Existence and optimality of competitive equilibria. (English) Zbl 0676.90001

Aliprantis, Charalambos D.; Brown, Donald J.; Burkinshaw, Owen

Berlin etc.: Springer-Verlag. xii, 285 p. DM 110.00/hbk (1989).

This monograph is concerned with the existence and optimality of competitive equilibrium with an infinite number of commodities, one of the frontiers areas of general equilibrium theory. Commodities are defined as physical goods which may differ in the location or time at which they are produced or consumed, or in the state of world in which they become available. If we allow an infinite variation in any of these contingencies, then we are led to consider economies with infinitely many commodities. The authors’ extension of the standard Arrow-Debreu-McKenzie model with a finite number of agents and commodities to economies with an infinite number of commodities constitutes the core of the book.
This book consists of five chapters. Chapter 1 reviews the main results established in the Arrow-Debreu-McKenzie model of general equilibrium. Chapter 2 briefly discusses the theory of topological Riesz spaces. In the following three chapters, the equilibrium analysis is conducted within the framework of dual topological Riesz spaces. Chapter 3 is concerned with the existence and optimality of exchange economies with an infinite number of commodities. Chapter 4 is addressed to a more demanding problem for production economies in the presence of an infinite number of commodities. Chapter 6 deals with the overlapping generations model with both an infinite (countable) number of agents and an infinite number of commodities. The authors present a new commodity-price space duality for the OGP model and consider a weaker notion of optimality - referred to Malinvaud optimality.
To sum up, this monograph seems to be a significant contribution to the literature on economies with an infinite number of commodities although the scope of the audience is a bit limited.

Reviewer: Y.Sakai

MSC:

91B50

General equilibrium theory

Keywords:

existence; optimality; competitive equilibrium; infinite number of commodities; Arrow-Debreu-McKenzie model; exchange economies; overlapping generations; Malinvaud optimality