Nonlinear Economic Dynamics

291 Pages·1997·11.387 MB·English

by  Professor Dr. Tönu Puu (auth.)

Nonlinear Economic Dynamics
Springer
Berlin
Heidelberg
New York
Barcelona
Budapest
Hong Kong
London
Milan
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Santa Clara
Singapore
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Tonu Puu
Nonlinear Economic
Dynamics
Fourth, Completely Revised and Enlarged Edition
With 119 Figures
Springer
Professor Dr. Tcinu Puu
Department of Economics
Umea University
S-90187 Umea, Sweden
ISBN -13: 978-3-642-64549-5 e-ISBN -13 :978-3-642-60775-2
DOl: 10.1007/978-3-642-60775-2
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Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Puu, Tonu: Nonlinear economic dynamics / Tonu Puu. - 4., compo rev. and enl. ed. - Berlin;
Heidelberg; New York; Barcelona; Budapest; Hong Kong; London; Milan; Paris; Santa Clara;
Singapore; Tokyo: Springer, 1997
ISBN-13:978-3-642-64549-5 Pp.
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Preface
This book was first pUblished in 1989 as volume 336 in the Springer series
"Lecture Notes in Economics and Mathematical Systems", and it reappeared
in a 2nd edition as a Springer monograph in 1991. After considerable revisions
it appeared in a 3rd edition in 1993.
The origin, still visible in the 3rd edition, was the joint work of the author
with Professor Martin J. Beckmann, documented in two co-authored mono
graphs "Spatial Economics" (North-Holland 1985), and "Spatial Structures"
(Springer-Verlag 1990). Essential dynamics had, however, been almost com
pletely lacking in these works, and the urge to focus the dynamic issues was
great. To fill this particular gap was the aim of the previous editions, and so
the spatial aspect provided core and focus.
In the present edition a substantial quantity of spatial issues have been
removed: All those that were dynamic only in the sense that structures were
characterized which were structurally stable, or robust in a changing world.
The removed material has meanwhile been published as a separate mono
graph under the title "Mathematical Location and Land Use Theory"
(Springer-Verlag 1996).
The space thus vacuated in the present edition has been filled by a new
chapter on the optimum search dynamics oft he monopolist, and by a consid
erably extended study of oligopoly dynamics. The chapters on business cy
cles were revised and extended, and only a very short chapter remains on the
business cycle diffusion in space. The focus thus shifted from the dynamic
aspect of spatial issues to nonlinearity in dynamical processes in economics
in general.
The most important change in the present edition, however, is the addition
of two introductory chapters on mathematical methods, one focusing differ
ential equations, the other focusing iterated maps. Some of the material, es
pecially that concerning perturbation methods, has never been available in
any book of this nature before.
VI Preface
In this way the book now becomes selfc ontained in a sense that it was not
before. It therefore is the hope of the author that the present edition, being
both more homogeneous and more selfc ontained, can serve as a textbook for
economics students of dynamical systems. To the same pedagogical purpose,
the number of illustrations has also been considerably expanded.
To the same end the author is preparing some of the software used in pro
ducing the illustrations for use by the readers on the PC, by making the
programs interactive and providing them with a user interface. They will
hopefully be included in later versions of this book. Meanwile, interested
readers can contact the author at the address tonu. puu@natek. umu. se
to obtain preliminary versions of the software .•
Simulations are essential, because with nonlinear systems, where closed
form solutions are generally not available, even theoretical science becomes
essentially experimental.
The author gratefully acknowledges the fmancial support of his research
by the Swedish Research Council for the Humanities and the Social Sci
ences, a support which definitely has been a necessary condition for this
work.
Umea in January 1997
TonuPuu
Contents
1 Introduction 1
1.1 Dynamics Versus Equilibrium Analysis 1
1.2 Linear Versus Nonlinear Modelling 2
1.3 Perturbation Methods 3
1.4 Structural Stability 4
1.5 Chaos and Fractals 5
1.6 The Choice of Topics Included 6
2 Differential Equations 9
2.1 The Phase Portrait 9
2.2 Linear Systems 16
2.3 Structural Stability 24
2.4 Limit Cycles 28
2.5 The HopfBifurcation 33
2.5 The Saddle-Node Bifurcation 35
2.7 Perturbation Methods: Poincare-Lindstedt 37
2.8 Perturbation Methods: Two-Timing 43
2.9 Forced Oscillators: van der Pol 52
2.10 Forced Oscillators: Duffmg 61
2.11 Chaos 68
2.12 A Short History of Chaos 71
3 Iterated Maps 77
3.1 Introduction 77
3.2 The Logistic Map 78
3.3 The Lyapunov Exponent 87
3.4 Symbolic Dynamics 90
3.5 Sarkovskii's Theorem and the Schwarzian Derivative 94
3.6 The Henon Model 96
VIII Contents
3.7 Lyapunov Exponents in 2D 100
3.8 Fractals and Fractal Dimension 103
3.9 The Mandelbrot Set 108
4 Monopoly 113
4.1 Introduction 113
4.2 The Model 115
4.3 Adaptive Search 118
4.4 Numerical Results 120
4.5 Fixed Points and Cycles 121
4.6 Chaos 125
4.7 A Case with Two Products 126
4.8 Discussion 130
5 Duopoly and Oligopoly 133
5.1 Introduction 133
5.2 The Cournot Model 134
5.3 Stackelberg Equilibria 137
5.4 The Iterative Process 138
5.5 Stability of the Coumot Point 139
5.6 Periodic Points and Chaos 140
5.7 Adaptive Expectations 142
5.8 Adjustments Including Stackelberg Points 145
5.9 Oligopoly with Three Firms 146
5.10 Stackelberg Action Reconsidered 149
5.11 The Iteration with Three Oligopolists 151
5.12 Back to "Duopoly" 152
5.13 Changing the Order of Adjustment 158
6 Business Cycles: Continuous Time 159
6.1 The Multiplier-Accelerator Model 159
6.2 The Original Model 160
6.3 Nonlinear Investment Functions and Limit Cycles 161
6.4 Limit Cycles: Existence 164
6.5 Limit Cycles: Asymptotic Approximation 167
6.6 Limit Cycles: Transients and Stability 172
6.7 The Two-Region Model 177
6.8 The Persistence of Cycles 178
6.9 Perturbation Analysis of the Coupled Model 180
6.10 The Unstable Zero Equilibrium 183
Contents IX
6.11 Other Fixed Points 185
6.12 Properties of Fixed Points 189
6.13 The Arbitrary Phase Angle 190
6.14 Stability of the Coupled Oscillators 192
6.15 The Forced Oscillator 194
6.16 The World Market 194
6.17 The Small Open Economy 196
6.18 Stability of the Forced Oscillator 196
6.19 Catastrophe 198
6.20 Period Doubling and Chaos 199
6.21 Relaxation Cycles 202
6.22 Relaxation: The Autonomous Case 203
6.23 Relaxation: The Forced Case 205
6.24 Three Identical Regions 207
,6.25 On the Existence of Periodic Solutions 209
6.26 Stability of Three Oscillators 214
6.27 Simulations 215
7 Business Cycles: Continuous Space 219
7.1 Introduction 219
7.2 Interregional Trade 220
7.3 The Linear Model 222
7.4 Coordinate Separation 224
7.5 The Square Region 226
7.6 The Circular Region 228
7.7 The Spherical Region 229
7.8 The Nonlinear Spatial Model 232
7.9 Dispersive Waves 234
7.10 Standing Waves 236
8 Business Cycles: Discrete Time 239
8.1 Introduction 239
8.2 Investments 240
8.3 Consumption 242
8.4 The Cubic Iterative Map 243
8.5 Fixed Points, Cycles, and Chaos 244
8.6 Formal Analysis of Chaotic Dynamics 251
8.7 Coordinate Transformation 251
8.8 The Three Requisites of Chaos 252
8.9 Symbolic Dynamics 253
X Contents
8.10 Brownian Random Walk 254
8.11 Digression on Order and Disorder 258
8.12 The General Model 259
8.13 Relaxation Cycles 260
8.14 The Slow Feed Back 264
8.15 The Autonomous Term: Changes of Fixed Points 264
8.16 The Autonomous Term: Response of the Chaotic Process 266
8.17 Lyapunov Exponents and Fractal Dimensions 269
8.18 Non-Relaxation Cycles 272
8.19 Conclusion 274
References 277
Index 281