Applications of Thom’s Transversality Theory and Brouwer Degree Theory to Economics

Abstract

Recently, techniques of differential topology and global analysis were introduced into the economics literature by Debreu [6] and Smale [20], [21]. The tools of differential topology enables us to investigate the local uniqueness and continuity of the economic equilibria as well as the existence problem. The existence problem has been extensively studied during the last 20 years (see Arrow and Hahn [2] for a comprehensive survey). The mathematical tools for the solution were provided by algebraic topology in the form of fixed point theorems. In this differential framework, one can also show that the equilibrium varies in a continuous and unique manner with respect to changes in the economic data of the model. Debreu [6] investigated these equilibrium properties for classical pure exchange economies with a finite number of agents and a finite number of consumption goods. His analysis is restricted to finite dimensional spaces in the sense that an economy is specified by a point of finite dimensional commodity space. Smale [20 ] extended this finite dimensional case to the case of allowing each agent’s utility function to vary arbitrarily for the same type model as Debreu [6].

This work was supported in part by NSF grant GS-18174 and in part by the Urban Institute, Washington, D. C. P. J. Kalman is visiting Harvard from SUNY at Stony Brook. We thank K. J. Arrow, M. Hirsch, G. Laroque, H. Wiesmeth and J. Wolf for helpful comments.