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* 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*)
> 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 (ii) f'(x) = 0° and f { ')> °> 0, tthen (ii) IfIf/"(**)= " unstable.
(iii) f (x )) = o0 and j (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 / " ( / ) 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, )
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, 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 l, A 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 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\ 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 1 or by by If Ipi
oscillating oscillating ifif pp 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 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 c0, (-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 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\(c) ( ( 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
>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 , o0. (i + + n)k n)k
(3.3.4)
When is also When k isisapproaching approaching zero, zero, 5'y (= (= /f(k) ( £ ) + &k) To 0. To prove this, use ds I dy ()
; t
92
ECONOMIC SYSTEMS SYSTEMS 3. ONE-DIMENSIONAL DYNAMICAL ECONOMIC
where we we use use s'(y) s'(j)) > 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> 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
