Competitive Equilibria: Convergence, Cycles or Chaos
The title of this monograph could have been "What does one do if Anything Goes"; a friend suggested that I should use it as a sub-title instead of the more prosaic one that I have used. There are two basic "Anything Goes" type of results which influence the role of dynamics in economic theory. The first is the Sonnenschein-Debreu-Mantel set of results which indicate that excess demand functions which satisfy only Walras law and the Homogeneity of degree zero postulate do not imply too many restrictions since almost any set of functions could be taken to excess demand functions; the second set of results are due to Boldrin and Montruchhio which show that any dynamical system (continuous or discrete) can describe the time evolution of the optimal paths of an infinite horizon discounted concave maximization problem subject to stationarity constraints. Thus loosely speaking, any dynamical system can be rationalized as occurring in the context of some maximization problem. These two sets of results have had a profound impact on economic theory since they seem to indicate that economic theory is unable to be definitive. A third type of "Anything Goes" result arise from the theory of dynamical systems itself, which economists do not perhaps refer to as much, is due to Smale. To describe the content of this result some notation becomes necessary. Let X be any C1 vector field in the unit simplex of dimension n | 1, (delta)n|1; then there exists a C1 vector field F = (Fi) in R of dimension n satisfying Fi = xiMi(x), Mij(x) 2, it would appear that anything goes on account of dynamical systems alone. Thus not only the theory of stability of competitive equilibrium, but the study of growth processes and even the study of dynamical systems and their long term behavior need to be handled and understood carefully. General results would be difficult to obtain; any result will require special and some times, what may appear, to be ad-hoc conditions. As the three sets of results mentioned above seems to indicate, we do not have any choice in the matter and we should thus address ourselves to the nature of conditions which might provide us with results of some interest. This monograph is directed towards this objective. Another aspect that we shall be concerned with is the robustness of results obtained. In economic theory we sometimes know signs of some terms; their magnitude is some thing beyond our grasp. Accordingly we need to worry about our results if they are dependant upon magnitudes of parameters. Or if this is some thing which we cannot overcome, we should look for some ways of obtaining such information. There should be thus better cooperation between researchers in economic theory and applied economics. Finally, it should not be surprising given Smale's results that we have devoted special attention to models where the dynamics involve motion on the plane. It turns out however that even here a rich variety of situations may be exhibited. We hope that this sets out the reason why such a study is being attempted. Currently, the study has been divided up into five chapters. Chapters 1 and 2 contain the basic tools of analysis; the first refers to continuous time processes whereas the second refers to discrete time processes. These chapters contain a summary of definitions and results and some applications of these results. The chapters are by no means a comprehensive account of non-linear dynamic systems; they are there to keep the study self-contained. Chapters 3 and 4 contain an analysis of Stability of Competitive Equilibria; the first refers to Walrasian Tatonnement processes while the second to Non-Walrasian or Non-tatonnement processes. We have tried to make the analysis in these chapters as exhaustive as possible, so that readers may understand and appreciate the different aspects of this problem. In short, we examine the workings of the so-called 'Invisible Han' and obtain conditions when the Invisible Hand is also successful in carrying out the tasks that we usually assume that it is capable of. Chapter 5, currently the last chapter, is devoted to processes of economic growth in one sector models. The aspect studied in some detail is the approach to the question of unemployment cycles.
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