Existence Theorems on Quasi-variational Inequalities over Banach Spaces and its Applications to Time-dependent Pure Exchange Economy

TL;DR

This paper establishes existence theorems for quasi-variational inequalities in Banach spaces and applies these results to demonstrate the existence of dynamic competitive equilibrium in a time-dependent pure exchange economy.

Contribution

It provides new existence results for quasi-variational inequalities with unbounded constraints and applies them to economic models involving dynamic exchange economies.

Findings

Solutions exist under upper semi-continuity and pseudomonotonicity assumptions.

Existence of solutions for inequalities with unbounded constraint maps.

Dynamic competitive equilibrium is demonstrated in a time-dependent exchange economy.

Abstract

We study a class of quasi-variational inequality problems defined over infinite dimensional Banach space and deduce sufficient conditions for ensuring solutions to such problems under the upper semi-continuity and pseudomonotonicity assumptions on the map defining the inequalities. The special structure of the quasi-variational inequality enables us to show the occurrence of solutions for such inequalities based on the classical existence theorem for variational inequalities. This special type of quasi-variational inequalities is motivated by the pure exchange economic problems and Radner equilibrium problems for sequential trading game. Further, we study the solvability of the specific class of quasi-variational inequalities on Banach spaces in which the constraint map may admit unbounded values. Finally, we demonstrate the occurrence of dynamic competitive equilibrium for a time-dependent pure exchange economy as an application.