One dimensional Kakutani's fixed point theorem in problems of fair division: a classroom capsule

Abstract

In this paper, we provide a simple proof of the one-dimensional version of the Kakutani s fixed point theorem, which is required to prove the existence of a Walrasian equilibrium in a two commodity multi-agent economy where preferences are convex. In a final section of this paper we apply the Kakutani s fixed point theorem to prove the existence of what we call an equitable equilibrium with rationing in a model where each individual is endowed with positive quantities of two goods. The solution is very similar to the solution due to Dreze (1975) except that ours is not a fixed price solution. It is a problem of resource allocation from an initial position and the solution we propose falls within the general category of non-Walrasian equilibria.