Maximum theorems for convex structures with an application to the theory of optimal intertemporal allocation

Abstract

A mathematical result often used in economic theory, is Berge's Maximum Theorem. This establishes continuity of the value function and upper semicontinuity of the maximizers' correspondence. However, the theorem requires the return function and the feasible correspondence to be continuous. For some applications in economics, it is difficult to justify these strong continuity requirements but quite possible to explain some ‘convex structures’ to the problem. The main purpose of this paper is to present a maximum theorem under convex structures but with weaker continuity requirements. We then illustrate the usefulness of our results by an application to a problem encountered in the theory of optimal intertemporalallocation.