J Glob Optim (2008) 40:87–97 DOI 10.1007/s10898-007-9199-0 An existence result of a quasi-variational inequality associated to an equilibrium problem Maria Bernadette Donato · Monica Milasi · Carmela Vitanza Received: 1 June 2007 / Accepted: 9 June 2007 / Published online: 20 July 2007 © Springer Science+Business Media, LLC 2007 Abstract In this paper we consider a Walrasian pure exchange economy with utility func- tion which is a particular case of a general economic equilibrium problem, without produc- tion. We assume that each agent is endowed with at least of a commodity, his preferences are expressed by an utility function and it prevails a competitive behaviour: each agent regards the prices payed and received as independent of his own choices. The Walrasian equilibrium can be characterized as a solution to a quasi-variational inequality. By using this variational approach, our goal is to prove an existence result of equilibrium solutions. Keywords Competitive equilibrium · Variational and quasi-variational inequality · Mosco’s convergence 1 Introduction The full recognition of the general equilibrium concept can be attributed unmistakably to Leon Walras [24]. Taking into account more aspects of a real economy, he obtains a system of equations, which he calls the “equations of exchange”: a solution to this system is an equilibrium for an exchange economy. The first rigorous result on the existence of general equilibrium is due to a series of papers by Wald [23]. Wald’s papers were of forbidding math- ematical depth, not only in the use of sophisticated tools, but also in the complexity of the argument. A help, finally, came from the development of a related line of research, as the Von Neumann’s theory of games. He deduced [20], an existence theorem from a generalization of the Brouwer’s fixed point theorem. With these foundations, plus the influence of the rapid M. B. Donato · M. Milasi · C. Vitanza (B ) Contrada Papardo, salita Sperone, 31, Messina, Italy e-mail: bdonato@dipmat.unime.it M. Milasi e-mail: monica@dipmat.unime.it C. Vitanza e-mail: vitanzac@unime.it 123