Topoloy and economics: the contributions of S. Smale

90 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 introduces 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 Stephan Smale follows this tradition. He uses topological tools to deepen and refine the results on existence and other properties of another type of economic equilibrium, the Walrasian equilibrium (Walrus [1874-77]), as formalized by Kenneth J. Arrow and Gerard Debreu [1954], and of non-cooperative equilibrium in game theory as formalized by Nash (1950). This article aim 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. Indeed, we now know that the space of all individual preferences, which define the individual optimization problems, is topologically nontrivial, and that its topological complexity is responsible for the impossibility of treating several individual preferences as if they were one, i.e., aggregating them (Chichilnisky, 1980; Chichilnisky and Heal, 1983). Because it is not possible, in general, to define a single optimization problem, other solutions are sought. This article will develop three solutions, discussed below. Because of the complexity arising from simultaneous optimization problems, economics differs from physics where many of the fundamental relations derive from a single optimization problem. The attempts to find solutions to conflicts among individual interests led to three different theories about how economies are organized and how they behave. These are general equilibrium theory, the theory of games, and social choice theory. Each of these theories leads naturally to mathematical problems of topological nature. Steve Smale has contributed fruitfully to the first two theories: general equilibrium theory and the theory of games. I will argue that his work is connected also with the third approach, social choice theory, by presenting in Section 4 results which link closely, and in unexpected ways, two seemingly different problems: the existence of a general equilibrium and the resolution by social choice of the resource allocation conflict in economics (Chichilnisky, 1991).