Bertrand-Edgeworth game under oligopoly. General results and comparisons with duopoly

This paper studies price competition among a given number of capacity-constrainedproducers of a homogeneous commodity under the efficient rationing rule andconstant (and identical) marginal cost until full capacity, whendemand is a continuous, non-increasing, and non-negative function defined onthe set of non-negative prices and is positive, strictly decreasing, twicedifferentiable and (weakly) concave when positive.The focus is on general properties of equilibria in the region of the capacityspace in which no pure strategy equilibria exist. We study how theproperties that are known to hold for the duopoly are generalized to theoligopoly and, on the contrary, what properties do not need to hold in oligopoly.Our inquiry reveals, among other properties, the possibility of an atom in the support of a firm smaller than the largest one and the properties that such an atom entails. Although the characterization of equilibria is far from being complete, this paper provides substantial elements in this direction. JEL: C72, D43, L13