Stochastic variational formulation for a general random time-dependent economic equilibrium problem
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Stochastic variational formulation for a general random time-dependent economic equilibrium problem Annamaria Barbagallo1 · Serena Guarino Lo Bianco2 Received: 16 January 2018 / Accepted: 5 March 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In the paper, in a Hilbert space setting, a random time-dependent oligopolistic market equilibrium problem in presence of both production and demand excesses is studied and the random time-dependent Cournot–Nash equilibrium principle by means of a stochastic variational inequality is characterized. Then, some existence results to such problem are established and the stochastic continuity of the equilibrium solution is proved. Moreover a simple numerical example illustrates the theoretical results. Keywords Random time-dependent Cournot–Nash equilibrium principle · Oligopolistic market equilibrium problem · Stochastic variational inequalities
1 Introduction The purpose of this note is to combine the new advances of the theory of stochastic and time-dependent variational inequalities with the Nash equilibrium game, and to propose an effective model of a oligopolistic market equilibrium problem. Taking into account these new tools, we are able to generalize results which have been obtained in the field of oligopolistic markets. In the last years many authors (see [8–13]) developed the study of stochastic variational inequalities and random equilibrium problems. Recently, a comprehensive study on the stochastic variational inequalities with anticipativity in a dynamic multistage setting is done in [19].
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Serena Guarino Lo Bianco [email protected] Annamaria Barbagallo [email protected]
1
Department of Mathematics and Applications “R. Caccioppoli”, University of Naples “Federico II, via Cintia, 80126 Naples, Italy
2
Department of Agricultural Science, University of Naples “Federico II”, via Università, 80055 Portici, NA, Italy
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A. Barbagallo, S. G. L. Bianco
In the last decade, the time-dependent variational formulation of the oligopolistic market equilibrium problem was introduced and intensively studied starting by [1]. In [3,4] the authors observed that 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, the presence of both production and demand excesses is a consequence of the fact that the physical transportation of commodities between a firm and a demand market is evidently limited, therefore, it is more realistic that some firms produce more fundamental good than they can send to all the demand markets and, on the other hand, some of the demand markets require more goods. The model in presence of uncertainty in which both production and demand excesses occur was analyzed in [2]. The development of the oligopolistic market equilibrium problem under conditions of uncertainty arises because th
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