# Increasing Returns and Efficiency

By Martine Quinzii | Go to book overview

2
NECESSARY CONDITIONS FOR EFFICIENCY

This chapter introduces the general equilibrium version of the marginal cost pricing principle which asserts that marginal cost pricing is a necessary condition for efficiency. We show that to every Pareto optimal allocation is associated a price system that supports this allocation in a sense which is made precise. Under certain conditions, this price system is such that the price of each good equals its marginal cost.

2.1 The Production Economy

The model that we consider is a natural extension of the standard general equilibrium model of a production economy ( Debreu 1959). The differences and subsequent difficulties that emerge arise from the fact that we allow for the possibility of increasing returns or, more generally, of non-convexities in the production sets of some firms.

We consider an economy with H goods. The production sector consists of m firms with production sets (Yj) j=1, . . ., m. The production set Yj of firm j is a subset of ℝH which describes the feasible production plans yj = (yj0, . . ., YjH) with the usual sign convention: yjh < 0 (> 0) indicates that good h is used as an input (is produced as an output). All production sets Yj have the following properties:

0 ∈ Yj. This simply means that inaction is possible: Using no inputs, firm j produces no output.

Yj is closed in ℝH. This assumption is mathematically convenient and does not impose a significant restriction from an economic point of view. It means that a limit of feasible production plans is feasible.

yj

Yj, , yj+z ∈ Yj (or equivalently ) where denotes the positive orthant of ℝH and the negative orthant.

This last condition is the assumption of free disposal which is technically convenient but restricts the set of goods that the model can incorporate: it asserts that it is possible to dispose of any quantity of any good without

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Increasing Returns and Efficiency

• 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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