Necessary and Sufficient Conditions for Uniqueness of a Cournot Equilibrium

Abstract

In this paper a theorem is developed giving necessary and sufficient conditions for the uniqueness of homogeneous product Cournot equilibria. The result appears to be the strongest to date and the first to involve both necessity and sufficiency. The theorem states than an equilibrium is unique if and only if the determinant of the Jacobian of marginal profits for firms producing positive output is positive at all equilibria. The result applies to the case where profit functions are twice differentiable and pseudoconcave, industry output can be bounded, the above Jacobian is non-singular at equilibria, and marginal profits are strictly negative for non-producing firms. The proof uses fixed point index theory from differential topology.