Existence and uniqueness of price equilibrium in oligopoly model with power demand

This paper provides a complete characterization of Pure Strategy Nash equilibria (PSNE) in an oligopoly model with power demand function recently proposed in the literature (Huang, 2004; Blavatskyy, 2018). This characterization shows that, if the marginal cost of production is strictly positive, then either there is a unique symmetric PSNE (when the demand function is sufficiently elastic) or there is no PSNE at all (when demand function is relatively inelastic). We also provide a novel interpretation of the unsatisfied demand as the demand of a social planner who partially supplies the market aiming at regulating the equilibrium price. Comparative statics results show that, if the marginal cost is zero, then the social planner can reduce the equilibrium price to zero without absorbing any additional demand. We also extend our analysis to the case when firms have different marginal costs and the elasticities of demand are firm specific. In particular, we show that the extended game has at least one PSNE when the firms’ demands are sufficiently elastic and when the unsatisfied demand is positive.