Analyzing strategic interaction in multi-settlement electricity markets: A closed-loop supply function equilibrium model

Transcription

1 Analyzng trategc nteracton n mult-ettlement electrcty market: A cloed-loop upply uncton equlbrum model A the preented by Steven Crag Anderon to The Commttee on Hgher Degree n Publc Polcy n partal ulllment o the requrement or the degree o Doctor o Phloophy n the ubect o Publc Polcy Harvard Unverty Cambrdge, Maachuett May 004

2 004 Steven Crag Anderon All rght reerved.

3 The advor: Wllam W. Hogan Steven Crag Anderon Analyzng trategc nteracton n mult-ettlement electrcty market: A cloed-loop upply uncton equlbrum model Abtract Mult-ettlement electrcty market typcally permt rm to bd ncreang upply uncton (SF n each market, rather than only a xed prce or quantty. Klemperer and Meyer (989 ngle-market upply uncton equlbrum (SFE model extend to a computable SFE model o a mult-ettlement market, that, a ngle orward market and a pot market. Spot and orward market upply and demand uncton are endogenouly under a cloed-loop normaton tructure wth ratonal expectaton. The cloed-loop aumpton mple that n choong ther pot market SF, rm oberve and repond optmally to the orward market outcome. Moreover, rm take the correpondng expected pot market equlbrum nto account n contructng ther orward market SF. Subgame-perect Nah equlbra o the model are characterzed

4 analytcally va backward nducton. Aumng ane unctonal orm or the pot market and an equlbrum electon mechanm n the orward market provde or numercal oluton that, ung mple emprcal benchmark, elect a ngle ubgameperect Nah equlbrum. Incentve or a uppler n the orward market decompoe nto three dtnct eect: a drect eect attrbutable olely to the orward market, a ettlement eect due to orward contract ettlement at the expected pot market prce, and a trategc eect arng due to the eect o a rm orward market actvty on the antcpated repone o the rm rval. Comparatve tatc analy examne the eect o mall parameter hock on the orward market SF. Shock that ncreae the elatcte o equlbrum upply and demand uncton tend to make rm more aggreve n the orward market, n that they bd hgher quantte at mot prce. Expected aggregate welare or the mult-ettlement SFE model ntermedate between that o the ngle-market SFE model and that o the perectly compettve cae. v

5 Table o Content Introducton.... Electrcty ector retructurng..... Scope and extent..... etructurng and economc ecency...3. Market power Denton and orgn Polcy repone Motvaton and obectve o the preent nvetgaton Modelng compettve electrcty market Market charactertc Applcaton o game theory Supply uncton A cloer look at market power Competng denton and the degree o market power Forward contractng and market power aement...8 Market power v. bddng baed on opportunty cot...3 Market power v. carcty Extng lterature...5 v

6 .5. Sngle-ettlement SFE model Mult-ettlement model Outlne o the the...38 The U.S. polcy repone to horzontal market power n electrcty generaton Htorcal development Merger Market-baed rate Dcuon Market power montorng and mtgaton Orgn Montorng and mtgaton n regonal market Aement A uppler orward market problem wth nancal contract The upply uncton bddng model: Notaton and termnology Tmng and normaton tructure o equental market Equlbrum concept Indutry tructure and rk preerence Prce Supply uncton...76 Provonal v. admble upply uncton...77 Imputed v. optmal upply uncton...80 Equlbrum upply uncton Quantte evenue Cot uncton Prot Demand uncton The nature o nancal orward contract Pong the orward market problem Solvng rm orward market problem va backward nducton Frt tage: The pot market Second tage: The orward market Dcuon Dervaton o the optmal orward market SF Frt tage: The pot market Second tage: The orward market...4 v

7 5 A mpled ane example Ane unctonal orm Implcaton or the pot market upply uncton Comparatve tatc or the pot market Implcaton or the optmal pot market prce uncton Implcaton or the orward market optmalty condton Concluon The demand de Modelng aumpton Prce-takng conumer Partal equlbrum analy A derved demand or electrcty Conumer optmzaton problem An expected utlty maxmzaton problem Approxmatng the expected utlty maxmzaton problem wth a mean-varance decon model Extence o a repreentatve conumer A normatve repreentatve conumer n the orward market A normatve repreentatve conumer n the pot market A potve repreentatve conumer n the orward market A potve repreentatve conumer n the pot market Summary and concluon Speccaton o unctonal orm or and φ Neceary and ucent condton or the repreentatve conumer to have an ane pot market demand uncton The repreentatve conumer producton uncton, ( q, T, or the amenty x The repreentatve conumer utlty uncton, φ ( x, or the amenty x...99, D p, T Condton or contency o D ( p ε and ( 6.5 A mple tochatc model or the pot market demand hock ε The repreentatve conumer optmzaton problem Spot market Forward market The relatonhp o demand hock and prce acro market...7 de ε ε dε The dervatve ( 0 0 v

8 6.7. The dervatve de( p p dp Properte o aggregate orward market demand (, Properte o 0 ( v D p ε... D p Properte o ε Properte o D ( p, ε = D p, e η ( η ( 7 The orward market upply uncton n the mpled ane example Equlbrum optmalty condton or the orward market Integratng prevou chapter reult concernng the uncton E ( p p and D 0 ( p The tructure o equaton (7. and (7...3 S p n equaton (7.8 and ( Iolatng the ( 7. Properte o the ytem (7.5 and (7.6 and extence and unquene o oluton Sngularte Soluton o the ytem (7.5 and (7.6 away rom the ngular locu Computatonal approache to olvng the derental equaton ytem characterzng the orward market SF Numercal ntegraton ung MATLAB Derence equaton approxmaton ung the Excel Solver: The dcrete Excel model Comparon o computatonal approache Qualtatve analy o the derental equaton ytem characterzng the orward market SF The parameter vector Θ The ngular qualnear ODE ytem, equaton ( The upper partton o the phae pace o the non-ngular ODE ytem, equaton (7.40 ( Prce relatonhp acro market Equlbrum n the orward market Equlbrum oluton o the derental equaton ytem Benchmarkng the dcrete Excel model Benchmarkng tep (pot market Benchmarkng tep (orward market Dcuon Comparatve tatc analy Computaton o orward market SF: Bae cae problem...30

9 7.6. Computaton o orward market SF: Tet cae problem eult and nterpretaton Comparon o expected aggregate welare under alternatve behavoral aumpton and market archtecture Dcuon, concluon, and urther reearch Motve or orward market actvty Eect o a uppler orward market actvty on equlbrum quantte Eect o a uppler orward market actvty on t rval prot Decompoton o uppler ncentve or orward market actvty Motve or orward market actvty by conumer Further reearch: elaxng retrcton mpoed n the model Number o compettor n Ane unctonal orm retrcton ole o perect obervablty o orward market acton Further reearch: Market power Appendx A: Proo that rm pot market upply uncton nterect t redual demand uncton exactly once Appendx B: Second-order ucent condton or the optmalty o the orward and pot market upply uncton B. Second-order condton or the optmalty o the pot market SF B. Second-order condton or the optmalty o the orward market SF Appendx C: Interpretaton o ( p ψ and the orward market equlbrum optmalty condton Appendx D: Computatonal detal o the pot market SFE under the mpled ane example D. Comparatve tatc o rm pot market SF lope β and parameter φ wth repect to the parameter c, c, and γ D.. The partal dervatve o β ( c, c, γ wth repect to c β c, c, γ wth repect to c D.. The partal dervatve o ( x

10 D..3 The partal dervatve o ( c, c, D..4 The partal dervatve o ( c, c, D..5 The partal dervatve o ( c, c, D..6 The partal dervatve o ( c, c, β γ wth repect to γ φ γ wth repect to c φ γ wth repect to c...39 φ γ wth repect to γ D. Comparng dervatve o rm pot market SF lope β and wth repect to the lope c and c o a rm own and the rm rval margnal cot uncton D.3 The geometry o the partal reacton uncton β ( β β = Appendx E: Computatonal detal o the dervaton o optmal orward market upply uncton and reult o numercal example...40 E. Supportng analy or text equaton (7.5 and ( E. Theory and computaton o ngularte n the ytem o text equaton ( E.3 The MATLAB ode5 olver...48 E.4 Numercal reult o comparatve tatc analy...4 Appendx F: Bae cae parameter value ued n the numercal example o the mult-ettlement SFE model...47 F. Spot market...48 F.. Prce and quantte...49 F.. Demand data F..3 Cot data...43 F..4 Spot market SF lope and related parameter...43 F..5 Dtrbutonal aumpton or pot market demand F. Forward market F.. Prce and quantte F.. Conumer rk preerence F.3 Summary Lterature cted x

11 Knowledge n the end baed on acknowledgement. Wttgenten, On Certanty Acknowledgement Scholarhp nomnally a oltary enterpre, but a network o communte utan ndvdual cholar n way both vble and nvble. The rt uch communty to whch I am grateul the Harvard Electrcty Polcy Group (HEPG, part o the Center or Bune and Government at Harvard Kennedy School o Government (KSG, under whoe aupce I wrote th the. I am ndebted to my HEPG colleague, pat and preent, or acltatng and encouragng my the reearch and, more broadly, or creatng and utanng a orum or polcy analy and delberaton havng demontrable mpact (to llutrate, try googlng electrcty polcy. Whle I grateully acknowledge the upport o the HEPG or th work, the reult and vew contaned heren are olely my own and do not necearly repreent thoe o any HEPG partcpant. Specal thank x

12 alo to Contance Burn, Energy Proect Coordnator, on whoe experence and udgment I reled contnually whle at KSG. I alo proted rom my contact wth the outtandng cholar and proeonal who erved a HEPG Vtng Fellow. In partcular, I thank o Baldck, Tor Johnen, Dougla N. Jone, Tare Krtanen, Juan oellón, and Fona Wool or rutul dcuon and helpul comment. In the coure o my reearch, I conronted a novel derental equaton ytem havng ome unuual properte. Seekng advce on the analy o th ytem, I conulted wth a number o cholar n the appled mathematc communty. I am grateul to Wllam Boert, Steve Campbell, Je Cah, Bob Froch, Dylan Jone, Nancy Kopell, Bela Palancz, Patrck aber, Gunther eßg, Werner henboldt, Larry Shampne, Bernd Smeon, Coln Sparrow, Peter Spellucc, and Allan Wttkop or harng workng paper, reerence, otware, and advce. Ther repone to my nqure were, wthout excepton, generou and thoughtul, depte my tatu a a dcplnary outder. Throughout th proect, I have beneted tremendouly rom nteractng wth a wdenng crcle o cholar and proeonal workng on and n the electrcty ndutry, n partcular oger Bohn, Fredel Bolle, Severn Borenten, Joe Bowrng, Jm Buhnell, Paul Centolella, chard Green, Ben Hobb, Hll Huntngton, Paul Jokow, Jm Krtkon, Davd Newbery, Davd akn, Alekandr udkevch, Carlo uín, Yve Smeer, Steven Stot, Shah Verma, and Frank Wolak. I am ndebted to the aorementoned ndvdual or data, advce and nght, ueul comment, and opportunte to preent and dcu work n progre. x

13 The Kennedy School communty nurtured my ntellectual development and ultmately, the work. I am thankul to Bll Clark, my ntal academc advor, or launchng my career a a cholar, or h manet concern or tudent welare, and or h nprng commtment to makng nnovatve and relevant cholarhp acceble to polcymaker. I am grateul alo to ob Stavn or advng my econd-year paper, and or gvng generouly o h tme and nght n a readng and reearch coure on envronmental and natural reource economc and polcy. ob breadth o experence and contrbuton n the phere o reearch and polcy n envronmental economc led me to apre to ncorporate a mlar portolo o proeonal actvte n my own career. From the gene o th work (a May 999 conerence call between Bll Hogan, oger Bohn then an advor to the Calorna Power Exchange and me to the preent document, Bll Hogan, char o my the commttee and an exceptonal mentor, provded an optmal mx o tmely and generou eedback, brllant nght, and unlaggng upport and encouragement toward my goal. It not poble to do utce here to my enormou ntellectual and peronal debt to Bll; th the would not have been wrtten wthout hm. Joe Kalt oned my commttee a an enthuatc upporter o th proect, and our converaton were alway tmulatng, ree-wheelng, and un. Joe puhed me to examne crtcally the conventonal aumpton o economc model o mperect competton and, even a I developed the theory, to look or emprcal puzzle and tetable hypothee. My nteracton wth Joe have planted eed or extenon o th work that wll be rcher both emprcally and conceptually, and I am grateul to hm or h contrbuton. In May 00 wth the wrtng well underway, Nolan Mller agreed to erve on my commttee. Nolan umped n and got up to peed on where I had been x

14 and hoped to be gong n remarkably hort order. I am ndebted to Nolan or teerng me out o game-theoretc blnd alley, emphazng the power o mple ntutve explanaton, and remndng me o the mportant role that the the play n hapng one proeonal opportunte. Nolan keen nght n theoretcal modelng, moreover, have helped me lay oundaton or the applcaton o th work to the analy o market power. Fnally, I thank Catherne Wolram, ormerly o the Harvard Economc Department, or advng me on my 998 dertaton propectu and agreeng to erve ntally on my the commttee; th early geture o upport meant a lot. Fellow traveler n academe that, tudent n KSG doctoral program and at other nttuton conttuted yet another communty o upport. In partcular, I am grateul to Davd Cah, J.. DeShazo, Alon Earle, Karen Fher-Vanden, Lucy Goodhart, Mram Jorgenen, Jm Moher, chard Newell, Todd Olmtead, Carlo uín, Todd Schatzk, Howard Shatz, Mchael Sheld, Davd Snelbecker, Krt Swope, ob Talerco, Carolyn Warner, and Jan Wrght or ther lvely camaradere, or empathy and optmm durng the rough tretche, or all I learned rom them over problem et and potluck dnner, and or ther contnung rendhp. Former houemate Carolyn Warner and Mchael Sheld deerve pecal recognton or graceully abdng my dometc oble whle cleverly concealng rom me any o ther. A nal communty that tood by me a I wrote that o my amly. They may at tme have wondered when the the marathon would end; at lat, I can anwer dentvely. Todd Anderon, my brother, and h we Ala were mot upportve and generou, knowng ntnctvely when to ak how thng were gong, and when to mply buy me a drnk. Lucy and Paul Thompon, my wonderul n-law, welcomed me warmly xv

15 nto the Thompon amly n 000, and oered u occaonal reuge rom cty le at ther Vermont home. To my parent, Carolyn and Jerry Anderon, I oer my grattude and apprecaton or ther acrce and teadat commtment to my educaton throughout my le. They ntlled n me a work ethc and generoty o prt that have been ntrumental n helpng me re to th challenge. In concluon, I would not have accomplhed th work wthout the contant love, upport, and encouragement o Janet Thompon, my we. Throughout, Janet labored a hard, not harder, or the Ph.D. a I dd, workng everal ob a well a ngle-handedly managng the quotdan chore o the houehold. In addton, he ptched n on the the tel, brngng to bear her gt wth language and her well-traned edtoral eye. She dd o cheerully, and tll had t wthn her to buoy my prt when thng looked bleak. A Goethe oberved, The um whch two marred people owe to one another dee calculaton. It an nnte debt, whch can only be dcharged through all eternty. I am tempted to add that one o the par wrtng a the then, a the ayng goe, pretty oon you re talkng about real money. And o t to Janet, and the plendd communty o our unon, that I dedcate th the. xv

16 xv To Janet, who alway knew

17 Concern or man hmel and h ate mut alway orm the che nteret o all techncal endeavor, concern or the great unolved problem o the organzaton o labor and the dtrbuton o good n order that the creaton o our mnd hall be a bleng and not a cure to manknd. Never orget th n the mdt o your dagram and equaton. Enten, Addre at the Calorna Inttute o Technology xv

18 ELECTICITY, n. The power that caue all natural phenomena not known to be caued by omethng ele. It the ame thng a lghtnng, and t amou attempt to trke Dr. Frankln one o the mot pctureque ncdent n that great and good man career. The memory o Dr. Frankln utly held n great reverence, partcularly n France, where a waxen egy o hm wa recently on exhbton, bearng the ollowng touchng account o h le and ervce to cence: Moneur Franquln, nventor o electrcty. Th llutrou avant, ater havng made everal voyage around the world, ded on the Sandwch Iland and wa devoured by avage, o whom not a ngle ragment wa ever recovered. Ambroe Berce, The Devl Dctonary It not rom the benevolence o the butcher, the brewer, or the baker, that we expect our dnner, but rom ther regard to ther own nteret. We addre ourelve, not to ther humanty but to ther el-love, and never talk to them o our own necete but o ther advantage. Adam Smth, The Wealth o Naton Introducton. Electrcty ector retructurng.. Scope and extent IN THE 980S, AND INCEASINGLY IN THE 990S, dozen o countre around the world ntated economc reorm or retructurng o ther electrcty ector. Thee countre launched ther reorm rom wdely dparate crcumtance, ncludng vared ncome level, producton and conumpton pattern, government role n the economy, legal and nttutonal ramework, and reource endowment. Depte th heterogenety, the traectory o electrcty retructurng ha been broadly mlar acro countre, typcally comprng the ollowng meaure (World Energy Councl 998; Grd 00:. The prvatzaton or corporatzaton o publcly-owned enterpre n the electrcty ector

19 . The vertcal dntegraton, or unbundlng o the ndutry generaton, tranmon, dtrbuton, and retalng egment 3. The deregulaton o the generaton and the retalng egment 4. The ntroducton o regulatory open acce rule or the tranmon egment In the Unted State, whch ha a htory o prvate ownerhp (though not excluvely o o electrc utlte, retructurng progreed through leglatve and regulatory ntatve on two urdctonal ront. On the wholeale level, the Energy Polcy Act o 99 (EPAct catalyzed the development o an open acce regme or the electrcty tranmon grd. Puruant to th leglaton, the Federal Energy egulatory Common (FEC (996d ued Order 888, mplementng open acce and encouragng the ormaton o ndependent ytem operator (ISO to manage the tranmon grd. Later, the Common Order 000 on egonal Tranmon Organzaton (TO (Federal Energy egulatory Common 999, 5 lad out an TO mnmum conguraton and urged (but dd not requre tranmon owner to cede control o ther tranmon aclte to TO. In t July 00 Notce o Propoed ulemakng (Federal Energy egulatory Common 00a, 3, the Common bult on t earler ntatve, propong to etablh a tandardzed tranmon ervce and eerence to the Common throughout th the denote the Federal Energy egulatory Common. A o th wrtng, the Common had approved the ollowng ve ISO: ISO New England (ISO-NE, Calorna ISO (CAISO, PJM Interconnecton (PJM or porton o the md-atlantc tate, Mdwet ISO (or Mdwet Independent Tranmon Sytem Operator (MISO, and New York ISO (NYISO. The Electrc elablty Councl o Texa (ECOT wa created n 996 by the Publc Utlty Common o Texa. ECOT contaned entrely wthn the tate o Texa, and hence not ubect to the Common plenary urdcton (Moore and Gooch 00,.

20 market degn to provde a level playng eld or all wholeale electrcty market partcpant. On the retal level, 3 tate retructurng ntatve began wth a 993 Calorna regulatory decon (Calorna Publc Utlte Common 993. A o Aprl 004, twenty-our tate and the Dtrct o Columba had enacted leglaton or ued regulatory order to permt retal acce to compettve electrcty uppler; more recently, however, even o thee tate delayed or upended ther plan or retal acce (Amercan Publc Power Aocaton 004, largely n repone to the turmol n Calorna market. 4.. etructurng and economc ecency The prmary ratonale or electrcty retructurng n mot countre ha been to reap welare gan by upplantng regulaton wth competton where t vable. Both theory and experence wth other ormerly regulated ndutre ugget that thee gan wll nclude ncreaed hort-run productve ecency, enhanced allocatve ecency through prcng that more cloely relect phycal and economc realty, and ncreaed dynamc ecency rom mproved ncentve or nvetment and nnovaton. One may gan ome perpectve on the magntude o potental ecency gan or the cae o the Unted State by notng that revenue rom electrcty ale to nal conumer n 000 totaled approxmately $8 bllon (Energy Inormaton Admntraton 00a, Table A5. By comparon, th amount exceeded recent U.S. annual pendng on automoble, 3 Whle wholeale and ntertate tranacton are ubect to regulaton at the ederal level by the Common, retal ale (.e., ale to nal conumer are under the urdcton o each tate publc utlty common (PUC or mlar regulatory body. 4 See Sweeney (00 and Blumten (00 or detaled analye o the Calorna experence. 3

21 telecommuncaton, or hgher educaton (Brennan et al. 996, 5. The net book value o electrc utlty plant owned by maor nvetor- and publcly-owned utlte provde a rough ndcator o the ze o the ndutry total captal tock. A o 996, th net book value wa approxmately $433.5 bllon. 5 A potentally gncant obtacle to realzng thee welare gan rom retructurng market power. Market power exerced by uppler typcally ental the wthholdng o output and an upward dtorton n the market prce. 6 Market power generally aocated wth varou orm o economc necency. Agan, t ntructve to conder potental ecency loe n term o productve, allocatve, and dynamc necence due to market power. Frt, market power tend to caue productve necence. To ee th, conder a mple example n whch a rm call t rm A exerce market power, retrctng t producton and drvng a wedge between the equlbrum prce and t margnal cot. 7 Suppoe that rm A rval do not exerce market power; they thereore chooe ther output level to equate prce and ther repectve margnal cot. In equlbrum, the margnal cot o rm A rval exceed that o rm A, o that aggregate output could be produced at lower total cot 5 A reported by Energy Inormaton Admntraton (997, 7 and Energy Inormaton Admntraton (00b, Table and (data or nvetor-owned utlte were lat avalable or Secton. below provde a more ormal denton o market power. Market power uually but not necearly aocated wth the wthholdng o output. Hogan (997 decrbe a alent excepton to th aocaton n an electrcty market ettng. In a tylzed electrcty network model wth locatonal margnal prcng, Hogan llutrate how tranmon network nteracton and contrant enable an owner o generaton plant at multple network locaton to exerce market power va ncreaed total output. In th event, prce ncreae at ome network locaton and decreae at other, whle total prot or the plant owner ncreae. 7 In th tylzed example, we aume trctly ncreang margnal cot uncton and gnore capacty contrant. A prot-maxmzng rm, naturally, wll alway chooe t output level to equate margnal revenue and t margnal cot. 4

22 producton were reallocated rom the rval to rm A. Second, market power alo create allocatve necence n that t generally lower aggregate quantte conumed, caung a deadweght lo to aggregate welare. 8 Fnally, market power create dynamc necency when market partcpant on both the upply and demand de o the market make nvetment decon baed on prce expectaton dtorted by market power. Temporal and (under locatonal prcng n a tranmon network patal dtorton n prce may are. ecallng the argument o note 6 above, thee patal prcng dtorton due to market power may go n ether drecton. Emprcal etmate o uch welare loe due to market power have been contentou, but many cae tude ugget that uch loe have been conderable. 9 Together, the potental magntude o the problem, controvere urroundng concept and methodology, and practcal dculte aocated wth aeng market power undercore the need or ubtantal urther reearch on th ue. The preent nvetgaton conttute one contrbuton toward mprovng the theoretcal oundaton o market power aement n electrcty market. 8 In the pecal cae o perectly elatc demand or electrcty, there no lo n allocatve ecency wth upply-de market power, rather, only a rent traner rom conumer to producer. 9 The Department o Energy (000 revew emprcal reearch on market power n the Unted Kngdom, the PJM Interconnecton, Calorna, and everal other U.S. tate. Partcularly noteworthy or the preent nvetgaton are Borenten, Buhnell, and Wolak (000, 33 ndng that rom June 998 through September 999, electrcty uppler n Calorna market receved revenue n exce o compettve level o $75 mllon. Thee author later nd (Borenten, Buhnell and Wolak 00, 396, moreover, that the problem worened by the ummer o 000, when (rom June to October the tate electrcty uppler receved $4.448 bllon n olgopoly rent. In a mlar ven, work by the Market Survellance Commttee o the Calorna ISO (000, 7 ound that or May and June 000, wholeale revenue n the Calorna pot market were 37% and 8% (repectvely n exce o revenue predcted under perectly compettve prcng. 5

23 . Market power.. Denton and orgn To economt, market power the ablty to alter protably prce away rom compettve level (Ma-Collel, Whnton and Green 995, A t relate to ndutry tructure, market power on the part o uppler commonly claed a ether horzontal or vertcal. Vertcal market power the ablty to engage n excluonary behavor conerred by one control o derent egment o the ndutry: generaton, tranmon, dtrbuton, and retal ervce. Horzontal market power, n contrat, the ablty to nluence prce wthn one o thee egment. Htorcally, mot o the world electrcty ndutre conted o vertcally ntegrated, publcly-owned and/or -regulated monopole wth excluve geographc ranche. In the Unted State, prvate ownerhp o electrc utlte ha been the norm, under whch tate regulatory common etablhed prce content wth a ut and reaonable tandard (ee, e.g., Phllp 993, 9, ch. 5. Under a compettve regme, n contrat, nteracton between competng generatng rm would determne prce endogenouly. In lght o th regulatory legacy, the deregulaton o generaton would endow thee utlte de acto vertcally-ntegrated regonal monopole wth conderable market power. The advantage o ncumbency enoyed by thee monopole would not necearly be overcome by the tmely entry o new compettor. 0 Whle ether uppler or demander may poe market power, we conder only upply-de market power n th nvetgaton. See alo note 6 above. It ometme argued that entry wll gncantly leen concern over horzontal market power, renderng t at bet a trantonal problem. In the abtract, th reaonng ha ome appeal. It oten the cae today, however, that ormdable entry barrer (e.g. local tng retrcton or new generaton and tranmon aclte characterze electrcty market n the Unted State, partcularly cloe 6

24 .. Polcy repone A number o promnent ndutry oberver have argued that ederal lawmaker have granted the FEC adequate authorty to addre vertcal market power n the U.S. electrcty ndutry. For example, Perce (996, 3 wrte that or anttrut purpoe, the FEC can now gnore the vertcal contrant on competton that were the prmary ocu o the FEC anttrut actvte durng the 980. A amended by the EPAct, the FPA [Federal Power Act] now gve the FEC regulatory tool that allow t to addre... [thee] vertcal contrant. Va t open-acce tranmon polce (Federal Energy egulatory Common 996d, 999 the Common ha, n act, brought thee tool to bear on vertcal market power concern. In the uture, the Common commtment to TO may reaonably be expected to mtgate ubtantally not elmnate any remanng vertcal market power problem. On the tate level, moreover, regulator (typcally, tate attorney general or PUC n many urdcton have nted upon the dvetture o vertcally-ntegrated utlte generaton aet a a qud pro quo or recovery o tranded cot, or unk cot n to urban center where electrcal load concentrated. One alo commonly oberve tranmon contrant n uch ettng, creatng o-called load pocket. Thu, the hort-run tranton may ndeed lat or ome tme, and may well be aocated wth conderable ecency loe a well a gncant traner to uppler wth market power. Th aement ha o ar proved perhap too optmtc, nce a the Common wrote later n t Order 000, we... conclude that opportunte or undue dcrmnaton contnue to ext that may not be adequately remeded by unctonal unbundlng [ee below]. We urther conclude that percepton o undue dcrmnaton can alo mpede the development o ecent and compettve electrc market. Thee concern... provde the ba or ung [Order 000] (Federal Energy egulatory Common 999, 65. Functonal unbundlng, requred by the Common Order 888, compre three retrcton on conduct or a vertcally-ntegrated utlty. The utlty mut ( take tranmon ervce under the ame tar a do other, ( pot eparate rate or generaton, tranmon, and ancllary ervce, and (3 rely on the ame electronc normaton network a do t tranmon cutomer when arrangng tranacton (Federal Energy egulatory Common 996d, 57. 7

25 exce o market prce. 3 A or horzontal market power, Perce (996, 3 alo remark that the FEC need to reocu t anttrut attenton on horzontal market power ue.... Indeed, horzontal market power n the ndutry generaton egment ha emerged a a central publc polcy concern n U.S. electrcty retructurng. For th reaon, th nvetgaton ocue excluvely on horzontal market power denoted herenater mply a market power n the electrcty ndutry generaton egment. The ollowng chapter, chapter, provde a more detaled account o the polcy repone n the U.S. to market power...3 Motvaton and obectve o the preent nvetgaton Below, we detal ome gap and nadequace n both the theoretcal oundaton or market power analy n retructured, compettve electrcty market and n the polcy ramework or addreng market power problem. In lght o the dramatc tructural change n the electrcty ndutry worldwde, the relevant theory need to be rened and extended. The hghly-tructured nttutonal envronment that neceary to coordnate ecently rm behavor n electrcty market create complex ncentve; thee ncentve render the characterzaton o market power n th context a dcult and unnhed tak. To advance the dcuon, a rutul tartng pont would be to lay the analytcal oundaton or denng and meaurng market power gven the archtecture o today compettve electrcty market. The preent work provde a coherent, tylzed, characterzaton o key ncentve that market partcpant ace n th 3 Whle mot commentator have taken the vew that thee regulator lack the authorty to compel dvetture, the propect o (at leat partal denal o tranded cot recovery nduced ome ntegrated utlte to dvet generaton aet. Such dvetture, o coure, can have both horzontal a well a vertcal compettve ramcaton. 8

26 envronment, whch neceary to gude the development o analytcal methodologe or emprcal analye o market power. Polcymaker and regulator may then brng uch method to bear n aeng the everty o market power, and n cratng approprate and welare-enhancng polcy repone. The preent nvetgaton ocue on the compettve mplcaton o a equence o market, an archtectural eature preent n many compettve electrcty market. Th partcular element o market degn create ntertemporal ncentve or market partcpant related, n general, to rk hedgng, peculaton, and trategc conderaton (ee, e.g., Allaz 987 the eect o whch are a yet poorly undertood. Th the examne the behavoral ncentve nduced by the archtecture o newly retructured electrcty market. In partcular, we derve prot-maxmzng upply uncton 4 equlbrum bd or electrcty uppler competng n equental orward and pot market. In a ere o numercal example, we examne how thee bd depend on underlyng economc charactertc o th envronment, and compute expected aggregate welare or th market ettng. Secton.3 and.4 below elucdate the cope o th nvetgaton n greater detal..3 Modelng compettve electrcty market.3. Market charactertc Compettve electrcty market n the U.S. hare ome alent market degn eature wth other around the world. Among thee common charactertc are the extence o orward market (n addton to the pot market or electrcty, gncant lexblty n 4 For more on upply uncton, ee ubecton

27 the orm o bd permtted rom uppler, uncertanty n demand, and determnaton o prce va a market-clearng compettve equlbrum. Frt, the degn o many o the world electrcty market nclude at leat one and ometme everal orward energy market. Such a market degn commonly reerred to a a mult-ettlement market. 5 When orward energy market clear cloe to real tme (e.g., one day ahead, they typcally rely on a market coordnator and compettor bd (rather than on blateral negotaton to et prce. Second, market rule n many electrcty market around the world permt gncant lexblty n rm upply bd, requrng only that bd take the orm o ncreang uncton rom prce to quantty. 6 Thrd, demand uncertanty n each perodc market 7 are rom uncertan weather condton, equpment alure, and other contngence. Market partcpant may make demand orecat to ad ther market decon makng, but thee orecat wll naturally be mperect. Lat, the pont at whch the aggregate upply uncton nterect aggregate demand normally determne the market-clearng prce n each market. The approach to modelng compettve electrcty market decrbed n the remander o th ecton and developed later n the the relect each o thee market charactertc. Other eature o electrcty market havng gncant compettve mplcaton nclude the nterconnected tranmon and dtrbuton network, ntertemporal 5 The moder mult-ettlement denote that the orward and pot market ental dtnct nancal ettlement (bllng and payment between buyer and eller n the repectve market. The cah low pad or receved by a partcpant n a partcular market ettlement, naturally, the product o the market-clearng prce and that market partcpant quantty bought or old. See note 9 below or urther detal. 6 Thee uncton are ometme alo requred to be contnuou. 7 Compettve electrcty market or energy typcally compre regular, perodc pot market (e.g., hourly or hal-hourly durng each day. Aocated wth each perod pot market may be one or more orward market a well a market or reerve (.e., generatng capacty. 0

28 contrant, and multple product and market. Th nvetgaton abtract, or mplcty, rom the complcaton aocated wth thee charactertc. We dcu brely below the mplcaton o thee mplyng aumpton. The tranmon and dtrbuton network neceary, o coure, or tranport and delvery o electrcty a well a or enurng relablty and qualty (e.g., voltage and requency tablty. Becaue network capablty lmted, the compettve prce o electrcty wll vary acro derent locaton (n addton to temporal varaton under locatonal margnal prcng. The model developed here mple th tuaton conderably. It may be nterpreted a a model analyzng competton at a ngle network locaton. Alternatvely, one may vew the preent work a modelng a completely uncongeted tranmon network whle alo gnorng tranmon loe. Electrcty generaton technologe exhbt to varyng degree numerou dynamc contrant retrctng the pattern and aocated cot o generaton plant producton over tme. Example o uch contrant nclude mnmum tme or tartup and hutdown (wth aocated cot, mnmum run tme, and ramp rate contrant. Startup cot mply that a currently dle unt may not nd t protable to begn operaton n a gven hour expected prce n the near term are nucent to cover t varable operatng cot a well a t tartup cot. amp rate contrant lmt the amount by whch a generatng unt can change t producton level rom one hour to the next. In practce, thee contrant have potentally gncant economc mplcaton or generatng unt operatng chedule. Proper analy o uch contrant complcated not only by ther ntertemporal nature, but alo becaue they ntroduce non-convexte nto the unt producton uncton. We abtract rom all uch complcaton by aumng (ee

29 ubecton 3..8 that cot uncton are trctly convex and that there are no ntertemporal operatng contrant o economc mportance. Th rather trong aumpton permt u to analyze each operatng perod ndependently. Fnally, compettve electrcty market compre multple product and market. A one alent example, n the early day o Calorna retructured market, there wa a total o eleven market or energy and ancllary ervce. 8 Degn n mot other regon do not nclude a many dtnct product market, although mot compettve electrcty market do eature, at a mnmum, both orward and real-tme (or pot energy market. The preent work preume the mplet market archtecture a ngle orward market and a pot market that permt u to examne the nluence o multple market on competton. Introducng addtonal product market (e.g., or ancllary ervce would ubtantally complcate the analy. An extenon o the preent model to a equence o two or more orward market n advance o the pot market would be relatvely traghtorward, at leat conceptually..3. Applcaton o game theory A a general matter, t natural to model nteracton among agent n dvere market ettng ung the tool o game theory. Th partcularly true n electrcty market, n whch market partcpant nteracton are hghly tructured and regularzed va market nttuton wtne the centrally-cleared market or electrcal energy and ancllary ervce organzed by varou ytem operator around the world. Such electrcty 8 Ancllary ervce reer to reerve generaton capacty, avalable on tmecale varyng rom ntantaneou to up to everal hour. Calorna orgnal market degn envoned our ancllary ervce traded n day-ahead and hour-ahead market and mbalance energy dpatched n real tme by the CAISO. The CAISO operated thee nne product market. In addton, the (ormer Calorna Power Exchange cleared day-ahead and hour-ahead market or energy, or a total o eleven product market.

30 market have, n eect, a well-dened et o player, and or each player, a trategy pace and a payo uncton; thee element o electrcty market degn are alo the bac conttuent element o any game-theoretc model. In electrcty market, the upply de ometme ucently concentrated that the preumpton o perectly compettve (.e., prce-takng behavor on the part o uppler eem napproprate. 9 Intead, model that permt olgopoltc nteracton or mperect competton are relevant n th context; thee model capture the ablty o ndvdual producer to nluence the market-clearng prce. Whle everal alternatve model o olgopoly behavor have been wdely appled, 0 the eature o compettve electrcty market revewed above trongly ugget that upply uncton equlbrum (SFE model are bet uted or modelng uch market, a the ollowng ubecton elaborate..3.3 Supply uncton In the upply uncton (SF model developed n th nvetgaton, player trategy pace are the et o trctly ncreang contnuou uncton rom prce to quantty. Snce orward market uch a the ormer Calorna Power Exchange (PX commonly requre a wth pot market that uppler bd be ncreang contnuou uncton or tep uncton, mult-ettlement market (ee ubecton.3. lend themelve to a neted SF model n whch rm bd SF n both the orward and the pot market. For 9 Apart rom concentraton, common eature o electrcty market generally vewed a contrbutng to market power are the nablty to tore electrcty economcally together wth the necety o ntantaneou upply and demand balance at every locaton n the tranmon grd, and demand nelatcty, partcularly n the hort run. 0 Well-known olgopoly model nclude thoe o Cournot, Bertrand, conectural varaton, and Bertrand-Edgeworth; ee Vve (999 or a comprehenve urvey o thee model and ther applcaton. 3

31 each market, the behavoral aumpton o SF n contrat to pure quantty or prce choce under Cournot or Bertrand competton uggetve o the range o tratege actually avalable to uppler n compettve electrcty market. A dcued urther below, the SF model explctly recognze and accommodate demand uncertanty: SF bd enable uppler to acheve an ex pot optmal outcome under any realzaton o uncertan demand. Fnally, prce ormaton n SF model occur content wth bac economc ntuton: the pont o nterecton o aggregate upply and aggregate demand determne the market-clearng prce. The above dcuon ugget that model baed on the SF behavoral aumpton poe a trkng vermltude to the charactertc o compettve electrcty market, and that SF model, thereore, are epecally well-uted to modelng uppler behavor realtcally n uch market. Moreover, we may extend the nglemarket SFE ramework developed n Klemperer and Meyer (989 emnal paper to a mult-ettlement market. Accordngly, we aume n th work that uppler bd (trctly ncreang SF n both the orward and pot market. In the ngle-market SFE ramework, rm equlbrum SF bd wll reult n ex pot optmal producton at any market-clearng prce. Put another way, or any realzaton o demand uncertanty, a rm that bd t equlbrum SF gven the SF o the The SFE ramework nherently more lexble than Cournot or Bertrand, allowng uppler to pecy through ther bd a chedule o quantte over a range o prce, rather than a xed quantty or prce. In th ene, we may vew SF-baed model a a generalzaton o the Cournot or Bertrand ramework. Such lexblty n rm tratege preent n context other than electrcty, a well. Namely, Klemperer and Meyer (989 cte a alent example the arlne ndutry n partcular, t computerzed reervaton ytem and management conultng. Th trong reult trctly true only n a ngle-market ettng, and mut be qualed omewhat n a mult-ettlement market context, a we dcu n ubecton

32 other bdder guarantee that t wll be called upon to produce t optmal quantty. emarkably, a rm equlbrum SF n the ngle-market SFE ramework dtrbutonree, n that t ndependent o the (non-degenerate probablty dtrbuton o the uncertan demand hock. Th property o SF may at rt appear counterntutve, but t attrbutable precely to the way n whch the SF contructed. A chapter 4 wll how, every eable value o the tochatc hock to demand correpond to a dtnct pont on the correpondng SF. 3 In contrat, the extent o the SF that, the doman o prce over whch the SF dened doe depend on the upport o the demand hock. Th a drect conequence o the clam above that every eable value o the demand hock correpond to a dtnct pont on each rm SF. Moreover, the expected value o prce, quantty, and prot aocated wth a gven SF do depend, a ntuton would ugget, on the probablty dtrbuton o the demand hock. In the mult-ettlement market ramework nvetgated here, th argument mut be moded. It natural, n th ettng, to take orward market equlbrum a contngent on the expected outcome n the pot market. Dong o, orward market SF then depend on the dtrbuton o the uncertan pot market demand hock. The orward market SF, thereore, no longer poe the dtrbuton-ree property exhbted by SF n a ngle-market ettng. A or pot market SF, once the orward market ha cleared, 3 Snce we wll aume the demand hock to have an atomle dtrbuton, any arbtrary value o the hock occur wth probablty zero. By contnuty, the value o the tochatc hock wthn an (arbtrarly mall nterval correpond to a partcular ecton o an SF; the probablty that the hock take on a value n th nterval trctly potve. Whether the probablty o a realzaton o the hock n th nterval large or mall that, the hock probablty dtrbuton nconequental; t matter only that th probablty trctly potve, that, that uch hock can occur. Gven uch a hock, rm repond va ther SF bd to maxmze prot. The SF, thereore, dened over equlbrum prce correpondng to the hock entre upport, the unon o all uch eable nterval. 5

33 the pot market eectvely a ngle market. Thu, a wth the ngle-market SFE, pot market SF n the mult-ettlement market wll agan be dtrbuton-ree..4 A cloer look at market power.4. Competng denton and the degree o market power Subecton.. appealed to a tandard text on mcroeconomc theory to dene market power a the ablty to alter protably prce away rom compettve level (Ma- Collel, Whnton and Green 995, 383; th the denton that we apply or eller throughout the preent work. Interetngly, ederal anttrut regulator and by reerence, the Common ue a omewhat more retrctve denton o market power, namely, market power to a eller the ablty protably to mantan prce above compettve level or a gncant perod o tme. 4 Stot (00, 366 explore the derng mplcaton o thee two denton, and argue (p. 368 that, under ether tandard, the goal hould never be the preventon o all market power. ather, regulator nevtably need to make a hard decon: How much market power too much? Indeed, the queton o the degree o market power under ether denton central to any welare-baed aement o market power that would balance ecency loe due to market power wth the drect and ndrect cot o market nterventon. Note that whle both o the above denton o market power reer to compettve 4 Department o Jutce (DOJ and Federal Trade Common (FTC (99, Sec. 0. Horzontal Merger Gudelne. The FEC Merger Polcy Statement (996 tate that the FEC wll ue the creenng approach o the DOJ/FTC Merger Gudelne to determne whether a merger wll reult n an ncreae n market power. 6

34 [prce] level, nether denton explct about what conttute uch level. 5 Ma- Collel, Whnton, and Green(MWG (995 dcuon permt u to make ome conceptual headway, although here, too, we are ultmately let wth unanwered theoretcal queton. From MWG (pp , we may ner that compettve prce level are thoe that clear the market n a compettve economy whch, n turn, one n whch all conumer and producer act a prce-taker. MWG elaborate that []or the prce-takng aumpton to be approprate, what we want that [conumer and producer] have no ncentve to alter prce that, taken a gven, equate demand and upply (empha n orgnal. For the purpoe o th nvetgaton o upply-de market power, we then conront two queton: 6. What conttute prce-takng behavor or upply? In other word, what the approprate perectly compettve behavoral benchmark (PCBB or a uppler?. What equlbrum prce reult rom uch prce-takng behavor? Gven an anwer to queton above, one ealy obtan the anwer to queton by computng the et o prce (not necearly unque that clear the market. Thu, queton an nteretng and mportant queton or market power analy. An approprate PCBB may erve, n partcular, a a oundaton or emprcal work aeng the everty o market power. Namely, by comparng oberved bd prce wth thoe mulated ung 5 emarkably, the Horzontal Merger Gudelne (Department o Jutce and Federal Trade Common 99 themelve al to upply any gudance or what conttute compettve [prce] level. 6 In th nvetgaton, we aume prce-takng demand whle permttng trategc behavor on the upply de. 7

35 the PCBB, we may ubect to the lmtaton o the partcular modelng ramework adopted hed lght on the queton o whether a uppler ha exerced market power. Intutvely, the PCBB depend on the ncentve, and hence the nttutonal envronment, that agent ace. In an dealzed ngle-perod, bd-baed market, 7 we may appeal to bac economc ntuton: the PCBB would be a prce bd o margnal cot or all quantte up to one producton capacty. In the mult-ettlement market ettng condered n th the, etablhng what conttute the PCBB a more ubtle and complex queton. One would generally need to conder (a we do here the eect o the orward market on pot market behavor a well a rm antcpaton and thu the nluence o the later pot market equlbrum on ther pror orward market behavor. The prncpal goal o the preent nvetgaton, thereore, to characterze and analyze the nter-market ncentve eect that ext n a mult-ettlement market. Achevng a old undertandng o uch eect the rt tep toward determnng a well-ounded and nternally content PCBB, a tak that tel beyond the cope o th work..4. Forward contractng and market power aement Long-term orward contract or energy generaton had been a common eature o the electrcty ndutry beore retructurng, and they contnue to play a role n today more compettve envronment. They have been ntrumental n provdng a ecure return on nvetment, thereby acltatng proect nancng. Once an nvetor commtted to a 7 Ignorng tart-up and no-load cot, and any other non-convexte o rm cot uncton. 8

36 proect, uch long-term contract alo help to allevate the hold-up problem. 8 Th ubecton outlne n more detal the extenon o market power analy to conder market or orward contract. Short-term (e.g., day-ahead orward contract or energy a more recent nancal nnovaton trade n centrally-cleared market organzed by mot U.S. ISO. Thee contract enable both hedgng o pot (e.g., real-tme prce or both buyer and eller, thu reducng rk, and nancal peculaton whch enhance lqudty n the hort run. I the orward market a credble prce benchmark, th can acltate development o uture and opton market or electrcty n the longer run. Thee market, n turn, are lkely to narrow pread n the varou market, and wll provde market partcpant wth more lexblty than would long-term blateral contract. Aumng a reaonably lqud market, t wll be eaer and more ecent or market partcpant to ue thee nancal dervatve rather than to renegotate a blateral contract when crcumtance change (nce contractual counterparte have oppong nteret n uch renegotaton. A market n hort-term contract can ulll the addtonal uncton o prce dcovery, allowng market partcpant to protably explot techncal lexblty. Contract alo upport generator chedulng and unt commtment, provdng a baelne or potentally protable rechedulng (e.g., through Schedule Adutment Bd n the (ormer Calorna PX. Mult-ettlement market that, orward and pot market that clear at dtnct pont n tme are a common eature o compettve electrcty market around the 8 The hold-up problem the ablty o opportuntc regulator or a monopontc buyer to approprate the carcty rent rom llqud xed aet (e.g., electrcty generaton plant once the nvetment unk by permttng pot prce to cover only margnal cot. 9

37 world. 9 The ollowng electrcty market eature a mult-ettlement market tructure (Jamab and Polltt 00, 7 8: Autrala (New South Wale, Queenland, Vctora, Canada (Ontaro, Colomba, England and Wale, France, Ireland, New Zealand, Nordpool (Fnland, Norway, and Sweden, and the Unted State (PJM, New York, New England, and propoed n the Mdwet (Mdwet ISO 004. The ntertemporal character o mult-ettlement market rae the ollowng polcy ue regardng degn and regulaton o thee market:. Both ex ante market degn and ex pot aement o electrcty pot market perormance need to take nto account (a how orward contractng change the expected payo rom (and hence ncentve or pot market actvty, and (b what thee eect mply or the aement o market power n a mult-ettlement market.. How may we evaluate the perormance o the orward market tel? For example, there a perectly compettve behavoral benchmark that apple to the orward market n olaton? Or, doe aeng market power n mult-ettlement market requre ont evaluaton o behavor n orward and pot market? Overall, the theoretcal oundaton or undertandng and aeng market power n mult-ettlement market weak and ncomplete. Queton uch a thee are only 9 In general, a mult-ettlement market a equence o market or a product that nclude. at leat one orward market, n whch buyer and eller may conclude nancal contract or later delvery, and. a pot market, whch clear contemporaneouly wth delvery o the product. Whle the approach outlned here could, n prncple, be extended to nclude two or more orward market, th the conder a ngle perod o orward tradng precedng the pot market. Market partcpant may tranact n both the orward and pot market, modyng ther orward poton n the later pot market, they chooe. In th the, we take orward contract to be legally bndng. 0

38 begnnng to be addreed by the relevant academc lterature (revewed n ecton.5 below. The practcal gncance o thee lacunae ha been partcularly acute n the context o Calorna electrcty market. The orgnal Calorna market degn o centrally-organzed orward and pot market or energy (and ancllary ervce wa an early and alent example o a mult-ettlement market. In th envronment, market power analye baed on the conventonal ngle-market model have been contentou and a target or crtcm. Quan and Mchael (00, 00, or example,... beleve that analye o the [Calorna] ISO and PX have oten reached concluon about market power on the ba o abtracton that obcure and mnterpret mportant apect o compettve behavor. One early tudy o the Calorna market by the Market Montorng Commttee (MMC o the Calorna Power Exchange (999 evnce the dculte to whch Quan and Mchael allude. The MMC tudy propoed (p. 58 to ae a rm perceved market power by calculatng the Lerner Index at each quantty level t bd [n the PX hourly energy aucton], and then averagng the Lerner Index value over the whole bd curve. Speccally, or each hour we ued the rm actual bd curve and our etmate o t margnal cot to calculate the weghted average gro margn. Thu, the MMC dene a PCBB or orward market bddng behavor baed on margnal producton cot. It denote th a the Bd-Markup Index (BMI, dened algebracally a ( BMI t (, ( qt, Max Q( t C q t = dq 0 ρ, where BMI ( t = Bd-Markup Index or hour t

39 Max Q( t = Maxmum quantty oered at any prce at or below $50/MWh n hour t ( qt, ρ = Bd prce at whch the rm oer quantty q n hour t C = etmated margnal cot o the rm. Elewhere n the report, the MMC recognze the potental mportance o opportunty cot ntroduced by the preence o the later pot market (ee, e.g., ther dcuon on pp. 3 and Moreover, the MMC ultmately cautou n drawng concluon regardng the exerce o market power n th novel and rapdly evolvng market envronment. The MMC explct choce o a margnal cot-baed benchmark a a PCBB would be placed on rmer conceptual ootng, however, upported by a ormal model. Borenten, Buhnell, and Wolak (00 alo tudy market power wthn the Calorna market. They argue that orce o arbtrage acro the pot and orward market wll tend to make prce n thee market converge, and nd that uch prce arbtrage upported by ther data. 30 Gven thee obervaton, the author ue o an etmated margnal producton cot uncton or ol-uel generaton 3 a the PCBB or energy bd nto ether market nternally content. In the preent work, we do not aume arbtrage n the ene o Borenten, Buhnell, and Wolak, but ntead take 30 Over ther ample perod o June 998 to October 000, the PX average prce wa not gncantly greater than the ISO average prce (Borenten, Buhnell and Wolak 00, 384. I one alo nvoke the ratonal expectaton aumpton, under whch agent (unoberved ex ante expectaton are content wth ex pot realzed prce dtrbuton, then we may conclude that or ther ample perod, pot and orward market prce are equal n expectaton. See alo Borenten, Buhnell, Knttel, and Wolram (00. 3 The author ocu on redual demand that, total demand net o demand met by non-ol uel generaton n the market power analy. The etmated margnal producton cot or ol-uel generaton account or generator ecency and avalablty, uel cot, and varable operatng and mantenance expene.

40 demand to be trctly rk avere 3 (whle aumng upply to be rk neutral. Under thee more general crcumtance, t no longer clear that margnal producton cot the approprate PCBB or the orward market. Mult-ettlement market rae ue o bddng baed on opportunty cot and carcty, each o whch dtnct rom market power a dened above. Dtnguhng thee ue both conceptually and emprcally ha been the ubect o much debate and conuon. In the paragraph below, we brely contrat thee concept wth market power. Market power v. bddng baed on opportunty cot. Margnal opportunty cot ( MOC or a rm the margnal revenue rom the hghet-valued alternatve ale opportunty or an ncrement o output. In electrcty market, uch outde opton that, alternatve market opportunte or a gven ncrement o generatng capacty are the rule rather than the excepton. Such poblte may be due mply to geography, uch a the propect o exportng power outde o a gven regonal market. Alternatvely, the archtecture o electrcty market may oer thee opportunte, or example, the chance wthn a gven regonal market to ell ancllary ervce (ee n. 8, ntead o ellng nto a orward energy market. Each uch alternatve opportunty aocated, at leat n prncple, wth an MOC. When uch opportunte ext, the conceptually approprate benchmark or aeng the compettvene o market behavor (e.g., a rm SF bd would be the greater o margnal producton cot (MPC and MOC (ee, e.g., Borenten, Buhnell and Wolak 000, Th aumpton motvate the dervaton (n chapter 6 o the preent work o an endogenou orward market demand uncton. In our ramework, we may model rk neutralty o demand a a lmtng cae by permttng the parameter capturng demand rk averon (ee ubecton 6.. to approach zero. 3

41 In the mult-ettlement SFE model, there a (probabltc opportunty cot n makng orward market commtment, even though nancal contract may be unwound, or revered, n the pot market. Such opportunty cot are becaue o the chance that a generator mght contract orward to ell quantty q at a contract prce p, whch may turn out to be le than the later pot prce, p. In a compettve equlbrum, we hould expect th rk to be relected n contract reervaton prce (.e., orward market bd, both or rm exercng market power a well a or perectly compettve rm. Market power v. carcty. In a gven compettve market equlbrum, the derence between a partcular generator revenue and t total varable cot commonly reerred to a carcty rent. Scarcty rent contrbute to coverng generator xed cot. They are partcularly mportant or peakng generaton capacty, whch operate or relatvely ew hour each year. I ucent over tme, carcty rent can alo provde the neceary ncentve or nvetment ether by extng market partcpant or new entrant n new generaton, tranmon capacty, or demand management technologe. Abent capacty wthholdng, however, there no welare lo aocated wth the extence o carcty rent (rather, only a wealth traner, and thereore, no exerce o market power. The preent model aume no generaton capacty lmt, and o wll not addre the tradtonal noton o carcty drectly. However, a trctly ncreang margnal producton cot uncton whch we do aume erve, n eect, a a ot capacty contrant: t ncreae the average opportunty cot (ee above o rm orward market poton. In th ene, then, carcty wll play a role n the mult-ettlement SFE model analyzed here. 4

42 .5 Extng lterature A n other ettng, mple model o quantty or prce choce under the Cournot or Bertrand ramework have ormed the ba o many tude o electrcty market competton. A ubecton.3.3 explaned, we may vew the SFE ramework a a generalzaton o thee mpler compettve model, and one, n partcular, havng a greater degree o vermltude to the archtecture o many compettve electrcty market. Accordngly, n th ecton revew o relevant lterature, we ocu prmarly on ngle-ettlement SFE model and tude o mult-ettlement market. Kamat and Oren (00 cte addtonal relevant ource and provde a ueul overvew o recent work on market power n compettve electrcty market..5. Sngle-ettlement SFE model Klemperer and Meyer (989 ( KM characterze a Nah equlbrum n SF under uncertanty or an olgopoly; thee equlbrum SF map market prce nto a level o output. In ther model, uppler bd SF once nto a pot market that cleared multaneouly wth phycal producton. A noted above n ecton.3, the advantage o upply uncton a tratege a oppoed to xed prce or quantte that uch uncton permt uppler to adut output optmally a a uncton o prce n the ace o changng or uncertan condton, or example, uncertanty n demand. KM prove the extence o a Nah equlbrum n SF or a ymmetrc olgopoly. I the upport o the tochatc demand parameter n th model unbounded above, 33 there ext a unque, lnear SFE. 33 And margnal cot and demand are ane or ucently large quantty and prce, repectvely; ee Klemperer and Meyer (989, 6. 5

43 Green and Newbery (99 and Bolle (99 were the rt author to apply KM SFE ramework to model (ngle-ettlement electrcty market. Green and Newbery analyze the Brth electrcty upply ndutry whch, or everal year ollowng the 990 prvatzaton o the Central Electrcty Generatng Board, prmarly compred two domnant generatng rm. They nd, lke KM, a range o SFE when the poble varaton n demand bounded. Th range narrowed, however, when they urther aume the rm to be capacty-contraned. The author mulate the Brth pot market, and nclude ome cenaro that allow or compettve entry. They nd, dconcertngly, that even the lower-prced equlbra reult n conderable welare loe. Entry doe caue ncumbent to bd omewhat lower prce, although the cot n welare term o the addtonal nvetment exceve. 34 Bolle (99 mlarly conder SF competton n an electrcty pot market, although he doe o or a hypothetcal market ettng. Lke the prevou author, he nd a contnuum o SF oluton. In contrat to Green and Newbery, Bolle mpoe no non-decreang contrant on the equlbrum SF whch he derve. In ome o Bolle cenaro, the equlbrum SF are ndeed downward-lopng. Th ugget that uch a non-decreang contrant a common eature n real-world electrcty market may ndeed be bndng on SFE oluton. 35 A later paper by Bolle (00 ntroduce prce-entve bd uncton or both upply- and (ome demand-de market partcpant competng n a ngle pot market. He model the remander o the demand-de entte a non-trategc and havng a tochatc level o demand. Bolle nd that, th non-trategc component o demand 34 See Mankw and Whnton (986 or a uller expoton o th phenomenon. 35 More recently, Baldck and Hogan (00 have characterzed the eect o non-decreang contrant on SFE. 6

44 ucently large, equlbrum prce may be conderably above margnal cot. In addton, under th condton, market partcpant mght employ mxed tratege. udkevch, Duckworth, and oen (998 develop a ueul extenon o the ngle-ettlement SFE model o KM and Green and Newbery (99 dcued above. Namely, the author relax KM convexty and derentablty aumpton on rm margnal cot uncton, permttng thee to be tep uncton. For mplcty, the author conder the cae o dentcal rm. The central analytcal reult o the nvetgaton an expreon or the market prce that reult rom a ymmetrc Nah equlbrum n SF. Th prce depend on the (tepped ytem margnal cot uncton, ntantaneou demand, the maxmum demand n the relevant perod, and the number o rm. udkevch, Duckworth, and oen ue electrcty upply and demand data rom Pennylvana (n 995 to nvetgate the properte o th model. The author compute the average prce markup over hort-run margnal cot that reult rom SF bddng n Nah equlbrum. They oberve that, whle markup do decreae wth the number o rm n, electrcty prce n the model reman gncantly hgher than the hort-run margnal cot o generaton, even or relatvely large n. A an example, lettng n = 0, average markup over margnal cot tll %. For xed n, the author alo nvetgate how markup vary wth ( the level o capacty non-avalablty, and ( the relatve error n the day-ahead demand orecat, ndng that markup ncreae monotoncally wth both o thee actor. They conclude that current Common polce and U.S. anttrut gudelne may not be adequate to mtgate market power n bd-baed, compettve electrcty market. A commonly-cted dculty n applyng SFE model to electrcty market 7

45 ther computatonal ntractablty, partcularly when attemptng to model tranmon network nteracton. To overcome th problem, ome author have degned electrcty market model that are readly computable. Day, Hobb, and Pang (00, or example, ntroduce a conectured upply uncton (CSF approach whch, whle t reemble an SFE model n ome repect, more cloely akn to a general conectural varaton model. A CSF or a gven generatng rm repreent t ubectve bele concernng the aggregate reacton o t rval to a change n the market prce. Baed on a thrteen-bu model o the England and Wale tranmon ytem, the author nd that the CSF approach yeld market prce that are generally more content (p. 8 wth thoe actually oberved n England and Wale, compared to the Cournot model. The CSF model, however, ubect to the ame crtcm a other conectural varaton model. Frt among thee an ncontency between conecture and rm actual tratege, abent an explct requrement o content conecture (ee, e.g., Brenahan 98, whch Day, Hobb, and Pang do not mpoe. Other hortcomng nclude retrctve unctonal orm aumpton (the author ue ane CSF, and arbtrarne n the choce o conectural parameter (.e., ether the lope or ntercept o the ane CSF a well a n the conectured value o the choen parameter. Day and Bunn (00 oer another computatonal modelng and mulaton approach to undertandng trategc behavor among SF bdder n electrcty market. ather than ung SF that are everywhere derentable, the author ue a grd o dcrete prce and capacty level or each compettor over whch they dene pecewe 8

46 lnear SF. 36 Snce ully lexble SF o th orm produce a nonconvergent cyclng o oluton, 37 they mpoe a bounded ratonalty contrant on generator behavor, under whch a rm change the prce o only one or two o t plant each day. Day and Bunn apply ther methodology to analyze the 999 generatng capacty dvetture n England and Wale. In mulatng competton among the three ncumbent generatng compane and two hypothetcal purchaer o varyng porton o the ncumbent capacty, the author nd that the ncreae n the number o compettor rom three to ve ha a marked mpact on bd-cot margn. Interetngly, whether ncumbent dvet 5% or 50% o generatng capacty to the two new compettor o econdary mportance n term o the eect on bd prce. The author alo emphaze the eect o varyng degree o demand elatcty, concludng that at low elatcte o demand (e.g., n the hort run, the dvetture o 40% o ncumbent capacty that occurred n 999 n England and Wale would not mtgate ncumbent market power. Speccally, they nd prce n th cae n exce o 0% above hort-run margnal cot. In the longer run, naturally, they expect hgher demand elatcte and market entry to exert downward preure on thee bd-cot margn. ecently, the Calorna ISO and London Economc Internatonal LLC (003 have developed a comprehenve methodology and computer model or evaluatng tranmon nvetment that ncorporate trategc SF bddng wthn a tranmon network. The central conceptual problem that the model addree the nterdependence 36 Th dcretzaton acltate the applcaton o an optmzaton procedure to derve generator SF, updatng each rm SF n ucceve perod to maxmze prot baed on t rval current acton. 37 Day and Bunn conecture that th phenomenon ndcatve o the extence o mxed-trategy, rather than pure-trategy, Nah equlbra. 9

47 o optmal path or generaton and tranmon nvetment (temporally and patally, gven a compettve envronment characterzed by decentralzed, market-baed decon makng. The man element o the author methodology are mulatng mport and export o power; modelng avalablty, commtment, and dpatch o hydroelectrc and thermal generaton; characterzng the entry o new generator over tme; and modelng market power. egardng market power, the author ncorporate two complementary approache to modelng generator trategc behavor:. A game-theoretc model o trategc bddng (n a dcrete trategy pace n whch rm conecture that rval current bd are uncton o prot-maxmzng bd rom prevou teraton. An emprcal approach that etmate the htorcal relatonhp between data characterzng the tate o the market 38 and oberved prce-cot markup The author propoe to apply the methodology to evaluate the benet o a propoed expanon o tranmon capacty on Path 6, the tranmon lnk connectng Southern and Central Calorna. Whle the preent work not yet computatonally olvable n a network ettng, the mult-ettlement SFE model compare equlbrum tratege that are mutually content n all tate o the world. That, n the equlbra we tudy, a rm conecture concernng t rval trategy concde precely wth the rval actual trategy. We do retrct the numercal analy o chapter 7 to the cae o ane pot 38 Thee data nclude or each hour and zone the redual upply ndex, the total uncommtted capacty o the larget uppler, the ytem load, and eaonal and zonal dummy varable (Calorna Independent Sytem Operator and LLC

48 market demand uncton, ane margnal cot, and ane pot market SF..5. Mult-ettlement model Mult-ettlement model can capture the potental or rm to take advantage o nteracton between the orward and pot market. The preent work propoe an extenon o the upply uncton equlbrum ( SFE ramework developed by KM to a mult-ettlement market, whereby demand n each market uncertan. To the author knowledge, the preent work the rt attempt to ue the SF behavoral aumpton n a equental market ramework. A ntuton would ugget, the reultng SFE n th ettng no longer characterzed by a ngle SF or each uppler, a n KM model; rather, a equence o SF one n each market conttute a uppler ubgame perect Nah equlbrum (SPNE trategy n th mult-market ramework. 39 Other author (e.g., Allaz 987; Allaz and Vla 993 conder how the ntroducton o one or more orward market, cleared n advance o the pot market, aect compettor behavor and market outcome. O partcular nteret the eect on the pot market equlbrum: namely, do orward tranacton make the pot market more or le compettve? Allaz and Vla (993 nd that orward market tradng detrmental or rm and benecal or conumer. Moreover, a the number o orward market 40 get large, the outcome n ther model approache the compettve oluton. More recently, Ferrera (003 derve a contratng reult or the cae o an nnte number o orward market. Namely, he nd a et o ubgame perect equlbra that 39 Accordngly, unle otherwe peced, the moder equlbrum denote, throughout th work, the equlbrum concept o ubgame perecton (ee ubecton That, the number o perod n whch there orward tradng. 3

49 can utan any outcome between perect competton and the Cournot outcome. Lookng at how orward market hape behavoral ncentve, Allaz (987, 8 argue that there are multple ratonale or takng a orward market poton. Speccally, he dente three derent motve peculatve, hedgng and trategc motve or orward market partcpaton: Speculatve motve are rom an attempt to prot rom prce derence between varou market, or example, between a orward market and the pot market. A pecal cae o peculaton pure nancal peculaton, n whch rm do not aume a poton n the pot market; ntead, they ettle any orward oblgaton nancally baed on the pot market outcome. Hedgng motve come about rom rk averon n the ace o uncertanty. Hedgng amount to purchang nurance, n other word, acceptng a lower expected return n order to acheve a reducton n the varance o return. Strategc motve lead market partcpant to aume orward poton n order to nluence the pot market equlbrum. Allaz (987, 4 (n. 43 oberve that, under uncertanty, thee motve can partly overlap n the ene that, or example, the total poton taken [n equlbrum] n the uture market le than the um o the trategc and hedgng poton taken eparately. Allaz ocu prmarly on trategc conderaton, notng that n an olgopoltc ettng wth perect oreght (no uncertanty, the trategc motve become the only ratonale or orward tradng. Th becaue n an equlbrum under perect oreght, the uture prce wll be equal to the pot prce. Thu, no prot wll be made between the orward and pot market, elmnatng any peculatve motve. Furthermore, 3

50 becaue there no uncertanty n Allaz model, there alo no need to hedge. Allaz (and Vla rely prmarly on an aumpton o Cournot conecture rather than the SF we ocu on n th work. 4 In the preent work, we conder SFE competton n a mult-ettlement market. We derve a ytem o ordnary derental equaton mplctly characterzng rm optmal orward market bd, gven expectaton concernng the pot market. Thee bd wll not, n general, be ex pot optmal gven the realzaton o pot market demand. ather, we wll nd an ex pot orward market optmum aumng an ex ante expected optmum n the pot market. 4 We would thu expect that or the mult-ettlement market wth SFE bddng, trategc motve or orward market partcpaton wll be preent. In equlbrum, our uppler alo have peculatve motve 43 or partcpatng n both the orward and pot market. Under our aumpton that uppler are rk neutral, however, uppler have no motve to hedge. Chapter 8 contnue th dcuon, comparng the reult obtaned rom the preent model wth Allaz prevou work cted above. Later work by Newbery (998 examne a mlar but dtnct equental market ettng o orward (or contract, n h termnology and pot market or electrcty n England. H prncpal ndng concern the eect o contract on entry. Namely, contetable entry and a lqud contract market can enhance ecency by reducng welare loe due to the market power o ncumbent. Alo, capacty 4 In h 987 the, Allaz doe examne everal other behavoral aumpton or pot market competton, though not ncludng SFE. 4 See ubecton or a more prece tatement o th noton o optmalty. 43 They are not purely nancal peculator, however, nce they produce and ell output n the pot market. 33

51 contrant tend to attract entry and wll ncreae competton to the extent that new entrant can et prce. Newbery conder general (.e., nonlnear SF and ha all demand pang through the pot market; that, the contract n h model are purely nancal n nature, a are thoe we tudy here. Wth repect to the preent work, t Newbery modelng o competton n the orward contract market that o partcular nteret. In th market, Newbery ha generatng rm makng take-t-or-leave-t oer o a xed contract quantty at a peced prce to conumer. That, rather than an SF n the orward market, rm oer a pont n prce-quantty pace. Newbery analyze ratonal expectaton equlbra aumng rk-neutral trader, whch together mply that the orward contract prce an unbaed etmate o the ubequent pot market prce. 44 Newbery acknowledge that more complex contractual orm are poble whch could erve to reduce rk or (rkavere marketer who have commtted themelve to ellng at xed prce and are thu expoed to nput prce rk. He conclude, however (p. 734, n. 4, that [l]ttle would be added to the equlbrum-electon tory by conderng more complex contract. 45 A more complex contractual envronment that, contract baed on SF bddng n the orward market ndeed appote or modelng mult-ettlement electrcty market, although th ental addreng the ue o equlbrum electon to whch Newbery allude. Demand or orward contract n the preent model uncertan, mplyng that 44 In equlbrum, he nd that the term oered by uppler wll be uch that conumer are nderent between buyng and not buyng the contract; he reolve th kne-edge cae n avor o conumer purchang the oered contract. 45 Many non-cooperatve game ncludng the one developed here have multple equlbra. Equlbrum electon reer to the proce o wnnowng down the et o thee equlbra perhap to a unque equlbrum by nvokng plauble ( ometme ad hoc crtera uch a a Pareto rankng o equlbra, Schellng (960 ocal pont, tablty conderaton, etc. 34

52 uppler mut ubmt SF bd n order to repond optmally to th uncertanty. Moreover, the ratonal expectaton aumpton that, the orward market prce equal to the expected pot prce wll not hold, abent ucent rk-neutral agent n the model. Lke Newbery (998, Green (999a alo examne orward (contract and pot market or electrcty n England and Wale. In h paper, duopoly electrcty generator each chooe a quantty o contract n a orward market whle holdng a conectural varaton concernng the compettor orward market repone, tel a contract quantty. In the ubequent pot market, each rm bd an SF. Green doe not ully motvate h choce o aymmetrc behavoral aumpton between the orward and pot market: the aumpton o SFE n the pot market relect nttutonal bddng rule or the (now deunct Electrcty Pool, whle the aumpton o quantty choce n the orward market appear arbtrary. It may be that electrcty contract n England and Wale tend to pecy xed quantte over a wde range o prce, but Green lent on whether th o. In the pot market, Green retrct attenton to lnear SF, and a n Newbery (998 above, ha all demand pang through the pot market. He conder uncertanty n an appendx to the paper (Green 999b n whch the ntercept o a lnear demand uncton tochatc when uppler chooe contract quantte, but th uncertanty reolved beore uppler chooe ther pot market SF. The preent ramework der n everal mportant repect rom Newbery (998 and Green (999a:. Here, we aume that rm bd SF n both the orward and pot market. The aumpton o SF bddng n the orward market relect actual bddng 35

53 protocol 46 n centrally-cleared compettve electrcty market, and thu arguably a more realtc treatment o orward contractng n actual electrcty market. We oberve that, gven the opportunty n uch market, rm chooe to bd an SF rather than a xed quantty or prce. 47 Accordngly, our behavoral model need, at the leat, to accommodate and ndeed, uty uch a choce. Newbery concern wth equlbrum electon noted above relevant to the preent work a well, a we wll alo encounter multple equlbra n the general cae tuded here. 48 A mpled example (ee chapter 5 n whch we retrct the analy to conder only ane pot market SF ha a unque oluton n the pot market. In the orward market, a numercal approach to equlbrum electon appear to yeld unque optma. We cannot ultmately guarantee, however, that the orward market SF that we compute conttute globally optmal acton or each rm, rather than merely local optma.. There a demand uncton n both the orward and the pot market; each o thee demand uncton, n turn, ubect to exogenou uncertanty a rm ubmt ther SF bd. Forward market demand endogenou to the expected pot 46 Thee protocol (e.g., Calorna Power Exchange Corporaton (000, Schedule 4, Bddng and Bd Evaluaton Schedule, Secton 3.4 commonly pecy that partcpatng trader or uppler mut ubmt a trctly ncreang, pecewe lnear bd uncton n the hourly orward energy market. Th uncton gve the quantty o energy that the bdder wllng to upply a a uncton o the marketclearng prce. 47 In addton, to the extent that blateral contract have the character o SF that, a contract quantty that ncreae wth prce uch contract would alo lend themelve to beng modeled va the SFE ramework. 48 Indeed, the problem o multple equlbra wll be aggravated by our aumpton o SF bddng n the orward a well a the pot market. A Newbery (998, 733 wrte, t would eem natural to model each market a a upply uncton equlbrum, but not only there typcally a contnuum o uch equlbra, to each pot market equlbrum there typcally a contnuum o contract market equlbra, creatng a double nnty o oluton. 36

54 market equlbrum and to conumer prvate gnal concernng the level o pot market demand. Spot market demand alo are endogenouly gven the technologcal attrbute o conumer and ther utlty uncton. 3. We do not entrely retrct ourelve to ane SF, a Green doe (though we only olve the aorementoned ane example numercally. More recently, Battone (00 dertaton examne the eect o torage, orward market, and trategc behavor on compettve electrcty market. The author ocue on characterzng and aeng the rk to whch market partcpant are expoed, and orward contract role n hedgng th rk. In a two-perod model comprng a orward and a pot market, Battone nd that by exercng market power, trategc uppler can ncreae rk or conumer whle ncreang prot. Smlar to the approach n chapter 6 o the preent nvetgaton, Battone derve an endogenou, downward-lopng, orward market demand uncton, aumng that conumer are rk avere and that they know the dtrbuton o pot market outcome. 49 The author long-run equlbrum concept 50 (Battone 00, ubec. 0.. together wth h allowance or market detablaton (ch. amount to a cloed-loop 5 (or eedback model o the orward and pot market, an normaton tructure whch we nvoke n the 49 The author dtnguhe between the behavor o conumer whoe load unreponve to prce rom thoe whoe load prce-entve. Gven a pot market prce dtrbuton, a conumer n the ormer cla maxmze the utlty o her total cot o electrcty, whle a conumer n the latter cla maxmze the utlty o her net benet gven a pot market demand uncton. In the preent work, we abtract rom th dtncton among conumer. 50 Th equlbrum provde or a contency condton to cloe the model, under whch generator and conumer compute dtrbutonal moment o pot prce that are content wth ( the dtrbuton o hydrologcal uncertanty and ( the market equlbrum proce o prce ormaton. 5 See ubecton 3.. below or urther dcuon o the cloed-loop concept. 37

55 preent work, a well. The preent nvetgaton dtnct rom Battone model, however, n ome undamental repect. Frt, we allow or demand uncertanty n both the orward and pot market, whle Battone take demand n each o thee market to be determntc. The ole ource o rk acng uppler hydroelectrc generator n Battone model nput prce rk n the orm o margnal water value, whch are modeled a tochatc due to uncertan uture hydrologcal condton. In the preent work, n contrat, we aume that cot uncton are determntc. Second, content wth the aumpton o uncertan demand, the preent nvetgaton pot competton n SF n each o the two market, whle Battone aume Nah-Cournot conecture n both market. A llutrated by the lterature begnnng wth Green and Newbery (99 and Bolle (99, the SFE ramework developed here naturally uted to model bd-baed, mult-ettlement electrcty market. In the preent work, we ocu on theoretcal oundaton, uppreng all but the eental nttutonal detal o actual electrcty market. The central reult o th nvetgaton are the dervaton and computaton o. trategc uppler optmal bddng tratege and. the optmal behavor or a prce-takng conumer wthn the mult-ettlement market ettng decrbed above..6 Outlne o the the The next chapter, chapter, provde a conce overvew o the evoluton o regulatory polcy toward market power n the U.S. electrcty ector. Chapter 3 then develop the mult-ettlement SFE model. Ater ntroducng key concept ued n the model and ome notaton, th chapter poe the orward market optmzaton problem or a duopoly 38

56 uppler bddng SF n both the orward and pot market. In chapter 4, we olve th optmzaton problem analytcally or the general cae. The equental nature o the problem ugget backward nducton a a oluton algorthm. Th chapter derve condton that mplctly characterze the rm pot and orward market SF. To obtan an explct oluton or the repectve market SF, chapter 5 ntroduce a number o mplyng aumpton wthn the model: ane margnal cot and pot market demand uncton and ane pot market SF bd. Whle thee mplcaton ental ome lo o generalty, they erve to harpen the model reult. Next, chapter 6 pece the charactertc and behavor o conumer. In partcular, gven rk-avere conumer, t decrbe how (tochatc orward and pot market demand uncton mght are endogenouly. elyng on the mplyng aumpton o chapter 5, chapter 7 derve a ngular qualnear ytem o ordnary derental equaton characterzng the orward market problem and examne the qualtatve properte o oluton. For a pecc numercal example, th chapter then perorm comparatve tatc analy wth repect to the model exogenou parameter, and compare welare reult o the multettlement SFE model wth thoe o alternatve compettve aumpton and market archtecture. Baed on thee reult, chapter 8 argue that we mght ueully vew orward market poton a trategc commtment. It decompoe the motve or orward market actvty by uppler n the mult-ettlement SFE model nto three dtnct eect: a drect eect, a ettlement eect, and a trategc eect. Th chapter alo outlne ome extenon that would enhance the model realm and hghlght avenue or urther reearch. Numerou appendce to the the collect proo and other mathematcal reult. 39

57 The work I have et beore me th... how to get rd o the evl o competton whle retanng t advantage. Alred Marhall We leglate agant oretallng and monopoly; we would have a common granary or the poor; but the elhne whch hoard the corn or hgh prce, the preventatve o amne; and the law o el-preervaton urer polcy than any leglaton can be. Emeron, Nature: addree, and lecture The U.S. polcy repone to horzontal market power n electrcty generaton THIS CHAPTE analyze how publc polcy partcularly on the ederal level ha reponded to horzontal market power a electrcty ndutry retructurng ha progreed. 5 Secton. revew the htorcal evoluton o publc polcy toward merger and market-baed rate n the electrcty ndutry. Next, ecton. ocue on the comparatvely recent development o market power montorng and mtgaton actvte, and examne current polce n the varou regonal market acro the Unted 5 Buhnell (003b; Heronymu, Henderon, and Berry (00; and oach (00 each provde a ueul revew and crtque o polce to addre market power n the varou context condered n th chapter. 40

58 State. Secton.3 conclude.. Htorcal development A early a We (975, tude o electrcty ndutry tructure and regulaton remarked on the potental or the exerce o market power, n the event that regulaton o generaton were relaxed. 53 In ther emnal 983 book, Market or Power: An Analy o Electrc Utlty Deregulaton, Paul Jokow and chard Schmalenee devote an entre chapter (ch. to the ubect, examnng hort- and long-run competton n generaton and related anttrut ue. They note (p. 98 that long-run propect or market orce to reduce extng level o concentraton eem dm, and, regardng remede or potental compettve problem n generaton, oberve that [e]xtng anttrut rule may not be well-uted to the problem poed by deregulaton n th ector; the eature o better rule are not apparent. But the need to create better rule beore deregulaton clear. Thee obervaton orehadow the extenve conceptual and polcy debate on market power and on the approprate polcy repone n the latter hal o the 990 a the retructurng proce unolded. Htorcally, publc polcy toward market power n the U.S. electrcty ndutry took hape n two dtnct arena: ( regulatory revew o utlty merger and acquton, and ( the ue o market-baed (a oppoed to regulated rate by utlte. We outlne below the evoluton o polcy and the aocated analytcal methodologe n both o thee arena. 53 We note (p. 65 that horzontal acquton by the larget utlte... could have erou antcompettve eect, although he cautouly optmtc, on the whole, about potental econome rom retructurng and rom vertcal unbundlng, n partcular. 4

59 .. Merger Puruant to ederal and tate anttrut tatue, a varety o regulator requre rm to demontrate that propoed merger or acquton would not gncantly ncreae the lkelhood o exerce o market power. On the ederal level, the Common aume the leadng role n revewng electrc utlte merger applcaton; t mut approve thoe that are content wth the publc nteret (Perce 996, 30. In addton, the Anttrut Dvon o the U.S. Department o Jutce (DOJ and the U.S. Federal Trade Common (FTC may conduct ndependent revew to determne whether a propoed merger content wth U.S. anttrut law. Cutomarly, t ha been the Anttrut Dvon that undertake uch aement. ather than conductng t own revew, however, the Anttrut Dvon ha n practce lmted t actvty n the electrcty ector to occaonal partcpaton n the FEC nvetgaton (Frankena and Owen 994, 3. In mot tate, publc utlty common or tate attorney general revew propoed merger eect on retal conumer and tate utlty regulaton (Dmuke and Dmuke 996. Although the doman o nttutonal reponblty or merger revew are welldened by tatute and reaonably ettled n practce, the aocated analytcal ramework or aeng market power n merger proceedng ha evolved over the year along wth electrcty market archtecture and tructure (Federal Energy egulatory Common 996, 998. The opnon o the Federal Power Common (FPC, the FEC predeceor, n the Commonwealth Edon Company cae o 966 (Commonwealth 54 wa an early 54 Federal Power Common (966, 96, a d ub nom. Utlty Uer League v. FPC, 394 F.d 6 (7 th Cr. 968, cert. dened, 393 U.S. 953 (969. 4

60 landmark n the applcaton o anttrut prncple to electrc utlte. In Commonwealth, the FPC et orth x crtera that would gude t evaluaton o propoed utlty merger:. The eect o the propoed acton on the applcant operatng cot and rate level. The contemplated accountng treatment 3. eaonablene o the purchae prce 4. Whether the acqurng utlty ha coerced the to be acqured utlty nto acceptance o the merger 5. The eect the propoed acton may have on the extng compettve tuaton 6. Whether the conoldaton wll mpar eectve regulaton ether by... [the Federal Power] Common or the approprate tate regulatory authorty For many year aterward, thee x o-called Commonwealth crtera were nluental n the FPC (and, ater 977, the FEC 55 treatment o merger. The FEC approach began to change n the 980 (ee Perce 996, 3 wth the recognton that greater competton n wholeale electrcty generaton would be both poble and ocally derable. 56 The prmary obtacle to uch competton wa utlte ablty to exclude potental compettor (other utlte and ndependent [.e., non-utlty] power producer (IPP rom ther market by denyng them equal acce to ther electrcty tranmon lne. Havng no tatutory authorty to dmantle th compettve 55 The FEC wa created through the Department o Energy Organzaton Act on October, 977. It nherted mot o the uncton o the Federal Power Common whch wa elmnated by th Act. 56 Compettve and regulatory development n the natural ga ndutry (n whch wholeale ale and ntertate ppelne were alo under Common urdcton were urther advanced (Natural Ga Polcy Act o ; Federal Energy egulatory Common 985, 99a; Natural Ga Wellhead Decontrol Act o Increangly, ndutry oberver cted the accumulatng experence and leon rom natural ga a a promng model or electrcty ndutry retructurng (Perce

61 obtacle drectly, the Common reorted to requrng merger applcant to provde open acce to ther tranmon ytem under Common-approved term ( open acce tranmon tar. In th way, the Common began to addre vertcal contrant to competton by exercng t condtonng authorty on a cae-by-cae ba. 57 Thee merger proceedng naturally raed horzontal compettve ue, a well, whch ocued attenton on merger compettve eect (recall tem 5 n the Commonwealth crtera above. In the wake o the EPAct, utlte began to undertake merger and acquton at unprecedented rate a they reacted to economc and nttutonal change wthn the ndutry. It wa only n 996 wth t Inqury Concernng the Common Merger Polcy Under the Federal Power Act and t ubequent Merger Polcy Statement that the Common explctly recondered t applcaton o the Commonwealth crtera (Federal Energy egulatory Common 996a, 996. In t Merger Polcy Statement, the Common aerted that t wll generally take nto account three actor n analyzng propoed merger: the eect on competton, the eect on rate, and the eect on regulaton. [Further, the Common ] analy o the eect on competton wll more precely denty geographc and product market and wll adopt the Department o 57 See Surratt (998, Moot (996, 4 4, and Perce (996, or conce revew o merger proceedng and the ubtantve ue nvolved durng the late 980 and early 990. Later, puruant to the 99 EPAct, the Common Order 888 (Federal Energy egulatory Common 996d requred utlte under the Common urdcton to le open acce tranmon tar. 44

62 Jutce/Federal Trade Common Merger Gudelne [DOJ/FTC Gudelne 58 ]... a the analytcal ramework or analyzng the eect on competton [p. 3]. 59 Appendx A o the Merger Polcy Statement et orth the Common Compettve Analy Screen, whch detal a tandard analytc method and data peccaton to allow the Common to quckly determne whether a propoed merger preent market power concern. 60 The methodology or evaluatng a propoed merger under the Compettve Analy Screen center on comparng emprcal meaure o market concentraton 6 wth threhold value drawn rom the DOJ/FTC Gudelne. The rt tep n the analy to dene relevant geographc and product market and to meaure concentraton n thoe market. The next tep to evaluate pot-merger concentraton level and the (pre- to pot-merger change n concentraton ung the DOJ/FTC Gudelne concentraton threhold to ndcate problematc merger. Numerou analyt have taken ue wth the Common contenton that t Merger Polcy Statement content wth the DOJ/FTC Gudelne. Cox (999, 8 note, or example, that the Merger Polcy Statement ha been crtczed or not ollowng the DOJ/FTC Gudelne cloely enough, partcularly wth repect to the method or denng the relevant market. Frankena (998a, 30 3 goe urther, outlnng ve 58 Department o Jutce and Federal Trade Common ( Echoe o Commonwealth crtera, 5, and 6 rom page 43 are apparent n th excerpt rom the Common Merger Polcy Statement. 60 Federal Energy egulatory Common (996a, App. A,. 6 Lke the DOJ/FTC Gudelne, the Common Compettve Analy Screen propoe to meaure market concentraton by computng the o-called Herndahl-Hrchman Index (HHI or the relevant geographc and product market. For any market, the HHI equal to the um o rm quared market hare. The HHI ha two appealng properte: ( t account or all rm n a gven relevant market, and ( t gve greater than proportonal weght to larger rm market hare. Stot (00, 344 explan the relatonhp between the HHI and the Cournot compettve model. 45

63 repect n whch the Common Appendx A methodology dverge rom the DOJ/FTC Gudelne. Becaue o thee dcrepance, he conclude, the Common Appendx A analy doe not conttute a relable ba or determnng the need or anttrut hearng or or ahonng approprate remede. Frankena clam elewhere (998b,, moreover, that [t]he Appendx A methodology or denng geographc market lead to ubtantal volaton o the compettve analy creen tandard or ome merger that would not create or enhance market power, and the Appendx A methodology produce no volaton or ome other potental merger that would n act create or enhance market power. 6 Fnally, Morr (000, 76 contend that the Common Appendx A methodology appear to overtate a merger potental antcompettve eect. Compared wth the reult o a market mulaton model 63 ung the ame data et, Morr nd that the Appendx A methodology dente potental compettve concern that appear not to ext. The author argue that the dcrepancy are becaue unlke the mulaton model the Common methodology or market power analy n the merger context nherently unrelated to the economc realte o the marketplace. Moreover, everal promnent ocal have opned that extng law and regulaton are nadequate to addre market power, hould t are n the coure o 6 In partcular, Frankena argue (998b, that the Appendx A analy could ealy produce mleadng reult wth repect to tuaton nvolvng ( tranmon contrant that lmt purchae rom multple eller and ( eller that ace opportunty cot. The ue o opportunty cot central to undertandng competton n mult-ettlement market, a explaned n ubecton.4. o the preent nvetgaton. 63 Morr ue a tandard producton cot model or an electrcty ytem: namely, a lnear program that compute the producton cot-mnmzng dpatch to aty exogenou demand, wth an explct repreentaton o the tranmon network phycal properte ncluded n the contrant et. 46

64 electrcty retructurng. For example, Joel Klen, a recent head o the DOJ Anttrut Dvon (under the Clnton Admntraton, noted that [t]he anttrut law provde ample authorty or the Jutce Department to challenge antcompettve conduct o varou ort, but we cannot challenge market tructure tel. In other word, to whatever extent retructured electrc power market are too hghly concentrated to yeld prcng at or near compettve level, the anttrut law provde no remedy (Klen 998. Klen deputy, A. Dougla Melamed, later oberved that [t]he anttrut law do not outlaw the mere poeon o monopoly power that the reult o kll, accdent, or a prevou regulatory regme. Anttrut remede are thu not well-uted to addre problem o market power n the electrc power ndutry that reult rom extng hgh level o concentraton n generaton or vertcal ntegraton (Melamed Thu, apart rom remedyng any hortcomng n analytcal methodology, the Common may requre new enorcement authorty to mtgate ome ntance o electrcty ector market power outde the context o merger revew. Falure to create uch authorty may eopardze the ecency gan rom electrcty ector retructurng whle creatng olgopoly rent to uppler wth market power. The comprehenve energy bll ntroduced n the Senate n February 004 (Senate 004 would reorm and clary the Common merger authorty n everal way, but doe not provde explct gudance on the conduct o market power analy n merger cae. Frt, the bll rae the monetary threhold or merger and acquton to be 64 Melamed word echo the U.S. Supreme Court opnon n a emnal anttrut cae, Unted State v. Grnnell Corp. (384 U.S. 563 (966, whch etablhed that market power ( monopoly power, n the Court language attaned only rom growth or development a a conequence o a uperor product, bune acumen, or htorcal accdent not obectonable under the U.S. anttrut tatute. 47

65 ubect to Common revew. Next, n evaluatng whether a merger or acquton n the publc nteret, the Common to conder adequate protecton o conumer nteret, contency o the tranacton wth compettve wholeale market, and the eect on the nancal ntegrty o the tranactng parte (among other crtera that the Common may deem content wth the publc nteret. In addton, the Common to develop procedure or expedted revew o merger and acquton, dentyng clae o thee tranacton that normally meet thee publc nteret tandard. The Common requred to report annually to Congre any condton mpoed n the precedng year on utlty merger and acquton, and uty thee under a publc nteret tandard. Fnally, the Secretary o Energy charged wth tudyng the extent to whch the Common authorty under ecton 03 o the Federal Power Act to revew utlty merger and acquton duplcated elewhere, and wth makng recommendaton to elmnate any unneceary duplcaton or delay n uch revew... Market-baed rate A combnaton o deregulatory leglaton 65 and techncal advance n natural ga-red generatng technologe acltated the growth o a compettve threat to ncumbent utlte cutomer bae rom IPP and exempt wholeale generator (EWG. 66 Whle the EPAct provded only that IPP and EWG could ell wholeale power to utlte, thee producer along wth large conumer (e.g., ndutral plant had natural ncentve 65 In partcular, the Publc Utlty egulatory Polce Act o 978, the Natural Ga Polcy Act o 978, the Natural Ga Wellhead Decontrol Act o 989, and the EPAct. 66 Surratt (998, 4. Under the EPAct (U.S. Code, vol. 5, ec. 79z-5a, an exempt wholeale generator denote an electrc power producer (a utlty alate or an ndependent that ell electrcty at wholeale and that the Common ha exempted rom the provon o the Publc Utlty Holdng Company Act o

66 to purue drect retal ale arrangement wth each other. Known a retal wheelng, realzng thee tranacton uually requred acce to utlte tranmon lne. Under the EPAct, the rght to compel utlte to provde uch (retal acce wa reerved to the tate. By md-993 le than one year ater paage o the EPAct at leat eght tate had leglatve or regulatory proceedng underway examnng the mert o retal wheelng (Anderon 993, 6 8. One avenue that utlte purued to meet th compettve threat wa to eek authorty rom the Common to ue market-baed rate (.e., unregulated rate or wholeale power ale. Market-baed rate gve utlte lexblty wth repect to rate level and tructure, whch would be eental n retanng cutomer that were able, ncreangly, to chooe ther electrcty uppler. A t noted n the Ocean State Power cae (Federal Energy egulatory Common 988, 6979, the Common ha dcreton to depart rom cot-baed ratemakng when neceary or approprate to erve a legtmate tatutory obectve o the Federal Power Act. Ocean State Power alo document the htorcal evoluton o the Common market-baed rate polcy and outlne n general term the Common threhold tet or permttng market-baed rate (or, a characterzed below, market-orented prcng : Generally, the Common can rely on market-orented prcng or determnng whether a rate ut and reaonable when a workably compettve market ext,... or when the eller doe not poe gncant market power.... A eller lack gncant market power the eller unable to ncreae prce by retrctng upply or by denyng the cutomer acce to alternatve eller. Lack o market power the key prerequte or allowng marketorented prcng (p (reerence omtted. 49

67 The Dowell Lmted Partnerhp proceedng (Federal Energy egulatory Common 990 ( Dowell helped to dene urther the ubtance o the Common market power tet or market-baed rate cae. The background o Dowell wa a compettve olctaton o bd or electrcal generatng capacty n 987 by Vrgna Electrc and Power Company ( Vrgna Power. Baed on the olctaton, Vrgna Power agreed to purchae capacty rom the Intercontnental Energy Corporaton ( Intercontnental, among other uppler. Intercontnental later agned t purchae agreement to the Dowell Lmted Partnerhp ( Dowell, and Dowell led the market-baed rate propoed n thee agreement n late 989 wth the Common. In t Dowell Order, the Common held that [t]here are everal actor that lead u to conclude that both Intercontnental and t ucceor, Dowell, lacked market power over Vrgna Power. Frt, Intercontnental dd not own or control, and wa not alated wth any entty that owned or controlled, tranmon aclte wthn or around the Vrgna Power ervce area, other than thoe neceary to nterconnect wth Vrgna Power or th ale. Thereore, Intercontnental wa not n a poton to prevent Vrgna Power rom reachng competng uppler.... Second, there no evdence that Intercontnental or Dowell wa a domnant rm n any generatng market that mght be relevant to provdng capacty and energy to Vrgna Power.... Thrd, there no evdence that ether Intercontnental or Dowell controlled reource that allowed t to erect any other barrer to potental competng generaton uppler (Federal Energy egulatory Common 990, Throughout the ndeed, untl the Order n Federal Energy egulatory Common (00c, the Common would grant market-baed rate to an applcant the eller [.e., the applcant] and t alate do not have, or have adequately mtgated, market power n generaton and tranmon and cannot erect other barrer to entry (p. 6969, echong the tructure o the Common market power tet n Dowell. Only the econd component o the market power tet, that or generaton market power, ha 67 Surratt (998, 4 7 and akn (998b, 7 8 trace the evoluton o the Common market power analy through t varou decon n market-baed rate proceedng ocung, n partcular, on generaton market power. 50

68 requred that the applcant perorm an analytcal tet. 68 Th analytcal tet or generaton market power ha come to be known a the hub-and-poke tet. The hub-and-poke tet begn a do the DOJ/FTC Gudelne, dcued above by denng relevant geographc and product market. A Bohn, Celeb, and Haner explan (00, 53 54, The hub-and-poke tet dene relevant geographc market a the combnaton o applcant detnaton market (the hub plu the et o all market that are drectly connected to the detnaton market (the poke.[ 69 ] Product market are generally dened a ntalled and uncommtted capacty. The tet nvolve a comparon o the hare o generaton reource controlled by the applcant and t alate to that o all owner o generaton wthn the relevant geographc market.... The Common ha generally nterpreted a market hare o le than 0% a evdence o a lack o horzontal market power. The Common own Merger Polcy Statement (Federal Energy egulatory Common 996, 0 decrbed the hortcomng o the hub-and-poke analy: [The hub-and-poke method] dene geographc market n a manner that doe not alway relect accurately the economc and phycal ablty o potental uppler to acce buyer n the market [I]t doe not account or the range o parameter that aect the cope o trade: relatve generaton prce, tranmon prce, loe, and tranmon contrant. Takng thee actor nto account, market could be broader or narrower than the rt- or econd-ter entte dented under the hub-and-poke analy.... In other word, mere proxmty not alway ndcatve o whether a uppler an economc alternatve. 68 A or the other two component o the market power tet, the market-baed rate applcant and t alate have led an open acce tranmon tar wth the Common, th ha been ucent to demontrate the abence (or mtgaton o tranmon market power. egardng barrer to entry, the Common rele on an applcant repreentaton and publc polcng (Federal Energy egulatory Common 00c, Generaton market power analye ometme reer to the uppler connected to the detnaton market by thee poke a rt-ter (or ter one uppler. Second-ter (or ter two uppler are thoe uppler drectly nterconnected wth the applcant and whch the cutomer n the detnaton market can reach va the applcant open acce tranmon tar. See Federal Energy egulatory Common (99b, 6757 and Dalton (997, 35. 5

69 The Common dd not elaborate t reaonng why, depte the decence that t noted n the merger context, the hub-and-poke analy contnued to be utable or marketbaed rate cae. In a concurrng opnon ome year later, FEC Commoner Wllam L. Maey oered h perpectve on the decence o the hub-and-poke analy (Federal Energy egulatory Common 000, : I have come to beleve that [hub-and-poke analy] an anachronm. Th method ocue olely on the market hare o the ndvdual eller ntead o the condton n the market. It aume that all eller that are drectly nterconnected wth the cutomer, and all eller drectly nterconnected wth the applcant or market-baed rate, can reach the market, and market hare are evaluated on that ba. Th a back o the envelope approach, more or le. It take lttle or no account o the mportant actor that determne the cope o electrcty market, uch a phycal lmtaton on market ze ncludng tranmon contrant, prce, cot, tranmon rate, and the varance o upply and demand over tme. The hub and poke much too prmtve or thee tme. Clearly, the Common mut develop a more ophtcated approach to market analy, and I would recommend that we proceed genercally to do o. Speakng to the Energy Bar Aocaton one year later (Maey 00, 6, Maey mpatence wth the Common wa palpable: Any market partcpant that cannot pa [the hub-and-poke] tet need a new lawyer. How accurate can th tet be? How much ath can tate commoner have n our market baed prcng polcy we tll ue th hore and buggy analytcal approach? elyng on the hub and poke heer olly. Indutry analyt outde o the Common have alo weghed n regardng the law n the hub-and-poke approach. For example, Bohn, Celeb, and Haner (00, 54 hare Maey mgvng. They note that the Common ha generally contrued a market hare o le than 0% n the hub-and-poke tet a a lack o evdence o market power, though th gure ha not erved a a brght lne tandard. The author argue that th threhold concentraton level undamentally arbtrary. Crtcally, t al to denty electrcty uppler havng lower market hare that, when market are tght, 5

70 may be able to exerce market power. Perhap the mot detaled and vocerou crtque o the hub-and-poke approach Stot (00,. He demontrate that the hub-andpoke tet lawed n the ollowng repect: It geographcal market denton account only or a actor that no longer relevant and or none o the actor that matter n a compettve market. It ue o uncommtted-capacty hare regter more market power when the market tel more compettve and le market power when t le compettve. Thu t oten read n revere the mpact o the market on the applcant. It take no account o the central market-power problem o electrcty market: the nelatcty o demand. It take no account o the thouand-old luctuaton n upply elatcty that concentrate and nteny market power durng a ew crucal hour. It take no account o uppler becomng pvotal to the market. It would allow a ngle uppler to pa t creen although t poeed enough market power to ngle-handedly double the average year-round prce n a market a well behaved a PJM. It would allow multple uppler to pa although they would be capable o detroyng any current power market. Stot conclude that []uch a creen mnorm, erve no ueul purpoe and hould be mmedately dcontnued (p.. epondng to the growng datacton wth and crtcm o t hub-and-poke analy o generaton market power n market-baed rate cae, the Common 53

71 concluded n a November 00 order (Federal Energy egulatory Common 00c, 6969 that becaue o gncant tructural change and corporate realgnment that have occurred and contnue to occur n the electrc ndutry, our hub-and-poke analy no longer adequately protect cutomer agant generaton market power n all crcumtance. The hub-and-poke analy worked reaonably well or almot a decade when the market were eentally vertcal monopole tradng on the margn and retal load were only partally expoed to the market. Th order alo ntroduced a new analytcal creen the Supply Margn Aement (SMA to replace the hub-andpoke analy. In eence, the SMA creen evaluate whether a market-baed rate applcant pvotal n the market, that, whether at leat ome o the applcant capacty needed to aty the market peak demand. I an applcant deemed pvotal, t doe not pa the creen and t pot market energy ale wll be prced ung cot-baed rate; moreover, the applcant mut publcze proected ncremental cot data to help buyer make ratonal purchang decon. The SMA creen apple to marketbaed rate applcaton and trennal revew o market-baed prcng authorty on an nterm ba pendng a re-examnaton o the Common method o market power analy. 70 Sale o energy n FEC-approved ISO or TO, however, are exempt rom the SMA creen. Whle generally acknowledgng t mprovement over the hub-and-poke tet, a ew author have called attenton to potental drawback o the SMA creen. ohrbach, 70 Accordng to the Common (Federal Energy egulatory Common 00c, 6969, the SMA creen mprove upon the Common ormer hub-and-poke analy n two repect. Frt, the SMA creen conder the eect o tranmon contrant on geographc market denton. Second, the creen etablhe a threhold baed on whether a rm pvotal n t market. 54

72 Klet, and Nelon (00, or example, contend that the SMA [creen] doe not adequately reolve a number o crtcal ue and rae new one (p.. They oberve that the SMA creen doe not requre that the potental exerce o market power be protable (p. ; hence, rm that would not prot they were to exerce market power would tll not pa the SMA creen. Bohn, Celeb, and Haner (00, 54 note that the SMA creen mprove on the hub-and-poke tet by modelng relevant tranmon contrant va total traner capablty (TTC. 7 Nonethele, the ue o TTC ha t own drawback: the decence o TTC and related metrc baed on traner capablty due to the realty o loop low n the tranmon ytem and t aocated economc eect are by now well-known (ee, e.g., Hogan 99, 5 6; Harvey, Hogan and Pope 997, 8. Bohn, Celeb, and Haner ugget everal renement to the SMA tet to account or actor that t currently gnore, ncludng the ollowng: durnal and eaonal demand varaton, mport capablty when the applcant control capacty outde o the market under tudy, multaneou mport lmt (whch are not accounted or by TTC, colluve exerce o market power, deratng ntalled capacty or unt outage, lexblty n generatng plant operaton (.e., dtnguhng plant that may readly vary ther output rom nlexble e.g., nuclear plant, retal 7 The North Amercan Electrc elablty Councl(NEC (996 denton o total traner capablty (TTC, n eence, a ollow: The amount o electrc power that can be tranerred over the nterconnected tranmon network n a relable manner baed on... the ollowng condton:. For the extng or planned ytem conguraton, and wth normal (precontngency operatng procedure n eect, all aclty loadng are wthn normal ratng and all voltage are wthn normal lmt.. The electrc ytem are capable o aborbng the dynamc power wng, and remanng table, ollowng a dturbance that reult n the lo o any ngle electrc ytem element, uch a a tranmon lne, tranormer, or generatng unt.... [See the cted ource or addtonal detal]. 55

73 load oblgaton, and compatblty wth the Common Appendx A methodology appled n merger proceedng. To provde a venue or dcuon o the mert o the SMA creen, the Common convened a Techncal Conerence n January 004 (Federal Energy egulatory Common 004. The Conerence agenda ncluded geographc market denton, accountng or tranmon lmtaton, the approprate nterm creen or generaton domnance, and approprate mtgaton meaure or utlte that al the generaton domnance creen. A companon FEC Order to the November 00 SMA Order propoe revng extng market-baed rate tar by explctly procrbng antcompettve behavor and the exerce o market power (Federal Energy egulatory Common 00b,. The propoed tar provon a ollow: A a condton o obtanng and retanng marketbaed rate authorty, the eller prohbted rom engagng n antcompettve behavor or the exerce o market power (p. 4. The Order contnue, denng thee term: Antcompettve behavor or exerce o market power nclude behavor that rae the market prce through phycal or economc wthholdng o upple. Such behavor may nvolve an ndvdual uppler wthholdng upple, or a group o uppler ontly colludng to do o. Phycal wthholdng occur when a uppler al to oer t output to the market durng perod when the market prce exceed the uppler ull ncremental cot.... Economc wthholdng occur when a uppler oer output to the market at a prce that above both t ull ncremental cot and the market prce (and 56

74 thu, the output not old (p Varou commenter crtczed the above tar provon or vague denton o economc and phycal wthholdng, 73 argung that ull ncremental cot, n the Common parlance, would need to account or opportunty cot due to multple market acro tme, pace, and varou product (e.g., energy v. generaton reerve, and voced ear that th new meaure would create ncreaed regulatory rk, deterrng needed nvetment and entry n the ndutry. Inormed by ntervenor comment, behavor oberved n the Wetern market o the Unted State (ee, e.g., Federal Energy egulatory Common 003a, accumulatng experence wth other U.S. electrcty market (partcularly n the Eat, and FEC publc conerence, the Common ued an Order n November 003 (Federal Energy egulatory Common 003b ( MB Tar Order condtonng new and extng market-baed rate tar on eller complance wth x Market Behavor ule, 74 ummarzed below:. Generaton unt chedulng, bddng, operaton, and mantenance n complance wth Common-approved rule and regulaton. Prohbton on market manpulaton, that, tranacton wthout a legtmate bune purpoe that are ntended to or oreeeably could manpulate market 7 In repone to everal procedural moton hortly ater th order, however, the Common deerred the eectve date o the propoed tar provon (Federal Energy egulatory Common 00a and granted rehearng o the order or urther conderaton (Federal Energy egulatory Common 00b. 73 A common aerton made by commenter wa that, due to varou techncal and nttutonal eature o the ndutry, prce-takng behavor would lkely be m-claed under the tar provon a economc and phycal wthholdng, thu nvtng the charge that the rm n queton had exerced market power. 74 Appendx A to Federal Energy egulatory Common (003b, pp

75 prce, condton, or rule 3. Provon o accurate, actual normaton n communcaton wth the Common, market montor, TO, ISO, and tranmon provder 4. Accurate and actual reportng o normaton to publher o electrcty or natural ga prce ndce (to the extent that a eller engage n uch reportng 5. etenton o data and normaton that explan prce charged or electrc energy and related product or a three-year perod 6. No volaton or colluon wth another party n volaton o a eller market-baed rate code o conduct The Common receved numerou requet or rehearng o t MB Tar Order, and n January 004, t granted rehearng o th Order or urther conderaton (Federal Energy egulatory Common 004d...3 Dcuon Buhnell (003b, ha noted that typcally, regulatory decon to grant market-baed rate authorty had a greater mpact on the progre o electrcty retructurng n the Unted State than dd merger approval. In the ormer ntance, many entte applyng or market-baed rate power marketer, or example were and are not ubect to tate-level regulaton o retal ale. Grantng authorty to thee market partcpant to charge market-baed rate or wholeale ale amounted to the removal o the only contrant on uch rm prcng behavor. A or merger, thee have been between regulated utlte, or the mot part, o that both mergng parte a well a the new potmerger entty are ubect to retal rate regulaton. Th not to ay that merger approval are nconequental a a matter o polcy. 58

76 Undong a utlty merger once t ha been conummated would lkely be mply neable. On the other hand, revokng a utlty market-baed rate authorty would be a relatvely traghtorward matter, entalng only an admntratve order. Puruant to Federal Energy egulatory Common (996c, eller wth market-baed rate authorty are requred every three year to update the market power analy underlyng the grant o uch authorty.. Market power montorng and mtgaton.. Orgn In repone to the Calorna Publc Utlte Common retructurng order n December 995, 75 Calorna three nvetor-owned utlte (IOU led applcaton wth the FEC or market-baed prcng authorty. Ctng the FEC growng concern wth the mplcaton o tranmon contrant or geographc market denton n the context o market power analy, the three utlte propoed n ther ont lng 76 to account or uch contrant n ther (orthcomng market power analye. In the event, one o the three Calorna IOU, Pacc Ga & Electrc (PG&E Co., ubmtted a eparate market power analy, whle the other two, Southern Calorna Edon (SCE and San Dego Ga and Electrc (SDG&E, conducted a ont tudy. 77 Sgncantly, apart rom market power analy and ome recommended market power mtgaton meaure, 75 Calorna Publc Utlte Common (995, a corrected by Calorna Publc Utlte Common ( Pacc Ga and Electrc Company (996. Federal Energy egulatory Common (996b elaborate the Common concern regardng tranmon contrant. 77 Thee analye are Pacc Ga & Electrc Co. (996, and Southern Calorna Edon and San Dego Ga and Electrc (

77 thee two lng each contaned a propoed montorng program or market power. PG&E propoal (996, 4 recommended that a montorng program be admntered and run by a Complance Dvon o the [Calorna] PX, mlar to the complance dvon that ext wthn the tock exchange, a well a the New York Mercantle Exchange. 78 Smlarly, SCE and SDG&E recommended that the Common requre that [a] three-year montorng program, admntered by the [Calorna] ISO, be put n place at the tme the PX begn operatng. The montorng program would be degned to collect normaton on market behavor and perormance that the Common could rely upon to evaluate complant, analyze propoal to ne tune operatng detal, and come to a nal concluon that the market perormance meet the Common tandard or ut and reaonable rate (996. In t December 8 order (Federal Energy egulatory Common 996e, 7 8, the Common requred that the three utlte le addtonal normaton on ther market power mtgaton plan, agreed wth SCE and SDG&E earler uggeton (996, tranmttal letter 6 7 to convene a techncal conerence on market power mtgaton opton, and drected the Calorna ISO to le a detaled montorng plan, addreng the ollowng conderaton: Who reponble or the montorng; What normaton would be collected; What the crtera or dentyng the exerce o market power would be; What report and normaton would be ubmtted to the Common; and 78 The Calorna Power Exchange (PX wa an ndependent, non-prot entty degned to manage the orward energy market n Calorna n conuncton wth the ISO. The PX upended operaton o t day-ahead and day-o market on January 3, 00 and led or bankruptcy protecton on March 9,

78 What mtgaton acton would be taken the exerce o market power dented. Wth th polcy decon, the uncton o market montorng wa born. 79 One ndutry oberver aw two prmary motvaton underlyng the Common charge to the Calorna ISO to nttutonalze a market montorng capacty n t emergng compettve market (Lock 998a. Frt, early deregulatory reorm n Chle and n England and Wale notwthtandng, the Common recognzed how lttle experence had been ganed, to date, wth the propoed aucton-baed market. Second, the Common wa cognzant that a argued n ubecton.. above the anttrut agence lacked the tatutory authorty to addre many market power concern, whle the Common tel dd not have the techncal capacty to perorm eectve montorng. Another analyt ha argued that the Common order o December 8, 996 (Federal Energy egulatory Common 996e gnaled a gncant change n the Common polcy toward market power, n that the Common ntended to ht t ocu rom an analy o market tructure to relance on mtgaton meaure to enure that generaton owner would not exerce market power (akn 998a. In the year ollowng that order, the Common mpoed a mlar market montorng requrement or the three ISO n the northeatern Unted State ISO-NE, NYISO, and PJM a they developed ther market archtecture. 79 Whle th repreented the rt ncarnaton o market montorng n the Unted State, Lock (998b, 8 note that Alberta, Canada requred a part o the Alberta Electrc Utlte Act (Alberta Electrc Utlte Act o , ecton 9((d that [t]he Power Pool Councl [o Alberta] hall... montor the perormance o the power pool and change the rule o the power pool, neceary, to promote an ecent, ar and openly compettve market or electrcty. The ncepton o and early experence wth Alberta market urvellance ytem dcued n Barker, Tenenbaum, and Wool (997,

79 The Common placed market montorng on a more ecure nttutonal ootng wth Order 000 on egonal Tranmon Organzaton (999 whch propoed, among many other provon regardng management o the tranmon grd, that TO perorm market montorng a one o ther core uncton : 80 Speccally, TO would be requred to: ( montor market or tranmon ervce and the behavor o tranmon owner and propoe approprate acton; ( montor ancllary ervce and bulk power market that the TO operate; (3 perodcally ae how behavor n market operated by other aect TO operaton and how TO operaton aect thoe market; and (4 provde report on market power abue and market degn law to the Common and aected regulatory authorte, ncludng pecc recommendaton (Federal Energy egulatory Common 999, 435. Each o the ve FECurdctonal U.S. ISO (ee note created a pecalzed entty to perorm the market montorng uncton 8 a the ISO along wth all FEC-urdctonal publc utlte undertook to comply wth Order 000. The authorty and reponblte o thee market montorng organzaton are mlar though not dentcal acro the ISO. In general, ISO tar empower the market montorng organzaton to perorm the ollowng tak: The obectve o market montorng to denty any exerce o market power, abue o market rule, or market degn law. To th end, montorng organzaton collect data on the operaton o all product market (e.g., energy, 80 Whereby ether the TO tel or an ndependent entty created by or under contract wth the TO may carry out the montorng actvte. 8 We reer to thee entte genercally a market montorng organzaton and denty the partcular organzaton wthn the varou ISO n the next ubecton below. 6

80 reerve o varou knd, capacty admntered by the repectve ISO and n ome cae, blateral market on an ongong ba. Market compettvene and economc ecency are the overarchng tandard o nteret to montorng organzaton, or whch they have developed a varety o ndcator (ee PJM Interconnecton 00 or an example o a comprehenve lt. Whle pecc methodologe and analytcal procedure vary among the ISO, common ndcator nclude aement o generaton ownerhp concentraton where the relevant market account or tranmon contrant ung HHI (ee note 6, comparon o bd and market prce to unt-pecc cot data (accountng or the prce o uel and other nput, and ometme ung cot-baed dpatch mulaton model, change n bddng behavor over tme, and declaraton o generaton unt avalablty. Take correctve acton, or example, ome montorng organzaton can make prce correcton reultng rom otware or data entry error. ecommend change n market rule or n market montorng procedure to the governng board or takeholder commttee whch, approved, are then led wth the Common or regulatory approval. At the Common or anttrut enorcement agence n nvetgaton that they may undertake. Wth t Standard Market Degn Notce o Propoed ulemakng ( SMD NOP (Federal Energy egulatory Common 00a, the Common propoed three mandatory market power mtgaton meaure and one uch voluntary meaure a component o all urdctonal utlte (and TO open-acce tranmon tar (p. 63

81 . The rt meaure target local market power poeed, n partcular, by generatng unt that mut run to upport relablty o the tranmon network. At tme when uch unt have market power, ther bd hould be capped. The econd montorng provon o the SMD NOP a aety-net bd cap o $,000/MWh to apply at all tme and locaton, ervng a a check on the degree o generator economc wthholdng. Thrd, the SMD NOP envon a reource adequacy requrement on a regonal ba to enure relablty. Th requrement doe not addre wthholdng drectly; rather, t degned to dmnh the ablty and ncentve o uppler to practce and prot rom ether phycal or economc wthholdng (p. 3. The ourth, voluntary, meaure ntended to apply at tme when non-compettve condton ext. Market operator would examne uppler bd and, wthholdng rather than carcty reponble or the level o uch bd, pobly mtgate thee bd. Certan predetermned condton or trgger, or nratructural contrant 8 could prompt the mpoton o uch a mtgaton meaure. epondng to extenve comment on t SMD NOP, the Common ued n Aprl 003 a Whte Paper on a Wholeale Power Market Platorm (Federal Energy egulatory Common 003d outlnng t von or urther electrcty ndutry retructurng and ketchng propoed change to the SMD NOP. In th Whte Paper, the Common emphazed the undamental balance that market power mtgaton meaure mut trke, namely, to protect agant the exerce o market power wthout uppreng prce below the level neceary to attract needed nvetment n new nratructure... (Federal Energy egulatory Common 003d, 8. Speccally, 8 Such a drought n a ytem relyng gncantly on hydropower reource (p

82 TO tar would be requred, at a mnmum, to lmt bddng lexblty n the preence o local market power and to prevent market manpulaton tratege (p. 8. Together wth the montorng and mtgaton provon o Standard Market Degn noted above, a recent nttutonal nnovaton at the Common may encourage the development and applcaton o a coherent analytcal ramework or curbng market power. Namely, n January 00, the Common created the Oce o Market Overght and Invetgaton (OMOI, a new montorng unt at the ederal level or energy market. The OMOI ha a t mon to to protect cutomer through undertandng market and ther regulaton, dentyng and xng market problem, and aurng complance wth Common rule and regulaton (Federal Energy egulatory Common 004b. Among the OMOI uncton are (Federal Energy egulatory Common 004a: Undertakng market reearch, modelng, and mulaton; mantanng data reource n upport o overght and nvetgatory actvte Conductng analye o energy market, provdng early warnng o vulnerable market condton, and propong approprate polce Invetgatng poble volaton o Common rule and regulaton, recommendng remede to addre volaton and, where authorzed, purung thee remede Mantanng a orum or reolvng dpute normally and advng the Common on queton o enorcement and complance.. Montorng and mtgaton n regonal market ISO ta and budget devoted to market power montorng and mtgaton have grown 65

83 markedly over tme a the extent and complexty o montorng ha ncreaed (Peteron et al. 00, 0. Table. below provde an overvew o market power montorng and mtgaton organzaton and the protocol or plan that they mplement n each o the x ISO n the Unted State See alo Goldman, Leeutre, and Bartholomew (004, Knzelman (00, Power Pool o Alberta (00, 4, Peteron et al. (00, and Energy egulator egonal Aocaton (00 or a more detaled dcuon and comparon o ndvdual ISO montorng actvte and experence. 66

84 TABLE.: OVEVIEW OF ISO MAKET POWE MONITOING AND MITIGATION PLANS CAISO ECOT ISO-NE MISO NYISO PJM Date that Montorng organzaton operaton commenced Internal External March 3, 998 July 3, 00 May, 999 February, 00 December, 999 Aprl, 997 Market Survellance Unt (MSU ( , Department o Market Analy (DMA (999 None Market Montorng and Market Power Mtgaton Secton None Market Montorng Unt (MMU Market Montorng Unt (MMU Market Survellance Commttee (MSC a Electrcty Overght Board (EOB b Publc Utlty Common o Texa (PUCT Market Overght Dvon Independent Market Advor (IMA c Independent Market Montor (IMM d Market Advor e None Montorng protocol( ISO Market Montorng & Inormaton Protocol (MMIP (Calorna Independent Sytem Operator 003b; ISO Enorcement Protocol (propoed (Att. B o Calorna Independent Sytem Operator 003a ECOT Protocol, Secton 7: Market Data Collecton and Ue (ECOT 00; Order Adoptng New 5.90, 5.9 and 5.40 (Publc Utlty Common o Texa 000 New England Power Pool and ISO New England, Inc. (003 Module D: Market Montorng and Mtgaton Meaure (propoed (Mdwet ISO 004; g Attachment S-: Independent Market Montor etenton Agreement (Mdwet ISO 00b ISO Market Power Mtgaton Meaure (New York Independent Sytem Operator 004b h PJM Market Montorng Plan (PJM Interconnecton 003 Note: a The MSC an external, ndependent market advory body contng o three (later ncreaed to our expert n anttrut economc and ndutral organzaton a well a utlty law, regulaton, and operaton. b Calorna Electrcty Overght Board (EOB wa etablhed by Chapter 854, Statute o 996 (AB 890, comprng tate leglator and appontee o Calorna governor. The EOB ntal tak wa to elect the Board o Drector or the CAISO and PX. In addton, the EOB overee the actvte o the CAISO, and conduct analy and drat recommendaton regardng market operaton, ytem relablty, and nratructure plannng. c The ISO-NE Board o Drector retan an Independent Market Advor (IMA to provde market analy and advce drectly to the Board on makng the ISO-NE market more compettve and ecent. ISO New England ([n.d.] decrbe the crcumtance n whch the ISO Market Montorng and Market Power Mtgaton Secton typcally ue the ervce o the IMA. 67

85 Note to Table. (cont d: d The MISO Independent Market Montor (IMM ha experence and experte approprate to the analy o compettve condton n market or energy, ancllary ervce, and tranmon rght.... The IMM adve the MISO and report to the Common regardng the nature and extent o, and any mpedment to, competton n and the economc ecency o the Mdwet ISO Market and Servce;... (Mdwet ISO 00a, ec. 4. and 4.3. e The Market Advor n the NYISO ha experence and experte approprate to the analy o compettve condton n market or electrc capacty, energy and ancllary ervce, and nancal ntrument uch a TCC.... The Market Advor report to the NYISO Board o Drector on the nature and extent o, and any mpedment to, competton n and the economc ecency o the New York electrc Market... (New York Independent Sytem Operator 999, ec. 4. and 4.3. In Federal Energy egulatory Common (004e, the Common ordered the CAISO to mody th propoed Protocol. The CAISO obected to the requred modcaton, however, and requeted rehearng and clarcaton o the Common Order (Calorna Independent Sytem Operator 004. g Fled on March 3, 004 wth eectve date o December, 004, pendng Common approval. h Th veron o the Market Power Mtgaton Meaure or the NYISO wa ubmtted a part o a complance lng to the Common (New York Independent Sytem Operator 004a, puruant to Federal Energy egulatory Common (004c. It ha an eectve date o May, 004, contngent on t acceptance by the Common..3 Aement The evolvng tandard or merger and market-baed rate cae revewed n ubecton. ugget not only a dynamc electrcty ndutry, but alo a lack o conenu both wthn the ndutry or between the ndutry and the Common regardng approprate crtera and methodology or market power analy. At th wrtng, method or market power aement n market-baed rate proceedng reman ubect to rehearng ome two-and-a-hal year ater the Common rt propoed revng thee method (Federal Energy egulatory Common 004d. Commoner Brownell concurrng opnon n Federal Energy egulatory Common (003c emphazed that bac theoretcal queton pert n th regard: I... have a undamental concern that we ve allowed market to orm wthout a ull apprecaton o what conttute a market let alone the market dynamc that oter a truly compettve market. For example, what dene a compettve market and what conttute carcty prcng? Thee queton reman largely unanwered (p. 6. The SMD proceedng ha arguably harpened the ocu o 68

86 the debate on market power and a hot o other market degn ue whle alo hghlghtng the gncance o regonal derence n economc tructure, market development, and tmng o reorm (Federal Energy egulatory Common 003d, 3. The unreolved problem n th proceedng nclude, or example, o-called eam ue between regonal market wth repect to market power mtgaton, 84 among other matter. A or montorng and mtgaton meaure n the varou regonal market, n Federal Energy egulatory Common (004e, the Common drected the CAISO to mody t recently-propoed Enorcement Protocol (ee Table. ntended to complement the extng MMIP to conorm t to the Common earler MB Tar Order (Federal Energy egulatory Common 003b. In repone, the CAISO ha requeted rehearng and clarcaton (Calorna Independent Sytem Operator 004 o the Common Order. In the Mdwet, the MISO Market Montorng and Mtgaton Meaure are one component o a recent tar lng (Mdwet ISO 004, on whch the Common ha yet to rule. The model developed n the ollowng chapter motvated by the gap n the theoretcal oundaton or market power aement cted above, emphazng the mportance o electrcty market archtecture. Ultmately, th reearch hould contrbute nght to help clary the ongong market power debate relected n the varou admntratve proceedng dcued here. 84 [M]tgaton tool whch vary by regon acro market eam have the potental to create enorcement problem and underable behavoral ncentve (Federal Energy egulatory Common 003d, 9. 69

87 The cence do not try to explan, they hardly even try to nterpret, they manly make model. By a model meant a mathematcal contruct whch, wth the addton o certan verbal nterpretaton, decrbe oberved phenomena. The utcaton o uch a mathematcal contruct olely and precely that t expected to work. John von Neumann Electrcty cannot be made at, mortared up and ended, lke London Monument, or the Tower, o that you hall know where to nd t, and keep t xed, a the Englh do wth ther thng, orevermore; t pang, glancng, getcular; t a traveller, a newne, a urpre, a ecret, whch perplexe them, and put them out. Emeron, Eay and Englh Trat 3 A uppler orward market problem wth nancal contract THIS CHAPTE ntroduce the SF bddng model. We begn n ecton 3. below by ntroducng eental notaton and termnology to develop the model o uppler behavor. Secton 3. examne the nature o nancal orward contract and the cahlow that they ntroduce n market partcpant optmzaton problem. Next, ecton 3.3 poe uppler orward market problem. Secton 3.4 conclude by decrbng the backward nducton oluton algorthm or th problem. 3. The upply uncton bddng model: Notaton and termnology Th ecton ntroduce nomenclature that we ue to dene the SF bddng model n the orward market; we develop th problem ormally n ecton

88 3.. Tmng and normaton tructure o equental market We nterpret the mult-ettlement SFE model a a two-tage game o complete 85 but mperect normaton. In perod, rm multaneouly ormulate ther orward market tratege that, ther SF bd; th market clear at t = wth the revelaton o the uncertan orward market demand uncton. Subequently, n perod, rm oberve orward market outcome and (agan, multaneouly ormulate ther SF bd or the pot market, whch clear when the uncertan pot market demand uncton revealed at t =. Fnally, producton take place n perod Fgure 3. below hghlght thee eature o the model: Perod : Formulate orward market SF bd (Frt tage game Perod : Formulate pot market SF bd (Second tage game Perod 3: Producton take place t FIGUE 3.: t = : Demand uncertanty n orward market reolved, clearng th market. Forward market prce and SF revealed, rom whch orward market quantte may be computed. t = : Demand uncertanty n pot market reolved, clearng th market. Spot market prce and SF revealed, rom whch pot market quantte may be computed. CONVENTIONS FO THE TIMING OF FOWAD AND SPOT MAKET BIDDING IN A SINGLE OUND OF THE MULTI-SETTLEMENT SFE MODEL We conder only a ngle round o play, contng o the ollowng equence o event (ee Fgure 3. above:. In perod, upply-de market partcpant ormulate an SF bd or the orward 85 Although a explaned below demand n both the orward and pot market uncertan, rm ue o SF a tratege and the extence o common pror probablty dtrbuton eectvely oet thee two ource o uncertanty. See ubecton 3.. below on Equlbrum concept or urther explanaton. 86 Snce we aume that rm commtment n the orward and pot market are bndng, rm do not ace any addtonal decon aocated wth producton n perod 3. Thu, we may neglect perod 3 or the purpoe o our analy. 7

89 market. At the end o the perod (t =, the market clear wth the revelaton o the orward demand uncton, whch et the orward market prce. Alo, rm orward market SF are revealed at th pont, rom whch rm orward market quantte may be computed.. The analogou equence o event occur n perod or the pot market. 3. Producton occur n perod 3. We aume that, a rm ace orward market competton n Perod, they begn the round wth no contractual poton ex ante. Fnally, (n ether market a marketclearng prce doe not ext or not unque, we aume that every rm then earn zero prot n that market. 87 Although the ngle round o the game depcted n Fgure 3. would n a typcal compettve electrcty market be repeated hourly, we abtract n th the or mplcty rom what, n realty, a repeated game. Th a trong mplcaton, a we thereby dpene wth undamental eature o repeated game that are generally compettvely gncant. Thee nclude threat and punhment tratege and evolutonary phenomena uch a learnng and reputatonal eect. Nonethele, the analy o the tatc (two-tage game an eental rt tep toward more realtc model o behavor n what, n realty, a dynamc ettng. The tmng o the mult-ettlement market game n Fgure 3. relect our aumpton that rm can oberve perod acton and outcome beore commttng to 87 We make th aumpton ollowng KM (Klemperer and Meyer 989, 47 (n. 8, who note that t enure, n the ngle-market SFE model, that uch outcome do not are n equlbrum. It not a crtcal aumpton, nce the equlbra that they conder reman equlbra or reaonable alternatve aumpton regardng rm payo n the ace o multple equlbra. We expect that th wll be the cae, a well, or the mult-ettlement SFE model examned here. 7

90 perod acton. Th eature o obervable acton and outcome n a mult-tage game mple a cloed-loop normaton tructure (ee Fudenberg and Trole 99, 30, n whch player can condton ther perod (pot market play on perod (orward market acton and outcome; we call the correpondng tratege cloed-loop tratege. In any cloed-loop equlbrum, 88 rm pot market bd gven any orward market bd and outcome mut be a Nah equlbrum o the pot market tage game. When choong ther orward market bd, rm naturally recognze that optmal pot market bd wll depend on orward market bd and outcome (ee Fudenberg and Trole 99, 3. Identyng the orm o th dependence and t mplcaton or the multettlement SFE model a gncant part o th chapter analy o the mult-tage game. Indeed, the cloed-loop aumpton the natural normaton tructure to aocate wth the mult-ettlement SFE model. In th model (all the more o nce n realty, th a repeated game ettng, rm wll recognze that optmal pot market acton or themelve and or ther rval wll depend on thoe n the orward market. The (polar alternatve to the aumpton o cloed-loop tratege would be to aume open-loop tratege, whch preuppoe that player oberve only ther own htory o play; accordngly, open-loop tratege depend only on tme. Open-loop tratege are generally eaer to analyze nce they produce mpler optmalty condton (wthout ntertemporal eedback term and nce the open-loop trategy pace oten much maller. Open loop tratege are alo oten computed a benchmark or examnng trategc eect, that, ncentve to nluence a rval uture acton through 88 Adaptng Fudenberg and Trole denton (99, 3, we take a cloed-loop equlbrum to mean a SPNE o a game n whch player can ( oberve opponent acton and realzaton o uncertan parameter ater each perod, and ( repond to thee revelaton n ther uture play. 73

91 one own current acton (Fudenberg and Trole 99, 3. The open-loop aumpton le realtc n th normaton-rch envronment, however, o that we ue the cloedloop aumpton excluvely n the preent work. 89 Aumng that rm orward market SF are perectly obervable may eem lke a trong aumpton; ndeed, ytem operator do not mply announce thee SF n the coure o market operaton. There are everal reaon, however, why obervablty may ndeed be a plauble aumpton wthn the context o compettve electrcty market: ( the long htory o economc regulaton wthn the ndutry ha generated a rch array o data and analye concernng producton technologe (peccally, cot and demand orecatng; ( the perodc nature o thee market provde an deal envronment or learnng about compettor hort- and long-run tratege; and (3 market authorte cutomarly make market data publcly avalable (albet wth a ew month delay and uually n aggregate orm rom whch at leat approxmate model o the behavor o one rval mght be nerred. 3.. Equlbrum concept Our ue o ubgame perecton a an equlbrum concept are becaue o the equental nature o the game depcted n Fgure 3.. In perod, rm antcpate that the repectve pot market SF choen later n perod wll be n Nah equlbrum 90 wth each other; 89 Beyond the extreme o open- and cloed-loop tratege, a more lexble and arguably more realtc aumpton regardng normaton tructure would be mperect obervablty. We ave th cae or uture work, however, and ocu n the preent model on the benchmark cae o perectly obervable acton. On the relatonhp between obervablty and trategc ncentve n dynamc game, ee the dcuon o ubecton For now, aume that there are multple Nah equlbra n the pot market ubgame, rm ucceully coordnate on the partcular pot market equlbrum to be antcpated (ee n. 3. We addre queton o equlbrum extence and unquene later n chapter 5 and 7. 74

92 mantanng th uppoton, rm contruct ther orward market SF. In a orward market equlbrum, thee orward market SF wll themelve be n Nah equlbrum wth each other (condtoned on the aorementoned pot market equlbrum. In perod, rm chooe ther pot market SF whch wll, n act a antcpated conttute a Nah equlbrum n the pot market ubgame. Fnally, or mplcty, we conder only pure trategy equlbra. A the oluton o the orward market problem (ee chapter 4 how, rm tratege depend, n general, on the probablty dtrbuton o pot market outcome 9 and alo on the relatve prot aocated wth thee outcome. In contrat, rm orward market acton are ndependent o the probablty dtrbuton o orward market outcome. That, a wth the SF n KM ngle-market SFE model, orward market acton n the mult-ettlement SFE model are ex pot optmal n every tate o the orward market. Snce we wll aume that all normaton publc (ncludng, n partcular, rm cot and the aorementoned probablty dtrbuton, there no ncomplete normaton. Becaue we aume the ue o cloed-loop tratege wth obervable acton and outcome, rm wll repond optmally both to the realzaton o random varable a well a to other acton n prevou perod. Th condton ucent to permt the ue o ubgame perecton n leu o perect Bayean equlbrum (PBE a our equlbrum concept. 9 I, n contrat to th ettng, rm 9 See ubecton 3..0 on Demand uncton below or more on how uncertanty enter th problem. 9 The equlbrum concept o PBE typcally apple to mult-tage game o ncomplete normaton. Becaue o ncomplete normaton, the bele o player need to be characterzed n equlbrum n addton to player tratege. The PBE concept (Fudenberg and Trole 99, 36 cont, then, o a et o tratege and bele uch that, at all tme, ( tratege are optmal gven the bele and 75

93 acton were not perectly obervable or probablty dtrbuton o the uncertan parameter depended on ubectve bele, then PBE mght be the approprate equlbrum concept Indutry tructure and rk preerence We aume that the ndutry a duopoly (n =, and that both producer are rk neutral. We ndex producer by =, ; unle otherwe peced, the ndex range over thee two value Prce Let m p be an arbtrary prce n market m, where m =, (or the orward and pot market, repectvely. 93 Denote an ex pot actual (or realzed prce n market m by a caret: p ˆ m Supply uncton The SF that we conder or each market wll map prce (pobly together wth other parameter, a wll be dcued below nto quantty uppled by the rm n queton. A n Klemperer and Meyer (989, 50 analy, we may ntutvely characterze a rm SF n a gven market a the et o the rm optmal 94 prce-quantty pont a t ( the bele are obtaned rom equlbrum tratege and oberved acton n accordance wth Baye ule. 93 We ue a upercrpt m =, a an ndex on everal market-pecc varable and parameter to aocate thee wth the orward and pot market, repectvely. The varable aterk * to denote optmalty, we wll ntroduce optmal prce uncton 76 m p a calar; later, ung m* p (ee chapter A we wll ee below, we employ a dtnct noton o optmalty n each o the two market n the model. Secton 3.4 epecally ubecton elaborate. For now, t uce to nterpret the quantte reultng rom SF mply a optmal n ome meanngul ene.

94 redual demand uncton vare due, ay, to exogenou uncertanty n demand aumng that t compettor trategy xed. Th property o SF mple that, ndependent o the tate o the world that ultmately realzed (we take p m a a convenent tate varable n the preent dcuon, the rm guaranteed to upply the optmal quantty gven th prce,, n the tage game n queton, t chooe t SF a t acton. By contructon, thereore, SF are nvarant to the tate o the world, and repreent ex-pot optmal acton n every tate o the world. The mult-ettlement market ramework, together wth the SPNE concept, requre that we ntroduce ome addtonal termnology to dtnguh the varou SF contruct that are n th problem. The ollowng dcuon dtnguhe SF along varou dmenon: provonal v. admble upply uncton mputed v. optmal upply uncton equlbrum upply uncton (appled only to optmal upply uncton We next motvate and dene each o thee term, and explan how thee varou type o SF are n the mult-ettlement SFE model. Provonal v. admble upply uncton. Th dtncton are n the multettlement SFE model due to our aumpton o cloed-loop tratege n SF, but doe not appear n the analogou ngle-market model o KM. For our two-market game, th dtncton between provonal and admble SF apple only to pot market SF. That, or the pot market we have both provonal and admble SF, whle or the orward market we have only admble SF. 77

95 Each rm conceve o t provonal pot market SF contemporaneouly (n perod wth the contructon o t orward market SF. Conceptually, we may contruct the provonal pot market SF or rm va the ollowng two-tep proce:. Fx a tate o the world n the orward market and mpute a pot market acton to.. Compute the (optmal pot market SF or. Then, we repeat tep and above or every poble tate o the world n the orward market. Each element o the et o pot market SF o computed then a proecton o rm provonal pot market SF nto the pot market prce-quantty plane, ndexed by the correpondng tate o the world that generated t. We now denote rm provonal pot market SF a Σ ( p ;. In th notaton, the ubcrpt =, ndexe rm and the upercrpt denote the pot market. The lt o argument or Σ, ( ; p, ndcate that thee argument wll nclude p n addton to other argument characterzng the orward market outcome that reman to be determned. Thu, by th (ncomplete peccaton, the dmenon o the doman o 78 Σ wll be greater than one. Th act a relecton o the cloed-loop property, dcued above, wth whch we have endowed thee tratege. In order or perod acton to depend optmally on event n perod, we mut permt the argument o Σ to relect thee perod event. Secton 3.3 below wll complete the peccaton o the argument o Σ ( p ; the cloed-loop SPNE o the mult-ettlement SFE model. approprately or A Fgure 3. depct, n perod and rm ormulate and ubmt to the market-clearng authorty ther orward and pot market SF, repectvely. In contrat to

96 the provonal SF dcued above, we dene an admble SF or rm (or, equvalently, rm bd a any SF n ether market that content wth exogenouly peced market rule that determne the allowable orm o bd. In gametheoretc term, thee market rule etablh the acton pace that rm may ue to partcpate n each market. Frm ubmt admble SF to the market-clearng authorty; m m we denote uch an admble SF mply a ( S p. 95 Here, the ubcrpt agan ndexe rm, whle the upercrpt m =, now denote the orward and pot market, repectvely. We aume that thee market rule requre rm SF ubmtted bd to be twce contnuouly derentable, 96 m trctly ncreang uncton S :, o that S m m m ( p 0 >. Thee uncton map market m clearng prce p m nto the quantty m ( S p that the rm wllng to upply (or, n prncple, purchae at th prce n market m To prevew the argument n ubecton 3.4.3, gven a parameterzaton and actual parameter value or optmal provonal pot market SF Σ ( p ; market SF are related to each other, ex pot, a ( ( ; 79, optmal admble and optmal provonal pot S p =Σ p p. 96 Pecewe derentablty (e.g., a pecewe lnear plne, a n the (ormer Calorna PX; ee note 46 or pecewe contnuty (e.g., a tep uncton a more lkely bd retrcton n actual electrcty market. We can, o coure, approxmate uch uncton arbtrarly cloely almot everywhere wth a contnuouly derentable uncton, o we ue the latter a an approxmaton o what realtc bd mght look lke. 97 Whereby negatve quantte would mply a net purchae, rather than a ale, by uppler. A Klemperer and Meyer (989, n. explan, retrctng rm n ther model to choong nonnegatve quantte at all prce would yeld the ame reult, but would complcate the analy by permttng redual demand uncton that are not everywhere derentable. We mlarly permt negatve quantte n ether market, n prncple. In the pecc numercal example o chapter 7 (ee, n partcular, problem (7.58 and (7.6, however, we exogenouly retrct pot market equlbrum quantte q to be nonnegatve, or mplcty. In contrat, orward market equlbrum quantte are not o retrcted. Becaue the orward market purely nancal n nature (ee ecton 3., negatve orward market quantte are unproblematc and are not precluded n the mult-ettlement SFE model. We wll ee n

97 The mult-ettlement SFE model ue provonal pot market SF beore reoluton o orward market uncertanty, and admble pot market SF ater reoluton o th uncertanty. To derve the provonal pot market SF n Perod, we ue mathematcal expectaton to accommodate pot market uncertanty, 98 whle we optmally account or orward market uncertanty va the orward market SF. Later, n Perod, orward market uncertanty ha been reolved and we then derve admble pot market SF that are optmal gven orward market acton and outcome. Note that rm do not actually ubmt the optmal provonal pot market SF to the marketclearng authorty; we compute them olely becaue, a we argue n ecton 3.3 below, optmal admble orward market SF depend on them. Imputed v. optmal upply uncton. Th dtncton are n the multettlement market SFE model, and alo n the ngle-market SFE model (e.g., that o KM. In determnng the Nah equlbrum n SF n each tage game o the multettlement SFE model, we pot that each rm aume an SF an mputed SF on the part o t compettor, and then determne t own optmal SF gven th aumpton. Th equence o tep o mputaton and optmzaton occur once or each o our two market: or the orward market n perod, and or the pot market n perod (ee Fgure 3.. So, or each rm and n each market, we wll have both mputed and optmal chapter 7 that, gven the lope retrcton S ( p 0 > noted above or the orward market, orward market SF over reaonable prce range tend, n any event, to produce potve orward market quantte. In prncple, market nttuton dene the crtera or admble SF, mpong addtonal retrcton apart rom ncreangne or example, pecewe lnearty, mnmum and maxmum prce m level, etc. on ther orm. Beyond the above denton o S, we do not mpoe any uch retrcton ex ante, but expect a KM nd that certan properte characterzng equlbrum SF wll emerge endogenouly. 98 Eq. (3.35 n ecton 3.3 wll make th noton more prece. 80

98 m m SF. We denote mputed SF wth tlde, and o wrte S ( p mputed admble SF n market m. Smlarly, Σ ( p ;, m=,, or rm denote rm mputed provonal pot market SF. For contency wth our aumed market rule, we aume here that the SF that rm mpute to ther rval wll be trctly ncreang n p, that, m m ( p 0 S > ( m, Σ ; > = and ( p Equlbrum upply uncton. We apply the moder equlbrum to optmal SF n ether market that alo conttute a Nah equlbrum that, a par o optmal SF, each o whch a bet repone to the other n all poble tate o the world. We add an upper bar to the notaton or an optmal SF to denote an equlbrum optmal SF. Thu n perod, we may derve rm equlbrum optmal provonal pot market SF, Σ ( p ; (, and t equlbrum optmal admble orward market SF, S p (aumng that uch equlbra ext. Analogouly, n perod, we may derve rm equlbrum optmal admble pot market SF, ( S p (agan aumng extence. Fnally, rm SPNE trategy or the mult-ettlement market SFE model cont o a et o SF, one or each market, namely,. an equlbrum optmal admble orward market SF, S ( p. an equlbrum optmal provonal pot market SF, Σ ( p ; argument 99 In the expreon Σ ( p ; p., the prme ( denote derentaton wth repect to the 8

99 That, or the tme beng, we dene (or now the SPNE a ollow: 00 { ( ( } SPNE or the two-player, S p, Σ p ;, =, (3. mult-ettlement market SFE game. In olvng the mult-ettlement SFE model, the natural ocu on the conttuent tratege o the SPNE (3.. Thu, where we may economze on termnology wthout ambguty, we uppre the decrptve moder equlbrum, optmal, and admble appled to SF. That, we conder SF to be equlbrum and optmal SF unle otherwe peced. Accordngly, n the orward market, we generally reer to an equlbrum optmal admble orward market SF a mply a orward market SF. In the pot market, n contrat, we reer to an equlbrum optmal (provonal or admble pot market SF a mply a pot market SF. Here, the provonal-admble dtncton hould be clear rom the context n whch the pot market SF appear, and rom the notaton ued. Nonethele, or clarty n what Σ ( p ; 00 We wll revt the SPNE denton (3. n ecton 3.3 below, once the peccaton o complete. 8

100 ollow, we add the moder provonal or admble to decrbe pot market SF where approprate. Table 3. below ummarze all o the dtncton among the varou SF contruct ntroduced above. TABLE 3.: A TAXONOMY OF SUPPLY FUNCTIONS IN THE MULTI-SETTLEMENT SFE MODEL FO FIM Perod Perod Forward market tage game problem: Each rm ormulate t orward market SF bd Aumed exogenouly: Computed endogenouly: Imputed provonal pot Optmal provonal pot market market SF Σ ( p ; SF Σ ( p ; Imputed admble orward Optmal admble orward S p S p market SF ( Forward market tage game equlbrum acton: market SF ( Equlbrum optmal admble orward market SF S ( p ( orward market SF, aumng equlbrum optmal provonal pot Σ p ; ( provonal pot market SF market SF ( Spot market tage game problem: Each rm ormulate t pot market SF bd Aumed exogenouly: Computed endogenouly: Imputed admble pot Optmal admble pot market S p S p market SF ( Spot market tage game equlbrum acton: SF ( Equlbrum optmal admble pot market SF S ( p ( admble pot market SF SPNE or the two-perod game (mult-ettlement market SFE: S p, Σ p ; { } Sequence o equlbrum optmal SF, one or each market: ( ( 3..6 Quantte Dene m q a a quantty uppled by rm n market m. Th quantty mply the m m m rm SF evaluated at ome prce n market m, that, q S ( p. Ung the mputed 83

101 admble SF or market m, we dene a correpondng mputed quantty or rm n m m m market m, q S ( p. Smlarly, dene rom market m equlbrum optmal m m m admble SF the equlbrum quantty, q S ( p. Fnally, we denote the ex pot actual (or realzed quantty awarded to rm n market m (not necearly an equlbrum quantty wth a caret: q ˆ m evenue Let revenue o rm n market m be m, o that = p q. (3. m m m 3..8 Cot uncton Let the cot uncton or rm producton be ( C q or q 0 (whereby producer cot uncton may der. We let th cot uncton pa through the orgn, o that we conder only varable cot. Let ( C q be twce derentable (except perhap at the orgn and be common knowledge. We aume that margnal cot C( q ncreang or potve quantte, that, C ( q 0 trctly > or q > 0. We aume urther that there are no capacty contrant on rm productve capacty; n other word, C( q reman nte or arbtrarly large q. Note that or mplcty, th ormulaton abtract rom the non-convexte ntroduced by tart-up cot, no-load cot, and ramp rate lmtaton. For any tate o the world, the argument o rm cot uncton, q, equal to 84

102 rm pot market redual demand uncton evaluated at the pot market clearng prce, p Prot We take prot n ether market to mean operatng prot, that, hort-term revenue le varable producton cot. Th conventon treat all xed cot a unk and thu rrelevant to the preent analy Demand uncton Th ubecton conder rt the pot market demand uncton, and then the orward market demand uncton. We denote the pot market demand uncton a (, D p ε, where we aume ε Ε to be an addtve tochatc hock to demand n the pot market. 0 That, we may wrte (, where we reer to 0 ( D ( p, ε, dene 0 ( uch that ( D p ε n addtvely eparable orm a (, ε 0 ( D p = D p + ε, (3.3 D p a the hape component o pot market demand. Gven D p a ( ( ε ( ε D 0 p D p, D 0, (3.4 D 0 0 = 0. That, D0 ( p pae through the orgn o the p -q plane. 0 Combnng eq. (3.3 and (3.4, we alo have that 0 KM alo rely upon th aumpton, but relax t or ome o ther comparatve tatc analy. 85

103 mplyng that ( 0, D ε ε, (3.5 ε the quantty-ax ntercept o (, D p ε. Let the upport o ε, Ε, be an nterval on the real lne, Ε ε, ε, ε < ε. The upper lmt o ε upport, ε, may be nnte, n whch cae Ε = ε,. A wth prce and quantte, let a caret ˆ denote the ex pot actual (or realzed value o the hock ε, ˆ ε. Fgure 3. below llutrate the relatonhp n eq. (3.3 (3.5. p Spot market D 0 ( p (, ˆ ε 0 ( D p = D p + ˆ ε 0 ˆ ε D D ( p ε ( p, ε,, 0 FIGUE 3.: THE SPOT MAKET DEMAND FUNCTION (, THE SHAPE COMPONENT OF SPOT MAKET DEMAND, D0 ( p D p ε FO ε = ˆ ε, AND 0 The hock ε alo hare th orgn. 86

104 A an example, uppoe that ( ε ( D0 p 0.0p, and a requred, ( D p, = 0.0p + ε. Then, we would have that D 0 0 = 0. The aumed unctonal orm (3.3 or (, D p ε ha mportant mplcaton or the analy. Frt, ollowng KM, the addtve hock ε ht but doe not rotate the pot market demand uncton (, zero, that, D p ε, and o we have that t cro-partal dervatve ( p, ε D p ε = 0. (3.6 Second, t alo ollow rom eq. (3.3 that 03 ( p, ε D D p D p p p (, ε = 0 (, ε o that the dervatve D ( p, ε and D0 ( p, (3.7 are nterchangeable. In chapter 6, we wll how endogenouly that pot market demand downward-lopng, that, ( ε ( ε D p, D p, p < 0. The pot market demand uncton, (, D p ε, are becaue o nal conumer wllngne to pay or energy-related ervce (e.g., or ether conumptve or productve purpoe that electrcty can provde. Subecton 6.6. explan how conumer utlty 03 Wth a lght abue o notaton, we ue a prme ( on the pot market demand uncton to ndcate partal derentaton wth repect to prce. A we do not need to reer to the partal dervatve wth repect to the tochatc hock ε, there no ambguty. 87

105 uncton gve re endogenouly to D ( p, ε. We alo aume 0 ( D p to be common knowledge and that the hock ε (due, or example, to varyng weather condton, economc actvty or other eect on conumpton drawn rom an exogenou, common knowledge dtrbuton. From chapter 5 onward, we retrct the analy and conder a mpled ane example, n whch we aume that the pot market demand uncton ane. Conder now the orward market demand uncton, whch we denote a (, 0 D p ε. Smlar to the pot market analy, we aume 0 ε Ε to be an addtve tochatc hock to demand n the orward market. That, we may wrte (, 0 D p ε n addtvely eparable orm a where we reer to 0 ( (, ε ( D p = D p + ε, ( D p a the hape component o orward market demand (whch we dene n eq. (3.9 below. A we how n chapter 6, n contrat to the tuaton n the pot market, the orward market demand uncton (, 0 orward market SF S ( p. The properte o (, 0 properte o S ( p ; moreover, the denton o (, 0 nvolved than the denton o (, D p ε above. 88 D p ε endogenou to the D p ε thereore depend on the D p ε omewhat more Beore dcung urther the properte o the uncton n eq. (3.8, conder the orward market SF, ( S p. For a varety o reaon, t may be the cae that, begnnng rom a gven ntal condton, we cannot dene a orward market SF S ( p

106 over all prce p. ather, the SF may have a retrcted doman o denton, ay, rom ome mnmum prce p to a maxmum prce p > p. In th cae, the doman o rm equlbrum orward market SF ( S p the nterval p, p ; we reer to th nterval a a doman retrcton on the uncton ( S p. 04 Becaue t endogenou, the orward market demand uncton D ( p, ε 0 nhert ( doman retrcton. Aume, thereore, that both SF ( S p S p and hence alo (, 0 D p ε are dened over the nterval p, p. Now aume ome reerence prce p 0 p, p contaned n the nterval over whch demand dened. In the ollowng, we dene the demand hock ε 0 n eq. (3.8 o that t equal to the demand uncton evaluated at the reerence prce p 0. To do th, aume a uncton (, 0 D p ε a n eq. (3.8, and dene the hape component o the orward market demand uncton 0 ( D p a ( (, ε (, ε D p D p D p ( Such doman retrcton may are or a varety o theoretcal or practcal reaon a the analy o chapter 7 make clear. A an example o the ormer, t may be the cae that, a we move along a pecc SF or a partcular rm, that rm econd-order condton (SOC or prot maxmzaton may be volated or prce above or below a certan level. The SF may not be contnued nto the regon n whch the SOC doe not hold; to prevent th, the doman o the SF mut be retrcted accordngly. Alternatvely, t may be that the rm SF become downward-lopng n p over certan prce range. An example o a practcal reaon or a doman retrcton are n chapter 7. There, we ee that the preence o ngularte may lmt the range o prce over whch we are able to ucceully numercally ntegrate the condton characterzng the orward market SF. 89

107 uch that D0 ( p 0 = 0. That, D0 ( p pae through the pont ( p, q ( p0,0 Combnng eq. (3.8 and (3.9, we alo have that (, =. 05 D p ε ε. (3.0 Let the upport o ε 0, Ε ε, ε, ε0 < ε0 Ε, be an nterval on the real lne, 0 0. The upper lmt o ε 0 upport, ε 0, may be nnte, n whch cae Ε = ε, 0. Agan, let a caret ˆ denote the ex pot actual (or realzed value o the hock ε 0, ε ˆ Fgure 3.3 below llutrate the relatonhp n eq. (3.8 (3.0 where, or concretene and eae o expoton, the gure aume that p 0 = p, (3. though a noted above, any p 0 p, p a utable choce. 05 The hock ε alo ha t orgn at q = The ue n the orward market o an arbtrary reerence prce p a generalzaton o the 0 approach ued or the pot market analy above. There, the pot market reerence prce mply zero, or D p, ε mplcty (compare, or example, eq. (3.5 and (3.0. The ane unctonal orm o ( aure u that or ntely-loped uncton D (, p ε, th uncton wll nterect the quantty ax (at ε. The ubcrpt 0 on ε ndcate that the orward market demand hock dened relatve to the 0 reerence prce p. 0 90

108 p p Forward market D D0 ( p ( p, ˆ ε D ( p = + ˆ ε p 0 = p 0 ˆ ε 0 D D ( p ε 0 ( p, ε,, 0 0 FIGUE 3.3: THE FOWAD MAKET DEMAND FUNCTION (, 0 p, p DEMAND 0 ( ˆ D p ε DEFINED ON FO ε0 = ε0 AND THE SHAPE COMPONENT OF SPOT MAKET D p, TAKING EFEENCE PICE 0 p TO BE EQUAL TO We may gve an example analogou to that ued n the dcuon o pot market demand. Namely, uppoe that ( ε ( D p e ( p p0 D p, = e + ε. Then, we would have that ( p p, and a requred, ( 0 ( p0 p0 D0 p0 = e = 0. The aumed unctonal orm (3.8 or (, 0 p D p ε ha mportant mplcaton or the analy. Frt, ollowng KM, the addtve hock ε 0 ht but doe not rotate the orward market demand uncton (, 0 D p ε, and o we have that t cropartal dervatve zero, that, 9

109 D ( p, ε 0 p ε 0 = 0. (3. Second, t alo ollow rom eq. (3.8 that 07 ( p, ε 0 D D p D p p p (, ε0 = 0 (, ε 0 o that the dervatve D ( p, ε 0 and D0 ( p, (3.3 are nterchangeable. In chapter 6, we wll how endogenouly that orward market demand downward-lopng under our D p, D p, p < 0. aumpton, that, ( ε0 ( ε0 A noted above, the orward market demand uncton, (, 0 D p ε, endogenou n the mult-ettlement SFE model. Forward market demand are due to the market actvty o rk-avere conumer n an uncertan envronment, who eek to buy orward contract or electrcty gven pot market demand (, that 0 ( D p ε. We aume D p common knowledge. Later, chapter 6 provde a ytematc analy o the provenance o the orward market demand uncton n the mult-ettlement SFE model (ncludng the dtrbuton o 0 properte dcued here. ε, and conrm that (, 0 D p ε ndeed ha the 07 Smlar to the notaton n the pot market, we ue a prme ( on the orward market demand uncton to ndcate partal derentaton wth repect to prce. A we do not need to reer to the partal dervatve wth repect to the tochatc hock ε, there no ambguty. 0 9

110 3. The nature o nancal orward contract The orward contract condered n the mult-ettlement SFE model are purely nancal n the ene that orward market poton nether commt rm to a partcular phycal producton chedule, nor commt purchaer to conume electrcty. ather, thee nancal contract repreent property rght to a cah low baed on ( contract quantty and ( relatve prce n the orward and pot market. 08 Frm may lqudate ther orward contract poton partally or completely n the pot market by repurchang the dered level o output at the pot market prce. 09 Content wth th denton, n the analytcal model developed n th ecton, orward contract poton q ˆ do not drectly enter rm cot uncton. ather, a the mult-ettlement SFE model wll make clear, rm pot market quantty produced, q ˆ, depend, through quantte q ˆ and q ˆ. Σ, on the orward market In a gven round o the mult-ettlement market, we dene the net cah low to rm rom a nancal orward contract old by rm a ( CF CF = p p q. (3.4 In eq. (3.4, each actor ( p p and q n CF and hence CF tel may be potve, negatve, or zero. Thu, CF > 0 n a gven round o the mult-ettlement 08 We aume thee property rght to be perectly and cotlely enorceable. 09 Many electrcty orward market relect th property: at leat at tradng hub, thee market tend to be lqud, oerng relable reale opportunte. 93

111 market, then contract holder pay CF to rm. I, n contrat, CF < 0, then rm pay CF to contract holder. The lterature on electrcty market commonly reer to th orm o contract a a two-way contract or derence, or CFD, where the term derence denote, naturally, the derence between the contract (or orward market prce, p, and the pot prce, p. A CFD a mple nancal ntrument degned to enable market partcpant to lock n a certan prce n the orward market or a quantty o electrcty. I exactly the orward contract quantty tranacted n the pot market, then the nancal outcome o that market round ndependent o a (uually more volatle pot market prce. 0 The bd-baed orward market examned n th nvetgaton eentally a double aucton or CFD, wth (n prncple both demand and upply bddng to tranact derent quantte, dependng on prce. To ocu attenton on th eental eature o the CFD, t helpul to conder eparately the three poble outcome rom rm perpectve: ( rm undercontracted ( q < q, ( rm ully contracted ( q = q, and (3 rm overcontracted ( q > q. We examne, n turn, each o thee outcome below, oerng an ntutve nterpretaton o each tranacton: 0 On th pont, ee paragraph below and alo Borenten et al. (000, 4.. Although development o an actve bd-baed demand de wthn compettve electrcty market ha htorcally lagged behnd that o the upply de, provon or prce-entve bd by demand-de agent are n place n many market around the world (ee, e.g., Internatonal Energy Agency 00,

112 . I 0 < q < q, we may nterpret the CFD a a xed-prce contract under whch rm and conumer tranact the rt q o output at p. Market partcpant then tranact the remanng porton o pot market output, q q, at p.. I q = q, we may nterpret the CFD a a xed-prce contract under whch rm and conumer tranact entre output o q at p. 3. I q > q, we may nterpret the CFD a a xed-prce contract under whch rm and conumer tranact output o q at p. Conumer then buy out o ther remanng contractual commtment o q q at a prce o p p, that, the to rm. 3 demand de make a buy-out payment o ( p p ( q q Alternatvely, we may vew th buy-out payment a two eparate tranacton. Under th nterpretaton, the demand de rt take ttle to t remanng contractual commtment o q q through a payment o p ( q q (thereby ulllng the orward contract. The demand de then reell th unwanted quantty on the pot market at the market-clearng prce, thereby recevng a. payment p ( q q I, ntead, we have that q < 0 < q, the nterpretaton o the aocated tranacton (though not the bac arthmetc change omewhat. Namely, n th cae, we may nterpret the CFD a a xedprce contract under whch rm purchae q orward contract rom conumer at p. Market partcpant then tranact the quantty q 3 In th cenaro, ( p p 0 q ( > at q <, rm wll pay demand de partcpant to reduce ther conumpton below the contracted quantty. That, the buy-out payment to rm gven by the product p p q q wll be negatve. ( ( 95 p.

113 In practce, becaue the orward and pot market clear at dtnct pont n tme, a upplyng rm perceve the cah low CF (ee eq. (3.4 rom the orward contract a comprng two eparate component. Namely, rm rt experence an nlow (aumng p > 0 o p q once the orward market clear at t =. Equaton (3. denoted th term a rm orward market revenue,, gven by = pq, (3.5 whch rm cah low n the orward market. Next, once the pot market clear at t =, rm ncur a contract ettlement payment o p q. (3.6 Th ettlement payment one component o rm cah low n the pot market (ee the ollowng ecton or more detal. Together, the derence o n eq. (3.5 and p q n (3.6 equal to CF rom eq. (3.4. We reer herenater to (nancal orward contract, orward contractng, etc. wth the undertandng that uch contract have the tructure o CFD a detaled n th ecton. 3.3 Pong the orward market problem To poe rm orward market problem n the mult-ettlement market SFE model wth orward contractng, t wll be ueul to begn by conderng rm acton n the pot market, and work backward rom there. Th approach relect the oluton algorthm o backward nducton whch we employ later n ecton 3.4. ecall rom ubecton 3.. that the cloed-loop normaton tructure poted or 96

114 our problem mple that rm are able to condton ther pot market play on orward market acton and outcome. Accordngly, rm recognze when choong ther orward market bd that, ultmately, pot market bd wll depend on thoe n the orward market. Th obervaton motvated the denton o rm provonal pot market SF, ( p ; Σ, a t perod characterzaton o t later pot market acton. 4 In the mult-ettlement SFE model, rm aware that the cloed-loop normaton tructure apple, a well, to t compettor, rm. Thereore, the partcular pot market SF that rm mpute n perod to rm when olvng t (rm own orward market problem wll lkewe be a provonal pot market SF. In ubecton 3..5, we denoted th SF a ( p ; Σ and aume t to be trctly ncreang n p. Gven th mputaton, rm wll conceve o t pot market redual demand uncton a pot market demand, (, D p ε, le rm mputed provonal SF, Σ ( p ; pot market-clearng equlbrum, then, rm pot market quantty. In any q wll le on th redual demand uncton at the market-clearng prce p. Thereore, we may dene, or any arbtrary ε and correpondng market-clearng p, 5 (, ε ( ; q D p Σ p. (3.7 4 In that dcuon, the placeholder n the argument lt o Σ repreented the (a-yetunpeced nluence o the orward market outcome on rm pot market acton. Later n th ecton, we wll be able to denty thee unknown argument rom the peccaton o the problem obectve uncton. 5 Note that we have dened q, or our preent purpoe, a a pont, not a a uncton. In ecton 4. below, we ue a renement o eq. (3.7 to contruct a provonal pot market SF or rm. 97

115 Now conder rm prot n the pot market, n the preence o nancal orward contract. 6 Secton 3. dcuon concernng thee contract hghlghted one component o thee prot, namely, the contract ettlement payment, p q (ee expreon (3.6, pad by uppler rm (or pq > 0 to conumer. There are two more contrbuton to rm pot market prot, namely, revenue rom ale o pot market output, and the producton cot o pot market output tel. In ubecton 3..7, we dened rm pot market revenue,, a (ee eq. (3. = pq, (3.8 and n ubecton 3..8, denoted rm producton cot a ( C q (3.9 or t pot market quantty, q. Beore brngng together the three conttuent term o rm prot n the pot market, p q, and ( C q rom expreon (3.8, (3.6, and (3.9, repectvely conder agan the above denton o rm equlbrum pot market quantty, q, that enter eq. (3.8 and (3.9. ecall that eq. (3.7 dened the quantty q n term o rm mputed provonal pot market SF, Σ ( p ;. The pot market prot computed ung th expreon or q lke the optmal admble pot market SF necearly contngent on the realzed orward market outcome. Untl we oberve 6 Green (999a tude the nteracton o contract and pot market (a ubecton.5. dcue. 98

116 th realzed orward market outcome, we may only expre rm pot market prot on a provonal ba, a well. For th reaon, we reer to th noton o pot market prot or rm a rm provonal pot market prot gven an mputed provonal pot market SF or rm, Σ, and denote th a π, whch we may wrte rom expreon (3.8, (3.6, and (3.9 a 7 π ( = pq C q. (3.0 In other word, π n eq. (3.0 rm perod concepton that, a t ormulate t orward market bd o t pot market prot. Subttutng or rom eq. (3.8, eq. (3.0 become ( π = p q p q C q. (3. Ung eq. (3.7 to ubttute or o π yeld q n eq. (3. and ncludng the unctonal argument π { p, Σ ( p ;, q, ε } = p D ( p ε Σ ( p p q C D ( p ε Σ ( p, ;, ;. (3. The SF Σ ( p ; arbtrary at th pont and thereore ncluded a an argument o n eq. (3.. The cot uncton and the pot market demand uncton are exogenouly π 7 The tlde on whch alo bear a tlde. π gne that th prot expreon a uncton o the mputed SF Σ, 99

117 xed 8 throughout the analy, and hence are not explctly repreented a argument o π. We may now characterze rm pot market optmum gven rm mputed provonal pot market SF. Equaton (3. gve an expreon or rm provonal pot market prot. For a Nah equlbrum n the pot market ubgame n any tate o the world, a neceary condton that or any gven demand hock ε, orward market quantty q, and mputed provonal pot market SF Σ ( p ; or rm, rm wll chooe an optmal that, provonal pot market prot-maxmzng prce, * p = p, n the pot market. 9 Let the optmal provonal pot market prot or rm, * π, be the maxmzed value o π at * p, that, { } { Σ ( } = Σ ( π ;, q, ε max π p, p ;, q, ε, * p or, ubttutng rom eq. (3. or π n the above equaton, π { ( ;, q, ε } Σ = * { p D ( p ε Σ ( p p q C D ( p ε Σ ( p } max, ;, ;. p (3.3 Let u now pecy the argument o the SF ( p ; Σ, =,. ecall that we wrote eq. (3.3 or a generc rm pot market optmum, gven a provonal pot 8 That, the demand uncton (, nclude a an argument o π. D p ε xed up to the tochatc hock ε, whch we do 9 For now, we mply aume the extence o a unque equlbrum prce conder th ue more ormally n ecton 4.. * p or each ε ; we 00

118 market SF or rm, Σ (, =,;. Whle we provde a more prece argument n ecton 3.4 and chapter 4 below, we argue ntutvely, at th pont a ollow. A neceary condton or the SF Σ ( p ; and Σ ( p ; to conttute a Nah equlbrum n the pot market ubgame wll be to aty eq. (3.3 or rm, =, (, gven any orward market outcome q and or any realzaton o the pot market demand hock ε. We are now n a poton to ak, on what addtonal varable or parameter, apart rom p, doe mputed provonal pot market SF, Σ, depend? By npecton o the rght-hand de o eq. (3.3, there are two poblte: the demand hock, ε, and the orward quantty, q ; we conder both o thee parameter below. Lookng rt at ε, we may rule th parameter out a a canddate or ncluon a an argument o Σ wth the ollowng reaonng. From the taxonomy o SF n ubecton 3..5, the proecton o ( p ; Σ nto the - p q plane S ( p, whch ha only p, and not ε, a an argument (that, S ( p mply a contnuou uncton n the p -q plane. The property that equlbrum SF yeld ex pot optmal quantte n all tate o the world 0 mple that S and hence Σ mut be optmal or all ε and or all orward market outcome. Thu, whle * p wll be (a argued above a uncton o ε, nclude S and hence ε a an argument o Σ wll not be uncton o ε. We conclude that we mut not Σ. 0 See Klemperer and Meyer (989, 50, and ecton 4. o the preent nvetgaton. 0

119 The orward quantty, q, alo appear a a parameter on the rght-hand de o eq. (3.3. The quantte q ncorporate normaton about both ( rm orward market acton (.e., ther SF bd and ( the realzaton o orward market uncertanty, ε 0, whle contanng no dentve normaton about the pot market outcome, ε. Under our aumpton o a cloed-loop normaton tructure n the mult-ettlement SFE model (ee ubecton 3.., rm can and ndeed, to enure ex pot optmalty n the pot market, mut condton ther pot market play on orward market acton and outcome. They do o by ncorporatng the approprate parameter rom the orward market a argument o ther pot market SF. From eq. (3.3, the approprate orward market parameter are precely the orward market quantte q. Becaue we mpoe eq. (3.3 or, =, ( n equlbrum, we mut nclude both rm orward market quantte n each uncton Σ ( p ; and Σ ( p ; wrte Σ ( p ; wth t complete lt o argument a (. In general, thereore, we Σ p ; q, q,, =,;, (3.4 For contency, note n rm problem that, ut a rm mpute Σ ( p ; pot market, rm wll alo mpute S ( p to rm or the (a we argue below to rm n the orward market. In analyzng the provonal pot market equlbrum, however, we do not requre rm mputed orward market SF, but mply rm orward market quantty, a mputed by rm. We denote th quantty, whch we have called rm mputed orward market quantty (mputed by rm a q = S p. It th quantty, ( along wth rm own optmal orward market quantty q. q, gven by q, that we nclude a one o the addtonal argument o ( p ; Σ, 0

120 rom now on, that, 3 Σ :. Havng peced the argument o Σ, we retate eq. (3.3 ung the parameterzaton o expreon (3.4, { { Σ ( ;,,, } = max (, Σ ( ;, p C D ( p, ε Σ ( p ; q, q }, π q q q ε p D p ε p q q p q * (3.5 and contnue wth the contructon o rm orward market problem. Gven that both rm and maxmze ther provonal pot market prot (.e., olve eq. (3.5, we may tate ontly neceary and ucent condton or a (pure trategy Nah equlbrum n provonal pot market SF: Σ =Σ Σ (3.6 Σ =Σ Σ (3.7 For any pot market Nah equlbrum, equaton (3.6 and (3.7 tate that the optmal SF Σ wll concde wth the mputed SF Σ, and we may dene uch an equlbrum optmal provonal pot market SF or rm a value o the argument o and ( ;, p q q Σ, o that we may alo wrte Σ. Thee equaton mut hold at all Σ and Σ a Σ ( ;, p q q Σ repectvely (, =, ;. I there ext multple Nah equlbra n pot market SF, we aume that rm ucceully coordnate on a ngle equlbrum, denoted a Σ ( =,. 3 We aume or now that uch an equlbrum ext and examne later a mpled example (ee chapter 5 or whch we may prove extence. 3 Th admttedly a trong aumpton. We could appeal ntead to renement o Nah equlbrum uch a ratonalzable tratege ; thee are tratege that are bet repone to bele that a 03

121 eplacng Σ ( ;, p q q wth ( ;, p q q Σ n eq. (3.5 at th Nah * equlbrum, we dene rm equlbrum optmal provonal pot market prot, π, a { { ( ;,,, } max (, ( ;, p C D ( p ε Σ ( p q q } π Σ q q q ε = p D p ε Σ p q q p q *, ;,. (3.8 By our aumpton, rm coordnate on a Nah equlbrum SF ( ;, p q q uncton no longer the arbtrary mputaton ( ;, p q q Σ. Th Σ, but a pecc uncton. We may thu re-expre ( ;,,, * π Σ q q q ε more uccnctly a π {,, q q ε } hence eq. (3.8 become * { } { } { } = Σ ( * p, 4 and π q, q, ε max π p, p ; q, q, q, ε (3.9 where {, ( ;,,, } (, ( ;, C D ( p ε Σ ( p q q π p Σ p q q q ε = p D p ε Σ p q q p q, ;,. (3.30 rm mght have about t rval tratege (Fudenberg and Trole 99, 49. Thee oluton concept tend to have lttle predctve power, however, and gven the repeated nteracton preent n real-world electrcty market (not modeled here, a ubecton 3.. explan, the emergence o ome degree o coordnaton on equlbra certanly plauble. In any event, n the mpled ane example that we olve n chapter 5, we wll demontrate the extence o a unque equlbrum n pot market SF, o that the coordnaton problem among multple equlbra doe not are. o Σ on 4 * edenng the argument o π (wth a lght abue o notaton and allowng the dependence * q to be ncorporated nto th redened uncton π. 04

122 Let the expected equlbrum optmal provonal pot market prot or rm be * ( π { } 0 q q ε ε E,,, (3.3 * where eq. (3.9 and (3.30 gve an expreon or π {,, q q ε }, and the expectaton n the expreon (3.3 taken wth repect to ε, condtonal on ε 0. The ratonale or ntroducng th expectaton a ollow. Frm ace pot market uncertanty emboded here n the demand hock ε a t contruct t orward market bd n perod. We aume that, beng rk neutral, the rm accommodate th uncertanty va mathematcal expectaton a n the expreon (3.3. Ater orward market uncertanty repreented here by ε 0 revealed, rm accommodate the remanng pot market uncertanty va t pot market SF bd, whch then ex pot optmal or all realzed value o ε gven a orward market outcome ε 0. Now let total prot or rm n the mult-ettlement SFE model, tot π gven an mputed orward market quantty or rm o q be the um o orward market revenue and expected equlbrum optmal provonal pot market prot rom the expreon (3.3, that, * ( { } 0 π tot = + E π q, q, ε ε. Ung eq. (3., we may rewrte the above equaton ubttutng p q or ncludng the argument o tot π : ( * { p, q, q, } = p q + E { q, q, } tot 0 0 (and π ε π ε ε. (3.3 05

123 In eq. (3.3, rm orward market quantty, q, equal to rm orward market redual demand uncton evaluated at p. The approprate redual demand uncton to. ue here that baed on rm mputed admble orward market SF, S ( p Namely, we dene 5 q D ( p, ε 0 S ( p, (3.33 at an arbtrary ε 0. Subttutng eq. (3.33 nto eq. (3.3 or place o q a an argument n eq. (3.3 yeld n q and ung S ( p { p, D ( p, 0 S ( p, S ( p, 0 } = p D ( p, ε 0 S ( p * + { D ( p S ( p S ( p } π ε ε tot ( π ε0 ε ε0 E,,,, (3.34 * where eq. (3.9 gve an expreon or π {,, q q ε }. Maxmzng eq. (3.34 wth repect to p wll conttute rm orward market obectve, gven 0 ε. We now characterze rm orward market optmum gven an mputed admble orward market SF or rm, S ( p. Eq. (3.34 gve an expreon or rm total prot. For a ubgame perect Nah equlbrum n the orward market problem n any tate o the world 0, ε, a neceary condton wll be that, gven S ( p 5 In eq. (3.33, we reer to the orward market demand uncton (, 0 ntroduced t n ecton 3..0, we have not yet dened explctly. A noted n that ecton, (, 0 D p ε, whch, though we D p ε endogenou to the mult-ettlement SFE model. Chapter 6 explan n detal how conumer acton gve re to (, 0 D p ε, and alo characterze t properte. 06

124 rm wll chooe an optmal that, total prot-maxmzng prce p p * = n the orward market. 6 Let the optmal total prot or rm, o tot π at p *, that, tot* π, be the maxmzed value { } { S ( } = p D ( p S ( p S ( p π, ε max π,, ε,, ε. tot* tot p {,,,, 0 0 } Subttutng rom eq. (3.34 or p D ( p S ( p S ( p above equaton, we get tot n the π ε ε { S (, 0 } = max p D ( p, 0 S ( p π ε ε tot* p * ( π { D ( p ε0 S ( p S ( p ε } ε0 + E,,,, (3.35 where, recallng eq. (3.9 and (3.30, { } { } = Σ ( π q, q, ε max π p, p ; q, q, q, ε (3.36 * p and where {, ( ;,,, } (, ( ;, C D ( p ε Σ ( p q q π p Σ p q q q ε = p D p ε Σ p q q p q, ;,. (3.37 We deer conderaton o equlbrum extence and unquene n the orward market ubgame and hence o the extence and unquene o ubgame perect Nah 6 For now, we mply aume the extence o a unque equlbrum prce conder th ue more ormally n ecton 4.. * p or each 0 ε ; we 07

125 equlbrum. Accordngly, we olve eq. (3.35 (3.37 gven an arbtrary mputaton, ( p S, or rm. Equaton (3.35 (3.37 compre the orward market problem tatement or rm. Beore dcung the oluton trategy or th problem, we brely revew and ummarze the oregong dervaton o thee equaton. A above, we tart beore the mpoton o equlbrum n the pot market namely, wth eq. (3. or π and revew the tep nvolved n developng eq. (3.35 (3.37. Examnng the three addtve term n eq. (3., we ee that the rt term repreent pot market revenue, the product o pot market prce and the redual demand (gven ε met by rm at that prce. The econd term rm contract ettlement payment at the pot market prce, p, wth holder o orward contract or q o output. The thrd term the cot o producton ncurred by rm or producng t pot market quantty, determned rom the rm redual demand uncton, gven ε and evaluated at p. We maxmze π wth repect to p to obtan * π, a on the let-hand de o eq. (3.5. Then, we mpoe a Nah equlbrum n pot market SF ( ;, p q q Σ n eq. * (3.8, whch yeld prot π a gven by eq. (3.9. Next, eq. (3.34 take the * condtonal expectaton E ( 0 π ε, and compute π tot a the um o orward market revenue the product o orward market prce and (gven ε 0 the redual demand met * by rm at that prce and E ( 0 yeld tot* π on the let-hand de o eq. (3.35. π ε. Fnally, we maxmze tot π wth repect to p to 08

126 To conclude th ecton, we retate the SPNE (expreon (3. n lght o the peccaton o Σ ( p ; a ( ;, p q q { ( ( } Σ, a ollow: SPNE or the two-player, S p, Σ p ; q, q, =, (3.38 mult-ettlement market SFE game. In the next ecton below, or concretene, we rewrte eq. (3.35 (3.37 or rm =, explan why the backward nducton oluton algorthm approprate, and how how t gve re to rm optmal SF, ( market problem. S p. Then, n chapter 4, we olve rm orward 3.4 Solvng rm orward market problem va backward nducton Frm orward market problem n the mult-ettlement market ettng to maxmze tot t total prot, π, gven S ( p or rm, n any tate o the world ε 0. We denote uch maxmzed prot a π, gven by eq. (3.35 (3.37, rewrtten below or = tot* and = : { S (, } = max p D ( p, S ( p π ε ε tot* 0 0 p * ( π { D ( p ε0 S ( p S ( p ε } ε0 + E,,,, (3.39 where { } { } = Σ ( π q, q, ε max π p, p ; q, q, q, ε (3.40 * p and 09

127 {, ( ;,,, } (, ( ;, C D ( p ε Σ( p q q π p Σ p q q q ε = p D p ε Σ p q q p q, ;,, (3.4 and ( p ; q, q Σ n eq. (3.40 and (3.4 rm equlbrum optmal provonal pot market SF. 7 Although not mmedately evdent rom eq. (3.39 (3.4, rm decon varable n perod are t orward market upply quantte or all eable prce p ; the locu o uch pont, at an optmum, the rm optmal SF, ( S p. Snce ( S p doe not appear explctly n the above equaton, t ueul to decrbe how th problem ormulaton, n act, ultmately yeld a uncton ( goal o th ecton. Note rt that the relatonhp ( S p. Th the q = S p (3.4 and (, ε ( q = D p 0 S p (3.43 are relected mplctly n eq. (3.40 and (3.4. Equaton (3.4 due to the denton o rm mputed admble orward market SF (ee ubecton Equaton (3.43 rom the market-clearng condton: p a market-clearng prce or the orward market, then rm orward market quantty, q, mut be equal to the rm 7 A we demontrate n chapter 5 below, the equlbrum SF Σ a uncton only o exogenou parameter that are common knowledge, o that rm may compute Σ n the coure o olvng t orward market problem. 0

128 redual demand uncton evaluated at p. Gven an mputaton S ( p and or any ε 0, the orward quantte q and q are uncton o p rom eq. (3.4 and (3.43. Frm compute rom eq. (3.39 (3.4 (or the aumed ε 0 t optmal prce ( p = p ε, the argmax or t orward market problem. Subecton 3.4. decrbe * 0 * how, by repeatng th computaton o p p ( ε 0 S p. 8 may contruct t optmal SF, ( = pontwe or all poble ε 0, rm The equental tructure o rm orward market problem ugget backward nducton a the approprate oluton algorthm. Indeed, a decrbed above, the rt backward nducton tep begn by olvng or rm optmal provonal pot market SF (parameterzed n term o the realzed orward market quantte, q ˆ and q ˆ. Then, we mpoe Nah equlbrum n the pot market, yeldng equlbrum pot market SF. Next, n the econd backward nducton tep, we contruct rm optmal admble orward market SF, gven the equlbrum pot market reult rom the rt tep. The ollowng two ubecton decrbe thee two backward nducton tep n more detal Frt tage: The pot market Conder rt the pot market. 9 Here, aumng a realzaton o the orward market demand hock ˆ ε 0 and realzed orward market quantte q ˆ and q ˆ, rm pot 8 Naturally, we wll alo aume that rm compute t optmal SF, ( S p n an analogou ahon. We then mpoe equlbrum n the orward market, gven the aumed pot market Nah equlbrum. The reultng tratege (.e., the equence o orward and pot market acton or each rm conttute a ubgame perect Nah equlbrum or the mult-ettlement SFE model. 9 Th ubecton ollow cloely the preentaton o Klemperer and Meyer (989, 5.

129 market redual demand at a prce p the derence between total demand n the pot market and the quantty that rm wllng to upply there at that prce. Thu rm commtted to a (trctly ncreang mputed provonal SF Σ ( p ; qˆ ˆ, q =Σ ( p ; qˆ ˆ, q, rm pot market redual demand uncton D ( p, ε ( p ; qˆ, qˆ Σ. Followng KM, nce ε a calar, the et o pont along rm pot market redual demand uncton atyng the rt-order condton (FOC correpondng to eq. (3.40 and (3.4 (xng q = qˆ and q = qˆ, a ε vare over all t poble value, a one-dmenonal uncton n p -q pace. I th uncton can be decrbed by an admble SF q ( qˆ ˆ, q ( p ; qˆ ˆ, q Σ that nterect each realzaton o rm pot market redual demand uncton once and only once, then by commttng to Σ, rm can acheve ex pot optmal adutment to the hock ε. In th cae, Σ rm unque optmal provonal SF or the pot market n repone to Σ. Frm may alo olve t veron o the pot market problem, whch we obtan rom eq. (3.39 (3.4 by nterchangng ubcrpt and throughout thee equaton. Frm olve t problem n the ame manner a dd rm, decrbed above, gven the mputed provonal pot market SF or rm, Σ ( p ; qˆ ˆ, q. Frm obtan Σ a t unque optmal provonal SF or the pot market n repone to Σ. Our earler aumpton that each rm mputed and optmal provonal pot market SF

130 concde at each p, ˆ ˆ q, and q ate the Nah equlbrum condton or the pot market; we denoted the equlbrum SF a Σ. For now, we aume that the et o pont yeldng equlbrum optmal provonal pot market prot π or rm (ee eq. (3.40 and (3.4 can be * decrbed by the provonal SF Σ and lkewe or rm and nvetgate later whether, under our hypothee, there ext equlbra n whch th ndeed the cae Second tage: The orward market In the econd tage o rm backward nducton algorthm, we move back n tme to perod, beore the orward market clear and beore revelaton o the uncertan demand hock ε Accordngly, we revert to the notaton or a-yet-unknown value o ε 0 and quantte q and q (to ndcate th, we wrte thee parameter now wthout caret and ue rm mputed orward market quantty, q. Conder the expreon or rm redual demand n the rt term o eq. (3.39 obectve uncton. Analogou to the tuaton n the pot market, rm orward market redual demand at any prce p the derence between total demand n the orward market and the quantty that rm wllng to upply there at that prce., Thu rm commtted to a (trctly ncreang mputed admble SF S ( p rm orward market redual demand uncton D ( p, 0 S ( p ε. 30 Note 9 apple here, a well. 3

131 Snce ε 0 a calar, the et o pont atyng the FOC correpondng to eq. (3.39 or maxmum total prot (gven S ( p, a ε 0 vare over all t poble value, a one-dmenonal uncton n p -q pace. I th uncton can be decrbed by an admble SF q S ( p that nterect each realzaton o rm orward market redual demand uncton once and only once, then by commttng to S, rm can acheve ex pot optmal (n the ene o eq. (3.39 (3.4 adutment to the hock ε. In th cae, S rm unque optmal admble SF or the orward market n 0 repone to S. Frm may alo olve t veron o the orward market problem, whch we obtan rom eq. (3.39 (3.4 by nterchangng ubcrpt and throughout thee equaton. Frm olve t problem n the ame manner a dd rm, decrbed above, gven the mputed admble orward market SF or rm, S ( p. Frm obtan S a t unque optmal admble SF or the orward market n repone to S. At th pont, we mpoe the Nah equlbrum condton or the orward market, whch that S and S concde at each p or =, ; we denote th equlbrum SF a S. For now, we aume that the et o pont yeldng maxmum total prot or rm gven S ( p (ee eq. (3.39 or π tot* can be decrbed by the admble SF S and lkewe or rm and nvetgate later whether, under our hypothee, there ext equlbra n whch th ndeed the cae. 4

132 3.4.3 Dcuon Each veron o the rt tage o the backward nducton problem (the pot market ee ubecton 3.4. aume a xed value o ε 0. Th tage neted wthn the problem econd tage (the orward market ee ubecton 3.4., n whch we contruct S n pontwe ahon by olvng the overall problem repeatedly or all eable ε 0, gven S. Th neted, herarchcal tructure yeld a orward market SF or rm that tot maxmze t total prot π or all eable ε 0. Eq. (3.39 dene π n term o the expected value o equlbrum optmal provonal pot market prot. To hghlght the tot* dtnct contrbuton o the pot and orward market to π, we could ay that S wll tot* yeld rm ex pot optmal total prot, π tot* * optmal provonal pot market prot, E( 0, aumng ex ante expected equlbrum π ε. Th the noton o optmalty exhbted by orward market SF n th the. The rm actual (.e., ex pot optmal pot market prot wll be determned by pot market SF bddng n perod. The relatonhp between the optmal provonal pot market SF and the optmal admble pot market SF hould now be clear. The optmal provonal pot market SF, ( ;, Σ p q q and Σ ( p ; q, q, are uncton o the orm Σ : 3 nce the orward quantte are tll unknown when contructng orward market bd. Once thee value o q and q have been revealed (a q = q = qˆ and q = q = qˆ, ay n perod, each rm may take thee value q ˆ and q ˆ nto account n contructng and ubmttng t optmal admble pot market SF whch, a market rule tpulate, have 5

133 the orm S ( p and ( S S p. Thee admble SF are uncton o the orm : (that, they le n the p -q plane; by contructon, they are alo the proecton (xng rm orward quantte at q ˆ and ˆ ( p ; q, q q o ( p ; q, q Σ and Σ onto thee plane. Algebracally, the relatonhp between thee two type o pot market SF, or realzed q ˆ and q ˆ, ( ( ; ˆ, ˆ S p =Σ p q q p (, =,;. (3.44 In th ene, then, the optmal provonal pot market SF ( ; ˆ, ˆ p q q wth the optmal admble pot market SF ( aumpton o a cloed-loop normaton tructure. Σ are content S p, relectng ubecton 3.. 6

134 Phloophy perectly rght n ayng that le mut be undertood backward. But then one orget the other claue that t mut be lved orward. Kerkegaard, Journal and Paper Sell when you can; you are not or all market. Shakepeare, A You Lke It 4 Dervaton o the optmal orward market SF THIS CHAPTE derve rm optmal orward market SF ung the backward nducton procedure ketched n ecton 3.4 above. Accordngly, ecton 4. below analyze the pot market n the rt tage o the problem. Secton 4. then devoted to the orward market n the econd tage o the problem. Th chapter ollow cloely the preentaton o Klemperer and Meyer (989, Frt tage: The pot market We begn by recatng the expreon or rm equlbrum optmal provonal pot market prot, π (eq. (3.40. We olve th equaton gven a realzed, arbtrary * orward market hock ε 0 = ˆ ε 0 and orward quantte q = qˆ and q = qˆ or rm 7

135 and, repectvely, and gven an aumed (though not yet realzed, arbtrary value o ε. A noted n ubecton 3.4., we alo aume that rm commtted to a (trctly ncreang mputed provonal SF Σ ( p ; qˆ ˆ, q =Σ( p ; qˆ ˆ, q market redual demand uncton then D ( p, ( p ; qˆ ˆ, q become ε Σ.. Frm pot Accordngly, rm provonal pot market prot maxmzaton problem { } { } ( π qˆ, qˆ, ε = max π p, Σ p ; qˆ, qˆ, qˆ, ε (4. * p and {, ( ;,,, } (, ( ;, C D ( p ε Σ( p qˆ ˆ q π p Σ p qˆ qˆ qˆ ε = p D p ε Σ p qˆ qˆ p qˆ, ;,. (4. The FOC o eq. (4. wth repect to p (aumng an nteror oluton {, Σ ( ; ˆ, ˆ, ˆ, ε } dπ p p q q q = 0, dp (, ε ( ; ˆ, ˆ ˆ ( ε ( = D p Σ p q q q { p C ˆ ˆ D p, p ; q, q } D ( p, ε ( p ; qˆ ˆ, q + Σ Σ (4.3 where prme on pot market demand and the SF denote dervatve wth repect to p. I the obectve uncton n eq. (4. globally trctly concave n p (Appendx B vere the econd-order condton, then eq. (4.3 mplctly determne, gven q ˆ and 8

136 ˆ * q, rm unque provonal pot market prot-maxmzng prce, p ( ε ; qˆ ˆ, q, or the aumed value o ε. The correpondng prot-maxmzng quantty ( ( ( ( ( D p ε ; qˆ, qˆ, ε Σ p ε ; qˆ, qˆ ; qˆ, qˆ q ε ; qˆ, qˆ. * * * * * The uncton p ( ε ; qˆ ˆ, q and q ( ; qˆ ˆ, q ε repreent n parameterzed orm rm et o ex pot optmal pont n the pot market (gven q ˆ and q ˆ a the rm pot * market redual demand uncton ht. I p ( ; qˆ ˆ, q ε partally nvertble 3 wth repect to ε, th locu can be wrtten a a uncton o pot market prce to quantty a ( ε * * ( ( ( q =Σ p ; qˆ, qˆ q p p ; qˆ, qˆ ; qˆ, qˆ, (4.4 * where ( p ( p ; qˆ ˆ, q ε * denote the partal nvere o p ( ; qˆ ˆ, q ε wth repect to ε. Snce D ( p ε ε, > 0, no two realzaton o rm redual demand uncton * can nterect; th condton, together wth unquene o p ( ; qˆ ˆ, q mple that ( p ; qˆ ˆ, q once or each ε or each ε Σ nterect rm redual demand uncton once and only * ε, at p ( ε ; qˆ ˆ, q. 3 Hence, ( p ; qˆ ˆ, q Σ rm optmal provonal pot market SF n repone to rm mputed provonal pot market SF, ( p ; qˆ, qˆ Σ. 3 We demontrate partal nvertblty n the context o a mpled ane example below (ee ecton See Appendx A or a proo o thee clam. 9

137 Let u rewrte eq. (4.3 o that t mplctly dene the uncton ( p ; qˆ ˆ, q Σ. Frt, however, we ollow Klemperer and Meyer (989, 50 and nvert the pot market demand uncton wth repect to ( D p, ε ε > 0. Let ε, notng that th nvere ext nce (, e Q p denote the value o the hock ε or whch total pot market demand ( that, e ( Q, p ate Q D p, e ( Q, p Q at prce p, =. To make explct the relatonhp between ε and the rm orward market poton ˆ ˆ q and q, we rt wrte the pot market-clearng condton gven Σ ( p ; qˆ ˆ, q and, rom eq. (4.4, ( p ; qˆ ˆ, q Σ a 33 ( ; ˆ, ˆ ( ; ˆ ˆ, Σ p q q +Σ p q q = Q. (4.5 Hence, rom the denton o the uncton (, e Q p and eq. (4.5, we have ( ˆ ˆ ( ˆ ˆ ε = e Σ p ; q, q +Σ p ; q, q, p. (4.6 Now, n eq. (4.3, replace ( ( ( ( ( q ε ; qˆ, qˆ D p ε ; qˆ, qˆ, ε Σ p ε ; qˆ, qˆ ; qˆ, qˆ (4.7 * * * 33 Note that eq. (4.5 repreent the pot market-clearng condton a rm would conceve t, n term o the SF that t mpute to rm, Σ, and t own optmal SF, Σ. Frm concepton o the pot market-clearng condton would be ymmetrc to eq. (4.5, and n any Nah equlbrum, thee two concepton o the pot market-clearng condton wll concde. 0

138 by ( p ; qˆ ˆ, q Σ and ue eq. (4.6 or (, Σ ( ; ˆ, ˆ +Σ( ; ˆ ˆ,, D p e p q q p q q p ( ;,, ε by * ε to replace D p ( ε qˆ qˆ o that eq. (4.3 become {, Σ ( ; ˆ, ˆ, ˆ, ε } dπ p p q q q = 0. dp ( p ; qˆ, qˆ qˆ p C ( p ; qˆ, qˆ D ( p, e Σ ( p ; qˆ ˆ, q +Σ( p ; qˆ ˆ, q, p Σ ( p ; qˆ ˆ, q =Σ + Σ (4.8 We aumed earler n eq. (3.6 that ( ε D p, p ε = 0, that, the hock ε tranlate the pot market demand uncton horzontally. We may thereore rewrte the (, ;, ;,, D p e p q q p q q p term Σ ( ˆ ˆ +Σ( ˆ ˆ, n eq. (4.8 mply a D0 ( p recallng eq. (3.7. Dong th and rearrangng eq. (4.8, we have or rm the mplct derental equaton { } ( ( ( ( Σ p ; qˆ, qˆ D p p C Σ p ; qˆ, qˆ =Σ p ; qˆ, qˆ qˆ 0. (4.9 Note that we could olve rm problem to obtan a reult completely ymmetrc to eq. (4.9 wth rm ubcrpt and nterchanged. The neceary Nah equlbrum condton n ether tage game that each rm optmal SF dentcal to the SF that t rval mpute to t. Gven that each rm SF ate t optmalty condton (e.g., eq. (4.9 or rm and lkewe or rm, the Nah equlbrum condton become a neceary and ucent condton or a (pure

139 trategy Nah equlbrum n SF. In the preent dervaton o the pot market provonal oluton n whch each rm mpute to t rval the rval optmal SF, th Nah equlbrum condton ( ˆ ˆ ( ˆ ˆ Σ p ; q, q Σ p ; q, q (, =,;, (4.0 where we have dened ( ; ˆ, ˆ p q q Σ (ee ubecton 3..5 a rm equlbrum optmal provonal pot market SF. Impoe th Nah equlbrum condton by recatng eq. (4.9 n term o thee equlbrum SF. 34 That, or each o the two rm, ubttute nto eq. (4.9 rom eq. (4.0 lettng, or rm, = and =, { } ( ( ( ( Σ p ; qˆ, qˆ D p p C Σ p ; qˆ, qˆ =Σ p ; qˆ, qˆ qˆ 0 (4. and, or rm, = and =, { } ( ( ( ( Σ p ; qˆ, qˆ D p p C Σ p ; qˆ, qˆ =Σ p ; qˆ, qˆ qˆ 0. (4. We call eq. (4. the equlbrum optmalty condton or rm equlbrum optmal provonal pot market SF Σ, mplctly denng th uncton (and mlarly or eq. (4. and Σ or rm. Fnally, recall our aumpton (ee note 90 that there are ˆ ˆ 34 For arbtrary value ε, q, and q, the rm repectve optmal pot market prce uncton * * p ( ε ; qˆ, qˆ and p ( ; qˆ, qˆ ( ( ( * * ˆ ˆ ˆ ˆ * p ε ; q, q p ε ; q, q p ε ; qˆ, qˆ * * ecton 5.4 or a mpled ane example that p ( ε ; qˆ, qˆ (and hence alo p ( ε ; qˆ, qˆ nvertble. ε mut concde n any pot market Nah equlbrum, that, =. We aumed n ecton 3. and wll prove n

140 multple Nah equlbra n the pot market ubgame, rm ucceully coordnate on a partcular pot market equlbrum to be antcpated. For purpoe o comparon wth prevou work, we make the temporary aumpton that the prce-cot margn p C ( ; ˆ, ˆ Σ p q q allow u to rearrange eq. (4. and (4. a are nonzero. Th ( p ; qˆ, qˆ qˆ Σ( ; ˆ ˆ, Σ Σ ( p ; qˆ ˆ, q = + D 0 ( p, (4.3 p C p q q and or rm, ( p ; qˆ, qˆ qˆ Σ( ; ˆ ˆ, Σ Σ ( p ; qˆ ˆ, q = + D 0 ( p. (4.4 p C p q q Comparng eq. (4.3 wth Klemperer and Meyer (989, 5 optmalty condton or the ymmetrc ngle-market SFE, namely, ( ( ( S p S ( p = + D p p C S p (, (4.5 we ee that, wth the excepton o the argument q ˆ and q ˆ n eq. (4.3 and the aumpton o ymmetrc rm (wth ymmetrc cot that underle eq. (4.5, the tructure o the two equaton dentcal. We have already argued that we may treat the hgher-dmenonal SF n eq. (4.3 a two-dmenonal proecton n the p -q plane, nce or any partcular teraton o eq. (4., the argument q ˆ and q ˆ are xed. Thu, the uncton Σ and Σ n eq. (4.3 are cloely analogou to the upply uncton S n eq. (4.5. We could vew KM optmalty condton (rewrtten above a eq. (4.5, 3

141 thereore, a mply a pecal cae o eq. (4.3 n whch qˆ = qˆ = 0 and rm are ymmetrc. We wll ee later when olvng explctly a mpled veron o eq. (4.3 that the SF oluton o the two equaton are ndeed cloely related. Th complete the rt backward nducton tage to nd the provonal oluton or the pot market. In the econd tage condered n the next ecton, we eek the oluton to rm orward market problem. 4. Second tage: The orward market In conrontng the econd tage o the backward nducton problem or rm, we move back n tme to perod, beore the orward market clear and beore revelaton o the orward market parameter ε 0, q, and q. Accordngly, we revert to the notaton or the not-yet-revealed value o thee parameter and wrte them now wthout caret. We rt recat the orward market problem by replacng the argument q and q n eq. (3.40 and (3.4 wth uncton o p ung eq. (3.43 and (3.4. Then, we olve th problem gven an aumed (though not yet realzed arbtrary value o ε 0. Wth thee ubttuton, eq. (3.39 (3.4 become { S (, } = max p D ( p, S ( p π ε ε tot* 0 0 p * ( π { D ( p ε0 S ( p S ( p ε } ε0 + E,,,, (4.6 where 4

142 { D ( p, S ( p, S ( p, } = π { p Σ p S ( p D ( p ε0 S ( p p D ( p, ε0 S ( p, ε } π ε ε * 0 { } max, ;,,, (4.7 and { p, Σ { p ; S ( p, D ( p, S ( p }, D ( p, S ( p, } = p D ( p, ε Σ p ; S ( p, D ( p, ε0 S ( p π ε ε ε 0 0 ( { } (, ε 0 ( ( ( ε { ( ( ε ( } p D p S p C D p, Σ p ; S p, D p, S p. 0 (4.8 Together, eq. (4.6 (4.8 conttute rm orward market optmzaton problem: maxmze total expected prot gven a value o the orward market demand hock, ε 0, and a (trctly ncreang mputed admble orward market SF or rm, S ( p by choong p. The FOC o eq. (4.6 (4.8 wth repect to p denotng 35 the obectve {,, ε } uncton o eq. (4.6 a π p S ( p and aumng an nteror oluton tot 0 35 Wth a lght abue o notaton nce we had earler dened π {,,, 0 } tot = π tot p q q ε (3.3 a a uncton o our argument. 5 (ee eq.

143 {, (, ε } d π p S p tot 0 dp = 0, (, ε ( 0 (, ε0 ( = D p S p + p D p S p { (, (, (, } dπ D p ε S p S p ε * 0 + E ε 0 dp (4.9 where the prme on orward market demand and SF denote dervatve wth repect to p. We may evaluate the dervatve nde the expectaton n eq. (4.9 by rt applyng the chan rule to the let-hand de o eq. (4.7: 36 { (, (, (, } * dπ D p ε0 S p S p ε dp { (, (, (, } * π D p ε0 S p S p ε dq = q dp { (, (, (, } * π D p ε0 S p S p ε dq + ε 0, q dp ε 0 (4.0 where we have ued the act that dε dp = 0 nce (a we wll ee n chapter 6 ε depend only on Perod (pot market uncertanty once ε 0 xed. * To evaluate the partal dervatve o π wth repect to q and q n eq. (4.0, we apply the envelope theorem to eq. (4.7. Th yeld 36 * ecallng eq. (3.43 and (3.4, we ee that the rt and econd argument o π are q and q, repectvely. It wll be ueul horthand n eq. (4.0 above to dene dervatve o to thee orward market quantte. * π wth repect 6

144 { D ( p, S ( p, S ( p, } * π ε0 ε q { { ( ( ε ( } (, ε0 (, ε } = π p, Σ p ; S p, D p, S p, 0 D p S p q (4. and { D ( p, S ( p, S ( p, } * π ε0 ε q { { ( ( ε ( } (, ε0 (, ε }. = π p, Σ p ; S p, D p, S p, 0 D p S p q (4. { } a Σ { } Suppreng the argument o Σ p ; S ( p, D ( p, ε 0 S ( p, or brevty, the rght-hand de o eq. (4. and (4. become, repectvely (ung eq. (4.8, { { ( ( ε ( } (, ε0 (, ε } π p, Σ p ; S p, D p, S p, 0 D p S p q (4.3 { } { } Σ Σ = p p C ( D ( p, ε Σ{ } q q and { { ( ( ε ( } (, ε0 (, ε } π p, Σ p ; S p, D p, S p, 0 D p S p q { } { } Σ Σ = p C ( D ( p, ε { }. Σ q q (4.4 7

145 Combnng eq. (4. and (4.3 and mplyng, we get { D ( p, S ( p, S ( p, } * π ε0 ε q ( ( ε { } { } Σ = p C D p Σ p,. q (4.5 Combnng eq. (4. and (4.4 and mplyng, we get { D ( p, S ( p, S ( p, } * π ε0 ε q ( ( ε { } Σ = p C D p Σ { },. q (4.6 ecallng eq. (4.4 (4.7 above and the aocated dcuon, now that we have derentated, we may replace the argument o the margnal cot uncton n eq. (4.5 { } and (4.6, D ( p, ε Σ { }, wth Σ p ; D ( p, ε 0 S ( p, S ( p Σ { }. Dong th, eq. (4.5 and (4.6 become { D ( p, S ( p, S ( p, } * π ε0 ε q { } Σ = p C Σ p ( { } q ; (4.7 and { D ( p, S ( p, S ( p, } * π ε0 ε q Σ = Σ p C ( { } q { }. (4.8 8

146 We may ubttute nto eq. (4.0 rom eq. (4.7 and (4.8 to obtan { (, (, (, } * dπ D p ε0 S p S p ε dp 0 { } { } Σ Σ = p C Σ + + p ε ( { } dq dq dq ε 0 q dp q dp dp. (4.9 Note that we may nterpret the econd bracketed term on the rght-hand de o eq. (4.9 a { } dq { } dq { } Σ Σ Σ + = q dp q dp p, (4.30 where the partal dervatve { } Σ p hold p contant. Agan ung eq. (3.43 and (3.4, we may expre the dervatve dq dp and dq dp n eq. (4.9 a: dq dp d = D ( p, ε0 S ( p = D ( p, ε 0 S ( p dp ; (4.3 and dq dp ( p ds = = S ( p. (4.3 dp Fnally, we ubttute eq. (4.3 and (4.3 nto eq. (4.9 and the reult, n turn, nto the orward market FOC (eq. (4.9 to obtan 9

147 (, (, ε tot d π p S p 0 dp = D ( p, ε0 S ( p + p D ( p, ε0 S ( p E { p C ( Σ{ } Σ { } Σ { } (, ε ( + + p D ( p, ε0 S ( p } ε0 = 0. Gven 0 D p 0 S p S p q q ε, the uncton D ( p, ε 0 ( and S ( are both contant a Hence, the lope o redual demand D ( p, ε 0 S ( p ε vare. nde the expectaton operator tel contant wth repect to ε (though the expectaton doe act upon whch premultple th term. Thereore, th term denotng the lope o redual demand may be treated a a contant n the above equaton, and taken outde o the expectaton. Ung th act and agan wrtng the argument o Σ { } and Σ { } explctly, th FOC become p, 30

148 (, (, ε d π p S p tot 0 dp (, ε ( 0 E ( ε0 (, ε0 ( = D p S p + p p D p S p { ( { ( ε ( 0 ( } { p ; S ( p, D ( p, ε S ( p } E p C Σ p ; D p, S p, S p = 0. Σ 0 q (, ε 0 ( D p S p Σ { p ; S ( p, D ( p, ε 0 S ( p } + S ( p ε 0 q (4.33 I the obectve uncton n eq. (4.6 globally trctly concave n p (Appendx B gve ucent condton or the econd-order condton to hold, then eq. (4.33 * mplctly determne rm unque prot-maxmzng prce, ( 0 o ε 0. The correpondng prot-maxmzng quantty The uncton * * p ( ε 0 and ( 0 ( (, ( ( ( D p ε ε S p ε q ε. * * * p ε, or each value q ε repreent n parameterzed orm rm et o ex pot optmal pont n the orward market a the rm orward market redual demand * uncton ht. I ( 0 orward market prce to quantty: p ε nvertble, 37 th locu can be wrtten a a uncton o 37 * We demontrate the nvertblty o p ( n ecton 5.4 below. 3

149 ( * * ( ( ( q = S p q p p, (4.34 * where ( p ( p denote the nvere o * p ( ε 0. Snce ( D p, ε ε > 0, no 0 0 two realzaton o rm redual demand uncton can nterect; th condton, * together wth unquene o ( 0 p ε or each 0 ε mple that ( redual demand uncton once and only once or each 0 ( S p nterect rm * ε, at ( 0 p ε. 38 Hence S p rm optmal admble orward market SF n repone to rm. mputed admble orward market SF, S ( p S p. Frt, Let u rewrte eq. (4.33 o that t mplctly dene the uncton ( however, we ollow Klemperer and Meyer (989, 50 and nvert the orward market demand uncton wth repect to ε 0, notng that th nvere ext nce ( D p, ε ε > 0. Let 0 0 (, e Q p denote the value o the hock ε 0 or whch total orward market demand p, that, (, ( e Q p ate Q D p, e ( Q, p ( ( Q at prce =. Now, n eq. (4.33, replace (, ( ( q ε D p ε ε S p ε (4.35 * * * See Appendx A or a proo o thee clam. 3

150 by S ( p, and ue e ( Q, p a dened above wth Q S ( p S ( p ( (,, D p e S p S p p * replace D p ( ε, ε by ( ( (4.33 become = + 39 to. Then, the FOC (, (, ε d π p S p dp = S tot 0 ( p ( 0 ( ( { ( ( } ( ε ( ( + p E p D p, e S p + S p, p S p E { p C Σ p ; S p, S p = 0. { p ; S ( p, S ( p } Σ q D p e S p + S p p S p Σ + (, ( (, ( p ; S ( p, S ( p S ( p ε 0 { } q (4.36 We aumed earler n eq. (3. that ( ε D p, p ε = 0, that, the 0 0 hock ε 0 tranlate the orward market demand uncton horzontally. We may thereore ( wrte, ( (, D p e S p + S p p mply a D0 ( p Makng th change n eq. (4.36 yeld, recallng eq. ( Th the orward market clearng condton a rm would conceve t. The argument o note 33 apple here, mutat mutand. 33

151 (, ( d π p S p tot dp = 0, ( E ( ( ( = S p + p p p D0 p S p E { p C ( Σ{ p ; S ( p, S ( p } Σ{ p ; S ( p, S ( p } q ( ( D 0 p S p Σ{ p ; S ( p, S ( p } + S ( p p q (4.37 where we now condton expectaton n eq. (4.37 on rm optmal prce ( p = p ε, thu uppreng explct dependence o the FOC on ε * 0 Fnally, we may rearrange eq. (4.37 a where we dene ( p ( ( E ( = ( + ψ ( S p D 0 p p p p S p p, (4.38 ψ a 40 Note that we may condton n eq. (4.37 on ether * ecton 5.4 that p ( nvertble, and hence abue o notaton n eq. (4.37 n expreng π (, ( tot p S p tot argument ( π ( p, S ( p, ε0 p or 0 ε under our aumpton (uted n p and ε 0 are one-to-one. We alo commt a lght a a uncton o only two rather than three, a n eq. (4.36 and the oregong analy. 34

152 ψ { ( { } { p ; S ( p, S ( p } ( p E p C Σ p ; S ( p, S ( p Σ D p S p q { p ; S ( p, S ( p } ( ( 0 Σ. + S ( p p q (4.39 Note that we could olve rm orward market problem to obtan a reult completely ymmetrc to eq. (4.38 and (4.39, but wth rm ubcrpt and nterchanged. The neceary Nah equlbrum condton n ether tage game that each rm optmal SF dentcal to the SF that t rval mpute to t. Gven that each rm SF ate t optmalty condton (.e., eq. (4.38 and (4.39 or rm n the orward market and lkewe or rm, the Nah equlbrum condton become a neceary and ucent condton or a Nah equlbrum n SF. In the preent dervaton o the orward market oluton, th Nah equlbrum condton where we have dened ( ( = ( ( (, =,; S p S p S p, (4.40 S p (ee ubecton 3..5 a rm equlbrum optmal admble orward market SF. Impoe th Nah equlbrum condton by recatng eq. (4.38 and (4.39 n term o thee equlbrum SF, that, ubttute nto thee equaton rom eq. (4.40 lettng = and =, yeldng 4 4 We may make an argument analogou to that n note 34 above that or an arbtrary value ε, 0 * * the rm repectve optmal orward market prce uncton p ( ε 0 and ( 0 35 p ε mut concde n any

153 where we redene ( p ψ ( ( E( = ( + ψ ( S p D p p p p S p p 0 ψ a { ( { } { p ; S ( p, S ( p } ( p E p C Σ p ; S ( p, S ( p Σ D p S p q { p ; S ( p, S ( p } ( ( 0 Σ. + S ( p p q, (4.4 eplacng ( S p wth rm equlbrum orward market quantty q, the above expreon become ψ { ( ( p E p C Σ { p ; q, q } { p ; q, q } Σ D p S p q { p ; q, q } ( ( 0 +. Σ S ( p p q (4.4 We ay that eq. (4.4 and (4.4 conttute the orward market equlbrum optmalty condton or rm equlbrum optmal admble orward market SF * * * orward market Nah equlbrum,.e., p ( ε0 p ( ε0 p ( ε0 * * wll prove n ecton 5.4 that p ( ε 0 (and hence alo ( 0 =. We aumed n ecton 3. and p ε nvertble. 36

154 ( S p. 4 Comparng the tructure o eq. (4.4 and (4.4 wth that o Klemperer and Meyer (989, 5 optmalty condton or the ngle-market SFE gven above a eq. (4.5 we ee that they der n three repect:. A wth the pot market oluton or rm (eq. (4.3, we derved eq. (4.4 and (4.4 or two aymmetrc rm wth aymmetrc cot uncton. Eq. (4.5 (rom KM, n contrat, aumed two ymmetrc rm.. In eq. (4.4, the expected pot prce E ( ( S p p play the role o margnal cot C n eq. (4.5. Th tructural mlarty ugget that we may nterpret the expected pot prce a a margnal opportunty cot to a (rk-neutral uppler o a partcular quantty contracted n the orward market. 3. Equaton (4.4 contan the term ( p ψ (ee eq. (4.4, whereby KM optmalty condton, eq. (4.5, ha no uch term. Appendx C provde an economc nterpretaton o ψ ( p. Namely, ( p ψ the expected change n rm equlbrum optmal provonal pot prot caued by a margnal change n p whle nettng out the expected change n t orward contract ettlement payment, ( p q, due to th change n p. In other word, ( p ψ capture the expected eect o a margnal change n p on rm pot market revenue 4 An analogou procedure would yeld the correpondng equlbrum optmalty condton or rm equlbrum optmal admble orward market SF ( S p. 37

155 le producton cot. 43 We may expre th nterpretaton o ψ ( p algebracally a ψ ( {,, ε } dπ q q dq p p p p dp dp * =E + E (, (4.43 a Appendx C demontrate. 44 The mert o th reult that the relatonhp o the optmalty condton, eq. (4.4 and (4.4, to the orgnal problem tatement, eq. (4.6 (4.8, then partcularly tranparent. Later n chapter 8, we alo denty ( p ψ a rm trategc eect, accountng, n part, or the rm partcpaton n the orward market. Equaton (4. and (4.4 (ung (4.4 and the analogou equaton or rm conttute a mxed ytem o derental equaton: partal derental equaton n { ;, Σ p q q } and total derental equaton n ( retrcton o q S ( p S p, wth the cro-equaton =,, =,;. From thee ytem, we oberve that the orward and pot market are coupled n at leat two way:. In general, rm equlbrum orward market quantte qˆ = q and qˆ = q enter both rm provonal pot market SF ( ;, p q q Σ a argument. 43 ecall that the denton o equlbrum optmal provonal pot market prot or rm rom eq. (3.9 and (3.30 ncluded not only the rm pot market revenue le producton cot, but alo the orward contract ettlement payment, n th cae ( pq. 44 In eq. (4.43, we have denoted the change n rm orward market quantty or a change n p a dq (, 0 ( 0 ( ( dp = D p ε S p = D p S p redual demand uncton, recallng eq. (3.43, (3.3, and (4.40., the lope o rm orward market 38

156 . Both the level uncton and the varou partal dervatve o rm provonal pot market SF ( ;, Σ p q q, enter the uncton ψ ( p, whch tel appear n rm orward market equlbrum optmalty condton n ecton 4.. In addton, we make explct a thrd relatonhp between the two market n chapter 5, where we etablh how the equlbrum pot market prce p depend on rm equlbrum orward market quantte q and q. Soluton to the aorementoned mxed ytem o derental equaton would be dcult to characterze n the general cae. Newbery (998, 733 antcpated th complexty, notng the double nnty o oluton that are when we permt a contnuum o pot market equlbra (characterzed by eq. (4. and (4. or every orward market equlbrum, themelve element n a contnuum. The contnuum o oluton n each market ext becaue each oluton correpond to a partcular ntal condton (or boundary condton n a contnuum o uch condton or each derental equaton. 45 In the ollowng chapter, we appeal to everal mplyng aumpton that render eq. (4.4 and (4.4 more tractable. 45 Thee contnua o ntal condton mght are, or example, due to phycal capacty contrant or lmt on nancal contractng related to credt rk. 39

157 Far better an approxmate anwer to the rght queton, whch oten vague, than the exact anwer to the wrong queton, whch can alway be made prece. J.W. Tukey, The Future o Data Analy Everythng hould be made a mple a poble, but not mpler. Enten 5 A mpled ane example THIS CHAPTE ntroduce an ane example that mple the pot market and ultmately, alo the orward market analy. Secton 5. below begn by ntroducng three aumpton regardng ane unctonal orm n the pot market, and ecton 5. explore the mplcaton o thee aumpton or the pot market SF. Secton 5.3 conduct comparatve tatc analy or pot market SF wth repect to cot and demand uncton parameter. Next, we nvetgate the mplcaton o the ane unctonal orm aumpton or optmal pot market prce and the orward market optmalty condton n ecton 5.4 and 5.5, repectvely. Secton 5.6 conclude. 40

158 5. Ane unctonal orm We now nvoke everal mplyng aumpton n order to carry the analy urther. From th pont orward, let u retrct ourelve to the cae n whch the ollowng three aumpton hold concernng the pot market: AFFINE SPOT MAKET DEMAND FUNCTION: The pot market demand uncton ane, havng the orm (, D p ε = γ p + ε. Thu, the pot market demand uncton D p D p p lope (, ε (, ε γ, where 0 γ >. 46 AFFINE MAGINAL PODUCTION COST FUNCTIONS: Each rm ha a quadratc producton cot uncton ( C q, gven by ( C q = c0q + cq, 0 q, where c0 0 and c > 0 ( =,. Margnal producton cot C ( q or each rm then alo ane: ( 0, 0 C q = c + cq q. (5. AFFINE SPOT MAKET SFS (EQUILIBIUM SELECTION: The provonal pot market SF ( p ; q, q Σ (,,; = are ane n p. That, ( ;, p q q Σ o the orm ( ;, Σ p q q = α + β p (, =,;, (5. 46 In ubecton 6.4.4, we relate the magntude o γ to parameter o conumer utlty uncton. 4

159 where α the quantty ax ntercept and β the lope o the (ane proecton o ( p ; q, q Σ onto the p -q plane. A prncpal goal o th chapter to nvetgate the eect that thee varou mplyng aumpton have on the pot market upply uncton, the optmal pot market prce uncton, and the orward market equlbrum optmalty condton. In ecton 5.3, we alo perorm comparatve tatc analy or the pot market n th ane cae. Whle Ane Spot Market SF may at rt appear to be a arly trong aumpton, there are two theoretcal ground or electng ane pot market SF or urther tudy. Frt, gven the cot uncton, the ane pot market SF the lmtng equlbrum acton a the range o uncertanty n pot market demand ncreae. Second, tablty argument avor the electon o the ane SF over alternatve trctly concave or trctly convex SF. We elucdate thee argument below. Fnally, apart rom thee theoretcal utcaton, the ane unctonal orm n the pot market mple the analy. Klemperer and Meyer (989, 6 (Propoton 4 how n ther ngle-market SFE analy that when the upport o the tochatc demand hock bounded above, there ext a contnuum, or connected et, o SFE contng o both trctly convex and trctly concave SF, a well a an ane SF n the nteror o the et (we call th the equlbrum et. 47 A we ncreae the upper endpont ε o th upport, the contnuum o equlbrum SF narrow a the mot concave and mot convex SF drop out o the 47 In addton, KM requre that demand and margnal cot uncton be lnear or ucently large prce and quantty. Whle ther proo alo aume ymmetrc rm, th property doe not appear to be neceary or ther reult (ee udkevch

160 equlbrum et. 48 In the lmt a ε, conderng the equence o equlbrum et aocated wth each value o ε, th equence converge to an equlbrum et havng a ngle element, the ane SF. For th reaon, KM conclude that []or unbounded upport, there ext a unque SFE and t lnear [or more generally, ane, gven ane margnal cot uncton wth trctly potve ntercept]. It traghtorward to how that th argument baed on the ngle-market SFE carre over to the pot market, a well, n the mult-ettlement market context. 49 We do not make the rather trong aumpton here that ε necearly unbounded. ather, we mply retrct ourelve under the Ane Spot Market SF aumpton to the cla o ane pot market upply uncton, notng that th aumpton become le retrctve the larger ε. More recent work on the tablty o SFE model ha hown that under plauble condton, (ngle-market non-ane SF are untable, a elaborated below. In partcular, under aumpton analogou to the Ane Spot Market Demand Functon and Ane Margnal Producton Cot Functon aumpton above, 50 Baldck and Hogan (00, 30 (Theorem 6 nd that ngle-market SFE comprng ether ( trctly concave SF or each rm or ( trctly convex SF or each rm are untable Th becaue at extreme value o the demand hock ε, the SF havng the greatet curvature volate the econd-order condton or prot maxmzaton beyond a certan pont n ther doman. 49 Whether th argument alo hold n the orward market wthn the mult-ettlement market, however, a matter or urther reearch; ee chapter And aumng, n addton (a we dd n ubecton 3..8, that uppler ace no bndng capacty contrant. 5 See Baldck and Hogan (00, 30 or detal. Thee author dene an untable SFE n the ollowng ntutve ene: An SFE untable when mall perturbaton to equlbrum SF elct bet repone rom rm that devate urther rom th equlbrum (wth repect to an approprate norm on the uncton pace o SF than do the orgnally aumed perturbaton. The author do not addre the cae o SFE n whch the concavty o the equlbrum SF vare acro rm (e.g., when ome rm have trctly 43

161 Baed on ther analy, t reaonable to conecture (although we do not prove th here that Baldck and Hogan aorementoned reult or the ngle-market ettng wll carry over (at leat to the pot market n the mult-ettlement market envronment. Thereore, tablty o the equlbrum a alent and derable charactertc, the ane pot market SF tuded here are alo thoe mot o nteret on tablty ground. Apart rom tablty conderaton, Green (996 ha made the cae that ane pot market SF may be reaonable approxmaton to the actual equlbrum SF, partcularly at certan demand level. 5 In addton, the Ane Spot Market SF aumpton naturally attractve, a t make the mult-ettlement SFE model more tractable analytcally. Fnally, th aumpton alo acltate comparon wth prevou work (e.g., Green 999a, whch ha mlarly ocued, or the mot part, on the ane or lnear cae. 5. Implcaton or the pot market upply uncton Th ubecton olve or the parameter α and explan how to olve or the β (ee eq. ( For concretene, we conduct the analy or rm. Begn by ubttutng rom eq. (5. or each rm ane pot market SF nto eq. (4.3, rm pot market concave SF whle other SF are trctly convex. Whether uch cae are not known, but they do, ther tablty properte are unknown. 5 For the cae o a ngle market wth margnal cot pang through the orgn, Green note (p. 09, n. 3 that the lope at the orgn o all SF through th pont equal to the lope o the unque lnear SF alo pang through the orgn. Thu, any ane SF approxmate an arbtrary (nonlnear equlbrum SF at low demand level. It may be hown that an analogou reult hold or pot market SF n a multettlement market, whereby the approxmaton vald n the neghborhood o the pont on the margnal cot uncton at the orward contract quantty (ee, e.g., Fgure The reult o th ubecton are content wth thoe o Green (999a, who examned a orward contract market ung conectural varaton nteractng wth a pot market ung SFE. 44

162 equlbrum optmalty condton. Dong o (and mpong Nah equlbrum n the orward market yeld, or all market-clearng p, { } ( ( β + γ p c + c α + β p = α + β p q 0. (5.3 Smplyng and collectng actor o p and contant term, we get ( c p ( c + c ( ( β β β α 0 = β cγ γ p α cγ c0γ q For th equaton to hold or any market-clearng prce p, the actor o p on ether de o th equaton mut be equal, a mut the contant term. Equatng thee term, denng the dmenonle parameter φ a φ (, =,;, (5.4 + c + ( γ β and olvng or α and or β (n term o β, we have 54 α = φq c β (5.5 0 and ( β = φ γ + β. (5.6 Conderng the (equlbrum orward market poton q and q, we ee that α n eq. (5.5 depend only on q and not on q, whle β doe not depend on ether rm 54 Equaton (5.5 and (5.6 are content wth Green (999a eq. (7 or the duopoly cae that he tude. 45

163 orward market poton. The nterpretaton that the ane pot market SF depend only on one own quantty awarded n the orward market and not on the compettor quantty. Th obervaton an ntance o Green (999a ndng concernng the eect o orward contract poton n h lnear SF model and the dtncton between quantte and tage game acton n the SFE ettng. A Green noted, [rm ] quantty decreang n [rm ] contract ale, but t [optmal pot market acton] t upply uncton not aected by them. 55 Our aumpton o ane SF crtcal to th property, however; non-ane pot market SF do depend on the rval orward market quantty. ewrtng eq. (5.6 or generc rm and and ung eq. (5.4, we nd that the parameter φ may alo be wrtten a φ = cβ, =,. (5.7 Note that φ a uncton only o pot market contant, and aumng that β > 0, eq. (5.4 mple, urther, that 0< φ <. (5.8 Ung eq. (5.5, we may rewrte eq. (5. or rm pot SF n term o β a 56 ( ;, ( Σ p q q = φq c β + β p. ( Green (999a, 6 (empha n orgnal. An ncreae n a rm orward market poton ( contract ale, n Green parlance decreae t rval quantty by depreng the equlbrum pot market prce, thereby callng orth le upply rom t rval, gven the rval xed pot market SF. On the nature o th eect n the preent model, ee ecton 5.4 below. 56 For contency, we mantan q a an argument o Σ, although we note that q doe not appear on the rght-hand de o eq. (5.9, a explaned above. 46

164 We may wrte analogou expreon that characterze rm pot SF by nterchangng ubcrpt and n eq. (5.5, (5.6, and (5.9: α = φ q c β, (5.0 0 ( β = φ γ + β, (5. and ( ;, ( Σ p q q = φ q c β + β p. (5. 0 The equaton (5.9 and (5. or the rm pot market SF ndcate that we may nterpret the dmenonle parameter φ ntroduced n eq. (5.4 a the partal dervatve o rm SF ( ;, p q q Σ wth repect to orward market quantty, that, Σ ( p ; q, q q = φ > 0,, =, ;. (5.3 In other word, we may contrue φ a the entvty, at the margn, o rm pot market quantty bd (at a gven prce p to change n t orward market quantty q. 5.3 Comparatve tatc or the pot market When olved multaneouly, eq. (5.6 and (5. yeld a quadratc orm n β and β, the lope o the repectve rm ane pot market SF, uch that β (,, = β c c γ (, =,;. Th ytem o β a pecal cae (or n = o the general n-rm model tuded by udkevch (999, n whch rm wth ane margnal cot bd ane SF nto a centrally-cleared market. For the duopoly cae tuded here, udkevch 47

165 reult mple that the quadratc orm n β and β ha exactly one root n whch both β and β are potve. Thu, there a unque oluton (, β β correpondng to a trctly ncreang pot market SF or each rm. 57 Gven that β (,, β c c γ alo have rom the denton o φ n eq. (5.4 that φ φ( c, c, γ =. =, we Table 5. below report the gn o the partal dervatve o ( c, c, ( c, c, β γ and φ γ a derved n Appendx D. va derentaton o eq. (5.6 and (5., a well a o denton (5.4 or φ. TABLE 5.: COMPAATIVE STATICS OF (,, β = β c c γ AND φ = φ( c, c, γ WITH ESPECT TO THE PAAMETES c, (SEE APPENDIX D. FO DETAILS c, AND = γ (,,; β c β c β γ < 0 < 0 > 0 φ < 0 c φ > 0 c φ < 0 γ The gn o the partal dervatve gven n Table 5. are nvarant wth repect to the parameter value c, c, and γ. The comparatve tatc eect n the table or ndcate that a ntuton mght ugget a ether rm margnal cot uncton or the pot market demand uncton become teeper, the pot market SF lope β β become 57 The exact analytcal expreon or ( c, c, β γ traghtorward but tedou to obtan rom eq. (5.6 and (5.; we do not requre t or the preent analy and o do not olve or t explctly here. 48

166 teeper. Moreover, agan ung eq. (5.4, (5.6, and (5., we may how that the ollowng nequalte obtan at all parameter value or the dervatve o wth repect to c and c (, =,; ; ee Appendx D. or detal: β and β β c > β c, (5.4 and β β > c c. (5.5 Inequalte (5.4 and (5.5 ndcate that the eect o changng rm own margnal cot uncton lope c on the lope magntude than ether β o pot market SF ( ;, p q q Σ greater n. the eect on (eq. (5.4, 58 or β when changng the correpondng parameter c or rval,. the eect o changng c on the lope (eq. (5.5. β o pot market SF Σ ( ;, p q q The general nght here content wth ntuton that a veron o dagonal domnance hold or a Jacoban matrx o dervatve o the orm 58 ecall rom Table 5. that both o thee eect are negatve. 49

167 β β c cn βn βn c cn, (5.6 n whch each dagonal element o the matrx (5.6 larger than the o-dagonal term n the ame row and column. 59 We next conder the relatonhp among the lope o margnal cot uncton c, lope o the pot market SF β, the parameter φ, and the dervatve ( ;, Σ p q q q rom eq. (5.3. Begn by conderng the cae o ymmetrc cot n whch c = c n the denton (5.4 or φ. In th cae, the ymmetrc orm o eq. (5.6 and (5. mply that we mut have β = β. From eq. (5.7, a a conequence, th ymmetrc cenaro mple urther that φ = φ. We may thereore wrte that c = c β = β φ = φ. (5.7 Moreover, ung the equaton o ecton 5., we may begn wth any one o the equaton n (5.7 to generate the other two equaton gven there. We thu may trengthen the mplcaton n tatement (5.7 to and only relatonhp a ollow: c = c β = β φ = φ. (5.8 Fnally, we may generalze the tatement (5.8 urther to nclude aymmetrc rm and. Conder the two aymmetrc cae c > c and c < c and the mplcaton o each or 59 The nequalte (5.4 and (5.5 mply that uch dagonal domnance hold or n =. We conecture that th property hold more generally or n >. 50

168 the relatve magntude o the β and the φ. Appealng to the gn o the partal dervatve rom Table 5., to eq. (5.3, and to nequalty (5.5 permt u to generalze (5.8 or the cae o aymmetry n the ollowng natural way: c > c β < β φ < Σ φ < Σ = = = = < > > q > q. (5.9 An mplcaton o the tatement (5.9 that, looely peakng, a hgh-cot rm le able to aect the quantty that t bd n the pot market (at any gven prce va t orward market poton than a low-cot rm. To put t another way, a a rm cot ncreae, t quantty bd nto the pot market, n equlbrum, become le entve to t orward market poton. 60 Fgure 5. below depct rm pot market upply uncton Σ ( ;, p q q content wth eq. (5.9 and (5. or each o the two rm Content wth the tatement (5.9, the termnology ued here o hgh-cot and low-cot rm denote, more precely, the lope c o a rm margnal cot uncton. 6 The geometry o Fgure 5. content wth Green (999a, 4 Fgure n whch he conder pot market competton (alo n ane SF n the preence o a orward contract market baed on conectural varaton. 5

169 c p + cq 0 Spot market β Σ ( p ; q, q c C ( q c 0 cβ q c β 0 0 α φ q q q FIGUE 5.: THE GEOMETY OF THE SPOT MAKET SUPPLY FUNCTION Σ ( p ; q, q O partcular nteret n Fgure 5. the relatonhp o the SF ( ;, p q q Σ to the margnal cot uncton C ( q and rm orward market quantty q. A Green (999a, 4 how, the uncton ( ;, Σ p q q nterect C ( q ( q, C ( q. Conequently, ncreang at the pont q tranlate the uncton Σ ( ;, p q q horzontally to the rght (recall eq. (5.3, ncreang rm pot market bd quantty at every prce p. 6 Another mplcaton o Fgure 5. geometry that rm bd t 6 Allaz and Vla (993 provde ueul ntuton or th eect o rm orward market poton ncreang ther pot market quantte (alo maneted n eq. (5.3. Namely, n a model havng a Cournot pot market, thee author nd (p. 4 that the decreae n prce neceary to ell [an] addtonal 5

170 pot market quantty below t margnal cot at quantte below q, and above t margnal cot at quantte above q. Fgure 5. above alo ueul to llutrate how the pot market geometry change wth hock to the underlyng parameter o nteret. 63 In partcular, conder the eect, n turn, o hock to the margnal cot uncton ntercept c 0 and c 0, the margnal cot uncton lope c and c, and the lope γ o the ane pot market demand uncton on the uncton depcted n Fgure 5. or rm. Aume, or mplcty, throughout th paragraph that rm orward market quantty xed at q. 64 Conderng rt an ncreae n the ntercept c 0, th hock nduce an upward tranlaton o both rm margnal cot uncton C ( q and pot market SF Σ ( p ; q, q. In contrat, a hock to c 0 leave the uncton n Fgure 5. unchanged. A hock to the lope c rotate the unt [on the pot market] doe not aect the revenue rom the orward ale. In other word, the margnal revenue uncton rotate counterclockwe about t prce ntercept, and the optmal pot market quantty ncreae. The ame eect preent n th SF-baed model. 63 Th dcuon rele on the comparatve tatc eect o Table 5. on β and the denton o rm margnal cot uncton C ( q (eq. (5. and SF ( ;, p q q Σ (eq. (5.9 and (5. above. See alo Table E. o Appendx E.4 or correpondng numercal reult rom an ane example. uncton ( and pot market SF ( prce ( 0 c. 64 A conequence o th aumpton that the pont o nterecton o rm margnal cot C q ;, Σ p q q reman xed at the quantty q = q, though the C q = c + cq at whch th pont o nterecton occur ht, naturally, wth hock to c or 0 53

171 uncton C ( q ( p ; q, q counterclockwe about t ntercept c 0, whle both rotatng the SF Σ counterclockwe and tranlatng t upward. I ntead we ncreae the lope c o rm margnal cot uncton, th leave the uncton C ( q whle rotatng the SF ( ;, p q q unchanged, Σ counterclockwe (but to a leer degree than gven a comparable hock to c, due to nequalty (5.4. Fnally, conder the eect o a hock to γ, the magntude o the pot market demand uncton lope. A hock that ncreae γ make th demand uncton (not hown n Fgure 5. le teeply-loped. Th hock lkewe make ( ;, p q q clockwe, but leave the uncton C ( q Σ le teeply-loped, rotatng th SF unchanged. Green (999a, 09 oberved that [a] general conecture mght be that a the pot market become more compettve, an uncompettve contract market wll have le mpact on t [ootnote omtted]. A a nal remark on the comparatve tatc reult o Table 5., we obtan reult rom the mult-ettlement SFE model that urther upport Green conecture above. Namely, conder agan the eect o a change n γ, the magntude o the pot market demand uncton lope. Increang γ lead both to ( le teeply-loped pot market SF Σ ( ;, p q q, and ( a decreae n φ, whch we may nterpret (rom eq. (5.3 a the entvty o (ether rm SF ( ;, p q q (own orward market quantty Σ to q. That, Table 5. reult or margnal change to γ mply that a change n the lope o pot market demand caung rm to behave more 54

172 compettvely n the pot market make pot market acton (and hence the pot market outcome le entve to orward market acton and outcome (and vce-vera. When rm compete n SF n the pot market, they compete n an nntedmenonal acton pace. In th cae, trategc nteracton n a duopoly cannot be completely characterzed by ung reacton uncton n the plane, whch aume a onedmenonal trategy (or acton pace or each rm. Even retrctng rm acton pace to ane pot market SF a we do n th chapter, uch acton pace are not onedmenonal, but two-dmenonal. In th ane cae, the rm acton pace compre, naturally, the lope and ntercept o the ane SF. A we have noted, we may olve eq. (5.6 and (5. or the SF lope β (,, β c c γ value or c, c, and γ, the SF lope =. That, gven exogenou β are ndependent o the ntercept α ; n partcular, the β are ndependent o orward market quantte q. Th property β β motvate the contructon below o what we call partal reacton uncton ( n the β - β plane. Thee partal reacton uncton capture that porton o rm repone to change n the parameter c, c, and γ relected n the lope o the ane pot market SF. 65 I we aume unctonal relatonhp between β and β, we may plot the partal reacton uncton ( β and ( or φ (, =,;, a done n Fgure 5. below. β ung eq. (5.6, (5., and ( Smlarly, Lauel (99 nterpreted the lope o an ane SF a the relevant trategc varable n a trategc nternatonal trade model. 55

173 Appendx D.3 demontrate rom thee three equaton that the partal reacton uncton ( ( β have the orm depcted n Fgure 5.. In partcular, each uncton β everywhere ncreang and concave n t argument β > 0, wth a potve β -ax ntercept (n the lmt. The lope ( β take on t maxmum value at the β -ax ntercept, decreang a β ncreae and gong to zero a β. Content wth thee relatonhp, we nd that or xed c, β bounded above by c. The unque nterecton o the partal reacton uncton ( β and ( orthant correpond, naturally, to rm equlbrum choce o β and β. β n the potve β = ( β ( β c + cγ γ ( + cγ Equlbrum ( β, β ( β γ + c γ ( + c γ c β = ( β FIGUE 5.: PATIAL EACTION FUNCTIONS ( 56 β β IN THE β - β PLANE: THE SPOT MAKET SUPPLY FUNCTION SLOPES β AND β AE STATEGIC COMPLEMENTS

174 Treatng the lope β a the trategc varable or each rm n the pot market, we may vew the β a trategc complement n the ene o Bulow, Geanakoplo, and Klemperer (985, nce ( 0 β > or, =, ;. The complementary relatonhp between β and β mple, or example, that rm were to chooe or whatever reaon a teeper SF (a lower β, rm bet repone would be to lkewe ubmt a teeper SF (a lower β. Smlarly, the bet repone to a latter SF lkewe an SF wth a latter lope. 5.4 Implcaton or the optmal pot market prce uncton The pot market-clearng condton, gven equlbrum orward quantte q and q and a realzaton o the pot market demand hock ε, ( p ; q, q ( p ; q, q D ( p, ε Σ +Σ =. (5.0 The pot market-clearng prce p atyng eq. (5.0 a uncton o both ε and q, * that, p p ( ε ; q, q. 66 Ung the Ane Spot Market Demand Functon and Ane Spot Market SF aumpton ntroduced at the outet o th chapter, eq. (5.0 become ( ( φq c0β βp φq c0β βp = γ p + ε whch, olvng or p, yeld 66 Where t caue no ambguty, we ue the more convenent notaton p n what ollow. 57

175 p ε φq φ q + c β + c β = 0 0 β + β + γ. (5. Gven our aumpton, we have rom eq. (5. that p φ = < 0, (5. q + + β β γ that, an ncreae n ether rm orward market poton decreae the equlbrum pot market prce, ceter parbu. For concretene, conder an ncreae n q, whch rom nequalty (5. caue a decreae n the equlbrum prce p. I rm ane pot market SF reman unchanged, nce th SF aumed to be trctly ncreang, the lower prce caue rm to reduce t pot market quantty oered. Th the ame a Green (999a, 6 obervaton or ane SF that one rm quantty decreang n the other rm orward market poton, although the rt rm SF unaected. Snce we have rom eq. (5. that p ane n ε, we conclude that ( ;, * p ε q q, n act, partally nvertble wth repect to ε n the mpled ane example. In a Nah equlbrum, th mple that our earler aumpton (ee ecton 4. * o the partal nvertblty o p ( ; ˆ, ˆ q q ε wth repect to ε uted or the mpled ane example. More generally, due to the contnuty o the underlyng derental equaton oluton n the ntal condton, 67 th property o partal nvertblty wll hold alo or pot market SF ucently cloe to the ane SF n eq. (5.9 and ( See ecton 7.. or a tatement o the relevant contnuty theorem or the orward market SF. A mlar argument hold here or the pot market SF. 58

176 eplacng q wth ( S p, =,, n eq. (5., we may alo wrte th equaton a a uncton o the orward market prce p, p ( ( ε φs p φs p + c0β+ c0β = β + β + γ. (5.3 It wll be ueul to mply eq. (5.3 and the expreon that ollow by denng ome addtonal notaton. Namely, let > 0 ω a β + β + γ (5.4 and ωb c β + c β, ( ung ubcrpt letter a and b to avod conuon wth rm and. The gn o ω a and ω b above ollow rom the analy o ecton 5. and 5.3 and our parametrc aumpton. Ung the notaton o eq. (5.4 and (5.5, we may recat eq. (5.3 a ( ( p = ω ε φs p φ S p + ω a b. (5.6 Fgure 5.3 below llutrate the clearng o the pot market and determnaton o the equlbrum prce p, aumng ane margnal cot and pot market demand uncton, and ane SF (a depcted n Fgure 5. or rm. 59

177 p Σ Spot market ( p ; q, q D ( p, ˆ ε ( ˆ ε * pˆ = p ; q, q Σ ( p ; q, q C ( q ( ;, Σ Agg p q q C ( q q 0 q qˆ qˆ qˆ Agg ˆ ε Σ, D, ε FIGUE 5.3: SPOT MAKET EQUILIBIUM ( ˆ, ˆ Agg q p ASSUMING AFFINE FUNCTIONAL FOMS, AND GIVEN FOWAD MAKET QUANTITIES q AND q AND A SPOT MAKET DEMAND SHOCK ˆ ε = ε Gven orward market quantte q and q and a pot market demand hock ε = ˆ ε, Fgure 5.3 llutrate how the rm pot market SF um horzontally to yeld the aggregate pot market SF Σ Agg ( p ; q, q. The nterecton o th uncton wth pot market demand (, D p ε, naturally, dene the equlbrum pont ( ˆ, ˆ Agg pot market. q p or the 60

178 eturnng to eq. (5.6 or p, we next compute the condtonal expectaton o th expreon. Condtonal on the orward market outcome o the demand hock 68 ε 0 and the correpondng market-clearng prce p, th expectaton ( p p 0 a ( p 0 S ( p S ( p E, ε = ω E ε, ε φ φ + ω b. (5.7 The quantte p and ε 0 are related, naturally, n any orward market equlbrum. Chapter 4 optmzaton problem or rm etablhed the extence o an optmal orward market prce uncton * * p ( ε 0 or rm, and ( 0 p ε nvertble, an optmal SF S ( p. 69 ecall alo that n Nah equlbrum, * * p ( ε 0 and p ( ε 0 * mut concde n a market-wde optmal orward market prce uncton, ( 0 p ε. Below, we etablh ucent condton nvolvng the orward market SF or the nvertblty * and derentablty o ( 0 p ε. Thee properte wll be ueul later n mplyng eq. (5.7. Conder equlbrum n the orward market. Gven orward market equlbrum SF S ( p and a demand uncton (, 0 D p ε wth arbtrary hock ε 0, the orward * market clearng condton (at a market-clearng prce p p ( ε 0 = ecall rom eq. (3.0 that the addtve orward market demand hock ε0 (, 0 0 D p ε equal to the orward market demand uncton (, 0 p. 0 = D p ε evaluated at the orward market reerence prce 69 Symmetrc reult obtan, naturally, or rm. 70 Under the aumpton o ecton 4., a unque market-clearng prce wll ext. 6

179 ( ( (, S p + S p = D p ε. (5.8 0 p ε or * Subttutng ( 0 p n eq. (5.8 and recallng the addtvely eparable orm (eq. (3.8 or the orward market demand uncton (, 0 (5.8 a ( ( ( ( ( ( * * * D p ε, we may recat eq. S p ε + S p ε = D p ε + ε. (5.9 Snce eq. (5.9 an dentty or each 0 D0 ( obtan ε, and aumng that S (, S (, and are derentable, we may totally derentate eq. (5.9 wth repect to ε 0 to * * * ( ( ( ( ( ( S p p ε dε + S p p ε dε = D p p ε dε + dε * Solvng the above equaton or p ( ε 0 p *, we get ( ε 0 = S p S p D p ( + ( ( 0. (5.30 We aume now that, n addton to beng derentable, the orward market SF S ( are alo trctly ncreang 7 whch, a chapter 6 wll how, ucent or D ( p, ε 0 p D * = 0 ( p < 0. Then, we have rom eq. (5.30 that ( 0 derentable and that p ε 7 Later, n the numercal example o chapter 7, we wll ee that there ext orward market SF atyng thee condton, thu utyng th aumpton. 6

180 or all 0 p * ( ε > ( * ε E. The nequalty (5.3 mple that the uncton ( 0 * that we may dene the uncton ep ( p = ε0 a the nvere o ( 0 * ( ( ( 0 p p ε nvertble, o p ε, that, ε = e p p p. (5.3 * Snce the relatonhp p p ( ε 0 = nvertble, p and ε 0 are one-to-one. In eq. (5.7, thereore, we need condton on only one o the two quantte p and ( p ε = e. Condtonng on 0 p p alone, we may wrte eq. (5.7 a E ( p p E( e ( p S( p S( p = ωa ε p φ φ + ω b. (5.33 Later, we ue eq. (5.33 at the outet o chapter 7 to mply the rm orward market equlbrum optmalty condton (ee alo ecton 5.5 below. In the mult-ettlement SFE model, t alo o nteret to determne how the expected pot market prce E ( p p vare wth margnal change n orward market outcome. To nvetgate th ue, we derentate eq. (5.33 wth repect to p to obtan ( de ( ε e ( p ( de p p de p dp d dp ( ( p p = ωa φ S p φs p ε 0. (

181 ( 0 de ε e p dε on the rght-hand de o eq. (5.34, we may condton In the term p ( on 0 ε ntead o p ( e p (recallng eq. (5.3 or eae o notaton. Makng th change and recognzng alo rom eq. (5.3 that * ( ( ε 0 dep p dp = dp dε 0, eq. (5.34 become ( * E ( ε ε ( ε 0 0 de p p d dp = ω a φs p φ S p dp dε0 dε0 * * Ung eq. (5.30 to ubttute or ( ( term, we have ( (. (5.35 p ε dp ε dε n eq. (5.35 and collectng ( ( ( ε ε0 ( p ( de p p de ε ε de ε ε 0 0 = ω a φ S ( p + φ S p ε0 ε0 dp d d d E. D 0 dε 0 ( (5.36 To mply eq. (5.36 urther, the next chapter develop an expreon or the dervatve d ( ε ε ε aumng ( a decompoton o ε nto conttuent tochatc E 0 d 0 parameter, and ( a relatonhp between conumer prvate normaton about the level o pot market demand, on the one hand, and orward market demand, on the other. In nterpretng the reult o th ubecton, t mportant to note that we have not yet peced the orward market demand uncton. In partcular, eq. (5.33 expree 64

182 the condtonal expectaton o p n term o the orward market SF, whoe dervaton n chapter 4 aumed the extence o a downward-lopng, twce-derentable orward market demand uncton. We revt th ue n chapter 6, n whch we explan how uch a orward market demand uncton could are, and analyze th uncton properte gven the attrbute o conumer. 5.5 Implcaton or the orward market optmalty condton The Ane Spot Market Demand Functon, Ane Margnal Producton Cot Functon, and Ane Spot Market SF aumpton permt u to mply rm equlbrum optmalty condton or t orward SF, eq. (4.4 and (4.4. From eq. (5., we have that Σ ( p ; q, q q = 0, and Σ ( p ; q, q q = φ. Ung thee expreon and ater ome mplcaton, eq. (4.4 become { φφ E( p p ( c0+ cs ( p E( p p p } S ( p = S ( p D 0 ( p E ( p p p (5.37 or all market-clearng prce p. 65

183 For purpoe o comparon wth prevou work, we make the temporary aumpton that the expreon n brace on the let-hand de o eq. (5.37 nonzero, that, ( ( ( ( φφ E p p c + cs p E p p p 0. ( Th aumpton permt u to rewrte eq. (5.37 a S ( p S ( p D 0 ( p E ( p p p ( ( ( E( =. (5.39 φφ E p p c0+ cs p p p p Examnng the rght-hand de o eq. (5.39, we ee that t depend on two prce derence:. The derence between the expected pot prce and the orward prce, E ( p p p. The derence between the expected pot prce and rm margnal cot o producng t contract quantty n the orward market, E ( p p ( c0 cs ( p + The tructure o eq. (5.39 reemble that o KM (ngle-market optmalty condton, namely, ( ( ( S p S ( p = + D p p C S p (. (5.40 The mlarty between eq. (5.39 and (5.40 partcularly apparent or the pecal cae n whch pot market demand (, D p ε very elatc, o that t lope 66 γ get large

184 n magntude, that, γ. A γ decreae, we have rom eq. (5.4 that φ 0 ( =,. Settng φ = 0 n eq. (5.39 a an approxmaton, we may then rewrte th equaton a ( p S S p D p p E p p ( = + 0 ( (. (5.4 Equaton (5.4 completely analogou to eq. (5.40, except that E ( ( place o C S( p p p appear n. 7 The tructural mlarty o thee equaton ugget that when pot market demand perectly elatc, the margnal opportunty cot o orward contract upply mply the expected pot market prce. We may alo derve a veron o eq. (5.4 more drectly we olve rm orward market problem (ee chapter 4 wth the mplyng aumpton that. the pot market prce random wth expectaton E ( p, and. uppler bd perectly elatc upply uncton that are ndependent o orward market outcome. We agan aume (a uted n chapter 6 a downward-lopng orward market demand uncton (wth hape component 0 ( ( S p. In th cae, eq. (5.4 become D p gven trctly ncreang orward market SF 7 Earler n chapter 4 (peccally, n dcung eq. (4.4 and (4.4, we took note o th analogy between expected pot prce n the orward market problem and margnal cot n the (ngle market pot market problem. 67

185 ( p E ( p S S p D p p ( = + 0 whch dentcal to eq. (5.4, except that E ( o mplyng aumpton above. ( p replace E (, (5.4 p p, a a conequence 5.6 Concluon Th chapter aumed that cot uncton, pot market demand uncton, and pot market SF have ane unctonal orm. Thee mplcaton, naturally, have conequence or both the pot and orward market whch we explored n th chapter. The next chapter, chapter 6, explan how the orward market demand uncton are and nvetgate t properte. Then, chapter 7 wll ntegrate the reult o the preent chapter, ung eq. (5.33 or E( p p to mply urther the orward market equlbrum optmalty condton (eq. (5.39 or rm, and analogouly or rm, rom whch we derve the orward market SFE. 68

186 I can get no remedy agant th conumpton o the pure; borrowng only lnger and lnger t out, but the deae ncurable. Shakepeare, Henry IV, Part Electrcty eem detned to play a mot mportant part n the art and ndutre. The queton o t economcal applcaton to ome purpoe tll unettled, but experment ha already proved that t wll... gve more lght than a hore. Ambroe Berce, The Devl Dctonary 6 The demand de CONSUMES purchae electrcty or conumpton n the pot market. In th chapter, we how under reaonable aumpton notably, conumer rk averon that conumer are alo actve n the orward market. Speccally, we derve here an endogenou aggregate orward market demand uncton or a repreentatve conumer. Moreover, th chapter tate ucent condton or th demand uncton whch we have denoted a D ( p, ε 0 to have the ollowng properte. 73 Frt, (, 0 D p ε lope downward at all prce p, derentable wth repect to both argument, and t hape 73 We already aumed ome o thee properte o (, 0 the model preented n th chapter a utyng thee aumpton. 69 D p ε n ecton We may vew

187 determntc and common knowledge. Second, (, 0 D p ε ha an addtve, exogenou, and tochatc component, ubumed n the hock ε 0, that ht (, 0 D p ε horzontally but doe not change the uncton hape (.e., rotate or deorm t. 74 The outlne o th chapter a ollow. We begn n ecton 6. by ntroducng ome undamental aumpton underlyng the demand-de model. Secton 6. motvate a neted optmzaton problem decrbng conumer behavor n each market, and ute a mean-varance approxmaton to conumer utlty maxmzaton problem n the orward market. Next, ecton 6.3 gve ucent condton or a repreentatve conumer to ext n the mult-ettlement SFE model. Secton 6.4 pece attrbute o the repreentatve conumer that are content wth an ane aggregate pot market demand uncton. Next, ecton 6.5 pece a mple model or the pot market demand hock ε. Secton 6.6 rerame the analy n term o a repreentatve conumer. It then derve the repreentatve conumer orward and pot market actvty a the oluton to her underlyng utlty maxmzaton problem. 75 In ecton 6.7, we how how demand hock and prce are related acro the two market. Fnally, ecton 6.8 characterze the eental properte o the aggregate orward market demand uncton. Several emprcal tude have ound that, a we mght expect, etmated electrcty orward market demand uncton are downward lopng, and are more elatc 74 Many o thee properte o the mult-ettlement SFE model orward market demand uncton are dentcal to thoe o KM ngle-market (.e., pot market demand uncton. The hared eature acltate the applcaton o KM SFE ramework to the orward market (n addton to the pot market n the preent work. 75 A an arbtrary conventon, we ue emnne pronoun to denote conumer. 70

188 than typcal etmate o pot market demand uncton. For example, Earle (000 tude the rt twenty month o operaton (.e., rom Aprl 998 to November 999 o Calorna compettve market. Earle nd a downward-lopng redual demand uncton 76 wth a medan elatcty o approxmately 0.; n 7% o the hour n h data et, the magntude o the redual demand elatcty exceed one. Such value o demand elatcty are ndeed markedly hgher than hort-run elatcte commonly meaured n pot electrcty market. The preent model endogenou determnaton o D ( p, ε 0 naturally permt uch elatcty, a well. 6. Modelng aumpton Th ecton outlne our aumpton regardng the attrbute o conumer and motvate ther optmzaton problem n the orward and pot market. 6.. Prce-takng conumer There are a total o J conumer actve n the market, ndexed by =,,, J. We aume that J large and xed. 77 Furthermore, each conumer a prce taker n both the orward and pot market (conumer may be actve n both market. 6.. Partal equlbrum analy Each conumer expendture on electrcty a mall racton o that conumer total expendture; th alo true wth repect to each conumer margnal expendture. In 76 Generatng unt n the Calorna market whch mut run due to engneerng contrant were bd nto the PX wth a (perectly elatc and non-trategc bd o zero dollar. Earle then ubtract uch bd rom total demand to obtan the redual demand uncton. 77 We neglect the poblty o entry and ext o conumer, wth the utcaton that thee acton are cotly. 7

189 addton, the electrcty market, a uch, mall relatve to the entre economy. Hence, prce o other good and ervce may be taken a approxmately contant a the prce o electrcty vare. In th ettng, Marhallan partal equlbrum analy (Marhall 90 mple or all conumer that ( we may neglect wealth eect on electrcty demand, and ( we may treat expendture on good and ervce (other than electrcty a expendture on a ngle compote commodty, termed the numerare commodty (Ma-Collel, Whnton and Green 995, 36 and denoted a m. Abent uncertanty, moreover, t reaonable under partal equlbrum aumpton (Ma-Collel, Whnton and Green 995, ec. 0.C to take conumer utlty uncton to be qualnear wth repect to th numerare (mplyng no wealth eect or electrcty demand, at leat n the hort run. We alo aume a utltaran ocal welare uncton. Together, qualnear utlty uncton and a utltaran ocal welare uncton mply that we may quanty change n ocal welare by meaurng change n aggregate urplu A derved demand or electrcty Demand or energy (or example, electrcty commonly condered a derved demand, a ether an nput to producton 79 or a mean to provde electrcty-dependent ervce 80 (herenater amente to conumer. The conequence or the analy o conumer behavor n the preent model that conumer utlty uncton do not depend drectly 78 Aggregate urplu (or Marhallan aggregate urplu rom conumpton o a commodty dened a the total utlty generated by conumpton o that commodty le t cot o producton (Ma- Collel, Whnton and Green 995, 36. Secton 7.7 ue aggregate urplu to compute ocal welare n the context o a pecc numercal example. 79 See, or example, Berndt and Wood (975 analy. 80 For example, lght, heat, ar condtonng, entertanment, etc. 7

190 on the amount o electrcty conumed, but rather on the level o amenty enoyed. A related element o the modelng ramework adopted here the aumpton that each conumer notonally produce her amenty n a gven market round ung nput o electrcty and other (unmodeled nput, or example, captal/durable good, labor/leure tme, aumed to be xed. 8 The amount o amenty produced by the conumer ubect to tochatc hock due, n turn, to envronmental or technologcal actor. We now ntroduce notaton to characterze conumer demand-de producton proce or her amenty. Dene the ollowng: q = conumer quantty o electrcty purchaed n the pot market and ubequently ued a an nput to amenty producton n a gven market round; 8 m = conumer conumpton o the numerare commodty m; 83 T T, T = tochatc producton hock wth upport T, T characterzng randomne n conumer producton proce due to envronmental or technologcal actor; x = level o amenty 84 enoyed by conumer ; and 8 The preent approach n the ame prt a Mchael and Becker (973 reormulaton o the theory o conumer behavor ung a houehold producton uncton. 8 An outcome n whch q < 0 correpond to conumer beng a net uppler o electrcty n the gven market round. Th poblty could be nterpreted a o-called net meterng, whereby conumer ownng electrcty generaton capacty may ell electrcty that they chooe not to conume. Whle the preent model permt th, n prncple, the partcular orward market equlbrum electon procedure employed n chapter 7 numercal example preclude uppler and conumer rom wtchng de n the pot market (but not n the orward market. See ubecton 7.6. or detal. 83 We aume m or convenence, to avod boundary complcaton. 73

191 ( : q, T = conumer producton uncton 85 relatng the nput q and the hock T to output. Conumer oberve the realzaton o the tochatc hock T beore electng the optmal level o the nput q to produce x. Aumng that no amenty wated (.e., produced but not enoyed, we may equate x and the amount produced a (, x = q T. (6. Next, aume that conumer derve utlty accordng to a utlty uncton W ( m, x rom two ource n the pot market: ( her conumpton m o the numerare commodty, and ( her enoyment o amenty repect to conumpton o the numerare commodty, x. 86 Let (, W m x be qualnear wth m, and let the contrbuton o 84 Snce we take the unt o the amenty x to be arbtrary or greatet generalty, the orgn o x alo arbtrary. Hence, x may be any real number. 85 We pecy the properte o the producton uncton below. Whle the uncton (, conumer-pecc, we uppre t ubcrpt to reduce notatonal clutter. The argument 74 q T q and ( q, T aocate th uncton wth conumer. From note 84 and eq. (6., producton ( q, T may be potve, negatve, or zero. 86 Whle the uncton (, to reduce notatonal clutter. The argument conumer. T o W m x conumer-pecc, we uppre t ubcrpt (a wth m and x o (, W m x aocate th uncton wth

192 x to utlty enter (, we may dene (, W m x a a uncton ( x W m x a (, φ ( W m x m x φ. 87 Wth the above aumpton, = +. (6. Turn now to the properte o the uncton and φ n expreon (6. and (6. above. Frt, let both and φ be twce contnuouly derentable n ther argument. Next, the conventonal neoclacal theore o producton and o demand a well a the preent modelng ramework ugget a number o a pror retrcton on the unctonal orm o both and φ. Namely, we aume the ollowng (lettng ubcrpt denote partal derentaton or the producton uncton : Producton (trctly ncreang n the nput ( q (or q ucently mall: 88 q, T > 0 (6.3 q Producton (trctly decreang n the producton hock T : ( q, T < 0 (6.4 T 87 Whle the uncton φ ( x conumer-pecc, we uppre t ubcrpt (a wth and x o φ ( aocate th uncton wth conumer. W to reduce notatonal clutter. The argument 88 We requre th qualcaton on value o the producton uncton (, q T, whch we pecy n ubecton x q to accommodate a quadratc unctonal orm or 75

193 The margnal product o the nput q nonncreang: ( q, T 0 (6.5 qq The margnal product o the nput q (trctly ncreang n T : ( q, T > 0 (6.6 qt ecall that by the argument o note 85, we have not retrcted producton to be nonnegatve. For example, gven the dervatve o above, x ( 0, T = mght be negatve or ucently large T, although th may not be an equlbrum outcome (ee ecton Now aume the ollowng regardng the uncton φ : Utlty (trctly ncreang n x : φ ( x > 0 (6.7 Margnal utlty nonncreang n x : φ ( x 0 (6.8 To provde ome ntuton or the applcaton o the demand-de producton model outlned above, conder the ollowng pecc example. Suppoe that conumer ha a derved demand or electrcty, q, to operate a houehold clmate control ytem producng the amenty o a comortable ndoor envronment or mply, comort, denoted a x. Conumpton o a greater amount o electrcty produce a hgher level o comort, but at a (weakly decreang rate, relectng dmnhng return n q content 76

194 wth nequalte (6.3 and (6.5 above. egardng the producton hock T, one mght nterpret th hock, roughly peakng, a ambent temperature ; t ueul, however, we contrue T more generally a any advere hock n the ambent envronment n real tme. In the preent example, ncreang T decreae comort, or any level o electrcty conumpton. 89 Th content wth nequalty (6.4 above. Fnally, a larger value o the hock T ncreae the margnal productvty o the electrcty nput, meanng that an ncrement n electrcty conumpton produce more equvalent comort at the margn, a nequalty (6.6 ndcate. 6. Conumer optmzaton problem Th ecton develop a model o conumer decon n both the orward and pot market. Th model aume that conumer decon maxmze her (expected utlty rom conumpton o electrcty and o the numerare commodty. Prevouly, we noted that conumer oberve the realzaton o the tochatc hock T beore makng her pot market conumpton decon. In modelng conumer pot market problem, thereore, we may take T a gven. In contrat, a conumer ace her orward market problem, the real-tme advere envronmental hock T a yet unoberved. It thereore approprate to treat T a tochatc when modelng conumer orward market decon makng. 89 Th lexble nterpretaton o T permt the demand-de producton model to apply to mot any redental (.e., conumptve or commercal (.e., productve ue o electrcty. 77

195 6.. An expected utlty maxmzaton problem Frt dene the ollowng addtonal notaton: ne w = conumer wealth endowment avalable or conumpton, not ncludng any proceed rom electrcty market actvty; n the partal equlbrum ramework, t cont o an endowment o the numerare commodty m (whereby the upercrpt ne on ne w ndcate non-electrcty q = conumer quantty o electrcty purchaed n the orward market 90 p m = the prce o the numerare commodty m A n prevou chapter, p and p are the electrcty orward and pot market prce, repectvely. We now dene conumer budget contrant or a gven market round. In word, th budget contrant enure that the um o conumer expendture doe not exceed the um o her wealth avalable or conumpton. In the mult-ettlement model, conumer ncur three dtnct expendture: p m = expendture on the numerare commodty m, m p q = expendture on electrcty contract n the orward market, and p q = expendture on electrcty n the pot market, 90 Note that the doman or q. Snce orward contract are ettled nancally and are not lnked to electrcty producton, t natural to permt conumer and or that matter, uppler a well both to buy ( q > 0, or a long poton and ell ( q < 0, or a hort poton n the orward market. 78

196 and ha two ource o wealth avalable or conumpton: ne w = non-electrcty wealth endowment, and p q = ettlement recept n the pot market gven orward contract q From the above dcuon, we may wrte conumer budget contrant algebracally a p m + p q + p q w + p q, ne m or collectng term n q, ( p m + p q w + p p q. (6.9 ne m In the budget contrant (6.9, we abtract rom cahlow arng rom hare that conumer may hold n the two uppler rm o the mult-ettlement SFE model. 9 We may ratonalze th aumpton n two way. The rt potental utcaton mply to aume that the rm are owned by agent other than the J conumer actve on the demand de o the model. The econd potental utcaton or th aumpton to permt uch hare ownerhp by conumer n the model, whle uppong urther that 9 Such hareholdng a tandard element o model o compettve equlbrum (Ma-Collel, Whnton and Green 995,

197 conumer gnore the eect o ther hare ownerhp on orward market behavor 9 due to bounded ratonalty. 93 Aume that conumer obectve n the pot market to maxmze her utlty uncton (, W m x through optmal choce o m and q or conumpton. For a gven hock T, and ung the maxmand n eq. (6., the producton contrant (6., and the budget contrant (6.9, we may wrte conumer pot market optmzaton problem a max m + φ ( x q m ( (.t. x = q, T (producton contrant pm+ pq w + p p q ne m (budget contrant. (6.0 Two mplcaton to problem (6.0 are poble. Frt, we may ubttute or x n th problem obectve uncton rom the producton contrant, nce t an equalty. Second, t evdent that the budget contrant, a well, wll hold wth equalty at any oluton to th problem. Conequently, we may olve the budget contrant a an equalty or m (takng p = wthout lo o generalty, and ubttute or th varable m 9 Only orward market behavor would be aected by the preence o cahlow rom hareholdng. In the orward market, uch cahlow would be a random varable (a uncton o pot market uncertanty that would covary wth the pot market prce p and hence aect the behavor o rk-avere conumer. In conumer pot market problem, n contrat, uch cahlow are treated a lump-um recept, xed or a gven p. A ubecton 6.6. below elaborate, conumer optmze n the pot market condtonal on p. 93 The bounded ratonalty o conumer n th ettng may be deended, n turn, by aumng that conumer hareholdng ntermedated (through, ay, mutual und. In uch a cae, the ntantaneou expoure o conumer to the cahlow o the electrcty uppler may be relatvely ntranparent. 80

198 n problem (6.0 obectve uncton. Wth thee two mplcaton, we may rewrte the pot market problem (6.0 a ( ( φ ( max w p q + p p q + q, T ne q. (6. Conumer ace problem (6. n perod (recall Fgure 3. ater the orward market ha cleared (revealng p and q, but beore the pot market ha cleared. Now conder conumer orward market decon n perod, gven that he wll ace problem (6. n perod. We need to augment problem (6. to provde a ba or her orward market decon makng. To do o, we add three eature to conumer problem:. We ntroduce uncertanty n the parameter T (content wth the dcuon at the outet o th ecton.. We permt conumer to agn a preerence rankng 94 to the optmal outcome o problem ( We allow conumer to maxmze th preerence rankng through her choce o orward quantty q (a a uncton o p, a we wll ee. 94 The uncton (, W m x ntroduced n eq. (6. account only or utlty rom conumpton o electrcty and the numerare commodty. In contrat, the preerence rankng ought here wll take nto account not only thee utlty term, but wll wegh ther value along wth change n wealth due to electrcty market actvty, a well, va the term pq p p q n problem (6.. and ( 95 Note that wth the ntroducton o uncertanty n T, the outcome o problem (6. are now themelve uncertan. 8

199 The optmal outcome rom problem (6. wll be n monetary unt ( wealth, 96 o t natural to aume that there ext a content unctonal repreentaton o conumer preerence rankng o th problem optmal outcome. We denote uch a uncton a V (, dened over orward and (optmal pot market outcome rom problem ( Applyng V ( to problem (6., we may wrte ( ( φ ( ne V max w p q + p p q + q, T q. (6. Conumpton o numerare produce utlty drectly, whle accordng to the demand-de producton model ntroduced n ecton 6..3, conumer ue electrcty a an nput to produce x, whoe enoyment then contrbute to her utlty. A an llutraton, let V ( be a negatve exponental uncton o the orm ( e λ z V z =, z, (6.3 wth rk averon parameter λ > 0. Note that V ( z trctly rk avere or all z, nce ( z 0 V <. Th orm o utlty uncton alo commonly reerred to a the contant abolute rk averon or CAA utlty uncton, nce the Arrow-Pratt abolute rk averon coecent, ra ( z, or the utlty uncton V ( z o eq. (6.3 contant: 96 Note that we may expre the optmzed value o (, ndrect utlty uncton. W m x a an equvalent money metrc 97 Whle the uncton V ( conumer-pecc, we uppre t ubcrpt (a wth, and φ to reduce notatonal clutter. The argument n V ( (ee problem (6. aocate th uncton wth conumer. We aume that V ( at leat twce derentable. W, 8

200 r A ( z ( z ( z V = λ. (6.4 V In eq. (6.4, denote the parameter λ a the CAA coecent or conumer. 98 For eae o preentaton, we contnue below to reer to the uncton V ( rather than ue explctly the negatve exponental unctonal orm o (6.3, although we wll appeal n what ollow to the properte o th unctonal orm. Aume that conumer maxmze her expected utlty o wealth, that, he maxmze (wth repect to q the expectaton E V ( n the orward market. ecatng problem (6. to relect th obectve, we have ( ( φ ( ne max E V max w p q + p p q + q, T q q. (6.5 We may mply problem (6.5 urther by notng the ollowng:. Both conumer non-electrcty wealth endowment ne w and (due to the prcetakng aumpton the term ( p p q are ndependent o q, the decon varable or the nner maxmzaton problem. Thereore, we may brng thee two term outde o the nner maxmzaton problem. 98 A ueul ntutve nterpretaton o the CAA coecent λ a ollow. Suppoe that conumer oered a lottery payng τ wth probablty and τ wth probablty. I th conumer ha a CAA utlty uncton (e.g., eq. (6.3 wth CAA coecent λ, t traghtorward to how that the value o τ or whch nderent between acceptng and not acceptng the gven lottery (approxmately the recprocal o λ, that, τ λ. In th ettng, we may nterpret τ a conumer rk tolerance. Pratt (964, 6 oer another characterzaton o the coecent o abolute rk averon n term o a probablty premum or acceptng a lottery. 83

201 . Smlar to the nttutonal tructure on the upply de, both the orward and pot market can accept bd rom conumer n the orm o a demand uncton, o that conumer choen quantty n market m, m q, may, n act, vary wth prce ( m=,. Snce conumer a prce taker, p m exogenou rom her perpectve. A a conequence, t approprate to condton market m obectve m p uncton on an arbtrary m p. Makng thee change n the orward market problem (6.5 yeld 99 ( ( max E ( max (, V w + p p q + φ q T p q p q q ne. (6.6 We now allow or aymmetrc normaton on the part o ndvdual conumer. Aume that, beore bddng n the orward market (.e., durng perod, each conumer oberve a prvate, random gnal η + that normatve concernng ubectve condtonal probablty dtrbuton o p gven p. 00 Hence, n problem (6.6, we condton expected utlty E V ( on η, a well, to obtan 99 Problem (6.6 ugget a natural way to ntroduce peculator nto the model, that, demandde agent who, rather than conumng electrcty, mply peculate on the derence between the orward and pot market prce. Namely, agent ĵ were a peculator, we would contran q ˆ = 0 and ( ( q T ˆ ˆ φ, = 0, nce by denton, the peculator ĵ doe not conume electrcty (or produce the amenty x and hence doe not partcpate n the pot market. Wth thee retrcton, problem (6.6 would max E ˆV p p qˆ p, whereby peculator ĵ chooe q ˆ to maxmze h become mply {( } q ˆ expected utlty o prot a a uncton o market demand uncton. p. By varyng 84 p, we would generate peculator ĵ orward 00 Snce only conumer oberve the gnal η, t reaonable to uppoe that market partcpant other than conumer treat η a tochatc. We may thnk o η a repreentng any propretary normaton avalable only to conumer, uch a compettve ntellgence on other market

202 ( ( ( η max E ( max (,, V w + p p q + φ q T p q p q q ne. (6.7 Becaue conumer a prce taker, the equlbrum pot market prce p doe not depend on q n eq. (6.7, but doe depend on both η and p. 6.. Approxmatng the expected utlty maxmzaton problem wth a meanvarance decon model In general, to compute the expectaton n problem (6.7 exactly, we would need to reort to numercal ntegraton, 0 nce the dtrbuton o the argument o V ( a non-trval tranormaton o the dtrbuton o T. Here we ollow a more tractable ( approxmate approach to problem (6.7 a mean-varance decon model 0 whch may yeld a reaonable approxmaton to the exact oluton o problem (6.7. There are everal ettng n whch the ue o a mean-varance decon model exactly content wth expected utlty maxmzaton, and other n whch a mean-varance model can erve, at the leat, a a good approxmaton o the expected utlty maxmzaton problem. Th ubecton examne thee ue urther, and ute the ue o a mean-varance approach to approxmate the problem (6.7. partcpant, market reearch, or pecalzed weather orecat that would help hape her pot prce expectaton or a partcular market round. 0 Alternatvely, one could alo apply Monte Carlo method to obtan an arbtrarly cloe approxmaton to an exact oluton. 0 That, a model n whch an agent decon are baed only on the mean and varance o the agent payo uncton and the orm o the agent utlty uncton. 85

203 Denote conumer payo rom a gven market round (gven p and η a z (ncludng, or convenence, the endowment w ne. In problem (6.7, thereore, ( z = z q the expreon wthn the brace, that, 03 ( ( φ ( ( ( ( η z max,, = z q w + p p q + q T p q p q ne. (6.8 Beore the pot market clear, revealng p, z tel a random varable whoe dtrbuton a tranormaton o the (unpeced dtrbuton o T. Ung the denton (6.8, we may wrte problem (6.7 concely a max q E ( ( q V z. (6.9 We rt note two cae n whch a mean-varance decon model exactly content wth expected utlty maxmzaton. The rt cae havng th property one n whch the underlyng utlty uncton ha a quadratc unctonal orm. Another example o uch exact contency when the utlty uncton o the negatve exponental orm and, n addton, the payo are normally dtrbuted (Freund 956, 55. Whle we could aume (recall eq. (6.3 that the utlty uncton V ( n problem (6.9 ndeed o the negatve exponental orm, the dtrbuton o payo z lkely 03 To reduce clutter n the ollowng analy, we uppre the dependence o z on caue no conuon. q where t 86

204 to be hghly non-normal. 04 Thu, gven the trong preme o Freund reult, t not reaonable to appeal to t here. Beyond the rather retrctve condton or exact contency between expected utlty maxmzaton and the mean-varance model, a growng trand o the nance lterature ha explored the condton under whch the mean-varance model erve a a reaonable approxmaton o expected utlty maxmzaton. In th work, the portolo electon problem ha naturally attracted much attenton. 05 Grauer and Hakanon (993, 859 urveyed th lterature and concluded that the conenu... that portolo choen on the ba o mean and varance can cloely approxmate portolo choen by maxmzng expected utlty, epecally when nvetor have mlar rk averon charactertc. More recent work (ee, e.g., Amlon 00 ha conrmed the earler ndng, 06 lendng upport to the argument that the mean-varance model oten lead to good approxmaton to the expected utlty maxmzaton reult or emprcal dtrbuton. 04 Note that we have not peced the dtrbuton o T tel, and moreover, do not need to do o or the preent analy. Snce we aumed n ecton 6..3 that T [ T, T ], however, whatever dtrbuton one mght chooe or T on th bounded upport would lkely be hghly non-normal. The related dtrbuton o conumer payo z would be a tranormaton o T dtrbuton. 05 Indeed, reearcher have examned th queton nce the mddle o the twenteth century: Markowtz (95 rt appled the mean-varance model to the portolo electon problem. Note, however, that the payo uncton (6.8 clearly not that o a portolo, whch would be mply a weghted um o (random aet return. Thereore, analytcal and emprcal reult rom the portolo electon context are not drectly applcable to the conumer orward market problem (problem (6.9 n the mult-ettlement SFE model. 06 Ung a htorcal dtrbuton o tock return (hown not to be multvarate normally dtrbuted, Amlon (00 examned the portolo electon problem. He ound certanty equvalent loe o only a ew percent or the mean-varance decon model compared to expected utlty maxmzaton or a wde varety o utlty uncton. 87

205 eturnng to the mult-ettlement SFE model (problem (6.9, we develop below a mple mean-varance decon model o conumer optmzaton problem n the orward market. Begn by wrtng the econd-order Taylor ere approxmaton to conumer utlty uncton V ( z n the neghborhood o the expected value o z, z ( z E, V ( z V ( z + ( z z V ( z + ( z z V ( z. (6.0 The expected value o th approxmaton E V ( z V ( z + V ( z Var( z. (6. The approxmaton (6. a wdely-ued peccaton o a mean-varance model (Levy and Markowtz 979. For our purpoe, however, we urther mply th model va an addtonal approxmaton. Namely, we approxmate the term V ( z n (6. wth a rt-order Taylor ere approxmaton n the neghborhood o an arbtrary pont 0 z n the upport o z that ucently cloe to but dtnct rom z. Thu, we have 0 0 ( ( ( 0 ( V z V z + z z V z. (6. Subttutng (6. nto (6. and rearrangng yeld E Dvdng by V ( z 0 ( ( ( ( ( Var ( V z V z zv z zv z V z z > 0 and wrtng the term z a E ( z, we have 88

206 Snce ( we choe ( ( 0 ( ( ( ( ( E V z V z V z ( 0 z + E z + Var z V z V z V z 0 z to be ucently cloe to. (6.3 z by aumpton above, and ( V ( z a mooth uncton (earler aumed to be o the negatve exponental orm, we may make the addtonal approxmaton that 0 ( ( 0 Subttutng the approxmaton (6.4 or V ( z de o (6.3 gve u ( ( 0 ( ( V z V z. (6.4 n only the lat term on the rght-hand ( ( ( E V z V z V z ( 0 z + E z + Var z 0 0 V z V z V z. (6.5 ecall that we dened conumer CAA coecent λ n eq. (6.4 gven a V z negatve exponental utlty uncton ( Settng z z (6.5 a 89 z e λ = (rom eq. (6.3, lettng z = z a ( z ( z V λ. (6.6 V = n eq. (6.6 to ubttute or V ( z V ( z E ( ( V z 0 ( ( V z n (6.5, we may wrte ( Var ( 0 z + E z z 0 0 V z V z 0 Multplyng both de o (6.7 by V ( z yeld λ. (6.7

207 λ E V z V z z V z V z E z Var z ( ( ( + ( ( ( Snce the expreon V ( z 0 0 ( 0 zv z and V ( z 0. (6.8 > 0 n (6.8 are contant, we may nterpret E V ( z a an ncreang uncton o the quantty E ( z Var( z Accordngly, maxmzng only the expreon E ( z Var( z λ λ. n (6.8 wth repect to q wll yeld the ame reult a maxmzng the entre rght-hand de o (6.8. Thereore, rom problem (6.9 (and recallng z z( q =, agan makng the dependence o z on q explct, we may wrte the optmal q = q rom the * ( maxmzaton o expected utlty E ( V z q approxmately a λ q arg max E V z q arg max E z q Var z q ( ( ( ( ( ( * q q. (6.9 The problem (6.9 aume that we wll nd approxmately the ame q by ( * ( maxmzng the expected utlty o a payo E ( V z q, a by ( maxmzng an ( addtvely eparable uncton o only the payo mean E ( Var ( ( z q and varance z q. The mean-varance model (6.9 ha the ollowng appealng properte:. It depend only on the rt two moment o the dtrbuton o z, and place no retrcton on the nature o th dtrbuton (e.g., z need not be even approxmately normally dtrbuted. 90

208 . A generalzed veron o the reult (6.9 would hold or other unctonal orm o V ( (e.g., thoe not havng the CAA property o eq. (6.4 n whch recallng eq. (6.5 and (6.6 wth z = z the contant λ would be replaced by V ( z V ( z. At th pont, we mply aume that the mean-varance decon model that underle (6.9 yeld acceptable approxmaton to conumer expected utlty maxmzaton problem over the doman o nteret. Naturally, when nterpretng the reult o the demand de analy, one hould bear n mnd the varou approxmaton n partcular, (6.0, (6., and (6.4 above nvoked n the coure o th dervaton. In accordance wth the above dcuon, we recat conumer expected utlty maxmzaton problem (6.7 a a mean-varance decon model over payo, o that problem (6.7 become 07 ( ( ( η ne max E ( max (,, w + p p q + φ q T p q p q q ( ( ( η λ ne Var w ( p p q max φ ( q,,. T p q p + + q (6.30 We later olve problem (6.30 or a repreentatve conumer n ecton 6.6. The next ecton determne ucent condton or the extence o a repreentatve conumer n the mult-ettlement SFE model. 07 Problem (6.30 eentally content wth Bolle (993 characterzaton o a conumer orward market problem, wth the addton o a tochatc hock T n the pot market. 9

209 6.3 Extence o a repreentatve conumer To mply the analy, 08 we now demontrate the extence o a notonal repreentatve conumer havng orward and pot market demand uncton that exhbt certan properte. It ueul to conder eparately the orward and pot market n th dcuon, and alo to dtnguh between two ene o a repreentatve conumer (ollowng Ma-Collel, Whnton and Green 995, 6 a potve repreentatve conumer (PC and a normatve repreentatve conumer (NC. Below, we explan normally the meanng o thee term, and then explore ucent condton or extence o a repreentatve conumer (n both the potve and normatve ene above n each o our two market. The ormer contruct, the PC, ntended to capture behavoral vermltude between all o the economy conumer, on the one hand, and the PC ( one ext, on the other. Inormally, 09 we may ay that there ext a PC we can pecy a utlty maxmzaton problem or a cttou ndvdual the putatve PC whoe oluton would generate the economy aggregate demand uncton. The latter contruct, the NC, preuppoe the extence o a PC (havng an aocated demand uncton, and n addton, requre that we be able to agn welare gncance to th demand uncton (Ma-Collel, Whnton and Green 995, 6 7. Note that the extence o an NC 08 The crucal mplcaton dered (n partcular, or the analy o ecton 6.6 below to abtract rom the dependence o the hape o aggregate demand on the lkely correlaton among conumer tochatc gnal η. In the preence o uch correlaton, the unctonal orm o D (, 0 p ε (recall eq. (3.8 would no longer be addtvely eparable. Potng the extence o a repreentatve conumer one mean o achevng th mplcaton. 09 Th normal denton taken rom Ma-Collel, Whnton, and Green (995, 6, who provde (n ther ecton 4.D a comprehenve overvew o repreentatve conumer theory, ncludng rgorou denton o the PC and the NC. The normal denton above uce or our purpoe. 9

210 mple the extence o a PC, o that t wll be ueul or our purpoe to conder rt the NC, a we do n ubecton 6.3. and 6.3. below A normatve repreentatve conumer n the orward market In ecton 7.7, we compute a welare meaure or the mult-ettlement SFE model whle potng a rk-neutral ocal planner. Under th aumpton, the pot market outcome contan all o the welare-relevant normaton. Notng that the eence o the NC to dene the attrbute o a cttou agent whoe preerence can erve a a meaure o aggregate welare, we need not conder the queton o the extence o the NC n the orward market A normatve repreentatve conumer n the pot market For mplcty, we rely on Ma-Collel, Whnton, and Green (995, 9 obervaton that the ollowng two condton are ucent or the extence o an NC:. Every conumer ndrect utlty uncton ( p, w, (, = ( + ( v ha the Gorman orm, that v p w a p b p w, (6.3 where p the vector o prce n the economy, v w total wealth, ( p, w ndrect utlty a a uncton o p and w, and a ( p and ( b p are uncton o p.. The ocal welare uncton utltaran. That, n the pot market o the mult-ettlement SFE model, we may tate the ollowng: the above condton and hold, then pot market aggregate demand may 93

211 alway be nterpreted a havng been generated by an NC (mplyng that the repreentatve conumer pot market demand uncton wll be welare-relevant. In ubecton 6.., we already aumed that the ocal welare uncton utltaran, thereby atyng condton above. A or condton, we argue that the model ntroduced n ecton 6. above mple that condton hold, a well. Namely, takng conumer pot market preerence to be qualnear wth repect to the numerare commodty m a we dd n eq. (6. mple that ndrect utlty v ( p, w wll be o the Gorman orm (eq. (6.3 wth b( p = (Ma-Collel, Whnton and p Green 995, 08 (n. 4. Snce condton and above hold, we conclude that we may nterpret any pot market aggregate demand uncton a havng been generated by an NC. m A potve repreentatve conumer n the orward market It may be hown that a the number o conumer J grow large, the nluence o any ndvdual conumer prvate gnal η wane. To put t another way, a J grow, the condtonal moment o hock to pot market demand condtonal on an ndvdual conumer gnal η approach the correpondng uncondtonal moment, aumed to be common knowledge. 0 I J ucently large o that the uncondtonal moment reaonably approxmate the condtonal moment o the demand hock, then we conecture that, at leat a an acceptable approxmaton, a PC ext n the orward market. 0 For a pecc model o the pot market demand hock 94 ε, ee ecton 6.5.

212 6.3.4 A potve repreentatve conumer n the pot market Snce the extence o an NC mple the extence o a PC, we conclude rom the argument o ubecton 6.3. above that there ext a PC n the pot market Summary and concluon Baed on the dcuon n the oregong ubecton, we aume now that. there ext an NC and hence a PC and n the pot market (ee ubecton 6.3. and and. there ext a PC n the orward market (ee ubecton For mplcty, we reer herenater to a repreentatve conumer or the multettlement SFE model, and denote th conumer by and lkewe, ubcrpt. The extence o the repreentatve conumer mple that we may olve utlty maxmzaton problem to obtan her orward and pot market demand uncton whch are, dentcally, alo aggregate demand uncton or the J conumer. 6.4 Speccaton o unctonal orm or and φ Th ecton eek to denty unctonal peccaton or. the repreentatve conumer producton uncton, x ( q, T. utlty uncton, φ ( x, or the amenty x =, and 95

213 that yeld a pot market demand uncton or (dentcally, the aggregate pot market demand uncton that content wth the ane pot market demand uncton n the mpled ane example rt ntroduced n chapter 5, (, D p ε = γ p + ε. (6.3 In eq. (6.3, γ > 0 a contant and ε a tochatc parameter (wth an a-yetunpeced dtrbuton. Begn by denng the compoton C o the uncton (, ( ( q, ( (, ( T φ q T φ q, T q T and ( φ a C. (6.33 x The analy o th ecton then proceed a ollow. Subecton 6.4. tate neceary and ucent condton on ( q, T demand uncton or denoted a (, C rom eq. (6.33 or the reultng pot market D p T to have the orm o the ane pot market demand uncton (6.3. Next, n ubecton 6.4. and 6.4.3, we pecy ndvdual unctonal orm or (, q T and ( φ that aty the a pror theoretcal retrcton o ubecton Fnally, ubecton then demontrate that the aumed unctonal orm o and φ are ucent to enure that (, orm o (, x D p T ha the D p ε n eq. (6.3. In addton, we ner a mple relatonhp between the tochatc parameter ε and T. See the Ane Spot Market Demand Functon aumpton, tated at the outet o chapter 5. Th alo the orm o the (ngle market demand uncton aumed by Klemperer and Meyer (989, 60 n ther Lnear Example. 96

214 6.4. Neceary and ucent condton or the repreentatve conumer to have an ane pot market demand uncton Th ubecton tate neceary and ucent condton or the repreentatve conumer to have an ane pot market demand uncton o the orm o eq. (6.3. Begn wth the repreentatve conumer pot market problem, that, the (dentcal nner maxmzaton problem o (6.30, condtonng on p (recallng ubecton 6.. argument and lettng = : ( φ ( ( max, q T p q p. (6.34 q Subttutng or the unctonal compoton ( denton n the expreon (6.33, problem (6.34 become ( ( ( q, T ( q, T φ = C rom the max C q, T p q p. (6.35 q The FOC correpondng to problem (6.35 Denng (, C ( q, T q p = 0. P q T a nvere pot market demand uncton parameterzed by the producton hock T, the FOC become (, ( q, T C P q T = p. (6.36 q 97

215 Next, denote the partal nvere o the uncton (, q a ( P ( p, T q P q T n eq. (6.36 wth repect to. The partal nvere o nvere demand wth repect to quantty mply (pot market demand uncton (alo parameterzed by T and denoted a (, D p T : We conclude that (, (, ( (, D p T P p T. (6.37 q q T and ( φ are uch that ha an ane pot market demand uncton o the orm o eq. (6.3 and only (, x (, γ ( D p T o the orm D p T = p + g T (6.38 (gven the denton n (6.33, (6.36, and (6.37, where γ > 0 contant and g ( ome derentable uncton o T. Note that or any uncton (, D p T havng the eparable ane orm o eq. (6.38, the partal nvere n eq. (6.37 ndeed ext The repreentatve conumer producton uncton, (, amenty x We now pecy a unctonal orm or (, q T, or the q T. Together wth a peccaton or φ ( x n the ollowng ubecton, thee example peccaton wll be ucent to Note that the notaton ( P n th expreon denote a partal nvere o q P wth repect to q, not partal derentaton. Followng eq. (6.38, we check whether th partal nvere n act ext. 98

216 enure that the reultant pot market demand uncton or content wth D ( p, ε n eq. (6.3. orm Let the repreentatve conumer producton uncton, (, q T, have the wth coecent 0,, 0 a ( q, ( ( T a0 + a q T q T, (6.39 a a a >. Gven the unctonal orm n eq. (6.39 or (, q T, the a pror retrcton (6.3 and (6.4 are ated or (takng q = q, optmal pot * market quantty 3 q a T <, (6.40 * a whle the a pror retrcton (6.5 and (6.6 alway obtan The repreentatve conumer utlty uncton, φ ( x, or the amenty x Let the repreentatve conumer utlty uncton or electrcty conumpton, φ ( x, be lnear n x, that, ( x bx, b 0 φ = >. (6.4 3 We revt the condton (6.40 n ubecton below. 99

217 The lnear unctonal orm n eq. (6.4 or φ ( x ucent or the a pror retrcton (6.7 and (6.8 to hold. The aumpton that φ ( x lnear n x a lmtng cae, ued here or mplcty wthout lo o generalty. A we may ner rom the development o the neceary and ucent condton n ubecton 6.4., there a tradeo n the degree o concavty n the uncton ( x cae o (, φ and (, q T (concavty wth repect to q, n the q T atyng thee condton. Hence, we may make ( φ concave whle preervng the dered properte o the compoton ( ( q, T multaneouly decreang the degree o concavty o (, uncton (, q T and ( x x φ by q T. For example, gven the φ rom eq. (6.39 and (6.4, uppoe that α parameterze a amly o par o uncton ( q, T ( q, T φ α α ( ( x φ ( x α = and =. Whle the example n the text aume α =, a par o uch uncton or any α > would alo yeld an ane pot market demand uncton or o the orm o eq. ( Condton or contency o (, D p ε and D( p, T A the analy n ubecton 6.4. demontrate, the orm o (, peccaton o (, or the uncton (, q T and ( x q T and ( D p T depend on the φ. Subttutng n the pot market problem (6.34 φ rom eq. (6.39 and (6.4, repectvely, yeld x α 00

218 max q a ( ( b a0 + a q T q T p q p. (6.4 The FOC (or an nteror oluton correpondng to problem (6.4 ( ( b a a q T p =. ( Solvng eq. (6.43 or the optmal 4 q = q * a a uncton o p and T yeld pot market demand uncton (, D p T, a q = q D ( p, T = p + T + * ab a. (6.44 By contructon, (, (6.38 where D p T n the expreon (6.44 ha the eparable ane orm o eq. γ = > 0 (6.45 ab and Becaue (, a g( T = T +. (6.46 a D p T n the expreon (6.44 a chedule o prce and correpondng optmal quantte (gven T, th uncton ueul n determnng when 4 We may alo ee rom eq. (6.43 that gven the aumed parameter retrcton, the econdorder ucent condton or a prot maxmum wll alo hold. 0

219 the nequalty (6.40 n ubecton 6.4. above ndeed hold. earrangng the expreon n (6.44, we have that q T = p a + * ab a. (6.47 Snce ( ab < 0, t ollow rom eq. (6.47 that > < a p = 0 q T = < > * a. (6.48 When p 0, the expreon (6.48 mple that the nequalty (6.40 volated. In th event, the a pror unctonal orm retrcton (6.3 and (6.4 do not hold. Whle we do not rule out the event p 0 n the mult-ettlement SFE model, we may chooe parameter value to render nonpotve prce a relatvely uncommon occurrence. Accordngly, we ay that under normal crcumtance, we have that p > 0, and thereore by the above argument, all o the a pror unctonal orm retrcton (6.3 (6.6 are normally ated. By denton, the only conumer n the repreentatve conumer model. Conequently, pot market demand uncton (, alo the aggregate pot market demand uncton, (, thee uncton are parameterzed derently by T and (, (, D p D p T D p T n eq. (6.44 dentcally D p ε, n eq. (6.3, although ε, repectvely. Thu we have ε = (6.49 0

220 or every prce p and producton hock T. From eq. (6.3, (6.38, and (6.44 (6.46, eq. (6.49 mple that we mut have the ollowng two parametrc retrcton or D ( p, ε and (, D p T to be mutually content: γ = γ = > 0 (6.50 ab and a ( ε = g T = T +. (6.5 a Gven a dtrbuton or T and the parameter o the producton uncton, eq. (6.5 ndcate that the dtrbuton o ε a mple tranlaton o the dtrbuton o T. In partcular, we may relate the upport o ε to that o T a ollow. ecallng that ε and ε are the lower and upper lmt o the upport o ε, Ε ε, ε, repectvely, thee lmt are gven by a ε = T + (6.5 a and ε = T + a a. (6.53 Fnally, to mply notaton n the remander o th chapter, we explot eq. (6.3 and (6.49 to rewrte eq. (6.44 or the optmal mply the pot market aggregate demand uncton (, * q (condtonal on D p ε, p and ε a 03

221 (, q D p ε = γ p + ε. (6.54 * Equaton (6.54 the orm o aggregate pot market demand that we poted n chapter 5 mpled ane example. In partcular, (, D p ε ane and downward-lopng. 6.5 A mple tochatc model or the pot market demand hock We now pecy a mple model or the pot market demand hock ε n term o tochatc gnal η. Ultmately, th model wll permt u to relate demand hock and prce acro the two market. Begn by ntroducng a random varable ν that revealed to at t = (ee Fgure 3., when the pot market clear wth (publc revelaton o the demand hock ε. Let ν be dened uch that a mple addtve relatonhp ext between the pot market demand hock ε on the one hand, and η and ν on the other. Namely, we have that ε ε η ν = +. (6.55 An ntutve nterpretaton o eq. (6.55 that ν a noe parameter whoe preence make gnal η an mperect gnal or ε. Now conder the probablty dtrbuton o η and ν. Let η and ν be ontly dtrbuted wth a tatonary dtrbuton uncton ( F η ν η ν, whch we,, aume to be common knowledge. Further, let η and ν be ndependent, o that, denotng the margnal dtrbuton o η and we have that F ( η ν F ( η F ( ν η,, ν η ν ν a F η ( η and ( F ν ν, repectvely, =. In ecton 6.., we took the tochatc 04

222 upport o η to be +, o that wth =, we have that η +, n prncple. 5 We now alo let the tochatc upport o ν be +, n prncple (ee note 5. From eq. (6.55 and rom ndependence, we then have that the upport Denote the mean o η and [ 0, varance a σ Var ( η and σ Var ( ν η moment ( E o ε, n prncple 6 E =. (6.56 ν a η E ( η and ν E ( ν ν, and denote ther, repectvely. Alo, dene the hgher Cov, ν, νσ ν ν. 7 In lght o the ndependence aumpton or η and ν, we may alo nterpret ν a that component o to the gnal η. ε that unexplaned by (or orthogonal 5 In practce, gven eq. (6.55, a nte upper lmt on the upport o both η and ν would delmt the extent o the correpondng pot market SF. 6 Equaton (6.5 and the upport o T T T a a [, ] [, =. ν, ν ε n eq. (6.56 mply that the upport o T, n prncple, 7 To ad ntuton concernng the hgher moment σ Cov ( ν, ν ν, ν 3 ( 3 ( V ν ν m m 3 3 σ = σ α +, where ( 3 α the coecent o kewne o ν,, we may how that m the k th k moment about the mean o m = σ Var ν ν, a dened above, and V ν σ ν ν the coecent o varaton o ν. ecallng that potvely-kewed dtrbuton correpond to α > 0, and 3 negatvely-kewed dtrbuton to 3 0 ν (o that ( α <, we may conclude the ollowng concernng gn ( ν, ν. I ν 0 and the dtrbuton o ν potvely kewed, then σ > 0.. I ν 0 and the dtrbuton o ν negatvely kewed, then σ < In all other cae, we may conclude only that gn ( σ = gn α + ( V ν ν ν, ν ν, ν, ν 3. σ : 05

223 We next derve expreon or ubectve condtonal moment o ε, condtonal on an arbtrary realzaton η o gnal. Frt, denote ubectve condtonal expectaton a E( ntroduced above, we have that E ε η where, rom eq. (6.55 and ung the notaton ( E ( ε η = η + ν η = η + ν. (6.57 The econd equalty n eq. (6.57 explot both the ndependence o η and ν and the common knowledge dtrbuton o ν. Smlarly, denote ubectve condtonal varance a Var( ε η, whch ( ε η (( η ν η ( ν η ( ν σ Var = Var + = Var = Var =. (6.58 ν We ue the reult o eq. (6.57 and (6.58 n ubecton 6.6. below to mply the expreon or contrbuton to aggregate orward market demand a we olve the repreentatve conumer maxmzaton problem n the mult-ettlement market ettng. 6.6 The repreentatve conumer optmzaton problem The equental tructure o the mult-ettlement market problem mple that, a on the upply de, backward nducton the approprate oluton algorthm. Accordngly, ubecton 6.6. conder the pot market n the rt tage o the backward nducton algorthm. Next, the econd tage o the algorthm, dcued n ubecton 6.6., addree the orward market. 06

224 6.6. Spot market The rt tage o the backward nducton algorthm to olve the repreentatve conumer pot market problem, that, the (dentcal nner maxmzaton problem o (6.30. We do o or a xed T, whch xe ε (by eq. (6.5, and or an arbtrary pot market prce p. Accordngly, we condton on p, and let = to obtan the pot market problem (ee problem (6.34 ( φ ( ( max, q T p q p. (6.59 q In preparaton or the orward market analy n the next ubecton, we may wrte problem (6.59 a ollow (ung eq. (6.39 (6.4, and (6.44 or (, φ ( x, and * q, repectvely: ( φ( ( φ ( ( * * max q,, T p q p = q T p q q a = b a0 + a( q T ( q T p q Solvng eq. (6.5 or T ε ( a a equaton above or T and mply to obtan ( p q T, * * * ap p T a a ab a = b a0 + + =, we may ubttute th expreon nto the thrd. ( φ( ( max a q, T p q p b a p q ( p = 0 + ε + a a b. (6.60 For notatonal convenence, dene a contant k a 07

225 a k b a0 + a. (6.6 We may wrte eq. (6.60 more compactly by ubttutng rom eq. (6.50 or ab and rom eq. (6.6 or b( a0 a a + to obtan ( φ( ( max γ q, T p q p = k p ε + q ( p. (6.6 The reult n eq. (6.6 wll be ueul n the orward market analy, to whch we now turn Forward market In the econd tage o the backward nducton algorthm, we analyze orward market problem whch, lettng =, the outer maxmzaton problem o (6.30. Subttutng ( rom eq. (6.6 or pot market urplu (, problem (6.30, we have φ q T p q (at an optmum n * * ( ( p γ ne w + p p q + k p ε + p ( η max E, q ( p γ ne w p p q k p ε p λ Var + ( + + ( η,. (6.63 Dtrbutng the expectaton and varance operator n the problem (6.63, th expreon become 08

226 ( γ ( ( ( p γ ne w + k+ q p η p p + p ε + η p ( max E, E, q λ {( q Var p ( η, p + Var p ε ( η, p ( p ( η p q ( p p ε ( η p + Var, Cov,, 4 ( ( ( ( ( + qγ Cov p, p η, p γ Cov,( p ε p η, p. The FOC wth repect to q or th maxmzaton problem (wth ome urther mplcaton ( ( η E p, p p { ( ( ( ( ( ( p p ( η p λ q Var p η, p Cov p, p ε η, p γ + Cov,, = 0. Solvng th condton or the optmal q a a uncton o p and η yeld q = q p (, η * ( p ( η p { ( ( η = E p, p p λ Var, ( p p ( p ( p ( p ( η p + λcov, ε η, γλ Cov,,. (6.64 Smplyng eq. (6.64 urther, we examne, n turn, the two covarance term and the expectaton and varance term on the rght-hand de o th equaton. To evaluate the two covarance term, we rt need to make explct the dependence o p on ε. 09

227 Content wth the tatement o the uppler orward market problem n eq. (4.6 (4.8, the repreentatve conumer olve her orward market problem aumng equlbrum n the pot market. Thereore, t approprate at th pont to take p n eq. (6.64 to be a market-clearng pot market prce, gven a pot market hock and the orward market outcome. Equaton (5.6 rewrtten below a eq. (6.65 gve an expreon or the market-clearng prce p a a uncton o ε and p : 8 ( ; (, ( a ( ( p p S p S p S p S p * ε = ω ε φ φ + ωb. (6.65 Now ubttute rom eq. (6.65 or p n Cov (, (, p p ε η p, the rt covarance term on the rght-hand de o eq. (6.64: ( p p ε ( η p Cov,, ( ( ( = Cov ω a ε φs p φs p ω + b, = Cov, ω a ε φs ( p φs( p ω b ε ( η, p + ( ωε ω ( ε η ( ωε a ω a ωb φs ( p φs( p ε η a a + Cov,, whch mple to ( ( p p ( p ( Cov, ε η, = ωa Cov ε, ε η ( ( ( + ω a ωb φs p φs p Var ε η. ( ω c β c β b ecall rom eq. (5.4 and (5.5 that, n eq. (6.65, ( +. That, 0 0 a ω and ω β + β + γ and a ω are uncton only o exogenou pot market parameter. b 0

228 Next, ubttute rom eq. (6.65 or ( ( p n Cov, ( p p η, p, the econd covarance term on the rght-hand de o eq. (6.64, and partally expand the quare ( p : ( p ( p ( η p = Cov ( ω a ε φs ( p φs( p + ω b, ωa ( ε + ω a ωb φs ( p φs( p ε + ω a ωb φs ( p φs( p ( η, p = Cov ( ωε a, ωa ( ε η + ( S( p S( p Cov,, Cov ωε a, ωa ωb φ φ ε η, whch mple to ( p ( p ( η p 3 = Cov (, ( Cov,, ωa ε ε η ( ( ( + ω ω φ φ ε η 3 a b S p S p Var. (6.67 Now conder the expectaton and varance term on the rght-hand de o eq. (6.64. We may evaluate thee term by takng the ubectve condtonal expectaton and varance o p n eq. (6.65, agan condtonal on η and p. Dong o yeld ( p ( p ( S ( p S( p E η, = ωa E ε η φ φ + ω b (6.68 and

229 ( p ( p ( p ( ( Var, Var, Var η ωa ε η ωa ε η = =. (6.69 Ung the reult o eq. (6.57 and (6.58 to mply eq. (6.68 and (6.69, thee latter equaton become ( p ( p S ( p S( p E η, = ωa η ν φ φ ω + + b (6.70 and ( p ( η p σν ω Var, a =. (6.7 Collectng the above reult, we ubttute rom eq. (6.66, (6.67, (6.70, and (6.7 nto eq. (6.64 to obtan q * ( p, η ν a { = ω η + ν φs ( p φ S ( p + ω p λσω a b ( ( + λω a Cov ε, ε η ( ( Var ( + λω a ωb φs p φs p ε η ( ( 3 λγω a Cov ε, ε η ( ( ( } 3 λγω a ωb φs p φs p Var ε η. Collectng lke term, th equaton become q * ( p, η ν a { = ω η + ν φs ( p φ S ( p + ω p λσω a b ( λω a + ( γωa Cov ε, ( ε η ( S( p S( p ( } + λω a γωa ωb φ φ Var ε η. (6.7

230 From the denton o ε n eq. (6.55, we may evaluate the covarance term on the rght-hand de o eq. (6.7 a (( ( ( Cov ε, ε η Cov η ν, η ν η = + + = Cov ( η + ην + ν, η + ν η = η Var( ν η + Cov ( ν, ν η. Ung the notaton or the dtrbutonal moment ntroduced n ecton 6.5, we may wrte th term a (( ε ε η η σ σ Cov, = + ν ν, ν. (6.73 Fnally, ubttutng rom eq. (6.73 and (6.58 or the covarance and varance term, repectvely, n eq. (6.7 and actorng ω a out o the brace, we have q p S p S p * (, η = η + ν φ ( φ ( + ωb λσω ν a ωa λω a + + ( γωa( ησσ ν ν, ν ( S( p S( p } + λσω ν. a γωa ωb φ φ p Collectng term n p and * η, we may wrte (, q p η a q * ( p, η ν { = + λσω ν ( ( ( a γω a φs p + φs p λσω a p λσω + λσω ν a γ ωa η ν γ ωa ω a ν, ν a ( ( ( } + ω + λ σ ω γ ω b ν. a a (6.74 3

231 Snce the orward market demand uncton o the repreentatve conumer dentcally alo the aggregate orward market demand uncton, denoted here a (, * q p η, we have (, η (, η * * q p = q p. (6.75 ecall that n chapter 3 (eq. (3.8, we expreed orward market demand a (, ε ( D p = D p + ε. ( Equaton (6.75 and (6.76 (ung eq. (6.74 are two derent parameterzaton o the ame aggregate orward market demand uncton. I we aume that there ext a uncton ( o the gnal vector e η we wll have that η uch that e η ( η ε 0 * (, ε0 (, η ( η (, η = or all relevant η, then D p = D p e = q p. (6.77 We may combne eq. (6.74 (6.77 to wrte 4

232 (, ε0 = (, η ( η = D0 ( p + eη ( η * = q ( p, η D p D p e p = + ( S( p + S( p + λσω λσω ν a γω a φ φ ν a ωa ν ν λσω ν, ν a { λσω ν ( ( a γω a η ν γωa + λσω a a ( } + ω b + λσ ν ω a γ ωa = { + λσ ν ω ( a γ ω a λσω ν ( ( φ ( ( p p 0 + ωa + { + λσω ( γω η + ν λσω a φ S p S p 0 S p S p + 0 ν a a λσω ν, ν a p0 + ( γωa ω a ( ( ν ( } + ωb φs p0 φs p 0 λσ ωa γ ω + a, where we recall rom ubecton 3..0 that p 0 an arbtrary reerence prce n the nterval p, p over whch we dened the orward market demand uncton. Comparng the econd and th equalte above, we have that D { ( p = + λσω ν a( γωa 0 λσω ν a p p 0 +. ωa ( ( φ ( ( φ S p S p 0 S p S p + 0 (6.78 5

233 Moreover, we may conrm that the uncton e η ( η ε 0 = ndeed ext, and n partcular, e η ( η = ε 0 λσω p = λσω ν, ν a 0 λσω ν ( ( a γωa η ν γωa ν a ωa ( ( ν ( } + ωb φs p0 φs p 0 λσ ωa γ ω + a. (6.79 Equaton (6.78 and (6.79 decompoe the orward market demand uncton (, ( D p e η η nto D p and. the prce-dependent hape component 0 (. the prce-ndependent tochatc hock ε ( η demand uncton. 0 e η = o the orward market Note that 0 ( D p n eq. (6.78 a determntc uncton o p ; ung th equaton, we may very that ( D p =. Equaton (6.79 ndcate that ( e η η depend on the realzaton o the gnal η and the expectaton ν, but not on the realzaton ν, nce ν revealed ater the orward market clear. In addton, t poble to how n eq. (6.78 and (6.79 that ( γω a > 0 and ( γω a value. component > 0 or all permble parameter We may urther decompoe the expreon or ε 0 n (6.79 nto a tochatc ( λσω γω η, ( ν a a λσω ν a 6

234 and a determntc component λσω λσω ν, ν a 0 ν + ( γ ωa ν a ωa ( ( ν ( } + ωb φs p0 φs p 0 λσ ωa γ ω + a. p (6.8 Note that the expreon (6.80 collect the actor dependent on η n eq. (6.79 or e η ( η, o that we may wrte ( η e η η a ( η ( γω + λσω ν a a λσω ν a e = > 0. (6.8 Fnally, conder how the gnal η aect the level o orward market demand. ( Applyng the chan rule to the uncton, ( D p e η η, we have that (, η ( η (, ε0 ( η D p e D p de η = 0 d η ε η. Ung eq. (6.76 and (6.8, we may conclude rom the above equaton that (, η ( η ( η D p e deη = > 0. (6.83 η dη The nequalty (6.83 ndcate that an ncreae n gnal to the rght. ( η ht D p, e η ( η 6.7 The relatonhp o demand hock and prce acro market Th ecton revt an expreon rom the prevou chapter (eq. (5.36 or de ( p p dp, whch we etablhed a a uncton o the dervatve 7

235 d ( ε ε0 dε 0. 9 In ubecton 6.7., we derve expreon or d ( E and lkewe or E ( d p p dp n ubecton The dervatve d ( E ε ε 0 dε 0 E 0 d 0 ε ε ε, Begn by takng condtonal expectaton o ε, condtonal on ε 0, rom eq. (6.55: ( ε ε0 ( η ε0 ( ν ε0 E = E + E. Th equaton an dentty n ε 0 o that we may derentate t wth repect to ε 0 to obtan Snce ( ε ε0 ( η ε0 ( ν ε0 d E de de = +. (6.84 dε dε dε ν exogenou, we have that E( ν ε E( ν 0 =, and hence d E ( ν ε0 d E ( ν dε = = 0. 0 dε0 Ung th reult, eq. (6.84 become mply To nd de ( term o ε 0 to obtan d E dε ( ε ε 0 d E ( η ε 0 =. (6.85 dε 0 0 η ε0 dε 0, we may olve the econd equaton n (6.79 or η n 9 We may nterpret th dervatve a the eect o orward market (publc normaton on expectaton concernng the level o pot market demand. 8

236 λσω η λ σ ω ε ν γ ω + ( ( ( γω } ( ν, ν a 0 = ν a 0 a + λσω ( ω ν a a γωa ωb φs p φ S p λσω ν. a a p (6.86 Takng condtonal expectaton o th equaton and derentatng wth repect to ε 0, we obtan ( 0 d E η ε λσω =. (6.87 dε + λσω γ ω ν a 0 ν a a ( Subttutng rom eq. (6.87 nto eq. (6.85, we alo have that ( 0 A eq. (6.88 ndcate, d ( d E ε ε λσω = > 0. (6.88 dε λσω γω 0 E 0 d 0 ν a + ν a a our aumpton. More peccally, d ( ( ε ε ε contant a ε 0 vare (all ele equal, gven E 0 d 0 ε ε ε a uncton o the repreentatve conumer attrbute, the varance o the underlyng tochatc parameter ν, and pot market demand and cot parameter. Comparng eq. (6.88 wth eq. (6.8, and notng (rom nequalty (6.8 that we may nvert ( ( ε ε0 d E = dε 0 e η η, we ee that e η ( η. 9

237 6.7. The dervatve E ( d p p dp Ung prevou reult n th chapter, we may mply the expreon or the dervatve de ( p p dp. Equaton (5.36 rt gave an expreon or th dervatve, whch we rewrte below a eq. (6.89: ( ( ( ε ε0 ( p ( de p p de ε ε de ε ε 0 0 = ω a φ S ( p + φ S p ε0 ε0 dp d d d E D. 0 dε 0 ( (6.89 Next, takng the dervatve o 0 ( D p rom eq. (6.78 wth repect to p, we obtan an : expreon or D0 ( p D ( S ( p S ( p =. ( λσω ν a γωa φ + φ + ωa 0 ( p λσω ν a Ung eq. (6.88 and (6.90 to ubttute or d ( repectvely, n eq. (6.89 yeld E 0 d 0, ε ε ε and D0 ( p 0

238 ( de p p dp λσω = ωa + λσω ν a φ S ν ( a γωa λσω + + λσω + ( γω ( p ν a φ S ν ( a γωa λσω ν λσω ν a a a ( p λσω ν ( ( ( a γω a φs p φs p ω a. λσω ν a Collectng term and mplyng, th become ( λσω ν ( ( ( ( a φ S p φ S p de p p dp = + +. (6.9 + λσω ( γω ν a a Aumng trctly ncreang orward market SF or all >, we conclude p, S ( p 0 that d p p E ( dp > 0. ( Properte o aggregate orward market demand D ( p,ε 0 The nal ecton o th chapter ummarze the alent properte o 0 ( D p, ε 0, and ther um, aggregate orward market demand D ( p, ε 0 = D ( p, e η ( η ( = D p + ε baed on eq. (6.76 (6.79 n ecton

239 6.8. Properte o D0 ( p Gven our parametrc aumpton and once we have determned the orward market SF S ( p (ee chapter 7 the hape component 0 ( D p o orward market demand (ee eq. (6.78 determntc, derentable, and common knowledge. Sgnng the reult o eq. (6.90 rom the prevou ecton (agan aumng S ( p 0 get + λσω ν a γωa φ + φ + ωa 0 ( p λσω ν a >, =,, we ( S ( p S ( p D = < 0. (6.93 From eq. (6.93 we conclude, under our parametrc aumpton and aumng trctly ncreang orward market SF, that D0 ( p and hence (, 0 D p ε are downwardlopng n p Properte o ε 0 Gven our parametrc aumpton, the hock ε 0 n eq. (6.79 a uncton o exogenou gnal η, and nclude both tochatc and determntc component a ndcated n the dcuon o that equaton. In addton, the tochatc component o ε 0 ha a tatonary, common knowledge dtrbuton ( 0 ( F η η ha thee properte. F ε ε, nce the dtrbuton 0 0 Moreover, 0 ( D p would be ane n =,, though we do not mpoe th ane retrcton here. p the uncton ( S p are ane or rm

240 Now conder the upport E o ε 0. In ubecton 6.., we aumed that the upport o any conumer gnal and thereore o η wa +. ecallng that ε 0 e η ( η = rom eq. (6.79, we may obtan an expreon or the mnmum value o the orward market demand hock denoted a ε 0 by ubttutng η = 0 nto eq. (6.79. Dong o yeld ε 0 = e η ( 0 λσω p = + λσω ν, ν a 0 ν ( γ ωa ν a ωa ( ( ν ( } + ωb φs p0 φs p 0 λσ ωa γ ω + a. (6.94 In other word, ε 0 bounded below by ome ε 0 rom eq. (6.94 n every round o the market. In term o the upport E, we have that ε0 E ε0, ε 0, (6.95 where ε0 < ε0 gven by eq. (6.94. The upport E n eq. (6.95 determne the extent o the orward market SF, that, the prce doman over whch they are dened. We need not pecy the upper lmt ε 0 o th upport; rom eq. (6.79 and gven η +, ε 0 may n prncple be nnte. I ε 0 ucently mall (and rom eq. In practce, however, a the next chapter dcue, we wll compute orward market SF * traectore over a nte nterval o prce p, p, where p = p ( ε 0, ε = e ( 0 η η, and the upport o 0, η, 0 < η <. η [ ] 3

241 (6.94, we can have ε 0 < 0, uppler orward market quantte wll be negatve, ndcatng purchae rather than ale n the orward market Properte o D ( p, ε 0 = D ( p, e η ( η The addtvely eparable unctonal orm D ( p, ε D ( p ubecton 3..0 and retated n eq. (6.76 above mple that (, (, ( 0 = + ε aumed n ( ( D p eη η D p eη η deη η = = 0, (6.96 p η p ε dη nce (, η ( η (, ε0 D p e D p = = 0. p ε p ε 0 0 The nterpretaton o eq. (6.96 that the gnal η ht the orward market demand uncton horzontally but doe not change th uncton hape. There at leat anecdotal emprcal evdence rom electrcty market (ee, e.g., Federal Energy egulatory Common 003a that generatng rm do requently take long poton n the orward market. In addton, recent theoretcal work (e.g., Hughe and Kao 997, 8; Prrong 000, 5 ha uggeted that, under a varety o crcumtance, uch behavor can ndeed be protable. A we ee n the pecc numercal example o chapter 7, however, ocung on trctly ncreang orward market SF over reaonable prce range tend to yeld potve orward market quantte (hort poton on the part o uppler wthn the preent model. 4

242 Equlbrum ha become a knd o holy acrament n economc and ha erouly dverted attenton rom the real world o Heracltean lux.... The economc ytem a tructure n pace-tme. Conequently, t evolutonary, ubect to contant and rreverble change. Kenneth Bouldng God doe not care about our mathematcal dculte. He ntegrate emprcally. Enten 7 The orward market upply uncton n the mpled ane example WE ETUN IN THIS CHAPTE to the upply-de analy o chapter 5. Secton 7. below mple urther rm equlbrum optmalty condton or the orward market, whle ecton 7. explore the extence and unquene properte o oluton to the reultng ytem o equaton and the eect o ngularte. Next, n ecton 7.3, we dcu two complementary numercal tratege or olvng th ytem. We develop qualtatve nght nto the phae pace o oluton n ecton 7.4 wth the help o numerou graphcal llutraton. Secton 7.5 then decrbe how we choe value o certan model parameter to enhance the vermltude o the model. Secton 7.6 preent an equlbrum electon procedure and conduct comparatve tatc analy to nvetgate the eect o parameter varaton on rm orward market SF. To conclude the 5

243 chapter, ecton 7.7 compare expected welare under the mult-ettlement SFE model wth that under alternatve behavoral aumpton and market archtecture. 7. Equlbrum optmalty condton or the orward market 7.. Integratng prevou chapter reult concernng the uncton E ( p p and D 0 ( p Begn by recallng eq. (5.37, rm equlbrum optmalty condton or the orward market under the aumpton o the mpled ane example, rewrtten a eq. (7. below: { φφ E( p p ( c0+ cs ( p E( p p p } S ( p = S ( p D 0 ( p E ( p p p. (7. The analogou condton or rm, by ymmetry, { φφ E( p p ( c0+ cs( p E( p p p } S ( p = S ( p D 0 ( p E ( p p p. (7. In the ollowng, we ue analytcal reult rom chapter 5 and 6 to ubttute or the uncton E ( p p and D0 ( p n eq. (7. and (7.. ecall eq. (5.33 or E( p p (rewrtten a eq. (7.3 below, E ( p p E( e ( p S( p S( p = ωa ε p φ φ + ω b, (7.3 6

244 where, alo n chapter 5, we dened ω ( β β γ a = + + and ω 0β 0β b = c + c. To evaluate eq. (7.3 n term o known contant and uncton o p, we need to evaluate ( the expectaton E ε e ( p p chapter 6, a decrbed below.. We may do o by appealng to varou reult rom Begn wth ecton 6.5 mple model or ε (eq. (6.55, ε η ν = +. (7.4 e p = ε yeld Takng expectaton o eq. (7.4 condtonal on ( 0 E ( ε ( E ( ( E( e ( p p η ep p ν ep p p = +, whch, nce ν exogenou, mply E( ( E( e p e ( p ε = η + ν. (7.5 p p Smlarly, takng expectaton o eq. (6.86 condtonal on e ( p p ε 0 = gve u E ( η ep ( p λσω = λσω ν aep ν γ ωa + λσω ( ν a γωa a ν (, ν a p ( p0 + ωb φs p φ S p ω ( ( λσω ( γω } ν. a a (7.6 Solvng the market-clearng condton or the orward market (eq. (5.9 or ε 0 p ( p = e and ubttutng p ε, we have * p or ( 0 7

245 ( ( ( 0 ( e p = S p + S p D p. (7.7 p Fnally, the hape component o orward market demand, 0 ( D { ( p = + λσω ν a( γωa 0 λσω ν a p p 0 +. ωa D p, (rom eq. (6.78 ( ( φ ( ( φ S p S p 0 S p S p + 0. (7.8 Combnng eq. (7.5 (7.8 to mply eq. (7.3 and collectng term yeld the dered reult, 3 E ( p p λσω ( φ ( ( φ ( ω ( γω ν = ν, ν ν a S p + S p + b + a + p ( γω + λσω ν a a σ σ ν. (7.9 We turn next to the uncton D0 ( p, the lope o the hape component o orward market demand. Derentatng eq. (7.8 wth repect to p, we have that 3 The dervatve o eq. (7.9 content wth E ( E ( 8 d p p dp rom eq. (6.9. Note alo that p p n eq. (7.9 (and ultmately, the orward market upply and demand uncton depend only on three moment o ν ν, σ, and σ ν ν, ν rather than on ν entre dtrbuton. For computatonal purpoe, we aume n Appendx F..5 that ν lognormally dtrbuted, whch permt u to expre σ a a uncton o the other two moment. We chooe the parameter ν and σ ν, ν ν, n turn, va an emprcally-baed benchmarkng procedure decrbed n ecton 7.5.

246 D + ( S ( p + S ( p + =. (7.0 λσω ν a γωa φ φ ωa 0 ( p λσω ν a The three equaton (7., (7., and (7.0 conttute a ytem o nonlnear ordnary derental equaton (ODE mplctly characterzng the orward market SF S ( p and S ( p a well a the lope o orward market demand, D0 ( p eq. (7.9 gve an expreon or E (, where p p. Each conumer 4 olve her orward market optmzaton problem (a n chapter 6 gven the two SF ( S p, and gven an equlbrum prce n both the orward and pot market. Each uppler (, =,, maxmze t prot, takng uppler SF a gven (the Nah aumpton, and alo takng conumer acton a gven. Each equaton n the ytem (7., (7., and (7.0 are rom the repectve optmzaton problem o the duopoly uppler and the repreentatve conumer. In order to olve th ytem numercally ung commercally-avalable derental equaton olver, however, we have ound t ueul to rearrange th three-equaton ytem by olatng the dervatve o the dependent varable S ( p and S ( p. 5 In addton, the mplcaton we undertake n the remander o th ecton are ueul n hghlghtng certan quadratc orm that characterze everal loc o nteret, a detaled n ecton 7.4 below. 4 ecall that eq. (7.0 above expree the lope o aggregate orward market demand, the um o ndvdual conumer orward market demand uncton. 5 We do th n ubecton 7.. or a retrcted veron o th three-equaton ytem. Whle the reultng expreon appear, anythng, more complcated than the orgnal ytem, the reved ytem doe have the vrtue o olatng the vector o upply uncton dervatve. 9

247 A a rt tep toward olvng the ytem (7., (7., and (7.0, we may reduce thee three equaton to a two-equaton ytem n S ( p and ( E ( p p by ung eq. (7.9 and (7.0 to ubttute or E ( S p and elmnate n p p and D0 ( p eq. (7. and (7.. Makng thee ubttuton and collectng term yeld the two equaton φω λ σω γ ω φ φ ( ( ( ( ( ( a a + ν a a S p + S p p ωa ( + cφφ λσω ν a γ ω + a σ + ωb + γ ωa ν σ ν ν, ν ( S ( p { } S ( p ( a ν a ( ( a ( { ( ( } S( p ( γωa λσω ν a φ( γωa φ φφ p + ω φ λ σ ω φφ + φ γ ω + φ + ωa φ λσ ν ω a φφ φ γ ω + a + φ { } ( φ + + λσ ν ωa( γωa γω σ ν, ν + c0φφ + λσω ν ( ( a γ ω a + ωa ωb + γ ωa ν σ ν { λσω ν ( ( } ( a φφ φ γω a φ S p + + ({ } S ( p + ( φ S ( p γω σ a ν, ν p + ωb + ( γ ωa ν 0 = ωa σ ν (7. and 30

248 ( { } S ( p ( ν ( ( ω φ λ σ ω φφ φ γ ω + + φ a a a ( + cφφ λσω ν a γ ω + a { } S ( p ( ν ( ( + ω φ λ σ ω φφ + φ γ ω + φ a a a { } ( a ν a ( a γω λσω φ γω φ φφ p σ ν, ν + c0φφ + λσω ν ( ( a γ ω a + ωa ωb + γ ωa ν σ ν + ( φ S ( p + ( φ + + λσω ν a( γωa { λσω ν ( ( } ( a φφ φ γω a φ S p + + γω + φω + λσω γ ω φ + φ ( ( a a ν ( ( ( ( a a S p S p p ωa σ + ωb + γ ωa ν σ ν ν, ν ( S ( p { } S ( p γω σ a ν, ν p + ωb + ( γ ωa ν 0. = ωa σ ν (7. In the next ubecton, we examne the tructure o eq. (7. and (7. and recat them n a orm more convenent or numercal oluton. 7.. The tructure o equaton (7. and (7. To make clear the tructure o eq. (7. and (7., dene ome addtonal notaton. Frt, let a upercrpt ++ be the matrx (or vector tranpoe operator. Let S ( p gven by ( ( ( ( S p S p S p p ++, (7.3, 3

249 be an ( n + x column vector o upply uncton, augmented by the ndependent varable p and the number The dervatve o S ( p wth repect to p, rom eq. (7.3, ( 0 ( = ( ( S ++ p S p S p. (7.4 Now dene P k, gven by (uppreng t dependence on notatonal mplcty P ++ ( Ck S ( p k p n the ollowng or, ( a the rt-order polynomal n the element o S ( p that multple the k th component o S ( p =. In eq. (7.5, dene ++ (recall eq. (7.4 n eq. (7. (or = and (7. (or (,,,3,4 C C C C C k k k k k (7.6 a an ( x n + column vector o contant, exogenou coecent C kl, (dened below, wth l =,, 3, 4 ndexng the element o the vector explctly, we have ( ( k k, k, k,3 k,4 C k. Wrtng out the polynomal P C S p + C S p + C p + C. (7.7 P k 6 Hence the upercrpt n the notaton S ( p. 3

250 We dene each coecent C kl, by comparng the denton o P k wth the repectve coecent n eq. (7. and (7.. 7 Ung the notaton P k, we may wrte eq. (7. and (7. more compactly a ( ( P P P (7.8 S p + S p + 3 = 0 and 7..3 Iolatng the S ( p ( ( P P P. (7.9 S p + S p + 3 = 0 n equaton (7.8 and (7.9 For computatonal purpoe, t ueul to recat eq. (7.8 and (7.9 o that each dervatve S ( p appear n only one equaton. Dong o yeld ( S ( p PP PP = PP PP ( and ( S ( p PP PP = PP PP, ( where we mpoe the retrcton that the determnant o the coecent matrx n eq. (7.8 and (7.9 nonzero, that, PP PP. (7. 0 Gven the retrcton (7., the two ytem [(7.8, (7.9] and [(7.0, (7.] are 7 For convenence, Appendx E. dene each coecent C explctly. kl, 33

251 equvalent n the ene that the et o oluton to each o thee two ytem concde. Note that the coecent o S ( p n eq. (7.0, ( nvere o the coecent o S ( p n eq. (7., ( that the coecent ( PP PP and ( PP PP, ut the addtve PP PP. An mplcaton PP PP vanh over the ame et o parameter value. Th property wll be mportant n the next ecton and n Appendx E. n characterzng properte o the phae pace that the oluton to the ytem (7.0 and (7. nhabt. Let both and k ndex element o the vector S ( p multplyng out the coecent o the orm ( k k ++ n eq. (7.4. By PP PP n eq. (7.0 and (7., Appendx E. make explct that thee coecent are quadratc orm n the element ++ o S ( p ( n x( n, a eq. (7.7, (7.0, and (7. mply. Next, let Q k be an + + ymmetrc matrx. We dene Q k mplctly below uch that t element are uncton o the coecent C kl,. In partcular, or a coecent ( k k PP PP n eq. (7.0 and (7., the ollowng relatonhp dene element o Q k n term o the coecent o the polynomal P and P k : ( Qk ( P Pk Pk P S p S p (7.3 From denton (7.3, we have that Q = Q. (7.4 Ung the notaton o eq. (7.3, we may rewrte eq. (7.0 and (7. a 34

252 ( Q ( ( ( Q3 ( S p S p S p = S p S p (7.5 and ( Q ( ( ( Q3 ( S p S p S p = S p S p , (7.6 whereby the condton (7. become ( Q ( S p S p (7.7 Equaton (7.4 mple, moreover, that ( Q ( ( Q ( S p S p = S p S p. (7.8 Appendx E. provde explct expreon or the element o the matrce Q k n eq. (7.5 and (7.6. Under the retrcton (7.7, equaton (7.5 and (7.6 conttute a tranormed veron o the orgnal ODE ytem (7. and (7. above characterzng rm optmal orward market acton. Later n th chapter, we compute numercal oluton o th ytem or a retrcted doman o prce p. The coecent matrce Q k n eq. (7.5 and (7.6 are uncton, ultmately, o parameter characterzng 8 uppler margnal cot, tochatc dtrbuton, and conumer technology, utlty and rk preerence. Hence, gven value or thee prmtve parameter, the element o Q k are mply known, exogenou contant. Equaton (7.5 and (7.6 conttute a coupled ytem o rt-order nonlnear ODE n 8 In part through the ntermedate varable γ, φ, φ, ω, and a ω. b 35

253 the equlbrum orward market quantte, q S ( p and q S ( p, wth ndependent varable p. The ytem need to be augmented by an ntal condton ( (,0,0 S p,0, S p, p to have a well-dened, unque oluton.9 We ue the term SF traectory (or mply, traectory to denote a curve ( S p n q -q -p pace (.e., ome ubet o 3 pang through ome ntal condton (,0 (,0 S,0 p, S p, p and olvng eq. (7.5 and (7.6 at every pont. The proecton o th SF traectory nto the p -q and - p q plane, n turn, are dentcally the SF S ( p and S ( p or rm and. Once we olve or the SF ( hape component o orward market demand, D0 ( p S p, we may compute the lope o the, rom eq. (7.0. Whle there are no known method o olvng a ytem o the orm o eq. (7.5 and (7.6 analytcally (Braun 993, 37, t poble to how that oluton to the ytem exhbt certan qualtatve properte. Alo, we may agn value to the exogenou parameter n the ytem and obtan numercal oluton. In ecton 7. below, we conder the properte o the ytem (7.5 and (7.6. Followng that, n the remanng ecton o th chapter, we olve the ytem (7.5 and (7.6 numercally over a retrcted doman, and examne n detal the qualtatve and quanttatve properte o uch oluton. 9 We encloe ntal condton n quotaton mark here nce uch a condton cutomarly denote the tate o a tme-dependent ytem at ome ntal tme o nteret t 0. Snce t the orward market prce, p rather than a tme coordnate that our ndependent varable n th tmele problem, the noton o an ntal tme doe not apply lterally here. Nonethele, we contnue to reer to ntal condton n th problem. On the extence and unquene o oluton, ee ubecton

254 7. Properte o the ytem (7.5 and (7.6 and extence and unquene o oluton 7.. Sngularte For th dcuon, t convenent to wrte the ytem (7.5 and (7.6 more compactly a ollow. Frt, augment the vector o the duopolt SF (.e., the dependent varable wth a thrd component 30 (only, dened a ( S p p, (7.9 3 whch we may derentate to yeld S ( p =. ( Next, ung eq. (7.9, dene S + a an ( n + xvector o the orm 3 ( ( ( ( 3 ( S p S p S p S p +, whch ha the dervatve wth repect to p o ( ( = ( ( S + p S p S p. (7.3 We may then wrte eq. (7.5, (7.6, and (7.30 n vector orm a the ytem 30 Or, an ( n + th component, or the general cae o n rm. 3 Th augmentaton o the vector o dependent varable permt u to uppre the explct appearance o p n the ODE ytem; uch ytem are commonly called autonomou ytem o ODE. Th tep helpul nce many theoretcal reult or ODE ytem are expreed wth reerence to uch autonomou ytem. 37

255 ( ++ where S ( p ( S ++ ( p S + ( p = S ++ ( p ( A G, (7.3 A an ( n x ( n + + matrx o the orm ( Q ( S p S p A ( S ( p = 0 S ( p QS ( p 0, ( ( G an ( ++ and S ( p n + xvector o the orm ( Q3 ( ( Q3 ( S p S p G ( S ( p = S p S p. (7.34 ( ++ Snce they contan quadratc orm, the matrx S ( p A and the vector G ( S ( p are each a quadratc uncton o the element o S ( p Sytem o the orm o eq. (7.3 are oten called qualnear becaue n th cae, ( we may wrte the general orm o an mplct ODE, F S ( p S ( p ( ( ( ( ++, + = 0, a ( ( ( ( ++, F S p S p A S p S p G S p = 0, (7.35 ( where the (mplct dervatve term S + ( p enter ++ S ( p, S + ( p The ytem (7.3 ngular at a pont ( ( ++ ( S ( p F lnearly. S p, S p, p. when the matrx A n eq. (7.33 not nvertble, or ngular. Th occur and only at 38

256 ( ++ leat one o the quadratc orm on the dagonal o S ( p pont, that, when A equal zero at that ( Q ( S p S p = 0 (7.36 or ( Q ( S p S p = 0, (7.37 or both. ecall rom eq. (7.8, however, that the locu o pont ( ( S p, S p, p at whch eq. (7.36 hold concde exactly wth the locu o uch pont at whch eq. (7.37 hold. Thu we may conder exactly one o eq. (7.36 and (7.37 to be redundant. We call pont n th locu the ngular pont, or ngularte, o the ODE ytem ( Pullng together th nomenclature, we may label eq. (7.3 a ngular qualnear ODE ytem. Geometrcally, the graph o each equaton (7.36 and (7.37 concde n a common graph: a quadratc urace. 33 Becaue th quadratc urace the locu o the ngular pont n th problem, we call th urace dened by eq. (7.36 and (7.37 the ngular locu. Inormally, we may thnk o mot pont on the ngular locu a that et o pont at whch, n the lmt, both rm orward market SF become nntely 3 To preerve ymmetry n the dcuon, however, we wll cutomarly contnue to reer to both o eq. (7.36 and (7.37 a characterzng the ytem (7.3 ngularte, although by the argument above, ether equaton (7.36 and (7.37, taken ndvdually, would uce to decrbe thee pont. In our earler notaton, a neceary and ucent condton or eq. (7.36 and (7.37 to hold PP PP = 0, the convere o the retrcton ( One generate quadratc (or quadrc urace by rotatng a conc ecton about an ax o ymmetry. On quadratc urace, ee Eve (987, 98 or a ueul taxonomy, a well a Weten (999a and Hlbert and Cohn-Voen (95 or addtonal llutraton. 39

257 loped. 34 Appendx E. examne n greater detal the theory and computaton o ngularte n the ytem ( Soluton o the ytem (7.5 and (7.6 away rom the ngular locu At pont not on the ngular locu dcued n ubecton 7.., 35 we wll have by denton that the convere o eq. (7.36 and (7.37 wll hold at all p, namely, (rewrtng the retrcton (7.7 ( Q ( S p S p (7.38 and (content wth eq. (7.8, ( Q ( S p S p ( ++ Under the condton (7.38 and (7.39, S ( p. (7.39 A nvertble, and we may wrte the ytem (7.3 n explct orm that, olvng explctly or the dervatve S ( p, =, a S S ( Q3 ( ( Q ( S p S p ( p =, ( S p S p ( Q3 ( ( Q ( S p S p ( p =, ( S p S p 34 Agnng prce p to the vertcal ax, a uual, nntely-loped SF would be parallel to the horzontal plane dened by the quantty axe q and q. See Table 7. on page 5 below or a more prece dcuon. 35 We reer to uch pont a beng away rom the ngular locu. 40

258 and S ( p =. (7.4 3 I the nequalte (7.38 and (7.39 hold or every pont on the SF o nteret, we obtan a non-ngular ODE ytem (7.40 (7.4. Gven the aorementoned nequalte, SF olvng (7.40 (7.4 do not nterect the ngular locu (eq. (7.36 and (7.37 dened n ecton 7... Snce the ytem (7.40 (7.4 non-ngular, we may appeal to the tandard theorem on extence, unquene and contnuty o oluton to ODE ytem (ee, e.g., Brkho and ota 989, ch. 6 (n partcular, Theorem,, 3, 8,, and applcable corollare. Thee theorem provde that, or the ytem (7.40 (7.4, a unque oluton ext perhap over a retrcted doman o p or any ntal condton. 36 Moreover, uch a oluton contnuou, and vare contnuouly wth the exogenou parameter o the problem. The ollowng ecton preent the computatonal method ued n th nvetgaton to olve the ytem (7.40 ( Techncally, the extence and unquene reult apply to a local oluton o the ytem (7.40 (7.4 n the neghborhood o a gven ntal condton ( (, 0,0,,0 S S p S p, p,0. By patng together uch local oluton, we may extend uch oluton to ome maxmal nterval o extence,0 (,0 J S, yeldng a reultng maxmal or global oluton on m ( m J S. See de la Fuente (000, 437. or detal. We wll not nvetgate the properte o oluton near the boundare o nterval,0 ( J S, and o do not need to dene them ormally here. m 4

259 7.3 Computatonal approache to olvng the derental equaton ytem characterzng the orward market SF We ued two dtnct approache to olvng the derental equaton ytem (7.40 (7.4 characterzng the orward market SF n the mult-ettlement SFE model: ( numercal ntegraton ung MATLAB (The MathWork 00, and ( a derence equaton approxmaton mplemented here ung Mcroot Excel Solver tool. Thee two approache are complementary n that each hghlght partcular properte o oluton to the ODE ytem. Th ecton provde detal on both o thee mplementaton o the mult-ettlement SFE model Numercal ntegraton ung MATLAB MATLAB oer everal derental equaton olver or numercal oluton o (nonngular problem o the orm (7.40 (7.4, together wth ymbolc algebra capablte (peccally, the Maple ymbolc algebra kernel (Mapleot We teted the perormance o each o MATLAB olver on the preent problem or reaonable range o parameter. The bet-perormng olver n term o both tablty and the range o prce over whch we could ntegrate ucceully named ode5. Appendx E. dcue the properte o MATLAB ode5 olver n greater detal. 38 Th olver ormed the core o the MATLAB-baed oluton to the ytem (7.40 (7.4, to whch we reer herenater a the MATLAB model. Gven an ntal condton 37 The MATLAB code ued n th the are avalable rom the author. 38 The olver ode5 worked bet wth the backward derentaton ormulae (BDF (rather than the numercal derentaton ormulae (NDF enabled. The BDF are alo commonly known a Gear method ; ee Gear (97. On the detal o and the dtncton between BDF and NDF, ee Shampne and echelt (997. 4

260 ( ( S p, S p, p,0,0,0 that, ntal quantte or each rm and a correpondng ntal prce 39 we may ue the MATLAB model to compute a traectory ( S p n that olve the ytem (7.40 (7.4. Proectng th traectory nto the p -q and p -q plane, n turn, yeld the SF ( S p and ( S p Derence equaton approxmaton ung the Excel Solver: The dcrete Excel model The econd computatonal approach that we employ n th nvetgaton to olvng the derental equaton ytem (7.40 (7.4 rele on a derence equaton approxmaton to th ytem. Snce th approach ue Mcroot Excel (Mcroot Corporaton 00 n partcular, Excel Solver tool, herenater mply the Excel Solver 4 we reer to th approach herenater a the dcrete Excel model. 4 In contrat to the MATLAB model requrement o an exogenouly-peced ntal condton, we ormulate the dcrete Excel model to elect endogenouly a (locally unque equlbrum traectory, a elaborated below. The dcrete Excel model compre a amly o doubly-neted optmzaton problem havng the general orm 39 ecall rom ubecton 7..3 that we mut pecy an ntal condton or an ODE to have a well-dened, unque oluton. The MATLAB model requre that th ntal condton be peced exogenouly. 40 A depcted, or example, n ubecton Fgure 7.7 below. 4 It appear that a mlar dcrete approxmaton o the ytem (7.40 (7.4 could alo have been mplemented and olved n MATLAB by explotng the capablte o the Optmzaton Toolbox, an add-on product or the MATLAB otware ute. Becaue the dcrete Excel model relatvely mple and eectve, however, we dd not attempt a MATLAB-baed dcretzaton o th problem. 4 The Excel le ued n th the are avalable rom the author. 43

261 mn/max ( ( S p dcretzed [Addtonal decon varable] Obectve uncton t.. Subgame-perect Nah equlbrum n Σ and S t.. Parameter Θ [Addtonal contrant] (7.43 where, a dcued urther below, the bracketed phrae n problem (7.43 ndcate (optonal addtonal element o the problem. We olve problem (7.43 ung the Excel Solver. 43 In the ollowng, we elaborate on the varou component o th problem. ( The element o prmary nteret n problem (7.43 are the dcretzed value o S p ( =, that repreent the quantte oered by rm over a peced vector o prce p. Thee prce-quantty par conttute a pecewe ane plne approxmaton to a mooth orward market SF or each rm. The [a]ddtonal decon varable noted n problem (7.43 could be, or example, parameter o the problem or whch market data and the lterature oer lttle quanttatve emprcal upport. Convertng uch parameter to decon varable n problem (7.43 would enable u to determne endogenou value or uch parameter n th problem oluton. We may olve problem (7.43 ung a varety o obectve uncton. Two ntutvely appealng choce or the obectve uncton would be 43 Accordng to Excel documentaton (n Excel, ee Help About Solver, [t]he Mcroot Excel Solver tool ue the Generalzed educed Gradent (GG nonlnear optmzaton code developed by Leon Ladon, Unverty o Texa at Autn, and Allan Waren, Cleveland State Unverty. 44

262 . The mnmzaton o the dcrepancy between endogenou model output (e.g., expected prce and quantte n each market, and correpondng emprcal reerence value 44 rom the lterature. The maxmzaton o expected aggregate welare, whch could be relevant a a benchmark or polcy analy In addton, we could denty other plauble canddate or obectve uncton that correpond to pecal cae o the mult-ettlement SFE model. For example, mnmzng the overall curvature (dened n ome meanngul way o the orward market SF mght be ued to denty orward market SF that are (nearly ane over a choen prce range. Choong an obectve uncton or problem (7.43 conttute an equlbrum electon rule that dente a ngle traectory (aumng a unque oluton or th problem rom the phae pace o SF traectore. Naturally, a derent obectve uncton would, n general, elect a derent SFE rom th phae pace. The upper-level contrant et o problem (7.43 tel a contraned equlbrum oluton o the mult-ettlement SFE model. The equlbrum contrant o Subgameperect Nah equlbrum n Σ and S reer to the (mpled ane pot and orward market equlbra decrbed n chapter 4 and 5. Th equlbrum compre each rm rt- and econd-order optmalty condton a well a lope retrcton on the orward market SF. Here, the orward market equlbrum SF are repreented by the pecewe S p (.e., ane approxmaton correpondng to the dcretzed decon varable ( quantte dened over a grd o xed prce. We compute the ubgame-perect Nah 44 For example, or Calorna electrcty market durng a partcular perod o nteret. See Appendx F or detal. 45

263 equlbrum ubect to choen parameter value element o the vector Θ and pobly to [a]ddtonal contrant. 45 Such addtonal contrant could nclude retrcton that enhance the vermltude o the model. Secton 7.5 and 7.6 below provde urther detal and pecc example o the applcaton o problem ( Comparon o computatonal approache The dcrete Excel model and the MATLAB model hare ome undamental mlarte. Lke any numercal ntegraton routne, the MATLAB model at t heart alo a dcretzaton o what away rom the ngular locu a contnuouly derentable problem. The maor algorthmc dtncton between the two approache le n the dcrete Excel model ncorporaton o equlbrum electon mplemented ung optmzaton problem havng the general orm o (7.43 not repreented n the MATLAB model. A a conequence, the Excel- and MATLAB-baed approache der n ther nput and output n way that are mportant or the preent nvetgaton. We revew thee dtncton below. For our purpoe, the dcrete Excel model oer two dtnct advantage over the MATLAB model decrbed n ubecton Frt, by allowng or an equlbrum electon procedure, the dcrete Excel model aord a ytematc mean o choong the ntal condton (,0,0,0,0 q, p and (, q p or each rm orward market SF.,0,0,0,0 Namely, the ntal quantte q = S ( p and q S ( p = appear a mply 45 To obtan numercal oluton, we agn value to the cot, demand, dtrbutonal, and rk parameter o the mult-ettlement SFE model, already ntroduced n chapter 3 through 6. Together wth ome new notaton, we collect thee parameter a element o a parameter vector Θ n ubecton 7.4. below. Appendx F explan the provenance o the partcular parameter value ued to conduct the comparatve tatc and welare analye o th chapter. 46

264 two o the endogenouly-determned decon varable n problem (7.43. The MATLAB model, n contrat, requre the uer to pecy exogenouly the ntal quantte or each rm. A a econd advantage, t traghtorward n the dcrete Excel model to mpoe explctly the contrant that orward market SF be trctly ncreang, wherea th not poble ung only MATLAB ODE olver (ee note 4, however. Fnally, we note that the dcrete Excel model permt the uer to adut both the (unorm tep ze and lke MATLAB the range o prce p condered. The dadvantage o the dcrete Excel model center around extence and unquene o oluton, and the eae wth whch we may olve the model to nd oluton. Frt and mot undamentally, a eable oluton to the optmzaton problem (7.43 cannot alway be ound or a gven et o contrant and decon varable. Tral and error 46 may be requred to denty a model or whch the Excel Solver can denty a eable oluton. I a eable oluton can be ound, the Solver can guarantee only a locally optmal oluton, not a globally optmal oluton due to the nonlnearty o problem (7.43. Accordngly, the dcrete Excel model oluton depend, n general, on the decon varable ntal value. 47 Fnally, the MATLAB ODE olver adut dynamcally the tep ze or numercal ntegraton to keep the dcretzaton error (ee note 380 below wthn acceptable lmt, whle the unorm tep ze n the dcrete Excel model xed by the uer. Th mple that, at certan pont, the approxmaton 46 That, tral and error wth repect to the ollowng attrbute: the contrant et, the et o decon varable, the grd o prce ued n the approxmaton, ntal value or the optmzaton, and parameter o the Excel olver (n Excel, ee Tool Solver Opton Help. 47 Away rom the ngular locu, thee queton o extence and unquene o oluton are due to the nonlnear optmzaton problem n the dcrete Excel model, rather than to theoretcal properte o the ODE ytem (7.40 (7.4. For (nonngular ODE ytem, we recall that the theorem noted n ubecton 7.. guarantee the extence and unquene o oluton to uch ytem. 47

265 (dcretzaton error o the dcrete Excel model can be relatvely large. 48 Provded, however, that the dcretzed SF n the choen prce range do not traddle any ngularte (not necearly the cae n the tral that we examne, the approxmaton error may n prncple be made arbtrarly mall by ncreang the number o prce tep, 49 or decreang the overall prce range over whch we compute the dcretzed SF. The MATLAB model (whch explot MATLAB graphc capablte welluted to nvetgate qualtatvely the phae pace and the properte o SF traectore tartng rom arbtrarly-peced ntal condton. Secton 7.4 preent qualtatve reult rom the MATLAB model; the analy emphaze the geometry o traectore, the ngular loc, and other alent eature o the phae pace. 7.4 Qualtatve analy o the derental equaton ytem characterzng the orward market SF We begn n ubecton 7.4. below by denng the general parameter vector Θ a well a a partcular vector bae Θ whoe element erve a our et o bae cae parameter value or the mult-ettlement SFE model. Subecton 7.4. then analyze qualtatvely the ngular qualnear ODE ytem, eq. (7.3. Followng that, ubecton explore n greater detal the non-ngular ODE ytem (7.40 ( Two example n whch th approxmaton error tend to be large n magntude are regon n whch an SF curvature large, and at pont on a egment o the ane approxmaton to an SF that are relatvely dtant rom the endpont o the egment (e.g., near the mdpont o uch a egment. At the egment endpont, n contrat, the approxmaton exact. 49 The current mplementaton o the dcrete Excel model characterze a rm orward market SF ung eleven ane egment connectng twelve prce-quantty par. Increang the number o prce tep poble, n prncple, though dong o would ncreae the ze o the problem and hence t computaton tme. 48

266 7.4. The parameter vector Θ We rt ntroduce ome new notaton to repreent the elatcty o pot market demand; denote th elatcty a e dem. Gven an emprcal mean reerence prce p and, mean empr quantty q or the pot market, 50 we may wrte, mean empr e dem n term o γ (, ε D p p 5 a e dem p γ =. (7.44, mean empr, mean qempr We ntroduce the parameter e dem n eq. (7.44 n order to conduct th chapter quanttatve analy n term o th ntutvely more appealng parameter. dened a Let Θ be the (general parameter vector or the mult-ettlement SFE model, Θ ( c c c c edem η ση ν σν λ 0 0. (7.45 The ten-element vector Θ collect the cot, dtrbutonal, and rk parameter already ntroduced n prevou chapter, along wth the demand elatcty parameter mmedately above. Now denote a e dem dened bae Θ the parameter vector Θ aumng bae cae value o each o t ten element. The bae cae value o the cot uncton parameter c 0 and c ( =, are baed on emprcal data rom Calorna electrcty market, 50 See the dcuon o Appendx F.. or data ource and value o, p mean and empr, q mean. empr 5 ecall that we wrote the orward market equlbrum optmalty condton (7. and (7. n ubecton 7.. above n term o the lope parameter γ rather than the elatcty e. dem 49

267 crca 999, a detaled n Appendx F..3. The bae cae value o the elatcty e dem, the our dtrbutonal parameter η, σ η, ν, and σ ν (ee ecton 6.5, and CAA parameter λ are endogenou to the benchmarkng procedure or the dcrete Excel model, decrbed n ubecton 7.5 below. Brngng together thee exogenouly- and endogenouly-determned parameter n th problem, the reultng bae cae parameter vector bae Θ, 5 to three gncant gure, bae c0 $5.60 MWh $30.50 MWh c 0 $ ( MWh c c $ ( MWh bae edem 5.95e-5 Θ =. (7.46 η 4640 MWh σ η.46e6 MWh ν 335 MWh σ ν 5.86e4 MWh λ 3.0e-4 $ Unle otherwe peced, the computaton n th ecton rely on the bae cae parameter vector bae Θ o eq. ( The ngular qualnear ODE ytem, equaton (7.3 To tudy the ngular qualnear ODE ytem (7.3, t ueul to begn by characterzng two type o loc n th ytem phae pace. Frt, there the ngular locu dened by eq. (7.36 and (7.37 and dcued n ubecton 7.. above. oughly 5 See alo eq. (F.39 and the aocated dcuon n Appendx F. 50

268 peakng, th the locu at whch (or mot pont n the locu ee Table 7. below both rm orward market SF become, n the lmt, nntely loped. For th reaon, we alo reer below to the ngular locu a the -locu. Second, we have the two loc at ( ++ whch, repectvely, each o the rt two element o S ( p (7.34, that, the locu repreented by the equaton G vanhe (ee eq. ( Q ( S p S p = 0 (7.47 and that correpondng to the equaton ( Q ( S p S p = 0. (7.48 For convenence, we reer to the loc (7.47 and (7.48 a the 0 -locu ( zero-one locu and the 0 -locu ( zero-two locu, nce at non-ngular pont n thee loc (agan, ee Table 7. below, we have that S ( p = 0 and S ( p 0 reerence, we collect th termnology n Table 7. below. =, repectvely. For eae o 5

269 TABLE 7.: LOCI OF INTEEST IN THE SINGULA QUASILINEA ODE SYSTEM (7.3 Name o the locu -locu (alo, ngular locu Equaton( characterzng the locu S ( p Q S ( p S ++ ( p Q S ++ ( p locu S ( p Q S ( p locu S ( p Q S ( p = 0 = 0 3 = 0 3 = 0 or S ( p 53 Properte ated by mot pont a on the locu, =, S S ( p ( p = 0 = 0 Note: a In the retrcton to mot pont, we exclude thoe pont lyng on the manold at whch ether ( the - and 0 -loc, or ( the - and 0 -loc nterect. We would need to determne the lope S ( p at uch pont on a cae-by-cae ba; the generalzaton n the rghtmot column o the table do not necearly apply. On the other hand, or pont at the manold o nterecton o the 0 - and 0 - loc (but not alo on the -locu, we have that S ( p = 0 and S ( p = (a the table ndcate. 0 We nclude the generalzaton n the rghtmot column o the table olely a an ad to ntuton, and emphaze that, wthout excepton, we characterze the loc ung the equaton n the mddle column o the table. Where approprate n the dcuon below, we reer genercally to the 0 -locu or the 0 -locu a a 0 -locu ( zero-eye locu. Th ubecton characterze each o Table 7. loc analytcally ung the taxonomy o Eve (987, 98 or quadratc orm, and plot ther graph ung MATLAB three-dmenonal vualzaton capablte (and aumng, unle otherwe peced, that bae Θ=Θ. Eve taxonomy aocate relatonhp among a quadratc orm coecent or example, the element o Q n each o Table 7. quadratc orm wth one o the eventeen type o quadratc urace. The taxonomy nvolve 53 ecall that thee two equaton are redundant; hence, we ue the conuncton or. 5

270 rank, determnantal, and egenvalue condton o the coecent matrce aocated wth each quadratc orm. Conder rt the -locu. Applyng the taxonomy o Eve (987, 98 to (ether equaton repreentng th locu, we may how that th locu a real ellptc (double cone. Fgure 7. below depct the -locu, conrmng th clacaton. FIGUE 7.: THE -LOCUS, A EAL ELLIPTIC (DOUBLE CONE, IN A NEIGHBOHOOD OF THE OIGIN 53

271 The graph depcted n Fgure 7., naturally, only a dcrete approxmaton made or the ake o vualzaton to a theoretcal real ellptc (double cone. The act that the two nappe o the cone do not appear to meet at a ngle pont the vertex but rather appear to nterect over a contnuum o pont merely an artact o th dcretzaton enng the reoluton o the lattce ued to vualze the cone hrnk the apparent contnuum at whch the nappe o the cone meet. Th behavor content wth the amlar theoretcal property that the cone two nappe meet at a pont. 54

272 Next, we examne the 0 -locu. The taxonomy o Eve (987, 98 mple rom the correpondng equaton that th locu a hyperbolod o one heet. Fgure 7. below depct the 0 -locu, conrmng th reult. FIGUE 7.: THE 0-LOCUS, A HYPEBOLOID OF ONE SHEET, IN A NEIGHBOHOOD OF THE OIGIN 55

273 Fnally, applyng the taxonomy o Eve (987, 98 to the equaton repreentng the 0 -locu, we nd that th locu alo a hyperbolod o one heet. Fgure 7.3 below depct the 0 -locu, agan conrmng th reult. FIGUE 7.3: THE 0 -LOCUS, A HYPEBOLOID OF ONE SHEET, IN A NEIGHBOHOOD OF THE OIGIN To emphaze the geometry o the three loc n Fgure 7. Fgure 7.3, we drew thee gure to a maller cale than would be approprate to depct equlbra n the Calorna electrcty market (ee Appendx F. or repreentatve orward market 56

274 quantte. Next, n Fgure 7.4 below, we upermpoe the graph o the loc hown n Fgure 7. Fgure 7.3, enlargng the cale o the plot, a well. FIGUE 7.4: THE -LOCUS (IN BLACK, THE 0-LOCUS (A TIANGULA MESH, AND THE 0 -LOCUS (IN GAY IN A (SMALLE NEIGHBOHOOD OF THE OIGIN In the next ew gure, or clarty, we uppre both 0 -loc and examne the relatonhp between varou SF traectore and the -locu. ecall, a Fgure 7. depct, that the -locu the black urace n Fgure 7.4 a real ellptc (double cone. Gven that the orentaton o the cone ax (a uncton o the parameter Θ 57

275 more nearly parallel wth the vertcal ( p ax than wth ether quantty ax, t natural to characterze th -locu a dvdng the phae pace nto three partton: the upper partton, the mddle partton, and the lower partton. Fgure 7.5 below portray the - locu along wth eparate SF traectore begnnng n each o thee three partton. 55 Mddle partton Upper partton Lower partton Mddle partton FIGUE 7.5: THE -LOCUS (BLACK SUFACE DIVIDING THE PHASE SPACE INTO UPPE, MIDDLE, AND LOWE PATITIONS, AND SF TAJECTOIES (BLACK CUVES MAKED WITH O BEGINNING IN EACH PATITION 55 We dene the partton o the phae pace a open et, bounded, n part, (a depcted n Fgure 7.5 by the -locu (and otherwe unbounded. By th denton, pont on the -locu tel belong to none o thee partton, and thereore each partton contan excluvely non-ngular pont. 58

276 Fgure 7.5 llutrate the three type o behavor that we have oberved or orward market SF traectore a they approach the -locu. 56 Namely, an SF traectory can be delected by, tranvere to, or aborbed by the -locu. 57 Fgure 7.5 depct three dtnct traectore n the neghborhood o the -locu, each o whch begn n a derent phae pace partton and each o whch exhbt one o the three behavor noted above. 58 We characterze thee behavor normally below. Conder rt the traectory depcted n the upper partton o Fgure 7.5, labeled a. Qualtatvely, we may ay that the -locu delect th traectory, that, the drecton o th traectory change abruptly n the vcnty o the -locu. Next, conder the traectory labeled a n Fgure 7.5, whch begn n the mddle partton at p = $,500 MWh and move up (.e., n the drecton o ncreang p and to the let rom there. Th traectory croe the -locu we ay t 56 The traectore depcted n Fgure 7.5 do not necearly aty the econd-order condton or optmalty or ether rm over the entre prce range; we ue thee traectore or expotory purpoe only. Th the cae (unle otherwe peced or SF and traectore portrayed n all o th ubecton gure. 57 Whle we do not clam that the above clacaton o behavor exhautve o all poblte, all traectore nvetgated n th tudy clearly ell nto one o thee three categore, a dened below. 58 The apparent correpondence n Fgure 7.5 between the partton and the three traectory behavor dcued here ncdental. For derent ntal condton or parameter value, we can nd traectore n each partton that behave derently. 59

277 tranvere to the -locu and contnue nto the lower partton. Cloer numercal examnaton o traectory reveal that the lope S ( p near the apparent nterecton wth the -locu are on the order o 0 3, that, thee lope are clearly nte. Th obervaton ugget that at the -locu, traectory encounter a and removable ngularty, 59 meanng that the magntude o the SF lope S ( p S ( p n eq. (7.40 and (7.4 are bounded along an SF traectory n the neghborhood o the ngularty. Conequently, the MATLAB ODE olver doe not al n th neghborhood, makng numercal ntegraton ung our model eable almot everywhere that, on both de o the -locu. 60 Th ndng upported by urther graphcal nvetgaton (not llutrated n Fgure 7.5, whch ndcate that the nterecton o th traectory wth the -locu alo cloe to a pont at whch the -, 0-, and 0 -loc all appear to nterect. Whle we would requre urther reearch to 59 A removable ngularty o a real uncton ( x a ngular pont x at whch we may agn 0 a value ( x uch that analytc, that, poee dervatve o all order and agree wth t Taylor 0 ere n the neghborhood o every pont (Weten 999b, 999c. 60 A more amlar example o a removable ngularty ound n Green (999a. In Green Fgure (p. 4, removable ngularte ext at the pont o nterecton o the margnal cot uncton and h pot market upply uncton, that, at quantte X and X or the upply uncton S(X and S(X, repectvely. To ee th analytcally, olve Green eq. (4 or dq dp to obtan The rato ( q ( p x ( p cq ( p ( ( dq q p x = b dp p c q p n the above equaton ndetermnate at the quantte X and X noted above (.e., at the removable ngularte, but t may be evaluated va L Hoptal ule. Smlarly, a removable ngularty ext at the orgn n Klemperer and Meyer (989 connected et o SFE; ee ther Fgure (reproduced a Fgure 7.0 below and ther eq. (5.. 60

278 corroborate numercally that th repreent an actual pont o nterecton, 6 thee obervaton ugget that the ngular pont at whch traectory croe the -locu belong to a cla o more complex ngularte. Thee more complex ngularte lkely der n mportant way rom other pont on the -locu, exempled by the poblty o traectore tranvere to the -locu at uch pont. Fnally, regardng the traectory labeled a n the lower partton o Fgure 7.5, we may ay that the -locu aborb th traectory. More precely, n th cae, the MATLAB olver al and numercal ntegraton halt (ee Appendx E.3 or detal when the traectory approache the -locu ucently cloely. Th numercal alure due, analytcally, to the dervatve S ( p that explode a traectory approache the -locu (recall Table 7. above. Whle a cloer analy o the actor governng traectore behavor n the neghborhood o the -locu let or urther reearch, 6 we make here a ew general obervaton on thee actor. In theoretcal term, the vector eld correpondng to an underlyng ODE ytem tangent to any arbtrary oluton traectory at all pont along the traectory. Accordngly, the parameter value that determne th vector eld wll clearly contrbute to determnng how traectore behave n derent regon o the phae 6 Or, practcally peakng, a mall neghborhood through whch the traectory and the varou urace pa, nce we are dealng nvarably wth approxmate numercal repreentaton o the underlyng theoretcal obect. 6 aza (00, 306 hghlght the dtncton made n the appled mathematc lterature between algebrac ngularte (where, n our ramework, ( S ( rge ( ( ++ p S ++ p (7.33 and (7.34, and geometrc ngularte (where ( S ( rge ( ( ++ p S ++ p G A, recallng eq. G A. Explorng th dtncton n the preent context may be a ueul pont o departure or uture work. See alo the dcuon o Appendx E., whch examne n greater detal the theory and computaton o oluton traectore n the neghborhood o th model ngularte. 6

279 pace. Each traectory that we compute n th chapter, naturally, a numercal approxmaton to an underlyng theoretcal traectory. Error tolerance and tep ze retrcton or numercal ntegraton wll play a role n determnng the extent, the accuracy o approxmaton near the ngularte, and perhap even whch o the three behavor dented above that a (numercal traectory exhbt. In ome cae, the numercal approxmaton may only approxmate the theoretcal traectory over a lmted range. For example, whle Fgure 7.5 howed that traectory wa apparently aborbed by the -locu, mply toppng hort beore reachng th locu, th behavor clearly attrbutable a MATLAB error meage report to a numercal rather than a theoretcal caue (.e., alure o the MATLAB olver. Thereore, although the numercal traectory top near the -locu, t certanly poble that the underlyng theoretcal traectory extend beyond th pont. Through a change o coordnate to remove the ngularty, t may alo be poble to extend uch a traectory numercally, through the -locu. 63 Agan, we reerve or uture reearch the exploraton o uch queton. A pecal cae the tuaton n whch uppler cot uncton and ntal condton are ymmetrc. A a pecc llutraton, dene a ymmetrc parameter vector ymm Θ a the vector bae Θ (recall eq. (7.46 wth rm replaced by a replca o rm n ymm bae ymm the bae cae, o that ( c = ( c and ( c ( c a 0 0 bae =. That, we dene ymm Θ 63 For example, nterchangng the dependent and ndependent varable would mply that SF lope would now approach zero at pont where they were ormerly unbounded. Accordngly, the ODE olver would no longer al at uch pont. The author ndebted to Allan Wttkop o Mapleot (Wttkop 00 or uggetng th approach to numercal ntegraton n the vcnty o uch ngularte, and or helpul dcuon concernng computatonal mplementaton ung MAPLE (Mapleot 00. 6

280 ymm c0 $5.60 MWh $5.60 MWh c 0 $ ( MWh c c $ ( MWh ymm edem 5.95e-5 Θ =. (7.49 η 4640 MWh σ η.46e6 MWh ν 335 MWh σ ν 5.86e4 MWh λ 3.0e-4 $ Alo take the ntal condton to be (,0, (,0,,0 [ e6 MWh,e6 MWh, $000 MWh] S p S p p =, ymmetrc acro the two rm. 63

281 Fgure 7.6 below llutrate the reultng traectory, ntegratng downward over the range [ $000 MWh, $000 MWh] p. FIGUE 7.6: WITH SYMMETIC SUPPLIES (THAT IS, ASSUMING ymm Θ=Θ FOM EQ. (7.49 AND SYMMETIC INITIAL CONDITIONS, THE SF TAJECTOY IS TANSVESE TO THE -LOCUS In th ymmetrc cae, we oberve that, lke traectory o Fgure 7.5, rm traectore are alo tranvere to the -locu. The SF traectory depcted n Fgure 7.6 appear to cro the -locu near the vertex o the double cone. 64 In analytcal term, 64 Whether th ndeed a general property o traectore n the ymmetrc cae a queton reerved or uture reearch. 64

282 we conecture that the numerator and denomnator o eq. (7.40 and ( wll go to zero at ame rate along an SF traectory a t approache the -locu. Th property mple that under ymmetry, all o the ngularte o the -locu become removable ngularte (ee note 59. eturn agan to the bae cae parameter portray varou traectore ( Θ bae and Fgure 7.5 above, whch S p rom derent ntal condton and range o ntegraton. In accordance wth the a pror lope contrant (due to market rule denng admble SF noted n ubecton 3..5, we are ntereted n locatng and characterzng a orward market SF or each rm that trctly ncreang. To mply the earch or trctly ncreang SF and or eae o expoton, we retrct the qualtatve analy or the remander o th ecton (and the numercal analy n the ret o th chapter to SF nhabtng the upper partton o the phae pace (ee Fgure 7.5 above. In the analy below, we are able to denty SF n the upper partton that lope upward, at leat over certan prce range. We may mplement the retrcton to conder only SF nhabtng the upper phae pace partton through udcou choce o the SF ntal condton. In partcular, we may explot the oberved emprcal regularty that an SF traectory begnnng at an ntal condton ( ( S S p, S p, p, 0,0,0,0 wthn the upper phae pace partton reman n the upper partton or any choen range o ntegraton Under ymmetry, thee two equaton are, o coure, dentcal. 66 Although th obervaton reman unproven a a theoretcal matter, we oberved th behavor n numercal tral, wthout excepton. Naturally, the peced range o ntegraton nclude prce,0 J S (ee note 36, the olver wll al to nd a oluton outde o the maxmal nterval o extence m ( 65

283 etrctng our attenton to the upper partton o the phae pace a ubtantve lmtaton n the cope o th analy, though extendng t to nclude SF n the other partton would be traghtorward. That, we could n prncple undertake an analy o traectore nhabtng the mddle and lower partton o the phae pace mlar to that conducted below or traectore n the upper partton. Indeed, prelmnary exploraton conrm that, a n the upper partton, there are regon wthn the mddle and lower partton n whch both rm SF lope upward. Moreover, t may be reaonable to uppoe that SF traectore lyng n thee other partton hare the charactertc o thoe traectore n the upper partton that we tudy here (e.g., the comparatve tatc properte dcued n ecton 7.6 below. We reerve or uture reearch, however, uch queton pertanng to SF traectore lyng n the mddle and lower partton o the phae pace, and do not conder urther thee traectore n the preent work. Accordngly, the next ubecton below retrct the analy to traectore nhabtng the upper partton whch, by contructon (recall note 55, contan only nonngular pont. In accordance wth th retrcton, we upplant the ngular qualnear ODE ytem (eq. (7.3 wth the non-ngular ODE ytem (eq. (7.40 (7.4 a the obect o our analy The upper partton o the phae pace o the non-ngular ODE ytem, equaton (7.40 (7.4 In th ubecton, we ocu on the porton o the 0 -loc and traectore that le wthn the upper partton o the phae pace depcted n Fgure 7.5. A argued at the cloe o at uch prce. Moreover, the olver may al to compute a numercal oluton a the theoretcal traectory approache the -locu ucently cloely rom wthn the upper partton. 66

284 the prevou ubecton, connng our attenton to the upper partton permt u to replace eq. (7.3 wth the non-ngular ODE ytem, equaton (7.40 (7.4. We alo urther enlarge the cale o the plot below to conder orward market quantte or each rm n the range [ e4, e4] MWh. Th range repreentatve o actual orward market quantte oberved n the Calorna electrcty market. Ung the ytem (7.40 (7.4, we tudy the upper partton o the phae pace to denty admble n partcular, trctly ncreang SF. S p that A ubecton 7.3. noted, or each rm, the orward market SF ( olve the ytem (7.40 (7.4 mply the proecton o the traectory ( S p n nto the p -q plane. Fgure 7.7 below how (a dahed lne thee planar proecton or rm and o an SF traectory ( 3 S p (the old lne lyng n the upper partton (that, above the -locu, the black urace n the gure. In the gure, we ee that the -locu delect th partcular SF traectory n the neghborhood o p = $40 MWh. 67

285 S ( p S ( p S ( p FIGUE 7.7: AN SF TAJECTOY ( S p (SOLID LINE, MAKED WITH O IN THE UPPE PATITION OF THE PHASE SPACE, ITS PLANA POJECTIONS THE SFS S ( p AND ( THE -LOCUS (BLACK SUFACE S p (DASHED LINES FO FIMS AND, AND 68

286 Fgure 7.8 below plot the proecton rom Fgure 7.7 the SF ( ( S p n a common prce-quantty plane. S p and 500 S (p S (p p ($/MWh S (p (MWh FIGUE 7.8: THE SFS S ( p AND ( OF THE SF TAJECTOY ( p - q PLANE S p OBTAINED FOM PLANA POJECTIONS S p IN FIGUE 7.7, PLOTTED IN A COMMON From Fgure 7.8, we ee that, or the partcular SF depcted, ( trctly ncreang, and ( S p everywhere S p trctly ncreang at all but the lowet prce (.e., trctly ncreang or p $44 MWh, approxmately. A the gure ugget, whether a partcular SF lope upward depend on the choen prce range or ntegraton a well 69

287 a on the ntal condton, or a gven parameter vector Θ (whereby cae. bae Θ=Θ, n th We now characterze more cloely the et o pont n the upper partton at whch the traectory S ( p uch that the SF ( S p are trctly ncreang. To do o, ome addtonal termnology wll be ueul. Namely, we dene a regon to be an open connected et o pont wthn any gven partton over whch the gn o both SF lope S ( p and S ( p are nvarant. From the denton o the 0 -loc n Table 7. n the prevou ubecton, t clear that wthn the gven partton, the 0 -loc conttute the boundare o the regon. In other word, wthn each partton, we wll have everal regon, demarcated by the 0 -loc and the -locu (recall Fgure 7.4 above. Whle we may urther ubdvde each partton o the phae pace nto regon, our ocu n th ubecton on the upper partton alone; we conder now the conttuent regon o th partton. To th end, Fgure 7.9 below rentroduce both o the 0 -loc (a hown, or example, n Fgure 7.4 above, emphazng va choce o ax cale the porton o thee loc lyng n the upper partton n a neghborhood o the orgn. The gure depct the -locu, a well; we may thnk o the -locu a the lower boundary (.e., n the p drecton o the upper partton. Content wth the prevou ubecton graphcal conventon, Fgure 7.9 portray the -locu n black, the 0 -locu a a trangular meh, and the 0 -locu n gray. 70

288 egon I egon II egon IV egon III FIGUE 7.9: THE UPPE PATITION COMPISES EGIONS I IV (SEE TEXT BELOW FO DETAILS, BOUNDED BY THE -LOCUS (IN BLACK, THE 0-LOCUS (A TIANGULA MESH, AND THE 0 -LOCUS (IN GAY Fgure 7.9 alo depct our regon n the upper partton, numbered I IV, delmted by a the varou loc and dened wth repect to the gn o the SF lope S ( p ollow: 7

289 egon I: S ( p > 0, S ( p > 0 egon II: S ( p < 0, S ( p > 0 egon III: S ( p < 0, S ( p < 0 egon IV: S ( p > 0, S ( p < 0 For mplcty, Fgure 7.9 doe not depct SF traectore n addton to the varou loc, and alo doe not attempt to depct or label the varou regon wthn the mddle or lower partton. A dcued above, the dtncton among the our regon labeled n Fgure 7.9 ollow rom the denton o the 0 -loc (recall, e.g., Table 7.. Snce we eek trctly ncreang SF or both rm, we can denty egon I rom the above denton a that porton o the phae pace that mot o nteret or the mult-ettlement SFE model. For expotory purpoe, however, we conder rt a varety o traectore havng, n general, both potvely- and negatvely-loped porton over derent prce range. Whle we have not oberved orward market SF traectore n the upper partton that cro the -locu, uch traectore can and do cro each o the 0 -loc, a we demontrate n th ubecton. I a traectory croe the 0 -locu (but not the 0 -locu at a partcular pont, or example, the gn o S ( p whle the gn o S ( p doe not change. 67 change at the crong pont, 67 Thee gn change are due, n turn, to change n the gn o the numerator and denomnator o the rato on the rght-hand de o eq. (7.40 and (7.4. In partcular, rom the denton o the varou loc, we have the ollowng gn relatonhp or any traectory: 7

290 The poblty that SF can have both potvely- and negatvely-loped ecton not novel n the SFE lterature. A example, we may cte two model o pot market SF competton or whch the equlbrum SF lope downward, at leat or ome prce. Frt, Klemperer and Meyer (989, 54 model generate a contnuum o SF a ther Fgure reproduced below a Fgure 7.0 depct. 68 In th contnuum o SF (ee Fgure 7.0 below, the ecton o the SF that ( le above the ( p, S = 0 locu, or ( le below the (, p S = locu, are decreang n prce p. Converely, the ecton o the SF lyng between thee two loc are ncreang n p. A noted n ubecton.5. above, a econd ntance o downward-lopng SF n the lterature Bolle (99, 99, who nd SF (n h Model B that are everywhere downward-lopng uncton o prce.. I the traectory croe the de o eq. (7.40 change gn locu, the numerator S ( p Q S 3 ( p 0 -locu, the numerator ( ( on the rght-hand I the traectory croe the S p Q S p on the rght-hand 3 de o eq. (7.4 change gn. 3. For two traectore on ether de o the -locu (and eparated only by th locu, the denomnator S ( p Q S ( p and S ( p Q S ( p on the rght-hand de o eq. (7.40 and (7.4 have oppote gn. Moreover, wthn the phae pace upper partton that we tudy here, the denomnator o the rato on the rght-hand de o eq. (7.40 and (7.4 are negatve and potve, repectvely; the gn o the numerator o thee rato then determne the gn o the lope S ( p and S ( p. 68 Aumng that the hock to the demand uncton n Klemperer and Meyer model ha nte upport. 73

291 p ( p, S = 0 ( p, S = S FIGUE 7.0: KLEMPEE AND MEYE S (989, 54 FIGUE DEPICTING THE, 0 p, S = LOCI (SOLID LINES, AND SUPPLY ( p S = AND ( FUNCTIONS (DASHED LINES SATISFYING THE DIFFEENTIAL EQUATION S ( p, S + D p p C S ( ( AND HAVING BOTH POSITIVELY- AND NEGATIVELY-SLOPED SECTIONS To provde addtonal nght nto the qualtatve behavor o oluton to the ODE ytem or the orward market, the ollowng ere o gure depct three derent example o traectore (or a varety o ntal condton nhabtng the upper partton. We how how thee traectore pa among the varou regon n th partton over the choen range o ntegraton, and examne how the path o each traectory correpond to change n the lope o each rm SF. A wth Klemperer and Meyer (989 and Bolle (99 or the pot market, the example preented below ndcate or the orward market that dependng on equlbrum electon and the prce doman condered nonnegatve contrant on SF lope could well be bndng n equlbrum. Th ugget, 74

292 urther, that uch SF lope contrant could be potentally mportant conderaton n market degn. Fgure 7. below portray an SF traectory n the prce range p [ 00,,500] $/MWh. Th traectory begn n egon IV at p = $00 MWh, pae through the 0- locu eparatng egon III and IV at approxmately p = $678 MWh, and end n egon III. Fgure 7. rotated o that the plane p = contant are perpendcular to the page, to acltate accurate readng o the prce p or pont along the SF traectory. 69 Fnally, note that egon I hdden on the other de o th gure, and not labeled. 69 An unortunate de eect o the perpectve o Fgure 7. that the q and q axe are 3 collnear n Fgure 7., although thee axe are, o coure, perpendcular n. To clary the perhap conung labelng o thee axe, the axe hare the lower lmt o x 0 4 MWh at the bottom o the gure. The q ax extend to the rght rom th pont, whle the q ax extend to the let (n each cae, to an upper lmt o x 0 4 MWh. 75

293 egon IV egon III egon II FIGUE 7.: AN SF TAJECTOY BEGINNING IN EGION IV AT p = $00 MWh, PASSING THOUGH THE 0-LOCUS AT APPOXIMATELY = $678 MWh, AND ENDING IN EGION III AT p = $,500 MWh p 76

294 Fgure 7. below plot the proecton o Fgure 7. SF traectory a the two rm SF n a common prce-quantty plane. 500 S (p S (p 000 p ($/MWh ( p S = 0 at p $678 MWh S (p (MWh FIGUE 7.: THE SFS S ( p AND ( TAJECTOY S ( p IN FIGUE 7. S p COESPONDING TO THE SF Note, n partcular, that the pont at whch the traectory n Fgure 7. pae through the 0 -locu concde wth the pont n Fgure 7. at whch ( the vertcal ( S ( p 0 S p bend back through = at p $678 MWh, and become downward-lopng. We preent another example portrayng a traectory on the other de o the upper partton. Fgure 7.3 below depct an SF traectory n the prce range 77

295 [ ] p 00,,500 $ MWh. Th traectory begn n egon II at p = $00 MWh, pae through the 0 -locu eparatng egon II and III at approxmately p = $,7 MWh, and end n egon III. Lke Fgure 7., Fgure 7.3 rotated o that the plane p = contant are perpendcular to the page, to acltate accurate readng o the prce p. 70 Fnally, note that egon I hdden on the other de o th gure, and not labeled. egon III egon II egon IV FIGUE 7.3: AN SF TAJECTOY BEGINNING IN EGION II AT p = $00 MWh, PASSING THOUGH THE 0 -LOCUS AT APPOXIMATELY = $,7 MWh, AND ENDING IN EGION III AT p = $,500 MWh p 70 Note 69 apple here, a well. 78

296 Fgure 7.4 below plot the proecton o Fgure 7.3 SF traectory a the two rm SF n a common prce-quantty plane. 500 S (p S (p 000 p ($/MWh ( p S = 0 at p $,7 MWh S (p (MWh FIGUE 7.4: THE SFS S ( p AND ( TAJECTOY S ( p IN FIGUE 7.3 S p COESPONDING TO THE SF The pont at whch the traectory n Fgure 7.3 pae through the 0 -locu concde wth the pont n Fgure 7.4 at whch ( ( S ( p 0 S p bend back through the vertcal = at p $,7 MWh, and become downward-lopng. Turnng now to th ubecton nal par o gure, Fgure 7.5 below portray an SF traectory n the prce range p [ ] 500,,500 $ MWh. Th traectory begn n 79

297 egon I at p = $500 MWh, rt pae through the 0 -locu eparatng egon I and II at p $,830 MWh, next pae through the 0 -locu eparatng egon II and III at p $,08 MWh, and end n egon III. egon III egon II egon IV egon I FIGUE 7.5: AN SF TAJECTOY BEGINNING IN EGION I AT p = $500 MWh, p $,830 MWh, PASSING 0 -LOCUS AT p $,08 MWh, AND ENDING IN PASSING THOUGH THE 0-LOCUS AT THOUGH THE EGION III AT p = $,500 MWh 80

298 Fgure 7.6 below plot the proecton o Fgure 7.5 SF traectory a the two rm SF n a common prce-quantty plane. 500 S (p S (p at p ( p S = 0 $,08 MWh 000 p ($/MWh 500 at p ( p S = 0 $,830 MWh S (p (MWh FIGUE 7.6: THE SFS S ( p AND ( TAJECTOY S ( p IN FIGUE 7.5 S p COESPONDING TO THE SF The pont at whch the traectory n Fgure 7.5 pae through the 0 - and 0 -loc concde wth the pont n Fgure 7.6 at whch S ( p and ( through the vertcal ( S ( p = at $,830 MWh 0 S p bend back p and S ( p = at 0 p $,08 MWh, repectvely, and become downward-lopng. Alo, each rm 8

299 econd-order condton or prot maxmzaton ated over the entre prce range o the SF n Fgure 7.6. The varou three-dmenonal gure above (.e., Fgure 7., Fgure 7.3, and Fgure 7.5, depctng the 0 -loc and the -locu are the analog to Klemperer and Meyer (989, 54 Fgure or the orward market n the (aymmetrc multettlement SFE model. Fgure o KM paper redrawn a Fgure 7.0 above depct varou SF olvng the derental equaton that characterze (ymmetrc pot market upply uncton n ther (ngle-market model, along wth the ( p, S = 0 and (, p S = loc analogou to the 0 -loc and the -locu dcued here. Fgure 7.0 uggetve o everal charactertc o KM SF. For example among other properte all SF pa through the orgn (a ngular pont wth a common lope, and any nonngular pont ha a unque SF pang through t. Such properte conttute the ba or KM characterzaton o ther SF, proo o extence, ymmetry, and unquene o SFE, and varou comparatve tatc reult. In our aymmetrc multettlement SFE model, on the other hand, we prove extence and unquene o oluton (or a gven ntal condton by appealng drectly to properte o (nonngular ytem o derental equaton. Becaue we cannot olve the ODE ytem (7.40 (7.4 explctly, we are only able n the preent work to conduct comparatve tatc analy numercally (ee ecton 7.6 below, rather than analytcally, a KM dd. In uture work, we may be able to characterze the SF traectore uch a thoe depcted n the gure o th ubecton more precely, and explot ther properte to prove addtonal reult o greater generalty than thoe documented here. Although the ODE ytem (7.40 (7.4 not analytcally tractable, oluton to the ytem lkely 8

300 poe ome properte that have not been explored here. For example, one conecture baed on numercal nvetgaton that, a p ncreae, the traectory enter egon III (o the upper partton n whch both SF are downward-lopng, the traectory reman n th regon orever. Another conecture that, or p ucently large, both SF are concave to the prce ax. Whle thee conecture are preently unproven, uture reearch could extend the catalog o uch regularte, make them more prece, and pobly prove them analytcally. The reult would be a rcher analytcal characterzaton o the connected et o traectore ( S p, whch mght be helpul n harpenng and extendng the generalty o the comparatve tatc and other reult preented n th work Prce relatonhp acro market We next nvetgate qualtatvely the relatonhp o orward market and expected pot market equlbrum prce or a range o orward market outcome. To do o, t wll be ueul to dene analytcally and graphcally an addtonal contruct or the orward market. Namely, denote a the arbtrage plane the et o orward market equlbrum pont ( ( S p, S p, p uch that the orward market prce p equal to the condtonal expectaton o the pot market prce, E ( p p, gven the prce p. To characterze th locu, we et p equal to E ( p p n eq. (7.9 and olve or p, yeldng the ollowng equaton o a plane n q -q -p pace the arbtrage plane, dened above: 83

301 p ω ( φ S ( p ( φ S ( p ω ( γ ω ν = ν, ν a + + b + a ( γωa σ σ ν. (

302 Fgure 7.7 below depct n q -q -p pace the arbtrage plane (n gray, an SF traectory (n the upper partton o the phae pace, and the -locu (n black n a neghborhood o the orgn. 7 FIGUE 7.7: THE ABITAGE PLANE (GAY SUFACE SEE EQ. (7.50, AN SF TAJECTOY (SOLID LINE, MAKED WITH O, AND THE -LOCUS (BLACK SUFACE IN A NEIGHBOHOOD OF THE OIGIN value 7 We plot the SF traectory or [ 3.95, 300] bae Θ p $/MWh and aumng bae cae parameter. For clarty, we do not plot the two 0 -loc n the above gure. 85

303 From the denton o the arbtrage plane n eq. (7.50, we may ner the ollowng relatonhp. For orward market equlbrum pont n q -q -p pace above the arbtrage plane, we have that p E( p p plane, we have that p E ( p p >, whle or uch pont below the arbtrage <. In the neghborhood o the ubet o q -q -p pace depcted n Fgure 7.7, we ee that the SF traectory everywhere above the -locu (content wth the traectory locaton n the upper partton. In contrat, the arbtrage plane everywhere below the -locu, tuatng t n the lower partton. Thee obervaton mply, urther, that the SF traectory n Fgure 7.7 le everywhere above the arbtrage plane, o that we conclude that the orward market equlbrum pont comprng th SF traectory are characterzed by the nequalty p > p p. (7.5 E ( Moreover, nequalty (7.5 apple along any SF traectory lyng n the upper partton that we may elect wthn the neghborhood o the orgn depcted n Fgure I nequalty (7.5 hold or all p along uch a traectory (a t wll n a moderatelyzed neghborhood o the orgn ee note 7, we have urther that E ( p E( p >. (7.5 The nequalte (7.5 and (7.5 ndcate that a rk-neutral, SF-bddng uppler n the mult-ettlement SFE model acng a downward-lopng orward market demand 7 At pont n the phae pace much more dtant rom the orgn (.e., or orward market quantte everal order o magntude larger, the arbtrage plane may cro one or both nappe o the - locu, and hence leave the lower partton o the phae pace. In uch a cae, the nequalty (7.5 may not hold at all pont along SF traectore n the upper partton. 86

304 uncton 73 wll not act o a to equalze the market prce n ether the ene E( p p p = or E( p E( p =. We may conclude that n th model, perect ntermarket prce arbtrage not a neceary mplcaton o prot maxmzaton. For uch arbtrage to obtan would requre, or example, that we ntroduce rk-neutral trader (wth no tradng lmt nto the model. 74 Inequalte (7.5 and (7.5 are natural reult or our bae cae traectory gven the aumpton o the mult-ettlement SFE model. To ee why, recall that n chapter 6, we aume that the repreentatve conumer rk avere. Accordngly, th conumer pay a rk premum to the (rk-neutral uppler n the orward market, leadng to orward market prce p n exce o condtonal expected pot market prce E ( p p. 73 And abtractng, a n the mult-ettlement SFE model, rom any rk-neutral trader. 74 Alternatvely, wthout ntroducng addtonal agent, t appear that permttng the repreentatve conumer to become progrevely le rk avere tend to equate E ( (and tend to make the orward market demand uncton (, 0 p p and D p ε approach the horzontal. That, prelmnary numercal mulaton or bae cae parameter value ugget that lm p E ( p p 0 + =. Whle content wth ntuton, explorng the generalty o th numercal λ 0 reult let or uture work. p 87

305 7.4.5 Equlbrum n the orward market To conclude the qualtatve graphcal analy o orward market competton n th ecton, we llutrate n Fgure 7.8 below the determnaton o the orward market * equlbrum prce p p ( ε 0 and demand. = va the nterecton o orward market aggregate upply p ($/MWh S (p S (p S Agg (p D (p,ε S x 0 4 (p (MWh FIGUE 7.8: FOWAD MAKET EQUILIBIUM FO EXAMPLE SUPPLY FUNCTIONS AND MEAN DEMAND SHOCK ε0 E ( ε0 EQUILIBIUM PICE p * ( p ε QUANTITY q S ( p = 5,488 MWh Agg Agg = 6,008 MWh, YIELDING AN 0 = $59.4 MWh AND AGGEGATE 88

306 Fgure 7.8 analogou to Fgure 5.3 or the pot market. In the gure above, aggregate orward market upply SAgg ( p nterect orward market demand (, 0 the mean orward market demand hock ε0 E ( ε0 correpondng to D p ε, gven bae Θ. Naturally, th nterecton dene the equlbrum or the orward market, at whch the equlbrum * prce p p ( ε 0 = $59.4 MWh and the aggregate quantty qagg SAgg ( p = 5,488 MWh Equlbrum oluton o the derental equaton ytem To conclude the dcuon o the qualtatve properte o ytem (7.40 (7.4, we nvetgate the extence o an equlbrum oluton to th derental equaton ytem. Frt, we dtnguh th new concept o an equlbrum oluton o a derental equaton ytem rom the noton o upply uncton equlbrum. ecall that Table 3. dened a mult-ettlement upply uncton equlbrum a equence o equlbrum (optmal SF { S (, ( p p ; } Σ, one or each market. 76 We alo mpoed the retrcton that thee SF mut be trctly ncreang n ther prce argument. Now, contrat upply uncton equlbra wth the concept o an equlbrum or teady-tate oluton o a derental equaton (DE ytem. For brevty, we reer to th concept a a DE equlbrum. Many applcaton o derental equaton to dynamc ytem ue tme a the ndependent 75 We choe the ntal quantte or the orward market SF n Fgure 7.8 o that thee SF lope upward, and alo to enure reaonable magntude or the orward market quantte, gven the mean orward market demand hock ε. Secton 7.6 preent a ytematc procedure or electng a partcular 0 par o orward market SF rom the connected et o SFE. 76 Where we later ound that Σ ( p ; Σ ( p ; q, q. 89

307 varable, rather than prce, a n the preent model. In tme-dependent problem, the ue o teady-tate a a ynonym or equlbrum relect the temporal nature o the concept o DE equlbrum. Namely, at a DE equlbrum o a tme-dependent problem, value o dependent varable are xed or all tme t begnnng wth the ntal tme. We may generalze th characterzaton o DE equlbrum or our problem o nteret a tatc problem n whch prce the ndependent varable. Namely, a DE equlbrum o the ytem (7.40 (7.4 or the orward market o the mult-ettlement SFE model 77 would aty S ( p 0 + = (7.53 or all p p,0, (aumng upward ntegraton rom an ntal prce,0 p. From the thrd component o the vector equaton (7.53, a DE equlbrum mut aty S ( p = ( or all p p,0, S p p. ecallng the denton 3 (, however, eq. (7.4 held that S ( p = or all 3 p. Equaton (7.4 thereby contradct eq. (7.54 and we conclude that the ytem (7.40 (7.4 ha no DE equlbra. Th reult not urprng, nce DE equlbra are not uually aocated wth non-autonomou equaton although they can occur (Jordan and Smth 999, 6. ecall 77 We need conder here only the orward market nce, n the mpled ane example or the p ; q, q Σ p ; q, q = 0 at all p (.e., a vertcal pot pot market, a uncton Σ ( uch that ( market SF ext only n the lmt a c. Thu, we conclude that there no DE equlbrum n the pot market or nte parameter value. 90

308 that the orgnal orm o the ytem (7.40 (7.4, eq. (7. and (7., wa ndeed a non-autonomou equaton, and convertng t to an autonomou ytem va eq. (7.9 doe not alter the preence (or abence, a the cae here o DE equlbra. We examne here, n pang, the extence o DE equlbra or the ytem (7.40 (7.4 nce uch equlbra and ther properte are a common component o the qualtatve analy o derental equaton. We emphaze that the nonextence o DE equlbra or our ytem nconequental or our purpoe. What o undamental nteret n the mult-ettlement SFE model, naturally, are the SF traectore and ther dependence on ntal condton and parameter value. Thee relatonhp are the ubect o comparatve tatc analy n ecton 7.6 below. Frt, however, n the ollowng ecton, we benchmark the dcrete Excel model to enure that t yeld reaonable numercal reult. 7.5 Benchmarkng the dcrete Excel model Th ecton decrbe how we benchmark the dcrete Excel model ung a repreentatve mult-ettlement market equlbrum. The purpoe o th benchmarkng procedure to agn value to certan parameter o the mult-ettlement SFE model that otherwe lack a plauble emprcal ba or quantcaton. Th procedure chooe thee parameter value uch that the mean equlbrum prce and quantte computed by the dcrete Excel model agree, to the extent poble, wth correpondng emprcally-baed reerence value rom the Calorna market. 78 In th way, the benchmarkng procedure 78 More undamentally, t the cae that not all parameter value Θ produce equlbra havng trctly ncreang SF n a ubet o nteret o q -q -p pace. Thereore, even our obectve doe not nvolve replcatng certan emprcal outcome, mply requrng that SF be trctly ncreang n a gven ubet o th pace place retrcton on the parameter vector Θ. 9

309 enhance the vermltude o oluton to the dcrete Excel model n a ene that we make more prece below. The benchmarkng procedure compre a lexcographc, two-tep herarchy. To ummarze th procedure, the rt benchmarkng tep produce numercally-computed expectaton o pot market prce and quantty that agree wth correpondng emprcal reerence value rom the Calorna market. The econd benchmarkng tep xe thee pot market expectaton rom the rt tep, and mlarly compute expectaton o orward market prce and quantty that agree a cloely a poble wth the correpondng emprcal reerence data rom Calorna. Both benchmarkng tep take a ther ba a veron o problem (7.43 rom ubecton 7.3. above. Each uch tep ental a revon o problem (7.43 n three repect:. Frt, we convert the parameter n problem (7.43 whoe value are to be determned to decon varable. That, we drop thee parameter rom the parameter vector Θ, yeldng a reduced parameter vector. We add thee ame parameter to the problem et o decon varable (.e., along wth the dcretzed SF ( S p, o that they become endogenou. 79 Subecton 7.5. and 7.5. below dcu the partcular parameter to be converted to decon varable n each benchmarkng tep. 79 In each o the optmzaton problem preented n ubecton 7.5. and 7.5. below (a well a n ecton 7.6 comparatve tatc analy, we ued the grd o orward market prce p = 0, 50, 500,,,750 $ MWh to compute the dcretzed SF S ( p. That, the dcretzed SF each cont o eleven ane egment connectng twelve prce-quantty par. Whle the dcrete Excel model permt the uer to adut both the (unorm tep ze and the range o prce the aorementoned grd o prce p yelded robut numercal reult n each ntance. p condered, 9

310 . We ue a the obectve uncton n problem (7.43 the mnmzaton (or each market, n turn o the um o quared proportonal devaton o expected prce and quantty rom the correpondng emprcal reerence value. The choce o obectve uncton conttute an equlbrum electon rule or electng a ngle orward market SF traectory (aumng a unque oluton or problem (7.43 rom the phae pace o SF traectore. 3. A approprate, we ntroduce addtonal contrant nto problem (7.43 that we call benchmarkng contrant. Thee addtonal contrant equate certan expected equlbrum prce and quantte computed va the dcrete Excel model wth correpondng emprcal reerence value. In the optmzaton problem that ollow, E ( and E ( Agg p denote the expected pot market prce q the expected aggregate (equlbrum pot market quantty. Thee expectaton account or both orward and pot market uncertanty; that, we compute thee expectaton wth repect to the tochatc parameter η and ν (ee ecton 6.5. We ue the dcrete Excel model to compute thee expectaton va dcrete approxmaton o the ont cumulatve dtrbuton uncton o thee parameter. greater detal. Subecton 7.5. and 7.5. below outlne thee two benchmarkng tep n 7.5. Benchmarkng tep (pot market ecatng problem (7.43 n accordance wth the dcuon o paragraph 3 above, we obtan a tep o the benchmarkng procedure the ollowng optmzaton problem: 93

311 mn ( S p ( dcretzed, 0, η 0, 0, ν 0 η σ ν σ ( E(, mean, mean empr Agg empr E p p q q +, mean, mean pempr q empr t.. Subgame-perect Nah equlbrum n Σ and S (0 t.. Parameter Θ \( η, σ η, ν, σν. (7.55 Problem (7.55 convert the parameter η, σ η, ν, and σ ν the mean and varance o the tochatc parameter η and ν n Θ to decon varable. 80 The obectve uncton o th problem the mnmzaton o the um o quared proportonal devaton o the expected pot market prce E ( quantty E ( Agg p and expected aggregate pot market q rom the correpondng emprcal reerence value p, mean and, q mean empr. 8 (0 The notaton \( η, ση, ν, σν empr Θ repreent the reduced parameter vector or problem ( The value choen or the element o th vector are, where poble, upported by emprcal data We make th choce o addtonal decon varable a the reult o expermentng wth varou ormulaton o the benchmarkng procedure. The peccaton o problem (7.55 yeld a eable oluton to th problem and, ultmately, reaonable bae cae value o all parameter, a dcued n ubecton below. 8 See Appendx F.. or detal on thee emprcal value. 8 That, Θ wth parameter η, 83 See Appendx F or detal. σ η, ν, and σ dropped. ν 94

312 ( ( ν Problem ( yeld optmal parameter value ( ( η, ( ( σ, ( ν ( η, and σ, whch we collect along wth other (xed parameter rom ( η ση ν σν (0 Θ \,,, n an ntermedate parameter vector produce optmal orward market SF ( ( ( Θ. 85 Th problem alo S p or rm =,. 86 Fnally, the optmzed obectve uncton value o problem (7.55 (.4e- approxmately zero, o that or practcal purpoe, we may conder the equalte (,, ( qagg q E mean empr procedure below. E mean p = p empr and = to hold. Th act wll be ueul n Step o the benchmarkng 7.5. Benchmarkng tep (orward market Agan recatng the general orm o problem (7.43, we obtan a tep o the benchmarkng procedure the ollowng optmzaton problem: 84 We olve problem (7.55 wthout ung automatc calng, one o the Excel Solver Solver Opton (Mcroot Corporaton 00 (n Excel, ee Tool Solver Opton. Scalng may be ueul n obtanng a eable oluton, partcularly when the underlyng matrce ued by the Excel Solver to repreent the optmzaton problem are poorly condtoned. 85 The upercrpt ( denote optmal value or benchmarkng tep (problem ( Optmal, o coure, only or problem (

313 mn ( S p ( dcretzed, 0, η 0, 0, ν 0, edem 0, λ 0 η σ ν σ ( E(, mean, mean empr Agg empr E p p q q +, mean, mean pempr q empr t.. Subgame-perect Nah equlbrum n Σ and S ( ( qagg t.. E p = p E = q, mean empr, mean empr ( η ση ν σν edem λ ( Parameter Θ \,,,,,. (7.56 Problem (7.56 related to problem (7.55 n our mportant way. Frt, (7.56 add two addtonal decon varable to thoe ued n (7.55, namely, the pot market demand elatcty e dem, and the repreentatve conumer CAA parameter λ. We ntroduce thee decon varable n problem (7.56 both to allow or maxmum lexblty n mprovng (7.56 obectve uncton, and becaue thee parameter have a rather tenuou emprcal ba, a Appendx F dcue. 87 Second, we ue value n ( Θ and ( ( S p rom problem (7.55 a ntal value or decon varable and or the ( parameter value \( η, ση, ν, σν, edem, λ Θ n problem (7.56. Thrd, problem (7.56 obectve uncton mnmze the um o quared proportonal devaton o expected prce E ( p and quantty E ( Agg q rom the correpondng emprcal reerence value p and, mean empr q or the orward market (rather than the pot market, a wa the, mean empr cae n problem ( Fourth and nally, we ntroduce the benchmarkng 87 ( Note that the only parameter n Θ let xed n problem (7.56 are the lope c and ntercept c o the rm margnal cot uncton See Appendx F.. or detal on thee emprcal value. 96

314 , contrant E ( = mean, and ( q q p p empr E mean Agg empr = n problem (7.56. ecallng that thee equalte held, eectvely, n the oluton to benchmarkng tep (problem (7.55, we may conder (7.56 a renement o (7.55, that, a renement n the drecton o a better t to the emprcal outcome n the orward market. At an optmal oluton to problem ( benchmarkng tep we have the optmal parameter value ( ( η, ( ( σ, ( ( η ν, ( ( σ, ( ( ν e, and ( ( dem λ, whch we collect along wth other (xed parameter rom ( η ση ν σν edem λ Θ \,,,,, n (another ntermedate parameter vector ( Problem (7.56 alo yeld optmal orward market SF ( ( S p or rm =,. ( Θ Dcuon The reult o the benchmarkng procedure outlned n th ecton nclude a vector o parameter value that we ue below a bae cae parameter value; that, we et ( Θ bae ( Θ =Θ. (7.57 Equaton (7.46 above gve value or the element o the reultng vector bae Θ, value whch are ntutvely reaonable and content wth a pror expectaton. In addton, the benchmarkng procedure yeld a correpondng et o (dcretzed orward market SF ( ( S p or ue a ntal condton n the comparatve tatc analy o ecton 7.6. Moreover, th procedure alo guarantee that the pot market benchmarkng 89 We olve problem (7.55 ung automatc calng n the Excel Solver (ee note

315 , mean, contrant E ( = and ( q q p p empr E mean Agg empr = rom problem (7.56 tll hold, eentally, n the bae cae problem analyzed n ubecton 7.6. below. 90 In th ene, the benchmarkng procedure enhance the vermltude o oluton to the dcrete Excel model. A noted at the outet o ecton 7.5, we may vew the benchmarkng procedure detaled n the oregong ubecton a a lexcographc approach to benchmarkng the model. Under th approach, we rt enure that the pot market benchmarkng contrant hold wth equalty. Then, or the orward market whle enorcng thee pot market contrant we eek the bet poble agreement between the model and tylzed realty. 7.6 Comparatve tatc analy Th ecton decrbe the comparatve tatc o a dcrete approxmaton to the ODE ytem (7.40 (7.4 n whch we nvetgate, n eect, the multaneou perturbaton o parameter and ntal condton or th ytem. Th analy ental, or each rm, a comparon o a bae cae SF (computed or bae cae parameter value wth a varety o tet cae SF, each correpondng to a certan parameter perturbaton. We may decompoe comparatve tatc analy o the mult-ettlement SFE model nto everal tep: 90 That, thee equalte hold to wthn the convergence crteron choen n the Excel Solver. 98

316 . Chooe a range o prce p, p (and a tep ze p over whch to olve a veron o problem (7.43, a derence equaton approxmaton to the orgnal ODE ytem (7.40 ( Fx two parameter vector, a bae cae vector and a perturbed tet cae vector. 3. Chooe an equlbrum electon rule, operatonalzed n problem (7.43 va the choce o obectve uncton, or electng a ngle orward market SF traectory rom the phae pace o SF traectore that olve th problem Solve problem (7.43 twce ung the choen obectve uncton, once or each parameter vector rom tep. The SF elected or each rm n th problem oluton wll, n general, der acro the bae cae and tet cae. Th mple, urther, that each rm ntal quantty wll alo typcally der acro the two cae. We may then compare each rm SF or the bae cae and the tet cae. In general, the drecton n whch a rm SF perturbed wll not be unorm acro all prce p p, p. That, we oberve not mply tranlaton o the SF, but alo rotaton and deormaton o thee uncton, leadng ater the perturbaton to hgher quantte 9 For contency, we chooe the ame grd o prce 7.5 benchmarkng procedure to dcretze the SF ( S p. p (ee note 79 a wa ued or ecton and S 9 ecall that problem (7.43 upper-level contrant Subgame-perect Nah equlbrum n mpoe the aumpton o the mpled ane example to olve the pot market problem. We requre no equlbrum electon procedure or the pot market under thee aumpton nce th ane equlbrum unque. Σ 99

317 at ome prce, and lower quantte at other prce. 93 The prmary ocu o th ecton comparatve tatc analy wll be to document and explan the oberved change n rm quantte under varou perturbaton to the ytem. Where helpul n developng ntuton, we alo dcu the eect on the SF lope. 94 The approach to comparatve tatc analy outlned above ncorporate both an equlbrum electon rule and a parameter perturbaton n the bae cae and tet cae problem. Th trategy combne two analytc technque or derental equaton that are uually treated eparately n the lterature: ( tablty analy, whch examne the eect o perturbaton o ntal condton, and ( tructural tablty analy, whch examne the eect o perturbaton o parameter. Fnally, nce we cannot olve the ODE ytem (7.40 (7.4 analytcally to obtan an explct expreon or the traectory ( S p, we mut conduct the comparatve tatc analy numercally rather than analytcally. Abent addtonal analytcal reult, 95 moreover, the comparatve tatc analy vald only locally, that, or a partcular bae cae parameter vector bae Θ. The outlne o th ecton a ollow. We ue the dcrete Excel model to compute the orward market SF or the bae cae parameter vector n ubecton 7.6., and then or varou perturbed parameter vector ( tet cae n ubecton Subect, o coure, to the contrant n problem (7.43 ncludng, n partcular, that both rm SF n both the bae and tet cae have non-negatve lope. 94 In general, there ext comparatve tatc eect on the SF hgher-order dervatve a well (curvature, etc., though t naturally more dcult to nd mple ntutve explanaton underlyng thee more ubtle eect. 95 The qualtatve analy o ubecton conclude wth ome conecture concernng more general properte o traectore S ( p. Further developng uch conecture, or example, may lead to more generally applcable comparatve tatc reult. 300

318 Subecton conclude, provdng ntutve nterpretaton o the oberved SF perturbaton Computaton o orward market SF: Bae cae problem We agan reormulate problem (7.43 to obtan the ollowng optmzaton problem (the bae cae problem or the comparatve tatc analy: mn ( S p ( dcretzed ( E(, mean, mean empr Agg empr E p p q q +, mean, mean pempr q empr t.. Subgame-perect Nah equlbrum n Σ and S t.. q 0 bae Parameter Θ. (7.58 Comparng problem (7.58 wth problem (7.56, we note our mportant dtncton. Frt, we drop all parameter rom (7.58 lt o decon varable, retanng a decon varable only the dcretzed orward market SF ( S p, =,. Second, problem (7.58 obectve uncton mnmze the um o quared proportonal devaton o expected prce and quantty rom the correpondng emprcal reerence value or the pot market. Thrd, becaue o problem (7.58 reved obectve uncton, we drop the, mean, contrant E ( = and ( q q p p empr E mean Agg empr = ued n (7.56. In addton, problem (7.58 xe the bae cae parameter vector a bae ( Θ =Θ (recallng eq. (7.57 rom problem (7.56. Fourth and nally, we ntroduce the contrant q 0 (or all tate o the world to preclude negatve pot market quantte or uppler rom arng n the model. Whle the contrant q 0 do not bnd n the optmal oluton to the bae cae problem (7.58, we cannot, ex ante, rule out the poblty that they wll bnd n one or 30

319 more o the tet cae condered below. For contency, we nclude thee (non-bndng contrant here n the bae cae problem. 96 The act that q 0 doe not bnd n the bae cae problem relect the potve orward market quantte q that reult, n equlbrum, rom the bae cae SF S ( p (ee Fgure 7.9 below elected va problem (7.58. ecallng the geometry o the pot market depcted n Fgure 5.3, uch quantte q > 0 tranlate rm pot market SF (and hence the aggregate pot market SF to the rght, ncreang the lkelhood that pot market quantte q are potve n equlbrum., mean, Note that becaue the contrant E ( = and ( q q p p empr E mean Agg empr = were ated n (7.56, problem (7.58 obectve uncton attan a mnmum o eentally zero (gven Excel convergence crteron. A a conequence, problem (7.56 and (7.58 have the ame optmal oluton. 97 For the orward market, n contrat, we nd the 96 ecall rom ubecton 3..5 that we dened each rm pot market SF Σ ( p ; q, q mplcty, a havng a range o, that,, or 3 Σ :. That ubecton contructon o th SF q =Σ ( p ; q, q reled on evaluatng rm margnal cot uncton C ( q quantty q reultng rom ( ;, Σ p q q. We dened the uncton C ( q ubecton 3..8, however, o optmalty o the uncton ( ;, p q q Σ ( p ; q, q = q < 0. Thu, contranng at each equlbrum only or q 0 (ee Σ not aured q to be non-negatve n problem (7.58 (and n problem (7.6 below enure the optmalty o the pot market quantte. The contrant q 0 mply that the duopoly uppler are precluded rom beng net demander n the pot market n th chapter numercal example. elaxng the contrant q 0 poble, n prncple, at the cot o ntroducng ome addtonal tructure nto the model. Namely, abent thee contrant, a uppler could become a net demander, and vce vera. Makng uch a cenaro operatonal computatonally would ental, or example, pecyng a utlty uncton or conumpton on the part o uppler, and converely, pecyng an electrcty producton technology or conumer. 97 We olve problem (7.58 wthout ung automatc calng n the Excel Solver (ee note

320 ollowng dcrepance between expected orward market prce and quantty rom the bae cae problem (7.58, on the one hand, and the correpondng orward market emprcal reerence value, on the other:, ( mean E p p empr = = $ MWh (7.59, ( qagg q mean E = 4,983 4, 033 = 950 MWh. (7.60 empr The agreement between E ( p and, mean p empr poor, but that between E ( Agg q and q reaonably cloe n relatve term: thee quantte der by only about 4%., mean empr We may vew the derence noted n eq. (7.59 and (7.60 above a a meaure o the devaton o the mult-ettlement SFE model rom the actual market. 303

321 Fgure 7.9 below plot the dcretzed orward market SF olvng problem (7.58, the comparatve tatc bae cae. Bae Cae orward market SF or rm and Prce ($/MWh S(p S(p Quantty (MWh FIGUE 7.9: BASE CASE FOWAD MAKET SUPPLY FUNCTIONS FO COMPAATIVE STATICS ANALYSIS The SF depcted n Fgure 7.9 or each rm are everywhere trctly ncreang 98 and moreover, yeld potve orward market quantte over the range o prce p depcted there. That, both uppler take hort orward market poton q > 0 n the bae 98 Content wth the contrant mplct n problem (7.58 that ( S p be trctly ncreang. A Fgure 7.9 ugget, th contrant bndng or each rm at the hghet prce level. 304

322 cae. 99 The trctly ncreang SF n Fgure 7.9 correpond to an SF traectory that le entrely n egon I o the phae pace upper partton (ee, e.g., Fgure 7.5 or p < $,830 MWh. Fnally, content wth ntuton, Fgure 7.9 how that the lowcot rm, rm, more aggreve n the orward market, bddng a larger orward market quantty at all prce p Computaton o orward market SF: Tet cae problem Let θ repreent an arbtrary element o Θ. We dene the tet cae or the parameter θ a a oluton o the mult-ettlement SFE model (a approxmated by the dcrete Excel model n whch the parameter θ and only that element perturbed rom t value n the bae cae vector bae Θ (recall eq. (7.46. We denote the reultng tet cae vector a the parameter vector tet Θ θ or the perturbaton o the parameter θ. The comparatve tatc analy decrbed below cont o perturbng each parameter θ n Θ rom t value n bae Θ wth a multplcatve hock o We dd o one parameter at a tme to obtan ten derent tet vector tet Θ θ. The tet cae problem or parameter θ agan rele on an optmzaton problem havng the general orm o problem (7.43. Begnnng wth problem (7.58, we replace bae Θ wth tet Θ θ to obtan a amly o tet cae problem, one or each parameter θ : 99 Allaz (99, 99. obervaton apropo, namely, that whether uppler are hort or long n the orward market entve, n partcular, to the type o conectural varaton a well a uppler atttude toward rk. 300 bae tet That, the parameter θ change by 0.% between the vector Θ and Θ. Through θ expermentaton, we nd that th mall multplcatve hock large enough to avod purou numercal reult, but mall enough to nterpret the change n the parameter a a margnal change. See Appendx E.4 or urther detal. 305

323 mn ( S p ( dcretzed ( E(, mean, mean empr Agg empr E p p q q +, mean, mean pempr q empr t.. Subgame-perect Nah equlbrum n Σ and S t.. q 0 tet Parameter Θ. θ (7.6 The only dtncton between the bae cae problem (7.58 and the tet cae problem (7.6 the perturbaton o the parameter θ, that, the ue o bae Θ veru tet Θ θ., Note that whle the two pot market benchmarkng contrant ( and (, E q mean Agg qempr E p = p empr mean = happen to hold n the bae cae problem (7.58, thee contrant are not explctly mpoed, ether n the bae cae problem (7.58 or the tet cae problem (7.6. In the oluton to the varou tet cae or arbtrary parameter perturbaton, thee contrant wll not necearly hold. For ucently mall parameter perturbaton, however, we would expect the obectve uncton n the problem (7.6 to be cloe to zero; th ndeed the cae eult and nterpretaton Table 7. below report the eect o perturbng each o the ten comparatve tatc parameter on rm orward market quantte, that, on the SF S ( p and S ( p computed rom problem ( Appendx E.4 report the numercal reult or the dcretzed SF S ( p and ( S p n the bae cae and tet cae. Thee reult orm the ba o the qualtatve eect reported n Table 7.. In the thrd and ourth column 30 We olve problem (7.6 wthout ung automatc calng n the Excel Solver (ee note

324 o the table, we ue the ymbol + and to denote an ncreae and a decreae, repectvely, n a rm quantty attrbutable to the perturbaton n queton. Except n two cae, the drecton that the SF ht n repone to a perturbaton monotone, that, unorm acro prce p n the choen range o ntegraton. The two exceptonal cae n whch the perturbaton o one or both SF not monotone are ( the change n ( q = S p due to perturbaton o 0 c, and ( the change n q S ( p = due to perturbaton o ν. The qualtatve eect n both o thee cae are an ncreae n the ndcated rm quantty at hgher prce and a decreae at lower prce, whch we ndcate n Table 7. wth the ymbol ±. In contrat, n each cae examned, the amount that the SF ht n repone to parameter perturbaton doe depend on prce, a evdent rom the quantte n the column o Table E. labeled (ee Appendx E.4. In general, thereore, thee parameter varaton change the lope a well a hgher dervatve o the SF. The preent dcuon o comparatve tatc eect largely abtract, however, rom uch hgher-order change n the SF, ocung ntead on the change n quantte ummarzed n Table 7. below. 307

325 TABLE 7.: COMPAATIVE STATICS ANALYSIS: EFFECTS OF PAAMETE PETUBATIONS ON FIMS QUANTITIES SUPPLIED IN THE FOWAD MAKET, q = S ( p AND q = S ( p Parameter Eect on θ a Decrpton b q = S p ( c Eect on q = S p c Prce-ax ntercept o rm 0 margnal cot uncton ± + c Prce-ax ntercept o rm 0 margnal cot uncton + + c Slope o rm margnal cot uncton c Slope o rm margnal cot uncton e Spot market demand elatcty + + dem η σ η ν σ ν λ Mean o repreentatve conumer gnal η Varance o repreentatve conumer gnal η Mean o pot market noe parameter ν Varance o pot market noe parameter ν epreentatve conumer parameter o contant abolute rk averon (CAA ± ( Note: a bae See eq. (7.46 or the bae cae value Θ o each parameter θ. b ecall rom eq. (7.46 that rm a low-cot rm and rm a hgh-cot rm. c The ymbol + and denote an ncreae and a decreae, repectvely, n a rm quantty (at all prce p attrbutable to the perturbaton under tudy. The ymbol ± denote an ncreae n the rm quantty at hgher prce p and a decreae n th quantty at lower prce. c 308

326 In the bulleted paragraph that ollow, we provde ntuton underlyng the comparatve tatc eect documented n Table 7. above. 30 Th dcuon rele upon everal properte o the mult-ettlement SFE model and t oluton. Among thee properte are the elatcte o upply uncton and demand uncton n each market, the aymmetry o the two rm, endogenety o orward market demand, and the rk preerence o the market partcpant. We appeal repeatedly to thee eature o the model n the ollowng dcuon. An ncreae n rm margnal cot uncton ntercept c 0 ha a prce-dependent eect on S ( p. Namely, ( q at hgher prce ( S p rotate clockwe, mplyng an ncreae n p, and a decreae n q at lower prce. The eect on S p, n contrat, monotone, htng th orward market SF to the rght. Conder rt rm. We begn by accountng or the rghtward ht n S ( p at hgher (ndeed, mot value o p, and then conder why the drecton o th ht may be revered or ucently low p. From the geometry o the pot market examned n chapter 5, an ncreae n c 0 wll ht Σ upward, leadng to hgher equlbrum prce p and lower quantte q or every realzaton o ε. Thee change due to ncreaed c 0 mply hgher pont elatcte o upply, o demand, and hence o redual demand. A a conequence, rm can ncreae t 30 Table E. o Appendx E.4 alo nclude numercal comparatve tatc reult or the eect o parameter varaton namely, or the parameter c and γ on SF lope β. Thee comparatve tatc eect are among reult reported n ecton 5.3 above or the pot market. 309

327 pot market quantty q wth proportonally lttle penalty n term o lower Under thee crcumtance, t tend to be protable or rm to ncreae t pot market quantty. One mean by whch t may do o to ncreae t orward market quantty q, nce q ht Σ to the rght, n equlbrum. Frm p. accomplhe th ncreae at leat or hgher value o p by htng ( S p to the rght. Next, we examne why the drecton o rm ncentve a ketched above mght be revered or ucently low p, caung ntead a letward ht n ( S p at uch prce (and leadng, n eect, to the clockwe rotaton o ( S p. The ncreae n 0 c mple a unorm ncreae n rm margnal cot, makng t a le aggreve compettor, maneted n part by the aorementoned upward ht n Σ. I, n addton, orward market demand weak, then the equlbrum orward market prce p wll be low. Th mple, n turn, that E ( decreang q to upport p alo low. Under thee crcumtance, return to rm rom p could outwegh the propect or ncreaed expected pot market prot decrbed n the oregong paragraph. Accordngly, at ucently low value o p, rm ht t orward market SF to the let. 303 S p. The net eect the clockwe rotaton o ( 303 Speccally, rom the c 0 cenaro n Table E. o Appendx E.4, th the cae or p $50 MWh. 30

328 Conder now the reacton o rm to the ncreae n c 0. Frm doe not change t pot market SF Σ drectly n repone to rm cot ncreae. 304 The upward ht n Σ, however, ht rm pot market redual demand uncton D upward, a well, ncreang or each ε the equlbrum prce that rm ace. In repone, t protable or rm to ncreae t equlbrum quantty q. Frm can do th (analogouly to the argument above or rm p by ncreang q. The rm doe o, n turn, by htng t orward market SF ( S p to the rght. Now compare the relatve repone o the two rm to the ncreae n c 0. Note rom Table E. n Appendx E.4 that rm ncreae t orward market quantty by a leer amount at each prce than doe rm, o that the overall eect o ncreang c 0 or rm to cede ome market hare to rm n both market. Fnally, we conder why ncreaed c 0 mght caue rm to decreae ( S p at low p, whle rm orward market SF ( S p, n contrat, ncreae over the entre range o p condered here. One conecture are, naturally, rom the aymmetry n rm margnal cot uncton. Snce rm the hgh cot rm, we have that Σ teeper than Σ. Th derental n the lope o the pot market SF mean that the lope o rm pot market redual 304 Though Σ doe ht to the rght wth the ncreae n q that we decrbe below, n accordance wth the analy o chapter 5. 3

329 demand uncton D have the oppote relatonhp. That, D teeper than D, and thu the magntude o rm redual demand elatcty tend to be greater than that or rm. The relatvely nelatc uncton D mple that rm more lkely than rm to prot rom decreang t pot market quantty, thereby drvng up the equlbrum prce p. In ome tate o the world namely, at low p, a argued above, where orward market margnal revenue are relatvely low t protable or rm to do ut th by decreang ( q = S p, thu htng Σ to the let. An ncreae n rm margnal cot uncton ntercept c 0 ht both rm orward market SF to the rght. For each rm, the argument here analogou to that or rm n the above dcuon regardng the eect o ncreaed c 0. Other thng equal, the ncreae n c 0 ncreae p. Both rm have an ncentve to prot rom ncreaed pot market prce by ncreang equlbrum quantte ncreang t orward market quantty ther orward market SF ( S p to the rght. q. Each rm can do th by q. The rm do o, n turn, by htng Smlar to the argument or c 0 above, Table E. n Appendx E.4 ndcate that rm ncreae t orward market quantty by a leer amount at each prce than doe rm. The overall eect o ncreang c 0, thereore, or rm to cede ome market hare to rm n both market. 3

330 An ncreae n rm margnal cot uncton lope c ht both rm orward market SF to the let. The eect o the ncreae n c nclude decreae n both β and β, that, teeper pot market SF Σ. Steeper SF Σ n eect, a counter-clockwe rotaton o thee uncton Σ are le elatc, and lead alo to le elatc pot market redual demand uncton. Such low elatcte n the pot market tend to make ncreae n the expected pot market prce E ( p protable. 305 In th cenaro, le elatc SF Σ mply that t would be protable or rm to decreae ther orward market quantte va a letward ht n ( S p. Th becaue even only a mall decreae n q wll drve E ( p markedly hgher, wth lttle change n q and relatvely lttle acrce n orward market revenue. An ncreae n rm margnal cot uncton lope c ht both rm orward market SF to the let. The eect o the ncreae n c are analogou to thoe or ncreaed c, dcued above. That, le elatc SF Σ mply that t would be protable or rm to decreae ther orward market quantte va a letward ht n ( S p. ncrement n 305 Note that wth the rotaton o the uncton c ncreae wth the ntal value o E ( p. Σ, the ze o the prce ncreae or a gven 33

331 An ncreae n the magntude o the pot market demand elatcty rm orward market SF to the rght. A the pot market demand elatcty e dem ht both e dem ncreae n magntude, both rm wll ace a lower penalty n the expected pot market prce E ( p rom expanon o ther repectve pot market output. Th change ncreae the elatcty o each rm pot market redual demand uncton, mplyng that greater expected pot market quantte are now protable. Gven that, or each rm, the rght, t optmal or rm to ncreae Th mple that ( S p ht to the rght. q ht q n repone to the ncreae n Σ to e dem. An ncreae n the mean η o the repreentatve conumer gnal ht both rm orward market SF to the rght. The parameter η doe not appear n the rm orward market equlbrum optmalty condton, eq. (7. and (7.. A a conequence, a hock to η whle holdng contant the ntal condton or the SF ( S p leave thee SF unaected. 306 Note, however, that an ncreae n η doe aect the uncondtonal expectaton o pot market prce E ( p and quantty E ( Agg q n the obectve uncton o the bae cae problem (7.58 and the tet cae problem (7.6. In th way, the equlbrum electon algorthm n thee problem depend on η 306 Th mut be the cae gven that we derved the optmalty condton or the orward market problem condtonal on p and η. Condtonng on the realzaton η render the oluton nvarant to change n η dtrbuton. 34

332 dtrbuton; n partcular, the equlbrum elected by th problem vare wth η. The mportant general reult here that apart rom equlbrum electon conderaton the orward market SF are ndependent o the dtrbuton o the gnal η. To undertand the ntuton behnd the rghtward ht n the SF oberved or an ncreae n η, begn by recallng the mple addtve relatonhp ε η ν = + among the mean o the tochatc parameter (ee eq. (6.55. From th equalty, an ncreae n η (holdng contant, or the moment, the orward and pot market SF ncreae both the expected pot market prce E ( the expected aggregate pot market quantty E ( q Agg nelatc pot market demand uncton (, E ( p much greater than that n E ( q Agg p and. Due, however, to the D p ε, the proportonal change 307 n, whch reman approxmately contant (and can hence be neglected n th dcuon. I we now olve the tet cae problem (7.6 gven the ncreae n η, the SF ( S p change o a to mnmze the obectve uncton o th problem. To mnmze th uncton, E ( p mut decreae to oet the ncreae n E ( occur, a noted above. To eect th decreae n E ( p p whch would otherwe, orward market S p. quantte mut ncreae, correpondng to rghtward ht n the SF ( 307 It the proportonal change n E ( p or E ( Agg o problem (7.58 and ( q that relevant or the obectve uncton

333 An ncreae n the varance σ η o the repreentatve conumer gnal ht both rm orward market SF to the rght. The ntuton n th cae very mlar to that n the precedng cae nvetgatng the eect o an ncreae n η. That, lke η, the parameter σ η doe not appear n the rm orward market equlbrum optmalty condton, eq. (7. and (7.. A a reult, a hock to σ η whle holdng contant the ntal condton or the SF ( S p leave thee SF unaected. A wth η, the ncreae n σ η doe aect the uncondtonal expectaton o pot market prce E ( p and quantty E ( Agg q n the obectve uncton o the bae cae problem (7.58 and the tet cae problem (7.6, o that the equlbrum elected by thee problem vare wth σ η. We may how numercally that an ncreae n σ η (holdng contant, at rt, the orward and pot market SF ncreae the uncondtonal expected pot market prce E ( p and decreae the uncondtonal expected pot market quantty E ( q Agg. Due to the nelatc pot market demand uncton (, D p ε, the proportonal change n E ( p agan much greater than that n E ( Agg q, whch reman approxmately contant (and can agan be neglected n th dcuon. Solvng the tet cae problem (7.6 gven the ncreae n σ η, the SF ( S p change o a to mnmze the obectve uncton o th problem. To mnmze th uncton, E ( p mut decreae to oet the ncreae n E ( p 36

334 whch would otherwe occur, a noted above. To eect th decreae n E ( p, orward market quantte mut ncreae, correpondng to rghtward ht n the SF ( S p. An ncreae n the mean ν o the pot market noe parameter ha a monotone eect on ( S p, htng th orward market SF to the rght. The eect on S ( p, n contrat, prce-dependent. Namely, ( S p rotate clockwe, mplyng an ncreae n q at hgher prce p, and a decreae n q at lower prce. Below, we rt explan the eect predomnant or both rm, at mot prce o ncreang q S ( p =, and then addre the queton o rm dtnct orward market behavor at low p. Snce ε η ν = +, the ncreae n ν ncreae expected pot market ( demand, htng the uncton E (, D p ε to the rght. Moreover, we may how numercally that the ncreae n ν ht the expected orward market ( demand uncton E (, 0 D p ε to the let at hgher (and ndeed, mot prce p, and to the rght at ucently low p (.e., at p $50 MWh, eectvely ( ( rotatng E D ( p, ε 0 counterclockwe. Thee change n E D ( p, ε 0 make th uncton more elatc and decreae the expected orward market prce E ( p. Content wth thee change, the ( S p alo become more elatc. 37

335 Moreover, or any realzaton o 0 ε, the uncton D ( p, ε 0 and ( negatvely related (ceter parbu, and thu the ( S p are S p tend to ht to the rght, n oppoton to the ht n D ( p, ε 0. The aorementoned ht n S ( p and (, 0 D p ε mply that the orward market equlbrum move toward the elatc range o both o thee uncton (.e., to lower value o the equlbrum orward market prce p. A a conequence, uppler may ncreae ther orward market quantte q wth lttle downward preure on p. Forward market revenue generally ncreae wth uch an ncreae n q, and thu a rghtward ht n ( S p tend to be protable or each uppler. Conder now rm dtnct reacton at low orward market prce p to ncreaed ν. ecall that rm and have aymmetrc cot uncton. A the hgher cot rm, rm a le aggreve compettor than rm. To be protable, rm requre a hgher equlbrum prce (n ether market than doe rm. When rm put downward preure on an already low orward market prce p by ncreang q, rm optmal repone may be to decreae q to upport p (and alo p. In ome tate o the world namely, at low p, where rm orward market margnal revenue are relatvely low t protable or rm to repond n exactly th way by htng ( S p to the let. Fnally, we compare the relatve repone o the two rm to the ncreae n ν. Table E. n Appendx E.4 ndcate that rm ncreae t orward 38

336 market quantty by a greater amount at each prce than doe rm. Accordngly, the overall eect on orward market competton o ncreang ν or rm to cede ome market hare to rm n the orward market. An ncreae n the varance σ ν o the pot market noe parameter ht both rm orward market SF to the let. The two uppler are rk neutral, and care only about an ncreae n σ ν through t eect on demand and on expected pot market prce (ee eq. (7.8 and (7.9. A rk-avere conumer, on the other hand, doe repond drectly to the change n σ ν, and a change n the orward market demand uncton wll aect the multaneouly-determned SF ( ν S p. By the above reaonng, t ueul to begn by conderng the eect o σ on orward market demand (, 0 D p ε. We can how numercally at bae cae parameter value that, or a gven realzaton o ε 0, ncreang σ ν ht (, 0 D p ε to the rght and make th demand uncton le elatc at all prce. The ntuton underlyng thee eect a ollow. Snce σ η held contant n th cenaro, the poted ncreae n σ ν ncreae the relatve rk o the pot market. Th change, n turn, lead a rk-avere conumer, ceter parbu, to reduce her expoure to the pot market prce. Accordngly, the conumer then demand hgher orward market quantte, and orward market demand become le prce-entve. 39

337 Content wth thee change, the ( Moreover, becaue D ( p, ε 0 and ( S p alo become le elatc. 308 S p are negatvely related (ceter parbu, the S ( p ht to the let. The aorementoned ht n ( (, 0 S p and D p ε mply that the orward market equlbrum move toward the nelatc range o both o thee uncton, and the equlbrum orward market prce p drven up. When both ( S p and (, 0 D p ε are nelatc, decreang ( S p ncreae markedly the equlbrum orward market prce p. Th change n ( hence the letward ht n ( S p ncreae rm orward market revenue, and S p protable or each uppler. An ncreae n the repreentatve conumer CAA parameter λ ht both rm orward market SF to the let. A n the above analy or hock to the parameter ν and σ ν, t ueul to begn analy o th cenaro by conderng the eect o λ on orward market demand (, 0 D p ε. Increaed λ mple that the repreentatve conumer more entve to rk. Snce rk n th problem may be proxed by σ ν, the eect o ncreaed entvty to σ ν qualtatvely equvalent to the eect o ncreaed σ ν (wth contant λ, analyzed above. 308 Except perhap at the hghet prce p. 30

338 Accordngly, we can how numercally at bae cae parameter value that, or a gven realzaton o ε 0, ncreang λ ht (, 0 D p ε to the rght and make th demand uncton le elatc at all prce. The ntuton here that an ncreangly rk-avere conumer demand hgher orward market quantte, and that orward market demand become le prce-entve a conumer rk averon ncreae. Content wth thee change, becaue (, 0 S ( p are negatvely related (ceter parbu, the ( The aorementoned ht n S ( p and (, 0 D p ε and S p ht to the let. D p ε mply that the orward market equlbrum move toward the nelatc range o both o thee uncton, and the equlbrum orward market prce p drven up. When both S ( p and D ( p, ε 0 are nelatc, decreang ( S p ncreae markedly the equlbrum orward market prce p. Th change n ( S p ncreae rm orward market revenue, and hence the letward ht n ( each uppler. S p protable or A a unyng ramework or undertandng the comparatve tatc reult documented n th ubecton, we may ocu on the eect o parameter hock on the elatcty o redual demand uncton n each market at the repectve equlbrum pont. From the qualtatve analy o th ubecton, we may conclude that the elatcty o pot market redual demand ncreae or ncreae n c 0 and e dem, whle th elatcty decreae or ncreae n c. Smlarly, the elatcty o orward market 3

339 redual demand generally ncreae or ncreae n η, σ η, and ν, whle th elatcty decreae or ncreae n σ ν and λ. Parameter change that ncreae the elatcty o redual demand n ether the orward or pot market tend, n general, to make rm more aggreve n the orward market n that they bd hgher quantte at each prce. That, rghtward ht n ( S p are the reult o uch change. The convere true or parameter change that decreae the elatcty o redual demand n S p. Th ether market. In other word, uch change caue letward ht n ( behavor content wth ntuton regardng a prot-maxmzng rm bet repone to uch hock. 7.7 Comparon o expected aggregate welare under alternatve behavoral aumpton and market archtecture We conclude th chapter by comparng expected aggregate welare or the multettlement SFE model wth that obtaned rom model employng alternatve behavoral aumpton and market archtecture. In partcular, we compare the mult-ettlement SFE model to two alternatve ngle-market model: 309. Sngle-market SFE: We aume away the orward market, and aume urther (a n the mult-ettlement SFE model that rm bd ane SF n the pot market Lke the mult-ettlement SFE model, both alternatve model aume duopoly uppler. 30 Th the cenaro that Klemperer and Meyer (989 examne. 3

340 . Perect competton (ngle market: Agan, we aume away the orward market, and moreover, aume that rm behave compettvely, bddng ther margnal cot uncton n place o upply uncton n the pot market. To compute a welare meaure or the mult-ettlement SFE model, we aume a rk-neutral ocal planner who aee welare ex ante under uncertanty ung the mathematcal expectaton o a utltaran ocal welare uncton. In the partal equlbrum ramework nvoked here, only electrcty produced and conumed (apart rom the numerare good m. Thereore, n ether the mult-ettlement SFE model or the two alternatve model noted above, expected aggregate welare E ( Agg component: W cont o two. the expected utlty E φ ( x o the repreentatve conumer conumpton o amenty x (produced ung electrcty a an nput, recallng eq. (6. and (6. and. the expected total cot o producton E C( q o the equlbrum quantty o electrcty q ued by, where E ( E ( E q C q C q C ( 0 q dq = = = = and eq. (5. gve each rm ane margnal cot uncton C ( q E ( Agg W the derence o the utlty and total cot term above, that, 3. Algebracally, 3 In eq. (7.6, we compute expectaton wth repect to both pot and orward market ource o uncertanty va the dcrete Excel model (ee ubecton

341 E E E 0 = ( ( q WAgg = φ x C ( q dq. (7.6 By denton, expected aggregate welare n eq. (7.6 doe not account or traner due, n partcular, to orward market actvty between conumer and producer. I uch dtrbutonal eect are alo o nteret to polcy maker, t traghtorward, or example, to ue the preent model to compute moment o the dtrbuton o conumer orward market payment to uppler. E ( Agg In Table 7.3 below, we ue eq. (7.6 to compute expected aggregate welare W or the mult-ettlement SFE model and the two alternatve model noted above. TABLE 7.3: EXPECTED AGGEGATE WELFAE E ( Agg W FO THE MULTI- SETTLEMENT SFE MODEL AND ALTENATIVE MODELS Model a Expected aggregate welare ($ b Percentage o expected aggregate welare n perectly compettve model Mult-ettlement SFE 30, % Sngle-market SFE 94, % Perect competton (ngle market Note: 39, % a bae Each model aume bae cae parameter Θ rom eq. (7.46. b We compute expected aggregate welare aumng the ollowng value or the parameter o the, φ : a =, 0 repreentatve conumer amenty producton uncton ( a = 40, a = 0.4, and b 5 (recall that b endogenou to the lope q T and utlty uncton ( x γ o the pot market demand uncton; ee ecton 6.4 or detal. The relatve welare rankng o the varou model doe not change, however, or alternatve choce o thee parameter. A ntuton would ugget, Table 7.3 ndcate that the perectly compettve cenaro ha the hghet value o expected aggregate welare. The mult-ettlement SFE model ha the next hghet gure or expected aggregate welare, and the ngle-market SFE the mallet. Comparng the mult-ettlement SFE model wth the ngle-market 34

342 SFE model, we ee that a we would expect or a one-hot equlbrum analy ntroducng a orward market ha a welare-enhancng eect. 3 Namely, expected aggregate welare or the mult-ettlement SFE model exceed that or the ngle-market SFE model by $ (.63%. ecall rom eq. (7.46 or bae Θ that the pot market demand uncton underlyng Table 7.3 cenaro nearly perectly nelatc (.e., e = 5.95e-5. I th uncton were more elatc, then the devaton o expected dem aggregate welare between each SFE cenaro n Table 7.3, on the one hand, and perect competton, on the other, would be greater. Fnally, we note that the welare-enhancng property o orward market content wth prevou lterature on mult-ettlement market revewed n chapter, n partcular, Allaz (987, Allaz and Vla (993, Powell (993, Green (999a, and Kamat and Oren (00. 3 Th concluon doe not necearly apply n a repeated game ettng. 35

343 [M]onopoly, n all t orm, the taxaton o the ndutrou or the upport o ndolence, not o plunder. John Stuart Mll, Prncple o Economy [I]ndutre der one rom the other, and the optmal mx o nttutonal arrangement or any one o them cannot be decded on the ba o deology alone. The central nttutonal ue o publc utlty regulaton reman... ndng the bet poble mx o nevtably mperect regulaton and nevtably mperect competton. Alred E. Kahn, The Economc o egulaton 8 Dcuon, concluon, and urther reearch THIS CHAPTE begn n ecton 8. below by examnng market partcpant motve or orward market actvty n the mult-ettlement SFE model. Next, ecton 8. hghlght potental avenue or uture reearch by oerng ome prelmnary conecture on the mplcaton o relaxng varou model retrcton. Secton 8.3 conclude the chapter by outlnng how the reult o the mult-ettlement SFE model mght be extended n urther reearch to contrbute to a ramework or market power analy. 8. Motve or orward market actvty ecallng rom ubecton.5. Allaz (987, 8 taxonomy o hedgng, peculatve, and trategc motve or orward market actvty, th ubecton examne whch o thee eect are preent n the mult-ettlement SFE model. Hedgng and peculatve 36

344 motve are relatvely tranparent n th model, and hence eay to denty. We devote mot o th ecton, accordngly, to analyzng trategc motve or orward market partcpaton by the duopoly uppler n the mult-ettlement SFE model. Becaue the uppler are rk neutral n the preent model, they do not exhbt hedgng motve. 33 In contrat, peculatve motve or uppler to partcpate n the orward market do ext, nce the condtonal expectaton o the orward contract cah low ( CF = p p q enter rm prot maxmzaton problem (eq. (3.39 ( Fnally, trategc motve are preent or uppler, a we explan below. To motvate the dcuon o trategc motve or uppler orward market actvty, conder Green (999a, 5 obervaton that a rk-neutral rm wll not want to ue a [orward] contract market unle th wll aect t rval trategy. By ellng orward, a rm can ncreae t equlbrum output, but t wll alo reduce the prce, ut a t had adopted a more aggreve trategy n the pot market. Snce the rm could have adopted uch a pot market trategy regardle o t poton n the [orward] contract market, there ha to be another mechanm at work to make ellng contract attractve. The opportunty to aect t rval trategy ut uch a mechanm. The mult-ettlement SFE model, however, more cloely reemble a 33 We could ntead aume that uppler are rk avere. It would then be approprate or uppler to maxmze a utlty uncton (e.g., o the mean-varance type rather than mply to maxmze prot. Th change n obectve uncton would produce hedgng motve or uppler. The ultmate eect on uppler orward market partcpaton would then lkely depend on the relatve degree o uncertanty n the orward and pot market. 34 Subecton 7.6. noted that q > 0 or bae cae parameter value over the range o orward market prce o nteret. We conclude that the condtonal expectaton E CF p = p E( p p q > 0 nce p E( p p > 0 rom nequalty (

345 varant o Green man model whch he develop n an appendx to h 999 paper (Green 999b. In Green alternatve model, buyer are rk avere, whch lead them n equlbrum to pay a hedgng premum to uppler n the orward market. A a conequence, the orward prce exceed the expected pot prce. Under thee crcumtance, Green (999b, 4 conclude that [t]he ablty to earn a hedgng premum gve another motve or ellng contract, o that a rm wll now hedge part o t output, even th doe not aect t rval trategy and reduce t pot market prot, n order to earn a hedgng premum. Due to the preence o rk-avere conumer and trategc uppler n the mult-ettlement SFE model, we nd a et o ncentve analogou to thoe n Green alternatve model wth rk-avere buyer. Namely, we nd that a rkneutral rm ha an ncentve to partcpate n the orward contract market, n part to earn a hedgng premum, and alo to aect t rval pot market tage game acton. In the ollowng ubecton, we explore how th latter trategc motve or uppler orward market actvty are n the mult-ettlement SFE model. 8.. Eect o a uppler orward market actvty on equlbrum quantte Gven an arbtrary SF or rm, S ( p, let rm bet repone to ( ( S p. 35 For a hock 0 ( ( S p S p δ or rm, o that δ >, dene rom ( ( ( S p be S p a bae orward market SF S p = S p + δ. (8. 35 Note that we reer here to an optmal though not necearly equlbrum SF or rm, and hence ue the notaton ( quantte. S p and q wthout the overbar that denote equlbrum uncton and 38

346 Gven the decompoton o ( hock S p n eq. (8., we examne the eect o a derental dδ to rm orward market SF bd, tranlatng ( S p to the rght. Content wth the development o the rm orward market optmzaton problem n chapter 4, aume urther that rm mpute to rm a xed (dequlbrum trategy o { (, ( ;, S p p q q } Σ. Conder rt the eect o the ht dδ n ( S p on orward market competton. Obvouly, th change ncreae rm orward market quantty q at each prce p. The rghtward ht n ( S p alo caue rm orward market redual demand uncton, D ( p, ε 0 D ( p, ε 0 S ( p ht to the let or xed 0 ε. 36 The uncton (, 0, to D p ε, o coure, the et o prcequantty tradeo that rm ace n the orward market. For the preent, uppoe that S ( p doe not change. For xed S ( p and a letward ht n (, 0 quantty-prce par (, D p ε, the q p acng rm move downward and to the let. Accordngly, rm orward market quantty q decreae, whle the orward market- * clearng prce p p ( ε 0 = decreae, a well. Next, we examne the eect o the rghtward ht n ( S p on pot market competton. A hown n chapter 5 (ee eq. (5.3 and Fgure 5., th ht n S ( p 36 From the orward market analy o chapter 7, the rghtward ht n ( D ( p, ε 0 both drectly, and through D (, 0 p ε. Snce (, 0 negatvely related to the SF S ( p (recall eq. (6.76 and (6.78, the rghtward ht n ( D ( p, ε 0, and hence D (, 0 p ε, to the let. 39 S p aect D p ε endogenou (and S p ht

347 tranlate, n turn, rm pot market SF ( p ; q, q Σ to the rght, ncreang the rm pot market quantty q or each p. Th change n ( p ; q, q Σ mple urther that rm pot market redual demand uncton, D ( p, ε ; q, q (, ε ( ;, D p Σ p q q, ht to the let or xed xed. The uncton D ( p, ε ; q, q ε (whereby (, D p ε, o coure, the et o prce-quantty tradeo that rm ace n the pot market. The aorementoned decreae n q alo ht rm pot market SF ( p ; q, q ( ;, Σ p q q and D ( p, ; q, q Σ to the let (ee eq. ( Snce both ε ht to the let, rm pot market quantty q decreae (a we may conrm rom the analy o chapter 5. That, we have that dq dδ ( S xed < 0. (8. The gn o the net eect o letward ht n ( ;, Σ p q q and D ( p, ; q, q ε on the pot market-clearng prce p, however, ambguou n the general cae. 38 In contrat, the gn o the eect o the rghtward ht n ( S p on rm pot market quantty q well-dened. Even p hould decreae, thereby puttng downward 37 Th the eect o rm orward market acton (the change n whch the ncrement dδ on rm pot market acton, Σ ( ;, p q q. See the quotaton rom Green (999a, 5 begnnng on page In partcular, th eect on p depend on the relatve magntude o φ and φ a well a on the partcular orward market SF elected. 330

348 preure on q, we may how rom the analy o chapter 5 that the net eect o ncreaed q on q potve, that, 39 dq dδ ( S xed > 0. (8.3 O coure, rm orward market SF ( S p need not and n general wll not reman xed (a n rm mputaton above n repone to dδ. ather, rm chooe t SF (gven the new SF or rm ( S p + dδ accordng to the orward market optmzaton problem detaled n chapter 4 and ketched or rm brely above. In dong o, rm ace a completely analogou et o ncentve a thoe decrbed prevouly or rm. Wthout repeatng chapter 4 analy, we next conder the lkely nature o rm bet repone to the ncrement dδ n rm orward market SF poted above. The precedng dcuon ndcated that an ncrement dδ caued both q and q to decreae, whle the correpondng quantte or rm ncreaed. In other word, wth a xed ( S p, rm would loe market hare n both market. A noted above, the eect o dδ on the pot market-clearng prce p (wth ( S p xed wa ambguou. We cannot be prece about rm bet repone to dδ wthout ( pecyng more 39 To how th, derentate eq. (5.9 totally wth repect to δ, ung eq. (5.3 or aumng a xed SF ( along rm SF ( ;, p q q S p or rm. I on the other hand p and p hould ncreae, then the movement Σ n the drecton o ncreang quantty renorce the rghtward ht n th uncton, reultng unambguouly n ncreaed q. 33

349 exactly the eect o dδ on the condtonal expectaton o p, ( xng the parameter vector Θ, and (3 pecyng the equlbrum electon rule. It appear unlkely, however, that rm bet repone to dδ would be to mantan ( S p xed a t loe market hare n both market. ather, a uggeted by the orward market equlbra examned n chapter 7, 30 rm wll lkely want to ncreae t orward market quantty at mot, not all, prce p n repone to dδ. Accordngly, we may approxmate rm optmal repone to the ncrement dδ by a mlar potve ncrement dδ n rm orward market SF ( S p. 3 Such an ncrement dδ alo ultmately ht rm pot market SF ( p ; q, q Σ to the rght. At ome pont, naturally, rghtward ht n both rm orward market SF wll drve down prce n both market 3 to a pont beyond whch urther ncreae n orward market quantty are not protable or ether rm. 33 At th pont, rm orward market 30 Namely, even or the markedly aymmetrc uppler rm tuded n th work, the bae cae SF S ( p or each rm tended to approach each other a p ncreaed, even or dparate ntal quantte. Moreover, n each o the comparatve tatc tet cae, each rm SF moved n the ame drecton n repone to parameter perturbaton at almot all prce level o nteret. See Table E.. 3 Whle we aume or mplcty that the approxmaton dδ contant or all optmal repone o rm need not, o coure, be contant wth p. p, the exact 3 Note that when both rm ncreae ther orward market quantte, both pot market SF and hence alo the aggregate pot market SF ht outward. A a reult, the eect on the pot market-clearng prce then unambguou: p decreae. 33 From eq. (C.9 n Appendx C, th optmal pont (or ether rm where the dervatve o orward market revenue wth repect to p and margnal expected optmal provonal pot market prot um to zero. 33

350 SF are the equlbrum orward market SF S ( p and ( S p rom the ubgame perect Nah equlbrum derved n chapter 4. The key reult rom the above dcuon are the nequalte (8. and (8.3 ndcatng the oppong eect on rm pot market quantte o a rghtward ht n S ( p (and analogouly or a rghtward ht n ( S p. Content wth the oppote gn o thee eect, we could characterze the trategc nteracton dcued above a a battle or expected market hare n the pot market, waged wth orward market SF. Moreover, rm market hare n the orward market are obvouly alo aected by rm relatve aggrevene n orward market bddng. Smlar to the reult o Allaz and Vla (993, 3, thereore, the potental n the mult-ettlement SFE model or orward tradng by both rm lead to a proner dlemma eect: each rm ha an ncentve to trade n the orward market, but when both rm do o, both end up wore o n that ther prce-cot margn are maller. Th eect due only to the potental or orward tradng by the duopolt, and ndependent o the partcular behavoral aumpton n ether the orward or pot market, provded that uch aumpton do not uppre orward tradng tel. 34 Over the relevant range o prce [ ] p 0,,500 $ MWh or the Calorna PX, chapter 7 equlbra howed that trctly ncreang orward market SF yeld potve orward market quantte. That, or the numercal example o the mult-ettlement 34 A the cae or Green (999a, 5 ndng that a rm havng Cournot conecture n the orward market and ung ane SF n the pot market wll ell no orward contract. 333

351 SFE model examned here, rm optmal behavor correpond to hort poton n the orward market. 35 It ntructve to note that the output-enhancng property (whch tranlate, generally, to ncreaed aggregate welare o orward contract doe not rely on thee beng vetng contract, that, contract whoe term and condton are ubect to regulatory control. ather, allocatng orward contract va a market-baed mechanm a n the orward market o the mult-ettlement SFE model ucent to realze welare benet rom uch contract. Moreover, n th modelng ramework, mpong prce control on orward contract would lead ( uch control are bndng to maller orward market poton by uppler. Th outcome, n turn, would reult n a lower level o expected aggregate pot market output than n the abence o uch prce control. Th ugget, urther, that uch regulaton o orward market contract may lead to lower level o aggregate welare. 36 Th argument, o coure, doe not mltate agant poble dtrbutonal ratonale or regulatory nterventon n the orward market. For example, unregulated orward market prce that are gncantly hgher than expected pot market prce would create large traner rom conumer to producer, whch may be poltcally underable. 35 The ndng that hort poton n the orward market are optmal or uppler.e., bae q > 0 may be contngent on our choce o bae cae parameter vector Θ ; ecton 7.6 comparatve tatc analy nvetgated model oluton wthn a mall neghborhood o th vector. Outde o th regon o the parameter pace, we may nd that we elect orward market SF uch that q < 0. Gven that we retrcted our attenton to trctly ncreang SF (n the upper partton, uch SF le n egon I, th reult doe not appear to be entve to the properte o the equlbrum electon procedure or the orward market. A Fgure 7.5 ugget, egon I contaned wthn the potve orthant o the phae pace. 36 A more ophtcated analy o the welare eect o orward contractng would requre a dynamc analy n a repeated game ettng. 334

352 8.. Eect o a uppler orward market actvty on t rval prot Th ubecton demontrate how orward market actvty by a uppler decreae t rval prot. We ue the ame technque a n ubecton 8.. o a perturbaton (not necearly n equlbrum o a rm orward market SF. ather than quantty eect, however, the queton o nteret here the eect o the SF perturbaton on the rval rm prot. For concretene, conder a hock δ > 0 to rm orward market SF. Dene a bae orward market SF ( ( Gven the decompoton o ( orward market SF bd tranlate ( S p S p δ or rm, o that ( ( S p = S p + δ. (8.4 S p n eq. (8.4, a derental hock dδ to rm S p to the rght. We examne the eect o th hock to ( S p on rm total prot π tot*, whereby th denote rm optmal (but not necearly equlbrum prot. In what ollow we aume, naturally, that rm mpute to rm the SF ( S p n eq. (8.4, and n addton mpute the equlbrum { } pot market SF Σ p ; S ( p, D ( p, ε 0 S ( p. Adaptng the orward market optmzaton problem or the mputaton (8.4, we may combne eq. (4.6 (4.8 to expre ollow: tot* π a a uncton o ( S p, δ, and ε 0 a 335

353 { S (,, } π δ ε tot* 0 ( ε0 ( δ * ( π { D ( p ε0 S ( p δ S ( p δ ε } ε0 = max p D p, S p p + E,,, +, (8.5 where { D ( p, S ( p, S ( p +, } { p p S ( p ( ( p D p S p D ( p, ε0 S ( p δ, ε } π ε δ δ ε * 0 { } = max π, Σ ; + δ,, ε0 δ, (8.6 and { { ( ( ( } (, ε0 ( δ, ε } p D ( p, p ; S ( p, D ( p, S ( p π p, Σ p ; S p + δ, D p, ε S p δ, 0 D p S p ( ε { δ ε0 δ } = Σ + (, ε0 ( δ ( ( ε { ( δ ( ε ( δ } p D p S p C D p, Σ p ; S p +, D p, S p. 0 (8.7 Takng the dervatve o eq. (8.5 wth repect to δ (ung eq. (8.6 and (8.7 and the envelope theorem, we get { S (,, } π δ ε tot* 0 δ { } { } dσ dσ = p + E p + p C ( q ε 0, dδ dδ (8.8 where we have abbrevated the argument o Σ n eq. (8.8 a. 336

354 Under the aumpton o the mpled ane example, we may evaluate the dervatve d { } Σ dδ n eq. (8.8 ung eq. (5.3 a { δ ε δ } { } ; (, (, ( dσ d Σ p S p + D p S p 0 = = φ. (8.9 dδ dδ * Ung eq. (8.9 and condtonng ntead on a market-clearng prce p p ( ε 0 may rearrange eq. (8.8 a =, we { S (,, } π δ ε tot* 0 δ ( ( ( ( = φ E p p E C q p p + E p p. (8.0 A n the dervaton o eq. (5.37 (whch reled on the mpled ane example, we may ( wrte the expected prce-cot margn E( E ( ( ( p p C q p n eq. (8.0 a ( φ ( 0 ( ( E p p E C q p = E p p c + cs p. (8. Ung eq. (8., we may then recat eq. (8.0 a { S (,, } π δ ε tot* 0 δ { φφ ( p p ( c0 cs ( p p ( p p } = E + + E. (8. 337

355 Whle the gn o the rght-hand de o eq. (8. depend on the partcular SF S ( p elected n the orward market, 37 we are able to determne th gn or cae o nteret by the ollowng argument. Frt, comparng eq. (7. and (7.4 n the prevou chapter, we have that ( p p ( c + cs ( p + p E( p p = S ( p Q S ( p φφ E 0. (8.3 Second, n note 67 o that chapter, we argued that everywhere wthn the phae pace upper partton on whch we ocu n th work, the quadratc orm ( Q ( S p S p ha a potve gn, 38 that, ( Q ( S p S p > 0. (8.4 Combnng the expreon (8. (8.4, we may conclude that * { S (,, } π δ ε tot 0 δ < 0, (8.5 whch ay that an ncreae n orward market actvty by rm decreae rm total tot* prot π at an optmum. We may nterpret nequalty (8.5 ung Trole (988 termnology rom h 37 In addton to the explct appearance o ( condtonal expected pot market prce E ( 38 Th becaue the equaton S ( p S ( p ++ Q ++ S p n eq. (8., recall rom eq. (7.9 that the p p tel depend on both rm orward market SF. 0 whle the upper partton le entrely on one de (the potve de o the ngular locu. 338 = characterze the ngular locu,

356 two-perod, two-rm model moded approprately 39 analyzng bune tratege and trategc nteracton. 330 In perod o Trole moded model, only rm the ncumbent preent n the market. 33 Frm chooe a varable, whch Trole call an nvetment, denoted a K ( K could be productve capacty, or example, although n general K mght be any varable aectng perod competton. In perod, rm oberve K and decde to ether enter, or not to enter, the market. Trole (988, 35 clae compettve cenaro n term o the eect o rm nvetment K on rm prot (n the entry-deterrence 33 cae. Denotng rm total prot a Π, Trole aocate the condton d dk Π < 0 ( In the ollowng account o Trole model, we exchange the (arbtrary ubcrpt and labelng Trole rm o that rm the ncumbent or contency wth the oregong analy o the mult-ettlement SFE model. The obectve here to how that nequalty (8.5 above cononant wth Trole analy n term o the eect o one rm rt-perod (e.g., orward market acton on t rval prot. 330 Trole analy expand on that o Fudenberg and Trole (984 and Bulow, Geanakoplo, and Klemperer (985. The clac example o uch a model a two-perod entry deterrence/accommodaton game between an ncumbent and a potental entrant, but the problem bac tructure apple to a conderable range o nteretng economc problem; n addton to Trole (988, ee Bulow, Geanakoplo, and Klemperer (985 or many other example. 33 The preence o only a ngle ncumbent rm n perod a crtcal dtncton between Trole model and the mult-ettlement SFE model o th the, n whch (a the chapter 7 varou numercal example how each rm actve n both the orward and pot market. In our ettng, both rm can and do make trategc choce n perod, and hence the ncumbent-entrant dtncton not relevant n the mult-ettlement SFE model. See alo note 39 regardng the labelng o the two rm n th dcuon o Trole model. 33 Denote rm total prot a Π. For the ncumbent rm to (ut deter rm entry, rm chooe K o that Π = 0. Hence, n the cae o entry deterrence n Trole model, t the eect o the ncumbent perod acton on the potental entrant total prot that determne the entry decon. 339

357 wth the cae n whch nvetment (.e., ncreang K make rm tough 333 n Trole termnology. I, on the other hand, we have that d dk Π > 0, (8.7 Trole characterze th tuaton a the cae o nvetment makng rm ot. 334 Makng the analogy between Trole ramework and the preent mult-ettlement SFE model (and arbtrarly takng rm to be the ncumbent n the latter model ee S p + δ a note 33, t natural to vew rm orward market actvty ( analogou to an nvetment or that rm, to ue Trole termnology. Appealng to th analogy, we ee rom nequalte (8.5 and (8.6 above that the prot dervatve * { S (,, } π δ ε δ and tot 0 dπ dk correpond to each other. In partcular, note that the gn o both o thee dervatve negatve. Thee obervaton ugget that we may alo apply Trole termnology to the mult-ettlement SFE model. Namely, we could nterpret the negatve eect on rm prot (.e., nequalty (8.5 a the orward market acton ( S p + δ makng rm tough (or, recallng note 333, dadvantagng rm. Th content wth the ntuton rom prevou chapter that ncreang δ (ceter parbu ht rm pot market SF { p ; S ( p δ, D ( p, ε S ( p δ } Σ + 0 to the rght, thereby makng that rm 333 Perhap more evocatvely, we mght ntead characterze tuaton o the nvetment K dadvantagng one compettor. dπ dk < 0 (nequalty (8.6 a a 334 Converely to note 333, we could ntead ay that exemple the cae o the nvetment K avorng one compettor. dπ dk > 0 (nequalty (

358 more aggreve n the pot market n that t bd a larger quantty at each prce. Naturally, ncreang δ make rm a more aggreve compettor n the orward market n the ame ene, a well. Competton n upply uncton dtnct, naturally, rom competton n quantte à la Cournot. Neverthele, the mult-ettlement SFE model alo relect Trole (988, 336 generalzaton that two-perod quantty game are oten more compettve than ther tatc (one-perod counterpart. We ee evdence o th more compettve property n the tendency o orward market actvty to ncreae one own pot market quantty (a n nequalty (8.3, a well a n the analy o expected welare o ecton 7.7. In thee welare computaton, we ound that expected aggregate welare o the mult-ettlement SFE model exceeded that or the ngle-ettlement SFE model, due, n part, to the larger expected pot market quantte n the mult-ettlement model Decompoton o uppler ncentve or orward market actvty In h model, Trole (988 emphaze the role o nvetment a commtment, n partcular, commtment that matter becaue o ther nluence on the rval acton (p. 33. Lkewe, we may ueully vew orward market poton n the multettlement SFE model a trategc commtment; thee mlarly nluence rval acton a we explan below. In th ubecton, we harpen the ocu on trategc conderaton and examne n more detal how trategc motve aect rm orward market decon. The analy here not undamentally new; rather we mply pare rm orward market equlbrum optmalty condton (5.37 n a new way. Namely, we decompoe a veron 34

359 o rm orward market equlbrum optmalty condton o a to emphaze rm mpact, va each market, on rm orward market acton. We begn by rewrtng rm orward market equlbrum optmalty condton (5.37 a eq. (8.8 below: { φφ E( p p ( c0+ cs ( p E( p p p } S ( p = S ( p D 0 ( p E ( p p p. (8.8 Equaton (8.8 a veron o rm rt-order neceary condton or t orward market optmzaton problem (eq. (4.6 (4.8 under the aumpton o the mpled ( ane example. 335 We may re-ntroduce the dervatve π, ( tot, d p S p dp rearrange the term n eq. (8.8, and decompoe the new expreon nto term that we label the drect eect, the ettlement eect, and the trategc eect, a ollow: Equaton (8.8 relect ( the ubttuton o S ( p or D (, 0 ( p S p market-clearng condton or the orward market, a well a ( the ubttuton o D0 ( p D ( p, ε 0 ε rom the or rom eq. (3.3. ecall alo that n the mpled ane example, rm margnal cot uncton and pot market SF a well a pot market demand uncton all poe ane unctonal orm. 336 The reader warned that the drect eect and the trategc eect dened n eq. (8.9 or the mult-ettlement SFE model are n the ame prt a but dtnct rom the drect eect and trategc eect dented n Fudenberg and Trole (984, 363 and later, Trole (988, ec In the preent work, we dene thee eect va derentaton wth repect to a prce p n eq. (8.9, content wth the dervaton o the SF. The other author cted motvate the denton o thee eect by derentatng wth repect to a (rm-pecc quantty. 34

360 (, ( d π p S p tot dp ( ( ( = S p + p D0 p S p Drect eect ( p p D 0 ( p S ( p E = 0. Settlement eect ( E( ( ( φφ S p p p c0+ cs p Strategc eect (8.9 Below, we dcu how each o the three conttuent eect n eq. (8.9 hapng rm orward market behavor are. We call the term ( + ( ( S p p D p S p 0 (8.0 on the rght-hand de o eq. (8.9 the drect eect nce t repreent rm repone to t orward market redual demand uncton, D ( p, ε 0 D ( p, ε 0 S ( p, conderng the orward market n olaton. Gven that rm ace th redual demand uncton, the expreon (8.0 the dervatve o rm orward market revenue, (, ε 0 ( p D p S p, wth repect to p (ung agan the ubttuton o note Beore mpong Nah equlbrum, rm olve t orward market problem, gven ε 0 (a * detaled n chapter 4, to yeld a rm-pecc optmal prce ( 0 * * * (ee note 4, rm contruct ther orward market SF uch that p ( ε0 = p ( ε0 p ( ε 0 p ε. In Nah equlbrum, naturally, whch we denote n equlbrum a mply the orward market prce p. Content wth prevou chapter conventon, we nterpret eq. (8.9 and the aocated analy a applyng to uch an equlbrum outcome, though we could ut a well recat the above dcuon n term o rm optmal though not * p ε. necearly equlbrum prce ( 0 343

361 The ettlement eect E ( p p D 0 ( p S ( p (8. on the rght-hand de o eq. (8.9 the expected change n rm ettlement payment p q made n the pot market or a margnal change n p, agan gven t orward market redual demand uncton (, 0 D p ε. The ettlement eect depend on the expected (optmal pot market prce E ( p p, condtonal on p. Fnally, the trategc eect ( E( ( ( φφ S p p p c0+ cs p (8. on the rght-hand de o eq. (8.9 are due to the conectured pot market repone o rm to rm choce o (optmal prce * p ( p ( ε 0 =. Lookng back at chapter 4 analy, we may how that the trategc eect o eq. (8. mply the expreon ( p ψ n eq. (4.4 under the aumpton o the mpled ane example. A oberved n Appendx C, we may nterpret ψ ( p, n turn, a the expected change n the derence between rm equlbrum pot market revenue and producton cot (evaluated at t equlbrum contract quantty ( S p or a margnal change n p. To undertand how the trategc eect are, begn by rewrtng eq. (5.3 or rm a eq. (8.3 below: Σ ( p ; q, q q = φ. (

362 Equaton (8.3 expree the margnal eect o change n rm equlbrum orward market quantty q (ceter parbu on t pot market quantty or an arbtrary prce p. 338 Frm pot market SF ( p ; q, q Σ appear n rm pot market redual demand uncton { } (, ε ; (, (, ε ( q = D p Σ p S p D p S p 0 (8.4 n problem (4.8 (takng S ( p = S ( p, n equlbrum. Frm conecture that, n equlbrum, rm repond to margnal change n p accordng to the uncton ( S p. Frm nduce rm n th way to change q (and hence q, a already decrbed n ubecton 8.. above. Th the heart o the trategc eect. Wth thee conderaton, eq. (8.3 and (8.4 mply that the change n q or a margnal change n p gven the uncton ( { } S p and Σ p ; S ( p, D ( p, ε 0 S ( p and holdng p xed dq q Σ dq = = φ S dp q q dp S, Σ, p p Σ, p S ( p, ( ecall that the mplcty o the (contant expreon or the dervatve n eq. (8.3 depended Σ p ; q, q beng ane n Σ p ; q, q, crtcally on the aumpton o ( p. In the ane cae, ( n act, ndependent o q. Th ugget that we extended the nvetgaton to nclude non-ane SF Σ ( p ; q, q, we would oberve an addtonal term n the trategc eect or rm. Th term would p ; q, q q = D p, ε S p alo changng wth p. correpond to a ht n Σ ( due to ( 0 ( 345

363 a relected n the expreon (8. or the trategc eect. Fnally, we note that the trategc eect proportonal 339 to rm (condtonal expected prce-cot margn E ( p p ( c0 + cs ( p n the pot market, evaluated at t equlbrum orward market quantty q = q ( p = S. I we rewrte rm FOC, eq. (8.9, ettng the trategc eect to zero, the reved FOC aumng zero trategc eect 340 a ollow: (, ( d π p S p tot dp Zero trategc eect ( ( ( = S p + p D0 p S p Drect eect ( p p D 0 ( p S ( p E = 0. Settlement eect (8.6 A ull accountng o the nluence o the trategc eect on the orward market equlbrum would requre computng new orward market SF rom the FOC (8.6 (and the ymmetrc condton or rm. In partcular, the uncton E ( (, 0 p p and D p ε, derved analytcally and computed numercally n chapter 6 and 7, both S p. depend endogenouly on the uncton ( 339 Wth contant o proportonalty φφ S ( p (gven 340 Th would be the cae the dervatve ( ;, or rm were equal to zero, mplyng that φ = φ = 0. p, rom the expreon (8.. Σ p q q q rom eq. (8.3 and t analog 346

364 8..4 Motve or orward market actvty by conumer To conclude th ecton, we turn brely to the demand de o the market to conder conumer motve or orward market partcpaton n the mult-ettlement SFE model. In contrat to the treatment o uppler, we take conumer to be rk avere n th model. Accordngly, conumer pay a rk (or hedgng premum to uppler n purchang orward contract at a prce p typcally n exce o E ( p p. Hedgng thu a motve or conumer partcpaton n the orward market, a chapter 6 analyzed n detal. Speculatve motve alo ext or conumer n the orward market. Aumng or a repreentatve conumer a orward contract quantty q, the condtonal expectaton o the term ( neted maxmzaton problem (6.30 (lettng p p q enter the obectve uncton o her = or the mult-ettlement SFE model. The preence o th term ndcate that the repreentatve conumer peculate on the (condtonal expected prce derence p p. Fnally, becaue conumer take prce a parametrc n problem (6.30, they do not behave trategcally n the preent model. Thereore, conumer ace no trategc motve or orward market partcpaton. 8. Further reearch: elaxng retrcton mpoed n the model 8.. Number o compettor n Throughout th nvetgaton, we have aumed that we have a duopoly on the upply de o the market that, n =. In th ecton, we conder what ncreang n would ental, and how the reult o the mult-ettlement SFE model mght be aected. The equlbrum optmalty condton or the cae o larger n are eay to expre. The maor tructural change to replace (n rm optmzaton problem the SF 347

365 S ( p and ( ;, Σ p q q wth the um S ( p and Σ ( ;, p q q, repectvely, relectng the acton o all o rm rval n redual demand uncton or each market. The dculte that are lkely to are or larger n appear to be largely computatonal. Prelmnary nvetgaton how that our choen numercal analy package, MATLAB (and the MAPLE ymbolc algebra kernel, 34 ha dculty olvng the pot market problem 34 ymbolcally n the ane cae or n > 6. Th may be becaue a oluton mply doe not ext, becaue the problem ll-poed gven the olver algorthm, or that the olver a currently congured unable to olve t. Numercal oluton, n contrat, may o coure be eaer to nd. We have not yet attempted to nd oluton o the orward market problem or n > rm. Whle vualzaton o uch traectore become more dcult or uch larger n, we antcpate no undamental obtacle to applyng the MATLAB- or Excelbaed model to cae o larger n. Obtanng a oluton or a larger value o n would oer a conderable mprovement n vermltude over the current duopoly cae n vew o the tructure o actual electrcty market. Moreover, a larger n would permt modelng o varou cenaro encompang rm entry, ext, generaton plant dvetture, and merger and acquton. Fnally, we are able to model cae n whch n get large, t would be nteretng to ee whether pot and orward market SF approach a compettve lmt. We mght obtan ome nght nto the nature o uch a compettve lmt rom the ymmetrc and computatonally ar mpler cae wth n dentcal rm. 34 The MathWork (00 and Mapleot ( In the ane cae, olvng th problem ental olvng a nonlnear ytem o algebrac not derental equaton. 348

366 8.. Ane unctonal orm retrcton Begnnng wth the mpled ane example o chapter 5, much o the analy retrcted to the cae n whch the pot market demand uncton, both rm margnal cot uncton, and pot market SF have an ane unctonal orm. Suppoe, n contrat, that we broaden the ocu rom ane pot market SF to conder alo non-ane SF arng rom the pot market equlbrum optmalty condton, eq. (4.3 and (4.4. In th cae, however, thee condton would no longer yeld a ytem o multaneou algebrac equaton or pot market SF lope (eq. (5.6 and (5.; ntead, a derental equaton ytem 343 wll characterze the pot market SF. Becaue n th cae the pot market SF wll no longer be unque, ue o equlbrum electon and coordnaton between rm would are n the pot market a well, leadng to the compound problem that Newbery (998, 733 ha characterzed a a double nnty o oluton. 344 Falure o the rm to coordnate ucceully on an equlbrum would not necearly lead to market ntablty, but t would mply that rm are almot certanly not upplyng ex pot optmal quantte, gven ther rval acton and realzaton o tochatc parameter. Provded that the rm realze that ther behavor uboptmal, we could urme that they mght engage n a heurtc earch proce n ther trategy pace n an attempt to mprove ther prot. We could go urther n generalzng the ane cae, and aume only trctly ncreang margnal cot and downward-lopng pot market demand. Under thee model. 343 Smlar to the orward market problem n the preent veron o the mult-ettlement SFE 344 Other crtera or equlbrum electon (e.g., Pareto optmalty, ratonalzablty recall n. 3 mght be nvoked, but the theory here generally nconcluve and omewhat controveral (ee Fudenberg and Trole 99,

367 aumpton, not only would we agan have a contnuum o nonlnear equlbrum SF n the pot market, but the extence o an ane equlbrum n pot market SF would not be aured. Nonethele, gven any derentable margnal cot and demand uncton, t traghtorward a an analytcal matter to generalze the (neceary equlbrum optmalty condton or both market. Then, aumng ome procedure or equlbrum electon n each market, t hould be poble to olve the reultng ytem o ODE numercally ung the method o chapter 7. Extreme unctonal orm, however or example, demand that too convex, or margnal cot that are too teep or nonconvex may caue a mult-ettlement market equlbrum wth trctly ncreang orward market SF not to ext, or to ext only on a harply retrcted prce doman or regon o the parameter pace ole o perect obervablty o orward market acton Begnnng wth the peccaton o the mult-ettlement SFE model normaton tructure n ubecton 3.., we have aumed throughout th the that rm equlbrum orward market acton that, the SF ( S p are perectly obervable a they ormulate ther pot market SF. In ecton 3.3, however, we howed that rm pot market SF a uncton o the pot market prce, p, and alo (n general each rm orward market quantty, q ( =,. Hence, we wrote rm (equlbrum pot market SF a a uncton ( ;, p q q Σ (, =,;. 345 From th peccaton, t clear that we may weaken the obervablty aumpton rom 345 Where rm repectve equlbrum orward market quantte concde wth the optmal a well a the mputed quantte, o that we may wrte q = q = q. 350

368 obervng orward market acton (the SF ( S p to mply obervng orward market equlbrum quantte q. Obervablty o the q crucal to the model, however. Weakenng the model aumpton urther n th repect by permttng le-thanperect obervablty o q would lkely have a crtcal eect on the model reult, partcularly on the trategc ncentve that are between market a dcued n ubecton 8..3 above. In th ubecton, we examne ome related lterature that ugget how ntroducng mperect obervablty o orward market outcome mght aect oluton o the mult-ettlement SFE model. Hughe and Kao (997 tudy a two-tage Cournot duopoly game wth orward contractng n the rt tage and producton n the econd tage. The author examne how obervablty o contract poton aect trategc and hedgng motve or orward contractng; they conder cae n whch the compettor are rk neutral, and n turn, rk avere. Table 8. below ummarze Hughe and Kao man reult. TABLE 8. MOTIVES FO FOWAD MAKET PATICIPATION AS A FUNCTION OF OBSEVABILITY OF CONTACT POSITIONS AND ISK PEFEENCES (Hughe and Kao 997 k Oberva- preerence blty o k neutral k avere contract poton Perectly obervable Strategc motve only Hedgng motve Strategc motve Not obervable No orward contractng a Hedgng motve Strategc motve b Note: a Hughe and Kao requre conecture to be content wth rm acton; accordngly, the only content conecture or the orward market n th cae q = q = 0. b Here, the trategc motve weaker than that n the cell above. 35

369 The ntuton or the abence o orward contractng n the (Not obervable, k neutral cell o Table 8. a ollow, lettng and ndex the two rm (, =, ; (Hughe and Kao 997, 5: I rm conecture that rm doe not take a orward poton, then rm ha no ncentve to devate rom th conecture. In eence, abent obervablty, there no mean or rm to alter rm bele. Fnally, the preence o a trategc motve or orward market actvty n the (Not obervable, k avere cell o Table 8. bear ome explanaton. To ee that a trategc motve preent n th cae depte unobervable contract poton, note that the rk-avere rm aware that rm expect t to hedge prce uncertanty va orward contract. A a conequence, rm redual demand uncton ht to the let, caung rm to concede market hare n the econd tage. In th way, hedgng behavor can have trategc conequence. 346 Conder now the mplcaton o the above ndng or the mult-ettlement SFE model ung upply uncton. The behavoral aumpton o upply uncton nvoked n the preent work gncantly more lexble than the Cournot conecture ued by Hughe and Kao. The lope o rm mputed orward market SF at an arbtrary prce p, S ( p, rm conecture regardng how rm would repond (locally, near p to a change n orward market prce. Under our aumed market rule, th lope may le anywhere on the potve real lne (ncludng zero. Th gncantly greater degree o lexblty may permt content conecture n the SFE cae where th wa not 346 Allaz (987, 4, n. 43 alo alluded to th phenomenon when he oberved that trategc and hedgng motve partly overlap. 35

370 poble under Cournot. 347 Wthout more careul nvetgaton we cannot be ure, but there ucent reaon to be keptcal that the No orward contractng reult n the (Not obervable, k neutral cell o Table 8. wll alo obtan under the SFE aumpton ued n the mult-ettlement SFE model. ather than potng uncertanty n demand a we do n the preent work, Hughe and Kao uppoe (n one part o ther analy that a par o Cournot duopolt ace uncertanty n ther cot that reolved ater the orward market clear, but beore rm act n the pot market. In th ettng, the author conder n turn the cae o perectly obervable orward contract poton, and unobervable contract poton. For the cae o perectly obervable contract poton, the author make a urther dtncton wth regard to rk preerence. For rm that are rk neutral to moderately rk avere, ellng orward contract optmal, wherea rm rk averon ucently great, rm buy orward contract. 348 For unobervable orward contract poton, on the other 347 A well-known, the Cournot model ha ncontent conecture. In contrat, gven contant margnal cot, the Bertrand model ha content conecture. See Brenahan (98 or detal. 348 Here, the ntuton that the trategc and hedgng eect acng each rm go n oppote drecton. The rm rky cot create varance n t prot. Th motvate the rm to decreae producton, whch t can do, eectvely, by buyng orward contract. On the other hand, the rm quantty decon are made, naturally, baed on th uncertan cot; thee quantty decon, n turn, aect prce, makng prce rky a well. Th eect on prce create an ncentve or the rk-avere rm to ell orward contract to lock n ale at a certan prce. Unlke the aorementoned eect on cot, th eect on prce alo naturally aect the prot o the rval, o that the prce rk ental a trategc a well a a hedgng component. When the rm rk averon ucently low, the trategc eect domnate, and the rm ell orward contract. When on the other hand the rm rk averon ucently hgh, hedgng the domnant eect, and the rm buy orward contract. Wth repect to hedgng, th behavor borne out on the demand de o the mult-ettlement SFE model. Speccally, ubecton dcue the eect o ncreang the repreentatve conumer rk averon coecent λ on the orward market equlbrum. There, we noted that ncreaed λ caue ( 0 D p, ε to ht to the rght, that, ncreaed conumer rk averon ncreae the demand or orward contract. 353

371 hand, Hughe and Kao nd that the trategc eect whle preent are ucently attenuated 349 o that, provded only that the rm rk avere, t buy orward contract. Beyond the eect o contractng, there a wder lterature on the crtcal role o obervablty on trategc ncentve n dynamc game that relevant to our model. Interetngly, ome author have generalzed the mple dchotomy between perectly obervable and unobervable acton by ntroducng noy obervaton o rt-perod acton, made operatonal va a random devaton between an oberved and an actual parameter value repreentng agent rt-perod acton. Bagwell (995, or ntance, ha analyzed two-perod Stackelberg game o quantty choce n whch the ollower obervaton o the leader choen quantty noy. He how that the pure trategy equlbrum et o th game concde wth that or the correpondng multaneou-move game (.e., the Cournot equlbrum, even or a very mall degree o noe. The mplcaton o th ndng that mperect obervablty can negate the commtment nherent n the leader acton or the econd perod tage game. In a related artcle, Magg (999 demontrate that permttng the leader n the aorementoned game to oberve prvate normaton (e.g., t own cot generally retore the Stackelberg outcome. 350 Magg (999, 556 provde the ntuton or th reult. Suppoe that the leader prvate normaton concern t type, or example, low-cot or hgh-cot. The ollower wll then ue the (noy obervaton o the leader quantty to attempt to ner the leader type, whereby both poolng and eparatng equlbra are, n general, poble. Gven that the ollower behave n th way, the leader then ha an ncentve to 349 ecall the dcuon o Table 8. above. 350 See Magg (999 or ome techncal qualcaton to th reult. 354

372 manpulate the ollower percepton o the leader type through t quantty choce. Speccally, the leader ncentve to produce more than the Cournot output, whch retore the Stackelberg rt-mover advantage. eturnng to the model n the preent work, the mult-ettlement market tuded here dtnct rom the Stackelberg equental-move ettng n that the ormer model compre two multaneou-move tage game. Whle equental move wthn each tage game would not be a realtc cenaro n electrcty market, the equence o tage game the orward and pot market a dynamc ettng n whch the nght o the Stackelberg game apply. Moreover, mperect obervablty o orward market poton would add realm to our ramework. In addton, we were to revt the model normaton tructure, a more plauble aumpton would be to have rm cot be prvate normaton. Wth thee change, the normaton tructure n the multettlement SFE model would parallel that n Magg (999 dcued n the oregong paragraph. In th new ettng we conecture that, ung an nght mlar to that o Magg, prvate cot normaton would oet mperect obervablty o that trategc ncentve are not mpared by mperect obervablty o orward market contract poton. The above dcuon o the obervablty o orward market contract poton ha mportant polcy mplcaton regardng regulatory rule or normaton dcloure n electrcty market. Gven the reult o ecton 7.7 welare analy that orward market are welare-enhancng, 35 the queton are o how dcloure polce or 35 Whle th reult doe not hold n all cae, t generally upported by the prevou reearch dcued n ecton.5.. For a counter-example, ee Ferrera (

373 orward market poton aect rm partcpaton n and hence the welare eect o orward market. In Table 8. ummary o Hughe and Kao (997 model, elmnatng normaton dcloure eectvely halt orward tradng n the rk-neutral cae and attenuate the trategc ncentve or orward tradng n the rk-avere cae. Th ugget (but doe not prove, a Hughe and Kao pont out (p. 30 that dcloure o orward contract poton wll be welare-enhancng. 35 A noted n the dcuon o Table 8. above, t appear unlkely that makng orward market poton unobervable n the SFE model wll completely elmnate orward contractng n the rk-neutral cae, although t may tll weaken the ncentve to contract. I o, then dcloure may not be a crtcal under the SFE aumpton a n the Cournot cae. 8.3 Further reearch: Market power The preent work only uggetve o the complexte that market power analy n real-world electrcty market mut conront, a Quan and Mchael (00, 06 attet (wth reerence to Calorna market: Over the coure o a day, a generator mut make (by our rough count at leat 480 prce bd decon at varou hour. Choong not to partcpate n certan market may requre a much thought, and be raught wth a much rk, a choong to bd n other. Snce generator actually bd hourly upply chedule wth up to 5 egment n many market, the potental prce decon over a day run nto the thouand. We alo count 46 capacty commtment decon over the day.... The analy o market power by eller n a ytem lke th o complex an endeavor that or all practcal purpoe t mpoble to perorm. 35 In more recent work, Hughe, Kao, and Wllam (00 examne the dcloure decon rom the rm perpectve n an aymmetrc duopoly model n whch only one o the two duopolt trade n the orward market. They analyze the tradeo that dcloure preent to a rm between explotng t normatonal advantage n the orward market, on the one hand, and nluencng the later producton decon o rval, on the other. They nd, unurprngly, that normed rm preer non-dcloure o orward market poton. In contrat, unnormed market partcpant (peccally, broker who are contraned to break even n ther tradng preer dcloure o orward contract. The author conder only n pang the ymmetrc cae o competng duopolt n the orward market. They conecture that ymmetry would only trengthen ncentve or non-dcloure on the part o normed market partcpant. 356

374 Quan and Mchael rather pemtc aement notwthtandng, the preent work ha broken new ground n undertandng trategc nteracton n a mult-ettlement market ettng. The decompoton and analy o the ncentve or orward market partcpaton a neceary and mportant rt tep toward market power analy n th novel nttutonal envronment. The next queton that we conront n th regard how the preent model mght be ued or augmented to etablh a orward market perectly compettve behavoral benchmark (PCBB or market power aement n the preence o rk-avere conumer. The only alternatve to orward market partcpaton that we repreent n the mult-ettlement SFE model, o coure, the opportunty to partcpate n the pot market. In th ettng, t natural to eek a PCBB n the orm o a margnal opportunty cot o orward market partcpaton nvolvng expected pot market return oregone through orward market actvty. Whle we reerve or uture reearch the development o an explct expreon or margnal opportunty cot, we may gan ome nght nto the nature o uch an opportunty cot by revtng the relevant dcuon rom chapter. We raed the queton n chapter o how to evaluate the compettvene o the orward market. In partcular, we aked whether aeng market power n multettlement market requred the ont evaluaton o behavor n both the orward and pot market, or whether we could analyze the orward market n olaton. We now revt th queton. In eq. (4.4, we ound that the expected pot market prce E ( p p play the role o margnal producton cot C ( S n eq. (4.5. Th tructural mlarty between the optmalty condton or the orward and pot market ugget a natural 357

375 nterpretaton o the expected pot market prce E ( p p a one contrbutng actor to the margnal opportunty cot o a uppler orward market actvty. In partcular, the ettlement eect dened n eq. (8.9 proportonal to the condtonal expectaton E ( p p. To evaluate E ( p p, naturally, we need to know the condtonal dtrbuton o p p, whch n turn requre chapter 5 analy o how the orward and pot market are coupled. 353 Th not the entre tory concernng margnal opportunty cot, however. From expreon (8.9 above, the trategc eect alo aect a rm orward market SF bd through antcpated pot market outcome. 354 Lke the ettlement eect, the trategc eect depend on E ( p p and, n addton, depend on rm expected margnal cot gven p. Thu, to compute the trategc eect, we agan requre normaton rom both the orward and pot market. We may conclude rom the denton o both the ettlement and trategc eect n expreon (8.9 that evaluatng the compettvene o a rm orward market behavor requre conderng behavor and market outcome n both the orward and pot market. Further reearch hould dene more precely the margnal opportunty cot o orward market actvty n th model, whch may then erve a an approprate PCBB n th two-market ettng. eq. ( For thee purpoe, we developed an expreon or E ( 358 p p n eq. (5.33, later mpled to whch, by npecton, we may aocate wth a change n orward market revenue rather than a change n the opportunty cot o orward market partcpaton. 354 Expreon (8.9 alo nclude the drect eect S ( p + p D 0 ( p S ( p

376 On the emprcal ront, one nteretng approach to meaurng market power n a dynamc game ettng that may prove ueul n uture work that o oeller and Sckle (000. Thee author derve and etmate econometrcally a tructural model o the European arlne ndutry that pot competton n capacte n the rt perod, and n prce n the econd. They nd that rm are gncantly le colluve n the two-tage model than under the one-tage peccaton. 355 They conclude that colluvene overetmated whenever competton naturally occur n two tage. Ther model, whch lnk theory to emprcal meaure o market power, may conttute a promng approach to dervng an emprcally-baed PCBB or the mult-ettlement SFE model. 356 Unlke n oeller and Sckle ramework, however, ntenty o competton n the preent SFEbaed model cannot be captured by a mple calar conduct parameter. Even n the cae o ane SF (a or the pot market SF n the mpled ane example, we requre two parameter to pecy unquely a rm acton. 355 Th obervaton content wth Trole (988, 336 generalzaton regardng two-perod quantty game. 356 In the preent ettng, o coure, we have ocued excluvely on unlateral market power rather than colluon. Alo, electrcty market are characterzed by repeated compettve nteracton o hgher requency than thoe that plaubly ext n the arlne ndutry. Accordngly, electrcty market are lkely to be more entve to dynamc eect than the arlne ndutry. 359

377 Prudence and utce tell me that n electrcty and team there more love or man than n chatty and abtnence rom meat. Chekhov, Letter to A.S. Suvorn Appendx A: Proo that rm pot market upply uncton nterect t redual demand uncton exactly once WE BEGIN by argung that, under our aumpton, rm redual demand uncton lope downward. Frt recall that, at the outet o text ecton 4., we dened rm pot market redual demand uncton (gven arbtrary q ˆ and q ˆ, and or a partcular ε a D ( p, ( p ; qˆ ˆ, q uncton a D ( p, ε ; qˆ ˆ, q ε Σ. For convenence, denote th redual demand, o that (, ε ; ˆ, ˆ (, ε ( ; ˆ, ˆ D p q q D p Σ p q q. (A. 360

378 Snce D ( p, ε < 0 (rom text ubecton 3..0 and ( ˆ ˆ p q q ubecton 3..5 by aumpton, 357 we have rom eq. (A. that ( ε ˆ ˆ Σ ;, > 0 (rom text D p, ; q, q < 0 (A. or all ˆ ˆ p, and gven ε, q, and q. We now prove that no two realzaton o rm redual demand uncton nterect. Aume, n contradcton, that two arbtrary redual demand uncton (, ˆ ; ˆ, ˆ ε and D ( p, ˆ ; qˆ ˆ, q D p q q ˆ ε ˆ ε. Algebracally, th aumpton ε do nterect at a prce p = p, where (, ˆ ε ; ˆ, ˆ = (, ˆ ε ; ˆ, ˆ D p q q D p q q, (A.3 and Fgure A. below depct th relatonhp graphcally. 357 Where prme ( denote derentaton wth repect to p. 36

379 p Spot market (, ˆ ε ; ˆ, ˆ D p q q p = p (, ˆ ε ; ˆ, ˆ D p q q FIGUE A.: THE FUNCTIONS D ( p, ˆ ε ; qˆ ˆ, q INTESECT AT PICE p 36 D, ε AND D ( p, ˆ ε ; qˆ ˆ, q = p (COUNTEFACTUAL CASE Eq. (A.3, however, contradct our aumpton n ubecton 3..0 o the text that ( ε D p, ε > 0 p, nce at the pont o nterecton we have (, ˆ ε ; ˆ, ˆ ε 0, and hence, rom (A., ( ˆ D p q q = D p, ε ε = 0. Thu, t mut be that no two realzaton o rm redual demand uncton nterect. Fnally, we how that ( p ; qˆ ˆ, q Σ nterect each redual demand uncton exactly once. ecall by the reaonng n text ecton 4. that, or each ε, there ext a unque optmal prce or rm. For any ( ˆ p. For example, lettng ε * = ˆ ε, we have that p = p ( ˆ ε ; qˆ ˆ, q ε = ˆ ε, there alo ext a unque redual demand uncton, nce D p, ε ε > 0 n the denton o rm redual demand uncton, eq. (A.

380 above. Thereore, the redual demand uncton D ( p, ˆ ; qˆ ˆ, q ( ˆ ε ; ˆ, ˆ * ε contan at prce p = p q q a unque ex pot prot-maxmzng pont or rm. Fgure A. below llutrate thee relatonhp. Becaue we have not yet characterzed the properte o the SF Σ ( p ; qˆ ˆ, q only ndcate t pont o nterecton wth D ( p, ˆ ; qˆ ˆ, q, the gure doe not depct t, but ε. Spot market p (, ˆ ε ; ˆ, ˆ D p q q Unque ex pot prot-maxmzng pont ( ˆ ˆ ˆ on D p, ε ; q, q, and alo the pont at whch Σ( p ; qˆ ˆ, q (not hown wll nterect D ( p, ˆ ε ; qˆ ˆ, q ( ˆ ε ; ˆ, ˆ p = p q q * ( ε ; ˆ, ˆ p q q * ˆ ε D ε, FIGUE A.: ( ; ˆ ˆ, Σ p q q INTESECTS D ( p, ˆ ; qˆ ˆ, q By contructon (ee text ecton 4., ( p ; qˆ ˆ, q ε EXACTLY ONCE Σ pae through each ex pot prot-maxmzng pont or rm (and only thee pont. Thereore, Σ ( p ; qˆ ˆ, q wll pa through the ex pot prot-maxmzng pont on D ( p, ˆ ε ; qˆ ˆ, q, whch le 363

381 * * at prce p = p ( ˆ ε ; qˆ ˆ, q. Snce p ( ˆ ; qˆ ˆ, q o Σ ( p ; qˆ ˆ, q and D ( p, ˆ ; qˆ ˆ, q ( ε unque, th pont o nterecton ε tel unque. A completely analogou proo apple n the orward market to how that S p nterect rm orward market redual demand uncton exactly once or each ε

382 Lke many bunemen o genu he learned that ree competton wa wateul, monopoly ecent. And o he mply et about achevng that ecent monopoly. Maro Puzo, The Godather Appendx B: Second-order ucent condton or the optmalty o the orward and pot market upply uncton WE FIST CONSIDE n ecton B. the econd-order ucent condton or optmalty n the pot market, ollowed n ecton B. by the analogou condton n the orward market. B. Second-order condton or the optmalty o the pot market SF The proo n th ecton parallel that o KM or ther Clam 7 (Klemperer and Meyer 989, 54. ecall rom text eq. (4.3 that the FOC or the provonal pot market problem (aumng an nteror oluton 365

383 {, Σ ( ; ˆ, ˆ, ˆ, ε } dπ p p q q q dp = 0, (, ε ( ; ˆ, ˆ ˆ { p C D ( p, ε ( p ; qˆ ˆ, q } D ( p, ε Σ ( p ; qˆ ˆ, q = D p Σ p q q q + Σ (B. where the prme on the pot market demand uncton and the SF denote dervatve wth repect to p. Derentatng eq. (B. agan to obtan the econd-order condton, we have {, Σ( ; ˆ, ˆ, ˆ, ε } ( dp = D ( p, ε Σ ( p ; qˆ, qˆ + { C D ( p, ε ( p ; qˆ ˆ, q D ( p, ε ( p ; qˆ ˆ, q Σ Σ } D ( p, ε Σ ( p ; qˆ ˆ, q + { p C D ( p, ε Σ( p ; qˆ ˆ, q } D ( p, ε Σ ( p qˆ ˆ q d π p p q q q ;,. (B. Smplyng and ung text eq. (4.4 (4.7 to replace D ( p, ε Σ( p ; qˆ ˆ, q ( p ; qˆ, qˆ wth Σ (where we have alo aumed Nah equlbrum between rm and, eq. (B. become 366

384 {, Σ ( ; ˆ, ˆ, ˆ, ε } ( dp = D ( p, ε Σ ( p ; qˆ ˆ, q d π p p q q q C ( p ; qˆ, qˆ D ( p, ε ( p ; qˆ ˆ, q { p C ( p qˆ ˆ q } D ( p ε ( p qˆ ˆ q Σ Σ + ;,, ;, Σ Σ. (B.3 I Σ ( p ; qˆ ˆ, q optmal gven Σ ( p ; qˆ ˆ, q, then t mut aty eq. (B., the optmalty condton or rm. Agan ung text eq. (4.4 (4.7 to replace (, ( ; ˆ D p ε Σ p q ˆ, q by ( p ; qˆ ˆ, q Σ n eq. (B., we get { ( ˆ ˆ } ( ( ˆ ˆ ;,, ;, ˆ ( ; ˆ ˆ p C Σ p q q D p ε Σ p q q = q Σ p q, q. (B.4 Derentatng both de o eq. (B.4 wth repect to p and rearrangng, we have { p C Σ( p ; qˆ ˆ, q } D ( p, ε Σ ( p ; qˆ ˆ, q = Σ ( p ; qˆ ˆ, q ( ( { C ˆ ˆ ˆ ˆ p q q p q q } D ( p ε ( p qˆ ˆ q ;, ;,, ;, Σ Σ Σ. (B.5 Subttutng eq. (B.5 nto eq. (B.3 yeld {, Σ ( ; ˆ, ˆ, ˆ, ε } ( dp = D ( p, ε Σ ( p ; qˆ ˆ, q d π p p q q q C ( p ; qˆ, qˆ D ( p, ε ( p ; qˆ ˆ, q ( ; ˆ ˆ p q, q { C ( p ; qˆ ˆ, q ( p ; qˆ ˆ, q } D ( p, ε ˆ ( p ; q, q Σ Σ Σ Σ Σ Σ ˆ. 367

385 Collectng actor o D ( p, ε Σ ( p ; qˆ ˆ, q n the above equaton, we have {, Σ ( ; ˆ, ˆ, ˆ, ε } ( dp = D ( p, ε Σ ( p ; qˆ ˆ, q + C Σ( p ; qˆ ˆ, q Σ ( p ; qˆ ˆ, q d π p p q q q { } ( ˆ ˆ ( ε ( ˆ ˆ ( ˆ ˆ C p ; q, q D p, p ; q, q Σ Σ Σ p ; q, q. (B.6 From eq. (B.6, we may conclude that any SF ( p ; qˆ ˆ, q Σ atyng the pot market FOC (eq. (B. that alo trctly ncreang (.e., ( p qˆ ˆ q Σ ;, > 0 over t doman part o an SFE. To ee th, note that, gven our parametrc aumpton and ( p qˆ qˆ Σ ;, > 0, we can gn the term n eq. (B.6 a {, Σ ( ; ˆ, ˆ, ˆ, ε } ( dp d π p p q q q ( ( ˆ ˆ D p, ε p ; q, q C ( p ; qˆ ˆ, q ( p ; qˆ ˆ, q = Σ + Σ Σ ( ; ˆ, ˆ (, ε C Σ p q q D p Σ ˆ ˆ ˆ ˆ p ; q, q Σ p ; q, q. + + ( ( + Thereore, or th p, we have that {, Σ ( ; ˆ, ˆ, ˆ, ε } ( dp d π p p q q q < 0. (B.7 Eq. (B.7 the econd-order ucent condton or p to be a global prot maxmum. 368

386 B. Second-order condton or the optmalty o the orward market SF The orward market FOC (text eq. (4.9, (, (, ε d π p S p tot 0 dp (, ε ( 0 (, ε0 ( = D p S p + p D p S p { (, (, (, } * 0 + E ε 0 = 0, dπ D p ε S p S p ε dp (B.8 where the prme on orward market demand and the SF denote dervatve wth repect to p. 358 Derentatng eq. (B.8 wth repect to p to obtan the econd-order condton, 359 we have (, (, ε ( dp = D ( p, ε0 S ( p + p D ( p, ε0 S ( p * d π { D ( p, ε0 S ( p, S ( p, ε } + E ε 0. ( dp tot d π p S p 0 (B.9 The rt term n eq. (B.9 negatve or trctly ncreang S ( p, but wthout urther retrcton on the unctonal orm o Σ ( p ; qˆ ˆ, q, Σ ( p ; qˆ ˆ, q, (, 0 S ( p, the econd and thrd term n eq. (B.9 are ndetermnate n gn. D p ε, and 358 Th ecton draw on numerou reult rom chapter 4 and And aumng unorm convergence o the expectaton ntegral n th equaton. 369

387 In th general cae, we can make no urther progre n gnng the term n eq. (B.9. We mut aume that the econd and thrd term n eq. (B.9 are uch that ( ( ( π,, ε < 0. Under thee aumpton, we conclude that tot d p S p 0 dp p a global prot maxmum or rm. In what ollow, we (oon retrct ourelve to the cae o the mpled ane example o chapter 5. To evaluate the dervatve nde the expectaton on the rght-hand de o eq. (B.9, rt recall text eq. (4.0: 360 { (, (, (, } * dπ D p ε0 S p S p ε dp { (, (, (, } * π D p ε0 S p S p ε dq = q dp { (, (, (, } * π D p ε0 S p S p ε dq + q dp. (B.0 We ultmately want to derentate eq. (B.0 agan wth repect to p, but rt ue ome reult rom chapter 4 and 5 to mply th equaton. Namely, rom chapter 4: * Eq. (4.5 and (4.6 gve expreon or the dervatve o π wth repect to q and q ; 360 * ecallng text eq. (3.43 and (3.4, we ee that the rt and econd argument o π are q * and q, repectvely. It wll be ueul horthand n eq. (B.0 to dene dervatve o π wth repect to thee orward market quantte. 370

388 Eq. (4.3 and (4.3 expre dq dp and dq dp n term o orward market SF. Makng thee ubttuton n eq. (B.0 yeld { (, (, (, } * dπ D p ε0 S p S p ε dp (, ε 0 ( D p S p { } Σ = p C ( D ( p, ε Σ{ } p q { } Σ + p C ( D ( p, ε Σ{ } S ( p, q (B. where we recall that, or eae o notaton, we ntroduced Σ { } { } ( ( ε ( n chapter 4, gven by { p ; S p, D p, S p } { p ; q, q } Σ Σ 0 = Σ. (B. Now, we retrct our ocu n th dcuon to the ramework o the mpled ane example o chapter 5. Under the aumpton that { ;, p q q} Σ (, =,; ane (recall the Ane Spot Market SF aumpton rom ecton 5., 36 we may evaluate the dervatve o { p ; q, q } Σ n eq. (B. a Σ { p ; q, q } q = 0 (B.3 and 36 Whle chapter 5 dcuon ue equlbrum quantte q and q, we ue here the analogou expreon n term o an arbtrary q and the mputed quantty or rm, q. 37

389 Σ { p ; q, q } q = φ. (B.4 Subttutng eq. (B.3 and (B.4 nto eq. (B. yeld { (, (, (, } * dπ D p ε0 S p S p ε dp (, ε ( 0 ( ( ε { } ( = p D p S p φ p C D p, Σ S p. (B.5 Derentatng eq. (B.5 wth repect to p, we have { (, (, (, } ( dp d π D p ε S p S p ε * 0 dp = D ( p, ε0 S ( p p D ( p, ε 0 S ( p dp d ( (, { } (, { } ( φ ( ε { } ( dp φ C D p ε Σ D p ε Σ dp dp ( ( S p p C D p, Σ S p. (B.6 We now apply everal reult rom chapter 4 to mply eq. (B.6. Begn by ( conderng the dervatve (, { } (B.6: d D p ε Σ dp on the rght-hand de o eq. d dp ( D ( p, ε Σ { } { } ( p, ε dσ p ; S ( p, D ( p, ε 0 S ( p dd = dp dp ; 37

390 d dp ( D ( p, ε Σ { } { } { } dp { } Σ Σ = D p S p dp (, ε dp p dp S p ( ( Σ D ( p, ε 0 S ( p. D ( p, ε 0 S ( p (B.7 Examnng varou term on the rght-hand de o eq. (B.6 and (B.7, we note the ollowng mplcaton under the aumpton o chapter 3 and 5: { } Σ p β gven the Ane Spot Market SF aumpton (text eq. (5. Σ D p, 0 S p = Σ q = 0 by derentatng text eq. { } ( ε ( { } (5., rewrtten n term o Σ { p ; q, q } Σ S p = Σ q = φ by adaptng text eq. (5.3 or = { } ( { } ( { } (, ε { } ( ( C D p Σ = C Σ = C q = c + cq recallng the 0 denton o { } aumpton (text eq. (5. (, ε { } Σ, 36 and the Ane Margnal Producton Cot Functon ( { } ( ( C D p Σ = C Σ = C q = c by the ame reaonng a above 36 Dened ymmetrcally to Σ { } n eq. (B.. 373

391 From the addtve eparablty o (, 0 (, ε 0 0 ( D p D p D p ε n text eq. (3.8, we may wrte By mlar reaonng a above, we may wrte D ( p, ε 0 D 0 ( p (, more mply a D p ε = γ gven the Ane Spot Market Demand Functon aumpton rom ecton 5. Ung thee reult to mply eq. (B.7, we get d dp dp ( D ( p, ε { } ( γ β φ S ( p Σ = +. (B.8 dp Ung the above reult along wth eq. (B.8 to mply eq. (B.6 yeld { (, (, (, } ( dp d π D p ε S p S p ε * 0 dp = D 0 ( p S ( p p D 0 ( p S ( p dp dp dp φ c γ + β φ S p S p dp dp ( ( ( ( ( φ p c + cq S p 0, or, rearrangng, 374

392 { (, (, (, } ( dp d π D p ε S p S p ε * 0 dp = D 0 ( p S ( p p D 0 ( p S ( p dp dp φ + c( γ + β S ( p cφ S ( p dp ( ( φ p c + cq S p 0. ecallng text eq. (5.4 or =, we may ubttute φ or + c( γ + β n the above equaton and collect term to obtan { (, (, (, } ( dp d π D p ε S p S p ε * 0 dp φ φ = D 0 ( p S ( p cφ S ( p dp φ ( ( φ ( ( p D p S p p c cq S + p 0 0. (B.9 ecall that we dened varant o text eq. (5.9, we may wrte a q a the pot market SF { p ; q, q } { ;, } ( 0 Σ, whch, ung a q Σ p q q = φq c β + β p. (B.0 Ung eq. (B.0, we may mply urther the ourth term on the rght-hand de o eq. (B.9. Namely, wrte the leadng actor o th term, φ p ( c0+ cq ( (, a ( { } φ p c cq φ p c c φq c β β p + 0 = , and collect term n p and c 0 to obtan 375

393 ( ( ( φ p c0 cq φ cβ p c0 cβ cφq + =. Ung text eq. (5.4 or = and lettng the rght-hand de o the above a q D ( p, ε 0 S ( p =, we may mply ( { } ( (, ( φ p c + cq = φφ p c + c D p ε S p (B. We now conoldate the reult n th ecton. Begn by ubttutng eq. (B. nto eq. (B.9: { (, (, (, } ( dp d π D p ε S p S p ε * 0 dp φ φ = D 0 ( p S ( p cφ S ( p dp φ p D 0 p S p ( ( ( ε ( ( p { c c D p S p } S ( p φφ +,. 0 0 Now ubttute th reult nto the expectaton term on the rght-hand de o eq. (B.9, ung D 0 ( p and D 0 ( p n place o D ( p, ε 0 and D ( p, ε 0 a beore:, repectvely, 376

394 (, (, ε ( dp = D 0 ( p S ( p + p D 0 ( p S ( p tot d π p S p 0 dp φ φ E D 0 ( p S ( p + cφ S ( p dp φ ( ( { ( ε ( } + p D p S p 0 ( p c c D p S p S ( p p } + φφ +,. 0 0 (B. * For an equlbrum p p ( ε 0 =, ecton 5.4 how that p and ε 0 are one-to-one. Hence, we may condton n eq. (B. on p ntead o on ε 0, a n eq. (B.9. Dtrbutng the expectaton operator nde o the brace n the above expreon and rearrangng, we have (, (, ε ( dp tot d π p S p 0 ( ( ( ( ( ( φ φ = D p S p E p p p D p S p 0 0 de p p D 0 ( p S ( p cφ S ( p dp φ ( ( p p { c c D ( p ε S ( p } S ( p φφ E +,. 0 0 (B.3 Conder now the gn o the ve term appearng on the rght-hand de o eq. (B.3. Baed on varou aumpton n the text, we may gn only the rt and ourth o thee term or the general orward market problem, a ndcated below: 377

395 (, (, ε ( dp tot d π p S p 0 ( ( ( ( ( = D 0 p S p E p p p D 0 p S p ( de p p φ φ D 0 ( p S ( p cφ S ( p dp φ? φφ ( E ( p p { c + c D ( p, ε S ( p } S ( p. 0 0?? + (B.4 The three term marked wth? on the rght-hand de o eq. (B.4 are o ndetermnate gn. 363 Becaue o thee analytcal ndetermnace, we retrct the quanttatve analy to an mputed admble orward market SF S ( p or rm (computed numercally, a doman o orward market prce p, and parameter value uch that the rght-hand de o eq. (B.4 negatve, o that (, (, ε ( dp tot d π p S p 0 < 0. (B.5 That, n text chapter 7 pecc numercal example or the orward market problem, we very numercally or the equlbra we tudy that the econd-order condton expreed by eq. (B.4 and nequalty (B.5 n act hold. 363 Note that we derved expreon or D 0 ( p, E ( p p, and E ( chapter 6 and 7, and can obtan an expreon or D 0 ( p by derentatng D0 ( p 378 d p p dp n text. Even we were to ue thee expreon to mply eq. (B.4, however, t would not elmnate the ndetermnacy o term gn n th equaton.

396 We conclude by notng that, under our aumpton, eq. (B.4 and nequalty (B.5 compre the econd-order ucent condton or p to be a global prot maxmum or rm. We may make a completely ymmetrc argument or rm global prot maxmum. 379

397 When you reach an equlbrum n bology you are dead. Arnold Mandell Appendx C: Interpretaton o ψ ( p and the orward market equlbrum optmalty condton THIS APPENDIX derve eq. (4.43 n the text, rewrtten below a eq. (C., ψ ( {,, ε } and provde an nterpretaton o ( p dπ q q dq p p p p dp dp * =E + E optmalty condton. Eq. (C. tate that ( p (, (C. ψ a well a o the orward market equlbrum ψ the expected change n rm equlbrum optmal provonal pot prot caued by a margnal change n p whle nettng out the expected change n t orward contract ettlement payment, ( p q, due to th change n p. In other word, ( p ψ capture the eect o a margnal change n 380

398 p on rm expectaton o pot market revenue le producton cot. Later n chapter 8, we alo denty ( p partcpaton n the orward market. ψ a rm trategc eect, accountng, n part, or the rm We retate the FOC or the orward market, text eq. (4.9, gven the equlbrum mpoed or th market later n text chapter 4. A explaned n that chapter, th ental. replacng D ( p, 0 S ( p ε pontwe (.e., or each 0 (recall text eq. (4.35 and the aocated dcuon,. replacng D ( p, ε 0 wth D0 ( p (ung text eq. (3.3, and ε wth S ( p 3. condtonng the expectaton on p rather than on 0 ε (ee note 40. Makng thee change to text eq. (4.9, we may retate th equaton a {, (, ε } d π p S p tot 0 dp ( ( ( = S p + p D p S p 0 { (, (, ε } * dπ S p S p + E p dp = 0. (C. Equatng eq. (C. and text eq. (4.37, a mpled veron o the ame FOC, we have that 38

399 { p ; S ( p, S ( p } ( { (, (, ε } * dπ S p S p S ( p + p D 0 ( p S ( p + E p dp = S ( p + p E ( p p D 0 ( p S ( p E { p C ( Σ{ p ; S ( p, S ( p } Σ{ p ; S ( p, S ( p } D 0 ( p S ( p q Σ + S p p. q (C.3 Next, recall text eq. (4.39 or ψ ( p, rewrtten below a eq. (C.4: ψ { ( { } { p ; S ( p, S ( p } ( p E p C Σ p ; S ( p, S ( p Σ D p S p q { p ; S ( p, S ( p } ( ( 0 Σ. + S ( p p q (C.4 Subttutng eq. (C.4 nto eq. (C.3, mplyng, and olvng or ψ ( p ψ { (, (, ε } dπ S p S p * ( p = E p + E p p D 0 p S p dp, we have that ( ( (. (C.5 Ung text eq. (3.43 and (3.4 to mply eq. (C.5 and mpong Nah equlbrum n the orward market, we may rewrte th equaton a 38

400 ψ ( {,, ε } dπ q q dq p p p p dp dp * =E + E (, (C.6 whch eq. (C. (and alo text eq. (4.43, the reult that we et out to how. To conclude th appendx, we provde an nterpretaton o the orward market equlbrum optmalty condton (4.4 n the text. Gven an arbtrary orward market demand hock ε 0, rm ace the orward market redual demand uncton (, ε 0 (. Frm orward market revenue (, 0 D p S p wrtten a (, ε = (, ε ( p ε may then be p 0 p D p 0 S p. (C.7 The dervatve o (, 0 p ε wth repect to p, rom eq. (C.7, ( p, ε 0 p (, ε ( 0 (, ε0 ( = D p S p + p D p S p. (C.8 Subttutng eq. (C.8 nto the econd equalty o text eq. (4.9, rm FOC or the orward market, we get ( p, ε d D ( p 0 S ( p S ( p π ε ε + E ε 0 = 0. (C.9 dp * {,,, } 0 p Eq. (C.9 a retatement o rm orward market FOC, text eq. (4.9. It the neceary condton or optmalty (aumng nterorty or the problem π ( S (, ε tot* 0 ( { } * ( ( ( ( max p, ε0 + E π D p, ε0 S p, S p, ε ε 0, p (C.0 383

401 whch tel a retatement o text eq. (4.6, the orgnal orward market problem. Eq. (C.9 ndcate that, gven rm mputed orward market SF S ( p hock 0 * ε, rm optmal prce p p ( ε 0 margnal change um to zero: and the demand wll be uch that the ollowng two. the margnal change n orward market revenue due to ncreaed p, and. the margnal change n expected optmal provonal pot market prot (whch nclude the orward contract ettlement payment due to ncreaed p. 384

402 I have yet to ee any problem, however complcated, whch, when you look at t n the rght way, dd not become tll more complcated. Poul Anderon Appendx D: Computatonal detal o the pot market SFE under the mpled ane example D. Comparatve tatc o rm pot market SF lope β and parameter φ wth repect to the parameter c, c, and γ TEXT EQ. (5.4 or the parameter φ φ (, =,;, (D. + c + ( γ β where under our parametrc aumpton, we note (recallng the expreon (5.8 n the text that 0< φ <. (D. ewrtng text eq. (5.6 (or (5. or a generc rm and ung eq. (D. or φ, we may wrte the lope o rm pot market SF β a 385

403 γ + β β =. (D ( γ β c Interchangng arbtrary ubcrpt and n eq. (D.3, we may expre the correpondng lope β or rm a γ + β β =. (D ( γ β c Clearly, we could olve eq. (D.3 and (D.4 or β (,, β c c γ = explctly. To gn the dervatve o β wth repect to the parameter c, c, and γ, however, t mpler to derentate eq. (D.3 and (D.4 mplctly, a done n ubecton D.. D..3 below. Ung thee reult and eq. (D., we may mlarly gn dervatve o ( c, c, φ = φ γ wth repect to thee parameter, a n ubecton D..4 D..6. A n the text, we aume throughout th appendx that β > 0, =,. 364 D.. The partal dervatve o ( c, c, β γ wth repect to c From eq. (D.3, we may partally derentate β (,, β c c γ ollow: = wth repect to c a 364 A noted n text ecton.3, there a unque oluton to eq. (D.3 and (D.4 or whch th the cae. Namely, udkevch (999 reult mple that the oluton to eq. (D.3 and (D.4 or β = β c, c, γ (, =, ; ha exactly one root n whch both β and β are potve. ( 386

404 387 ( ( ( ( ( ( (,,, c c c c c c c c c c c β γ β β β γ β γ β γ β γ β β γ β γ β = = = + + whch, ung the denton (D., we may wrte a ( c c β β φ γ β = +. (D.5 From eq. (D.4, we may derentate β wth repect to c to obtan ( ( ( ( (,,, c c c c c c c c c c c β γ β β β γ β γ β γ β β γ β = = + + = + + whch, agan ung the denton (D., we may wrte a c c β β φ =. (D.6 Subttutng eq. (D.6 nto eq. (D.5 yeld

405 β c β = φ φ γ + β c ( whch, olvng or β c, become ( + φ γ β = c φφ β. (D.7 ewrtng text eq. (5.6 or generc rm and, we get ( β = φ γ + β. (D.8 We may ubttute rom eq. (D.8 to mply eq. (D.7 a β ( β = c φφ. (D.9 Snce the ubcrpt n the expreon (D. arbtrary, we have that φφ 0 >. (D.0 Gven nequalty (D.0, we conclude rom eq. (D.9 that β c < 0. (D. D.. The partal dervatve o ( c, c, β γ wth repect to c From eq. (D.6, by ymmetry, the partal dervatve o β (,, β c c γ c = wth repect to 388

406 β c β = φ c. (D. Lettng = n nequalty (D., we have that β c < 0 whch, together wth eq. (D., mple that β c < 0. (D.3 D..3 The partal dervatve o ( c, c, β γ wth repect to From eq. (D.3, we may partally derentate β (,, β c c γ ollow: γ = wth repect to γ a β γ β = ( c, c, γ γ β β + + c( γ + β ( γ + β c + γ γ =, c ( γ β + + whch mple to β + β γ = γ + c γ + β (. Ung the denton (D., th equaton become β γ β = φ +. (D.4 γ Interchangng arbtrary ubcrpt and n eq. (D.4, we may wrte th equaton a 389

407 β β = φ +. (D.5 γ γ Subttutng eq. (D.5 nto eq. (D.4 yeld β γ β = φ + φ +, γ whch, olvng or β γ, yeld β γ ( + φ φ = φ φ. (D.6 Ung nequalty (D.0, we conclude rom eq. (D.6 that β γ > 0. (D.7 We collect the gn o the dervatve o β n nequalte (D., (D.3, and (D.7 n the rt column o Table 5. n the text. D..4 The partal dervatve o ( c, c, We may partally derentate φ φ( c, c, γ ollow: φ γ wth repect to c = rom eq. (D. wth repect to c a β ( γ + β + c φ c = c ( c γ β + +, whch, ung the denton (D., become 390

408 φ β = φ γ + β + c. (D.8 c c Subttutng eq. (D.9 nto eq. (D.6 and the reult, n turn, nto eq. (D.8 or β c yeld ( β φ = φ γ + β + c φ, c φφ whch we may wrte a ( + ( φ γ β c β φ φ = ( φφ. (D.9 c φφ γ + β Solvng eq. (D.8 or ( β γ + β = φ, we may ubttute or th expreon n the lat term n bracket on the rght-hand de o eq. (D.9 to obtan φ c ( + φ γ β = ( φφ cβφφ φφ. (D.0 Next, on the rght-hand de o eq. (D.0, we ( ubttute or φ( γ β (D.8 n the rt actor, and ( rearrange the bracketed expreon to obtan + rom eq. φ β φ = φφ ( φ + cβ c. (D. φφ By rearrangng text eq. (5.7, we nd that φ + c β = whch, ubttuted nto eq. (D., yeld mply 39

409 βφ ( φφ φ = c φφ. (D. Snce the ubcrpt n the expreon (D. arbtrary, we have that φφ 0 > n eq. (D.. ecallng the nequalty (D.0 and our parametrc aumpton, both the numerator and the denomnator o the rato on the rght-hand de o eq. (D. are potve. We conclude rom eq. (D., thereore, that φ < 0. (D.3 c D..5 The partal dervatve o ( c, c, We may partally derentate φ φ( c, c, γ ollow: φ γ wth repect to c = rom eq. (D. wth repect to c a β c φ c = c c + + ( γ β, whch, ung the denton (D., become φ β = cφ c c. (D.4 Interchangng arbtrary ubcrpt and n eq. (D.9, we may wrte β ( β = c φφ. (D.5 Subttutng eq. (D.5 nto eq. (D.4 yeld 39

410 ( β φ = cφ, c φφ whch mple to φ c c ( β φ = φφ. (D.6 ecallng nequalty (D.0 and our parametrc aumpton, both the numerator and the denomnator o the rato on the rght-hand de o eq. (D.6 are potve. We conclude rom eq. (D.6, thereore, that φ > 0. (D.7 c D..6 The partal dervatve o ( c, c, We may partally derentate φ φ( c, c, γ ollow: φ γ wth repect to = rom eq. (D. wth repect to γ a γ β c + φ γ γ = c γ β + + (, whch become, ung the denton (D., φ β = cφ +. (D.8 γ γ Interchangng arbtrary ubcrpt and n eq. (D.6, we may wrte 393

411 β γ ( + φ φ = φ φ. (D.9 Subttutng eq. (D.9 nto eq. (D.8 yeld ( + φ φ φ = cφ +, γ φφ whch mple to ( + φ φ c φ = γ φ φ. (D.30 ecallng nequalty (D.0 and our parametrc aumpton, both the numerator and the denomnator o the rato on the rght-hand de o eq. (D.30 are potve. We conclude rom eq. (D.30, thereore, that φ < 0. (D.3 γ We collect the gn o the dervatve o φ n nequalte (D.3, (D.7, and (D.3 n the econd column o Table 5. n the text. D. Comparng dervatve o rm pot market SF lope β and β wth repect to the lope c and c o a rm own and the rm rval margnal cot uncton Th ecton prove nequalte (5.4 and (5.5 n the text, rewrtten below a eq. (D.3 and (D.33 (, =, ; : 394

412 β c > β c, (D.3 and β β > c c, (D.33 whereby nequalte (D.3 and (D.33 obtan at all parameter value content wth our parametrc aumpton. Equaton (D.9 gve an expreon or β c. To how that nequalte (D.3 and (D.33 hold, we rt need to derve expreon or β c and β c rom the analy n ecton D.. Subttutng eq. (D.5 nto eq. (D., we may wrte that ( β β = φ, c φφ or β ( β φ = c φφ. (D.34 Interchangng arbtrary ubcrpt and n eq. (D.34 yeld β ( β φ = c φφ. (D.35 Now we may demontrate that nequalty (D.3 hold. Suppoe, n contradcton, that t doe not, that, 395

413 β c β c. (D.36 Subttutng eq. (D.9 and (D.34 nto nequalty (D.36 yeld ( ( β β φ φφ φφ, whch we may mply and rearrange a β φ ( β. (D.37 Solvng eq. (D.8 or the rato β φ yeld β γ β φ = +. (D.38 Subttutng eq. (D.38 nto nequalty (D.37, we nd that ( γ β ( β +, or mplyng, ( γ γ β 0 +. (D.39 Snce nequalty (D.39 ale gven our parametrc aumpton that γ > 0 and β > 0, we have a contradcton. Thu, the uppoton (D.36 ale and we conclude that nequalty (D.3 (dentcal to nequalty (5.4 hold or our parametrc aumpton. 396

414 We turn next to nequalty (D.33, and how that t hold. Suppoe, n contradcton, that t doe not, that, β β c c. (D.40 Subttutng eq. (D.9 and (D.35 nto nequalty (D.40 yeld ( ( β β φ φφ φφ, whch mple to φ. (D.4 Snce nequalty (D.4 ale gven the expreon (D., we have a contradcton. Thu, the uppoton (D.40 ale and we conclude that nequalty (D.33 (dentcal to nequalty (5.5 alo hold under our parametrc aumpton. D.3 The geometry o the partal reacton uncton β =( β In text ecton 5.3, we dened the partal reacton uncton ( β plane. Ung eq. (D.3 above or β a β, we may wrte ( β β n the β - γ + β β = ( β (, =,;. (D.4 + c + ( γ β The preent ecton demontrate that the uncton ( properte clamed n text ecton 5.3 and depcted n text Fgure 5.. β n eq. (D.4 have the 397

415 We rt how that each uncton ( t argument 0 β >. Takng the dervatve o ( β everywhere ncreang and concave n β wth repect to β, we get ( β ( γ β ( γ β + c ( γ + β + c + + c =, whch mple to From eq. (D.43, the econd dervatve ( β ( β = > 0. (D.43 + c ( γ + β c ( β = < 0. (D.44 + c ( 3 γ + β From eq. (D.43 and (D.44 and gven our parametrc retrcton, ( ncreang and concave n t argument β > 0. Next, conder how the uncton ( β β everywhere + β behave a 0. From eq. (D.4, γ + β lm ( lm β = + + β 0 0 β + c + ( γ β, or mply γ lm ( β = > 0. (D.45 + c γ + β 0 From eq. (D.43, 398

416 lm ( β = lm + + β 0 β 0 + c + ( γ β, whch A 0 β lm = > 0. (D.46 + β 0 ( β ( + c γ +, eq. (D.45 and (D.46 ndcate that ( ntercept γ ( + c γ wth a potve lope ( c γ eq. (D.4, β approache a potve β -ax +. β a We now examne the lmtng behavor o the uncton ( β. From γ + β lm ( β = lm β + c + β ( γ β = lm, β + c γ + β whch mply lm ( 0 β = >. (D.47 c β From eq. (D.43, β lm ( β = β lm + c + ( γ β, whch yeld 399

417 ( β lm = 0. (D.48 β A β, eq. (D.47 ndcate that ( β approache c (an upper bound, by nequalty (D.43, whle by eq. (D.48, the lope ( β goe to zero. Fnally, we note that the above properte guarantee a unque nterecton o the partal reacton uncton ( β and ( β n the potve orthant correpondng, naturally, to rm equlbrum choce o β and β. 365 That, there a unque oluton (, β β correpondng to a trctly ncreang pot market SF or each rm. 365 See alo note

418 Someone told me that each equaton I ncluded n the book would halve the ale. Stephen Hawkng, A Bre Htory o Tme Appendx E: Computatonal detal o the dervaton o optmal orward market upply uncton and reult o numercal example E. Supportng analy or text equaton (7.5 and (7.6 THIS SECTION provde ome algebrac detal and explct parameter denton or the dervaton o text eq. (7.5 and (7.6. Gven the denton n text ecton 7., we may recat text eq. (7. and (7. the rm repectve orward market equlbrum optmalty condton a ( + ( + + ( + C,S ( p C,S ( p C,3 p C,4 S ( p + C3,S ( p + C3,S ( p + C3,3 p + C 3,4 = 0 C S p C S p C p C S p,,,3,4 (E. and 40

419 ( + ( + + ( + C,S ( p C,S ( p C,3 p C,4 S ( p + C3,S ( p + C3,S ( p + C3,3 p + C 3,4 = 0. C S p C S p C p C S p,,,3,4 (E. We may dene the coecent C kl, n eq. (E. and (E. by drect comparon wth text eq. (7. and (7. a ollow: C C C C C C C C C C C C ( ( a ν, a a ( a ( ν, a a ( a ν a( a = φω φ + λ σω γ ω,,,3 ν, ν,4 = φω a + λσω ν a γ ωa ωb + γ ωa ν σ ν, ν,,3,4 0 3, 3, 3,3 = φω φ + λ σω γ ω φ γ ω λ σ ω γ ω = +, σ ( (, = cφφ + λ σωa( γω a + ωa( φ { λσ ν ω ( ( }, a φφ φ γ ω + a + φ = ωa( φ { λσ ν ω ( ( }, a φφ + φ γ ω a + φ = ( ( γωa { λσω ν ( }, a + φ γω a + φ + φφ = c φφ + λσω ν a( γ ωa σ ν, ν + ωa ωb + ( γ ωa ν { λσω ν ( ( }, a φφ φ γωa φ σ + + ν = ( φ + λσ ν ω (, a γ ω + a = φ, (E.3 γω a =, ωa σ ν, ν = ωb + ( γ ωa ν, σ ν 3,4 and 40

420 C C C C C C C C C C C C { } ( ( ( a ν a a, φφ λσω ν ( a γ ω a ωa( φ λσ ν ω a ( φφ φ( γ ωa φ (( γωa λσω a φ( γωa φ φφ φφ λσω ν a( γ ωa = c + + { }, + + = { ν + + } +, = c + σ ν + ωa ωb + ( γ { ( ( }, ν ωa ν λ σ ν ω, a φφ + φ γ ω a + φ σ ν = φωa( φ λσω ν (, a γ ω + a = φωa( φ + λσω ν (, a γ ω a = φ ( γ ωa + λσ ν ω (, a γ ω a σ ν, ν = φω a + λσω ν ( a γ ω a ωb + ( γ ωa ν, σ ν = φ, = ω φ λ σ ω φφ + φ γ ω + φ,,,3,4 0,,,3,4 3, = ( φ + + λσ ν ωa( γ ωa, (E.4 3, γωa 3,3 =, ωa σ. σ ν ( ν, ν 3,4 = ωb + γ ωa ν We next how that the coecent o the orm PP PP n text eq. (7.0 k k ++ and (7. are quadratc orm n the element o S ( p, where ( ( ( ( S p S p S p p ++ (E.5 rom text eq. (7.3. Ung the denton o P k rom text eq. (7.7, we may expre the coecent n text eq. (7.0 and (7. n term o the coecent C, above a kl 403

421 ollow. In text eq. (7.0, PP PP ( ( ( ( ( ( ( ( PP PP = C S p + C S p + C p + C,,,3,4 C S p + C S p + C p + C,,,3,4 C S p + C S p + C p + C,,,3,4 C S p + C S p + C p + C,,,3,4, (E.6 or expandng the product, PP PP = ( C C C C S ( p,,,, ( C,C, C,C, S ( p ( C,3C,3 C,3C,3 p ( C,C, C,C, C,C, C,C, S ( p S ( p ( C,C,3 C,3C, C,C,3 C,3C, S ( p p ( C,C,3 C,3C, C,C,3 C,3C, S ( p p ( C,C,4 C,4C, C,C,4 C,4C, S ( p ( C,C,4 C,4C, C,C,4 C,4C, S ( p ( C,3C,4 C,4C,3 C,3C,4 C,4C,3 p ( C,4C,4 C,4C, (E.7 In text eq. (7., PP PP ut the addtve nvere o PP PP gven by eq. (E.6, o we have rom eq. (E.7 that 404

422 PP PP = ( C C C C S ( p,,,, ( C,C, C,C, S ( p ( C,3C,3 C,3C,3 p ( C,C, C,C, C,C, C,C, S ( p S ( p ( C,C,3 C,3C, C,C,3 C,3C, S ( p p ( C,C,3 C,3C, C,C,3 C,3C, S ( p p ( C,C,4 C,4C, C,C,4 C,4C, S ( p ( C,C,4 C,4C, C,C,4 C,4C, S ( p ( C,3C,4 C,4C,3 C,3C,4 C,4C,3 p ( C,4C,4 C,4C, (E.8 In text eq. (7.0, PP PP 3 3 ( ( ( ( ( ( ( ( PP PP = C S p + C S p + C p + C 3 3,,,3,4 C S p + C S p + C p + C 3, 3, 3,3 3,4 C S p + C S p + C p + C 3, 3, 3,3 3,4 C S p + C S p + C p + C,,,3,4, (E.9 or expandng the product, 405

423 PP PP = ( C C C C S ( p 3 3, 3, 3,, ( C,C3, C3,C, S ( p ( C,3C3,3 C3,3C,3 p ( C,C3, C,C3, C3,C, C3,C, S ( p S ( p ( C,C3,3 C,3C3, C3,C,3 C3,3C, S ( p p ( C,C3,3 C,3C3, C3,C,3 C3,3C, S ( p p ( C,C3,4 C,4C3, C3,C,4 C3,4C, S ( p ( C,C3,4 C,4C3, C3,C,4 C3,4C, S ( p ( C,3C3,4 C,4C3,3 C3,3C,4 C3,4C,3 p ( C,4C3,4 C3,4C, (E.0 In text eq. (7., PP PP 3 3 ( ( ( ( ( ( ( ( PP PP = C S p + C S p + C p + C 3 3,,,3,4 C S p + C S p + C p + C 3, 3, 3,3 3,4 C S p + C S p + C p + C 3, 3, 3,3 3,4 C S p + C S p + C p + C,,,3,4, (E. or expandng the product, 406

424 PP PP = ( C C C C S ( p 3 3, 3, 3,, ( C,C3, C3,C, S ( p ( C,3C3,3 C3,3C,3 p ( C,C3, C,C3, C3,C, C3,C, S ( p S ( p ( C,C3,3 C,3C3, C3,C,3 C3,3C, S ( p p ( C,C3,3 C,3C3, C3,C,3 C3,3C, S ( p p ( C,C3,4 C,4C3, C3,C,4 C3,4C, S ( p ( C,C3,4 C,4C3, C3,C,4 C3,4C, S ( p ( C,3C3,4 C,4C3,3 C3,3C,4 C3,4C,3 p ( C,4C3,4 C3,4C, (E. The rght-hand de o each o eq. (E.7, (E.8, (E.0, and (E. are ndeed ++ quadratc orm n the element o S ( p ( n x( n. Below, we pecy explctly the + + ymmetrc coecent matrce Q k aocated wth thee our quadratc orm. Text eq. (7.3, rewrtten below a eq. (E.3, ( Qk ( P Pk Pk P S p S p ++ ++, (E.3 dened mplctly the coecent matrx Q k wth reerence to the quadratc orm PP PP (rom the let-hand de o eq. (E.7, (E.8, (E.0, and (E.. ecallng k k the our quadratc orm n text eq. (7.5 and (7.6 o the orm ( Q k ( S p S p Q 3 whch we now pecy., each ha a correpondng coecent matrx Q, Q, Q 3, and Denote the element n row x and column y o the coecent matrx Q k a xy q k. In 407

425 term o the coecent C kl, gven n the expreon (E.3 and (E.4, the element o Q are a ollow (ee the rght-hand de o eq. (E.7: Dagonal element o Q,,,, q = C,C, C,C, 33 q = C,3C,3 C,3C,3 44 q = C,4C,4 C,4C,4 O-dagonal element o Q q = C C C C q = q = ( C,C, + C,C, C,C, C,C, 3 3 q = q = ( C,C,3 + C,3C, C,C,3 C,3C, 4 4 q = q = ( C,C,4 + C,4C, C,C,4 C,4C, 3 3 q = q = ( C,C,3 + C,3C, C,C,3 C,3C, 4 4 q = q = ( C,C,4 + C,4C, C,C,4 C,4C, = = ( C,3C,4 + C,4C,3 C,3C,4 C,4C,3. q q (E.4 Snce Q = Q, the element o Q are mply the addtve nvere o the expreon n (E.4 above (ee the rght-hand de o eq. (E.8. Smlar to the above, n term o the coecent C kl, gven n the expreon (E.3 and (E.4 we wrte the element o Q a ollow: 408

426 Dagonal element o Q,,,, q = C,C, C,C, 33 q = C,3C,3 C,3C,3 44 q = C,4C,4 C,4C,4 O-dagonal element o Q q = C C C C q = q = ( C,C, + C,C, C,C, C,C, 3 3 q = q = ( C,C,3 + C,3C, C,C,3 C,3C, 4 4 q = q = ( C,C,4 + C,4C, C,C,4 C,4C, 3 3 q = q = ( C,C,3 + C,3C, C,C,3 C,3C, 4 4 q = q = ( C,C,4 + C,4C, C,C,4 C,4C, = = ( C,3C,4 + C,4C,3 C,3C,4 C,4C,3. q q (E.5 In term o the coecent C kl, gven n the expreon (E.3 and (E.4, the element o Q 3 are a ollow (ee the rght-hand de o eq. (E.0: 409

427 Dagonal element o Q 3, 3, 3,, q3 = C,C3, C3,C, 33 3 q3 = C,3C3,3 C3,3C,3 44 q3 = C,4C3,4 C3,4C,4 O-dagonal element o Q 3 q = C C C C q3 = q 3 = ( C,C3, + C,C3, C3,C, C3,C, 3 3 q3 = q3 = ( C,C3,3 + C,3C3, C3,C,3 C3,3C, 4 4 q3 = q3 = ( C,C3,4 + C,4C3, C3,C,4 C3,4C, 3 3 q3 = q3 = ( C,C3,3 + C,3C3, C3,C,3 C3,3C, 4 4 q 3 = q3 = ( C,C3,4 + C,4C3, C3,C,4 C3,4C, = 3 = ( C,3C3,4 + C,4C3,3 C3,3C,4 C3,4C,3. q q (E.6 In term o the coecent C kl, gven n the expreon (E.3 and (E.4, the element o Q 3 are a ollow (ee the rght-hand de o eq. (E.: 40

428 Dagonal element o Q 3, 3, 3,, 3 q = C,C3, C3,C, q = C,3C3,3 C3,3C, q = C,4C3,4 C3,4C,4 O-dagonal element o Q 3 q = C C C C 3 q = 3 q = ( C,C3, + C,C3, C3,C, C3,C, q = 3 q = ( C,C3,3 + C,3C3, C3,C,3 C3,3C, q = 3 q = ( C,C3,4 + C,4C3, C3,C,4 C3,4C, q = 3 q = ( C,C3,3 + C,3C3, C3,C,3 C3,3C, q = 3 q = ( C,C3,4 + C,4C3, C3,C,4 C3,4C, = 3 = ( C,3C3,4 + C,4C3,3 C3,3C,4 C3,4C,3. q q (E.7 xy Some mplcaton o the element q k o the coecent matrce Q k n the expreon (E.4 (E.7 would be poble we were to ubttute the denton o the coecent C kl, rom the expreon (E.3 and (E.4. Thee mplcaton are xy nucent, however, to uty recatng the element q k n (E.4 (E.7 n term o xy the underlyng prmtve parameter. A done above, repreentng the q k a uncton o the coecent C kl, relatvely tranparent and convenent or our purpoe. The MATLAB code ued to olve the ytem n text eq. (7.5 and (7.6 ue th repreentaton o the problem parameter n the rm orward market equlbrum optmalty condton. 4

429 E. Theory and computaton o ngularte n the ytem o text equaton (7.3 In text ecton 7.., we clamed that the ytem (7.3 an example o a ngular qualnear ODE ytem. In th ecton, we explan why th termnology approprate. Sngular ODE bear a cloe reemblance to but are dtnct rom o-called derental-algebrac equaton (DAE, commonly expreed n the orm (aber and henboldt 00, 89 (, ( x m x = x x 0, n m = g x (E.8 where ( x, x m n m. 366 To nvetgate the dtncton between ngular ODE and DAE, rt conder the general orm or an mplct ODE, ( F x, x = 0, (E.9 n n n where (lettng r x or notatonal clarty F ( xr, : a ucently mooth uncton. 367 I the (partal dervatve DF ( x, r nvertble at a pont (, r x r, then eq. (E.9 clearly reducble to an explct ntal-value problem wth ntal condton x( 0 = x0 and x( 0 = r0 other hand DF ( x, r not nvertble at (, r 0 0, and the tandard theory o ODE apple. I on the x r, then eq. (E.9 ether a DAE or a A aber and henboldt (00, 90 pont out, DAE need not be clearly dvble nto derental and algebrac component (a the cae n the ytem (E.8 havng m derental equaton and the n m algebrac equaton. Whle (E.8 a amlar orm or DAE rom numerou F x, x = 0, a we dcu urther applcaton, they may alo have the more general mplct orm o ( below. 367 Th dcuon ollow cloely that o aber and henboldt (00, 90. 4

430 ngular ODE. The dtncton between the two le n how the ngularty o DF ( x, r at ( x, r aect the total dervatve o ( x, r 0 0 a DF ( x, r F wth repect to both argument, denoted. Equaton (E.9 claed a a DAE and only two condton hold:. The dervatve DF ( x, r urectve, depte the ngularty o D ( x, r 0 0 (Note that when DF ( x, r nvertble, D ( x, r uncton map onto.. The rank o D ( x, r o (, x r. 0 0 r r In all other cae partcularly when (, pont ( x, r at whch D ( x, r 0 0 r r F. 0 0 F urectve,.e., the F contant (and hence not ull on ome neghborhood r x r may be approxmated arbtrarly cloely by 0 0 F nvertble eq. (E.9 a ngular ODE. Now conder condton and above wth repect to the mplct equaton n the text, the ytem (7.35, or the problem at hand. Whether the rt condton above ated depend, n general, on the parameter o the problem, o t wll be eaer to proceed by examnng the econd condton above. eplacng x wth ( (E.5 and now lettng r S + ( p = S ( p S ( p ++ a ( S, r S ++ (rom eq., we may wrte F ( x, r F. Let p 0 be a prce, and chooe the augmented vector o SF ( ( ( ( S p S p S p p + uch that the pont + S ( p0 le on the ngular 43

431 locu (ee text ubecton Next, let S0 S ( p 0 be the correpondng doubly-augmented vector o SF evaluated at p 0, and dene r0 S + ( p0 uch that r ++ ( 0, 0 DF S r ngular and o rank k 0. Then, t clear rom the geometry o the ngular locu that although Dr ( S0, r0 p p arbtrarly cloe to F not nvertble, there ext prce p (denng S00 S ( p 00 and r00 S + ( p00 or ++ + whch DrF ( S00, r00 nvertble. Hence, there no neghborhood o S ( p0 or (, ++ whch the rank o r ( DF S p r contant at k 0 throughout. We conclude that the econd condton above (.e., that concernng contant rank neceary or the ytem (7.35 n the text to be a DAE doe not hold. Th conrm our clacaton (n text ecton 7.. o the ytem (7.35 a a ngular ODE rather than a DAE. Sngular ytem o ODE are a relatvely recent reearch ocu n mathematc. 370 For example, aber (989 wa the rt ytematc tudy o ngular qualnear (and related ODE ytem (aber and henboldt 00, 34. aber and henboldt conecture (p. 34 that th paucty o attenton may be due to a lack o apprecaton or the connecton between ngular DAE and ngular ODE. Such DAE do are naturally, or example, n the theory o electrcal network, n low problem, and 368 Here, we have ued 3 ( S p p rom text eq. ( ecall rom text ubecton 7.. and Fgure 7. that the ngular locu o the ytem (7.35 a quadratc urace n (,, S S p -pace, a ubet o 370 aber and henboldt (00, 34. Thee author alo note, however, that the cae o calar ngular ODE wa analyzed at leat a early a

432 n platcty theory. 37 Under a geometrc reducton procedure, 37 the DAE that characterze uch phenomena may be recat a ngular ODE, and a uch, are oten more amenable to analy. For our purpoe, we note that no prevou economc applcaton o ngular ODE are known to the preent author, whether n the lterature on upply uncton equlbra or, more broadly, n the eld o game theory or ndutral organzaton. The preent nvetgaton thu ugget that the theoretcal and numercal tool developed n aber and henboldt (00 may nd a new area o applcaton n olvng mult-ettlement market SFE model. Preentng the detal o aber and henboldt (00, ch. VII and XIV analy o ngular qualnear ODE would take u too ar aeld, o we mply tate ther eental reult wthout proo, noar a they apply to the ytem (7.3 n the text, our problem o nteret. Frt, denote a ngular pont 373 ( ( ( ( S p S p S p p at a prce p 0 a a mple ngular pont the ollowng two condton hold: Among other eld; ee aber and henboldt (00, 33 or relevant reerence. 37 aber and henboldt (00. See ther chapter IV or detal o th reducton procedure Augmented by a the nal element o the vector S ( p0 45, or compatblty wth ytem (7.3 (7.34 n the text. 374 egardng thee condton, recall the ollowng denton rom the theory o lnear tranormaton (de la Fuente 000, 3. Let X and Y be two vector pace dened over the ame eld F, and let T : X Y be a lnear uncton. Then:. The range o T, rge T, the ubet o Y gven by ( { ( } rge T = T X = y Y : y = T X or ome x X.. The kernel (or null pace o T, ker T, the ubet o X gven by ( { ( } T = T 0 = x X T x = 0. ker :

433 ( ++ Condton : A S ( p0 dm ker = Condton : ( ( S p0 rge ( S ( p0 G A Conder whether text eq. (7.3 (7.34 that characterze our problem aty thee two condton. We argue, rt, that no pont n the ngular locu o the ytem (7.3 n the text ate Condton above. Th becaue at every ngular pont n our problem, both text eq. (7.36 and (7.37 hold, mplyng that ++ ( S ( p0 dm ker A =. (E.0 Thereore, even beore conderng Condton, we may conclude nce Condton volated everywhere that pont n the ngular locu o the ytem (7.3 n the text are not mple ngular pont. Examnng Condton above or completene ake, th condton mple that any pont lyng on the manold at whch the ngular locu ( ++ nterect the graph o ether o the rt two term o the vector G S ( p0 375 alo ++ not a mple ngular pont. That, Condton requre that, or S ( p0 to be a mple ngular pont, t neceary that ( Q ( S p S p (E. and In word, rge T the et o vector y Y or whch T( X = y ha at leat one oluton, whle ker T the et o oluton to the homogeneou lnear ytem T( x = ecall that thee graph are alo quadratc orm. 46

434 ( Q ( S p S p (E. Mot, but not all, ngular pont n the preent problem do aty eq. (E. and (E., a text Table 7. explan. 376 aber and henboldt (00, Theorem 39. an extence theorem or oluton to ngular ODE n the neghborhood o mple ngular pont. It pot the extence o two dtnct oluton n uch a neghborhood whoe (ont graph doe not cro (.e., not tranvere to the ngular locu. In contrat, the extence theory or oluton to ngular ODE n the neghborhood o ngular pont that are not mple, accordng to aber and henboldt (00, 33, much more nvolved and vrtually untouched n the publhed lterature.... The problem when ( ++ [ dm ker A S ( p0 ] or [ ( ( S ++ p0 rge ( S ++ ( p0 G A ] are open. ecallng eq. (E.0, th tatement apple to the ytem (7.3 n the text. Apart rom extence theory or oluton, aber and henboldt (00, ch. XIV alo outlne a computatonal approach or olvng ngular ODE. Ther procedure baed on computatonal method or nonlnear algebrac equaton, that, equaton ytem lackng a dynamc component. Th procedure explot a reparameterzaton o the problem that render the equaton computable n the neghborhood o the (ertwhle ngularty. It wa orgnally developed n earler work by thee author (aber and henboldt 994a, 994b, and remarkably, applcable not only to mple ngular pont, but alo to more complex ngularte (aber and henboldt 00, 483. The 376 Condton above, o coure, uperluou n th cae nce pont on our ngular locu do not aty Condton. A noted above, the ngular locu contan only ngular pont that are not mple. 47

435 algorthm ha not yet, to the author knowledge, been mplemented ung tandard numercal analy otware package uch a MATLAB, Maple, or Mathematca. 377 Such an eort would be worthwhle to the extent that computng oluton near the ngularte o ytem uch a (7.3 n the text o concern. E.3 The MATLAB ode5 olver The MATLAB ODE olver ued n th nvetgaton named ode5. Th olver perormed qute well or the qualtatve and numercal nvetgaton o text chapter 7, permttng the author to compute SF traectore qute cloe to the ngular locu. Nonethele, t mportant to keep n mnd when nterpretng the MATLAB-baed reult o text chapter 7 that aber and henboldt procedure or olvng ngular ODE dcued n ecton E. above not relected n the olver ode5. That, the algorthm n ode5 may not be ully robut n the preence o ngularte. A a conequence, n the neghborhood o ngularte, t a pror unclear whether a partcular traectory relect underlyng theoretcal properte o the ngular ODE ytem, or whether charactertc o an SF traectory mght only be artact o the olver algorthm tel. Becaue th nvetgaton doe not explore n detal traectore behavor n the neghborhood o ngularte, we need not explore th ue urther here. The ode5 olver perormed bet wth the backward derentaton opton enabled, whch explot the o-called backward derentaton ormulae (BDF. To 377 The author ndebted to Werner henboldt or makng avalable ome FOTAN code tll under development or oluton o ngular DAE. The perormance o thee code on the problem at hand ha not yet been nvetgated; th, too, a matter or uture reearch. 48

436 undertand the eental o the BDF, rt dene S S ( p upply uncton: ( ( ( ( S = S p S p S p = a the vector o rm. (E.3 Next, expre the ytem (7.40 (7.4 n the text n vector orm a ( (, S p = g S p. Index the terate n the numercal approxmaton to the traectory ( upercrpt t, and thu wrte the t th terate o th approxmaton a (, t, t, S p. S p wth a Next, dene the backward derence operator o order 0,, nductvely a ollow: 0, t, t S = S ; For a partcular ˆ, t, t t= t {( S, p t= 0} +, t, t, t S = S S., tˆ+, tˆ+ t = tˆ, the mplct ormula or ( S, p 378 and a tep ze h (Shampne and echelt 997, ( gven the t ˆ + terate k m, tˆ+, tˆ+, tˆ+ S h g S, p = 0. (E.4 m= m 378,0,0 The rt uch terate, o coure, a gven ntal condton (, S p. 49

437 The MATLAB ode5 olver approxmate the mplct nonlnear equaton (E.4 wth mpled Newton teraton tartng wth the predcted value S k, tˆ+ m, tˆ = S. m= 0 The routne ode5 a varable order olver, meanng that the olver vare the order k o the nte derence ued to compute S, tˆ + va eq. (E The choce o order ental, n general, a tradeo between ecency (peed o computaton and tablty (roughly peakng, the property that mall perturbaton n the ntal condton lead to mall devaton n the equence o terate. In th work, we nd va expermentaton that a maxmum order o k max = 5 contently produce table oluton o the ytem (7.40 (7.4 n the text, o we ue th value a the deault maxmum order or all reult. Shampne and echelt (997, characterze the routne ode5 a havng a qua-contant tep ze h (ee eq. (E.4. By th they mean that the tep ze held contant [by the olver] durng an ntegraton unle there good reaon to change t. The good reaon, n th ntance, would be to mantan the local dcretzaton error wthn dered tolerance (ee note 380 below. In text ubecton 7.4., or example, we noted that when the SF traectory aborbed by the -locu (a wth traectory n text Fgure 7.5, the MATLAB olver al and numercal ntegraton halt. Here, olver alure ndcate that t no longer numercally eable or the olver to mantan multaneouly the ollowng two condton: 379 In addton, MATLAB permt the uer to pecy the maxmum order k to be equal to any max nteger between and 5 (ncluve. 40

438 . enorce the choen tolerance on numercal error 380 by electng a ucently mall tep ze, and. keep the tep ze large enough 38 to make acceptable progre n the ntegraton. E.4 Numercal reult o comparatve tatc analy Table 7. n text ubecton report the qualtatve eect on rm orward market quantte, q = S ( p and q S ( p =, o perturbng each o the ten element n the parameter vector Θ ; we reer to the tudy o uch eect a comparatve tatc analy (ee text ecton 7.6. The qualtatve eect documented n text Table 7. are baed on the numercal reult o chapter 7 dcrete Excel model. Th ecton report thee numercal reult. Table E. below ummarze the comparatve tatc reult produced by the dcrete Excel model. Each tet cae reported n the table ha row headng or the top 380 Any MATLAB olver (agan, we ue the olver ode5 wll compute a numercal approxmaton to the true oluton ( S (, (, p S p p to the ytem n text eq. (7.40 (7.4. Naturally, numercal error nherent n th numercal approxmaton. Numercal error are o two type, dcretzaton error, and roundo error (Moler and Moler 003, ecton 6.3. The ormer depend on the underlyng derental equaton ytem and the choen numercal method, whle the latter a uncton o computer otware and hardware. Ung current computng platorm, roundo error only lkely to become mportant very hgh accurace are requeted or the nterval o ntegraton very large. Through adutment o the tep ze h, the olver algorthm control the (local dcretzaton error (related to the order k o the numercal method, mantanng t wthn precrbed tolerance. The hgher the order, the maller the local dcretzaton error. A dcued above, we ue the hghet poble order, k = 5, n the k + 6 BDF. Gven a tep ze h or tep t, the local dcretzaton error O t ( ht = O( ht, whch lkely to be acceptably mall. We ued MATLAB deault error tolerance or relatve and abolute error; ee the program documentaton (The MathWork 00 or detal. For n-depth treatment o error analy n numercal ntegraton, conult Butcher (987 or Harer, Noerett and Wanner ( The mnmum tep ze a parameter n MATLAB ODE olver that the model uer may vary. The deault mnmum tep ze 4 ~0. 4

439 our row n the table labeled θ, θ bae, δ mult corner o Table E.. We dene thee headng a ollow:, and θ tet (ee the upper let θ : Comparatve tatc parameter (that, the element o Θ : c 0, c 0, and o orth; ee Table 7. n the text or an explanaton o each parameter bae θ : The bae cae value o each comparatve tatc parameter θ, baed on emprcal data rom the Calorna PX, crca 999 (ee Appendx F or detal. mult δ : The multplcatve hock relatng the bae cae parameter value to the tet cae parameter value (ee eq. (E.6 below. The reult o Table E. below aume that mult δ =.00 (E.5 or each comparatve tatc cenaro, correpondng to a 0.% ncreae n the parameter under conderaton. 38 tet θ : The tet cae value o the comparatve tatc parameter θ, that, tet mult bae θ = δ θ. (E.6 A explaned below, the body o Table E. ha a par o column correpondng to each o the ten tet cae n whch we perturb the parameter n Θ ( c 0, c 0, and o orth 38 ecall that each parameter n Θ enter the underlyng equlbrum optmalty condton (text eq. (7. and (7. hghly nonlnearly and through multple pathway. It thereore not urprng that S p can vary wth not only the magntude but alo the gn o the comparatve tatc eect on ( mult ucently large varaton n the magntude o the multplcatve hock δ. The gn o the comparatve tatc eect reported n Table E. below are vald at leat or mall to moderate hock n the nterval [.00,.0] mult δ (.e., hock o 0.% to % o bae cae value, and uually or a much larger range o mult δ. 4

440 ndvdually, a well a a column o bae cae value (the thrd column o the table or β and the dcretzed SF ( S p. The row n the body o Table E. compre three ecton, a ollow: The upper ecton o the body o the table (contng o two row only contan value o the pot market SF lope β and β or the bae cae and each tet cae, ncludng (n the column labeled the abolute change n β between the tet cae and the bae cae. The mddle and lower ecton o the body o the table contan, repectvely, quantte (n MWh denng rm and rm dcretzed SF n the prce range o [ ] p 0,,750 $ MWh wth a tep ze o p = $50 MWh. For eae o readablty, we denote rm dcretzed SF a S_t n the table. Here, =, ndexe the uppler rm, whle t = 0,,,, ndexe the pont, or quantte, at whch we evaluate each rm dcrete SF. (The notaton S_0 repreent, naturally, rm ntal quantty. The mddle and lower ecton o the column labeled n Table E. gve or rm and, repectvely the abolute change n the repectve dcretzed SF (.e., change n quantty between the tet cae and the bae cae. For example, n the ecton o Table E. correpondng to c 0 (on the rt page o the table, conder the pont on the dcretzed orward market SF S_ at p = $500 MWh, or S ( 500, n our cutomary notaton. For th pont on rm SF, we have that (to our decmal place 43

441 ( 500 S ( 500 = S Tet cae Bae cae = MWh MWh = MWh. For purpoe o the comparatve tatc analy, we gnore n Table E. below the lowet and hghet pont on each dcretzed SF, nce the rt- and econd-order optmalty condton or the SF are not mpoed at thee pont. 383 Fnally, the abbrevaton NS ued n the row o headng n Table E. denote No Scalng. Th degnaton ndcate that automatc calng wa not ued n the dcrete Excel model n producng a comparatve tatc cenaro that o labeled (ee note 84 n chapter That, we gnore the pont S_0 and S_ ( =, on the dcretzed SF at prce p = $0 MWh and p = $,750 MWh, repectvely. 384 A t happen, we dd not ue calng n any o the cenaro reported n Table E.. 44

442 TABLE E.: COMPAATIVE STATICS ESULTS θ Spot mkt. c0 c0 c c e dem bae θ lope & Bae cae E-05 mult δ wd. mkt. value.00 NS.00 NS.00 NS.00 NS.00 NS tet θ quantte E-05 p β β S_ S_ S_ S_ S_ S_ S_ S_ S_ S_ S_ S_ S_ S_ S_ S_ S_ S_ S_ S_ S_ S_ S_ S_ Note: NS: No automatc calng ued to produce cenaro. 45

443 TABLE E.: COMPAATIVE STATICS ESULTS (CONT D η σ ν σ λ η NS.00 NS.00 NS.00 NS.00 NS Prce vector ν 46

444 [T]he trouble about argument, they an t nothng but THEOIES, ater all, and theore don t prove nothng, they only gve you a place to ret on, a pell, when you are tuckered out buttng around and around tryng to nd out omethng there an t no way TO nd out.... There another trouble about theore: there alway a hole n them omewhere, ure, you look cloe enough. Mark Twan, Tom Sawyer Abroad Appendx F: Bae cae parameter value ued n the numercal example o the mult-ettlement SFE model THIS APPENDIX explan the provenance o the bae cae parameter value ued or the qualtatve and quanttatve analy o text chapter 7. A the ctaton below ugget, the choen parameter value are baed (very roughly on Calorna ol-red generaton capacty durng the nterval June 998 to September 999, whch we call the reerence perod. 385 Generatng unt n the Calorna market whch mut run due to engneerng contrant were bd nto the PX wth a (perectly elatc and non-trategc bd o zero dollar; thee unt were largely thoe ung non-ol uel generaton technologe: hydroelectrc, nuclear, and geothermal plant. Thee unt almot never et the market- 385 Unle otherwe noted, data reported apply to th reerence perod. 47

445 clearng prce and mght a a rt approxmaton (ollowng Borenten, Buhnell and Wolak 00 be treated a bddng non-trategcally. The analy that ollow net out the load erved by thee non-ol uel unt and ocue on the racton o the market erved by ol-red unt. We dcu pot market parameter n ecton F. below, ollowed by thoe parameter relevant to the orward market n ecton F.. In clong, ecton F.3 ummarze the numercal ndng o th appendx n the bae cae parameter vector bae Θ. F. Spot market Fgure F. below depct rm margnal cot uncton C( q demand uncton (, and the pot market D p ε gven a demand hock ε, ung text chapter 5 ane aumpton. The gure alo depct an emprcal reerence prce p and emprcal, mean empr (aggregate reerence quantty q, mean or the pot market. Subecton F.. below empr provde value o, p and q mean rom the lterature., mean empr empr 48

446 p Spot market, mean p empr ( C q = c + c q 0 γ c c 0 c 0 c (, D p ε = γ p + ε ( C q = c + cq 0, mean q empr ε q FIGUE F.: F.. SPOT MAKET GEOMETY Prce and quantte The ollowng parameter are avalable drectly rom the lterature:, mean p = $6.54 MWh : Mean Calorna ISO pot market prce durng the empr reerence perod, averaged over all hour and the two zone NP5 and SP5 (Borenten, Buhnell and Wolak 000, 3, mean q = 4,955 MWh : Mean aggregate pot market demand acng ol-red empr unt n the Calorna ISO ytem durng the reerence perod, averaged over all hour (Buhnell 003a 49

447 , mean q empr, tot = 6,5 MWh : Mean aggregate pot market demand acng all generatng unt n the Calorna ISO ytem durng the reerence perod, averaged over all hour (Borenten, Buhnell and Wolak 00, 393 F.. Demand data In text chapter 7, we denoted a e dem the prce elatcty o pot market demand acng ol-red unt n the Calorna ISO ytem durng the reerence perod (evaluated at the emprcal reerence prce p, mean and emprcal (aggregate reerence quantty empr q, mean empr. One emprcally-baed approxmaton o the pot market demand elatcty Buhnell and Manur (00, 9 etmate o e = Unortunately, th value o e dem dd not lead to eable oluton o the dcrete Excel model when appled to the benchmarkng procedure o text ecton 7.5. A a conequence, we permtted dem e dem to be endogenou n the benchmarkng procedure, and decrbe here how we obtaned the value o e dem ultmately ued n the analy o text chapter 7. The benchmarkng procedure o text ecton 7.5 center around a equence o two optmzaton problem:. Benchmarkng tep (text problem (7.55. Benchmarkng tep (text problem ( Buhnell and Manur cauton agant nterpretng the reult o ther calculaton a elatcte, per e, nce retal prce to conumer were beng deregulated durng the perod that they tudy, and the prce that conumer thought that they aced a conumpton decon were made, o coure, unoberved. Moreover, thee author emprcal work baed on data only or the San Dego area durng the perod Augut and September 000, rather than data or the Calorna market a a whole. 430

448 Tral and error wth varant o text problem (7.55 lead to ung e = a (0 an element o the reduced parameter vector \( η, ση, ν, σν hence alo a the ntal value o dem Θ n th problem, and e dem n text problem (7.56. The oluton to text problem (7.56 n tep o the benchmarkng procedure yeld an endogenou value o e dem, ( ( dem e-5 e =, (F. that we may then ncorporate nto the bae cae parameter vector bae Θ. The elatcty n eq. (F. practcally equal to zero, and hence probably maller n magntude than would be realtc or the Calorna electrcty market. It, however, the endogenou value o e dem that yelded the bet t o prce and quantte n the benchmarkng procedure. Aumng ane pot market demand a n the mpled ane example o text chapter 5, and ung value o ( ( e, dem p, mean empr, and q rom th and the prevou, mean empr ubecton, we may compute the correpondng lope γ o the ane pot market demand uncton acng ol-red unt a 387 (to ve gncant gure γ ( ( edem ( (, mean dq qempr 4,955 MWh e-5 MWh = = = = (F., mean dp p $ empr $MWh 6.54 MWh 387 Mnor dcrepance n numercal reult are due to roundng. 43

449 F..3 Cot data We model the aggregate margnal cot uncton or ol-red unt n the Calorna ISO ytem durng the reerence perod a comprng only two (hypothetcal rm, labeled and, n accordance wth the duopoly model developed n the the. Baed on the aggregate margnal cot uncton or ol-red unt n Fgure o Borenten, Buhnell, and Wolak (000, we nd graphcally the ollowng parameter value or the ntercept and lope o the two hypothetcal rm margnal cot uncton: $ c 0 = 5.6 ; (F.3 MWh $ c 0 = 30.5 ; (F.4 MWh $MWh c = ; (F.5 MWh and $MWh c = (F.6 MWh The parameter value n eq. (F.3 (F.6 mply that rm a low-cot rm and rm a hgh-cot rm n the ene that c < c and c 0 < c 0. We ue thee value rom eq. (F.3 (F.6 n the bae cae parameter vector bae Θ. F..4 Spot market SF lope and related parameter ecallng the analy o text ecton 5., we may olve the par o equaton ( γ + β c( γ β 43 (,,; β = = + +

450 or the lope β and β o the pot market SF. Evaluatng thee lope at the value γ, c, and c rom eq. (F., (F.5, and (F.6 above, we have (to ve gncant gure MWh β =.4830 (F.7 $MWh and MWh β = (F.8 $MWh Gven eq. (F. and (F.3 (F.8, we may compute ω a and ω b (ee text eq. (5.4 and (5.5 a ω a = β + β + γ =, MWh MWh MWh $MWh $MWh $MWh or $MWh ω a = 0.08, (F.9 MWh and ωb = c β + c β 0 0 $ MWh $ MWh = , MWh $ MWh MWh $ MWh or 433

451 ω = 39.0 MWh. (F.0 b F..5 Dtrbutonal aumpton or pot market demand The oluton to text problem (7.56 correpondng to tep o the benchmarkng procedure yeld optmal value o the mean η and varance conumer gnal η, a well a the mean ν and varance σ η o the repreentatve σ ν o the pot market noe parameter ν. To ve gncant gure, thee optmal value are a ollow: ( η ( = 4,64.4 MWh, (F. ( ( σ =.4567e6 MWh, (F. η ( ν ( = MWh, (F.3 and ( ( σ = 58,605 MWh. (F.4 ν From eq. (F. and (F.4, the tandard devaton o η and ν are ( σ ( η =,567.4 MWh (F.5 and ( σ ( ν = 4.08 MWh. (F.6 The value n eq. (F. (F.4 are optmal n the ene that they olve text problem, (7.55, enurng alo that the two benchmarkng contrant ( E p p empr mean = and 434

452 , ( qagg q E mean empr = mpoed n that problem are met. Fnally, we ncorporate the optmal value rom eq. (F. (F.4 nto the bae cae parameter vector bae Θ. We now conder the hgher moment ( Cov, ν, νσ ν ν, and make ome addtonal dtrbutonal aumpton that permt u to compute σ a a uncton o ν, and σ ν rom eq. (F.3 and (F.4. Namely, we aume now that ν lognormally dtrbuted, and dene moment o the natural logarthm o ( µ mean o ln ν ν ν ν, ln ( ν, a and σ ( ν varance o ln. Then, a a uncton o thee parameter, the probablty denty uncton o n term o the parameter µ and σ ν, ( ν ν, ( ln ν µ ν ( ν exp = νσπ σ. (F.7 A a uncton o the dtrbutonal parameter µ and σ n eq. (F.7, we may how (ee, e.g., Hatng and Peacock 975 that ν ha a mean ν o σ ν = exp µ +, (F.8 a tandard devaton σ ν o 435

453 ( ( ( σ ν = exp µ exp σ exp σ, (F.9 a varance σ ν o ( ( exp exp νσ = µ + σσ, (F.0 and a coecent o kewne α 3 o ( ( α3 = exp σ+ exp σ. (F. From note 7 n text chapter 6, we may expre the hgher moment σ a, ν ν 3 σ =, ( σ ν ν 3 ν α +, (F. Vν where α 3 the coecent o kewne and V ν the coecent o varaton o ν, that V ν σ ν. (F.3 ν To expre σ n term o the underlyng dtrbutonal parameter µ and, ν ultmately n term o ν and (F.3: ν σ ν begn by ubttutng or σ and V ν n eq. (F. rom eq. 3 ν σ =, ( σ ν ν α 3 ν +. (F.4 σν Next, ubttute nto eq. (F.4 rom eq. (F.8 (F. to obtan 436

454 { ( ( } ( ( { 3 exp exp exp exp ν, νσ = µ + σσ σ σ + σ exp µ + +, exp( µ exp( σ exp( σ whch mple to 5 (. (F.5 exp 3 exp ν, νσ = µ + σ σ Solvng eq. (F.8 and (F.0 or µ and σ yeld σ ln ( ν µ = ν ln + ν (F.6 and σ σ ν = ln +. (F.7 ν Subttutng eq. (F.6 and (F.7 nto eq. (F.5, we have σ ν, ν σ 5 ν σ ν σ ν = exp 3 ln ( ν ln + ln exp ln + + +, ν ν ν whch we may mply a 388 σ 3 σν σ ν σ ν = σ 0 ν, ν ν + + > ν ν ν. (F The potvty o σ ollow rom ν > 0, recallng that ν lognormally dtrbuted. ν, ν 437

455 In term o the coecent o varaton σ V ν (ee eq. (F.3, we may wrte eq. (F.8 a ( ( 3 σ V V V 0 ν, ν ν ν ν ν = + + >. (F.9 Ung value o the mean and tandard devaton o ν rom eq. (F.3 and (F.6, we may compute V ν rom eq. (F.3 a V ν σν 4.08 MWh = = (F.30 ν MWh Subttutng rom eq. (F.6 and (F.30, eq. (F.9 become (to ve gncant gure 3 ( ( ( ( σ = 4.08 MWh , ν, ν or ( 3 σ = e7 MWh. (F.3 ν, ν F. Forward market Fgure F. below depct a repreentatve orward market demand uncton (, 0 D p ε. The gure alo depct an emprcal reerence prce p and emprcal (aggregate, mean empr reerence quantty q, mean empr. Subecton F.. below provde a value o p baed on, mean empr the lterature, and explan how we compute q. Unle otherwe noted, chapter 7, mean empr numercal analy conder orward market SF over the range [ ] p 0,,750 $ MWh. (F.3 438

456 Th prce range n (F.3 alo nclude the prce $,500 MWh, whch wa the applcable (otware-mpoed Calorna PX prce cap a o March 999 (Market Montorng Commttee o the Calorna Power Exchange 999, p Forward market (, ε ( D p = D p + ε 0 0 0, mean p empr p0 ε, mean qempr 0 q FIGUE F.: F.. FOWAD MAKET GEOMETY Prce and quantte The ollowng parameter are avalable drectly rom the lterature: 389 We emphaze that the retrcton n the range o orward market prce condered, [ ] p 0,,750 $ MWh, or computatonal purpoe only; t apple, n partcular, to the pecc numercal example o text ecton 7.6. Text ubecton 3..5 denton o ( p tll apple. I dered, we may pecy the nterval o ( S p to nclude negatve prce. S p a a uncton over p over whch we compute the uncton 439

457 , mean p = $6.60 MWh : Mean Calorna PX uncontraned orward market prce empr durng the reerence perod, averaged over all hour (Borenten, Buhnell and Wolak 00, 393, mean q empr, tot =,579 MWh : Mean aggregate orward market demand n the Calorna PX (acng all generatng unt durng the perod Aprl 998 to Aprl 999, 390 averaged over all hour (Calorna Power Exchange 999, 7 We now compute the emprcal (aggregate reerence quantty q or the orward, mean empr market, correpondng to the (aggregate reerence demand level acng only ol-red generatng unt. ecallng the mean hourly pot market demand q = 4,955 MWh,, mean empr aume that the racton o th quantty that tranacted n the orward market gven by the rato q q. We may then compute, mean, mean empr, tot empr, tot q a, mean empr q q,579 MWh = = ( 4,955 MWh = 4,033 MWh. (F.33 6,5 MWh, mean, mean, mean empr, tot empr qempr, mean qempr, tot F.. Conumer rk preerence Text ubecton 6.. dened the parameter λ a the contant abolute rk averon or CAA coecent or conumer. The purpoe o th ubecton to determne an approprate value o the CAA coecent, λ, or the repreentatve conumer ntroduced later n text chapter 6. In the abence o data on conumer rk averon n the context o electrcty market, we turn to other economc ettng to provde a ba 390 Th perod doe not concde exactly wth the reerence perod, but we take th a a utable approxmaton o average orward market demand durng the reerence perod. 440

458 or quanttatve etmate o λ. It reaonable to uppoe that conumer rk preerence n electrcty market are comparable to thoe governng behavor n market or other good and ervce. Accordngly, th ubecton brely urvey the lterature on emprcal etmate o the CAA coecent rom a varety o economc context, and place thee etmate on a comparable ba. Table F. below report the reult o numerou emprcal tude o agent rk preerence, conducted n a wde varety o economc ettng (notably, the agrcultural ector, whch ha oten been tuded n th context. The rghtmot column o the table gve the etmate o the CAA coecent λ computed n each tudy, expreed n unorm unt o ($ 999 or purpoe o comparon acro the tude. 39 Table F. lt the varou tude n order o ncreang CAA coecent (.e., ncreang rk averon. 39 Note 98 n the text provde an ntutve nterpretaton o the CAA coecent λ. 39 See the note to Table F. or detal o currency and current-to-contant dollar converon, where applcable. 39 When Table F. report a range o value or λ (ee the repectve orgnal tude or detal, the geometrc mean o the endpont o th range ued to order the tude. 44

459 TABLE F.: EMPIICAL ESTIMATES OF THE CONSTANT ABSOLUTE ISK AVESION (CAA COEFFICIENT λ (IN ODE OF INCEASING λ Ctaton Buccola (98 Bar-Shra, Jut, and Zlberman (997 Len (00 Ozanne (998 Zachara and Grube (984 Smmon and Pomareda (975 Kramer and Pope (98 Love and Buccola (99 Brnk and McCarl (978 Beetma and Schotman (00 Wol and Pohlman (983 Chava and Holt (996 Antle (987 Note: reerence year. Obect o tudy and data reerence year a Proceng tomato producer n Calorna, USA 979 Farmer n Arava regon, Irael 978 Lowland crop and lvetock armer n Norway 996 Crop and lvetock armer n the USA 964 Experment at Agronomy South Farm, Urbana, IL, USA 97 Crop armer n Mexco exportng to the USA 97 Feld crop armer n Kern County, CA, USA 974 Corn and oybean armer n Iowa, USA 967 Large corn belt cah gran armer n the USA 975 Televon game how contetant n the Netherland 996 A dealer n USA Treaury Bll aucton 977 Corn and oybean armer n the USA 967 ce armer n Aurepalle Vllage, Inda 979 λ (orgnal unt Contant dollar and currency converon actor b ($000 PPF 979 = 66 λ ($ e-7.e-6 4.5e-6 $ PPF 978 = 58.3e-6.44e-6 NOK PPF 996 = (NOK/$ e-5 4.8e-4 $ PPF 964 = e-5 9.e-5.5e-5 3.5e-3 PPF 97 = 30.7 $ 9.3e Peo PPF 97 = (Peo/$ $ PPF 974 = $ PPF 967 = e-4 e $ PPF 975 = Gulder CPI-U 996 = 56.9 c CPI-U 999 = (Gulder/$ $ 4.4 d $ PPF 967 = PPF 979 = 66 upee (upee/$ 979 a Where data are drawn over multple year, we ue the mdpont o th tme nterval a the data 44

460 Note to Table F. (cont d: b For agrcultural tude, we convert dollar rom the data reerence year to 999 dollar ung the prce ndce or prce pad by armer or all commodte, ervce, nteret, taxe, and wage rate or the relevant year (Economc eport o the Predent 99, Table B-98 or data reerence year pror to 975; Economc eport o the Predent 003, Table B-0 or data reerence year 975 and later. For year t, we denote th prce ndex a PPF t, and note that t normalzed ung PPF 99 = 00. For 999, recallng that the CAA coecent reported n the rghtmot column o Table F. have unt o ($ 999, we have PPF 999 = 5. For non-agrcultural tude, we report the approprate converon actor n the ourth column o Table F.. We ue purchang power party exchange rate rom the Penn World Table (Heton, Summer and Aten 00 to perorm currency converon to current dollar n the data reerence year. See below or an example o the ue o the varou converon actor. c CPI-Ut the conumer prce ndex or all tem n year t, where CPI-U = 00 (Economc eport o the Predent 003, Table B-60. d Becaue th partcular tudy addree the behavor o a Treaury bll dealer, the converon actor 4.4 the approxmate return on $ nveted at the average annual Treaury bll rate begnnng n 977, and compounded annually untl 999 (Internatonal Monetary Fund 003, Treaury Bll ate. A an example o the converon ued n Table F., conder Smmon and Pomareda 975 tudy o Mexcan armer that export to the USA. The author report a value o ( λ = 0.5 Peo 97, whch we convert to unt o ($ a ollow: Peo 3.9 PPF 0.5 Peo = 9e-4 $. (F ( ( $ 97 5 PPF999 The CAA coecent λ n Table F. (n the rghtmot column le n the nterval [ 6.9e-7, 5.9 ], a range o nearly 7 order o magntude, wth geometrc mean λ geom ( = $ 999. (F.35 The economc agent whoe rk preerence are characterzed n Table F. tend to be maller-cale (n term o revenue, or example and, plaubly, le nancally ophtcated than mot o the electrcty conumer partcpatng n the Calorna PX. Thu, we would expect the electrcty conumer that we wh to model here to be le rk avere, on average, than the average agent characterzed n Table F.. Takng the geometrc mean λ geom o the coecent λ n Table F. to be repreentatve o the 443

461 agent characterzed n the table, the above obervaton ugget that the repreentatve conumer CAA coecent λ related to λ geom a ollow: ( λ < λgeom = $ 999. (F.36 Baed on the conderable range o CAA coecent reported n Table F., we aume, more peccally, that λ and λ geom der by one order o magntude. We aume, et thereore, that a a rough etmate ( λ o λ, we may ue the value et ( ( λ = $ 999. (F.37 Becaue o the approxmate nature o the above dcuon, we ue eq. (F.37 a merely an ntal condton or λ n text ecton 7.5 benchmarkng procedure ung the dcrete Excel model. That, we ue the value ( λ n eq. (F.37 a an element o the parameter vector et (0 Θ n text problem (7.55, and hence alo n ( Θ, the vector o ntal value or text problem (7.56. The oluton to th benchmarkng problem (tep o the benchmarkng procedure yeld an endogenou value o λ only lghtly derent et rom ( λ above namely, opt ( ( λ = $ 999. (F.38 ecallng the llutratve nterpretaton o the CAA coecent rom note 98 n chapter 6, we oberve that the value o ( λ opt n eq. (F.38 correpond to a rk tolerance parameter τ o 444

462 τ = = ( $ λ $ ( 999 For a rk-avere conumer wth CAA coecent λ gven n eq. (F.38, the nterpretaton o the rk tolerance τ that the conumer (approxmately nderent between acceptng and not acceptng a lottery oerng even odd over payo o τ = ( $ and = ( $ We ncorporate the value ( λ opt 999 τ 999 rom eq. (F.38 nto the bae cae parameter vector bae Θ. F.3 Summary Collectng the numercal reult documented n th appendx, we may wrte the bae cae parameter vector bae Θ a (roundng reult to three gncant gure bae c0 $5.60 MWh $30.50 MWh c 0 $ ( MWh c c $ ( MWh bae edem 5.95e-5 Θ =. (F.39 η 4640 MWh σ η.46e6 MWh ν 335 MWh σ ν 5.86e4 MWh λ 3.0e-4 $ For eae o reerence, the vector bae Θ n eq. (F.39 alo appear n the text a eq. (

463 The melanchola o everythng COMPLETED! Netzche, Beyond Good and Evl Lterature cted Alberta Electrc Utlte Act o Alberta Statute and egulaton. Allaz, Blae Strategc orward tranacton under mperect competton: The duopoly cae. Ph.D. d., Department o Economc, Prnceton Unverty Olgopoly, Uncertanty and Strategc Forward Tranacton. Internatonal Journal o Indutral Organzaton 0 (June: Allaz, Blae and Jean-Luc Vla Cournot Competton, Forward Market and Ecency. Journal o Economc Theory 59 (February: 6. Amercan Publc Power Aocaton Iue Summare o State etructurng Law. Avalable rom Amlon, Henrk. 00. Comparon o Mean-Varance and Exact Utlty Maxmzaton n Stock Portolo Selecton. Lund Unverty, Department o Economc, Workng Paper Sere (No. 4, Lund, Sweden. 446

464 Anderon, John A The Compettve Sourcng o etal Electrc Power: An Idea Who [c] Tme Ha (Fnally Come. Preented at Utlty Drector Workhop, September 0, Wllamburg, VA. Antle, John M Econometrc Etmaton o Producer k Atttude. Amercan Journal o Agrcultural Economc 69 (Augut: 509. Bagwell, Kyle Commtment and Obervablty n Game. Game and Economc Behavor 8 (February: Baldck, o and Wllam Hogan. 00. Capacty Contraned Supply Functon Equlbrum Model o Electrcty Market: Stablty, Non-decreang Contrant, and Functon Space Iteraton. Workng Paper, Program on Workable Energy egulaton (POWE (PWP-089, Unverty o Calorna Energy Inttute, Berkeley, CA (December. Barker, Jame, Jr., Bernard Tenenbaum, and Fona Wool egulaton o Power Pool and Sytem Operator: An Internatonal Comparon. Energy Law Journal 8 (: Bar-Shra, Z.,.E. Jut, and D. Zlberman Etmaton o Farmer k Atttude: An Econometrc Approach. Agrcultural Economc 7 (December:. Battone, Stephen. J. 00. Apect o rk management n deregulated electrcty market: Storage, market power and long-term contract. Ph.D. d., Management Scence, Unverty o Canterbury, New Zealand. Beetma, oel M. W. J. and Peter C. Schotman. 00. Meaurng k Atttude n a Natural Experment: Data rom the Televon Game Show Lngo. The Economc Journal (October: Berndt, Ernt. and Davd O. Wood Technology, Prce, and the Derved Demand or Energy. evew o Economc and Stattc 57 (Augut: Brkho, Garrett and Gan-Carlo ota Ordnary Derental Equaton. 4th ed. New York: John Wley & Son. Blumten, Carl, L. S. Fredman, and. J. Green. 00. The Htory o Electrcty etructurng n Calorna. Center or the Study o Energy Market Workng Paper Sere (CSEM WP 03, Unverty o Calorna Energy Inttute, Berkeley, CA (Augut. Bohn, Jame, Metn Celeb, and Phlp Haner. 00. The Degn o Tet or Horzontal Market Power n Market-Baed ate Proceedng. The Electrcty Journal 4 (May: Bolle, Fredel. 99. Supply Functon Equlbra and the Danger o Tact Colluon: The Cae o Spot Market or Electrcty. Energy Economc 4 (:

465 Who Prot From Future Market? o Studen 39 (3 4: Competton wth Supply and Demand Functon. Energy Economc 3 (May: Borenten, Severn, Jame B. Buhnell, and Frank A. Wolak. 00. Meaurng Market Inecence n Calorna etructured Wholeale Electrcty Market. Amercan Economc evew 9 (December: Borenten, Severn, Jame Buhnell, Chrtopher. Knttel, and Catherne Wolram Learnng and Market Ecency: Evdence rom the Openng o Calorna Electrcty Market. Prelmnary Drat (November Tradng Inecence n Calorna Electrcty Market. Workng Paper, Program on Workable Energy egulaton (POWE (PWP-086, Unverty o Calorna Energy Inttute, Berkeley, CA (October. Borenten, Severn, Jame Buhnell, and Frank Wolak Dagnong Market Power n Calorna Deregulated Wholeale Electrcty Market. Workng Paper, Program on Workable Energy egulaton (POWE (PWP-064, Unverty o Calorna Energy Inttute, Berkeley, CA (Augut. Braun, Martn Derental equaton and ther applcaton: An ntroducton to appled mathematc. Fourth Edton. New York: Sprnger-Verlag. Brennan, Tmothy J., Karen L. Palmer, aymond J. Kopp, Alan J. Krupnk, Vto Staglano, and Dalla Burtraw A hock to the ytem: etructurng Amerca electrcty ndutry. Wahngton, DC: eource or the Future. Brenahan, Tmothy F. 98. Duopoly Model wth Content Conecture. Amercan Economc evew 7 (December: Brnk, Lar and Bruce McCarl The Tradeo Between Expected eturn and k Among Cornbelt Farmer. Amercan Journal o Agrcultural Economc 60 (May: Buccola, Steven T. 98. Portolo Selecton Under Exponental and Quadratc Utlty. Wetern Journal o Agrcultural Economc 7 (July: Bulow, Jeremy I., John D. Geanakoplo, and Paul D. Klemperer Multmarket Olgopoly: Strategc Subttute and Complement. Journal o Poltcal Economy 93 (June: Buhnell, Jame. 003a. Electronc mal to author, May 3. Buhnell, Jm. 003b. Lookng or Trouble: Competton Polcy n the U.S. Electrcty Indutry. Center or the Study o Energy Market Workng Paper Sere (CSEM WP 09, Unverty o Calorna Energy Inttute, Berkeley, CA (June. 448

466 Buhnell, Jame B. and Ern Manur. 00. The Impact o etal ate Deregulaton on Electrcty Conumpton n San Dego. Workng Paper, Program on Workable Energy egulaton (POWE (PWP-08, Unverty o Calorna Energy Inttute, Berkeley, CA (May. Butcher, J. C The numercal analy o ordnary derental equaton: unge- Kutta and general lnear method. New York: John Wley & Son. Calorna Independent Sytem Operator. 003a. ISO Enorcement Protocol. Fled wth the Federal Energy egulatory Common, Docket No. E , Amendment No. 55 to the CAISO Tar (July.. 003b. ISO Market Montorng & Inormaton Protocol. Fled wth the Federal Energy egulatory Common, Docket No. E , Amendment No. 55 to the CAISO Tar (July equet or ehearng and Moton or Clarcaton o the Calorna Independent Sytem Operator Corporaton. Fled wth the Federal Energy egulatory Common, Docket No. E , Calorna Independent Sytem Operator Corporaton. Wahngton, DC (March. Calorna Independent Sytem Operator and London Economc Internatonal LLC A Propoed Methodology or Evaluatng the Economc Benet o Tranmon Expanon n a etructured Wholeale Electrcty Market. (February 8. Avalable rom Calorna Power Exchange Electrcty Market o the Calorna Power Exchange. Annual eport to the Federal Energy egulatory Common. Market Complance Unt (July 30. Calorna Power Exchange Corporaton FEC Electrc Servce Tar No.. Fled wth the Federal Energy egulatory Common (Augut. Calorna Publc Utlte Common Calorna Electrc Servce Indutry: Perpectve on the Pat, Stratege or the Future. ( Yellow eport. Avalable rom Preerred Polcy Decon. Docket No (December Order Adoptng Correcton to Decon. Docket No (January 0. Chava, Jean-Paul and Matthew T. Holt Economc Behavor Under Uncertanty: A Jont Analy o k Preerence and Technology. evew o Economc and Stattc 78 (May:

467 Cox, Alan J Merger, Acquton, Dvetture, and Applcaton or Market- Baed ate n a Deregulatng Electrc Utlty Indutry. The Electrcty Journal (May: Dalton, John C Aeng the Compettvene o etructured Generaton Servce Market. The Electrcty Journal 0 (Aprl: Day, Chrtopher J. and Derek W. Bunn. 00. Dvetture o Generaton Aet n the Electrcty Pool o England and Wale: A Computatonal Approach to Analyzng Market Power. Journal o egulatory Economc 9 (March: 3 4. Day, Chrtopher J., Benamn F. Hobb, and Jong-Sh Pang. 00. Olgopoltc Competton n Power Network: A Conectured Supply Functon Approach. IEEE Tranacton on Power Sytem 7 (Augut: de la Fuente, Angel Mathematcal method and model or economt. New York: Cambrdge Unverty Pre. Dmuke, Davd E. and Kmberly H. Dmuke Electrc M&A: A egulator Gude. Publc Utlte Fortnghtly 34 (January : Earle, obert L Demand Elatcty n the Calorna Power Exchange Day-Ahead Market. The Electrcty Journal 3 (October: Economc eport o the Predent. 99. Wahngton, DC: GPO (February. Economc eport o the Predent Wahngton, DC: GPO (February. Energy Inormaton Admntraton, U.S. Department o Energy Fnancal tattc o maor U.S. nvetor-owned electrc utlte 996. DOE/EIA- 0437(96/. Wahngton, DC: GPO (December. Avalable rom 00a. Electrc Power Annual 000. DOE/EIA-0348(000/. Wahngton, DC: GPO (Augut. Avalable rom 00b. Fnancal tattc o maor U.S. publcly owned electrc utlte 000. DOE/EIA-0437(00. Wahngton, DC: GPO (November. Avalable rom Energy Polcy Act o U.S. Code. Vol. 5, ec. 79z-5a and Vol. 6, ec. 796( 5, 84 l, and elewhere. Energy egulator egonal Aocaton, Lcenng/Competton Commttee. 00. Montorng, Meaurng and Aurng the Compettvene o Energy Market. Preented at 5th Annual Energy egulatory Conerence, Energy egulator egonal Aocaton, December 3 5, Soa, Bulgara. 450

468 ECOT. 00. ECOT Protocol. (July. Avalable rom /dratercotprotocol.htm. Eve, Howard Analytc Geometry. In CC Standard Mathematcal Table, 8th ed., edted by W. H. Beyer. Boca aton, FL: CC Pre. Ferrera, Joe Lu Strategc Interacton Between Future and Spot Market. Journal o Economc Theory 08 (January: 4 5. Frankena, Mark W. 998a. Analyzng Market Power Ung Appendx A o FEC Merger Polcy Statement: atonale, elablty, and eult. CCH Power and Telecom Law (January/February: b. Geographc Market Delneaton or Electrc Utlty Merger. Fled wth the Federal Energy egulatory Common, Docket No. M , eved Flng equrement Under Part 33 o the Common egulaton: Notce o Propoed ulemakng. Wahngton, DC (Augut. Frankena, Mark W. and Bruce M. Owen Electrc Utlty Merger: Prncple o Anttrut Analy. Wetport, CT: Praeger. Freund, udol J The Introducton o k nto a Programmng Model. Econometrca 4 (July: Fudenberg, Drew and Jean Trole The Fat-cat Eect, the Puppy-dog Ploy, and the Lean and Hungry Look. AEA Paper and Proceedng 74 (May: Game Theory. Cambrdge, MA: MIT Pre. Gear, C. Wllam. 97. Numercal ntal value problem n ordnary derental equaton. Prentce-Hall ere n automatc computaton. Englewood Cl, NJ: Prentce-Hall. Grd, Dean. 00. Power and Ga egulaton Iue and Internatonal Experence. Drat Workng Paper. Wahngton, DC: World Bank (Aprl. Goldman, Charle, Berne C. Leeutre, and Emly Bartholomew A evew o Market Montorng Actvte at U.S. Independent Sytem Operator. LBNL Berkeley, CA: Ernet Orlando Lawrence Berkeley Natonal Laboratory (January. Grauer, obert. and Nl H. Hakanon On the Ue o Mean-Varance and Quadratc Approxmaton n Implementng Dynamc Invetment Stratege: A Comparon o eturn and Invetment Polce. Management Scence 39 (July:

469 Green, chard Increang Competton n the Brth Electrcty Spot Market. Journal o Indutral Economc 44 (June: a. The Electrcty Contract Market n England and Wale. Journal o Indutral Economc 47 (March: b. Supplementary Materal or chard Green, The Electrcty Contract Market n England and Wale, Journal o Indutral Economc 47 (March: Avalable rom Green, chard J. and Davd M. Newbery. 99. Competton n the Brth Electrcty Spot Market. Journal o Poltcal Economy 00 (October: Harer, E., S. P. Noerett, and G. Wanner Solvng Ordnary Derental Equaton I: Nont problem. nd ed. Sprnger Sere n Computatonal Mathematc 8, Vol.. New York: Sprnger-Verlag. Harvey, Scott M., Wllam W. Hogan, and Suan L. Pope Tranmon Capacty eervaton and Tranmon Congeton Contract. eved veron (March 8. Avalable rom Hatng, N. A. J. and J. B. Peacock Stattcal dtrbuton: A handbook or tudent and practtoner. London: Butterworth. Heton, Alan, obert Summer, and Bettna Aten. 00. Penn World Table Veron 6.. Center or Internatonal Comparon at the Unverty o Pennylvana (CICUP. Avalable rom Heronymu, Wllam H., J. Stephen Henderon, and Carolyn A. Berry. 00. Market Power Analy o the Electrcty Generaton Sector. Energy Law Journal 3 (: 50. Hlbert, D. and S. Cohn-Voen. 95. Geometry and the Imagnaton. New York: Chelea Publhng Co. Hogan, Wllam W. 99. Contract Network or Electrcal Power Tranmon. Journal o egulatory Economc 4 (September: A Market Power Model wth Strategc Interacton n Electrcty Network. The Energy Journal 8 (4: Hughe, John S. and Jenner L. Kao Strategc Forward Contractng and Obervablty. Internatonal Journal o Indutral Organzaton 6 (November: 33. Hughe, John S., Jenner L. Kao, and Mchael Wllam. 00. Publc Dcloure o Forward Contract and evelaton o Propretary Inormaton. evew o Accountng eearch 7 (December:

470 Internatonal Energy Agency. 00. Competton n Electrcty Market. Energy Market eorm Sere. Par: OECD. Internatonal Monetary Fund Internatonal Fnancal Stattc Onlne. Avalable rom ISO New England. [n.d.]. Procedure or Contactng the Market Advor to the ISO-NE Board o Drector. Market Montorng and Mtgaton Group. Avalable rom /Procedure_or_Contactng_Market_Advor.doc. Jamab, Toora and Mchael Polltt. 00. Benchmarkng and egulaton: Internatonal Electrcty Experence. Utlte Polcy 9 (September: Jordan, D. W. and. P. Smth Nonlnear ordnary derental equaton: An ntroducton to dynamcal ytem. Oxord Appled and Engneerng Mathematc. New York: Oxord Unverty Pre. Jokow, Paul and chard Schmalenee Market or power: An analy o electrc utlty deregulaton. Cambrdge, MA: MIT Pre. Kamat, anh and Shmuel S. Oren. 00. Two-Settlement Sytem or Electrcty Market: Zonal Aggregaton Under Network Uncertanty and Market Power. Workng Paper, Program on Workable Energy egulaton (POWE (PWP-09, Unverty o Calorna Energy Inttute, Berkeley, CA (February. Knzelman, Gregory L. 00. Comparon o Market Power Mtgaton Employed by PJM and ISO-NE (Internal memorandum to Wllam F. Young. Wahngton, DC: Hunton & Wllam, June 5. Klen, Joel I Makng the Tranton rom egulaton to Competton: Thnkng About Merger Polcy Durng the Proce o Electrc Power etructurng. FEC Dtnguhed Speaker Sere, January, Wahngton, DC. Klemperer, Paul D. and Margaret A. Meyer Supply Functon Equlbra n Olgopoly Under Uncertanty. Econometrca 57 (6: Kramer, andall A. and ulon D. Pope. 98. Partcpaton n Farm Commodty Program: A Stochatc Domnance Analy. Amercan Journal o Agrcultural Economc 63 (February: 9 8. Lauel, Dder. 99. Strategc Commercal Polcy evted: A Supply-Functon Equlbrum Model. Amercan Economc evew 8 (March: Levy, H. and H. M. Markowtz Approxmatng Expected Utlty by a Functon o Mean and Varance. Amercan Economc evew 69 (June:

471 Len, Gudbrand. 00. Non-parametrc Etmaton o Decon Maker k Averon. Agrcultural Economc 7 (May: Lock, ener. 998a. Power Pool & ISO. Publc Utlte Fortnghtly 36 (March : b. Survellance o Compettve Electrcty Market: A New Paradgm n Anttrut egulaton? The Electrcty Journal (March: 7 7. Love, H. Alan and Steven T. Buccola. 99. Jont k Preerence-Technology Etmaton wth a Prmal Sytem. Amercan Journal o Agrcultural Economc 73 (Augut: Magg, Govann The Value o Commtment wth Imperect Obervablty and Prvate Inormaton. AND Journal o Economc 30 (Wnter: Mankw, N. Gregory and Mchael D. Whnton Free Entry and Socal Inecency. and Journal o Economc 7 (Sprng: Mapleot. 00. MAPLE 8. Waterloo, Ontaro. Market Montorng Commttee o the Calorna Power Exchange Second eport on Market Iue n the Calorna Power Exchange Energy Market. Prepared or the Federal Energy egulatory Common. (March 9. Market Survellance Commttee o the Calorna ISO An Analy o the June 000 Prce Spke n the Calorna ISO Energy and Ancllary Servce Market. (September 6. Markowtz, Harry. 95. Portolo Selecton. Journal o Fnance 7 (March: Marhall, Alred. 90. Prncple o Economc, An Introductory Volume. 9th ed. New York: Macmllan. Ma-Collel, Andreu, Mchael D. Whnton, and Jerry. Green Mcroeconomc Theory. New York: Oxord Unverty Pre. Maey, Wllam L. 00. I the FEC Keepng It Part o the egulatory Bargan? Preented at The 4th Annual Mdwet Energy Conerence, Energy Bar Aocaton, February 8, Kana Cty, Mour. Melamed, A. Dougla Statement o A. Dougla Melamed. Beore the Subcommttee on Energy and Power, Commttee on Commerce, U.S. Houe o epreentatve. Electrcty Competton: Market Power, Merger and PUHCA. 06th Cong., t e. May 6. Mchael, obert T. and Gary S. Becker On the New Theory o Conumer Behavor. Swedh Journal o Economc 75 (December:

472 Mcroot Corporaton. 00. Mcroot Excel 00. edmond, WA. Mdwet ISO. 00a. Attachment S: Independent Market Montorng Plan. Fled wth the Federal Energy egulatory Common, Docket No. E and E , Mdwet Independent Tranmon Sytem Operator, Inc. (December b. Attachment S-: Independent Market Montor etenton Agreement. Fled wth the Federal Energy egulatory Common, Docket No. E and E , Mdwet Independent Tranmon Sytem Operator, Inc. (December Mdwet ISO Open Acce Tranmon and Energy Market Tar. Fled wth the Federal Energy egulatory Common, Docket No. E , Mdwet Independent Tranmon Sytem Operator, Inc. (March 3. Moler, Cleve B. and Kathryn A. Moler Numercal computng wth MATLAB. Avalable rom Moore, Bll and Cecly Gooch. 00. Electrc etructurng Leglaton n Texa. Energy Indutry, etructurng, Fnance, Merger, and Acquton Commttee Newletter (May. Avalable rom Moot, John S A New FEC Polcy or Electrc Utlty Merger? Energy Law Journal 7 (: Morr, John Fndng Market Power n Electrc Power Market. Internatonal Journal o the Economc o Bune 7 (: Natural Ga Polcy Act o U.S. Code. Vol. 5, ec. 330 et eq. Natural Ga Wellhead Decontrol Act o U.S. Code. Vol. 5, ec. 330 et eq. New England Power Pool and ISO New England, Inc Market ule - NEPOOL Standard Market Degn, Appendx A - Market Montorng, eportng, and Market Power Mtgaton. Fled wth the Federal Energy egulatory Common (FEC Electrc ate Schedule No. 7 (June. New York Independent Sytem Operator, Inc Market Montorng Plan. Fled wth the Federal Energy egulatory Common, Docket No. E , OA and E , New York Independent Sytem Operator, Inc. (July

473 . 004a. Complance Flng and Notce o Implementaton o the New York Independent Sytem Operator, Inc. n Docket No. E and E Fled wth the Federal Energy egulatory Common, Docket No. E and E , New York Independent Sytem Operator, Inc. (March.. 004b. ISO Market Power Mtgaton Meaure. Fled wth the Federal Energy egulatory Common, Docket No. E and E , New York Independent Sytem Operator, Inc. (March. Newbery, Davd Competton, Contract, and Entry n the Electrcty Spot Market. AND Journal o Economc 9 (Wnter: North Amercan Electrc elablty Councl Gloary o Term. Avalable rom Ozanne, Adam Uncertanty, Dualty and Perverty: An Emprcal Tet o the Schultz-Baron Hypothe. Appled Economc 30 (Aprl: Pacc Ga and Electrc Company Market Power Analy o Pacc Ga and Electrc Company n Support o Jont Applcaton. Fled wth the Federal Energy egulatory Common, Docket No. E , Pacc Ga and Electrc Company, San Dego Ga & Electrc Company, and Southern Calorna Edon Company (July 9. Pacc Ga and Electrc Company, San Dego Ga & Electrc Company, and Southern Calorna Edon Company Jont Applcaton o Pacc Ga and Electrc Company, San Dego Ga & Electrc Company, and Southern Calorna Edon Company or Authorty to Sell Electrc Energy at Market-Baed ate Ung a Power Exchange. Fled wth the Federal Energy egulatory Common, Docket No. E , Pacc Ga and Electrc Company, San Dego Ga & Electrc Company, and Southern Calorna Edon Company (Aprl 9. Peteron, Paul, Bruce Bewald, Lucy Johnton, Etenne Gonn, and Jonathan Wallach. 00. Bet Practce n Market Montorng: A Survey o Current ISO Actvte and ecommendaton or Eectve Market Montorng and Mtgaton n Wholeale Electrcty Market. Prepared or Maryland Oce o People Counel et al. Cambrdge, MA: Synape Energy Economc and eource Inght (November 9. Phllp, Charle F., Jr The regulaton o publc utlte: Theory and practce. Arlngton, VA: Publc Utlte eport, Inc. Perce, chard J. Jr. 99. Ung the Ga Indutry a a Gude to econttutng the Electrcty Indutry. eearch n Law and Economc 3: Anttrut Polcy n the New Electrcty Indutry. Energy Law Journal 7 (:

474 Prrong, Crag Manpulaton o Power Market. John M. Oln School o Bune, Wahngton Unverty, St. Lou, MO (March 4. PJM Interconnecton, L.L.C. 00. eport to the Federal Energy egulatory Common: Aement o Standard, Indce and Crtera. Market Montorng Unt (Aprl Attachment M: PJM Market Montorng Plan. Fled wth the Federal Energy egulatory Common, FEC Electrc Tar, Sxth eved Volume No. (March 0. Powell, Andrew Tradng Forward n an Imperect Market: The Cae o Electrcty n Brtan. Economc Journal 03 (March: Power Pool o Alberta. 00. Economc Wthholdng n the Alberta Energy Market. Market Development (March 4. Pratt, John W k Averon n the Small and n the Large. Econometrca 3 (Augut: 36. Publc Utlty Common o Texa Order Adoptng New 5.90, 5.9 and 5.40 a Approved at the Augut 0, 000 Open Meetng and Publhed n the Texa egter on Augut 5, 000 (Proect 08, Subtantve ule. Chapter 5. Electrc (Augut. Publc Utlty Holdng Company Act o 935 (PUHCA U.S. Code. Vol. 5, ec. 79 et eq. Publc Utlty egulatory Polce Act o 978 (PUPA U.S. Code. Vol. 6, ec. 60 et eq. Quan, Ngyuen T. and obert J. Mchael. 00. Game or Opportunte: Bddng n the Calorna Market. Electrcty Journal 4 (January/February: aber, Patrck J Implct Derental Equaton Near a Sngular Pont. Journal o Mathematcal Analy and Applcaton 44 (December: aber, Patrck J. and Werner C. henboldt. 994a. On Impae Pont o Qualnear Derental-Algebrac Equaton. Journal o Mathematcal Analy and Applcaton 8 (January: b. On the Computaton o Impae Pont o Qua-Lnear Derental- Algebrac Equaton. Mathematc o Computaton 6 (January: Theoretcal and Numercal Analy o Derental-Algebrac Equaton. In Handbook o Numercal Analy. Vol. 8, edted by P.G. Carlet and J. L. Lon. New York: North-Holland. 457

475 akn, Davd B. 998a. ISO Market Montorng and Mtgaton: The Addctve Allure o Electrc Prce egulaton. Preented at Fteenth Plenary Seon, Harvard Electrcty Polcy Group, January 9 30, San Dego, CA.. 998b. ISO: The New Anttrut egulator? The Electrcty Journal (Aprl: 5 5. aza, cardo. 00. Stablty Iue n egular and Noncrtcal Sngular DAE. Acta Applcandae Mathematcae 73 (September: oach, Crag. 00. Meaurng Market Power n the U. S. Electrcty Bune. Energy Law Journal 3 (: 5 6. oeller, Lar-Hendrk and obn C. Sckle Capacty and Product Market Competton: Meaurng Market Power n a Puppy-dog Indutry. Internatonal Journal o Indutral Organzaton 8 (Augut: ohrbach, John, Andrew Klet, and Blake Nelon. 00. Can FEC Solve It Market Power Problem? Supply Margn Aement Doen t Seem to Be a Promng Frt Step. The Electrcty Journal 5 (Aprl: 0 8. udkevch, Alekandr Supply Functon Equlbrum n Power Market: Learnng All the Way. TCA Techncal Paper 99-70, Tabor Caraman & Aocate, Cambrdge, MA (December. udkevch, Alekandr, Max Duckworth, and chard oen Modelng Electrcty Prcng n a Deregulated Generaton Indutry: The Potental or Olgopoly Prcng n a Poolco. The Energy Journal 9 (3: Schellng, Thoma C The trategy o conlct. Cambrdge, MA: Harvard Unverty Pre. Shampne, Lawrence F. and Mark W. echelt The MATLAB ODE Sute. SIAM Journal on Scentc Computng 8 (January:. Smmon, chard L. and Carlo Pomareda Equlbrum Quantty and Tmng o Mexcan Vegetable Export. Amercan Journal o Agrcultural Economc 57 (Augut: Southern Calorna Edon Company and San Dego Ga & Electrc Company Supplement o San Dego Ga & Electrc Company, and Southern Calorna Edon Company to Applcaton or Authorty to Sell Electrc Energy at Market- Baed ate Ung a Power Exchange. Fled wth the Federal Energy egulatory Common, Docket No. E , Pacc Ga and Electrc Company, San Dego Ga & Electrc Company, and Southern Calorna Edon Company (May

476 Stot, Steven. 00. An Analy o FEC Hub-and-Spoke Market-Power Screen. Contract No Prepared or Calorna Energy Overght Board (September. Avalable rom FEC-Hub+Spoke.pd Power Sytem Economc: Degnng Market or Electrcty. New York: Wley-Intercence. Surratt, Walter The Analytcal Approach to Meaurng Horzontal Market Power n Electrc Utlty Market: A Htorcal Perpectve. The Electrcty Journal (July: 33. Sweeney, Jame L. 00. The Calorna Electrcty Cr. Stanord, CA: Hoover Inttuton Pre: Stanord Inttute or Economc Polcy eearch. The MathWork, Inc. 00. MATLAB (Student Veron, eleae. Natck, MA. Trole, Jean The theory o ndutral organzaton. Cambrdge, MA: MIT Pre. U.S. Department o Energy - Oce o Economc, Electrcty and Natural Ga Analy and Oce o Polcy Horzontal Market Power n etructured Electrcty Market. Wahngton, DC (March. U.S. Department o Jutce and U.S. Federal Trade Common. 99. Horzontal Merger Gudelne (eprnted n 4 Trade eg. ep. (CCH 3,04. Wahngton, DC (Aprl, a amended Aprl 8, 997. U.S. Federal Energy egulatory Common Order No. 436: Fnal ule. Docket No. M (Part A D (33 FEC 6,007, egulaton o Natural Ga Ppelne Ater Partal Wellhead Decontrol. Wahngton, DC (Aprl Order Acceptng Amendment to Power Sale Agreement. Docket No. E (44 FEC 6,6, Ocean State Power. Wahngton, DC (Augut Order Acceptng ate or Flng, Notng Interventon, and Grantng and Denyng Waver. Docket No. E (50 FEC 6,5, Dowell Lmted Partnerhp. Wahngton, DC (February a. Order No. 636: Fnal ule. Docket No. M and M (59 FEC 6,030, Ppelne Servce Oblgaton and evon to egulaton Governng Sel-Implementng Tranportaton Under Part 84 o the Common egulaton, egulaton o Natural Ga Ppelne Ater Partal Wellhead Decontrol. Wahngton, DC (Aprl b. Order on ate Flng. Docket No. E (58 FEC 6,34, Entergy Servce, Inc. Wahngton, DC (March

477 . 996a. Inqury Concernng the Common Merger Polcy Under the Federal Power Act. Docket No. M (6 F Wahngton, DC (January b. Order Acceptng or Flng and Supendng Propoed Tar, Conoldatng Docket, and Etablhng Hearng Procedure. Docket No. EC et al. (74 FEC 6,069, Wconn Electrc Power Co., Northern State Power Co. (Mnneota, Northern State Power Co. (Wconn, and Cenergy, Inc., et al. Wahngton, DC (January c. Order Condtonally Acceptng or Flng Propoed Market-Baed ate, and etrctng Ablty to Sell at Market-Baed ate. Docket No. E and E (76 FEC 6,33, Delmarva Power and Lght and Atlantc Cty Electrc Company. Wahngton, DC (September d. Order No. 888: Fnal ule. Docket No. M and M (75 FEC 6,080, Promotng Wholeale Competton Through Open Acce Non-dcrmnatory Tranmon Servce by Publc Utlte, ecovery o Stranded Cot by Publc Utlte and Tranmttng Utlte. Wahngton, DC (Aprl e. Order Provdng Gudance and Convenng a Techncal Conerence. Docket No. E (77 FEC 6,65, Pacc Ga and Electrc Company, San Dego Ga & Electrc Company, and Southern Calorna Edon Company. Wahngton, DC (December Polcy Statement Etablhng Factor the Common Wll Conder n Evaluatng Whether a Propoed Merger Content wth the Publc Interet. Docket No. M , Inqury Concernng the Common Merger Polcy Under the Federal Power Act. Wahngton, DC (December eved Flng equrement. Docket No. M (63 F Wahngton, DC (Aprl Order 000: Fnal ule. Docket No. M (89 FEC 6,85, egonal Tranmon Organzaton. Wahngton, DC (December Order Acceptng or Flng eved ate Tar and Code o Conduct (Commoner Maey, concurrng. Docket No. E (93 FEC 6,93, Sthe Edgar LLC et al. Wahngton, DC (November.. 00a. Notce o Extenon o Tme. Docket No. EL , Invetgaton o Term and Condton o Publc Utlty Market-Baed ate Authorzaton. Wahngton, DC (November

478 . 00b. Order Etablhng eund Eectve Date and Propong to eve Market-Baed ate Tar and Authorzaton. Docket No. EL (97 FEC 6,0, Invetgaton o Term and Condton o Publc Utlty Market- Baed ate Authorzaton. Wahngton, DC (November c. Order On Trennal Market Power Update and Announcng New, Interm Generaton Market Power Screen and Mtgaton Polcy. Docket No. E et al. (97 FEC 6,9, AEP Power Marketng Inc., et al. Wahngton, DC (November a. Notce o Propoed ulemakng. Docket No. M (00 FEC 6,38, emedyng Undue Dcrmnaton through Open Acce Tranmon Servce and Standard Electrcty Market Degn. Wahngton, DC (July b. Order Grantng ehearng or Further Conderaton. Docket No. EL , Invetgaton o Term and Condton o Publc Utlty Market-Baed ate Authorzaton. Wahngton, DC (January a. Fnal eport on Prce Manpulaton n Wetern Market: Fact-Fndng Invetgaton o Potental Manpulaton o Electrc and Natural Ga Prce. Wahngton, DC (March.. 003b. Order Amendng Market-Baed ate Tar and Authorzaton. Docket No. EL and EL (05 FEC 6,8, Invetgaton o Term and Condton o Publc Utlty Market-Baed ate Authorzaton. Wahngton, DC (November c. Order Seekng Comment on Propoed evon to Market-Baed ate Tar and Authorzaton. Docket No. EL (03 FEC 6,349, Invetgaton o Term and Condton o Publc Utlty Market-Baed ate Authorzaton. Wahngton, DC (June d. Whte Paper: Wholeale Power Market Platorm. Docket No. M Wahngton, DC (Aprl a. About FEC: Oce o Market Overght and Invetgaton - What We Do. Avalable rom 004b. Market Overght and Invetgaton. Avalable rom 004c. Order Acceptng Tar Flng Subect to Modcaton. Docket No. E and E (06 FEC 6,, New York Independent Sytem Operator, Inc. Wahngton, DC (February.. 004d. Order Grantng ehearng or Further Conderaton. Docket No. EL , Invetgaton o Term and Condton o Publc Utlty Market-Baed ate Authorzaton. Wahngton, DC (January 4. 46

479 . 004e. Order on Tar Amendment No. 55. Docket No. E (06 FEC 6,79, Calorna Independent Sytem Operator Corporaton. Wahngton, DC (February Supplementary Notce o Techncal Conerence on Supply Margn Aement Screen and Alternatve. Docket No. PL et al., Conerence on Supply Margn Aement et al. Wahngton, DC (January 9. U.S. Federal Power Common Opnon and Order Authorzng Merger. Docket No. E-775 (36 FPC 97, Commonwealth Edon Company and Central Illno Electrc and Ga Company. Wahngton, DC. U.S. Senate Energy Polcy Act o th Cong., d e., S Vve, Xaver Olgopoly prcng: Old dea and new tool. Cambrdge, MA: MIT Pre. We, Leonard W Anttrut n the Electrc Power Indutry. In Promotng competton n regulated market, edted by Almarn Phllp. Wahngton, DC: Brookng Inttuton. Weten, Erc W. 999a. Quadratc Surace. From MathWorld A Wolram Web eource. Avalable rom 999b. eal Analytc Functon. From MathWorld A Wolram Web eource. Avalable rom 999c. emovable Sngularty. From MathWorld A Wolram Web eource. Avalable rom Wttkop, Allan. 00. Electronc mal to author, November. Wol, Charle and Larry Pohlman The ecovery o k Preerence rom Actual Choce. Econometrca 5 (May: World Energy Councl The Benet and Decence o Energy Sector Lberalaton: Current Lberalaton Statu. Volume II. London. Avalable rom /open.plx?le=publcaton/deault/current_cl/cltoc.htm. Zachara, Thoma P. and Arthur H. Grube An Economc Evaluaton o Weed Control Method Ued n Combnaton wth Crop otaton: A Stochatc Domnance Approach. North Central Journal o Agrcultural Economc 6 (January: