Nonlinear Economic Dynamics

231 Pages·1993·4.435 MB·English

by  Professor Dr. Tönu Puu (auth.)

T6nuPuu
Nonlinear Economic
Dynamics
Third, Revised and Enlarged Edition
With 93 Figures
Springer-Verlag
Berlin Heidelberg New Yark
London Paris Tokyo
Hong Kong Barcelona
Budapest
Professor Dr. Tonu Puu
Department of Economics
Umea University
S-90187 Umea, Sweden
ISBN-13: 978-3-642-97452-6 e-ISBN-13: 978-3-642-97450-2
DOl: 10.1007/978-3-642-97450-2
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© Springer-Verlag Berlin· Heidelberg 1989, 1991, 1993
Softcover reprint of the hardcover 3rd edition 1993
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PREFACE
The present study deals with nonlinear economic dynamics, with which the author has
been concerned the last years. It grew out from the joint work by Professor Martin
Beckmann and the present author on nonlinear statics in spatial economics, Beckmann
and Pull, "Spatial Economics" (N orth-Holland 1985), later followed by its companion,
Beckmann and Puu "Spatial Structures" (Springer-Verlag 1990). The first mono
graph mentioned contains sections on price waves and business cycles, but in a linear
format. The rest is static theory.
The author has finally come to the conviction that linear dynamic modelling has very
little to yield. This is due to the poor set of alternatives -decay or explosion of motion
-pertinent to linear models. Therefore, the present work centres on non-linearity.
Another distinction is that only purely causal models are dealt with, as those formatted
as inter-temporal equilibria hardly belong to the more restricted field of dynamics.
The spatial origin is visible in the choice of models. Chapters 1 and 2 summarize the
work by the author on the structural stability of continuous spatial market eqUilibrium
models. Chapter 3 deals with a re-formulation oft he ingenious population growth and
diffusion model invented by the young Hotelling in 1921. Chapter 4 is a detailed
digression on business cycle models in a continuous spatial format with inter-regional
trade.
However, one of the lessons from modem dynamics is that it is hard to get ahead with
partial differential equations. What can be achived by approximations in terms of
ordinary differential equations in finite dimension and recurrence maps on Poincare
sections has been demonstrated by Lorenz. Accordingly we first discretize the business
cycle model with respect to space in Chapter 5 and with respect to time in Chapter 6.
The concluding Chapter 7 deals with Coumot duopoly.
VI
The tools of analysis are basically classical perturbation methods originating with
Poincare. Occasionally, more exotic beasts from the mathematical zoo are encountered,
such as catastrophe, or detenninistic chaos.
It is worthwhile noting that the chaotic regime, along with quasi-periodicity and fre
quency locking, turns up in the most traditional multiplier-accelerator models of the
business cycle, provided there is a non-linear investment function, and inter-regional
trade. Thus, there is no need to stretch economic principles to fit the most popular
chaotic models. Chaos is inherent in the existent body of economic theory, and the
mathematics needed dates back to 1945, thus being much prior to Lorenz, ROssler,
Shaw, and the discrete logistic mapping.
The present monograph was first published in 'Lecture Notes in Economics and
Mathematical Systems 336" (Springer-Verlag 1989) under the title 'Nonlinear
Economic Dynamics'; and upon reprinting (Springer-Verlag 1991) it was revised and
somewhat extended. The present version is substantially enlarged, including much more
material particularly on the topic of chaotic systems. The art-work has been extended
too, so that the present volume includes more than 90 illustrations.
The research work documented here would have been impossible without the generous
financial support over many years by 'The Swedish Council for Research in the
Humanities and the Social Sciences':
Umed in July 1992.
TonuPuu
CONTENTS
INTRODUCTION - NONLINEAR ECONOMIC DYNAMICS ........... .
I DYNAMICS VERSUS EQUILIBRIUM ANALYSIS .................. .
2 LINEAR VERSUS NONLINEAR MODELLING ....................... 2
3 THE TOOLS OF ANALYSIS ........................................................ 4
3.1 Perturbation Methods ............................................................. 4
3.2 Structural Stability and Catastrophe ...................................... 5
3.3 Chaos and Fractals .................................................................. 6
4 THE CHOICE OF MODELS ........................... .............................. 8
CHAPTER 1 - SPATIAL P ATTERN FORMATION .......................... 10
I SCIENTIFIC EXPLANATIONS .................................................... 10
1.1 Spatial Patterns ....... ..... ..... ..... ..... ............................................. 10
1.2 Types of Scientific Explanation ................................... ......... II
1.3 Teleological Explanation as Shorthand for Causal............... II
1.4 The Case of Minimal Action .. ............................................... 12
1.5 Hexagonal Cell Formation ...................................................... 13
2 OPTIMAL PATTERNS ................................................................. 14
2.1 Tessellations ............................................................................ 14
2.2 The Isoperimetric Problem ..................................................... 14
2.3 Average Distance .................................................................... 15
3 STRUCTURALLY STABLE PATTERNS ................................... 16
3.1 Dangers of Optimality ............................................................ 16
3.2 Structural Stability of Cell Aggregates .................................. 17
3.3 Structural Stability of Flows .................................................. 18
3.3.1 The Flow Model .............................................................. 18
3.3.2 The Structure of Flow Portraits ..................................... 19
3.3.3 Perturbations ................................................................... 20
3.3.4 Topological Equivalence ................................................. 21
VIII
3.3.5 Structural Stability .......................................................... 21
3.3.6 The Character of Stable Flows ....................................... 22
3.3.7 Economic Interpretation ................................................. 23
3.4 Transitions Between Stable Patterns ...................................... 24
4 CONCLUSION ............................................................................... 30
APPENDIX ........................................................................................ 31
CHAPTER 2 - THE GENESIS OF ECONOMIC CENTRES 32
lONE DIMENSION ......... ................................................................ 33
2 TWO DIMENSIONS: CIRCULAR MARKETS ............................ 36
3 TWO DIMENSIONS: TRIANGLES, SQUARES, HEXAGONS .. 38
4 CHANGING POPULATION DENSITy....................................... 40
5 CONCLUSION 43
CHAPTER 3 - POPULATION DYNAMICS ........................................ 44
I THE ORIGINAL HOTEL LING MODEL .................................... 44
1.1 Stationary Solutions ................................................................ 46
1.2 Stability ................... ........................................................... ..... 47
1.3 Discrete Case ........................................................................... 49
2 GROWTH ........................................................................................ 50
2.1 Production ............................................................................... 51
2.2 Pure Growth: Stationary Solutions ......................................... 52
2.3 Pure Growth: Stability ............................................................ 53
3 DIFFUSION .......................................................................... .......... 54
4 GROWTH AND DIFFUSION ....................................................... 56
4.1 Stationary Solutions in One Dimension ................................. 57
4.2 Amplitude and Period ............................................................ 58
4.3 Stability ................................................................................... 60
4.4 Dynamics ................................................................................. 63
5 STRUCTURAL STABILITY ......................................................... 64
5.1 Stabilizing the Original Hotelling Model.............................. 66
5.2 Stabilizing the Model with Production .................................. 71
6 CONCLUSION ............................................................................... 73
7 APPENDIX: MODEL WITH ENDOGENOUS CAPITAL .......... 74
7.1 Dynamics of Capital and Labour .......................................... 74
7.2 Bifurcations: Geometric Aspects ............................................ 78
7.3 Bifurcations: Computational Aspects ..................................... 81
7.4 Diffusion ................................................................................. 85
IX
CHAPTER 4 - BUSINESS CYCLES: CONTINUOUS TIME 86
I THE MULTIPLIER-ACCELERATOR MODEL ........................ 86
1.1 The Original Model ................................................................ 87
1.2 Nonlinear Investment Func.tions and Limit Cycles .............. 89
1.2.1 Limit Cycles: Existence ................................................... 93
1.2.2 Limit Cycles: Asymptotic Approximation .. ....... ... ...... ... 97
1.2.3 Limit Cycles: Transients and Stability......... ... ... ... ... ... ... 102
2 SPATIAL MODELS ....................................... ............. ... ... ... ... .... ... 106
2.1 Interregional Trade ......... ................. .... .... .... ..... .... .......... ... ..... 107
2.2 The Linear Model .. ..... ....................... ... ......... .... .... .... ... ... ... .... 109
2.3 Coordinate Separation ............................................................. III
2.3.1 Example: Square Region ................................................. 113
2.3.2 Example: Circular Region ............................................... 118
2.3.3 Example: Spherical Region ............................................. 120
2.4 Nonlinear Spatial Model ......................................................... 124
2.4.1 Example: Dispersive Waves ............................................. 125
2.4.2 Example: Standing Waves ................................................ 127
CHAPTER 5 - BUSINESS CYCLES: DISCRETE SPACE ................. 129
I THE TWO-REGION MODEL 129
1.1 The Persistence of Cycles 130
1.2 Perturbation Analysis ............................................................. 132
1.3 The Unstable Zero Equilibrium ............................................ 135
1.4 Other Fixed Points ................................. .... .... ..... ... .... ... ... ... .... 136
1.5 Properties of Fixed Points .. ............. ... ........ .... .... .... ... ... ... ... ... 140
1.6 The Arbitrary Phase Angle .................................................... 142
1.7 Stability ................................................................................... 143
2 THE FORCED OSCILLATOR ...................................................... 145
2.1 The World Market .................................................................. 146
2.2 The Small Open Economy ...................................................... 147
2.3 Stability ................................................................................... 148
2.4 Catastrophe . .... .... ........................... .... ........... .... .... ...... ... ... ....... 150
2.5 Quasiperiodic Motion ........ ..... .... .... .... .... .... ... .... .... ... ..... ... ... ... 151
3 RELAXATION CYCLES .............................................................. 153
3.1 Relaxation Oscillations: The Autonomous Model................. 155
3.2 Relaxation Oscillations: The Forced System ......................... 157
4 THREE IDENTICAL REGIONS .................................................. 159
4.1 On the Existence of Periodic Solutions ................................. 162
4.2 Stability ................................................................................... 167
4.3 Quasiperiodicity and Chaos .................................................... 168
x
CHAPTER 6 - BUSINESS CYCLES: DISCRETE TIME 170
1 FIRST DISCRETE MODEL .......................................................... 170
1.1 Investments .............................................................................. 170
1.2 Consumption ........................................................................... 171
2 THE CUBIC ITERATIVE MAP ................................................... 173
2.1 Fixed Points, Cycles, and Chaos ........................................... 173
2.2 Formal Analysis of Chaotic Dynamics .................................. 181
2.2.1 Co-ordinate Transformation .......................................... 181
2.2.2 The Three Requisites of Chaos ...................................... 182
2.3 Symbolic Dynamics ................................................................. 183
3 BROWNIAN RANDOM WALK ................................................... 184
4 DIGRESSION ON ORDER AND DISORDER ............................ 187
5 THE GENERAL MODEL ............................................................. 188
5.1 Relaxation Cycles ................................................................... 190
5.2 Other Cycles ............................................................................ 195
5.3 The Slow Feed Back ............................................................... 196
5.3.1 Changes of the Fixed Points .......................................... 197
5.3.2 Response of the Chaotic Process ................................... 198
6 CONCLUSION ............................................................................... 201
7 APPENDIX: DIGRESSION ON THE RATIONALE OF THE
CUBIC ................................................................................................ 203
CHAPTER 7 - COURNOT DUOPOLY 205
1 DUOPOLY ...................................................................................... 205
2 THE COURNOT MODEL ............................................................ 206
3 ADJUSTMENT BY TAKING TURNS ........................................ 210
4 SIMULTANEOUS ADJUSTMENT .............................................. 213
5 CONCLUSION ............................................................................... 217
REFERENCES ..................... ... ..... ..... .... .... ..... ..... .... ..... ........................... 218
INTRODUCTION
NONLINEAR ECONOMIC DYNAMICS
1 DYNAMICS VERSUS EQUILIBRIUM ANALYSIS
Dynamic analysis in economics is as old as economics itself. A glance at
the subject index in Schumpeter (1954) is sufficient to convince you about
this. Even dynamic mathematical models are fairly old. The cobweb model
of price adjustments for instance dates back to 1887.
Throughout the history of economics there has been a competition
between the dynamic and the equilibrium outlooks. As an alternative to
a truly causal or recursive dynamics there is the concept of an equilibrium
balance of forces.
In general the equilibrated forces are results of optimizing behaviour, and
therefore the epistemological polarity - causal versus teleological - is
involved. (Moreover, expectations of the actions of others as well as the
optimality of one's own belong to the concept of equilibrium.)
Certain controversies in the history of economics reflect this polarity of
different philosophical outlooks. One example is provided by those who
objected the Marshallian concept of market equilibrium on the grounds
that price could not be determined both by cost and utility at once. These
objections need not be ascribed to mathematical ignorance. Another
example is the more recent discussion on recursive versus interdependent
systems in econometrics.