Supply Function Equilibria with Capacity Constraints and Pivotal Suppliers*

Transcription

1 Supply Function Equilibria with Capacity Contraint and Pivotal Supplier* Talat S. Genc a and Stanley S. Reynold b April 2010 Abtract. The concept of a upply function equilibrium (SFE) ha been widely ued to model generator bidding behavior and market power iue in wholeale electricity market. Oberver of electricity market have noted how generation capacity contraint may contribute to market power of generation firm. If a generation firm rival are capacity contrained then the firm may be pivotal; that i, the firm could ubtantially raie the market price by unilaterally withholding output. However the SFE literature ha not properly analyzed the impact of capacity contraint and pivotal firm on equilibrium prediction. We characterize the et of ymmetric upply function equilibria for uniform price auction when firm are capacity contrained and how that thi et i increaing a capacity per firm rie. We provide condition under which aymmetric equilibria exit and characterize thee equilibria. In addition, we compare reult for uniform price auction to thoe for dicriminatory auction, and we compare our SFE prediction to equilibrium prediction of model in which bidder are contrained to bid on dicrete unit of output. Keyword: upply function equilibrium, pivotal firm, wholeale electricity market JEL code: D43, L11, L94 * We are grateful for comment and uggetion from the correponding editor, anonymou referee, participant at the Southwet Economic Theory Conference at UC Irvine, the Supply-Function Equilibrium Workhop at the Univerity of Auckland, and eminar at Southern Methodit Univerity, the Univerity of Arizona and the Univerity of Guelph. a Department of Economic, Univerity of Guelph, Ontario, Canada N1G 2W1, tgenc@uoguelph.ca b Department of Economic, Eller College of Management, Univerity of Arizona, Tucon, AZ 85721, reynold@eller.arizona.edu

2 1 Introduction The upply function equilibrium (SFE) concept ha become a widely ued approach to tudy the exercie of market power by eller in multi-unit auction environment. SFE model aume that each eller ubmit a upply function for diviible output to the auctioneer, who et a uniform market clearing price. Klemperer and Meyer (1989) (hereafter KM) characterize upply function equilibria in environment for which product demand i uncertain. They how that there are multiple equilibria when the range of demand variation i bounded. Roughly peaking, thee equilibria are contained in a range of price between the Cournot price and the competitive price. The SFE concept ha found it widet application in the analyi of wholeale electricity auction. Many of thee auction are run a uniform price, multi-unit auction in which power eller ubmit offer chedule indicating their willingne to upply. Example of application of the SFE concept to wholeale electricity market include Green and Newbery (1992), Newbery (1998), Rudkevich et al (1998), Baldick and Hogan (2002), and Baldick, Grant and Kahn (2004). Thee paper conider a variety of extenion of the KM model, including production capacity contraint, cot aymmetrie, potential entry, multi-tep cot function, and forward contracting. Recent aement of wholeale electricity market performance have emphaized the role of the extent of exce production capacity in the market and the ability of firm to influence the market price by withholding production (ee Jokow and Kahn (2002), Borentein, Buhnell and Wolak (2002), Perekhodtev et al (2002), and Wolak (2009)). The term pivotal firm ha been ued to decribe a upplier that i able to dictate the price in the auction by withholding ome portion of it production from the auction. One or more pivotal firm are mot likely to be preent when demand (or, load) i near it peak, when available production capacity in the market i limited relative to the peak demand, and when firm capacitie are aymmetrically ditributed. While prior application of the SFE approach to electricity market have conidered a variety of extenion of the baic SFE model, thee application have not adequately addreed the impact of production capacity contraint nor have they examined the potential role of pivotal firm. In thi paper we formulate a imple model for which the notion of a pivotal firm ha a natural interpretation. We aume that demand i uncertain and perfectly inelatic up to a 1

3 maximum price. We focu on the cae in which firm marginal cot are identical and contant up to production capacity. In the ymmetric model, firm have identical capacitie a well a cot. In the aymmetric model, firm have differing capacitie. A in other SFE model with bounded demand variation, there i a continuum of equilibria. The introduction of production capacity contraint into a SFE model change the analyi in a fundamental manner. The SFE model without capacity contraint utilize quaiconcavity of firm profit function and firt-order neceary condition to determine optimal price-quantity pair along with the curvature of the equilibrium upply function. When a firm rival have capacity contraint the firm profit function need not be quai-concave in price. In addition to a locally optimal price-quantity pair aociated with the SFE neceary 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 firm and the et of upply function equilibria. In ymmetric and aymmetric verion of the model we how that the preence of pivotal firm alter the et of equilibria. The ize of the equilibrium et depend on obervable market characteritic uch a the amount of indutry exce capacity, the demand ditribution, and the number of firm. We how that a the amount of indutry exce capacity fall the et of ymmetric upply function equilibria become maller; the equilibria that are eliminated are the lowet-priced, mot competitive equilibria. We alo how that if the demand ditribution i concentrated near it maximum value then there are aymmetric equilibria in which the maximum price occur with probability one. In prior analye baed on the SFE approach, neither a change in the demand ditribution nor in the amount of exce capacity have any impact on the predicted et of equilibrium upply function. Thi i the firt paper in the literature howing that both of thee factor do influence the et of equilibria. Relatively competitive upply function trategie are eliminated a equilibrium trategie when pivotal upplier are preent. Other contribution of thi paper are (1) a full characterization of the et of ymmetric upply function equilibria for a model with capacity contraint and, (2) a demontration of exitence of aymmetric equilibria with price equal to the price cap for ome configuration of the demand ditribution and capacitie. 2

4 Furthermore we compare our SFE prediction to equilibrium prediction of model in which bidder are contrained to bid on dicrete unit of output. Section 2 review the relevant literature. Section 3 decribe the model formulation and explain preliminary reult. In ection 4 we conider the role of capacity contraint and pivotal upplier in equilibrium prediction for a ymmetric model. We how how the failure of quaiconcavity of profit a a function of price change the problem of finding an optimal upply repone to rival upply function trategie. We provide a complete characterization of ymmetric upply function equilibria for all parameter configuration with a general demand ditribution, which i the main contribution of thi paper. We alo decribe two type of aymmetric equilibria that may arie. Section 5 provide a dicuion of everal iue, including aymmetric duopoly, increaing marginal cot, a comparion of SFE reult with reult from dicrete-unit bidding model, and equilibrium election. Section 6 conclude. 2 Background 2.1 Pivotal Supplier and Market Power Wholeale electricity market are particularly vulnerable to upplier market power. Thee market are often organized a auction in which firm ubmit offer chedule. Buyer in thee auction typically have little or no price enitivity, o that the price elaticity of demand i cloe to zero. For example, buyer may be electricity ditributor that are obligated to erve whatever quantity (load) their cutomer requet at a fixed retail price. Electricity production by power generation firm i limited by capacity contraint. In many part of the world, wholeale market are linked together via a tranmiion grid. However, the ability of buyer to acquire power from ditant generation firm may be limited by tranmiion capacity contraint. Taken together, thee characteritic of wholeale electricity market can yield ignificant market power for individual firm when demand i near it peak level. By unilaterally withholding output, a firm may be able to achieve a large price increae for it output, becaue of the combination of zero or low demand elaticity and limited production capacity of rival firm. For example, Borentein, Buhnell and Wolak (2002) etimate that 50 percent of electricity wholeale market expenditure in California in ummer 2000 could be attributed to exercie of market power by generation firm. 3

5 The concept of a pivotal firm embodie an extreme type of market power that may be preent in wholeale electricity market. Suppoe that market demand i perfectly inelatic up to a maximum price, or price cap, and firm have capacity contraint. If the market demand quantity exceed the um of production capacitie of a firm rival with poitive probability then that firm i aid to be a pivotal firm. The firm i pivotal in the ene that it can move the market price to the price cap with poitive probability by withholding output at price below the cap. We develop a model in which the pivotal firm concept play a crucial role in determining the et of equilibria. The Reidual Supply Index (RSI) i a market power index that wa devied by the California Independent Sytem Operator (ee Sheffrin (2001, 2002)). RSI i calculated a the ratio of reidual upply (total upply minu larget eller capacity) to the total demand. Uing ummer 2000 peak hourly data from the California Power Exchange, Sheffrin (2001) found a negative correlation between the Lerner Index and RSI. She find that when RSI i about 1.2, the average price-cot markup i zero. Wolak (2009) tudie the performance of the New Zealand wholeale electricity market. He invetigate the unilateral market power problem, and in particular explain how pivotal upplier emerge and exercie market power. He provide empirical evidence on the ability and incentive to exercie unilateral market power by pivotal upplier. He write, (on page 153), We find that when a upplier i a pivotal it offer price are higher by economically ignificant magnitude. He meaure pivotal frequency of power producer in the New Zealand market. For example, he report (on page 163), for each year of the ample period from January 1, 2001 to June 30, 2007, that Meridian (Energy) ha by far the highet pivotal and net pivotal frequency. For all but 2001, it wa pivotal in more than 50 percent of the half-hour period of the year. Next i Contact (Energy) with annual pivotal frequencie that range from 10 to 20 percent. Genei annual pivotal frequency range from lightly more than 3 percent to lightly more than 10 percent. Mighty River Power ha the lowet annual pivotal frequency of the four upplier. Concern about pivotal upplier are the bai for ome energy policy proviion. For intance, the U.S. Federal Energy Regulatory Commiion (FERC) may block a generation company from charging market-baed rate for energy if the company fail a creening tet for market power. One of the tet ued by FERC i the pivotal upplier creen; a generation upplier 4

6 i deemed pivotal, and therefore fail the tet, if peak demand cannot be met in the relevant market without production from the upplier capacity Equilibrium Model Klemperer and Meyer (1989) formulate a model in which each eller ubmit a upply function for diviible output a a function of the market price. There i no private information in their model, but the level of demand i uncertain at the time eller ubmit upply function. A uniform market price i determined by the interection of the realization of the demand function and the aggregate upply function. The neceary condition for equilibrium yield a ytem of differential equation that equilibrium upply function mut atify. If the range of demand variation i bounded then there i a continuum of equilibria; maximum price aociated with thee equilibria range from the competitive price to the Cournot price. Green and Newbery (1992) applied the SFE model to analyze competition in the Britih wholeale electricity pot market. Thi market wa run a a uniform price auction in which power eller ubmit offer chedule. Green and Newbery argue that in a ymmetric model, firm hould elect the ymmetric equilibrium that yield the highet profit. Uing demand and cot parameter to reflect condition in the early year of the England and Wale wholeale electricity market, they how that at the mot profitable ymmetric SFE, the two dominant bulk electricity firm were predicted to chooe upply function that yield price above marginal cot and caue ubtantial deadweight loe. Their SFE predicted price were well above oberved price. Wolfram (1999) perform a more detailed analyi of pool outcome in the England and Wale market. She alo find that the mot profitable ymmetric SFE yield predicted price that are ubtantially above actual pool price. Green and Newbery (1992), Newbery (1998), and Baldick and Hogan (2002) formulate SFE model that include production capacity contraint for firm. They argue, quite correctly, that if a olution of the SFE differential equation ytem violate any firm capacity contraint, then that olution cannot be a SFE (ee Green and Newbery (1992, pp ) and Baldick and 1 The following quote i from FERC Order No. 697 (2007), pp : The econd creen i the pivotal upplier creen, which evaluate the potential of a eller to exercie market power baed on uncommitted capacity at the time of the balancing authority area annual peak demand. Thi creen focue on the eller ability to exercie market power unilaterally. It examine whether the market demand can be met abent the eller during peak time. A eller i pivotal if demand cannot be met without ome contribution of upply by the eller or it affiliate. 5

7 Hogan (2002, p. 12)). However, thee paper neglect another effect of capacity contraint. Even when capacity contraint are non-binding for all firm at market clearing price for a propoed vector of upply function trategie, thee contraint have an impact on a firm profitability of changing it upply function. A firm may find it profitable to change it upply by withholding output and bidding up the price, given that it rival are capacity contrained. Prior analye of SFE model have focued on local optimality condition for a firm upply repone to rival upply trategie. Local optimality condition yield the ytem of differential equation that a SFE mut atify. However, when there are capacity contraint it i important to check for global optimality of a firm upply repone. Rival capacity contraint may yield a non-quaiconcave payoff function for which local optima need not be globally optimal. We develop the implication of thi obervation in Section 4 below. Uing a formulation imilar to our, Holmberg (2008) aume that demand exceed total capacity with poitive probability and how that there i a unique ymmetric SFE with upply function that reach the price cap when output i equal to capacity. Our analyi aume that demand i tochatic but never exceed total indutry capacity.. In Section 4 we how that the et of ymmetric equilibria hrink a total capacity fall, though in general, multiple equilibrium remain. Our aumption that total capacity exceed peak demand permit our reult to be compared to reult from dicrete-unit bidding model of electricity market, uch a Fabra et al (2006). Delgado and Moreno (2004) provide another avenue for eliminating multiplicity of equilibria in a SFE model. They how that the only equilibrium in a upply function model with a ingle, certain demand function i the equilibrium that yield the Cournot price when the Coalition-proof Nah equilibrium refinement i ued. However thi reult change when output i ubject to capacity contraint. Delgado (2006) how that the preence of ufficiently aymmetric capacity contraint may increae the number of Coalition-proof Nah equilibrium outcome; the new outcome involve price below the Cournot price. Fabra, von der Fehr and Harbord (2006) compare dicriminatory veru uniform-price electricity auction in which firm ubmit offer to upply one or more dicrete unit of output. Their model of a uniform-price auction hare many feature of our model. We compare reult 6

8 from their dicrete unit analyi to thoe from our analyi of upply function with infinitely diviible output in Section A Supply Function Equilibrium Model 3.1 Model Formulation We aume that the level of demand i θ, a random variable that i independent of the market price. The level of demand i ditributed according to F( θ ) with upport, θ, θ, where 0 θ < θ. We aume that the probability denity function i trictly poitive on the interior of it upport. Demand i perfectly inelatic for price up to an exogenou market reerve price, p. Thi reerve price could repreent either buyer maximum willingne to pay or, a price cap impoed by regulator. 2 There are n > 2 firm. We aume a imple cot tructure in which each firm ha contant marginal cot of production, c, up to it capacity contraint, where 0 c< p. The capacity contraint for firm i i K i. Total capacity for all firm i, K n K i= 1 i. Throughout mot of thi paper we aume that firm have equal capacitie, with Ki = k for i= 1,... n. Each firm i i aumed to chooe a upply function, ( p ), from the et S, which i defined a the et of non-decreaing, right continuou, and piecewie-differentiable function mapping R + into [0, k ]. Firm are aumed to elect upply function imultaneouly prior to the realization of demand. The auctioneer (or, ytem operator) contruct an aggregate upply function and determine a market clearing price baed on the realization of demand. Let p( θ ) be the market clearing price when the level of demand i θ. If firm i ubmit ( p ), then i i obligated to produce up to i ( p( θ )) when the level of demand i θ and will be paid p( θ ) per unit for it production. 3 i i 2 See von der Fehr and Harbord (1993) for a dicuion of market reerve price for wholeale electricity market. 3 Our uncertain demand formulation may alo be interpreted a a model of time-varying demand with long-lived bid, a explained in Green and Newbery (1992). For intance, our model could be applied to approximate an electricity auction in which each firm ubmit a upply chedule for an entire day, and in which the day i divided into 48 one-half hour interval, with a clearing price etablihed for each one-half hour interval baed on demand for each interval. See Baldick and Hogan (2002) and Fabra, et al (2006) for dicuion of hort-lived v. long-lived bid in electricity auction. 7

9 Right continuity of upply function i more general than the aumption of differentiable upply function that i employed in mot SFE analye. 4 Right continuity allow for two type of upply trategie that are important to conider when capacity contraint are preent. Firt, it allow a trategy of offering a firm ome or all of a firm capacity at the market reerve price. We how that thi type of trategy may be part of an equilibrium when firm are capacity contrained. Second, it allow for a dicontinuou jump in quantity upplied up to full capacity for price above thoe on the equilibrium path of price. Let S( p) j ( p) and S i ( p) S( p) i ( p). If θ > S( p) for p [0, p] then we j aume that the price i p when demand i θ and S( p) unit are provided to buyer. Otherwie, m a uniform market clearing price, p ( θ ), i etablihed when demand i θ that atifie m (3.1) p ( θ) = min{ p [0, p]: θ S( p)}. It i poible that there i exce upply at a market clearing price. We aume a pro-rata on the margin (PRM) rule for allocating exce upply. Thi rule i commonly ued in electricity auction. Under PRM demand i firt allocated to offer below the clearing price. Next, the remaining demand (quantity demanded at the clearing price le upply quantity offered at price below the clearing price) i allocated to firm according to a pro-rata rule. If 0 p i the clearing price then the PRM rule yield ale for firm i a follow: (3.2) q = p + S p i 0 0 i ( ) ( θ ( )) p p S p S p 0 0 i ( ) i ( ) 0 0 ( ) ( ) where 0 ( p ) = lim 0 ( p ) and S 0 ( p ) = lim S 0 ( p ). i p p i p p 3.2 Supply Function Equilibrium A upply function equilibrium (SFE) i a Nah equilibrium in upply function trategie. In thi ub-ection we provide preliminary reult on equilibria with differentiable upply 4 Right continuity i the analog of left continuity of a bidder demand function in a ale auction. Kremer and Nyborg (2004) allow bidder to ue left continuou demand function in their analyi of the effect of alternative allocation rule. Our interet i in the interaction of capacity contraint and upply trategie, rather than the impact of alternative allocation rule. 8

10 function, under an aumption that firm capacitie are large enough o that production capacitie have no impact on the et of equilibria. Aumption A1. Each ubet of n 1 firm ha ufficient capacity to meet maximum demand. That i, ( n 1) k θ. Dicontinuitie in upply function are important to conider when A1 i relaxed, a we explain in Section 4 below. Conider the profit for firm i in the event that demand i θ, given that rival firm j chooe upply function j ( p) S, for j i. If the clearing price i p and firm i upplie the reidual demand, S ( p), then it profit i: θ i (3.3) π ( p, θ ) = ( p c)[ S ( p)] i θ i We eek a upply function ( p ) for firm i that ha the property that the clearing price p i maximize π ( p, θ ) in (3.3) with ( p) = S ( p), for each poible θ. If it rival upply i i θ i function are differentiable then the neceary condition for an (interior) optimal price for θ for each firm i yield a ytem of ordinary differential equation for upply function: (3.4) j i i ( p) j '( p) =, i= 1,..., n ( p c) Thi i a imple verion of the differential equation ytem that ha been intenively tudied in the SFE literature. 5 If all firm utilize a common upply function, ( p ), then the ytem of equation in (3.4) implify to the following ingle differential equation: (3.5) ( p) '( p) = ( n 1)( p c) There i a continuum of olution to (3.5) of the form: (3.6) θ p c i( p) = n p' c 1 n 1 5 Thi verion i imple becaue there i no demand function term, due to perfectly inelatic demand, and becaue marginal cot i contant. 9

11 We depict thee upply function in Figure 1. The parameter p ' i a boundary price, repreenting the equilibrium price for the highet poible demand realization, for a particular equilibrium. Supply function ( ) defined in (3.6) i a ymmetric SFE trategy for each boundary price p ' ( c, p ]. 6 Note that the et of ymmetric upply function equilibria i independent of the ditribution of demand quantitie. (3.7) The expected payoff for a firm when all n firm chooe ( p) in (3.6) i: p c n w ( p') = ( ' ) E[ θ ] n n 1 θ The upply function olution in (3.6) are a pecial cae of reult in Rudkevich et al (1998). They derive SFE olution for a ymmetric model with zero demand elaticity and general cot function (including multi-tep marginal cot function). Following Klemperer and Meyer (1989), the neceary condition for maximization of π ( p, θ ) at each θ are alo ufficient if each firm extend it upply function in (3.6) linearly in i p for price above p ' up to it capacity. However, thi reult depend critically on aumption A1. If A1 doe not hold, then π ( p, θ ) need not be quai-concave in p and there may be a econd local maximum at a price above i p '. Thi point i developed further in ection 4. It i poible that there are aymmetric equilibria, in addition to the ymmetric equilibrium upply function in (3.6). Let xi = i ( p') be the upply quantity offered by firm i at boundary price p' and let of ODE ytem (3.4) are a follow: max x be the larget of firm upply quantitie offered at p '. Solution (3.8) where 1 p c n 1 θ p' c i θ i( p) = + x n p' c n p c p ' and ( x1, x2,..., x n) atify, (3.9A) p ' ( c, p ], 6 Since we allow upply trategie that are right continuou, the Bertrand trategy of offering unit at marginal cot, c, i alo an equilibrium. Symmetric equilibrium trategie are of the form, offer zero unit for ale for price le than c, and offer y unit for ale for price of c or higher, where y > θ /( n 1). 10

12 (3.9B) (3.9C) n xi = θ, i= 1 max ( θ ) n ( n 1) x / n n θ θ θ The upply function defined in (3.8) collape to a ymmetric upply function if firm have equal output at the boundary price. 7 Condition (3.9C) i required in order for upply function to be non-decreaing in price. If θ > 0 then aymmetric upply function equilibria exit; if θ = 0 then the only upply function equilibria are ymmetric equilibria Capacity Contraint and Pivotal Supplier In thi ection we relax the aumption that any group of n 1 firm ha enough capacity to meet maximum demand. Intead we adopt, Aumption A2. ( n 1) k < θ < nk. We ay that firm i i pivotal conditional on θ if ( n 1) k< θ. When thi inequality hold, firm i would be able to achieve a market price of p when demand i θ by ubmitting a upply function with ome portion of it capacity bid in at price p, and ell at leat θ ( n 1) k unit. We ue the term pivotal firm to decribe a firm that i pivotal with poitive probability. In thi ection we focu on the ymmetric cae in which each firm ha the ame capacity; in thi cae, if one firm i pivotal then all are pivotal. In what follow we examine how the preence of capacity contraint alter the et of equilibria. Conider a modified verion of the trategy defined in (3.6). 7 (3.8) i obtained by olving the ytem of linear differential equation (3.4) along with boundary condition. For a derivation ee Genc and Reynold (2004) for two firm. They form an equilibrium ince the profit function are twice differentiable and trictly concave in price. The expreion in (3.9C) obtained by taking the derivative of the upply function and equating it to zero. 8 Thi correpond to a reult in Klemperer and Meyer (1989). In their model with downward loping demand they how that aymmetric equilibria are ruled out if the lowet value of the demand hock yield zero quantity demanded at price equal to marginal cot. 11

13 1/( n 1) θ p c if c p p ' (4.1) *( p) < = n p ' c k if p ' p The modified upply function trategy in (4.1) define upply quantitie for price both above and below the boundary price. In particular, (4.1) pecifie that a firm produce output equal to it production capacity for price at or above the boundary price. Thi i the mot aggreive upply behavior poible for price above the boundary price, and therefore yield the lowet poible reidual demand for rival firm for price above the boundary price. Alo, note that if all n firm chooe the upply function (4.1) and if demand i θ then the market clear at the boundary price, p', even though there i exce upply at price p'. Thi i true becaue the limit of *( p ) a p approache p' from below i θ / n, and the PRM allocation rule give firt priority to upply offer below the clearing price. Suppoe that a firm rival adopt trategy ( ), defined in (4.1) with boundary price,. If the firm bet repone i to chooe ( ) with the ame boundary price, then thi trategy i a ymmetric equilibrium trategy. In a lemma ued to prove Propoition 1, we how that if the firm bet repone differ from ( ), then it bet repone i of the following form: (a) follow ( ) for low price up to ome quantity, q, (b) upply quantity q for price in the interval, ( ), ) and, (c) upply quantity k at the price cap. For q [ 0, θ ( n 1) k] the expected payoff aociated with thi trategy i defined a, (4.2) θ ( q) θ ( q) 1 2 w( q; p ') = [( p( θ ) c)( θ / n)] df( θ ) + ( p( θ ) c) q df( θ ) + θ θ ( q) 3 ( p' c) q df( θ ) + [( p c)( θ ( n 1) k)] df( θ ) θ ( q) θ ( q) 2 3 θ θ ( q) 1 where p' i the boundary price aociated with rival trategy, * ( ), quantitie θ1, θ2, θ3 are defined a, 12

14 nq θ1( q) = θ and price are given by, if nq θ otherwie n ( ) q n ( ) θ if q+ θ θ θ n n 2( q) = θ otherwie q+ ( n 1) k if q+ ( n 1) k θ θ3( q) = θ otherwie n 1 p ( θ θ ) c= ( p ' c ) forθ θ θ1 θ [, ), n 1 ( θ q) n p( θ ) c= ( p' c) forθ [ θ1, θ2). ( n 1) θ The boundary price p' i included a a parameter argument for payoff function w( ) to emphaize the dependence of thi payoff on the boundary price aociated with the trategie ued by rival firm. Thi payoff function alo depend on other parameter, uch a capacity per firm, the number of firm, and the demand ditribution. The function w( q; p') i continuou and bounded in q for q [ 0, θ ( n 1) k]. Therefore, w( q; p') attain a maximum value with repect to q, for each boundary price, ] : (4.3) max q [ 0, θ ( n 1 ) k ] v ( p ') w ( q ; p ') Propoition 1. Under Aumption A2 the et of ymmetric equilibrium upply function i given by ( )= /( ) for, ) with boundary price in the interval [ p, p], where the lower bound of the interval i defined by, (4.4) = {, ]: ( )= ( )}. Proof. See the Appendix. 13

15 Price above are outide the et of price oberved in equilibrium. The form of the upply function for price above matter only in o far a it influence the incentive of rival firm. The pecification of * (.) in (4.1) provide the lowet poible incentive for a rival firm to deviate from * (.), and therefore thi form i conitent with the larget poible et of ymmetric equilibria. The lower bound p of the interval of equilibrium boundary price i defined in (4.4). Thi lower bound i determined a the mallet boundary price p' at which the function w ( p') croe the function v( p ') from below. The function w ( p') indicate the payoff per firm when all firm chooe upply function * ( ) in (4.1) with boundary price p'. Thi payoff function i independent of capacity per firm, k. However, an increae in k doe affect v( p '), ince a higher capacity per firm reduce reidual demand for any one firm at price greater than or equal to p'. A higher value of k caue v( p ') to fall at each boundary price p', and thi hift down yield a lower croing point with capacity per firm i ummarized in the following propoition. w ( p'). Thi effect of an increae in Propoition 2. Under Aumption A2 the lower bound of the et of ymmetric equilibrium boundary price i decreaing in capacity per firm, k. Proof. See the Appendix. The et of ymmetric equilibria depend on the number of firm and on the ditribution of demand quantitie, a well a on capacity per firm. We do not have analytical reult for the impact of change in n and F( θ ). Intead we provide numerical reult in Table 1 to illutrate the impact of change in n and F( θ ) on the equilibrium et for an example. The table i baed a on demand ditribution F( θ ) = θ, for θ [ 0, 1], where a i a poitive parameter. A larger value of parameter a indicate that demand i more concentrated near it maximum poible of unity. Numerical reult are baed on the aumption that total capacity i held fixed at 125 percent of the maximum demand quantity. 14

16 Table 1 Ratio of Minimum Boundary Markup to Maximum Markup for Symmetric Supply Function Equilibria* a (ditribution parameter) n * Table entrie provide the value of, ( p c)/( p c), in percentage term. For each value of n, the lower bound of ymmetric equilibrium boundary price rie a a rie. That i, a demand become more concentrated near it maximum value, relatively competitive upply trategie are eliminated a equilibrium trategie and the et of ymmetric equilibria hrink. Holding both total capacity and a fixed, the lower bound of equilibrium boundary price fall a n increae. A in the model of Section 3, if θ > 0 then there are aymmetric equilibria, in addition to the ymmetric equilibria characterized in Propoition 1. The introduction of capacity contraint alter the et of aymmetric equilibria in two poible way. Conider firt aymmetric upply function defined in (3.8). Obviouly, output at the boundary price mut be le than capacity for each firm (o that max x k ). In addition, aymmetric upply function defined by (3.8) are equilibria only for boundary price that atify a condition analogou to the condition inide curly bracket in (4.4). That i, a boundary price p' aociated with an aymmetric equilibrium mut be high enough o that no firm would wih to witch to a different upply function that would yield clearing price above p'. The analye of von der Fehr and Harbord (1993) and Fabra et al (2006) for procurement auction with dicrete unit of output ugget a econd type of aymmetric equilibrium that may arie. Suppoe that the demand ditribution i concentrated near the maximum demand quantity, o that the following aumption i atified. 15

17 Aumption A3. ( n 1) k < θ < θ < nk. Under Aumption A3, a ingle firm i aured of poitive reidual demand at any price up to the market reerve price, for each poible demand realization. Propoition 3. Under Aumption A3 there exit aymmetric equilibria in which: a) one firm offer no upply for price below the market reerve price, and offer it entire capacity at the market reerve price, b) each of the other n 1 firm offer their entire capacity at ome price below the market reerve price, and c) the uniform market clearing price i equal to the market reerve price for each demand realization. Propoition 3 i baed on the following two obervation. Firt, the high-price firm ha no incentive to deviate from it upply trategy a long a rival firm offer their capacity at or below price p% atifying, ( p% c) k = ( p c)( θ ( n 1) k). Second, the low-price firm have no incentive to deviate ince each ha ale equal to their capacity at the maximum poible price with probability one. The aymmetric equilibria of Propoition 3 yield the highet poible indutry profit; all potential urplu i extracted in the form of profit for firm. Note, however, that the ditribution of profit may be quite uneven acro firm. If the lowet poible demand quantity i only lightly greater than capacity for one firm, then the high-price firm will earn profit that i a mall fraction of the profit earned by each other firm in equilibrium. 9 Earlier in thi ection we howed that the et of ymmetric upply function equilibria expand a capacity per firm, k, increae (Propoition 2). Propoition 3 indicate that aymmetric equilibria exit with price equal to the price cap if Aumption A3 hold. Suppoe that Aumption A3 hold, and k i increaed, holding the demand ditribution and the number of 9 Note that the aymmetric equilibria of Propoition 3 rely on our definition of the et of upply trategie, which permit dicontinuou upply function. 16

18 firm fixed. Then at ome point a k rie, Aumption A3 will no longer hold and the aymmetric equilibria with price equal to the price cap will no longer exit. 5 Extenion and Dicuion 5.1 Aymmetric Capacitie Concern about market power in wholeale electricity market have often focued on the larget firm (e.g., ee Lave and Perekhodtev (2001)). Firm with the greatet generation capacity appear to be mot likely to be able to force up the market price by withholding production. However there i relatively little analyi of model in which firm have aymmetric capacitie in the SFE literature. 10 An exception i a recent paper by Anderon and Hu (2008) that report SFE reult for particular configuration of aymmetric capacitie and cot. They propoe a new numerical method to calculate aymmetric SFE uing an approximation technique for olving differential equation. They implement their method on everal numerical example. We focu on duopoly market in which firm differ only in capacitie. We aume that firm 1 ha more capacity than firm 2. We alo aume that K1+ K2 θ > K2. 1 If K > K θ then ymmetric upply function equilibria are feaible, ince each firm ha enough capacity to meet at leat one-half of maximum demand. Symmetric equilibria are defined in Propoition 1, uing k = K2. Propoition 2 indicate that the lower bound of the et of 1 ymmetric equilibrium boundary price i decreaing in K 2 for K > K > θ There may alo be aymmetric equilibria, depending on the value of θ. If K < θ then the only poible pure trategy equilibria are aymmetric. Equation (3.8) pecifie mooth upply function that can be aymmetric equilibrium trategie when firm have different output at the boundary price. In addition to the condition in (3.9), trategie would need to atify the following two condition in order to be aymmetric equilibrium trategie: a) x1 K1 and x2 K Green and Newbery (1992) dicu equilibria for aymmetric duopoly. They write that it i, more difficult to olve for the pair of (differential) equation for the aymmetric equilibrium than the ingle equation for the ymmetric equilibrium, and the ret of the paper will retrict attention to the ymmetric cae. Baldick and Hogan (2002) alo conider aymmetrie in capacitie and cot function. However they note that the differential equation approach of olving for upply function may not be effective becaue the reulting upply function often fail the non-decreaing property. 17

19 b) The boundary price p ' i high enough o that neither firm wihe to deviate to a trategy that yield price above p '. 1 If K < θ then condition (3.9C) require that the minimum demand quantity atifie, 2 2 θ θ ( θ 2 K ), ince the mallet equilibrium output for firm one at the boundary price i, x = θ K There may alo be aymmetric equilibria in which the equilibrium price i equal to the market reerve price with probability one, a in Propoition 3. If θ > K2 then there i an equilibrium in which firm one ubmit a upply function with zero output for price below the market reerve price and output equal to capacity at the market reerve price. Firm two ubmit a upply curve uch that firm one ha no incentive to offer output at lower price than firm two offer it capacity. If θ > K1 then there are aymmetric equilibria in which either firm play a high-price trategy and it rival play a low-price trategy. 1 We ummarize the preceding two paragraph a follow. If K < θ then exitence of a pure trategy aymmetric equilibrium depend on the upport of the demand ditribution. Specifically, the following condition mut be atified for exitence of a pure trategy equilibrium: (5.1) θ min{ K2, θ ( θ 2 K2)}. If (5.1) doe not hold then a pure trategy equilibrium doe not exit. It i poible that mixed trategy equilibria exit, but we do not attempt to identify or characterize uch equilibria Symmetric Firm with Step Marginal Cot In preceding ection we aumed that all firm have a common, contant marginal cot c for production up to capacity. Our approach can be extended to the cae in which firm have multi-tep marginal cot function. Rudkevich et al (1998) derive SFE reult for a multi-tep marginal cot model with identical firm. Genc (2003) examine the impact of pivotal firm on the et of upply function equilibria when each firm ha a two-tep marginal cot chedule 11 The condition, θ θ ( θ 2 K ) i obtained by plugging n=2 and x = θ K into (3.9C). 2 2 max 18

20 compried of low-cot (bae load) generation unit and higher cot (peak load) unit. We ummarize two key finding from Genc (2003). Firt, with riing marginal cot tep, even the mot competitive ymmetric SFE yield poitive profit ince market clearing price are above marginal cot for low cot unit. A a conequence, the preence of pivotal firm may not have any effect on the et of upply function equilibria. Thi i in contrat to the contant marginal cot cae, in which the preence of pivotal firm alway eliminate the mot competitive upply function equilibria (the lower bound on equilibrium markup i alway poitive if there are pivotal firm). Second, the fraction of total capacity that i compried of low cot unit ha an impact on the et of upply function equilibria. By analyzing a erie of numerical example, Genc (2003) find that the lower bound for the SFE initial price i non-increaing in the proportion of total capacity that i compried of bae load unit. 5.3 Offer to Supply Dicrete Unit v. Supply Function A upply function equilibrium formulation provide an approach for characterizing upply function trategie for infinitely diviible output in n-firm procurement etting. The preent paper focue on how capacity contraint and pivotal upplier contribute to market power in a SFE model. An alternative approach i to model procurement a a n-firm game in which each firm ubmit offer to upply one or more dicrete unit of output. von der Fehr and Harbord (1993), Anwar (1999), and Fabra et al (2006) utilize thi approach. We compare our reult for ymmetric capacitie to thoe of Fabra et al (2006), who conider a dicrete unit model of a uniform price auction with n firm, contant marginal cot up to capacity, perfectly inelatic demand up to a price cap, and uncertain demand. Under Aumption A1, a dicrete unit model yield a unique price prediction regardle of the number of unit per firm; namely, the Bertrand equilibrium with price equal to marginal cot. In contrat, a SFE model ha multiple equilibria that include the Bertrand equilibrium a well a other equilibria with poitive markup over marginal cot. Equilibria with poitive markup in the diviible output model are ometime referred to a implicitly colluive equilibria. Under Aumption A3, pure trategy equilibria of the dicrete unit model are decribed in Propoition 3; thee equilibria yield price equal to the market reerve price with probability one. Thee are the only pure trategy equilibria for the dicrete unit model; there are no 19

21 ymmetric equilibria in pure trategie. Under Aumption A3, our model with diviible output yield the aymmetric equilibria decribed in Propoition 3, ymmetric equilibria decribed in Propoition 1, and other aymmetric equilibria with mooth upply function. Equilibrium price for the diviible output model are either equal to or lower than thoe for the dicrete unit model. If Aumption A2 hold, but not A3, then there i not a pure trategy equilibrium for the dicrete unit model. If each firm upplie a ingle unit of output then there i a unique ymmetric mixed trategy equilibrium with offer trictly above marginal cot. If each firm i permitted to ubmit offer for multiple unit of output then there may be a mixed trategy equilibrium, but characterization or computation of uch an equilibrium would appear to be quite challenging. If Aumption A2 hold, but not A3, then Propoition 1 decribe ymmetric equilibria of the diviible output model; in addition there may be aymmetric equilibria in mooth upply trategie. Expected payoff per firm in a ymmetric equilibrium of the diviible output model i greater than or equal to expected payoff in the mixed trategy equilibrium for the ingle-unit offer model. 5.4 Dicriminatory v. Uniform-price Auction A dicriminatory (pay-a-bid) auction format i an alternative to the uniform-price format. Under Aumption A1 there i a unique pure trategy SFE under dicriminatory auction rule. The unique equilibrium involve each firm offering their capacity at a price equal to marginal cot (ee Wang and Zender (2002) for the cae of a ale auction). Supply function equilibria under uniform-price rule include the equilibrium with price equal to marginal cot, a well a other (implicitly colluively) equilibrium with price greater than marginal cot. So, under Aumption A1, SFE price for uniform-price auction are greater than or equal to SFE price for dicriminatory auction. Genc (2009) examine upply function equilibria for dicriminatory auction under Aumption A2, with the additional aumption that demand i uniformly ditributed. He how that a pure trategy SFE doe not exit. He characterize a ymmetric mixed trategy SFE in which firm mix over horizontal upply function. Thi mixed trategy equilibrium i baed on the ecurity profit a firm can achieve by offering it capacity at the price cap; ecurity profit i equal to: ( ) {0, ( 1) }]. Under uniform-price auction rule, a firm can 20

22 alo achieve profit by offering it capacity at the price cap. A a conequence, expected profit per firm in any upply function equilibrium under uniform-price auction rule i greater than or equal to expected profit per firm in equilibrium under dicriminatory rule. A corollary to thi profit reult i that expected equilibrium price under uniform-price rule are greater than or equal to expected equilibrium price under dicriminatory rule. 5.5 Equilibrium Selection Under Aumption A1, the ymmetric model ha a continuum of Pareto-ranked ymmetric equilibria. If the lower bound of the upport of the demand ditribution i poitive, then there are alo multiple aymmetric equilibria that, for a given vector of boundary output ( x, x,..., x ), are Pareto-ranked. Suppoe that the amount of exce capacity i reduced, o that 1 2 n Aumption A2 hold and A3 doe not hold. Then ome equilibria are eliminated; upply trategie that yield low price and low expected profit are no longer equilibrium trategie. If firm are able to coordinate on the mot profitable upply function equilibrium then the fact that ome le profitable upply trategie are eliminated a equilibria when exce capacity i reduced would be irrelevant for oberved upply behavior. 12 However, there are two reaon to quetion the ability of firm to coordinate on the mot profitable SFE. Firt, a noted in Section 2.2, both Green and Newbery (1992) and Wolfram (1999) found that the mot profitable SFE predicted price were ubtantially above actual pool price for the England and Wale wholeale electricity market. Forward contract and the preence of a regulator with anti-trut power may explain ome of the difference between actual and highet-profit SFE price, but probably not all of the difference. Second, evidence from laboratory experiment on coordination game how that human ubject conitently fail to coordinate on the mot profitable Nah equilibrium in ome environment (ee Van Huyck et al (1990), and Devetag and Ortmann (2007)). Failure become more likely a the number of player rie and the complexity of the environment increae. Suppoe that Aumption A3 hold. By Propoition 3 there are aymmetric equilibria for which the price i equal to p with probability one. Expected profit for the firm that offer it 12 If total capacity i low enough o that a hortage occur with poitive probability, a Holmberg (2008) aume, then there i a unique ymmetric equilibrium and the election problem diappear. 21

23 upply at the market reerve price (the high price firm) in any uch equilibrium i, H H w = ( p c)( E( θ) ( n 1) k). If w ( p) > w then there i another equilibrium, the ymmetric SFE with boundary price equal to p, that yield an expected price le than p and that i not a Pareto dominated equilibrium. 13 Alo note that continuity of ( ) in the boundary price implie that if ( )> then ( )> for in a neighborhood of the price cap. 6. Concluion The upply function equilibrium (SFE) model ha been applied to numerou analye of generator bidding behavior and electricity market power iue. The important role that pivotal upplier play in wholeale electricity market ha been documented empirically. However SFE model in which pivotal upplier can play a role have not been conidered in the literature. We examine the connection between capacity contraint and the et of upply function equilibria. We include production capacity contraint in our model to allow for the poibility that a ingle firm i pivotal; that i, the firm can unilaterally move the price to the market reerve price with poitive probability by withholding output, ince rival are capacity-contrained. While other SFE analye have conidered capacity contraint, we argue that thee analye failed to fully account for the impact of thee contraint due to a focu on local, rather than global, optimality condition. We characterize the et of ymmetric upply function equilibria for a model with capacity contraint. We how that when the contraint atify a pivotal upplier condition, the et of ymmetric equilibria i increaing in the amount of capacity per firm. We alo how that the et of ymmetric equilibria depend on the demand ditribution and on the number of firm. Thi dependence of the equilibrium et on the demand ditribution i in contrat to SFE model without capacity contraint, for which the equilibrium et i independent of the demand ditribution. 13 Whether or not thi payoff inequality hold depend on n, k, and the demand ditribution. For example, uppoe that =3, =3/8, and load i uniformly ditributed with =0.8, and =1.0. Then ( ) 0.246( )> =0.15( ). Another poible diadvantage of the equilibrium with price equal to the price cap i that it may lead to an anti-trut or regulatory invetigation of poible colluion. Thi iue i outide the cope of our formal analyi, but it may be a potentially important factor in ome wholeale electricity market that operate with a price cap. 22

24 We how that aymmetric equilibria exit for ome configuration of the demand ditribution and capacitie. If the lower bound of the upport of demand quantitie i ufficiently high, then there are aymmetric equilibria for which the price i equal to the market reerve price with probability one. There are a variety of poible extenion of our analyi. More general cot function that better approximate cot of electricity generation from variou plant type could be introduced. We could alo conider the role that price-enitive demand might play. We conjecture that the pirit of our reult would carry through to more general model. That i, production capacity contraint hape the et of equilibrium upply function via their role in limiting rival ability to repond to a firm high price-low output upply trategy. Although our model provide ome tool to ae market power iue in power market, we alo did not analyze the likely effect of other factor uch a tranmiion contraint, and uncertain congetion and outage on upply function equilibrium prediction in the preence of pivotal upplier. Thee iue are direction for future reearch. Appendix equilibria. The following lemma i ueful for characterizing the et of ymmetric upply function Lemma. Suppoe that rival of firm one chooe upply function * ( ) in (4.1) for ome p ' ( c, p). Let ŝ( ) be a upply function choen from et S by firm one, and define quantity q' by, q ' ˆ( p '). a) If θ - ( n 1) k q' k then expected profit aociated with ŝ( ) for firm one i le than or equal to expected profit aociated with * ( ) for firm one. b) If 0 q' < θ ( n 1) k then expected profit aociated with ŝ( ) for firm one i le than or equal to expected profit aociated with % ( p; q') S, where: 23