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)
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