Information aggregation in a prediction market for climate outcomes

2018·0.35 MB·English

by  Elmira Aliakbari

EnergyEconomics74(2018)97–106
ContentslistsavailableatScienceDirect
Energy Economics
journal homepage: www.elsevier.com/locate/eneeco
Information aggregation in a prediction market for climate outcomes
ElmiraAliakbaria,RossMcKitrickb,⁎
aFraserInstitute,Vancouver,BC,Canada
bDepartmentofEconomics,UniversityofGuelph,Guelph,ON,Canada
a r t i c l e i n f o a b s t r a c t
Articlehistory: Twoformsofuncertaintyinclimatepolicyarethewiderangeofestimatedmarginalcostsanduncertaintyover
Received16November2017 credibilityofrivalinformationsources.Weshowhowarecently-proposedsolutiontothefirstproblemalsohelps
Receivedinrevisedform13February2018 addressthesecond.Thepolicyisanemissionstaxtiedtoaveragetemperatures,coupledwithpermitsthat
Accepted1June2018
exempttheemitterfrompayingthetaxinafutureyear.Ithasbeenshownthattheresultingtaxpathwillbecor-
Availableonline5June2018
relatedwithfuturemarginaldamages.Ithasbeenconjecturedthatthepermitpriceswillyieldunbiasedforecasts
JELclassification: oftheclimate,which,iftrue,wouldaddresstheseconduncertainty.Weconfirmtheconjecturebydescribinga
tradingmechanismthatconvergesonunbiasedforecastsiftradersarerisk-neutral.Riskaversionslowsdown
Q54
Q58 butdoesnotpreventconvergence.Wealsoshowthattheforecastsaremorelikelytobesufficientstatistics
H23 thestrongertheconsensusonclimatescience.
G13 ©2018ElsevierB.V.Allrightsreserved.
Keywords:
Climatechange
Uncertainty
Carbontax
Tradablepermits
State-contingentpricing
Predictionmarkets
1.Introduction temperature-indexed carbon tax (McKitrick, 2011) that presents
emitterswithadynamicpricingruleratherthanacommitmenttoa
Apolicymakerwantingtoaddresstheclimatechangeissuefaces specificpricepath.Onanex-postbasis,theruleyieldsapricepath
manyinformationproblems.Thispaperexaminesnovelsolutionsto highlycorrelatedwiththeunobservabletruemarginaldamagestrajec-
twoofthemostacute:uncertaintyregardingmarginaldamages of tory,thuscontributingtoresolutionofthefirstformofuncertainty,
greenhousegases,andincompleteorbiasedselectionofinformation sincefirmssubjecttothetaxmustformunbiasedexpectationsabout
forpresentationtopolicymakers.Thefirstderivesinpartfromscientific futurevaluesandplanaccordingly.Aproposedmodificationtothe
uncertaintyoverEquilibriumClimateSensitivity(ECS),whichisdefined policyinstrumentcanalsoaddressthesecondinformationproblem.
astheprojectedlongtermclimateresponsetodoublingtheconcentra- Tofacilitateexpectations-formation,Hsu(2011)proposedpairingthe
tionofCO2intheatmosphere,afterenoughtimehaspassedforallcom- taxwithasequentialfuturesmarketfortradablecertificates,eachof
ponentsoftheclimatesystemtoadjust.Awidely-usedstudybyRoeand whichwouldexempttheholderfrompayingthetaxonatonneofemis-
Baker(2007)yieldedanECSdistributionwitha5thpercentileof1.72°C sionsinaspecifiedfutureyear.Heconjecturedthatsincefirmshavea
anda95thpercentileof7.14°C(IWG2010).Applyingthisdistribution financialincentivetogettheforecastsright,thepricepaththusgener-
anda3%discountrate,IntegratedAssessmentModels(IAMs)generate atedwouldprovidethemostobjectiveandinformativeforecastoffu-
arangeofmarginaldamageestimatesspanning-$22to$727pertonne turemarginaldamagesand,byimplication,futureclimatewarming,
ofCO2(IWG,2013).Hence,basedonmainstreamclimateandeconomic sinceinvestorswillhaveanincentivetouseallavailableinformation,
modelling,theoptimalclimatepolicyseemstoliesomewherebetween i.e.toavoidcherry-picking.
asmallsubsidyfor,andaneffectivebanon,allemissions—aratherun- Thispaperisconcernedwiththeconditionsunderwhichtheper-
helpfulstateofaffairstosaytheleast. mitstradingmechanismwouldyieldinformativeandunbiasedfore-
Past attempts to reduce the uncertainty of climate policies are castsoffutureclimatestates.Werefertothetemperature-indexed
reviewed in Section 2. The approach considered herein involves a pricingruleasastate-contingentcarbontax,andthetradingsystem
asafuturesmarketforexemptionpermits.Sinceprojecteddamages
⁎ Correspondingauthor. areexpectedtooccurratherfarinthefuture,theycannotbedirectly
E-mailaddress:[email protected](R.McKitrick). measuredandwilldependstronglyonthemodeler'schoiceofECS,
https://doi.org/10.1016/j.eneco.2018.06.002
0140-9883/©2018ElsevierB.V.Allrightsreserved.
98 E.Aliakbari,R.McKitrick/EnergyEconomics74(2018)97–106
amongotherparameters.Policymakersmustthereforerelyondamage outcomefailstotakeaccountofsomerelevantinformationaboutthefu-
estimatesthatrestonexpertopinion,whichraisesthepossibilityof turepathoftheclimaterequiresanassumptionthatthereisnocurrent
biassincescientistswillbeawarethattheirparameterselectionshave consensusaboutclimatescience.
implicationsforpolicyoutcomesoverwhichtheymayhavepreferences TheFV96frameworkassumestradersarerisk-neutral.Weextend
(Johnston,2012,InterAcademyCouncil,2010).Apredictionmarketthat thetheorybyallowingfirmstoberisk-averse. Weshow thatrisk-
generatesastate-contingentcarbontaxpathbasedonexpectations aversionslowsdowntheconvergenceprocess,sothemarketoutcome
amongagents who haveafinancialincentivetogettheirforecasts ismoresusceptibletotheinfluenceofuninformedtraders.
rightwouldpotentiallyresolvethisproblembycreatinganincentive Predictionmarketswerefirstproposedin1988byresearchersatthe
toformunbiasedclimateprojections.Anidealauctionpriceshould UniversityofIowainterestedinpredictingUSpresidentialoutcomes
thereforebebothanunbiasedestimateoftheactualfutureprice(and (Segol,2012).Themarketsweredesignedbasedontheinformational
hencetheactualfutureclimatestate),andalsoasufficientstatistic roleofpricesandtheefficientmarkethypothesissuggestedbyHayek
(makingefficientuseofallavailableinformation). (1945)andFama(1970).Theyarenotmagicdevicesforgenerating
WemodelapredictionmarketimplementingtheMcKitrick(2011) error-freeinformationaboutthefuture,sincetheyarestillsusceptible
andHsu(2011)state-contingentcarbontax/predictionmarketsystem toallthebiases,wishfulthinkingandotherformsofirrationalitythat
in an auction framework developed by Kyle (1985) and Foster and enterhumanjudgment.Theiradvantageisthattheycreateafinancial
Viswanathan (1996, herein FV96). Three types of traders are as- incentiveforparticipantstobeobjectiveandtouseallavailableinfor-
sumed to participate. Risk-neutral firms subject to the emissions mation,andthisleadsustoaskhowtheymightbeharnessedtoyield
tax are assumed to have private access to noisy signals about the unbiasedandefficientclimateforecasts.Althoughmarketefficiency
likely future climate state and hence the likely future tax rate, and assumptions have been exposed to some critiques, especially from
tomakebidsforpermitsbasedontheirprofit-maximizingstrategies. behavioral economists, other researchers (Wolfers and Zitzewitz,
Also,anunspecifiednumberofuninformedtradersgeneratenoisein 2004;Bergetal.,2008)haveshownthatpredictionmarketsperform
the market by trading based on purely random signals about the wellandgenerateaccurateforecasts.ArecentsurveybyGraefe(2016)
futureclimatestateratherthaninformativeprivatesignals.Amarket compares prediction markets to avarietyof pollingandprediction
makerwhoonlyseestheaggregateorderflowbutdoesnotobserve methodsininternationalelectionsandshowsthattheyconsistently
individualbidsorthenumberofuninformedtradersclearsthemar- providethemostaccurateforecasts(eventhough,asnotedinGraefe,
ketineachroundofbidding,therebygeneratingapricesignalwhich 2017,theycanstillfail,astheydidintheBrexitandUS2016Presidential
isincorporatedintoinformationsetsbytradersinsubsequenttrading electioncases).Thismotivatesourinterestinusingthemforpredicting
rounds. climatechange,whichisanapplicationofconsiderableinternational
Interestingly,thistradingset-upcorrespondstothatofanewly- importance.
proposedclimatepredictionmarkettoberunbyaUK-basedfinancial Recent work by Atakan and Ekmekci (2014) has shown that a
firm(WintonCapital,2017).Traderswillbeabletobuyandsellsecuri- common-valuemarketmayfailtoaggregateinformationifthebuyers
tieswhoseunderlyingvaluewillbetiedtofuturevaluesofpublicly- mustsimultaneouslychooseanaction,thebidpayoffisjointlydeter-
available global temperature measures. Though there is no state- minedbytheactionchosenandthestateoftheworld,andtheprice
contingentpolicyneedingtobehedged,themarketdesignerswere oftheobjectisindependentofthestateoftheworld.Inourset-up
motivatedbyconcernsthatbiasedinformationinterferedwithoptimal buyersdonotcommittoanyactionotherthanholdingthepermit,the
policyplanning,asking:“Arepeoples'positionsdrivenbywhatthey valueofwhichisfullydeterminedbythestateoftheworldindependent
wanttobetrueorwhattheybelievetobetrue?”(WintonCapital, ofanysubsequentactionchosenbyabidder.Consequently,thefailure
2017)Theproposedmarketallowinformedanduninformedpeopleto ofinformationaggregationinAtakanandEkmekci(2014)doesnot
tradecertificatestiedtofutureclimatestates,withWintonactingasa occurinourmodel.However,ifthepolicygamehereinweremodified
non-profitmarketmaker. suchthateverybiddermustsimultaneouslycommittoanactionto
Ourpaperprovidessomeevidencethatsuchamarketwillinfact betakenaftereachauction,thevalueofwhichwoulddependonthe
aggregateinformationinausefulway.Employingapropositiondue stateoftheworldafterthepriceisrevealed,thenwewouldneedto
to FV96 we confirm part of the conjecture of Hsu (2011) for the takeintoaccountthepossibilitythatanequilibriummayemergein
caseofrisk-neutraltraders:suchapredictionmarketwouldyieldan whichnoinformationisaggregated.
unbiasedforecastofthefuturetaxrate,andhencethestateofthe Theassumeddamagefunctionunderlyingourmodeliscontinuous
climate.Itislessclearthatthepredictionmarketwilluseallavailable (seeAssumption4inMcKitrick,2011)asistypicallyassumedinclimate
informationefficiently.Interestingly,one'sbeliefaboutthelevelof policy models. Lemoine and Traeger (2012, 2016) have explored a
consensusaroundclimatesciencestronglyinfluencesone'sinterpreta- modelwithoneormorepotentialdiscontinuitiesinthestatefunction,
tionofthemarketoutcome.Thequestionofthedegreeofconsensusin reflectingconcernsaboutpossible“tippingpoints”orbifurcationsin
climatescience,andwhatspecificpointsexpertsactuallyagreeon,is theclimatesystematwhichanirreversibletransitiontoanewstate
itselfcontroversial(Berryetal.,2016;Tol,2014).Wetreatthenotion functiontakesplace.Suchtippingpointsareassociatedwithchanges
ofscientificconsensusascorrespondingtoahighlevelofcorrelation inthepollutantstock(concentration),whichisaheavilysmoothed
of informed traders' signals, and hence of their beliefs. In other functionofannualemissionsinthecaseofcarbondioxide.Thepossible
words, a strong scientific consensus means that regardless of the existenceofafuturetippingpoint,thetiminguntilitisencountered,
sourceaninformedtraderdrawsupon,itisthesameinformationas andthemagnitudeofthediscontinuityitmightimpose,areadditional
other traders are receiving. We find that, even in the presence of formsofuncertaintythatwedonotaddressherein.Theprediction
noisyuninformedtraders,themarketpricewillconvergetoasuffi- marketweproposemayhelpreducetheseuncertaintiessincetraders
cientstatistic(inotherwords,anexpectationthatefficientlyusesall willprofitfromcorrectlyanticipatinganyformofchangeinthestate
availableinformation)if,atthestartoftrading,privatebeliefsofin- variable,butestablishingthisformallyisbeyondthecurrentlimitsof
formedtradersarehighlycorrelated,orinotherwords,ifthereisa ouranalysis.
trulystrongconsensusonthescientificissues,sothattradersseeking Thepaperisorganizedasfollows.Section2providestheliterature
credibleinformationareeffectivelydrawingthesamesignalsregard- reviewonthistopic.Section3presentsthemodel,assumptions,and
lessofsource.Inthiscase,thetradingprocesswedescribewillyield definitions.Section4describesthelinearequilibriumofthemodeland
apricepaththatincorporatesallrelevantinformationandnotrader itsnecessaryandsufficientconditions.Section5presentstheresults.
wouldbeabletoimproveonthemarketpriceforecastusinghisorher Section6extendstheexistingmodeltoincluderiskaversionandthe
privateinformationset.Conversely,toarguethatthepredictionmarket finalsectionbrieflysummarizestheconclusions.
E.Aliakbari,R.McKitrick/EnergyEconomics74(2018)97–106 99
2.Theuncertaintyproblem rateτ(s )whichisunique,sos ⟹τ andviceversa.FollowingHsu
T T T T
(2011),insteadofpayingthetaxinyearT,firmshavetheoptionto
Thereisalargeliteratureonpricingcarbondioxide(CO )emissions submitemissionallowancesthattheyalreadypurchasedinprevious
2
asawayofaddressingtheglobalwarmingexternality.Usingclassical years, each of which allows a one tonne exemption from paying
theory,thepriceshouldbeequaltomarginalsocialdamagesfromemis- thetax.Weassumeallowances(whichwealsocallpermits)dated
sions.However,inthecontextofglobalwarming,thereissignificant foronevintageTcannotbebankedforuseinanotheryear.Therefore,
uncertaintyregardingthemagnitudeoftheeffectofgreenhousegas thespotpriceofapermitp infutureyearTmustequalthetaxrate,
T
emissionsonglobalwarmingandthemagnitudeoftheimpactthat sop =τ (s ).2Sincethefunctionalformofτ isknown,thepermit
T T T T
globalwarminghasontheeconomy.Henceidentifyingtheoptimal pricealsorevealss.
T
policyhasprovedtobechallenging. Atcurrenttimetconsideramarketfortradablepermitsforvintage
Nordhaus(1991,1993a,1993b)introducedtheDynamicIntegrated yearT.Weassumetherearethreetypesoftradersinthismarketwho
ClimateandEconomics(DICE)modeltohelpstudythemechanisms willtradeaspecificpermitoverh=1,2,…,kauctionrounds.First,
connectingCO2emissions,atmosphericconcentration,changesintem- thereareMrisk-neutraltraderswhopossessdisparateprivateinforma-
perature,andchangesinoutputandconsumption,andtheeffectsover tionaboutthetrueclimaticstateinperiodT,andthusaboutthetrue
timeofthesethingsonwelfare.Usingagrowthmodelhefoundanop- priceofapermit.Eachtrader'sinitialprivateinformationisdenoted
timalpathforemissionabatementandtheassociatedcarbontaxcondi- g (T)where i= 1, 2, …,M.The private signal can consist bothof
i0
tionalonassumptionsaboutkeyparametervalues.Eversince,many publicly-available information and privately-acquired information.
DICE-styleIntegratedAssessmentModels(IAMs)havebeendeveloped. Second, there are several uninformed traders who do not possess
TheadvantageoftheIAMapproachisthatityieldsaspecificpricepath. anyprivateinformationandtradeasasingleentityinthismarket.
Amajordisadvantageistheassumptionthatallthekeyparameter Thismeansthatasingleaggregatebidemergesfromthisgroup.The
valuesareknownatthestart,andnonewinformationwillbeobtained assumption thatuninformed traders are present allows the results
overtime,sothereisnoroleforlearning.Theoptionoflearningwould toberobusttothepossiblepresenceofagentswhoactonsignalsthat
implythatafixedpathisnotprescribedatthestart,onlyastartingvalue haveno validinformationalcontent. Aswe will discussbelow, the
plusaruleforassimilatingnewinformation. assumption is also necessary to avoid the Grossman and Stiglitz
KellyandKolstad(1999)relaxedtheno-learningassumptionbyin- (1980)paradoxthat,ifonlyinformedtradersarepresentinamarket,
corporatingBayesianupdatingabouttherelationshipbetweentemper- informationwillnotactuallybecollectedandused.Third,thereisa
aturechangesandgreenhousegaslevelsintheDICEmodel.Theirmain setofrisk-neutralmarketmakerswhocanbeconsideredaslargepri-
resultwasthatitwouldtakebetween90and160yearsfortheuncer- vatefinancialinstitutionswhoseroleistoclearthemarketatzeroex-
taintytoresolvesufficientlytorejectanincorrectpolicychoice.Leach pectedprofit.Themarketmakersareresponsibleforfacilitatingtrades
(2007)addeduncertaintyoverthetimelagoftemperaturechange inthismarket.
andfounditcouldtakehundredsorthousandsofyearsuntiluncertainty Atthebeginningofthefirstauctionround(h=1),Minformedtraders
isresolvedsufficientlytorejectanincorrectpolicystance. receivedifferentnoisysignalsregardingthetruestatevariableinafuture
McKitrick(2011)proposedanalternativeapproachthatusesthein- dateT.Usingtheirprivatesignalstheyeachformdifferentexpectations
formationimplicitintheclimatestatefunctionbytyingtheemissions p^ aboutthefutureprice.Then,informedtradersanduninformedtraders
iT
taxlinearlytoanobservedmeasureofatmospherictemperatures.In simultaneouslychoosethenumberofpermitstheywanttobuyandsub-
thiswaypricesarecontinuallyupdated,soemittersmustformexpecta- mittheirquantities(marketorders)tothemarketmakers.Wedenotethe
tionsaboutthefuturepathofwarming(andtheemissionstax)rather informedtraders'marketordersandtheuninformedtraders'aggregate
thanbeinggivenadeterministicschedulebypolicymakers.Ifglobal marketorderasq ðp^ Þandu ,respectively.Sinceweareonlyconsider-
ih iT Th
warmingisexpectedtoberapid,taxeswillbeexpectedtogouprapidly, ingasinglevintageperiodT,wherevertheTsubscriptisnotneededwe
andviceversa.Agentswouldneedtoacquirecredibleforecastsofcli- willsuppressitforclarity.Atthisstage,whenchoosingtheirpermitquan-
mateoutcomesinordertomakelong-terminvestments,andwould tity,theonlyinformationinformedtradershaveistheirinitialsignal,so
haveanincentivetoidentifythemostaccurateinformation. theirinformationsetisΩ (g ).Marketmakersobservetheaggregate
ih i0
ofinInveosrtdoerrrtisokp,Hrosvuid(e20o1b1je)cptirvoepoinsefodrmthaetiiodneaaonfdcaolulopwlinfgorthheesdtgaitneg- quantities(yh¼ΣMi¼1qihþuTh)withoutseeingindividualorderssepa-
rately,andtheysetapricedeterminedbyacompetitiveprocessthat
contingent pricing rule with a market for permits, each of which yieldszeroexpectedprofitsforthemarketmakers,implyingthatitbe-
exemptstheholderfrompayingthetaxonaunitofemissionsinafu- comesequaltoE[p|y].3Everyoneobservesthepriceandthetotalquan-
tureperiod.Themarketforfutureexemptioncertificateswouldmoti- titiestraded.Next,Tmahrketmakerstradeaquantitym = −y.
h h
vatebuyerstoseekinformationaboutthelikelyfutureclimatestate,
Inthesubsequentauctions,informedtraderswilltradebasedon
andthetradingprocesswouldaggregatetheinformationintoanequi- theirinitialsignalsandwhattheylearnbyobservingpreviousrounds.4
libriumprice.Thiswouldprovidethebestpossibleforecastsoffuture
Inotherwords,theirinformationsetconsistsoftheirinitialsignals,past
tax rates, as long as the trading process yields objective, unbiased
prices,pastmarketorders,andpastquantitiestradedbythemselves,
priceexpectations.Thepricepath,inturn,wouldyieldunbiasedfore- whichwewriteasΩi(gi0,p−h,y−h,qi,−h)where–hdenotesallprevious
castsofthefuturetemperaturepath.Ourmodelhereinfocusesonthe
rounds. Market makers do not observe the individual quantities;
questionofwhethersuchamechanismwouldyieldunbiasedforecasts
whentheyreceivetheaggregatequantities,theyupdatetheirbeliefsre-
andmakeefficientuseofallinformationavailabletotraders.
gardingtheirestimateofp andsetthepriceaccordingly.Hence,their
T
informationsetconsistsofthecurrentandpastaggregateorderflows
3.Themodel andalsopastprices,ΩMM(yh,y−h,p−h).Theprocessoftradinggoesfor
FollowingMcKitrick(2011),atafuturedateT,astatevariables ,
T
whichisanon-manipulablemeasureoftheclimatestate,1willbere- 2 SinceτTisalinearandknownfunctionofsT,tradingincontractsforthefuturecarbon
vealed.Basedonthisobservation,thepolicymakerwillimposeatax taxwouldyieldthesamefindingastradingincontractsforthefutureglobaltemperature.
3 Themarketmaker'sexpectedprofitisE(π)=−yh(E(pT)–pk)wherekreferstothe
lastroundoftheauctions.Thereforeateachactionroundhwhereh≠k,apriceequalto
1 McKitrickproposesthemeantemperatureoftropicaltroposphereasmeasuredby theexpectedvalueofapermitgiventheobservedorderflow,pH=E(pT|yh),wouldyield
weathersatellitesasthestatevariable.Eachsystemformeasuringthestatevariablehas zeroexpectedprofitsformarketmakers.
itsstrengthsandweaknessesbutwesetthatissueasideforthepresentstudy.SeeCederlöf 4 Buyingpermitsonafuturesmarkethelpsinvestorshedgeagainstcostsoffuturecli-
etal.(2016)foracomparativesurveyofclimatedatasetsandtheirrelativeuses. matepolicy.
100 E.Aliakbari,R.McKitrick/EnergyEconomics74(2018)97–106
k rounds where k is somefinite, largenumber.At this point when knowinganyotherfunctionsofg ,g ,…,g willnotimprovethe
10 20 M0
consideringthetradingprocessforthevintageyearT,weignoreinfor- estimateofp .Asmentionedbefore,thenormaldistributionofthe
T
mationspilloversthatmightexistfromclimaticpredictionmarkets randomvariablep withmeanρandvarianceΣisknowntoallpartici-
T
forothervintages.However,thisinformationcanbeconsideredpart pants.Therefore,allthatisneededtopredictp isθg.Infact,knowingθg
T
ofthepublicly-availableinformationusedbytradersandtherefore wouldprovideuswiththebestestimateofthetruevalueofpermitp as
T
wouldnothaveanyeffectonourmodel. itisasufficientstatisticforalltheavailableinformation.Thisassump-
Thetradingsetupcorrespondstothatappliedinwell-knownfinan- tionsimplifiesthecomplexmodelandwillbeusedinthefollowing
cialmarketssuchastheNewYorkStockExchange(NYSE)andTokyo discussion.
StockExchange.TheclosestapplicationofthissetupistheNYSEcall Itishelpfultoclarifytheset-upthusfarusinganumericalexample.
auctionopeningwheremarketmakersindicateapricerange,traders Supposethatintheyear2018anauctionisheldforyear2025permits.
submittheirmarketorders,andatlastmarketmakersadjusttheinitial Theactualpricethatwewillobservein2025isdenotedP andisa
2025
prices.Inourmodelwehavetakenintoaccountstrategicbehaviorof drawfromN(25,50),whereρequals25andΣequals50.Thisinforma-
theinformedtradersgiventheirprivateinformation,meaningthat tioniscommonknowledge.Theactualpricein2025willbe25+ε.No
theyarenotpricetakersandtheyconsidertheeffectsoftheiractions oneknowsthisamount,howeverinformedtradershaveprivatesig-
onthepriceandinformationalcontentofthepricewhenchoosing nalsaboutε.Therefore,tradersusingtheirsignalswillhavedifferent
theiractions.Sincewehaveafinitenumberoftraders,itisnotreason- expectations regarding P , such as (for example) 20, 22, 25, 27,
2025
abletoassumepricetakingbehavior.Forinstance,ifinformationis and29.
costlytoobtain,theremightbesomelargefirmsthatcaninvestsuffi- Now,supposethatapricep emergesfromtheauctionfor2025
k
cientamountstoacquireexclusiveinformation,leadingthemnotto allowances.Ifbothp and ρþθg arethesame,forinstanceifthey
k
behaveaspurepricetakers. bothequal$25.50,thenthepricearisingfromtheauctionisasufficient
Choosingthenumberofpermitstomakeavailableforavintageyear statisticforalltheavailableinformation.Auctionparticipantscannotdi-
Tisacriticalmatter.Thenumberofavailablepermitsshouldbelarge rectlyverifywhetherthisisthecasebecausetheyonlyobservetheauc-
enoughtocreatearealmarketsothattraderswouldhaveenoughin- tionpricep.Suppose,forinstance,thatρþθg¼25:50butp =$25.80.
k k
centivetoobtaininformationandengageintrading.However,itcannot Thisimpliestheauctionpricehasnotusedalltheavailableinformation.
besolargeastoexceedtheemissionlevelassociatedwiththetaxthat What we will show subsequently is that both p = $25.50 and
k
period,otherwisethepricewouldgobelowthetaxrateandconverge p =$25.80areunbiasedforecastsofP giventheorderflow.The
k 2025
onzero,defeatingthepurposeofthehybridsystem(Hsu,2011).In differencebetweenthetwopricesreflectshowmuchoftheirprivate
ordertoavoidthis,apolicymakerwouldfirstneedtohaveanestimate signalsagentschoosetorevealthroughtradingactivity.Bylooking
oftheemissionlevelinafutureyearT,e,andthensetthenumberof justattheauctionpricep thereisnowaytoknowwhetheritisasuf-
T k
availablepermitstobeafractionofthat. ficientstatistic.Sufficiencycanonlybeinferredbasedontheinforma-
Inclusionofanappropriatediscountrateforfutureprofitswillnot tionstructureofthemarket.ItwillturnouttodependcriticallyonΩ ,
0
changetheresultsinourmodel.Inthecaseofalong-termprediction namelytheinitialcorrelationofthesignalstradersdrawupon,which
market,weexpecttraderstoforecastanddiscounttheirfuturetax we interpret as a measure of thedegreeof scientific consensus on
liability,sothatthepriceofapermitrevealsthediscountedexpected climatechange.
liability. Uninformedtraderssubmitaquantityu atroundhthathasanor-
h
FollowingFV96,weassumethatthepriceofapermitforfutureyear maldistributionwithmeanzeroandvarianceσ2.Weassumeunin-
u
Tisarandomvariablewithanormaldistributionwithknownmean formedtraders'orderu tobeindependentofalltheotherrandom
h
ρ(T)andvarianceΣ(T),sop~N(ρ,Σ).The“true”priceofapermitisare- variables.Theyhavenoprivateinformationandnothingcanbelearned
T
alizationoftherandomvariablep thatisfixedandunknowntoallthe aboutp fromtheiractionsduringthetradingprocess.
T T
traderspriortoT,however,informedtradersobserveanoisysignalre- Theithinformedtrader,givenhisprivateinformationandwhathe
gardingthistruevalue.5 learnsfrompastprices,submitsaquantityq atroundh.Marketmakers
ih
Weassumethatthesignalvectorg=[g ,g ,…,g ]isdrawnfrom observetheaggregatequantityy submittedbybothinformedandun-
10 20 M0 h
amultivariatenormaldistributionwithvariance-covariancematrixΨ informedtraders,updatetheirbeliefsregardingtheinitialsignalsusing
0
andzeromeans.Theformofalldistributions,includingp ,isknown g =E[g |y ,…,y ],6andsetthecompetitivepriceatperiodhusing
T ih i0 1 h
to all tradersin this prediction market.We assumethat all signals p =E[g |y ,…,y ],meaningthatmarketmakersearnzeroprofits
hT i0 1 h
havethesameinitialvarianceΛ ,theinitialcovarianceΩ between onaverage,byassumption.
0 0
anytwosignalsisthesame,andthecrosscovariancewiththetrue Inthismarket,informationgetsfullyaggregatedifthemarketprice
price of the permit c is the same for all signals. As shown in the containsinformationfromallmarkettraderssothateachinformed
0
Appendix,thesethreeassumptionsimply: trader finds his own private information, g , redundant. In other
ih
words,privateinformationgetsfullyaggregatedifaroundhoccursin
Eðp jg ;g ;…;g Þ¼ρþθg ð1Þ which
T 10 20 M0
wheregistheaveragesignalandθisaconstant.Thismeansthatacon- phT ¼E½pTjy1;…;yh(cid:2)¼EðpTjg10;g20;…;gM0Þ¼ρþθg ð2Þ
stantmultipleoftheaveragesignal,θg,isasufficientstatisticforallthe
Therefore,iftheauctionpricebecomesequaltoρþθg,itbecomesa
informationknownbytraders(allthesignals)topredictp .Inother
words, θg uses all the available information (all the signTals), and perfectaggregatorofinformation,andconsequentlyanidealestimateof
p.Inthiscase,thepricecontainsalltheavailableinformationandbe-
T
comesfullyrevealing,withoutcontaininganyerrorinpredictingp .7
T
5 ThedistributionofpossiblefuturetemperaturesmaynotbeGaussian.RoeandBaker Notethattradersdonotnecessarilyknowiftheobservedpricep
hT
(2007)pointoutthatinthestandardrepresentationofclimatesensitivity,ifthefeedback
parameterisnormallydistributedtheequilibriumclimatesensitivity(ECS)couldhavefat
tails.Weitzman(2009)usedthisresulttoarguethataninsurancepremiumagainstcatas- 6 Marketmakersneedtoupdatetheirestimateregardinginitialsignalssothattheycan
trophe approaches infinite present value. But Roe and Bauman (2013) criticized haveabetterestimateofthetruepriceofapermitpT.Inparticular,price,whichisthe
Weitzman'sargumentbecauseitignoresthetime-to-equilibrium,whichgoesupwith marketmakers'updatedestimateofpTgiventheirobservedorderflow,isrelatedtothe
thesquareofclimatesensitivity.Therealizedwarmingassociatedwiththefatuppertail marketmakers'estimateofsignalsinthefollowingway(FV96)
XM
osefqEuCeSnwtloy,ueldvetankuenadtehroaushaignhdEyCeaSr,sthtoemdiastterirbiaultizioen(RofoeteamnpdeBraautumreasn,o2n0t1h3e,pti.m6e53s)c.aCleono-f phT¼Mθi¼1gcιh
themarketweconsiderwouldnotnecessarilydepartfromaGaussianprofile. 7 Inparticular,weassumethattheidealestimateofpT,ρþθg,isindeedequaltopT.
E.Aliakbari,R.McKitrick/EnergyEconomics74(2018)97–106 101
satisfies(2)ornot.Theyonlyobservethepriceinthismarket,notthe totalquantitiesinthemarket.TheBertrandauctionwouldresultina
underlying“true”price ρþθg.Asaresult,themagnitudebywhich pricewhichyieldszeroprofitsformarketmakers,indicatingthatprice
thepriceatanauction,whichisalsoanestimateofp,differsfromρþθ becomesequaltothepermit'sexpectedvalue.
T
g;showsthelevelofnoise,andisofinterestinthisanalysis.Wedefine FV96useddynamicprogrammingandbackwardinductiontochar-
noiseΣ asthefollowing: acterizealinearequilibriuminthissetting.Proposition1restatestheir
h
resultinthecontextoftheapplicationdevelopedherein,showingthe
Σh¼Varðρþθg−phTÞ ð3Þ existenceofanequilibrium.
Σhmeasuresthemagnitudeofnoiseatroundhinanauctionsystem 4.1.Proposition1
conveyinginformationregardingtheexpectedpriceofthepermit.In
otherwords,itmeasuresinformativenessateachroundhintheclimate ThereexistsarecursivelinearMarkovequilibriumthatsatisfies
predictionmarket.IfΣhequalszero,thatwouldmeanthatthepriceat conditions1and2.Theequilibriumoccurswhereallinformedtraders
roundhhasbecomeexactlyequaltoρþθg,meaningthatalltheinfor- i=1,2,…,Msubmitbidsoftheformq =β g foralltradingperiods
ih h ih
mationhasbeenaggregatedandincorporatedintotheprice.Inother h=1,2,…,k,andpricesbecomeequaltoph=ph−1+λhyh,wherethe
words,theauctionpriceisanunbiasedestimateofthetruevaluepT, parametersβhandγharedefinedasfollows:
thoughitcontainssomenoiseandthemagnitudeofthisnoiseshows
theextenttowhichpricesbecomeinformativeinthismarket.8 β ¼ ηh–λhΨh
Afterhroundsmarketmakersobserve(y ,…y )andlearnaboutpri- h λ ½1þð1þϕ ðM–1ÞÞð1–ððλ Ψ =θÞÞ(cid:2)
1 h h h h h
vatesignals,soatroundh,theithinformedplayerhasaninformational
advantagerelativetomarketmakersthatisequalto: γ ¼ð1–2μhλhÞð1–ðλhβhðM–1Þ=θÞÞ
h 2λ ð1–μ λ Þ
h h h
g ¼g −E½g jy ;…;y (cid:2) ð4Þ
ih i0 i0 1 h (cid:6) (cid:7)
(cid:4) (cid:5) λ β 2
FV96alsodefinethefollowingvariancecovariancesinordertomea- αh−1¼ ηh–λhβh½1þðM−1Þϕh(cid:2) βhþαh 1− hθ h½1þðM−1Þϕh(cid:2)
suretheremaininginformation:
(cid:8) (cid:9)
(cid:4) (cid:5) λ β ðM−1Þ
Λh¼Varðgi0jy1;…;yhÞ¼VarðgihÞ ð5Þ Ψh−1¼ ηh−λhβh½1þðM−1Þϕh(cid:2) γh−λhγhβhþβh 1− h hθ
(cid:2) (cid:3) (cid:2) (cid:3) (cid:8) (cid:9)(cid:8) (cid:9)
Ω ¼Cov g ;g jy ;…;y ¼Cov g ;g ð6Þ λ β λ β ðM−1Þ
h j0 i0 1 h jh ih þΨ 1− h h½1þðM−1Þϕ (cid:2) 1− h h −λ γ
h θ h θ h h
whereiandjaretwodifferenttraders. (cid:8) (cid:9) (cid:8) (cid:9)
λ β ðM−1Þ λ β ðM−1Þ 2
4.Theequilibrium μh−1¼−λhγ2hþγh 1− h hθ þμh 1− h hθ −λhγh
(cid:2) (cid:3)
WehaveusedtheFV96andKyle(1985)auctionmodeltocharacter- α λ2 β2λ2
izeapermitstradingsystemthatwouldimplementtheHsu(2011) δh−1¼δhþ θh2hσ2uþαh hθ2hðM−1ÞVar gjh−1jgi0;y1;…;yh−1
(cid:2) (cid:3)
exemption allowance proposal. Now we need to characterize the β2λ2
equilibriumthatemergesandexplorethepropertiesoftheresulting þαh hθ2h½ðM−2ÞðM−1Þ(cid:2)Cov gjh−1;grn−1jgi0;y1;…;yh−1
prices as they relate to the question of whether such a market
(cid:2) (cid:3)
wouldyieldvalidclimateforecasts.FollowingKyle(1985)andFV96, Λ2 −Ω2
aBayesianNashequilibriumexists ifthere is avectorof strategies Var gjh−1jgi0;y1;…;yh−1 ¼ h−1Λ h−1
h–1
(q ,q ,…,q ,p)suchthat:
1 2 M (cid:2) (cid:3)
Ω ðΛ –Ω Þ
1. Forreachtradingroundh=1,2,…,kandforeveryinformedtraderi, Cov gjh−1;grh−1jgi0;y1;…;yh−1 ¼ h–1 Λh–1 h–1
h–1
i=1,2,…,M,analternativestrategyq′=(q ′,q ′,…,q ′)wouldre-
i i1 i2 ih
sultinalowerorequalexpectedprofitforfirmigiventhisinforma- ϕ ¼Ωh–1
tionset,thepricingrule,andstrategiesoftheotherinformedtraders: h Λh–1
(cid:4) (cid:4) (cid:5) (cid:5)
E πh q1;q2;…;q0i;…;qM;p jgi0;qi;−h;y−h η ¼ θ ½1þðM−1Þϕ (cid:2)
(cid:4) (cid:5) h M h
≤E πhðq1;q2;…;qi;…;qM;pÞjgi0;qi;−h;y−h ð7Þ (cid:8) (cid:9)
Mλ β
i.e.Theoptimalstrategyoftraderiatperiodhshouldbebestnomat- Σh¼ 1− θh h Σn−1
terwhatstrategiesheplayedinpreviousperiods;and
2. Thepriceatroundh=1,2,…,kbecomesequaltotheexpectedvalue Λ ¼Λ −MλhβhΣ
ofthepermitgiventheobservedorderflowuptoh;andgiventhe h h−1 θ2 θ h−1
strategiesoftheinformedtraders.
M λ β
Ω ¼Ω − h hΣ
p ¼E½p jy ;y ;…;y (cid:2)forh¼1;2;…;k ð8Þ h h–1 θ2 θ h−1
hT T 1 2 h
Thisisamarketefficiencyconditionthatmakespricebecomeanun- whereαh=Ψh=μh=δh=0.
Asecondorderconditionensuringthattheinformedtraders'utility
biasedestimateofthefuturepriceandhencethefuturetaxrate.This
ismaximizedalsoholds:
conditioncanbeviewedasbeingaresultofaBertrandauctionamong
atleast two risk-neutral market makers who can observe only the
λ ð1−μ λ ÞN0
h h h
8 FV96assumedρtobeequaltozerosothattheydefinedΣhasthevarianceof(θg−phTÞ. Proof:seeproofofproposition1,FV96Appendix.
102 E.Aliakbari,R.McKitrick/EnergyEconomics74(2018)97–106
Thestatedconditionsarethenecessaryandsufficientconditionsfor fourdifferentinformationstructuresamongtraders.Specifically,they
theMarkovequilibriumtohold.Thestatedrecursionsweresolvednu- consideredthefollowingcasesfortheinitialcorrelationamongsignals:
mericallyinFV96usingabackwardinductionalgorithmandtheequi- veryhighpositive(0.9999),whichcorrespondstoidenticalinforma-
libriumparameterswerecalculatedfordifferentcorrelationstructures tion,lowpositive(0.1818),zero,andlownegative(−0.2857).The
betweentheinitialsignals. evolutionofΣ,whichisthevarianceofthenoiseinthismarket,was
Beforeproceedingtothenextsection,itisworthclarifyingtherole computedforthefourdifferentcases.Fig.1isdrawnusingdatafrom
ofuninformedtradersinthismodel.Asstatedintheaboveequilibrium, FV96, leaving out the negative correlation case since it does not
pricesarealinearfunctionoftheorderflowsandbecomeequalto: applyinthismodel.ItshowshowΣchanges,orinotherwordshow
informativepricesbecome,afterfourauctionsunderdifferentsignal
ph¼ph−1þλhyh ð9Þ structures.
AsFig.1shows,thehighertheinitialcorrelation,thelowertheter-
Inperiodh=1wewillhave: minalvarianceandthereforethehigherthepriceinformativeness.The
(cid:2) (cid:3) amountofreductioninsigmaisverydramaticwhentradershaveiden-
p1¼p0þλ1y1¼ρþλ1 ∑Mi¼1β1gi1þu1 ð10Þ ticalinformation,indicatingthatmostoftheinformationgetsreleased
afterrunningonlyfourrounds.Butevenwhenthereisnoinitialagree-
P ! ment(corr=0)thevariancefallsbymorethanhalfafterfourrounds.
⟹ p ¼ρþλ β M Mi¼1gi1þu ð11Þ ResultsfromFig.1implythattraderswithalmostidenticalinformation
1 1 1 M 1
inthepredictionmarketwouldfacemorecompetitionandtrademore
intenselyinearlyrounds,causingmoreinformationtogetreleased.
⟹ p1¼ρþλ1ðθgþu1Þ ð12Þ However,traderswithmoreheterogeneousinformationfacelesscom-
petitionbecausepartoftheirinformationisuniqueandgivesthem
Therefore,intheequilibrium,asstatedbeforeandascanbeseen somemonopolypower.Asaresult,theyhavelessincentivetotradeag-
fromEq.(12),priceisanoisysignalofθg –asufficientstatisticfor gressivelyinearlyauctions,causinglessinformationtogetreleased
alltheavailableinformation-andthesourceofthenoiseiscoming overfourauctions.
fromtheuninformedtraders'marketorder,u .Morespecifically,unin-
1
formedtradersinthismodelactasacamouflageforinformedtraders' 3. An auction price p is a sufficient statistic (Eq. (1) holds) for p
kT T
informationandhelpinformedtradershidetheirinformationfrom onlywheninformedtradershavealmostidenticalprivateinformation
marketmakerswhentradingwiththem.Ifweremoveuninformed regardingthestatevariable.
tradersfromthemodel,thepricebecomesfullyrevealingasitincor-
poratesandrevealsalltheavailableinformation.Inotherwords,re- Itwasshownbeforethatθgisasufficientstatisticfortheavailable
moving uninformed traders from the model implies that prices in informationtopredictpT(Eq.(1)).Inotherwords,onceθgisknown,
financialmarketsalwaysrevealalltheavailableinformationperfectly. nofurtherinformationcanbegainedaboutpTbyknowinganyindi-
ThiswouldleadtoaparadoxpointedoutbyGrossmanandStiglitz, vidualsignalsg1,g2,…,gM.Therefore,anauctionpricepkTbecomesa
1980.Whentheequilibriumpriceisalwaysaperfectaggregatorof sufficient statistic for pkTwhen it becomes equal to ρþθg. As was
information, no traders would have any incentive to collect costly showninFig.1,pkTbecomesequaltoρþθgorequivalentlythelevel
informationastheyknowtheycannotearnareturnontheirinforma- ofnoisebecomesequaltozeroevenafteronly4tradingroundswhen
tiongathering.Therefore,noonewouldgatherinformationonwhich informedtradershadalmostidenticalinformationinthemarket.But
pricesarebased. simulationsinFV96showthat,withweakercorrelationsignals,Σcon-
vergestozerobutitremainspositiveevenafter800auctionrounds.
5.Propertiesofthemarketoutcome Thismeansthatrepeatedtradingroundsmaynotcompensateforinfor-
mationheterogeneityifpriorbeliefsaresufficientlyuncorrelated.
Inthissection,wepresentthreepropertiesofthemarketoutcome Insummary,theclimatepredictionmarketwehavedescribedfora
thatemergefromtheanalysisoftheBayesianNashtradingequilibrium. futureperiodTyieldsanunbiasedestimateofthetruefutureprice,
and hence of the climate state. Also, the level of consensus about
1. TheauctionpricepkT=E[pT|y1,y2,…,yk],isanunbiasedestimateofthe climate science strongly influences theeffectiveness ofthe market.
truefuturepricepTandhenceofsT.Theunbiasednessofthepriceholds Thehigherthecorrelationofinformationsignalspriortotrading,the
because: moreinformationisaggregatedbythepriceforecastduringtrading.
Addingmoreroundsoftradinghelpsincreasetheaccuracyofprices
EðpkTÞ¼E½pTjy1;y2;…yk(cid:2)¼E½pT(cid:2) ð13Þ evenwhensignalsareuncorrelated.Putanotherway,theargument
thatapriceemergingfromanallowanceauctionasdescribedherein
Therefore,theunbiasednessconditionE(p −p)=0holds.Inany systematicallyignoresimportantinformationaboutthefutureclimate
specificoutcome,pricesarenotnecessarilykTequalTtothetruevalue, reliesontheassumptionthatthereisnoscientificconsensusaround
climate change. Belief that such a consensus exists implies that
becausetheycontainsomenoise,buttheaverageoftheexistingnoise
thepriceemergingfromthemarketwoulduseallavailableinforma-
isequaltozero.
tion and therefore would bethemost informative climate forecast
possible.
2. Thestrongerthescientific“consensus”aboutglobalwarming,themore
accuratewillbetheforecastsimpliedbytheauctionpricesignals.More
6.Extensiontoriskaversefirms
formally,thehigherthecorrelationofsignalsreceivedbytraders,and
hencethemorecorrelatedtheirbeliefs,theclosertheauctionprice
Thepreviousresultsrelyontheassumptionthattradersarerisk-
getstoρþθg,meaningitisasufficientstatistic,oronethatusesall
neutral.HereweextendtheframeworkbyconsideringMinformed
availableinformationefficiently.
agents who possessdifferentprivateinformation andwhoarerisk
averse,usingthefollowingCARA(constantabsoluteriskaversion)util-
This was demonstrated by FV96 through numerical simulations
ityfunctionswith“A”asariskaversioncoefficient:
oftheequilibriumconditions.FV96consideredthecaseoffourauc-
tions(k=4)andthreeinformedtraders(M=3)inordertosolve
theirmodel.Theyfixedthetotalavailableinformationandexamined UðWNþ1Þ¼−eð−AWNþ1Þ
E.Aliakbari,R.McKitrick/EnergyEconomics74(2018)97–106 103
0.9
0.8
0.7
0.6
0.5
Sigma Corr=0.9999
0.4
Corr=0.1818
0.3 Corr=0.000
0.2
0.1
0
1 2 3 4
Periods
Fig.1.Evolutionofsigmawith4periods,wherezeroindicatesconvergenceonafull-informationoutcome.(Datasource:FV96).
whereW representsterminalwealth.Thefollowingpropositiones- Therefore,
N+1
tablishestheexistenceofatradingequilibriuminthiscase.
0 1
6.1.Proposition2 B C
B σ2 C
Σ ¼Σ B(cid:8) (cid:9) u C
Thereisanequilibriumwhereinformedtraderssubmittheoptimal 1 0@ βM 2Σ þσ2A
θ 0 u
orderq,thatisalinearfunctionofinitialsignalsandisgivenby:
i
q ¼βg
i i0 As was proved, the ith informed risk averse trader would
trade q =βg inthefirstauctionwhereβ¼ η :Since
andthepriceissetaccordingtothefollowinglinearrule: i i0 2λþKλðM−1ÞþAϑ
Aϑinthedenominatorisapositiveterm,comparingtheoptimalde-
!
XM mandinthecaseofriskaversetraderswiththecaseofrisk-neutral
p¼χ q þu ¼λy traders indicates that risk averse traders would trade smaller
i
i¼1 amounts,causingpricestobelessinformative.Asitwasshown,Σ1
isgivenby:
whereβandλaregivenby:
0 1 2 3
η
β¼ B C 6 7
2λþKλðM−1ÞþAϑ B σ2 C 6 σ2 7
Σ ¼Σ B(cid:8) (cid:9) u C¼Σ 6(cid:8) u (cid:9) (cid:8) (cid:9) 7
1 0@ βM 2 A 04 η 2 M 2 5
Σ þσ2 Σ þσ2
BM θ 0 u 2λþKλðM−1ÞþAϑ θ 0 u
Σ
θ 0
λ¼(cid:8) (cid:9)
BM 2
Σ þσ2
θ 0 u
∂Σ
Therefore,intheaboveequation 1ispositiveandisgivenby:
∂A
TheproofisintheAppendix.
Tocheckhowinformativepricesbecomeafteronlyoneauction (cid:8) (cid:9)
roundweneedtofindΣ1whichisequaltoVar(pT|y)bydefinition. −2 M 2Σ β(cid:4)Σ σ2(cid:5) !
∂Σ ∂Σ ∂β θ 0 0 u −ϑη
Σ1¼VarðpTjyÞ¼VarðgθjyÞ¼Σ0(cid:2)1−corr2θg;y(cid:3)¼ ∂A1¼ ∂β1(cid:3)∂A¼ "(cid:8)βM(cid:9)2Σ þσ2#2 (cid:3) ½2λþKλðM−1ÞþAϑ(cid:2)2
θ 0 u
(cid:8) (cid:9)
! 0 (cid:8) βM (cid:9)21 2 M 2Σ 2σ2ϑηβ
B Cov gθ; ðgθÞþu C θ 0 u
Σ0 1−VaCroðvθðgθÞgv;ayrÞð2yÞ ¼Σ0BB@1− Σ0Var(cid:8)βθθMðgθÞþu(cid:9) CCA ¼ "(cid:8)βθM(cid:9)2Σ0þσ2u#2½2λþKλðM−1ÞþAϑ(cid:2)2
0 1
(cid:8) (cid:9)
B βM 2Σ 2 C
B θ 0 C Astradersbecomemoreriskaverse,theywilltradesmallerquanti-
¼Σ0BB@1− "(cid:8)βM(cid:9)2 #CCA tiesandasaresult,thepermitpricewouldbecomeinformativeata
Σ Σ þσ2 slowerrate.Inotherwords,weexpecttoseealowerrateofreduction
0 θ 0 u
inΣwhenaddingriskaversiontothemodel.
104 E.Aliakbari,R.McKitrick/EnergyEconomics74(2018)97–106
7.Conclusion AppendixA
Inthispaperwedevelopedtheideaofintegratingastate-contingent ProofofEq.(1):
emission pricing rule with tradable permit markets as a way of
obtaininganaccurateforecastofthefutureclimatestate.Wemodeled LetXandYbetworandomvariableswithnormaldistributions.The
tradingofpermitsthatexemptemittersfrompayingatemperature- conditionalexpectationofXgivenY=yisthefollowingformula:
indexedtaxamongtraderswithdifferentprivatesignalsandinvesti-
gatedpriceformationinthismarket.Withriskneutraltraders,we E½XjY¼y(cid:2)¼m þΣ Σ−1ðY−m Þ
x XY XY Y
usedtheFV96frameworktoshowhowinformationgetsefficientlyag-
gregatedandincorporatedintoprices,makingthemunbiasedforecasts wherem =E[X],m =E[Y],andΣisthecovariancematrix.Giventhe
x Y
ofthefutureclimateoutcomes.Also,wedemonstratedthattheinitial aboveformulaandthenormalityassumptionswewillhave:
structureofprivateinformationplaysanimportantroleinthelevelof (cid:4) (cid:5)
competitionamongtradersandconsequentlytheextenttowhichprices EðpTjg10;g20;…;gM0Þ¼mPT þΔ00Ψ−01 g−mg 0
becomeinformative.Inparticular,thehigherthelevelofscientificcon-
sensus(meaningthehigherthecorrelationofprivatesignalsreceived p has a normal distribution with mean ρ and variance Σ . In
T 0 0
byinformedtraders),themorecompetitiontradersface,causingmore theabove equation, g is thevector of signals withzero means. The
informationtogetreleased.Withalmostidenticalinformation,ittakes diagonal elements ofΨ−1areallΛ (initial varianceofsignals)and
0 0
veryfewauctionroundsforalltheinformationtogetincorporated the off diagonal elements of Ψ−1 are all Ω (initial covariance be-
0 0
intothemarketprice.Ontheotherhand,traderswithmoreheteroge- tween any two signals. Δ is thecovariancevector between signals
0
neousinformationfacelesscompetitionastheyhavesomemonopoly and thetrueprice of a permitwith c asits element, meaning that
0
power.Theytradelessintensely,causinginformationtogetreleased Δ ′ = [c ,c ,…,c ]. Therefore the entries of the matrix Δ ′Ψ−1 are
0 0 0 0 0 0
moreslowlyovertime.Weshowedthat,inthiscase,addingmultiple all the same and equal to c [Λ + (M − 1)Ω ]. We showtheterm
0 0 0
roundsofbiddingwouldhelpincreasethepriceinformativeness.We c [Λ +(M−1)Ω ]byς.Thus,
0 0 0
alsoextendedtheexistingmodeltoincluderiskaversetraders.Inthis (cid:8)P (cid:9)
X
case,theytradesmallerquantitiesandasaresult,thepermitprice Eðp jg ;g ;…;g Þ¼ρ þς g ¼ρ þMς gi0 ¼ρ þθg
wouldbecomeinformativeataslowerrate. T 10 20 M0 0 i0 0 M 0
Notethatweareassumingthereisaminimalfeedbackbetween
policyandthestatevariable.Itisconceivablethatifasteepwarming whereθ=Mς,whichisasetofweightsimpliedbyconditioningon
pathledtoapathofsteeply-risingemissiontaxes,andtheresponse thesignals.
tothepolicycausedthewarmingpathtodrop,therecouldemerge
multipleequilibriumpaths.Thisisnotlikelytohappen,however, ProofofProposition2
duetotheunresponsivenessoftheclimateeventomajorpolicyini-
tiatives.Wigley(1998)andLomborg(2016)modeledtheclimatic Theithinformedtradermaximizeshisexpectedutilitygivenhis
effectsof,respectively,theinternationalcommitmentsunderthe initialsignalandgiventheassumptionthatothertradersfollowtheir
KyotoandParisclimateaccordsandshowedthattheybarelybudged optimalstrategies.Therefore,atroundh=0,hesolvesthefollowing
thetemperaturepaththrough2100.Consequently,anyindividual problem:
nationimplementingastate-contingentemissionpricewouldlikely
notexpectitspolicyresponsestohavealargeeffectonthestate Maxqi E½−eð−AW1Þjgi0(cid:2)¼−E½eð−AWo−AqiðpT−pÞÞjgi0(cid:2)
variable. h (cid:2) (cid:3) i
These results are important because they provide a feasible ¼−e−AWoE e pT−λqi−λ∑i≠jqj−λu ð−AqiÞjgi0 ð14Þ
mechanismforaddressingtwoseriousimpedimentstoclimatepol-
P
icyformation:uncertaintyabouttheoptimalfutureemissionprice giventhatp¼λð Mi¼1qiþuÞand∑i≠jqj¼βgj0
givenanavailableinformationset,anduncertaintyaboutwhether
Weassumethatallinformedtradershavethesameinitialwealth
the information set itself is biased or incomplete. The combined
W .UsingtheLinear-Normal-Exponentialmodelproperties,wewill
0
state-contingent tax/exemption permit auction would yield a
havethefollowing:
forward-looking price path that correlates to the unobservable
h (cid:2) (cid:3) i
itnotkenr-otwemthpeocrualrrmenatrgfoinrmalodfatmhaeguensdpeartlhy,inwgitshtaotuetfaunrecgtiuolnatcoornnneeecdtiinngg E exp pT−λqi−λ∑i≠jqj−λu ð−AqiÞjgi0
(
theclimatetoemissions.Thepermitsauctionwouldyieldunbiasedpre- h(cid:2) (cid:3) i
dictionsofthefutureclimatestate.Anditwouldmakemoreefficientuse ¼exp E pT−λqi−λ∑i≠jqj−λu ð−AqiÞjgi0 ð15Þ
ofallavailableinformation,themorecorrelatedareprivateinformation )
signals,inotherwordsthegreaterthestrengthofthescientificconsensus 1 h(cid:2) (cid:3) i
onclimatechange.Iftheconsensusisweak,informationwouldstillag- þ2Var pT−λqi−λ∑i≠jqj−λu ð−AqiÞjgi0
gregate,onlymoreslowly,butthatwouldcorrespondtothecasein
whichthebasisforanypolicystanceislikewiseweakened.Finally,risk-
Therefore,maximizingEq.(14)isequivalenttomaximizing:
aversionamongtraderswouldslowtherateofinformationaggregation
20 1 3
butwouldnotpreventit.
X
E4@p −λq−λ q−λuAð−AqÞjg 5
T i j i i0
Acknowledgment i≠j
20 1 3
TheauthorsthanktheInstituteforNewEconomicThinkingforfi- X
nancialsupport. þ1Var4@p −λq−λ q−λuAð−AqÞjg 5 ð16Þ
2 T i j i i0
i≠j
Declarationsofinterest
sincetheexpressioninEq.(16)isamonotoneincreasingtransforma-
None. tionoftheexpressionin(15).
E.Aliakbari,R.McKitrick/EnergyEconomics74(2018)97–106 105
P
FromEq.(16): AsbothpTandyð¼ Mi¼1βgi0þuÞarenormallydistributed,E[pT|y]
20 1 3 becomeslinearinyandλisgivenby:
X
E4@pT−λqi−λ qj−λuAð−AqiÞjgi05 Covðp ;yÞ
i≠j λ¼ VarðTyÞ
8 9
< X h i=
¼−Aq E½p jg (cid:2)−λq−λ E βg jg ð17Þ where
i: T i0 i j0 i0 ;
i≠j P !
y¼XM βg þu¼βM Mi¼1gi0 þu¼βMðgθÞþu
MultivariatenormalityimpliesthatbothE[p|g ]andE[g |g ]are i0 M θ
T i0 j0 i0 i¼1
linearing ,thereforeEq.(17)becomesequalto:
i0
⟹
−Aq½ηg −λq−λβðM−1Þφg (cid:2)
i i0 i i0
(cid:8) (cid:9) (cid:8) (cid:9)
where Cov p βMðg(cid:2)θÞþu Cov g(cid:2)θβMðg(cid:2)θÞþu
T θ θ
λ¼ ¼
η¼CoV(cid:6)vaðrpðTg;i0gÞi0Þ¼CoVva(cid:7)ðrθðgg;i0gÞi0Þ¼θCovV(cid:8)aΣrMðggi0i0;Þgi0(cid:9)¼Mθ ½Λ0þΛðM0−1ÞΩ0(cid:2) ¼(cid:2)VβaθMr(cid:3)ðð2βVβMθaMr=ðVθgÞaθ(cid:2)ðrÞg(cid:2)ðþθg(cid:2)ÞθVþÞaruðÞuÞ¼VðaβrMððβ=θβMÞθM2=ΣθΣÞ0ð0þg(cid:2)θσÞþ2u uÞ
θ ðM−1ÞΩ
¼ 1þ 0
M Λ
0 Thiscompletestheproof.
and
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