Nonlinear Dynamical Systems in Economics PDF
Preview Nonlinear Dynamical Systems in Economics
^ SpringerWienNewYork CISM COURSES AND LECTURES Series Editors: The Rectors Giulio Maier - Milan Jean Salengon - Palaiseau Wilhelm Schneider - Wien The Secretary General Bemhard Schrefler - Padua Executive Editor Carlo Tasso - Udine The series presents lecture notes, monographs, edited works and proceedings in the field of Mechanics, Engineering, Computer Science and Applied Mathematics. Purpose of the series is to make known in the international scientific and technical community results obtained in some of the activities organized by CISM, the International Centre for Mechanical Sciences. INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES COURSES AND LECTURES - No. 476 NONLINEAR DYNAMICAL SYSTEMS IN ECONOMICS EDITED BY MARJI LINES UNIVERSITY OF UDINE, ITALY SpringerWien NewYork This volume contains 90 illustrations This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. © 2005 by CISM, Udine Printed in Italy SPIN 11494331 In order to make this volume available as economically and as rapidly as possible the authors' typescripts have been reproduced in their original forms. This method unfortunately has its typographical limitations but it is hoped that they in no way distract the reader. ISBN-10 3-211-26177-X SpringerWienNewYork ISBN-13 978-3-211-26177-4 SpringerWienNewYork PREFACE Many problems in theoretical economics are mathematically formalized as dynam ical systems of difference and differential equations. In recent years a truly open approach to studying the dynamical behavior of these models has begun to make its way into the mainstream. That is, economists formulate their hypotheses and study the dynamics of the resulting models rather than formulating the dynamics and studying hypotheses that could lead to models with such dynamics. This is a great progress over using linear models, or using nonlinear models with a linear approach, or even squeezing economic models into well-studied nonlinear systems from other fields. There are today a number of economic journals open to publishing this type of work and some of these have become important. There are several societies which have annual meetings on the subject and participation at these has been growing at a good rate. And of course there are methods and techniques avail able to a more general audience, as well as a greater availability of software for numerical and graphical analysis that makes this type of research even more excit ing. The lecturers for the Advanced School on Nonlinear Dynamical Systems in Economics, who represent a wide selection of the research areas to which the the ory has been applied, agree on the importance of simulations and computer-based analysis. The School emphasized computer applications of models and methods, and all contributors ran computer lab sessions. The exigencies of space left us no room to include the related exercises and software, but you can get a taste of those (and access to a wealth of other useful material) by referring to contributors home pages. The volume is structured as follows. The first three chapters are introductory (though not necessarily elementary). The first provides a quick introduction to nonlinear analysis: a short review of what is useful from linear systems theory in the analysis of nonlinear systems through first-order approximations; the essential theorems useful for local analysis; definitions and terminology for stability analy- sis; a discussion of limit sets and local bifurcation theory. The second chapter is a discussion of chaos and complexity at an intermediate level of difficulty. Typi cal examples of systems with chaotic trajectories are provided in order to discuss deeper issues including chaotic attractors as a form of global stability, random versus deterministic chaotic series, predictability of chaotic systems, statistical predictability of chaotic systems and financial and economic implications of de terministic chaos. The section on complexity focuses on the cellular automata approach, considering complexity classes, predictability and agent-based modeling in economics. The third chapter is an introduction to a relatively new line of research in economics, the ergodic approach, which investigates the probabilistic properties of dynamical systems. The basic concepts of elementary measure the ory are used to understand the dynamics of nonlinear models. Concepts such as invariant, ergodic, absolutely continuous and natural measures are explained with simple examples. The issue of deterministic chaos and randomness is discussed from the point of view of predictability, by means of the notion of metric entropy. These three chapters, coming from, quite different approaches, give a broad intro duction to definitions, concepts and methods that are useful for the more applied chapters that follow. The final four chapters are applications of local and global bifurcation theory to models coming from different approaches and fields in economics. In Chapter 4 the local approximation is used to understand the dynamics in two versions of one of the models currently dominating macroeconomics, the Overlapping Genera tions (OLG) model. From the basic 1-dimensional Diamond model, with standard choices of functions, hypotheses are altered to develop two other models, each il lustrating problems that arise once the basic hypotheses are abandoned. A second type of OLG model is studied, in 2 and 3 dimensions, for which the Neimark- Sacker bifurcation is typical and invariant curves are common limit sets. Chapter 5 focuses on some very interesting work in modeling the dynamics of financial markets with heterogeneous agents, an approach receiving much attention in the field. The first part employs the cobweb model with rational versus naive agents to demonstrate a rational route to randomness as well as the existence of a ho- moclinic orbit. The second part develops an asset pricing model in which agents switch between different forecasting or trading strategies, using an evolutionary fitness measure. Prices and beliefs co-evolve over time, leading to instability and complicated price fluctuations. Chapter 6 is dedicated to complex dynamics in models from oligopoly theory, which is one of the fields that pioneered in the ap plication of nonlinear dynamics. A model of duopolists with nonlinear reaction curves gives rise to a period-doubling scenario to chaos and the coexistence of cycles and complicated basins of attraction. The model is extended to include adaptive expectations, and stability is lost through the Neimark-Sacker bifurca tion giving rise to a very complicated structure of periodic ArnoVd tongues ob servable in the two-parameter bifurcation diagram. The critical line approach is used to define the absorbing areas. In Chapter 7 an excellent review of defini tions and properties concerning noninvertible maps is followed by a description of the method of critical lines and curves in determining the trapping region, with examples of global bifurcations causing nonconnected basins of attraction. These methods are applied to a Cournot duopoly game with best reply, naive expectations and adaptive behavior, and to a duopoly game with gradient dynamics. Finally^ the related phenomena of chaos synchronization and riddled basins are studied in a dynamic brand competition model with market shares and marketing effort. The organization of a learning experience such as the Advanced School is amazingly complex in itself and I wish to thank Prof Manuel Velarde, current Rector of CISM, for his support at all stages, the CISM staff for their patience and competence, my fellow lecturers for their efforts in preparing presentations, lab sessions, and the chapters that follow, and of course, the students, who gave themselves up for an entire week to the joys of nonlinearity. Marji Lines CONTENTS Introductory notes on the dynamics of linear and linearized systems by M. Lines and A. Medio 1 Complex and chaotic dynamics in economics by D. Foley 27 Ergodic approach to nonlinear dynamics by A, Medio 67 Local bifurcation theory apphed to OLG models by M. Lines 103 Heterogeneous agent models: two simple examples by C. Hommes 131 Complex oligopoly dynamics by T. Puu 165 Coexisting attractors and complex basins in discrete-time economic models by G. Bischi and F. Lamantia 187 Introductory Notes on the Dynamics of Linear and Linearized Systems Marji Lines and Alfredo Medio Department of Statistics, University of Udine, Udine, Italy Abstract In the following we provide terminology and concepts which are central to understanding the dynamical behavior of nonlinear systems. The first four sec tions are a necessarily very brief introduction to the dynamics of linear systems, in which we concentrate on those aspects most useful for acquiring a sense of the basic behaviours characterising systems of differential and difference equations. The last four sections introduce basic notions of stability, the linear approximation and the Hartman-Grobman Theorem, the use of the Centre Manifold Theorem, local bifurcation theory.* 1 Linear systems in continuous time In this section we discuss the form of the solutions to the general system of linear differ ential equations x = Ax X G E^ (1.1) where A is a m x m matrix of constants, also called the coefficient matrix. An obvious solution to equation (1.1) is x{t) = 0, called the equilibrium solution because if x = 0, i: = 0 as well. That is, a system starting at equilibrium stays there forever. Notice that if A is nonsingular, x = 0 is the only equilibrium for linear systems like (1.1). Nontrivial solutions will be of the form x{t) = e^'u (1.2) where ix is a vector of real or complex constants and A real or complex constants. Differ entiating (1.2) with respect to time, and substituting into (1.1), we obtain Xe^^u = Ae^^u which, for e^^ ^0, implies (A-A/)ix-0 (1.3) where 0 is an m-dimensional null vector. A nonzero vector u satisfying (1.3) is called an eigenvector of matrix A associated with the eigenvalue A. Equation (1.3) has a nontrivial solution w 7^ 0 if and only if det{A - A7) - 0 (1.4) *For suggestions on further reading and an extended bibliography please see Medio and Lines, Nonlinear Dynamics: A Primer^ Cambridge: Cambridge University Press, 2001.
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