Spatial oligopolistic equilibria with arbitrage, shared resources, and price function conjectures

38 Pages·2004·0.3 MB·English

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