Optimal Bank Regulation in the Presence of Credit and Run Risk

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Optimal Bank Regulation in the Presence of Credit and Run Risk Anil K. Kashyap, Dimitrios P. Tsomocos, and Alexandros P. Vardoulakis 2017-097 Please cite this paper as: Kashyap,AnilK.,DimitriosP.Tsomocos,andAlexandrosP.Vardoulakis(2017). “Optimal Bank Regulation in the Presence of Credit and Run Risk,” Finance and Economics Discussion Series 2017-097. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2017.097. 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.

Optimal Bank Regulation ∗ In the Presence of Credit and Run Risk AnilKKashyap† DimitriosP.Tsomocos‡ AlexandrosP.Vardoulakis§ August29,2017 Abstract We modify the Diamond and Dybvig (1983) model of banking to jointly study various regulationsinthepresenceofcreditandrunrisk. Bankschoosebetweenliquidandilliquidassets on the asset side, and between deposits and equity on the liability side. The endogenously determinedassetportfolioandcapitalstructureinteracttosupportcreditextension, aswellas to provide liquidity and risk-sharing services to the real economy. Our modifications create wedges in the asset and liability mix between the private equilibrium and a social planner’s equilibrium. Correcting these distortions requires the joint implementation of a capital and a liquidityregulation. Keywords: BankRuns,CreditRisk,LimitedLiability,Regulation,Capital,Liquidity JELClassification: E44,G01,G21,G28 ∗Revisedversionof“Howdoesmacroprudentialregulationchangebankcreditsupply?",NBERWorkingPaperNo. 20165. WearegratefultoSakiBigio(discussant), DongBeomChoi(discussant), EmmanuelFahri(discussant), John Geanakoplos,ToddKeister,FrankSmets(discussant),AdiSunderam(discussant)andseminarparticipantsatnumerous institutionsandconferencesforcomments.KashyapthankstheInitiativeonGlobalMarketsattheUniversityofChicago BoothSchoolofBusiness, theHoublonNormanGeorgeFellowshipFund, andtheNationalScienceFoundationfora grant administered by the National Bureau of Economic Research for research support. Kashyap’s disclosures of his outsidecompensatedactivitiesareavailableonhiswebpage. Allerrorshereinareours. Theviewsexpressedinthis paperarethoseoftheauthorsanddonotnecessarilyrepresentthoseofFederalReserveBoardofGovernors,anyonein theFederalReserveSystem,theBankofEngland,oranyoftheinstitutionswithwhichweareaffiliated. †University of Chicago Booth School of Business, United States and Bank of England; email: anil.kashyap@chicagobooth.edu ‡SaïdBusinessSchoolandSt.EdmundHall,UniversityofOxford,UnitedKingdom; email:dimitrios.tsomocos@sbs.ox.ac.uk §BoardofGovernorsoftheFederalReserveSystem,UnitedStates;email:alexandros.vardoulakis@frb.gov

1 Introduction Financial intermediaries, hereafter banks, perfom various socially useful functions. These include providing liquidity (Diamond and Dybvig, 1983), facilitating credit extension to fund productive investment (Diamond, 1984), and improving risk-sharing (Benston and Smith, 1976; Allen and Gale, 1997, 2004). Banks’ asset portfolio composition and liabilities structure interact to allow them to perform these services. However, these same interactions can also be a source of fragility. Transforming illiquid long-term assetsinto liquidshort-term claims, such as demandabledeposits, is desirable, but exposes banks to the possibility of a run which can be disastrous for the bank, its borrowersanditsdepositors. Likewise,fundingriskyloansthroughbothdebtandequityimproves risk-sharing (and potentially raises growth), but can lead to socially wasteful bankruptcy costs. Finally, the presence of short-term liabilities can generate better incentives for banks to monitor borrowers and honor their liabilities (Calomiris and Kahn, 1991; Diamond and Rajan, 2001), but createsrunriskthatlong-termfundingavoids. In this paper we expand the Diamond and Dybvig (1983) model of banking to incorporate all of the three aforementioned banking functions and explore whether private decisions result in an efficientlevelofintermediation. Ouranalysisisrelatedtotheemergingliteratureexploringoptimal macroprudentialregulationtoaddressvariousinefficiencies,suchasaggregatedemandexternalities in the presence of nominal rigidities (Farhi and Werning, 2016), pecuniary externalities operating throughcollateralconstraints(BianchiandMendoza,forthcoming)orfire-salesexternalities(Stein, 2012). However, our focus is different as we are interested in identifying externalities that pertain tobankschoicesassociatedwithendogenouscreditriskandrunrisk. Bankers have a comparative advantage at intermediating funds, but their incentives to monitor theirinvestmentcandifferfromtheirinvestorsduetoprivatebenefitsthatareavailabletothem. On onehand,afragilefundingstructureandrunriskcanbeoptimaltodisciplinethebankerandalign theincentivesformonitoring. Short-termdebtis,thus,preferredtolong-termfundingforitsdisciplining function even when both are able to provide liquidity services through retrading in capital markets. Ontheotherhand,afragilefundingstructurecanmisaligntheincentivesbetweenbankers and debt-holders in the presence of credit risk and, hence, result in both distorted asset holdings and a capital structure. Contrary to the case of pure run risk, equity financing has an advantage over short-term debt in dealing with externalities from the management of credit risk. Overall, run and credit risk endogenously interact to determine banks’ asset portfolio and capital structure, which, in turn, has implications for the level of intermediation and the allocation of benefits from intermediation. We make five modifications to the original Diamond-Dybvig model to capture the aforementionedinteractionofcreditandrunrisk.First,bankloansarerisky. Theriskarisesbecauseborrowers use the bank funding to invest in a technology whose payoff is uncertain and whose results are privateinformationfortheborrower. Because loans are risky, borrowers can default due to insufficient funds. This can potentially cause the bank to be unable to fully repay depositors who incur additional bankruptcy costs. Our 2

secondmodificationistoassumethatbothbanksandborrowersaresubjecttolimitedliability. Third, the private information about a borrower’s success leads banks to have to monitor the borrower. Thebanksarerunbybankerswhoseektomaximizethevalueofdividendstheyreceive fromthebanks. Absentmonitoringthebankersenjoyaprivatebenefit,butwithoutmonitoringtheir borrowerswillneverrepaytheirloans. Fourth, afullsetinstrumentstoinsureagainstallrisksintheenvironmentisunavailable. Both loan and deposit contracts cannot be made contingent on the aggregate realization of risks. As a result, credit risk occurs in equilibirium. Moreover, contracts can be incomplete such that not all actions of borrowers are ex-ante contractible. For example, a more comprehensive debt contract wouldnotonlyspecifyaninterestrateandadebtamount, butalsothecompositionandamountof assetsthatcanbeseizedifdefaultoccurs. Likewise,bankerschoosehowmuchequitytocontribute to the bank in addition to accepting deposits. Not only the asset portfolio, but also the capital structureofthebankisendogenouslydetermined. Fifth, we assume that depositors receive signals about the value of loans that the bank can recover before they are due. This interim liquidation value is available to pay depositors who are seekingtowithdraw. Wesupposethatthedepositorsmakeadecisionwhethertorunbasedonthese signals. Our assumption about the nature of these signals means that the decision to run depends ontheassetandliabilitystructureofthebank,andthevalueofthesignal. Thereisauniquesignal thresholdthatdetermineswhetherthereisarun. There are three important consequences of these modifications. First, they create an environment where the level of credit risk and run risk in the economy are endogenously determined and interact. Second,thebank’schoicesofboththemixbetweenthelevelofliquidandilliquidassets andbetweendebtandequitydifferfromwhatasocialplannerwouldselect. Theprivateequilibrium featuresexcessivelevelsoflendingrelativetoliquidassetholdingandmoredebtfinancingrelative to equity financing. Because of the market incompleteness there is not a unique social planner’s allocation. Instead the preferred allocations will depend on the planner’s weights on the different actors in the economy. Loosely speaking, when the planner favors the savers, she will choose to limit risk while emphasizing liquidity provision for depositors. Alternatively, if the planner is primarily looking out for borrowers, the allocations are arranged to control run risk while increasing lending. The third outcome in the model is that regulations akin to those embedded in the new Basel regulations for liquidity and capital can be studied to see if they could align the private asset and liability mix with the social efficient one. In particular, we study two capital regulations, one that ties capital requirements to the riskiness of bank assets and a second leverage requirement that is determined by the total scale of bank assets. We also look at a pair of liquidity regulations. One, akin to the so-called liquidity coverage ratio, makes the bank hold more short-term liquid assets whenitusesmorerunnablefunding. Theother,liketheso-callednetstablefundingratio,requires thebanktoincreaseitslongtermfundingtomatchitslongertermassets. Although regulations can individually reduce the probability that a run occurs and improve 3

welfare, they affect the asset mix and the liability mix in different ways. Capital regulations result inmorelending, butinlowerliquidasset holdings thanitissociallyoptimal. Incontrast, liquidity regulations reduce lending, but leave the level of capital below the social optimum. Because the private allocations diverge from the socially optimal allocations in two ways, no single regulation is sufficient to implement the social optimum; we show that at least two tools are needed. Yet, theoptimalregulatorymixcannotarbitrarilyincludeanytwotools, becausesomeregulationsmay be redundant. For example, we find that the two liquidity regulations cannot be jointly binding. Nevertheless, combinations of a capital and a liquidity regulation are feasible and are sufficient to implementthesocialplanner’ssolution. Aspecialcaseariseswhenbankershaveamplewealthtoinvestinsomuchbankequity,which pushes their economic surplus down to zero in equilibrium. Therein, planning outcomes are decentralized with only one regulation, which depends on the deadweight losses in bankruptcy. If thelatterarelow,thenonlyliquidityregulationisneeded,whileforhighones,capitalregulationis used. The remainder of the paper is separated into four parts. In section 2, we describe the model andshowtheprivatelyoptimalchoicesforthebank, thesaversandtheentrepreneurs. Insection3 westudytheefficientallocationschosenbyasocialplannerandderiveexpressionsforthewedges between the private and social decisions. In section 4, we explore how regulation can be used to correcttheprivateinefficiencies. Thefollowingsectionanalyzesaspecialcasewherelessregulation isneeded. Section6concludesbysummarizingthemainfindings,reiteratingtheintuitionforthem, anddescribingafewdirectionsforfutureresearch. Additionalderivationsandmodelextensionsare relegatedtoanonlineappendix. 2 Model Themodelconsistsofthreeperiods,t ={1,2,3},featuresasingleconsumptiongoodandincludes three types of (representative) agents; an entrepreneur (E), a saver (S) and a banker (B). The entrepreneur has access to a productive, but illiquid, risky technology. The entrepreneur’s primary decisionishowmuchofherownmoneytoallocatetotheprojectandhowmuchtoborrow. Fundsinvestedatdate1yieldanuncertainpayoffA ·F(·)atdate3dependingontherealization 3s ofstates, whereF isaconcaveproductionfunctionandA aproductivityshock. States={g,b} 3s occurswithprobabilityω andthesestatesrepresentagoodandabadrealizationoftheshock,i.e., 3s A >A . The project delivers no output at date 2 but it can be liquidated. The liquidation value, 3g 3b ξ,isuncertainandindependentoftheproductivityshock.1 Thebankermanagesaninstitutionwhichwecallabankthatactsasanintermediarybetweenthe 1Thediscretestatespacefortheproductivityshockisnotimportantforourresults,butitfacilitatesthecomputation ofthenumericalequilibria. Asdescribedbelow,allagentshavelinearpreferenceatt=3,sothattheycareonlyabout theexpectedpayoffsandnotthestatebystatepayoffs. Moreover,anothershockwillberealizedatt=2,whichfollows acontinuousdistributionandisindependentoftherealizationoftheproductivityshock. Thus,fromtheperspectiveof t=1,thereisan“infinite"dimensionalstatespaceinthefuture.Wearemoreprecisebelow. 4

entrepreneurs and savers. The bank is funded partly from the banker’s endowment and by raising addtional funds from the saver. The funds raised at date 1 are invested into either a liquid storage assetorinaloantotheentrepreneur,whichthebankcanrecallintheintermediateperiod. Moreover, the banker decides whether to monitor the entrepreneur’s project at t = 3 or not. Monitoring is importantbecausetheproductivityshockisprivateinformationtotheentrepreneur. Thesaverhasalargeendowmentatdate1thatisusedtofundinitialconsumptionandsavings. The savers have uncertain future consumption needs and, as in Diamond and Dybvig (1983), after date 1 some fraction will need to consume at t=2 and the rest can wait to consume at date 3. The saverinvestsinbankdepositsorholdsaliquidstorageasset. Thedepositsaredemandable,which isimportanttoprovideincentivestothebankertomonitoraswedescribeindetaillater.2 (cid:104) (cid:105) The liquidation value, ξ, follows a uniform distribution U ∼ ξ,ξ with 0 ≤ ξ < 1 < ξ and ∆ =ξ−ξ. The fact that ξ can exceed 1 will be important in what follows. We assume that long- ξ term loans are callable in which case the entrepreneur forfeits the portion of the project that is fundedbytheloan. Moreover,whenaprojectisliquidatedityieldsanimmediategrossreturnξ. Theliquidationvaluecanbejustifiedinseveralways. Forinstancetheincompleteprojectcould have a secondary use in the interim period because it can be used in conjunction with alternative short-term technology. Or we could assume that it can be sold to some outside investors as in Shleifer and Vishny (1992). In other words, ξ does not strictly represent the salvage value of the long-term investment, as for example in Cooper and Ross (1998), but rather the liquidation/resale valueoflong-terminvestment. ξhastobehighenoughthatthebankcanalwayswithstandapanic for some realizations. Yet, ξ has to be low enough that the bank may run out of liquidity even if a panicdoesnotoccur. Wedescribetheimportanceoftheseboundsinsection2.4.3 Sections2.1-2.4describeindetailtheagents’optimizationproblemsintheprivateequilibrium. Asweintroducetheagents’problems, weemphasizethereasonswhyindividualagentswillmake choicesthatwoulddifferfromasocialplanner. Section2.5discussesthekeymodelingassumptions. 2Weassumethatsaverscannotbuyequityinthebankinordertosimplifytheexpositionofourbaselinemodel. In theonlineappendix,wepresentamorecomplicatedmodelwherewherethebankraisebothinsideequityfrombankers andoutsideequityfromsavers. Therein,thebanksharespurchasedbysaversaretradableinafrictionlessmarketinthe intermediateperiodand,thus,alsoprovideliquidityservices. Althoughbankequitycanalsoprovideliquidityservices, becauseitcanbetradedinasecondarymarketsimilartoJacklin(1987), anall-equityfundingstructurewouldnotbe optimalineveninthisrichersetupduetothedisciplinaryroleofrunnabledebt.Overall,themainresultsfromthemodel inthebodyofthepapercontinuetohold. 3Our model can easily be adjusted to make the liquidation value depend on the expected value of the loans, i.e, ξ·E sV 3 I s (1+rI), whereV 3 I s isthepercentagerepaymentontheloangivenby(15)laterandrI istheloanrate. Then, ξwouldcapturethefraction(between0and1)oftheexpectedvaluethatcanbeobtainedatliquidation. Theexpected valueiscomputedoverthepossiblerealizationsofstatesinthelastperiodforknownprobabilitiesω3s. Theliquidation value,ξ·E sV 3 I s (1+rI),wouldvarybecauseξvaries. Giventhattheexpectedvalueofloansishigherthanone,thetwo approacheswouldyieldqualitativelysimilarresults. Alternatively,wecouldhaveassumedthatξdoesnotvary,butthe probabilitydistributionω˜ 3s variesasinGoldsteinandPauzner(2005). Then,theliquidationvaluewouldcontinuously vary with the realization of the true probability distribution ω3s because E sV 3 I s (1+rI) varies. The upper and lower dominanceregionsintheincompleteinformationgamewouldstillbeendogenouslydeterminedinthesecases. 5

2.1 Savers The savers are endowed with eS and eS at t =1 and t =2, respectively. In the initial period, they 1 2 invest in bank deposits, D, and can additionally save by investing in the liquid storage technology, LIQS. A portion of savers, δ, receive a preference shock to consume in the intermediate period, 1 while the rest, 1−δ, want to consume att =3. The preference shock is private information, i.i.d. andisnotcontractibleex-ante. Depositsaredemandable, earlywithdrawalsareservicedsequentiallyandtheinterestrates, rD 2 and rD, for withdrawals at t = 2 or t = 3 respectively, are uncontingent. This contract structure 3 creates the possibility of a run, since patient savers may choose to demand their deposits early dependingontheirowninformationandtheirexpectationsabouttheactionsofotherpatientsavers; every(patient)depositorreceivesanoisysignalatt =2abouttheliquidationvalue,ξ,ofthebank’s ∗ loans and there is a threshold, ξ , determining whether a patient saver decides to run or keep her depositsinthebank. Theprobabilityofarunwillbeuniqueanddependonfundamentalssimilarly toGoldsteinandPauzner(2005).4 In order to facilitate the exposition of the model, while retaining precision, we denote all variablesthatarenot(pre-)determinedatt =1asfunctionsoftheliquidationvalue, ξ, andtheportion ofsaverswhodecidetowithdrawatt=2,λ∈[δ,1]. Inequilibrium,eitherallsaverschoosetowithdraw,λ=1,oronlytheimpatientsaverswithdraw,λ=δ. However,theout-of-equilibriumbeliefs, which play an important role in the determination of the run probability derived below in section 2.4, depend on the conjectured portion of savers withdrawing. This conjecture can be anywhere betweenδand1. It is instructive to review the different possible scenarios separately. If there is no run, only impatient depositors withdraw and they receive the full amount of promised payment, D (cid:0) 1+rD(cid:1) . 2 Patient depositors’ repayments are determined as a function of the technology shock in the next period. Inarun,alldepositorsattempttowithdrawandthereisprobabilityθ(ξ,1)thatadepositor is served.5 Conditional on the bank surviving to t = 3 patient depositors receive their promised payment in full or in part if the bank defaults, VD(ξ,δ)D (cid:0) 1+rD(cid:1) . The percentage repayment on 3s 3 4Bank-runsinourmodelcanalsobepanicbasedratherthanpurelyinformationbasedasinChariandJagannathan (1988), Jacklin and Bhattacharya (1988), Allen and Gale (1998), Uhlig (2010), Angeloni and Faia (2013), Boissay, CollardandSmets(2016). Inotherwords,abank-runcanoccurduetoacoordinationproblemamongdepositorseven ifthebankissolventinthelong-run. Similarly,abank-runcanalsooccurbecausetheinformationaboutfundamentals is very bad. In determining the optimal ex-ante decisions, it is important to know what determines panics. In the Diamond-Dybvig model panics are a multiple equilibrium outcome. Cooper and Ross (1998), Peck and Shell (2003) andKeister(2015)supposeinsteadthattheprobabilityofabank-runisdrivenbysunspots. Inourearlierworkingpaper Kashyap, Tsomocos and Vardoulakis (2014), in Gertler and Kiyotaki (2015) and in Choi, Eisenbach and Yorulmazer (2016)theprobabilityofarunisdeterminedbyanexogenousfunctionofkeyfundamentals. EnnisandKeister(2005) take an axiomatic approach to equilibrium selection and link the probability of a particular equilibrium being played to appropriately defined incentives of agents. Instead, we use the global games approach developed by Morris and Shin (1998) and applied to banks runs by Goldstein and Pauzner (2005) to derive a unique probability of run which dependsonfundamentals.AlthoughwemaintaintheykeyassumptioninGoldsteinandPauzner(2005)toobtainaunique equilibrium,weamendtheirapproachbyintroducingnoisysignalsonadifferentvariablesothattheupperdominance regionisendogenouslyderivedratherthanassumed. RochetandVives(2004)andVives(2014)alsotakeaglobalgame approach,butdelegatethewithdrawaldecisiontoa(deposit)fundmanagerwithasimplerpayofffunction. 5Thisprobabilityisdeterminedbyequation(12),derivedinsection2.2.Inarunallsaversattempttowithdraw,λ=1. 6

period3depositwithdrawals,VD(ξ,δ),isgivenbyequation(16)thatisderivedbelow. Depositors 3s have to pay an additional cost, c , per unit of promised payments to receive a payment when the D bankdefaults. Thus, the net repayment on deposit is (cid:0) VD(ξ,δ)−c ·I (cid:1) D (cid:0) 1+rD(cid:1) , where I is an indicator 3s D d 3 d functionthattakesthevalueof1whenthebankdefaults. Weproceedbyformallypresentingsavers’problem. Savers’consumptionatt =1isgivenby c =eS−D−LIQS. (1) 1 1 1 Given the various cases that occur in the latter periods when the savers may or may not be patient,choosetowithdraworrun,andbepaidornotinarun,itishelpfultointroducesomefurther notation. Denote by j = i,p the saver’s type, which is realized at t = 2. They will be impatient (cid:104) (cid:105) (j=i)withprobabilityδandpatient(j= p)withprobability1−δ. Arunoccurswhenξ∈ ξ,ξ ∗ . In this case, all savers attempt to withdraw, but only a fraction of them are repaid. We denote by I an indicator which takes the value of 1 if an individual saver is repaid, where the (endogenous) θ probabilityofrepaymentisθ(ξ,1). Then,theconsumptionofasaveroftype j isgivenby c (j,I )=I ·D (cid:0) 1+rD(cid:1) +LIQS+eS, (2) ts θ θ 2 1 2 wherets=2for j=iandts=3sfor j= p,becausepatientsaversstillonlyconsumeatt =3and willtransfertheirresourcesfromperiod2to3usingthestoragetechnology. WealsodefineanindicatorI whichtakesthevalueof1ifanagentoftype j withdrawswhen w (cid:104) (cid:105) a run does not occur, i.e., when ξ∈ ξ ∗,ξ , and is 0 otherwise. Her consumption is denoted by c (j,I ). Althoughinequilibriumpatientsaverswilltruthfullyreporttheirtypeandonlyimpatient ts w saverswillwithdraw,weneedtocontemplatedeviationswhereapatientsavercouldopttowithdraw andshowthatsuchdeviationsneverareoptimal(seesection2.4). Theconsumptionofanimpatient saveris,then,givenby c (i,I =1)=D (cid:0) 1+rD(cid:1) +LIQS+eS. (3) 2 w 2 1 2 Theconsumptionatt=3ofapatientsaverwhochoosestowaitorwithdraware,respectively,given by c (p,I =0)= (cid:0) VD(ξ,λ)−c ·I (cid:1) D (cid:0) 1+rD(cid:1) +LIQS+eS, (4) 3s w 3s D d 3 1 2 or c (p,I =1)=D (cid:0) 1+rD(cid:1) +LIQS+eS. (5) 3s w 2 1 2 Notethatinequilibrium,λ=δin(4). Finally,short-sellingofdepositsandtheliquidassetisnotallowed. Soinsolvingthemodelwe add the following constraints (with the associated Lagrange multipliers indicated in parentheses): D≥0 (ν );andLIQS≥0 (ν ). D 1 LIQS 1 The savers choose the level of deposits and their holdings of the liquid asset to maximize their 7

utilitysubjecttoconstraints(1)-(5). Theexpectedutilityofarepresentativesaverisgivenby  run norun   (cid:122) (cid:125)(cid:124) (cid:123) (cid:122) (cid:125)(cid:124) (cid:123)   (cid:90) ξ∗ dξ (cid:90) ξ dξ   US=U (c )+ ∑ E U (c (j,I );j) + E U (c (j,I );j) . (6) 1 1 j,θ t ts θ j,s t ts w t=2,3  ξ ∆ ξ ξ∗ ∆ ξ      Conditional on a run occurring, patient savers compute the expected utility from remaining patient or attempting to withdraw and probabilistically receiving payment on their deposits (E ). j,θ If a run does not occur, patient savers compute the expected utility from withdrawing early, and from receiving the state contingent payment on deposits (E ). Note that we have indexed the j,s utility function by the timet and agent type j. Impatient savers receive utility only att =2 which is discounted to the present by β < 1, while patient savers receive utility only at t = 3 which is discountedtothepresentbyβ2. Moreover,weassumethatsavershavequasi-linearpreferences;at t=1 and at t =2 (for j =i) savers have concave utilityU, while savers have linear preferences at t =3(for j= p).6 Anindividualsavertakestheprobabilityofbeingrepaidinarun,θ(ξ,1)andthepercentagerepayment,VD(ξ,δ),asgiven. Theseobjectsdependontheaggregatebankportfolioandwesuppose 3s thattheindividualsaverissufficientlysmallsoastonotaccountforherimpactonthem. Asocial plannerwouldinternalizetheeffectofthechoices. Theoptimalsupplyofdepositsbysaversisgivenby:   run norun,impatient DS:−U 1 (cid:48)(c 1 )+ (cid:0) 1+r 2 D(cid:1)     t (cid:122) = ∑ 2,3 (cid:40) (cid:90) ξ ξ∗ θ(ξ,1)·E (cid:125) j (cid:124) U t (cid:48)(c ts (j,1);j) d ∆ ξ ξ (cid:41) (cid:123) + (cid:122) δ (cid:90) ξ∗ ξ U 2 (cid:48)(c (cid:125) 2 (cid:124) (i,1);i) d ∆ ξ ξ (cid:123)      +(1−δ) (cid:90) ξ ∑ω U(cid:48)(c (p,0);p)· (cid:0) VD(ξ,δ)−c ·I (cid:1) ·(1+rD) dξ +ν =0. (7) ξ∗ s 3s 3 3s 3s D d 3 ∆ ξ D (cid:124) (cid:123)(cid:122) (cid:125) norun,patient Condition (7) says that savers equate the marginal utility of forgone consumption at t =1 to the expected marginal utility gain from holding deposits in the future. In a run, all savers withdraw; theirmarginalutilitydependsontheirtype, j, andtheprobabilitythattheyarerepaid, θ(ξ,1). Ifa rundoesnotoccur,impatientsaversarefullyrepaidatthepromisedrate,1+rD,whilepatientsavers 2 do not withdraw and receive the uncertain deposit payoff,VD(ξ,δ)· (cid:0) 1+rD(cid:1) , minus any marginal 3s 3 bankruptcycosts. Saversmaywanttoself-insureandholdtheliquidasset. Theoptimalliquidholdings,LIQS,are 1 6Thelinearityofutilitiesinthefinalperiodisnotimportantforourresultsandwehaveassumeditforsimplicityof exposition.Wediscussfurtherthisassumptioninsection2.5. 8

givenby:  run norun   (cid:122) (cid:125)(cid:124) (cid:123) (cid:122) (cid:125)(cid:124) (cid:123)   (cid:90) ξ∗ dξ (cid:90) ξ dξ   −U(cid:48)(c )+ ∑ E U(cid:48)(c (j,I );j) + E U(cid:48)(c (j,I );j) +ν =0. (8) 1 1 t=2,3  ξ j,θ t ts θ ∆ ξ ξ∗ j,s t ts w ∆ ξ  LIQS 1     Condition(8)saysthatholdingtheliquidassetallowsthesavertoself-insureagainststatesthatshe is not repaid in a run, i.e., compared to using a deposit as in (7) the liquid asset always delivers futureconsumption,butdoessobyforgoingthehigheruncertainreturnthatdepositspromise. Moreover,notethattheequilibriumoutcomeswillbeincentivecompatible,i.e.,apatientsaver willnothaveanincentivetomisrepresenthertypeandwithdrawforξ>ξ ∗ . Thisisguaranteedby thewaytherunthresholdisdetermined,whichwedescribeindetailinsection2.4. Finally,weneedtospecifywhatwouldhappenifthesaverschosetoavoidusingthebank. One possibility is that they save using only the liquid asset. We refer to this as autarky. The liquidity choice in this case is the solution toU 1 (cid:48)(eS 1 −LIQS 1 ,aut )+∑ t=2,3 E j U t (cid:48)(eS 2 +LIQ 1 S,aut ;j)=0. These holdingsimplyautilityinautarkyUS,aut,whichisausefulbenchmarkforgaugingthebenefitsthat intermediation delivers through liquidity provision. A second alternative is that the savers could attempttodirectlylendtoentrepreneurs(assumingthattheywouldalsohavetomonitorafterdoing so). We denote the resulting level of utility by byUS,dl. The participation constraint of savers is, then,givenby: (cid:16) (cid:17) US≥max US,aut,US,dl . (9) Giventhatthisconstraintwillmostlynotbindfortheresultswepresent,wereportthedetailed problemwhensaverslenddirectlytoentrepreneursintheonlineappendix. Wewillbeexplicitabout theoccasionsthattheconstraintbinds. 2.2 BankersandBanks The banker makes all investment and funding decisions to maximize her own utility. Att =1, she isendowedwitheB anddecideshowmuchequity,E,toputintothebank. Herutilityisgivenby: norun (cid:122) (cid:125)(cid:124) (cid:123) UB=γ·U (cid:0) eB−E (cid:1) + (cid:90) ξ ∑ω DIV (ξ,δ) dξ , (10) 3s 3s ξ∗ s ∆ ξ whereDIV arethedividendsinstatesatt =3. 3s The banker trades off foregoing current consumption to investing in equity and receiving dividends in the future. The banker has also quasi-linear preferences and the same utility function at t =1 as the savers, but unlike the saver never needs to consume in the interim period. We have maintained the linearity of preferences att =3, though we restrict parameters so that savers never 9

insurethebankeragainstfutureuncertainty.7 Additionally, the banker chooses how many deposits to raise, D, and using the total funds, the bankerinvestsintheliquidassets,LIQ ,andilliquidloans,I. Theloancontractisuncontingentand 1 requiresapaymentof1+rI perdollaroflendingatt =3. Asalreadymentioned,loansarecallable at any time before maturity at which case the entrepreneur surrenders the projects funded by these loansanddoesnothaveanobligationtorepaythematt =3. Thebalancesheetconstraintatt =1isgivenby: BS:I+LIQ =D+E. (11) 1 Wedefineψ tobethemultiplieronthebalancesheetconstraint(11),whichrepresentstheshadow BS valueoffunding,i.e.,theendogenouscostofexpandingassetsbyraisingaunitoffunds. Thebalancesheetandprofitsaftert=2dependontherealizationofξandthenumberofpeople withdrawing, λ. If a bank-run occurs then the bank is liquidated and the proceeds are distributed accordingtosequentialserviceconstraint. Thus,theprobabilitythatanysaverisservedisequalto LIQ +ξ·I 1 θ(ξ,λ)= . (12) λ·D·(1+rD) 2 If the bank survives the run, it will have to recall and liquidate a portion y(ξ,λ) of its loan portfoliotoservetheearlywithdrawalsgivenby λ·D·(1+rD)−LIQ +LIQ (ξ,λ) y(ξ,λ)= 2 1 2 , (13) ξ·I where LIQ (ξ,λ) ≥ 0 are the liquid holdings carried over to the third period. Our assumptions 2 regardingthedistributionofξleadbanktoholdinsufficientliquidassetstoserviceallearlydeposit withdrawalsevenwhenonlytheimpatientsaverswithdraw. Sothebankisalwaysplanningtocall some loans. In principle, the bank could want to liquidate its whole loan portfolio and carry the proceeds forward using the storage technology, but this would only be the case if the realization of ξ is higher than the expected return from holding the loan to maturity, which we have excluded by assumption. As a result, y(ξ,λ) will take interior values between zero and one, and it will be decreasinginξandincreasinginλforapre-determinedbankportfolio. Conditional on the bank surviving, the dividends depend on the portion of the portfolio liqui- 7The difference between the banker and the savers expected utility is that the former values future consumption morethancurrentorinotherwordsγ<1. AssigningtothebankerthesameutilityfunctionrequireshighenougheB or low enough γ such that she would be willing to invest enough of her own wealth in equity to provide risk-sharing benefits to savers. We do the second because we want the banker endowment to represent only a small part of the totalendowmentintheeconomy,withthevastmajorityaccruingtothesavers. Forγ=1/β2,suchthatsaversandthe bankerdiscountthefuturethesameway, andforlogarithmicutility, wecanobtainthesameequilibriumforbanker’s wealthe˘B=E+(eB−E)/(β2γ),whereE istheequilibriumvalueofcontributedequity. Finally,giventhatbankersare protectedbylimitedliabilityandthatfutureendowmentsarenotcontractible,quasi-linearpreferencesallowustoexclude finalperiodendowmentsfromouranalysisbysettingthemtozero. 10

datedy(ξ,m)andaregivenby DIV (ξ,λ)=(1−y(ξ,λ))·VI (ξ,λ)·I· (cid:0) 1+rI(cid:1) +LIQ (ξ,λ)−VD(ξ,λ)·(1−λ)·D· (cid:0) 1+rD(cid:1) , 3s 3s 2 3s 3 (14) whereVI (ξ,λ)isthepercentagerepaymentontheremainingriskyloans. Itisgivenby 3s (cid:34) A F (cid:0) IE+(1−y(ξ,λ)·I) (cid:1)(cid:35) VI (ξ,λ)=min 1, 3s (15) 3s (1−y(ξ,λ))·I·(1+rI) andVD(ξ,λ)istherepaymentrateondeposits. Itisgivenby 3s (cid:34) (1−y(ξ,λ))·VI (ξ,λ)·I· (cid:0) 1+rI(cid:1) +LIQ (ξ,λ) (cid:35) VD(ξ,λ)= 1, 3s 2 . (16) 3s (1−λ)·D· (cid:0) 1+rD(cid:1) 3 In other words, bank profits in (14) are equal to the revenue received from the repayment on the outstanding loans plus any liquid assets carried forward minus the repayment on the deposits that werenotwithdrawnearly. Equation(15)saysthattheloanisfullyrepaidwhentherevenueavailable to the entrepreneur, which is derived from the own funds invested by the entrepreneurs, IE, and bank loans, is higher than the outstanding loan obligation; otherwise the entrepreneur defaults and thebankseizeseverythingthatisavailable. Equation(16)saysthatlatedepositorsarerepaidinfull when the value of bank assets is higher than the promised deposit payments; otherwise the bank defaultsanddepositorsdividetheassetsinapro-ratafashion. After the run uncertainty has been resolved and the true value of ξ is learned, the banker can choose to monitor the borrower to learn the true value of the productivity shock at t =3, which is private information to the entrepreneur. Alternatively, the banker can forgo the monitoring and enjoyaprivatebenefitfromrunningthebank. Wefollowthelongtraditionintheliteratureassuming thatmonitoringiscostlyforthebankerbecauseshewouldhavetogiveupaprivatebenefitshewould otherwise receive from managing the bank (see, for example, Holmström and Tirole, 1997). This assumption creates an ex-post moral hazard problem in which the banker will choose to monitor onlyiftheexpecteddividendsarehigherthantheprivatebenefit. Ifthebankeroptsnottomonitor, theentrepreneurwouldalwaysreportthelowestrealizationoftheproductivityshockanddefaulton theloan.8 Thebankerwillchoosetomonitorforallξ≥ξ ∗ ifthefollowingincentivecompatibility constraintissatisfied: IC:∑ω DIV (ξ ∗,δ)−PB≥0, (17) 3s 3s s wherePBistheprivatebenefit. The first term in the IC constraint is the expected payoff to the banker if she monitors when ξ=ξ ∗ . We take the expectation because the banker has to decide whether to monitor before she 8Theproductivityleveliscommonacrossprojects. Therefore,asinDiamond(1984),monitoringcostsareconserved byhavingabankmonitorallborrowers,relativetohavingindividuallendersmonitorindividualborrowers. Thus,the bankmonitoringexpandsthesupplyofcredit. 11

learnsthetruevalueofA . Thesecondtermistheprivatebenefit. Ifthebankerdoesnotmonitor, 3s then the entrepreneur reports the lowest realization for A , defaults on the loan repayment and 3s forces the bank to default on its deposits (so that bank equity is worthless). It suffices that the IC constraint is satisfied for ξ=ξ ∗ , because expected dividends are increasing in ξ, thus the banker willalwayshaveanincentivetomonitorifthereisnorun. Thebankandthedepositorsmaywanttowriteadepositcontractnotonlyonthedepositsrate(s) andtheamountofdeposits,butalsooverallthefactorsaffectingtheriskinessofthedeposits. These risksaregovernedbyallaspectsofthebank’sbalancesheet,inparticular,itschoiceofleverage(or equivalentlyacapitalratio),itsassetallocationbetweenloansandliquidassets(i.e.,aliquidityratio) anditsmaturitymismatch(i.e.,anetstablefundingratio). However,suchcomprehensivecontracts may not be possible for a number of reasons and do not resemble observed deposit or unsecured funding arrangements in reality.9 As a result, the bank would be tempted to deviate in the way it choosesitsleverage,liquidityandmaturitymismatchafterithasenteredintoadepositcontractand receivedthedeposits. Technically,thislackofcommitment meansthatthebankwilloptimizeonly overstatesoftheworldinwhichitissolventbecauseitisprotectedbylimitedliability. Likewise, it will only internalize how it affects the supply of deposits when it chooses the contract terms (cid:0) D,rD,rD(cid:1) . The bank does understand that taking more risk increases the cost of raising deposits, 2 3 and would ideally want to promise depositors that it will behave prudently. But, after the deposit contracthasbeensigned,thebankhasanincentivetodeviatetowardslendingmore,holdingfewer liquidassetsandraisinglessequity. Depositors have rational expectations and ex-ante require that the bank offers higher deposit rates to compensate for the anticipated risk-taking due to the lack of commitment. In contrast, a socialplannerwouldrecognizethatthebank’sinsolvencyadverselyimpactssavers,andwouldaccountforthisinmakingallocations. Webelievethatincompletecontractingisanimportantfeature of reality when financial institutions have a rich balance sheet and their activities expose savers to credit risk. Nevertheless, we also examine the case where comprehensive contracts, specifying the full set of choices made by the banker, can be written. We denote by I an indicator function, c which takesvalue oneif deposit contractsare comprehensiveand zeroif they areincomplete. Our conclusionsregardingtheneedforbankregulationholdforbothcases. Oneforceinthemodelthatpartiallydisciplinesthebankeristhepossibilityofabankrun. The banker will internalize how her investment and funding choices affect the probability of a run via condition(32)(thatisderivedbelow),andhencetheprobabilitythatshewillmakeprofits. Similarly, the banker understands that her ex-post incentives to monitor need to be consistent with condition (17); otherwise depositors would anticipate that the banker will not have an incentive to monitor andwouldalwaysrunatt =2drivingthebanker’srentstozero. Inthisrespect,therunriskcreates 9For example, Dewatripont and Tirole (1994) argue that individual depositors are sufficiently small and diverse to enforcecomprehensivecontractswhichdisciplineallbankingchoices. SeealsoStiglitzandWeiss(1981),Matutesand Vives(2000),BoydandDeNicoló(2005)amongothersformodelswithrisk-takingincentiveswhenloancontractsare not comprehensive. Contrary to these papers, which maintain the price-taking assumption for the borrowing rate, we allowborrowertooptimallychooseallthetermsspecifiedinthecontract. 12

anincentiveforthebankertomonitoritsborrowersandtoprudentlychooseitscapitalstructureand amountoflendingatt =1(seeCalomirisandKahn,1991,DiamondandRajan,2000,2001). Overall, the banker will understand how the investment and funding decision matter for future behaviorbysaversandwilltakeequations(32)and(17)asadditionalconstraintsinheroptimization problem,butsheneglectstheothereffectsofherdecisionsonsaversandentrepreneursutilitiesgiven by(6)and(26). In solving for bank’s optimal choices, we will focus on equilibria such that the bank is always solvent in state g and always defaults in state b for all realizations of ξ≥ξ ∗ .10 Substituting into (10) equations (13), (14), (15), (16), the banker optimizes over the risky loan, I, the liquid asset holdings,LIQ andLIQ (ξ,δ)foreachξ,theequitycontributed,E,therunthreshold,ξ ∗ ,thelevel 1 2 of deposits, D, and the deposit rates, rD and rD. She takes (11), (17) and (32) as constraints in her 2 3 problem. Thelastconstraintistheglobalgamecondition,GG,whichdeterminestherunthreshold is derived in section 2.4 below. Due to limited liability the banker will only consider the states in whichsheissolvent. Theoptimalityconditionforloans,I,is: dUB dIC dGG dDS −ψ +ψ +ψ +ψ ·I =0, (18) dI BS IC dI GG dI DS dI c where dUB/dI = (cid:82)ξ (cid:8) ω (cid:0) 1+rI(cid:1)(cid:9) dξ/∆ is the marginal effect of investment on banker’s share ξ∗ 3g ξ of profits and ψ , ψ , ψ and ψ are the multipliers on constraints (11), (17), (32) and (7) BS IC GG DS respectively. Theexpression(18)saysthatoptimalleveloflendingisdeterminedbyhavingthebankertrade off the marginal return accruing to her against the shadow cost of funding additional lending and the way it affects the incentive compatibility and the run threshold determination constraints. As already mentioned, the banker only internalizes states where she is solvent due to limited liability. Finally, the banker considers how her investment decisions affects the deposit supply only when the level of investment is a contractual deposits term at which she can commit to. Equation (18) correspondstotheloansupplyschedule,denotedbyLS,offeredtoentrepreneurs. Theoptimalityconditionforfirstperiodliquidassets,LIQ ,is: 1 dUB dIC dGG dDS −ψ +ψ +ψ +ψ ·I =0, (19) dLIQ BS ICdLIQ GGdLIQ DSdLIQ c 1 1 1 1 where dUB/dLIQ = (cid:82)ξ (cid:8) ω (cid:0) 1+rI(cid:1) /ξ (cid:9) dξ/∆ is the marginal effect of liquidity on banker’s 1 ξ∗ 3g ξ share of profits. The optimal choice of liquid assets is governed by the same considerations to determine optimal lending. The only difference is that the marginal return on the liquid assets is scaledbytheliquidationvalueξ,becausethebankneedstoliquidate1/ξfewerloanstoserveearly 10Equivalently, entrepreneurs default on their loan in state b and deliver fully in state g. We have also solved the modelwithmorestatesfortherealizationoftheproductivityshock, suchthatentrepreneurs’defaultdoesnotneedto coincidewithbanks’default. Giventhatourresultcontinuetohold,wehavechosentopresentthemodelwithtwolevel ofproductivitytosimplifytheanalysisandpresentthemorecomplicatedcaseinanonlineappendix. 13

withdrawalsforeachadditionalunitoftheliquidasset. ∗ Thebankerwilloptimallychoosetherunthreshold,ξ ,whichyields: dUB dIC dGG dDS +ψ +ψ +ψ ·I =0, (20) dξ ∗ ICdξ ∗ GG dξ ∗ DS dξ ∗ c where dUB/dξ ∗ =−∑ s ω 3s DIV 3s (ξ ∗,δ)/∆ ξ . In making this choice, (20) says that the banker bal- ∗ ancesthereductionindividendsbecauseofamarginallyhigherξ againsttheeffectfromrelaxing theICandGGconstraints(andDSiftherunthresholdisadepositcontractterm). Theoptimalchoiceofliquidityholdings,LIQ (ξ,δ)ismadeaftertherununcertaintyisresolved 2 and depends on the realization of ξ. As a result, the banker will only consider the effect on (her) profits,butnottheeffectontherunthresholdduetotheinabilitytocommit. Banksmaywantpatient investors to think that they will hold liquid assets from t =2 to t =3 to reduce the probability of a run, but if the bank survives, then banks may not have an incentive to hold liquid assets because theyonlycareaboutstatesinwhichtheyaresolvent(unlessthedepositcontractspecifiesthelevel of second period liquid assets for every realization of ξ). Under incomplete contracts, the banker will carry liquidity in period 3 only if the liquidation value is higher than the expected loan return in the states that the bank is solvent, i.e., if ξ>ω (1+rI). In the equilibria we examine this is 3g never the case, because ξ<1+rI, so it is optimal for the bank to recall loans only to serve early withdrawalsandnottohoardliquidity. Theoptimalityconditionwithrespecttocontributedequity,E,is: dUB +ψ =0, (21) dE BS where dUB/dE = −γ·U(cid:48)(cid:0) eB−E (cid:1) . Condition (21) says that injecting more equity requires the bankertogiveupconsumptionintheinitialperiodinexchangeforincreasingthefundsofthebank. Note the condition does not include a term for the effect of additional equity on constraints GG or IC (as well as DS for comprehensive contracts). This is true because, E does not appear directly in (32), (17) or (7), but this doesn’t mean that equity is irrelevant for their determination. On the contrary, equity issuance can affect the run probability, the incentives to monitor and the deposit supplythroughitsjointdeterminationwiththeotherequilibriumvariables. Condition(21)governstheshadowcostoffundsψ , whichisinverselyrelatedtotheamount BS ofequitythebankerputsinthebank. Inbankingmodelswithoutendogenouscreditorrunrisk,the higherfundingcostsofinjectingmoreequitywouldfeedinhigherloanratesandlowerinvestment. Thisdoesnotneedtobetruewhenequitychangesthelevelofcreditandrunriskasinourmodel; higherequityandcostoffundingcanbecompatiblewithlowerloanratesandmoreinvestment. Finally, the banker chooses the deposit contract (cid:0) D,rD,rD(cid:1) which needs to lie on the deposit 2 3 14

supplycurve(7). Theoptimaldepositcontractsatisfiesthefollowingfirst-orderconditions: dUB dIC dGG dDS +ψ +ψ +ψ +ψ =0 (22) dD BS IC dD GG dD DS dD dUB dIC dGG dDS +ψ +ψ +ψ +ν =0, (23) drD ICdrD GG drD DS drD r 2 D 2 2 2 2 dUB dIC dGG dDS +ψ +ψ +ψ =0, (24) drD ICdrD GG drD DS drD 3 3 3 3 where dUB/dD=− (cid:82)ξ (cid:8) ω (cid:0) 1+rI(cid:1)(cid:0) δ (cid:0) 1+rD(cid:1)(cid:1) /ξ+(1−δ) (cid:0) 1+rD(cid:1)(cid:9) dξ/∆ captures the effect ξ∗ 3g 2 3 ξ of deposits, dUB/drD = − (cid:82)ξ (cid:8) ω (cid:0) 1+rI(cid:1) (δ·D)/ξ (cid:9) dξ/∆ the effect of the early deposit rate 2 ξ∗ 3g ξ and dUB/drD =− (cid:82)ξ {ω (1−δ)·D}dξ/∆ the effect of the late deposit rate—the three deposit 3 ξ∗ 3g ξ contract terms— on banker’s profits, respectively. Finally, ν is the multiplier on non-negativity rD 2 constraintrD≥0,whichwediscussbelow. 2 Condition(22)canbeeasilyinterpreted. ψ istheshadowbenefitofraisinganadditionalunit BS ofdeposits. Whenadepositisaccepted,itentailspayingtheinterestrateonlatewithdrawalsto1−δ depositorsandliquidatingthelongtermassettoservicetheearlywithdrawalsofδdepositors,andit alterstherunthresholdandtheincentivecompatibilityconstraint. Similarlytotheotherdecisions, the banker considers the effect of repaying deposits on her profits only in states that she expects to be solvent. The last term captures the effect on the privately optimal supply of deposits and is presentevenifdepositcontractsarenotcomprehensive. Conditions(23)and(24)canbesimilarly interpreted with the difference that deposit rates do not entail a direct balance sheet cost and, thus, ψ doesnotappearintherespectiveoptimalityconditions. BS Werestrictdepositratestobepositive,whichcanbeparticularlyimportantforthechoiceofrD 2 in(23). Absentconstraints,thebankermaywanttoofferanearlydepositratethatisnegative,since thiswouldallowhertoreducetheprobabilityofarun. Suchrun-preventingdepositcontractshave been studied for example in Cooper and Ross (1998). In our model, however, runnable deposits are important to discipline the banker and there are limits to how low the early deposit rate can be set both because of the disciplinary role and because savers can stop using the bank if the rates become too low. In the numerical examples we present, rD hits the non-negativity constraint both 2 intheprivateandplanningequilibria,butwehavealsosolvedforcaseswhereitisallowedtotake negativevalues. Theimplicationsofourmodelforthedistortionsbetweentheprivateandplanning equilibria as well as the effects and desirability of regulation continue to hold under a negative depositrateforearlywithdrawals. Wepresenttheseresultsintheonlineappendix.11 The banker is willing to intermediate funds between savers and entrepreneurs if the utility she 11See also Keister (2015) for a model with flexible deposit contracts, i.e., the payment that a depositor receives is determinedbythebankasabestresponsetorealizedwithdrawalsintheintermediateperiod.Runsinhisframeworkare partialinthesensethatthebankcanalterpaymentstostopwithdrawalsbypatientdepositorsandavoidliquidationonce therunstateisrevealed. 15

obtainsishigherthattheutilityisautarky,i.e.,ifthefollowingparticipationconstraintsissatisfied: UB≥UB,aut, (25) whereUB,aut =γ·U (cid:0) eB−LIQB(cid:1) +LIQB. Inautarky,theconsumptionofthebankeratt=1isequal 1 1 toherendowment,eB,minusanyholdingoftheliquidasset,LIQB,carriedforwardtot =3. LIQB 1 1 isthesolutiontoequationγ·U(cid:48)(cid:0) eB−LIQB(cid:1) =1ifpositiveandzerootherwise.12 1 2.3 Entrepreneurs Entrepreneurs have the rights to real projects that are in elastic supply, require a unit of funding at t =1,areinfinitelydivisiblewhenliquidated,andmatureatt =3. Entrepreneurshaveendowment eE intheinitialperiodandborrowIfromthebankatinterestraterI. DenotebyIE theownfundsput into the real projects att =1. Then, E consumes eE−IE and takes a bank loan which depends on boththeloanamount(orequivalentlytheloan-to-valueratio, LTV =I/ (cid:0) I+IE(cid:1)(cid:1) ,andaloanrate, rI. For simplicity, we will assume that the entrepreneur is risk-neutral and that she derives utility only from consumption at t =3. Hence, she will invest all her endowment in the risky project as longasthereturnishigherthatthereturnonthestoragetechnologywhichhaszeroyield. Finally, E isprotectedbylimitedliabilitywhenprojectsmatureandloansaredue. Ifarundoesnotoccur, shewillrepaytheoutstandingloans,1−y(ξ,δ),notrecalledatt=2onlyiftheinvestmentpayoffis higherthanthecontractualloanobligation. Inarun,allprojectsfundedbybankloansareliquidated (y(ξ,δ)=1forallξ<ξ ∗ )andtheentrepreneurcanonlyproduceusingherowncapitalcommitted att =1. Hence,theutilityofanindividualentrepreneuris:  norun run   (cid:122) (cid:125)(cid:124) (cid:123) (cid:122) (cid:125)(cid:124) (cid:123)  UE =∑ω  (cid:90) ξ (cid:2) A F (cid:0) IE+(1−y(ξ,δ))I (cid:1) −(1−y(ξ,δ))I(1+rI) (cid:3)+ dξ + (cid:90) ξ∗ A F (cid:0) IE(cid:1)dξ   . 3s 3s 3s s   ξ∗ ∆ ξ ξ ∆ ξ      (26) The loan contract is comprehensive and E optimally chooses a combination of I and rI that lie on the loan supply curve (18) to maximize (26). In addition to the loan contract terms, E’s utility and the loan supply curve offered by the bank to each individual entrepreneur depend on a set of aggregate bank variables that the entrepreneur takes as given. These aggregate variables include 12Theoutsideoptionisimportantbecausetheplannerwilldrivethebankertoherparticipationconstraint. Offering the most competitive lending terms to entrepreneurs would require the intermediation of deposits from savers. Thus, assuming that entrepreneurs can freely choose the banker that offers the best terms, the outside option for the banker isherutilityinautarky. Alternatively,wecouldhaveassumedthatentrepreneursarecaptiveofbankersand,hence,the outsideoptionisequaltotheutilitytheywouldobtainbylendingtoentrepreneursusingonlytheirowncapital.Wederive theconditionsforthiscaseinanonlineappendix. Weshouldnotethatfortheequilibriumweexamine,entrepreneurs wouldnotborrowfrombankers,unlessthelatterraisedepositstoreducefundingcosts. Thereasonisthatentrepreneurs wouldobtainahigherutilityinvestingonlyoutoftheirownfunds.Thus,theautarkicutilityistherelevantoutsideoption forbankersundereitherassumption. 16

the probability of bank run, which depends on ξ ∗ , y(ξ,δ), and the shadow values ψ , ψ , and BS IC ψ . Although the individual loan characteristics will matter for the aggregate bank variables in GG equilibrium, each individual entrepreneur is small compared to the aggregate bank portfolio such thatsheneglectstheeffectoftheloantermsonthem. Combining the optimality conditions with respect to the loan terms I and rI, we obtain the optimalloandemand,LD,oftheentrepreneur: (cid:34) (cid:35) LD: (cid:90) ξ (1−y(ξ,δ)) A F(cid:48)(cid:0) IE+(1−y(ξ,δ))·I (cid:1) −(1+rI)+I· ∂LS (cid:18) ∂LS (cid:19)−1 dξ =0. (27) ξ∗ 3g ∂I ∂rI ∆ ξ The first two terms in the entrepreneur’s loan demand (27) schedule correspond to the profit margintotheentrepreneur,givenbythedifferencebetweenthemarginalproductofinvestmentand thegrossloanrate. Limitedliabilitymeansthattheentrepreneuronlycaresaboutthestatesinwhich shedoesnotdefault,i.e.,sheconsiderstheprofitmarginonlyinstateg. Thethirdtermcapturesthe dependenceoftheloanrateontheloanlevelI. Althoughtheentrepreneurcaresonlyaboutthestates in which she is solvent, her loan demand is influenced by the states in which she defaults because thebankcaresaboutthiswhentheinterestrateisdetermined. Becauseentrepreneurialdefaultand bankdefaultoccuratthesametime,∂LS/∂I=0fromtheperspectiveofanindividualentrepreneur who takes the aggregate portion of loans recalled, y(ξ,δ), and the other aggregate bank variables as given.13 A social planner would instead take full account of how a default by the entrepreneur influencestheothertwoagents. Finally, the entrepreneur is willing to borrow from the bank if her utility is higher than the utilityfromjustinvestingherownfundsintheproject,i.e.,ifthefollowingparticipationconstraints issatisfied: UE ≥UE,aut, (28) where UE,aut = ∑ s ω 3s A 3s F (cid:0) IE(cid:1) . Constraint (28) implies that there is at least one state that the entrepreneurdoesnotdefaultonherloan. 2.4 GlobalGameandBank-runThreshold We conclude our description of the model by examining the incentives of patient savers to run or not. As already mentioned, we take all variables that are not predetermined at t=2 to be functions of the realization of ξ and the number of people that choose to withdraw, λ∈[δ,1]. Patient savers receive at t=2 private signals x =ξ+ε, where ε are small error terms that are independently and i i i uniformlydistributedover[−ε,ε]. Focusingonthresholdstrategies,anindividualpatientsaverwill run if the private signal realization is lower than a threshold, x ≤x∗, and will not run otherwise. i ∗ Thethresholdforthestrategiesimpliesathresholdforfundamentalsξ . Thenumberofsaversthatwithdrawunderthresholdstrategyx∗atagivenleveloffundamentals 13Thisisnotgenerallytrueifthereareadditionalstatessuchthatthebankremainssolventevenifentrepreneursdefault ontheirloans.Asalreadymentioned,expandingourmodeltoaccountforsuchoutcomesisnotimportantforourresults. 17

ξis   1 ifξ<x∗−ε    λ(ξ,x∗)= δ+(1−δ)Prob(x ≤x∗) ifx∗−ε≤ξ≤x∗+ε, (29) i     δ ifξ>x∗+ε where Prob(x ≤x∗)=(x∗−ξ+ε)/2ε. The number of savers withdrawing is decreasing in ξ, so i thebankisliquidatedinarunonlyifξ≤ξ ∗ whereξ ∗ istheuniquesolutiontoθ (cid:0) ξ ∗,λ (cid:0) ξ ∗,x∗(cid:1)(cid:1) =1 (seeequation(12)): λ (cid:0) ξ ∗,x∗(cid:1) D(1+rD)=LIQ +ξ ∗I 2 1 ε (cid:2) (1+δ)D(1+rD)−2·LIQ (cid:3) +x∗(1−δ)D(1+rD) ⇒ξ ∗= 2 1 2 (30) 2εI+(1−δ)D(1+rD) 2 Nextconsiderthedecisionofanindividualpatientsavertowithdrawgivenherexpectationabout thetotalnumberofpeoplewithdrawingandthesignalshereceives. Foranyλandξsuchthatthebanksurvivestherun,i.e.,λ≤(LIQ +ξ·I)/ (cid:0) D (cid:0) 1+rD(cid:1)(cid:1) ,equa- 1 2 tions (4) and (5) give the period 3 consumption of a patient saver who waits, c (p,I =0), and 3s w withdraws,c (p,I =1). Thedifferencebetween(4)and(5)arisesbecausethepersonwhowaits 3s w will receive a late deposit payment, while the other person will get her deposits early and transfer themtoperiod3usingtheliquidasset. Theexpectedutilitydifferentialbetweenwaitingandwithdrawingconditionalonthebanksurvivingtherunis∑ s {ω 3s (cid:0) V 3 D s (ξ,λ)−c D ·I d (cid:1) ·D· (cid:0) 1+r 3 D(cid:1) }−D· (cid:0) 1+rD(cid:1) . 2 Ontheotherhand,inarun,i.e.,forλ≥(LIQ +ξ·I)/ (cid:0) D (cid:0) 1+rD(cid:1)(cid:1) ,apatientsaverwhowaits 1 2 consumesLIQS+eS,whileapatientsaverwhoattemptstowithdrawconsumesD (cid:0) 1+rD(cid:1) +LIQS+ 1 2 2 1 eS with probability θ(ξ,λ) and LIQS+eS, otherwise. The expected utility differential between 2 1 2 waitingandwithdrawingisLIQS+eS−θ(ξ,λ)·(D (cid:0) 1+rD(cid:1) +LIQS+eS)−(1−θ(ξ,λ))·(LIQS+ 1 2 2 1 2 1 eS)=−θ(ξ,λ)D (cid:0) 1+rD(cid:1) ,whereθ(ξ,λ)=(LIQ +ξ·I)/(λ·D (cid:0) 1+rD(cid:1) ). 2 2 1 2 Overall, theutilitydifferentialbetweenwaitingandwithdrawingwhenfundamentalsareξand λsaverswithdrawisgivenbythefollowingpiecewisefunction:      ∑ (cid:8) ω 3s (cid:0) V 3 D s (ξ,λ)−c D ·I d (cid:1) ·D· (cid:0) 1+r 3 D(cid:1)(cid:9) −D· (cid:0) 1+r 2 D(cid:1) if D LI · Q (1 1 + + ξ rD ·I ) ≥λ≥δ   s 2 ν(ξ,λ)= .      − LIQ 1 +ξ·I ·D· (cid:0) 1+rD(cid:1) if 1≥λ≥ LIQ 1 +ξ·I  λ·D·(1+rD) 2 D·(1+rD) 2 2 (31) To understand the decision to run, consider an individual patient saver who receives signal x. i The agent will use the signal to update her beliefs about the realization of ξ. Given the distributional assumptions we make (both ξ and ε are uniformly distributed), the posterior distribution of i ξ given x is ξ|x ∼U[x −ε,x +ε]. This implies that the utility differential between waiting and i i i i withdrawing fora patientsaver whoreceives signalx as afunction ofthe cutoffvalue forrunning i 18

is 1 (cid:90) xi+ε ∆(x,x∗)= ν(ξ,λ(ξ,x∗))dξ. i 2ε xi−ε Consider next an agent who receives a signal equal to the threshold x∗. This agent by definition is indifferent between waiting and withdrawing, i.e., ∆(x∗,x∗) =0. The posterior distribution of λ(ξ,x∗)forthisagentisuniformover[δ,1].14 Asξdecreasesfromx +εtox −ε,λincreasesfrom i i δ to 1. Changing variables and taking the limit ε→0, which implies that x∗ →ξ ∗ , provides the ∗ indifferenceconditionthatdeterminestheuniquevaluefor ξ intheglobalgame: (cid:90) θ∗(cid:20) (cid:21) GG: ∑ (cid:8) ω · (cid:0) VD(ξ ∗,λ)−c ·I (cid:1) ·D· (cid:0) 1+rD(cid:1)(cid:9) −D· (cid:0) 1+rD(cid:1) dλ 3s 3s D d 3 2 δ s − (cid:90) 1 LIQ 1 +ξ ∗I ·D· (cid:0) 1+rD(cid:1) dλ=0 (32) θ∗ λ·D·(1+r 2 D) 2 whereθ∗= (cid:0) LIQ +ξ ∗I (cid:1) / (cid:0) D·(1+rD) (cid:1) .15 1 2 AsinGoldsteinandPauzner(2005),ourmodelexhibitsone-sidedstrategiccomplementarities, i.e., ν in (31) is monotonically decreasing in λ whenever it is positive. We refer the reader to Goldstein-Pauznerforadetailedproofofexistenceanduniquenessoftheequilibriumrunthreshold. In contrast to their setup, we obtain well-defined upper and lower dominance regions under our assumptions for the liquidation value ξ, with each patient agent’s best action being independent of her belief concerning other patient agents’ behavior. The existence of these regions is critical for obtainingarunthreshold. LD The lower dominance region is defined by a threshold ξ for fundamentals such the every individual patient depositor will run on the bank irrespective of what other patient depositors do whenξ<ξ LD . Thisthresholdisgivenbyξ LD = (cid:0) δ·D·(1+rD)−LIQ (cid:1) /I. Inotherwords, when 2 1 theliquidationvalueturnsouttobesolowthattheimpatientdepositorscannotbefullyrepaid,then thepatientdepositorswillalwaysrun. UD The upper dominance region is defined by a threshold ξ for fundamentals such that every individual patient depositor will not run on the bank when ξ > ξ UD , irrespective of what other patient depositors do. This threshold is given by ξ UD = (cid:0) D·(1+rD)−LIQ (cid:1) /I. This condition 2 1 says that the liquidation value is so high that even if everyone were to run the bank would be able topaythem. Inthatcase,runningmakesnosense. In the equilibria we consider we verify that ξ<ξ LD <ξ ∗ <ξ UD <ξ. The conditions that are neededtoestablishthetworegionsarenotveryrestrictive. Becausethereisaggregateuncertainty about the liquidation value and the loans may be worth more than their face value if liquidated, thebankwillholdfewerliquidassetsthanthepredictedwithdrawalsbyimpatientdepositors. This 14ThisistruebecauseProb(λ(ξ,x∗)≤N)=1−Prob (cid:0) ξ≤ξ∗+ε−(N−δ)/(1−δ)2ε (cid:1) =1−(ξ∗+ε−(N−δ)/(1− δ)2ε−ξ∗+ε)/(2ε)=(N−δ)/(1−δ),henceλ(ξ,x∗)∼U[δ,1]. 15Equation (32) is sufficient to guarantee that a patient saver will not withdraw if a run does not occur; only impatient savers withdraw in equilibrium. In other words, her incentive compatibility constraint ∑s (cid:8) ω3s · (cid:0) V 3 D s (ξ,δ)−cD (cid:1) ·D· (cid:0) 1+r 3 D(cid:1)(cid:9) −D· (cid:0) 1+r 2 D(cid:1) ≥0isalwayssatisfiedasitispositiveforξ∗ andincreasingin ξ. 19

establishes the lower dominance region. Moreover, if the liquidation value is high enough and/or ifthebankhassufficientequity,thenitwouldbeabletorepayalldepositorsearlywithoutrunning outoffunds. Thiswillguaranteetheupperdominanceregion. 2.5 DiscussionofModelingAssumptions Before analyzing the model’s properties, it is helpful to clarify the role that the various modifications we have made to the standard Diamond-Dybvig model play in our analysis. There are three important changes that are essential for our results and several lesser alterations that are made to simplifytheanalysisandexposition. PrivateBenefitandMonitoring. Onecriticalchangeistheassumptionthatthebankershavean outsideoptionwhichdepositorsmusttakeintoaccountinprovidingfunding. Wehaveintroduced this consideration by assuming that the realized productivity of entrepreneurial projects is private information, hence there is a need for monitoring. However, bankers are willing to monitor only whentheprofitsaccruingtothemarehigherthantheirprivatebenefits. Demandabledepositsexert disciplinebecausedepositorswouldruniftheyexpectedthattheincentivecompatibilityconstraint of bankers to be violated ex-post. These adjustments are important in generating endogenous run risk. IncompleteMarketsforAggregateRisk. Thesecondfundamentaladaptationistheassumption thatrealeconomicactivityissubjecttoaggregateproductivityriskandagentscannotwritecontingentcontractsontherealizationoftheproductiveshockinstates. Theuncontingentdebtcontracts couldbesetsothattheywouldberisklessbyrestrictingtheloanamountsothattheborrowercould repayinallstatesoftheworld. Thisisnotprofit-maximizingandinsteadthebankiswillingtotake some credit risk. In addition to the aggregate risk, the liquidation value of long-term investment isuncertain(anduninsurableanduncontractible). Thecombinationofthesemodificationscreates endogenous risk of a run. The technical assumptions about the signals regarding the liquidation valuemeansthatthe probabilityofarunisa uniquelydefinedasafunctionof fundamentals. This assumptionisalsoimportanttogenerateendogenouscreditrisk. Banker as an Agent and Banker’s Wealth. Our third important modification is the assumption that the intermediaries are run by bankers who want to maximize their own utility rather than the utility of depositors and also enjoy private benefits from operating the bank. This assumption is importanttojustifyshort-termfundingasadisciplinedeviceandgeneratedivergentincentivesdue to credit risk. We suppose that in our baseline case that bankers can earn profits. As mentioned already,itispossiblethatthebankersaresowealthythattheydesiretolendsomuchthatprofitsare driventozero. Thisreducesoneofthedistortionsinthemodel,but,asweshowinsection5,there isstillscopeforbankingregulation. There are several other modifications that we make to the Diamond-Dybvig set up that are for convenienceandarenotessentialfortheresults. Quasi-linearPreferencesandEquityFinancing. Wehaveassumedquasi-linearpreferencesfor savers and bankers to simplify the exposition of the model. Our results would also hold under 20

concave utilities in period 3 and, arguably, they would be stronger given that the stability of the banking sector would interact positively with risk-aversion. Our results go through provided that bankersarenotmorerisk-aversethansavers,sothatbankersarewillingtoinjectequityandarenot insured by savers. Nevertheless, quasi-linear preferences are important to simplify the solution of theincompleteinformationgamewhensaversareallowedtoalsoinvestinbankequityasexplained in the online appendix. To facilitate a comparison to that case we have maintained the assumption inourbaselinemodel. BankruptcyCosts. Theintroductionofbankruptcycostsessentiallygivesbanker’sanadvantage at investing in entrepreneurs’ projects and introduces a risk-sharing role of equity.16 Although assuming zero bankruptcy costs would be inconsequential for most of our analysis, the level of these costs matters when bankers have ample wealth and planning equilibria can be implemented withoneregulationasweexplaininsection5. Finally, it is not necessary that the value of liquidity provision arises only for the reasons emphasizedbyDiamondandDybvig. Wecouldchangeagents’preferencestoreducethecomplexity ofourmodel. Thefirstdrawbackofdoingsoisthatwewouldneedtointroduceanothersourceof outflowsthatthebankexperiencesintheintermediateperiodsuchthatthelowerdominanceregion in (27) is well defined. Such outflows could result, for example, from tax obligations; yet, certain assumptionsabouttheseniorityoftheseoutflowsandshort-termdebtwouldhavetobemade. The seconddrawbackisthatwewouldnotbeabletostudytheeffectofregulationonliquidityprovision, whichwouldmatterforthewelfareimplicationsofourmodel. 3 Efficient Allocations Bankers internalize how their investment and capital decisions change the probability of a run and choosethedeposittermsoptimallygiventhesupplyscheduleofferedbysavers. However,bankers may still have an incentive to take risk to exploit their limited liability and choose banking allocations that maximize their own utility at the expense of the other agents. Savers and entrepreneurs are sufficiently small to internalize how their own decisions matter for aggregate bank allocations drivingrunriskandcreditrisk. Inordertoexaminehowtheseexternalitiesdistorttheefficientallocationsweconsiderasocialplannerwhointernalizestheeffectsofinvestmentandcapitaldecisions onallagents,butstillisconstrainedbythemarketstructureoftheeconomy. Wewillshowthatthere aretwomajordistortedmarginsinbanker’sprivatedecisions.17 Section3.1setstheplanner’sproblemandderivesthesociallyefficientoptimizationmargins. Section3.2derivesexpressionsforthe distortions between the private and social optimization margins. Section 3.3 presents a numerical 16SeealsoAllen,CarlettiandMarquez(2015)whoalsointroducebankruptcycoststoendogenizethecostofequity anddepositfinanceforbanks. 17InamodelwithDiamond-Dybvigpreferencesandcompleteassetmarketsforaggregaterisk,AllenandGale(2004) showthatequilibriumallocationsunderfinancialintermediationareconstrainedefficient.Inourframework,thepresence ofincompletemarkets,incompletecontractsandlimitedliabilitymakestheassetandcapitalstructureofbanksmatterfor equilibriumoutcomesandbankrisk. Theoptimalityconditionsofasocialplannerwilldifferfromthoseintheprivate equilibriumandwelfareimprovementsarepossible. 21

solutiontothemodelanddescribeshowtheprivatelyandsociallyallocationdiffer. 3.1 SocialPlanner The social planner chooses banking assets, {I,LIQ ,LIQ }, banking liabilities, (cid:8) D,EB(cid:9) , the run 1 2 1 threshold, ξ ∗ , savers’ liquidity holdings, (cid:8) LIQS(cid:9) , and interest rates, (cid:8) rI,rD,rD(cid:9) , to maximize the 1 2 3 followingsocialwelfarefunction: Usp=w UE+w US+w UB, (33) E S B wherew ,w andw aretheweightsassignedtothethreeagents,whicharepositiveandsumupto E S B 1. Agents’utilitiesaregivenby(26),(6)and(10). Itwillbeusefulinwhatfollowstointroducesome additional notation. Define the set of the aforementioned optimizing variables as X. The planner will optimally choose variables X ∈X subject to a set of constraints B(X), which are described below,i.e.,theplanner’sproblemis max Usp X s.t. B(X)≥0. (34) The planner is constrained by the market structure of the economy, i.e., she cannot use lumpsum transfers to allocate resources across agents,18 and needs to respect: the individual budget constraints (1), (2), (3), (4), (5); the balance sheet constraints (11), (12), (13), (14); the private incentives to default, i.e., constraints (15) and (16); the banker’s incentive compatibility constraint (17); the global game constraint (32); and the fact that liquidity, deposits, equity and interest rates cannotbenegative. Moreover,theplannertakesthedepositsupplyandloandemandschedules(7) and(27)asadditionalconstraints. Yet,inprinciple,shedoesn’tneedtorespectthem,whichmeans that(7)and(27)donotneedtoholdwithequalityintheplannersolution. Forexample,theplanner could choose deposit or loan rates that do not necessarily satisfy all these conditions with equality andimplementtheresultingallocationsbychoosinginstruments,suchasPigouviantaxesoninterest income/expenses, that distorts (7) or (27).19 If these conditions do not hold with equality, then the Lagrangemultipliersassociatedwiththemarezero. Giventhatourfocusisonbankingregulation, wewillimposetheprivatedepositsupplyandloandemandschedulesasequalitiesintheplanner’s problem. Hence, in our baseline analysis the planner is essentially choosing a set of allocations that need to satisfy the pricing equations given by the deposit supply and loan demand schedules. We relax this assumption in the online appendix and show that our conclusions on the need for 18Giventheabsenceoflump-sumtransfers,wecannotunambiguouslyconstructawelfarecriteriumtomaximizethe totalsurplus.Thus,weassignweightsfordifferentagentsinasocialwelfarefunctionandstudydifferentconstellationsof theseweights.Althoughweremainagnosticabouttheoriginofsuchweights,wediscussthepotentialpoliticaleconomy considerationsofregulation. 19FarhiandWerning,2016,andBianchiandMendoza,forthcoming,considersuchtaxestoimplementtheconstrained efficientallocations.Notethatthetaxescanalsotakenegativevalues,inwhichcasetheyareinterpretedassubsidies. 22

banking regulation continue to hold. To summarize, the set B(X) includes constraints (1)-(5), (7), (9),(11)-(17),(25),(27),(28),(32). We report the planner’s first order conditions in a compact form, because the detailed expressionsarelongandnotparticularlyenlightening. Thefirst-orderconditionwithrespecttoavariable X ∈Xwill,ingeneral,takethefollowingform: dUh dBS dIC dGG dLD dDS ∑ w +ζ +ζ +ζ +ζ +ζ =0, (35) h dX BS dX IC dX GG dX LD dX DS dX h={E,R,B} where ζ , ζ , ζ , ζ , ζ and ζ are the multipliers on (11), (17), (32), (27) and (7), respec- BS IC GG LD DS ES tively. Thefirsttermin(35)captureshowvariableX mattersfortheweightedutilitiesofagentswhere w¯ =w +ζ and ζ , h={E,S,B} are the multiplier on E’s, S’s and B’s participation conh h PC,h PC,h straints given by (28), (9) and (25), respectively. The second, third and fourth terms capture the effect of variable X on the balance sheet, the banker’s incentive compatibility and the global game constraints,whilethelasttwotermscapturehowvariableX changestheloansdemandanddeposit supplyschedules.20 3.2 PrivateversusSocialdecisions Inthissection,wecomparetheallocationschosenbyprivateagentstotheefficientallocationschosenby thesocial plannerabove. We identifytwo distortedmarginsof optimizationin thebanker’s private decisions; first, a distorted asset mix, and, second, a distorted liabilities mix. The former captures the way the banker and the planner choose between investing in the risky loans or the liquidasset. Thelattercapturesthechoiceoffundingusedforinvestment.21 Toseewhytherearetwoindependentintermediationmargins,observethatthebanker’soptimizingbehaviorintheprevioussectionyieldssevenoptimizingconditionfor{I,LIQ ,E,D,rD,rD,ξ ∗}. 1 2 3 Moreover,therearefourLagrangemultipliers{ψ ,ψ ,ψ ,ψ }associatedwithfourconstraints. BS IC GG DS Giventhattheconstraintsbind,asisthecase,onecanusefouroftheoptimizingconditiontodeterminethemultipliers. Inparticular,butnotexclusively,use(18)todetermineψ ,(20)todetermine IC ψ , (24) to determine ψ , and (21) to determine ψ . Moreover, use the four constraints to pin GG DS BS downI,rD,E andξ ∗ ,and(23)topindownrD,asfunctionofLIQ andD. Thelattertwovariables 3 2 1 are determined by (19) and (22). Alternatively, we could have expressed everything in terms of functions I and E – or in fact in terms of any other combination of one of the liabilities and one of the assets. So a natural way to think of the two “free" banking choices is that the proportions of liquidtoilliquidassetsanddepositstoequityarethecriticalendogenousobjectsinthemodel. The samelogicappliestosolveforthefreevariablesintheplanner’sproblem.22 20Notethatthemultipliersintheplanner’ssolutionaredenotedbyζratherthanψintheprivateequilibrium,because thetwowillbedetermineddifferently. 21Athirddistortedintermediationmarginarisesifweallowsaverstoalsobuybankequity(seetheonlineappendix). 22IntheexpressionsbelowtheLagrangemultipliersareconsideredtobeattheirequilibriumvaluesandarenotsub- 23

The asset mix distortion is derived by combining the investment and liquid asset optimality conditionsofthebanker, (18)and(19), andoftheplanner, equation(35)forX =I andX =LIQ , 1 respectively. Thebanker’sinvestment-liquiditymargin,ILIQ ,is,then,givenby B dUB dUB (cid:20) dIC dIC (cid:21) (cid:20) dGG dGG (cid:21) (cid:20) dDS dDS (cid:21) − +ψ − +ψ − +ψ − ·I =0. dI dLIQ IC dI dLIQ GG dI dLIQ DS dI dLIQ c 1 1 1 1 (36) In contrast the socially optimal investment-liquidity margin, ILIQ , will include additional sp termscapturinghowbankingdecisionsalsoaffectsaversandentrepreneurs: (cid:20) dUh dUh (cid:21) (cid:20) dIC dIC (cid:21) (cid:20) dGG dGG (cid:21) ∑ w − +ζ − +ζ − h dI dLIQ IC dI dLIQ GG dI dLIQ h={E,R,B} 1 1 1 (cid:20) (cid:21) (cid:20) (cid:21) dDS dDS dLD dLD +ζ − +ζ − =0. (37) DS dI dLIQ LD dI dLIQ 1 1 Wecangroupthedifferencesbetweenthebanker’sandtheplanner’smargininthreecategories. First, the planner considers the direct effect of a portfolio shift from liquid asset to risky loans on weightedsocialwelfareratherthanononlythewelfareofthebanker. Thiscanbeseenbycomparing thefirsttermin(37),∑ h w¯ h [dU h /dI−dU h /dLIQ 1 ],tothefirsttermin(36),dU B /dI−dU B /dLIQ 1 . Second,theplannerconsidershowthewelfareofallagents,notonlyofthebanker,mattersforthe level of multipliers on constraints GG, IC and DS.23 In other words, the planner internalizes how picking the run threshold, how relaxing the banker’s incentive compatibility constraint and how moving along the deposit supply curve also affects entrepreneurs’ and savers’ welfare. Third, the planner internalizes how the choice of investment and liquidity affects the deposit supply and loan demandschedules. Thebankeronlypricestheeffectofinvestmentandliquidityondepositsupply whendepositcontractsarecomprehensive,i.e.,I =1. c Forfurtherreference,denotethesumofthesedistortionsintheinvestment-liquiditymarginby ILIQ ,suchthat wedge ILIQ =ILIQ +ILIQ . (38) sp B wedge Theliabilitiesmixdistortioncanbederivedsimilarlybycombining(21)and(22)fortheprivate margin,andequations(35)forX =E andX =Dfortheplanner’smargin. Wecallthistheequitydeposits margin, ED. The differences between the private margin, ED , and the social margin, B ED , fall in the same categories described above for the investment-liquidity margin. A subtle sp distinction is that the banker internalizes the effect of deposit taking on deposit supply even when stitutedoutfollowingthestrategyoutlinedabove.Wereportinanonlineappendixtheintermediationmarginsexpressed onlyintermsofallocationssuchthatthemultipliersaresubstituted. Theseexpressionareconvolutedanddonotprovideadditionalintuition. Thus,wehaveoptedtopresenttheintermediationmarginswithoutsubstitutingthemultipliers herein. 23Forexample, themultiplierψ in(36)onlydependsondUB/dξ∗, whilethemultiplierζ in(37)dependson GG GG ∑h w¯ h dU h /dξ∗.Thiscanbeseenfromtheoptimalityconditionsfortherunthreshold,(20)and(35)forX=ξ∗.Similarly fortheothertwomultipliers. 24

deposit contracts are incomplete. For further reference, denote the sum of these distortions in the equity-depositsmarginbyED ,suchthat wedge ED =ED +ED . (39) sp B wedge In the next section we present a numerical example of the private and planning equilibria and discusshowtheallocationsdifferinreferencetotheaforementionedintermediationmargins. 3.3 Numericalexample The full set of parameters we used to solve the model is shown in Table 1. The parameterization should be taken more as an illustrative example to highlight the mechanisms in the model rather than as a realistic calibration of the economy attempting to make quantitative statements about the absoluteoptimallevelofbankingregulations. Wehaveexperimentedwithvariousotherparameter choicesandthefindingsthatweemphasizearequiterobust. Ourmodelwouldrequiresomeobviousmodificationstouseitforquantitativepolicyanalysis. Forexample,allliabilitiesinourmodelareunsecured,whileinpracticecertaintypesofdepositsare insured. Deposit insurance, even partial, would reduce the market discipline exerted by depositors andhencethecreditriskpremiaindepositratesbringingthemclosertowhatisobservedinreality. Moreover, it is not clear whether the various capital regulations in practice (Basel requirements, stress tests, restrictions on dividend payouts) are indeed binding and whether one should be calibrating to match a regulated economy rather than an unregulated private equilibrium. Finally, the assumptionoflinearityofutilitiesinthirdperiodconsumption,whichsimplifiesthecomputationof therunthresholdsignificantly,aswellasthefinitehorizonofthemodelmakedepositorswillingto acceptahigherprobabilityofaruniftheywererisk-averseoriftherewasacontinuationvaluefor thebank. Onecouldaddconvexbankruptcycoststomimicahigherdegreeofrisk-aversionaswell as model the continuation value, but we have not done so because it is not important to make our fundamentalanalyticpoints. Withthesecaveatsinmind,letuscallattentiontosomeoftheconsiderationsthatwetookinto accountwhilechoosingthemodelparameters. First, the probabilities of default and losses given default will determine the amount of default risk that the bank is facing. We opt to have entrepreneurs and banks default in the bad state irrespectiveoftherealizationoftheliquidationvalueintheintermediateperiod. Second, the bank is profitable enough, and the initial equity of the banker and her preference for current consumption are such that she voluntarily uses some of her endowment to buy more equityinthebank. Inourbaselineequilibrium,thebankerenjoysapositiveeconomicsurplusfrom intermediating. In section 5 we examine a case that the banker has sufficient initial wealth so that she invests in the bank up to the point that the economic surplus accruing to her is driven to zero, i.e., she enjoys the same utility as in autarky. We believe that this is not realistic, but describing this case isstill useful to highlightthat the justification forbanking regulation is not tocapture the 25

economicsurplusofbankers,butrathertoimproveallocativeandproductiveefficiency. Third, the liquidity provision by the bank leads savers not choose to additionally self-insure by holding the liquid asset. When savers self-insure, the banking sector is under-performing as a providerofliquidityand,hence,intermediation,andregulationsthatmakebanksmorestablewould have an additional positive effect. In our baseline parameterization we want to mute this channel and make it harder for regulation to improve economic outcomes. However, our results hold even whensaversself-insureintheprivateequilibrium. Fourth,wehavechosentheparameters,amongthem,mostimportantly,theliquiditypreference shock, the distribution of the liquidation value and risky technology payoffs, such that the bank holds a portfolio of both liquid assets and risky loans, and also liquidates part of its risky holdings toserveearlywithdrawals. Fifth,wehavechosenlogarithmicutilityforperiod1andperiod2consumption,whilewespecializetheproductionfunctiontobeF= (cid:0) I+IE(cid:1)α (cid:96)1−α= (cid:0) I+IE(cid:1)α ,withα<1andentrepreneurial skills’supplynormalizedto(cid:96)=1.24 Beforepresentingthemodelsolutionandexplaininghowtheprivateandsocialequilibriadiffer, webrieflydescribesomeregulatoryratiosandriskmetricsthatwehaveconstructedtofacilitatethe analysis. The capital adequacy ratio (CR) is equal to the value of equity divided by the level of risky loans. Wehavenormalizedtherisk-weightonloanstoone,whileliquidassetsreceivearisk-weight ofzero: E CR= . (40) I Theleverageratio(LevR)includesboththeriskyandliquidholdingsandisgivenby: E LevR= . (41) I+LIQ 1 Theliquiditycoverageratiotakesthe(lowest)liquidationvalueofthebank’sportfolioinarun relativetorunnableliabilities:25 LIQ +ξ·I 1 LCR= . (42) D· (cid:0) 1+rD(cid:1) 2 Finally,wecomputeanetstablefundingratiowhichiscomputedasthefractionilliquidassets 24TheoriginalDiamond-Dybvigframeworkrequirestherelativerisk-aversioncoefficienttobehigherthanone. This isnotnecessarywhentheliquidationvalueoflong-terminvestmentcanbelowerthanoneaspointedoutbyCooperand Ross(1998). Inaddition, theshareofincomefortheriskytechnologyaccruingtoentrepreneurialhumancapital(set to0.25)ischosentoreflectestimatesfromtheliterature. Gollin(2005)findsthattheshareofprofitsinentrepreneurial activitiesis0.10.Therestistheshareoflaborandcapital.Inoursetting,laborfromworkersisnotmodeled,andweare interestedintheshareoftheremainingoutputwhichisdistributedtoentrepreneursandsuppliersofcapital. Settingthe shareofcapitalrelativelytolaborto0.30,whichisstandardintheliterature,givearelativeshareforentrepreneurialand capitalprofitsof0.1/(0.1+0.9·0.3)=0.28and(0.9·0.3)/(0.1+0.9·0.3)=0.72,respectively. 25Itisnotobviouswhethertocounttheportionoftheloansthatarealwaysavailableasbeingliquidornot. Ourresults areverysimilarifweexcludethemfromthenumeratorofthisregulation. 26

fundedbyrelativelystablesources: E+(1−δ)·D NSFR= . (43) I (cid:16) (cid:17) Moreover, the probability of a run is computed as q= ξ ∗−ξ /∆ and can be further disag- ξ gragated into a fundamental-driven and a panic-driven component. The probability that depositors (cid:16) (cid:17) runonlybecausefundamentalsturnouttobebadisqf = ξ LD−ξ /∆ . ξ Finally, we compute a measure of the liquidity provision delivered by the bank. As already mentioned, savers expected utility must be higher than in autarky. However, the bank can make thishappenindifferentways. Forexample,itcouldofferhighercompensationforpatientsaversin exchangeforlowerliquidityprovisiontoimpatientones. We,thus,separatelycomputetheexpected utilityofimpatientsaverswhenthebankintermediatesrelativetotheirutilityinautarkyasameasure ofliquidityprovision: (cid:82)ξ∗ E U (c (i,I );i) 1 dξ+ (cid:82)ξU (c (i,1);i) 1 dξ Liq.Prov.= ξ θ 2 2 θ ∆ ξ ξ∗ 2 2 ∆ ξ . (44) (cid:16) (cid:17) U eS+LIQS,aut ;i 2 2 1 Table 2 reports the equilibrium values of some main variables of interest along with the computedmetricsfortheprivateequilibrium(PE),andthesocialplanner’ssolutionfordifferentweights onE,S,andB.Wehavesetbanker’sweightto0.2andalsosetalowervalueof0.2fortheweights oftheothertwoagents. Aswewillexplaininmoredetaillater,thischoiceisnotimportantforthe generality of the results. We have normalized the utility of all agents in the private equilibrium to onewhenwecomputethewelfarechange. %∆Usp and%∆Ssp arethepercentagechangeinsocial welfaregiventheweightsandthechangeintotal(unweighted)utilityfromtheprivateequilibrium, respectively. Note that we report two types of private equilibria in Table 2: one where the funding contracts with depositors are incomplete, and another where the banker can write comprehensive contracts. Westartbydiscussingthefirstcase,andexaminethesecondattheendofthissection. We focus the analysis around the two intermediation margins derived in section 3.2. First, the plannercorrectsthedistortionintheassetmixbetweenriskyloansandliquidassets. Duetolimited liability and incomplete contracting the banker has an incentive to tilt her portfolio towards risky loans, which have a higher payoff in the states where B is solvent, while fully internalizing the effectoftheassetmixontherunprobabilityandthestabilityofbankingprofitsaccruingtoher. The planner also internalizes the effect of the bank’s choices on savers and entrepreneurs and chooses a more liquid asset mix. This can be seen from the big increase in liquid assets in the planner’s solutionforallweightsandthehigherliquiditycoverageandnetstablefundingratios. The funding mix is also distorted. The banker prefers to fund herself with deposit rather than equity because by levering up she can exploit her limited liability. Also, deposits carry a liquidity premium, in that the deposit rate reflects more than credit risk and time-preference. The banker herself has no preference for liquidity given that she doesn’t want to consume in the intermediate 27

period and would like to extract any liquidity premium by using deposit funding. The planner insteadprefersmoreequityfundingandamorestablecapitalstructure. Hence,theplannerchooses highercommonequityandhighercapitaladequacy. Bydoingso,theplanner’sallocationsallowthe banktooperatewithalargerbalancesheet. Despiteahighercapitalratio,thebankoperateswitha lowerleverageratioanddeliversitsintermediationservicesmoreefficiently. The more stable asset and funding choices of the planner come from the desire to reduce run risk. Theprobabilitythatarunoccursdropsintheplanner’ssolution,asdoestheprobabilityoffundamentalruns. Overall, thisimprovesthestabilityofthebankingsectorandenhancesthestability ofrealeconomicactivityastheentrepreneurseesherfundingbeingwithdrawnlessfrequently.26 Moreover, the planner reduces maturity transformation in relative terms, which can be seen by the higher risk weighted capital and net stable funding ratios. However, the reduction in maturity transformation is not necessarily accompanied by a drop in the level of credit extension. For sufficientlyhighw ,theplannerchooseshigherinvestmentthanintheprivateequilibrium. Indeed,we E show that increasing the equity in the bank, without requiring more liquidity, leads to more credit extension through a combination of channels described in detail in sections 4.1 and 4.2 where we studycapitalandleveragerequirementsinisolation. Onthecontrary,creditextensionislowerwhen thesaverisfavoredandtheplannerforcesthebanktoholdmoreliquidassets. Overall,privateallocationscanexhibitbothunder-andover-investmentindicatingthatthesourceofinefficiencyisnot thetotalleveloflending,butitsrelationtotheholdingofliquidassetsandthefundingstructureof thebank. Liquidity provision is also higher in the planner’s solution. In fact, liquidity provision as measuredin(44)isbelowoneintheprivateequilibrium,whichsuggeststhatthebankeroffersattractive enough returns to patient depositors to induce them to accept lower utility when they turn out to beimpatient–thedeposittermsstilldeliverhigheroverallutilitythaninautarky. Thismaynotbe surprisinggiventhatthebankerfavorslendingoverholdingliquidity,sothebankfindsitmoredifficulttoprovideinsurancetoimpatientsavers. Insteadtheplannerdeliversmoreliquidityprovision to impatient depositors without sacrificing long-term returns to patient savers.27 This is possible becausetheplanner’schoicesleadtoalargeroveralllevelofintermediation. Theenhancedstabilityofboththeassetportfolioandcapitalstructureofthebankarebeneficial 26Thefactthatpanic-drivenrunsoccurwithnon-negligibleprobabilitysuggeststhatgovernmentguarantees,suchas depositinsuranceorimplicitbailoutsubsidies,maybeusefulpolicyinterventions. Wehaveabstractedfromintroducing government guarantees in the model for two reasons. First, it would not unambiguously improve outcomes as in the originalDiamond-Dybvigset-upbecauseofrisk-takingincentives(see,forexample,KarekenandWallace,1978,Cooper and Ross, 2002, Admati et al., 2012). Second, designing deposit insurance when runs have both a fundamental and panicriskcomponentisfarfromstraightforward. Suchanexerciseisnottrivialandisbeyondthescopeofthecurrent paperwhichaimstoidentifythebankingexternalitiesarisingfromincompletecontractingforcreditandrunrisk(Allen etal.,2015,studygovernmentguaranteeswithinaglobalgamesframeworkandasimplerbankingsectorthattheonein ourpaper).Inthesametoken,wedonotstudyemergencyliquidityassistancefromaLenderofLastResort(Rochetand Vives,2004)orsuspensionofconvertibility(EnnisandKeister,2009),whichwouldalsorequirenon-trivialmodifications inthemodelwepresent.SeealsoKeister(2015)forananalysisofefficientbailouts,whichshouldbecomplementedwith prudentialregulation.Webelievethattheseareimportantavenuesforfutureresearchinmodelsthatfeatureanelaborate bankingsectorsubjecttobothcreditandrunrisklikeours. 27Thoughnotshowninthetable,theutilityofthepatientsaversishigherthanintheprivateequilibrium. 28

tosaversandentrepreneurs. However,thebankerisworse-off;inourmodelthebankerinternalizes all the effects that matter for her welfare and optimally chooses more risk to maximize her own utility. Theplannercaresabouttheexternalitiestotheotheragentsaswellandweakenstheability of the banker to take advantage of her limited liability. In fact, the banker’s welfare always drops in the planner’s solutions and drives the banker to her participation constraint, unless the weight on the banker is (unreasonably) high. We do not explore such planning equilibria because they cannot be implemented with banking regulation, which always drives bankers utility down, as it will become more clear in section 4. Nevertheless, the planner not only increases social welfare (Usp), which depends on Pareto weights, but also the overall surplus in the economy, which is captured by the change in Ssp. The planner could improve the welfare all agents if she had access toare-distributive,non-distortionary(lump-sum),taxsystemtotransferresourcesacrossagents. In section5,weexamineequilibriawhereaverywealthybankerinjectsenoughequityintothebankto drivetheeconomicprofitstozero. Inthiscase,weshowthataParetoimprovementovertheprivate equilibriumispossible. Finally,theplannerchoosesdifferentlevelsofcapitalandliquiditydependingonwhichagents she favors most. When the weight on the entrepreneur is higher, the planner chooses more capital to support higher lending. This is beneficial for the entrepreneur, but results in lower liquidity provision, whichisrelativelybadforsavers. Onthecontrary, whentheweightonsaversishigher, theplannerchoosesamoreliquidassetmix. Weshouldnotethatneitherthebankernortheplannerholdexcessliquidity,i.e.,LIQ (ξ,δ)=0 2 for all ξ. Holding excess liquidity could be desirable in order to eliminate the probability of a run altogether. If the liquidation value of the bank for the lowest possible realization of ξ was higherthatthetotalrunnabledepositobligations,i.e.,LIQ +ξ·I≥D(1+rD),thenonlyimpatient 1 2 depositors would withdraw. The excess liquidity carried over to period 3 would then be LIQ = 2 LIQ −δ·D(1+rD) ≥ (1−δ)D(1+rD)−ξ·I. Such run-proof equilibria may not be desirable 1 2 2 when the lowest liquidation value of long-term investment is small or when savers are not very risk-averse. Afairamountofliteraturehasfocusedonrun-proofequilibria,whichnaturallyrestrict creditintermediation(seeCooperandRoss,1998,EnnisandKeister,2006,DiamondandKashyap. 2016). Run-proof contracts require certain assumptions to be optimal and our work has, instead, focusedonoptimalpolicyinthepresenceofbothrunriskandcreditrisk. Beforeturningtotheimplementationoftheplanningoutcomes,wediscusstheprivateequilibriumwhendepositcontractsarecomprehensive,i.e.,I =1inthebanker’soptimizationconditions. c TheresultsarereportedinthethirdcolumninTable2. Comprehensivedepositcontractsallowthe bankertocommit,butshestillchoosesthelevelsofriskthatimprovesherownutilityanddoesnot internalize all the effects on other agents. Acting in her own interests, the banker still chooses a riskier asset and liabilities structure than a planner would. The social planner instead internalizes the effects of all decisions on all agents utility, and choose a less risky asset mix and liabilities structure. The gapbetween the socialand private equilibriumgrows with commitment, because in thiscasethebankeroffersacontractthatisevenmoreskewedtowardsherinterestsattheexpense 29

of the other agents. Indeed, as the table shows the banker’s choices result in lower welfare for entrepreneurs and savers compared to the private equilibrium with incomplete deposit contract, and theparticipationconstraintofsaversstartsbinding. 4 Regulation We now explore how the planner’s solution can be decentralizing via various regulatory interventions, which tighten the regulatory ratios (40)-(43). Sections 4.1-4.4 discuss the effects when the toolsareusedinisolation. Section4.5discusseshowtheregulationscanbeoptimallycombinedto implementtheplanner’ssolutionasaprivateequilibrium. Table3reportstheresultsforthevarious regulations. 4.1 CapitalRequirements Capital regulation requires the bank to hold a certain percentage of equity for every unit of risky loansextendedanditformallyamountstoincreasingCRinequation(40). Mandatinghighercapitalrequirementsreducestheabilityofthebankertotakeriskthroughdepositfunding. Inmodelswherethebankcannotraiseadditionalequity,strictercapitalrequirements (mechanically)result ina dropin creditextension (see, for example, Corbae andD’Erasmo, 2014, Clerc et al. 2015 and the references therein). More generally, one could allow banks to raise both equityanddeposits. Then, capitalregulationhasaneffectiftheModigliani-Millertheoremisviolated. Despiteabstractingfromanytaxadvantagesofdebt,whichisthemostcommonviolationput forward, our environment breaks Modigliani-Miller in various ways, even holding the probability thatarunoccursconstant. First,depositscarryaliquiditypremium.28 Second,theavailableequity capitalisnotperfectlyelasticallysuppliedand,from(21),thebankerrequiresahigherreturnifshe contributes more equity. Third, default is costly and there are positive bankruptcy costs. The first two frictions push for higher cost of funding, while the last for lower, when capital requirements increase. Although the overall partial equilibrium effect, fixing the probability of a run and the liquiditypremium,seemsambiguous,itisplausibletosupposethatthethreeforcesresultinhigher fundingcostsandlowerlendingforlowenoughbankruptcycosts. Butourmodelfeaturesadditionalchannelswhichpushuplending. Thefirstimportantchannel is that higher capital reduces the probability of a run. This makes savers more willing to make deposits and the entrepreneurs more inclined to borrow.29 Second, substituting equity financing for deposit financing on the margin allows the bank to hold less liquidity to serve the impatient households. This would incrementally free up resources to be invested in risky loans. Finally, the 28SeeVandenHeuvel(2008)andHanson,KashyapandStein(2011)forestimatesoftheliquiditypremiumforbank deposits. 29In a model where the bank is funded only with deposits, Ennis and Keister (2006) show that shifting the asset mix towards more illiquid loans would result in a lower probability of being repaid given that a run occurs, which counterbalancestheincreaseincreditextensionwhentheprobabilityofarundecreases.Thisdoesnotneedtobetruein ourmodel,becausetheincreaseinthecreditextensionisfundedbymorecapital. 30

reduced demand for deposits suppresses incrementally the deposit rate, other things equal, due to impatientsavers’liquiditydemand.30 Accounting for all these considerations, lending rises when capital requirements increase.31 There are other noteworthy general equilibrium effects that also arise. For example, the cost of funding decreases which also allows for lower loan rates.32 Moreover, the lower probability of a run allows for more deposit taking, which pushes the deposit rate up; the banker continues to try to take advantage of her limited liability and funding higher investment exclusively with equity is expensive. Finally,capitalrequirementsdonotnecessarilyneedtoresultinlowerliquidityholdings. Theopportunitycostofliquidatingriskyloansincreasesbecausetheloansarefundedwithamore expensivesourceoffinancingonthemargin. Hence,thebankchoosestoliquidateasmallerportion ofitsloanportfoliointheintermediateperiodforanyrealizationoftheliquidationvalue(thoughthis is not shown in Table 3). This result is why the holdings of liquid assets also increase with higher capitalrequirements,butthebankstillholdslessliquiditythattheplannerwould. Thisimpliesthat capitalandliquidityregulationcouldbeoptimallycombinedasweshowinsection4.5. Although credit extension goes up, the regulation has two important implications. First, the higher credit extension needs to be funded exclusively with more equity. Second, liquid asset holdings also rise. As we will discuss in the following section, this is not the case under the risk-insensitive leverage regulation, because that type of rule allows the banker to offset lending increaseswithreductionsinliquidassets. Onnet,runriskfallsbecausethebankusesalowerpercentageofdepositfunding. Thereduction inrunriskisbeneficialforsaversandentrepreneurs. Thelevelofdepositsneednotfall,becausethe bankissaferoverallandhenceitstotalbalancesheetgrows. Thismeanthatliquidityprovisioncan bemaintained. However, thebankerismadeworse-offwithhighercapitalrequirements, whichis notsurprisinggiventhatshecouldhavevoluntarilychosenmorecapitalifitwasbeneficialforher. Ofnote,thelevelofcapitalrequirementsthatmaximizessocialgainswithoutviolatingbankers’ participationconstraints, ishigherthanthecapitalratiointheplanner’ssolution. Aswediscussin section 4.5, the planner uses multiple tools to implement the socially optimal allocations. If a regulator is limited to one tool, that tool must be used more aggressively than if several can be deployed,soit’svaluewill“over-shoot”thelevelthataplannerwillpick. Asweseebelow,thisis trueforallthetools. 30Begenau(2015)alsoshowsinarealbusinesscycleframeworkthatthefallinthedepositrate,whencapitalrequirementsincreaseandsaversvaluetheliquidityservicesofdeposits,canpushtheoverallcostoffundingdownandresultin highercreditextension. Thestrengthofthismechanismismitigatedwhensaverscanalsopurchasebankequity(seethe extendedmodelintheonlineappendix). 31Wehaveexperimentedwithseveralversionsofthemodelandparameterizationsandthisconclusionisveryrobust. 32ThefirstorderconditionwithrespecttoE,(21),becomesψ =γ·U(cid:48)(eB−E)−ψ .Althoughmoreequitypushes BS CR thecostoffundingup,asmeasuredbythebankersmarginalutility,theLagrangemultiplieronthecapitalrequirement, ψ ,operatesintheoppositedirectionandinequilibriumitdominates. CR 31

4.2 LeverageRequirements Leverage regulation ties the level of capital to the overall size of the bank’s balance sheet and it formallyamountstodecreasingLevRinequation(41). Leverage requirements operate through the same channels as capital requirements, i.e., they reduce the ability of the banker to take risk through deposit funding. Overall, they push credit extension up, but there is a critical difference. Risk-weighted capital regulation requires banks to holdmorecapitalonlyagainstriskyloans,whileleverageregulationmandatesmorecapitalagainst allassetsand,hence,itdoesnotdirectlyaffectthemarginalchoicebetweeninvestmentandliquidity. Althoughthebankerisrequiredtooperatewithasaferliabilitystructure,shecantilttheassetmixto reduceliquidassetsandraiselending. Asaresult,creditextensionshouldincreaserelativelymore fromtighteningtheleveragerequirementsthanfromraisingrisk-weightedcapitalrequirements. Theincreaseinassetilliquidityincreasesfundamentalrunrisk. Thedropinliquidholdingsand lowerdemandfordepositsmakessaversworse-off,whoarepushedtotheirparticipationconstraint forsmallchangesintheregulation. Entrepreneursaremarginallybetter-offduetothehighercredit extension, but the drop in social welfare suggests that leverage requirements would not be used in isolation in this economy. This conclusion is not robust to some reasonable modifications of the environment–seeforexampletheextensionintheonlineappendix. However,eveninthemodelas itstands,belowweshowthatleverageregulationcanbecombinedwithotherregulationstoimprove economicoutcomes. Aswithalltheregulationsweconsider,thebankerisworse-off. 4.3 LiquidityRequirements A liquidity-coverage-ratio regulation requires that the immediately available funding for the bank isatleastacertainpercentageofrunnabledebt(deposits)and,inourmodel,itformallyamountsto increasingLCRinequation(42).33 Mandating that the bank holds more liquidity changes the trade-off between investing in risky loans and liquid assets, since liquid assets count fully towards this regulation and loans do not. Lookedatinisolation,thisregulationreducestheincentivetofundloansthroughdeposits. Liquidityrequirementsaregoodtoolsforraisingliquidityandreducingcreditextension. However, they erode bank profitability and make it harder to raise equity. The amount of equity falls as does the capital adequacy ratio (despite the decrease in credit extension). The bank switches to more deposit financing to compensate for lower equity financing and leverage is higher than both thePEandSPoutcomes. Overallthisregulationhastheabilitytoreduceriskontheassetside,but itresultsinhigherriskontheliabilityside.34 33We focus on a form of liquidity coverage regulation, but the results in this section hold more generally for other types of liquidity regulation, such simple restrictions on the ratio of liquid to illiquid assets (LIQ /I) or reserve ratio 1 requirements(LIQ /D). 1 34Theliteraturehasstudiedadditionalmarketfailuresthatjustifytheregulationofbanks’liquidity.InAllenandGale (2004)andDiamondandRajan(2011)theneedforpolicyinterventionstemsfromthepresenceoffire-sales,inFarhi, GolosovandTsyvinski(2009)liquidityregulationtacklesinefficientrisk-sharingduetohiddentrades,whileDiamond andKashyap(2016)showthatliquidityrequirementsareimportanttodeterrunriskwhendepositorshaveincomplete 32

However, the lower probability of runs is beneficial to both savers and entrepreneurs, and the high level of deposits improves liquidity provision. Yet, the decrease in credit extension results in higher benefits for savers than for entrepreneurs compared to the planner’s solution, while the oppositeholdforcapitalandleverageregulations. Finally,thebankerseesherwelfaregoingdown forthesamereasondescribedbefore. 4.4 NetStableFundingRequirements Thistypeofregulationrequiresthatthebankfundsacertainpercentageofilliquidassetswithlongterm,stablesourcesoffinancing,whichinourmodelareequityandtheportionofdepositsthatwill notbewithdrawn. Formally,theregulationsetsahighervalueforNSFRinequation(43). The effects of NSFR regulation are parallel to the LCR. Investment goes down, liquidity improves,andthebankreliesmoreondepositfundingratherthanequitycapital. Similarly,theprobabilityofbankrunsgoesdown. A generalized version of the NSFR could be calibrated to look more like capital or liquidity regulation. Suppose that the relative weights on equity capital and stable deposits that appear in the numerator of the NSFR could vary. Figure 1 shows the change in credit extension for this generalized version of the net stable funding ratio where equity and long-term deposit funding are weighteddifferently,i.e.,NSFR=(E+w·(1−δ)·D)/I with0<w≤1. Dependingontheweight on deposits, an increase in the NSFR would resemble more closely the effects of capital versus liquidity regulations. In particular for a low weight on deposits, the NSFR results in higher credit extension similarly to capital requirements, while for higher weights credit extension decreases similarlytoliquidityrequirements. 4.5 OptimalRegulatoryMix This section examines whether and how regulation can be combined to implement the social planner’ssolutionasaprivateequilibrium. Thesocialplannersolvesforallocationswithouttakinginto consideration how the optimal behavior of the bank will change, or in other words the first-order conditions of the banker (adjusted for regulatory interventions) are not taken as additional constraintsin(34)–thoughthebanker’sparticipationconstraintmustbesatisfied. Hence,theplanner’s allocationsarecomputedwithouttyingtheplannertospecifictools. Thiswaywehavebeenableto clearlyidentifythedistortedmarginsbetweentheprivatelyandsociallyoptimaldecisionsinsection 3.2. Therestofthesectionshowshowtheregulatorytoolsstudiedabovecanbecombinedtocorrect forthedistortedbankingdecisionsderivedasthewedgesinconditions(38)and(39). Forthatpurposeweset-upanaugmentedplannerendowedwithcertainregulatorytools. LetT betheavailablesetofregulatorytoolswhichwillincludeatleastthefourregulationsstudiedabove and possibly others. For each T ∈T there is a regulatory constraint RC(T,X)≥0, which ties the tool with the endogenous variables X ∈X (for example, constraints (40)-(43)). It is important to informationaboutbanks’assetportfolios.Finally,Calomiris,HeiderandHoerova(2015)discusshowliquidityregulation canprovidetherightincentivesformanagingrisksunderdepositinsuranceguarantees. 33

note that the regulatory constraints are defined as inequalities, i.e., the planner can tighten them, butnotloosenthem. Letψ bethemultipliersthatthebankerintheprivateequilibriumassignsto T constraintRC(T,X)≥0. Underregulation,theoptimizationmarginschangeto: (cid:20) (cid:21) dRC(T,X) dRC(T,X) dRC(T,X) ILIQT:ILIQ B +∑ψ T dI − dLIQ A ILIQ + dE (1−A ILIQ ) =0, (45) T 1 (cid:20) (cid:21) dRC(T,X) dRC(T,X) dRC(T,X) EDT:ED B +∑ψ T dE (1+A ED )− dD + dI A ED =0, (46) T whereA andA aregivenby(A.9)and(A.11)intheonlineappendix. ILIQ ED The tools-augmented planner’s problem, akin to a Ramsey planner’s problem in the public financeliterature(see,forexample,LucasandStokey,1983),isderivedinanonlineappendix.35 To implementtheequilibriumallocationsofthesocialplanner,denotedbyXsp,theavailabletools,T, havetobechosensuchthat,first,XspsatisfytheregulatoryconstraintsRC(T,Xsp)=0,and,second, theintermediationmarginsintheassociatedequilibriumarethesameastheintermediationmargins of the planner. Essentially, this means that the additional terms in (45) and (46) need to equal the wedgesderivedin(38)and(39). Inmatrixform,thiscanbewrittenas: ∆RC·Ψ=WD , (47) sp whereΨistheTx1vectorofthemultiplierontheTregulatoryconstraints,WD isthe2x1vector sp ofthewedgesinthetwointermediationmarginsevaluatedattheplanner’sequilibriumvalues,and ∆RC is the 2xT matrix of the partial derivatives of the relevant variables for each intermediation marginontheTregulatoryconstraints. Thesederivativesarealsoevaluatedattheequibriumvalues forthevariablesXsp andforthelevelsofthetoolsTsp,whichimplicitlysolveRC(Tsp,Xsp)=0. Hence, it suffices to find two regulatory tools such, first, the matrix ∆RC is invertible, and, second, all elements in Ψ are positive. We will now explain these conditions in more detail and providetheunderlyingeconomicintuition. Given that the banks’ asset and liability mix are each distorted, two tools are generally needed to implement the planner’s allocations. The exception would be if the distortions turn out to alter bothmixesinidenticalways. Inthis,measurezero,casethewedgeswouldbe“equal”toeachother in equilibrium. Moreover, the optimization variables should not load on the regulatory constraints in a collinear way, or, in other words, the matrix ∆RCsp should not be singular. This means that the choice of one tool should not determine the level of another tool and, hence, there are enough degreesoffreedomtocorrectbothofthedistortions. 35TheproblemintheRamseyliteratureistomaximizeasocialwelfarefunctionsubjecttoalltheconstraintsconstitutingacompetitiveequilibriumforthepurposeoffinancinggovernmentexpenditurewithdistortionarytaxation.Although thepurposeofouraugmentedplannerisdifferent,themethodologytooptimallychoosethelevelofinstrumentsthatshe isendowedwithisthesame. 34

Finally, the regulatory tools should be jointly binding, which means that the multipliers ψ T should be strictly positive. The reason is that quantity regulations as in (40)-(43) mandate a minimum level of capital, liquidity or combinations of assets and liabilities, and the bank cannot be forcedtooperateatalowerlevel. Thisbecomesimportantwhentools,whicharebindingbythemselvesintheprivateequilibrium,arecombined. In order to examine which regulations can be jointly binding, we set the level of tools to the regulatory ratios in the planner’s solution and compute the vector of multipliers Ψ for all possible combinations. It turns out that many pairs of regulations can be combined to deliver the planner’s allocations. As long as one of the pair is a capital tool (CR or LevR) and the other a liquidity tool (LCR or NSFR) then the regulations will replicate the planner’s allocations. This result is intuitive. Theplannerwantstoholdmoreliquidityandmorecapitalthanintheprivateequilibrium. Liquidityrequirementscanforcemoreliquidityinthebank,butatthecostofreducingcapitalratios or, equivalently, increasing leverage. Hence adding a capital or leverage requirement can correct forthe(unintended)consequencesofliquidityregulation. Yet,twoliquiditytoolscannotbejointly bindingastheyreinforceeachotherandmovethekeyvariablesinthesamedirection.36 Thesame problem arises if only the two capital regulations are deployed. Otherwise, the palnner’s outcome showninthelastcolumnofTable3canbedeliveredbyanyofthefourcombinationsofthatinvolve asinglecapitalregulationalongwithasingleliquidityregulation. 5 Social outcomes, regulation and banker’s wealth abundance Until now, our analysis has focused on equilibria where banking is sufficiently profitable that the banker’sutilityisaboveherreservationvalue. Wehaveshownthatregulationdepletesthissurplus, but improves economic efficiency and increases the total surplus created. This raises the question ofwhetherregulationisneededjustbecausethebanker,actinginherowninterests,maximizesthe surplusaccruingtoher. Althoughregulationhasredistributiveaspects,weshowinthissectionthat itisbeneficialevenwhenthebankerisreceiveszeroeconomicsurplusintheprivateequilibrium. Onewaytoclarifytheseissuesistoconsiderabankerwhoiswealthyenoughthatthemarginal value of consumption at t=1 (or the relevant outside option in a richer model) is sufficiently low. Inthiscase,thebankerwillinvestinsomuchbankequitythatthebanker’sparticipationconstraint is binding in the private equilibrium. Table 4 reports the private and planning outcomes for two levelsofthebankruptcycost. Aswediscussbelow,thelevelofthebankruptcycostisnotimportant to obtain divergent private and social outcomes, but matters for the kind of regulation that can 36Our results are consistent with the analysis in Checchetti and Kashyap (2016), who show that LCR and NSFR regulationsalmostsurelywillneverbindatthesametime. However, thecollinearityoftheCRandLevRregulations maybespecifictoourmodel. Ifthebankthatcouldchoosebetweenmoretypesofassetswithdifferentlevelsofrisk, ortoholdoff-balancesheetassets,thisresultmaynohaveobtained–thoughthiswouldnotlikelydelivertheplanner’s allocations. Although we can only speculate at this point, we believe that such modifications are important avenues for future research. Other papers that study the use of capital and liquidity requirements include Walther (2016) and KaraandOzsoy(2016)inthepresenceoffiresaleexternalities,BoissayandCollard(2016)whentheinterbankmarket cannotefficientlyallocateresources,andVandenHeuvel(2017)whoquantifiesthewelfarecostsofcapitalandliquidity requirementsinaneoclassicalgrowthmodel. 35

decentralizetheplanner’ssolution. Asinthegeneralcasewhenthebanker’sutilityexceedsherreservationvalue,theplannercontinuestofavoralowerrunprobability. Socialwelfareimprovesandthetotalsurplusishigherthan in the private equilibrium. Moreover, Pareto improvements are possible, whereby both savers and entrepreneurs are better-off compared to the private equilibrium. Nevertheless, the choice of the asset mix and of the liabilities structure depends on the agent who is favored more and the level ofdeadweightlossesinbankruptcy. Thetensionsarisebecauseinthisequilibriumthebankercannot simultaneously increase both capital and liquidity. The banker is already at her participation constraint in the private equilibrium, asking her to contribute more equity requires a reduction in liquidity so the lower return to equity is counterbalanced by the positive effect of holding fewer liquidassets(andviceversa). Theplannercannotrequirebothhighercapitalandliquiditywithout violatingthebanker’sparticipationconstraint. Ifbankruptcycostsarelow,therisk-sharingeffectofhighercapitalislessstrongandtheplanner chooses more liquid assets. As a result, investment goes down and savers enjoy most of the gains fromtheintervention,whileentrepreneursareslightlybetter-offformostweightsandworse-offfor loww . Inthiscase,planningoutcomescanbefullydecentralizedwithjustoneliquidityregulation. E If bankruptcy costs are high, raising capital requirements is a more efficient way to improve social welfare and investment goes up. This outcome is equally beneficial for entrepreneurs and savers. Finally,planningoutcomescanbefullydecentralizedwithjustonecapitaltool. For all of the prior results, where the banker enjoys positive economic surplus in the private equilibrium, both a capital tool and a liquidity tool are needed for decentralization irrespective of thelevelofthebankruptcycost. 6 Conclusions Banks perform important services for the real economy using both sides of their balance sheet. However,theprivatebankingequilibriamaynotbesociallyoptimalandregulatingbankingactivitiescanimprovesocialwelfare. Wehaveexaminedhowmanyregulationsthatareoftendiscussedin policydiscussionsperforminarelativelyfamiliarmodelofbanking. WestartedfromtheDiamond and Dybvig (1983) benchmark precisely because it is so thoroughly studied. The modifications that we made trade-off tractability to keep the model relatively simple, against our preference for additionalrealisticforcesthatthebaselinemodelexcludes. Ourmodificationsgenerateendogenouscreditriskinbanks’portfoliosaswellastheriskofan endogenous funding run. This simple pair of features interact in interesting and unexpected ways. We draw several general lessons from the model that we believe will carry over to many other models. First,weidentifytwogeneralintermediationmarginsthataredistorted,i.e.,therelativeamounts of liquid and illiquid assets and the mix of deposits and equity. The way that banks privately set these margins diverges from what a social planner would choose, because bankers do not fully 36

internalize the effects of their choices on savers and entrepreneurs. In particular, a social planner chooses relatively more liquidity and equity than the banker. As a result, the planner reduces run risk, improves the provision of liquidity, and guarantees a more stable extension of credit and real productioncomparedtotheprivateequilibrium. Thesetwodistortionswillbepresentifweexpand the set of assets that banks can invest in or the types of funding sources, but additional ones may alsoarise. Second, the two wedges between the private and social choices are not collinear. Thus, more that one regulatory tool is needed to implement the socially optimal allocations. Optimal policy in models without both distortions can be misleading. For example, if the liability structure is constrained, say because deposit levels are exogenously determined and equity is fixed, studying assetallocationsanddistortionsbecomesmucheasier. But,regulation,ifanyisneeded,willamount to fixing liquidity ratios. Similarly, shutting down the liquidity demand and liquidity risk makes it easier to focus on the optimal capital structure and level of investment. But, regulation, if again anyisneeded,wouldamounttofixingcapitalratios. Instead,whenbothsidesofthebank’sbalance sheetareendogenouslydeterminedthedistortionsfromeachsideinteractandacombinationofboth capitalandliquidityrequirementsemergeintheoptimalregulatorymix. Third, the political economy aspects of regulation deserve attention. Our bankers internalize how their decisions matter for run risk and choose funding contracts optimally to maximize their own welfare. Their distorted choices, from a social point of view, have real macroeconomic consequences. Regulation improves aggregate welfare, but reduces the rents accruing to bankers. If possible, therefore, banks’ incentives to engage in regulatory arbitrage would be strong. The lack of regulatory arbitrage in the model we have studied is one of its main shortcomings. Moreover, regulating capital and/or liquidity is beneficial for both savers and entrepreneurs, but the relative benefitsofthetypeofregulationdiffer. Saversgainmorewithliquidityregulationgiventhatithas a bigger effect of liquidity provision, while entrepreneurs gain more with capital regulation given thatitallowsformorecreditextension. There are other interesting avenues to extend our model, some of which we have already been mentionedandareanalyzedintheonlineappendix. Onefurtherdirectionwouldbetoallowbanksto issue long-term debt together with demandable deposits and equity. Including loss-absorbing debt instrumentsintheregulatorymixcouldintroduceadditionalwaystotacklewithrunriskandcredit risk. Butitwouldnotconstituteafullremedybyitselfduetothedisciplinaryrolethatdemandable liabilities play. Moreover, our model is flexible enough to incorporate fire-sale dynamics by endogenizing the liquidation value of long-term investment. Although this would introduce pecuniary externalitiesasanadditionalreasonwhyprivateallocationsareinefficient,itwouldnotqualitatively overturnourmainconclusions;theassetandliabilitysidedistortionswouldbesimilar. Finally,one couldenrichthesetofriskyinvestmentsfromwhichabankercouldchooseand, thus, increasethe scope for asset substitution. Setting the (relative) risk-weights in capital requirements to capture socialriskswouldbe,then,highlyimportant. 37

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Tables and Figures eS=2.50 A =3.40 ω =60% α=0.77 1 3g 3g eS=0.80 A =0.80 PB=0.20 γ=0.10 2 3b eE =0.22 ξ=1.20 δ=0.50 ρ=1.00 eB=0.30 ξ=0.01 β=0.70 c =1% D Table1: Parameterization. PE PE SPforweights(w ,w ) E S Incomplete Compr/ve Contracts Contracts (0.2,0.6) (0.3,0.5) (0.4,0.4) (0.5,0.3) (0.6,0.2) I 0.745 0.745 0.734 0.736 0.739 0.743 0.747 LIQ 0.166 0.164 0.276 0.275 0.273 0.271 0.268 1 D 0.679 0.676 0.778 0.778 0.778 0.777 0.776 E 0.233 0.232 0.232 0.233 0.235 0.237 0.239 CR 0.312 0.312 0.316 0.316 0.317 0.319 0.320 LevR 0.255 0.256 0.229 0.230 0.232 0.234 0.236 LCR 0.256 0.253 0.364 0.363 0.361 0.358 0.355 NSFR 0.768 0.766 0.846 0.845 0.843 0.842 0.839 rI 1.796 1.797 1.750 1.749 1.748 1.747 1.746 rD 1.278 1.272 1.549 1.548 1.547 1.545 1.541 3 q 0.408 0.409 0.369 0.369 0.369 0.369 0.370 q 0.187 0.188 0.121 0.122 0.123 0.125 0.127 f Liq.Prov. 0.949 0.943 1.219 1.218 1.217 1.214 1.210 %∆UE - -0.02% 1.04% 1.05% 1.07% 1.09% 1.11% %∆US - -0.07% 3.43% 3.42% 3.41% 3.38% 3.34% %∆UB - 0.02% -1.45% -1.45% -1.45% -1.45% -1.45% %∆Usp - -0.03% 1.97% 1.74% 1.50% 1.27% 1.04% %∆Ssp - -0.02% 1.00% 1.01% 1.01% 1.01% 1.00% Table 2: Privately versus Socially Optimal Solutions. The table reports private equilbria under incomplete and comprehensive contracts. The welfare changes are computed over the level of welfare in the private equilibriumwithincompletecontracts,whichisnormalizedtooneforeachagent. 41

PE CR LevR LCR NSFR SP I 0.745 0.769 0.751 0.701 0.717 0.739 LIQ 0.166 0.246 0.161 0.283 0.282 0.273 1 D 0.679 0.764 0.676 0.772 0.777 0.778 E 0.233 0.251 0.236 0.212 0.222 0.235 CR 0.312 0.326 0.314 0.303 0.309 0.317 LevR 0.255 0.247 0.259 0.216 0.222 0.232 LCR 0.256 0.332 0.249 0.376 0.372 0.361 NSFR 0.768 0.823 0.764 0.854 0.851 0.843 rI 1.796 1.745 1.795 1.765 1.757 1.748 rD 1.278 1.504 1.272 1.536 1.547 1.547 3 q 0.408 0.374 0.408 0.373 0.370 0.369 q 0.187 0.140 0.190 0.115 0.117 0.123 f Liq.Prov. 0.949 1.173 0.941 1.209 1.219 1.217 %∆UE - 1.12% 0.02% 0.70% 0.89% 1.07% %∆US - 2.87% -0.07% 3.21% 3.39% 3.41% %∆UB - -1.45% -0.01% -1.45% -1.45% -1.45% %∆Usp - 1.31% -0.02% 1.27% 1.42% 1.50% %∆Ssp - 0.85% -0.02% 0.82% 0.94% 1.01% Table3: Singleregulationsversusplanner’ssolutionfor(w ,w )=(0.4,0.4). Regulationissetatitshighest E S levelsuchthattherearegainsinsocialwelfare,whileagents’participationconstraintsaresatisfied. Figure 1: The figure shows the response of credit extension for different levels of the deposit weight in the NSFR. The horizontal axis represents the number of successive times the NSFR is tightened. The first iterationcorrespondstothecompetitiveequilibriumlevelwherethetoolisnotbinding. 42

c =1%&eB=0.33 c =5%&eB=0.33 D D PE SPforweights(w ,w ) PE SPforweights(w ,w ) E S E S (0.2,0.6) (0.4,0.4) (0.6,0.2) (0.2,0.6) (0.4,0.4) (0.6,0.2) I 0.827 0.787 0.792 0.798 0.757 0.786 0.789 0.795 LIQ 0.154 0.198 0.195 0.191 0.150 0.147 0.145 0.141 1 D 0.701 0.726 0.726 0.724 0.666 0.675 0.675 0.674 E 0.279 0.259 0.261 0.265 0.241 0.257 0.259 0.263 CR 0.338 0.329 0.330 0.332 0.319 0.328 0.329 0.330 LevR 0.285 0.263 0.265 0.268 0.266 0.276 0.278 0.280 LCR 0.231 0.283 0.280 0.275 0.237 0.230 0.227 0.221 NSFR 0.762 0.790 0.788 0.786 0.759 0.758 0.756 0.754 rI 1.755 1.757 1.756 1.754 1.795 1.780 1.779 1.777 rD 1.331 1.400 1.399 1.394 1.295 1.319 1.317 1.313 3 q 0.396 0.388 0.388 0.389 0.413 0.408 0.408 0.409 q 0.192 0.168 0.169 0.172 0.194 0.195 0.196 0.198 f Liq.Prov. 0.992 1.069 1.067 1.061 0.914 0.933 0.931 0.926 %∆UE - -0.01% 0.02% 0.05% - 0.30% 0.32% 0.34% %∆US - 0.85% 0.83% 0.78% - 0.28% 0.26% 0.22% %∆UB - 0.00% 0.00% 0.00% - 0.00% 0.00% 0.00% %∆Usp - 0.51% 0.34% 0.19% - 0.22% 0.23% 0.25% %∆Ssp - 0.28% 0.28% 0.28% - 0.19% 0.19% 0.19% Table4: PrivatelyversusSociallyOptimalSolutions: Zeroeconomicsurplustobankers. 43

Optimal Bank Regulation In the Presence of Credit and Run Risk AnilKKashyap DimitriosP.Tsomocos AlexandrosP.Vardoulakis Online Appendix Appendix A reports derivations and results that were omitted in the main body of the paper. SubsectionA.1providesadditionaldetailsaboutthecomputationoftherunthresholdintheincomplete information game described in section 2.4. Section A.2 reports the detailed expressions for theintermediationmarginsinsection3.2. SectionA.3derivestheproblemofthetools-augmented Ramsey planner described in section 4.5. Section A.4 derives the equilibrium conditions when the bankfundslendingwithherownfunds. SectionA.5presentstheproblemwhensaverslenddirectly to entrepreneurs. Section A.6 reports the planning outcomes when the planner can use other tools to distort the deposit supply and loan demand schedules. Section A.7 presents the privately and sociallyoptimalsolutionswheninterestratesareallowedtotakenegativevalues. Appendix B presents an extension of the model where savers can also purchase equity in the bank and the probability of bankruptcy in the final period conditional of the bank surviving the run is endogenous. A Additional derivations and computations A.1 Runthreshold This section provides details about the calculation of the run threshold in equation (32). The utility differential between waiting and withdrawing depends on the expected repayment on deposits, which in turn is a function of the expected repayment on bank loans. Moreover, they both vary as theportionofdepositors,λ,variesfromδtoθ∗. Theentrepreneuralwaysdeliversinthegoodstate oftheworldandthebankissolventwhentheportionofdepositorswithdrawingisδ. However,for givenξ=ξ ∗ ,bankprofitsfallasλincreasesandthereisaλ ˜ ∈(δ,θ∗)suchthatthebankisinsolvent, i.e.,VD(ξ ∗,λ)<1,forλ>λ ˜ . Thethresholdλ ˜ iscalculatedasthesolutiontoequation 3g (cid:16) (cid:17) (cid:16) (cid:17) 1−y(ξ ∗,λ ˜) ·I· (cid:0) 1+rI(cid:1) +LIQ (ξ ∗,λ ˜)− 1−λ ˜ ·D· (cid:0) 1+rD(cid:1) =0. (A.1) 2 3 The bank is always insolvent in the bad state of the world, but the entrepreneur may not be. The reason is that the entrepreneur’s loan obligation decreases as λ increases and the bank recalls TheviewsexpressedinthispaperarethoseoftheauthorsanddonotnecessarilyrepresentthoseofFederalReserve BoardofGovernors,anyoneintheFederalReserveSystem,theBankofEnglandFinancialPolicyCommittee,oranyof theinstitutionswithwhichweareaffiliated. 1

ˆ moreloans. Asaresult,thereisathresholdλsuchthattheentrepreneurrepaysfullyherremaining loansinthebadstate,i.e.,VI (ξ ∗,λ)=1,forλ>λ ˆ . Thethresholdλ ˆ iscalculatedasthesolutionto 3b equation (cid:104)(cid:16) (cid:17)(cid:105) (cid:16) (cid:17) A ·F 1−y(ξ ∗,λ ˆ) − 1−y(ξ ∗,λ ˆ) ·I· (cid:0) 1+rI(cid:1) =0. (A.2) 3b Takingintoconsiderationthesetwothresholds,condition(32)canbewrittenas: (cid:90) λ˜ (cid:90) θ∗ (1−y(ξ ∗,λ))·I· (cid:0) 1+rI(cid:1) ω D· (cid:0) 1+rD(cid:1) dλ+ ω dλ δ 3g 3 λ˜ 3g 1−λ (cid:90) λˆ A ·F (cid:2) (1−y(ξ ∗,λ))·I+IE(cid:3) (cid:90) θ∗ (1−y(ξ ∗,λ))·I· (cid:0) 1+rI(cid:1) 3b ω dλ+ ω dλ 3b 3b δ 1−λ λˆ 1−λ − (cid:20)(cid:90) θ∗ ω dλ+ (cid:90) θ∗ ω dλ (cid:21) ·c ·D· (cid:0) 1+rD(cid:1) − (cid:90) θ∗ D· (cid:0) 1+rD(cid:1) dλ− (cid:90) 1θ∗ D·(1+rD)dλ=0. λ˜ 3g δ 3b D 3 δ 2 θ∗ λ 2 (A.3) When computing the derivatives of (A.3) with respect to the choice variables, the banker and the ˜ ˆ planner explicitly consider how they affect the two thresholds λ and λ. The respective derivatives arecomputedbytotallydifferentiating(A.1)and(A.2). A.2 Intermediationmargins Thissectionpresentsthedetailedexpressionsfortheintermediationmarginsderivedinsection3.2. The approach proceeds by using first-order conditions to solve for and substitute out the Lagrange multipliers, such that the final remaining first-order conditions are only expressed in terms of allocations. First,use(20)and(24)toexpressψ andψ intermsofallocationsandψ suchthat: GG DS IC ψ =A +Γ ψ (A.4) GG GG GG IC ψ =A +Γ ψ (A.5) DS DS DS IC where dUBdDSI −dUBdDS dICdDSI −dICdDS drD dξ∗ c dξ∗ drD drD dξ∗ c dξ∗ drD A =− 3 3 and Γ =− 3 3 (A.6) GG dGGdDSI −dGGdDS GG dGGdDSI −dGGdDS drD dξ∗ c dξ∗ drD drD dξ∗ c dξ∗ drD 3 3 3 3 dUB +A dGG dIC+Γ dGG drD GG drD drD GG drD A =− 3 3 and Γ =− 3 3 (A.7) DS dDS DS dDS drD drD 3 3 Substitutein(18)and(19)thevaluesforψ ,ψ andψ from(21),(A.4)and(A.5),respec- BS GG DS 2

tively,togettheinvestment-liquiditymarginintheprivateequilibrium: dUB dUB (cid:18) dUB dUB(cid:19) − +(1−A ) + ILIQ dI dLIQ dLIQ dE 1 1 (cid:18) (cid:19) (cid:18) (cid:19) dGG dGG dDS dDS +A − A +A − A I =0, (A.8) GG ILIQ DS ILIQ c dI dLIQ dI dLIQ 1 1 where dIC+dGGΓ +dDSΓ I A = dI dI GG dI DS c . (A.9) ILIQ dIC + dGG Γ + dDS Γ I dLIQ1 dLIQ1 GG dLIQ1 DS c Similarly, combine (21), (22), (18), (A.4) and (A.5) to get the equity-deposit margin in the privateequilibrium: dUB dUB (cid:18) dUB dUB(cid:19) − +A + ED dE dD dI dE (cid:18) (cid:19) (cid:18) (cid:19) dGG dGG dDS dDS −A − A −A − A I =0, (A.10) GG ED DS ED c dD dI dD dI where dIC+dGGΓ +dDSΓ A = dD dD GG dD DS . (A.11) ED dIC+dGGΓ +dDSΓ I dI dI GG dI DS c The same process is followed to derive the investment-liquidity and equity-deposit margins in theplanner’ssolution,whichare,respectivelygivenby: (cid:18) dUh dUh (cid:19) (cid:18) dUh dUh(cid:19) ∑w − +(1−∆ )∑w + h ILIQ h dI dLIQ dLIQ dE h 1 h 1 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) dGG dGG dDS dDS dLD dLD +∆ − ∆ +∆ − ∆ +∆ − ∆ =0, GG ILIQ DS ILIQ LD ILIQ dI dLIQ dI dLIQ dI dLIQ 1 1 1 (A.12) and (cid:18) dUh dUh(cid:19) (cid:18) dUh dUh(cid:19) ∑w − +∆ ∑w + h ED h dE dD dI dE h h (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) dGG dGG dDS dDS dLD dLD −∆ − ∆ −∆ − ∆ −∆ − ∆ =0, (A.13) GG ED DS ED LD ED dD dI dD dI dD dI where dIC+dGGZ +dDSZ +dLDZ ∆ = dI dI GG dI DS dI LD , (A.14) ILIQ dIC + dGG Z + dDS Z + dLD Z dLIQ1 dLIQ1 GG dLIQ1 DS dLIQ1 LD 3

dIC+dGGZ +dDSZ +dLDZ ∆ = dD dD GG dD DS dD LD , (A.15) ED dIC+dGGZ +dDSZ +dLDZ dI dI GG dI DS dI LD (cid:104) (cid:105) ∑ h w h d d U rD hd d D ξ∗ S−d d U ξ∗ hd d D rD S+ (cid:0)d d L r D I (cid:1)−1d d U rI hd d L ξ D ∗ d d D rD S ∆ =− 3 3 3 , (A.16) GG dGGdDS−dGGdDS+ (cid:0)dLD(cid:1)−1dLDdGGdDS drD dξ∗ dξ∗ drD drI dξ∗ drI drD 3 3 3 dICdDS−dICdDS+ (cid:0)dLD(cid:1)−1dLDdIC DS drD dξ∗ dξ∗ drD drI dξ∗ drI drD Z =− 3 3 3 , (A.17) GG dGGdDS−dGGdDS+ (cid:0)dLD(cid:1)−1dLDdGGdDS drD dξ∗ dξ∗ drD drI dξ∗ drI drD 3 3 3 ∑ h w h d d U rD h +∆ GG d d G rD G d d I r C D +Z GG d d G rD G ∆ =− 3 3 and Z =− 3 3 , (A.18) DS dDS DS dDS drD drD 3 3 ∆ =− ∑ h w h d d U rI h +∆ GG d d G r G I and Z =− d d I r C I +Z GG d d G r G I . (A.19) LD dLD LD dLD drI drI As discussed in the paper, the private intermediation margins differ from the planner’s in a number of ways. Most importantly, the banker does not care how her choices directly affect the utility of savers and entrepreneurs. Thus, additional terms enter into the planner’s solution, which capture the direct total effect of the banking choices governing credit and run risk on savers’ and entrepereneurs’welfare. Forexample,thederivativesdUj/dξ ∗ , j=S,E,arepresentin(A.12)and (A.13), but not in (A.8) and (A.10). These derivatives introduce a wedge between the private and socialintermediationmargins. Additionally,thebankerisprotectedbylimitedliabilityandwillnot internalize all effects when contracts are incomplete, i.e., I =0 in (A.12) and (A.13) contrary to c theplanner’sintermediationmarginswheretherelevanttermsarealwayspresent. A.3 Tools-augmentedplanner Inthissectionwespecifytheproblemofthetools-augmentedplannerandshowthat(47)is(generically)anecessaryandsufficientconditionsuchthatthesocialplanner’ssolutiondescribedinsection 3.1 can be decentralized as a private equilibrium by using regulatory tools T ∈T. The toolsaugmented planner not only chooses optimally allocations and prices X ∈X, but also the level of toolsT ∈Tandthemultipliersψ ,whicharetheshadowvaluesthatthebankassignstoconstraints T RC(T,X)≥0inthenewequilibrium. Herproblemis: max Usp X,T,ψ T s.t. B(X)=0, RC(T,X)≥0, ILIQT(T,X,Ψ T )=0,IDT(T,X,Ψ T )=0, (A.20) 4

whereILIQT andIDT aretheregulation-distortedmarginsgivenby(45)and(46). Thefirst-orderconditionwithrespecttoX (similartofirst-ordercondition(35))are: dUh dBS dIC dGG dLD dDS ∑ w +ζ +ζ +ζ +ζ +ζ h dX BS dX IC dX GG dX LD dX DS dX h={E,R,B} dRC dILIQT dIDT +∑ζ +ζ +ζ =0, (A.21) T dX ILIQ dX ID dX T where ζ , ζ and ζ are the multipliers the tool-augmented planner assigns to regulatory con- T ILIQ ID straintsRC(T,X)andthethreeregulation-distortedintermediationmargins. Thefirst-orderconditionswithrespecttotheleveloftoolsT are: dRC dILIQT dIDT ζ +ζ +ζ =0, (A.22) T dT ILIQ dT ID dT andchoosingoptimallythemultipliersψ yields: T dILIQT dIDT ζ +ζ =0. (A.23) ILIQ dψ ID dψ T T Toprovesufficiency,(47)impliesthatthereneedtobetworegulatorytoolssuchthatasolution tomultipliersψ canbeobtained. Inturn,thismeansthattherearetwofirst-orderconditionsofthe T form in (A.22) and two of the form in (A.23). Conditions (A.23) can be written in matrix form as transpose(∆RC)·transpose([ζ ζ ])=0. Giventhat∆RC isinvertible,ζ =ζ =0. Thus, ILIQ ID ILIQ ID the only solution is one where all ζ , ζ and ζ are zero and the first-order conditions (A.21) T ILIQ ED coincidewiththefirst-orderconditions(35)ofthesocialplanner. Toprovenecessity,supposethat(47)doesnotholdorinotherwordsthespanof∆RCislessthan wedge two. Usingconditions(A.21)wecanderiveintermediationmarginsILIQ =ILIQ +ILIQ tap sp tap wedge and ID =ID +ID for the tool-augmented planner, where the wedges are linear combitap sp tap nation of one multiplier ζ , ζ and ζ . The social planner’s and tools-augmented planner’s T ILIQ ID solutions coincide if both wedges are zero, which in principle is possible because there are three multipliers, hence three degrees of freedom. However, equations (A.22) and (A.23) remove two degrees of freedom. Hence, it is not possible to replicate the social planner’s solution with fewer thattwoindependenttools,whichleadstoacontradiction. A.4 Equilibriumwithoutdepositintermediation The participation constraint (25) of the banker supposes that utility in autarky is the outside option. As already mentioned, we could consider the utility that the banker obtains by lending to entrepreneurs using only her own funds as an outside option. This section derives the conditions for the alternative outside option and shows that for the equilibrium considered in section 3.3 the autarkicutilityistherelevantoutsideoption. 5

Theutilityofthebankerwholendstotheentrepreneuronlyusingherowncapitalandnottaking depositsis UB,n=γ·U (cid:0) eB−In(cid:1) +∑ω VI,nIn(cid:0) 1+rI,n(cid:1) , (A.24) 3s 3s s where In is the loan to E, rI,n the loan rate, and VI,n = min (cid:0) 1,A F(IE+In)/(In(1+rI,n)) (cid:1) the 3s 3s percentagerepaymentontheloan. TheoptimalchoiceofIn yields: −γ·U(cid:48)(cid:0) eB−In(cid:1) +ω (cid:0) 1+rI,n(cid:1) +ω F(cid:48)(cid:0) IE+In(cid:1) =0, (A.25) 3g 3b consideringthatE defaultsinthebadstateoftheworld. Anindividualentrepreneurchoosesaloanrateandloanamountthatsatisfytheloansupplyby thebanker(A.25)tomaximizeherutilitygivenby: UE,n=∑ω (cid:2) A F (cid:0) IE+In(cid:1) −In(cid:0) 1+rI,n(cid:1)(cid:3)+ . (A.26) 3s 3g s Given that an individual entrepreneur does not internalize how her loan demand affects the shadowcostoffundsforthebanker,i.e.,thefirsttermin(A.25),butdoesinternalizehowitaffects therepaymentindefault,E(cid:48)sloandemandscheduleisgivenby: ω (cid:2) A F(cid:48)(cid:0) IE+In(cid:1) − (cid:0) 1+rI,n(cid:1)(cid:3) +ω InA F(cid:48)(cid:48)(cid:0) IE+In(cid:1) =0. (A.27) 3g 3g 3b 3b Conditions(A.25)and(A.27)yieldasolutionfortheloanamountandtheloanrate. TableA.1 belowcomparestheprivateequilibriawhenbanksintermediatedepositsandwhentheydonot. The percentagechangeinwelfareforEandBiscalculatedovertheutilitylevelinautarky(normalizedto one). Theparticipationconstraintofentrepreneursisviolatedwhenthebankerdonotraisedeposits tolowerlendingrates. Loanrate Loanamount EVI %∆UE 3b Intermediation 1.796 0.745 0.454 0.55% Nointermediation 2.014 0.276 0.560 -4.57% TableA.1: Privateequilibriumsolutionsunderdepositandnodepositintermediation. A.5 Directlending This section derives the conditions for direct lending to entrepreneurs by savers and computes the equilibriumoutcomesfortheparameterizationinsection3.3. Directlendingrequirestheindividualsaverstobeabletomonitortheentrepreneur. Denoteby MC the monitoring cost to an individual saver, which can be higher or equal to the cost for the banker, i.e., her private benefit. Att =1, an individual saver can invest in the liquid asset, LIQdl, 6

orlendtotheentrepreneur,Idl,atinterestraterI,dl. Intheintermediateperiod,shewouldliquidate all her loans if she turns out to be impatient. Otherwise, the saver waits until the final period and receivesthepercentagerepaymentontheloansshemade. Herutilityunderdirectlendingisgiven by: (cid:16) (cid:17) (cid:90) ξ (cid:16) (cid:17)dξ (cid:16) (cid:17) US,dl =U cdl +δ U cdl;i +(1−δ)∑ω U cdl;p , 1 1 2 2 ∆ 3s 3 3 ξ ξ s where cdl =eS−Idl−LIQdl, cdl =eS+LIQdl+ξ·Idl, and cdl =eS+LIQdl+(VI,dl −c ·I )· 1 1 2 2 3 2 3s D dl Idl·(1+rI,dl)−MC. Moreover,VI,dl =min[1,A ·F(IE+Idl)/(Idl·(1+rI,dl))] is the percentage 3s 3s repaymentontheloanandI istheindicatorfunctionfordefault. dl Under the assumption that an individual saver lends to an individual entrepreneur, the former willinternalizehowherloanextensionaffectstheexpecteddeliveryindefault(muchlikethebanker does). Hence,theoptimalchoiceoflending,Idl,yields: (cid:16) (cid:17) (cid:90) ξ (cid:16) (cid:17)dξ (cid:104) (cid:16) (cid:17) (cid:16) (cid:16) (cid:17) (cid:17)(cid:105) −U(cid:48) cdl +δ ξ·U(cid:48) cdl;i +β2(1−δ)· ω 1+rdl +ω A F(cid:48) IE+Idl −c =0 1 1 2 2 ∆ 3g 3b 3b D ξ ξ (A.28) wherewehaveusedthefactsthatU(cid:48)(·;p)=β2 andthattheentrepreneurwoulddefault,inequilib- 3 rium,inthebadstate. Similarly,theoptimalchoiceofliquidholdings,LIQdl,yields: (cid:16) (cid:17) (cid:90) ξ (cid:16) (cid:17)dξ −U(cid:48) cdl +δ U(cid:48) cdl;i +β2(1−δ)+ν =0, (A.29) 1 1 2 2 ∆ LIQDL ξ ξ whereν istheLagrangemultiplierontheconstraintLIQDL≥0. LIQDL Theutilityofanindividualentrepreneur (cid:104) (cid:16) (cid:17) (cid:16) (cid:17)(cid:105) UE,dl =δ·UE,aut+(1−δ)·ω A F eE+Idl −Idl· 1+rdl , (A.30) 3g 3g given that E invests all of her wealth in the project, i.e., IE =eE. With probability δ an individual entrepreneur has her project liquidated and continues to produce only with her own capital. As a result,sheenjoysthesameutilityasinautarky. Withprobability1−δ,thesaverdoesnotliquidate theprojectandtheentrepreneurdefaultsinthatbadstate. Theentrepreneurchoosestheloanamount, Idl, and the loan rate, rdl, that satisfy (A.28) to maximize (A.30). Consistent with our analysis in therestofthepaper,theentrepreneurinternalizeshereffectonthemarginalpayoffaccruingtothe saver, but takes the other forces determining saver’s costs of funds (marginal utilities at t =1 and t =2)asgiven. Thus,theoptimalloandemandbytheentrepreneuris: (cid:104) (cid:16) (cid:17) (cid:16) (cid:17)(cid:105) (cid:16) (cid:17) ω A F(cid:48) eE+Idl − 1+rdl +ω A F(cid:48)(cid:48) eE+Idl Idl =0. (A.31) 3g 3g 3b 3b 7

Conditions(A.28),(A.29)and(A.31)jointlydetermineIdl,LIQdl andrdl inequilibrium. Using theparameterizationinsection3.3andsettingMCequaltoPB,theutilityofsaversis0.07%higher under bank intermediation compared to direct lending which, in turn, is higher than the utility in autarky. By increasing MC we can obtain equilbria where direct lending delivers lower utility to savers and eventually is dominated by autarky. In addition, the utility of entrepreneurs is higher thaninautarky,thustheyarewillingtoborrowdirectlyfromsavers. A.6 Additionaldistortionarytools This section extends the analysis in section 3.3 by allowing the planner to use tools to distort the depositsupplyandloandemandschedulesofsaversandentrepreneurs. Weconsidergenerictools, τ for the deposit supply schedule, and τ for the loan demand schedule, and discuss how they DS LD canbeimplementedinpractice. Thedepositsupplyschedule(7)thattheplannerfacesbecomes: (cid:34) (cid:40) (cid:41) (cid:35) −U(cid:48)(c )+ (cid:0) 1+rD(cid:1) ∑ (cid:90) ξ∗ θ(ξ,1)·E U(cid:48)(c (j,1);j) dξ +δ (cid:90) ξ U(cid:48)(c (i,1);i) dξ 1 1 2 t=2,3 ξ j t ts ∆ ξ ξ∗ 2 2 ∆ ξ +(1−δ) (cid:90) ξ ∑ω U(cid:48)(c (p,0);p)· (cid:0) VD(ξ,δ)−c ·I (cid:1) ·(1+rD) dξ +ν +τ =0. (A.32) ξ∗ s 3s 3 3s 3s D d 3 ∆ ξ D DS Totheextentthatsaverssupplydeposits,i.e.,ν =0,theplannercandistorttheirwillingnessto D holddepositsatgivendepositratesbyvaryingthelevelofthedistortionarytoolτ . Inotherwords, DS the planner can set τ (cid:54)=0, which implies that (A.32) stops being a constraint in her optimization DS problem (33) and ζ =0 in (35). The intervention can be implemented, for example, either as a DS tax on the supply of deposits att =1 or as a tax on the interest income accruing to late depositors att =3whenthebankissolvent. Inthefirstcase,thetaxcanbecomputedas−τ /U(cid:48)(c ),while DS 1 1 in the second as −τ / (cid:0) ω (1−q)·(1−δ)·U(cid:48)(c (p,0);p)·(1+rD) (cid:1) . If τ <0, then a tax is DS 3g 3 3g 3 DS levied,whileτ >0impliesasubsidy. Weassumethattheplannerrebatesthetaxproceedsbackto DS thesameagentsinthesameperiodinalump-sumfashioninordertoneutralizeanyincomeeffects. Similarly,theloandemandschedule(27)becomes: (cid:34) (cid:35) (cid:90) ξ (1−y(ξ,δ)) A F(cid:48)(cid:0) IE+(1−y(ξ,δ))·I (cid:1) −(1+rI)+I· ∂LS (cid:18) ∂LS (cid:19)−1 dξ +τ =0. ξ∗ 3g ∂I ∂rI ∆ ξ LD (A.33) The planner can distort the willingness of entrepreneurs to borrow by varying the level of the distortionary tool τ . such that if τ (cid:54)=0, then ζ =0 in (35). The intervention can be imple- LD LD LD mented with a tax on loan repayment in the good state of the world, which can be computed as (cid:18)(cid:90) (cid:19) −τ / ξ (1−y(ξ,δ))(1+rI)dξ/∆ . If τ <0, then a tax is levied, while τ >0 implies a LD ξ∗ ξ LS LS 8

subsidy. A tax can also be implemented with restrictions on the maximum loan-to-value ratio for entrepreneurialloans,i.e.,I≤LTV/(1−LTV)·IE whereLTV istheloan-to-valuelimit. Then,τ LD isthevalueoftheLagrangemultiplierontheLTV constraint. Notethatthislimitisimposedonthe entrepreneurratherthanthebanker,becausetheobjectiveistodistorttheloandemandschedule. Table A.2 below reports the planning equilibria under two sets of weights when distortionary tools are available (using the parameterization discussed in section 3.3). Comparing the planning outcomes with and without distortionary tools, we can observe that the planner can improve social gains if she is endowed with more tools. The reason is that both savers and entrepreneurs do not internalize how their behavior affects the aggregate bank variables, and most importantly the probabilityofarun. Nevertheless, there are three important observations about this extension of the model. First, banking regulation is still needed to implement socially optimal outcomes. The additional distortionary tools affect the deposit supply and loan demand schedules, but do not correct for the distortions in the banker’s optimization condition. Capital and liquidity regulation are required for the latter. Second, the use of the distortionary tools has implications for the allocation of social gains. Whilethebankerremainsatherparticipationconstraint,eitherthesaverortheentrepreneur canbemadebetter-offwhenthesetoolsareusedcomparedtothesocialplanningoutcomeswithout them. Third, a tax that restricts the supply of deposits can be beneficial for savers. In particular, the bank has to offer higher deposit rates to attact deposits and the smaller reliance on deposits in combination with capital and liquidity regulations improves the bank’s stability. Liquidity provision is lower, but this does not hurt savers overall because they are able to self-insure by holding the liquid asset. The benefits are smaller when savers are not allowed to self-insure. Moreover, entrepreneurs are worse-off because the level of funds channeled through the bank goes down and theyaredriventotheirparticipationconstraint. Asaresult,τ cannotbecombinedinthisexample LD withτ becausetherearenoadditionalsocialgainstobemade. LD A.7 Negativeinterestrates Thissectionrelaxestheassumptionaboutthenon-negativityoftheearlydepositrate,rD,andshows 2 that our conclusions about the necessity for capital and liquidity regulations carry over. Table A.3 reportstheprivateandsociallyoptimaloutcomesfornegativerD. 2 Negative early deposit rates reduce the probability of a run, since both the savers’ incentive to run and the bank’s liquidity needs are lower. The banker will weigh the reduction in the run probability to the potential increase in late deposit rates when choosing to set early deposit rates negative. However,thebankerisnotabletodecreaserD allthewaytothelevelthattheprobability 2 of a run is zero, because she would either need to offer very high rD, which eliminates her own 3 profits, or violate the participation constraint of savers. In the private equilibrium in Table A.3 saversaredriventotheirparticipationcontraint(9). Somerelyallowingfornegativeratesdoesnot allowtheprivatesectortodeliverrun-freebanking. The planner can reduce the early deposit rate all the way to the point that runs are ruled out. 9

PE SPfor(w ,w )=(0.4,0.4) SPfor(w ,w )=(0.6,0.2) E S E S Notools τ (cid:54)=0 τ (cid:54)=0 Notools τ (cid:54)=0 τ (cid:54)=0 DS LD DS LD I 0.745 0.739 0.550 0.744 0.747 0.550 0.724 LIQ 0.166 0.273 0.221 0.286 0.268 0.221 0.192 1 D 0.679 0.778 0.510 0.794 0.776 0.510 0.685 E 0.233 0.235 0.261 0.236 0.239 0.261 0.231 CR 0.312 0.317 0.474 0.317 0.320 0.474 0.320 LevR 0.255 0.232 0.338 0.229 0.236 0.338 0.253 LCR 0.256 0.361 0.444 0.369 0.355 0.444 0.291 NSFR 0.768 0.843 0.938 0.851 0.839 0.938 0.793 rI 1.796 1.748 1.828 1.774 1.746 1.828 1.638 rD 1.278 1.547 2.321 1.609 1.541 2.319 1.215 3 q 0.408 0.369 0.207 0.364 0.370 0.207 0.399 q 0.187 0.123 0.043 0.117 0.127 0.044 0.166 f Liq.Prov. 0.949 1.217 0.933 1.254 1.210 0.933 0.987 %∆UE - 1.07% -0.55% 0.49% 1.11% -0.54% 3.07% %∆US - 3.41% 13.73% 4.01% 3.34% 13.72% 0.25% %∆UB - -1.45% -1.45% -1.45% -1.45% -1.45% -1.45% %∆Usp - 1.50% 4.99% 1.51% 1.04% 2.13% 1.60% %∆Ssp - 1.01% 3.91% 1.02% 1.00% 3.91% 0.62% τ - - -0.321 - - -0.321 - DS τ - - - 0.017 - - -0.068 LD TableA.2: PrivatelyversusSociallyOptimalSolutionwhenadditionaldistortionarytoolsareavailable. Doing so requires the liquidation value of the bank’s assets to exceed the total value of runnable liabilities for any realization of the liquidation value, i.e., (LIQ +ξ·I)/(D(1+rD))≥1. This is 1 2 exactlytheconditionthatLCRmustequal1in(42). Anyexcessliquidityontopofwhatisneededto serveearlywithdrawalswouldthenbecarriedovertothefinalperiodusingthestoragetechnology, i.e.,LIQ =LIQ −δ·D(1+rD). FortheplanningequilibriumreportedinthelastcolumninTable 2 1 2 A.3theplannerdoesnotcarryoverexcessliquidity,becausesheisabletoeliminaterunsbydriving theearlydepositrateverynegative. Asaresult,theliquiditytheplannerneedstoholdissmall,yet theLCRgoestoitshighestlevel. Althoughliquidityprovisionislower,thesavergainsfurtherfrom thereductionintherunprobability. Andmostofthegainsaccruetotheentrepreneur,sincethelower amount of liquidity needed to control run risk allows for more investment. The further increase in the late deposit rate and the decrease in the loan rate, makes it more difficult to raise equity from the banker without violating her participation constraint.1 The social planner’s allocations force the banker to invest in more equity than she would do voluntarily. Hence, to decentralize this allocation,capitalregulationwouldalsobeneeded. Thereforejustlikeinthebaselinemodelinthe bodyofthepaper,theprivateequilibriumisinefficientandonecapitalandoneliquidityregulation 1Keepinmindthatalltheutilitylevelsinthetablearenormalizedtooneintheprivateequilibrium. Thebanker’s utilityskyrocketswhennegativeratesareallowed. Sothelargedropforthesocialplanner’sallocationscomebecause thestartingpointforthebankerissofavorable. 10

isrequiredtomatchthesocialplanner’sallocations. PE SP I 0.870 1.492 LIQ 0.123 0.015 1 D 0.742 1.343 E 0.251 0.164 CR 0.288 0.110 LevR 0.253 0.109 LCR 0.282 1.000 NSFR 0.715 0.560 rI 1.675 1.313 rD -0.370 -0.976 2 rD 1.030 3.644 3 q 0.193 0.000 q 0.099 0.000 f Liq.Prov. 0.946 0.724 %∆UE - 23.74% %∆US - 29.14% %∆UB - -27.01% %∆Usp - 15.75% %∆Ssp - 8.63% TableA.3: PrivatelyversusSociallyOptimalSolutionsforrD<0. Theplanningoutcomesareforweights 2 (w ,w )=(0.4,0.4). We have added a fixed number (equal to 1) to the utility of impatient depositors, E S because it takes negative values for q=0 as early consumption, c (i), is less than 1. This does not affect 2 marginaldecisionsandequilibriumoutcomes,butitallowstheeasycomparisonoftheLiq.Prov.ratioacross equilibria,whichwouldotherwisehaveanegativevalueforq=0. 11

B Extendedmodel Thissectionextendsthebaselinemodelsothatsaverscanalsopurchaseequityinthebankandthe probability of bankruptcy in the final period conditional on the bank surviving the run is endogenous. The first modification implies that the banker and the planner have an alternative source of funding apart from the equity contributed by the banker and deposits offered by savers. We will refer to equity contributed by bankers and savers as “inside” and “outside” equity, respectively. Theintroductionofanadditionalsourceoffundingaddsanotherintermediationmarginforbanking decisions. We show that this margin is also distorted and that a planner would need an additional toolontopofacapitalandaliquidityregulationtofullyimplementasolutionwithpositiveoutside equity. Thesecondmodificationallowsustoexaminehowregulationdifferentiallyaffectsrunriskand credit risk. To do so, we introduce a third "medium" state for the realization of the productivity shock in the final period, which is between the level in the good and the bad state. Thus, the state spaceatt =3iss∈{g,m,b}andtheproductivityrealizationsatisfyA >A >A . Wefocuson 3g 3m 3b cases in which entrepreneurs default in states m and b, while they fully repay in state g. The bank issolventisstateganddefaultsinstateb,whilethebankruptcydecisiondependsontherealization (cid:16) (cid:17) of ξ in state m. Hence, there is a threshold ξ ˆ ∈ ξ ∗,ξ such that the bank is solvent in state m ˆ only if the realization of the liquidation value is higher than ξ. The threshold is endogenous and depends on the balance sheet of the bank. Thus, it plays a critical role in the expected probability of bank default and the benefit of raising equity to reduce expected bankruptcy costs.2 We also considerageneralspecificationforbankruptcycostsandintroduceinvestmentadjustmentcostsfor entrepreneurswhentheirloansarerecalledandinvestmentliquidated. These modifications allow us to study equilibria where the planner chooses positive outside equity and there is room for redistributive effects of regulation. To avoid repeating ourselves, we onlypresenttheequationswherethesemodificationsenter. B.1 Modifiedsavers’problem As in the baseline model, savers invest in bank deposits and the liquid asset at t =1 to maximize their lifetime expected utility (6). But, they can additionally buy bank (outside) equity shares, ES, inaprimarymarketatapricePpershare. Equityisvaluablebecauseofthedividendspaidoneach share,DPS (ξ,λ),att =3. Recallthatinthebaselinemodelwedidnotdistinguishbetweenbank 3s profits and dividends per share given that the banker is the sole equity-holder. We will be precise 2Havingonlythreelevelsofproductivitydoesnotchangethefundamentaleconomicoutcomesintheextendedmodel. Inparticular,evenwithmanymoreoutcomesfortechnologytherearefundamentallyonlythreedifferenttypesofoutcomes. Forsomerealizationsofproductivitytheresourcesaresufficientsothattheloansarefullypaidandinthiscase thedepositsarealsofullypaid. Conversely, itispossiblethattheinvestmentoutcomeissopoorthattheloanrepaymentissolowthatthedepositorscanneverbefullypaid. Finally,thereareinterimcaseswheretheloansmaynotbe completelypaid,butthebankstillcanfullypaydeposits. Nothingintheanalysiswouldchangeifwehadmanymore states. 12

about how the dividends per share are determined in the modified banker’s problem below. As a result,thebudgetconstraintatt =1–equation(1)–becomes: c =eS−D−P·ES−LIQS. (B.1) 1 1 1 Each share can be re-traded in a secondary market as a price P (ξ,λ). In a run, equity is sec worthless,i.e.,P (ξ,1)=0andDPS (ξ,1)=0becausethebankisliquidated. Patientsaverswill sec 3s enterthesecondarymarkettobuyequityfromimpatientsavers. Thepatientsavers’totalfundsare the sum of their new endowment, eS, and their liquid holdings carried over from the first period, 2 LIQS. The patient savers total equity holdings after trading are ES (ξ,λ). Thus, the net purchase 1 sec isP (ξ,δ)·(ES (ξ,λ)−ES)andtheremainingresourcesaretransferredtot =3usingthestorage sec sec technology. Conditionalonarunocurring,theconsumptionofasaveroftype jisstillgivenby(2). However, the consumption of an impatient saver when a run does not occur –equation (3)– is now givenby: c (i,I =1)=D (cid:0) 1+rD(cid:1) +LIQS+P (ξ,λ)+eS. (B.2) 2 w 2 1 sec 2 Similarly, the consumption at t = 3 of a patient saver who chooses to wait or withdraw – equations(4)and(5)respectively–willbegivenby3 c (p,I =0)=ES (ξ,λ,I =0)DPS (ξ,λ)+P (ξ,λ) (cid:0) ES−ES (ξ,λ,I =0) (cid:1) 3s w sec w 3s sec sec w + (cid:0) VD(ξ,λ)−c (D)·I (cid:1) D (cid:0) 1+rD(cid:1) +LIQS+eS, (B.3) 3s D d 3 1 2 or c (p,I =1)=ES (ξ,λ,I =1)DPS (ξ,λ)+P (ξ,λ) (cid:0) ES−ES (ξ,λ,I =1) (cid:1) 3s w sec w 3s sec sec w +D (cid:0) 1+rD(cid:1) +LIQS+eS. (B.4) 2 1 2 Since the individual saver takes the bank dividends as given, the optimal decision to purchase equity,ES,intheprimarymarketischosensothat norun (cid:122) (cid:125)(cid:124) (cid:123) (cid:40) (cid:41) (cid:90) ξ 1 ES:−P·U(cid:48)(c )+ ∑ E U(cid:48)(c (j,I );j)·P (ξ,δ) dξ +ν =0, (B.5) 1 1 t=2,3 ξ∗ j t ts w sec ∆ ξ ES 3In the extended model we consider a more general function for the bankruptcy costs given by cD(D)=cD·Dφ D, φ ≥0. Themoregeneralspecificationcanenhancetherisk-sharingroleofequitybecausedifferentallocationsimply D differentmarginalbankruptcycosts. Savers’takecD(D)asgivensinceitisafunctionofthetotaldepositsinthebank. Hence,thedepositsupplyequation(7)hasthesamefunctionalformintheextendedmodelwiththedifferencethatthe marginalcostdependsonDinequilibrium.Thebankerandtheplanneraccountforthisdependance. 13

where ES stands for equity supply and ν is the multiplier on the no short-sale constraint ES ≥ ES 0. Condition (B.5) says that savers equate the marginal utility of lost consumption from buying one bank share at price P to the expected marginal utility gain from the value of the share in the future,P (ξ,δ). Theshareonlyhasanyvalueifthebanksurvivesarun,sinceotherwiseequityis sec worthless. Thevalueofequitythatemergesfromthesecondarymarkettradingsatisfies P (ξ,δ)=∑ω DPS (ξ,δ), (B.6) sec 3s 3s s i.e., the secondary equity price is equal to the expected value of future dividends because patient savershavelinearutilityatt =3andtheiroutsideoptionpayszerointerest. Ifarundoesnotoccur, impatientsaversselltheirbanksharestopatientsavers. Marketclearingrequirestheequityholdings ofeachindividualpatientsaveratt =3besuchthatES (ξ,δ)=ES/(1−δ). sec Before turning to the modified problems of the banker and the entrepreneur, it is easy to show thattheglobalgameanalysisinsection2.4remainsintact(onceweaccountfortheadditionalstate m). Thereasonisthatbecauseofquasi-linearutilitiestheexpectedutilitydifferentialbetweenwaiting and withdrawing conditional on the bank surviving the run –upper part in (31)– is not affected bythedecisiontopurchaseoutsideequity. Tobemoreprecise,theexpectedutilitydifferentialisthe differenceinexpectedconsumptionin(B.3)andexpectedconsumptionin(B.4),whichdifferintwo ways. Onearisesfromthedifferentequityholdingaftersecondarytradingforasaverthatwaitsand asaverthatwithdraws,andtheothercomesbecausethepersonwhowaitswillreceivealatedeposit payment, while the other person will get her deposits early and transfer them to period 3 using the liquid asset. However, the demand for equity in the secondary market determines the secondary equitypriceP (ξ,λ)=∑ω DPS (ξ,λ). Substitutingthesecondarypricein(B.3)and(B.4),the sec 3s 3s s expectedutilitydifferentialbetweenwaitingandwithdrawingconditionalonthebanksurvivingthe runis∑{ω (VD(ξ,λ)−c (D)·I )D (cid:0) 1+rD(cid:1) }−D (cid:0) 1+rD(cid:1) .4 3s 3s D d 3 2 s B.2 Modifiedbanker’sproblem The banker makes the same decisions as in the baseline model, but additionally needs to decide how much outside equity to raise from savers. The equity shares in the bank will be split between the banker and savers and the respective holdings are denoted by EB and ES. At this point we distinguishbetweentheinitialequity, EB, thatthebankerholds, andtheadditionalequity, EB, that 0 1 4The linearity of utility from consumption at t =3 simplifies the run decision substantially since the expected utility differential between waiting and withdrawing, given by equation (31), depends only on predetermined variables and not on actions taken after the run decision, such as trading in the secondary equity market. The terms ∑ω3s E s S ec (ξ,δ,I w)(DPS 3s (ξ,δ)−Psec(ξ,δ)), for both I w=0 and I w=1, in patient agents’ period 3 expected utility s dropout.Thisisanoutcomeofthelinearpreferenceatt=3andistrueforanyportionofsaversλdecidingtowithdraw. As a result, the computation of the run threshold in the global game is largely simplified, because the distribution of equity holdings between patient savers that choose to withdraw and those that choose to wait does not matter for the utilitydifferential. 14

she decides to put into the bank at price P by participating in the primary equity market at t =1. This distinction was inconsequential in the baseline model that the banker is the sole owner of the bank. Hence, the total share holdings of the banker are EB =EB+EB. Pinning down the share of 0 1 ownership is important because the profits accruing to the banker depend on her relative holdings, EB/(EB+ES),orinotherwordsshewillreceiveadividendpershareforeachoftheEB sheholds. Thebanker’sutility–equation(10)–changesto: norun (cid:122) (cid:125)(cid:124) (cid:123) UB=γ·U (cid:0) eB−P·EB(cid:1) +EB (cid:90) ξ ∑ω DPS (ξ,δ) 1 dξ. (B.7) 1 ξ∗ s 3s 3s ∆ ξ Thebankertradesoffforegoingcurrentconsumptiontoinvestinginequityandreceivingdividends inthefuture. Notethatthebankergetstoconsumehershareofdividendsonlyifthebanksurvives therun. Thebanker“buys"additionalequityatthesamepriceatwhichsheissuesequitytosavers. Thebalancesheetconstraintatt =1–equation(11)–becomes: BS: I+LIQ =D+CEQ, (B.8) 1 whereCEQ=P· (cid:0) ES+EB(cid:1) +EB isthetotalcommonequity. 1 0 Raisingoutsideequitydoesnotaffectthebalancesheetconstraintsatt =2,thustheprobability that a depositors is served, θ(ξ,λ), in a run are given by (12) and the fraction of loans recalled, y(ξ,λ),whenarundoesnotoccuraregivenby(13). Thedividendspersharearethetotaldividends dividedbythetotalnumberofshares,i.e., DIV (ξ,λ) 3s DPS (ξ,λ)= , (B.9) 3s EB+ES whereDIV (ξ,λ)aregivenby(14). 3s Moreover, the banker will choose to monitor if her share of the dividends rather than total dividendarehigherthantheprivatebenefit. Thus, theincentivecompatibilityconstraint–equation (17)–becomes: IC: EB∑ω DPS (ξ ∗,δ)−PB≥0. (B.10) 3s 3s s Finally,thesecondmodificationtothebaselinemodelimpliesanendogenousbankruptcythresh- ˆ old,ξ,instatemisdeterminedbythefollowingequation: (cid:16) (cid:16) (cid:17)(cid:17) (cid:16) (cid:17) (cid:16) (cid:17) 1−y ξ ˆ ,δ VI ξ ˆ ,δ I (cid:0) 1+rI(cid:1) +LIQ ξ ˆ ,δ −(1−δ)D (cid:0) 1+rD(cid:1) =0. (B.11) 3m 2 3 We now turn into describing how the optimality conditions are altered and what are the new optimalityconditionswithrespecttooutsideequityandtheequityprice. Wewillfocusattentionto incompletefundingcontracts,i.e.,depositcontractsspecifythetuple(D,rD,rD)andequitycontracts 2 3 15

specifythetuple(EB,ES,P) 1 The marginal effect of investment on banker’s utility in the optimality condition for loans, I –equation(18)–becomes:   nodefault,stateg nodefault,statem (cid:122) (cid:125)(cid:124) (cid:123) (cid:122) (cid:125)(cid:124) (cid:123) d d U I B = EB E + B ES      (cid:90) ξ∗ ξ (cid:8) ω 3g (cid:0) 1+rI(cid:1)(cid:9) ∆ 1 ξ dξ+ (cid:90) ξˆ ξ (cid:8) ω 3m A 3m F(cid:48)(cid:0) IE+(1−y(ξ,δ))I (cid:1)(cid:9) ∆ 1 ξ dξ      . (B.12) Asshownin(B.12),limitedliabilitymeansthatthebankerstillonlyinternalizesstateswheresheis solvent. Similarly, the marginal effect of investment on banker’s utility in the optimality condition for firstperiodliquidassets,LIQ –equation(19)–becomes: 1   nodefault,stateg nodefault,statem (cid:122) (cid:125)(cid:124) (cid:123) (cid:122) (cid:125)(cid:124) (cid:123) d d L U IQ B 1 = EB E + B ES      (cid:90) ξ∗ ξ(cid:26) ω 3g (cid:0) 1+rI(cid:1)1 ξ (cid:27) ∆ 1 ξ dξ+ (cid:90) ξˆ ξ(cid:26) ω 3m A 3m F(cid:48)(cid:0) IE+(1−y(ξ,δ))I (cid:1)1 ξ (cid:27) ∆ 1 ξ dξ      . (B.13) Theoptimalchoiceoftherunthreshold–equation(20)–becomes 1 dIC dGG −EB∑ω DPS (ξ ∗,δ) +ψ +ψ =0. (B.14) 3s 3s ∆ ICdξ ∗ GG dξ ∗ s ξ Theoptimalchoiceofliquidityholdings,LIQ (ξ,δ),att=2aftertherununcertaintyisresolved 2 isgivenby: EB (cid:32) (cid:34) VI (cid:0) 1+rI(cid:1)(cid:35) (cid:34) VI (cid:0) 1+rI(cid:1)(cid:35) (cid:33) ω 1− 3g +ω 1− 3m ·(1−I ) +νLIQ2(ξ,δ)=0, ∀ ξ≥ξ ∗, EB+ES 3g ξ 3m ξ d (B.15) whereνLIQ2(ξ,δ) isthemultiplierontheshort-saleconstraintLIQ (ξ,δ)≥0. 2 Turning to the deposit contract, the marginal effects of the deposit contract terms on banker’s utilityintheoptimalityconditions(22)to(24)–become:  nodefault,stateg (cid:122) (cid:125)(cid:124) (cid:123) d d U D B =− EB E + B ES      (cid:90) ξ∗ ξ (cid:40) ω 3g (cid:0) 1+rI(cid:1)δ (cid:0) 1+ ξ r 2 D(cid:1) +(1−δ) (cid:0) 1+r 3 D(cid:1) (cid:41) ∆ 1 ξ dξ  nodefault,statem (cid:122) (cid:125)(cid:124) (cid:123) + (cid:90) ξˆ ξ (cid:40) ω 3m A 3m F(cid:48)(cid:0) IE+(1−y(ξ,δ))I (cid:1)δ (cid:0) 1+ ξ r 2 D(cid:1) +(1−δ) (cid:0) 1+r 3 D(cid:1) (cid:41) ∆ 1 ξ dξ      , (B.16) 16

  nodefault,stateg nodefault,statem (cid:122) (cid:125)(cid:124) (cid:123) (cid:122) (cid:125)(cid:124) (cid:123) d d U r 2 D B =− EB E + B ES      (cid:90) ξ∗ ξ(cid:26) ω 3g (cid:0) 1+rI(cid:1)δ· ξ D (cid:27) ∆ 1 ξ dξ+ (cid:90) ξˆ ξ(cid:26) ω 3m A 3m F(cid:48)(cid:0) IE+(1−y(ξ,δ))I (cid:1)δ· ξ D (cid:27) ∆ 1 ξ dξ      , (B.17)   nodefault,stateg nodefault,statem (cid:122) (cid:125)(cid:124) (cid:123) (cid:122) (cid:125)(cid:124) (cid:123) dUB EB   (cid:90) ξ 1 (cid:90) ξ 1   dr 3 D =− EB+ES    ξ∗ {ω 3g (1−δ)·D} ∆ ξ dξ+ ξˆ {ω 3m (1−δ)·D} ∆ ξ dξ   =0. (B.18) Wenowturntothedecisionsintheprimaryequitymarket. Buyingmoreequitynotonlyrequiresthebankertogiveupconsumptionintheinitialperiodin exchange for a higher share of future dividends, but it also changes the mix of inside and outside equitywhichmattersfortheincentivestomonitorthroughequation(B.10),Inaddition,thebanker understandshowputtingmoreofherownequitychangestheincentivesofsaverstobuyequityand toholddeposits. Thelatterwillbepricedbythebankertotheextentthatcontractsarecomprehensive. Overall,theoptimalityconditionwithrespecttoinsideequity–equation(21)–becomes: −γ·P·U(cid:48)(cid:0) eB−P·EB(cid:1) + ES (cid:90) ξ ∑ω DPS (ξ,δ) 1 dξ+ψ ·P+ψ dIC +ψ dES =0. 1 EB+ES ξ∗ s 3s 3s ∆ ξ BS ICdE 1 B ESdE 1 B (B.19) where ψ is the multiplier on the equity supply schedule (B.5) offered by savers, satifying the ES complementarityslacknessconditionψ ·ν =0 ES ES Finally,thebankeralsochooseshowmuchoutsideequitytoraisefromsavers,ES,andtheprice at which the bank will issue equity in the primary market, P. As was the case for inside equity, thesechoiceswillmatterfortheincentiveofsaverstobuyequityasdescribedintheequitysupply schedule(B.5). TheoptimalityconditionsforES andP,whichdonothaveacounterpartinthebaselinemodel, are: EB (cid:90) ξ dξ dIC dES dDS − ∑ω DPS (ξ,δ) +ψBS·P+ψ +ψ +ψ I =0, (B.20) EB+ES ξ∗ s 3s 3s ∆ ξ ICdES ESdES DSdES c −γ·EB·U(cid:48)(cid:0) eB−P·EB(cid:1) +ψ · (cid:0) EB+ES(cid:1) +ψ dES +ψ dDS I =0. (B.21) 1 1 BS 1 ES dP DS dP c Conditions (B.20) and (B.21) can be easily interpreted. Selling equity to the savers delivers the shadow benefit of more equity but reduces the banker’s share of future dividends, thus changing the incentive to monitor. This combination moves the banker to a different point in the equity 17

supply schedule of the savers. Finally, a higher equity issuance price affects the banker’s current consumption negatively, because she has to pay this price, but has a positive balance sheet effect andallowsthebankertomoveatadifferentpointonthesavers’equitysupplyschedule. B.3 Modifiedentrepreneurs’problem Asinthebaselinemodel,theentrepreneurusesherowncapitalandborrowsfromthebanktoinvest intheproject. Theloancontractspecifiestheloanamountandtheloanrate. Asalreadymentioned, the entrepreneur repays her loan only if state g realizes. We, additionally, introduce adjustment costs when part of entrepreneur’s initial investment is liquidated. These costs are paid by the entrepreneurintheintermediateperiodandareafunctionoftherequiredadjustment,c (y(ξ,δ)·I)= I c (y(ξ,δ)·I)φ I,wherec >0,φ ≥0. Forsimplicity,weassumethatE paysthesecostsoutofnew I I I endowment,eE,shereceivesatt =2. NotethatE cannotinvestinmorelong-termprojectsatt =2, 2 thussheconsumesatt=3whatisleftoftheperiod2endowmentafterpayingtheadjustmentcosts. Hence,theutilityofanindividualentrepreneur–equation(26)–becomes:  norun run   (cid:122) (cid:125)(cid:124) (cid:123) (cid:122) (cid:125)(cid:124) (cid:123)  UE =∑ω  (cid:90) ξ (cid:2) A F (cid:0) IE+(1−y(ξ,δ))I (cid:1) −(1−y(ξ,δ))I(1+rI) (cid:3)+ dξ + (cid:90) ξ∗ A F (cid:0) IE(cid:1)dξ   3s 3s 3s s   ξ∗ ∆ ξ ξ ∆ ξ      (cid:90) ξ dξ +eE− c (y(ξ,δ)·I) . (B.22) 2 I ∆ ξ ξ (cid:124) (cid:123)(cid:122) (cid:125) adjustmentcosts Theoptimalloandemandofanindividualentrepreneur–equation(27)–becomes: (cid:34) (cid:35) LD: ω (cid:90) ξ (1−y(ξ,δ)) A F(cid:48)(cid:0) IE+(1−y(ξ,δ))·I (cid:1) −(1+rI)+I· ∂LS (cid:18) ∂LS (cid:19)−1 dξ 3g ξ∗ 3g ∂I ∂rI ∆ ξ (cid:90) ξ dξ − y(ξ,δ)·c(cid:48)(y(ξ,δ)·I) =0. (B.23) I ∆ ξ ξ Thesecondlinein(B.23)showstheimpactofinvestmentonthemarginaladjustmentcosts. The firstlinehasthesametermsasinthebaselinemodelbutthereisasubtledifference. Entrepreneurial andbankdefaultdonotnecessarilyoccuratthesametimegiventhatthebankissolventinstatem ˆ forξ>ξ,whiletheentrepreneuralwaysdefault. Hence,∂LS/∂I (cid:54)=0astheentrepreneurpricesthe recovery value of her investment in state m. The partial derivatives of the loan supply curve with respecttotheloancharacteristics,takingallaggregatevariablesasgiven,are: ∂LS = (cid:90) ξ (cid:8) ω A (1−y(ξ,δ))F(cid:48)(cid:48)(cid:0) IE+(1−y(ξ,δ))I (cid:1)(cid:9) 1 dξ, (B.24) 3m 3m ∂I ξˆ ∆ ξ 18

∂LS (cid:90) ξ 1 = {ω } dξ. (B.25) ∂rI ξ∗ 3g ∆ ξ B.4 Modifiedplanner’sproblemandintermediationmargins Theplanner’sproblemissimilartotheonedescribedinsection3.1. Theonlydifferenceisthatthe planner will also internalize the effect of her decisions on the equity supply schedule and will also have two additional optimality conditions for ES and P (the secondary equity price is substituted outinthebudgetsets). Thus,thegenericfirst-orderconditionfortheplanneris: dUh dBS dIC dGG dLD dDS dES ∑ w +ζ +ζ +ζ +ζ +ζ +ζ =0, h dX BS dX IC dX GG dX LD dX DS dX ES dX h={E,R,B} where ζ is the multipliers on the equity supply schedule (B.5), satifying the complementarity ES slacknessconditionζ ·ν =0. ES ES Theabilitytochoosethelevelofoutsideequityintroducesanadditionalintermediationmargin. To see this fix the assets mix, i.e., the investment-liquidity margin, and also fix the liabilities mix, i.e., the equity-deposits margin. Then, one can additionally use the balance sheet and incentive compatibility constraints to express all variables in terms of the amount of outside equity issued. In other words, the banker can scale up or down the level of credit extension and bank size by choosing different levels of outside equity even if the marginal relationship between liquid and illiquid assets and between equity and deposits is fixed. One way to express this margin is to combine the optimality conditions for outside equity and inside equity. We will refer to it as the equity-mixmargin,denotedbyEE. Thebankerandtheplannerwillhavedifferentincentiveswhen choosingbetweeninsideandoutsideequity. Hence,thereisanadditionalwedgebetweentheprivate andplanningsolutionontopofILIQ andED inequations(38)and(39): wedge wedge EE =EE +EE . (B.26) sp B wedge The wedge in (B.26) represents a distortion in the equity mix or, as discussed above, in the scaleofcreditintermediationchosenbythebankerversustheplanner. Indeed,weshowinthenext sectionthatusualprudentialtoolsareinsufficienttocorrectthisthirdmargin,thoughtheILIQand EDmarginscanbecorrected. TheEE marginitcanbeaddressedwithtargetedcorrectivetaxes. B.5 Numericalexample This section presents a numerical example for the equilibrium in the extended model. Table B.4 shows the parameterization of the exogenous variables, which have been chosen such that it is optimal for the private economy and the planner, at least for some weights, to invest in outside 19

equity. Table B.5 reports the private equilibrium as well as the planner’s solutions for different weightsinthesocialwelfarefunction. TableB.6reportstheeffectsofindividualregulations. The planner chooses to raise outside equity as long as the weight on entrepreneurs is high enough (w ≥0.5 in this example). The reason is that raising outside equity reduces the reliance E on deposits, which reduces the need for holding liquidity and allows for more credit extension. In addition,thelowerdemandfordepositssuppressesdepositratesandallowstheplannertosetlower loan rates given the intermediation spread required to satisfy the banker’s incentive compatibility constraint. These effects are beneficial for entrepreneurs, but reduce savers’ utility. Hence, the plannerwillchoosetoraiseoutsideequitywhenw ishighenough. Forlowerw ,theplannerwill E E notchoosetoraiseoutsideequityandtheanalysisisthesameasinthebaselinemodel. Therestoftheconclusionsderivedinsections3.3and4continuetoholdintheextendedmodel. To summarize a few, the planner chooses both higher liquidity and capital ratios to address the distorted investment-liquidity and equity-deposits margins.5 The run probability goes down and liquidity provision is higher in the planner’s solution. As in the baseline model, the welfare of savers and entrepreneurs improves, while the banker is driven to her participation constraint. The total surplus created by the planner is positive. Moreover, the planner chooses higher common equitycapital,fewerliquidassetholdingsandhigherinvestmentwhentheweightonentrepreneurs ishigher,andviceversa. Finally,theimpactofindividualregulationsissimilartothatinthebaseline model. Extending the analysis in section 4.5 to three intermediation margins, three independent tools are, in principle, needed to replicate the planner’s solution when ES >0. However, the tools need to be jointly binding, which is not the case for any of the combinations of the four capital and liquidity regulations discussed in section 4.5. Instead, corrective (Pigouvian) taxes can be used in combination with a capital and a liquidity tool to replicate the planner’s solution. These taxes can affect marginal decisions, but the tax proceeds are assumed to be fully rebated to agents in a lump-sum fashion in order to eliminate any income implications. Despite the fact that such taxes mayseemunrealisticfromthelensofactualpolicyimplementation,theycanpointtothedirection that the additional distortion operates. For example, a capital requirement and a liquidity tool can be combined with a corrective tax levied on inside equity to push relatively more outside equity intothebankandbringthescaleofcreditintermediationdowntodesirablelevels. Alternatively, a leveragerequirementandaliquiditytoolcanbeusedincombinationtoacorrectivetaxonthetotal sizeofthebank(orjustdeposits)topushthescaleofcreditintermediationdown. Overall,thethird intermediationmargindeterminesthescaleofcreditintermediation,becausethebankercandecide on the level of equity issued to scale up their balance sheet. Given regulations that pin down the othertwomargins,atargetedtoolisneededtocontrolthesizeofthebank. 5For wE ∈[0.2,0.4], where the planner sets ES=0, the capital ratio in the planner’s solution is lower than in the private equilibrum. This does not mean that the planner chooses a lower capital ratio compared to an LCR regulated economy,sincethedropincapitalisduetothebigincreaseinliquidity. 20

Tables eS 2.95 ρ 1.00 ω 65% 1 3g eS 1.10 γ 0.10 ω 30% 2 3m eE 0.05 A 3.30 c 2.5% 1 3g D eE 0.01 A 1.15 φ 0.50 2 3m D eB 0.20 A 0.70 c 2.5% 3b I EB 0.13 α 0.75 φ 3.0 0 I δ 0.50 ξ 1.20 PB 0.14 β 0.70 ξ 0.10 TableB.4: Parameterization. PE SPforweights(w ,w ) E S (0.2,0.6) (0.3,0.5) (0.4,0.4) (0.5,0.3) (0.6,0.2) I 0.895 0.831 0.838 0.845 0.847 0.854 LIQ 0.085 0.243 0.239 0.233 0.175 0.172 1 D 0.789 0.907 0.906 0.904 0.812 0.815 CEQ 0.191 0.167 0.170 0.174 0.209 0.211 EB/(ES+EB) 0.997 1.000 1.000 1.000 0.671 0.685 CR 0.213 0.201 0.203 0.206 0.247 0.247 LevR 0.194 0.156 0.158 0161 0.205 0.206 LCR 0.221 0.360 0.356 0.351 0.320 0.315 NSFR 0.654 0.747 0.744 0.741 0.727 0.724 P 0.958 1.000 1.000 1.000 1.018 0.998 rI 1.650 1.672 1.668 1.664 1.665 1.662 rD 1.161 1.342 1.339 1.335 1.127 1.136 3 q 0.482 0.450 0.450 0.450 0.431 0.433 q 0.224 0.139 0.142 0.145 0.157 0.160 f Liq.Prov. 0.884 1.011 1.009 1.007 1.013 1.010 %∆UE - 0.84% 0.91% 0.98% 1.36% 1.38% %∆US - 3.19% 3.16% 3.10% 2.70% 2.66% %∆UB - -0.22% -0.22% -0.22% -0.22% -0.22% %∆Usp - 2.04% 1.81% 1.59% 1.45% 1.32% %∆Ssp - 1.27% 1.28% 1.29% 1.28% 1.27% TableB.5: PrivatelyversusSociallyOptimalSolutions. 21

PE CR LevR LCR NSFR SP I 0.895 0.902 0.913 0.861 0.862 0.847 LIQ 0.085 0.108 0.111 0.217 0.216 0.175 1 D 0.789 0.797 0.816 0.897 0.897 0.812 CEQ 0.191 0.213 0.209 0.181 0.181 0.209 EB/(ES+EB) 0.997 0.844 0.912 1.000 1.000 0.671 CR 0.213 0.236 0.229 0.210 0.210 0.247 LevR 0.194 0.211 0.204 0.168 0.168 0.205 LCR 0.221 0.249 0.249 0.338 0.337 0.320 NSFR 0.654 0.678 0.676 0.731 0.730 0.727 P 0.958 0.918 0.889 1.000 1.000 1.018 rI 1.650 1.623 1.635 1.655 1.654 1.665 rD 1.161 1.134 1.181 1.320 1.319 1.127 3 q 0.482 0.459 0.463 0.452 0.452 0.431 q 0.224 0.202 0.204 0.154 0.154 0.157 f Liq.Prov. 0.884 0.937 0.933 0.998 0.998 1.013 %∆UE - 1.27% 0.91% 1.10% 1.10% 1.36% %∆US - 1.08% 1.13% 2.89% 2.87% 2.70% %∆UB - -0.22% -0.22% -0.22% -0.22% -0.22% %∆Usp - 0.92% 0.75% 1.37% 1.37% 1.45% %∆Ssp - 0.71% 0.60% 1.25% 1.25% 1.28% Table B.6: Single regulations versus planner’s solution for (w ,w )=(0.5,0.3). Regulation is set at its E S maximum level such that there are gains in social welfare, while the banker’s participation constraint is satisfied. 22