DiamondandDybvig’sClassicTheoryofFinancialIntermediation:What’sMissing?
EdwardJ.GreenSeniorResearchOfficerResearchDepartment
FederalReserveBankofMinneapolisandAdvisor
FinancialMarketsand
PaymentsSystemRiskDepartmentFederalReserveBankofChicago
PingLin
AssistantProfessor
DepartmentofEconomicsLingnanUniversityHongKong
Abstract
Thearticleshowsthatinafinite-traderversionoftheDiamondandDybvigmodel(1983),theexanteefficientallocationcanbeimplementedasauniqueequilibrium.Thisissoeveninthepresenceofthesequentialserviceconstraint,asemphasizedbyWallace(1988),wherebythebankmustsolveasequenceofmaximizationproblemsasdepositorscontactitatdifferenttimes.Athree-traderexamplewithconstantrelativerisk-aversionutilityisusedinordertoillustrateclearlytherequirementsthatthesequentialserviceconstraintimposesonimplementation.
TheviewsexpressedhereinarethoseoftheauthorsandnotnecessarilythoseoftheFederalReserveBankofMinneapolisortheFederalReserveSystem.
Historyisrepletewithinstancesinwhichaseeminglyhealthyeconomyhasplungedintodifficulty,investorshavebecomesuddenlyinsistentonexercisingcontrac-tualoptionstomitigatetheirindividualrisks,andfinan-cialintermediarieshaveconsequentlybecomeunabletohonoralltheircommitments.Todesigngoodpolicytopreventormitigatesuchcrises,economicpolicymakersneedtomakeajudgmentaboutcausality.Isfinancialin-termediation’sstructureunstableandthuscausingthebroadereconomicdifficulty?Oristheobservedinstabil-ityamongfinancialintermediariesmerelyreflectingthatbroadereconomicdifficulty?DouglasDiamondandPhilipDybvig(1983;reprintedinthisissueoftheQuar-terlyReview)haveprovidedaclassictheorytoformalizethefirstpossibility,theideathatfinancial1intermedia-tion’sstructurecauseseconomiccrises.WedemonstrateherethatDiamondandDybvig’stheoryisincomplete,sothatthesecondpossibilityremainsaliveone.Furtherworkisneededtodeterminewhat’smissinginDiamondandDybvig’stheoryandultimatelytoprovidepolicy-makerswithabetterunderstandingoffinancialinterme-diation.
AsNeilWallace(1988,pp.4,8–9)pointsout,theenvironmentDiamondandDybvigconsiderhasfourkeyingredientsofanactualbankingsystem:uncertain-tyaboutpeople’spreferencesforexpenditurestreams,whichproducesdemandforliquidassets;privacyofin-formationaboutthesepreferencesaftertheyhavebeenrealized(informationaboutpeople’stypes,whethertheyarepatientorimpatienttoconsume);asequentialserviceconstraint,orarulethatspendingbydifferentpeoplemustoccursuccessively;andrealinvestmentprojectsthatarecostlytorestartiftheyareinterrupted.DiamondandDybvigarguethatwelfare-maximizingagentsinsuchanenvironmentwillselectabankingarrangementthatresemblesademanddepositcontractwhichcanim-plementtheefficientallocationinanequilibrium.More-over,unlesstheeconomyalsohaseitherdepositinsur-anceorasuspensionofpaymentscontingencysuchasexisteduntilthe1930sintheUnitedStates,financialin-termediationviademanddepositcontractswillhaveabankrunequilibrium.ThatisthesenseinwhichDia-mondandDybvigthinkthatfinancialintermediation’sinstabilitycanbeacause,ratherthanmerelyasideef-fect,ofbroadeconomiccrises.
DiamondandDybvigmaketheirfundamentalpointinabenchmarkmodelwhichhasnoaggregateuncertaintyaboutthenumberofagentswhoareimpatienttocon-sume(andthuswanttowithdrawtheirdepositsearly).DiamondandDybvigthenshowthatasuspensionofpaymentsschemecaneliminatethebankrunequilibriuminthebenchmarkmodelandthat,withaggregateuncer-tainty,whenthesuspensionofpaymentsschemedoesnotwork,adepositinsurancearrangementcandothetrick.ThecontractualarrangementsconsideredbyDiamondandDybvigarelimitedtoaspaceoffeasiblearrange-mentsthat,astheypointout,istoonarrowtoimplementanefficientallocationintheenvironmentwithaggregateuncertainty.Specifically,DiamondandDybvigassumethatthebankingarrangementmustgivealldepositorswhodemandearlywithdrawalsthesameamountofcon-sumption,namely,thesociallyefficientamountcalculat-edbasedonthetrueparameter(thefractionofimpatientdepositors),nomatterhowmanydepositorsactuallyclaimtobeimpatient.IntheDiamondandDybvigmod-el,althoughtheconsumptiongiventoindividualdeposi-torsvarieswiththeirownclaimedtypes,theamountforeachtypedoesnotdependonthefullinformationcom-municatedtothebankbyalldepositorscollectively.Letuscallthisapproachsimplecontracting.
WesuggestthatDiamondandDybvig’stheoryisin-completeinanimportantrespect.Inparticular,wearguethatthedemanddepositcontractconsideredbyDiamondandDybvigisonlyoneofthefeasiblearrangementsintheenvironmentoftheirmodel.Weshowthatacontrac-tualarrangementexiststhatimplementsanefficientallo-cationintheirenvironmentwithaggregateuncertainty,butwithoutabankrunequilibrium.2
Insteadofrestrictingattentiontoonlysimplecontract-ing,weallowthebanktousemorefullytheinformationreportedbyalldepositorsregardingtheirpreferences.Thebankingarrangementinourmodelspecifiescon-sumptionsforeachdepositor,ortrader,ofeachtypeun-derallpossibleconfigurationsoftypesreportedtothebank.Infact,foreachvectorofmessagesthatthetraderssendtothebank,reportingtheirtypes,ourarrangementassignstraderstheefficientallocationcomputedfortheentirereportedeconomy.Moreover,underthisarrange-ment,individualtradersalwaysfinditintheirbestinter-esttotruthfullyreporttheirtypes.Hence,theefficiental-locationforthemodel’strueeconomyprevailsintheuniqueequilibrium.(Recallthat,inDiamondandDyb-vig’senvironmentwithaggregateuncertainty,thecon-tractualarrangementthattheystudyhasmultipleequilib-ria,andgenerally,allareinefficient.)Also,inourmodel,asinDiamondandDybvig’s,thebankingarrangementcontainselementsofademanddepositcontract;tradershavethefreedomtochooseeithertoconsumeearly(byclaimingtobeimpatient)ortowaittoconsumewhentheirassetsmature(byclaimingtobepatient).However,ourarrangementwillneverinvolvebankruns.Thesepropertiesholdevenwhenthebankfacesthesequentialserviceconstraint.Inlightofourfindings,fromamech-anismdesignapproach,DiamondandDybvig’sbankrunequilibriumappearstobeanartifactoftheirsimplecon-tractingapproachratherthanagenuinefeatureoftheeco-nomicenvironmentthattheyhavemodeled.
Ouranalysisdoesnotdiminishthefundamentalim-portanceofDiamondandDybvig’sinsightregardingfi-nancialinstability,butwethinkitshowstheneedtosyn-thesizethatinsightwithfurtherideasinordertofullyunderstandfinancialinstability.Inlightofthestrikinglyoppositefeaturesofourmodel’sresultsandtheU.S.his-toryofbankruns,wewonder,whatmightpreventra-tionalagentsintheDiamondandDybvigenvironmentfromusingefficientarrangements,ormechanisms,suchastheoneinourmodel?WhatwouldleadtheminsteadtoadoptthepotentiallydestabilizingdemanddepositcontractconsideredbyDiamondandDybvig?Ourpur-posehereissimplytoraisesuchquestions,nottoanswerthem.OurresultsimplythatenvironmentalfeaturesfromwhichDiamondandDybvig’smodelabstractsarecru-cialtoafullunderstandingofbankinginstability.Attheendofthearticle,wereflectonouranalysistoidentifysomepromisingcandidatesforfurther,complementaryresearch.
TheModel
Themodelweuseisafinite-traderversionoftheDia-mondandDybvigmodelwithaggregaterisk.
ConsiderapopulationofItraders,eachofwhomisendowedwithoneunitofadivisiblegood.Thegoodcanbeeitherconsumedatdate0ortransformedintoaconsumptiongoodavailableatdate1.Foreachunitofthegoodusedasaninput,thetransformationtechnologyproducesR>1unitsofconsumptiongoodatdate1.Atthebeginningoflife(date0),alltradersareuncer-tainabouttheirpreferencesoverconsumptionstreams.Withprobabilityp,atraderbecomespatient(type1)andvaluesthesumofdate0anddate1consumption.Withprobability1−p,atraderbecomesimpatient(type0)andvaluesdate0consumptiononly.Here,asinDia-mondandDybvig’smodel,theutilityfunctionisv(cforanimpatienttraderandv(c)forapatienttrader,0)wherec0+c1tdenotesconsumptionatdatet.
Forsimplicity,weassumethatthepopulationislimit-edtothreetraders,eachhavingtheutilityfunctionv(c)=c1−γ/(1−γ)withariskaversionparameterofγ>1.Re-sultsderivedinthissimplesetup,however,alsoholdinmoregeneralsettings.Theeconomiccontentoftheas-sumptionthatγ>1isthattraders’relativeriskaversionisgreaterthan1everywhere.
Alltraderslearnwhichtypetheyare(patientorim-patient)atdate0.Typesareprivateinformation.Inau-tarky,apatienttraderwouldhavehigherutilityexpostthananimpatienttraderbecausepatienttradershaveanopportunitytoapplytheintertemporaltransformationtechnologytotheirendowments.Becausetradersareriskaverse,theywouldliketoenterexanteintoanar-rangementtoinsurethemselves(andso,oneanother)againstpreferenceshockrisk.
Thus,toprotectthemselvesagainstpreferenceshocks,thetradersatthebeginningofdate0(beforeanyonelearnstheirtypes)pooltheirresourcesandsetupabank,whichisactuallyaclublikearrangementamongthetrad-ers.Thebylawsofthebankspecifyarule,accordingtowhicheachtraderwillreceiveconsumptionthatmayde-pendonthetrader’sreport,ormessage,tothebankaboutthetrader’sprivatelyobservedtype.Amessageof0sentbyatradermeansthatthetraderisimpatient,whileamessageof1meansthetraderispatient.Thetradersareassumedtospecifythebylawstomaximizetheexantetotalutilityofalltraders.Thebylawsthusstipulatearesourcedistributionrule,whichspecifiestheamountofconsumptionthateachtraderreceivesateachdateforeachpossibleconfigurationofreportedtradertypes.Wewillshowthatthisrulehasanequaltreatmentproperty:Ineachstateofnature,alltype0tradersshouldreceiveidenticalconsumptionc0atdate0,andalltype1tradersshouldreceiveidenticalconsumptioncconsumptionfortype1tradersshould1atdate1.(Thedate0beze-robecauseR>1.)
Sofar,theonlysignificantdifferencebetweenoursettingandDiamondandDybvig’sisthesizeofthepop-ulation.DiamondandDybvigconsideraninfinitepopu-lation.Incontrast,wehaveonlythreetraders,andindi-vidual-levelrandomnessimpliesthatourmodelalwayshasaggregateuncertainty.
AnotheraspectofourmodeldiffersfromDiamondandDybvig’s:thesequentialserviceconstraint.Diamond
andDybvigdiscussthisconstraintinformally,butdonotmodelitexplicitly.Wedo.Inourmodel,duringdate0,tradersarriveatthebankinrandomorder.Alltradersob-servetheirownarrivaltimes.Theyalsoobservewhetherthey,themselves,arethefirst,thesecond,orthethirdtoarriveatthebank.3Theresourcedistributionrule(stip-ulatedinthebankbylaws)specifiesthattheithtradertoarriveatthebanksendsamessagemwhenapproachingit,withmstandingi∈{0,1}fortobeingthebanki=0im-patientandmtraders’i=1forbeingpatient.Thebankthendis-tributesthepooledresourcesonthebasisofthemessagesithasreceived.
Formally,wemodelsequentialserviceinthefollow-ingway.Letxi(m)denotetheconsumptiongiventothetraderwho4istheithtoarriveatthebank,wherem≡(m1,m2,m3).Thesequentialserviceconstraintrequiresthatforanym,x1(m)=x1ifm1=0andx2(m)=x2(m1)ifm2=0.Thatis,theconsumptiongiventotheithtraderwhoreportsbeingimpatientmustnotdependoninformationfromtraderswhoarrivelater,sincethosetradershavenotyetcommunicatedtheirinformationtothebank.Sinceapatienttraderdoesnotconsumeuntildate1,afteralltradershavesenttheirmessagestothebankatdate0,theconsumptiongiventoapatienttradercanbedeterminedonthebasisofalltraders’messages.BankingWithout
aSequentialServiceConstraint
First,weconsiderthemodelenvironmentwithoutas-sumingasequentialserviceconstraint.Thatis,weas-sumethateachtrader’sconsumptioncanbemadetode-pendonthereportsofallthreetraders.Wecharacterizeanoptimalresourcedistributionrule,andweshowthatthisruleinducestruth-tellingastheuniquereportingde-cisionofrational,optimizingtraders.Ifabankrunisin-terpretedasaninefficientequilibriuminwhichtraderswhoareactuallypatientmisrepresentthemselvesasbe-ingimpatient,thennobankruncanoccurwhenthisop-timalruleisadopted.
Webeginourstudybyabstractingfromthesequen-tialserviceconstraintbecausethisenvironmentispre-ciselythefinite-traderanalogoftheformalDiamondandDybvigmodelenvironment.Thus,theresultsuggeststhatDiamondandDybvig’sadhocfocusonaparticularclassofrulesregardingdemanddepositcontracts,withorwithoutsuspensionofpayments,iscrucialtotheirfindingofadilemmaofhavingtochoosebetweeneco-nomicinefficiencyandbankinginstability(theexistenceofabankrunequilibrium).
Anotherreasonforbeginningourstudywithoutthesequentialserviceconstraintistoexhibit,inassimpleasettingaspossible,thelogicofourmainargument.Theargumenthastwoparts.First,weimaginethattraders’typesarepublicinformation,andwecharacterizetheop-timalresourcedistributionrulethatwouldusethisinfor-mationdirectly.Second,wetakeintoaccountthefactthattheresourcesmustbedistributedonthebasisoftraders’unverifiableandunfalsifiablereports,ratherthanonthebasisofthetruesituation.Thus,therulethatwehavecharacterizedinthehypotheticalenvironmentwithpublicinformationcanonlybeusedintheprivate-infor-mationenvironmentiftraderscanbetrustedtotellthetruthvoluntarily.Thatis,theruleisusableonlyif,what-evertheirtypes,traderscanachievehigherutilityby
truth-tellingthanbylying.Weshowthatanenvironmentwithoutthesequentialserviceconstrainthasaverystrongtruth-tellingincentive:eachtraderdoesbestbytellingthetruth,regardlessofwhetherornotothertrad-ersaretruthful.(Wewillderivearesultinthesamespir-it,althoughslightlyweaker,whenwetakeaccountofthesequentialserviceconstraint.)Clearly,withthisunam-biguousincentive,alltraderswilltellthetruth,sotherecanbenobankrunequilibrium.
Todevelopthisargument,supposethattradersreceiveconsumptionaftertheyallhavereportedtheirtypestothebank.(Thatis,ignorethesequentialservicecon-straint.)Also,supposethatthetruestateofnature,ω,isknown—or,equivalently,thattheprofilemoftraders’messagestothebankisidenticaltoω.Wewillcharac-terizetherulethatmaximizesthesumoftraders’exanteexpectedutilitylevels.
Thetricktosolvingthismaximizationproblemistomaximizethesumoftraders’expostutilitiesineachstateofnatureandnotethattheproblemsofmaximiz-ingexanteandexpostutilityhavethesamesolutioninthisenvironment.Letθ(ω)denotethenumberofpatienttradersinagivenstateofnatureω.Expostefficiencyrequiresthattheendowmentgood’smarginalutilitytoanimpatienttraderequalthattoapatienttraderineachstateofnature,(1)
v′(c0(θ(ω)))=Rv′(c1(θ(ω)))
andthatthefollowingresourceconstraintbesatisfied:5(2)
[I−θ(ω)]c0(θ(ω))+R−1θ(ω)c1(θ(ω))=I.
Equations(1)and(2)determinethefunctionsc)completely.
0(ω)andc1(ωFortheassumedutilityfunctionv(c)=c1−γ/(1−γ),itisstraightforwardtosolvethesetwoequations:(3)c0(θ)=I/[I+θ(R(1/γ)−1−1)]
and(4)
c1/γ1(θ)=IR/[I+θ(R(1/γ)−1−1)].Thiscompletesthefirststageofourargument.
Nowwemustundertakethesecondstage,toshowthattraderswouldalwayschoosetoreporttheirtypestruthfullyifthisrule(basedontheirreportsratherthanontheirtruestates)weretodeterminetheirconsump-tion.SinceR>1,wehavethatc1(θ)isgreaterthanccan0(θ)andthatbothincreasewithθ.Thepatienttraderstakeadvantageofthetransformationtechnology,sotheyeachreceivemoreconsumptionthandotheimpa-tienttraders.Furthermore,instatesofnatureinwhichthenumberofimpatienttradersissmaller,moreendow-mentgetstobetransferredtodate1consumption,en-ablingbothtypesoftraderstoconsumemore.
Atdate0,whentraderslearntheirtypes,theysendmessagesreportingtheirtypestothebank.Thebankcalculatesthevalueofθ(ω)basedonthesereportsandthendistributesresourcesaccordingtotheconsumptionsderivedabove.Regardlessofwhetherornotthecalcu-latedvalueofθisactuallythetruevalue,eachtraderhas
theincentivetotruthfullyreporthisorherowntype,whatevermessagesothertraderssend.
Toseethis,considerseparatelyeachofthetwopos-sibletypesoftraders.Ifatraderistype0,thenthetraderreceivesc0(θ)atdate0ifheorshesendsamessageof0,butreceivesnoconsumptionuntildate1ifheorshesendsamessageof1.Sinceanimpatienttradervaluesdate0consumptiononly,thistraderstrictlypreferstotellthetruth.Nowconsideratraderoftype1.Regardlessofwhatmessagesothertraderssend,thepatienttraderre-ceivesc(θ−1)if1(θ)ifheorshesendsamessageof1andc0heorshesendsamessageof0.Aswehaveexplainedabove,c1(θ)>c1(θ−1)>c0(θ−1).Thus,apa-tienttraderpreferstosendatruthfulmessageaswell.Therefore,thebankingarrangementherehastheproper-tythattruth-tellingisthestrictlydominantstrategyforalltraders.RogerMyerson(1991)hasshownthatapro-fileofstrictlydominantstrategiesforamechanismistheuniqueBayesianNashequilibriumofthemechanism.Therefore,here,noalternative,inefficientbankrunequi-libriumcanexist.
BankingWithaSequentialServiceConstraintThesimplemodelofbankingstudiedaboveabstractsfromkeyfeaturesofanactualbank:usually,tradersdonotallcontactthebankatthesametime,andthebankmustdealpromptlywithtraderswhocontactitearly.Anactualbankisthusconstrainedfrommakingitstreat-mentofearlytraderscontingentoninformationyettobeprovidedbylatertraders,especiallyiftheearlytraderswanttomakewithdrawals.6Nowwemodifythemodeltomakeitmorerealisticinthissense.Werequiretraderstocontactthebanksequentiallyduringdate0,accordingtothesequentialserviceconstraintformalizedabove.Thegenerallogicofourargumenthereisparalleltothatjustused.Wefirstdefinetheefficientallocationinthiseconomyassumingthattradersreporttheirtruetypestothebank.Thenweprovethatunderthespeci-fiedruleofdistributingresources,itisinthebestinter-estsofthetraderstotruthfullyreporttheirtypestothebankandthatthesymmetricefficientallocationcanbeimplementedastheperfectBayesianNashequilibriumofthemechanism.
TheBank’sPlanningProblem
Anallocationinthisthree-tradereconomywithsequen-tialserviceisalistofconsumptionbundlesxi(m)foralliandmwhichthebank(here,thesocialplanner)mustchoose.Anallocationisfeasibleif,ineachstateofna-ture,thetotalamountofconsumptionthepatienttradersreceivefromthebankequalsRtimestheamountofre-sourcesavailableafterthebankgivesconsumptiontoimpatienttradersinthatstateofnature:(5)
i∈I,mi=1
xi(m)=R[I−
i∈I,mi=0
xi(m)]
forallm.Thisistheeconomy’sresourceconstraint.Thebankmustchooseanallocationtomaximizethesumofexpectedutilityofalltraders.Theefficientallo-cationisthusobtainedby
(6)
max(1−p)v(x1(0))+pEm2,m3v(x1(1,m2,m3))
+(1−p)Em1v(x2(m1,0))+pEm1,m3v(x2(m2,1,m3))+(1−p)Em1,m2v(x3(m1,m2,0))
+pEm1,m2v(x3(m1,m2,1))
subjecttotheresourceconstraint(5).
Theformoftheaboveoptimizationproblem(anad-ditivelyseparableobjectivefunctionintermsofv(xi)andtheresourceconstraintindistinctstatesofnature)im-pliesthattheproblemcanbesolvedusingtheusualdy-namicprogrammingtechniques:maximizingtotalutilityimpliesoptimizingalongeachpathofrealizationsoftradertypes.Forinstance,attheefficientallocation,xmustbesuchthatitmaximizestheutilityofthe1(0)firsttradertoarriveatthebank,v(x1(0)thesecondand),plusthesumoftheexpectedutilitiesofthirdtraderscondi-tionalonm1=0.Therefore,inwhatfollows,wewillsolvethebank’sproblemusingtheusualbackwardin-ductionprocedure.Specifically,westartbyderivingtheoptimalconsumptionforthetraderwhoisthelasttoarriveatthebankandthenmoveontooptimizationproblemsfortraderswhoarrivedearlier.Eachproblemissolvedbasedontheinformationreportedbythetrad-ersastheyarriveatthebank.Wethenshowthatthetradershavetheincentivetotruthfullyrevealtheirtypeswhentheymakedecisions,soprivacyofinformationisactuallynotabindingconstraintinthebank’splanningproblem.
ConsumptionfortheLastTrader
Westart,again,withthelasttradertoarriveatthebank.Lety(m1,m2)denotetheamountofendowmentthebankhasleftwhenthelasttraderarrives.Thatis,I−yhasbeengivenouttopreviouslyarrivingtraders,whohavesentmessagesofm1andmlast2.Thebank’sdecisionproblemhereissimple.Ifthetraderispatient(m1),thenthattraderatdate1willreceiveRy/θ,thetrad-3=er’sshareoftheremainingendowmenttransformedbytheRtechnology,whereθ≡mlasttraderisimpatient,then1+mthe2+mbank3.If,instead,theneedstoim-mediatelyassignthattraderconsumptionatdate0,de-notedbyx3(m1,mof2,0),bybalancingthetrader’smarginalutilitywiththatthepatienttraders.Thus,x3(m1,msatisfiesthefollowing:72,0)(7)
x3(m1,m2,0)=argmaxx3v(x3)
+(m1+m2)v(R(y−x3)/(m1+m2)).
Thefirst-orderconditionisthusv′(x3)=Rv′(R(y−x3)÷(m1+m2)).SinceR>1,v′(x3)>v′(R(y−xtheassumptionthatthetraders’relative3)/(mrisk1+maver-2)).Undersionisgreaterthan1,thefirst-orderconditionalsoim-pliesthatv′(xfollowing3) Thefollowingboundsapplytox3(m1,m2,0):(8) y/(m1+m2+1) x3(m1,m2,0)=y(m1,m2)/[1+(mγ)−11+m2)R(1/]. ConsumptionfortheSecondTrader Nowconsiderthetraderwhoissecondtoarriveatthebank. TheamountofendowmentthebankhasavailablenowiseitherIorI−xtraderispatient.1(0),dependingonwhetherornotthefirstIfthesecondtraderispatient,thenatdate1thattraderwillreceivehisorhershareofRtimestheamountoftheendowmentnotdistributedtoimpatienttraders,equallydividedamongallthepatienttraders.Otherwise,thebankmustassignthetraderdate0consumptionimmediately.Below,wederivetheoptimalconsumptionforthesecondtraderwhenthattraderisoftype0,x2(m1,0). WithaPatientFirstTrader Supposethatthefirst-arrivingtraderclaimstobepatient(m1=1).Indeterminingthevalueofx2(1,0),thebankmusttakeintoconsiderationthepossibletypeofthethirdtrader,whoisyettoarrive,aswellasthefirsttrad-er,whohasbeenwaitingtoconsumeatdate1.Supposethatxp,2(1,0)isgiventothesecondtrader.Withprobabil-itythethirdtraderispatient;thenthattraderandthefirsteachgetR[I−x2(1,0)]/2atdate1.Withprobability1−p,thethirdtraderisimpatient;thenthattraderre-ceivesatdate0theamountxandthe3(1,0,0),asdeterminedop-timallyinequation(7),firsttraderreceivestheamountofx1(1,0,0)=R[I−x2(1,0)−xGiventheseprobabilities,thebankchooses3(1,0,0)]atdate1.xmaximizethetotalutilityofallthreetraders:2(1,0)to(10) maxx2(1,0)v(x2(1,0))+2pv(R[I−x2(1,0)]/2) +(1−p)[v(x3(1,0,0))+v(x1(1,0,0))]. Thefirst-orderconditionforthismaximizationprob-lem,aftertheenvelopetheoremisapplied,is(11) v′(x2(1,0))=Rv′(x1(1,0,0)) −pR(v′(x1(1,0,0)) −v′(R[I−x2(1,0)]/2)). Forv(c)=c1−γ/(1−γ),thesolutionofequation(11)canbefound,via(9),tobe(12)x2(1,0)=I/(1+A1/γ) where(13) A≡(1−p)(1+R(1/γ)−1)γ+p2γR1−γ.WithanImpatientFirstTrader Nowsupposethatthefirsttradertoarriveatthebankisimpatient,sothattheamountoftheendowmentavail-abletothebankwhenthesecondtraderarrivesisI−x1(0). Indecidingonx2(0,0),theconsumptiontobegiventothesecond-arrivingtraderwhenthattraderistype0,thebankmaximizesthesumoftheexpectedutilityofthesecondandthirdtraders:(14) V00(x1(0))≡maxx2(0,0)v(x2(0,0)) +(1−p)v(x3)+pv(Rx3) wherexfirsttrader3≡andI−xx1(0)−x2(0,0).Withx1(0)giventothe2(0,0)tothesecond,theconsumptionforthethirdtraderisI−x[I−x1(0)−x2(0,0)ifthattraderisimpatientandRfirst-ordercondition1(0)−x2(0,0)]ifthetraderispa-tient.Theforx2(0,0)is,thus,(15) v′(x2)=(1−p)v′(I−x1(0)−x2) +pRv′(R[I−x1(0)−x2]) which,forv(c)=c1−γ/(1−γ),hasthefollowingsolution:(16)x2(0,0)=[I−x1(0)]/(1+B) where(17) B≡[1−p+pR1−γ]1/γ.Whatistheoptimalconsumptionforthesecond-arrivingtraderifthattraderispatient?Withprobabilityp,thethirdtraderispatient;thenboththesecondandthirdtradersreceiveR[I−x1(0)]/2atdate1.Withprob-ability1−p,thethirdtraderisimpatient;thenthethirdtrader’soptimaldate0consumptionisxreceivesR[I−x3(0,1,0)andthesecondtradertotalutilityofthe1(0)−xsecond3(0,1,0)]atdate1.Thus,theandthirdtraders,conditionalonthesecondbeingpatient(andthefirstbeingimpatient),is(18) V01(x1(0))≡2pv(R[I−x1(0)]/2) +(1−p){v(x3(0,1,0))+v(R[I−x1(0) −x3(0,1,0)])}. ConsumptionfortheFirstTrader Nowconsiderthetraderwhoisthefirsttoarriveatthebank. Ifthistraderisimpatient,thenthebankchoosesxmaximizethesumofallthreetraders’expectedutili-1(0)toties:(19) maxv(x1(0))+(1−p)V00(x1(0))+pV01(x1(0)). Usev(c)=c1−γ /(1−γ)and(9)towritethefirst-orderconditionforthisoptimizationproblemas (20) x2−11(0)=(1−p)([I−x1(0)]/(1+B))−γ +(1−p)pR(R[I−x11(0)]/(1+B−))−γ+p(1−p)R(R[I−x(1/γ)1(0)]/(1+R1−))−γ+p2R(R[I−x1(0)]/2)−γ. Thisyieldsthefollowingsolution:(21) x1(0)=I/{1+[pA+(1−p)(1+B)γ]1/γ}. SincebothAandBdecreaseinp,x1(0)isanincreas-ingfunctionofp.Theintuitionforthisisasfollows.Asprises,thetraderswhoarriveafterthefirsttraderaremorelikelytobepatient.SincetheconsumptionofthesepatienttraderscanbesupportedbytheRtransfor-mationtechnology,xordertobalancethemarginal1(0)shouldincreaseaccordinglyinutilityofthecurrentimpa-tienttraderwiththatoftheselaterarrivals.Similarly,theoptimalconsumptionforthesecondtraderderivedearlier,x2(1,0),alsoincreaseswithp.Thesepropertieswillbeusedlater,intheproofofProposition1.TheSymmetricEfficientAllocation Theoptimalconsumptionsoftradersineverystateofnaturehavebeenderived.Thebankingarrangement(themechanism)mustdistributetheresourcesaccordingtothesederivedconsumptionsandthetraders’reportedtypes.Nowweshowthatwiththespecifiedmechanism,tradersinthismodelwilltruthfullyreporttheirtypesandthatthistruthfulcommunicationconstitutestheuniqueperfectBayesianNashequilibriumofthemechanism.Toprovethis,weusestandardbackwardinductionreason-ing. First,considerthetraderwhoarrivesatthebanklast.Accordingtothelemma,thisthirdtraderalwayspreferstotellthetruthregardlessofthemessagessentbyprevi-ouslyarrivingtraders. Next,considerthetraderwhoarrivesatthebanksec-ond.Sinceanimpatienttraderneverclaimstobepatient,weonlyneedtoconsiderwhathappenswhenthesecondtraderispatient.Beforethistrader’sarrival,theavailableendowmentiseitherI−xornotthefirsttraderisimpatient.1(0)orI,dependingonwhetherSupposethefirsttrad-erisimpatient.Canthesecondtraderbenefitfromlying,claimingtobeimpatienttoo?Ifthetradertellsthetruth,heorshereceivesxonthe2(0,1,mreported3)atdate1,thevalueofwhichdependstypeofthethirdtrader.Ifthesecondtraderchoosestolie,thetraderreceivesx2(0,0)rightaway,atthetimeofarrival.Thus,thetraderwillchoosetotellthetruthifandonlyif(22)v(x2(0,0))≤E[v(x2(0,1,m3))] where(23) E[v(x2(0,1,m3))]= (1−p)v(R[I−x1(0)−x3(0,1,0)])+pv(R[I−x1(0)]/2). Similarly,ifthefirsttradertoarriveispatient,thenthe(patient)secondtraderwilltellthetruthifandonlyif (24)v(x2(1,0))≤E[v(x2(1,1,m3))]where(25) E[v(x2(1,1,m3))]= (1−p)v(R[I−x3(1,1,0)]/2)+pv(R). Notethat,sincethethirdtraderneverliesabouthisorhertype,wecanusetheobjectiveprobabilityofbecom-ingapatienttrader,p,toevaluatetheexpectedutilitiesin(22)and(25). Nowconsiderthefirsttradertoarriveatthebank.Thistraderispatientandreceivesxm1(0)forlyingandx1(1,2,m3)otherwise.Truth-tellingthusrequiresthat(26)v(x1(0)) E[v(x1(1,m2,m3))]= (1−p)2v(R[I−x2(1,0)−x3(1,0,0)])+p2v(R)+(1−p)p(v(R[I−x2(1,0)]/2)+v(R[I−x3(1,1,0)]/2)). Thefollowingresult,provedintheAppendix,showsthatboththesecondandfirsttradersstrictlyprefertotellthetruthabouttheirtypesbecausetheyanticipatetruth-fulcommunicationbythethirdtrader.Thus,truthfulre-portingbyalltradersistheonlyequilibriumoutcomethatresultsfrombackwardinduction.Hence,itistheuniqueperfectBayesianNashequilibriumofthemech-anism. PROPOSITION1.Assumethatv(c)=c1−γ/(1−γ)forγ>1.Thenincentiveconditions(22),(25),and(26)holdforallp∈[0,1]andforR>1. Finally,wepresentthefollowingresult,whichcorre-spondstoapartialsuspensionofpaymentsschemeun-dertheoptimalbankingarrangementinourmodel.(SeeWallace1990.) PROPOSITION2.Forallp∈[0,1],R>1,andγ>1,thereisx1(0)>x2(0,0)>x3(0,0,0). Proof.Fromtheexpressionsofxthat1(0)andx2(0,0)derivedearlier,weknow(28) x2(0,0)=[I−x1(0)]/(1+B) ={[pA+(1−p)(1+B)γ]1/γ/(1+B)}x1(0). Thus,x(1+B1(0)>x)γ.The2(0,0)isequivalenttopA+(1−p)(1+B)γ Q.E.D. AccordingtoProposition2,ifalltradersdemandear-lywithdrawalinourmodel,thenthetraderswhoarriveatthebankearlierreceivemoreconsumptionthanthosewhoarrivelater.(Thispropertyalsoholdsforotherpathsofrealizationsoftypes,suchasthoseinwhichtwoofthethreetradersclaimtobeimpatient.)Notethatsuchapartialsuspensionofpaymentsoccurswithpositiveprob-abilityinourmodel,althoughabankrunneveroccurs. Conclusion Inafinite-traderversionofthemodelofDiamondandDybvig(1983;reprintedinthisissue),wehaveshownthattheexanteefficientallocationcanbeimplementedasauniqueequilibrium.Inthemechanismwehavecon-sidered,truth-tellingisthestrictlydominantstrategyforalltradersintheenvironmentwithaswellaswithoutsequentialservice.Alltradersprefertruth-tellingevenwhenthereisasequentialserviceconstraintbecausetheyexpect(correctly)thatthosewhoarriveatthebanklaterwilltellthetruth.Therefore,inourmodel,unlikeinDia-mondandDybvig’s,thereisnobankrunequilibrium.(TheseresultsalsoholdinmoregeneralsettingswithItradersandgeneralutilityfunctions,aswehaveshowninGreenandLin1999.) DiamondandDybviginterprettheirmodelasanex-planationofthenumerousobservedbankrunsinU.S.history.Wearenotclaiming,ofcourse,thatbankrunsdonotactuallyoccur.Rather,wearesimplytryingtoshowthatwithinthebasicframeworkofDiamondandDybvig—evenwiththesequentialserviceconstraint—anarrangementexiststhatimplementstheefficiental-locationwithoutleadingtobankruns.Inthemodel,ra-tionalagentshavenoreasontobypassthisoptimalarrangementandinsteadchooseanotherthatmightpro-ducebankrunsinequilibrium.Therefore,wethinkthatsomethingessentialhasbeenneglectedinthebasicDia-mondandDybvigenvironmentinordertohaveatheo-rythatmatchesU.S.history. Ourmodeldoesnotcaptureallofthekeyfeaturesofanactualbankingsystemeither.Oneobviouslymissingfeatureisthebankingsystem’songoingnature.Ifthepopulationofaneconomyhasanoverlapping-genera-tionsstructure,thennotraderisthelasttoarriveatthebank,soourbackwardinductionargumentmaynotwork.Thesameproblemarisesifthesizeofthepopu-lationisnotobservabletoindividualtraders,sothatnooneiscertainwhetherheorsheisthelastoneinline.Whenthesefeaturesarepresent,thevalidityofournobankrunresultneedstobereconsidered.Finally,alsoabsentinourmodelaretheincentiveproblemsamongbankofficialstomanageresourcesinawaythatmaxi-mizestheutilityoftheirdepositors.Suchincentiveprob-lemscanresultfromincompleteinformation.9Thebank-ingcontractinourmodelabstractsfromtheseincentiveproblems,whichmightbeareasonsuchacontractisnotcommonlyobserved.Allofthesemissingfeaturesareworthinvestigatingaswetrytoimproveourunderstand-ingofbankinginstability. *TheauthorsthankLingnanUniversityforprovidingfinancialsupport.TheviewsexpressedherearethoseoftheauthorsandnotnecessarilythoseoftheFeder-alReserveBanksofMinneapolisorChicagoortheFederalReserveSystem. 1 DiamondandDybvig’sworkisoneofthepivotalcontributionstoalargelit-eratureonbankingcontractsandbankruns.OthersuchcontributionsincludetheworkofJohnBryant(1980),CharlesJacklin(1987),andNeilWallace(1988). 2 Ourmechanismdesignapproachingeneralinvolvessupposingthattheformalmodeloftheeconomicenvironmentsucceedsincapturingthesignificantrelevantconstraintsonhowcontractualarrangementscanbestructuredintheactualecono-myandthensolvingtheoptimizationproblemofdesigninganefficientarrangementsubjecttothoseconstraints. 3 Thereare6(3!)possibleordersofarrivalofthethreetraders,andweassumethateachoccurswithprobability1/6.Thisformulationofsequentialserviceisrelat-edtothenightcampingstorytoldbyWallace(1988).However,inWallace’sfor-malizationofsequentialservice,neitheratrader’stimeofarrivalnoratrader’splace inlineisinthetrader’sowninformationset.UnderWallace’sassumption,theback-wardinductionreasoningweuselatermaynotnecessarilywork.InGreenandLin1999,werelaxtheassumptionthattradersknowtheirplacesintheorderofarrival.However,westillassumericherinformationaboutarrivaltimethanWallacedoes. 4 Inprinciple,xi(m)shouldspecifyquantitiestobeconsumedatbothdates0and1.Forsimplicity,wewillsupposethatatraderwhosendsamessageof0(whoreportsbeingimpatient)willbepermittedtoconsumeonlyatdate0andthatatrad-erwhosendsamessageof1(whoreportsbeingpatient)willbepermittedtocon-sumeonlyatdate1.Thefollowingdiscussionshouldmakeitclearthattheoptimalresourcedistributionrulemusthavethisfeature,evenifitwerenotimposedbyas-sumption.WeexplicitlyderivesucharesultinGreenandLin1999. 5 Theseconditionsalsoimplyexanteefficiency.SeeGreenandLin1999.6 ThisfeatureplaysakeyroleinDiamondandDybvig’sintuitivediscussionoftheirmodel,anditisformalizedbyWallace(1988),whoderivesfurtherconsequen-cesfromit. 7 Ifalltradersclaimtobeimpatient(m1+m2+msox3=0),thenthethirdtraderjustconsumestheendowmentavailable,3(0,0,0)=I−y(0,0). 8 Toseethis,letΦ(R)≡Rv′(Rc).ThenΦ′=Rcv″(Rc)+v′(Rc),whichisnega-tive,sincecv″(c)/v′(c)≤−1forallc.Thus,Φ(R)<Φ(1);thatis,Rv′(Rc) 9 TheseproblemshavebeenusedbyDouglasDiamond(1984)andStephenWil-liamson(1987)toarguefortheefficiencyofstandarddebtcontracts.AsynthesisoftheirmodelswithDiamondandDybvig’smightproduceabankrunequilibriumun-deranefficientcontract. Appendix ProofofProposition1 Hereweprovetheprecedingtext’sProposition1,thatboththefirstandsecondtraderstoarriveatthebankprefertotellthetruthabouttheirtypesbecausetheyexpectthethirdtradertodoso. PROPOSITION1.Assumethatv(c)=c1−γ /(1−γ)forγ>1.Thenincentiveconditions(22),(25),and(26)holdforallp∈[0,1]andforR>1. Proof.Notethatforv(c)=c1−γ/(1−γ),v′(c)=Ψ(v(c))forallc,whereΨ(x)=[(1−γ)x]γ/(γ−1)forΨ′<0andΨ″>0. Wefirstprovethat(22)holds.By(16),I−x1(0)−x2(0,0)=R[I−x1(0)]/(1+B−1) 1−(1/γ) +1) >[I−x1(0)]/2>[I−x1(0)]/(1+B−1). Fromtheseandthefirst-orderconditionforxgetthat2(0,0),equation(15),we(A2) v′(x2(0,0))>(1−p)v′(I−x1(0)−x2(0,0)) +pv′(R[I−x1(0)−x2(0,0)]). Usingv″<0andtheabovederivations,wehavethat(A3) v′(x2(0,0))>(1−p)v′(R[I−x1(0)−x3(0,1,0)]) +pv′(R[I−x1(0)]/2). RewritingtheaboveinequalityintermsoffunctionΨyieldsthat(A4) Ψ(v(x2(0,0))) >(1−p)Ψ(v(R[I−x1(0)−x3(0,1,0)]))+pΨ(v(R[I−x1(0)]/2)). SinceΨisconvex,wehavethat (A5) Ψ(v(x2(0,0))) >Ψ((1−p)v(R[I−x1(0)−x3(0,1,0)]) +pv(R[I−x1(0)]/2)) which,alongwithΨ′<0,impliesthat(A6) v(x2(0,0))<(1−p)v(R[I−x1(0)−x3(0,1,0)]) +pv(R[I−x1(0)]/2). Thisprovesthat(22)holds. By(12),wehavethatx2(1,0)=I/(2+R(1/γ)−1)>I/3atp=0.Sincex =R(1/γ)−1[I−x2(1,0)]/(1+R(1/γ)−1) and (A8) R[I−x3(1,1,0)]/2=R(1/γ)−1I/(1+2R(1/γ)−1). Sincex2(1,0)>I/(2+R1−(1/γ)),weknowthat(A9) I−x2(1,0)−x3(1,0,0)=R[I−x3(1,1,0)]/2. Usingthefirst-orderconditionforx2(1,0),wefindthat(A10)v′(x2(1,0))>(1−p)v′(R[I−x2(1,0)−x3(1,0,0)]) +pv′(R[I−x2(1,0)]/2). Theabovederivations,alongwithv″<0,implythat(A11)v′(x2(1,0))>(1−p)v′(R[I−x3(1,1,0)]/2)+pv′(R).Hence, (A12)Ψ(v(x2(1,0))) >(1−p)Ψ(v(R[I−x3(1,1,0)]/2))+pΨ(v(R))>Ψ((1−p)v(R[I−x3(1,1,0)]/2)+pv(R)). SinceΨ′<0, (A13)(1−p)v(R[I−x3(1,1,0)]/2)+pv(R)>v(x2(1,0)).Therefore,(25)holds. Theincentivecompatibilitycondition(26)forthefirsttrad-ertoarriveatthebankcanbesimilarlyprovedtoholdbyusingtheconvexityofΨ.Theproofislengthyandthusomit-tedhere. 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