# Quasivariational Inequalities for a Dynamic Competitive Economic Equilibrium Problem

- Maria Bernadette Donato
^{1}, - Monica Milasi
^{1}and - Carmela Vitanza
^{1}Email author

**2009**:519623

https://doi.org/10.1155/2009/519623

© Maria Bernadette Donato et al. 2009

**Received: **3 February 2009

**Accepted: **12 October 2009

**Published: **15 October 2009

## Abstract

The aim of this paper is to consider a dynamic competitive economic equilibrium problem in terms of maximization of utility functions and of excess demand functions. This equilibrium problem is studied by means of a time-dependent quasivariational inequality which is set in the Lebesgue space . This approach allows us to obtain an existence result of time-dependent equilibrium solutions.

## 1. Introduction

The theory of variational inequality was born in the 1970s, driven by the solution given by G. Fichera to the Signorini problem on the elastic equilibrium of a body under unilateral constraints and by Stampacchia's work on defining the capacitory potential associated to a nonsymmetric bilinear form.

It is possible to attach to this theory a preliminary role in establishing a close relationship between theory and applications in a wide range of problems in mechanics, engineering, mathematical programming, control, and optimization [1–4]. In this paper, a dynamic competitive economic equilibrium problem by using a variational formulation is studied. It was Walras [5] who, in 1874, laid the foundations for the study of the general equilibrium theory, providing a succession of models, each taking into account more aspects of a real economy. The rigorous mathematical formulation of the general equilibrium problem, with possibly nonsmooth but convex data, was elaborated by Arrow and Debreu [6] in the 1954. In 1985, Border in [7] elaborated a variational inequality formulation of a Walrasian price equilibrium. By means of the variational formulation, Dafermos in [8] and Zhao in [9] proved some qualitative results for the solutions to the Walrasian problem in the static case. Moreover, Nagurney and Zhao [10] (see also Zhao [9], Dafermos and Zhao [11]) considered the static Walrasian price equilibrium problem as a network equilibrium problem over an abstract network with very simple structure that consists of a single origin-destination pair of nodes and single links joining the two nodes. Furthermore, the characterization of Walrasian price equilibrium vectors as solutions of a variational inequality induces efficient algorithms for their computation (for further details see also Nagurney's book [12, Chapter 9], and its complete bibliography).

In [13] it was proven how, by introducing the Lagrange multipliers, a general economic equilibrium with utility function can be represented by a variational inequality problem. In recent years, some papers have been devoted to the study of the influence of time on the equilibrium problems in terms of variational inequality problems in suitable Lebesgue space [14–22]. We refer the interested reader to the book [23] where a variety of problems arising from economics, finance, or transportation science are formulated in Lebesgue spaces. In this paper, we have focused on the generalization of the dynamic case of the competitive economic equilibrium problem studied, in the static case, in [24–26].

The paper is organized as follows. In Section 2 we introduce the evolution in time of the competitive economic equilibrium problem in which the data depend on time and we show how the governing equilibrium conditions can be formulated in terms of an evolutionary quasivariational inequality. By means of this characterization, in Section 3, we are able to give an existence result for the equilibrium solutions by using a two-step procedure. Firstly, we give the existence and uniqueness to the equilibrium consumption and for this equilibrium we achieve a regularity result. Then we are able to prove the existence of the competitive prices.

## 2. Walrasian Pure Exchange Model

During a period of time , , we consider a marketplace consisting of different goods indexed by , , and agents indexed by .

the price vector at the time
. We assume that the *free disposal* of commodities is assumed, that is, the a priori exclusion of negative prices. We choose the vectors
,
and
in the Hilbert space
and
in
.

For each and , is a closed and convex set of .

We assume that the utility function, for each agent , satisfies the following assumptions:

() for all : for all a.e. ; moreover for all such that in , for all , it results in .

If there is , the solution to maximization problem (2.6), we pose and .

Then the definition of the dynamic competitive equilibrium problem for a pure exchange economy takes the following form.

Definition 2.1.

Our purpose is to give the following characterization.

Theorem 2.2.

Proof.

Now, we will prove the theorem by means of the following steps.

**(**

**)**For all , is a solution to the problem (2.6) if and only if is a solution to the variational problem

namely, the variational inequality (2.13).

Hence is a solution to the problem (2.6).

From (2.28) it derives that in , that is, assumption is contradicted. Hence from (2.26) it results in that a.e. for all .

Condition (2.32) contradicts the assumption and the estimate (2.24) is proved.

that is, is a solution to maximization problem (2.6) against the assumption on .

that is, the well-known Walras law.

If the above estimate does not hold.

If by the choice of it results in that the estimate is false. Then (2.40) cannot occur and we get a.e. , for all .

## 3. Existence Results

In this section we are concerned with the problem of the existence of the dynamic competitive equilibrium, by using the variational theory.

### 3.1. Existence and Regularity of the Equilibrium Consumption

Now, our goal is to give a regularity result for the evolutionary variational inequality (3.1), in particular, we prove that is continuous on . In order to achieve the continuity result, we need to recall the concept of set convergence in the sense of Mosco (see, e.g., [27]).

Definition 3.1 (see [27]).

Let be an Hilbert space a closed, nonempty, convex set. A sequence of nonempty, closed, convex sets converges to as that is, if and only if

(M1) for any there exists a sequence strongly converging to in such that lies in for all

(M2) for any weakly converging to in , such that lies in for all , then the weak limit belongs to .

Definition 3.2 (see, e.g., [28]).

(M3) the sequence strongly converges to in for any sequence strongly converging to .

Theorem 3.3 (see, e.g., [28]).

Theorem 3.4.

For all strongly converging to , then in Mosco's sense.

Let fixed and let be a sequence such that . We prove that in Mosco's sense, that is, it is enough to show that (M1) and (M2) hold.

Hence, (M1) holds.

condition (3.19) is contradicted. Then for all a.e. .

hence . So condition (M2) holds.

Then we have proved that for all such that , it results in that converging to in Mosco's sense.

Theorem 3.5.

Let be an affine operator of form (3.40). Then is continuous on .

Proof.

We have that in the sense of (M3). In fact,

Hence, we have proved that for all strongly converging to , strongly converges to , then is continuous on .

### 3.2. Existence of Competitive Prices and Existence of Equilibrium

In order to prove an existence result of solutions to (3.31), we recall the following.

**Theorem 5.1 of [ 15 ]**.

*Let*

*be a real topological vector space and let*

*be a convex and nonempty*.

*Let*

*be such that for all*

*and there exist*

*nonempty, compact and*

*compact such that for every*,

*there exists*

*with*,

*there exists*

*such that*

Theorem 3.6.

with if . Then has compact closure in .

Theorem 3.6 is the -version of Ascoli's theorem, due to Riesz, Fréchet, and Kolmogorov (see, e.g., [30]). Now, we can prove the following.

Theorem 3.7.

Let us consider evolutionary variational inequality (3.31). There exists at least one solution to (3.31).

Proof.

is continuous. By [15, Theorem 5.1] the evolutionary variational inequality (3.31) admits a solution.

Finally, we have following existence result of dynamic competitive equilibrium for a pure exchange economy.

Theorem 3.8.

namely, there exists at least a dynamic competitive equilibrium.

## Declarations

### Acknowledgment

The authors wish to express their gratitude to Professor A. Maugeri for his very helpful comments and suggestions.

## Authors’ Affiliations

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