The duality and Lagrangean theory provide interesting contributions, absolutely necessary for the better understanding and handling of an increasing number of equilibrium problems whether in the static case or in the evolutionary case: the traffic equilibrium problem, the financial equilibrium problem, the spatially distributed markets equilibrium problem, the oligopolistic market equilibrium problem, the mixed equilibrium problem, the pollution control problem, the vaccination problem, and the Walras equilibrium problem (see e.g. [1], [2], [3], [4], [5], [6], [7], [8], [9] and [10]). Each of the aforementioned problems can be transformed into a variational problem on a convex subset KK of a suitable functional space, for which an impressive quantity of results in terms of existence, calculation of the solutions, stability and sensitivity analysis hold. This study is also undertaken by applying a suitable duality and Lagrangean theory in order to obtain necessary and/or sufficient optimality conditions. The Lagrangean theory is useful not only to provide a qualitative analysis of equilibria, but also to calculate the solutions.
In this paper, a dynamic competitive equilibrium for a Walrasian pure exchange economy has been considered. This problem was introduced by Walras in [11]. The equilibrium model is built through the maximization of a utility function from which a nonpositive excess demand function is derived. Still now the mathematical study of the Walras equilibrium represents an active and interesting research topic (see e.g. [12], [13], [14], [15], [8], [16], [17] and [18]). In this paper, we introduce a continuum model of a competitive equilibrium for a pure exchange economy in the finite interval [0,T][0,T] and we give a dynamic formulation of the equilibrium conditions. In this evolutionary market the aim of each agent is not to maximize his utility at the fixed instant tt, but to maximize it globally in the whole period of time. During the interval of time [0,T][0,T] each agent trades his own commodities with the other agents taking into account his own budget set: the amount that each agent pays for acquiring the goods in the whole period [0,T][0,T] is at most the amount that each agent receives, during the period [0,T][0,T], as his initial endowment. Then, mathematically the equilibrium is reformulated in terms of a maximization problem of an integral utility function on a feasible set, in which the constraint is given in an integral form and in terms of a nonpositive excess demand function. First, we prove that the equilibrium is a solution to a suitable evolutionary quasi-variational inequality which is set in the Lebesgue space L2([0,T],R)L2([0,T],R). Our main result in this paper is to obtain the Lagrangean variables relative to the equilibrium, by applying the Lagrangean theory in infinite dimensional normed spaces. These variables allow us to have a complete description of the behaviour of the economic market during the evolution in time. We would like to emphasize that, since the budget constraint is expressed in an integral form, it is not possible to obtain the equivalence between the evolutionary variational problem and a pointwise variational problem in a finite dimensional space. So we cannot calculate equilibrium solutions by solving the pointwise variational inequality. Our study will also be useful to overcome the difficulty of computing equilibrium solutions. The paper is organized as follows: in Section 2, we introduce the evolution in time of a competitive equilibrium problem for a pure exchange economy. In Section 3, we prove that the evolutionary equilibrium is a solution to a suitable evolutionary quasi-variational inequality and, by using this variational formulation, we apply the Lagrangean theory to our problem. Finally, the equilibrium is characterized in terms of a quasi-variational inequality and in terms of the Lagrange multipliers. In Section 4, we conclude the paper with some remarks and, in order to support the theoretical results, we provide a numerical example. So this numerical example will provide evidence of the efficacy of our main result.
In this section, we would like to point out the practical aspects of the theoretical results obtained in the previous section. In this paper, it has been shown that a dynamic competitive equilibrium, under suitable assumptions, represents a solution to an evolutionary quasi-variational inequality. By means of this mathematics formulation, by applying the Lagrangean theory, we have proven the existence of Lagrangean multipliers. The interest of this study consists in the fact that, first of all we can provide a qualitative analysis of the dynamic equilibrium, since the Lagrangean multipliers are able to give a complete description of the behaviour of the evolutionary market, and, furthermore, we can calculate the equilibrium solutions. More precisely, we were able to obtain an equilibrium solution by solving the system provided by Theorem 3. Furthermore, it is important to observe that in our model it is not possible to obtain equivalence between the evolutionary problem and a pointwise problem and this is a difficulty in the calculus of equilibrium solutions. We were able to overcome this difficulty by using Theorem 5.
