Cournot Competition, Financial Option markets and Efficiency

Transcription

1 ournot ompetton nancal Opton markets and Effcency by Bert WILLEMS ublc Economcs enter for Economc Studes Dscussons aper Seres (DS) November 2004

2 ournot ompetton nancal Opton markets and Effcency by Bert WILLEMS Econometrcs enter for Economc Studes Dscussons aper Seres (DS) November 2004

3 ournot ompetton nancal Opton Markets and Effcency Bert Wllems ABSTRAT: Allaz and Vla (1993) show that the exstence of futures markets ncreases the effcency of markets n a ournot settng. Ths paper looks at the effcency effect of fnancal optons n a smlar framework. It shows that also the exstence of fnancal optons makes markets more effcent; though to a smaller extent than futures. Ths s partcularly relevant for markets wth market power and costly storage lke the electrcty market. Keywords: utures markets Optons markets ournot Market power Electrcty Arbtrage JEL: 72 D43 G13 L13 L50 L94 Introducton Several countres recently decded to lberalze ther electrcty markets and to organze competton n electrcty generaton. They assumed that economes of scale and entry barrers n the generaton sector were suffcently small to make competton vable. In practce the generaton market s not always very compettve. Generators often succeed n drvng up prces sgnfcantly above compettve levels. Ths also happens n markets wth low levels of market concentraton. The two man reasons for market power n the electrcty sector are the non-storablty of electrcal energy and the low demand elastcty 1. Both market characterstcs make that unlateral wthholdng of producton output can be hghly proftable for frms. Especally n perods of peak demand generators are often producng close to the techncal maxmum output of ther generaton plants whch nevtably leads to steep supply functons. The resdual demand functons faced by the generators are therefore often steep as well. Generaton market power becomes even more pronounced because techncal constrants n the transmsson network do not allow generators to effectvely compete wth each other. urthermore generaton plants are captal ntensve and nvestments requre a long lead tme. rces above margnal producton costs have been shown to exst n several markets. Borensten et al. (2002) fnd a sgnfcant departure from compettve prcng durng hgh demand summer months n the alfornan electrcty market. Also Joskow and Kahn (2002) show that hgh prces n summer 2000 reflected n part the exercse of market power by supplers. Wolak and atrck (2001) argue that the two largest generators n the Katholeke Unverstet Leuven and Unversty of alforna Energy Insttute. The author would lke to thank Severn Borensten Jm Bushnell Hans Degryse Gudo epermans Stef roost Ana Rvas atrck Van ayseele rank Verboven and rank Wolak and the semnar partcpants n Leuven and Berkeley for ther valuable comments. 1 Ths effect s even worse because changes n the real-tme wholesales prce are not passed through to consumers. Instead consumers pay a flat rate whch s changed only nfrequently and does not reflect the scarcty of electrcty n certan perods. Hence consumers do not have the rght ncentves to reduce ther demand when the market s tght.

4 2 early England and Wales market were able to obtan prces for ther output substantally above ther margnal cost of generaton. Wolfram (1999) shows that the Brtsh electrcty prces were above the perfectly compettve prces. omparng dfferent electrcty markets n the US Bushnell et al. (2004) show that alforna had a relatvely unconcentrated generaton market 2 but that the lack of long term contracts led to hgh prce-cost margns n the summer of Wth long term contracts generators sell part of ther electrcty ex-ante at a locked-n prce. As a result generators wll behave more compettvely n the spot market. 3 The ntuton s that of the durable goods monopolst n oase s conecture. 4 See gure 1. Graph A shows the proft maxmzng prce p M for a monopolst who sells only n the spot market has producton costs () q and faces an nverse demand functon pq (). The monopolst wll set a prce such that hs margnal revenue s equal to hs margnal costs. Graph B shows the same stuaton for a monopolst who sgned long term contracts for k unts of electrcty. In the spot market (the second stage) k unts wll therefore dsappear both at the demand M and the supply sde. The proft maxmzng prce s equal to p and s lower than p M. In the frst stage the contractng stage consumers wll take nto account that the prce n the spot market wll be equal to p M. They wll only buy long run contracts for electrcty at the prce p M as they would lose money otherwse. Hence the M monopolst wll receve the prce p n the producton as wells as n the contractng stage. The fact that the prce should be the same n the contractng and the producton stage s called the perfect arbtrage condton. M p p Graph A pq () Graph B p pq ( + k) c'( q ) M p c'( q + k) q q k gure 1 Effect of an ex ante contract on the behavor of a monopolst. The study of Bushnell et al. hghlghts the mportance of long term contracts n electrcty markets. There s however no consensus on the role of long term contracts n electrcty markets. Hstorcally polcy makers have been opposed to long term contractng n electrcty markets. They feared that long term contracts between ncumbent generators and retalers mght slow down entry n the generaton market. They also assumed that long term contractng would decrease the transparency and the lqudty of the spot markets. 5 Illqud spot markets would lead to neffcent real tme producton decsons and would also make 2 The Herfndahl-Hrschman Index (HHI) for alforna New England and JM (ennsylvana New Jersey and Maryland) were and The most extreme form of long term contractng between a retaler and a generaton frm s of course a merger. 4 See oase (1972). 5 There s no theoretcal evdence that forbddng long-term contractng wll ncrease lqudty on spot markets. Wth lqud long term contracts nformaton becomes publc sooner than wth spot markets. Ths mght ncrease lqudty. Long term contracts make spot markets also more compettve; prces wll therefore be closer to compettve levels and less prone to manpulaton.

5 3 entry more dffcult. A small entrant wll have to rely on the spot market to balance the dfference between the energy sold and the energy produced. urrently polcy makers are changng ther mnd and are becomng more favorable towards long term contracts. They hope that long term contracts wll ease entry n the generaton market by reducng the rsk for entrants and wll reduce market power n the spot market. Long term contracts wll also help retalers who sell electrcty at fxed regulated prces to hedge ther prce rsks. Nowadays there s a debate whether they should mpose the usage of long term contracts or whether generators and retalers wll sgn the rght amount of long term contracts on ther own. See for example ret and abra (2004). If they decde to ntervene they need to make several choces. A frst choce s on whom to mpose the oblgatons: on the generators or on the retalers? Generators can be oblged to sell a part of ther producton capacty and retalers can be oblged to contract a fracton of ther estmated demand ex-ante. An oblgaton to retalers mght help entry n generaton (long run effcency) but could place retalers n a bad barganng poston n the contract market when there s no entry n the short run. A second choce s how the quanttes of the contract should be specfed. ontracts could specfy a constant quantty over tme a predetermned load-shape 6 a specfc fracton of system demand or a quantty whch depends on the spot prce. Two standard contracts whch we consder n ths paper are the futures contract and the call opton contract. In a futures contract the quantty whch needs to be delvered s fxed. In a call opton ths quantty depends on the spot prce: t ncreases wth the spot prce. Optons mght have some advantages compared wth futures. Optons allow generators and retalers to hedge quantty rsks whle futures can only be used to hedge prce rsks. 7 Gven that electrcty cannot be stored very easly quantty rsks are very mportant n the electrcty market and optons play therefore an mportant role. Market power s most pronounced durng perods of peak demand and s characterzed by hgh spot prces. Retalers mght sgn opton contracts to counter the market power of generators durng these perods 8. The electrcty sector s characterzed by a lot of mssng markets. Often optons are used to correct these problems. A thrd choce s whether these oblgatons should take the form of fnancal contracts where the seller and the buyer transfer a sum of money at the end of the contract or of physcal contracts where a specfc generaton plant s assocated wth a specfc contract. Appendx II comes back to the dfference between fnancal and physcal optons. 6 A load-shape specfes a certan level of demand as a functon of tme. As a lot of retal consumers do not have real-tme meters these load-shapes are often used for accountng purposes. 7 To llustrate ths consder a retaler who has the oblgaton to serve a number of customers at a fxed retal prce. Wthout a long term contract the retaler buys hs electrcty on the wholesales spot market and faces two types of rsks. He faces a prce rsk because he does not know whch prce he wll pay on the wholesales spot market and a quantty rsk because he does not know how much he wll need to buy as demand depends on for example weather factors. Wthout quantty rsk the retaler would be able to hedge hs entre rsk by sgnng futures contracts for the quantty he s expected to delver. Wth quantty rsk the retaler needs also to buy some call optons whch he wll use when demand s hgher than expected. 8 Also the regulator can use optons to am ts regulaton more precsely at perods of hgh demand mnmzng ts nterventon n the market.

6 4 Ths paper In ths paper we wll look at the strategc effects of fnancal call optons n a ournot game. We assume that there are only two markets: a fnancal call opton market wth an exogenously determned strke prce and a spot market. In our set-up frms decde themselves how many optons they sell. In the model there s no uncertanty so hedgng s not an ssue. The number of generators s assumed to be fxed. Hence we do not look at the entry decson of new generaton frms. The paper s an extenson of Allaz and Vla (1993). They showed that n a ournot game frms have a strategc reason to sell futures contracts because futures contracts serve as a commtment devce for the frms to obtan a larger market share n the spot market. Sellng futures leads to a prsoners dlemma type of problem. All frms sell futures and as a result the spot prce wll decrease. We wll use a smlar framework as Allaz and Vla to analyze fnancal call optons nstead of futures contracts. Several papers have crtczed the assumptons of the Allaz and Vla paper. As we make the same assumptons these crtcsms are also applcable to the current paper. Allaz and Vla assume ournot competton n the spot market. The actons of the players are strategc substtutes. Mahenc and Salane (2004) show that n a Bertrand game players wll take opposte postons n the spot market: they buy ther own output nstead of sellng t. The result of Allaz and Vla depends also on the assumpton that the number of futures contracts a frm sgns s observable by all frms. Hughes and Kao (1997) show that f the contract poston s not observed by other players then frms have no longer an ncentve to sell futures contracts. 9 A thrd key assumpton n the model s perfect arbtrage between the contractng stage and the spot market. In practce arbtrage n the electrcty market s however far from perfect. The precse reason for mperfect arbtrage and how one should model ts effects remans for further nvestgaton. 10 Le oq (2003) looks whether long term contracts make tact colluson between the generators more lkely. As generators nteract whch each other on an hourly bass tact colluson s certanly an ssue that has to be addressed. She shows that some long term contracts mght stmulate tact colluson leadng to hgher prces. nally Thlle (2003) showed that the results of Allaz and Vla are weakened when storage s possble. However f there would exst storage n the electrcty market market power would be less of an ssue n the frst place. Relaton wth hao and Wlson The paper s closely related to recent work of hao and Wlson (2004). They argued that generators should be oblged to sell physcal call optons to retalers. They see several reasons for ths: In the long run electrcty markets are contestable and thus more compettve. hyscal call optons mght have better strategc effects than futures contracts. 11 The regulaton of market power mght be easer wth physcal call optons than wth futures. And physcal delvery makes sure that generaton s effectvely bult. 9 They show however that under uncertanty and rsk averson there are agan strategc reasons to buy or sell futures contracts. The behavor of the frms wll depend on the type of uncertanty that the frms face. If demand levels are uncertan generators wll sell futures contracts. If cost levels are uncertan generators wll buy futures contracts. 10 Arbtrage mght be hndered by lack of nformaton perceved or real regulatory restrctons on arbtrage and entry barrers n the arbtrage market. 11 See also Appendx II.

7 5 hao and Wlson dscuss several nterestng deas but make several modelng assumptons whch make a straghtforward comparson of futures and call optons dffcult. They assume perfect regulaton of the number of optons that generators have to sell and free entry n the contractng stage. However wth these two assumptons a lot of contract types wll gve the perfect compettve outcome. A comparson of futures contracts and call optons s not possble. In our paper we assume a fxed number of frms and that generators decde themselves about the number of optons they sell. A second problem wth ther paper s that t makes non-standard assumptons on the type of optons and on the type of competton whch makes a comparson wth the Allaz and Vla model very hard. hao and Wlson assume that generators bd lnear supply functons n the producton stage and sell bundles of physcal optons contracts n the contractng stage. Each bundle conssts of one opton contract of each possble strke prce. Gven these assumptons the ntuton of the Allaz and Vla model does no longer have to be vald. 12 We assume ournot competton n the contractng stage and generators wll sell fnancal call optons wth only one exogenously determned strke prce. We thnk that a sngle opton s more realstc than a lnear bundle of optons. It s consstent wth what we see n some electrcty markets where only a small number of call optons s actvely traded. The reason why we do not observe a larger number of opton contracts s that these markets would not be lqud enough and that transacton costs would be too hgh. 13 ournot game wth futures contracts Ths secton explans the standard ournot game and the ournot game wth futures contracts (.e. the Allaz and Vla model). It presents the set up of the model and the defnton of the man varables. The next secton then contnues wth the ournot game wth fnancal optons. Our paper consders an olgopoly wth two frms {12}. rm produces q unts at a producton cost ( q ) ' wth 0 0. Total producton of both frms s equal to q + q and the spot prce = p( q +q ). '' 1 2 Standard ournot game We start wth the standard ournot game wthout futures contracts for whch we wll use the superscrpt. The proft of a frm s equal to ts revenue mnus producton costs: π ( q ; q ) = q ( q ) (1) 12 An example: In the Allaz and Vlla model generators play ournot n the second stage. Ther actons are therefore strategc substtutes. In the hao and Wlson model they use lnear supply functons. Dependng on the slope of the supply functons and the demand functon ther actons mght be strategc substtutes or complements. 13 Wth rsk-averse market partcpants and no transacton costs one would expect a complete set of markets. Ths would mean that there s a market for each possble contngency. Hence we would see an opton market for each possble strke prce.e. a contnuum of optons markets. Wth transacton costs market partcpants trade off the gans of trade and the transacton costs they would ncur. If two fnancal contracts are relatvely good substtutes then often only one wll be traded as most gans of trade can be made n that way whle a second market would greatly ncrease transacton costs. See Suenaga and Wllams (2004).

8 6 In a ournot game frm maxmzes ts proft by settng ts producton quantty q takng nto account that the prce depends on the ont producton of the frms. = p (q + q ). All frms set ther producton level q smultaneously. The Nash equlbrum of ths game s the ntersecton of the reacton functons q ( q ) of the players = q q ( q ) argmax π ( q q ) or later reference the equlbrum producton quanttes and spot prce wll be denoted by q q and eq eq eq = p( q + q ). (2) eq eq To llustrate our paper we wll use a numercal example n whch the two frms have quadratc cost functons 2 2 ( q ) =.1q and ( q ) =.2 q and the nverse demand functon s equal to q () = 1 q. gure presents the ournot equlbrum of ths game graphcally. It shows the strategy space of the generators. The curved lnes represent the so-proft lnes of each frm. π ( q q ) = (1 q q ) q ( q ) = cte They are the ndfference curves of each frm. The dashed lnes are the contour lnes for frm 2 and the sold lnes for frm 1. The reacton functons of frm 1 and 2 cross ther so-proft lnes where they are horzontal and vertcal respectvely. The Nash equlbrum s the ntersecton of the two reactons functons. To smplfy the notaton we use vector notaton: q = ( ). (3) ( 2) = 1 q q q ournot - Equlbrum q eq ( 1) = 2 q q q gure 2 Standard ournot equlbrum ournot game wth futures contracts If there are futures contracts then we need to model the game wth two stages: a contractng stage and a producton stage. gure 3 shows the tmng of the game. ontractng Stage Generators sell k utures at a prce Generators learn each other s contractng poston roducton Stage Generators sell q electrcty on spot market gure 3 Tmng of the ournot game wth futures TIME

9 7 In the frst stage the contractng stage generators sell futures contracts to retalers. The generators sell the contracts n a ournot fashon:.e. they decde smultaneously about the number of contracts k they sell takng the number of contracts of the compettor as gven. Each futures contract s a two-sded nsurance contract whch nsures the prce of one unt of electrcty. If the spot prce s above the futures prce then the generator wll refund the retaler the dfference of the spot prce and the futures prce. If the spot prce s below the futures prce then the retaler wll pay the generator the dfference between the futures prce and the spot prce. The total payment of generator s thus k ( ). 14 After the frst stage and before the second stage each frm learns the contract poston of the other frms. In gure 3 ths happens at tme = 1.5. There s therefore perfect nformaton at the begnnng of the second stage. In the second stage the producton stage the frms smultaneously set ther producton level q n a ournot fashon. Each frm wll take ts own and ts compettor s contractng poston as gven. rm s proft s equal to revenue n the spot market mnus producton costs and payments related to the futures contracts. The superscrpt denotes the game wth futures contracts. π ( q ; q k ) = q ( q ) k ( ) = ( q + q ) We wll solve the game by backward nducton and derve frst the Nash Equlbrum n the second stage of the game as a functon of the number of futures sold n the frst stage q e q ( k ). Ths s schematcally presented n gure 4. After dervng the second stage equlbrum we wll derve the reduced proft functon of the frst stage and solve the equlbrum of the frst stage k. (4) Stage 1 ontractng Stage Reduced ay Off ( q ) q wth erfect arbtrage = = p( q + q ) q = q ( k k ) Stage 2 roducton Stage ay Off q ( q) k( ) wth = p( q + q ) Nash Equlbrum k = k Backward Inducton Nash Equlbrum q = q ( k k ) gure 4 ournot game wth futures contracts and backward nducton Second Stage rm maxmzes ts proft by settng ts producton q takng nto account the number of futures contracts whch tself and ts compettor own. The best response functons of the second stage are q q ( q k ) = argmax π ( q q k ) Note that frm s second stage reacton functon does not depend on the number of futures that frm owns as also ts proft functon (4) does not depend on t. (5) 14 Ths dscusson explans the futures contract as a fnancal nsurance contract. An alternatve explanaton consders the futures contract as a physcal contract where the generator sells k unts n the frst stage at a prce q k unts n the second stage at a prce and produces q. Generator s proft becomes k + ( q k ) ( q ) whch s equvalent wth equaton (4). It can be shown that fnancal and physcal futures are equvalent but that ths s not the case for fnancal and physcal optons.

10 8 The fact that frm owns futures contracts changes ts ncentves to produce n the second stage. rm needs to refund buyers of the futures contract for hgh strke prces. It has therefore less nterest n hgh spot prces and produces more n the second stage of the game. q ( q k) 0 k Hence ownng futures contracts makes a frm more aggressve n the second stage.e. t produces more and ts reacton functon moves outwards. Ths effect s based on exactly the same ntuton as oase s conecture as explaned n gure 1. gure 5 shows ths effect n the numercal example. The sold and the dashed curved lnes are the so-proft lnes of frm 1 when k unts were sold n the contractng stage. ( π ( q q k ) = constant ) and when no futures were sold ( π 1 ( q1 q2) = constant ) respectvely. The best reacton functon of frm 1 q1 = ( 2 k1) crosses ts so-proft lnes where they are horzontal. It shfts to the rght when more futures contracts are sold. (6) q orward - Reacton functon 1 ( 2 1) = 1 q q k q k 1 q eq q ( k ) q ( q k ) = q 2 2 k gure 5 Second stage equlbrum n the ournot game wth futures contracts The second stage Nash equlbrum s defned by the ntersecton of the best response functons. Gven that k k futures contracts are sold n the frst stage the equlbrum quanttes are q ( k k ) and the equlbrum prce s p ( k k ). gure 5 shows further how the reactons functons of both frms shft out wth larger numbers of forwards. The ntersecton of the reacton functons s the second Nash equlbrum. Note that the equlbrum quantty that s produced by frm depends on the number of forwards sold by both frms. rst Stage In the frst stage the frms maxmze ther proft (4) q ( q ) k ( ) takng nto account that q s determned by the second stage behavor of the frms q q ( k k ) and that the prce s determned by the demand functon = p ( q1 + q2). We now need to defne how the futures prce k ( ) depends on the number of futures sold by the generators.e. the nverse demand functon for futures n the contractng stage. (7) = Allaz and Vla assume perfect arbtrage between the contractng and the producton stage. Ths means that there s no proft to be made by arbtragng between the spot market and the futures market.e.

11 9 = k ( k) p ( k k) (8) Arbtrageurs are not modeled n the Allaz and Vla paper and they wll not be modeled n ths paper ether. uture work could look at alternatve assumptons for the demand functon for futures. Note that (8) mples that arbtrageurs correctly antcpate the strategc effects of the futures contracts on the spot prce. Defne the frst stage reduced pay-off functon of generator as Π ( k k). Ths s the proft frm wll obtan when the players sell k futures contracts n the frst stage and play ournot n the second stage: Π ( k k ) = q ( q ) k ( ) where q and are determned by the followng equatons: q = q ( k k ) = p( q q ) = The frst equaton of (10) s the second stage equlbrum condton the second equaton s the nverse demand functon for electrcty n the spot market and the thrd equaton s the arbtrage condton. Note that gven perfect arbtrage a frm has no pecunary reason to buy or sell futures contracts. There s only a strategc reason to sell futures. Ths can easly be shown by substtutng the perfect arbtrage condton n the obectve functon of the frm. The obectve functon of the frm depends only ndrectly on the number of futures sold. By sellng more futures n the frst stage a generator can change the second stage equlbrum. gure 6 shows ths for frm 1. In gure 6A frm 1 does not sell futures; whle n gure 6B t sells k futures. By sellng futures total producton n the second stage s ncreased leadng to a lower prce. Ths nfluences frm 1 s proft negatvely. However sellng futures ncreases the market share of frm 1 whch ncreases proft. Graph a orward - Equlbrum Graph b orward - Equlbrum (9) (10) k q1 1( q1) = fxed q ( q ) eq q q ( k q) gure 6 Impact of sellng more futures by frm 1 on the second stage equlbrum. At the optmal number of futures k ( k ) both effects are balanced. Ths trade-off defnes the frst stage reacton functons of both frms ( gure 7). k ( k ) = arg max Π ( k k ) k The equlbrum n the frst stage s determned by the ntersecton of the frst stage reacton functons. The equlbrum number of futures contracts are k and k. (11)

12 10 k 2 k 1 k2 ( ) k 2 k 2 k1 ( ) k 1 eq k 1 gure 7 Equlbrum n the frst stage of the ournot game wth futures contracts gure 8 shows the second stage equlbrum when the generators sell the equlbrum futures quanttes k = ( k k ). In the equlbrum pont both reacton functons are tangent to the so-proft lnes of ther compettor. or further reference we defne the Allaz and Vla spot prce as: AV eq = p ( k k ) (12) orward - Equlbrum 1 ( 2 1 ) = 1 q q k q k 1 q eq ( k ) eq ( q) q ( q k ) = q ournot game wth optons gure 8 Allaz and Vla equlbrum Ths secton dscusses the ournot game wth fnancal optons. We wll use the superscrpt O to denote ths game. As already mentoned above we assume that only fnancal call optons wth an exogenously fxed strke prce S are traded. The call opton s a one-sded nsurance contract whch nsures retalers aganst prce ncreases above the strke prce S. If the spot prce s above the strke prce then the generator wll refund the retaler the dfference between the spot prce and the strke prce: S. When the spot prce s below strke prce then there s no payment. In short the generator pays the retaler the amount V( ) wth V ( ) = max{ S0} The ournot game wth fnancal optons wll be modeled smlarly to the ournot game wth utures contracts. In the frst stage the contractng stage frms sell fnancal call optons n a ournot fashon: rm sells k optons at a prce. After the frst stage each frm learns about the contract poston of the other frms. In the second stage the producton stage the frms smultaneously decde about ther producton level n the spot market. (13)

13 11 Second Stage : reacton functons rm s proft s equal to the sum of ts proft n the physcal market (revenue n the spot market mnus producton costs) and ts proft n the fnancal market: O π ( q ; q k ) = q ( q ) + k ( V( )) wth (14) = ( q + q ) The best response functon of frm maxmzes proft (14). O O q q ( q ; k ) = argmax π ( q q k ) (15) gure 9 shows the reacton functons of the standard ournot game (graph A) the second stage of the ournot game wth futures (graph B) and the second stage of the ournot game wth optons contracts (graph ). The - 45 degrees downward slopng lne n graph s the set of producton quanttes at whch the spot prce s equal to the strke prce ( pq ( + q ) = S). or producton quanttes above ths lne the spot prce s below the strke prce pq ( + q ) < S behaves as a standard Allaz and Vla futures contract. The best response functon of the frm s the best response functon of a frm who sold k futures contracts q ( q ; k). Ths s the South-West corner n gure 9 gure 9 ( pq ( + q ) < S).. or producton quanttes below ths lne the spot prce s above the strke prce pq ( + q ) > S. We wll now descrbe the best response functons of the generators. If the spot prce s above the strke prce n whch case the opton s sad to be n-the-money the opton ( pq ( + q ) > S). If the spot prce s below the strke prce n whch case the opton s sad to be out-the-money the opton has no strategc effect. The decson of frm s not nfluenced by the number of optons that t sold. Its best response functon s equal to q ( q ) the standard ournot reacton functon. Ths s the north-east corner n If the spot prce s equal to the strke prce a generator wll sell a quantty between the optmal quantty wth futures and the standard ournot quantty. ournot a b utures c Opton q O 2 q ( ) q ( q k) q k q ( q ) q ( + q) = S q ( + q) < S q ( 1+ q2) > S k1 k1 gure 9 Reacton functon of the standard ournot game the ournot game wth futures and the ournot game wth opton contracts. ormally the best response functon for a frm that sold k opton contracts s descrbed by S q ( q) I q ( q) + q > q O q ( q k) = q ( q k) I q ( q k) + q < q S q q otherwse S where q = p 1 ( S) s the demand level at whch the spot prce s equal to the strke prce. Equaton (16) descrbes the three parts of the reacton functon n gure 9. S (16)

14 12 Second stage: Nash Equlbrum The Nash equlbrum n the second stage of the game s defned by the ntersecton of the reacton functons. If the strke prce s hgher than the ournot prce eq p < S then ndependently of the number of optons sold n the frst stage the second stage equlbrum s the standard ournot equlbrum. Oeq eq q ( k k ) = q The optons are out the money and have no strategc effect. Ths s shown graphcally n gure 10 whch shows that the number of optons sold by frm 1 has no effect on the second stage equlbrum. a b c k 1 k1 k1 (17) k 2 k2 k2 q eq q eq q eq pq ( + q ) = S pq ( + q ) = S pq ( + q ) = S gure 10 Impact of optons sold by frm 1 on the second stage equlbrum Hgh Strke prce. If the strke prce s below the ournot prce eq p > S then the second stage equlbrum q Oe q ( k ) depends on the number of optons sold n the frst stage. gure 11 shows the mpact of the number of optons sold by generator 1 on the second stage equlbrum. a b c k1 k1 k1 AS ( k ) AS ( k ) k2 k2 k2 q eq q eq q eq pq ( + q ) = S pq ( + q ) = S pq ( + q ) = S gure 11 Impact of optons sold by frm 2 on the second stage equlbrum Low Strke prce. eq If a small number of optons s sold ( ( k k ) > S ) then the optons wll be n-the-money n the second stage and have the same strategc effect as futures contracts. See gure 11A. The second stage equlbrum wth optons s the same as wth futures contracts: Oeq q ( k) = q ( k) If a large number of optons s sold n the frst stage ( ( k k ) < S ) then generators wll ncrease ther output n the spot market untl the prce drops to the strke prce of the opton. None of the frms wll ncrease ther output further as the opton s out-the-money. There s a set of equlbrums AS ( k ) n the second stage of the game Oeq q ( k) AS ( k) each wth a spot prce = S and where the output of both frms s n between the reacton functon of the standard ournot game and the ournot game wth futures. ormally AS ( k ) s descrbed as follows: (18) (19)

15 13 q ( 1 q2) S + = AS ( k) = q q1 ( q2) q1 q1 ( q2 k1) (20) q2 ( q1) q2 q2 ( q1 k2) If a lot of optons are sold we wll say that the opton market s flooded. See gure 11B and gure 11. gure 12 shows the second stage equlbrum as a functon of the number of optons sold by both frms and of the strke prce. In gure 12A the strke prce s hgh and the generators play the standard ournot equlbrum. In gure 12B the strke prce s low and the equlbrum type depends on the number of optons sold n the frst stage. k 2 a S p eq > b S < p eq k 2 LOODING Oeq q ( k) AS ( k) Oeq eq q ( k) = q p ( k) = S k 1 NO LOODING Oeq q ( k) = q ( k) k 1 gure 12 Equlbrums n the second stage as a functon of the number of optons sold n the frst stage k and the spot prce S. rst Stage : roft As before we assume that there s perfect arbtrage between the contractng stage and the producton stage; the prce of the opton s therefore equal to the pay-out of the opton: = V( ) (21) Gven arbtrage the reduced frst-stage proft functon of the frm smplfes to O Π ( k k ) = q ( q ) wth = p( q + q ) (22) where the quanttes are determned by the second stage Nash equlbrum. The proft of a frm depends only ndrectly on the number of optons t sells through the dependences of q and q on the number of opton contracts sold. rst Stage: Equlbrum The fact that there are several Nash equlbrums n the second stage f a lot of optons are sold and the strke prce s low (See gure 12B) complcates the soluton of the frst stage. As there s no obvous focal pont n the second stage t s hard to predct how the second stage behavor of the players depends on ther frst stage commtments. In partcular frms can use one second stage equlbrum as a punshment equlbrum for msbehavor of a frm n the frst stage and play another equlbrum when both frms behave. unshment strateges are credble as they are Nash equlbrums n the second stage of the game. 15 In order to narrow down the number of equlbrums we assume that generators co-ordnate on the equlbrum wth the hghest prce. We thnk ths s a vald assumpton because of the followng reasons: 15 Appendx III shows the large set of equlbrums of the game.

16 14 rst the low prce equlbrums rely on the fact that both generators extensvely flood the market n the frst stage of the game.e. the number of optons they sell by far exceeds ther actual producton n the second stage. Ths type of behavor s very rsky. It s however a Nash Equlbrum because the generators perfectly coordnate ther producton n the second stage on one of the nfnte number of equlbrums and make sure that the spot prce s exactly equal to the strke prce. If the generators do not succeed n co-ordnatng on a Nash Equlbrum n the second stage they mght end up losng a lot of money. urthermore f we add some uncertanty to the model ths extensve floodng becomes even more rsky. Second as generators play the game regularly t s lkely that generators wll learn how to play the equlbrum wth the hghest prce and a lower rsk. Before we derve and prove the equlbrum of the game we frst dscuss the results. gure 13 shows the equlbrum spot prce as functon of the strke prce of the fnancal opton. If the strke prce s above the ournot prce then optons are never n-the-money and they have no strategc effect. We obtan the ournot outcome. (Segment A n gure 13) If the strke prce s below the prce of the equlbrum that would preval wth futures (cfr. Allaz and Vla) then the generators co-ordnate on the Allaz and Vla equlbrum. The optons are n-the-money and have the same strategc effect as futures. (Segment B n gure 13) If the strke prce s between the Allaz and Vla and the standard ournot prce then the frms wll make sure that the spot prce s equal to the strke prce. (Segment n gure 13) eq A AV B 45 AV eq S gure 13 rce n the equlbrum as a functon of the strke prce. gure 13 clearly shows that the prce wth futures s (weakly) lower than wth optons. Optons have the same effect as futures when ther strke prce s low. In the appendx we wll prove that gure 13 represents ndeed the equlbrums of the game. The proof conssts of the followng four theorems: Theorem 1: If the strke prce s above the standard ournot prce (S > p ) then the ournot equlbrum s the equlbrum of the game. Theorem 2: If there exsts an equlbrum wth a spot prce above the strke prce then t has to be the Allaz and Vla equlbrum. Theorem 3: If the strke prce s below the Allaz and Vla prce eq then the Allaz and Vla equlbrum s an equlbrum. Theorem 4: If the strke prce s between the Allaz and Vla prce and the standard ournot prce then floodng s an equlbrum. The spot prce becomes equal to the strke prce. AV

17 15 Theorem 1 defnes the equlbrum prce when the strke prce s above the ournot prce. Theorem 2 and 3 can be used to show that for a strke prce below the Allaz and Vla prce the Allaz and Vla equlbrum s an equlbrum (Theorem 3) and that there are no other equlbrums whch would gve a hgher spot prce. (Theorem 2) Theorem 2 and 4 can be used to show that for ntermedate prces the generators wll flood the market and set the spot prce equal to the strke prce (Theorem 4) and that there are no other equlbrums whch would gve a hgher spot prce. (Theorem 2) oncluson Ths paper dscusses the effcency effects of optons n a ournot olgopoly extendng the work of Allaz and Vla. Instead of lookng at futures contracts t looks at fnancal call optons. It s assumed that only optons wth one specfc strke prce are traded. Ths strke prce s exogenous n the model. Optons mght be better than futures because they allow the players to hedge quantty rsk they allow more precse regulaton of market power and they solve some of the problems whch are assocated wth the mssng markets n the electrcty market. Ths paper does not look at these effects and concentrates on the strategc effects of optons when generators decde endogenously on the number of optons. We show that optons make markets more compettve but to a smaller extent than futures. The precse effects depend on the strke prce of the optons. If the strke prce s hgh fnancal optons have no effect on the effcency of markets because they are out the money. If the strke prce s ntermedate frms wll sell a lot of optons n the frst stage and flood the market. The equlbrum prce s equal to the strke prce. or low strke prces optons have the same strategc effect as futures frms wll sell the same number of futures as n the Allaz and Vla model. In order to restrct the number of equlbrums n the game we assumed that generators wll co-ordnate on the equlbrums whch lead to the hghest prce n the frst stage of the game. In theory they could also co-ordnate on other equlbrums but we thnk these equlbrums are less lkely to occur n practce. References Allaz B. and Vla J. L. (1993). "ournot ompetton orward Markets and Effcency." Journal of Economc Theory Borensten S. Bushnell J. B. and Wolak. A. (2002). "Measurng Market Ineffcences n alforna's Restructured Wholesale Electrcty Market." Amercan Economc Revew 92(5) Bushnell J. Mansur E. and Sarava. (2004). "Market Structure and ompetton: A ross-market Analyss of U.S. Electrcty Deregulaton." Rep. No. SEM W-126 Unversty of alforna Energy Insttute. hao H.-. and Wlson R. (2004). "Resource Adequacy and Market ower Mtgaton va Opton ontracts." Unversty of alforna Energy Insttute ower onference. oase R. H. (1972). "Durablty and Monopoly." Journal of Law & Economcs 15(1) ret A. and abra N. (2004). "apacty Markets for Electrcty." Rep. No. SEM W-124 Unversty of alforna Energy Insttute. Hughes J. S. and Kao J. L. (1997). "Strategc orward ontractng and Observablty." Internatonal Journal of

18 16 Industral Organzaton 16(1) Joskow. L. and Kahn E. (2002). "A Quanttatve Analyss of rcng Behavor n alforna's Wholesale Electrcty Market Durng Summer 2000." Energy Journal 23(4) Le oq. (2003). "Long-Term Supply ontracts and olluson n the Electrcty Markets." Rep. No. 552 Stockholm School of Economcs. Mahenc. and Salane. (2004). "Softenng ompetton Through orward Tradng." Journal of Economc Theory 116(2) Suenaga H. and Wllams J. (2004). "The Natural Number of orward Markets for Electrcty." Unversty of alforna Energy Insttute ower onference. Thlle H. (2003). "orward Tradng and Storage n a ournot Duopoly." Journal of Economc Dynamcs & ontrol 27(4) Wolak. A. and atrck R. (2001). "The mpact of Market Rules and Market Structure on the rce Determnaton rocess n England and Wales Electrcty Market." Rep. No. Workng paper 8248 NBER. Wolfram. D. (1999). " Measurng Duopoly ower n Brtsh Electrcty Spot Market." Amercan Economc Revew 89(4)

19 17 Appendx I : roofs Theorem 1 If the strke prce s above the standard ournot prce (S > p ) then the ournot equlbrum s the equlbrum of the game. The second stage equlbrum s equal to the standard ournot equlbrum ndependent of the number of optons sold Oeq q ( k k ) q eq = eq Ths s therefore also the equlbrum of the two-stage game. Ths result s trval. Theorem 2 If there exsts an equlbrum wth a spot prce above the strke prce then t has to be the Allaz and Vla equlbrum Any equlbrum of the two-stage game has to be an equlbrum n the second stage of the game. The spot prce can therefore only be above the strke prce f the players do not flood the market and sell a small number of optons. See gure 12. Assume there exsts an equlbrum of the frst stage of the game kˆ such that the players do not flood the market n the second stage( ( k 1 k 2) > S ). We have to prove that ths equlbrum s the Allaz and Vla equlbrum kˆ eq = k. Oeq Π ˆ ( k ) Gven that kˆ s an equlbrum of the game kˆ s locally optmal for frm wth other words = 0. k As only a small number of optons are sold the proft of frm s the same as when the frm would sell futures nstead of optons: Hence k s also the local optmum of ˆ Oeq ˆ Π ( ) ( ˆ k = Π k) Π ( ˆ k ) : Π ( ˆ k ) = 0 k If the functon Π ( k k) s concave n k t follows that k ˆ s also a global optmum of ˆk s the Nash Equlbrum of the ournot game wth futures contracts: kˆ eq = k. (23) (24) Π ( k ). Hence Theorem 3 If the strke prce s below the Allaz and Vla prce S < equlbrum. We wll prove that playng k = k eq AV then the Allaz and Vla equlbrum s an s an equlbrum of the frst stage of the game when the players play a punshment equlbrum n the second stage of the game. The punshment equlbrum ensures that unlateral floodng of the market s not optmal. We wll now explan the punshment equlbrum and then we show that t s an equlbrum n the frst stage. gure 14 shows the strategy space of second stage of the game. In the frst graph both frms sell the equlbrum quanttes k. In the second graph frm 1 devates from the frst stage equlbrum and floods the

20 18 market. If the market s flooded there are an nfnte number of equlbrums. In the punshment equlbrum the frms co-ordnate on that equlbrum whch gves the devator the smallest market share. Note that once frm 1 has flooded the market sellng even more optons has no effect on the second stage equlbrum. a b qk ( ) k qk ( ) k dev eq AS ( k k ) q eq q eq θ 1 Equlbrum gure 14 Devaton of k Unlateral Devaton by frm 1 ormally f frm unlaterally floods the market the frms wll co-ordnate on the punshment equlbrum q ( k ) n the second stage where q ( k ) s defned by the followng two equatons We wll now verfy that k pq ( + q ) = S = q q ( q k ) (25) s an equlbrum n the frst stage gven the punshment equlbrum q ( k ) the second stage. gure 15 shows the frst stage strategy space. We need to prove that when player 2 plays k t s optmal for player 1 to play k. 2 1 If frm 1 does not flood the market.e. t sells fewer optons than n the pont B then ts proft Π O ( k k eq ) s the same as wth future contracts. O = Π 1 2 Π ( k k ) ( k k ) If frm 1 floods the market by sellng more optons than n pont B then t wll obtan the same proft as n pont B; ts proft Π O ( k k eq ) becomes: O B = Π 1 2 Π ( k k ) ( k k ) Ths s a drect consequence of the fact that once the market s flooded sellng more optons has no effect on the second stage equlbrum and hence the proft level of frm 1. As k maxmzes Π ( k k eq ) frst stage devaton s not proftable for frm 1. 1 (26) (27) n

21 19 k 2 k2 ( k1) = k LOODING 2 p ( k) = S k ( k ) = k 1 k 2 A B NO LOODING k eq k dev 1 1 k 1 gure 15 Devaton from frst type of equlbrum Theorem 4 If the strke prce s between the Allaz and Vla prce and the standard ournot prce < S < than floodng s an equlbrum. * We wll prove that any q A( S) B as and eq s a second stage equlbrum outcome where AS ( ) and B are defned q ( 1 q2) S + = AS ( ) = q q1 ( q2) q 1 (28) q2 ( q1) q 2 ' 1( q1) ( q1 + q2) B = q ' (29) 2( ) ( + ) AS ( ) s the set of producton quanttes where each player produces more than hs ournot best response functon and where the spot prce s equal to the strke prce. AV s the set of producton quanttes where for each generator the spot prce s hgher than hs margnal cost. or all producton quanttes n B each generator would lke to obtan a larger market share gven constant prces. We wll assume for now that both frms sell a very large number of opton contracts k n such way that when one frm unlaterally decdes to sell fewer contracts the remanng number of contracts stll floods the market. 16 If a lot of optons are sold n the frst stage then the set of equlbrums n the second stage AS ( k ) becomes equal to AS ( ) * Any element q A( S) lm AS ( k) = AS ( ) k s thus a second stage equlbrum of the game. We wll now prove that extensvely floodng of the market (k ) s a frst stage equlbrum. Therefore we need to make an assumpton on the equlbrum the generators wll co-ordnate on n the second stage of the * game; also off-the-equlbrum path. We assume that the players co-ordnate on q as long as t s a second stage equlbrum and co-ordnate on any other second stage equlbrum when q * s not an equlbrum n the second stage. B (30) 16 Analytcally ths means that (0 k ) < S and ( k 0) < S.

22 20 gure 16A shows the strategy space of the second stage when the players both extensvely flood the market. We wll now prove that t s not proftable for frm 1 to devate from ths equlbrum. or small devatons n the * frst stage the frms wll stll play the equlbrum q = ( q q ) n the second stage as shown n gure 16B. * * rm 1 wll thus not ncrease ts proft. or larger devatons the set of second stage equlbrums becomes smaller AS ( k) AS ( ). As a result the equlbrum q * s no longer possble. In any of the second stage equlbrums frm 1 s market share decreases whle frm 2 s market share ncreases. Ths has a negatve mpact on frm 1 s proft as we assumed that q * B. Also large devatons are thus not proftable. See gure 16. a b c AS ( ) AS ( k ) AS ( k ) q * eq q eq q * q q eq q * Equlbrum Small Unlateral Devaton Bg Unlateral Devaton gure 16 Unlateral devaton by frm 1. To conclude the proof we need to show that the set AS ( ) B s not empty for < S <. Take * producton quanttes q = ( q q ) such that each frm produces more than the ournot quantty and less than Allaz and Vla quantty * * eq * eq and such that the spot prce equal s equal to the strke prce wth q ( ) = S. S q q q ( k ) (31) * * = S q q q Gven that the strke prce s between the Allaz and Vla prce and the ournot prce ( < S < follows that Hence a q * S eq eq q q + q S eq eq q q1 ( k ) + q2 ( k ) * whch satsfes (31) and (32) certanly exsts. We now wll prove that q A( S) B * q * * q s an element of AS ( ) when q ( q ). Ths s the case gven the followng dervaton: * eq eq * ( ) ( ) = (1) (2) AV AV (32) (33) q q q q q q (34) The frst step s vald because the reacton functon s downward slopng and the second step s vald because eq q s the ournot equlbrum. * q s an element of B when the spot prce s larger than the margnal cost. Ths s the case gven the followng dervaton: ' * ' ( ) ( ) (1) (2) AV q q S (35) eq eq ) t

23 21 The frst step s vald because the margnal cost functon s upward slopng and the second step s vald because generators wll choose n the ournot game wth futures an allocaton where the prce s above the margnal costs. Ths proves the exstence of the equlbrums. Appendx II : Dfference of nancal and hyscal all Optons As mentoned before we wll now dscuss the dfference of fnancal and physcal contracts on market power. Most authors mplctly assume that both contracts are equvalent. Ths s true for futures contracts but t s not the case for opton contracts as we wll show now. Above we dscussed fnancal opton contracts; we wll now dscuss physcal opton contracts and show the dfference wth fnancal optons. The man dfference s that a physcal opton s assocated wth a specfc generaton plant. If a retaler buys a physcal opton n the contractng stage he takes an opton on one MW of producton capacty of a specfc plant.e. he reserves the generaton capacty. In the producton stage the retaler can decde to run hs plant and produce electrcty. or ths electrcty he wll pay the strke prce of the opton S. The generators can only use the generaton plants whch are not reserved by retalers to sell electrcty n the spot market. They are thus not allowed to use the generaton plants that were already commtted n the physcal call optons. As a result of sellng physcal optons a generator does no longer decde on the producton level of all hs plants. He sold the rght to set the producton level of the reserved generaton plants to the retalers. Retalers wll turn on these generaton plants when the spot prce ncreases above the strke prce S. The advantage of physcal optons s that because specfc producton plants are assgned to the contracts the probablty that electrcty s physcally delvered ncreases. hyscal opton contracts can be modeled n a two-stage game as we dd wth fnancal optons. In the frst stage generators sell physcal call optons and assgn some producton capacty to ther commtment. In the second stage retalers decde about the producton level of the plants they reserved and generators decde about the producton level of the plants whch were not reserved. Market clearng wll determne the spot prce for electrcty. Because retalers turn on plants when the spot prce s above the strke prce the producton level of a generator ncreases wth the spot prce. Hence a generator wll produce more when the sales of the other generators n the market decrease. When generators decde about the producton level of the unreserved plants n the spot market they wll take nto account that the total producton level of the other generators s not constant. The resdual demand functon that a generator faces becomes flatter when other generators sell physcal call optons. 17 Ths leads to a more compettve behavor by the players n the second stage. The best response functon of a generator s a functon of the optons sold by the frm tself and by compettors. Note that ths s dfferent wth fnancal optons. There the best response functon depends only on the number of optons sold by the frm tself. The prce of physcal optons s determned by arbtrage. The value of a reservaton of one MW of producton capacty s equal to the spot prce mnus the strke prce when the opton s n the market and zero when t s out-the-market. 18 = max{ S 0} 17 See also hao and Wlson (2004). 18 Because physcal call optons can only be offered when they are backed by generaton capacty arbtrageurs cannot sell physcal call optons n the contractng stage as they do not own producton capacty. Retalers themselves wll therefore need to arbtrage between the two markets.