Supply Function Equilibria with Capacity Constraints and Pivotal Suppliers* Talat S. Genca and Stanley S. Reynoldsb Revised, January 2008 Abstract. The concept of a supply function equilibrium (SFE) has been widely used to model generators’ bidding behavior and market power issues in wholesale electricity markets. Observers of electricity markets have noted how generation capacity constraints may contribute to market power of generation firms. If a generation firm’s rivals are capacity constrained then the firm may be pivotal; that is, the firm could substantially raise the market price by unilaterally withholding output. However the SFE literature has not properly analyzed the impact of capacity constraints nor has it considered the impact of pivotal firms on equilibrium predictions. We characterize the set of symmetric supply function equilibria when firms are capacity constrained and show that this set is increasing as capacity per firm rises. We also provide conditions under which asymmetric equilibria exist and characterize these equilibria. Keywords: supply function equilibrium, pivotal firm, wholesale electricity market JEL codes: D43, L11, L94 _____________ * We are grateful for comments and suggestions from the corresponding editor, anonymous referees, participants at the Southwest Economic Theory Conference at UC Irvine, the Supply-Function Equilibrium Workshop at the University of Auckland, and seminars at Southern Methodist University, the University of Arizona and the University of Guelph. a Department of Economics, University of Guelph, Ontario, Canada N1G 2W1, email: [email protected] b Department of Economics, Eller College of Management, University of Arizona, Tucson, AZ 85721, email: [email protected] Introduction The supply function equilibrium (SFE) concept has become a widely used approach to study the exercise of market power by sellers in multi-unit auction environments. SFE models assume that each seller submits a supply function for divisible output to the auctioneer, who sets a uniform market clearing price. Klemperer and Meyer (1989) (hereafter KM) characterize supply function equilibria in environments for which product demand is uncertain. They show that there are multiple equilibria when the range of demand variation is bounded. Roughly speaking, these equilibria are contained in a range of prices between the Cournot price and the competitive price. The SFE concept has found its widest application in the analysis of wholesale electricity auctions. Many of these auctions are run as uniform price, multi-unit auctions in which power sellers submit offer schedules indicating their willingness to supply. Examples of applications of the SFE concept to wholesale electricity markets include Green and Newbery (1992), Newbery (1998), Rudkevich, et al (1998), Baldick and Hogan (2002), and Baldick, Grant and Kahn (2004). These papers consider a variety of extensions of the KM model, including production capacity constraints, cost asymmetries, potential entry, multi-step cost functions, and forward contracting. Recent assessments of wholesale electricity market performance have emphasized the role of the extent of excess production capacity in the market and the ability of firms to influence the market price by withholding production (see Bushnell, Knittel and Wolak (1999), Joskow and Kahn (2001), Borenstein, Bushnell and Wolak (2002), and Perekhodtsev, et al (2002)). The term “pivotal firm” has been used to describe a supplier that is able to dictate the price in the auction by withholding some portion of its production from the auction. One or more pivotal firms are most likely to be present when demand (or, load) is near its peak, when2available production capacity in the market is limited relative to the peak demand, and when firms’ capacities are asymmetrically distributed. While prior applications of the SFE approach to electricity markets have considered a variety of extensions of the basic SFE model, these applications have not adequately addressed the impact of production capacity constraints nor have they examined the potential role of pivotal firms. In this paper we formulate a simple model for which the notion of a pivotal firm has a natural interpretation. We assume that demand is uncertain and perfectly inelastic up to a maximum price. We focus on the case in which firms’ marginal costs are identical and constant up to production capacity. In the symmetric model, firms have identical capacities as well as costs. In the asymmetric model, firms have differing capacities. As in other SFE models with bounded demand variation, there is a continuum of equilibria. The introduction of production capacity constraints into a SFE model changes the analysis in a fundamental manner. The SFE model without capacity constraints utilizes quasi-concavity of firms’ profit functions and first-order necessary conditions to determine optimal price-quantity pairs along with the curvature of the equilibrium supply function. When a firm’s rivals have capacity constraints the firm’s profit function need not be quasi-concave in price. In addition to a locally optimal price-quantity pair associated with the SFE necessary condition, a pivotal firm may have a global profit optimum at the maximum price, or price cap. By withholding output, a pivotal firm can unilaterally move the market price to the price cap. We examine the connection between pivotal firms and the set of supply function equilibria. In symmetric and asymmetric versions of the model we show that the presence of pivotal firms alters the set of equilibria. The size of the equilibrium set depends on observable market characteristics such as the amount of industry excess capacity, the demand distribution, and the number of firms. We show that as the amount of industry excess capacity falls the set of symmetric supply function equilibria becomes smaller; the equilibria that are eliminated are the lowest-priced, most competitive equilibria. We also show that if the demand distribution is3concentrated near its maximum value then there are asymmetric equilibria in which the maximum price occurs with probability one. Section 2 reviews the relevant literature. Section 3 describes the model formulation and explains preliminary results. In section 4 we consider the role of capacity constraints and pivotal suppliers in equilibrium predictions for a symmetric