Discriminatory versus Uniform-Price Electricity Auctions with Supply Function Equilibrium

Transcription

1 Dscrmnatory versus Unform-Prce Electrcty Auctons wth Supply Functon Equlbrum Talat S. Genc a December, 007 Abstract. A goal of ths paper s to compare results for dscrmnatory auctons to results for unform-prce auctons when supplers have capacty constrants. We have a pretty good understandng of what equlbrum results look lke for the unform-prce auctons. But an unresolved problem s what happens when a dscrmnatve aucton s run and supplers have capacty constrants. We formulate a supply functon equlbrum (SFE) model n contnuous offer schedules wth nelastc, tme varyng demand and wth sngle step margnal cost functon to compare two aucton nsttutons n the presence of capacty constrants. We show that payments made to the supplers n the unque equlbrum of the dscrmnatory aucton can be less than the payments n the unform-prce aucton, dependng on whch unform-prce aucton equlbrum s selected. For the hgh demand and/or low excess capacty cases we also characterze mxed strategy supply functon equlbrum under the dscrmnatory aucton. Keywords: Supply functon equlbrum, contnuous offer schedules, electrcty markets, unform-prce aucton, dscrmnatory aucton. I am grateful to Stanley S. Reynolds for helpful dscussons and suggestons n mprovng the manuscrpt. I thank the assocate edtor and the anonymous referees for constructve remarks. I also thank Bram Cadsby, and the partcpants at the semnars at Unversty of Toronto and HEC-Montreal for useful comments. Ths work s supported by the Socal Scences and Humantes Research Councl of Canada Grant CID a Assstant Professor of Economcs, Department of Economcs, Unversty of Guelph, Guelph, ON, Canada, NGW; emal: tgenc@uoguelph.ca; phone: ext:5606

2 . Introducton The dscrmnatory or pay-as-bd aucton nsttuton was adopted by the Brtsh Regulatory Authorty n the England and Wales wholesale electrcty market n March 00. In some US electrc power markets (e.g., Calforna power market) smlar dscussons are beng held for changng the aucton format from unform to dscrmnatory prcng wth the supposton that the dscrmnatory aucton mght acheve more effcent outcomes. In these envronments, whch aucton nsttuton yelds hgher profts for sellers, or lower average prces for buyers stll reman open questons. Effcency results and welfare rankngs of these auctons n the electrcty markets are beng examned by varous researchers such as Fabra, von der Fehr and Harbord (Ref. ), Federco and Rahman (Ref. ), Rassent et al. (Ref. 3), Fabra (Ref. 4), Holmberg (Ref. 5), and Son, Baldck, Lee and Sddq (Ref. 6) under varous assumptons. A goal of ths paper s to compare results for dscrmnatory auctons to results for unform-prce auctons when supplers have capacty constrants. Take the case of constant margnal cost up to capacty, for example. We have a pretty good understandng of what equlbrum results look lke for the unform-prce auctons. But an unresolved problem s what happens when a dscrmnatve aucton s run and supplers have capacty constrants. Fabra, von der Fehr and Harbord (Ref. ) artcle s partcularly mportant and s closely related to the research reported here. Most of the analyss of Fabra et al. s based on the assumpton that bds are short-lved and are dscrete step supply offers. However, n some markets supplers can submt what amount to contnuous, pece-wse lnear, postvely-sloped supply functons wth quantty choces that determne the elbow ponts [see e.g., Wolfram s (Ref. 7) descrpton of the England and Wales aucton n the early 990 s]. Strateges of ths type are also descrbed by Hortacsu and Puller (Ref. 8) for the Texas electrcty balancng market. Fabra et al. s analyss and conclusons about the relatve utlty of dscrmnatory versus unform-

3 prce electrcty auctons are relevant and valuable n markets characterzed by short-lved bds and rules requrng dscrete supply functons. However, contnuous supply functon equlbrum (SFE) outcomes are qute dfferent n markets characterzed by long-lved bds. Genc and Reynolds (Ref. 9) developed a contnuous SFE model that proves the exstence of multple pure strategy equlbra under the unform-prce aucton nsttuton n markets of ths type. Ths stands n marked contrast to Fabra et al. step functon approach whch, for example, fnds no pure strategy equlbrum n a parameter regon of consderable emprcal nterest. The model presented n ths paper dffers from the current lterature n many ways. To see the dfferences, I explan closely related papers n detal. Federco and Rahman (Ref. ) compare the two aucton formats for two extreme cases: perfect competton and monopoly market structures. Under complete nformaton over costs and fxed-perfectly-nelastc demand, they fnd that these two auctons result n the same prces and payoffs. However these results change f demand s nelastc and uncertan. Rassent et al. (Ref. 3) expermentally analyze market outcomes of the dscrmnatory and unform auctons. Each seller has multple technologes wth fxed capactes and submts step functon offer schedule to the market. They fnd that changng aucton format from unform to dscrmnatory leads to sgnfcant electrcty prce ncreases n the off-peak and shoulder perods. They also fnd that the prce varablty from tradng perod to tradng perod s lower under the dscrmnatory aucton than under the unform-prce aucton when there s a greater excess capacty durng the peak perod. Wolfram (Ref. 0) conjectures n favor of the unform-prce aucton n the England Wales Electrcty Pool. Kahn et al. (Ref. ) reject the dea of swtchng to the dscrmnatory aucton, and clam that the dscrmnatory aucton may cause neffcences, because generators wll no longer bd at ther margnal costs, and the tact colluson that exsts wthn the unform aucton may persst n the dscrmnatory aucton. Evans and Green (Ref. ) report that after shftng the aucton format to dscrmnatory prcng wholesale electrcty prces have decreased n England and Wales. However, Bower (Ref.

4 3) and Newbery (Ref. 4) argue that ths decrease s due to the asset dvesttures, ncrease n mports and excess market capacty. The other related research s the Holmberg (Ref. 5) paper, whch s a valuable contrbuton. Although there are many dfferences between hs paper and ths paper, some of Holmberg results could be vewed as complementary to my results, snce they deal wth a stuaton capacty shortage wth postve probablty. Holmberg compares the aucton formats usng completely nelastc demand. He assumes contnuous margnal costs and derves SFE wth the condton that demand exceeds total avalable ndustry capacty wth postve probablty. He clams that pure strategy equlbrum may not exst n the dscrmnatory aucton, f demand has an ncreasng hazard rate and the margnal costs are constant, and concludes that average prces are weakly lower n the dscrmnatory aucton. However, he does not deal wth characterzaton of mxed strategy equlbrum n the dscrmnatory aucton. Anwar (Ref. 5) also compares the dscrmnatory and unform-prce auctons. He studes equlbra n mult-unt common value aucton model that sometmes provdes a postve resdual market demand to supplers. He consders dscrete step supply offers, and assumes uncertan demand. Each frm has the same constant margnal cost up to capacty. He shows that when demand s hgh and frms face some resdual demand, the unform aucton leads to hgher prces than the dscrmnatory aucton. Fabra, von der Fehr and Harbord (Ref. ) analyze a game-theoretc model n whch frms wth asymmetrc capactes and costs submt dscrete unt offer schedules to the auctoneer. Most of ther analyss assumes a completely nelastc demand wth a fxed market reserve prce, constant margnal cost of producton, and producton capacty constrants. For the uncertan demand and perfectly symmetrc case, they fnd that expected payments to supplers are the same for both auctons. They also fnd that for low demand realzatons, equlbrum s both unque and dentcal; the equlbrum s bddng at the margnal cost of the neffcent suppler for the two aucton formats. For the asymmetrc duopoly case, n the dscrmnatory aucton they fnd that 3

5 there s no pure strategy equlbrum but only n mxed strateges. Son, Baldck, Lee and Sddq (Ref. 6) compare performance of two strategc players, one s wth large capacty the other s wth small capacty, under both aucton formats n a short term electrcty market game. Players bd energy blocks and face nelastc or elastc demand n the aucton. They show that contrary to the revenue equvalence theorem players expected total revenues are hgher under unform prcng than under the pay-as- bd prcng. They smulate the model results and compute the mxed strategy equlbrum under the dscrmnatory aucton by applyng the Lemke and Hawson algorthm, and dscuss the lkely effects of transmsson constrants and multple players on ther fndngs. In the paper I extend the SFE approach presented by Genc and Reynolds (Ref. 9) to compare dscrmnatory versus unform-prce electrcty auctons. I assume that demand s tme dependent (equvalently, stochastc) and s perfectly nelastc up to a prce cap. I also assume that total ndustry capacty s greater than or equal to the peak demand. I compare the two aucton formats when each frm s margnal cost of producton s constant up to ts producton capacty. I consder symmetrc (and asymmetrc, whenever possble) Nash equlbrum n contnuous supply functon strateges n olgopoly. In ths basc model, I fnd that n the dscrmnatory aucton, the optmal equlbrum supply functon s unque and supplers bd compettvely. However, n the unform-prce aucton there s a contnuum of equlbra as n other SFE models (such as Baldck and Hogan (Ref. 6), Day, Hobbs and Pang (Ref. 7), Anderson and Phlpott (Ref. 8), Klemperer and Meyer (Ref. 9)) n whch equlbrum prces range from margnal cost to the prce cap. Therefore, each player s proft n the unform aucton s always weakly greater than the proft n the dscrmnatory aucton at any tme durng the tradng perod. I also fnd that n the sngle-step margnal cost case, the functonal form of the demand s rrelevant to the equlbrum strateges n both aucton nsttutons. These results are consstent wth the work of Back and Zender (Ref. 0), Wang and Zender (Ref. ). 4

6 In addton ths paper makes the followng contrbutons. Frst, our fndngs are applcable to wholesale electrcty auctons that are conducted under both long-lved and short-lved aucton rules (see Secton ). Second, we consder the effects of capacty constrants and pvotal supplers. We demonstrate that there s no pure strategy equlbrum n the dscrmnatory aucton; however equlbrum strateges n the unform-prce aucton are n pure strateges and multple. By the pvotal supplers analyss Genc and Reynolds (Ref. 9) show that some of the proposed SFE n the lterature are not equlbrum under the unform-prce aucton. In the current paper, I show that ths result holds even under the dscrmnatory aucton (for example, Wang and Zender type of equlbra can be ruled out). Thrd, n the asymmetrc capacty model, for the hgh demand and/or low excess capacty cases we also characterze the mxed strategy Nash equlbrum n supply functons under the dscrmnatory aucton. Fourth, gven that bddng rules as well as aucton formats are market desgn ssues, we dscuss that as a market polcy the auctoneer should promote contnuous supply functon bddng. Ths polcy would be useful for both the auctoneer and the supplers especally for equlbrum predctons, snce the step functon bddng predcts mxed strategy equlbrum whch s hard to compute and undesrable from the operatonal pont of vew (see Son et al. (Ref. 6)).. A Symmetrc Supply Functon Equlbrum Model Let n be the number of supplers and Nt () = t be the tme dependent and perfectly nelastc market demand (load duraton curve) up to some exogenous choke prce p, where t [0,] s tme n a tradng perod. The determnstc varaton n demand over tme s mathematcally equvalent to Klemperer and Meyer s (Ref. 9) model of uncertan demand wth bounded varaton. For ths load functon the correspondng stochastc demand would be GQ ( ) = N ( Q) n whch load quantty Q has support [0,]. Let K > 0 be capacty for frm, Demand changes over perods n a predctable way are also used by several authors ncludng Anderson and Xu (Ref. ), and Newbery (Ref. 3). 5

7 and assume that K = K N(0), that s the total ndustry capacty can meet the peak load. Also let zs ( ( pt ( ))) be pece-wse contnuous margnal cost functon, where s ( p ) s frm s supply functon. Each suppler s requred to submt a supply functon pror to the realzaton of the load to maxmze hs proft; gven that rvals supply functons are fxed. Market clears at the lowest prce that equates aggregate supply functon to demand. Each frm s pad at that prce for all capacty offers n the unform-prce aucton. Each frm s pad at ts bd prce for the quanttes suppled n the dscrmnatory aucton. In case of excess supply at the clearng prce, we assume a pro-rata on the margn rule for allocatng excess supply (see Genc and Reynolds (Ref. 9)). If there s excess demand we assume there s no trade n that perod. Let p (), p (0) be prces at tmes and 0, respectvely. Let T( p) = s ( p) be the tme at whch prce p would clear the market, and β be the aucton format parameter. Specfcally let β = refer to the unform aucton, and β = 0 refer to the dscrmnatory aucton. I assume players are symmetrc, and I characterze pure strategy symmetrc SFE. (Asymmetrc equlbrum n mxed strateges wth asymmetrc players s consdered n Secton 3.) I assume that each suppler has a constant margnal cost of producton up to capacty, that s zsp ( ( )) = c, where 0 c< p. Note that our model could be appled to two dfferent types of aucton rules: long-lved and short-lved auctons that have been conducted n wholesale electrcty markets. Aucton wth long-lved bds: Each suppler must submt a sngle supply functon that s bndng for an entre day. Baldck and Hogan (Ref. 6, n ther secton 4.) refer to ths as aucton rules requrng consstency of bds over the tme horzon. Supplers have a common understandng of what the load wll be for each perod (e.g., ¼ hour) of the day, but they cannot submt dfferent supply functons for dfferent (¼ hour) perods. The auctoneer determnes a clearng prce for each (¼ hour) perod, and each suppler s oblgated to produce up to ther supply quantty at the 6

8 clearng prce for each ¼ hour) perod. The load functon N(t) can be nterpreted as a contnuous approxmaton to the 96 load quanttes for ¼ hour perods, ordered from hghest to lowest. Aucton wth short-lved bds: Supplers are permtted to submt a supply functon for each (¼ hour) perod. The stochastc load nterpretaton of our model would be applcable f supplers faced some uncertanty about the load quantty for these (¼ hour) perods at the tme they submt ther supply functons. There would be a dfferent load rato (.e., rato of mnmum possble load to maxmum possble load) for each (¼ hour) perod. Proft maxmzaton problem of a frm n both aucton formats can be formulated by usng several methods, below we use a specfc one. The other methods, such as the one n Wang and Zender (Ref. ), lead to the same SFE condtons. Analogous to the method descrbed n Rudkevch (Ref. 4), assume that for each tme perod t, and for each ncrement of capacty ds( p) offered to the market at the prces range from p to p + dp, a frm s pad the bd prce p and a prce that s proporton of the market prce and the bd prce. Then the nfntesmal revenue flow wll be, dr( t, p) = [ p + β ( P( t) p)] ds( p). Hence, the tme t revenue for a frm s, Pt () rt ( ) = sp ( ())[ p() + β( Pt ( ) p())] + [ x+ β( Pt ( ) x)] dsx ( ). Subtractng tme t cost and P() ntegratng tme t proft over total tradng perod (.e., [0,]) wll result n total proft. After some algebrac manpulatons (by usng several ntegraton by parts) one can obtan the total proft for p(0) p(0). a frm durng the tradng perod: π(.) = ( p z(.)) s( pt ) ( pdp ) ( β) T( ps ) ( pdp ) p() p() Denote rvals total supply q(p), then the frm s optmzaton problem s mn( π (.)) s.. t ds dp = u( p), q( p) + s( p) = N( t), T( p()) =, T( p(0)) = 0, u( p) 0. 7

9 where the suppler can only choose a prce at some tme t ndrectly, va ther choce of supply functon. Lagrangan: L(.) = ( p z) s( p) T ( p) + ( β ) T( p) s( p) + ψ( p)[ s ( p) u( p)] + λ( p)[ q( p) + s( p) N( t)] Termnal functon and necessary condtons are as follows. Termnal functon: l(.) = s( p()) µ + s( p(0)) η. Lagrange-Euler equatons: d L = L dp s s, d L L = dp T T. Pontryagn optmalty prncple: L= mn u 0 L(.). Transversalty condtons: L l =, s p= p() s( p()) L l =. s p= p(0) s( p(0)) Complementarty condtons: s( p ()) µ = 0, µ 0, s( p(0)) η = 0, η 0. After solvng Lagrange-Euler equatons and Pontryagn prncple, and rearrangng terms we obtan the equaton s( p) + ( p z) s ( p) = ( β ) s( p) + N ( t)[( p z) T ( p) + ( β ) T( p)] () When β = and players are symmetrc, for any N(t), we obtan s( p) s ( p) =, (snce ( n )( p z) T ( p) N ( t) = s ( p) n). When β = 0 and players are symmetrc and Nt ( ) = t, then () results n s( p) n s ( p) =. Therefore, we obtan the followng proposton. ( n )( p z) Proposton.: Symmetrc equlbrum supply functons must satsfy the followng condtons: For the unform-prce aucton ( n )( p( Q) z( Q/ n)) p un ( Q) = (), Q 8

10 for the dscrmnatory aucton where Q s the ndustry supply. ( n )( p( Q) z( Q/ n)) p dsc( Q) = n( Q) (3), Proposton.: For symmetrc and constant margnal cost case (and non-bndng capacty constrants), SFE s unque and the equlbrum market prce s equal to the margnal cost of producton n the dscrmnatory aucton and SFE s multple and the range of equlbrum prces s n between margnal cost and prce cap n the unform-prce aucton. Proof: In the case of zq ( / n) = c, that s the margnal cost of producton s constant, where Q (0,), the soluton of () satsfes, wth the ntal condton p ( N(0)) p (0), un un pun c s( pun ) = n pun (0) c /( n ) (4). Ths s the commonly studed unform-prce aucton SFE n the lterature. In (4) a frm s equlbrum supply functon s ndexed by the ntal prce, p (0) ( c, p], and (4) does not volate the capacty constrants. (4) can also be wrtten n as nverse ndustry equlbrum supply functon, un p ( Q) c ( p (0) c) Q un n = + un (5). For the dscrmnatory aucton the soluton of (3) yelds, Q + / n ( n ) / n 0 n 0 pdsc( Q) = ( Q ) [ a + c( x ) dx], where a 0 s an ntegraton constant. Ths equaton can also be wrtten as p ( Q) c a ( Q ) dsc + / n = +, where a s another constant. For a =, p ( Q) = c. For a > 0 as Q, p ( Q). For a < 0 as Q, p ( Q). 0 dsc dsc dsc Hence there s only one economcally plausble soluton, whch s pdsc( Q) = c (6). 9

11 One can also show that ths s the only unque soluton by usng the ntal condton pdsc( N() = 0) = c. Note that f the ntal prce at tme was c m ε, where ε 0 then p ( Q) = c ( Q) dsc + / n m ε. For a fxed ε as Q, p ( Q) m, hence ε = 0 must hold. dsc Multplcty of the equlbra n the unform aucton stems from the fact that there are multple functons that pass through the proft maxmzng ponts. Bds that are always nframargnal are rrelevant to determne equlbrum payoffs. Corollary.: In the symmetrc frms wth constant margnal cost case, each player s proft n the unform aucton s weakly greater than the proft n the dscrmnatory aucton at any tme durng the tradng perod. Ths corollary s a drect result of the Proposton.. Because n the unform aucton, the market prce s always greater or equal to the margnal cost of producton, hence prce-cost markup s greater than or equal to zero, and players submt non-negatve and non-decreasng supply functons. Whereas n the dscrmnatory aucton, prce-cost markup s always equal to zero, no matter what proper supply functons are submtted proft for a frm becomes zero. Intutvely, Bertrand type equlbrum prevals because margnal cost s constant,.e., ndependent of the level of output. It appears that equlbrum supply functons are ndependent of demand functonal form for both aucton formats n the sngle-step margnal cost case, as long as demand s a If, as n my formulaton, total capacty s greater than or equal to max load, then a soluton to the dfferental equaton (3) must have pq ( = ) = z(/ n) ; that s, the boundary prce assocated wth the max load must equal the margnal cost (mc). When ths prce condton s satsfed, p ( Q = ) s equal to zero/zero; but usng l Hoptal s rule one can show that p ( Q = ) = z (/ n)/( n(n )). A boundary prce n excess of margnal cost yelds a stuaton n whch p ( Q = ) s equal to a postve number dvded by zero, whch s undefned. Ths dscusson explans why Wang and Zender fnd that when margnal cost s constant, the only equlbrum s to supply all unts at the constant margnal cost. 0

12 dfferentable functon of tme. As t can be seen from the equaton () that n the unform aucton ( β = ), s( p) = q ( p)( p z) holds for any Nt ( ), snce T ( p) N = s ( p) + q ( p), where q( p ) s the rvals total supply functon. In the dscrmnatory aucton ( β = 0 ), the equaton () takes the form of ( p zq ) ( p) = N T( p). For example, f demand was Nt ( ) = e t, then by usng the equaton t (), N T( p) = ns( p)ln( ns( p)) holds for the symmetrc case. Then the soluton of ths ordnary t t dfferental equaton (ODE) yelds p ( Q) c a (ln Q) + dsc / n = +. Usng the smlar argument that we used n the above proof, there s a unque soluton of ths expresson, whch only admts the constant a = 0, hence p ( Q) = c. If demand was, say Nt ( ) = N(0) l t, where l > 0, then the dsc soluton of () for β = 0 yelds to, wth symmetrc players, p ( Q) c a (ln( Q/ N(0)) dsc + / n = + 3, whch has agan an economcally meanngful soluton only f a 3 = 0. Thus Q ( N(), N(0)), pdsc( Q) = c. In the dscrmnatory aucton modfcaton of the demand functonal form may change only the form of the second term n the nverse ndustry equlbrum supply functon, pdsc ( Q ). But the coeffcent of the second term always gets to zero for an economcally meanngful soluton. 3. Extensons 3. Symmetrc Frms wth Capacty Constrants In ths secton we consder the role of capacty constrants n equlbrum predctons. The followng subsecton regardng unform-prce aucton s n the sprt of Genc and Reynolds (Ref. 9) and I wll extend ther approach to ncorporate dscrmnatory aucton. Wthout loss of generalty let s make the followng transformaton of tme perod n the demand formulaton so that we obtan one more varable n the analyss. That s, let tme t lτ so that demand takes

13 the form of N( τ ) = lτ, where l = N() N(0) 3, whch we call load factor and l (0,), τ [0,]. As defned earler, the capacty ndex s k K / N(0), where K = K and K s the frm s total capacty. Also let margnal cost of producton be constant up to the capacty. Capacty constrants may elmnate some of the equlbra proposed n equatons (5) and (6). Genc and Reynolds have studed ths only for the unform-prce aucton. In ther analyss, some supplers may tend to wthhold some of ther capacty from producton, when demand s hgh and/or excess capacty s low n the market. These supplers are called pvotal supplers. Formally, a pvotal suppler j may wthhold capacty durng the tradng perod τ, f K < N ( τ ). When ths nequalty holds, pvotal suppler j may sell the quantty j N( τ ) K j at the prce cap p, and ncrease ts proft. If K < N () holds, then each j suppler s pvotal durng the entre tradng perod. For the sake of smplcty, we wll consder a smple devaton whch mples that whenever rval frms do not have enough capacty to meet demand, a pvotal suppler wll emerge and meet the resdual demand at the prce cap p, and he wll not supply any quantty below that prce. There are three exogenous parameters ( kln,, ) that determne parameter regons. If l < ( n ) k/ n then each frm s pvotal at all tmes τ n [0,]. We refer to ths as totally pvotal 4 (TP) regon, formally TP {( kln,, ) n, k [, n/( n )), l (0, ( n ) k/ n) }. Second stuaton s that each frm s pvotal for some tradng perod, whch we call partally pvotal (PP) case, formally PP {( kln,, ) n, k [, n/( n )), l [ ( n ) k/ n, ) }. A thrd state s called never pvotal (NP) case, n whch n- combnaton of frms have enough capacty 3 Ths demand s qualtatvely same as the demand N(t)=-t. By transformng the demand to N ( τ ) = lτ, we extend the dmenson of the parameters space from to 3 to more extensvely research the effects of capacty constrants n equlbrum predctons. Also ths transformaton helps us better compare the equlbrum predctons of ths paper wth that of Fabra et al. (Ref. ). 4 Here parameter regons are n the same ven of the regons depcted n Fgure Two of Genc and Reynolds. The dfference s that the load rato curve s downward slopng of ts argument k n the above defnton.

14 to meet the peak load N(0). Formally, NP {( kln,, ) n, k n/( n ), l (0,)} >, n whch the pvotal suppler has no gan to devate from the canddate optmal supply functon. 3.. Role of Capacty Constrants n the Unform-prce Aucton Gven the functonal forms we have assumed for demand and cost, we calculate, for each possble ntal prce, the SFE equlbrum prces and proft per frm p ( τ) c ( p (0) c)( lτ) n un = + un, n+ pun c l SFE ( (0) )( ( ) ) Π un = ( p( τ) c)( N( τ)/ n) dτ = (7), nn ( + ) l τ = 0 where SFE Π un s the proft per frm assocated wth a canddate SFE. The smple type of devaton nvolves a supply functon wth no unts offered for prces below the maxmum prce, p, and all unts up to capacty offered at the prce p. We compare the proft assocated wth ths smple type of devaton wth the proft from canddate SFE. Let the parameters be n the totally pvotal regon; ( kln,, ) TP. Then the resdual demand at tme t for a devatng frm at prce p s, [ N( τ ) ( n ) K/ n] > 0. Total proft for the devatng frm s, D ( p c)[ n( l) ( n ) k] Π un = [ N( τ) ( n ) K / n]( p c) dτ = (8). n 0 Devaton proft exceeds proft assocated wth the canddate SFE f Π <Π, or equvalently SFE un D un f, p(0) c< φ( k, l, n)( p c), where φ n+ ( kln,, ) [ nn ( )( ll ) ( n ) kl]/[( ( l) ] + (9). If the markup at the ntal prce for a canddate SFE s less than the fracton φ of the maxmum markup, p c, then the canddate SFE s not an equlbrum. Equaton (9) provdes a suffcent condton for elmnatng certan canddate supply functon equlbra as equlbra. 3

15 Remark 3.: The market power ndex (.e., φ functon) s decreasng n the capacty ndex k, decreasng n the load factor l, and decreasng n the number of supplers n for ( kln,, ) TP. We omt the proof of ths remark snce t s analogous to the Proposton n Genc and Reynolds (The dfference s, n Genc and Reynolds the ndex s ncreasng n the load factor l). The economc nterpretaton of ths proposton s straghtforward. Now let the parameters be n the partally pvotal regon; ( kln,, ) PP. Then smlar to the above remark t can be easly shown that market power ndex for the PP regon s decreasng n the capacty ndex k, decreasng n the load factor l, and decreasng n the number of supplers n. Some of the results we fnd above may be easly compared to those of Fabra et al. (Ref. ) for the case of a unform prce aucton wth long-lved bds and two symmetrc supplers accordng to the above parameter regons. Frst, n the never pvotal (NP) regon, each frm has enough capacty to serve the entre peak load. Fabra et al. fnd a unque Nash equlbrum n whch each suppler bds at margnal cost. Ths Bertrand-lke result s a much more compettve predcton than the contnuous SFE approach yelds. As defned n equaton (4) supply functon equlbra always nvolve market clearng prces above margnal cost, and nclude the least compettve SFE (wth ntal prce equal to the market reserve prce, p ). The Fabra et al. predcton for the NP regon holds regardless of how many dscrete unts each suppler s permtted to submt offers for. Second, n the totally pvotal (TP) regon, nether frm has enough capacty to serve the off peak load; at least some of the capacty of each frm s requred to meet demand at all tmes durng a tradng perod. Fabra et al. fnd multple pure strategy equlbra, but all such equlbra yeld a market clearng prce equal to the market reserve prce, p. Ths s a more collusve predcton than that of the contnuous SFE model, for whch equlbrum prces are contaned n an nterval wth upper bound equal to p. Thrd, n the partally pvotal (PP) 4

16 regon, any sngle frm s pvotal for part, but not all, of the tradng perod. For step functon bddng, Fabra et al. fnd that a pure strategy equlbrum does not exst for parameters n the PP regon [see Lemma 3, Fabra et al., p. 35]; the equlbrum s n mxed strateges. Fabra et al. do not have characterzaton results for the mxed strategy equlbrum, except that the equlbrum prce dstrbuton approaches margnal cost prcng as parameters approach the NP regon and that the equlbrum prce dstrbuton approaches prcng at the market reserve prce wth probablty one as parameters approach the TP regon. In the contnuous SFE model there are multple equlbra. The least compettve equlbrum yelds an nterval of prces wth upper bound p. The most compettve equlbrum has prces above margnal cost; how far above margnal cost depends on load factor and capacty ndex parameters. We state that predcted market clearng prces for the step functon model may be ether hgher or lower than equlbrum SFE market clearng prces dependng on parameter values, when parameters are n the PP regon. 3.. Role of Capacty Constrants n the Dscrmnatory Aucton In ths secton we show that the only pure strategy equlbrum n the dscrmnatory aucton may be ruled out because of the capacty constrants and the role of pvotal supplers. Ths suggests characterzaton of mxed strategy equlbrum n the dscrmnatory aucton wth capacty constraned frms. Suppose that parameters (k, l, n) are n the NP regon, n that case equaton (3) does not volate capacty constrants and admts a unque soluton. That s, nverse ndustry supply functon s dfferentable and the soluton s pdsc( Q) = c. Snce (n-) rval supplers have enough capacty to meet the hghest level of load, there wll be no ncentve to devate from margnal cost prcng. If a suppler offers a prce above c then other frms already have ncentves to undercut ths suppler s bd slghtly so that they ensure to be called out. Also snce no suppler 5

17 faces postve resdual demand, any margnal prce (.e., the prce of the margnal unt that equates supply to demand) above the margnal cost s elmnated as pure strategy equlbrum. When the parameters (k, l, n) are n the TP or the PP regons, then there s an ncentve to devate from margnal-cost-prcng. That s whenever capacty constrants bnd, locally optmal offer schedules would not be globally optmal because proft functon of devatng suppler becomes dscontnuous and non-concave n prce. We dscuss ths next. Suppose that the parameters (k, l, n) are n the TP regon, whch means that the rval frms do not have enough capacty even to meet the off-peak load N() = l. Each frm s some porton of capacty s needed. Then a pvotal suppler at tme τ faces the resdual demand N( τ ) K > 0 at the prce p. Its devaton proft at tme 0 s j D Π dsc (.) = ( p cdq ) = ( p cl ). l Obvously, SFE Π > Π, where SFE proft s Π = 0 snce p ( Q) = c for all Q K. Hence D dsc SFE dsc dsc dsc margnal cost prcng cannot be equlbrum and the devaton leads to the margnal prce p. If parameters (k, l, n) are n the PP regon, then a devatng frm s proft at the tradng perod zero s ˆ D SFE Π dsc = ( p c) dq= [ K( n )/ n]( p c) > 0=Πdsc. K( n ) n Hence durng the perods 0 throughτ, where N( τ ) = ( n ) K / n, the margnal cost prcng suggested by pdsc( Q) = c for all Q K s not part of the equlbrum. The rvals total capacty s not enough to meet the maxmum load, wth a postve probablty a frm wll face a resdual demand n the PP regon, wth sure probablty a frm wll face a resdual demand n the TP regon. But ths s true for each frm n the dscrmnatory aucton; hence there wll not be a pure strategy equlbrum. Thus, we have the followng proposton. 6

18 Proposton 3.: If parameters (k, l, n) are n the TP or the PP regons, n whch the demand s hgher than the rvals total capacty for at least some tradng perod, then devaton ncentves rule out the only equlbrum canddate n the dscrmnatory aucton. 3. Mxed Strategy SFE wth Asymmetrc Supplers n the Dscrmnatory Aucton Horzontal Supply Functons n TP Regon: In the above secton we have seen that smple devaton rules out the only canddate pure strategy SFE when the parameters are n the TP regon. It s also not optmal when all frms offer ther generaton unts at the prce cap snce devaton s proftable. In ths secton we prove that t s equlbrum n the mxed strateges to bd a sngle prce for the entre capacty of a suppler. Assume n= asymmetrc supplers that face tme-varyng load N( τ ) = lτ such that K+ K N(0), K N() and K N(). Also assume that frm chooses a flat prce p (whch s a contnuous random varable) n (, c p ) for hs entre capacty wth the probablty dstrbuton F( p ). F( p ) s an atomless dstrbuton functon wth support [ c, p ]. Let p be the prce chosen by frm for all of hs capacty offered at tme t n the tradng perod. Let k and k be capacty ndces of frms and, respectvely. Note that k = K, =,, snce N (0) =. These parameters naturally lead to the TP set. Proposton 3.: Assume that supplers =, are allowed to submt step supply functon bds. Let k = k+ k. Any prce offer p (, c p) for the entre capacty of suppler ( =,) wth the probablty dstrbuton ( p c)( k + l/) ( p p) k F( p ) = ( p c)( k + l/) consttutes a mxed strategy Nash equlbrum n the TP regon under the dscrmnatory aucton. The auctoneer s total expected payment to the supplers s ( p c)( l k) + c( l/) durng the tradng perod. Proof: The expected proft of suppler durng the entre tradng perod s, 7

19 p p () t π ( p ) = [ p ( t) c] k df( p) dt + [ p ( t) c][ N( t) k ] df( p) dt 0 p ( t) 0 c = [ F( p )][ p c] k + F( p )[ p c][ k l/ ] n whch we used the boundary condtons F( p ) = and Fc ( ) = 0. Above the frst expresson s the proft that stems from when the rval frm chooses a prce above p. In that case frm dumps hs entre capacty to the market. The second expresson of the proft s regardng the possblty that frm chooses a prce below p, and then frm meets the resdual demand, whch s equal to the expected load ( l /) mnus the rval s supply, k. In equlbrum π ( p) = 0 holds and t mples the probablty dstrbuton functon ( p c)( k l/) f( p ) = ( p c) ( k + l/) (0), and the cumulatve dstrbuton functon ( p c)( k + l/) ( p p ) k F( p ) = ( p c)( k + l/) (). Substtutng ths probablty nto the proft functon gves π ( p) = ( p c)( k l/), whch s ndependent of p (, c p) and suppler can randomze the prce, n ths nterval, wth the probablty n (). Snce the game s symmetrc we can perform a smlar analyss for player and obtan the probablty functons, ( p c)( k l/) f( p ) = ( p c) ( k + l/) ( p c)( k + l/) ( p p ) k F( p ) =, ( p c)( k + l/). Player s proft wll be π ( p) = ( p c)( k l/), whch s ndependent of p (, c p), and he can randomze any prce n ths nterval. Now we need to show that these mxed strateges form equlbrum. In the above we have shown that the strateges satsfy the frst order necessary condtons. Next we wll prove that f 8

20 suppler uses mxed strateges then suppler j s ( j ) best response cannot be ncreasng step supply functon. The followng proof s n the sprt of Anwar (Ref. 5). Assume that frm chooses hs mxed strateges n the prce range (, c p ) accordng to the above probablty functon. Let ( bj, q j) j be the sequence of bd prces and quanttes for suppler, where m qj = k, and... m j= p b > b > > b > c. Also let b= ( b j ) j. Then the total expected proft for frm durng the tradng perod s, t p b m m π( b) = qj( bj c) df( p) + qj( bj c) + [( N( t) k) ( k q) ]( b c) df( p) dt 0 b j= j c = t p m b m + q ( b c) df( p) + q ( b c) df( p) j j j j t b j= j b = b m + qj( bj c) + [( N( t) k) ( k q q) ]( b c) df( p) dt c j= p b b m m m r m + qj( bj c) df( p) + qj( bj c) df( p) + qj( bj c) df( p) t j j r 3 j r m b = b = = b = r bm m + ( Nt ( ) k) ( k qj) ( bm cdf ) ( p) dt, c j= where the frst ntegraton refers to the load nterval for whch the proft s calculated. The second ntegraton along wth the summaton represents expected proft from the sales of each unt of quantty q j at a prce of b j, gven the prce chosen by rval. If frm chooses a prce (wth some probablty) above the frm s maxmum bd prce, then frm sells each unt of capacty at varyng bd prces. Frm also sells some quantty to the market snce nether frm has enough capacty to meet even the off-peak load. The nterpretaton of other expressons s also smlar. Note that the above proft functon s separable n terms of bd prces for the bd steps. That s, π( b) = π( b j ), where j 9

21 t m m π( b) = q( b c) + F( b) qj( bj c) qj( bj c) + [( N( t) k) ( k q) ]( b c) dt 0 j= j= t m m + q( b c) + F( b) qj( bj c) qj( bj c) dt t j= j= +... m m + q( b c) + F( b) qj( bj c) qj( bj c) dt. t m j= j= Ths proft expresson smplfes to t [ ] π ( b) = [ F( b)] q ( b c) + F( b)( b c) ( N( t) k ) ( k q ) dt, 0 where Fb ( ) s as n () wth b = p. We take the frst dervatve of the proft functon and obtan, t k ( N( t) k) dt qdt + q( k + l/) dπ( b) 0 t q( k + kt + l/) k( kt t+ lt /) = =. b k + l/ k + l/ Let Y k + kt + l/ and Z kt t + lt. / Frst note that f t = 0 then Z = 0, and f t > 0 then Z > 0. Next, l l Z Y = ( k)( t) ( t) = ( t)[ k ( + t)] > ( t)[ k l] 0, wth strct nequalty for t <. The nequaltes nvolvng Z and Y, combned wth the condtons q dπ( b) < k and k + l/> 0, mply that < 0. db Ths dervatve mples that frm should reduce ts bd prce from b to b. Then frm wll have m- bd prces. Repeatng the above process (m-) tmes results n a sngle prce offer, whch s n the nterval (, c p ), for all unts of hs capacty. Thus, optmal response to the rval s 0

22 sngle-prce-offer s to submt one prce for all unts of generaton. If we perform the smlar analyss for frm, we obtan that frm also submts a sngle prce offer for hs entre capacty. Gven that 0 < F( p), F( p) <, π( p) and π ( p) are twce dfferentable and concave n prces and when suppler uses mxed strateges then suppler j s ( j ) best response cannot be ncreasng step supply functon, we conclude that ( p c)( k + l/) ( p p) k F( p ) = ( p c)( k + l/) consttutes a mxed strategy Nash equlbrum n the TP regon under the dscrmnatory aucton for =,. The auctoneer s total expected payment to the players wll be, π ( p ) + π ( p ) + c N( t) dt= ( p c)( l k) + c( l/), where the frst term s frm s proft, 0 the second term s frm s proft, and the fnal term s the total cost of producton n the entre tradng perod. Smooth Supply Functon Case: Now consder the case n whch supplers are allowed to bd n contnuous supply functon strateges (nstead of step supply functons), and a rval suppler uses a mxed strategy. Then we clam that one cannot construct a smooth ncreasng offer functon that yelds hgher expected proft than the expected proft from any horzontal offer. The ntuton behnd that s that by choosng a suffcently large number of steps, one could come arbtrarly close to the profts obtanable from a smooth supply functon by usng a mult-step offer functon. But n the proof of Proposton 3. we show that mult-step offer functon yelds lower proft than a sngle step offer functon. Thus, the optmal response to the rval s mxed strategy cannot be a smooth supply functon.

23 Horzontal Supply Functons n PP Regon: Now consder the case such that for =,, lτ = N( τ ) k, f τ (0, t] and N( τ ) < k, f τ (,] t. Also assume that N(0) k+ k. Ths case represents the PP set. Proposton 3.3: Assume that supplers are allowed to submt step supply functon bds. Let k be the capacty ndex of suppler ( =,) and let k = k+ k. There exsts a mxed strategy Nash equlbrum n the dscrmnatory aucton n the PP regon through the prce p (, c p) offered for the entre capacty of suppler wth the probablty functon, F( p ) = ( p c)( t+ lt l/ + tk) ( p p)( t+ lt / l/ + tk) ( p c)( t+ lt l/ + tk). Proof: Suppose that (suppler beleves that) suppler chooses a prce p (, c p) wth probablty F( p ) and suppler smultaneously responds to rval s strategy wth a prce p for hs entre capacty. Then the expected proft of suppler durng the tradng perod s, t p p () t π( p) = [ p( t) c] kdf( p) + [ p( t) c][ N( t) k] df( p) dt 0 p ( t) c p p () t + [ p ( t) c] N( t) df( p) + 0 df( p) dt t p () t c = [ F( p )][ p c] kt+ F( p )[ p c][ t lt / tk ] + + [ F( p)][ p c][ l/ t lt / ], where t solves for lt = k. In equlbrum π ( p) = 0 holds and t mples the probablty and the cumulatve dstrbuton functons ( p c)( A B) f( p ) = ( p c) ( B) ( p c)( B) ( p p )( A), respectvely,, F( p ) = ( p c)( B) where A = t+ lt / l/+ tk, and B t lt l/ tk = + +.

24 Snce the game s symmetrc we can perform smlar analyss for player and obtan the probablty functons, ( p c)( A B) f( p ) = ( p c) ( B) ( p c)( B) ( p p)( A) and F( p) =, ( p c)( B) where A = t+ lt / l/+ tk, and B s as above. Next we wll show that f suppler uses mxed strategy then suppler s best response cannot be ncreasng step supply functon n prces. Assume that frm chooses hs strateges n the prce range (, c p ) accordng to the above probablty functon. We form the horzontal supply strateges as follows. For smplcty assume that frm chooses two prce-quantty pars, ( bj, q j) j, j =,, where m= qj = k, and j= p b > b > c. Also let b= ( b j ) j. One can easly extend ths analyss to m> prce-quantty pars, as we dd above. The total expected proft for frm durng the tradng perod s, t p ( b) = [ q( b c) + ( k q)( b c) ] df( p) 0 b b + ( )( ) + [( ( ) ) ( ) ]( ) ( ) c π k q b c N t k k q b c df p dt t p b + q ( b c) + ( k q )( b c) df( p) + ( k q )( b c) df( p) [ ] t b b b + ( Nt ( ) k)( b cdf ) ( p) dt c t3 p b + [[ N( t) ( k q)]( b c) + ( k q)( b c) ] df( p) + ( k q)( b c) df( p) dt t b b p b + N()( t b c) df( p) + N()( t b c) df( p) dt. t3 b b 3

25 The above proft functon s separable n terms of bd prces for the steps. That s, π ( b) = π ( b j ). After some smplfcatons we wrte the proft functon n terms of b as j follows: t π ( b) = F( b) [ N( t) k ( k q )]( b c) dt 0 t3 + [ Fb ( )] [ Nt ( ) k]( b cdt ) + [ Fb ( )] tq( b c), 3 t where Fb ( ) s as n above wth b = p. We take the frst dervatve of the proft functon and obtan, dπ db t ( b) 0 = f( b) [ N( t) k ( k q )]( b c) dt t + Fb ( ) [ Nt ( ) k ( k q)] dt+ tq[ f( b)( b c) + Fb ( )] 0 t3 + [ f( b)( b c) + F( b)] [ N( t) ( k q )]( b c) dt. t We fnd that [ f( b)( b c) + F( b)] = ( B A)/ B< 0, where A, B are as defned above and B A = t k + lt t l <, and B > 0. Also Nt ( ) = k q and Nt ( ) = k hold. [ /] / 0 Hence, n the above the frst two terms are postve and the rest s negatve. If k q s large, dπ( b) then the frst two terms approach to zero snce t 0. Therefore < 0. Subsequently we db conclude that offerng the entre capacty at a sngle prce s more proftable than usng a step functon wth multple bd prces for the quantty steps. 4. Conclusons We examne generators bddng behavor n the unform and dscrmnatory prce auctons under varous assumptons on equlbrum bddng types, and capacty constrants. We dscuss the 4

26 dffcultes, from operatonal and computatonal ponts of vew, of the mxed strategy equlbrum n the market desgns allowng dscrete offers, even though both types of bddng rules have dfferent predctons. Some of our results n the basc model are consstent wth the current lterature. We show that the dscrmnatory aucton SFE s unque, but the equlbrum s multple n the unform aucton. When capacty constrants bnd and pvotal supplers face postve resdual demand we obtan that there s no pure strategy SFE n the dscrmnatory aucton. In the mxed strategy supply functon analyss we examne the nature of a best response to a rval strategy that s a mxed strategy over horzontal supply functons. We show that offerng all of the capacty at a sngle prce s more proftable than usng a step functon wth multple bd prces for the steps. We conclude that t s not clear whether the dscrmnatory aucton format s tractable for the supplers and/or the auctoneer or whether t entals low prces for the consumers. REFERENCES. FABRA, N., VON DER FEHR, N., HARBORD, D., Desgnng Electrcty Auctons, Rand Journal of Economcs, vol. 37:, pp. 3-46, FEDERICO, G., and RAHMAN, D., Bddng n an Electrcty Pay-as-Bd Aucton, Journal of Regulatory Economcs, vol. 4:, pp. 75-, RASSENTI, S., SMITH, V., WILSON, B., Dscrmnatory Prce Auctons n Electrcty Markets: Low Volatlty at the Expense of Hgh Prce Levels, Journal of Regulatory Economcs, vol. 3, pp. 09-3, FABRA, N., Tact colluson n repeated auctons: unform versus dscrmnatory, Journal of Industral Economcs, vol. 5:3, pp. 7-93, HOLMBERG, P., Comparng Supply Functon Equlbra of Pay-as-Bd and Unform-prce Auctons, PhD thess, chapter 4, Uppsala Unversty, Economcs Department; SON, Y.S., BALDICK, R., LEE, K.H, SIDDIQI, S., Short-Term Electrcty Market Aucton Game Analyss: Unform and Pay-as-Bd Prcng, IEEE Transactons on Power Systems, vol. 9 (4), pp ,

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28 . ANDERSON, E.J., and XU, H., Supply functon equlbrum n electrcty spot markets wth contracts and prce caps, Journal of Optmzaton Theory and Applcatons, vol. 4(), pp , NEWBERY, D.M., Competton, contracts and entry n the electrcty spot market, Rand Journal of Economcs, vol. 9, pp , RUDKEVICH, A., Supply Functon Equlbrum: Theory and Applcatons, Proceedngs of the 36 th Hawa Internatonal Conference on System Scences,