Math.Program.,Ser.B101:57–94(2004) DigitalObjectIdentifier(DOI)10.1007/s10107-004-0537-4 BenjaminF.Hobbs·Jong-ShiPang Spatial oligopolistic equilibria with arbitrage, shared resources, and price function conjectures(cid:1) Received:November11,2002/Accepted:April9,2004 Publishedonline:7July2004–©Springer-Verlag2004 Abstract. Thispaperconsidersequilibriaamongmultiplefirmsthatarecompetingnon-cooperativelyagainst eachothertosellelectricpowerandbuyresourcesneededtoproducethatpower.Examplesofsuchresources includefuels,powerplantsites,andemissionsallowances.Theelectricpowermarketisaspatialmarketon anetworkinwhichflowsareconstrainedbyKirchhoff’scurrentandvoltagelaws.Arbitragersinthepower marketerasespatialpricedifferencesthatarenon-costbased.Powerproducerscancompeteinpowermarkets a`laCournot(gameinquantities),orinageneralizationoftheCournotgame(termedtheconjecturedsupply functiongame)inwhichtheyanticipatethatrivalswillrespondtopricechanges.Ininputmarkets,producers eithercompetea`laBertrand(price-takingbehavior)ortheycanconjecturethatpricewillincreasewithcon- sumptionoftheresource.Thesimultaneouscompetitioninpowerandinputmarketspresentsopportunitiesfor strategicpricebehaviorthatcannotbeanalyzedusingmodelsofpowermarketsalone.Dependingonwhether theproducerstreatthearbitragerendogenouslyorexogenously,wederivetwomixednonlinearcomplemen- tarityformulationsoftheoligopolisticproblem.Weestablishtheexistenceanduniquenessofsolutionsas wellasconnectionsamongthesolutionstothemodelformulations.Anumericalexampleisprovidedfor illustrativepurposes. 1. Introduction Competition is being introduced into network-based industries throughout the globe withtheobjectivesoflowercostsandimprovingproductqualityandinnovation[26]. Examples of such industries include telecommunications, transportation, and natural gas.Aswithothernetwork-basedindustries,restructuringoftheelectricpowergenera- tionsectorhasintroducedmarketforcesintoanindustrythatwaseithersubjecttoprice controls and/or public ownership. In many power markets in Europe, NorthAmerica, andSouthAmerica,theresulthasbeenloweredpricesforconsumers[17]. But in a few places (notably California), supply shortages and inability of con- sumers to adjust their consumption to prices that vary greatly from hour to hour have B.F. Hobbs: Department of Geography and Environmental Engineering, The Johns Hopkins University, Baltimore,MD21218-2682,USA.e-mail:[email protected] SupportwasprovidedbytheNationalScienceFoundationundergrantECS-0080577. J.-S.Pang:DepartmentofMathematicalSciences,RensselaerPolytechnicInstitute,Troy,NY12180-3590, USA.e-mail:[email protected] Thisauthor’sresearchwaspartiallysupportedbytheNationalScienceFoundationundergrantsECS-0080577 andCCR-0098013. (cid:1) TheauthorsthankChrisDay,FiekeRiekers,andAdrianWalsfortheircollaboration.Theyareparticularly indebtedtoGrantRochforwritingAMPLandMATLABcodesforsolvingthenumericalexamplereported inthepaper. 58 B.F.Hobbs,J.-S.Pang resulted in large price increases. In California, a large share of the increases experi- enced in 2000–2001 has been blamed on market power: the ability of firms to cause pricestodeviatefromcompetitivelevelsbymanipulatingoutputsorbids,orbyother means[4,19].Onewaytoexercisemarketpoweristowithholdoutput,forinstanceby declaringgeneratorstobeunavailablebecauseofequipmentfailures.Pricescanalsobe affectedbystrategicbidding,bymanipulationofmarketsforneededinputssuchasfuel oremissionsallowances,andbydeliberatecongestionordecongestionofthenetwork [3].However,despitetheinfamous“GetShorty”,“DeathStar”,andothersophisticated strategiesdetailedintheinfamousEnronmemos,mostofthemarketpowerproblems experienced by California were due to simple economic and physical withholding of capacity [31]. Economic withholding occurs when generation capacity is made avail- ableonlyathighprices;physicalwithholdingiswhencapacityunavailableatanyprice, forexamplebydeclaringanoutageformaintenanceatatimewhendemandishigh.It has been argued that economic withholding in California was facilitated by contrived shortagesofnaturalgasandNO allowancesatcrucialtimes[18,20,31]. x Exercise of market power affects not only prices, profits, and consumer welfare, butcanalsodecreaseproductiveefficiency(ifpriceincreasesencouragegenerationby high-costsmallerfirms)whileimpactingtheenvironment(ifoutputsshiftamonggen- eratorswithdifferentemissionrates).Marketpowerisuniversallyviewedasoneofthe most serious imperfections in new power markets [14, 25, 30].Therefore, models for projectingpricesandothermarketoutcomesshouldexplicitlyconsiderthepotentialfor strategicbehaviorbypowerproducers. Manysuchmodelshavebeendeveloped(seethereviewsin[9,11,19]).Mostare basedonthecalculationofNashequilibriaforasingletimeperiodforonecommodity (electricenergy).ThemostcommonNashgamessimulatedarethoseinwhichthefirm’s strategic variable is either sales and/or production quantities (Cournot games) or bid functions(supplyfunctiongames).Anewapproachistheconjecturedsupplyfunction [11](similartothenotionofconjecturalvariations,[13]).Itcanbeviewedasageneral- izationoftheCournotmodel,inthatgeneratorscanconjecturethatrivalswillrespond in some a priori way to price changes rather than acting as if they assume that rivals willholdtheirquantitiesfixed.Approachesotherthanmodelingthathavebeenusedto project market outcomes under conditions of market power include experiments with livesubjects[24]andsimulationmodelsbasedonautomata[5,10]. Mostmodelsconsideranundifferentiatedenergymarket.However,inactualpower markets,power(anditsprice)isdifferentiatedbybothtime(e.g.,houroftheday),space (locationonthepowernetwork),andeven“greenness”(renewableversusfossil-fuelled ornuclearenergy).Furthermore,powermarketsarestronglylinkedtomarketsforother commodities,includingancillaryservices(suchasoperatingreserves),fuels,andemis- sionsallowances.Interactionsamongenergymarketsseparatedinspace,time,orquality andbetweenmarketsforenergyandothercommoditiesmaypresentadditionaloppor- tunitiestoexercisemarketpower.Theseopportunitieswouldbeoverlookedifmarkets weretobeanalyzedseparately. Therehavebeensomemodel-basedanalysesoftheseinteractions.Forinstance,by selectivelycongestingordecongestingpowerlines(i.e.,forcingflowconstraintstobe bindingorslack),strategicpowerproducerscanisolatesub-marketsfromcompetition ordeprivetransmissionownersofrevenuetheywouldnormallyreceive[2,7,27].Large Spatialoligopolisticequilibria 59 hydropowerproducerscanshifttheirgenerationfrompeakperiodstooffpeakperiodsin ordertocausepricespikesduringthetimesofhigherdemand[6].Strategicbehaviorin greenandnon-greenenergymarketshasalsobeenmodeled[1].Finally,asmentioned above,ithasbeenarguedthatcreationbypowerproducersofartificialshortagesininput markets(naturalgasandNO emissionsallowances)contributedtothehighCalifornia x pricesof2000–2001.Surprisingly,however,modelershavepaidlittleattentiontostra- tegicmanipulationofinputandpowermarkets(see[22]foranexception).Thus,there isaneedfordevelopment,analysis,andapplicationofsuchmultimarketmodels. Inthispaper,weproposemodelsthatrepresentthelinkagesbetweenspatiallysep- arated markets for electricity and the resource inputs required for power production. These models explicitly recognize opportunities for simultaneous exercise of market power in more than one market. The models include representations of the following featuresofpowermarkets: • Thepredominanceofbilateraltradingofpowerbetweenproducersandconsumers, as opposed to POOLCO-type arrangements, in which a single auctioneer buys all powerfromproducersandthenresellsittoconsumers; • Strategicbehaviorinenergymarkets,representedbyCournotandconjecturedsupply functiongames[11]; • Demandsbypowerproducersforscarcetransmissionservicesthatareallocatedby anindependenttransmissionsystemoperator(ISO); • PowertransmissionflowsamongspatialsubmarketsthataregovernedbyKirchhoff’s voltageandcurrentlaws,modeledusingalinearizedDCnetwork(asin[15]); • Arbitragebymarketersamongdifferentpowermarkets;and • Competitionbypowerproducersandarbitragersforresourceinputs,includingvar- ioustypesoffuelandemissionsallowances. The models are introduced in Section 2, which describe each market participant’s problem,includingrepresentationsoftheproblemsfacedbytheISO,allocatorsofinput resources,arbitragers,andpowerproducers.Twoversionsofthepowerproducermodel arepresented.Inthefirst,producersanticipatehowpowerwillberedistributedbyar- bitragers among spatially separated markets as a result of changes in prices.This can be viewed as a Stackelberg game between producers (Stackelberg leaders) and arbitr- agers(Stackelbergfollowers).Inthesecondversion,producersinsteadviewarbitrage as exogenous. These two types of games have previously been analyzed for the case of pure energy markets in [23]; this paper extend those results to consider how pro- ducersandarbitragersinteractininputmarketsaswellasenergymarkets.Themarket model is completed by imposing a set of consistency (market clearing) conditions on theparticipants. Theremainderofthepaperanalyzescertainpropertiesofthemodels.Solutionexis- tenceanduniquenessisfirstaddressedfortwovariantsofthesemodels(Subsections3.1 and 3.2). Generalizing the conjectured supply function models of [11], both variants assume energy producers anticipate that rivals will adjust their production and input consumptionlinearlyifenergyandinputprices,respectively,changefromtheirequilib- riumlevels.Onevariantassumesthatthelinearadjustmentisdefinedbyafunctionwith afixedpriceinterceptandaslopethatdependsonthemarketprice.Theothervariant, which is simpler to analyze, assumes instead that the slope is fixed and that the price 60 B.F.Hobbs,J.-S.Pang interceptdependsinsteadonthemarketprice.Furtherresultsareobtainedforthespecial caseinwhichthearbitragerdoesnotutilizeresources,andproducersusethemonlyfor production (Subsection 3.3). Section 4 provides a simple computational example that illustratesthemodelandtheeffectofresourceconstraintsandexpectationsconcerning resourcepricesonthesolution.Asetofconclusionsclosesthepaper(Section5). 2. Themathematicalmodel Thespatialoligopolisticcompetitionmodelconsideredhereinismadeupoffourmajor components, each describing the behavior of an essential player in the market that is modeledbyalinearizedDCnetwork[15].UndertheNashequilibriumconcept,these componentsareconcatenatedtoyieldamixedlinearornonlinearcomplementarityfor- mulationoftheoverallmodel. Therearefourtypesofplayers:theproducers,theISO,theinputresourceallocator, andthearbitrager.Theproducersarefirmsthatgenerateandsellacommodity(electric power);theirprimarydecisionvariablesarethesalesateachpointofconsumptiononthe network,andgenerationbyeachoftheirpowerplantsthataredistributedonthenetwork. These firms utilize market-allocated resources in their operations, the consumption of whichisdeterminedbythesalesandgeneration.Inaddition,thefirmsneedtopaythe ISOfortheiruseofthenetworktotransferpowerfromgeneratorstoconsumers.The marketpriceofpowerateachlocationisdeterminedbyanaffinedemandfunction.In ordertorepresentafirm’sexpectationsconcerningtheirrivals’reactionstoprices,the modeladoptsasupplyfunctionconjecturethatusesafirst-orderapproximationnearthe equilibriumfortherelationshipbetweenpriceanditsrivals’salesintheregion. Theproducingfirmsareawareofthepresenceofanarbitragerinthemarket,who tradesbutdoesnotproducepower.Weconsidertwoalternativemodels:oneinwhich thefirmstreatthearbitrager’sactionsendogenouslyandtheotherexogenously.Inaddi- tion to the existence and uniqueness of solutions to the resulting models, we are also interestedintheconnectionbetweentheirsolutions. 2.1. Thepricefunctionconjectures Webeginthemathematicalformulationofthemodelwiththeintroductionoftwosetsof pricefunctionconjectures:oneforthepriceofpowerandtheotherfortheresourceprice. Wecouldalsoincludeafirm’sconjectureconcerningthepriceoftransmissioncharged bytheISO;butsincethetreatmentofthisextensionissimilar,wedonotdevelopthe latteradditionalconjectureinthemodel. The model postulates an affine demand function at each network node for power. In region i = 1,... ,n, the power price p anticipated by producing firm f ∈ F is fi an affine function of the total sales S by all the firms plus the arbitrage amount a i fi anticipatedbyfirmf,whereF isafiniteindexsetcontainingthelabelsofthefirms. Specifically,wehave P0 p ≡ P0− i (S +a ), ∀i = 1,... ,n, (1) fi i Q0 i fi i Spatialoligopolisticequilibria 61 whereP0 andQ0 arepositiveconstants.Bydefinition, i i (cid:1) S ≡ s . i hi h∈F In the supply function conjecture model, the firms are assumed to anticipate that a deviation of the power price from its equilibrium level will stimulate a deviation in supplyfromrivalfirmsfromitsequilibrium.Inparticular,themodelpostulatesthatthe rivalfirms’sales (cid:1) s−fi ≡ shi h(cid:4)=f,h∈F arerelatedtothepricep viathelinearexpression: fi s−fi = s−∗fi +βfi(pi∗,s−∗fi)(pfi −pi∗), (2) ∗ ∗ where(p ,s )isanequilibrium(price,sales)pairthatisexogenoustofirmf’sprofit i −fi maximizationproblembutisendogenoustothemarket,andthefunctionβ (x,y)isof fi oneoftwoforms:(a)apositiveconstantβ ,or(b)arationalfunctiony/(x−α )for fi fi somepositiveconstantα .Incase(a),wehave(afterrearrangement) fi (cid:2) (cid:3) ∗ pfi = pi∗− sβ−fi + β1 s−fi. (3) fi fi Withthetermintheparenthesisontherightasexogenoustothefirmbutendogenous to the market, this expresses firm f’s anticipated price p as a linear function of its fi rivals’saless−fi withthefixedslopeβf−i1 andavariableintercept(variablerelativeto themarket).Incase(b),wehave p∗−α pfi = αfi + i ∗ fi s−fi, (4) s −fi which expresses firm f’s anticipated price p as a linear function of its rivals’sales fi s−fi with the fixed intercept αfi and a variable slope (variable relative to the market, but exogenous to the producer).A noteworthy observation about the fixed-slope con- jecture (3) is that two extreme specifications of β yield two well-known models in fi the literature. Indeed, when βfi = 0, we get s−fi = s−∗fi, which corresponds to the Cournot (fixed quantity) competition model. On the other hand, when β = ∞, we fi getp = p∗,whichcorrespondstotheperfectcompetitionmodel.Bylettingβ be fi i fi a finite, positive constant, we obtain a range of models that vary between these two well-knowncases. Ingeneral,bysubstituting(2)into(1),weobtain (cid:4) (cid:5) P0 p = P0− i s +s∗ +β (p∗,s∗ )(p −p∗)+a fi i Q0 fi −fi fi i −fi fi i fi i 62 B.F.Hobbs,J.-S.Pang whichyields P0 P0− i [s +s∗ −β (p∗,s∗ )p∗+a ] i Q0 fi −fi fi i −fi i fi p = i (5) fi P0 1+ i β (p∗,s∗ ) Q0 fi i −fi i Thisistheeffectivedemandfunctionthatfirmf usesinitsprofitmaximizationproblem. It shows the reaction of price to a firm’s supply decision s , accounting for both the fi responseofconsumersandtheconjecturedresponseofrivalsuppliers.Partofthegoal oftheanalysisistodeterminetherangeofvaluesα inavariable-slopesupplyfunction fi conjecture(4)thatwillensurethewell-definednessoftheoverallmodel.Thefollowing simplelemmaisusefulinthesubsequentanalysis. Lemma1. Let y β(x,y) ≡ , (x,y) ∈ (cid:7)2. x−α ForeverycompactsubsetU ⊂(cid:7)2,thereexistsascalarα¯ >0suchthatforeveryαsuch that |α| > α¯, β is a well-defined Lipschitz continuous function on U with a Lipschitz constantτ(α)>0satisfying lim τ(α) = 0. |α|→∞ Proof. Since U is bounded, it is clear that for all α less than a threshold value, the denominatorinβ(x,y)isnonzeroforall(x,y)inU.Fixanysuchα.Foranytwopairs (cid:10) (cid:10) (x,y)and(x ,y )inU,wehave (cid:6) (cid:7) y y(cid:10) y−y(cid:10) 1 1 − = −y(cid:10) − x−α x(cid:10)−α x−α x(cid:10)−α x−α (cid:6) (cid:7) y−y(cid:10) x−x(cid:10) = −y(cid:10) . x−α (x(cid:10)−α)(x−α) Therefore, (cid:8) (cid:8) (cid:6) (cid:7) (cid:8)(cid:8)(cid:8) y − y(cid:10) (cid:8)(cid:8)(cid:8) ≤ [|y−y(cid:10)|+|x−x(cid:10)|] 1 + |y(cid:10)| . x−α x(cid:10)−α |x−α| |x−α||x(cid:10)−α| (cid:10) (cid:10) Since(x,y)and(x ,y )arebounded,theexistenceofthedesiredτ(α)followsreadily. (cid:12)(cid:13) Inadditiontothesupplyfunctionconjecture(1)intheenergycommoditymarket, we also postulate that each firm has a conjecture regarding the response of the input resourcepricetochangesinthequantityofresourcethefirmconsumes.Specifically,for resourcej = 1,... ,m,theresponsethatfirmf anticipatesintheresourcepriceρ fj inreactiontochangesinfirmf’sconsumptionofresourcej isgivenbythefollowing first-orderapproximationaroundtheequilibriumresourceconsumption: ρ = ρ∗+σ (r −r∗ ), fj j fj fj fj Spatialoligopolisticequilibria 63 where σ is a nonnegative constant and the quantities with the asterisk indicate that fj they are variables exogenous to the generation firms but endogenous to the market. If σ = 0, then this represents a Bertrand game, in which firms are price-takers with fj respecttotheresourceprice.Moregenerally,however,theyanticipatethataresource’s priceincreasesiftheyconsumemoreofit. Inessence,theaboverelationshipisafixed-slope,variable-interceptresourceprice functionconjecture.Wecouldalsotreatavariable-slope,fixed-interceptresourceprice functionconjecturebyaneasyextensionoftheanalysisthatfollows.Similarly,wecould alsoincludeafunctionalconjectureonthetransmissionfee(asin[16]).Forsimplicity, allthesevariationsandextensionsareomitted. 2.2. TheISO’sproblem Letw ∈(cid:7)nbedefinedasthevectoroftransmissionfeeschargedbytheISOforuseof thetransmissionnetwork,whereelementw isthefeefortransmitting1unitofenergy i fromthenetworkhubtonodei.InalinearizedDCnetwork[29],theprincipleofsuper- positionappliessuchthatthecostoftransportingpowerfromitoi(cid:10)equals−wi+wi(cid:10). As a result, the choice of hub is arbitrary.This unusual property (which, for instance, (cid:10) (cid:10) impliesthatthecostfromi toi isthenegativeofthecostfromi toi)differentiates electricpowermarketsfromspatialmarketsforothercommodities. ∗ The ISO is assumed to set the fees w in order to efficiently clear the market for transmissioncapacity.Alternatively,itmightbeassumedthatthereisacompetitivemar- ketfortransmissioncapacityinwhichtransmissionservicesareallocatedtothosefirms that value them the most. Either assumption can be shown to be equivalent to model- ∗ ∗ ing the ISO as a “price-taker” with respect to w [9]. Thus, taking w as exogenous to his problem, the ISO solves the following linear program to determine the energy commodityflowsy ∈(cid:7)ninorderto maximizeyTw∗ subjectto Hy ≤ h, whereH ∈(cid:7)(cid:7)0×nisthetechnologicalmatrixoftheISOandh∈(cid:7)(cid:7)0 isagivenvector. Inwords,theISOmaximizesitsrevenue,subjecttotransmissionconstraints.Examples ofsuchconstraintsinapowersystemcouldincludethermallimitsonflowsinindividual powerlines,constraintsonlinearcombinationsofsuchflows(linearizationsofso-called “nomograms”),andlinearrepresentationsofflowcontroldevicessuchasphaseshifters. Forsimplicity,wehaveformulatedtheaboveproblemusingonlyinequalityconstraints; equalityconstraintsdonotaffectthesubsequentresults.Theoptimalityconditionofthe abovelinearprogramis:thereexistsavectorz∈(cid:7)(cid:7)0 suchthat w∗ = HTz and 0 ≤ z ⊥ h−Hy ≥ 0, wherethenotationz⊥wmeansthatthevectorszandwareorthogonal. 64 B.F.Hobbs,J.-S.Pang 2.3. Theresourceallocator’sproblem TheroleoftheresourceallocatorisanalogoustothatoftheISOinthefollowingmanner. TheresourceallocatorcanbeviewedasaWalrasianauctioneerwhosetsthepriceofthe resourcetoclearthemarket,orasasocialplannerwhoallocatestheresourcetothose whovaluetheresourcethemost.Asanexample,agovernmentenvironmentalagency might auction off a fixed amount (here designated as d) of emissions allowances that generatorsneedtooffsetthepollutantstheyemitasresultofthepowerproductionpro- cess.Inthemodel,theresultingsalesofallowanceswouldberepresentedbythevector u.Alternativelyabrokermightbeapurefacilitatoroftradesof,say,naturalgasamong buyers and sellers, in which case the amount of gas d he contributes to the market is zero.Inthatcase,negativeelementsofuwouldrepresentsalesbyotherparties,while positiveelementswouldbepurchases. ProceedinginamannersimilartotheISOmodeldevelopment,wetaketheresource priceρ∗ ∈(cid:7)masexogenoustotheresourceallocator’sproblem.Therefore,theallocator solvesthefollowinglinearprogramtodeterminetheresourcedistributionsu ∈ (cid:7)m in orderto maximizeuTρ∗ subjectto Du ≤ d, whereD ∈ (cid:7)(cid:7)1×m isthetechnologicalmatrixoftheresourceallocatorandd ∈ (cid:7)(cid:7)1 is thefixedvectorofresourcesthattheallocatorcancontributetothemarket.Thesecon- straintscanbeviewedasmarketclearingconditionsthatensurethatthenetamountof resourcesallocatedtousersDudonotexceedtheresourcesavailabled.LiketheISO’s problem, we have formulated the resource allocator’s problem using only inequality constraints.Theoptimalityconditionofthelatterlinearprogramis:thereexistsavector v ∈(cid:7)(cid:7)1 suchthat ρ∗ = DTv and 0 ≤ v ⊥ d−Du≥ 0, While the resource allocator’s linear program may seem rather simplistic, it actually includessomeusefulspecialcases.Forinstance,intherealisticsituationwheretheonly constraintontheresourcesisthatthereisafixedamountavailablefordistribution,the constraint of the above resource problem becomes u not exceeding a given constant, whichcaneasilybehandledbyourtreatment. Amoregeneralformulationthatisastraightforwardextensionoftheabovemodel isthefollowing.Theallocatorcouldhaveaccesstoanelasticsupplyoftheresourcethat couldbeusedtomeetthegenerators’demands;i.e.,dcouldbeavariablewithaconvex cost function. This could be used to simulate fuels markets, where higher fuel prices stimulateincreasedsupply. Ofcourse,becauseofthelinearityoftheDCloadflowmodel,thisframeworkcan alsobeusedtomodeltheISOasanallocatorofthetransmissionresource.However,we treatthetransmissionmarketseparatelybecausewewishtoanalyzetheroleofarbitrage amongthespatiallyseparatedenergymarketsthattransmissionconstraintscreate. Spatialoligopolisticequilibria 65 2.4. Thearbitrager’sproblem Thearbitragerisassumedtobeanentitythatcanbuypowerinonelocationandsellitin another.TheonlycostthearbitragerincursistheISO’stransmissionfeesbetweenthe twolocationsandthecostofinputresourcesrequiredforthattransaction.Thearbitrager is assumed to be a price taker in all markets.As a result of this frictionless arbitrage, pricedifferencesinthenetworkmustreflectthecostoftransmission(includingperhaps thecostofinputresources). Inthespecialcaseinwhicharbitragerequiresnoresourcesotherthantransmission capacity(asassumedin[16]),animplicationofthelinearDCtransmissionrepresenta- tionintheISO’smodelisthatsucharbitragewillforcethedifferenceinpricepi(cid:10) −pi (cid:10) betweenanytwonodesiandi ofthenetworktopreciselyequalthecostoftransmission fromi toi(cid:10) (i.e.,−wi +wi(cid:10)).Ofcourse,thisisnotgenerallytrueofarbitrageamong commodities;inthemoregeneralcase,thepricedifferenceisconstrainedtobenomore thanthecostoftransportingpowerfromonelocationtotheother,andcouldbeless. This efficient arbitrager is modeled as follows. Taking p∗ ∈ (cid:7)n, w∗ ∈ (cid:7)n, and ρ∗ ∈ (cid:7)m as exogenous to his problem, the arbitrager solves the following linear pro- gramtodeterminethearbitragequantitya ∈ (cid:7)n andresourceusagera ∈ (cid:7)m inorder to maximizeaT(p∗−w∗)−(ra)Tρ∗ subjectto Ga = Ge0 and Eaa = ra +ωa, whereωa ∈(cid:7)misthepre-allocatedresourcesownedbythearbitragerande0 ∈(cid:7)nisa givenvector.Thefirstconstraintisageneralizationoftheconditionthatifanarbitrager sellsenergy,itmustalsobuyanequalamount(onnet,thearbitragerneithergenerates norconsumesenergy).WeassumethroughoutthatthematrixG ∈ (cid:7)(cid:7)a×n hasfullrow rank.Thesecondconstraintdeterminestheamountofresourcesthearbitragermustbuy asafunctionoftheamountofenergyarbitraged. Substitutingthevariablera =Eaa−ωa intotheobjectivefunction,lettingλbethe dualvariableoftheconstraintGa = Ge0,andwritingtheoptimalityconditionofthe abovesimplelinearprogram,weobtain: (cid:9) (cid:10) (cid:2) (cid:3) (cid:2) (cid:3) 0 −GT a (Ea)Tρ∗−p∗+w∗ = . G 0 λ Ge0 WeshouldpointoutthatunliketheISO’sproblemandtheresourceallocator’sprob- lem,thetechnologicalconstraintinthearbitrager’sproblemGa = Ge0 isanequality. In fact, inequality constraints in the latter problem would pose a technical difficulty whenweconsidertheoptimalityconditionsofthearbitrager’sproblemendogenousto thefirms’problems,whichwewilldescribenext.Suchinequalityconstraintspresentno difficultyintheexogenous-arbitrageversionofthefirms’problems;seeSubsection2.8. 66 B.F.Hobbs,J.-S.Pang 2.5. Generationfirm’sproblem:Endogenousarbitrage Takingallquantitieswithanasteriskasexogenoustoitsproblem,firmf solvesaconvex quadraticprogramtodetermineitssalessf ≡(s )∈(cid:7)n,generationgf ≡(g )∈(cid:7)n, fi fi resourceusagerf ≡ (r ) ∈ (cid:7)m,andanticipatedarbitrageamountaf ≡ (a ) ∈ (cid:7)n fj fi inorderto (cid:1)n maximize [s p −c (g )−(s −g )w∗] fi fi fi fi fi fi i i=1 (cid:1)m − [ρ∗+σ (r −r∗ )]r j fj fj fj fj j=1 (cid:1)n (cid:1)n subjectto s = g , fi fi i=1 i=1 Bfsf +Efgf = rf +ωf, 0 ≤ gf ≤ CAPf, 0 ≤ sf, (cid:9) (cid:10) (cid:2) (cid:3) (cid:2) (cid:3) 0 −GT af (Ea)Tρ∗−pf +w∗ = , G 0 λ Ge0 Q0−[s +s∗ −β (p∗,s∗ )p∗+a ] and p = i fi −fi fi i −fi i fi , fi Q0 i +β (p∗,s∗ ) P0 fi i −fi i ∀i = 1,... ,n, whereωf ∈(cid:7)misthepre-allocatedresourcesavailabletofirmf,Bf andEf aretech- nologicalmatrices,CAPf ∈(cid:7)nisthevectorofgenerationcapacities,andc :(cid:7)→(cid:7) fi isthegenerationcostfunction,whichisassumedtobeconvexbutnotnecessarilylinear throughout the paper. This cost function excludes the cost of resources that are con- sumedduringtheproductionprocess.Forinstance,itiscommontoincludefuelcosts inc (g )(bymultiplyingthefueluseperunitbythefuelprice);however,iffuelis fi fi modeledasaresourcemarket,thenitwouldnotbesoincluded. Thefirm’sobjectivecanbeinterpretedasfollows.Thegenerator’sprofitdependson revenuesfromsellingenergy,thecostsofgeneratingandtransmittingenergy,andthe net revenue from buying or selling in the input resources markets. For instance, input resourcesexpenditurescouldincludethecostoffueloremissionsallowancesconsumed in generation.As explained in Subsection 2.1, the price of resource j is a first-order ∗ approximation of how the equilibrium price ρ is conjectured by firm f to change if j ∗ f changestheamountitconsumesfromtheequilibriumvaluer .Theasterisksuper- fj scriptsindicatethatthosevariablesarefixedparametersfromthepointofviewofthe firmf,eventhoughtheyarevariablesfromthepointofviewofthemarket. Turning to the constraints, the first one ensures that sales are balanced by genera- tion. The second defines the net amount of input resources rf that the firm needs to buyfromtheresourceallocator.Examplesofsuchresourcesincludefuelandemissions