Existence and Optimality of Competitive Equilibria

By C.D. Aliprantis, D.J. Brown, O. Burkinshaw
Springer-Verlag

This monograph is a systematic exposition of the authors' research on general equilibrium models with an infinite number of commodities. It is intended to serve both as a graduate text on aspects of general equilibrium theory and as an introduction, for economists and mathematicians working in mathematical economics, to current research in a frontier area of general equilibrium theory. To this end, we have provided two introductory chapters on the basic economic model and the mathematical framework. The exercises at the end of each section complement the main exposition.

Chapter one is a concise but substantiative discussion of the questions of existence and optimality of competitive equilibria in the Walrasian general equilibrium model of an economy with a finite number of households, firms and commodities. Our extension of this model to economies with an infinite number of commodities constitutes the core material of this book and begins in chapter three. Readers familiar with the Walrasian general equilibrium model as exposited in [13], [23] or [52] may treat chapter one as a handy reference for the main economic concepts and notions that are used throughout the book.

Chapter two is an introduction to the theory of topological Riesz spaces. These spaces are the mathematical structures we use to model economies with an infinite number of commodities. Chapter two should be readily accessible to readers familiar with the duality theory of locally convex spaces, say at the level of [13], [33], [58] or [62]. Comprehensive discussions of the theory of topological Riesz spaces can be found in the books [6], [8] and [63].

In chapters three and four we again address the questions of the existence and optimality of Walrasian equilibria for economies with a finite number of households and firms, but with an infinite number of commodities. In particular, chapter three is concerned with pure exchange economies and chapter four considers production economics.

The final chapter, chapter five, is our most original and significant contribution to the literature on economies with an infinite number of commodities. Building upon the analysis of the previous chapters we address the questions of existence and optimality of a competitive equilibrium in the overlapping generations model. This is an intertemporal model of exchange where there are a countable number of generations each finitely lived. Hence, formally this is an economic model with both an infinite (countable) number of agents (households) and an infinite number of commodities. Chapter five may be read immediately after chapter three.

C. D. ALIPRANTIS, D. J. BROWN, AND O. BURKINSHAW
November, 1988