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

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.


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 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.

Dynamic Oligopolistic Market Equilibrium
Let us consider m firms P i , i 1, . . ., m, that produce only one commodity and n demand markets Q j , j 1, . . ., n, that are generally spatially separated.Assume that the homogeneous commodity, produced by the m firms and consumed by the n markets, is involved during a time interval 0, T , T > 0. Let p i t , i 1, . . ., m, denote the nonnegative commodity output produced by firm P i at the time t ∈ 0, T .Let q j t , j 1, . . ., n, denote the nonnegative demand for the commodity at demand market Q j at the time t ∈ 0, T .Let x ij t , i 1, . . ., m, j 1, . . ., n, denote the nonnegative commodity shipment between the supply market P i and the demand market Q j at the time t ∈ 0, T .In particular, let us set the vector x i t x i1 t , . . ., x in t , i 1, . . ., m, t ∈ 0, T as the strategy vector for the firm P i .Finally, let us introduce the production and demand excesses.Let i t , i 1, . . ., m, be the nonnegative production excess for the commodity of the firm P i at the time t ∈ 0, T .Let δ j t , j 1, . . ., n, be the nonnegative demand excess for the commodity of the demand market Q j at the time t ∈ 0, T .
Let us group the production output into a vector-function p : 0, T → R m , the demand output into a vector-function q : 0, T → R n , the commodity shipments into a matrix-function x : 0, T → R mn , the production excess into a vector-function : 0, T → R m , and the demand excess into a vector-function δ : 0, T → R n .

2.1
Hence, the quantity produced by each firm P i , at the time t ∈ 0, T , must be equal to the commodity shipments from that firm to all the demand markets plus the production excess, at the same time t ∈ 0, T .Moreover, the quantity demanded by each demand market Q j , at the time t ∈ 0, T , must be equal to the commodity shipments from all the firms to that demand market plus the demand excess, at the same time t ∈ 0, T .Furthermore, we assume that the nonnegative commodity shipment between the producer P i and the demand market Q j has to satisfy time-dependent constraints, namely, there exist two nonnegative functions x, x : 0, T → R mn such that 0 ≤ x ij t ≤ x ij t ≤ x ij t , ∀i 1, . . ., m, ∀j 1, . . ., n, a.e. in 0, T .

2.2
For technical reasons, let us assume that

2.5
Furthermore, let us associate with each firm P i a production cost f * i , i 1, . . ., m, and assume that the production cost of a firm P i may depend upon the entire production pattern, namely: Similarly, let us associate with each demand market Q j , a demand price for unity of the commodity d * j , j 1, . . ., n and assume that the demand price of a demand market Q j 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 g * i , i 1, . . ., m, that denotes the storage cost of the commodity produced by the firm P i and assume that this cost may depend upon the entire production pattern, namely: Finally, let c ij , i 1, . . ., m, j 1, . . ., n, denote the transaction cost, which includes the transportation cost associated with trading the commodity between firm P i and demand market Q j .Here we permit the transaction cost to depend upon the entire shipment pattern, namely: Hence, we have the following mappings:

2.10
The profit v * i t, x t , t , δ t , i 1, . . ., m, of the firm P i at the time t ∈ 0, T 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 K * in an equivalent way.By virtue of 2.1 we can express i t in terms of p i t and x ij t and δ j t in terms of q j t and x ij t , namely: x ij t , i 1, . . ., m, a.e. in 0, T , x ij t , j 1, . . ., n, a.e. in 0, T .

2.13
We can observe that K includes the presence of both production and demand excesses described in K * .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

2.17
Now let us consider the dynamic oligopolistic market, in which the m 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 t ∈ 0, T .We seek to determine a nonnegative commodity distribution matrix-function x for which the m firms and the n 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 .

2.20
In Section 4 we will prove that, under the assumptions i , ii , iii on the profit function v, 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.x * ∈ K is a dynamic oligopolistic market problem equilibrium in presence of excesses if and only if, for each i 1, . . ., m, j 1, . . ., n and a.e. in 0, T , there exist x * ij t − q j t 0, ν * j t ≥ 0.

2.25
The terms λ * ij t , ρ * ij t , μ * i t , ν * j t are the Lagrange multipliers associated to the constraints 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 t ∈ 0, T , we have: a if λ * ij t > 0 then, by using 2.22 , we obtain x * ij t x ij t , namely, the commodity shipment between the firm P i and the demand market 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.

Lagrange Theory
Let us present the infinite dimensional Lagrange duality theory which represents an important and very recent achievement see 16-18 .At first, we remember some definitions and then we give some duality results see 15-17 .Let X denote a real normed space, let X * be the topological dual of all continuous linear functionals on X, and let C be a subset of X.Given an element x ∈ Cl C , the set: If C is convex, we have see 23 : Following Borwein and Lewis 14 , we give the following definition of quasi-relative interior for a convex set.Definition 3.1.Let C be a convex subset of X.The quasi-relative interior of C, denoted by qri C, is the set of those x ∈ C for which T C x is a linear subspace of X.
If we define the normal cone to C at x as the set: the following result holds.
Using the notion of qri C, in 17 , the following separation theorem is proved.
Vice versa, let one suppose that there exist ξ / θ X * and a point x 0 ∈ X such that ξ, x ≤ ξ, x 0 , for all x ∈ C, and that Cl T Now, let us present the statement of the infinite dimensional duality theory.
Let X be a real linear topological space and S a nonempty convex subset of X; let Y, • Y be a real normed space partially ordered by a convex cone C and let Z, • Z be a real normed space.Let f : S → R and g : S → Y be two convex functions and let h : S → Z be an affine-linear function.
Let us consider the problem where K {x ∈ S : g x ∈ −C, h x θ Z }, and the dual problem where We say that Assumption S is fulfilled at a point x 0 ∈ K if and only if it results in where 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 x 0 ∈ K, then also problem 3.6 is solvable, the extreme values of both problems are equal, and if x 0 , u, v ∈ K × C * × Z * is the optimal point of problem 3.6 , it results in: u, g x 0 0.

3.8
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, x 0 ∈ K is a minimal solution to problem 3.5 if and only if there exist u ∈ C * and v ∈ Z * such that x 0 , u, v is a saddle point of the Lagrange functional 3.9 , namely: and, moreover, it results in u, g x 0 0. 3.11

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 v, 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.

4.4
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 x * ∈ K be a solution to the variational inequality 2.20 .Setting

4.7
Now, we remember Lemma 4.2 in 1 that holds when production excesses occur.
Lemma 4.4.Let x * ∈ K be a solution to the variational inequality 2.20 .Setting 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 x * ∈ K be a solution to the variational inequality 2.20 .Setting and, as a consequence, by summing over i 1, . . ., m and j 1, . . ., n, integrating on 0, T and using the conditions 2.24 and 2.25 , it results, for each x * ij t q j t − q j t dt x ij t − q j t dt ≥ 0.

4.13
Hence, we obtain 2.20 .Vice versa, let x * ∈ K 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.

4.17
Before starting with the proof let us observe the following: Moreover, and, analogously,

4.22
As a consequence, we have

4.23
We note that being −∂v i t, x * t /∂x ij 0, a.e. in E 0 ij , for all i 1, . . ., m, for all j 1, . . ., n.We will prove that

4.25
Abstract and Applied Analysis 19 In fact, it results in Taking into account Theorems 3.7 and 3.8, if we consider the Lagrange function, x ij t − q j t dt,

4.32
As a consequence, we have

Existence Results
This section is devoted to show some results for the existence of solutions to the dynamic oligopolistic market equilibrium problem in presence of excesses.
Let us recall some definitions see 24 .Let X be a reflexive Banach space, let K be a subset of X, and let X * be the dual space of X. Definition 5.1.A mapping A : K → X * is strongly monotone on K if and only if for all u, v ∈ K, there exists ν > 0 such that Au − Av, u − v ≥ ν u − v 2 K .
Definition 5.2.A mapping A : K → X * is pseudomonotone in the sense of Karamardian K-pseudomonotone if and only if for all u, v ∈ K, Definition 5.4.A mapping A : K → X * is pseudomonotone in the sense of Brezis Bpseudomonotone if and only if: 1 for each sequence {u n } weakly converging to u in short u n u in K and such that lim Let, now, K be a convex subset of X.
Definition 5.5.A mapping A : K → X * is lower hemicontinuous along line segments, if and only if the function ξ → Aξ, u − v is lower semicontinuous for all u, v ∈ K on the line segments u, v .
Definition 5.6.A mapping A : K → X * is hemicontinuous in the sense of Fan Fhemicontinuous if and only if for all v ∈ K the function u → Au, u − v is weakly lower semicontinuous on K.
Let us recall that in the Hilbert space is its duality mapping, where φ ∈ L 2 0, T , R k * L 2 0, T , R k and y ∈ L 2 0, T , R k .We are able to show the following existence result.

5.5
If A is B-pseudomonotone or F-hemicontinuous, or assuming that A is K-pseudomonotone and lower hemicontinuous along line segments, then the variational inequality admits a solution.
Proof.Let us note that K is clearly a nonempty, closed, convex, and bounded subset of L 2 0, T , R mn , and, consequently, it is a weakly compact subset of L 2 0, T , R mn .Then, the claim is achieved by applying Theorems 2.6 and 2.7 and Corollary 3.7 in 24 .

Regularity Results
In the following, we want to establish conditions under which the solutions to the dynamic oligopolistic market problem with both production and demand excesses are continuous with respect to time.

Set Convergence
Let us remember the classical notion of convergence for subsets of a given metric space X, d , which was introduced in the 1950s by Kuratowski see 25 , see also 26, 27 .
Let {K n } n∈N be a sequence of subsets of X.Let us remember that where eventually means that there exists δ ∈ N such that x n ∈ K n for any n ≥ δ, and frequently means that there exists an infinite subset N ⊆ N such that x n ∈ K n for any n ∈ N in this last case, according to the notation given above, we also write that there exists a subsequence Now, we are able to recall the Kuratowski's convergence of sets.Definition 6.1.We say that the sequence {K n } n∈N converges to some subset K ⊆ X in Kuratowski's sense, and we briefly write Thus, in order to verify that K n → K, it suffices to check that i K ⊂ d − lim n K n , that is, for any x ∈ K there exists a sequence {x n } n∈N eventually in We observe that the set convergence in Kuratowski's sense can also be expressed as follows.
Remark 6.2.Let X, d be a metric space and K a nonempty, closed, and convex subset of X.A sequence of nonempty, closed and convex sets K n of X converges to K in Kuratowski's sense, as n → ∞, that is, K n → K, if and only if K1 for any x ∈ K, there exists a sequence {x n } n∈N converging to x ∈ X such that x n lies in K n for all n ∈ N, K2 for any subsequence {x n } n∈N converging to x ∈ X such that x n lies in K n , for all n ∈ N, then the limit x belongs to K.
The following lemma, that now we prove, assures that the feasible set K of the dynamic oligopolistic market problem in the presence of both production and demand excesses satisfies the property of the set convergence in Kuratowski's sense.Lemma 6.3.Let x, x ∈ C 0 0, T , R mn , p ∈ C 0 0, T , R m , q ∈ C 0 0, T , R n , and let {t k } k∈N be a sequence such that t k ∈ 0, T , for all k ∈ N, and t k → t, with t ∈ 0, T , as k → ∞.Then the sequence of sets x ij t ≤ q j t , ∀j 1, . . ., n ,
Proof.In the first part, we prove the condition K1 .Let {t k } k∈N be a sequence such that t k ∈ 0, T , for all k ∈ N, and t k → t, with t ∈ 0, T , as k → ∞.By virtue of the continuity of x, x, p, q, it follows that x t k → x t , x t k → x t , p t k → p t , q t k → q t , as k → ∞, respectively.Let x t ∈ K t be fixed and let us note that, for i 1, . . ., m and j 1, . . ., n, and if As a consequence, there exists an index ν 1 such that for k > ν 1 we get a ij t k ≥ 0, ∀i 1, . . ., m, ∀j 1, . . ., n.

6.6
We remark lim where is the production excess function.Then, there exists an index ν 2 such that for k > ν 2 we have Obviously if k ≤ ν, for 6.12 we get x t k ∈ K t k .Instead, for k > ν, since for 6.6 , min{x ij t − x ij t , x ij t k − x ij t k , a ij t k } ≥ 0, for all i 1, . . ., m, for all j 1, . . ., n, we obtain x ij t − i t i t p i t k , ∀i 1, . . ., m, 6.17 and, making use of 6.10 , we obtain x ij t q j t k − q j t δ j t m i 1 x ij t q j t k − m i 1 x ij t − δ j t δ j t q j t k , ∀j 1, . . ., n.

6.18
Abstract and Applied Analysis Hence x t k ∈ K t k , for all k ∈ N, and it results in x ij t .

6.19
Then, the proof of the condition K1 is completed.Now let us prove condition K2 .Let {t k } k∈N be a sequence such that t k ∈ 0, T , for all k ∈ N, and t k → t, with t ∈ 0, T , as k → ∞.Let {x t k } k∈N be a sequence, such that x t k ∈ K t k , for all k ∈ N, and converging to x t , as k → ∞.We need to prove that x t ∈ K t .
Since x t k ∈ K t k , for all k ∈ N, it results in x ij t k ≤ q j t k , ∀j 1, . . ., n, ∀k ∈ N.

6.20
Passing to the limit as n → ∞ and taking into account the continuity assumption on the functions x, x, p, q, we obtain x ij t ≤ q j t , ∀j 1, . . ., n.

6.21
As a consequence x t ∈ K t , and, hence, the condition K2 is achieved.

Continuity Theorems for Equilibrium Solutions
In order to show the continuity result for the dynamic oligopolistic market equilibrium solution in presence of both production and demand excesses, we present the following result see e.g., 2, Corollary 3.1 .x ij t ≤ q j t , ∀j 1, . . ., n .

6.23
The continuity of solutions to evolutionary variational inequalities with respect to time under the only assumption of continuity on the data has been proved in several papers see for instance 8-12 .Now, in our case, by applying Theorem 4.2 in 12 and taking into account Lemma 6.3, we obtain the following result.Theorem 6.5.Let one assume that the production function p, the demand function q, and the capacity constraints x and x are continuous on 0, T .Moreover, let one assume that the function −∇ D v is a strictly pseudomonotone and continuous on 0, T .Then the unique dynamic market equilibrium distribution in presence of both production and demand excesses x * ∈ K is continuous on 0, T .

Numerical Example
Let us present a numerical example about the dynamic oligopolistic market equilibrium problem in presence of both production and demand excesses.

Abstract and Applied Analysis
Let us consider four firms and four demand markets, as in Figure 1.Let x, x ∈ L 2 0, 1 , R 16 be the capacity constraints such that, a.e. in 0, 1 , Let p ∈ L 2 0, 1 , R 4 be the production function such that, a.e. in 0, 1 , and let q ∈ L 2 0, 1 , R 4 be the demand function such that, a.e. in 0, 1 , As a consequence, the feasible set is x ij t ≤ q j t , ∀j 1, . . ., 4, a.e. in 0, 1 .

7.4
Let us consider the profit function v ∈ L 2 0, 1 × L 2 0, 1 , R The dynamic oligopolistic market equilibrium distribution in presence of excesses is the solution to the evolutionary variational inequality: In order to compute the solution to 7.8 we make use of the direct method see 28-30 .We consider the following system: 8x x ij t ≤ q j t , ∀j 1, . . ., 4, 7.9 and we get the following solution, a.e. in 0, 1 :

Concluding Remarks
In this paper, we have considered the variational formulation for the dynamic oligopolistic market equilibrium problem in presence of both production and demand excesses.The very general model allows to study all the economic periods that a market can be gone through.In this way, the previous models presented in 2 and in 1 are improved.The equilibrium conditions are given according to the well-known dynamic Cournot-Nash principle and by means Lagrange multipliers which allow to point out the importance of the excesses in the equilibrium solutions.Making use of the variational formulation, which expresses the equilibrium conditions, the equivalence between the two equilibrium definitions is proved.In particular, the evolutionary variational inequality allows to obtain very important theoretical results for equilibrium solutions.More precisely, under general assumptions, the existence of equilibrium solutions is guaranteed.Moreover, after that the powerful property of the set convergence in Kuratowski's sense has been proved for the constraint set, a continuity result for the equilibrium solution has been obtained.

Lemma 4 . 1 .
Let x * ∈ K be a solution to the variational inequality 2.20 and let one set
then, taking into account 2.22 , λ * It is worthy to underline that in Definition 2.3, even if in 2.21 -2.25 the unknown Lagrange variables λ * ij , ρ * ij , μ * i , ν * j appear, they do not influence the equilibrium definition because the following equivalent condition in terms of evolutionary variational inequality holds.
* i t ν * j t − λ * ij t ∂v i t, x * t /∂x ij , namely, μ * i t ν * j t − λ * ij t is equal to the marginal profit, * ∈ K 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: Proof of Theorem 2.4.Let us assume that x * ∈ K is an equilibrium solution according to Definition 2.3.Then, taking into account that λ * ij t x ij t −x * ij t 0 and ρ * ij t x * ij t −x ij t 0, a.e. in 0, T , we have for every x ∈ K, a.e. in 0, T , ∀j 1, . . ., n, ∀i 1, . . ., m, ∀j 1, . . ., n, R mn : x ij t ≤ x ij t ≤ x ij t , ∀i 1, . . ., m, ∀j 1, . . ., n, K t k • denotes the Hilbertian projection on K t k .