A topological analysis of competitive economies

This thesis deals with some central issues in the theory of competitive economies and with the intrinsic topological nature that they exhibit. The topics discussed focus on economic models with a multitude of agents and many commodities and are divided in an extended introduction and four chapters. In Chapter 2 we prove a convexity result for the range of finitely-additive vector measures and correspondences with values in a locally convex space. These results are based on a new topological reformulation of the so-called saturation property which, in some form, generalizes the notion of non-atomicity of a measure space and plays an important role in many economic problems. Chapter 3 investigates what topological assumptions on the commodity space are essential to formulate and study the problem of existence of competitive equilibrium in a Walrasian competitive economy with many commodities. It includes a general theorem on the existence of equilibrium prices in abstract markets with infinite dimensional commodity spaces that is inspired by Nikaido's work of 1959. In Chapter 4 we study a coalitional model of an exchange economy in which coalitions are represented as the elements of a topological Boolean ring, allocations are finitely additive vector measures and the commodity space is an ordered locally convex space. In this general framework we will use the results of Chapter 2 to provide a topological condition for a perfectly competitive economy and use it to prove two extensions of classical theorems on the veto power of coalitions (Schmeidler 1972 and Vind 1972). This will allow us to conclude that the economic power of coalitions is an essentially topological property. Finally, Chapter 5 studies the competitive objection mechanism in a saturated economy with a separable Banach space of commodities whose positive cone has non-empty interior. By doing so we will provide a new characterization of Mas-Colell's bargaining set in the infinite dimensional setting. We will then introduce stronger notions of bargaining set that allow to extend our analysis to the case of markets with imperfections