General Equilibrium

Problems and Prospects

Frank Hahn, Fabio Petri

• Publisher: Routledge
• Year: 2003
• ISBN:
  9781134433612
• Edition: 1
• Topic: Commerce
• Subtopic: Commerce Général

In recent years certain leading figures in the world of economics have called the usefulness of general equilibrium theory into question. This superb new book brings together leading economic theorists with important contributions to the ongoing debate.
General equilibrium theorists including Michio Morishima, Michael Magill and Martine Quinzii debate strengths, weaknesses and possible futures with leading thinkers such as Herb Gintis, Pierangelo Garegnani and Duncan Foley, who seek to explain the rejection of general equilibrium. Uniquely, none of the contributors portray general equilibrium theory as the perfect guide to market economies actual behaviour, but rather illustrate that there is insufficient acquaintance with existing alternatives and that general equilibrium theory is often used as an ideal 'benchmark'.

1
Existence of GE

Are the cases of non-existence a cause for serious worry?*

Elvio Accinelli^†

* Work for the XII Workshop on “General Equilibrium: Problems, Prospects and Alternatives” 4–11 July 1999, at Certosa di Pontigniano, Siena, Italy. The author is grateful to Paulo K. Monteiro for helpful comments that were very important for this work, and thanks Celina Gutierrez and Fabio Petri for several comments and suggestions.

† Fac. de Ingeniería, IMERL, CC 30. Montevideo Uruguay. E-mail: [email protected]

In this work, we attempt to characterize the main theoretical difficulties in the proof of the existence of competitive equilibrium in infinite dimensional models. We shall show cases in which it is not possible to prove the existence of equilibrium and some others in which, although the existence of equilibrium can be proved, the equilibrium prices seem not to have natural economic interpretation. Nevertheless in pure exchange economies, most of these difficulties may be avoided by mild restrictions on the model. In productive economies new specific problems appear, for instance non-convexity of the production sets or non-boundedness of the feasible allocation sets. To prove the existence and the efficiency of the equilibrium in productive economies we need some strong hypotheses about the technological possibilities of each firm.

1.1 INTRODUCTION

The existence of Walrasian equilibrium can be established in considerable generality. In this work we will offer an examination of the existence problems in infinite dimensional economies. Although the title of this work does not suggest any restrictions to some special models, we limit ourselves to consider economies in infinite dimensional spaces; the advantage of this approach is its generality, and the cost is the necessity to follow the methodology of functional analysis. The well known difficulties that arise in finite dimensional cases to prove the existence of equilibrium, like the non-convexity of preferences or other of this kind, survive in infinite dimensional cases, and at the same time new problems appear. A successful formulation of this question not only requires novel technical arguments and a modification of our finite dimensional intuition, but also a new mathematics. Every economic problem requires its own appropriate mathematical treatment. For instance, to prove the existence of Walrasian equilibrium in a finite dimensional model, fixed point arguments were required; to prove the existence of the equilibrium in more realistic models, like infinite economies, i.e. economic models that consider the consumption space as a subset of an infinite dimensional vectorial space, new and more sophisticated mathematical arguments, specially new topological arguments, are required. So, it may be that to obtain success in providing correct answers to new and old questions yet not answered in economics, new mathematical arguments will be necessary. The economic problem determines the mathematical tool that is applied to obtain the solution and its precise formulation; this does not mean that we have to take a mathematical tool and then look for application.

Looking at commodities as physical goods which may differ by date or by the states of the world in which they become available, and allowing infinite variation in these contingents, the generalization of the classical model of GE to models of economies with infinitely many commodities looks natural.

While the extension of the classical general equilibrium model to an infinite dimensional setting gives new answers to relevant questions of the economic theory, new theoretical challenges appear; for example, new cases where it is still now theoretically impossible to prove the existence of the Walrasian equilibrium or where the mathematical interpretation or economic meaning of this equilibrium is not concrete or natural. Recent researches done by different authors tried to obtain new theoretical tools or more general conditions for existence of the Walrasian equilibrium on these models. For instance, the condition in preferences defined in (Mas-Colell, 1986) and known as properness plays an important role in compensating for the absence of interior points in positive cones in some Banach spaces. We shall analyze this and other conditions that play analogous roles.

For finite dimensional theory the finite dimensional linear space, Rⁿ is a canonical space, the situation changes drastically in more general conditions. By contrast with the finite dimensional cases there is not a canonical infinite dimensional commodity space. Different e...

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright Page
• Contents
• List of contributors
• Acknowledgements
• Introduction
• Introduction
• 1 Existence of GE: are the cases of non-existence a cause for serious worry?
• 2 Learning in intertemporal equilibrium models and the sunspot case
• 3 Disequilibrium and stability
• 4 Statistical equilibrium in economics: method, interpretation, and an example
• 5 Savings, investment and capital in a system of general intertemporal equilibrium
• 6 Some implications of endogenous contract enforcement for general equilibrium theory
• 7 Macroeconomics and general equilibrium
• 8 ‘Classical’ vs. ‘neoclassical’ theories of value and distribution and the long-period method
• 9 Endogenous uncertainty and rational belief equilibrium: a unified theory of market volatility
• 10 Incentives and the stock market in general equilibrium
• 11 Intermediation, the stock market and intergenerational transfers
• 12 The sequential indeterminacy problem
• 13 General equilibrium and the destiny of capitalism à la Schumpeter
• 14 A ‘Sraffian’ critique of general equilibrium theory, and the classical-Keynesian alternative
• 15 Competitive equilibrium and non-cooperative game theory: noise and bounded rationality
• 16 Applications of the classical approach
• 17 General equilibrium: problems, prospects and alternatives – an attempt at synthesis
• 18 General equilibrium: problems, prospects and alternatives – final discussion
• Index