Variational Problems and Application to a Competitive Equilibrium Problem

Competitive equilibrium problems are usually studied by means of fixed point theory. Alternatively, it is possible to study these problems by means of a variational approach. The variational inequality theory was introduced by Fichera and Stampacchia, in the early 1960’s, in connection with several equilibrium problems originating from mathematical physics. This theory turned out to be an innovative and powerful methodology for the study of several kind of equilibrium problems.

Here we consider a new formulation of a competitive equilibrium in terms of a suitable quasivariational inequality involving multivalued maps. More precisely, it is considered a pure exchange economy where the consumer’s preferences are represented by quasiconcave and non-differentiable utility functions. By relaxing concavity and differentiability assumptions on the utility functions, the subdifferential operator of the utility function, required in the variational problem, is suitably replaced by a multimap involving the normal operator to the adjusted sublevel sets. Using this variational formulation, we are able to prove the existence of equilibrium points by using arguments from the set-valued analysis and non-convex analysis.