Limited Deposit Insurance Coverage and Bank Competition

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Limited Deposit Insurance Coverage and Bank Competition Oz Shy, Rune Stenbacka, and Vladimir Yankov 2014-99 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

∗ Limited Deposit Insurance Coverage and Bank Competition OzShy† RuneStenbacka‡ FederalReserveBankofBoston HankenSchoolofEconomics VladimirYankov§ BoardofGovernorsoftheFederalReserveSystem October15,2014 Abstract Deposit insurance designs in many countries place a limit on the coverage of deposits in each bank. However, no limits are placed on the number of accounts held with different banks. Therefore, under limited deposit insurance, some consumers open accounts with different banks to achieve higher or full deposit insurance coverage. We compare three regimes of deposit insurance: No deposit insurance, unlimited deposit insurance, and limited deposit insurance. We show that limited deposit insurance weakens competition among banksandreducestotalwelfarerelativetonoorunlimiteddepositinsurance. Keywords: Limiteddepositinsurancecoverage,depositrates,bankcompetition. JELClassificationNumber: G21. (Draft=deposit-136.tex2014/10/1517:23) Note: Thispapercontainshyper-referencesforeasiernavigation. Ifyoureadthisarticleona computer,youcanuseALT-leftarrow(Windows)orCommand-leftarrow(Mac)togoback tothereferringpageafterclickingonanyhyper-reference. ∗We held discussions and received most helpful comments from John Driscoll, Huberto Ennis, Michal Kowalik, NedPrescott, RafaelRepullo, JonathanRose, andAlexandrosVardoulakis, aswellasparticipantsatseminarsgiven attheFDIC,2014EuropeanMeetingoftheEconometricSociety,and2014ConferenceoftheFederalReserveSystem CommitteeonFinancialStructureandRegulation. Theviewsexpressedinthispaperarethoseoftheauthorsanddo notnecessarilyrepresenttheviewsoftheFederalReserveBankofBostonortheFederalReserveSystem. †E-mail:Oz.Shy@bos.frb.org.ResearchDepartment,FederalReserveBankofBoston,600AtlanticAvenue,Boston, MA02210,U.S.A. ‡E-mail:Rune.Stenbacka@hanken.fi.HankenSchoolofEconomics,P.O.Box479,00101Helsinki,Finland. §E-mail: Vladimir.L.Yankov@frb.gov. BoardofGovernorsoftheFederalReserveSystem,20thStreetandConstitutionAvenueN.W.,Washington,D.C.20551.

1. Introduction During the Free Banking Era and the Great Depression banks faced deposit runs, where small depositorssimultaneouslywithdrewtheirdepositstriggeringilliquidityanddefaultonotherwise healthy financial institutions. The financial crisis of 2008 brought a new type of “bank runs”, which involved the non-traditional “shadow” banking system, and where financial institutions ran on other financial institutions.1 The most significant institutional change since the Great Depression that prevented the traditional type of bank runs was the presence of deposit insurance. This paper focuses on two aspects of the design of the deposit insurance that have not received muchattentionintheacademicliteratureandtheimportanceofwhichbecameevidentduringthe 2008financialcrisis. The first aspect of the deposit insurance design is that insurance is partial in the sense that it haslimitedcoverage. Thesecondaspectisthatthedepositinsurancelimitappliestooneinstitutionperdepositoraccountbutisunlimitedwithrespecttothenumberofaccountswithdifferent banksallofwhicharesubjecttothesamedepositinsurancelimit. Ourpaperaddressesthequestionofhowlimiteddepositinsurancecoverageaffectstheintensityofcompetitioninthedeposit market. We also explore the effects of limited deposit insurance on consumer welfare as well as totalwelfarecomparedwithsystemsofunlimitedornodepositinsurance. Ourstudyinitiallydocumentsafewstylizedfactsonthedemandformultipledepositaccounts across different banks. We demonstrate that wealthier U.S. households hold multiple deposit accountswithmultipledepositinstitutions. Thedemandformultipleaccountscorrelatespositively with the financial wealth of U.S. households. Further, the average amount deposited in accounts thatexceedthedepositinsurancelimitisapproximatelyatmostthreetimesthedepositinsurance limit,thus,makingitfeasiblefordepositorswithpartiallyinsureddepositaccountstoachievefull insurance by distributing their deposits among several banks. We further document that smaller banks, which are deemed riskier, attract more insured brokered certificates of deposits as compared to larger banks. During the recent financial crisis, however, both small and large banks experiencedanequallylargeincreaseintheshareofinsuredbrokereddeposits. 1SeeGorton(2010)andGorton(2012)foranalysisoftherecentfinancialcrisisinhistoricalperspective. 1

We next develop a stylized theoretical model of deposit market competition with the feature thatsomeconsumersdiversifytheirfundsacrossdifferentbanksinordertoqualifyforcomplete deposit insurance coverage. We establish that a system with limited deposit insurance coverage softens deposit market competition as compared to systems with unlimited or no deposit insurance. Wefurthershowthatlimiteddepositinsurancereducesconsumerwelfareandtotalwelfare notonlybyinducingdepositorstobearcostsofopeningseveralaccounts,butalsobyweakening competitioninthedepositmarket. We build on an extensive literature which has examined the role of deposit insurance for social welfare. Following the seminal contribution by Diamond and Dybvig (1983), the literature has typically analyzed deposit insurance systems within the framework of models focusing on bank runs. Diamond and Dybvig (1983) demonstrated how the interaction between pessimistic depositor expectations may generate bank runs as an inefficient Nash equilibrium, and how depositinsurancesystemscaneliminatesuchinefficientequilibria. Subsequently,animportantand extensive category of studies, exemplified by Keeley (1990), Matutes and Vives (2000), and Shy andStenbacka(2004),hasexploredtheconsequencesofimperfectcompetitionfordepositsonthe risk-takingincentivesbybanks. Forexample, MatutesandVives(2000)characterizeindetailthe roles played by limited liability, deposit insurance with complete coverage, and deposit market competition for the determination of risk-taking by banks. Also, Matutes and Vives (1996) characterizehowthewelfareimplicationsofdepositinsurancewithcompletecoveragedependonthe marketstructureofthebankingindustry. Furthermore, theoretical studies regarding the effects of deposit insurance have typically focusedoncompletedepositinsurancewithunlimitedcoverage. OneexceptionisManz(2009),who characterizes the optimal level of deposit insurance coverage as well as its determinants. However,Manz(2009)doesnotanalyzetheeffectoflimiteddepositinsurancecoverageonthedemand formultipledepositaccountsandthecompetitioninthedepositmarket. Empirical studies have presented cross-country evidence regarding the effects of deposit insurancecoverageondepositrates. PenatiandProtopapadakis(1988)analyzemoralhazardissues generatedbydepositinsurance. Demirgu¨c¸-KuntandHuizinga(2004)exploitcross-countrydiffer- 2

ences regarding the country-specific features of deposit insurance to conclude that the existence ofanexplicitinsurancepolicylowersdepositrates,whileatthesametimeitalsoreducesmarket discipline on bank risk taking. Bartholdy, Boyle, and Stover (2003) present evidence that the risk premium is on average over 40 basis points higher in countries without deposit insurance than in countries with deposit insurance. Bartholdy, Boyle, and Stover (2003) argue that the risk premium is a non-linear function of the deposit insurance coverage, a feature which they interpret to mean that the market recognizes that extended deposit insurance coverage makes the moral hazard problems more severe. Pennacchi (2006) shows that the combination of a deposit insurancedesignwhichfacilitatescompleteinsurancecoveragethroughmultipledepositaccountsand mispriceddepositinsurancepremiahasgivenbanksacompetitiveadvantageovermoneymarket fundsinprovidingsafehavenassetclasses. Since Merton (1978), who applied option pricing to characterize the arbitrage free pricing of deposit insurance premia under costly supervision, the debate on the deposit insurance design has focused on formulating actuarially fair premia that correctly reflect the credit risk that individualbanksface. Thisdebatewasintheearly1990saccompaniedwiththeintroductionofcapital requirementsimposedbytheBaselregulation, whichfocusedoncontrollingtheindividualbank creditrisk. Sincethefinancialcrisis,theparadigmofbothcapitalrequirementsandthedesignof depositinsurancepremiashiftedtoanalyzethepricingthesystemicriskoffinancialinstitutions, see Pennacchi (2009). However, neither of these studies nor the policy debate has focused on the effectofthepartialinsurancedesignonbankcompetition. Itshouldbeemphasizedthatourstudyanalyzestheeffectsofdepositinsurancewithlimited coverage on deposit market competition without explicitly modeling banks’ risky lending decisions. Abstractingfrommoralhazardissues,wedevelopastylizedmodelinordertohighlightin atransparentwayhowdepositinsurancesystemswithlimitedcoverageinducesomeconsumers to diversify their deposits across several banks.2 Our normative analysis is restricted to the investigationofhowdepositinsurancesystemswithlimitedcoverageaffectbankprofits,consumer 2Anumberofimportantstudies,forexample,Hellwig(1998)andWinton(1997),haveanalyzedtheperformanceof thebankingsystemfromtheperspectiveofdiversificationofeconomy-widerisks.Thesestudieshavetypicallyfocused onbanks’lendingactivities. Inourmodelthediversificationiscausedbythelimitedcoverageofdepositinsuranceas someconsumerssplittheirfundsacrossseveralbanks. 3

welfare, and total welfare. We do not attempt to address the more challenging issue of how to characterize the socially optimal design of deposit insurance. Instead, the goal of this study is to pointoutsomedistortionsthatarisefrompartialinsuranceanddonotariseinsystemswithnoor unlimiteddepositinsurance. The paper is organized as follows. Section 2 presents some empirical facts regarding the implementation of deposit insurance in the United States. Section 3 constructs a model of deposit marketcompetition. Section4analyzesequilibriumdepositratesandprofitsaswellasconsumer and total welfare in the absence of deposit insurance. Section 5 introduces unlimited deposit insurance. Section 6 analyzes equilibrium deposit rates and profits as well as consumer and total welfare with limited deposit insurance. Section 7 presents the main results of our analysis by comparing the performance of the banking industry under the three regimes of deposit insurance. Section 8 extends the model to independent bank failures. Section 9 outlines some further extensions. Finally,Section10presentssomeconcludingcomments. 2. Deposit Insurance and Demand for Multiple Deposit Accounts: Empirical Facts SinceitsestablishmentwiththepassingoftheBankingActin1933,theFederalDepositInsurance Corporation (FDIC) in the United States was designed to insure bank deposits up to a certain dollaramount,calledthedepositinsurancelimit.3 Therationaleforthelimitedinsurancedesign is twofold: to guarantee financial stability by preventing bank runs, and to provide incentives to monitorthebanks. The intention behind the partial deposit insurance coverage is to protect small and unsophisticatedinvestors,whileatthesametimetoexposethewealthierandbetterinformedinvestorsto theindividualbank’screditrisk. Beingexposedtoabank’screditrisk,thewealthierandmoresophisticatedinvestorsareexpectedtoimposemarketdisciplineonbanksbywithdrawingdeposits from banks with lower asset quality. However, the deposit insurance design gives the option to 3Partialdepositinsuranceisalsothenorminmostcountrieswithexplicitdepositinsurance. AsurveybytheIMF Garcia (2000) documents that out of the 78 countries with explicit deposit insurance in 2000, 68 had implemented limiteddepositinsuranceandonly10countrieshadunlimiteddepositinsurance. 4

thesewealthyinvestorstoextendtheinsurancecoverageorevenachievecompletedepositinsurance by opening multiple deposit accounts with different banks. To achieve full insurance, the numberofaccountscanbecomputedbydividingtotaldepositamountsbythedepositinsurance limit.4 The FDIC does not provide any official explanation for how the deposit insurance limit was determined and to what extent the two rationales for its design were met. Table 1 displays the historical values of the deposit insurance limit both in nominal terms at the time they were set and in real values measured in 2010 dollar amounts. Table 1 shows that for the average U.S. householdthedepositinsurancelimitwasalwayssufficienttocovertheaveragefinancialwealth held in deposits and most part of the total financial wealth. Similarly, Figure 1 shows the time series behavior of the real values of the deposit insurance limit, the average deposit, and total financialwealthduringtheperiodsbetweentheinsurancelimitadjustments. Table1: FDICinsurancelimits1934-present Year Limit(nominal) Limit(real) Fin.wealth(real) Deposits(real) 1934 2,500 40,218 n/a n/a 1935 5,000 78,434 n/a n/a 1950 10,000 89,460 119,581 20,439 1966 15,000 99,497 184,555 37,293 1969 20,000 117,384 194,933 39,321 1974 40,000 174,658 181,028 47,361 1980 100,000 261,263 208,522 49,177 2008 250,000 250,000 370,674 69,176 NOTE:Allrealvaluesarecomputedusingtheconsumerpriceindexforallitemswithbaseyear2008, thefinancialwealthanddepositsaretheaveragerealvaluesperU.S.household. SOURCE: TheFDIC,“ABriefHistoryofDepositInsuranceintheUnitedStates”,FREDdatabase,Census BureauandFinancialAccountsoftheUnitedStates. Although the deposit insurance limit once set was continuously eroded by inflation, it was 4For example, a depositor with $1 million could fully insure this amount under the current insurance limit by splittingtheamountequallyinaccountswithfourdifferentbanks. InAugust2013therewere6,938FDIC-insuredinstitutionsintheU.S.whichatthecurrentinsurancelimitof$250,000wouldallowanindividualtobefullyinsuredup to$1,734,500,000bysplittingthetotalamountacrossall6,938insuredinstitutions.Inaddition,theFDICwouldinsure amountsuptotheinsurancelimitperdepositor,perinsuredbank,foreacheligibleaccountownershipcategory. Eligibleaccountcategoriesincludesingleaccounts,certainretirementaccounts,jointaccounts,revocabletrustaccounts, irrevocabletrustaccounts, employeebenefitplan accounts, corporation, partnership, unincorporatedassociationaccountsandgovernmentaccounts. 5

always reset to levels that guaranteed proper coverage of the average deposit balances. In this respect, the deposit insurance design achieved its goal of protecting the small uninformed and unsophisticatedinvestors.5 Regarding the second objective that targets wealthy and sophisticated investors to discipline banks, it can be argued that a design with an upper limit on deposit insurance coverage generates a strong demand for multiple deposit accounts. Whereas we do not address the question of how well large and sophisticated investors imposed market discipline on banks, we argue that threefactorshavecontributedtotheincreasingdemandforimproveddepositinsurancecoverage by these investors: First, real economic growth has increased the average incomes and financial wealthofmanyU.S.householdsabovethelevelsobservedinthe1970sand1980s. Second,growth inincomesandfinancialwealthhavebeendisproportionatelyhigherforthewealthiestU.S.households,seePikettyandSaez(2003). Finally,Figure1showsthatinflationovertheperiodfrom1980 until2008reducedinhalftheeffectivedepositinsurancecoverage,therebyincreasingthefraction ofwealthyhouseholdsnotfullyinsured.6 Inordertoobtainanestimateofthemagnitudeofthedemandformultipledepositaccounts, we use publicly available data on the average deposit balances from the regulatory reports of FDICinsuredcommercialbanksandcombinethesedatawithsurveydataonindividualdepositor balancesfromtheSurveyofConsumerFinances. Fordataonbankaccounts, weusethepublicly available data on the total number and the total balance of deposit accounts above the deposit 5During the recent financial crisis, the insurance limit was deemed insufficient to guarantee the stability of the payment system and the FDIC implemented the Transaction Account Guarantee (TAG) program that fully insured non-interestbearingtransactiondepositaccounts.Interestbearingdepositaccountssuchasinterestcheckingaccounts, moneymarketdepositaccounts,timedepositsandcertificatesofdepositwerekeptsubjecttothelimiteddepositinsurance. Aspartoftheextraordinarymeasures,thedepositinsurancelimitwhichwasraisedto$250,000onOctober 3,2008from$100,000limitwhichhadbeeninplacesince1980. WhiletheTAGprogramwastemporaryandexpired onDecember31,2012,thenewdepositinsurancelimitwassetpermanentlyto$250,000withthepassageoftheDodd- FrankWallStreetReformandConsumerProtectionActonJuly21,2010. 6Furtherindirectevidencefortherisingdemandformoreextensivedepositinsurancethroughmultipleaccounts withdifferentbanksisthecreationofamarketthatspecializesincollectingdepositsexceedingtheinsurancelimitand allocatingthemoverthenecessarynumberofdifferentbankstoachievefulldepositinsurancecoverage.Forexample, the Certificate of Deposit Account Registry Service (CADR) allows individuals, companies, non-profits, and public fundstoinvestlargeamountsinoneaccountwhichCADRsplitsandplacesinanetworkofover3,000participating FDICinsuredcommercialbanks.TheCDARismanagedbyPromontoryInterfinancialNetworkandisprotectedbyU.S. patentsUS7376606,US7440914,US7596522. Formoredetailsseewww.cdars.com. CADRactsasatwo-sidedplatform connectinginvestorsseekingcompleteinsurancecoverageoftheirinvestmentswithFDICinsuredcommercialbanks seekingfunds.DepositscollectedandreallocatedthroughtheCADRareaccountedforasbrokereddeposits. 6

Figure1: Thedepositinsurancelimit,averagehouseholdfinancialwealthanddeposits(in2008USD) Deposit insurance limit 500 Average financial assets Average deposits 250 100 40 30 20 1935 1950 19661969 1974 1980 2008 NOTE:Allrealvaluesarecomputedusingtheconsumerpriceindexforallitemswithbaseyear2008, financialwealthanddepositsaretheaveragerealvaluesperU.S.household. SOURCE: FDIC, “A Brief History of Deposit Insurance in the United States”, FRED database, Census BureauandFinancialAccountsoftheUnitedStates insurancelimittoestimatethedistributionofaverageuninsureddepositaccountbalances.7 Figure 2 plots the historical variation of the distribution of the average deposit account balances of large denomination accounts at FDIC-insured commercial banks. In addition, Figure 3 plots the empirical cumulative density function of the average account balance held in deposit accounts exceeding the deposit insurance limit of $100,000 in the second quarter of 2008, just a quarterpriortotheincreaseinthedepositinsurancelimitto$250,000. Approximately,60percent ofthelargedenominationdepositaccountswerebelowthenewdepositinsurancelimitandmost of the accounts were within two times the new deposit insurance limit. It is evident from these two figures that for most of the time since the deposit insurance limit was set to $100,000 in 1980 7ThedatacomesfromtheregulatoryfilingsofU.S.commercialbankscalledtheReportsonIncomeandCondition or“CallReports”whichcontainquarterlydataonbanks’balancesheetandincomestatements. Thedataarepublicly availableatFederalFinancialInstitutionsExaminationCouncilhttps://cdr.ffiec.gov/public. 7

Figure2: Theinter-quartilerangeofaveragepartially-insureddepositaccountbalances1986–2006 4 3.5 3 2.5 2 1.5 1 Interquartile P(25)/P(75) Median Jan86 Jan88 Jan90 Jan92 Jan94 Jan96 Jan98 Jan00 Jan02 Jan04 Jan06 NOTE: Thefigureplotstheinter-quartilerangeofaveragebalancesinlargedenomination,partiallyinsured,depositaccountsasafractionoftheinsurancelimitof$100,000fortheperiod1986Q2to2006Q1. Theaverageaccountbalanceforeachbankiscomputedasthetotalamountofdepositaccountsexceeding $100,000(itemrcon2710)dividedbythenumberofsuchaccounts(itemrcon2722). SOURCE: ReportsonIncomeandCondition(CallReports) and until its revision in 2008, large denomination partially insured deposit accounts were within twoorthreetimesthedepositinsurancelimit. Fact1. For the period 1986–2008, the average balance of large denomination, partially-insured, deposit accountswaswithintwoorthreetimesthedepositinsurancelimit. TheempiricalFact1isastatementabouttheobserveddistributionoftheaveragesizeoflarge denomination, partially-insured, deposit accounts. Because we do not have information on how many of the existing deposit accounts below the deposit insurance limit are owned by the same individual or firm, we can only make statements regarding the deposit accounts that have not been distributed into multiple institutions. The evidence suggests that, on average, the balance leftuninsuredinlargedenominationdepositaccountscouldbespreadovertwoorthreebanksto 8

Figure3: Empirical cumulative density of average account balances held in deposit accounts exceeding $100,000in2008Q2 1 0.9 0.8 0.7 0.6 57.9 % of accounts below $ 250,000 0.5 ← $ 235,000 median account balance 0.4 0.3 0.2 0.1 0 100 150 200 250 300 350 400 450 500 Average account balance in thousands NOTE:Thefigureplotstheempiricalcumulativedensityfunctionofaveragedepositaccountbalance fordepositaccountsexceeding$100,000reportedbyallFDICinsuredUScommercialbanksin2008Q2. The variable is constructed from the Call Reports as the ratio of the total deposit amount in accounts exceeding $100,000 (item rconf051) to the number of such accounts (item rconf052). As compared to Figure2, hereweusethereviseditemsintheCallreports–itemrconf051replaceditemrcon2710and itemrconf052replaceditemrcon2722in2006.ThesenewreportingitemsontheCallreportsalsoreflected thechangeintheFDIClimit.TheFDIClimitwasraisedto$250,000onOctober3,2008. SOURCE: ReportsonIncomeandCondition(CallReports) achievefulldepositinsurance. Further evidence regarding the demand for multiple deposit accounts in order to optimize the deposit insurance coverage can be obtained by examining the share of insured brokered deposits.8 Commercial banks are required to report the total amount of brokered deposits on their balancesheetaswellasabreakdownofthisamountintoinsuredanduninsured. Figure4plotsthe time series variation of the share of insured brokered deposits on the books of three size classes of banks—small banks with assets below the 75th percentile, medium large banks with assets 8ForalegaldefinitionofbrokereddepositsseeFDIC(2011)whichwascommissionedasaresponsetoregulation introducedbytheDodd-FrankAct. 9

between the 75th percentile and the 99th percentile, and large banks with assets in the top one percentileofassets. Wesummarizetheseobservationsinthefollowingempiricalfact. Figure4: Shareofinsuredbrokereddeposits Shareofinsuredbrokereddeposits 0.9 Smallbanks:below75pctile Mediumlargebanks:75-99pctile 0.8 Largebanks:Top99pctile 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Jul82 Jan85 Jul87 Jan90 Jul92 Jan95 Jul97 Jan00 Jul02 Jan05 Jul07 Jan10 Jul12 NOTE:Computedastheratiooftotalinsuredbrokereddeposits(itemrcon2343)andthetotalamount ofbrokereddeposits(itemrcon2365). SOURCE: ReportsonIncomeandCondition(CallReports) Fact2. Formostoftheperiod1982–2008,smallerbanksattractedalargershareofbrokeredinsureddeposits comparedwithmediumandlargesizebanks. Asaggregatedefaultriskincreasedattheonsetofthefinancial crisis,thedemandfordepositinsuranceincreasedatbanksofallsizes. We can think of three reasons that explain the fact that smaller banks carried a higher share of insured deposits. First, on average, smaller banks are more volatile as these banks operate in limitedgeographicareasandhavemuchlessscopefordiversificationcomparedwithlargebanks operatinginmultiplegeographicalmarkets. Consequently,thesebanksrelyonretaildepositfunding and rarely borrow from the wholesale funding markets. Second, larger banks are implicitly 10

covered by a too-big-to-fail guarantee which is hard to measure, but lowers the perceived likelihood of default. Finally, large banks are more likely to attract larger clients with larger deposit accounts and serve as their primary account custodians. Smaller banks, on the other hand, due to their larger number and the symmetric treatment by the deposit insurance limit, could serve assecondaryaccountsofdepositorswhowanttoachievehigherdepositcoveragebydistributing theirdepositsamongmultiplebanks. Attheonsetofthe2008financialcrisis,theshareofinsured brokereddepositsincreasedinalltypesofbanks,andthemostpronouncedincreasewasrecorded in large banks. The evidence suggests that the demand for high deposit insurance coverage increasedduringthatperiod. Shifting our attention to the depositors, the Survey of Consumer Finances (SCF) provides evidence regarding the demand for multiple deposit accounts. The survey collects information on the size and allocation of financial assets over different financial institutions from a representative sample of U.S. households. In particular, it surveys households regarding the different bank accounts they have with different financial institutions and their corresponding balances. In Figure5,weexaminetheallocationofcertificatesofdepositsoverdifferentbankaccountsinthe2007 SCF.9 While a large fraction of households maintain a single account with a single commercial bank, there is also large fraction of wealthy U.S. households maintaining deposit accounts with multiple depository institutions. Households with higher total financial wealth in the form of depositsabovethedepositinsurancelimitarealsomorelikelytoholdmultipledepositaccounts withdifferentbanks. Weattributepartofthedemandformultipledepositaccountstothedemand forlargerinsurancecoverage. 9Householdsinthe2007SurveyofConsumerFinancesaregroupedintengroupsaccordingtotheirtotalfinancial wealthdepositedwithFDICinsuredcommercialbanks.Thefirstgrouparehouseholdswithfinancialwealthwithinthe depositinsurancelimitof$100,000. Thesecondgrouparehouseholdswithdepositsexceedingthedepositinsurance limit but no more than two times the deposit insurance limit. The rest of the groups are households with deposits nolargerthanthecorrespondingmultipleofthedepositinsurancelimitandgreaterthanthewealthofthepreceding group. The tenth group are households with deposits at commercial banks exceeding $1,000,000 (or ten times the depositinsurancelimit).Thehouseholds’depositwealthasmultiplesoftheFDIClimitisplottedagainsttheallocation ofcertificatesofdeposits(CDs).Onthelefty-axis,weplotthenumberofcommercialbankswhereCDsareheldat(red diamonds). Ontherighty-axis,weplotthenumberofCDcontracts(greycircles). Thesizeofthemarkerforboththe numberofcommercialbanksandthenumberofcontractsistherelativefrequencyweightedbyaggregatevolumeof depositsoftheobservedallocations. InthepubliclyavailableversionoftheSCF,households’depositallocationsinto differentnumberofcommercialbankaccountsanddifferentnumberofCDcontractsaccountsaretopcodedat10and 20,respectively. 11

Figure5: Depositinsurancelimit,financialwealthanddepositallocations )01 ta dedoc pot( stcartnoc DC :sknab fo rebmuN 01 8 6 4 2 1 02 81 61 41 21 01 8 6 4 2 1 )02 ta dedoc pot( stcartnoc DC fo rebmuN < 1 2 3 4 5 6 7 8 9 10 >10 Total household deposit wealth: Multiples of the FDIC limit Number of banks: CD contracts Number of CD contracts SOURCE: SurveyofConsumerFinances,2007 Fact3. According to the Survey of Consumer Finances, a large fraction of wealthy households maintain multiple deposit accounts with multiple depository institutions. There is a strong positive correlation between the average number of CD accounts, the average amount deposited, and the number of banks these accountsareheldwith. 3. A Model of Bank Competition 3.1 Banks There are two financial institutions (“banks” in what follows) that pay interest on deposit accounts. Letr andr denotetheinterestratespaidbybankAandbankB,respectively. Oneach A B $1 deposit, a bank earns ρ by lending the money to a risky project or by investing the money in other ways (bonds, stocks, credit default swaps, real estate, and other derivatives).10 The project 10Thebanks’projectreturn(ρ)andtheinterestratespaidtoindividualdepositors(r Aandr B)couldalsobeviewed asrealrates.Infact,atthetimeofcompletingthisarticle(October2014),theinflationrateintheUnitedStatesexceeds 2 percent, whereas interest rates on deposit accounts are below 1 percent. Therefore, our analysis does not rule out 12

(and hence the investing bank) fails with probability φ meaning that the expected net return to bank A and B on a $1 deposit is (1−φ)(ρ−r ) and (1−φ)(ρ−r ), respectively. Therefore, a A B bank that fails loses its entire amount of deposits and is not able to pay back the principal and the promised interest to depositors. For reasons of tractability, we will focus on perfectly correlated default risks for banks, but in Section 8 we extend the model to cover independent failure probabilitiesacrossbanks. 3.2 Depositors Eachconsumerisendowedwith$2,andthisendowmentisinitiallydepositedeitherinbankAor in bank B. Each consumer has the option to shift the entire deposit ($2) or part of it to the rival bank. Openinganewaccountiscostlytodepositors,butitallowsdepositorstotransfermoneyto thecompetingbank. Depositors are differentiated with respect to two characteristics: their history and the costs associated with opening a new account. We refer to consumers who initially have their entire $2 deposited with bank A (bank B) as type A (type B) depositors. Type A (similarly, type B) depositors are indexed by their costs of opening a new account with a different bank s, where 0 ≤ s ≤ n. Moreprecisely,thecostofopeninganewaccounttoaconsumerindexedsisσs,where σ > 0 is a parameter capturing the magnitude of the cost of switching all or part the deposits. We can also interpret the parameter σ as a measure of the intensity of deposit rate competition betweenthebanks(wherelowervaluesofσimplymoreintensecompetition). Further,weassume these switching costs to be uniformly distributed.11 As shown in Figure 6, depositors with low s have a higher incentive to open a new bank account than depositors with a high s. A type i, i = A,B, depositor who is indifferent between opening and not opening a new bank account is denotedinFigure6bys ,i = A,B. i negativerealinterestrates. 11Throughout this paper, we use “switching cost” and “cost of opening a new account” interchangeably because initially each consumer has one account with one bank only. Also, there is ample evidence that switching costs are empirically significant in banking markets and that the switching costs are differentiated across consumers; see, for example,Shy(2002),Kim,Kliger,andVale(2003),andYankov(2014). 13

Figure6: Divisionoftypeidepositorsbetweenthosewhoopenanddonotopenanewbankaccount. (cid:45) s 0 Openasecondbankaccount s i Donot n 3.3 Assumptions We analyze three regimes of deposit insurance and compute the equilibrium deposit rates under each regime. In order to facilitate the formal analysis of the effects of partial deposit insurance on competition, we have to impose some technical conditions on the relationship between the returnonthebanks’outsideinvestmentprojectρandthebankruptcyprobabilityφ. Thefollowing conditionsaresufficientforensuringinteriorequilibriummarketshares: ASSUMPTION 1. Thereturnona$1investmentbyabankisbounded. Formally, 2nσ nσ(2+φ) −1 < ρ < −1. φ(2−φ) φ(2−φ) Assumption 1 is needed in Section 6 (limited deposit insurance). The lower bound on ρ ensures existenceofequilibriumwhensomedepositorssplittheirsavingsbetweentwobanks. Theupper boundensuresthatsomeconsumerschoosenottodosoduetosufficientlyhighswitchingcosts, as reflected by the parameter σ. Note that the interval where ρ is bounded is nonempty as its lengthequalsnσ/(2−φ) > 0. 4. No Deposit Insurance Withnodepositinsurance,consumerslosetheirentiredeposit(s)withprobabilityφ. TheexpectedutilityofatypeAdepositors ∈ [0,n](initiallyinvestedinbankAonly)isgiven 14

by12 (cid:40) (1−φ)2r −φ2 ifdoesnotopenasecondbankaccount A u (s) = (1) A (1−φ)2r −φ2−σs ifopensasecondaccountandtransfers$2tobankB. B Note that (1) ignores a potential third option where a type A depositor opens a second account with bank B but transfers less than $2 (thereby keeping a positive balance with both banks). In theabsenceofdepositinsurance(andalsounderunlimitedinsurance),thisoptionisnotbeneficial becauseadepositorthatmaintainstwoaccountsshouldtransfertheentireamounttothebankthat paysthehighestinterest. The first term in the first row in (1) , (1 − φ)2r , is the expected interest payment on the $2 A depositkeptinbankA. Thesecondterm,φ2,reflectstheexpectedlossofadepositresultingfrom a failure of bank A. The second row is very similar to the first one, except that the depositor holds the entire $2 with bank B instead of bank A. The additional term, σs, measures the cost of openinganaccountwithbankB bornebyatypeAdepositorindexedbys. Theparameterσ > 0 capturesthemagnitudeofthiscost,and,likeswitchingcosts,itcanbeviewedasameasureofthe intensity of deposit market competition (where low values of σ are associated with more intense competition). Thecaseσ = 0impliesthatalldepositors canopenasecondaccount atnocost. In contrast,higherlevelsofσ makesthisoperationmorecostlyandalsowidensthevariationofthis cost across depositors (thereby enhancing differentiation across depositors with different values ofs). Similarto(1),theexpectedutilityofatypeB depositors ∈ [0,n](initiallyinvestedinbankB only)isgivenby (cid:40) (1−φ)2r −φ2 ifdoesnotopenasecondbankaccount B u (s) = (2) B (1−φ)2r −φ2−σs ifopensasecondaccountandtransfers$2tobankA. A The utility function (1) implies that a type A depositor s opens an account with bank B and transfers the entire $2 deposit if (1−φ)2r −φ2−σs > (1−φ)2r −φ2. Similarly, the utility B A 12Wedonotassumeanyreservationutilitytoallowforpossiblereallossesfromcheckingandsavingsaccounts.For example, thedeposit ratesintheUnitedStatesare presentlybelow1-percentwhereastheinflation rateexceeds1.5percentsin2014. Theunderlyingassumptioninthispaperisthattheexpectedlossfromstoringlargesumsofmoney “underthemattress”exceededthelossfrombankaccountsattimeswheredepositratesarebelowtheinflationrate.Of course,analternativeapproachwouldbetoimposeaparticipationconstraintforthedepositors. 15

function (2) implies that a type B depositor s opens an account with bank A and transfers the entire$2depositif(1−φ)2r −φ2−σs > (1−φ)2r −φ2. Therefore,typeAdepositorswho A B open a second bank account (with bank B) and transfer their deposits are characterized by an idiosyncraticswitchingcostssmallerthanathresholds : A    0 ifr A ≥ r B   2(1−φ)(r B −r A ) σn s < s A d=ef σ ifr B − 2(1−φ) < r A < r B (3)   σn    n ifr A ≤ r B − 2(1−φ) . Accordingto(3),typeAdepositorswhofacehighcostofopeninganewaccount(s > s )decide A not to open a new account. Similarly, type B depositors who open a new bank account with bank A and transfer their deposits are characterized by an idiosyncratic switching cost s smaller thanathresholds : B    0 ifr B ≥ r A   2(1−φ)(r A −r B ) σn s < s B d=ef σ ifr A − 2(1−φ) < r B < r A (4)   σn    n ifr B ≤ r A − 2(1−φ) . Thenatureofthethresholdsdefinedin(3)and(4)impliesthatifs > 0thens = 0andifs > 0 A B B then s = 0. Intuitively, type B depositors will open a new bank account (with bank A) only if A bank A pays a higher deposit rate than bank B, r > r , while type A depositors, in this case, A B wouldlosefromopeninganaccountwithbankB. Withnolossofgenerality,wederivetheequilibriumdepositratesbyexaminingthecasewhere r ≥ r so that s = 0. In this case, the total volumes of deposits maintained by bank A and A B A bank B are 2(n + s ) and 2(n − s ), respectively. Therefore, the optimization problem facing B B bankAistotaketheinterestratesetbybankB asgivenanddecideonitsinterestrater inorder A to maximize π = (1−φ)(n+s )2(ρ−r ), where ρ−r is the profit per dollar deposited and A B A A 1−φ is the probability that this bank does not fail. Similarly, bank B determines its interest rate r inordertomaximizeπ = (1−φ)(n−s )2(ρ−r ). B B B B Maximizing profits subject to the thresholds s and s given in (3) and (4), the equilibrium A B 16

interestratesandtheresultingprofitlevelsarefoundtobe σn rN = rN = ρ− and πN = πN = σn2, (5) A B 2(1−φ) A B where the superscript “N” refers to equilibrium values with no deposit insurance. It should be pointed out that, with no deposit insurance, depositors do not benefit from opening a second bank account (sN = sN = 0) because, in a symmetric equilibrium, banks pay the same deposit A B rate. Next, consumer welfare with no deposit insurance is defined by cwN = nu +nu , where n A B isthenumber(measure)ofconsumersofeachtype. Substituting(5)into(1)and(2)yields (cid:20) (cid:18) (cid:19) (cid:21) σn cwN = 4n (1−φ) ρ− −φ . (6) 2(1−φ) Finally, we define total welfare as the sum of consumer welfare and profits of the banks and we subtracttheexpectedbailoutcostsassociatedwiththeprevailingsystemofdepositinsurance(di). Of course, with no deposit insurance di = 0. Hence, from (5) and (6), with no deposit insurance, totalwelfare(wN)isgivenby wN = cwN +πN +πN = 4n[(1−φ)ρ−φ]. (7) A B Fromthedepositrateequilibrium(5),aswellaswelfareexpressions(6)and(7),wecanapply straightforwarddifferentiationtodrawthefollowingconclusions: Result1. Supposethatbanksoperatewithoutanydepositinsurance. (a) The equilibrium interest rates (rN and rN), consumer welfare (cwN), and total welfare (wN) increase A B inresponsetoanincreaseinbanks’investmentreturn(ρ),whereasbanks’equilibriumprofits(πN and A πN)areinvariant. B (b) An increase in consumers’ cost of opening a new bank account (σ) reduces the equilibrium deposit rates(rN andrN)andconsumerwelfare(cwN),itincreasesbanks’profits(πN andπN),whereastotal A B A B welfare(wN)isinvariant. (c) The equilibrium deposit rates (rN and rN), consumer welfare (cwN), and total welfare (wN) decrease A B inresponsetoanincreaseinbanks’failureprobability(φ),whereasbanks’equilibriumprofits(πN and A 17

πN)areinvariant. B Result 1(a) reveals that competition between banks guarantees that the gains from higher investmentreturnsforbanksthatdonotfailflowtothedepositorsintheformofhigherdepositrates. The intuition behind Result 1(b) can be formulated as follows. An increase in the switching cost parameter σ implies that banks have stronger market power that leads to lower equilibrium deposit rates and higher profits. Such an increase in σ induces a redistribution of surplus from consumerstobanks. However,becauseallindividualsdepositalltheirfundswiththetwobanks, thisredistributionisneutralfromtheperspectiveoftotalwelfare. Result 1(c) characterizes the equilibrium response to a more fragile banking industry. The qualitative findings reported in Result 1(c) are the mirror image of those reported in Result 1(a). This feature reflects the fact that banks’ expected returns (1−φ)ρ are multiplicative with (1−φ) andρasfactorsandthereforedeclinewiththedefaultprobabilityφ. Furthermore,itshouldbeemphasizedthattheassumedBernoullidistributionofassetreturnsdoesnotallowustodistinguish an increase in default risk from a decrease in expected asset returns. This feature is important fortheconclusionthatequilibriumdepositratesfallwithanincreaseinthedefaultprobabilityφ. Subsection9.1demonstratesthatthisfeatureneednotholdtrueifwefocusonamean-preserving distributionofassetreturns. 5. Unlimited Deposit Insurance Weshiftourattentiontoanenvironmentwithunlimiteddepositinsurance,thatis,asystemsuch that all bank accounts are insured to their full amount. In this case, consumers do not face any riskassociatedwiththeirdeposits. Inaneventofabankfailingtomeetitsobligation,depositors receivetheirprincipalandthepromisedinterestfromtheinsuringagency. The deposit insurance system can be viewed as a redistributive taxation system. Following anestablishedtradition,weassumethatitisfundedbyalumpsumtaxsothatwecandisregard potential distortions created by this form of taxation. Of course, such distortions could easily be incorporated into the analysis by multiplying the raised tax with a multiplier (larger than one) thatrepresentsthesocialcostsassociatedwiththosedistortions. 18

Underunlimiteddepositinsurance,consumers’expectedutilities(1)and(2)aresimplifiedto (cid:40) 2r ifdoesnotopenasecondbankaccount A u (s) = (8) A 2r −σs ifopensasecondaccountandtransfers$2tobankB. B (cid:40) 2r ifdoesnotopenasecondbankaccount B u (s) = (9) B 2r −σs ifopensasecondaccountandtransfers$2tobankA. A The utility function (8) implies that a type A depositor s opens a new account with bank B (and transfers the entire $2 deposit) if 2r −σs > 2r . Similarly, the utility function (9) implies B A that a type B depositor s opens an account with bank A (and transfers the entire $2 deposit) if 2r − σs > 2r . Therefore, with unlimited deposit insurance, the thresholds (3) and (4) are A B transformedtobe     0 ifr A ≥ r B   0 ifr B ≥ r A   2(r −r ) σn 2(r −r ) σn s d=ef B A ifr − < r < r and s d=ef A B ifr − < r < r A B A B B A B A σ 2 σ 2   σn   σn  n ifr A ≤ r B −  n ifr B ≤ r A − . 2 2 (10) Applying an optimization procedure analogous to the previoussection, we now find that the equilibriumdepositratesandtheresultingequilibriumprofitsunderunlimiteddepositinsurance aregivenby σn rU = rU = ρ− and πU = πU = (1−φ)σn2, (11) A B 2 A B where the superscript “U” denotes equilibrium values under unlimited deposit insurance. Note that sU = sU = 0, because depositors cannot benefit from opening a second account if all banks A B offer the same interest rate and if all banks are insured to the full amount. Substituting the equilibriumdepositrates(11)into(8)and(9)yieldstheconsumerwelfare (cid:16) σn(cid:17) cwU = nu +nu = 4n ρ− . (12) A B 2 Next,unliketheconfigurationwithnodepositinsuranceanalyzedintheprevioussection,the presenceofdepositinsuranceintroducesaneconomy-widecostoffundingsuchaninsurancesystem. Thus,theexpectedcostofthedepositinsurancesystemshouldbesubtractedfromconsumer 19

welfareorbanks’profitinordertoobtaintherelevantexpectedtotalwelfare. Theexpectedbailout costofdepositinsuranceis (cid:16) σn(cid:17) diU = φn2(1+rU)+φn2(1+rU) = φ4n 1+ρ− . (13) A B 2 Equation(13)capturesformallytheexpectedcostofbailingouttwofailingbanks. Thisexpected bailout cost is the product of the failure probability (φ), total amount deposited in the two banks ($4n),andthepromisedinterestpayment. Finally, the expected total welfare is obtained by subtracting the expected bailout costs (diU) fromthesumofexpectedconsumerwelfareandindustryprofits. Hence, wU = cwU +πU +πU −diU = 4n[(1−φ)ρ−φ]. (14) A B From the deposit rate equilibrium (11), the welfare expressions (12) and (14), as well as the bailoutcost(13),wecanconductordinarycomparativestaticstodrawthefollowingconclusions: Result2. Supposeallbankaccountsarecoveredbyunlimiteddepositinsurance. (a) The equilibrium interest rates (rU and rU), consumer welfare (cwU), bailout costs (diU), and total A B welfare (wU) all increase in response to an increase in banks’ investment return (ρ), whereas banks’ equilibriumprofits(πU andπU)areinvariant. A B (b) Anincreaseinconsumers’costofopeninganewbankaccount(σ)reducestheequilibriuminterestrates (rU andrU),bailoutcosts(diU),andconsumerwelfare(cwU);itincreasesbanks’profits(πU andπU), A B A B whereastotalwelfare(wU)isinvariant. (c) Anincreaseinbanks’failureprobability(φ)reducestheequilibriumprofits(πU andπU)andtotalwel- A B fare(wU);itincreasesbailoutcosts(diU),whereasequilibriuminterestrates(rU andrU)andconsumer A B welfare(cwU)areinvariant. Result2(a)verifiesthatcompetitionbetweenbanksensuresthatthegainsfromhigherbanks’ investment returns flow to depositors in the form of higher deposit rates also with unlimited depositinsurance. Inthisrespect,itisqualitativelyidenticaltoResult1(a)withtheexceptionthat ahigherreturnalsoimplieshigherbailoutcosts. The intuitive explanation for Result 2(b) is identical to that for Result 1(b). The new element 20

included in Result 2(b) is that the induced reduction in deposit rates also reduce the expected bailoutcosts. Finally, Result 2(c) formalizes the very intuitive idea that, with unlimited deposit insurance, depositorsareperfectlysecuredagainstincreasesinbanks’failurerate. 6. Limited Deposit Insurance In order to exhibit the economic mechanisms in a very transparent way, we introduce a particularly simple form of limited deposit insurance: Each account is insured up to $1 worth of deposits.13 By opening a second account, and bearing the cost σs, a consumer can benefit from complete depositinsurance. Moreprecisely,throughdiversificationbyallocating$1toeachbank,adepositor’sentirewealthwouldbefullyinsured. Incontrast,maintainingasinglebankaccountwould saveadepositorthecostσs,butwouldleave$1(outof$2)uninsured. Thus,withlimiteddeposit insuranceadepositorfacesthefollowingtradeoff: Toacceptexposuretotheriskofabankfailure while avoiding the cost σs of opening a new account or to diversify away the risk caused by a potentialbankfailurebybearingthecostassociatedwithopeningasecondaccount. Underlimiteddepositinsurance,consumers’expectedutilities(1)and(2)aremodifiedto14 u (s) = A    1r A +(1−φ)1r A −φ1 doesnotopenasecondbankaccount  1r +1r −σs opensasecondaccountandtransfers$1tobankB (15) A B   1r +(1−φ)1r −φ1−σs opensasecondaccountandtransfers$2tobankB. B B 13The assumption that the insurance limit equals exactly half of the initial deposit amount saves us a tremendous amount of algebra, because under the computed equilibrium deposit rates, low-cost consumers who open a second accountwilltransferexactlyhalftheirinitialdeposittothesecondaccounttherebymaintainingfullinsurancecoverage. Assumingotherwisewouldgenerateoscillationswiththefeaturethateachbankattemptstoattractconsumersto transferdepositamountsexceedingtheinsurancecoverage.Priceoscillationsarecommonlyreferredtoas“Edgeworth PriceCycles,”andoccurinoligopoliessellinghomogeneousproductsorservices. MaskinandTirole(1988)tacklethis problembyusingaMarkovPerfectEquilibrium,whichisbeyondthescopeofourpaper. 14Forthesakeofsimplicity, thespecificationoftheutilityfunctions(15)and(16)isincompleteastheyomitother possibletransfersoflowerthan$1andamountsstrictlybetween$1and$2.AppendixAindeedshowsthat,inequilibriumwithalimiteddepositinsurance,consumerswhoopenasecondaccountwilltransferexactlytheamountofthe depositinsurancelimit,whichis$1. 21

u (s) = B    1r B +(1−φ)1r B −φ1 doesnotopenasecondbankaccount  1r +1r −σs opensasecondaccountandtransfers$1tobankA (16) B A   1r +(1−φ)1r −φ1−σs opensasecondaccountandtransfers$2tobankA. A A The expected utility (15) demonstrates the consequences of limited deposit insurance. Without diversification, a type A depositor is guaranteed a riskless return of r on a $1 deposit only. The A excess deposit of $1 will provide a return only with probably 1 − φ, whereas the depositor will lose the $1 principal with probability φ. These features are captured by the first row in (15). The second row in (15) shows that this depositor can eliminate all risks by opening a second account and splitting the initial deposit amount into two separate bank accounts that do not exceed the insurancelimit. Lastly,thethirdrowin(15)capturesadepositorwhoopensasecondaccountand completely transfers the entire initial deposit to the new account. In this case, opening a second account would not result in any risk reduction for this consumer because the transfer still leaves $1uninsured(withadifferentbank). Theutilityfunction(15)impliesthatatypeAdepositorsopensanaccountwithbankB (and transfers$1)ifr +r −σs > r +(1−φ)r −φ. Therefore, withlimiteddepositinsurance, (3) A B A A and(4)become     0 ifr A ≥ r 1 B − + φ φ r −(1−φ)r +φ s A d=ef  B σ A if rB+ 1− φ− φ σn < r A < r 1 B − + φ φ   n ifr ≤ rB+φ−σn A 1−φ     0 ifr B ≥ r 1 A − + φ φ r −(1−φ)r +φ and s B d=ef  A σ B if rA+ 1− φ− φ σn < r B < r 1 A − + φ φ (17)   n ifr ≤ rA+φ−σn. B 1−φ Figure 7 illustrates how the two types of consumers allocate their deposits between one or two accounts. 22

Figure7: Division of type A (top) and type B (bottom) depositors between those who open and do not openasecondbankaccount. (cid:45) s 0 PrimarybankaccountA($1) s A BankAonly($2) n SecondarybankaccountB ($1) (cid:45) s 0 PrimarybankaccountB ($1) s B BankB only($2) n SecondarybankaccountA($1) InviewofFigure7,thebanks’profitfunctionsaregivenby π = (1−φ)(ρ−r )[2(n−s )+s +s ] (18) A A A A B π = (1−φ)(ρ−r )[2(n−s )+s +s ]. B B B B A Theterminthebracketsinthefirstrowin(18)isthedemandfacingbankA,whichconsistsofthree components: First, bank A maintains the volume $2(n−s ) of deposits from type A depositors A who remain loyal and do not open a second account. Second, the bank keeps the volume $s of A depositsfromtypeAdepositors,whodecidetosplittheirresourcesbetweenthetwobanks. And, third, bank A attracts the volume $s of type B depositors, who each decide to diversify $1 to B bank A. Substituting (17) into (18), we find the equilibrium profit-maximizing deposit rates and theassociatedequilibriumprofitsunderlimiteddepositinsurancetobe 2σn 4(1−φ)σn2 rL = rL = ρ− and πL = πL = , (19) A B 2−φ A B 2−φ wherethesuperscript“L”denotesequilibriumvalueswithlimiteddepositinsurance. Next, substituting (19) into (17) shows that the equilibrium thresholds determining market segmentationaregivenby φ(cid:16) 2σn (cid:17) sL = sL = 1+ρ− (20) A B σ 2−φ The thresholds (20) are proportional to the cost of opening a new account at which the depositor is indifferent between diversifying $1 to the rival bank in order to qualify of complete deposit insurance or remaining loyal to its present banking relationship. For depositors with a 23

cost of opening a new bank account exceeding this threshold, the benefit from a complete depositinsuranceareinsufficienttojustifythecostofdiversificationacrosstwobanks,whereasthe opposite holds true for costs below this threshold. Technically, Assumption 1 guarantees that 0 < sL = sL < n. In particular, Assumption 1 implies that in equilibrium with limited deposit A B insurance, the benefits of full deposit insurance exceed the cost of opening a second account for somedepositors,morepreciselyforthosewithrelativelylowswitchingcosts. Formally, by combination of consumers’ utility functions (15) and (16), the equilibrium deposit rates (19), and the associated equilibrium segmentation thresholds (20), we find aggregate consumerwelfareunderlimiteddepositinsurancetobe sL sL (cid:90)A (cid:90)B cwL = cwL +cwL = (rL +rL −σs)ds+ (rL +rL −σs)ds+ (21) A B A B A B 0 0 (cid:124) (cid:123)(cid:122) (cid:125) Utilityofdepositorswithtwoaccounts +(n−sL) (cid:2) rL +(1−φ)rL −φ (cid:3) +(n−sL) (cid:2) rL +(1−φ)rL −φ (cid:3) = A A A B B B (cid:124) (cid:123)(cid:122) (cid:125) Utilityofdepositorswithoneaccount φ2(cid:16) 2nσ (cid:17)2 (cid:16) 2nσ (cid:17) = 1+ρ− +2n(2−φ) 1+ρ− −4n σ 2−φ 2−φ The first component in the first row of (21) is the sum of utilities of depositors with idiosyncratic switching costs below the thresholds (sL,sL) who open two accounts. The component in the A B second row is the sum of utilities for those depositors who do not open a second account, and thereforedonotbearcostsofopeninganewaccount. Next,consideringthemarketsegmentationofdepositorsintodepositorswithasingleaccount and depositors with two accounts (Figure 7), we compute the expected cost to the deposit insurance fund of bailing out failing banks by summing the effective deposit insurance coverage for eachofthesesegments. Hence, (cid:16) (cid:17) diL = φ(sL +sL)(2+rL +rL)+φ (n−sL)(1+rL)+(n−sL)(1+rL) A B A B A A B B (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) Depositorswithtwoaccounts Depositorswithoneaccount φ2(cid:16) 2σn (cid:17)2 (cid:16) 2σn (cid:17) = 2 1+ρ− +2φn 1+ρ− . (22) σ 2−φ 2−φ 24

Thefirstterminthefirstrowin(22)istheexpectedcostofbailingouttypeAandtypeBdepositors who split their $2 evenly between the two banks. For these accounts, the deposit insurance fund coversthefullamountandpromisedinterestrates. ThesecondtermappliestotypeAandtypeB depositorswhodonotopenasecondaccount,inwhichcaseonly$1isinsured(outofatotalof$2 deposit). Using(19),(21),and(22),totalwelfareunderlimiteddepositinsuranceisgivenby wL =cwL+πL +πL −diL = A B φ2(cid:16) 2σn (cid:17)2 =4n((1−φ)ρ−φ)− 1+ρ− . (23) σ 2−φ (cid:124) (cid:123)(cid:122) (cid:125) sL The total welfare can be decomposed into two terms. The first term measures the total expected valueofinvestingtheaggregateamountofdeposits4nintheriskyportfolioofbanksassets. The expectedreturnperdollarofinvestmentis(1−φ)ρ−φ. The second term in (23) takes into account the deadweight losses of the aggregate switching costs incurred by the mass of depositors who open two deposit accounts with the two banks to achieve full deposit insurance. To see this observe that, under limited deposit insurance, sL A depositors of type A and sL depositors of type B bear the costs of opening a second account. In B viewofFigure7,theaggregatecostsofopeningasecondaccountarethereforecomputedtobe (cid:90) s1 (cid:90) s2 φ2(cid:16) 2σn (cid:17)2 SL = σs ds+ σs ds = 1+ρ− . (24) σ 2−φ 0 0 Thiscostisacomponentofconsumerwelfare(21)whichalsoappearsintheaggregatewelfare(23). Asthenextsectionshows,thisaggregateswitchingcostplaysakeyrolewhendistinguishingthe regime with limited deposit insurance from those associated with either no or unlimited deposit insurance. 7. A Comparison of Three Regimes of Deposit Insurance Wearenowreadytocharacterizetheeffectsoflimiteddepositinsurancecoverageonequilibrium depositrates,associatedindustryprofits,consumerwelfare,bailoutcostsandtotalwelfarebased on a comparison among the investigated three deposit insurance regimes (no insurance, unlim- 25

ited, andlimitedinsurance). Westartbyfocusingontotalwelfare. Comparing(7), (14), and(23), yieldsthefollowingresult: Result3. A regime with limited deposit insurance coverage yields lower total welfare than either no or unlimited deposit insurance. Formally, wL < wU = wN. Moreover, the reduction in total welfare caused bylimiteddepositinsurancecoverageequalsthedepositors’aggregatecostsofopeningasecondaccount. The second part of Result 3 can formally be verified by adding depositors’ aggregate cost (24) to (23),whichyieldswL+SL = wU = wN. In our model, the regimes with no deposit insurance and unlimited insurance are efficient fromtheperspectiveoftotalwelfare. Undertheregimewithlimiteddepositinsurance,consumers withsufficientlylowswitchingcostshaveanincentivetoopenasecondbankaccountinorderto obtaincompletedepositinsurance. But,theswitchingcostsassociatedwithopeningnewaccounts generateasocialdeadweightloss. By comparing the equilibrium deposit rates (5), (11), and (19), we obtain the relationship that rU −rN = nσφ/[2(1−φ)] > 0,foreachbankk = A,B. FurthermorerN −rL = nσ(2−3φ)/[2(1− k k k k φ)(2−φ)] > 0,ifφ < 2/3. Acomparisonof(5),(11),and(19)alsoimplies,foreachbankk = A,B, thatπU −πN = −n2σφ < 0andπN −πL = −n2σ[3−4/(2−φ)] < 0,ifφ < 2/3. Theseinequalities k k k k provethefollowingresults: Result4. (a) Asystemwithlimiteddepositinsurancecoveragesoftenscompetitioninthedepositmarket compared with no deposit insurance if the probability of bank default is not too high. Furthermore, competition is always more intense with unlimited than with no deposit insurance. Formally, rU > k rN > rL,foreachbankk = A,B,ifφ < 2/3. k k (b) The nature of the deposit insurance system determines the banks’ equilibrium profits according to the followingrelationship: πL > πN > πU,foreachbankk = A,B,ifφ < 2/3. k k k AccordingtoResult4(a), limiteddepositinsurancecoveragesoftensdepositratecompetition between banks as long as the probability of bank default is not too high. This feature can be explained according to the following mechanism. Limited deposit insurance relaxes competition forconsumerswithlowswitchingcosts. Fortheseconsumers,thebenefitsassociatedwithdeposit 26

insurance outweigh the loss from lower deposit rates. In fact, our formal model endows each bank with some monopoly power over the rival bank’s depositors with low switching costs. In equilibrium,eachbank’sreceivesareciprocalmassofswitchingdepositorsoverwhichbothbanks gainsomemonopolypower. Result4(a)couldalsobeexplainedbyreferencetothefactthatthedifferentdepositinsurance systems induce different demand elasticities. To see this, we compare the nature of the demand functionswithnodepositinsuranceandwithlimiteddepositinsurancecoverage. Forthepurpose of this argument we focus on bank A. For the case of no deposit insurance the demand function for bank A is given by 2(n+s ) = 2n+ 4(1−φ)(rA−rB) (see Section 4), whereas the demand with B σ limited deposit insurance is given by 2(n − s ) + s + s = 2n + (2−φ)(rA−rB) (see Section 6). A A B σ By comparing these two demand functions we can conclude that the demand function is less sensitive to a change in the deposit rate difference r − r in the regime with limited deposit A B insurance precisely if φ < 2/3. Banks exploit the feature with lower elasticity of demand under limited deposit insurance by paying lower deposit rates. Furthermore, the probability of bank default affect the elasticity of demand in each of the considered systems of deposit insurance in suchawaythatthedepositraterankingreportedinResult4(a)holdtrueaslongasφ < 2/3. 15 Limited deposit insurance coverage essentially relaxes deposit market competition by inducing some depositors to transfer money between banks in order to improve their insurance coverage. From a theoretical perspective, this mechanism resembles how information exchange between lenders (who have established customer relationship) softens lending rate competition by improving banks’ ability to target their poaching activities towards specific borrowers from the rivalbank.16 In addition, Result 4(a) captures the idea that consumers can benefit more from deposit rate competition in a system with unlimited deposit insurance compared with a system offering no deposit insurance. This can be explained as follows. In these two regimes banks compete for deposits in a symmetric way with the only difference that bank competition is supported by a 15Placedwithintheframeworkofahorizonwherebankbankruptciesarenottoofrequenttherestrictionthatφ<2/3 seemsempiricallyplausible. 16Formaltwo-periodmodelscapturinghowinformationexchangesoftenscompetitioninlendingmarketshavebeen developedinBouckaertandDegryse(2004)andGehrigandStenbacka(2007). 27

transfer from the insurance agency to depositors under unlimited deposit insurance, and this transferintensifiesthecompetitionbetweenbankswhichresultsinhigherdepositrates. Result4isillustratedinFigure8, whichshowsasimulationofhowequilibriumdepositrates andprofitsdependonthesystemofdepositinsurance.17 Inparticular,Figure8demonstratesthat limiteddepositinsuranceleadstohigherindustryprofitsthanunlimitedornodepositinsurance simplybecausebothbankspaylowerinterestondepositaccounts. Figure8: Consumerwelfare,banks’profit,anddepositinsurancebailoutcostasfunctionsofthreeregimes ofdepositinsurance. 2.0 Consumer welfare (cw) 1.5 Deposit rates (r) Banking industry profit (pA+pB) Deposit insurance bailout cost (di) 1.0 0.5 0.0 None Limited Unlimited Deposit insurance regime The following result summarizes our comparison of the three regimes of deposit insurance withrespecttoconsumerwelfareandthecostofbailingoutbanks: Result5. (a) Consumerwelfareincreaseswiththelimitondepositinsurance. Formally,cwN < cwL < cwU. (b) Expectedcostofbailingoutbanksincreaseswiththelimitondepositinsurance. Formally,diN < diL < diU. 17Simulationsarebasedonthefollowingparametervalues: Returnonbank’sinvestmentρ = 1.07, banks’failure probabilityφ=0.05,costparameterσ=0.2,andinitialmeasureofdepositorswitheachbankn=0.5. 28

From Result 3 and Result 4, we can directly conclude that consumers are better off with unlimited (U) compared with limited (L) deposit insurance coverage. That is, because diU > diL and πU < πL, it cannot hold true that wU > wL unless it also holds true that cwU > cwL . In k k other words, consumers unambiguously benefit from unlimited compared with limited deposit insurancecoverage. When comparing limited (L) deposit insurance coverage with no (N) deposit insurance, we can first make use of Result 3 and Result 4 to conclude that the introduction of limited deposit insuranceimposeslossesonsocietyintheformofexpectedbailoutsoronconsumersintheform of switching costs or lower deposit rates. In particular, we know from Result 3 that the sum of theselossesexceedsthebenefitstobanksassociatedwithlimiteddepositinsurance. Thisexplains whyconsumerwelfareishigherunderlimiteddepositinsurancecomparedwithnoinsuranceas illustratedinFigure8. Result 5(b) does not require a formal proof. It captures the intuitive idea that the expected bailoutcostsincreaseasafunctionoftheinsurancecoverage. Overall, in light of Result 3, Result 4, and Result 5 we can draw the conclusion that limited deposit insurance introduces a redistribution of surplus between banks and depositors. Limited depositinsurancecoveragepromotesmarketpowerofbanksoverconsumerswithsmallswitchingcostsandthismechanismisthesourceoftheredistribution. Furthermore,wehaveestablished that the benefit to banks falls short of the costs to consumers and society when the bailout costs are taken into account. Thus, limited deposit insurance generates a social deadweight loss (costs ofopeningsecondbankaccounts)comparedwithsystemsofunlimitedornodepositinsurance. 8. Independent Bank Failures Our analysis so far has focused on perfectly correlated default risks for banks. This section explores the robustness of our results regarding this assumption by analyzing the configuration where banks face independent default risks. For simplicity, we restrict ourselves to symmetric banks facing identical default risks, measured by the bankruptcy probability φ. Under such circumstances, both banks fail with probability φ2, only one bank fails with probabilities φ(1 − φ) 29

and(1−φ)φ,respectively,andnonefailswithprobability(1−φ)2. We proceed by examining each of the three deposit insurance regimes separately, and show that the equilibria derived under correlated default risks are identical to the equilibria under independentdefaultrisks. 8.1 IndependentBankFailures: NoDepositInsurance Section4establishedthat,inequilibrium,depositorsdonotopenasecondaccount. Furthermore, according to Section 4, if a consumer opens a second account, this consumer transfers the full volumeofdeposits,i.e.$2,tothebankthatpaysthehigherinterest. Underindependentbankfailures,wenowexaminethepossiblecase(notcoveredinSection4) inwhichsomeconsumersopenasecondaccountandtransferhalfoftheamount,sotheymaintain $1 with each bank as a diversified portfolio bearing independent risks. In this case, the utility function(1)becomes    (1−φ)2r A −2φ ifdoesnotopenasecondbankaccount;  u A (s) = (1−φ)2(r A +r B )+(1−φ)φ(r A −1) ifopensasecondaccountand (25)   +φ(1−φ)(r −1)+φ2(−2)−σs transfers$1tobankB. B The first row in (25) is the same as in (1). It characterizes the utility of type A depositors, who keeptheirentiredepositwithbankA. Thesecondalternativein(25)(thesecondandthirdrows) captures the expected return associated with opening up a second bank account. The consumer earnsr +r interestifneitherbankAnorbankB fails,whichhappenswithprobability(1−φ)2. A B If only bank B fails (with probability (1−φ)φ), the consumer earns interest r from bank A, but A loses the $1 deposit with bank B. If only bank A fails (probability φ(1−φ)) the consumer earns interest r from bank B, but loses $1 deposit with bank A. Finally, the consumer loses all his $2 B depositsifbothbanksfail(withprobabilityφ2). Comparingthetwoutilitiesin(25)revealsthattypeAdepositorswhoopenasecondaccount andtransfer$1tobankB arecharacterizedbyaswitchingcostlowerthanathresholds : A (1−φ)(r −r ) s < s d=ef B A , (26) A σ 30

where we do not display the corner solutions for the sake of brevity. The value of s in (26) A is proportional to that in (3). This implies a type A consumer opens a second account only if r > r . However,inthiscase,theconsumerisbetterofftransferringthewholedeposit($2)from B A AtoB,whichreplicatestheanalysisinSection4undercorrelatedbankfailures. 8.2 IndependentBankFailures: UnlimitedDepositInsurance Under unlimited deposit insurance, consumers do not bear any risk and therefore will not open a second account unless the rival bank offers a higher interest. Hence, the analysis of Section 5 applies also to the case of independent bank failures. Still, it is worthwhile to check whether the expectedcostofbailingoutbanksunderindependentfailuresisthesameaswithcorrelatedbank failures,computedin(13). The expected total bailout cost under unlimited deposit insurance with independent failures isgivenby diU = φ2[2n(1+r )+2n(1+r )]+φ(1−φ)[2n(1+r )]+(1−φ)φ[2n(1+r )]+(1−φ)20 A B A B = 2nφ[2(1+ρ)−σn], (27) wherethesecondrowisobtainedbysubstitutingtheequilibriuminterestratesfrom(11)intothe first row. The first row in (27) sums up four terms: The expected cost of bailing out two failing banks,expectedcostofbailingoutbankAonly,expectedcostofbailingoutbankB only,andzero costofnotbailingoutanybank(ifbanksdonotfail). Comparing (27) with (13) reveals that the expected bailout cost is the same independently of whetherwefocusonindependentbankfailuresorperfectlycorrelatedfailures. 8.3 IndependentBankFailures: LimitedDepositInsurance InviewofFigure7,withlimiteddepositinsurance,s ands low-costdepositorsopenasecond A B accountanddeposit$1witheachbank. Therefore,theequilibriumderivedinSection6holdsalso underindependentfailures. The expected bailout cost to support limited deposit insurance with independent failures is 31

givenby diL = φ2[s (1+r +1+r )+(n−s )(1+r )+s (1+r +1+r )+(n−s )(1+r )] A A B A A B B A B B +φ(1−φ)[s (1+r )+(n−s )(1+r )+s (1+r )] (28) A A A A B A +(1−φ)φ[s (1+r )+(n−s )(1+r )+s (1+r )]. B B B B A B Thefirstrowin(28)istheexpectedinsurancecostofbailingouttwofailingbanks,whereinview ofFigure7,s typeAands typeB consumerssplittheirdepositsbetweentwobanks,whereas A B n−s andn−s consumersleavetheirentiredeposit$2inasinglebankaccountwithonlyhalfof A B this amount being insured under limited deposit insurance. The second row is the expected cost of bailing out bank A only, where s type B depositors also keep $1 of their deposits. Similarly, B thethirdrowistheexpectedcostofbailingoutbankB only. Substituting the equilibrium interest rate (19) and the segmentation thresholds (20) into (28) reveals that the expected insurance cost under independent failures (28) is the same as under perfectlycorrelatedfailures(22). Theanalysisinsubsections8.1,8.2and8.3yieldsthefollowingconclusion. Result6. All the results derived under the assumption that the bank failures are perfectly correlated also apply to a model where the bank failures are realized as independent events. In particular, the expected depositinsurancebailoutcostsarealsothesame. 9. Extensions 9.1 Mean-PreservingAssetReturns For reasons of a tractability we have focused on a simple return structure which does not make it possible to distinguish between an increase in default risk (increasing φ) and a decrease in the expected return on banks’ assets. This section conducts a comparative statics analysis to explore how an increase in the probability of default φ (without changing the expected return on assets) affectstheequilibriumdepositratesandprofits. We focus on a mean-preserving spread of the asset returns modeled as a process (ρ˜,φ˜) such thatφ > φ˜ and(1−φ˜)ρ˜= (1−φ)ρ. Amean-preservingspreadincreasestheprobabilityofdefault, 32

but it keeps the expected return unchanged. We can characterize the effects of an increase in the probabilityofdefaultontheequilibriumdepositratesacrossthethreedepositinsuranceregimes accordingtothefollowingresult(proofisprovidedinAppendixB). Result7. Consider a mean-preserving spread of the asset returns of banks. An increase in riskiness leads to (a) higher deposit rates and unchanged profits in equilibrium with no deposit insurance; (b) unchanged depositratesandlowerprofitsinequilibriumwithunlimiteddepositinsurance;(c)higherdepositratesand lowerprofitsinequilibriumwithlimiteddepositinsurance. The importance of this result is that, in our model, banks do not have incentives to increase risk eveninthepresenceofdepositinsurance. Sinceweabstractfromissuesrelatedtomoralhazard, this feature of the model allows us to focus on comparing the degree of competition for deposits underthethreedepositinsuranceregimes. 9.2 MultipleBankAccounts Solving the general problem where depositors may hold different levels of wealth that would require opening multiple bank accounts is beyond the scope of this paper. In fact, such a model should probably be designed for the purpose of using numerical simulations of a nationwide wealthdistributionamongdepositors,ratherthanforobtainingclosed-formsolutionsasweoffer inoursimplifiedmodel. Therefore,thissectionsketchesonlyonewayinwhichthedemandside couldbeformulatedwhenaconsumerhasalargesumofmoneythatmustbedepositedinmore thantwobankaccountsinordertosecure100-percentdepositinsurance. Suppose that there is a large number of banks and that all banks pay the same interest rate, r. Consider a depositor with d dollars. Let λ(1+r) denote the deposit insurance limit. If d ≤ λ, the depositor is fully insured and therefore does not have to open a second account. However, if d > λ,thedepositormaybenefitfromopeningadditionalaccounts. LetI d=efint[(d−λ)/λ]andM d=ef(d−λ)modλbetheintegerandtheremainderpartsoftheratio of a depositor’s total wealth less than the deposit limit to the deposit limit, respectively. Define 33

twothresholdsofthecostofopeninganadditionalbankaccountby φλ(2+r) 2φM(1+r) sλ d=ef and sM d=ef . (29) σ σ AppendixCprovesthefollowingresult: Result8. Adepositorwithawealthlevelofdandcostofopeningeachadditionalaccountgivenbyσswill (a) notopenanyadditionalaccountifs ≥ max{sλ,sM}; (b) openI additionalaccountsifsM ≤ s < sλ; (c) openI +1additionalaccountsifs < min{sλ,sM}. Asexpected,thenumberofadditionalaccountsincreaseswhenthecostofopeningeachaccount declines(lowervaluesofσ). Higherdepositlimit(higherλ)andhigherinterest(higherr)would inducemoreconsumerstoopenadditionalaccounts. 10. Conclusion Thisstudycomparedtheperformanceofasystemwithlimiteddepositinsurancecoveragetothe performanceofsystemswithunlimitedornodepositinsurance. Inordertoachievethisgoal,we havedevelopedastylizedmodeltohighlightinatransparentwayhowadepositinsurancesystem with limited coverage induces some consumers to diversify their deposits across several banks. Within such a framework, we demonstrate that limited deposit insurance coverage softens competition among banks, thereby introducing a redistribution of surplus from depositors to banks. Furthermore, we establish that the benefits to banks of limited deposit insurance fall short of the costs to consumers and society when bailout costs are taken into account. Thus, limited deposit insuranceleadstoalossintotalwelfarecomparedwithunlimitedornodepositinsurance. The simple model we have designed abstracts from many important issues, and could be extendedindifferentdirections. Mostimportantly,weabstractfrommoralhazardissuesassociated withthelendingorinvestmentdecisionsofbanks. Modelsincorporatingmoralhazardassociated with banks’ lending/investment activities typically emphasize that deposit insurance offers an optionvalueforbanksandthatthisoptionvalueismonotonicallyincreasingasafunctionofthe insurancecoverage. Inourmodel,thevaluetothebanksofthedepositinsuranceisverydifferent 34

in nature, because limited deposit insurance coverage is more profitable to banks than unlimited depositinsuranceduetothesofteningofdepositcompetition. Further,wedonotformallyaddressthefollowingquestion: Aredepositorsalwaysguaranteed toreceivetheinsuredamountinthecaseofbankfailure? Thisneednotalwaysbethecasebecause theFDICdoesnothavesufficientreservestobailoutallbanks. However,recentexperienceshows thatgovernmentstendtousetaxpayermoneytobailoutbankswhentheinsuranceagency(such as the FDIC) does not have sufficient funds to cover bank losses.18 But, of course, the funding of such bailout programs would cause distortions which would affect welfare evaluations. The welfareanalysiscouldbeextendedtoincorporatethesocialcostsofsuchdistortions. Forreasonsoftractability,wehavefocusedondepositorsdifferentiatedbythecostsassociated withopeninganewaccount,buthomogeneouswithrespecttothevolumeoftheirdeposit($2). A natural extension would be to analyze a deposit market where consumers are differentiated also withrespecttotheiravailablefunds. Thiswouldmakethewelfareanalysismorecomplicatedas some consumers would not be affected by the deposit limit at all, whereas others would benefit fromopeningmultipleaccountsinordertoqualifyforcompletedepositinsurance. It should be emphasized that we have focused on an economy with the feature that the consumers have to deposit their money in a bank, and that they have access to no outside option like a shadow banking system. This restriction is increasingly severe in light of the increase of the institutional cash pools. Actually, as Pozsar (2013) argues, the institutional cash pools have expandedtosuchanextentintheU.S.thatdividingtheaverageinstitutionalcashpoolintofully FDIC-insuredsliceswouldrequiremorebanksthanthereisintheU.S. Finally,wehaverestrictedourattentiontoanevaluationoflimiteddepositinsurancecoverage bycomparingitwithsystemswithunlimitedornodepositinsurance. Clearly,apromisingdirection for extending our approach would be to characterize the socially optimal deposit insurance coverage. Withsuchanapproachitwouldbepossibletomorefundamentallycharacterizewhich particularfactorsdetermineoptimaldepositinsurancepolicy. 18SeeaMay28,2013WallStreetJournalarticlebyAlexPollockentitled“DepositsGuaranteedUpto$250,000–Maybe,” whichdiscussesthelegalquestionwhetherFDICinsuredaccountsarebackedbythe“fullfaithandcreditoftheUnited StatesGovernment.Further,CooperandKempf(2013)exploretheeffectsoforderlyliquidationoffailingbanksonthe emergenceofbankrunsundercircumstanceswheredepositinsurancepolicieshavenocommitmentpower. 35

Appendix A Existence and Uniqueness of an Equilibrium with Limited Deposit Insurance The derivation of the equilibrium interest rates (19) under limited deposit insurance ignored the possibility that depositors who open a second account may benefit from transferring more than $1 (deposit insurance limit). Such an allocation was considered in the third rows of (15) and (16) in which depositors transfer their full $2 initial endowment to the rival bank and maintain zero balancewiththeirinitialbankaccount. Ourfirstobservationisthatinanysymmetricequilibriumwherebankspaythesameinterest on deposits (so that r = r ), depositors who open a second account transfer exactly $1. This is A B becauseanyotherwayofdistributingthe$2totalamountbetweenthetwobanksdoesnotresult in higher expected interest payment but increases the risk by leaving some amount uninsured. Therefore, to prove that the derived deposit rates (19) constitute a Nash equilibrium, we only needtoruleoutadeviationwhere,say,bankB raisesthedepositrateabovetheequilibriumlevel (19)inordertoattracttypeAdepositorstotransfer$2tobankB insteadofjust$1. Thisappendix showsthatsuchandeviationisnotprofitableforbankB. LetbankA’sdepositrate(rL)begivenby(19). Then,inordertoattracttypeAdepositorswho A openanaccountwithbankB totransfer$2insteadof$1,bankB hastoraiseitsdepositratetor(cid:48) B satisfying 1rL +1r(cid:48) −σs < 1r(cid:48) +(1−φ)1r(cid:48) −φ1−σs. This basically says that the expected A B B B utility captured by the third row in A’s utility function (15) exceeds that captured by the second row. Substituting(19)forrL yields A r +φ (2−φ)(ρ+φ)−2nσ r(cid:48) > r d=ef A = . (A.1) B (cid:98)B 1−φ (1−φ)(2−φ) ForthisdeviationtobeprofitableforbankB,theinterestr paidtodepositorscannotexceed (cid:98)B thereturnρthatbankB earnsona$1investment,sothatr < ρ. However,itcanbeshownthat (cid:98)B 2nσ r < ρ ifandonlyif ρ < −1, (A.2) (cid:98)B φ(2−φ) whichcontradictsAssumption1. ThiscompletestheproofshowingthatbankB willnotdeviate fromtheequilibriuminterestrate(19). 36

Finally, notethatthisresultalsoshowsthatbankscannotprofitfrompricediscriminationbetweendepositorswhomaintainbalanceswithinthedepositinsurancelimitandthosethatmaintainbalancesabovethedepositinsurancelimit(byofferingthemtwodifferentdepositrates). Appendix B Proof of Result 7 Proof. Wefocusonamean-preservingspreadandexploretheeffectofanincreaseinbanks’probability of default on equilibrium deposit rates and profits across the three regimes of deposit insurance: 1. Nodepositinsurance: • deposit rates: By taking the ratio of the equilibrium deposit rates under the meanpreserving spread and the benchmark, we obtain that r˜N = 1−φ > 1 or r˜N > rN. rN 1−φ˜ Withnodepositinsurance,bankscompensatedepositorsforthehigherdefaultriskby offeringhigherdepositrates. • profits: profitsareinvarianttoamean-preservingspreadπN = σn2 2. Unlimiteddepositinsurance • depositrates: Depositratesarehigherunderamean-preservingspreadasρ˜= 1−φρ > ρ 1−φ˜ andtheequilibriumdepositraterU = ρ−σn doesnotdependontheprobabilityofbank 2 default. • profits: Profitsarelowerunderamean-preservingspreadπU = (1−φ)σn2 . 3. Limiteddepositinsurance • deposit rates: Deposit rates are higher under a mean-preserving spread as r˜L −rL = (cid:104) (cid:105) φ˜−φ ρ− 2σn 1−φ˜ > 0. 1−φ 2−φ2−φ˜ • profits: Profits are lower under a mean-preserving spread as the profit function πL = 1−φ4σn2 isastrictlydecreasingfunctioninφandφ˜> φ 2−φ 37

Appendix C Proof of Result 8 To prove Result 8, note that a consumer s will open one additional new account (call it a second account)if (1−φ)(d−λ)(1+r)−φ(d−λ)+λ(1+r) < (1−φ)(d−2λ)(1+r)−φ(d−2λ)+2λ(1+r)−σs, (C.1) yieldings < sλ. Thefirsttwotermsontheleftsideof(C.1)aretheexpectedgrossbenefitfromthe above-the-limit deposit d−λ, which is uninsured. The third term is the safe gross return on the insuredamount,λ. Next,aconsumersopens2additionalaccounts(thirdaccount)if (1−φ)(d−2λ)(1+r)−φ(d−2λ)+2λ(1+r)−σs < (1−φ)(d−3λ)(1+r)−φ(d−3λ)+3λ(1+r)−2σs, (C.2) yieldingagains < sλ. Next,aconsumerswithN −1accountsopensanNthaccountif (1−φ)[d−(N −1)λ](1+r)−φ[d−(N −1)λ]+(N −1)λ(1+r)−(N −2)σs < (1−φ)(d−Nλ)(1+r)−φ(d−Nλ)+Nλ(1+r)−(N −1)σs, (C.3) yielding again s < sλ. Finally, a consumer s opens an additional account just to deposit the remainder,M,if (1−φ)M(1+r)−φM < M(1+r)−σs, (C.4) yieldings < sM. References Bartholdy,Jan,GlennBoyle,andRogerStover.2003. “DepositInsuranceandtheRiskPremiumin BankDepositRates.” JournalofBankingandFinance27(4):699–717. Bouckaert,JanandHansDegryse.2004. “SofteningCompetitionbyInducingSwitchinginCredit Markets.” JournalofIndustrialEconomics52(1):27–52. Cooper, Russell and Hubert Kempf. 2013. “Deposit Insurance and Orderly Liquidation without Commitment: Can We Sleep Well?” National Bureau of Economic Research, Working Paper No.19132. 38

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