Uniqueness and characterization of capacity constrained Cournot–Nash equilibrium

Abstract

We add capacity constraints to a multi-market Cournot model in which asymmetric firms have linear demand functions. We show that the problem is equivalent to maximizing a concave objective function over a convex region which ensures the existence of a unique capacity constrained Cournot–Nash equilibrium.

Introduction

In this paper we demonstrate the existence and the uniqueness of the Nash equilibrium when capacity constraints are added to a multi-market Cournot model. Understanding the role of capacity constraints (that may appear exogenous when competition in quantities takes place) is critical in many industrial sectors, such as the cement or the pulp industry, in which the importance and the irreversibility of investments on production capacity impose long-term decisions. In the proposed extension of the Cournot model, we consider an asymmetric oligopoly (non-identical production costs and capacities) selling a homogeneous good in a multi-market context. We assume linear demand functions that may be different on each market. Furthermore, we assume positive and convex production costs. We show that this setting allows for taking into account the case of multiple production plants per firm, each firm facing a transportation problem when delivering the different markets.

Despite the simplicity of this setting, this problem is not solved analytically in the literature. Nevertheless, since producers incur different costs and supply two or more markets, the way quantities should be allocated given the existence of a capacity constraint is not a priori obvious. Indeed, when considering capacity constraints in Cournot settings, most Industrial Organization models are restricted to the analysis of a single-market duopoly, thereby putting aside the difficulty that arises in a multi-market context.

The fundamental impact of the introduction of capacity constraints in a price competition model has been studied [5]: it is shown that in a two-stage game with choice of capacity followed by price competition, and under some rationing rules, the equilibrium is identical to the one obtained from competition in quantity without capacity constraints. Nevertheless, other economic analysis show that it can be interesting to introduce capacity constraints when firms compete in quantity depending on the timing of the game and on the way information is considered. For example, in two-periods Cournot models, focusing on the impact of the strategic choice of production capacity on entry deterrence [1], or studying the choice of capacity based on priors on the demand, this choice being followed by a capacity-constrained Cournot competition, once the information is acquired [3]. All these contributions are carried out considering a single-market duopoly.

Another stream of the literature focuses on computing equilibria in more complex settings than those presented in the simple theoretical models of imperfect competition. Using the fact that Nash equilibria satisfy variational inequalities [2], algorithmic methods of convergence have been developed in order to deal with oligopolistic models. Concerning the introduction of capacity constraints in a Cournot model, the case of the electricity market has been examined [4], the constrained equilibrium being computed using fixed point algorithms [6]. Although these kinds of approaches, which rely on convergence algorithms, are able to solve numerically the problem that we consider in this article, they have the disadvantage of being a “black box”, thus suffering from a lack of interpretability.

In this note, we show that the problem can be reduced to that of finding the maximum of a concave objective function over a convex region. This allows us to demonstrate the existence of a unique capacity constrained Cournot–Nash equilibrium.