An Equilibrium Existence Theorem

Bewley 14, Theorem l] proved an infinite dimensional equilibrium existence theorem which is a significant extension of the classical finite dimensional theorem of Arrow and Debreu [ 11. The assumptions on technology and preferences are natural and applicable in a wide variety of cases. The proof is based on a limit argument that makes direct use of the existence of equilibrium in the finite dimensional case. This paper establishes the existence of equilibrium under assumptions which are essentially the same as those given by Bewley, with the additional assumption that the preference orderings of consumers are representable by real valued utility functions. This approach is related to the welfare approach of Negishi [9] and Arrow and Hahn [2, Chap. 51 in the finite dimensional case and simplifies the approach originally adopted by Bewley [5]. In addition to making clear the role played by each of the assumptions in establishing the existence of equilibrium, this approach has the merit of constructing directly a certain real valued function that is maximised at an equilibrium, a result that provides a powerful tool in the analysis of qualitative properties of an equilibrium. A model of resource allocation in continuous time over an infinite horizon that may be viewed as an application of the model that follows is given in 181.