On solving dynamic oligopolistic market equilibrium problems in presence of excesses

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1 Communications in Applied and Industrial Mathematics, ISSN , e-415 DOI: /journal.caim.415 On solving dynamic oligopolistic market equilibrium problems in presence of excesses Annamaria Barbagallo 1 and Paolo Mauro 2 1 Department of Mathematics and Applications R. Caccioppoli, University of Naples Federico II, via Cintia Naples, Italy annamaria.barbagallo@unina.it 2 Department of Mathematics and Computer Science University of Catania viale Andrea Doria n Catania, Italy mauro@dmi.unict.it Communicated by Nicola Bellomo Abstract This paper deals with the dynamic oligopolistic market equilibrium problem in presence of excesses. In particular, besides presenting the problem in which only the production excess is considered, the demand excess is also discussed. In both cases, the variational formulation of the equilibrium conditions is given. Taking into account the continuity property of equilibrium solutions, it is possible to introduce a discretization procedure in order to present a computational method for the dynamic oligopolistic market equilibrium problems in presence of excesses. Moreover, a convergence result is proved. Finally, the performance of our algorithm is shown on some examples. Keywords: Dynamic oligopolistic market equilibrium problem, production excess, demand excess, evolutionary variational inequality, discretization procedure. AMS Subject Classification: 49J40, 58E35, 65K10, 91A10 1. Introduction. The main purpose of this paper is to provide a computational method for solving oligopolistic market equilibrium problems in presence of excesses. In the literature, these problems are modelled as evolutionary variational inequalities as well as it happens for the most important time-dependent equilibrium problems [1 8]. Moreover, many problems arising from the theory of population dynamics and other physics phenomena are studied not only by means of partial differential equations (for further details see Received Accepted Published Licensed under the Creative Commons Attribution Noncommercial No Derivatives

2 A. Barbagallo et al e.g. [9 16]), but also by means of evolutionary variational inequalities (see for instance [17,18]). It is worth to perform a little history of the problem in question. The oligopoly problem, namely the problem regarding the equilibrium between a finite number of firms producing homogenous commodities in a noncooperative manner, has its origin in Cournot who first studied this problem in [19]. Cournot investigated competition between only two producers while Nash, one century later, generalized the problem to n agents, each acting in his own self-interest, which has come to be called a noncooperative game (see [20,21]). The time-dependent formulation of the oligopolistic market equilibrium problem has been presented, for the first time, in [22]. Since 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, as M.J. Beckmann and J.P. Wallace pointed out in [23], it is fundamental to study of the dynamic behaviour of the model. Moreover, an other important aspect is that the 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, for instance [24]). In the study of evolutionary variational inequalities the search of the Lagrange variables is of topical importance and, to this regard, we mention the paper [25] where the authors applied the infinite-dimensional duality results developed in [26] to overcome the difficulty of the voidness of the interior of the ordering cone which defines the constraints and so proved the existence of Lagrange variables which permit to describe the behaviour of the market. Moreover, in [25] 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 [27], an oligopolistic market equilibrium model with an explicit long-term memory has been considered. In [1] and in [2] the variational formulation of the oligopolistic market equilibrium problems in presence of production excesses and both production and demand excesses are introduced. The equilibrium conditions in terms of the well known dynamic Cournot-Nash principle are analyzed, and then the equilibrium conditions will be expressed also by using the Lagrange multipliers and the infinite-dimensional duality theory. The equivalence between the two conditions is proved because they are both expressed by an appropriate evolutionary variational inequality. Moreover, thanks to the variational formulation, some existence and regularity results for the equilibrium solutions are shown. In this paper we propose an algorithm for the computation of equilibria, by means of discretization methods. The continuity of the solution to 2

3 DOI: /journal.caim.415 dynamic oligopolistic market equilibrium problems in presence of excesses, proved in [1] and in [2], allows us to consider a partition of the time interval and hence to reduce the infinite-dimensional problem to some finitedimensional problems that can be solved by means of a known method. Since in literature some numerical schemes for the calculation of solutions to dynamic equilibrium problems are available (see for instance [22,28 32]), our results seem to have a particular relevance. More specifically, we propose a discretization procedure which reduces the infinite-dimensional problem to the calculus of solutions to finite-dimensional variational inequality. In particular, taking into account of the continuity, we discretize the time interval [0, T ] and then we can compute, by means of the extragradient method in Marcotte s version, the solution of the finite-dimensional oligopolistic market equilibrium problems obtained using the discretization. At last, we construct an approximation solution with linear interpolation. In order to obtain numerical results, we make use of a MatLab computation. Moreover, we present some numerical examples which show the efficiency of the algorithm. In fact, we compute the equilibrium solutions by using also the direct method (see [33 35]) and we obtain the same numerical results. At last, we compare the numerical equilibrium solutions for the problem in presence only of production excesses and the one in presence of both production and demand excesses. The paper is organized as follows: in Section 2 we recall the dynamic oligopolistic market equilibrium problem in presence of production excesses and then both production and demand excesses, then in Section 3 we describe an algorithm and we prove a convergence result, and finally in Section 4 we report the computational results. 2. Dynamic oligopolistic market equilibrium problem. Let us consider, in a period time [0, T ], T > 0, m firms P i, i = 1,..., m, that produce one only commodity and n demand markets Q j, j = 1,..., n, that are generally spatially separated. Let p i, i = 1,..., m, be the nonnegative commodity output produced by firm P i at the time t [0, T ]. Let q j, j = 1,..., n, be the nonnegative demand for the commodity at demand market Q j at the time t [0, T ]. Let x ij, i = 1,..., m, j = 1,..., n, be the nonnegative commodity shipment between the supply market P i and the demand market Q j at the time t [0, T ]. In the following, we set x i = (x i1,..., x in ), i = 1,..., m, 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 com- 3

4 A. Barbagallo et al modity shipments into a matrix-function x : [0, T ] R mn +. We assume that the nonnegative commodity shipment belongs to L 2 ([0, T ], R mn + ). 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, f i = f i (t, x). 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, in general, upon the entire consumption pattern, namely, d j = d j (t, x). Moreover, let g i, i = 1,..., m, denote 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, g i = g i (t, x). 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, c ij = c ij (t, x). Hence, we have the following mappings, f, g : [0, T ] L 2 ([0, T ], R mn + ) L 2 ([0, T ], R m + ), d : [0, T ] L 2 ([0, T ], R mn + ) L 2 ([0, T ], R n +), c : [0, T ] L 2 ([0, T ], R mn + ) L 2 ([0, T ], R mn + ). The profit of the firm P i, i = 1,..., m, at the time t [0, T ] is, (1) v i (t, x) = d j (t, x)x ij f i (t, x) g i (t, x) c ij (t, x)x ij, namely, it is equal to the price that the demand markets are disposed to pay minus the production costs, the storage costs and the transportation costs. The aim of the model for this problem in which the m firms supply the commodity in a noncooperative fashion, is to maximize its own profit function considered the optimal distribution pattern for the other firms. We want to determine a commodity distribution matrix-function for which the m firms and the n demand markets will be in a state of equilibrium as defined later. Suppose also that the following assumptions on the profit function v 4

5 DOI: /journal.caim.415 ( ) vi (given by (1)) and on D v = x ij i = 1,..., m j = 1,..., n hold: i. v(t, x) is continuously differentiable a.e. in [0, T ], ii. D v is a Carathèodory function such that (2) h L 2 ([0, T ]) : D v(t, x) mn h x mn a.e. in [0, T ], iii. v i (t, x) is pseudoconcave with respect to the variables x i, i = 1,..., m, namely the following holds a.e. in [0, T ] : v (3) (t, x 1,..., x i,..., x m ), x i y i 0 x i v i (t, x 1,..., x i,..., x m ) v i (t, x 1,..., y i,..., x m ). In oligopolistic markets the pseudoconcavity assumption (3) is satisfied mainly because explicit conditions on the data (i.e. production cost, demand price, storage cost, transaction cost) are not given Dynamic oligopolistic market equilibrium problem in presence of production excesses. At first, we want to consider the case in which some of the amounts of the commodity available are sold out whereas for a part of the producers can occur an excess of production (see [1]). Let ɛ i, i = 1,..., m, be the nonnegative production excess for the commodity of the firm P i at the time t [0, T ]. Let us group the production excess into a vector-function ɛ : [0, T ] R m +. The following feasibility condition holds, a.e. in [0, T ] : (4) p i = x ij + ɛ i, i = 1,..., m, namely the quantity produced by each firm P i at the time t [0, T ] must be equal to the sum of the commodity shipments from that firm to each demand markets plus the production excess at the same time t [0, T ]. We suppose that ɛ L 2 ([0, T ], R m + ). As a consequence, we have p L 2 ([0, T ], R m + ). In virtue of (4) since the production excess ɛ i at the time t [0, T ] is nonnegative, we have (5) x ij p i, i = 1,..., m, a.e. in [0, T ]. 5

6 A. Barbagallo et al (6) Taking into account (5), the feasible set for the distribution pattern is: { K = x L 2 ([0, T ], R mn ) : x ij 0, i = 1,..., m, j = 1,..., n, a.e. in [0, T ], } x ij p i, i = 1,..., m, a.e. in [0, T ]. Now, we give the dynamic oligopolistic market equilibrium conditions in presence of production excesses by using Lagrange variables associated to the constraints. Definition 2.1. 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 exists λ ij L2 (0, T ), µ i L2 (0, T ), such that (7) v i(t, x ) x ij + µ i = λ ij, (8) ( v i(t, x ) ) + µ i x x ij = 0, ij (9) λ ijx ij = 0, λ ij 0, (10) µ i x ij p i = 0, µ i 0. The terms λ ij and µ i are the Lagrange multipliers associated to the constraints x ij 0 and to x ij p i, respectively. Making use of the infinite-dimensional duality theory (see [26,36 38]), it is possible to prove the following equivalence (see Theorem 2.2 in [1]). Theorem 2.1. x K is a dynamic oligopolistic market equilibrium in presence of production excesses if and only if it satisfies the evolutionary variational inequality (11) T 0 m i=1 v i(t, x ) x ij (x ij x ij)dt 0 x K, 6

7 DOI: /journal.caim.415 where K is given by (6). Moreover, the equivalence between the evolutionary variational inequality (11) and the dynamic Cournot-Nash principle, which we recall in the following, has been proved (see Theorem 3.1 in [22]). Definition 2.2. x K is a dynamic oligopolistic market equilibrium in presence of production excesses if and only if for each i = 1,..., m and a.e. in [0, T ] we have (12) v i (t, x ) v i (t, x i, ˆx i ), a.e. in [0, T ], where x i = (x i1,..., x in ), a.e. x i+1,..., x m). in [0, T ] and ˆx i = (x 1,..., x i 1, Hence, 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, such that the m firms try to maximize their own profit at the time t [0, T ] Dynamic oligopolistic market equilibrium problem in presence of production and demand excesses. Now we recall the dynamic oligopolistic market equilibrium problem in presence of both production and demand excesses introduced in [2], namely we allow that for a part of the producers can occur an excess of production whereas for a part of the demand markets the supply can not satisfy the demand. In order to clarify the presence of both excesses we consider some concrete economic situations. During an economic crisis period the presence of production excesses can be due to a decrease in market demands and, on the other hand, the presence of demand excesses may occur when the supply can not satisfy the demand especially for fundamental goods. Moreover, since the market model presented in this paper evolves in time, the presence of both excesses can be consequence of the fact that the physical transportation of commodity between a firm and a demand market is evidently limited, therefore there exist some time instants in which, though some firms produce more commodity than they can send to all the demand markets, some of the demand markets require more commodity. In addition to what we have considered previously, let us denote by δ j, j = 1,..., n, the nonnegative demand excess for the commodity of the demand market Q j at the time t [0, T ]. Let us group the demand excess into a vector-function δ : [0, T ] R n + and let us suppose that the 7

8 A. Barbagallo et al following feasibility conditions hold, a.e. in [0, T ] : (13) p i = x ij + ɛ i, i = 1,..., m, (14) q j = m x ij + δ j, j = 1,..., n. i=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 ] and the quantity demanded by each demand market Q j at the time t [0, T ] must be equal to the commodity shipments from that firm to all the demand markets plus the demand excess at the same time t [0, T ]. Taking into account (13) and (14), since the production excess ɛ i at the time t [0, T ] and the demand excess δ j at the same time t [0, T ] are nonnegative, we have (15) x ij p i, i = 1,..., m, a.e. in [0, T ], (16) m x ij q j, j = 1,..., n, a.e. in [0, T ]. i=1 Moreover, we assume that the nonnegative commodity shipment between the producer firm P i and the demand market Q j has to satisfy time-dependent constraints, namely, there exist two nonnegative functions x, x L 2 ([0, T ], R mn + ) such that (17) x ij x ij x ij, i = 1,..., m, j = 1,..., n, a.e. in [0, T ] that define the limits in transportation. Let us assume that ɛ L 2 ([0, T ], R m + ), and δ L 2 ([0, T ], R n +). As a consequence, we have p L 2 ([0, T ], R m + ), and q L 2 ([0, T ], R n +). According to (15), (16) and (17), the feasible set for the distribution pattern is: 8

9 DOI: /journal.caim.415 { (18) K = x L 2 ([0, T ], R mn ) : x ij x ij x ij, i = 1,..., m, j = 1,..., n, a.e. in [0, T ], x ij p i, i = 1,..., m, a.e. in [0, T ], } m x ij q j, j = 1,..., n, a.e. in [0, T ]. i=1 Let us present the dynamic oligopolistic market equilibrium condition in presence of 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 exists λ ij L2 (0, T ), ρ ij L2 (0, T ), µ i L 2 (0, T ), ν j L2 (0, T ) such that (19) v i(t, x ) x ij + ρ ij + µ i + ν j = λ ij, (20) λ ij(x ij x ij) = 0, λ ij 0, (21) ρ ij(x ij x ij ) = 0, ρ ij 0, (22) µ i x ij p i = 0, µ i 0, ( m ) (23) νj x ij q j = 0, νj 0. i=1 The terms λ ij, ρ ij, µ i, ν j are the Lagrange multipliers associated to the constraints x ij x ij, x ij x ij, x ij p i and to m x ij q j, respectively. i=1 9

10 A. Barbagallo et al Also in this case, the previous equilibrium conditions is equivalent to an evolutionary variational inequality (see Theorem 2.2 in [2]). Theorem 2.2. x K is a dynamic oligopolistic market equilibrium in presence of both production and demand excesses if and only if it satisfies the evolutionary variational inequality (24) T 0 m i=1 where K is given by (18). v i(t, x ) x ij (x ij x ij)dt 0 x K, Finally, we have that the state of equilibrium of the dynamic oligopolistic market in presence of both production and demand excesses is the same Cournot-Nash principle as the one considered in the previous model. For existence and regularity results, consult [1,2,39,40]. 3. Solution method. Let us introduce a method to solve dynamic oligopolistic market equilibrium problems in presence of excesses expressed by evolutionary variational inequalities. We suppose that the assumptions which ensure the continuity of dynamic oligopolistic market equilibrium solution are satisfy. As a consequence, (11) and (24) hold for each t [0, T ], namely (25) D v(t, x ), x x 0, x K, t [0, T ] where K = { x R mn : x ij 0, i = 1,..., m, j = 1,..., n, x ij p i, for the problem with only production excesses and { K = i = 1,..., m x R mn : x ij x ij x ij, } i = 1,..., m, j = 1,..., n, x ij p i, i = 1,..., m m x ij q j, i=1 j = 1,..., n }, 10

11 DOI: /journal.caim.415 for the problem with both production and demand excesses. In the following, applying a discretization procedure and making use of the extragradient method in Marcotte s version, we compute the solutions of the evolutionary variational inequality which expresses the dynamic oligopolistic market equilibrium conditions in presence of excesses. Let us consider a partition of [0, T ], such that: 0 = t 0 < t 1 <... < t r <... < t N = T. For each point t r, r = 0, 1,..., N, of the partition, we consider the finite-dimensional variational inequality (26) D v(t r, x (t r )), x(t r ) x (t r ) 0, x(t r ) K(t r ). We compute now the solution to the finite-dimensional variational inequality (26) making use of a modified version of the extragradient method introduced by Marcotte in [41]. The algorithm starting from any x 0 (t r ) K(t r ) and a fixed number α 0 > 0, iteratively updates x k+1 (t r ) from x k (t r ) according to the following projection formulas (27) x k+1 (t r ) = P K(tr)(x k (t r ) α k ( D ( x k (t r )))), (28) x k (t r ) = P K(tr)(x k (t r ) α k ( D (x k (t r )))) for k N, where α k is chosen as following { α k 1 (29) α k = min 2, x k (t r ) x k (t r ) mn }. 2 ( D (x k (t r ))) ( D ( x k (t r ))) mn If D v is a monotone and Lipschitz continuous mapping, then, the convergence of the scheme is proved. This method was improved by Tinti in [42]. After the iterative procedure, we get the dynamic equilibrium solution through a linear interpolation of the computed static equilibrium solutions Convergence analysis. In this section, we investigate on the convergence of algorithms presented in the previous section. We prove that under suitable assumptions, the sequence, generated by the algorithm, converges in L 1 -sense to the dynamic oligopolistic market equilibrium solution in presence of excesses. Let us assume that all hypotheses which ensure the continuity of solution to (25) and the convergence of the Marcotte s extragradient method to compute solutions to finite-dimensional variational inequalities hold. Let 11

12 A. Barbagallo et al us introduce a sequence {π s } s N of (not necessarily equidivided) partitions of the time interval [0, T ] such that π s = (t 0 s, t 1 s,..., t Ns s ), where 0 = t 0 s < t 1 s <... < t Ns s = T. We consider a sequence of equidivided partitions, in the sense that k s := max{ t r s t r 1 s : r = 1, 2,..., N s }, approaches zero for s +. By the interpolation theory, we know that if we construct the approximate solution to (25) by means of Hermite s polynomial, using known values of the solution x, the sequence converges uniformly to the exact solution. We do not use Hermite s polynomial, but we consider an approximation by means of piecewise constant functions and we can prove that the convergence is in L 1 -sense. Let us consider the approximate solutions to (25) given by the following formula: N s (30) x s = x (t r s)χ [t r 1 s,t r[, s r=1 where x (t r n) is the solution to the finite-dimensional variational inequality which is obtained to (25) for t = t r s, which can be compute by means of the Marcotte s extragradient method. Let us estimate the following integral T 0 N s x r=1 T = x (t r s)χ [t r 1 s,t r[ dt s mn 0 N s r=1 N s t s r r=1 t s r 1 x χ [t r 1 s Ns,t r s[ r=1 x x (t r s) mn dt. x (t r s)χ [t r 1 s,t r s[ mn dt Being x uniformly continuous, it results that for every ε > 0 there exists δ > 0 such that if t [t r 1 s, t r s] satisfies the condition t t r s < δ it results 0 x x (t r s) mn < ε T, for r = 1, 2,..., N s, s N. Choosing s large enough in such way that k s < δ, we reach T N s N s (31) x x (t r ε s)χ [t r 1 s,t r[ dt < s mn T (ts r t s r 1) = ε. r=1 Taking into account (31), we conclude that the sequence (30) converges in L 1 -sense to the dynamic oligopolistic market equilibrium solution in presence of excesses. r=1 12

13 DOI: /journal.caim Numerical results. Let us consider some numerical examples of the dynamical oligopolistic market equilibrium problem in presence of excesses consisting of three firms and four demand markets, as in Figure 1. At first, we suppose that there are only production excesses. P 1 P 2 P 3 Q 3 Q 4 Q 1 Q 2 Figure 1. Network structure of the numerical dynamic spatial oligopoly problem. Let p C([0, 1], R 3 ) be the production function such that, in [0, 1], p = ( 4t 3t 4t ) T, As a consequence, the feasible set is { K = x L 2 ([0, 1], R 3 4 ) : x ij 0, i = 1, 2, 3, j = 1, 2, 3, 4, a.e. in [0, 1], 4 } x ij p i, i = 1, 2, 3, a.e. in [0, 1]. Let v C 1 (L 2 ([0, 1], R 3 4 ), R 3 ) be the profit function defined by v 1 (t, x) = 4x x x x x 11 x 12 6x 13 x 14 +3tx tx 12 + tx 13 + tx 14, v 2 (t, x) = 5x x x x x 21 x 22 2x 23 x 24 +2tx tx tx tx 24, v 3 (t, x) = 10x x x x x 11 x 32 2x 33 x 34 2x 12 x 31 + tx tx tx tx 34. Then, the operator D v C(L 2 ([0, 1], R 3 4 ), R 3 4 ) is given by 8x 11 4x t 8x 12 4x t 12x 13 6x 14 + t 12x 14 6x 13 + t D v = 10x 21 2x t 4x 22 2x t 4x 23 2x t 4x 24 2x t. 20x 31 2x 12 + t 8x 32 2x t 8x 33 2x t 10x 34 2x t Moreover, it is possible to verify that D v is a strongly monotone operator (for the proof, you can see similar calculations in [1] and [2]). The dynamic oligopolistic market equilibrium distribution in presence of excesses is the solution to the evolutionary variational inequality: (32) i=1 v i(t, x ) x ij (x ij x ij)dt 0, x K. 13

14 A. Barbagallo et al Taking into account the direct method (see [33 35]), we consider the following system 8x x 12 3t = 0, 4x x 12 4t = 0, 12x x 14 t = 0, 6x x 14 t = 0, 10x (33) x 22 2t = 0, 2x x 22 2t = 0, 4x x 24 3t = 0, 2x x 24 2t = 0, 20x x 12 t = 0, 8x x 11 2t = 0, 8x x 34 10t = 0, 2x x 34 3t = 0 We get the following solution, in [0, 1], 1 6 t 5 12 t 1 18 t 1 18 t x 1 = 9 t 4 9 t 2 3 t 1 6 t, t 5 24 t t 1 19 t that belongs to the constraint set K, then it is the equilibrium solution. Moreover, the production excesses, in [0, 1], are given by ( 119 ɛ = 36 t t 2843 ) T 1140 t. Now, we solve the numerical problem using the generalized Marcotte s version of the extragradient method. This method is convergent for the properties of D v. Then, we can compute an approximate curve of equilibria, by selecting t r { k 20 : k {0, 1,..., 20}}. With the help of a Mat- Lab computation and choosing the initial point x 0 (t r ) = 2 3 t r 1 3 t r 1 2 t r t r 1 2 t r 1 10 t r 1 2 t r 1 8 t r t r 1 3 t r 2 5 t r 1 8 t r in order to start the iterative method, we obtain the equilibrium solutions for every time instant. Setting R(x k (t r )) = x k (t r ) x k 1 (t r ) the difference between two approximations of the equilibrium solution in the time instant t r and making use of the stopping criterion R(x k (t r )) , for r = 0, 1,..., 20, we obtain the approximate curve of equilibria shown in Figure 2. Now, we analyze the dynamic oligopolistic market equilibrium problem in presence of both production and demand excesses. 14

15 DOI: /journal.caim.415 Numerical solution x* 11 x* 12 x* 13 x* 14 x* 21 x* 22 x* 23 x* 24 x* 31 x* 32 x* 33 x* Time Figure 2. Curves of equilibria of problem with only production excesses. Let x, x C([0, 1], R 3 4 ) be the capacity constraints such that, in [0, 1], t 1 50 t t 2t 4t 6t t x = t t, x = 4t 3t 5t t t 1 2 t 0 t t t 3t Let p be the same production function considered in the previous example and let q C([0, 1], R 4 ) be the demand function such that, in [0, 1], Hence, the feasible set is { K = x L 2 ([0, 1], R 3 4 ) : q = ( 6t 4t 4t 8t ) T. x ij x ij x ij, i = 1, 2, 3, j = 1, 2, 3, 4, a.e. in [0, 1], 4 x ij p i, i = 1, 2, 3, a.e. in [0, 1], } 3 x ij q j, j = 1, 2, 3, 4, a.e. in [0, 1]. i=1 Let us consider the same profit function v. In order to compute the solution to (32) we make use of the direct method (see [33 35]). We consider again the system (33) and we get the same solution. But we observe that x 15

16 A. Barbagallo et al does not belong to the constraint set K because x 33 > t. Let us consider now the set { K = x L 2 ([0, 1], R 3 4 ) : x ij x ij x ij, i = 1, 2, 3, j = 1, 2, 3, 4, (i, j) (3, 3), a.e. in [0, 1], x 33 = t, a.e. in [0, 1], 4 x ij p i, i = 1,..., 3, a.e. in [0, 1], } 3 x ij q j, j = 1,..., 4, a.e. in [0, 1] i=1 and the system 8x x 12 3t = 0, 4x x 12 4t = 0, 12x x 14 t = 0, 6x x 14 t = 0, 10x (34) x 22 2t = 0, 2x x 22 2t = 0, 4x x 24 3t = 0, 2x x 24 2t = 0, 20x x 12 t = 0, 8x x 11 2t = 0, x 33 = t, 2x x 34 3t = 0 We can observe that 1 6 t 5 12 t 1 18 t 1 18 t x 1 = 9 t 4 9 t 2 3 t 1 6 t, t 5 24 t t 1 10 t is a solution, in [0, 1], since 8x x 34 10t < 0. Now, we are able to compute the production and the demand excesses of each firm in [0, 1] : ( 119 ɛ = 36 t t 161 ) T ( t, δ = 360 t t t 691 ) T 90 t, With the same method and initial data of previous example it is possible to compute the numerical solution, so we get the approximate curve of equilibria shown in Figure 3. Comparing the numerical results, we can remark that the equilibrium solutions are different and dependent on the presence of the excesses. The difference of the equilibrium solutions between the two examples is pointed 16

17 DOI: /journal.caim.415 Numerical solution x* 11 x* 12 x* 13 x* 14 x* 21 x* 22 x* 23 x* 24 x* 31 x* 32 x* 33 x* Time Figure 3. Curves of equilibria of problem with both production and demand excesses. out by the presence of capacity constraints. If we did not consider the capacity constraints, the equilibrium solutions computed through the direct method, would be the same and belonging to the same set K. The presence of the capacity constraints, like we see, forces the solution to belong to the face K. In this examples we observe that the numerical method fits very well the approximate solution with the exact solution computed through the direct metod. This verifies even the good convergence of the numerical method. REFERENCES 1. A. Barbagallo and P. Mauro, Evolutionary variational formulation for oligopolistic market equilibrium problems with production excesses, J. Optim. Theory Appl., vol. 155, pp. 1 27, A. Barbagallo and P. Mauro, Time-dependent variational inequality for an oligopolistic market equilibrium problem with production and demand excesses, Abstr. Appl. Anal., no , F. Raciti and L. Scrimali, Time-dependent variational inequalities and applications to equilibrium problems, J. Global Optim., vol. 28, pp , F. Raciti, Equilibrium conditions and vector variational inequalities: a complex relation, J. Global Optim., vol. 40, pp , M. B. Donato, M. Milasi, and C. Vitanza, A new contribution to a dynamic competitive equilibrium problem, Appl. Math. Letters, vol. 23, pp ,

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