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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 651975, 35 pages
http://dx.doi.org/10.1155/2012/651975
Research Article

Time-Dependent Variational Inequality for an Oligopolistic Market Equilibrium Problem with Production and Demand Excesses

1Department of Mathematics and Applications “R. Caccioppoli”, University of Naples “Federico II”, Via Cintia 80126 Naples, Italy
2Department of Mathematics and Computer Science, University of Catania, Viale A. Doria, 95125 Catania, Italy

Received 2 February 2012; Accepted 20 March 2012

Academic Editor: Kanishka Perera

Copyright © 2012 Annamaria Barbagallo and Paolo Mauro. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The paper is concerned with the variational formulation of the oligopolistic market equilibrium problem in presence of both production and demand excesses. In particular, we generalize a previous model in which the authors, instead, considered only the problem with production excesses, by allowing also the presence of demand excesses. First we examine the equilibrium conditions in terms of the well-known dynamic Cournot-Nash principle. Next, the equilibrium conditions will be expressed in terms of Lagrange multipliers by means of the infinite dimensional duality theory. Then, we show the equivalence between the two conditions that are both expressed by an appropriate evolutionary variational inequality. Moreover, thanks to the variational formulation, some existence and regularity results for equilibrium solutions are proved. At last, a numerical example, which illustrates the features of the problem, is provided.

1. Introduction

The aim of this paper is to introduce a time-dependent variational formulation for the dynamic oligopolistic market equilibrium model in presence of both production and demand excesses. Moreover, in line with [1], we want to eliminate the serious drawback present in [2] where the authors made the unreasonable assumption that the production of a given commodity could be unbounded, by making possible any commodity shipment from a firm to a demand market. This is not possible because the amount of a commodity that the producers can offer is limited as a consequence of finite resources. Therefore, it can happen that some of the amounts of the available commodity are sold out so that can occur an excess of demand, whereas for a part of the producers can occur an excess of production. The question about the unbounded production had already been solved in [1], but the presence of both excesses realizes a more complete study of the problem. In order to clarify the presence of both production and demand excesses we consider some concrete economic situations. During an economic crisis period the presence of production excesses can be due to a demand decrease in demand markets and, on the other hand, the presence of demand excesses may occur when the supply cannot satisfy the demand especially for fundamental goods. Moreover, since the market model presented in this paper evolves in time, the presence of both production and demand excesses is a consequence of the fact that the physical transportation of commodity between a firm and a demand market is evidently limited, therefore, there can exist some time intervals in which some of the demand markets require more commodity, though some firms produce more commodity than they can send to all the demand markets. For these reasons the new model results in a more realistic generalization of the ones presented in [1, 2], where the authors studied the case without presence of excesses and the case with the only presence of production excesses, respectively.

The equilibrium formulation fits in the light of a dynamic noncooperative behaviour. The first author who treated the most trivial example of noncooperative behaviour between two producers of a given commodity, nowadays called duopoly problem, was Cournot (see [3]). Later Nash, in [4, 5], extended this concept by introducing agents in his model nowadays called noncooperative game, each acting according to his own self-interest.

In order to study the time-dependent behaviour of the model, we will afford this study by considering the evolution of the market in time and, as a consequence, all the variables present in this model, such as the costs, the commodity shipments, and the excesses depend on time. As Beckmann and Wallace pointed out, for the first time, in [6], “the time-dependent formulation of equilibrium problems allows one to explore the dynamics of adjustment processes in which a delay on time response is operating.” Of course a delay on time response always happens because the processes have not an infinite speed. Usually, such adjustment processes can be represented by means of a memory term which depends on previous equilibrium solutions according to the Volterra operator (see, e.g., [7]). The time-dependent process was one of the main features of the paper [2] where, in particular, the authors studied the variational formulation and proved the existence and regularity of a dynamic equilibrium solution. Moreover, the regularity property allows to provide a computational procedure to compute the equilibrium solutions (see, e.g., [812]).

In [13] the authors, through the notion of quasi-relative interior of sets (see [14]), applied the infinite dimensional duality results developed in [1518] to overcome the difficulty of the voidness of the interior of the ordering cone which defines the constraints of the problem and so proved the existence of Lagrange variables which permit to describe the behaviour of the market. Moreover, in [13] some sensitivity results have been obtained each of them showing that small changes of the solution happen in correspondence with small changes of the profit function. In [7, 19], an oligopolistic market equilibrium model with an explicit long-term memory has been considered. Then, in [19], the Lipschitz continuity of the solution, which depends on the variation rate of projections onto time-dependent constraints set, is shown and the existence of Lagrange multipliers is provided.

In the first part of the paper, we present the equilibrium conditions in presence of excesses according to the well-known Cournot-Nash principle. This is a practical equilibrium definition that expresses that each firm acts trying to maximize its own profit. The second one, more theoretical, is shown through the Lagrange variables that better emphasize, in particular, the presence of excess constraints. The equilibrium conditions established in terms of Lagrange variables do not arouse any concern because we will prove that their presence is not influential in the definition of equilibrium because we can characterize such equilibrium conditions by means of an evolutionary variational inequality that does not contain the Lagrange variables. Another thing to notice is that the equilibrium conditions, provided with the help of the duality theory, are equivalent to the dynamic Cournot-Nash equilibrium principle because we can prove that they are both equivalent to the same evolutionary variational inequality. Such variational formulation gives us a powerful tool for the study of the existence, the regularity, and the calculus of equilibrium solutions. In particular, we show that the constraint set satisfies the property of set convergence in Kuratowski’s sense which has an important role in order to guarantee the continuity of equilibrium solutions. Moreover, the continuity property is very useful in order to introduce a numerical scheme to compute equilibrium solutions (see [20, 21]).

The outline of the paper is as the following. In Section 2, we describe the model of the dynamic oligopolistic market equilibrium problem in presence of both production and demand excesses and we show the equilibrium conditions making use of both Cournot-Nash principle and Lagrange multipliers. In Section 3, we recall the new infinite dimensional duality theory requested to show the existence of Lagrange variables. In Section 4, after showing some preliminary lemmas, we give the proof of the characterization of the dynamic oligopolistic market equilibrium conditions established in terms of Lagrange variables by means of an evolutionary variational inequality, so we can derive their equivalence with the dynamic Cournot-Nash principle. In Section 5, after recalling some preliminary definitions, we give some existence results. Section 6 is devoted to provide a regularity result for the equilibrium solution after proving that the constraint set of commodity shipments satisfies the requirements of the set convergence in Kuratowski’s sense. Finally, in Section 7, we provide a numerical example of a dynamic oligopolistic market equilibrium problem in presence of production and demand excesses and underline some important features of the problem.

2. Dynamic Oligopolistic Market Equilibrium

Let us consider firms , that produce only one commodity and demand markets , , that are generally spatially separated. Assume that the homogeneous commodity, produced by the firms and consumed by the markets, is involved during a time interval , . Let , , denote the nonnegative commodity output produced by firm at the time . Let , , denote the nonnegative demand for the commodity at demand market at the time . Let , , , denote the nonnegative commodity shipment between the supply market and the demand market at the time . In particular, let us set the vector , , as the strategy vector for the firm . Finally, let us introduce the production and demand excesses. Let , , be the nonnegative production excess for the commodity of the firm at the time . Let ,, be the nonnegative demand excess for the commodity of the demand market at the time .

Let us group the production output into a vector-function the demand output into a vector-function , the commodity shipments into a matrix-function , the production excess into a vector-function , and the demand excess into a vector-function .

Let us assume that the following feasibility conditions hold: Hence, the quantity produced by each firm , at the time , must be equal to the commodity shipments from that firm to all the demand markets plus the production excess, at the same time . Moreover, the quantity demanded by each demand market , at the time , must be equal to the commodity shipments from all the firms to that demand market plus the demand excess, at the same time .

Furthermore, we assume that the nonnegative commodity shipment between the producer and the demand market has to satisfy time-dependent constraints, namely, there exist two nonnegative functions such that

For technical reasons, let us assume that As a consequence, we have

Then, the set of feasible vectors is

Furthermore, let us associate with each firm a production cost ,, and assume that the production cost of a firm may depend upon the entire production pattern, namely: Similarly, let us associate with each demand market , a demand price for unity of the commodity and assume that the demand price of a demand market may depend upon the entire consumption pattern, namely: Moreover, since we allow production excesses and, consequently, the storage of commodities, we must consider the function ,  that denotes the storage cost of the commodity produced by the firm and assume that this cost may depend upon the entire production pattern, namely: Finally, let , denote the transaction cost, which includes the transportation cost associated with trading the commodity between firm and demand market . Here we permit the transaction cost to depend upon the entire shipment pattern, namely: Hence, we have the following mappings: The profit , of the firm at the time is, then, namely, it is equal to the price that the demand markets are disposed to pay minus the production cost, the storage cost and the transportation costs.

Now, we can rewrite in an equivalent way. By virtue of (2.1) we can express in terms of and and in terms of and , namely: Then, the equivalent constraint set becomes We can observe that includes the presence of both production and demand excesses described in .

Then, the production costs, the demand price, and the storage costs, by virtue of (2.12) and taking into account (2.6), (2.7), and (2.8), become and, analogously, the profit function (2.11) becomes

Let us denote by and let us assume the following assumptions: (i) is continuously differentiable for each , a.e. in ,(ii) is a Carathéodory function such that (iii) is pseudoconcave with respect to the variable , a.e. in .

For the reader’s convenience, we recall that a function , continuously differentiable, is called pseudoconcave with respect to , a.e. in (see [22]), if the following condition holds, a.e. in : Now let us consider the dynamic oligopolistic market, in which the firms supply the commodity in a noncooperative fashion, each one trying to maximize its own profit function considered the optimal distribution pattern for the other firms, at the time . We seek to determine a nonnegative commodity distribution matrix-function for which the firms and the demand markets will be in a state of equilibrium as defined below. In fact, we can consider different, but equivalent, equilibrium conditions each of them illustrates important features of the equilibrium.

The first one makes use of the dynamic Cournot-Nash principle (see [2]).

Definition 2.1. is a dynamic oligopolistic market equilibrium in presence of excesses if and only if for each and a.e. in one has where With the same technique used in [2, Theorem 3.1] it is possible to prove that under the assumptions (i), (ii), (iii) on , Definition 2.1 is equivalent to an evolutionary variational inequality, as the following result shows.

Theorem 2.2. Let one suppose that assumptions (i), (ii), (iii) are satisfied. Then, is a dynamic oligopolistic market equilibrium in presence of excesses according to Definition 2.1 if and only if it satisfies the evolutionary variational inequality

In Section 4 we will prove that, under the assumptions (i), (ii), (iii) on the profit function , Definition 2.1 is equivalent to the equilibrium conditions defined through Lagrange variables which are very useful in order to analyze the presence of both production and demand excesses.

Definition 2.3. is a dynamic oligopolistic market problem equilibrium in presence of excesses if and only if, for each and a.e. in , there exist such that

The terms , , , are the Lagrange multipliers associated to the constraints , and , respectively.

They, as it is well known, have a topical importance on the understanding and the management of the market. In fact, at a fixed time , we have: (a)if then, by using (2.22), we obtain , namely, the commodity shipment between the firm and the demand market is minimum,(b)if then, taking into account (2.22),   and, making use of (2.21), it results , namely, is equal to the marginal profit,(c)if then, by using (2.23), we obtain , namely, the commodity shipment between the firm and the demand market is maximum,(d)if then, making use of (2.23), and, taking into account (2.21), we get , namely, is equal to the marginal profit,(e)if then, for the condition (2.24), we have , namely, there is no production excess,(f)if , as a consequence of (2.24) we get and, for the condition (2.21), , namely, is equal to the marginal profit,(g)if then, for the condition (2.25), it results , namely, there is no demand excess,(h)if , as a consequence of (2.25) we obtain and, for the condition (2.21), , namely, is equal to the marginal profit.

It is worthy to underline that in Definition 2.3, even if in (2.21)–(2.25) the unknown Lagrange variables , , , appear, they do not influence the equilibrium definition because the following equivalent condition in terms of evolutionary variational inequality holds.

Theorem 2.4. is a dynamic oligopolistic market equilibrium in presence of excesses according to Definition 2.3 if and only if it satisfies the evolutionary variational inequality:

Taking into account Theorems 2.2 and 2.4, the equivalence between Definitions 2.1 and 2.3 is proved.

Finally, we observe that also in the case in which the production is bounded and we are in presence of excesses, the meaning of Cournot-Nash equilibrium does not change.

3. Lagrange Theory

Let us present the infinite dimensional Lagrange duality theory which represents an important and very recent achievement (see [1618]). At first, we remember some definitions and then we give some duality results (see [1517]).

Let denote a real normed space, let be the topological dual of all continuous linear functionals on , and let be a subset of . Given an element , the set: is called the tangent cone to at .

If is convex, we have (see [23]): where .

Following Borwein and Lewis [14], we give the following definition of quasi-relative interior for a convex set.

Definition 3.1. Let be a convex subset of . The quasi-relative interior of , denoted by , is the set of those for which is a linear subspace of .

If we define the normal cone to at as the set: the following result holds.

Proposition 3.2. Let be a convex subset of and . Then if and only if is a linear subspace of .

Using the notion of , in [17], the following separation theorem is proved.

Theorem 3.3. Let be a convex subset of and . Then, there exists , , such that Vice versa, let one suppose that there exist and a point such that , for all , and that . Then, .

Now, let us present the statement of the infinite dimensional duality theory.

Let be a real linear topological space and a nonempty convex subset of ; let be a real normed space partially ordered by a convex cone and let be a real normed space. Let and be two convex functions and let be an affine-linear function.

Let us consider the problem where , and the dual problem where is the dual cone of .

We say that Assumption S is fulfilled at a point if and only if it results in where .

Remark 3.4. If , then Assumption S holds, because .

Remark 3.5. If Assumption S holds, , and , then .

Remark 3.6. If Assumption S holds, then .

The following theorem holds (see [16]).

Theorem 3.7. Under the above assumptions, if problem (3.5) is solvable and Assumption S is fulfilled at the extremal solution , then also problem (3.6) is solvable, the extreme values of both problems are equal, and if is the optimal point of problem (3.6), it results in:

Using Theorem 3.7, we are able to show the usual relationship between a saddle point of the so-called Lagrange functional: and the solution of the constraint optimization problem (3.5) (see [16]).

Theorem 3.8. Let one assume that the assumptions of Theorem 3.7 are satisfied. Then, is a minimal solution to problem (3.5) if and only if there exist and such that is a saddle point of the Lagrange functional (3.9), namely: and, moreover, it results in

4. Proof of Existence of Lagrange Variables

In this section, making use of the infinite dimensional Lagrange duality theory shown in Section 3, we will prove that equilibrium conditions (2.21)–(2.25) can be equivalently expressed by the evolutionary variational inequality (2.20). As a consequence, we determine under assumptions (i), (ii), (iii) on the profit function , the equivalence with dynamic Cournot-Nash equilibrium conditions (2.18).

In order to prove Theorem 2.4, let us show some preliminary results. At first we recall Lemma  3.7 in [13] for the capacity constraints of the commodity shipments.

Lemma 4.1. Let be a solution to the variational inequality (2.20) and let one set Then, one has

Now, we recall Lemma  4.1 in [1] that holds when production excesses occur.

Lemma 4.2. Let be a solution to the variational inequality (2.20). Setting one has

With the same technique used for proving Lemma 4.2, we can obtain the following analogous result that holds when demand excesses occur.

Lemma 4.3. Let be a solution to the variational inequality (2.20). Setting one has

Now, we remember Lemma  4.2 in [1] that holds when production excesses occur.

Lemma 4.4. Let be a solution to the variational inequality (2.20). Setting one has

Finally, by proceeding as in Lemma 4.4 we can prove the following analogous result that holds when demand excesses occur.

Lemma 4.5. Let be a solution to the variational inequality (2.20). Setting one has

Now we are able to prove Theorem 2.4.

Proof of Theorem 2.4. Let us assume that is an equilibrium solution according to Definition 2.3. Then, taking into account that and , a.e. in , we have for every , a.e. in , and, as a consequence, by summing over and , integrating on and using the conditions (2.24) and (2.25), it results, for each Hence, we obtain (2.20).
Vice versa, let be a solution to (2.20) and let us apply the infinite dimensional duality theory. First of all, let us prove that the Assumption S is fulfilled.
Let us set, for , we must show that if belongs to , namely: with , , ,, , for all , and then is nonnegative.
Let us set Before starting with the proof let us observe the following: and also Moreover, and, analogously, Now we observe that, for Lemmas 4.1, 4.2, 4.3, 4.4, and 4.5, we get As a consequence, we have We note that being , a.e. in , for all , for all . We will prove that In fact, it results in By virtue of the previous remarks, conditions (4.4) and (4.6), Lemmas 4.1, 4.2, 4.3, 4.4, and 4.5, for the conditions of belonging to the tangent cone, we get the first inequality of (4.25) and, with analogous considerations, we get the second inequality of (4.25).
Therefore, thanks to (4.25) and (4.24), we have that is nonnegative.
Taking into account Theorems 3.7 and 3.8, if we consider the Lagrange function, we have that there exist , such that for all and, moreover, Then, for conditions (