Nonlinear Dynamics And Economics

This 1997 book presents developments in nonlinear economic dynamics along with related research from other fields, including mathematics, statistics, biology, and physics.

1. 1 Introduction In economics, one often observes time series that exhibit different patterns of qualitative behavior, both regular and irregular, symmetric and asymmetric. There exist two different perspectives to explain this kind of behavior within the framework of a dynamical model. The traditional belief is that the time evolution of the series can be explained by a linear dynamic model that is exogenously disturbed by a stochastic process. In that case, the observed irregular behavior is explained by the influence of external random shocks which do not necessarily have an economic reason. A more recent theory has evolved in economics that attributes the patterns of change in economic time series to an underlying nonlinear structure, which means that fluctua tions can as well be caused endogenously by the influence of market forces, preference relations, or technological progress. One of the main reasons why nonlinear dynamic models are so interesting to economists is that they are able to produce a great variety of possible dynamic outcomes - from regular predictable behavior to the most complex irregular behavior - rich enough to meet the economists' objectives of modeling. The traditional linear models can only capture a limited number of possi ble dynamic phenomena, which are basically convergence to an equilibrium point, steady oscillations, and unbounded divergence. In any case, for a lin ear system one can write down exactly the solutions to a set of differential or difference equations and classify them.

The plan to publish the present book arose while I was preparing a joint work with Gunter Gabisch (Gabisch, G. /Lorenz, H. -W. : Business Cycle Theory. Berlin-Heidel berg-New York: Springer). It turned out that a lot of interesting material could only be sketched in a business cycle text, either because the relevance for business cycle theory was not evident or because the material required an interest in dynamical economics which laid beyond the scope of a survey text for advanced undergraduates. While much of the material enclosed in this book can be found in condensed and sometimes more or less identical form in that business cycle text, the present monograph attempts to present nonlinear dynamical economics in a broader context with economic examples from other fields than business cycle theory. It is a pleasure for me to acknowledge the critical comments, extremely detailed remarks, or suggestions by many friends and colleagues. The responses to earlier versions of the manuscript by W. A. Barnett, M. Boldrin, W. A. Brock, C. Chiarella, C. Dale, G. Feichtinger, P. Flaschel, D. K. Foley, R. M. Goodwin, D. Kelsey, M. Lines, A. Medio, L. Montrucchio, P. Read, C. Sayers, A. Schmutzler, H. Schnabl, G. Silverberg, H. -\'\!. Sinn, J. Sterman, and R. Tscherning not only encouraged me to publish the book in its present form but helped to remove numerous errors (not only typographic ones) and conceptnal misunderstandings and flaws. Particular thanks go to G.

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.

Advances in physics, computers, and mathematics have made it possible to illustrate an astonishing array of potential behavior that can occur when nonlinear interactions are present. As Prigogine explains from a physicist's perspective, the fundamental role of instability and bounded rationality provide more precise understanding for evolution and changes. This volume considers these developments from various fields in the context of economic science. The work starts with a general non-mathematical discussion, introducing the major themes--nonlinearity, dynamical systems, and evolution in economic processes. The work continues with nonlinear analysis of macroeconomic growth and fluctuations. It describes analyses of economic adaptation, learning, and self-organization. The volume also scrutinizes a specific market--equities using nonlinear analysis, controlled experiments, and statistical inference when nonlinearity plays an essential role in data generation. The volume closes with an historical reflection by Richard Goodwin and a roundtable discussion on basic issues and new challenges in nonlinear economic dynamics.

Brock, Hsieh, and LeBaron show how the principles of chaos theory can be applied to such areas of economics and finance as the changing structure of stock returns and nonlinearity in foreign exchange.

Complex dynamics constitute a growing and increasingly important area as they offer a strong potential to explain and formalize natural, physical, financial and economic phenomena. This book pursues the ambitious goal to bring together an extensive body of knowledge regarding complex dynamics from various academic disciplines. Beyond its focus on economics and finance, including for instance the evolution of macroeconomic growth models towards nonlinear structures as well as signal processing applications to stock markets, fundamental parts of the book are devoted to the use of nonlinear dynamics in mathematics, statistics, signal theory and processing. Numerous examples and applications, almost 700 illustrations and numerical simulations based on the use of Matlab make the book an essential reference for researchers and students from many different disciplines who are interested in the nonlinear field. An appendix recapitulates the basic mathematical concepts required to use the book.