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 the types of equilibrium solution that can occur. 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. Moreover, there can be at most one supply function equilibrium with the property that all but one of the firms are at their capacity limits when demand is at its highest level. We also propose a new numerical approach to find asymmetric supply function equilibria. We use a scheme which approximates general supply function equilibria using piecewise linear supply functions 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 of this method using GAMS, to demonstrate that the approach is effective in practice. Our method is superior to approaches based on numerical methods for ordinary differential equations, or on iteration procedures in the function space of admissible supply functions.