Analytic Solutions for Supply Function Equilibria: Uniqueness and Stability

Supply Function Equilibria (SFE) o¤er an attractive equilibrium concept for an electricity Pool in which all suppliers receive the market clearing price and are an important tool for examining market power. It is helpful to have analytical solutions available for simple models to explore market behaviour and to check computational solutions. This note derives analytic solutions for the symmetric case of linear and quadratic costs, and where each …rm has an identical set of constant but di¤erent marginal cost technologies, as in most practical applications to data. Such stepped marginal cost schedules can replicate general marginal cost schedules to any desired degree of accuracy and hence symmetric SFEs can be solved analytically by piecing together recursively de…ned supply functions for general cost functions. The paper discusses the question of the uniqueness and stability of these symmetric solutions, but notes that …nding asymmetric analytic solutions is generally di¢ cult. It collects together and extends results scattered in earlier working papers to make them more accessible.