Variational Problem, Generalized Convexity, and Application to a Competitive Equilibrium Problem

Abstract

The main purpose of this article is to present a new formulation of a competitive equilibrium in terms of a suitable quasivariational inequality involving multivalued maps. More precisely, a pure exchange economy is considered where the consumer's preferences are represented by utility functions that we assume to be generalized concave and non-differentiable. In the concave context, we have characterized the equilibrium by means of a variational problem involving the subdifferential. Now, by relaxing concavity and differentiability assumptions on utility functions, the subdifferential operator of the utility function is replaced by a suitable multimap involving a new concept, recently introduced in [1]: the normal operator to the adjusted sublevel sets. Thanks to this variational formulation we are able to achieve the existence of equilibrium points by using arguments of the set-valued analysis. Finally, we provide some example of utility functions which verify our assumptions.