Core and Walras equilibria in the subadditive economies

The model of large scale economies, and the search of equilibria for them, is one of the central problem in Mathematical Economics. After the framework proposed by Aumann, to represent perfect competition assuming that the space of agents is a non-atomic measure space, several extensions have appeared in the literature. Some authors have considered finitely additive models, particularly in the coalitional sense. More recently finitely additive economies with innite dimensional commodity space have been investigated from the individualistic point of view. In this paper we turn our attention again to the classical nite dimensional commodity space, but whith a more general structure for the set of agents: instead of the finite additivity of its structure, we shall assume that on only a capacity is given. To compensate fro the loss of additivity we have to assume some further conditions. First we adopt a model with correlated goods assuming that the components of the initial endowement are Co-monotone, that is to say that agents that have more of one good, also have more of the others. This can be assumed precisely when the market treats goods that are correlated. Besides we consider almost simple economies; this is an assumption inspired by the confederate models. More precisely we assume that the whole population can be decomposed into finitely many coalitions, and that agents within the same coalition have the same initial endowement and the same criteria of preference. This is not the same as considering a finite set of agents. In the first two sections we remind the properties of the monotone integral, and re-adapt the main economic concepts to a subadditive model. In the third section we discuss competitive equilibria and compare them with the core of the economy: we prove that competitive equilibria belong to the core, and conversely, that some particular allocations in the core are in fact competitive equilibria. Finally we provide a particular model where the initial endowement turns out to be an equilibrium.