Equilibria in abstract economies with a continuum of agents with discontinuous and non-ordered preferences

This paper focus on the problem of the existence of an equilibrium in abstract economies and exchange economies. Spanning over the literature we have managed to extend and generalize some previous results. In particular, we generalize the main theorem of Yannelis (1987) on the existence of an equilibrium in an abstract economy with a continuum of agents, by allowing for discontinuous preferences. As a corollary of this result, we extend the finite agent Cournot–Nash equilibrium existence theorems with discontinuous preferences (e.g., Reny, 1999; Bareli and Meneghel, 2013; He and Yannelis, 2016; among others), to a continuum of agents. We also obtain an existence theorem for an abstract economy which allows for a convexifying effect on aggregation and nonconvex strategy and constraint sets. Furthermore, our new main theorem is used to prove the existence of a Walrasian equilibrium with a continuum of agents with discontinuous, non-ordered, interdependent and price-dependent preferences and thus extending the results of Aumman (1966) and Schmeidler (1969).