Numerical calculation of an asymmetric supply function equilibrium with capacity constraints

Abstract

Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in AbstractProducers submit committed supply functions to a procurement auction, e.g. an electricity auction, before the uncertain demand has been realized. In the Supply Function Equilibrium (SFE), every firm chooses the bid maximizing his expected profit given the bids of the competitors. In case of asymmetric producers with general cost functions, previous work has shown that it is very difficult to find valid SFE. This paper presents a new numerical procedure that can solve the problem. It comprises numerical integration and an optimization algorithm that searches an end-condition. The procedure is illustrated by an example with three asymmetric firms.

Supply function equilibria: Step functions and continuous representations

“…For example, numerical instabilities often arise in computation especially when mark-ups are small (Baldick and Hogan, 2002;Holmberg, 2008). Our analysis amplifies this observation, namely, the relationship we establish between the discrete and continuous cases relies on mark-ups that are positive and bounded away from zero.…”

“…Because of a singularity at zero mark-up, equilibrium bid-curves tend to be numerically unstable and easily non-monotonic near such points (Baldick and Hogan,28 2002; Holmberg, 2008). We have the same experience with our stepped offer curves.…”

“…The first order conditions for the Nash equilibrium for each demand realization satisfy a set of linked differential equations, which under various simplifying assumptions can be solved analytically, although for realistic specifications of costs numerical integration is normally required (Anderson and Hu, 2008;Baldick and Hogan, 2002;Holmberg, 2008). This approach opened the way for a large number of papers deriving solutions under various assumptions.…”

“…6 Closed form solutions are also available for symmetric firms and perfectly inelastic demand (Rudkevich et al, 1998;Anderson and Philpott, 2002). The literature on numerical algorithms for finding SFE of markets with asymmetric firms and general cost functions (Holmberg, 2008;Anderson and Hu, 2008) is particularly relevant to our investigation.…”

“…This is related to the instability near zero supply that has been observed when continuous SFE are calculated by means of standard numerical integration methods (Baldick and Hogan, 2002;Holmberg, 2008).…”