Finding Supply Function Equilibria with Asymmetric Firms

Firms compete in supply functions when they offer a schedule of prices and quantities into a market; for example, this occurs in many wholesale electricity markets. We study the equilibrium behaviour when firms differ, both with regard to their costs and their capacities. We characterize strong equilibrium solutions in which, given the other players' supply functions, optimal profits are achieved for every demand realisation. If the demand can be low enough for it to be met economically with supply from just one firm, then the supply function equilibria are ordered in a natural way. We consider equilibria in which, for the highest levels of demand, all but one of the firms have reached their capacity limit. We show that there can be at most one supply function equilibrium with this property. We also propose a new numerical method to find asymmetric supply function equilibria, using piecewise-linear approximations and a discretization of the demand distribution. We show that this approach has good theoretical convergence behaviour. Finally, we present numerical results from an implementation using GAMS to demonstrate that the approach is effective in practice.