欧洲央行-金融中介流动性的宏观经济学（英）

EUROPEANCENTRALBANKEUROSYSTEMDavidePorcellacchia,KevinD.SheedyThemacroeconomicsofliquidityinfinancialintermediationDisclaimer:ThispapershouldnotbereportedasrepresentingtheviewsoftheEuropeanCentralBank(ECB).TheviewsexpressedarethoseoftheauthorsanddonotnecessarilyreflectthoseoftheECB.AbstractInfinancialcrises,thepremiumonliquidassetssuchasUSTreasuriesincreasesalongsidecreditspreads.Thispaperexplainsthelinkbetweentheliquiditypremiumandspreads.Wepresentatheoryofendogenousbankfragilityarisingfromacoordinationfrictionamongbankcreditors.Thetheory’simplicationsreducetoasingleconstraintonbanks,whichisembeddedinaquantitativemacroeconomicmodeltoinvestigatethetransmissionofshockstospreadsandeconomicactivity.Shocksthatreducebanknetworthexacerbatethecoordinationfriction.Inresponse,bankslendlessanddemandmoreliquidassets.Thisdrivesupbothcreditspreadsandtheliquiditypremium.Bymitigatingthecoordinationfriction,expansionsofpublicliquidityreducespreadsandboosttheeconomy.Empirically,weidentifyhigh-frequencyexogenousvariationinliquiditybyexploitingthetimelagbe-tweenauctionandissuanceofUSTreasuries.Wefindacausalefectonspreadsinlinewiththecalibratedmodel.Keywords:bankruns,bank-lendingchannel,liquidassets.JELCodes:E41,E44,E51,G01,G21.Non-technicalsummaryIntimesoffinancialstress,banksfindithardtofundthemselvesandcreditbecomesmoreexpensive.Thispaperdocumentsthatfinancialturmoilisalsoassociatedwithanincreaseinthepremiumonliquidassets,suchasUSTreasuryBills.Whydoesthevalueofliquidassetsgoupwhencreditistight?Theanswertothisquestionhasimplicationsfortheefectivenessofexpansionsinliquid-assetsupply,afrequently-deployedpolicyresponsetofinancialstress.Toanswerit,wedevelopanovelfinancialfrictionexplainingthisempiricalrelationship.Then,weembeditinastandardmacroeconomicmodeltostudyimplicationsfortheeconomyandpolicy.Maturitytransformation,akeyfunctionoffinancialintermediation,resultsinamismatchonbanks’balancesheets.Thiscreatestheconditionsforcoordinationfailuresintheformof“runs”bypanickedcreditors.Thepotentialforrunsleadsinvestorstopricerunriskinthedebtofintermediaries.Becauserunsareself-fulfillingphenomena,itisdiculttopindowntheriskofruns.Todothis,wedepartfromtheassumptionofcommonknowledgeacrossbankcreditors,acommonapproachtolimittheirabilitytocoordinateonarbitrarybehaviour.With this,wefindthattheintensityofthefrictiondependsonbankbalance-sheetfundamentals.Inparticular,bankscanmitigaterunriskbyholdingmoreliquidassetsorbyhavingmorenetworth(i.e.,equity).Thus,ifashockreducestheirnetworth,banksdemandmoreliquidassets tokeeptheirrunriskincheck.Thisfrictiongeneratesacountercyclicalliquiditypremium.Withthecoordinationfrictionembeddedinastandardrealbusinesscyclemodel,wecanstudyitsrolequantitativelyinthetransmissionofmacroeconomicshocksandpolicy.Thefrictionamplifiesshocks:abadshockthatreducesbanks’networthincreasesbanks’fundingcostsonaccountofheightenedrunrisk.Higherbankfundingcostsreducethesupplyofcreditandtherebydrivedowninvestment.Moreover,thefrictionpropagatesshocksthroughtimebymakingitharderforbankstomakeprofitsandthusaccumulatenetworth.Themodel’sliquidassets,definedasassetsthatkeeptheirvalueincaseofasystemicfinancialcrisis,arethemonetaryandfiscalliabilitiesofthegovernment.Anincreaseinthesupplyofliquidassetscrowdsinprivateinvestmentbyexpandingcreditsupply.Thisisbecausebanksholdtheadditionalliquidassets,andtheresultingreductioninrunriskisreflectedinbetterfundingtermsforbanks.Therealefectsofliquiditysupplyimplythatitcanbeusedasastabilizingtoolinthefaceofshocks.Ifthegovernmentrespondstodisruptionstofinancialintermediationbyaccommodatingtheincreaseddemandforliquidassets,itcandampentheamplificationofFinally,wetestthemodelempirically.Thekeyimplicationofthemodelisthatahighliquiditypremiumpushesupbank-fundingcosts.WeruntheanalysisatdailyfrequencyandusethequantityofUSTreasuriesoutstandingasaninstrumentfortheliquiditypremium.TheinstrumentispredeterminedatdailyfrequencyonaccountofthetimelagbetweenauctionandissuanceofUSTreasuriesandthereforeitisvalid.Withthiseconometricstrategy,wefindasignificantpositiveefectoftheliquiditypremiumonbank-fundingcostswhichisquantitativelyinlinewiththecalibratedmodel.1IntroductionDisruptionstofinancialintermediationmakecreditmoreexpensiveandtherebyharmtheeconomy.Thispatternmotivatedtheintroductionofaspecificbankingfrictioninmacroeco-nomicmodels.Intheirseminalcontribution,GertlerandKiyotaki(2010)introduceaproblemofmoralhazardbetweenbanksandtheircreditors.Consequently,banks’abilitytofundthem-selvesislimitedbythevalueoftheirequity.Theresultingleverageconstraintleadstoapowerfulimpactofbanknetworthonmacroeconomicoutcomesviacreditspreads.Thisexplainsthegeneralobservationofplummetingbankvalues,higherbank-fundingcostsandincreasedcreditspreadsinfinancialcrises.1However,thisapproachtobankingissilentonwhyweobservesoaringdemandsforliquidityandhenceliquiditypremiumsintimesoffinancialstress.Weobserveaheightenedliquiditypremium,definedasthediferencebetweenthe3-monthgeneral-collateralreporateandthe3-monthtreasury-billrate,duringbankingcrises.2Figure1showsthisfortheglobalfinancialcrisis.3Moresystematically,thispaperdocumentsapositiverelationshipovertimebetweentheliquiditypremiumandbanks’fundingcosts,asmeasuredbythediferencebetweenthe3-monthLIBORandthe3-monthreporate.Figure2showsthepositivecorrelationbetweenthesetwovariables.4Sincepolicymakersoftenreacttobankingcrisesthroughexpansionsofliquidity,itiscrucialtounderstandthecausesoftheempiricalrelationshipbetweentheliquiditypremiumandfundingcosts.5Existingresearch(KrishnamurthyandVissing-Jorgensen,2012;Nagel,2016)hasshowntheliquiditypremiumrespondstogovernmentpolicies.Thefactsdocumentedinthispapersuggestthattightbankfundingdrivesdemandforliquidassets.Thisisconsistentwithaviewthatscarceliquidityimpairsbanklendingintimesofstress,suggestingachannelthroughwhichagreatersupplyofpublicliquiditycanbenefittheeconomy.Motivatedbythis,wedotwothingsinthepaper.First,wedevelopanovelfinancialfrictionbasedoncoordinationfailureamongbankcreditors.Liquid-assetholdingsandbanknetworthbothmitigatethecoordinationfrictionandaresubstitutes.Hence,whennetworth1Figure9andFigure10inappendixAshowstheaveragedynamicsofthesevariablesinbankingcrisesasidentifiedbyBaronetal.(2021).2Thisdefinitionofliquiditypremiumisstandardintheliterature(Nagel,2016).Morediscussiononthispointisprovidedinsection7.3Figure11inappendixAzoomsinontherecentperiod(2019–2023).4Figure12showsthatthepositivecorrelationholdsbothinexpansionsandrecessions.5Thereisadebateintheliteratureontherealefectsofliquiditypoliciesandthechannelsthroughwhichtheyoperate(Kuttner,2018).fundingspread(p.p.)2.522110.5fundingspread(p.p.)2.522110.500Figure1:Globalfinancialcrisis.FundingFundingspread(p.p.)Liquiditypremium(p.p.)2006200720082009Figure2:May1991–June2023.00.20.40.60.81liquiditypremium(p.p.)Note1:Fundingspreadis3-month(3M)LIBORminus3Mgeneral-collateral(GC)reporate.Liquiditypremiumis3MGCreporateminus3MT-billrate.Note2:USmonthlydata.DatasourcesinappendixB.isscarce,asinafinancialcrisis,banksdemandmoreliquidity.Thisexplainsahighliquiditypremium.ItalsoimpliespolicycanstabilizetheeconomybyappropriatelysupplyingliquidSecond,wetestwhetherthedatasupportsthemodel’smechanism.Inparticular,themodelimpliesthatanincreaseintheliquiditypremiumpushesupthebank-fundingspread.Thisisbecauseitinducesbankstoeconomizeonholdingliquidassets.Toidentifyexogenousvariationintheliquiditypremium,weusethequantityofoutstandingUSTreasuriesasaninstrumentalvariable.TheinstrumentisstronglyrelevantandpredeterminedatdailyfrequencygiventhelagofafewdaysbetweentheauctionandissuanceofTreasurysecurities.Wefindasignificantpositiveefect.Maturitytransformation,akeyfunctionoffinancialintermediation,resultsinamismatchonthebalancesheetsofbanks.6Thiscreatestheconditionsforcoordinationfailuresinthemarketfordeposits(DiamondandDybvig,1983).7Suchcoordinationfailurestaketheformof“runs”bypanickedcreditorsandplayedacentralroleintheglobalfinancialcrisisin2007,thecrisisofUSmoney-marketfundsin2020andthe2023regionalbankingcrisis(Shin,2009;Bernanke,2010;Lietal.,2020;Choietal.,2023).6Forsimplicity,weuse“banks”asagenerallabelforfinancialintermediariesand“deposits”fortheirshort-termdebt.Theanalysisappliesmorebroadlytofinancialintermediarieswithamaturitymismatchontheirbalancesheet.7Perfectdepositinsurancerulesoutcoordinationfailuresinthesemodels.However,intheperiod1984–2023Q3depositsmadeup73%ofbanks’liabilitiesandonly62%ofdepositswereinsuredonaverage.Thesevaluesarerespectively79%and57%in2023Q3(datasource:FDICQBP).Thispapermodelsthedepositsmarketasacoordinationgame.Strategiccomplementari-tiesimplythatunderperfectinformationtherearemultipleequilibria.However,adeviationfromcommonknowledgeacrossdepositors,whichweintroducefollowingthelargeliteratureonglobalgames(MorrisandShin,2003),leadstoauniqueequilibrium.Intuitively,withoutcommonknowledgeitisimpossiblefordepositorstocoordinateonarbitraryequilibria.Intheresultinguniqueequilibrium,depositorsdemandalevelofcompensationthatiscommensuratetoabank’sfragility,definedastheminimumshareofdepositorsthatmustnotrunforthebanktosurvive.Ifthebankofersaninsucientdepositrate,thendepositorsruneventhoughthebankissolvent.Intuitively,banksmustcompensatedepositorsforrunrisk.However,aslongasthebankofersasucientlyhighdepositrate,noruntakesplacebecausenodepositorhasanincentivetostarttherunthattheyfear.Bankfragility,theheartofthecoordinationfriction,isendogenous.Itisafunctionofthebank’sbalance-sheetfundamentals.Inparticular,moreleveredbanksandbankswithfewerliquidassetsasashareoftotalassetsaremorefragile.Therefore,theyfacehigherfundingcosts.Inotherwords,thecoordinationfrictionresultsinamappingfromhighercapitalandliquidityratiosintoalowerfundingspread.Thecapitalandliquidityratiosarebankchoices.Inequilibrium,thesechoicestradeofthereturnsonilliquidassetsagainsttheincreasedfundingcostsduetomorefragility.Withthecoordinationfrictionembeddedinastandardrealbusinesscyclemodel,wecanstudyitsrolequantitativelyinthetransmissionofmacroeconomicshocks.Thebankingfrictioncanbecalibratedusingobservationsontheaveragesizeoftheliquiditypremium,thecreditspread,andbanks’returnonequity.Theparametersofthemacroeconomicmodelaresetfollowingtheliterature.Thefrictionamplifiesshocksthatafectbanks’networth.Bymakingitmorecostlyforbankstofundthemselves,areductioninnetworthweakensthesupplyofcreditandreducestheeconomy’soutput.Thefrictionamplifiestheefectonoutputofcapital-destructionshocks,commonlystudiedintheliteratureonfinancialcrises,byaboutonethirdonimpact.Atlongerhorizons,theamplificationisgreater.Thispersistencecomesfrombanks’fundingcostsrisingalongsidecreditspreads,implyingbanks’networthisrebuiltveryslowlyincontrasttomodelswithaleverageconstraint.Furthermore,theincreaseinfragilityduetoscarcernetworthgivesbanksanincentivetodemandmoreliquidassets.Thisgeneratesacountercyclicalliquiditypremium.Monetaryandfiscalliabilitiesofthegovernmentarethenaturalsourceofliquiditysupply.Bankscreateliquidassetsforothersectorsoftheeconomybuttheycannotproduceassetsthatmaintaintheirvalueincaseofasystemicrun.8Therefore,therelevantsupplyofliquidassetsisapolicyvariable.Inthemodel,anincreaseinthesupplyofliquidassetsisexpansionary.Theliquidassetsareabsorbedbybanks’balancesheetsandreducetheirfragility.Withlowerfragility,bankshaveaccesstofundingonbettertermsandthusfinditoptimaltolendmore.Inotherwords,thesupplyofliquiditycrowdsinprivateinvestment.Inthecalibratedmodel,ashockthatreducestheliquiditypremiumby15basispointsleadstoanexpansionofcreditsupplyreducingcreditspreadsby24basispoints.Thisgeneratesa2-percentincreaseininvestmentonimpact,withGDPalsogoingupbyaquarterofonepercent.Moreover,thesupplyofliquiditycanbeusedasastabilizingpolicytoolinthefaceofshocks.Ifthegovernmentrespondstodisruptionstofinancialintermediationbyaccommodatingtheincreaseddemandforliquidassets,itcandampentheamplificationofshocks.Wetestthekeyimplicationofthemodel:anincreaseintheliquiditypremiumcausesanincreaseinthefundingspread.9Theeconometricchallengeistofindexogenousvariationintheliquiditypremium.OurstrategyistoruntheanalysisatdailyfrequencyandusethequantityofoutstandingUSTreasuriesasaninstrument.Theinstrumentisstronglyrelevanttotheliquiditypremium.Asforitsvalidity,thequantityoftreasuriesispredeterminedatdailyfrequencybecausethereisalagofafewdaysfromauction,whereitisdetermined,toissuance.Moreover,weincludeascontrols80lagsoffinancialandeconomicvariablesavailableatdailyfrequency,suchasthedollarexchangerateandtheliquiditypremiumitself.Thiscleanstheautocorrelationoutoftheerrortermandensuresthereisnoendogeneityoftheinstrumentdrivenbyconfoundingvariablesorreversecausality.Afterall,iftheerrortermonlycontainsanon-autocorrelateddailyshock,itcannotdriveavariabledeterminedonapreviousday.Theempiricalresultisarobustly-significantpositiveefectoftheliquiditypremiumonthefundingspread.A1-basis-pointincreaseintheliquiditypremiumcausesthefundingspreadtoincreasebyabout1basispoint.Thisisinlinewiththesizeofthecorrespondingefectinthecalibratedmodel.Asarobustnesscheck,weefectivelysplitthesamplebetweenexpansionsandrecessions.Wefindnoevidenceofadiferentsizeoftheefectaccordingtothestateofthe8ThisisrelatedtotheseminalfindinginHolmstrmandTirole(1998)ofaroleforpublicliquiditysupplyinthepresenceofaggregaterisk.9Wemeasurethefundingspreadasthediferencebetweenthe3-monthLIBORandthe3-monthGCreporate.Morediscussiononthemeasurementisprovidedinsection7.economy.Literaturereview.Anextensiveliteraturebuildsmacroeconomicmodelsaroundaleverageconstraintonbanks(GertlerandKiyotaki,2010;GertlerandKaradi,2011;HeandKrishna-murthy,2013;BrunnermeierandSannikov,2014;Boissayetal.,2016;Phelan,2016;KaradiandNakov,2021;VanderGhote,2021;Fernndez-Villaverdeetal.,2023).10Thisfriction,basedonmoralhazard,doesnotnaturallygivearoletobanks’liquid-assetholdingsunlikethispaper’sfrictionbasedoncoordinationfailure.Moreover,modelswiththemoral-hazardfrictiongeneratelimitedshockpropagationbecauseadverseshockstobanknetworthpushupbankprofitabilitybyincreasingcreditspreadswithlittleresponseoffundingcosts.Also,theystruggletomatchtheobservedprocyclicalityofbanks’bookleverage(NuoandThomas,2017).Thecoordinationfrictionisanimprovementonthelattertwocounts,too.Inthispaper’smodel,shockpropagationisstrongbecausethepositiveefectofhighercreditspreadsonbankprofitabilityafteradverseshocksislargelyofsetbyincreasedfundingcosts.Andwefindthatleverageisprocyclicalbecausefragilityiscountercyclical.Inthispaper,banksdemandliquidassetstomitigatetheriskofcoordinationfailuresamongtheircreditors.Thisisanovelsourceofdemandforliquidassetsinthemacroeconomicliterature.11Theexistingliteraturepositsanexogenousriskthatbankcreditorswithdrawtheirfunds.Banksdemandliquidassetsasaprecautiontolimittheamounttheymustborrowfromthecentralbankatapunitiveinterestrate(Poole,1968;Arceetal.,2020;BianchiandBigio,2022)ortheamountofassetstheymustsellatfire-saleprices(Drechsleretal.,2018;d’AvernasandVandeweyer,forthcoming;Li,forthcoming)ifhitbyanadverseliquidityshock.Inourmodel,theriskofdeposit-holderwithdrawalsisafullyendogenousfunctionofbankfundamentals.12Studiesevaluatingquantitative-easingprogrammes,recentexamplesofpoliciesthatincreasedthesupplyofliquidassets,findreductionsininterest-ratespreadsinlinewithourmodel(Gagnonetal.,2011;KrishnamurthyandVissing-Jorgensen,2011).Morerecently,10Thefirstpapertoderivealeverageconstraintonbanksfromamoral-hazardproblemisHolmstrmandTirole(1997).BrunnermeierandPedersen(2008)derivealeverageconstraintbasedonvalueatrisk.11Astrandofthebankingliteratureformalizesthisinstaticpartial-equilibriummodels(RochetandVives,2004;Ahnert,2016).12Areduced-formapproachtothedemandforliquidassetsiscommoninstudiesoftheefectsofliquiditysupply(KrishnamurthyandVissing-Jorgensen,2012;BenignoandBenigno,2022;Angeletosetal.,2023).Suchapproachmaymissimportantcharacteristicsofdemandforliquidassetssuchasthesubstitutabilityofliquidityandbankcapital,whichisafeatureofourmodelandwhichDeYoungetal.(2018)findsempirically.AcharyaandRajan(2022)andDiamondetal.(2023)havesoundedacautionarynoteontheefectsofliquid-assetsupplyinthecontextofQE.Theformerstressesthatsomeofthebenefittobankfragilityofadditionalliquiditysupplyisundonebybankstakingonextraleverage.Thisresultconformstothispaper’smechanism.Thelattercontributionfindsempiricallythatliquid-assetholdingsincreasebanks’marginalcostoflending.Theauthorssuggestthereasonforthismaybelimitedbalance-sheetspaceduetoregulation.Whiletheefectofregulationisbeyondthescopeofourpaper,thedrivingforcebehindourpaper’sresults,i.e.thepositiveefectofliquid-assetholdingsbybanksonthedemandfortheirdebt,isnotconsideredinDiamondetal.(2023).Banks’vulnerabilitytorunshasbeenfirstformalizedinDiamondandDybvig(1983).Thatpaperillustratesthepossibilityofruns,butitdoesnotspeaktotheirdeterminantsbecauseithasmultipleequilibria.Aliteratureinmacroeconomicshasadoptedthemultiple-equilibriumapproachtostudytheefectsofbankruns(GertlerandKiyotaki,2015;Gertleretal.,2016,2020;AmadorandBianchi,forthcoming).Alimitationofthisapproachistheneedtoassumeanarbitraryrelationshipoftheprobabilityofrunswithfundamentals.Becauseofthislimitation,theroleofliquidityinthedeterminationofrunriskdoesnotemerge.LeveragingtheoreticalresultsfromCarlssonandvanDamme(1993),GoldsteinandPauzner(2005)showthatasmalldeparturefromperfectinformationproducesauniqueequilibriuminabank-rungame.Thisisanattractivefeaturebecausetheevidencepointstoastrongrelationshipbetweenpoorbankfundamentalsandbankingcrises(Gorton,1988;Baronetal.,2021).Alargeliteratureinbankingusesvariationsofsuchsecond-generationbank-runmodelstostudyoptimalpolicy(Vives,2014;Kashyapetal.,2024;Ikeda,2024).Ourpaperisthefirsttointegrateasecond-generationbank-runmodelinamacroeconomicframework.13Outlineofthepaper.Thecoordinationgameamongdepositorsislaidoutinsection2.Thisresultsinaconstraintonbankbehaviour,whichisintegratedinastandardmacroeconomicmodelinthefollowingtwosections.Insection5,wediscusspropertiesofbanks’demandforliquidassets.Themodeliscalibratedandquantitativeexperimentsarecarriedoutinsection6.Insection7,theempiricalresultsofthestudyarereported.Theappendicescontain:(A)figures,(B)detailsaboutdatasources,(C)proofs,(D)steady-stateresults,and(E)thefullmodel13Asmallstrandofthebankingliteraturehasstudiedtherelationshipofbankrunswithselectedmacroeconomicvariables(EnnisandKeister,2003;Martinetal.,2014;Porcellacchia,2020;MattanaandPanetti,2021;Leonelloetal.,2022).solution.2CoordinationgameThissectionsetsupthecoordinationgameplayedbybankdepositors.Itsolvesfortheuniqueequilibrium,whichimpliesarelationshipbetweenbanks’balancesheetsandtheinterestratesrequiredtoinducehouseholdstoholddeposits.Sincebanksanticipatetheoutcomeofthecoordinationgame,intheremainderofthepaperthisrelationshipconstrainsthechoicesmadebybanks.Theeconomycontainsaunitcontinuumofbanks(moregenerally,financialintermediaries)indexedbyb2[0,1].Depositsatbankbaredemanddepositspayinginterestratejbifheldtothenexttimeperiod,butwithanoptiontowithdrawondemand.Whilereferredtoas‘demanddeposits’,thisbankdebtcanbeinterpretedmorebroadlyasshort-termunsecuredborrowinginmoneymarketsthatisfrequentlyrolledover.Acoordinationgameamongdepositorsisplayedineachdiscretetimeperiod,buttimesubscriptsareomittedinthissectiongiventheessentiallystaticnatureofthegame.Justbeforethecoordinationgamebegins,alldepositsDb≥0atbankbareheldequallybyaunitcontinuumofhouseholdsindexedbyh2[0,1].Expectedpayofsinthenexttimeperiodarediscountedatrateρbyallhouseholds.14Bankfragility.Beforehouseholdsdecidewhethertoholddepositsinthecoordinationgame,banksmakeportfolioandleveragedecisions.BankbchooseshowmuchtoinvestinilliquidandliquidassetsAbandMbrespectively,wherethenotionofliquidityisdefinedbelow.Takingasgivennetworth(equity)Nb,thesechoicesresultindepositcreationuptoalevelofdepositsDbconsistentwiththebalance-sheetidentityAb+Mb=Db+Nb.Thesedepositsareinthehandsofhouseholdsatthepointwherethecoordinationgameamongdepositorsisplayed.Ifapositivefraction1-HbofhouseholdschoosesnottoholddepositsDbatbankb,thebankmustmakeatotalpayment(1-Hb)Dbtothesehouseholdsbydisposingofsomeassets.ThefullvalueMboftheliquidassetsacquiredearliercanbeobtainedatthispoint,butdisposalofilliquidassetsAbduringthecoordinationgameonlyrecoversafractionλoftheirvalueatacquisition.Iftheproceedsoftheseassetliquidationsareinsucienttocoverthewithdrawals,14Sincethereisacontinuumofbanks,depositorbehaviourcanbeanalysedasifhouseholdswereriskneutralandρtakenasgiven.Inthefullmodel,thecommondiscountrateρisanendogenousvariable.thenthebankfails.Theconditionforfailureisgivenby(1-Hb)Db>λAb+Mb.(1)Theparameterλ2[0,1]measurestheliquidityofassetsAbrelativetothebenchmarkoftheperfectlyliquidassetMb.Rearrangingtheconditionaboveandusingthebalance-sheetidentity,bankbdoesnotfailifHb≥Fb,wherefragilityFbisgivenbyIfnetworthispositive,fragilityisanumberbetween0and1-λ,andgreaternetworthlowersfragility.IncreasedholdingsofliquidassetsMbreduceabank’sfragilitywhereitisinitiallypositive,whileholdingmoreilliquidassetsAbraisesfragilitywhenitisbelow1-λinitially.Anoteworthyfeatureoffragilityisthatitcanbeexpressedintermsoffamiliarliquidityandcapitalizationratios,respectivelyHence,thebank’sscaleplaysnoroleindeterminingitsfragility.NoticethattheilliquidityofassetsAbiskeytotheexistenceofacoordinationproblem.Ifthefullvalueofanyassetscanalwaysberealized(i.e.,λ=1),thenbankswithpositivenetworthcanneverbefragile.Itisalsoimportantthatbanks’portfoliochoiceismadebeforepeopledecidewhethertoholddeposits:onceilliquidassetsarefundedbydepositcreation,thereisstrategiccomplementarityindepositors’holdingdecisions.Thistimingassumptioncouldcapturethefactthatbankscreatedepositswhentheymakealoanandthensomeoneintheeconomymustbewillingtoholdthedepositsifthebankistoavoidhavingtodisposeofassets.Moregenerally,itcouldbeinterpretedasamismatchbetweenthetimingofcapitalinvestment,whichistypicallylong-term,andbanks’moreshort-termfundingsources.Structureofthegame.Independentlyforeachbankb,householdsmakeasimultaneousbinarychoicewhethertoholddepositsDbuntilthenextperiod.15ThischoiceiscapturedbytheindicatorfunctionHbh,whichequals1ifhouseholdhholdsand0ifitchoosestowithdraw.Withdrawinghouseholdsreceivefundsinthesametimeperiod.16Holdingbankdepositsexposeshouseholds