Discrete Dynamical Systems, Bifurcations and Chaos in Economics (eBook)

Cover 1
Contents 8
Preface 6
Difference equations in economics 8
Difference equations and economic analysis 12
Overview 18
Scalar linear difference equations 24
Linear first-order difference equations 26
Some concepts 32
Stabilities 38
Stabilities of nonhyperbolic equilibrium points 53
On dissipative maps 60
Linear difference equations of higher order 65
Equations with constant coefficients 71
Limiting behavior 78
One-dimensional dynamical economic systems 90
A model of inflation and unemployment 91
The one-sector growth (OSG model) 94
The general OSG model 99
The overlapping-generations (OLG) model 104
Persistence of inequality and development 108
Growth with creative destruction 111
Economic evolution with human capital 117
Urbanization with human capital externalities 123
The OSG model with money 129
The OSG model with labor supply 137
Time-dependent solutions of scalar systems 146
Periodic orbits 166
Economic bifurcations and chaos 196
Period-doubling bifurcations 9
Aperiodic orbits 9
Some types of bifurcations 174
Liapunov numbers 181
Chaos 188
Business cycles with knowledge spillovers 197
A cobweb model with adaptive adjustment 204
Inventory model with rational expectations 206
Economic growth with pollution 213
The Solow and Schumpeter growth oscillations 216
Money, growth and fluctuations 224
Population and economic growth 230
Higher dimensional difference equations 238
Phase space analysis of planar linear systems 239
Autonomous linear difference equations 251
Nonautonomous linear difference equations 259
Stabilities 272
Liapunov's direct method 281
Linearization of difference equations 287
Conjugacy and center manifolds 292
The Henon map and bifurcations 300
The Neimark-Sacker (Hopf) bifurcations 307
The Liapunov numbers and chaos 312
Higher dimensional economic dynamics 316
An exchange rate model 317
A two-sector OLG model 321
Growth with government spending 326
Growth with fertility and old age support 331
Growth with different types of economies 338
Unemployment, inflation and chaos 342
Business cycles with money and capital 346
The OSG model with heterogeneous households 352
Path-dependent evolution with education 369
Appendix 385
Proving proposition 7.81 385
Proving proposition 7.9.1 390
Epilogue 396
Appendix 402
Matrix theory 402
Systems of linear equations 408
Metric spaces 409
The implicit function theorem 412
The Taylor expansion and linearization 419
Concave and quasiconcave functions 421
Unconstrained maximization 426
Constrained maximization 429
Dynamical optimization 433
Bibliography 438
Index 450