Increasing Returns and Efficiency

By Martine Quinzii | Go to book overview

5
EXISTENCE

5.1 Single Firm Existence Theorem

It may seem surprising that we have discussed the normative properties of a marginal cost pricing equilibrium before establishing conditions under which such an equilibrium exists. In analyzing the normative properties, however, we have already obtained an existence result: under the assumptions of Theorem 4.1 an equilibrium exists for every income map. We shall now show, however, that the assumptions of this theorem are unnecessarily restrictive for the purpose of proving existence.

The methods of proof which have been successful for proving existence of equilibrium for a convex economy do not directly apply to non-convex economies. In particular, Debreu's method of proof ( 1959) which (essentially) consists in applying the Kakutani fixed point theorem to the excess demand correspondence p

Z (p)p′ (prices which maximize the value of excess demand) cannot be used. Without the assumption of convexity of production sets, the supply correspondence (productions where the vector normal to the boundary of the production set is colinear to the price p), is generally non-convex.

The method of proof used by Negishi ( 1960) and Arrow and Hahn ( 1972) which consists in applying a fixed point theorem to the boundary of the feasible utilities can be applied only if the assumptions of Theorem 4.1 hold. This proof requires that the allocations associated with a given vector of utility levels be convex, which is generally not the case (except if it is unique as assumed in Theorem 4.1).

A modification of this approach, which applies in the case where there is a single aggregate production set, has been introduced by P. Beato ( 1976, 1982) and R. Mantel ( 1979) and later improved by B. Cornet ( 1982, published 1990). It consists in applying a fixed point theorem to the boundary of the production set. Let us explain heuristically the idea behind this approach.

Suppose that there is only one firm in the economy. Consider the part of the boundary of the set (w + Y) which is in

. First, one has to show that this boundary is homeomorphic to the simplex of ℝH. Then,

-77-

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Increasing Returns and Efficiency
Table of contents

Table of contents

  • Title Page iii
  • Preface v
  • Contents vii
  • 1: Introduction 3
  • Notes 10
  • 2 - Necessary Conditions for Efficiency 13
  • 3 - Marginal Cost Pricing Equilibrium 27
  • 4 - Efficiency 41
  • 5 - Existence 77
  • 6 - The Core 101
  • Notes 152
  • 7 - Conclusion 153
  • References 155
  • Index 161
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