Supply Function Equilibria in Oligopoly under Uncertainty

TLDR

In oligopoly with uncertain demand, supply‑function strategies allow firms to adapt better than fixed price or quantity, and uncertainty reduces the multiplicity of equilibria. The study models an oligopoly under uncertain demand where each firm selects a supply function linking quantity to price. The model specifies firms choosing supply functions and analyzes comparative statics on costs, demand, uncertainty, and firm number, extending to differentiated products. The authors prove a Nash equilibrium exists for symmetric oligopolies with homogeneous goods, show conditions for uniqueness, and find that equilibrium supply functions become steeper when marginal costs are steeper, when there are fewer firms, more differentiation, or greater uncertainty at high prices, making competition resemble Cournot, while flatter functions bring it closer to Bertrand.

Abstract

We model an oligopoly facing uncertain demand in which each firm chooses as its strategy a relating its quantity to its price. Such a strategy allows a firm to adapt better to the uncertain environment than either setting a fixed price or setting a fixed quantity; commitment to a supply function may be accomplished in practice by the choice of the firm's size and structure, its culture and values, and the incentive systems and decision rules for its employees. In the absence of uncertainty, there exists an enormous multiplicity of equilibria in supply functions, but uncertainty, by forcing each firm's supply function to be optimal against a range of possible residual demand curves, dramatically reduces the set of equilibria. Under uncertainty, we prove the existence of a Nash, equilibrium in supply functions for a symmetric oligopoly producing a homogeneous good and give sufficient conditions for uniqueness. We perform comparative statics with respect to firms' costs, the industry demand, the nature of the demand uncertainty, and the number of firms, and sketch the extension to differentiated products. Firms' equilibrium supply functions are steeper with marginal cost curves that are steeper relative to demand, fewer firms, more highly differentiated products, and demand uncertainty that is relatively greater at higher prices. The steeper are the supply functions firms choose in equilibrium, the more closely competition resembles the Cournot model (which exogenously imposes vertical supply functions-fixed quantities); with flatter equilibrium supply functions, competition is closer to the Bertrand model (which exogenously imposes horizontal supply functions-fixed prices).