DisclosuretoaCredulousAudience:TheRoleofLimitedAttention
DavidHirshleiferSonyaSeongyeonLimSiewHongTeoh
Wemodellimitedattentionasincompleteusageofpubliclyavailableinformation.In-formedplayersdecidewhetherornottodisclosetoobserverswhosometimesneglecteitherdisclosedsignalsortheimplicationsofnon-disclosure.Theseobserversmaychooseexantehowtoallocatetheirlimitedattention.Inequilibriumobserversareunrealisticallyoptimistic,disclosureisincomplete,neglectofdisclosedsignalsincreasesdisclosure,andneglectofafailuretodisclosereducesdisclosure.Regulationrequiringgreaterdisclosurecanreduceobservers’beliefaccuraciesandwelfare.Disclosureinonearenaaffectsperceptionsinfundamentallyunrelatedarenas,owingtocuecompetition,salience,andanalyticalinterference.Disclosureinonearenacancrowdoutdisclosureinanother.
FisherCollegeofBusiness,TheOhioStateUniversity,Columbus,OH43210-1144;Hirshleifer:hirshleifer2@cob.osu.edu,614-292-5174,
http://www.cob.ohio-state.edu/fin/faculty/hirshleifer/;Lim:lim.128@osu.edu,614-292-9772;Teoh:teoh2@cob.osu.edu,614-292-6547
WethankPaulFischer,JackHirshleifer,BoyanJovanovic,JohnKagel,DanLevin,BarbaraMellers,ourdiscussantAmritaNain,andparticipantsatseminarsatDukeUniversity,NewYorkUniversity,PennsylvaniaStateUniversity,OhioStateUniversity,andattheUniversityofMichiganSummerFinanceConferenceatEstesPark,Coloradoforveryhelpfulcomments.
DisclosuretoaCredulousAudience:TheRoleofLimitedAttention
Wemodellimitedattentionasincompleteusageofpubliclyavailableinformation.In-formedplayersdecidewhetherornottodisclosetoobserverswhosometimesneglecteitherdisclosedsignalsortheimplicationsofnon-disclosure.Theseobserversmaychooseexantehowtoallocatetheirlimitedattention.Inequilibriumobserversareunrealisticallyoptimistic,disclosureisincomplete,neglectofdisclosedsignalsincreasesdisclosure,andneglectofafailuretodisclosereducesdisclosure.Regulationrequiringgreaterdisclosurecanreduceobservers’beliefaccuraciesandwelfare.Disclosureinonearenaaffectsperceptionsinfundamentallyunrelatedarenas,owingtocuecompetition,salience,andanalyticalinterference.Disclosureinonearenacancrowdoutdisclosureinanother.
1Introduction
Intheclassicmodelsofvoluntarydisclosureofverifiableinformation,observersexhibitextremeskepticismaboutthosewhodonotrevealwhattheyknow(seeGrossman(1981)andMilgrom(1981)).Thisskepticismistherationalresponseofobserverstotheincen-tiveofapartywithadverseinformationtowithholdit.Inpractice,observersdotendtobeskepticalofthosewhofailtodisclose.However,thefurtherimplicationofthesemodelsthattherewillbefulldisclosureisinpracticeoftenviolated,asevidencedbythefactthatonoccasionfirmscollapsewhenconcealedadverseinformationcomestolight.Severalextensionstothebasictheoryallowforthewithholdingofinformation.1Disclosurecostsprovideaninnocentreasonfornon-disclosure,i.e.,areasonotherthanthepossessionofanadversesignal.Sodisclosurecostsmakeobserverssomewhatlessskepticalofnon-disclosure.Thus,aninformedplayerwithasufficientlyfavorablesig-naldiscloses,whereasifhissignalisbelowsomecutoffhewithholdsit(seeJovanovic(1982),Verrecchia(1983)).However,inthesemodelsfulldisclosureisstillapproachedasdisclosurecostsbecomessmall.2
Inthispaper,incompleteskepticismderivesfromthelimitedattentionandcognitiveprocessingpowerofobservers.Limitedattentionsometimescausesobserverstoignoreusefulsignals.Itcanalsocauseobserverstofailtotakeintoaccounttheimplicationsofanabsenceofasignal—thataninformedplayerhasdeliberatelywithheldrelevantinformation.Thislackofskepticismweakensthepressureoninformedplayerstodisclose.Asaresult,evenwhentherearenoexogenouscostsofdisclosureandthedisclosingplayersurelyisinformed,inequilibriumdisclosureisincomplete.Ontheotherhand,alackofattentiontodisclosedsignalscanencouragegreaterdisclosure,byreducingthereputationalpenaltytolowtypesofdisclosing.
Ourassumptionoflimitedattentionisintendedtocapturetwostylizedfacts.Firstistheobviousfactthathumaninformationprocessingpowerislimited,whichfollowsfromthephysicalanddesignconstraintsofthehumanbrain.Alargeliteraturein
TheseincludethemodelsofJovanovic(1982),Verrecchia(1983),FishmanandHagerty(1989),DarroughandStoughton(1990),andTeohandHwang(1991).2
Alternatively,FishmanandHagerty(2003)examineasettinginwhichinformedfirmssetpricesanddecidewhetherornottodiscloseasignalaboutthequalityofitsproduct,andasubsetofcustomerspossessinformationcomplementarytothedisclosedsignal.Thisleadstodifferentialupdatingbycustomersbaseduponthedisclosedsignal,andthepossibilitythatinequilibriumfirmsdonotdisclose.
1
1
psychologystudieslimitedattention,asdiscussedinSection2.Thesecondisthatpeopleoftendonotseemtobeasskepticalofthemotivesofinterestedpartiesasrationalitywouldseemtorequire.Wewillarguethatsuchcredulityisanaturalconsequenceoflimitedattention.Forexamplesandmarketevidencebearinguponlimitedattentionandcredulity,seeAppendixA.
Whencognitiveresourcesarelimited,anindividualmayfailtoupdateadverselybaseduponobservableevents,andespeciallybaseduponnon-events,suchasthefailureofaninformedpartytodisclose.Drawingacorrectinferencefromnon-disclosurerequiresbothfocusingattentiononthisnon-event,andpayingenoughattentiontothedisclosuregametoreasonoutitsstrategicimplications.Anindividualwhodoesnotdirecthisattentioninthisfashionmayfailtoupdatehispriorbeliefatall.
Inthispaperweconsidertheequilibriumbehaviorofaninformedplayerorplayerswhentheaudiencetheyfaceissubjecttolimitedattention.Webeginwithabasicmodelwithasinglearenaofpossibleinformationdisclosure.Theinformedplayerunderstandsthatexogenousfractionsofhisaudienceignoredisclosedsignals,aphenomenonwecallcueneglect;andignoretheimplicationsofnon-disclosure,whichwecallanalyticalfailure.Asaresultofanalyticalfailure,inequilibriumthereisapoolofnon-disclosingtypeseventhoughthecostofdisclosureiszero.Perhapmoresurprisingly,cueneglectencouragesdisclosure,becausethemarginaltypetakeslessofareputationalhitfromdisclosing.Theoveralloutcomeisintuitive:owingtolimitedattention,inequilibriumthereisonlypartialdisclosure,andonaveragethereisalsoexcessiveoptimismaboutthequalityoftheinformedplayer.However,theanalysisshowsthatthisfindingdoesnotderivefromtherawfactoflimitedattention,butfromatendencyforobserverstoattendmorefullytodisclosedsignalsthantoafailuretodisclose.Wefurtherexploretheeffectsofgovernmentimposeddisclosureregulation,andofvariationinobserverattention,ontheprecisionandbiasinobserverperceptions.
Furthermore,weextendthebasicmodeltoasettinginwhichindividualscanchooseupfronthowcarefullytoattendtodisclosedsignalsortowardthefailuretodiscloseinanarena.Wefindthatthemaininsightsandspecificresultsofthebasicsettingextendtoasettingwithendogenousallocationofattention.
Wealsoexamineasettingwithtwoarenasofdisclosureinwhichdifferentinformedplayerscancompetefortheattention(ortoavoidtheattention)ofobservers.Thisleadstoeffectswhichwecallcuecompetition,andanalyticalinterference.Cuecompetitionisthetendencyforobservationofadisclosureinonearenatodistractobservers’attentionfromdisclosureintheotherarena.Forexample,theannouncementofanacquisition
2
maydistractinvestorsfromthefactthatafirmhasjustmissedanearningsforecast.Analyticalinterferenceisthetendencyfordisclosureinonearenatodistractob-serversfromtakingintoaccountappropriatelytheinformationimplicitinthefactofnon-disclosureintheotherarena.Forexample,theannouncementoflargeearningssurprisesbyfirmsinoneindustrymaydistractinvestorattentionfromadelayintheissuanceofanearningsforecastbyafirminanotherindustry.
Weexaminetheimplicationsoftheseeffectsforseveralfurtherissues.Firstiswhetherregulationrequiringdisclosureinonearenacausestheinformedplayerintheotherarenatodiscloselessfrequently(‘crowdingout’).Secondiswhetherrequiringfulldisclosureinbotharenasincreasestheaccuracyofperceptions,andsocialwelfare.Thirdiswhetherthereisanoptimallevelofrequireddisclosurethatexceedswhatinformedplayerswouldvoluntarilyprovidebutwhichfallsshortoffulldisclosure.Fourthisthecross-arenacontagionofnewsannouncementsonobserverperceptions(ormarketprices)inotherarenas.Lastistheeffectofdisclosureregulationonobserverwelfareindifferentarenas.
Althoughourapplicationoflimitedattentioninthispaperistothetheoryofopti-maldisclosure,thesimplemodellingapproachweprovideisreadilyapplicabletootherproblemsininformationeconomics.Wedescribesomesuggestedfurtherdirectionsinwhichtheapproachcanbetakenintheconcludingsectionofthepaper.
Previousworkonlimitedattentionandeconomicdecisionshasfocusedmainlyontheimperfectrationalityofmanagersorotherorganizationaldecisionmakers(see,e.g.,theearlydiscussionofMarchandSimon(1958)).AsSimon(1976)stated:“...thescarceresourceisnotinformation;itisprocessingcapacitytoattendtoinformation.”Severalpapershaveanalyzedtheallocationofmanagerialattentionacrossactivities.3Ourapproachdiffersinfocusingonageneralaudienceofobservers.Inaddition,wedescribehowtointerpretourmodelassumptionsinthecontextofasecuritymarketsetting.Thus,ourapproachlendsitselftothestudyofhowlimitedattentionaffectsthepricingofassets.
Inconcurrentwork,HirshleiferandTeoh(2002)examinetheconsequencesoflim-itedinvestorattentionforfinancialreporting.Theiranalysistakesasgiventhatallrelevantinformationispubliclyavailable(eitherthroughdisclosureorspontaneously).Theyfocusontheeffectsofadditionallyreportinginformationaspartofearningsinafirm’sfinancialstatements.Incontrast,ouranalysisfocusesonthedecisiontodiscloseinformationwhichotherwisewillnotbepubliclyavailable.
3
SeeRadner(1975),RadnerandRothschild(1975),Gifford(1992),andGiffordandWilson(1995).
3
PerhapsthemostcloselyrelatedpapertothisoneisthatofMilgromandRoberts(1986).Theyshowthattheextremeskepticismresultsofpastliteratureextendtoasettinginwhichtheinformedplayercandiscloseasettowhichhissignalbelongs(ratherthantheprecisevalueofthesignal).Ifobserversarerational,thereisstillfulldisclosure.However,ifthereareunsophisticatedobserverswhoareinsufficientlyskeptical,disclosurecanbeincomplete.Theirpaperalsoanalyzestheconditionsunderwhichcompetitionamongself-interestedinformedpartiesinprovidinginformationcanleadtoaccuratedecisions(anissuewedonotaddress).
AkeydifferenceinourapproachfromthatofMilgromandRobertsisthatweanalyzeaspecificsourceofunsophisticatedbehavior,limitedattention.Thus,wemodelbothincompleteskepticismaboutnondisclosure,andfailuretoincorporatedisclosedsignals.4Afurtherdifferenceinapproachthatderivesfromourfocusonlimitedattentionisthatweanalyzethecarryoverofdisclosurebetweendifferentinformationalarenas(aswiththe‘crowdingout’effect),andhowregulationinonearenaaffectsdisclosure,beliefs,andwelfareinanother.Thus,adistinctiveaspectofourapproachisouranalysisofexclusionaryeffectswhereinasalientdisclosureattractsattentionawayfromanotherdisclosure;andofinterferencebetweenattentiontoadisclosedsignalinonearenaandattentiontotheimplicationsofafailuretodiscloseinanother.
Somereadersmayquestionwhetherlimitedattentionaffectsmarketprices.Whatishardtocontestisthatboththepubliccommentsofpolicymakersandactualregula-tionsreflectconcernsaboutprotectinginvestorswithlimitedattentionandprocessingpower.Forexample,therearerulesspecifyingnotjustthatcertaininformationitemsberevealedinafirm’sfinancialstatement,butwhereonthefinancialstatementtheseitemsmustbeplaced(aswithrulesonthereportingofcomprehensiveincome;seeHirstandHopkins(1998)).Furthermore,abitterfightbetweenregulators(includingformerSECchairArthurLevitt)andfirmshasconcernedwhetheremployeestockoptioncom-pensationneedstobemerelydisclosedinafootnote,orshouldbeintegratedaspartofreportingearnings(see,e.g.,Mayer(2002)).Thus,ataminimumitseemsusefultoassessrigorouslytheimplicationsofaviewthatformspartofthebasisforexistingpolicy.
Theremainderofthepaperisstructuredasfollows.Section2discussesthepsy-chologyoflimitedattentionandsalience.Section3describesthebasicsettingwitha
Thelattereffectinfluencesthenatureofourresults.WhereinMilgromandRobertsfulldisclosureintheirbasicmodelrequiresthattheobserverbesmartenoughtodrawextremeskepticalinferences,inoursettingthereisfulldisclosureevenwhensomeobserversarecredulous,ifinattentiontodisclosedsignalsissufficientlystrong.
4
4
singlearenaofdisclosure.Section4describesequilibriuminthebasicmodel.Section5analyzesasettingwhereobserversattendtooneoftwoarenasofinformationdisclosure.Section6examinestheeffectsofregulationondisclosure,beliefs,andwelfare.Section7extendsthebasicsettingtoallowforanendogenousdecisionbyobserversofhowmuchtoattendtodisclosedsignalsortothefactofnon-disclosureinanarena.Section8concludes.
2PsychologicalFindingsonLimitedAttention
Limitedattentionisanecessaryconsequenceofthevastamountofinformationavailableintheenvironment,andoflimitstoinformationprocessingpower.Inthefaceofcognitiveconstraints,attentionmustbeselectiveandrequireseffort(substitutionofcognitiveresourcesfromothertasks);see,e.g.,Kahneman(1973).Severalwell-knowndecisionbiasesareprobablycloselyrelatedtolimitstoattention,suchasthephenomenonofnarrowframing(asreviewedinRead,Loewenstein,andRabin(1999)),whichinvolvesanalyzingproblemsintooisolatedafashion.
Attentionisrequiredbothfortheencodingofenvironmentalstimuli(suchasacor-porateinformationdisclosure),andtheprocessingofideasinconsciousthought(asintheanalysisofacorporatedisclosureorofafailureofacompanytodisclose).Asdis-cussedinFiske(1995),theencodingprocessinvolvestakingexternalinformationandrepresentingitinternallyinawaythatenablesitsuse.Consciousthoughtinvolvesafocusonparticularideasormemoriestotheexclusionofothers.Forexample,ifanindividualisfocusedonunderstandingtheimplicationsofadisclosurebyonefirm,hisabilitytostudyanotherfirmmaybereduced.
Somestimulitendtobeperceivedandencodedmoreeasilyorretrievablythanothers.Thesalienceofastimulusisits‘prominence,’tendencyto‘standout’,oritsdegreeofcontrastwithotherstimuliintheenvironment.Forexample,anunusuallylargeearningssurpriseishighlysalientforinvestors.Theeffectsofsalienceare“robustandwide-ranging”(FiskeandTaylor(1991),ch.7),withinfluenceonjudgmentsaboutcausality,importanceofthestimulus,andhowextremeitis.Wereflectsalienceinourmodelasinfluencingtheprobabilitythatasignalwillbeattendedto,andtheprobabilitythatanindividualwillanalyzecorrectlytheimplicationsofafailuretodisclose.
Reasonablyenough,astimulusisalsomoresalientifitisgoal-related;e.g.,anin-dividualinagroupbecomesmoresalientifyoulearnthatsheistobeyournewboss.However,attentiontostimulicanbemisdirectedinmanyways,andthisaffectsjudg-5
ments.Seeminglytrivialmanipulationsofsaliencehaveimportanteffectsonjudgments(see,e.g.,TaylorandFiske(1978)Sect.IV).Attentionisalsodrawntovividstimuli.5Incontrast,peopletendtounderweightabstract,statistical,andbase-rateinformation(see,e.g.,KahnemanandTversky(1973)andNisbettandRoss(1980)).Inviewofthesefindings,inourmodelwedonotassumethattheamountofattentionthatobserversdirecttowardasignalcorrespondsperfectlywithitseconomicimportance.
Howattentionisdirectedinconsciousthoughtdependsontheeasewithwhichmemoriesareaccessed–whatTverskyandKahneman(1973)refertoasavailability.Forexample,adisclosurebyafirmthatreceivesheavyplayinthenewsmediamay,throughemphasisandrepetition,bemoreretrievablethanastorythatreceivesonlyamention.Intheavailabilityheuristicindividualsassessthefrequencyorlikelihoodofaphenomenonaccordingtotheirabilitytoretrieveconfirmatoryexamplesfrommemory.Totheextentthatsomeideasorfactscontainhooksthatmakethemmore‘available’thanothers,attentionalbiasesthereforeinducebiasedbeliefs.
Theseconsiderationssuggestthatinacorporatesetting,disclosuresbyfirmsthatareinthenewsalot(largerfirmsorfirmsin‘fashionable’sectors),orare‘proximate’andaffect-linkedforobserverswhoconsumethefirms’products(suchasentertainment,sports,orautomobilefirms)maybeparticularlysalient.Basedonvividness,wewouldalsoexpectmoreattentiontosimpledisclosuresthantothosethatarehardtoprocess.Aliteratureinpsychologyhasexaminedhowsubjectslearnbyobservationovertimetopredictavariablethatisstochasticallyrelatedtomultiplecues(see,e.g.,KruschkeandJohansen(1999)).Apervasivefindingisthatanimalsandpeopledonotachievecorrectunderstandingofthecorrelationstructure.Instead,cuecompetitionoccurs:salientcuesweakentheeffectsoflesssalientones,andthepresenceofirrelevantcuescausessubjectstouserelevantcuesandbaserates(unconditionalfrequencies)less.Thepresenceofmultiplecuesalsocausespeopletomakeanalyticalerrors,suchas‘learning’overtimetouseirrelevantcues.
3TheEconomicSettingoftheBasicModel
TheInformedPlayer
Theinformedplayerobservesasignalθontheinterval[θ,θ].Hedecideseither
Vividnessisgreatestforconcretedescriptionsandscenarios,personalstoriesaboutindividualex-periences,informationthatfallsintoaneasilysummarizedpattern,stimulithattriggeremotionalre-sponses,orwhicharemore‘proximateinasensory,temporalorspatialway’(NisbettandRoss(1980),p.45).
5
6
todiscloseorwithholdhissignal;ifhediscloseshemustbetruthful.Wefollowtheconventionthataplayerwhoisindifferentalwaysdiscloses.
UninformedObserversThereisacontinuumofuninformedobservers.Limitedattentionhastwoeffects.First,fractionαW∈[0,1]oftheobserversarerationallyskepticalaboutthemotivesofanon-disclosing(Withholding)informedplayer,withtheremainingfraction1−αWinattentive(analyticalfailure).Anindividualwhoisinattentiveinthisfashiondoesnotupdatehisbeliefs.Thismayoccurbecausehesimplydoesnocognitiveprocessing.Alternatively,hemaynotethefactthatinformationwaswithheld,butfailtotakethefurthercognitivestepofattributingthiswithholdingtothestrategicincentivesoftheinformedparty–credulousbehavior.Undereitherinterpretation,theinattentiveobserver’sbeliefremainsathisprior.
Wedonotbyanymeanswishtoimplythataccountingforthepossibilityofstrategicbehaviorbyaninformedpartyisaprofoundlydifficultconceptualleapformostpeople.Whenattendingcarefully,mostpeopleareadeptinrecognizingsuchpossibilities.How-ever,evenintelligentpeoplewillsometimesneglectfairlyobviouspointsinanygivenarenaofaction.Timeandcognitiveresourcesarelimitedandtheuniverseofpossiblesignalsandconsiderationstoattendtoislarge.6
ThesecondeffectoflimitedattentioninthemodelisthatafractionαD∈[0,1]attendtoinformationthatisdisclosed;theremainingfraction1−αDofobserversfailtoattendtothedisclosure(cueneglect).WeassumethatαD≥αW,basedonthenotionthatdisclosureissalient,andthereforecallsattentiontoitselfmorestronglythanafailuretodisclose.Thisisconsistentwithpsychologicalevidencethatpeopletendtobemoreinfluencedbytheinformationreflectedintheoccurrenceofaneventthanthenon-occurrence(see,e.g.,Newman,Wolff,andHearst(1980),andNisbettandRoss(1980)).
Thesharedpriorbeliefofobserversabouttheinformedplayer’stypehasdensityf(θ)anddistributionfunctionF(θ).Thepublicinformationsetφisequaleitherto(D,θ)(knowledgethatinformationwasdisclosed,andthattherevealedvaluewasθ)orelsetoW(knowledgethatinformationwasnotdisclosed).
Theaveragepopulationbeliefaboutthetypeoftheinformedplayeristheaverage
Broadlysupportiveofthisargument(thoughnotspecificallyatestofit)isevidencethatpeopletendtounderweighttheprobabilitiesofeventcontingenciesthatarenotexplicitlyavailableforconsideration.Forexample,peopletendtounderstimatetheprobabilityof‘othercauses’inalistofpossiblecausesofanevent(Fischoff,Slovic,andLichtenstein(1978)).
6
7
ofthecredulous/inattentiveandtherationalbeliefs,
ˆD≡(1−αD)E[θ]+αDθθ
ˆW≡(1−αW)E[θ]+αWθˆρ(W),θ
(1)
whereahatdenotesanaverageobserverperception,andaρsuperscriptindicatesan
attentivebelief.7
Theinformedplayer’sobjectiveindecidingwhetherornottodiscloseistoachievethehighestpossibleaverageperceptionamongobservers.Forexample,inacorporatedisclosurecontext,thiswouldamounttomaximizingthecurrentstockprice.Ashasbeenfoundinseveralmodelsofmarketequilibriumwhensomeinvestorshaveimper-fectlyrationalbeliefsandothershaverationalbeliefs,equilibriumstockpricesreflectaweightedaverageofthebeliefsofbothgroupsoftraders(see,e.g.,Daniel,Hirsh-leifer,andSubrahmanyam(2001)),wheretheweightsreflecttherelativenumbers,risktolerances,andperceivedrisksofindividualswithdifferentbeliefs.
4
4.1
EquilibriumintheBasicModel
CharacterizingtheEquilibrium
Asisstandardinseveraldisclosuremodels,thebehavioroftheinformationrecipientscanbeviewedinaverysimpleway.Weassigntheinformedplayertheobjectiveofmakingtheaveragebeliefsofobserversbeasfavorableaspossible.8Weproposeathresholdequilibriuminwhichthereisacriticalsignalvalueθ∗suchthataninformedplayerdisclosesifandonlyifhistypeθ≥θ∗.
Ifthefirmdoesnotdisclose,thentheaverageexpectationamongtheaudienceoftheinformedplayer’ssignalisaweightedaverageoftheinattentive/credulousexpectation
ˆρ(W)=E[θ|θ<θ∗].SotheaverageperceptionsE[θ],andtherationalexpectationθ
Equation(1)hastwopossibleinterpretations.Oneisthatdifferentindividualsbynaturehave
differentdegreesofattentivenesstowardsdifferentinformationsignalsavailableintheenvironmentandtotheopportunitiesoftheinformedplayertoengageinstrategicdisclosurebehavior.Theotheristhatobserversareexanteidentical,buteachhasaprobabilityofbeinginattentiveorattentivetowardenvironmentalinformationandtowardthestrategicincentivesofothers.8
Thiscanbeviewedasareducedformofasettinginwhichobserverstakeactionsbasedupontheirbeliefsthataffecttheinformedplayer.Forexample,inacorporatedisclosuresetting,investorswoulduseincorrectbeliefsintheirsecuritytradingdecisions.Theresultingstockpricewouldbeofconcerntotheinformedplayer(acorporatemanager).
7
8
giventhattheinformedplayerwithholds,orthathediscloseshistypeθ,are
ˆD=(1−αD)E[θ]+αDθ=θ−(1−αD)(θ−E[θ])θ
ˆW=(1−αW)E[θ]+αWE[θ|θ<θ∗].θ
(2)(3)
Whenanabove-averagetypediscloses,limitedattentiondetractsfromhisreputation
(thesubtractedtermontheRHS),whereaslimitedattentionenhancesthereputationofadisclosingbelow-averagetype.
Theequilibriumthresholdvalueθ∗makestheinformedplayerjustwillingtodisclose,
(1−αD)E[θ]+αDθ∗=(1−αW)E[θ]+αWE[θ|θ<θ∗],or
θ∗=γE[θ]+(1−γ)E[θ|θ<θ∗],
where
(4)(5)
αD−αWγ≡.(6)
αD
Theparameterγcanbeviewedasameasureoftheexcessattentionpaidtoanarenawhenasignalisdisclosedratherthanwithheld.Wedescribethepossibleequilibriaasfollows.
Proposition1Forallparametervalues,anequilibriumexists.1.IfαW≥αD>0,theuniqueequilibriumentailsfulldisclosure.
2.If0<αW<αD,theninequilibriumthereexistsathresholdvalueθ∗,θ<θ∗ (7) increasesmonotonicallywithθ.Thus,thereareuptothreepossibletypesofequilibrium: 1.Alltypesdisclose;2.Notypesdisclose;and3.Aplayerdisclosesifandonlyifhistypeequalsorexceedsacriticalvalueθ∗,θ<θ∗≤θ.Inaproposedequilibriumwithnodisclosure(2.),theperceptionofatypethatwithheldwouldbeE[θ],soanytypeθ>E[θ]wouldprefertodisclose.Thisbreakstheproposedequilibrium,soonly(1.)and(3.)areviableequilibriumcandidates. 9 IfαW=αD,thenγ=0,andthereisfulldisclosure(1.),becauseequation(5)canonlybesatisfiedbyθ∗=θ.IfαW>αD,thenγ<0,andthereisnoθ∗satisfying(5);theinformedplayeralwayspreferstodisclose.ItremainstobeshownthatifαW<αD,fulldisclosure(1.)isnotanequilibrium,sothatonlypossibility(3.)remains,andthatequilibriumexists.TheproofisinAppendixB.Intuitively,whenαW<αD,theexpectedreputationalpenaltyonalowtypeforfailingtodiscloseissosmallthatsuchatypestrictlypreferstowithholditssignal.Finally,thecriticalvalueθ∗ 4.2ComparativeStaticsontheAmountofDisclosure Attentionbyobserverstothewithholdingofinformation,αW,andattentiontodis-closure,αD,haveopposingeffectsontheincentiveoftheinformedplayertodisclose.Attentiontoafailuretodiscloseencouragesdisclosurebecauseoftheincreasedskep-ticismtowardtheinformedplayerwhowithholds.Incontrast,attentiontodisclosurediscouragesdisclosurebythemarginaltype.Sinceθ∗ TounderstandtheeffectofvaryingαD,considerthecriticaltypeθ∗.HigherαDincreasesthefractionofobserverswho,whenhediscloses,perceivehistypeasθ∗ dγ <0,dαW 10 dγ >0.dαD (8) Applying(3)and(2),let ˆD−θˆWθ G(t,γ)≡=t−γE[θ]−(1−γ)E[θ|θ Anequilibriumthresholdθ∗satisfiesG(θ∗,γ)=0.Forastableequilibrium,G1(θ∗,γ)>0,sothatamarginalincreaseinthethresholdencouragesdisclosurebythemarginaltype.Underthemarketperceptionsassociatedwithsuchamarginalincrease,thefirmthenpreferstodisclosureatacriticalthresholdbelowtheincreasedthreshold.SinceG(θ,γ)<0foragivenγ∈[0,1]andG(θ,γ)>0,thereexistsatleastonestableequilibriumintheinterval[θ,θ].Toderivecomparativestaticsofθ∗withrespecttoγintheneighborhoodofastableequilibrium,wedifferentiatebothsidesofG(θ∗(γ),γ)≡0withrespecttoγ: dθ∗ 0=G1(θ,γ)+G2(θ∗,γ) dγdθ∗∗ =G1(θ,γ)−E[θ]+E[θ|θ<θ∗],so dγ dθ∗E[θ]−E[θ|θ<θ∗] =>0.dγG1(θ∗,γ) ∗ (10) Thelastinequalityholdsforstableequilibria(G1(θ∗,γ)>0)sinceE[θ]>E[θ|θ<θ∗]. Forexample,undertheuniformdistributionf(θ)=1/(θ−θ),by(5), θ∗= γθ+θ,γ+1 (11) whichisincreasinginγsinceθ>θ.Itfollowsbythechainruleand(8)thatdθ∗/dαW<0,anddθ∗/dαD>0. Proposition2Undertheassumptionsofthebasicmodel,intheneighborhoodofastableequilibrium: 1.TheamountofdisclosureincreasesuniformlywiththefractionofobserverswhoareattentiveaboutthewithholdingofinformationαW;2.TheamountofdisclosuredecreasesuniformlywiththefractionofobserverswhoattendtodisclosureαD. 4.3AccuracyofObserverPerceptions Wenowexaminehowattentionaffectstheaccuracy(biasandmeansquarederror)ofobservers’averageperception. 11 4.3.1Optimism Observersonaveragetendtobeoptimisticaboutthequalityoftheinformedparty ˆ]>E[θ]).Ontheonehand,foragiventhresholdθ∗,credulityaboutnon-disclosure(E[θ increasestheaverageperception—inattentiveindividualsperceivethetypeofanon-disclosinginformedplayerasonaverageE[θ]insteadofθ<θ∗.Ontheotherhand,inattentiontodisclosuretendstodecreaseaverageperception—inattentiveindividualsperceivethetypeofanon-disclosinginformedplayerasonaverageE[θ]insteadofthedisclosedθ>θ∗,whichonaveragemustbegreaterthanE[θ].However,solongasαW<αD,thefirsteffectdominates(seeAppendixB).Wethereforehave: Proposition3IfdisclosureisincompleteandifαD>αW,thenobserversareon ˆ]>E[θ]).IfαD=αW,thenaverageoveroptimisticabouttheinformedsignal(E[θ observersonaveragecorrectlyassessthequalityoftheinformedplayer. Althoughthereisanaveragetendencytowardoptimism,owingtocueneglectaverageinvestorperceptionsaretoopessimisticwhenθissufficientlyhigh. Similarpointsmayapplymorebroadlyinsettingswhereaninformedplayerhassomediscretioninwhathetellsanobserverwithlimitedattention.Psychologistshavefoundthatindividualsonaveragetendtoexhibitunrealisticoptimismaboutthelike-lihoodofexperiencingfavorablepersonaloutcomes.9Unrealisticoptimismmayresultinpartfromlimitedattention.Lifeeventsaresubstantiallyinfluencedbythestrategicrevelationpoliciesofinterested,informedparties.Participantsinbusinessandpersonalrelationshipsoftenconcealtheirlackofcommitment;asaresult,alltoooftenpeopleareshockedwhentheylosetheirjobsorlifepartners‘outoftheblue.’Ouranalysissug-geststhatcredulityaboutthestrategicincentivesofothersmaybeasourceofunrealisticoptimism.10,11 AdamSmith(1776),inregardto“thegreaterpartofmankind,”referredto“Theirabsurdpre-sumptionintheirowngoodfortune...”SeealsotheexperimentalworkofWeinstein(1980,1982).10 KennedyandDimick(1987)findthat48%ofcollegeathletesinrevenue-producingsportsexpecttoplayprofessionally,whiletheactualfigureis2%.Collegesmayhaveanincentivetoallowathletestobelievetheyhavearealshotatgoingprofessional,ratherthandisclosingadverseinformationaboutlikelihoodofsuccess.11 Togiveaconjecturalhintabouthowsuchanissuecouldbemodelled,considerasocialexchangesettinginwhichtwoindividualsplayarepeatedPrisoners’Dilemmaoveraninfinitenumberofperiods.Insuchasettingtheremaybeatriggerstrategyequilibriumenforcingcooperation(C)asopposedtodefection(D).However,inroundtonepartyIreceivesprivateinformationaboutthevalueofanexternaloptionthatwillbecomeavailableinroundt+k,wherek>0isknowntoall.Toexploittheexternaloptionhemust,attimet+k,abandontheexistingrelationship,i.e.,hemustplayDatroundt+kandatalllaterrounds.Iftheexternaloptionissufficientlyfavorable,itpaysforhimtodo 9 12 Differentiatingtheaverageoptimism(49)withrespecttoαWandαD,itisnothardtoverifythatmoreattentiontodisclosedsignalsdiscouragesdisclosure,andtherebyincreasesoptimism;andgreaterattentiontonon-disclosuredecreasesoptimism.ˆ]−E[θ]Proposition4Intheneighborhoodofastableequilibrium,averageoptimismE[θ ˆ−θ]/dαD>0)anddecreaseswithincreasesasattentiontodisclosureincreases(dE[θ ˆ−θ]/dαW<0).attentiontonon-disclosure(dE[θ Inaddition,greateruncertaintybyobserversabouttheinformedplayer’sinformation increasesoptimism.ItisnothardtoshowthatifthedensityofθishorizontallystretchedbyafactorofK,thenaverageoptimismisalsomultipliedbyK.4.3.2 MeanSquaredError Themeansquareddeviationoftheaverageperceptionoftheinformedtypefromtheactualtype, θ∗ 2 ˆ−θ)2]=E[(θ(1−αW)E[θ]+αWE[θ|θ<θ∗]−θf(θ)dθ + θθ (1−αD)E[θ]+αDθ−θ 2 f(θ)dθ,(12) θ∗ isameasureoftheinaccuracyofobserverperceptions. Proposition5Ifthereisahigherprobabilitythatobserversattendtodisclosedinfor-mationthanthattheyattendtothefactthatinformationiswithheld,αD>αW,then intheneighborhoodofastableequilibrium: 1.ThemeansquarederroroftheaverageobserverperceptionsasanestimateofthetruetypeisdecreasinginαW.2.ThemeansquarederroroftheaverageobserverperceptionsasanestimateofthetruetypecaneitherincreaseordecreaseasαDincreases. so.I’ssignalprovideshimwithsuperiorinformationinroundtaboutwhetherhewilllaterabandonthecurrentrelationship.IftheuninformedpartyUinfersthatIissufficientlylikelytoabandon,thenthetriggerstrategyequilibriumbreaksdownandUdefectsimmediatelyaswell.SuchbreakdownofcooperationiscostlytoI,whopreferstoreaptherewardsoftheoldrelationshiplonger.Thus,underlimitedattention,Imaybenefitfromconcealingfavorableexternaloptions,therebyencouragingUtobeoptimisticaboutI’scommitmenttotherelationship. 13 TheproofiscontainedinAppendixB. Intuitively,byProposition2,themoreattentiveobserversareaboutnon-disclosure,themoredisclosureoccurs.Greaterdisclosuremakesbeliefsonaveragemoreaccurate,implyingalowermeansquarederror.ItistruethatperceptionsaresometimesinaccurateevenafterdisclosuresolongasαD<1.However,sinceαD>αW,thefrequencyofinaccurateperceptionsislowerwhentheinformedplayerdisclosesthanwhenhedoesnotdisclose.Furthermore,wheninvestorsarerationallyskeptical,theinferencedrawnfromnondisclosure,E[θ|θ<θ∗],isalesspreciseestimateofθthantheinferencethatcanbedrawnfromrationallyattendingtodisclosure,theactualvalueofθ. Incontrast,thecomparativestaticswithrespecttoαDisambiguousevenintheuni-formcase.Intuitively,themoreattentiveobserversaretodisclosure,thelessdisclosureoccurs(forreasonsdescribedearlier).Themarginaltypefindsdisclosurelessattractiveifthisdisclosureisnoticed,ashistypeisbelowaverage.Forreasonsdiscussedintheprecedingparagraph,lessdisclosuretendstoincreasethemeansquarederroroftheaverageinvestorperceptionifαD>αW.However,acountervailingforceisthathigherαDincreasestheprobabilitythatanindividualincorporatesthedisclosedinformationintohisbeliefs. 5CompetingAttentionalDemandsandSalience Whenindividualshavelimitedcognitivecapacityitisimpossibletoattendtoalldecision-relevantinformationofdifferentformsandderivedfromdifferentsources.Theallocationofattentionacrosssignalsorstrategicconsiderationswillingeneralbebiasedbythesalienceofdifferentaspectsofthedecisionenvironment.AsdiscussedinSection2,thereisextensiveevidencefromthepsychologyliteraturethatpeopleattendmoretosomestimulithanothers,ofteninwaysthatdonotcorrespondtodifferencesintheinformativenessorusefulnessofthedifferentsignals.Wenowexploreasettinginwhichdisclosureornon-disclosureineacharenainfluencesperceptionsanddisclosurebehaviorintheother. Inthissettingtherearetwoarenasofdisclosure,i=AorB.Theinformedplayerinarenaiobservesasignalθi,whereθi≤θi≤θi.Disclosuredecisionsineacharenaaretakensimultaneously.Anobserverwhoattendstodisclosureortonon-disclosureinagivenarenaupdateshisbeliefrationally,whereasanobserverwhofailstoattendtoarenaiholdstohisprioraboutθi. Weconsideraverysimpleformoflimitedattention:individualscanattendtoone 14 ortheotherarena,butnotboth.Thisspecificationcapturesthenotionofinformationoverloadinasimpleway.Borrowingfromtheliteratureinexperimentalpsychologyonmultiplecuelearning(seeSection2),wecallthetendencyforaninformationdisclosureinonearenatodistractobserversfromattendingtoadisclosureintheotherarenacuecompetition.Wecallthetendencyforaninformationdisclosureinonearenatodistractobserversfrominferringthereasonforanactionorfailuretoactinanotherarenaanalyticalinterference. 5.1TheBasicModelwithCompetingInformationSources ForeacharenaAandB,wewillfirstshowthatanequilibriumofthesortdescribedintheprecedingsectionapplies.Limitedattentiondeterminesthefractionoftheobservers WDWD whoarecredulouswithrespecttoeacharena,αA,αA,αBandαB,endogenously. Iftheindividualisfacedwithnodisclosureineitherarena,weassumethatheattendstooneortheotherarenawithequalprobability.Ifthereisdisclosureaboutonearenabutnotanother,thentheeffectofthedisclosureonattentiontotheotherarenaisassumedtoberelatedtothesalienceoftheinformationdisclosure. Weallowdifferentarenasofdisclosuretohavedifferentlevelsofsalienceorvividness(seeSection2).Forsimplicitytheamountofattentionalinterferencebetweenarenasdependsonlyonwhetherdisclosureoccurred,notonthesignalrealization.ThesaliencesofarenasAandB,denotedsAorsB,helpdeterminetheprobabilitythatindividualswillattendtoeacharena. Intheabsenceofanyattention-grabbingevents,theprobabilityofanindividualattendingtoAversusBwouldbe.5.IfthearenaAsignaliswithheldthearenaB,signalisdisclosed,arenaBislikelytocaptureagreatershareofobserverattention.Toreflectthehighersalienceofoccurrencethannon-occurrenceofanevent,weassumethattheprobabilitythatanindividualattendstoarenaAortoBis αA(WA,DB)=.5(1−sB)αB(WA,DB)=.5(1+sB). (13) Thus,asthesalienceofdisclosureinarenaBincreases,itrobsmoreattentionfromthenon-disclosingarenaA.Ifthesalienceofthedisclosedinformationis0thiseffectvanishes(Bgetsonlyits50:50shareofattention).However,assaliencerisesto1the 15 probabilityofattendingtoAdiminishestozero.Symmetrically,weassumethat αA(DA,WB)=.5(1+sA)αB(DA,WB)=.5(1−sA). (14) Ifthereisdisclosureinbotharenas,thenitisassumedthattheprobabilitythatanindividualattendstoAortoBis αA(DA,DB)=.5(1+sA−sB)αB(DA,DB)=.5(1−sA+sB). (15) Thereisagreatertendencytoattendtothemoresalientarena,andifthedifferenceinsaliencebetweenthetwoarenasismaximal(1−0=1),thenanobserverattendstothemoresalientdisclosurewithcertainty. Inequilibrium,theinformedplayerinarenaitakesthestrategyoftheotherinformed ∗ player(i.e.,playeri’sthresholdvalueθi)asgiven.Eachinformedplayerthereforetreats DW,ortonon-disclosure,αi,inthefractionofobserverswhowillattendtodisclosure,αi hisarenaasgiven.Thus,wecanapplytheequilibriumoftheprevioussectiontoeach ∗ ,asafuncitonoftheporposedofthearenas,toderivethethresholdvalueinarenai,θi ∗∗ criticalvalueintheotherarenai,θi.GivencriticalvalueθB,theprobabilitythatanobserverattendstothefactthatinformationiswithheldinarenaAis WαA=.5Pr(WB)+.5(1−sB)Pr(DB) ∗∗ =.5[Pr(θB<θB)]+.5(1−sB)[Pr(θB>θB)]∗=.5[1−sB+sBF(θB)]. (16) Similarly,forarenaB, W∗ αB=.5[1−sA+sAF(θA)]. (17) ∗ ,theprobabilitythatagivenobserverGivenaproposedthresholdinarenaB,θB attendstodisclosureinarenaAis DαA=αA(DA,WB)Pr(WB)+αA(DA,DB)Pr(DB) ∗∗ )+.5(1−sB+sA)[1−FB(θB)]=.5(1+sA)FB(θB ∗ =.5(1+sA−sB[1−FB(θB)]). (18) TheprobabilitythatanobserverisattentivetodisclosureinBisderivedsimilarly: ∗D )]).=.5(1+sB−sA[1−FA(θAαB (19) 16 DW From(16)-(19),αi>αiwhensi>0.Weproposeanequilibriuminwhicheach ∗∗ informedplayerfollowsathresholddisclosurerule,withcutoffsθAandθB.Wedeterminetheequilibriumineacharenatakingthecutoffintheotherarenaasgiven.Weseekasetofself-confirmingcutoffvaluesthatsatisfythebasicmodelequilibriumconditionstogetherwith(16)-(19). However,ingeneraltheremaybemultipleequilibria.Evenifexantethearenasaresymmetric,theremaybeasymmetricequilibria.Highdisclosureinonearenacandistractfromtheother,leadingtolowerdisclosureintheother.Laterwewillshowuniquenessbydirectcalculationinthecaseofauniformdistributionoftypes;andinthecaseofequalsaliences,wewillsolveforasymmetricequilibrium. Theequilibriumconditionforinformedplayeritobejustwillingtodisclose,asin(5),isthat ∗∗θi=γiE[θ]+(1−γi)E[θ|θ<θi], DWαi−αi γi≡,i=AorB,Dαi where(20)(21) WD andwhereαi,αiaretheprobabilitiesthatindividualsattendtoeitherthewithholdingofinformation,orthedisclosureofinformation,inarenai. ∗WD Equations(20)and(21)describeθAintermsofαAandαA.Buttheseinturn ∗∗∗ arebothfunctionsofθB.Thus,wecansolveforareactioncurveθA(θB).Similarly, ∗∗ wecansolveforthereactioncurveθA(θB).Togetherthesereactioncurvesdetermineequilibriumvaluesforthetwodisclosurethresholds. Weconsiderthecaseofuniformdistributions,andwithoutlossofgeneralitysetθA=θB=0,andθA=θB=1(arescalingandtranslation).By(6)and(11), ∗WDθi(αi,αi) DW αi−αi=D,W2αi−αi i=A,B.(22) WenowsolvefortheattentionparametersinarenaA(theα’s)intermsofthe ∗ tendencyfordisclosureinarenaB,asmeasuredbythethresholdinthatarena,θB.By(17),thisgivesthereactioncurvefortheinformedplayerinarenaA,andasimilarderivationgivesthecurveforarenaB: sA∗∗ (23)θA(θB)=∗ 1+2sA−sB+sBθB sB∗∗ .(24)θB(θA)=∗ 1+2sB−sA+sAθAFigure1showsreactioncurvesfordifferentparametervalues,andtheequilibriadeter-minedbytheintersectionsofthesecurves. 17 InsertFigure1Here ∗ Combiningequations(23)and(24)givestheequilibriumdisclosurethresholdθAasarootofthequadraticequation ∗2∗ sA(1+2sA−sB)θA+[(1+2sA−sB)(1+2sB−sA)+sB2−sA2]θA−sA(1+2sB−sA)=0. (25) ∗∗ Thisequationhasonlyonerootbetween0and1.Since0<θA<1,by(24),θBisalsobetween0and1.Wethereforehave: Proposition6Ifthetypesoftheinformedplayersintwoarenasaredistributeduni-formlyon[0,1],thenthereisauniqueequilibrium.Inthisequilibrium,thereispartialdisclosure. Example:Wesolvefortheequilibriumwhensalienceissymmetric(s≡sA=sB)andtypesareuniformon[0,1].Letthecommonthresholdvaluebeθ∗,andletsaliencebes.Ifs=0,then(25)reducestoalinearequationwithsolutionθ∗=0,implyingfulldisclosure.Intuitively,s=0causesαD=αW=.5,sothattheadvantagetoaverylowtypeofwithholdinginthehopeofnotbeingattendedtoisjustoffsetbytheadvantagetodisclosinginthehopeofnotbeingattendedto.Fors>0,θ∗isthepositivesolutionto(25), −(1+s)+(1+s)2+4s2∗ θ=.(26) 2s √∗∗ Foralls>0,θ>0,implyingpartialdisclosure.Ifs=1,θ=−1+2≈.414. ∗∗ By(23),dθA/dθB<0,sothereactioncurvesaredownwardsloping.Inotherwords,disclosuresinthetwoarenasarestrategicsubstitutes.Itfollowsthatregulationthatforcesmoredisclosureinonearenathanwouldhaveoccurredinequilibriumcrowdsoutdisclosureintheotherarena.Intuitively,reducedattentiontoBreducesthepressureonamarginalBplayertodisclose. Proposition7Supposethattheprobabilitydistributionsoftypesoftheinformedplayersinthetwoarenasareuniformon[0,1].Thenregulationthatforcesgreaterdisclosurein ∗ onearena(byreducingθiinarenai)causestheinformedplayerintheotherarenato ∗ discloselessinformation(i.e.,inarena∼ithedisclosurethresholdθ∼iincreases). 5.2Cross-ArenaContagionofNewsAnnouncements Thissubsectionexaminestheeffectofdisclosureversusnon-disclosureinarenaAontheexpectedperceptionbyobserversofthesignalvalueinafundamentally-unrelated 18 arenaB.Appliedinastockmarketsetting,theresultswederivedescribehowanannouncementaboutonestock,suchasanearningsforecastordividendannouncement,affectsthepriceofanotherstock. ˆB|DA,θ]andE[θˆB|WA],wheretheexpectationistakenwithThus,wecalculateE[θ respecttothepossibleoutcomeforθB.TheaverageperceptioninarenaBiftheinformed ˆW,is,inanalogytoequation(3),playerinarenaBwithholds,θB θB∗ f(θB)WW∗WW (1−αB)E[θB]+αBE[θB|θB<θB]=(1−αB)E[θB]+αBθBdθB.(27)∗F(θ)θBBIfhediscloses,thentheaverageperceptionis ˆD=(1−αD)E[θB]+αDθB.θBBB (28) TakingexpectationsoverthepossiblevaluesofθB,wefindtheexpectedperceptionby observersoftheinformedplayerinarenaBconditionalonthebehavioroftheinformedplayerinarenaA.Recallingournotationφ=Wor(D,θ),thisis θB∗ f(θB)ˆB|φA]=F(θ∗)(1−αW)E[θB]+αWE[θθdθBBBBB∗F(θ)θBB θB DD +(1−αB)E[θB]+αBθBf(θB)dθB. ∗θB DW=(αB−αB) ∗θB {E[θB]−θB}f(θB)dθB+E[θB]. (29) θBWD TheeffectofφAonperceptionsinarenaBcomesfromφA’seffectonαB,αB,and ∗ therebyonθB;attentioninarenaBdependsonthedisclosurechoiceandoutcomein WD arenaA.ToconditiononDA,wesubstituteαBfrom(17)andαBfrom(19)when∗θA=θ(implyingcertaintyofdisclosureinarenaA).Thisyields W αB(DA)=.5(1−sA)DαB(DA)=.5(1+sB−sA). (30) WD Similarly,toconditiononWA,wesubstituteαBfrom(17)andαBfrom(19)when∗θA=θ(implyingnodisclosureinarenaA).Thisyields W αB(WA)=.5DαB(WA)=.5(1+sB). (31) ˆB|DA]withE[θˆB|WA],weobtain:ComparingE[θ 19 Proposition8Disclosureinonearenacausestheexpectedperceptionofthetypeintheotherarenatoincrease. Thetwoarenashavenofundamentalrelationship(θAandθBareindependent).Cross-effectshereareinducedbytheattentionalrelationshipbetweenthetwoarenas.Moregenerally,iftherearemanyarenas,disclosureinanysinglearenamayhavelittleeffectonanotherarenaunlessthereissomekindofattentionallinkagebetweenthetwo.Theattentionallinkagecouldderivefromafundamentalrelationship,orcouldbeentirelysuperficial(aswithtwofirmswithsimilar-soundingnames).12 Whentherearemanyarenas,thecrosseffectbetweenattentionallyrelatedonesisprobablypositive.Insteadofdistracting,disclosureinagivenarenaprobablycallsfurtherattentiontoafewattentionallyrelatedarenas,whiledistractingslightlyfromalargenumberofmoredistantlyrelatedarenas.Thus,althoughweregardtheimplicationderivedhereregardingthecontagioneffectsofnewsannouncementsashavingconceptualinterest,empiricalpredictionsrelevantforstockmarketcontagionwillprobablyrequireanalysisofatleastthreearenas. 6 EffectsofRegulationonBeliefAccuracyandWel-fare Weexamineherehowdisclosureregulationaffectstheaccuracyofinvestorperceptions,thedegreeofoptimism,andwelfare.Ifgovernmentsimplymandatesdisclosureandthatmandateisalwaysobeyed,ineffectthedisclosurethresholdissetbelowθ.However,oftenamorerealisticdescriptionofthelegal/regulatoryenvironmentisthatonlypartialdisclosureisenforced,implyingathresholdatanintermediatevaluebetweenθandθ.Thethreatofliabilityforthefailuretodisclosecanencourageafirmtodoso(see,e.g.,Skinner(1994)),butmaynotenforcecompletedisclosure.Evenwhendisclosureisclearlymandated,firmsmaybalancetheriskofliabilityagainstpossiblebenefitsofwithholdingbadnewsfromthemarket.Afirmmayintentionallychoosetoviolateadisclosurerule.Insomecasestheremaybelegaluncertaintyastowhetherdisclosureoftheinformationitemismandatory.Firmswillbepressuredtodisclosemorebadnewswhenthereisahigherprobabilityoflegalliabilityfornon-disclosure,andwhenthe Rashes(2001)providesevidenceofstockmarketpricingerrorsbaseduponinvestorsbeingconfusedbysimilaritiesinnamesandtickersymbolsofdifferentstocks,aclearsymptomthatlimitedattentioncausesinappropriatecontagionacrossarenas. 12 20 expectedpenaltiesarehigher.Thus,thelegal/regulatoryenvironmentinvolvesdifferenteffectswhichcanbeviewedasadjustingthelevelofthethresholdindirectly. Weassumethatwelfareismeasuredbytheaccuracyofinvestorperceptions.Ifthereisonlyasinglearena,wedefinewelfareasthenegativeofthemeansquarederror, ˆ−θ)2],W≡−E[(θ (32) ˆistheaverageperceptionoftypeconditionalonthedisclosuredecision.whereθ Whentherearemultiplearenas,thevalueofbeliefaccuracywillgenerallyvarydependingonthearena’ssizeandcharacteristics.13WethereforeconsiderasocialwelfarefunctionthatallowsforunequalweightsonperceptionerrorsbetweenarenasAandB, ˆA−θA)2]−(1−λ)E[(θˆB−θB)2],W≡−λE[(θ (33) ˆAandθˆBaretheaverageperceptionsoftypeinthetwoarenasconditionalonwhereθ thedisclosuredecision,andλmeasurestherelativeimportanceofthetwoarenas.Webeginbydescribingtheeffectsofregulationwhenthereisonlyasinglearena.Proposition9Undertheassumptionsofthebasicmodel,ifthereisahigherprobabilitythatobserversattendtothedisclosedinformationthanthattheyattendtothefactthatinformationiswithheld,αD>αW,then: 1.Supposethatθ∗isimposedbyregulation(insteadofsatisfyingtheindifferencecondition(5)).Thenthemeansquarederroroftheaverageobserverperceptionasanestimateofthetruetypedecreases,andsocialwelfareincreaseswiththeamountofrequireddisclosure,i.e.,themeansquarederrorincreaseswithθ∗.2.Anexogenousdecreaseinthedisclosurethresholdreducesoptimismwhenθ∗ 13 21 Theaccuracyofobservers’beliefsincreaseswiththeamountofdisclosure.Considernowaregulationthatimposesalevelofdisclosureθ∗.Intuitively,higherθ∗(aweakerdisclosurerequirement)increasestheprobabilitythatobserversholdtotheirpriorbeliefs(sinceαD>αW)insteadofupdatingappropriately.Italsotendstomakeevenfullyrationalinferencesoftypemorenoisybyincreasingthesetoftypesthatarenotrevealed(seeAppendixB,ProofofProposition9,Part1). Whentherearemultiplearenas,wecanexaminetheeffectsofachangeinthedisclosureruleinoneofthearenas,orinbothsimultaneously. Proposition101.Ifthereisdisclosureintwoarenas,thenanexogenousincrease inthedisclosurethreshold(implyingalowerprobabilityofdisclosure)inonearenacausesaveragebeliefstobemoreaccurate(lowermean-squared-error)intheotherarenaintheneighborhoodofastableequilibrium.2.Anexogenousincreaseinthedisclosurethresholdinonearenareducesoptimismintheotherarenaintheneighborhoodofastableequilibrium.3.Forceddisclosureinonearenacanreducewelfarebydiscouragingdisclosureintheotherarena.4.Ifthereisdisclosureintwoarenas,thenasimultaneousexogenousincreaseinthe ∗∗ =θB)cancauseaveragebeliefsineachdisclosurethresholdinbotharenas(θ∗=θA ˆ−θ)2]/∂θ∗canbenegativeareatobecomeeithermoreorlessaccurate,i.e.,∂E[(θ orpositive.Thus,eventhoughtherearenocostsofdisclosure,forceddisclosureinbotharenascanreducewelfare.TheproofofParts1,2,and3areinAppendixB.Intuitively,thereisacrowdingouteffectofdisclosure.Forexample,ifregulatorsimposehigherdisclosureinarenaA,thenthisdistractsobserversfromarenaB.Thelowerattentiontoeitherdisclosureornon-disclosureinarenaBmakesbeliefsinarenaBlessaccurate,whichcanreducewelfare,especiallyifarenaBismoreimportantthanarenaA. Thissuggestsanimportantlimitationtodisclosurepolicy.Itisimpossibletoregulatedisclosureinallarenas.Evenifhighdisclosurewererequiredofallfirms,thereisprivateinformationinotherpartsofeconomicandsociallife.Theeffectofimposingdisclosurerequirementsinsomearenasbutnotothersmayberedirectingattentionratherthanimprovingtheaccuracyofperceptionsoverall.Thissuggeststhatarelevantinputforregulatorydecisionsistheimportanceofdifferentarenasforthedecisionsofobservers. 22 ToprovePart4,weconsiderthecaseofsymmetricsalienceandcalculatethemeansquarederrorofbeliefsasafunctionofthecommoncutoffθ∗.Whentheθi’saredistributeduniformlyon[0,1],bydirectcalculationthemeansquarederrorcanincreaseordecreasewiththecommoncutoffθ∗.Forexample,whensA=sB=0.99andθ∗=0.4, ˆ−θ)2]/∂θ∗≈−0.021,butwhensA=sB=0.1,thederivativeispositive(≈0.025).∂E[(θ Part3indicatesthatdisclosureinalessimportantarenamaydistractsomuchfromdisclosureintheimportantarenathatwelfaredeclines.ConsiderthesymmetriccasewheresA=sB=s.Usingtheattentionprobabilitiesgiveninequations(16)-(19),wecandifferentiatewelfarewithrespecttothedisclosurethresholdforA(holdingconstant∗θB);detailsareprovidedinAppendixB,proofofPart3.Usingthesymmetricsolution ∗ givenin(26),∂W/∂θAasafunctionofsandλisshowninFigure2. InsertFigure2Here ∗ AsseeninFigure2,∂W/∂θAispositivewhenthesaliencesishighandtheweight ∗ )candecreasewelfarewhenonarenaAislow.ForceddisclosureinarenaA(lowerθA theimportanceofarenaBishigh(lowerλ)andthesalienceishigh.AhighsalienceofdisclosureinarenaA,combinedwithgreaterdisclosureinA,makeswithholdingmoreattractiveinarenaB.Therefore,forceddisclosureinAcanhaveanegativeoveralleffectbydiscouragingdisclosureinthemoreimportantarena. InPart4therearecountervailingeffects.Ontheonehand,greaterdisclosureinonearenahasadirecttendencytoincreasetheprecisionofobserverbeliefsinthatarena.Ontheotherhand,itdistractsattentionfromtheotherarena,whichtendstoreduceprecisionthere.Overall,jointlyforcingincreaseddisclosurecanreducebeliefprecisionsandthereforewelfare,asillustratedinFigure3. ∗∗ LettingθA=θB≡θ∗,inFigure3∂W/∂θ∗ispositiveforhighvaluesofsandnegativeforlowvaluesofs(calculationsinAppendixB). InsertFigure3Here Intuitively,althoughforcingdisclosureinbotharenasdirectlyincreasestheamountofinformationavailable,italsomakespeoplelessattentivetoagivendatum.Ifsalienceishighenough,thesecondeffect(lowerattention)canoutweighthefirst(moreinforma-tion/disclosure). 23 7OptimalAllocationofAttention Evenboundedlyrationalobserversmay,througheffort,tryespeciallyhardtoattendtothosesignalsthatofferhighreturntoattention.14Thissectiongeneralizesthebasicsingle-arenamodelofSection3toallowindividualstodecideexantehowmuchattentiontodevotetoeitherdisclosedinformationorthestrategicimplicationsofnon-disclosure.Weexaminewhethertheequilibriuminasettingwithendogenousallocationofattentionhasimplicationssimilartothoseofthebasicmodel. ThefocusoftheanalysishereisonthefirstarenaA,butweincludeasecondarenaBtogiveindividualsanopportunitycostofattendingtoarenaA.IndividualsexantealsohaveachoicewithininarenaAastohowmuchattentiontodevotetodisclosedinformationversusthefailureofinformationtobedisclosed. Therearetwostagesinthemodel.Inthefirststageidenticalobserverschoose WD attentionprobabilitiesαA,αA,asdefinedearlier,andtheprobabilityofattendingtoasecondindependentarenaB,αB.Thischoiceisnotobservabletotheinformedplayer,althoughinequilibriumheknowswhatthechoicewillbe. Inthesecondstage,attentionoutcomesarerealized,soeachobservereitherattendsordoesnotattendtoeacharenaandformsbeliefsaccordingly.TheinformedplayerobserveshisprivatesignalaboutarenaA,anddecideswhetherornottodisclose.Atthispoint,thedecisionproblemoftheinformedplayerisidenticaltothatoftheinformed ∗ isdeterminedasinthebasicplayerinthebasicmodel.SothedisclosurethresholdθA WD modelasafunctionofαAandαA. Alsoatthesecondstage,eachobservermakesaprojectchoicebasedonhisbeliefsatthattime.Thus,atthefirststagetheindividualallocatesattentionsoastoincreasethequalityofhislaterprojectchoice.LetθAbeanobserver’snetpayofffromadoptingthearenaAproject.Weassumethatifheisindifferent,headoptstheproject.Thus,hewouldliketoadoptifandonlyifθA≥0.Foralgebraicsimplicity,weassumethatE[θA]=0,sothatanindividualwhoendsupnotattendingtoarenaAisjustwillingtoundertaketheproject.Similarresultsapplymoregenerally. Alsoforsimplicity,weassumethatthereisnodisclosuregameinarenaB.Instead,aninformationsignalbecomespublicspontaneouslyandwithcertainty.WethereforeassumethattheBcomponentoftheobservers’objectiveisequaltoαBK,whereK>0isaconstant.15 Conlisk(1996)considersthedecisionofaboundedlyrationaldecisionmakerofhowcarefullytodeliberatewhenmakingcorrectdecisionsiscostly.15 ItiseasytoderivethisifthereisaninvestmentprojectrelatedtoarenaBwithnetvaluekθB,where 14 24 ∗ Fromtheanalysisofthebasicmodel,θA ∗ toafailuretodisclosure,herejectstheprojectsinceE[θA|θA<θA]<0.Asdiscussedabove,wheneveranindividualfailstoattend(eithertoadisclosureortoafailuretodisclose),headopts.Thus,hisexpectedpayofffromhisarenaAprojectchoicesgiven DW hisattentionprobabilitiesαAandαAisD∗∗ΠA=(1−αA)Pr(θA>θA)E[θA|θA>θA] DW∗∗W +αAPr(θA>0)E[θA|θA>0]+(1−αA)Pr(θA<θA)E[θA|θA<θA]+αA·(0) θAθAθA∗DDW)=(1−αAθAf(θA)dθAθAf(θA)dθA+αAθAf(θA)dθA+(1−αA) = θA∗ θA θA∗ DθAf(θA)dθA−αA 0 0 ∗θA WθAf(θA)dθA+(1−αA) θA ∗θA θAf(θA)dθA (34) θAAddingtothistheexpectedprofitfromthearenaBproject,anobserver’soverall firststageoptimizationproblemis max θA D θAf(θA)dθA−αA 0 DαWA,αA,αB θA∗ ∗θA W θAf(θA)dθA+(1−αA) ∗θA θAf(θA)dθA+αBK, θAsubjecttotheattentionallocationconstraint WDG(αA,αA,αB)≤1, (35) whereG(·,·,·)isweaklyincreasingineachofitsarguments.IfarenaAiscomplex andnon-salient,arelativelylargereductioninαBmayberequiredtomaintaingivenlevelsoftheαA’s.Iftheindividualhasgoodcontroloverhisattention,theα’smaybehighlysubstitutable.Ontheotherhand,ifvividnessandsaliencegrabpeople’sattentionwithoutconsciousvolition,thedifferentα’smaybehighlycomplementaryinG,sothatagivenincreaseinoneoftheα’srequiresarelativelylargedecreaseinoneormoreoftheothers. WeconsideratractablefunctionalformforG, W2D2WD )+αB,,αA,αB)=σW(αA)+σD(αAG(αA (36) E[θB]=0andk>0.AttendingtoBallowsanobservertoobtainonaveragekE[θB|θB≥0]byinvesting whendoingsoisprofitable,insteadofalwaysinvestingforanaveragepayoffof0.Inthissetting,arenaBcontributesαBkE[θB|θB≥0]totheobserver’sexpectedprofits,sothatK=kE[θB|θB≥0]. 25 whereσW,σD,andσBareexogenousparameterswhichmeasuretheopportunitycostofdirectingattentiontoaparticulartarget.Substitutingout W2D2 αB=1−σW(αA)−σD(αA), WD wewritethefirstorderconditionsoftheoptimizationwithrespecttoαAandαA,supressingAsubscripts,as 0 0=(−θ)f(θ)dθ−2KσDαD θ∗θ∗ 0=(−θ)f(θ)dθ−2KσWαW, θ sothat αDαW 0 1=(−θ)f(θ)dθ 2KσDθ∗ θ∗ 1=(−θ)f(θ)dθ.W2Kσθ (37) Thedisclosurethresholdisdeterminedbyequations(5)and(6)ofthebasicmodel appliedtoarenaA.From(37),theratioαW/αDdoesnotdependonK.Therefore,wecanensurethatαB>0byselectingKsufficientlylargewithoutaffectingtheequilibriumdisclosurelevel.Also,forappropriatevaluesofparametersσWandσD,αD>αW,ensuringthattheequilibriuminvolvesonlypartialdisclosure.Differentiating(37)withrespecttoσWandσDrespectively: dαDdσWdαWdσD Sinceθ∗<0, dαdθ =−sign dσWdσWW∗dαdθsign=sign. dσDdσDsign D θ∗f(θ∗)dθ∗ = 2KσDdσWθ∗f(θ∗)dθ∗ =− 2KσWdσD. ∗ (38) Thethresholdvaluefordisclosuredependsontheattentionlevels,θ∗=θ∗(αW,αD),whereby(6)and(10), ∂θ∗∂θ∗ >0,<0.(39)∂αD∂αW 26 Takingthetotalderivativeofθ∗withrespecttotheexogenousparametersσWandσD, dθ∗∂θ∗dαD∂θ∗dαW =+dσW∂αDdσW∂αWdσWdθ∗∂θ∗dαD∂θ∗dαW =+DDDdσ∂αdσ∂αWdσD From(38)-(41), ∗W dθdαdαD sign=−sign=sign dσWdσWdσW ∗DW dαdθdαsign=sign=sign dσDdσDdσD Differentiating(37)withrespecttoσWandσDrespectively: dαW dσW 1 = 2K(σW)2 θ∗ (40)(41) (42) 1dαD =dσD2K(σD)2 Sinceθ<0, ∗ θ0 θ∗f(θ∗)dθ∗ θf(θ)dθ− 2KσWdσW (43) θ∗ θ∗f(θ∗)dθ∗ θf(θ)dθ+. 2KσDdσD 0 θ∗ θf(θ)dθ<0and θ∗ θ θf(θ)dθ<0,(42)and(43)implythat dθ∗ >0.dσW dθ∗ <0,dσD Thus,thecomparativestaticswhenindividualshaveanallocationchoicearesimilartothosewithexogenousα’s,asdescribedinSection4.2,Proposition2.SinceanincreaseinσDincreasestheopportunitycostofattendingtodisclosure,itcauseslessattentiontodisclosure,andalowerthreshold(moredisclosure).AnincreaseinσWmakesattend-ingtonon-disclosuremorecostly,whichreducesattentiontowithholding,andtherebyincreasesthedisclosurethreshold(lessdisclosure).Thus,variationinthecostofattend-ingtodifferentarenas(theσ’s)leadstocomparativestaticsontheamountofdisclosureessentiallyidenticaltothoseinthebasicmodelvaryingtheα’s. Toanalysismatchesthatofthebasicmodelevenmorecloselyintheextremespe-cialcaseofperfectcomplementaritybetweenbotharenasofattention,andbetweenwithholdinganddisclosing: WDW2D2G(αA,αA,αB)=max[σW(αA),σD(αA),σBαB]. (44) 27 Underperfectcomplementarity,itisoptimaltoallocateattentionsoastoequatethethreecomponentsinthemaximization, W2D2 σW(αA)=σD(αA)=σBαB=1; reallocatingattentionbydecreasingoneofthesecomponentswouldnotleadtoany correspondingincreaseintheothercomponents,itwouldmerelythrowawayattentionalresources.Thus,theoptimalattentionlevelsare αW=√1σW , 1 αD=√. σD (45) Eachoftheα’sin(45)isafunctiononlyofitscorrespondingexogenousσ.Itfollowsthatallpropositionsofthemodelwithexogenousattentionlevelsareconsistentwiththeper-fectcomplementaritycase,withcomparativestaticsonα’sinterpretedascorrespondingvariationsinσ’sasin(45). 8Conclusion Inthispaperwehavemodelledlimitedattentionasincompleteusageofpubliclyavail-ableinformation.Informedplayersdecidewhetherornottodiscloseinformationtoanaudienceofobserverswhosometimeneglecteitherdisclosedsignalsortheimplicationsofnon-disclosure.Inequilibrium,wefoundthatobserversareunrealisticallyoptimistic,disclosureisincomplete,neglectofdisclosedsignalsincreasesdisclosure,andneglectofafailuretodisclosereducesdisclosure.Wealsofoundthattheseinsightsextendtoasettinginwhichobserverschooseexantehowtoallocatetheirlimitedattention.Inasettingwithmultiplearenasofdisclosure,wefoundthatdisclosureinonearenaaffectsperceptionsinfundamentallyunrelatedarenas,owingtocuecompetition,salience,andanalyticalinterference;andthatdisclosureinonearenacancrowdoutdisclosureinanother. Wehavealsoconsideredtheimplicationsoflimitedattentionandtheresultingcredulityofinvestorsfordisclosureregulation.LawandregulationintheU.S.requirefirmstorevealinformationinfinancialreports,andtodiscloseotherrelevantinforma-tion.Thiscontrastswiththeappealinglysimplepolicyimplicationoftheclassicmodelsofdisclosure.Inthesemodels,rationalobservers,throughappropriateskepticism,inducefulldisclosure,andfulldisclosureisagoodthing.Inextensionswithcostlydisclosure,therecommendationisalmostassimple:forceadditionaldisclosureonlyifthesocialbenefitsexceedthecosts. 28 Incontrast,limitedattentionsuggeststhatthebalanceofconsiderationsismorecomplex,evenifdisclosureiscostless.Ontheonehand,informedpartiesmaycon-cealinformationinthehopeofexploitingtheinattentionandcredulityofobservers.Thisputsregulationofdisclosureonthetable.However,wefind,paradoxically,thatregulationsdesignedtoforcegreaterdisclosurecanmakeperceptionslessaccurate.Forexample,wefindthatforceddisclosureinonearenacancrowdoutdisclosureinanother,andtherebycanreducewelfare.Thus,evenifforceddisclosureinonearenacreatesben-efitstoobserversinthatarena,thereisnopresumptionthatforceddisclosureissociallydesirable. Furthermore,forcingsimultaneousdisclosureinmultiplearenascanalsoreducewel-fare,foratleasttworeasons.First,greaterforceddisclosurecanincreasewhatwecallanalyticalinterference,whereinadisclosedsignalinonearenadistractsobserversfromanalyzingthereasonsforafailureofaplayertodiscloseintheotherarena.Second,andthispointappliesevenifcompletedisclosureisenforcedinbotharenas,greaterdis-closurecancausegreatercuecompetitionbetweendisclosures—observershavetroublepayingattentiontobothsignals.Eventhoughmoreinformationispubliclyavailable,observerperceptionsmayonaveragebelessaccurate. Thus,todeterminewhetherforcinggreaterdisclosureinoneorinmanyarenaswillimprovewelfare,policymakershaveachallengingtask.Evaluatingsuchapolicyrequiresanassessmentoftherelativeimportanceofthedifferentarenas,theprecisionoftheinformationthatmightbedisclosed,andthesalienceofthedifferentarenas.Atleastthreesignificantconsiderationsnotcapturedinourmodeldeservefurtherexploration.First,salienceheredependsonlyonthearenaofdisclosure.Inpractice,extremesignalsmaybemoresalient.Thiseffectmayfurtherdiscouragedisclosure,asthecriticalminimumtypetodiscloseisbydefinitionatanoticeablelowerextremeifhediscloses. Second,theanalysisofmanyarenas(morethantwo)whentherearefundamentalaswellasattentionalrelationshipsbetweenarenasmayofferinsightsintocontagioninevaluationsofdifferentsecurities,productcategories,orpoliticalideas. Finally,aninformedpartymaymisrepresenthisinformationby‘disclosing’anincor-rectvalueforhisinformationsignal.ThisissueishighlightedbyrecentU.S.corporateaccountingscandalssuchasthoseinvolvingEnronandWorldCom.Limitstoinvestorattentionpresumablyaffecttheincentivesforfirmstoengageinfraudormildershadingofthetruth.Aninterestingfurtherdirectionwouldbetoanalyzehowdifferentregula-torypoliciesinfluencetheincentivesforfirmstoexploitinattentiveobserversthrough 29 misrepresentationaswellassimplynotdisclosing. Owingtothesimplicityofourmodellingapproach,itcaneasilybeadaptedtoanalyzetheeffectsoflimitedattentioninothersettingsinvolvingstrategicbehaviorandasymmetricinformation.Possibleapplicationsincludeequilibriuminsignallinggames,adverseselectioninmarkets,andthedesignofoptimalcontractsbetweenaprincipalandanagent.Webelievethatsuchaprogramforresearchisanalyticallytractable;whetheritwillbeempiricallyvalidatedremainstobeseen. Weclosebyemphasizingthatthegeneralapproachtolimitedattentionofferedheremaybeapplicabletoawiderangeofhumantransactions.AsarguedinAppendixA,therearevariousinteractionsinwhichinformedplayersseemtotakeadvantageofinattentiontomanipulatetheperceptionsofothers.Somepossibledirectionsthatmeritfurtherexplorationincludethereportingoffirms’financialconditiontoinvestors,theadvertisingofproductstoconsumers,thepresentationofinformationbypoliticalactivists,andthepresentationofpersonalinformationbyindividualsintheireverydaylives.Theapproachdescribedhere—inwhichobserversarelimitedintheirabilitytoattendtosignalsorfeaturesofthestrategicenvironment,salienceparametersinfluencewhichsignalsorenvironmentalfeaturesareattendedtomore,andoccurenceofaneventismoresalientthannon-occurence—maybehelpfulincapturingminimallytheeffectsoflimitedattentionintheseandothercontexts. 30 AppendixA:DoLimitedAttentionandCredulityAffectMarkets? Casualobservationsuggeststhatlimitedattentionandcredulityareimportant.Ratherthanfocusingondetailed,carefulanalysisofissues,politiciansandpoliticalpressuregroupsinvestheavilyin‘soundbites’or‘photoops’designedtounderscoreasimple,vividmessage.Relatedly,theidea‘rationalignorance’ofvotersisconsistentwithlimitstoattentionandprocessingpower.Manyproductadvertisementsaredesignedtoengageviewers’attentionandemotions,ratherthantopresentastrongdocumentarycaseinsupportofclaimsaboutqualityandprice.16Despitetheevidentpossibilityofinterestedmotives,conartistsrecurrentlyseducethefoolishwithget-rich-quickscams.Soataminimumthereisanextremetailofcredulousindividuals. Experimentalstudieshaveprovidedstrongevidencethatsubjectsfrequentlyfailtodiscountadequatelyforthemotivesofinterestedparties.Subjectsfailtoadjustproperlyforadverseselectionproblemsevenafternumerousrepetitions.17Inaproductmarketexperiment,Lynchetal(1991)findsomeevidenceofsellermisrepresentation,andthatinexperiencedbuyersoverpay.Somebuyersofferahighpricetothesamesellerimmedi-atelyafterbeing‘rippedoff’byalowqualityproduct.However,sincethesellerschoosetoproducealmostentirelylow-qualityproducts,buyernaiveteiseliminatedthroughexperience. Especiallygermaneforthecurrentpaperisevidencethatindividualsareinsuffi-cientlyskepticalaboutfirms’motivesforrefrainingfromdisclosinginformation.IntheexperimentsofForsythe,Isaac,andPalfrey(1989)andKingandWallin(1991),thesellerofanassetcanchoosewhetherornottodisclosetheasset’svalue;anydisclosuremustbetruthful.Forsythe,IsaacandPalfreyfindthatinearlyexperimentsbuyersoverpayinthe‘blindbid’auctionsthatresultfromasellerchoosingnottodisclose;andthatsellersprofitablyexploitedthisbuyernaivete.Withmanyrepetitionstherewasgradualunravellingtowardfullerdisclosure.However,intheversionoftheKingandWallin(1991)experimentsinwhichtherationalequilibriumentailsfulldisclosure,neitherfulldisclosurenorextremeskepticismwascloselyapproached. HansonandKysar(1999)reviewevidencefromtheconsumermarketingandconsumerpsychologyliteratures,which,theyclaim,indicatesthatsellerssuccessfullymanipulateconsumerperceptionsoftheirproducts.17 Foranexcellentreviewofthisevidence,seeKagelandLevin(2002)ch.1.D.Forexample,buyersdonotadjustproperlyforthewinner’scurseinbilateralbargaininggames(SamuelsonandBazerman(1985),Ball,Bazerman,andCaroll(1991),andHoltandSherman(1995)).Ball,BazermanandCarollfindthatveryfewsubjectslearnthroughexperiencetoadjustproperly. 16 31 Forsythe,Lundholm,andRietz(1999)findinanexperimentalassetmarketthatwhenthereisnocommunication,buyerspaymorefortheassetthanthevalueofthelowest-qualityasset,anapparentfailureofbuyerstotakeadverseselectionfullyintoaccount.Furthermore,whentherules(knowntoall)allowsellerstoengagein“cheaptalk”(fraudulentclaims)aboutthequalityofthesecurity,buyersoftenoverpay.Underthe“antifraud”regimeinwhichsellerscandiscloseasetofqualitiesthatincludesthetruequality,outcomesareclosertothefulldisclosureequilibrium,butbuyersarestillsometimesinsufficientlyskeptical. Thereisalsoevidenceoflimitedattentioninpractice.Mathios(2000)examinedtheeffectoftheNutritionLabelingandEducationActonpurchasesofsaladdressing.Thislegislationmandatedthelabellingofinformationaboutfatcontent.Hefoundthateventhoughtherewasvoluntarylabeling(mostlyoflow-fatbrands)priortotheregulation,afterdisclosurewasmandatedfattierdressingslostmarketshare.Ifconsumershadbeenhighlyattentiveandskeptical,withoutregulationtheywouldhavealreadyinferredahighfatcontentamongnon-disclosingproducts. Thereisevidencethatthereannouncedgainsfromdebt-equityswapsinquarterlyearningsannouncementsweresignificantlyrelatedtomeanabnormalreturns(Hand(1990)).Amir(1993)foundthatfootnotedisclosureofpost-retirementbenefitswasunderweightedbyinvestorsuntilthepolicydiscussionsleadinguptoSFAS106,whichmadethelong-termcostsofthesebenefitsmoresalient.InAboody(1996),stockpricesintheoilandgasindustryweightedwrite-downinformationmoreheavilywhenitwasexplicitlyrecognizedaspartofearningsthanwhenitwasmerelydisclosedtoinvestorsseparately.SchrandandWalther(2000)provideevidencethatmanagersselecttheformoftheprior-periodearningsbenchmarkwhenannouncingearnings.Priorperiodgainsweremorelikelytobeannouncedthanpriorperiodlossesinthesample,apparentlytolowerthebenchmarkforcurrent-periodevaluation. Financialmarketsalsoprovideindirectevidenceoflimitedattentionandexcessivecredulity.Investorsmaketradesbaseduponanonymousinternetchat,sothatcheaptalkappearstomoveprices.18Rationalinvestorsshouldunderstandthatanonymousin-ternetchatcommentsmaybedesignedtomisleadothersinordertomoveprices.Shiller Thiscreatesopportunitiesfor‘pumpanddump’manipulationstrategieswhereinanindividualbuysthestock,startsrumorstodriveuptheprice,andthensells.Inonerecentcasea14-year-oldspreadfavorablerumorsaboutthinlytradingstocksusingnumerousfictitiousnames,andimmediatelysoldthestocks(Bloomberg,9/21/00).TheSECallegedthathemade$272,826inillegalprofitsonstockshetoutedandsold,thoughhistotalgainswereseveraltimeslarger. 18 32 (2000)describesseveralepisodesindicativeofinvestornaivete.19AbodyofevidencereviewedinDaniel,Hirshleifer,andTeoh(2002)suggeststhatcredulityofinvestorsaboutthemotivesoffirms,analysts,andbrokersexplainsseveralgeneralpatternsininvestortradingandcapitalmarketprices.20Suchevidenceissurprisingsincemispricinginsecuritiesmarketsofferspotentialprofitopportunitiestosmarttraders.Somehavearguedthatinattentiontoinformationaboutdeferredexecutivestockoptioncompen-sationcontributedtotheinternetbubbleofthe1990s.21 Alsoindicativeoflimitedattentionisevidencethatthestockmarketreactstore-publicationofnewsthatisalreadypubliclyavailable(seeHoandMichaely(1988),HubermanandRegev(2001)),andtocorporatenamechanges(Cooper,Dimitrov,andRau(2001));andreactsmorestronglytopublicnewsannouncementsiftheoutletisTheNewYorkTimes(Klibanoff,Lamont,andWizman(1999)).Grinblatt,Masulis,andTitman(1984)proposeaninvestorattentionhypothesistoexplainthepositiveaveragemarketpricereactiontostocksplits(seeminglyanirrelevantchangeinunits).Indi-vidualsparticipateinonlyasubsetofstocks,whichmaybeaconsequenceoflimitedattention.Consistentwithinvestorsformingimperfectlydiversifiedportfolios,thereisamarketpremiumforsecurity-specificrisk(seeBessembinder(1992)andMalkielandXu(2000)).Finally,BarberandOdean(2001)findthatindividualinvestorstendtobepurchasersofstockon‘highattentiondays’–daysinwhichabnormalnewsaboutafirmisannounced,daysfollowingextremepricemoves,anddaysinwhichastockhashighabnormaltradingvolume. AppendixB:Proofs ProofofProposition1:WenowshowthatifαW<αDthereisnoequilibriumwithfulldisclosure.Themostskepticalinferenceinanyequilibriumthatcouldbedrawn Theseincludethebizarreclaimsandself-promotionmethodsusedbythefinancialforecasterJosephGranville,whosepronouncementscausedlargemovementsintheDowJonesIndustrialAverage.ShillerdescribesthehistoryofPonzischemes,suchasthe1996-7episodeinAlbania,whichledtoriotingandtheresignationofthePrimeMinisterandhiscabinet.Morebroadly,Shillerarguesthatshiftsinthefocusofpublicattentioncausestockmarketboomsandovershooting.20 Thesepatternsincludeafailureofmarketpricestodiscountadequatelyforincentivesoffirmstomanageearnings(see,e.g.,Sloan(1996),Teoh,Welch,andWong(1998)),andtheapparentfailureofthemarkettodiscountfullyfortheincentiveoffirmswithprivateinformationtosellshareswhenthesesharesareovervaluedbythemarket(seeLoughranandRitter(1995)),andtobuyshareswhensharesareundervalued(seeIkenberry,Lakonishok,andVermaelen(1995)).21 Thereisevidencethatmarketpricesfailtodiscountsufficientlyforthedilutioninvalueimplicitinoutstandingmanagerialstockoptions.Thepubliclyavailableinformationthatafirmhasalargeoverhangofexecutivestockoptioncompensationisanegativepredictorofsubsequentabnormalstockre-turns(GarveyandMilbourn(2001)).Thisillustratesbothlimitedattention,andalsoperhapscredulityonthepartofinvestorsabouttheincentiveofmanagerstomanageearningsupward. 19 33 ˆ=θ.Thus,observers’perceptionofawithholdingaboutnon-disclosurewouldbeθ playersatisfies ˆW≥(1−αW)E[θ]+αWθ.θ Theperceptionupondisclosingforatypeθ=θ+is ˆD=(1−αD)E[θ]+αD(θ+).θ Thus, ˆW−θˆD≥(αD−αW)(E[θ]−θ)−αDθ >0, wherethelastinequalityholds,forgivenαWandαD,bychoosingtosatisfy (αD−αW)(E[θ]−θ)0<<. αD Thus,inanyequilibriumthereexistsasetoftypeswithθ<θ+,smallthatprefernottodisclose.Furthermore,solongasγ<1,θ∗ ProofofProposition3: θ∗θˆ−θ]={(1−αW)E[θ]+αWE[θ|θ<θ∗]−θ}f(θ)dθ+E[θ{(1−αD)E[θ]+αDθ−θ}f(θ)dθ θ (46) (47) (48) ={(1−αW)E[θ]+αWE[θ|θ<θ∗]}F(θ∗)− θθ∗ θ∗ θf(θ)dθ θθ∗ +(1−αD)E[θ][1−F(θ∗)]−(1−αD)E[θ]− θf(θ)dθ ={(1−αW)E[θ]+αWE[θ|θ<θ∗]}F(θ∗)−E[θ|θ<θ∗]F(θ∗) +(1−αD)E[θ][1−F(θ∗)]−(1−αD){E[θ]−E[θ|θ<θ∗]F(θ∗)}=(αD−αW)F(θ∗)(E[θ]−E[θ|θ<θ∗]).ˆ−θ]>0.SinceαD>αWandE[θ]>E[θ|θ<θ∗],E[θProofofProposition4: ∗ˆ−θ]dE[θ∗∗DW∗∗dθ=−(E[θ]−E[θ|θ<θ])F(θ)+(α−α)f(θ)(E[θ]−θ)W dαWdα ∗ˆ−θ]dE[θ∗∗DW∗∗dθ=(E[θ]−E[θ|θ<θ])F(θ)+(α−α)f(θ)(E[θ]−θ)D. dαDdα (49) 34 SinceE[θ]>E[θ|θ<θ∗],αD>αW,E[θ]>θ∗,dθ∗/dαW<0anddθ∗/dαD>0,ˆ−θ]/dαW<0andE[θˆ−θ]/dαD>0E[θ ProofofProposition5:By(3)–(4),wecanrewritetheMSEas θ∗ 2 DD∗2ˆE[(θ−θ)]=(1−α)E[θ]+αθ−θf(θ)dθ + θθ 2 (1−α)(E[θ]−θ)f(θ)dθ. D (50) θ∗ ˆ−θ)2]onlythroughθ∗,SinceαWaffectsE[(θ ˆ−θ)2]ˆ−θ)2]dθ∗dE[(θdE[(θ =, dαWdθ∗dαW where ˆ−θ)2]D∗dE[(θD∗D∗ =2αF(θ)αθ+(1−α)E[θ]−E[θ|θ<θ] dθ∗ >0. Thus, Sign By(6)and(10), ˆ−θ)2]dE[(θdαW =Sign dθ∗dαW (51) (52) . dθ∗/dγdθ∗ =−<0dαWαD forastableequilibrium. ToseehowthemeansquarederrorvarieswithαD,substituteθ∗oftheuniform[0,1]distributioncasefrom(11)intotheMSEformula,(12),anddifferentiatingwithrespecttoαD.Thisyields D2ˆdE[(θ−θ)]α 8(1−αD)4−3(αW)2(1−αW)2−8(1−αD)3(3−2αW)≡DDW4dα6(2α−α) +4(1−αD)26−8αW+3(αW)2−(1−αD)8−16αW+21(αW)2−6(αW)3,ˆ−θ)2]/dαD=−0.0354938when(αW,αD)=whichcanbeeitherpositiveornegative:dE[(θ ˆ−θ)2]/dαD=0.000708395when(αW,αD)=(0.4,0.95).(0.4,0.5),anddE[(θ ∗∗ ProofofProposition8:ByProposition1,E[θB]>θB>0.Therefore,ifθB<θB,E[θB]−θB>0,whichimpliesthattheintegralontheRHSofthefinalequationin(29) ∗ isincreasingwithθB.Furthermore,by(30)and(31),thecoefficientonthisintegral, DWDW (DA)−αB(DA)=αB(WA)−αB(WA)=.5sB,αB 35 doesnotdependonφA.Thus,substitutingφA=WAorDAinto(29),weseethatˆB|DA]>E[θˆB|WA]ifandonlyifθ∗(DA)>θ∗(WA).LetγB=(αD−αW)/(αD).E[θBBBBBSince γB(DA)=sB/(1+sB−sA)>sB/(1+sB)=γB(WA), ∗∗ˆB|DA]>E[θˆB|WA].θB(DA)>θB(WA).ThusE[θ ProofofProposition9:Part1: ˆ−θ)2]E[(θ ==+ θθ∗ (1−α)E[θ]+αE[θ|θ<θ]−θ 2 θ θθ∗ WW∗ 2 f(θ)dθ+ θ 2 (1−αD)(E[θ]−θ)f(θ)dθ θθ∗ θ∗ (1−αW)E[θ]+αWE[θ|θ<θ∗] θ∗ f(θ)dθ−2(1−αW)E[θ]+αWE[θ|θ<θ∗] θf(θ)dθ (53) θ2f(θ)dθ+ 2 (1−αD)(E[θ]−θ)f(θ)dθ. θθ∗ ˆ−θ)2]W∂E[θ|θ<θ∗]∂E[(θWW∗ =2(1−α)E[θ]+αE[θ|θ<θ]αF(θ∗)∗∗∂θ∂θ ∗θ∗∂E[θ|θ<θ]2 θf(θ)dθ+(1−αW)E[θ]+αWE[θ|θ<θ∗]f(θ∗)−2αW ∂θ∗θ −2(1−αW)E[θ]+αWE[θ|θ<θ∗]θ∗f(θ∗) 2 ∗2∗D∗ +(θ)f(θ)−(1−α)(E[θ]−θ)f(θ∗).(54)SinceαW<αD,by f(θ∗)∗∂E[θ|θ<θ∗]∗ =(θ−E[θ|θ<θ])and ∂θ∗F(θ∗)θ∗ θf(θ)dθ=E[θ|θ<θ∗]F(θ∗), θ ˆ−θ)2]∂E[(θ∗W2∗∗2W2D2∗2 =f(θ)[1−(1−α)](θ−E[θ|θ<θ])+[(1−α)−(1−α)](θ−E[θ])>0. ∂θ∗Part2: ˆ−θ]=(αD−αW)F(θ∗)(E[θ]−E[θ|θ<θ∗])E[θˆ−θ]∂E[θ =(αD−αW)f(θ∗)(E[θ]−θ∗)>0,whenE[θ]>θ∗.∗∂θ 36 (55) ProofofProposition10,Part1:Themeansquarederrorinarenaiis θi∗ 22WW∗ˆE[(θi−θi)]=(1−αi)E[θi]+αiE[θi|θi<θi]−θif(θi)dθi θi+ θi (1− ∗θi D αi)(E[θi] 2 −θi)f(θi)dθi, (56) WD∗∗ whereαi,αi,andθiarefunctionsofθ∼i,thecriticalvalueintheotherarena.The ∗ derivativeofmeansquarederrorinarenaiwithrespecttoθ∼icanbewrittenas: Wˆi−θi)2]∂αi∂E[(θW∗2∗ =−2(1−α)(E[θ]−E[θ|θ<θ])Pr[θ<θ]iiiiiii∗∗∂θ∼∂θi∼i θi D∂αiD −2∗(1−αi)(E[θi]−θi)2f(θi)dθi ∗∂θ∼iθi ∗∂θi∗WW∗∗2+f(θ)α(2−α)(θ−E[θ|θ<θ])iiiiiii∗∂θ∼i ∗∂θi∗DWDW∗2+f(θ)(α−α)(2−α−α)(θ−E[θ]).iiiiiii∗∂θ∼i (57) WD∗ Byinspectionofequations(16)-(19),thederivativesofαiandαiwithrespecttoθ∼iare WD∂αi∂αi∗ ==0.5sf(θ(58)∼i∼i)>0.∗∗ ∂θ∼∂θi∼i Also,∗∗DW∗ ∂θi∂θi∂γi∂θiα−αii∗ ==−0.5s∼if(θ∼<0.(59)i)∗∗D2 ∂θ∼i∂γi∂θ∼i∂γi(αi) ∗ Therefore,intheneighborhoodofastableequilibrium(∂θi/∂γi>0),themeansquarederrorofarenaidecreaseswiththethresholdoftheotherarena∼i, ˆi−θi)2]∂E[(θ <0.∗∂θ∼i ProofofProposition10,Part2: DWˆ∂E[θi−θi]∂αi∂αi∗∗=−F(θi)(E[θi]−E[θi|θi<θi])∗∗∗∂θ∼i∂θ∼i∂θ∼i ∗ DW∗∗∂θi+(αi−αi)f(θi)(E[θi]−θi)∗∂θ∼i∗ DW∗∗∂θi =(αi−αi)f(θi)(E[θi]−θi)∗<0 ∂θ∼i 37 (60) Thelastlinefollowsfromequations(58)and(59). ProofofProposition10,Part3:ToprovePart3,wedifferentiatethewelfare ∗ functionwithrespecttoθAwhenθAandθBaredistributeduniformlyover[0,1].Theattentionprobabilitiesaregiveninequations(16)-(19). 1∂W∗2=(−9λ(θ)A∗∂θA48 ∗3∗∗3∗2∗2∗ −2s−1+(θB)+λ4−12θA−(θB)+9(θA)+3(θA)θB ∗∗∗∗∗∗∗∗2 +s2{2(θB−1)[3(θB−1)θB+θA+θAθB+θA(θB)]∗2∗3∗222∗+λ[−3+12(θB)−6(θB)−9(θA)+18(θA)θB∗3∗∗∗∗22∗2 )]}).(θB)(θB)+14θA−24θAθB−2θA+3(θA (61) ∗ Usingthesymmetricsolutiongivenin(26),∂W/∂θA=0.02244when(s,λ)=(0.8,0.2),anditbecomes−0.01663when(s,λ)=(0.8,0.9). CalculationsUnderlyingFigure3:WedifferentiatethewelfareWwithrespecttothecommonthresholdθ∗whereθAandθBaredistributeduniformlyover[0,1] ∂W1∗2∗∗2∗32∗∗2∗3∗4 =−9θ−2s[2−12θ+9(θ)+4(θ)]+s[−3+22θ−45(θ)+24(θ)+5(θ)].∂θ∗48Thederivative∂W/∂θ∗ispositiveforhighvalueandnegativeforlowvaluesofs.Forexample,∂W/∂θ∗=0.0114whens=0.8,and∂W/∂θ∗=−0.0079whens=0.2. 38 References Aboody,D.,1996,Recognitionversusdisclosureintheoilandgasindustry,JournalofAccountingResearch34,21–32.Amir,E.,1993,Themarketvaluationofaccountinginformation:thecaseofpostre-tirementbenefitsotherthanpensions,AccountingReview68,703–724.Ball,S.B.,M.H.Bazerman,andJ.S.Caroll,1991,Anevaluationoflearninginthebilateralwinner’scurse,OrganizationalBehaviorandHumanDecisionPro-cesses48,1–22.Barber,B.andT.Odean,2001,Allthatglitters:Theeffectofattentionandnewsonthebuyingbehaviorofindividualandinstitutionalinvestors,UniversityofCalifornia,Davis.Bessembinder,H.,1992,Systematicrisk,hedgingpressure,andriskpremiumsinfuturesmarkets,ReviewofFinancialStudies5,637–67.Boot,A.W.A.andA.V.Thakor,2001,Themanyfacesofinformationdisclosure,ReviewofFinancialStudies14,1021–1057.Conlisk,J.,1996,Whyboundedrationality?,JournalofEconomicLiterature34,669–700.Cooper,M.J.,O.Dimitrov,andP.R.Rau,2001,ARose.combyanyothername,JournalofFinance56,2371–2388.Daniel,K.D.,D.Hirshleifer,andA.Subrahmanyam,2001,Overconfidence,arbitrage,andequilibriumassetpricing,JournalofFinance56,921–965.Daniel,K.D.,D.Hirshleifer,andS.H.Teoh,2002,Investorpsychologyincapitalmarkets:Evidenceandpolicyimplications,JournalofMonetaryEconomics49(1)/CarnegieRochesterSeriesinPublicPolicy56,139–209.Darrough,M.andN.Stoughton,1990,Financialdisclosurepolicyinanentrygame,JournalofAccountingandEconomics12,219–244.Fischoff,B.,P.Slovic,andS.Lichtenstein,1978,Faulttrees:Sensitivityofestimatedfailureprobabilitiestoproblemrepresentation,JournalofExperimentalPsychol-ogy:HumanPerceptionandPerformance4,330–334.Fishman,M.andK.Hagerty,1989,Disclosuredecisionsbyfirmsandthecompetitionforpriceefficiency,JournalofFinance44,633–646.Fishman,M.J.andK.M.Hagerty,2003,Mandatoryvs.voluntarydisclosureinmarketswithinformedanduninformedcustomers,JournalofLaw.forthcoming.Fiske,S.andS.Taylor,1991,SocialCognition(2nded.).(McGraw-Hill,NewYork).Fiske,S.T.,1995,.Socialcognition,.InAdvancedSocialPsychology,pp.149–194.(McGraw-Hill,NewYork).Forsythe,R.,R.M.Isaac,andT.R.Palfrey,1989,Theoriesandtestsof‘blindbidding’insealed-bidauctions,RandJournalofEconomics20,214–238. 39 Forsythe,R.,R.Lundholm,andT.Rietz,1999,Cheaptalk,fraud,andadversese-lectioninfinancialmarkets:Someexperimentalevidence,ReviewofFinancialStudies12,481–518.Garvey,G.T.andT.T.Milbourn,2001,Dostockpricesincorporatethepotentialdilutionofemployeestockoptions?,Workingpaper,WashingtonUniversity,St.Louis.Gifford,S.,1992,Allocationofentrepreneurialattention,JournalofEconomicBe-haviorandOrganization19,265–284.Gifford,S.andC.Wilson,1995,Amodelofprojectevaluationwithlimitedattention,EconomicTheory5,67–78.Grinblatt,M.,R.W.Masulis,andS.Titman,1984,Thevaluationeffectsofstocksplitsandstockdividends,JournalofFinancialEconomics13,97–112.Grossman,S.J.,1981,Theinformationalroleofwarrantiesandprivatedisclosureaboutproductquality,JournalofLawandEconomics24,461–483.Hand,J.R.M.,1990,Atestoftheextendedfunctionalfixationhypothesis,AccountingReview65,740–63.Hanson,J.D.andD.A.Kysar,1999,Takingbehavioralismseriously:Someevidenceofmarketmanipulation,HarvardLawReview112,1420–1572.Hirshleifer,D.andS.H.Teoh,2002,Limitedattention,informationdisclosure,andfinancialreporting,FisherCollegeofBusiness,TheOhioStateUniversity.Hirst,D.andP.E.Hopkins,1998,Comprehensiveincomereportingandanalysts’valuationjudgments,JournalofAccountingResearch36,47–75.Supplement.Ho,T.andR.Michaely,1988,Informationqualityandmarketefficiency,JournalofFinancialandQuantitativeAnalysis5,357–386.Holt,C.A.andR.Sherman,1995,Naivebiddingandthewinner’scurseinauctionswithindependentcommonvaluecomponents,U.Virginia,DiscussionPaper.Huberman,G.andT.Regev,2001,Contagiousspeculationandacureforcancer,JournalofFinance56,387–396.Ikenberry,D.,J.Lakonishok,andT.Vermaelen,1995,Marketunderreactiontoopenmarketsharerepurchases,JournalofFinancialEconomics39,181–208.Jovanovic,B.,1982,Truthfuldisclosureofinformation,BellJournalofEconomics13,36–44.Kagel,J.H.andD.Levin,2002,.Biddingincommon-valueauctions:Asurveyofexperimentalresearch,.InJ.KagelandD.Levin,eds.,CommonValueAuctionsandtheWinner’sCurse,Chapter1.(PrincetonUniversityPress,Princeton,NewJersey).forthcoming.Kahneman,D.,1973,AttentionandEffort(Prentice-Hall,EnglewoodCliffs,NewJersey). 40 Kahneman,D.andA.Tversky,1973,Onthepsychologyofprediction,PsychologicalReview80,237–25l.Kennedy,S.R.andK.M.Dimick,1987,Careermaturityandprofessionalsportsexpectationsofcollegefootballandbasketballplayer,JournalofCollegeStudentDevelopment28,293–297.King,R.R.andD.E.Wallin,1991,Voluntarydisclosureswhenseller’slevelofinfor-mationisunknown,JournalofAccountingResearch29,96–108.Klibanoff,P.,O.Lamont,andT.A.Wizman,1999,Investorreactiontosalientnewsinclosed-endcountryfunds,JournalofFinance53,673–699.Kruschke,J.K.andM.K.Johansen,1999,Amodelofprobabilisticcategorylearning,JournalofExperimentalPsychology:Learning,Memory,andCognition25,1083–1119.Loughran,T.andJ.Ritter,1995,Thenewissuespuzzle,TheJournalofFinance50,23–52.Lynch,M.,R.M.Miller,C.R.Plott,andR.Porter,1991,.Productquality,informa-tionalefficiency,andregulationsinexperimentalmarkets,.InR.M.Isaac,ed.,ResearchinExperimentalEconomics,Volume4.(JAIPress,Greenwich,Connecti-cut).Malkiel,B.G.andY.Xu,2000,Idiosyncraticriskandsecurityreturns,SchoolofManagement,UniversityofTexasatDallas.March,J.andH.Simon,1958,Organizations(Wiley,NewYork,NY). Mathios,A.D.,2000,Theimpactofmandatorydisclosurelawsonproductchoices:Ananalysisofthesaladdressingmarket,JournalofLawandEconomics43,651–677.Mayer,J.,2002,Theaccountants’war:ArthurLevittwantedtoreinintheauditors.buthedidn’treckononcongress,TheNewYorker,April22&29.Milgrom,P.andJ.Roberts,1986,Relyingontheinformationofinterestedparties,RANDJournalofEconomics17,18–32.Milgrom,P.R.,1981,Goodnewsandbadnews:Representationtheoremsandappli-cations,BellJournalofEconomics12,380–391.Newman,J.,W.Wolff,andE.Hearst,1980,Thefeature-positiveeffectinadulthumansubjects,JournalofExperimentalPsychology:HumanLearningandMemory6,630–650.Nisbett,R.andL.Ross,1980,HumanInference:StrategiesandShortcomingsofSocialJudgment(Prentice-Hall,EnglewoodCliffs,NJ).Radner,R.,1975,Abehavioralmodelofcostreduction,RandJournalofEconomics6,196–215.Radner,R.andM.Rothschild,1975,Ontheallocationofeffort,JournalofEconomicTheory10,358–376. 41 4B10.80.6RCA00.4RCA1RCA2RCB10.2RCB0RCB24A 0.20.40.60.810.04W0.02cccccccccccc4A 0-0.020.20.4s0.60.8100.210.80.60.4OFigure2:WelfareeffectofexogenousincreaseinthedisclosurethresholdinarenaAThegraphshowsthederivativeofwelfarewithrespecttoanexogenousincreaseinthedisclosurethresholdinarenaA,asafunctionofthecommonsalienceofdisclosures,andtheweightλonarenaAinthesocialwelfarefunction. 0.02W0.01ccccccccc4 00.60.20.4s0.60.8100.20.4O10.8Figure3:WelfareeffectofexogenousincreaseinthecommondisclosurethresholdinarenasAandB ThegraphshowsthederivativeofwelfarewithrespecttoanexogenousincreaseinthecommondisclosurethresholdforarenasAandB,asafunctionofthecommonsalienceofdisclosures,andtheweightλonarenaAinthesocialwelfarefunction. 因篇幅问题不能全部显示，请点此查看更多更全内容