Oligopoly with Hyperbolic Demand: A Differential Game Approach

Convex demand functions, although commonly used in consumer theory and in accordance with a large amount of empirical evidence, are known to be problematic in the analysis of firms’ behavior; therefore, they are rarely used in oligopoly theory, due to the possible lack of concavity of the firms’ profit functions and the indeterminacy arising in the limit as marginal costs tend to zero. We investigate a dynamic oligopoly model with hyperbolic demand and sticky price, characterizing the open-loop optimal control and the related steady-state equilibrium, to show that the indeterminacy associated with the limit of the static model is indeed confined to the steady state of the dynamic model, while the latter allows for a well-behaved solution at any time during the game. Although the feedback solution cannot be analytically attained since the model is not built in linear-quadratic form, we show that analogous considerations also apply to the Bellman equation of the individual firm.