Linear Supply Function Equilibrium: Generalizations, Application, and Limitations. Ross Baldick, Ryan Grant, and Edward Kahn

Transcription

1 PWP-078 Lnear Supply Functon Equlbrum: Generalzatons, Applcaton, and Lmtatons Ross Baldck, Ryan Grant, and Edward Kahn August, 000 Ths paper s part of the workng papers seres of the Program on Workable Energy Regulaton (POWER). POWER s a program of the Unversty of Calforna Energy Insttute, a multcampus research unt of the Unversty of Calforna, located on the Berkeley campus. Unversty of Calforna Energy Insttute 539 Channng Way Berkeley, Calforna

2 Lnear Supply Functon Equlbrum: Generalzatons, Applcaton, and Lmtatons Ross Baldck Department of Electrcal and Computer Engneerng, Unversty of Texas, Austn Ryan Grant Analyss Group/Economcs, San Francsco, CA Edward Kahn Unversty of Calforna Energy Insttute, Berkeley, CA and Analyss Group/Economcs, San Francsco, CA Abstract August, 000 We consder a supply functon equlbrum (SFE) model of nteracton n an electrcty market. We assume a lnear demand functon and consder a compettve frnge and several strategc players all havng capacty lmts and affne margnal costs. The choce of SFE over Cournot equlbrum and the choce of affne margnal costs s revewed n the context of the exstng lterature. We assume that bd rules allow affne or pecewse affne non-decreasng supply functons and extend results of Green and Rudkevtch concernng the lnear SFE soluton. An ncentve compatblty result s proved. We also fnd that a pecewse affne SFE can be found easly for the case where there are non-negatvty lmts on generaton. Upper capacty lmts, however, pose problems and we propose an ad hoc approach. We apply the analyss to the England and Wales electrcty market, consderng the 996 and 999 dvesttures. The pecewse affne SFE solutons generally provde better matches to the emprcal data than prevous analyss.

3 Lnear Supply Functon Equlbrum: Generalzatons, Applcaton, and Lmtatons Ross Baldck, Ryan Grant, and Edward Kahn August, 000. Introducton Ths paper explores the lnear verson of the supply functon equlbrum (SFE) model. The general SFE approach was ntroduced by Klemperer and Meyer (989) and appled by Green and Newbery (99) to the electrcty ndustry reforms n England and Wales (E&W). Green (996) used a lnear verson of ths model and appled t to prospectve dvesttures of generaton assets mandated by the regulator of the electrcty ndustry n E&W. We offer a generalzaton of Green s model and extend the applcaton to subsequent changes n the horzontal structure of the electrcty market n E&W, beyond those studed by Green. Before explorng these ssues, t s worthwhle addressng two threshold questons. Frst, what does SFE offer beyond the tradtonal Cournot framework? Second, why use the lnear form of SFE rather than the more general formulaton? Why Not Cournot? SFE competes wth the Cournot model as a practcal tool for studyng olgopoly n the electrcty ndustry. Recent reforms of the electrcty ndustry around the world have stmulated numerous studes of olgopoly behavor n restructured electrcty markets. Papers of ths knd have been publshed reflectng ssues n Scandnava, Span, New Zealand, and U.S. electrcty markets, partcularly Calforna. All of these papers rely on the Cournot framework. SFE s attractve compared to Cournot because t offers a more realstc vew of electrcty markets, where bd rules typcally requre supplers to offer a prce schedule that may apply throughout a day, rather than smply put forth a seres of quantty bds over a day. In the Cournot framework, prce formaton depends exclusvely on the specfcaton of the demand curve (and of the specfcaton of a compettve frnge.) The market prce s determned by the ntersecton of the aggregate quantty offered and the demand curve. It s notorously dffcult to specfy the market demand curve n electrcty, due to low short-run elastcty and nexperence wth market competton n Department of Electrcal and Computer Engneerng, Unversty of Texas, Austn Analyss Group/Economcs, San Francsco, CA Unversty of Calforna Energy Insttute, Berkeley, CA and Analyss Group/Economcs, San Francsco, CA For Scandnava, see Andersson and Bergman (995). For Span, see Alba et al (999), Ramos et al (998) and Rver et al (999). For New Zealand, see Read and Scott (996) and Scott (998). In all three of these countres hydro storage plays an mportant role. For the US, see Borensten and Bushnell (999). US markets typcally nvolve network congeston ssues. Network congeston s treated n an olgopoly context by Hogan (997) and by Borensten, Bushnell and Stoft (to appear).

4 electrcty. As a result, prce predctons from Cournot models are not partcularly relable. The SFE approach s not mmune to the problem of senstvty to specfcaton of the market demand and ths ssue s dscussed further below. However, the SFE model formulaton at least offers the possblty of developng some nsght nto the bddng behavor of frms, partcularly n markets where they are constraned to bd consstently over an extended perod of tme. One recent example of ths applcaton s the use of the SFE framework by the Market Montorng Commttee of the Calforna Power Exchange (Bohn, Klevorck, and Stalon, 999). Therefore, the case for the SFE approach s ts realsm regardng the bddng behavor of frms n electrcty markets. SFE can estmate equlbrum mark-ups. Equlbrum mark-ups are less transparent n the Cournot framework than n the SFE framework. Why Use the Lnear Model? In the SFE model, functonal forms must be specfed for demand, cost, and supply functons. We frst dscuss demand. A partcularly smple form s to assume a lnear demand functon; that s, at each tme the demand as a functon of prce has a non-zero ntercept and a constant negatve slope. 3 Assumng that the slope s ndependent of tme greatly smplfes the model. Most authors use lnear demand functons wth demand slope ndependent of tme. 4 Next we consder the margnal cost as a functon of producton. There are a range of possble functonal forms. The smplest non-trval case s an affne functon wth zero ntercept or, equvalently, all cost functons havng the same non-zero ntercept. Followng the lterature, we wll refer to margnal costs that are affne wth zero ntercept as lnear margnal costs. In some models, there s a compettve frnge as well as strategc frms. The functonal form for the strategc frms and the frnge can be dfferent. Two further decsons concern whether or not the strategc frms are assumed to be symmetrc and whether or not the strategc frms and frnge have maxmum capacty lmts. Fnally, consder the supply as a functon of prce. Agan, there are a range of possble functonal forms. Typcal applcatons use a form for the supply functon that s smlar to the assumed form of the (nverse) margnal cost functon. If the cost functons are symmetrc then the equlbrum supply functons often turn out to be symmetrc. For example, Frame and Joskow (998) offer the followng observaton made n the context of revewng a partcular Cournot model mplementaton n electrcty, We are not aware of any sgnfcant emprcal support for the Cournot model provdng accurate predctons of prces n any market, let alone an electrcty market. 3 We follow the conventon of callng ths specfcaton a lnear demand functon although t s more precsely descrbed as affne. In dscussng cost and supply functons, we reserve the word lnear for affne functons wth a zero ntercept. 4 A computatonal advantage of the Cournot framework s that constant elastcty demand curves are straghtforward to represent, as n Borensten and Bushnell (999), p.30. 3

5 Assumng lnear demand and affne margnal costs greatly smplfes the SFE mathematcs. For example, Turnbull (983) analyzes an asymmetrc two frm model wth lnear demand, affne margnal costs, and affne supply functons. The resultng condtons for the SFE are straghtforward to solve. Green and Newbery (99) (GN) generalze the lnear demand and lnear margnal cost asymmetrc two frm model by analyzng strategc frms havng quadratc margnal costs. Ths requres numercal soluton of dfferental equatons and s undertaken n the nterest of greater realsm (p.94). For the duopoly structure examned, GN report results prmarly for the case of symmetrc strategc frms. As the structure of the electrcty ndustry n E&W has changed, realsm suggests that the symmetry assumpton and the duopoly assumpton both need to be dropped. The asymmetrc duopoly case s also solved by GN and by Laussel (99). Nether of the latter two papers requre lnearty. However, there does not appear to be any other results on asymmmetry beyond the duopoly case for non-lnear margnal costs. The great advantage of the SFE wth lnear margnal costs over the more general form s the ablty to handle asymmetrc frms when there are more than two strategc frms. Green (996) does not emphasze ths property, but t turns out to be useful n practce. As noted above, the general SFE requres solvng a set of dfferental equatons. Ths s suffcently dffcult that most authors typcally rely on the case of symmetrc frms. For practcal applcatons, the asymmetrc case s more nterestng. Ths motvates the use of the lnear model for the asymmetrc, multple strategc frm ndustry we consder. Organzaton of paper The remander of the paper s organzed as follows. Secton ntroduces the affne margnal costs case. We characterze the affne SFE and prove an ncentve compatblty result regardng the prce ntercept of the equlbrum supply bds. Secton 3 ntroduces capacty constrants. These have been addressed for strategc frms by GN under the assumpton that the cost functons are dentcal for each frm. We address the case of capacty constrants, both for the strategc frms and frnge, where there are asymmetrc costs. Ths case s more realstc for the England and Wales market subsequent to dvestture. We show that ths stuaton s very much more complex than when ths stuaton arses n the Cournot framework. Secton 4 apples these methods to recent structure and prce changes n the electrcty market n England and Wales. The theoretcal ssues dscussed n Sectons and 3 are llustrated by the numercal examples ntroduced n secton 4. Secton 5 offers some conclusons.. The Affne Margnal Cost Case As dscussed above, the SFE models reported n the lterature typcally assume that the bdders margnal cost functons have zero ntercept, or, equvalently, assume that all have a common ntercept. Ths makes the SFE easer to fnd. Occasonally authors attempt to defend the plausblty of ths assumpton. We argue that, at least for electrcty, ths assumpton s nether plausble nor practcally useful. 4

6 What s Ganed The requrement that margnal cost functons have zero ntercept s compared to a case where they are allowed to have a postve ntercept and the more realstc case of nonlnear margnal costs. Fgure llustrates these two approxmatons to a typcal margnal cost curve characterstc of electrcty generaton frms. The two approxmatons wll be used n Secton 4 below. If the margnal costs curves are equal at full producton, as llustrated n fgure, then assumng that the margnal costs pass through zero wll overestmate profts compared to the true functon. The affne case wll typcally underestmate profts. The lne through zero s lkely to be partcularly unrealstc when the actual supply of a frm s from the low end of ts capacty. Our examples n Secton 4 wll show cases of ths knd. Prce (Pounds/MWh) Capacty (GW) Actual PG/NP_0 PG/NP_+ Fgure. Lnear Approxmatons to a Convex Margnal Cost Curve Formulaton We begn wth the same general form of the demand curve as n Green, namely: D(p,t) = N(t) γp. () The underlyng load-duraton characterstc s specfed by the functon N(t). Furthermore, for each tme t, the demand D s lnear n p wth slope dd/dp = γ. The coeffcent γ s assumed to be postve. The total cost functon for frm, C, =,,n, s gven as a strctly concave quadratc functon of producton. Ths form results n an affne margnal cost functon for each frm:, q 0, C (q ) = ½ c q + a q, (), q 0, dc /dq (q ) = c q + a, (3) wth c > 0 for each frm for strctly convex costs. In contrast to prevous analyss n the lterature, we allow the a to be non-zero and to be specfc to each frm. 5

7 We assume that the market rules specfy that the supply functon of each frm s affne; that s, of the form:, q (p) = β (p α ). (4) The parameters α and β are chosen by frm subect to the requrement that β be nonnegatve. Strctly speakng, we should modfy the supply functon (4) so that t s always non-negatve; however, we wll ntally assume that the prces are such that no supply functons ever evaluate to beng negatve. We wll subsequently revst ths assumpton and generalze the allowed form of the supply functon to pecewse affne nondecreasng functons. Soluton The basc equaton governng the SFE soluton s provded by Green (996) as hs equaton (4), whch we quote here for reference:, q (p) = (p dc /dq (q (p)))( dd/dp + dq (p)/dp). Any soluton to these coupled dfferental equatons such that each q s non-decreasng over the relevant range of prces s an SFE. These equatons do not nvolve the loadduraton characterstc N(t). Substtutng from (), (3), and (4) above nto Green s equaton (4), notng that dq /dp = β, we obtan:, β (p α ) = (p c β (p α ) a )(γ + β ). (5) Assumng that the bd functon must be consstent across all tmes, ths equaton must be satsfed at every realzed value of prce p. If there are at least two dstnct values of prce that are realzed, then (5) can only be satsfed for all the realzed prces f t s dentcally true. That s, we can equate coeffcents of lke powers of p on the left and rght hand sde of the equaton. Equatng coeffcents of p, we obtan Green s equaton (6):, β = ( c β ){γ + Σ β }. (6) Equatng coeffcents of the constant terms we obtan:, α β = (a c β α ){γ + Σ β }. (7) Both (6) and (7) must be satsfed wth non-negatve values of β for each frm for an affne SFE to exst. Substtutng from (6) for β for each nto the left hand sde of (7) yelds:, α ( c β ){γ + Σ β } = (a c β α ){γ + Σ β }. 6

8 Snce the requred soluton must satsfy, β 0 then, f γ > 0, we have that γ + Σ β > 0 and we can cancel ths factor on both sdes. Rearrangng the resultng expresson yelds α = a for each frm. Conversely: f α = a for each frm and f (6) s satsfed wth non-negatve β, then (5) s satsfed and the resultng α and β specfy an affne SFE. Rudkevch (999) shows that there s exactly one non-negatve soluton to (6) and presents an teratve scheme n the specal case of all frms havng zero ntercept (that s,, a = 0) for fndng the soluton to (6). The teratve scheme begns wth each frm bddng compettvely. That s, each frm ntally bds β = /c. Rudkevch shows that f each frm at each teraton updates ts value of β so as to fnd the proft maxmzng value for frm, gven the values of β for the other frms from the prevous teraton, then the sequence of terates converges to the optmal soluton of (6). In the affne case, we can slghtly generalze Rudkevch s scheme to magne each frm bddng values of β and α at each teraton, wth β and α chosen by frm at each teraton to maxmze profts gven the most recent bds of the other frms. By the same argument as prevously, the optmal value of α at each teraton wll satsfy α = a. Rudkevch s result therefore also provdes an explanaton for how the frms could arrve at the affne SFE, gven that they all begn by bddng compettvely. In Appendx we generalze Rudkevch s result to show that f n< + mn + cγ then Rudkevch s update s a contracton map and so the + c γ teratve scheme converges to the unque optmum from any startng pont. The condton n< + mn + cγ + c γ s always satsfed for n =,, but for larger values of n t depends on the cost functon and demand functon. If the condton s satsfed, then Rudkevch s teratve scheme converges to equlbrum even f, for example, some frms begn by bddng compettvely whle others begn wth non-compettve bds. The above dscusson and unqueness result are both predcated on the assumpton that bds are affne. If non-affne functons may be bd then there are n general multple SFE. Aggregate demand and supply Now consderng the aggregate demand and prce as a functon of tme, we have: t, N(t) γp(t) = Σ β (p a ), = p(t) Σ β - Σ β a, where the sum s over all. So, 7

9 t, p(t) = (N(t) + Σ β a )/(Σ β + γ). (8) Incentve compatblty The condton α = a for each frm means that t s ncentve compatble for bdders to reveal the ntercepts of ther margnal cost functons. Ths generalzes a smlar result n Rudkevch (999), whch was proven for the case where margnal costs had ntercept a = 0. A straghtforward economc nterpretaton of ths result s that, at low output levels, the bdders have no motve to exaggerate ther cost snce they have no nfra-margnal capacty. Interestngly, however, the result that α = a for each frm does not rely on low output levels and prces ever beng realzed. The result only depends on ther beng at least two, perhaps hgh, prces beng realzed. Low demand and prce levels As remarked, f prces fall below the value of a for a frm then the affne functonal form (4) wll requre negatve producton, whch s outsde the regon of valdty of the cost functon specfcaton (). A mnor generalzaton of the affne supply functon would allow for pecewse affne, non-decreasng bd supply functons. That s, we now assume that the bd rules allow for pecewse affne, non-decreasng bd supply functons. We can construct a canddate SFE n pecewse affne supply functons by pecng together several supply functons. In each pece, we use the optmalty condtons (6) to evaluate the slope of the supply functons of the frms that are actually generatng. 5 So long as the resultng composte supply functons are all non-decreasng then the canddate s truly an SFE. We llustrate ths approach for n = frms and consder pecewse affne supply functons of a form such as: % &K 0, f p a, q, ( p) = β ( p a), f a p p, ( 9) 'K β ( p a), f p> p, where p a s an approprate break - pont and β and β specfy the slopes of the supply functons n the regons a p p and p> p, respectvely. In general, there may be ust one break-pont, two break-ponts as n (9), or more than two break-ponts. The break-pont at a for frm reflects the fact that for prces below ths level t cannot be optmal for frm to produce anythng. 6 Wth the two frm case as an example, let us assume that a < a and seek the locaton of the common break-pont p' n each supply functon. Notce that for prces up to a there wll be no producton by ether frm. 5 In fact, we must modfy the optmalty condtons by recognzng that a pecewse affne supply functon s not dfferentable at ts break-ponts. However, ts drectonal dervatves exst and for our specfcaton of the slopes we fnd that Green s equaton (4) s satsfed n each drecton. 6 We are gnorng ssues such as start-up costs. 8

10 For prces between a and a, only frm wll generate. For ths range of prces, we can consder the SFE where frm s the only frm. Substtutng nto (6) yelds γ β =. Also, β = 0, snce frm s not producng n ths regon. These values + cγ apply for prces up to a. That s, the hgher break-pont n equaton (9) s at p = a for ths example. At hgher prces than a, both frms wll generate and the resultng slopes of the demand functons are gven by the smultaneous solutons of: γ β γ β β ' = + ' + ' and β ' = + c( γ + β ') + c( γ + β '). By wrtng the condtons n ths way, lemma n Appendx shows that β ' β and β ' β. Because β ' β and β ' β, the resultng composte supply functons of the form (9) are non-decreasng, as llustrated n fgure for a frm. In fgure, the ntercept s at a = 6/MWh, whle the break-pont s at p' = /MWh. In general, f there are n frms wth dfferent ntercepts a then there wll be n peces n the supply functon of each frm wth break-ponts at a, =,,n. The composte supply functon so defned s nondecreasng. In prncple, smlar analyss apples to the case where there are non-zero mnmum capacty constrants for a frm under the assumpton that the frm cannot choose to take all machnes out of servce and generate zero. q (p ) 0 8 Slope β ' 6 4 Slope β a p p ' Fgure. Illustraton of pecewse affne, non-decreasng supply functon. 3. Maxmum Capacty Constrants In the applcaton dscussed below, we encounter maxmum capacty constrants on prcetakng bdders. (Analogous ssues apply n representng maxmum capacty constrants on the output of strategc frms.) Green and Newbery (99) dscuss how to treat capacty constrants when cost functons are dentcal amongst frms. However, ther 9

11 arguments do not appear to generalze to the case where frms have asymmetrc costs. We take a dfferent approach here. We use the functon D r to represent the resdual demand resultng from a prce takng frnge wth capacty constrants. Followng Bushnell (998), f the demand s lnear then the resultng resdual demand D r can be modeled as a pecewse lnear functon. In partcular, f the margnal cost of the frnge at ts full producton s p' then there are γ and γ' such that: ddr = dp % & ' γ, f p p, γ, f p> p. The slope γ' for prces above p = p' s due to the demand alone when the frnge s at capacty, whle the composte slope γ for prces below p = p' s due to the combned effect of demand and the compettve frnge when the frnge s also margnal. (See Bushnell (998) secton 4 for dervaton of ths functonal form for the resdual demand.) We have that 0 < γ' < γ. That s, the combnaton of demand and margnal frnge capacty s more elastc than when the frnge s at ts capacty. A straghtforward approach to ths case would be agan to post a canddate SFE n pecewse affne functons of the form (9) as we dd n the prevous secton. We solve for the slopes of the supply functons n each pece usng (6) wth the approprate demand slope for the pece. In contrast to the case consdered n the prevous secton, however, t wll be the case that β ' < β because of the relatonshp between γ and γ'. Ths means that the canddate supply functons so constructed wll not be non-decreasng. In partcular, there s a drop n the supply at p = p'. The drop n the supply functon volates typcal pool rules. Moreover, even f the pool rules were to allow such bds, there can be two ntersectons of the demand and aggregate supply curve. It s not clear that one of the ntersectons would be preferred by all of the frms to the other ntersecton. The proft functon of each frm can have two local maxma, one n each of the two prce regons, p p' and p p'. Dfferent frms may have ther global maxmum n dfferent regons. That s, the proposed supply functons may not be an equlbrum even under the (unrealstc) assumpton that non-monotonc bds were allowed. A smlar observaton about proft functons was made n Bushnell (998). If we knew a pror that demand (and prces) were low then we could delete the part of the supply functons for p p'. On the other hand, f we knew that the demand and prces were hgh we could delete the part of the supply functon for p p'. Ths latter approach was taken by Green (996) n modelng the frnge n the Brtsh pool. However, our nterest s n the case where we antcpate that prce wll vary from below p' to above p'. Unless the prce umps from below p' to above p', we are faced wth decdng on a strategy for values of prce near to p'. Unlke Green s equaton (4), whch ( 0) 0

12 made no reference to the load-duraton characterstc N(t) of the demand, condtons for an equlbrum n pecewse affne supply functons requres knowledge of N(t). As an ad hoc approach that avods the need to specfy N(t), we propose a non-decreasng approxmaton to the prevous functon. For prces p > p' we mantan the functonal form β '(p a ) where the β ' satsfy (6) for demand slope γ'. For prces p sgnfcantly below p' we mantan the functonal form β (p a ) where the β satsfy (6) for demand slope γ. For prces ust below p = p' we rearrange Green s equaton (4) to obtan slopes β '' for yet another affne pece of the supply functon such that the supply functon s contnuous at p = p'. That s, we post a supply functon of the form: 7 q, ( p) = % &K 'K β ( p a), f p p'', β ''( p α ''), f p'' < p p, β '( p a ), f p> p, where: α '' s the soluton to β ''( p' α '') = β '( p' a), so that q( p) s contnuous at p= p', and p '' s the soluton to: β ( p '' a ) = β ''( p '' α ''), so that q ( p) s contnuous at p= p ''. ( ) Fgure 3 llustrates ths supply functon wth a = 6/MWh, p '' = 0.4/MWh, and p' = /MWh. 0 q (p) Slope β ' Slope β '' Slope β p ' p ' a p Fgure 3. Illustraton of supply functon (). 7 It s also possble to consder a supply functon that matches the functonal form β (p a ) at p = p' by modfyng the functon for prces above p'. However, ths results n a greater supply and t s presumably n all the strategc players nterests to keep supply lower.

13 As dscussed above, we propose to evaluate the slopes β '' usng Green s equaton (4) so that the slopes of the affne supply functon are optmal for prces nfntesmally below p = p'. However, Green s equaton (4) s not convenently expressed to evaluate the optmal slopes of supply functons because the slopes of supply functons for several frms are added together. We rearrange Green s equaton (4) nto the standard form of a vector dfferental equaton: dq = f ( q, p), where: dp q s a column vector of supply functons for all frms, dq s the dervatve of the vector q, and dp f s a vector functon. To make ths rearrangement, note that: dq " dp T dq = I, ( ) dq dp dp #! n $ # where bold-face s the column vector of all ones and bold-face I s the dentty matrx. T T We can wrte down the nverse of the matrx n brackets: I = I. n Substtutng nto Green s equaton (4) we obtan: q( p) " " p C q p dq T " '( ( )) ddr = I dp! n $# + q p dp, ( ) # # p C '( q ( p)) =!!! n q( p) p C q p '( ( )) T " ddr I n $# +, ( 3) q p n dp # ( ) p C '( q ( p))! n whch s now n the standard form of a dfferental equaton. 8 We evaluate (3) at two prces: nfntesmally above p' and nfntesmally below p'. If we consder canddate pece-wse affne supply functons that are contnuous at the boundary p = p' then the values of the supply functon at these two prces wll be the same. The only dfference n (3) across the boundary from above p' to below p' s due to the change n the slope of the $ # " # $ # $ # 8 Ths form allows numercal ntegraton to evaluate the nonlnear SFE for arbtrary demand and cost functons. In our numercal experments, we found that nonlnear solutons are always unstable.

14 demand functon from γ' to γ. The change n slope of the supply functons across the boundary s therefore gven by: β'' β' = γ + γ ' or, β '' = β ' γ γ '. n n 6 (4) Although the supply functon slopes β '' are only vald at the prce ust nfntesmally below p', we nevertheless propose usng ths slope to on wth the part of the supply functon for low prces. If all of the slopes β '' n the above expresson evaluate to beng non-negatve then we post a supply functon of the form (). Snce γ > γ', we have that β > β ' > β '' so that the stuaton llustrated n fgure 3 shows the slopes wth the correct relatve magntudes. If any partcular canddate slope β '' turns out to be negatve then we modfy () to be: % &K 'K β ( p a), f p p'', q, ( p) = β '( p' a), f p'' < p p, β '( p a), f p> p. Ths makes the supply functon constant between p '' and p' and stll guarantees that the supply functon s non-decreasng. ( 5) We emphasze that ths set of supply functons cannot be an equlbrum n nonlnear supply functons snce for prces suffcently below p' but above p '' the suggested demand slopes cannot satsfy Green s equaton (4). Nevertheless, we clam that the general form of ths supply functon s a reasonable one n practce, for three basc reasons. The frst reason s that, for the cases consdered n secton 4, we dscretze the tme dmenson farly coarsely. Under these crcumstances, we fnd that prces near to the break-pont prce p' do not arse for the cases we consder. That s, the prce does ump from well below p' to well above p' and the value of the supply functon s rrelevant n the vcnty of p'. The good fortune that prces near to p' are not realzed s partally an artfact of the coarseness of our tme dscretzaton. That s, we must n general consder the case where prces are realzed n the vcnty of p'. We consder how sub-optmal t s for frm to use slope β '' for prces p '' < p p' compared to frm usng an optmal response to the other frms, wth slope say β *, gven that each other frm mantans slope β ''. We construct a smple argument that suggests that ths sub-optmalty s small. Ths s the second reason why the pecewse affne functon s reasonable. Ths argument also suggests that although the proposed functon s not an equlbrum n nonlnear supply functons t s nevertheless lkely to be close to an equlbrum n pece-wse affne supply functons. We frst note that the loss n proft for frm s quadratc n the dfference β * β ''. The value β * β '' s zero for prce p = p' and ncreases approxmately lnearly as prces decrease from p = p'. Assumng that the demand curve s approxmately lnear n t when 3

15 the prces are n the range p '' < p p ', we can argue that the loss of proft s then approxmately cubc n the duraton of tme that p '' < p p '. If ths duraton s relatvely small then the effect on the total profts ntegrated over tme wll be small. In summary, the choce of supply functon slope β '' s sub-optmal but the loss of profts s relatvely small compared to unlaterally devatng from ths strategy. Ultmately, ths argument ponts to the fact that n the case of capacty constrants, t s not possble to fnd an optmal strategy that s ndependent of the load-duraton characterstc N(t). That s, the choce of the slopes n the pecewse affne functons wll be affected by the amount of tme that prces are realzed n the vcnty of the break-pont prce. For frms faced wth makng a decson about ther supply functons, the lack of optmalty may be relatvely nconsequental because t s lkely to persst over a relatvely short tme. We wll see that ths s true for the cases consdered n secton 4. The thrd reason for the reasonableness of the choce of supply functons s that calculaton of the optmal response by a frm requres estmaton of the slope of the aggregate supply functon of the rest of the frms. Generalzng the dscusson n Rudkevch (999), we can magne frm fttng a pece-wse affne curve to the observed aggregate supply functon. If the prce p' where the frnge reaches capacty s known publcly, then t s reasonable to for frms to estmate dfferent slopes for the parts of the supply functon above and below prce p = p'. It may even be reasonable for frms to estmate slopes for prces above p', well below p', and n the vcnty of p' as requred for the functonal form n (). However, f only a few data ponts are observed t wll not be possble to estmate parameters of functonal forms wth a large number of parameters. That s, pecewse affne functons wth known break-ponts may be a practcal lmt to the ablty to estmate functons. 4. Applcatons In ths secton we show how to apply the prevous results to modelng the prce effects of structural change n the E&W electrcty market. Our obectve s to llustrate how the SFE enhancements descrbed n Sectons and 3 mprove the numercal ft of such models to actual market experence. We begn by characterzng the structural changes n the E&W electrcty market. Next, we reconsder Green s forecast of the effects of dvesttures made by Natonal Power (NP) and Power Gen (PG) n 996. Ths dscusson llustrates the mportance of fttng the demand curve properly and the role of capacty constrants. We examne the effect of the 999 dvesttures of Natonal Power (NP) and Power Gen (PG). We show that a lnear SFE that does not use postve cost ntercepts for the strategc frms wll predct prces that are too low n the post-dvestture case. We also consder two specal ssues n ths context. Frst we examne the role of earn out payments as an explanaton of post-996 prcng behavor. Fnally we llustrate the effect of frnge capacty constrants on the supply curves of strategc frms, showng that the sze of ths effect s small, as argued above. Table summarzes the structure of the cases examned n ths secton. 4

16 Table. Cases Examned and Ther Features Method Structural Change Table or Fgure 996 Dvesttures Capacty Lmts frms 3 frms Tables 4,5,6,7 Demand Curve Specfcaton 999 Dvesttures Postve vs. Zero 3 frms 5 frms Tables 8,9,0 Intercept Earn Out Table Pastng Problem Fgure 5 The E&W Electrcty Market Market power has been a constant theme n the regulaton of the electrcty ndustry n England and Wales. 9 Ths has motvated regulatory nterventon and responses by both ncumbents and entrants. The two domnant generators, Natonal Power (NP) and Power Gen (PG), have retred very substantal amounts of capacty, nvested n new combned cycle gas turbnes (CCGTs) and been requred to dvest generaton to new entrants. Addtonal entry has come prmarly from CCGT proects. Nuclear output also grew over tme. All of ths change on the supply sde has occurred n the face of very sluggsh growth n demand (about.5% per year). Table summarzes the changes n capacty by ownershp category over the perod , wth some comments about the prevous hstory. Ths table forms the capacty bass for the LSFE estmates reported below. Table. Supply Mx by Frm and Fuel Type Category Comment MW MW MW Nuclear Brtsh Energy Magnox Electrc Interconnector France +Scotland 300 IPP Eastern Natonal Power 9 Vestng GW CCGT retrements Power Gen 9 Vestng GW CCGT retrements AES w/o IPPs 3945 Edson Msson Energy w/o Frst Hydro For recent statements of these ssues see Offer (998, 999) and Newbery (995, 999) among others. 5

17 Nuclear generaton expanded n 994 when Szewell B, the last nuclear plant constructed n E&W came nto servce. In 996, the government owned Nuclear Electrc was restructured nto the prvatzed Brtsh Energy and Magnox Electrc, whch remans n government hands. Independent power producers (IPPs) all rely on combned cycle gas turbne (CCGT) technology. In the frst "dash for gas" from , IPPs contracted wth regonal dstrbutors (Newbery, 999, p.4f). By 995 about 5000 MW of IPP capacty was operatng. Table shows that ths nearly doubled by 999. Table also ndcates substantal reductons n capacty by NP and PG over the 990s. When NP and PG were frst created n 989, NP had 9,486 MW and PG had 9,80 MW (Newbery, 999, p.0). Each added about 3,00 MW of CCGT capacty and together the two frms retred more than 0,000 MW. 0 These retrements were n addton to dvesttures of more than 6000 MW to Eastern n 996 and the 999 dvesttures to AES and Edson Msson Energy ndcated n Table. Table 3 shows the cost and avalablty assumptons used n our calculatons. The cost ntercept s assumed to be zero for Nuclear and the Interconnector, a common ntercept of /MWh s assumed for the generators wth coal-fred plant only (Eastern, AES and EME) and 8/MWh for generators wth CCGT plant (IPP, NP, and PG). These margnal cost estmates are based on Bunn and Day (999). The cost at maxmum capacty follows Green (996) for generators wth thermal plant. These costs can be thought of as ether open cycle turbne costs or the costs of coal plant runnng very few hours and recoverng start up and no load costs over that short perod. The avalablty estmates n Table 3 for all fossl fuel generators are generc. The estmates for nuclear are set to reproduce recent producton levels. The hgh avalablty for the Interconnector reflects the multplcty of resources avalable from Scotland and France. The cost and avalablty data n Table 3 are combned to produce the cost slope parameters c that are used to solve for the equlbrum prces and mark-ups. These parameters also depend upon the capacty of each frm. Appendx gves the numercal values used n the results reported below. 0 Accordng to NGC (999) there were 5,5 MW of Dsconnectons between 99 and 999 (see Table 3. and 538 MW of Decommssonngs (see Table 3.) for NP and PG together. GN use 8.5/MWh as the ntercept of the margnal cost functon for coal plant (p.94), based on coal prces quoted n generators prospectus. In the years snce the flotaton of NP and PG, coal and gas prces have declned substantally. 6

18 Table 3. Cost and Avalablty Parameters Category Avalablty Cost Intercept ( /MWh) Cost at Maxmum Capacty ( /MWh) Nuclear 83.0% Interconnector 98.0% IPP 85.0% Eastern 85.0% Natonal Power 85.0% Power Gen 85.0% AES 85.0% Edson Msson Energy 85.0% Fttng Demand Curves A maor problem assocated wth practcal use of both SFE and Cournot models s the representaton of demand. The plausblty of prce forecasts wth these models depends substantally on how the demand curve s specfed. Although there s lttle demand-sde response n the E&W market, low values of demand elastcty have typcally yelded poor fts to the observed data. For example, when GN used very low slopes for the lnear demand curve they estmated prces that were very much hgher than what was subsequently observed. Even at a hgh slope (-0.5 GW/( /MWh)) the predcted prces are much hgher than what was observed. A good part of the problem of representng demand nvolves how the demand curve s anchored n prce-quantty space. GN use an estmate of pre-competton margnal costs and producton to anchor ther demand curve. Green (996) uses a demand curve n the form of Eq. () above, wth N(t) specfed as a cubc load duraton curve and a slope parameter of -0.5 GW/( /MWh). Ths value s consderably hgher than what other authors use. GN use 0.5 as ther central case; Bushnell (998) uses 0.; Bunn and Day (999) use values between 0.0 and 0.0. We wll follow GN and report cases for demand slopes of 0.5, 0.5 and 0.. Our procedure takes advantage of the Pool prce hstory. Fgure 4 shows both the load duraton curve and the prce duraton curve for The prces shown n ths fgure are the System Margnal Prce (SMP). We use ths data to anchor the demand curves used n our smulatons. The anchor pont for each of the nne perods we consder s taken to be the (p,q) par correspondng to the decles of the prce and load duraton curves. Ths procedure assumes that past prce behavor represents a set of expectatons that s famlar to strategc frms and acceptable to regulators. We gnore other prce elements,.e. the uplft and capacty charges. 7

19 Fgure 4. Load & Prce Duraton Curves MW $/MWh Load Prce Tme For the condtons studed by GN and Green (996), the prce-takng producers were never margnal. Therefore, Green (996), for example, reduces the load duraton curve by hs estmate of constant nframargnal producton and apples the LSFE to the resdual demand. 3 By 999, however, growth n the capacty of IPPs and the ncreased avalablty of nuclear plant made the prce-takers margnal durng low demand perods. Incorporatng ths effect ntroduces the need to model capacty lmts explctly. The methods dscussed n Secton 3 are mplemented n our estmates. Examples of how ths works are gven below. We also dscuss the plausblty of the ad hoc pecewse affne model (). Revstng the 996 Dvesttures In ths secton, we examne the 996 dvesttures. Our chef nterest s n showng how capacty constrants and demand curve representaton affect numercal estmates. Ths s done by re-examnng Green (996), whch we refer to as G96. We begn by reproducng G96 results at the same range of demand slopes used by GN. Table 4 reports the GN results and correspondng estmates from the G96 model. Ths table also reports the demand at each demand slope. 3 We followed the same approach n secton 3 for notatonal convenence. However, here we wll generally report the actual demand and supply, not the resdual demand. 8

20 Demand Slope (GW/( /MWh)) Table 4. Duopoly Game GN Prce ( /MWh) G96 Prce ( /MWh) G96 Demand (Average GW) Whle the prces n GN and G96 are smlar for each demand slope, GN appears to have shfted the demand curve for each demand slope case, snce they report the same output n all cases. Table 4 s based on a lteral nterpretaton of the G96 model,.e. no shfts n the demand curve, and ndcates sgnfcant demand level dfferences. 4 Average demand levels of 33 GW are consstent wth late 990s observatons. However, both G96 prces and demand levels are substantally too hgh at a demand slope of 0.0, and move n more reasonable drectons as the demand slope ncreases. Table 5 reports tests of the G96 model where PG and NP dvest 5% of ther capacty to frms that each bd strategcally. Ths amounts to about 4.5 GW for NP and GW for PG. Table 8 shows the same knds of demand effects seen n Table 4. It also shows that at demand slopes of less than 0.5, the new frms would produce more than ther capacty. There s no noton of capacty n these models, whch amounts to the assumpton of potentally nfnte supply from any producer. Wthout an explct mechansm to ntroduce maxmum capacty lmts, any verson of the SFE approach can result n the knd of anomaly llustrated n Table 5. Demand Slope (GW/( /MWh)) Table 5. G96 Dvestture Cases Dvest Prce ( /MWh) Dvest Demand (GW) Maxmum Output of Dvested Plant (GW) Tables 4 and 5 show that the case that s closest to observed prces and physcally realzable quanttes s the one that Green reports,.e. the hgh demand slope case. It produces an expected prce declne of about 5% compared to the correspondng duopoly case. The hgh demand slope s dffcult to ustfy, however, snce electrcty demand s extremely nelastc n the short-run. Most studes of electrcty markets usng equlbrum approaches adopt a low elastcty parameterzaton. Ths msmatch between G96 and standard ntuton motvates a further examnaton of what mght mprove the realsm of results at lower demand slopes. Table 6 shows the effect of ntroducng capacty lmts and adustng the ntercept of Green s demand functon. These estmates are based on 5000 MW of IPPs, 3 GW of 4 We report the realzed demand as the sum of the 5 GW nframargnal supply and the duopoly supply. 9

21 capacty for NP and 8.8 GW for NP n the duopoly case. Average and Adusted Demand s n each case the average of the duopoly and dvestture cases. Demand Slope (GW/ ( /MWh)) Duopoly Prce ( /MWh) Table 6. G96 LDC and Capacty Lmts Dvest Prce ( /MWh) Average Demand (GW) Adusted Demand (GW) Duopoly Prce ( /MWh) Dvest Prce ( /MWh) G96 Load Duraton Curve Intercept Adusted LDC Comparng Tables 4, 5 and 6 requres some care. The nframargnal prce takng capacty s about 000 MW greater n the Table 6 cases than the Table 4 and 5 cases. 5 At hgh demand slope and wthout adustng the demand curve ntercept, ths results n large percentage dfferences n the duopoly prce ( 7./MWh at a demand slope of 0.5 GW/( /MWh) n Table 4 versus.6/mwh at the same slope n Table 6) and n the dvestture prce ( 3.5/MWh at a demand slope of 0.5 GW/( /MWh) n Table 5 versus 9.8/MWh at the same slope n Table 6). At low demand slope (agan wthout adustng the demand curve ntercept) the changes work dfferently. The duopoly prce drops about 0% (from 66.7/MWh to 53.9/MWh at a demand slope of 0.0 GW/( /MWh) and from 4./MWh to 33.3/MWh at a demand slope of 0.5 GW/( /MWh)), but the dvestture prce ncreases by about 0% (from 39./MWh to 43.9/MWh at a demand slope of 0.0 GW/( /MWh) and from 30.7/MWh to 33.3/MWh at a demand slope of 0.5 GW/( /MWh)). The dvestture prce ncrease effect seems strongly ted to enforcng capacty lmts n Table 6, because Table 5 showed the bggest problem n ths regard,.e. output greater than capacty, at low demand slope. The obectve of the Intercept Adusted LDC cases n the rght hand panel of Table 6 s to llustrate the dramatc effect on prce and quantty that the shfts of the demand curve acheve. The case wth demand slope of 0.5 GW/( /MWh) nvolved a reducton of 4000 MW of the ntercept term n Green s specfcaton of the load duraton curve. The adustment s 7000 MW for the case of demand slope of 0.0 GW/( /MWh). These adustments result n average demands at vrtually the same level as the case of demand slope of 0.5 GW/( /MWh) that s shown on the left hand sde of Table 6. Prces n the duopoly and dvestture cases are more reasonable wth these adustments, although stll not where they should be. A fnal set of tests nvolves usng our demand curve nstead of the G96 curve. It also examnes both the zero and the postve ntercept cases (compared to zero only n Table 6). Table 7 summarzes these tests. It should be compared to the rght hand panel of Table 5 Ths s due to G96 smply subtractng an estmate of nframargnal supply from the load duraton curve versus explctly representng the capacty and producton of these producers. 0

22 6 to show the ncremental effect of usng a demand curve anchored at hstorcal prces and quanttes. Broadly speakng, the Table 7 tests are mprovements compared to the rght hand panel of Table 6. For the zero ntercept representaton, prces decrease n all cases. Wth postve ntercept, the prces are lower for the 0. GW/( /MWh) demand slope, but they are hgher for the demand slope 0.5 GW/( /MWh) compared to the rght hand panel of Table 6. Table 7. Ftted LDC Demand Slope Duopoly prce Dvest to 3 Average GW GW/( /MWh) ( /MWh) prce ( /MWh) Zero Intercept Cost Intercepts Earn Out Table 7 ncludes the effect of the earn out payments n the cost ntercept case. The role of cost ntercepts s dscussed n more detal n the next secton. The 999 Dvesttures In 999, NP and PG each dvested about 4000 MW of coal-fred plant. Ther motvaton was to meet regulatory requrements assocated wth ther proposed vertcal mergers. Table shows that the U.S. frms AES and Edson Msson Energy (EME) acqured the dvested plant. Both frms already had generaton assets n the E&W market; IPPs for AES and both IPPs and the pumped storage plants for EME. 6 We use the LSFE framework to assess the prce mplcatons of these dvesttures, and to compare the performance of the affne case wth the case where the margnal cost curves must go through the orgn. Table 8 shows the results of usng the data n Table and two versons of the cost curve for the strategc generators wth gas and coal-fred plant. The cases labeled Zero Intercept assume that all cost curve pass through the orgn. The Postve Intercept cases assume the cost ntercept values n Table 3. Our calculatons assume that both AES and EME bd strategcally. Because the prce changes from dvestture are realzed over tme, we have used the level of IPP capacty (.e 7339 MW) for the 3 frm game and the level of capacty (.e. 97 MW) for the 5 frm game. 6 For smplcty, the pumped storage plant s neglected n these estmates.

23 Demand Slope (GW/ ( /MWh)) Average Prce ( /MWh) Table 8. Base Case Prce Results ( /MWh) 5 Frms 3 Frms Hgh Prce Low Prce Average Hgh Prce ( /MWh) ( /MWh) Prce ( /MWh) ( /MWh) Low Prce ( /MWh) Zero Intercept Postve Intercept Table 8 shows for each case the tme-weghted average prce, the hghest prce and the lowest prce. The prce change predctons of dvestture,.e. the dfference n average prces, are greater at low elastcty than at hgh elastcty regardless of the cost curve characterzaton. As expected, the bggest dfferences between the postve and zero ntercept representaton are at the low end of the prce dstrbuton. In realty, the SMP has dropped substantally n the power year. Prevously, SMP had been reasonably stable n the 3-4/MWh range over many years. The declne s proected to be about 0% from the prevous level (Standard and Poor s, 000a,b). Ths suggests that the estmates based on the postve ntercept cases are better than those based on the zero ntercept cases. Tables 9 and 0 show the quantty behavor (MW produced per hour) of the frms for the cases wth demand slope of 0.5 GW/( /MWh) for each cost curve characterzaton. Table 9a shows that the frnge frms (Nuclear, Interconnector and IPP) are margnal durng perods 8 and 9. Ths s ndcated by shadng the perods and ndcatng n bold that the frnge frm producton s below the maxmum levels acheved n hgher demand perods. In Table 9b, the frnge frms are margnal n perod 7 as well as perods 8 and 9. Table 9a. Zero Intercept Case: 3 Strategc Frms Tme Perod Average Prce ( /MWh) Nuclear 8,707 8,73 8,73 8,73 8,73 8,73 8,73 8,73 8,73 8,50 Interconnector 3,075 3,36 3,36 3,36 3,36 3,36 3,36 3,36 3,36,589 IPP 5,966 6,39 6,39 6,39 6,39 6,39 6,39 6,39 5,779 4,4 Eastern 3,584 5,695 5,93 4,393 3,849 3,457 3,6,75,49,683 NP 6,49 0,740 9,086 7,686 6,733 6,048 5,45 4,768 4,44 3,659 PG 6,36 0,530 8,86 7,496 6,566 5,898 5,37 4,650 4,03 3,50

24 Table 9b. Zero Intercept Case: 5 Strategc Frms Tme Perod Average Prce ( /MWh) Nuclear 8,643 8,73 8,73 8,73 8,73 8,73 8,73 8,73 8,73 7,93 Interconnector 3,039 3,36 3,36 3,36 3,36 3,36 3,36 3,36,987,43 IPP 7,655 8,6 8,6 8,6 8,6 8,6 8,6 7,607 6,48 5,35 Eastern 3,0 5,350 4,73 3,568 3,088,743,443,8,89,576 AES,88 3,35,704,58,954,736,546,333, NP 4,75 7,893 6,63 5,537 4,79 4,56 3,79 3,69 3,85,73 PG 4,48 7,568 6,3 5,7 4,56 4,05 3,608 3,409,994,556 EME,88 3,35,704,58,954,736,546,333, Tables 9a and 9b show that the coal based strategc frms produce even when the frnge frms are margnal. The prces n those perods, however, are below ther margnal costs;.e. the assumed ntercept of /MWh. The zero ntercept formulaton forces the plants to operate at prces that are below the ntercept of ther true margnal cost curves. By contrast, Tables 0a and 0b show producton by coal based strategc frms declnng toward zero as prce falls toward the margnal cost mnmum. Table 0a. Postve Intercept Case: 3 Strategc Frms Tme Perod Average Prce ( /MWh) Nuclear 8,73 8,73 8,73 8,73 8,73 8,73 8,73 8,73 8,73 8,73 Interconnector 3,36 3,36 3,36 3,36 3,36 3,36 3,36 3,36 3,36 3,36 IPP 6,86 6,39 6,39 6,39 6,39 6,39 6,39 6,39 6,39 5,760 Eastern 3,005 5,695 4,993 4,06 3,367,894,48,009,35 30 NP 6,50 0,683 9,074 7,634 6,653 5,948 5,335 4,63 3,65,637 PG 6,06 0,50 8,859 7,453 6,495 5,807 5,09 4,5 3,565,54 Table 0b. Postve Intercept Case: 5 Strategc Frms Tme Perod Average Prce ( /MWh) Nuclear 8,73 8,73 8,73 8,73 8,73 8,73 8,73 8,73 8,73 8,73 Interconnector 3,36 3,36 3,36 3,36 3,36 3,36 3,36 3,36 3,36 3,36 IPP 8,6 8,6 8,6 8,6 8,6 8,6 8,6 8,6 8,6 7,03 Eastern,96 5,086 3,903 3,085,58,8,780, AES,480 3,78,56,989,630,37, NP 4,636 8,34 6,75 5,664 4,93 4,39 3,98 3,396,656,05 PG 4,40 7,9 6,45 5,390 4,685 4,78 3,738 3,3,58,96 EME,480 3,78,56,989,630,37,