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Supply Function Equilibria in Oligopoly under Uncertainty

Paul D. Klemperer
Margaret A. Meyer

Abstract

We model an oligopoly facing uncertain demand in which each firm chooses as its strategy a "supply function" 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).