Introduction

In the competitive model, where production sets are convex, firms are assumed to maximise profit at given prices. In the non-convex case, this assumption is known to be inadequate. Profit maximisation may lead to unbounded outputs and, more generally, the supply correspondence which assigns profit-maximising production plans to prices may be neither convex valued nor upper hemicontinuous. Beyond these problems, even in the convex case, this behaviour often lacks in realism: many producers announce prices and satisfy the demand which materialises at these prices, instead of choosing optimal quantities in reaction to prices (formed on commodity exchange for instance).

In the present paper, we introduce axiomatically a concept of equilibrium, which combines the following two properties: (i) producers announce prices for their outputs and satisfy the demand which materialises at these prices and (ii) these output prices are ‘competitive’. We first prove that under the assumption of convexity, these equilibria coincide with the usual competitive equilibria, thus deserving the label of the ‘competitive equilibria with price-taking agents’. In the general case, allowing for non-convex technologies, we then prove the existence of competitive equilibria with quantity-taking producers where conditions (i) and (ii) are satisfied for each producer.

Condition (i) is a condition of voluntary trading on prices and quantities: the output must be such that, at the given prices, it is not more profitable for the producers to produce less. Thus, at an equilibrium, producers maximise profit subject to a sales constraint.