Understanding Financial Crisis

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Thispageintentionallyleftblank UNDERSTANDINGFINANCIALCRISES UnderstandingFinancialCrisesFRANKLINALLENandDOUGLASGALE1 3GreatClarendonStreet,OxfordOX26DPOxfordUniversityPressisadepartmentoftheUniversityofOxford.ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship,andeducationbypublishingworldwideinOxfordNewYorkAucklandCapeTownDaresSalaamHongKongKarachiKualaLumpurMadridMelbourneMexicoCityNairobiNewDelhiShanghaiTaipeiTorontoWithofficesinArgentinaAustriaBrazilChileCzechRepublicFranceGreeceGuatemalaHungaryItalyJapanPolandPortugalSingaporeSouthKoreaSwitzerlandThailandTurkeyUkraineVietnamOxfordisaregisteredtrademarkofOxfordUniversityPressintheUKandincertainothercountriesPublishedintheUnitedStatesbyOxfordUniversityPressInc.,NewYork©FranklinAllenandDouglasGale,2007ThemoralrightsoftheauthorshavebeenassertedDatabaserightOxfordUniversityPress(maker)Firstpublished2007Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmitted,inanyformorbyanymeans,withoutthepriorpermissioninwritingofOxfordUniversityPress,orasexpresslypermittedbylaw,orundertermsagreedwiththeappropriatereprographicsrightsorganization.EnquiriesconcerningreproductionoutsidethescopeoftheaboveshouldbesenttotheRightsDepartment,OxfordUniversityPress,attheaddressaboveYoumustnotcirculatethisbookinanyotherbindingorcoverandyoumustimposethesameconditiononanyacquirerBritishLibraryCataloguinginPublicationDataDataavailableLibraryofCongressCataloginginPublicationDataDataavailableTypesetbyNewgenImagingSystems(P)Ltd.,Chennai,IndiaPrintedinGreatBritainonacid-freepaperbyBiddlesLtd.,King’sLynn,NorfolkISBN978–0–19–925141–413579108642 PrefaceThisbookhasgrownoutofaseriesofpaperswrittenoveranumberofyears.Ourpaper“LiquidityPreference,MarketParticipation,andAssetPriceVolatility”wasactuallybegunbyoneofusin1988,althoughitappearedin1994.Ourinterestinbankrunsandfinancialcrisesbeganwith“OptimalFinancialCrises,”andthisledtofurtherstudiesonthewelfareeconomicsofcrises.Eachpaperseemedtoleavequestionsunansweredandledtonewpaperswhichledtonewquestions.WhenoneofuswasinvitedtogivetheClarendonLecturesinFinance,itseemedtherighttimetobeginthetaskofsynthesizingtheideaspresentedinavarietyofdifferentplacesusingdifferentmodels.Ouraimwastomaketheseideasaccessibletoawideraudience,includingundergraduatesandpolicymakersincentralbanksandinternationalorganizations,andalsotoputtheminacoherentframeworkthatmightmakethemmoreusefulforgraduatestudentsandresearchers.Thisisfarfrombeingthelastwordonthesubject,butitmayprovideasetoftoolsthatwillbehelpfulinpursuingthemanyopenquestionsthatremain.Overtheyearswehavehadmanyopportunitiestopresentourworkinsemi-narsandatconferencesandhavebenefitedfromthecommentsandsuggestionsofmanyeconomists.Inparticular,wewouldliketothankViralAcharya,ChristinaBannier,MichaelBordo,PatrickBolton,MikeBurkart,MarkCarey,ElenaCarletti,MichaelChui,MarcoCipriani,PeterEnglund,PrasannaGai,GaryGorton,AntonioGuarino,MartinHellwig,MarcelloPericoli,GlenHog-garth,JamieMcAndrews,RobertNobay,ÖnürOzgur,JoãoSantos,MassimoSbracia,HyunSongShin,GiancarloSpagnolo,XavierVives,DavidWebb,AndrewWinton,andTanjuYorulmazer.WehaveincludedsomeofthesetopicsingraduatecourseswetaughtatNewYorkUniversityandPrincetonUniversity.Oncewebeganwritingthebook,wewerefortunatetohavetheopportunitytopresentlectureseriesattheBankofEngland,theBancad’Italia,theStockholmSchoolofEconomics,andtheUniversityofFrankfurt.WedevelopedanundergraduatecourseonfinancialcrisesatNYUbasedonthemanuscriptofthebook.Weareverygratefultotheundergraduateswhoallowedustoexperimentonthemandrosetothechallengepresentedbythematerial.AntonioGuarinousedseveralchaptersfor viPrefaceanundergraduatecourseatUniversityCollegeLondonandofferedusmanycommentsandcorrections.Wearesurethereareotherswhomwemayhaveforgotten,butwhosecontributionsandencouragementhelpedusgreatly.Thankstoallofyou. Contents1.Historyandinstitutions11.1Introduction11.2HistoricalcrisesinEuropeandtheUS21.3Crisesandstockmarketcrashes51.4Currencyandtwincrises91.5Crisesindifferenteras101.6Somerecentcrises141.6.1TheScandinaviancrises141.6.2Japan151.6.3TheAsiancrisis151.6.4TheRussiancrisisandlongtermcapitalmanagement(LTCM)161.6.5TheArgentinacrisisof2001–2002171.7Thecostsofcrises181.8Theoriesofcrises191.9Concludingremarks232.Time,uncertainty,andliquidity272.1Efficientallocationovertime272.1.1Consumptionandsaving272.1.2Production362.2Uncertainty402.2.1Contingentcommoditiesandrisksharing402.2.2Attitudestowardrisk442.2.3Insuranceandriskpooling482.2.4Portfoliochoice492.3Liquidity522.4Concludingremarks573.Intermediationandcrises583.1Theliquidityproblem593.2Marketequilibrium603.3Theefficientsolution643.4Thebankingsolution723.5Bankruns74 viiiContents3.6Equilibriumbankruns763.7Thebusinesscycleviewofbankruns823.8Theglobalgamesapproachtofindingauniqueequilibrium903.9Literaturereview943.10Concludingremarks964.Assetmarkets994.1Marketparticipation994.2Themodel1034.3Equilibrium1044.3.1Market-clearingatdate11074.3.2Portfoliochoice1094.4Cash-in-the-marketpricing1104.5Limitedparticipation1144.5.1Themodel1164.5.2Equilibrium1174.5.3Equilibriumwithfullparticipation1204.5.4Fullparticipationandasset-pricevolatility1204.5.5Limitedparticipationandasset-pricevolatility1214.5.6MultiplePareto-rankedequilibria1234.6Summary1245.Financialfragility1265.1Markets,banks,andconsumers1285.2Typesofequilibrium1325.2.1Fundamentalequilibriumwithnoaggregateuncertainty1335.2.2Aggregateuncertainty1355.2.3Sunspotequilibria1405.2.4Idiosyncraticliquidityshocksforbanks1425.2.5Equilibriumwithoutbankruptcy1445.2.6Completeversusincompletemarkets1465.3Relationtotheliterature1475.4Discussion1486.Intermediationandmarkets1536.1Completemarkets1556.2Intermediationandmarkets1646.2.1Efficientrisksharing1656.2.2Equilibriumwithcompletefinancialmarkets167 Contentsix6.2.3Analternativeformulationofcompletemarkets1706.2.4Thegeneralcase1726.2.5Implementingthefirstbestwithoutcompletemarkets1776.3Incompletecontracts1816.3.1Completemarketsandaggregaterisk1826.3.2Theintermediary’sproblemwithincompletemarkets1866.4Conclusion1887.Optimalregulation1907.1Capitalregulation1917.1.1Optimalcapitalstructure1947.1.2Modelswithaggregateuncertainty1997.2Capitalstructurewithcompletemarkets2017.3Regulatingliquidity2047.3.1Comparativestatics2067.3.2Toomuchortoolittleliquidity?2097.4Literaturereview2137.5Concludingremarks2138.Moneyandprices2168.1Anexample2188.2Optimalcurrencycrises2238.3Dollarizationandincentives2268.4Literaturereview2288.5Concludingremarks2329.Bubblesandcrises2359.1Agencyproblemsandpositivebubbles2379.1.1Therisk-shiftingproblem2389.1.2Creditandinterestratedetermination2439.1.3Financialrisk2459.1.4Financialfragility2469.2Bankingcrisesandnegativebubbles2479.2.1Themodel2479.2.2Optimalrisksharing2489.2.3Optimaldepositcontracts2519.2.4Anassetmarket2529.2.5Optimalmonetarypolicy2569.3Concludingremarks258 xContents10.Contagion26010.1Liquiditypreference26310.2Optimalrisksharing26610.3Decentralization26810.4Contagion27410.4.1Theliquidation“peckingorder”27510.4.2Liquidationvalues27610.4.3Buffersandbankruns27710.4.4Manyregions28010.5Robustness28010.6Containment28110.7Discussion28210.8Applications28410.8.1UpperandWorms(2004)28410.8.2DegryseandNguyen(2004)29010.8.3Cifuentes,Ferrucci,andShin(2005)29110.9Literaturereview29310.10Concludingremarks295Index299 1Historyandinstitutions1.1INTRODUCTIONWhathappenedinAsiain1997?CountriessuchasSouthKorea,Thailand,Indonesia,Singapore,andHongKongwhoseeconomieshadpreviouslybeentheenvyoftheworldexperiencedcrises.Banksandotherfinancialintermedi-arieswereputundergreatstrainandinmanycasescollapsed.Stockmarketsandcurrenciesplunged.TheirrealeconomieswereseverelyaffectedandtheirGDPsfellsignificantly.Whatwerethecausesofthesedramaticevents?Tomanypeoplethesecriseswereanewphenomenon.TherehadbeencrisesinothercountriessuchasMexicoandBrazilbutthesecouldbeattributedtoinconsistentgovernmentmacroeconomicpolicies.Inthosecasestaxesweretoosmallrelativetogovernmentexpenditurestomaintainafixedexchangerate.ThiswasnotthecasefortheAsiancrisis.Othercauseswerelookedforandfound.TheinstitutionsinthesecountrieswerequitedifferentfromthoseintheUS.Manyhadbank-basedfinancialsystems.Therewaslittletransparencyeitherforbanksorcorporations.Corporategovernanceoperatedinaquitedifferentway.Inmanycasesitdidnotseemthatmanagers’interestswerealignedwiththoseofshareholders.InsomecountriessuchasIndonesiacorruptionwasrife.Thesefactorswereseenbymanyasthecauseofthecrises.However,theyhadallbeenpresentduringthetimethatthesecountriesweresosuccessful.Othersblamedguaranteestobanksandfirmsbygovernmentsorimplicitpromisesof“bail-outs”byorganizationssuchastheInternationalMone-taryFund(IMF).Ratherthaninconsistentmacroeconomicpoliciesbeingtheproblem,badmicroeconomicpoliciesweretheproblem.Eitherwayitwasgovernmentsandinternationalorganizationsthatweretoblame.Inthisbookwewillarguethatitisimportantnottotaketoonarrowaviewofcrises.Theyarenothingnew.Theyhavenotbeenrestrictedtoemergingeconomieseveninrecenttimes.TheScandinaviancrisesoftheearly1990’sareexamplesofthis.Despitehavingsophisticatedeconomiesandinstitutions,Norway,SwedenandFinlandallhadseverecrises.TheseweresimilarinmanywaystowhathappenedintheAsiancrisisof1997.Bankscollapsed,assetprices 2Chapter1.Historyandinstitutionsplunged,currenciescameunderattackandtheirvaluefell.Outputwasseverelyaffected.Takinganhistoricalviewtheperiodfrom1945–1971wasexceptional.Therewerenobankingcrisesanywhereintheworld,apartfromoneinBrazilin1962.Therewerecurrencycriseswhenexchangerateswerepeggedatthewronglevelsbutthatwasall.Goingbacktothefirsthalfofthetwentiethcenturyandbeforethereweremanyexamplesoffinancialcrises.Thestockmarketcrashof1929,thebankingcrisesoftheearly1930’sandtheGreatDepressionweresomeofthemostdramaticepisodes.Thereweremanyothers,particularlyintheUSinthelasthalfofthenineteenthcenturywhenithadnocentralbank.InEuropecrisesweremuchlessfrequent.TheBankofEnglandhadlearnedtopreventcrisesandthelastonetherewastheOverend&Gurneycrisisof1866.Othercentralbanksalsolearnedtopreventcrisesandtheirincidencewassignificantlyreduced.PriortothatcriseswereendemicinEuropeaswell.ParticularlyaftertheexperienceoftheGreatDepressionintheperiodpriorto1945–1971,criseswereperceivedasamarketfailure.Itwaswidelyagreedtheymustbeavoidedatallcosts.ThereformoftheFederalReserveSystemintheearly1930’sandtheextensiveregulationofthefinancialsystemthatwasputinplaceintheUSwerepartofthismindset.Inothercountriesfinancialregulationwentevenfarther.Governmentscontrolledtheallocationoffundstodifferentindustriesthroughstate-ownedbanksorheavilyregulatedbanks.Thisextensiveregulationwasthecauseofthevirtualdisappearanceofbankingcrisesfrom1945–1971.However,theeliminationofcrisescameatacost.Becauseoftheextensiveregulationandgovernmentinterventionthefinancialsystemceasedtoperformitsbasicfunctionofallocatinginvestment.Thereweremanyinefficienciesasaresult.Thisledtocallsforderegulationandthereturnofmarketforcestotheallocationofinvestment.Asaresultcriseshavereturned.Bordoetal.(2000)findthatthefrequencyofcrisesintherecentperiodsince1971isnotthatdifferentfromwhatitwasbefore1914.Westartinthischapterwithanhistoricalreviewofcrisesandtheinstitutionsinvolved.Thisprovidesabackgroundforthetheoriesthataresubsequentlydeveloped.1.2HISTORICALCRISESINEUROPEANDTHEUSPriortothetwentiethcenturybankingpanicsoccurredfrequently.Kindle-berger(1993,p.264)inhisbookrecountingthefinancialhistoryofWesternEuropepointsoutthatfinancialcriseshaveoccurredatroughly10year 1.2HistoricalCrisesinEuropeandtheUS3intervalsoverthelast400years.Panicsweregenerallyregardedasabadthingbecausetheywereoftenassociatedwithsignificantdeclinesineconomicactiv-ity.Overtimeoneofthemainrolesofcentralbankshasbecometoeliminatepanicsandensurefinancialstability.Ithasbeenalongandinvolvedprocess.Thefirstcentralbank,theBankofSweden,wasestablishedover300yearsagoin1668.TheBankofEnglandwasestablishedsoonafter.Itplayedanespe-ciallyimportantroleinthedevelopmentofeffectivestabilizationpoliciesintheeighteenthandnineteenthcenturies.ThelasttruepanicintheUKwastheOverend&Gurneycrisisof1866.InhisinfluentialbookLombardStreet,Bagehot(1873)laidouthisfamousprinciplesofhowacentralbankshouldlendtobanksduringacrisis.•Lendfreelyatahighrateofinterestrelativetothepre-crisisperiodbutonlytoborrowerswithgoodcollateral(i.e.anyassetsnormallyacceptedbythecentralbank).•Theassetsshouldbevaluedatbetweenpanicandpre-panicprices.•Institutionswithoutgoodcollateralshouldbeallowedtofail.Bordo(1986)documentsthatfortheperiod1870–1933therewereveryfewbankingpanicsintheUK,Germany,andFrance.Kindleberger(1993)pointsoutthatmanyBritisheconomistsascribetheabsenceofcrisesintheUKtocentralbankingexperiencegainedbytheBankofEnglandandtheirabilitytoskillfullymanipulatediscountrates.However,Francealsoexperiencednofinancialcrisesfrom1882–1924despiteleavingitsdiscountrateconstantformanydecades.KindlebergersuggeststhatFrancewasperhapsstabilizedbyEngland.TheUStookadifferenttack.AlexanderHamiltonwasinfluencedbyBritishexperiencewiththeBankofEnglandandaftertherevolutionadvocatedalargefederallycharteredbankwithbranchesalloverthecountry.ThisledtothefoundationoftheFirstBankoftheUnitedStates(1791–1811)andlatertheSecondBankoftheUnitedStates(1816–1836).However,therewascon-siderabledistrustoftheconcentrationofpowertheseinstitutionsrepresented.InareportontheSecondBank,JohnQuincyAdamswrote“Powerforgood,ispowerforevil,eveninthehandsofOmnipotence”(Timberlake1978,p.39).Thecontroversycametoaheadinthedebateonthere-charteringoftheSecondBankin1832.AlthoughthebillwaspassedbyCongressitwasvetoedbyPresidentJacksonandthevetowasnotoverturned.Sincethentherehasbeenastrongbiastowarddecentralizationofthebankingsystemandanaver-siontopowerfulinstitutionsofanykind.TherewasnocentralbankintheUSfrom1836until1914. 4Chapter1.HistoryandinstitutionsThroughoutthenineteenthcenturytheUSbankingsystemwashighlyfrag-mentedandunlikeeveryotherindustrializingcountrytheUSfailedtodevelopnationwidebankswithextensivebranchnetworks.PriortotheCivilWar,stateswerefreetoregulatetheirownbankingsystemsandtherewasnonationalsys-tem.Manystatesadopteda“freebanking”systemwhichallowedfreeentry.Therewereseriousbankingpanicsin1837and1857andbothwerefollowedbydepressionsandsignificanteconomicdisruption.TheadventoftheCivilWarin1861andtheneedtofinanceitsig-nificantlychangedtheroleoftheFederalGovernmentinthefinancialsystem.TheNationalBankActsof1863and1864setupanationalbank-ingsystem.Theygrantedlimitedpowerstobanks.Inparticular,the1864Actwasinterpretedasconfiningeachtoasinglelocation.Whenthequestionofwhetherbankscouldholdequityarose,theSupremeCourtruledthatsincethe1864Acthadnotspecificallygrantedthisrighttheycouldnot.ThecreationoftheNationalBankingsystemdidnotpreventtheproblemofpanicsandtheassociatedeconomicdisruptionanddepressions.Therewerepanicsin1873,1884,1893and1907.Table1.1,whichisfromGorton(1988),showsthebankingcrisesthatoccurredrepeatedlyintheUSduringtheNationalBankingErafrom1863–1914.ThefirstcolumnshowsthebusinesscyclesidentifiedbytheNationalBureauofEconomicResearch(NBER).Thefirstdateisthepeakofthecycleandthesecondisthetrough.Thesecondcolumnshowsthedateonwhichpanicsoccurred.InabankingpanicpeopleworryTable1.1.NationalBankingErapanics.NBERcyclePanicdate%
(Currency/%
PigPeak–Troughdeposit)∗iron†Oct.1873–Mar.1879Sep.187314.53−51.0Mar.1882–May1885Jun.18848.80−14.0Mar.1887–Apr.1888NoPanic3.00−9.0Jul.1890–May1891Nov.18909.00−34.0Jan.1893–Jun.1894May189316.00−29.0Dec.1895–Jun.1897Oct.189614.30−4.0Jun.1899–Dec.1900NoPanic2.78−6.7Sep.1902–Aug.1904NoPanic−4.13−8.7May1907–Jun.1908Oct.190711.45−46.5Jan.1910–Jan.1912NoPanic−2.64−21.7Jan.1913–Dec.1914Aug.191410.39−47.1∗Percentagechangeofratioatpanicdatetopreviousyear’saverage.†Measuredfrompeaktotrough.(AdaptedfromTable1,Gorton1988,p.233) 1.3CrisesandStockMarketCrashes5aboutthesoundnessofthebankstheyhavedepositedtheirfundsin.Asaresulttheywithdrawtheirmoneyandholditintheformofcurrency.Thethirdcolumnshowsthepercentagechangeintheratioofcurrencytodeposits.Itisameasureoftheseverityofabankingpanic.Thehigherthechangeinthecurrency/depositratio,themoreseriousisthecrisis.Itcanbeseenthatthepanicsof1873,1893,1896,and1907wereparticularlysevere.Thefinalcolumnshowshowmuchtheproductionofpigironchangedfromthepeakofthecycletothetrough.GDPfiguresforthisperiodhavenotbeenreliablycompiled.EconomichistoriansoftenuseproductionofpigironasaproxyforGDP.Thefinalcolumnisthereforemeanttoindicatehowserioustherecessionswere.Itcanbeseenthatthetroughsoccurringafterthepanicsof1873,1890,1893,1907,and1914wereparticularlysevere.Afterthecrisisof1907,aEuropeanbankersummedupEuropeanfrustrationwiththeinefficienciesoftheU.S.bankingsystembydeclaringtheUSwas“agreatfinancialnuisance”(StudenskiandKrooss1963,p.254).Theseverityoftherecessionfollowingthe1907bankingpanicledtoadebateonwhetherornotacentralbankshouldbeestablishedintheUS.TheNationalMonetaryCommissioninvestigatedthisissueandfinallyin1914theFederalReserveSystemwasestablished.TheinitialorganizationoftheFederalReserveSystemdifferedfromthatofatraditionalcentralbankliketheBankofEngland.Ithadaregionalstructureanddecisionmakingpowerwasdecentralized.Duringtheyearsafteritscre-ationitdidnotdeveloptheabilitytopreventbankingpanics.In1933therewasanothermajorbankingpanicwhichledtotheclosingofbanksforanextendedperiodjustafterPresidentRoosevelttookoffice.TheproblemsfacedbythebankingsystemledtotheGlass–SteagallActof1933,whichintroduceddepositinsuranceandrequiredtheseparationofcommercialandinvestmentbankingoperations.TheBankingActof1935extendedthepowersoftheFed-eralReserveSystemandchangedthewayitoperated.ThesereformsfinallyeliminatedtheoccurrenceofbankingpanicsalmostseventyyearsafterthishadhappenedintheUK.1.3CRISESANDSTOCKMARKETCRASHESSofarwehavefocusedonbankingcrises.Oftenbankingcrisesandstockmarketcrashesarecloselyintertwined.Forexample,Wilsonetal.(1990)considerfourmajorbankingpanicsaccompaniedbystockmarketcrashesintheUSduringtheNationalBankingEra.ThesearethecrisesofSeptember1873,June1884,July1893,andOctober1907. 6Chapter1.HistoryandinstitutionsWhywastherealinkbetweenbankingpanicsandstockmarketcrashes?Asmentionedabovebankswerenotabletoholdequitysoitmightbethoughtthatmovementsinthestockmarketwouldbeindependentofbanks’pol-icies.Infactthiswasnotthecase.Toseewhy,itisnecessarytohavesomeunderstandingofthelinkbetweenbanksandthestockmarketduringthisperiod.Banksmustholdliquidreservesincasecustomerswishtowithdrawcashfromtheiraccounts.Allbanksholdsomereservesintheformofcurrency.Inadditionalargeproportionofreserveswereheldintheformofinterbankbalances.Inpractice,mostbankshaddepositsinNewYorkCitybanks.Thereasonbanksheldinterbankdepositsratherthancashwasthattheypaidinterest.TheNewYorkCitybankscouldpayattractiveratesofinterestbecausetheylentalargeproportionofthesefundsinthecallloanmarketatthestockexchangeinNewYork.Theloanswereusedtobuystocksonmargin(i.e.thestockswereboughtwithborrowedmoney).Theywerereferredtoascallloansbecausetheywerepayableondemand.Theborrowerscouldeitherobtainfundstorepaytheircallloansbytakingoutotherloansorifnecessarytheycouldsellthesecuritiestheoriginalcallloanswereusedtopurchase.ThesecallloansconstitutedalargepartofNewYorkbanks’assets.Forexample,Sprague(1910,p.83)reportsthatonSeptember12,1873,31percentofNewYorkbanks’loanswerecallloans.AgriculturewasmuchmoreimportantduringtheNationalBankingErathanitistoday.DuringtheSpringplantingandAutumnharvestingbanksinfarmingareasrequiredcash.BecauseoftherandomnatureofthesedemandsforcashitwasdifficultfortheNewYorkCitybankstoplanwithcertaintywhattheirliquidityneedswouldbe.WhenliquidityneedswerehightheNewYorkCitybankswoulddemandrepaymentoftheircallloans.Theborrowersmightbeforcedtosellthesecuritiestheyhadpurchasedonmargin.Awaveofsellingcouldcausepricestofallifthoseparticipatinginthemarketonthebuysidehadlimitedamountsofcash.Inotherwordstherecouldbeacrashinprices.Wilson,Sylla,andJonesinvestigatestockreturnsandtheirvolatilitydur-ingthepanicandcrashperiodsof1873,1884,1893,and1907.Table1.2showsthe25lowestand25higheststockmonthlypricechangesbetween1866and1913.Fouroftheeightlowestreturnsoccurringduringthisperiodwereduringpanicmonths.ApartfromMay1880,whichisnotassociatedwithabankingpanic,alltheothersfromtheninelowestreturnsarearoundpanics.Noticealsofromthehigheststockreturnsthatthereissometen-dencyforstockstorallytwoorthreemonthsafteracrisis.December1873 1.3CrisesandStockMarketCrashes7Table1.2.The25lowestand25higheststockpricechanges1866–1913(fromTable1ofWilsonetal.1990).YearMonthLowestRankYearMonthHighestreturnreturn190710−10.8514%118791010.8824%19073−9.79872190169.967818937−9.434031873129.538518935−8.89934190148.4437187310−8.67215189198.060518845−8.557561900117.851218805−7.91377189917.692318739−7.75008190687.407419078−7.48099187786.9869189011−7.335010189856.812018776−7.173011189396.686918774−7.058812189786.6852189912−6.7308131896116.666719017−6.7251141908116.606618967−6.609215188486.406718699−6.4913161885116.313118846−6.4171171898126.308418769−6.012718187796.122418772−5.944119188115.9574190711−5.8052201904105.9423189512−5.6911211900125.938719036−5.5556221885105.882418968−5.538523189555.698019119−5.420124188275.689318773−5.204525188585.5710isthethirdhighestreturn,September1993istheeleventhhighest,andAugust1884isthefifteenthhighest.Itisnotjuststockswherethiseffectisfound.Bondsandcommercialpapershowsimilarpatternsofreturns.Returnsarelowduringthepanicandthenreboundinthemonthsafterthepanic.Table1.3showsthetop50monthsofvolatilityforstocksbetween1866and1913.Thesevolatilitiesarecalculatedbyincludingtheannualizedstandarddeviationofreturnsusingthecurrentmonthandnineoftheprevious11monthswiththetwodiscardedbeingtheoneswiththehighestandlowestreturns.Thegreatestvolatilityseemstooccurintheyearfollowingthepanicwithpeakstockpricevolatilitycoming2–7monthsafterthepanic. 8Chapter1.HistoryandinstitutionsTable1.3.Thetop50monthsofvolatilityforstocks1866–1913(fromTable5ofWilsonetal.1990).RankStocksYearMo.Stocks11908516.243321908615.680131908715.623941908415.459051908215.050961908115.017971901715.007881878114.2182918771014.19601018771214.19211118771114.18411218731214.1461131908313.9722141901813.76951519011013.7645161877813.7459171877913.72381819071213.5497191893913.5273201908913.0782211908813.0658221901913.0519231878213.02062418961112.8153251894412.5641261901512.4214271894312.38362819011112.3543291891912.2079301884812.1837311898512.04303219011212.0014331901611.9526341902111.8947351878311.84153618931211.8154371874811.8127381902211.8042391880511.7880401898611.7863 1.4CurrencyandTwinCrises9Table1.3.(Continued)RankStocksYearMo.Stocks411874711.7802421874611.7571431874511.7442441874411.71324518931111.7040461874311.5068471894211.5040481874211.4914491901411.4480501902311.44221.4CURRENCYANDTWINCRISESManyofthecrisesinthenineteenthandearlytwentiethcenturywereinter-nationalinscope.Forexample,thecrisisof1873hadanextensiveimpactinAustriaandGermanyaswellasintheUSandinanumberofemergingcoun-triessuchasArgentina.Infactthe1873crisisendedawaveoflendingthatoccurredinthe1850’sand1860’stofinancerailroadsinLatinAmerica(BordoandEichengreen1999).Theseinternationaldimensionsledtoaflowoffundsbetweencountriesandthisinturncouldcauseacurrencycrisis.Whenbankingcrisesandcurrencycrisesoccurtogetherthereissaidtobeatwincrisis.PriortotheFirstWorldWarcountrieshadastrongcommitmenttothegoldstandard.Ifacountrysufferedanoutflowoffundsitmightleavethegoldstandardbutitwasgenerallyexpectedtoresumeaftersometimehadpassed.Thislessenedtheeffectofcurrencycrisesasinvestorsbelievedthevalueofthecurrencywouldeventuallyberestored.Betweenthewars,commitmenttothegoldstandardwasweakened.Asaresultbankingandcurrencycrisesfrequentlyoccurredtogether.Thesetwincrisesaretypicallyassociatedwithmoresevererecessionsthanbankingorcurrencycrisesoccurringontheirown.AftertheSecondWorldWartheBrettonWoodssystemoffixedexchangerateswasestablished.Strongbankingregulationsandcontrolswereputinplacethateffectivelyeliminatedbankingcrises.Currencycrisescontinuedtooccur.Duetotheextensiveuseofcapitalcontrolstheirnaturechanged.Duringthisperiodtheyweretypicallytheresultofmacroeconomicandfinancialpoliciesthatwereinconsistentwiththeprevailingexchangerate.AfterthecollapseoftheBrettonWoodssystemintheearly1970’sbankingcrisesandtwincrisesreemergedascapitalcontrolswererelaxedandcapitalmarketsbecameglobal. 10Chapter1.Historyandinstitutions1.5CRISESINDIFFERENTERASBordoetal.(2000,2001)haveaddressedthequestionofhowrecentcrisessuchastheEuropeanMonetarySystemcrisisof1992–1993,theMexicancrisisof1994–1995,theAsiancrisisof1997–1998,theBraziliancrisisof1998,theRussiancrisisof1998,andtheArgentiniancrisisof2001comparewithearliercrises.Theyidentifyfourperiods.1.GoldStandardEra1880–19132.TheInterwarYears1919–19393.BrettonWoodsPeriod1945–19714.RecentPeriod1973–1997Asweshallseethereareanumberofsimilaritiesbetweentheperiodsbutalsosomeimportantdifferences.Theyconsider21countriesforthefirstthreeperiodsandthenfortherecentperiodgivedatafortheoriginal21aswellasanexpandedgroupof56.Thefirstissueishowtodefineacrisis.Theydefineabankingcrisisasfinancialdistressthatissevereenoughtoresultintheerosionofmostorallofthecapitalinthebankingsystem.Acurrencycrisisisdefinedasaforcedchangeinparity,abandonmentofapeggedexchangerateoraninternationalrescue.Thesecondissueishowtomeasurethedurationofacrisis.TodothistheycomputethetrendrateofGDPgrowthforfiveyearsbefore.ThedurationofthecrisisistheamountoftimebeforeGDPgrowthreturnstoitstrendrate.Finally,thedepthofthecrisisismeasuredbysummingtheoutputlossrelativetotrendforthedurationofthecrisis.Figure1.1showsthefrequencyofcrisesinthefourperiods.Comparingthedatawiththeoriginal21countriesitcanbeseenthattheinterwaryearsaretheworst.ThisisperhapsnotsurprisinggiventhatthiswaswhentheGreatDepressionoccurred.Bankingcriseswereparticularlyprevalentduringthisperiodrelativetotheotherperiods.ItcanbeseenthattheBrettonWoodsperiodisverydifferentfromtheotherperiods.Asmentionedabove,aftertheGreatDepressionpolicymakersinmostcountriesweresodeterminednottoallowsuchaneventtohappenagainthattheyimposedsevereregulationsorbroughtthebanksunderstatecontroltopreventthemfromtakingmuchrisk.Asaresultbankingcriseswerealmostcompletelyeliminated.TherewasonetwincrisisinBrazilin1962butapartfromthattherewerenootherbankingcrisesduringtheentireperiod.TherewerefrequentcurrencycrisesbutaswehaveseentheseweremostlysituationswheremacroeconomicpolicieswereinconsistentwiththelevelofthefixedexchangeratessetintheBrettonWoodssystem. 1.5CrisesinDifferentEras1114BankingcrisesCurrencycrises12TwincrisesAllcrises10864Frequency(annual%probability)201880–19131919–19391945–19711973–19971973–1997(21countries)(56countries)Figure1.1.Crisisfrequency,1880–1997(fromFigure1ofBordoetal.2001).InterestinglythemostbenignperiodwastheGoldStandarderafrom1880to1913.Herebankingcrisesdidoccurbutwerefairlylimitedandcurrencyandtwincriseswerelimitedcomparedtosubsequentperiods.Sincetheglobalfinancialsystemwasfairlyopenatthistime,theimplicationisthatglobalizationdoesnotinevitablyleadtocrises.Therecentperiodisnotasbadastheinterwarperiodbutisneverthelessfairlybad.Bankingandtwincrisesaremorefrequentthanineveryperiodexcepttheinterwaryearsandcurrencycrisesaremuchmorefrequent.Thisisespeciallytrueifthesampleof56countriesisusedasthebasisofcomparisonratherthanthe21countriesusedintheotherperiods.Thecountriesthatareaddedtocreatethelargersamplearemostlyemergingcountries.Thissuggeststhatemergingcountriesaremorepronetocrisesandparticularlytocurrencycrises.Figure1.2confirmsthis.Itbreaksthesampleintoindustrialcountriesandemergingmarkets.Inrecentyearsemergingcountrieshavebeenparticularlypronetocurrencycrisesandtwincrises.TheotherinterestingobservationfromFigure1.2isthatduringtheinterwarperioditwastheindustrialcountriesthatwereparticularlyhardhitbycrises.Theywereactuallymorepronetocurrencyandtwincrisesthantheemergingcountries.Table1.4showstheaveragedurationanddepthofcrisesbrokenoutbytypeofcrisisandforthedifferentperiodsandsamples.PerhapsthemoststrikingfeatureofTable1.4istheshortdurationandmildeffectofcrisesduringthe 12Chapter1.HistoryandinstitutionsIndustrialcountries30BankingcrisesCurrencycrises25Twincrises20Allcrises1510501880–19131919–19391945–19711973–1997Emergingmarkets30BankingcrisesCurrencycrises25Twincrises20Allcrises1510501880–19131919–19391945–19711973–1997Figure1.2.Frequencyofcrises–distributionbymarket(fromFigure2ofBordoetal.(2000)).BrettonWoodsperiod.Theseconddistinctivefeatureisthattwincrisesaremuchworsethanothercrisesintermsoftheoutputlost.Asmightbeexpectedduringtheinterwarperiodtheeffectofcriseswasmuchmoreseverethanintheotherperiods.Althoughtheydidnotlastlongerthecumulativelossinoutputishigherthanintheotherperiods.DuringtheGoldStandardErathedurationandcumulativelosswerenotremarkablecomparedtotheotherperiods.Inrecentyearstwincriseshavelastedforaparticularlylongtimeandthelostoutputissignificant. 1.5CrisesinDifferentEras13Table1.4.Durationanddepthofcrises(fromTable1ofBordoetal.2001).Allcountries1880–19131919–19391945–19711973–19971973–199721nations56nationsAveragedurationofcrisesinyearsCurrencycrises2.61.91.81.92.1Bankingcrises2.32.4a3.12.6Twincrises2.22.71.03.73.8Allcrises2.42.41.82.62.5Averagecrisisdepth(cumulativeGDPlossin%)Currencycrises8.314.25.23.85.9Bankingcrises8.410.5a7.06.2Twincrises14.515.81.715.718.6Allcrises9.813.45.27.88.3Notes:aindicatesnocrises.Source:Authors’calculations.Finally,Figure1.3showstheeffectofcrisesonrecessions.ItcanbeseenthatrecessionswithcriseshaveamuchhigherlossofGDPthanrecessionswithoutcrises.Thiswasparticularlytrueintheinterwarperiod.Alsotheaveragerecoverytimeissomewhathigherinrecessionswithcrisesratherthanrecessionswithoutcrises.Insummary,theanalysisofBordoetal.(2000,2001)leadstoanumberofconclusions.Bankingcrises,currencycrises,andtwincriseshaveoccurredunderavarietyofdifferentmonetaryandregulatoryregimes.Overthelast120yearscriseshavebeenfollowedbyeconomicdownturnslastingonaveragefrom2to3yearsandcosting5to10percentofGDP.Twincrisesareassociatedwithparticularlylargeoutputlosses.Recessionswithcrisesweremoreseverethanrecessionswithoutthem.TheBrettonWoodsperiodfrom1945to1971wasquitespecial.Countrieseitherregulatedbankbalancesheetstopreventthemfromtakingverymuchriskorownedthemdirectlytoachievethesameaim.Thesemeasuresweresuccessfulinthattherewerenobankingcrisesduringthistimeandonlyonetwincrisis.Theinterwarperiodwasalsospecial.Bankingcrisesandcurrencycriseswerewidespread.Moreovertheoutputlossesfromtheseweresevereparticularlywhentheyoccurredtogetherandtherewasatwincrisis.Themostrecentperioddoesindeedappearmorecrisispronethananyotherperiodexceptfortheinterwaryears.Inparticular,itseemsmorecrisispronethantheGoldStandardEra,whichwasthelasttimethatcapitalmarketswereasglobalizedastheyarenow. 14Chapter1.HistoryandinstitutionsGDPlosswithandwithoutcrises30.025.020.015.010.0PercentageGDPloss5.00.01880–19131919–19391945–19711973–19971973–1997(21nations)(56nations)Averagerecoverytimewithandwithoutcrises5.04.03.0Years2.01.00.01880–19131919–19391945–19711973–19971973–1997(21nations)(56nations)Figure1.3.Recessionswithandwithoutcrises(fromFigure2ofBordoetal.2001).1.6SOMERECENTCRISESNowthatwehaveseenacomparisonofrecentcriseswithcrisesinothereras,itisperhapshelpfultoconsidersomeofthemorerecentonesingreaterdetail.WestartwiththosethatoccurredinScandinaviaintheearly1990’s.1.6.1TheScandinaviancrisesNorway,FinlandandSwedenexperiencedaclassicboom–bustcyclethatledtotwincrises(seeHeiskanen1993andEnglundandVihriälä2006).InNorwaylendingincreasedby40percentin1985and1986.Assetpricessoaredwhileinvestmentandconsumptionalsoincreasedsignificantly.Thecollapseinoil 1.6SomeRecentCrises15priceshelpedburstthebubbleandcausedthemostseverebankingcrisisandrecessionsincethewar.InFinlandanexpansionarybudgetin1987resultedinmassivecreditexpansion.Housingpricesrosebyatotalof68percentin1987and1988.In1989thecentralbankincreasedinterestratesandimposedreserverequirementstomoderatecreditexpansion.In1990and1991theeconomicsituationwasexacerbatedbyafallintradewiththeSovietUnion.Assetpricescollapsed,bankshadtobesupportedbythegovernmentandGDPshrankby7percent.InSwedenasteadycreditexpansionthroughthelate1980’sledtoapropertyboom.Inthefallof1990creditwastightenedandinterestratesrose.In1991anumberofbankshadseveredifficultiesbecauseoflendingbasedoninflatedassetvalues.Thegovernmenthadtointerveneandasevererecessionfollowed.1.6.2JapanInthe1980’stheJapaneserealestateandstockmarketswereaffectedbyabubble.Financialliberalizationthroughoutthe1980’sandthedesiretosupporttheUnitedStatesdollarinthelatterpartofthedecadeledtoanexpansionincredit.Duringmostofthe1980’sassetpricesrosesteadily,eventuallyreachingveryhighlevels.Forexample,theNikkei225indexwasaround10,000in1985.OnDecember19,1989itreachedapeakof38,916.AnewGovernoroftheBankofJapan,lessconcernedwithsupportingtheUSdollarandmoreconcernedwithfightinginflation,tightenedmonetarypolicyandthisledtoasharpincreaseininterestratesinearly1990(seeFrankel1993;Tschoegl1993).Thebubbleburst.TheNikkei225fellsharplyduringthefirstpartoftheyearandbyOctober1,1990ithadsunkto20,222.Realestatepricesfollowedasimilarpattern.Thenextfewyearsweremarkedbydefaultsandretrenchmentinthefinancialsystem.Threebigbanksandoneofthelargestfoursecuritiesfirmsfailed.Therealeconomywasadverselyaffectedbytheaftermathofthebubbleandgrowthratesduringthe1990’sand2000’shavemostlybeenslightlypositiveornegative,incontrasttomostofthepost-warperiodwhentheyweremuchhigher.UsingtheaveragegrowthrateofGDPof4percentfrom1976–1991,thedifferencebetweentrendGDPandactualGDPfrom1992–1998isaround¥340trillionorabout68percentofGDP(MikitaniandPosen2000,p.32).1.6.3TheAsiancrisisFromtheearly1950’suntiltheeveofthecrisisin1997the“Dragons”(HongKong,Singapore,SouthKorea,andTaiwan)andthe“Tigers”(Indonesia,Malaysia,thePhilippines,andThailand)wereheldupasmodelsofsuccessful 16Chapter1.Historyandinstitutionseconomicdevelopment.Theireconomiesgrewathighratesformanyyears.Aftersustainedpressure,theThaicentralbankstoppeddefendingthebahtonJuly2,1997anditfell14percentintheonshoremarketand19percentintheoffshoremarket(FourçansandFranck2003,Chapter10).ThismarkedthestartoftheAsianfinancialcrisis.ThenextcurrenciestocomeunderpressurewerethePhilippinepesoandtheMalaysianringitt.ThePhilippinecentralbanktriedtodefendthepesobyraisinginterestratesbutitneverthelesslost$1.5billionofforeignreserves.OnJuly11itletthepesofloatanditpromptlyfell11.5percent.TheMalaysiancentralbankalsodefendedtheringittuntilJuly11beforelettingitfloat.TheIndonesiancentralbankdefendedtherupeeuntilAugust14.TheDragonswerealsoaffected.AtthebeginningofAugust,SingaporedecidednottodefenditscurrencyandbytheendofSeptemberithadfallen8percent.Taiwanalsodecidedtolettheircurrencydepreciateandwerenotmuchaffected.HongKong’sexchangerate,whichwaspeggedtothedollarcameunderattack.However,itwasabletomaintainthepeg.InitiallytheSouthKoreanwonappreciatedagainsttheotherSouthEastAsiancurrencies.However,inNovemberitlost25percentofitsvalue.BytheendofDecember1997whichmarkedtheendofthecrisisthedollarhadappreciatedby52,52,78,107,and151percentagainsttheMalaysian,Philippine,Thai,SouthKorean,andIndonesiancurrencies,respectively.Althoughtheturbulenceinthecurrencymarketswasoverbytheendof1997,therealeffectsofthecrisiscontinuedtobefeltthroughouttheregion.Manyfinancialinstitutions,andindustrialandcommercialfirmswentbankruptandoutputfellsharply.Overall,thecrisiswasextremelypainfulfortheeconomiesinvolved.1.6.4TheRussiancrisisandlongtermcapitalmanagement(LTCM)In1994JohnMeriwetherwhohadpreviouslyworkedforSalomonBrothersandhadbeenaverysuccessfulbondtraderfoundedLTCM.InadditiontoMeriwether,theotherpartnersincludedtwoNobel-prizewinningeconomists,MyronScholesandRobertMerton,andaformervice-chairmanoftheFederalReserveBoard,DavidMullins.Thefundhadnoproblemraising$1.3billioninitially(seehttp://www.erisk.com/Learning/CaseStudies/ref_case_ltcm.aspandLowenstein2000).Thefund’smainstrategywastomakeconvergencetrades.Theseinvolvedfindingsecuritieswhosereturnswerehighlycorrelatedbutwhosepriceswere 1.6SomeRecentCrises17slightlydifferent.Thefundwouldthenshort(i.e.borrow)theonewiththehighpriceandusetheproceedstogolongintheonewiththelowprice.TheconvergencetradesthatweretakenincludedthesovereignbondsofEuropeancountriesthatweremovingtowardsEuropeanMonetaryUnion,andon-the-runandoff-the-runUSgovernmentbonds.Sincethepricedif-ferencesweresmallthestrategyinvolvedalargeamountofborrowing.Forexample,atthebeginningof1998thefirmhadequityofabout$5billionandhadborrowedover$125billion.Inthefirsttwoyearsthefundwasextremelysuccessfulandearnedreturnsforitsinvestorsofaround40percent.However,1997wasnotassuccessfulwithareturnof27percentwhichwasaboutthesameasthereturnonequitiesthatyear.BythistimeLTCMhadabout$7billionundermanagement.Meriwetherdecidedtoreturnabout$2.7billiontoinvestorsastheywerenotabletoearnhighreturnswithsomuchmoneyundermanagement.OnAugust17,1998Russiadevaluedtheroubleanddeclaredamoratoriumonabout281billionroubles($13.5billion)ofgovernmentdebt.Despitethesmallscaleofthedefault,thistriggeredaglobalcrisiswithextremevolatilityinmanyfinancialmarkets.ManyoftheconvergencetradesthatLTCMhadmadestartedtolosemoneyastheflighttoqualitycausedpricestomoveinunexpecteddirections.BySeptember22,1998thevalueofLTCM’scapitalhadfallento$600million.GoldmanSachs,AIG,andWarrenBuffetofferedtopay$250milliontobuyoutthepartnersandtoinject$4billionintothebusinesssothatitwouldnotbeforcedtoselloutitspositions.EventuallytheFederalReserveBankofNewYorkcoordinatedarescuewherebythebanksthathadlentsignificantamountstoLTCMwouldpay$3.5millionfor90percentoftheequityofthefundandtakeoverthemanagementoftheportfolio.ThereasontheFeddidthiswastoavoidthepossibilityofameltdowninglobalassetmarketsandthesystemiccrisisthatwouldfollow.1.6.5TheArgentinacrisisof2001–2002Duringthe1970’sand1980’sArgentina’seconomydidpoorly.Ithadanumberofinflationaryepisodesandcrises.In1991itintroducedacurrencyboardthatpeggedtheArgentinianpesoataone-to-oneexchangeratewiththedollar.Thisusheredinaperiodoflowinflationandeconomicgrowth.Despitethesefavorabledevelopments,anumberofweaknessesdevelopedduringthisperiodincludinganincreaseinpublicsectordebtandalowshareofexportsinoutputandahighconcentrationoftheseinalimitednumberofsectors(seeIMF2003). 18Chapter1.HistoryandinstitutionsInthelasthalfof1998anumberofeventsincludingthecrisisinBrazilandtheresultingdevaluationandtheRussiancrisistriggeredasharpdownturninArgentina’seconomy.Thepublicdebtthegovernmenthadaccumulatedlimitedtheamountoffiscalstimulationthatthegovernmentcouldundertake.Alsothecurrencyboardmeantthatmonetarypolicycouldnotbeusedtostimulatetheeconomy.Therecessioncontinuedtodeepen.Attheendof2001,itbegantobecomeclearerthatArgentina’ssituationwasnotsustainable.Thegovernmenttriedtotakeanumberofmeasurestoimprovethesituationsuchasmodifyingthewaythatthecurrencyboardoperated.Exportersweresubjecttoanexchangeratethatwassubsidizedandimporterspaidatax.Theeffectofthesekindsofmeasureswastolowerconfidenceratherthanraiseit.DespiteanagreementwiththeIMFinSeptember2001toinjectfundsof$5billionimmediatelyandtheprospectofanother$3billionsubsequentlythesituationcontinuedtoworsen.Therewereanumberofattemptstorestructurethepublicdebtbutagainthisdidnotrestoreconfidence.DuringNovember28–30therewasarunonprivatesectordeposits.Thegovernmentthensuspendedconvertibilityinthesensethatitimposedanum-berofcontrolsincludingaweeklylimitof250pesosontheamountthatcouldbewithdrawnfrombanks.InDecember2001,theeconomycollapsed.Industrialproductionfell18percentyear-on-year.Importsfellby50per-centandconstructionfell36percent.InJanuary2002thefifthpresidentinthreeweeksintroducedanewcurrencysystem.Thisinvolvedmultipleexchangeratesdependingonthetypeoftransaction.InFebruarythiswasabolishedandthepesowasallowedtofloatanditsoonfellto1.8pesostothedollar.Overallthecrisiswasdevastating.RealGDPfellbyabout11percentin2002andinflationinApril2002wentto10percentamonth.Thegovernmentdefaultedonitsdebt.Althoughtheeconomystartedtorecoverin2003andhasdonewellsincethen,itwillbesometimebeforeitretainsitspre-crisisactivity.1.7THECOSTSOFCRISESThereisalargeliteratureonthecostsofcrisesandtheirresolution(see,e.g.Bordoetal.2001;Hoggarthetal.2002;RoubiniandSetser2004;Boydetal.2005;andHonohanandLaeven2005).Muchofthedebatehasbeenconcernedwithhowexactlytomeasurecosts.Alargepartoftheearlyliteraturefocusedonthefiscalcosts.Thisistheamountthatitcoststhegovernmenttorecapitalizebanksandreimburseinsureddepositors.However,thesearemostlytransfers 1.8TheoriesofCrises19ratherthantruecosts.Thesubsequentliteraturehasfocusedmoreonthelostoutputrelativetoabenchmarksuchastrendgrowthrate.Therearetwoimportantaspectsofthecostsofcrises.Thefirstisthehighaveragecostandthesecondisthelargevariationintheamountofcosts.Boydetal.(2005)estimatetheaveragediscountedpresentvalueoflossesinanumberofdifferentways.Dependingonthemethodusedthemeanlossisbetween63percentand302percentofrealpercapitaGDPintheyearbeforethecrisisstarts.Thedistributionoflossesisverylarge.InCanada,France,Germany,andtheUS,whichexperiencedmildnonsystemiccrises,therewasnotanysignificantslowdowningrowthandcostswereinsignificant.How-ever,attheotherextremetheslowdownanddiscountedlossinoutputwereextremelyhigh.InHongKongthediscountedPVoflosseswas1,041percentofrealoutputtheyearbeforethecrisis.Itisthelargeaveragecostsandtheveryhightailcostsofcrisesthatmakespolicymakerssoaversetocrises.Thisiswhyinmostcasestheygotosuchgreatlengthstoavoidthem.However,itisnotclearthatthisisoptimal.Therearesignificantcostsassociatedwithregulationstoavoidcrisesandinmanycasescrisesarenotverycostly.Animportantthemeofthisbookisthatthecostsofavoidingcrisesmustbetradedoffagainstthecostsofallowingcrises.1.8THEORIESOFCRISESThecontrastbetweenthemajorityviewconcerningthecauseofcrisesinthe1930’sandtheviewofmanytodayisstriking.Inthe1930’sthemarketwastheproblemandgovernmentinterventionthroughregulationordirectownershipofbankswasthesolution.Todaymanyarguethatinconsistentgovernmentmacroeconomicpoliciesormoralhazardinthefinancialsystemcausedbygovernmentguaranteesisattherootofrecentcrises.Heretheviewisthatgovernmentisthecauseofcrisesandnotthesolution.Marketforcesarethesolution.Inthisbookweaimtoprovidesomeperspectiveonthisdebatebydevelop-ingatheoreticalapproachtoanalyzefinancialcrises.Ineachchapterwewilldevelopthebasicideasandthenprovideabriefaccountofthetheoreticalandempiricalliteratureonthetopic.WestartinChapter2withsomebackgroundmaterial.Inparticular,wereviewthebasicsoftime,uncertainty,andliquidity.FormanyreaderswhoarequitefamiliarwiththismaterialitwillbebettertoproceedstraighttoChapter3.Forthosewhoarenot,orwhoneedarefresheronmodelsofresourceallocationovertimeandwithuncertainty,Chapter2willprovide 20Chapter1.Historyandinstitutionsanintroduction.Thefirstpartofthechapterdevelopsbasicideasrelatedtoconsumptionandsaving,andproduction,suchasdatedcommoditiesandforwardmarkets.Thesecondpartconsidersuncertaintyandintroducesstatesofnature,contingentcommodities,completemarkets,andArrowsecurities.Attitudestowardrisk,andtherolesofinsuranceandriskpoolingarealsointroduced.Thefinalpartofthechapterconsidershowliquidityandliquiditypreferencecanbemodeled.Chapter3considersintermediation.Inordertounderstandhowbank-ingcrisesariseitisfirstnecessarytodevelopatheoryofbankingormoregenerallyofintermediation.Theapproachadoptedistomodelintermedi-ariesasprovidingliquidityinsurancetoconsumers.Usingthisfoundationtwoapproachestocrisescanbedeveloped.Bothviewsofcriseshavealonghistory.Oneview,wellexpoundedbyKindleberger(1978),isthattheyoccurspon-taneouslyasapanic.ThemodernversionwasdevelopedbyBryant(1980)andDiamondandDybvig(1983).Theanalysisisbasedontheexistenceofmultipleequilibria.Inatleastoneequilibriumthereisapanicwhileinanotherthereisnot.Thebusinesscycletheoryalsohasalonghistory(see,e.g.Mitchell1941).Thebasicideaisthatwhentheeconomygoesintoarecessionordepressionthereturnsonbankassetswillbelow.Giventheirfixedliabilitiesintheformofdepositsorbondstheymayunabletoremainsolvent.Thismayprecipitatearunonbanks.Gorton(1988)showedempiricallythatintheUSinthelatenineteenthandearlytwentiethcenturies,aleadingeconomicindicatorbasedontheliabilitiesoffailedbusinessescouldaccuratelypredicttheoccurrenceofbankingcrises.ThesecondpartofChapter3developsthisapproachtocrises.Oneofthemostimportantcausesofcrisesisadramaticcollapseinassetprices.Oneexplanationforthisdropinprices,whichisthebasisforthebusinesscycleviewofcrisesexaminedinChapter3,isthatexpectedfuturecashflowsfall.Anotherpossibilityisthatpricesarelowbecauseofashortageofliquidity.Chapter4investigatestheoperationofassetmarketswhereassetpricevolatilityisdrivenbyliquidityshocks.ThemodelissimilartothatinChapter3,excepttherearenointermediaries.Inadditionthereisafixedcosttoparticipatinginmarketsandthiscanleadtolimitedmarketparticipation.Whenliquidityisplentiful,assetpricesaredrivenbyexpectedfuturepayoffsintheusualway.However,whenthereisascarcityofliquiditythereis“cash-in-the-marketpricing.”Inthiscase,anasset’spriceissimplytheratiooftheamountsoldtotheamountofcashorliquiditythatbuyershave.Expostbuyerswouldliketohavemoreliquiditywhenthereiscash-in-the-marketpricing.Exantetheybalancetheopportunitycostofholdingliquiditywhenliquidityisplentifulagainsttheprofitstobemadewhenliquidityisscarce.Thistheoryof 1.8TheoriesofCrises21assetpricedeterminationisconsistentwithsignificantassetpricevolatility.ItisshownthattherecanbemultiplePareto-rankedequilibria.Inoneequilibrium,thereislimitedparticipationandassetpricesarevolatile.Inanother,whichisPareto-preferred,thereiscompleteparticipationandassetpricesarenotveryvolatile.Althoughinsomecrisestheinitialtriggerisalargeshock,inothersitappearsthetriggerisasmallevent.Forexample,intheRussiancrisisof1998discussedabove,themoratoriumondebtpaymentsthattriggeredthecrisisinvolvedatinyproportionoftheworld’sassets.Neverthelessithadahugeimpactontheworld’sfinancialmarkets.Therewassubsequentlyaperiodofextremeturbulenceinfinancialmarkets.UnderstandinghowthistypeoffinancialfragilitycanariseisthetopicofChapter5.RatherthanjustfocusingonbanksasinChapter3,oronmarketsasinChapter4,heretheinteractionofbanksandmarketsisconsidered.Themarketsareinstitutionalmarketsinthesensethattheyareforbanksandintermediariestosharerisksandliquidity.Individualscannotdirectlyaccessthesemarketsbutinsteadinvesttheirfundsinbanksthathaveaccesstothemarkets.AsinChapter4,thekeytounderstandingtheformofequilibriumistheincentivesforprovidingliquiditytothemarket.Inorderforbankstobewillingtoholdliquidity,theopportunitycostofdoingthisinstateswheretheliquidityisnotusedmustbebalancedbytheprofitstobemadewhenliquidityisscarceandthereiscash-in-the-marketpricing.Itispossibletoshowthatifsucheventsarerarethenverylargechangesinpricescanbetriggeredbysmallchangesinliquiditydemand.Thesepricechangescancausebankruptcyanddisruption.Thereisfinancialfragility.WhileinChapters3–5thefocusisonunderstandingthepositiveaspectsofhowvarioustypesofcrisiscanarise,inChapter6wedevelopageneralframeworkforunderstandingthenormativeaspectsofcrises.Themodelisabenchmarkforinvestigatingthewelfarepropertiesoffinancialsystems.Simi-larlytoChapter5,therearebothintermediariesandmarkets.However,whereasinChapter5marketswereincompleteinthesensethathedgingopportunitieswerelimited,hereweassumefinancialmarketsarecomplete.Inparticular,itispossibleforintermediariestohedgeallaggregaterisksinthefinancialmarkets.UndertheseidealcircumstancesitcanbeshownthatAdamSmith’sinvisiblehandworks.Theallocationofresourcesisefficientinthefollowingsense.Ifthecontractsbetweenintermediariesandconsumersarecompleteinthattheycanalsobeconditionedonaggregaterisks,thentheallocationis(incentive)efficient.Manycontractsobservedinpracticebetweenintermediariesandconsumerssuchasdebtanddepositcontractsareincomplete.Providedfinancialmarketsarecomplete,thenevenifcontractsbetweenintermediariesandconsumersareincomplete,itcanbeshowntheallocationisconstrainedefficient.Inother 22Chapter1.Historyandinstitutionswords,aplannersubjecttothesameconstraintsintermsofincompletecon-tractswithconsumerscouldnotdobetter.Whatismoreitisshownthattheequilibriumwithincompletecontractsofteninvolvestherebeingfinancialcrises.Forexample,ifabankusesadepositcontractthentherecanbeabank-ingcrisis.Thisdemonstratesthatcrisesarenoteverywhereandalwaysbad.Insomecasestheycanincreaseeffectivecontingenciesandimprovethealloca-tionofresources.Ofcourse,wearenotsayingthatcrisesarealwaysgood,onlythatinsomecasestheycanbe,inparticularwhenfinancialmarketsarecom-pleteandcontractsbetweenintermediariesandconsumersareincomplete.Iffinancialmarketsareincompletethencrisescanindeedbebad.Forexample,asmentionedthefinancialfragilityconsideredinChapter5occursbecausemar-ketsareincomplete.ThusthecontributionofChapter6istoidentifywhentherearemarketfailuresthatpotentiallyleadtoalossofwelfare.Havingidentifiedwhenthereisamarketfailure,thenaturalquestionthatfollowsiswhetherthereexistpoliciesthatcancorrecttheundesirableeffectsofsuchfailures.ThisisthetopicofChapter7.Twotypesofregulationareconsidered.Thefirstistheregulationofbankcapitalandthesecondistheregu-lationofbankliquidity.Simpleexampleswithconstantrelativeriskaversionconsumersareanalyzedwhenfinancialmarketsareincomplete.Itisshownthattheeffectofbankandliquidityregulationdependcriticallyonthedegreeofrelativeriskaversion.Whenrelativeriskaversionissufficientlylow(below2)increasinglevelsofbankcapitalabovewhatbankswouldvoluntarilyholdcanmakeeverybodybetteroff.Forbankliquidityregulation,requiringbankstoholdmoreliquiditythantheywouldchoosetoiswelfareimprovingifrelativeriskaversionisabove1.Theinformationalrequirementsforthesekindsofinterventionarehigh.Thusitmaybedifficulttoimprovewelfarethroughthesekindsofregulationasapracticalmatter.TheanalysisinChapters6and7stressestheabilityofinvestorstosharedifferentrisks.Risksharingtotheextentitispossibleoccursbecauseofexplicitcontingenciesincontractsoreffectivecontingenciesthatcanoccurifthereisdefault.Liquidityisassociatedwithsuppliesoftheconsumptiongood.Therehasbeennoroleformoneyorvariationsinthepricelevel.Chapter8considerstheeffectofallowingformoneyandthedenominationofdebtandothercontractsinnominalterms.Itisshownthatifthecentralbankcanvarythepricelevelthenthisprovidesanotherwayforrisktobeshared.Thisistrueforriskssharedwithinacountry.Itisalsotrueforriskssharedbetweencountries.Byvaryingtheexchangerateappropriatelyacentralbankcanensureriskissharedoptimallywiththerestoftheworld.However,suchinternationalrisksharingcreatesamoralhazardbecauseofthepossibilitythatacountrywillborrowalotindomesticcurrencyandthenexpropriatethelendersbyinflatingawaythevalueofthecurrency. 1.9ConcludingRemarks23Thefinaltwochaptersinthebookconsidertwoformsofcrisisthatappeartobeparticularlyimportantbutwhichwerenotconsideredearlier.Inmanyinstancesfinancialcrisesoccurafterabubbleinassetpricescollapses.HowthesebubblesformandcollapseandtheireffectonthefinancialsystemisthesubjectofChapter9.ThemostimportantrecentexampleofthisphenomenonisJapanwhichwasdiscussedabove.Inthemid1980’stheNikkeistockindexwasaround10,000.Bytheendofthedecadeithadrisentoaround40,000.AnewgovernoroftheBankofJapanwhowasconcernedthataloosemonetarypolicyhadkindledprospectsofinflationdecidedtoincreaseinterestratessubstantially.Thisprickedthebubbleandcausedstockpricestofall.Withinafewmonthstheyhadfallenbyhalf.Realestatepricescontinuedtoriseforoverayearhowevertheythenalsostartedtofall.Fifteenyearslaterbothassetpricesandrealestatearesignificantlylowerwithstocksandrealestateataroundaquarteroftheirpeakvalue.Thefallinassetpriceshasledtoafallingrowthandabankingcrisis.Japanisbynomeanstheonlyexampleofthisphenomenon.ItcanbearguedtheAsiancrisisfallsintothiscategory.IntheUStheRoaring1920’sandtheGreatDepressionofthe1930’sareanotherexample.TheAsiancrisisillustratedanotherimportantphenomenon,contagion.TheepisodestartedinThailandandspreadtomanyothercountriesintheregionincludingSouthKorea,Malaysia,Indonesia,HongKong,thePhilippinesandSingapore.InterestinglyitdidnotaffectTaiwannearlyasmuch.Otherregions,particularlySouthAmerica,werealsoaffected.Understandingthecontagiousnatureofmanycriseshasbecomeanimportanttopicintheliterature.Thereareanumberoftheoriesofcontagion.Oneisbasedontradeandreallinks,anotherisbasedoninterbankmarkets,anotheronfinancialmarketsandoneonpaymentssystems.ContagionthroughinterbankmarketsisthesubjectmatterofChapter10.1.9CONCLUDINGREMARKSThewordcrisisisusedinmanydifferentways.Thisnaturallyraisesthequestionofwhenasituationisacrisisandwhenitisnot.Itisperhapshelpfultoconsiderthedefinitionofcrises.Accordingtothedictionary(dictionary.com)acrisisis:1.(a)theturningpointforbetterorworseinanacutediseaseorfever(b)aparoxysmalattackofpain,distress,ordisorderedfunction(c)anemotionallysignificanteventorradicalchangeofstatusinaperson’slife 24Chapter1.Historyandinstitutions2.thedecisivemoment(asinaliteraryplot)3.(a)anunstableorcrucialtimeorstateofaffairsinwhichadecisivechangeisimpending;especially:onewiththedistinctpossibilityofahighlyundesirableoutcome(b)asituationthathasreachedacriticalphase.Thisgivesarangeofthesensesinwhichthewordisusedingeneral.Withregardtofinancialcrisesitisalsousedinawiderangeofsituations.Bankingcrisesusuallyrefertosituationswheremanybankssimultaneouslycomeunderpressureandmaybeforcedtodefault.Currencycrisesoccurwhentherearelargevolumesoftradeintheforeignexchangemarketwhichcanleadtoadevaluationorrevaluation.Similarlyitisusedinmanyothersituationswherebigchanges,usuallybad,appearpossible.Thisisthesenseinwhichweareusingthewordinthisbook.Historically,thestudyoffinancialcriseswasanimportantfieldineco-nomics.Theeliminationofbankingcrisesinthepost-warperiodsignificantlyreducedinterestincrisesanditbecameanareaforeconomichistorians.Nowthatcriseshavereemergedmuchworkremainstobedoneusingmodernthe-oreticaltoolstounderstandthemanyaspectsofcrises.Thisbookisdesignedtogiveabriefintroductiontosomeofthetheoriesthathavebeenusedtotryandunderstandthesecomplexevents.Thereisasignificantempiricalliteratureonfinancialcrises.Muchofthisworkisconcernedwithdocumentingregularitiesinthedata.Sincethetheoryisatarelativelyearlystagethereisrelativelylittleworktryingtodistinguishbetweendifferenttheoriesofcrises.Inthechaptersbelowthehistoricalandempiricalworkisdiscussedasabackgroundtothetheory.Muchworkremainstobedoneinthisareatoo.Thereisatendencyinmuchoftheliteratureoncrisestoarguethattheparticulartheorybeingpresentedis“THE”theoryofcrises.Aseventhebriefdiscussioninthischapterindicatescrisesarecomplexphenomenainpractice.Oneofthemainthemesofthisbookisthatthereisnoonetheoryofcrisesthatcanexplainallaspectsofthephenomenaofinterest.Ingeneral,thetheoriesofcrisesthatwewillfocusonarenotmutuallyexclusive.Actualcrisesmaycontainelementsofsomecombinationofthesetheories.REFERENCESBagehot,W.(1873).LombardStreet:ADescriptionoftheMoneyMarket,London:H.S.King.Bordo,M.(1986).“FinancialCrises,BankingCrises,StockMarketCrashesandtheMoneySupply:SomeInternationalEvidence,1870-1933,”inF.CapieandG.Wood References25(eds.),FinancialCrisesandtheWorldBankingSystem.NewYork:St.Martin’sPress,1986,190–248.Bordo,M.andB.Eichengreen(1999).“IsOurCurrentInternationalEconomicEnvironmentUnusuallyCrisisProne?”Workingpaper,RutgersUniversity.Bordo,M.,B.Eichengreen,D.KlingebielandM.Martinez-Peria(2000).“IstheCrisisProblemGrowingMoreSevere?”Workingpaper,UniversityofCalifornia,Berkeley.Updatedversionofpreviouspaper.http://emlab.berkeley.edu/users/eichengr/research.html–thislinkalsoallowsyoutodownloadthefigures:scrolldownthepageuntilyoureachthelinktothepaperandfigures–seealsohttp://www.haas.berkeley.edu/arose/BEKSc.pdfBordo,M.,B.Eichengreen,D.KlingebielandM.Martinez-Peria(2001).“IstheCri-sisProblemGrowingMoreSevere?”EconomicPolicy,April2001,53–82+WebAppendix.Boyd,J.,S.Kwak,andB.Smith(2005).“TheRealOutputLossesAssociatedwithModernBankingCrises,”JournalofMoney,Credit,andBanking37,977–999.Bryant,J.(1980).“AModelofReserves,BankRuns,andDepositInsurance,”JournalofBankingandFinance4,335–344.Diamond,D.andP.Dybvig(1983).“BankRuns,DepositInsurance,andLiquidity,”JournalofPoliticalEconomy91,401–419.Englund,P.andV.Vihriälä(2006).“FinancialCrisesinDevelopedEconomies:TheCasesofFinlandandSweden,”Chapter3inLarsJonung(ed.),Crises,MacroeconomicPerformanceandEconomicPoliciesinFinlandandSwedeninthe1990s:AComparativeApproach,forthcoming.Fourçans,A.andR.Franck(2003).CurrencyCrises:ATheoreticalandEmpiricalPerspective,Northampton,MA:EdwardElgar.Frankel,J.(1993).“TheJapaneseFinancialSystemandtheCostofCapital,”inS.Takagi(ed.),JapaneseCapitalMarkets:NewDevelopmentsinRegulationsandInstitutions,Oxford:Blackwell,21–77.Gorton,G.(1988).“BankingPanicsandBusinessCycles,”OxfordEconomicPapers40,751–781.Heiskanen,R.(1993).“TheBankingCrisisintheNordicCountries,”KansallisEconomicReview2,13–19.Hoggarth,G.,R.Reis,andV.Saporta(2002).“CostsofBankingSystemInstability:SomeEmpiricalEvidence,”JournalofBankingandFinance26,825–855.Honohan,P.andL.Laeven(2005).SystemicFinancialCrises:ContainmentandResolution,Cambridge,UK:CambridgeUniversityPress.IMF(2003).“LessonsformtheCrisisinArgentina,”Washington,DC:IMF,http://www.imf.org/external/np/pdr/lessons/100803.pdf.Kindleberger,C.(1978).Manias,Panics,andCrashes:AHistoryofFinancialCrises,NewYork:BasicBooks.Kindleberger,C.(1993).AFinancialHistoryofWesternEurope(secondedition),NewYork:OxfordUniversityPress.Lowenstein,R.(2000).WhenGeniusFailed:TheRiseandFallofLong-TermCapitalManagement,NewYork:RandomHouse. 26Chapter1.HistoryandinstitutionsMikitani,R.andA.Posen(2000).Japan’sFinancialCrisisanditsParallelstoU.S.Experience,Washington,DC:InstituteforInternationalEconomics,SpecialReport13.Mitchell,W.(1941).BusinessCyclesandTheirCauses,Berkeley:UniversityofCaliforniaPress.Roubini,N.andB.Setser(2004).BailoutsorBail-Ins?RespondingtoFinancialCrisesinEmergingEconomies,Washington,DC:InstituteforInternationalEconomics.Sprague,O.(1910).AHistoryofCrisesUndertheNationalBankingSystem,NationalMonetaryCommission,WashingtonDC:U.S.GovernmentPrintingOffice.Studenski,P.andH.Krooss(1963).FinancialHistoryoftheUnitedStates(secondedition),NewYork:McGrawHill.Timberlake,R.(1978).TheOriginsofCentralBankingintheUnitedStates,CambridgeMA:HarvardUniversityPress.Tschoegl,A.(1993).“ModelingtheBehaviourofJapaneseStockIndices,”inS.Takagi(ed.),JapaneseCapitalMarkets:NewDevelopmentsinRegulationsandInstitutions,Oxford:Blackwell,371–400.Wilson,J.,R.SyllaandC.Jones(1990).“FinancialMarketPanicsandVolatilityintheLongRun,1830–1988,”inE.White(ed.),CrashesandPanics,Illinois:Dow-JonesIrwin,85–125. 2Time,uncertainty,andliquidityFinancialeconomicsdealswiththeallocationofresourcesovertimeandinthefaceofuncertainty.Althoughweusetermslike“presentvalues,”“statesofnature,”and“contingentcommodities”toanalyzeresourceallocationinthesesettings,thebasicideasareidenticaltothoseusedintheanalysisofconsumerandproducerbehaviorinordinarymicroeconomictheory.Inthischapterwereviewfamiliarconceptssuchaspreferences,budgetconstraints,andproductiontechnologiesinanewsetting,whereweusethemtostudytheintertemporalallocationofresourcesandtheallocationofrisk.Weusesimpleexamplestoexplaintheseideasandlatershowhowtheideascanbeextendedandgeneralized.2.1EFFICIENTALLOCATIONOVERTIMEWebeginwiththeallocationofresourcesovertime.Althoughweintroducesomenewterminology,thekeyconceptsarethesameasconceptsfamiliarfromthestudyofefficientallocationina“timeless”environment.Weassumethattimeisdividedintotwoperiods,whichwecanthinkofasrepresentingthe“present”andthe“future.”Wecalltheseperiodsdatesandindexthembyt=0,1,wheredate0isthepresentanddate1isthefuture.2.1.1ConsumptionandsavingSupposeaconsumerhasanincomestreamconsistingofY0unitsofahomo-geneousconsumptiongoodatdate0andY1unitsoftheconsumptiongoodatdate1.Theconsumer’sutilityU(C0,C1)isafunctionofhisconsumptionstream(C0,C1),whereC0isconsumptionatdate0andC1isconsumptionatdate1.Theconsumerwantstomaximizehisutilitybutfirsthastodecidewhichconsumptionstreams(C0,C1)belongtohisbudgetset,thatis,whichstreamsarefeasibleforhim.Thereareseveralwaysoflookingatthisquestion.Theyallleadtothesameanswer,butitisworthconsideringeachoneinturn. 28Chapter2.Time,Uncertainty,andLiquidityBorrowingandlendingOnewayofposingthequestion(ofwhichconsumptionstreamstheconsumercanafford)istoaskwhethertheincomestream(Y0,Y1)canbetransformedintoaconsumptionstream(C0,C1)byborrowingandlending.Forsimplicity,wesupposethereisabankthatiswillingtolendanyamountatthefixedinterestratei>0perperiod,thatis,thebankwilllendoneunitofpresentconsumptiontodayinexchangeforrepaymentof(1+i)unitsinthefuture.SupposetheconsumerdecidedtospendC0>Y0today.ThenhewouldhavetoborrowB=C0−Y0inordertobalancehisbudgettoday,andthisborrowingwouldhavetoberepaidwithinterestiBinthefuture.Theconsumercouldaffordtodothisifandonlyifhisfutureincomeexceedshisfutureconsumptionbytheamountoftheprincipalandinterest,thatis,(1+i)B≤Y1−C1.Wecanrewritethisinequalityintermsoftheconsumptionandincomestreamsasfollows:1C0−Y0≤(Y1−C1).1+iConversely,iftheconsumerdecidedtoconsumeC0≤Y0inthepresent,hecouldsavethedifferenceS=Y0−C0anddeposititwiththebank.Wesupposethatthebankiswillingtopaythesameinterestratei>0ondepositsthatitearnsonloans,thatis,oneunitofpresentconsumptiondepositedwiththebanktodaywillbeworth(1+i)unitsinthefuture.Theconsumerwillreceivehissavingswithinterestinthefuture,sohisfutureconsumptioncouldexceedhisincomeby(1+i)S,thatis,C1−Y1≤(1+i)S.Wecanrewritethisinequalityintermsoftheconsumptionandincomestreamsasfollows:1C0−Y0≤(Y1−C1).1+iNoticethatthisisthesameinequalityaswederivedbefore.Thus,anyfeasibleconsumptionstream,whetheritinvolvessavingorborrowing,mustsatisfythesameconstraint.Wecallthisconstrainttheintertemporalbudgetconstraintandwriteitforfuturereferenceinaslightlydifferentform:11C0+C1≤Y0+Y1.(2.1)1+i1+i 2.1EfficientAllocationOverTime29C1(1+i)Y0+Y1Y0+Y1/(1+i)C0Figure2.1.Intertemporalbudgetconstraint.Figure2.1illustratesthesetofconsumptionstreams(C0,C1)thatsatisfytheintertemporalbudgetconstraint.Itiseasytoseethattheincomestream(Y0,Y1)mustsatisfytheintertemporalbudgetconstraint.IfthereisneitherborrowingnorlendinginthefirstperiodthenC0=Y0andC1=Y1.Theendpointsofthelinerepresentthelevelsofconsumptionthatwouldbepossibleiftheindividualweretoconsumeasmuchaspossibleinthepresentandfuture,respectively.Forexample,ifhewantstoconsumeasmuchaspossibleinthepresent,hehasY0unitsofincometodayandhecanborrowB=Y1/(1+i)unitsofthegoodagainsthisfutureincome.Thisisthemax-imumhecanborrowbecauseinthefuturehewillhavetorepaytheprincipalBplustheinterestiB,foratotalof(1+i)B=Y1.SothemaximumamounthecanspendtodayisgivenbyY1C0=Y0+B=Y0+.1+iConversely,ifhewantstoconsumeasmuchaspossibleinthefuture,hewillsavehisentireincomeinthepresent.Inthefuture,hewillgethissavingswithinterest(1+i)Y0plushisfutureincomeY1.SothemaximumamounthecanspendinthefutureisC1=(1+i)Y0+Y1.Supposenowthatconsumptioninthefirstperiodisincreasedby
C0.Byhowmuchmustfutureconsumptionbereduced?Everyunitborrowedinthefirstperiodwillcost(1+i)inthesecondbecauseinterestmustbepaid.Sothedecreaseinsecondperiodconsumptionis
C1=−(1+i)
C0.This 30Chapter2.Time,Uncertainty,andLiquidityshowsthathecanaffordanyconsumptionstreamonthelinebetweenthetwoendpointswithconstantslope=−(1+i).(SeeFigure2.1.)Wehaveshownthatanyconsumptionstreamthatcanbeachievedbyborrowingandlendingmustsatisfytheintertemporalbudgetconstraint.Con-versely,wecanshowthatanyconsumptionstream(C0,C1)thatsatisfiestheintertemporalbudgetconstraintcanbeachievedbysomefeasiblepatternofborrowingorlending(saving).Toseethis,supposethattheintertemporalbudgetconstraintissatisfiedbytheconsumptionstream(C0,C1).IfC0>Y0weassumetheconsumerborrowsB=C0−Y0.Inthefuturehehastorepayhisloanwithinterest,soheonlyhasY1−(1+i)Blefttospendonconsumption.However,theintertemporalbudgetconstraintensuresthathisplannedfutureconsumptionC1satisfiesC1≤(1+i)(Y0−C0)+Y1=Y1−(1+i)(C0−Y0).SotheconsumercanborrowBunitstodayandrepayitwithinteresttomorrowandstillaffordhisplannedfutureconsumption.TheothercasewhereC0≤Y0ishandledsimilarly.Thus,wehaveseenthattheincomestream(Y0,Y1)canbetransformedintotheconsumptionstream(C0,C1)throughborrowingorlendingattheinterestrateiifandonlyifitsatisfiestheintertemporalbudgetconstraint(2.1).WealthandpresentvaluesAnotherwayofthinkingaboutthesetofaffordableconsumptionstreamsmakesuseoftheconceptofpresentvalues.Thepresentvalueofanygoodistheamountofpresentconsumptionthatsomeonewouldgiveforit.Thepresentvalueofoneunitofpresentconsumptionis1.Thepresentvalueofoneunitoffutureconsumptionis1/(1+i),becauseoneunitofpresentconsumptioncanbeconvertedinto1+iunitsoffutureconsumption,andviceversa,throughborrowingandlending.Thus,thepresentvalueoftheincomestream(Y0,Y1),thatis,thevalueof(Y0,Y1)intermsofpresentconsumptionis1PV(Y0,Y1)≡Y0+Y11+iandthepresentvalueoftheconsumptionstream(C0,C1)is1PV(C0,C1)≡C0+C1.1+i 2.1EfficientAllocationOverTime31Theintertemporalbudgetconstraintsaysthatthepresentvalueoftheconsumptionstream(C0,C1)mustbelessthanorequaltothepresentvalueoftheconsumer’sincomestream.Thepresentvalueoftheincomestream(Y0,Y1)isalsocalledtheconsumer’swealth,denotedbyW≡Y10+Y1.Theintertemporalbudgetconstraint1+iallowstheconsumertochooseanyconsumptionstream(C1,C2)whosepresentvaluedoesnotexceedhiswealth,thatis,C1C0+≤W.(2.2)1+iDatedcommoditiesandforwardmarketsThereisstillathirdwaytointerprettheintertemporalbudgetconstraint(2.1).Wearefamiliarwiththebudgetconstraintofaconsumerwhohastodividehisincomebetweentwogoods,beerandpizza,forexample.Thereisapriceatwhicheachgoodcanbepurchasedandthevalueofconsumptioniscalcu-latedbymultiplyingthequantityofeachgoodbythepriceandaddingthetwoexpenditures.Theconsumer’sbudgetconstraintsaysthatthevalueofhisconsumptionmustbelessthanorequaltohisincome.Theintertemporalbudgetconstraint(2.1)canbeinterpretedinthiswaytoo.Supposewetreatpresentconsumptionandfutureconsumptionastwodifferentcommoditiesandassumethattherearemarketsonwhichthetwocommoditiescanbetraded.Weassumethesemarketsareperfectlycompetitive,sotheconsumercanbuyandsellasmuchashelikesofbothcommoditiesattheprevail-ingprices.Theusualbudgetconstraintrequirestheconsumertobalancethevalueofhispurchasesandexpendituresonthetwocommodities.Ifp0isthepriceofpresentconsumptionandp1isthepriceoffutureconsumption,thentheordinarybudgetconstraintcanbewrittenasp0C0+p1C1≤p0Y0+p1Y1.Supposethatwewanttousethefirst-periodconsumptiongoodasournumeraire,thatis,measurethevalueofeverycommodityintermsofthisgood.Thenthepriceofpresentconsumptionisp0=1,sinceoneunitofthegoodatdate0isworthexactlyone(unitofthegoodatdate0).Howmuchisthegoodatdate1worthintermsofthegoodatdate1?Ifitispossibletoborrowandlendattheinterestratei,thepriceofthegoodatdate1willbedeterminedbyarbitrage.Ifp1,thenanyonecanmakearisklessarbitrageprofitby1>1+isellingoneunitoffutureconsumptionforp11,usingtheproceedstobuy1+iunitsofpresentconsumption,andinvestingthe1unitsattheinterestratei1+i 32Chapter2.Time,Uncertainty,andLiquiditytoyield(1+i)1=1unitoffutureconsumption.Thisstrategyyieldsaprofit1+iofp1atdate0andhasnocostsincetheunitoffutureconsumption1−1+ithatissoldisprovidedbytheinvestmentatdate0.Suchariskfreeprofitisincompatiblewithequilibrium,sinceanyonecanusethisarbitragetogenerateunlimitedwealth.Thus,inequilibriumwemusthavep11≤.1+iAsimilarargumentcanbeusedtoshowthatifp1,itispossibleto1<1+imakeariskfreeprofitbyborrowing1unitsofpresentconsumption,buying1+ioneunitoffutureconsumptionatthepricep1,andusingittorepaytheloanatdate1.Thus,inequilibrium,wemusthavep1.1≥1+iPuttingthesetwoarbitrageargumentstogether,wecanseethat,ifitispossibletoborrowandlendattheinterestrateiandpresentconsumptionisthenumeraire,theonlypricesconsistentwithequilibriumarep0=1andp1.1=1+iSubstitutingthepricesp1intothebudgetconstraint0=1andp1=1+iabove,weseethatitisexactlyequivalenttotheintertemporalbudgetconstraint(2.1).Borrowingandlendingataconstantinterestrateisequivalenttohavingamarketinwhichpresentandfutureconsumptioncanbeexchangedattheconstantprices(p0,p1).Thesamegooddeliveredattwodifferentdatesistwodifferentcommoditiesandpresentandfutureconsumptionare,infact,simplytwodifferentcommoditieswithtwodistinctprices.Fromthispointofview,theintertemporalbudgetconstraintisjustanewinterpretationofthefamiliarconsumer’sbudgetconstraint.ConsumptionandsavingSincetheconsumer’schoicesamongconsumptionstreams(C0,C1)arecom-pletelycharacterizedbytheintertemporalbudgetconstraint,theconsumer’sdecisionproblemistomaximizehisutilityU(C0,C1)bychoosingaconsump-tionstream(C0,C1)thatsatisfiesthebudgetconstraint.WerepresentthisdecisionproblemschematicallybymaxU(C0,C1)C1s.t.C0+=W.1+iNotethathereweassumethebudgetconstraintissatisfiedasanequalityratherthananinequality.Sincemoreconsumptionispreferredtoless,thereisnolossofgeneralityinassumingthattheconsumerwillalwaysspendasmuchashecanonconsumption.ThesolutiontothismaximizationproblemisillustratedinFigure2.2,wheretheoptimumoccursatthepointonthebudgetconstraintwheretheindifferencecurveistangenttothebudgetconstraint. 2.1EfficientAllocationOverTime33C1(C0,C1)0(W,0)C0Figure2.2.Consumptionandsaving.Theslopeofthebudgetconstraintis−(1+i)andtheslopeoftheindifferencecurveattheoptimumpointis∂U(C∗,C∗)∂C001−,∂U(C∗,C∗)∂C101sothetangencyconditioncanbewrittenas∂U(C∗,C∗)∂C001=(1+i).∂U(C∗,C∗)∂C101Itiseasytointerpretthefirst-orderconditionbyrewritingitas∂U∗∗∂U∗∗(C0,C1)=(1+i)(C0,C1).∂C0∂C1Thelefthandsideisthemarginalutilityofconsumptionatdate0.Therighthandsideisthemarginalutilityof(1+i)unitsofconsumptionatdate1.Oneunitofthegoodatdate0canbesavedtoprovide1+iunitsofthegoodatdate1.Sothefirst-orderconditionsaysthattheconsumerisindifferentbetweenconsumingoneunitatdate0andsavingituntildate1whenitwillbeworth(1+i)unitsandthenconsumingthe(1+i)unitsatdate1.AnalternativetothegraphicalmethodoffindingtheoptimumistousethemethodofLagrange,whichrequiresustoformtheLagrangeanfunction
1L(C0,C1,λ)=U(C0,C1)−λC0+C1−W1+i 34Chapter2.Time,Uncertainty,andLiquidityandmaximizethevalueofthisfunctionwithrespecttoC0,C1,andtheLagrangemultiplierλ.Anecessaryconditionforamaximumat(C0∗,C1∗,λ∗)isthatthederivativesofL(C0,C1,λ)withrespecttothesevariablesshouldallbezero.Then∂L∗∗∗∂U∗∗∗(C0,C1,λ)=C0,C1−λ=0,∂C0∂C0∂L∗∗∗∂U∗∗λ∗(C0,C1,λ)=C0,C1−=0,∂C1∂C01+i∂L∗∗∗∗1∗(C0,C1,λ)=C0+C1−W=0.∂λ1+iThefirsttwoconditionsareequivalenttothetangencyconditionderivedearlier.Toseethis,eliminateλ∗fromtheseequationstoget∂U∗∗∗∂U∗∗C0,C1=λ=(1+i)C0,C1.∂C0∂C0Thelastofthethreeconditionssimplyassertsthatthebudgetconstraintmustbesatisfied.Asbefore,theoptimum(C0∗,C1∗)isdeterminedbythetangencyconditionandthebudgetconstraint.Clearly,theoptimalconsumptionstream(C0∗,C1∗)willbeafunctionoftheconsumer’swealthWandtherateofinteresti.Ifthepatternofincomewere(W,0)insteadof(Y1,Y2)thevalueofwealthwouldbethesameandhencethebudgetlinewouldbethesame.Sothesamepoint(C1∗,C2∗)wouldbechosen.Infact(Y1,Y2)couldmovetoanyotherpointonthebudgetlinewithoutaffectingconsumption.Onlysavingsorborrowingwouldchange.Ontheotherhand,anincreaseinWtoW,say,willshiftthebudgetlineoutandincreaseconsumption.ThecaseillustratedinFigure2.3hasthespecialpropertythatthemarginalrateofsubstitutionisconstantalongastraightlineOAthroughtheorigin.TheslopeofthebudgetlinedoesnotchangesointhiscasethepointoftangencymovesalongthelineOAasWchanges.Inthisspecialcase,C1isproportionaltoW.Problems1.Anindividualconsumerhasanincomestream(Y0,Y1)andcanborrowandlendattheinterestratei.Foreachofthefollowingdatapoints,determinewhethertheconsumptionstream(C0,C1)lieswithintheconsumer’sbudget 2.1EfficientAllocationOverTime35C1A0(W,0)C0Figure2.3.Theeffectofanincreaseinwealth.set(i.e.whetheritsatisfiestheintertemporalbudgetconstraint).(C0,C1)(Y0,Y1)(1+i)(10,25)(15,15)2(18,11)(15,15)1.1(18,11)(15,15)1.5(10,25)(15,15)1.8Drawagraphtoillustrateyouranswerineachcase.2.Anindividualconsumerhasanincomestream(Y0,Y1)=(100,50)andcanborrowandlendattheinterestratei=0.11.Hispreferencesare˙representedbytheadditivelyseparableutilityfunctionU(C0,C1)=logC0+0.9logC1.ThemarginalutilityofconsumptioninperiodtisdlogCt1=.dCtCtWritedowntheconsumer’sintertemporalbudgetconstraintandthefirst-orderconditionthatmustbesatisfiedbytheoptimalconsumptionstream.Usethefirst-orderconditionandtheconsumer’sintertemporalbudgetconstrainttofindtheconsumptionstream(C0∗,C1∗)thatmaximizesutility.Howmuchwilltheconsumersaveinthefirstperiod?Howmuchwillhissavingsbeworthinthesecondperiod?CheckthathecanaffordtheoptimalconsumptionC∗inthesecondperiod.1 36Chapter2.Time,Uncertainty,andLiquidity2.1.2ProductionJustaswecancasttheconsumer’sintertemporaldecisionintothefamiliarframeworkofmaximizingutilitysubjecttoabudgetconstraint,wecancastthefirm’sintertemporaldecisionintotheformofaprofit-orvalue-maximizationproblem.Imagineafirmthatcanproduceoutputsofahomogeneousgoodineitherperiodsubjecttoaproductiontechnologywithdecreasingreturns.Outputatdate0isdenotedbyY0andoutputatdate1isdenotedbyY1.ThepossiblecombinationsofY0andY1aredescribedbytheproductionpossibilitycurveillustratedinFigure2.4.Y10Y0Figure2.4.Theproductionpossibilitycurve.Notethefollowingpropertiesoftheproductionpossibilitycurve:•thecurveisdownwardslopingtotherightbecausethefirmmustreduceoutputtomorrowinordertoincreasetheoutputtoday;•thecurveisconvexupwardbecauseofthediminishingreturns–asthefirmdecreasesoutputtoday,eachadditionalunitofpresentoutputforegoneaddslesstooutputtomorrow.TheproductiontechnologycanberepresentedbyatransformationfunctionF(Y0,Y1).Apairofoutputs(Y0,Y1)isfeasibleifandonlyifitsatisfiestheinequalityF(Y0,Y1)≤0.ThefunctionFissaidtobeincreasingifanincreaseinY0orY1increasesthevalueF(Y0,Y1).ThefunctionFissaidtobeconvexif,foranyoutputstreams 2.1EfficientAllocationOverTime37(Y0,Y1)and(Y0,Y1)andanynumber0YL,thatis,incomeishigherinthe“high”state.Anindividual’spreferencesoveruncertainconsumptionoutcomescanberepresentedbyautilityfunctionthatisdefinedonbundlesofcontingentcommodities.LetU(CH,CL)denotetheconsumer’sutilityfromthebundleofcontingentcommodities(CH,CL),whereCHdenotesfutureconsumptioninstateHandCLdenotesfutureconsumptioninstateL.Later,weintroducethenotionofastate’sprobabilityanddistinguishbetweenanindividual’sprobabilitybeliefsandhisattitudestorisk.Here,anindividual’sbeliefsabouttheprobabilityofastateandhisattitudestowardsriskaresubsumedinhispreferencesoverbundlesofcontingentcommodities.CompletemarketsTherearetwoequivalentwaysofachievinganefficientallocationofrisk.Oneapproachtotheallocationofriskassumesthattherearecompletemarketsforcontingentcommodities.AneconomywithcompletemarketsisoftenreferredtoasanArrow–Debreueconomy.InanArrow–Debreueconomy,thereisamarketforeachcontingentcommodityandaprevailingpriceatwhichcon-sumerscantradeasmuchofthecommodityastheylikesubjecttotheirbudgetconstraint.LetpHandpLdenotethepriceofthecontingentcommoditiescor-respondingtostatesHandL,respectively.Theconsumer’sincomeconsistsofdifferentamountsofthetwocontingentcommodities,YHunitsofthecon-sumptiongoodinstateHandYLunitsoftheconsumptiongoodinstateL.Wecanusethecompletemarketstovaluethisuncertainincomestreamandtheconsumer’swealthispHYH+pLYL.Thentheconsumer’sbudgetconstraint,whichsaysthathisconsumptionexpendituremustbelessthanorequaltohiswealth,canbewrittenaspHCH+pLCL≤pHYH+pLYL.Theconsumercanaffordanybundleofcontingentcommodities(CH,CL)thatsatisfiesthisconstraint.Hechoosestheconsumptionbundlethatmaximizes 42Chapter2.Time,Uncertainty,andLiquidityhisutilityU(CH,CL)subjecttothisconstraint.Weillustratetheconsumer’sdecisionprobleminFigure2.7.CL(CH,CL)0CHFigure2.7.Maximizingutilityunderuncertainty.ArrowsecuritiesTheassumptionofcompletemarketsguaranteesefficientallocationofrisk,butitmaynotberealistictoassumethateverycontingentcommodity,ofwhichtherewillbeahugenumberinpractice,unlikeinourexample,canliterallybetradedattheinitialdate.Fortunately,thereexistsanalternativeformulationwhichisequivalentintermsofitsefficiencyproperties,butrequiresfarfewermarkets.Moreprecisely,itrequiresthatsecuritiesandgoodsbetradedonspotmarketsateachdate,butthetotalnumberofspotmarketsismuchlessthanthenumberofcontingentcommodities.ThealternativerepresentationoftheallocationofriskmakesuseoftheideaofArrowsecurities.WedefineanArrowsecuritytobeapromisetodeliveroneunitofmoney(oranabstractunitofaccount)ifagivenstateoccursandnothingotherwise.Intermsofthepresentexample,therearetwotypesofArrowsecurities,correspondingtothestatesHandLrespectively.LetqHdenotethepriceoftheArrowsecuritycorrespondingtostateHandletqLdenotethepriceoftheArrowsecurityinstateL.Inotherwords,qHisthepriceofoneunitofaccount(one“dollar”)deliveredinstateHatdate1andqListhepriceofoneunitofaccountdeliveredinstateLatdate1.AconsumercantradeArrowsecuritiesatdate0inordertohedgeagainstincomerisksatdate1.LetZHandZLdenotetheexcessdemandfortheArrowsecurities 2.2Uncertainty43correspondingtostatesHandLrespectively.1IfZs>0thentheconsumeristakingalongposition(offeringtobuy)intheArrowsecurityandifZs<0theconsumeristakingashortposition(offeringtosell).Weassumetheconsumerhasnoincomeatdate0–thisperiodexistsonlytoallowindividualstohedgerisksthatoccuratdate1–sotheconsumerhastobalancehisbudgetbysellingonesecurityinordertopurchasetheother.SupposethattheconsumerchoosesaportfolioZ=(ZH,ZL)ofArrowsecurities.Thenatdate1,oncethetruestatehasbeenrevealed,hisbudgetconstraintwillbepˆHCH≤pˆHYH+ZH,(2.3)ifstateHoccurs,andpˆLCL≤pˆLYL+ZL,(2.4)ifstateLoccurs.TheconsumerwillchoosetheportfolioZandtheconsump-tionbundle(CH,CL)tomaximizeU(CH,CL)subjecttothedate-0budgetconstraintqHZH+qLZL≤0andthebudgetconstraints(2.3)and(2.4).SinceZH=pˆH(CH−YH)andZL=pˆL(CL−YL),thebudgetconstraintatdate0isequivalenttoqHpˆH(CH−YH)+qLpˆL(CL−YL)≤0orqHpˆHCH+qLpˆLCL≤qHpˆHYH+qLpˆLYL.ThislooksjustlikethestandardbudgetconstraintinwhichweinterpretpH=qHpˆHandpL=qLpˆLasthepricesofcontingentcommoditiesandCHandCLasthedemandsforcontingentcommodities.Nowthatwehaveseenhowtointerprettheallocationofriskintermsofcontingentcommodities,wecanusethestandardframeworktoanalyzeefficientrisksharing.Figure2.8showsanEdgeworthboxinwhichtheaxescorrespondtoconsumptioninstateHandconsumptioninstateL.Acompet-itiveequilibriuminwhichconsumersmaximizeutilitysubjecttotheirbudgetconstraintleadstoanefficientallocationofcontingentcommodities,thatis,anefficientallocationofrisk.Asusual,theconditionsforefficiencyincludetheequalityofthetwocon-sumers’marginalratesofsubstitution,butheretheinterpretationisdifferent.1EachagentbeginswithazeronetsupplyofArrowsecurities.Thenagentsissuesecuritiesforonestateinordertopayfortheirpurchaseofsecuritiesintheother.ThevectorZrepresentstheagent’snetorexcessdemandofeachsecurity:acomponentZsisnegativeifhissupplyofthesecurity(equalsminustheexcessdemand)ispositiveandpositiveifhisnetdemandispositive. 44Chapter2.Time,Uncertainty,andLiquidity0BCL0ACHFigure2.8.Efficientallocationofriskbetweentwoagents.Weareequatingthemarginalratesofsubstitutionbetweenconsumptioninthetwostatesratherthantwodifferent(physical)goods.Themarginalrateofsubstitutionwillreflectanindividual’ssubjectivebeliefabouttheprobabilityofeachstateaswellashisattitudetowardrisk.2.2.2AttitudestowardriskTocharacterizeindividuals’attitudestowardriskweintroduceaspecialkindofutilityfunction,whichwecallavonNeumann–Morgenstern(VNM)utilityfunction.Whereasthestandardutilityfunctionisdefinedonbundlesofcon-tingentcommodities,theVNMutilityisdefinedonquantitiesofconsumptioninaparticularstate.VonNeumannandMorgensternshowedthat,undercer-tainconditions,arationalindividualwouldactsoastomaximizetheexpectedvalueofhisVNMutilityfunction.IfindividualssatisfytheassumptionsoftheVNMtheory,theywillalwaysmakechoicessoastomaximizethevalueoftheirexpectedutility.Toseewhatthismeansinpractice,letU(C)denotetheVNMutilityofconsumingCunitsofthegoodatdate1andsupposethattheprobabilityofstatesoccurringisπs>0fors=H,L.Thentheexpectedutilityofaconsumptionplan(CH,CL)isπHU(CH)+πLU(CL).Thedecisionproblemoftheconsumerweencounteredabovecanbere-writtenasmaxπHU(CH)+πLU(CL)s.t.qHCH+qLCL≤qHYH+qLYL. 2.2Uncertainty45Thefirst-orderconditionsforthisproblemareπsU(Cs)=µqs,fors=H,L,whereU(Cs)isthemarginalutilityofconsumptioninstatesandµ(theLagrangemultiplierassociatedwiththebudgetconstraint)canbeinterpretedasthemarginalutilityofmoney.Noticethatthemarginalutilityofconsumptioninstatesismultipliedbytheprobabilityofstatesanditistheproduct–theexpectedmarginalutility–whichisproportionaltothepriceofconsumptioninthatstate.Thenthefirst-orderconditioncanbeinterpretedassayingthattheexpectedmarginalutilityofoneunitofconsumptioninstatesisequaltothemarginalutilityofitscost.Wetypicallythinkofindividualsasbeingriskaverse,thatis,theyavoidriskunlessthereissomeadvantagetobegainedfromacceptingit.Theclearestevi-denceforthispropertyisthetendencytobuyinsurance.Wecancharacterizeriskaversionandanindividual’sattitudestoriskgenerallyintheshapeoftheVNMutilityfunction.Figure2.9showsthegraphofaVNMutilityfunction.Utilityincreaseswithincome(themarginalutilityofconsumptionisposi-tive)buttheutilityfunctionbecomesflatterasincomeincreases(diminishingmarginalutilityofconsumption).AVNMutilityfunctionhasdiminishingmarginalutilityofconsumptionifandonlyifitisstrictlyconcave.Formally,aVNMutilityfunctionUisstrictlyconcaveif,foranyconsumptionlevelsCandC(C=C)andanynumber0tU(C)+(1−t)U(C).(2.5)UU(C)0CFigure2.9.ThevonNeumann–Morgensternutilityfunction. 46Chapter2.Time,Uncertainty,andLiquidityConcavityoftheVNMutilityfunctioncanbeinterpretedasanattitudetowardsrisk.Toseethis,supposethatanindividualisofferedagambleinwhichhereceivesCwithprobabilitytandCwithprobability1−t.IfhisVNMutilityisstrictlyconcave,hewillprefertoreceivetheexpectedvaluetC+(1−t)Cforsure,ratherthantakethegamble.Thisisbecausetheexpectedutilityofthegamble(therighthandsideoftheinequality)islessthantheutilityofthesurething.Theutilityfunctionwillalwaysbestrictlyconcaveiftheindividualexhibitsdiminishingmarginalutilityofincome.Anindividualwhosatisfiestheassumptionofdiminishingmarginalutilityofincomeissaidtoberiskaverse.(Drawthegraphofautilityfunctionwithincreasingmarginalutilityofincomeandseehowthecomparisonofthetwooptionschanges.Anindividualwiththesepreferencesiscalledarisklover.)Theconclusionthenisthat,facedwithachoicebetweenariskyincomedistributionandadegeneratedistributionwiththesameexpectedvalue,ariskaverseindividualwillalwayschoosetheonewithoutrisk.Inwhatfollows,weassumethattheVNMutilityfunctionisconcaveand,hence,individualsareriskaverse.Wehaveseenthatriskaversionisassociatedwiththecurvatureoftheutilityfunction,inparticular,withthefactthatthemarginalutilityofincomeisdecreasing.Mathematically,thismeansthatthesecondderivativeoftheutilityfunctionU(C)islessthanorequaltozero.ItwouldbetemptingtotakethesecondderivativeU(C)tobethemeasureofriskaversion.Unfortunately,theVNMutilityfunctionisonlydetermineduptoanaffinetransformation,thatis,foranyconstantsαandβ>0,theVNMutilityfunctionα+βUisequivalenttoUintermsoftheattitudestoriskthatitimplies.Thus,wemustlookforameasurethatisindependentofαandβ>0.Twosuchmeasuresareavailable.OneisknownasthedegreeofabsoluteriskaversionU(C)A(C)=−U(C)andtheotheristhedegreeofrelativeriskaversionU(C)CR(C)=−.U(C)Thereisasimplerelationshipbetweenthedegreeofriskaversionandtheriskpremiumthatanindividualwilldemandtocompensatefortakingrisk.Sup-posethatanindividualhaswealthWandisofferedthefollowinggamble.Withprobability0.5hewinsasmallamounthandwithprobability0.5helosesh.Sincetheexpectedvalueofthegambleiszero,theindividual’sexpectedincome 2.2Uncertainty47isnotchangedbythegamble.SinceariskaverseindividualwouldratherhaveWforsurethanhaveanuncertainincomewiththesameexpectedvalue,hewillrejectthegamble.Theriskpremiumaistheamounthewouldhavetobegiveninordertoacceptthegamble.Thatis,asatisfiestheequation11U(W)=U(W+a−h)+U(W+a+h).22ATaylorexpansionoftherighthandsideshowsthat,whenhissmall,12U(W)≈U(W)+U(W)a+U(W)h.2Thus,U(W)h2h2a≈−=A(W).U(W)22Sotheriskpremiumaisequaltothedegreeofabsoluteriskaversiontimesone-halfthevarianceofthegamble(ameasureoftherisk).Asimilarinter-pretationcanbegivenforthedegreeofrelativeriskaversionwhenthegambleconsistsofwinningsofhWand−hWwithequalprobability.Ifthedegreeofabsoluteriskaversionisconstant,theVNMutilityfunctionmusthavetheformU(C)=−e−AC,andA>0isthedegreeofabsoluteriskaversion.Ifthedegreeofrelativeriskaversionisconstantanddifferentfrom1,then11−σU(C)=C,1−σwhereσ>0isthedegreeofrelativeriskaversion.Thisformulaisnotdefinedwhenthedegreeofrelativeriskaversionisσ=1;however,thelimitingvalueoftheutilityfunctionasσ→1iswelldefinedandgivenbyU(C)=lnC,wherelnCdenotesthenaturallogarithmofC.Thehigherthedegreeof(relativeorabsolute)riskaversion,themoreriskaversetheindividualwiththeVNMutilityfunctionU(C)is. 48Chapter2.Time,Uncertainty,andLiquidity2.2.3InsuranceandriskpoolingReturningtotheexampleofefficientrisksharingstudiedearlier,wecanusetheassumptionthatconsumershaveVNMutilityfunctionstocharac-terizetheefficientrisksharingallocationmoreprecisely.SupposethattherearetwoindividualsAandBwithVNMutilityfunctionsUAandUBandincomedistributions(YAH,YAL)and(YBH,YBL)respectively.Iftheefficientallocationofconsumptioninthetwostatesisgivenby{(CAH,CAL),(CBH,CBL)},thentheconditionforequalitybetweenthemarginalratesofsubstitutioncanbewrittenasU(CAH)U(CBH)A=B.U(CAL)U(CBL)ABTheprobabilitiesdonotappearinthisequationbecause,assumingAandBhavethesameprobabilitybeliefs,theyappearasmultipliersonbothsidesandsocancelout.Itisinterestingtoconsiderwhathappensinthecasewhereoneoftheconsumersisriskneutral.WesaythataconsumerisriskneutralifhisVNMutilityfunctionhastheformU(C)≡C.Ariskneutralconsumercaresonlyabouttheexpectedvalueofhisincomeorconsumption.Inotherwords,hisexpectedutilityisjusttheexpectedvalueofconsumption.SupposethatconsumerBisriskneutral.Thenhismarginalutilityisidenticallyequalto1ineachstate.Substitutingthisvalueintotheefficiencycondition,weseethatU(CAH)A=1,U(CAL)AwhichimpliesthatCAH=CAL.AlltheriskisabsorbedbytheriskneutralconsumerB,leavingconsumerAwithacertainlevelofconsumption.Riskneutralityisaveryspecialproperty,buttherearecircumstancesinwhichriskaverseconsumerscanachievethesameeffects.First,letuscon-sidermorecarefullythewayinwhichtheoptimalconsumptionallocationdependsonincome.First,notethattheefficiencyequationimpliesthatU(CAH)CALifandonlyifCBH>CBL.Thisimmediatelytellsusthattheoptimalconsumptionalloca-tiondependsonlyontheaggregateincomeofthepair.LetYs=YAs+YBsfor 2.2Uncertainty49s=H,L.Thenfeasibilityrequiresthattotalconsumptionequalstotalincomeineachstate:CAs+CBs=Ys,fors=H,L.Thus,theconsumptionofeachconsumerrisesifandonlyifaggregateincomerises.Thispropertyisknownascoinsurancebetweenthetwoconsumers:theyprovideinsurancetoeachotherinthesensethattheirconsumptionlevelsgoupanddowntogether.Inparticular,iftheaggregateincomeisthesameinthetwostates,YH=YL,thenCAH=CALandCBH=CBL.Soifaggregateincomeisconstant,theconsumptionallocationwillbeconstanttoo,nomatterhowindividualincomesfluctuate.Whenthereareonlytwoconsumers,constantaggregateincomedependsonaratherremarkablecoincidence:whenA’sincomegoesup,B’sincomegoesdownbythesameamount.Whenthereisalargenumberofconsumers,however,thesameoutcomeoccursquitenaturally,thankstothelawoflargenumbers,aslongastheincomesofthedifferentconsumersareassumedtobeindependent.Thisis,infact,whatinsurancecompaniesdo:theypoollargenumbersofindependentrisks,sothattheaggregateoutcomebecomesalmostconstant,andthentheycanensurethateachindividualgetsaconstantlevelofconsumption.Supposethereisalargenumberofconsumersi=1,2,...withrandomincomesthatareindependentlyandidenticallydistributedaccordingtotheprobabilitydistributionYHwithprobabilityπHYi=YLwithprobabilityπL.Thenthelawoflargenumbersensuresthattheaverageincomeisequaltotheexpectedvalueofanindividual’sincomeY¯=πHYH+πLYLwithprobabilityone,thatis,withcertainty.Thenaninsurancecompanycouldensureeveryoneaconstantconsumptionlevelbecausetheaverageaggregateincomeisalmostconstant.2.2.4PortfoliochoiceTheuseofArrowsecuritiestoallocateincomeriskefficientlyisaspecialcaseoftheportfoliochoiceproblemthatindividualshavetosolveinordertodecidehowtoinvestwealthinanuncertainenvironment.Wecangainalotofinsightintotheportfoliochoiceproblembyconsideringthespecialcaseoftwosecurities,oneasafeassetandtheotherariskyasset. 50Chapter2.Time,Uncertainty,andLiquidityAsbefore,weassumetherearetwodatest=0,1andtwostatess=H,Landasingleconsumptiongoodateachdate.SupposethataninvestorhasaninitialincomeW0>0atdate0andthathecaninvestitintwoassets.Oneisasafeassetthatyieldsoneunitofthegoodatdate1foreachunitinvestedatdate0.Theotherisariskyasset:oneunitinvestedintheriskyassetatdate0yieldsRs>0unitsofthegoodinstates=H,L.Weassumethattheinvestor’sriskpreferencesarerepresentedbyaVNMutilityfunctionU(C)andthattheprobabilityofstatesisπs>0fors=H,L.Theinvestor’sportfoliocanberepresentedbythefractionθofhiswealththatheinvestsintheriskyasset.Thatis,hisportfoliowillcontainθW0unitsoftheriskyassetand(1−θ)W0ofthesafeasset.Hisfutureconsumptionwilldependonhisportfoliochoiceandtherealizedreturnoftheriskyasset.LetCHandCLdenoteconsumptioninthehighandlowstates,respectively.ThenCs=RsθW0+(1−θ)W0,fors=H,L.Theinvestorchoosestheportfoliothatmaximizestheexpectedutilityofhisfutureconsumption.Thatis,hisdecisionproblemismaxθπHU(CH)+πLU(CL)s.t.Cs=RsθW0+(1−θ)W0,s=H,L.SubstitutingtheexpressionsforCHandCLintotheobjectivefunctionweseethattheexpectedutilityisafunctionofθ,sayV(θ).Theoptimalportfolio0<θ∗<1satisfiesthefirst-orderconditionV(θ∗)=0,or(CπHUH)(RH−1)+πLU(CL)(RL−1)=0.TheoptimumisillustratedinFigure2.10.Thesetofattainableconsump-tionallocations(CH,CL)isrepresentedbythelinesegmentwithendpoints(W0,W0)and(RHW0,RLW0).Iftheinvestorputsallofhiswealthinthesafeassetθ=0,thenhisfutureconsumptionineachstatewillbeCH=CL=W0.Ifheputsallhiswealthintheriskyasset,thenhisfutureconsumptionwithbeRHW0inthehighstateandRLW0inthelowstate.Ifheputsafractionθofhiswealthintheriskyasset,hisconsumptionbundle(CH,CL)isjustaconvexcombinationofthesetwoendpointswithweights1−θandθrespectively.Inotherwords,wecantraceoutthelinejoiningthesetwoendpointsjustbyvaryingtheproportionoftheriskyassetbetween0and1.Theoptimalportfoliochoiceoccurswheretheinvestor’sindifferencecurveistangenttotheconsumptioncurve.Thetangencyconditionisjustageometricversionofthefirst-orderconditionabove. 2.2Uncertainty51CL(CH,CL)0CHFigure2.10.Optimalchoiceofportfoliowithtwoassets.Dependingontheinvestor’sriskpreferencesandtheratesofreturn,theoptimalportfoliomayconsistentirelyofthesafeasset,entirelyoftheriskyasset,oramixtureofthetwo.Itisinterestingtoseeunderwhichconditionseachofthesepossibilitiesarises.Toinvestigatethisquestion,weneedtofindoutmoreabouttheslopesoftheindifferencecurvesandthefeasibleset.Theslopeofthefeasiblesetiseasilycalculated.Comparetheportfolioinwhichallincomeisinvestedinthesafeassetwiththeportfolioinwhichincomeisallinvestedintheriskyasset.ThechangeinCHis
CH=W0−W0RHandthechangeinCLis
CL=W0−W0RL.Sotheslopeis
CLW0−W0RL1−RL==.
CHW0−W0RH1−RHTheslopeisnegativeifRL<1>.πLU(W0)RH−1πLU(RLW0)ThelefthandinequalitycanbesimplifiedtoπH1−RL>πLRH−1orπHRH+πLRL>1.Inotherwords,theinvestorwillholdapositiveamountoftheriskyassetifandonlyiftheexpectedreturnoftheriskyassetisgreaterthanthereturntothesafeasset.Thismakessensebecausethereisnorewardforbearingriskotherwise.Therighthandinequalityimpliesthat1−RL>0,RH−1orRL<11unitsofthegoodatdate2.Weassumethereturnofthelongassetisknownwithcertainty.Thisassumptionsimplifiestheanalysisandallowsustofocusattentionontheothersourceofuncertainty,thatis,uncertaintyaboutindividualtimepreferences.Thereisatrade-offbetweenanasset’stimetomaturityanditsreturn.Thelongassettakestwoperiodstomature,butpaysahighreturn.Theshortassetmaturesafteroneperiodbutyieldsalowerreturn.Thistrade-offischarac-teristicoftheyieldcurveforbondsofdifferentmaturities,whereweseethatbondswithshortmaturitiestypicallyhavelowerreturnsthanbondswithlongmaturities.Thehigherreturnsonthelonger-datedassetscanbeinterpretedbothasarewardfortheinconvenienceofholdingilliquidassetsandasareflectionofthegreaterproductivityofroundaboutmethodsofproduction.LiquiditypreferenceWemodelpreferenceforliquidityastheresultofuncertaintyabouttimepreference.Imagineaconsumerwhohasanendowmentofoneunitofthegoodatdate0andnothingatthefuturedates.Allconsumptiontakesplaceinthefuture,atdates1and2,buttheconsumerisuncertainabouttheprecisedateatwhichhewantstoconsume.Moreprecisely,weassumetherearetwotypesofconsumers,earlyconsumerswhoonlywanttoconsumeatdate1andlateconsumerswhoonlywanttoconsumeatdate2.Initially,theconsumerdoesnotknowhistype.Heonlyknowstheprobabilityofbeinganearlyoralateconsumer.Letλdenotetheprobabilityofbeinganearlyconsumerand1−λtheprobabilityofbeingalateconsumer.Theconsumerlearnswhetherheisanearlyorlateconsumeratthebeginningofdate1.Uncertaintyabouttimepreferencesisasimplewayofmodelingwhateconomistscalla“liquidityshock,”thatis,anunanticipatedneedforliquid-ityresultingfromaneventthatchangesone’spreferences.Thiscouldbean 54Chapter2.Time,Uncertainty,andLiquidityaccidentthatrequiresanimmediateexpenditure,thearrivalofanunexpectedinvestmentopportunity,oranunexpectedincreaseinthecostofanexpend-iturethatwaspreviouslyplanned.Wecanthinkofλasmeasuringthedegreeofaconsumer’sliquiditypreference.Otherthingsbeingequal,hewillwanttoearnthehighestreturnpossibleonhisinvestments.Butifheisuncertainaboutthetimingofhisconsumption,wewillalsocareaboutliquidity,thepossibilityofrealizingthevalueofthisassetatshortnotice.Ifλisone,theconsumer’sliquiditypreferencewillbehigh,sincehecannotwaituntildate2toearnthehigherreturnonthelongasset.Ifλiszero,hewillhavenopref-erenceforliquidity,sincehecanholdthelongassetwithoutinconvenience.Forλbetweenzeroandone,theconsumer’suncertaintyaboutthetimingofhisconsumptionposesaproblem.Iftheconsumerknewthathewasalateconsumer,hewouldinvestinthelongassetbecauseitgivesahigherreturn.Ifheknewthathewasanearlyconsumer,hewouldholdonlytheshortassetinspiteofitslowerreturn.Sincetheconsumerisuncertainabouthistype,hewillregretholdingtheshortassetifheturnsouttobealateconsumerandhewillregretholdingthelongassetifheturnsouttobeanearlyconsumer.Theoptimalportfoliofortheconsumertoholdwilldependonbothhisriskaversionandhisliquiditypreferenceandonthereturntothelongasset(theslopeoftheyieldcurve).InvestmentunderautarkySupposetheconsumerhasaperiodutilityfunctionU(C)andletC1andC2denotehisconsumptionatdate1(ifheisanearlyconsumer)anddate2(ifheisalateconsumer).Thenhisexpectedutilityfromtheconsumptionstream(C1,C2)isλU(C1)+(1−λ)U(C2).Hisconsumptionateachdatewillbedeterminedbyhisportfoliochoiceatdate0.Letθdenotetheproportionofhiswealthinvestedintheshortasset.Recallthathehasaninitialendowmentofoneunitofthegoodatdate0soheinvestsθintheshortassetand1−θinthelongasset.Thenhisconsumptionatdate1isgivenbyC1=θ,sincehecannotconsumethereturnstothelongasset,whereashisconsumptionatdate2isgivenbyC2=θ+(1−θ)R, 2.3Liquidity55sincethereturnstotheshortassetcanbere-investedatdate1andconsumedatdate2.NotethatC11unitsofthegoodatdate2;ifthelongassetisliquidatedprematurelyatdate1thenitpays01thenthelongassetdominatestheshortassetandnoonewillholdtheshortassetatdate0.Inthatcase,earlyconsumerswillbeofferingthelongassetforsalebuttherewillbenobuyersatdate1.ThenthepricemustfalltoP=0,acontradiction.Ontheotherhand,ifP<1,thentheshortassetdominatesthelongassetandnoonewillholdthelongassetatdate0.Earlyconsumerswillconsumethereturnsontheshortassetatdate1,realizingareturnof1>P;lateconsumerswilltrytobuythelongassettoearnareturnofR/P>R.Sincenoonehasanyofthelongasset,thepricewillbebiduptoP=R,anothercontradiction.Thus,P=1inequilibrium.Atthisprice,thetwoassetshavethesamereturnsandareperfectsubstitutes.Theagent’sportfoliochoicebecomesimmaterial.Theagent’sconsumptionisc1=x+Py=x+y=1atdate1andyc2=x+R=(x+y)R=RP 3.2MarketEquilibrium63atdate2.ThentheequilibriumexpectedutilityisλU(1)+(1−λ)U(R).Thislevelofwelfareservesasabenchmarkagainstwhichtomeasurethevalueofhavingabankingsystemthatcanprovideinsuranceagainstliquidity(preference)shocks.ThevalueofthemarketItisobviousthattheinvestorisatleastaswelloffwhenhehasaccesstotheassetmarketasheisinautarky.Typically,hewillbebetteroff.Toshowthis,wehavetocomparethemarketequilibriumallocation(c1,c2)=(1,R)withthesetofallocationsthatarefeasibleinautarky.AswesawinChapter2,iftheconsumerdoesnothaveaccesstotheassetmar-ketandisforcedtoremaininautarky,hisconsumptionasanearlyconsumerwouldbeequaltohisinvestmentyinthesafeasset,andhisconsumptionasalateconsumerwouldbeequaltothereturnRx=R(1−y)onhisinvestmentinthelongassetplushisinvestmentinthesafeassetywhichcanbere-investedattheseconddate.Thus,thefeasibleconsumptionbundleshavetheformc1=yc2=y+R(1−y)forsomefeasiblevalueofybetween0and1.ThesetofsuchconsumptionbundlesisillustratedinFigure3.1.c2MarketallocationRFeasiblesetunderautarky101c1Figure3.1.Comparisonofthemarketallocationwiththefeasiblesetunderautarky. 64Chapter3.IntermediationandCrisesAsthefigureillustrates,themaximumvalueofearlyconsumptionisattainedwheny=1andc1=1andthemaximumvalueoflateconsumptionisattainedwheny=0andc2=R.Themarketallocation(c1,c2)=(1,R)dominateseveryfeasibleautarkicallocation,thatis,itgivesstrictlygreaterconsumptionatonedateandtypicallygivesgreaterconsumptionatboth.Thus,accesstotheassetmarketdoesincreaseexpectedutility.3.3THEEFFICIENTSOLUTIONThemarketprovidesliquiditybyallowingtheinvestortoconverthisholdingofthelongassetintoconsumptionatthepriceP=1attheseconddate.Becausetheassetmarketisperfectlycompetitive,theinvestorcanbuyandsellanyamountoftheassetattheequilibriumprice.Thismeansthatthemarketisperfectlyliquidinthesensethatthepriceisinsensitivetothequantityoftheassetthatistraded.However,itturnsoutthattheprovisionofliquidityisinefficient.Wewilldiscussthereasonsforthisingreaterdetaillater,buttheshortexplanationforthisinefficiencyisthatthesetofmarketsintheeconomydescribedaboveisincomplete.Inparticular,thereisnomarketatdate0inwhichaninvestorcanpurchasethegoodfordeliveryatdate1contingentonhistype.Ifsuchamarketexisted,theequilibriumwouldbequitedifferent.Wetakeasourdefinitionoftheefficientprovisionofliquidity,thelevelofwelfarethatcouldbeachievedbyacentralplannerwhohadcommandofalltheallocationdecisionsintheeconomy.Tobeginwith,weassumethattheplannerhascompleteinformationabouttheeconomy,includingtheabilitytotellwhoisanearlyconsumerandwhoisalateconsumer.Thisimportantassumptionwilllaterberelaxed.Thecentralplannerchoosestheamountpercapitaxinvestedinthelongassetandtheamountpercapitayinvestedintheshortasset.Thenhechoosestheconsumptionpercapitac1oftheearlyconsumersatdate1andthecon-sumptionpercapitac2ofthelateconsumersatdate2.Thecentralplannerisnotboundtosatisfyanyequilibriumconditions.Heisonlyconstrainedbytheconditionthattheallocationhechoosesmustbefeasible.Atdate0thefeasibilityconditionissimplythatthetotalinvestedpercapitamustbeequaltothepercapitaendowment:x+y=1.(3.1)Attheseconddate,thefeasibilityconditionisthatthetotalconsumptionpercapitamustbelessthanorequaltothereturnontheshortasset.Sincethefractionofearlyconsumersisλandeachoneispromisedc1theconsumption 3.3TheEfficientSolution65percapita(i.e.perheadoftheentirepopulation)isλc1.Thenthefeasibilityconditionisλc1≤y.(3.2)Ifthisinequalityisstrict,someofthegoodcanbere-investedintheshortassetandconsumedatdate2.Thus,thetotalavailableatdate2(expressedinpercapitaterms)isRx+(y−λc1).Thetotalconsumptionpercapita(i.e.perheadoftheentirepopulation)atdate2is(1−λ)c2sincethefractionoflateconsumersis1−λandeachofthemispromisedc2.Sothefeasibilityconditionis(1−λ)c2≤Rx+(y−λc1),whichcanalsobewrittenasλc1+(1−λ)c2≤Rx+y.(3.3)Theplanner’sobjectiveistochoosetheinvestmentportfolio(x,y)andtheconsumptionallocation(c1,c2)tomaximizethetypicalinvestor’sexpectedutilityλU(c1)+(1−λ)U(c2),subjecttothevariousfeasibilityconditions(3.1)–(3.3).Thislookslikeamoderatelycomplicatedproblem,butitcanbesimplifiedifweusealittlecommonsense.Thefirstthingtonoteisthatitwillneverbeoptimaltocarryoveranyoftheshortassetfromdate1todate2.Toseethis,supposethaty>λc1sothatsomeofthegoodisleftoveratdate1.Thenwecouldreducetheamountinvestedintheshortassetatdate0byε>0sayandinvestitinthelongassetinstead.Atdate2,therewouldbeεlessoftheshortassetbutεmoreofthelongasset.ThenetchangeintheamountofgoodsavailablewouldbeRε−ε=(R−1)ε>0,soitwouldbepossibletoincreasetheconsumptionofthelateconsumerswithoutaffectingtheconsumptionoftheearlyconsumers.Thiscannothappeninanoptimalplan,soitfollowsthatinanyoptimalplanwemusthaveλc1=yand(1−λ)c2=Rx.Thus,oncexandyaredetermined,theoptimalconsumptionallocationisalsodeterminedbytherelationsyc1=;λRxc2=.1−λ 66Chapter3.IntermediationandCrises(Recallthatyisthereturnontheshortassetperheadoftheentirepopulation,whereasc1istheconsumptionofatypicalearlyconsumer,soc1isgreaterthany.Similarly,theconsumptionc2ofatypicallateconsumerisgreaterthanthereturnonthelongassetperheadoftheentirepopulation.)Ifwesubstitutetheseexpressionsforconsumptionintotheobjectivefunctionandusethedate-0feasibilityconditiontowritex=1−y,wecanseethattheplanner’sproblemboilsdowntochoosingyintheinterval[0,1]tomaximize
yR(1−y)λU+(1−λ)U.(3.4)λ1−λIgnoringthepossibilityofaboundarysolution,wherey=0ory=1,anecessaryconditionforanoptimalchoiceofyisthatthederivativeofthefunctionin(3.4)bezero.Differentiatingthisfunctionandsettingthederivativeequaltozeroyields
yR(1−y)U−UR=0,λ1−λor,substitutingintheconsumptionlevels,U(c1)=U(c2)R.(3.5)Thereareseveralinterestingobservationstobemadeaboutthisfirst-ordercondition.First,thevalueofλdoesnotappearinthisequation:λdropsoutwhenwedifferentiatetheobjectivefunction.Theintuitionforthisresultisthatλappearssymmetricallyintheobjectivefunctionandinthefeasibilityconditions.Anincreaseinλmeansthatearlyconsumersgetmoreweightintheobjectivefunctionbutitalsomeansthattheycostmorepercapitatofeed.Thesetwoeffectscanceloutandleavetheoptimallevelofconsumptionunchanged.TheinefficiencyofthemarketsolutionThesecondpointtonoteconcernsthe(in)efficiencyofthemarketsolution.Thesetoffeasibleconsumptionallocationsfortheplanner’sproblemisillus-tratedinFigure3.2.Foreachchoiceofyintheintervalbetween0and1,theconsumptionallocationisdefinedbytheequationsyc1=;λR(1−y)c2=.(1−λ) 3.3TheEfficientSolution67c2R/(1–λ)MarketallocationREfficientallocationgiventhesepreferences1011/λc1Figure3.2.Comparisonofthemarketallocationwiththeefficientallocation.Theconsumptionoftheearlyconsumersismaximizedbyputtingy=1,inwhichcase(c1,c2)=(1/λ,0).Similarly,theconsumptionofthelatecon-sumersismaximizedbyputtingy=0,inwhichcase(c1,c2)=(0,R/(1−λ)).Sincetheequationsforconsumptionarelineariny,wecanattainanypointonthelinesegmentjoining(1/λ,0)and(0,R/(1−λ)).Thisfeasiblefrontierisdescribedinthefigure.Theefficientpointisdeterminedbythetangencyoftheconsumers’indifferencecurvewiththefeasiblefrontier.Dependingontheconsumers’preferences,thepointoftangencycouldoccuranywherealongthefeasiblefrontier.Themarketallocationoccursonthefeasiblefrontier.Simplyputy=λandweget
yR(1−y)(c1,c2)=,=(1,R).λ(1−λ)Thisallocationcouldbeefficientbuttypicallyitwillnotbe.Toseethis,supposethatbysomechancethemarketequilibriumresultedinthesameallocationastheplanner’sproblem.Thenthefirst-ordercondition(3.5)becomesU(1)=U(R)R. 68Chapter3.IntermediationandCrisesInsomespecialcasesthisconditionmaybesatisfied.Forexample,supposethattheinvestorhasalogarithmicutilityfunctionU(c)=lncsothatU(c)=1/c.Substitutingthisexpressionintheprecedingequation,thelefthandsidebecomesU(1)=1andtherighthandsidebecomesU(R)R=(1/R)R=1.Inthisparticularcase,themarketprovisionofliquidityisefficient:acentralplannercoulddonobetterthanthemarket.Forotherutilityfunctions,thiswouldnotbethecase.Suppose,forexample,thattheinvestor’sutilityfunctionbelongstotheconstantrelativeriskaversionclass11−σU(c)=c1−σwhereσ>0isthedegreeofrelativeriskaversion.ThenU(c)=c−σandsubstitutingthisintothenecessaryconditionforefficiency,weseethatthelefthandsidebecomesU(1)=1andtherighthandsidebecomesU(R)R=R−σR=R1−σ.Exceptinthecaseσ=1(whichcorrespondstothelogarithmiccase),R1−σ=1andtheallocationchosenbytheplannermustbedifferentfromthemarketallocation.Soforanydegreeofriskaversiondifferentfrom1theplannerachievesastrictlybetterlevelofexpectedutilitythanthemarket.LiquidityinsuranceAthirdinsightthatcanbederivedfromthefirst-ordercondition(3.5)concernstheprovisionofinsuranceagainstliquidityshocks,thatis,theeventofbeinganearlyconsumer.Evenintheefficientallocationtheindividualfacesuncertaintyaboutconsumption.Thefirst-orderconditionU(c1)=RU(c2)impliesthatc11soitisclearlyworsetobeanearlyconsumerthanalateconsumer.Ariskaverseinvestorwouldliketohavemoreconsumptionasanearlyconsumerandlessasalateconsumer,assumingthattheexpectedvalueofhisconsumptionremainsthesame.Aninterestingquestioniswhethertheplannerprovidesinsuranceagainsttheliquidityshockbyreducingthevolatilityofconsumption,thatis,increasingconsumptionofearlyconsumersandreducingconsumptionoflateconsumersrelativetothemarketsolution.InFigure3.2,theconsumptionallocationsthatlietotherightofthemarketallocationallsatisfyc1>1andc21andc2=R(1−y)/(1−λ)RU(R).AsufficientconditionisthatcU(c)bedecreasinginc,whichisequivalenttosayingthattherelativeriskaversionisgreaterthanone:cU(c)η(c)≡−>1.U(c)Ifthisinequalityisreversedandη(c)<1,earlyconsumersgetlessandlateconsumersgetmorethaninthebenchmarkcase,thatis,c1=y/λ<1,c2=R(1−y)/(1−λ)>R.Theseresultshaveanintuitiveexplanation.Thelogarithmicutilityfunction,whichhasadegreeofrelativeriskaversionequaltounity,markstheboundarybetweentwodifferentcases.Ifrelativeriskaversionequalsone,asinthenat-urallogcase,themarketoutcomeisefficientand,inparticular,themarket’sprovisionofliquidityisoptimal.Ifrelativeriskaversionisgreaterthanone,themarketprovisionofliquidityisinefficient.Anefficientallocationshouldprovidemoreinsurancebyincreasingtheconsumptionoftheearlyconsumersandreducingtheconsumptionofthelateconsumers.Ifrelativeriskaversionislessthanone,thereisparadoxicallytoomuchliquidityinthesensethatefficiencyrequiresareductionintheconsumptionoftheearlyconsumersandanincreaseintheconsumptionofthelateconsumers.Thislastresultalertsustothefactthatinsuranceiscostly.Inordertoprovidemoreconsumptiontotheearlyconsumers,itisnecessarytoholdmoreoftheshortassetand,hence,lessofthelongasset.Sincethelongasset’sreturnisgreaterthantheshortasset’s,theincreaseintheamountoftheshortassetintheplanner’sportfoliomustreduceaverageconsumptionacrossthetwodates.Aslongasrelativeriskaversionisgreaterthanone,thebenefitfrominsuranceisgreaterthanthecost,atleasttostartwith.Iftherelativeriskaversionislessthanone,thebenefitsofgreaterinsurancearenotworththecostand,indeed, 70Chapter3.IntermediationandCrisestheefficientallocationrequirestheplannertoincreasetheriskbornebytheinvestorsinordertocapturetheincreasedreturnsfromholdingthelongasset.Whydotheinvestorsholdthewrongportfoliointhemarketsolution?TheirportfoliodecisionisdependentuponthemarketpriceforthelongassetP=1.Thispriceisdeterminedbytheconditionthatinvestorsatdate0mustbewillingtoholdbothassets.Themarketdoesnotrevealhowmuchinvestorswouldbewillingtopayfortheassetcontingentonknowingtheirtype.Consequently,thepricedoesnotreflectthevalueofbeingabletosellthelongassetasanearlyconsumerorbeingabletobuyitasalateconsumer.CompletemarketsWementionedearlier(inSection3.2)thepossibilityofintroducingmarketsthatwouldallowindividualagentstotradeatdate0claimsondate-1con-sumptioncontingentontheirtype,earlyorlate.Theexistenceofsuchmarketswouldachievethesameallocationofriskandthesameportfolioinvestmentasthecentralplanner.Unlikethemodeleconomyinwhichtherearenocon-tingentmarkets,aneconomywithmarketsforindividualcontingencieswouldsignalthecorrectvalueofeachassettothemarket,inparticularthevalueofliquidity,andensurethattheefficientallocationwasachieved.Toseethis,supposethatanindividualcanpurchasedate-1consumptionatapriceq1ifheisearlyandq2ifheislate.Notethatthesepricesaremeasuredintermsofthegoodatdate0.Theimplicitpriceofgoodsatdate2intermsofgoodsatdate1isp=P/R,asusual.Thenthebudgetconstraintforanindividualatdate0isq1λC1+q2p(1−λ)C2≤1.(3.6)Thelefthandsiderepresentsthepresentvalue(atdate0)ofexpectedcon-sumption(sincethereisnoaggregateuncertainty,eachindividualonlypaysfortheexpectedvalueofhisdemandforgoodsateachdate).Withprobabil-ityλhedemandsC1unitsofdate-1consumptionandthepresentvalueofλC1isq1λC1.Similarly,withprobability1−λhedemandsC2unitsofdate-2consumptionandthepresentvalueof(1−λ)C2isq2p(1−λ)C2.Theindividualchooses(C1,C2)tomaximizeλU(C1)+(1−λ)U(C2)subjectto(3.6)andthesolutionmustsatisfythefirst-orderconditionsλU(C1)=µq1λand(1−λ)U(C2)=µq2p(1−λ) 3.3TheEfficientSolution71whereµ>0istheLagrangemultiplierassociatedwiththeconstraint.ThenU(C1)q1=.U(C2)q2pSincetheinvestmenttechnologyexhibitsconstantreturnstoscale,theequi-libriumpricesmustsatisfytwono-arbitrageconditions.Toprovideoneunitofthegoodatdate1,itisnecessarytoinvest1unitintheshortassetatdate0.Thus,therearezeroprofitsfrominvestingintheshortassetifandonlyifq1=1.Similarly,toprovideoneunitofthegoodatdate2,itisnecessarytoinvest1/Runitsinthelongassetatdate0.Thus,therearezeroprofitsfrominvestinginthelongassetifandonlyifpq2=1/R.ThisimpliesthatU(C1)=R,U(C2)theconditionrequiredforefficientrisksharing.PrivateinformationandincentivecompatibilitySofarwehaveassumedthatthecentralplannerknowseverything,includingwhetheraninvestorisanearlyorlateconsumer.Thisallowstheplannertoassigndifferentlevelsofconsumptiontoearlyandlateconsumers.Sinceaninvestor’stimepreferencesarelikelytobeprivateinformation,theassumptionthattheplannerknowstheinvestor’stypeisrestrictive.Ifwerelaxthisassump-tion,werunintoaproblem:iftimepreferencesareprivateinformation,howcantheplannerfindoutwhoisanearlyconsumerandwhoisalateconsumer?Theplannercanrelyontheindividualtruthfullyrevealinghistypeifandonlyiftheindividualhasnoincentivetolie.Inotherwords,theallocationchosenbytheplannermustbeincentive-compatible.Inthepresentcase,itisquiteeasytoshowthattheoptimalallocationisincentivecompatible.First,theearlyconsumershavenoopportunitytomisrepresentthemselvesaslateconsumers.Alateconsumerisgivenc2atdate2andsincetheearlyconsumeronlyvaluesconsumptionatdate1hewouldcertainlybeworseoffifhewaiteduntildate2toreceivec2.Thelateconsumerposesmoreofaproblem.Hecouldpretendtobeanearlyconsumer,receivec1atdate1andstoreituntildate2usingtheshortasset.Toavoidgivingthelateconsumeranincentivetomisrepresenthispreferences,hemustreceiveatleastasmuchconsumptionastheearlyconsumer.Thismeansthattheallocationisincentivecompatibleifandonlyifc1≤c2.(3.7) 72Chapter3.IntermediationandCrisesFortunatelyforus,ifweconsultthefirst-orderconditionfortheplanner’sproblem,equation(3.5),wefindthatitimpliesthatc11andU(c)<0sothattheoptimalallocationisautomaticallyincentive-compatible.Theallocationthatoptimizesthetypicalinvestor’swelfaresubjecttotheincentiveconstraint(3.7)iscalledincentive-efficient.Inthiscase,becausetheincentiveconstraintisnotbindingattheoptimum,theincentive-efficientallocationisthesameasthefirst-bestallocation.Althoughtheincentive-compatibilityconditiondoesnothaveanyeffectontheoptimalrisk-sharingallocation,privateinformationplaysanimportantroleintheaccountofbankingthatwegiveinthesequel.Inparticular,thefactthatabankcannotdistinguishearlyandlateconsumersmeansthatallconsumerscanwithdrawfromthebankatdate1andthisisacrucialfeatureofthemodelofbankruns.3.4THEBANKINGSOLUTIONAbank,bypoolingthedepositors’investments,canprovideinsuranceagainstthepreferenceshockandallowearlyconsumerstosharethehigherreturnsofthelongasset.Thebanktakesoneunitofthegoodfromeachagentatdate0andinvestsitinaportfolio(x,y)consistingofxunitsofthelongassetandyunitsoftheshortasset.Becausethereisnoaggregateuncertainty,thebankcanoffereachconsumeranon-stochasticconsumptionprofile(c1,c2),wherec1istheconsumptionofanearlyconsumerandc2istheconsumptionofalatecon-sumer.Wecaninterpret(c1,c2)asadepositcontractunderwhichthedepositorhastherighttowithdraweitherc1atdate1orc2atdate2,butnotboth.Thereisassumedtobefreeentryintothebankingsector.Competitionamongthebanksforcesthemtomaximizetheexanteexpectedutilityofthetypicaldepositorsubjecttoazero-profit(feasibility)constraint.Infact,thebankisinexactlythesamepositionasthecentralplannerdiscussedintheprevioussection.Atdate0thebankfacesabudgetconstraintx+y≤1.(3.8)Atdate1,thebankfacesabudgetconstraintλc1≤y.(3.9)Recallingthatitisneveroptimaltocarryconsumptionoverfromdate1todate2byholdingtheshortasset,wecanwritethebudgetconstraintforthe 3.4TheBankingSolution73bankatthethirddateas(1−λ)c2≤Rx.(3.10)Formally,thebank’sproblemistomaximizetheexpectedutilityofthetypicaldepositorλU(c1)+(1−λ)U(c2)subjecttothebudgetconstraints(3.8)–(3.10).Wedonotexplicitlyimposetheincentive-compatibilityconstraintbecause,aswesawpreviously,thesolutiontotheunconstrainedoptimizationproblemwillautomaticallysatisfytheincentiveconstraintc1≤c2.Sothebankisabletoachievethefirst-bestallocationonbehalfofitscustomers.Itisworthpausingtonotehowthisaccountofbankbehaviorimplementsthreeofthefourelementsofbankingtheorymentionedatthebeginningofthischapter.•Itprovidesamodelofthematuritystructureofbankassets,inwhichlessliquidassetsearnhigherreturns.Inthiscase,therearetwobankassets,theliquidshortasset,whichyieldsareturnof1,andtheilliquidlongasset,whichyieldsareturnofR>1.•Itprovidesatheoryofliquiditypreference,modeledasuncertaintyaboutthetimingofconsumption.Thematuritymismatcharisesbecauseaninvestorisuncertainofhispreferencesoverthetimingofconsumptionatthemomentwhenaninvestmentdecisionhastobemade.•Itrepresentsthebankasanintermediarythatprovidesinsurancetodepos-itorsagainstliquidity(preference)shocks.Bypoolinghisresourceswiththebank’sandacceptinganinsurancecontractintheformofpromisesofconsumptioncontingentonthedateofwithdrawal,theinvestorisabletoachieveabettercombinationofliquidityservicesandreturnsoninvestmentthanhecouldachieveinautarkyorintheassetmarket.Thepropertiesoftheefficientallocation,derivedintheprecedingsection,ofcourseapplytothebankingallocation,sowewillnotrepeatthemhere.Instead,wewanttofocusonthepeculiarfragilityofthearrangementthatthebankhasinstitutedinordertoachieveoptimalrisksharing. 74Chapter3.IntermediationandCrises3.5BANKRUNSAtthebeginningofthischapter,wementionedfourcontributionstobankingtheorymadebytheseminalpapersofBryant(1980)andDiamondandDybvig(1983).Wehavediscussedthefirstthreeandnowweturntothefourth,namely,theexplanationofbankruns.Inthissection,wedevelopamodelofbankrunsaspanicsorself-fulfillingprophecies.Laterweshallconsiderbankrunsastheresultoffundamentalforcesarisinginthecourseofthebusinesscycle.Supposethat(c1,c2)istheoptimaldepositcontractand(x,y)istheoptimalportfolioforthebank.Intheabsenceofaggregateuncertainty,theportfolio(x,y)providesjusttherightamountofliquidityateachdateassumingthattheearlyconsumersaretheonlyonestowithdrawatdate1andthelateconsumersallwithdrawatdate2.Thisisanequilibriuminthesensethatthebankismaximizingitsobjective,thewelfareofthetypicaldepositor,andtheearlyandlateconsumersaretimingtheirwithdrawalstomaximizetheirconsumption.Sofar,wehavetreatedthelongassetascompletelyilliquid:thereisnowaythatitcanbeconvertedintoconsumptionatdate1.Suppose,instead,thatthereexistsaliquidationtechnologythatallowsthelong-terminvestmenttobeterminatedprematurelyatdate1.Moreprecisely,weassumethat•ifthelongassetisliquidatedprematurelyatdate1,oneunitofthelongassetyieldsr≤1unitsofthegood.Undertheassumptionthatthelongassetcanbeprematurelyliquidated,withalossofR−rperunit,thereexistsanotherequilibriumifwealsoassumethatthebankisrequiredtoliquidatewhateverassetsithasinordertomeetthedemandsoftheconsumerswhowithdrawatdate1.Toseethis,supposethatalldepositors,whethertheyareearlyorlateconsumers,decidetowithdrawatdate1.Theliquidatedvalueofthebank’sassetsatdate1isrx+y≤x+y=1sothebankcannotpossiblypayallofitsdepositorsmorethanoneunitatdate1.Intheeventthatc1>rx+y,thebankisinsolventandwillbeabletopayonlyafractionofthepromisedamount.Moreimportantly,allthebank’sassetswillbeusedupatdate1intheattempttomeetthedemandsoftheearlywithdrawers.Anyonewhowaitsuntilthelastperiodwillgetnothing.Thus,giventhatalateconsumerthinkseveryoneelsewillwithdrawatdate1itisoptimalforalateconsumertowithdrawatdate1andsavetheproceedsuntildate2.Thus,bankrunsareanequilibriumphenomenon.Thefollowingpayoffmatrixillustratesthetwoequilibriaofthiscoordinationgame.Therows 3.5BankRuns75correspondtothedecisionofanindividuallateconsumerandthecolumnstothedecisionofalltheotherlateconsumers.(Note:thisisnota2×2game;thechoiceofcolumnrepresentstheactionsofallbutonelateconsumer.)Theorderedpairsarethepayoffsforthedistinguishedlateconsumer(thefirstelement)andthetypicallateconsumer(thesecondelement).RunNoRunRun(rx+y,rx+y)(c1,c2)NoRun(0,rx+y)(c2,c2)Itisclearthatif00.depositorcanbewrittenπU(y+rx)+(1−π){λU(c1)+(1−λ)U(c2)}.Nowsupposethatweincreaseybyasmallamountε>0anddecreasexbythesameamount.Weincreaseλc1byεandreduce(1−λ)c2byRε.Thisinsuresthefeasibilityconstraintsaresatisfiedateachdate.Thenthechangeinexpectedutilityis(y+rx)(1−r)ε+(1−π)U(cπU1)−U(c2)Rε+o(ε).Theoptimalportfoliomustthereforesatisfythefirst-ordercondition(y+rx)(1−r)+(1−π)U(cπU1)=(1−π)U(c2)R.Ifπ=0thenthisreducestothefamiliarconditionU(c1)=U(c2)R.TheserelationsaregraphedinFigure3.4.Thelatterconditionholdsaty∗whiletheformerholdsaty∗∗.Thus,thepossibilityofarunincreasesthemarginalvalueofanincreaseiny(theshortassethasahigherreturnthanthelongassetinthebankruptcystateifr<1)andhenceincreasestheamountoftheshortassetheldintheportfolio.TheoptimaldepositcontractOurnexttaskistoshowthatabankrunispossiblewhenthedepositcontractischosentosolvethebank’sdecisionproblem.Tomaximizeexpectedutility,thebankmustchoosethedepositcontract(c1∗,c2∗)tosatisfythefirst-orderconditionU(c∗∗1)=RU(c2).(3.11) 3.6EquilibriumBankRuns79U
πU
(y+rx)(1–r)+(1–π)U
(c1)(1–π)U
(c2)R(1–π)U
(c1)y*y**Figure3.4.Thedeterminationoftheoptimalportfoliowhenbankrunsarepossible.Thiscondition,whichisfamiliarfromourcharacterizationofthefirstbest,playsacrucialroleindeterminingwhetherthebankissusceptibletoruns.Aswesawearlier,ifrelativeriskaversionisgreaterthanone,thenthesolutionofthefirst-ordercondition(3.11)mustsatisfytheinequalityc1>1.Thisconditionimpliesthereexiststhepossibilityofarun.Ifallthedepositorstrytowithdrawatdate1,thetotaldemandforconsumptionisc∗>1but1themaximumthatcanbeprovidedbyliquidatingallofthelongassetis1.However,therewillbenothingleftatdate2sothedepositorsarebetteroffjoiningtherunthanwaitinguntildate2towithdraw.Inwhatfollows,weassumethattheagent’spreferencessatisfythecondi-tionthat•relativeriskaversionisgreaterthanone,thatis,U(c)c−>1,∀c>0.U(c)Tosimplifythecharacterizationoftheequilibrium,weonlyconsiderthespecialcaseinwhichthelongasset,whenliquidatedprematurely,yieldsasmuchastheshortasset.Inotherwords,•theliquidationvalueofthelongassetisr=1. 80Chapter3.IntermediationandCrisesThisimpliesthatthelongassetdominatestheshortassetso,withoutessentiallossofgenerality,wecanassumeinwhatfollowsthattheentirebankportfolioisinvestedinthelongasset.Intheeventofabankrun,theliquidatedvalueofthebank’sportfolioisoneunitofthegood,soeverydepositor’sconsumptionisalsooneunitofthegood.Ifthebankissolvent,thedepositorsreceivethepromisedconsumptionprofile(c1,c2).Sincethesequantitiesonlyapplyintheeventthatthebankissolvent,theyarechosentomaximizethetypicalconsumer’sexpectedutilityintheeventthatthebankissolvent.ThedepositcontractmustsolvethedecisionproblemmaxλU(c1)+(1−λ)U(c2)s.t.Rλc1+(1−λ)c2≤R.Toseewhythebudgetconstrainttakesthisform,notethatthebankhaspromisedatotalofλc1unitstotheearlyconsumersandthisrequiresthebanktoliquidateλc1unitsofthelongassetatdate1.Theamountofthelongassetleftis(1−λc1)andthisproducesR(1−λc1)unitsofconsumptionatdate2.Thus,themaximumamountthatcanbepromisedtolateconsumers(1−λ)c2mustbelessthanorequaltoR(1−λc1).Ineffect,oneunitofconsumptionatdate1isworthRunitsofconsumptionatdate2.EquilibriumwithoutrunsSofarwehaveassumedthatarunoccurswithprobabilityπandthatthebanktakesthispossibilityasgiveninchoosinganoptimaldepositcontract;however,thebankcanavoidarunbychoosingasufficiently“safe”contract.Rememberthatourargumentfortheexistenceofarunequilibriumatdate1wasbasedontheassumptionthatc1>1.Thus,ifallthelateconsumersjointherunonthebankatdate1thereisnowaythatthebankcanprovideeveryonewithc1.Infact,thebankwillhavetoliquidateallitsassetsandeventhencanonlygiveeachwithdrawer1,theliquidatedvalueofitsportfolio.Moreimportantly,sincethebank’sassetsareexhaustedatdate1,anyonewaitinguntildate2towithdrawwillreceivenothing.Inordertoremovethisincentivetojointherun,thebankmustchooseadepositcontractthatsatisfiestheadditionalconstraintc1≤1.IfwesolvetheproblemmaxλU(c1)+(1−λ)U(c2)s.t.Rλc1+(1−λ)c2≤Rc1≤1 3.6EquilibriumBankRuns81wefindthesolutionc1∗∗,c2∗∗=(1,R).Inthiscase,thebankwillbeabletogiveeveryonethepromisedpaymentc1atdate1andifanylateconsumerswaituntildate2towithdrawtherewillbeenoughleftovertopaythematleastR>1.Moreprecisely,if1−εofthedepositorswithdrawatdate1,thebankhastoliquidate1−εunitsofthelongasset,leavingεunitsofthelongassettopaytotheremaininglateconsumers.Theneachconsumerwhowithdrawsatdate2willreceiveεR/ε=R>1.AcharacterizationofregimeswithandwithoutrunsIfthebankanticipatesarunwithprobabilityπ,thenwithprobabilityπthedepositor’sconsumptionis1,regardlessofhistype.Withprobability1−πthereisnorunandwithprobabilityλthedepositorisanearlyconsumerandhisconsumptionisc∗andwithprobability1−λheisalateconsumerandhis1consumptionisc∗.ThepossibleoutcomesareillustratedinFigure3.3(above).2TheexpectedutilityofthetypicaldepositorwillbeπU(1)+(1−π)λU(c∗)+(1−λ)Uc∗,12andwehaveshownthatthebank’schoiceofportfoliox,y=(1,0)anddepositcontractc∗,c∗willmaximizethisobjective,takingtheprobabilityπ12ofarunasgiven.Alternatively,ifthebankchoosesadepositcontractthatavoidsallruns,theexpectedutilityofthetypicaldepositorisλU(c∗∗)+(1−λ)U(c∗∗)=λU(1)+(1−λ)U(R).11Whetheritisbetterforthebanktoavoidrunsoraccepttheriskofarunwithprobabilityπdependsonacomparisonoftheexpectedutilitiesineachcase.Precisely,itwillbebettertoavoidrunsifπU(1)+(1−π)λU(c∗)+(1−λ)Uc∗>λU(1)+(1−λ)U(R).12Noticethatthelefthandsideisaconvexcombinationofthedepos-itors’utilityU(1)whenthebankdefaultsandtheirexpectedutilityλU(c1∗)+(1−λ)Uc2∗whenthebankissolvent.Now,theexpectedutilityfromthesafestrategyλU(1)+(1−λ)U(R)liesbetweenthesetwovalues:U(1)<λU(1)+(1−λ)U(R)<λU(c∗)+(1−λ)Uc∗.12 82Chapter3.IntermediationandCrisesλU(1)+(1–λ)U(R)πU(1)+(1–π){λU(c1*)+(1–λ)U(c2*)π0πFigure3.5.Determinationoftheregionsofπsupporting(respectivelynotsupport-ing)runs.Sothereexistsauniquevalue0<π0<1suchthatπλU(c∗)+(1−λ)Uc∗=λU(1)+(1−λ)U(R)0U(1)+(1−π0)12andthebankwillbeindifferentbetweenthetwostrategiesifπ=π0.Obvi-ously,thebankwillpreferrunsifπ<π0andwillprefernorunsifπ>π0.ThesetworegionsareillustratedinFigure3.5.Wehaveshownthataslongastheprobabilityofabankrunissufficientlysmall,therewillexistanequilibriuminwhichthebankiswillingtoriskarunbecausethecostofavoidingtherunoutweighsthebenefit.Inthatcase,therewillbearunifthesunspottakesthehighvalueandnototherwise.Thereisanupperbound(lessthanone)totheprobabilityofarun,however.Iftheprobabilityofarunistoohigh,thebankwilltakeactiontodiscouragearunandthedepositorswillfinditoptimaltowithdrawatthecorrectdate.Notethatwehavenotspecifiedwhatthesunspotis.Itcouldbeanypubliclyobservedrandomvariablethattakesonaparticularvaluewithprobabilityπ<π0.Ifsuchavariableexists,thendepositorscaninprinciplecoordinateonthisvariabletosupportanequilibriumbankrun.3.7THEBUSINESSCYCLEVIEWOFBANKRUNSTheprevioussectionshaveoutlinedaDiamond–Dybvigstyleaccountofbankrunsinwhichextrinsicuncertaintyplaysacrucialrole.Runsoccurinthis 3.7TheBusinessCycleViewofBankRuns83frameworkbecauseoflateconsumers’beliefs.Ifallthelateconsumersbelievetherewillbearun,theywillallwithdrawtheirmoneyinthemiddleperiod.Iftheydonotbelievearunwilloccur,theywillwaituntilthelastperiodtowithdraw.Inbothcases,beliefsareself-fulfilling.Inthelastsectionweusedtheterminologyofsunspotstoexplainhowcoordinationoccurs.Traditionalaccountsofbankrunsoftenreferredto“mobpsychology”asthemotiveforthebankrunor“panic.”Thisviewofbankrunsaspanicshasalonghistorybutitisnottheonlyview.Analternativeviewofbankrunsisthattheyareanaturaloutgrowthofweakfundamentalsarisinginthecourseofthebusinesscycle.Aneconomicdownturnwillreducethevalueofbankassets,raisingthepossibilitythatbankswillbeunabletomeettheircommitmentsinthefuture.Ifdepositorsreceiveinformationaboutanimpendingdownturninthecycle,theywillanticipatefinancialdifficultiesinthebankingsectorandtrytowithdrawtheirfunds.Thisattemptwillprecipitatethecrisis.Accordingtothisinterpretation,crisesarenotrandomevents,butarationalresponsetounfoldingeconomiccircumstances.Inotherwords,theyareanintegralpartofthebusinesscycle.InChapter1webrieflydiscussedfinancialcrisesintheUSduringtheNationalBankingErafrom1865to1914.Gorton(1988)conductedanempir-icalstudytodifferentiatebetweenthesunspotviewandthebusinesscycleviewofbankingcrisesusingdatafromthisperiod.Hefoundevidenceconsistentwiththeviewthatbankingpanicsarepredictedbythebusinesscycle.Itisdifficulttoreconcilethisfindingwiththenotionofcrisesas“random”events.Table3.1showstherecessionsandcrisesthatoccurredintheUSduringtheNationalBankingEra.Italsoshowsthecorrespondingpercentagechangesinthecurrency/depositratioandthechangeinaggregateGDP,asproxiedbythechangeinpigironproductionduringtheseperiods.Thefiveworstreces-sions,asmeasuredbythechangeinpigironproduction,wereaccompaniedbycrises.Inall,crisesoccurredinsevenoftheelevencycles.Usingtheliabilitiesoffailedbusinessesasaleadingeconomicindicator,Gortonfindsthatcriseswerepredictableevents:wheneverthisleadingeconomicindicatorreachedacertainthreshold,apanicensued.ThestylizedfactsuncoveredbyGortonthussuggestthat,atleastduringtheUSNationalBankingEra,bankingcriseswereintimatelyrelatedtothebusinesscycleratherthansomeextraneousrandomvariable.CalomirisandGorton(1991)considerabroadrangeofevidencefromthisperiodandconcludethatthedatadonotsupportthe“sunspot”viewthatbankingpanicsarerandomevents.Amongotherthings,theyfindthatforthefiveepisodestheyfocuson,stockpricesfellbythelargestamountbyfarduringthepre-panicperiods.Inthissection,weadaptourmodeltoallowustoconsiderthefundamentalorbusinesscycleviewofbankingcrises.Inparticular,insteadofassumingthe 84Chapter3.IntermediationandCrisesTable3.1.NationalBankingErapanics.NBERCyclePanicdate%
(Currency/%
PigPeak–Troughdeposit)∗iron†Oct.1873–Mar.1879Sep.187314.53−51.0Mar.1882–May1885Jun.18848.80−14.0Mar.1887–Apr.1888NoPanic3.00−9.0Jul.1890–May1891Nov.18909.00−34.0Jan.1893–Jun.1894May189316.00−29.0Dec.1895–Jun.1897Oct.189614.30−4.0Jun.1899–Dec.1900NoPanic2.78−6.7Sep.1902–Aug.1904NoPanic−4.13−8.7May1907–Jun.1908Oct.190711.45−46.5Jan.1910–Jan.1912NoPanic−2.64−21.7Jan.1913–Dec.1914Aug.191410.39−47.1∗Percentagechangeofratioatpanicdatetopreviousyear’saverage.†Measuredfrompeaktotrough.(AdaptedfromTable1,Gorton1988,p.233).longassethasacertainreturn,weassumethatthereturnisrisky.HerewearefollowingtheapproachdevelopedinAllenandGale(1998)(cf.alsoBryant1980).•Thelongassetisaconstantreturnstoscaletechnologythattakesoneunitofthegoodatdate0andtransformsitintoRHunitsofthegoodatdate2withprobabilityπHandRLunitswithprobabilityπL.Ifthelongassetisprematurelyliquidated,oneunitoftheassetyieldsrunitsofthegoodatdate1.WeassumethatRH>RL>r>0.Anintermediarytakesadepositofoneunitfromthetypicalconsumerandinvestsitinaportfolioconsistingofyunitsofthesafe,shortassetandxunitsoftherisky,longasset,subjecttothebudgetconstraintx+y≤1.Inexchange,theintermediaryofferstheconsumeracontractpromisingc1unitsofconsumptionifhewithdrawsatdate1andc2unitsofconsumptionifhewithdrawsatdate2.Asbeforeweassumethattheintermediarycannotobservetheconsumer’stype(i.e.earlyorlate)andsocannotmakethecontractcontingentonthat.Amorestringentrequirementisthattheintermediarycan-notmakethedepositcontractcontingentonthestateofnatureor,equivalently,thereturntotheriskyasset. 3.7TheBusinessCycleViewofBankRuns85Freeentryandcompetitionamongtheintermediariesleadsthemtomaximizetheexpectedutilityoftheircustomers.Thisimpliesthattheinter-mediarieswillreceivezeroprofitsinequilibrium.Inparticular,thisrequiresthattheconsumersreceivetheentirevalueoftheremainingassetsatdate2.Becausetheterminalvalueoftheassetsisuncertain,theintermediarywillpromisealargeamountthatwillcertainlyexhaustthevalueoftheassetsatdate2.Withoutlossofgeneralityweputc2=∞and,inwhatfollows,wecancharacterizethedepositcontractbythesingleparameterc1=d,wheredstandsforthefacevalueofthedepositatdate1.IntroducingriskintheformofrandomassetreturnsdoesnoteliminatetheDiamond–Dybvigphenomenonofbankrunsbasedonself-fulfillingexpect-ationsorcoordinationonsunspots.Infact,theDiamond–DybvigmodelisjustaspecialcaseofthecurrentmodelwithRH=RL.Inordertodistinguishthisaccountofbankruns,wesimplyruleouttheDiamond–Dybvigphenomenonbyassumptionandconsideronlyessentialbankruns,thatis,runsthatcannotbeavoided.Looselyspeaking,weassumethatifthereexistsanequilibriuminwhichthereisnobankrunaswellasoneormorethathavebankruns,thentheequilibriumweobserveistheonewithoutabankrunratherthantheonewithabankrun.Supposethatthebankhaschosenaportfolio(x,y)andadepositcontractdatdate0.Atdate1thebudgetconstraintrequiresλd≤yandwecanassume,withoutlossofgenerality,thattheintermediaryalwayschooses(x,y)anddtosatisfythisconstraint.Otherwise,theintermediarywillalwayshavetodefaultandthevalueofdbecomesirrelevant.Conse-quently,theconsumptionofthelateconsumers,conditionalonnorun,willbegivenby(1−λ)c2s=Rs(1−y)+y−λd.Thisisconsistentwithnorunifandonlyifc2s≥dord≤Rs(1−y)+y.Thislastinequalityiscalledtheincentiveconstraint.Ifthisinequalityissatisfied,thereisanequilibriuminwhichlateconsumerswaituntildate2towithdraw.Sinceweonlyadmitessentialruns,thenecessaryandsufficientconditionforabankrunisthattheincentiveconstraintisviolated,thatis,d>Rs(1−y)+y. 86Chapter3.IntermediationandCrisesSinceRH>RL,thisconditiontellsusthattherecanneverbeanessentialruninstateHunlessthereisalsooneinstateL.Thereisnopointchoosingdsolargethatarunalwaysoccurs,sowecanrestrictattentiontocasesinwhicharunoccursinstateLifitoccursatall.Therearethenthreedifferentcasesthatneedtobeconsidered.Inthefirst,theincentiveconstraintisneverbindingandbankruptcyisnotapossibility.Inthesecondcase,bankruptcyisapossibilitybutthebankfindsitoptimaltochooseadepositcontractandportfoliosothattheincentiveconstraintis(just)satisfiedandthereisnobankruptcyinequilibrium.Inthethirdcase,thecostsofdistortingthechoiceofdepositcontractandportfolioaresogreatthatthebankfindsitoptimaltoallowbankruptcyinthelowasset-returnstate.CaseI:TheincentiveconstraintisnotbindinginequilibriumInthiscase,wesolvetheintermediary’sdecisionproblemwithouttheincentiveconstraintandthencheckwhethertheconstraintisbindingornot.Theinter-mediarychoosestwovariables,theportfolioyandthedepositcontractdtomaximizethedepositor’sexpectedutility,assumingthatthereisnobankrun.Withprobabilityλthedepositorisanearlyconsumerandreceivesdregardlessofthestate.Withprobability1−λ,thedepositorisalateconsumerandthenhisconsumptiondependsonthereturntotheriskyasset.Thetotalconsump-tioninstatesisequaltothereturntotheriskyassetplustheremainderofthereturnsfromthesafeassetaftertheearlyconsumershavereceivedtheirshare,thatis,Rs(1−y)+y−λd.Theconsumptionofatypicallateconsumerisjustthisamountdividedbythenumberoflateconsumers1−λ.Thus,theexpectedutilityis
RH(1−y)+y−λdλU(d)+(1−λ)πHU1−λ
 RL(1−y)+y−λd+πLU.1−λThisexpressionismaximizedsubjecttothefeasibilityconstraints0≤y≤1andλd≤y.Assumingthattheoptimalportfoliorequiresinvestmentinbothassets,i.e.0RdL(1−y)+y, 90Chapter3.IntermediationandCrisesandU∗∗∗>U∗∗.Thefirstconditionguaranteesthattheincentiveconstraintisviolated,sothattheintermediarymustdefaultinstateL,andthesecondcondi-tionguaranteesthatdefaultispreferredtosolvency.Otherwisethebankprefers(d∗∗,y∗∗)andthereisnodefault.3.8THEGLOBALGAMESAPPROACHTOFINDINGAUNIQUEEQUILIBRIUMSection3.6demonstratedhowthesunspotapproachallowedacompletedescriptionofanequilibriumwithbankruns.Theweaknessofthisapproachisthatitdoesnotexplainwhythesunspotshouldbeusedasacoordinationdevice.Thereisnorealaccountofwhattriggersacrisis.Thisisparticularlyaproblemifthereisadesiretousethetheoryforpolicyanalysis.CarlssonandvanDamme(1993)showedhowtheintroductionofasmallamountofasymmetricinformationcouldeliminatethemultiplicityofequilib-riaincoordinationgames.Theycalledthegameswithasymmetricinformationaboutfundamentalsglobalgames.Theirworkshowedthattheexistenceofmul-tipleequilibriadependsontheplayershavingcommonknowledgeaboutthefundamentalsofthegame.Introducingnoiseensuresthatthefundamentalsarenolongcommonknowledgeandthuspreventsthecoordinationthatisessentialtomultiplicity.MorrisandShin(1998)appliedthisapproachtomodelsofcurrencycrises.RochetandVives(2004)andGoldsteinandPauzner(2005)haveappliedthesametechniquetobankingcrises.InthissectionwepresentasimpleexampleoftheglobalgamesapproachprovidedbyAllenandMorris(2001).Therearetwodepositorsinabank.Depositori’stypeisi.Ifiislessthan1,thendepositoriisanearlyconsumerandneedstowithdrawhisfundsfromthebank.Ifiisgreaterthanorequalto1,heisalateconsumerandhasnoliquidityneeds.Inthiscaseheactstomaximizehisexpectedreturn.Ifadepositorwithdrawshismoneyfromthebank,heobtainsaguaranteedpayoffofω>0.Ifbothdepositorskeeptheirmoneyinthebankthenbothobtainρwhereω<ρ<2ω.Ifadepositorkeepshismoneyinthebankandtheotherdepositorwithdraws,hegetsapayoffof0. 3.8TheGlobalGamesApproachtoFindingaUniqueEquilibrium91Notethattherearefourstatesofliquiditydemand:bothareearlyconsumersandhaveliquidityneeds,depositor1onlyisanearlyconsumerandhasliquidityneeds,depositor2onlyisanearlyconsumerandhasliquidityneeds,andbotharelateconsumersandhavenoliquidityneeds.Ifthereiscommonknow-ledgeoffundamentals,andatleastonedepositorisanearlyconsumer,theuniqueequilibriumhasbothdepositorswithdrawing.Butifitiscommonknowledgethatbothdepositorsarelateconsumers,theyareplayingacoord-inationgamewiththefollowingpayoffs.(Thefirstelementrepresentsthepayofftotheplayerchoosingtherowstrategyandthesecondelementisthepayofftotheplayerchoosingthecolumnstrategy.)RemainWithdrawRemain(ρ,ρ)(0,ω)Withdraw(ω,0)(ω,ω)Animportantfeatureofthiscoordinationgameisthatthetotalpayoffswhenonlyonepersonwithdrawsearlyarelessthanwhenbothpeoplewithdrawearly.Onesetofcircumstanceswherethiswouldarise,forexample,iswhenthebankcanclosedownaftereverybodyhaswithdrawn,butwhenanybodykeepstheirmoneyinthebankthenextracostsareincurredtostayopenandthebank’sassetsaredissipatedmore.Withcommonknowledgethatneitherinvestorisanearlyconsumer,thisgamehastwoequilibria:bothremainandbothwithdraw.Wewillnextconsiderascenariowhereneitherdepositorisanearlyconsumer,bothknowthatnooneisanearlyconsumer,bothknowthatbothknowthis,andsoonuptoanylargenumberoflevels,butnonethelessitisnotcommonknowledgethatnooneisanearlyconsumer.Wewillshowthatinthisscenario,theuniqueequilibriumhasbothdepositorswithdrawing.Inotherwords,beliefsaboutothers’beliefs,orhigher-orderbeliefsastheyarecalled,inadditiontofundamentals,determinetheoutcome.Hereisthescenario.Thedepositors’types,1and2,arehighlycorrelated;inparticularsupposethatarandomvariableTisdrawnfromasmoothdistri-butiononthenon-negativenumbersandeachiisdistributeduniformlyontheinterval[T−ε,T+ε],forsomesmallε>0.Giventhisprobabilitydis-tributionovertypes,typesdiffernotonlyinfundamentals,butalsoinbeliefsabouttheotherdepositor’sfundamentals,andsoon.Toseewhy,recallthatadepositorisanearlyconsumerifiislessthan1.Butwhendobothdepositorsknowthatbothiaregreaterthanorequalto1sotheyarelateconsumers?Onlyifbothiaregreaterthan1+2ε.Thisisbecausebothplayersknowsthattheother’siiswithin2εoftheirown.For 92Chapter3.IntermediationandCrisesexample,supposeε=0.1anddepositor1has1=1.1.ShecandeducethatTiswithintherange1.0−1.2andhencethat2iswithintherange0.9−1.3.Onlyif1≥1.2doesdepositor1knowthatdepositor2isalateconsumer.Whendobothinvestorsknowthatbothinvestorsknowthatbothiaregreaterthanorequalto1?Onlyifbothiaregreaterthan1+4ε.Toseethis,supposethatε=0.1anddepositor1receives1=1.3.ShecandeducethatTiswithintherange1.2−1.4andhencethatdepositor2’ssignaliswithintherange1.1−1.5.However,ifdepositor2receives2=1.1,thenhesetsapositiveprobabilityofdepositor1having1withintherange0.9−1.3asabove.Onlyifdepositor1’ssignalisgreaterorequalto1+4εwouldthispossibilitybeavoidedandbothwouldknowthatbothknowthatbotharelateconsumers.Aswegoupanorderofbeliefstherangegoesonincreasing.Henceitcanneverbecommonknowledgethatbothdepositorsarelateconsumersandhavenoliquidityneeds.Whatdothesehigher-orderbeliefsimply?Itissimplesttoconsiderwhathappenswhenεisverysmall.Inthiscase,sinceTissmoothlydistributedtheprobabilityoftheotherdepositorhavinganiaboveorbelowapproaches0.5ineachcaseasε→0(seeFigure3.7).Wewilltakeitas0.5inwhatfollows.(AnalternativeapproachistoassumeTisuniformlydistributedinwhichcaseitisexactly0.5evenawayfromthelimitofε→0–seeMorrisandShin(2003).)Howdodepositorsbehaveinequilibrium?Observefirstthateachdepositorwillwithdrawifi<1sothedepositorisanearlyconsumer.Whataboutifi≥1?Giventhestructureofthemodelwithapersonbeinganearlyconsumerwheni<1andalateconsumerwheni≥1,themostnaturalstrategyforadepositortofollowistochooseastrategyofremainingonlywheni>kTProbabilitydensityfunctionofT
–T
T
+TFigure3.7.Theprobabilityoftheotherdepositor’sibeingaboveorbelow0.5asε−→0. 3.8TheGlobalGamesApproachtoFindingaUniqueEquilibrium93forsomek≥1andwithdrawingotherwise.Supposedepositor1followsthisstrategy.Considerwhathappenswhen2=k.GivenourassumptionsaboutεbeingsmallandTbeingdrawnfromasmoothdistribution,depositor2deducesthatthereisa0.5probabilitythat11unitsofthegoodatdate2.Thereturnstobothassetsareassumedtobecertain.Themodelcaneasilybeextendedtoallowforuncertaintyaboutassetreturns(seeAllenandGale1994),butherewewanttofocusonothersourcesofuncertainty.Theassetmarketcontainsalargenumber,strictly,acontinuum,1ofexanteidenticalconsumers.Eachconsumerisendowedwithoneunitofthegoodatdate0andnothingatdates1and2.Thereisnoconsumptionatdate0andconsumersinvesttheirendowmentinaportfoliooflongandshortassetstoprovideforfutureconsumptionatdates1and2.Eachconsumerlearnsatdate1whetherheisanearlyconsumerwhoonlyvaluesconsumptionatdate1oralateconsumerwhoonlyvaluesconsumptionatdate2.Ifaconsumerexpectsc1unitsofconsumptionatdate1whenhe1Werepresentthesetofagentsbytheunitinterval[0,1],whereeachpointintheintervalisadifferentagent.WenormalizethemeasureoftheentiresetofagentstobeequaltooneandmeasurethefractionofagentsinanysubsetbyitsLebesguemeasure.Theassumptionofalargenumberofindividuallyinsignificantagentsensuresperfectcompetition,thatis,noonehasenoughmarketpowertoaffecttheequilibriumtermsoftrade. 104Chapter4.AssetMarketsisanearlyconsumerandc2unitsofconsumptionatdate2whenheisalateconsumer,hisutilitywillbearandomvariableU(c1)withprobabilityλ,u(c1,c2)=U(c2)withprobability1−λwheretheutilityfunctionU(c)hastheusualneoclassicalpropertieswithU(c)>0andU(c)≤0.Theprobabilityofbeinganearlyconsumerisdenotedbyλ>0.Theonlyaggregateuncertaintyconcernsthedemandforliquidity.Weassumethatλisarandomvariable.Forsimplicity,supposeλtakestwovalues:λHwithprobabilityπλ=λLwithprobability1−πwhere0<λL<λH<1.Atdate0,individualsknowthemodelandthepriordistributionofλ.Atdate1,theyobservetherealizedvalueofλanddiscoverwhethertheyareearlyorlateconsumers.4.3EQUILIBRIUMWhenatypicalconsumermakeshisplansatdate0,hedoesnotknowwhetherstateHorstateLwilloccurandhedoesnotknowwhetherhewillbeanearlyoralateconsumer;buthedoesknowtheprobabilityofeachoftheseeventsandrationallytakesthemintoaccountinmakinghisplans.Moreprecisely,heknowsthatthereareessentiallyfouroutcomes:heiseitheranearlyconsumerinstateH,alateconsumerinstateH,anearlyconsumerinstateL,oralateconsumerinstateL.Eachstates=H,Loccurswithprobability1/2andthen,conditionalonthestate,theconsumerbecomesanearlyconsumerwithprobabilityλsandalateconsumerwithprobability1−λs.TheprobabilitiesofthefouroutcomesaregiveninTable4.2.Table4.2.Probabilitydistributionofindividualoutcomes.EarlyLateStateH1λ1H(1−λH)221λ1StateLL(1−λL)22 4.3Equilibrium105Althoughthereisonlyasinglegood,wedistinguishgoodsbythedateandstateinwhichtheyaredelivered.Sinceallconsumptionoccursatdates1and2andtherearetwostatesofnature,HandL,therearefourcontingentcom-modities,oneforeachorderedpair(t,s)consistingofadatet=1,2andastates=H,L.Aconsumerwillhaveaconsumptionbundlethatconsistsofdiffer-entquantitiesofallfourcontingentcommodities.Letc=(c1H,c2H,c1L,c2L)denotetheconsumptionbundleobtainedinequilibrium,wherectsdenotesconsumptionatdatetinstates,thatis,consumptionofthecontingentcom-modity(t,s).Figure4.1illustratestheoutcomesfortheindividualconsumerandtheconsumptionassociatedwitheachoutcome.Theexpectedutilityassociatedwithaconsumptionbundlec=(c1H,c2H,c1L,c2L)isgivenby11λHU(c1H)+(1−λH)U(c2H)+λLU(c1L)+(1−λL)U(c2L).22IfanindividualisanearlyconsumerandthestateisHthenheconsumesc11HandgetsutilityU(c1H).TheprobabilityofthishappeningisλH.The2firsttermistheexpectedutilityfrombeinganearlyconsumerinstateH.Theothertermsarederivedsimilarly.Alloftheconsumer’sdecisionsareassumedtomaximizethevalueofhisexpectedutility.Theconsumptionbundleobtainedbyatypicalconsumerdependsonhisportfoliodecisionatdate0andtheassetpricesobservedatfuturedates.Sup-posetheconsumerinvestsx≥0unitsinthelongassetandy≥0unitsintheshortasset,wherex+y=1.Theconsumer’sportfoliowillproduceyunitsofthegoodatdate1and(1−y)Runitsofthegoodatdate2.Anearlyconsumerc1HλH1–λHHc2H1/21/2Lc2L1–λLλLc1Lt=0t=1t=2Figure4.1.Informationstructureoftheassetmarketmodel. 106Chapter4.AssetMarketswillwanttoconsumeasmuchaspossibleatdate1andalateconsumerwillwanttoconsumeasmuchaspossibleatdate2.Weassumethereisanassetmarketatdate1.Thepriceofthelongasset,measuredintermsofthatgoodatdate1,isdenotedbyPsinstates=H,L.Thepresentvalueoftheinvestor’sportfolioatdate1instateswillbew1s=y+Psx.(4.1)NotethatifthepriceofthelongassetisPs,thentheimplicitpriceoffutureconsumptionintermsofpresentconsumptionisps≡Ps/R.Theinvestor’swealthintermsofconsumptionatdate2is
w1sy=+xR.(4.2)psPsFigure4.2illustratesthedeterminationofatypicalprofileofconsumptionateachdateandstate.Notethatconsumptionisdeterminedbytheportfoliodecisionatdate0,thatis,thechoiceofxandy,andtheassetpricesineachstate.Sinceconsumerstakepricesasgiven,theconsumerhasdeterminedhisconsumption(contingentonthestate)oncehehaschosentheportfolio(x,y).Thedeterminationofequilibriumfortheassetmarketmodelcanbebrokendownintotwosteps.First,wecandeterminetheassetpricesineachstateatdate1takingasgiventheportfoliodecisionsmadeatdate0.Then,foranypairofprices(PH,PL),wecandeterminetheoptimalportfolioatdate0,thatis,theportfolio(x,y)thatmaximizesexpectedutility.Tobecertainthatwehavefoundanequilibrium,wehavetocheckthatthesetwostepsareconsistent,thatc1H=y+PHxPHc2H=(y/PH+x)Rx+y=1PLc2L=(y/PL+x)Rc1L=y+PLxt=0t=1t=2Figure4.2.Assetreturnsintheassetmarketmodel. 4.3Equilibrium107is,thattheportfolioschosenatdate0willleadtotheexpectedpricesatdate1andthattheexpectationofthosepriceswillleadtothechoiceofthesameportfoliosatdate0.Webeginwiththeanalysisofmarket-clearingatdate1.4.3.1Market-clearingatdate1Supposethatalltheconsumershavechosenthesameportfolio(x,y)atdate0.Thebudgetconstraintensuresthatx+y=1soinwhatfollowswelet1−ydenotetheinvestmentinthelongasset.Also,becausethetruestateisknownandtheanalysisappliesequallytobothstates,wecansuppressthereferencetothestateandletPdenotethepriceofthelongassetandλthefractionofearlyconsumers.ThepricePisdeterminedbydemandandsupply.Thesupplyofthelongassetcomesfromtheearlyconsumers.Anearlyconsumerwantstoliquidateallhisassetsinordertoconsumeasmuchaspossibleatdate1.Inparticu-lar,theearlyconsumerswillinelasticallysupplytheirholdingsoftheshortasset,whatevertheprice.SothereisaverticalsupplycurveandthequantitysuppliedisS=λ(1−y)becausethereareλearlyconsumersandeachofthemhas1−yunitsofthelongasset.Thedemandforthelongassetcomesfromthelateconsumers,butthedeterminationofdemandisalittlemoresubtlethanthedeterminationofsupply.Becausethelateconsumersdonotwanttoconsumeuntildate2,theyhaveanon-trivialdecisionaboutwhichassetstoholdbetweendate1anddate2.Becausethetruestateisknownatdate1thereisnouncertaintyand,inparticular,thereturnsofthetwoassetsareknownforcertain.Oneunitofthegoodinvestedintheshortassetatdate1willproduceoneunitatdate2.Oneunitofthegoodwillpurchase1/Punitsofthelongassetatdate0andthiswillproduceR/Punitsofthegoodatdate2.Consumerswillholdwhicheverassethasthehighestreturns.Therearethreecasestobeconsidered.1.IfR/P<1,thereturnonthelongassetislessthanthereturnontheshortasset,andnoonewillwanttoholdthelongassetbetweendates1and2.2.IfR/P=1,theone-periodholdingreturnsontheshortandlongassetsareequalizedatdate1.Thenlateconsumersshouldbeindifferentbetweenholdingthetwoassetsatdate1. 108Chapter4.AssetMarkets3.IfR/P>1,theone-periodholdingreturnonthelongassetisgreaterthanthereturnontheshortassetandnoonewillwanttoholdtheshortassetbetweendate1anddate2.InCase1,thedemandforthelongassetiszero.InCase2,thedemandforthelongassetisperfectlyelastic,atleast,uptothemaximumamountthatthelateconsumerscouldbuy.InCase3,thelateconsumerswillwanttoholdonlythelongasset,andtheywillsupplytheirholdingsoftheshortassetinelasticallyinexchangeforthelongasset.Sincethereare1−λlateconsumersandeachholdsyunitsoftheshortasset,thetotalsupplyoftheshortassetis(1−λ)y.IfthepriceofthelongassetisPthenthenetdemandforthelongassetwillbe∗(1−λ)yD(P)=.PThusthedemandcurvewillhavetheform0ifP>R,D(P)=[0,D∗(R)]ifP=R,∗D(P)ifPD∗(R)thentheintersectionoccursatthedownward-slopingsectionofthedemandcurveandthepricesatisfiesS=D∗(P).SubstitutingforthevaluesofSandD∗(P)weseethatPisdeterminedbytheequation(1−λ)yλ(1−y)=Por(1−λ)yP=.λ(1−y)Puttingtogetherthesetwocases,wecanseethat(1−λ)yP=minR,.(4.3)λ(1−y) 4.3Equilibrium109Thispricingformulaillustratestheimpactofliquidityonassetpricing.Whenliquidityisplentifulthenthepriceofthelongassetissimplythediscountedpayoffwherethediscountrateistheopportunitycostgivenbythereturnontheshortasset.Whenliquidityisscarcethepriceofthelongassetisdeterminedbythecashinthemarket.Theearlyconsumersexchangealloftheirholdingsofthelongassetinexchangefortheconsumptiongoodsgivenbythelateconsumersholdingsoftheshortasset.PRD*(P)=(1–λ)y/P0λ(1–y)QFigure4.3.Demandforandsupplyofthelongassetatdate1.4.3.2PortfoliochoiceNowconsidertheinvestmentdecisionoftheconsumersatdate0.TakingasgiventheassetpricesPHandPLineachstateatdate1,theinvestorswillchoosetheportfolio(y,1−y)tomaximizetheirexpectedutility,
wsEλsU(ws)+(1−λs)U,(4.4)pswherews=y+Ps(1−y)andps≡Ps/R.Anequilibriumconsistsofapairofassetprices(PH,PL)andaportfoliochoiceysuchthatpricesaregivenbytheequation(4.3)andtheportfoliomaximizes(4.4)atthegivenprices. 110Chapter4.AssetMarkets4.4CASH-IN-THE-MARKETPRICINGUsingthemodeldescribedabove,wecanshowhowliquiditypreferenceaffectsthepricesofassets.NoaggregateuncertaintyConsiderfirstthecasewherethereisnoaggregateuncertainty,soλH=λL.Sincethetwostatesareidenticalwecanreasonablyassumethattheassetpriceisthesameineachstate,say,PH=PL=P.Aswesawinthepreviouschapter,whenthereisnouncertaintyabouttheassetpriceP,theonlypossibleequi-libriumvalueisP=1.Atanyotherprice,oneofthetwoassetsisdominatedandwillnotbeheldatdate0,whichimpliesthattheassetmarketcannotclearatdate1.WhenP=1,thetwoassetshavethesamereturnatdate0andtheinvestor’swealthatdate1isindependentofhisportfoliochoiceatdate0.Ifheisanearlyconsumer,hisconsumptionatdate1isc1=1−y+Py=1and,ifheisalateconsumer,hisconsumptionatdate2isyc2=1−y+R=R.PThisconsumptionallocationisfeasiblefortheeconomyiftheaverageinvestmentintheshortassetsatisfiesy=λ.Thenλc1=λ=yand(1−λ)c2=1−yRasrequired.AggregateuncertaintyNowsupposethatthereisaggregateuncertaintyaboutthetotaldemandforliquidity,asrepresentedbyfluctuationsinλs.Thisimpliesnon-zeroasset-pricevolatilityinthesensethatPHλL,wecanseethatPH1unitsofthegoodatdate1).Butifnoneoftheshortassetisheldatdate0,thepriceoftheassetatdate1willbezero.Thiscontradictionimpliesthatthemarket-clearingpricesmustsatisfyPH0.•ConsumershavetheusualDiamond–Dybvigpreferences,thatis,theyareearlyconsumerswithprobabilityλiorlateconsumerswithprobability1−λi:Ui(c1)w.pr.λiui(c1,c2)=Ui(c2)w.pr.1−λiwhere0<λi<1fori=A,B.Theylearnwhethertheyareearlyorlateconsumersatdate1.•TypeAisassumedtobelessriskaversethantypeB.•Eachinvestorhasaninitialendowmentofoneunitofthegoodatdate0andnothingatfuturedates.•Thereisasingleconsumptiongoodateachdateandtherearetwoassets:–theshortassetproducesoneunitofconsumptionatdatet+1foreachunitinvestedatdatetandtheamountchosenbyeachinvestorisdenotedyi;–thelongassetproducesR>1unitsofconsumptionatdate2foreachunitinvestedatdate0andtheamountchosenbyinvestorsoftypeiisdenotedxi.•Investorscanholdtheshortassetwithoutcost.However,thelongassetcanonlybetradedifaninvestorpaysafixedcostofe≥0utilsandentersthemarketforthelongasset. 4.5LimitedParticipation1174.5.2EquilibriumIndividualdecisionsThedecisiontreefacedbyatypicalinvestorisillustratedinFigure4.6.Sincethedecisionsareessentiallythesameforbothtypes,wesuppresstheisubscriptinthissection.U(y+PHx)–eEarlyLatePHU{(y/PH+x)R}–eHighInLowPLU{(y/PL+x)R}–eEarlyLateOutU(1)U(y+PLx)–et=0t=1t=2Figure4.6.Consumer’sdecisiontreewithparticipationdecision.Atdate0,aninvestorfirstdecideswhetherornottoentertheassetmarket.Ifhedecidesnottoenter,hecanonlyinvestintheshortasset.Sincethereturnontheshortassetisone,hisconsumptionwillbeequaltohisinitialendowment,i.e.c=1,andhisutilitywillbeU(1),whetherheisanearlyorlateconsumer.Ifhedecidestoentertheassetmarket,hemustpaytheentryfeee(inutils)andthendivideshisendowmentbetweenaninvestmentofxunitsinthelongassetandyunitsintheshortasset.Atdate1,theinvestorlearnsthetruestateofnatureandwhetherheisanearlyorlateconsumer.Ifheisanearlyconsumer,heliquidateshisportfolioandconsumestheliquidatedvalue.IfthepriceofthelongassetisPsinstates=H,Lthentheearlyconsumer’snetutilitywillbeU(y+Psx)−e.Otherwise,heisalateconsumerandherollsoverhisassetsuntilthefinalperiodwhenheyconsumestheliquidatedvalueandreceivesanetutilityU+xR−e.PsGiventhattheinvestordecidestoenterthemarket,hechooseshisportfoliox,ytomaximizehisexpectedutility,takingasgiventhepricesPHandPLthatwillprevailineachstate.Usingx=1−y,wecanwritehisdecision 118Chapter4.AssetMarketsproblemas∗(PUH,PL)=maxEλsU(y+Ps(1−y))−e0≤y≤1

 y+(1−λs)U+(1−y)R.PsItisoptimalfortheinvestortoenterthemarketifU∗(PH,PL)≥U(1).Marketclearingatdate1Letnidenotethenumberofinvestorsoftypeiwhoenterthemarketandletyidenotetheportfoliochosenbyinvestorsoftypeiwhochoosetoenter.Themarket-clearingconditionsareessentiallythesameasforthemodelwithonetype.Sincetheargumentisthesameforbothstates,wesuppressthesubscriptsinwhatfollows.Theearlyconsumersofbothtypeswillsupplytheirholdingsofthelongassetinelastically.Thetotalnumberofearlyconsumersoftypeiisni,thenumberofentrants,timesλi,thefractionofearlyconsumers.Sinceeachearlyconsumeroftypeisupplies1−yi,thetotalsupplyofthelongassetwillbeS=nAλA1−yA+nBλB1−yB.IfP=Rthenthelateconsumersareindifferentbetweenholdingtheshortandthelongassetbetweendate1anddate2.IfPUi(1)=⇒ni=NiandU∗(PiH,PL)Ui(1).iIfitisstrictlyoptimaltoenterthemarketwhentheentrycostiszerothenitmustalsobestrictlyoptimaltoenterthemarketforallsufficientlysmallentrycostse>0.Thus,wehaveprovedthefollowingproposition.Proposition1Foralle>0,sufficientlysmall,anequilibriumnA,yA,nB,yB,(PH,PL)mustentailfullparticipation,thatis,nA=NAandnB=NB.4.5.4Fullparticipationandasset-pricevolatilityToillustratefurtherthepropertiesofthefull-participationequilibrium,weuseanexampleinwhichtheinvestorsoftypeAhavearandomliquidityshockandinvestorsoftypeBhaveanon-randomliquidityshock:k>0ifs=HλAs=0ifs=LandλBH=λBL=λB. 4.5LimitedParticipation121WhenthereisfullparticipationthepricingformularequiresPs=min{R,Ys/Ss}.Nowask→0,soλAH→0,andtype-AinvestorsbecomemoreandmorecertainofbeinglateconsumersyA→0soYs(1−λB)yB→=Q¯,SsλBxBsay,whereQ¯isaconstantsinceλBisaconstant.Inthelimit,Ps=minR,Q¯.RecallthatR>1foralls.ThenQ¯>1impliesPs>1foralls.Thisimpliesthattheshortassetisdominatedbythelongassetatdate0sonoonewillholdtheshortasset;butthisisnotconsistentwithequilibrium.Ontheotherhand,ifQ¯<1thenPs<1foralls.Thismeansthattheshortassetdominatesthelongassetatdate0,whichisalsoinconsistentwithequilibrium.Therefore,Ps=Q¯=1fors=H,L.Thus,asliquidityshocksamongthetypeAsbecomesmall,thetypeBsareabletoabsorbtheshocks,whichhavenoeffectonprices.Alongthesamelines,itcanbeseenthatasNB→∞,holdingNAconstant,Pstendstothesamelimit,andasimilarresultholds.AsthenumberoftypeBsgoesup,theamountofliquidityinthemarketbecomeslargerelativetothefluctuationsinliquiditytrading,andthisdampensvolatility.4.5.5Limitedparticipationandasset-pricevolatilityContinuingwiththespecialcaseconsideredintheprecedingsection,supposethatthereexistsanequilibriuminwhichonlytypeAinvestorsenterthemarket.Thatis,nA=NAandnB=0.WithonlytypeAinthemarketatdate1,therewillbenodemandfortheshortassetinstateLandPL=R.ThepriceinstatePHmustsatisfythefirst-orderconditionthatmakesinvestorswillingtoholdbothassetsatthemarginindate0.Inthelimit,ask→0,investorsoftypeAarelateconsumerswithprobabilityoneandinvestalloftheirwealthinthelongasset.So,inthelimit,alateconsumerreceivesutilityUA(R)regardlessofthestate.Themarginalutilityofincomeisalsoconstantacrossstates,sotheinvestorbehavesasifheisriskneutral.Theninordertobewillingtoholdbothassets,theexpectedreturnsmustbeequalovertwoperiods,thatis,1R1R=+.2PH2Thelefthandsideisthereturntooneunitinvestedinthelongasset.Therighthandsideistheexpectedreturntooneunitinvestedintheshortassetatdate0. 122Chapter4.AssetMarketsToseethis,supposethattheinvestorholdshiswealthintheformoftheshortassetatdate0andinvestsinthelongassetatdate1(thisisalwaysweaklyoptimalandwillbestrictlyoptimalifPse0,itisnotworthwhileforthetype-Binvestorstoenteriftheyanticipatethehighdegreeofvolatilityassociatedwiththelimitedparticipationequilibrium.Thus,foranyentrycoste11unitsatdate2.Claimsonthelongassetcanbetradedatdate1.Thereistheusualtrade-offbetweenliquidityandreturns:long-terminvest-mentshavehigherreturnsbuttakelongertomature(arelessliquid).Bycontrast,thereisnorisk–returntrade-off:weassumethatassetreturnsarenon-stochasticinordertoemphasizethat,inthepresentmodel,financialcrisesarenotdrivenbyshockstoassetreturns.Thereisacontinuumofexanteidenticalconsumers,whosemeasureisnormalizedtounity.Eachconsumerhasanendowment(1,0,0)consistingofoneunitofthegoodatdate0andnothingatsubsequentdates.Therearetwo(expost)typesofconsumersatdate1:earlyconsumers,whovalueconsumptiononlyatdate1;andlateconsumers,whovalueconsumptiononlyatdate2.Ifλdenotestheprobabilityofbeinganearlyconsumerandctdenotesconsumptionatdatet=1,2,thentheconsumer’sexanteutilityisE[u(c1,c2)]=λU(c1)+(1−λ)U(c2). 5.1Markets,Banks,andConsumers129U(·)isaneoclassicalutilityfunction(increasing,strictlyconcave,twicecontinuouslydifferentiable).Itisimportantinwhatfollowstodistinguishbetweentwokindsofuncertainty.Intrinsicuncertaintyiscausedbystochasticfluctuationsintheprimitivesorfundamentalsoftheeconomy.Anexamplewouldbeexogenousshocksthataffectliquiditypreferences.Extrinsicuncertaintybydefinitionhasnoeffectonthefundamentalsoftheeconomy.Anequilibriumwithnoextrinsicuncertaintyiscalledafundamentalequilibrium,becausetheendoge-nousvariablesarefunctionsoftheexogenousprimitivesorfundamentalsofthemodel(endowments,preferences,technologies).Anequilibriumwithextrinsicuncertaintyiscalledasunspotequilibrium,becauseendogenousvari-ablesmaybeinfluencedbyextraneousvariables(sunspots)thathavenodirectimpactonfundamentals.Acrisiscannotoccurinafundamentalequilibriumintheabsenceofexogenousshockstofundamentals,suchasassetreturnsorliquiditydemands.Inasunspotequilibrium,bycontrast,assetpricesfluctu-ateintheabsenceofaggregateexogenousshocks,andcrisesappeartooccurspontaneously.Therearethreesourcesofintrinsicuncertaintyinthemodel.First,eachindividualconsumerfacesidiosyncraticuncertaintyaboutherpreferencetype(earlyorlateconsumer).Second,eachbankfacesidiosyncraticuncer-taintyaboutthenumberofearlyconsumersamongthebank’sdepositors.Forexample,differentbankscouldbelocatedinregionssubjecttoindepen-dentliquidityshocks.Third,thereisaggregateuncertaintyaboutthefractionofearlyconsumersintheeconomy.Tobeginwith,weignorethebanks’idiosyncraticuncertaintyandfocusonindividualidiosyncraticuncertaintyandaggregateuncertainty.Aggregateuncertaintyisrepresentedbyastateofnaturesthattakesontwovalues,HandL,withprobabilityπand1−πrespectively.Theprobabilityofbeinganearlyconsumerinstatesisdenotedbyλswhere0<λL≤λH<1.Weadopttheusual“lawoflargenumbers”conventionandassumethatthefractionofearlyconsumersinstatesisidenticallyequaltotheprobabilityλs.NotethatthereisaggregateintrinsicuncertaintyonlyifλL<λH.IfλL=λH=λthenthereisnoaggregate(intrinsic)uncertaintyandthestateofnaturesrepresentsextrinsicuncertainty.Alluncertaintyisresolvedatdate1.Thetruestatesispubliclyobservedandeachconsumerlearnshistype,thatis,whetherheisanearlyoralateconsumer.Anindividual’stype,earlyorlate,isprivateinformationthatonlytheindividualcanobservedirectly. 130Chapter5.FinancialFragilityTherearenoassetmarketsforhedgingagainstaggregateuncertaintyatdate0;forexample,therearenosecuritiesthatarecontingentonthestateofnatures.Atdate1,thereisamarketinwhichfuture(date-2)consumptioncanbeexchangedforpresent(date-1)consumption.Ifpsdenotesthepriceofdate2consumptionintermsofpresentconsumptionatdate1,thenoneunitofthelongassetisworthPs=psRatdate1instates.Marketsareincompleteatdate0,becauseoftheinabilitytohedgeuncer-taintyaboutthestates.Atdate1marketsarecompletebecausealluncertaintyhasbeenresolved.Weassumethatmarketparticipationisincomplete:financialinstitutionssuchasbankscanparticipateintheassetmarketatdate1,butindividualconsumerscannot.Banksarefinancialinstitutionsthatprovideinvestmentandliquidityservicestoconsumers.Theydothisbypoolingtheconsumers’resources,investingtheminaportfolioofshort-andlong-termassets,andofferingconsumersfutureconsumptionstreamswithabettercombinationofassetreturnsandliquiditythanindividualconsumerscouldachievebythemselves.Bankscompetebyofferingdepositcontractstoconsumersinexchangefortheirendowmentsandconsumersrespondbychoosingthemostattractiveofthecontractsoffered.Freeentryensuresthatbanksearnzeroprofitsinequilib-rium.Thedepositcontractsofferedinequilibriummustmaximizeconsumers’welfaresubjecttothezero-profitconstraint.Otherwise,abankcouldenterandmakeapositiveprofitbyofferingamoreattractivecontract.Anythingaconsumercando,thebankcando.Sothereisnolossofgeneralityinassumingthatconsumersdeposittheirentireendowmentinabankatdate0.Consumerscannotdiversifybyspreadingtheirmoneyacrossmorethanonebank.Thebankinvestsyunitspercapitaintheshortassetand1−yunitspercapitainthelongassetandofferseachconsumeradepositcontract,whichallowstheconsumertowithdraweitherd1unitsatdate1ord2unitsatdate2.Withoutlossofgenerality,wesetd2=∞.Thisensuresthatconsumersreceivetheresidueofthebank’sassetsatdate2.Withoutthisassumptionthedepositors’expectedutilitywouldnotbemaximized.Thedepositcontractischaracterizedbythepromisedpaymentatdate1.Inwhatfollowswewritedinsteadofd1.Ifpsdenotesthepriceoffutureconsumptionatdate1instateθ,thenthevalueofthebank’sassetsatdate1isy+psR(1−y).Notethat,unliketheDiamond–Dybvigmodel,thevalueoftheseassetsdoesnotdependonwhetherthebankisindefaultornot.Becausethereisacompetitivemarketonwhichassetscanbesold,thebank’sportfolioisalwaysmarked-to-market.Inwhatfollows,weassumethatbankrunsoccuronlyiftheyareunavoidable.Inotherwords,lateconsumerswillwithdrawatdate2aslongasthebankcan 5.1Markets,Banks,andConsumers131offerthemasmuchatdate2astheearlyconsumersreceiveatdate1.Intheeventofbankruptcy,thebankisrequiredtoliquidateitsassetsinanattempttoprovidethepromisedamountdtotheconsumerswhowithdrawatdate1.Ifthebankdoesnothaveenoughresourcestopayalloftheearlywithdrawersanamountd,thentherewillbenothingleftatdate2forlateconsumerswhodonotjointherun.Hence,inequilibrium,allconsumersmustwithdrawatdate1andeachconsumerwillreceivetheliquidatedvalueoftheportfolioy+psR(1−y).Underwhatconditionswillabankbeforcedtodefaultonitsdate1promisesandliquidateitsassets?Atdate1,alluncertaintyisresolved.Peoplefindouttheirindividualtypesandtheaggregatestatesisrealized.Earlyconsumerswillwithdrawtheirdepositsfromthebankforsure.Lateconsumershavetheoptionofleavingtheirmoneyinthebank,buttheycouldalsopretendtobeearlyconsumers,withdrawtheirfundsatdate1,andcarrythemforwardtodate2usingtheshortasset.Lateconsumerswillbewillingtowaituntildate2towithdrawiftheyareconfidentthattheywillreceiveatleastdunitsofthegoodatdate2.Otherwise,theywillrunonthebankandwithdrawdatdate1.Thecheapestwaytosatisfythelateconsumers,sothattheywaituntildate2towithdraw,istogivethemdatdate2.Ifthelateconsumersreceiveexactlydatdate2,thepresentvalueoftheclaimsonthebankisλd+(1−λ)psd.Ifλd+(1−λ)psd≤y+psR(1−y),(5.1)itispossibletopaythelateconsumersatleastdsothattheywillbewillingtowaituntildate2towithdraw.Otherwise,thebankcannotpossiblyhonoritspromisetogiveeachearlyconsumerdand,atthesametime,givelateconsumersatleastd.Arunonthebankatdate1isinevitable.Ifcondition(5.1)issatisfied,thedepositcontractissaidtobeincentive-compatible,inthesensethatitisoptimalforthelateconsumerstowithdrawatdate2.Weoftenrefertotheinequalityin(5.1)astheincentiveconstraint,althoughitalsoassumesthatthebank’sbudgetconstraintissatisfied.Ifthebankchooses(d,y)atdate0,thedepositor’sconsumptionatdatetinstatesisdenotedbycts(d,y)anddefinedbydif(5.1)issatisfiedc1s(d,y)=y+psR(1−y)otherwise,y+psR(1−y)−λsdif(5.1)issatisfiedc2s(d,y)=(1−λs)psy+psR(1−y)otherwise. 132Chapter5.FinancialFragilityIf(5.1)issatisfiedthenc1sissimplythepromisedamountd.Understandingc2sisalittlemorecomplex.Thepresentvalueofthefirm’sassetsatdate1isy+psR(1−y).Ofthis,λsdispaidouttoearlyconsumerssotheremaindery+psR(1−y)−λsdisavailabletopayforthelateconsumers’consumption.Sincethepriceoffutureconsumptionisps,wedividebypstofindthetotalamountofconsumptionavailableatdate2.Thereare1−λslateconsumers,sowehavetodivideby1−λstofindtheconsumptionofanindividuallateconsumer.Thus,eachlateconsumerreceives[y+psR(1−y)−λsd]/[(1−λs)ps]atdate2.If(5.1)isnotsatisfied,thenthereisacrisisandallconsumers,bothearlyandlate,trytowithdraw.Thebankgoesbankruptandisliquidatedfory+psR(1−y).Theearlyconsumersconsumetheirshareatdate1whilethelateconsumerscarryovertheirsharetodate2usingtheshortasset.Sincetheydonothaveaccesstotheassetmarket,wedonotdividebyps.Usingthisnotation,thebank’sdecisionproblemcanbewrittenasmaxE[λU(c1s)+(1−λ)U(c2s)](5.2)s.t.0≤d,0≤y≤1.Abank’schoiceofdepositcontractandportfolio(d,y)isoptimalforthegivenpricevectorp=pH,pLifitsolves(5.2).Aswesawintheprecedingchapter,theassetmarketatdate1canonlyclearifthepriceoffutureconsumptionislessthanorequalto1.Ifps>1,theassetpriceisPs=psR>Randthebankswouldonlybewillingtoholdtheshortasset.Thenthemarketforthelongassetcannotclear.Ifps=1thenbankswillbeindifferentbetweenholdingtheshortassetandthelongassetbetweendates1and2,sinceinbothcasesthereturnononeunitofthegoodinvestedatdate1isoneunitofthegoodatdate2.Ontheotherhand,ifps<1thennooneiswillingtoinvestintheshortassetatdate1andeverythingisinvestedinthelongasset.Thisisconsistentwithmarketclearing,becausethereisnostockoftheshortassetunlesssomeonechoosestoinvestinitatdate1.Proposition1Foranystates,theassetmarketatdate1clearsonlyifps≤1.Ifps=1,banksarewillingtoholdbothassetsatdate1.Ifps<1,thenonlythelongassetisheldbybanksatdate1.5.2TYPESOFEQUILIBRIUMInordertounderstandthedifferentformsthatequilibriumcantake,itishelpfultoconsidersomesimpleexamples. 5.2TypesofEquilibrium133Example1U(c)=ln(c);R=1.5;0.8ifs=Lλs=0.8+εifs=H;(π,1−π)=(0.35,0.65).5.2.1FundamentalequilibriumwithnoaggregateuncertaintyWestartbyconsideringthecasewhereε=0,soλH=λL=0.8andsrep-resentsextrinsicuncertainty,andlookatthefundamentalequilibrium,wheresplaysnorole.Ifthereisnouncertainty,thebankcanpromisethedeposit-orsanyconsumptionallocationthatsatisfiesthebudgetconstraintandtheincentiveconstraintbyputtingtheearlyconsumers’promisedconsumptionc1equaltodandputtingthelateconsumers’consumptionc2equaltotheresidualvalueoftheportfolio.Hence,inthefundamentalequilibriumwedonotneedtoconcernourselveswiththeformofthedepositcontract.Wecanassume,withoutessentiallossofgenerality,thatthebankoffersthedepositorsaconsumptionbundle(c1,c2).Sincethereisnoaggregateuncertainty,banksusetheshortassettoprovideconsumptionfortheearlyconsumersatdate1andwillusethelongassettoprovideconsumptionforthelateconsumersatdate2.Sincethefractionofearlyconsumersis0.8,theamountoftheshortassetintheportfoliomustsatisfy(0.8)c1=yorc1=y/(0.8).Similarly,theamountofthelongassetmustsatisfy(0.2)c2=R(1−y)orc2=R1−y/(0.2),where1−yistheamountinvestedinthelongassetandR=1.5isthereturnonthelongasset.Thebanks’decisionproblemthenismax0.8U(c1)+0.2U(c2)s.t.0≤y≤1c1=y/(0.8)c2=R(1−y)/(0.2).Substitutingtheexpressionsforc1andc2fromthebudgetconstraintsintotheobjectivefunctionandusingU(c)=ln(c),theproblemreducesto
yR(1−y)max0.8ln+0.2ln.0.80.2 134Chapter5.FinancialFragilityAssuminganinteriorsolutionwithregardtoy,thenecessaryandsufficientfirst-orderconditionforasolutiontothisproblemis0.80.2=,y1−ywhichcanbesolvedfory=0.8.Itfollowsthat,yc1==1;0.8(1−y)c2=1.5=1.5;0.2andE[u(c1,c2)]=0.8ln(1)+0.2ln(1.5)=0.081.Aswehaveseenbefore,whenthereisnouncertaintyaboutthefuture,theassetmarketsatdate0canonlycleariftheassetpriceatdate1satisfiesPH=PL=1.Thisistheonlypriceatwhichbanksarewillingtoholdbothassetsatdate0becauseitistheonlypricethatequalizestheholdingreturnsofthetwoassetsbetweendate0anddate1.Ineachstates,theassetpricePs=1impliesthatoneunitinvestedinthelongassetatdate1hasareturnofR=1.5.Thus,thereturnonthelongassetdominatesthereturnontheshortassetbetweendate1anddate2andbankswillonlyholdthelongassetatdate1.TheequilibriumthatwehavedescribedforExample1isautarkic.Eachbankcanprovidetherequiredconsumptionforitsdepositorsateachdatewithoutmakinguseoftheassetmarketatdate1.Thisisnottheonlyfundamentalequilibrium,infacttherearemany,buttheydifferonlyintheinvestmentsmadebybanksandnotwithrespecttotheaggregateinvestment,consumption,andexpectedutilityofthedepositors.Wehavearguedthatbanksareindifferentbetweenholdingthetwoassetsatdate0,whichyieldthesamereturnatdate1,inanyfundamentalequilibrium.Soanyportfolioisoptimalforthem.Aslongastheaggregateoraverageportfoliosatisfiesy=0.8,thereisnoreasonforbanksnottoholddifferentindividualportfolios.Forexample,supposethatafraction0.8ofthebanksholdonlytheshortassetandafraction0.2holdonlythelongasset.Atdate1,abankholdingonlytheshortassetwilluse80%ofitsportfoliosatisfyingtheearlyconsumerswhowanttowithdrawandwiththe 5.2TypesofEquilibrium135remainderwillbuy0.2unitsofthelongassettoprovidethelateconsumerswithconsumptionof1.5atdate2.Banksthathaveonlyheldthelongassetwillliquidate80%oftheirportfolioinordertoobtainconsumptionfortheirearlyconsumersandwillholdontotheremaining20%toprovideforconsumptionofthelateconsumersatdate2.Althoughtheassetmarketisusedinthisexample,itisnotessential.Abankcanstillachievethefirstbestwithoutusingthemarket.Ifbanksreceiveidiosyncraticliquidityshocks,ontheotherhand,theywillhavedifferentliquid-ityneedsatdate1.Somebankswillhavesurplusliquidityandotherbankswillhavealiquiditydeficit.Thesesurplusesanddeficitscanonlyberemovedbyusingthemarket.Attheendofthischapter,wewillconsideracasewheretheassetmarketplaysanessentialrole,butforthemomentwecontinuetoignoreidiosyncraticshocks.5.2.2AggregateuncertaintyNowsupposethatε=0.01,sothatλL=0.8andλH=0.81.Uncertaintyaboutthestatesgivesrisetoaggregateintrinsicuncertainty.Althoughtheamountofaggregateuncertaintyissmall,theequilibriumwefindfortheeconomywithaggregateuncertaintyisquitedifferentfromthefundamentalequilibriumdiscussedabove.Wecaneasilyidentifysomereasonsforthedra-maticeffectofasmalldegreeofaggregateuncertaintyintheinelasticsupplyofanddemandforliquidity.Supposethatallbanksmakeidenticalchoicesatdate0.Weshalllaterseethatthisassumptionisnotplausible,butitwillservetoillustratesomeimportantpoints.Iftheinvestmentintheshortassetisyunits,thenthetotalamountofthegoodavailableforconsumptionintheeconomyatdate1isyunitspercapita,independentlyoftheaggregatestateandtheprevailingassetprices.Inthissense,theaggregatesupplyofliquidityisinelastic.Wedenotetheaggregatesupplyofliquidityatdate1byS(P)anddefineitbyputtingSs(P)=yforallpricelevelsPandstatess.Supposethateachbankalsochoosesadepositcontractthatpromisesdunitsofthegoodtoanydepositorwhowithdrawsatdate1.Assumingtherearenobankrunsand,hence,nodefaults,theonlydemandforliquidityatdate1istoprovidegoodstotheearlyconsumers.Sincethefractionofearlyconsumersis0.8instates=Land0.81instates=H,theaggregateamountofthegoodneededbybanksis(0.8)dwithprobability0.65and(0.81)dwithprobability 136Chapter5.FinancialFragility0.35.LetDs(P)denotetheaggregatedemandforliquiditywhentheassetpriceisPandthestateiss.Then(0.8)difs=L,Ds(P)=(0.81)difs=H.Thedemandispriceinelastic,becausethepromisedpaymentdisinelastic,buttheaggregatedemandvarieswiththestate.Marketclearingrequiresthatthedemandforliquiditydoesnotexceedthesupply,thatis,Ds(Ps)≤Ss(Ps)foreachstates=H,L.Ifthemarketclearsinthehighstates=H,however,therewillbeanexcessofliquidityinthelowstates=L,because(0.8)d<(0.81)d≤y.Aswehaveseeninourearlierdiscussionsofequilibriumintheassetmarket,whenthereisexcessliquidityatdate1thepriceofthelongassetwillrisetoPL=R=1.5.Onlyatthispricewillthereturnsontheshortandlongassetbeequalized,sothatbanksarewillingtoholdtheshortassetuntildate2.Butbanksmustalsoholdbothassetsatdate0andtheywillnotdothatiftheshortassetisdominatedbythelongasset.IfPL=1.5theshortassetisundominatedonlyiftheassetpriceislowerinthehighstate.Infact,wemusthavePH1.Sotheinelasticityofdemandandsupplymeansthatevensmallaggregateliquidityshockswillcauselargefluctuationsinassetprices.Wehaveshownthat,evenintheabsenceofdefault,theequilibriumwithasmallamountofaggregateuncertaintyε=0.01willbequitedifferentfromthefundamentalequilibriuminthelimiteconomywhereε=0.Ifweallowforthepossibilityofdefault,wediscoveranotherdifference:banksthatstartoutlookingverysimilarwilladoptquitedifferentinvestmentandrisk-sharingstrategies.Toseethis,supposetothecontrarythatallthebankscontinuetomakethesamechoicesatthefirstdate.Asweassumedabove,eachbankinvestsyunitsofthegoodintheshortassetandoffersadepositcontractpromisingdunitsofconsumptiontodepositorswhowithdrawatdate1.Wehavearguedthatabankwilldefaultifandonlyifitfailstosatisfytheincentiveconstraint(5.1)atdate1.Butifeverybankhasmadethesamechoicesatdate1,eithertheywillallviolatetheincentiveconstraintornonewill.Ifeverybankdefaults,therewillbenoonetobuythelongassetsthatmustbeliquidatedatdate1andtheassetpricewillfalltozero.Thiscannotbeanequilibrium,however.Atapriceofzero,someonewouldbetemptedtodeviate,chooseyanddsothat 5.2TypesofEquilibrium137bankruptcyisavoided,andmakealargecapitalgainbypurchasingthelongassetswhentheirpricefallstozero.So,inordertohavedefaultinequilibrium,equilibriumwillhavetobemixedinthesensethattherearetwotypesofbanks,callthemsafebanksandriskybanks,thatmakeverydifferentchoicesatdate0.Safebanksholdalotoftheshortassetandofferdepositcontractspromisinglowpaymentsatdate1.Riskybanksholdalotofthelongassetandofferdepositcontractspromisinghighpaymentsatdate1.Whenliquiditydemandsarelow(s=L)thesafebankshaveexcessliquiditywhichtheysupplytothemarketbybuyingthelongasset.Riskybanksobtaintheliquiditytheyneedtohonortheirdepositcontractsbysellingthelongasset.Whenliquiditydemandsarehigh(s=H)themarketforthelongassetislessliquidbecausethesafebanksmustdevotemoreoftheirliquiditytosatisfyingtheneedsoftheirowncustomers.Thisliquidityshortageleadstoadropinthepriceofthelongasset,whichforcestheriskybankstogobankruptandliquidatetheirstocksofthelongasset.Theincreaseinthesupplyofthelongassetcanleadtoasharpdropinprices.Inthiscasethereis“cash-in-the-market”pricing.Thesafebanksholdjustenoughliquidityinexcessoftheircustomers’needstoenablethemtobuyupthelongassetatafiresaleprice.Thelowpricecompensatesthemforthecostofholdingtheextraliquiditywhenliquiditydemandsarelowandpricesarehigh.Toseehowsuchanequilibriumoperatesindetail,considerthefollowingexampleinwhichtheparametervaluesarethesameasinExample1exceptthatnowε=0.01ratherthanε=0.ThisvariantwithaggregateuncertaintyisreferredtoasExample1A.Example1AU(c)=ln(c);R=1.5;α=0.8;π=0.35;ε=0.01.Then0.8withprobability0.65;λs=0.81withprobability0.35. 138Chapter5.FinancialFragilityTheassetpricePsisrandomandtakesthevaluespHR=(0.430)(1.5)=0.645ifs=H;Ps=pLR=(0.940)(1.5)=1.410ifs=L.Nextwedescribethebehaviorofthesafeandriskybanks.Aproportionρ=0.979ofthebanksadoptthesafestrategyofavoidingbankruptcyanddefault.Specifically,theychoosetoinvestalargeamountySintheshortassetandpromisealowamountdStodepositorswhowithdrawearly.yS=0.822;dS=0.998.Bothchoicesmakeiteasierforthebanktosatisfytheincentiveconstraint.Oncethesechoicesaremade,thepricesandbudgetconstraintsdeterminethedepositors’consumptionateachdateandineachstate:cS,cS=(0.998,1.572)ifs=H;S1H2Hcs=cS,cS=(0.998,1.461)ifs=L.1L2LNotethatearlyconsumersreceivethesamelevelofconsumption(dS)ineachstate.ThelateconsumershavedifferentlevelsofconsumptionineachstatebecausetheyreceivetheresidualvalueoftheportfolioandthisdependsontheassetpricePs.Notethatthelateconsumersarebetteroffinthehighstate:thelowassetpriceallowsthebanktopurchasefutureconsumptioncheaply.Alternatively,onecanthinkofthebankasmakingcapitalgainsfrombuyinguplongassetscheaplyatdate1andsellingthemmoredearlyatdate2.Aproportion1−ρ=0.021ofthebanksadopttheriskystrategythatwillleadtodefaultandbankruptcywhenpricesfallinstates=H.Unlikethesafebanks,theychoosealowlevelofyRandpromiseahighlevelofconsumptiontoearlywithdrawers.yR=0.016;dR=1.405.Bothofthesechoicesmakeithardertosatisfytheincentiveconstraint.TheconsumptionlevelsdeterminedbythesechoicesandbythebudgetconstraintsarecR,cR=(0.651,0.651)ifs=H;R1H2Hcs=cR,cR=(1.405,1.486)ifs=L.1L2LInthelowstate,thebankissolventandcangivebothearlyandlateconsumerswhatwaspromised.Inthehighstate,bycontrast,thebankgoesbankrupt 5.2TypesofEquilibrium139anddefaultsonitspromisesandtheearlyandlateconsumersbothreceivetheliquidatedvalueofthebank’sportfolio.Inequilibrium,individualsmustbeindifferentbetweendepositingtheirmoneyinasafeorriskybank.Otherwise,onetypeofbankwillattractnocustomers.Ifwecomputetheexpectedutilityofconsumptionfordepositorsinthesafeandriskybanks,wefindthatbothareequalto0.078.Tocheckthatthesechoicesareconsistentwithequilibrium,thefirstthingweneedtoshowisthatmarketsclearatdate1.Considerwhathappensinstates=L.Thesafebankscansatisfytheliquiditydemandoftheirdepositorsfromtheirownholdingsofthesafeasset.AfractionλL=0.8oftheircustomersareearlyconsumers,sothedemandforliquidityisλLdS=0.8×0.998=0.798.ThenthesafebankswilleachhaveyS−λLdS=0.822−0.798=0.024unitsofthegoodleftover.Sincetheproportionofsafebanksisρ=0.979,thetotalamountofexcessliquidityisρ(yS−λLdS)=0.979×0.024=0.023.SincePL=0.940<1theshortassetisdominatedbetweendate1anddate2,sothesafebankswillonlywanttoholdthelongasset.Accordingly,theysupplytheentire0.023unitsofthegoodinexchangeforthelongasset.Nextconsidertheriskybanks.Sincetheyholdnoneoftheshortasset,theymustsellpartoftheirholdingsofthelongassetinordertoprovidethepromisedliquiditytotheirearlyconsumers.ThedemandforliquidityfromtheircustomersisλLdR=0.8×1.405=1.124.Theyhave0.016fromtheirholdingoftheshortassetsotheyneed1.124−0.016=1.108.Sincetheproportionofriskybanksis1−ρ=0.021thetotaldemandforliquidityis(1−ρ)λLdR=0.021×1.108=0.023.Thusdemandforliquidityequalssupplywhenthestates=L.Instates=H,thesafebanks’supplyofliquiditycanbecalculatedinthesameway,simplytakingnotethattheproportionofearlyconsumersisnowλH=0.81.ThesupplyofliquidityisρyS−λSHd=(0.979)(0.822−0.81×0.998)=0.013.Theriskybankshave(1−yR)=0.987unitsofthelongasset.Becausetheyarebankrupt,theymustliquidatealltheirassetsandthismeanstheymustsupplythewholeamountofthelongassetatdate1.Implicitly,theyaredemandingPH=0.645unitsofliquidityinexchange.Sincethereare(1−ρ)=0.021riskybanks,thetotaldemandforliquidityis(1−ρ)P1−yR=0.021×0.645×0.987=0.013.HThus,theassetmarketclearsinstates=Htoo. 140Chapter5.FinancialFragilityToshowthatmarketsclearatdate0,itisnecessarytoshowthattheportfolioheldbyeachbankisoptimal.Thisrequiresustocheckthefirst-ordercondi-tionsthatshowtheeffectofasmallchangeintheportfolioonconsumptionineachstateandonexpectedutility.Thisisarathercomplicatedexercisethatwewillnotpursuehere.Theexamplehasshownthatasmallincreaseinε=λH−λLleadstoverydifferentpricebehaviorcomparedtothefundamentalequilibrium.Italsoleadstothepossibilityofbanksdefaultinginequilibrium.Aswehavearguedabove,substantialassetpricevolatilityisageneralpropertyofequilibriawithintrinsicuncertainty,howeversmallthatuncertaintymaybe.Thisleadsustoasktwoquestions.First,whatsortofequilibriumwouldweobserveifweletε>0convergetozero?Second,wouldthelimitofthissequenceofequilibriabeanequilibriumofthelimiteconomywhereε=0?Theanswertobothquestionsisthattheequilibriumwithintrinsicuncertaintyconvergestoasunspotequilibriumofthelimiteconomy.Inthissunspotequilibrium,pricesfluctuateeventhoughthereisnointrinsicaggregateuncertaintyintheeconomy.5.2.3SunspotequilibriaTodescribethelimitoftheequilibriawithintrinsicuncertainty,wereturntoExample1.Theparametersareidenticalbutanotherequilibriumisfound.TodistinguishthiscasewerefertoitasExample1S.Example1STheassetpriceisrandompHR=(0.432)(1.5)=0.648ifs=H;Ps=pLR=(0.943)(1.5)=1.415ifs=L.Theproportionofsafebanksisρ=1andtheirchoicesareyS=0.8;dS=1.Thesechoicesandtheequilibriumpricesdeterminetheconsumptionineachstateateachdate:cS,cS=(1.0,1.5)ifs=H;cS=1H2H(5.3)scS,cS=(1.0,1.5)ifs=L.1L2L 5.2TypesofEquilibrium141Notethatconsumptionisexactlythesameasinthefundamentalequilibrium.Theexpectedutilityobtainedfromthisconsumptionis,asweclaimedearlier,0.081.Riskybanksontheotherhand,continuetochoosealowinvestmentintheshortassetyRandpromisehighconsumptiontoearlywithdrawersdR:yR=0;dR=1.414andthecorrespondingconsumptioniscR,cR=(0.648,0.648)ifs=H;cR=1H2HscR,cR=(1.414,1.500)ifs=L.1L2LNowtheriskybanksinvestnothingintheshortassetandenterbankruptcyinthehighstates=H.Ifweweretocalculatetheexpectedutilityofthisconsumptionplan,wewouldfindthatitequalstheexpectedutilityofthesafebanks,0.081.Thus,consumersareindifferentbetweenthetwotypesofbanks,eventhoughinequilibriumtherearenoriskybankssinceρ=1.Itmayseemoddthatitisoptimalforbankstoadoptariskystrategyeventhoughnoneattempttodoso.Infact,thisisanecessaryrequirementforanyequilibriumthatisthelimitofequilibriaasεconvergestozerofromabove.Intheequilibriawithε>0,apositivefractionofthebankswererisky,implyingthatitmustbeoptimalforriskybankstooperate.Asεbecomesvanishinglysmall,sodoesthefractionofriskybanks,butitremainsoptimalinthelimitforriskybankstooperate,eventhoughnonechoosetodoso.Theabsenceofdefaultinthelimitequilibriumresultsfromthefactthat,aswesawinthefundamentalequilibrium,bankscanachievethefirstbestwithoutusingtheassetmarket.Theonlywayforeveryonetogetthefirstbestexpectedutilityisforallbankstochoosethesafestrategy.Eventhoughitisoptimalforasinglebank(ofnegligiblesize)toadopttheriskystrategy,ifapositivemeasureofbanksweretodoso,thedepositors’welfarewouldbereducedbelowthefirst-bestlevel.Thus,inthelimiteconomywithε=0,equilibriumrequiresρ=1.Attheallocationgivenin(5.3),depositorsbearnoriskandsoareapproxi-matelyriskneutralinthefaceofsmallrisks.Theywillbeindifferentbetweenholdingabitmoreorlessofeachassetifandonlyifthefollowingconditionissatisfied:RRπ+(1−π)=R.(5.4)PHPL 142Chapter5.FinancialFragilityTherighthandsizeisjusttheexpectedreturnatdate2toaninvestmentofoneunitinthelongassetatdate0.Thelefthandsideistheexpectedreturntothefollowingstrategy:investoneunitintheshortassetatdate0andusethereturnsatdate1tobuyasmuchofthelongassetaspossible.Withprobabilityπthehighstateoccurs,thepriceisPH,andthestrategyresultsinthepurchaseof1/PHunitsofthelongasset;withprobability1−πthelowstateoccurs,thepriceisPL,andoneobtains1/PLunitsofthelongasset.EachunitofthelongassetyieldsRatdate2sotheexpectedreturnfromtheinvestmentstrategyisequaltothelefthandsideof(5.4).Inotherwords,theexpectedreturnfromholdingtheshortassetisthesameastheexpectedreturnfromholdingthelongassetifandonlyif(5.4)holds.Itcaneasilybecheckedthat0.65×1.50.35×1.5+=1.5,1.4150.648sotheassetmarketdoesclearatdate1inthelimitingsunspotequilibrium.Therearemanyothersunspotequilibria.Infact,anypairofpricesbetween0and1.5correspondstoasunspotequilibriumifitsatisfies(5.4).Suchalargesetofpricevectorscanclearthemarketandsupportanequilibriumpreciselybecausethemarketisnotneededwhenthereisnoaggregateuncertainty.Theonlyfunctionofthepricesistomakesure(a)thatbanksarewillingtoholdtheappropriateamountsofbothassetsatdate0and(b)thatnoneofthemwanttousetheassetmarket.Condition(a)isguaranteedbythefirst-ordercondition(5.4)andcondition(b)isguaranteedbythefactthatbankshavejustenoughliquiditytopayofftheirearlywithdrawersatdate1andsohavenothingtotradeatdate1.ThefactthatPs0)thefundamentalequilibriumisnowrobust.Itisthelimitequilibriumasε0.Weconsiderthecasewithcompletemarketsandcontrastitwiththeincompletemarketscaseinthenextchapter.REFERENCESAllen,F.andD.Gale(1998).“OptimalFinancialCrises,”JournalofFinance53,1245–1284.Allen,F.andD.Gale(2000a).“FinancialContagion,”JournalofPoliticalEconomy108,1–33.Allen,F.andD.Gale(2000b).“OptimalCurrencyCrises,”CarnegieRochesterSeriesonPublicPolicy53,177–230.Allen,F.andD.Gale(2000c).“BubblesandCrises,”TheEconomicJournal110,236–256.Allen,F.andD.Gale(2004).“FinancialFragility,Liquidity,andAssetPrices,”JournaloftheEuropeanEconomicAssociation2,1015–1048.Azariadis,C.(1981).“Self-FulfillingProphecies,”JournalofEconomicTheory25,380–396.Bernanke,B.andM.Gertler(1989).“AgencyCosts,NetWorth,andBusinessFluctuations,”AmericanEconomicReview79,14–31.Bernardo,A.andI.Welch(2004).“FinancialMarketRuns,”QuarterlyJournalofEconomics119,135–158.Bryant,J.(1980).“AModelofReserves,BankRuns,andDepositInsurance,”JournalofBankingandFinance4,335–344.Calomiris,C.andG.Gorton(1991).“TheOriginsofBankingPanics,Models,Facts,andBankRegulation.”InFinancialMarketsandFinancialCrises,editedbyR.Hubbard.Chicago,IL:UniversityofChicagoPress.Calomiris,C.andJ.Mason(2000).“CausesofU.S.BankDistressDuringtheDepression.”NBERWorkingPaperW7919.Cass,D.andK.Shell(1983).“DoSunspotsMatter?”JournalofPoliticalEconomy91,193–227.Chari,V.andP.Kehoe(2000).“FinancialCrisesasHerds.”FederalReserveBankofMinneapolisWorkingPaper.Diamond,D.andP.Dybvig(1983).“BankRuns,DepositInsurance,andLiquidity,”JournalofPoliticalEconomy91,401–419.Farmer,R.(1999).TheMacroeconomicsofSelf-FulfillingProphecies.CambridgeandLondon:MITPress.Geanakoplos,J.(1990).“AnIntroductiontoGeneralEquilibriumwithIncompleteAssetMarkets,”JournalofMathematicalEconomics19,1–38.Gorton,G.(1988).“BankingPanicsandBusinessCycles,”OxfordEconomicPapers40,751–781. 152Chapter5.FinancialFragilityGottardi,P.andA.Kajii(1995).“GenericExistenceofSunspotEquilibria:TheRealAssetCase,”UniversityofPennsylvania,CARESSWorkingPaper95/12.Gottardi,P.andA.Kajii(1999).“TheStructureofSunspotEquilibria:TheRoleofMultiplicity,”ReviewofEconomicStudies66,713–732.Hart,O.(1995).Firms,ContractsandFinancialStructure.Oxford:OxfordUniversityPress.Kindleberger,C.(1978).Manias,Panics,andCrashes:AHistoryofFinancialCrises.NewYork,NY:BasicBooks.Kiyotaki,N.andJ.Moore(1997).“CreditChains,”JournalofPoliticalEconomy99,220–264.Lagunoff,R.andS.Schreft(2001).“AModelofFinancialFragility,”JournalofEconomicTheory99,220–264.Magill,M.andM.Quinzii(1996).TheoryofIncompleteMarkets,VolumeI.CambridgeandLondon:MITPress.Postlewaite,A.,G.MailathandL.Samuelson(2003).“SunkInvestmentsLeadtoUnpre-dictablePrices.”UniversityofPennsylvania:http://www.ssc.upenn.edu/˜apostlew/.Schnabel,I.andH.Shin(2004).“LiquidityandContagion:TheCrisisof1763,”JournaloftheEuropeanEconomicAssociation2,929–968.Shell,K.andA.Goenka(1997).“WhenSunspotsDon’tMatter,”EconomicTheory9,169–178. 6IntermediationandmarketsIntheprecedingchapterwestudiedfinancialfragilityfromthepointofviewofpositiveeconomics,thatis,tryingtounderstandthefactorsthatgiverisetofinancialcrisesandthereasonswhyafinancialsystemmightbesensitivetosmallshocks.Wenotedinpassingthatfinancialcrisesmightbeinefficient,butthiswasnotthefocusofouranalysis.Nowitistimetoturntonormativequestionsandtrytounderstandwhyfinancialcrisesarea“badthing.”Tounderstandwhyfinancialcrisesarea“badthing,”webeginbyaskingadifferentquestion“Underwhatcircumstancesarefinancialcrisesefficient?”Wetakethisindirectapproachforseveralreasons.•First,wewanttochallengetheconventionalwisdomthatfinancialcrisesarealwaysandeverywherea“badthing.”Itmaywellbetruethatfinancialcrisesimposesubstantialcostsontheeconomy.Certainlytherehavebeenmanyhistoricalepisodesinmanycountriesthatsuggestthecostsoffinancialcrisescanbeverysubstantial.Atthesametime,anyregulationofthefinancialsys-teminvolvescosts.Themostimportantofthesecostsarethedistortionsimposedonthefinancialsystembyaregulatoryregimethatrestrictswhatfinancialinstitutionsmayandmaynotdo.Tomeasurethecostsandbene-fitsofanypolicy,weneedtohaveaclearunderstandingoftheconditionsforefficiencyinthefinancialsystem,includingtheconditionsforefficientfinancialcrises.•Asecondreasonforstudyingtheconditionsunderwhichcrisesareefficientisthatknowledgeoftheseconditionsmaysuggesttechniquesformanagingcrisesandreducingtheircosts.Casualobservationsuggeststhatthecentralbankingtechniquesdevelopedoverthepasttwocenturiesaremainlytheresultoftrialanderror,withoutmuchrigoroustheorybehindthem.Thefirststepindevelopinganoptimalfinancialstabilitypolicyistocarryoutathoroughwelfareanalysisoffinancialcrises,includingtheconditionsforefficientfinancialcrises.•Athirdreasonforadoptingtheproposedapproachisthateconomistshaveawelldevelopedsetoftoolsforstudyingoptimaleconomicsystems.So 154Chapter6.IntermediationandMarketsitissomewhateasiertoaddressnormativequestionsbycharacterizingtheconditionsforefficiencyratherthanthereverse.Oneofthelessonsofthischapteristheimportantroleofmissingmarkets.Althoughitmaynothavebeenobviousatthetime,thepropertiesofthemodelstudiedinChapter5dependcruciallyontheabsenceofcertainfinancialmarkets,specifically,marketsforArrowsecurities.1Aswesaw,depositcontractscommitintermediariestoprovideeachdepositorwhowithdrawsatdate1afixedamountofconsumption,independentlyofthestateofnature.Ifthedemandforliquidityishigh,theonlywayfortheintermediarytoobtainenoughliquiditytomeetitscommitmentsisthroughassetsales.Fromthepointofviewoftheintermediary,sellingassetsisunfortunatefortworeasons.First,theintermediarymaybeforcedtodisposeofassetsat“fire-sale”prices.Depositorsreceivelowerpayoutsasaresult.Second,ifalargenumberofintermediariessellatthesametime,thesellingpressurewilldrivepricesdownfurther,forcingtheintermediariestounloadevenmoreassets,andworseningthecrisis.Thesetwoeffectstogetherexplaintheinefficiencyandseverityofthefinancialcrisis.Therearetwotypesofincompletenessintheprecedingchapter’sanalysisoffinancialfragility.Wesaythatacontractiscompleteiftheoutcomeis(inprinciple)contingentonallstatesofnature.Depositcontractsarenotcompleteinthissensebecausetheamountofconsumptionpromisedtowithdrawersatdate1isfixed,i.e.notcontingentonthestateofnature.WesaythatmarketsarecompleteiftherearemarketsonwhichintermediariescantradeArrowsecur-itiesforeachstateofnature.Thesemarketsallowintermediariestopurchaseliquiditycontingentonthestateofnature.IftherearenomarketsforArrowsecurities,thenmarketsareincomplete.Ofthetwotypesofincompleteness,itistheincompletenessofmarketsthataccountsfortheinefficiencyoffinancialcrises.Toseetheimportanceofmissingmarkets,supposethatmarketsforArrowsecuritieswereintroducedtothemodeloffinancialfragility.Ifanintermediaryanticipatedashortageofliquidityinaparticularstateofnature,itwouldbepossibletopurchaseArrowsecuritiesthatpayoffinthatstateinordertoprovideextraliquidityandavoidtheneedforassetsales.Thiswouldcutthelinkbetweenthedemandforliquidityandthesaleofassets.Asaresult,thepricingofassetswouldbeinsulatedfromliquidityshocksandthedestabilizingeffectsofincompletecontractswouldbereducedifnotavoidedaltogether.1AnArrowsecurityisapromisetodeliveroneunitofaccount(one“dollar”)ifaspecifiedstateofnatureoccursandnothingotherwise.TheconceptsofArrowsecuritiesandcompletemarketsarereviewedinChapter2. 6.1CompleteMarkets155AnotherbenefitofmarketsforArrowsecuritiesisimprovedrisksharing.Theyallowtheintermediarytopayforliquidityinthestatewhereitisneededbysellingliquidityintheotherstate.Ineffect,theintermediaryistransferringwealthfromastatewherethemarginalutilityofconsumptionislowtoastatewhereitishigh.Thisiswhatefficientrisksharingrequires.Assetsales,bycontrast,forcetheintermediarytoreduceconsumptioninthestatewheremarginalutilityisalreadyhigh,increasingthevariationinconsumptionacrossstatesandresultingininefficientrisksharing.Intherestofthischapter,weexploretheimplicationsofcompletemarketsfortheefficiencyoftheincidenceoffinancialcrises.Undercertainadditionalassumptions,whichareanalogoustotheassumptionsofthefundamentaltheoremsofwelfareeconomics,weshallshowthatcompletemarketsguaranteetheefficiencyoflaisser-faireequilibrium.Inthiscontextatleast,afinancialcrisisdoesnotrepresentamarketfailureand,hence,doesnotprovideareasonforgovernmentinterventionorregulation.6.1COMPLETEMARKETSSincethetimeofAdamSmith,economistshavebeenfascinatedwiththeprop-ertiesofadecentralizedmarketsystem.Overthelasttwocenturies,theyhaverefinedtheirtheoreticalunderstandingoftheconditionsthatmustbesatisfiedifthemarketallocationistobeefficient.Theconditionsarenotinnocuous:•marketsmustbeperfectlycompetitive;•theremustbenoexternalities;•theremustbenoasymmetricinformation(moralhazardoradverseselection);•theremustbenotransactioncosts;•andmarketsmustbecomplete.Fromourperspective,thecriticalconditionisthecompletenessofmarkets.Technically,wesaythatmarketsarecompleteifitispossibletotrade,atasinglepointintime,allthecommoditiesthatwilleverexist.Indefiningcommod-ities,wedistinguishgoodsthataredeliveredatdifferentdatesorindifferentplacesasdifferentcommodities.Wealsodistinguishcontingentcommodities,thatis,goodswhosedeliveryiscontingentontheoccurrenceofanuncertainevent.Inorderformarketstobecomplete,itmustbepossibletotradeallthecommodities,so-defined,atasingledate.Itisbothastrengthandaweaknessofthetheorythatthedefinitionofcommoditiesisverybroad.Ontheonehand,theassumptionofcompletemarketsmayseemlikea“tallorder.”Mosteconomistswouldnotclaimthat 156Chapter6.IntermediationandMarketsmarketsareevenapproximatelycompleteandthisisaweaknessofthetheory.2Ontheotherhand,theassumptionofcompletemarketsiscrucialbecauseitallowsustoextendthetheorytocovermanynewphenomenajustbydefin-ingcommoditiesappropriately.Theeconomicprinciplesthatweredevelopedoriginallytoexplainexchangeofordinarygoodsinspotmarketscanbeusedtostudyallocationdecisionsacrossspaceandtimeandinthefaceofuncertainty.Conversely,wecancharacterizemanymarketfailuresasexamplesof“missingmarkets.”Finally,theefficiencyofeconomieswithcompletemarketssuggeststhattheinventionofnewmarketsmayprovidearemedyforcertainmarketfailures.Togetanideaoftheroleofthecompletemarketsassumptionintheanalysisofefficientmarketsandlaythegroundworkforourlateranalysisoffinancialcrises,itishelpfultoconsiderasimpleexampleandaskwhatacompletesetofmarketswouldlooklikeinthiscase.CommoditiesAsusual,weassumetherearethreedates,t=0,1,2andasingle,all-purposegoodateachdate.Therearetwostatesofnature,denotedbys=H,L.Atdate0thestateisunknown,althoughindividualsknowthetrueprobabilityπsofeachstates.Thetruestateisrevealedatthebeginningofdate1.Sincethestateisunknownatdate0,wecannotmakethedeliveryofthegoodcontingentonthestateatdate0;sothereisasinglecommodityatdate0,thegooddeliveredindependentofthestateatdate0.Atdatest=1,2,thestateisknownwithcertaintysowecandefinecontingentcommoditiescorrespondingtoeachstate,thegooddeliveredatdatet=1,2instates=H,L.Thus,therearefivecommoditiesinall,thesinglenoncontingentcommodityatt=0andthefourcontingentcommodities(t,s)=(1,H),(2,H),(1,L),(2,L)atthesubsequentdates.ThiscommodityspaceisillustratedinFigure6.1.ConsumptionSupposethataconsumerhasanendowmentofoneunitofthegoodatdate0andnothingatdates1and2.Intermsofourdefinitionofcommodities,the2Marketscanbeeffectivelycompleteeveniftheredoesnotexistacompletesetofmarketsforallpossiblecontingentcommodities.Dynamictradingstrategiesusingalimitedsetofsecuritiesallowinvestorstosynthesizeamuchlargersetofderivatives.Itispossiblethatinsomecircum-stancessuchstrategiescaneffectivelycompletethesetofmarkets.Whetheragoodapproximationtocompletemarketsisachievedisultimatelyanempiricalissuewhichweareunabletoresolvehere;butwebelievethatincompletenessisanimportantissueevenforfinancialinstitutionsandsophisticatedinvestors. 6.1CompleteMarkets157Commodity(1,H)Commodity(2,H)s=HCommodity0s=LCommodity(1,L)Commodity(2,L)t=0t=1t=2Figure6.1.Commodityspacewithtwostatesandthreedates.endowmentconsistsofavectore=(1,0,0,0,0),indicatingthattheconsumerhasoneunitofthesinglecommodityatdate0andnoneofthecontingentcommoditiescorrespondingtodates1and2andthestatesHandL.Wesupposeasusualthattheconsumeronlyvaluesconsumptionatdates1and2andthathispreferencesarerepresentedbyaVNMutilityfunctionU(c1)+βU(c2),wherec1denotesconsumptionatdate1andc2denotesconsumptionatdate2.Ifwedenoteconsumptionatdatetinstatesbyctsthentheconsumer’sconsumptionbundleisdescribedbyavectorc=(0,c1H,c2H,c1L,c2L)andtheexpectedutilityofthisbundleisπH{U(c1H)+βU(c2H)}+πL{U(c1L)+βU(c2L)},whereπsdenotestheprobabilityofstatesoccurring.Sincemarketsarecomplete,theconsumercanpurchaseanyconsumptionbundlec=(0,c1H,c2H,c1L,c2L)thatsatisfieshisbudgetconstraintatdate0.Sinceonlyrelativepricesmatter,thereisnoessentiallossofgeneralityinchoosingthegoodatdate0asthenumeraireandsettingitspricep0equaltoone.Lettingptsdenotethepriceofoneunitofthecontingentcommodity(t,s),wecanwritetheconsumer’sbudgetconstraintasp1Hc1H+p2Hc2H+p1Lc1L+p2Lc2L≤p01=1.Therighthandsideisthevalueoftheconsumer’sendowmentandthelefthandsideisthevalueofhisconsumptionbundle.ThislooksjustlikethestandardbudgetconstraintandthatisoneofthegreatstrengthsoftheArrow–Debreumodel:byreinterpretinggoodsatdifferentdatesandindifferentstatesofnatureasdifferentcommodities,wecanextendthestandardbudgetconstraintandindeedthetheoryofcompetitiveequilibriumtodealwithuncertaintyandtime. 158Chapter6.IntermediationandMarketsProductionInthesameway,wecanreinterpretthestandardtheoryofproductiontoexplaintheallocationofinvestment.Theproductiontechnologyforthisecon-omyconsistsoftheinvestmentopportunitiesprovidedbythetwoassets,theshortassetandthelongasset.Asusual,oneunitofthegoodinvestedintheshortassetatdatetproducesoneunitatdatet+1,wheret=0,1,andoneunitinvestedinthelongassetatdate0producesRs>1unitsofthelongassetinstatesatdate2;however,sincetheseinvestmenttechnologiesproducecontingentcommodities,somecareisrequiredintheirinterpretation.Let’sstartwiththeshortasset.Asingleunitinvestedatdate0producesoneunitofthegoodatdate1,independentlyofthestate.Intermsofourdefinitionofcommodities,oneunitofthegoodatdate0producesoneunitofthecontingentcommodity(1,H)andoneunitofthecontingentcommodity(1,L).Wecanrepresentthistechnologybytheproductionvectora0=(−1,1,0,1,0),wheretheentry−1denotesaninputofoneunitatdate0andtheotherentriesdenoteoutputsofcontingentcommodities.Investmentintheshortassetatdate1issimilar,exceptthatthetruestateisalreadyknown,soboththeinputandtheoutputarecontingentonthestate.Forexample,ifoneunitisinvestedatdate1instateH,theinputconsistsofoneunitof(1,H)andtheoutputconsistsofoneunitof(2,H).Thus,wehavetwotechnologiesatdate1,oneforeachstate,representedbytheproductionvectorsa1H=(0,−1,1,0,0)anda1L=(0,0,0,−1,1).Thelongassetrepresentsatechnologythatcanonlybeusedatdate0,soitissimplertoanalyze.Oneunitinvestedatdate0producesRs>1unitsatdate2instates,sotheoutputisabundleofcontingentcommodities,namely,RHunitsof(2,H)andRLunitsof(2,L).Thistechnologycanberepresentedbythevectora2=(−1,0,RH,0,RL).Sinceeachoftheseproductiontechnologiesoperatessubjecttoconstantreturnstoscale,inacompetitiveequilibriumtheprofitsderivedfromeach 6.1CompleteMarkets159processwillbezero.Ifprofitswerepositive,aprofit-maximizingfirmwouldbetemptedtooperateataninfinitescale,whichisinconsistentwithmarketclearing.Iftheprofitsperunitwerenegative,thefirmwouldshutdownandearnzeroprofits.Sothefirmcanonlyoperateatapositivescaleiftheprofitperunitiszero.Becauseequilibriumprofitsarezero,itdoesnotmatterwhoundertakesthedifferentproductionactivities.Wecanassumethatarepresen-tativefirmdoesthisorwecanassumethatindividualconsumersdoit.Withcompletemarketstherearemanydegreesoffreedom.Herewewillassumethatarepresentativefirmmakesallproductiondecisions.Therepresentativefirmchoosesalevelofinvestmentineachoftheactivitiesdescribedabove.Lety0denoteinvestmentintheshortassetatdate0,lety1sdenoteinvestmentintheshortassetatdate1instates=H,L,andletxdenoteinvestmentinthelongassetatdate0.Eachactivitymustyieldnon-positiveprofits,so−1+p1H+p1L≤0,(6.1)withequalityify0>0,−p1s+p2s≤0,(6.2)withequalityify1s>0fors=H,L,and−1+p2HRH+p2LRL≤0,(6.3)withequalityifx>0.Theseconditionsarenecessaryandsufficientforprofitmaximization.Oneinterestingpointtonoteisthat,eventhoughthetruestateisunknownwhenallthedecisionsaremadeatdate0,thefirmdoesnotfaceanyuncertainty.Thisisbecausethefirmbuysandsellscontingentcommoditiesatknownpricesatdate0.Forexample,inthecaseofinvestmentintheshortassetatdate0,thefirmbuysoneunitofthegoodatdate0asaninputandsellsoneunitofeachofthecontingentcommoditiesatdate1.Thecostoftheinputis1andtherevenuefromsellingtheoutputsisp1H+p1L,sotheprofitisp1H+p1L−1.Thisprofitis“realized”atdate0,whenthecommoditiesaretraded,ratherthanatdate1whentheoutputsareproduced.EquilibriumTherequirementsforcompetitiveequilibriumarethat,atthegivenprices,(a)eachconsumerchoosesaconsumptionbundlethatmaximizeshisexpectedutility,subjecttohisbudgetconstraint; 160Chapter6.IntermediationandMarkets(b)therepresentativefirmchoosesinvestmentsinthevariousactivitiestomaximizeprofits;and(c)thechoicesoftheconsumersandthefirmareconsistentwithmarketclearing(i.e.demandequalssupply).Forthepurposesofillustration,letusassumetherearetwotypesofconsumers,AandBwithpreferencesdescribedbyVNMutilityfunctionsUA(c1)+βUA(c2)andUB(c1)+βUB(c2).Formally,acompetitiveequilib-riumconsistsofapricevectorp∗=(1,p∗,p∗,p∗,p∗),aconsumption1H2H1L2Lbundleci∗=(0,ci∗,ci∗,ci∗,ci∗)forconsumersi=A,Bandavectorof1H2H1L2LinvestmentsI∗=(y0∗,y∗,y∗,x∗)suchthat1H1L(a)ci∗maximizesconsumeri’sexpectedutilitysubjecttohisbudgetconstraint,foreachi=A,B;(b)I∗satisfiesthezeroprofitconditions;and(c)allmarketsclear.Sincealltradingoccursatdate0,marketclearingisachievedatdate0aswell.Theactualdeliveryofgoodstakesplacelater,butthedecisionshavealreadybeenmadeatdate0andtheymustbeconsistent.Themarketforthegoodatdate0willclearifx∗+y∗=2;0thetwoconsumerssupplytheirone-unitendowmentsandthefirmdemandsinputsx∗andy∗forinvestmentinthelongandshortassets,respectively.At0date1,therearetwocontingentcommodities,oneforeachstates.Forthecorrespondingmarketstoclearatdate0requirescA∗+cB∗+y∗=y∗,1s1s1s0foreachs=H,L;herethefirmsuppliesy0∗unitsofthegood(thepayofffromtheinvestmentintheshortasset)anddemandsy∗unitstoinvestintheshort1sassetandtheconsumerdemandsc∗unitstoconsume.Atdate2,thereare1sagaintwocontingentcommodities,oneforeachstates.Forthecorrespondingmarketstoclearatdate0requirescA∗+cB∗=y∗+R∗2s2s1ssx,foreachs=H,L;thefirmsuppliesy1∗s+Rsx∗unitsofthegood(thereturnoninvestmentinthelongassetatdate0andtheshortassetatdate1)andtheconsumersdemandcA∗+cB∗unitsofthegoodforconsumption.2s2s 6.1CompleteMarkets161Example1Toillustratethemodelinthesimplestpossibleway,consideraRobinsonCrusoeeconomyinwhichthereisasingletypeofconsumerwithendowmente=(1,0,0)andpreferencesgivenbythefamiliarCobb–DouglasversionoftheVNMutilityfunctionU(c1)+βU(c2)=lnc1+lnc2.AssumethatthestatesareequiprobableπH=πL=0.5andthereturnsonthelongassetare(RH,RL)=(3,0).Sincethereisasinglerepresentativeconsumer,thereisauniquePareto-efficientallocationforthiseconomy,namely,theallocationthatmaximizestherepresentativeconsumer’sexpectedutility.ThefirstfundamentaltheoremofwelfareeconomicsensuresthatacompetitiveequilibriumisParetoefficient.Usingthisfact,wecansolveforthecompetitiveequilibriumintwostages.First,wesolvefortheuniqueefficientallocation.Second,wefindthepricesthatsupportthisallocationasanequilibrium.Supposethataplannerchoosesanattainableallocationthatmaximizestheexpectedutilityoftherepresentativeconsumer.Heinvestsinxunitsinthelongassetandyunitsintheshortasset.Anecessaryconditionformaximizingexpectedutilityisthat,ineachstates,theconsumptionbundle(c1s,c2s)maximizestheconsumer’sutilitylnc1+lnc2subjecttothefeasibilityconditionsc1s≤y,andc1s+c2s≤y+Rsx,fors=H,L.Theallocationofconsumptionbetweendates1and2dependsonthevalueofRs,thereturntothelongasset.IfRsishigh(Rsx>y),itisoptimaltoputc1=yandc2=Rsx.IfRsislow(Rsx1unitsofthegoodatdate2foreveryunitinvestedatdate0.Asusual,thereisacontinuumofeconomicagentsatdate0,eachofwhomhasanendowmentofoneunitofthegoodatdate0andnothingatfuturedates.Atdate1,eachagentlearnswhetherheisanearlyconsumer,whoonlyvaluesthegoodatdate1,oralateconsumer,whoonlyvaluesthegoodatdate2.Theprobabilityofbeinganearlyconsumeris0<λ<1.Expost,thefractionofearlyconsumersintheeconomyisassumedtobeλaswell.Themaindifferencebetweenthecurrentmodelandthoseusedinthepre-cedingchaptersliesinthespecificationofuncertainty.Weassumethattheeconomyisdividedintotworegions,labeledAandB.Exantethetworegionsareidentical,withthesamenumberofidenticalagentsandthesameassets. 6.2IntermediationandMarkets165Therearetwoaggregatestatesofnature,denotedbyHLandLH.Eachstateisequallylikely,thatis,eachoccurswithprobability0.5.InstateHLthefractionofearlyconsumersinregionAisλHandthefractionofearlyconsumersinregionBisλL,where0<λL<λH<1.InstateLHthefractionsarereversed.Weassumethat1λ=(λH+λL),2sothatthefractionofearlyconsumersineachstateisλ.Thisalsoimpliesthattheprobabilityofanyinvestorbecominganearlyconsumerisalsoλ.Theinvestors’attitudestowardriskarerepresentedbyacommonVNMutilityfunction.Ifaninvestorconsumescunitsofthegoodattheappropriatedate,hisutilityisU(c),whereU(·)satisfiesalltheusualproperties.Alluncertaintyisresolved,asusual,atthebeginningofdate1whenthetruestateHLorLHisrevealedandeachinvestorlearnshistype,earlyorlate.TheuncertaintyandinformationstructureareillustratedinFigure6.3.λA=λH>λB=λLs=(H,L)s=(L,H)λA=λL<λB=λHt=0t=1t=2Figure6.3.LiquidityshocksofgroupsAandBinstates(H,L)and(L,H).6.2.1EfficientrisksharingSupposeacentralplannerwasresponsibleformakingthedecisionsinthiseconomy.Wouldthedivisionofinvestorsintotworegionsmatter?Evidentlynot.Theinvestorsareexanteidenticalandthenumberofearlyconsumersisthesameineachstate,sothereisnoreasonwhytheplannershouldpayanyattentiontotheregiontowhichaninvestorbelongs.Theplannerwillallocateafractionyoftheendowmenttotheshortassetandafraction1−ytothelongassetandofferc1unitsofthegoodatdate1totheearlyconsumersandc2unitsofthegoodatdate2tothelateconsumers,independentlyofthestate.Thesevariablesarechosentomaximizetheexpectedutilityoftherepresentative 166Chapter6.IntermediationandMarketsinvestorλU(c1)+(1−λ)U(c2),subjecttotheusualfeasibilityconditionsλc1=yand(1−λ)c2=(1−y)R.Asusual,thefirstbestischaracterizedbythefeasibilityconditionsandbythefirst-orderconditionU(c1)=RU(c2).Sincethefirst-orderconditionimpliesthatc1λc1=y.Thereisnotenoughliquidityatdate1toprovidetheearlyconsumerswiththepromisedamountofconsumption.Similarly,ifthefractionofearlyconsumers 6.2IntermediationandMarkets167islow,then(1−λL)c2>(1−λ)c2+(λ−λL)c1=(1−y)R+y−λLc1.Theintermediaryhastoomuchliquidityatdate1andnotenoughtodistributeatdate2.So,clearly,theintermediarycannotachievethefirstbestinautarky.6.2.2EquilibriumwithcompletefinancialmarketsTheintermediary’sproblemiseasilysolvedifweintroducemarketsinwhichtheintermediarycaninsureagainstgettingahighliquidityshock.Sincethereisnoaggregateuncertainty,wheneversomeintermediariesgetahighliquidityshockandhencehavetoolittleliquiditytogivetheirdepositorsthefirst-bestconsumptionc1,therewillbeintermediariesintheotherregionwithalowliquidityshockandtoomuchliquidityatdate1.Theseintermediarieswouldbehappytolendtotheliquidity-constrainedintermediariesinordertogetmoreconsumptionatdate2whentheywillneedit.Inordertoimplementthefirstbest,weneedcompletemarkets.Therearefivecontingentcommodities,thegoodatdate0,thegoodatdate1instatesHLandLH,andthegoodatdate2instatesHLandLH.Takingthegoodatdate0asthenumeraire(p0≡1),letptsdenotethepriceofthegoodatdatetinstates,wheret=1,2ands=HL,LH.Theno-arbitrageconditionfortheshortassetatdate0impliesthatp1HL+p1LH=1.Thesymmetryofthetwostatessuggeststhatthereexistsasymmetricequilibriuminwhichp1HL=p1LH.Puttingtogetherthesetwoconditionsgives1p1HL=p1LH=.2Similarly,weassumethatp2HL=p2LHandusetheno-arbitrageconditionforthelongasset,p2HL+p2LHR=1,toconcludethat1p2HL=p2LH=.2RThenwithoutriskofconfusionwecanwritep1forthepriceofthegoodatdate1andp2forthepriceofthegoodatdate2ineitherstate.SupposetheintermediaryinRegionApromisesaconsumptionprofile(c1,c2)thatis 168Chapter6.IntermediationandMarketsindependentofthestate.Theintermediary’sbudgetconstraintisp1HLλHc1+p1LHλLc1+p2HL(1−λH)c2+p2LH(1−λL)c2=1⇐⇒p1(λH+λL)c1+p2(1−λH+1−λL)c2=11⇐⇒λc1+(1−λ)c2=1.RTheanalogouscalculationforintermediariesinRegionByieldsanidenticalbudgetconstraint.Theneachintermediaryischoosingaprofile(c1,c2)tomaximizetheexpectedutilityλU(c1)+(1−λ)U(c2)subjecttothebudgetconstraintaboveandthisrequiresthatthefirst-orderconditionU(c1)=RU(c2)besatisfied.Inotherwords,thefirst-bestconsumptionprofilesatisfiestheintermediary’sbudgetconstraintandsatisfiesthefirst-ordercondition,henceisoptimal.Itisalsoeasytocheckthatthezero-profitconditionsaresatisfied.Thisallocationisfeasiblefortheeconomy,aswehavealreadyseeninouranalysisoftheplanner’sproblem.Somarketswillclearwheneachintermediarychoosesthefirst-bestconsumptionprofileandthecorrespondingproductionplan.Thus,completemarketsallowpreciselythetransfersbetweenstatesthatarenecessaryforfirst-bestrisksharing.Example2Intheprecedingsketchoftheintermediary’sproblemweassumedthatconsumersreceivedthesameconsumptionindependentlyofthestate.Wewillobtainthisconditionaspartofthesolutiontotheintermediary’sproblemusingthefollowingparameters:R=3;1−5U(c)=−c;5λH=0.6,λL=0.4;πHL=πLH=0.5.Supposethat1p1HL=p1LH=p1=2and1p2HL=p2LH=p2=.6 6.2IntermediationandMarkets169WelookattheproblemfromtheperspectiveofanintermediaryinRegionAandlet(c1H,c2H)(resp.(c1L,c2L))denotetheconsumptionprofilepromisedwhenthefractionofearlyconsumersisλH(resp.λL)forthatintermediary.Theintermediaryneedstomaximizetherepresentativedepositor’sexpectedutility1{0.6U(c1H)+0.4U(c2H)+0.4U(c1L)+0.6U(c2L)}2subjecttotheintermediary’sbudgetconstraintp1(0.6c1H+0.4c1L)+p2(0.4c2H+0.6c2L)=1.Thefirst-orderconditionsforthisproblemareµU(c1H)=U(c1L)=µp1=2µU(c2H)=U(c2L)=µp2=6whereµistheLagrangemultiplierforthebudgetconstraint.Substitutingtheformulaformarginalutilityintothefirst-orderconditionsweget−6−6µ(c1H)=(c1L)=µp1=2−6−6µ(c2H)=(c2L)=µp2=.6Thenc1H=c1L=c1andc2H=c2L=c2,thatis,consumptionisindependentofthestate,asweshouldexpectinanefficientallocation,andwecansolvethesefirst-orderconditionsfortheoptimalconsumptionratioc2√6=3=1.201.c1Usingtherelationshipbetweenc1andc2andthebudgetconstraintwecansolveforp1(0.6c1+0.4c1)+p2(0.4c2+0.6c2)=p1c1+p2c211=c1+c22611=c1+(1.201)c1=1.26 170Chapter6.IntermediationandMarketsThisequationcanbesolvedforc1c1=1.428andthenweusetheoptimalratiotofindc2c2=1.201c1=(1.201)(1.428)=1.715.6.2.3AnalternativeformulationofcompletemarketsAnalternativetoassumingacompletesetofmarketsforcontingentcommod-itiesatdate0istoallowtradetooccursequentially.AsdescribedinSection6.1,marketsaresequentiallycomplete,inthesimple,two-state,three-periodmodel,iftherearetwoArrowsecuritiesatdate0andspotandforwardmarketsforthegoodatdate1.Wefirstdescribethespotandforwardmarketsatdate1,afteralluncertaintyhasbeenresolved,andthenwedescribetheArrowsecuritymarketsatdate0.Atdate1weassumethereisaforwardmarketinwhichintermediariescantradethegoodatdate1forpromisestodeliverthegoodatdate2.Letpdenotethevalueofoneunitofthegoodatdate2intermsofunitsofthegoodatdate1.Inotherwords,pisthepresentvalueofoneunitofthegoodatdate2.SinceoneunitofthelongassetyieldsRunitsofthegoodatdate2,thevalueofoneunitoftheassetatdate1willbepR.Supposethatanintermediaryhasaportfolio,includingtradesinArrowsecuritiesmadeatdate0,thatisworthwsinthestatewheretheliquidityshockisλs,s=H,L.Theintermediary’sbudgetconstraintisλsc1+p(1−λs)c2=wsandtheintermediarywillchoose(c1,c2)tomaximizeλU(c1)+(1−λs)U(c2)subjecttothisbudgetconstraint.LetV(p,ws;λs)denotethemaximizedvalueofthisutilityfunction.InSection6.1weassumedthattheforwardpricepwasstate-dependent.Herethetwostatesaredifferentfromthepointofviewofanindividualinter-mediarybutidenticalfromamacroeconomicpointofview.Inotherwords,theproportionsofearlyconsumersintheintermediaryvaryfromstatetostatebuttheproportionofearlyconsumersintheeconomyasawholeisthesame.Sincethetwostatesthatcanoccuratdate1areidenticalfromanaggre-gatepointofview,weconsiderasymmetricequilibriuminwhichtheprice 6.2IntermediationandMarkets171pisindependentofthestate.Thismeansthatthereturnstothetwoassetsarecertainatdate0.Inordertopersuadeintermediariestoholdbothassets,theone-periodreturnsmustbeequal,which,aswehaveseeninChapter4,requiresp=1/R.Thenthevalueoftheintermediary’sportfolioatdate1isindependentoftheamountinvestedintheshortandlongassets.Nowconsidertheintermediary’sproblematdate0.AnArrowsecurityatdate0promisesdeliveryofoneunitofthegoodinonestateatdate1andnothingintheotherstate.Sinceweareconsideringasymmetricequilibrium,thepricesofthetwosecuritiesshouldbeequal,andwithoutlossofgeneralitywecannormalizethemtoequal1.LetzsbetheamountoftheArrowsecuritythatpaysoffinthestatewheretheintermediary’sliquidityshockisλs.Notethatsreferstotheintermediary’sstate,nottheaggregatestate,butsincethetwoareperfectlycorrelated,thisshouldnotleadtoanyconfusion.Withoutlossofgenerality,wecanassumethattheintermediaryinvestsallofitsdepositsintheshortandlongassets,sothetradesinArrowsecuritiesmustbalance,thatis,zH+zL=0.Inotherwords,theintermediarybuysArrowsecuritiesinonestateandsellstheminanother.Thevalueoftheintermediary’sportfolioatdate1willbews=y+p(1−y)R+zs=1+zsfors=H,L.Clearly,theintermediarycanattainanypatternofdate-1payoffs(wH,wL)thatsatisfywH+wL=2.SotheintermediaryshouldchooseaportfolioofArrowsecurities(zH,zL)tomaximizetheexpectedvalueoftheindirectutilityfunctionV(p,ws;λs)ssubjecttothebudgetconstraint1(wH+wL)=1.2Thesolutionofthisproblemimpliesthatthemarginalutilityofconsumptionmustbethesameineachstate,thatis,wH=wL,whichinturnimpliesthattheconsumptionallocation(c1,c2)willbethesameineachstate.Theimportanceofthisalternativeapproachusingsequentialtradeisthatitmakesclear(a)thatcompletemarketsforcontingentcommoditiesarenot 172Chapter6.IntermediationandMarketsstrictlynecessaryformarketstobeeffectivelycompleteand(b)onecanachievethesameresultswithfewermarkets.Ingeneral,ifthereareSstatesandT+1periodsanduncertaintyisresolvedatdate1,thereareST+1contingentcommoditiesandhenceST+1marketsintheArrow–Debreumodel.Thesequentialtrademodel,bycontrast,requiresSArrowsecuritiesandmarketfortheT+1datedcommoditiesineachstate,sothenumberofmarketsisS+T+1.Whenthenumberofstatesanddatesislarge(ortherearemanyphysicalgoodsineachdateandstate),thedifferencebetweenthetwoapproachesbecomesevengreater.6.2.4ThegeneralcaseThetwo-statemodeldevelopedinSection6.2isquitespecial,notonlybecauseitisrestrictedtotwostates,HLandLH,butespeciallyasregardsthelackofaggregateuncertainty.Theargumentbasedonthisspecialcaseisquitegeneral,however.AllenandGale(2004)presentageneralmodelofaneconomywithfinancialmarketsforaggregaterisksandintermediariesandestablishsimilarresults.ItisbeyondthescopeofthischaptertodomorethangivesomeoftheflavoroftheAllen–Galeanalysis.Wecandothisbyextendingthetwo-statemodeltoamoregeneralenvironmentwithaggregateuncertaintyaboutbothassetshocksandliquidityshocks.Tosimplifythenotation,wecontinuetoassumethattherearetworegions,AandB,thattheprobabilityofbeinganearlyconsumermaybedifferentineachregion,butthattheassetreturnsarethesameineachregion.Thereisassumedtobeafinitenumberofstatesindexedbys=1,...,Sandeachstatesoccurswithprobabilityπs>0.Therearethreedates,t=0,1,2,thestateisunknownatdate0andthetruestateisrevealedatthebeginningofdate1.Atdate1,eachagentlearnswhetherheisanearlyorlateconsumer.TheutilityofconsumptionisrepresentedbyacommonVNMutilityfunctionU(c).Theprobabilityofbeinganearlyconsumercandependonboththestatesandtheregioni.Weletλi(s)denoteboththeprobabilityofbeinganearlyconsumerandtheproportionofearlyconsumersinregioniinstates.Therearetwoassets,shortandlong.Thereturntotheshortassetisalwaysone,independentlyofthestate,butthereturntothelongassetmaydependonthestate.WeletR(s)denotethereturntooneunitinvestedinthelongassetifthestateiss.Marketsplayanessentialroleonlyifthereisheterogeneityamongintermedi-aries;otherwisetherearenogainsfromtrade.WefocusontheextremecasewheretherearedistinctintermediariesforregionsAandB.Anintermediaryoftypeidrawsallitsdepositorsfromregioni.Sincetheproportionofearly 6.2IntermediationandMarkets173consumerscanbedifferentineachregion,intermediariescaninsuretheriskofliquidityshortagesbytradingonmarketsforcontingentcommodities,buyingliquidityinstateswhereitexpectshighdemandandsellingitinstateswhereitexpectslowdemandforliquidity.Thesharingofriskisintermediatedbythemarkets,sotherisksharingacrossregionsisnotimmediatelyapparent.Whattheintermediaryappearstobedoingisdemandinganoptimalconsumptionprofileforitsdepositors,subjecttoabudgetconstraint;however,theeffectisthesameasiftheintermediariesindifferentregionswerewritingoptimalrisk-sharingcontractswitheachother,promisingtosharetheavailableliquidityinthewayacentralplannerwould.Theintermediarytakesanendowmentofoneunitfromeachagentatdate0andoffersinexchangec1(s)unitsofthegoodinstatestoeveryonewhowith-drawsatdate1andc2(s)unitsofthegoodinstatestoeveryonewhowithdrawsatdate2.Sincetheintermediarycannottellwhetherthepersonwithdraw-ingisanearlyoralateconsumer,theintermediaryhastoensurethatlateconsumershavenoincentivetopretendthattheyareearlyconsumersandwithdrawatdate1.Soweassumethateveryconsumptionplansatisfiestheincentiveconstraintc1(s)≤c2(s)fors=1,...,S.(6.4)Aconsumptionplanc={(cS1(s),c2(s))}s=1specifiesaconsumptionprofilec(s)=(c1(s),c2(s))foreverystates.Theconsumptionplaniscalledincentive-compatibleifitsatisfiestheincentiveconstraint(6.4).Thereisasinglecommodityatdate0andacontingentcommodityforeverystates=1,...,Sanddatet=1,2.Letthecommodityatdate0bethenumeraireandletp1(s)denotethepriceofthegoodinstatesatdate1andp2(s)denotethepriceofthegoodinstatesatdate2.Theintermediaryisassumedtobeabletotradeallthecontingentcommoditiesatdate0.Theintermediaryreceivesoneunitofthecommodityatdate0andusesthistopurchasethecontingentcommoditiesthatithaspromisedtothedepositors.Theintermediary’sbudgetconstraintcanbewrittenasSp1(s)λic1(s)+p2(s)(1−λi(s))c2(s)≤1,(6.5)s=1wherethelefthandsideisthecostoftheconsumptionplanandtherighthandsideisthevalueofasingleagent’sdeposit.Freeentryandcompetitionrequiretheintermediarytomaximizetheexpectedutilityoftherepresentativedepositor,sotheintermediary’sdecisionproblemistochooseanincentivecompatibleconsumptionplanc={c(s)}to 174Chapter6.IntermediationandMarketsmaximizetheexpectedutilityoftherepresentativedepositorSπs{λi(s)U(c1(s)+(1−λi(s)U(c2(s))}s=1subjecttothebudgetconstraint(6.5).Asbefore,theexistenceofacompletesetofmarketsmakesthephysicalassetsredundantinthesensethatanintermediarydoesnotneedtoholdthem.Someonemustholdthem,however,toproduceoutputsofthegoodsatdates1and2butwecanassumethatarepresentativefirmchoosestoinvesty0intheshortassetatdate0,xinthelongassetatdate0andy1(s)intheshortassetatdate1instates.Sincetheseinvestmentsearnzeroprofitsinequilibrium,itreallydoesnotmatterwhomakestheinvestments.LetI=y0,y1(s),xdenotethevectorofinvestmentsundertaken.Thezero-profitconditionsareanalogoustotheoneswederivedbefore.Sinceinvestmentintheassetsissubjecttoconstantreturnstoscale,positiveprofitsareinconsistentwithequilibriumandpositiveinvestmentwilloccuronlyiftheprofitsarenon-negative(i.e.zero).Oneunitinvestedintheshortassetatdate0yieldsoneunitatdatesineachstate,sothezero-profitconditionisSp1(s)≤1,(6.6)s=1withequalityify0>0.Thelefthandsideof(6.6)isthevalueoftheoutputsatdate1andtherighthandsideisthevalueoftheinputatdate0.Theinequalitystatesthatinvestingintheshortassetatdate0yieldsnon-positiveprofitsandtheprofitsmustbezeroifinvestmentispositive.Similarly,oneunitinvestedintheshortassetatdate1instatesyieldsoneunitatdate2instates,sothezero-profitconditionisp2(s)≤p1(s),(6.7)withequalityify1(s)>0,fors=1,...,S.Again,thelefthandsideof(6.7)isthevalueofoutputsandtherighthandsideisthevalueofinputs.Finally,oneunitinvestedinthelongassetatdate0yieldsR(s)unitsatdate2instates,sothezero-profitconditionisSp2(s)R(s)≤1,(6.8)s=1withequalityifx>0. 6.2IntermediationandMarkets175Allcontingentcommoditiesaretradedatdate0.Subsequently,intermedi-ariessimplyfulfillthecommitmentstheyenteredintoatdate0.Ifthemarketsclearatdate0,thesubsequentexecutionofthesetradesmustbefeasible.First,considerthemarketforthecommoditydeliveredatdate0.Thesupplyofthecommodityisequaltotheendowmentofthegood.Thedemandforthecommodityequalstheinvestmentintheshortandlongassetatdate0.Thenmarketclearingrequiresx+y0=1.(6.9)Thereisacontingentcommodityforeachstatesatdate1.Thedemandforthiscommodityequalsthetotalconsumptionoftheearlyconsumersinthetworegionsplustheinvestmentintheshortasset.Thesupplyofthiscommodityequalstheoutputfromtheinvestmentintheshortassetatdate0.Thenmarketclearingrequires1{λA(s)cA1(s)+λB(s)cB1(s)}+y1(s)=y0,(mc2)2foreachs=1,...,S.Notethatonthelefthandsidethetotaldemandforconsumptionistheaverageofconsumptioninthetworegions,becausehalfthepopulationisineachregion.Thereisacontingentcommodityforeachstatesatdate2.Thedemandforthiscommodityisequaltothetotalconsumptionofthelateconsumersfromthetworegions(thereisnoinvestmentatdate2).Thesupplyistheoutputfromtheinvestmentintheshortassetatdate1plustheoutputfromtheinvestmentinthelongassetatdate0.Marketclearingrequires1{(1−λA(s))cA2(s)+(1−λB(s))cB2(s)}=R(s)x+y1(s),(6.10)2foreachs=1,...,S.Intheeconomyjustdescribed,theincentiveconstraint(6.4)maybebinding.Ifitisbinding,thenthefirst-bestallocation(theallocationacentralplannerwouldchooseifhehadcompleteinformationabouttheagents’types)maynotbefeasible.Wetaketheviewthataregulatororplannerhasnomoreinfor-mationthanthemarketandsotheconsumptionplansimplementedbytheplannermustalsosatisfytheincentiveconstraint.Inthatcase,thefirstbestisnottheappropriatebenchmark.Weshouldinsteadask:“Whatcouldacentralplannerachieveifhehadthesameinformationasthemarketandweresubjecttothesameincentiveconstraint?”Thissortofreasoningleadstothecon-ceptofincentiveefficiency.Anallocationconsistingoftheconsumptionplans 176Chapter6.IntermediationandMarketsc=(cA,cB)andtheinvestmentplanIisattainableifitsatisfiesthemarket-clearingconditions(6.9)–(6.10).Anattainableallocation(c,I)isincentivecompatibleiftheconsumptionplanssatisfytheincentiveconstraint(6.4).Anincentive-compatibleallocation(c,I)isincentiveefficientiftheredoesnotexistanincentive-compatibleallocationc,Ithatmakesbothregionsofinvestorsbetteroffexante.Inotherwords,theplannercannotdobetterthanthemarketwhentheplannerhasthesameinformationasthemarket.AllenandGale(2004)showthatanyequilibriumallocationisincentiveefficientaslongasmarketsarecompleteandinvestorsarerestrictedtousingintermediariestoaccessthemarkets.Wesketchtheargumenthere.Supposethat(c∗,I∗)isanequilibriumallocationandp∗istheequilibriumpricevector.Ifbothregionscanbemadebetteroffbychoosinganincentive-compatibleallocation(c,I),thenitmustbethecasethatcAandcBarenotwithintheintermediaries’budgetsets,thatis,S∗(s)λ∗∗∗p1ici1(s)+p2(s)(1−λi(s))ci2(s)>1(6.11)s=1foreachi=A,B.Foralltheinvestmentactivities,eitherprofitsarezeroortheinvestmentleveliszero.Thisimpliesthatthesumofprofitsiszero:SSp∗(s)−1y∗∗10+p2(s)−p1(s)y1(s)s=1s=1S+p∗(s)R(s)−1x=0.(6.12)2s=1RearrangingthisequationwegetSSp∗(s)y∗10−y1(s)+p2(s)R(s)x+y1(s)=x+y0,(6.13)s=1s=1andsubstitutingfromthemarket-clearingconditions(6.9)–(6.10)intoequation(6.13)wehaveS
1p∗(s){λ}1A(s)cA1(s)+λB(s)cB1(s)2s=1S
1+p∗(s){(1−λ}=12A(s))cA2(s)+(1−λB(s))cB2(s)2s=1 6.2IntermediationandMarkets177or1S∗(s)λ∗p1i(s)ci1(s)+p2(s)(1−λi(s))ci2(s)=12i=A,Bs=1contradictingtheinequality(6.11).6.2.5ImplementingthefirstbestwithoutcompletemarketsInveryspecialcases,itmaybepossibletoimplementthefirstbestwithoutcompletemarkets.Toseethis,wereturntothespecialmodelwithtworegions,AandB,andtwostates,HLandLH,andsupposethatthereisanassetmarketatdate1.Anintermediarycouldsellsomeofthelongassetinordertogetadditionalliquidityinthe“high”stateandbuythelongassetwiththeexcessliquidityinthe“low”state.Inthespecialcaseoflogarithmicutilityandnoaggregateuncertainty,thisisenoughtoachievethefirstbest.SupposetheVNMutilityfunctionisU(c)=ln(c)andtheplanner’sproblemistomaximizeλln(c1)+(1−λ)ln(c2)subjecttotheconstraintsλc1=y(1−λ)c2=R1−y.Thesolutiontothisproblemistheallocationthatgivesearlyconsumersc1∗=1andlateconsumersc2∗=R.Toachievethisconsumptionprofile,theinvestmentintheshortassetatdate0mustbey∗=λ.Thenλc∗=λ=y∗1and(1−λ)c∗=(1−λ)R=R1−y∗.2Thisistheallocationthatacentralplanner,whocouldmovegoodsbetweentheregions,wouldimplement.Cantheintermediariesdothesamebytradingontheassetmarketatdate1? 178Chapter6.IntermediationandMarketsIntheabsenceofaggregateuncertainty(remember,intermediariesfrombothregionstradeintheeconomy-wideassetmarket)theassetpriceatdate1mustbeP=1.Otherwise,intermediarieswouldnotbewillingtoholdbothassetsatdate0.Supposethattheintermediariesinvesty=λintheshortassetand1−yinthelongassetandpromisetheearlyconsumersc1=1atdate1.Thenatdate1,anintermediaryinthehighstatewillneedλHc1=λHunitsofthegoodbutonlyhasy=λunitsoftheshortasset.SoitneedsanadditionalλH−λunitsofthegood.Togetthis,itmustsellλH−λunitsofthelongasset(remembertheassetpriceisP=1).Anintermediaryinthelowstatehasy=λunitsoftheshortassetbutonlyneedsλLc1=λLunitsofthegoodtopaytheearlyconsumers.SincePc2∗/R,asinthecaseoftheutilityfunction(6.14)inExample4,thenthepresentvalueofconsumptionfortheearlyconsumersisgreaterthanthepresentvalueofconsumptionforthelateconsumers.Anincreaseintheproportionofearlyconsumerswillincreasethepresentvalueoftotalconsumption,soinordertosatisfythebudgetconstrainttheintermediarywillhavetoreducesomeone’sconsumption.Sinceearlyconsumersarepromisedafixedamountd,theintermediarywillendup 6.3IncompleteContracts187givinglesstothelateconsumers,assumingitcandosowithoutcausingarun.Assumingthatc1=d,thelateconsumersreceivec2sinstates=H,L,where(1−λsd)Rc2s=.1−λsWithanassetpriceP=1,theintermediary’swealthatdate1isy+P1−y=y+1−y=1.Becausetheintermediary’swealthatdate1isindependentofy,anychoiceofyisoptimalfortheintermediaryatdate0.Iftheintermediarygivesλsdtotheearlyconsumersithas1−λsd(inpresentvalue)forthelateconsumers.Butoneunitatt=1willbuyRunitsatt=2,sotheintermediarycanbuy(1−λsd)Runitsofconsumptionatt=2andsincethereare1−λslateconsumerseachofthemwillreceive(1−λsd)R/(1−λs).Thelateconsumersreceivethesameamountineachstateifandonlyifd=1.Ingeneral,thelateconsumer’sconsumptionvarieswiththestateandtheintermediary’sproblemistochoosedandytomaximize

 1(1−λHd)R(1−λLd)RλU(d)+(1−λH)U+(1−λL)U,21−λH1−λL(6.15)subjecttotheincentiveconstraintc2s≥d,whichweshallassumeissatisfied.Sinceeverychoiceofyisoptimal,theintermediaryonlyhastooptimizewithrespecttod.Thefirst-orderconditionforasolutiontothisproblemis
1(1−λHd)R−λHRλU(d)+(1−λH)U21−λH1−λH
(1−λLd)R−λLR+(1−λL)U1−λL1−λL=0whichsimplifiesto

 λH(1−λHd)RλL(1−λLd)RU(d)=RU+U.(6.16)2λ1−λH2λ1−λLThisisanalogoustotheusualconditionU(c1)=RU(c2),exceptthattheterminbracesontherighthandsideisaweightedaverageofthedifferentmarginalutilitiesineachstateatdate2. 188Chapter6.IntermediationandMarketsExample4(Continued)Iftheintermediarycannotachievethefirstbest,whatshoulditdo?Toillustrate,wegobacktotheparametersofExample4andcalculatetheexplicitsolution.Thefirst-ordercondition(6.16)tellsusthat
−6
−6−6(1−0.6d)3(1−0.4d)3d=30.6+0.4,0.40.6whichcanbesolvedford=1.337.Thisimpliesthattheconsumptionofthelateconsumersis(1−λHd)R(1−(0.6)(1.337))(3)c2H===1.485(1−λH)0.4inthehighstateand(1−λLd)R(1−(0.4)(1.337))(3)c2L===2.327.(1−λL)0.6Noticethatalthoughthelateconsumersdoquitewellinthelowstate,thisdoesnotcompensateforthelowconsumptioninthehighstate.Ifwecalculatetheequilibriumexpectedutility(6.15)explicitly,weget−1−5−1−5−1−5(0.5)(1.337)+(0.5)(0.4)(1.485)+(0.6)(2.327)555=−0.030,whereasthefirstbestis−1−5−1−5(0.5)(1.428)+(0.5)(1.715)=−0.024.55Finally,noticethattheincentiveconstraintc2s≥dissatisfiedineachstate,eventhoughwedidnotimposeitonthesolution.Thisjustifiesourassumptionthattheincentiveconstraintissatisfied.Moreover,itshowsthatitisoptimaltohavenodefaultinequilibrium.6.4CONCLUSIONWecansummarizethechapter’sconclusionsbrieflyasfollows.Aslongaswehavecompletemarketsforhedgingaggregateriskandintermediariescanuse References189completecontingentrisk-sharingcontracts,theequilibriuminalaisser-faireeconomywillbeincentiveefficient.Ifintermediariesareforcedbytransactioncoststouseincompletecontracts,theequilibriumwillbeconstrainedefficient.Ineithercase,itiswrongtosuggestthatfinancialcrisesconstituteasourceofmarketfailure.Acentralplannersubjecttothesameinformationalcon-straintsorthesametransactioncostscouldnotdobetterthanthemarket.Ifcontractsarecomplete,thereisneveranyneedtodefault.Theintermediarycanachievethesameendsbysimplyalteringthetermsofthecontract.Incom-pletecontracts,ontheotherhand,distortthechoicesthatanintermediarywouldotherwisemake;relaxingtheseconstraintsbydefaultinginsomestatesofnatureallowstheintermediarytoprovidethedepositorwithsuperiorrisksharingand/orhigherreturns.Whethertheintermediarychoosestodefaultornot,itschoicesmaximizethewelfareofitsdepositors.Sincemarketsarecomplete,pricesgivetherightsignalstointermediariesandguidethemtochoosetheefficientallocationofriskandinvestments.Itisonlywhenmar-ketsareincompletethatweencounterinefficienciesthatcouldinprinciplebecorrectedbygovernmentregulationandleadtoapotentialimprovementinwelfare.Weexplorethescopeforwelfare-improvingregulationandtheformittakesinthenextchapter.REFERENCESAllen,F.andD.Gale(2004).“FinancialIntermediariesandMarkets,”Econometrica72,1023–1061.Zame,W.(1993).“EfficiencyandtheRoleofDefaultwhenSecurityMarketsareIncomplete,”AmericanEconomicReview83,1142–1164. 7OptimalregulationForthemostpart,thedevelopmentoffinancialregulationhasbeenanempiri-calprocess,amatteroftrialanderror,drivenbytheexigenciesofhistoryratherthanbyformaltheory.AnepisodethatillustratesthecharacterofthisprocessistheGreatDepressionintheUS.ThefinancialcollapseintheUSwaswidespreadanddeeplydisruptive.Itledtosubstantialchangesinthelawsregulatingthefinancialsystem,manyofwhichshapeourcurrentregulatoryframework.TheSECwasestablishedtoregulatefinancialmarkets.InvestmentandcommercialbankingweresegregatedbytheGlass–SteagallAct(subsequentlyrepealedandreplacedbytheGramm–Leach–BlileyActof1999).TheFederalReserveBoardreviseditsoperatingproceduresinthelightofitsfailuretopreventthefinan-cialcollapse.TheFDICandFSLICweresetuptoprovidedepositinsurancetobanksandsavingsandloaninstitutions.Lookingback,thereisnosignofformaltheoryguidingthesechanges.EveryoneseemstohaveagreedtheexperienceoftheGreatDepressionwasterrible;soterriblethatitmustneverbeallowedtohappenagain.Accordingtothismindset,thefinancialsystemisfragileandthepurposeofprudentialregulationistopreventfinancialcrisisatallcosts.Whydoesthemindsetofthe1930’scontinuetoinfluencethinkingaboutpolicy?Whatdoespolicymakingcontinuetobeanempiricalexercise,withlittleattentiontotheroleoftheory?Thisempiricalprocedureisunusual.Indeed,theareaoffinancialregulationissomewhatuniqueintheextenttowhichtheempiricaldevelopmentshavesofaroutstrippedtheory.Inmostareasofeconomics,whenregulationbecomesanissue,economistshavetriedtoidentifysomespecificmarketfailurethatjustifiestheproposedintervention.Sometimestheyhavegonefurtherandhavederivedtheoptimalformofregulation.Thishasnotbeentheusualprocedurewithfinancialregulation,however.ThepurposeofthischapteristoshowhowtheframeworkdevelopedinChapter6canbeusedasthebasisforanalyzingoptimalregulation.Thewidespreadperceptionthatfinancialsystemsare“fragile,”togetherwithmanyhistoricalepisodesoffinancialinstability,hascreatedapresumptionthatregu-lationisrequiredtopreventcostlyfinancialcrises.Inthepreviouschapterwearguedtothecontrarythat,underconditionsanalogoustotheassumptions 7.1CapitalRegulation191ofthefundamentaltheoremsofwelfareeconomics,alaisser-faireequilibriummaybeefficient.Theoccurrenceofdefaultandfinancialcollapseinequilib-riumdoesnotnecessarilyindicateamarketfailure.Ifaplannerusingthesamecontractingtechnologycandonobetter,wesaythatequilibriumisconstrainedefficient.Unlesstheauthoritieshaveaccesstoasuperiortechnology,interven-tionisnotjustifiedwhentheincidenceoffinancialinstabilityisconstrainedefficient.Toprovideajustificationforregulationofthefinancialsystem,wefirstneedtoidentifyasourceofmarketfailure(constrainedinefficiency).Thenweneedtoidentifyapracticalpolicythatcanremedyoratleastamelioratethatfailure.Inthischapterweundertaketwopolicyexercises.Firstwelookatthepotentialbenefitsofregulatingcapitalstructure.Thenwelookatthepotentialbenefitsofregulatingliquidity.Ineachcase,weareinterestedindeterminingwhetheralaisser-faireequilibriumisconstrainedefficientand,ifnot,whatcanbedoneaboutit.Ourviewisthatitisnotenoughmerelytoshowthatthereexistsawelfare-improvingpolicy.Wealsoneedtocharacterizethepolicyandshowthatitcanbeimplemented.Abadlydesignedinterventioncouldmakethingsworse.Ifthewelfare-improvingpolicyistoocomplicatedordependsoninformationthatisunlikelytobeavailabletothepolicymaker,suchmistakesarelikely.7.1CAPITALREGULATIONCapitaladequacyrequirementsarerulesthatspecifyaminimumlevelofcapitalthatabankmustmaintaininrelationtoitsassets.Thisrulemaytaketheformofasimplefractionoftheassetsoramorecomplicatedformula.Capitaladequacyrequirementsareoneofthemostimportantinstrumentsofbankregulation.ThefirstBaselAccordimposeduniformcapitaladequacyrequirementsonthebanksofallthesignatorycountries.AsecondBaselAccordintroducesmoresophisticatedmethodsofdeterminingtheappropriatelevelofcapitalforbanks,buttheideathatbanksmustbecompelledtoholdtheappropriatelevelofcapitalremainsabasicprincipleoftheregulatorysystem.Theseaccordsprovideanexampleofregulationthatisempiricallyratherthantheoreticallymotivated.Practitionershavebecomeexpertsatthedetailsofahighlycomplexsystemforwhichthereisnowidelyagreedrationalebasedineconomictheory.Whatistheoptimalcapitalstructure?Whatmarketfailurenecessitatestheimpositionofcapitaladequacyrequirements?Whycan’tthemarketbelefttodeterminetheappropriatelevelofcapital?Wedonotfindgoodanswerstothesequestionsinthetheoreticalliterature. 192Chapter7.OptimalRegulationIntheliteratureoncapitaladequacy,itisoftenarguedthatcapitaladequacyrequirementsarenecessarytocontrolthemoralhazardproblemsgeneratedbytheexistenceofdepositinsurance.Depositinsurancewasintroducedinthe1930’stopreventbankrunsor,moregenerally,financialinstability.Becausebanksissueinsureddebt-likeobligations(e.g.bankdeposits)theyhaveanincentivetoengageinrisk-shiftingbehavior.Inotherwords,thebankhasanincentivetomakeexcessivelyriskyinvestments,becauseitknowsthatintheeventoffailurethelossisbornebythedepositinsurancefundandintheeventofsuccessthebank’sshareholdersreaptherewards.Theexistenceofbankcapitalreducestheincentivetotakerisksbecause,intheeventoffailure,theshareholderslosetheircapital.Thus,capitaladequacyrequirementsareindirectlyjustifiedbythedesiretopreventfinancialcrises.Alargeliteratureinvestigatestheeffectofcapitaladequacyrequirementsonrisktaking.Whiletheeffectofcapitaladequacyrequirementsisusuallytodecreaserisktaking,thereverseisalsopossible(see,e.g.KimandSantomero1988;FurlongandKeeley1989;GennotteandPyle1991;Rochet1992;andBesankoandKanatas1996).Theincentivetotakerisksmayalsobeoffsetbythelossofchartervaluewhenafirmgoesbankrupt(see,e.g.Bhattacharya1982).Thiseffectwillbesmallerthemorecompetitivethestructureofthebankingmarket.Keeley(1990)hasprovidedevidencethatthesharpincreaseinbankfailuresintheUSintheearly1980’swasduetoincreasedcompetitioninthebankingsectorandtheassociatedfallinchartervalues.Itappearsfromourreviewoftheliteraturethatthejustificationforcapitaladequacyrequirementsisfoundintheexistenceofdepositinsurance.Itcouldbearguedthatanimportantquestionisbeingbeggedhere:onebadpolicy(depositinsurance)doesnotjustifyanother(capitaladequacyrequirements).Evenifitisassumedthatdepositinsurancepreventsfinancialcrises,itisnotclearwhyweshouldwanttoreducetheincidenceoffinancialcrises,stilllesseliminatethemaltogether.AswedemonstratedinChapter6,theincidenceoffinancialcrisesmaybesociallyoptimalinalaisser-fairesystem.Andifnot,forexample,iffinancialcrisesinvolvedeadweightlosses,itshouldberecog-nizedthatregulationalsoinvolvesadministrativecostsanddistortseconomicdecisions.Anyanalysisofoptimalpolicymustweighthecostsandbenefitsofregulation.Thiscanonlybedoneinamodelthatexplicitlymodelsthepossibilityofcrises.Hellmanetal.(2000)isanexceptionintheliteratureoncapitaladequacyrequirements.Ratherthansimplytakingtheexistenceofdepositinsuranceasgiven,theauthorsalsoexaminewhathappensintheabsenceofdepositinsurance.Intherestoftheliterature,therationalefordepositinsuranceandinparticularitsroleinpreventingfinancialcrisesisdiscussedbutnotexplicitlymodeled.Intheabsenceofexplicitmodelingofthecostsof 7.1CapitalRegulation193financialcrises,itisdifficulttomakeacasefortheoptimalityofinter-vention.Asacorollary,itisdifficulttomakeacaseforcapitaladequacyrequirementsasameansofoffsettingtherisktakinggeneratedbydepositinsurance.AllenandGale(2003)arguethat,intheabsenceofawelfare-relevantpecuniaryexternality,bankswillchoosethesociallyoptimalcapitalstruc-turethemselves,withoutgovernmentcoercion.Foralongtime,policymakershavetakenitasaxiomaticthatcrisesarebestavoided.Bycontrast,inAllenandGale’sframework,alaisser-fairefinancialsystemwithcompletemarketsachievesaconstrained-efficientallocationofriskandresources.Whenbanksarerestrictedtousingnoncontingentdepositcontracts,defaultintroducesadegreeofcontingencythatmaybedesirablefromthepointofviewofopti-malrisksharing.Farfrombeingbestavoided,financialcrisescanactuallybenecessaryinordertoachieveconstrainedefficiency.Bycontrast,avoidingdefaultiscostly.Itrequireseitherholdingaverysafeandliquidportfolio(andearninglowerreturns),orreducingtheliquiditypromisedtothedepositorsattheintermediatedate.Inanycase,thebankoptimallyweighsthecostsandbenefitsandchoosestheefficientlevelofdefaultinequilibrium.Ourargumentisthatavoidanceofcrisesshouldnotbetakenasaxiomatic.Ifregulationisrequiredtominimizeorobviatethecostsoffinancialcrises,itneedstobejustifiedbyamicroeconomicwelfareanalysisbasedonstandardassumptions.Furthermore,theformoftheinterventionshouldbederivedfrommicroeconomicprinciples.Afterall,financialinstitutionsandfinancialmarketsexisttofacilitatetheefficientallocationofrisksandresources.Apolicythataimstopreventfinancialcriseshasanimpactonthenormalfunctioningofthefinancialsystem.Anygovernmentinterventionmayimposedeadweightcostsbydistortingthenormalfunctioningofthefinancialsystem.Oneoftheadvantagesofamicroeconomicanalysisoffinancialcrisesisthatitclarifiesthecostsassociatedwiththesedistortions.Inadditiontotheincentivefunction,discussedabove,bankcapitalhasanothermainfunction.Thisistherisk-sharingfunction.Capitalactsasabufferthatoffsetsthelossesofdepositorsintheeventofabankfailureandallowsanorderlyliquidationofthebank’sassets,thusavoidingtheneedtodisposeofassetsat“firesale”prices.Thesefunctionsofbankcapitalexplainwhyshareholdersanddepositorsshouldcareaboutthebank’scapitalstructure,buttheydonotexplainwhygovernmentsneedtoregulatecapitalstructure.Totheextentthatcapitalstruc-tureaffectstheefficiencyofrisksharingorthebank’sincentivetotakerisk,thecostsandbenefitsshouldbeinternalizedbythebank’sobjectivefunction.Intheabsenceofsomesortofexternalitynottakenintoaccountbythebanks,thereisnoobviousreasonwhythebank,lefttoitsowndevices,shouldnotchoose 194Chapter7.OptimalRegulationthe(socially)optimalcapitalstructure.Inotherwords,wehavenot(yet)iden-tifiedasourceofmarketfailurethatgivesrisetoaneedforinterventionbytheregulator.Incompletemarketsprovideonepossiblejustificationforcapitalregulation,inthesensethatpecuniaryexternalitieshaveanimpactonwelfarewhenmar-ketsareincomplete,andinthatcaseregulationofcapital(oranythingelse)canpotentiallyimprovewelfare.Inthefollowingsections,weadaptthemodelfromthepreviouschaptertoshowthat,whenmarketsareincomplete,thereisaroleforbankcapitaltoimproverisksharingandaroleforgovernmentinterventiontoimprovewelfare.7.1.1OptimalcapitalstructureAsusualtherearethreedatest=0,1,2andanall-purposegoodthatcanbeusedforconsumptionorinvestment.Therearetwoassets,ashortassetrepresentedbyastoragetechnologythatyieldsoneunitatdatet+1foreachunitinvestedatdatet,andalongassetrepresentedbyaconstantreturnstoscaletechnologythatyieldsR>1unitsofthegoodatdate2foreachunitinvestedatdate0.Thereisacontinuumofidenticalinvestorsatdate0eachofwhomhasanendowmentofoneunitofthegoodatdate0andnothingatfuturedates.Atdate1eachconsumerlearnswhetherheisanearlyconsumer,whoonlyvaluesconsumptionatdate1,oralateconsumer,whoonlyvaluesconsumptionatdate2.Theprobabilitythataninvestorbecomesanearlyconsumeris0<λ<1.Theinvestors’attitudestoriskarerepresentedbyaVNMutilityfunction.Ifcistheinvestor’sconsumption,hisutilityisU(c),wherethefunctionU(·)satisfiestheusualneoclassicalproperties.Therearetwogroupsofconsumers,groupAandgroupB,andexactlyhalfoftheconsumersbelongtoeachgroup.Therearetwoaggregatestatesofnature,denotedby(H,L)and(L,H).Eachstateisequallylikely,thatis,eachoccurswithprobability0.5.Instate(H,L)thefractionofearlyconsumersingroupAisλHandthefractionofearlyconsumersingroupBisλL,where0<λL<λH<1.Instate(L,H)thefractionsarereversed.Thenthefractionofearlyconsumersineachstateisgivenby1λ=(λH+λL).2Theinvestors’attitudestowardriskarerepresentedbyaVNMutilityfunc-tion.Ifaninvestorconsumescunitsofthegoodattheappropriatedate,hisutilityisU(c),whereU(·)satisfiesalltheusualproperties. 7.1CapitalRegulation195Alluncertaintyisresolvedatthebeginningofdate1,whenthetruestateisrevealedandeachinvestorlearnshistype.Inthepreviouschapterweassumedthatmarketswerecomplete.Specifically,weassumedtheexistenceoftwoArrowsecuritiesatdate0,whichallowthetransferofwealthbetweenstates(H,L)and(L,H),andanassetmarketonwhichthelongassetcanbetradedatdate1.Hereweassumethatmarketsareincomplete.Specifically,weassumethattherearenoArrowsecurities,butthereisanassetmarketatdate1.Theincompletenessofmarketswouldhavenoeffectontheallocationofriskifintermediariesservedarepresentativesampleofthepopulation.Instead,weassumethatintermediariesdrawtheircustomersfromeithergroupAorgroupB,butnotboth.OneinterpretationofthisassumptionisthatgroupsAandBcorrespondtodifferentregionsandthatintermediariesarerestrictedbylawtooperateinonlyoneregion.Inanycase,theheterogeneityofintermediariesgivesrisetogainsfromrisksharingwhichcannotberealizedbecausemarketsareincomplete.Apartfromtheincompletenessofmarkets,wedonotimposeanyfrictionsonthemodel.Inparticular,intermediariesareallowedtousecompleterisk-sharingcontracts.Anintermediarytakesadepositofoneunitofthegoodfromeachconsumeratdate0andinvestsitinaportfolio(x,y)consistingofxunitsofthelongassetandyunitsoftheshortasset.Inexchange,theconsumergetsaconsumptionstream(c1H,c2H,c1L,c2L),wherec1Histheconsumptionpromisedifhewithdrawsatdate1whentheproportionofearlyconsumersisλH,c2Histheconsumptionpromisedifhewithdrawsatdate2whentheproportionofearlyconsumersisλH,andsoon.Inordertodiscusscapitalstructure,weintroduceaclassofriskneutralinvestorstoprovidecapitaltointermediaries(seeGale2003,2004).Eachinvestorisassumedtohavealargeendowmentofthegoodatdate0andnothingatdates1and2.Theinvestorsareriskneutral,buttheirconsump-tionmustbenon-negative(otherwise,theinvestorscouldabsorballriskandthefirstbestwouldbeachieved).Capitalisassumedtobeexpensiveinthesensethatinvestorsdemandahigherreturnthantheintermediary’sinvest-mentopportunitiescanprovide.Wemodeltheopportunitycostofcapitalbyassumingthatinvestorsareimpatient:oneunitofconsumptionatdate0isworthρ>Runitsoffutureconsumption.Foreveryunitofcapitalinvestedintheintermediaryatdate0theinvestorswilldemandanexpectedreturnofρunitsinthefuture.Sincetheintermediary’sinvestmentscannotyieldareturnhigherthanR,theintermediaryhastotransfersomeofthedepositors’returnstotheinvestorstocompensatethemfortheuseoftheircapital.Eventhoughcapitaliscostly,itisoptimaltoraiseapositiveamountofcapitalbecauseitallowsforimprovedrisksharing. 196Chapter7.OptimalRegulationTheintermediaryoffersinvestorsacontract(e0,eH,eL),wheree0denotestheamountofcapitalprovidedbytheinvestorsatdate0andeHandeLdenotethereturnspaidtotheinvestorswhenthefractionsofearlyconsumersareλHandλL,respectively.Withoutlossofgenerality,wecanassumethateHandeLarepaidatdate2becauseequilibriumrequiresthatthedate1priceofdate2consumptionp≤1,sothegoodisalwaysatleastascheapatdate2asatdate1.Investorswillsupplycapitaltothebankonlyifthereturnscovertheiropportunitycost,thatis,1(eH+eL)≥ρe0.(7.1)2Sincethereisalargenumberofinvestors,eachofwhomhasalargeendowment,competitionamonginvestorsimpliesthattheygetnosurplusinequilibrium,thatis,theinequality(7.1)holdsasanequation.Sowecanassumewithoutlossofgeneralitythattheintermediarychoosesaportfolio(x,y),capitalstruc-ture(e0,eH,eL),andconsumptionplan(c1H,c2H,c1L,c2L)tomaximizetheexpectedutilityofthetypicaldepositorsubjecttotheinvestors’participationconstraint(7.1)andthefeasibilityconstraints.Atdate0thetotalinvestmentisconstrainedbythedepositor’sendowmentandthecapitalsuppliedbytheinvestors:x+y≤e0+1.Atdate1,theintermediary’sbudgetconstraintisλsc1s+(1−λs)pc2s+pes≤y+Pxfors=H,LwhereP=Rpisthepriceoftheassetatdate1.Giventhereisnoaggregateuncertaintythepricepwillbedeterminedintheusualway.Inorderforthebankstobewillingtoholdbothassetsbetweendates0and1theymustbeindifferentbetweenthemso1p=;P=1.RSincegroupsAandBaresymmetric,andineachstateofnatureonegrouphasahighproportionandonehasalowproportionofearlyconsumers,themarket-clearingconditionsatdate1anddate2are1(λHc1H+λLc1L)=y2 7.1CapitalRegulation197and1((1−λH)c2H+eH+(1−λL)c2L+eL)=Rx.2Fromthesecondofthetwobudgetconstraints,wecanseethatifeHandeLarebothpositive,thenthefirst-bestrisksharingmustbeachieved,thatis,c2H=c2L.Otherwise,onecanincreaseexpectedutilitybyreducingesinonestateandincreasingitintheother.Forexample,supposethatc2H0.1−λH1−λLThemarginalvalueofinsuranceiszerowhenrisksharingiscompletewhereasthemarginalcostofcapitalispositive.Soitisneveroptimaltoholdenoughcapitaltoachievecompleterisksharing.Consequently,esmustbezeroinatleastonestate,theoneinwhichconsumptionc2sislower.Forexample,supposethatσ>1.Thenweknowbytheusualargumentthatc1s>pc2sandaverageconsumptionislowerwhentheproportionofearlyconsumersishigh.Thentheoptimalcapitalstructureshouldincreaseconsumptioninthehighstateandreduceitinthelowstate.Soe1H=0andeL=ρe0.2Proposition1Supposeconsumershaveaconstantdegreeofrelativeriskaversionσandlet(e0,eH,eL)betheoptimalcapitalstructure,wheree0>0.TheneH>eL=0ifσ<1andeL>eH=0ifσ>1.Inthelastchapterwesawthat,whenthereisnoaggregateuncertainty,aPareto-efficientallocationgiveseveryconsumeraconsumptionallocation(c1,c2)thatisindependentofthestateofnature.Whethertheefficientallo-cationcanbeachieveddependsonthecostofcapitalρ.Ifρisveryhigh,inrelationtoR,theoptimalcapitalstructurewillentailasmallinfusionofcapitale0atdate0,theintermediary’sabilitytosmoothconsumptionbetweenstates 198Chapter7.OptimalRegulationwillbelimited,andthefirstbestwillnotbeattained.Itistemptingtocon-cludeincaseslikethisthatthemarkethasfailedbecausethemarketoutcomeisnotPareto-efficient;butthisassumesthattheplannerisnotsubjecttothetransactioncostsandotherfrictionsthatpreventmarketsfrombeingcom-plete.Beforewedecidethatthemarkethasfailedandthatsomeinterventionisrequired,weshouldaskwhetherthecentralplannercoulddobetterifhewereconstrainedtouseonlythetradingopportunitiesavailabletothemarketparticipants.Forexample,itisclearthataplannercanimproveonthelaisser-faireallocationbytransferringgoodsfromintermediarieswhosedepositorshavealowmarginalutilityofconsumptiontointermediarieswhosedepos-itorshaveahighmarginalutilityofconsumption.Indoingso,theplannerisperformingthefunctionofthemissingmarketsforArrowsecuritiesthatallowintermediariestotransferwealthacrossstatesandachievethefirstbest.Butifthemarketparticipantsarepreventedbytransactioncostsorotherfric-tionsfrommakingthesetrades,perhapstheplannerwillbetoo.Thissuggeststhattheappropriatetestformarketfailureistoaskwhetheraplannercouldimproveonthelaisser-faireallocationusingthesametechnologyavailabletothemarketparticipants.Itisnotentirelyclearwhatthetechnologyavailabletotheplannershouldbe,butoneapproachwouldbetorestricttheplannertoalteringdecisionsmadeatdate0andrequiringhimtoallowthemarkettodeterminetheallocationatdates1and2intheusualway.Thiswouldensurethatwearenotgrantingtheplanneraquestionabletechnologicaladvantageoverthemarket.Wewillsaythatalaisser-faireallocationisconstrainedefficientiftheplannercannotimproveonitmerelybychangingtheallocationatdate0andleavingthemarkettodeterminethefutureallocation(GeanakoplosandPolemarchakis1986).Sincetheintermediarychoosesanoptimalcapitalstructureandanopti-malinvestmentportfolioandconsumptionplan,theplannercandobetterthantheintermediaryonlyifhechangestheequilibriumprice.Withoutachangeinprice,thechoiceoftheplannerisidenticaltothechoicesetoftheintermediaries.Thenitiseasytoseethatforcingtheintermediarytoadoptadifferentcapitalstructurecannotimprovewelfarebecausewithnoaggregateuncertaintytheequilibriumassetpriceisdeterminedbytheconditionthattheratesofreturnonthetwoassetsshouldbeequalizedintheusualway.Inotherwords,theequilibriumpriceisindependentofthecapitalstructure.Sincethereisnopecuniaryexternalitythatcanbeexploitedbytheregulator,forcingtheintermediarytoraisemoreorlesscapitalcanonlydistorttheoptimaldecision.Proposition2Underthemaintainedassumptions,thelaisser-faireequilib-riumisconstrainedefficientandwelfarecannotbeimprovedbychangingtheequilibriumcapitalstructure. 7.1CapitalRegulation1997.1.2ModelswithaggregateuncertaintyThecasewehaveexaminedisveryspecial.BecausetheassetpricePisaconstantatdate1,itisdeterminedbytherequirementthatthereturnsontheshortandlongassetsbeequal.Thismeansthatnochangeinportfoliosorcapitalstructureatdate0canhaveanyeffectonpricesatdate1and,aswehaveseen,pricechangesaretheessentialingredientofanyimprovementinwelfare.Inmodelswithaggregateuncertainty,theassetpricefluctuatesbetweenstates.Althoughthereisafirst-orderconditionthatconstrainsthedistributionofprices,capitaladequacyrequirementscanhaveaneffectonequilibriumpricesandhencehavesomeimpactonwelfare.Sinceequilibriawithincompletemarketsaretypicallynotconstrainedefficient,thesechangesinassetpricescanbemanipulatedtoincreasewelfare.Thecrucialquestion,however,iswhatkindofcapitalregulationwillleadtoanimprovementinwelfare.Itisnotobviousthatrequiringintermediariestoholdmorecapitalwillbebeneficial.Infact,simpleexampleswithnopathologicalfeaturescangiverisetothesurprisingconclusionthatincreasingcapitallowerswelfareandreducingcapitalincreaseswelfare.Toillustratetheseresults,wedescribeamodelstudiedbyGaleandÖzgür(2005).Themodelisidenticaltotheonedescribedaboveexceptforthestructureofliquidityshocks.Allindividualsandintermediariesareexanteidentical.Expostthereareintermediariesoftwotypes.Onetypeconsistsentirelyofearlyconsumersandtheothertypeconsistsoflateconsumers.Therearetwo(aggregate)statesofnature,HandL,whichoccurwithprob-ability0.5.Theproportionofearlyconsumersinstatesisdenotedbyλs,where1>λH>λL>0.Theproportionofintermediariesconsistingofearlycon-sumersinstatesisλsandtheprobabilitythatanyintermediaryhasonlyearlyconsumersisλstoo.Ifλiistheproportionofearlyconsumersinintermediaryithen1w.pr.λsinstates,λi=0w.pr.1−λsinstates,fors=H,L.Theexistenceofaggregateuncertaintyrequiressomeadditionalcomplexityintheintermediaries’contractswithconsumersandinvestors.Specifically,thepaymentsmadetobothgroupswilldepend,ingeneral,onboththeintermediary’sstate(1or0)andontheaggregatestate(HorL).Aconsumerdepositshisentireendowmentwithasingleintermediarywhooffershimaconsumptioncontractc={(c1s,c2s):s=H,L}inexchange,wherec1sisconsumptionofferedtoanearlyconsumeratdate1instatesandc2sistheconsumptionofferedtoalateconsumeratdate2instates. 200Chapter7.OptimalRegulationTheintermediarywritesacontracte={(e0,e1s,e2s):s=H,L}withinvestors,wheree0≥0istheamountofcapitalinvestedatdate0,e1s≥0istheamountofthegoodpromisedtoinvestorsinstatesifallthedepositorsareearlyconsumersande2s≥0istheamountofthegoodpromisedinstatesatdate2ifallthedepositorsarelateconsumers.Inordertoreducethevolatilityofconsumption,thecapitalstructureeshouldbechosensothatpaymentstoinvestorsoccurwhenconsumptionishighandnotwhenitislow.Sincetherearefourpossiblepaymentoppor-tunities,thisleavesalotofscopefordesigningtheoptimalrisksharingarrangements.Asusual,becausecapitaliscostly,itisnotoptimaltoelim-inatethefluctuationsinconsumptionaltogether.Thismeansthatchangesinpriceswillhaveincomeeffectsthatcanincreasetheexanteexpectedutilityofdepositors.Sincetheintermediariesareassumedtochoosetheircapitalstructureopti-mally,takingasgiventhepricescorrespondingtoeachstateofnature,itisclearthatcapitalregulationcanimprovewelfareonlybychangingtheequilibriumprices.Theimpactoftheseincomeeffectsmaybecomplex.Forexample,whenanintermediaryhasonlyearlyconsumers,itsellsitsholdingofthelongassettomeetitsobligationstoitsdepositors.Anincreaseinassetpriceswillthereforeraisetheirconsumption.Foranintermediarythathasonlylateconsumers,theeffectwillbereversed.Iflateconsumersaredoingbetteronaveragethanearlyconsumers,theneteffectonexanteutilitymaybebeneficial.Butthenweneedtoconsiderthepossibilitythatanincreaseinassetpricesinonestatemaynecessitateareductioninanother.Ultimately,thequestionis:whatchangeincapitalstructurewilleffectivelyincreasewelfare?TheanswerfoundbyGaleandÖzgürdepends,notsurprisingly,onthedegreeofrelativeriskaversion.Theyconsideramodelwithconstantrelativeriskaversionandsolveforequilibriumnumericallyforvariousparametricassumptions.Theyfindthat,ifthedegreeofriskaversionishighenough(greaterthanσ≈2),areductioninbankcapitalreducespricevolatilityandincreaseswelfare.Forlowerriskaversion(i.e.lowerthanσ≈2),anincreaseinbankcapitalincreasesvolatilityandwelfare.Theintuitionbehindtheseresultsappearstobethatforcingbankstoraisecostlycapitalwillraiseboththeirinvestmentintheshortassetandinthelongasset,butitraisesinvestmentintheshortassetlessthaninvestmentinthelongasset.Thisisbecausethebankistryingtominimizethecostofcapitalbyinvestingexcesscapitalinthehigher-yieldingasset.Asaresultofthisshiftinportfoliocomposition,theassetmarketbecomeslessliquidandthisreducesassetpricesinthehighstateandincreasesvolatilityoverall.Itisnotknownhowrobusttheseresultsarewhenthemodelspecificationisalteredorgeneralized,buteveniftheresultsturnouttobespecialtheyreinforce 7.2CapitalStructurewithCompleteMarkets201thelessonthat,whengeneral-equilibriumeffectsareinvolved,itisverydifficulttopredictthemacroeconomiceffectofchangesincapitalstructuresacrossthefinancialsystem.Untilwehaveageneraltheorytoguideus,cautioninpolicymakingwouldseemtobeadvisable.7.2CAPITALSTRUCTUREWITHCOMPLETEMARKETSThefunctionofbankcapitalintheprecedingsectionistoallowrisksharingbetweenriskneutralinvestorsandriskaversedepositors.Theinvestors’returnsareconcentratedinthosestateswherethedemandforliquidityislowrelativetothesupply.Byvaryingtheinvestors’returnsacrossstates,itispossibletoreducethefluctuationsinthedepositors’consumption,inotherwords,itispossibletoprovidedepositorswithinsuranceagainstliquidityshocksorassetreturnshocks.Anoptimalcapitalstructureisonewaytoprovidethiskindofinsurance,butitisnottheonlyway.Ifmarketsatdate0werecomplete,insurancecouldbeprovidedthroughthemarketsandtheneedforcapitalwouldbeeliminatedentirely.Letthegoodatdate0bethenumeraireandletptsdenotethepriceofoneunitofthegoodatdatet=1,2instates=H,L.Anintermediarywillwantliquidityatdate1ifitsdepositorsareearlyconsumersandatdate2iftheyarelateconsumers.Sowhattheintermediarywantsisanoptiononthegoodateachdate.Whatwillthisoptioncost?Sincetheprobabilityofhavingdepositorswhoareearlyconsumersisλsinstates,thecostoftheoptionshouldbeλsp1sfordate1and(1−λs)p2sfordate2.Iftheintermediaryusesthecompletemarketstoobtainanoptimalrisk-sharingcontractforitsdepositors,itwillofferaconsumptionplan(c1H,c2H,c1L,c2L)tomaximizetheexpectedutility11{λHU(c1H)+(1−λH)U(c2H)}+{λLU(c1L)+(1−λL)U(c2L)}22subjecttothebudgetconstraintλHp1Hc1H+(1−λH)p2Hc2H+λLp1Lc1L+(1−λL)p2Lc2L≤1.Similarly,investorscanusemarketstospreadtheirconsumptionoverthethreedates.Asusual,thereisnolossofgeneralityinassumingthattheyconsumeonlyinthefirstandlastperiods,sotheywillchooseabundle(e0,eH,eL)wheree0istheamountofthegoodsuppliedatdate0andesistheconsumptionat 202Chapter7.OptimalRegulationdate2instates,tomaximize1{eH+eL}−ρe0(7.2)2subjecttothebudgetconstraintp2HeH+p2LeL≤e0.(7.3)Becausemarketsarecomplete,investmentsintheshortandlongassetsyieldzeroprofits.Itdoesnotmatterwhomakestheinvestments,sincetheyaddnothingtowealthorthepossibilityofrisksharing,sowecan,withoutlossofgenerality,assumethatallinvestmentsaremadebyarepresentativefirm.Inequilibrium,thefirmwillbuythegoodssuppliedbytheintermediariesandtheinvestorsatdate0,thatis,1+e0,investthemintheshortandlongassets,andusethereturnsfromtheseassetstosupplygoodstotheinvestorsandtheintermediaryatdates1and2.Inadditiontothezero-profitconditions,theusualmarket-clearingconditionsmustbesatisfied(seeChapter6).Becausetheusualassumptions,includingtheassumptionofcompletemar-kets,aresatisfied,thefundamentaltheoremsofwelfareeconomicsensurethatanequilibriumisPareto-efficient(orincentiveefficientiftheincentivecon-straintsarebinding).Thus,withcompletemarkets,wegetefficientrisksharingbetweeninvestors,ontheonehand,andtheintermediariesandtheircus-tomers,ontheother.Althoughitisintermediatedbythemarket,theprovisionofinsuranceissimilartowhathappensinamodelwithcapitalstructure.Theinvestorssupply“capital”atdate0thatmustbeinvestedinrealassetsinordertoprovidefutureconsumption.Theytaketheirreturnsintheformofconsumptionatdate2.Becausetheyareriskneutral,theywillconsumeonlyinthestatewherethepricep2sisaminimum,leavingmoreconsump-tionforthedepositorsinotherstates.Thisbunchingofconsumptionbytheinvestorsinasinglestateallowsthedepositorstosmooththeirconsumptionacrossstatesand,inparticular,toconsumemoreinthestatewithahighcostofconsumption.Theexistenceofcompletemarketsnotonlyprovidesaperfectsubstituteforoptimalcapitalstructure,thusmakingcapitalredundant,italsomakestheoptimalcapitalstructureindeterminate.Thisisbecauseanycapitalstructurecanbeundonebytransactionsinthemarket.Supposethat(eˆ0,eˆH,eˆL)isanactionthatmaximizes(7.2)subjectto(7.3).Becausetheobjectivefunctionandtheconstraintarebothlinearin(e0,eH,eL),theoptimaltrademustsatisfy1eˆH+ˆeL−ρeˆ0=02 7.2CapitalStructurewithCompleteMarkets203andp2HeˆH+p2LeˆL−ˆe0=0.Nowsupposethattheintermediaryadopts(eˆ0,eˆH,eˆL)asitscapitalstructure.Becausetheintermediaryhasaccesstocompletemarkets,theonlyeffectofcapitalstructureisontheintermediariesbudgetconstraint.Thiscapitalstruc-tureisoptimalfortheintermediaryinthesensethatitminimizesthecostsubjecttotheinvestors’participationconstraint.Becausethecostiszero,itdoesnotaffectthesetofconsumptionplanstheintermediarycanaffordandtheoptimalplan(c1H,c2H,c1L,c2L)willsatisfythebudgetconstraintλHp1Hc1H+(1−λH)p2Hc2H+λLp1Lc1L+(1−λL)p2Lc2L+p2HeˆH+p2LeˆL−ˆe0≤1.Finally,ifboth(eˆ0,eˆH,eˆL)and(e0,eH,eL)maximizetheinvestors’objec-tivefunction(7.2)subjecttothebudgetconstraint(7.3),thensodoes(e0,eH,eL)−(eˆ0,eˆH,eˆL),sowecanassumethattheintermediarieschoosetotrade(e0,eH,eL)−(eˆ0,eˆH,eˆL)inequilibrium.Thenthecombinedeffectoftheoptimalcontractbetweentheintermediariesandtheinvestors(eˆ0,eˆH,eˆL)andthenettradeinthemarkets(e0,eH,eL)−(eˆ0,eˆH,eˆL)ispreciselyequivalenttothetrade(e0,eH,eL)intheoriginalequilibrium.Thus,theoptimalcapitalstructureisindeterminate.ThisissimplyaversionoftheModigliani–Millertheorem.Intheanalysisoftheprecedingsection,therearetwosourcesofliquidityforintermediariesthatsufferabadliquidityshockatdate1.Oneisthecapitalstructurenegotiatedwithinvestors,whichallowspaymenttoinvestorstobereducedintheeventofabadliquidityshock;theotherisassetsalestootherintermediaries.Thecapitalstructureischosenoptimallybytheintermediaries,sothisisnotasourceofinefficiency.Rathertheincompletenessofmarketsforcesbankstosellassetsatpricesthataredeterminedexpostbythedemandforandsupplyofliquidityineachstate.Aswehaveseen,theexpostprovisionofliquiditymaybeinefficientbecauseintermediariesendupsellingtheirassetsatalowpriceinstateswherethemarginalutilityoftheirdepositorsishigh,theoppositeofwhatgoodinsurancerequires.Bycontrast,whenmarketsarecompleteatdate0,theintermediarycantransferwealthacrossstatesattheprevailingpricesandthisensuresthatthemarginalratesofsubstitutionacrossstatesareequalizedforalldepositors.Itisworthnotingthattheheterogeneityofintermediariesexpostiscrucialforthisresult.Ifintermediarieswereidenticalatdate1therewouldbenogainsfromtradeandhencenoneedformarkets.Toputitanotherway,markets 204Chapter7.OptimalRegulationarealwayseffectivelycompleteinaRobinsonCrusoeeconomy.Somarketscanbeincompletewithoutanyeffectonefficiencyifeachintermediaryhasarepresentativesampleofconsumersateachdate.Theindeterminacyofcapitalstructuredoesnotsurvive,however:iftherearenomarketsforcontingentcommoditiesthecapitalstructureisdeterminatebecauseitisonlythroughthecapitalstructurethatefficientrisksharingcanbeachieved.7.3REGULATINGLIQUIDITYNowweturntothestudyofliquidityregulation,thatis,thepossibilityofimprovingwelfarebyregulatingtheamountoftheshortassetheldinequilib-rium.Tokeepthingssimple,weeliminatetheriskneutralinvestors,sothereisnoprovisionofcapital.Otherwise,theassumptionsarethesameasinSection7.1.TheexampleisbasedononeinAllenandGale(2004).Apartfromtheincompletenessofmarkets,wedonotimposeanyfrictionsonthemodel.Inparticular,intermediariesareallowedtousecompleterisk-sharingcontracts.Anintermediarytakesadepositofoneunitofthegoodfromeachconsumeratdate0andinvestsitinaportfolio(x,y)consistingofxunitsofthelongassetandyunitsoftheshortasset.Inexchange,theconsumergetsaconsumptionstream(c1H,c2H,c1L,c2L),wherec1Histheconsumptionpromisedifhewithdrawsatdate1whentheproportionofearlyconsumersisλH,c2Histheconsumptionpromisedifhewithdrawsatdate2whentheproportionofearlyconsumersisλH,andsoon.Becausethereisnoaggregateuncertainty,wecanassumethepriceoftheassetatdate1isindependentofthestate,thatis,PHL=PLH=P.Sinceequilibriumrequiresthatbothassetsareheldatdate0,theone-periodholdingreturnsonbothassetsmustalsobethesameatdate0.ThereturnontheshortassetisequaltooneandthereturnonthelongassetisequaltoP,sotheequilibriumpricemustbeP=1.IfP=1isthepriceofRunitsofthegoodatdate2,thepriceofoneunitisp=1/R.Becauseofthesymmetryofthemodel,wefocusonasymmetricequilib-riumanddescribethebehaviorofarepresentativeintermediary.Althoughintermediariesareheterogeneous,theysolveessentiallythesamedecision 7.3RegulatingLiquidity205problem.Eachhasanequalprobabilityofhavingahighoralowpropor-tionofearlyconsumers,anditisthenumberofearlyconsumers,λHorλL,thatmatterstotheintermediary,notthestateofnature(H,L)or(L,H).Sowecandescribetheintermediaries’decisionproblemintermsoftheinter-mediary’s“state”HorL,meaningthenumberofearlyconsumersisλHorλL.Atdate0,anintermediarytakesinadepositofoneunitfromeachcon-sumerandinvestsitinaportfolio(x,y)andacontingentconsumptionplanc=(c1H,c2H,c1L,c2L).Sinceallintermediarieschoosethesameportfolioandconsumptionplan,wecandescribeanallocationbyatriple(x,y,c).Anallocation(x,y,c)isattainableifitsatisfiesthemarket-clearingconditionsateachdate.Atdate0thisrequiresthattotalinvestmentequaltheendowmentofgoods:x+y=1.Atdate1,thesupplyofgoodsisy(sincePc2/R.Intuitively,ifthepresentvalueofearlyconsumptionisgreaterthanthepresentvalueoflateconsumption,anincreaseintheproportionoflateconsumerswillincreasethepresentvalueoftotalconsumption.Inordertosatisfyitsbudgetconstraint,theintermediarywillhavetoreduceaverageconsumption.Ifc1Rc2forallpairs(c1,c2)satisfyingthefirst-ordercondition(7.5).Conversely,consumptionishigherinstateHthaninstateLifc1⇐⇒Rσ>1⇐⇒σ>1.RThepresentvalueofconsumptionishigheratdate1thanatdate2ifandonlyifthedegreeofrelativeriskaversionisgreaterthanone.Proposition4Iftheconsumers’degreeofrelativeriskaversionisaconstantσ,thenc1>Rc2forallpairs(c1,c2)satisfyingthefirst-ordercondition(7.5)ifandonlyifσ>1.Theroleofriskaversionindeterminingtheconsumptionallocationhasanintuitiveinterpretation.Thefirst-orderconditionimpliesthatmarginalutil-ityisloweratdate2thanatdate1.Inotherwords,c1islessthanc2.Otherthingsbeingequal(i.e.holdingconstanttheexpectedvalueofconsumption)ariskaverseconsumerwouldprefertoreducetheuncertaintyabouthislevelofconsumption.Otherthingsarenotequal,ofcourse.Inordertoreduceconsumption-riskitisnecessarytoholdmoreoftheshortassetandlessofthelongasset.Amoreliquidportfolioyieldsloweraveragereturnsandprovidesloweraverageconsumption.Giventhistrade-offbetweenconsumptionriskandaverageconsumptionlevels,weshouldexpectthedegreeofriskaversiontoaffecttheintermediary’schoice.Themoreriskaversetheconsumer,themorehevaluesinsuranceandthelowertheaveragelevelofconsumptionheiswillingtoacceptinordertosmoothconsumptionoverthetwoperiods.Thecriticalvalueσ=1correspondstothecaseinwhichitisoptimaltoequalizethepresentvalueofconsumptionbetweenthetwoperiods:c1=c2/R.Ifriskaversionislessthanone,thehighreturnsfromdelayingconsumptionout-weighthevalueofinsuranceandtheoptimalconsumptionallocationgives 208Chapter7.OptimalRegulationearlyconsumptionalowerpresentvaluethanlateconsumption.Ifriskaver-sionisgreaterthanone,thevalueofinsuranceoutweighsthereturnfromdelayingconsumptionandtheoptimalconsumptionallocationgivesearlyconsumptionhigherpresentvaluethanlateconsumption.Figure7.1illustratestherelationshipbetweenriskaversionandconsump-tionrisk.Whenthedegreeofrelativeriskaversionisverylow,itisoptimaltotakeahighriskofbeinganearlyconsumerwhogetsverylowconsumptioninordertohaveachanceofbeingalateconsumerwhogetsaveryhighcon-sumption.Bycontrast,whenthedegreeofrelativeriskaversionishigh,thedifferencebetweenconsumptionatthetwodatesisreducedtothepointwherethepresentvalueoffutureconsumptionislowerthanthevalueofpresentconsumption.2.521.52/c11Rc0.501234s–0.5Figure7.1.Illustrationofrelationshipbetweenσandtheratioofthepresentvaluesc1andc2/RforR=2.Thecorrespondencebetweenriskaversionandthepresentvalueofcon-sumptionatthetwodatestranslatesimmediatelyintoacorrespondencebetweenriskaversionandtheslopeoftheconsumptionfunctionsrelatingthevalueofλandthelevelofconsumption.Ifσisgreaterthanone,bothc1andc2declineasλincreases.Ifσislessthanone,c1andc2increaseasλincreases.Proposition5Supposetheconsumers’degreeofrelativeriskaversionisaconstantσandletc=(c1H,c2H,c1L,c2L)betheoptimalconsumption 7.3RegulatingLiquidity209allocationchosenbytherepresentativeintermediary.Thenc1H1.Figure7.2illustratestheoptimallevelsofconsumptionatdate1anddate2asafunctionofλ,thefractionofearlyconsumers,whenthedegreeofrelativeriskaversionisgreaterthanone.Foreachvalueofλ,consumptionatdate1islessthanconsumptionatdate2,buttheratioofc2toc1islessthanR=2,sothepresentvalueofconsumptionatdate1isgreaterthanthepresentvalueofconsumptionatdate2.Thus,asλincreases,consumptionatbothdatesfalls.2.521.512c,1c0.500.250.50.751λ–0.5–1Figure7.2.Illustrationofrelationshipbetweenλand(c1,c2)forσ=2,R=2.Intermsofwelfare,ifσ>1,consumersarebetteroffwhenthenumberofearlyconsumersislowand,ifσ<1,consumersarebetteroffatbothdateswhenthenumberofearlyconsumersishigh.Thispropertyofequilibriumisthekeytounderstandingthewelfareeffectsofanyinterventioninthemarket.7.3.2Toomuchortoolittleliquidity?Inthelastchapterwesawthat,whenthereisnoaggregateuncertainty,aPareto-efficientallocationgiveseveryconsumeraconsumptionallocation(c1,c2)thatisindependentofthestateofnature.Theprecedinganalysisshowsthat,intheabsenceofArrowsecuritiesthatallowwealthtobetransferredacrossstates,theequilibriumconsumptionallocationwilldependonthefractionofearly 210Chapter7.OptimalRegulationconsumersandhenceonthestate.SoanunconstrainedcentralplannercancertainlyachieveahigherlevelofwelfarethantheintermediariescanintheabsenceofArrowsecurities;however,aswearguedinSection7.1,therelevantquestioniswhethertheequilibriumisconstrainedefficient.Itiswellknownthatmodelswithincompletemarketsaretypicallynotconstrainedefficient,soourpresumptionisthatbymanipulatingdecisionsatdate0theplannercanpotentiallyimproveonthelaisser-faireallocation.Moreprecisely,thereexistsawelfare-improvingintervention,butitisnotobviouswhatitis.Thisisanimportantdistinctionforthepolicymaker:itisnotsufficienttoknowthatthereexistssome(possiblycomplex)policythatwillimprovewelfare.Thepolicymakerneedstoknowwhattodo;otherwise,hemaymakethingsworse.Ourobjectiveinthissectionistocharacterizethewelfare-improvingpolicies.Asweshallsee,eveninthecontextofthissimpleexample,itisnoteasytosaywhattherightpolicyis.Inwhatfollows,weassumethattheplanner’sinterventionislimitedtocon-trollingtheportfoliochoicesoftheintermediaries,specifically,theamountoftheshortassettheyhold.Theplanner’sabilitytoimproveonthelaisser-faireallocationdependsuponthepossibilityofchangingtheequilibriumprices.Inalaisser-faireequilibrium,theintermediarieschooseportfoliosandcon-sumptionplansoptimally,takingasgiventheassetpricestheyfaceinthefuture.Itisrationalforintermediariestotreatpricesasparametersbeyondtheircontrolbecausethenumberofintermediariesissolargethatnosingleintermediarycanhaveasignificantimpactonthemarket-clearingassetprice.Aplanner,ontheotherhand,doesnottakefuturepricesasgiven.Althoughintermediarieswillmakefutureconsumptiondecisionstakingpricesasgiven,theplanneranticipatesthathisinfluenceovertheintermediaries’portfoliodecisionswillhavesomeimpactonprices.Itisthroughtheeffectofportfoliochoicesonpricesthattheplannercanpotentiallyimprovethewelfareoftheintermediaries’depositors.Supposethattheplannerrequiresintermediariestoholdmoreoftheshortassetintheirportfolios.Thisactionwillhavetwoimmediateeffects.First,itwillhaveadirecteffectonportfolios,increasingyandreducingx.Second,itwillchangethemarket-clearingassetpricesatdate1.Presumably,increasingtheamountofliquidityandreducingthestockofthelongassetwillincreasetheassetprice.Theconsumers’welfaredependsonlyonconsumption,sotheeffectofanychangeinportfoliosandpricesonwelfarewillbeindirect.Thechangeintheportfolioandtheassetpricesatdate1willshifttheintermediaries’budgetconstraintsatdate1,causingthemtochoosedifferentconsumptionplans.Weknowthatallfeasibleportfolios(x,y)havethesamemarketvalueatdate1becausethelongandshortassetshaveequalreturns.Thus,toafirstapproximation,asmallchangeintheportfoliohasnoimpactonthe 7.3RegulatingLiquidity211intermediary’sbudgetconstraint.Achangeinprices,ontheotherhand,willhaveanimpactontheintermediary’sbudgetconstraint,infact,itwillhavebothsubstitutionandincomeeffects.Inanalyzingtheimpactofthepolicyinterventiononconsumers’welfare,weonlyneedtopayattentiontoincomeeffects.Theenvelopetheoremassuresusthat,sincetheintermediarychoosestheconsumptionplantomaximizethetypicalconsumer’sexpectedutility,asmallmovementalongthebudgetconstraintwillhavenoimpactonexpectedutility.Soitisonlytheincomeeffectofthepricechangethatisrelevantforwelfareanalysis.SupposethatachangeinyincreasesP(andp)ineachstate.Whatwillbetheincomeeffectofthischange?ConsiderthebudgetconstraintoftheintermediaryinstateH,λHc1+(1−λH)pc2H=y+pRy.Anincreaseinpincreasesthelefthandbecauseitincreasesthepresentvalueoftheconsumptionprovidedatdate2.Anincreaseinpalsoincreasestherighthandsidebecauseitincreasesthepresentvalueofthereturntothelongasset.Thus,theincomeeffectofanincreaseinpisgivenbyRy−(1−λH)c2H.Moreprecisely,thisistheamountbywhichwecouldincreaseexpenditureontheearlyconsumersandstillbalancethebudgetinstateH.Similarly,theincomeeffectofthepricechangeinstateLisgivenbyRy−(1−λL)c2L.Nowthemarket-clearingconditionatdate2requiresthat1[(1−λH)c2H+(1−λL)c2L]=Ry,2so,inthiscase,theincomeeffectsinthetwostatessumtozeroRy−(1−λH)c2H+Ry−(1−λL)c2L=0.Theincomeeffectofapricechangeraisesconsumptioninonestateandlowersconsumptionbyanequalamountintheotherstate.Whenmarketsarecomplete,sothatmarginalutilityofconsumptionisthesameineachstate,atransferofconsumptionfromonestatetoanotherhasnoeffect.Whenmarketsareincomplete,bycontrast,themarginalutilityofconsumptionistypicallyhigherinonestatethanintheotherandthismakesitpossibleforincomeeffectstoincreaseexpectedutility.Suppose,forexample,thatthedegreeofrelativeriskaversionisgreaterthanone,sothatconsumptionateachdateishigherinstateLthaninstateH.ThisimpliesthatRy−(1−λH)c2H>0>Ry−(1−λL)c2L, 212Chapter7.OptimalRegulationsotheincomeeffectofanincreaseinpispositiveinstateHandnegativeinstateL.Furthermore,themarginalutilityofconsumptionforearlyconsumersishigherinstateHthaninstateL.Thus,ifweassumethatthechangeinrealincomeisreflectedintheconsumptionofearlyconsumersineachstate,thegaintotheearlyconsumersinstateHwillmorethanoffsetthelosstotheearlyconsumersinstateL.Formally,(cU1H)Ry−(1−λH)c2H>−U(c1L)Ry−(1−λL)c2L.Byasimilarargument,wewouldgettheoppositeresultifwestartedbyassumingthatthedegreeofrelativeriskaversionwaslessthanone.Thereisnolossofgeneralityinassumingthatonlytheconsumptionoftheearlyconsumerschanges.Bytheenvelopetheorem,wecannotdobetterbydividingthechangeinconsumptionbetweenearlyandlateconsumers.Thus,wehaveanecessaryandsufficientconditionforanimprovementinwelfarefromanincreaseinpwhichwestateasthenextproposition.Proposition6Startingatthelaisser-faireequilibrium,anincreaseinp(orP)iswelfareimproving(i.e.willincreasedepositors’expectedutility)ifandonlyifthedegreeofrelativeriskaversionisgreaterthanone.Itremainstoconnectthechangeinpriceatdate1tothechangeinportfoliosatdate0.Intuitively,weexpectanincreaseinytobeassociatedwithanincreaseinP(orp)becauseitincreasesthesupplyofthegoodatdate1anddecreasesthesupplyofthelongasset.However,inthelaisser-faireequilibriumintermediariesareindifferentbetweenthetwoassetsatdate0andthequantitytheyholdintheirportfoliosisdeterminedbytherequirementsformarketclearingatdate1.Ifconsumptionatdate1goesup,theamountoftheshortassetmustincreasetoprovidethatconsumption.Sowhetheranincreaseinpisassociatedwithanincreaseinydependsonthereactionoftheintermediary’sconsumptionplans.Whatwecansayisthatforanysmallchangeinptherewillbeanequilibriuminwhichintermediaries’portfoliochoicesareconstrainedappropriately.Irwinetal.(2006)haveextendedthesimpleexampleforconsideringtheregulationofliquidityconsideredinAllenandGale(2004)andthischapter.TheyshowthatwhileminimumliquiditycontrolscanleadtoParetoimprove-mentswhenintermediariesarehomogeneousexante,thisisnotthecasewhentheyareheterogeneousexante.Inthiscaseotherpoliciessuchasstate-contingenttaxandtransferschemesorstate-contingentlenderoflastresortpoliciesarenecessaryforimprovementsinefficiency.Whatlessonscanwedrawfromthekindofexerciseconductedinthissection?Forstarters,apolicymakerwhowishestoincreasewelfareneedsto 7.5ConcludingRemarks213havedetailedknowledgeabouttherisk-sharingarrangementsundertakenbythefinancialsector.Moregenerally,theeffectsofincreasedliquidityinthemarketareindirectandworkthroughthegeneral-equilibriumdeterminationofassetpricesandconsumptionplans.Soittakesalotofinformationaboutthestructureofthemodelandtheequilibriumtopredicttheeffectofpolicyonequilibrium.Ifthisistrueinafairlytrivialexample,onewouldexpecttheproblemsfacingapolicymakerinthe‘real’worldtobequitechallenging.7.4LITERATUREREVIEWTherehavebeenanumberofgoodsurveysandoverviewsofbankingregula-tion.TheseincludeHerringandSantomero(2000),Santos(2001),FreixasandSantomero(2004),Barthetal.(2006).Forthisreason,thissectionwillbekeptshort.Thereiswidespreadagreementthatthemostimportantrationaleforbank-ingregulationisthepreventionofsystemicrisk.However,asdiscussedinitiallyinthischapterthereisnotagreementaboutthenatureofthemarketfailurethatleadstothissystemicrisk.Thepoliciesthathavebeenusedtotryandlimitsystemicriskincludecapitaladequacyratios,liquidityrequirements,reserverequirements,depositinsurance,andassetrestrictions.Anotherimportantmotivationforregulationisconsumerprotection.Conflictofinterestrulesandinterestrateceilingsonloansareexamplesofpoliciesaimedatprotectingconsumers.Otherpolicies,suchascompetitionpolicy,aredirectedatindustrygenerallybutenhancetheefficiencyofthebankingsystem.Thegovernmentalsotriestoimplementbroadersocialobjectives,suchasthepreventionofmoneylaunderingthroughreportingrequirementsforlargecashtransactions.DewatripontandTirole(1994)havepointedtoanothercategoryofrationaleforjustifyingbankingregulation.Bankers,likethemanagersofanyothercoporation,needtobemonitoredbyinvestors.Bankdepositorsareparticu-larlyunsuitedforthisrolebecausetheytypicallyhavelimitedresourcesandlimitedexperience.Regulationcanbeasubstituteformonitoringtoensurethebankactsintheinterestsofthedepositors.7.5CONCLUDINGREMARKSInthischapter,wehavearguedthatthefirststepinfindingoptimalregula-torypoliciesistheidentificationofmarketfailure(s).ThemodelconsideredinChapter6providesconditionsunderwhichmarketforcesleadtoanefficient 214Chapter7.OptimalRegulationallocationofresources.Moreover,theoptimalallocationcaninvolvefinancialcrises.Soitisnotthecasethateliminatingsystemicriskisalwaysoptimal.Acarefulanalysisofthecostsandbenefitsofcrisesisnecessarytounderstandwheninterventionisnecessary.Thisanalysisistypicallymissingfrompro-posalsforcapitaladequacyregulationssuchastheBaselAccords.Incompletefinancialmarketsprovideoneplausiblesourceofmarketfailureandapossiblejustificationforcapitalregulation.Theformthisregulationtakesiscomplexandinformationallyintensive,however,soitisnotclearthatthisprovidesthebasisforapracticalpolicy.Onealsoneedstokeepinmindthecontinuingfinancialinnovationthatallowsbankstohedgerisksinevermoresophisti-catedways.Whetherthismakestheassumptionofcompletemarketsrealisticisanopenandimportantempiricalquestion.AllenandGale(2006)containsafurtherdiscussionofsomeoftheseissues.Regulationisonlyoneofthewaysthatgovernmentsinterveneinthefinan-cialsystem.Theotherimportantwayisthroughtheactionsofthecentralbank.Thenextchapterconsiderstheroleofmonetarypolicy.REFERENCESAllen,F.andD.Gale(2003).“CapitalAdequacyRegulation:InSearchofaRationale,”inEconomicsforanImperfectWorld:EssaysinHonorofJosephStiglitzeditedbyR.Arnott,B.Greenwald,R.KanburandB.Nalebuff,Cambridge,MA:MITPress,83–109.Allen,F.andD.Gale(2004).“FinancialIntermediariesandMarkets,”Econometrica72,1023–1061.Allen,F.andD.Gale(2006).“SystemicRiskandRegulation,”inR.StulzandM.Carey(eds.),FinancialRiskandRegulation.Cambridge,MA:NBER..Barth,J.,G.CaprioJr.andR.Levine(2006).RethinkingBankingRegulation:TillAngelsGovern,Cambridge,NewYorkandSydney:CambridgeUniversityPress.Besanko,D.andG.Kanatas(1996).“TheRegulationofBankCapital:DoCapitalStandardsPromoteBankSafety?”JournalofFinancialIntermediation5,160–183.Bhattacharya,S.(1982).“AspectsofMonetaryandBankingTheoryandMoralHazard,”JournalofFinance37,371–384.Dewatripont,M.andJ.Tirole(1994).ThePrudentialRegulationofBanks,Cambridge,MA:MITPress.Freixas,X.andA.Santomero(2004).“RegulationofFinancialIntermediaries:ADis-cussion,”inCreditIntermediationandtheMacroeconomy,ModelsandPerspectives,editedbyS.Bhattacharya,A.BootandA.Thakor,OxfordandNewYork:OxfordUniversityPress.Furlong,F.andM.Keeley(1989).“CapitalRegulationandBankRisk-Taking:ANote,”JournalofBankingandFinance13,883–891. References215Gale,D.(2003).“FinancialRegulationinaChangingEnvironment,”inT.CourcheneandE.Neave(eds.),FramingFinancialStructureinanInformationEnvironment.Kingston,Ontario:JohnDeutschInstitutefortheStudyofEconomicPolicy,Queen’sUniversity.Gale,D.(2004).“NotesonOptimalCapitalRegulation,”inP.St-AmantandC.Wilkins(eds.),TheEvolvingFinancialSystemandPublicPolicy.Ottawa:BankofCanada.Gale,D.andO.Özgür(2005).“AreBankCapitalRatiosTooHighorTooLow:RiskAver-sion,IncompleteMarkets,andOptimalCapitalStructures,”JournaloftheEuropeanEconomicAssociation3,690–700.Geanakoplos,J.andH.Polemarchakis(1986).“Existence,Regularity,andConstrainedSuboptimalityofCompetitiveAllocationsWhentheAssetMarketIsIncomplete,”inW.Heller,R.Starr,andD.Starrett(eds.),EssaysinhonorofKennethJ.Arrow:Volume3,Uncertainty,information,andcommunication.Cambridge,NewYorkandSydney:CambridgeUniversityPress,65–95.Gennotte,G.andD.Pyle(1991).“CapitalControlsandBankRisk,”JournalofBankingandFinance15,805–824.Hellmann,T.,K.Murdock,andJ.Stiglitz(2000).“Liberalization,MoralHazardinBanking,andPrudentialRegulation:AreCapitalRequirementsEnough?”AmericanEconomicReview90,147–165.Herring,R.andA.Santomero(2000).“WhatisOptimalFinancialRegulation?”TheNewFinancialArchitecture,BankingRegulationinthe21stCentury,editedbyB.Gup,Westport,Connecticut:QuorumBooks,51–84.Irwin,G.,V.Saporta,andM.Tanaka(2006).“OptimalPoliciestoMitigateFinancialStabilitywhenIntermediariesareHeterogeneous,”workingpaper,BankofEngland.Keeley,M.(1990).“DepositInsurance,Risk,andMarketPowerinBanking,”AmericanEconomicReview80,1183–1200.Kim,D.andA.Santomero(1988).“RiskinBankingandCapitalRegulation,”JournalofFinance43,1219–1233.Rochet,J-C.(1992).“CapitalRequirementsandtheBehaviourofCommercialBanks,”EuropeanEconomicReview36,1137–1178.Santos,J.(2001).“BankCapitalRegulationinContemporaryBankingTheory:AReviewoftheLiterature,”FinancialMarkets,InstitutionsandInstruments14,289–328. 8MoneyandpricesIntheprecedingchapters,wehaveassumedthatintermediariesofferdepositors“real”contracts,thatis,contractsthataredenominatedintermsofgoods.Forexample,anintermediaryoffersitsdepositorsadepositcontractthatpromisesc1unitsofthegoodifthedepositorwithdrawsatdate1andc2unitsofthegoodifthedepositorwithdrawsatdate2.Inpractice,thetermsofdepositcontractsanddebtcontractsingeneralarewrittenin“nominal”terms,thatis,theyspecifypaymentsintermsofmoney.Economistsoftenjustifythesubstitutionof“real”for“nominal”contractsbyclaimingthatmoneyisa“veil”thathidestherealityweareinterestedin,namely,thegoodsandservicesthatfirmsproduceandindividualsconsume;butthefactthatcontractsarewrittenintermsofmoneyhassomeimportantimplicationsthatarenotcapturedby“real”models.Themostimportantfeatureofnominalcontractsisthatchangesinthepricelevel(thegenerallevelofpricesmeasuredintermsofmoney)changetherealvalueofthecontract.A.C.Pigouwasoneofthefirsttopointouttheimpactofthepricelevelontherealvalueofdebt.Moreprecisely,outsidemoney,thepartofthemoneysupplythatconstitutesaclaimonthegovernment,representspartoftheprivatesector’snetwealth.Afallinthepricelevelbydefinitionincreasestherealvalueofoutsidemoneyand,consequently,increasestheprivatesector’srealwealth,thatis,itswealthmeasuredintermsofgoodsandservices.This“wealtheffect”subsequentlycametobeknownasthe“Pigoueffect.”PigouusedittocriticizeKeynes’argumentthatageneralfallinpriceswouldhavenoeffectondemandforgoodsandservices.Keynesreliedonthefamiliarhomogeneitypropertyofdemandfunctions:becausedemandandsupplydependonlyonrelativeprices,anequalproportionatechangeinpricesandwagesshouldhavenoeffectondemandandsupply.Pigouarguedtothecontrarythatafallinthegenerallevelofpriceswouldincreaseperceivedwealthandthatthismightleadconsumerstodemandmoregoodsandservices.AnimportantelementofPigou’sargumentwasthedistinctionbetweeninsideandoutsidemoney.OutsidemoneyconsistsofcurrencyanddepositswiththeFederalReserveSystem,sometimescalledthemonetarybaseorhigh-poweredmoney.Insidemoneyconsistsofbankdepositsandotherliabilitiesof Chapter8.MoneyandPrices217thebankingsystem.Insidemoneyconsistsofobligationsoftheprivatesectortoitselfwhereasoutsidemoneyrepresentsanobligationofthegovernmenttotheprivatesector.Afallinthepricelevelwillincreasethevalueofprivatedebts,thusreducingthenetwealthofthedebtor;buttherewillbeanequalandoppositeeffectonthewealthofthecreditor.Whenaggregatedovertheentireprivatesector,thesedebtsandcreditswillcancelout,thusleadingtoazeroeffectonthenetwealthoftheprivatesectorasawhole.Thisleavesthewealtheffectofpricechangesonoutsidemoneybutsincethequantityofoutsidemoneyisusuallysmallthiswealtheffectisalsosmall.Toafirstapproximation,thismaybeareasonableapproachinnormaltimes,butincasesoffinancialcriseswherethechangeinthepricelevelislarge,asitwasduringtheGreatDepressionofthe1930’s,itcanbemisleading.Therea-sonisthataverylargefallinthepricelevelmaymakeitimpossibleforfirmsandindividualstopaytheirdebts,inwhichcasetheymaybeforcedtodefaultandseektheprotectionofbankruptcy.Ifbankruptcyhadnorealeffectonthevalueofthecreditors’claims,itmightnotmattertoPigou’sargument;butinpracticebankruptcyofteninvolveslargedeadweightcosts.Theseincludenotonlythelegalcostsofliquidationbutalsothelossoforganizationalcapitalthatoccurswhenanenterpriseceasestobeagoingconcernandthedisloca-tionthatcanspreadthroughouttheeconomywhileproductiveactivitiesarebeingreorganized.Thesedeadweightlosses,whichcanamounttoasignificantfractionofGDP,areoneofthereasonswhyfinancialcrisesareregardedwithsuchhorrorbypolicymakers.Justasafallinthegeneralpricelevelhastheeffectofincreasingthelevelofrealindebtednessintheeconomy,anincreaseinthepricelevelhastheeffectofreducingthereallevelofindebtedness.Oftengovernmentsburdenedwithalargenationaldebthaveresortedtotheexpedientofreducingitbycreatinginflation.Thisisoftenreferredtoas“monetizing”thedebt,sincetheinflationaryprocessbeginswithanincreaseinthemoneysupplyandthefinaleffectistoreducetherealvalueofthedebtandtoincreasethequantityofmoneybalancesheldbyindividuals.Itdoesnotrequirehyperinflationtohaveasignificanteffectontherealvalueofthenationaldebt.SteadybutmodestinflationhadaverysignificantimpactontherealvalueoftheUSnationaldebtleftattheendoftheSecondWorldWar,forexample,andmanyothercountrieshadasimilarexperience.Morerecentepisodesalsoillustratetheimportanceofnominaldebtcon-tracts.AgoodexampleistheAsiancrisisof1997.Manyofthecountriesaffectedbythatcrisishadlargeamountsofexternaldebtdenominatedintermsofdol-lars.Foreignlenderswereunwillingtoacceptdebtissuedintermsofthelocalcurrencybecausetheydidnottrustthegovernmenttomaintainitsvalue.Bycontrast,thesamelendersfeltsomewhatprotectediftheirinvestmentwas 218Chapter8.MoneyandPricesdenominatedindollars.OneexampleofacountryaffectedbytheexistenceofforeigncurrencydebtwasThailand.ManyThaifirmshadborrowedindollarstomakeinvestmentsintheThaieconomy.ThevalueoftheThaicurrency,thebaht,waspeggedtothedollar,whichmayhavemadetheloansseemlessriskytotheborrowers.However,whenthedollarbegantoriseintermsofothercur-rencies,thebahthadtorisewithit,thusmakingThaiexportsmoreexpensiveandcausingthebalanceoftradetodeteriorate.Speculatorsanticipatedthatthegovernmentwouldhavetodevaluethebahtandbegantosell.Thecurrencyattackledtoabalanceofpaymentscrisisandthebahtwasdevalued.ThislefttheThaibusinessesthathadborrowedlargeamountsofdollarsinatenuousposition.Theirrevenuesweredenominatedinbahtandtheirdebtsindollars.Thedebtshadgrownrelativetotheirabilitytopayandmanyhadtodefault.AnotherillustrationoftheimportanceofnominalcontractscomesfromJapan,whichsufferedfromlowinvestmentandslowgrowththroughoutthe1990’sasaresultoftheburstingofanasset-pricebubblein1990.Thegov-ernmentofJapanadoptedKeynesianremedies(governmentexpenditureonconstructionprojectsandlowinterestrates)withlittleeffect.Atonepoint,therewasaconsiderablefearthatdeflation(ageneralfallinthepricesofgoodsandservices)mightcauseaseriousproblem.SincemanyJapanesefirmswereheavilyindebtedandmanybanksweresaddledwithnon-performingloans,thisfearofdeflationwasquitereal.Fortunately,thedeflationwasnottoosevereandthisepisodepassedwithoutfurthermishap,butitprovidedanobjectlessonontherelevanceofthepricelevelwhendebtisdenominatedintermsofmoney.Inthischapterwewanttofocusonsomeofthemorebenignaspectsofthepricelevelwhendebtisdenominatedinnominalterms,inparticular,weshowhowvariationsinthepricelevel,byvaryingtherealvalueofdebt,canintro-duceadesirablelevelofcontingencyinrisk-sharingcontracts.Simpledebtcontractspromisefixedrepayments,independentlyofthestateofnature.Anoptimalrisk-sharingcontract,ontheotherhand,willtypicallymakepaymentscontingentonthestate.Byvaryingprices,acontractthatisfixedinnominaltermscanbemadecontingentinrealterms.Thisincreaseinthecontingencyofthecontractmayimproverisksharingundercertainconditions.8.1ANEXAMPLEWeillustratetheroleofpricelevelvariabilityinsupportingefficientrisksharingbypresentingasimpleexamplebasedononeinAllenandGale(1998).Asusual,weassumethattimeisdividedintothreeperiodsordates,indexedby 8.1AnExample219t=0,1,2.Thereisasinglegood,whichcanbeusedforconsumptionorinvestment,ateachdate.Therearetwo(real)assets,ashortasset,representedbyastoragetechnologywhichproducesoneunitofthegoodatdatet+1foreveryunitinvestedatdatet,andalongasset,representedbyalong-terminvestmenttechnologywhichproducesR>1unitsofthegoodatdate2foreveryunitinvestedatdate0.Thereisalargenumberofidenticalconsumersatdate0,eachwithanendowmentofoneunitofthegoodatdate0andnothingatdate1.Theconsumersaresubjecttoliquiditypreferenceshocks:theyareeitherearlyconsumerswhoonlyvalueconsumptionatdate1orlateconsumerswhoonlyvalueconsumptionatdate2.Therearetwostatesofnature,s=H,L,withprobabilitiesπHandπL,respectively.Theprobabilitythatanindividualbecomesanearlyconsumerdependsonthestate.Letλsdenotethefractionofearlyconsumers,whichequalstheprobabilityofbecominganearlyconsumer,instates=H,L.Weassumedthat0<λL<λH<1.Alluncertaintyisresolvedatthebeginningdate1,whenthetruestateofnatureisrevealedandeachconsumerlearnswhetherheisanearlyoralateconsumer.Weassumethatfreeentryandcompetitionforceintermediariestomaximizetheexpectedutilityofthetypicaldepositor,subjecttoazeroprofitconstraint.Theintermediarytakesadepositofoneunitfromeachdepositoratdate0andinvestsitinaportfolio(x,y)consistingofxunitsofthelongassetandyunitsoftheshortasset.Inexchange,theintermediaryoffersarisk-sharingcontractc=(c1H,c2H,c1L,c2L)thatpromisesctsunitsofconsumptiontoaconsumerwhowithdrawsatdatet=1,2instates=H,L.IfU(c)denotesaconsumer’sutilityfromconsumingcunitsofthegood,theexpectedutilityofthiscontractisπs{λsU(c1s)+(1−λs)U(c2s)}(8.1)t,sandacompetitiveintermediarywillchoosetheportfolio(x,y)andthecon-sumptionallocationctomaximize(8.1)subjecttothefeasibilityconstraintsx+y≤1;(8.2)λsc1s≤y,∀s=H,L;(8.3)λsc1s+(1−λs)c2s≤y+Rx,∀s=H,L.(8.4)Supposethattheportfolio(x,y)hasbeenchosenandconsidertheoptimalallocationofconsumptioninagivenstates.Theconsumptionallocation(c1s,c2s)hastomaximizeexpectedutilityinthatstate,thatis,maximizeλsU(c1s)+(1−λs)U(c2s) 220Chapter8.MoneyandPricessubjecttothefeasibilityconditions(8.3)and(8.4).Weknowthatc1smustbelessthanorequaltoc2s;otherwisewecouldincreaseexpectedutilitybyusingtheshortassettoshiftconsumptiontodate2.Wealsoknowthatifλsc1sp2s.Thenominalreturntoholdingtheshortassetuntildate2isp2s−p1s<0,sothebankshouldbewillingtosellallofthegoodsitproducesfromtheshortassetandthisiswhathappensinequilibriumsinceλsc1s=y.Ontheotherhand,ifλsc1sK>YL>0. 8.3DollarizationandIncentives227Theprobabilityofsuccessdependsontheefforttakenbytheentrepreneur.Iftheentrepreneurtakesefforttheprobabilityofsuccessisπ>0;otherwiseitiszero.ThecostofeffortisC.SupposethattheentrepreneurfinanceshisprojectbyissuingrealordollarbondswithfacevalueD.Foreigninvestorsareassumedtoberiskneutralandthereturnonthesafeassetisone,sotheforeigninvestorsarewillingtolendKifandonlyiftheexpectedrepaymentisequaltoK.Iftheentrepreneurtakesnoeffort,theoutputwillbeYLwithprobabilityone,sothemosttheinvestorswillreceiveisYLwhichislessthanK.Sotheinvestorswillnotbewillingtobuythebondsissuedbytheentrepreneurunlesstheyaresurethathewillmakeaneffort.Nowsupposethattheentrepreneurmakesaneffort.Iftheprojectissuc-cessful,hecanrepayDCtheentrepreneuriswillingtoundertaketheprojectwithcostlyeffortandeveryoneishappy.Nowsupposethattheentrepreneurborrowsinthedomesticcurrencyandtheexchangerateeiscontrolledbythegovernment.Exante,thegovernmenthasanincentivetosaythatitwillmaintaintheexchangerateinordertoencourageforeigninvestment.Expost,iftheinvestmentprojectsareunsuc-cessful,ithasanincentivetoreducetheexchangeratesothatthefacevalue 228Chapter8.MoneyandPricesofthedebtisonlyeD=YL.ThisallowstheentrepreneurtoretainhisprivatebenefitBfromavoidingdefaultandstillallowstheforeigninvestorstoretaintheirclaimtotheoutputYLinthelowstate.Itmightbethoughtthatsincetheforeigninvestorsreceivethesamepaymentineachstate,theydonotcarewhetherthedomesticcurrencyisdevaluedornot.Theyshouldcarebecausetheentrepreneur’sincentiveshavechanged.Sincehereceivestheprivatebene-fitBinanyevent,itnolongeraffectshiswillingnesstotakeeffort.ThenetgaintotakingeffortisnowπYH+(1−π)YL−K0.17.Theexpectedpayoffof1.5ontheinvestmentin1unitofthesafeassetisthesameasontheinvestmentof1/1.5unitsoftheriskyasset.Theriskyassetismoreattractivetotheborrowerthough.Withthesafeassettheborrowerobtains0.17andthelenderobtains1.33.Withtheriskyassettheborrowerobtains0.67whilethelenderobtains0.25×1.33+0.75×1×(1/1.5)=1.5−0.67=0.83.Theriskofdefaultallows0.5inexpectedvaluetobeshiftedfromthelendertotheborrower.Thisistheriskshiftingproblem.Ifthelendercouldpreventthe 9.1AgencyProblemsandPositiveBubbles241borrowerfrominvestingintheriskyassethewoulddosobuthecannotsincethisisunobservable.Whatistheequilibriumpriceoftheriskyassetgiventhisagencyproblem?Inanequilibriumwherethesafeassetisused,thepriceoftheriskyasset,P,willbebidupsinceitisinfixedsupply,untiltheexpectedprofitofborrowersisthesameforboththeriskyandthesafeasset:
10.25×6−1.33+0.75×0=1.5−1.33PsoP=3.Thereisabubblewiththepriceoftheriskyassetabovethebenchmarkof1.5.Theideathatthereisarisk-shiftingproblemwhenthelenderisunabletoobservehowtheborrowerinveststhefundsisnotnew(see,e.g.JensenandMeckling1976andStiglitzandWeiss1981).However,ithasnotbeenwidelyappliedintheassetpricingliterature.Insteadofthestandardresultincorporatefinancetextbooksthatdebt-financedfirmsarewillingtoacceptnegativenetpresentvalueinvestments,themanifestationoftheagencyproblemhereisthatthedebt-financedinvestorsarewillingtoinvestinassetspricedabovetheirfundamental.Theamountofriskthatisshifteddependsonhowriskytheassetis.Thegreatertheriskthegreaterthepotentialtoshiftriskandhencethehigherthepricewillbe.Toillustratethisconsiderthepreviousexamplebutsupposethereturnontheriskyassetisamean-preservingspreadoftheoriginalreturns,asshowninTable9.2.Nowthepriceoftheriskyassetisgivenby
10.25×9−1.33+0.75×0=1.5−1.33PTable9.2.AssetSupplyInvestmentatdate1Payoffatdate29withprob.0.25Risky1PR=0withprob.0.75ER=2.25 242Chapter9.BubblesandCrisessoP=4.5.Moreriskisshiftedandasaresultthepriceoftheriskyassetisbiduptoanevenhigherlevel.Itisinterestingtonotethatinboththestockmarketboomofthe1920’sandtheoneinthe1990’sthestocksthatdidbestwere“high-tech”stocks.Inthe1920’sitwasradiostocksandutilitiesthatwerethestarperformers(seeWhite1990).Inthe1990’sitwastelecommunications,mediaandentertainment,andtechnologystocksthatdidthebest.Itispreciselythesestockswhichhavethemostuncertainpayoffsbecauseofthenatureofthebusinesstheyarein.Oneofthecrucialissuesiswhythebanksarewillingtolendtotheinvestorsgiventhechanceofdefault.ToseethisconsideragainthecasewherethepayoffsontheriskyassetarethoseinTable9.1andP=3.Inthiscasethequantityoftheriskyassetpurchasedwhensomebodyborrows1is1/P=1/3.Intheequilibriaconsideredabovetheinvestorsareindifferentbetweeninvestinginthesafeandriskyasset.Supposeforthesakeofillustrationthefixedsupplyoftheriskyassetis1.Theamountoffundsdepositorshaveis10andthenumberofborrowersis10.IntheequilibriumwhereP=3,3oftheborrowersinvestintheriskyassetand7inthesafeinorderforthefixedsupplyof1unitoftheriskyassettobetakenup.Inthiscase30percentofborrowersareinriskyassetsand70percentareinsafeassets.Abank’sexpectedpayofffromlendingoneunitisthengivenbythefollowingexpression.Bank’sexpectedpayoff=0.3[0.25×1.33+0.75×(1/3)×1]+0.7[1.33]=1.11.Thefirsttermisthepayofftothebankfromthe30percentofinvestorsintheriskyasset.Ifthepayoffis6,whichoccurswithprobability0.25,theloanandinterestisrepaidinfull.Ifthepayoffis1,whichoccurswithprobability0.75,theborrowerdefaultsandthebankreceivestheentireproceedsfromthe1/3unitownedbytheborrower.Thepayoffisthus(1/3)×1.The70percentofinvestorsinthesafeassetareabletopayofftheirloanandinterestof1.33infull.Ifthebankingsectoriscompetitivethereceiptsfromlending,1.11,willbepaidouttodepositors.Inthiscaseitisthedepositorsthatbearthecostoftheagencyproblem.Inorderforthisallocationtobefeasiblemarketsmustbesegmented.Thedepositorsandthebanksmustnothaveaccesstotheassetsthattheinvestorswhoborrowinvestin.Clearlyiftheydidtheywouldbebetterofftojustinvestinthesafeassetratherthanputtheirmoneyinthebank. 9.1AgencyProblemsandPositiveBubbles2439.1.2CreditandinterestratedeterminationThequantityofcreditandtheinterestratehavesofarbeentakenasexogenous.Thesefactorsareincorporatedintheexamplenexttoillustratetherelationshipbetweentheamountofcreditandthelevelofinterestrates.WestartwiththesimplestcasewherethecentralbankdeterminestheaggregateamountofcreditBavailabletobanks.Itdoesthisbysettingreserverequirementsanddetermin-ingtheamountofassetsavailableforuseasreserves.ForeaseofexpositionwedonotfullymodelthisprocessandsimplyassumethecentralbanksetsB.Thebankingsectoriscompetitive.Thenumberofbanksisnormalizedat1andthenumberofinvestorsisalsonormalizedto1.EachinvestorwillthereforebeabletoborrowBfromeachbank.Thereturnonthesafeassetisdeterminedbythemarginalproductofcapitalintheeconomy.Thisinturndependsontheamountoftheconsumptiongoodxthatisinvestedatdate1intheeconomy’sproductivetechnologytoproducef(x)unitsatdate2.ThetotalamountthatcanbeinvestedisBandtheamountthatisinvestedatdate1intheriskyassetsincethereis1unitisP.Hencethedate1budgetconstraintimpliesthatx=B−P.Itisassumedf(x)=3(B−P)0.5.(9.1)Providedthemarketforloansiscompetitivetheinterestrateronbankloanswillbesuchthatr=f(B−P)=1.5(B−P)−0.5.(9.2)Atthislevelborrowingandinvestinginthesafeassetwillnotyieldanyprofitsforinvestors.Ifrwaslowerthanthistherewouldbeaninfinitedemandforbankloanstobuythesafeasset.Ifrwashigherthanthistherewouldbezerodemandforloansandnobodywouldinvestinthesafeassetbutthisisacontradictionsincef(0)=∞.TheamounttheinvestorswillbepreparedtopayfortheriskyassetassumingitspayoffsareasinTable9.1isthengivenby
10.25×6−r+0.75×0=0.PUsing(9.2)inthis,P=4(B−P)0.5. 244Chapter9.BubblesandCrisesSolvingforPgives√P=8(−1+1+0.25B).(9.3)WhenB=5thenP=4andr=1.5.TherelationshipbetweenPandBisshownbythesolidlineinFigure9.1.Bycontrollingtheamountofcreditthecentralbankcontrolsthelevelofinterestratesandthelevelofassetprices.Notethatthisrelationshipisdifferentfromthatinthestandardassetpricingmodelwhenthepriceoftheriskyassetisthediscountedexpectedpayoff.2.25PF=.rThiscaseisillustratedbythedottedlineinFigure9.1.Acomparisonofthetwocasesshowsthatthefundamentalisrelativelyinsensitivetotheamountofcreditcomparedtothecasewherethereisanagencyproblem.Changesinaggregatecreditcancauserelativelylargechangesinassetpriceswhenthereisanagencyproblem.7Agency6Fundamental54P3210024680.40.81.21.62.42.83.23.64.44.85.25.66.46.87.27.6BFigure9.1.Creditandassetprices(fromFigure1ofAllenandGale2004). 9.1AgencyProblemsandPositiveBubbles2459.1.3FinancialriskTheprevioussectionassumedthatthecentralbankcoulddeterminetheamountofcreditB.Inpracticethecentralbankhaslimitedabilitytocon-troltheamountofcreditandthismeansBisrandom.Inadditiontheremaybechangesofpolicypreferences,changesofadministration,andchangesintheexternalenvironmentwhichcreatefurtheruncertaintyaboutthelevelofB.Thisuncertaintyisparticularlygreatincountriesundergoingfinanciallib-eralization.Inordertoinvestigatetheeffectofthisuncertaintyanextraperiodisaddedtothemodel.Betweendates1and2everythingisthesameasbefore.Betweendates0and1theonlyuncertaintythatisresolvedisaboutthelevelofBatdate1.Thusbetweendates0and1thereisfinancialuncertainty.TheuncertaintyaboutaggregatecreditBatdate1causesuncertaintyaboutpricesatdate1.Giventhatinvestorsareborrowingfrombanksatdate0inthesamewayasbeforethispriceuncertaintyagainleadstoanagencyproblemandriskshifting.Thepriceoftheriskyassetatdate0willreflectthispriceuncertaintyandcanleadtheassetpricetobeevenhigherthanatdate1.Supposethatthereisa0.5probabilitythatB=5anda0.5probabilitythatB=7atdate1.Thenusing(9.2)and(9.3)thepricesandinterestratesareasshowninTable9.3.Table9.3.ProbabilityBPr0.5541.50.575.271.14Thepricingequationatdate0is
10.5×5.27−r0+0.5×0=0,P0wherer0,thedate0interestrate,isgivenby(9.2)withBandPreplacedbyB0andP0.Substitutingforr0andsimplifying5.270.5P0=(B0−P0).1.5TakingB0=6andsolvingforr0andP0givesr0=1.19P0=4.42. 246Chapter9.BubblesandCrisesAswhentheuncertaintyisduetovariationsinassetreturns,thegreaterthefinancialuncertaintythegreaterisP0.ConsiderameanpreservingspreadonthefinancialuncertaintysothatTable9.3isreplacedbyTable9.4.Table9.4.ProbabilityBPr0.543.141.810.585.861.03Inthiscaseitcanbeshownr0=1.27P0=4.61.Therisk-shiftingeffectoperatesforfinancialriskinthesamewayasitdoesforrealrisk.Althoughtheexpectedpayoffatdate2isonly2.25thepriceoftheriskyassetatdate1inthelastcaseis4.61.Thepossibilityofcreditexpansionoveraperiodofyearsmaycreateagreatdealofuncertaintyabouthowhighthebubblemaygoandwhenitmaycollapse.Thisisparticularlytruewheneconomiesareundergoingfinancialliberalization.Asmoreperiodsareaddeditispossibleforthebubbletobecomeverylarge.Themarketpricecanbemuchgreaterthanthefundamental.9.1.4FinancialfragilityTheexamplesintheprevioussectionillustratedthatwhatisimportantindeterminingtheriskyasset’spriceatdate0isexpectationsaboutaggregatecreditatdate1.Ifaggregatecreditgoesupthenassetpriceswillbehighanddefaultwillbeavoided.However,ifaggregatecreditgoesdownthenassetpriceswillbelowanddefaultwilloccur.Theissuehereiswhatisthedynamicpathofaggregatecredit.Thepointisthattheexpectationofcreditexpansionisalreadytakenintoaccountintheinvestors’decisionsabouthowmuchtoborrowandhowmuchtopayfortheriskyasset.Ifcreditexpansionislessthanexpected,orperhapssimplyfallsshortofthehighestanticipatedlevels,theinvestorsmaynotbeabletorepaytheirloansanddefaultoccurs.InAllenandGale(2000)itisshownthatevenifcreditisalwaysexpandedthentheremaystillbedefault.Infactitisshownthattherearesituationswheretheamountofcreditwillbearbitrarilyclosetotheupperboundofwhatisanticipatedandwidespreaddefaultisalmostinevitable. 9.2BankingCrisesandNegativeBubbles2479.2BANKINGCRISESANDNEGATIVEBUBBLESIntheprevioussectionwefocusedonhowassetpricescouldgettoohighbecauseofanagencyproblembetweenlendersandthepeoplemakinginvest-mentdecisions.Inthissectionweconsiderwhathappenswhenassetpricesarelowrelativetothefundamental.Animportantfeatureofmanyofthehistoricandrecentbankingcrisesisthecollapseinassetpricesthataccompaniesthem.ThepurposeofthissectionistoconsiderthisphenomenonusingthemodelofAllenandGale(1998,2004).Westartbydevelopingasimplemodelandderivetheoptimalallocationofresources.Ifthereisamarketforriskyassetsthatallowsbankstoselltheirassetsthentheallocationisnotefficient.Thesimul-taneousliquidationofallbanks’assetsthataccompaniesacrisisleadstoaneg-ativebubbleandinefficientrisksharing.However,byadoptinganappropriatemonetarypolicyacentralbankcanimplementtheoptimalallocation.9.2.1ThemodelTimeisdividedintothreeperiodst=0,1,2.Therearetwotypesofassets,asafeassetandariskyasset,andaconsumptiongood.Thesafeassetcanbethoughtofasastoragetechnology,whichtransformsoneunitoftheconsump-tiongoodatdatetintooneunitoftheconsumptiongoodatdatet+1.Theriskyassetisrepresentedbyastochasticproductiontechnologythattrans-formsoneunitoftheconsumptiongoodatdatet=0intoRunitsoftheconsumptiongoodatdatet=2,whereRisanon-negativerandomvariablewithRHwithprobabilityπR=RLwithprobability1−π.Atdate1depositorsobserveasignal,whichcanbethoughtofasaleadingeco-nomicindicator,similarlytoGorton(1988).ThissignalpredictswithperfectaccuracythevalueofRthatwillberealizedatdate2.Initiallyitisassumedthatconsumptioncanbemadecontingentontheleadingeconomicindicator,andhenceonR.Subsequently,weconsiderwhathappenswhenbanksarerestrictedtoofferingdepositorsastandarddepositcontract,thatis,acontractwhichisnotexplicitlycontingentontheleadingeconomicindicator.Thereisacontinuumofexanteidenticaldepositors(consumers)whohaveanendowmentof1oftheconsumptiongoodatthefirstdateandnoneatthesecondandthirddates.Consumersareuncertainabouttheirtimepreferences.Somewillbeearlyconsumers,whoonlywanttoconsumeatdate1,andsome 248Chapter9.BubblesandCriseswillbelateconsumers,whoonlywanttoconsumeatdate2.Atdate0consumersknowtheprobabilityofbeinganearlyorlateconsumer,buttheydonotknowwhichgrouptheybelongto.Alluncertaintyisresolvedatdate1wheneachconsumerlearnswhetherheisanearlyorlateconsumerandwhatthereturnontheriskyassetisgoingtobe.Forsimplicity,weassumethatthereareequalnumbersofearlyandlateconsumersandthateachconsumerhasanequalchanceofbelongingtoeachgroup.Thenatypicalconsumer’sexpectedutilitycanbewrittenasλU(c1)+(1−λ)U(c2)(9.4)wherectdenotesconsumptionatdatet=1,2.Theperiodutilityfunc-tionsU(·)areassumedtobetwicecontinuouslydifferentiable,increasingandstrictlyconcave.Aconsumer’stypeisnotobservable,solateconsumerscanalwaysimitateearlyconsumers.Therefore,contractsexplicitlycontingentonthischaracteristicarenotfeasible.Theroleofbanksistomakeinvestmentsonbehalfofconsumers.Weassumethatonlybankscanholdtheriskyasset.Thisgivesthebankanadvantageoverconsumersintworespects.First,thebankscanholdaportfolioconsistingofbothtypesofassets,whichwilltypicallybepreferredtoaportfolioconsistingofthesafeassetalone.Second,bypoolingtheassetsofalargenumberofconsumers,thebankcanofferinsurancetoconsumersagainsttheiruncertainliquiditydemands,givingtheearlyconsumerssomeofthebenefitsofthehigh-yieldingriskyassetwithoutsubjectingthemtothevolatilityoftheassetmarket.Freeentryintothebankingindustryforcesbankstocompetebyofferingdepositcontractsthatmaximizetheexpectedutilityoftheconsumers.Thus,thebehaviorofthebankingindustrycanberepresentedbyanoptimalrisk-sharingproblem.Avarietyofdifferentrisk-sharingproblemscanbeusedtorepresentdifferentassumptionsabouttheinformationalandregulatoryenvironment.9.2.2OptimalrisksharingInitiallyconsiderthecasewherebankscanwritecontractsinwhichtheamountthatcanbewithdrawnateachdateiscontingentonR.Thisprovidesabench-markforoptimalrisksharing.Sincetheriskyassetreturnisnotknownuntiltheseconddate,theportfoliochoiceisindependentofR,butthepaymentstoearlyandlateconsumers,whichoccurafterRisrevealed,willdependonit.Letyandx=1−ydenotetherepresentativebank’sholdingoftherisky 9.2BankingCrisesandNegativeBubbles249andsafeassets,respectively.Thedepositcontractcanberepresentedbyapairoffunctions,c1(R)andc2(R)whichgivetheconsumptionofearlyandlateconsumersconditionalonthereturntotheriskyasset.Theoptimalrisk-sharingproblemcanbewrittenasfollows.maxE[λU(c1(R))+(1−λ)U(c2(R))]s.t.(i)y+x≤1;(ii)λc1(R)≤y;(9.5)(iii)λc1(R)+(1−λ)c2(R)≤y+Rx;(iv)c1(R)≤c2(R).Thefirstconstraintsaysthatthetotalamountinvestedmustbelessthanorequaltotheamountdeposited.Thereisnolossofgeneralityinassumingthatconsumersdeposittheirentirewealthwiththebank,sinceanythingtheycandothebankcandoforthem.Thesecondconstraintsaysthattheholdingofthesafeassetmustbesufficienttoprovidefortheconsumptionoftheearlyconsumersatdate1.Thebankmaywanttoholdstrictlymorethanthisamountandrollitovertothefinalperiod,inordertoreducetheuncertaintyofthelateconsumers.Thenextconstraint,togetherwiththeprecedingone,saysthattheconsumptionofthelateconsumerscannotexceedthetotalvalueoftheriskyassetplustheamountofthesafeassetleftoveraftertheearlyconsumersarepaidoff,thatis,(1−λ)c2(R)≤(y−λc1(R))+Rx.(9.6)Thefinalconstraintistheincentivecompatibilityconstraint.ItsaysthatforeveryvalueofR,thelateconsumersmustbeatleastaswelloffastheearlyconsumers.Sincelateconsumersarepaidoffatdate2,anearlyconsumercannotimitatealateconsumer.However,alateconsumercanimitateanearlyconsumer,obtainc1(R)atdate1,andusethestoragetechnologytoprovidehimselfwithc1(R)unitsofconsumptionatdate2.Itwillbeoptimaltodothisunlessc1(R)≤c2(R)foreveryvalueofR.Thefollowingassumptionsaremaintainedthroughoutthesectiontoensureinterioroptima.ThepreferencesandtechnologyareassumedtosatisfytheinequalitiesE[R]>1(9.7)andU(0)>E[U(RE)R].(9.8) 250Chapter9.BubblesandCrisesThefirstinequalityensuresapositiveamountoftheriskyassetisheldwhilethesecondensuresapositiveamountofthesafeassetisheld.Anexaminationoftheoptimalrisk-sharingproblemshowsusthattheincentiveconstraint(iv)canbedispensedwith.Toseethis,supposethatwesolvetheproblemsubjecttothefirstthreeconstraintsonly.Anecessarycondi-tionforanoptimumisthattheconsumptionofthetwotypesbeequal,unlessthefeasibilityconstraintλc1(R)≤yisbinding,inwhichcaseitfollowsfromthefirst-orderconditionsthatc1(R)≤c2(R).Thus,theincentiveconstraintwillalwaysbesatisfiedifweoptimizesubjecttothefirstthreeconstraintsonlyandthesolutionto(9.5)isthefirst-bestallocation.ItcanbeshownthatthesolutiontotheproblemisyRxc1(R)=c2(R)=y+Rxif≥,(9.9)λ1−λyRxc1(R)=y/λ,c2(R)=Rx/(1−λ)if<,(9.10)λ1−λy+x=1(9.11)(cE[U1(R))]=E[U(c2(R))R].(9.12)(SeeAllenandGale1998foraformalderivationofthis.)TheoptimalallocationisillustratedinFigure9.2.Whenthesignalatdate1indicatesthatR=0atdate2,boththeearlyandlateconsumersreceiveysinceyisallthatisavailableanditisefficienttoequateconsumptiongiventheformoftheobjectivefunction.Theearlyconsumersconsumetheirshareλyatdate1withtheremaining(1−λ)ycarriedoveruntildate2forthelateconsumers.AsRincreasesbothgroupscanconsumemoreuntily/λ=Rx¯/(1−λ).ProvidedR<(1−λ)y/λx≡Rtheoptimalallocationinvolvescarryingoversomeofct(R)c2(R)d=y/λc1(R)yR=(1–λ)y/λxRFigure9.2.Optimalrisksharing. 9.2BankingCrisesandNegativeBubbles251theliquidassettodate2tosupplementthelowreturnsontheriskyassetforlateconsumers.WhenthesignalindicatesthatRwillbehighatdate2(i.e.R≥(1−λ)y/λx≡R),thenearlyconsumersshouldconsumeasmuchaspossibleatdate1whichisy/λsinceconsumptionatdate2willbehighinanycase.Ideally,thehighdate2outputwouldbesharedwiththeearlyconsumersatdate1,butthisisnottechnologicallyfeasible.Itisonlypossibletocarryforwardconsumption,notbringitbackfromthefuture.Toillustratetheoperationoftheoptimalcontract,weadoptthefollowingnumericalexample:U=ln(ct);EU=0.5ln(c1)+0.5ln(c2);(9.13)2withprobability0.9;R=0.6withprobability0.1.Fortheseparameters,itcanreadilybeshownthat(y,x)=(0.514,0.486)andR¯=1.058.Thelevelsofconsumptionarec1(2)=1.028;c2(2)=1.944withprobability0.9;c1(0.6)=c2(0.6)=0.806withprobability0.1.ThelevelofexpectedutilityachievedisEU=0.290.9.2.3OptimaldepositcontractsSupposenextthatcontractscan’tbeexplicitlyconditionedonR.Letddenotethefixedpaymentpromisedtotheearlyconsumersatdate1.Initiallyweassumethereisnomarketforthelongtermasset.Ifthebankisunabletomakethepaymentdtoallthoserequestingitatdate1thentheshort-termassetthatitdoeshaveissplitupequallybetweenthem.Sincethebankingsectoriscompetitiveandtheobjectiveofthebankistomaximizetheexpectedutilityofdepositorsthelateconsumerswillalwaysbepaidwhateverisavailableatthelastdate.Inthatcase,inequilibriumtheearlyandlateconsumerswillhavethesameconsumption.Itcanstraightforwardlybeseenintheexamplethattheoptimalallocationcanbeimplementedusingadepositcontract.Todothisthebankchooses(y,x)=(0.514,0.486)andsetsd=1.028.Anythingleftoveratdate2isdistributedequallyamongtheremainingdepositors.WhenR=RH=2thebankusesalltheshortassettopayouttoitsearlyconsumersandthey 252Chapter9.BubblesandCrisesreceivec1(2)=d=0.514/0.5=1.028.Thelateconsumersreceivec2(2)=0.486×2/0.5=1.944.WhenR=RL=0.6thelateconsumerscancalculatethatifalltheearlyconsumersweretoreceived=1.028andexhaustalltheshortassetthentheamountremainingforeachlateconsumeratdate2wouldbe0.6×0.486=0.583<1.028.0.5Thussomeofthelateconsumerswillalsowithdrawatdate1.Thismeansthebankwillnotbeabletosatisfyallthosewithdrawing.Sincethereisnomarketforthelongasset,whathappensisthattheavailableproceedsaresplitequallyamongthoseaskingtowithdraw,asexplainedabove.Supposeα(0.6)lateconsumerswithdrawearly,thentheλ+α(0.6)withdrawingearlyandthe1−λ−α(0.6)withdrawinglatewillhavethesameexpectedutilitywhenyRx=.λ+α(0.6)1−λ−α(0.6)Substituting0.5140.6×0.486=0.5+α(0.6)1−0.5−α(0.6)andsolvingforα(0.6)givesα(0.6)=0.138.Hencetheconsumptionofeverybodyis0.5140.6×0.486c1(0.6)==c2(0.6)==0.806.0.5+0.1381−0.5−0.138InAllenandGale(1998)itisshownmoregenerallythattheoptimalallocationcanbeimplementedusingadepositcontract.9.2.4AnassetmarketSupposenextthatthereisacompetitivemarketforliquidatingthelong-termassetforpriceP.Ifthebankcanmakethepaymentdtothedepositorswhorequesttowithdrawatdate1thenitcontinuesuntildate2.Butifthebankisunabletodothisthenitgoesbankruptanditsassetsareliquidatedanddistributedonaproratabasisamongitsdepositors.Thenthestandarddeposit 9.2BankingCrisesandNegativeBubbles253contractpromisestheearlyconsumerseitherdor,ifthatisinfeasible,anequalshareoftheliquidatedassets.Theparticipantsinthelong-termassetmarketarethebanks,whouseittoobtainliquidity,andalargenumberofwealthy,riskneutralspeculatorswhohopetomakeaprofitincasesomebankhastoselloffassetscheaplytogetliquidity.Thespeculatorsholdsomecash(thesafeasset)inordertopurchasetheriskyassetwhenitspriceatdate1issufficientlylow.Thereturnonthecashislow,butitisoffsetbytheprospectofspeculativeprofitswhenthepriceoftheriskyassetfallsbelowitsfundamentalvalue.Supposetheriskneutralspeculatorsholdsomeportfolio(ys,xs).Theycannotshortsellorborrow.Inequilibriumtheywillbeindifferentbetweentheportfolio(ys,xs)andputtingalltheirmoneyintheriskyasset.TheimpactofintroducingtheassetmarketcanbeillustratedusingFigure9.3.Thegraphsinthisfigurerepresenttheequilibriumconsump-tionlevelsofearlyandlateconsumers,respectively,asafunctionoftheriskyassetreturnR.ForhighvaluesofR(i.e.R≥R∗),thereisnopossibilityofabankrun.Theconsumptionofearlyconsumersisfixedbythestandarddepositcontractatc1(R)=dandtheconsumptionoflateconsumersisgivenbythebudgetconstraintc2(R)=(y+Rx−d)/(1−λ).ForlowervaluesofR(RR∗thespeculatorscontinuetoholdbothassetsandareindifferentbetweenthem.Sinceoneunitofthesafeassetisworth1inthelastperiod,thefundamentalvalueofeachunitoftheriskyassetisR/1=R.ForRR¯andp1(R)c1(R)=D=p2(R)c2(R)forRR¯,andp1(R)c1(R)=D=p2(R)c2(R)forR1aftertwoperiods.Thelongassethasahigherreturnifheldtomaturity,butliquidatingitinthemiddleperiodiscostly,soitisnotveryusefulforprovidingconsumptiontoearlyconsumers.Thebankingsectorisperfectlycompetitive,sobanksofferrisk-sharingcontractsthatmaximizedepositors’exanteexpectedutility,subjecttoazero-profitconstraint.Usingthisframework,weconstructasimplemodelinwhichsmallshocksleadtolargeeffectsbymeansofcontagion.Moreprecisely,ashockwithin 262Chapter10.Contagionasinglesectorhaseffectsthatspreadtoothersectorsandleadeventuallytoaneconomy-widefinancialcrisis.Thisformofcontagionisdrivenbyrealshocksandreallinkagesbetweenregions.Aswehaveseen,oneviewisthatfinancialcrisesarepurelyrandomevents,unrelatedtochangesintherealeconomy(Kindleberger1978).Themodernversionofthisview,developedbyDiamondandDybvig(1983)andothers,isthatbankrunsareself-fulfillingprophecies.Thedisadvantageoftreatingcontagionasa“sunspot”phenomenonisthat,withoutsomerealconnectionbetweendifferentregions,anypatternofcorrel-ationsispossible.Sosunspottheoriesdonotprovideacausallinkbetweencrisesindifferentregions.Weadoptthealternativeviewthatfinancialcrisesareanintegralpartofthebusinesscycle(Mitchell1941;Gorton1988;AllenandGale1998)andshowthat,undercertaincircumstances,anyequilibriumofthemodelmustbecharacterizedbycontagion.Theeconomyconsistsofanumberofregions.Thenumberofearlyandlateconsumersineachregionfluctuatesrandomly,buttheaggregatedemandforliquidityisconstant.Thisallowsforinterregionalinsuranceasregionswithliquiditysurplusesprovideliquidityforregionswithliquidityshortages.Onewaytoorganizetheprovisionofinsuranceisthroughtheexchangeofinterbankdeposits.SupposethatregionAhasalargenumberofearlyconsumerswhenregionBhasalownumberofearlyconsumers,andviceversa.SinceregionsAandBareotherwiseidentical,theirdepositsareperfectsubstitutes.Thebanksexchangedepositsatthefirstdate,beforetheyobservetheliquidityshocks.IfregionAhasahigherthanaveragenumberofearlyconsumersatdate1,thenbanksinregionAcanmeettheirobligationsbyliquidatingsomeoftheirdepositsinthebanksofregionB.RegionBishappytooblige,becauseithasanexcesssupplyofliquidity,intheformoftheshortasset.Atthefinaldate,theprocessisreversed,asbanksinregionBliquidatethedepositstheyholdinregionAtomeettheabove-averagedemandfromlateconsumersinregionB.Inter-regionalcrossholdingsofdepositsworkwellaslongasthereisenoughliquidityinthebankingsystemasawhole.Ifthereisanexcessdemandforliquidity,however,thefinanciallinkagescausedbythesecrossholdingscanturnouttobeadisaster.Whilecrossholdingsofdepositsareusefulforreallocatingliquiditywithinthebankingsystem,theycannotincreasethetotalamountofliquidity.Iftheeconomy-widedemandfromconsumersisgreaterthanthestockoftheshortasset,theonlywaytoprovidemoreconsumptionistoliquidatethelongasset.Thereisalimittohowmuchcanbeliquidatedwithoutprovokingarunonthebank,however,soiftheinitialshockrequiresmorethanthisbuffer,therewillbearunonthebankandthebankisforcedintobankruptcy.Banksholdingdepositsinthedefaultingbankwillsufferacapitalloss,whichmaymakeitimpossibleforthemtomeettheircommitmentstoprovideliquidityintheirregion.Thus,whatbeganasafinancialcrisisinone 10.1LiquidityPreference263regionwillspreadbycontagiontootherregionsbecauseofthecrossholdingsofdeposits.Whetherthefinancialcrisisdoesspreaddependscruciallyonthepatternofinter-connectednessgeneratedbythecrossholdingsofdeposits.Wesaythattheinterbanknetworkiscompleteifeachregionisconnectedtoalltheotherregionsandincompleteifeachregionisconnectedwithasmallnumberofotherregions.Inacompletenetwork,theamountofinterbankdepositsthatanybankholdsisspreadevenlyoveralargenumberofbanks.Asaresult,theinitialimpactofafinancialcrisisinoneregionmaybeattenuated.Inanincompletenetwork,ontheotherhand,theinitialimpactofthefinancialcrisisisconcentratedinthesmallnumberofneighboringregions,withtheresultthattheyeasilysuccumbtothecrisistoo.Aseachregionisaffectedbythecrisis,itpromptsprematureliquidationoflongassets,withaconsequentlossofvalue,sothatpreviouslyunaffectedregionsfindthattheytooareaffected.Itisimportanttonotetheroleofafreeriderprobleminexplainingtheprocessofcontagion.Crossholdingsofdepositsareusefulforredistributingliquidity,buttheydonotcreateliquidity.Sowhenthereisexcessdemandforliquidityintheeconomyasawholeeachbanktriestomeetexternaldemandsforliquiditybydrawingdownitsdepositsinanotherbank.Inotherwords,eachbankistryingto“passthebuck”toanotherbank.Theresultisthatalltheinterbankdepositsdisappearandnoonegetsanyadditionalliquidity.Theonlysolutiontoaglobalshortageofliquidity(withdrawalsexceedshortassets),istoliquidatelongassets.Aswehaveseen,eachbankhasalimitedbufferthatitcanaccessbyliquidatingthelongasset.Ifthisbufferisexceeded,thebankmustfail.Thisisthekeytounderstandingthedifferencebetweencontagionincompleteandincompletenetworks.Whenthenetworkiscomplete,banksinthetroubledregionhavedirectclaimsonbanksineveryotherregion.Everyregiontakesasmallhit(liquidatesasmallamountofthelongasset)andthereisnoneedforaglobalcrisis.Whenthenetworkisincomplete,banksinthetroubledregionhaveadirectclaimonlyonthebanksinadjacentregions.Thebanksinotherregionsarenotrequiredtoliquidatethelongassetuntiltheyfindthemselvesonthefrontlineofthecontagion.Atthatpoint,itistoolatetosavethemselves.10.1LIQUIDITYPREFERENCEInthissectionweusethestandardelementstomodelliquidityrisk.Therearethreedatest=0,1,2.Thereisasinglegood.Thisgoodcanbeconsumedorinvestedinassetstoproducefutureconsumption.Therearetwotypes 264Chapter10.Contagiont=012Shortliquidasset:111(storage)LongIlliquidasset:1R=1.5r=0.4(liquidate)Figure10.1.Theshortandlongassets.ofassets,ashortassetandalongassetasshowninFigure10.1.Theshortassetisrepresentedbyastoragetechnology.Oneunitoftheconsumptiongoodinvestedinthestoragetechnologyatdatetproducesoneunitoftheconsumptiongoodatdatet+1.InvestmentinthelongassetcanonlytakeplaceinthefirstperiodandoneunitoftheconsumptiongoodinvestedinthelongassetatthefirstdateproducesR>1unitsofoutputatthefinaldate.Eachunitofthelongassetcanbeprematurelyliquidatedtoproduce0λ¯,inwhichcaseitwillneedmorethanytosatisfythedemandsoftheearlyconsumers.Itcanmeetthisexcessdemandbyliquidatingsomeofthelongasset,butthenitwillnothaveenoughconsumptiontomeetthedemandsofthelatecon-sumersatdate2.Infact,ifrissmallenough,thebankmaynotbeabletopaythelateconsumersevenc1.Thenthelateconsumerswillprefertowith-drawatdate1andstoretheconsumptiongooduntildate2,thuscausingabankrun.Thereisnooverallshortageofliquidity,itisjustbadlydistributed.Onewaytoallowthebankstoovercomethemaldistributionofliquidityisbyintrodu-cinginterbankdeposits.UsingthedatafromExample1,wefirstconsiderthelogisticsofimplementingthefirstbestusinginterbankdepositswhenthereisacompletenetworkofinterbankrelationships.Thenweconsiderthecaseofanincomplete,connectednetwork.Example2(Acompletenetwork)Supposethattheinterbanknetworkiscompleteandthatbanksareallowedtoexchangedepositsatthefirstdate.ThiscaseisillustratedinFigure10.3.Eachregionisnegativelycorrelatedwithtwootherregions.Thepayoffsonthesedepositsarethesameasforconsumers.Foreach1unitdepositedatdate0,thebankcanwithdraw1atdate1or1.5atdate2.Weareinterestedinseeinghowthefirst-bestallocationcanbeimplemented.Supposeeachbankholdsaportfolio(y,x)of(0.5,0.5).Ifeverybankinregioniholdszi=(λH−λ)/¯2=0.25/2=0.125depositsineachoftheregions 270Chapter10.ContagionABDCFigure10.3.Completenetwork.j=i,theywillbeabletosupplytheirdepositorswiththefirst-bestallocationnomatterwhetherstateS1orS2occurs.Atdate1thestateofnatureSisobservedandthebankshavetoadjusttheirportfoliostosatisfytheirbudgetconstraints.Iftheregionhasahighdemandforliquidity,λi=λH=0.75,itliquidatesallofitsdepositsinotherregions.Ontheotherhand,ifithasalowdemandforliquidity,λi=λL=0.25,itretainsthedepositsitholdsintheotherregionsuntilthefinaldate.SupposethatthestateS1occursandconsiderthebudgetconstraintofabankinregionAwithahighdemandforliquidityasshowninFigure10.4.Thefirstthingtonoticeisthatitsdepositof0.125withabankinregionC,whichalsohashighdemand,cancelswiththeclaimonitof0.125fromregionC.Inadditionitmustpayc1=1tothefractionλH=0.75ofearlyconsumersinitsownregionforatotalof0.75.Ontheothersideoftheledger,ithasy=0.5unitsoftheshortassetandclaimsto2×0.125=0.25depositsinregionsBandD.Thus,ithasenoughassetstomeetitsliabilities.ThesameanalysisholdsforabankinregionC.NextconsiderabankinregionB,whichhaslowliquiditydemand.Itmustpayc1=1toafractionλL=0.25oftheirowndepositorsandredeem2×0.125=0.25depositsfromthebanksintheregionsAandCwithhighliquiditydemand.Ithasy=0.5unitsoftheshortassettomeetthesedemands,soitsassetscoveritsliabilities.AsimilaranalysisholdsforabankinregionD.Atdate2,allthebanksliquidatetheirremainingassetsanditcanbeseenfromFigure10.4thattheyhavesufficientassetstocovertheirliabilities.ConsiderabankinregionAorCfirst.Itmustpayc2=1.5tothefraction1−λH=0.25ofearlyconsumersinitsownregionforatotalof0.375.ThebanksinregionsBandDhavetotalclaimsof2×0.125×1.5=0.375.Thetotalclaimsonitare0.75.Ontheothersideoftheledger,ithas1−y=0.5unitsofthelongassetthatgivesapayoffof0.5×1.5=0.75.Thus,ithasenoughassetstomeetitsliabilities.ThesameanalysisholdsforabankinregionC.BanksinregionsBandDmusteachpayc2=1.5toafraction1−λL=0.75oftheirowndepositorssotheirtotalliabilitiesare0.75×1.5=1.125.Intermsoftheirassetstheyhave0.5ofthelongassetwhichhasapay-offof0.5×1.5=0.75anddepositsinbanksinregionsAandCworth 10.3Decentralization271Date1:Demand=0.750.125Demand=0.25ABSupply=0.5Supply=0.5Crossdepositscancel0.1250.125Demand=0.25DCDemand=0.75Supply=0.5Supply=0.50.125Date2:0.1875Demand=0.375Demand=1.125ABSupply=0.75Supply=0.750.1875Crossdepositscancel0.1875Demand=1.125DCDemand=0.375Supply=0.75Supply=0.750.1875Figure10.4.TheflowsbetweenbanksinstateS1withacompletenetworkstructure.2×0.125×1.5=0.375.Soitstotalassetsare0.75+0.375=1.125andthesecoveritsliabilities.Thus,byshufflingdepositsamongthedifferentregionsusingtheinterbanknetwork,itispossibleforbankstosatisfytheirbudgetconstraintsineachstateSandateachdatet=0,1,2whileprovidingtheirdepositorswiththefirst-bestconsumptionallocationthroughastandarddepositcontract.Example3(Anincompletenetwork)Theinterbanknetworkinthepreced-ingsectioniscompleteinthesensethatabankinregionicanholddepositsineveryotherregionj=i.Insomecases,thismaynotberealistic.Thebankingsectorisinterconnectedinavarietyofways,buttransactionandinformationcostsmaypreventbanksfromacquiringclaimsonbanksinremoteregions.Totheextentthatbanksspecializeinparticularareasofbusinessorhavecloserconnectionswithbanksthatoperateinthesamegeographicalorpoliticalunit,depositsmaytendtobeconcentratedin“neighboring”banks.Tocapturethiseffect,whichiscrucialinthesequel,weintroducethenotionofincompletenessoftheinterbanknetworkbyassumingthatbanksinregioniareallowedtoholddepositsinsomebutnotalloftheotherregions.Forconcreteness,weassumethatbanksineachregionholddepositsonlyinoneadjacentregion, 272Chapter10.ContagionasshowninFigure10.5.ItcanbeseenthatbanksinregionAcanholddepositsinregionB,banksinregionBcanholddepositsinregionCandsoon.Thisnetworkstructureagainallowsbankstoimplementthefirst-bestallo-cation.Themaindifferencebetweenthiscaseandthepreviousoneisthatinsteadofdepositing0.125intwobanks,eachbankdeposits0.25inonebank.Thetransfersatdates1and2arethenasshowninFigure10.6.OneinterestingfeatureofthenetworkstructureinFigure10.5isthat,althougheachregionisrelyingonjustitsneighborforliquidity,theentireeconomyisconnected.RegionAholdsdepositsinregionB,whichholdsABDCFigure10.5.Incompletenetworkstructure.Date1:Demand=0.750.25Demand=0.25ABSupply=0.5Supply=0.5Demand=0.25DCDemand=0.75Supply=0.5Supply=0.50.25Date2:Demand=0.375Demand=1.125ABSupply=0.75Supply=0.750.3750.375Demand=1.125DCDemand=0.375Supply=0.75Supply=0.75Figure10.6.TheflowsbetweenbanksinstateS1withanincompletenetworkstructure. 10.3Decentralization273depositsinregionC,andsoon.Infact,thisisunavoidablegiventhenet-workstructureassumed.ConsiderthealternativenetworkstructureshowninFigure10.7.RegionAholdsdepositsinregionBandregionBholdsdepositsinregionA.Likewise,regionCholdsoneunitofdepositsinregionDandregionDholdsoneunitofdepositsinregionC.ThisnetworkstructureismoreincompletethantheoneinFigure10.2andthepatternofholdingsinFigure10.5isincompatiblewithit.However,itispossibletoachievethefirstbestthroughthepatternofholdingsinFigure10.8.ThisistrueeventhoughABDCFigure10.7.Aseparatedincompletenetworkstructure.Date1:Demand=0.750.25Demand=0.25ABSupply=0.5Supply=0.5Demand=0.25DCDemand=0.75Supply=0.5Supply=0.50.25Date2:0.375Demand=0.375Demand=1.125ABSupply=0.75Supply=0.75Demand=1.125DCDemand=0.375Supply=0.75Supply=0.750.375Figure10.8.TheflowsbetweenbanksinstateS1withaseparatedincompletenetworkstructure. 274Chapter10.Contagiontheeconomyisdisconnected,sinceregionsAandBtradewitheachotherbutnotwithregionsCandDandregionsCandDtradewitheachotherbutnotwithregionsAandB.Again,thesepatternsdonotseemtohaveanysignifi-canceasfarasachievingthefirstbestisconcerned;buttheyturnouttohavestrikingdifferencesforcontagion.10.4CONTAGIONToillustratehowasmallshockcanhavealargeeffect,weusethedecentral-izationresultsfromSection10.3.ThenweperturbthemodeltoallowfortheoccurrenceofastateS¯inwhichtheaggregatedemandforliquidityisgreaterthanthesystem’sabilitytosupplyliquidityandshowthatthiscanleadtoaneconomy-widecrisis.ThenetworkstructureisassumedtobegivenbyFigure10.5.Thecor-respondingallocationrequireseachbanktoholdaninitialportfolioofinvestments(y,x)andofferadepositcontract(c1,c2),where(y,x,c1,c2)isthefirst-bestallocation.Inordertomakethisdepositcontractfeasible,therepresentativebankineachregionholdsz=0.25depositsintheadjacentregion.Notethatzistheminimalamountthatisneededtosatisfythebud-getconstraints.Itwillbecomeapparentbelowthatlargercrossholdingsofdeposits,whileconsistentwiththefirstbestinSection10.3,wouldmakethecontagionproblemworse.Now,letustaketheallocationasgivenandconsiderwhathappenswhenwe“perturb”themodel.ByaperturbationwemeantherealizationofastateS¯thatwasassignedzeroprobabilityatdate0andhasademandforliquiditythatisveryclosetothatofthestatesthatdooccurwithpositiveprobability.Specifically,theliquidityshocksareshowninTable10.2.InstateS¯,everyregionhasthepreviousaveragedemandforliquidityλ¯exceptforregionAwherethedemandforliquidityissomewhathigherλ¯+ε.TheimportantfactisthattheaveragedemandforliquidityacrossallfourregionsisslightlyhigherthaninthenormalstatesS1andS2.SincetheabnormalstateS¯occurswithnegligibleprobability(inthelimit,probabilityzero)itwillnotchangetheallocationatTable10.2.Regionalliquidityshockswithperturbation.ABCDS1λH=0.75λL=0.25λH=0.75λL=0.25S2λL=0.25λH=0.75λL=0.25λH=0.75S¯λ¯+ε=0.5+ελ¯=0.5λ¯=0.5λ¯=0.5 10.4Contagion275date0.InstatesS1andS2thecontinuationequilibriumwillbethesameasbeforeatdate1;instateS¯thecontinuationequilibriumwillbedifferent.Inthecontinuationequilibriumbeginningatdate1,consumerswillopti-mallydecidewhethertowithdrawtheirdepositsatdate1ordate2andbankswillliquidatetheirassetsinanattempttomeetthedemandsoftheirdeposit-ors.Earlyconsumersalwayswithdrawatdate1;lateconsumerswillwithdrawatdate1ordate2dependingonwhichgivesthemthelargeramountofcon-sumption.Becausewewanttofocusonessentialbankcrises,weassumethatlateconsumerswillalwayswithdrawtheirdepositsatdate2ifitis(weakly)optimalforthemtodoso.Banksarerequiredtomeettheirpromisetopayc1unitsofconsumptiontoeachdepositorwhodemandswithdrawalatdate1.Iftheycannotdoso,theymustliquidatealloftheirassetsatdate1.Theproceedsoftheliquidationaresplitprorataamongdepositorsintheusualway.Ifthebankcanmeetitsobligationsatdate1,thentheremainingassetsareliquidatedatdate2andgiventothedepositorswhohavewaiteduntildate2towithdraw.Intherestofthissection,wedescribethecontinuationequilibriumatdate1instateS¯,assumingtheactionsconsistentwiththefirstbestatdate0.10.4.1Theliquidation“peckingorder”Atdate1abankcanfinditselfinoneofthreeconditions.Abankissaidtobesolvent,ifitcanmeetthedemandsofeverydepositorwhowantstowithdraw(includingbanksinotherregions)byusingonlyitsliquidassets,thatis,theshortassetandthedepositsinotherregions.Thebankissaidtobeinsolventifitcanmeetthedemandsofitsdepositsbutonlybyliquidatingsomeofthelongasset.Finally,thebankissaidtobebankruptifitcannotmeetthedemandsofitsdepositorsbyliquidatingallitsassets.Thesedefinitionsaremotivatedbytheassumptionthatbankswillalwaysfinditpreferabletoliquidateassetsinaparticularorderatdate1.Wecallthisthe“peckingorder”forliquidatingassetsanditgoesasfollows:first,thebankliquidatestheshortasset,thenitliquidatesdeposits,andfinallyitliquidatesthelongasset.Toensurethatthelongassetisliquidatedlast,weneedanadditionalassumption,Rc2>(10.6)rc1whichismaintainedinthesequel.Sincethefirst-bestconsumptionallocation(c1,c2)isindependentofr(thisvariabledoesnotappearinthefirst-bestprobleminSection10.2)wecanalwaysensurethatcondition(10.6)issatisfied 276Chapter10.Contagionbychoosingrsufficientlysmall.Itcanbeseenthat(10.6)issatisfiedinourexamplesinceR/r=1.5/0.5>1.5/1=c2/c1.Eachofthethreeassetsoffersadifferentcostofobtainingcurrent(date1)consumptionintermsoffuture(date2)consumption.Thecheapestistheshortasset.Oneunitoftheshortassetisworthoneunitofconsumptiontodayand,ifreinvestedintheshortasset,thisisworthoneunitofconsumptiontomorrow.Sothecostofobtainingliquiditybyliquidatingtheshortassetis1.Similarly,byliquidatingoneunitofdeposits,thebankgivesupc2unitsoffutureconsumptionandobtainsc1unitsofpresentconsumption.Sothecostofobtainingliquiditybyliquidatingdepositsisc2/c1.Fromthefirst-orderconditionu(c1)=Ru(c2),weknowthatc2/c1>1.Finally,byliquidatingoneunitofthelongasset,thebankgivesupRunitsoffutureconsumptionandobtainsrunitsofpresentconsumption.SothecostofobtainingliquiditybyliquidatingthelongassetisR/r.Thus,wehavederivedthepeckingorder,shortassets,deposits,longassets:c2R1<<.c1rInordertomaximizetheinterestsofdepositors,thebankmustliquidatetheshortassetbeforeitliquidatesdepositsinotherregionsbeforeitliquidatesthelongasset.Theprecedingargumentassumesthatthebankinwhichthedepositisheldisnotbankrupt.Bankruptcyrequiresthatallassetsofthebankruptinstitutionbeliquidatedimmediatelyandtheproceedsdistributedtothedepositors.Sotheprecedinganalysisonlyappliestodepositsinnon-bankruptbanks.10.4.2LiquidationvaluesThevalueofadepositatdate1isc1ifthebankisnotbankruptanditisequaltotheliquidationvalueofallthebank’sassetsifthebankisbankrupt.Letqidenotethevalueoftherepresentativebank’sdepositsinregioniatdate1.Ifqi0issufficientlylargeandR>1issufficientlylow,then,inanycontinuationequilibrium,thebanksinallregionsmustgobankruptatdate1instateS¯.Wehaveillustratedthispropositionforanumericalexample.AmoreformalversionofthepropositionisprovidedinAllenandGale(2000).10.4.4ManyregionsSofarwehaveonlyconsideredthecaseoffourregionsA,B,C,andD.However,itcanbeseenfromthefinalstepoftheexampleabovethattherewouldbecontagionnomatterhowmanyregionsthereare.WhenBankDgoesbankruptalltheotherregionsalsogobankrupt.Thesameargumentwouldholdwithathousandormanythousandsofregions.Thus,eventhoughtheinitialshockonlyoccursinoneregion,whichcanbeanarbitrarilysmallpartoftheeconomy,itcanneverthelesscausebanksinallregionstogobankrupt.Thisiswhycontagioncanbesodamaging.Proposition3Proposition2holdsnomatterhowmanyregionsthereare.10.5ROBUSTNESSTheincompletenessofnetworksisimportantforthecontagionresult.IntheexampleweconsideredlastwheretherewascontagionwhennetworkswereasinFigure10.5itcanbeshownthatifthereisacompletenetworkasinFigure10.3,thereisnocontagion.Thekeydifferenceisthatnowwithacom-pletenetwork,eachbankdeposits0.125intwootherbanksratherthan0.25injustonebank.Example7ε=0.10,R=1.2Theanalysisisthesameasbeforeexceptwhenweareconsideringwhetherthereiscontagion.BankBandD’sdepositclaimsof0.125onBankAarenowworth0.76×0.125=0.095.BanksB,C,andDhaveclaimsof0.5intotalfromtheirearlyconsumersand0.125fromthethreebanksthathavedepositsinthem.Theyhaveliquidityof0.5fromtheirshortassetand0.125fromtwooftheirdepositsand0.095fromtheirdepositinBankA.Theyneedtoliquidate 10.6Containment281enoughofthelongassettoraise0.125−0.095=0.03.Givenr=0.4theamountliquidatedis0.03/0.4=0.075.Theythushave0.5−0.075=0.425ofthelongassetremaining.Atdate2theywillhave0.425×1.2=0.51fromthisholdingofthelongasset.Theycandistributec2=0.51/0.5=1.02tothelateconsumers.Sincethisisabovethec1=1theywouldobtainiftheypretendedtobeearlyconsumers,thereisnorunonBanksB,C,andDandnocontagion.Thereasonthereisnocontagioninthecasewithacompletenetworkisthattheliquidationofassetsisspreadamongmorebankssotherearemorebufferstoabsorbtheshock.Thismeansthatbanksdonothitthediscontinuityassociatedwitharunwhenallthebank’sassetsareliquidatedatasignificantlossandasaresultthereisnocontagion.Wehaveillustratedthisfortheexample.AllenandGale(2000)againshowthatasimilarresultholdsmoregenerally.Completenetworksarelesssusceptibletocontagionthanincompletenetworks.10.6CONTAINMENTThecriticalingredientintheexampleofcontagionanalyzedinSection10.4isthatanytworegionsareconnectedbyachainofoverlappingbankliabilities.BanksinregionAhaveclaimsonbanksinregionB,whichinturnhaveclaimsonbanksinregionC,andsoon.Ifwecouldcutthischainatsomepoint,thecontagionthatbeginswithasmallshockinregionAwouldbecontainedinsubsetofthesetofregions.ConsidertheincompletenetworkstructureinFigure10.7andtheallocationthatimplementsthefirst-bestallocationforourexample,whichisshowninFigure10.8.TheallocationrequiresbanksinregionsAandBtohaveclaimsoneachotherandbanksinregionsCandDtohaveclaimsoneachother,butthereisnoconnectionbetweentheregion{A,B}andtheregion{C,D}.IfstateS¯occurs,theexcessdemandforliquiditywillcausebankruptciesinregionAandtheycanspreadtoregionB,butthereisnoreasonwhytheyshouldspreadanyfurther.BanksinregionsCandDaresimplynotconnectedtothetroubledbanksinregionsAandB.Comparingthethreenetworkstructureswehaveconsideredsofar,com-pletenetworksinFigure10.3,incompletenetworksinFigure10.5,andthedisconnectednetworkstructureinFigure10.7,wecanseethatthereisanon-monotonicrelationshipbetweencompletenessorincompletenessofnetworksandtheextentofthefinancialcrisisinstateS¯.WiththecompletenetworkstructureofFigure10.3thecrisisisrestrictedtoregionA,withthenetwork 282Chapter10.ContagionstructureinFigure10.5thecrisisextendstoallregions,andwiththenetworkstructureinFigure10.7thecrisisisrestrictedtoregionsAandB.Itcouldbearguedthatthenetworkstructuresarenotmonotonicallyordered:thecompletenetworkdoescontaintheothertwo,butthepathsinthenetworkinFigure10.7arenotasubsetofthenetworkinFigure10.5.ThiscouldbechangedbyaddingpathstoFigure10.2,butthentheequilibriumofFigure10.7wouldalsobeanequilibriumofFigure10.5.Thisraisesanobviousbutimportantpoint,thatcontagiondependsontheendogenouspatternoffinancialclaims.AnincompletenetworkstructureliketheoneinFigure10.5mayprecludeacompletepatternoffinancialconnectednessandthusencour-agefinancialcontagion;butacompletenetworkstructuredoesnotimplytheopposite:eveninacompletenetworktheremaybeanendogenouschoiceofoverlappingclaimsthatcausescontagion.Infact,thethreeequilibriaconsid-eredsofarareallconsistentwiththecompletenetworkstructure.Thereareadditionalequilibriafortheeconomywiththecompletenetworkstructure.Likethethreeconsideredsofar,theyachievethefirstbestinstatesS1andS2,buthavedifferentdegreesoffinancialfragilityintheunexpectedstateS¯,dependingonthepatternsofinterregionaldepositholding.WhatisimportantaboutthenetworkstructureinFigure10.5,then,isthatthepatternofinterregionalcrossholdingsofdepositsthatpromotesthepos-sibilityofcontagion,istheonlyoneconsistentwiththisnetworkstructure.Sinceweareinterestedincontagionasanessentialphenomenon,thisnet-workstructurehasaspecialrole.Thecompletenetworkeconomy,bycontrast,hasequilibriawithandwithoutcontagionandprovidesaweakercaseforthelikelihoodofcontagion.10.7DISCUSSIONTheexistenceofcontagiondependsonanumberofassumptions.Thefirstisthatfinancialinterconnectednesstakestheformofexanteclaimssignedatdate0.Theinterbankloannetworkisgoodinthatitallowsreallocationofliquidity.Butwhenthereisaggregateuncertaintyaboutthelevelofliquiditydemandthisinterconnectednesscanleadtocontagion.Itisnotimportantthatthecontractsaredepositclaims.Thesameresultholdswithcontingentordiscretionarycontractsbecausetheinterbankclaimsnetout.Exantecon-tractswillalwaysjustnetout.Spilloverandcontagionoccurbecauseofthefallinassetvaluesinadjacentregions,nottheformofthecontract.Theinterbanknetworkoperatesquitedifferentlyfromtheretailmarketinthisrespect. 10.7Discussion283Notethatifthereisanexpostloanmarketsocontractsaresignedatdate1ratherthandate0thentherecanbethedesirablereallocationofliquiditybutnocontagion.Thereasonthattherewillbenocontagionisthattheinterestrateintheexpostmarketmustcompensatelendersforthecostofliquidatingassetsbutatthisrateitwillnotbeworthborrowing.However,therearetheusualdifficultieswithexpostmarketssuchasadverseselectionandthe“hold-up”problem.Ifthelongassethasariskyreturnthenexpostmarketswillalsonotbeoptimal.Wehavesimplifiedtheproblemconsiderablytoretaintractability.Inpartic-ular,byassumingstateS¯occurswithzeroprobability,weensurethebehaviorofbanksisoptimalsincetheinterbankdepositsandtheresultingallocationremainsefficient.WhenS¯occurswithpositiveprobabilitythetrade-offswillbemorecomplex.However,providedthebenefitsofrisksharingarelargeenoughinterbankdepositsshouldbeoptimalandthisinterconnectionshouldleadtothe(lowprobability)possibilityofcontagion.WhenS¯occurswithpositiveprobabilityabankcanpreventrunsbyholdingmoreoftheliquidasset.There’sacosttothisinstatesS1andS2though.WhentheprobabilityofS¯issmallenoughthiscostisnotworthbearing.Itisbettertojustbeartheriskofcontagion.Thefocusinthispaperisonfinancialcontagionasanessentialfeatureofequilibrium.Wedonotrelyonargumentsinvolvingmultipleequilibria.Theaimisinsteadtoshowthatundercertainconditionseverycontinu-ationequilibriumatdate1exhibitsfinancialcontagion.Nonetheless,therearemultipleequilibriainthemodelandifoneissodisposedonecanusethemultiplicityofequilibriatotellastoryaboutfinancialcontagionasasunspotphenomenon.Forsimplicity,wehaveassumedthatthelongassethasanon-stochasticreturn,butitwouldbemorerealistictoassumethatthelongassetisrisky.Wehaveseenatmanypointsinthisbook,suchasChapter3onintermediation,thatwhenthelongassetisriskyitisnegativeinformationaboutfuturereturnsthattriggersbankruns.Inthepresentframework,uncertaintyaboutlongassetreturnscouldbeusedbothtomotivateinterregionalcrossholdingsofdepositsandtoprovokeinsolvencyorbankruptcy.Theresultsshouldbesimilar.Whatiscrucialfortheresultsisthatthefinancialinterconnectednessbetweentheregionstakestheformofclaimsheldbybanksinoneregiononbanksinanotherregion.Wehaveshowninthischapterthatasmallshockinasingleregionorbankcanbringdowntheentirebankingsystemnomatterhowbigthissystemisrelativetotheshock.Whatiskeyforthisisthenetworkstructureforinter-bankdeposits.Iftherearerelativelyfewchannelsofinterconnectednessthencontagionismorelikely.Clearlytransactioncostsmeanitismuchtoocostly 284Chapter10.Contagionforeverybanktoholdanaccountwitheveryotherbanksonetworksarecom-plete.However,onelowcostequivalentofthisistohaveacentralbankthatisconnectedwitheveryotherbank.Thetheoryherethusprovidesarationaleforcentralbanks.10.8APPLICATIONSInthissectionwetrytobridgethegapbetweenthetheorydevelopedintheprecedingsectionsandempiricalapplications.Themodelwehavepresentedheredescribesanartificialandhighlysimplifiedenvironmentinwhichitispossibletoexhibitthemechanismbywhichasmallshockinoneregioncanbetransmittedtootherregions.Thisresultdependsonanumberofassumptions,someofthemquiterestrictive,andsomeofthemmadeinordertokeeptheanalysistractable.Inanycase,themodelisquitefarremovedfromtheworldinwhichpolicymakershavetooperate.Nonetheless,althoughthemodelonlyprovidesanextremelysimplifiedpictureoftheentirefinancialsystem,thebasicideascanbeappliedtorealdatainordertoprovidesomeindicationoftheprospectsforcontagioninactualeconomies.Therehavenowbeenseveralstudiesofthissort,carriedoutondatafromdifferentcountries.AnexcellentsurveyofthisliteratureiscontainedinUpper(2006).Herewestartwithoneparticularlytransparentexample,whichwillservetoillustratewhatcanbedoneinpractice.10.8.1UpperandWorms(2004)Inanimportantstudy,UpperandWorms(2004),henceforthUW,usedataoninterbankdepositstosimulatethepossibilityofcontagionamongthebanksintheGermanfinancialsystem.TheUWmodelisverysimple.Thereisafinitenumberofbanksindexedbyi=1,...,n.Eachbankihasalevelofcapitaldenotedbythenumberci≥0.Foranyorderedpairofbanksi,jthereisanumberxij≥0thatdenotesthevalueoftheclaimsofbankionbankj.Theseclaimsmayrepresentdepositsinbankjorbankloanstobankjorsomecombination.Thenumbersciandxijrepresentthedataoninterbankrelationshipsandthatwillbeusedasthebasisfortheanalysisofcontagion.Theothercrucialparameterthatplaysaroleintheprocessofcontagionisthelossratioθ.Whenabankgoesbankrupt,itsassetswillbeliquidatedinordertopayoffthecreditors.Typically,theassetswillbesoldforlessthan 10.8Applications285their“book”oraccountingvalue.Thereasonswhyassetsaresoldatfiresalepricesarenumerousandwellknown.Somepotentialbuyers,perhapsthosewhovaluetheassetshighly,maybeunabletobidforthembecauseofalackofliquidity.Oritmaybethatassetsaresoldinhaste,beforeeverypotentialbuyerisreadytobidonthem,leadingtoanon-competitivemarketfortheassetsonwhichbuyerscanpickuptheassetsforlessthanthefairvalue.Thereisalsotheproblemofasymmetricinformation.Ifpotentialbuyershavelimitedinformationaboutthequalityoftheassets(andsomebankassets,suchasloans,arenotoriouslyhardtovalue),fearofadverseselectionwillcausebuyerstoincorporatea“lemons”discountintheirbids.Partofthelossfromsellingtheassetsatfiresalepriceswillbeabsorbedbythebank’scapital,butthelossesmaybesogreatthattheliquidatedvalueoftheassetsislessthanthevalueofthebank’sliabilities.Inthatcase,partofthelosseswillbepassedontothecreditors,whoonlyreceiveafractionofwhattheyareowed.Thelossratioθmeasuresthefractionofthecreditors’claimthatislostbecauseoftheliquidation.Theprocessofcontagionbeginswiththefailureofasinglebank.Ifthelossratioispositive,allthecreditorswillloseapositivefractionoftheclaimsagainstthefailingbank.Ifthislossisbigenough,someoftheaffectedbanksmayfail.Thisisthefirstroundofcontagion.Thefailureoftheseadditionalbankswillhavesimilareffectsontheircreditors,someofwhommayfailinthesecondround.Theseeffectscontinueinroundafterroundasthecontagionspreadsthroughoutthefinancialsector.Thisrecursivedescriptionofcontagionsuggestsanalgorithmforcalculatingtheextentofcontagionfromthefailureofasinglebank.Thefirststepofthealgorithmtakesasgiventhefailureofagivenbankandcalculatestheimpactonthecreditorbanks.Supposethatbankjfailsandbankiisacreditorofbankj.Banki’sclaimonbankjisxijandafractionθofthisislost,sobankisuffersalossofθxij.Ifthelossθxijislessthanbanki’scapitalci,thebankcanabsorbthelossalthoughitscapitalwillbediminished.Ifthelossisgreaterthanthevalueofthebank’scapital,however,thebank’sassetsarereducedtothepointwheretheyarelowerthanitsliabilities.Whenassetsarelessthanliabilities,thebankisinsolventandmustdeclarebankruptcy.Sothefailureofbankjwillcausethefailureofbankiifandonlyifthelossθxijisgreaterthanitscapitalci:θxij>ci.Thebanksthatsatisfythisinequalityconstitutethefirstroundofcontagionfromthefailureofbankj. 286Chapter10.ContagionLetI1denotethesetofbanksithatfailedinthefirstroundofcontagion,plustheoriginalfailedbankj.ThefailureofbankjhasspreadbycontagiontoallthebanksinthesetI1butthecontagiondoesnotstopthere.BanksthatdidnotfailasaresultoftheoriginalfailureofbankjmaynowfailbecauseofthefailureofbanksinI1.SupposethatidoesnotbelongtoI1.ForeveryjinI1,bankilosesθxijsoitstotallossisj∈I1θxij.Thenbankiwillfailinwhatwecallthesecondroundifandonlyifθxij>ci.j∈I1LetI2denotethesetofbanksthatfailinthesecondroundplusallthebanksincludedinI1.Wecancontinueinthiswaycalculatingroundafterroundoffailuresuntilthesetoffailedbanksconverges.LetIkdenotethesetofbanksthatfailinthek-thandearlierrounds.Sincethereisafinitenumberofbanks,contagionmustcometoanendafterafinitenumberofperiods.ItiseasytoseefromthedefinitionthatifIk=Ik+1thenIk=Ik+forevery>0.Theinterestingquestionishowfarthecontagionwillgo.Therearevariouswaysofmeasuringtheextentofcontagion.UWreportvariousmeasures,includingthetotalnumberoffailedbanksandthepercentageofassetsinfailedbanks.Theprocedureforestimatingtheextentofcontagionisasfollows.Chooseanarbitrarybankiandassumethatitfails,calculatethesetofbanksthatwillfailasaresultofthefailureofbanki(theproceduredescribedabove),andthenrepeatthisprocedureforeverypossiblechoiceofthestartingbanki.Theextentofcontagionistheleastupperboundoftherelevantmeasure,say,thetotalnumberoffailedbanksorthepercentageofassetsinfailedbanks,overallstartingvaluesi.Inotherwords,wechoosetheinitialfailedbanktomaximizethemeasureofcontagionandcallthatmaximizedvaluetheextentofcontagion.Thedatarequiredforthisexercisearenotavailableincompletelydisag-gregatedformsoUWwereforcedtoapproximateitusingpartiallyaggregateddata.GermanbanksarerequiredtosubmitmonthlybalancesheetstotheBun-desbankinwhichtheyreportinterbankloansanddepositsclassifiedaccordingtowhethertheircounterpartyisaforeignordomesticbank,abuildingsociety,ortheBundesbank.Savingsbanksandcooperativebankshavetoreportinadditionwhetherthecounterpartyisagiroinstitutionoracooperativecentralbank.Allbanksarealsorequiredtodividetheirlendingintofivematuritycat-egories.Whatarenotreportedaretheactualbilateralpositionsbetweenpairsofbanks.Thisinformationhastobeinterpolatedfromtheaggregatesreportedbyindividualbanks.Nonetheless,thedivisionofinterbankloansanddeposits 10.8Applications287intosomanycategoriesallowsUWtoconstructanumberofmatricescor-respondingtolendingbetweendifferenttypesofbanksindifferentmaturityclasses.Sinceweareultimatelyconcernedonlywiththeaggregateamountofbilaterallending,thesecalculationswouldbeofnointerestifwecouldobservebilateralexposuresdirectly.Sincewecannotobservebilateralexposures,theadditionalinformationonthebreakdownoflendingbyindividualbanksisusefulinestimatingtheunobservedbilateralexposures.UWuseacomplex,recursivealgorithmwhichusesthesumoflendingandborrowingbyeachbankineachofseveralcategories(definedbytypeofcounterpartyandmaturity)toestimatethebilateralexposuresforthosecategories.Byapplyingthispro-ceduretothe25separatematricescorrespondingtolendingbetweendifferenttypesofbanksandindifferentmaturities,UWareabletogetmorepreciseestimatesofthetruebilateralexposures,becausemanybanksareactiveonlyinsomeofthesecategories.Theresultingmatricesarethensummedtogiveaggregatebilateralexposures.Sincetheyonlyhavedataondomesticbanksanddomesticbranchesofforeignbanks,UWeliminateexposurestoforeignbanks,buildingsocietiesandtheBundesbank.Theyareleftwithaclosedsysteminwhichassetsandliabilitiessumtozero.The“fullinformation”methodshowsthatthepatternsoflendingandbor-rowingarequitedifferentfordifferentclassesofbanks.Amongtheirbroadconclusionsaboutthestructureoflendingarethefollowing.•Bankstendtohavelargerclaimsonbanksinthesamecategory.Forexample,commercialbankstransactmuchmorewithothercommercialbanksthanonewouldexpectfromthebaselinematrix.•Similarly,theheadinstitutionsofthetwogirosystems(cooperativebanksandsavingsbanks)havealargeproportionoftheloansanddepositsintheindividualbanksineachgirosystem.Therearealmostnodepositsheldbetweenindividualbanksatthebaselevelofthesamegirosystem.•Therearethereforetwotierstothebankingsystem,thelowertierconsistingofsavingsandcooperativebanks,theuppertierconsistingofcommercialbanks,theheadbanksofthegirosystems(Landesbankenandcoopera-tivecentralbanks)plusavarietyofotherbanks.Whereasthelowertierbankshavelittleexposuretobanksinthesametier,theuppertierbankshavetransactionswithavarietyofotherbanksincludingbanksinothercategories.Thetwo-tiersystemfallsbetweenthecompletenetworkandtheincompletenetworkemphasizedinthetheorymodel.Asanetworkitisincompletebutthehubsintheuppertierplayanimportantroleinintegratingthesystem.Inaddition,thetheoreticalmodelassumesthatallbanksareexanteidentical, 288Chapter10.ContagionbankAbankBUppertierbankDbankCLowertierbankD1bankD2bankD3bankC1bankC2bankC3Figure10.9.Two-tierstructureofGermaninterbankholdings(Figure4inUpperandWorms2004).whereasthekeytounderstandingthetwotiersysteminGermanyisthedif-ferenceinsizesandspecializedfunctionsofthebanksineachtier.Astylizedpictureofthetwo-tiersystem,reproducedfromUW,isshowninFigure10.9.Asmentionedabove,thelossratioisakeyparameterindeterminingthepossibilityofcontagion.SincedataonlossratiosapplicabletotheGermanbankingsystemarenotavailable,UWconsiderarangeofvaluesofθandcalculatetheincidenceofcontagion,asmeasuredbyavarietyofstatistics,foreachvalueofθ.Recallthatwearereferringtothemaximumincidencehere,thatis,themaximumextentofcontagionassociatedwiththefailureofasinglebank.WereproduceinFigure10.10therelationshipbetweenthelossratioandboththemaximumnumberofbanksandthemaximumpercentageoftotalassetsaffectedusingthefullinformationmatrixofbilateralexposures.TherelationshipbetweenthelossratioandtheextentofcontagionhasseveralimportantfeaturesthatcanbeseeninFigure10.10orintheunderlyingcalculations.•Thereisalwayscontagionforanylossratio.Infact,thereare17banksthatfailinthefirstround,independentlyofwhichbankischosentofailfirst.Theseareallsmallbanksandtheassumptionsonwhichtheinterpolationofinterbankexposuresisbasedmaybeunrealisticintheircase.•Thereappearstobeacriticalvalueofθ(around40%)atwhichtheextentofcontagionresultingfromasinglebankfailureincreasesverysharply.•Forverylargevaluesofθ(possiblyunrealisticallylargevalues),thecontagionextendstomostofthefinancialsystem. 10.8Applications289Numberofbanksaffected3000Maximumeffect20001000Averageeffect00.20.40.60.8Lossratiou%oftotalassetsaffected1008060Maximumeffect40Averageeffect2000.20.40.60.8LossratiouFigure10.10.Lossratioandtheseverityofcontagionintheabsenceofa“safetynet”(Figure5inUpperandWorms2004).UWprovideanumberofotherresults,relatingtothedynamicsofcontagionandthedisparateimpactofcontagionondifferenttypesofbanks.TheUWmethodologyprovidesamethodforestimatingtheincidenceofcontagionusinghypotheticalvaluesofthelossratioandestimatesofbilateralexposuresbasedonavailabledata.Itprovidesaquantitativeassessmentofthefinancialfragilityofthebankingsystemaswellassomeinterestinginsightsintothesensitivityoftheresultstodifferentstructuralparametersofthesystem.Atthesametime,theapproachhasanumberoflimitations.•UWfocusoninterbankholdings,butthereareothersourcesofinstability,forexample,shocksoriginatingoutsidethefinancialsystem,thatmayleadtocontagion.•UWinterprettheiralgorithmforcalculatingtheextentofcontagionasadynamicprocessinwhicheachroundcanbethoughtofasadifferenttimeperiod.Inatrulydynamicmodel,bankswouldbeabletochangetheirportfoliosineachperiodasthecontagionprogressed.Itisnotclearhow 290Chapter10.Contagionthiswouldaffecttheanalysis.Ontheonehand,banksmightbeabletotakedefensiveactiontoprotectthemselvesagainstcontagion.Ontheotherhand,eachbank’sattempttodefenditself,forexample,bywithdrawingfundsfromotherbanks,mayactuallyacceleratetheprocessofcontagion.•Arelatedpointisthattheanalysisassumesthatassetpricesandinterestratesremainconstantthroughouttheprocess.Iflargescaleliquidationofassetsistakingplace(orisanticipated),theremaybeastrongimpactonassetprices.Afallinassetpricesmayincreasethevulnerabilityofbanksbyreducingtheircapital.Again,thismayacceleratetheprocessofcontagion.10.8.2DegryseandNguyen(2004)AnotherinterestingstudyinthisveinhasbeencarriedoutbyDegryseandNguyen(2004),henceforthDN,fortheBelgianbankingsystem.ThemostinterestingfeatureofDNisthatitusesdataoninterbankloansanddepositsfortheperiod1993–2002.Thisallowstheauthorstostudytheevolutionoftheriskofcontagionoveratenyearperiod.Intheyearsbetween1998and2001,thebankingsystemexperiencedsubstantialconsolidationwhichchangedthestructureoftheindustryaswellasinterbankexposures.Atthebeginningoftheperiod1993–2002,theBelgianbankingsystemcouldbecharacterizedasacompletenetwork,inwhichallbankshavemoreorlesssymmetricexposures.Bytheendoftheperiod,itresembledanincompletenetworkwithmultiplemoneycenters,inwhichthemoneycenterbankshavesymmetriclinkstootherbanksandthenon-moneycenterbanksdonothavelinkswitheachother.DNsimulatetheriskofcontagionatthebeginningandendoftheperiod1993–2002usingavarietyofvaluesoflossgivendefault(LGD),whichistheircounterparttothelossratioθinUW.Theyfindthattheriskofcontagionhasfallenovertheperiodandisquitelowbytheend.Evenwithanunrealis-ticallyhighLGDof100%,theycalculatethatbanksrepresentinglessthanfivepercentoftotalassetswouldbeaffectedbycontagionfollowingthefailureofasingleBelgianbank.Thus,thebankingsystemhasbecomelesscompleteovertheperiodduringwhichtheriskofcontagionhasfallen.ThisstandsincontrasttotheresultofAllenandGale(2000),whichsuggeststhatcom-pletefinancialnetworksaremorestablethanincompletefinancialnetworks.Itmustberemembered,however,thattheassumptionsoftheAllen–GaleresultarenotsatisfiedbytheBelgianbankingsystem.AllenandGale(2000)assumesthatallbanksareexanteidentical,whereastheBelgianbankingsys-temcontainswellcapitalizedmoneycenterbanksinadditiontosmallerbanks.Also,theAllen–Galeresultclaimsthatgreatercompletenessincreasedstability 10.8Applications291ceterisparibus.TheBelgianbankingsystembycontrastexperiencedsubstan-tialchangesbetween1993and2002intermsofthenumber,size,andbalancesheetsofthebanks.Nonetheless,theestimatedstabilityofthebankingsystemisastrikingresult.DNmakeanumberofotherinterestingobservationsaboutthestructureoftheBelgianbankingsystem.Theypayparticularattentiontotheinter-nationalnatureofthebankingbusinessandthefactthatBelgianbanksarewellintegratedintheinternationalbankingsystem.ThisleadsthemtodistinguishanalyticallybetweencontagionhavingasourceoutsidetheBelgianbankingsystemfromcontagionoriginatinginthefailureofaBelgianbank.Itturnsoutthattheextentofcontagioncausedbythefailureofaforeignbankissomewhatlargerthanthatcausedbythefailureofadomesticbank.Thisresulthastobequalified,however,becausethemostimportantforeignbanksareverylarge,wellcapitalizedandhaveveryhighcreditratings,sotheprobabilityoffailureiscorrespondinglysmall.AnotherinterestingconsequenceoftheintegrationofBelgianbanksintheinternationalbankingsystemisthatseveralofthesebankshaveverylargeoperationsoutsideofBelgium.Asaresult,theirassetholdingsarelargerela-tivetothesizeoftheBelgianeconomy.Ourdiscussionofcontagionsofarhastakennoaccountofthesafetynetprovidedbygovernmentsandcentralbanks.IntheBelgiancase,theverysizeofsomeofthebanksmightmakeitdifficulttoputtogetherarescuepackageforoneoftheselargebanksifitweretofinditselfinfinancialdistress.AlthoughUWestimatedthattheextentofcontagionmightbelarge,atleastifthelossratiowerelargeenough,theexistenceofasafetynetmightstopcontagionintheearlystages,beforeitreachesthecriticalmassneededtospilloverintoalargepartofthefinancialsector.Bycontrast,smallcountrieswithverylargeinternationalbanksmaynothavetheresourcestostopthisprocessintheearlyrounds.AsDNshow,theriskofcontagionappearstobesmallevenintheabsenceofasafetynet;butthismaynotbetrueofothersmallcountries,wheresomebanksareliterally“toolargetosave.”10.8.3Cifuentes,Ferrucci,andShin(2005)InourdiscussionofUWwepointedoutthatnoaccountwastakenofpriceeffects,thatis,assetpricesareassumedtobeunaffectedbybankfailuresandtheprocessofcontagion.DNandmostotherstudiesofthistypemakethesameassumption.Weconjecturedthatifpriceeffectswereimportant,thedownwardpressureexertedonassetpricesbyliquidationsand/orhoardingofliquiditywouldreducebankcapitalandacceleratethespeedandextentofcontagion.Cifuentesetal.(2005),henceforthCFS,havedevelopedamodel 292Chapter10.Contagionofcontagioninwhichpriceeffectsplayanimportantrole.Inadditiontotheusualmatrixofinterbankclaims,CFSaddanassetmarketwithadownwardslopingdemandcurveforassets.Inthismarket,therearetwochannelsforcontagion.Thefirstisthroughtheusualbilateralexposuresintheinterbankmarket;thesecondisthroughtheeffectofassetpricechangesonbankcapital.Everybankisassumedtosatisfyacapitaladequacyrequirement.Whenbankcapitalistoolowrelativetothevalueofassets,thebankmustsellsomeassetsinordertosatisfythecapitaladequacyconstraint(itisnotpossibletoraiseadditionalcapital,atleastintheshortrun).Thebankwillfirsttrytosellliquidassets,whosepriceisassumedtobefixed,butifitstillcannotsatisfythecapitaladequacyconstraintitwillhavetoselltheilliquidasset.Themarketfortheilliquidassethasadownwardslopingresidualdemandcurve,sothemoreoftheilliquidassetissoldbythebanks,thelowertheprice.Contagionthroughinterbankexposuresworksintheusualway.Onebankfailurecreatesalossforthecreditorbanksandreducestheircapital.Ifthelossisbigenough,capitalbecomesnegative,assetsarelessthanliabilities,thebankfails,andthelossesspillovertootherpreviouslyunaffectedbanks.Thenewchannelisdifferent.Whenonebankfails,otherbankssufferlossesthatreducetheircapital.Ifthecapitaladequacyconstraintwasbinding,thiswouldputthecreditorbankinthepositionofhavingtosellassetstoreducethemarked-to-marketvalueofitsassets.Atfirst,itmaybepossibletosatisfythecapitalassetconstraintbysellingliquidassets,buteventuallyitwillbenecessarytosellilliquidassets.Ifseveralbanksdothis,theassetpriceisreducedandthishasaneffectonbanksingeneral.Otherthingsbeingequal,areductioninassetpricesreducestheamountcapitalineachbank,possiblycausingittoviolatethecapitaladequacyconstraint.Thosebanksforwhomtheconstraintisviolatedwillbeforcedtosellassetsthemselves,thusincreasingthedownwardpressureonassetprices.ThisallhasafamilyresemblancetothestorytoldinChapter5andindeeditisverysimilar.ThenoveltyoftheCFSapproachisthatitcombinestheassetpricechannelwiththeinterbankborrowingandlendingchanneltogetamorepowerfuleffect.Thetwochannelsrunsidebyside,eachreinforcingtheother.CFSdonotcalibratetheirmodeltorealworlddata,buttheydosimulatethebehaviorofthemodelforreasonableparametervaluesandfindthatthepriceeffectsgreatlyamplifytheextentofcontagionforappropriateparametervalues.Theanalysisprovidesimportantinsightsintothefactorsthatcanincreasethelikelihoodandextentofcontagionandshouldprovideaguideforfutureresearch. 10.9LiteratureReview29310.9LITERATUREREVIEWThereareanumberofdifferenttypesofcontagionthathavebeensuggestedintheliterature.Thefirstiscontagionthroughinterlinkagesbetweenbanksandfinancialinstitutions.Thesecondiscontagionofcurrencycrises.Thethirdiscontagionthroughfinancialmarkets.InadditiontothesurveysbyMasson(1999)andUpper(2006)alreadymentioned,DeBandtandHartmann(2002),Karolyi(2003),andPericoliandSbracia(2003)containsurveysofthisliterature.ClaessensandForbes(2001)andDungeyandTambakis(2005)containanumberofpapersonvariousaspectsofinternationalcontagion.Giventhelargenumberofrecentsurveysthissectionwillberelativelybrief.Banksarelinkedinseveralwaysincludingpaymentssystemsandinter-bankmarkets.Theselinkagescanleadtoaproblemofcontagion.Westartbyconsideringmodelsofpaymentsystemcontagion.BuildingonalocationalmodelofpaymentsystemsdevelopedbyMcAndrewsandRoberds(1995),FreixasandParigi(1998)haveconsideredcontagioninnetandgrosspaymentsystems.Inanetpaymentsystembanksextendcredittoeachotherwithinthedayandattheendofthedaysettletheirnetposition.Thisexposesbankstothepossibilityofcontagionifthefailureofoneinstitutiontriggersachainreaction.Inagrosssystemtransactionsaresettledonaone-to-onebasiswithcentralbankmoney.Thereisnoriskofcontagionbutbankshavetoholdlargereservebalances.Anetpaymentsystemispreferredwhentheprobabilityofbankshavinglowreturnsissmall,theopportunitycostofholdingcentralbankmoneyreservesishigh,andtheproportionofconsumersthathavetoconsumeatanotherlocationishigh.Freixas,ParigiandRochet(2000)usethismodeltoexaminetheconditionsunderwhichgridlockoccurs.Theyshowthattherecanbegridlockwhenthedepositorsinonebankwithdrawtheirfunds,anticipatingthatotherbankscannotmeettheirnettingobligationsifalltheirdepositorshavealsowithdrawntheirfunds.RochetandTirole(1996a)considertheroleofthetoo-big-to-failpolicyinpreventingcontagion.Furfine(2003)considersinterbankpaymentflowsintheUSandconcludesthattheriskofcontagionfromthissourceissmall.Asdiscussedaboveatlength,AllenandGale(2000)focusonachannelofcontagionthatarisesfromtheoverlappingclaimsthatdifferentregionsorsectorsofthebankingsystemhaveononeanotherthroughinterbankmar-kets.Whenoneregionsuffersabankingcrisis,theotherregionssufferalossbecausetheirclaimsonthetroubledregionfallinvalue.Ifthisspillovereffectisstrongenough,itcancauseacrisisintheadjacentregions.Inextremecases,thecrisispassesfromregiontoregionandbecomesacontagion.EisenbergandNoe(2001)derivevariousresultsconcerningtheinterconnectednessof 294Chapter10.Contagioninstitutions.Aghionetal.(1999)alsoconsideramodelofcontagionthroughinterbankmarkets.Intheirmodeltherearemultipleequilibria.Inoneequi-libriumthereareself-confirmingbeliefsthatabankfailureisanidiosyncraticeventandintheotherthereareself-fulfillingbeliefsthatabankfailuresig-nalsaglobalshortageofliquidity.LagunoffandSchreft(2001)studythespreadofcrisesinaprobabilisticmodel.Financiallinkagesaremodeledbyassumingthateachprojectrequirestwoparticipantsandeachparticipantrequirestwoprojects.Whentheprobabilitythatone’spartnerwillwith-drawbecomestoolarge,allparticipantssimultaneouslywithdrawandthisisinterpretedasafinancialcrisis.RochetandTirole(1996b)usemonitor-ingasameansoftriggeringcorrelatedcrises:ifonebankfails,itisassumedthatotherbankshavenotbeenproperlymonitoredandageneralcollapseoccurs.Dasgupta(2004)usesaglobalgamesapproachtoshowhowauniqueequilibriumwithcontagioncanarisewhenbanksholdcrossdeposits.AllenandCarletti(2006)showhowcontagioncanoccurthroughthemarketforcreditrisktransfer.VanRijckeghemandWeder(2000)documentlinkagesthroughbankingcentersempirically.IyerandPeydró-Alcalde(2006)consideracasestudyofinterbanklinkagesresultingfromalargebankfailureduetofraud.Thereisagrowingliteratureoncontagiouscurrencycrisesandinternationalcontagion.Masson(1999)providesagoodoverviewofthebasicissues.Hedistinguishesbetween“monsoonal”effects,spilloversandpurecontagion.Monsoonaleffectsoccurwhentherearemajoreconomicshiftsinindustrialcountriesthatimpactemergingeconomies.Spilloversoccurwhentherearelinksbetweenregions.Purecontagioniswhenthereisachangeinexpectationsthatisnotrelatedtofundamentalsandisassociatedwithmultipleequilibria.Eichengreenetal.(1996)andGlickandRose(1999)provideevidencethattradelinkagesareimportantfactorsinthespreadofmanycurrencycrises.Kaminskyetal.(2003)consideralonghistoryofcontagionacrossbordersandconsiderwhycontagionoccursinsomecasesbutnotinothersimilarsitua-tions.PickandPesaran(2004)considersomeoftheeconometricissuesthatariseindistinguishingcontagionfrominterdependence.Thereareanumberofpapersthatconsidercontagionthroughfinancialmarkets.KingandWadwhani(1990)considerasituationwhereinformationiscorrelatedbetweenmarkets.Pricechangesinonemarketareperceivedtohaveimplicationsforassetvaluesinothermarkets.Calvo(2002)andYuan(2005)considercorrelatedliquidityshocksasachannelforcontagion.Whensomeinvestorsneedtoobtaincashto,forexample,meetamargincalltheymayliquidateinanumberofmarketssotheshockisspread.KodresandPritsker(2002)useamulti-assetrationalexpectationsmodeltoshow 10.10ConcludingRemarks295howmacroeconomicriskfactorsandcountry-specificasymmetricinforma-tioncancombinetoproducecontagion.KyleandXiong(2001)presentamodelofcontagioninfinancialmarketsduetotheexistenceofawealtheffect.PavlovaandRigobon(2005)provideatheoreticalmodelofcontagionofstockmarketpricesacrosscountriesarisingfromwealthtransfersandportfolioconstraints.10.10CONCLUDINGREMARKSContagionisoneofthemostimportanttopicsintheareaoffinancialcrises.Theideathatshockscanspreadandcauseagreatdealmoredamagethantheoriginalimpactisonethatisextremelyimportantforpolicymakers.Itisusedtojustifymuchoftheinterventionandregulationthatisobserved.Aswehaveseencontagiontakesmanyforms.Althoughthereisalargeliteratureonthistopic,muchworkinthisarearemainstobedone.Thesameistrueofallthetopicscoveredinthisbook!REFERENCESAghion,P.,P.Bolton,andM.Dewatripont(1999).“ContagiousBankFailures,”workingpaper,PrincetonUniversity.Allen,F.andE.Carletti(2006).“CreditRiskTransferandContagion,”JournalofMonetaryEconomics53,89–111.Allen,F.andD.Gale(1998).“OptimalFinancialCrises,”JournalofFinance53,1245–1284.Allen,F.andD.Gale(2000).“FinancialContagion,”JournalofPoliticalEconomy108,1–33.Calvo,G.(2002).“ContagioninEmergingMarkets:WhenWallStreetisaCarrier,”ProceedingsfromtheInternationalEconomicAssociationCongress,vol.3,BuenosAires,Argentina2002.AlsoinG.Calvo,EmergingCapitalMarketsinTurmoil:BadLuckorBadPolicy?Cambridge,MA:MITPress2005.Calvo,G.andE.Mendoza(2000a).“ARationalContagionandtheGlobalizationofSecuritiesMarkets,”JournalofInternationalEconomics51,79–113.Calvo,G.andE.Mendoza(2000b).“ACapital-MarketsCrisesandEconomicCollapseinEmergingMarkets:AnInformationalFrictionsApproach,”AmericanEconomicReview90,59–64.Cifuentes,R.,G.Ferrucci,andH.Shin(2005).“LiquidityRiskandContagion,”JournaloftheEuropeanEconomicAssociation3,556–566.Claessens,S.andK.Forbes(eds.)(2001).InternationalFinancialContagion,Norwell,MA:Kluwer. 296Chapter10.ContagionDasgupta,A.(2004).“FinancialContagionthroughCapitalConnections:AModeloftheOriginandSpreadofBankPanics,”JournaloftheEuropeanEconomicAssociation6,1049–1084.DeBandt,O.andP.Hartmann(2002).“SystemicRiskinBanking:ASurvey,”inFinan-cialCrises,Contagion,andtheLenderofLastResort:AReader,C.GoodhartandG.Illing(eds.),Oxford:OxfordUniversityPress.Degryse,H.andG.Nguyen(2004)“InterbankExposures:AnEmpiricalExaminationofSystemicRiskintheBelgianBankingSystem,”NationalBankofBelgium,NBBWorkingPaperNo.43.Diamond,D.andP.Dybvig(1983).“BankRuns,DepositInsurance,andLiquidity,”JournalofPoliticalEconomy91,401–419.Dungey,M.andD.Tambakis(eds.)(2005).IdentifyingInternationalFinancialContagion,Oxford:OxfordUniversityPress.Eichengreen,B.,A.Rose,andC.Wyplocz(1996).“ContagiousCurrencyCrises:FirstTests,”ScandinavianJournalofEconomics98,463–484.Eisenberg,L.andT.Noe(2001).“SystemicRiskinFinancialSystems,”ManagementScience47,236–249.Freixas,X.andB.Parigi(1998).“ContagionandEfficiencyinGrossandNetInterbankPaymentSystems,”JournalofFinancialIntermediation7,3–31.Freixas,X.,B.Parigi,andJ.Rochet(2000).“SystemicRisk,InterbankRelationsandLiquidityProvisionbytheCentralBank,”JournalofMoney,Credit&Banking32,611–638.Furfine,C.(2003).“TheInterbankMarketDuringaCrisis,”JournalofMoney,CreditandBanking35,111–128.Glick,R.andA.Rose(1999).“ContagionandTrade:WhyAreCurrencyCrisesRegional?”Chapter9inP.Agénor,M.Miller,D.VinesandA.Weber(eds.),TheAsianFinancialCrisis:Causes,ContagionandConsequences,Cambridge,U.K.:CambridgeUniversityPress.Gorton,G.(1988).“BankingPanicsandBusinessCycles,”OxfordEconomicPapers40,751–781.Hicks,J.(1989).AMarketTheoryofMoney,NewYork:ClarendonPress;Oxford:OxfordUniversityPress.Iyer,I.andJ.Peydró-Alcalde(2006).“InterbankContagion:EvidencefromRealTransactions,”workingpaper,EuropeanCentralBank.Kaminsky,G.,C.Reinhart,andC.Vegh(2003).“TheUnholyTrinityofFinancialContagion,”JournalofEconomicPerspectives17,51–74.Karolyi,G.(2003).“DoesInternationalFinancialContagionReallyExist?”InternationalFinance6,179–199.Kindleberger,C.(1978).Manias,Panics,andCrashes:AHistoryofFinancialCrises,NewYork,NY:BasicBooks.King,M.andS.Wadhwani(1990).“TransmissionofVolatilityBetweenStockMarkets,”ReviewofFinancialStudies3,5–33. References297Kodres,L.andM.Pritsker(2002).“ARationalExpectationsModelofFinancialContagion,”JournalofFinance57,768–799.Kyle,A.andW.Xiong(2001).“ContagionasaWealthEffect,”JournalofFinance56,1401–1440.Lagunoff,R.andS.Schreft(2001).“AModelofFinancialFragility,”JournalofEconomicTheory99,220–264.Masson,P.(1999).“Contagion:MonsoonalEffects,SpilloversandJumpsBetweenMultipleEquilibria,”Chapter8inP.Agénor,M.Miller,D.VinesandA.Weber(eds.),TheAsianFinancialCrisis:Causes,ContagionandConsequences,Cambridge,UK:CambridgeUniversityPress.McAndrews,J.andW.Roberds(1995).“Banks,PaymentsandCoordination,”JournalofFinancialIntermediation4,305–327.Mitchell,W.(1941).BusinessCyclesandTheirCauses,Berkeley:UniversityofCaliforniaPress.Pavlova,A.andR.Rigobon(2005).“WealthTransfers,Contagion,andPortfolioConstraints”NBERWorkingPaperNo.W11440.Pericoli,M.andM.Sbracia(2003).“APrimeronFinancialContagion,”JournalofEconomicSurveys17,571–608.Pick,A.andM.Pesaran(2004).“EconometricIssuesintheAnalysisofContagion,”CESifoWorkingpaperSeriesNo.1176.Rochet,J.andJ.Tirole(1996a).“InterbankLendingandSystemicRisk,”JournalofMoney,CreditandBanking28,733–762.Rochet,J.andJ.Tirole(1996b).“ControllingRiskinPaymentSystems,”JournalofMoney,CreditandBanking28,832–862.Upper,C.(2006).“ContagionDuetoInterbankCreditExposures:WhatDoWeKnow,WhyDoWeKnowIt,andWhatShouldWeKnow?”workingpaper,BankforInternationalSettlements.Upper,C.andA.Worms(2004).“EstimatingBilateralExposuresintheGermanInter-bankMarket:IsthereaDangerofContagion?”,EuropeanEconomicReview48,827–849.VanRijkeghem,C.andB.Weder(2000).“SpilloversThroughBankingCenters:APanelDataAnalysis,”IMFWorkingPaper00/88,Washington,D.C.Yuan,K.(2005).“AsymmetricPriceMovementsandBorrowingConstraints:ARationalExpectationsEquilibriumModelofCrisis,Contagion,andConfusion,”JournalofFinance60,379–411. Thispageintentionallyleftblank IndexAdams,JohnQuincy3Bossons,J.125Agénor,P.296,297Boyd,J.18,19,228Aghion,P.294Brennan,M.101Allen,F.84,90,95,103,124,128,147,172,BrettonWoodsPeriod1945–197110176,184,193,204,212,214,218,224,Bryant,J.20,58,59,74,84,94,95,96,228,231,237,238,239,246,247,250,147,149252,262,280,281,290,293,294Buffersandbankruns277Alonso,I.95BusinesscycleviewofbankrunArgentinacrisisof2001–200217–1882,95Arnott,R.214Arrowsecurities146,150,42Arrow–Debreueconomy41Callloans6Asiancrisisof19971,15,260–261,217–218Calomiris,C.83,94,96,147,231AssetpricebubblesCalvo,G.260,294agencyproblems237Campbell,J.100bankingcrises247Capitalregulation191negative236Capitalstructurepositive236Modigliani–Millertheorem203riskshifting237optimal194withoutCentralBankintervention254withcompletemarkets201Azariadis,C.147CaprioJr.,G.214Carey,M.214Carletti,E.294Bagehot,W.3Carlsson,H.90,95BankcapitalCash-in-the-marketpricing102,incentivefunction193110–114risk–sharingfunction193Cass,D.147BankofEngland3Chang,C.233BankofSweden3Chang,R.230,231Bankruns74Chari,V.94,95,147empiricalstudies96Chatterjee,K.97BankingActof19355Chui,M.229,230Bankingandefficiency72–73Cifuentes,R.291Bannier,C.93Claessens,S.293Barth,J.213Competitiveequilibrium159Benefitsoffinancialcrises153Completemarkets41,70Bernanke,B.147Cone,K.94Bernardo,A.148Constantinides,G.98Bertaut,C.101Constrainedefficiency182,198Besanko,D.192Consumptionandsaving27,32Bhattacharya,S.94,95,192,214ContagionBillofexchange126asymmetricinformation260Blume,M.101BelgianbankingsystemBolton,P.295290–291Boot,A.214currencycrises261,294Bordo,M.2,3,9–12,16empiricalstudies284–292 300IndexContagion(contd.)Fama,E.100financialmarkets294–295Farmer,R.147Germanbankingsystem284–290FDIC190incompleteinterbankmarkets274Ferrucci,G.291overlappingclaims260FinancialCrisisof1763126paymentssystems293Financialfragility126priceeffects291–292FirstBankoftheUnitedStates3Contingentcommodities40FirstBaselAccord191Corsetti,G.231Fisher,S.125Costsoffinancialcrises18–19,153Fixedparticipationcost101–102Courchene,T.215Flood,R.229Crashof19292Forbes,K.293Crashof1987100Forwardmarketsanddatedcommodities31Creditandinterestratedetermination243Fourçans,A.16,229Crisesandstockmarketcrashes5Franck,R.16,229Crisesindifferenteras10Frankel,J.15,235Crockett,J.125Freixas,X.94,213,293Currencycrisesandtwincrises229Friedman,M.58,96Friend,I.101,125FSLIC190Dasgupta,A.294Fullparticipationequilibrium120DeBandt,O.293Fundamentalequilibrium129,141,148deNeufvilleBrothers126,147Furfine,C.293DeNicoló,G.231Furlong,F.192Deflation218Degryse,H.290Desai,M.258Dewatripont,M.98,213,295Gai,P.229,230Diamond,D.20,58,59,74,94,95,96,130,Gale,D.84,95,103,124,128,147,172,176,147,149,228,229,230,255,262184,193,195,199,204,212,214,218,Diamond–Dybvigpreferences150,116224,226,228,231,232,237,238,246,Dollarization231–232247,250,252,262,280,281,290,293Dollarizationandincentives226–228Galindo,A.231,232Dornbusch,R.125Garber,P.229Drees,B.235Geanakoplos,J.147,198Dungey,M.293GeneralequilibriumwithincompletemarketsDybvig,P.20,58,59,74,94,95,96,130,147,147149,230,255,262Gennotte,G.192Dynamictradingstrategies156Gertler,M.147Glass–SteagallActof19335,190Glick,R.294Economywidecrises149GlobalgamesapproachtocurrencycrisesEfficientallocationovertime27230Efficientrisksharing165Globalgamesequilibriumuniqueness90,95Eichengreen,B.9,25,294Goenka,A.147Eisenberg,L.293GoldStandardEra1880–191310Endogenouscrises148Goldstein,I.90,95Englund,P.14,235Goodhart,C.296Equilibriumbankruns76Gorton,G.4,20,83,94,95,96,147,231,239,Essentialbankruns85247,262Excessvolatilityofstockprices100Gottardi,P.147ExchangeRateMechanismcrisis229Gramm–Leach–BlileyAct190Extrinsicuncertainty129,148GreatDepression2,190,217 Index301Green,J.57Kaminsky,G.230,231,294Greenwald,B.214Kanatas,G.192Guiso,L.100,101Kanbur,R.214Gup,B.215Karolyi,G.293Keeley,M.192Kehoe,P.147Haliassos,M.101,125Keynes,J.M.216Hamilton,Alexander3Kim,D.192Hansen,L.98Kindleberger,C.2,3,20,58,126,147,Harris,M.98235,262Hart,O.147King,M.100,294Hartmann,P.293Kiyotaki,N.147Heiskanen,R.14,235Klingebiel,D.25Heller,W.215Kodres,L.260,294Hellman,T.192Krooss,H.4Hellwig,M.95Krugman,P.229Herring,R.213Kwak,S.25Hicks,J.261Kyle,A.295Hoggarth,G.18Honohan,P.25,233Hubbard,R.151,233Laeven,L.25Lagunoff,R.148,294Leape,J.100Illing,G.296Leiderman,L.231,232IMF25Leroy,S.100IncentivecompatibilityandprivateLevine,R.214information71Limitedmarketparticipation100–102,Incentivecompatible131114–124Incentiveconstraint131Liquidationpeckingorder275Incentiveefficiency72,175Liquidityinsurance68Incompletecontracts154Liquiditypreference53,59–60Incompletemarkets154Liquiditytrading100Inefficiencyofmarkets66LongTermCapitalManagement(LTCM)Insidemoney21616,127Insuranceandriskpooling48Lowenstein,R.16Interbanknetworkcomplete263,269incomplete263,271Magill,M.147Intrinsicuncertainty129,148Mailath,G.152Irwin,G.212Mankiw,N.100Iyer,I.294Marion,N.229Ize,I.233Martinez–Peria,M.25Mas–Collel,A.57Jacklin,C.94,95Mason,J.96,147Jagannathan,R.95Masson,P.261,293,294Japaneseassetpricebubble15,235McAndrews,J.293Jappelli,T.125Meckling,W.241Mendoza,E.260Jensen,M.241Merton,R.100Jones,C.6,26Mikitani,R.15Jonung,L.259Miller,M.296,297Mishkin,F.236Kahn,C.94Mitchell,W.20,58,262Kajii,A.147Moneyandbankingcrises228–229 302IndexMoneyandrisksharing218–223Reinhart,C.230,231,234,296Moore,J.147Reis,R.25Morris,S.90,92,93,95,230,259Rigobon,R.295Murdock,K.215Riskaversion45absolute46relative46Nalebuff,B.214Riskpooling55NationalBankActsof1863and18644Roberds,W.293NationalBankingEra4,83Rochet,J.90,94,95,192,293,294Neave,E.215Rogoff,K.234Nguyen,G.290Rose,A.294,296Noe,T.293Roubini,N.18,231RussianCrisis127Obstfeld,M.230RussianCrisisof199816Optimalcurrencycrises223–226Optimalmonetarypolicy256OptimalrisksharingthroughinterbankSaid,Y.258markets266Samuelson,L.152Outsidemoney216Samuelson,W.97Overend&Gurneycrisis2,3Santomero,A.192,213Özgür,O.199Santos,J.213Saporta,V.25,215Panic-basedruns94Savastano,M.234Parigi,B.293Sbracia,M.93,293Parke,W.100Scandinaviancrises1,14–15,235–236Participationandasset-pricevolatilitySchnabel,I.126,147120–121Schreft,S.148,294Pauzner,A.90,96Schwartz,A.58,96Pavlova,A.295SecondBankoftheUnitedStates3Pazarbasoglu,C.235SecondBaselAccord191Pericoli,M.293SeparationTheorem39Pesaran,M.294Sequentialserviceconstraint94Pesenti,P.231Sequentiallycompletefinancialmarkets170Peydró-Alcalde,J.294Setser,B.18Pick,A.294Shell,K.147Pigou,A.C.216Shiller,R.100Polemarchakis,H.198Shin,H.90,92,93,95,126,147,230,291Porter,R.100Smith,B.25,228,233Portfoliochoice49Sprague,O.6,96Posen,A.15St-Amant,P.215Postlewaite,A.95,148,259Starr,R.215Prati,A.93Starrett,D.215Prescott,E.98Stiglitz,J.215,241Pritsker,M.260,294Studenski,P.4Production36Stulz,R.98,214Pyle,D.192Sunspotequilibria129,140,147,148Sunspotsandbankruns76–77,94Sylla,R.6,26Quinzii,M.147Rajan,R.94,233,229Takagi,S.25,26Realbusinesscycle147Tambakis,D.293Regulationofliquidity204Tanaka,M.215 Index303Thakor,A.94,214Wallace,N.94,98Tillman,P.93Wealthandpresentvalues20Timberlake,R.3Weber,A.296,297Tirole,J.213,293,294Weder,B.260,294Tschoegl,A.15,235Weiss,A.241Turnovsky,S.98Welch,I.148Twincrises9,230–231West,K.100Whinston,M.57White,E.242Uncertainbankrunsandequilibrium76–82Wicker,E.96Upper,C.284,293Wilkins,C.215Wilson,J.5–8Valueofthemarket63Winton,A.94VanDamme,E.90,95Worms,A.284VanRijckeghem,C.260,294Wyplocz,C.296Vegh,C.296Velasco,A.230,231Vihriälä,V.14,235Xiong,W.295Vines,D.296,297Vissing–Jorgensen,A.101Vives,X.90,95,226,232Yuan,K.294VonNeumann-Morgensternutilityfunction44Zame,W.186Wadhwani,S.294Zeldes,S.100