policy implications by Haibin Zhu Monetary and Economic Department Abs - PDF document

No 107 policy implications by Haibin Zhu Monetary and Economic Department Abstract This paper proposes a model in which bank runs are closely related to the state of the business cycle. The benchmark model shows that, in a market economy, there are welfare losses due to the existence of bank runs. Extensions of the model explore the welfare effects of various government policies. The results suggest that an interest-cap deposit insurance scheme is an efficient policy to prevent bank runs, while other policies, including the suspension of convertibility, a penalty on short-term deposits and full-coverage deposit insurance schemes, will all have adverse side effects. JEL Classification Numbers: G21, G28, E5 Keywords: bank runs, bank regulations, suspension of convertibility, deposit insurance, capital requirement. BIS Working Papers are written by members of the Monetary and Economic Department of the Bank for International Settlements, and from time to time by other economists, and are published by the Bank. The papers are on subjects of topical interest and are technical in character. The views expressed in them are those of their authors and not necessarily the views of the BIS. Copies of publications are available from: Bank for International Settlements Information, Press & Library Services CH-4002 Basel, Switzerland E-mail: publications@bis.org Fax: +41 61 280 9100 and +41 61 280 8100 This publication is available on the BIS website (www.bis.org Bank for International Settlements 2001. All rights reserved. Brief excerpts may be reproducedor translated provided the source is cited. ISSN 1020-0959 1Introduction1Thebankingsectorisvulnerabletobankrunsbecause,bynature,banksissueliquidliabilitiesbutinvestinilliquidassets.Whenabankrunoccurs,agentsrushtothebanksandwithdrawtheirfundsasquicklyaspossible.Banksaredrivenintobankruptcyduetoliquidityproblems.Thebreakdownofthebankingindustrydistortscapitalallocationandinmostsituationsaddsdownwardpressuretotherealeconomy.Historically,bankrunsoccurredfrequentlyinEuropeinthe19thcentury,andplaguedtheUnitedStatesuntilthereformoftheFederalReserveSystemafterthecrisisof1933.Overthepasttwoorthreedecades,thebankrunphenomenonhashitmostemergingcountries(seeLindgrenetal1996).RecentworkbyKaminskyandReinhart(1999,2000)suggestsanewphenomenonsincethe1980sinthatbankrunshavealsoplayedaveryimportantroleintheso-called\twincrises"episodes.Giventhefrequentoccurrenceofbankrunsandtheassociateddestabilisingcosts,variouspolicyinstrumentshavebeenimplementedtoavoidtheundesirablephenomena.Inearlytime,policy-makerspaidmoreattentiononcrisisresolution,or,howtostopabankrunsonceitoccurs.Suchpolicyinstrumentsincludethesuspensionofconvertibilityofdepositsandapenaltyonshort-termdeposits(seeDwyerandGilbert1989).Morerecently,thepolicymakershaveshiftedtheiremphasistocrisisprevention,or,howtominimisethepossibilityofbankrunswithouthurtingthebankingsector'sroleinprovidingliquidity.Theproposedpoliciesincludeholdingappropriateprovisionsandcapitalreserves,strengtheningbanks'self-regulation,anddesigningdepositinsuranceschemes(seeFSF2001).Inthispaper,Itrytoexplorethewelfaree®ectsofthesepolicyinstruments.Thequestionwillbeaddressedintwoparts.First,whatarethee®ectsofthesepoliciesinstoppingbankruns?Second,andmoreimportantly,whataretheexantee®ectsofthesepoliciesiftheirexistencehasbeentakenintoaccountbythemarket?Tostarttheanalysis,itisimportanttoexplainthemicroeconomicunderpinningsofbankruns.Therearetwogeneralviews.Onegroupofeconomists,includingDiamondandDybvig(1983),1ThispaperispartofmythesisatDukeUniversity.IthankCraigFur¯ne,PhilipLowe,EnriqueMendoza,PietroPeretto,KostasTsatsaronis,DiegoValderramaandLinZhouforhelpfulcomments.IamalsogratefulforhelpfulsuggestionsfromtheseminarparticipantsatDukeUniversity,CityUniversityofHongKong,theBISandtheInternationalConferenceonEconomicGlobalisation.Allerrorsremainmine.1 CooperandRoss(1998),ChangandVelasco(2000,2001),Park(1997)andJeitschkoandTaylor(2001),considerbankrunsasself-ful¯llingprophecies,unrelatedtothestateoftherealeconomy.Thereexisttwoequilibriainthebankingsector.Ontheonehand,ifnoagentexpectsthatabankrunwillhappen,therisk-sharingmechanismprovidedbythebankingsectorfunctionswellandtheeconomicresourcesareallocatedinane±cientway.Ontheotherhand,ifallagentsanticipateabankrun,thentheyallhavetheincentivetowithdrawtheirdepositsimmediatelyandabankrunoccursasexpected.Whichofthetwoequilibriahappensdependsontheexpectationsofagents,which,unfortunately,arenotaddressedintheirmodels.Thesecondview,asre°ectedintheempiricalstudiesofGorton(1988),CalomirisandGorton(1991),andrecentworkbyAllenandGale(1998),Zhu(2001)andGoldsteinandPauzner(2000),considersbankrunsasaphenomenoncloselyrelatedtothestateofthebusinesscycle.AllenandGale(1998)showthatthebusinesscycleplaysanimportantroleingeneratingbankingcrises.Theyalsoshowthatbankrunscanbe¯rst-beste±cientandcentralbankinterventionmaybeundesirableinsomesituations.Zhu(2001)developsatwo-stagemodelinwhichagentsmakewithdrawaldecisionssequentially.Heshowsthatbankrunshappenonlywhenagentsperceivealowreturnonbankassets,andbanksmaydeliberatelychooseabank-runcontractoverarun-proofalternative.GoldsteinandPauzner(2000)showthatwhenagentsreceiveslightlynoisysignalsregardingthefundamentals,theeconomywillfeatureauniqueequilibriuminwhichtheoccurrenceofbankrunsisdeterminedbythestateofthebusinesscycle.ThispaperfollowsthebusinesscycleoriginmodeldevelopedinZhu(2001)forthreemajorreasons.First,themodelpredictsthattheoccurrenceofbankrunsisrelatedtoeconomicfunda-mentalsratherthana\sunspotphenomenon".Thispredictionisconsistentwithrecentempiricalstudies.Second,themodelfeaturesauniqueequilibriumandtheprobabilityofbankrunscanbeendogenouslydetermined.Thispropertyeliminatestheundesirableindeterminacyintheanalysis.Third,andmostimportantly,thetwo-stageframeworkinthemodelallowsustostudyhowthebanksandagentswillreacttothegovernmentpolicies(or,theexantee®ects),whichisabsentfrommuchoftheexistingliterature.Thebenchmarkmodelillustratesthatthebankingsectorprovidesarisk-sharingmechanismagainsttheuncertaintyindepositors'liquidityneeds.However,itcanalsobethesourceofinsta-2 bilitybecauseofthepotentialbankrunproblems.Hence,theequilibriuminamarketeconomyissuboptimal.Extensionsofthemodelexplorethewelfareimplicationsofvariousgovernmentpolicies.Themainresultsareasfollows.First,suspensionofconvertibilityofdepositsisbothexpostandexanteine±cientinpreventingrunsbecauseitcannotdistinguishbetweenthosewithtrueliquidityneedsandthosewhoarerunningonthebanks.Therefore,althoughbankrunsaresuccessfullystopped,itisverylikelythatsomeagentswithtrueliquidityneedscannotwithdrawtheirdepositsinatimelymanner,whileotheragentswhodonothavegenuineliquidityneedswillhavetheirdepositsrepaid.Second,taxationonshort-termdeposits,althoughitcana®ectboththequantityandcompo-sitionofearlywithdrawals,introducesinvestmentdistortionstotheeconomyandturnsouttobeine±cientaswell.Third,depositinsuranceisanexposte±cientpolicyinpreventingbankruns,butitisexanteine±cientduetothe\moralhazard"problems.Becausethedepositinsuranceauthoritycannotmonitorthebanks'investmentbehaviourperfectly,banksalwayshavetheincentivetobehaveaggressivelybyo®eringhighinterestrates.However,thispaperproposesthatsubstitutingthefull-coveragedepositinsuranceschemewithaninterest-capdepositinsuranceschemecanovercomethismoralhazardproblemandhelptheeconomytoachievethesociallyoptimaloutcome.Fourth,impositionofacapitalrequirement,orequivalentlyancapital/assetratiorequirement,isane±cientpolicytopreventbankrunsinthelimit.Asthecapitalrequirementincreases,themarketequilibriumgraduallyconvergestothesocialoptimum.Theproblemis,however,thatthecapitalrequirementmightbeveryhigh.Theremainderofthispaperisorganisedasfollows.Section2developsthebenchmarkmodelandde¯nesthecompetitiveequilibriuminthemarket.Section3analysesthewelfarepropertiesofthecompetitiveequilibriumincomparisonwithtwotypesofsociallyoptimalallocations.Section4discussesthewelfaree®ectsof¯vedi®erentpolicies:suspensionofconvertibilityofdeposits,taxa-tiononshort-termdeposits,full-coveragedepositinsuranceschemes,interest-capdepositinsuranceschemesandcapitalrequirements.Section5concludes.3 2BenchmarkmodelThebenchmarkmodelisbasedontheframeworkdevelopedinZhu(2001).Therearethreeperiods:T=0;1;2.Twoinvestmenttechnologiesareavailableinperiod0:astoragetechnologyandariskytechnology.Thestoragetechnologyisriskless:ityieldsaconstantreturnof1inperiod1or2.Theriskyassetyieldsalong-termreturnof~Rwhichhasasupport[0;1].Besides,theriskyassetisilliquidinthatitsliquidationvalueinperiod1is(1¡¿)~R.2Theriskyassetismoreproductiveinthelongrunbutlesse±cientintheshortrunontheassumptionthat(1¡¿)E(~R)1E(~R).3Therearealargebut¯nitenumber(N)ofexanteidenticalagents.Eachagentisendowedwithoneunitofgoodsinperiod0.Agentsaresubjecttoapreferenceshockintheinterimperiod.Afraction(®)oftheseagentsturnouttobeimpatient,implyingthattheyderiveutilityfromperiod1consumptiononly;theothers(1¡®)willbepatient,whoonlycareforperiod2consumption.Theirutilityfunctionsareu1(c1;c2)=u(c1)(2.1)u2(c1;c2)=u(c2)(2.2)respectively,whereu(¢)satis¯esu(0)=0,u0(¢)�0andu00(¢)0.Thebankingsectorisperfectlycompetitive.Inperiod0,bankscompetewitheachotherbyo®eringdemand-depositcontractswhichspecifyashort-terminterestrate(r1)andalong-terminterestrate(r2).Individualagentsthendecidewhethertodeposittheirendowmentswiththebankornot.4Afterreceivingthedeposits,eachbankchoosesitsoptimalportfolioallocationbetweenthesafeasset(1¡i)andtheriskyasset(i).Intheinterimperiod,theuncertaintiesinconsumertypesandassetreturnsareresolved.Eachagentlearnshisownpreferencetype,andhealsoreceivesapubliclyobservablesignal(s)that2Thedeterminationofliquidationvalueisexogenousinthispaper.Someexistingpapers,suchasKrugman(1998b)andBackusetal(1999),mayshedlightonfuturestudyinthisdirection.Thispaperemploystheproportionalliquidationvalueoutoftwoconsiderations.First,duetoaliquiditycrunch,theassetsarealwayssoldatalowerprice.Second,sincetheinformationisperfectinthismodel,theliquidationvalueofanassetshouldbeassociatedwithitstruevalue.3Thisisanecessarycondition.Asu±cientconditionisE[u[(1¡¿)~R]]u(1)E[u(~R)],whereu(¢)istheutilityfunctionforarepresentativeagent.4Tosimplifythealgebra,Iassumeeachagenthasonlytwochoices:eithertodepositallhisendowmentsortodepositnothing.Themainconclusionsinthispaperremainrobustwhenagentsareallowedtodepositafractionoftheirendowments.Besides,ifagentsareindi®erentbetweentwocontracts,theyrandomlypickupthedepositbank.4 correctlyrevealsthereturnontheriskyasset(s=R).5Basedonthepublicandprivateinformation,agentsmakewithdrawaldecisionssequentially.Thissequentialdecisionrulehasatwofoldmeaning.First,atacertaintime,onlyoneagentisallowedtowithdrawhisdepositfromthebank.Second,eachagentcanobservetheactionsofallagentsaheadofhimbeforehechooseshisaction.Bankspaytheshort-terminterestratetotheearlyconsumersaccordingtothe\¯rstcome,¯rstserved"ruleinperiod1.6Inperiod2,alllateconsumerssharetheremainingassetsofthebankswithamaximumpaymentofr2.Lemma1(Uniquenessofequilibriumoutcome)Giventhesequentialdecisionrule,foracertaindemand-depositcontract(r1;r2)andportfoliostructure(i),thereisauniqueequilibriumoutcome.Equilibriumaggregateearlywithdrawalamount,L,7equals®whenR¸r1¡1+iiandL=1otherwise.Proof:seeAppendixA.Lemma1statesthat,underagivencontract,abankrunhappensifandonlyifagentsperceivealowreturnonbankassets.Theintuitionisasfollows:ifpatientagentsknowthattheeconomyisinagoodstate,theywillhavenoincentivetorunonthebank.Thesequentialdecisionruleprovidesamechanismthroughwhichthepatientagentscancoordinatetheiractions.The¯rstpatientagent,knowingthatthefollowerscanobservehisaction,willchooseto\wait"andsendthesignal\IamapatientagentandIamexpectingtheotherpatientagentsnottorunonthebank".Observingthissignal,thefollowersalsochooseto\wait".Therefore,thesequentialdecisionruleprovidesanequilibrium-selectionmechanismintheDiamond-DybvigframeworkandonlytheParetoe±cientoutcomeischoseninequilibrium.Lemma1alsoimpliesthattheprobabilityofbankrunscanbeendogenouslydeterminedbycontractvariables.Speci¯cally,therewillbenobankrunsundercontractsthatfeaturer1·1¡i.Thisisquiteintuitivebecausetherun-proofcontractspenalisesu±cientlyearlywithdrawalsandthereforeneverhavetheliquidityproblemintheinterimperiod.Inatwo-stagegame,banksareallowedtochoosetheinterestrates(r1andr2)andportfolio5Thispaperonlystudiesthecaseinwhichtheinformationisperfect.Whentheinformationisimperfect,thebanksaremorevulnerabletoruns,butthewelfareanalysisinthepaperremainsvalid.SeeZhu(2001).6Thesequentialserviceconstraintisnotessentialinthemodel.AsshowninZhu(2001),removingthissequentialserviceconstraintdoesnotchangetheequilibriumproperties.7Lisde¯nedastheproportionofearlywithdrawerswithabank.5 structure(i)takingintoaccountthewithdrawalstrategiesofagentsasspeci¯edinLemma1.Inacompetitivemarket,thecompetitiveequilibriumcanbede¯nedbyusingthebackwardinductionmethod.De¯nition1Thecompetitiveequilibriumcontract(r¤1;r¤2;i¤)inthemarketeconomyshouldsatisfythefollowingconditions:1.Withdrawaldecisionsunderacertaincontractareasspeci¯edinLemma1.2.i¤ischosentomaximisethebank'sexpectedpro¯tforagiven(r1,r2).3.Agentsdeposittheirendowmentswiththebankthato®ersthebestcontract(withthehighestexpectedutility).4.Interestrates(r¤1,r¤2)arechosentomaximisethebank'sexpectedpro¯t.Inamarketeconomy,theaboveproblemisequivalentto\bankschoosethebestcontractforarepresentativeagentsubjecttotheconstraintthatexpectedpro¯tiszero".8TheresultisintuitiveandcanbeeasilyshownbytheDualityTheorem.Arepresentativebankmustmakezeropro¯tinthemarketequilibrium.Ifthepro¯tispositive,atleastonebankwilldeviatebyo®eringahigherinterestrateanditwillwinalldeposits.Thisbid-upprocesswillcontinueuntilallbanks'pro¯tsaredriventozero.Thiszero-pro¯tpropertyimpliesthat,inamarketeconomy,thee®ectivelong-terminterestrateisdeterminedbydividingtheremainingresourcesamonglateconsumersandthereforeisstate-contingent.9Inotherwords,¯nancialintermediariesonlyprovidearisk-sharingmechanismagainsttheidiosyncraticriskbutcannotdiversifytheaggregateriskintheeconomy.Accordingly,thebanks'optimisationproblemisasfollows:Lemma2Theequilibriumcontractinamarketeconomycanbecharacterisedbythefollowingproblem:maxr1;iRR¤01¡i+iR(1¡¿)r1u(r1)f(R)dR+R1R¤[®u(r1)+(1¡®)u(re2(R))]f(R)dR(2.3)s:t:1¡i¸r1®8Noticethatthebank'sexpectedpro¯tisinverselyrelatedtoarepresentativeagent'sexpectedutility.9AsdiscussedinSection4,whenthebankshavetheirownstake(capitalassetsordepositinsurancepremium),thespeci¯cationofthelong-terminterestratebecomesmuchmoreimportant.Bankswillmakezeroexpectedpro¯tratherthanmakezeropro¯tinallstates.6 R¤´r1¡1+iire2=1¡i+iR¡r1®1¡®Theobjectivefunctionconsistsoftwoparts.The¯rstpartreferstothecasewheretheassetreturnislowandacompletebankrunisunavoidable.Banksliquidatealltheirassetsandpaythedepositorsaccordingtothesequentialservicerule.Onlyafraction(1¡i+iR(1¡¿)r1)ofagentsarefullypaidandtherestreceivenothing.Thesecondpartcorrespondstothecaseinwhicheveryoneknowsthattheeconomyisinagoodstateandnobankrunoccurs.Banksdonotneedtoliquidatetheirproductiveassetsintheinterimperiod.Impatientagentsreceivetheshort-terminterestrateandpatientagentssharetheremainingassetsinperiod2.The¯rstconstraintisthebudgetconstraintinperiod1,whichrequiresthattheminimumholdingofsafeassetsbetheexpectedshort-termliabilitiestoimpatientagents.Thisisquiteintuitivebecauseifthebanksholdfewersafeassets,thentheywillalwayshavetoliquidatesomeriskyassetsintheinterimperiod.Sincetheriskyassetislessproductiveintheshortrun,thisportfoliochoiceisneveroptimal.Anotherimplicationfromthe¯rstconstraintisthatthebanksmighthaveanincentivetoholdextrasafeassets(1¡i¡r1®).Holdingextraliquidityhastwooppositee®ects.(1)Itcouldbewelfare-improvingfortworeasons.First,sincetheprobabilityofdefault(bankruns)isdecreasingini,anextraliquidityholdingwillmakethebankingsectorlessvulnerabletoruns.Second,ifabankrunhappens,holdingmoresafeassetswillreducetheliquidationcosts.(2)However,holdingextraliquidityiscostlyinthatwhennobankrunoccurs,thelateconsumerswillreceivealowerpaymentbecausethesafeassetislessproductiveinthelongrun.Howmuchextraliquiditythebanksshouldholddependsonwhiche®ectplaysadominantrole.Thesecondconstraintde¯nesthethresholdreturnbelowwhichabankrunisinevitableasindicatedbyLemma1.Thethirdconstraintspeci¯esthepayo®forthelateconsumersbyusingthezero-pro¯tproperty.3SociallyoptimalallocationInthissection,Iintroducetwotypesofsociallyoptimalallocationsandcomparethemwiththeequilibriumoutcomeinamarketeconomy.Giventheexistenceoftwokindsofrisks(theidiosyn-7 craticriskinpreferencetypeandtheaggregateriskinassetreturn)intheeconomy,asocialplannercanprovidearisk-sharingmechanismthroughsettingupanationalbankandallocatingthere-sourcesaccordingtothetrueliquidityneeds.Dependingondi®erenttypesofrisksbeinginvolved,Ide¯netwotypesofsociallyoptimalallocations.De¯nition2(First-bestallocation)A¯rst-bestallocationisde¯nedinthefollowingproblem:maxr1;i®u(r1)+(1¡®)u(r2)(3.1)s:t:1¡i¸r1®r2=1¡i+i¢E(~R)¡r1®1¡®Inthe¯rst-bestallocation,thesocialplannerpayseachagentaccordingtohistruepreferencetype.Nocostlyliquidationoccursintheinterimperiod.Thesocialplannerchoosestheoptimaldemand-depositcontractandinvestmentportfoliotomaximisetherepresentativeagent'sexpectedutility.Theconstraintequationsrefertothebudgetconstraintsforthesocialplannerinperiod1andperiod2,respectively.Animportantfeatureofthe¯rst-bestallocationisthat,intheperiod2,thesocialplanneractuallyprovidesasmoothingdeviceacrossthestateofthebusinesscycleasthelong-terminterestrateisnon-state-contingent.Onepossibleexplanationisthatthesocialplannercansubsidisetheinterestpaymentsinbadstatesbyusingthepro¯tsingoodstatesifithasenoughreserveassets.10Asaresult,boththeidiosyncraticriskandtheaggregateriskarediversi¯ed.Lemma3The¯rst-bestallocationischaracterisedby:u0(rf1)=E(~R)¢u0(rf2)(3.2)if=1¡rf1®(3.3)rf2=if¢E(~R)1¡®Proof:seeAppendixB.10Anotherpossibleexplanationisthateachbankisconfrontedwithindividualinvestmentrisk.Therefore,thereisnoaggregateinvestmentriskintheeconomy.Asocialplannercano®erarisk-sharingschemeamongallbanksbyusingthepro¯tsfromhigh-returnbankstosubsidisethelow-returnbanks.8 Equation(3.2)isthefamiliarEulerequationforanoptimalcontract,whichbalancesthemarginalcostandmarginalbene¯tofchanginginterestratesinequilibrium.Equation(3.3)im-pliesthatthesocialplannershouldholdaminimumamountofsafeassetsforinterimpayment,andinvestalltheremainingdepositsinthemoreproductivetechnology.Thisisnotsurprisingbecausethereisnobankruninthesociallyoptimalcontractandthereforeextraliquidityholdingisundesirable.Ifwenowimposetherestrictionthatthesocialplannercannotsmooththeconsumptionacrossthestateofthebusinesscycle,thesociallyoptimalallocationisde¯nedasfollows:De¯nition3(Second-bestallocation)Asecond-bestallocation11isde¯nedinthefollowingprob-lem:maxr1;iER[®u(r1)+(1¡®)u(r2(R))](3.4)s:t:1¡i¸r1®r2(R)=1¡i+iR¡r1®1¡®Inthesecond-bestallocation,allagentswithdrawtheirdepositsaccordingtotheirtrueliquidityneeds.Theriskinidiosyncraticpreferencetypesisdiversi¯edthroughthecontract.However,theaggregateriskstillexistsbecausetheamountoftotalassetsvariesacrossthestateofthebusinesscycle.Incontrasttothe¯rst-bestcontract,thelong-terminterestpaymentisstate-contingent,andthesocialplannerisrequiredtobreakevenineverystate.Thesecond-bestallocationshouldsatisfythefollowing¯rst-ordercondition:u0(rs1)=E[u0(rs2(R))¢R](3.5)Proposition1Comparingtheabovethreeallocations,arepresentativeagentobtainsthehighestwelfareinthe¯rst-bestallocationandthelowestwelfareinthecompetitiveequilibriumcontract.Inotherwords,theequilibriuminthemarketeconomyissuboptimal.Proof:seeAppendixC.11Iammisusingtheterm\second-best"inthispaper.Itisdi®erentfromthestandardde¯nitioninwelfareeconomics.HereIamusingittoemphasisethefactthatthisallocationdiversi¯esonlytheidiosyncraticriskintheeconomy.9 Thisoptimalitysequenceisnotsurprising.Figure1providesanintuitiveexplanation.Whilethe¯rst-bestallocationprovidesaninsuranceagainstboththeidiosyncraticriskandtheaggregaterisk,thesecond-bestallocationonlydiversi¯estheidiosyncraticrisk.Inamarketeconomy,thebankingsectoralsoinsuresagainsttheidiosyncraticrisk.Thisrisk-sharingmechanismfunctionswellwhentheeconomyisinagoodstate.However,whentheassetreturnturnsouttobelow,allagentshavetheincentivetorunonthebanksandthisrisk-sharingmechanismbreaksdown.Duetotheexistenceoftheliquidationcosts,theeconomysu®erslargewelfarelosses.Thisdestabilisatione®ect,whichisrelatedtothefragilityofthebankingsector,partiallycancelsouttherisk-sharingbene¯tandmakesthecompetitiveequilibriuminferiortothesecond-bestoptimum.Forillustration,Iprovideanumericalexample.Supposethattheproportionofimpatientagents(®),thedistributionoftheassetreturn(~R),theliquidationcost(¿),andtheformoftheutilityfunctionareasfollows:®=0:5,~R»lognormal(0:25;0:52),¿=0:5,u(c)=(c+1)1¡¯¡11¡¯,where¯=2FromLemma3,itiseasytode¯nethe¯rst-bestallocation.Theoptimalcontractfeaturesrf1=1:0160,rf2=1:4317andif=0:4920.TheexpectedutilityforarepresentativeagentisE(Uf)=0:5464.Similarly,thesecond-bestallocationcanbesolvedfromproblem(3.4).Thecontractischar-acterisedbyrs1=1:0335andis=0:4832.Thelong-terminterestrateisstate-contingentandisdeterminedbydistributingtheremainingassetsamonglateconsumers.ArepresentativeagentreceivesanexpectedutilityofE(Us)=0:5295,whichislowerthanunderthe¯rst-bestallocation.Theequilibriumcontractinamarketeconomy,whichisde¯nedinproblem(2.3),canbesolvedbyusingthegrid-searchingmethod.Inthegivenexample,itfeaturesrd1=0:83,12id=0:38andE(Ud)=0:5164.Inequilibrium,bankschooseabank-runcontract(bankrunsoccurwhen12Theshort-terminterestrateislowerthan1inthecompetitiveequilibrium.Thisispartlybecausetheliquidationvalueisverylowinthismodel.Banksareprovidingarisk-sharingmechanismtotheagentsbyo®eringthemasmootherconsumptionpath.10 R0:5526)insteadofarun-proofcontract.Besides,thebanksarewillingtoholdextrasafeassets(1¡i¡r1®�0)inequilibriumtomitigatetheshockscausedbybankruns.Notsurprisingly,thecompetitiveequilibriumoutcomeisinferiortothesecond-bestoptimum.4WelfareanalysisofvariouspoliciesInthissection,Iextendthebenchmarkmodelanddiscussthewelfaree®ectsof¯vedi®erentgovernmentpolicies.4.1SuspensionofconvertibilityofdepositsSuspensionofconvertibilityofdepositsallowsthebankstosuspendpaymentwhentheearlywith-drawalsreachacertainlevel(®N)intheinterimperiod.Inthe19thandearly20thcentury,thispolicywasusedveryoftenduringbankingpanics(seeDwyerandGilbert1989).Eveninrecent¯nancialcrises,somesimilarpolicieswereadoptedastemporarycrisismanagementmeasures.Forexample,Malaysiadecidedtoimposewiderangecapitalcontrols13soonaftertheoccurrenceofthe1997EastAsiancrisis.DiamondandDybvig(1983),intheirclassicpaper,claimthatsuspensionofconvertibilityofdepositscanbeusedtoachieveoptimalrisk-sharing.Theyarguethatthesuspensionpolicyputsarestrictiononagents'expectations,eliminatespossiblebankrunsandthereforeachievesthesocialoptimum.However,thisconclusionisnotvalidinthismodel.Infact,thewelfaree®ectofthispolicysolelydependsonthedecisionsequenceinperiod1.Inmostsituations,thispolicyturnsouttobeine±cient.Proposition2Inamarketeconomy,suspensionofconvertibilityofdepositsisbothexpostandexanteine±cienttopreventbankruns.Proof:accordingtothepolicy,theaggregatewithdrawalinperiod1cannotexceed®¢N.Thepayo®functionforagenti,whoobservesanaggregateearlywithdrawalLiaheadofhim,is:r1(R;Li)=(r1ifLi®0ifLi¸®ifwithdraw13ThegovernmentannouncementinSeptember1998prohibitedcitizensfromtakingmorethanUSD100outofMalaysia.11 r2=1¡i¡r1®+iR1¡®ifwaitApatientagent'sstrategydependsonthereturnonassetsandhispositioninthequeue.IfR¸r1¡1+ii,healwayschoosestowait;ifRr1¡1+ii,hechoosestowithdrawhisdepositwhenfewerthan®¢Nagentshavewithdrawntheirdepositsandwaitotherwise.Therefore,di®erentdecisionsequenceswillleadtodi®erentpolicye®ects.Letus¯rstconsiderthebestscenariounderwhichinperiod1agentsmakedecisionsaccordingtothefollowingsequence:allimpatientagentsmakewithdrawaldecisionsaheadofallpatientagents.Giventhisdecisionsequence,suspendingconvertibilitycanachievethesecond-bestallocation.WhenR¸r1¡1+ii,allpatientagentschoosetostayandnobankrunoccurs.WhenRr1¡1+ii,onlyimpatientagentsareabletowithdrawtheirdepositsandthepatientagentsareforcedtowaitunderthesuspensionpolicy.Asaresult,thedestabilisatione®ectcausedbybankrunsiscompletelyremovedwithoutanysidee®ects.Thenewequilibriuminsurescompletelyagainsttheidiosyncraticriskandachievesthesecond-bestoptimum(exantee®ect).Butthisconclusionisnolongervalidwhenthedecisionsequenceisdi®erent.Consideranotherextremecaseinwhichthedecisionsequenceistheoppositetothatabove:allpatientagentsmakewithdrawaldecisionsaheadofimpatientagents.Inthiscase,thedepositsuspensionpolicyturnsouttobeine±cient.Inparticular,whenRr1¡1+ii,thepolicycauseslargewelfarelossesfortheimpatientagents.Withoutthesuspensionpolicy,bankrunshappenandthe¯rst1¡i+iR(1¡¿)r1agentsreceivetheshort-terminterestrate,r1.Withthesuspensionpolicy,however,onlythepatientagentsareabletowithdrawtheirdepositsfromthebanksinperiod1(notice\withdraw"istheoptimalstrategyforeachpatientagentinthissituation).Althoughthebanksnolongersu®erliquidationcosts,inperiod2theremainingassetsareuselesstotheimpatientagentswhohavebeenforcedtostay.Thesuspensionpolicyleadstoaseveremisallocationofassets.Tosummarise,thesuspensionpolicyhastwoe®ects.Ontheonehand,itprotectsthebanksfromrunsandminimisestheliquidationcosts.Ontheotherhand,itbringsaboutthe\misallocatione®ect"assomeimpatientagentsmaybeprohibitedfromwithdrawinginperiod1.Conditionalonthedecisionsequencespeci¯edinthemodel,the\misallocatione®ect"maybecomeveryseriousandoutweighthebene¯tinmostcases.Theunderlyingreasonfortheexistenceofthe\misallocatione®ect"isthatthesuspension12 policycanonlycontrolthequantityofaggregateearlywithdrawalsbutcannotdistinguishthetrueliquidityneedsofindividualagents.Therefore,itcannotguaranteethattheresourcesareallocatedinthecorrectway.Inmostcases,thispolicyisnote±cientinpreventingbankpanics.144.2Taxationonshort-termdepositsAsecondpolicythathasbeenusedduringbankingcrisesistheimpositionoftaxes(orpenalties)onearlywithdrawals.Itisarguedthatthispolicywillindirectlypreventbankpanicsbyincreasingthecostofearlyconsumption.Inpractice,Chileimposedataxonshort-termcapitalout°owsintheearly1990s.Thequestionis:isthispolicyane±cientwaytopreventbankruns?Supposethegovernmentimposesataxratetonshort-termdeposits,andthenreturnsthecollectedtaxestoallagentsasalumpsumtransfer.Obviously,underthistaxregime,itistheafter-taxpayo®thata®ectsthepatientagents'withdrawaldecisions.AsshowninAppendixD,underacertaincontract(r1;r2;i),whenR¸R¤´r1(1¡t)(1¡®)+r1®¡1+ii,orequivalentlyr1;after¡tax·r2;after¡tax,allagentswillreporttheirtypestruthfullyandnobankrunhappens.However,whenRR¤,\waiting"becomesadominatedstrategyandabankrunoccurs.Ifnobankrunhappens,anearlyconsumerreceivestheafter-taxincomeofr1(1¡t)+r1¢Lt,inwhichr1(1¡t)istheafter-taxshort-terminterestrate,andr1¢Ltisthegovernmenttransfer.Instead,alateconsumerwouldreceivere2(R;L)+r1¢Lt,whichconsistsofthelong-terminterestpaymentandthegovernmenttransfer.Ontheotherhand,ifabankrundoeshappen,aproportion(p=1¡i+iR(1¡¿)r1)oftheagentswhoreceivetheshort-terminterestpaymentspaythetax.Afterthegovernmentredistribution,theincomedistributionisasfollows:aproportionpofagentsreceiver1(1¡t)+p¢r1t,andtherestoftheagentsreceivethegovernmenttransferofp¢r1t.Thebanks'optimisationproblemisthereforeasfollows:maxr1;iRR¤0[p¢u(r1(1¡t)+p¢r1t)+(1¡p)u(r1t)]f(R)dR(4.1)+R1R¤[®u(r1(1¡t)+®r1t)+(1¡®)u(1¡i+iR¡r1®1¡®+®r1t)]f(R)dRwherepandR¤arede¯nedasabove.Thispolicydi®ersfromthesuspensionofconvertibilityofdepositsinthatthesuspension14Inthegivenexample,undertheworstscenario,thecompetitiveequilibriumunderthesuspensionpolicyfeaturesr1=0:80,i=0:32andE(U)=0:5133.Arepresentativeagentisevenworseo®underthenewenvironment.13 policyimposesadirectquantitycontrolonearlywithdrawals,whilethetaxpolicycancontrolboththequantityandthecompositionofearlywithdrawalsthroughitse®ectontheafter-taxinterestpayment.Therefore,thetaxpolicywillnotbringaboutthe\misallocatione®ect".However,therearetwodisadvantageswiththetaxpolicy.First,itcannoteliminatethepossibilityofbankrunsunlessthetaxrateisextremelyhigh.Second,andmoreimportantly,theearlyconsumptiontaxwillintroducedistortionstobanks'investmentdecisions.Inparticular,banksnowfaceadi±culttrade-o®.Iftheywishtomaintaintheafter-taxshort-terminterestrateattheinitiallevel,theyhavetoinvestlessintheriskyassets,thereforesu®eringtheproductivityloss.However,iftheywishtomaintaintheinvestmentinthemoreproductiveassets,theafter-taxshort-terminterestratehastobereduced,therebydamagingtherisk-sharingbene¯t.Thiscon°ictwillusuallyleadtounderinvestmentandinvolvewelfarelosses.Proposition3Anearlywithdrawaltaxisbothexpostandexanteine±cientandisalwaysdom-inatedbythesecond-bestallocation.Proof:seeAppendixE.Iusethegivennumericalexampletoillustratetheexantee®ectsofthetaxpolicy.Figure2re°ectshowtheequilibriumafter-taxshort-terminterestrate,theprobabilityofbankruns,andexpectedutilitychangeunderdi®erentlevelsoftaxes.Thenumericalresultssuggest:(1)Asthetaxincreases,thebefore-taxshort-terminterestrate(thedashedline)increases,buttheafter-taxshort-terminterestrate(thesolidline)inequilibriumalmostremainsthesame.However,theheavyburdeninshort-termliabilityforcesthebankstoinvestmoreintherisklessassetsandlessinthemoreproductiveassets.Theunderinvestmentphenomenonleadstowelfarelossesintheeconomy.(2)Theimpositionofearlywithdrawaltaxesdoesnotimprovethewelfareofarepresentativeagent.AsFigure2shows,thewelfareincreasesinsigni¯cantlywhenthetaxisatalowormediumlevel,muchinferiortothe¯rst-bestandsecond-bestoutcomes.Whenthetaxrateisveryhigh,theequilibriumoutcomeisevenworsethanthebenchmarkresult,suggestingthatthedistortione®ectismuchhigherwhenthetaxrateishigher.(3)Thetaxpolicyisnotverysuccessfulinpreventingbankrunsexante.Infact,theprobabilityofbankrunsinequilibriumevenincreaseswhenthetaxrateisnotveryhigh.14 (4)Intheextremecaset=1,nobankrunoccursinequilibrium.Thisisquiteintuitive.Becauseifanagentchoosestowithdrawearly,hewillloseeverythingandonlyreceivethegovernmenttransfer.Thusnopatientagentwilldisguisehimselfasanimpatienttype.Asaresult,bankrunsarecompletelyeliminated.Buttheextremelyhightaxalsobringssevereinvestmentdistortiontotheeconomy.Onaggregate,theequilibriumoutcomeismuchworsethanthebenchmarkcase.4.3Full-coveragedepositinsurance(FCDI)schemeTheroleofdepositinsurancehasbeenaverycontroversialtopic.Insomeearlywork(DiamondandDybvig1983),depositinsuranceisconsideredasane±cientpolicytoachievethesocialoptimum.However,continuingresearchsuggeststhatweshouldbemorecautious.CooperandRoss(1998)pointoutthatdepositinsuranceschemeseliminatethepossibilityofbankrunsbutatthesametimereduceagents'incentivetomonitorthebanks.Krugman(1998a),afterthe1997EastAsiancrisis,arguesthattheimplicitdepositinsurancepolicycausesaseveremoralhazardproblemandleadstoimprudent\overinvestment",whichisthecoreelementintheeconomiccrash.ArelateddebateistheroleoftheIMF.Someeconomists(Sachs1998,RadeletandSachs1998)arguethatalenderoflastresortisane±cientwaytopreventself-ful¯lling¯nancialpanics;thereforetheIMFshouldbeexpandedandalargeramountoffundsshouldbeprovidedmorequicklywhen¯nancialcrisesoccur.Attheotherextreme,Schwartz(1998)andCalomiris(1998)criticisetheIMFforactingaslenderoflastresort,arguingthatsuchactioncausesmoralhazardproblemsandinthelongrunincreasesthefragilityoftheworld¯nancialsystem.TheysuggestthatIMFbailoutschemesshouldbeavoided.Thissectionexplorestheexpostandexantee®ectsofanFCDIscheme,orablanketguaranteescheme.UnderanFCDIscheme,thecentralbank(orapublicauthority)guaranteesdepositorsthepromisedinterestrateswhenthebanksareinsolvent.Andthefundingsourcecomesfromtheinsurancepremiumpaymentbythebanks.15UnderanFCDIscheme,thepayo®functionforeachagentis:re1=r1re2=r2(4.2)15Iassumethebankspaythepremiumoutoftheirownfundsratherthanoutofdeposits.Obviously,anydepositinsuranceschemewithoutanyinsurancepremiumrequirementonthebanksisexanteine±cientandunsustainableduetothemoralhazardproblems.15 wherere1andre2aretheexpectedpayo®anagentcanreceiveifhechoosestowithdrawearlyortowait,respectively.Obviously,whenr1·r2,nopatientagenthastheincentivetomisreporthispreferencetype.Asaresult,nobankrunhappensinperiod1.Lemma4Afull-coveragedepositinsuranceplanisexposte±cientinthatitcaneliminatebankrunsandavoidcostlyliquidation.AmoreinterestingproblemiswhethertheFCDIschemeisexantee±cient.Toputitanotherway,howwillthebanksrespondtotheFCDIschemeinchoosingthedepositcontractandportfoliostructure?Andhowhighaninsurancepremiumshouldbechargedinordertokeeptheplansustainable?I¯rststudythebehaviourofbanksunderagiveninsurancepremium±.Thebanks'problemisasfollows:maxr1;r2®u(r1)+(1¡®)u(r2)(4.3)s:t:1¡i¸r1®(4.4)R¤=r1®+r2(1¡®)¡1+iiR1R¤i(R¡R¤)¢f(R)dR¸±(4.5)i=argmaxR1R¤i(R¡R¤)¢f(R)dRTheobjectivefunctionre°ectsthefactthatthereisnobankruninperiod1whenanFCDIplanexists.Allimpatientagentsgetaconsumptionofr1andallpatientagentsreceiver2.The¯rstconstraintistheusualbudgetconstraintinperiod1.Thesecondconstraintspeci¯esthethresholdreturnR¤belowwhicharescuepackageisneeded.Itcomesfromtheconditionthat1¡i+iR¤¡r1®1¡®=r2.WhenR¸R¤,banksareabletoearnpositivepro¯ts.WhenRR¤,banksareinsolventandarescuepackageisimplementedbythecentralbank.Thethirdconstraintistheincentiveconstraintforbankstojointheinsuranceplan,whichstatesthatbanksmustbeabletoearnenoughpro¯tstocovertheinsurancepremiumpayment.Thelastconstraintdeterminesthechoiceofportfoliostructure,whichmaximisesthebanks'expectedpro¯tsinequilibrium.Lemma5Inproblem(4.3),bothrestriction(4.4)andrestriction(4.5)arebinding.16 Proof:seeAppendixF.Lemma5statestwoimportantfeaturesoftheequilibrium.First,bankschoosetoholdnoexcessliquidityandmaximisetheirinvestmentsinthemoreproductivetechnology.Second,inthecompetitivemarket,thenetpro¯tforarepresentativebankiszero.Anotherfeatureoftheequilibriumcontractunderthedepositinsuranceschemeisthatthelong-termpaymentforapatientagentisnotstate-contingent.Hence,thesystemprovidesarisk-sharingmechanismagainstbothidiosyncraticriskandaggregaterisk.Thenewquestionis:canthispolicybeusedtoachievethe¯rst-bestoptimum?Unfortunately,theanswerisnobecauseofthemoralhazardissues.BasedonLemma5,itisstraightforwardtode¯netheequilibriumcontractunderacertainFCDIscheme.Lemma6Theequilibriumcontract(r1;r2)inproblem(4.3)shouldsatisfythefollowingconditions:u0(r1)=u0(r2)¢E[RjR¸R¤](4.6)Z1R¤(1¡r1®)(R¡R¤)¢f(R)dR=±(4.7)whereR¤=r2(1¡®)1¡r1®.Equation(4.6)isthefamiliarEulerequation.Theintuitionisasfollows.Iftheshort-terminterestrateisreducedbyanamountof4r,theagentwillsu®eralossofu0(r1)¢4rintheshortrun.ButthelongrunpaymentwillbeincreasedbyE[RjR¸R¤]¢4r.16Inequilibrium,themarginalcostandmarginalbene¯tshouldbeequalised.Noticethedi®erencebetweenthe¯rst-orderconditionsinthe¯rst-bestallocationandtheFCDIsystem.Inthe¯rst-bestenvironment,themarginalrateoftransformationisdeterminedbytheunconditionalmeanofassetreturn.Inthelattercase,however,thebanksdonotcareaboutthelossesforthecentralbank.Therefore,themarginalrateoftransformationisrelatedtotheconditionalmeanofassetreturn.This\extortione®ect",whichreferstothefactthatbanksignorethenegativeexternalityofhigherbailoutcostsando®ertoohighinterestrates,preventstheeconomyfromachievingthe¯rst-bestoptimum.16Considerthepro¯tfunctionforthebanks.BankslosenothingwhenRR¤(centralbankwillbailout)andgainthepro¯tswhenR¸R¤.The4rassetswillbringthebanksanexpectedpro¯tofE[RjR¸R¤]¢4r.17 Lemma7Underthefull-coveragedepositinsuranceplan,the¯rst-bestallocationisfeasiblebutnotchoseninthemarketeconomy.Proof:I¯rstshowthatthe¯rst-bestallocationcontract(rf1,rf2,if)isfeasiblewhenthecentralbankchargesaninsurancefeeof±f=R1E(~R)if[R¡E(~R)]¢f(R)dR.UnderthisFCDIscheme,1.Bankscanmakeenoughpro¯tstocovertheinsurancepremiumpayment.FromLemma3,rf2(1¡®)=if¢E(~R)andif=1¡rf1®,thereforeR¤=rf2(1¡®)1¡r1®=E(~R).Fromthede¯nitionof±f,theexpectedpro¯tforthebanksequalstheinsurancefeepayment.2.ThebailoutcostscanbecoveredbytheinsurancepremiumpaymentbacauseZR¤0[rf1®+(1¡®)rf2¡(1¡if+ifR)]¢f(R)dR=ZE(~R)0if[E(~R)¡R]¢f(R)dR=Z1E(~R)if[R¡E(~R)]¢f(R)dR=±fUnfortunately,the¯rst-bestallocationwillnotbechoseninthemarketeconomy.Fromequation(4.6),itisobviousthatbankswillnotchoosethe¯rst-bestcontract.Instead,bankswillo®erthedepositorshigherinterestratesandtheexpectedbailoutcostsforthecentralbankarehigherthan±f.TomaintainacredibleFCDIscheme,thecentralbankhastochargeahigherinsurancepremium±¤thatisabletocoveritsbailoutcosts.Inequilibrium,ZR¤0i(R¤¡R)¢f(R)dR=±¤Combiningwiththeequilibriumconditions(4.6)and(4.7),theself-sustainablemarketequilib-rium(r¤1;r¤2;i¤;±¤)underanFCDIschemeischaracterisedby:8��������:r¤2(1¡®)=(1¡r¤1®)E(~R)u0(r¤1)=u0(r¤2)¢E[RjR¸E(~R)]±¤=RE(~R)0i¤[E(R)¡R]¢f(R)dRi¤=1¡r¤1®Proposition4Afull-coveragedepositinsurancebringsstabilityintothebankingsectorandim-provesthewelfareofinvestors.However,itcannotachievethe¯rst-bestoptimumduetothe\moralhazard"problem.18 Istillusethenumericalexampletoillustratethewelfaree®ectsoftheFCDIscheme.Figure3showsthatthe¯rst-bestallocationisfeasiblebutwillnotbechosenwhen±=±f.Theasteriskrepresentsthe¯rst-bestallocation,underwhichthedepositinsuranceauthority'sbailoutcostscanbefullycoveredbytheinsurancepremiumpayment.Butindividualbankshavetheincentivetoo®erahigheraverageinterestrate17toinvestorsandtheFCDIschemeisnolongersustainable.Speci¯cally,when±=±f=0:1407,theequilibriumcontractwillber1=0:85,r2=1:8044,i=0:575,E(U)=0:5514andtheexpectedlossforthecentralbankis0:0656.Theagentsarebettero®thanunderthe¯rst-bestoptimum,ofcourse,atthecostofhugede¯citstothecentralbank.Figure4showsthecontractsthatthebankswillo®erunderdi®erentinsurancefees.Whentheinsurancefeeincreases,thebankswillo®eraloweraverageinterestrate,thecentralbank'sbalancesheetimproves,andthewelfareforarepresentativeagentislower.Inthegivenexample,theequilibriumcontractinwhichthepremiumpaymentscanfullycoverthebailoutcostsis:±¤=0:1677,r¤1=0:8321,r¤2=1:6993,i¤=0:584andE(U)¤=0:5419.Itisinferiortothe¯rst-bestallocationduetotheexistenceofmoralhazardproblems.4.4Interest-capdepositinsurance(ICDI)schemeDuetotheexistenceofmoralhazardproblemsundertheFCDIscheme,policymakershavebeenlookingfordi®erentvariantsofdepositinsuranceplanstomitigateortoremovethisadversee®ect.TwomajorvariantswereproposedintherecentreportbytheFinancialStabilityForum(2001):alimited-coveragedepositinsuranceschemeandcoinsurance.Thelimited-coveragedepositinsuranceschemeprotectstheprincipalandinterestofeachdepositoruptoacertainlimit.18Thecoinsurancesystemspeci¯esthatonlyaproportionofdeposits(includinginterest)areprotected.Inthissubsection,Iproposethatasimilarvariantofdepositinsurancescheme,whichIrefertoasthe\interest-capdepositinsurance"(ICDI)scheme,canremovethemoralhazardproblemsandachievethe¯rst-bestoptimum.UnderanICDIscheme,themaximumbasicprotectioneachdepositorcanreceiveishisprincipal17De¯nedas®r1+(1¡®)r2.18Forexample,themaximumprotectionforeachdepositorisUSD100,000intheUnitedStatesandCAD60,000inCanada.19 andacertainamountofinterestthatdoesnotexceedapredeterminedcap(r¡1).Inotherwords,themaximumpaymentadepositorcanreceiveuponbankdefaultisr.19Asshownbelow,awell-designedICDIcanovercomethecon°ictofinterestsbetweenthecentralbankanddepositbanks,andachievethe¯rst-bestsocialoptimum.Bysettingtheinterestcaponprotection,thedepositinsuranceauthorityindirectlyimposesacapontheinterestratethatadepositbankwillo®ertoagents.Proposition5Aninterest-capinsuranceschemeise±cientinpreventingbankrunsandcanachievethe¯rst-bestsocialoptimum.Inparticular,anICDIschemewithr=rf2anddepositinsurancepremium±f=R1E(~R)if[R¡E(~R)]¢f(R)dRisabletoachievethe¯rst-bestsocialoptimum.AppendixGshowsthat,underthisICDIscheme,bankswillchoosethe¯rst-bestcontract(r1=rf1,r2=rf2).Ontheonehand,thebankswillnotchoosealowerr2becausetheyalwayshavetheincentivetomaximisetheutilisationofdepositinsurance.Moreimportantly,ontheotherhand,thebankshavenoincentivetoincreasetheinterestrateo®erasundertheFCDIscheme.UndertheFCDIscheme,thebankschoosetoincreasethelong-terminterestrateanddecreasetheshort-terminterestrate(tomaintainexpectedzeropro¯t).Themarginalcostofthelowershort-terminterestratewillbecompensatedbythefactthatthelong-terminterestrateishigherinallstates.However,undertheICDIscheme,thisincentivenolongerexistsbecausearepresentativeagentcannotearnahigherlong-terminterestratewhenbanksareinsolventduetotheexistenceofacoveragelimit.Therefore,theinitialmoralhazardproblem,inwhichbanksincreasethecentralbank'sbailoutcoststhrougho®eringhigherinterestrates,nolongerexistsunderthespeci¯cICDIscheme.20OneimportantimplicationfromAppendixGisthattheICDIschemeshouldcoverboththeprincipalandpartof(orall)interestratepayments.Thisisnotsurprising.Themaximumpro-tectionmustbegreaterthantheshort-terminterestratetoinducetheagentsnottorunonthebanks.2119Inthismodel,thisICDIschemeisactuallythesameasthelimited-coveragedepositinsurancescheme.Theydi®erwhenagentsareheterogeneous.20However,weshouldbecautiousaboutthisresultinpractice.Thisconclusionisbasedontheassumptionthatallbanksarefacedwiththesameproductivityshock.Inreality,consideringthefactthatbanksarealsoconfrontedwithidiosyncraticproductivityshocks,auniforminterestcapisnotabletocatchthisheterogeneity.21Inpractice,toreducethebailoutcostsofdepositinsuranceschemes,ortoincreasethelargeagents'incentiveto20 4.5CapitalrequirementAnotherwidelyusedtoolinbankregulationistheimpositionofacapitalrequirement.Ingeneralterms,acapitalrequirementspeci¯eshowmuchequityabankshouldholdforeachunitofdeposits.Thisequitycanbeinvestedineithertechnologyandcanbeusedtorepaythedepositorswhenassetreturnsarelow.Throughoutthispaper,Iuse·torepresentthecapitalrequirementforeachunitofdeposit.22Fromthede¯nition,thecapitalrequirementrelatestothebanks'ownfunds.Supposeindividualbanksinvest1¡i+·inthesafeassetsandiintheriskyassets.Theexistenceofcapitalreducestheprobabilityofbankrunsbecauseitincreasesthebanks'solvencyability.FollowingLemma1,equilibriumaggregateearlywithdrawal,L,equals®whenR¸r1¡(1¡i+k)iandL=1otherwise.Therearethreepossibleoutcomes(seeFigure5):(1)RR1,whereR1´r1¡(1¡i+k)iWhenthereturnonriskyassetsisverylow,thebankdefaultisunavoidable.Allagentsrushtobanksinthehopeofretrievingpartof(orall)theirdeposits.Abankrunhappensandbankslosetheirentirecapital.(2)R1·RR2,whereR2´r1®+r2(1¡®)¡(1¡i+k)iWhenthereturniswithintheintermediatelevel,thebanksarenotinimmediatedangerofdefaultandpatientagentsarewillingtowaitintheinterimperiodandnobankrunhappens.Butbankshavetousealltheircapitaltorepaythedemand-depositcontracts.Patientagentsreceiveapaymentofre2=1¡i+k+iR¡r1®1¡®,whichishigherthanr1butlessthanthepromisedlong-terminterestrate,r2.(3)R¸R2Whenthereturnonriskyassetsishigh,alllateconsumersreceivethepromisedinterestrate,r2,inthelongrun.Bankrunsneveroccur.Thepayo®functionforbanksissubtler.De¯neR3´r1®+r2(1¡®)¡(1¡i)i.WhenR2·R·R3,bankshavetousepartoftheircapitalcollateraltopaythelateconsumers.WhenR¸R3,thecapitalcollateralisuntouchedandthebanksearnpositivepro¯ts.Combiningtheaboveanalysis,underacertaincapitalrequirement(·),theoptimisationproblemmonitorthebanks,thedepositinsuranceauthoritymaychoosealowerinterestcaporthecoinsurancescheme.22Obviously,acapitalrequirementof·isequivalenttoacapitalratioof·1+·becauseassets=liabilities=deposits+capital.21 forarepresentativebankis:maxr1;r2ZR101¡i+·+iR(1¡¿)r1u(r1)¢f(R)dR(4.8)+ZR2R1[®u(r1)+(1¡®)u(re2(R))]¢f(R)dR+Z1R2[®u(r1)+(1¡®)u(r2)]¢f(R)dRs:t:imaximisesZR20¡·¢f(R)dR+Z1R2i(R¡R3)¢f(R)dR(4.9)ZR20¡·¢f(R)dR+Z1R2i(R¡R3)¢f(R)dR=·[E(~R)¡1](4.10)1¡i+·¸r1®(4.11)Theobjectivefunction,whichspeci¯estheexpectedutilityforarepresentativeagent,consistsofthreepartsthatcorrespondtothreepossibleoutcomes.Equation(4.9)statesthatthechoiceofportfoliostructureshouldmaximisethebanks'expectedpro¯ts.Equation(4.10)istheincentiveconstraintforindividualbanks,whichsuggeststhattheexpectedpro¯tsforbanksshouldcovertheopportunitycostsofthecollateralassetsinequilibrium.Sincetherisk-neutralbankerscanalwaysinvesttheirequityassetsinthemoreproductivetechnologyandobtainanexpectedreturnofE(~R),thebankingsectorshouldassurethemthesamepayo®.Equation(4.11)speci¯estheminimumholdingofrisklessassets.Thewelfaree®ectsofthispolicy,accordingly,aredi®erentindi®erentstatesoftheeconomy:{WhenR¸R2,bothidiosyncraticriskandaggregateriskareeliminated.Agentsreceiveconstantpaymentsbasedontheirtrueliquidityneeds.Thebankingsectorfunctionswellandnodefaultoccurs.{WhenR1·RR2,impatientagentsreceiveaconstantpayo®butthepatientagentsreceiveastate-contingentpayo®.Onlytheidiosyncraticriskisdiversi¯ed.Thebankingsectordefaultsinperiod2butnobankrunhappensinperiod1.{WhenRR1,bankrunshappen.Banksgointobankruptcyandtherisk-sharingmechanismbreaksdown.Thedestabilisatione®ectleadstocostlyliquidationandwelfarelosses.Insummary,impositionofacapitalrequirementgivesthebanksapartialdefenseagainstbankrunswithoutcausingnewdistortions.Itcannoteliminatetheoccurrenceofbankruns,though.However,asthecapitalrequirementincreases,thethresholdreturnsR1andR2decrease,22 suggestingthattheprobabilityofdefaultandtheprobabilityofbankrunsaresmaller.Thereforetherisk-sharingbene¯tdominatesandthemarketequilibriumgraduallyconvergestothe¯rst-bestoptimum.Proposition6As·increases,theequilibriumcontractinamarketeconomyconvergesgraduallytothe¯rst-bestoptimum.Proof:seeAppendixH.Figure6providestheresultsfromnumericalsimulations.Thehorizontalaxisrepresentsthecapitalratio(·1+·).Asthecapitalratioincreases,theequilibriumcontractinthemarketeconomy,includingtheinterestratestructure,andinvestmentportfolio,convergestothe¯rst-bestoptimum.Besides,thebankingsectorbecomesmorestableasahighercapitalrequirementisimposed.However,apotentialproblemwiththecapitalrequirementpolicyisthespeedofconvergence.Theoretically,themarketequilibriumconvergestothesocialoptimumonlyasthecapitalratioapproaches100%.Inthegivennumericalexample,thewelfareinthemarketequilibriumreachesthesecond-bestoptimumwhenthecapitalratioisabout13%,andisequivalenttotheoutcomeundertheFCDIscheme(±¤=16:77%)whenthecapitalratiois32.43%.Thisrequirementisobviouslyveryhigh.5ConclusionThispaperproposesamodelinwhichbankrunsarecloselyrelatedtothestateofthebusinesscycle.Extensionsofthemodelstudythewelfaree®ectsof¯vedi®erentpolicies:suspensionofconvertibilityofdeposits;taxationonshort-termdeposits;full-coveragedepositinsuranceschemes;interest-capdepositinsuranceschemesandcapitalrequirements.Theresultssuggestthat,inacompetitivemarket,aninterest-capdepositinsuranceschemecanbeusedtoachievethe¯rst-bestsocialoptimum.Thelimitedguaranteeremovesthebanks'incentivetoo®erinterestratesthataretoohigh,thereforeavoidingthemoralhazardissuesconfrontedbyafull-coveragedepositinsuranceschemeandrestoringtheeconomytothe¯rst-bestoptimum.Mystudyalsosuggeststhattheotherpoliciesdesignedtopreventbankrunswillhaveadversesidee®ects.First,suspensionofconvertibilityofdepositscanonlycontrolthequantityofearly23 withdrawalbutcannotdistinguishthetrueliquidityneedsofdepositors.Althoughsuchapolicycansuccessfullystopabankrun,theresourcesmightbeallocatedinanine±cientwaybecause,whentheassetreturnislow,somedepositorswithtrueliquidityneedswillnotbeabletowithdrawtheirfundswhileotherdepositorswhodonothavegenuineliquidityneedswillobtainaccesstotheirfunds.Second,taxationonearlywithdrawalsintroducesinvestmentdistortionsintotheeconomy.Thetaxationpolicyreducesthedepositors'incentivetorunonthebanks,butmeanwhilethebankshavetoreducetheirinvestmentinthemoreproductiveassets.Third,thefull-coveragedepositinsuranceschemeisexposte±cientbutexanteine±cientinpreventingbankruns.Ontheonehand,theblanketguaranteepromisesthelateconsumersalong-termpaymentthatisatleastasgoodastheshort-termpayment,therebysuccessfullyprotectingthebanksfromruns.Ontheotherhand,thisschemewillcausemoralhazard,whicharisesfromthecon°ictofinterestsbetweenthedepositinsuranceauthorityandindividualbanks.Finally,acapitalrequirementprovidesapartialcushionagainstbankrunswithoutintroducinganynewdistortiontotheeconomy.However,themarketequilibriumconvergestothesocialoptimumonlyasthecapitalratioapproaches100%.Apossibleextensionofthispaperistorelaxtheassumptionofarepresentativebanksetting.Banksinthispaperarehomogeneousandfacethesameaggregateuncertainty.Byallowingforbothaheterogeneousbankingindustryandtheexistenceofaninterbankcreditmarket,wecanstudyhowarunonanindividualbankspreadsacrossthewholebankingindustry,andhowtheliquidationvalueofbankassetsisendogenouslydetermined.Thislineofresearchispotentiallyverypromising.24 References[1]Allen,FranklinandDouglasGale(1998):\OptimalFinancialCrises",JournalofFinance,volLIII,August,pp1245-84.[2]Backus,David,SilverioForesiandLiurenWu(1999):\LiquidityandContagioninFinancialMarkets",workingpaper.[3]BasleCommitteeonBankingSupervision(1988):\InternationalConvergenceofCapitalMea-surementandCapitalStandards".[4]Calomiris,Charles(1998):\TheIMF'sImprudentRoleasLenderofLastResort",CATOJournal,vol17,no3,pp275-94.[5]Calomiris,CharlesandGaryGorton(1991):\TheOriginsofBankingPanics,Models,Facts,andBankRegulation",inGlennHubbard(ed):FinancialMarketsandFinancialCrises(Uni-versityofChicagoPress,Chicago).[6]Chang,RobertoandAndresVelasco(2000):\Banks,DebtMaturityandFinancialCrises",JournalofInternationalEconomics,vol51,pp169-94.[7]||(2001):\AModelofFinancialCrisesinEmergingMarkets",QuarterlyJournalofEco-nomics,vol116no2,May,pp489-517.[8]Cooper,RussellandThomasWRoss(1998a):\BankRuns:DepositInsuranceandCapitalRequirements",workingpaper.[9]||(1998b):\BankRuns:LiquidityCostsandInvestmentDistortion",JournalofMonetaryEconomics,vol41,pp27-38.[10]Diamond,DouglasWandPhilipHDybvig(1983):\BankRuns,DepositInsurance,andLiquidity",JournalofPoliticalEconomy,vol91,pp401-19.[11]Dwyer,GeraldPandAltonGilbert(1989):\BankRunsandPrivateRemedies",FederalReserveBankofSt.LouisReview,May/June,pp43-61.25 [12]FinancialStabilityForum(2001):\GuidanceforDevelopingE®ectiveDepositInsuranceSys-tems".[13]Goldstein,ItayandAdyPauzner(2000):\DemandDepositContractsandtheProbabilityofBankRuns",workingpaper.[14]Gorton,Gary(1988):\BankingPanicsandBusinessCycles",OxfordEconomicPapers,vol40,pp751-81.[15]Jeitschko,ThomasDandCurtisRTaylor(2001):\LocalDiscouragementandGlobalCollapse:ATheoryofInformationAvalanches",AmericanEconomicReview,vol91,no1,March,pp208-24.[16]Kaminsky,GracielaLandCarmenMReinhart(1999):\TheTwinCrises:theCausesofBankingandBalance-of-PaymentProblems",AmericanEconomicReview,vol89,no3,June,pp473-500.[17]||(2000):\OnCrises,Contagion,andConfusion",JournalofInternationalEconomics,vol51,no1,June,pp145-68.[18]Krugman,Paul(1998a):\WhatHappenedtoAsia?",mimeo,MIT.[19]||(1998b):\Fire-SaleFDI",NBERConferenceonCapitalFlowstoEmergingMarkets.[20]Lindgren,Carl-Johan,GillianGarciaandMatthewSaal(1996):\BankSoundnessandMacroe-conomicPolicy",IMF.[21]Park,Sangkyun(1997):\Risk-takingBehaviorofBanksunderRegulation",JournalofBank-ing&Finance,vol21,pp491-507.[22]Radelet,StevenandJe®reySachs(1998):\TheOnsetoftheEastAsianFinancialCrisis",HIIDworkingpaper.[23]Sachs,Je®reyD(1998):\AlternativeApproachestoFinancialCrisesinEmergingMarkets",HIIDworkingpaper.[24]Schwartz,AnnaJ(1998):\TimetoTerminatetheESFandtheIMF",mimeo.26 [25]Zhu,Haibin(2001):\BankRunswithoutSelf-Ful¯llingProphecies",inThreeEssaysonFi-nancialCrises,PhDdissertation,DukeUniversity.27 Figure1Comparisonofthreedi®erentallocations-R6Payo®First-bestallocationr1c1c2r2-R6Payo®Second-bestallocation´´´´´´´´´´´´´´´´´´r1c1c20-R6Payo®Equilibriumallocationinamarketeconomy0r1½½½½½½½½½½½½©©©©©©Aggregateresourcewhenreturnislow(consumptioniseitherr1or0)¾R¤c1c228 Figure2Marketequilibriaunderdi®erentearlywithdrawaltaxes00.10.20.30.40.50.60.70.80.910.511.52Early withdrawal taxr1........BenchmarkBefore-tax r1After-tax r100.10.20.30.40.50.60.70.80.9100.20.40.60.8Early withdrawal taxR*.......Benchmark00.10.20.30.40.50.60.70.80.910.50.520.54Early withdrawal taxE(U)First-bestSecond-bestBenchmark29 Figure3First-bestallocationisfeasiblebutnotchosenundertheFCDIscheme(insurancepremium±f=R1E(~R)if[R¡E(~R)]¢f(R)dR=0.1407)0.20.40.60.8111.21.41.61.82r1Average interest rate0.20.40.60.811.21.41.61.82r1R* (bailout when R)0.20.40.60.810.40.450.50.55r1E(U)0.20.40.60.81-0.5-0.4-0.3-0.2-0.100.1r1CB profit30 Figure4OptimalcontractsunderFCDIschemes00.050.10.150.20.250.311.522.5Insurance feeAverage interst rates.......First-best00.050.10.150.20.250.30.50.550.60.650.7Insurance feeE(U)00.050.10.150.20.250.3-1-0.500.5Insurance feeNet payment31 Figure5Payo®functionsundercapitalrequirements-60RPro¯t´´´´´´´´´´´´´´´´¡·R2R1R3i(R¡R3)Bank'spro¯tfunction-60RPayo®R1R2R3r1r2EarlyconsumptionLateconsumptionBankruns¾-©©©©©©©©©©©©©©©©©©Totalliquidityduringcrises¾Agent'spayo®function32 Figure6Optimalcontractsundercapitalrequirements00.10.20.30.40.50.60.811.2Capital ratior1.......First-best00.10.20.30.40.501234Capital ratior2.......First-best00.10.20.30.40.50.40.5Capital ratioSafe asset.......First-best00.10.20.30.40.50.30.40.50.6Capital ratioR100.10.20.30.40.501234Capital ratioR200.10.20.30.40.50.50.550.6.......First-best---Second-bestCapital ratioE(U)33 AppendixAProofofLemma1TheprooffollowsZhu(2001).Impatientagentsalwayschoosetowithdrawearly.Apatientagent,ifhechoosestowithdrawearly,willreceiver1;Ifhechoosestowait,theexpectedpayo®is:re2(R;L)=8��:min(1¡i+iR¡r1L1¡L,r2)whenr1L·1¡imin(1¡i¡r1L+i(1¡¿)R(1¡L)(1¡¿),r2)whenr1L�1¡i(A.1)whereListheaggregateearlywithdrawal.SimplealgebrashowsthatwhenRr1¡1+ii,thepayo®functionre2(R;L)r1forallL.WhenR¸r1¡1+ii,itistruethatre2(R;®)¸r1.UsingtheconclusioninZhu(2001),abankrunhappensifandonlyifRr1¡1+ii.BProofofLemma3Rewriteproblem(3.1)asmaxr1maxi·1¡r1®®u(r1)+(1¡®)u(1¡i+i¢E(R)¡r1®1¡®)Foragivenr1,bankswillchoosetomaximisetheirholdingsofriskyassets.Thisisbecause:dUfdi=u0(r2)¢[E(~R)¡1]�0Thereforei=1¡r1®andproblem(3.1)isequivalentto:maxr1®u(r1)+(1¡®)u(r2)(B.1)s:t:r2=(1¡r1®)¢E(~R)1¡®The¯rst-orderconditionisu0(r1)=E(~R)¢u0(r2)(B.2)34 CProofofProposition1Iprovethepropositionintwosteps.1.The¯rst-bestallocationissuperiortothesecond-bestallocation.Supposethesecond-bestallocationischaracterisedbyrs1andis.Byusingthesamer1andi,arepresentativeagentobtainsanexpectedutilityof®u(rs1)+(1¡®)u[E(rs2)]undera¯rst-bestcontract.Duetoconcavityoftheutilityfunction,arepresentativeagentisbettero®inthe¯rst-bestcontractenvironment.Andfromthede¯nitionof¯rst-bestallocation,®u(rf1)+(1¡®)u(rf2)¸®u(rs1)+(1¡®)u[E(rs2)].Therefore,the¯rst-bestoptimumissuperiortothesecond-bestoptimum.2.Thesecond-bestallocationissuperiortotheequilibriumallocationinthemarketeconomy.Supposethattheequilibriumcontractinthemarketeconomyischaracterisedbyr1andi(r2isstate-contingent).TheexpectedutilityforarepresentativeagentisUd=ZR¤01¡i+iR(1¡¿)r1u(r1)f(R)dR+Z1R¤[®u(r1)+(1¡®)u(re2(R))]f(R)dRwherere2=1¡i+iR¡r1®1¡®.Nowconsiderthatthesocialplanneradoptsthesamecontractinthesecond-bestcontractenvironment.TheexpectedutilityforarepresentativeagentisUs=Z10[®u(r1)+(1¡®)u(re2(R))]f(R)dRTherefore,Us¡Ud=ZR¤0[®u(r1)+(1¡®)u(re2(R))¡1¡i+iR(1¡¿)r1u(r1)]f(R)dR�ZR¤0[®u(r1)+(1¡®)u(1¡i+iR¡r1®1¡®)¡1¡i+iRr1u(r1)]f(R)dR=ZR¤0[(1¡®)u(1¡i+iR¡r1®1¡®)¡1¡i+iR¡r1®r1¢u(r1)]f(R)dR�0wherethelastinequalitycomesfromtheconcavitypropertyfortheu(¢)functionandu(0)=0.Therefore,thesecond-bestoptimummustbesuperiortothemarketequilibrium.35 DWithdrawaldecisionsundertaxationonshort-termdepositsGiven(r1;r2;i),theafter-taxconsumptionfrom\wait"and\withdraw"is:c1=r1(1¡t)+r1Ltc2=1¡i+iR¡r1L1¡L+r1Ltwhenr1L1¡ic2=1¡i+i(1¡¿)R¡r1L1¡L+r1Ltwhenr1L¸1¡i(1)WhenR�R¤´r1(1¡t)(1¡®)+r1®¡1+ii.c2(L=®)=1¡i+iR¡r1®1¡®+r1®t�r1(1¡t)(1¡®)1¡®+r1®t=r1(1¡t)+r1®t=c1thereforepatientagentschoosetowaitwhenR�R¤.(2)WhenRR¤.Similarly,itcanbeshownthatc2(L=®)c1,and@c2(L)@L0forL¸®.Thusc2(L)c2(®)c1forallL¸®.Asaresult,abankrunhappens.EProofofProposition3Supposetheequilibriumcontractunderacertainearlywithdrawaltax,(r1;i),hasbeende¯nedbysolvingproblem(4.1).De¯neR¤=r1(1¡t)(1¡®)+r1®¡1+iiandp=1¡i+iR(1¡¿)r1.Inequilibrium,bankrunshappenifandonlyifRR¤.Theequilibriumwelfareis:ZR¤0[pu(c12)+(1¡p)u(c22)]f(R)dR+Z1R¤[®u(c11)+(1¡®)u(c21)]f(R)dRwherec11=r1(1¡t)+®r1t;c21=1¡i+iR¡r1®1¡®+®r1t,c12=r1(1¡t)+pr1t;andc22=pr1t.Nowconsideracontractinthesecond-bestallocationsetting.Letr¤1=r1(1¡t)+®r1t,i¤=i;thelong-termpaymentisthereforer¤2(R)=1¡i+iR¡r¤1®1¡®=1¡i+iR¡r1®1¡®+®r1t=c2136 ThesetwocontractsleadtothesameallocationofresourceswhenR�R¤.However,whenRR¤,thecontractinthesecond-bestenvironmentisbetterbecause®u(r¤1)+(1¡®)u(r¤2)�pu(c12)+(1¡p)u(c22)forallRR¤.Therearetworeasons.First,thetotalresourcesunderthesecond-bestallocationarehigherbecausethereisnoliqui-dationcost.Speci¯cally,®r¤1+(1¡®)r¤2=1¡i+iRandp¢c12+(1¡p)c22=1¡i+iR(1¡¿).Second,thesecond-bestallocationdividestheresourcesmoreevenly.Noticethatr¤1�r¤2andc12�c22whenRR¤.From1¡i¸r1®)p�r1®+iR(1¡¿)r1�®,wehaver¤1¡c12=(®¡p)r1t0Thusr¤1c12andr¤2�c22.Fromtheconcavityoftheutilityfunction,thecontractinthesecond-bestenvironmentmustbebetter.23Combiningtheaboveresultssuggeststhat,foranycontractunderthetaxpolicy,therealwaysexistsabettercontractinasecond-bestenvironment.Theearlywithdrawaltaxisexanteine±cient.FProofofLemma5(1)DenoteE(¦)=R1R¤i(R¡R¤)¢f(R)dR,d[E(¦)]di=Z1R¤(R¡1)f(R)dR¡i(R¤¡R¤)@R¤@if(R¤)=Z1R¤(R¡1)f(R)dR�0Asaresult,imustbechosenatitsmaximumvalue,1¡r1®,intheequilibriumcontract.(2)Theconstraint(4.5)shouldbebinding,too.Otherwisethebankscano®erabettercontracttodepositorsandattractmoredepositsbybiddinguptheinterestrates.Thisprocesscontinuesuntilthenetpro¯tisdrivendowntozero.GProofofProposition5Whenr=rf2and±f=R1E(~R)if[R¡E(~R)]¢f(R)dR,thebankscanchoosefromtwotypesofcontracts.23Concavityofu(¢)implies:ifx1�y1�y2�x2,then®u(x1)+(1¡®)u(x2)¯u(y1)+(1¡¯)u(y2)forany®,¯solongas¯y1+(1¡¯)y2=®x1+(1¡®)x2.37 (1)r2·rf2=r,inwhichtheICDIisactuallyanFCDIscheme.maxr1;r2®u(r1)+(1¡®)u(r2)s:t:1¡i¸r1®R¤=r1®+r2(1¡®)¡1+iiR1R¤i(R¡R¤)¢f(R)dR¸±i=argmaxR1R¤i(R¡R¤)¢f(R)dRFollowingAppendixF,bothrestrictionsarebinding.Thereforei=1¡r1®andR1R¤(1¡r1®)(R¡R¤)¢f(R)dR=±.Thissuggeststhatbothiandr1canbedeterminedoncer2ischosen,becausedUdr2=@U@r2+@U@r1¢dr1dr2=(1¡®)u0(r2)+®u0(r1)¡R1R¤(1¡®)f(R)dRR1R¤®Rf(R)dR¸[u0(rf2)¢E(RjR¸R¤)¡u0(rf1)]¢1¡®E(RjR¸R¤)=u0(rf2)[E(RjR¸R¤)¡E(R)]¢1¡®E(RjR¸R¤)�0The¯rstinequalitycomesfromthefactthatr2·rf2andr1¸rf1.Asaresult,bankswillchoosethe¯rst-bestcontractr1=rf1,r2=rf2amongcontractsofthistype.(2)r2¸rf2=r.Theoptimisationproblemforthebanksis:maxr2RR¤10[®u(r1)+(1¡®)u(r)]f(R)dR+RR¤R¤1[®u(r1)+(1¡®)u(1¡i+iR¡r1®1¡®)]f(R)dR+R1R¤[®u(r1)+(1¡®)u(r2)]f(R)dRs:t:1¡i=r1®R¤1=r1®+r(1¡®)¡1+iiR¤=r1®+r2(1¡®)¡1+iiR1R¤i(R¡R¤)¢f(R)dR=±38 Theobjectivefunctionre°ectsthefactthatthemaximumamountofpaymentadepositorcanreceiveisrwhenthebankisinsolvent.Andsimilarly,boththebudgetconstraintandtheincentiveconstraintarebinding;thereforetheonlychoicevariableisr2.Aftersomealgebra,itisshownthatdUdr2=@U@r2+@U@r1¢dr1dr2=R1R¤(1¡®)f(R)dRR1R¤®Rf(R)dR¢[u0(r2)Z1R¤®Rf(R)dR+ZR¤R¤1®Ru0((1¡r1®)R1¡®)f(R)dR¡®u0(r1)]1¡®E(RjR¸R¤)¢[u0(r)Z1R¤1Rf(R)dR¡u0(r1)]1¡®E(RjR¸R¤)¢[u0(rf2)E(R)¡u0(rf1)]=0wherethe¯rstinequalitycomesfromthefactthatr2¸rand(1¡r1®)R1¡®�rforR2[R¤1;R¤],andthesecondinequalityusesthefactthatr1·rf1whenr2¸rf2.Therefore,the¯rst-bestcontractisalsochosen.Combiningtheresults,undertheICDCwithr=rf2and±f=R1E(~R)if[R¡E(~R)]¢f(R)dR,thebankswillchoosethe¯rst-bestcontractandtheeconomyachievesthe¯rst-bestsocialoptimum.HProofofProposition6Theproofisdividedintothefollowingsteps.{Step1:i=1+·¡r1®.De¯neE(¦)astheexpectedpro¯tforthebanksasspeci¯edinequation(4.9).Itiseasytoshowthat@E(¦)@i=Z1R2(R¡1)f(R)dR�0Therefore,equation(4.11)isbinding.Accordingly,theincentiveconstraint(equation4.10)isrewrittenas:Z1R2(1+·¡r1®)(R¡R2)f(R)dR¡·¢E(R)=0(H.1)39 TheLagrangeequationfortheproblemis:MAXU(·)=maxr1;r2ZR10r1®+(1+·¡r1®)R(1¡¿)r1u(r1)¢f(R)dR+ZR2R1[®u(r1)+(1¡®)u((1+·¡r1®)R1¡®)]¢f(R)dR+Z1R2[®u(r1)+(1¡®)u(r2)]¢f(R)dR+¸fZ1R2(1+·¡r1®)(R¡R2)f(R)dR¡·¢E(R)gwhereR1=r1(1¡®)1+·¡r1®,R2=r2(1¡®)1+·¡r1®,and¸istheLagrangemultiplier.MAXU(·)istheindirectutilityfunction.{Step2:The¯rst-orderconditionsfortheaboveproblemare:@MAXU(·)@r1=ZR10f(1+·¡r1®)R(1¡¿)r1u0(r1)¡(1+·)(1¡¿)Ru(r1)r21gf(R)dR+®u0(r1)¡ZR2R1®Ru0[(1+·¡r1®)R1¡®]f(R)dR¡(1¡®)¿u(r1)f(R1)(1¡®)(1+·)(1+·¡r1®)2¡¸®Z1R2Rf(R)dR=0(H.2)@MAXU(·)@r2=Z1R2(1¡®)[u0(r2)¡¸]f(R)dR=0)u0(r2)=¸(H.3){Step3:As·!1,theequilibriumconvergestothe¯rst-bestallocation.When·!1,byusingR1!0,R2!0,andR3!1,equation(H.2)canbewrittenas®u0(r1)¡¸®E[~R]=0)u0(r1)=¸E[~R]Combinedwithequation(H.3),weobtainthefamiliarEulerequationu0(r1)=E[~R]¢u0(r2).Furthermore,Ishowthatr2=(1¡r1®)E[~R]1¡®satis¯estheincentiveconstraint(equationH.1)as·!1.Asi=1+·¡r1®,r2=(1¡r1®)¢E(R)1¡®,when·!1,Z1R2(1+·¡r1®)(R¡R2)f(R)dR¡·E(R)40 !Z10(1+·¡r1®)Rf(R)dR¡r2(1¡®)¡·E(R)=(1¡r1®)E(R)¡r2(1¡®)=0FromLemma3,itissafetoconcludethatthemarketequilibriumconvergestothe¯rst-bestallocationasthecapitalrequirementincreases.41 Recent BIS Working Papers No Title Author December 2001 Bank runs without self-fulfilling prophecies Haibin Zhu December 2001 Macroeconomic news and the euro/dollar exchange rate Gabriele Galati and Corrinne Ho September 2001