Nonlinear Economic Dynamics

This book deals with the effects of nonlinearity on dynamic economic models. Topics from both micro- and macroeconomics are dealt with. The text is illustrated by extensive illustrations.

Klappentext

This book is a general inquiry into the effects of nonlinearity on dynamic economic models, focusing phenomena such as limit cycles, quasiperiodicity, bifurcations, and chaos. Topics from both micro- and macroeconomics are dealt with, for instance the search dynamics of a monopolist facing multiple optima, the dynamics of Cournot oligopoly, and, above all, business cycle theory, including mostly original research results. Models in discrete time as well as in continous time are focused, and this completely rewritten fourth edition is provided with an extensive mathematical introduction, dealing with both differential equations, and with discrete maps, so making the book selfcontained, and even explaining tools such as perturbation methods or the computation of Lyapunov exponents.

Inhalt
1 Introduction.- 1.1 Dynamics Versus Equilibrium Analysis.- 1.2 Linear Versus Nonlinear Modelling.- 1.3 Perturbation Methods.- 1.4 Structural Stability.- 1.5 Chaos and Fractals.- 1.6 The Choice of Topics Included.- 2 Differential Equations.- 2.1 The Phase Portrait.- 2.2 Linear Systems.- 2.3 Structural Stability.- 2.4 Limit Cycles.- 2.5 The Hopf Bifurcation.- 2.5 The Saddle-Node Bifurcation.- 2.7 Perturbation Methods: Poincaré-Lindstedt.- 2.8 Perturbation Methods: Two-Timing.- 2.9 Forced Oscillators: van der Pol.- 2.10 Forced Oscillators: Duffing.- 2.11 Chaos.- 2.12 A Short History of Chaos.- 3 Iterated Maps.- 3.1 Introduction.- 3.2 The Logistic Map.- 3.3 The Lyapunov Exponent.- 3.4 Symbolic Dynamics.- 3.5 Sarkovskii's Theorem and the Schwarzian Derivative.- 3.6 The Hénon Model.- 3.7 Lyapunov Exponents in 2D.- 3.8 Fractals and Fractal Dimension.- 3.9 The Mandelbrot Set.- 4 Monopoly.- 4.1 Introduction.- 4.2 The Model.- 4.3 Adaptive Search.- 4.4 Numerical Results.- 4.5 Fixed Points and Cycles.- 4.6 Chaos.- 4.7 A Case with Two Products.- 4.8 Discussion.- 5 Duopoly and Oligopoly.- 5.1 Introduction.- 5.2 The Cournot Model.- 5.3 Stackelberg Equilibria.- 5.4 The Iterative Process.- 5.5 Stability of the Cournot Point.- 5.6 Periodic Points and Chaos.- 5.7 Adaptive Expectations.- 5.8 Adjustments Including Stackelberg Points.- 5.9 Oligopoly with Three Firms.- 5.10 Stackelberg Action Reconsidered.- 5.11 The Iteration with Three Oligopolists.- 5.12 Back to Duopoly.- 5.13 Changing the Order of Adjustment.- 6 Business Cycles: Continuous Time.- 6.1 The Multiplier-Accelerator Model.- 6.2 The Original Model.- 6.3 Nonlinear Investment Functions and Limit Cycles.- 6.4 Limit Cycles: Existence.- 6.5 Limit Cycles: Asymptotic Approximation.- 6.6 Limit Cycles: Transients andStability.- 6.7 The Two-Region Model.- 6.8 The Persistence of Cycles.- 6.9 Perturbation Analysis of the Coupled Model.- 6.10 The Unstable Zero Equilibrium.- 6.11 Other Fixed Points.- 6.12 Properties of Fixed Points.- 6.13 The Arbitrary Phase Angle.- 6.14 Stability of the Coupled Oscillators.- 6.15 The Forced Oscillator.- 6.16 The World Market.- 6.17 The Small Open Economy.- 6.18 Stability of the Forced Oscillator.- 6.19 Catastrophe.- 6.20 Period Doubling and Chaos.- 6.21 Relaxation Cycles.- 6.22 Relaxation: The Autonomous Case.- 6.23 Relaxation: The Forced Case.- 6.24 Three Identical Regions.- 6.25 On the Existence of Periodic Solutions.- 6.26 Stability of Three Oscillators.- 6.27 Simulations.- 7 Business Cycles: Continuous Space.- 7.1 Introduction.- 7.2 Interregional Trade.- 7.3 The Linear Model.- 7.4 Coordinate Separation.- 7.5 The Square Region.- 7.6 The Circular Region.- 7.7 The Spherical Region.- 7.8 The Nonlinear Spatial Model.- 7.9 Dispersive Waves.- 7.10 Standing Waves.- 8 Business Cycles: Discrete Time.- 8.1 Introduction.- 8.2 Investments.- 8.3 Consumption.- 8.4 The Cubic Iterative Map.- 8.5 Fixed Points, Cycles, and Chaos.- 8.6 Formal Analysis of Chaotic Dynamics.- 8.7 Coordinate Transformation.- 8.8 The Three Requisites of Chaos.- 8.9 Symbolic Dynamics.- 8.10 Brownian Random Walk.- 8.11 Digression on Order and Disorder.- 8.12 The General Model.- 8.13 Relaxation Cycles.- 8.14 The Slow Feed Back.- 8.15 The Autonomous Term: Changes of Fixed Points.- 8.16 The Autonomous Term: Response of the Chaotic Process.- 8.17 Lyapunov Exponents and Fractal Dimensions.- 8.18 Non-Relaxation Cycles.- 8.19 Conclusion.- References.