134REVIEWOFECONOMICSTUDIESOurmainresultisthatthefeedbackeffectfromthe - PDF document

134REVIEWOFECONOMICSTUDIESOurmainresultisthatthefeedbackeffectfromthenancialmarkettotherealinvestmentdecisiongeneratesequilibriawherethespeculatorsellsthestockevenwhenhehappenstobeuninformed.Whenhedoesthat,hereducestheinformativenessofthepriceandtheefciencyoftheresultinginvestmentdecision.Sellingintheabsenceofinformationisprotableinourmodelfortworeasons.First,thepricequotedbythemarketmakerreectsthepossibilitythatthespeculatorisinformedandthatthemarketwillimprovetheefciencyoftheinvestmentdecisionandincreasethevalueoftherm.Whenthespeculatorknowsthatheis,infact,uninformed,heknowsthatthemarketwillnotimprovetheallocationofresources.Thus,hecansellatapricethatishigherthantheexpectedvalue(conditionalonhimknowingthatheisuninformed).Second,thespeculatorcanprotfromtheeffectthathistradewillhaveontherm’sinvestmentdecision.Inparticular,hecanprotbyestablishingashortpositioninthestockandsubsequentlydrivingdowntherm’sstockpricebyfurthershortsales.Thermwillinferthatthelowerpricemayreectnegativeinformationabouttheinvestmentopportunityandthereforewillnotinvest.Onaveragethisisthewrongdecisionandwillinitselfreducethevalueoftherm.Thisenablesthespeculatortocoverhisshortpositionatalowercostandmakeaprot.Duetothenatureofthisstrategy,werefertoitasManipulationoccursinourmodelifinformationconveyedbysellingpressureinthenan-cialmarketissufcienttoleadtocancellationoftheinvestmentproject.Thisstrongfeedbackeffectwillexistwhen(i)theex-anteNPVoftheinvestmentisnottoohigh;(ii)theuncertaintyregardingtheoutcomeoftheinvestmentislarge;or(iii)thermexpectsthatusefulinformationislikelytocomefromthenancialmarket.Thus,insuchcases,ourmodelpredictsalargesell-ingpressurefollowedbycancellationofinvestmentprojects.Interestingly,weshowinthepaperthatwhenthefeedbackeffectfromthenancialmarkettotherealeconomyisverystrong,equilibriaofthetradinggamewillfeaturemanipulation.Finally,weshowthatmanipulationisprotableonlyviasellorders.Thisisbecausethetwosourcesofprotbehindthemanipulationstrategy,asdescribedabove,cannotgenerateprotswithbuyorders.First,theallocationalroleofpricesalwaysincreasesthevalueoftherminex-pectation.Thus,conditionalonbeinguninformed,thespeculatorexpectsthatthevalueofthermislowerthanwhatisreectedinthemarketprice.Thiscreatesanincentiveforhimtosell,nottobuy.Second,whilethespeculatorcanpotentiallymanipulatetherm’sinvestmentdecisionwitheitherbuyordersorsellorders,thiswillonlybeprotablewithsellorders.Buyingthestockwithnoinformationwillleadtooverinvestment,whichwillonlydecreasethevalueofthespeculator’saccumulatedlongposition.Thisisbecauseoverinvestment—justlikeunderinvestment—isinefcientandleadstoareductioninrmvalue.Thus,thespeculatorcanbenetfrommanipu-latingthepriceanddistortingtherealinvestmentdecisiononlywhenheaccumulatedashortposition.Aninterestingimplicationfromthisisthatthereisasymmetrybetweenbuyordersandsellorders.Asaresult,sellordersarelessinformativeaboutthefundamentals,andtheresponseofthepricetosellordersislessstrong.Inourmodel,thefeedbackeffectisgeneratedbymanagerslearningfromthestockpricewhentakinginvestmentdecisions.ThismechanismhasreceivedempiricalsupportintherecentworkofDurnev,MorckandYeung(2004),Luo(2005),andChen,GoldsteinandJiang(2007).Itisimportanttonote,however,thatlearningbymanagersisnotnecessaryforourresults.Alternatively,feedbackmayarisefromtheeffectofthestockpriceontherm’saccesstocapital1.Theassumptionthatstock-marketparticipantshaveinformationaboutsomeaspectsoftherm,whichisnotavailabletotherm’smanagers,ismadeinpapersbyDowandGorton(1997),SubrahmanyamandTitman(1999,2001),DowandRahi(2003),andothers.Theinitialpublicoffering(IPO)literaturehasalsousedsuchanassumptioninexplainingthedeterminantsofunderpricing.See,forexample,Rock(1986),BenvenisteandSpindt(1989),BenvenisteandWilhelm(1990),andBiais,BossaertsandRochet(2002).2008TheReviewofEconomicStudiesLimited 136REVIEWOFECONOMICSTUDIESKhannaandSonti(2004),particularlybecauseinourmodelthefeedbackeffectandthepositionofthetraderarederivedendogenously.IncontrasttoKhannaandSonti(2004),thetraderinourmodeltradeswithaviewtodistortingtherm’sinvestmentdecision.Thisallowsustoderivenewresultsforallocationalefciencyandtheroleofshortsalesinmanipulation.Hirshleifer,SubrahmanyamandTitman(2006)discusstheroleoffeedbackinamodelwithirrationaltradersandshowthatsuchtradersmaysurviveinnancialmarketswhentheirtradesaf-fectrmvalue.Attari,BanerjeeandNoe(2006)exploretradingbyinstitutionalinvestorsaroundcorporatecontrolchangesandpointtothepossibilityofmanipulationwhenavalueenhancingactionistakenbyalargetraderconditionalonstockpricemovements.Apricedropleadstoin-creasedrmvalue,becauseittriggersshareholderactivism.Thiscontrastswithourresultwhereapricedrophasanegativeimpactonrmvaluerenderingmanipulationprotableforsellersonly.Theremainderofthepaperisorganizedasfollows:Section2presentsthemodelset-up.InSection3wederiveasabenchmarktheequilibriumtradingstrategieswhenrmvalueisindepen-dentofstockpricemovements.Section4thenintroducesfeedbackandshowsthatmanipulationarisesinequilibrium.InSection5,weanalysethepossibilityofequilibriawithoutmanipula-tioninthepresenceoffeedback.Section6analysesthedifferencebetweenseller-initiatedandbuyer-initiatedmanipulationandshowsthatthelatterisnotprotable.InSection7wecon-sidertherobustnessofthemanipulationequilibriumtotheintroductionofmorestrategictradersandtotheendogeneityofliquiditytrading.Section8concludes.Allproofsarerelegatedtothe2.THEMODELThemodelhasfourdatesandarmwhosestockistradedinthenancialmarket.Therm’smanagerneedstotakeaninvestmentdecision.In0,arisk-neutralspeculatormaylearnprivateinformationaboutthestateoftheworldthatdeterminestheprotabilityoftherm’sinvestmentopportunity.Tradinginthenancialmarketoccursin1and2.Inadditiontothespeculator,twoothertypesofagentsparticipateinthenancialmarket:noisetraderswhosetradesareunrelatedtotherealizationofandarisk-neutralmarketmaker.Thelattercollectstheordersfromthespeculatorandthenoisetradersandsetsapriceatwhichheexecutestheordersoutofhisinventory.Theinformationofthespeculatormaygetreectedinthepriceviathetradingprocess.In3,themanagertakestheinvestmentdecision,whichmaybeaffectedbythestockpricerealizations.Finally,alluncertaintyisrealizedandpay-offsaremade.Wenowdescribetherm’sinvestmentproblemandthetradingprocessinmoredetail.Therm’sinvestmentproblemSupposethatthermhasaninvestmentopportunitythatrequiresaxedinvestmentattheamount.Therm’smanageractsintheinterestofshareholdersandchooseswhetherornottoinvestwiththeobjectivetomaximizeexpectedrmvalue.Supposealsothatthermcannancetheinvestmentwithretainedearnings.Thermfacesuncertaintyoverthequalityoftheavailablein-vestmentopportunity.Therearetwopossiblestatesthatoccurwithequalprobabilities.Wedenotethevalueofthermifitinvestsinthehighstate;thevalueofthermifitinvestsinthelowstateFirmvaluecanthenbeexpressedasafunction(
,oftheunderlyingstateandtheinvestmentdecision6.Inapreviousversionofthispapertherm’svaluewasderivedendogenouslybasedonaset-upwherethermcouldspendasunkcosttoenteraproductmarketofuncertainsizeinwhichitwouldthenbeamonopolist.Firmvalueinsuchaset-upisincreasinginmarketsize.See,forexample,Sutton(1991)foramodelofthiskind.2008TheReviewofEconomicStudiesLimited 138REVIEWOFECONOMICSTUDIESthetrade,(ii)aninvestmentstrategybythermthatmaximizesexpectedrmvaluegivenallotherstrategies,(iii)aprice-settingstrategybythemarketmakerallowshimtobreakeveninexpectation,givenallotherstrategies.Moreover,(iv)thermandthemarketmakeruseBayes’ruleinordertoupdatetheirbeliefsfromtheorderstheyobserveinthenancialmarket.Finally,(v)allagentshaverationalexpectationsinthesensethateachplayer’sbeliefabouttheotherplayers’strategiesiscorrectinequilibrium.3.EQUILIBRIUMINAMODELWITHNOFEEDBACKAsabenchmark,weconsiderinthissectionthecasewherethereisnofeedbackfromthenancialmarkettotherm’sinvestmentdecision.Wecalltheresultinggamethe“no-feedbackgame”.Specically,weassumethatthermisfullyinformedaboutthestateoftheworld,andthereforeinvestswhenthestateishighanddoesnotinvestwhenitislow.Thus,theinvestmentdecisionin3isnotaffectedbythetradingoutcomesinthenancialmarketin1andIntheno-feedbackgame,thevalueofthermiswhenthestateoftheworldishighand0whenthestateoftheworldislow.Whenthespeculatorispositivelyinformed;thatis,when,heknowsthatthevalueofthermis.Whenthespeculatorisnegativelyinformed,thatis,when,heknowsthatthevalueofthermis0.Finally,whenthespeculatorisuninformed,thatis,when,hedoesnotknowthevalueoftherm,butknowsthatinexpectationitis Themarketmakeralsostartswiththeexpectationthatthevalueofthermis andupdatesthisexpectationaftereveryroundoftrade.Theexistenceofequilibriumintheno-feedbackgamecanbeeasilyshownwithanexam-ple.Infact,weknowthattheno-feedbackgamehasmultipleequilibria.Forbrevity,wedonotdevelopaparticularequilibriumhere.Instead,weestablishsomeresultsonthestrategyofthespeculatorinanyequilibriumoftheno-feedbackgame.Then,inthenextsection,wedemon-stratehowthisstrategychangesinthepresenceoffeedback.Thefollowinglemmacharacterizesthestrategyofthepositivelyinformedspeculatorandthenegativelyinformedspeculatorinanyequilibriumoftheno-feedbackgame.Buildingonthislemma,thenextpropositionestablishesanimportantresultregardingthestrategyofthespeculatorwhenheisuninformed,whichisthefocusofourpaper.Notethatweallowformixed-strategyequilibria.Infact,formanyparametervaluesourmodeldoesnothaveapurestrategyequilibrium.Lemma1.Inanyequilibriumoftheno-feedbackgamethetradingstrategyoftheposi-tivelyinformedspeculatorisasfollows:Int,hebuyswithprobabilityanddoesnottradewithprobability,where.Int,hebuyswithprobabilityifpmixesbetweenbuying,selling,andnottradingifpThetradingstrategyofthenegativelyinformedspeculatorisasfollows:Int,hesellswithprobabilityanddoesnottradewithprobability,where.Int,hesellswithprobabilityifpandmixesbetweenbuying,selling,andnottradingifpThetradingstrategiesareratherintuitive.Thespeculatortradestomakeaprotonhisinformationandthusbuyswhenhehaspositiveinformationandsellswhenhehasnegativeinformation,withthefollowingtwoexceptions.First,in2,heisindifferentbetweenhispos-sibleactionsifhisinformationwasrevealedtothemarketmakerafter1trading.Inthiscase,8.Alternatively,wecouldsetupano-feedbackgame,inwhichthermalwaysinvests,regardlessoftheinforma-tiongeneratedbythenancialmarket.Thismighthappenifmanagersreceivedprivatebenetsfromtheinvestment.Allourqualitativeresultsgothroughunderthisalternativespecication,althoughsomequantitiessuchasequilibriumpricesandprotswouldchange.2008TheReviewofEconomicStudiesLimited 140REVIEWOFECONOMICSTUDIESfeedbackfromthenancialmarkettotheinvestmentdecision.Wecalltheresultinggamethe“feedbackgame”.Notethatinthisgamethestockpricehastoreectthisfeedback,andthisiswhatmakessolvingthetradingmodelmuchmoreinvolvedthanintheprevioussection.Thepresenceofafeedbackeffectaltersthebehaviourofthespeculatorinafundamentalway.Wewillshowthatduetothefeedbackeffectthespeculatorhasanincentivetotradeinevenwhenheisuninformed.Inparticular,inthersttradinground,hehasanincentivetosellintheabsenceofinformationwiththeaimofestablishinganinitialshortpositionfromwhichhecanprotoncehehasdrivendownrmvaluethroughfurthershortsalesinthesecondround.Givenitsnature,werefertothistradingstrategyas.InSection6,wediscussindetailwhymanipulationthroughbuyordersisnotanoptimaltradingstrategyinthissetting.Acrucialconditionfortheuninformedspeculatortobeabletoprotfromthemanipulativetradingstrategyisthatthemanagerdecidestocanceltheinvestmentfollowingsellorders.Other-wise,thetradingoftheuninformedspeculatordoesnotaffectrealinvestmentandrmvalue,aswasthecaseintheno-feedbackgameoftheprevioussection.Inequilibrium,thisimpliesthattheinformationcontentofasellorderisstrongenoughtojustifythecancellationoftheinvestment,eventhoughitisknownthatthesellordercouldbegeneratedbyboththenegativelyinformedandtheuninformedspeculator.Inparticular,thefollowingconditionwillbeusedbelow: 2VŠ+(1Š Thisconditionimpliesthattheprobabilityofinformedtradeissufcientlyhighsothatthermoptimallyrejectstheprojectafterordersthatdonotdistinguishbetweenthenegativelyinformedandtheuninformedspeculator.Thefollowingpropositionshowsthatinthepresenceoffeedback,thereexistsanequilib-riumwheretheuninformedspeculatoremploysamanipulativestrategyandsellsin1.Asintheprevioussection,thisisamixed-strategyequilibrium.Proposition2Inthepresenceoffeedback,ifaresatised,thenthereexistsanequilibriuminwhichtheuninformedspeculatorsellswithstrictlypositiveprobabilityintCharacterizationofequilibriumstrategiesWenowturntocharacterizethestrategiesinanequilibriumthatfeaturesmanipulation.Westartwiththespeculator’stradingstrategyandcontinuewiththerm’sinvestmentdecision.1,thespeculatorbuysifhehaspositiveinformation.Ifhehasnegativeinformationornoinformation,heeithersellsordoesnottrade.Weusetodenotetheprobabilitythathedoesnottrade,conditionalonbeingnegativelyinformedoruninformed,andso1istheprobabilitythathesellsinthiscase.Notethatisthesamefortheuninformedandthenegativelyinformedspeculator.Thisisafeatureoftheequilibrium,notaconstraint.Formally,10.Inapreviousversionofthepaper,westudiedfeedbackwhenthemanagermayhavesomeprivateinformation.Inthatversion,theset-upofthetradingmodelwaslessrichthanitishere.Theresultswereverysimilar.11.Notethateventhoughthenegativelyinformedanduninformedspeculatorsgetdifferentprots,theyfaceanidenticaltradeoffbetweensellingandnottradingin1.Thus,forthemixingequilibriumtoobtain,oneequationhastoholdforbothtypesofspeculatorstobeindifferent.Thisiswhywecancharacterizeanequilibriumwithoneparameterthatdeterminesthemixingprobabilitiesforbothspeculators.Sincebothspeculatorsusethesamemixingprobabilities,theconditionforthisequilibriumtoholdis(4).2008TheReviewofEconomicStudiesLimited 142REVIEWOFECONOMICSTUDIESTheprotfrommanipulationThekeyfeatureoftheequilibriumdescribedaboveisthattheuninformedspeculatorsellsin1withpositiveprobability(i.e.1).Thefollowingcorollarycharacterizeshisstrategymoreprecisely.ItfollowsdirectlyfromtheproofofProposition2.Corollary1.IntheequilibriumcharacterizedinSection,whenV V,theuninformedspeculatorsellsintwithprobability(i.e.).WhenV V,hesellsintwithaprobabilitythatisstrictlyabove (i.e. Wenowexplore,inmoredetail,theprotthattheuninformedspeculatorgainsfromsellingAftersellingin1,theuninformedspeculatorfacesthreepossiblepricepaths.Withprob- ,thefactthatthespeculatormustbeeitheruninformedornegativelyinformedisrevealedtothemarketmakerintherstround(2or1).Inthiscase,given(4),theupdatedNPVoftheinvestmentfromthepointofviewofthemarketmakerandtherm’smanagerisnegative.Thus,pricesdropto0andsodoesthevalueoftherm.Asaresult,alongthispath,thespeculatormakesaprotof0.Withprobability ,thespeculator’sidentityasuninformedornegativelyinformedisre-vealedonlyinthesecondround(2or1).Inthiscase,oneunitofhisshortpositionisestablishedatthepositivepriceandtheotherunitisestablishedatapriceof0.Sincethermdoesnottaketheinvestmentafterthispricepaththevalueofthermis0andtheuninformedspeculator’sprotisFinally,withprobability ,thespeculator’sidentityasuninformedornegativelyinformedisnotrevealedineitherroundoftrade(0).Inthiscase,thespeculatorestablishesoneunitofhisshortpositionatapriceofandanotherunitatapriceof.Giventhispath,andfrom(3),therminvests.Conditionalonthespeculator’sinformation(theexpectedrmvalueis .Notethatisequal :given0andthetradingprocessrevealsnonewinformationregardingtheprotabilityoftheproject,sotheexpectedvaluefromthepointofviewofthemarketmaker .Thus,alongthispath,thespeculatormakesaprotof .Thisprotisstrictlypositive.Thisisbecausethereectsthepossibilitythat2tradingwillrevealprivateinformationandthereforeimproveresourceallocation(i.e.leadthemanagertorejecttheinvestmentinthelowstateoftheworldwithpositiveprobability).Thus,isgreaterthanthevalueoftheinvestmentabsentanyinformation Aftermultiplyingtheprotsbytherespectiveprobabilities,wegetthattheexpectedprotoftheuninformedspeculatorfromsellingin1is 3p1(0)Š 1 .Thisprotcanbeattributedtotwodifferentsources;bothgeneratedbytheallocationalroleofprices.First,thespeculatorcansellin1ataprice .Fromtheperspectiveoftheuninformedspeculator,hecanprotfromsellingatthisprice.Thisisbecausethepricereectsthepossibilitythat2priceswillhaveanallocationalroleandthereforeincreasermvalue.However,giventhatthespeculatorknowsthatpriceswillnotactuallyfullthisrole.Thiseffectgeneratesanexpectedprotof 13(p1(0) forthespeculator.Second,inourmodel,thespeculatorstartswithnoinitialposition.Thus,histradingstrategyentailsshortsellingthestockin1andthenshortsellingagainin2.Thisgeneratesaprotfromdistortingtherm’sinvestmentdecisiontowardsrejectingtheprojectwhenitshouldacceptit.Specically,bysellinginthespeculatordrivesdownthepriceofthestockanddecreasesthevalueoftherm.Thisisbecausethermrespondstoalowstockpricebyrejectingtheinvestmentproject,eventhoughtheoptimalinvestmentpolicyconditionalonthespeculator’sinformation()wouldhave2008TheReviewofEconomicStudiesLimited 144REVIEWOFECONOMICSTUDIESmanipulationunprotable.Still,ourresultssuggestthatinvestmentefciencycanbeimprovedbyregulationthatdecreasestheincentiveofuninformedspeculatorstomanipulatetheprice.Onesuchregulationistoimposeacostonshortsales.Ingeneral,suchacosthastwoop-positeeffectsontheefciencyofinvestmentdecisions.First,itmightdecreaseefciencybyde-creasingtheincentiveofthenegativelyinformedspeculatortoshortsell,thuspreventingnegativeinformationfromgettingintothepriceandguidingrms’investments.Second,itmayincreaseefciencybydecreasingtheincentiveoftheuninformedspeculatortomanipulatetheprice.Ausefulobservationisthat,inourmanipulationequilibrium,thenegativelyinformedspeculatorexpectstomakeahigherprotfromshortsellingthantheuninformedspeculator.Thisisbecausethermsometimesinvestsafterthespeculatorshortsells,inwhichcasethenegativelyinformedspeculatorneedstocoverashortpositionatavalueof,whiletheuninformedspeculatorneedstocoverashortpositionatanexpectedvalue (seetheproofofProposition2).Thus,bysettingtheshortsellingcostatanintermediatelevel,regulatorsmaybeabletodrivetheun-informedspeculator,butnotthenegativelyinformedspeculator,outofthemarket.Inthiscase,investmentefciencywillbeimproved.Notethatacompleteevaluationofsuchapolicywillneedtoconsiderotherimplicationsitmayhaveontheequilibriumstrategiesofspeculators.Suchananalysisisbeyondthescopeofourpaper.Finally,anotherpossibilityisthatthermitselfcouldtrytocommittoaninvestmentpol-icythatwouldreducetheprotabilityofmanipulativetradingwithoutremovingtheallocationalbenetsthatinformedtradehas.Onewayofachievingthismaybeforthermtoinvestwhentheorderowismoderatelynegative,buttorejecttheinvestmentwhentheorderowishighlynega-tive.Insuchacase,theuninformedspeculatorwouldbeexposedtolosseswhenheshortsellsandmayrefrainfrommanipulation.Thermmaybeabletocommittosuchapolicybydesigningacompensationcontractthatrewardsthemanagerwhenheinvests,butthatisalsolinkedtoper-formance.Therewardsforinvestmentshouldbechosensuchthatthemanageroverinvestswhentheorderowismoderatelynegative,butrejectstheprojectafterastronglynegativeorderow.Indecidingonitscommitmentpolicy,thermwouldhavetoweighthegainfromunderminingmanipulationagainstthelossofmakinginteriminefcientinvestments.EmpiricalimplicationsInthissubsectionwepointoutsomeempiricalpredictionscomingoutofourmanipulationequi-librium.First,wediscusspredictionsregardingtheinformativenessoforderowsandprices.Then,wediscussanimplicationfortheeffectofsellingpressureonrealinvestments.Finally,wediscusspredictionsregardingwhenthemanipulationequilibriumisexpectedtooccur.4.4.1.Theinformativenessoforderßowsandprices.Aninterestingfeatureofourmanipulationequilibriumistheasymmetrybetweenbuyordersandsellorders.Asaresultofthisasymmetry,pricesaremoreinformativeaboutfundamentalsfollowingapositiveorderowthanfollowinganegativeorderow.Thisissummarizedinthefollowingcorollary,whichfollowsdirectlyfromtheproofofProposition2.Corollary3.ConsidertheequilibriumcharacterizedinSectionInt,whereXInt,afterQ13.Inapreviousversionofthispaper,westudiedanextensionwithmultiplermsthatexhibitproductioncom-plementarities.Insuchasetting,nancialmarketscandecreasetheoverallexpectedvalueoftherm,asthedifferentmanagerscannotcoordinatetochoosethecollectivebestresponse.2008TheReviewofEconomicStudiesLimited 146REVIEWOFECONOMICSTUDIESconditionforthemanipulationequilibriumissatised.First,condition(12)ismorelikelytobesatisedwhentheinvestment’sex-ante falls.Thisisbecausethentheinvestmentislesspromisingandthusmorelikelytobeabandonedaftersellingpressure.Second,(12)ismorelikelytobesatisedwhenthereismoreuncertaintyovertheprospectsoftheinvestment.Inthiscase,indicationfromthenancialmarketthatthestateoftheworldmightbelowismorelikelytoleadtocancellationoftheinvestment.Third,(12)ismorelikelytobesatisedwhenishigher.Thisisbecauseahighthatthestockmarketismoreinformativeabouttheinvestmentproject’sfundamentals,andthisstrengthensthefeedbackeffect.Notethatcanbethoughtofasameasureoftheamountofinformationthatmarketshave,whichisnototherwiseavailabletodecision-makersintherealeconomy.TheIPOliteraturehasdiscussedtypesofinformationthatsatisfythiscriterion(seeRock,1986).Forexample,somespeculatorsarelikelytohaveinformationaboutacompetitorthatcouldhaveasignicantimpactontherm’sproduct.Alternatively,somespeculatorsmayknowbetterthanthermtheappropriateratetodiscounttherm’scashows.Inmergersandacquisitions,marketparticipantscanhaveusefulprojectionsontheprospectsforfuturesyner-gies(seeLuo,2005).SubrahmanyamandTitman(1999)suggestthatsomemarketparticipantsmayhaveusefulinformationaboutthepotentialofarm’snewproduct.Finally,fornanciallyconstrainedrms,investmentdecisionsareeffectivelybeingmadebyprovidersofcapitalsuchasbanks.Inthiscasethefeedbackeffectisexpectedtobeparticularlystrong(Bakeretal.,2003)becauseoutsideprovidersofcapitalcanlearnfromtheinformationinthemarketonmanyaspectsoftherm’slife.5.THE(IM)POSSIBILITYOFEQUILIBRIUMWITHNOMANIPULATION:THECASEOFSTRONGFEEDBACKIntheprevioussection,wecharacterizedanequilibriumwithmanipulationwhentherm’sin-vestmentdecisioniscontingentonthestockprice.Aninterestingquestioniswhetherallequilib-riainsuchagameexhibitmanipulation;thatis,whether,inagamewithfeedback,theuninformedspeculatoralwayssellswithpositiveprobabilityin1.Wenowaddressthisquestion.SupposethespeculatorweretotradeaccordingtotheequilibriumstrategiescharacterizedinSection4.1whenheispositivelyornegativelyinformed,butwouldnottradein1whenheisuninformed.Giventhis,theuninformedspeculator’sbestresponsewouldbetosellin1.Toseethis,notethat,takingotherstrategiesasgiven,increasesifthemarketmakerbelievesthattheuninformedspeculatordoesnotsellin1.Thisisbecause,intheabsenceofmanipu-lation,pricesperformabetterallocationalrole,sotheexpectedvalueofthermincreases.Thisincreasestheincentivefortheuninformedspeculatortosellin1(relativetotheequilibriumcharacterizedinSection4.1),andsomanipulationmustoccur.Inequilibrium,however,thingsbecomemorecomplicated,becausewecannottakeasgiventhestrategiesofthespeculatorwhenheispositivelyornegativelyinformed.Forexample,anincreaseinreducestheincentivesforthepositivelyinformedspeculatortobuyin1,whichmayleadhimtomixbetweenbuy-ingandnottradingin1.Thermwillthencanceltheinvestmentlessoften,becausethepositivelyinformedspeculator’stradeisnowclosertothatofthenegativelyinformedspeculator,anditisthereforemoredifculttoextractnegativeinformationfromtheprice.Thisreducestheincentiveoftheuninformedspeculatortomanipulate,sinceheendsupholdingashortpositioninarmthatinvestsandthushasvalue.Duetosuchissues,wecannotgenerallyruleoutequilibriawithoutmanipulationinagamewithfeedback.Wecan,however,rulethemoutwhenthefeedbackeffectissufcientlystrong,14.Thiseffectisstrengthenedbythefactthat,underthebeliefthattheuninformedspeculatordoesnottradein1,otherpriceschangeinawaythathisprotfromwaitingandtradingonlyin2decreases.2008TheReviewofEconomicStudiesLimited 148REVIEWOFECONOMICSTUDIESregardlessofwhetherhisactionleadstoadecreaseortoanincreaseintherm’sinvestment.Thisisbecausetheeffectofthenancialmarketontherm’sinvestmentisbasedonlearning.Ifthemanagertriestolearnfromthenancialmarketwhenthespeculatorhappenstobeun-informed,themanagerisessentiallymisledtothinkingthatthereisinformation,andthushisinvestmentdecisionisdistorted.Asaresult,thevalueofthermdecreases.Thespeculatorcanprotfromdecreasingthevalueofthermonlywhenheaccumulatesashortposition,notwhenheaccumulatesalongposition.Overall,ourmodelsuggeststhatthereisanasymmetrybetweenbuyerandsellerinitiatedtrades:manipulationofpricesviareductionofrms’valuescanoccuronlywithseller-initiatedtrades.ThisalsoreinforcestheregulatoryimplicationsexploredinSection4.3:Sincebuyer-initiatedmanipulationcannotbeoptimal,thereissomejusticationforregulators’focusonshort7.ROBUSTNESSISSUESInthissectionweanalysetherobustnessofthemanipulationequilibriumcharacterizedinSection4.1totwomodicationsofthemodel.First,inSection7.1,weextendthemodeltoincludeadditionalstrategictraders,whoareneverinformedaboutthestateoftheworld.Second,inSection7.2,weendogenizethenoisetraders’motivefortradingbymodellingthemasrisk-averseagentswhotradeinordertohedgeafutureriskexposure.AdditionaluninformedtradersSofar,weassumedthatthereisonlyonestrategictraderinthenancialmarket.Sincethisspec-ulatormakesaprotfromsellingevenwhenhefailstogetaninformativesignal,thequestionariseswhetheranyotheruninformedstrategictradercoulddothesame.ThismightinterferewiththeequilibriumcharacterizedinSection4.1byreducingtheexpectedprotsoftheoriginalspec-ulator.Inthissubsectionweextendthemodeltoincludeadditionalstrategictraders(hencefortharbitrageurs)whoareneverprivatelyinformedaboutthetruestate.ThefollowingpropositionshowsthatthemanipulationequilibriumcharacterizedinSection4.1isrobusttotheinclusionofsucharbitrageursinthemodel.Proposition5.Whenthereareadditionaltraderswhoareneverinformed,thenthereex-istsanequilibriuminwhichtheynevertradeandthepotentiallyinformedspeculatorfollowsthestrategycharacterizedbyTheintuitionbehindthisresultisasfollows.Anarbitrageurcannotconditionhistradesonthesignalthatthespeculatorhas.Ifanarbitrageurtriestomanipulatethepricebysellingin1,hethereforeendsupsellingsometimeswhenthespeculatorispositivelyinformed,andthereforehelosesmoneytohimbyaffectingthepriceinanunfavourabledirection.Asaresult,arbitrageursmakeanoveralllossfromtradingandthusprefernottotrade.EndogenousnoisetradeThissectionextendstheanalysisofSection4toallowforfullyrationalnoisetraders.modelnoisetradersasrisk-averseagentswhotradeinordertohedgeafutureriskexposurethatiscorrelatedtotherm’sliquidationvalue.ThisapproachfollowsSpiegelandSubrahmanyam15.Theadditionaltradersintroducedintheprevioussubsectionarenotincludedhere.2008TheReviewofEconomicStudiesLimited 150REVIEWOFECONOMICSTUDIESfundamentalvalueofthesecurity.Noisetradersthereforearewillingtotradeifthereductioninriskexposureisworthmorethantheexpectedtradingloss.Ifthisisthecase,theirorderowsareessentiallythesameasintheprevioussections,andthusthepreviousresultsextendtoequilibriainwhichnoisetradeisendogenized.Toclarifyourmodellingchoicesinthissubsection,weshouldemphasizethatthepurposeoftheanalysishereistoshowthattheequilibriumderivedunderexogenousnoisetradeisrobusttotheintroductionofendogenousnoisetraders.Thus,wechosethesimplestframeworktodemonstratethispoint.Onefeatureofthisframeworkisthatnoisetradershaveexposuretotherandomvariable,whichisuncorrelatedtothevariableaboutwhichthereisprivatelyinformed.Aswasnotedintheliteraturebefore(seeDowandRahi,2003),whetherahedger’sexposureiscorrelatedtothevariableaboutwhichthereisprivatelyinformedtradehascrucialimplicationsfortheequilibriumtradingbehaviour.OneeffectthatcouldcomplicatetheanalysisifnoisetradershadexposuretoistheHirshleifereffect,accordingtowhichinformedtradeleadstorevelationofinformationinthepriceanddestroyshedgingopportunities.Thisissuebe-comesevenmorecomplicatedwithmultipleroundsoftrade(likeinourpaper),sinceitimpliesthattheamountoftradeforthepurposeofhedgingmightchangefromoneperiodtoanotherasmoreinformationgetsrevealedintheprice.Thus,toavoidtheseissues,whicharenotthefocusofourpaper,weassumedthatnoisetradershaveexposureto,whichisuncorrelatedto.Ananalysisoftheeffectofmanipulationonliquidityinthepresenceofexposuretobothaninterestingdirectionforfutureresearch.8.CONCLUSIONSItiscommonlybelievedthatnancialmarketsprovideinformationthatguidesrealinvestmentdecisions.Yet,mostofthetheoreticalanalysisontheinformationalcontentofpricesinnancialmarketsassumesthatthevalueoftradedrmsisexogenousandthusthatrealinvestmentsarenotaffectedbytheinformationcontainedinprices(e.g.Kyle,1985).InthispaperweexplicitlyincorporatethefeedbackeffectfromthenancialmarkettotherealeconomyinatradingmodelàlaKyle(1985).Weshowthatthepresenceofafeedbackeffectchangesthescopeforspeculationinafundamentalway.Specically,westudythebehaviourofaspeculatorwhomayormaynotbeinformedaboutthestateoftheworld.Inamodelwithoutfeedback,thisspeculatorwouldbuythestockwhenheispositivelyinformed,sellthestockwhenheisnegativelyinformed,andnottradewhenheisuninformed.Oncefeedbackisintroduced,thespeculatormayhaveanincentivetosellevenwhenheisuninformed.Sellingisprotablefortworeasons.First,knowingthatheisuninformed,thespeculatorknowsthatthestockmarketwillnothaveabenecialallocationalrole.Sincethisisnotknowntoothermarketparticipants,hecansellthestockatapricethatishigherthanthetrueexpectedvalue.Second,thespeculatorcanprotfromtheeffectofhistradeontherm’sinvestmentdecision.Specically,hecanestablishashortpositioninthestockandthendrivethepricedownwithfurthersellorders.Thedecreaseinthepricewillleadtocancellationofrealinvestmentprojectsandtoareductionintherealvalueoftherm.Thiswillenablethespeculatortoprotonhisshortposition.WerefertothistradingstrategyasTheimportanceofidentifyingthemanipulationstrategygoesbeyondthecontributiontotheunderstandingoftradingdynamicsinnancialmarkets.Duetothefeedbackeffectfromthenancialmarkettotherealeconomy,thevalueofrmsdecreaseswhenspeculatorsengageinmanipulation.Thus,manipulationreectsalimitationthatisinherentinthebenecialalloca-tionalroleofprices.Yet,despitethisfeature,nancialmarketsstillimprovetheoverallefciencyofinvestmentdecisions.Afterall,ifthiswasnotthecase,thenpriceswouldlosetheirallocationalrole,andthepossibilitytomanipulatethemwoulddisappear.Animplicationofourpaperisthat2008TheReviewofEconomicStudiesLimited 152REVIEWOFECONOMICSTUDIESIfhedeviatesandbuysin1,histotalexpectedprotsareatleast 13(2V+Šp1(1)ŠE[p2|u2=1,Q1=1])+ p2|u2=1,Q1=0]).(18)Thus,theexpectedgainfromdeviatingisatleast 3V+Š 13p1(1)Š 13p1(0)+ p2|u2=1,Q1=Š1].(19)Thepositivelyinformedspeculatorwilldeviatewhenthisispositive.SincePr[0,weknowthat p2|u2=1,Q1=Š1]�p1(Š1).Thus,itsufcestoshowthatThisisalwaystruesinceandwhen1,then.Thus,thepositivelyinformedspeculatorbuysin1withpositiveprobability.Similarly,wecanshowthatthenegativelyinformedspeculatorsellswithpositiveprobability.ProofofPropositionWeshowthatfortheuninformedspeculator,tradingin1generatesastrictlynegativetotalprot.Supposethattheuninformedspeculatorbuysin1.Thetotalorderowin1canbe0,1,or2.Becausearestrictlybelow1,weknowthat +2andp1(2 .Wealsoknowthat .Thus,sincetheuninformedspeculatorexpectsthevalueofthermtobe ,buyingin1generatesastrictlynegativeprotonhis1transaction.Then,ifhedoesnottradein2,histotalprotremainsnegative.Ifhebuysagainin2,theexpectedpriceonlyincreases,andthushistotalprotremainsnegative.Ifhesellsin2,histotalprotprotp2|u2=Š1,Q1]Šp1(Q1).Weknowthatthiscannotbepositive.Moreover,giventhatthepositivelyinformedspeculatorisexpectedtobuyin2,sellingin2pushestheexpectedpricestrictlybelow,sotheexpectedprotisnegative.Asaresult,inequilibrium,theuninformedspeculatorneverbuysin1.Similarly,wecanshowthatheneversellsinProofofPropositionThetradingstrategiesinourproposedmanipulationequilibriumaregivenbyequations(5)–(11).Ourproofthatthisisanequilibriumhasthreeparts.InPartA,wedeterminemarketpricesandtherm’sinvestmentstrategygiventheequilibriumtradingstrategies.InPartB,weshowthat2tradingstrategiesareoptimalgivenmarketpricesandtherm’sinvestmentdecision.InPartC,weshowthat1tradingstrategiesareoptimalgivenmarketprices,therm’sinvestmentdecision,andtakingintoaccountthefuture2strategies.A.MarketpricesandtheÞrmÕsinvestmentTheinvestmentpolicyofthermistoinvestwhentheexpectedvalueofinvestmentispositiveandnottoinvestwhenitisnegative.Theexpectedvalueiscalculatedbasedonpriorinformationandonobservedorderows.Giventheequilibriumstrategiesandassumptions(3)and(4),thermdoesnotinvestwhen{Š,orwhen.Otherwise,therminvests.Thepricein2is0whenthermisnotgoingtoinvest,andreectstheexpectedvalueoftheinvestmentotherwise.Thus,usingBayesianupdating,2pricesare{Š{Š1and0and =V,p2(0,1)=V++2(1Š )µ + )µ1pricesarethengivenbytheexpected2prices: 13p2(1,0)+ 23V )µ 1 3V+ 13 )µ 13 2008TheReviewofEconomicStudiesLimited GOLDSTEIN&GUEMBELALLOCATIONALROLEOFPRICES155thattheinequalitybecomesstrictifthespeculatorstrictlypreferstobuyin2andifinvestmenthappenswithpositiveprobabilityafterwards.InLemma3,weshowthatthepositivelyinformedspeculatorstrictlypreferstobuyin2afterbuyingin1.ThisgeneratesinvestmentwithpositiveprobabilitysinceinvestmentalwayshappensafterThus,wegetthatthefollowingstrictinequalityholds:holds:|s=
,u1=1]E[|s=h,u1=1].(27)Finally,weconcludethispartoftheproofbystatingthreelemmasthatcharacterizethebehaviourofspeculatorsinacandidateequilibrium,wheretheuninformedspeculatordoesnotsellin1.TheproofsoftheselemmasappearaftertheproofofProposition3.Lemma2.Inanycandidateequilibrium,wheretheuninformedspeculatordoesnotsellint andVissuf-cientlysmall,thefollowingholds:ThenegativelyinformedspeculatordoesnotbuyintWhenp,thenegativelyinformedspeculatordoesnotbuyintAfterQ,thenegativelyinformedspeculatoralwayssellsintLemma3.Inanycandidateequilibrium,whereoff-the-pathbeliefssatisfytheintuitivecriterion,theuninformedspeculatordoesnotsellint andVissufcientlysmall,thefollowingholds:ThepositivelyinformedspeculatorbuysintafteranorderowofQinvestmentoccursfollowingQThepositivelyinformedspeculatordoesnotsellintAfterQ,thepositivelyinformedspeculatoralwaysbuysint.AfterQ,thepositivelyinformedspeculatoralwaysbuysint,aslongastheuninformedspeculatorbuysintinequilibrium.Lemma4.Inacandidateequilibrium,wheretheuninformedspeculatordoesnotsellint andVissufcientlysmall,theuninformedspeculatordoesnotbuyintafterQB.Nottradingvs.sellingTheexpectedprotoftheuninformedspeculatorfromnottradingin1iss|s=
,u1=0],whichcanbewritten 13E[2|s=
,u1=0,Q1+ 13E[2|s=
,u1=0,Q1=0]+ 2|s=
,u1=0,Q1=1].(28)Ifhesells,hegetsatleast 13(p1(Š2)+p1(Š1)+p1(0 onhis1transaction,andthen 13E[2|s=
,u1,Q1+ 13E[2|s=
,u1,Q1=0]+ 2|s=
,u1=Š1,Q1=Š2],(29)onhis2transaction.Ifhesellsand1or0,theprothecanmakeonhis2transactionisatleastashighastheprothemakesafterhedoesnottradeand1or0.Ifhesellsand2,hemakesatleast0onhistransaction.Thus,hestrictlyprefers1to0when 13(p1(Š2)+p1(Š1)+p1(0 V� 2|s=
,u1=0,Q1=1].(30)Giventhatpricesarenevernegative,andtakingasufciently ,itissufcienttoshowthatthat2|s=
,u1=0,Q1=1]p1(0).(31)c
2008TheReviewofEconomicStudiesLimited GOLDSTEIN&GUEMBELALLOCATIONALROLEOFPRICES157holdsforasufciently .Themoreinterestingcaseiswhentheuninformedspeculatorsellsin2afterInthiscase,his2protisatmost .Thus,weneedtoshowthat 13(p2(2,0)+p2(2,Š1)+p2(2,Š2p1(2 FromLemma2weknowthatthenegativelyinformedspeculatordoesnotbuyin1.Thus,iftheuninformedspeculatorbuysin1inequilibrium,2canresultfromboththepositivelyinformedandtheuninformedspecu-lator.Then,fromLemma3,thepositivelyinformedspeculatorbuysin2.Thus,giventhatisaweightedaverageofpossible2prices,sellingin2willpushthepricedown,implyingthattheL.H.S.of(39)isatmost0,andthusthat(39)holds.ProofofLemmaPart1:Bywayofcontradiction,supposethatthenegativelyinformedspeculatorbuysin1.Inthatperiod,totalorderowwillbe0,1,or2.Sincepricesareeither0orpositive,andthevalueoftherm,conditionalon,iseither0(ifthermdoesnotinvest)ornegative(iftherminvests),hemaximizeshisexpectedprotbysellinginAsaresult,themaximumtotalprotfrombuyingin1is 3(E[p2|Q1=0,u2Šp1(0 13(E[p2|Q1=1,u2Šp1(1 p2|Q1=2,u2=Š1]Šp1(2)).(40)If,instead,thespeculatordeviatestonottradingin1andsellingin2,hisexpectedprotis 13(E[p2|Q1=0,u2ŠE[V|s=l,Q1=0,u2)+ 13(E[p2|Q1=1,u2ŠE[V|s=l,Q1=1,u2)+ p2|Q1=Š1,u2=Š1]ŠE[V|s=l,Q1=Š1,u2=Š1]).(41)Weknowthatpricesarealways0orpositive,andrmvalueconditionaloniseither0ornegative.Wealsoknowthatisstrictlypositiveandthatinvestmenthappensafter0,generatinganegativevalueofThus,thesumofthersttworowsin(40)isstrictlysmallerthanthesumofthersttworowsin(41).Also,thelastrowin(41)isatleast0.Thus,toshowthatthenegativelyinformedspeculatorwoulddeviatefrombuyingin1tonottrading,weneedtoshowthatthelastrowin(40)isatmost0.Thatis,theprotfrombuyingin1andthenselling2cannotbestrictlypositive.Here,weneedtodistinguishbetweentwocases.First,if,following2,thenegativelyinformedspeculatorsellswithprobability1,theexpectedpriceinequilibriumfollowingasellordercannotincrease.Second,if,following2,hesellswithprobabilitylessthan1,itmeansheisindifferentbetweensellingandeitherbuyingornottrading.Sincewealreadyknowthat,conditionalon,buyingornottradingcannotgenerateastrictlypositivecontinuationprot,hiscontinuationprot,andthushistotalprot,cannotbestrictlypositive.Part2:FromPart1weknowthatthenegativelyinformedspeculatordidnotbuyin1.Supposethathedidnottradein1.Then,hiscontinuationprotfrombuyingisisVŠp2|s=l,u2=1]0,withstrictinequalitywhentherminvestsaftersome.Alternatively,thecontinuationprotfromsellingisisp2ŠV|s=l,u2=Š1]0,withstrictinequalityiftherminvestsfollowingsome{Š.Hence,aslongasinvestmenthappensafteri.e.aslongas0),sellingin2isstrictlypreferredoverbuying.Now,supposethatthenegativelyinformedspeculatorsoldin1.Hiscontinuationprotfrombuyingisisp2|u2=1]0,withstrictinequalitywhentherminvestsaftersome.Alternatively,thecontinuationprotfromsellingisisp2Š2V|s=l,u2=Š1]0,withstrictinequalityiftherminvestsfollowingsome{Š.Hence,asbefore,aslonginvestmenthappensaftersome,sellingin2isstrictlypreferredoverbuying.Part3:0,weknowfromPart2thatthenegativelyinformedspeculatordoesnotbuyin2after0.Wenowshowthathestrictlypreferstosellin2thannottotrade.Supposethathedidnottradein1.Then,hiscontinuationprotfromsellingisisp2ŠV|s=l,u2=Š1],whilehiscontinuationprotfromnottradingis0.Thecontinuationprotfromsellingisstrictlygreatersinceinvestmenthappensafter0generatingapositivepriceandanegativevalue.Now,supposethatthenegativelyinformedspeculatorsoldin1.Hiscontinuationprotfromsellingsellingp2Š2V|s=l,u2=Š1],whilehiscontinuationprotfromnottradingisisV|s=l,u2=0].Aftercomparing2008TheReviewofEconomicStudiesLimited GOLDSTEIN&GUEMBELALLOCATIONALROLEOFPRICES159Iftheuninformedspeculatordoesnottradein2,histotalprothis0,whichisclearlybelow .Ifhebuys,hepaysaprice,andgetsanassetwithamaximumvalue (conditionalonhisinformation),soforasufciently ,hisprotisbelow .Iftheuninformedspeculatorsellsin2hismaximumexpectedprotis .Notethat1isconsistentwithapositivelyinformedspeculatorwhoeitherboughtordidnottradein1.Weknowthatthepositivelyinformedspeculatoralwaysbuysin2afterbuyingin1.Ifhedidnottradein1,weneedtodistinguishbetweenthecaseinwhichhestrictlypreferstobuyin2andthecaseinwhichhedoesnot.Inthesecondcase,weknowfromPart1ofthislemmathatinvestmentdoesnothappenafteranyconsistentwith1.Thus,0,andtheuninformedspeculatormakesnoprotfromsellingin2.Ifthepositivelyinformedspeculatorstrictlypreferstobuyin2,bothcannotbegreater ,as1and2areinconsistentwiththepositivelyinformedspeculator.Moreover,hastobebelow,since1and0isconsistentwiththeuninformedspeculatorwhenhedoesnotsellin1.Thus, )) 3(V++ .Thisimpliesthat,forasufciently (44)holds.Part3:Thisfollowsdirectlyfromthersttwopartsofthelemmaandthefact(whichfollowsfromtheinvestmentstrategiesprovenabove)thatafter0therminvests.ProofofLemmaGiventhatthespeculatorisuninformed,thevalueofthermis or0.Thus,forasufciently ,theuninformedspeculatorchooseshis2transactiontomaximizethe2prot(i.e.effectofthetransactiononthevalueofhisexistingpositionisnegligible).WeknowfromLemmas2and3thatthenegativelyinformedspeculatoralwayssellsandthepositivelyinformedspeculatoralwaysbuysafter0.Thus,iftheuninformedspeculatorbuysafter0,thepricehepaysat1and2isaweightedaverage .Sincethevalueoftheasset,givenhisinformation,doesnote ,hemakesalossonhis2transaction.Thus,heprefersnottobuy.ProofofPropositionSupposethattheuninformedspeculatorbuysin1.Sincepricesareeither0orpositive,andtheexpectedvalueoftherm,conditionalon 0iseither0(ifthermdoesnotinvest)ornegative(iftherminvests),hecannotmakeastrictlypositiveprotunlesshesellsin2.Sellingin2generatesatotalprotof 13(E[p2|Q1=0,u2Šp1(0 13(E[p2|Q1=1,u2Šp1(1 p2|Q1=2,u2=Š1]Šp1(2)).(45)First,notethatthethirdtermof(45)cannotbestrictlypositive.Ifitisstrictlypositive,thenthenp2|Q1=2,u2=Š1]�0.Given 0andthatat2thespeculatorhasalongposition,thisimpliesthattheuninformedspeculatorandthenegativelyinformedspeculatorstrictlyprefertosellin2.Asaresult,sellingpushesthepriceweaklybelow,andsosop2|Q1=2,u2=Š1]Šp1(2)0.Now,fortheuninformedspeculatortomakeapositiveprot,eithereitherp2|Q1=0,u2=Š1]Šp1(0�)0orrp2|Q1=1,u2=Š1]Šp1(1�)0,orboth,musthold.Thisimpliesthatinvestmenthappensaftersomenodesfollowing0and/or1,andthusisstrictlypositive.Butthen,theuninformedspeculatorisstrictlybetteroffdeviatingtonottradingin1andsellingin2,inwhichcasehemakesaprotof 13(E[p2|Q1=0,u2ŠE[V|s=
,Q1=0,u2)+ 13(E[p2|Q1=1,u2ŠE[V|s=
,Q1=1,u2)+ p2|Q1=Š1,u2=Š1]ŠE[V|s=
,Q1=Š1,u2=Š1]).(46)Thisisstrictlygreaterthan(45)becausethethirdrowin(46)isatleast0,thelasttermin(45)isatmost0,theexpectedvalueisatmost0,andisstrictlypositive.ProofofPropositionSupposethattherearearbitrageurs.Ifallthearbitrageursnevertradeinequilibrium,thenallotherplayersareinagamethatisidenticaltotheonesolvedforinProposition2.Hence,toshowthattheequilibriuminProposition2isstillanequilibriumwhenthearbitrageursarepresent,weonlyneedtoprovethatthearbitrageur’sbestresponsetothespeculator’sandtherm’sstrategiesisnottotradeatanynodeofthegame(knowing2008TheReviewofEconomicStudiesLimited GOLDSTEIN&GUEMBELALLOCATIONALROLEOFPRICES1610,thenbuyingin2generatesthefollowingprotforthearbitrageur: 2 V+Š 23V+Š )µ VŠ 13(V++p2(0,1) +V)+ )( Šp2(0,1 13 2(2VŠ Thiscanberewrittenas 2 VŠ 13V+Š )µ VŠ 13(V++p2(0,1) +V)+ )( ))Sellingin2generatesthefollowingprot: 13 2(p2(0,1) Thus,protsarestrictlynegativeifthearbitrageurbuysorsellsfollowing0,sohewouldprefernottotrade.1,thenbuyingin2generatesthefollowingprotforthearbitrageur: )µ 2 3VŠ 13V+Š 13p2(1,0)+ )µ 23VŠŠ 13V+Š Sellingin2generatesthefollowingprot: )µ 13(V++p2(1,0 Thus,protsarestrictlynegativeifthearbitrageurbuysorsellsfollowing1,sohewouldprefernottotrade.1:Giventhat1,itfollowsthat{Š2,orareequalto0regardlessof2orderow.Moreover,thermendsupnotinvesting,soitsvalueisalso0.Thus,thearbitrageurmakesaprotof0.1,thespeculatormusthavesubmitted1,andthusispositivelyinformed.Ifthearbitrageurthenbuysordoesnottradein2,histotalprotis.Ifhesellsin2,histotalprotis Thus,inthisnode,hechoosestosellin2,andhistotalprotisgivenby(57).Wewillconsiderthisindeterminingthetotalexpectedprotthatthearbitrageurmakesbysellingin0,thenbuyingin2generatesthefollowingprotforthearbitrageur: )µ( +V+p2(0,1 V+p2(0,1 )µNottradingin2generatesthefollowingprot: 133V++4(1Š + )µThisisatleastashighastheprotfrombuying,sowecanignorethepossibilitythatthearbitrageurbuysininthisnode.Sellingin2generatesthefollowingprot: 132(0,1) +VŠ4V+)Š2(1Š )µNow,supposethearbitrageurchooses0after1and0.Then,thetotalexpectedprotfrom1isgivenby 13 2(p1(1)+ 13p2(1,0)ŠV+)+ 13 )µ 118(3V++4(1Š 2008TheReviewofEconomicStudiesLimited 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