Spot Market Copetitio i the UK Electricity Idustry Nils-Herik M. vo der Fehr Uiversity of Oslo David Harbord Market Aalysis Ltd 2 February 992 Abstract With particular referece to the structure of the UK idustry, price copetitio i a deregulated wholesale arket for electricity is odelled as a sealed-bid, ultiple-uit auctio with a rado uber of uits. It is argued that uder the existig regulatory rules oe ust expect volatile prices, above argial-cost pricig ad iefficiet despatchig. Evidece fro the pricig perforace of the UK idustry is preseted ad show to be copatible with the odel predictios. We also discuss alterative regulatory rules ad show that offerig to supply at argial cost ca be iduced as a doiat strategy for all firs, thereby securig efficiet despatchig. JEL Classificatio Nubers: D44. L94, L0, L5 Keywords: Multi-uit auctios, electricity spot arkets, regulatory refor The first author has beefited cosiderably fro cotact with the ESRC project "The Regulatio of Firs with Market Power ad wishes to thak its participats, ad i particular Joh Vickers, for helpful discussios. The secod author's work was partially supported by Lodo Ecooics Ltd. We are grateful to Friedel Bolle, Robi Cohe, Fi Førsud, Joh Kay, Kai-Uwe Küh, Meg Meyer, Jorge Padilla, Toy Curzo Price, Lars Sørgaard, workshop participats at Nuffield College ad two aoyous referees for valuable coets ad suggestios o earlier versios. Fiacial support fro NORAS (Norges råd for avedt safusforskig) is gratefully ackowledged (NHF).
. Itroductio At the core of the recetly deregulated ad privatized UK electricity idustry is the wholesale spot arket. Before each period that the arket is ope, the geeratig copaies (the geerators) subit iiu prices at which they are willig to supply power. O the basis of these offer prices, the Natioal Grid Copay, which plays a cetral role as coordiator ad which is resposible for ruig the trasissio grid, draws up a least-cost pla of geeratig uits for despatchig i the ext period. This rak order, together with dead, deteries which uits will actually be despatched. 2 Payets to supplyig uits, or sets, are based o a syste argial price, deteried as the offer price of the argial operatig uit i every period. The particular orgaizatio of the electricity spot arket akes stadard oligopoly odels iadequate as aalytical tools. We propose istead to odel this arket as a sealed-bid, ultiple-uit auctio. I the first stage of the odel, firs siultaeously subit offer prices at which they are willig to supply their (give) capacities. As i the UK idustry, firs (geerators) ca subit differet offer prices for each idividual plat or geeratig set, i.e. firs offer step supply fuctios. Sets are the raked accordig to their offer prices (i.e. a supply fuctio is costructed). I the fial stage, dead is realized ad the syste argial price is deteried by the itersectio of dead ad supply; that is, by the offer price of the argial operatig uit. It turs out that pure-strategy equilibria do ot always exist i such a odel. The reaso is basically the sae as that i stadard oligopoly odels of capacity-costraied price copetitio (Kreps ad Scheika, 986). Sice, whe dead is sufficietly large, a fir is uable to serve the whole arket at the copetitive price, there is a icetive to raise bids above argial cost, ad thus the copetitive outcoe caot be a equilibriu. It ca the be show that for a rage of dead distributios o other pure-strategy cobiatios costitute a equilibriu either. We believe that this result does ot ecessarily reflect a iadequacy of our odellig approach, but rather suggests that there is a iheret price istability i the preset regulatory set up. Ideed, our epirical evidece would see to cofir that experietatio ad abrupt chages i pricig strategies is a feature of the ew idustry. This particular result (the o-existece of equilibriu i pure strategies) also casts soe doubt o the relevace of the odel aalyzed by Richard Gree ad David Newbery (99) (see also Bolle,990, ad Newbery, 99). These authors argue that the step-legth, i.e. the size of idividual geeratig sets, is sall eough to justify approxiatig the step-supply schedules by sooth (i.e. cotiuously differetiable) fuctios, thus applyig the supplyfuctio fraework developed by Kleperer ad Meyer (989). As we deostrate however, the particular types of equilibria they derive do ot geeralize to a odel where sets are of positive size. Although theirs is a seeigly very useful cotributio, it reais a ope questio whether the biddig strategies of the firs will differ sigificatly if they are forced For details o the UK electricity idustry, ew ad old, see Vickers ad Yarrow (99), Gree (99a) ad Jaes Capel & Co.(990). 2 Allowaces for particular circustaces such as trasissio costraits ad syste stability cosideratios ay lead to deviatios i despatchig fro that which follows fro the rak order. Further details are provided i appedix B.
to provide a step fuctio, or whether they are allowed to provide a sooth schedule (Gree ad Newbery, 99, footote 2, page 5). 3 Nevertheless the ost iportat result, iefficiet pricig, turs out to be robust to alterative fors of odellig. Ideed, we fid a eve stroger tedecy tha Gree ad Newbery towards above argial-cost pricig. Thus the cojecture that the Bertrad outcoe is ulikely i the preset istitutioal set-up of the UK electricity idustry, eve if there is o collusive behaviour, sees to be strogly supported. I additio, our odel suggests that high-cost sets ay be bid i at lower offer prices tha lower-cost sets ad thus be despatched before these ore efficiet uits. Hece despatchig ay be iefficiet i the sese that overall ecooic geeratio costs are ot iiized. A iportat advatage of our fraework is that it akes it possible to odel explicitly the role of the grid copay (the auctioeer), ad the use isights fro the auctio literature to study the effects of differet pricig rules, i.e. the rules deteriig the prices paid to differet supplyig uits. With a pricig rule like the oe used i the preset UK electricity idustry, the gae correspods to a first-price sealed-bid auctio. However, by lettig the syste argial price be deteried by the offer price of oe of the argial o-operatig sets, the gae correspods to a (geeralized) secod-price sealed-bid auctio, ad i this case offerig to supply at argial cost ca be show to be a doiat strategy for each fir. This result is i accord with what is typically foud i the literature o optial auctios where it is well kow that secod-price (or Vickery) auctios lead to higher reveues for the auctioeer tha do first-price auctios (Myerso (98) or Maski ad Riley (989)). The reaider of the paper is orgaized as follows. Our auctio odel of the U.K. electricity spot arket is preseted i sectio 2 ad the aalyzed i sectio 3. Sectio 4 the relates the odel to epirical evidece fro the UK electricity idustry. I sectio 5 we cosider a alterative regulatory rule. Sectio 6 cotais a short suary ad coclusios. 2. The Model Before the actual opeig of the arket, N idepedet firs siultaeously subit offer prices at which they are willig to supply electricity fro each of their geeratig uits, or sets. O the basis of these bids, the arket orgaizer (or auctioeer ) draws up a rakig of uits, i.e. a arket supply curve is costructed. Whe the arket opes, dead is deteried as a rado variable idepedet of price, ad the auctioeer, by callig suppliers ito operatio, equates dead ad supply. Operatig uits, i.e. uits actually supplyig output, are paid the syste argial price which is equal to the offer price of the argial operatig uit. It is assued that geerators have costat argial costs, c 0, =,2,...,N, at productio levels below capacity, while productio above capacity is ipossible. We let the idex rak firs accordig to their argial costs, i.e. c c +. The total capacity of fir is give by k, =,2,...,N. The capacity of fir cosists of sets where k i is the capacity of the ith set, i =,2,...,, ad = k i. We let M deote the total uber of sets, i.e. M =. i k 3 Gree ad Newbery also assue dowward slopig dead curves, whereas copletely ielastic dead would see to be ore appropriate for the UK idustry. Bolle (990) proves that i the latter case, o equilibriu exists i the supply-fuctio odel. 2
Firs ca subit differet bids for each of their sets. If two or ore sets (of ay fir) are offered at the sae price, they are equally likely to be called ito operatio. We cosider a gae G with N+ players: N suppliers ad Nature. The ove order is as follows: Stage : The suppliers siultaeously subit offer prices =,..., N, cn p<. p i p, i =, 2,...,, r Stage 2: Sets are raked accordig to their bids such that if the bid of the set with rak r is p r ad that of the set with rak r is p r ad p < p r the r < r. If sets are offered at the sae price p, the these sets are desigated ubers r, r+,...,r+-, with (argial) probabilities /, for soe r {,2,..., M + }. Stage 3: Nature chooses a uber d d, d [ 0, K] probability distributio G( d ). 4 Let K 0 = 0 r j ad K = k, r =,2,..., M r j= Let ρ { r K d}, N K = k, accordig to soe =, where k r is the capacity of the set with rak r. r = ax r <. The all sets with rak r =,2,...,ρ- get paid p ρ k while set ρ ρ gets p d Kρ. Let s be the actual supply of fir, ρ r i.e. s = δ ( ) ( ) r r k + δ ρ d K = ρ, where δ ( r) is if the set with rak r belogs to ρ fir ad zero otherwise. The payoff to fir is the p c s, =,2,..., N. All players are assued to be risk eutral ad hece ai to axiize their expected payoff i the gae. All aspects of the gae, as well as the players argial costs ad capacities ad the probability distributio G( d ), are assued to be coo kowledge. Note that firs offer prices are costraied to be below soe threshold level p <, sice otherwise, i cases whe there is a positive probability that all sets will be called ito operatio, expected payoffs could be ade ifiitely large. A atural iterpretatio of p is that it is a (regulated) axiu price, either officially, or as perceived by the geerators (i.e. firs believe that the regulatio authorities will effectuate price regulatio if the price rises above p ). A alterative iterpretatio is that p is a reservatio price, below which dead is copletely ielastic. Our odel ay be cosidered as a special case of a auctio, i fact, as a first-price, sealedbid, ultiple-uit auctio with a rado uber of uits, where all uits are sold siultaeously (McAfee ad McMilla, 987, Hausch, 986). It is sealed-bid because of the siultaeous ove structure ad first-price i the sese that the arket price is deteried by the argial successful supplier. I Sectio 5, we show that this iterpretatio is particularly coveiet for aalyzig alterative pricig rules. 4 Note that to assue d [ 0, K] is without loss of geerality sice supply is liited to K ad, thus, dead will have to be ratioed if it icreases beyod K. I particular, G( d) will typically have a ato at d = K, reflectig the fact that ratioig ay occur with positive probability. I additio, the preset UK electricity-supply idustry is characterized by sigificat excess capacity, ad this is likely to reai true for the foreseeable future. Hece d < K would appear to be the relevat case. 3
3. Aalysis I this sectio we characterise the Nash-equilibria of the odel preseted i Sectio 2. Most of the discussio will cetre o the duopoly case, for which we are able to derive explicit results. Apart fro beig the relevat case for the UK electricity idustry (see the discussio i Sectio 4), explicit forulae for optial strategies are difficult to derive i the ore geeral oligopoly case. Hece our discussio of oligopoly i this sectio is i ost cases liited to poitig out where ad how the duopoly results geeralize. We start by presetig a result o the types of pure-strategy equilibria that ca occur. 5 Propositio : I a pure-strategy equilibriu, (geerically) at ost oe fir will deterie the syste argial price with positive probability. Reark: By geericity is here eat that firs have differet argial costs, i.e. c cˆ for all ˆ. If firs have idetical argial costs there ay exist Bertrad-type equilibria where firs subit offer prices equal to the argial costs of each set, ad i which ore tha oe fir ows sets which with positive probability ay becoe the argial operatig uit. The ituitio uderlyig the result is the followig. A player who ows a set which has a positive probability of becoig the argial operatig uit, will always wat to icrease the offer price of that set by soe sall aout towards the ext higher bid, sice that does ot affect the rakig, but icreases the fir s payoff i the evet that this is the argial set. O the other had, it caot be optial to subit a offer price equal to or just above that of a set of aother player, sice as log as the offer price is above argial cost (which it will be for at least oe fir), profits ca be icreased by udercuttig the rival slightly, thereby icreasig the probability of beig called ito operatio, without sigificatly reducig the price received i ay state. These two opposig forces destroy ay cadidate for a purestrategy equilibriu i which two or ore firs both have sets which with positive probability will becoe the argial operatig uit. Propositio iplies that the types of pure-strategy equilibria that ay exist are very restricted, ad, furtherore, it rules out the existece of pure-strategy equilibria for a wide rage of dead distributios (we retur to this latter poit i later sectios). Sigificatly, it follows that the types of equilibria foud by Gree ad Newbery (99) i their odel, do ot geeralize to the case where idividual geeratig sets are of positive size. The reaso that such equilibria exist i their supply-fuctio fraework is that whe idividual sets are of size zero (the cost fuctio is cotiuously differetiable everywhere), the gai fro udercuttig ay idividual rival set is egligible, ad thus the secod part of the above arguet does ot apply. I the followig sectio we cosider circustaces uder which pure-strategy equilibria will exist, as well as presetig exaples of ixed-strategy equilibria whe pure strategy equilibria are o-existet. The existece, ultiplicity ad the type of equilibria will be see to deped crucially o the support of the dead distributio. We will therefore distiguish betwee three geeric cases, which, for N = 2, are defied as follows: Case (Low dead): Supp G() = [ d, d] [0,i{ k, k2}], i.e. a sigle fir ca supply the whole of dead. 5 All proofs have bee relegated to the appedices. 4
Case 2 (High dead): ( ) 2 Supp G = [ d, d] [ax{ k, k }, K], i.e. both firs will be producig with probability oe. Case 3 (Variable dead): d d > ax{ k, k2}, i.e. there is positive probability for both the evet that a sigle fir ca supply the whole of dead ad the evet that both firs will have to be called ito operatio, irrespective of their bids. 3. Low-dead periods We begi by cosiderig the case where, with probability, dead is less tha the capacity of the sallest fir. This case turs out to correspod to the stadard Bertrad odel of oligopoly ad thus there is a uique equilibriu outcoe i which both firs offer to supply at a price equal to the argial cost of the least efficiet fir: Propositio 2: If supp G() [0,i{ k, k2}] the there exist pure-strategy equilibria, i all of which the syste argial price equals the argial cost of the least efficiet fir, c 2, ad oly fir produces. 6 Reark: Whe k k 2 G such that Pr( d k) = E d k p c p c, 7 but other equilibria ay exist also (see the ext sectio). >, such equilibria cotiue to exist for all ( ) ad [ ] [ ] 2 2 The arguet i the proof is idetical to that of the stadard Bertrad odel. Sice, with probability, dead ca be covered by oe fir, there will be copetitio to becoe the sigle operatig fir. I particular, a fir will always udercut its rival so log as its rival s bids are above its ow argial costs. Thus ay equilibriu ust have the ost efficiet fir (fir ) subittig offer prices for a capacity sufficiet to cover dead, at or below the argial cost of the least efficiet fir. Sice i this rage, fir s profit is icreasig i its ow offer price, these bids ust equal c 2. We coclude that i low dead periods, the syste argial price is bouded above by the argial costs of the less efficiet fir. A siilar result ca be show to hold i the oligopoly odel. If, with probability, dead is less tha the total capacity of the ost efficiet firs ( < N), the i ay equilibriu syste argial price caot exceed the argial cost of the + st ost efficiet fir. However, as we show i sectio 3.3, pure-strategy equilibria will geerally ot exist i this odel. 3.2 High-dead periods I this sub-sectio we assue that with probability both firs will be called ito operatio. Sice the high-pricig fir will be operatig for sure, ad i equilibriu firs ever subit equal bids (see Propositio ), its profit will be icreasig i its ow offer price. Thus, the extree opposite to the result of the previous sectio holds; whereas i low-dead periods the syste argial price equals the argial cost of the least efficiet fir, i high-dead periods it always equals the highest adissible price. 6 To avoid o-existece, we ipose the tie-breakig rule that fir is called ito operatio with probability wheever the firs' offer prices tie at c 2. This captures the idea that the ost efficiet fir argially uderbids its rival, while siplifyig the descriptio of the equilibriu. 7 E is the expectatios operator. 5
Propositio 3: If supp G() [ax{ k, k2}, K] all pure-strategy equilibria are give by offerprice pairs ( p, p2) satisfyig either p = p ad p2 b2 or p 2 = pˆ ad p b for soe < p, i =,2. bi Reark: If k > k 2 ( k < k 2 ), the all ( p, p 2) such that p = pad p 2 b 2 ( p2 = pad p b) cotiue to be equilibria for all G ( ) such that Pr( d > k2) = ( Pr( d > k) = ). That is, a sufficiet coditio for the existece of this type of equilibriu is that, with probability, dead is greater tha the capacity of the saller fir. As already oted, the ituitio for the result is straightforward. The high-biddig fir will always deterie the syste argial price. Therefore its payoff is icreasig i its ow offer prices ad profit axiizatio requires biddig at the highest adissible price. The lowbiddig fir is idifferet betwee ay offer price lower tha that of the high-biddig fir. However, to esure that the high-biddig fir does ot deviate, the low-biddig fir has to bid low eough so that the high-biddig fir s payoff fro udercuttig is less tha the payoff eared i equilibriu. Thus the upper boud o the low-biddig fir s offer price. I all of the equilibria characterized by Propositio 3, the syste argial price equals the highest adissible price. The low-biddig fir is despatched with full capacity while the high-biddig fir supplies the residual dead. It follows that both firs prefer equilibria where they act as the low-biddig fir, sice the received price is the sae while a fir s output is greater i the equilibriu i which it is raked first. Note that soe of these equilibria ivolve iefficiet despatchig: the high-cost fir ay be the fir with the lowest bid ad thus will be despatched with its total capacity, while the low cost fir is oly despatched with part of its capacity. I such equilibria, geeratio costs are ot iiized. It is easy to see that i the oligopoly case we get a correspodig result: Wheever dead is such that the highest-biddig fir deteries the syste argial price with probability, ay vector of offer prices such that the highest-pricig fir subits p while the rest bid sufficietly below this, will be a equilibriu. 3.3. Variable-dead periods We tur ow to the iterediate case i which there is a positive probability of either fir becoig the argial fir, whatever their offer prices (i.e. rak). It is clear that offer-price pairs like those i Propositio 3 caot costitute equilibria i this case sice the low-biddig fir will ow always wish to icrease its offer price; i doig so it thereby icreases the syste argial price i the evet that it becoes the argial operatig fir. I fact, we have the followig result: Propositio 4: If d d > ax{ k, k2}, where [ d, d ] is the support of the dead distributio G (), the there does ot exist a equilibriu i pure strategies. This result follows directly fro Propositio. Sice the rage of possible dead distributios exceeds the capacity of the largest fir, it follows that for ay strategy cobiatio there is a positive probability that sets of either fir will be the argial operatig uit. We ca the apply the result of Propositio ; there caot exist pure strategy 6
equilibria for which ore tha oe fir has a positive probability of deteriig the syste argial price. 8 I the reaider of this sectio we cosider a exaple where for all, =, i.e. each fir ows oly oe set, or ca subit oly oe offer price for the whole of its capacity, ad we characterize ixed-strategy equilibria for both the duopoly ad the oligopoly cases. 9 I the duopoly case we are able to show that there exists a uique ixed-strategy equilibriu, ad we derive the explicit for of the two players strategies. I particular, we fid that the lowest price i support of the players strategies is strictly greater tha the argial cost of the least efficiet fir, ad that this lowest price is a icreasig fuctio both of the highest possible price p, the probability that both firs will be operatig (i.e. dead), ad the argial cost of the least efficiet fir. We coclude this sectio by characterizig the (uique) equilibriu of the syetric oligopoly odel ad deostrate that the (expected) syste argial price is a decreasig fuctio of the uber of firs i the idustry. The aalysis is cosiderably siplified by restrictig attetio to the followig special case: All firs are assued to have equal capacities oralized to, ad dead is discrete ad distributed o {,2,..,N} with probabilities π = Pr( d = ), =,2,...,N, with Pr( d = ) 0 ad π =. Sice the ai results carry over to the ore geeral odel, for the rest of this sectio we cocetrate o this special case. Assue N = 2 ad defie π π. Without loss of geerality, oralize c to zero ad let c c 2. The assuptio o the support of the dead distributio i propositio 4 ow correspods to the case where 0 < <, i.e. the evets that oe ad two firs are called ito operatio both occur with positive probability. Defie p c l e whe π = p c 2 F ( p) = 2π π p c π π whe π 2π p c + 2π 2 () 8 I the oligopoly case this result is oly "geerically" true. By "geeric" we ea here that two assuptios ust be fulfilled. First, at least two firs ust have positive probability of becoig the argial fir irrespective of the rak order. Secod, if is the iiu ad M the axiu uber of firs that ca be called ito operatio, the the M+st ost efficiet fir ust have higher argial cost tha the th ost efficiet fir. If these coditios are ot fulfilled, the there are (ultiple) pure strategy Bertrad-type equilibria where the syste argial price equals the argial cost of the M+st ost efficiet uit, ad oly (a subset of) the M ost efficiet uits supply (see vo der Fehr (990)). 9 Aalyses of ixed strategies i odels with a siilar structure to the odel preseted here ca be foud i Shiloy (977), Varia (980), ad Padilla (990). See Bradeburger (990) for a discussio of the iterpretatio of ixed-strategy equilibria as equilibria i beliefs. 7
p l e whe p < p,ad π = p+ [ e ] c 2 F2 ( p) = 2π π π p π + whe p < p,ad π 2π p+ [ απ ] c 2π 2 (2) F2 ( p ) =, ad where α π π π π 2π p p c +c w he π = e 2 = π π 2π [ p c] + c w he π π 2 where l( e). We ca the prove the followig result: (3) Propositio 5: Whe N = 2 there exists a uique ixed-strategy Nash equilibriu i which player i s strategy is give by: Play p [ p, p] accordig to the probability distributio Fi ( p ), where F( p ), i =, 2 are give by () ad (2), ad p is give by (3). i Fro () ad (2) it follows that whe the probability that both firs will be operatig is large (i.e. π is sall), ore probability ass is placed o higher prices (ad vice versa). The ituitio for this is straightforward: the icetive to raise the offer price is checked by the likelihood of edig up as the higher pricig fir ad ot beig called ito operatio. Whe π is sall however, it is very likely that both firs will be despatched, i.e. there is a substatial probability that a fir will be operatig eve if it offers to supply oly at a very high price. Thus, for sall π both firs will ted to subit high bids. I particular, the followig is easily deostrated: li p c π =, ad (4) li p p π 0 = (5) (Note that the liit i (5) correspods to the case discussed i sectio 3., while the liit i (6) correspods to that discussed i sectio 3.2.) Fro (2) oe sees that the higher-cost fir s ixed strategy distributio has a ass poit at the highest adissible price, p. Furtherore, fro () ad (2) it follows that p p, p, F2( p) F( p), i.e. the high-cost fir s strategy profile first-order stochastically doiates the strategy profile of the low-cost fir. Thus, the high-cost fir will geerally (i.e. i 8
expected ters) subit higher bids tha the low-cost fir. We have ot bee able to fid a algebraic expressio for the probability that the high-cost fir subits a bid below that of the low-cost fir, but a lower boud ca be established by cosiderig the probability that p2 < p c. Whe π =/2, this reduces to e 2 Pr( p2 < p c) = [ l( + )] (6) 2 α where α p / c. If α = 5 (0), i.e. p is 5 (0) ties the argial cost of the high-cost fir, this probability equals 2% (27%). Thus, although the typical outcoe is that the high-cost fir prices above the low-cost fir, there is a potetially sigificat positive probability that the high-cost fir subits the lowest offer price ad thus becoes the oly operatig fir. Therefore we ay coclude that, as i the case discussed i sectio 3.2, the regulatory rule, as it is odelled here, is ot ex-post efficiet. We ow exted the aalysis of the duopoly case to the oligopoly odel uder the assuptio that firs have equal argial costs which, without loss of geerality, are oralized to zero. We cosider a syetric odel sice this is the oly case i which it is possible to characterize equilibria i ay detail. We assue that there is a positive probability that all uits will be called ito operatio, i.e. π N > 0, sice otherwise, give the syetry assuptio, oly the perfectly copetitive outcoe would be possible (see sectio 3.3 ad footote 8). We obtai the followig result: Propositio 6: Assue c = 0, =,..., N. The there exists a uique syetric ixedstrategy equilibriu for the gae G i which each player plays prices p [ p, p] accordig to the probability distributio F( p ) where F( p ) is the solutio to ad α( F( p)) F ( p) = Ω ( p, F( p)) (7) pβ ( F( p)) N α( q) π b( i ; N, q) i= i N β( q) πibq( i; N, q). i= b(; i N, q ) is the desity fuctio of the bioial probability distributio with paraeters N ad q, B(; i N, q) is oe ius the correspodig cuulative bioial probability distributio, ad B B/ q. Furtherore, F( p ) = 0, F( p ) =, ad p > 0. q Fro the uiqueess of the solutio to (7), it follows that p is decreasig i p. Note that Bq (; in, q) is always decreasig i i for sufficietly sall q. For larger q, Bq ( in ;, q) is icreasig (decreasig) i i for sall (large) i. bi ( ; N, q), as a fuctio of i, is shaped as a iverse V. Thus, reducig π i for sall i ad icreasig π i for larger i, typically icreases Ω( p, F( p)) for give p. Therefore oe would expect that p is larger the ore probability weight there is o π i for large i s. Oe has the followig liitig results: 9
li p = 0 (8) π li p π N = p (9) Oe questio of particular iterest is how the uber of suppliers i the arket will affect the price structure. There are i geeral two differet ways of aalyzig this. We could either thik of a situatio where, for a give level of dead, additioal firs are itroduced ito the arket, i.e. total capacity is icreased, or a situatio i which existig firs are split up ito saller uits, i.e. a give total capacity is divided betwee a larger uber of firs. If the questio of priary iterest is the orgaizatio of the deregulated structure of a existig idustry, the latter approach sees the ost atural ad this is what will be cosidered here. We aalyze a particular exaple, where π i = /N, by coparig the outcoe for differet N s. By substitutig for π i ad solvig (7), we have the followig: Result: Whe : c = 0, ad i: π i = / N, N p F( p) = l e N p (0) p N = pe, ad () p Ε p= e N Thus, both p ad Ε p are decreasig i N. That is, prices will ted to be lower o average i a ore frageted idustry. The ituitio for this ay be explaied as follows: By icreasig its offer price a fir reduces the probability that it will receive a positive payoff. O the other had, subittig a high offer price icreases, i expected ters, the syste argial price. The syste argial price effect, however, beefits the fir oly whe it happes to be the argial fir, a evet which is less likely the ore firs there are i the idustry. This ituitio also suggests that i the geeral odel with ulti-uit firs, prices will ted to be higher tha i the odel i which these sae uits act idepedetly. As idicated above, raisig the offer price of oe uit will have a exteral effect o other uits i that it icreases the expected syste argial price. A ower who cotrols ay uits will iteralize part of this exterality ad will thus have a additioal icetive to icrease their offer prices. I particular, this coordiatio icetive is stroger the ore uits a ower cotrols. It therefore sees reasoable to coclude that for a give uber of geeratig sets i the idustry, the syste argial price will be a decreasig fuctio of the uber of owers, or geerators cotrollig the sets, i.e. the idustry cocetratio ratio. N (2) 4. The UK Electricity Idustry I this sectio we preset epirical evidece o the biddig behaviour of the ajor geerators i the U.K. electricity idustry 0. Sice our odel of price copetitio i the 0 See Vickers ad Yarrow (99), Gree (99 a) ad Jaes Capel & Co. (990) ad Holes ad Plaskett (99) for descriptios of the ew UK electricity idustry. Vickers ad Yarrow (99) i 0
electricity spot arket is obviously too abstract ad siplified to be tested directly agaist the epirical evidece, our purpose here is rather to deostrate that the types of strategic behaviour we have idetified i our odel are at least cosistet with actual historical biddig behaviour, ad that our ost iportat coclusio for policy purposes, viz. that bids will ted to be well above geeratio costs, is supported by the evidece. 4.. Structure of the U.K. idustry There are three ai geeratig copaies i the syste for Eglad ad Wales: Natioal Power with approx. 52% of the geeratig capacity of the old Cetral Electricity Geeratig Board, PowerGe with 33% ad Nuclear Electric with 5%. I additio there are soe suppliers i Scotlad ad o the cotiet coected to the syste i Eglad ad Wales. These, however, are for the tie beig of ior iportace. Natioal Power ad PowerGe are private copaies (with the goveret as a iority shareholder), while Nuclear Electric is publicly owed. Nuclear Electric s productio is copletely based upo uclear power. It therefore fuctios etirely as a baseload producer ad its capacity is bid i at (virtually) zero. Thus there are i reality oly two sigificat strategic players i the electricity spot arket. The electricity spot arket has bee desiged very uch like a ulti-uit auctio. Each day the geeratig copaies subit bids to the Natioal Grid Copay which give the iiu prices at which they are willig to supply electricity fro each geeratig uit (or geset ). 2 A erit order is the costructed fro the bids, with sets raked i ascedig order, ad a despatch schedule deteried to atch supply ad predicted dead for each half-hour of the followig day (this is called the ucostraied schedule ). Syste argial price (SMP) the ajor copoet of the price paid to each despatched geset is deteried by the bid price of the argial despatched set. I ost half-hour periods (Table A periods), each despatched geset is paid, i additio to syste argial price, a capacity eleet, iteded to reflect the probability of loss of load, i.e. a power shortage. They ay also receive uplift payets. I Table B periods, whe there is expected to be a excess of ruig, partly-loaded capacity, capacity payets are ot ade, ad oly the icreetal bid prices are used to deterie syste argial price. There are further additioal coplexities to the syste. These have bee described elsewhere (c.f. NGC (99), Gree (99a), Gree (99b), Jaes Capel & Co. (990)), ad further details are provided i appedix B. At vestig, o March 3 990, cotracts for differeces were placed betwee the two ajor geerators ad the regioal electricity supply copaies coverig approxiately 85% of the geerators capacities. 3 These are optio cotracts uder which the differece betwee the spot, or pool price of electricity ad the cotract strike price is paid to the purchaser (i.e. the regioal electricity copay) o a specified uber of uits. These optio cotracts have sigificatly reduced the icetives of the geerators to bid pool prices above the level of particular provide a discussio of a broad rage of issues relevat for the evaluatio of the deregulatio of the UK electricity idustry. Gree (99b) clais to have foud bids to be at or ear geeratio costs o ost of the two ajor geerators' geeratig uits. However the recet report by the regulator (OFFER (99)) cites bids well above estiates of 'avoidable geeratio costs' as a cause of cocer i its attept to evaluate how well copetitio i the ew syste is workig. 2 Aogst a great deal of other iforatio - see appedix B ad NGC (99) for further details. 3 Details ay be foud i the share offer prospectus, Kleiwort Beso Ltd. (99); Powell (99) also cotais a discussio of the iportace of cotracts the ew British electricity idustry.
cotract strike prices, sice ay differece betwee the pool price ad cotract strike prices is paid back to the regioal supply copaies i the for of a differece payet o the aout of capacity cotracted for. Oe would therefore ot expect to see the type of ocopetitive biddig behaviour predicted by the theoretical odel irrored i the historical biddig data. 4 By March 3 99 however, a proportio of these cotracts had expired (approx. 5%), ad the rest are due to expire by March 993. With cotract coverage lowered to about 70% of the geerators capacities, strategic or ocopetitive biddig behaviour becoes ore likely, ad so oe expects to see i the first year of operatio of the ew syste, bids reflectig geeratio costs sice cotract strike prices were chose to represet expected argial geeratio costs ad after February/March 99 a possible chage i regie to ore aggressive, ocopetitive biddig. It is precisely this kid of chage i regie that we see reflected i the data to April 3 99, ad which is described here. I appedix B we describe i soe detail the operatio of electricity pool, as operated by the Natioal Grid Copay. We also provide iforatio cocerig our data sources ad our aalysis of the bid data. The iterested reader is referred there for this iforatio ad we tur here directly to a discussio of the results of our epirical ivestigatios. 4.2. Geerator biddig behaviour I the figures below, two differet ways of describig the biddig behaviour of Natioal Power ad PowerGe for the period fro July 990 to April 99 are depicted. Figures ad 2 show the actual bids 5 of the two ajor geerators for each level of output o particular weekdays of the year, i.e. the geerators supply curves. Figure 3 o the other had represets the average weekly bids of each geerator for geeratig sets of a particular size ad fuel type. The forer are thus sapshots of geerator bids at particular poits i tie, while the latter gives a loger-ter picture of geerator biddig behaviour over the period. Figure a shows the supply curves of both geerators o July 2 990. Sice PowerGe s capacity is approxiately 64% of Natioal Power s, its supply curve becoes vertical uch earlier, at approxiately 2,000 MGWhs. Figures b ad c copare the geerators estiated cost curves to their supply curves. Sice the geerators cost curves were costructed by suig over all of their capacity, while o a give day soe capacity will be declared uavailable (due to aiteace, etc.), it is to be expected that the supply curves becoe approxiately vertical before the cost curves do. Apart fro this however, the geerators see to have bee biddig a very close approxiatio to their cost curves. 6 Thus durig the first 8 to 0 oths of the operatio of the ew syste, the evidece sees to suggest that the two ajor suppliers were biddig copetitively, i.e. at cost. The figures for 22 February 99 begi to idicate a differet patter. PowerGe s supply curve o this day lies uiforly above Natioal Power s (fig. 2a), with arked differeces i bids i 2,000 to 0,000 MGWh rage. I figures 2b ad 2c we see that PowerGe s supply curve also lies uiforly above its cost curve, while Natioal Power s supply curve is below its cost curve fro 0 to approxiately 0,000 MGWhs, the rage of output covered by its 4 vo der Fehr ad Harbord (992) odel the strategic icetives of the geerators with siple oeway ad two-way differece cotracts. 5 See appedix B for a discussio of how these were costructed. 6 Figures Ba, Bib ad Bc i appedix B show siilar coparisos for a early witer day, Noveber 22 990. Agai the geerators' bids closely reflect their costs at all levels of output. 2
large, coal-burig sets (oce adjustets for availability have bee ade), ad thereafter is above it. 7 While these figures do ot provide eough data to allow us to reach ay fir coclusios, they do see to idicate a chage i the patter of biddig behaviour. I particular the figures for February idicate ore sophisticated patters of biddig behaviour tha siply biddig i at cost. This is cofired by a exaiatio of the weekly averages of bids o gesets of a particular size ad fuel type over the etire period fro May 990 to April 30 99. 8 I figure 3 average weekly bids for Natioal Power s ad PowerGe s large coal sets are show. 9 It is apparet that at aroud week 40 (early Deceber 990) both geerators altered their biddig behaviour sigificatly, ad i opposite directios. Natioal Power s bids o its large coal sets drop draatically fro a average of approxiately 4/MGWh to well below 0/MGWh i alost all weeks, ad its largest coal sets were occasioally bid i at below 2 /MGWh. PowerGe, o the other had, icreased its bids o its large coal sets by a average aout of approxiately /MGWh. This patter of bids reaied stable fro Deceber 990 to the ed of April 99. 20 As oted above, a possible explaatio for these results is that sice i the first part of this period cotract coverage for each geerator was approxiately 85% of their capacities, cotract strike prices put a dowward pressure o spot prices, while i the latter part, whe cotract coverage was less, this pressure was eased. However, cotracts oly started expirig right towards the ed of the period (March 3 99), while the chage of patter occurs i Deceber 990. A alterative explaatio is give by our odel. The first period ore or less coicides with the war seaso, ad based o the odel predictios we expect to observe prices closer to costs whe dead is low. Thus for dead levels below approxiately 27,500 MGWhs, which is typical durig war seasos, large coal sets are alost exclusively the argial sets which deterie syste argial price. Furtherore, sice i the war seaso dead ay fall very low at ight ad i early orig, there will be strog copetitio to be despatched (see the discussio i sectios 3. ad 3.2). I the colder seaso, however, dead is always so high that sets with low rak will ever becoe argial. Therefore, fro Deceber 990 owards, PowerGe s large coal sets were deteriig syste argial price over a large uber of periods, while Natioal Power s large coal sets were beig bid i low eough so that they were certai to be 7 The figures for April 8 99 i appedix B tell a siilar story. I figure B2a PowerGe's supply curve lies sigificatly above Natioal Power's up to approxiately 8000 MGWhs of output, ad is uiforly above its cost curve (fig. B2b). Natioal Power's supply curve for April 8 shows a siilar patter to that of February 22. The first 8000 MGWhs of output have bee bid i below cost, ad the reaider at offer prices above geeratio costs 8 I figure 5, week 5 is the week begiig May 990. 9 C5 idicates a coal set of greater tha 500 MGWhs ad C6 a coal set of greater tha 600 MGWhs. We cocetrate o coal sets sice for other sets the picture is blurred by the frequet chages i iput prices, i.e. prices of oil ad gas. 20 I appedix B figures B3 through B6 give soe further evidece. Figures B3 ad B4 copare bids o large coal sets to the geerators' average costs. As is evidet, Natioal Power's bids fell sigificatly below costs fro Deceber 990, while PowerGe's bids reaied above average costs ad icreased after Noveber 990. Fially, figures B5 ad B6 show the average weekly bids of Natioal Power ad PowerGe respectively over all coal sets ad all sets, for the etire period. It is clear that the picture for other sets is siilar to that for the coal sets, although the forer is soewhat blurred by the frequet chages i oil ad gas prices. 3
despatched. This type of biddig behaviour, the, has the flavour of the equilibria described i sectio 3.3. 5. A Alterative Payoff Rule As we have show i sectio 3, firs will i geeral choose bids greater tha their argial costs, ad thus the syste argial price will ted to exceed the argial costs of each of the operatig uits. Furtherore, sice less efficiet sets ay subit lower offer prices tha ore efficiet sets, iefficiet despatchig ay result. It is therefore a iterestig questio whether the regulatory rule ca be odified so as to iduce truthful revelatio of costs ad, as a result, efficiet despatchig. I this sectio we show that by extedig a isight o optial auctios due to Vickrey (96), such a odificatio is ideed possible. The gae G ay be iterpreted as a first-price, sealed-bid, ultiple-uit auctio with a rado uber of uits. I particular, the syste argial price is deteried by the offer price of the argial operatig set, ad thus a fir s bids will deterie the price received i the evet that oe of its sets is the argial operatig uit. The fudaetal isight of Vickrey (96) was that by akig the price received by a fir idepedet of its ow offer price, argial cost pricig ca be iduced as a doiat strategy for all firs. The reaso for this is that i such a set-up a fir ca oly ifluece its ow payoff to the extet that it affects the probability of beig called ito operatio. A fir will prefer to be operatig for all realizatios of dead such that its payoff is positive, ad will prefer ot to operate wheever its payoff is egative. Therefore, offerig to supply at a price equal to argial cost becoes a doiat strategy because it axiizes the probability of beig called ito operatio wheever the fir s payoff is expected to be o-egative. I a stadard Vickrey auctio, price is deteried by the argial usuccessful, i.e. ooperatig, player. To geeralize this result, we ust costruct a echais which is both icetive copatible ad idividually ratioal. This ca be doe by lettig the price paid to fir be deteried by the itersectio of dead with the residual (i.e. et of the capacity of fir ) supply curve. Cosider therefore a slight variatio of the gae G where the oly chage ivolves the payoff rule: The itersectio of the dead ad the supply curves deteries which uits will be called ito operatio. All operatig uits are paid fir-specific prices deteried by the itersectio of the dead ad the respective residual supply curves if such a itersectio exists, ad set equal to pˆ ax{ c } otherwise. Call this gae Ĝ. The the followig result holds: Propositio 7: The gae Ĝ has a uique doiat strategy equilibriu i which pi = c, =,2,..., N Reark: Other Nash equilibria typically exist. However, sice offerig to supply at argial cost (weakly) doiates all other strategies, we cosider this the atural focal poit, ad thus base our discussio solely o this equilibriu 2. I Ĝ, as opposed to G, despatchig is efficiet sice firs are always despatched i order of icreasig argial cost. Thus, our alterative regulatory rule leads to iiizatio of real geeratio costs. 22 2 I additio, sice all other strategies are weakly doiated, o other equilibria are strategically stable i the sese of Kohlberg ad Mertes (986) (while it is easy to check that the equilibriu i questio is stable); this is aother reaso for ot cosiderig the. 4
I additio to techical efficiecy, oe ight ask how total (expected) payets to the geerators copare i the two auctios. Deote by Ε C total expected payets i G, ad by ΕĈ total expected payets i Ĝ, respectively. It is easy to verify that reveue equivalece holds whe valuatios are draw fro the sae distributio (if we let ˆp = p ), as we would except fro the Reveue-Equivalece Theore (Vickrey (96), McAfee ad McMilla (987)). For exaple, i the syetric oligopoly odel cosidered i sectio 3.3, Ε C = Nπ p (sice players receive the sae payoff whichever of the prices i the support of the strategies they play, ad, i particular, the profit fro playig p is π N p ). O the other had, Ε Cˆ = Nπ ˆ N p (sice payets are zero whe soe firs do ot operate ad ˆp to each of the N firs otherwise). It turs out to be difficult to establish the sig of ΕC ΕCˆ i the geeral odel. However, i the duopoly case oe ca show that Ε C is ever saller tha Ε Ĉ. This is obvious i the cases discussed i sectios 3. ad 3.2. I the case aalyzed i sectio 3.3, Ε C ca be foud by cosiderig ϕ ( p) ad 2 ϕ ( p) as p p, where ( ) ϕ i p is the profit of fir i fro playig p, i =, 2, fro which it follows that Ε C = π Kc+ [ π ]2 p, where α αl +, whe π = α + e 2 K = 2π π π α α, whe π 2π α + απ 2 (3) α = p c π, ad απ π ad Ε Cˆ = πc+ [ π]2 p. Now, K, ad K = whe α = ad is icreasig i α. For a give α, K is axiized at π = /2 ad K e. We suarize the duopoly result i the followig propositio: Propositio 8: Whe N = 2, ΕĈ is a lower boud for Ε C. Such a iproved pricig perforace echoes the result i the optial-auctio literature that secod-price sealed-bid auctios yield higher payoffs to the auctioeer tha do first-price sealed-bid auctios (McAfee ad McMilla (987), Myerso (98), ad Maski ad Riley (989)). Thus, soe of the first-price/secod-price copariso results foud i the auctio literature exted to this settig as well. We coclude that (disregardig collusio ad log-ter cotracts) a istitutioal setup which iduces firs to ake offer prices equal to argial costs is perfectly possible eve π 2π 22 Efficiecy cosideratios i electricity supply idustries are cosiderably coplicated by the etwork characteristics of such idustries, ad beig able to rak geeratig uits accordig to productio costs is oly a ecessary coditio for short ru efficiecy. For a treatet of efficiecy ad optial pricig i electrical etworks, see Boh, Caraais, ad Schweppe (984). 5
whe firs are capacity-costraied, soethig which sees ot to have bee appreciated i the literature. As such, it also shows that applyig results fro stadard oligopoly odels (such as those foud i for exaple Kreps ad Scheika (983) ad Tirole (988), ch. 5) ca be isleadig as a descriptio of the outcoe of copetitio i the U.K. electricity arket. 6. Coclusio I this paper price copetitio i the deregulated wholesale arket for electricity for Eglad ad Wales has bee aalyzed as a first-price, sealed-bid, ultiple uit auctio. I doig so, we have deostrated that uder the existig istitutioal set-up there is likely to be both iefficiet despatchig ad above argial cost pricig, eve i the absece of collusio ad log ter cotracts. While these poits have bee argued elsewhere (see for istace Vickers ad Yarrow (99) or Gree (99a)), the arguets have bee largely iforal ad usually based upo stadard odels of oligopoly pricig, ad hece soewhat icoclusive. A ajor purpose of the preset paper has bee to address these issues i a foral odel specifically desiged to capture the essetial eleets of the ew electricity pricig syste i the Uited Kigdo. To our kowledge Newbery ad Gree (99) (see also Newbery (99)) is the oly other odel specifically desiged to study the biddig behaviour of the geerators uder the ew U.K. syste. 23 While our coclusios echo theirs i ay respects, our results have also cast soe doubt upo the type of equilibriu aalysis they have eployed, i.e. Kleperer ad Meyer s (989) supply fuctio equilibriu odel. This is because the equilibria foud uder the assuptio that firs subit sooth, i.e. cotiuously differetiable, supply fuctios do ot appear to geeralize to the case where supply fuctios ust be discrete step fuctios, eve whe the step-legth ca be ade very sall. Ideed, we have foud that for a wide rage of dead distributios pure-strategy (i.e. supply fuctio) equilibria will ot exist i this case. It is therefore reassurig to fid that Gree ad Newbery s ost sigificat coclusio for policy purposes, viz. above argial cost pricig, is also a property of the odel aalysed here, ad hece does ot deped upo the particular assuptios they ipose. I sectio 4 we have preseted epirical evidece o the biddig behaviour of the two ajor geerators i the U.K. idustry which has teded to support the coclusios of our theoretical odel. While ot claiig to have tested the odel i ay sese, we have bee able to deostrate that at the very least the odel is ot cotradicted by the epirical evidece, ad that the biddig strategies of the geerators ay be viewed, at least i part, as coforig to the types of strategies described by the theory. While our epirical coclusios, i particular bids greater tha geeratio costs, do ot agree with those of Gree (99b), they have bee cofired elsewhere, sigificatly i the recet report of the regulator OFFER (99). There thus ow exists serious evidece, both theoretical ad epirical, that copetitio i the ew electricity supply idustry for Eglad ad Wales ay ot be achievig the purposes for which it was origially desiged, i.e. the efficiet geeratio of electricity, sold at copetitive prices to cosuers. 23 While the odel of Bolle (990) is very close that of Gree ad Newbery (99) i ay respects, its purpose is soewhat ore geeral. 6
While the aalysis preseted here would appear to be useful i providig a fraework for studyig pricig behaviour i the deregulated U.K. electricity idustry, the iportace of our coclusios is liited by the extet to which they do ot take ito accout opportuities for collusive behaviour betwee the geerators, or the effects of log-ter cotracts betwee suppliers ad purchasers (or third parties). These probles call for further research. 24 24 I a copaio paper - vo der Fehr ad Harbord (992) - the odel of the preset paper is exteded to accout for the presece of log-ter cotracts. 7
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