Trad Forward te Brazla Eletrty Maret Paulo Couto Adre Ross de Olvera Te paper odels te terato betwee a otrat ad pot aret wose features are borrowed fro te Brazla eletrty aret. Te spot aret s odeled as a rado eas tat yelds spot pres of eletrty. Te otrat aret s oprsed of supplers, osuers, ad areters. Supplers ad osuers are pre taers, wle areters ave aret power. It s sow tat, we te uber of osuers reases te otrat aret, t s possble for te pre of te eery tey buy forward to derease, eve f tere s oly a oopolst areter te aret. Moreover, te quatty of eery traded te otrat aret approaes te total aout of eery avalable te syste (et of eery sold to aptve osuers) we te uber of areters reases wtout boud. Feld of Resear: Eoos Keywords: Brazla eletrty setor, forward arets, rs averso, areters, Courot equlbru JEL Classfato Nubers: L13, L94 G10, C61, C7 Paulo Couto Afflato: Aout Departet, Uversdade de Brasíla, Brazl. Address: Uversdade de Brasla, Capus Dary Rbero, FACE, Departaeto de Cêas Cotábes e Atuaras - CCA, Brasla D.F. 70910-900, Brazl. E-al: outo@ub.br Adre Ross de Olvera Afflato: Fae ad Eoos Departet, Woodbury Sool of Busess, Uta Valley Uversty. Address: 800 Uversty Parway, Woodbury Busess Sool, Uta valley Uversty, Ore, UT 84097. E-al: aolvera@uvu.edu 1
1. Itroduto Trad Forward te Brazla Eletrty Maret I ts paper we vestate te terato betwee forward ad spot arets a odel based o te Brazla eletrty aret. Altou t draws ts a features fro te Brazla ase, our odel s of terest ad of tself, se t brs ew eleets to te aalyss of otrat arets, partular te role played by areters. Te Brazla eletrty setor uderwet two ajor overauls te last two deades. Te frst started 1995, we Law # 8.987, ow as te Coessos Law, was passed by te Brazla Coress. By establs te leal fraewor to reulate te oesso of publ serves, t usered a ew era te eletrty setor w several dstrbuto ad a few eerato opaes were prvatzed, a reulator ad yste operator were reated, ad a wolesale aret was strutured. Ts ew fraewor was desed to proote opetto eerato ad oeralzato, ad provde ope aess to te trassso ad dstrbuto rds, wle eep dstrbuto ad trassso uder (etve) reulato. Te seod overaul too plae te wae of a eery rss 001-0 tat fored te federal overet to tae drast easures to urtal osupto, ad te advet of a ew overet, 003, tat ae to power wt a prora all for refor of te eletrty setor. Its aeda ae to fruto wt te troduto of te so-alled New Eletrty Setor Model, 004. Ts ew odel aed several aspets of te oral des of te Brazla eletrty aret, but t ept ope a otrat aret were free (.e. ot aptve) osuers ad eerators ould trade eletrty forward. Ts opeed te door for areters, aets wo purase ad resell eery ad/or elp lose deals betwee buyers ad sellers, to eter te aret. I Brazl, tese areters a be eter depedet or afflated wt eerators ad/or dstrbutors. Te presee of areters s ot a exlusve feature of te Brazla eletrty aret. Several oter arets aroud te world, le te PJM ad te Texas arets te Uted States, also ave areters. Tere s o sortae of papers te lterature tat study te terfae betwee spot ad otrat arets. Soe of te are teoretal papers terested te eeral features of ts terato. Oters are applatos to dfferet produt arets le tat of eletrty. We wll revew ts lterature te ext seto. Our paper dffers fro te reet lterature two a respets. Frst, t does ot allow supplers (eerators) to set quattes or subt supply sedules te spot aret. Tey atually do t ave ay otrol over te spot pre. Seod, bot supplers ad osuers of eletrty beave opettvely te otrat aret. Ts s otrast wt te reet lterature o futures ad forward arets, w aowledes tat ay oodtes traded o tose arets are ot produed opettvely, but several eletrty arets te uber of supplers ad osuers s atually relatvely lare. I te Brazla aret, for stae, tere are urretly 579 free osuers ad 495 eerators. I our odel, all te aret power te otrat aret belos to te areters, aoter ovelty trodued by our paper. Te paper s orazed as follows. Seto provdes a bref lterature revew, wle seto 3 desrbes te a features of te Brazla eletrty aret. Seto 4 develops te bas fraewor of aalyss ad presets te a fds. It s dvded to two sub-setos, oe tat vestates te ase of a oopolst areter, ad te oter osders te ase of several
areters play a Courot ae. Seto 5 oludes ad te Appedx presets te a proofs.. Lterature Revew Te seal result te teoretal lterature about te terato betwee spot ad forward arets s Allaz ad Vla [1993]. Tey develop a eeral odel (w apples to several stuatos, ot oly eery arets) to sow tat forward arets a eere eve te absee of uertaty. Ter a odel s a two-perod ae were (duopoly) produers frst buy or sell forward (bd ad observable) otrats ad te, te seod perod, play a Courot ae quattes pot aret. A ey assupto of ter odel s perfet forest, w etals o arbtrae, tat s, te forward pre s equal to te pre tat wll obta te spot aret. Tey sow frst tat produers ave stro etves to sell forward part of ter produto, for we oe of te sueeds be te oly produer to trade forward, e reatly beefts fro do so. Trad o te forward aret, owever, s a prsoers dlea for te produers, se bot ed up worse off we tey trade forward. Tey also sow tat Courot spot arets wt forward arets are effet te lt, as te uber of trad perods oes to fty. Several later papers ae te assuptos of te odel used by Allaz ad Vla [1993] ad sow tat ter a oluso tat forward arets are soal desrable eve te absee of uertaty ay ot old. Mae ad Salaé [004], for stae, sow tat a odel wt pre-sett duopolsts wt dfferetated produts, forward trad results produers buy forward ter ow produto, so tat equlbru pres are reased opared to te ase wtout forward trad. Gree ad Le Coq [010] try to aswer a dfferet questo, aely ow te let of otrats affets te possblty of olluso a repeated pre-sett ae. Tey olude tat frs a always sustoe ollusve pre above aral ost f tey sell te rt uber of otrats, watever ter dsout fator. As te durato of otrats reases, owever, olluso beoes ore dffult to susta. Tere s also a lare u of te lterature tat fouses o te eletrty setor. Te seal paper ts are Gree ad Newbery [199], te frst oe to apply te oept of supply futo equlbru developed by Kleperer ad Meyer [1989] to eletrty arets. I ter odel, eerators subt upply sedule of pres for eerato ad reeve te aretlear pre, w vares wt dead. Tey sow tat te Nas equlbru supply sedules yelds a arup o aral ost ad substatal deadwet losses, ad use ter fds to expla te early outoes observed te Brts eletrty spot aret. Powell [1993] odels te otrat aret Brta, were faal otrats ow as otrats for dfferees (CfDs) are traded. Dead for eletrty oes ostly fro dstrbuto opaes wt ea-varae utlty. Geerators are pre setters te otrat aret ad quatty setters te spot aret. Hs a olusos are te follow: We eerators are o-ooperatve bot arets, te opettve result (aral ost pr ad otrat pre equal to expeted spot pre) ay eere; we eerators ollude bot arets, spot pres are above aral osts, future pres are above expeted spot pres, ad ed s oly partal; we eerators ollude oly te otrat aret, ed ay be lower stll (we rs averso s suffetly low). Oter early otrbutos to te study of te UK eletrty aret are vo der Fer ad Harbord [1993] ad Wolfra [1998]. 3
Gree [1999] s aoter portat referee ts lterature. He odels te eletrty aret te UK as a two-stae ae of pot aret ad a ed otrat (CfDs) aret, just le Powell [1993]. Geerators stratees te spot aret are dfferet, owever. Tey sultaeously subt supply futos 1 ad te Pool (aret operator) osders bds ased order. Te otrat pre s detered by a arbtrae odto, w states tat t ust equal te expeted spot pre, ve te uber of otrats sold. Te a olusos of te paper are: (a) A fr wt Bertrad ojetures wll over all of ts expeted output te otrat aret ad wll sell at aral osts bot arets; (b) A fr wt Courot ojetures wll sell o otrats equlbru ( te lear odel ase); ore eerally, a rs-eutral fr wll ot wat to use te otrat aret uless ts wll affet ts rvals stratees; () Geerators ay over ost of ter output te otrat aret ad stll rase pres above ter aral osts; (d) If buyers are rs averse, te otrat pre ay exeed te expeted spot pre, reas te eerators etve to sell te otrat aret. More reet otrbutos to te lterature are Busell [007] (US aret), Carreta ad Esposa [010] (Spas aret), ad Adlov [010]. Te lterature o te wors of te Brazla eletrty aret s ostly Portuuese ad does t o u beyod provd aouts of te storal evoluto of te eletrty setor ad desrb te urret syste. Exeptos are Dutra ad Meezes [005], wo study te propertes ad outoes of te autos arred out te reulated part of te Brazla otrat aret, ad Wola [008], wo presets a proposal for sort-ter pre deterato te wolesale aret. 3. Te Brazla eletrty aret Oe of te a features of te New Eletrty Setor Model, trodued 004, s te exstee of two separate eery trad evroets. I te frst oe, aed te Reulated Cotrat Evroet (RCE), eery s sold by eletr utltes, depedet power produers, self-eerators ad power areters, ad te oly buyers are dstrbuto opaes, wo are requred to otrat ter etre foreast dead for aptve osuers. Cotrats are autoed off over te wt delvery dates of oe, tree, ad fve years after te date of te auto, ad separate autos for ew ad exst eletrty 3. Cotrats for ew eletrty are loer (durato of ore ta 15 years) ta tose for exst eletrty (et years). Tere are also aual adjustet autos were dstrbuto opaes a buy addtoal eery we ter foreasts are off te ar. Mareters are oly allowed to partpate tese adjustet autos te reulated evroet. Te seod trad evroet s alled te Free Cotrat Evroet (FCE), ad brs toeter eletr utltes, depedet power produers, self-eerators, areters, porters, exporters, ad free osuers (tose tat do ot eed to buy power fro dstrbuto opaes, typally dustral ad oeral frs). Buyers ad sellers are free to eter blateral otrats ad eotate pres, quattes ad delvery dates ad odtos. Mareters 1 Gree (1999) wors wt lear supply futos ost of te te. Abete de Cotrataão Reulada ad Abete de Cotrataão Lvre, respetvely, Portuuese. 3 New eletrty refers to power to be eerated by plats yet to be bult, ad exst eletrty refers to power eerated by exst plats. 4
a be eter depedet or afflated wt eerators ad/or dstrbutors. Tey ay eter purase ad resell eery or oly elp lose deals betwee buyers ad sellers. Te FCE, also ow as te free aret Brazla eletrty setor parlae, as bee row steadly te past few years. It ossted of aroud 1,100 free ad speal 4 osuers 011, w aouted for approxately 8% of total osupto te Brazla eletrty syste (ABRACEEL [011]). Dfferees betwee te eery otrated ad te eery effetvely produed or osued by aret partpats are lqudated te spot aret at te so-alled Lqudato of Dfferees Pre. I otrast wt oter spot arets aroud te world, o sort-ter eery trad taes plae te Brazla aret. It s purely a eas for ultlateral lear of eery surpluses or defts. Geerators, partular, do ot dede ow u eery to produe. Tat s detered by te syste operator based o a dya prora odel explaed below. Te spot pre s oputed weely (by subaret) ad s based o te aral operatoal ost of te syste, wt lower ad upper bouds set by te reulator. Se te Brazlyste s prepoderatly ydroeletr, te spot pre s oputed by toast dya prora alort tat sees to fd te optal balae betwee us water today ad stor t for future use. To use as u water as possble today to produe eletrty s te best sort ter soluto, but tat would rease te lelood of eletrty sortfalls te future. O te oter ad, to oserve water today by eep reservors full s te ost relable soluto, but t requres er teral eerato ad, tus, er eletrty osts ad pres. 4. Model ad fds I ts seto, we are terested vestat te pat of power areters a eletrty aret wt te aratersts of te Brazla aret. I order to do tat, we eed to odel two separate but terled arets, te otrat ad te spot aret. I addto, we eed to tae to aout te fat tat te otrat aret s atually dvded to two subarets, a reulated (te RCE) ad a free aret (te FCE). Let s start wt te otrat aret. Free osuers do ot buy te reulated aret, so we do t eed to odel ay terato betwee tat ad te free aret as far as dead s oered. As for te supply sde, we ae te assupto tat eerators sell bot (sub) arets, frst te reulated ad te te free aret. We a desos about ow u to sell te latter, tey tae ter otets te reulated aret as ve. Eve tou t s ot etrely realst, ts assupto aes sese f we tae to aout te fat tat te Brazla aret eerators subt bds autos arred out wt te RCE, ad eter lo ter otrats wt dstrbuto opaes f ter bds are suessful. We ted to vestate te possble opportutes for strate beavor avalable to supplers as a result of ter presee bot arets future wor. Te spot aret 5 s odeled as a eas tat yelds a rado spot pre. Ts pre s tur depedet o te dead for eletrty foreast by te syste operator. Altou ot a perfet represetato of realty, ts way of odel te spot aret bears out te a aratersts of te alulato of te spot pre te Brazla ydroeletr-doated 4 Speal osuers are tose ettled to buy eery fro etvzed soures (wd, sall ydroeletr plats, boass ad solar). 5 Eve tou teally t s ot a aret, we wll otue to use ts ter. 5
syste. I addto to te foreast dead, puts to te alort used by te Brazlyste operator to opute te spot pre are stoast varables su as te level of water reservors, preptato, evaporato, ad oter uses of water (rrato, water supply et.) 6. I wat follows, we frst study te ase were tere s le areter preset te otrat aret, ad te 4.1 Cotrat aret wt oe areter Tere are two perods our odel. I perod 0, a forward otrat aret wt eletrty supplers (dexed by ), osuers (dexed by ), ad oe areter opes. Te syste operator publly aoues te foreast dead to be used te alulato of te spot pre perod 1 7, ad te areter buys forward otrats fro supplers ad sells te to osuers at a preu. Te spot aret opes perod 1, we dfferees betwee observed ad otrated quattes of eletrty are settled at te spot pre. Forward otrats are also settled perod 1. As etoed above, te spot aret s a rado eas tat yelds pot pre p. We odel te spot pre as a rado varable e p a - bq e, (1) were e Q s te foreast dead ad e s a orally dstrbuted rado varable wt ea ad varae s. Bot supplers ad osuers are rs averse ad ave eatve expoetal - a utlty futos ve by u( p) - e p, were p s proft. Te proft futo of osuer s ve by were ( ) ( ) ( ) p r f R - p R - y - q y - f R, () q s te pre of a ut of otrated eletrty as quoted by te areter to te osuer, r s te ve retal pre of ts produt, te osuer to produe ( ) R s te atual aout of eletrty used by f R uts of ts produt, f s ts produto futo, s ts ostat aral (ad averae) produto ost, ad y s te quatty t buys forward. Reveue () s equal to te output te osuer produes fro a volue R of eletrty (reall tat a free osuer te forward aret s a produer ts produt aret) tes te retal pre of ts produt. We oralze te aral produto ost to zero, ad so te ost part of osuer s proft ludes oly te ost of buy eery te spot aret ad te ost of buy t te otrat aret. Note tat te quatty t buys te spot aret s te dfferee betwee ow u eletrty t atually osues ad ow u t buys forward. Se te osuer s utlty futo as a eatve expoetal for, ts axzato proble a be expressed ters of te ertaty equvalet easure: 6 For a detaled exposto of te stoast dual dya prora based alorts used by te Brazla syste operator, see Maera et al. [008]. 7 A alteratve but equvalet assupto would be tat te dead for eletrty foreast by aret aets (supplers, osuers, ad areters) odes wt tat used by te syste operator te alort tat deteres te spot pre. 6
ax E a - (3) ( p ) V ar ( p ) s were ts oeffet of rs averso. Before we a opute te expeted value ad te varae of osuer s proft, we eed to uderstad ow t fors ts expetatos about ts sales te produt aret. It would be pratal to odel ea osuer s produt aret, so we assue t a perfetly foreast ow u of ts produt t wll be produ ad sell perod 1. Hee R s ve 8, ad ( p ) ( )- ( - ) - ( ) ( ( ) ) ( ) p - ( - ) - - s E r f R p R y q y V ar Var r f R p R y q y R y e were p a - bq s te expeted value of te spot pre. Te soluto to proble (3) a be easly alulated: y p q a σ R R p q a σ a σ A B q, (5) p 1 were A R ad B. Note tat A > 0 ad B > 0. Te suppler s a pre taer bot te spot ad otrat arets. Aordly, ts proft futo s ve by: ( ) were y s te quatty of output sold forward, quoted by te areter to te suppler, (4) p p F - y q y - v F, (6) q s te ut pre of otrated eletrty F s te suppler s atual eletrty output et of ts v s ts ostat aral (ad averae) ost. sales te reulated aret, ad I te forulato above, te suppler eeds to ow ow u t wll atually be requred to produe by te syste operator perod 1 order to fure out te aout of eery to sell forward. We ae te splfy assupto tat supplers are syetr ad so ea wll produe a quatty of eletrty equal to te foreast dead dvded by te uber of supplers. e Tus ( ) F Q - F for all, were R R F s te eletrty te suppler sold te reulated aret. Moreover, we a, wtout loss of eeralty, set aral osts to zero. Note tat, ve te defto of F, we ave F Q e 1 1 R ( ) F Q e R F, ad so F R. Te suppler s proble a ow be expressed ters of te ertaty equvalet easure: a ( ) ( ( ) ) ax p F - y q y - Var p F - y q y, (7) 8 A equvalet assupto would be tat e R as ea R ad varae 0. 1 1 1 7
were te suppler s oeffet of rs averso. Te soluto to ts proble s y q p a σ F F p a σ q a σ C D q, (8) p 1 were C F - ad D. It a be easly see tat D > 0. Te areter s a oopolst te otrat aret. It quotes ell pre to osuers ad a buy pre to supplers. It s rs eutral ad tus wats to axze ts profts, ve by te spread d q - q tes te quatty traded y. I our odel, all trades o trou areters, ad so y y s y. Se quatty deaded s equal to quatty suppled te 1 1 otrat aret, we ave 1 1 B B s a ad F 1 1 -, were A A R p 1, σ a A Bq C Dq å å, C å C F - å, D å D å, å F. 1 p 1 1 1 Te areter solves te follow axzato proble: ax( q - q ) y s. t. q - q ³ 0, s a 1 1 1 s a 1 1 (9) R 1 å R, Te proposto below follows fro te soluto to (9). Te proof a be foud te Appedx. Proposto 1.1: Te equlbru quattes ad pres a forward aret were (a) supplers ad osuers are pre taers, (b) te areter as oopoly power, () supplers ave te sae oeffet of rs averso, ad (d) osuers ave te sae oeffet of rs averso, are te follow: æ ö æ ö q p R, q p F -, è ø è ø æ R s öæa a ö R F d, y R - ad y F -, è øè ø (10) 1 s s were a a " 1, K, ad a a 1,, " K. Upo speto, we a edately see tat te forward pre of eery sold (by supplers) s lower ta te expeted spot pre, wereas te pre of eery bout (by 8
osuers) s er ta te expeted spot pre. Aordly, supplers sell forward less ta ter (et) produto ad osuers buy forward less ta ter osupto of eletrty. Rs-averse aets wat to ede aast rs. I our odel, tey do tat te forward aret, ad ay fator tat reases te rs (of be exposed to te spot aret) or aes te aet ore rs-averse reases ts dead for ed, affet te forward pre aordly. Terefore te follow results sould oe as o surprse: () Te forward pre pad by (to) a osuer (suppler) s er (lower) te ore rs averse t s. Ts aes sese beause a ore rs-averse aet asss ore value to less exposure to te spot aret. () Te forward pre pad by (to) a osuer (suppler) s er (lower) te larer te varae of te spot pre, se ts eas ore rs. () Te forward pre pad by (to) a osuer (suppler) reases (dereases) wt total atual osupto (produto) for a fxed uber of osuers (supplers). We averae osupto (produto) s er, ea osuer (suppler) dvdually as to trade ore eery te aret, ad ts reases ts rs of exposure. A ore terest result a be obtaed by lett, te uber of osuers, rease wle total dead for eletrty does ot ae. Ts orrespods to tuato were soe aptve osuers rate fro te reulated aret to te (free) otrat aret. Se does t ae, ad, by defto, F R, te pre of eery sold forward dereases after te rato taes plae. O te oter ad, te beavor of te pre of eery bout forward depeds o wat appes to te rato R. If t s larer after te rato, te q reases. If t s lower, dereases. Te latter s urprs result, se te areter as oopoly power te otrat aret. Te explaato s tat se averae osupto dereases, te averae osuer s exposed to less spot pre rs. As a osequee, te elastty of dead for otrats reases, for rs sar beoes less portat to te averae osuer. Let s ow tur to te areter. Frst ote tat spread d s strtly postve, ad, as expeted, reases wt te deree of rs averso of supplers, wt tat of osuers, ad wt te varae of te spot pre. Moreover, se alf of te syste s (et) eery s traded te otrat aret 9, te areter s proft s equal to æ R s öæa a öær ö æ s öæa a ö p d y R d 4. We tere s rato to te free è øè øè è øè ø aret, wat appes to ts proft depeds o te beavor of averae osupto. Te ext seto dsusses a odel were tere s ore ta oe areter. q 9 æ Te frst way to see ts s: F ö F F y º y F - F - 1 1 è ø æ s R ö R R F y º å y å R - R -. 1 1 è å å. Aoter way to see t s: 9
4. Cotrat aret wt ore ta oe areter Aord to Proposto 1.1, te oopolst areter obtas trtly postve spread ad, osequetly, aes postve proft trou ts operatos te forward aret. Ts sould ete oter frs to eter te aret as areters. Te stuato were tere are ay areters s te fous of ts seto. Tere are ow H detal areters ad tey play a Courot ae. Mareter s proft futo s ( ) p q - q y, were Mareter as to solve te follow proble, were y å y : ax y ( - ) y s te quatty of eery traded by areter. q q y s. t. y A - Bq C Dq H 1 (11) Ts proble s equvalet to æa - y y - C ö ax - y B D è ø were te ostrat as already bee plued to te objetve futo. (1) Proposto.1: Te equlbru quattes ad pres a forward aret were (a) eerators ad supplers are pre taers, (b) tere are ay areters wo play a Courot ae, () all eerators ave te sae oeffet of rs averso ad (d) all supplers ave te sae oeffet of rs averso, are ve by: æ öæ R ö æ öæ F ö q p, q p -, H 1 H 1 è ø è è ø è æ s öæa a ö r s R d F, y R, H 1 - (13) è øè ( H 1) F F æ H ö y F -, y ad y F ( H 1) H 1 H 1 è As far as ow tey deped o derees of rs averso ad varaes s oered, pres ad quattes bout ad sold forward ave slar propertes to tose tey featured te oopolst aret ase, so we wll ot oet o te. As a atter of fat, te results of Proposto.1 bol dow to tose of Proposto 1.1 we H 1. Te total aout of eletrty traded trou forward otrats s aa less ta te (et) eery avalable, but o loer exatly equal to alf of t. We tur our atteto to te effets of a larer uber of areters o te equlbru values of te varables. Frst, t s easy to see tat te pre of eery bout forward dereases wt te uber of areters. Tat s exatly wat a Courot odel sould yeld: Te ore areters tere are, te stroer te opetto betwee te, ad ts drves dow te pre tey are osuers. I te lt, tey a are o ore ta te expeted spot pre. 10
Aaloously, te pre of eery sold forward reases wt te uber of areters, aa as a osequee of te eaed opetto betwee areters. I te lt, te expeted spot pre s aeved. It oes as o surprse tat te spread ared by areters oes to zero as te uber of areters reases wtout boud. We a also see edately upo speto of te forulas for y ad y tat, as te uber of areters reases, te eery sold forward by uppler approaes ts (et) produto, wle te eery bout forward by a osuer approaes ts atual osupto. Ts s a trval osequee of te fat tat, se te pre pad by osuers dereases ad te pre reeved by supplers reases wt te uber of areters, eerators ad supplers are faed wt stroer etves to ede ter postos te otrat aret. Fally, we sould eto tat, as te uber of areters reases, te portfolos eld by dvdual areters sr sze ad te total aout of eery traded te otrat aret oves toward te avalable (et) eery te syste. Ts was expeted, se bot osuers ad supplers are forward trad alost all te eery tey eed or ave, respetvely. Ts dates tat te role played by te spot aret teds to ds due to reas opetto betwee areters. 5. Colusos I ts paper, we odeled te terato betwee areters, supplers, ad osuers a eletrty forward aret we tere are o bds allowed te spot aret. Ts s essetally ow te Brazla eletrty aret s set up, wt te spot pre be oe of te outputs of a dya prora alort wose objetve s to fd te optal balae betwee us water today ad stor t for future use. We frst obta results tat are stadard te lterature. Forward pres pad by osuers are reas ter deree of rs averso ad te varae of te spot pre, wle pres supplers sell ter eletrty forward for derease wt tose sae dators. We also sow tat a oopolst areter wll be able to are pres tat yeld a postve spread, ad tat t reases wt te rs averso of supplers ad osuers, as well as wt te varae of te spot pre. Oe of our ost portat otrbutos oes fro te aalyss of wat appes we te uber of osuers te otrat aret reases, w our odel orrespods to a searo were aptve osuers rate fro te reulated aret to te otrat (free) aret ad tere s o ae te total dead for eletrty. We pot out tat oe of te osequees s tat te pre of eery sold forward dereases, w, oter ts be equal, urts supplers. As for te effet o te pre of eery bout forward, t depeds o te beavor of averae osupto. If averae osupto s larer after te rato, te te pre osuers pay reases. If averae osupto s lower, te t dereases. Te seod possblty s a o-stadard result, se te areter as oopoly power te otrat aret. To uderstad t, ote tat we averae osupto dereases te averae osuer s exposed to less spot pre rs. Ts eas tat rs sar beoes less portat to, ad so areters fae reased opetto fro te spot aret, w dlutes ter aret power. Ts s sue tat, despte ts portae to ay eletrty arets, s ot addressed te lterature. For stae, Gree [1999], Allaz ad Vla [1993] ad Powell [1993] all odel te eerat setor as a duopoly. As for te dead sde of te otrat aret, Gree [1999] 11
assues buyers detere te aret-lear pre, Powell [1993] assues tey set quattes, ad Allaz ad Vla [1993] odels te as speulators. Aoter portat otrbuto of our paper s to sow tat te total aout of eery traded te otrat aret approaes te syste s avalable eery (et of reulated trades) as te uber of areters reases. Ts eas tat spot arets ay beoe less portat we tere s reased opetto betwee areters te otrat aret, ad s soet eletrty reulators sould ertaly reo wt. Fally, we also sow tat te pre of eery bout forward dereases wt te uber of areters. Te ore areters tere are, te stroer te opetto betwee te, ad ts drves dow te pre tey are osuers. I te lt, tey a are o ore ta te expeted spot pre. Aaloously, te pre of eery sold forward reases wt te uber of areters, ad equals te spot pre te lt. Tere are ay ways w our odel a be proved. We loo forward to te opportuty of vestat ssues su as te strate terato betwee free ad reulated (were te eletrty deaded by aptve osuers s traded) otrat arets, pre opetto betwee areters, ad oters. Appedx Proof of Proposto 1.1: Se A Bq C Dq -, we a rewrte te proble (9) as æa - C D ö ax - q - q C Dq B B è s. t. q - q ³ 0 ( ) (14) We wll frst solve te uostraed proble ad te sow tat te ostrat s satsfed at te optu. But frst let s sow tat te objetve futo s oave. Let æ A - C D ö ( A - C ) C T ( q ) º - q - q ( C Dq ) B B è ø B ( A - C ) D DC D q - q - ( q ) - Cq - D( q ), B B B Se D > 0, B > 0, we ave - q B B B T - D - D < 0. ( q ) B T ( A C ) D DC D - - q - C - Dq 1
Tus te frst order odto s bot eessary ad suffet for a axu. Te frst order odto for ts proble s ve by ( A - C ) D DC D - - q - C - Dq 0 B B B D AD - DC - BC Þ q Dq B B Ts equato a be solved to obta ad so A D - DC - BC A D - C ( D B ) q, (15) æ D ö D( D B ) B D B è A - C D æa D - C ( D B ) ö A - C A D - C ( D B ) q - - B B D( D B ) è B B( D B ) ( A - C )( D B ) - A D C ( D B ) A D AB - BC B( D B ) B( D B ) AD B( A - C ) B( D B ) Te odto q q s satsfed f A D - C ( D B ) A D B( A - C ) D( D B ) B( D B ) Se B > 0 ad D > 0, ts s equvalet to [ A D - C ( D B )] B [ A D B( A - C )] D Û - - < - A BD BCD CB A D A BD BCD A BD BCD A D CB Û > Now let s use te splfy assuptos tat supplers ave te sae oeffet of rs averso,.e. a a " 1, K,, ad tat osuers also ave te sae oeffet of rs p averso,.e. a a " 1, K,. Te A R, B, p C F -, D, ad so 0, (16) 13
æ p öæ öæ ö æ p öæ öæ ö - è øè øè ø è øè øè a s ø A BD CBD A D CB R F æ öæ ö æ öæ ö p p R F - è øè ø è øè a s ø ( R F æ öæ ö ép p ùæ öæ ö ) - a a a a a a è s øè s ø ê s s ú s ë û è øè s ø æ p öæ ö æ p öæ ö R F - è øè ø è øè a s ø æ ( R F) ö æ ö æ p öæ öæ ö a a è s ø è s ø è øè øè a s ø æp öæ öæ ö æ ö æp öæ ö - R è øè øè ø è ø è øè a s ø æ p F ö æ öæ ö - a è s ø èa s øè a s ø æ öæ ö æ ö æ ö ( R F) R F > 0 è øè ø è ø è ø w proves tat te restrto s satsfed at te optu. Let s o ba to odto (15). It a be rewrtte as q æ p öæ ö æ p öæ ö R - F - è øè ø è øè ø æ öæ ö è øè ø R p F F p p - - a a a a a æ ö æ ö è ø è ø ( R - F ) F p p - a a ( s ) ( a ) ( s ) æ öæ ö è øè ø ( R - F) æ ö p F æ ö - è ø è ø æ öæ ö æ öæ ö è øè ø ( s ) ( ) ( s ) ( s ) è øè ø 14
( ) æ R - F ö p - F æ ö è ø è ø Slarly, (16) a be expressed as q Se R to (5) to et: (17) æ p öæ ö æ öæ p p ö R R - F è øè è øè æ öæ ö è øè R p ( R - F) p p a a ( s ) ( a ) ( s ) a a ( s ) æ ö æ ö è è æ ö p æ ö R ( R - F) è è ø æ öæ ö æ öæ ö è øè è øè æ ö R - F p R è æ ö è ø (18) F, we a wrte Slarly, plu (17) to (8) to obta a q p R æ s ö è ad q p - F æ ö è ø. Now plu (18) p æ 1 öé æa s öù y R - êp R è ø ê ê ë è øú û p p R R - - R R - p æ 1 öé æa s öù y F - êp - F è ø ê ê ë è øú û p p F F - - F F - 15
Fally, te spread a be alulated as æ ö æ ö d q - q p R p F - è ø è ø æ s öæra Fa ö è øè ø w, ve tat R F, bols dow to d æ öæ ö æ öæ ö R s a a F s a a è øè ø è øè ø. Proof of Proposto.1: Proble (1) a be rewrtte as æa D BC ( D B ) y ö ax - y y BD è Te frst order odto for ts proble s: ( D B ) AD BC - ( D B ) y - y 0 BD BD Þ - ( D B ) y A D BC - ( D B ) y - ( D B ) y 0 ( ) Þ y ( D B ) A D BC - ( D B ) y y A D BC j ¹ Þ y - ( D B ) j j ¹ å j j ¹ j å å (19) (0) Se te areters are syetr, y y ( H - 1) y ad å y j H H j ¹. Terefore A D BC ( H - 1) y y - ( D B ) H H( AD BC ) H( H - 1) y Þ y - ( D B ) H ( H - 1) H( A D BC ) Þ y y ( D B ) éh ( AD BC ) ùé ù H( AD BC ) Þ y úê ê ( D B ) úêh 1 ú ë ûë û ( H 1)( D B ) AD BC ad y ( H 1)( D B ) Now we plu te forulas for A, B, C ad D to (1) to et (1) 16
y were we used te fat tat R æ p ö æ p ö R F - è ø è ø æ ö ( H 1) è R p F p - a a ( s ) a a ( s ) æ ö ( H 1) è æ ö F a a s s è F æ H ö ad y F, æ ö ( H 1) H 1 H 1 è è F. Te ext step s to alulate te pres: () ad q p æ H ö R - F A y H 1 - è B R æ H öæ ö æa s öæ 1 ö p - R p R H 1 H 1 è øè è è ø (3) q æ H ö p F F H 1 - y - C è ø D p æ 1 ö - F H 1 è ø æa s öæ 1 ö p - F H 1 è øè Te spread a ow be easly alulated: æ ö æ ö æ 1 ö æ 1 ö d q - q p F p F - H 1 H 1 è ø è ø è ø è ø æ s öæa a ö F è H 1 øè ø (4) 17
Quattes otrated by supplers ad osuers a be obtaed by us (5), (8), (3) ad (4): ad p 1 æ a 1 y F p F æ s öæ ö - - H 1 è è ø è ø p p F F F - - F - ( H 1) ( H 1) p 1 æ a 1 y R p R æ s öæ ö - a a s s H 1 è è ø è ø p p R R R - - R - ( H 1) ( H 1) Fally, we a e our alulatos as follows: y æ F ö F H F æ ö F - - F ( H 1) H 1 H 1 è è y æ R ö R æ H ö R - R R ( H 1) - H 1 H 1 è è F æ H ö F H 1 H 1 è å å 1 1 å å 1 1 H H å y å 1 1 (5) (6) 18
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