Mixed variational inequalities and economic equilibrium problems

I V Konnov,E O Volotskaya

doi:10.1155/s1110757x02106012

Abstract

We consider rather broad classes of general economic equilibrium problems and oligopolistic equilibrium problems which can be formulated as mixed variational inequality problems. Such problems involve a continuous mapping and a convex, but not necessarily differentiable function. We present existence and uniqueness results of solutions under weakened P‐type assumptions on the cost mapping. They enable us to establish new results for the economic equilibrium problems under consideration.

Highlights

Variational inequalities (VIs) are known to be a very useful tool to formulate and investigate various economic equilibrium problems
Note that most of works on mixed variational inequality problem (MVI) are traditionally devoted to the case where G possesses certain strict monotonicity properties, which enable one to present various existence and uniqueness results for problem (1.1) and suggest various solution methods, including descent methods with respect to a so-called merit function; for example, see [22]
We briefly outline two economic equilibrium models which can be formulated as MVI of form (1.1)

Summary

Introduction

Variational inequalities (VIs) are known to be a very useful tool to formulate and investigate various economic equilibrium problems. It clearly reduces to the usual (single-valued) VI if f ≡ 0 and to the usual convex nondifferentiable optimization problem if G ≡ 0, respectively It can be considered as an intermediate problem between single-valued and multivalued VIs. Note that most of works on MVIs are traditionally devoted to the case where G possesses certain strict (strong) monotonicity properties, which enable one to present various existence and uniqueness results for problem (1.1) and suggest various solution methods, including descent methods with respect to a so-called merit function; for example, see [22]. Note that most of works on MVIs are traditionally devoted to the case where G possesses certain strict (strong) monotonicity properties, which enable one to present various existence and uniqueness results for problem (1.1) and suggest various solution methods, including descent methods with respect to a so-called merit function; for example, see [22] These properties seem too restrictive for economic applications, where order monotonicity type conditions are used. A function f : U → R is (a) convex if and only if ∂f is monotone; (b) strictly convex if and only if ∂f is strictly monotone; (c) strongly convex with constant τ > 0 if and only if ∂f is strongly monotone with constant τ > 0

Economic equilibrium models

Technical preliminaries

General existence and uniqueness results

Application to the Walrasian equilibrium model

Application to the oligopolistic equilibrium model

Concluding remarks