Abstract:
In an oligopoly under uncertainty, the concept and application of a supply function equilibrium (SFE) offers advantages over other forms of equilibria previously considered. Supply functions occur when players offer bids into the market as a schedule of price-quantity pairs, and are particularly representative of the behaviour of generators in electricity markets. A Nash equilibrium in such functions forms the basis of the framework, which is typically investigated whilst seeking ex post optimal solutions with respect to the realization of stochastic demand shocks. Such strong SFE satisfy the monotonicity requirements of offer curves in this context. The supply function approach allows flexibility in the presence of uncertainty but can be difficult to apply computationally. Numerical methods to search for candidate SFE have been explored in the strong case and presented in the literature. We consider an extension of the SFE framework to a more general weak setting in which ex ante optimal (in expectation) solutions are sought. In such cases where the conditions for strong SFE produce non-monotonic offer curves, a weak SFE may be constructed to meet monotonicity whilst maintaining best response conditions with regard to the demand distribution. Examples of such weak SFE are not prevalent in the literature and we seek to deepen our understanding of this concept through the development of a numerical method in the general setting. We develop a suite of tools, SCOPE (Supply Curve OPtimization and Equilibrium), to address the problem of computing best response curves (BRCs) and SFE in the general setting. An overview of SFE in the weak setting is presented with discussion around the important features of such a concept. A complementarity approach is considered, but we conclude that the equilibrium problem is better approximated through optimization. As a result, we demonstrate the ability to produce strong and weak BRCs and SFE under certain simplifying assumptions and provide a base for future development.