# Differential Equations, Stability and Chaos in Dynamic Economics (Advanced Textbooks in Economics)

To answer this question usually requires the arbitrary specification of a dynamic principle, usually expressed in the form of a differential equation or dynamical system. The most obvious example here is the Samuelsonian adjustment process of standard general equilibrium theory. If all goods in the economy are weak gross substitutes, the process of equation 1 demonstrates global stability. That is, the economy will approach an equilibrium from any initial conditions. In fact Sonnenschein and others have shown that any reasonable excess demand function 2 can be thought of as being derived from the maximisation of standard utility and profit functions.

As will be shown below, this implies a wide range of alarming dynamic behaviour for the economy. More recent equilibrium enthusiasts have sought to avoid this kind of problem, and they have discovered an important aid in achieving this objective in the form of the Nash equilibrium concept. The textbook version of the Cournot duopoly model usually depicts two reaction curves intersecting at the unique Nash equilibrium. In addition a step-like adjustment path to equilibrium is depicted see figure 1 which the two firms are supposed to follow should they be out of equilibrium.

Not if we are to take the Nash concept seriously. Given time series data, the objective is t o locate a long-run equilibrium relationship in those data. The possibility that there may be no such relationship, or that it may be too complicated for easy mathematical representation] is usually taken as a counsel of despair. The textbook Cournot model. It could be argued that dynamic theory is the appropriate basis for time series econometrics. The former offers explanations in terms of intention, while the latter focuses on temporal cause.

The two are not inconsistent and might, for example, be thought to come together in models based on intertemporal optimisation or dynamic programming. The attribution of intentions to individuals is of course quite plausible, but economists have on occasions carried their enthusiasm for equilibrium to the point that they have found it necessary to attribute intentions to the entire economy.

This strange notion can be perceived, for example, in saddlepoint RE models, which are discussed below. From the point of view of applications, most of the central ideas can be found in the work of Duffing, van der Pol, Rayleigh and above all, PoincarC.

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Nonetheless, developments in the theory of dynamical systems, and in particular the theory of chaos, are turning the balance of power in economics back towards the dynamic approach. This is well illustrated by the books reviewed here, though, it must be admitted, they do occasionally echo to the undignified sound of economists jumping on to passing bandwagons. It focuses on linear systems until the final chapter, which, among other things, introduces the notion of chaos.

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Applications are to the regulation of firms, investment theory, macroeconomic policy, capital theory and finance, and are of varying degrees of interest. Barnett et al.

It includes papers, of varying quality, on sunspots, bubbles including hyperinflation , chaos and nonlinear econometrics. The last section is somewhat out of place, since it contains papers which though interesting in themselves, do not, in any significant way, pick up the theoretical issues raised in the other sections.

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## MAT (Differential Equations: Dynamics and Chaos) - Course Information

Velupillai is a collection of essays in honour of Richard Goodwin, who must surely be counted the doyen of dynamicists. There is a biographical section on Goodwin, including a piece by Velupillai, in a scholarly style all too rare these days. This predator-prey type of model has fascinated the political left see, for example, Desai, , though this fascination is probably more mathematical than empirical since, formally speaking, it is the capitalists who correspond to the prey and the workers to the predators.

There is also a section on multisectoral macrodynamics, which deals with linear models and is therefore differentiated sharply from most of the papers in the Barnett et al. Chiareffaand Puu are Springer-Verlag lecture notes.

The former is slightly more elementary, starting with linear dynamics and including discussions of the Goodwin model, cobweb models, chaos and perfect foresight models. The latter discusses structural stability, topological equivalence and chaos, and contains applications to population dynamics, business cycles and spatial models.

Their behaviour is extremely simple and is well discussed in Brock and Mafliaris. Any solution path of 2 either converges to a point the origin , or diverges to infinity, or is a cyclical path surrounding the origin.

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This property is known as structural instability and is returned to in section 3 below. Type iv behaviour entails the central notion of chaos, which is discussed in all the volumes reviewed here. It has been suggested that, in high dimension nonlinear systems, chaos is, in a sense, typical.

If this is correct it has some alarming implications for economics.

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The world is obviously nonlinear, but the complexities which nonlinearity introduces have encouraged economists in their conviction that difficulties ignored are difficulties resolved. Acting in this spirit they traditionally have recourse to the method of local linearisation, which relies on a theorem due to Hartman. This theorem ensures that the linearisation of 3 obtained by standard calculus methods has in most cases a phase portrait which locally is qualitatively equivalent in a topological sense to the phase portrait of the original system. While this result can be of great significance if used carefully, there are several difficulties with it which seem to have escaped the attention of most economists.

The equivalence of phase portraits only holds near the point at which the linearisation is carried out; away from that point it may break down. Accordingly, the focus is on the methods, and every new mathematical technique introduced is followed by its application to select economic models.

## Economics and Mathematics

The mathematical methods coveredc range from elementary linear difference and differential equations and simultaneous systems to the qualitative analysis of non-linear dynamical systems. Stability considerations are stressed throughout, including many advanced topics. All chapters conclude with two sections on miscellaneous applications and exercises and further remarks and references. In total the reader will find a valuable guide to over selected references that use differential equations, stability analysis and chaotic dynamics.

Graduate students in economics with a special interest in economic theory, economic researchers and applied mathematicians will all benefit from this volume. Linear differential equations