Topology and Economics: The Contribution of S. Smale

Abstract

Classical problems in economics are concerned with the solutions of several simultaneous nonlinear optimization problems. one for each consumer or producer. all facing constraints posed by the scarcity of resources. Often their interests conflict. and it is generally impossible to find a single real-valued function representing the interests of the whole of society. To deal with this problem. John Von Neumann introduced the theory of games. He also defined and established the existence of a general economic equilibrium, using topological tools [Von Neumann. 1938]. The work of Stephen Smale follows this tradition. He uses topological tools to deepen and reline the results on existence and other properties of another type of economic equilibrium, the Walrasian equilibrium (Walras [1874-77]), as formalized by Kenneth J. Arrow and Gerard Debreu [l954], and of non-cooperative equilibrium in game theory as formalized by Nash (1950). This article aims to show that topology is intrinsically necessary for the understanding of the fundamental problem of conflict resolution in economics in its various forms; and to situate Smale's contribution within this perspective. The study of conflicts of interests between individuals is what makes economics interesting and mathematically complex.