Equilibrio dinamico per i mercati oligopolistici

We consider the dynamic oligopolistic market equilibrium problem introduced in the chapther one, in which the equilibrium conditions are equivalently expressed in terms of an evolutionary variational inequality. We give existence theorems and apply the infinite dimensional duality theorem developed in chapter two, obtaining the existence of Lagrange variables, which allow to describe the behaviour of the market. Moreover, we present some sensitivity results. We remark that the variational inequality formulation plays a fundamental role in order to achieve all the above results. The oligopoly problem, which consists of a finite number of firms involved in the production of homogenous commodities in a noncooperative manner, is one of the classical problems in economics, dating to Cournot, who first studied this problem. In particular, Cournot investigated competition between two producers, the so-called duopoly problem, and is credited with being the first to study noncompetitive behaviour. In his treatise, the decisions made by the producers are said to be in equilibrium if no one can increase his income by unilateral action, given that the other producer does not alter his decision. Subsequently, to fundamental Cournot contributions, the problem has been studied extensively, both with respect to the definition and the search of equilibrium solutions, using, for example, variational inequality theory, as well as in a dynamical context, in order to understand the interactions among firms. Nash in turn generalized Cournot's concept of equilibrium for a behavioral model consisting of n agents or players, each acting in his own self-interest, which has come to be called a noncooperative game. It has been showed that Nash equilibria satisfy variational inequalities, under the assumption that each ui is continuously differentiable on X and concave with respect to xi. Karamardian relaxed the assumption of compactness of X and provided a proof of existence and uniqueness of Nash equilibria under the strong monotonicity condition. Moreover, if the operator satisfies the strict monotonicity condition then the uniqueness of the solution is guaranteed, provided that the equilibrium exists. In our work for the sake of generality, we consider the oligopoly problems, operating under the Nash equilibrium concept of noncooperative behaviour, as a problem of the game theory. In particular, we consider the arms as players and their commodity production output and shipments as their strategies. The oligopolistic market is considered in an interval of time [0; T] and, hence the profit and the commodity shipments depend on time. The reason for considering the evolution in time of the oligopolistic market remains in the fact that the static case considers only a representation of an instantaneous status of the market during evolution in time. In such a way it is not possible to take into account the dynamic of market adjustment processes in which a lag of time response is operating, whereas the dependence on time of the profit allows to remedy to the lack of adjustments, for example by means of a memory term. We prove that the time-dependent oligopolistic market equilibrium conditions can be directly incorporated into an evolutionary variational inequality and this result allow to apply the powerful methods and techniques of variational inequalities theory to solve the problem. In fact, we can give some existence theorems with and without mono Duality theory for the dynamic oligopolistic market equilibrium problem assumptions. Then we can search the Lagrange variables associated to the problem and overcome, using the new results obtained in chapther tree the difficulty that the ordering cone which defines the cone constraints is empty. Finally, in the chapter four, we can present also some results of sensitivity analysis, which allows to know how the solution changes under controlled variation of the data. At last, we apply the sensitivity analysis to an example.