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A competitive economic equilibrium model integrated with exchange, consumption, and production is considered. Our goal is to give an existence result when the utility functions are concave, proper, and upper semicontinuous. To this aim we are able to characterize the equilibrium by means of a suitable generalized quasi-variational inequality; then we give the existence of equilibrium by using the variational approach.

In this paper a competitive economic equilibrium model integrated with exchange, consumption, and production is considered. In 1874, Walras [

Recently, an alternative approach to the study of general economic equilibrium has been considered in terms of variational inequalities. Many economic equilibrium models, including the general equilibrium model of Arrow-Debreu, can be formulated as variational inequalities and/or complementarity problems. Variational theory was introduced in the early 1960s with the works of Fichera and Stampacchia; they study equilibrium problems arising from elastoplastic theory and from mechanics. Subsequently this theory was applied in different kinds of equilibrium problems and, now, represents a powerful tool for the study of a large class of equilibrium problems arising in mechanics, physics, optimization and control theory, operations research, and several branches of engineering sciences. For the state of art about this topic, we refer the reader to [

In particular, Jofré et al. in [

More precisely, an economic market with

Finally, we would like to stress that in the classical literature the so-called survivability assumption is required: each agent has, at the beginning, a positive quantity of each commodity (see, e.g., [

The plan of the paper is as follows. Firstly, for the reader's convenience, we recall some basic definitions and properties which will be useful in the sequel. After, we introduce the competitive economic equilibrium model integrated with exchange, consumption, and production and we reformulate it in terms of a generalized quasi-variational inequality. Finally, we use this approach to investigate the existence of equilibrium.

It is worth to mention that this model allows to consider a wide class of utility functions, which are only convex, proper, and upper semi-continuous.

In the whole section

A multivalued map (multimap)

compact if its range

Let

Assume that

for every

Then the multimap

Let

Here it follows some properties of the subdifferential.

Let

for

Let

Then there exist

We consider a marketplace consisting of two types of agents:

“for all agents

Each agent

We denote by

We notice that in the considered model, when the required demand of a commodity is not satisfied, the consumer does not suffer any loss. In this market producers act to maximize their profit

Letting

for all

for all

Denoting with

“find

Let assumptions (H1)-(H2) be satisfied. A solution to (

We divide the proof in several steps.

It follows directly from the definition of the variational inequality (

Conversely, let

Indeed, by Step 1, variational inequality (

Then we can conclude that

If

By equilibrium condition (

This section concerns the study of existence of solutions to the generalized quasi-variational inequality (

“find

Let assumptions (H1)-(H2) be satisfied. Then there exists a solution of (

We prove that all the hypotheses of the existence result of Theorem

Since, from (H2), the map

Indeed the map

To apply the mentioned theorem we only need to verify that for any

Follows directly from the definition of

First of all we prove that

Hence,

Moreover, from Step 3,

Hence, we can apply Theorem

Let assumptions (H1)-(H2) be satisfied, let

Fix

Then, we can conclude that

In conclusion, directly from Theorems

Let assumptions (H1)-(H2) be satisfied, let

We observe that if each agent is endowed with each commodity

In fact, for all

If each agent is endowed with all commodities there exists a competitive equilibrium

The main result of this paper has been to obtain the existence of a competitive economic equilibrium for a model integrated with exchange, production, and consumption. In order to obtain a wide applicability in the economic framework, care was taken to keep a level of generality on the assumptions of the market. In particular utility functions which are proper, convex, and upper semi-continuous have been considered, hence without assuming any differentiability. These assumptions allow us to consider a wide range of utility functions frequently used in the economic literature.

A class of utility functions most widely used in economics consists of

Another very interesting class of utility functions is represented by

The functions, before mentioned, satisfy the assumptions we have taken in this paper; in fact they are proper, convex, and upper semi-continuous. However, these functions are not always differentiable on all their domain. They are defined and continuous on all of

To conclude we want to stress that, in our opinion, by using the variational approach, the generalized quasi-variational inequalities are especially suitable to handle equilibrium problems for a market of exchange, consumption, and production, for it allows to take into account a wide class of models.