENT SOLUTIONS SOLUTIONS OF OF SCALAR SCALAR SYSTEMS SYSTEMS

150

we conclude that: (i) x2 x2 isisan anasymptotically asymptotically stable stable fixed point for 11 < ao <3;3;and and (i) x2 point for for a >> 3. 3. an unstable unstable fixed point x2 isis an point of of period period 2 for We now show that the map does not have any period period point for ae (—i, 3). A point x0 is periodic of period 2 if x(2)= x(O) and x(1) x(O). period if x(2) = x(o) and x(l) * x(o). a e (-1, point x0 is periodic We have 2 x(1) = ax(O) (o), x(l)=ax(o)-ax —

x(O)= ax(1)— 2 (l). x(o)=ax(l)-ax

(4.2.1)

Subtracting the second second equation equation of of equation equation (4.2.1) (4.2.1) from from the the first first yields yields

x(i) -— x(o) = a(x(l) — x(o)) + a(xQ) x(o)). x(l) - — - fl(x(l) - x(0)) a{x{\) —- x(O)Xx(1) x(0)Xx(l) ++ x(o)). Since Since x(1) x(l) * x(o), x(o), from from the above equation we obtain

=

xQ) + x(o)

1

+ a

(4.2.2) (42.2)

Adding the two equations equations of equation (4.2.1) (4.2.1) gives x(O))22 + x(l) ++ x(o) x(0) == a(x(1) a(x(l) + x(o)) x(0)) -— a(x(1) a(x(l) ++ x(o)) + 2ax(1)x(O). 2ax(l)x(o). x(1)

Substituting equation (4.2.2) (4.2.2) into the above above equation equation yields xQ)x(o) x(l)x(0) = =

1+a a2

(4.2.3) (4.2.3)

.

We can determine x(1) x(l) and and x(O) x(o) by by solving solving the the quadratic quadratic equation equation

1+a

z 22—- — z a a

Hence

+

1+a^ =0. = 0. a2 a

(42.4) (4.2.4)

PERIOD-DOUBLiNG BIFURCATIONS 4.2. PERIOD-DOUBLING

=

a ++1l + +

151 151

2a — 3 yja—2-2a-3 *2a

aa ++11—- 7« 2 —- 2a 2a—3 -3 x4 (a) = x(0) = *— . 2a n

fn\

2a

x(l) and x(O) x(o) are We see see that that no no periodic periodic orbit orbit of Both xQ) are fixed fixed points pointsof of /f.. We period for a <3,3,and period 22 exists exists for andexactly exactly exists exists for for aa>>3.3.That Thatis, is,we wehave havetwo two branches of when aa>> 3. 3. At At aa == 3, 3, we obtain branches of periodic periodicpoints pointsofof f(x, f(x, a) when x2(a) = x4(a). x2{a) == xx3(a) x4(a). 3{a) =

It should should be noted noted that that we we may may directly directly get get solution solution (4.2.4) (4.2.4) by dividing dividing the the 2 equation equation x == f2(x) f (x) with with xx —- f(x). f(x). asymptotically stable stable ifif The 2 -cycle is asymptotically <1,

or or 2 — 2x4)< 1. -—1< I < aa2(1 ( l -—2 x2x3)(l 3)(l-2x4)
Substituting the the solutions the above Substituting solutions into into the above inequalities, inequalities, we we have have

3 << aa <

.

At At aa=I+iJ& = 1 + V6,wehave we have f'(x3)f'(x4)=

—1.

conclude that -cycle becomes becomes We conclude that the the 2 -cycle -cycle isis attracting. attracting.IfIf aa>>11+ V6, the 22 -cycle unstable. As the of the the parameter parameter a changes changes its its values, values, the the qualitative qualitative behavior behavior of solutions also also changes. changes.Let Letus usdenote denotethe thevalue valueof ofthe theparameter parameteraa at the kk -th solutions point where the changes occurs as ak. We have the following table.

152 152

SOLUTIONS OF OF SCALAR SCALAR SYSTEMS SYSTEMS 4. TIME-DEPENDENT SOLUTIONS as aachanges changes Table Table 4.2.1: 4.2.1: The Thebehavior behaviorofof/f = ax(l ar(i — — 9x)as

interval parameter interval 0 << a a<\ cI \
3

1< a < i + \
type of behavior zero equilibrium point is is stable asymptotically stable positive equilibrium point stable is asymptotically stable period- 22 solution solution is stable asymptotically stable

critical critical value value of of a aa0o=l =1

a, == 33 a1 a2 == 11 + A/6 a2

Definition value a0 a0 e I/ isisa abjfurcation bifurcationpoint pointofofthe theone-parameter one-parameter Definition 4.2.1. 4.2.1. A value family thereare areintervals intervals family f/ ififthere JJ,cI, t,
a0 e Jj, i=1,2ori=I,2,3, i = 1, 2 or i = 1, 2, 3, a0eJ,,

and continuous functions functions x.: xt: Jt —> —» R, graphs are are branches branches of fixed fixed and continuous R, whose whose graphs points such that points or or periodic periodicpoints pointsofof/ f such

x.(a0)=x1(a0), a0, i * j . *,-(0o) = Xj(ao\ xXa) * Xj{a), a * a0, The value a0 a0 isis called called the the bjfurcation bifurcation value value or or critical critical value value of of the the parameter. parameter. We require require JJ,; to to have have nonzero nonzero length, length, and its We and allow allow aa0 to be be one one of of its 0 to endpoints. Hence, may have have the the form form endpoints. Hence, J, may {b, c), (b,

[a (b, a0]. a0]. [a0, b), or (b, 0, b),

When i/ = 1,1, 2, then J1 J, = JJ22 and all points x,(a), /i = 1,1, 2, have have the the same same period. period. When points x1(a), When i/ == 1,1, 2, 3, then then = = [a0, [a0, b) or or (b, (b,a0] a0] Jl = {b, (b, c), J .j2 =3.j3 2=J 2, 3, may have period equal or double double the and the points of xt(a), i =- 2, the period period of of ^(a). x(aj) on on the the plane, plane, representing representing the parameter parameter a along along the the x1 (a). We We depict depict (a, x(a)) axis. The The pictorial pictorial display display obtained obtained by by plotting plotting all allbranches branchesof ofstationary stationary horizontal axis.

4.2. PERIOD-DOUBLING BIFURCATIONS BIFURCATIONS

153

states and periodic points isis called the bifurcation diagramofof/ f on /. states and periodic points called the bifurcation diagram At a I. At there is frequently frequently an an exchange exchange of of stability stability between between the the bifurcation 0 there bifurcation point point aa0 different branches. In particular, the bifurcation taking place at a value of a is different branches. In particular, the bifurcation taking place at a value of a is branches is is double double called period-doubling period-doubling if the period of the points of one of the branches the period of the points of of another. another. With regard to the logistical map, if we continue continue to to increase increase the the value value of of a, a, we will find the next critical value a3 which corresponds to the appearance of we will find the next critical value a, which corresponds to the appearance of aa prime period- 4 solution, solution, as as shown shown in in figure figure 4.2.1. 4.2.1.

0.8 0.7 0.6 0.5 0.5

a 2.2 2.2

2.4 2.4

2.6 2.6

2.8 2.8

33

3.2 3.2

3.4 3.4

Figure 4.2.1: 4.2.1: The Thebifurcation bifurcationdiagram diagramforfor2 2< aa < 3.5 3.5

The next bifUrcation value a4 bifurcation value aA corresponds to a prime period- 8 solution. solution. In In addition, itit can can be proved that for for

ae{a(a2, ), 2,a2a3), the the prime prime periodperiod- 44 solution solution is stable stable while while the the periodperiod- 2 solution solution becomes becomes

unstable. Similarly, Similarly, for for

a€ ae

(a3, a4),

the prime prime periodperiod- 8 solution solution is is stable while the prime prime periodperiod- 4 solution solution stable while

becomes unstable. As this process continues, there is a sequence of bifurcation becomes unstable. continues, there bifurcation values of parameter {at}^=0 with the following property: for for

154

SOLUTIONS OF OF SCALAR SCALAR SYSTEMS SYSTEMS 4. TIME-DEPENDENT SOLUTIONS

ae (ak, ak+l), of all all periods periods the prime period- 2k is stable, while the periodic solutions of 2,---,2k'1

become unstable. This phenomenon is known as the period-doubling bjfurcation bifurcation is called cascade of of route to to the the chaos. chaos.The Theentire entiresequence sequence {ak {a,t}7=0 is route called aa cascade bifurcations. btfurcations. The sequence biffircationsends ends atat the the value value which which is sequence of period-doubling period-doubling bifurcations approximately approximately a == 3.56994••• ,, where the map has the periodic solutions solutions of all periods as well as some some aperiodic aperiodic solutions. solutions. The last last situation situation is is ofien often described described as chaotic behavior behavior or or chaos. chaos. The Thelast lastperiod period that thatcan canarise ariseininthis thisbifUrcation bifurcation process process is is period period 3. 3. There There are are three three important important features features of of this this route route to to chaos. chaos. The periods fmishes with 3; the order is known as Sharkovsky's order, which The periods finishes with 3; the order is known as Sharkovsky's order, which will is only which is is stable in will be be described described below. below. There There is only one one periodic periodic solution solution which stable in each of the interval (ak, ak+l). The third is that the sequence {ak has the each of the interval (ak, ak+l). third is that the sequence {ak\=0 property remarkable property bk bk — bk ~ bkl 'x == 85 == 4.66920. lim 4.66920. bk+l — - °° bk+1 - bk

i >

The constant constant S 8 isiscalled calledMyrberg Myrberg's number or or Feigenbaum Feigenbaum's's number. number. ItIt 's number discovered by by Myrberg Myrberg and and rediscovered rediscovered by by Feigenbaum.5 Feigenbaum.5 Figure 4.2.2 was first discovered is the bifurcation bifurcation diagram diagram of

f(x, a) == ar(i ax{\ -— 9. x). f(x, a)

Exercise 4.2 1 Determine bifurcationofofthe the following followingmaps mapsf(x, f(x, /J.) p) and Determine the the period-doubling period-doubling bifurcation and investigate the stability stability of of the the periodic periodicorbits orbitsof ofperiod period 22::

(i)p—x2; (0 M - *2; -ju + + x2; (ii) —p

5

(1958, 1959, 1959,1963) Myrberg (1958, 1963) and Feigenbaum (1978).

4.2. PERIOD-DOUBLING BIFURCATIONS BIFURCATIONS

155 155

1l 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2

2.5 2.5

33

3.5 3.5

4 4 a

<xc(l -- x) x) Figure 4.2.2: 4.2.2: The The bifurcation bifurcationdiagram diagramofof/f = ax(i

(iii) lit + x3; (Hi) /£c x3;

-

(iv) jux3 + + (i(l - //)*. p)x.

4.3

Aperiodic orbits

o(x0) ofofthe thedynamical dynamical system system The orbit 0(x0)

x(t-i-l)=f(x(t)) x(f + l) = /(*(*)) states. These states states may may not not be be all all distinct. distinct. For For instance, instance, they they has infinitely many states. are all equal when when x0 is a fixed point and there are only p distinct points when there are only distinct points when x0 O(x0) is is periodic periodic of p. the orbit orbit 0(x0) ofperiod period p.

Definition 4.3.1. said to to be be a limit limitpoint pointof of O(x 0(x0) 4.3.1. A point x is said 0) if there exists a subsequence

=o,i,...} MO:* 0,1,-} {x(tk): k = of 0(x0) ofO(x such that 0) such

156

4. TIME-DEPENDENT SOLUTIONS SOLUTIONS OF SCALAR SCALAR SYSTEMS SYSTEMS

jx(tk) —Dx(c)

x I -> - » °°. —*00 as as kk—>

O(x0) isisthe theset setof ofall alllimit limitpoints pointsof ofthe theorbit. orbit. The ///K/Y i(x 0 ) of the orbit 0(x0) limit se? set L(x0) Definition 4.3.2. 4.3.2. An orbit 0(x0) O(x0) is is said said to to be be asymptotically asymptotically stationary stationary if its Definition An orbit limit set is aa stationary stationary state, state, and and asymptotically asymptotically periodic limit set is aa limit periodic ifif its its limit such that periodic orbit. orbit. An An orbit orbit O(x 0(x0) 0) such x{t + M=x(t) p) = x{t) x(t+ for some andsome some pp 1> 1 is said said to for some tt >11and to be eventually eventually stationary stationary ifif p == 1 and 1

eventually periodic 1. eventually periodicifif p >>1. Every eventually eventually stationary stationary (eventually (eventually periodic periodic orbit) orbit) isisasymptotically asymptotically Every stationary (asymptotically instance, for stationary (asymptotically periodic). periodic). The The converse converse isis not not true. true. For instance, difference equation the difference

x{t + \) = x{t){\-x{t)\ x(t+1)=x(t)(1—x(t)), the orbit 1

1

3

2

4

16

x(0) jc(l) -—, w = l2 >, xIj)=—, w = 4I ', x{2) w = lfi »,

-

is asymptotically stationary eventually is asymptotically stationary(with (withthe the limit limit point point 0 ), but not eventually stationary. Example Example Let

x(t ++ 1) \) == 4x(t)(1 4x{t){l-x{t)). x(t — x(t)). The orbit o(l/2) orbit 0(1/2)

x(0) = i ,

x(l)=l, = x{3)=- = 0, x(1)=1, x(2) x(2)=x(3)=".=0,

is eventually = 2, 1). eventually stationary stationary (t(t = 2, p p == 1).

4.3. APERIODIC ORBITS ORBITS

157

beproved proved that whenever whenever l(x It can be L(x0) has only only finitely finitelymany manypoints pointsp, p, they 0) has actually of the the system. actually constitute constitute aa periodic periodicorbit orbitwith withperiod period p p of aperiodicififitsitsset setL(x0) L(x0) isisnot notfinite. finite. Definition 4.3.3. 4.3.3. The The orbit orbit 0(x0) O(x0) isisaperiodic Example Figure4.3.1 4.3.1provides providesnumerical numerical evidence evidence that that the the orbit of the dynamical Example Figure system f{x)=4x(l-x), f(x) = 4x(1 — starting at xx0 =0.3 analyze aperiodic aperiodic behavior behavior later later on. on. 0.3 is aperiodic. We will analyze 0 =

x(t + 1) 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0.2 0.2

0.4 0.4

0.6 0.6

0.8 0.8

1

(x) x<*)

Figure 4x(l -— x) x0 = = 0.3 0.3 Figure 4.3.1: 4.3.1:The Theorbit orbitof of f(x) f(x) == 4x(1 x) with x0

isdifficult difficult to toestablish establish theoretically the aperiodic character character of It is of an orbit. The following result ensures the existence of aperiodic orbits for dynamical systems in following of aperiodic orbits in the real line. Theorem 4.3.1. 4.3.1. (Li-Yorke6) (Li-Yorke6) Let bean aninterval interval and and f: / : I/ —* - > /I be becontinuous. continuous. Let/I be / has a periodic orbit of period 3. Then / has a periodic Assume that Assume that f has a periodic orbit period 3. Then f has a periodic orbit orbitof of 61

The proof proof isis referred referred to to Li Li and and Yorke Yorke (1975), (1975),or orBlock Blockand andCoppel Coppel (1992). (1992). The

4. TIME-DEPENDENT SOLUTIONS SOLUTIONS OF SCALAR SCALAR SYSTEMS SYSTEMS

158 158

every infinite set contained in such that that every every orbit orbit every period period and and there there is is an an infmite set SS contained in S such starting from from aa point point of of S isisaperiodic. aperiodic.

Example Consider [o, 1] Example Consider /f:: [o, [0, 1] l] —* -» [0, l] defmed defined by

3x

for 0 < x < - , 3

17 17

8x

9

3

9 1/9

11

2

3

3

for—cx—, for - < x < —,

3

3 3 for—<x1. for — < x < 1. 33

The map is continuous and

41)=i, x == 1/9 is periodic 4.3.1, there there is an periodic of period period 3. According According to theorem theorem 4.3.1, infinite set S that point of of S isisaperiodic. infinite set that the the orbit orbit of of every every point aperiodic. Determining Determining SS isis not not easy. easy. Hence, Hence,

ExampleConsider Considerthe thedifference differenceequation equationgenerated generatedby bythe the tent tent fbnction function Example 2x

T(4=

for 0 < x < - , 22

2(l f o r - < x <1.l . 2(1-x) —4 for'< 22 This may also be be written in in the the compact compact form form

T(x)=1—2x—-1-. = 1- 2 J C 2

We first that the periodic points of of period period 2 are first observe observe that periodic points are the fixed fixed points points of T2 T2 which is given by

for-<*<-,

4 he

for — < x < —, A

H—

A

A

IA

IA

t'JI—' al—'

A

for - < x < - . 4 2 -t

I

4(1 K -x) J

4'

A

I

k

IN) I

-I,

2

2J j'—

IN)

a

I

IA

2(l -2x)

0

for 0 < x < —, 4

a

4x

159 Ui

4.3. APERIODIC ORBITS

+

k

x{t + 3)

Figure 4.3.2: Fixed points of T3

IN)

-S

C

H

bo

a Ca

—1

WCD

p

-t

There are four fixed points, 0, 0.4, 2 / 3 ^ and 0.8, two of which, 0 and 2/3 are fixed points of T. Hence, {0.4, 0.8} is the only cycle of T. As shown in figure 4.3.2, {2/7, 4/7, 6/7}, is a 3-cycle.

160

SOLUTIONS OF SCALAR SYSTEMS SYSTEMS 4. TIME-DEPENDENT SOLUTIONS

special case case of aa theorem theorem published published in in 1964 1964 by the the The Li-Yorke theorem is a special Ukranian mathematician A.N. Sharkovsky. Sharkovsky introduced a new ordering of that if of the the positive positive integers integers >> in in which which number number 33 appears appears first. first. He He proved proved that if m and kk > m and f/ has hasa k a -periodic k -periodicpoint, point,then thenititmust musthave haveananinm-periodic -periodic point. point. Here, the notation in means Here, the notation kU> >m means that that kk appears appears before before in m in Sharkovsky's Sharkovsky's ordering. Sharkovsky's ordering ordering is is defmed defined as follows follows 3>5>7> >2"

> 2x3 > 2x5 > 2x7 > 2

> 2"x3 > 2"x5 > 2"x7

>2>1

We first list timesthe theold oldintegers, integers, list all allthe the odd oddintegers integersexcept except11and and 2, 2, then 22 times 222

2 times the old integers, etc. This in descending descending order This is is followed followed by bypowers powers of of 2 in endingin 1. ending in 1. 7 Theorem 4.3.2. (Sharkovsky's (Sharkovsky's theorem ) Let interval in (finite or Theorem 4.3.2. theorem7) Let/ I be be an an interval in R R (finite infinite). Let

f:I-*I be a continuous hasaaperiodic periodic point of period then itit must must possess continuousmap. map.Let Let /f has period k, then m. a periodic periodic point pointof ofperiod periodmin for for all all m in with k >>m. simply method method for for checking checking complicated complicated behavior behavior of of the the This theorem gives aa simply dynamical system. with the f(x) dynamical system. If If we we plot plot the the bisector y == x with the graph graph of of the themap map f(x) 3 3 and if the graphs graphs of the bisector graph of the third and graph third iterate iterate /f3 (x), (4, and bisector and and /f3 (x) (4 the equilibrium equilibrium points, points, then then there there isis at at least least one one intersect in some points that are not the periodic point of period 33 and periodic point and consequently consequently there are are periodic periodic points points of of all all other other periods. the existence periodic points points of of periods. Similarly, Similarly, one one can can check check virtually virtually the existence of of other other periodic 3 reasonably small periods. periods. Figure Figure 4.3.3 4.3.3 illustrates illustratesthis thismethod. method. The The function fimction/f3 is reasonably small is shown as a thick line; / in a thin line; and the identity shown identity function function in a dashed line. intersections correspond correspond to tofixed fixedpoints pointsof of /f.. The Two intersections Theremaining remaining six sixintersections intersections indicate that there are two prime period- 3 solutions. It should be remarked remarked that Sharkovsky's Sharkovsky's theorem theorem does not extend to two or or higher dimensions. dimensions. ItIt is known higher known that one can can construct construct aa continuous continuous map with aa periodic point of 5 but but not not with with periodic periodic point 3, 3, ororwith withaaperiodic periodicpoint point of of period period 11 but not with period 5, etc. with period etc. 7

The proof is referred to Block and Coppel (1992).

4.3. APERIODIC ORBITS

161

1

0.8

0.6

0.4

0.2 0.2

0.2 0.2

0.4 0.4

0.6 0.6

0.8 0.8

11

Figure 4.3.3: 4.3.3:Plot Plotofoff{x) f3(x) for /f = Figure = 4.91x(1 4.91x(l —4 - x)

Consider a dynamical system: /f:: 1/ — —> and /1 is > /I,, where where /f isiscontinuous continuous and is a dynamical system: bounded interval. We shall say that / is chaotic in / in the Li-Yorke sense if bounded interval. We shall say that f chaotic in I in the Li-Yorke sense if f/ has aa periodic periodic point period 3. 3. Notice Notice that as aa consequence consequence of of has point inin /I ofof period that as Sharkovsky's theorem periodic orbit for every every integer p. Sharkovsky's theorem/ f has a periodic orbit of of period periodpp for integer p. Moreover, Li and Yorke proved that there is an infinite (actually, uncountable) set Moreover, Li and Yorke proved that there is an infinite (actually, uncountable) set SS cczIIsuch suchthat thatfor for every every x e SS is aperiodic and unstable.

Example Let Let / = 2\x\ - 1 f=2k1—1 and [—1, 1].Then Then/f has [-1, l]. hasaaperiodic periodic point point of of period 3, 3,

1751 III 1

9'9'9 9'9'9

-

Hence /f isischaotic chaoticininthe theLi-Yorke Li-Yorke sense. sense.

4. 4. TIME-DEPENDENT SOLUTIONS SOLUTIONS OF OFSCALAR SCALAR SYSTEMS SYSTEMS

162

Theorem positive integer integer r, r, Theorem 4.3.2. 4.3.2.(a(aconverse converse of ofSharkovsky's Sharkovsky's theorem8) theorem8) For any positive there continuous map . : /I— > /I on the the closed closed interval interval /I such such that that f,. fr there exists exists aa continuous map /fr: has a point point of ofminimal minimal period period r but butno nopoints points of ofminimal minimal periods periods s, s, for all has positive integers that precede rrininthe theSharkovsky's Sharkovsky's ordering, ordering, i.e., ss > r. positive integers s that r.

The following theorem solve the problem of how many many The following theorem by by Singer Singer tries fries to to solve the problem attracting periodic points points aa differentiable differentiable map map can can possess. possess.

Theorem Let/f:: /I— —,» /I be Theorem 4.3.3. 4.3.3. (Singer's (Singer's theorem9) theorem9) Let be aa map defined on the closed interval /I such derivativeof of f/ suchthat that the the Schwarizian Schwarizian derivative interval

Sf(x)c 0, Sf{x)<0, for all x e I. / . IfIff/has has n criticalpoints pointsininI,/then , thenforfor everypositive positive integer, integer, k, k, for n critical every the map map /f has period- k solutions. the hasat atmost most nn + + 22 attracting attracting periodsolutions.

Exercise 4.3 4.3 1 Show Show that that for forthe the map map

1

f(x) = 2x — [2x], f{x)=2x-[2x], where less than than or equal equal to x, the where [x] [x] is the the greatest greatest integer integer less theorbit orbit 0(3 0(3/8) is / 8) is eventually stationary and the the orbit orbit o(i / 24) isiseventually eventually stationary O(l/24) eventually periodic. periodic.

2 Let /,

\{2x12x ^ '~[2-2x X

forO<x<0.5, for O.5cxl. for0.5<jc
Find a periodic periodic orbit orbit of ofperiod period 33 for f. to / .Can Canthe theresult result of ofLi-Yorke Li-Yorke be beapplied applied to Find

3 Let 88

9

See Elaydi (1996). SeeElaydi(1996). Singer (1978). (1978).

ORBITS 4.3. APERIODIC ORBITS

163 163

f

Show that / has has aperiodic aperiodicorbits orbitsinin[—1, [-1, l].

4.4

Some Some types types of of bifurcations

This section provides some typical examples of biftircations. bifurcations. Example Consider Consider f(x,ju) = jux-x\ arethe thesolutions solutions of Fixed points pointsof of /f are x = / a - x3. Hence, x1 x,(//) is always always aa fixed fixed point. point. No Noother other fixed fixed point point exists exists for (p) = = 00 is Whenp crosses JJ. crosses 1, two more stationary statesappear, appear,namely namely pfi < 1. 1. When 1, two more stationary states

From

fx=M~ 3x2, we see that that 0 0 isisaarepeller repeller for forIpi \[i\ >>11 and aa sink sink for for |//| << 1. At fj. have 1. At p == 1,1, we have / x (0, = 1. l. Substituting x2(ju) x3(ju) respectively into weconclude conclude (p) respectively 1) = into fx, we (o, l) Substituting x2 (u) and and x3 for pfiee (i, (l, 2), 2), the thepoints points of ofthe the two two branches branches are sinks for ji e (i, (l, 2) 2) and and that that for are sinks pE source for 2. At At p ju == 1, source for ju p >> 2. 1, we have an exchange exchange of stability stability between between *,(//) x1(p) and x2(ju), xx3(p). periodic orbit orbit of of period period 22 arises. Since x2(p), At (i p ==-\, —1, a periodic Since the the 3{/i). At derivative of is (2a (2a ++ 3)2, 3)2, which is larger than than 11 for for derivative ofthe the second seconditerate iterateofof/ f is is larger is always aa source. There is is no exchange of ofstability at —1, the periodic orbit is at pfi >> -1, - ! . Figure 4.4.1 is the bifurcation bifurcation diagram for the map. p// == —1.

164

4. TIME-DEPENDENT SOLUTIONS SOLUTIONS OF OF SCALAR SCALAR SYSTEMS SYSTEMS

x

µ ~M

Figure 4.4.1: Part Part of of the the bifurcation bifurcationdiagram diagramofof /f == fjx —— x2

Example Let

f(x,p)=x2 f(x, fi) = x2

-—p. ju-

Fixed pointsof of /f are Fixed points aregiven given by by

x12(p) =

1

±

/i

p.

The fixed fixed points pointsexist existfor for//p >—0.25. -0.25. At At ju0 == —0.25, -0.25, x1(—0.25)= x, (- 0.25) =x2(—0.25). x2(- 0.25).

Hence, //0 isis aabifurcation bifurcation point. point.The Thetype typeofofbifurcation bifurcationtaking takingplace placeat at /ia is Hence, p0 4U0 fold, as as illustrated illustrated in in figure figure 4.4.2. 4.4.2. Along Along the branch (x1 (x, (//), /i), called supercritical supercritical fold, (p), p), l.That is, x1(p) xx(/i) are sources all /u - 0.25. Along Along (x2{/i), //) fx > >1.Thatis, sourcesfor forall p >—0.25. 2 = 11 -— (1 (i ++ 4a) 4a)2, fx = ,

4.4. SOME TYPES OF BIFURCATIONS BIFURCATIONS

165

xx22(ju) are sinks for (ti) are

/ie (

-K'3'\

1

and sources and sources for for //p

> 3/4. >3/4.

x

µ

2 Figure — pju Figure 4.4.2: 4.4.2: Supercritical Supercriticalfold foldbifurcation bifurcationfor for/ f = x —

Example Let /f(x,p)=x2 ( * , ju) = x2 +p. + MFixed points aregiven given by pointsof of /f are

1

/i

The fixed fixed points 0.25. At Atp0 fi0 = 0.25, pointsexist existfor for//p<0.25.

166 166

TIME-DEPENDENT SOLUTIONS OF SCALAR SYSTEMS SYSTEMS 4. TIME-DEPENDENT JC, (0.25) = x2(o.25). x2 (0.25). x1(o.25)

is aa bifurcation bifurcation point. point. The type of of bifurcation bifurcation taking ju0 isis Hence, ju0 is Hence, p0 taking place place at p0 illustrated in called subcritical fold, called fold, as illustrated in figure figure 4.4.3. 4.4.3. The The fixed fixed points points of of the the upper upper branch are always sources. sources. The fixed points of the lower branch are sinks for for

(

3

1'\

- 3 / 4. and sources sourcesfor forjup < <—3/4.

x

µ

Figure 4.4.3: Figure 4.4.3: Subcritical Subcriticalfold foldbifurcation bifurcationfor for/ f = x2 + p

Let Example Let f{x, ju) = jux

-x\

*[(//) fixed point point independent independent of No other other x1 (p) = 0 0 is a fixed of ju. p. No

fixed exists for for fixed point point exists When ju crosses 1, two more fixedpoints pointsappear appear for pju>> 1,1, pH < 1. 1. When p crosses 1, two more fixed

4.4. SOME TYPES OF BIFURCATIONS

167

—1.

Hence, ju p == 11 is a biftircation point. The The type type of of bifurcation biftircationtaking takingplace placeatat fip = Hence, bifurcation point. = 11 called supercritical is called supercritical pitchfork. pitchfork. The Thesubcritical subcriticalpitchfork pitchforkcase caseisiswhen when ju p <1.1.

x

µ

Figure 4.4.4: 4.4.4: Supercritical Supercriticalpitchfork pitchforkbifurcation bifurcationfor for/ f = px Figure jux —-

Example Consider the logistic map again

f(x, p) f(x, ju)==pr(i /*(l —- 4.4 We know that the two branches branches of of fixed fixed points points *,(//) = 0, 1

x 2 (//) = l - - , 'II,

at //p == 1. 1. Hence, p == 11 is aa bifurcation at Hence, // bifurcation point. point. This type of of bifurcation bifurcation is is called transcriticctl transcritical (see called (seefigure figure4.4.5). 4.4.5).

meet

168

4. TIME-DEPENDENT SOLUTIONS SOLUTIONS OF OF SCALAR SCALAR SYSTEMS SYSTEMS

x

µ

Figure 4.4.5: Transcritical Transcritical bifurcation biftircationfor for /f == jux(l -—x) 4

summary, we see see that that transcritical transcritical bifurcation bifurcation takes In summary, takes place place at at p// == //0 when two branches of fixed fixed points (or (or periodic points of the the same same period) exist exist when in open interval interval J, ju0 e J, and they meet at p ju = = p0. juo. Fold bifUrcation bifurcation is in an open observed branches of fixed fixed points (or periodic same observed when when two branches periodic points points of the same in an interval |// ,//J (supercritical case) or (//,,//„] period) exist only period) exist only in an interval [a0, 0 p1) (supercritical case) or (p1, po I (subcritical bifurcation requires three (subcritical case) case) and andthey theyequal equalatatju p= = ju p0.o. Pitchfork bifUrcation branches of fixed fixed points (or (orperiodic periodic points points of ofthe thesame sameperiod). period). Two Twoof ofthem them branches in an interval [*/<,,//,) (supercritical case) or (^,// ] (subcritical exist only exist only in an interval [a0, p1) (supercritical case) or (a1, p0] 0 (subcritical case). The Thethree threebranches branches meet meet case). The third third exists existsininan anopen openinterval intervalJ,J, ju p00 G e IJ.The /i = = // 0 . It can can be shown that at ,a

p0)=l necessary for for p0 //0 totobe bea afold, fold,transcritical, transcritical,ororpitchfork pitchforkbifUrcation. bifurcation. The The is necessary is not not sufficient. sufficient. This This is is verified verified by by condition is

f(x,p)=(l+p2)x+x3. At p jU== 0, is aa unique unique fixed fixedpoint pointx1 x](o) (o) = = 0. 0. We We have At 0, there is

4.4. SOME SOME TYPES TYPES OF OF BIFURCATIONS BIFURCATIONS

169

f'(O,O)= 1.

But ju p = 00 is not aa biflircation But bifurcation point. point. The above examples are are typical typical cases cases of of bifttrcation bifurcation types. We We provide provide the the necessary conditions conditions for the four necessary four types types of of biflircations. bifurcations. We require the fimction function fix, p), n\ its first and second partial derivatives, and f^ to be continuous. f(x,

Table 4.4.1: The conditions for for the four types of bifürcations'° bifurcations1'

L

/»

fold

l1

transcritical

l1

00

* 00

pitchfork

l1

00

0

0

0

otherat(x0,p0) other at (x0, ju0) cO0 supercritical fxcf/i < supercritical —

>0

U * < > , / „ * o, fXMfxxx < o supercritical

periodperioddoublmg doubling

-l

—1

r = /»/„+2 ft/l*O,s

= 2fxa

2

+ 3 / ^ 5*0, 0, rs rs c<0,0,supercritical supercritical +

Exercise 4.4 4.4 1 Study Study the bifurcation diagram diagram of

1

f(x,p)=p—x2. f{x,ju) = ju-x2. particular, determine determine the fold fold biflircation bifurcation and In particular, and investigate investigate the the stability stability of of the two branches of the fixed fixed point. point. 2 Study Study the the biftircation bifurcation diagram of

f(x,p)=jtr—x2. f{x, ju) = px-x2. In particular, particular, determine determine the the transcritical transcritical bifUrcation bifurcation and investigate investigate the the stability stability of the branches of of the the fixed fixed point. point.

10 10

The The table is is referred referred to to Martelli Martelli (1999: (1999: 92). 92). The examples examples of this section section are are also also cited from the same same source.

170

TIME-DEPENDENT SOLUTIONS OF SCALAR SYSTEMS 4. TIME-DEPENDENT

3 Study Study the the bifurcation bifurcation diagram diagram of of

f(x, /i) = —,tr -jux + x\ f(x, p)= + x3. determine the pitchfork pitchfork bifurcation bifurcation and investigate investigate the stability stability of In particular, particular, determine of the fixed fixed point. branches of Studythe thebifurcation bifurcation diagram of 4 Study 2 f(x,p)=(x+p2 _1)(x2 f(x, M)={X + M ~ l)(*2—2x—p)+x. -2x-/i)+x.

— 1. Determine Determine type type of of bifurcation bifurcationpoint pointwhen whenfip = —I.

4.5 Liapunov numbers Liapunov number number is important important for for measuring measuring the complexity complexity of solutions solutions The Liapunov behavior of difference difference equations. It is known that chaotic dynamics is characterized by an exponential exponential divergence divergence of case of the the oneoneby an of initial initial close close points, points. hi In the the case dimensional discrete discretemap mapof ofan aninterval interval(a, (a, b) b) into itself itself

x(t ++ 1)\)=f{x{t)), = Liapunov exponent exponent is divergence of two orbits orbits starting starting with with the Liapunov is a measure of the divergence slightly JC0 isis aa point point of of aa periodperiod- k slightly different different initial initial conditions conditions xx0 and xx0 0 and 0 ± So. If x0 orbit and if we start the orbit x0 ± <50, then after one iteration orbit from a nearby nearby point point x0 distance between between the two is is approximated by by the distance <,j'i ? H / ' ( * o K ==M050. Nowhere M0 Mo isisthe themagnification magnification factor factor for for the the first first step. step. At the second step, = M181 = M1M080.

in this manner, we conclude that the total magnification magnification factor over one Continuing in orbitisisthe the product, product, M0M1 M0Ml Mk_v We Weare areconcerned concernedwith with cycle of the periodperiod-kk orbit Mk_l. geometric average average of the factor factor the geometric

4.5. 4.5. LIAPUNOV NUMBERS

171 171

lk {M (A4'0A1, 0Mr-MkJ ,

which by taking logarithms leads to to the the arithmetic arithmetic average avenge

forstability of ofa periodic orbit is isthat the average average magnification magnification The condition for factor is less less than than 1, which which is is equivalent equivalent to to say say that that the the orbit orbit is is stable stable (unstable) (unstable)ifif factor Aninterpretation interpretation of the theLiapunov Liapunov exponent exponent is the measure of the measure of 2X << (>) 0. An information loss during the process of of iteration. iteration. Definition 4.5.1. beaasmooth smooth map map on onRRand andx0 x0be beaagiven giveninitial initial point. point. Definition 4.5.1. Let Let /f be The Liapunov givenby by Liapunovexponent exponentA,(x 2(x0) ofaamap map/f isisgiven 0) of

2(xj =

k—*co

k

provided the limit exists. In the case when any of the derivatives is is zero, set 2(x0) =

defined as as the the exponent exponent of of the the Liapunov Liapunov exponent, exponent, The Liapunov number L(x0) L(x0) isisdefmed The whenever the latter exists L{xo)=exp(A(x L(x0) o)).)). exp(2(x0 In fact, fact, we wemay may consider consider two two orbits orbits{ir(t)} {x(t)} and and{x(t)} {x(t)} originating originating from fromtwo two nearby points x0 and x0. x0.At At the the nnth thiteration iteration they they are are separated separated by an amount \x(t)-x(t)\ = \f'(xo)-f'(xo)\. Taking xx0 constant, expanding ) in inaaTaylor Taylor series series around around x0 x0 and and Taking as aa constant, expandingf'(x f' 0(is) 0 as retaining only the first-order term, we have

x{t) -— x(t) « (f'(x0)f'(x(l))) - f'{x(t f'(x(t -— l)))fo —- JC0),

172

4. 4.TIME-DEPENDENT SOLUTIONS SOLUTIONS OF OF SCALAR SCALAR SYSTEMS SYSTEMS

thederivative derivative is is different different from zero, zero, the theapproximation approximation can canbe be and, providing that the made very accurate accurate by taking |x0 —- xQ\ sufficiently small. Asymptotically, we have xM \rm\x{t)-x{t)\~e" (x0-x—0). x0). — —

I

We thus thus can can interpret that that 2(x0) A(x0) is is the the (local) average, average, asymptotic, We

exponential rate rate of divergence divergence of nearby nearby orbits. Usually, itit is difficult exponential orbits. Usually, difficult to exactly calculate Liapunov exponents. But there are are exceptions. exceptions. Example Consider Consider the the tent tent map map Example

ifx2-, x<-, if 2

px px p(1

-- *4) iifx> f*>! 2 ' .

We have

ifxJ-, 2

pP

if x < - ,

—p -P

ifx>—,

and T 7^(l/2) contain the (1 / 2) is is undefined. undefined. For For any any orbit orbit of the tent map that that does not not contain the point 1/2, we we have have point 1/2,

2(x0)= lnp. This holds for / 2. forall x0 x0 that thatare arenot noteventually eventuallyequal equaltoto11/2.

4.5.2. An An orbit orbit Definition 4.5.2. {x(l), x(2), x(3),

.

is called asymptotically periodic if if there exists aaperiodic orbit

4.5. LIAPUNOVNEJMBERS LIAPUNOV NUMBERS

173

{y(l), y(2),y(3\y(2), y(3), Ml),

such that

\im\x(t)-y(t)\ = — = 0. beaamap map of R R and andlet Definition 4.5.3. 4.5.3.Let Let /f be

O(x0)={x(0),x(l),x{2),-} 0(x0) = {x(O), x(1), x(2), be a bounded Theorbit orbit is is chaotic in the sense bounded orbit orbitof of /f.. The senseofYorke" of Yorke" if: if: (i) (i) O(x 0(x0) 0) 0. is not asymptotically asymptotically periodic; periodic;(ii) (ii)no noLiapunov Liapunovexponent exponentisis0;0;and and(iii) (iii) A(x 2(x0)> 0) > 0. In general, there are are very few few examples examples where the the Liapunov exponents can can be computed exactly. exactly, In hi most most cases cases one onecan canjust just compute compute them them numerically. numerically. Figure Figure computed Liapunov exponents 4.5.1 depicts depicts the 4.5.1 the Liapunov exponentsofof the the logistic logisticmap map ff == jux(\-x), ,uv(i — x), for ju seen that approaching 3.6, 3.6, the theexponents exponents become become 4]. ItIt can can be be seen that as p[i isisapproaching p e [2.8, 4j. The system system becomes becomes chaotic chaotic as as shown shown in infigure figure4.5.1. 4.5.1. positive. The

λ 0.5 0.5

2.8

3.2

3.4

3.6 3.6

3.8 3.8

-0.5 -0.5

-1 -1

-1.5 -1.5

Figure 4.5.1: The TheLiapunov Liapunov exponents exponents of of the the logistic logistic map H U

SeeAlligoode/a/. See Alligood et aL (1997). (1997).

4 4

jU µ

4. TIME-DEPENDENT SOLUTIONS OF SCALAR SCALAR SYSTEMS SYSTEMS

174

Example market) We We now now introduce introduce another economic Example (a (a model model of of chaotic labor market) mechanism of proposed by mechanism of economic economic chaos chaos in in labor labor market. market. The The model model is is proposed Bhaduri to demonstrate demonstrate the existence of chaos chaos when when wage bargain takes takes place Bhaduri the existence wage bargain specification of the Phillips Phillips curve curve while while firms firms maximize maximize according to the usual specification profit.12 The coordination between between profit-maximizing profit-maximizing firms profit.'2 The lack lack of coordination firms and and wagebargaining workers irregular dynamic dynamic path. In this model, model, the the bargaining workers may may result result in an irregular settlement takes according to the usual specification specification of the Phillips wage settlement takes place place according curve. agents are are endowed endowed with perfect foresight foresight in sense that they they curve. All All agents with perfect in the the sense correctly anticipate the short-term rate of inflation. Neutrality of money is thus correctly anticipate the short-term rate inflation. Neutrality thus built into into the model by assumption. assumption. There is no misperception of the real wage There misperception of the real rate on on the part of of the concerned agent. agent. linear Phillips Phillips curve'3 curve13 Consider the following linear

w(t+l)—w(t) wit +1) - wit) aL(t) Lit) , , . —b, a,b>O, w = - i 4') ; N - b, a,b>0, w[t) N

..... (4.5.1) (4.5.1)

where Lit) is is the the level level of of employment, employment, and N isisthe thelevel levelof offull full employment. employment. where L(t) Perfect foresight resulting in consistently correct short-term anticipation of the Perfect foresight resulting consistently correct short-term anticipation of inflation rate by the workers implies that workers bargain inflation rate bargain for real wages. wages. The The variable w wage. The The unique unique rate rate of of w ininequation equation (4.5.1) (4.5.1) is is interpreted interpreted as the real wage. unemployment u that keeps the real wage rate at a stationary stationary level level is given given by by L* . L' bb u =1——=—. =1 = —. N aa N *

U

Perfect foresight foresight on on the the part part of the firms ensures Perfect ensures that they also anticipate anticipate correctly the real real wage rate and maximizes maximizes profit profit by by equating equating the the real wage rate correctly the marginal marginal product product of labor labor which which is is assumed assumed to to be be diminishing diminishing linearly. linearly. to the That is PL{t), a,/3>0. wit) = a —- ,13L(t), w(t) = a, /3> 0.

(4.5.2)

Inserting equation (4.5.2) into equation (4.5.1) yields cia

a

^-2w(t). \t). w(t +1) = {i + £!L _ bU) w(t+l)=I1+——blw(t)---——-w K W fiN fiN ) M pN PN 12 12

See Bhaduri (2002). (2002). Sec Phillips (1954). There There are are various various forms forms of of the so-called Phillips curve, curve, see, for See Phillips (1954). so-called Phillips for instance, Goodwin (1967), Friedman (1968), and and Hahn and and Solow Solow (1985). (1985). 13 13

4.5. LIAPUNOV NUMBERS LIAPUNOVNTJMBERS

175 175

Introduce the following linear transformation transformation x(t)= w

kw(t) a ^ , Jfc B -2-, l1+a'k-b + ak-b PN fiN

then the difference difference equation for the wage rate is transformed into into — x(t)), Ax{t)(l ~ *('))> 1 > x(t)> x{t) > 0. 0. x(t + + i) l) = Ax(t)(I

the much much studied studied logistic logistic map. map. The The reader reader isis encouraged encouraged to to interpret interpret This is the of the model. economic implications of Example (rational (rationalchoice choice and anderratic erratic behavior) behavior) We We now now introduce introduce aabehavioral behavioral 14 14 It is assumed that preference is model of chaos by Benhabib and Day. model chaos by Benhabib and Day. assumed that preference is changeable as people experience more. Let us begin describing describing optimal optimal behavior behavior of the consumer function consumer in in t by maximizing the Cobb-Douglas utility function

u(t) = subject to the budget constraint

Pxx + Pyy = m, where and yy denote denotethe theconsumption consumption of of two two goods, goods, px and p3, py their prices, where x and m isisthe theincome. income.The Thedemand demandfunctions functions from from the the optimal optimal problem problem are are given given and m by m

x{t)==a—, a—, y(t) = x(t) Px Pr

in

(1 —

a)—. Py

Assume that the parameter parameter aa ininthe theutility utilityfUnction function depends depends on on past past choices choices in in following way the following a{t + 1) l) = = bx(t)y(t). bx(t)y{t). a(t + Substituting the preference change equation into the demand functions yields 14 14

This example is based on Benhabib and Day (1981).

4. TIME-DEPENDENT TIME-DEPENDENT SOLUTIONS SOLUTIONS OF OF SCALAR SCALAR SYSTEMS SYSTEMS

176

bm

x(t + i) = —x(t)y(t), px

y(t + i) = (i

—

py

We now now concentrate concentrate on on x. x. For For simplicity, simplicity, we we normalize normalize the the prices prices We F r o m = 1. From Px = =Py=lx{t)+y{t)=m y(t)= m x(t)+ demand equation we get get and the demand equation for for x, we

x{t x(t + bmx(t)(m — x(t)) f(x(t)). + i)=l)=bmx{t)(m-x{t))=f{x{t)). is The unique nontrivial fixed point is 2

, bm2—1 bm - 1 x= bm *

x

=

bm

We require bm2 bm2 > 1 so that x* is positive. It is straightforward straightforward to show show that that the 12; the themaximum maximum consumption consumption is is the map map/f has its maximum maximum at at x == m /2; 1

m]

—

bm bm3

2)

44

the other hand, as as xx <m, m,m, i.e., On the other hand, m,we wehave havebm3 bm%/4 /4 < i.e., bm2 bm2 < 4. 4. InIn summary, we should should require require < 4. 11cbm2 < bm2 <4. solutions. Introduce to have meaningfUl meaningful solutions. x(t) in m

we have

4.5. LIAPUNOV NUMBERS

177 177

z(t+l)=az(t)(l—x(t)), 1
4.6 Chaos gives another another definition definition of ofchaos. chaos. This section gives

Definition 4.6.1. The The map map /f on Definition 4.6.1. onaametric metric space space X X isissaid saidtotopossess possesssensitive sensitive dependence on initial initial conditions, conditions,ififthere thereexists exists£e> X > 0 such such that that for for any anyx0 x0 ec X dependence on and any containing x0 x0 there there exists exists y0 yoeE U andk ee Z + such that U and any open openset set U containing k k d(fk(xo), d{f {x0),ffk(y0))> {y0))>e. e.

Example Consider

x(t+1)=cx(t), + l ) = cx(t), c>1. c>\. x(t Lety0 =x0+&Then Let y0 = x0 + S. Then

f'&0) - ft(x) = Hence

fI() how small to oc, «>, regardless of of how small S8is. is. will will increase increase to to °° oc as as tt goes to said to to map on space (jf, (x, d). be onaa metric metric space d).15 Then /f is is said be aa map be (topologically) (topologically) transitive transitive if if for any any pair of of nonempty open sets sets U andV, V, there there be nonempty open U and exists a positive such that exists positive integer integer & k such

Definition 4.6.2. Let Let /f Definition 4.6.2.

15

tometric metric space space are referred to toappendix appendix A.3. A.3. ' The The concepts related to

178 178

4. TIME-DEPENDENT SOLUTIONS OF SCALAR SYSTEMS

fk(U)flV fk(u)nV*0.o. map a point wanders all all over over the the space space X X where transitive map point wanders where its its orbit orbit Under a transitive x. gets as close as we wish from every point in x. —>X,X, where whereXXisisa ametric metricspace. space.Then Then the the map f/ isis / : XX—> Theorem 4.6.1. Let Let f: transitive hasaadense denseorbit. orbit.Furthermore, Furthermore, closedinterval intervalininR,R,then then f/ transitive ifif itithas if ifX Xisisclosed is transitive if if and and only only ifif itit has has aa dense dense orbit. orbit. The following defmition definition of chaos is due to Devaney.'6 Devaney.16 Definition 4.6.3. 4.6.3. A map f: f:X^>X, where X is aa metric metric space, space, is is said said to be be Definition A map X —* X, where X is chaotic ifif (i) (i) /f isistransitive; periodicpoints pointsPP is dense dense in in X; X; and transitive; (ii) (ii) the set of periodic and (iii) f/ has hassensitive sensitivedependence dependenceon oninitial initial conditions. conditions.

It should should be noted that it has has been been proved proved now that that conditions conditions (i) (i) and and (ii) (ii) in in defmition 4.6.3 4.6.3 imply condition (iii) of sensitive dependence on initial conditions; definition conditions; but no other two conditions imply the third. The proof of the following following theorem is referred to to Elaydi.17 Elaydi.17

continuousmap mapon onaametric metricspace space(X, (x, d). Theorem 4.6.2. 4.6.2.Let Let/ f: X -» X X be a continuous Theorem : X d). If f/ isistransitive points isisdense, dense,then then/f possesses transitiveand andits itsset set of of periodic periodic points possesses sensitive sensitive dependence on chaotic. dependence on initial initialconditions, conditions,i.e., i.e.,/ f isis chaotic. following theorem intervals in The following theorem demonstrates demonstrates that that for for continuous continuous maps maps on intervals transitivity implies implies that that the the set set of of periodic periodic points is dense. R, transitivity

Theorem .J —* J be map on on an an interval Theorem 4.6.3. 4.6.3. Let Let f:f:J->J be aa continuous continuous map interval JJ (not (not necessarily fmite) in in R. If f/ is istransitive, necessarily finite) transitive,the theset setofofperiodic periodicpoints pointsisisdense, dense,that that is, is, chaotic. f/ isischaotic.

Example (chaos (chaos in in aa demand demand and and supply supply model) model) We We now now demonstrate demonstrate chaos chaos in in We consider consider that that supply supply involves a time time lag. lag. a simple simple demand-supply demand-supply model.18 model.18 We 16

Devaney (1992) Devaney (1989). See also Peitgene/a/. Peitgenetal. (1992) Section 3.5 in Elaydi Elaydi (2000). (2000). 18 This example example draws draws heavily heavily on Hommes Homines (1991: section section 5.1). 5.1). See See also also Shone Shone(2002: (2002: section 8.1). 8.1). 17

4.6. CHAOS

179 179

supply increases increases slowly, slowly, partly because because of of start-up start-up costs costs and and fixed fixed At low prices supply costs of production. production. Supply might also increase only slowly slowly at high prices, say, say, because This suggests because of capacity capacity constraints. constraints. This suggests a S5 -supply -supply curve. curve. The arctan arctan function function exhibits exhibits such such aaSS-shape. -shape.Let Letususexpress expresssupply supplyq5qs(t) as asfUnction function of of

expected price p0 (t) by expected pe(t) qS(t) qs(t) == arctan(upe(t)). arctan(jupe{t)).

The origin is an inflection point. As As shown shown in in figure figure 4.6.1, 4.6.1, the theparameter parameterju p inflection point. determines the steepness steepness of of the the SS -shape. -shape. The The higher higher the thevalue valueof of ju p the determines the the steeper steeper the curve.

qs 1.5

1

as µ : 0.5 → 4 0.5

( )

q s = arctan µp e -4

-2

2

4

pe

-0.5

-1

-1.5

Figure 4.6.1: Specified relations relations between between expected expected price price and and supply supply

simplicity, assume assume that that demand demand isis aa linear linear function function of ofactual actual prices, prices,i.e. i.e. For simplicity,

qd(t)__a_bp(t), b>O. qd{t)=a-bp{t), b>0. We assume that price expectation is formed as follows follows

180

SOLUTIONS OF OF SCALAR SCALAR SYSTEMS SYSTEMS 4. TIME-DEPENDENT SOLUTIONS

pe(t + i)= ,tpQ)+(l — 2)pe(t). It can can be be shown shown that that under under these these specifications, specifications, the the market market condition condition that that the the following difference difference equation equation demand equals supply is expressed by the following a2

Aarctan(W(t))

b

b

(4.6.1)

We will demonstrate demonstrate dynamic model by simulation. In dynamic behavior behavior of of the model by simulation. In the remainder of this this section, we fix: = 0.3, 0.3,bb==0.25 0.25and andconsider considera a as as aa remainder of section, we fix: A X= bifurcation parameter for different values of ju. In the case of ju = 0.5, figure biftircation parameter for different values of p. In case p = 0.5, bifurcation diagram for for 4.6.2 depicts the biftircation [-1.25,1.25]. ae [—1.25, 1.25]. We see that /. that there there is is aa unique unique fixed fixedpoint pointfor forthe themap map f.

p e* 2

1

-1 -1

0.5 0.5

-0.5 -0.5

1

a

-1 -1

-2

Figure 4.6.2: Bifurcation 0.5, ae a e[—1.25,1.25] [-1.25,1.25] BifUrcationdiagram diagramfor for fip == 0.5,

Let to 33and anddepict depict the thebifurcation bifurcation diagram diagram with Let us rise rise the the value value of of fip to with aa as the bifurcation bifurcation parameter as in figure 4.6.3. For low values of a, there parameter as in figure 4.6.3. For low values of a, there is is aa

CHAOS 4.6. CHAOS

181 181

unique fixed fixed point. point. Around Around a = —0.9 unique -0.9 a period-doubling bifurcation bifurcation occurs. occurs. The The stable orbit remains until a reaches 0.9 and then a period-halving occurs. stable orbit remains until a reaches 0.9 and then a period-halving occurs. Thereafter the system settles settles down down again again to to aa unique unique stable stable equilibrium. equilibrium. The The Thereafter the system diagram is symmetrical about origin because because of the the characteristic characteristic of diagram is symmetrical about the the origin of the function. arctan ftrnction. In figures figures 4.6.5 to to 4.6.6, 4.6.6, we we depict depict respectively respectively the the bifurcation bifurcation diagrams diagrams when H = 4, u=4,

// = 4.5. ,u=4.5.

We see that within the the period-four orbit orbit chaos chaos occurs. occurs. Figure 4.6.4 4.6.4 depicts diagram when when fip = 3.5. Figure depicts the the bifurcation bifurcation diagram 3.5. The The figure figure shows a doubling into a period-four orbit, which which then then turns turns into a shows doubling bifurcation bifurcation into period-four orbit, finally aa stable stable equilibrium. equilibrium. period-two orbit and finally

pe *

1

0.5

a

-1

-0.5

0.5

1

-0.5

-1 ace [—1.25, 1.25] Figure 4.6.3: [-1.25,1.25] 4.6.3: Bifurcation Bifurcation diagram diagramfor// forp = 3, 3, a

182

OF SCALAR SCALAR SYSTEMS SYSTEMS 4. TIME-DEPENDENT SOLUTIONS OF

p

I

1

r.s a

-0.5 -0.5

Figure 4.6.4: 3.5, ae [—1.25,1.25] [-1.25,1.25] 4.6.4: Bifurcation Bifurcation diagram diagramfor// forp = 3.5,

p e*

1.5 1 0.5

a -1

-0.5

0.5

1

-0.5 -1 -1.5 [-1.25,1.25] Figure 4.6.5: Bifurcation Bifurcation diagram diagramfor// forp = 4, 4, a ee [—1.25, 1.25]

4.6. CHAOS CHAOS

183 183

p e* 1.5 1 0.5

-1

-0.5

0.5

1

-0.5 -1 -1.5

ae [—1.25, 1.25] Figure 4.6.6: Biffircation diagram for forp Bifurcation diagram ju == 4.5, ae [-1.25,1.25]

a

Chapter Chapter 5 Economic bifurcations and chaos chaos This chapter chapter applies applies the the concepts concepts and theorems theorems of the previous chapters to analyze different economics. The models in this this chapter chapter exhibit exhibit periodic, periodic, different models models in in economics. The models aperiodic, or chaotic behavior. Section 5.1 studies a model of of endogenous endogenous business business cycles in the presence of knowledge spillovers. Many economic indicators, indicators, such such as as GDP, exhibit exhibit asymmetry asymmetry as if they they repeatedly repeatedly switch switch between between different different regimes. regimes. For instance, instance, itit has been found found that that (i) (i) positive positive shocks shocks are are more more persistent persistent than than negative shocks in the United States and France; (ii) negative shocks are more negative shocks in the United States and France; (ii) negative shocks are more persistent persistent than than negative negative shocks shocks in in the the United United Kingdom Kingdom and and Canada; Canada; and and (iii) (iii) there there is almost no asymmetry in persistence in Italy, Japan, and (former) Germany. The is almost no asymmetry in persistence in Italy, Japan, and (former) Germany. The model in this section provides some insights into the well-observed asymmetric model in this section provides some insights into the well-observed asymmetric nature of business business cycles. nonlinear cobweb cobweb model model with with nature of cycles. Section Section 5.2 5.2 studies studies aa nonlinear normal demand and supply, naive expectations and adaptive production normal demand and supply, naïve expectations and adaptive production adjustment. The exhibits horseshoes. horseshoes. Section Section 5.3 5.3 examines examines an an inventory inventory adjustment. The model model exhibits model with rational expectations. In section 5.4, we discuss an economic growth model with rational expectations. In section 5.4, we discuss an economic growth model with pollution. The model is an extension of the standard neoclassical model with pollution. The model is an extension of the standard neoclassical growth model unique stable stable equilibrium growth model which which has has aa unique equilibrium point. point. Chaos Chaos exists exists in in the the model because of the effects of pollution upon production. It is know that the model because of the effects of pollution upon production. It is know that the neoclassical growth theory based on the Solow growth model focuses neoclassical growth theory based on the Solow growth model focuses accumulation engine of of growth, growth, while while the theneo-Schumpeterian neo-Schumpeterian growth growth accumulation as as an an engine theory stresses innovation. Section 5.5 studies a model to capture these two theory stresses innovation. Section 5.5 studies a model to capture these two mechanisms within the same framework. The model generates an unstable mechanisms within the same framework. The model generates an unstable balanced growth path and the economy achieves sustainable growth cycles, balanced growth path and the economy achieves sustainable growth cycles, moving back and forth between the two phases - one is characterized by higher moving back and forth between the two phases one is characterized by higher output growth, higher investment, no innovation, and a competitive market output growth, higher investment, no innovation, and a competitive market structure; the other by lower output growth, lower investment, high innovation, structure; the other by lower output growth, lower investment, high innovation, and a more monopolistic structure. Section 5.6 identifies economic fluctuations and a more monopolistic structure. Section 5.6 identifies economic fluctuations in a monetary economy within the OLG framework. Section 5.7 shows chaos in in a monetary economyofwithin the OLG framework. Section 5.7 shows chaos in a model of interaction economic and population growth. a model of interaction of economic and population growth. 185 185

186

5.1

5. 5. ECONOMIC BIFURCATIONS AND CHAOS

Business cycles cycles with with knowledge knowledge spillovers

nowintroduce introduce aa model model of ofendogenous endogenous business business cycles cycles in in the thepresence presence of We now knowledge spillovers. ' Many Many economic economic indicators, such as GDP, GDP,exhibit exhibit as if if they repeatedly repeatedly switch switch between different different regimes. For instance, instance, in asymmetry as in their study of changes in in GDP, Hess and andIwata fmd find that that (i) (i)positive positive shocks shocks are their study of changes are shocks in inthe the United United States States and and France; France; (ii) (ii) negative negative more persistent than negative shocks shocks are are more more persistent persistent than than negative negative shocks shocks in in the theUnited United Kingdom Kingdom and shocks and(iii) there is isalmost almost no noasymmetry asymmetry in inpersistence persistence in inItaly, Italy, Japan, Japan, and Canada; and and (former) Germany.2 Germany.2 The provide some some insights insights into into wellwell(former) The model model is is supposed supposed to provide ofbusiness cycles. observed asymmetric nature of Time discretely extends from to infinity. infinity. There is of firms, is aa continuum of from zero to each indexed by i and the total population is normalized to 1. At the end of of each indexed by / and the total population is normalized to 1. At the end each period, where 8S ee (o, 1), of of randomly each period, aaconstant constantfraction fraction1 1— - 8, S, where (0, l), randomly chosen chosen firms disappear disappear and and are replaced replaced by by new new ones. ones. Each and Each firm firm is is risk risk neutral neutral and firms attempts to to maximize with yy e (o, i) attempts maximize the discounted discounted sum of expected expected profits profits with (0, l) being the subjective discount factor. factor. Defme Define 1

as effective discount either active active or as the effective discount factor. factor.In In any period, period, each each firm firm is either or inactive and engages engages in in at most most one one project project at at aatime. time. Each Each project project is inactive is characterized determines the characterizedby by its its quality, quality, either either high high or or low, low, that that determines the cost cost of of production. The cost of of production is is kkififthe theproject project isisof oflow low quality quality and zero zeroifif it is is of ofhigh high quality. quality. Upon Upon entering entering the themarket, market, each each new new firm firm has hasaalow-quality low-quality it and must must decide decide either either to to adopt adopt the the project project as asititisisor orto toinnovate innovate it. it. project project at at hand hand and If a firm decides to adopt, it immediately becomes active and produces until If a firm decides to adopt, it immediately becomes active and produces until it it disappears; if firm decides decides to to innovate innovate (and (and each each firm firm can can innovate innovate only only once), once), it disappears; if aa firm it can upgrade upgrade the the project project to tohigh high quality quality in in the the next next period period while whileititmust must stay stay inactive inactive can for that period. Let for that period. Let

n(t)e[0,l-S], n(t)E [o, 1—8], denote the fraction and fraction of firms that that decide decideto to innovate innovate in inperiod period t, and x{t)e [0, 1], l], x(t)e [o,

1 basedon onIshida Ishidaand and Yokoo Yokoo (2004). (2004). Thissection sectionisisbased 'This (1997). Hess and Iwata (1997).

22

5.1. BUSiNESS BUSINESS CYCLES CYCLES WITH WITH KNOWLEDGE KNOWLEDGE SPILLOVERS SPILLOVERS

187 187

beginning of of period period t. The the fraction of firms firms with aa high-quality high-quality project project at the beginning The law of motion motion of of x(t) x(t) is is

x(t ++ i) l)=S(x(t) = 8(x(t) ++n(t)). Assume that the productivity of each firm depends on the fraction of firms with a high-quality high-quality (innovated) (innovated) project projectatatthe thebeginning beginningof ofeach eachperiod.3 period.3 Each firm is (t) denote is equally equally productive productive in in any any given given period. period. Let Lety1 y^t) denote the output level level of of firm i in period period /t and output and

y(t)= h(x(t)), y,{t)=h{x{t)), where

h: [o, i]—* [o,

1],

is continuous and strictly strictly increasing increasing in in x and and

h(O)=9e{O,1), h(1)=1. Firm Firm i's i 'sprofit profit is is = h{x{t))-k, h(x(t)) — Ic, xi{t) =

if its project is of low quality and the profit is (t) = h(x(t)), *,(<)=

ifif it is of high quality. If a firm firm chooses chooses to adopt, adopt, the expected gain is *,(/ + s)= E,YFh{x{t

+ s)) -

* ~P

If a firm chooses to innovate, innovate, the the expected expected gain is is

3

This assumption (1991, 1993), 1993), and and Gale Gale (1996). (1996). assumption isis accepted accepted by, by, for for instance, instance, Durlauf Durlauf (1991,

188 188

5. 5. ECONOMIC BIFURCATIONS BIFURCATIONS AND AND CHAOS CHAOS

+

4=

+ 4).

chooses not to innovate if and only The firm chooses only if

(x(t ++ ,))

— 1l~

> P

5=1

which can be rewritten as k

1-/i To examine examine h(x(t)) h(x(t)) k>1(1 k/(\—- /3) 0) under

6e < h(x{t)) < i x(t) Ee {o, [o, i}, l], we we may may consider consider three three cases. cases. for any x(t)

Casel1 6>kl{\P) Case 9>k/(1—/3) n(t) == 0. 0. Hence, Hence, It is never profitable to innovate, nQ)

x(t+l)=âv(t) for x0 x0 e [O, l], which whichimplies impliesx(t) x(t)—* —> 0 as tt —, —> oo• oo. The project project of of every every firm firm is is of [o, i], low quality in the end.

Case2 kI(1—/3)>1 kl{\- P)>\ Each firm firm always always chooses chooses to innovate innovate

n(t) = \-5, n(t)=l—5, for any t.t. Hence

5.1. BUSINESS CYCLES WITH KNOWLEDGE KNOWLEDGE SPILLOVERS

189 189

lj. The system has a globally asymptotically stable steady state. x0 e [O, for x0 [o, ii. Case3 6
k

\-P

Consider as aa fimction function of k. k. ItItcan can be be seen seen that that c(k) c(k) continuously continuously increases increases Consider c as with k for for with 0

l

k

. 1.

1-fl \-p

From strictly increasing From the the definition defmition of of cc and strictly increasing h, h, we we see see that that each each new new firm firm in in period tt chooses x(t) < c c (i.e., period chooses to innovate innovate ifif x(t) (i.e., n(t) n(t) = = 1 —- S8)) and chooses not to and chooses innovate ifif x(f) (i.e.,n(t) n(t)== 0). The innovate x(t) >cc(i.e., The economy economy is is said said to to be in the contraction phase x(f) c< c; c; in in the the expansion expansion phase summary, the the law law of of phase if x(t) phase ifif x{t) x(t) >c.c.InInsummary, motion x(f) is characterized characterized by the the following following piecewise piecewise linear linear difference difference motion of of x(t) equation 1

1&(t) + 8(1 — a) ,

fL (x(t)), if x(t) <

if xv) c. •

(5.1.1)

lead to an an asymmetric asymmetric periodic periodic cycle which the the Equation (5.1.1) (5.1.1) can can lead cycle in which the contradiction contradiction phase phase alternate alternatewith witheach eachother otherasymmetrically.4 asymmetrically.4 expansion and the cycle Let us first examine a type of periodic cycle {p(i), p(2), ..., p(m

—

i), p(in)}

such that

()
(5.1.2) (5.1.2)

This section is only only concerned concerned with some some simple simple cases. cases. Methods Methods for for aa comprehensive comprehensive examination of this type of equations is referred to, for instance, Nagumo and Sato (1972).

190 190

5. ECONOMIC BIFURCATIONS BIFURCATIONS AND AND CHAOS CHAOS

for for some some natural natural number number in. m. Such a cycle cycle may may be be generally generally illustrated illustrated as in in figure 5.12. figure 5.1.2.

450

f

x(t) C

cycle Fig. 5.1.1: A periodic cycle

To compute compute p(1), p(l), we we need need to calculate calculate

fm(p(o) /"(/>(!))= A . under equation (5.1.2). As ftm (p(1)) = fR

°

f11 (p(1)),

we see that fm(p(1)) = p1 px becomes becomes

+82(1_5m1)=p1.

5.1. BUSINESS CYCLES WITH KNOWLEDGE SPILLOVERS

191 191

We thus solve

(5.1.3)

p1 =

For this to be consistent, consistent, we we should should require requirep1 pxininequation equation(5.1.3) (5.1.3)totosatisf5,' satisfy

c c, From (p(1)) <

C,

we have have

= 5m-2p1 + 5 — +8-

c >

=

—

3m-i

ST* = S - \ - J

5m-1

+ +S s

-Lm• Lm.(5.1.4) (5.1.4)

From

fR(c) we we have

.A 8

8-8" = Rm\-8"

(5.1.5) (5-1.5)

The two conditions conditions c c, fR(c) /T 2 (/>(i))
are thus represented by by conditions conditions (5.1.4) (5.1.4) and and (5.1.5). (5.1.5). That That isis Lm
Lm >>0,0, (Lm, R^] isiswell-defmed. well-defined. Conversely, Conversely, suppose suppose Lm Lm < As Rm Rm——Lm ccc < As

be satisfied. Further, defme define 3c such that

192

5. ECONOMIC BIFURCATIONS AND CHAOS CHAOS

fr(e) = Solving this equation, we have , _ c - 5 + 5m-x

We can show that the mapping fm restricted to the interval interval mapping ftm

TT=[Sc,3], = [&, c], maps T into into itself) itself) where where fm\T well-defined (that (that is, is, fm maps is well-defined

linear with with a is linear

T. We We see see that slope Sm and a unique constant slope unique fixed fixed point point p, p1 Ee T.

Lm
^
S 5

1+5 \ +S

Exercise 5.1 Exercise 5.1 1 Similar Similar to to the theprocedures procedures for for examining examining formula formula (5.1.2), (5.1.2), examine of examine aa type of periodic cycle {p(i), p(2), ..., p(m

—

1),

p(m)}

such that p{\) > p(2) > p(m - 1) > c> p{m). p(1)>p(2)"•>p(m—l)c>p(m).

5.2. A COBWEB MODEL WITH ADAPTIVE ADJUSTMENT ADJUSTMENT

5.2

193 193

A cobweb model model with with adaptive adaptive adjustment

A nonlinear cobweb model with normal demand and supply, supply, naïve naive expectations expectations and adaptive production adjustment is recently proposed by Onozaki Let and adaptive production adjustment is recently proposed by Onozaki etetal.5 al.5Let us consider a market of a single commodity. In period 1, a supplier decides us consider a market of a single commodity. In period t, a supplier decides his his production x(t+1). + i). Nevertheless, Nevertheless,this thislevel levelmay maynot not be be equal production x(t equal to to the the profit profit i) which maximum x(t + +1) which he he calculates calculates and and uses uses as as aatarget targetofofadjustment. adjustment. Suppose Suppose that that the the calculation calculation is is made made under under the thequadratic quadratic cost costfunction function 2 bx2 / 2, b> 0 with with the the naïve price expectation means that that his price bx /2,b>0 naive price expectation (which (which means price The profit profit expectation for current price expectation for the the next next period period isis equal equal to to the the current price p(t)). p(t)). The maximum level level of of output is is given given by by

It is assumed assumed that that the the producer producer will will adjust adjust his his production production according according to the the It is following following hedging rule in the uncertain economy

x(t + i) l) = x(t) + a(x(t + l) - x(t)), where a e (0, (0, i) l) is is the the speed speed of of adjustment. adjustment. Suppose Suppose that there are are nn identical identical suppliers in the market. The aggregate supply is thus thus given given by by

X{t)=nx{t). x(t) =

elasticity of Assume a monotonic Assume monotonic demand demand function function with with constant constant price price elasticity 11/3(41>0)

Price clears the market in each period, i.e. i.e.

X{t)=Y{t). x(t) = Y(t).

5

This section et al. at (2000). This section is based based on Onozaki Onozaki et (2000). See See also alsoHommes Hommes (1994), (1994), Gallas Gallas and and Nusse (1996), Goeree type of nonlinear nonlinear models. models. An An Nussc (1996), Goeree and and Hommes Hommes (2000), (2000),for for this this type extension of the model to two two types types of of producers, producers, cautious cautious adapters adapters and and naïve naive optimizers optimizers (2003). is carried out by Onozaki Onozaki et et al. at (2003).

194

5. ECONOMIC BIFURCATIONS AND CHAOS

isstraightforward straightforward to show show that under under the the above above specifications, specifications, the motion of It is the aggregate supply is given by

x(t+l)=(i—a)x(t)+ acn

bX (t)

linear transformation transformation Introduce aalinear

(

z(t)H—I

x(t).

\cn

Then, the above equation is transformed into into

+ i)1) = (l - a)z(t) + -j^r = f(z(t), f(z(t\ a, a, ji), p), (a, p)e (o, (0,1)x(O, l)x(o,oo). -). z(t + (i — (a, /3)€ +

(5.2.1) The map has aa unique unique fixed are The map /f has fixedpoint, point,z*f == 1.1. The first and second derivates are

f=1—a—

7 aMi+M0 z

z

z€

which implies strictlyconvex convexand andunimodal unimodal function function on on R++ with which implies that that /f isisa astrictly its minimum minimum at the critical point

9[afl

ii(i+p)

[1-a

The fixed point is aa repeller if

f(z)=\-a-ap<-\, 1 — a — a/i <—1, or equivalently 2

^ ^ a

a

< p.

(5.2.2) (5.2.2)

5.2. A COBWEB MODEL WITH ADAPTIVE ADJUSTMENT

195

can be be demonstrated demonstrated that for sufficiently sufficiently large ji, the map map For this map, it can large /3, /f exhibits exhibitsa ahorseshore. horseshore.By Byaahorseshoe horseshoe itit means means here here aa compact compact invariant invariant set on which topologically conjugate conjugate to to the the one-sided full-shift full-shift which some some iterate iterate of of /f isistopologically on two symbols. The existence of a horseshoe horseshoe is assured assured by that of a transverse homoclinic map is is said exhibit topological said to to exhibit topologicalchaos chaosifif itit has has aa homoclinic point. point. A A map horseshoe or, alternatively, if the topological entropy of the map is positive. horseshoe alternatively, if the topological entropy of the map is positive. It It should be remarked that aamap maprestricted restricted ononhorseshoes horseshoes behaves behaves in a should be remarked that complicated way, existence of horseshoes horseshoes itself does does not not assume assume complex complex complicated way, the existence dynamics in the long run; the system system may eventually eventually settle dynamics settle down down to to a periodic following theorem motion horseshoes are are present.6 present.6 In motion even ifif horseshoes In the following theorem proved proved by Onozaki et al, an attractor attractor is contains aa dense dense orbit Onozaki et a!, is said said to to be be strange strange if it contains orbit with with positive Liapunov Liapunov exponent. exponent. positive 5.2.1. For (0, l), Theorem 5.2.1. For any any a e (o, i), there is generically generically aa positive positive measure measure set set of of + parameter values such that exhibits parameter valuesofof/?, /3, EE cc R , such thatfor forevery eveiy/?/3e cE, E, the themap map /f exhibits

a strange strange attractor. attractor. The results results can be be demonstrated demonstrated numerically. 0.7, The numerically. For For instance, instance, for for a ==0.7, figure 5.2.1 depicts depicts the Liapunov Liapunov exponent exponent for different values /?. Figure Figure figure 5.2.1 for different values of /3. 5.2.2 depicts the biftircation bifurcation diagram the map map with with regard regard to /3 /? diagram ofof the (1,5 < / ? < 4.7) with a=0.7. a = 0.7.

5.3 Inventory Inventory model model with with rational rationalexpectations7 expectations7 This section section introduces introduces aa disequilibrium disequilibrium inventory inventory model. model. The actual actual labor labor This L(t), isisgiven givenby bythe theshort short side side of of the the labor labor market market employed, L(t), s L{t)=mm{Ld(t\L L(t)= (t) {t)l

and11(t) If(t) are arerespectively respectivelydemand demandand andsupply supply of of labor. labor. Assume Assume Ld(t) and where II' L'{t)=c, c, 66

See Block See

and Coppel (1992) (1992) and de Melo and van Strien (1993) (1993) for issues related to

and topological chaos. chaos. horseshoes and 7

This example example is given in in Hommes Hommes (1991: (1991: chapter chapter 28). See also Shone Shone (2002: section

7.9). 7.9).

196 196

5. 5. ECONOMIC BIFURCATIONS AND CHAOS

0

—I

—2

—3

22

2.5 2.5

33

3.5 3.5

44

p

4.5 4.5

Figure 5.2.1: The Liapunov Liapunov exponent exponent for for 1,5 1,5
j& 2

2.5 2.5

33

3.5 3.5

44

4.5 4.5

ji <4.7 4.7with withaa= = 0.7 Figure 5.2.2: 5.2.2: The bifurcation bifurcationdiagram diagramfor for1,5 1,5< /1

p

INVENTORY MODEL WITH RATIONAL EXPCTATIONS EXPCTATIONS 5.3. INVENTORY

197

d s where c is where is constant. Let yydQ) {t) and yyS(t) {t) denote respectively aggregate demand and aggregate aggregate supply. inventories l(t) positive when when there is and supply. The The level level of inventories 1(t) is positive excess demand, otherwise it is zero

1(t) = max{O,

—

yd(t)}

d We denote expected expected aggregate aggregate demand demandby by E[y E(y"(t)) (t)) and and the desired level of

inventories by I" (t). We Weassume assume perfect perfect foresight, foresight, i.e. i.e. Id (t).

E(y'(t))= yd(t) Suppose

jdQ) = where J3 parameter. Production Production isis proportional proportionaltoto labor labor employed, employed,SL(t). 1(t). fi isis a parameter.

We thus have have

ys(t) = I(t — i) + 1(t), yd(t)= (t))+ jd (t)rz (i + /3)E(y"(t)). Setting these two equations to equal each other other yields

(i + fl)E(/(t)) - A^l>. I(t -1)

,5.3,, (5.3.1)

8 Hence labor labor demand demand is is given by

L"(t) =

(i + J3)E(/(t)) — I(t

—

1)}

(5.3.2)

Aggregate demand demand is is assumed assumed to to be be aa linear linear ftinction function of of labor labor employed employed

yd(t)= + bL(t). yd{t)=a a + bL(t).

5. ECONOMIC BIFURCATIONS ANIJ AND CHAOS CHAOS

198

We is greater We assume assume that that the thelabor laborproductivity productivity is greater than than the themarginal marginal propensity 5 >> b. We have completed the model. We now b. We consume, 8 completed the model. We now show propensity to consume, of the system that the evolution of system can be described described by aa difference difference equation for I./ . d First, (t), which implies e (o, First, we weare areconcerned concerned with with L(t) L(t) = = L (t), implies L(t) L(t)s (0, c). c). From equation (5.3.2), we have

t'

(1 +

/3)/ (t)— i(t 5S

—1) = (i

+ fiXa + bL(t))— i(t —1)

5

above equation equation for for L Solve the above

i(t — 1) 5—b(i+fl)

(i + fl)a

—

Assume

5—b(l+fl)>O. 8 - b{\ + p) > 0. As L(t)c c ccisisguaranteed L(t)e(o, (0,c), c), then then L(t) l(t)< guaranteed by —

Ti

Similarly,the thecondition conditionl(t) L(t)> Similarly, > 0 is is guaranteed guaranteed ifif

>I(t—l),

72>l{t-\), 72 where y2 72

ea(l -i-fl). = a{l + p).

E (o, c) is if Hence, L(t) L(t)e (0, c) isguaranteed guaranteed if

>I(t—1)>Ti.

y2>l{t-l)>y]. 72

We need to toconsider the following three three situations situations

5.3. INVENTORY MODEL WITH RATIONAL EXPCTATIONS 199 RATIONAL EXPCTATIONS 199 (i) 1(t — i)

(ii) 72 y2 >>l{t-l)>y and I(t — i)> l; 7i; and (iii) 1(t l{t— 1)-\)>y 72•2. (i) 1(t .—i)x (x)l{t-\)
71

In this case, L(t) = = c. Then

1(t) =

— y"(t) = 1(t —1) + & — a — c.

(5.3.3)

(ii) 72 y2 > l{t —- l) >7iyx > 1(t In this case case d 1(t) = yS(t) — = 1(1 —-1)+SL{t) i) + 1(t) —- aa —- bL(t) i{t)=/(t) - yy"(t) {t)=i(t

=

I(t

- i) ÷s

—

br(1 + L

- I(t

—

8—b(I+/3)

1)1 — a. — a.

j

From the the above above equation, equation, we we obtain obtain ()_ {)

—b/31(t—1) -bpl(t-\) aS/i 8b{\ + p) S—b(i÷fl) S—b(1+/J)

(5.3.4) (5.3A)

(iii) I(t —i) 722 (m)l{t-\)>y Under these circumstances, circumstances, we we have have

L(t)= L"(t)= o. We We thus thus have have

1(t) = yS(t) — y"(t) = 1(t

—

i)

—

a.

Combining all all these these three three cases, cases, then then Combining

i(t) =

f(I(t

—

1)),

is aapiecewise function function is

(5.3.5) (5.3.5)

200

5. ECONOMIC BIFURCATIONS AND CHAOS l{t —i)+ - l) + &— 8c - a—c, a - c, I(t

i(t)=

i(t —i)

-bpi{t-\) ( aSp y2>I(t—1)>y1, 8-b{\ + p) 8-b(l + p)'

(5.3.6)

J(t—1)y2. I(t-l)>y 2.

I(t—1)—a,

The equilibrium is defined equilibrium investment investment is defined only for for72 y2>> l(t - 1i)> ) > Ti' yx, which is 1(t — is given by the following condition _b/31*

y ,=>/-=— *# S—K 8-b <J - 6(1 + yff) + ^ - 6(1 aS/3

—

aS/i

Since S8 >> b, b, we have 1* /* >> 0.0. We now provide two numerical examples. First, we specify aa=0.2, = 0.2, b=0.75, 6 = 0.75, c=1, c = 1, 8=1, £ = 1, /3=0.2. P = 0.2. Under these specifications, equation (5.3.6) becomes

I(t —- 1i)) ++ 0.05, 1(t f/(r 0.05, / ( f—- li)) <0.14, 0.14, 1(t) 1(t — i) c 0.24, -1.5l{t —1) - 1 ) ++ 0.4, 0.4, 0.24 0.24 .c 0.24. I(t—1)0.24.

(5.3.7)

We illustrate of map (5.3.7) the illustrate the behavior behavior of (5.3.7) as asin infigure figure 5.3.1. 5.3.1. Although Although the (/* ==0.1.6), thefigure figureititis isnot not clear clear whether whether this this equilibrium level exists (1* 0.1.6), from the In fact, as shown in figure 5.3.2, the system fixed point will be approached. fixed point will be approached. In fact, as shown in figure 5.3.2, the system is is in the the present case. case. chaotic in 'We We still assume assume 6 = 0.75, c=1, c = 1, 8=1, 8 = 1, a = 0.2, b=0.75, a=0.2,

but leave leave /3 >9 unspecified. We consider /3 P as asaabifUrcation bifurcation parameter. We allow allow torange over over the interval interval (o, (0,1/3), of11/3 /? to /3 1 / 3), the value of / 3 is is determined determinedfrom fromthe the condition

8—b(1+/3)=0. 8 - 6(1 + P) = 0.

5.3. INVENTORY MODEL WITH RATIONAL EXPCTATIONS

201 201

I (t + 1)

0.25 0.2 0.15 0.15 0.1 0.1 I (t )

0.05 0.05

0.1 0.1

0.12 0.14 0.14 0.16 0.16 0.18 0.18 0.12

0.2 0.2

0.22 0.24 0.24 0.22

Figure 5.3.1: The The dynamics with /3 P = 0.2

In constructing the the bifurcation diagram, diagram, we we notice notice that that with with unspecified unspecified /3, /?, equation (5.3.6) (5.3.6) becomes becomes equation

I(t—1)+o.os, l(t-\)+ 0.05, 1(t)

0.75 /31(t —

- 1) < 0.2^ 0 - A , I(t—1)o.2j30

i) + 0.2/3 —

I(t — i) — 0.2, l(t-l)0.2,

<1(t

—

i) <

— i) 1(t /(/-l)>0.2y5 0.

where where

=1-0.75^.

Figure 5.3.2 5.3.2 depicts the the bifurcation bifurcation diagram. diagram. Although Although the the system system exhibits exhibits chaos over certain range for /3, /?, there thereare areregular regular alternating alternating behavior. behavior.

202

5. ECONOMIC BIFURCATIONS AND CHAOS CHAOS I

0.3 0.3

0.25 0.25 0.2 0.2

0.15 0.15

0.15

0.2

0.25

0.3

β

0.05 0.05 fi as as biftircation bifurcation parameter Figure 5.3.2: The bifurcation biftircation diagram diagram with with /1

5.4 Economic Economicgrowth growthwith withpollution pollution 54 model is is due due to to Day.8 Day.8 First, we have the following The following model following relations Y(t) = c(t) +

K(t + i) = 1(t),

S(t) = Y(t)— Y{t) - C(t) 1 > s >0, > 0, s(t) c(t)== sY{t\ 1>

n>0, where Y(t) output, C(t) investment, S(t) is savings, where Y(t) is is output, c(t) isis consumption, consumption,l(t) 1(t) is is investment, s(t) is thetotal total labor labor force, force, and ss and andnnarearerespectively respectivelythe thefixed fixedrate rateof ofsavings savings N(t) isisthe and the the population populationgrowth growthrate. rate.The Theproduction productionfunction fbnctionF(K(t), F(K(t),N(t)) N(t))(= (= Y(t)) Y(t)) is is linear-homogeneous. Introduce

K(t) N(ty N(t)

See Day (1982).

5.4. ECONOMIC GROWTH WITH POLLUTION

203 203

Hence, we have

k(t+1)= ( )

S

1+n

f(k(t))

1+ n

where f(k) f(k) = F(k, where F(k, i). l).The Theproduction productionfunction function isistaken taken on on the the following following form form

f(k) = /){t){m-k(t)Y, (t)(m — k(t) f{k)=Bk k{t)<m, where reflects the k(t))r reflects the where B, B, fi, ft, 7, y, and and m m are areconstant. constant. Here, Here, the the term term (m (m — - k(t))1 influence of pollution on per-capita output. As the capita intensity increases, influence of pollution on per-capita output. As the capita intensity increases,

pollution resources have have to to be be sacrificed pollution increases increases as as well. well. Suppose Suppose that that resources sacrificed in in order order to to avoid this pollution. Substituting f into the equation for capital accumulation avoid this pollution. Substituting / into the equation for capital accumulation yields

k(t 1) = = nkfi{t)(m-k{t)Y, — k{t ++ \) where where

sB sB n = l+n . \ +n

If we choose choose /1 fi == 7y == in m == I,\,

then the equation equation becomes becomes

i) = nk{t){m -— k(t)). k(t k(t)). k{t + l) This is formally identical identical with the the logistic logistic equation. equation. Hence, Hence, all all the the properties properties of of the the identified in Day's growth growth model. logistical map can be identified now consider consider the the general general equation equation We now fi — (t){m-k{t)y. k(t ++ i) l) == ñk'° nk(t)(m

5. ECONOMIC BIFURCATIONS AND CHAOS

204

To apply apply the Li-Yorke Li-Yorke theorem theorem to this equation, equation, we consider consider the following following c three different Let k" k be bethe thecritical criticalpoint pointof ofthe the map, map, i.e., i.e., the value value of different values values of of k. Let k(t) in the next k(t) that that implies implies the the highest highest possible possible capital capital intensity intensity in next period. period. We We determine k" kc by bysolving solvingthe thefirst-order first-ordercondition condition

ak(t÷1) =

-

- k(t)Y - jk'(t)(m

--

0.

We have

k'=

k

Jim

y+J3

As shown shown in in figure figure 5.4.1, 5.4.1, ifif we wemake makeBBsufficiently sufficiently large, large, it is always always c lower than than the fixed point k\ Next let kb be be the the result of the possible to possible to make make kk' lower point k*. let k" backward iteration

kb

f1(kC)

b ck*, k" When k
m denote the kh will will be smaller smaller than than k''. kc. Finally, let kktm denote maximum attainable capital intensity, intensity, i.e., the intersection intersection of the graph of the map map with with the the abscissa. abscissa. Variations in in BB eventually eventually imply that the the graph graph of of p r m c ñ'k'3 (m — nk {m —kk) isisstretched stretchedupward upwardsuch suchthat thatktm k isisthe theforward forwarditeration iterationof ofk''k

f(kc)=

sB

1+n

As ktm is mapped mapped to the origin, the following following relations relations between between these these values values of of km is k hold hold

O /(*")< k"
Hence, the the requirements of the the Li-Yorke theorem are are satisfied. The model model is Hence, requirements of Li-Yorke theorem satisfied. The chaotic in the Lie-Yorke sense sense for for appropriate values values of of the the parameters.

5.4. ECONOMIC GROWTH WITH POLLUTION

205

1

t

--3-

/ /

/ 1' k"

kc

km

k (t)

Figure 5.4.1: Day's Day's growth growth model model with with pollution pollution

5.5

The Solow-Schumpeter growth oscillations oscillations

As shown shown in chapter chapter 3, 3, the the neoclassical neoclassical growth growth theory theory based based on the the Solow Solow growth model model focuses focuses accumulation accumulation as engine of of growth, growth, while while the the neoneogrowth as an engine Schumpeterian growth growth theory theory stresses innovation. Recently, Recently, Matsuyama Matsuyama Schumpeterian

capture these these two two mechanisms mechanisms within within the the same sameframework.9 framework.9 proposes a model to capture The model model generates generates an an unstable unstable balanced balanced growth growth path path and and the theeconomy economy The achieves sustainable growth cycles, moving back and forth between the two two achieves sustainable growth cycles, moving back forth between the phases - one one is characterized characterized by higher higher output output growth, growth, higher higher investment, investment, no no

9

This section is is based based on on Matsuyama Matsuyama (1999). (1999). The model is is based based on on the the lab lab equipment equipment model model of of Rivera-Batiz Rivera-Batiz and Romer Romer (1991) (1991) and and the the search-based search-based model model of of technology technology evolution evolution by Jovanovic Jovanovic and Rob (1990). (1990). It It should should be be noted noted that that iimovations innovations cycles cycles are are also also generated generated by many many other other models, models, for for instance, instance, by by Deneckere Deneckere and and Judd Judd (1992) (1992) and and Gale (1996). (1996). Matsuyama Matsuyama (2001) (2001) proposed proposed another another model model totoendogenously endogenously determine determine Gale consumers within the framework framework of of an an infinitely infinitely lived lived agent agenteconomy. economy. See See behavior of consumers also Mitra (2001) (2001) for discussing conditions for topological chaos chaos with regard regard to the model model also by Matsuyama. Matsuyama. by

206

5. 5. ECONOMIC BIFURCATIONS AND CHAOS

innovation, and competitive market market structure; structure; the other by bylower lower output output innovation, and a competitive structure. growth, lower investment, high innovation, and aa more monopolistic structure. There is a single final good, taken as a numeraire; it is competitively There is a single final good, taken as a numeraire; it is competitively produced and orinvested. invested. Let LetK(t K(t—- li)) be bethe the fmal final good good produced and can can either either be be consumed consumed or left unconsumed in period t — 1 and made available for use in production left unconsumed in period t - 1 and made available for use in production in in t. /. Labor is supplied in amount amount N. Labor supplied inelastically inelastically in Labor goes directly directly into into fmal final goods production. Capital Capital K(t K(t -—i) mustbe be first first converted convertedinto into aa variety varietyof of l ) must goods production. differentiated are aggregated the differentiated intermediate intermediate products; these intermediates intermediates are aggregated intothe composite by by aa symmetric CES. The final goods production function function is is z (r) z(t)

Y(t) = = A0NVa jxa(z,t)dz, t)dz, Y{t)

(5.5.1)

0

where x(z, of variety variety z in x(z, it) t) is is the input input of inperiod period t

a=l—-2-, a =\ , ae(l,oo), <7e(l,°°), o and z(t)] is and [o, [0, Z(t)\ is the range of of intermediates available at at t.

The market The market structure structure of of the theintermediate intermediate sector sector is is described described as asfollows. follows.

Prior to it,t, the the economy economy developed developed all all the the intermediates in the therange, range, Prior to [o, i)], with z(o)>> 0.0. These [0, z(t — -1)], with z(o) These old oldintermediates intermediates are aremanufactured manufactured by converting aa units units of capital capital into into one one unit unit of ofan anintermediate, intermediate, and and sold sold converting In addition, the intermediates of ofvariety variety competitively ininit.t. In competitively zZE e

[z(t—i),z(t)], [z(t -1), z(t)l

may be by their innovators be introduced introduced and and sold sold exclusively exclusively by innovators in in t.t. Innovating

new requireFF units units of capital capital per per variety. variety. The The process process of new intermediates intermediates require manufacturing new intermediates intermediatesrequires requires aa units output. Let Let r(t) manufacturing new units of of capital per output. r(t) denote the theprice of of capital. The marginal cost of of manufacturing intermediates intermediates in in t is is equal equal to ar(t). ar(t). The The old old products are are supplied supplied at atthe the marginal marginal costs costs c p(z, = ar(t), ze[0,Z(f-l)]. it) = p"(t) [o, zQ — 1)1. p{z,t)=p (t)=ar{t),

All the new products are sold at at

5.5. THE 5.5. THE SOLOW AND AND SCHUMPETER SCHUMPETER GROWTH GROWTH OSCILLATIONS OSCILLATIONS207 207

'U'

p(z, t) = pm(t) =

r(t), z e {Z(t — i), Z(t)]. l ze[z(t-l\z(f)l

Since all we all the the intermediates enter symmetrically in inthe the fmal final goods production, we have

£ = *"(')

=

(5.5.2)

=a

where c t) = pc(t) x(z, x{z,t)=p (t),

z

[o, zfr — 1)], zs[0,Z(t-l)],

m p(z, t) = ptm [z(t — i), Z(t)]. p(z,t)=p (t), (t), z ze[z(t-l),Z(t)].

The one-period monopoly enjoyed enjoyed by by the The one-period monopoly theinnovator innovator yields the themonopoly monopoly profit m + F), r(axm pmxm -—r(ax + F), n == p?flxm

at t.t. The The profit is is negative negative ififand and only only if if

F xm(t)<(c_1)_. a entry conditions conditions ensure ensure The free entry

axm{t)<{a-\)F, (a — 1)F, z(t) z(t — i), z{t)>z{t-\), {axm(t) —

(a — 1)F][z(t) — z(t

(5.5.3) (5.5.3) —

I)] = 0.

The resource constraint constraint on capital capital in t is isgiven by by K(t 1)== axC(t)Z(t ax°{t)z{t —- 1)++ (axm(t) (ax-it) + + F)(Z(t) F ) ( Z ( / )—- z(t Z(t —- l)). K{t —- i) i)). Under equations (5.5.2) (5.5.2) and and(5.5.3), (5.5.3), the theabove above constraint constraint becomes becomes

208

5. ECONOMIC BIFURCATIONS AND AND CHAOS CHAOS 5. m

= aa^x axc {t) =

{t) = = m i n j | | - | , 6OF\, K(t—

K li) z(t) = Z(t) = z(t Z(t— - l) i) ++ max-fo, ^ ~ ' -— 6Z{t -— l)l, oF

where where S6 = a

a

(5.5.4) (5.5A) (5.5.5) (5.5.5)

. Under equations (5.5.3)-(5.7.5), (5.5.3)-(5.7.5), the the total total output output is given givenby by

\A\9oFZit -— l)faKa(t y / v = TA[0cFZ(t Y(t) = — i) [AK(t-l)

- 1) K{t— - 1) 0oFZ(t—- l), 1) if KQ i) ) >0oFZ(t - l), if OaFZ(t — i),

5

(5.5.6)

where we use — A—

z(r)

4 ( aN f(z, t)th = (xc(t)rz(t(/-l)+(r-(/)f(z(/)-Z(/-l)). — 0± (x" (t)t(z(t)— z(t —1)).

Equations (5.5.5) and (5.5.6) tell that if if K(t

—

i)

z(t-1)

OaF,

the the resource base of of the economy, economy, K, K, isistoo toosmall small relative relative to to the thenumber of of the

Z,and andthere thereisisno noinnovation. innovation.The Theeconomy economy fri inthis this situation situation existing products, Z, is If is called called the the Solow regime. If K(t

—

1)

z(t—l)

> OaF,

some new newproducts products are are introduced; introduced; the be in inthe theRomer Romer some the economy economy is is said said to be regime. regime. To themodel, model, ititisisassumed assumed that that the theeconomy economy carries carries over over aaconstant constant To close close the fraction of ofits its output output to tothe thenext nextperiod, period,i.e. i.e. fraction

K(t) K(t) ==sY(t), sY{t),

5.5. THE SOLOW AND SCHUMPETER GROWTH OSCILLATIONS 209 209

where ss isisaaconstant. constant. The Thesystem system has hasaaunique unique solution solution for forany anypositive positive initial initial condition, condition, K0 Ko and andZ0. Z o .SetSet

—

K(t) K{t)

6oFZ{t)'

Then the be expressed as the system can be as the the one-dimensional difference difference equation equation

\sAka(t-\), k(t) =

f(k(t -i))=

sAk(t 1 + O(k(t

1) I)

— —

ifif/fc(f-l)
i)—

i)'

if k(t—l)>1.

The critical critical value value of of k, k, denoted denoted by (= 1), 1), which which separates separates the theSolow Solow and by k (= and Romer regimes is is equal to one. If If sA < 1, sA<1, hasaa unique steady state the system has

cE. k* ==(sA)c (sAf < k. innovation, all all goods are are competitively competitively supplied supplied and the economy economy Without innovation, and the does not not grow: the the steady steady is is aa neoclassical neoclassical stationary stationary path. path.If If sA > 1, sA>1, steady state, state, given given by the steady by ** — ... , sA—l &4 - 1 7=1+ >k, k =1 + 9 >k, 0 inthe theRomer Romer regime. regime. New Newproducts products are areintroduced introduced steadily, andZZ is in is steadily, and and K K and atthe thesame rate. It is aa balanced growth path, characterized characterized by grow at It is by

k

K(t K(t ++ l)=sAK(t). \) = sAK(t). Figures 5.5.1-3 5.5.1-3 depict the dynamicsof Figures of

_____

5. ECONOMIC BIFURCATIONS AND CHAOS

210

k(t) = f(k(t

—

1)).

Figure 5.5.1 5.5.1 depicts the case case of of sA sA << 1. 1. For any initial condition, condition, after after at most one period, innovation stops and and the economy economy is is trapped into into the the Solow Solow regime. regime.

k The The Solow Solow Regime Regime

k

i)

C

The Romer Romer Regime Regime

Figure 5.5.1: The growth by the Solow mechanism with sA sA c<11

Figures 5.5.2 and 5.5.3 both depict the case of sA sA >> 1,1, for which

«\ 1—9 \-e sA sA

They differ They differ in the local local stability stability of of the the steady steady state. state. Figure Figure 5.5.2 5.5.2 illustrates illustrates the the case of 11 c< sA sA c<99— - 1. 1. The equilibrium equilibrium point kA:** is locally locally unstable unstable because because of —1.

(k**), f(k)] The interval |/2(£**), /(^**)J represents the trapping region, i.e., the region that the economy enters eventually and once entered entered will never leave. leave. The The trapping trapping

5.5. THE SOLOW AND SCHUMPETER GROWTH OSCILLATIONS 211 211

region covers covers both both the Solow Solow and and Romer Romer regimes. regimes. If If the the economy economy starts starts with with aa region small may stay small k0, k0, the economy economy may stay in the the Solow Solow regime regime for for some some periods, periods, but eventually to enter eventually accumulates accumulates enough enough capital capital to enter the the Romer Romer regime, regime, and and innovation begins. The The economy innovation begins. economy begins begins cycling cycling between between the two two regimes. regimes. Formally, it can be proved that if

lcsA <9—I, \<sA<0 -\, there are period period 2— 2 - cycles, k(t) k{f) fluctuates fluctuates forever forever between between the Solow and Romer regimes for almost all initial conditions.

The Solow Regime Regime

Regime The Romer Regime

Figure 5.5.2: The Solow-Schumpeter < 9—- 1 Solow-Schumpeter mechanisms mechanisms with with 1 < sA <9 1

Figure 5.5.3 depicts the case of sA sA > > 98 —- 11. . We have I **\

- 1 < / ' ( £ " ))cO. < 0.

The The steady steady state state is globally globally stable. stable. In the long long term, term, the the economy economy will will forever stay in in the the Romer Romer regime. regime.

212

______________J 5. ECONOMIC BIFURCATIONS AND CHAOS

k The Solow Solow Regime Regime

The Romer Regime Regime

Figure 5.5.3: The Schumpeterian growth with sA S —1 1 sA> >6

Exercise Exercise 5.5 5.5 1 Consider Considerthe theMatsuyama Matsuyamamodel modelpresented presentedininthis thissection. section. Prove: Prove: (i) (i)IfIf

sAd, sA<\, then for for any k0 k0 ee R+, /c(t) k(t)<< îç kc for all all 1,t, and and lim,^ k(t) k(t)= k*; (ii)If (ii)If = k*;

l<sAcO— 1, 1 < sA < 9 - 1, there there are areperiodperiod- 22 cycles, fluctuates fluctuates forever forever between between the the Solow Solow and andRomer Romer regimes for If foralmost all initial conditions; and (iii) If

sA>'G—l, sA>6-\, then, for exists aa fI such forany k0 k0 ee R+, there exists suchthat that {k(t); it t'}c c,f{kc)\ {k{t);t>t'}^[k

limk(t) = k**. \\mk{t)=k". 1-400

5.6. MONEY, MONEY, GROWTH AND FLUCTUATIONS

5.6

213

Money, growth and fluctuations fluctuations

section is is to to identify identify endogenous endogenous fluctuations fluctuations in in aamonetary monetary economy.'0 economy.10 This section The economy is is inhabited inhabited by by an an infmitely infinitely lived lived representative representative agent agent with with perfect perfect foresight, who maximizes

YiJ

m(t)), m(t)\

subject to the flow flow budget constraint constraint

M(t)= P(t)(y —- c(t))+ c(t)) +H(t)+ H{t) +M(t— M(t- i),l), M(—l) M(-1) given given M{t) =P(t)(y where where U: R+2 —>RR isisthe theone oneperiod periodutility utilityfunction, function,which whichisisstrictly strictlyconcave, concave, increasing in both arguments, arguments, /3 /? Ee (o, (0, l) theconstant constant i) is the discount discount factor; factor; y isisthe consumption level level endowment of perishable consumption consumption good; good; c(t) isis consumption

M(t)

is real balance, M(t) isisnominal nominal money money holdings, holdings, and P(t) P(t) isisthe theprice pricelevel. level.The The considers {P(t)}0 {P(t)}~ independent independent of of his his own own money money holdings. holdings.At Atthe thebeginning beginning agent considers

of period period t,/, the the agent agent receives receives H(t) Hit) units unitsofofpaper papermoney moneyfrom fromthe thegovernment government though a "helicopter "helicopter drop", drop", also also considered considered to be be independent independent of of his his money money though / > /,5, ? , which whichimplies implies holdings. The money money supply supplygrows growsatatthe therate rate/ p>

H(t)=(,ll—1)M(t—1). Thus Thus the the markets markets clear clear when when

M(t) M{t)==,I/M0, n'M0, c(t) c{t)== y, for for all t.t. An equilibrium equilibrium point of of the the economy economy isis given given by by aa nonnegative nonnegative sequence sequence of of real real balances balances satisfying satisfying

/3UC {y,m(t m{t++1))m(t \))m{t++i)l)==pn(t){Ujy, fJm{t)[Uc (y, m(t)) - Um (y, m{t)) ], m(t)) m(t))], — Um(y, }

This section section isis based based on on Matsuyama Matsuyama (1991). (1991). 10This

(5.6.1 (5.6.1)

214

5. ECONOMIC BIFIJRCATIONS BIFURCATIONS AND AND CHAOS CHAOS

as well as the transversality condition

= 0. O. \\mP'Uc{y,m{t))m{t) mQ))m(t) = t—>oo

The steady state is ,n(t) m(t) = - m or

(M0

P(t)=pl

*

m for all t where where m* m >> 00 satisfies satisfies

(p

m*)= /JUm(Y, tn),

—

exists uniquely.1' uniquely.11 the utility function function is is taken on the following form For simplicity, assume that the I \

—Q+q)

[g(c)m]-

Eg

U(c, m)= m) = — U(c,

l+i7 1 + 7]

-, if 1777 * - l ,

(5.6.2) (5.62)

log g(c)+ g(c) + log log m, m,

where gg>0,g'>O,and > 0, g' > 0, and where

<1,

and /7 ^ isisaaparameter parameter satisfying satisfying

2)

The elasticity elasticity of of intertemporal intertemporal substitution substitution of of real real balances balances is is equal to The

11

Strictly Strictly speaking, speaking, m(t) m(t)=0 or P(t) P(t) == oo °° isis another another candidate candidate for the steady steady state. state. = 0 or for the

Nevertheless, for for the the utility utility function function used used below, below,this thiscase caseisisruled ruledout. out. Nevertheless,

5.6. MONEY, GROWTH AND FLUCTUATIONS

215 215

cr = (n + 2)-\ Substituting equation (5.6.2) into into equation (5.6.1) yields + 5y'q p(t)(1 — + i) p(t + l) == (i(1 + SrP(t)(l - p(t)r - F(p(t)), F(p(t)\ p(t) p(t)e (0,1), E (0, i),

(5.6.3)

for all tt and and

g(y) gg(y)m(t) l

18 p

For the new variable variable p, p, the thetransversality transversality condition condition becomes 0. l i m / ? y ( ; ) = 0.

unique steady steady state state is is The unique S P =-, 1+5 *

also rule rule out outthe thecase caseofof77 7]=- 0.0.Inhithe theremaining remaining ofofthis thissection, section,we we We also 12 require: require: 77 77 >> 0. 12 then If 'y77== 1,1, then

FF == (i(l ++ S)p(t)(l-p(t)\ 8)p(t)(1 — p(t)), p(th p(t) E (0,1). (o, i). is again again the thewell-studied well-studied logistic logistic map. map.There There are areperiodic, periodic, aperiodic aperiodic and This is solutions to tothis this system. system. chaotic solutions is straightforward straightforward to It is to check checkthe thefollowing followingproperties propertiesofofF: F: (1)

F(0) = F(1) =

0;

(2) FFhas hasaasingle singlepeak peak at

12 12

Under Under this this requirement, requirement,the thetransversality transversalitycondition condition

satisfied for pp Ee (o, (0,i).l).

lim,_^ fit fS'pn(i) = 0 is automatically

5. ECONOMIC BIFURCATIONS AND CHAOS

216 17

1+i7 1 + 7/

F isisstrictly strictlyincreasing increasing on on{o, [o, p), and strictly decreasing on (p, l]; i]; (3) F'(O) = (i +

5)1/17

> 1;

(4)

F'(p')= 7] 17

and (5) Let —1.

If 58 < (>) (>)A(ij), A{rj),

F maps [o, into itself itselfifif8S
= 0, A(^) > 0 =0, for all all17. TJ. for

Proposition 5.6.1. (i) IfIf 0 < c S8 217, 5.6.1. (i) < 2TJ,then thenfor forall allp0 p0ee(o, (0, 1), l),

iimFt(pc)= pt; (ii) If 0 c< 2q then a period2 2cycle 2TJ 5, < 8, then a periodcycleexists, exists,and andthe theset setp0 p0eG(o, (0, i) l) such that Ft (p0) N, isisatatmost mostcountable. countable. Furthermore, Furthermore, if F'(p 0) converges, N, 217 2r]

<8

5.6. MONEY, MONEY, GROWTH AND FLUCTUATIONS

217

the set of initial initial prices that lead to equilibrium equilibrium points along which the price level fluctuates forever, T N, fUll Lebesgue measure. fluctuates forever, V is , isofoffiillLebesgue measure. Proof: (i) Introduce G(p)_= {F2(p)/p}n1.

Clearly, = for some Clearly, GG(p) ( / ? * ) = 1. 1. If ULfi) G{j>)=\ 1

pE (O,1)\jp}, pe{0,l)\{p}, then there exists exists aa periodperiod-22cycle cycleofofF. F. From From the the definitions definitionsofofFF and and G, G, we then there we have

G(p)= (1 + 8)2(1

G'(p) = (1+

—

p)(1

—

+

+ 8)(l -

s)2+1/n[i = (i + +

IJO

- i} —

—

-

[2+ iJp}.

Let —

1+2,7 We We see that that C? G' isis strictly strictly increasing increasing in in (o, (0, p), strictly decreasing (p, i], l], and G'{p) <0.0.Thus ThusG'(p)c G'(p)<00inin(o, (0, l)|\{p}, or (3 G isis strictly strictly decreasing decreasing in in (o, (0, i). l).

Therefore, j isisthe that FF has Therefore, p* theonly onlysolution solution of of (3(p) G(p) == 1. l. This This implies implies that has no no n periodcycle. From From the the theorem theoremofofCoppel, Coppel,'3which whichstates states thatif if/ f is is a period- 22 cycle. that continuous map of of aa compact intervaltoto itself itself that that has has no period2, the continuous map compact interval period- 2, (x)} converges sequence {f' to aa fixed fixed point point of of /f for {/'(*)} converges to for every every xx ininthe theinterval. interval. (0, 1) l) can asymptotically, we have have As no no p € (o, can approach approach 0 asymptotically,

13

See Coppel Coppel (1955). (1955). See

5. ECONOMIC BIFURCATIONS AND AND CHAOS

218

lim F'(p)= p* limF'(p)= p t—>«>

forevery for every pp in in (o,i). (0, l). (ii) For the existence to show show that that G(p) G(p) = 11 existence of periodperiod- 22 cycles, cycles, itit is sufficient sufficient to has a solution solution in in (o, 1)\ [p*\. {p}. As (0, l)\ G(O + )>O,

G(I) = O, G(1)=0,

the intermediate (o, p*) such that intermediate value value theorem theorem implies that there there exists exists pc e (o,

if

-2] >0.

(i +

For the the proof proof of of countability countabilityofofN, N, let let pp be be aa point point in N N and and p°° p°°be be the the limit limit point of the the sequence sequence starting startingatat p. From the continuity of F continuity of FçlimF' (p))= F(p°°),

p°°

we we see see = 0 or p°° =p. - p . pp°° =Oorp 00

00

Since 0 c< 2iy 2T] << SS

*

implies that both 00 and p* are are locally unstable, unstable,

(p) cannot l i m , ^ F' F'(p) cannotapproach approachthem themasymptotically. asymptotically. Therefore Therefore

N= ={qe for a finite T\ {q e (o, (0, l)| FT{q) = 0 or or p* p forafmiteT},

which since property (2) of of F implies which is at at most most countable, countable, since property (2) implies that, that, for for any any z,z, there most two two x 's that solve zz = F(x). The secondhalf halffollows follows from there are at most The second from thethe fact that if

___ 5.6. MONEY, GROWTH AND FLUCTUATIONS S
219

N* (o, 1)\N. N* = (O, l)\N.

A*(q) (77) satisfying Proposition 5.6.2. 5.6.2. For any /7 r] > > 0, 0, there exists a value A Proposition 2TJ
such that that aa periodperiod-3 cycle of of FF exists A* (77). 3 cycle exists if 5> 8 >A(77). Proof: Introduce

G(p) =

Then G ( O + ) > 11 and G(p*) G[p)=l. =

1.

From the the intermediate intermediate value value theorem, theorem, it is From

sufficient (o, p) such such that G(pC) G(pc) > A(TJ),

p)

then there exists pc e (o, (o, p'j such that F(pc) =-11 so that

or G(/? C ) < 1. From the continuity of G G on 8, there exists a A*(rj)< A(TJ) such such c A(/7)

that G(p c ) < c 1. 5.6.1. 1. That That 2/7 277c
The proposition guarantees guarantees the existence existence of of chaos. chaos.

5.7

Population and and economic economic growth growth

The model model in in this this section section isisbased basedon onsection section6.2 6.2ininThang.14 Zhang.14 The The model model isis Haavelmo in in continuous continuous form. Its Its discrete discrete form form was was examined examined by by constructed by Haavelmo modern mathematics mathematics for for one-dimensional one-dimensional mappings. mappings. First, First, Stutzer,15 by applying modem Stutzer,'5 consider aa macroeconomic macroeconomicgrowth growthmodel modelproposed proposedby byHaavelmo16 Haavelmo16 consider

14 14

15

Section Section 6.2 in in Zhang Zhang (1991). (1991).

Stutzer Stutzer (1980). (1980). 16 Haavelmo (1954). (1954). 16llaavelmo

220

5. 5. ECONOMIC BIFURCATIONS AND CHAOS a,P>Q, Y=ANa, Y = ANa, A > 0, 0 < a < 1, a,fi>0, A>0, Oca
= N\a-^-\,

(5.7.1) (5.7.1)

thepopulation, population, Y is is the thereal real output, output, and anda, a, j?, /?, a and andA A are where N isisthe where constant parameters. Substituting

Y = ANa into the differential equation equation yields yields 1

BN '" ( N=N(aN = N\a- —

I.

(5.7.2) (5.7.2)

isaageneralization generalization of the familiar familiar logistic form widely see that the growth law is We see inbiological biological population used in population and and economic economic analysis. analysis. It It is is not not difficult difficult to to see see that that the dynamics of this system are simple. If the initial condition satisfies

/ aA\1/(2—a)

N(0)>(c) —

then both N Nand andYYwill willdecrease decrease(increase) (increase)monotonically monotonically until until approaching approaching their their

unique equilibrium equilibrium point, derivatives by first unique point, respectively. respectively.IfIf we we replace replace time time derivatives by first differences and accept discrete time, then equation (5.7.2) becomes

N(t+1)=N(4(l+a)_ which can can be further further simplified as which as x(t ++1) i) ==(1+ (1 + a)x(t)(l —- x112 xm (t)) (0) == F(x(t}, a, a,0.5), 0.5),

in which the new variable x(t) x(t)isisdefined defined by by the the transformation transformation in

[Au; a)] 1/(1 - a) x{t), 11(1 — a)

N(t)

=

(5.7.3) (5.7.3)

5.7. POPULATION AND ECONOMIC GROWTH

221

with aa = 11/2. / 2. ItIt can with can be be shown shown that that none noneof ofthe thequalitative qualitativeproperties properties of of the the system system affected by the particular particular choice choice of 00 << aa < The term term chaotic chaotic dynamics dynamics are affected c 1. The refers to the dynamic behavior of certain equations F which possess: (a) nonrefers dynamic behavior certain equations which possess: aa nondegenerate nn -period point for and(b) (b)ananuncountable uncountableset set degenerate for each each n >1,1,and

Sc 5 e J=[O,1] J = [0,1] containing no periodic containing no periodic points points and and nonoasymptotically asymptotically periodic periodic points. points. The The trajectories of such such points pointswander wanderaround aroundininJJ "randomly". trajectories of For each value of a, a, equilibrium equilibrium points points are are given given by by the the intersection intersection of the graph of F(x(t); For each each value value of of a, a, there graph of F(x(t); a) a) with with the the 45 45-degree -degree line. line. For there are are two two equilibrium equilibrium points points

x0=0, x0 = 0,

xo =

The point x0 = 00 is unstable and repels repels nearby nearby points. points. The The local local stability stability of the the x0 = other one can be be determined determined by by linearization linearization at at the the equilibrium equilibrium point. point. We We have have

F'(x0; a) =

1

—

= 0(a).

(5.7.4) (5.7.4)

The 0(a) determines the local stability of of JC x0. When 00
o0

attracts attracts nearby points points in in an exponential, exponential, monotonic fashion. fashion. When When

> 09 >> — attracts nearby nearby points points in in aadamped dampedoscillatory oscillatory manner. manner. When When - 1,1, x0 xQ attracts

x0 is is neither neither stable stable nor unstable. unstable. Finally, Finally, if tel> |#| > 1,1, x0 x0 isis unstable. unstable. These These o6 ==1,1, x0 when behaviors occur when

0
222

BIFURCATIONS AND AND CHAOS CHAOS 5. ECONOMIC BIFURCATIONS

of aa ininexcess into two two 44-period excess of of about about 4.8, 4.8, and andeach each 2-period 2 -periodpoint pointbiflircates bifurcates into -period points, producing an stable cycle cycle of of length length four denoted by x4 *01>

4

4

x x 44}. 43, 42' 02> 03> A

A

Figure 5.7.1 illustrates the phenomenon.

(t ++ 11) xx()t 1l

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

A A 0.2 0.2

0.4 0.4

0.6 0.6

0.8 0.8

11

(t ) xx()

Figure 5.7.1: 5.7.1: The 4-period 4-period orbit 4.9 Figure orbit for for aa =- 4.9

pitchfork bithrcation bifurcation process process continues parameter a increases, increases, This pitchfork continues as as the parameter producing non-degenerate (k == 2,...). 2,...). These orbits are called called producing non-degenerate orbits orbits of of length length 2k 2k (k -period orbit. orbit. ItIt can can be be shown shown that all the harmonics occur prior harmonics of the 22 -period the parameter parameter a reaching reaching 5.54, 5.54, although although how to the how much much prior prior to to this this value value is is not within which which aa stable stable orbit of length length k first first appears appears known. Thus, the range range of of a, within and later becomes unstable and bifurcates to a 2k k -period orbit, decreases in and later becomes unstable and biftircates 2k k -period orbit, decreases in length as the the parameter parameter aa increases increases to to aalimiting limiting value value ac <5.54. < 5.54. The The range range of < a 5.75 < 5.75 is termed chaotic region.As Asthe theparameter parameteraaenters entersthis thisregion, region, ac
5.7. POPULATION AND ECONOMIC GROWTH

223 223

a near near 5.540. 5.540. This, This, then then gives gives rise rise to to orbits orbits of of periods periods 3k 3k (k (k == 2, 2, ...) ...) via the the pitchfork process just described. In fact, fact, if we can locate the 3 -period orbit, the LiLiYorke theorem demonstrates that for any F(x(t); a) in which a non-degenerate theorem demonstrates that for any F(x(t); in non-degenerate 33-period orbit arises, arises, there must must also also exist exist non-degenerate non-degenerate points all periods, periods, as as period points of all well as an an uncountable uncountable set set of of periodic periodic (not (not asymptotically asymptotically periodic) periodic) points points whose whose 17 trajectories wander randomly randomly throughout throughoutthe the domain domain of of F. trajectories wander F. 17 Our (kir dynamic economic system system satisfies satisfies the requirements in Li-Yorke theorem theorem for for some some economic the requirements in the Li-Yorke values of a. a. This This guarantees guarantees the the existence existence of of chaotic chaotic behavior behavior as illustrated illustrated in values 5.7.2. figure 5.7.2.

x(t ) 1

0.8

0.6

0.4

0.2

t 20 20

40 40

60 60

80 80

100 100

120 120

140 140

Figure 5.7.2: The existence of chaos chaos for for a == 5.75 0.4 5.75 with with x0 x0 == 0.4

The existence existence of of chaos chaos implies implies that that no no one one can can precisely precisely know know what what will will The happen in in society society in in the the future, future, except except that that itit will will be be changing. changing. To Toillustrate illustrate why why no one one can can precisely precisely foresee foresee the the consequences consequences of of the the intervention intervention policy, policy, let let us us try try to to find find out out what what happen happen to to the the system system when when itit starts starts from from two two different different but very we consider the case case of of a == 5.75. very near near states. states. In figure figure 5.7.3, 5.7.3, we consider the 5.75. Let Let us us consider consider two cases cases of of x0 x0 == 0.400 0.400 and x0 x0 == 0.405 0.405 over over 100 100 years. years. It can can be be seen seen that that the the two two behaviors behaviors are are varied varied over over time. time.

17

' Li Liand andYorke Yorke(1975). (1975).

224

5. 5. ECONOMIC BIFURCATIONS AND CHAOS

xt [0.400]

xt [0.405]

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2 2 0 20

4 0 40

660 0

880 0 1100 00

(a) x0 = .400 = 00.400

2 00 200 4400 6 600 8 080 1 100 (b) x0 = .405 = 00.405

Figure 5.7.3: 5.7.3: The dynamics with different initial conditions conditions

We calculate calculate the the difference difference x[t, x[t,0.400] 0.400]—- x[t, x[t, 0.405] 0.405] between between the the two two paths paths started at x0 x0 = = 0.400 and x0 x0 = 0.405 0.405 over 100 100 years. years.

x[t,[t , 00.400] .400]

xAf, [t , 00.405] .405]

11

0.5 0.5

t

20

40

60

80

-0.5 -0.5

-1 Figure Figure 5.7.4: 5.7.4: Small Small differences at the beginning signify much

100

5.7. POPULATION AND ECONOMIC GROWTH GROWTH

225

In growth rate rate a exceeds hi summary, summary, as the autonomous autonomous growth exceeds aa certain certain value, value, the the ceases being being approached approached monotonically, monotonically, and and an anoscillatory oscillatory approach approach steady state ceases occurs. occurs. If aa isisincreased increased further, further, the the steady steady state state becomes becomes unstable unstable and and repels repels nearby points. points.As As aa increases, one can can find fmd aa value nearby increases, one value of aa where where the the system system possesses a cycle cycle of period for arbitrary arbitrary kk (see (see figure figure 5.7.5). 5.7.5). Also, Also, there there exists exists possesses period k for initial conditions conditions from from which which emanate emanate trajectories trajectories that that an uncountable number of initial fluctuate in a bounded and aperiodic fashion and are indistinguishable from fluctuate in a bounded and aperiodic fashion and are indistinguishable aa realization of some stochastic (chaotic) process.

x 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0

a 2

2.5 2.5

3

3.5 3.5

4

4.5 4.5

5

Figure 5.7.5: The map of bifurcations biflircations for for aa Ee [2,5.75] [2,5.75]

5.5 5.5

Chapter 6

Higher dimensional difference difference equations In this chapter, we study the dynamics of difference equations of the form x = /f(x1(t), 2,•.•, n. x.(t *,(* + + ]i)) = W 4 x2(t), i(t\ -•••, , *„('))» ii == 1,1, 2,..., n.

form The system can be expressed in the vector form x(t + l) = = /(*(/)). f(x(t)). x(/

the chapter 6.1 studies phase space space analysis analysis of of We organize organize the chapter as follows. follows. Section Section 6.1 studies phase planar This section depicts behavior of the planar linear difference difference equations. equations. This depicts dynamic dynamic behavior the system when the characteristic system characteristic equation equation has two distinct distinct eigenvalues, or repeated repeated eigenvalues, or complex conjugate eigenvalues, conjugate eigenvalues. eigenvalues. Section Section 6.2 6.2 studies studies autonomous autonomous linear difference equations. This This section section provides provides a procedure linear difference equations. procedure of finding finding general general solutions of the system. Section 6.3 studies nonautonomous linear difference solutions of the system. Section 6.3 studies nonautonomous linear difference equations. This This section section provides provides aa procedure procedure of of finding general solutions solutions of of the equations. finding general system. We We also examine system. examine a few few models models of of economic economic dynamics. dynamics. They include include aa dynamic input-output input-output model model with with time time lag lag in in production, production, aa cobweb cobweb model in two dynamic two interrelated markets, duopoly oligopoly with with 3 firms, and a model duopoly model, a model oligopoly international trade between two countries. This section also shows how the oneof international dimensional difference expressed in in multimultidimensional differenceequation equationofof higher higher order order can can be expressed Section 6.4 defines defines concepts of stabilities stabilities and dimensional equations of first order. Section section also also provides provides conditions conditions for for stability stability relations among these concepts. This section instability of of difference difference equations. equations. Section Section 6.5 studies studies Liapunov's Liapunov's second second or instability method or direct direct method. The theory theory of of Liapunov Liapunov functions functions is aa global global approach approach method 227

6. HIGHER DIMENSIONAL DIFFERENCE EQUATIONS 6. EQUATIONS

228

toward determining asymptotic behavior of ofsolutions. Section 6.6 studies the theory of linearization of difference difference equations. There are aretwo two possible possible ways ways to to simplify linearization of dynamical systems: one is is to totransform transform one complex complex system system to toanother another one one which which is much easier toanalyze; analyze; and and the the other other approach approach is isto toreduce reduce higher higher dimensional dimensional easier to problems to lower helps us to problems to lower ones. ones. The Thecenter center manifold manifold theorem theorem helps toreduce reduce dimensions of dynamical dynamical problems. Section 6.7 6.7defines the theconcept concept of ofconjugacy conjugacy dimensions of and shows the howto toapply apply the thecenter centermanifold manifold theorem. theorem. Section Section 6.8 6.8studies studies the shows how Hénon Henon map, map, demonstrating demonstrating biflircations bifurcations and andchaos chaosofofplanar planardifference differenceequations. equations. Section 6.9 6.9 studies the Neimark-Sacker (Hopf) biftircation. bifurcation. This section identifies identifies Section the Hopf the Hopf bifurcation bifurcation in inthe thediscrete discrete Kaldor Kaldor model. model. Section Section 6.10 6.10introduces introduces the Liapunov numbers numbers and and discusses discusses chaos chaos for for planar planar dynamical dynamical systems. systems.

6.1 Phase space analysis of planar linear 6.1 linear systems systems This section This section examines examines stability stability properties properties of ofsecond-order second-order linear linear autonomous autonomous systems x1(t + l) i) == a11x1(t) + l2x2(t), xl(t + anxl(t)+a

+ l) i) = x2(t + 22x2(t), x2(t + a2lxl(t)+a = a21x1(t)

or

x(t+1)=Ax(t), x(t + l)=Ax(t),

(6.1.1)

If If Ax* Ax* =- x* or

(A_I)x* =0, {A-l)x'=0, x* is is an anequilibrium equilibrium point. point. IfIf(A(A—-1) nonsingular, then is the theonly only i) is nonsingular, then x* x == 0 0 is ofsystem system (6.1.1). (6.1.1).On Onthe theother otherhand, hand,ifif(A(A——i)i) isissingular, singular,then then equilibrium point of there is isaa family of ofequilibrium points. points. In In the the latter latter case, case, let let

y(t)==x(t)— y{t) x{t)-x, we obtain obtain

PHASE SPACE SPACE ANALYSIS ANALYSISOF OFPLANAR PLANARLINEAR LINEARSYSTEMS SYSTEMS 229 6.1. PHASE

y(t+1)=Ay(t). y{t + l)=Ay{t). Thus the stability stability properties properties of an an equilibrium equilibrium point x 5* 00 are the same same as as those of of the the equilibrium equilibrium point point x* = 0. 0. We can thus assume that x* == 00 is the only only those equilibrium point of system (6.1.1). Let

J == P'AP P-'AP be the Jordan form of A. A. Introduce Introduce

y(t) = r1x(t). Then .1) is transformed into Then system system (6.1 (6.1.1) into ] y(t ++ i)l) = Jy(t), y0 yo(=pxo). (= P1x0).

(6.1.2) (6.1.2)

the qualitative qualitative properties properties of systems systems (6.1.1) (6.1.1) and and (6.1.2) (6.1.2) are are identical, identical, itit is As the sufficient for us to be concerned only with with equation equation (6.1.2). (6.1.2).ItItcan canbe beseen seenthat that JJ sufficient for concerned only We examine examine these these cases cases separately. separately. has only three canonical forms. We Case CaselI When A has two distinct eigenvalues, p1 p2 When A has two distinct eigenvalues, pl ^ p2

J = diag[p1, p2]. J = diag[p,, p2]. The solution of system (6.1.2) is given by The solution of system (6.1.2) is given by = 1, 2.

=

Thus Thus ^ 2 0 i=l,2. _ I f 2 Y2(tLI.&')Xa 1

y1Q)

. A j Y10 ^io

We We conclude conclude

=

1

2

230

6. HIGHER HIGHER DIMENSIONAL DIMENSIONAL DIFFERENCE DIFFERENCE EQUATIONS EQUATIONS

1.y2(t)

Jo

>

if It'd different combinations combinationsof of pp1 and p2. Figures 6.1.1 to 7.6.5 illustrate different p2. x and

x2

Figure 6.1.1: 6.1 .1:pp1 c p2 p2 <1, < 1,asymptotically asymptoticallystable stablenode node Figure x < x2

xi

> 1,1, unstable node Figure > p2 p2 > Figure 6.1.2: 6.1.2: pp1 x >

231 6.1. PHASE SPACE SPACE ANALYSIS ANALYSIS OF OF PLANAR PLANARLINEAR LINEARSYSTEMS SYSTEMS 231 x2

x1

Figure 6.1.3: 00< < p1 p{ <1, < 1, p2 p2 >>1,1, saddle saddle (unstable) (unstable) Figure6.1.3:

x2

x1

p2 <1, < 1,asymptotically asymptotically stable stable node node pl == p2 Figure 6.1.4: 00 << p1

232

6. HIGHER DIMENSIONAL DIMENSIONAL DIFFERENCE EQUATIONS x2 I

1

-

\

Figure 6.1.5: p1 = = 1,1, p2 p2
Case 2

A has hasrepeated repeated eigenvalues eigenvalues When A PP = P1 Pi== Pi^ we have

J= J=I

rp P

1

[Opp 0

The solution of system (6.1.2) is given by y1(t) =

y2(t) = p'y20. Thus Thus

lim^ff = = 0. o. (t)

+ tp'y20,

6.1. 6.1. PHASE PHASE SPACE SPACE ANALYSIS ANALYSISOF OFPLANAR PLANARLINEAR LINEARSYSTEMS SYSTEMS 233 We have two cases as illustrated in figures 6.1.6 and 7.1.7.

x2

>1

= p2 p2 << 1,1, asymptotically stable stable node node Figure 6.1.6: 6.1.6: pp1 x =

x2 /

\ /

S /\

—

m

S—

m

*

S

- >

x1

S

/

1'

*

5

S

/

S

S

S

Figure 6.1.7: p2 == 1,1, degenerate case case (unstable) (unstable) 6.1.7: p1 pl == p2

6. HIGHER DIMENSIONAL DIFFERENCE EQUATIONS

234 Case Case 33

When A A has hascomplex complexconjugate conjugate eigenvalues eigenvalues

=o±i/i,

p]2

p*Q,

we have

a

p fi

J-fl -B a a

Suppose £ == [i[l / ] T is the eigenvector eigenvector of of A corresponding corresponding to

pp1 a + tp. =a+i/31 l = may then be given by by In this case, a solution of of system (6.1.2) may + ipy[1J.

y(t) = (a

(6.1.3)

Introduce e(9== tan"1

r = 4a2 + fl2\

£ .

Then this solution can cannow nowbe be written as as

[1]rt(coste y(t)= [1][r(cose y{t)=\ \[r{cosd++isin6)}' isind)]'==\ \r'(cost8++isintO) isintd) Ucost9

Tin

1

,[ 1 = r'\C0St + ir\ &mt \ = u(t) + iv(t), =rI I+lrI \_—sint9J [_cosf#J [—srnt9j [cost9j .

.

.

where

,[costol

u(t)=r u(t)=rI W

[—smt9j

I

, v(t)=rI v(t) = r'\ [cost9 W

[cos^J

(6 L4)

-

6.1. PHASE SPACE SPACE ANALYSIS ANALYSIS OF OFPLANAR PLANARLINEAR LINEARSYSTEMS SYSTEMS 235 235

One can show One show that u(t) u(t) and and v(t) v(t) are aretwo twolinearly linearly independent independent solutions solutions of system (6.1.2). Hence we do not need to consider the solution generated by by p and system solution generated £,. A general solution may then be given by

\y^y\ = a u/t\ L^2(?)J '

I

+ a V/A = 2

J

a cosW + +a a2 \ a1 cost9 2si

sint8

L— a1 sint9+ [-a^sintd +a2cost8 a2cost0j'

values y10, Given initial values yl0, we obtain

—

yM Ly2&)j

—r

cost9 + cost9 + y20 y20 sint9 sintd —y10 sint9 + y20 cost6 ~y w smtd yl0 y10

If we introduce introduce

where

=

/

2

+

2 \1/2

then we have y1(t) = rtr0r'rcos(tO yl{t) = ocos(t0-y), —

y2(t) = —r'r0 y2{t) = -r'rsinQe osin{t0-y). — 4. Using polar coordinates coordinates we we can can now now write write the the solution solution as as

R(t) =

=—(te—y). If rr <1, < 1,we wehave haveananasymptotically asymptoticallystable stablefocus, focus,as asillustrated illustrated in in figure figure 6.1.8. 6.1.8. If rr >>I,1,we wehave havean anunstable unstable focus focus as as shown shown in in figure figure 6.1.9. 6.1.9. When When r == 1,\, we we This case is depicted in figure figure r0. This obtain a center where orbits are circles with with radii radii r0. 6.1.10.

236

6. HIGHER DIMENSIONAL DIFFERENCE EQUATIONS EQUATIONS x2

Pm

Figure Figure 6.1.8: 6.1.8: \p\ <1, < 1,asymptotically asymptotically stable stable focus focus

Figure 6.1.9: 1, unstable focus 6.1.9: |/?| > 1, focus

PHASE SPACE SPACE ANALYSIS ANALYSIS OF OF PLANAR PLANARLINEAR LINEARSYSTEMS SYSTEMS 237 6.1. PHASE 237 x2

x1

Figure 6.1.10: IpI \p\ = = 1,1, center center (stable) (stable)

Example Sketchthe thephase phase space space portrait of Example Sketch

x(t + i) l) = Ax(t), Ax(t), where

[11

A=I A=

1

1

.25 0.25 LO

11

are p1 pl =1.5 and and p2 == 0.5; 0.5; the corresponding eigenvectors The eigenvalues eigenvalues of of A A are are _1)T = (2 If, £=(2 =(2 -If. £=(2 i)T,

Hence Hence

238

6. HIGHER DIMENSIONAL DIMENSIONAL DIFFERENCE DIFFERENCE EQUATIONS EQUATIONS p1.5

P-1AP=J=[

0

0.5

Let x(t) x{t) = Py(t). Py(f). We obtain

y(t + Jy(t) is depicted as in figure 6.1.11. We may The phase space portrait of y(t + i) l) = >/y(/) also describe the the phase phase space space portrait portrait of of the the original original system with x(t) = Py(t), x{t)=Py(t), as in figure figure 6.1.12. 6.1.12.

y1

Figure 6.1.11: Saddle of the canonical canonical form, form, y(t ++ l) 1) = Jy(t) Jy(t)

239 6.1. PHASE SPACE SPACE ANALYSIS ANALYSIS OF OF PLANAR PLANARLINEAR LiNEARSYSTEMS SYSTEMS 239 x2

x1

l) = Ax(t) Ax(t) Figure 6.1.12: Saddle of the original original system, system, x(t ++ i)

Exercise 6.1 6.1 Exercise Depictthe thephase phasespace spacediagram diagram and and determine determine the the stability stability of of the equation equation I1Depict x(t + l) =Ax(t), Ax{t), x(t +i)= A isisgiven given by where A ro0.5 .5

(i)

0

L o0

(ii) (ii)

-0.5 [—0.5

°0

ri1 L

(iii) (iii)

0.5J' o.sJ'

-0.5 L—o.5

'1

1

-0.5 —O.5j' 0.5 0.5 11

240

6. HIGHER DIMENSIONAL DIFFERENCE EQUATIONS

6.2

difference equations Autonomous linear difference

now study the the following following linear linear homogenous homogenous difference difference equations equations We now x(t + + i)= l)=Ax(t), Ax(t),

(6.2.1)

where

/\

I

f\ I\

f\\T

A isis aan nxn realnonsingular nonsingular matrix. matrix. A A solution solution of of system system (6.2.1) (6.2.1) is is an an A x n real satisfies this system system for for all all /it >0.0.AAgeneral generalsolution solutionisisaasolution solution expression that satisfies the system. A particular solution isisone onethat that satisfies satisfies that contains all solutions of the particular solution x0 == x(t0). x(t0). The problem problem of fmding finding aa particular particular solution solution with with initial condition condition x0 an initial and

specified specified initial initial conditions conditions is called called an initial value problem. problem. It can be seen that that the the solution of system (6.2.1) has the form has the x{t) x(t) = A'-'°x0. Theorem 6.2.1. There exists a fundamental set, set, denoted by {x1(t), x2(t), ..., n{t)l {x x(t),x 2(t),-,x

for system (6.2.1). A A general solution is is given given by by of solutions for

x(t)=Ec,X,(t), c1eR, i=1,",n. ( o ! > , , ( ) , J > ' c

e R

/ = 1

Along with the the homogeneous nonhomogeneous system

system

(6.2.1), (6.2.1),

we we

x{t + \)=Ax{t)+B{t), x(0) = = x0, x 0 , tf == 00,, 1, l , ... -. x(t + i) = Ax(t) + B(t), x(o) The initial value problem (6.2.2) has has a unique solution given given by

x(t) = A'x3 +

t = 0,

consider

the

(6.2.2)

6.2. AUTONOMOUS LINEAR DIFFERENCE EQUATIONS

241 241

We see see that the main problem is to calculate calculate At. A'. There There are are some some algorithms algorithms 1 for computing At. A'. Here, Here,we weintroduce introduce the thePutzer Putzeralgorithm.' algorithm. Let Letthe thecharacteristic characteristic equation of of A A be equation

a1p"' =

0,

where 1. Let where a0 a0 = 1. PP. As.'". P2' A A..

the eigenvalues eigenvalues of A A (some (some of ofthem them may maybe berepeated). repeated). The The following following formula formula determines AA' be

A' =z,1(t where

M(O)=J, M{0) = l, M(k)=JTJ(A —p1i),

u1(t) =

Example Example Solve X(t-

+Hi)) = Ax(t), AX(t)

where

412 A=0 2—4. 016 1

"4

1

2

A= 0

2

-4

0

1

6

The algorithm algorithm isis referred referred to to section section 3.1 3.1 in inElaydi Elaydi(1999). (1999).

6. 6. HIGHER DIMENSIONAL DIMENSIONAL DIFFERENCE DIFFERENCE EQUATIONS EQUATIONS

242

The three eigenvalues eigenvaluesof of A A are 4 A = P2 Pi = = Pi == 4. -

Hence, Hence, we have

M(0)=/, M(O)= i,

011 "o

2"

M(\)=A-4I = 0 —2 I —4 -4 M(1)=A—41=

011

0

3

22

000

"0 0 0" M(2) = (A- 4l)h ) = 0 0 0 M(2)=(A—41)M(1)= 0 0 0

000

u1(t)= «,(') = 4',

t4''

u2(t)=

41-1-1(/41)

u3(t)=

t(t

041-2

the above above calculation calculation results results to to Applying the

A'

J(t)M{j-\),

get we get

4' A' = 00 A' 0

t4'' t4'

—t4' -t4'

l

4' + 2n4'' 4'

4' 4'-2t4'~

Consequently

x(t) = A'x3 x{t)=A'x 0 the solution. is the

2t4'' 2t4' l

t4'~

6.2. AUTONOMOUS LINEAR DIFFERENCE EQUATIONS

243 243

now apply apply the the Jordan Jordan to to solve solve system system (6.2.1). (6.2.1). First, we consider the case We now A isissimilar similarto tothe the diagonal diagonal matrix that A

D == diag[p,], diag\p\ p, are arethe theeigenvalues eigenvalues of ofA.2 A.2 That That is, is, there there exists a non-singular non-singular matrix where p. matrix P such that

P'AP D. P^AP = = D. From From AP AP == PD, we have have A4t1

=

of P. We is the eigenvector where 4t, £ is is the the ith column column of We see see that that 4t, £ is eigenvector of A A corresponding to to the the eigenvalue eigenvalue pp1. corresponding i. From

P1AP P~XAP = =D and PP being beingnon-singular, non-singular, we we have = PDP~\ PDP'. A= Consequently, we we have = Pdiaglp'^-'. A' == PD'P1 PD'P-' = A'

(6.2.3)

Substituting equation (6.2.3) into into

x(t) = x{t)=A'-'°x 0,

with tt00 -= 00 yields the general solution x(t) = Pdiag[p,' }P'x0. 0. 22

(6.2.4) (6.2.4)

A.1 for for an introduction to matrix matrixtheory. theory.Itit isis known knownthat that aa nxn appendix A.I introduction to nxn Refer to appendix diagonalizable if and linearly independent independent eigenvectors. eigenvectors. matrix is diagonalizable and only only ifif itit has has n linearly

244

6. HIGHER DIMENSIONAL DIFFERENCE EQUATIONS EQUATIONS

As Pdiag\p',]=

[/?,£, p 2 % 2 ,

...,,

pjn],

the general solution (6.2.4) can can also also be be expressed expressed by by x(t) x{t) == a . p f t + a2p&

+ anp'Jn,

(6.2.5)

where a = P1x0. P~lx0.

After After having having calculated calculated the eigenvalues eigenvalues and and eigenvectors, eigenvectors, we may may directly directly determine aa by byequation equation(6.2.5) (6.2.5)through through the theinitial initialconditions conditions without without calculating calculating

Example Find Example Find the the general solution and the initial value problem of x(t+l)=Ax(t) x{t + l) = Ax(t)

221 2 2 1 0 A= A = 11 3 1 , x(0)= *(o) = 1 122 1 2 2 0 0

3

1

,

1

0

The three eigenvalues eigenvalues of of matrix matrix A A are are p1 = 55,, p2 1. A A == pp33 == 1. Correspondingly, we we can can fmd find three three linearly linearly independent independentvectors3 vectors3 11 1 1 , e2— 1, £ = 00 11 —1 -1

00

1

e=

3

,

1

1

—2 -2

It It should should be be noted noted that ifif the the matrix matrix has has repeated repeated roots, then it is is diagonalizable diagonalizable if it is is T normal, that that isistotosay, say,ififALA normal, A == AAT.

6.2. AUTONOMOUS LINEAR LINEAR DIFFERENCE DIFFERENCE EQUATIONS EQUATIONS

245

ItIt should should be noted noted that that there there are are infmite infinite choices choices for for £2 and £, because because of multiplicity of the corresponding eigenvalues. The general solution is a15' +

x(t)=

+

+

a2 a3

a15' — 2a3 a2 — 2a3 al 5' —- £12

The solution solution of the initial initial value value problem problem is is solved solved by by substituting substituting the the initial initial condition x0 and then then solving solving a,. a,. We into the the above equation and We calculate calculate x0 into 1

11

1

2

11

2'

Hence

i,

1, ~

1

51 -1 —i 5' 5 +1 5' 5t —1 -1 5'

+i.

A'x00 and and equation (6.2.3) to solve the initial We may also equation (6.2.3) initial value value also use use x(t) x(t) = = A'x problem. We get the same solution by calculating calculating

1 1 1 0 1 -1

0" 1 =>P~] = 2

11 4 3 4 4

11 ~2 1

11 1

~2" ~4 1 1 2

H 4

reader isis asked asked to to check check the the result. result. The reader A may may not not be bediagonalizable diagonalizable when when AA has hasrepeated repeated eigenvalues. eigenvalues. The matrix A There is something something close close to to diagonal diagonal form form called called the Jordan Jordan canonical canonical form of of aa is expressed square matrix. matrix. A A basic basicJordan Jordan block associated associatedwith withaavalue valuepp is

246

EQUATIONS 6. HIGHER DIMENSIONAL DIFFERENCE EQUATIONS

p1 0 ... o p 1 0•••0 0 p 1 ... o op1...0

00 o0

J= 0 0

0 0

0 ... p 1 0 ... o p

0 000

The Jordan Jordan canonical canonical form form of aa square square matrix matrix isis compromised compromised of of such such Jordan Jordan blocks. blocks. 6.2.2. (the (the Jordan Jordan canonical canonical form) matrix A A is is similar similar to aa Theorem 6.2.2. form) Any Any nxn nx n matrix form given by Jordan form J = diag[j^,

J2,

Jk], \ < k < n ,

basic Jordan Jordan block and where each .1, J, is an stxs, x s, basic >

s == fl. n. S

i=1 i L-H=\ i

l

I

A is is similar similar totoJ.J under AP = = J. We have have Assume that that A underP, F, i.e., i.e., P~ P1AP = FJP'. PJP~l. A= Hence,

A' PJ'P~\ A' == PJ'P1. It can be seen that J' =diag[j{,Jt2,--,J[\

\
We can as can write write J, as =

+

—00 for for all all kk Si, > s,,,we wehave have where N1 Nt is an s, x st nilpotent matrix. matrix. Using Using N* NJ' =

6.2. AUTONOMOUS LINEAR DIFFERENCE EQUATIONS EQUATIONS

« , V %r /

t-2

Pi

00 k

p\

.

\p'r-+l

[Jp;-i

t

[Si

=(p11+N,)' = (A'

/

[St

/

Lit 0 0

00

p/i A;

The general solution of equation (6.2.1) (for t0 t0 == 00 ) is now given by = PJ'P1x0 PJ'P~]x0 = = PJ'a, PJ'a, a = P'x0. P'lx0. x(t) = A'x A'x0 0 = Corollary 6.2.1. Assume that A A is is any nx « x n« matrix. matrix. Then Then Corollary 6.2.1. =0, limA'' =0, I—

p\ < for all of A. A. ifif and only if jpl c 1 for all eigenvalues eigenvaluespp of

Exercise 6.2 6.2 1 Use Use the the Putzer Putzer algorithm algorithm to evaluate A' A' 'ri1

(i) (i) A=[2 -2 1

11

4 ;

J

2 11

(ii)A= A = 00 (ii) 4 -4

-1 —1 0

4—455

22 Solve the following systems with the Putzer algorithm xl{t + i)= l) = —x1(t)+ -xl(t)+x x1(t xjt), 2{t), W

x (t ++ i)= 1) = 2x2(t), 2x2(t), x1(0)=1, xx(0) = 1, x2(0)=2; x2(0) = 2;

x2(t 2

(ii) Ax(t) where (ii) x(t x(t + 1) l) = V4JC(*)

c

4."?

247 247

6. HIGHER DIMENSIONAL DIFFERENCE EQUATIONS EQUATIONS

248

-2 - 2 —2 1 —2

1

A= 00

00

0 2 02

"ir —1,x0= 1. 1

_ j

33

00

3 Use Use formula formula (6.1.5) (6.1.5) to find the solution solution of of x(t + i)l) == Ax(t) 2 3 0" 230 "o" (i)A= 4 4 3 1 0 0, x(0)= 1; = 006 0 0 6 0 0

3

0

1 o r 101 1 22 33 0 0 3

(ii) (ii)

1

003

,

'0" 0 x(0)= (o) = 00 11

Nonautonomous linear difference difference equations equations

6.3

This section nonautonoinous homogenous section fir firstst examines the nonautonomous homogenousc4fference difference equations equations

x(t + = A(t)x(t), + i) l)=A(t)x{t),

(6.3.1) (6.3.1)

where where T x2(t), ..., xn(tj)T x(t) x(t) == (x1(t), (Xl{t),x 2{t),-,x n{t))

and x n nonsingular and A(t) A(t) isisaannxn nonsingularmatrix matrixfunction. function. Theorem 6.3.1. 6.3.1. For For each each x0 x0 es R" and t0t0 e Z + there there exists exists aa unique unique solution solution Theorem x(t, x0) x0) of ofequation equation(6.1.1) (6.1.1)satisfying satisfying the the initial initial condition. condition.

Let Xx{t), x2(t), X2(t), ..., xjt) Xn(t) be nnsolutions solutionsofofequation equation(6.3.1). (6.3.1). They They are are said said Let x1(t), to they are are linearly lthearlyindependent independentfor forallallt t>0.0.Otherwise, to be linearly linearly independent independent if they Otherwise, they they are are said said to to be be linearly linearlydependent. dependent. A A set set

{x1(t), x2(t), ..., xjt)}, {Xxif),X 2{f),-,X n{f%

6.3. NONAUTONOMOUS LINEAR DIFFERENCE EQUATIONS

249 249

of any nn linearly linearly independent independent solutions solutions is said to be aafundamental fundamental set set of of solutions, and the matrix

x(t) =

.x2(t), •..,

fundamental matrix. is called a fundamental matrix. We We have have 0

as x1 (t), as X^t),

x2 (it), ..., Xn(t)(it) are linearly linearly independent. independent.As Aseach each Xx.t{t)(it) satisfies X2(t), satisfies equation (6.3.1), we have

x(t A(t)X(t). X(t + i) l) == A{t)x{t).

(6.3.2) (6.3.2)

In fact, the above equation has has aa unique solution. solution.

Theorem 6.3.2. 6.3.2. There is aa unique unique solution solution X(t) x(t) ofofequation equation (6.3.2) (6.3.2) with with X(to) = I. The The general general solution solution of of equation equation (6.3.1) (6.3.1) isis given given by by

x(t) X{t)c, x{t) == X(t)c, where cc ee R'1. (6.3.1) is R". The The nonhomogeneous nonhomogeneous system system corresponding corresponding to equation equation (6.3.1) given by

x(t + g(t), A{t)x{t)+g{t), x{t + + i) \) == A(t)x(t) where A(t) and g(t) A(f) isisaa nx nxnn nonsingular nonsingularmatrix matrix function function and g(t)se R". Theorem6.3.3. 6.3.3.Any Any solution solution x(t) x(f)ofofequation equation(6.3.3) (6.3.3)can canbe be written written as Theorem x(t) = x(t)c + x{t)=x{t)c

x(t),

(6.3.3) (6.3.3)

250

6. 6. HIGHER DIMENSIONAL DIFFERENCE DIFFERENCE EQUATIONS EQUATIONS

X(t) isisa aftmdamental fundamental matrix matrixofofthe thecorresponding corresponding homogeneous homogeneous system system where X(t) (6.3.1) (t) isis a particular particular solution solution of of equation equation (6.3.3) (6.3.3) for for an anappropriate appropriate (6.3.1) and and x,, xp(t) choice of of the the constant constant vector c.c. choice

Example (a with time time lag in (a dynamic dynamic input-output input-output with in production) production) There There are two two industries in the the economy. economy. Each Each industry industry produces produces only only one onehomogenous homogenous industries commodity; each input ratio for the the production production of its its commodity; each industry industryuses uses aa fixed fixed input ratio for output; production production in industry is subject subject to constant constant return return to scale. scale. In In output; in every every industry order to produce produce each commodity, the input input for for the the i/ th th order each unit unit of of the jj ththcommodity, (t) indicate the fmal commodity is fixed amount, amount, denoted denoted by by atJ. Let d.dt{f) final commodity is aa fixed demand for the / th commodity in t. Assume that there is a one-period lag in there is a one-period lag in demand i th commodity t. Assume demanded in period determines not the the current current production so that the amount demanded period t determines denote the output output of the i/ th output but the output output of of period period (t(/ + i). l). Let xxt{t) (t) denote output Then, the input-output input-output model is is represented by industry. Then, a1x1(t)+ a,2x2(t)+ d,(t). x(t x,{t ++ i) =l)=a nxl{t)+a i2x2(t)+d l{t).

form, we we have have In matrix form,

x(t ++ i) l) = Ax(t) Ax{t) ++ d{t), where

x=[x1 * = [*i x2f, x2f, A=[a4L2. A = [atJ\2><2. Here, we specify

d(t) = [a' d{t)=k

s>\ a'I,

where 58isisa apositive positivescalar. scalar.We Wenow nowneed needtotodetermine determinethe theparticular particularintegrals. integrals. Following the method of of undetermined coefficients, we try solutions of the form form —

x2{t)\ [x2(t)j

—

[b18'l [b [b2a'j' 2S'

where kbtare areunknown unknownparameters parameters to tobe bedetermined. determined. Substituting Substituting this this into into the the input-output model yields

6.3. NONAUTONOMOUS LINEAR DIFFERENCE EQUATIONS =

[b1

[b2j

[a11

a12 1[bi

[a21

a22J[b2j

+

[lJ

251 251

S'.

- A A is nonsingular, we we solve the particular integral as follows follows If II—

*,(')' [xi(t)1

[8—a11 S-a,,

Lx2(t)J

[ -a,

—a12 -al2

aflj an.

S8 — —

[1j

S'.

It is straightforward fmd the general general solution solution of of x(t straightforward to find x{t ++ 1)l) == Ax(t). Ax{t).

(A cobweb cobweb model model in intwo twointerrelated interrelated markets) markets) We We now nowconsider consider aa Example (A 4 with time lags.4 Let subscripts model of two corn and hog markets of two hog - markets lags. Let subscripts 1 and and 22 denote corn corn and hog, variables Xj and andV1 Yj denote variables pj p1 the price price of goods goods jj, , variables X1 stand for the demand demand and supply stand supply of of goods goods jj,, respectively. The corn market is described by

b,l>0, >0, Xl{t)=al-b,pl{t), b

ir,(t)= 1), d1 >0, (/) = c1 ++ d1p1(t rf,/»1(f-i), dx>o, c1

—

x1(t)= X]{t)=Y]y(t). {t). the market equilibrium equilibrium condition condition From the

x1(t)= x{t), X,{t)=Y we have

_a1—c1 _fpi(t_l). The hog hog market is is described by

X2{t)=a2-b—b2p2(t), b2>Q, X2(t)=a2 b2 >0, 2p2(t),

Y2(t)=c2 +d2p2(t—l)+ep1(t—l), Y2 (/) = c2 + d2p2(t - l) + ePl{t - l), dd2 0, eecO, < 0, 2 >>0, 4 Anearly early example example of ofthe the corn-hog corn-hog cycle cycle was wasgiven given by byEzekiel Ezekiel (1938) (1938)and andWaugh Waugh(1964). (1964). "An This version of of the model comes comes from from Shone Shone (2002: (2002: 346-348). 346-348).

_________

6. HIGHER DIMENSIONAL DIFFERENCE EQUATIONS EQUATIONS

252

.xjt) X2(t) = = Y2(t). From the market condition

xjt)= X 2(t)=Y2(t), we have have

p2(t) =

a2—c2 —

d2 —pjt

e

—

i) — —p1(t — i).

The system The system consists consists of of two-dimensional two-dimensional difference difference equations equations for for p1px(t) {t) and P p22(t). (t). The fixed point is determined by *

aL-cL _ a1—c1

d1

p1=~ P\ =

A

b1P1

a—c — c, 2

2

aa \ ~ c\ ** _=_ a1—c1

*

d

e

*

=

*

2

x1(t)= p1(t)— 12;.

Then the dynamics are given by

[xj(t+1)1

[x2(i+I)f

o°

e e

d. b2

The two eigenvalues are

d.

j=1,2.

* 2

Introduce

- Ad1

T +4

rxi(t)

a—c 2

2

b2+d2

e

b2+d2

*

LINEAR DIFFERENCE DIFFERENCE EQUATIONS EQUATIONS 253 6.3. NONAUTONOMOUS LINEAR 253 condition for for stability stability in inthe thecorn corn market market isisId1 \dx Ib^ 1. The Thecondition condition for for The condition lb1 < c 1. in the hog market is Id2 \d2I lb2 b2\ <1. < 1. stability in Example Example (duopoly (duopoly model) We We consider aa duopoly model. model. There There are are two two firms, firms, 11 2,producing producingaasingle singlecommodity. commodity. The The demand demand ftxnction function in in the market is and 2,

p{t) = 9-Q{t), p(t)=9—Q(t),

where pp isisthe and Q theprice priceof ofthe the commodity commodity and Q is is the total supply supply Q(t)= q1(t)+ q2(t), Q(t)=q 1{t)+qM is Suppose that that firm firm j 's's total cost where q1Q) q^t) is firm firm j 's 's output. output. Suppose cost is where TC1(q1)=

assume that in intime time itt aa firm firm chooses its itsown own output output atattime timeitt so so as as toto We assume thatthe the rival rival of the the firm maximize profits at t under the maximize its profits the assumption assumption that firm will will choose the same choose the same output output level level it chose chose in time time t/ — - 1.1. Profits (= (= revenue revenue minus given by cost) are given by (9— q1(t)— q2(t

—

*2 (0 == (9 (9 —- q1(t
aq, (it)

=9

—

q1(t

—

i)

—

2q2(t)

—

6q2(t) = 0.

dci2(t)

The Cournot-Nash by Cournot-Nash solution solution isisthus thus given given by 1

(it—i),

254

6. HIGHER DIMENSIONAL DIFFERENCE EQUATIONS EQUATIONS

91 is The unique fixed point is (q;, q;)= (1, 1).

Solving the difference difference equations for for initial initial values values (q13, \q]0, qq20), we obtain obtain 20), we

i( ii

( iY (

q2(t)= 1— ft(0=l-|~)

+

i( iY

^"(-^

(q10

(q10

+ q20)+

+

+

The system converges to the equilibrium point. Example (oligopoly with with 3 finns) firms) We We consider consider a oligopoly oligopoly model with three three Example (oligopoly and 3, 3, producing producing a single single commodity. commodity. The The demand demand function ftinction in the firms, 1,1, 2, and market is

= 9-Q{t), p(t)=9—Q(t), where pp isisthe theprice price of of the the commodity commodity and the total supply and Q isis the

Q(t)= q1(t)+ q2(t)+ q3(t), where q1(t) q^t) is firm firm jj 's output. where output. Suppose Suppose firm firm jj 's total cost is 2 . TCj{gj)=3q3q2.

We firm chooses chooses its own so as to We assume assume that that in in time time t aa firm own output output at time t so thatthe the rivals rivals of of the firm will maximize assumption that will maximize its its profits profits at at t under the assumption Profits (= revenue minus choose the the same output level level itit chose choose same output chose in time t -— 1. 1. Profits cost) are given

6.3. NONAUTONOMOUS LINEAR DIFFERENCE DIFFERENCE EQUATIONS EQUATIONS

255 255

7r1(t) = (9 *i (0 = (9 — - q1(t) <7i (') —- q2(t 92 (' —- 1)— 1) - q3(t 93 (' —- 1))q1(t) 1) h (0 —- 3q1 3?,(t), (0, 2r2(t) * 2 (') == (9 (9 —- q1@ 9i(' —" 1)— 1)" q2(t) 9 2 (0 —" 9 3 (' —- 1))q2(t) !))
= (9 .7r3(t) ^3 (0 = (9

<72 ( '—" Ii)) — q3(t))q3(t) - q1(t 9i (' — " i) 1) — - q2(t

—

—

3q2(t).

3

=

Differentiate the profit functions Differentiate functions (t)

9—

— 2q2 (t

—

i) —

W = 99_— q1(t (/_—l}i)_—22q2(t) (/)_

—

0

q3

(t

—

i) —

0,

(r_—l)_ 3 =3 = 0, 1)—

dq2 (t)

=9 — 3q3

q1(t —i)— q2Q —1)—2q3(t)—3 = 0.

(t)

The Cournot-Nash solution is thus given by 1

i)

—

—

1

1

1

1

i),

-..q1(t —I)——q3(t—1), i). = 3 - ^-^(r - 1) - -|qr2(r —- 1).

The unique fixed point is is

(3 33 . *. *\ ,\ (3 Wi' q2, q3j= equations for for initial initial values values (q10, [ql0, q20, q20, q30), we obtain Solving the difference equations

1(I)' 2

+q20 3

+q20+q30 2

3

— q20

—

)'

3 q10 + 2q20 3

—

256

6. HIGHER DIMENSIONAL DIFFERENCE EQUATIONS

The system does not converge to the equilibrium point. It is important to note that the duopoly dynamics with with the same same demand demand and and total cost is stable; if the number of firms total cost functions functions is stable; if number of firms is larger larger than than 3, then then dynamics is unstable. Theorem 6.3.4. 6.3.4.(variation (variation of ofconstant constant formula) formula) The The unique unique solution solution of of the the initial initial Theorem problem x(t + l) = A{t)x(t) + g(t), x{t x(t0)=x0, o) = x0, is given by

(t,x )Jf[A(i)\ + [ilAo)]xo Xo )

x(t, X o =

r=t\i=r+\ i=r+1

hi particular, constant, we have particular, when when A isisconstant, x(t, x0) =

At_tax0

+ r=t0

Example Apply Apply theorem theorem 6.3.4 6.3.4 to to solve solve Example

x0, + 1)l)=Ax(t)+g(t), x(0) == x0, = Ax(t) + g(t), x(O) x(t + where

[22 111 0 2

A= A=[

[ti

[1

gQ)=[11, x(O)=[0

Using the Putzer algorithm, we obtain

At = [21 A' =

[00

t2'1 2t 2'

LINEAR DIFFERENCE DIFFERENCE EQUATIONS EQUATIONS 257 6.3. NONAUTONOMOUS LINEAR 257 Hence

2' x(t) = [2'

tf'-'lM+ ^[2'-^ t2thl[tl 2'>-'j[1j [0 ' JUJ r=o[ ^ L o0

=

2' + t2'1

—

r 2 {t-r-l)Z'- Jr r — (t — 21_)_1

4.

2' —1

(check the the last last step). step).

Example Consider thetrade tradebetween between two two countries, countries, i = = 1,1, 2. 2. For Forcountry country i, z, we we Considerthe have have

i. + .x1(t)— *11(t), 1'(t)= Yi{t) = c,(t)+ Ci{t)+I,+X i{t)-M,{t),

(6.3.4) (6.3.4)

are respectively respectively national national income income (in I,, x, (t), M1 where yY^t), (t), C. Ct(t), (t\ /,-, ^(f), M ; (/)(t)are (inperiod period t), total consumption, (fixed) net investment, exports, and imports. According consumption, (fixed) net investment, exports, and imports. According t), total consumption of ofdomestic domestic products definitions, country to the defmitions, country i/ 's'sconsumption — D,(t) />,(*)== c(t) C,(0-A/,(r).

thedomestic domestic consumption, andimports, imports,M, M(t), ofeach each Assume that the consumption, Dj (t), and t (t),of

are proportional to the the country's national one at period period t/ ++1 are proportional to national income income one country at country 1

That is time period earlier. That is D1(t)= a11Y(t), M1(t) M1{t) == a21Yj(t), a21Y1{t), i \

i \

D2{t)=a22Y2{t), M M2(t)= 2(t)=aa12Y2(t). nY2{t).

(63 5) (6.3.5)

consists of ofthe two two countries, countries, each country's country's exports exports equals equals the theother other As the world consists country's imports, imports, i.e., i.e., X1 Xx == M andX2 X2 ==M1 M, in t.t. Substituting Substituting country's M22 and D, —= C. C, —- Al,, A/,, Xl == M2, X2 == M, into equation equation (6.3.4) (6.3.4) yields the the global global economic economic model model and equation (6.3.5) into

6. 6. HIGHER DIMENSIONAL DIFFERENCE DIFFERENCE EQUATIONS EQUATIONS

258

[all

[Yj(t+i)1

+ i)J - [a21 a21

a \[Y2{t)\

a22 22

['I

[I2\ + L'2

By the variation of constant formula formula we solve Y(t) = A')'Q + dA'l\ Y(t)=A%+f

(6.3.6) (6.3.6)

1)T

Itcanbeshownthatif can be shown that if where /* = (/,, I2 )T. It an + a21 <<1,1, a12 al2 + a22 < a11 + a21 + a22 c 1,1,

(6.3.7)

then for all allthe theeigenvalues eigenvaluespp of Ipi < c i.5 of A, A, \p\ 1.5Hence, Hence, under under conditions conditions (6.3.7)

limA' 0. lim^' ==0. /-*oo to the so-called Neumann's expansion, On the other hand, according to expansion, we we have have

=

(i

—

Hence

limY(t) = (i

—

equations of higher order. It examined one-dimensional one-dimensional difference difference equations We have examined can be be shown shown that that aaone-dimensional one-dimensional difference difference equation equation of higher higher order order can canbe be expressed in inmulti-dimensional equations equations of offirst first order. order. informs forms Let us consider difference equations in

5

condition implies implies that the thesum sumof ofdomestic domestic consumption consumption D. £>.(? l) and andimports imports This condition (t + i) i.e. M(t less than thanthe thenational nationalproduct productY^t) i(t) of period t/ ++11 is less of period period t,t, i.e. Mf(t ++i)l)ininperiod

D(t +

M(t + i)c v,(t).

259 6.3. 6.3. NONAUTONOMOUS L[NEAR LINEAR DIFFERENCE DIFFERENCE EQUATIONS EQUATIONS 259

x(t ++i)l)=f{x{t),x{t-l)). = f(x(t), x(t — i)). x(t

(6.3.8) (6.3.8)

equation the the state state of ofthe the orbit orbit atattime timeitt ++11depends depends directly directly from from its itsstate state at at In this equation time tt and and tt — - 1.1. We Wecall call equation equation (6.3.8) (6.3.8) aa discrete discrete dynamical dynamical system system with with a time delay of of aa one-unit one-unit unit. unit. We can eliminate the delay by increasing the dimension of by of y(t ++ i)l) == x(t). We We can can rewrite rewrite equation equation (6.3.8) (6.3.8) as asfollows follows the system. Set Set yQ = f(x(t), f{x{t),y{t% x{t ++ l) 1) = y(t)), x(t y(t + l) = x{t). y(t+1)=x(t). We might might express express the the above above system system in invector vector form form We

z(t + i) = where where (x,y), = (f(x, {f(x, y), z ==(x, y), x). x). y), g = complicated cases are are also possible; but we can make similar transformations. More complicated instance, consider For instance, x{t++I) =l)=ax{t){\-x(t-2)). ax(t)(l — x(t — 2)). x(t The dynamical system contains contains a delay delay of two-time two-time units. units. We can can replace replace it it with with aa system with no delay. Set Set three-dimensional system y{t + l) = x{t), z(t + \) = y{t). We obtain

x{t + \)=ax{t){\-z{t)), y{t + \) = x{t), y(t+1)=x(t), + O=y(t). The above example can be generalized. Assume x(t + + i) l) == f(x(t), f{x{t),x{t-l\x{t-2)). x(t — i), x(t — 2)). x(t

260

6. HIGHER DIMENSIONAL DIFFERENCE EQUATIONS

and z(t + i) l) = y(t). >>(?). We obtain 1) = x(t) and Set y(t + l) x(t +

1)

= f(x(t), y(t), z(t)),

y(t+i)=x(t), z(t + l) = y{t). l)=y(t). We can also write the system in vector form form w(t

+ 0= g(w(t)).

Hence, always assume assume that x(t + +1) depends directly x{t). Hence, we we can always i) depends directly only only on x(t). Systems l) depends Systems with with delay, delay, i.e., i.e., when when x(t x(t + i) depends directly directly on on one one or more states of form x(t x(t—- k), k > arereplaced replacedbybyhigher-dimensional higher-dimensionalsystems systemswith withno nodelay. delay. 1,1,are the form

Exercise 6.3 6.3 Solvethe thedifference difference equations equations 1 Solve x1(t + i) = 2x1(t) + 3x2(t) +

1,

x2(t ++ l) xl{t)+4x -l l)== x1(t)+ x2(o)= x2(t 4x2(t), 2(t), x1(0) = 0, * 2(0) =—1. 2 Let Let

x(t + i) = ax(t

—

i)(i — x(t — 2)).

Replace this one-dimensional one-dimensional system three-dimensional system Replace system with with a three-dimensional system with with no delay. delay. Develop aamathematical mathematical model model for for aaforeign foreign trade tradeamong amongthree threecountries countries using using 3 Develop an argument similar in the the example example of of trade trade model model between between two an argument similar to to that that used used in countries in section 6.2. 4 Let Let

0.2x(t —- l)y(t \)y{t —-1), x{t ++ l) i) = 2x(t) -— O.2x(t i), l) = y(t ++ 1) = y(t) y{t)+OAx(t-\)y(t-l). + O.lx(t — l)y(t — 1).

6.3. NONAUTONOMOUS LINEAR DIFFERENCE DIFFERENCE EQUATIONS EQUATIONS

261 261

Replace the the two-dimensional system having aa delay of of aa one-time one-time unit unit with with aa fourfourno delay. dimensional system with no tosolve solvethe thefollowing following difference difference equations equations 5S Apply Apply theorem theorem 2.6.4 2.6.4to (i) x(t ++ 2) 2)++ 8x(t &x(t++1)l)++12x(t) I2x(t) == 0; 0; 5x(t + i) (ii) x(t x{t++2)2)— - 5x{t l) ++ 4x(t) 4x{t)== 4'. 4'.

Stabilities

6.4

We now study the thevector vector difference difference equation

x(t = f(x(t), t), x(t0) = x0, x{t + + i) l)=f(x{t),t), x{to)=x o,

(6.4.1)

where

x(t)e R" x(t)eR",

f: f:R"xZ+-^R".

We assume assume f/ isiscontinuous continuousininx.x.Equation Equation (6.4.1) (6.4.1) isissaid said to tobe beautonomous autonomous or time-invariant ifif the the variable variable itt does does not notappear appear explicitly explicitly in in the theright-hand right-hand side side of time-invariant the equation

f(x(t), t) == f(x(t)). f(x(t),t) f(x(t)). said to to be beperiodic periodic if if for forall all ne n E Z, It is said

f(x, t ++N) = f(x,t N)=f(x), for some some positive positive integer. integer.AApoint pointx" i ininR'1 pointof of R" isissaid said to to be bean anequilibrium equilibrium point equation (6.4.1) (6.4.1) ifif

f(x*, x, f(x*,t)=x', for all t > t0.t0We can always the origin. If x* is not the the origin, let . We can alwaysassume assume x* to to be be the is not

262

EQUATIONS 6. HIGHER HIGHER DIMENSIONAL DIMENSIONAL DIFFERENCE EQUATIONS — y{t) = x(t)x(t)-x. y(t) =

Then equation (6.4.1) becomes

y(t ++ i) 1)== f(y(t) f{y(t) + +x',t)-xt= x = g(y{t\ y(t g(y(t), t).

(6.4.2) (6.4.2)

corresponds to now introduce introduce some some stability stability Notice Notice that that y^ = 0 corresponds to x == x*. We now notations of x*.6 Definition 6.4.1. 6.4.1. (i) (i) The Theequilibrium equilibrium point point x* of equation equation (6.4.1) said to be be Definition (6.4.1) is is said stable ififgiven > 0 and and t0 t0 >0 0there thereexists exists 88== S(e, such that stable givense> 8(e, tt0)> 0) > 0 such

implies

jx{t,x o)-x'j<e xl <S Hx(t, xe)— uniformly stable stable if 8 may maybe bechosen chosen independent independent of unstable if for all t > tt0, of tt0, if 0, unjformly 0, unstable 7 is not not stable.7 stable. it is said to ju(t0) such that pc0 — — xx* < ju (ii) x* is is said to be be attractive attractive ifif there thereexists exists jU p = p(t0) H

implies x0) = x*, iiinx(t,?, x0) t

uniformly attractive attractive if the independent of of t0. t0. unjformly the choice choice of of fi p isisindependent

66 7

notations are aredefined defined in insection section 2.3. 2.3. ItIt should be noted that some notations symbol || || means means the the Euclidean norm in in this book. x, The symbol Euclidean norm book. For For a vector x, H

=

= (1412, IMhlS,-*. )' ' for for a matrix A, A,114 \A\\= 2 2

\P{ATA)]'\

P{A) = Mzr{|A|, 2Xis is an an eigenvalueof eigenvalue ofA}. A]. p(A)

where

6.4. 6.4. STABILITIES STABILITIES

263 263

(iii) x* is said said to be be asymptotically asymptoticallystable stable ifif itit isis stable stable and andattractive, attractive, and andun(formly uniformly (iii) asymptotically stable ifif itit is uniformly stable asymptotically stable stable and and uniformly attractive. attractive, stableififthere thereexists existsS 8> (iv) (iv) JC* is said said to to be be exponentially exponentially stable > 0,0, M M >> 0, 0, and and TJ e (o, (0,i)l) such suchthat that

\\x{t,t0)? — X *< | <MO; M | X 0 -. 0 ) -x*O

-X'\]'~'\

whenever

k —- xH x\\ << 8. (v) A A solution solution x(t, x(t, x0) x0) isisbounded boundedifif for forsome some positive positive constant M, M, (v) \\x{t, xo)\\< M lk(t,

for all ft >t0,t0where may depend onon each , whereMM may depend eachsolution. solution. for all // == oo or or in (iv) °°, the corresponding stability stability property property (iv) 85 == oo, If in (ii) and and (iii) (iii) fi is said to be global. global. We We illustrate illustrate the the concept concept of of stability stability in inphase phase space space as as in infigure figure 6.4.1. 6.4.1

x2

x1

equilibrium Figure 6.4.1: Stable equilibrium

264 264

6. 6. HIGHER HIGHER DIMENSIONAL DIMENSIONAL DIFFERENCE DIFFERENCE EQUATIONS EQUATIONS

Example Consider Consider Example x2cos9), GE 9e (o, (0, 2,r). 2K). f{xx,x2x2)=(x1cos9 )= (x, cos9 —- x2sin9, x2 sin9, x1sinO x, sin6 ++ x2cosO), f(x1, is a counterclockwise counterclockwise rotation (0, 24, 2K), This is rotation of of the the plane plane of of an an angle angle 9. 9. Since 99ee (o, x0 of of the the plane plane isis moved moved from fromits itsinitial initial position position except except the the origin origin 0, 0, every point x0 Setting which is the only only fixed fixedpoint pointofof /f.. Setting x0 ==(*,(()), x 2 (o))*O, (x1(O), x2(0)) 0, we obtain obtain x(1) x(l) == f(x(O)) /(x(o)) as as we x,(l) == x1(O)cos9 x1(0)cos6> —- x2(0)sin9, x2(0)sin6>, x1(1) x2(l) == x1(0)sino x1(o)sin9 ++ x2(0)cose. x2(0)cos<9. x2(1)

from x(l) x(I) with with aa rotation rotation of of 9, 9, or state x(2) x(2) = f(x(1)) /(x(l)) can be obtained obtained from or The new state Hence x0 with a rotation from x0 rotation of of 29. Hence x2(0)sin2<9, x1(1)= x1(0)cos2o —- x2(O)sin2e, x,(0)sin26> + x2(0)cos2e. x2(0)cos26>. x2(l) == x1(O)sin29 x(f) isisobtained obtained from fromx0 x0 with withaarotation rotation of of tO t9 In general, x(t) xl(t) = x1(O)coste x1(0)cost9 — - x2(0)sintO, x2(0)sint9, x1fr)= x2(t) = = x1(0)sintG x,(0)sin/6' + xx2(0)cost6. x2(t) 2(0)cos?<9. have We thus have ll.t(t)_

=

- x2(0)sint9}2 x2(0)sin/6>}2 + {x1(O)sinz'9 {x1(o)sin/<9 + x2(O)cost9}

—

STABILITIES 6.4. STABILITIES 1, fixed point. Hence, all states states x(t), x(t), t == 0, 0,1, where x == 00 is the unique fixed

265

ofofthe the

O(x0) are are at the same same distance distance from from the the fixed fixed point point x*. The origin is is thus thus orbit 0(x0) stable. stable. above definitions, definitions, some of of the the stability stability properties properties automatically automatically impiy imply In the above one of of more more of ofthe theothers. others.Figure Figure 6.4.2 6.4.2 shows showsthe therelations relations among among these these concepts. concepts. In general, general, none none of of the the arrows arrows can can be bereversed. reversed. For For instance, instance, the the solution solution of of x(t ++ i)= l) = x(t) is is given given by x(t, x(t, x0) x 0 ) = x0. x0. This solution solution is is uniformly uniformly stable but not asymptotically stable.

exponentially stable

asymptotically stable stable'

uniformly asymptotically stable stable

uniformly attractive

attritive ive uniformly stable stable uniformly

stable stable.

Figure 6.4.2: Relations of of stabilities stabilities

Example Example Consider8 Consider8

\x2(t).

of this this equation equation is is given by The solution of

1

1). This This and and the the following following examples examplesare areaccording accordingto toElaydi Elaydi(1999: (1999: 160-6 160-61).

(6.4.3)

6. HIGHER DIMENSIONAL DIMENSIONAL DIFFERENCE DIFFERENCE EQUATIONS EQUATIONS

266 266

\2/ / \/ (tYt—l\(t--4\

/

2

2

)

t—to —l

2

)

If |xo| is sufficiently sufficiently small, then limx(t) = 0.

solution is attractive. attractive. However, uniformly attractive. attractive. For if Thus the zero solution However, it is not uniformly so chosen chosen that that 8 >00isisgiven givenand and t0 t0 isis so 8> 2 1)82 {t0++l)S >2, 2, (t0

then for \xo\ =—S8

l. The zero solution is stable. For given ee >>00 and t0 choose t0 > 0, choose

8=e. l+t0 If1x01 |xo| If

then c< 8, then

\x{to+l,X + 1, x0)l < e, a)\<£, lx(to t0. Since 8 depends depends on on the the choice choice of of t0, t0, the zero solution is stable but for all t > t0. not uniformly stable. stable.

Example The The equilibrium equilibrium point point (1, (l, o) 0) of Example

At + l) = jAt), e{t + l)

r>0, r > 0,

O82,r,

(6.4.4)

6.4. STABILITIES STABILITIES

267

is attractive but not stable. The solution of the equation is

r(t) = 9(t) =

'1—2' a2t

We have

Limr(t) == 1,l, LimO(t) = 2,r, i-*oo

0. Now ifib if r0 * 0, 0O = 0, 0, then with 0O * 0. 0,90

r(t)= rf', 9(t)= r(t)=rf,

0,

which converges converges to to (i, (l, o). 0). However, if

00=ar, 6^ = an, 0
Figure 6.4.4: Attractive but not stable equilibrium

268 268

6. 6. HIGHER DIMENSIONAL DIMENSIONAL DIFFERENCE DIFFERENCE EQUATIONS EQUATIONS

Under Under some some additional additional conditions, conditions, the the arrows arrows in in figure figure 6.4.2 6.4.2 may may be bereversed. reversed. For instance, instance, for for linear linearsystems, systems,uniformly uniformlyasymptotically asymptotically stable stable 4-> <-> uniformly uniformly autonomous systems, systems, stable stable4-> <-> uniformly stable, asymptotically stable stable stable; for autonomous <-> uniformly asymptotically stable, and attractive <-> uniformly attractive. attractive. asymptotically stable, and attractive 4-> 4—> We provide some results about about stability stability of of the the following following linear linear autonomous autonomous system9 system9

x(t = A(t)x(t), tt>to>O. t0 0. x{t ++ 1) \)=A{t)x(t),

(6.4.5) (6.4.5)

0. It should We A(t) to be for any should be be We always always assume assume A(t) be non-singular non-singular for any t > t0 t0 >0.

remarked that for equation (6.4.5), (6.4.5), every local local stability property of the zero solution corresponding global global stability stability property. property. implies the corresponding

Theorem 6.4.1. 6.4.1. (i) If Theorem

X t . K W ^ l , ^j^k,

t>t0,

system (6.4.5) (6.4.5) is uniformly stable; and (ii) if then the zero solution of system

for some v> v > 0, 0, then then the the zero solution solution of system system (6.4.5) is uniformly for some asymptotically stable. Theorem 6.4.2. 6.4.2. Consider Consider Theorem x{t + l) Ax(t), n>to>0, 1) = Ax(t), t0 o, x(t

(6.4.6)

where A A isisconstant. constant.The Thefollowing followingstatements statementshold: hold: (i) The is stable stable ifif and and only only if if p(A) The zero zero solution solution of system system (6.4.6) (6.4.6) is p(A) <1 1and andthe the 10 and semisimple; and eigenvalues of unit modulus are semisimple;'° (ii) The zero stableifif and and only only if (ii) zero solution solution of system system (6.4.6) (6.4.6) is asymptotically asymptotically stable

p{A)<\.1. p(A)<

9

The stability properties in the remainder of this section is referred referred to Elaydi (1999: 168172). 172). 10 '° An eigenvalue is said said to be be seinisimple semisimple ifif the corresponding corresponding Jordan Jordan block block isis aa diagonal. diagonal.

6.4. STABILITIES STABILITIES

269 269

Theorem 6.4.3. Consider Consider Theorem 6.4.3.

x(t x{t + + i) \)=A = 2x2x{t),

n>to>O,

(6.4.7)

x 2 real real matrix. matrix. The The zero solution where A is aa constant constant 22x2 solution of system system (6.4.7) (6.4.7) is is asymptotically stable if and only if \trA\< 1 + detA <2. < 2. ItrAI<1

Exercise Exercise 6.4 6.4 1 Find Findfixed fixedpoints pointsof ofthe thesystem system x1(t + 1) = ax1(t)(l

—

x1(t)

—

= bx1(t)x2(t). x2(t + i) \) = bXl{t)x2(t).

Show that that the the system system 2 Show + i) x1(t)cosl — 1, xx{t + l) = x,(/)cosl - xx2(t)sinl 2(t)s\nl + 1, x2(t)cosl + 2. 1) = x1(t)sinl xx2(t x,(?)sinl + x,(/)cosl 2. 2(t + l) has one one and and only only one one fixed fixed point. point.

3 Find Find fixed fixedpoints points of ofthe the systems systems and and discuss discuss their stability stability

;)

(a) f(x1, (a) / ( * 1 , X 2 ) = (0.5x1 ( 0 . 5 J C+1 +1, 10.5; , 0 . 5 *+ 2); 2 + 2);

(b) f(x1,x2)=(2x1 +1,2; +2). 4 Discuss solution of of x(t ++ i) Discussstability stability properties properties of the zero solution l) = Ax(t), where

[—i -1

551

(i) A(t)=L -0.5 2 1.5

1 (ii) A(t)= —1.5 -1.5 —0.5 -0.5 0.5

1

1

1

—1 -1

1.5

00

EQUATIONS 6. HIGHER DIMENSIONAL DIFFERENCE EQUATIONS

270

Using theorem theorem 6.4.2 determine whether the zero zero solution solution of of 5 Using 6.4.2 to to determine whetheroror not not the x(t + 1) l) = A(t)x(t) is uniformly stable or uniformly asymptotically stable stable when: when: t

(t)=[lt \+t -1

(u)A{t) (ii) A(t) ==

1 t +\ 1

0 1 0 1 .

—

—sm(t)

4

2

1

0 0

55

t+2 t +2 t +l t+1 (iii) = (iii) A{t) A(t)=

00 1

t +l t+1

- sin —cos(t)

4* 4

0

0 o0 11 00 0 1 01

6.5 Liapunov's direct method Since the theory of linearization is a local local theory, itit does does not not address address global global issues. issues. In this section, we discuss discuss another another approach, approach, known known as asLiapunov Liapunov's's second second method or direct direct method." method}1 The The method method determines determines the the stability stability or or instability instability of of aa critical critical point by constructing a suitable auxiliary function. The theory of Liapunov point constructing suitable auxiliary fimction. The theory of Liapunov functions is a global approach toward determining determining asymptotic behavior of of solutions. solutions. thnctions Basically, the method is a generalization of two physical principles for conservative systems, potential energy local systems,namely, namely,(i) (i)aa rest rest position position isis stable stable ifif the the potential energy isis aa local minimum, otherwise it is unstable, and (ii) the total energy energy is a constant during any motion. The Liapunov function function shows initial values values from aa large large region region motion. The Liapunov shows that that initial converge to an equilibrium point. Consider the autonomous difference equation

= f(x(t)), f{x{t)), x(t + i) =

(6.5.1)

where 1

' The The method is referred referred to as as aa direct direct method method because because no no knowledge of the the solution of the system of of difference equations isis required. required.

6.5. LIAPUNOV'S DIRECT DIRECT METHOD METHOD

/:G->/?",

271 271

GczR",

We assume that x is is aa fixed point point of ofequation equation (6.5.1). (6.5.1). That That is is is continuous. We

f(x)= i. V: R" R"—, —>RRbebea real-valued a real-valuedfunction. function.The Thevariation variationofofVVrelative relativetoto Let V: equation (6.5.1) is is defined as

AV{x)=v{f{x))-V{x), AV(x) = V(f(x)) — V(x), and V(x(t + i)) —- F(x(0). V(x(t)). V(f(x(t))) -— V(x(t)) == V(x(t AV(x(t)) = = V(f(x(t))) + 1))

If AAF(x) V (9 < 0, then then VV isisnonincreasing nonincreasing along along solutions solutions of of equation equation (6.5.1). (6.5.1). The The

function V isissaid Liapunovfunction functionon onaasubset subset/H function said to to be a Liapunov / of of R" if (i) V on H, and F isiscontinuous continuous on (ii) AV(x) AV(x) <0,0,whenever whenever xx and andf(x) f(x) belong belongto toH. H. (ii)

radius r and Let B(x, B(x, r) r)denote denotethe theopen open ball in R" of radius andcenter center x defined defined by

B(x, r)

{y E

—


We denote r) with with B(r). The denote B(O, J3(0, r) The real-valued real-valued function function V is is said said to to be bepositive positive definite at x if and (i) (0 v(x*)=0, and (ii) F(x) > 0 for for all (ii) V(x)> allxe xe B\x*', B(x, r), xx^x*

for 0. for some somerr > >0.

12 If V isisaaLiapunov Theorem Theorem6.5.1. 6.5.1.(the (theLiapunov Liapunov stability stability theorem) theorem)12 Liapunov function function for

equation (6.5.1)inin aa neighborhood neighborhoodHHof of the the equilibrium equilibriumpoint, point,x*, x, and and FV is equation (6.5.1) definite with respect to to x*, then x* is stable. If, in in addition positive defmite

12 12

The proof is The is referred to to section section 4.5 4.5 in in Elaydi Elaydi (1999). (1999).

272 272

6. HIGHER HIGHER DIMENSIONAL DIMENSIONAL DIFFERENCE DIFFERENCE EQUATIONS EQUATIONS 6. AF(x)<0 AV(x)cO

whenever f(x) e H H and whenever x,x, f(x)e and xx * x', then x* is asymptotically asymptotically stable. Moreover, Moreover, if G = H = R"and and ifG=H=R" ||jcj| —* -» cc, oo, -» cc °° as as Ikil V(x) -4

(6.5.2) (6.5.2)

then then x* is globally asymptotically stable. stable.

For illustration, let us consider a planar system with x* = 00 as the equilibrium Suppose that equation equation (6.5.1) (6.5.1) has aa positive positive definite definite Liapunov Liapunov function function V V point. Suppose defined on B(r). the graph graph of of V inin aa3 3-dimensional defined on B(r). Figure Figure 6.5.1 6.5.1 illustrates illustrates the -dimensional coordinate system. If we now let e > 0, B(e) then contains one the level level coordinate system. If we now let s> 0, B(s) then contains one of the say curves of of V, say

V{x) == c. v(x) The level curve V(x) v(x) == cc contains contains the theball ballB(S) B(S)forforsome someS Swith with0 0< c8 S< e. a IfIfaa The starts at at x0 x0 Ee B(S), then then v(x0) v(xo)
V{x(t)) 0. Thus, Thus, the the solution solution x(t, x(t, x0) x0) will stay forever forever in ball B(s). B(e). for all will stay in the ball Consequently, the zero solution is stable.

a} for some Theorem 6.5.2. 6.5.2. If If VV isisaaLiapunov Liapunovfunction function on on the the set set p e /?" H > o\ a >> 0, 0, and if if condition condition (6.5.2) (6.5.2) holds, then all all solutions solutions of equation equation (6.5.1) (6.5.1) are are bounded. Consider Example Consider l) = x2(t), x2(t), x1Q Xl{t + 0=

av(t) 2k

)

DIRECT METHOD METHOD 6.5. LIAPUNOV'S DIRECT

273

V( v(x1, Xl,xx2) 2)

x2

xi

Figure 6.5.1: 6.5.1: A Liapunov fimction function Figure

There are three equilibrium points,

(0,0), (±fl*,±p*), (o,o), if if a >> 1, 1, where whereP* jf = J(a - \)l p. Consider the stability of the equilibrium point (0, 0). (o, o). Let

v(x(t))=x? (t) + is continuous and and positive defmite definite on R2 This is a2

Av(x(t)) AK(*(,)) = f-r—? =

(a2 —

-

a2 1,< according 1, according theorem 6.5.1, uniqueequilibrium equilibriumpoint point(o, (0, 0) If a2 to to theorem 6.5.1, thetheunique stable. o) is stable. Since

V(x) v(x) —*

oo as as

Ikil

>

6. HIGHER DIMENSIONAL DIFFERENCE DIFFERENCE EQUATIONS

274

according to theorem 6.5.2, all solutions are bounded. Since

AV(x(t))= AV(x(t)) = o0 for all points on the x1 -axis, theorem theorem 6.5.1 6.5.1 fails fails to determine asymptotical asymptotical stability x, -axis, stability

for this this equation. equation. In fact, fact, it can be shown shown that that in in the the case case of of aa22 == 1,1, the zero

is not asymptotically asymptotically stable; stable; in in the the case case of of a2 a2 c<1,1, the origin is solution is the origin asymptotically stable; stable; in the case of origin is of a2 a2 >> 1,1, the stability of origin indeterminable.'3 indeterminable.13

Example Consider Consider 2 x,(t i) == 2x2(t) 2x2(t)-2x — Xl(t + l) 2{t)x {t),

x {t + l)^-x](t)+xl(t)x22(t). + 0= ix1Q)+

x2(t 2

There are three equilibrium points

(0,0), (0,0),

.

Let v(x(t)) ++4x22(t). V(x(t)) == x?(t) x2(t) This is continuous and positive definite defmite on R2 2

AV(Xl ( 4 xx2(t)) = Ax\ (t)x zxv(x1(t), 2 (/)) =

(t)[x2 (t) ++ x\ (/) —-1].

If x2(t)+x+22{t)<\, x?(t)

1,

then av(x) AV(x) 0. < 0. conclude that zeroisisstable. stable. then WeWe conclude that thethezero 13 13

The results are based See section results are based on LaSalle's LaSalle's invariance invariance principle. principle. See section 4.5 4.5 in in Elaydi Elaydi (1999). (1999).

6.5. 6.5. LIAPUNOV'S LIAPUNOV'S DIRECT METHOD

275 275

o)T (a, Of For any a, the thesolution solution with with an an initial initial value value of of x0 x0 =- (a, any real real a, is periodic with period 2 and with orbit period and with orbit iT in Jr {[a pg Oj07,[a/27\ F

,

and a solution solution with with an an initial initial value value of of xx0 (o, af a)T is also periodic periodic with with period period 2. 0 = (0, Hence, the zero solution solution cannot cannot be be asymptotically asymptotically stable. 6.5.3. IfIf AV(x) AV(x) isispositive positive definite definite in in aaneighborhood neighborhood of the origin and Theorem 6.5.3. —>00 with with V(a,)> V{ai) >0,0,then thenthethezero zerosolution solutionofofequation equation there exists a sequence aa, —, 14 unstable. (6.5.1) is unstable.'4 Example Consider Consider 1 xx(t + + i) \) = = 4x2(t) 4x2{t)-2x x,(t — 2(t)x l(t),

xx2(t -x,{t)+xx{t)x22{t). 2{t + + \) = —x1(t)+ Define

v(x(t))=

Then AV(x1(t), 3x?(t)+ \6xlit)xl{t)+ 4x?{t)x22{t) > 0,ififx,(t) AV(xM xx2(t))= *,(*) *o. 0. 2{t)) = 3x?(t)+ theorem 6.5.3, we we see see that that the the zero zero is unstable. From theorem

asymptotical stability of the In section 6.2, we mentioned that the condition for asymptotical linear system is that p(A) < 1. system is that p(A) < 1.

Exercise 6.5 6.5 Exercise 1 For For

14

The conclusion conclusion of the theorem AV(x) is negative theorem is is also also true true ifif AV(x) negative definite definiteand andv(a v(a,) c 0. j) <

6. 6. HIGHER DIMENSIONAL DIMENSIONAL DIFFERENCE DIFFERENCE EQUATIONS EQUATIONS

276

+ 22(t) + + xf{t), x,(t ++i)l) ==X] (t)+x x1(t x1(t)

x2(t ++ i)= l) = x2(t), x2{t), x2(t function introduce the Liapunov function V{x{t))==x, (t) x,{t) {t). v(x(t)) + X2 +x2(t). the origin is is unstable. unstable. Show that the

2 For

xjt + 0= x1(t)— x2(t +i)=l)=x x2(t), x2(t + 2{t), introduce the the Liapunov function function (t). v(x(t)) = (t) ++xl{t). V(x{t))=xt{t) Show Show that that the theorigin origin isisstable. stable.

Consider the theplanar planar system system 3 Consider

x1(t+1)=

X2(t)

1+x?(t) *?(/)'

x(t)

Find the the equilibrium points points and anddetermine determine their their stability. stability.

6.6

Linearization of ofdifference difference equations equations Linearization

We consider thenonlinear nonlinear systems systems of ofdifference difference equations equations consider the

x{t + + l)=A{t)x{t)+g{x{t),t), x(t i) = A(t)x(t) + g(x(t), t),

(6.6.1)

LINEARIZATION OF DIFFERENCE EQUATIONS 6.6. LINEARIZATION

277 277

x n nonsingular for all all te t Z+ and where A(t) A(t)isisaannxn nonsingular matrix for and

g:Z+xG->R",

GcR'1, GczR",

is aa continuous fbnction. function. Its corresponding linear system is x(t +

(6.6.2)

A(t)x(t).

as aa perturbation We may consider system system (6.6.1) (6.6.1) as perturbation of ofsystem system (6.6.2). (6.6.2). System (6.6.1) may arise from from the linearization of nonlinear equations of the form form

= f(x(t),t),

(6.6.3)

where

f:txG—>R", ", GcR", GcR", is continuously differentiableatatananequilibrium equilibrium point continuously differentiable point x' i((=0) - 0 )ininthis thissection. section. We We

require for all all t ee Z + .The require /f(O, ( 0 , t) i) = = 00 for TheJacobina Jacobina matrix matrix of off/ isisdefined definedasas

af(x,t) t) a/.fc

af(x,t) dfi(x,t)

,) af2(x,t)

ax2 3x2

ax11n

af2(x,t)

aL(x,t)

ax, 3x,

ax2

ax11

ax,

ax2

dx1

af(x,t)1 cx dx

1x=O

L

dx

Let af(o, and 3/(0, t)/ax ?)/3x == A(t) and g(x(t), t) = f(x(t), t) — A(t)x(t). Then equation equation (6.6.3) (6.6.3) may be written in inthe form of ofsystem (6.6.1). In Inthe case that equation (6.6.3) (6.6.3) is autonomous, autonomous, x(t x(t + + l) i) = = f(x(t)), we have have

278

6. HIGHER DIMENSIONAL DIFFERENCE DIFFERENCE EQUATIONS EQUATIONS 6.

x(t + x(t + l)=Ax(t)+g(x(t)), i) = Ax(t) + g(x(t)),

(6.6.4)

where = [(0) /'(o) and where A = and g = f~Ax. g=f—Ax. Since f/ isisdifferentiable differentiable at 0, 0, it follows that Since

g(x) g ( x )== o(x) o ( x )asa s | y -—> >0,

—*

0

as as

—>

0.

IxH

15 Theorem 6.6.1. Assume that

g(x(t), t) = uniformly as ||x| —* —» 0. 0. If the the zero zero solution solution of of the thehomogeneous homogeneous system system (6.6.2) is as IIxD uniformly asymptotically stable, the the zero solution of the nonlinear system (6.6.1) is stable. exponentially stable. Corollary 6.6.1. If p(A) p{A) c<1,1, then the zero zero solution solution of ofequation equation (6.6.4) (6.6.4) is Corollary 6.6.1. then the is exponentially stable. Example Examinethe thestability stabilityof ofthe thezero zero solution solution of ofthe the planar planar system system Example Examine x1(t +

X2&

1)

=

+ 0=1

x2Q)),

f2(xl(t), x2&)).

As 15 15

The The proof of the theorem is referred referred to section 4.6 in Elaydi Elaydi (1999).

LINEARIZATION OF DIFFERENCE EQUATIONS 6.6. LINEARIZATION

aj 3/ dx

'ML ML (0,0) (0,0) --

dxx

[00 aal oJ' b O l'

ax22 dx

af2 ML ML ax, dxl

279

ax22 dx

the system may be rewritten as -axc1v2(t)xl(t) 2{t)xl{t)

—

+1)1[o alrxj(t) (t +2)j[b b oiLx2(t) 0||*2(/)J + [x2At-

1+x?(t) *?(/) — bx1

(t)

(6.6.5)

or or

+ i)\)=Ax{t)+g{x{t)). = Ax(t) + g(x(t)). x(t x{t +

The eigenvalues of A A are ± 4ab. If c 1, eigenvalues of If yfab < 1, the zero solution solution of the linear is asymptotically asymptotically stable. is continuously continuously differentiable differentiable at the the part is stable. Since Since g{x(t)) g(x(t)) is origin, the zero zero solution solution of of equation equation (6.6.5) (6.6.5) isis exponentially exponentially stable. stable. Example ExampleIntroducing Introducingaadelay delayof oftime time 11into intothe thePielou Pieloulogistic logisticequation equation

x(t+1)= av(t) x(t+1)=—^ 1 + /Jx(t)' 3(

yields the following following difference delay equation

x(t+1)= x{t + 1) =

ax{t)

, a>O, a > 0, J3 > 0.

1+flx(t— 1 + fitc(t - 1)' l)

Let

a—l a-\ y(t)=x(t)— v{t) = x(t)Then the above equation becomes

6. HIGHER DIMENSIONAL DIFFERENCE EQUATIONS EQUATIONS

280

v(t , y(t( + 1)

—1)yQ—1) ^_ay{t)-{a-\)y{t-\) aa+(a—1)y(t—1) + (a-l)y(t-l) "

~

To change the the above above equation equation into into aa planar planar system, system, introduce introduce

.x1(t)=y(t

—

i), x2(t)=yfr).

Then

Xjf)

aX2{t)-(a-l)xx{t) a + (a-\)Xx(t)

X2(t-

(6.6.6)

By linearizing equation (6.6.6) around (o, o), we (0, 0), we have

x(t = Ax(t) + X{t ++i)\)=AX{t)+g{x{t)), where 00

A= 1—a

11

aa 0

g(x) =

—

aflK1(t)X2(t) a(a + flX1(t)) a{a + —

The eigenvalues of A A are areinside insidethe the unit unit disk disk ifif and only if

00<<

a—l

aa

<1, < 1,

which is always always held Consequently, the solution of equation equation which held since since a >>\.1. Consequently, the zero solution (6.6.6) is is asymptotically asymptotically stable. stable.

Theorem 6.6.2. Theorem 6.6.2. The Thefollowing following statements statements hold If p(A) p(A) = (i) If = 1,1, then the zero solution of equation (6.6.4) may be stable or unstable;

LINEARIZATION OF DIFFERENCE EQUATIONS 6.6. LINEARIZATION

281 281

(i) If p(A)> o(x) as as IxD|x|—> P(A) >I 1and andg(x) g(x)is is o(x) —>oo, °°, then thenthe thezero zerosolution solutionofofequation equation (6.6.4) is unstable.

Exercise 6.6 6.6 1 Determine Determinethe thestability stabilityof ofthe thezero zerosolution solution of ofthe the equation equation 13

x(t 2)— l)+ —x(t + l) —x(t)= x{t + + l)x(t)+ lWf) 0. i + x(t v+ y-\/ + 1—x(t) \j = 0. v + 2) i - -x(t 2 v 6

2 Find Find the the equilibrium equilibrium points points and and determine determine theft their stability stability of the system 2

2

x1(t x2(t)+ x3(t), x1(t + 0=—x1(t)— l) = -x^(t)-x 2(t)+x 3(t),

x2(t x2(t) + x2(t + + 0= l) =x1(t) xl{t)-x—2(t) + x3(t), x3{t), l ( +i)=x1(t)—x2(t)+-1-x3(t). ])() () () x3(t X3{t+ )-Xi(t)-X2(t) + -Xs{t).

3 Determine of the the zero solution Determine conditions conditions for the asymptotical asymptotical stability stability of solution of the the system

ax(t) ,(t) ax, 1

x2(t +

6.7

1)

+ x2(t)

2(y— x1(t)][I +

= [bx2(t)

= [bx2{t)-xl{t)][\

x1(t)].

Conjugacy and center center manifolds manifolds

There are are two two possible possible ways ways to to simplify dynamical systems: one is to transform transform one complex system to another one which is much easier to analyze; and the other is to reduce higher dimensional problems to to lower ones. The center manifold dimensional problems manifold theorem helps us to reduce dimensions of dynamical problems. A change of variable helps us to reduce dimensions of dynamical problems. A change of variable may may reduce a system to a simple one. This idea is made more precise with the concept of of reduce a system to a simple one. This idea is made more precise with the concept conjugate maps. The concept of conjugacy arises in many subjects of mathematics. conjugate maps. The concept of conjugacy arises in many subjects of mathematics, In is aa hi linear linear algebra, algebra, the the natural natural concept concept isis linear linear conjugacy. conjugacy. Thus, Thus, ififx1 xl == Ax Ax is linear map and x = Cy is a linear change of coordinates for which C has an linear map and x = Cy a linear change coordinates for which C an -variablesisisgiven givenas asfollows follows inverse, then the map on the yy -variables

282

6. 6. HIGHER DIMENSIONAL DIFFERENCE EQUATIONS EQUATIONS

l 1 = C~ C1Ax = cC'x1 yx = x1 = Ax==C1ACy. C'UCy.

Thus, the the matrix matrix for the the map map in inthe the y -variables -variables is is C1AC. are C lAC. As Aslong longas asthe themaps maps are defined on on the the same same space, space, aaconjugacy conjugacy can canbe beconsidered considered aachange changeof ofcoordinates coordinates of the variables on on the the space on onwhich the thefunction function acts. one-one.J> be Definition 6.7.1. and JJbebetwo twointervals intervals and : / — J continuous, be continuous, Definition 6.7.1.Let Let/I and 0: 1 —* J > and to-one, and that 0<j>is is a conjugacy between J and to-one, and onto. onto. We We say that a conjugacy between G : IG: —,/ — F:J -> / if F:J—*I Ø(G(x)) = F(Ø(x)),

for Themaps maps FFand andUGare are saidtotobe beconjugate conjugateby by 0. 0. for all all xx e I./. The said fixed point pointof of G U ifif and only ifif Let T F7"and by 0. 4 Then x is a fixed andU Gbe beconjugate conjugate by =
fixed point pointof of G U if if is a fixed F. Similarly, Similarly, xi,, xp is a fixed if and only if fixed point point of of F.

yP=Axp) is a fixed point F. point of of F.

Example Let U(x) = — 9, xe G{x) = 4x(1 4x{l-x), xe[0,l], EQ 1], 2 F{y) = 2y —1, -l, je[-l,l], F(y)=2y2 [—1,1], jy=Ø(9=1—2X. ; = ^(x) = l - 2 x .

i] ljtoto 11—1, continuous, onto from [o,[O, We see see that that 0
6.7. CONJUGACY AND CENTER CENTER MANIFOLDS MANIFOLDS

283 283

8 X + 1. ) = 8JC2 -—8x+1. F(Ø(x))—8x2

Hence the two maps maps F and andGGareareconjugate conjugateininthe thegiven givenintervals intervalsby bythe themap mapçb. <j>. Since is invertible, have Since 0<j> is invertible, we we alsoalso have G == Ø' (j)'1oFoØ, o p o ^, U

where the symbol o° stands for the composition of of ftinctions. functions. introduce a way way to to transforming transforming aa higher higher dimensional dimensional problem problem to aa We now introduce A) lower one. one. Consider Considerthe the m in-parameter -parametermap mapfix, f(x, 2)

f:R"xRm

-* —,

R", with with xe R"and andAc AeRm Rm xc]?"

where / f is is Cr ((r r >3)3 on ) on some sufficientlylarge largeopen openset setininR"R"x xRm. Rm. Let Let where some sufficiently (x0, AQ fixed point point of i.e. (x0, k) )bebeaafixed of /f,, i.e.

f(x0, f\xo> k) ==

x x0. o-

determined from from the the We know that the stability stability of of hyperbolic hyperbolic fixed fixedpoints pointsofof/f isisdetermined stability of the fixed points under the linear map

J=

,t0).

However, However, the the situation situationisis different differentififone oneofofthe theeigenvalues eigenvaluesp p of of J.J lies on the unit cycle, that is, \p = 1. There are three cases in which the fixed point point the unit cycle, that is, = 1. There are three cases in which the fixed (x0, AQ) is nonhyperbolic: (i) J has one real eigenvalue equal to 1 and the other (x0, *) is nonhyperbolic: (i) J has one real eigenvalue equal to 1 and the other eigenvalues are off the unit circle; (ii) J has one real eigenvalue equal to —1 and eigenvalues are off the unit circle; (ii) J has one real eigenvalue equal to —1 and the other eigenvalues are off the unit circle; and (iii) J has two complex conjugate the other eigenvalues are off the unit circle; and (iii) J has two complex conjugate eigenvalues modulus 1 and the other eigenvalues are off the unit circle. To analyze eigenvalues modulus 1 and the other eigenvalues are off the unit circle. To analyze behavior of of the the system system in in these these cases, cases, we we need need to to introduce introduce aa rudiment rudiment of of center center behavior manifold theory. manifold theory. change of of variables, variables, we we may may assume assume without without loss loss of of generality generality that By aa change Let us temporarily suppress the parameter p. Then the map f(x, X) 2) can x0 = 0. x0 = Let us temporarily suppress the parameter ju. Then the map fix, can be written as

6. HIGHER DIMENSIONAL DIFFERENCE EQUATIONS EQUATIONS

284

yy»Apxpy z

z

f(y,z), Z),

+ +

g(y, z), 4, i-> B^z + g{y, BqxqZ +

where

p+q=n, p + q = n,f(O,O)=g(O,O)=O, /(0, 0) = g(0, 0) = 0,Df(O,O)=Dg(O,O)=O, and .J J has has the the form form

nxn

1

o

Bqxqj'

where all eigenvalues of A of B eigenvalues of A lie lie on on the the unit unit circle and all of the eigenvalues eigenvalues of are off off the the unit unit circle. circle. Observe Observe that that system system (6.7.1) (6.7.1) corresponds corresponds to the system system of difference equations equations y(t Ay(t) + f(y(t), z(t)), y{t ++1) =l)=Ay{t) +f(y(t),z(t)), z(t i) = Bz(t)+g+{y{t\z(t)). z{t ++ \)=Bz{t) g(y(t), z(t)).

{

The following following theorems are are referred to Cam'6 Carr.16

Theorem Theorem 6.7.1. 6.7.1. There There is aa CCrcenter centermanifold manifold for for system system (6.7.1) (6.7.1) that that can can be be represented locally as Mc ={(y,z)e = {(y, Z)GRPXR":

z=h(y), Z = h{y), jyj<S, \y\ < S, h(o) = 0, Dh(O)=O}, Dh(o) = o}, h(O)=O,

for a sufficiently sufficiently small small 5. S. Furthermore, Furthermore, the the dynamics dynamics restricted restricted to Mc are given given locally by the map

y F-* Ay + f(y, h(y)), yyeR". e R".

(6.7.3)

This theorem asserts the existence of a center manifold, manifold, i.e., i.e., a curve z == h(y) h(y) which the dynamics dynamics of system system (6.7.1) (6.7.1) is given given by system system (6.7.3). (6.7.3). The next on which The next 16 16

See See Carr (1981).

AND CENTER MANIFOLDS 6.7. CONJUGACY AND

285 285

theorem shows that the the dynamics on on the the center manifold determines determines completely completely the the dynamics of of system system (6.7.1).

Theorem Theorem 6.7.2. 6.7.2. IfIfthe thefixed fixed point point (o, (0, o) 0) of system (6.7.3) is stable, asymptotically asymptotically stable, or o) of system (6.7.1) is or unstable, then the the fixed point (o, (0, 0) is correspondingly correspondingly stable, stable, asymptotically stable, stable, or or unstable. unstable.

Theorem Theorem 6.7.3. 6.7.3. For Forany anysolution solution (y(t), (y(t), z(t)) z(t))of ofsystem system (6.7.1) (6.7.1) with an an initial initial point z0) in in a small neighborhood neighborhood around (y0, z0) (y0, around the the origin, origin, there there exists exists aa solution solution w(t) of system (6.7.3) and positive positive constants constants L, /3 1 such that fi > >1 yO)— w(t)j \y{t)-w{t)\
\z(t) - h(w(t))I h{w{t))\ L/Y, < L/3 forailte , for all t e IzQ)—

Z\

The now is how h(y). The question question now how totocalculate calculate the thecenter centermanifold, manifold, z == h[y). Substituting Substituting z = h(y) into system (6.7.1) yields 1) = Ay(t) + h(y(t))), y(t ++ \)=Ay{t) + f(y(t), f(y(t),h(y(t))), Bh(y(t)) t+))+g g(y(t), h(y(t))). (y(t),h(y(t))). z(t ++ 1) =l)=Bh(y(

We also have

z(t ++ i) i)) = h(Ay(t) h(y(t)))), 1) == h(y(t h(y{t ++ 1)) h(Ay(t) + + f(yQ), f(y(t), h(y{t))%

where we we use use the the first first equation equation in in system system (6.7.4). (6.7.4). Equating Equating this equation equation and the in system (6.7.4) we we have second in

4(h(y))= h[Ay ®(h{y)) = h[Ay++f(y, f{y, h(yfl]— h(y))]- Bh(y)— Bh{y) - g(y, g{y, h(yfl h(y))==0.0.

(6.7.5) (6.7.5)

We may this solution solution via via power power series. series. ItIt can be shown may approximate approximate this shown that that although the choice notunique, unique, any any two twocenter center manifolds manifolds of a given given fixed fixed although choice of of h isisnot may differ differ only in transcendentally transcendentally small point may small terms. terms. The Taylor Taylor series series expansion expansion any two two center center manifolds manifolds must agree agree in in all allorders. orders. The Thefollowing following theorem theorem of any the approach.'7 approach.17 justifies the

17

See See Can Carr 1981). 1981).

6. HIGHER DIMENSIONAL DIFFERENCE EQUATIONS

286

Theorem Theorem 6.7.3. 6.7.3. Let P: ¥ :R'° Rp—* -> K' Rq be a C1map mapwith with

'v(o) =

= o.

Suppose that

=

as as

x -> 0o

for some some rr >1. > 1. Then Then

h(y)=P(y)+q(y) asy—*O, as j , where

number K if there is aa positive number ATsuch such that that

in a small small neighborhood neighborhood of of zero. zero.

Example Consider Consider

[xi(t +

-—1 1

[x2(t +

0

00

[x1(t)x2(t)

o - 1 Lx2&)i

The center manifold is

[

(6.7.6)

x?(t)

2.

The center manifold is

={(x1,x2)E RxR: x2 =h(x1),

h(0)=h'(O)=O}.

Mc = {(x,, x 2 )e RxR: x2 = h{xx\ \x\ < 8, h(o)= h'{o)=o}.

The function h satisfies

The function h satisfies h[Ax1 + f(x1, h(x1 ))]

—

Bh(x1) h(x1)) = 0 x,) —- g(x1, g(x1, A(x,))

6.7. CONJUGACY AND CENTER MANIFOLDS

287

where {t)x2{t), ff = X]x1(t)x2(t),

g = xf{t).

With equation (6.7.5), (6.7.5), the the above equation becomes x1h(x1)] + h[- x, ++ *,*(*,)] + i'h(x1) Afo) — - xf == 0. 0.

(6.7.7) (6.7.7)

Let us assume that h(x1) takes the h{xx) takes the form form h(x1) = c1x? + c2x\ ++o(x*). h{xl)=c 1xf + Substituting Substituting

2

c1x1 cxx\

this equation into equation (6.7.7) (6.7.7) yields

1 1 4\ —c2x1 +—c1x1 +—c2x1 - c2x\3 + -c,x, 2 + -c2x\ —x1 - x\2 + 0{x\) = 0. 2 2

Hence Hence 2

c1=—, c , = | , c2=O. c2=0. 3

We thus thus obtain obtain We

The The map mapon on the the center manifold is is given by by

x1(t + 1) i)= -x1(t)++ |x 1 3 (?) + O(x,5). = -*,(/) X](t + atzero zero isisgiven given by by Schwarzian derivative derivative at The Schwarzian

s(fxo)

-

=

f'(O)

3[f'(O)J L/'(0)J

= -4<0.

(6.7.8) (6.7.8)

288

6. HIGHER DIMENSIONAL DIFFERENCE EQUATIONS EQUATIONS

the origin origin isis asymptotically asymptotically stable under equation (6.7.8). This implies that Hence the asymptotically stable stable under under equation equation (6.7.5). We We illustrate illustrate the the local local the origin origin isis asymptotically dynamics as as in in figure figure 6.7.1.

M

x1

Figure 6.7.1: An An asymptotically stable center manifold

We now now illustrate illustrate the the case that the the system system depends on on aa vector vector of of parameters parameters m 2Aee Rm. Then system (6.7.2) takes the form R . Then system (6.7.2) takes the form

y(t y{t + + i) = Ay(t) \)=Ay{t)+f(y{t\z{t\X), + f(y(t), z(t), 2), z(t + i) l)=Bz{t)+g{y{t),z{t),X), = Bz(t) + g(y(t), z(t), 2), z{t +

(6.7.9)

where where f/ and andggare areCCfunctions functions (r >3), 3), in some neighborhood of

(o, 0, 0), f(0, 0, o) == g(0 g(0, )0, 0) = 0, Df(0, 0, o)= Dg(o, 0, o)== 0. (0,0,0), /(0,0,0) 0 ) 0)=0, D/(0,0,0)=Dg(0,0,0) 0. To fmd we consider consider 2A as find the the center manifold of system (6.7.9), we as aa ftmction function of time time and and rewrite rewrite system system (6.7.9) (6.7.9) as as

y(t = Ay(t) z(t), 2(t)), y{t + + 1) \)= Ay{t)++ f(y(t), f{y(t),z{t),A{t)), 2(t+1)=2(t), A(t + l) = A(t), z(t i) == Bz(t) + g(y(t), z(t), 2(t)). z{t + + \) Bz{t)+g{y(t\z{t\A{t)\ The center manifold now nowtakes the the form of

(6.7.10)

6.7. CONJUGACY AND CENTER MANIFOLDS Mc = {{y, A,z):z

289

= h[y, A), \y\ < Sx, \X\ < S2, h(0,0)=0, h(0, 0) = 0, Dh(0,O)=0}. Dh{0, 0) = o}. (6.7.11)

Substituting = h(y, 2) X) into intosystem system (6.7.10) (6.7.10) yields yields Substituting for for z = y(t + 1) f{y{t\ h(y(t), h{y{t\ 2(t)), A(t)\ 2(t)), A(t% i) = Ay(t) + + f(y(t), z{t + 1) h[Ay{t) + f(y(t), h(y(t), 2(t)), X{t)\ 2(t)), Mf)\ 2(t)] z(t 1) = h[Ay(t) + f(y(t),

ti$)\

(6.7.12)

= Bh(y(t), Bh{y(t), A{t)) g{y(t), h(y{t\ Mf% 2(t)). A(t)). 2(t)) + g(y(t), h(y(t), 2(t)), The latter equations lead to the fimction function ®(h(y, 2D= A)) =h[Ay h[Ay + + f(y, f(y, h(y, X), 2]— A] - Bh{y, h(y, X), 2), 2), Bh(y, A) 2) + g{y,h{y,X),X) = Q. h(y, 2), 2)=o. + g(y, For instance, and 2Aare areone-dimensional, one-dimensional,we wemay may take instance,ifif yy and cxy2 + c22y c2Ay + + c322 c3X + h(y, h(y, A) 2) = c1y2 to approach h. approach h.

6.7 Exercise 6.7 1 Calculate Calculate center center manifolds manifolds near the origin origin and and describe describe the biftircations bifurcations of the origin: origin: 1

x1(t i (t) ~ xx1(t)xjt), *i (' + !) ~T*i (0 - xx2(t)— 0==——x1(t)— l [t)xl it), (i) \ 2

x2(t{t++1) *2 ^ 2 (t) +x?(t); xf (/); 0== - - * i W + Axjt)+

+

0=

Ax1(t),

xjt(' ++ l)=*2 x1(t)x2(t). 0= x2(t)+ *2 (')+*! (0*2 (') 6.8

The Henon bifurcations Hénon map and bifurcations

Consider a two-dimensional map

290

EQUATIONS 6. HIGHER DIMENSIONAL DIFFERENCE EQUATIONS

f:R2—*R2, f(xì,x2)eCr, f:R2 ^R2, /(x,,x2)eCf,

r>5. r5.

Let x* = = (x*, x*2j be be a fixed Let; fixed point and the the Jacobian matrix

J= we deduce deduce the the following: following: Using the center manifold theorem we J has has an an eigenvalue eigenvalue equal equal to 1,1, then thenwe we have have aa saddle saddle node node bifurcation; bifurcation; (i) If .1 hasananeigenvalue eigenvalueequal equaltoto—1, - 1 ,then thenwe wehave haveperiod-doubling period-doubling bifurcation; bifurcation; (ii) If JJ has and J has has two two complex complex conjugate conjugate eigenvalues eigenvalues of modulus modulus 1, 1, then (iii) If .J then we we have have a new type of bifurcation, called Neimark-Sacker (Hopf) bifurcation. bifurcation, Neimark-Sacker (Hopt) bifurcation. Henon map map to to illustrate illustrate the the saddle saddle node nodeand andthe theperiod periodbifurcation. bifurcation. We use the Hénon 1976, the French Astronomer Michael Hénon Henon suggested suggested a simplified simplified model In 1976, model for the dynamics of the the Lorenz Lorenz system.18 system.18 The Henon map is given by The Hénon map

x(t+1)=1_ax2(t)+y(t), x(t + l) = \-ax2(t)+y(t), y(t +

= bx(t).

(681)

(681}

We explain a few properties of the the Henon Hénon map. We also denote the Henon Hénon map by by

X(t + l) = HX(t), x(t+l)=Hx(t), [x(t P /

[YV)J

[

bx

It is known known that the Hénon Henon map map contracts contracts areas for |&|
x(t+1)=1_ax2(t).

'Henon Hénon (1976).

18

6.8. THE HENON MAP AND BIFURCATIONS BIFURCATIONS

291 291

To see of the the Jacobian matrixofof H. H. If To see this, this, we we find find the the determinant determinant of Jacobian matrix det DIII < I for all (x, y), the map is area contracting. From vector calculus |det£>#j 1 for (x, the map is area contracting. From vector calculus we we know maps an an infinitesimal infinitesimal rectangle rectangle atat (x, into an know that that H maps (x,y)y) with with area area dxdy drdy into an infinitesimal parallelogram with area

det DH(x, y)dx \det DH(x,y)dxdy\. Thus, if < 1, \detDH(x,y)\
then H H isisarea areacontracting. contracting. As

[-2ax ii \-2ax 1" DH{x,y) = b

oj' 0

we have b. = —-b. det DH(x, y) =

Hence, c 1, \b\ < 1, the theHénon Henonmap mapisisarea areacontracting. contracting. Hence, ifif IbI

We show show that that the the Hénon Henon map map isisinvertible. invertible. To To see see this, this, one one may may decompose decompose H into three simple maps T1 as follows into three simple maps Tj as follows 1

1=1

[yJ

[xl

22

[I—ax 1 - ax +yj +y rbxl bx y.

i'

= y X

As As shown in figure 6.8.1, 77j is an area-preserving area-preserving bending map, map, T2 T2 contracts contracts

in and 1T3 rotates rotates by 90°. 90". The Thecomposite compositetransformation transformation in the the xx direction, direction, and T o T2 T3 i ° T2 o° T\

e yields m the Hénon Henon map.

292 292

6. HIGHER DIMENSIONAL DIFFERENCE EQUATIONS EQUATIONS 6.

T3> ° T2o° T17J Figure 6.8.1: The decomposition decomposition of of the the Henon Hénonmap, map, H == T3 °T2

to show show that that all all T1 T, are are invertible.'9 invertible.19 Hence Hence It is straightforward to =

T11 oT21

i.e.

19

x X

1 I

21'

y—l+axj y - 1 + ax'

—II

Lx/b I

21 LYJ

I

L and and T3

m [Y Ly J

y\

=

lx

HENON MAP AND BIFURCATIONS 6.8. THE HENON 1

[

[x1

HiJ=[ Y

—

1

b a2 +71y +xj

one-to-one, H isisalso As all T1 Tj are one-to-one, alsoone-to-one. one-to-one.

If a ^ 0, H hasafixedpointX* has a fixed point X * if

a

—

(1

—

b

A fixed point is is determined by by 1—ax2 1 - ax +y=x, + y = x,

bx=y. bx — y. We see that that x isisgiven given by ax2 ax + (i

—

b)x

—1=0.

If \-b

we see that that H has two fixed points

b—1+b0 b-\ 2a b-l + bn — b_l+bob 2a

,

where

bf ++ 4a. 4a. b0 = ^(l —- b)2

b —1—b0 b-\-h 2a X" b-\-b n A. 2 - b_i_bob 2a

293 293

294

EQUATIONS 6. HIGHER DIMENSIONAL DIFFERENCE EQUATIONS

It can be shown that if a t- 0 and

ere

-

b)2

(l

4

then X* is is asymptotically asymptotically stable stableand andX* 4 2 is is a saddle. At

2

)

we have

b- 1 b—i 2a — *; = b-\ — b_lbb 2a matrix at at this this equilibrium equilibrium point point has has an an eigenvalue eigenvalue equal equal to to Moreover, the Jacobian matrix 1. By the center manifold manifold theorem, theorem,there thereisisaasaddle saddlenode nodebifurcation bifurcationatatX*4.x. For a fixed fixed value value of of the the parameter parameter bb e (o, (0, l), i), X*x loses loses its stability stability and becomes a saddle point at

a == 331

\-b

and new stable stable 22 -cycle -cycle appears. appears. The reason reason is that one one of of the the eigenvalues eigenvalues of and a new —1. InInthe = 0.3, period-doubling DH\X\) will decrease decrease and and pass DH(X) will pass —1. the case case of of b = 0.3, a period-doubling cascade figure 6.8.2) 1.06. cascade starts starts at at a = 0.3675 (see figure 6.8.2) and and ends ends at at a ~ 1.06. Beyond a ~ 1.06, a strange attractor appears. Figure 6.8.3 strange Beyond 1.06, strange attractor appears. Figure 6.8.3 shows a strange attractor = 0.3. 0.3. Zooming into the strange strange attractor of of the the Henon Hénon map map when when a = 1.4 1.4 and b = attractor, we can see that there are six parallel curves. If we zoom further Ibrther on the top three curves, we can see that they are really six curves grouped the same way as the first batch. This "self-similarity" "self-similarity" continues continuestotoarbitrarily arbitrarilysmall smallscales.2° scales.20 20 20

Benedick and Carleson (1991) demonstrate that this strange attractor attractor is the closure of a the unstable unstable manifold. manifold. branch of the

THE HENON MAP AND 6.8. THE ANDBIFURCATIONS

0.88 0.88 0.9 0.9 00.92 . 9 2 0.94 0.94 0.96 0.96 0.98 0.98

1.02 1.02

0.29 0.29

0.28 0.27

Figure = 0.3675 and b = = 0.3 Figure 6.8.2: 6.82: A 2 -cycle for for a = 0.3

0.4 0.4

0.2

-1

-0.5

0.5

1

-0.2

-0.4 -0.4 Figure 6.8.3: The 1.4 and b = = 0.3 0.3 The Henon Hénon attractor attractorfor for a = = 1.4

295 295

296

6.9

6. 6. HIGHER DIMENSIONAL DIFFERENCE EQUATIONS

(Hopf) bifurcations bifurcations The Neimark-Sacker (Hop!)

For illustration of the Hopfbifi.ircation, Hopf bifurcation, let us consider an example. Consider the family of maps

rx1(t+O1 / LX2V+l)i

2

I

cos/9 —sinfllrx1(tfl -sin/9 \rcosp rx1(t)1 /\I=fAI[X21/)J . /\I' sin/? cos/9 ii x2(t) X, (6.9.1) (6.9.1)

where /3 /? = /3(2) fi{£) is aa smooth J3(o)
J = (l + X)

cos/? — sin/? -sinfl] sin/? cos/3 cos/? [sm/i

The matrix has eigenvalues = (i +

with =

1

+

4

Al.

Hence, at A X == 0, 0, the eigenvalues cross unit circle. circle. Clearly, Clearly, the origin is is Hence, the eigenvalues cross the the unit the origin asymptotically X >> 0. 0, we may asymptotically stable stable for for A bifurcation when when X A = 0, 0. To analyze the bifurcation by introducing introducing fk in polar polar coordinates coordinates (r, (r, d) e) by write the map map f2 x1(t)= r(t)cos 9(t), x1 (t) = r(t)cose(t),

x2(t) r(t)smd{t). x2 (t) = r(t)sin 9(t).

Under this transformation, transformation, equations (6.9.1) become

9(t + i)= e(t)÷ /1. p.

(6.9.2) (692)

The form of equations (6.9.2) enables us to detect the presence of an invariant invariant circle by solving

6.9. THE NEIMARK-SACKER (HOPF) BIFURCATIONS

297 297

r == (i(\ + + 2)r — r3. X)r-r\ The invariant circle circle isis of of radius radius

r* = VI. The circle when 2A crosses circle appears appears when asshown shown in infigure figure 7.9.1. 7.9.1. For For crossesthe the value value 00 as <0 0 the origin is stable.The The instance instancethat that A A becomes A< is asymptotically asymptotically stable. becomes positive, positive, attracting (asymptotically (asymptotically stable) circle circle the origin loses its stability to give rise to an attracting with r == -JA. with radius radius r The dynamics circle are aredetermined determined by themap map dynamics on on this this circle by the fl. The P, which which is a rotation by an angle /3 /? in the the counterclockwise counterclockwise direction. direction. 69 E—* — i » 90 ++ /3, bifurcation). This phenomenon is called a Neimark-Sacker Neitnark-Sacker bifurcation bjfurcation (or (or aa Hopf Hopfbjfurcation). The above above analysis analysis is isactually actually valid valid for foraacertain certain class class ofoftwo-dimensional two-dimensional maps with one one parameter. Consider Consider aatwo-dimensional two-dimensional map map

f:R2XR—*R2, f(xi,x2,2)eCr, r5. f:R2xR->R2, f(x,,x r>5. 2,A)eC, Suppose that (i) (o, o) 0)be be aafixed fixed point; and (i) the origin x* == (0,

(ii) the theJacobian Jacobian matrix matrix 2*) J = Dj(x',X) J

have two complex complex conjugate conjugate eigenvalues eigenvalues

p(2) = and p(2), p{A), where where

r(0)=1, r(0) = l,

, 0{O)=0O,

1); and (which imply \p(o)\ == 1);

(iii) e"°° eM°

00

for k = 2, 3, 4, for = 1,1, 2, 4,5, 5,i.e., i.e., p(0) p(o)isisnot notaalow lowroot rootof ofunity. unity.

298

DIFFERENCE EQUATIONS EQUATIONS 6. HIGHER DIMENSIONAL DIFFERENCE

0.02

0.1

0.01 -0.15-0.1-0.05

0.05 0.1 0.15

-0.1 -0.015 -0.01 -0.005

0.005 0.01

-0.01

-0.2

a) 2A < < 0: approaching the origin b) 2 A= =0

0.1 -0.2

-0.1

0.1

0.2

-0.1 -0.2 -0.3 --0.4 0.4

c) 2A > 0: an attracting circle bifurcation21 Figure 6.9.1: Supercritical Neimark-Sacker bifurcation2'

If the system satisfies satisfies these three conditions, conditions, then in then (1) by a change of basis in R , we may assume, assume, without without loss loss of generality, that R2, 2

Df(0,Q,0)=(l J == Df(0, 0, o)= (1

cos/7 —sinfl] -sin/? + 2)[cosfl sin/?

cos/? cosfl

(2) from (iii), takes the (iii), by by aa change changeof ofcoordinates coordinateswe wemay mayassume assumethat thatthe themap map/ f takes form 21 21

simulation is for /? = 0.3, a) a) X 0 . 1 , b)2 b) X == 0, 0, and and c) X == 0.06. 0.06. The simulation for /3 2 = -—0.1, c)2

6.9. THE NEIMARK-SACKER (HOFF) (HOPF) BIFURCATIONS

[xl f[ hj=NAL 9 A>^ x2

299 299

(6.9.3) x2

where

[eJ

,6.9.4, L e + /3(2) + G(2)

with F(O) F(o) * 0. Moreover, Moreover, the the radius radius of of the the invariant invariant circle circle isis given by by -JA/I F(2). F(A). satisfies assumptions assumptions (i)-(iii). Theorem 6.9.1.(Neimark-Sacker22) (Neimark-Sacker22) Suppose Theorem 6.9.1. Supposethat that/f satisfies Then, for hasanan invariantclosed closedcurve curve enclosing enclosing the the Then, for sufficiently sufficientlysmall smallA,2, ffhas invariant origin AI F(2)> F(A) >0.0.Moreover, Moreover, this this curve curve is is attracting if F(o) > 0; 0; it is origin ifif 2/ if F(0)> repelling 0. repelling ifif F(o) F(0) << 0. Example (Hopf (Hopfbiftircation bifurcation in in the theKaldor Kaldor model23) model23) The Kaldor Example Kaldor model model is describe by Y(t 1 )—- Y(t) = a{I(Y(t), K(t)) K(t))-S(Y(t), a K(t)), Y(t + i) — s(Y(t), K(t))} K(tD} =aF(Y(t), l(Y(t),K(tj)-SK(t), K(t + i) — K(t) = = J(Y(t), K(t)) — in which variables and and parameters parameters are are define define as as Y(t) and and K(t) K(t) == output level and capital respectively; Y(t) capital stock stock in in period period t, respectively; IK <0); < 0); l(Y, K) = investment function (IY >> 0, 'K 24 Ø);24 < 1, SK SK >> 0); S(Y, K) (0 < Sr <1, K) == savings function function(0

22 22

SeeRasband(1990). See Rasband (1990). model was was proposed proposed by Kaldor Kaldor (1940). (1940). Kaldor's Kaldor's contribution contribution was was in in 23The The original original model conjunction of Kalecki Kalecki (1937, (1937, 1939), 1939), who investigated investigated similar conjunction with the work of similar models models but concentrated on different different aspects of stability. stability. The analysis below is based on section 3.3 3.3 concentrated in Lorenz Lorenz (1993). (1993). For For the the analysis analysis and and behavioral behavioral interpretations interpretations of model for for in of the model see also also Chang Chang and and Smyth Smyth (1971) (1971) and and Gabisch Gabisch and and Lorenz Lorenz (1989). (1989). continuous case, see 24 The assumption of o f SK SK >> 0 is not not convincing. convincing. In Chang Chang and and Smyth Smyth (1971), (1971), itit is T h e assumption assumed Ass we assumed SSK 0. A w e require require 'K IK — - SK SK <0, < 0, the thedifferent different signs signs do d o not not affect affect our our K <<0. analytical conclusion. conclusion.

23

300

6.HIGHER DIMENSIONAL DIFFERENCE EQUATIONS EQUATIONS 6.

and 88 == aa positive adjustment parameter parameter and andcapital capital depreciation depreciation rate, rate, a and positive adjustment respectively. atleast least one one equilibrium equilibrium point. point. The The determinant determinant Suppose Suppose that that the system has at K*) are and trace of the Jacobian at aa fixed point \Y*, K*) are det J = {aFy + l\lK +1-S)aFKIy, +IXIK +l—8)—aFKIY, trJ = aFy + IK —8+2. - 5 + 2. trf=aFY+IK

The eigenvalues are complex conjugate conjugateif if

/

tr.J detJ>i—

\2

Assume that a0 Assume that the the inequality inequality holds. holds.A A Hopf Hopfbifurcation bifurcationoccurs occursatataavalue valueaa = = a0 if det

= 1,

where

an0 =

8-IK Fy{l K+\-S)-FKIr' +1—8)—FKJY

The modulus modulus crosses crossesthe theunit unit circle circlewith withnonzero nonzerospeed speedwhen whenthe theparameter parameteraa is is 25 changed changed25 K dlp(a)IcliJJetJ S-I .8'K

da

25

da da

2a 2a00

The modulus of the complex eigenvalues, if if they exist, of the characteristic equation, p2 ++ap+b=0 ap + b = 0 p2

is given by -Jb.

6.9. THE NEIMARK-SACKER (HOPF) (HOPF) BIFURCATIONS BIFURCATIONS

301 301

We may apply apply the the preceding preceding procedure procedure to to describe describe the the Hopf Hopf bifurcation bifurcation in in the the discrete Kaldor model. model.

Exercise 6.9 6.9 1 Analyze Analyzethe thebifbrcation bifurcation structure structureof ofthe themap map

[xjt+l)1 I

/

2

2

cos/? —sinfllrx1(t) -sin/7 \rcosfl . II / sin/? cosfijLx2v cos/? Lslnfl

2 Consider Consider the the discrete discrete predator-prey predator-prey system

rx1(t +

')l

Lx2(t+OJ=

—

I

( ) —) xl(t)x2(t) I

-x,[t)x2{t)

denotes the the prey prey population population atat generation generationt t and and x2(t) where xx1(t) x2(t) denotes denotes the x{t) denotes predator population at generation t. Show that a nontrivial fixed point predator population at generation t. Show that nontrivial fixed point of of the map undergoes aa Neimark-Sacker bifbrcation. bifurcation.

6.10 6.10 The The Liapunov Liapunov numbers numbers and and chaos chaos is well characterized by by an It is well known known that that chaotic chaotic dynamics dynamics is is characterized an exponential exponential divergence of of initially initially chose chose points. points. In In the case maps, the the divergence case of of one-dimensional one-dimensional maps, Liapunov exponent exponent isis aa measure Liapunov measure of the the divergence divergence of two two orbits orbits starting starting with with slightly different different initial initial conditions. conditions. For For aa map map on on R2, each orbit orbit has has two Liapunov Liapunov slightly R2, each numbers that measure the rates of separation from the current orbit point along two orthogonal directions. These directions are determined by the dynamics of the map. The first first direction direction is is the the direction direction along along which which the theseparation separation between between nearby nearby points is the greatest. The second is the direction of greatest separation, chosen points the greatest. The second is the direction of greatest separation, chosen perpendicular first. The stretching stretching factors perpendicular to the first. factors in each of these two directions directions are are numbers of of the orbit. To the Liapunov Liapunov numbers To formally formally define define Liapunov Liapunov numbers, numbers, we introduce D D(t) = f(x0), {l)=D

302

6. HIGHER DIMENSIONAL DIFFERENCE EQUATIONS

to denote denote the Let U be be the the unit unit circle circle the Jacobian Jacobian matrix matrixofofthe the/t-th -th iterate iterateofof/ f.. Let centered on orbit. Then Then D(,)(U) D^(ll) isis an ellipse. This centered on the the first first point point of of the orbit. an ellipse. This is a consequence of the fact fact that the linear transformation transformation of consequence of aa circle circle is an ellipse. The axis is is longer longer than than 11 in the the expanding expanding direction direction and and shorter shorter than than 1 in the in the axis contracting direction. 1

Definition 6.10.1. be aa smooth smooth map map on on R2, R2, and and let let D, D, == DfZ(xO). D ,(x0). For Definition 6.10.1.Let Let/ f be thelength lengthofofthe thek -th k -thlongest longestorthogonal orthogonalaxis axisofofthe the 2, , let rk^ bebethe 1, 2, ..., let kk == 1, k ellipse D, Dt(u) for an an orbit orbit with with initial initial point point x0. x0. Then Then r ^'' measures measures the the ellipse (u) for iterations. The The contraction or or expansion nearthe theorbit orbit that that at 0 during contraction expansion near at xx0 during the the first first tt iterations. k -th Liapunov number number of this trajectory is

L(k)x0 == JimV/^, L(k)x0 if the limit exists. The k 4/i -th Liapunov Liapunov exponent of the trajectory trajectory that that starts starts at at xx0 0 is h(k)(x0) = ln(L(k)(x0)), \n(L(k)(x0)), k == 1,2. \,2. h(k)(x0)= Clearly, LQ) l(l) >L(2) L{2)and andh(l) h{l) h(2). > h{2). Definition beaasmooth smooth map on R2. An orbit Definition 6.10.2. 6.10.2.Let Let/f be on R2. orbit {x0,

x1, x2,

asymptotically periodic convergesto toaaperiodic periodicorbit orbitas ast —* t - » oc; °°; that is, there is asymptotically periodic ifif itit converges exists a periodic orbit \y0, yy1, y2, l, y2, lim||x(/)-— y(t)\\ limDx(t)

suchthat that .} }such

=0,0, as tt—3 - » °°. =

Definition 6.10.3. 6.10.3. Let mapofofR2, R2,and andletlet{x0, {x0,x1, xx, x2, x2, Definition Let f/ bebea amap Theorbit orbitisissaid saidto to be be chaotic chaotic if: if: (i) (i) orbit of of f./ . The \XQ, x1, X j , x2, X2, {x0,

J

.} be

bounded a bounded

THE LIAPUNOVNIJMBERS LIAPUNOV NUMBERS AND AND CHAOS CHAOS 6.10. THE

303 303

not asymptotically asymptotically periodic; periodic; (ii) Liapunov number exact, and and (iii) (iii) the the is not (ii) no Liapunov number is exact, Liapunov > 0. Liapunov exponent exponent satisfies satisfies h(\) h(1)> the Definitions 6.10.1 6.10.3 extend extend to in m -dimensional case Rm, with the Defmitions 6.10.1 and and 6.10.3 case Rm, word word "sphere", and the word word "ellipse" "ellipse" replaced replaced by word "circle" replaced by the word the word "ellipsoid".

Chapter Chapter 7

Higher Higher dimensional dimensional economic economic dynamics This chapter chapter applies applies the concepts concepts and and theorems theorems of the the previous previous chapter chapter to to This examining behavior behavior of of different different economic economic systems. systems. Section 7.1 examining 7.1 studies Dornbush's exchange exchange rate model. model. We We show show how how aamonetary monetary expansion expansion will will Dornbush's an immediate immediate depreciation depreciation of the currency currency and sustain sustain the inflation inflation as result in an the price price level level gradually gradually adjusts adjusts upward. upward. Section Section 7.2 7.2studies studiesaatwo-sector two-sector OLG OLG model with the Leontief production functions. The economy produces model with the Leontief production functions. The economy produces two, consumption and investment, investment, goods; it has has two, two, consumption consumption and and investment, investment, consumption sectors. We provide provide conditions conditions when when the the system system isisdeterminate determinate or orindeterminate. indeterminate. Section 7.3 7.3 introduces introduces aaone-sector one-sector real real business business cycle cycle model model with with mild mild increasing increasing returns-to-scale with government spending. Section 7.4 introduces endogenous returns-to-scale with government spending. Section 7.4 introduces endogenous fertility and and old old age age support support into into the the OLG OLG model. model. Section Section 7.5 7.5 examines examines aa model model to to fertility capture the historical evolution of population, technology, and output. The capture the historical evolution of population, technology, and output. The economy evolves evolves three three regimes regimes that that have havecharacterized characterized economic economic development: development: economy from a Malthusian regime (where technological progress is slow and population from a Malthusian regime (where technological progress is slow and population growth prevents any sustained rise in income per capita) into a post-Malthusian growth prevents any sustained rise in income per capita) into a post-Malthusian regime (where (where technological technological progress progressrises risesand andpopulation population growth growth absorbs absorbs only only part part regime of output growth) to a modern growth regime (where population growth is reduced of output growth) to a modem growth regime (where population growth is reduced and income income growth growth is is sustained). sustained). The The model model isis defined defined within within the the OLG OLG framework and framework with a single good and it exhibits the structural patterns observed overhistory. history. with a single good and it exhibits the structural patterns observed over Section 7.6 examines a model of unemployment and inflation. We demonstrate that Section 7.6 examines a model of unemployment and inflation. We demonstrate that the model which is built on the well-accepted assumptions behaves chaotically. the model which is built on the well-accepted assumptions behaves chaotically. Section 7.7 7.7 provides provides aa model model ofoflong-run long-runcompetitive competitive two-periodic two-periodic OLG OLG model model Section with money and capital. Section 7.8 introduces heterogeneous groups to the OSG with money and capital. Section 7.8 introduces heterogeneous groups to the OSG model. Section Section 7.9 7.9 examines examines interdependence interdependence between between economic economic growth growth and and model. human capital accumulation within the OSG modeling framework. human capital accumulation within the OSG modeling framework. 305 305

306

7.1 7.1

7. HIGHER DIMENSIONAL ECONOMIC DYNAMICS DYNAMICS

An exchange rate model An

We now now consider the thefollowing following Dornbush Dornbush exchange-rate exchange-rate model1 model1

yd(t)= j — p(t))— / ( f ) = 8(e(t)+ S{e(t) +pp(t)) - a(r(t)— a(r(t) - p(t p(t++0+ l) +p(t)), p(t)), p(t + i) — p(t) = m—p(t)=Øy—ArQ),

—

(7.1.1) (7.1.1) (7.1.2) (7.1.2)

y),

(7.1.3) (7-1-3) (7.1.4)

r(t)=7+e(t+i)—e(t), r{t) = 7 + e(t + l)-e(t),

where 2,?'X,are parameters, where 5, S, p, a, a, a, y, y, m, m, 0, <j>, r positive are positive parameters, (t) yd(t)andand p(t)p(t) areare logarithms of domestic domestic aggregate aggregate demand domestic price level at time time logarithms demand and the domestic t; t; r is the the domestic domestic nominal nominal interest interest rate; the logarithm logarithm of the rate; and and e(t) is the exchange rate, that is, is,the the price price in in home home currency currency of of one oneunit unit of offoreign foreign money. money. rise in in eeimplies implies that that domestic domestic money money depreciates. depreciates. Here, and p are the A rise Here, yy and logarithms of the domestic domestic commodity 7 is is logarithms commodity supply supplyand andforeign foreignprice pricelevel; level;and and?' interest and in m is is the thelogarithm logarithm of of the nominal stock the foreign nominal rate of interest Equation (7.1.1) implies that home aggregate aggregate demand of money. Equation demand is a decreasing function of the the home-to-foreign home-to-foreign price expected real function price ratio ratio and and of of the expected real rate of and services services drives drives interest. Equation (7.1.2) tells that excess demand for goods and is a standard Keynesian LM schedule that price inflation. Equation (7.1.3) price inflation. Equation (7.1.3) a standard Keynesian LM schedule that relates the demand for money to real money and the nominal interest rate. relates demand for money to real money and the nominal interest rate. an arbitrage arbitrage condition condition in asset asset markets which says that the Equation (7.1.4) is an interest rate will will exceed exceed the the foreign foreign interest mterest rate by by an anamount amount exactly exactly home interest theexpected expected rate rate of ofdepreciation depreciation in in the thehome home country. country. The Themodel model equal to the equal consists of four four equations equations with with four four endogenous endogenous variables, variables, describing describing the consists evolution of aa small small open open economy economy with with perfect perfect foresight. foresight. Here, Here, by by perfect perfect evolution foresight it means that the the rate of of inflation inflation individuals individuals expect expect equals foresight equals the actual rate rate

i)

—

p(t) =

(P(t +

i) J

P(t)

— 1,

for small changes in the actual level level of prices prices P(t). P(t). the behavior behavior of of the the model, model, we we first first eliminate eliminate two two variables variables and To analyze the thedynamics dynamics to toaatwo-dimensional two-dimensional problem. problem. We Wecan caneliminate eliminate yd(t) reduce the (t) and 1

Themodel model isisproposed proposed by byDornbusch Dornbusch (1976), (1976), for foran anopen open economy economy adaptation adaptation of ofthe the ISISThe LM structure. The Theexample example isisfrom from Azariadis Azariadis (1993: (1993: 46-51). 46-51).

7.1. AN EXCHANGE RATE RATE MODEL

307

rQ) by respectivelyequation equation (7.1.1) (7.1.1) into into equation and r(t) by substituting substituting respectively equation (7.1.2) (7.1.2) and equation (7.1.4) into equation (7.1.3). (7.1.3). We We thus thus have have (i + i)— (i — — aa)p{t ac)p(t + ac ++ aö)p(t)= (l l) - (l - aa aS)p(t) = a5p-- aar*—-czae(t aae(t++1), (S + (8 + cr)ae(t) c)ae(t)++ce45 i), A.e(t+1)—Ae(t) =p(t)+ le(t + l) - Ae{t) =p{t)+ -Xr -m.

(7.1.5)

The system has aa unique equilibrium equilibrium point point

y + or 8

e =p -

(7.1.6)

(7.1.6)

p*=Ai+m_q:?v.

p =

m -

At At equilibrium, equilibrium, we wealso also have have

yd(t) =7. r(t) r{t) == 7r, yd{t)=y.

At equilibrium equilibrium the domestic domestic price price level level is an anincreasing increasing function function of the the exogenous variables m and exogenous variables and 7, F,aadecreasing decreasingffinction function of ofy; y;the theexchange exchange rate rate is is

an increasing functionofofminand and7, 7, a decreasing functionof of p. ji We increasing function decreasing function Wemay may transform dynamical system (7.1.5) into a homogeneous one by introducing —

x1(t) eQ) — e, x,{t)=e{t)-e,

x2{t)=p{t)-p. pQ)

Under the transformation, system system (7.1.5) (7.1.5) becomes becomes

1

[xj(t+i)1 [x2Q

+ l)j

1

1

—

[xj(t)1 X aS ad ++ ac/A ao/A 1— 11—ac 1-ac 1 - aa 1 - aa aS ad

or inmatrix form form or in x(t x(t ++1)l) == Ax(t). Ax{t).

The is The characteristic characteristic equation equation is

M

17 (7. (7.1.7) . )

___________

7. 7. HIGHER DIMENSIONAL ECONOMIC DYNAMICS DYNAMICS

308

z(p)=(p z[p) = (p-p p2) = p2 -—(TraecA)p [TraecA)p++DetA, DetA, pj(p —- p2)=p2 — l){p where = p1 />, ++p2 p 2=-22 TraecA =

DetA==p1p2=l— DetA = = p,p2 - 1 yir2

—

aS + cia/A 1-au 1 - ao a"^ f

aS" 55 + — \\

aS ad r. 2X) X) 2(1—as) X{\-aS)

1-au laa{ l-aa{

T

We calculate calculate 2 _4DetA=[ A=(TraecA)2 A = V{TraecA) - ADetA = a\-^—{s 7

[ l1—au ao-l

z(i)= (1— p1)(l

—

+ -Yl + 4 , 2 2(1—aS) AjJ A(l

—______

+ p3(1 + p2)=4

i

If If 11—- ac ao >> 0, 0, i.e. i.e.

J_ a<—, 1

C

the of A the eigenvalues eigenvalues are real (because of A >> 0), 0 ), and and lie lie on on opposites opposites of of11 (because (because

z(l) << 00 ).). The The sum sumof ofthe thetwo two eigenvalues is is less less than than 2 2 (because (because of z(1) eigenvalues of TraceA < 2 ).IfIfthe theequilibrium equilibrium point pointisisaasaddle, saddle,we weneed needthe theeigenvalues eigenvalues to tolie lie c 2). on different different sides sidesofof—1 - 1, , that is on is z(—1)>O z(-l)>0 »

a>-r——i——r. a> (2+21)(s+2a)

We conclude conclude that coefficient a smallenough enough (price (price adjustment adjustment is We that if the coefficient a isissmall is sufficiently low),the the steady steadystate state of of the the system sufficiently low), system is aa saddle saddle point. point. In Inthe the

remaining this section, we require require aa totobe remaining of of this section, we besufficiently sufficiently small small so sothat that the thetwo two eigenvalues satisfy: satis&: p1 px >> 1 and and \p2\ <1. < 1. can be be written as as The general solution can 1

7.1. AN AN EXCHANGE RATE MODEL e(t) =

+

+

=

p(t) = P +c,A + ++c2p2, P\t)

309 309

718

1)T the eigenvector eigenvector associated associated with eigenvalues, pp,, for where (hH, \f is the with the eigenvalues, / ; for We are are interested interested in in aa special special path path which which is is not not explosive explosive and and aa i/ = 1,1, 2. We situation in which for given level level of of price price the the exchange exchange rate rate adjusts adjusts as situation in which for the given the system on on the the unique convergent path. To exclude explosive needed to keep the cx = 0. 0. Then c2 c2 can canbe be determined determined by paths, we take c1

= p(0)

—

is straightforward to to show that equations equations (7.1.7) can can be be written as as It is

e(t)=e* +h2]{p(t)-p'). e(t)=e is direct to to check ^ < by It is <00 by Ah, Ak

=p2h2,

and IL'2! \p2\ < equation thus thus describes describes the the negatively negatively sloped sloped straight straight line line < 1. 1. The equation = (e*, \e , p"). through the equilibrium equilibrium point point S s = We apply this this equation equation to examine examine impact impact of changes changes in in some some We may apply m on onthe thedynamics dynamics of of the thesystem. system. An An increase increase in in in m will shift shift parameters, say, in equilibrium values equilibrium point. Equations (7.1.5) the equilibrium (7.1.5) show show that that the equilibrium values of the the variables will equal amount, to S = p) in two variables will rise rise by an equal amount, from from Ss =- \e (e,, p j to = (e, (e, AS) in the long term. As price is assumed to be sticky, the adjustment process is the long term. price is assumed to be sticky, the adjustment process depicted figure 7.1.1. 7.1.1. A A monetary monetary expansion expansion results results in an an immediate immediate depicted as in figure depreciation currency. In initial stage, the exchange exchange rate overshoots overshoots its depreciation of of the currency. long-run equilibrium equilibrium value, that inflation inflation is actually actually accompanied accompanied by aa long-mn value, so so that gradual currency during during the transition. transition. This happens as the the gradual depreciation depreciation of the currency presumed output prices prices puts puts the the fUll full burden burden of of the theimmediate immediate presumed stickiness stickiness of of output the exchange exchange rate. rate. The Theinstantaneous instantaneous depreciation depreciation produces produces aa adjustment on adjustment on the disequilibrium in the goods disequilibrium goods market market which which adjusts adjusts over over time time as as the output output price sudden depreciation, depreciation, and with output output prices prices being being adjusts. Following Following the adjusts. the sudden and with constant, domestic domestic goods goods become become cheaper cheaper relative relative to to foreign foreign goods. goods. This This leads leads constant, an excess excess demand demand for for domestic domestic output gradual increase increase in in domestic domestic to to an output and and to to aa gradual prices. prices.

7. HIGHER DIMENSIONAL ECONOMIC DYNAMICS

310

pp .the the convergent path

p

V

e{t + \)=e{t)

pQ)

>

pp

e(t +1):P e(t)

ee

ee Figure 7.1.1: The The overshooting adjustment process process

7.2

Nr

e

A two-sector OLG model model

This section section deals deals with with aatwo-sector two-sector overlapping overlapping generations generations model model with with fixed fixed coefficients by Ralf.2 Ralf.2 The The economy economy produces produces two, two,consumption consumption and and coefficients proposed proposed by investment, investment, goods; goods; it has has two, two,consumption consumption and andinvestment investment sectors. sectors. The The investment goods is taken in investment goods taken as as the thenumeraire. numeraire. Individuals Individuals live live two twoperiods periods - in each period, period, there there are areold old and andyoung young people. people. People People work work only only in in the thefirst first period period lives supplying supplying one unit unit of of labor labor inelastically. inelastically. There There are N(t) N(t) young young of their lives individuals young individual w(t) the representative representative young individual splits his his income income w(t) individuals at it;t; the consumption, c(t), c(t), and saving, saving, s(t). The The young young agent's agent's problem problem is is between consumption, defined as defmed iviax(c2(t)+ Max[cx{t)+ cr(t ++ l))"'', ye(—oo,O) ye (-°°, 0) sQ) = w(t) — p(t)c(t) s(t)=w(t)-p(t)c(t)

s.t:

2

/ ,x c(t+l)=

r(t l)s(t) r(r + lW/)

p(t+l)

See See Ralf (2001). For For two two sector growth models, see see also Galor Galor (1992) and and section 15.5 15.5 in Azariadis (1993). inAzariadis(1993).

7.2. A TWO-SECTOR OLG OLG MODEL MODEL

311 311

where r(t +1) + i) are are interest paymentson on saving, saving,and andp(f) p(t) is is the price where interest payments price of the the consumption goods investment goods. consumption goods in in terms terms of the investment goods. Suppose Suppose the the elasticity elasticity of of substitution between consumption of different different periods 1 a --y—1 1

r-Y

is restricted restricted to be less than 11 in in absolute absolute value. value. The The optimal optimal saving function function is given by w(t) 1 + (r(t +

l)p(t)Ip(t + 1))b0

(7.2.1)

where

—7

70—

y—1

Once 1), w(t), w(t), and Once p(t), p(t + + l), and r(t + i) l) are determined, we uniquely determine determine well as as c(t) s(t), as well c(f) and c(t c(t ++ i). l). Using respectivelytoto denote Using subscripts subscripts c and and I/ respectively denote the consumption consumption and investment sectors, we have the production functions of the two sectors

c(t) = = Fc(Kc(t), Nc(t)) = = min{Kc{t), acNc{t)}, C(t) i(t) = N1(t)) 1, ,(t)) -= ,umin{K,(t), fimm{K,(t), a1N,(t)}, a.N^t)}, p // >> 1,

capital and and labor where Kc, Kt, Nc and Nc denote denote capital labor inputs inputs of of sector sector jj,, j = i, c, the aggregated aggregated consumption, l(t) isis the the total total investment. investment. c, C(t) c(t) isis the consumption, and and 1(t) Introduce

K(t)

N.(t)

N1(t)

N(t)

We have n

fl:tkct+ nl(t)k,(t) = = k(t). nc(t)kc(t)+n1(t)k,(t)

(7.2.2)

312 312

ECONOMIC DYNAMICS DYNAMICS 7. HIGHER DIMENSIONAL ECONOMIC

Efficient production in the two sectors sectors requires requires Efficient

K1(t)= KJ(t) =a1N1(t). aJNJ(t). That is is

k]{t) == a1. a j. k1(t) sector takes takes place place with with the the capital capital intensity intensity Production in the consumption goods sector higher goods sector, sector, if ac > a,. «,. Perfect Perfect competition competition in in higher than than in the investment investment goods the markets yields yields =

w(t)

ccc

+

(7.2.3) (7.2.3)

w(t) w(t)==a1p a,/! —- a1r(t). a,r(t).

Investment is given as output of the the investment investment goods goods sector, sector, i.e. i.e. i(t) = a,pn1 (t),

where where ii = I11N. Capital stock stock is is assumed assumed to to depreciate depreciate fully fully after after one oneperiod. period. / N. Capital output of of the theinvestment investment goods goods sector sector can canbe beused used ininthe thefollowing following period. period. The output Hence

k(t +i) =

1(t)

= a,,un,(t).

On On the the other other hand, the saving in period equalsthe theinvestment investment in in period period t,t, i.e. i.e. period t equals e(t). s{t)=i{t). S@)=

We have thus built the the two-sector two-sector model. model. From From We k(t *(/+!) = a,pn,(t), a,fm,{f), kj(t) = a1, atj,

and and equations equations (7.2.2), (7.2.2), we w ehave have

___= 7.2. A TWO-SECTOR OLG MODEL

k(t + i) = paa, (k(t) — ae),

313 313 (7.2.4) (7-2-4)

where 11 a =

a,

-ac

solve By equation (7.2.1), we solve

- if11o.

(7.2.5)

On the other hand from from equations equations (7.2.3), (7.2.3), we we solve solve w(t) = = aaia(p(t) w(t) — c{p{t)-ju), r(t) r{t) == a(a,p a{aiJu-a — cp{t)).

From the equation for for w w ininequations equations (7.2.6) (7.2.6) and

s(t)= s{t) = apn1(t), a,nnt(t), «,(/) =

a{k{t)-ac),3

we obtain obtain we

w(f) _ ac 5(t) s{t)

p{t)-ju

2

.^-.

p

i)

From the equation for for rr ininequations equations(7.2.6) (7.2.6)

p(t+1) PJt + l) —7 r(t

+

1)7

p(t+1) Pit + l)

_ —7

a(a1p

—

cçp(t7 + i))\\-

r(/ + 1) cApc,n - acP{t + 1)) Substituting equations (7.2.7) and (7.2.8) into equation (7.2.5) yields Substituting equations (7.2.7) and (7.2.8) into equation (7.2.5) yields

1

This This relation relation isis obtained obtained from from equation equation (7.2.2) (7.2.2) and and kj == ap Ij == c, c, i.i.

(/.Z.o)

__________ 314

7. HIGHER DIMENSIONAL ECONOMIC ECONOMIC DYNAMICS DYNAMICS

(tsl'l— I

ci, + g(p(t), k(t)) k(t))

,

-

1/10

ap(t)[

cM

(7.2.9) (7.2.9)

J

equations The dynamics of the economy is now described by the motion of equations (7.2.4) and (7.2.9). Suppose and (7.2.9). Suppose ac >> a.. at. Then, for the variables to be positive, we should add the following constraints on the solution

ac >k(t)>a',, > k{t)>ai,

)c p(t)

(7.2.10) (7 2 10) ' -

£%JJ

If ac
a" or,

P{)

(7.2.11) (7 2 U)

tLk(t)

--

ac

From (7.2.4), we we see From equation equation (7.2.4), see that that k(t) k(t) is independent independent of p(t). The The of p(t). equation has a unique steady state

*

'

=

1-- l/pcza, \l /iaat

From equation equation (7.2.9), (7.2.9), a steady state state of price is determined by *

1 sy *% jtk //If Q\p) = ajp -\— — ^ — - — 1 £Z[,llk -a'jij a[juk -acJu]

1/ye

-/ia.=0.

Under constraints constraints (7.2.10) (7.2.10) or or (7.2.11), (7.2.11), the equation equation

cI(p)=0

(7.2.12) (7.2.12)

7.2. A TWO-SECTOR OLG OLG MODEL MODEL

315 315

has a unique has unique solution.4 solution.4 It is is straightforward straightforward to calculate calculate the the two twoeigenvalues, eigenvalues, Pj, at the steady state

p1 /?, ==crpcç ajua, >0, > 0, P2

Jacp acP

_acp*/y
-/&) -juk )

jua,

In economics, a concept called indeterminancy indeterminancy is currently often often used. This concept concept is related to to stability stability of of steady steady state. state. IfIfaasteady steadystate stateisisasymptotically asymptotically stable, for any given initial stable, for initial capital capital stock stock the the economy economy will will eventually eventually approach approach the steady state. state. Hence, Hence, the the agent agent isis not not able abletotodistinguish distinguish rationally rationally between between the different different perfect perfect foresight foresight paths. paths. This This property property isis called calledindeterminancy. indeterminancy. Determinacy of the perfect foresight foresight dynamics requires that the steady steady state state is is aa saddle or a source. If a >> 0, saddle or source. If 0, i.e., i.e., aa,t >> ac, the system system isis always always determinate, determinate, since and it is is aa since p1 /?, is greater greater than 1; 1; the the steady steady state state isis aasource source ififp2 p2c <—- 1 and saddle point ifif pp2 >— - 1.1. If a c<0,0, i.e. i.e. saddle point 2 > 1

a,


the steady steady state state may may be be either either indeterminate indeterminate or or determinate, determinate, depending depending on on the the parameters.55 parameters.

7.3 7.3

Growth with government government spending spending

This section introduces introduces aa one-sector one-sector real real business business cycle cycle model model with with mild mild increasing increasing 6 returns-to-scale with government government spending. spending.6 Theeconomy economyhas has aa continuum returns-to-scale with The continuum of identical firms in in the economy, identical competitive competitive firms economy, with the total total number number normalized normalized to

4

the value value of of Q(p) at This can be proved by calculating calculating the at the the boundaries boundaries of the the range range of of and then applying applying the mean value value theorem. theorem. definition and 5 When the parameters parameters are specified, specified, we may may have have /?, == —1. - 1 . PeriodPeriod-22 solutions solutions appear. appear. It It should should be be noted noted that that in in aa two-sector two-sector model model of of optimal optimal growth growth with with infinite infinite horizon horizon by Nishimura and Yano Nishimura and Yano (1995), (1995), the the optimal optimal time path path of of capital capital accumulation accumulation may may be be chaotic for any any rate rate of of time time preference, preference, ififthe theproduction production function function in in the theconsumption consumption chaotic takes place place with with aa higher higher capital capital intensity intensity than in in the investment investment sector. sector. goods sector takes 6 section is is based based on on aamodel model developed developed by by Guo Guo (2004), (2004), which which isis influenced influenced by by This section Devereux et et aL al. (1996, (1996, 2000). 2000). Devereux

7. HIGHER DIMENSIONAL ECONOMIC DYNAMICS 7. DYNAMICS

316

Each firm firm produces produces output output y(t) y(t) according according totoa aconstant constantreturn-to-scale return-to-scale one. Each technology

y{t) =x(t)ka(t)hl_a(t), x{t)ka{t)ti-a{t),0< 0<«
U w(t) =

(7.3.1)

h(t)

x(t) changes changesaccording according to to Assume that x(t) x(t) = a*(t)h{l-a)*(t), x(t)=k

z

me

the degree degree of of productive productive externalities. externalities. Under Under the externalities, externalities, the function becomes production fbnction where where x isis

y(t) =

(t),

(7.3.2) (7.3.2)

where where

zk=aO+x), zh_=(l—aXI+z). Zt = a(l + Z\ Xh = ^~ «)(l + Z\ The The economy economy is populated populated by aa unit unit measure measure ofofidentical identical infinitely infinitely lived lived household, each endowed endowed with one one unit unit ofoftime. time.The Therepresentative representative household household household, maximizes maximizes aa discounted discounted stream stream of ofexpected expected utilities7 utilities7

7 should be be noted noted that that we weare areonly onlyconcerned concerned with with the thecase caseofofperfect perfectforesight foresight ItIt should dynamics with full full capital capital utilization. utilization. Guo Guo (2004) (2004) examines examines other other cases, cases, for for instance, instance, dynamics

when capital capital may may not notbe befully fully utilized. utilized. when

7.3. GROWTH WITH GOVERNMENT SPENDING 1+c

00

317

0<0<1, a0, q>0,

, 0 < 6> < 1, ( 7 > 0 , 77 > 0 ,

:+ a where 06 isisthe thediscount discount factor, factor, c(t) c(t) is is consumption, consumption, and denotes the the inverse inverse of and oa denotes the intertemporal elasticity of of substitution for for labor supply. The The budget constraint is is

cl++i{t)+tif)=w{t)h{t) +r(t)k(t), r{t)k{t), 1(t) + r(t) = w(t)h(t) + i(t) is investment investment and lump-sum tax. tax. We We may may rewrite rewrite the the budget where 1(t) and t(t) r(t) isisaalump-sum constraint as

c,+i{t)+T{t)=y(t), c1 + i(t)+ r(t)= y(t), where we we use use

y{t) = ^t)h{t)+r{t)k(t). y(t) + r(t)k(t). = w(t)h(t) The motion of capital is given by (i

k(t +

—

o)k(t)+

1(t),

k0 being being given, (0<8<J< thecapital capitaldepreciation depreciation rate. rate. The Thefirstfirstk0 given, where where 88 (0< c 1)1)isisthe for the the optimization optimization problem problem are aregiven givenby8 by8 order conditions for

7jc(t)h°(t)=w{t), 17c(t)h"(t)= w(t), =

o[(1

k(t+1) linALo. =0.

lim

'-*

£-*OO

H

+ r(t +

(733)

-

(7.3.3)

c{t) c(t)

Assume government spending, spending, g(t), is constant. constant. The The period period government government Assume that government budget constraint is f(t). g = r(t). 81

See appendix appendix A.9 A.9 for for dynamical dynamical optimization. optimization. See

7. 7. HIGHER DIMENSIONAL ECONOMIC DYNAMICS DYNAMICS

318

The aggregate resource constraint for by for the economy isisgiven given by c(t) k(t ++i)— (i — 5)k(t) + g = = y(t). c{t)++k{t \)-(\-S)k{t)+g

It can the sense that we can be be shown shown that this this equation equation is is redundant redundant in the we can canobtain obtain it the motion through the household budget, the motion of ofcapital capital and and marginal marginal conditions conditions (7.3.1). (7.3.1). We now show that the expressed by byaa two-dimensional two-dimensional dynamical dynamical the dynamics dynamics isisexpressed system. First, from equations (7.3.1) and (7.3.2), (7.3.2), and and 17c(t)hCQ) = w(t)

in equations (7.3.3),we wesolve solve h(t) h(t) as a function ifinctionof of kit) k(t) and and c(t) equations (7.3.3), c(t)as asfollows follows 1/(xh-1-a)

h(t)

2

- —k15c(t)] ^1

.

= [1

(7-3.4)

(7.3.4)

From equations (7.3.3), (7.3.3), we wedirectly directly have have

ö+ c(t + + i) l) == [(i [(l —- S + akXt~x(t(t ++ \)hx-(t (t ++ l))]ft
(7.3.5)

where we use use

ay(t+1)

k(t+1) equation (7.3.2). (7.3.2). From From the thehousehold householdbudget budgetand and gg == t(t), we and equation we solve

i(t) = g. '(0 = y(t) y(t)—- ccQ) (t) -— gSubstituting this equation into the the equation of of motion of of capital yields

k(t (t) (t) — c(t) +i) 1) == (i(1 — - 5)k(t) 5)k{t) + + k*> [t)h*> - c{t) - g, k{t + — g,

(7.3.6) (7.3.6)

we use where we useequation equation (7.3.2). (7.3.2). Substituting Substituting equation (7.3.4) into into equations equations (7.3.5) (7.3.5) and (7.3.6), we we obtain obtainthe the two equations equationswhich whichcontain containonly only kk and andcc

GOVERNMENT SPENDING SPENDING 7.3. GROWTH WITH GOVERNMENT yi (t) mroY1 {t)

i) = (1(1 —" S)k{t) k(t 8)k(t) + 7]0C*° *(/ + 1)

- C{t) g, c(t) -— g,

—

\%6c{t), °u{t ++ 1))]&(t),

(t + c(t [(1— c{t + i) l) = [(l - S8 + a7]ocz'{t

319

(7.3.7)

where \Zo

7

-

= do —

_[

Xh

Z'h -1-a '

17o _ =I

'/o —

17

\-a

It should should be noted noted that that the the right-hand right-hand side side of of the the first first equation equation in in equations equations c(t + i) l) and and k(t k(t ++i). l).ItItisisstraightforward straightforward to to show show that that the the system system (7.3.7) contains contains c(t has a unique unique non-trivial non-trivial steady steady state state (k*, [k*, c*). c). The The Jacobian Jacobian matrix cj is matrix atat [k*, (F, c*) given by p2 Pi + P2P3

p1

JtJ= = pip3

1

p4

p4

where

=1-8—

Pl=l-S-

(i +

+(1_g/y*)(1+a)j P2=S

+ ^ -

(9o+8)LtYl+a)+zh —1—a] (zh—1-a)(1+9O) p4 =1—

+ 8)zh (ih —1— a)(1 +

where where 60 = 1/9 1 / 6—- 1.1. Local Local indeterminacy indeterminacy requires requires that that both both eigenvalues eigenvalues of of .JJ

are less than 11 in in modulus. modulus. ItIt can can be be shown shown that that aanecessary necessary condition condition for for the the equilibrium point to display indeterminacy indeterminacy is is Xh > 1 +

320

7.4

7. 7. HIGHER DIMENSIONAL ECONOMIC DYNAMICS DYNAMICS

Growth with fertility and old Age support

section introduces introduces endogenous support into the theOLG OLG This section endogenous fertility fertilityand and old old age age support model introduced in section section 3.3. 3.3. 9 In the the literature, assumptions about about model introduced literature, two two assumptions population population growth growth and and economic economic conditions conditions are are made; made; the the first first is is to assume that in the utility function of the children are consumption goods and appear in function the parents; the 10 second is is to to assume that children children are valued as a source of old age support. This are valued of support.'° assumption is is made based on onthe theobservation observation that that countries countries with with more more developed developed assumption or better social security programs tend to have lower population financial markets financial markets or better social security programs tend to population growth rates than those that do do not. For Forinstance, instance, in in developing countries, children and prior prior to to often start start contributing contributing to often to family family income income while while living living at at home home and adulthood. Here, assume that individuals give constant fraction fraction of of their their adulthood. Here, we we assume that individuals give aa constant to theft their parents parents in support, and income to income in the the form form of of old old age age support, and that that parents parents incur incur aa time time cost of of raising raising children children which is increasing increasing in inthe the number number of ofchildren. children. cost which is The number agents at time young agents time tt is denoted denoted by N(t) N(t) and andthere there are The number of young N(-1) 1)agents N(— agentsin inthe the initial initial old old generation. generation. The The initial initial old old generation generation is endowed anaggregate aggregate initial > 0. 0.All Allyoung youngagents agentsare areidentical identical with an initial capital capital stock stock of of A^(o) K(O)> and they are areendowed endowed with with one one unit of of labor labor when when young, young, and andhave have no noother other endowments of at any date. There ofgoods or or assets at There isisaasingle single good good in inthe the economy, economy, which is is produced produced using capital and andlabor as as inputs. inputs. For Forsimplicity, simplicity, it it is is assumed assumed that individuals care only about old age consumption, denoted by c; their consumption, denoted by c; their lifetime lifetime individuals only about old consumption utility is assumed to be increasing in consumption utility is is given givenby by u(c), u(c), which is be increasing in c, c, and and concave. Agents work and old. Agents work and raise children children when young, and and are areretired retired when old. Young remit aa fraction, fraction,//, p, of Young agents agents remit of their their income income to their parents parents as old old age age support, and save save the rest rest which which is equal support, equal to to (i(l —- p) jii) of of the income. income. Each Each young young have n(t) n{t)children. children. A A time time cost cost of ofraising raising children, children, 7(n(t)), y(n(t)), agent decides to agent decides to have satisfies j/(0) = 0, y'>O, f> 0, y"O. y" > 0. 7(0)r0, All agents take the the wage rate, w(t), wit),atatperiod perioditt as as given. The The saving 5(t) s(t) is is given by by = (1 — a)w(t)(i — y(n(t))). 5(t) s(t)=(l-a)w(t)(l-r{n(t)))-

9

( 7(7.4.1) A1)

The Themodel model isisproposed proposed by byChakrabarti Chakrabarti (1999). (1999). The The first approach approach is is accepted by, by, for for instance, instance, Barro and andBecker Becker (1989), (1989), Becker Becker etetal.al. (1990), Galor and and Weil (1996); the second approach by, for instance, Raut and andSrinivasan Srinivasan (1994). Chakrabarti tried to combine the two ideas within the same framework. 10 10

7.4. GROWTH WITH FERTILITY AND OLD OLD AGE SUPPORT SUPPORT

321 321

Old age consumption is thus given by

c(t 1) = r(t + l)s(t) 1)s(t) + + an(t)w(t y(n(t + l))), c{t + l) r{t + an(t)w{t + + i)(i l)(l —- y{n(t

where r isis the therate rateofofinterest. interest. The Theproduction production isis carried carried out out according according to the the constant returns to scale production fimction function

F(t) = F(K(t),

(i

—

where K isisthe thecapital capital stock. stock. The The intensive intensive production production function f[k), thnction is is given given by by f(k), where K

N(1-y) Assume that the production function function is of the CES form

f(k)=(ak" f{k) = (akp +/3)"", + p)'p, a,/3>O, a,p>Q, —oo
K(t 1) = K{t ++ l) = s(t)N(t). s{t)N{t). The labor and and capital capital markets markets are areperfectly perfectly competitive. competitive. The The marginal marginal The labor conditions are are

r(t) = =f'(k(t)) fikit)) == ak^ (t)[ak" (/)[«*"(t)(t)+ + p\»-\ r(t) + w(t) w(t)==f(k(t)) /(*(,))-- k(t)f'(k(t)) k{t)f'(k{t))= = /3[ak»(t) + / t ' " , ^ 0.

(7.4.2)

The The young young agent's agent's problem problem is is to to maximize maximize u(c(t u(c(t ++1)) choosing n(t) n(f) i)) by choosing subject to c(t a)r(t + 1)w(t)(l c(t ++ 1) ! )== (i( !—- a)r{t

—

7(n(t))) + an(t)w(t +

i)(i — y(n(t +

i))).

7. HIGHER DIMENSIONAL ECONOMIC DYNAMICS

322

w and andr rasasgiven, given,the theproblem problemofofoptimization optimization has has an an interior interior optimum optimum With w determined by - (l - a)r(t a)r{t ++ 1)w(t)y'(n(t)) l)w(t)f{n(t)) ++ aw(t aw{t ++ i)(i l)(l —- 7(n(t y(n(t ++ i))) l))) == 0. 0.

— (1 —

(7.4.3) (7.4.3)

(7.4.3) yields Substituting equations (7.4.2) into equation (7.4.3)

JKK

—

+

aw(k(t + I))[l

-

—

7(n(t + 1))]

-

(i {\-a)w(k(t))f{n(t))

744

which states states that that the the rate rate of return which return on capital capital is equal equal to the rate rate of of return return on on children." children.11 By By the the defmition, definition, we have K(t + i) i)[i —- y(n(t + l))}k(t l) == N(t N{t ++ l)[l y{n{t + l))]k(t ++ l). i). With this equation and equation (7.4.1), (7.4.1), we we can can rewrite rewrite K(t K(t ++ \) = s(t)N(t) s(t)N{t)

as —

(i

—

a)w(t)(l

—

7(11(t)))

l—y(n(t+1))n(t)

(7.4.5)

In the remainder of of this section, we specify the cost fimction function as follows follows

y(n(t)) = cn2(t). We We also also introduce introduce aa new new variable variable

z(t)

11

rate of of return return on onchildren children 7'f{n(t)) given by byequation equation (7.4.3), (7.4.3), which which equals equals the the The rate (n(t)) isis given

right-hand right-hand side side of ofequation equation (7.4.4). (7.4.4).

____

WITH FERTILITY FERTILITY AND AND OLD OLD AGE AGE SUPPORT SUPPORT 7.4. GROWTH WITH

323 323

Multiplying both equations (7.4.4) k(t + +1) Multiplying both sides sides of of equations (7.4.4) and and (7.4.5) (7.4.5) by by k(t i) and (7.4.5), we we obtain obtain utilizing equation (7.4.5), 2

/ ,\ 1— — cn2(t) cn (t) z(t+1)=a z { t + x ) = a

._ . .. (7.4.6) (7A6)

2

2cn (t)-

Applying equation (7.4.6) to equation (7.4.5) yields

I— -

cn2(t 1) = a0 a0 cn\t + + i)

(1 —

(747)

where

a0

(i

(

—

a

ç 2ov

{z(t),n(t) n(t)}. equations (7.4.6) (7.4.6) and and (7.4.7) (7.4.7) describe describe the the sequence sequence of of{z(t), Difference equations }. A fixed point of the system is determined by

p(z)_z—a

1 — cn2

2cn

1— cn2 —

a0

=0,

(7.4.8) (7.4.8)

z-l-1 I/p-I

(i

—

(7.4.9) (7/1.9)

Proposition 7.4.1. 7.4.1. If p> p >0,0,then thenequations equations(7.4.8) (7.4.8) and and (7.4.9) (7.4.9) have have aa unique unique Proposition solution. solution.

The proof is left to the reader. equations (7.4.8) (7.4.8) and and (7.4.9) (7.4.9) The reader. When When p >> 0,0, both equations 7.4.1. define downward downward sloping slopingloci, loci,as asdepicted depicted in infigure figure7.4.1. define The linearized linearized system system of of equations equations (7.4.8) (7.4.8) and and (7.4.9) (7.4.9) atat aafixed fixed point point isisgiven given The by by 0o

nzl—p nzl-p

ap 1 + z apl+z

- ^ -4 zz

aa

ccn n

1—p \-p pen1 pcn2

324

DIMENSIONAL ECONOMIC ECONOMIC DYNAMICS 7. HIGHER DIMENSIONAL n

—1/2 .,-1/2

W(z)=0 )=0

C

=0

Figure 7.4.1: The unique unique fixed fixed point point for for p >> 0

We have T 1 = Trace.J

TraceJ =

DetJ=

zZ

i—p

a a

pcn2 pen

i—p pcn2 pen 11+ zz

the case caseof of pp >> 0, In the 0, TraceJ < 0 and DetJ < 0, J has one positive and one TraceJ
TcD<1+D. T < D < 1 + D. We conclude conclude that that the positive positive eigenvalue is strictly less than one. It can be proved thatif 0

The steady state is aa saddle. saddle.When When p >> 0.5, 0.5, the fixed fixed point can be either aa sink sink or or aa saddle saddle point. point.

7.4. GROWTH WITH FERTILITY AND OLD AGE SUPPORT SUPPORT

325

Proposition Proposition 7.4.2.12 7.4.2.n If If p >> 00 and and sufficiently then, (i) sufficientlylarge large(with (withfixed fixeda,a,a), a), then,

there is a critical there critical value of c, c, denoted by for c > (c) (<) c5, c , the fixed by cc,, such that that for point is a saddle (sink); (ii) At At c = = cc,, aa flip bifurcation point (sink); (ii) bifurcation occurs. Thus, for cc in a solutions of of period period 2 can neighborhood of c5, c , periodic solutions can be be observed. observed.

theconfiguration configurationofofequation equation(7.4.8) (7.4.8)isisthe thesame sameas asin inthe the In the case of p c<0,0,the case of p> locus p >0;0;thethe locusdefmed definedbybyequation equation(7.4.9) (7.4.9)has hasthe thebell-shape, bell-shape, and and attains a unique (local) maximum at at the the value value I1

z =

z =

p

.

P

As demonstrated in figure figure 7.4.2, 7.4.2, there are four possible cases. These cases can can be be identified identified as as follows. follows. For figure 7.4.2a 7.4.2a

a

C

c

a

p

0.1

0.25

0.5

—0.5

0.1 0.25 0.5 -0.5

(4,ii;) (z*, «,*)

\Z2,n2)

(0.965, 0.443) (58.592, 0.058) (0.965,0.443) (58.592,0.058)'

For figure 7.4.2b For figure 7.4.2b a C a c 0.1 0.25 0.1 0.25

a

p

a p 0.5 —2 0.5 - 2

(z*, «;) (z'2, «*) (0.884, 0.463) (4.469, 0.210) (0.884,0.463) (4.469,0.210)'

For figure 7.4.2c 7.4.2c

a c a (z*, «*) p 0.6062 —4 0.1 0.25 0.25 0.6062 - 4 (1.418, 0.369) 0.369)' aa

C

For figure 7.4.2d

121

aa

cc

a

pp

0.1

0.1 0.1

0.4

—4 -4'

The The proof of this proposition is referred to Chakrabarti (1999).

326

ECONOMIC DYNAMICS DYNAMICS 7. HIGHER DIMENSIONAL ECONOMIC n

11

C

=0

=0

z\ —i/p -\lp z\ (a)4—1/p
z

(b) -1/p < z* < z\ (b)—1/pcz
n

11

C-0.5

z

zl

C

p(z)=0

-0.5

=0 1=0

z

z

(c) f >—llp (c) z*>-\lp

z

(d) no no fixed fixed point (d)

Figure 7.4.2: 7.4.2: Fixed Fixed points for for p << 00

straightforward to check the the eigenvalues eigenvalues for for the above above examples. examples. Further It is straightforward examination of of the the model is is referred to Chakrabarti. Chakrabarti. examination

Exercise Exercise 7.4 7.4 11 Prove 7.4.1. Proveproposition proposition7.4.1.

7.5. GROWTH WITH DIFFERENT DIFFERENT TYPES TYPES OF OFECONOMIES ECONOMIES

7.5

327 327

Growth with different different types typesof ofeconomies economies

This section introduces a growth model, proposed by Galor and and Weil, that captures 13 ofpopulation, population,technology, technology,and andoutput. output.13 The economy the historical historical evolution evolution of The economy evolves three regimes that have characterized economic development: from a evolves three regimes have characterized economic development: from Maithusian regime (where technological progress is slow Malthusian regime technological progress slow and andpopulation population growth growth sustained rise income per per capita) capita) into into aapost-Maithusian post-Malthusian regime regime prevents any sustained prevents rise in income (where progress rises rises and population growth absorbs absorbs only only part of (where technological technological progress population growth output output growth) growth) to aa modem modern growth growth regime regime (where (where population population growth is reduced reduced is sustained). The Themodel is is defined defined within the the OLG OLGframework framework and income growth is with aa single good. The Theproduction uses uses land land and andefficiency efficiency unit unit of of labor labor as asinputs. inputs. The units is The supply supply of land land isisexogenously exogenously fixed. fixed. The The number number of efficiency efficiency units endogenous. endogenous. The output produced at time tt is Y(t) = Ha(t)(A(t)xy_a,

where X and H(t) H(t)are arethe thequantities quantities ofofland landand andefficiency efficiency units units of oflabor labor where X and employed in in production at at t,t, 0 < a < A(t) > 00 is is endogenously t. The Theoutput output per perworker at at itt is determined technological level at it. is

y(t) =

y(h(t), x(t)),

(h,x)x>0 where Yh yh > 0, yx >0 > 0 for for any any (h, x)» 0 where , H AX h = —, x = , N N TV N

where N(t) N(t) isisthe thetotal totallabor labor force force atatt.t.Suppose Supposethat thatthere there are areno noproperty property rights rights over land land and andthe thereturn return to toland land isisthus thus zero. zero.The Thewage wageper perefficiency efficiency unit unitof oflabor laborisis over therefore equal to its average avenge product 7

f\\1—a

w(t) M>)= iri =

13 13

w(x(t), h(t)). =w(x(t),h(t)).

(7.5.1) (7.5.1)

See See Galor and and Weil (2000). (2000). Rather Rather than than analyzing analyzing behavior behavior of ofthe the difference difference equations, equations, section shows shows how howto toconstruct construct aadiscrete discretedynamical dynamical system system which which explains explains very very this section complicated complicated economic economicphenomena. phenomena.

328

7. HIGHER DIMENSIONAL DIMENSIONAL ECONOMIC DYNAMICS

Each individual born at period period t — - 11 lives two periods. In the first period, they fraction of their their parents' time. time. In In the the second second period, period, they they allocate allocate their their consume a fraction endowed endowed one unit unit of of time time between between child-rearing child-rearing and and labor labor force force participation. participation, In hi each period t a generation that consists of N(t) identical individuals joins the each period t a generation that consists A^) identical individuals the labor force. The The utility is is represented by

u(t)== c}-y{t)[w(t + \)n{t)h{t++ l)] r , c(t) > c > 0,

u(t)

where Z c isis aasubsistence subsistence level, level, n(t) n{t) isis the the number number of of children children of of individual individual t, h{t + i) l) is the level level of of human capital of each child, child, and l) is the wage per per h(t and w(t + 1) efficiency unit time t + 1. The utility utilityfrmnction function is monotonically monotonically efficiency unit of of labor labor at at time + 1. and strictly strictly quasi-concave. Let Let increasing and 'r0 + re(t ++1)\) to+te(t

be the the total total time time for for aamember member ofofgeneration generation itt of raising raising a child child with level of with a level education quality e(t e(t + i). l). Define Define potential potential income income as asthe theamount amountthat thatgeneration generationitt would earn if they devoted devoted theft their entire entire time time endowment endowment to to labor labor force force participation. participation. is, potential potential income income is is given given by by w(t)h(t). w(t)h(t). This This income income isisdivided divided between between That is, and working. working. Hence, Hence, in in the the second second period period of of life, life, the the individual individual daces daces child-rearing and constraint the budget constraint + l)}+ i)} + c(t) w(t)h(t)n(t)[r0 + ve(t re{t + c{t) < w(t)h(t). w(t)h(t). w\(t)h(t)n(t)[T0 +

It is is assumed assumed that that the the level level of of human human capital capital of ofmembers members of of generation generation it,/, h(t h(t ++ I), l), is an an increasing increasing function function of of theft their education education e(t e(t ++ 1) l) and a decreasing decreasing function function of rate of of progress progress in in the the state state of of technology from from period period it/ to to itt ++11 the rate

8{t+ l)

=

A(t+1)—A(t)

4)

A(t)

That That is is

h(t+l)=h(e(t+l),g(t+l)), h{t + l) = h{e(t + \),g{t + l)), h,he,hg,he>0, >0, hg,hee<0,

y{e,g)>0. V(e,g)0.

7.5. GROWTH WITH DIFFERENT TYPES OF OF ECONOMIES

329

erosion effect effect is assumed assumed to become higher as aa result The erosion result of of technological technological progress, i.e. progress, ay(h(t), x(t)) dy(h(t\x(tj) > ag(t) It is further assumed that

r A ( 0 , o) 0 )—- *zh(O, ( 0 , 0o)) <<0. 0. It is assumption, there exists exists a value of g is straightforward straightforward to show that under this assumption, such such that —

= o.

zh(o,

Denote z(t ) and Denote z(t)

izthe thelevel levelofofpotential potentialincome income and andthe thelevel levelofofpotential potential

income income at which which the the subsistence constraint is just binding; binding; that is

z(t) = h(t)w(t), z= z(t)=h(t)w(t), z=

C

By By equation (7.5.1), (7.5.1), we we have ZQ) z{t) ==

h'(t»-*{t) == ha(e(t), h"(e(t), g(ty-"(t) (t) = z(e(t), z(e(t), gQ), g(t), x(t)).

(7.5.2) (7.5.2)

Members Members of generation generation t/ choose choose n(t) n{t) and and h(t+1) h(t +1) to to maximize maximize their intertemporal utility function subject to the budget constraints. It can be proved that the the optimal optimal solution isis characterized by the the following solution

Ir

, if

r0 0)' te(g(t ++ l)) t0 ++ re(gQ

1—?/z(t) \-clz{t) if r0 re(g(t + 0)' To ++te(g(t

{e(t+l')=0,

ifg(t if + i) = e(g(t +i)), ), e'>O e'>0 if if g(/

(7.5.3) (7.5.3)

(7.5.4) (7.5.4)

330

ECONOMIC DYNAMICS DYNAMICS 7. HIGHER DIMENSIONAL ECONOMIC

where e(g) is implicit function ifinctionbetween betweenee and and g, i.e. where is an implicit i.e. (to+ve)h = th(e, ih(e,g). g). + > g. We have described the behavior described behavior of of the the producers producers and and consumers. consumers. We We now now 14 describe technological change by the following equation'4 equation A& +0— A(t) g(eQ)), g(0), g' > 0, g" c 0. g(t + 1) 1 ) . A(f + l}~)A{f) == g(e(t)), g(0\ g' > 0, g" < 0.

(7.5.5) (7.5.5)

The size of working population at time time t ++11 is determined by

N(t + I)l) = n(t)N(t), n{t)N(t),

(7.5.6)

where N0 No isishistorically historically given. given. Utilizing Utilizing X(t) {>

=

A(t)X

N(t)'' N(t)

and equations (7.5.5) and (7.5.6), we have

/ ,\ 11+g(t+1) + git +l) / N x(t+l)= x(t). n(t) n{t)

Substituting equations (7.5.3) and (7.5.5) into the above equation yields Substituting equations (7.5.3) and (7.5.5) into the above equation yields

+ re(g(e(t)))J[i + g(e(t))] x(t + 1) =

x(t),

if zQ)

7

+

+

1—clz(t) l-c/z(t)

(7.5.7) (7.5.7) x(t), if z@) <

The construction of the model model isis thus thus completed. completed. The Thesystem systemconsists consistsof ofequations equations The (5.5.2)-(7.5.7). In hi the dynamical analysis, analysis, the the economy economy isis divided divided into into two two regimes: regimes: 14 14

ItIt should be noted that Galor and Weil also discussed discussed implications implications of of introducing introducing N(t) into the technological progress function function g. g.

7.5. GROWTH WITH DIFFERENT TYPES OF OF ECONOMIES

331 331

the subsistence regime characterized andmodem modernregime regimecharacterized characterized characterized by by z(t)<'z z(t) i' and by z(t) Although the complicated, itit will long space space to the analysis analysis is is not not complicated, will take take aa long to by z(t)>> z. Although examine. The reader is encouraged to analyze the behavior of model, and then to examine. The reader is encouraged to analyze the behavior of model, and then to the analysis by by Galor Galor and andWeil. Weil. read the

7.6

and chaos chaos Unemployment, inflation and

nonlinear outflow outflow rate from introduces a worker flow model with a nonlinear This section introduces unemployment. The model isis proposed The unemployment unemployment U(t) U(t) at unemployment. The proposed by byNeugart.15 Neugart.15 The followsthe thefollowing following identity identity period tt follows U(t i(L -— U(t)) o(t)u(t), u(t ++ l)-U(t)= 1)— u(t) = i(L u(t)) -— o(t)u(t),

(7.6.1)

the (exogenous) (exogenous) inflow inflow rate, force, and o(f) where I/ >> 00 denotes the rate, L = 1 the labor force, 0(t) is the the outflow outflow rate from from unemployment. unemployment. Here, the outflow outflow rate is is defined defined as as the the fraction of jobs that come to the market to jobs searchers, searchers, i.e. i.e. ofjobs market at at time time t to 1

1\

- u(t)+J{t) d(1

.1(t) —

u(t))

where the the parameter 00 << d << 11 gives the as aa constant fraction the on-the-job searchers as fraction 16 and j{f) denotes job creations. We specify of employed workers employed workers and 1(t) denotes job creations.'6

j(t) =

+ y(m

—

where Js is the the job jobcreation creation due duetotothe thestructural structural characteristics characteristics of of the theeconomy economy y{m— - ir(t)) #(/)) describes describes the cyclical cyclical component and y(m and componentof ofjob job creations, creations,where whereyy is a positive parameter, the exogenous exogenous money money growth rate, and and .lr(t) 7t{t) the the inflation inflation positive parameter, m the rate at at it./. Hence, Hence, the the outflow rate rate from from the the unemployment can canbe beexpressed expressed

15 15

See Neugart(2004). (2004). See Neugart It should be be remarked that in in Neugart (2000), the the parameter is is treated as as an an endogenous an endogenous endogenous variable; in the appendix appendix to Neugart Neugart (2004), (2004), the the parameter parameter ii is treated as an variable. 16 16

7. HIGHER DIMENSIONAL ECONOMIC DYNAMICS DYNAMICS

332

+y(m-it(t)) u(t) +

d(1

—

u(t))

Substituting the above equation into into equation equation (7.6.1) (7.6.1) yields yields

u(t

- e) +

-

+

A(U(t),

(7.6.2)

Assume Assume that firms cannot cannot raise prices prices to the same same extent extent as as nominal nominal wages wages increase is reflected in the requirement of 8> increase at at it.t. This This assumption assumption is reflected in requirement of 5 >11 in the the following formation of of inflation inflation rate rate dynamics dynamics "

+

=

(7.63) (7.6.3)

, W

P

)

where 2t(t) expectedinflation inflationrate rate 7t(t) is is the the inflation inflation rate rate at at it,t, ,re 7te(t) is is the the expected at at t, 1, wb(t) {t) Wb

is the bargained price determined determined real wage (which (which bargained wage rate at at it, t, and and wp is the price ju)y,where where 00 ca < //c <1 1isisthe thefixed fixedmark-up mark-upand andyy== 1isisthe theconstant constant equals (i(l——,u)y, equals marginal productivity). productivity). The bargained bargained real realwage wageWb wb(it) (t) is given by 1

—

(i— b)U(t),

where 00 .c< bb c<11 is the "reservation wage". wage". The The inflation expectation expectation is is formed by

ge(t) an{t —- 0± l) + (i(l —- a)ne(t —1), - l), 7te{t) = air(t

where where 00
+1) =

From this equation and

+ 1)) +

p / *-( (1-b)uQ i » M * ++ ii)n

.6.4) (7 (7.6.4)

7.6. UNEMPLOYMNT, INFLATION AND CHAOS

333 333

7te(t + 0= l) = an(t) + (i (l —- a)n'{t), we have c /

i\

/\

/,

\ ei\

— (i— U-(\-b\j(t

afr(t)+ Sn\t + 1) = a;r(/) + (i(1 —- a)n (t) + —— + 0=

—

i-ji

+ + i) \) '-.

Substituting this equation into into equation (7.6.4) for 1,/, we solve + i) n{t +

+f

= jUod + {0

+ (1 (1-a)U) + J

JU - 0

A(u(t), (7.6.5)

(7.6.5) where where 1—b,

bQ=\-b,

juo=l-ju.

The dynamical system consists of equations (7.6.2) and (7.6.5), that is The dynamical system consists of equations (7.6.2) and (7.6.5), that is

u(t + i) = A(U(t),

2r(t)),

+ i) = (r4u(t),

(7.6.6) (7.6.6)

behavior in the the same same way way as as we we did did for for the the Henon Hénon map map in the We may analyze its behavior previous previous chapter. chapter. A fixed point is given by

U=A(Ujr), ir=c(u U = k(U,n), 7t = <&{U,n). It is imposedthat thatat ataasteady steadystate, state,nI ==m, is imposed m, the inflation equals the money growth rate. It It can be shown that the system has aaunique steady steady state state (u*,

provided that

[1u + mp0(i

—

a)

_____

7. HIGHER DIMENSIONAL ECONOMIC DYNAMICS

334 334

The Jacobian Jacobian matrix matrix JJ isisgiven given by

J=

a,, [a21 a,,

ay a121 a.

where where

a11

dA(U,,r) dA(u, n) = au du

= 1—i

aA(u,47t) _ dAJU,

—

.

=

a12—

a21 Q-y]

(u* + d(l

—

yU'

ac4u,,r) ^

*

Sji

dU r)

a22

—

1(

au dU

It is straightforward straightforward to calculate the two eigenvalues. eigenvalues. We now provide some numerical examples examples carried carried out out by Neugart. numerical Neugart. As As discussed discussed by by Neugart, Neugart, it is reasonable to choose the following parameters

a = 0.5, 0.5, 8 == 2, 2, b == 0.5, 0.5, y7 == 0.5. 0.5.

(7.6.7) (7.6.7)

Under (7.6.7), if we further further specify

d == 0.01, 0.03, iJ = 0.13199, 0.01, m w == 0.03, then one eigenvalue bifurcation occurs. It can be eigenvalue is is equal equal to to —1; - 1 ; a period-doubling bifurcation shown that under (7.6.7), the eigenvalue with a positive root never never crosses crosses 1. 1. shown that (7.6.7), eigenvalue a positive root Figure 7.6.1 depicts bifurcation bifurcationdiagram diagramofof U U over I/ under under(7.6.7) (7.6.7) and and

d= = 0.01, = 0.03 •17 ." 0.01, m =

The plot and and the the following following one one are are from from Neugart Neugart (2004). (2004). Neugart provides provides more more charts for different values values of of the the parameters. parameters.

7.6. UNEMPLOYMNT, INFLATION AND CHAOS

335

Increases from 1/ = 0.13199 will lead lead to to further further biflircations. bifurcations. Mathematically, Mathematically, the the Henon map. map. dynamic mechanism is similar to bifurcations observed for the Hénon

U

0.13 0.13

0.14 0.14

00.15 .15

0.16 0.16

0.17 0.17

018 0.18

1

Figure 7.6.1: Bifurcation diagram diagram of unemployment over the inflow rate rate

Figure 7.6.2 demonstrates demonstrates the existence existence of of an an attractor attractor with with the the inflation inflation rate rate over the unemployment rate when the parameters are specified specified as in (7.6.7) and

d == 0.01, 0.0l, m m == 0.03, 0.03, i/ = 0.18. 0.18. attractor looks like a Phillips curve. Nevertheless, The attractor Nevertheless, in in this this model, model, there there is not a stable trade-off trade-off between inflation rate unemployment rate. system stable between the the inflation rate and unemployment rate. The system shifts up and down down aa negatively negatively sloped sloped "curve" "curve" erratically. erratically. A pair pair of of inflationinflationshifts unemployment rates rates today cannot cannot tell where the economy will will be be in in the the long long term. term.

7.7 7.7

Business cycles with money and capital capital

This section provides aa model of of the OLG OLG model model with with money money and and capital.'8 capital.18 18 18

The model and analysis in in this section are based on Jullien (1988). A model in the same The spirit is is proposed proposed by by Grandmont Grandmont (1985), even though though Grandmont's model does not take take spirit account of capital accumulation. accumulation. See See also also Tirole Tirole (1985), (1985), Farmer Farmer (1986), (1986), and and von von Thadden Thadden account (1999). As far as technical issues issues related to existence of chaos are concerned, the reader is is (1999). also referred referred to to Yokoo Yokoo (2000). (2000). also

7. HIGHER DIMENSIONAL ECONOMIC DYNAMICS DYNAMICS

336

Inflation* 0.03

0.025 0.025

".*.

00.02 .02

S

0.015 0.015

.4'..

0. 01 0.01

0.005 0.005 0.02 0.02

0.04

0.06

0.08 Q.OB

0.1 0.1

Figure 7.6.2: 7.6.2: An attractor

A single single good good isis produced produced and and each each agent agent lives lives two two periods. periods. The The young young generation sells one unit of labor (t), generation sells labor at at aa real realwage, wage,w(t), w(t),consumes consumesquantity, quantity,c1cx(t), saves s(t), by holding money money and and capital. capital. Let Let r(t + saves the real quantity, quantity, s{t), + 1) l) denote the the real rate of interest between t/ and t ++1.1. The old generation spends all its savings its savings from the previous period on consumption, cr2 (t). consumer maximizes maximizes previous period on consumption, c2(t). A typical consumer his utility subject to the budget constraint as follows AJax Max U(c1Q), U(c,(t), c2(t)) c2{t)) S

s(t) w(t), 1)s(t), '" c2Q)rQ+ c2{t)
Ci{t)>0,

11=1,2. = 1,2.

Under the consumer's has aa unique Under standard standard assumptions, assumptions, the consumer's decision decision problem problem has unique solution characterized by by the the saving 1)), which satisfies solution characterized saving function function s(w(t), s(w[t), r(t + lj), satisfies o0 < < s(w, s(w, r) c< w.

Production is made through aa neoclassical neoclassical constant constant return to scale scale technology. Output per capita, y(t), y(t), is is aa function function of capital intensity, k(t), i.e. i.e.

yQ) =

where /f isisincreasing, increasing,strictly strictly concave concave on R+ and C2 C 2 on on R++, and satisfies satisfies

BUSINESS CYCLES WITH MONEY AND CAPITAL 7.7. BUS1NESS

337

[0, l[, limf(k) = oo, limf'(k)e [o, i{, li

k-oo

k-*O

kf{k) kf' (k) is non-decreasing, and

lim f(k)-kf'(k)= iimf(k)kJ'(k)= +00, llimf(k) i m / ( * ) -—*kf'(k) /'(*) = 0. =0. k—+O

Obviously, if

/ = Aka, 0 < a < 1, the above above properties properties of aresatisfied. satisfied. The TheCES CESproduction production functions functions with an of /f are also have have these properties. We also have elasticity of substitution larger than 11 also

r{t)= f(kit)), r(t) = f'(k(t)), w(t) f(k(t))-k(t)fik(t))=W((k(t))). w(t) = = f(k(t)) — k(t)f'(k(t)) w((k(t))). 1 increasing, and It is known knownthat that W Wisis CC' on R++, increasing, andmaps maps R14 itself + into itself. At date 0, the the economy economy isis endowed endowed with with aa fixed fixed quantity quantity of capital capital k0 and a quantity of pit) and and m(t) mit) (MI (MIp(t)) p(t))denote denoterespectively respectivelythe theprice price quantity of money moneyM. M. Let p(t) money. A perfect foresight sequenceof of of goods and the real money. foresight equilibrium equilibrium isisaasequence

{/ML.. {*(')L.. M')L.. M')L-, achieves aa competitive competitive temporary temporary equilibrium equilibrium with perfect foresight foresight at at each each which achieves date. perfect foresight foresight equilibrium equilibrium is described described by sequence of {&(?)},>„, and date. A perfect by a sequence {m(t)},>c.., such that that M')L~ such

+ k(t + 1) S{w(k(t)), f'(k(k f'(k(k ++ i))), 1))), m(t) + i) = s(w(k(t)),

(7.7.1)

m(t)>0, m{t + + l) m(t i) = 1))m(t), k(t)>0, = f'(k(t f'(k(t ++ l))m(t), k(t)> 0, m(t) 0, kk00, M, given.

(7.7.2)

Equation (7.7.1) equalizes the demand and and supply supply of of assets; assets; equation equation (7.7.2) (7.7.2) Equation equalizes the the interest interest rate rate on money and interest rate on i) and equalizes money p(t)I p{t)l p(t + +1) interest rate on capital capital f'(k(t +1)). We call an equilibrium non-monetary or monetary according f'(k(t + i)). We call an equilibrium non-monetary monetary according to

338

7. HIGHER DIMENSIONAL ECONOMIC DYNAMICS

in0 m = 00 or in0 m0 >> 0. By By equations equations (7.7.1) (7.7.1) and (7.7.2), (7.7.2), we can can replace replace equation equation 0 = (7.7.1) with

m(t s(w(k(t)), f(k(k f'(k(k ++i)))f'(k(k m(t + i) 1)++ k(t *(/ ++ 1)f'(k(k l)f'{k{k ++1)) 1))== S{w(k(t)\ l)))f'(k(k ++i)). 1)). backward dynamics dynamics as as It can be shown that the above equation determines aa backward k(t) == A(k(t ++ i), l),m(t m(t++1)). l)).

(7.7.3)

According to the properties properties of the saving and production production functions, According functions, itit can can be 1 increasinginineach each of of its its arguments, arguments, and (resp. proved that is C', C ,increasing proved that A is and A tends to 0 (resp. infinity) when k goes to 0 (resp. infinity) with the fixed m. A perfect foresight infmity) when k goes to 0 (resp. infinity) with the fixed m. A perfect foresight (backward) mapping mapping equilibrium can be described by a two-dimensional (backward) (k(t), m&)) i), m(t + 1)), m(t))= =[A(k(t [A(*(,+ +1), m(t + 1)), y

^ y

1)). + l)). - O(kif +m(t 1),+m{t

(7.7.4) (7.7.4)

We may express express the the system system in in vector vector form form as as

x(t) = cI(x(t + I)),

where x = (Ic, (k, m). interested in a monetary economy. economy. A A steady steady state state in). Here, we are interested (k*, ,n*) \k\ rn) is determined determined by by fT(k*)= f'[k*)= 1 and m +

= s(W(k),

1).

We assume assume sfrv(k*), s\f¥{k j , 1)> l)> k*. Then the the conditions conditions for the existence existence of a unique unique monetary steady steady state state are are satisfied. satisfied. Under this this assumption, assumption, there there exists exists at at least least one one 19 inefficient non-monetary steady state (Ic> k* )19 and we denote denote the the lowest lowest capital capital inefficient steady state (k > k* ) and intensive one by ks, i.e. i.e. 19 19

A A non-monetary non-monetary steady verifies s(w(k), s(w(k),f'(k)) k. It is inefficient inefficient ifif k >> k*, steadystate state kk verifies f' (k)) = = k.

where s(W(k*), s(ff(&*), l)i)>> k. k .For Forsufficiently sufficiently where

large k, k, w(k) W{k)c
7.7. BUSINESS CYCLES WITH MONEY AND CAPITAL

339 339

ks=inf{k>k'\A(k, 0)=k}. =inf{k >k*IA(k,O)=k}. Here, we we are areinterested interested in inperiodic periodic perfect perfect foresight foresight equilibrium. equilibrium. To To analyze analyze of the system, it can be proved that all the periodic orbits must global behavior global behavior system, can be proved that all the periodic orbits must 1 belong to a C' C invariant invariantcurve. curve.This Thiswill willenable enableus usto tofocus focus the thedynamical dynamical analysis analysis in one-dimension. one-dimension. on the curve in 7.7.1. There There exists a compact h, decreasing decreasing Theorem 7.7.1. compact set set K e R^+ and and a function fimction h, C\ from to R++ such that if if we define the sets sets and C', from R++ to and

r'={(k,m)e R÷21k=h(m)}, 2 T+={{k,m)eR = m)e +\k>h{m)\ 1k >

T_={(k, = {(k, m)s Rl\k
t

« limm(t, x) = °°,

jXEL e r _ < =<=>iimm(t,x)=O, > limm(t, x) = 0, xeT m(t, t being sufficiently large. xc F <^>m(t, x) x)e E K Kforfor t being sufficiently large. The proof referred to Jullien. We We depict the contents of the theorem theorem as proof is referred to Jullien. the contents as in figure 7.7.1. The real quantity of money increases infinitely along the orbit of aa figure The quantity of money increases infinitely along the of point greater than Xx and decreases to along the the orbit orbit of ofaasmaller smaller point. point. point greater and decreases to 0 along an orbit Between Between these these two two behaviors behaviors it is possible possible to to exhibit exhibit aa set of points with an bounded andinfinity. infinity. ItItcan canbe beshown shownthat thatall allcycles cyclesmust mustbelong belongto to bounded away away from 00 and the curve curve T. F. map Introduce aa one-dimensional map

0:

—> Rj., —*

m m —> —7—7m —* ,. ,.

f

1 The map, map, 0, <j>, is C . The cycles equivalent are equivalent the cycles The is C'. The cycles of 0ofare to thetocycles of Q of in Q thein the sensesof of

340

7. HIGHER DIMENSIONAL ECONOMIC DYNAMICS DYNAMICS m

m m

F k*

ks

k

Figure 7.7.1: The partition of R++

cl"(x)=x The following theorem is referred to Jullien. Theorem 7.7.2. 7.7.2. A sufficient sufficient condition condition for for the the existence existenceof of aa cycle cycle of of order order 22 is that 2

l k* + 2S 2SR(W(k*), 2k*SW(W(k*), s{w{k'),0— l)-k'+ s(w(k), R{w(k'\ l)+ w(w(k'\ 0—)-JTjf*
f"(k

verify the The proof is to veriQv theexistence existenceof of solutions solutions of of
should be noted noted that that Jullien Jullien also discusses discusses the conditions conditions for It should for the the existence existence of of which guarantees guarantees economic economic chaos. periodic points of period period 3, which

THE OSG OSG MODEL 341 7.8. THE MODEL WITH WITH HETEOGENEOUS HETEOGENEOUSHOUSEHOLDS HOUSEHOLDS 341

7.8

The OSG model with heterogeneous households households

This section explores explores the theinterdependence interdependence between between income income distribution, distribution, wealth wealth This section distribution, and and economic growth growth within within the theOSG OSGmodeling modeling framework.2° framework.20 The key The key feature of this theory is is that itit groups the the population population into into different different groups, groups, whose whose saving behavior behavior are homogenous homogenous within within each each group and and are are consumption consumption and saving different the groups. different among the production side of our model OSG model in section section The production model is the same as in the OSG Each type type 3.2. The population population is is classified classifiedinto intotwo twogroups, groups,indexed indexedby byjj == 1,1, 2. Each fixed number, number, denoted by N1. Nj. The Theaggregated aggregated labor labor force, force, consumers has a fixed of consumers N, isisgiven given by N = h1N}+ h,N2, where hj are the level level of human human capital capital of group jj, , j ==1,1,2. Production in of group 2. Production period inputs amount and amount amount of labor labor services, It period t uses uses inputs amount of of capital, capital, K(t), K(t), and services, N. It supplies amount goods, F(t). The The production production process process is described described by aa supplies amount of of goods, neoclassical production function function

F(t)=F{K(t),N). F(t) = F(K(t), N). The real interest rate and and the the wage of of labor are are given given as asbefore before by

r(t)+S r(t) + t=f'{k(t)), = w1(t) = h1[f(k(t))

- k(t)f'(k(t))],

(7.8.1) ( 7 8 1 }

Letk1(t) kj{t)denote denote per percapita capita wealth wealth of where k(t) = K{i)l K(t)/ N. Let of group group jj in period t.t. where to the the definitions, we we have According to

kl{t)N,+k K(t) == k1(t)N1 + k2(t)N2. 2{t)N1. Dividing the two two sides sides of ofthe theabove aboveequation equationby byNN yields 20 20

The represented, for instance, by Kaldor Kaldor (1940), Pasinetti Pasinetti The post-Keynesian post-Keynesian approach approach is represented, (1960, 1974), Sato Sato (1966), (1966), Samuelson Samuelson and and Modigliani Modigliani (1966), (1966), Salvadori Salvadori (1991), (1991), and and (1960, 1974), Panico and and Salvadori Salvadori (1993). The The recent literature on on growth growth and and distribution distribution is is referred referred to, for forinstance, instance,Becker Becker(1980), (1980),Sorger Sorger (2002), (2002), Nishimura Nishimura and andShimomura Shimomura (2002), Ghiglino and Sorger (2000), and Zhang (2005c). and Sorger (2002), and

342

ECONOMIC DYNAMICS DYNAMICS 7. HIGHER DIMENSIONAL ECONOMIC

k{t)=k k(t) = k1(t)n1 +2(t)n2, l(t)nl+k where nj = NjIN. Asn1 rijare areconstant, constant, we we see see that that k(t) k{i)isisaaftinction function of ofk1(t) kjyt) IN. As k2(t). With With equations equations (7.8.1), r(t) and Wj(t) functions of and k2(t). and (7.8.1), we we see that that rQ) w1(t) are fUnctions of *,(/) and and k2(t). k2(t). k1(t)

(t), from from the Group jj 's's per capita Group capita current current income, income, y1 yj{t), theinterest interest payment, payment, (t), and (t), is r(t)kj(t), andthe thewage wage payment, w1 Wj(t), is defined by

yJ(t)=r{t)k r(t)k1(t)+ J(t)+wW1(t). j(t). The per capita capita disposable income defined as the sum sum of of the income of of consumer consumer jj is is defined current income and the wealth available for purchasing consumption goods current income the wealth available for purchasing consumption goods and saving yj(t) = yjif) + kj(t) = (1(I + r(t))k1(t) r(t))kj(t)++W1(t), wjit), j ==1,l, 2. = y1(t) + k1(t) 2. The disposable income is used for saving and consumption. At each point of time, a consumer consumer would would distribute distributethe the total total available available budget budget among among saving, saving, Sj(t), s1(t), and consumption of goods, c1 Cj (t). (t).The Thebudget budgetconstraint constraintisisgiven givenby by cj(t) + sJ(t) = = y5'1(t). + s1Q) J(t).

(7.8.2) (7.8.2)

the two two variables variables subject subject to the the disposable disposable In each period, consumers decide the income. assume that that utility utilitylevel, level,U1 Uj(t), dependent on the the consumption consumption income. We assume (it), is is dependent ofcommodity, commodity, c1 Cj{t), andthe thelevel level of of saving, Sj(t), asfollows follows level of (it), and (t), as Uj(t) = cf (t)Sj' (0, U1(t)=cf(t)s'(t),

£j, *j >0, > 0, e1 £, + Zj =1, =1, j=1,2, j = 1, 2,

(7.8.3)

where ^ and and Aj are respectively respectively group propensities to hold group j's j 's propensities to consume and to hold wealth. Maximizing U1 U} subject subject to tobudget budgetconstraints constraints (7.8.2) (7.8.2) yields yields cJ(t) = = Zjyj(t),

sJ(t) = = 2)1(t). AJyj(t). s1(t)

(7.8.4) (7.8.4)

7.8. 7.8. THE OSG MODEL MODEL WITH WITH HETEOGENEOUS HETEOGENEOUSHOUSEHOLDS HOUSEHOLDS 343 343 Per capita in period the saving made in in period capita wealth wealth of of group group jj in period t + + 11 is equal to the t. t. That is kj(t ++ l)= k1(t i) =Sj{yj(t)), s1(5'1(t)), 7j = 11,, 22..

(7.8.5)

themotion motion of ofthe the system. system. By Byequations equations (7.8.2) (7.8.2) and and(7.8.1), (7.8.1), control the These mappings control we have yj(t) == *,/(*(')) ++ [kj(t)—- h1k(t)jf'(k(t)) hjk(t)\f{k{t))++ Skj{t), j ==1,1,2,2,

(7.8.6) (7.8.6)

where y>j(t) arefunctions functionsofof andk2k2(t). {t). where S = 11 -— S5k• that j), (t) are k1k^t) (t) and k. We We see that As output is is either consumed or or saved, the the sum sumof of net saving and and consumption consumption equals output. That is C{t) ++ S(t) -— K(t) + +S SkK(t) = F(t), F(t), C(t) kK(t) =

(7.8.7)

C(t)isisthe thesum sumofofconsumption consumption and where C(t) S(t)—- K(t) + StK(t) S(t) + (9kK(t) is the thesum sumof ofnet saving saving of of the two groups

C(t) = S(t)

—

bK(t) = E(s1(t)

—

It can can be be shown shown that that equation equation (7.8.7) is is redundant redundant in the the sense that itit can canbe be derived from the other equations in the system. We thus omit equation (7.8.7) in derived the other equations the system. We omit equation later discussions. As yj(t) are fimctions functions of £,(?) k2(t), we see that difference difference equations equations (t) are (t) and and k2 (t), we (f) and and /c2 k2(f). It is straightforward straightforward totoshow (7.8.5) (7.8.5) determine determine the the evolution evolutionof of kkx(t) (t). It show variable in in the the system system is is uniquely uniquely determined determined as as aa function function of of k1(t) kx{f) that any any other variable and k (f). It is sufficient for us to examine behavior of difference equations (7.8.5). and k22 (t). It is sufficient for us to examine behavior of difference equations (7.8.5). We are arenow nowconcerned concerned with withdynamic dynamic properties properties of ofequations equations (7.8.5). (7.8.5). We and equations (7.8.6) into equations (7.8.5) yields Sjit) Substituting s1 (t) = Ajpjit) (t) and =

344

7. DYNAMICS 7. HIGHER DIMENSIONAL ECONOMIC DYNAMICS

kj{t ++ 1) = 21h1f(k(t)) Ajhjf{k{t))+ +Lk1(t) [kj(t) + A^t), j =j i,=2.1, 2. i) = — - hjkit^f'ikit))+ 21&1(t), (7.8.8) A steady state is is defined by by k1

the above equations as We rewrite the as

k1f'(k) +

—

kf'(k)) =

where

j=1,2.

1

the above equations, we we exclude the the possibility of of kj = From the = 0 0 for any any jj for k >0.0.Rearrange Rearrangethe theabove aboveequations equations as asfollows follows k>

f(k) -

)J = k..

(7.8.9)

f(k)— - kf'(k)> kf'(k) > 0 > 0. To guarantee guarantee As for k > As /f isis strictly concave, concave, f(k) 0 for 0. To > 0,0, we weshould should require require > Sj-f{k)>0, j j=l,2. = l,2. 81—f'(k)>O, Define

k=max{k:f'(k)=Sj}. f(k)= Because Because ft?(k) oc, f"{k)<0, lim/'(0) == oo, lim/'(0) lim c 0, limf'(O) [(0) == 0,0, k—,O *>o

k—*-I-c.o * > +

7 .8.THE THEOSG OSGMODEL MODEL WITH WITH HETEOGENEOUS HETEOGENEOUS HOUSEHOLDS HOUSEHOLDS 345 345 7.8. we see that uniquely defined. defined. Obviously Obviously that k isisuniquely

Sj ~ /'(*) > 0, for k > k, j=1,2, j = \, 2, fork>k, 81—f'(k)>O, and 8j —- f'(k)<0 f'(k)<

We should > k. k. should require require k > 0 for forsome somej j ifif k < k. k. We

two sides of of each each equation equation by by n} and then adding the two Multiplying the two sides (7.8.9) yields yields equations in equations (7.8.9) +

n2h2

—f'(k)

—f'(k)A

k

)

k>k,

(7.8.10)

where we we use usen1k1 w,jfc, + n2k2 n2k2 = k and neglect neglect the trivial trivial solution solution k = 0. 0. We now now show that this equation has has a unique unique solution solution for for k >> k. Introduce +

*(*) =

+

ui2h2

52—f'(k))k.. k

)

(7.8.11)

(k) —>oc + oo k —> k from the right and <&(+°°) —1, —> we — 1,conclude we conclude As $k)—>+ As the right and 4(+oo)—> that that as as k —, k from oc'). Here, we use (&) = 0 has [k, + °oj. has at at least least one one solution solutionfor for kk ee (k,

fk —> 0 as k —> °°, M k 0

lim k

hfflk_*ool

calculate We calculate

- [(k))2 + (s -[(k))2 ( 2-f(k)f +

—

+

8,-fikV We see that if

n2h2

A(k)

52-f'(k)) k2

S2-f'(k)

k2

- f'(k))ft'(k) (7.8.12)

7. 7. HIGHER DIMENSIONAL ECONOMIC DYNAMICS DYNAMICS

346

A(k)=f—/cf+k2f"O, 0 for e \k, + » )It. should It should noted even if the term then <£>'(k) then w(k) 0
f-kf'+k2f" f—kf'+k2f" mayhold. hold.If If is negative, negative, O'(k) W(k) <00may A(k)>0,ke(k,+oc), ke(k,+oo), A(k)0,

(7.8.13)

<£>(k) 0 has a unique solution. is straightforwardto to check that the equation '14k) = 0= has a unique solution. It isIt straightforward check that if if F F takes takeson onthe theCobb-Douglas Cobb-Douglasform, form,then then(7.8.13) (7.8.13)holds. holds.IfIfthe theproduction productionftinction function form takes on the CES form

ff ==

+ A(ak'+iy,

where aa c<1,1,and AA areare positive, andaaand and positive,we wecalculate calculate

I

akC+1 aka + 1

I

>0.

We see that when the production thnction function takes takes on onthe the Cobb-Douglass Cobb-Douglass form form or orthe the CES form, the system system has hasaaunique uniqueequilibrium. equilibrium. For Forany anypositive positive solution solution CES form, the

k* (> (>k), (7.8.9).We We get get rr and by k* k), we solve solve k1 kx and k2 k2 by by equations equations (7.8.9). and Wj by equations (7.8.1), by equations (7.8.6), and and c c1} and and s1 (7.8.1), 5), j y by Sj by byequations equations (7.8.4).

Proposition equilibrium point. point.IfIf Proposition7.8.1. 7.8.1.The Thedynamic dynamic system system has has at atleast least one equilibrium

A(&)>0, ke(k,+oo), ke(k, +°°), A(k)0,

If the system has a unique unique equilibrium equilibrium point, point,denoted denotedby by k'. F. If 1

h2n2

+ n2n2 h2n2

(7.8.14)

7.8. 7.8.THE THEOSG OSGMODEL MODELWITH WITHHETEOGENEOUS HETEOGENEOUS HOUSEHOLDS HOUSEHOLDS 347 347 then the unique equilibrium equilibrium point point isis stable. stable. The proof of the stability 1. stability condition condition isis provided providedininAppendix AppendixA.7. A.7.1. the difference differencebetween betweenA1/i,— - A2 A? is small, small, then then the system system is stable. stable. For If the

instance, as/f'' >0,0,wewesee instance, as seethat thatifif A,A2S>A1-A1, A1A28>A1 (7.8.14) always always holds. holds. ItIt should should be noted noted that that even even ifif condition condition (7.8.14) (7.8.14) isis condition (7.8.14) not held, held, itit isis possible possible that that the the unique unique equilibrium equilibrium point point isis stable. stable. From equations (7.8.9), (7.8.9), we we get

(A1-k)&-kr)

- f'i

-

k

f'

positive at the equilibrium point, we see that the kf and Sj — - / ' are positive equilibrium point, As f/ —- kf'

sign of of k1 £, l \ —- /c2 k2fh2 is the same as that A?. This relation tells that if one one that of \ -— k• group's group's propensity propensity to to save save isis higher higher than than the the other other group, group, then wealth wealth per qualified qualified capita of of the former than that that of the latter. capita former is greater greater than latter. Hence, Hence, in the long long term, term, differences in in wealth per qualified differences wealth per qualified capita capita are are determined determined by by differences differences in in propensities to save among different groups. It should be noted that the sign of propensities to save among different groups. It should be noted that the sign of and is not determined solely by A1 For instance, if A1 — A2 > 0 — k2 &, — k2 is not determined solely by A^—— A^. For instance, if A, — A^ > 0 and h2 —- \ >> 0, 0, k1 &, -— '2 k2 < possible.InInfact, fact,from fromequations equations (7.8.9), (7.8.9), we we have have <00isispossible.

kk

1

kk

_('A1kkk - (k

At equilibrium, we have j A

(('

;

f = kj I Aj. From equations (7.8.9), we have

A

-h2)Q+Sk)-(kk -hA)(1+f'))(5

-A1J'i £-V)'

We can obtain obtain explicit explicit conclusions conclusions about signs of differences differences in key We can about the the signs in the key variables only when one one group does not have lower lower level level of of the the propensity propensity to save save and lower than the the other and lower level level of human human capital capital than other group. group. That That is A1 A\ > A^ and

348

DYNAMICS 7. HIGHER DIMENSIONAL ECONOMIC DYNAMICS

hh1 Otherwise,the thecomparative comparativeeconomic economicconditions conditions of ofthe thetwo twogroups groupsare are Otherwise, x >k.h^. ambiguous. ambiguous. examine impact impact of changes changes in one group's propensity propensity to save and We now examine level level of human human capital capital upon upon the the income income and andwealth wealthdistribution distribution and and national national economic product. Taking derivatives of equation (7.8.12) with respect }\ yields derivatives of equation (7.8.12) with respect k yields dk

(L_ f)'

(21—k)n1n2h2

dk

(7815)

4>' < 0. 0. It can be shown that if /I, > (<) /^, we have where cb' (cz)k,

J--s2<(>)o, L-sl>{<)o, kk

kk

and if J^ =22, =^, wehave we have f/k—S1 f/k-Sj =0.Since Since =0. andifA1 j-f>0,

Sj-f>0, 1—1>0

from d\ >> (=, 0 if if \ >> (<)A2. (<)/l 2 . As from equation equation (7.8.15) (7.8.15)we we conclude concludethat that dkl dk/dk (=, <) c) 0 group 1 's level of human capital is increased, if group 1 's propensity to save save is is group 1 's level of human capital is increased, if group 1 's propensity to higher (lower) than group 2 's, then k increases (decreases). We see that higher (lower) than group 2 's, then k increases (decreases). We see that improvement group's human human capital capital does does not not necessarily necessarily lead lead to to improvementinin one one group's increases in the wealth per qualified capita capita in in the economy. economy. The The difference difference in in the the preferences preferences affects affects change change directions directions in in k. k. From equations (7.8.9), we obtain obtain

dk1k1 ^L =k dk

+ (f [L _ §\ ek2f"h1 f"^

dk dk

h1

dk2(f5"1 k2f"h2

dk

dk lemma summarizes summarizes our our analytical analytical conclusions. conclusions. The following lemma Lemma 7.8.1. 7.8.1.As Asgroup group 11's'shuman humancapital capital isis increased, increased, (i) if group 11 's propensity propensity Lemma to save is higher than group > A^), group 2 's (i.e., (i.e., \ > group 1l's'swealth wealth per per capita capita may may 22), group increase capita increases; increases; (ii) increase or or decrease decrease and and group group 22 's wealth per capita (ii) if A, = = ^, capita increases but group 2 's's wealth wealth per per capita capita isis not not then group 11 's wealth per capita

____

7.8. THE OSG MODEL MODEL WITH WTTHHETEOGENEOUS HETEOGENEOUSHOUSEHOLDS HOUSEHOLDS 349 affected; and group 1 's's wealth wealth per capita capita may be be increased increased or or affected; and (iii) (iii) if X, << X^, A2, group wealth per per capita capita increases. decreased and group 2 's's wealth group 11 becomes becomes more effective, effective, group 2 's wealth wealth per per capita capita We see that as group term. This This conclusion conclusion does does not not hold hold increases, at least does not decrease, in the long term. when the economy has more than 2 groups. economy has more than From

K = kN, F=JN, F = JN, r-I-Sk—f, r + 5k= / ' , w1=h1(f—kjj, w, = * , ( / - kf'\ K=kN, „

Yi=J' C1= A

_

XJ '

we we get

1ldK dK _ ldk 1 dk N1 N, Kdh1 K dh,~ kdk k dh, NN' }_dF__f_dk_ Nj_ r + ~F~dh,~ f~dh, ~N' Fdh1 fdh1 N dr dk dr _ dk — Hh,~ ~d\' dIi1' dw1

dk di'v2

=(f—kr)—kkf"-4-, dk

= —

2 dh, dh, dk dk 1 dy _ 1 dcj dkj dc1 } = 1 1 =_ 1l dk1

yj dh,

Cj dh, c1dk

kj dh,

rate From lemma 7.8.1, we can judge the signs of these variables. We note that the rate of interest may be either either increased or decreased when group 11 's human capital capital is is improved. This will be confirmed by simulation results later on. We obtain the improved. This will confirmed simulation results We obtain the following results. Lemma 7.8.2. 7.8.2. As As group group 11's's propensity propensity to to save save increases, increases, (i) (i) if group group 11 's 's Lemma propensity (i.e., A1 X, > X^), group 1 's's wealth per propensity to to save save is higher than group 2 's's (i.e., capita may either increased increased or decreased and wealth per capita capita capita may be either or decreased and group group 22 's's wealth increases; (ii) (ii) ififA1 X, == A^, group group 1 's's wealth wealth per capita increases and group per capita increases and group 22 's

7. HIGHER DIMENSIONAL ECONOMIC DYNAMICS

350

wealth per capita and (iii) (iii) ifif k^ < A2, 's wealth per capita wealth capita is not affected; affected; and k^, group 1 's capita increases and group 22 's'swealth wealthper percapita capita decreases. decreases. Lemma Lemma7.8.3. 7.8.3.As Asgroup group1 1's'spopulation populationincreases, increases,(i)(i)ififA1 \ * k^, group group 1 's's wealth

decreases and group 2 's's wealth wealth per per capita capita increases; increases; (ii) (ii) ifif A1 ^ = k^, per capita decreases is not not affected. affected. wealth per capita is

multiple groups groups of consumers. consumers. We We We now extend the two-group model model to multiple may assume that that the the population is N0 may generally generally assume population is No and and the the population population can can be be classified into M groups, indexed by j, according to theft preferences, wealth, classified into M groups, indexed by j , according to their preferences, wealth, human capital, and and social social status. status. We We have have N N0 M.Two Twoextreme extremecases casesare are human capital, o >M. M N0o and and M M = 1.1. Let the number of group group j beNj. M= =N number of aggregated labor labor be N1. The aggregated force . / V is given by force N is given M

N0 =

where where hj are the level level of of human human capital. capital. We We may may also also interpret interpret the parameter parameter

hj as the work work time time of of group group jj.. The neoclassical production fUnction is F(t) == F(K(t), N). N). Let Let K, K, r, w1, function is wJt

and 5k Sk (0 < 5k < 1) be bedefined defined as asbefore. before. The The marginal marginal conditions conditions are then given <1) by by

r(t)+ = r(t)+Sk=fik(t)), h1w(t), Wj(t)=hjw{t), w(t) f(kQ)) — k(t)f'(k(t)),

/ N. The income of group j is j is where k = K KIN. Theper per capita capita,disposable disposable income of group

;1(t)=y1(t)÷k1(t), yj(t) = yj(t)+kj(t), jj=1,2,•••,M, = l,2,-,M, where

y1(t)= yJ(t) = r(t)kj(t)+wJ(t).

7.8. THE THE OSG MODEL MODEL WITH WITHHETEOGENEOUS HETEOGENEOUSHOUSEHOLDS HOUSEHOLDS351 351 The budget constraint is is given by by C]{f) ++s1Q) S}{t)== yj{t). c1Q)

j

Consumer j maximizes the the utility level Uj (0 = cf (t)s} (/), £,, Aj > 0, j = >0,

l,2,-,M,

to the Theoptimal solutions are subject to the budget constraints. The are c1Q) = s1Q) = Ajyj(t). = £jyj(t), e1;1(t), Sj cj(t) (t) = As equations (7.8.8), we we describe the the wealth accumulation by by

kj (t + + 0= l) =23h1f(k(t))÷ Ajhjf{k(t)) + [kj (t) - hjk(t%fik(t)) AjSkj (t), k1(t [k1(t)— h1k(t)j21f'(k(t))++21&1(t), j=\,2,....M.M. j=l,2 A steady state is is defined by by kj = Xfrfik) + + {k; — - h1k)21f'(k) hjk^f = 21h1f(k) + Aft1. We rewrite the the above equations as as

k1f'(k) + h1(f(k)

=

—

where 8j = 11// Aj —- 6> 5 >0. 0.Rearrange Rearrangethe theabove aboveequations equationsas asfollows follows (f(k)-kf'(k)\_ f(k)—kf'(k) \hj~kj. { 81—f'(k) Sj-f(k) .

Define k

(7.8.16)

352

7. 7. HIGHER DIMENSIONAL DIMENSIONAL ECONOMTC ECONOMIC

DYNAMICS DYNAMICS

As before, we can before, we can show show that that k is uniquely uniquely defined. defined. We require require kk > > k.k. Multiplying the two two sides then adding Multiplying the sides of each each equation equation by by ii1 rij and then adding the two two yields equations in equations (7.8.16) yields

—

J Sj

—f'(k) - f(k)

=

1,

k > k,

((7.8.17) 7.8.17)

where we use

V nJk. == kk i—ij j J and neglect the the trivial trivialsolution solution kk = = 0. 0. Like equation (7.8.11), we introduce

-

"A -1. -p(k)

If

A(k)O, ke(k, 0 has a unique solution. the equation equation<£>(k) $k) = 0=has a unique solution. Proposition 7.8.2. The dynamic dynamic system system has least one one equilibrium equilibrium point. Proposition 7.8.2. has at least point. If If A(k) >0,0, k ee (r, [k,++ °°\ the system has a unique equilibrium. equilibrium. zx(k) We omit analyzing stability properties of the equilibrium equilibrium because it is difficult difficult The model model of three groups is simulated with a wide to fmd find explicit explicit conclusions. conclusions. The three groups simulated with range of the parameters and the unique equilibrium is stable. Nevertheless, I did not succeed in constructing a Liapunov function to confirm confirm the stability. For illustration, simulate the the model modelwhen when M M= aggregated labor laborforce force N N is we simulate - 3. 3. The aggregated is

N=

7 .8.THE THEOSG OSGMODEL MODEL WITH WITH HETEOGENEOUS HETEOGENEOUS HOUSEHOLDS 353 7.8. HOUSEHOLDS 353 where hj is the Suppose that the the level level of of human humancapital capitalof ofgroup group jj,, j ==1,1, 2, 3. Suppose function takes the Cobb-Douglas form. The production production ffinction function and production ftinction and marginal conditions are are

F=AKaNfl,

r=—, w= K

/3h.F

-'

N

where A isisthe thetotal totalproductivity. productivity. Let LetK1 Kj(t) and V1 Yj(t) be respectively where (t) and (t) be respectively the capital stocks current income capital stocks owned owned by by group group j and the the current income level level of of group group jj.. The and total current income Y(t) are are total capital capital K(t) and income Y(t)

K(t) =

Y(t) =

where

K kJ(t)N YçJ(t) = (t)N1, K1j(t) (t) = = k1 j, j{t)NJ. = y1y(t)N1. consumers' behavior behavior are are described described by The consumers' s1(t) =

c1Q) =

To simulate model, we we speciQi specify the the groups' groups' human human capital, capital, preference preference simulate the the model, follows parameters and the other parameters as follows = 0.05, 0.34, A = = 1.3, 1.3, Sk = a == 034,

k h = K k =

3

1

1

0.3 0-3,

= u 21

1 ,

22

23

^0.8 0.8 0.65 0.65,

> 00.3 .3,

'r5 N N2 = 5. N1

2

N3

1

=

(7.8.18)

v 10 y

class - with the highest level level of human capital and highest Group 11 is the rich class propensity toto own in the the total total propensity own wealth. wealth. The The population populationshare shareofof the the rich rich in population is only population only 11/16 / 16percent. percent.Group Group22 is the the middle middle class class - with the middle level human capital capital and and middle-level middle-level propensity propensity to own wealth. wealth. The The level of human to own population of this group is 5/16 percent. Group 3 has the lowest level population of this group is 5/16 percent. Group 3 has the lowest level of human capital capital and the the lowest lowest propensity propensity to to own own wealth. wealth. Under Under (7.8.18), (7.8.18), we we human calculate the equilibrium values of the variables variables

354

DYNAMICS 7. HIGHER DIMENSIONAL ECONOMIC DYNAMICS

F = 18.424, 18.424, Ic k ===4.558, 4558, r = 0.1249, 0.1249, (kIc1 \ kIc22 kIc33

^31.334" 31.334 3.373 , = 3.373 > 1.941 ,

w w22

N ^3.317 3.317 1.105 , = 1.105 :0.333 0.333

c1

c2 c2 c3

'7.713" 7.713 = 1.175 0.385

see that in in the the long long term term the the differences differences in in living living conditions conditions are are great great We see among the groups. the distribution of wealth, wealth, the the levels among the groups. Table Table 7.8.1 7.8.1 gives gives the distribution of levels of current among the three groups. Group 11 's current income income and levels of consumption consumption among share of the population is only 6.25 percent share the population only 6.25 percent of the the total total population, population, but its its

shares of wealth, current income and consumption are respectively respectively 62.49, 62.49, 9.25, 9.25, and 38.30 percent; percent; group 3 's share share of population population is 62.5 percent, its shares is 62.5 percent, shares of of wealth, current income income and and consumption consumption are respectively respectively 3.87, 3.87, 19.11, wealth, current 19.11, and 19.12. 19.12. Table 7.8.1: Distribution of wealth, income, and consumption consumption group 11 group 22 group 3 group3

population 6.25 % 31.25% 62.5 % 62.5%

wealth 62.49 % 33.64 % 33.64% 3.87%

income 9.25 % 41.44% 19.11%

consumption 38.30 % 42.58 % 42.58% 19.12%

The three eigenvalues are given by p, == 0.857, 0.857, p2 p2 = = 0.734, pp3 = 0.342. 0.342. 3 = The steady state is stable. stable. Figure Figure 7.8.1 7.8.1 illustrates illustrates the motion of per per capita capita The steady the motion wealth of the three groups. We see that the growth process is characterized wealth groups. growth process characterized of of convergence competition. We convergence under under perfect perfect competition. We can can illustrate illustrate dynamics dynamics of the other variables in the system. system. As Asthe thedynamic dynamic system system has has aa unique unique equilibrium, equilibrium, we may examine impact impact of changes in the parameters. First, First, we examine impact of change in human capital. We fix the parameter \ . We increase parameter values values as as in in (7.8.18) (7.8.18) except except h1. /z, /z, = 33 to hj \\ == 3.5. We calculate the new equilibrium values as h1 from h1

k:

3.5:

AF=1.l30, AF = 1.130, AK=23.458, AK = 23.458, Ar=—0.004, Ar = - 3.004, N N 'M, '4.557 Aw, '1.122" '0.61 1.122 4.557 Aw1 0.611T Ac1 M 2 == 0.014 , Aw2 == 0.017 , Ac 0.007 Ac22 = 0M07 M2 Ak3

0.003, 0.003

W , AAw3 3,

0.005

Ac3

0.005, 0.005 (7.8.19)

7.8.THE THEOSG OSGMODEL MODELWITH WITHHETEOGENEOUS HETEOGENEOUSHOUSEHOLDS 355 355 7.8. k j (t ) 30

k1 (t )

25 20 15 15

1 0
55

1

kKit) 2 (t )

. . .

1

k^ 3 (t ) 10

5

15 15

20 20

25 25

t

of per per capita capita wealth wealth of of the the three three groups groups Figure 7.8.1: Dynamics of difference between the equilibrium equilibrium values of the In (7.8.19), we denote the difference variables at the new equilibrium equilibrium and variables and old old one one state state by by A. From (7.8.19), we see that group 11 's human human capital capital is increased, increased, the total output, output, the total total wealth wealth that as group and per capita wealth of the three groups, the wage rates, and the levels and per capita wealth of three groups, the wage rates, and the levels of consumption are all increased. society as a whole whole consumption are all increased. Hence, Hence, every every group group and and the society benefit improvement of benefit from from human human capital capital improvement of group group 1. 1. This This property property does does not propensities to save and lower hold for the other two groups which have lower propensities 1. As As demonstrated demonstrated in in (7.8.20) (7.8.20) which hold hold levels of human capital than group 1. under (7.8.18) except for hj, as group 2 's's human human capital capital is is improved, improved, group 3 which has the lowest level of human capital and lowest propensity propensity to save will of wage wage rate, rate, per per capita capita wealth wealth and and consumption consumption level. level. not benefit at all in terms of benefit but group loses. As h2 increases, increases, group group 1 and and group group 2 benefit group 3 loses. ^: k:

1 => 1.5 =>

AF 3.874, AA: 1sF = 3.874, AK = 27.73, Ar = 0.004, 0.004

Ak^ Ak1 Ak2 = Ak3

^: h3:

(

0.600 ^ 1.167

.002, v -—00.002

' AAw1 wi

,,

Aw2 = Aw33 , v Aw

0.530

' 0.148 0.148 Ac Ac22 = 0.848 A c -0.005, 0.005 , Ac3s y v — Ac1 A
f —0.046 -0.046 ,

-—00.005 .005

0.3^0.8^ 6.305, AA: AK = 26.250, Ar == 0.021, 0.02 1, AF == 6.305,

7. HIGHER DIMENSIONAL ECONOMIC DYNAMICS DYNAMICS

356

Ak1 ' 4.776 4.776N 'Ak: a12 = Ak, -0.046 , = —0.046 Ak3 ^ 0.289 0.289

Aw1 —0.256 Aw, '-0.256" = -0.085 —0.085 Aw, = Aw3 Aw, 0.484

,

(Ac: Ac1 1.176 -0.024 Ac, Ac2 = = —0.024 [Ac,) Ac3 , 0.574 0.574 ,

.

(7.8.20)

This describes that as describes a situation situation that as the themiddle middle class class gets gets more more educated, educated, the middle class class becomes becomes richer richer and and the middle therich richclass classbecomes becomes richer, richer, the thepoor poor class class becomes poorer. poorer. Similarly, Similarly, as as group group 3 improves its human group 2 's becomes improves its human capital, capital, group to deteriorate; but the other two groups' groups' living living conditions conditions living conditions tend to tend to to be improved. The effects as of change change in inthe thetechnological technological parameter parameter A are arecalculated calculated as effects of follows

A: A: 1.3 => 1.4:

AP=0.717, AF = 0.717, AK=24.765, AR 24.765, Ar=—0.009, Ar = - 0.009, N '0.129" (AAc1 ^0.801^ Ak1 3.724 Aw1 0.129 0.801 'A^ '3.724 (AW: C: Ak2 0.043 , Ac2 Ac, == 0.161 0.161 Ak, = 0.401 , Aw, =- 0.043 = 0.401 0.023 Aw33/ 0.013^ , Ac3 Ak3 0.013 0.023 Aw AC v0.026y 3,

As A the total total output, the total and per of the A increases, increases, the output, the total wealth wealth and percapita wealth wealth of three groups, the thewage wage rates, rates, and andthe thelevels levels of ofconsumption consumption are areall allincreased. increased. society as asaawhole whole benefit benefit from the the technological technological change. change. Every group and the society The effects of changes in the propensities to save of the three groups are of the three groups are The effects of changes in the propensities illustrated as in in(7.8.21): (7.8.21): 0.8 => 0.82 => AF == 0.780, 0.780, AK A£ = = 25.308, 25.308, Ar Ar = =—0.010, -0.010, 0.448^ 6.220N '"Aw," Aw1 0.140 (Ac, Ac1 0.448 '6.220 '0A40" 0.0211 Ak2 == 0.042 , Aw, = = 0.047 , Ac, Ac2 = Ak, = 0.02 Ak3 0.007 0.014 iAc Ac33 / 0.014 Ak, 0.007^ Aw3 Aw, Ak1 Akt

A2: X,:

0.65 => 0.68 => AF=0.322, AF = 0.322, AK=21.430, AK = 21.430, Ar=—0.004, Ar = -0.004, Ak1 —0.651 '-0.65f (M: Ak2 Ak, == 0.649 , Ak3 [Ak 1 0.003 0.003 j 3/

A3:: 0.3 0.4 => 0.3^0.4:

Aw1 'AW:

Aw2 Aw, Aw3 Aw,

0.058 '0.058" = = 0.019

0.006 0.006;

'-0.160^ —0.160

Ac1 (AcA ,

Ac, Ac2 Ac3 Ac,

= =

0.088

^ 0.006 J

7.8. 7.8.THE THEOSG OSGMODEL MODELWITH WITHHETEOGENEOUS HETEOGENEOUS HOUSEHOLDS HOUSEHOLDS 357 357

AF=0.1l9, AF = 0.1 19, AK=19.764, AK--= 19.764, Ar=—0.002, Ar = -0. 002, Aw1 0.021 Ac1 i"o.o2r ' -—0.062 0.062N 'AkA '-—0.253 0.253" (Aw^ Ak1 'Ac," Aw2 = 0.007 , Ac2 Ak2 Ak2 = Ac2 == 0.003 = 0.006 , Aw2

I 0.118 0.118 J

Ak3

A W , Aw3 3,

0.002 0.002,

AC
(7.8.2 1) (7.8.21)

^ 0.026 ,

As group 's propensity to save save increases, increases, the the total total output, output, the the total total wealth and group 1 's propensity to and

per of the the wage and the per capita capita wealth wealth of the three three groups, groups, the wage rates, rates, and the levels levels of of consumption all increased. increased. As As group group 2 's's propensity consumption are all propensity to to save save increases, increases, group group 3 will will benefit benefit in in terms terms of of per per capita capita wealth, wealth, wage wage rate, rate, and and consumption consumption level level but group group 1 's consumption consumption level and per per capita capita wealth wealth will will be be reduced. reduced. but level and 1

propensity will will not not benefit benefit the the rich rich group. group. Similarly, an increase in group 33 's's propensity

As is changed, As each each group's group's labor labor force force is changed, the the total total output, output, wealth wealth and and distribution of wealth wealth and and income income will will be be changed. We illustrates the effects effects of of distribution of changed. We illustrates the change in each group's labor labor force force as as in in (7.8.22) (7.8.22) N1:

AF=1.354, AF - 1.354, AK=24.386, AK == 24.386, Ar=—0.004, Ar = -0. 004, Ac ' —0.667 - 0.667" 'Aw," ^0.060" Ak1 Aw1 0.060 Ac1 'AkA ' —0.164 - 0.164" Ac2 = 0.010 M 2 == 0.017 , Aw Ak2 Aw22 = 0.020 , Ac2 = Aw3 Ac33 M Aw3 0.006, Ak33 , 0.006 0.006 v 0.003 , v Ac N . 5 ==> N2: 5.5 2: S5 = > 5

f 0

AF 1, AF = 0.777, AK AA: = 20.602, Ar = 0.00 0.001, Ak1 A/c22 = AA =

Ak3

,M3^

< 0.129 0.129" —0.03 -0.03

-0.001y v —0.001

Aw1 'Aw,"

,,

Aw2 Aw2 Aw3 ,Aw3.

( 0.032 N

Ac1 'AcA

' -—0.010 0.010"

= -—0.004 0 . 0 0 4 ,, = —0.001 -0.001,

Ac22 = Ac -0.001 = —0.001 Ac3 {Ac,)

—0.001 -0.001,

Ac1 (AcA

' 0.065 ^

N3: AF = 0.3 83, AK Ar = 0.002, AF 0.383, AA: == 19.225, Ar 0.002 Aw1 Ak1l Ak < 0.263 0.263 " 'Aw,' ' -—0.021 0.021" -0.007 Aw2 = Ak - 0 . 0 0 5 , Aw2 Ak2 = —0.007 2 = —0.005

Ak3

—0.001y r o.ooi

Aw3

—0.002 ^-0.002^

,

Ac2 Ac2

Uc3J Ac3

=

= -—0.003 0.003

(7.8.22)

-—0.003 0.003,

the rich rich group group has has more more labor labor force, force, the the living living conditions conditions of of all all the the We see that ifif the groups will be be improved. improved. IfIf group group 22 or group 33 has groups will has more more labor labor force, force, only only group benefits and both both group group 2 and and group group 33lose loseininper percapital capitalterms. terms. In In our our model, model, 1 benefits growth in the population reduces per capita output of the national national economy; economy; but this does not mean mean that that every every group's group's wealth wealth and/or and/or consumption consumption level level are are reduced reduced does

358

DIMENSIONAL ECONOMIC DYNAMICS DYNAMICS 7. HIGHER DIMENSIONAL

the population population growth. growth. Some Some group(s) group(s) may may benefit benefit from from the thepopulation population from the growth.

7.9

Path-dependent evolution evolution with education education Path-dependent

model of of economic economic growth growth with with endogenous endogenous physical physical capital capital This section builds aa model Our model model isis built built upon the three main models accumulation.21 Our and human capital accumulation?' Solow's one-sector doing model, model, and and the the one-sector growth growth model, model, Arrow's Arrow's learning learning by by doing growth model model with with education education -- in the growth growth literature.22 literature.22 We will Uzawa-Lucas's growth when the the main main mechanisms mechanisms of of economic economic growth growth in in these these three three models models show that when integrated into into aasingle singleframework, framework, they they will willproduce producecertain certaineconomic economic are integrated phenomena that that none none of of these these models models can can produce. produce. phenomena Many aspects of the model are the same as the OSG model in section 3.3. The economy has an infinite infinite fbture. future. We represent represent the passage sequence of passage of of time time in a sequence periods, numbered numbered from from zero zero and and indexed indexed by by itt = 0,1, 2,.... There is one education = 0, 1, 2 sector sector in the system. We assume a homogenous homogenous and and fixed fixed national national labor labor force force N. The labor force is distributed distributed between economic activities, teaching and study. We assume markets and select commodity commodity to serve as as assume perfect perfect competition competition inin all all markets and select to serve numeraire, with the other other prices prices being being measured measured relative relative to its its price. price. We We numeraire, with all the introduce F(t) of the the production production sector sector at at time time t;t; F(t) == output level of H(t) capital of the population; H(t) == level of the human capital labor force capital stocks stocks employed employed by the the production production Nj(t) and K force and and capital N, (it) and K,t(t) (it) = labor sector; Ne(t)(it) and Ke(t) teachers and and capital capital stocks stocks employed employed by the the (t) == number of teachers sector; education sector; w(t) and r(t) == wage rate and rate of interest. The production process is is described by

F{t)= F(t)= AK?(t)(Hm{t)TNl(t)J', A,a,/3>0, A, a, /1> 0, aa+P + fi ==1,\, A, a and and/3/?are arepositive positiveparameters parameters and and TT isisthe thework worktime. time. The The marginal marginal where A, conditions are given by 21 21 22 22

The model is aa discrete discrete version version of of the the continuous continuous model model proposed proposed by byZhang Zhang(20Usd). (2005d). The See (1965). See See also Zhang (2005a). See Arrow (1962) and Uzawa (1965).

7.9. PATH-DEPENDENT EVOLUTION WITH EDUCATION

r(t) == (i(1 -— T^K^t),

w(t) w(t) == (i(1—- r)pAHm{t)kr,

359 359 (7.9.1) (7.9.1)

rate of of capital, capital,and and rr isis the K.t I Hm"TN TN1, Sk is is depreciation depreciation rate the tax tax rate on k, = K where k. j, 8k the producers. We by k(t), We denote denote per per capita capita wealth wealth by k(t), i.e., i.e., k(t) k(t) = K(t)/N. K.{i)lN. Per Percapita capita current current income from the interest payment, r(t)k(t), and the wage payment, income from the interest payment, r(t)k(t), and the wage payment, w(t), is is defmed by defined by

y(t) + w(t). y{t) ==r(t)k(t) r{t)k(t)+w(t). We call y(t) y{f) the thecurrent currentincome incomeininthe thesense sense that that itit comes comes from from consumers' consumers' wages wages and current earnings from ownership of wealth. The per capita disposable disposable income is defined defined as the sum of the the current current income income and the wealth available available for purchasing purchasing consumption goods goods and and saving saving by by

y{t) == y(t) y{t)+k(t). + k(t). disposable income incomeisis used used for for saving and consumption. In each each period, period, a The disposable saving and consumption. In available budget budget among consumer distribute the the total available among saving, saving, s(t), s(t), consumer would would distribute consumption of goods, c(t). The budget constraint is given by consumption of goods, c(t). The budget constraint is given by c(t) + s(t) =

c(t) + s(t) = y{t). At each each point of of time, time, households households decide decide the two variables variables subject subject to the the disposable income.We We assume assumethat thathouseholds' households'utility, utility,U(t), U(t), is dependent on the disposable income. consumption level asfollows follows consumption levelofofcommodity, commodity,c(t), c(t), and the the saving, saving, s(t), s(t), as

U(t) = The consumer is choose his most most preferred preferred bundle bundle of consumption consumption and The consumer is to choose saving, (c(t), s(t)), saving, s(t)), under underhis hisbudget budgetconstraint. constraint.The Theoptimal optimal solution solution is given by c(t) =

s(t) = 25'(t).

(7.9.2)

+ 11 is equal to According to the the definitions, definitions,the thehousehold's household's wealth wealth in in period period t + the saving saving made made in in period period t

360

7. HIGHER DIMENSIONAL ECONOMIC ECONOMIC DYNAMICS DYNAMICS 7.

+ I) l) = = s(t) s(t) = k(t + kQ = Ay(t).

(7.9.3) (7.9.3)

As the the disposable disposable income is aa fimction function of the the rate rate of ofinterest, interest, wage wage rate, rate, and and the the wealth in period period t, we we see wealth see that the the value value of of the theright-hand right-handside side of ofequation equation (7.9.3) (7.9.3) is determined in period period t.t. We now now model the the education education sector sector and and accumulation accumulation of of human human capital. It is are two twosources sources of ofimproving improving human human capital, capital,through through education education assumed that there are We will combine combine the the two twoforces forces in inaasingle singleequation. equation. and learning by producing. We is given by We propose that human capital dynamics is

1)-H(t)= H(t + i)— H(t)= W

(t)TNeY ^MH(t)TN ef(H(t)TeNf

'

+

N

NH*(t) NH (t) (7.9.4)

8h (>0) (> 0) is the the depreciation rate of of human where 7Te is the time used for education, 5h fiv, and and fle fie are arenon-negative non-negative parameters. parameters. Equation Equation (7.9.4) (7.9.4) is is aa capital, ve, vt, a capital, e, flu, ae, synthesis and generalization generalization of ideas about about human human capital capital synthesis of Arrow's Arrow's and Uzawa's ideas m m accumulation. The Theterm term veK"' (HmTN€ (H TNef" (H TeNf' describes the contribution contribution to to accumulation. describes the human capital capital improvement improvement through account of learning learning by human through education. education. We We take take account in human capital accumulation by by the vtFI H'T. IH". doing effects in the term v,F We assume that education is free for students. The The total tax tax income income is is used for the teachers and and the the capital capital stocks stocks employed employed by by the theeducation education sector. sector. The paying the education sector's budget is is given by w(t)TNv(t) + (r(t) +S wfr)TNy(t)+ (r(t)+ Sk)KC(t) = tF{t). zF(t). k )Ke(t) =

(7.9.5) (7.9.5)

The education education sector sector distributes distributesits itstotal totalresource resourcetF(t) 'rF(t)to to decide decide the number number of teachers, Ne(t), level of of capital capital stocks, stocks, Ke Ke(t), to maximize teachers, Ne (t), and the level (t), to maximize the output of the education sector m (HmTNe VeKae-(H TNe veK2

(HmlNr.

The education sector's problem problem is is formed formed as as follows follows (t)zNTe, (t)(Hm (t)TNe (Hm (H'{t)T Max v.K:-{t)(H-{t)lN.f tNf,

to constraint (7.9.5). (7.9.5). The Theoptimal solution solution is is given given by by subject to

EVOLUTION WITH EDUCATION 7.9. PATH-DEPENDENT EVOLUTION K K(t) =

e\

/

(tee

£*£] + /3j(r(t)+

() e\ /

fivtF(t)

_ (cc

+

361 361

(796)

(796)

We see that Ke is the is negatively related to to the rate of of interest and positively positively to the budget, and andthe thenumber number of ofteachers teachers is isnegatively negatively related to education sector's total budget, the wage rate rate and andpositively positively related related to tothe thetotal total budget. budget. Labor Labor force force and andcapital capital arefrilly fully employed stocks are

K(t). N,(t)+N.(t)=N, K,(t) +Ke(t)= Ke(t) =K(t). N1(t)+ N, K(t)+ We have thus built the the model. model. We We now now examine examineproperties properties of ofthe the dynamic dynamic system. system. From equations (7.9.1) and (7.9.6), (7.9.6), we obtain K(t)K (t)=

«M™,HmV

jV-A^*

N= e

+PV)(r(t)+Sk)

r0fl

( 7(7.9.7) 97)

where we use use where m f, F= = TN,H TN1Hmf,

in which in which r o =(a.+A)(l-r). — Pr). ((Xe + equations (7.9.7) N,we wesolve solve From equations (7.9.7) and and N N,t ++ N = N, Nee =

N, = r0/IN - rj+py

,

e

=

top+pvt

From From 0

= (l - t)aAk; r + Sk =(l—r)a4k7 r+ok

(7.9.7), we wehave have and equations (7.9.7),

(7.9.8)

DIMENSIONAL ECONOMIC DYNAMICS 7. HIGHER DIMENSIONAL

362

= aerKj(t) K.(t)=a'iK'(t\

(799) (7.9.9)

a;

where From K K.t + + Ke where we use use the the defmition definition of of Ic.. kr From Ke = =K K and andequation equation (7.9.9), (7.9.9), we we solve K.(t) = a.k(t), = aek(t), Ki{t)=a ik{t), K,(t)=a.k(t),

(7.9.10) (7-9.10)

where

a'r0N _O^N_ , + aer

a;

a; +

The capital distribution between the two two sectors is islinear linear functions functions of of the the total total m capital. As ki ==K. KiITH Ni, we As Ic. I THtmN1, we obtain

kt

, k(t)=n. 'H"'(t) I

(7.9.11)

a fimction Wesee see that k.(t) *,.(') is is a function of k(t) k(t) and and H(t). H{$). where «,=
Lemma 7.9.1. given levels levelsof ofwealth wealthand andhuman humancapital, capital,kit) k(t)and andH(t), H(t),in in For any given Lemma 7.9.1. For the other in the period t,t, all period all the other variables variables in thesystem system are areuniquely uniquely determined determined in in this this period. The The values values of of the the variables variablesare aregiven givenasasfunctions thnctionsof ofk(t) k(t)and and H(t) H(t) by by the the period. K. (t) and following N1 and and Ne following procedure: procedure: N, Ne by byequations equations (7.9.8) (7.9.8) —, —> K,(t) and Ke Ke(t) (t) by by m and F F ==TN,Hmf equations equations (7.9.10) (7.9.10)—* -> kk1fr) by equation equation (7.9.11) (7.9.11)—> -> f/ == Akff Ak" and TN c —+ rk ++ w Ic : — and j); == y ++ A —» r(t) r(t) and w(t) w(?) by by equations equations (7.9.1) (7.9.1)—* -^ y = = rk w and c and 5s 1 = cesA. equations(7.9.2) (7.9.2)—* -> [/ c^i . by equations U=

We now show show how to to determine determine k(t) //(?) in inany anyperiod. period. From From the k(t) and and H(t) the definitionsof of;, definitions y, we wehave have y = rk +w + k. rk + w+

7.9. PATH-DEPENDENT EVOLUTION WITH EDUCATION

363 363

Substituting equations (7.9.1) and and (7.9.11) into the the above equation yields yields

+ /Jnff)(l

—

a r)AH't (t)k (t)k0(t)+ (t) + (1— (l - 88jk(t). k)k(f\

(7.9.12) (7-9.12)

From From equations equations (7.9.3), (7.9.3), (7.9.4), (7.9.4), (7.9.10), (7.9.10), (7.9.12), (7.9.12), and and lemma lemma 7.9.1, 7.9.1, it is straightforward to show showthat thatthe themotion motionof ofk(t) k(t) and H(t) straightforward to H(t)isisexplicitly explicitly given given by bythe the equations following two two difference equations kQ + 1) = 2*Hflm(t)ka(t)+ Sc(t), XHPm{t)ka{t) + <*(/), k(t

H(t + l) = veka- (?)i/^ +A)m (?) + v'H^-"ka(t) H(t

+ 8*H(t), S*H(t),

(7.9.13)

where S = A(l - Sk), 8* El—b;, = 1 - Sh and and + /Jnff)(1

—

=

._

-

N

TN n? . _ VA vtAmrf *

I

N

'

Lemma 7.9.1 7.9.1 guarantees guaranteesthat thatifif we we know know values values of of k(t) k(t) and H(t), Lemma H(t), then then we we explicitly solve Hence, to to can explicitly solve all all the the other othervariables variablesas asfunctions functionsofofk(t) k(t) and and H(t). Hence, examine dynamic properties of the whole system, it is sufficient to examine the examine dynamic properties of the whole system, it is sufficient to examine the properties of ofequations equations (7.9.13). (7.9.13). dynamic properties of equations (7.9.13) is is given by A steady state of Ic k == XH^k" a

H == y> -77

+ Sk, a, + (A+A>

"+ + v'H^-'k"

+ + 8*H. 8*H.

(7.9.14)

From the the first equation in in equations (7.9.14), (7.9.14), we we solve solve

( k=2hHm, AbEl

1* I

(7.9.15) (7-9.15)

Substituting equation equation (7.9.15) (7.9.15) into into the the second second equation equation in equations equations (7.9.14) Substituting yields

364

DYNAMICS 7. HIGHER DIMENSIONAL ECONOMIC DYNAMICS

0(H)^^e(H) + + ^(H)-S 't'(H) *(H) —h=O, = 0,

(7.9.16) (7.9.16)

where

xe Xe

= (ae + fiv + pe)m - 1, x, x• = m - K - 1 .

We see that the number of economic equilibrium points is equal equal to to the the number number for00<< H < °°. As shown in 7.9.1, solutions of the of solutions the equation, equation, <&(//) D(H) == 0 0 for H
ofXe xe > andX1 xt >> 0, 0, the the once. Figure Figure 7.9. 7.9.1b depictsthe thecase caseof > 00 and horizontal axis only once. lb depicts function monotonically andititpasses passesthe thehorizontal horizontal axis axisonly only once. once. function monotonically increases increases in in H H and > 0and andXxt < ( xe < xt >0). > 0). depicts the case of Figure 7.9.1 7.9.lcc depicts of xe >0 <00 (Xe <00 and and X, wecheck check the theconditions conditions in infigure figure 7.9.1. 7.9.1. The Thefollowing following hi appendix 7.1, we In appendix system are are determined determined by by the the proposition shows that the properties of the dynamic system returns to to scale scale parameters, parameters,Xe xe and X1. xr two returns (1)IfIfXe xe < xi < 0, the thesystem system has hasaaunique unique stable stable Proposition 7.9.1. 7.9.1. (1) and X• <00 and <0, (2) If If xe >> 00 and xt > > 0, thesystem system has hasaaunique uniqueunstable unstable (2) and x1 0, the xe > < 0 (x > 0),the thesystem system has has equilibrium; and (3) (3) IfIfXe equilibrium; >00 and Xxt <0 <00 and x, >0), (Xee < equilibrium; equilibrium;

either no equilibrium, one equilibrium point, or two equilibrium equilibrium points. points. If If itit has has two two equilibrium is stable stable equilibriumpoints, points,the theequilibrium equilibriumpoint pointwith withlow low(high) (high)level levelof ofH H is (unstable). We only interpret interpret the thestability stability condition condition xe < 0. From From the We only <00 and and x, < <0. the definitions of x and x , we may interpret x and x respectively definitions of Xee and t we may interpret Xee and t respectively as as measurements of return to scales of the education sector and the industrial sector measurements of return to scales of the education sector and the industrial sector in the thedynamic dynamic system. system. When WhenXe xe (<(>)0, (< (>) 0, we wesay saythat that the theeducation education sector sector to scale in the dynamic economy. displays decreasing (increasing) returns displays decreasing (increasing) returns to scale in the dynamic economy. if the education education and and the the production production sectors sectors display display Proposition Proposition 7.9.1 7.9.1 shows that if decreasing returns, then the dynamic system has aa unique stable equilibrium. equilibrium. twosectors sectors exhibit exhibit decreasing decreasing returns scale, the thesystem system will will the two If the returns to to scale, in the long term. In Inaatraditional traditional society society like the one one equilibrium in approach to its equilibrium mainly come come from from constructed by Adam Smith where increases in human capital mainly labor and traditional traditional education, education, the economic system be division of division of labor the economic system tends tends to to be dominated by stability. In Inaanewly newly industrializing industrializing economy, economy, education education may may dominated by stability. bevery very exhibit increasing exhibit increasingreturns returnsto to scale scale and and learning learning by by doing doing may may not not be

7.9. PATH-DEPENDENT EVOLUTION WITH EDUCATION

365

effective effective in improving improving human human capital. capital. The The economy economy may may have have multiple multiple equilibrium points. We We will will demonstrate that ifif the the society equilibrium points. demonstrate that society fails fails to explore explore increasing increasing returns from education when the system system is characterized characterized by multiple multiple equilibrium points, its development equilibrium points, development is not sustainable. sustainable. When When the system has aa unique its behavior behavior is easy unique equilibrium, equilibrium, its easy to to determine. determine. When When itit has has multiple multiple equilibrium We now equilibrium points, points, its behavior behavior is is path-dependent. path-dependent. We now demonstrate demonstrate dynamics of the nonlinear nonlinear system system with with multiple multiple equilibrium equilibrium points. points.

unique equilibrium

unique

H a) the both exhibit decreasing returns

H b) the both exhibit increasing returns

w(H) two equilibrium points

H c) the one sector exhibits decreasing returns and the other increasing returns Figure Figure 7.9.1: 7.9.1: The The two two sectors sectors exhibit exhibit different different returns returns to scale effects effects

7. HIGHER DIMENSIONAL DIMENSIONAL ECONOMIC DYNAMICS

366

specify the parameters as follows follows We specify T0=l, =0.15, r=0.08, r = 0.08, 2=0.75, A = 0.75, = 0.3, N=1, N = l, A=2, A = 2, T0 a =0.3, =1, Ti e=0.15, (7.9.17)

ae =0.35, Pe ==0.55, =0.55, =0.06, vt ==0.01, = 0.35, fle 0.55, # , = 0.55, ve = 0.06, v. 0.01,

Sk=0.05, SSh=O.03, = 03, in=1. m = \. ,r=0.3, h=0.03, 71 5k0.05, The population devotes 15 15 percent of the total time time to to education education and and the the tax tax rate rate is 8 percent. percent. For For illustration, illustration, we specify specify the the parameter parameter values values related related to to is Pe == 0.55 and /?„ = 0.55. An productivity of the education productivity education sector sector by by aaee == 0.35, fle in qualified teachers or diligent students would improve the total output increase in education sector. sector. It It should should be be remarked remarked that that the the education education sector sector exhibits exhibits of the education diminishing returns teaching, student's student's effort, effort, or or material material input. input. Under Under diminishing returns in in teaching, we have have (7.9.17), we xe = 0.45, 0.45, xx•t = -0.3, N, ==0.93. 0.93. Xe —0.3, N Ne = 0.07, 0.07, N, e = The economy exhibits increasing returns to scale in the education sector but decreasing returns in the production sector. About 92 percent of the labor force force is engaged in production and 8 percent in education. The system has two is engaged production and 8 percent in education. system two equilibrium points points {k1,HH1)= (k2, H (181.469,19.255). (k1, (31.782, 3.372), 3.372), (k2, H2)= 1) = (31.782, 2) =(181.469,19.255). At the the equilibrium equilibrium point point (1c2, (k2, HH2) the economy economyhas has higher higher levels levels of per 2) the capita capital and human capital than at the equilibrium equilibrium point (kx, H point (k1, H1). We can can x). We also calculate calculate the equilibrium values the other other variables variables at at the the two two the equilibrium values of the equilibrium points. The simulation results are provided in Table 7.9.1. 7.9.1.

Table 7.9.1: The values of the key variables at the two equilibrium points

F

r

(*„ H (k1, H1) x)

10.85 10.85

0.11 0.11

(k2,H2) (*2, H2)

61.98 61.98

0.11

w W 23.44 764.03 764.03

y 3' 27.05 784.72

y5, 58.84 966.19

cC

U

14.71 14.71

33.53

241.54

550.61 550.61

We see that except the interest rate, the difference difference between between the two values variable at at the the two two equilibrium equilibrium points points is is great. great. Our Our model model demonstrates demonstrates of any variable

7.9. PATH-DEPENDENT PATH-DEPENDENT EVOLUTION EVOLUTION WITH EDUCATION EDUCATION

367

possibility equilibrium points same type type of of economy. economy. Our Our possibility of of two two equilibrium points for for the the same analytical that two two seemingly seemingly identical identical regions regions may may follow follow analytical results results show show that radically prosperity, the other other to to radically different different development development paths, paths, one one leading leading to to prosperity, stagnation. Taiwan for this this stagnation. Taiwan and and Mainland Mainland China China may may provide provide aa proper proper case for result. similar backgrounds backgrounds in terms of cultural cultural heritages, heritages, result. Although Although they they had similar values (except level of education), Taiwan and Mainland China had had experienced experienced totally industrialization during the period of 1950-1980 1950-1980 - the totally different different paths of industrialization former had rapidly moved to the high equilibrium point, while the latter latter had had former had rapidly moved the high equilibrium point, the remained near the low equilibrium. See Zhang for analyzing these dynamical remained near equilibrium. Zhang for analyzing these dynamical 23 processes. processes.23 system evolves evolves when it starts starts from from aa state state far far away away We now show how the system equilibrium. We with five five different different initial initial states states as as from equilibrium. We simulate simulate the the model model with illustrated in five initial initial states states are marked as 1, 1, ...,, 5. illustrated in figure figure 7.9.2. 7.9.2. The The five are marked 5. The approach the the figure shows shows that that the the processes processes starting starting from from states states 1, ..., 4,, approach figure equilibrium point (k //,) with low levels of consumption and human capital. it H1) with low levels of consumption and human capital. equilibrium point (k1, The process starting infinite in the long long term. term. It It passes passes The process starting from from state state 55 will will go go infinite in the above the equilibrium point (k , H ). We may interpret this case as sustainable 2 2 above the equilibrium point (k2, H2). We may interpret this case as sustainable development as it will not end up at the "poverty trap" (kx, / / , ) . It is important development as it will not end up at the "poverty trap" (k1, H1). It is important to note that the process started at state 1 will end up at the poverty trap, though to note that the process started at state 1 will end up at the poverty trap, though the initial level of wealth is high. In contrast, the process started at state 5 is the initial level of wealth is high. contrast, the process started at state 5 is sustainable, even though the initial level of wealth is low. As human capital is sustainable, even though the initial level of wealth is low. As human capital is the key key factor factor for for increasing increasing returns, returns, the the economy economy with with low low level level of of human human the capital cannot capital cannot escape escape poverty povertytraps trapsininthe the long longterm termeven evenififitit has has aa plenty plenty of of material wealth. wealth. An implication implication of the above above results results is is that that even even big big donations donations from from An of the international little in helping helping them them from from international organizations organizations to poor countries may matter little avoiding avoiding poverty poverty in in the the long long term. term. For For instance, instance, let let us us consider consider aa case case that that a poor country country receives receives donation donation of of aa discrete discrete quantity quantity of of capital capital from from the the World World Bank. Bank. If If economy is near its low equilibrium equilibrium and high level level the economy and the the donation donationraises raises A k: to a high near to to 175 175 without without much much improving improving the the level level of of human human capital, capital, the theeconomy economy near would enjoy temporary high levels of income and consumption, but it will return would enjoy temporary high levels of income and consumption, but it will return back to its poverty state in the long term. back to its poverty state in the long term. We study the impact of system, the We now now study the impact of change change in in the the tax tax rate. rate. In In our system, the tax tax income is totally spent on education. We may thus interpret increases in the tax rate encouraging education education by government. Intuitively, Intuitively, an increase in tax tax rate rate as encouraging by the government. an increase may either increase or decrease output because education costs resources and may have little impact impact on human human capital capital improvement. improvement. We show under under what what have little We now now show conditions an increase in the tax rate may promote the economy. Taking derivatives weget getthe theimpact impact on on the the number number of of teachers teachers of equations (7.9.8) (7.9.8) with with respect respectto to T, r, we

23 23

Zhang (2003a). Zhang

368

DYNAMICS 7. HIGHER DIMENSIONAL ECONOMIC DYNAMICS H

25

5

20 4

(k 2 , H 2 )

15 15

2

10 10 3

(k1 , H1 )

5

1

25

50

75

100 100

125 125

150 150

175

k

200 200

Figure 7.9.2: Path-dependent dynamic processes processes

dNe dt dv

dt dv

1

\-t [1—v

increased,the thenumber numberofofteachers teachersisisincreased increased and and the the number number of of workers workers As rt isisincreased, is reduced. share of of educational educational expenditure expenditure in the total output output reduced. An increase in the share make some some people people to to shift shift from from the the production production sector sector to the the education education tends to make sector. Taking derivatives derivatives of of equation equation(7.9.16) (7.9.16)with withrespect respecttototv yields

d®_dH_ = (ae{pv+ae) dH dt tjav0+av) \ (at0 + aet) dHdv C

1 dN. dN,

Pv dNe +ab")W Cl) + N dt Ne dv

+

C

C

+ab2+ot

I

dv

I

where

bb 1 == b

2

-

l1 dX

dAh "

\

dt

-

—i)

ce+/Jn,

1

(1—v)18'

> -- k ++ A)(^gy - aJ3) n, dv dt — {at + aet){tofi + + vflj n• (av00 + l dn

(7.9.18)

7.9. PATH-DEPENDENT EVOLUTION WITH EDUCATION

369

We notice that the sign of the right-hand side of equation (7.9.18) is ambiguous. ambiguous. It should be remarked that that (1) (1) ififXe xe < 0 and x < 0, dO/dH < 0; (2) If xe > 0 and <0 and xt <0, dC/dH <0; (2) If Xe x, > 0, d4/dH>0; d&l dH > 0; and and (3) (3) If If xe > 0 and x, < 0 (Xe<0 ( xe < 0 and x,>0), x, > 0 ), x.>0, d'D / dH < (>) 0 at at the pointwith with low low (high) (high)level levelofof H. H. By d<$>/dH<(>)0 the equilibrium equilibrium point By

k= = 2hHm, A.hHm, we get

mkdH

dk

— = Nc/v dv dr H dx

hk. bk.

positive, then per per capita capita wealth wealth will will definitely definitely be be dH /I dv dr isisnegative negative and andb1 bx positive, If dH reduced as the society society spends more more resources resources on on education. education. By equations (7.9.10), we obtain

dK,(idk — =

dv

Jr dK, dKe dv

{ k dr ( 1 dk

1

1—v

1

Ks >

a'v0+cçv) ar + a r

\-r aid, + a

0

avo+aev)

e

C

increased, the the capital capital used used by by the the education education sector sector will be be We see that if kk isisincreased, increased, even level of of capital capital employed employed by the the production production sector sector increased, even though the level may be reduced. As the conclusions from the comparative statics analysis are ambiguous, we illustrate impact of changes in the tax rate by simulation. Here, we are interested in the path-dependent case. We still specify specify the parameter values as in equations (7.9.17) t. Let us us consider consider the case case that that the the expenditure expenditure on (7.9.17) except except the the tax tax rate v. education 8.5 percent, percent, that that is, is, educationisis increased increasedfrom from88 percent percentofof the the GDP GDP to to 8.5 r: 0.08 => 0.085. We calculate v: =' 0.085. N, == 0.925. 0.925. N Ne = 0.075, 0.075, N, e = We see that the number of of teachers is is increased. increased. The system system has two two equilibrium equilibrium points given by (ku HH1)= (k2, H2)= H2) =(143.511,15.346). (143.511,15.346). (k1, (34.172, 3.654), 3.654), (k2, l)= (34.172, As the government further encourages education, the equilibrium equilibrium point with downgraded. The low level of human capital is upgraded; while the other one is downgraded.

370 370

7. HIGHER DIMENSIONAL ECONOMIC DYNAMICS DYNAMICS

distance between the the two equilibrium equilibrium points is is shortened. shortened. We We can can also also calculate calculate equilibrium values values of of the the other other variables variables at at the the two two equilibrium equilibrium points. points. The The the equilibrium simulation results are provided provided in symbol + (—) (-) in in the table table simulation results are in table 7.9.2. The symbol the difference difference between between the the variable variable values values in in table table 7.9.2 7.9.2 and and table table 7.9.1 7.9.1 isis means the (negative). We see that the output output at the the equilibrium equilibrium point with with lower lower positive (negative). level of human capital is increased as the tax rate is increased; increased; but the output at the other other equilibrium equilibrium point point isis reduced. reduced.

Table 7.9.2: The values of of the key variables variables at at the the two two equilibrium equilibrium points points

F

r

w w

yY

(k1, H1)

11.7(+)

O.12(+) 0.12(+)

27.4(+)

3l.4(+) 31.4(+)

y 65.5(+) 65.5(+)

{k2, H2) H2) (k2,

49.0(-) 49.O(-)

O.l2(+) 0.12(+)

484.O(-) 484.0(-)

500.4(-)

644.O(-) 644.0(-)

c

C

16.4(+) 16.4(+) 160.9(-) 160.9(-)

Figure 7.9.3 shows shows how the equilibrium equilibrium points are are shifted shifted - the points with larger sizes is the new steady states and the other two points with smaller sizes states. The initial states in figure 7.9.3 are correspondingly the are the old steady states. same as in figure 7.9.2. It is important to note that the path started at initial state 4 will will not notend end up up poverty poverty trap; trap; instead instead the economic development is sustainable of strengthened strengthened education education policy. policy. as a consequence of The identification of poverty identification of poverty traps traps in in theoretical theoretical models models has has important important stable poverty, poverty, implications for policy making. When an economy is trapped into stable it carmot cannot escape traps by small policy intervention or foreign aids. The economy economy needs either large short-term short-term policy interventions or large amount of foreign foreign aids to sustain sustain economic economic progress. Moreover, even large interventions interventions may may fail fail if the society does not learn, learn, for instance, instance, about about how how to improve improve skills, how live society how to live inequality caused caused by rapid rapid economic economic growth, and how to to handle handle with with class class with inequality mobility due to differences differences in in human human capital. capital. In In an an evolutional evolutional economy, economy, as as mobility demonstrated in our example, it is is possible possible that national national growth is is affected affected by demonstrated government education other parameters parameters remain fixed and and the the government educationpolicy. policy. IfIf the the other remain fixed government encourages relatively government encourages education, education, our our example example indicates indicates that that itit isis relatively easier for for the the economy economy to to escape escape poverty poverty trap trap with with the the encouraging encouraging policy. policy. We We easier see initial conditions conditions matter. matter. see that that initial We now study the impact impact of of change change in in the the propensity propensity to to save. save. From From equations equations (7.9.8), we see that change in the preference preference has has no impact on the labor distribution, dA = 00 and dIV1 / dA = = 0. i.e., dNvl/ dX dNJdX 0. Taking derivatives derivatives of equation (7.9.16) with respect to A X yields yields

EVOLUTION WITH WITH EDUCATION EDUCATION 7.9. PATH-DEPENDENT EVOLUTION

371 371

H

25 5

20

4

15 15

2

2 (()k k 2 2, ,HH22)

10 10 3

5

1

(k1 , H1 )

k

50

100 100

150 150

200

250

Figure 7.9.3: The impact of change in the education policy

JO dH CftbdHtrfte+O(Di dH dX dHd2 2/3(1—8) < 0 and x, <0, < 0,the theequilibrium equilibrium to save save increases, increases, (1) (1)ififXe xe <0 As the propensity to capital increase; increase; (2) (2) ififXe xe >> 0 and and x1 xt > 0, the the equilibrium equilibrium level level of of level of human capital 0 and and xx1 0 (x 0 and 0), human andand(3) (3) If xIf; humancapital capitalwill willdecrease; decrease; >0 (Xe and x,x, >>0), e > t <<0 e <<0 the equilibrium level the equilibrium level of of human human capital capital may may either either increase increase or or decrease, decrease.hi hi a traditional society where no sector exhibits increasing returns to scale, an increase increase traditional propensity to save will increase increase human in the propensity human capital. capital. On On the the other other hand, hand, in in an equilibrium is located economy where its current equilibrium located at the higher higher equilibrium equilibrium level level of of anincrease increaseininthe thepropensity propensitytotosave savemay mayreduce reducethe thelevel levelof ofhuman humancapital. capital. H, an

By k=AhHm, weget m

By k = AhH , we get

dk

dk dX d2

mkdH mk dH dX H d2

k

k Xp{\-S)' 2/3(1—8)

372

ECONOMIC DYNAMICS DYNAMICS 7. HIGHER DIMENSIONAL ECONOMIC

dH /I dA dA>> 0, an an increase increase in in the the propensity propensity to save save will will increase the level of per If dH capita wealth. remarked that ifif no no sector sector exhibits exhibits increasing increasing sector, sector, capita wealth. It should be remarked Adam Smith's argument argument that saving saving benefits the nation nation as well well as as the the individuals individuals hold. But if if dH dH /I cIA dA <<0, 0, we see see that that increases increases in in the the propensity propensity to save save may may hold. But reduce the equilibrium equilibrium wealth small. Malthus Malthus argued argued that that higher higher reduce the wealth ifif kkl/ At; is small. propensity economic growth growth due due to to decreasing decreasing returns returns propensity to save may not benefit the economic to scale in population growth; Keynes argued that a high propensity to save will not benefit economic growth because of loss of the total output due to unemployment. unemployment. benefit Our model model shows shows that aa high high propensity propensity to tosave savemay mayharm harmnational nationaleconomic economic growth even if we assume assume that that the the population population isis constant constant and and labor labor force force isisfUlly fully employed. employed. By equations (7.9.10), we obtain

1 dK, dKt _ J_dK^ _ }_dk_ K, dt dt ~ kdr k dr' K1 Jr ~ Kv dr 1

1

employed by the two sectors sectors are changed changed in the same same direction direction The capital stocks employed m we have as that of k. By By k.(t)=n,k/Hm, k,(t) - ntk I H , wehave asthatofk.

k dA d2

Ap{\ — - a) 5)

>0.

AKa mN^f, , we get From equations equations (7.9.1) (7.9.1)and and FF = AK"{H

dr . dk, dk n — = H-Br—'-<0, dA dA dA d2 ]_dw__ mdH m_dH_ dkj_ ldw dk. : +a w~d~A~~H~dA ~dA' wd2 HdA dA }_dF__fim_dH_ a_dk_ fiJidN, dNt 1 dFJimdHctdk F dA~ H dA kkdA dA N1d2 N, dA FdA HdA interest will will definitely definitely decrease decrease as as the the propensity propensity to save save isis We see that the rate of interest increased. If dH //dA>0, the wage rate also increases. The impact on the output is dA> 0, the wage rate also increases. The impact on the output is ambiguous even when I dA >>00 because becausethe thenumber numberof ofworkers workers isis reduced. reduced. when dH IdA For illustration, illustration, we simulate simulate the the model. model. We Westill stillspeciQv specify the the parameter parameter values as in (7.9.17) (7.9.17) except the propensity propensity to save save A. A. Let Let us us consider consider the the case case

7.9. PATH-DEPENDENT EVOLUTION EVOLUTION WITH WITH EDUCATION EDUCATION

373

is increased increased from from 0.75 0.75 to 0.76, that is, that the propensity to save is k: 0.75 0.75 => => 0.76. 0.76. The two equilibrium points are given now by by 2: 15.61 i). (k1, H1) (*„ H = (42.217, 4.189), (k2, (k2, H2) H2)== (157.315, (157.315,15.611). 1) =

We We can also also calculate calculate the equilibrium equilibrium values values of the other variables variables at the the two equilibrium points. points. The The simulation simulation results results are are provided provided in in table table 7.9.3. 7.9.3.

7.9.3: The The values values of of the the key key variables variables at at the the two two equilibrium equilibrium points points Table 7.9.3:

F (k1, #H1) 1

(k2, #H2) 2 )

w W

r

0.ll(-) 0.1l(-) 0.1l(-) 0.ll(-)

13.7(+) l3.7(+) 51.3(-)

;y

3'

c

C

36.9(+)

41.5(+)

83.7(+)

512.4(-)

529.6(-)

686.7(-)

20.l(+) 20. l64.9(-)) 164

Figure 7.9.4 shows the simulation simulation results - the points with larger sizes is the new steady states and the other steady states and the other two two points points with with smaller smaller sizes sizes are are the the old old steady steady states.

H 25

5 20

4 2

15

10

(k 2 , H 2 ) 3

5

1

({kk1v,HH1 ) 25

50

75

100

125 125

150 150

175

k 200

Figure 7.9.4: The impact of change in the propensity to save save

374

7. HIGHER DIMENSIONAL ECONOMIC DYNAMICS

We see that like the encouraging education policy in the previous discussion, aa higher propensity propensity to save makes it easier for a poor society society to escape poverty trap. suggests a way to escape escape poverty traps. The way is is to to strengthen strengthen the the Our model suggests propensity wealth and encourage encourage education. education. These propensity to to own wealth These policies policies would would make economic growth sustainable. Foreign aids are important important for for rapid industrialization as the aids aids may may help help the the economy economy to to move move to to aanecessary necessary state state for for sustainable sustainable economic development. The economic miracles of Japan, Taiwan, Korea, and economic development. The economic miracles of Japan, Taiwan, Korea, and Singapore after the Second World War are all characterized by high saving rates, Singapore after the Second World War are all characterized by high saving rates, encouraging education encouraging education policy, policy, and and proper proper international internationalenvironment environmentin inthe the form form of of foreign aids or/and foreign investment. Mainland China is currently repeating foreign aids or/and foreign investment. Mainland China is currently repeating similar processes processes by by heavily investing in in education, education, saving saving as as much much as as possible, possible, and and similar heavily investing attracting aids and foreign investment as much as possible. attracting aids and foreign investment as much as possible.

Appendix proposition7.8.1 7.8.1 A.7.1: Proving proposition The Jacobian matrix at equilibrium is [A1a11

,?1a121

f

n2

where

a11 =

+ S + —a12 l = 21 —(

(f )

—

^ + —a12, kf')— A,

nn22

an =fo-kk)n2f",

a12 =

(k1 —

a21 = (k2

f'

—

h2k)n1f",

a22 == f ++ S = 1—(f - ( /—- 4Jj.?!?_ kffe + 8++ ^a21 =1 + ^a2V a22 nx

k

1c22

111

r\ n1

According to rij, we we have have According tothe thedefinitions defmitionsofofkk and and n1, (Ic1 —

+(Ic2 (k2 -— x +

h,k)n 0. h2k)n2 = 0. 2 =

(A.7.l.1)

A.7.1. A.I. 1. PROVING PROPOSITION PROPOSITION 7.8.1 7.8.1

375

From and From this this equation, equation, we have: have: a12a21 aua2] < 0. We see see that that (&, l\k) and < 0. (k1 — h1k) h2k) have the opposite signs if (k (k22 -— h2k) 0. k^-hfi* 0. — kk

we have have From equations (7.8.9) we ' I/i.,

k—h.k=I

f'

where Sj — - / ' >0. > 0. We We see see that that if S2 > Su k2 —k/c - hjc (and ff—/cS2) - kS2) must must be negative. If If 82 =
(and (and f/ —- kS2)) will equal zero. zero. As

S. S /17 Aj

82 >> Sj means means A\ >> A2. X\. In the the remainder remainder of the the proof, proof, we werequire require X\ >> A2. A \ . We A\. can similarly prove the case of X\ < A2. we have Under k\ >> A \ , we

\ A;

- \ < 0, ^f - S2 < 0, -^ - A > 0, L - S, > 0, a12cz0, an < 0, a21>0. o21 > 0. A:

k

k

The The characteristic characteristic equation equation is is given by

p2 —- Trace(J)p Trace(j)p ++ Det(J) Det{j) M(p) == p2 where where

X\au + + A2a22, k\a22, Trace(j) == A1a11 Trace(J) Det{j)=A\A - a12a21). — Det(J)= 21k(a11a22 2{aua22

(A.7.1.2) (A.7.I.2)

7. HIGHER DIMENSIONAL ECONOMIC DYNAMICS DYNAMICS

376

From equations 1.1) and (A.7. 1.2) we calculate equations (A.7. (A.7.1.1) (A.7.1.2)

Det{j) = A1X2 \f

+ 5 + ^-au

A122(f' +

II / ' + 8 + ^a2l

I - ana. (A.7.l.3) (A.7.1.3)

> 0,

where we also use

na n1a12 1

12

n2

+|

nna _Q 22a2l21=0. «!

f'

From Sj — - / ' > 0, we obtain

f')>

11 >> Xj{\ ++ / ' ) > 0. 0.

From equation 1.3), we have equation (A.7. (A.7.1.3), have

0cDetJc1. 0 < DetJ < 1. straightforward to calculate It is straightforward

M(l) = 11—- Trace(.J) Trace(j)++Det(J) Det(j) == +

_[ + /') - {^kaa12 ] >0, > 0, - f)(S2 - f') a21) 12 + ?£-a2

=

-

whereweuse where we use Sj —[>0 — f > 0and and =

+

a21

a12

<0. nl

+

-aa21 21 -

n,

n2

(A.7.1.4)

PROPOSITION 7.8.1 7.8.1 A.7.1. PROVING PROPOSITION

377 377

From

Det{j)>Trace{j), \I ++Det(J)> Trace(J), 00
(A.7.1.5) (A.7.1.5)

we see > Trace(J). Trace(j). We now > 0. now try to to find fmd conditions conditions for for Trace(j) Trace(J) > 0. First we we see 22 > have Trace(j) Trace(.J) == {\ ++ A,)(/' a) — —2){k2 )(f' ++S)-(\-A

-h2k)n2f", h2k)n2f",

—

implies where we use use equations equations (A.7.1.3) (A.7.1.3)and and(A.7.1.4). (A.7.1.4).As AsA(k) A(k)>00implies

k2f"

+

from the above bequality inequality we we have have from

Trace(J)

+

+ a)

-

)k2n2f"

-

-

-

As - ^k.nj" > 0,0, - ^ — A2)k2n2f"> Trace(j) guaranteed if Trace(J)>>00 isisguaranteed if

+ -X^VHU- - A 0. > 0. a)- U -k)kn2(+-r) + ^)(/' + S)From

_J_ / f<82 we have

(A.7.1.6) (A.7.l.6)

7. HIGHER DIMENSIONAL ECONOMIC DYNAMICS DYNAMICS

378

((i ((1 ++

V2

- lJkn2'

+ a)-

+

where we use 1. nx + + h2n2 h2n2 = 1. kn1

If

''

+

h2n2 h2n2

h2n2 l

(A.7.I.7)

)

then (A.7.1.6) holds. We see that if the difference difference \ -— X^ large, (A.7.1.7) (A.7.1.7) k isis not large,

holds. In summary, under under A(k) 1.7) we A(k) >0 0and and(A.7. (A.7.1.7) wehave have

0< pxp2) <<1,1, 0 > Trace(J) Trace{j) (= (= p1 px ++ p2) /?2) >> 0. 0.

(A.7.1.8)

We also have

M(-1) = 11++ Det(J)+ Det{J) + Trace(J)> Trace{j) > 0, 0, M(1)> M(l) > 0. 0. M(—1)= We calculate + + 1 Tracej2 Traced— -4DetJ ADetJ== {Ai + Atf{f ++ Sf + \ ^ aa12n +

+

+ +

a21J

a12

+

+ Fi2k nn

)(p+

a21]2

n.

+ a12

-

n21a2

a21J.

+

^^a a21]2 2l

A.7.1. PROViNG PROVING PROPOSITION PROPOSITION 7.8.1 7.8.1

379

The sign of TracaJ2 the case of TraceJ2—- 4DetJ ADetJ is ambiguous. In the Trace.J21— > 0, TraceJ - 4Det.J ADetJ >

the two eigenvalues eigenvalues are real and and distinct. distinct. The The two two eigenvalues eigenvalues have the same same sign. From (A.7.1.8), (A.7.1.8), we have 1. 0 „ p2 p2 << 1. 0<

The equilibrium point is a sink and the adjustment adjustment is monotone. monotone. If If Tracef2 < 0, TraceJ2—- 4DetJ ADetJ <

the two eigenvalues are imaginary and the eigenvalues have modulus less than 1. 1. The equilibrium 1.7) equilibrium point is is aa steady steady focus. focus. We We thus thus conclude conclude that that under under(A.7. (A.7.1.7) the unique equilibrium equilibrium point point isis stable. stable.

A.7.2: Proving Proving proposition proposition 7.9.1 7.9.1 First, we fmd find conditions conditions such such that that

* ( / / ) = 0, has positive solutions. We exclude the case of xe Xe

= x, = = 0. 0.

It is > 0 and is easy easy to to check check that that ififXe xe > and x• x, >> 00 (which guarantees guarantees O(o) < 0,

<00 and xt < (P'(H)>O for H H>0) cO0 (which O(°o) > 0 and and '(//) > 0 for > 0 )ororXe xe < (which guarantees guarantees <&(0)>0, O(OO)<00 and and > 00 ), ), the system system has unique ($o)> 0, 1b(oo)< cv(H)< 00 for has aa unique <0 equilibrium point. We now show show that that in inthe thecases casesofofXe xe > 0 and Xxt <0 positive equilibrium (or Xe xe <0 < 0 and and X• x, > 0), the system has two equilibrium points, or one equilibrium equilibrium (or point, or no one. We just prove and x, <0. < 0. The other case can can be be similarly similarly prove one onecase, case,Xe xe >> 00 and and 4)(oo)> checked. Since Xe This implies and Xx, <0, i(o)>> 00 and O(°°) > 0. checked. Since xe > > 00 and < 0, <&(o) 0. This implies that that <&(/f) == 00 has has either either no no solution solution or or multiple multiple solutions. solutions. Since Since

7. HIGHER DIMENSIONAL ECONOMIC ECONOMIC DYNAMICS DYNAMICS

380

+ x.4),,

ff4)' =

4),<Xare positive, wewe seesee thatthat 4)' <&'(//) (H) may be be either positive or or <£>and >, are positive, may either positive where 4)e e and negative, depending depending upon the values values ofofXe xe and x,. xt. If 4)(H) <&(//) = 00 has more more than than negative, two positive solutions, there are at least two positive values of H such that two positive solutions, there are at least two positive values of H such that <&'(//) = 0.Since Since 4)'(H)=o. dH4)

dH dH

>0,

strictly holds for for H >> 0, strictly holds 0, it is impossible for for

tb'(H)= ) =o, 0, to have more than one one solution. solution. Accordingly, Accordingly, 4)(H) <&{H)= 0 has either no solution or one solution, for the solution, or two two solutions. solutions. A A necessary necessary and and sufficient sufficient condition condition for ff* existence of two equilibrium points isis that that there there exists exists aa value value of of H* such that existence of equilibrium points <&(//')< 0. 0. The Jacobian matrix is given by

J=

a+/38 aeH4)eaff4),

The characteristic equation is given by

M(p) == p2 Trace{j)p ++ Det(J), Det(j), p2 —- Trace(J)p where

+8*, Trace(j) = a + p8 + (j3v + fie)m®e +(flm—,r)4), + [fim-n^, + S\ Det{j) == ({J3r++ pe),n4)e K ++(Jim (pm—- *r)O, + S')(a ){a+ 138) Det(J) -(\-S)pm{a + - 8)flm(cc4)e e<&e+o®,). We see that that Trace(j) rrace(J) >> 0. 0. We calculate

A.7.2. PROVING PROPOSITION 7.9.1 7.9.1

381 381 ))2

[fim—-ir)4), *)*, +8* + 0. > 0. As

Trace(jf -— 4Det(j) Trace(J)2 4Det(J)>> 0, 0, that the the eigenvalues eigenvalues are are real real and and distinct. distinct. We We calculate calculate we conclude that M(1) = 1 M(l) - Trace(J)+ Trace{j) + Det(J) Det{j) == -— {x + x,C1)(l xiOi )(l -— S)fi, 1 — o)p, e®e + (xfte

(A.7.2.1)

in which we use

H=4)eH+(DiH+8*H, H = <&eH + 0>,// + S*H,

1-5>0. \-3>Q.

We rewrite as follows follows rewrite Det(j) Det(.J) as 0(l—- 5)(Xe(be 5)(xe®e ++x1t,) xpt) Det(j) = = {/3v+l3 Det(J) (Jim — / r ) ^—- ji(i + /Jee )m
(A.7.2.2) (A.7.2.2)

+ ?-p(\-S\ where we use where we use

* . + * , = Sh at at the steady state. As 5*

/3(1

8)

Si,

/38

we can always assume assume this this term term to be be positive positive ifif the the depreciation depreciation rate rate of of human human capital is assumed less than a. assumed less than <00 arid <0. and xx,i < 0. From From equations equations (A.7.2.1) (A.7.2.1) We now examine examine the the case caseof ofXe xe < (A.7.2.2), evidently evidently and (A.7.2.2), 0, M(0)= M(0) =Det(J)> Det(j) >0. 0. M(l)>0, are real and distinct and As the eigenvalues are

382

7. HIGHER DIMENSIONAL ECONOMIC DYNAMICS

px+ p2 = TraceJ > 0, p1p2=DetJ>O, pxp2 = DetJ > 0, p1+p2=Trace.J>O, we conclude conclude that > 0, two eigenvalues eigenvalues are are that they they are are both both positive. positive. As As M(l) M(1)> 0, the two Since on the same side side of of +1. Since

Trace{j)=1+ = l +a a+188 + p8+ + (A+ + A>»*.+ +(,8mtfm-x)*, - Sk Trace(J) — (18., , s, + x,4>, - m\a,) < 2, = 1 + a + PS + xeQ>e+x,01

()

(A.7.2.3)

where we use 8h'

1+a+flö<2,

we conclude conclude that 0 < pp1, < 1.We Wethus thusproved provedthat thatififXe xe < xt < 0, the cO0 and X <0, p2 x, p 2 ci. unique equilibrium is stable. > 00 and From equations (A.7.2. I), now consider consider the the case case of of xe > and X• We now JC, >> 0. 0. From (A.7.2.1), This implies implies that that the the two two distinct distinct eigenvalues eigenvalues are are on on the the two sides of M(1) M(l) << 0. This +1, i.e., +1, i.e., p1 /?] <<11 and and p2 p2 > > 1. 1. The unique equilibrium equilibrium is unstable. Furthermore, it straightforward to confirm confirm the following following relation is straightforward

Trace{j) + Det(j) == M(1) A/(l) ++ 2Trace(j). M(-1) M(— 1) = = 1 + Trace(J) + Det(J) 2Trace(J). 1

Substituting (A.7.2.3) into into the above equation equation yields yields Substituting equations equations (A.7.2.1) (A.7.2.1) and and (A.7.2.3) the above

50) +2(1 ++ a+ a + po)— M(-1) = {xe0e ++ xpi ){\ + +a a++ 8/3)+2(1

+ a4',)

=(2—Sh—mc5,X1+a+fl8) [(l + ae +p., + pv + fi,)(l + a + dp)m - 2ae\n<S>e

[(l + m-n)(l + a + 80) +[(i+m—7rXi+a+8/3)—2ma]4,, where we use 0 ^ + 0b.; == 6h 8h and and the defmitions definitions of of xe and xr As it is reasonable to assume the following three terms to be positive

2-8h-m8h>>0, 0, 2—Sh—mSh

{\ + at+fl > 0, +flv+/Je)(1+a+5fl)m—2ae v+ fl,)(l + a + S0)m - 2ae >0, (\ + m- x){\ + a + 8p)- 2ma > 0,

(A.7.2.4) (A.7.2.4)

PROVING PROPOSITION PROPOSITION 7.9.1 7.9.1 A.7.2. PROVII4G

383 383

we see M(—\) > 0. Since Since see that that M(—l)>O.

M(1)cO, M(—1)>'O, we have p1 px Ee (—1, (-1, l). Hence, the equilibrium equilibrium point under (A.7.2.4) (A.7.2.4) is saddle i). Hence, is a saddle point. We now now check check the the case case of ofXe xe > 00 and and x• xt < 0, when when there there are are two two We <0, > 00 and x• xt <0, < 0,can canbe besimilarly similarly proved. proved. equilibrium points. The The other other case, case, xe > equilibrium First, we note that

+ X.(I), <(>)0

Xe(Pe

the equilibrium equilibrium point with the lower lower (higher) (higher) level level of H*. Let H1 H, and and Hh Hh at the point with (H1
xXe4)e < 0. ee+ + x;4>,<0. It same as as in in the case of Xe xe <0 < 0 and ft can be seen that the stability conditions are the same xi < 0. We thus have 00<< p1, ply p2 p2 <1. < 1. At At Hh Hh <0. X1 Wethushave r d)

+ +

v (f)

> 0. 0 >

The stability xt > 0. 0. We thus > 0 and stability conditions conditions are are the the same same as as in in the the case case of of xe > and X1 conclude that the equilibrium point with the higher value of H is unstable. the equilibrium point with the higher value of H is unstable.

Chapter Chapter 8

Epilogue To conclude this study, study, we we mention mention two two important important issues, issues, time time scales scales and and economic economic structures, for understanding economic economic evolution. evolution. As time time passes, passes, economic economic issues issues with with which which economists economists are concerned have are concerned shifted. time of of Adam Adam Smith, Smith, the the economic economic variables variables that that shifted. Even Even since since the the time economists have dealt with appear to have have been been invariant. invariant. But the in which the ways ways in which economists these variables are combined and the speeds at which they change have constantly varied and the dominant economic doctrines have shifted shifted over time and space. The complexity of economic reality is constantly increasing modern time. This This is is complexity of economic reality constantly increasing in modern partially because of the expanded capital and knowledge stocks of mankind. partially because of the expanded capital and knowledge stocks of mankind. Knowledge, in in fields fields of of philosophy, philosophy, arts, arts,literature, literature, music, music, technology technology and andsciences, sciences, expands man's imagination and extends possibilities of human action, expands man's imagination and extends possibilities of human action, not not to mention that the knowledge knowledge reservoir reservoir can can directly directly satisfy satisfy the the desires desires of of an an unlimited unlimited number of people at the same time. the same time. Knowledge Knowledge is not only power and and sources sources of money, but also the most durable capital goods for human mind. Increases in money, but also the most durable capital goods for human mind. Increases in machines, housing and infrastructures has enriched human environment, increased machines, housing and infrastructures has enriched human environment, increased accessibility enlarged variety human behavior. behavior. The The accessibility to to various various locations, locations, and and enlarged variety of of human explosion resulted in very explosionofof knowledge knowledgeand and capital capital inin modern modem times times has has resulted in very complicated action fields. fields. complicated human human action

Time is at the center of the the chief chief difficulty difficulty of almost almost every every economic economic problem. problem. is becoming increasingly The role of time in decision-makings and action becoming increasingly The role of time in decision-makings and action complicated difficult complicated as variety of action and social networks are expanding. It is a difficult issue to decide decide the length length of of time time which which affects affects aaspecial special decision decision making making since since issue each kind of human decisions decisions are are made made with with different different time time scales scales and and two two persons persons may have different different time scales with regard to the same same kind kind of of decision decision making. making. Because of the high variety of human behavior and time scales, in order to analyze single person's economic economic behavior behavior as a whole whole we have have to to conduct conduct the the analysis analysis a single within a framework framework with varied time scales. scales. Human Human behavior is is connected connected in in direct direct 385 385

386

8. EPILOGUE

or indirect ways in human action fields; but we may miss interdependence interdependence between between some elements if we do not properly recognize the role of time. time. Another dimension Another dimension in analysis is space. space. Man, Man, action, action, capital, capital, knowledge knowledge and and time can only if we locate can become become culturally culturally and and socially socially meaningfUl meaningful only locate them them over over space. Each human being is born into a unique existence and each piece of land has its unique attributes in affecting affecting human human action. action. Space means individual Space means characteristics accordingly requires classification. This is particularly particularly characteristics and and accordingly requires refined refined classification. important in analyzing modern economies. Fast technological changes, important analyzing modem Fast technological changes, richness richness of of material living conditions, conditions, complicated complicated international international interactions, and many other interactions, and modern phenomena have increased complexity complexity of spatial spatial economies. economies. The The modem phenomena have increased subsystems such as ecological, economical economical and social social subsystems, subsystems, which which could could be be once decomposable decomposable as separate elements in analyzing the social social system system at at least least in in short terms over a homogenous space, have to be treated as a part of the whole short terms over homogenous space, have to treated a part of the whole system. economic relations recognized if we don't don't explicitly explicitly system. Some Some economic relations cannot cannot be be recognized if we introduce spatial and temporal dimensions. It will take some time for what is is introduce spatial and temporal dimensions. It will take some time for what happening in a scientific lab to affect economic reality. Without spatial dimension, happening in a scientific lab to affect economic reality. Without spatial dimension, we can hardly hardly analyze analyze actual actual processes processes of, of, for for instance, instance, how how Japanese Japanese economy economy we can may actually affect affect the the world world economy. economy. In In fact, fact, the the choice choice of of spatiotemporal spatiotemporal scale scale may actually is aa delicate delicate and and obligatory obligatory process process and and must made before before actual actual study study of of any any is must be be made special economic economic problem. problem. The The explicit explicit awareness awareness of of this this necessity necessity isis important important for for special understanding both economic reality and structure of economics. For instance, for understanding both economic reality and structure of economics. For instance, for human life what is good to one's taste (assessment on a short timescale) may be human life what is good to one's taste (assessment on a short timescale) may be harmful to one's health (assessment on a longer timescale). One can hardly explain harmfUl to one's health (assessment on a longer timescale). One can hardly explain differences between between Keynes Keynes and and Schumpeter's economic visions without differences Schumpeter's economic visions without differentiating their temporary scales. Temporal scales in the economist's vision differentiating their temporary scales. Temporal scales in the economist's vision have complicated interdependent relationships with actual analyses and abstraction have complicated interdependent relationships with actual analyses and abstraction of reality. reality. of We are in in an an era era of ofhigh higheconomic economiccomplexity. complexity. This This implies implies that that economic economic decisions have large context context in in which which internal internal structures structures of decisions have to to be made within a large subsystem and connections connections of different different subsystems subsystems have be taken taken into into each subsystem have to be dynamic framework. framework.The The bringing-up bringing-upof ofchildren, children,lower lower account within aa genuinely dynamic higher education, education, family family structure, structure, and connected in a and higher and family family values values are are all connected subtle and complicated complicated way in economic networks. We have to consider reciprocal reciprocal of social social and and economic economic factors factors rather rather than than considering considering relations of different aspects of these facts facts in inisolation. isolation. Simple Simple one-sector one-sector growth growth models models without without economic economic these structures will hardly provide any useful information about the complexity structures will hardly provide usefUl information about the complexity of of modern economies. We need to to enlarge enlarge analytical analytical frameworks to handle handle multiple multiple hierarchical levels, levels, multiple multiple space space degrees degrees and and multiple multiple time time scales. scales. hierarchical system is composed of many many people, people, and and the psychology of An economic economic system composed of psychology of people relations (which are are reflected reflected in in values, values, institutions institutions and and customs) customs) people and the relations among people are constantly changing. The difficult difficult task is to find out out whether whether or or not there are durable (if not permanent) patterns or orders orders in human behavior and in in human societies and to explicitly construct descriptions (usually, models) for these orders if they exist. In order to construct a comprehensive theory it is necessary to

8. EPILOGUE

387 387

understand general general patterns patterns of of people's people's behavior in a society over time and space. understand difficult task is how to construct such such aa comprehensive comprehensive economic economic theory. theory. The difficult It is significant to examine examine economic economic systems spatiotemporal structural systems with with aaspatiotemporal structural vision. The key key words words are space, time, and structure. It is hard to give aa precise vision. The structure. It precise concept of structure. Here, a structure structure means aa sum sum of ofelements elements and andrelationships relationships concept between those elements. elements. In other words, structure structure stands for the way way the the elements elements constituent parts and constituent parts of a whole are arranged with respect to each other. Structure represents represents aa whole whole in which which each each element element depends depends on on the the others others by by virtue virtue of its 1 relation with them. According to Thorn, structure is defined as a spatiotemporal relation Thom,1 structure is defined spatiotemporal morphology described significant spatial morphology described by significant spatial discontinuities discontinuities and and by by the the syntax syntax that determines how these sets of discontinuities form into relatively relatively stable systems. In evolving structures time. The The structure structure includes includes properties, properties, evolving structures relations relations depend depend on time. which are properties of the whole rather than only properties of its component which are properties of the whole rather than only properties of its component parts. Any changeininone oneelement element or orone onerelationship relationship will will cause cause aamodification modification in in parts. My change other elements or relationships. By means of the cooperation of the individual parts other elements or relationships. By means of the cooperation of the individual parts of different different subsystems subsystems new present in in the the of new properties properties may may emerge emerge that that are are not not present subsystems. Economic evolution involves not only changes in variable levels and subsystems. Economic evolution involves not only changes in variable levels and functions in organizational organizational structures concern the way elements elements are are ftinctions but but also also in structures that that concern the way connected within subsystems, the way subsystems are embedded in large ones, and connected within subsystems, the way subsystems are embedded in large ones, and way that the way that organizational organizational structures structures emerge emerge or or disappear. disappear. As mentioned in in the the the As mentioned introduction, advances in theory of complex systems provide promising ways for introduction, advances in theory of complex systems provide promising ways for understanding the dynamics of structural changes in socio-economic systems. understanding the dynamics of structural changes in socio-economic systems. Theory systems provides many deep deep insights insights into into structural structural evolution. evolution. Theory of of complex complex systems provides many The modern study of economic chaos permits the discovery of chaotic economic The modem study of economic chaos permits the discovery of chaotic economic structures disguised by very complicated fluctuations. The concept of structural structures disguised by very complicated fluctuations. The concept of structural stability in theory complex of systems is essentially significant in the of stability in theory of complex systems is essentially significant in the study study of structural evolution. structural evolution.

Hierarchy is is a main Hierarchy main character character of ofeconomic economic structures. structures. Economic Economic systems systems consist hierarchical structuring structuring among the component component parts. Hierarchy Hierarchy here consist of a hierarchical 2 means, , aa set Chinese boxes of of aaparticular particular kind. kind. means, following following Herbert Herbert Simon Simon2, set of Chinese Opening Opening any given box discloses discloses a whole small set of boxes; and opening any one of these these component component boxes turn. Power Power distribution distribution is boxes discloses discloses aa new new set set in turn. is an important indicator instance, in common common situations situations the important indicator of of this this structure. structure. For instance, the state organizes country, and the the regional regional governments governments in turn turn organizes the the regions regions within within the the country, organize the lower-levels within them. Each society is characterized by its own organize the lower-levels within them. Each society characterized by hierarchical structures may hierarchical structure. structure. In In social social evolution, evolution, these structures may be either stable or unstable, depending on material, affective, cognitive, and spiritual, factors. In the unstable, depending affective, factors. traditional societies economic structures often remained quite stable many traditional societies economic structures often remained quite stable over over many generations; in modern societies structural changes may occur several times within generations; in modem societies structural changes may occur several times within short period period of of time. time. aa short

Thom (1977). 'Thorn Simon (1973). Simon (1973).

22

388

8. EPILOGUE

Hierarchy Hierarchy is not only only the the character character of of human human societies; societies; even even sciences sciences exhibit exhibit hierarchical structures. Dawkins sees scientific theories and and areas areas as a hierarchical hierarchical scientific theories phenomena.3 structure, on different levels, corresponding to levels of description of phenomena? Philosophers and scientists have Philosophers and some some scientists have sought sought ultimate ultimate reality reality in in the the structure structure of of matter increasingly finer scales in order to to provide provide the the most most elementary elementary mailer at increasingly fmer scales in order explanation, while universe in explanation, while astronomers astronomershave havesought soughtthe the structure structureof of the the universe natural science, science, the the complexities complexities of ecosystems ecosystems are are increasingly wider domains. In natural explained by examining those of organisms, organisms, organisms are explained explained by referring explained them back to the the growth growth of of spatially spatially organized organized proteins proteins and and other other macromolecule, macromolecule, the complex organization organization of organisms is explained back to the linear linear complexity complexity of their DNA DNA code, code, the the complexity complexity of DNA DNA isis referred referred back back to to combinations combinations of simpler atoms, atoms, and and so so on. on. We Weshould shouldhave havenational nationalmacroeconomics macroeconomics based based on on simpler regional economics; regional economics should be referred back to urban and rural regional economics; regional economics should be referred back to urban and rural economics; spatial spatial economics economics referred referred back back to to family-level family-level and and company-level company-level economics; economics. In a broader perspective, psychology and behavior sciences should be be economics. In a broader perspective, psychology and behavior sciences should the starting point of microeconomics. Chemists will to explain psychological the starting point of microeconomics. Chemists will to explain psychological processes in in terms terms of of natural natural laws. laws. The The processes processes can can be be further further going going on. on. Darwin's Darwin's processes remarks that it is not necessary to refer every phenomenon back down this remarks that it is not necessary to refer every phenomenon back down this chain chain of of reductions in order to understand it. hi natural sciences, chemistry be reductions in order to understand it. In natural sciences, chemistry can can be considered as 'fixed parameter' parameter' for for the thepurpose purpose ofofunderstanding understanding DNA. DNA. In In considered as aa 'fixed economics, macroeconomics can be considered as 'given' for labor economics and economics, macroeconomics can be considered as 'given' for labor economics and family is obviously grand theory, obviously important important to construct aa grand theory, which which family economics. economics. ItIt is to construct 4 connects all the levels within a compact framework. connects all the levels within a compact framework.4

Connections between between levels levels in a social Connections social hierarchy are usually not not simple. simple. An An economic different scales. variables and economic hierarchical hierarchical system system may may operate operate on on different scales. Its Its variables and substructures different process rates. Since higher levels substructures may may operate or change in different quickly affect affect low low levels levels in in the the hierarchical hierarchical structure, structure, higher higher usually strongly and quickly levels be changed changed in in lower lower frequencies. frequencies. But this asymmetry asymmetry in levels usually usually tend tend to be change speeds is not not always always held. held. To To study study the thehierarchical hierarchical nature nature of of complex complex change speeds systems, we have to accept a different perspective - aa different different spatiotemporal scale scale used. There are gaps between any two levels of social hierarchy. hierarchy. For instance, instance, we we may reasonable understanding understanding of single male or female female behavior and we we may have aa reasonable know how men and women get married and form families. families. But the functioning functioning of families is far more complicated. Micro level phenomena such as family ties have have families is far more complicated. Micro level phenomena such as family ties significant implications for macro economies. An economic theory without significant implications for macro economies. An economic theory without endogenous family structure structure can hardly explain explain modern economic reality reality since since on on endogenous family can hardly modem economic one hand family been structures have affected by economic development, on the one hand family structures have been affected by economic development, on the 3

Dawkins (1986). (1986). "The The foundation foundation for for aa grand grand economic economic theoiy theory has has been been laid laid through through my my efforts over years. I have to 'modernize' have first first applied applied nonlinear nonlinear science science to economics economics (Zhang, (Zhang, 1991) 1991) to 'modernize' the the vision vision and methodology methodology of of economic economic analysis analysis and and then then have have developed developed the the analytical analytical framework issues with with the the vision vision of framework with time and space space for analyzing analyzing various economic economic issues nonlinear science science (Zhang, (Zhang, 1996, 1996,2000, 2000,2001, 2001,2002, 2002,2003b, 2003b,2005b). 2005b).

4

8. EPILOGUE

389

other hand is the consequences other hand economic economic development development is consequences of cooperative cooperative (and (and competitive) behavior among family members. All these intrinsic intrinsic difficulties difficulties related to economic structures structures heavily affect the the efficiency Multiple levels have to be described efficiency of modeling economic systems. Multiple described in in long-term studies. internal structures structures to to long-term studies. This This requires requires economic economic theory theory to to have internal represent complexity of subsystems subsystems and represent the complexity and connections connections of of the the subsystems. subsystems. Such Such structural models find structural models will will eventually eventually turn turn out out to to be be complicated. complicated. Indeed, Indeed, we we may may fmd able out some special characteristics of the system under consideration and thus are able to simplify the analysis. For instance, some hierarchical systems are decomposable, decomposable, at least in short short timescales. This means that it is is possible possible to to effectively effectively isolate isolate and and describe a part of the system for a given timescale. We may analyze behavior of the independent subsystem hierarchy to independent subsystemininisolation isolationfrom fromthe therest rest of of the the hierarchy to which which itit belongs. A study of dynamics of a particular process on a particular level thus belongs. A study of dynamics of a particular process on a particular level can can thus be conducted by taking behavior of higher levels as fixed and 'enslaving' behavior be conducted by taking behavior of higher levels as fixed and 'enslaving' behavior of the the low low levels levels as as structurally structurally determined determined flows. chosen of flows. In In other other words, words, for for the the chosen time scale the behavior of higher levels are so slow that they can be effectively time scale the behavior of higher levels are so slow that they can be effectively negligible and the of lower lower levels levels are are so so fast fast that that perturbations perturbations generated generated negligible and the behavior behavior of by the behavior of lower levels can also be effectively neglected. For instance, by the behavior of lower levels can also be effectively neglected. For instance, an an economic analysis time scale scale short short enough enough to to assume assume economic analysismay may be be conducted conductedinin aa time changes in in ecological ecological processes average out out noise noise changes processes negligible negligible and and long long enough enough to to average from processes occurring at individual levels. It should be remarked that this from processes occurring at individual levels. It should be remarked that this method might might be invalid especially especially in 'revolutionary' periods. such critical critical method be invalid in 'revolutionary' periods. At At such points, neither the dynamics of higher levels nor the perturbations generated the points, neither the dynamics of higher levels nor the perturbations generated by by the behavior of lower levels are negligible. The model used to describe the dynamic behavior of lower levels are negligible. The model used to describe the dynamic interaction the chosen chosen subsystem subsystem is no longer longer able able to to provide provide reliable reliable interaction of of the is no 5 information about possible behavior of the subsystem. information about possible behavior of the subsystem.5 An important feature of of economic structures isis that that they An important feature economic structures they are are intrinsically intrinsically Individuals, groups groups or or clubs, clubs, regions regions and and nations, nations, even even as as complicated at each level. Individuals, develop under under practically practically similar similar conditions, conditions, are never exactly exactly the the same. same. they develop are never Detailed evolution have examples of an intrinsic intrinsic Detailed studies studies of their theft evolution have provided provided many many examples complexity. complexity. For instance, instance, random random fluctuations fluctuations in in tastes tastes may may affect affect microeconomic evolutionary large scale. The economic economic structure structure represents represents the evolutionary processes processeson on aa large values and principles principles of the economic organization. organization. The system may be analyzed analyzed The system by dividing the whole system into different different levels, each representing a subsystem, subsystem, which consists of relatively relatively uniform uniform elements elements that that interact interact with with each each other other either either in simple or complicated ways. To find and describe these interactions are the key key simple complicated ways. describe interactions elements for analyzing order and disorder at any given level. Economists have elements for analyzing order and disorder given level. Economists have sought structural structural invariants invariants on levels. The construction of of aa sought on macro, macro, meso meso and and micro micro levels. The construction theory with structure is not arbitrary and gratuitous. We first have to determine theory with structure is not arbitrary and gratuitous. We first have to determine issues under under examination, examination, scales scales (both variables, time time and and space) space) and and domains, domains, issues (both of of variables, and analytical methods. Here, when assuming the habitual three-dimensional and analytical methods. Here, when assuming the habitual three-dimensional 5

Synergetic Synergetic economics economics by Zhang (1991) deals with nonlinear economic dynamics with different time time scales scales and and speeds speeds of of changes.

390

8. EPILOGUE

representation of of space, space, with with time time as a fourth representation fourth dimension, dimension, scale is defined defined as the the is agreed not to to try try to to distinguish distinguish the the smallest volume within the interior of which itit is nonuniformity of property being shortest interval interval of time nonuniformity of a property being measured measured and and as the shortest during which it is agreed not to try to distinguish variations of a given property. The defined as the greatest volume and the longest time interval over which domain is defined the study will be extended. extended. For instance, instance, the whole economy economy can be studied studied by employing several treated as a employing several scales. scales.The The variables variablesused usedatat one one scale scale may may be be treated with the the first first by by taking taking averages averages of of larger larger coarser scale, macroscopic in comparison with volumes and longer intervals intervals of time. In building building a sophisticated sophisticated economic theory, one has has to to construct, construct, without without making making any any mistakes, mistakes, aa long long chains chains of of assertions, assertions, has has one to be aware of what one is doing at each step of the construction process, and has to to be aware of what one is doing at each step of the construction process, and has to speculate about where one is going. The constructor has be to able to guess what is speculate about where one is going. The constructor has be to able to guess what is true and what is false at each level and be able to judge what is useful and what true and what is false at each level and be able to judge what is useful and what is is not in in the the whole whole framework. not framework.

Appendix The appendix is arranged as follows. A.! A.I introduces introduces matrix matrix theory. theory. A.2 A.2 shows shows how how to solve linear equations, based on matrix theory. A.3 introduces metric spaces and concepts and theorems related to metric spaces. A.4 defines some basic some basic concepts concepts in study study of of fimctions functions and states the implicit fbnction function theorem. A.5 gives a general expression general expression of the Taylor Expansion. Expansion. A.6 is concerned concerned with convexity convexity of sets sets and and functions functions and and concavity concavity of functions. functions. A.7 A.7 shows shows how how to to solve unconstrained maximization maximizationproblems. problems.InIn A.8, A.8, we introduce unconstrained introduce conditions conditions for for constrained maximization. maximization. A.9 A.9 introduces introduces theory theory of ofdynamic dynamic optimization. optimization.

A.1 A.1 Matrix theory We present some some important important concepts concepts and theorems from from linear linear algebra algebra and and matrix matrix theory. Some Some elementary elementary concepts, concepts, such as as identity identity matrices matrices and and null nullmatrices, matrices, operations, and and proofs proofs of of theorems theorems are are omitted.1 omitted.1 matrix operations, Let A be vector xx in Let vectors vectors A be aa nonempty nonempty set set of of vectors vectors in in R11. R". AA vector in R't R" is and scalars linearly dependent on the set A. if there exist vectors y1, y2, set A. if there exist vectors yx, y2,..., ym a1, ..., aa,,, suchthat that a,, aa2, m such 2,...,

x= For any any nonempty nonemptyset setAA of of vectors vectorsininR", R', (A) (A} isisthe theset setof ofall all vectors vectors in R"

that (A) is aa subspace of R". that are are dependent dependent on on A. A. {A} subspace of A vector R". A vector of of the the form form linear combination. combination. AA set vectors is a basis basis of of the the is called aa linear set A A of vectors

11

This Gilbert and and Gilbert (1970). See See also also Chiang Chiang This part on on matrix matrix theory theory is is based based on on Gilbert Gilbert (1970). (1984), Berman and Plemons (1979), Zhang (1999fz), and Peterson and Sochachi (2002).

391 391

APPENDIX

392

subspace U if (i) A If U isis any subspace U A "spans" "spans" U U and and (ii) (ii)AAisislinearly linearly independent. independent. If any subspace of R'7, number of of vectors vectors in in aa basis basis of of U isis called of subspace of R", the number called the dimension dimension of U and andisisabbreviated abbreviated as dim(u). The The dimension dimension of is n. n. as dim(U). of R" W is Let

u = {uvu2,...,ur} U={Ui,U2,...,Ur} be a set of vectors in R" and

r = ft, K2, ...,*;} be a set of vectors in (U), a matrix of transition from U to V is a matrix be a set of vectors in (t/j, a wa/rix: of transition from C/ to F is a matrix A=

such that

such that

r, = £ > / / „ y = 1,2, ...,*. Definition A.1.1. A square matrix A if and and only only ifif A A is Definition square matrix ^ = [ a sj| ^ is is nonsingular if is aa matrix of transition from one basis basis of R" to another basis of R'2. R". A square square matrix matrix that is not nonsingular is called singular. singular.

We denote the identity identity matrix by /„ (= [^Jjnxn where Sy is the the Kronecker Kronecker delta). For any x n rectangular any m mxn rectangular matrix, matrix, if if the the maximum maximum of of linearly linearly independent independent rows that can be found in such a matrix is r, the matrix is said to be be of of rank rank r,r, rows that can be found in such a matrix is r, the matrix is said to denoted by Rank(A) Rank(A) or Rank A. number of of denoted by A. The The rank rank also also tells tells us the maximim maximim number linearly independent independent columns columns in in the the same same matrix. matrix.As As aa square squarematrix matrixhas has nn linearly independent rows (or (orcolumns), columns),ititmust mustbebeofofrank rankofofn.n.IfIf AA is mx n matrix matrix over over independent rows R and n, then and PPisisany anyinvertible invertible nx nxn, thenwe we have have Rank(A) = Rank(AP). Rank{A) = Rank(AP).

A.1. MATRIX THEORY A.I.

393 393

Definition A.1.2. An nxn nxn matrix B isis an ofthe the nxn nxn matrix Definition A. 1.2. An an inverse of matrix AA == [aUjnrn [aiJ\nm if AB = In= B A .

Also a square matrix matrix isis called called invertible invertibleifif itit has has an an inverse. inverse. Theorem A.1.1. A.1.1. An nx nxnn matrix matrixAAisisinvertible invertible ififonly only ifif AA isisnonsingular. nonsingular. The The inverse of an invertible invertible matrix matrix isis unique. unique.

If If AA= = [ay L I

itsunique uniqueinverse inverseisisdenoted denoted K'. If is invertible, invertible, its bybyA~\

A1, A2, A' A' •..,' An are

square invertible invertible matrices matrices ooff order order n over square over R", R " ,then thenA,A2 AXA2

m

is invertible invertible

and

(4A2 [A1A2 A

m

) = A

m

A

2

Ax .

For any mxn matrix matrixA,A,the thetranspose transposeofofAAisisdenoted denotedby byAT. AT. IfIfa asquare squarematrix matrix A is and A is invertible, then AT AT isis also also invertible and

(AT)1 =(A-'f. The following concept concept isis only only referred referred to square square matrices.

Definition 4.1.3. Definition A.1.3. The The determinant determinantof of the the square square matrix matrix A A == Lx j det(A) defined defined by

det(A)=

where ^

1)'a,1

a11

denotes denotes the the sum sum of of all all terms terms of of the the form form

(—1)'a,1 a11

is is

the scalar

APPENDIX

394

assumes all all possible possible permutations permutations of of the the numbers numbers of the numbers as Ii' y',,12 j 2 ..., ..., jinn assumes numbers as 1, 22,, . ... the number used to carry . . n, and and the exponent exponent t is the number of interchanges interchanges used 1, carry

j 2 ..., in y'j, j n into the 1, 2, 2,... n. n. the natural order 1, Ii' 12

The with det(A). The notations notations det detAA and and \A\ are are used used interchangeably interchangeably with det(^4). When When nn=2 = 2 andn=3, and n = 3,wehave we have

4.,

a11 au

fl

«21

a21

a22 a 22

a

fl

a12 12

aa13 13

= a21 «21

a22 «22

a23 =a11a22a22 ana22a22 -+ «23 =

a31 fl 31

a32 G 32

a33 a33

= =

a11 u ^3x3

=

— a31a22a13 —

a12 12

= a11a22 —- a21a12, a2]al2, «11«22

=

a32a23a11

—

a12a23a31

+ aa13a32a21 13 a 32 a 21

(A.1.I) (A.1.1)

a33a12a21.

Definition A == \an I Definition A.1.4. The minor of of the the element element a;, in A 'J

L

is the determinant determinant

'J -hixn

of the 1) submatrix of A obtained by deleting row i and l)x(« —— l) My of the (n — — 1)x(n submatrix of A obtained by deleting row i and column i of A. The cofactor, denoted by A column j of A. The cofactor, denoted by Aij, of afJ in .4 = [ay \mn is the product of (— +j) of(-lf

and that is, and M^, that is,

+j) Ay=(-lf Mr = (—

The adjoint of of A, ^4, denoted denoted by adi(A), adj(A\ isis given given by adj(A)= [Aü.I:n.

Theorem Theorem A.1.2. A.1.2. If A A ==[a..] [aJ If L

, then

,

'J Jiixn

a det(A), /,i, k = 2, ..., a!lAkl nAx ++ aE2Ak2 1, 2,..., 0/2^*2 ... ++ aajnAkn = 1, mAn = = 5ik det(^), a

alIAlk u A + a2JA2k «2^2*

», 8 8jk det(A), j, k = 1, 2, ..., n. + anjA,,k = « n ; 4* = jk det{A), j , k = \,2,..., n.

formula and equations equations (A.1.1) (A. 1.1)with with S8,, 1, we can calculate the By the above formula it = 1, value of det(A) det(^4) of of any any dimensional dimensional matrix, in principle. principle.For Forinstance, instance,when when nn == 4, we have

A.1. A. 1. MATRIX THEORY + a21A21 det(A4x4)= = aa11A11 a^A2X + + aa31A31 + uA u 3lA2l +

395

a^A^,

x 3 matrices. where Ail are are calculated from the corresponding 33x3 is be shown shown that that ifif A = = la,, It can be is invertible, then [a..Ii j| nnj = 1 A1 adj(A). A'1 =—\-radj{A). V det (A) det(^) '

Definition A.1.5. If A Definition A. 1.5. If is nx A is nxnn matrix, matrix,an an eigenvector eigenvector of A A isis aanonzero nonzero column column vector vector v in R" such that

Av=pv Av = pv for some somescalar scalar p; p; the for the scalar scalar pp isiscalled calledan aneigenvalue eigenvalueof A. of A.

Theorem If A A is an nxn matrix, of A Theorem A.1.3. A. 1.3. If matrix, aa number number p isisan aneigenvalue eigenvalue of A if and only if

-— A) A) == 0. 0. The equation —

A) = 0

the characteristic equation of of the matrix is called called the characteristic equation matrix A. A Upon Upon expanding expanding the the of degree degree nn in in p. The determinant, det(p/nxn -— A), we will have have a polynomial polynomial of The called the the characteristic characteristic polynomial polynomial of matrix A. of the the matrix A. polynomial is called An An nxn « x « matrix matrix B5 isissaid to be said to be similar similar to the nxn nxn matrix matrix A A ififthere there is is an an invertible n x n matrix matrix P such invertible nxn such that B = P1AP. B P'XAP.

A square matrix is said to be diagonalizable diagonalizable ifif itit isis similar similar to to aa diagonal diagonal matrix. matrix. It can be proved that that a square square matrix matrix A isis diagonalizable basis for for diagonalizable ifif only only if there is a basis consisting of eigenvectors of A. Not square matrix is not diagonalizable. There R" consisting eigenvectors of square canonicalform form of ofaasquare square something close close to to diagonal diagonal form form called called the Jordan Jordan canonical is something matrix. A expressed A basic basic Jordan Jordan block associated associatedwith withaavalue valuepp isis expressed

APPENDIX

396 p 1 0 p10

Op1 0 p 1

000 000

0

0

0

0

0

0

0

p

11

0

0

0

0

pp

The Jordan Jordan canonical canonical form form of aa square square matrix matrix isiscompromised compromised of of such such Jordan Jordan blocks. blocks. Theorem Theorem A.1.4. A.1.4. Suppose Suppose that A A is an nxn nxn matrix matrixand andsuppose suppose that that

det(pJ —-A)={pA)= (p detO?/

—

p. p1)"(p

P2)

where

Pi' P2' are distinct is similar to a distinct roots of of the the characteristic characteristic polynomial polynomial of A. A. Then Then A A is similar to matrix of the form B1l B

0

...

0

00

B2 B 2

0

00 0 0

BBkL

where each each B, B. is an m, x in, where TM, matrix of the form J,I ^

00

...

0

0 o'

j1 k

...

o

0

00

...

J,

and each .J, basic Jordan Jordan block block associated associatedwith with p,. pt. Jt is a basic

A.2. SYSTEMS OF LINEAR EQUATIONS A.2.

397

A.2 Systems A.2 of linear Systemsof linear equations equations A system of linear equations is aaxx + ai2x2

+ ainxn = bj, jj=1,2,...m, = 1, 2 , ... m ,

or in the matrix form A™*,*, = *W-

(A.2.1) (A.2.1)

set of of values values of of xx that A solution of the system is a set that satisfies satisfies

Ax=b. Ax = b. In this system, A and b A isiscalled calledthe the coefficient coefficient matrix, x the thematrix matrixof of unknowns, and the matrix of constants. Z>] is is called called the the augmented augmented matrix matrix of of the the constants. The The matrix matrix [A, [^4, b] system. system.

Theorem Theorem A.2.1. A.2.1. The The system Ax Ax = b has a solution if and only if Rank({A,b]) Rank{[A,b]) == Rank(A). Rank(A).

Theorem Theorem A.2.2. If Rank([A,b]) = Rank(A) = r, then the solution to Amxnxixn =

can be expressed expressed in interms ofnn—— r parameters. termsof Theorem A.2.3.Consider Consider aasystem system of of linear linear equations AnxnxlM = Theorem A.2.3. = bnxl. If det(A) * 0, det(^) 0,

then the unique solution is given by

APPENDIX

398

x == Aib. A~lb. Theorem TheoremA.2.4. A.2.4.Consider Consider aasystem system of of linear linear equations equations =

If det(A) by Asx{A) 0, 0, then thenthe theunique unique solution solution of ofthe the system system isis given given by

X=

Xj=-

4

where ^

bA. k=lkkJ det(A) det(^)

'

'

'

are cofactors cofactors of A. A. are

formula is called Cramer's Cramer's Rule. The above formula Rule. We Wenote that

column of of A A by determinant of the is the determinant the matrix matrix obtained obtained by by replacing replacing the the j1th' h column the column of constants b. b.

A.3 Metric A.3 Metric spaces spaces Metric spaces are used to represent states of economic systems. A metric measures twostates. states. the "distance" "distance" between two

ofaa set set X X and Definition A.3.1. A.3.1.AA metric metricspace spaceisisaa pair pair (X, (x, d) consisting consisting of and a Definition fimction function

d:XxX—*[0,oo) called aa metric such that (i) if xx == y; (i) d(x, d(x, y)= y)=00 ififand and only only if (ii) d(y, x); (ii) d(x, d(x, y) y) == d(y, x);

(iii) y) + d(y, 4 d(x, 4. (in) d(x,d{x,y)+d{y,z)
A.3. METRIC SPACES

399 399

from the the set integers to to X. X. The A sequence sequence s is isaafUnction function from setof ofnonnegative nonnegative integers n th point point of the sequence is the point sn. A subsequence of a sequence ss is any nth sequence is strictly increasing fUnction from sequence p of the form pj = pa where a is isaa strictly function from > 00 there A Cauchy Z + to to Z + . A Cauchy sequence sequence is is aasequence sequence {xn} such such that that given given ee > there positive integer integer k such exists a positive such that Xm)< C

n and k. whenever n and m mare aregreater greater than k. Definition A.3.2. A.3.2. A A metric metric space space X isiscomplete sequence in in X Definition complete if ifevery every Cauchy Cauchy sequence X point in in X. converges to a point X.

A point point pp isisaalimit limitpoint point of ofaasubset subset YYof ofXXprovided provided that that there there is isaa sequence ofpoints pointsin inYVwhich whichconverges convergestotop.p.AAsubset subsetYVof of X is sequence of isclosed closed if if itit contains all limit points. denoted by cl(Y) is is the set set of ofall all contains all its its limit points. The The closure closure of of 7 V denoted by cl(Y) limit points pointsof of Y. V. The The ball ballof ofradius radiusrr centered at x isis the set centered at

B(x,r)= {y,ye X,d(x,y)cr}. B{x,r)={y,yeX,d(x,y) 0 0 such A is open if for iscontained contained in may depend depend on x. x. A Asubset subset that B(x, r) that the the ball ball B(x, r) is in U. U. The radius radius rr may A CI(A) = = X. A of of a metric metric space spaceX X is is said saidto tobe bedense densein inX X if if cl(A) X. Another Another way wayof of saying this is that A is dense in X if every ball centered at a point of X contains saying this is that A is dense in if every ball centered point of X contains a point of ofA, A, no nomatter matterhow howsmall smallthe the radius radius of of the the ball. ball. Theorem Theorem A.3.1. A.3.1. (1) If V is V is an open set. (1) If 7 isaaclosed closed set set then then XX— - Y (2) If U U is a closed closedset. set. (2) If U isisan anopen openset set then then XX—- U

Suppose thatXX and and Y V are from X Suppose that are metric metric spaces. spaces. A A fUnction function /f from X totoVYisis continuous if for for every every open open subset subset V V of of Y V the set open continuous if set / ~ '(v) ( K )isisanan opensubset subset of of X. a homeomorphism continuous X. f/is is a homeomorphismififf /is is continuousand andinvertible, invertible,and and the the inverse inverse of of /f isiscontinuous. continuous.

APPENDIX

400

Theorem A.3.2. A function Theorem A.3.2. A function /f from X X totoYYisiscontinuous continuous ifif and and only only if if given given

any point point xx in any in X X and andany anysequence sequence {*„}} converging to xx then then the thesequence sequence to f(x). {/(*„)} f(x). )} converges to

of X, X, then Theorem A.3.3. A.3.3. IfIf /f isiscontinuous continuous and A A isis aa subset subset of then

f(cl(A)) c cl(f(A)). f(cl{A))
int(S)= cl(x — s). iat{s) =x —X-cl(X-S). A neighborhood neighborhoodofofaapoint pointxx in in aa metric metricspace spaceXX isis aa subset subsetNN such such that that xx belongs to to the the interior interiorofofX. X. The exterior of ofaasubset subsetSS of a metric metricspace space XX is the belongs set ext(S) = = x X— cl(s). ext{s)

The boundary of of aa subset subset S of space X X isisthe of aa metric space the set set

as — s). dS = cl(S)ncl(X cl{s)r\cl{X-S). A subset subset S of of aametric metric space space XXisisdisconnected disconnectedififthere thereexists exists disjoint disjoint open open sets sets U and with S such andVFeach eachhaving havingnonempty nonempty intersection intersection with such that that S5 isiscontained contained in in the union union of of U and and V. V. A A subset subset 5S ofofaametric metric space space X X isisconnected connected if if itit is is not not disconnected. disconnected.

Theorem space XX and Theorem A.3.4. A.3.4. If N JVisisaasubset subset of of a metric metric space and SS isisaaconnected connected subset of X exterior of of N N isisnonempty, X whose whoseintersection intersectionwith withboth both the the interior interior and the exterior nonempty, with the the boundary boundary of N N isisnonempty. nonempty. then the intersection intersection of of SS with A subset subset K ofofaametric metricspace space XX isiscompact compactififevery everysequence sequence of of points in K K has a convergent metric space space X X isislocally convergent subsequence. subsequence. A metric locallycompact compact ifif each each point point x in A subset subset KK of a metric space X X in XXhas hasa aneighborhood neighborhood which which is compact. compact. A metric space is bicompact if given any collection of open sets whose union contains K there bicompact given collection of open sets whose union contains K finite number of these sets whose union also contains K. exists a finite K.

A.3. METRIC SPACES

401 401

Theorem 4.3.5. A.3.5. A A subset subset K K of ofaametric metric space space X isiscompact compact if if and and only only if it is bicompact. Theorem that f/ isisa acontinuous Theorem A.3.6. A.3.6. Suppose Suppose that continuousfhnction function from aa metric metric space X to metric space space Y. to a metric Y. Suppose Supposethat thatCC isis aa connected connectedsubset subsetofofXXand andKK isis a compact subsetofofX. X. Then(l) Then (1) f(c) f(C) isisconnected; and(2) (2) f{K) f(K) isiscompact. compact subset connected; and compact. Theorem Theorem A.3.7. A.3.7. Suppose Suppose that that gg isisa acontinuous continuousreal-valued real-valued function function defined defined on a compact subset K K of of aa metric metricspace. space.Then Thenthere thereexist existpoints pointsx xand andy yinin KK such compact subset such that g(K) g(K) isiscontained contained in in the the interval interval [g(x), [g(x), g(y)]. Thus Thus g achieves achieves its its maximum minimum. and its minimum.

A function (x, d) d) to aa metric (x, D) D) is A function /f from from aa metric metric space space (X, metric space space (X, unjformly continuous continuousifgiven if given ee > > 00 there exists S8>>00 such uniformly there exists suchthat that d(x1, x2)c S implies that

D(f(x1), f(x2)) < e. that / isisa acontinuous Theorem A.3.8. A.3.8. Suppose Suppose that continuousfUnction function from aa compact compact metric space X totoaa metric space metric space space Y. Y. Hence Hence / isisuniformly uniformly continuous. continuous. Theorem A.3.9. that X and Theorem A.3.9. Suppose Suppose that andYYare arecompact compactmetric metricspaces. spaces. Let Let

f:X-*Y be continuous, one-to-one,and andonto. onto.Then Thenthe theinverse inverseofof/ / isiscontinuous continuous, one-to-one, continuous and and be hence / isisa ahomeomorphism. hence homeomorphism.

A.4 A.4 The Theimplicit implicit function function theorem theorem First we state a few theorems from from analysis.

APPENDIX

402

and V2 are two two normed Definition Definition A.4.1. A.4.1. Suppose Suppose that that Vx and V2 are normed linear linear spaces spaces with with respective norms || j| and and1|| ||2. Then

that x e e Vx and is continuous at x0 for all all e >> 0 x0 e Vx if if for > 00 such that and 0 there exits aa S >

\x - Xo\[ 6 S k—xoIl1e implies that

\\F(X)-F( IIF(x) — Xo]\2<£. set U And F F is is said to to be becontinuous continuous on the set Uee Vx if if it it is iscontinuous at at each point xe writeFFe c(u). x e U, U, andwe and we write e c(u). Theorem A41. Theorem A.4.1.(the (theintermediate-value intermediate-valuetheorem) theorem)IfIfthe thefUnction function

f:[a, i4—> R f:[a,b]->R is continuous and

f(a)< O
f(c)=O. Definition Definition A.4.2. A.4.2.The Thefunction functionf:R"->R"is f: —> transformation Df(X0) a linear transformation Df(x0) that satisfies satisfies

lim = IhI—,0

— f(x0) — ^0 + h) +h)-f{x o)-Df(x o)h\_ _=Q

Ihi

is differentiable at

x0 e /?" if there is

A.4. THE IMPLICIT FUNCTION THEOREM

403

linear transformation transformationvv is is called calledthe thederivative derivativeofof/ f at x0. The linear x0. The following theorem gives gives us us a method following theorem method for for computing computing the the derivative derivative in in

coordinates. coordinates.

Theorem A.4.2. A.4.2. IfIf the the function function f:/ : R" / ? "—* - » R" is differentiable differentiable at at x0 x0 e R", then partial derivatives derivatives df/dxj aji ax1 all the partial all exist exist at x0 x0 and and for all xxe R"

Df(x0)x =

a adifferentiable Df isisgiven xn Thus if f/ is is differentiable flinction, function, the derivative derivative Df given by by the n nxn Jacobian matrix

Df== Df

aXj

k defmed in For U an D"f(xo) anopen open subset subset of R", R", the thehigher higher order order derivatives derivatives D f(x0) are are defined

a similar way. Definition A.4.3. A.4.3. Suppose that f: U—> R'7

djfferentiableononU. U.Then Thenfef is differentiable

] C1 (U) ififthe thederivative derivativeDf Df is continuous on U. C (u) continuous on

We can define define

/ e Ck(U), C (u\ k == 2,3, 2,3,fe k

in a similar manner. manner.

that f:/ :U Theorem Theorem A.4.3. A.4.3. Suppose Suppose that that U U isisananopen opensubset subsetofofR?I R" and that ( / —> ->/?". Then f/ e C1 C}(u) andonly onlyififthe thepartial partialderivatives derivatives df/dxj Then (u) ififand I U. continuous on U.

all exist and are are all exist

APPENDIX

404

It can can be be shown shownthat that/f e C (u) ififand andoniy onlyififthe the partial partial derivatives derivatives C"k (if)

ax. dx..Ji

,

ik

1,..., kk exist continuous on on U. =1, and are continuous with 1,i, Ji,---, jIkk = exist and Theorem 4.4.4. theorem)Let Let U U be an open open set in in R" and A.4.4. (the (the inverse inverse function function theorem)

f: U-> R'1 be If a point be a C" Ck function function with with kk >1.1. If point 1€ x eU U isissuch such that that the the nxn nxn matrix matrix Df(i) open neighborhood neighborhoodVVofofxI in in U such Df(x) isisinvertible, invertible,then then there there is an open such that that

f:Vf(V) is invertible with a Ci" inverse. C* inverse.

The inverse function theorem theorem implies impliesthat that ifif the the matrix matrix /^(o) fjO) isisnonsingular, inverse function nonsingular, then there is a locally defined smooth function function

x=g(y), x = g(y), g:R'7—>R" g:R"^R" such that

f(g(y)) = y, for all all y ininsome origin of of R". The function for some neighborhood neighborhood of the origin function g isis called called the the inverse function functionfor for/f and inverse andisisdenoted denoted by

g=f'.

g = /"' If If

A.4. THE IMPLICIT FUNCTION THEOREM THEOREM

y=g(x), y = g(x),

g:R"^Rm

and and

z=f(y),f:Rm —*R" z = /(y),/:*"->** superposition are two maps, then their superposition

fog h= = fog is a a map

if"k, z= h{x), R" —, = h(x), -^R defined by the formula formula

h(x) = f(g(x)). Let fy (y) denote the Jacobian matrix /f evaluated Rmm evaluatedatataapoint point y eysR

\

j

Jkxm

1

We similarly define hx(x) as (9 as

L

a map We consider a (x, F(x,y), {x, y)F—* y)n> F(x,y),

where " v, D"l F:R"xRm F:R"xR

- D«

405 405

APPENDIX

406

of (x, y) == (o, = 0. is a smooth map defined defined in a neighborhood neighborhood of (0, o) 0) and and F(O,O) F(0,0) = 0. Let

the matrix matrix of of first derivativesofofFF with with respect to x Fx(0,0) denote the first partial partial derivatives respect to evaluated at (o, (0, o) 0)

V

dx.J

M

J

(x,y)=(o,o) A.j-Ho.0)

Theorem A.4.4. (the implicit fUnction If the function theorem) theorem) If the matrix matrix Fx(0,0) is

nonsingular, then then there there is aa smooth locally defmed defined fUnction function y == f(x),

f:R"^>Rm

suchthat such that

F(x, f(x)) = 0, F{x,f{x))=0, for all all xx in ofthe theorigin originofofR". R'. Moreover for in some some neighborhood neighborhood of

j(o) = — degree of of smoothness smoothnessof ofthe thefunction fUnction/ f is that of of F. The degree is the same as that F. Theorem A.4.5. theorem) Let Let U be Theorem A.4.5. (the (the submanifold submanifold theorem) bean anopen open set set in in R'7 R" and and let let

f:U -, R" f:U^>R" differentiable function fUnctionsuch suchthat thatDf(x) Df(x) has rank rank p whenever be a differentiable whenever

f(x)=0. f(x) = 0. -1 Then (o) isisan an (n -— p)— p)- dimensional manifold in R". Then /f1(0)

Morse Lemma. Let /f:]?" : R" ->R -> R

A.4. THE IMPLICIT FUNCTION THEOREM

407

be a sufficiently sufficiently differentiable differentiable function. function. If x* is a nondegenerate nondegenerate critical point of /f,, that of that is is

Df(x)= o, D/(x')=0, and the Hessian matrix

[a2f(x*)

is nonsingular, then then there is nonsingular, there is aa local local coordinate coordinate system system (y1, (yu

••,

in a

neighborhood of x* x with neighborhood £/U of with yj(x*)= 0, y,(x')=o, for all i, z", such such that

+ 1=1

i=1+k

U. The is: is thethenumber for all yy eeU. Theinteger integerk A numberofofnegative negativeeigenvalues eigenvaluesof ofthe theHessian Hessian matrix.

Sard's Sard's Theorem. Theorem.Let Let UUbebean anopen openset set in R" and let f: f : UU —* -^ Rp fbnction.Let LetCC be the set set of of critical criticalpoints pointsofof/f,, that be a differentiable differentiable function. that is, is, the the set of all all xe xe U U with with

rankDf(x) << p. rankDf{x) p.

Then f(c) f(C) hasmeasurezeroin Then has measure zero in R". Rp.

APPENDIX

408

A5 The A.5 TheTaylor Taylorexpansion expansion and and linearization linearization Given fimctionf(x), f(x), the Taylor Given aa successively successively differentiable differentiable one-variable one-variable function the Taylor expansion around around a point x* gives the series expansion

f"(x)(*)2 + 2!

(X)(XX*t+R(X)

a polynomial higher powers powers (than (than n) n) of appears on the polynomial involving involving higher of (x (JC — — JC x0) 0 ) appears right. fimction,f(x, f(x, y), right. For a two-variables two-variables function, y), the Taylor expansion around a point point where where

(x*, y*) [x*, y*j is given by y*)(x — x*)+ f(x, y)= y) = f(x*, f{x\ y)+ fx{x', y){x f{x, - x*)+ fy{x',

)(y y'){y

-— / ) +

i[f(x*, y)(x — ^ k ( * \ yfc - x)2 x'l ++ 2 / v ( r \ /y*)(x ) ( * —- / ) ( y - /y)+ ) + / ^ f r ' , / ) ( y -— y*)2] yj] —

+ ... + R{x, y).

linearization of of aa ftinction is obtained by simply dropping all all terms The linearization function is obtained by simply dropping terms of the function. function. For For instance, instance, the the linear linear order higher than one from the Taylor series of the approximation of gives approximation of aa one-variable one-variablefunction functionf(x) f(x) gives

f(x) = f(x)+ p(x*)(x

-

In the case of two variables

f(x*, / *) x*)+ fy(x*, fx(x, y*)(x f(x, y)= y) = f(x*, /(*> ) + /*(**, / ) ( * -— x')+ fy{x', y*){y -— y ) . We now give the Taylor expansion for any dimension around the origin. origin. Let U be the origin be aaregion region in in R" containing containing the origin xx == 0. 0. We denote denote the the set of of all all continuous functions functions

/ : U -» Rm

A.5. THE TAYLOR EXPANSION AND LINEARIZATION

409 409

by c° (u, Rm) by C°(u, Rm) and differentiable ftinctions functions with and the theset setofofall alldifferentiable with continuous continuous first first m Rm). we will use the use C k (u, Rm) derivatives by C' C'(u, ). Analogously, Analogously, we Rm) to indicate indicate the derivatives (u, R m Rm) functions with C k (u, R ) with a fUnctions withcontinuous continuousderivatives derivativesup upthrough throughorder orderk.k. If If f eG Ck

sufficiently large large k, k, the function function / f is is called called smooth. smooth. A A C°° C°° function function has has sufficiently

(u, Rm) continuous partial derivatives derivatives of of any any order. order. Any Any function continuous partial function f e C" C k (u, Rm) can can be be represented near x = represented near =00ininthe theTaylor Taylorexpansion expansion k

f(x) /(*)= =

1

ç!

...

+

where

= o\\xf+1) ) == o\\x(), i1 + + ... + ^ /, + ii22 + +i,,, /„, R(x) =

|i| II namely

H4 —*0 asas \\x\\ x —>0 - > 0, lf

oxHk

Here, we we give give precise precisedefinitions defmitionsofofO0and ando.o.Let Let/ f and and in which |x|= which HI = -JxTx. Here, g be betwo twogiven givenfunctions. functions. We We say say that

f(x) = 0(g(x)), as asx-^0, x —4 0, f(x)=O{g(x)), thereare areconstants constantsaa> if there > 0 and and A A> > 00 such that If(x)I |JC| < < a. a. We say that for N for

f(x) == o(g(x)), —* 0, f{x) o(g{x)),as xasx^O, if for forany anyee> if > 0Othere there isisa a 88>0 > 0 suchthat such that

APPENDIX

410

f(x)I for Ext |JC|
11

U

<

i

MA, At

Jt"i

i ...

xIn «

AM

f(x) atatany to xx = converges to f(x) anypoint point xx sufficiently sufficiently close to = 0.

A.6 Concave A.6 Concaveand andquasiconcave quasiconcave functions functions Consider Consider / f:: Q Q -» R 7? where Q isisaadomain domain that that isis aa convex convex subset of /?" (such as Q. = = R" or = {XE

Definition A.6.1. The function function

defined onthe theconvex convexset setQQ cc R" is cowcove concave ifif defined on (I — / ( « ' + (l - a)x) «)x) >W(x') qf{x<)++ (i(1—- a)f(x), a)/(x),

(A.6. 1) (A.6.1)

for all xx and and x'e Q for all i l and and all a ee [o, [0, 1]. lj. If the inequality is strict for all x' * x and thenwe wesay saythat thatthe the ftmction is strictly concave. all (o, l), i), then all aa e (0, function is strictly concave.

1 aillustrates illustratesaa strictly strictly concave concave function function of variable. Figure lb Figure A.6. A.6. la Figure A.6. A.6.1b depicts a concave but but not not strictly strictly concave concave function. function.

A.6. CONCAVE AND QUASICONCAVE FUNCTIONS

411 411

f(x) fix)

f(x) /M (1 a)f(x) qf{x')+(l-a)f(x)

+ (1 — a)f(x) =W(x') cf(x')+(\-a)f{x) a) =

f(x) f Va) ——

I

f(x)

x x' x x' xa xa fbnction b) concave but not strictly concave function a) strictly concave function function Figure 1: Concave functions A.6.1: Figure A.6. functions

Condition (A.6. 1) is equivalent to the following property (A.6.1)

aix']

(A.6.2)

"C any collection collection of of vectors vectors x' e A for any A and and numbers numbers

aO, i=1,2,•••,M such that

=1.

The properties convexityand andstrict strictconvexity convexityforfora afunction function/ f are defmed properties of convexity defined analogously but with with the the inequality in equation equation (A.6.1) (A.6. 1)reversed. reversed.Note Notethat that/ f is analogously but inequality in is concave if and only if —f - / is convex. For a strictly convex function, a straight straight line connecting any two points in its graph should lie entirely above its graph, as as shown shown in figure A.6.2.

APPENDIX

412

f(x) ctf(x') + (1 — a)f(x)

--

f(x) f(Xa)

I

I I

I

I

I

I I

X

I I

X'

Figure A.6.2: Strictly convex functions functions

Theorem A.6.1. A.6.1.Assume Assumef fee C\ C1. The Thefunction function/ f: if Theorem : Q -> R is concave if only if

f(x f{x ++ z) z)
(A.6.3) (A.6.3)

for all

xeQ, zeR", z e R", x+zeQ. x+ zeQ.

XGQ,

The function function /f isisstrictly for all all x e Q. strictlyconcave concaveififinequality inequality (A.6.3) (A.6.3) holds strictly strictly for and all z 0.0. Condition (A.6.3) says says that that any any tangent tangentto to the the graph graph of of aa concave concavefunction function f/ Condition (A.6.3)

must corresponding must lie weakly weakly above above the graph graph of of f./ .The The correspondingcharacterization characterization of convex and strictly strictly convex functions functions simply entails reversing the direction of the inequality in (A.6.3). Definition A.6.2. The N x N matrix NxN matrix AAisisnegative negativeseinidefinite semidefiniteif ZTAZ 0 z'Az<0

(A.6.4)

A.6. CONCAVE CONCAVE AND AND QUASICONCAVE QUASICONCAVE FUNCTIONS FUNCTIONS

413 413

for all is If the inequality is strict for all all z ^ f l0, for allzz ee R". If inequality is strict for , then then the matrix matrix A is negative definite. definite. Reversing Reversing the the inequalities inequalitiesin in condition condition (A.6.4), (A.6.4), we we get negative get the the positive definite matrices. concepts of positive semidefinite semidefinite and andpositive Theorem A.6.2. A.6.2.Assume AssumefeC f 2.C2. The Theorem The function function f:Q.->R

is concave if and only for every every xe x e Q. 11.IfIf concave if only if ifD2f(x) D2f(x) isisnegative negative semidefinite semidefinite for D2f(x) every xe D2f(x) isisnegative negative definite definite for every x eQ, Q,the thefunction functionisisstrictly strictlyconcave. concave.

Definition A.6.3. The The function function /f::QQ—> set, Q Q cc R",2 —> R, R, defmed defined on on the convex convex set, Definition A.6.3. quasiconcave if its upper contour is quasiconcave contour sets sets

{Xe Q:f(x)t} {xeQ:f{x)>t} are is,is, if iff(x) f(x') > t implythat that are convex convexsets; sets;that that f(X)> tt and f(x') t imply

f(ax + + (i (l - a)x') > t, f(ax

(A.6.5)

for any any tt ee R, and a a ee[o, R,x,x,X' x' e 11, Q., and [0, ii. l]. If the concluding inequality inequality in (A.6.5) is is for whenever xx * x X and and aaee[o, then we we say saythat that /f isisstrictly strict whenever [0, i}, l], then strictlyquasiconcave. From the above defmition, function is is quasiconcave quasiconcave iff iff definition, we see that the function

f(ox ++(1 (l -— a)x') « > ' ) >Min{f(4, M i n { / ( 4f(x')}, , f(ca

(A.6.6)

for all X, Q, and and a a e [o, for all x, x'X'eeQ, [o, 1]. l]. We see that aaconcave concave function function is quasiconcave. quasiconcave. But the converse is is not true. true. Concavity Concavity is is aa stronger stronger property than quasiconcavity. quasiconcavity. Note that f/is is quasiconveX qiiasiconvexififitsitslower lowercontour contoursets setsare are convex; convex; that is if

22

The set D Q ccS " is convex if The set

(it ++(1(l —- a)x' a)x' Ee £2 ax

x, x'xtee Q, Q, and and aa e [0, [o, lj. whenever x,

APPENDIX

414

f(x) that f(oa f(ar + (l(i -— a)x')
-— x) 4 > 0,0,

(A.6.7)

whenever

f(x') f(x),

x, x'e Ii If for all all A;,x'eQ. Vf(x)(x' (^' -— x) 0, 9>>0,

whenever/(x') f(x')> f(x) then f/ isisstrictly whenever / ( x ) and andxx ^ x, then strictlyquasiconcave. quasiconcave. In the other other direction,ifif /f isisstrictly andifif V/(x) Vf(x) * 0 for all Jx Ec Q, direction, strictly quasiconcave quasiconcave and for all Q ,then then Vf(x)(x' — -x)>0 x)> 0 Vf(x)(x'

wheneverf(x') f(x')>f(x) whenever f(x) and andxx * x. x. Theorem AssumefeC f 2.C2. The function —* if Theorem A.6.4. A.6.4. Assume function f: /:Q— > R is quasiconcave quasiconcave iff for every every XE for xe Q, negativesemedefinite semedefinite in in the the subspace subspace Cl, D2f(x) D2f(x) isisnegative

R":Vf(x)z=O}, [z e R" : Vf{x)z = o}, that is, if and only if zTD2f(X)z zTD2f(x)z < 0

whenever

(A.6.8)

A.6. CONCAVE AND QUASICONCAVE QUASICONCAVE FUNCTIONS FUNCTIONS

415

Vf(x)z V/(*)z = 0, matrix D2f(x) every xxeQ.. E Q. If the Hessian matrix negativedefmite definite ininthe the subspace subspace for every D2f(x) isisnegative

{zee R" : Vf{x)z = 0 } {z

f

for every x e Q, then / is strictly quasiconcave.

A.7 Unconstrained A.7 Unconstrained maximization maximization Definition Definition A.7.1. A.7.1. Consider Consider the the function function f: / ' :R" R"—* -» R. The The vector vector xx* is a local local maximizer an open openneighborhood neighborhood 2 cc R", maximizer ofof/ f ifif there is an of of x*,x, Q. R",such suchthat that

f(x) f(x) take 1) forevery everyx ex Qe .Q. f(x) f(x) everyxexeR" r (i.e., if we can take for I f If/(**)> / ( * ) for for every Q = R"), then we say that x x* is a global maximizer maximizer (or simply a maximizer). maximizer). The The concepts concepts

of local local and and global global minimizers minimizers are are defined defined analogously. analogously.

is aa local Theorem A.7.1. A.7.1. Suppose Supposethat that/ f isis differentiable andthat thatx* x ee R" is Theorem differentiable and local maximizer or local localminimizer minimizerofof/f.. Then maximizer or Then

at .rx* forevery for every i,/', /i = 1,1, 2,

.

The condition for every condition df/dx, every /i can can be beexpressed expressed in in more more concise concise ax1 = 00 for notation Vf(X*)= o.

APPENDIX

416

A vector is called vector x e R" R" such such that that Vf(x*) V/(JC* ) = 0 0 is called a critical critical point. point. The The above above theorem tells that every maximizer or minimizer is a critical point. The converse theorem maximizer or minimizer is a critical point. The converse does not hold. For instance, consider

f(x1, = f{xl,x2)=xf

—

-x\.

the origin originwe wehave have V/ Vf = 0. point, but it is neither a At the 0. Thus the origin is a critical point, following second-order local maximizer nor minimizer. The following second-order conditions provide provide the sufficient sufficient condition for maximization. maximization. Theorem TheoremA.7.2. A.7.2.Suppose Suppose that that the the function function

f:R"->R

is C2 C 2 andthat and that

Vf(x*)= V / ( x ' ) = 0o.. 2 D2f(x*), If x* eE R" isa the nxn matrix, f\x\ is (i) If is alocal local maximizer, maximizer, then the matrix, the the Hessian D negative semidefinite. negative definite, definite, then then semidefinite. (ii) (ii) If the the Hessian Hessian D2f(x*) D2f[x) is negative

R"isisaalocal localmaximizer. maximizer. JC* e W 2 D2f(x*) It should should be noted that in f[x ) is negative negative in the the borderline borderline case in in which which D

semidefinite but not negative defmite, definite, we we cannot cannot assert assert that that x* is a local maximizer. Consider, for instance

f = x3. 2 We have have D D2f(0) negative demidefinite; demidefmite;but butx* x = 0 is neither a local maximizer f(o) is negative nor a local minimizer of this function. function. replace "negative" "negative" by by "positive" in in the the above above theorem, theorem, the the same same is is true true If we replace 2 D2f(x*) for local local minimizers. minimizers.AAcritical criticalpoint point of /f for for x*i of forwhich which the the Hessian Hessian D f[x*) is

indefiniteisiscalled calledaasaddle saddlepoint pointofof/f.. A saddle point xx* is is aa minimizer minimizerofof/ f in indefinite saddle point directions and and maximizer maximizerof of /f ininother some directions otherdirections. directions.For Forinstance, instance, the the origin origin is is point of a saddle point

A.7. UNCONSTRAINED MAXIMIZATION MAXIMIZATION

f(x1, J \Xl ,

X2)=

417

X x?t — X2

as the Hessian is

[22 [o0

0 —2 -2

A saddle point is local maximizer nor is neither aa local nor aa local local minimizer. minimizer.

Theorem A.7.3. Theorem A.7.3. Let

f:R" —>R be a C2 whose domain domain isisan anopen openset setQ. U in in R". C 2 flinction function whose R". Suppose Suppose that *

— ( x ' ) = 0 at x* foreveryi,i=1,2,•••. . forevery j , / = 1, 2, ax,j dx

(i) IfIf the the n«leading leadingprincipal principalminors minors of of the Hessian D2f(x*) D2f[x') alternate in sign

IfflI<0,

/„

fn

Jl\

J22

/ll

/l2

/l3

/21

J22

J23

ft

732

/33

/31

strict maximizer maximizerofof /f . at x*, then x* is a strict the nnleading leadingprincipal principalminors minors of of the Hessian D2f(x*) D2f[xJ are all positive (ii) IfIf the

> 0,

at

/„

fn

21

722

Jl 1

^22

/23

731

^32

/33

1, of f. x*,then thenx*x*isisaastrict strict minimizer minimizer of /.

2 D2f(x*) (iii) If some of the nn leading leadingprincipal principal minors minors of of the the Hessian Hessian D f[x') violate the

patterns in (i) (i) and and (ii) (ii) at x*, then x* is a saddle saddle point pointof of f. sign patterns /.

APPENDIX

418

Theorem Any critical criticalpoint pointx*i e R" of a concave fUnction/ f isis aa global Theorem A.7.4. A.7.4. Any concave function global maximizer.

Similarly, any any critical criticalpoint pointofofaaconvex convexfunction fttnction/ f is a global minimizer of of Similarly, global minimizer

f. A.8 A.8 Constrained Constrained maximization maximization Consider the the problem problemof ofmaximizing maximizing/f under under mmequality equalityconstraints constraints

Maxf(x) Max f(x) XE R

xsR"

2, •", m, s.t.: = bb1, s.t: gj(x) = = 1,\,2,-,m, Jt j =

(A.8.l) (A.8.1)

where the the fUnctions functions

f, / i

S\ > '"'» 8m

defined on R11. We generally generally assume assume nn> m. m. Let Let us introduce are defined R". We introduce the constraint set

ci{xe

=b1, j=1,2,",m

The feasible point x* e Q 1) if Q isisa alocal /oca/constrained constrainedmaximizer maximizerininproblem problem(A.8. (A.8.1) there exists an open neighborhood of x*, say M cc Q Q,, such say M suchthat that

f(x) f(x) MnQ. /(**) > f(x)forall for all Xe xe M n Q. The point point x* x is a global The global constrained constrainedmaximizer maximizer if itit solves solves problem problem (A.8.1). (A.8.1). The The first-order condition is stated as follows. first-order follows. A.8.1. Suppose Suppose that Theorem A.8.1. that the theobjective objective and and constraint constraint fUnctions functions of problem problem (A.8. 1) are are differentiable differentiable and that x* e QQisisaalocal (A.8.1) localmaximizer. maximizer. Assume Assume also also that that the mxn mxn matrix matrix

A.8. CONSTRAINED MAXIMIZATION

Lax. a*,

419 419

mxn

m.3 Then Then there are numbers Pj e R, R,one onefor foreach eachconstraint, constraint, such such that has rank in.3

af(*) =

xi

ad

£Pl*M

,.12...,*

i = 1, 2,

(A.8.2) (A.8.2,

J=l

or, in in more concise notation

Vf(x)= referred to as Lagrange Lagrange multipliers. The numbers pj, are referred

Often the first-order conditions conditions (A.8.2) (A.8.2) are presented presented in a different different way. way. Given Often the and variables xxeE R'1 R" and

P = (Pi, P2,

Pm)'

Lagrangian function we can define the Lagrangian function

L(x, p)= f(x)— Jpjgj{x). L(x,p)=f{x)-Y

(A.8.3) (A.8.3)

j

constrained maximizer (and if the the theorem says says that that if if x* is a local constrained The above theorem constrain qualification is is satisfied), then for some some values values

A> P2' A.

Pm' >

all of of the partial derivatives of of the Lagrangian function function are arenull; null; that that is

says that that the constraints constraints are independent independent at at This is called the constraint qual(fIcation. qualification. It says x'.

APPENDIX

420

aL(x*,p)0 dx,

i=1,",n, j=l,•••,m.

There is also a second-order theory for for problem problem (A.8.3). (A.8.3). Suppose Supposethat that atatx* x There second-order theory the constraint constraint qualification qualification is satisfied satisfied and that that there there are areLangrange Langrange multiplies multiplies satisfying (A.8.2). If x* is a local maximizer, then

p)= D2f(x*)_

is negative semidefinite on the subspace

vgj(x*)z R": Vgj {z e R": (x* )z = o, 0, forallj}. for all j }.

In the other i eeQQand and ifif other direction, direction, ifif x* andsatisfies satisfiesthe thefirst-order first-order condition, condition, and D;L[x*, p) isisnegative negative definite definite on the subspace {Ze [ze

R": vg1(x)z_—o, Vgj (x' )z = 0, forallj}, for all j},

then x* is a local maximizer.

local constrained constrained minimizer localconstrained constrained maximizer maximizer of As aa local minimizerofof/f isisaalocal of —f, - / , theorem theorem A.8. A.8.11 and the discussion discussion of of second-order second-order conditions conditions above above are are also also applicable to the characterization of local constrained minimizers. We now We now generalize generalize our analysis analysis to to problems problems that that may may have have inequality inequality constraints. Consider Considerthe theproblem problemofofmaximizing maximizing/ f under constraints. under in m equality equality constraints constraints and q inequalities and inequalities

Maxf(x) Max f(x) re xeR" R"

g1(x) = in, j = 1, 2, s.t, st: * > ) = W = 1 . 2 . , » , k=1,2,.•• hk(x)
where the flmctions functions

(Ag4) (A.8A)

A.8. CONSTRAINED MAXIMIZATION

421

are defined defined on on R". R". We We generally generallyassume assumen> n mm++ q. We say are say that that the the constraint constraint qualificationisissatisfied satisfiedatatx*x ee Q that hold hold at at x* x with qualification Q, ififthe the constraints constraints that withequality equality are independent, that if, if, if the vectors in {vg1 [x ), for all all j }u {Vh all *}, {Vgj (x*), {vhkk (x"), for all

j

are linearly independent.

that the objective Theorem A.8.2. A.8.2. (the (the Kuhn-Tucker Kuhn-Tucker conditions) conditions) Suppose Suppose that objective and and constraint functions differentiable and that x* e Q Clisisaalocal local ftmctions of problem (A.8.4) are differentiable maximizer of the problem. Assume also that the constraint constraint qualification qualification is satisfied. satisfied. R, one onefor for each each equality equality constraint, constraint, and R+, Then there are multipliers Pj e R, and pPk k e R÷, one for each inequality constraint, such that (i) ,m (i) forevery for every /i=l,•••,m = 1,

af(x*) +

=

EPk

at),

i =

n,

(A.8.5)

or in more moreconcise concise notation notation Vf(x*)= V/(**)=j>,Vg,(** j=\

(ii) for every k k = = 1, I, (ii)

—

i.e., i.e.,

k=\

,qq

ck) = o,

that does not hold with equality. pk == 00 for every constraint Ac & that equality. Pk

The above theorem is also applicable to local local minimizers, with the only change change

of the is now being that the restriction on on all being that the sign sign restriction all of the multipliers multipliers is now Pk pk <0,0,

k=l,",q.

APPENDIX

422

Theorem A.8.3. = 0) and and A.8.3. Suppose Suppose that there are that there areno noequality equality constraints constraints (i.e., in m= that every inequality constraint constraint kk isisgiven givenby byaaquasiconvex quasiconvex function function hk. hk. Suppose also that the the objective function function satisfies satisfies

x)> 0 for any anyxxand andx'I with f(x')>> f(x), f(x), V/(*X*' with f{x') -x)>0 Vf(xXx' — then if (i) and (ii) in in Q.satisfies if x* ee Q satisfies the theKuhn-Tucker Kuhn-Tucker conditions conditions (conditions (conditions (i)

theorem A.8.2), and is the constraint constraint qualification qualification holds at x , it andififthe itfollows that x is a global maximizer. maximizer.

If we have

Vf(x)(x' — 4 < 0, Vf(x)(x'-x)<0,

for anyxx and andx'I with for any multipliers have havethe with f(x') f(x')<
A.9 Dynamical A.9 Dynamicaloptimization4 optimization4 Consider state variable variable x(t) x(t) which whichbelongs belongsto toan aninterval interval/ I of of R; its Consider aa state itsmotion motion by through time is governed by

+ 1) = f(x(t), c(t)), x(t + l)=f{x(t),C{t)),

(A.9. 1) (A.9.1)

where

C(t) =

(ci1 (t), c2 (t),

c11 (t))

Assume that that the the initial initial stock stock x x00 is given and that that the vector of is the of control variables. Assume the set in t,t, denoted by 'V(x(t)), set of feasible decisions decisions in of feasible *¥(x(t)), depends depends on on the the level level of the

stock x(t), C(t)e C(t)e'P(x(t)). *F(*(/)) AAsequence stock sequence (x(t), c(t)),, tf==0, 0 ,i,l , - , 4

Appendix A.9 A.9 is ismainly mainly referred referred to toappendix appendix A.5 A.5in inde deIa laCroix Croix and andMichel Michel (2002). (2002).

DYNAMICAL OPTIMIZATION A.9. DYNAMICAL

423

(A.9. I), the the initial initialcondition, condition,and and C(t)e C(t) *?(*(*)) 'Y(x(t)) is called called aafeasible feasible which satisfies satisfies (A.9.1), trajectory starting from from x0. x0. Consider a payoff function, function, depending on the the stock stock variable variable and and the the decisions decisions at each date as

u(x(t), C(t)) U(x(t), C] c1(t), U(x(t), C(t)) = = U{x{t), (t), cc2(t), , cn{t)\ 2{t), -•.., Given a positive discount discount 8, the the objective objective is is to to maximize maximize the the discounted discounted flow flow of payoff payoff

cfr)), xit), C{t)\

(A.9.2)

1=0

on the set of feasible trajectories starting from from x0. x0.

'P(x) is non-empty, and for for all Assumption A.9.1. A.9.1. For For all all elements, elements,x,x, of of /, I, *?(*) non-empty, and Assumption vectors subset *¥(x) defined and i.e., vectors CC in the subset 'P(x) of R", f(x, and belongs belongs to to /, I, i.e., f(x, C) c) isis defined dynamics are are defined definedon on /. I. the dynamics

Assumption A.9.2. A.9.2. For For C C given, the f(x, C) c) and U{x, u(x, C) c) are Assumption the functions functions f(x, are nondecreasing decreasingwith withrespect respecttotox.x.The Theupper upperbound boundofoff(x, f(x, C) c) with respect to C in respect to in »P(JC) P(x) is finite finite

0(x)= sup sup f{x, /. f(x, C)e I. 'P (x) CE¥(X)

The upper upper bound boundof of U(x, U(x, C) c) with respect respecttoto CC in P(x) *F(x)isisfinite finite

U(x,C)eJ. u{x) = sup sup U(x, C) e / . Cc 'P (x)

()

For any initial initial condition condition x0, x0, there there exists exists aa constant constant b0 b0 and and aa scalar scalar So >> S such

that the sequence sequence 1(t) x{t) defined defined by by + i) =

Vt o,

APPENDIX

424 satisfies

80'u(i(t)) b0,0, Vt 0. 8'Ax(t))0. Proposition 1 PropositionA.9.1. A.9.1.(the (theconvergence convergenceofofthe theinfmite infinite sum) sum)Under Underassumptions assumptions A.9. A.9.1 A.9.2, every every discounted discounted sum sum of of feasible feasible payoff payoff is defined defined and and has has values values in in and A.9.2,

{} Ru{-oo} T

C(t))= = lim lim ^ < 1=0

C(t))e Ru{-oo}.

~ > + ™ 1=0

A.9.3. For For all all xx0 Assumption A.9.3. I,there thereexists existsaafeasible feasible path path 0 e I, (x(t), c(t))1>0

x0 such such that that the thesequence sequence starting at x0

C(t)) ES1U(x(t) c(t)) 1=0

is bounded below below when when TT—* —>+ oo• + °°.

1 -- A.9.3, Proposition A.9.2. A.9.2. (the (thevalue valueflinction) function)Under Underassumptions assumptionsA.9. A.9.1 A.9.3, the function V(x0)= C(t)) feasible from from ^0 , V{xo) = sup | V t / ( x ( 4 c(t)); C(0); (x(t), (*(/), C(0)

definedon on/I and satisfies V* Vxee I is defined and satisfies v(x) = (x, C)); V{x) = sup{g(x, sup{g(x, c) C) + c5v(f SV{f(x, C));C C ee V{x)}. Proposition A.9.3. Proposition A.9.3.(the (thecharacteristics characteristics of ofoptimal optimal trajectories) trajectories) Under assumptions assumptions (x*(t), c(t)) feasible path (**(?), C*(t)) starting from x'o = - x0 x0 is is optimal optimal if if and A.9.1 - A.9.3, a feasible only if we have have for all all t

A.9. DYNAMICAL OPTIMIZATION (t)) + + ov(x* V(x* {t))=u{x'(t),C' (t)) = U(x* (t), 1)). (t)) V{x Sv{x (t(t + + l)).

425 (A.9.3) (A.9.3)

We now examine examine the same same problem problem in this this section section in in aa different different way. way. We We 1) and the the initial condition are satisfied and maximize (A.9.2) (A.9.2) such such that that (A.9. (A.9.1)

C(t)e P(x(t)) C(t)s Wlxlt)) by introducing the Lagrangian.

Definition A.9.4.1. (x(t), C(t)) C(t)) of period period t is obtained obtained as the A.9.4.1. The The Lagrangian Lagrangian Lt(x(t), sum of the payoff payoff U(x, u(x, c) C)with withthe theincrease increaseininthe thevalue value of of the stock L, (x(t\ (x(t), C{t)) C(t)) = l)f(x(t), C(t)) -— q(t)x(t). = U(x(t), C(t)) + Sq{t ÷ + l)f(x(t), a(t)x(t).

supportedby by aa sequence A feasible feasible trajectory trajectory\x'(t), (i (t), C'(t)) c (t)) is supported A sequence of shadow shadow prices (q(t)) if, for every integer t 0, the Lagrangian L1 (x(t), C(t)) attains prices if, for every integer t > 0, the Lagrangian L, (x(t), C(t)) attains its its maximum (i (t), C'(t)) C(t)) on the the set set of ofvectors vectors {x(t), (x(t), C(t)) which verify verify maximum atat\x*(t),

x(t)e I,I, C(t)e 'P(x(t)). x(t)e We then have L

forall for all

c(t))=

]ktax

LI (x(t), C(t)),

t t>0. 0.

Assumption Assumption A.9.4. A.9.4. The The set set AAofofthe thefeasible feasibletriplets tripletsofofpayoff, payoff,current current stock, stock, and and resulting stock is is convex. A isis the x, y) y) of resulting stock convex. Formally, Formally, A the set set of of elements elements (a, x, of Rx IxJ for which there exists CE 'P(x) such that R x / x / for which there exists C e ^(x) such that

aU(x,C), y=f(x,C).

a
APPENDIX

426

Proposition A.9.4 Proposition A.9.4(the characteristics of (thecharacteristics ofoptimal optimal trajectories)5 trajectories)5 (it) trajectory (x (t), C*(t)j starting from x0 Let us consider a feasible feasible trajectory \x'(t), x0 for for which xx'(t) 1 -- A.9.4, A.9.4, the the trajectory is interiortoto /I for is interior for all all itt > 0. 0. Under Under assumptions assumptionsA.9. A.9.1 trajectory C (t)) is optimal (i (it),C*(t)) [x*(t), optimal if and only if if there there exists exists aa sequence sequence of of shadow shadow prices prices

is supported sequence of (q(t)) such that: (i) the the trajectory trajectory \x*(t\ supported by by the sequence (q(t)) such (it),C'(t)) C (t)) is shadow (9^)); and (ii) (ii) for for any any other other feasible feasible trajectory trajectory (x(t), (x(t), C(t)) C(t)) starting starting shadow prices prices (q(t)); from x0 x0 such that

C(t)),

is finite, we have have

limS'q(t)(x(t)— limS'q(t)(x(t) - x x(t)) (t)) >0.0.

(A.9.4)

The condition condition (A.9.4) (A.9.4) is the the transversality transversality condition. condition. It means means that that the the discounted value value of of the the optimal capital is exhausted in the long run and discounted optimal capital exhausted in and that that the the value of of any of other feasible stock should should be be greater greater than than or or equal equal to to that that of of the the value feasible stock optimal stock. stock.

5

The necessary necessary condition condition is given given by by Michel Michel (1990) (1990) and and the sufficient sufficient condition condition is is standard.

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Index Balanced growth path 209 209 RJ. 93, 93,320 Barro, R.J. 320 Basic Jordan block 245,395 245,395 Basin of of attraction attraction 41 41 320 Becker, G.K. 320 G.S. 320 320 Becker, G.S. Becker, R.A. R.A. 341 341 Becker, R. 97 Benabou, R. Benedicks, M. M. 294 294 Benedicks, BenhabibJ. 174 Benhabib, J. 174 Berman,A.391 Berman, A. 391 Bernstein, D.J. DJ. 140 140 Bernstein, Bertinelli, L. 112 112 Bertinelli, L. Bhaduri,A. 174 Bhaduri, A. 174 Bifurcation 4,152,289 Bifurcation 4, 152, 289 diagram 153, 163,201 diagram 153,163,201 flip 47 169 period-doubling 148, 148,169 saddle node 294 169 subcritical fold 166, 166,169 subcritical pitchfork 167, 167, 168,222 168, 222 supercritical fold 164, 168 164,168 supercritical pitchfork 167, 169 169 transcritical 167, 167,169 value 152 Bisector 160 160 D. 112 112 Black, D. Blanchard, 0.93 Blanchard, 0.93 Block, L.S. 154, 154,160,195 160, 195 Bolton, P. 97 Bordered Hessian 89

Abe, N. 97 Abell, M.L. 2 Abel's lemma 56 Ability distribution 108 108 Absorbing interval, interval, see see abbtracting abbtracting interval Adaptive adjustment 193 193 18; see also Adaptive expectation 18; price dynamics hypothesis 18, 18, 80 Adjoint of matrix 394 development phase phase 111 111 Advanced development 100 Aghion, P. 97, 100 173 Alligood, K.T. 173 Amortization 19 19 Andronov 33 Andronov Aperiodic 135 135 Aperiodic orbit 155 155 Arbitrage 306 condition 18, 18,306 opportunity 119 Arctan function function 179 Area contracting 291 Arrow, K.J. 358 Attracting 29 globally 29 Attracting interval 49 Attractive 262 uniformly 262 Attractor 29 global 29, 140 29,140 Autonomous 15, 261 15,261 Azariadis,C.4,306,310 Azariadis, C. 4,306,310 439

440

Branch of fixed points 148 148 Braselton, J.P. J.P. 2 Budget constraint 98 Burmeister, E. 118 118 Business cycles 186 Capital accumulation 90 Carleson, L. 294 Carr, 284-5 Car, J.J. 284-5 Carvalho, Carvaiho, L.A.V. 145 Carvalho's lemma 145 145 Carvaiho's lemma Cascade of bifurcation 154 Casoratian 55 Catastrophe 4 Center manifold 284 theorem 281, 294 281,294 CES production function function 132,206, 321, 337, 342 321,337,342 R. 320, 320,325 Chakrabarti, R. 325 Champ, B. B. 125 125 Champ, Chang, W.W. 299 Change speed 388 asymmetry 388 388 3,135,154,303 Chaos 3, 135, 154, 303 in a demand and supply model 178 in the Li-Yorke sense sense 161 161 in the sense of Yorke 173 topological 195 195 equation 60, 60, 82, 82,124, Characteristic equation 124, 380, 395 380,395 Characteristic root 60 Chiang, A.C. A.C. 33, 33,80, 80,391 Chiang, 391 Chiarella, C. 4 Chinese box 387 Cobb-Douglass production function 132, function 85, 85,132, 342 120, 175 utility fUnction function 85, 85,120,175 23; see stair Cobweb diagram 23; diagram Cobweb model model 41 41 in two interrelated markets

INDEX 251 251

of demand and supply supply 33 33 with the normal-price expectation 36 with adaptive adjustment 193 193 with adaptive expectation 37 Cofactor 394 Cofactor394 Collatz conjecture 140 Comparative statics analysis 369 Competitive equilibrium 121 121 3, 389; see also Complexity 3,389; nonlinear theory Composition of fimction function 283 Concave 111,410 Concave strictly strictly 410 410 Conjugate 282 topologically 195 195 qualification 419 Constrained qualification Control, see external force force Converse of Sharkovsky's theorem 162 162

Convex 111 111 strictly 411 411 Coppel, W.A. 217 Coppel,W.A.217 195 Coppel, W.L. 160, 160,195 Corn-hog cycle cycle 251 251 Cournot-Nash solution 253 253 Crammer's rule 398 Critical value, see see bifUrcation bifurcation value OSU model Current income in the OSG 85, 120, 342, 358 85,120,342,358 Damped 35 388 Dawkins, R. R. 388 Day, R.H 174,202 174,202 Day's growth Day's growth model model 203 422 De la Croix, D. 93, 93,422 De Melo, W. 195 DeMelo,W. 195 De Vault, R. R. 53 53 Deflation 119 119 Degree of smoothness 129 Delay 259 Dendrinos, Dendrinos, P.S. P.S. 4

INDEX Denecker, R. 205 205 Derivative 403 403 Determinant of matrix 291, 393 291,393 Determinacy 315 315 178 Devaney,R. 178 Development reap 115 115 Devereux, M.B. M.B. 315 Diagonalizable 243, 243,395 395 Diamond, P. P. 93 93 Difference 13 13 first first 13 13 quotient 13 13 Difference equation v, 1, 1,13; Difference 13; see also map higher dimensional 227 linear 22 homogenous st-order homogenous fir first-order 15, 240 15,240 nonhomogeneous firstfirstorder 16, 248 16,248 k-thnonhomogeneous k-thorder 54 second-order 67 linearized 2 nonlinear 2, 30,403 Differentiable 30, 403 Dimension of space 392 Diminishing return 366 Discrete dynamical system 14 14 dynamics 13 time time 13 13 variable variable 13 13 Disequilibrium inventory model 195 120, 128, Disposable income 85, 85,120,128,

342,350,358 Dobell,A.R. 118 118 Dornbusch, R. 306 Dornbusch exchange-rate model 306 Duopoly model 253 Durlauf, S. 97, 97,187 187 Dynamical optimization 422

441 441

characteristics of optimal trajectories 425 425 convergence of the infinite infinite sum 424 feasible trajectory 423 value fUnction function 424 Dynamical system 21; 21; see also map Dynamical theory theory 33 nonlinear 33 Economic chaos 4 Economic development 4,97; 4, 97; see also economic growth sustainable 367, 370 367,370 Economic growth growth 4,202,205 4,202,205 with government spending 315 315 with pollution 202 Economic miracle miracle Japan 374 Korea 374 Singapore 374 Taiwan 374 Economics 4 360,386 Education 360, 386 policy 370 sector 360 Eigenvalue 234,243,395 Eigenvalue 234,243,395 semisimple 268 268 234,243,395 Eigenvector 234, 243, 395 T.S. 106 106 Either, T.S. Elasticity of substitution between consumption 311 311 between the two types of labor 111 111 for labor supply 317 of real balances 214 Elaydi,S.N.41,43,56,71,141, Elaydi, S.N. 41,43, 56, 71, 141, 163, 178,241,265,268,271, 163,178,241,265,268,271, 274,278 Enns, R.H. 2 Equations of general Riccati type 64 Equations of Riccati type 64

442 Equilibrium point22,261; 22,261;see see also Equilibrium point also fixed point Erratic 175 175 Eventually 25 equilibrium point point 25 25 fixed point 25 periodic orbit orbit 156 156 periodic point point 138 138 stationary orbit 156 156 stationary point point 25 25 Expectations-augmented 80 Explosive 35 External force 54 Externality in education 109 human capital 112 112 productive 316 Ezekiel, M. 251 Ezekiel,M. 251

Farmer, R.E.A. 335 Fast 4 Feigenbaum, M.J. 154 Feigenbaum's number, number, see Myrberg' s number Myrberg's Ferguson, B.S. 4 60 Fibonacci sequence 60 First-order condition for maximization 89 89 Fiscal 122 122 103 Fisher equation 103 22,137 Fixed point 22, 137 P. 44 Flaschel, P. Forcing term, term, see see external force force Franke, R. 4 207 Free entry 207 Freeman, S. S. 125 125 Freeman, Friedman, M. 174 174 Function 399; see also concave, convex analytical 410 continuous 399 401 uniformly 401 smooth 409

INDEX Fundamental matrix 249 Fundamental set of solutions 55, 55,249 Fundamental theorem 57 Gabisch, G. 299 Gale, D. 122, 187, 205 Gale,D. 122,187,205 Gallas, J.A.C. 193 193 Galor, 0. 93, 310, 320, 325 0.93,310,320,325 Gambler's ruin 69 Gandolfo, G. 36, 75 GarcIa-Pefialosa, Garcia-Penalosa, C. 106 106 GDP 1,1,186 186 General OSG model 88; see also OSG model General solution solution 21 21 Genotype selection selection model model 141 141 Ghiglino, C. C. 341 341 Gilbert, J. 391 Gilbert,! 391 Gilbert, L. L. 391 391 Goeree, J.K. 193 193 Goodwin, R.M. R.M. 174 174 Government spending spending 315 315 Grand theory 388 388 Grandmont, J.M. J.M. 335 335 Gross domestic domestic product, see see GDP GDP Guo, J.T. 3315-5 15-5 Guo,J.T. Haavelmo, T. 219 Haavelmo,T.219 Hahn, F.H. 174 Hahn,F.H. 174 Harmonics of of the the 2-period 2-period orbit orbit 222 Head, A.C. A.C. 315 315 Hénon, Henon, M. M. 290 290 Hénon Henon map 289 289 Hess, G. 186 Hessian 407,414,416 407,414,416 Hessian Heterogeneous household household 351 351 middle class class 353 353 poor class 353 353 rich class 353 353 Hierarchy 387 in science science 388 388 social 388 388 Homeomorphism 399; see also

iNDEX INDEX

443 443

fimction function 193, 195 Hommes, C.H. 178, 178,193,195 Hopf bifig cation 296 bifurcation Horseshore 195 195 Howitt, P. 97, 100 Howitt,P.97,100 97, 106,112,341, Human capital capital 97,106,112,341, 358 formation 97 97 Hyperbolic 31 31

Jacobian 124, 291, 380, 403 Jacobianl24,291,380,403 Jordan block 268 395 Jordan canonical form 245, 245,395 form 229,243 229,243 Jordan form Jovanovic, B. 205 Judd, K. 205 Jullien, B. 8. 335 Jurgens,FI. Jurgens, H. 178 Jury, E. 74 Jury,E.

Implicit function unction theorem 402 theorem 129, 129,402 Income distribution 341 341 Incommensurable period 3; see also quasiperiodic hideterminancy Indeterminancy 315 315 97,106 Inequality 97, 106 80,118 Inflation 80, 118 anticipated 125 125 Initial value problem 21, 54, 240 21,54,240 of aa matrix matrix 74 74 Inner of Innovation 101,205, 101,205,207 Innovation 207 cycle 205 205 Input, see external force Input output with time lag in production 250 Intermediate good 101,106 101, 106 sector 206 sector Intermediate value theorem theorem 219,402 Invariant 41 41 set 195 Invariant interval 50 Inventory 39, 39,195 195 cycle 75 Inverse fimction function theorem 404 Irie, K. 75 IS-LM 306 Ishida,J. 186 Ishida, J. 186 Iterate first first 15 15 t-th 15,41 15,41 Iwata, S. 186 186

341 Kaldor, N. 299, 299,341 Kaldor model 299 Kalecki, M. 299 Keynes 100, 386 the propensity to save 372 on the Knowledge, 385 spillover 186 143 Kocic, V.L. 53, 53,143 Kronecker delta 392 Kuhn-Tucker condition 421 421 Kulenovic, M.R.S. 50

143 Ladas, G. 50, 53, 53,143 Lagarias, J.C. 140 Lapham, B.J. 315 Lapham,BJ. Learning by doing 109 producing 360 Labor market 101, 174 101,174 107 skilled 102, 102,107 unskilled 102, 107 102,107 Lagged supply function 33 Lagrange multiplier 419 Lagrangian 89, 89,419,425 Lagrangian 419,425 LaSalle's principle 274 18 Learning 18 Lebesgue measure 217 154,223 Li, T.Y. 154, 223 160,204,223 Li-Yorke theorem 160,204,223 Liapunov 33 Liapunov Liapunov Liapunov direct method 270

444 exponent 170, 195, 301 exponent 170,195,301 function 270 function number 170, 301 170,301 second method method 270 270 stability theorem theorem 271 271 Lifetime utility in the OLG model 93 Lim, G.C. 4 Limit point 155 155 set 156 Limiting behavior 67 Linear algebra 391 391 Linear equations 397 augmented 397 coefficient matrix 397 coefficient of constants 397 397 of unknowns, unknowns, 397 397 Linearity 3,4 Linearity 276,408 Linearization 276, 408 136 Linearized oscillation theorem 136 Linearized stability stability 31 31 Linearly independent independent 55, 55,248,391 248, 391 Logistical map 4, 4,25,139,153,167, 25, 139, 153, 167, 177,203 177,203 Lorenz, H.W. H.W. 4,299 4,299 trap 111 111 Low-growth trap

Mainland China 374 Mathus on on the propensity to to save save 372 Maithusian 321 Malthusian321 Map2l Map 21 chaotic 178 178 dissipative 50 126, 350 Marginal conditions 88, 88,126,350 Marginal cost 296 Marginal propensity to consume 72 Marginal return to human capital 98 Martelli, M. 169 Martelli,M. Marx 100 Mathematica 2 Matrix 391 391 identity 391 391

INDEX invertible 392 negative definite 413 413 negative semidefmite 412 null 391 391 operation operation 391 391 positive definite 413 413 positive semideflnite semidefmite 413 413 rank 392 square 392 nonsingular 392 singular 392 theory 391 391 Matsuyama, K. 97, 205, 213 Matsuyama,K. 97,205,213 Matsuyama model 212 Maximization 415 constrained 418 418 critical point point 415 415 global 415,418 415,418 local 415, 418 415,418 strict 417 unconstrained 415 415 McGuire, G.C. 2 Method of undetermined undetermined coefficient 58 58 Metric space 398 complete 399 normed 402 Metzler, L.A. 75 Metzler equation 75 Michel, P. 422, 422,426 426 Minimization 415 constrained 418 critical point 415 418 global 415, 415,418 local4lS, 418 local 415,418 strict 417 417 unconstrained 415 Mitra, T. 205 Moav, O. 0. 97 Modem Modern growth regime 327 Modigliani, F. F. 341 341 Modigliani, Monetary economy 119, 213 119,213

INDEX

445

expansion 82, 309 82,309 policy 122 Money 118 118 demand for 118 118 equilibrium 122 122 inside 122 outside 122 125 fiat 118, 118,125 neutrality of 174 174 outside 118 118 supply 213 Monopoly 101,207 profit 207 rent rent 101 101 Morse lemma 407 Mortgage 20 Multiple equilibrium points 365 Multiplier of of the the orbit 140 140 Murphy, K. 320 K. 320 Myrberg, Myrberg, P.J. P.J. 154 154 Myrberg's Myrberg's number number 154 154

OLG OLG model model 93 93 old age support 320 smail-open small-open 97 97 two-sector 310 310

Nagomo, J. 189 Nagomo,J. 189 Native expectation expectation 193 193 Neimark-Sacker bifl.ircation, bifurcation, see Hopf bifurcation bifurcation M. 37 37 Nerlove, M. Neugart,M.331,334 Neugart, M. 331, 334 Newton-Raphson method 39 Nishimura,K. 315,341 Nishimura, K. 315, 341 money 80 80 Nominal money 30 Non-hyperbolic 30 Nonautonomous 15 Nondegenerate critical point 407 Nonlinear dynamics 33 Nonlinear dynamics Nonlinear theory Nonlinear theory 44 Nonlinearity Nonlinearity 6 6 Norm 27 Norm 27 Euclidean 27 Normal price 36 Nusse, H.E. 193 193

OSG model 82 82 with endogenous labor supply

Okuguchi, Okuguchi, K. K. 75 75

Oligopoly with 33 firms firms 254; see also also duopoly OSG One-sector growth model, see OSG model Onozaki, Onozaki, T. T. 193 193 city size size 113 113 Optimal city earning 113 113 Optimal level of net earning Orbit 15,22 aperiodic 157 157 asymptotically periodic 172 172 asymptotically stationary 156 eventually 156 Order 6 387 Organism 387 71,135 Oscillation 71, 135

Oscillatory 135 135 non- 135 non-135 strictly 135 135 126 126 with heterogeneous households households 341 341 with money 119 Overlapping-generations model, see OLG model Panico, C. 341 341 Pasinetti, L.L. 341 Path dependent 358 Peitgen,H.0.178 Peitgen,H.O. 178 Perfect foresight foresight 118, 174, 174,337 337 Period 13, 13,136 136 Period-doubling 47 Periodic 135, 135,261 261 asymptotically 302 302 point of minimal period k 136 136 97 Persistence of inequality 97 391 Peterson, G.L. G.L. 391

446 Phase contraction 189 189 expansion 189 189 A. W. 174 174 Phillips, A.W. 174,335 Phillips curve 174, 335 80; see also also Phillips Phillips Phillips relation 80; curve Pielou's 279 Pielou's logistical logistical equation equation 42, 42,279 Piketty, T. T. 97 97 Plemons,R.J. 391 Plemons, R.J. 391 Poincare Poincaré 3 Pontryagin Pontiyagin 3 Population 219 Portfolios 118 definite 271 271 Positive definite Positive innerwise 74 Post-Keynesian 341 341 Post-Keynesian Post-Malthusian regime regime 327 327 Post-Malthusian Poverty 97,367 trap Poverty trap 97, 367 Power distribution distribution 387 387 Power Preference 97,353 Preference 97, 353 175 change 133, 133,175 Price adjustment adjustment 39 39 Price Price Price dynamics dynamics 18 18 Prime 136 Prime period periodkk 136 Principal minor 417 Propensity to consume in the OSG model 85, 127, 133 85,127,133 own wealth in the OSU OSG 85,127,133 model 85, 127, 133 save 372 use leisure time in the OSG model 127, 133 127,133 Puu,T.4 Puu, T. 4 Putzer algorithm 241, 241,256 Putzer 256

Qualitative Qualitative 33 Quantitative 33 Quasiconcave 410 Quasiconvex411 Quasiconvex 411 Quasiperiodic 33 Quasiperiodic

INDEX Rabbit problem 60 109 R&D 103, 103,109 Ralf, K. 310 Rait Rasband, S.N. 299 Rational choice 175 175 expectation 195 195 Raut, L.K. 320 Repeller; see source Return to scale 118, 316, 364 constant 97, 97,118,316,364 decreasing 364 increasing 315, 364 315,364 Risk neutral 186 186 Rivera-Batiz, L.A. 205 Rob, R. 205 Romer, P.M. 205 Rome regime 208 Rosser, J.B.Jr. 4 Route to chaos 154 154 Rural infrastructure 112 S-shaped S-shaped 109 109 Saddle point 416 X. 93 93 Sala-i-Martin, X. Samuelson, P.A. P.A. 93, 93,341 Samuelson, 341 multiplier-accelerator Samuelson multiplier-accelerator model Salvadori,N. Salvadori,N. 341 341 Sandefur, J.T. 44 Sandeflir, Sard's theorem 407 Sato, K. 341 341 Sato, S. 189 Sauer, T. 173 Saupe,D. 178 178 Schumpeter 100, 100,386; 386; see also innovation, Schumpeterian creative destruction destruction Schumpeterian creative destruction

100 Schur-Cohn criterion 74 derivative 48, 48,143 Schwarzian derivative 143 Search-based model of technological

INDEX evolution 205 205 Second-order condition condition of of maximization 90 Second-order linear autonomous systems 228 228 Seigniorage 119 Self-organization 6 Self-similarity 294 Semiasymptotically stable from from the left left 43 43 from from the right right 43 43 Semistable from from the left left 43 43 from from the right right 43 43 Semmler, W. 4 Semmier, Sensitive dependence on initial condition 177 177 Sequence 399 Cauchy 399 subsequence 399 Set 399; see also metric space ball 399 boundary 400 closed 399 closure 399 compact 399 399 connected 400 400 convex 410 dense 399 disconnected 400 open 399 Sexual division of labor 4 Sharkovsky's 160 order 154, 154,160 theorem 160 160 Shimomura, K. 341 341 Shone, R. 2,4, 2,4,178,195,251 178, 195, 251 Sieg, G. 193 193 Similar 395 Simon, H.A. 387 Simplicity 33 Singer, D. D. 162 162 Singer, 162 Singer's theorem 162

447 447

Sink 30 Slow 4 Smith 385 on the propensity to save 372 Smyth, D.J. DJ. 299 299

J.S. 391 391 Sochacki, J.S. Socioeconomic process 44 Solow,R. 101,174 Solow, R. 101, 174 Solow regime 208 Solow-Schumpeter growth oscillation 205 Solow model 118 118 54,240 Solution 54, 240 complementary 57 bounded bounded 263 263 general 57, 240 57,240 linear independent 248 particular 58, 240 58,240 Sonis, M. 4 Sorger,G. 341 Sorger, ci. 341 Source 28 Space 386 Spatiotemporal scale 386 320 Srinivasan, T.N. 320 Stability 44 140,262 Stable 140, 262 140, 263 asymptotically 30, 30,140,263 exponentially 263 locally 27 uniformly 261 261 uniformly asymptotically 263 Stair diagram 23 23 Stair-step diagram 23; see stair diagram Stationary Stationary point point 22; 22; see see also fixed fixed point Stochastic 101 101 noise 4 Strange 195 195 Structure 3 macroscopic 4, 389 meso 389 meso389

448 389 4,389 microscopic 4, Structural change 6 Structural invariant 389 Structural stability 387 Stutzer, M. 219 Stutzer,M. Subjective discount factor factor 186 186 Submanifold theorem 406 Sudden change 33 Superposition of maps 405 of maps 405 Superposition principle 57 Tamura, R. 320 Taylor expansion 408 408 Taylor series 171,410 Tax 367 Teacher 360 Technological change 109 Technology 97, 97,109,132 109, 132 Tent equation 26 Tent map 172 Thorn, Thom, R. 387 Threshold 189 Time distribution 127 127 lag 178 leisure 127 work 127 127 Time-invariant; see see autonomous autonomous Time series 24 Time-variant; see see nonautonomous nonautonomous Tirole, Tirole, J. J. 335 335 Tobin,J. 125 Tobin,J. 118, 118,125 .Tobin effect 125 Tobin effect Trade cycle 4 Trade model for two countries 257 Trajectory 22; see also orbit Trajectory Transitive 177 Transversality condition condition 214-5,426 214-5,426 Transversality

U-shaped 109 Unemployment 80, 80, 331 331 Uniform 35 Unpredictability 27

INDEX 28, 140, 262; see also Unstable 28,140,262; stable Urban 112 dynamic model 112 112 external diseconomies 112 112 pattern formation 4 Urbanization 112 Full 114 no- 114 no-114 partial 114 Utility 112 expected 316 in the OSG model 85, 85, 89, 120, 342 120,342 Uzawa, H. 358 358

van Strien, 5. S. 195 195 Variation of constant formula 256 von Thadden, L. 335 Wage bargain 174 Waugh,F.V.251 Waugh, F.V. 251 Wealth distribution 341 341 Weil, D.N. 320, 325 320,325 Whole 3; 3; see also also structure structure

Yano, M. 315 Yano,M. 315 Yokoo, M. 186, 193, 335 Yokoo,M. 186,193,335 Yorke, J.A. 213 Yorke,J. A. 154, 154,213

Zeira, J. J. 97 97 Zhang, F.Z. Zhang, F.Z. 391 391 Zhang, W.B. 4, 118, 126,219.341, 126,219. 341, Zhang,W.B.4, 367, 388, 358 367,388,358

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