Ranking Supply Function and Cournot Equilibria in a Differentiated Product Duopoly with Demand Uncertainty

Ismail Saglam

doi:10.3390/g9030060

Abstract

In this paper, we provide a welfare ranking for the equilibria of the supply function and quantity competitions in a differentiated product duopoly with demand uncertainty. We prove that the expected consumer surplus is always higher under the supply function competition, irrespective of whether the (duopolistic) products are substitutes, complements, or independent. Numerical simulations suggest that if the products are either complements or independent, or if they have an extremely low degree of substitution, then the supply function competition can always be Pareto superior to the quantity competition in terms of the producers’ and consumers’ welfares. Moreover, if the degree of product substitution is not extremely low, then the supply function competition can be Pareto superior to the quantity competition if and only if the size of the demand uncertainty is sufficiently large to exceed a critical level. We find that this critical level of demand uncertainty becomes higher when the duopolistic products are less differentiated. Additionally, this critical level is nonincreasing both in the marginal cost of producing a unit output and in the own-price sensitivity of each inverse demand curve when all other parameters are fixed. Our results imply that in electricity markets with differentiated products, the regulators should not intervene to impose the quantity competition in favor of the supply function competition unless the degree of product substitution is sufficiently high and the predicted demand fluctuations are sufficiently small.

Highlights

The supply function competition that was originally developed by Grossman (1981) [1] could find applications in oligopolistic industries only after Klemperer and Meyer (1989) [2], who eliminated the problems with the multiplicity of supply function equilibria by introducing an exogenous uncertainty about the demand functions faced by oligopolists
We characterize the unique and symmetric Nash (1950) [14] equilibrium obtained under the supply function and quantity competitions, and we show that the expected consumer surplus under the supply function competition is always higher than that under the quantity competition independently of the size of the demand uncertainty and any other attributes of the industry
We present in Propositions 1 and 2 the characterizations of the symmetric equilibrium obtained under each form of competition, and by calculating the expected welfares of the producers and consumers at each of these equilibria, we first study how they would respond to changes in various model parameters

Summary

Introduction

The supply function competition that was originally developed by Grossman (1981) [1] could find applications in oligopolistic industries only after Klemperer and Meyer (1989) [2], who eliminated the problems with the multiplicity of supply function equilibria by introducing an exogenous uncertainty about the demand functions faced by oligopolists. Saglam (2018) [11] found that if the demand uncertainty in the industry is sufficiently large with respect to the number of firms, the size of the product markets, and the marginal cost of a unit output, the supply function competition can be ex ante more desirable for the producers. If the degree of product substitution is not extremely low, the expected producer profits under the supply function competition can be lower than those under the quantity competition in situations for which the size of the demand uncertainty is below a critical level We find that this critical level of demand uncertainty becomes higher when the duopolistic products are less differentiated. It is assumed that the form of the cost, demand, and inverse demand curves; the parameters c, β, γ, b, and g; and the density f (α) and its support are commonly known by both firms

Results

Quantity Competition

Supply Function Competition

Welfare Ranking

Conclusions