Collective Moral Hazard and the Interbank Market

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Collective Moral Hazard and the Interbank Market Levent Altinoglu and Joseph E. Stiglitz 2020-098 Please cite this paper as: Altinoglu, Levent, and Joseph E. Stiglitz (2020). “Collective Moral Hazard and the Interbank Market,” Finance and Economics Discussion Series 2020-098. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2020.098. 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.

Collective Moral Hazard and the Interbank Market Levent Altinoglu and Joseph E. Stiglitz∗ September 2020 Abstract Theconcentrationofriskwithinfinancialsystemisconsideredtobeasourceofsystemic instability. Weproposeatheorytoexplainthestructureofthefinancialsystemandshowhow italterstherisktakingincentivesoffinancialinstitutions. Webuildamodelofportfoliochoice andendogenouscontractsinwhichthegovernmentoptimallyintervenesduringcrises. Byissuing financial claims to other institutions, relatively risky institutions endogenously become large and interconnected. This structure enables institutions to share the risk of systemic crisis in a privately optimal way, but channels funds to relatively risky investments and creates incentives even for smaller institutions to take excessive risks. Constrained efficiency can be implementedwithmacroprudentialregulationdesignedtolimittheinterconnectednessofrisky institutions. ∗Levent Altinoglu: Federal Reserve Board of Governors. Joseph E. Stiglitz: Columbia University. A previous version of this paper was circulated under the title, “Collective Moral Hazard, Risk Sharing, and Concentration in the Financial System.” The views expressed in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or of anyone elseassociatedwiththeFederalReserveSystem. WewouldliketothankGadiBarlevy,GiacomoCandian,Eduardo Davila, Joao Gomes, Martin Guzman, Tomohiro Hirano, Gazi Kara, Alexandros Vardoulakis, and participants of seminarsandconferencesattheCentralBankofChile,ColumbiaUniversity,theFederalReserveBoardofGovernors, FinanceForumMadrid,GRIPSTokyo,IFABSOxford,InternationalEconomicAssociationWorldCongress,theNorth AmericanEconometricSocietySummerMeetings,andtheRIDGEworkshoponfinancialstability. Financialsupport fromINETisgratefullyacknowledged.

1. INTRODUCTION Asalientfeatureofthefinancialsystemsofadvancedeconomiesisthepredominanceofafewlarge financialinstitutionswhoarehighlyinterconnectedwithmanysmallerinstitutions. Thisstructure, sometimes referred to as core-periphery or hub-and-spoke, is a source of systemic instability as it concentrates resources in systemically important financial institutions (henceforth SIFIs), leaving the rest of the system vulnerable to their failure. Indeed, this concentrated structure is widely seen as a contributing factor for financial crises of late, and has received considerable attention from policymakers as a result. What leads to such a structure to arise in the first place? Why do financial institutions concentrate risk in a manner that generates systemic instability, rather than spreadingriskacrossthefinancialsystem? Addressing these questions requires an understanding of the interaction between the portfolio choices and risk sharing incentives of financial institutions in a setting with heterogeneous agents and bilateral exposures. Modeling these elements jointly poses methodological challenges, and as a result the literature has typically sought to make progress on one dimension or another. The literatureoncollectivemoralhazard(e.g. DavilaandWalther(2020),Keister(2016),Bornsteinand Lorenzoni (2018), Farhi and Tirole (2012), Acharya and Yorulmazer (2007)) has shed important light on how government intervention affects the portfolio choices of financial institutions, but has stopped short of explaining the structural features of the financial system, or takes as given the existence of institutions which are ‘too big to fail’. By contrast, the literature on endogenous network formation has broken new ground in this regard but often takes financial contracts to be exogenous,ordoesnotconsiderhowthisstructureaffectsagents’portfoliochoices. Inthispaper,westudytheinteractionbetweentheportfoliochoicesoffinancialinstitutionsand their risk sharing incentives. To this end, we construct a parsimonious model of portfolio choice and systemic crisis with two key ingredients: a government which optimally intervenes during crises and is subject to limited commitment; and an endogenous interbank market through which financial institutions can exchange endogenous financial contracts. Our framework is tractable enoughtoprovideananalyticalcharacterizationofequilibriaandwelfare. Our paper makes three contributions. First, we offer a new theory of the structure of the financial system.1 In our setting, the concentrated structure that emerges in equilibrium enables financial institutions to share the risk of systemic crisis in a privately optimal way. During a crisis,thegovernmentoptimallybailsoutanyinstitutionwhichissufficientlylargeorinterconnected 1Undoubtedly, there are a multitude of factors which likely play a role in shaping the structure of the financial system,includingeconomiesofscale,regulation,andtechnology,tonameafew. Wesetasidetheserelevantconsiderationstofocusontheroleofrisksharingandsystemiccrises. 1

in equilibrium. Interconnected SIFIs arise endogenously because they are the only private agents that can insure other financial institutions against systemic crises, since their liabilities are implicitly guaranteed by the government. Risk sharing between these SIFIs and non-SIFIs generates a core-periphery structure in financial markets. Despite its parsimony, the predictions of our model are therefore consistent with this and other important qualitative features of the data, discussed in section1.2. Our second contribution is to show that the financial structure that emerges in equilibrium alters the risk taking behavior of financial institutions in two ways. First, the interbank market channels funds to investment opportunities with high upside risk. As a result, the institutions whichbecomelargeandinterconnected(SIFIs)arerelativelyrisky. Second,theimplicitinsurance offered by a SIFI’s liabilities enables smaller, peripheral institutions to take excessive risks, even though they do not directly benefit from bailouts themselves. In this manner, the excessive risk taking that characterizes SIFIs propagates to other financial institutions by means of interbank financial markets. Thus, our theory shows that the systemic instability generated by the financial systemismoreseverethanpreviouslyunderstood. Third, we show that this equilibrium structure of the financial system is suboptimal from a welfare point of view. We characterize the optimal regulatory policy required to implement constrainedefficiencyandshowthatitismacroprudentialinnatureanddiscouragesinterbanklending to institutions with risky portfolios. Moreover, we show that restrictions on bank leverage are insufficient to implement constrained efficiency. We thus provide a rationale for some post-crisis regulationsdesignedtolimitsizeandinterconnectedness,butshowthattheymaybeinadequatein importantrespects. Our theory is guided in part by important qualitative features of data, which we discuss in section 1.2. Various financial markets are highly concentrated and feature a pronounced coreperiphery structure in which the small number of large and interconnected institutions at the core holdriskierassets. Moreover,theseSIFIstypicallybenefitfromanimplicitgovernmentguarantee whichlowersthecostoftheirliabilities. Thislowercostisconsistentwiththeevidencethat,historically,thecreditorsandcounterpartiesofsystemicallyimportantfinancialinstitutions(SIFIs)have been among the main beneficiaries of government bailouts, despite not being bailed out directly themselves.2 These empirical observations suggest an important role for implicit guarantees for understandingthestructureofthefinancialsystem. Relativetotheliteratureoncollectivemoralhazard(e.g. DavilaandWalther(2020),Farhiand Tirole(2012),AcharyaandYorulmazer(2007)),thekeynewelementinourmodelisaninterbank 2For example, according to the Financial Crisis Inquiry Commission report into the 2008 financial crisis, of the $182billionbailoutfundsissuedtoAIGfromtheUSgovernment,abouthalfwaspassedontoitsmajorcreditorsand othercounterparties. SeeUnitedStates(2011),page377. 2

marketthroughwhichfinancialinstitutionscanexchangeendogenousfinancialcontracts. Weview this very broadly as capturing the various markets through which financial institutions interact, such as derivatives, insurance, equity, syndicated loan, corporate debt, deposit, or money markets. Weneverthelessadopttheterm‘interbankmarket’foreaseofexposition. We consider an environment with three dates (0, 1, and 2) in which a risk averse household ownsmanyfinancialinstitutions,whichwecallbanks. Bankshaveaccesstotwoprojectsatdate0 whichconvertphysicalcapitalintoaconsumptiongoodafteroneperiod: a‘prudent’projectwhich is common to all banks, and a risky project which is specific to each bank. While the prudent project has a risk-free return, the returns to the risky project are subject to an aggregate shock at date 1. All risky projects are excessively risky from a social perspective: any bank’s risky project offers the same expected return as the prudent project, but with higher risk. We assume banks are heterogeneous with respect to the projects to which they have access. Specifically, each bank differs in how exposed its risky project is to the aggregate shock. At date 1, after the resolution of uncertainty, banks have access to a risk-free continuation project which pays out at date 2. In addition, the household owns outside,‘traditional’ firms who make less productive use of capital, similartoLorenzoni(2008). To invest in either project at date 0, a bank can raise funds from two sources: the household and other banks. Banks can borrow funds from the household subject to a limited commitment problem, through the use of a state-contingent debt contract similar to Lorenzoni (2008). Because of limited commitment, the privately optimal state-contingent debt contract limits the borrowers’ ability to efficiently allocate funds across states of the world, as banks may not be able to fully insureexanteorraisefundsexpostagainstlossesfromriskyinvestments. In addition to borrowing from the household, banks can raise funds from one another in an interbank financial market – meant broadly to capture any of the financial markets through which financialinstitutionsexchangerisk–byenteringintobilateralinterbankfinancialcontractsatdate 0. We assume the contracting environment between any two banks is free of commitment or enforcement problems, so that interbank contracts are complete and channel funds to the investment opportunitieswhichmaximizethejointsurplusoftheborrowerandlender.3 At date 0, each bank decides how to allocate its portfolio across the available investment projects (prudent and risky) and financial claims issued by other banks on the interbank market. Assets are priced by the stochastic discount factor of the risk averse household and accordingly reflectariskpremium. At date 1, the aggregate shock to risky projects is realized and financial claims are settled. 3Whileourbroadresultswouldholdwithincompleteinterbankcontracts,theassumptionthatthesecontractsare completeisusefultohighlightthatanyconstrainedinefficiencyofprivaterisksharingbetweenbanksdoesnotderive fromimperfectionsininterbankfinancialmarkets. 3

Bankscanthentradephysicalcapitalinaspotmarketinordertoinvestinthecontinuationproject. In the bad state of the world, a bank holding risky assets incurs losses which it must finance by selling physical capital. If the aggregate losses of the banking sector as a whole are sufficiently large – that is, if banks’ collective exposure to risky assets is sufficiently high – then the economy entersintoacrisisinwhichbanksareforcedtofireselltheircapitalholdingstothelessproductive traditionalsector. Next,weintroduceabenevolentgovernmentwhichseekstomaximizehouseholdwelfareusing taxes and transfers which are available only after the aggregate shock at date 1 is realized. The governmenttakesagents’optimizingbehaviorasgivenandfacesinformationfrictionswhichimply bailouts can be only imperfectly targeted across banks. In a crisis, the government always finds it optimal to bail out banks in order to prevent the inefficiencies associated with the fire sale of capital to the traditional sector. The government bails out only the most critical banks so as not to incentivize excessive risk taking by other banks. The government cannot commit to a policy whichissuboptimalexpost. Moreover,sinceabailoutoccursonlywhenmanybanksarefailingat thesametime,thebailoutpolicyintroducesastrategiccomplementarityinbanks’date0portfolio choices. As a result of the strategic complementarity, there are two subgame perfect Nash equilibria. In the ‘prudent equilibrium’, all banks undertake prudent investments, and so crises and bailouts never occur in equilibrium. In the ‘risky equilibrium’, both risk sharing and risk taking are constrained inefficient. The interbank market channels funds to the investment opportunities with the highest upside risk. As a result, the banks with relatively risky investment opportunities become excessivelylargeandinterconnected,suchthattheybenefitfromanimplicitgovernmentguarantee on their assets. The safety provided by the implicit guarantee drives a wedge between the private and social value of financial claims issued by risky banks. In turn, these SIFIs invest in their risky project,indirectlyexposingallotherbankstopreciselytheriskiestprojectsintheeconomythrough theirholdingsofSIFIliabilities. Thecore-peripherystructureofinterbankmarketplaysacrucialroleininsuringnon-SIFIbanks againstsystemiccrises. Whenabankholdsariskyasset,itbearscrisisrisk–theriskthatitincurs a loss during a crisis, precisely when the the stochastic discount factor is highest. Banks are unwilling to hold excessively risky assets in the absence of some form of insurance against this risk. Thegovernmentprovidessuchinsuranceintheformofbailouts,butonlytoSIFIs. Whilesmaller,peripheralbanksdonotdirectlybenefitfromgovernmentguarantees,theybenefit indirectly through the interbank market by investing in the liabilities of SIFIs. In a crisis, a SIFI forgoes some of the bailout funds it receives from the government to pay its claimholders a higher rate of return than what it earns on its own assets. This insures claimholders against losses fromtheSIFI’sinvestmentsduringcrises,makingriskyassetsappearsaferfromtheperspectiveof 4

each individual bank. The insurance value provided by SIFI liabilities is reflected in a lower risk premium, consistent with empirical evidence. As a result of this insurance, even smaller banks whodonotdirectlybenefitfromthegovernmentguaranteemaytakeexcessiverisks. Toelucidatetheroleoftheinterbankmarket,weanalyzetwobenchmarkvariantsofthemodel. Inthefirst,weconsideraspecialcaseinwhichthereisnointerbankmarketandshowthatthereis never excessive risk taking in equilibrium. Without an interbank market in which banks can share the proceeds of bailouts widely, the government’s optimal bailout policy is sufficient to eliminate the risky equilibrium. In the second variant of the model, we vary the degree of household risk aversionandshowthat,underriskneutrality,banksneverundertakeexcessiveriskandSIFIsnever arise in equilibrium. Indeed, it is the insurance provided by SIFI claims that makes excessively riskyinvestmentsworthwhileforotherbanks. The risky equilibrium is associated with strictly lower household welfare due to excessive consumption volatility. The source of the constrained inefficiency is a soft budget constraint externality, common to models with strategic complementarities, in which agents do not internalize how their collective exposure to the aggregate shock reduces household consumption in the bad statethroughthelump-sumtaxesneededtofinancebailouts.4 Aregulatorcanaddresstheconstrainedinefficiencythroughexanteinterventioninbanks’date 0portfoliodecisionsthroughtheuseoftaxincentivesorquantityrestrictionsininterbankfinancial marketswhichdistortportfoliochoicesawayfromclaimsissuedbybankswithriskyportfolios,in ordertopreventthesebanksfrombecomeexcessivelylargeandinterconnected. Wethusprovidea rationaleformacroprudentialpoliciesdesignedtoreducetheinterconnectednessoflargeandrisky institutions. Importantly, we show that the focus on limiting bank leverage, while helpful, may be inadequate, and that greater attention should be devoted to reducing interconnectedness and size directly. 1.1. Relatedliterature Our paper relates most closely to the collective moral hazard literature, particularly Farhi and Tirole (2012), Davila and Walther (2020), Keister (2016), and Acharya and Yorulmazer (2007), whoshow thatbailoutsmay createinefficientincentives forrisktaking through leverage,maturity mismatch, or by undertaking similar projects. Other papers highlighting the significance of timeinconsistencyingovernmentinterventionsareFreixas(1999),ChariandKehoe(2016),Holmstrom and Tirole (1998), Nosal and Ordonez (2016), Schneider and Tornell (2004), Dell’Ariccia and Ratnovski(2019),andMorrisonandWalther(2018). 4Forinstance,seeFarhiandTirole(2012). 5

Relative to this literature, we examine collective moral hazard on a different margin of agents’ portfolio decisions: their risk sharing incentives. The key new element in our framework is an interbank market through which banks can exchange endogenous financial contracts. We show that interconnected SIFIs arise endogenously in this market due to banks’ inefficient risk sharing incentives,andthatthismayleadevenperipheral,non-SIFIbankstotakeexcessiverisk. Inaddition, ourstylizedmodelgeneratesempiricallyrelevantfeaturesofinterbankfinancialmarkets. Our paper is also related to a growing literature on ex ante inefficiencies and macroprudential policy,particularlyLorenzoni(2008),DavilaandKorinek(2017),Bianchi(2016),andBianchiand Mendoza (2018). In our model, the government’s optimal bailout policy completely eliminates inefficiencies related to pecuniary externalities, but replaces them with an inefficiency deriving from the effect of strategic complementarities on the hardness of the household budget constraint. InBornsteinandLorenzoni(2018),expostgovernmentinterventiondoesnotleadtoinefficientex ante incentives. This is not the case in our setting because ex post intervention alone cannot fully eliminatetheinefficienciesassociatewithexcessiverisktaking. Several papers examine optimal policy for regulating systemically important financial institutions, such as Freixas, Parigi, and Rochet (2000), Freixas and Rochet (2013), and Davila and Walther (2020). While these papers take as given that large banks exist, SIFIs emerge endogenously in our model as the result of private risk sharing arrangements. We also show that the inefficienciesassociatedSIFIsareintimatelylinkedwiththerisksharingincentivesofallbanks. There is a growing literature which analyzes the endogenous formation of financial networks, including Acemoglu et al. (2014), Chang (2019), Di Maggio and Tahbaz-Salehi (2014), Elliot et al. (2014), Elliot et al. (2018), Erol (2018), Kanik (2019), Leitner (2005), Shu (2019). Our model includes an element of endogenous network formation which depends on a strategic complementarity in banks’ portfolio choices. Moreover, to solve for equilibrium, we do not need to keep track of the full structure of the underlying network; solving for a few features of the network is sufficient to characterize allocations. As a result, our model is tractable enough to yield analytical characterizationsofequilibriaandwelfare. The rest of the paper is organized as follows. We first summarize motivating empirical evidence. Then we introduce the model and characterize the optimizing decisions of private agents. Wethenintroduceagovernmentlackingcommitmentatdate0andcharacterizetheexpostefficient policy of transfers. After solving for general equilibrium, we setup the problem of a constrained plannerandcharacterizetheinefficienciesinthecompetitiveequilibrium. Finally,weanalyzehow macroprudentialpoliciescanimplementconstrainedefficiency. 6

1.2Motivatingempiricalevidence Here, we present a brief review of the empirical evidence on the structure of interbank markets, with a focus on three ‘stylized facts’. The overall picture painted by these facts is one of a highly concentrated financial system in which a small number of large and interconnected institutions hold riskier assets, and benefit from an implicit government guarantee which lowers the cost of theirliabilities. Thefirststylizedfactisthatinterbankfinancialmarketstypicallyexhibitastrongcore-periphery structure, in which a few highly interconnected institutions at the core interact with the many sparsely connected institutions in the periphery. This has been shown for a wide range of markets includinginter-dealermarketsforcorporatebonds,over-the-counterderivativesmarkets,interbank markets,andfedfundsmarkets.5 The second fact is that these large and interconnected financial institutions often benefit from an implicit government guarantee of their assets or liabilities. Moreover, this guarantee lowers their costs of funding on deposit or wholesale funding markets, and lowers their cost of insurance viacreditdefaultswapsorputoptionsonequityprices.6 The third fact is that these large and interconnected institutions often make riskier investments than those in the periphery. Afonso et al. (2015), and several papers cited therein, show that the anticipationofgovernmentsupportisassociatedwithincreasedrisktaking. Moreover,Elliottetal. (2019) provide evidence that banks who are more interconnected also undertake more correlated risks. Consistent with these three features of the data, our model will endogenously feature a coreperipherystructureintheinterbankmarketinwhichlarge,interconnectedbanksatthecorebenefit from an implicit government subsidy and undertake riskier investments. In addition, the liabilities of these SIFIs will command a lower risk premium, reflecting the insurance value provided by the implicitgovernmentguarantee. 5For example, see Di Maggio et al. (2015) for evidence of a core-periphery structure in the inter-dealer market forcorporatebonds,Peltonenetal. (2014)andVuillemeyandBreton(2013)forover-the-counterderivativesmarkets, Boss et al. (2004), Chang et al. (2008), Craig and von Peter (2014), and van Lelyveld and in ’t Veld (2014) for interbank markets, and Afonso and Lagos (2015), Allen and Saunders (1986), Bech and Atalay (2010) for the fed fundsmarket. 6See Kelly et al. (2016) for evidence of the size of implicit government guarantees from out of the money put options,andVeronesiandZingales(2010)fromdataoncreditdefaultswapsforthelargestfirmsfrom2008Paulson plan,andLucasandMcDonald(2006)andLucas(2019)forthesizeofguaranteesgovernment-sponsoredenterprises. SeealsoO’HaraandShaw(1990),BakerandMcArthur(2009),andDemirguc-KuntandHuizinga(2013). 7

Figure1: ModelEnvironment 2. MODEL There are three periods: dates 0, 1, and 2. All uncertainty is resolved at date 1. There are four types of agents: a representative household, banks, traditional firms, and later we introduce a government. The household owns N banks, where I denotes the set of banks. Each representative bank i consists of a continuum of identical, atomistic banks. There are two goods, a consumption good and a capital good. The consumption good can be costlessly converted one-for-one into the capital good at any date. Capital can be converted into the consumption good only via investment projects, which are available only to investors. Banks pay out dividends to the household only at date2. Figure 1 illustrates the general environment. The risk averse representative household owns banks and traditional firms, the latter of which always makes less productive use of capital. Each bank has access to a prudent (risk-free) project and a risky project, which are subject to an aggregate shock at date 1. Banks are heterogeneous in how exposed their risky projects are to the aggregate shock. Banks can raise funds to invest in these projects from the household, via an optimal state-dependent debt contract, or from one another via optimal bilateral interbank financial contracts. Finally,wewilllaterintroduceagovernment. Wenowdiscusseachagentinmoredetail. 2.1. Household The representative, risk averse household gets utility from consuming the consumption good according to u(·), where u(·) is twice-differentiable, u(cid:48)(·) > 0, u(cid:48)(cid:48)(·) < 0, and u(·) satisfies the 8

Inadaconditions. Eachperiodthehouseholdisendowedwitheunitsoftheconsumptiongood. At date1(afteruncertaintyisresolved),thehouseholdalsohasaccesstoarisklessstoragetechnology, which we call bond B .7 At date 0, the household is offered a state-contingent financial contract 1 (cid:0) di, (cid:8) di(s),di(s) (cid:9) (cid:1) by each bank i, which consists of a loan di from the household to bank i at 0 1 2 s 0 date0,andasetofstate-contingentpayments (cid:8) di(s),di(s) (cid:9) fromthebankbacktothehousehold 1 2 s at dates 1 and 2, where states are indexed by s. Let fi be an indicator function taking a value of 1 0 if the household accepts bank i’s contract. In addition, the household faces lump-sum taxes T in t eachperiodt. Appendix1specifiesthehousehold’sprobleminmoredetail. Thehouseholdsolvesaconsumptionsaving and portfolio allocation problem, taking as given the financial contracts that banks offer, to maximize expected utility E[u(c )+u(c (s))+u(c (s))], subject to each period’s budget con- 0 1 2 straints. Thefirst-orderconditionsfor fi andthedate1bondholdingsB are 0 1 u(cid:48)(c )di ≥E (cid:2) u(cid:48)(c (s))di(s)+u(cid:48)(c (s))di(s) (cid:3) (1) 0 0 1 1 2 2 u(cid:48)(c (s))=u(cid:48)(c (s)). (2) 1 2 Condition (1) implies that the household accepts bank i’s contract only if the expected discounted return promised by the contract exceeds the marginal utility of consumption at date 0. Condition (2)equatesmarginalutilityacrossdates1and,inanystate. 2.2. Investmentprojectsandaggregaterisk At date 0, banks have access to investment projects which convert capital at date 0 into the consumption good at date 1. Each bank has access to a prudent project, which is common to all banks, and a risky project which is specific to each bank. We assume that a bank cannot directly invest in another bank’s risky project; rather, each bank can directly invest in the prudent project ortheirownriskyproject.8 Theprudentprojectyieldsarisk-freereturnatdate1ofR >0unitsoftheconsumptiongood C for each unit of capital invested in the project. On the other hand, each bank’s risky project yields a risky return Ri (s) > 0 at date 1, which varies across states of the world s and across banks i. A 7The date 1 risk-free bond is not necessary for the model’s results, but improves tractability by allowing the householdtocompletelysmoothconsumptionexpostbetweendates1and2. 8This assumption captures the notion that there may be limitations in a bank’s ability to replicate the business modelorinvestmentopportunitiesofotherbanks,duetoconsiderationsrelatedtobanks’businessmodels,geographic exposures,regulatoryconstraints,etc. However,aswewillsee,banksinthemodelwillbeabletogenerateexposure toeachothers’riskyprojectsbytradingfinancialclaims. 9

Ourassumptions willimplythatrisky projectsareexcessively risky: theirexpectedreturnsdo not sufficientlycompensateinvestorsfortherisktheyentail. Weelaborateonthisbelow. TheonlysourceofriskintheeconomyisanaggregateshockR (s)tothereturnsonallbanks’ A risky projects at date 1. The aggregate shock can take a high value or a low value s∈H, L, where R (H)>R (L)>0andE[R (s)]=R . Moreprecisely,thereturnonbanki’sriskyprojectatdate A A A C 1isgivenby Ri (s)=ρ iR (s)−µ i. (3) A A µi is a constant which we assume to be µi =R (cid:0) ρi−1 (cid:1) . This constant simply adjusts the return C of i’s risky project to ensure that all risky projects have an expected return of E (cid:2) Ri (s) (cid:3) =R (the A C return on the prudent project). Thus, each bank’s risky project entails more risk than the prudent project, but does not compensate the investor for that risk. By construction, risky projects are therefore‘excessivelyrisky’fromasocialperspective. Banks are heterogeneous in the riskiness of projects to which they have access, through the parameters ρi >1. The parameter ρi determines how exposed the returns of bank i’s risky project are to the aggregate shock. A bank with a higher ρi is riskier in the sense that the risky project to which it has access has a higher variance. One may interpret ρi as capturing factors inherent to bank i’s business model which expose it to aggregate risk.9 The parameters ρi are thus a reducedformwaytocapturetheheterogeneityintheinherent‘riskiness’ofbanks. 2.3. Marketforphysicalcapital The spot market for capital features the potential for fire sales due the presence of the less productive traditional firms.10 At date 0, each bank decides how much capital ki to invest in 0 projects and how to allocate its capital across the prudent and risky projects. The fraction of its portfolio bank i invests in the prudent project is denoted ωi, with the 1−ωi being invested in i’s riskyproject. Turning to date 1, bank i must pay a unit maintenance cost γ <1 on its capital holdings ki for 0 thecapitaltoremainproductiveatdate1. Atdate1,eachbankihasaccesstoarisklessinvestment, which we call the continuation project, which transforms one unit of capital into one unit of the consumption good at date 2. Each household also owns a so-called traditional firm which has access to a less productive, but riskless investment technology at date 1 only. An investment of kT(s) of capital in a traditional firm at date 1 produces F(kT(s)) units of the consumption good at 1 1 9Inpractice,thiscouldarisefromanybehaviorwhichincreasesthebank’sportfolioreturnsingoodstatesofthe worldbutmagnifieslossesinbadstates,suchasmaturitymismatch,leverage,relianceonwholesalefunding,having runnableliabilities,orsimplyhavingaccesstoprojectsorassetswithahighermarketbeta. 10Weintroducetheselessproductivetraditionalfirmsintothemodeltocapturesomenotionofcapitalmisallocation. 10

date 2, where F(·) is strictly concave, and F(cid:48)(·) is bounded from above by 1 and below by q<1. Since F(·) is strictly concave and F(cid:48)(0)<1, traditional firms make less productive use of capital atdate1thaninvestmentbanks,foranylevelofinvestment. Assumption1: WeassumethatF(0)=0,F(cid:48)(·)>0,F(cid:48)(0)<1,andF(cid:48)(·)≥q,whereγ <q<1. Let q(s) denote the price of capital in date 1 state s, which is determined in a competitive market. Because the consumption good can be costlessly converted one-for-one into the capital good at any date, q(s) ≤ 1 by arbitrage.11 At date 1 state s, traditional firms choose their date 1 capital holdings kT(s) to maximize F(kT(s))−q(s)kT(s). If q(s) ≥ 1, the manager optimally 1 1 1 chooseskT(s)=0,whereasifq(s)<1,thenkT(s)ischosentosatisfythefirstordercondition12 1 1 F(cid:48)(kT(s))=q(s). (4) 1 Finally, we allow for the possibility of government transfers to banks at date 1. Let gi(s,ki,ωi) 0 denote a subsidy to i’s date 1 return on its capital, as a function of the state of the world and bank i’sdate0portfolio. 2.4. Financialmarkets We allow agents to share resources at dates 0 and 1 in two ways. First, the household can save intheconsumptiongoodbylendingittobanksatdate0. Bylendingtobanksthroughtheuseofan optimal state-contingent debt contract, the household obtains a set of claims on banks’ portfolios returns. However,alimitedcommitmentproblembetweenthehouseholdandbanksconstrainsthe allocationoffundsbetweentheseagents–thisisthekeyfrictioninthemodel. In addition to borrowing from the household, we allow banks to borrow from one another in a financial market which opens at date 0, which we refer to as the interbank market.13 In the interbank market, banks can trade claims on each other’s portfolio returns in the form of statecontingent financial contracts. An interbank financial contract issued by bank j to bank i is a debt 11Weassumethattheeconomybeginswith0unitsofthecapitalgoodsatdate0,whichpinsdownthedate0price ofcapitalat1,whilethepriceofcapitalatdate2is0. 12More precisely, traditional firms’ objective function is to maximize the household’s utility E (cid:2) m (s) (cid:0) F(kT(s))−q(s)kT(s) (cid:1)(cid:3) . But since traditional firms only buy capital at date 1 after uncertainty is re- 0 2 1 1 solved,thehousehold’sstochasticdiscountfactordoesnotaffectitsinvestmentdecision,anditsobjectivesimplifies to maximizing F(kT(s))−q(s)kT(s). The first order condition follows from the assumptions that F(·) is strictly 1 1 concaveandF(cid:48)(0)=1. 13We view the interbank market broadly as capturing a wide range of financial markets through which financial institutionscangainexposuretooneanother,suchascorporatedebt,commercialpaper,repo,derivatives,insurance, wholesalefunding,orequitymarkets. 11

contractwhichspecifiesadate0loan(cid:96)ij fromito j,andastate-contingentrepaymentrij(s)atdate 1from j toi,perunitof(cid:96)ij.14 2.5. Banks Bank budget constraints at date 0 At date 0, each bank is endowed with n units of the consumptiongood. Itcanborrowfromthehouseholdbyofferingthehouseholdafinancialcontract (cid:0) di, (cid:8) bi(s),bi(s) (cid:9) (cid:1) which consists of a date 0 loan di from the household to bank i, and a set 0 1 2 s 0 of state-contingent returns (cid:8) bi(s),bi(s) (cid:9) at dates 1 and 2 from the bank back to the household.15 1 2 s In addition, the bank may choose to raise funds from other banks at date 0 by offering another bank an interbank financial contract. An interbank financial contract between bank j and bank i specifiesthedate0initialinvestment(cid:96)ji from j toiinunitsofcapital,andasetofstate-contingent repaymentsrji(s)atdate1fromibackto j,whicharechosenoptimally. At date 0, bank i can use its internal funds and debt to finance capital holdings of size ki and 0 makeloans(cid:96)ij tootherbanks j,subjecttoadate0budgetconstraintgivenby ki +∑(cid:96)ij ≤n+di +∑(cid:96)ji. (5) 0 0 j j Given its capital holdings, the bank also decides how to allocate its capital across the prudent project versus its risky project. Let ωiε[0,1] denote the fraction of bank i’s capital holdings ki invested in the prudent project; then 1−ωi is the fraction of i’s capital invested in its risky 0 project. Let ωi denote the one-by-two vector [ωi 1−ωi] and let Ri(s) denote the two-by-one vector[R Ri (s)]T,sothatthedate1returnonbanki’sprojectsaregivenbythescalarωiRi(s). C A Bank budget constraints at date 1 At date 1, bank i must pay the unit maintenance cost γ <1 on its capital holdings, where R ≥γ. Once the state of the world is realized at date 1, C banki’sdate1fundsaregivenbythesumofitsportfolioreturnsωiRi(s)ki,thevalueofitscapital 0 holdings q(s)ki, and any government transfers gi(s,ωi)ki net of its capital maintenance costs and 0 0 its debt repayment to the household and other banks. The bank then chooses how much capital ki(s) to hold and invest in the continuation project at date 1 subject to its date 1 budget constraint 1 instates. 14Thenetworkstructureofinterbankfinancialclaimswillbepinneddownbythesetofinterbankfinancialcontracts whichemergeinequilibrium. 15More precisely, the contract defines state-contingent payments (cid:8) di(s),di(s) (cid:9) at dates 1 and 2 from the bank 1 2 s back to the household. To simplify the notation, we redefine the contract in terms of returns bi(s)≡ d 1 i(s)+d 2 i(s) and 1 n+di 0 bi(s)≡ d 2 i(s) ,whichscaletheserepaymentsbythebank’stotalnetliabilitiesatdates0and1,respectively. 2 ki(s) 1 12

Let θi(cid:0) s,ωi,gi(cid:1) denote bank i’s rate of return on it’s own physical capital holdings in state s, k given the allocation ωi of its capital across projects and any government gi subsidy to i. θi(s, j) (cid:96) denotestherateofreturnonbanki’sloantobank j instates,giventhecontract (cid:8) rij(s) (cid:9) . s θ i(cid:0) s,ω i,gi(cid:1) ≡q(s)+ωiRi(s)−γ−bi(s)+gi(s,ω i) (6) k 1 θ i(s, j)≡rij(s)−bi(s) (7) (cid:96) 1 Definethetotalrateofreturnonbanki’sassetsatdate1instatesasθi(s)≡ θ k i(s,ωi,gi)k 0 i+∑j θ (cid:96) i(s,j)(cid:96)ij . k 0 i+∑j (cid:96)ij Wecanwritebanki’sdate1budgetconstraintinstatesas (cid:16) (cid:17) q(s)ki(s)≤θ i(cid:0) s,ω i,gi(cid:1) ki +∑θ i(s, j)(cid:96)ij−∑ rhi(s)−bi(s) (cid:96)hi+bi(s)ki(s) (8) 1 k 0 (cid:96) 1 2 1 j h Finally, in period 2, investment bank i pays dividends πi(s) back to the household, which are 2 determinedbybanki’sfinalprofitsatdate2netofdebtrepaymentstothehousehold. π i(s)=ki(s)−di(s) (9) 2 1 2 2.5.1. Contractingenvironmentbetweenthehouseholdandbanks Atdate0,eachbankimayofferthehouseholdacontractwhichspecifiesaninitialloandi from 0 the household and a set of state-contingent returns (cid:8) bi(s),bi(s) (cid:9) to the household at dates 1 and 1 2 s 2. We assume that both the household and banks have a limited ability to commit to honoring the contractatdates1and2. Namely,atdates1and2,thebankchooseswhethertohonorthecontract ornot. Ifthebankdoesnotpay,itmakesthehouseholdatake-it-or-leave-itofferregardingthedate 1 and 2 payments. If the household refuses the offer, the bank is liquidated. The liquidation value of a bank depends on its date 0 portfolio choice and the price of capital in state. The contracting environmentandliquidationvalueofbanksarespelledoutinfurtherdetailinappendix2. This limited commitment problem imposes no-default constraints on the optimal contracts whichensurethatagentsneverdefaultinequilibrium. Theno-defaultconstraintsaregivenby 0≤bi(s)≤q(s)−γ (10) 1 0≤bi(s)≤Γ. (11) 2 To entice the household to accept the contract, bank i’s contract must satisfy a participation con- 13

straint,whichisthehousehold’soptimalitycondition(1). Anadditionalsetofassumptionswillbe useful. Assumption2: Weassumethat a)γ−Γ−RL >0,R +Γ≥1,andR −γ+Γ<ρi(R −R (L))forρ ≡min (cid:8) ρi(cid:9) . A C C C A i b)1−Γ>F(cid:48)(0) c) (cid:0) F(cid:48)(kT(s))−Γ (cid:1) kT(s)isincreasinginkT(s). 1 1 1 Assumption2(a)makesputsboundsonsizeofprojectreturnsrelativetocosts;assumption2(b) ensuresthatthebanksalwaysmakemoreproductiveuseofcapitalthantraditionalfirmsatdate1; andassumption2(c)willhelpusruleoutmultipleequilibriainthedate1marketforcapitalinany givenstate. 2.5.2. Contractingenvironmentbetweenbanks Each bank i may raise funds from any other bank h by offering an interbank financial contract at date 0. Recall that an interbank financial contract between bank h and bank i consists of a date 0 investment (cid:96)hi of capital from h to i, and a set of state-contingent repayments at date 1 given by rhi(s)(cid:96)hi plusthevalueofthecapitalthathinitiallyinvestedini,givenbyq(s)(cid:96)hi. Tosimplifytheexposition,weassumetherearenoincentiveorenforcementproblemsorother frictions between banks. Because banks never have an incentive to default on interbank claims, there are no no-default conditions that needed to be imposed on the interbank contract. The interbank contract between banks h and i simply has to satisfy a bank participation constraint to incentivizebankhtolendtobanki. (cid:16) (cid:110) (cid:111) (cid:17) uhi (cid:96)hi, rhi(s) ≥u¯h (12) s The value u¯h is the equilibrium marginal benefit to bank h of investing in its next best alternative , whileuhi isthevalueofbankhifitacceptsbanki’scontract,whileu¯h isbankh’sreservationvalue – i.e. the value of bank h if it invests in its next best alternative (either lending to another bank, or investing in a project on its own behalf). Therefore, the participation constraint says that bank i must choose a set of state contingent returns to bank h which yields a benefit at least equal to h’s outside option. The exact form of this constraint will be derived later from each bank’s first order conditionforlendingtoanotherbank. 2.6. Banks’optimizingbehavior Wecannowputtheseelementstogethertosolveeachbank’soptimizationproblem. Atdate0, eachinvestorichoosesthefinancialcontract (cid:0) di, (cid:8) bi(s),bi(s) (cid:9)(cid:1) withthehousehold,thefinancial 0 1 2 14

contract (cid:8) (cid:96)ji,rji(s) (cid:9) with each other investor j, how much to lend to other investors (cid:8) (cid:96)ij(cid:9) , s j investmentlevelski,ki(s),andportfolioallocationωi acrossprojects,tomaximizethevalueofits 0 1 investment bank E (cid:2) m (s) (cid:0) 1−bi(s) (cid:1) ki(s) (cid:3) . Here, m (s) denotes the stochastic discount factor 0 2 2 1 2 at date 2 given state s, and reflects the risk aversion of the household. This problem is subject to budget constraints (5) and (8), no-default constraints for the household contract (10) and (11), the household participation constraint (1), the other banks’ participation constraints for each j (12), andnon-negativityconstraintsoncapitalholdingsandinterbankloanski,ki(s),(cid:96)ij ≥0 ∀ j. 0 1 The full optimization problem and its solution are given in detail in appendix 3. In what follows,wecharacterizebanks’optimizingbehavior. 2.6.1. Date0portfoliochoice Atdate0,bankidecideshowtoallocateitsfundsacrossitsavailableinvestmentopportunities: claimsissuedbyotherbanks,orcapitaltobeinvestedintheprudentprojectoritsriskyproject. In deciding which assets to hold at date 0, the bank compares the expected discounted value of each asset. Let θi(s) denote the state-dependent return to i from investing in some asset a (a project or a an interbank claim). Because each bank is owned by a risk averse household, the bank discounts the returns by the household’s stochastic discount factor at date 1.16 Therefore, an asset’s value canbedecomposedintotheexpecteddiscountedreturnplusariskpremiumcomponent. E (cid:2) m (s)θ i(s) (cid:3) = E[m (s)]E (cid:2) θ i(s) (cid:3) +Cov (cid:0) m (s),θ i(s) (cid:1) 1 a 1 a 1 a (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) valueof asset expecteddiscountedreturn riskadjustment Bank i prefers to invest in asset a rather than another asset b if and only the expected discounted returnstoinvestinginaexceedsthatofb. E (cid:2) m (s)θ i(s) (cid:3) ≥E (cid:2) m (s)θ i(s) (cid:3) . (13) 1 a 1 b Inwhatfollows,welayoutafewresultsregardingtheinvestmentbehaviorofbanksthatwillhelp characterize equilibria later on. Proposition 1 shows that each bank is always at a corner solution initsdate0portfolioallocationdecision. Proposition1: Cornersolutionsinportfoliochoice Eachbank’sportfolioallocationchoiceisacornersolution. Namely,foranybanki, a)Eitherki =0and(cid:96)ij >0forsomej;or ki >0and(cid:96)ij =0forall j. 0 0 b)Ifki >0,theneitherωi =0orωi =1. 0 16We will show in general equilibrium that the Lagrange multiplier z (s) on a bank’s date 1 budget constraint is 1 equaltothehousehold’sstochasticdiscountfactorm (s). 1 15

Proof: Foraformalproof,seeappendix12. Part(a)ofProposition1saysthatbankieitherinvestsincapitalonitsownbehalfanddoesnot lendfundstoanyotherbank,orthebanklendsfundstoatleastoneotherbankanddoesnotinvest in capital on its own behalf. Part (b) says that if the bank chooses to invest in capital on its own behalf, then it either invests all of its capital in the risky project or it invests all of its capital in the prudent project.17 Proposition 1 follows from the linearity of the bank’s portfolio allocation problem, which arises from the constant returns-to-scale of each bank’s investment technologies the absence of idiosyncratic risk, which implies that there is no diversification benefit from investing indifferentassets. Intermediariesandinvestingbanks Acorollaryofthispropositionisthat,inequilibrium, all banks can be divided into two groups: banks who invest all of their funds in a project in their own behalf, and banks who forgo their own projects in order to intermediate funds to those investing banks. Henceforth, we refer to banks who invest as investing banks, and those banks who intermediate funds between the household and investing banks as intermediaries. J and L are defined as the sets of investing banks and intermediaries, respectively. Because (15) holds for all investingbanks,investingbanksallholdidenticalportfoliosinequilibrium. Banks endogenously sort themselves into these groups in general equilibrium based on the investmentopportunitiesavailabletothem. Inequilibrium,theonlybankswhoinvestinaprojectare thosewithaccesstotheprojectswiththehighestexpecteddiscountedreturnsE (cid:2) m (s)θi(cid:0) s,ωi,gi(cid:1)(cid:3) . 1 k DefinethissetofbanksbyW.18 Therefore,thesetofinvestingbanksJisgivenbyJ=W. Allother banks become intermediaries who forgo their own investment projects in favor of intermediating fundsbetweenthehouseholdandtheseinvestingbanks(orotherintermediaries).19 2.6.2. Choiceofwhichinterbankcontractstooffer Theanalysisaboveimpliesthatanybankh’sparticipationconstraint(12)takestheform (cid:104) (cid:105) (cid:104) (cid:105) E z (s)θ h(s,i) ≥E z (s)o¯h(s) (14) 1 (cid:96) 1 where o¯h is the bank h’s opportunity cost of funds, which is determined in general equilibrium. Thus, bank h accepts an interbank contract offered by bank i if and only if the value of contract exceeds that of h’s opportunity cost of funds. This opportunity cost of funds will depend on all of 17Withoutlossofgenerality, weignoretheknife-edgecasesinwhichoneoftheseconditionsholdswithequality, inwhichcaseinteriorchoicesmayalsobeoptimal. 18Moreprecisely,W ≡ (cid:8) w|E (cid:2) z (s)θw(s,ωw,gw) (cid:3) ≥E (cid:2) z (s)θi(cid:0) s,ωi,gi(cid:1)(cid:3) ∀i ∈ I (cid:9) . 1 k 1 k 19ThesetofintermediarybanksisgivenbythecomplementofJ. 16

itsinvestmentopportunities(includinginterbankcontractsofferedbyotherbanks)andgovernment transfers. 2.6.3. Optimal contract with household Proposition 1 characterizes the optimal contract between banks and the household. Intuitively, the optimal contract provides some risk sharing betweenbanksandthehousehold,wherebythestate-contingentreturnbi(s)paidtothehousehold 1 ishighifthebank’sportfolioreturnishigh. However,thisrisksharingislimitedbytheno-default constraints. Moreover,sinceeachrepresentativebankiconsistsofacontinuumofatomisticbanks, wehavethatzi(s)isthesameacrossallbanks.20 1 Proposition2: Optimalfinancialcontractswithhousehold Givenavectorofequilibriumprices,anindividuallyoptimalfinancialcontractsatisfiestheconditionsforeachs: bi(s)=Γ,bi(s)=0ifzi < zi 1 (s) ,bi(s) ∈ [0,q(s)−γ]ifzi < zi 1 (s) ,andbi(s)= 2 1 0 m (s) 1 0 m (s) 1 2 2 q(s)−γ ifzi > zi 1 (s) ,where zi 1 (s) = 1−Γ ,andzi = max{E[zi 1 (s)θ k i(s,ωi,gi)], maxj {E[zi 1 (s)θ (cid:96) i(s,j)]}} . 0 m 2 (s) m 1 (s) q(s)−Γ 0 1−E[m 1 (s)bi 1 (s)] Proof: Seeappendix4. 2.7. Privatelyoptimalinterbankfinancialcontracts Wenowcharacterizeinterbankfinancialcontractsinpartialequilibrium. Toobtainfundsonthe interbankmarket,banksmustessentiallycompetewithoneanotherforfundsbyofferingcontracts withthemostfavorableterms. Wesupposethatthemarketforinterbankfundsatdate0isperfectly competitive.21 Givenperfectcompetition,interbankcontractsarepinneddownbytheopportunity costofbanks’fundsineachstate.22 Thepropositionbelowstatescharacterizesinterbankcontracts as a function of banks’ collective choices (cid:8) ωi(cid:9) , which will be determined in general equilibi∈I rium. Toputitbriefly,interbankcontractsarepinneddownbytheopportunitycostoffundsofthe investingbanksw ∈ W. 20Sincebanksareatomistic,theirinvestmentdecisionsthemarketvariablesdeterminedingeneralequilibrium,such asthedate1priceofcapitalq(s). Asaresult,zi(s)isthesameacrossallbanksi. 1 21Wesupposethateachbankiconsistsofacontinuumofidenticalatomisticbanks,andthatthereisfreeentryatdate 0. Wemodelcompetitionbetweenbanksasastaticgamebetweentheseatomisticbanksbankswhoofferacontract (cid:8) (cid:96)hi,rhi(s) (cid:9) toaprospectiveinvestorhatdate0. Bankhthenevaluateseachofferedcontractbasedonitsrisk-return s profileandacceptsthatwhichhasthehighestexpecteddiscountedreturns. WesolvefortheNashequilibriumofthis gameandsummarizethekeyresultshere. 22Allbankswillhavethesameopportunitycostoffundsinequilibrium,implyingthatthereisonlyonecontractto solve for. This follows from the constant returns-to-scale of all banks’ projects and the fact that interbank financial contractsarefrictionless. 17

Proposition3: Interbankfinancialcontractsinpartialequilibrium Foreachintermediarybankh ∈ L,interbankcontractsarepinneddownbythereturnoncapital θw(s,ωw,gw)ofinvestingbanksw ∈ W,sothatθw(s,ωw,gw)=θh(s)forallstatesoftheworld. k k (cid:96) Moreover,thestate-contingentreturnpaidbythiscontractsisgivenbytheunitreturnonbankw’s investmentprojectr(s)=ωwRw(s)+q(s)−γ+gw(s,ωw). Proof: Seeappendix5. To understand intuitively how we arrive at Proposition 3, recall that bank h only accepts a contract offered by bank i if the present discounted value of the contract exceeds h’s opportunity costoffunds. Becausethereisonlyaggregateriskintheeconomy,abankwwithaccesstothemost privately valuable investment project can always design an interbank contract which incentivizes the prospective lender to accept. In this way, these banks w can out-compete all others for funds ontheinterbankmarket.23 The optimal contract outlined in Proposition 3 is effectively an equity contract in which an investor purchases a claim on the portfolio returns of the issuing bank, where the return on the claimperfectlyreflectstheportfoliorisksoftheissuer. Interbankrisksharingisthereforeefficient in a partial equilibrium sense: the interbank market channels funds to the most privately valuable assets. However,wewillseethatingeneralequilibrium,theprivatevalueofriskyassetscandiffer fromtheirsocialvalue. 2.8. Investmentatdate1 In order to evaluate the date 1 spot market for capital, we first characterize aggregate net investment in capital at date 1. Define K ≡ ∑ ki and K (s) ≡ ∑ ki(s) to be the aggregate capital 0 i 0 1 i 1 holdings of the banking sector at dates 0 and 1, respectively. In appendix 6, we show that banks’ netaggregateinvestmentincapitalatdate1isgivenby (cid:20) θw(s,ωw,gw) (cid:21) K (s)−K =K k −1 . (15) 1 0 0 q(s)−Γ Equation(15)saysthatthebankingsector’snetaggregateinvestmentisgivenbytheaggregaterate of return on capital holdings at date 1, discounted by the cost of capital at date 1. At date 1, the 23TheresultsinProposition3implythatwedonothavetokeeptrackofthefullstructureoftheunderlyingnetwork ofinterbankclaimsingeneralequilibriuminordertosolvefortheallocationofresourcesandwelfare,afeaturewhich greatlyimprovesthemodel’stractability. Theequilibriumnetworkstructureofinterbankclaimsmattersonlyinsofar asitdetermineswhichbanksareinvestingbanksinequilibrium. 18

aggregate rate of return on banks’ date 0 capital holdings K is given by the rate of return earned 0 by bank w’s assets θw(s,ωw,gw). Since banks do not pay out dividends at date 1, this return is k invested in capital at date 1. The cost of capital at date 1 is given by the spot price q(s) net of the date2repaymenttothehouseholdbi =Γ. 2 2.8.1. Date1spotmarketforcapital Wenowanalyzeinpartialequilibriumthedate1spotmarketforcapital. Themarketforcapital features two possible regimes at date 1: normal times or a crisis. Normal times are characterized by a net positive investment in capital by the banking sector as a whole. This occurs only if the aggregate losses of the banking sector are not too high. In the good state, banks’ portfolio returns arehighandsotheyincreasetheirinvestmentincapital. Inthebadstate,capitalisreallocatedfrom banksfacingnetlossestothosenot,butotherwiseremainsentirelywithinthebankingsector. Incontrast,acrisisischaracterizedbyanetsaleofcapitalfromthebankingsectortotraditional firmsduetoafiresaleexternality. Thisoccursinthebadstatewhenthebankingsector’saggregate losses are high – therefore, the precondition for a crisis to occur in the bad state is for the banking sectortohavelargeholdingsofriskyassetsexante. Inthebadstateatdate1,thebankingsectoris facing a net aggregate loss which needs to be financed. Because there are insufficient funds in the bankingsectortocoverbanks’aggregatelosses,banksareforcedtoliquidatetheircapitalholdings tothetraditionalsectoratafiresaleprice. Theseresultsaresummarizedinthepropositionbelow. Proposition4: Date1marketforcapital A)Inequilibriumwehave q(s)=F(cid:48)(kT(s)) 1 kT(s)=max{0,K (s)−K }. 1 1 0 B)Moreover,K (s)−K <0ifandonlyifs=L andωw =0. 1 0 Proof: Seeappendix7. 2.8.2 Date 1 bond market clearing The date 1 market for bonds clears when supply equals demand. The demand is derived from the household’s date 1 budget constraint: D (s) = B e +∑ fidi(s)−T −c (s). ThenthesupplyofbondsB (s)adjuststoclearthemarket. 1 i 0 1 1 1 1 2.9. Government’sproblem Wenowintroduceabenevolentgovernmentwhichseekstomaximizehouseholdwelfareusing 19

unittransfersgi(s,ωi)toeachbank,whicharefinancedbylump-sumtaxesT (s)onthehousehold 1 at date 1.24 We analyze the government’s optimal policy at date 1 after the resolution of uncertainty; the government therefore takes all date 0 variables as given.25 The government chooses thesetaxesandtransferstomaximizehouseholdwelfaresubjecttoabudgetconstrainteachperiod, whichatdate1isgivenby ∑kigi(s,ω i)=T (s). (16) 0 1 i The government fully internalizes how its actions at date 1 affect those of private agents, and therefore takes the equilibrium conditions which determine agents’ date 1 and date 2 choices as constraintswhensolvingitsproblem. Weassumethatthegovernmentfacestwotypesoffrictions,whichwediscussingreaterdetail in appendix 9A. First, we rule out counterfactual situations in which the government bails out banks even in the absence of a crisis. Moreover, we assume that the government cannot bail out a bankunlessitcanverifythatthebankisfacinganetloss.26 Assumptionsofthistypearecommon in models of bailouts, and prevent the government from using transfers as a way to enable banks to circumvent their financial constraints and increase investment in non-crisis states (for example, seeAcharyaandYorulmazer(2007)). Second, we assume that the government and banks have asymmetric information about the returns on banks’ portfolios, which will imply that government transfers can be only imperfectly targetedacrossbanksatdate1. Moreprecisely,thegovernmentcanonlyverifyabank’sreturnson its own investments, but cannot verify a bank’s returns from its interbank claims.27 See appendix 9Aformorediscussion. Thepropositionbelowcharacterizesthesolutiontothegovernment’sproblem,whichwederive formallyinappendix8. 24Theassumptionthatbailoutsarefinancedbylump-sumtaxesratherthangovernmentbonds, andthepreclusion of date 2 transfers are without loss of generality. This is because date 1 transfers are sufficient to maximize the government’s objective, and because government bonds B (s) are supplied perfectly elastically and therefore adjust 1 inequilibriumtofacilitateperfectconsumptionsmoothingacrossdates1and2,irrespectiveofthesizeoflump-sum taxes. 25Weruleouttaxesandtransfersatdate0inordertoexcludemacroprudentialpolicyfornow,allowingustoisolate theeffectsofexpostinterventionsontheequilibrium. Wewillsolveforthejointlyoptimalmacroprudentialandex postinterventionslater. 26Theseassumptionscanbeinterpretedasastand-inforthepoliticalconstraintsthatgovernmentsfacewhenconsidering direct transfers to the private sector, or for the the distortionary effect of government intervention or the taxesrequiredtofinancebailouts. Forinstance,in2008,theUSTreasuryfacedconsiderablepoliticaloppositionand pressurefromthepressagainstthefiscalmeasuresithadproposedfortherescueofthefinancialsector. 27This assumption is similar to that in Farhi and Tirole (2012) in which the government has an imperfect ability to verify bank losses, and can be interpreted as capturing the greater difficulty that bank regulators and supervisors oftenhaveinverifyingafinancialinstitution’slossesfromoff-balancesheetexposures,whichareoftencomplexand opaqueinpracticeandarefrequentlyassociatedwithinterbankfinancialclaims. 20

Proposition5: Government’sexpostoptimalbailoutpolicy A)ThetotalsizeoftheoptimalbailoutisgivenbyG(s,ωw):  K (cid:0) q(s)−Γ−Rw(s) (cid:1) for s=Land ωw =0 G(s,ω w)= 0 A . 0 otherwise B) The bailout is distributed arbitrarily across the set of investing banksW. Each bank i ∈ W receives gi(s,ωw)ki, where ∑ gi(s,ωw)ki = G(s,ωw). Banks outside of W do not receive a 0 i∈W 0 bailout,sothatgi(s,ωw)=0foralli ∈/ W. Proof: Seeappendix8. Part (A) of the proposition establishes the size of the aggregate bailout to the banking sector, whereas part (B) establishes how these funds are distributed across banks. First consider part (A). TheoptimalaggregatebailoutG(s,ωw)istheminimumaggregatetransfertothebankingsectorto ensureK (s)=K ,i.e. thatallcapitalremainswithinthebankingsectorratherthanbeingfire-sold 1 0 to the traditional sector. Combining this optimal policy with the expression (15) for net aggregate investment in capital at date 1 shows that the optimal bailout puts a floor on the aggregate return on aggregate capital K equal to q(s)−Γ. Therefore, capital is never misallocated and we always 0 haveq(s)=1andkT(s)=0inequilibrium.28 1 Part (B) shows that the government distributes the bailout arbitrarily across investing banks.29 How exactly the bailout is distributed across investing banks is indeterminate and allocatively irrelevant. Putdifferently,ourresultsarerobusttoanysuchdistributionthatthegovernmentmight choose.30 While we sketch a proof of this in appendix 8, the intuition is that the perfect risk sharing that occurs between banks via the interbank market implies that, in general equilibrium, the bailout is perfectly shared across all banks regardless of the government’s choice of how to initiallydisburseitacrossinvestingbanks.31 Thegovernmentbailouteffectivelyservesasanimplicit,freeputoptiononriskyassets,conditional on the banking sector as a whole having sufficiently great exposure to the aggregate shock. Note that the optimal bailout policy features a kink: a bailout occurs if and only the banking sec- 28This is sufficient to maximize household welfare ex post since at date 1, given date 0 variables, the only inefficiencywhichreduceswelfareisthepotentialmisallocationofcapitalwhichoccursduringacrisis. 29Thegovernmentdoesnotbailoutintermediarybankssinceitcannotverifytheirlosses,asdescribedabove. See appendix9Aforadiscussionofthisassumption. 30This means that, in general, the government is not necessarily expected to bail out only the riskiest or most interconnectedbankexante. 31Weshowinsection4,thatinaversionofthemodelwithoutaninterbankmarket,thisdistributionmatters.Indeed, thegovernment’soptimalpolicyisthentodisbursethebailouttoonlyasmallsubsetofinvestingbanks,soasnotto incentivizeotherbankstotakeexcessiverisks.Inthatsetting,thispolicyissufficienttoeliminateexcessiverisktaking inequilibrium. 21

tor’s aggregate losses at date 1 are sufficiently large to cause a crisis. This kink will introduce a strategiccomplementarityinbanks’date0investmentdecisions. In general, the government’s optimal bailout policy at date 1, characterized above, is not the same as that which the government would choose at date 0. However, the government cannot credibly commit ex ante at date 0 to implementing a bailout policy which is suboptimal ex post at date1. Asaresult,thislackofcommitmentgeneratesatime-consistencyconstraintwhichisatthe heartofthecollectivemoralhazardliterature. Alternative government policies In appendix 9B, we discuss the implications of possible alternativegovernmentpolicies,includingthenotionofrandomizedbailoutsputforthinNosaland Ordonez(2016). 3. GENERAL EQUILIBRIUM In the preceding sections, we characterized the equilibrium conditions for all date 1 and date 2 variables as a function of banks’ date 0 portfolio choices. It remains to jointly determine the set of investing banks W and their date 0 investment choices ωw ∀w ∈ W in general equilibrium. The government’s optimal bailout policy introduces a strategic complementarity in banks’ date 0 investment actions: a bank’s expected discounted payoff to investing in an asset at date 0 depends on the portfolio choices of all other banks due to the possibility of a bailout.32 In this section, we characterizeeachbank’sbestresponsefunctionsandsolveforthesubgameperfectNashequilibria. An equilibrium is given by a vector of prices q(s), financial contracts di, di(s), di(s) , port- 0 1 2 si folio and investment decisions for the investment banks ki,ωi, {ki(s)} , consumption and in- 0 1 si vestment decisions for the household c , c (s), c (s) ,{kT(s)} ,{fi}, bonds B (s) , and bailout 0 1 2 s 1 s 0 i 1 s policy and lump-sum taxes (cid:8)(cid:8) gi(s) (cid:9) ,{T (s))} (cid:9) , such that the household’s, banks’, traditional 1 i 1 s firm’s,andgovernment’sbehaviorisoptimalgiventheirconstraints,andcapitalandgoodsmarkets clearinallperiodsandstates. 32Thisisduetoakinkinthegovernment’soptimalpolicyoutlinedinProposition5: abailoutispositiveifandonly ifaggregatebankinglossesarelargeenoughtocauseacrisis. 22

3.1. Bankbestresponsefunctionsatdate0 Given the government’s optimal policy, we now characterize each investing bank i’s best response function for its date 0 portfolio choices. We begin with bank i’s choice ωi – the fraction of itscapitalthatbankiinvestsintheprudentprojectasopposedtoitsriskyproject. Recallfrom(13) thatbankichoosesωi =0ifandonlyifthediscountedvalueofreturnsfrominvestingintherisky projectexceedthoseoftheprudentproject. Inequilibrium,thisconditionreducesto E[m (s)]R <E (cid:2) m (s) (cid:0) Ri (s)+gw(s) (cid:1)(cid:3) (17) 1 C 1 A where the return Ri (s)+gw(s) on i’s risky project in state s consists of the fundamental return A Ri (s)andthegovernmentsubsidytoinvestingbanksgw(s).33 A We can split the investment decision of a bank into two cases: when conditions for a bailout to occur in the bad state of the world are satisfied or not, where these conditions are defined in Proposition 4. First, supposed that the conditions for a bailout in the bad state do not hold. Then gw(s)=0, and banks are forced to fully internalize the riskiness of risky projects. Since the risky project is excessively risky, (17) does not hold and banks choose ωi = 1 to invest only in the prudentproject. Now suppose that the conditions for bailout to occur during a crisis hold. In this case, the government’s optimal bailout policy in Proposition 5 implies that (17) holds, and so banks choose ωi =0toinvestonlyintheirrisky projects. Theimplicitputoptionontheriskyassetprovidedby the government places a floor on the risky asset’s return. Therefore, we can summarize bank i’s bestresponsefunctionforωi asfollows.  1 if gw(L,ωw)=0 ω i({ω w} )= (18) wεW 0 otherwise Wenowturntoabank’schoiceofwhethertoinvestinaninterbankclaimorinvestsinaproject. Inequilibrium,bankichoosestoinvestinaprojectifandonlyitsexpecteddiscountedreturnfrom investinginaprojectisatleastasgreatasthatofotherinvestingbanksw ∈ W. E (cid:2) m (s)θ i(cid:0) s,ω i,gi(cid:1)(cid:3) ≥E[m (s)θ w(s,ω w,gw)] (19) 1 k 1 k Moreover, given the definition of θw(s,ωw,gw) and the government’s optimal bailout policy, this k condition is satisfied if and only if ωiRi(s)=ωwRw(s). Therefore, bank i is an investing bank in equilibrium if and only if its portfolio returns are equal to that of other investing banks. This pins 33Recallthat,becauseofperfectinterbankrisksharing,regardlessofhowthegovernmentinitiallydistributesbailout fundsacrossinvestingbanks,allultimatelyreceivethesameunitsubsidygw(s)inequilibrium. 23

downthesetW ofinvestingbanks. 3.2. SubgameperfectNashequilibria Wefirstusethebanks’bestresponsefunctionstoshowthattherearetwoequilibriawhichvary by banks’ date 0 portfolio decisions ωi, both of which are characterized by herding behavior in date 0 investment. The following lemma shows that there is an equilibrium in which all banks adopt only the prudent investment, and an equilibrium in which all investors adopt only the risky investment. Lemma1: TwosubgameperfectNashequilibria There are two subgame perfect Nash equilibria: a ‘prudent’ equilibrium in which all banks invest in only safe assets (the prudent project or interbank claims on the prudent project), and a ‘risky’ equilibrium in which all banks invest only in the riskiest assets. Given equilibrium conditions established so far, each equilibrium can be fully described by the investment choice of investingbanksωw andthesetofinvestingbanksW. Prudentequilibrium: ωw =1 ∀w ∈ W andthesetW isnon-empty. Riskyequilibrium: ωw =0 ∀w ∈ W andw ∈ W ⇐⇒ ρw =ρ¯. Proof: Seeappendix10. 3.2.1. Prudentequilibrium Intheprudentequilibrium,allbanksminimizetheirportfolioexposurestotheaggregateshock by investing only in safe assets, i.e. the prudent project on interbank claims which yield the same returnastheprudentprojectineachstate. Asaresult,inthisequilibrium,neithercrisesnorbailouts ever occur in equilibrium. Moreover, no bank is systemically important from the government’s perspective. As a result, no bank has incentive to deviate from investing in safe assets at date 0 to investing in a risky project, since any losses incurred during in the bad state would be fully borne bythebank. Moreover,intheprudentequilibrium,whichbanksareinthesetW ofinvestingbanks in equilibrium is both indeterminate and inconsequential for output and welfare (beyond thatW is non-empty).34 34Toseewhy,recallthatProposition3establishedthatequilibriuminterbankcontractsareeffectivelyequityclaims on the issuer which pay the return on the portfolio of the issuer. Since investing banks invest only in the prudent project in the prudent equilibrium, all interbank claims pay the prudent return in all states. Therefore, each bank is indifferentintheprudentequilibriumbetweeninvestingintheprudentprojectandinvestinganyinterbankclaim,and sothestructureofinterbankfinancialclaims (cid:110)(cid:8) (cid:96)ij(cid:9) (cid:111) areindeterminate. Furthermore,thestructureoftheseclaims j i is allocatively irrelevant in the prudent equilibrium, as it has no bearing on aggregate investment or consumption at anydateorstate. 24

3.2.2. Riskyequilibrium Intheriskyequilibrium,ontheotherhand,allbanksmaximizetheirexposuretotheaggregate shock. ThesetofinvestingbanksW isgivenbyonlythosebankswhohaveaccesstoriskyprojects with the greatest exposure to the aggregate shock, i.e. W consists only of all banks i for whom ρi =ρ¯ ≡max (cid:8) ρj(cid:9) . These banks invest exclusively in their risky projects. In turn, they finance j these investments by issuing claims on these risky investments which are held by all other banks in the economy. In this way, all other banks become exposed to the riskiest project available by forgoingtheirownprojectsinfavorofbuyingclaimsontheriskiestbanks’portfolios.35 Theriskyequilibriumthusfeaturesaconcentrationoffundsandcapitalatdate0intheriskiest banks in the economy. These banks endogenously become highly interconnected with the rest of the banking sector through interbank financial contracts. From the perspective of the government, theseriskybanksare‘systemicallyimportant’inthattheyaretoobigandtoointerconnectedtofail. Namely, in the bad state, since losses incurred by these banks on their risky assets are sufficiently largetocauseacrisis,thegovernmentalwaysbailsthemout. The government guarantee enjoyed by a SIFI w has two effects on its portfolio: the guarantee not only increases the expected return of its assets E (cid:2) θw(s,ωw,gw) (cid:3) by putting a floor on their k lossesinthebadstate,butitalsolowerstheriskpremiumofitsassetsbyincreasingthecovariance Cov (cid:0) m (s),θw(s,ωw,gw) (cid:1) betweentheportfolioreturnoftheSIFIandthehousehold’sstochastic 1 k discountfactor. SinceinterbankclaimsissuedbyaSIFIareessentiallyclaimsonthetotalportfolio return of the SIFI so that θw(s,ωw,gw) = θi(s,w), it follows that the implicit guarantee reduces k (cid:96) theriskpremiumontheseclaims(i.e. itincreasesCov (cid:0) m (s),θi(s,w) (cid:1) ),makinginterbankclaims 1 (cid:96) issued by SIFIs relatively safe assets with a relatively high expected return. The safety provided by interbank claims issued by SIFIs is what gives rise to the risk sharing motive on the interbank market. 3.2.3. Risksharingintheriskyequilibrium Thesystemicallyimportantbanksthatemergeintheriskyequilibriumplayanimportantrolein allocatingriskacrosstheinterbankmarket. Intheriskyequilibrium,somebanks(theSIFI)benefit directly from the government guarantee while the rest do not. Because the government guarantee reduces the risk premium Cov (cid:0) m (s),θw(s,ωw,gw) (cid:1) associated with the SIFI’s portfolio, there 1 k is scope for banks to share risk with the SIFI by exchanging interbank contracts at date 0 in a way which reduces the covariance between each bank’s return and the stochastic discount factor. Indeed,theinterbankcontractsthataretradedintheriskyequilibriumfacilitatepreciselythiskind ofrisksharingbetweentheSIFIandnon-SIFIbanks. 35GivenourcharacterizationofequilibriuminterbankcontractsinProposition3,thestructure(cid:96)ij foralliand j of interbankclaimsareneitherdeterminatenorallocativelyrelevantbeyonddescribingwhichbanksareinsetW. 25

When a SIFI w sells a financial claim to another bank i, in the bad state of the world, the SIFI paystobankinotonlythelowreturnRw(L)earnedonitsriskyinvestment,butitalsoforgoessome A of bailout funds transferred by gov gw(L,ωw)K and pays bank i a higher amount, so that bank i 0 earns a return of Rw(L)+gw(L,ωw) instead of just Rw(L). This partially insures the holder of the A A claim, bank i, against losses from the SIFI’s investments during crises. This insurance motive is reflectedinthelowerriskpremiumoffinancialclaimsissuedbytheSIFI. Corollary1: Interbankrisksharingintheriskyequilibrium The interbank market in the risky equilibrium features risk sharing between the SIFI and non- SIFI banks. Consider a financial claim issued by SIFI w to a non-SIFI bank i. In a crisis state of the world, the SIFI gives up some of the bailout funds it receives from the government and pays a return of Rw(L)+gw(L,ωw) to bank i, which exceeds the return Rw(L) it received on its own A A investments. This amounts to insurance provided by the SIFI to bank i. This insurance reduces the date 0 riskiness of bank i’s portfolio, increasingCov (cid:0) m (s),θi(s,w) (cid:1) . Therefore, by buying a 1 (cid:96) claim issued by the SIFI on the interbank market, bank i’s portfolio becomes relatively safer and yieldsahigherexpectedreturn. This risk sharing motive for investing in interbank claims issued by the SIFI is reflected in the low risk premium of these claims. In this way, the SIFI acts as an intermediary insurer whereby it benefitsdirectlyfromthegovernmentguarantee,andeffectivelyinsuresotherbanksagainstlosses during a crisis through the interbank market. The safety offered by SIFI liabilities makes risky assetsmoreattractivetoallbanks. Tosummarize,thestructureoftheinterbankmarketintheriskyequilibriumaffectsrisktaking in two ways: it channels funds to the riskiest projects, and it provides incentives for peripheral banks to hold excessively risky assets, even though they do not benefit directly from an implicit guarantee. Importantly, in our parsimonious environment, the emergence of SIFIs is necessary and sufficient to support excessive risk taking in equilibrium. This is because SIFIs are the only private agents that can provide insurance against crisis risk, as they are only agents who benefit directly from gov guarantee. Indeed, this insurance is necessary to incentivize non-SIFI banks to holdexcessivelyriskyassets. Weexplorethesepointsfurtherinthetwobenchmarkversionsofthe modelnext. 3.3. Welfare-rankingtheequilibria Let ex ante welfare in the prudent and risky equilibria respectively be denoted by Φ¯ and Φ˜, so that Φ¯ ≡ u(c¯ )+u(c¯ )+u(c¯ ) and Φ˜ ≡ u(c˜ )+E[u(c˜ (s))]+E[u(c˜ (s))]. It is straightfor- 0 1 2 0 1 2 ward to show that household welfare is strictly greater in the prudent equilibrium, Φ˜ < Φ¯. The 26

risky equilibrium is associated with lower welfare because the household’s consumption is more volatile. This simply reflects that aggregate output at date 1 is more exposed to aggregate risk in the risky equilibrium (without having a higher mean), and is therefore second-order stochastically dominatedbyaggregateoutputintheprudentequilibrium. 4. TWO BENCHMARK ECONOMIES To further elucidate the role of the interbank market in facilitating risk sharing and creating the collectiveexcessiverisktaking,weanalyzetwovariantsofthemodel. Benchmark 1: Model without interbank market In the first benchmark, we consider a specialcaseofthebaselinemodelinwhichweshutdownthemarketforinterbankfinancialclaims. In this setting, there is a unique subgame perfect Nash equilibrium in which all banks undertake only prudent investments. The reason for this is that, to support risk taking in equilibrium, the insurance benefits of government guarantees need to be shared widely across banks – otherwise not enough banks will be exposed to the aggregate shock to trigger a crisis and bailout in the bad state. Without an interbank market to facilitate risk sharing between banks, the sole benefactors of a bailout are those that the government bails out directly. By concentrating the bailout on small number of investing banks, the government is able to force the majority of banks to internalize the riskiness of their investments, eliminating the risky equilibrium.36 Thus, the interbank market is the means by which banks can ensure that the benefits of implicit guarantee are shared widely enoughtosupportcollectiveinvestmentinriskyassets. Benchmark2: Varyingthedegreeofriskaversion Inthesecondbenchmark,weillustrate how the risk sharing between SIFIs and non-SIFIs per se leads to excessive risk taking. (This benchmarkisanalyzedindetailinappendix13.) Wemodifythemodelsothatonlytherisksharing roleoftheinterbankmarketaffectsbanksportfoliochoices,andthenperformacomparativestatic exerciseinwhichwevarythedegreeofriskaversionofthehousehold. Underriskneutrality,eachbanki’sbestresponsefunctionistoalwaysinvestinprudentassets. Since agents do not value risk sharing, the insurance provided by claims on SIFIs has no value. As a result, no bank ever undertakes a risky investment in equilibrium. As the household’s risk aversionincreases,thesafetyofferedbyaninterbankclaimissuedbyaSIFIisvaluedmorehighly, 36Intheabsenceofaninterbankmarket,howthegovernmentdistributesthebailoutacrossinvestingbanksisnow relevantforagents’behavior. Thegovernment’soptimalpolicyisthereforeamodifiedversionofthatinProposition 5 in which the bailout is concentrated on only a small subset of investing banks, to discourage the remainder from undertakingriskyprojectsexante. 27

increasing the safety premiumCov (cid:0) m (s),θi(s,w) (cid:1) commanded by these claims. If the insurance 1 (cid:96) value of these claims is sufficiently high, then other banks forgo their prudent projects in favor of investing in these claims. Protected by the government guarantee, SIFIs in turn invest in risky projects. Hence, banks undertake excessive risks only when the insurance provided by these SIFI claimsissufficientlyhigh. 5. SOCIAL PLANNER’S PROBLEM In this section, we characterize the constrained efficient allocation. Consider the problem of a socialplannerwhoseekstomaximizehouseholdwelfareandfacesthesameconstraintsasprivate agentsandthegovernment. Theplanner’sproblemistochooseconsumptionandinvestmentplans for all agents, the allocation of funds across banks, and taxes and transfers, subject to the limited commitment problem between banks and the household, and the constraint that the allocation of capitalatdate1isdeterminedinaspotmarket.37 Moreover,theplannermustrespecttheinability of the government to credibly commit at date 0 to policies which are suboptimal at date 1, which implies the planner faces the same time consistency constraint faced by the government discussed insectiononthegovernment’sproblem. Therefore, the only ways in which the planner’s problem differs from those of private agents is that the planner internalizes the effect of contracts and portfolio choices on the price of capital, and also internalizes the effect of government transfers on the softness of the household budget constraintviataxes. Weformalizeandsolvethefullplanner’sprobleminappendix11. 5.1. Socialplanner’ssolution In appendix 11, we show that the planner’s optimal transfer policy at date 1 in the planner’s solution coincides with the government’s optimal bailout policy in Proposition 5, which simply reflectsthatthisbailoutpolicyeliminatestheinefficienciesrelatedtothefiresaleexternality.38 37Theplanner’schoicesaresubjecttobanks’participationconstraintsforinterbankclaims,asymmetricinformation between the household and banks which prevents households from contracting directly on banks’ portfolio choices andforcesthemtocontractonbanks’expostreturnsinstead. 38Theplanner’soptimalityconditionsimplythat,asinthecompetitiveequilibrium,wealwayshavebi(s)=1−γ 1 and bi(s)=Γ. Therefore, the household contract is constrained efficient. Henceforth, we impose this result on the 2 planner’soptimalityconditions. 28

There are three margins through which the planner’s optimality conditions differ from those of private agents in the competitive equilibrium: the aggregate leverage (or date 0 investment) of the banking sector, the exposure of investing banks to the aggregate shock, and the risk sharing arrangements of banks. The planner internalizes the effect of each margin on the likelihood and size of a bailout, and the effects that a bailout has on the softness of the household budget constraintthroughlump-sumtaxes,whicharerequiredtofinanceanybailout. Inaddition,theplanner internalizesthenetworkeffectsofinterbanklendingoneachbank’sdate0investment. Wecharacterize the constrained efficient allocation below; the full planner’s solution is analyzed in detail in appendix11. 5.1.1. Socialvaluationofassets We discussed in section 2.6.1. how private agents value assets based on the standard asset pricing condition. From the planner’s first order conditions, we can see the value to the planner of anassetwithstate-dependentreturnsR(s)isgivenby (cid:20) (cid:21) ∂T (s) 1 E[m (s)R(s)]−E m (s) . (20) 1 1 ∂ωi (cid:124) (cid:123)(cid:122) (cid:125) privatevaluation (cid:124) (cid:123)(cid:122) (cid:125) socialcost Hence,thesocialvalueofanassetadjuststhevaluetoprivateagentsforthesocialcostofinvesting intheasset. Inturn,thissocialcostreflectshowinvestingintheassettightensthehouseholdbudget constraintinthebadstatethroughthelump-sumtaxesneededtofinanceagovernmentbailout. In the competitive economy, the prudent equilibrium is constrained efficient – the allocation linesupwiththatoftheplanner. Ontheotherhand,relativetotheconstrainedefficientallocation, theriskycompetitiveequilibriumfeaturesthreemarginsofinefficiency: over-borrowing,excessive risktaking,andconstrainedinefficientrisksharing,thelastofwhichisthefocusofourpaper. 5.2. Constrainedinefficientrisksharing Inthissection,weshowthatrisksharingisconstrainedinefficientintheriskyequilibrium. The plannercaresaboutinterbankrisksharingtotheextentthatitinfluencestheallocationofriskacross heterogeneousbanksatdate0.39 Recallthatwedefinedthetotalrateofreturnonbanki’sportfolio by θi(s), given after equation (7). The ratio θi(H) of its portfolio returns in each state corresponds θi(L) theextenttowhichbanki’sportfolioofassetsisexposedtotheaggregateshock. (Moreprecisely, 39The planner does not care about interbank risk sharing beyond its implications for date 0 portfolio choices. Namely, the ex post distribution of net worth across banks is irrelevant for output and welfare due to the constant returns-to-scaleofthecontinuationprojectandthefactthatalluncertaintyisresolvedatthebeginningofdate1 29

θi(H) =1ifandonlyifbankihasnonetexposuretotheaggregateshock,whilethisratiodiverges θi(L) from 1 as this net exposure increases.) Let M and m denote the ‘riskiest’ and ‘safest’ banks, i.e. thebankswiththegreatestandleastnetexposuretotheaggregateshock,respectively.40 Itfollows θM(H) θm(H) that ≥ . θM(L) θm(L) Condition for efficient risk sharing The condition for constrained efficient risk sharing is that all banks have the same exposure to the aggregate shock as the safest bank. More formally, takingasgiveneachbanki’schoiceofrisktakingωi,thisconditionis θi(H,ωi) θm(H,ωm) ∀i ∈ I = ≥1. (21) θi(L,ωi) θm(L,ωm) Intuitively, constrained efficiency requires that each bank use interbank contracts to minimize its exposureto theaggregateshock. Moreover, theplanner would havebanks undertakeonlyprudent θi(H) investments, implying that under constrained efficiency, = 1 for all i. This condition is θi(L) satisfiedintheprudentequilibrium. In the risky equilibrium, by contrast, private risk sharing is constrained inefficient. Indeed, there is a drastic divergence between the risk sharing behavior of banks and that desired by the planner: private risk sharing arrangements in the risky equilibrium maximize banks’ exposure to the aggregate shock, rather than minimizing it as in the planner’s solution. The lemma below conveysthisstarkresult. Lemma2: Constrainedinefficientrisksharingintheriskycompetitiveequilibrium Fix each bank i’s portfolio choice of ωi. Then risk sharing in the risky equilibrium is characterizedby θi(H,ωi) θM(H,ωM) ∀i ∈ I = (22) θi(L,ωi) θM(L,ωM) θM(H,ωM) θm(H,ωm) where > . θM(L,ωM) θm(L,ωm) Moreover,intheriskyequilibriumeachinvestingbankundertakesonlyriskyprojects(ωi=0), θi(H) implyingthat >1foralli. θi(L) Importantly, the constrained inefficiency of interbank risk sharing does not derive from any imperfections in interbank financial markets. Rather, the source of this constrained inefficiency is thestrategiccomplementarityinbanks’portfoliochoices,whichcancollectivelyincentivizebanks to become exposed to the aggregate shock, leading to a government bailout in the bad state of the world. A bailout is ultimately funded by lump sum taxes on the household, which reduces 40ThesebanksaredefinebyM≡ (cid:110) i: max θj(H) (cid:111) andm≡ (cid:110) i: min θj(H) (cid:111) . j∈I θj(L) j∈I θj(L) 30

household consumption in the bad state. Ex ante, private agents do not internalize how their exposure tothe aggregateshock, whether throughtheir holdingsof risky interbank claims orfrom investingintheirownriskyprojects,affectsthebudgetconstraintofthehouseholdinthebadstate of the world. As a result, this externality generates a welfare loss in the form of excessively high consumptionvolatilityexante. Moreover, it is precisely the concentration of resources in the riskiest banks that allows the banking sector as a whole to maximize its exposure to the aggregate shock. By issuing claims to the rest of the banking sector and effectively acting as an intermediary between the government andotherbanks,theSIFIsensurethattheinsurancebenefitsofgovernmentguaranteesonitsassets areshared. Thissmoothsthereturnsofallbanksacrossstates,incentivizingthemtoinvestinrisky assetsinthefirstplace. 5.3. Over-borrowingandexcessiverisk-taking Inadditiontoinefficientrisksharing,theriskyequilibriumfeaturesover-borrowingandexcessive risk taking at date 0. Over-borrowing is reminiscent of Lorenzoni (2008), and arises because agents do not internalize how aggregate leverage and investment affect the softness of the household budget constraint in a crisis through the taxes needed to finance bailout.41 Excessive risk taking is a feature shared with other papers in the literature on collective moral hazard, including FarhiandTirole(2012)andAcharyaandYorulmazer(2007). 6. OPTIMAL MACROPRUDENTIAL POLICY We now discuss how the constrained efficient allocation can be implemented using portfolio taxes andinterventionsininterbankfinancialmarketsatdate0. 6.1. Regulationoftheinterbankmarket A regulator can implement constrained efficiency of risk sharing through appropriate taxes on holdings of claims on banks with risky portfolios, in order to prevent these banks from becoming excessively interconnected or large at date 0. Proposition 6 characterizes the taxes on interbank claimswhicharenecessaryandsufficienttoimplementconstrainedefficiencyofrisksharing. 41An important difference with Lorenzoni (2008) is that while excessive leverage in that paper derives from a pecuniaryexternality,theoptimalbailoutpolicyinoureconomycompletelyeliminatesthissourceofinefficiency. In itsstead,thebailoutpolicyintroducesthe‘softbudgetconstraint’externalitydescribedabove. 31

Proposition6: Optimaltaxesoninterbankclaims Taxes τ ij on each bank i’s holdings (cid:96)ij of claims on each bank j’s portfolio can implement (cid:96) ij constrainedefficiencyofinterbankrisksharing,whereτ isgivenby (cid:96) E (cid:2) m (s)θj(cid:0) s,ωj,gj(cid:1)(cid:3) τ ij(cid:0) s,gj(s,ω j) (cid:1) =E (cid:2) u(cid:48)(c (s))gj(s,ω j) (cid:3) − 1 . (23) (cid:96) 1 1−(1−γ)E[m (s)] 1 These taxes distort private portfolio choices away from investing in claims issued by banks withriskyportfolios,topreventthesebanksfrombecometoointerconnectedandlargetofailfrom ij theperspectiveofthegovernment. Thefirsttermofτ reflectsthewelfarecostsofbankiholding (cid:96) aclaimon j’sportfoliofromincreasingthebailouttobank j. Thesecondtermistheshadowvalue ofbank j’sfundsatdate0.42 Threefeaturesofthesetaxesoninterbankclaimsareworthemphasizing. First,thetaxonifor investing in a claim issued by j is increasing in the riskiness of issuer j of the claim, captured by the exposure of j to the aggregate shock through the dependence of gj on ωj. Second, the full set (cid:26) (cid:27) (cid:110) (cid:111) ij of taxes τ between all bank pairs addresses the indirect exposures of each bank to the (cid:96) j i aggregate shock through the network effects of their higher order interbank linkages.43 Third, the taxes are macroprudential in nature, and depend on the aggregate exposure of the banking sector asawholeingeneralequilibrium.44 Theseresultsrationalizetheuseofmacroprudentialtoolsdesignedtoreducethesystemicconsequences of interconnectedness in the financial system.45 However, the many policies that have thus far been proposed or implemented may be insufficient in some important respects to implement constrained efficiency. Unlike the taxes given in (23), the proposed policies often do not take into account the full network of higher-order exposures to risky assets, nor are they generally macroprudentialinnature. 42Thistermadjuststhetaxinawaywhichcapturesthebenefitofinterbankclaimswhichrelaxtheconstraintsof bankswithprudentportfolios. (cid:26)(cid:110) (cid:111) (cid:27) 43Thesetoftaxes τ ij fullyaddressestheconsequencesofanyhigher-ordernetworkexposuresofeachbank (cid:96) j i totheaggregateshock–i.e. theindirectexposureofbank jtotheaggregateshockthroughthehigher-orderexposures ofitscounterparties’counterparties. 44Thesetaxesarenon-zeroonlywhenaggregateexposuresofthebankingsectorasawholetotheaggregateshock aresufficientlylargetotriggeracrisisandabailoutinthebadstate,sinceotherwisegj(s)=0foralls. 45Anexampleofsuchregulationsarethelimitationsoncounterpartyexposuresofbanks,whichwereproposedby theBaselIIIframeworkoftheBaselCommitteeonBankingSupervisionandhavebeenadoptedbybankregulatorsin severaladvancedeconomies. 32

6.2. Regulationofrisktakingandaggregateleverage Turningtotherisktakingmarginofbanks’portfoliochoice,excessiverisktakingcanbeeliminated using a date 0 tax τi on banks’ investments in their risky projects, while a tax τi on investment ω k at date 0 – or equivalently, a tax on borrowing from the household – can implement constrained efficiencyinaggregateinvestmentorleverage. τ i (cid:0) s,gi(s,ω i) (cid:1) =τ i(cid:0) s,gi(s,ω i) (cid:1) =E (cid:2) u(cid:48)(c (s))gi(s,ω i) (cid:3) ≥0 (24) ω k 1 Thesetaxesreflectthewelfarecostthatcrisesandgovernmentbailoutsimposeonthehouseholdin certain states. In this setting, we do not need all three taxes outlined above in order to implement the constrained efficient allocation. In particular, the taxes on either risk sharing or risk taking alonearesufficienttoimplementconstrainedefficiency. 6.3. Practicalconsiderations In our setting, interventions in leverage or investment alone are never sufficient to implement constrainedefficiency. Thishighlightsthattheconstrainedinefficienciesinoursettingderivefrom banks’ exposures on their asset side of the balance sheet to risky investments. Leverage only exacerbatestheseinefficienciestotheextentthattheyalreadyexistinequilibrium. In order to address the inefficiencies associated SIFIs, policy must address banks’ risk sharing incentives in interbank financial markets rather than simply curtailing the risk taking behavior of SIFIs themselves. Nevertheless, the policies required to fully implement the constrained efficient allocation in this setting are likely to be unfeasible in practice due to the nature and amount of informationrequiredofaregulator. 7. CONCLUSION We analyzed the efficiency of risk sharing between banks in a setting where the government has limited commitment. We have shown implicit government guarantees, which are ex post efficient, generatestrategiccomplementaritiesinbanks’portfoliochoices. Intheriskyequilibrium,implicit guarantees distort private risk sharing incentives of banks in a way which concentrates resources in the riskiest banks, who in turn become excessively interconnected. By issuing claims on their 33

portfolios,theseintermediatetheinsurancebenefitsofimplicitguaranteestothebankingsectoras a whole. This smooths the returns of banks across states, facilitating collective risk taking. Thus, the constrained inefficiency in private risk sharing arrangements transforms a risk-shifting problem of an individual bank into a collective risk shifting problem involving the banking sector as a whole. In this way, the inefficient risk sharing arrangements which emerge in interbank financial markets exacerbates the inefficiencies associated with the collective moral hazard problem, generatingawelfarelossfromexcessiveconsumptionvolatility. Wecharacterizedoptimalmacroprudential policy and provided a rationale for macroprudential interventions interbank financial marketswhichdisincentivizestheinterconnectednessoflargeandriskyinstitutions. 34

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APPENDICES APPENDIX1: Householdoptimizationproblem At date 0, the household solves a consumption-saving and portfolio allocation problem, given the financial contracts available to it. Namely, it chooses consumption at each date and in each state {{c (s)} } ,andhowtoallocateitsdate0savingsacrossinvestmentbanksi,describedbyweights t s t the indicator functions fi which take the value of 1 if the household accepts bank i’s contract and 0 0 otherwise. Given di, the total amount of funds the household invests in bank i is given by fidi, 0 0 0 andaggregatedate0savingisthenΣ fidi.46Wefurtherassumethatbankscannotcommitatdate0 i 0 0 toinvesting inparticular projectsat date1. Therefore,the householdhas noinformation onwhich projectseachbankwillinvestinatdate1. Asaresult,thehouseholdchooses fi basedonlyonthe 0 contract (cid:0) di, (cid:8) di(s),di(s) (cid:9) (cid:1) offeredbyeachbank. 0 1 2 s At date 1, the household also chooses its date 1 bond holdings to maximize expected utility subjecttoitsbudgetconstrainteachperiod. max E[u(c )+u(c (s))+u(c (s))] 0 1 2 {{ct(s)} s } t ,{f 0 i} i ,{B 1 (s)} s c +Σ fidi ≤e −T (25) 0 i 0 0 0 0 c (s)+B (s)≤e +∑ fidi(s)−T −q(s)kT(s) (26) 1 1 1 0 1 1 1 i c (s)≤e +B (s)+∑ fidi(s)+Π (s)−T (27) 2 2 1 0 2 2 2 i Here, Π (s) is the date 2 profits of all banks and traditional firms in state s, and T are lump-sum 2 taxes. Let e =e =e . Also assume that e is sufficiently large that non-negativity constraints for 0 1 2 c ,c ,andc areneverbinding. Thefirst-orderconditionsfor fi andthedate1bondholdingsare 0 1 2 0 u(cid:48)(c )di ≥E (cid:2) u(cid:48)(c (s))di(s)+u(cid:48)(c (s))di(s) (cid:3) (28) 0 0 1 1 2 2 u(cid:48)(c (s))=u(cid:48)(c (s)) (29) 1 2 46The household’s problem is equivalent to a consumption CAPM in which the household simultaneously solves a consumption-savings and portfolio allocation problem, in which it chooses total savings and the share of savings allocatedtoeachbanki. 38

APPENDIX2: Contractingenvironmentbetweenthehouseholdandbanks At date 0, each bank i may offer the household a contract which specifies an initial loan di from 0 the household and a set of state-contingent repayments (cid:8) di(s),di(s) (cid:9) to the household at dates 1 1 2 s and 2. We assume that both the household and banks have a limited ability to commit to honoring the contract at dates 1 and 2. Namely, at dates 1 and 2, the bank chooses whether to honor the contract and make payments di(s) and di(s) to the household. If the bank does not pay, it makes 1 2 the household a take-it-or-leave-it offer regarding the date 1 and 2 payments. If the household refuses the offer, the bank is liquidated. Upon liquidation, the household cannot seize any of the bank’s date 1 net returns, but can seize the bank’s net capital holdings ki +∑ (cid:2) (cid:96)ij−(cid:96)ji(cid:3) . Notice 0 j that this implies the household cannot seize the capital holdings bank i which are owed to other banks (cid:96)ji.47 Therefore, the net capital holdings of bank i which can be seized by the household in the event of liquidation consist of the bank’s own capital holdings ki(s) less the capital it owes on 1 the interbank claims it issued ∑ (cid:96)ji, plus any capital owed to it by other banks ∑ (cid:96)ij. In addition, j j we assume the household can seize a fraction Γ <1 of the bank’s profits date 2 profits, where Γ satisfies ΓA < q. Any profits not seized by the household is retained by the bank. (While bank profitseventuallyfindtheirwaytotothehouseholdintheformofdividendsatdate2,thisgeneral equilibriumresultisnotinternalizedbytheatomistichouseholds). Any assets that the household seizes can be converted to capital and invested in the date 1 project,afterincurringthemaintenancecostγ. Therefore,thevaluetothehouseholdofaliquidated bank i at date 1 is (q(s)−γ) (cid:0) ki +∑ (cid:2) (cid:96)ij−(cid:96)ji(cid:3)(cid:1) , and at date 2 it is ΓAki(s). Then investor i never 0 j 1 defaultsinequilibriumifandonlyifthefollowingconditionsaremet: ineachperiod,thevalueof repaymentdoesnotexceedtheliquidationvaluetothehouseholdofbanki. (cid:32) (cid:33) di(s)+di(s)≤(q(s)−γ) ki +∑ (cid:2) (cid:96)ij−(cid:96)ji(cid:3) (30) 1 2 0 j di(s)≤ΓAki(s) (31) 2 1 Similarly,thehouseholdcanalwayswalkawayfromthecontractwithoutconsequence. Therefore, thehouseholddoesnotdefaultinequilibriumifandonlyiftwoconditionshold. 0≤di(s)+di(s) (32) 1 2 47Recall that we assumed bank i is contractually obligated to return to j its borrowed capital (cid:96)ji. Then one can interpret this assumption as a pari passu clause in the debt contract in which the claims of one creditor on bank i’s assetsshouldrespectthoseofothercreditors. 39

0≤di(s) (33) 2 We can scale the contract by the value of i’s net capital holdings at dates 0 and 1 in units of the numeraire, so that the contract is denoted (cid:0) di, (cid:8) bi(s),bi(s) (cid:9)(cid:1) where bi(s) and bi(s) are given by 0 1 2 1 2 bi(s)≡ d 1 i(s)+d 2 i(s) andbi(s)≡ d 2 i(s) . Thenwecanrewritetheno-defaultconstraints(30)-(33)as 1 n+di 2 ki(s) 0 1 0≤bi(s)≤q(s)−γ (34) 1 0≤bi(s)≤ΓA. (35) 2 To entice the household to accept the contract, bank i’s contract must satisfy a participation constraint,whichisthehousehold’soptimalitycondition(1). APPENDIX3: Bankoptimizationproblems Wecannowputtheseelementstogethertosolveeachbank’soptimizationproblem. Atdate0, eachinvestorichoosesthefinancialcontract (cid:0) di, (cid:8) bi(s),bi(s) (cid:9)(cid:1) withthehousehold,thefinancial 0 1 2 contract (cid:8) (cid:96)ji,rji(s) (cid:9) with each other investor j, how much to lend to other investors (cid:8) (cid:96)ij(cid:9) , s j investmentlevelski,ki(s),andportfolioallocationωi acrossprojects,tomaximizethevalueofits 0 1 investment bank. Here, m (s) denotes the stochastic discount factor at date 2 given state s, and 2 reflectstherisk-aversionofthehousehold. max E (cid:2) m (s) (cid:0) 1−bi(s) (cid:1) ki(s) (cid:3) (36) 0 2 2 1 subjecttobudgetconstraints ki +∑(cid:96)ij ≤n+di +∑(cid:96)ji (37) 0 0 j j (cid:16) (cid:17) q(s)ki(s)≤θ i(cid:0) s,ω i,gi(cid:1) ki +∑θ i(s, j)(cid:96)ij−∑ rhi(s)−bi(s) (cid:96)hi+bi(s)ki(s) (38) 1 k 0 (cid:96) 1 2 1 j h no-defaultconstraintsforthehouseholdcontract 0≤bi(s)≤q(s)−γ (39) 1 0≤bi(s)≤Γ (40) 2 40

the household participation constraint, where we have combined the household’s optimality conditions(1)and(2) (cid:32) (cid:33) u(cid:48)(c )di ≥E (cid:2) u(cid:48)(c (s))bi(s) (cid:3) ki −∑(cid:96)hi+∑(cid:96)ij (41) 0 0 1 1 0 h j andtheotherbanks’participationconstraintsforeach j uji(cid:0) (cid:96)ji, (cid:8) rji(s) (cid:9) (cid:1) ≥u¯j (42) s andnon-negativityconstraintsoncapitalholdingsandinter-bankloans. ki,ki(s),(cid:96)ij ≥0 ∀ j (43) 0 1 Let zi,zi(s),λ ¯i(s),λ i (s),µ¯i(s),µi(s), and νji denote Lagrange multipliers on the date 0 bud- 0 1 get constraint (37), the date 1 budget constraint (38), the upper and lower bounds on bi(s), the 1 upper and lower bounds on bi(s), and bank j’s participation constraint (42) respectively. Also, let 2 g(cid:48)(s,ki,ωi) denote the derivative of the government transfer gi to bank i with respect to ωi, which 0 represents how a marginal increase in ωi affects the bailout that i receives conditional on i being bailed out. (Importantly, this may in general depend on not only the state of the world and i’s investment, but also on the investment decisions ωj of all other banks j.) Because the household hasaccesstoarisklessbondatdate1withgrossreturn1,andalluncertaintyisresolvedindate1, wewillhaveinequilibrium u(cid:48)(c (s))=u(cid:48)(c (s)). (44) 2 1 Theoptimalityconditionsarethengivenby ∂Li ≤0 ⇐⇒ zi (cid:18) 1 E (cid:2) u(cid:48)(c (s))bi(s) (cid:3) −1 (cid:19) +E (cid:2) zi(s)θ i(cid:0) s,ω i,gi(cid:1)(cid:3) ≤0 (45) ∂ki 0 u(cid:48)(c ) 1 1 1 k 0 0 ∂Li ≤0 ⇐⇒ m (s) (cid:0) 1−bi(s) (cid:1) ≤zi(s) (cid:0) q(s)−bi(s) (cid:1) (46) ∂ki(s) 2 2 1 2 1 (cid:32) (cid:33) ∂Li (cid:20) u(cid:48)(c (s)) (cid:21) ≤0 ⇐⇒ 1 zi −zi(s) ki −∑(cid:96)hi+∑(cid:96)ij ≤λ i(s)−λ i(s) (47) ∂bi(s) u(cid:48)(c ) 0 1 0 1 0 1 0 h j 41

∂Li ≤0 ⇐⇒ (cid:2) zi(s)−m (s) (cid:3) ki(s)≤µ i(s)−µ i(s) (48) ∂bi(s) 1 2 1 1 0 2 ∂Li (cid:34) ∂θi(cid:0) s,ωi,gi(cid:1)(cid:35) ≤0 ⇐⇒ E zi(s)ki k ≤0 (49) ∂ωi 1 0 ∂ωi ∂Li ≤0 ⇐⇒ E (cid:2) zi(s)θ i(s, j) (cid:3) ≤zi (cid:18) 1−E (cid:20) u(cid:48)(c 1 (s)) bi(s) (cid:21)(cid:19) (50) ∂(cid:96)ij 1 (cid:96) 0 u(cid:48)(c ) 1 0 ∂Li ∂uji(cid:0) (cid:96)ji, (cid:8) rji(s) (cid:9) (cid:1) ≤0 ⇐⇒ −ν ji s ≤π(s)zi(s)(cid:96)ji (51) ∂rji(s) ∂rji(s) 1 APPENDIX4: Optimalhouseholdcontract Noticefrom(46)and(48)thatwhentheoptimalityconditionforki(s)holds,thatforbi(s)cannot 1 2 holdsincebi(s)≤Γ<1andq(s)≤1. Therefore,giventhatinequilibriumtheoptimalitycondition 2 for ki(s) holds, we have zi(s) ≥ m (s). Although bi(s) ∈ [0,Γ] when zi(s) = m (s), we assume 1 1 2 2 1 2 for simplicity it is at its upper bound in this situation. (This does not affect our main results.) Consequently,wealwayshaveacornersolutionforbi(s)asitissetatitsmaximum. 2 bi(s)=Γ (52) 2 And since m (s) > 0 by the Inada condition of u(·), it follows that zi(s) > 0, so that i’s date 1 2 1 budgetconstraintalwaysbindsinequilibrium. Notice from i’s optimality condition for bi(s), the household’s optimality condition for the 1 u(cid:48)(c (s)) bond and the definition of the stochastic discount factor m (s) = 2 , we can write (30) the 2 u(cid:48)(c ) 0 optimalityconditionforbi(s)as 1 zim (s)≤zi(s) (53) 0 2 1 Then bi(s) is set at its maximum q(s)−γ (a corner solution) if and only if zi > zi 1 (s) = 1−Γ , 1 0 m (s) q(s)−Γ 2 at its minimum 0 (corner solution) if and only if zi < 1−Γ , and is indeterminate if and only if 0 q(s)−Γ zi = 1−Γ . Proposition1characterizestheindividuallyoptimalfinancialcontractinlightofthese 0 q(s)−Γ conditions. 42

APPENDIX5: ProofofProposition3 Proof: The proof relies on two results. First, perfect competition between atomistic banks implies in each state interbank contract issued from any i to h equates the return on the contract to h to the return on i’s assets in each state, such that θh(s,i)=θi(s). Second, Proposition 1 showed that (cid:96) each bank is always at a corner solution in its portfolio choice. This implies only one contract accepted: the contract with highest private valuation E (cid:2) z (s)θh(s,i) (cid:3) . It follows that θh(s,i) = 1 (cid:96) (cid:96) θw(s,ωw,gw),whereW ≡ (cid:8) w|E (cid:2) z (s)θw(s,ωw,gw) (cid:3) ≥E (cid:2) z (s)θi(cid:0) s,ωi,gi(cid:1)(cid:3) ∀i ∈ I (cid:9) . k 1 k 1 k APPENDIX6: Aggregateinvestmentatdate1 In order to evaluate the date 1 spot market for capital, we first characterize aggregate net investment in capital at date 1. Consider net aggregate investment by all banks in state s at date 1, definedasthedifferencebetweenaggregatecapitalholdingsatdate1andaggregatedate0holdings of capital, K (s)−K , where we have defined K ≡∑ ki and K (s)≡∑ ki(s) to be the aggregate 1 0 0 i 0 1 i 1 capital holdings of the banking sector at dates 0 and 1, respectively. We can write aggregate net investmentinstatesas K (s)−K =∑∆i(cid:0) s,ω i,gi(cid:1) (54) 1 0 i where ∆i(cid:0) s,ωi,gi(cid:1) ≡ ki(s)− (cid:2) n+di(cid:3) = ki(s)− (cid:2) ki +∑ (cid:96)ih−∑ (cid:96)hi(cid:3) denotes the difference be- 1 0 1 0 h h tweenbanki’schoiceofdate1capitalki(s)anditsdate0fundsn+di availableforinvestmentin 1 0 anyasset.48 This object can be derived from each bank i’s date 1 budget constraint in state s, after imposing the partial equilibrium characterization of optimal interbank contracts given in Proposition 3 θi(s)=θ j (s)=θw(s,ωw,gw)foralli, j inthesetofintermediarybanksL. (cid:96) (cid:96) k (cid:20) θw(s,ωw,gw) (cid:21) K (s)−K =∑∆i(s,ω w,gw)=K k −1 (55) 1 0 0 q(s)−Γ i Equation(55)saysthataggregatenetinvestmentincapitalbythebankingsectoratdate1isgiven by the aggregate rate of return on capital holdings at date 1, discounted by the cost of capital at date 1. At date 1, the aggregate rate of return on banks’ date 0 capital holdings K is given by 0 the rate of return earned by bank w’s assets θw(s,ωw,gw). Since banks do not pay out dividends k at date 1, this return is invested in capital at date 1. The cost of capital at date 1 is given by the 48To see this, first note that we can re-write aggregate date 0 holdings of capital as ∑ ki = ∑ (cid:2) n+di +∑ (cid:0) (cid:96)ji−(cid:96)ij(cid:1)(cid:3) =∑ (cid:2) n+di(cid:3) +∑ ∑ (cid:0) (cid:96)ji−(cid:96)ij(cid:1) =∑ (cid:2) n+di(cid:3) =∑ (cid:2) ki +∑ (cid:96)ih−∑ (cid:96)hi(cid:3) . Given ou i r 0 defi 0 j i 0 i j i 0 i 0 h h initionofDi(cid:0) s,ωi,gi(cid:1) ,itfollowsthataggregatenetinvestmentcanbewrittenas∑ ki(s)−∑ ki =∑ Di(cid:0) s,ωi,gi(cid:1) . i 1 i 0 i 43

spot price q(s) net of the date 2 repayment to the household bi =Γ. Therefore, the aggregate net 2 investmentinnewcapitalbythebankingsectoratdate1isgivenby(55). APPENDIX7: ProofofProposition4 ProofofPart(A) Recallthattheassumptionthattheconsumptiongoodcanbecostlesslyconvertedintothecapitalgoodone-for-one,butnotviceversa,impliesq(s)≤1. Thisalsoimpliesthataggregateinvestment cannot be negative in equilibrium, i.e. kT(s)+∑ (cid:0) ki(s)−χ(s)ki(cid:1) ≥0. If aggregate invest- 1 i 1 0 mentisstrictlypositive,thenq(s)=1byarbitrage,andsoequation(4)impliesthatkT(s)=0since 1 1=F(cid:48)(0). If, on the other hand, aggregate investment is 0, then we have kT(s)=∑ (cid:0) ki −ki(s) (cid:1) . 1 i 0 1 These two cases imply that q(s) = F(cid:48)(kT(s)) and kT(s) = max (cid:8) 0,∑ ki −ki(s) (cid:9) . Assumption 1 1 1 i 0 1 impliesthatγ <q<q(s). Therefore,inequilibrium,wehave q(s)=F(cid:48)(kT(s)) 1 kT(s)=max{0,K (s)−K }. 1 1 0 Q.E.D. ProofofPart(B) Recallthatthereturntoi’sriskyprojectisgivenby Ri (s)=ρ iR (s)−µ i. (56) A A Weassumedthat µi =R (cid:0) ρi−1 (cid:1) ,implyingthatwehave C Ri (L)=ρ i(R (L)−R )+R . A A C C FirstweshowthatwehavemisallocationifandonlyifRw(s)<bw(s)+γ−ΓA. Supposebw(L)=0, 1 1 thenRw(s)<γ−ΓA. Rw(L)<?γ−ΓA A ρ i(R (L)−R )+R <?γ−ΓA A C C This is true by Assumption 2 that that R ≥γ and R +ΓA≥1, which implies that R >γ−ΓA, C C C andforeventhesmallestρi,wehaveR −(γ−ΓA)<ρi(R −R (L)). C C A 44

Next,weshowthatmisallocationdoesnotholdforR ,i.e. C R ≥γ−ΓA. C This is true by Assumption 2. Since this holds for Rw(L) but not R this condition holds for any A C equilibriumvalueofb (s). 1 Now we show that there is no misallocation if and only if Rw(s)≥q(s)−ΓA. First we show thatthisholdsforR . C R ≥q(s)−ΓA C ThisholdsalreadybyAssumption2thatR +ΓA≥1. C WenowshowthatthisdoesnotholdforRw(L). A Rw(L)<?q(s)−ΓA A ρ iR (L)+R (cid:0) ρ i−1 (cid:1) <?q(s)−ΓA A C This is already satisfied by Assumption 2 and that q>γ, which implies q(s)≥q>γ. Therefore, theseconditionsholdwhichcaseholdsforanyequilibriumvalueofb (s). 1 Therefore,inequilibrium,K (s)−K <0ifandonlyifs=L andωw =0. Q.E.D. 1 0 APPENDIX8: Derivingthegovernment’soptimalbailoutpolicy ProofofPart(A): At date 1, the government solves its problem taking date 0 variables as given. First substitute out of the household’s date 1 budget constraint lump sum taxes T =K gw using the governments 1 0 bindingbudgetconstraint. c (s)+B (s)≤e +∑ fidi(s)−K gw−q(s)kT(s) (57) 1 1 1 0 1 0 1 i Recallthatweruledoutcounterfactualsituationsinwhichthegovernmentbailsoutbanksoutside of a crisis. We can also substitute out total date 2 dividends to the household, which are equal to theprofitsofbanksandtraditionalfirmsatdate2πT(s)=F(kT(s)),andcombinethehousehold’s 2 1 date1and2budgetconstraintsbysubstitutingoutB (s). 1 45

c (s)+c (s)≤e +∑ fidi(s)+∑ (cid:0) A−bi(s) (cid:1) ki(s)+π T(s)+e +∑ fidi(s)−K gw−q(s)kT(s) 1 2 2 0 2 2 1 2 1 0 1 0 1 i i i (58) Recall that household’s optimality condition u(cid:48)(c (s))=u(cid:48)(c (s)) implies that in equilibrium we 1 2 always have c (s)=c (s). Use the definitions of bi(s)≡ d 1 i(s)+d 2 i(s) , and result that in equilib- 1 2 1 k 0 i−∑h (cid:96)hi+∑j (cid:96)ij riumwehavebi(s)=q(s)−γ,bi(s)=ΓA,and fi =1foralli. Recallalsothatwehaveaggregate 1 2 0 interbankliabilitiessatisfy∑ ∑ ((cid:96)hi−(cid:96)ih)=0,andthatbanks’aggregatedate1holdingsofcapital i h θw(s,ωw,gw) are given by K (s)=K k . Finally, we plug in the two conditions characterizing the date 1 0 q(s)−ΓA 1marketforcapital,q(s)=F(cid:48)(K −K (s))andπT(s)=F(K −K (s)). 0 1 2 0 1 2c (s)≤e +(q(s)−γ)K +A(1−Γ)K (s)+F(K −K (s))+e −K gw−q(s)(K −K (s)) 1 2 0 1 0 1 1 0 0 1 (59) Below we characterize the bailout per unit of capital gw, but this is equivalent to characterizing the total bailout G=gwK . (This is because, as we show below, the distribution of bailout funds 0 acrossinvestingbanksisallocativelyirrelevant.) Given this equation, we want to find the total derivative of c (s) with respect to gw when the 1 conditionsforamisallocationofcapitalatdate1aresatisfied–namely,whenωw =0ands=L. d2c (s) dq(s) dK (s) dK (s) dq(s) dK (s) dq(s) 1 =K +A(1−Γ) 1 −F(cid:48)() 1 −K −K +q(s) 1 +K (s) dgw 0 dgw dgw dgw 0 0 dgw dgw 1 dgw dq(s) dK (s) 1 =K (s) +A(1−Γ) 1 dgw dgw dK (L) Claim: 1 >0 dgw Proof: Wefirstshowthat d (cid:104) ωwR C +(1−ωw)Rw A (s)+gw(cid:105) >0 dgw q(s)−ΓA d (cid:20) ωwR +(1−ωw)Rw(s)+gw(cid:21) 1 (cid:0) ωwR +(1−ωw)Rw(s)+gw(cid:1) ∂kT(s) C A = − C A F(cid:48)(cid:48) 1 dgw q(s)−ΓA q(s)−ΓA (q(s)−ΓA)2 ∂gw with ∂kT(s) ∂ ∂ 1 = [K −K (s)]=− K (s) ∂gw ∂gw 0 1 ∂gw 1 Therefore,wehave 46

dK (L) K 1 0 = (cid:20) (cid:21) dgw q(s)−ΓA− (ωwR C +(1−ωw)Rw A (s)+gw) K F(cid:48)(cid:48) (q(s)−ΓA) 0 dK (L) For 1 >0itsufficestoshowthat dgw (cid:0) ωwR +(1−ωw)Rw(s)+gw(cid:1) q(s)−ΓA− C A K F(cid:48)(cid:48) >0 0 (q(s)−ΓA) whichholdsbecauseF(cid:48)(cid:48)(·)<0. Q.E.D. Itfollowsthat ∂q(s) =−F(cid:48)(cid:48)dK 1 (L) >0,sinceF(cid:48)(cid:48)(·)<0. Thuswehave ∂gw dgw d2c (s) dq(s) dK (s) 1 1 =K (s) +A(1−Γ) >0 dgw 1 dgw dgw Sowhenωw=0ands=L,thenhouseholdwelfareisincreasingingw whenkT(s)>0. Hence, 1 when there is a misallocation of capital at date 1, the government sets g at the minimum to ensure thatcapitalisnomisallocatedtothetraditionalsector. Thisoptimalchoiceofgw thereforesatisfies kT(s)=0andisgivenby 1  q(s)−ΓA−Ri (s) fori=w, s=Land ωi =0 gi(s,ω w)= A 0 otherwise It follows that the total bailout is given by K gi(s,ωw)=K (cid:0) q(s)−Γ−Rw(s) (cid:1) . This proves part 0 0 A (A)ofProposition5. ProofofPart(B): With regard to part (B), first recall that the government cannot verify the losses that a bank incurs on its interbank claims. As a result, the government does not bail out any intermediaries in equilibrium. Howdoesthegovernmentprefertodistributethebailoutacrossinvestingbanks? Firstnotethat any bailout that satisfies the conditions above will prevent a misallocation of capital ex post, regardlessofhowitisdistributedacrossinvestingbanks. Nevertheless,inprinciple,howthebailout is distributed across banks may affect banks’ ex ante incentives. Below, we sketch a proof that equilibrium allocations are unaffected by the government’s choice of how to distribute the bailout acrossinvestmentbanks. Thisreliesonthecharacterizationofgeneralequilibriuminsection3. Fix some rule for how the government distributes its bailout across investing banks. Perfect risk sharing between banks ensures that any bailout is perfectly shared across banks, regardless of how the government initially disburses it across investing banks. Then the rule affects neither aggregate investment at date 1 (since the aggregate bailout ensures that capital remains entirely 47

withinthebankingsector)noragent’sexanteincentives(sincethisisdeterminedbytheaggregate bailout and banks’ risk sharing arrangements). Moreover, the arguments laid out in appendix 10 forauniqueriskyequilibriumthenfollow. Hence,thisruledoesnotaffectwelfare,andistherefore indeterminate. For the sake of example, suppose that the alternative rule of thumb is one in which the government bails out only the least risky investing bank. One might think that we would have a risky equilibrium in which only the least risky investing bank j becomes a SIFI. However, in general equilibrium,bank j wouldhaveincentivetodeviateandlendtoanyriskierbankh. Thisisbecause interbankcontractswouldensurethat j getsahigherupsideinthegoodstatefrombankh’sriskier investment, while the downside risk is still protected by the government bailout. So then bank h would become a SIFI, while bank j becomes an intermediary bank. So our original conjecture that j is a SIFI cannot be an equilibrium. Therefore, in any equilibrium with risk taking, there is a unique investing bank, and this is always the riskiest bank, regardless of rule of thumb for how the government disperses the bailout across investing banks off-equilibrium. Then all our results regarding the risky equilibrium go through regardless of the choice of rule-of-thumb. Hence, our choiceofrule-of-thumbiswithoutlossofgenerality. APPENDIX9: Discussionofgovernmentproblem A.Discussionofthefrictionsfacedbythegovernment An important assumption in the literature on collective moral hazard, and also in our model, is that bailouts cannot be perfectly targeted across banks (e.g. see Farhi and Tirole (2012)). If bailouts could be perfectly targeted to any bank in the financial system, the government could always design a transfer scheme which punishes SIFIs, thereby getting rid of the moral hazard problem (for example, by bailing out all banks except for the SIFIs). In practice, however, there are frictions which prevent the government from doing this, be it informational frictions, political constraints, etc. In the model, we impose a straightforward assumption which can capture this. While our results do not depend on the precise nature of this assumption, it is an empirically plausibleandtractablewaytogenerateimperfecttargeting. Our assumption is that it is difficult for the government to verify the losses that a bank incurs on its holdings of interbank claims. This assumption captures the fact that it is difficult for the government to identify banks’ bilateral exposures during a crisis, due to the complexity of interbank markets and thefact thatthese marketsare typicallyover-the-counter. Indeed,the lossesthat financial institutions incurred in 2008 from their (frequently off-balance-sheet) exposures to other banks on interbank markets were difficult to verify externally, and often these institutions did not themselvesknowtheextentoftheseexposuresinthemidstofthecrisis. 48

In the model, this assumption implies that, in general equilibrium, bailouts can be only imperfectly targeted to the SIFIs themselves, who are more central in the network and therefore hold relatively fewer interbank claims in equilibrium. Nevertheless, the results would hold under a broadclassofalternativeassumptionstotheextentthatbailoutscannotbeperfectlytargeted. B.Alternativegovernmentpolicies The bailout policy outlined in Proposition 5 calls for the government to transfer resources to SIFIsduring a crisis, financed bylump-sum taxeson thehousehold. Here,we consideralternative governmentpoliciesandtheirimplications. 1. Randomizedbailouts An alternative type of government policy is analyzed in Nosal and Ordonez (2016). In that paper, the government faces uncertainty about whether a crisis is systemic, and therefore delays intervenion to attain more information. This forces banks to internalize the riskiness of their investments to some extent, mitigating the ex ante moral hazard problem. In our setting, there is no such uncertainty; the government knows with certainty whether there is a crisis, and so this mechanism is not at play. Moreover, given the inefficiencies associated with a crisis, it would be suboptimal (and therefore not credible) for the government not to intervene during a crisis with positiveprobability. Nevertheless, a government could conceivably choose to randomize which investing banks it bails out during a crisis. In our setting, however, the interbank contracts which emerge in equilibrium would ensure that the bailout to any individual bank would be shared more broadly. Put differently risk sharing between banks in the interbank market always ensures that the benefits of bailouts are shared perfectly. Therefore, the interank market ensures that no amount of bailout randomizationcaneliminatethecollectivemoralhazardproblem. That being said, in practice, a lack of confidence in the government’s ability to carry out its optimalbailoutpolicycouldmitigaterisktakingexante. 2. TransferofcapitalfromSIFIstonon-SIFIs One alternative policy would be for the government to simply transfer capital from SIFIs to other banks during a crisis, in a way which keeps production at the first best ex post and eliminatestherisktakingincentiveofSIFIsexante. Itisimportanttonote,however,thatthiswouldbe isomorphic to a bailout of non-SIFI banks. To see why, suppose that, in a crisis, the government obtains the capital of the SIFIs (either through expropriation, or by purchasing the capital at some price) and grants it directly to non-SIFI banks. In a crisis, non-SIFI banks are also, in aggregate, facing losses. Therefore, these non-SIFI banks would be forced to liquidate these capital holdings tothetraditionalsector,andwewouldstillendupwithamisallocationofcapital. Thisisbecause, in the bad state of the world, there are losses, incurred from risky investments, that need to be 49

absorbed by some agents in the economy. In order to prevent a misallocation of capital, the governmentwouldneedtocoverlossesofotherbanksviaatransferfinancedbytaxingthehousehold. Thisiseffectivelyabailoutofnon-SIFIbanks. However,recallfromsection2.9thatthegovernmentcannotbailoutbankswhoselossesitcannot verify. Because the government cannot verify exposures from interbank claims, it would then be infeasible for the government to bail out non-SIFI banks, as these banks are facing losses only from their holdings of interbank claims. These frictions prevent the government from perfectly targeting bailouts to non-SIFI banks. Otherwise, the government could simply design a bailout of all banks except for the SIFIs, without ever having to directly reallocate capital across banks. As wediscussedaboveinPart(A),thisdoesnothappeninpracticeforvariousreasons. APPENDIX10: ProofofLemma1 We prove Lemma 7 by backward induction. We have already characterized banks’ optimal decisions at dates 1 and 2. Given these, we also characterized each investing bank’s best response function for its date 0 portfolio choice. We now prove that, given these best response functions, thereexistexactlytwosubgameperfectNashequilibria. Recall that, to complete the characterization of general equilibrium, it remains to determine the investment choices ωw of investing banks, and to determine which banks are in the set W of investingbanksinequilibrium. Oncethesearedeterminedjointly,theinvestmentchoicesωi ofall other banks (i.e. banks in the set L=I/W, who simply invest in the liabilities of investing banks) areirrelevantfortheallocation. Proof: Theproofisinthreeparts. Inallcases,wemakeuseofthebestresponsefunctions  1 if gw(L,ωw)=0 ω i(cid:0) {ω w} (cid:1) = . w∈W 0 otherwise Claim (i): {ωw =1 ∀w ∈ W} is an equilibrium. This is the ‘prudent’ equilibrium, as all banksundertaketheprudentinvestment. Proof: Wewillshowthat,whenallinvestingbanksinsetW chooseωw =1,thenbankw ∈ W has no incentive to deviate from ωw =1. Suppose that all investing banks choose ωw =1. Recall from the government’s optimal bailout policy that when all investing banks are exposed to risky projects,thenthereisneverabailoutinthelowstateatdate1,i.e. gi(s,ωw)=0. Thebestresponse functionforωw thenimpliesthatbankwfindsitoptimaltosetωw =1. Also, recall in that we showed in the partial equilibrium characterization of optimal interbank contractsthatthesetofinvestingbanksJisgivenbyJ=W ≡ (cid:8) w|w ≡max E (cid:2) z (s)θi(cid:0) s,ωi,gi(cid:1)(cid:3)(cid:9) . i∈M 1 k In this case when ωw =1 ∀w ∈ W, all banks are invested in only to prudent assets, so that that 50

E (cid:2) z (s)θi(cid:0) s,ωi,gi(cid:1)(cid:3) is the same for all banks i. Therefore, the structure of interbank lending in 1 k this equilibrium, and therefore the set of investing banks W, is indeterminate – in this prudent equilibrium, we can have any combination of banks investing in the prudent project on their own behalf, with rest of banks investing in their liabilities. W is non-empty, so that at least one bank investsintheprudentprojectinequilibrium. Claim (ii): {ωw =0 ∀w ∈ W} is also an equilibrium, where w ∈ W ⇐⇒ ρw =ρ¯. This is the‘risky’equilibrium,asallinvestingbanksinvestintheriskiestprojectavailable. j Proof: We will show that, when all banks set ω =0, then bank i has no incentive to deviate C from ωi = 0. Suppose that all investing banks choose ωw = 0. Recall from the government’s C optimal bailout policy that when all investing banks are exposed to risky projects, then there is a bailout in the low state at date 1 given by gˆi(s,ωw) = q(s)−ΓA−Ri (s). The best response A functionforωw thenimpliesthatbankwfindsitoptimaltosetωw =0. Again, we showed that interbank contracts in equilibrium are such that the set of investing banksJ isgivenbyJ=W ≡ (cid:8) w|w ≡max E (cid:2) z (s)θi(cid:0) s,ω¯i,gi(cid:1)(cid:3)(cid:9) . Sincez (s)=m (s)isproi∈M 1 k 1 1 portional to u(cid:48)(c (s)) and in this case θi(cid:0) s,ωi,gi(cid:1) =Ri (s)+gi(s,ωi)=ρiR (s)−µi+gi(s,ωi), 1 k A A it is easy to show that E (cid:2) z (s)θi(cid:0) s,ωi,gi(cid:1)(cid:3) is monotonically increasing in ρi. This is because: 1 k (i) u(·) is strictly concave; (ii) the variance of Ri (s) is increasing in ρi, while its mean is inde- A pendent of ρi; and (iii) the government’s optimal gi(s,ωi) bounds θi(cid:0) s,ωi,gi(cid:1) from below by k 1−Γ. Therefore,E (cid:2) z (s)θi(cid:0) s,ωi,gi(cid:1)(cid:3) ishighestforthebankwiththegreatestpotentialexposure 1 k totheaggregateshock,ρi =ρ¯. Hence,W ={w ∈ W |ρw =ρ¯},i.e. onlybankswithaccesstothe riskiest projects invest in equilibrium, while the rest of banks invest in the liabilities of these risky banks. Claim(iii): Therearenootherequilibria. Proof: Suppose for the sake of contradiction that some {ωw} is an equilibrium, where w∈W {ωw} (cid:54)={ωw =1 ∀w ∈ W}and{ωw} (cid:54)={ωw =0 ∀w ∈ W}. Thegovernment’sopw∈W w∈W timal bailout policy implies that, in any equilibrium, either gw(L,ωw)=1−ΓA−Ri (L) for some A w ∈W (i.e. acrisisandbailoutoccursinthebadstate)orgw(s)=0foralls(i.e. acrisisandbailout never occur). Take the lattercase inwhich we alwayshavegw(s)=0. Thenall investing bankw’s best response functions favor investing only in the prudent project by setting ωw =1. Moreover, this is consistent with having gw(s) = 0. So we must have {ωw} = {ωw =1 ∀w ∈ W}, wεW whichcontradictsthepremisethatthisequalitydoesnothold. Sothiscannotbeanequilibrium. Now suppose that we have a bailout in the bad state. Then the best response function of each investing bank implies all investing banks invest only in the risky their risky projects by choosing ωw =0, which is consistent with having a bailout in the bad state. So we must have {ωw} = w∈W {ωw =0 ∀w ∈ W},whichcontradictsthepremisethatthisequalitydoesnothold. Sothiscannot be an equilibrium either. Therefore, any equilibrium must be either the prudent equilibrium in 51

which{ωw =1 ∀w ∈ W},ortheriskyequilibriuminwhich{ωw =0 ∀w ∈ W}. Q.E.D. UniquenessofSIFI Although the results above imply that, in the risky equilibrium, the SIFI is always the riskiest bank(i.e. thebankwiththehighestρi),itmaybeinstructivetoreiteratewhythisisnecessarilythe case. Supposewehaveanequilibriumwithrisktakinginwhichbank j istheonlyinvestingbank, whereρj <ρh forsomeh(i.e. bank j isnottheriskiestbank). Canthisbeanequilibrium? Given that bank j is the only investing bank, it will be bailed out in the bad state. All other banks have incentive to lend their funds to bank j in order to benefit from the bailout in the bad state. Bank j in turn invests in its risky project. Indeed, other banks may not have incentive to deviate and lend toadifferentbank(sinceitmaynotbebailedout)orinvestinitsownproject. (Thiswouldindeed be the case if the government announced in advance that it would bail out the least risky investing bank.) However,exante,bank jhasincentivetodeviateandlendallofitsfundstotheriskiestbankh. This is because, giving the perfect risk sharing facilitated by interbank contracts, it would benefit from a higher upside in the good state, and still benefit equally from the bailout in the bad state. Therefore,thiscannotbeanequilibrium. Indeed,theonlyriskyequilibriumfeaturepreciselybank hastheuniqueinvestingbank. APPENDIX11: Fullplannerproblem Theplanner’sproblemistochoosec (s), fi,B (s),di,bi(s),bi(s),(cid:96)ji,rji(s),ki,ki(s),ωi,T (s), t 0 1 0 1 2 0 1 1 andgi(s,ωi)forallbanksi, j,allstatessandallperiodst tosolve max E[u(c )+u(c (s))+u(c (s))] 0 1 2 s.t. c +Σ fidi ≤e −T (60) 0 i 0 0 0 0 c (s)+B (s)≤e +∑ fidi(s)−T (s)−q(s)kT(s) (61) 1 1 1 0 1 1 1 i 52

c (s)≤e +B (s)+∑ fidi(s)+Π (s) (62) 2 2 1 0 2 2 i Finaldividendpayout(includingdividendfromtraditionalfirms) Π (s)=∑ (cid:0) A−bi(s) (cid:1) ki(s)+F(kT(s)) 2 2 1 1 i budgetconstraints ki +∑(cid:96)ij ≤n+di +∑(cid:96)ji (63) 0 0 j j (cid:16) (cid:17) q(s)ki(s)≤θ i(cid:0) s,ω i,gi(cid:1) ki +∑θ i(s, j)(cid:96)ij−∑ rhi(s)−bi(s) (cid:96)hi+bi(s)ki(s) 1 k 0 (cid:96) 1 2 1 j h no-defaultconstraintsforthehouseholdcontract 0≤bi(s)≤q(s)−γ (64) 1 0≤bi(s)≤ΓA (65) 2 theotherfirms’participationconstraintsforeach j uji(cid:0) (cid:96)ji, (cid:8) rji(s) (cid:9) (cid:1) ≥u¯j (66) s andnon-negativityconstraintsoncapitalholdingsandinterbankloans. ki,ki(s),(cid:96)ij ≥0 ∀ j (67) 0 1 assetprices q(s)=F(cid:48)(kT(s)) 1 kT(s)=max{0,K (s)−K }. 1 1 0 thegovernment’soptimalbailoutpolicy   (cid:0) q(s)−bw(s) (cid:1) ∑ ki −∑ q(s)−bw 2 (s) X fors=L kwgw(s,ω w)= 2 i 0 i q(s)−bi 2 (s) 0 0 otherwise 53

where (cid:16) (cid:17) (cid:16) (cid:17) X ≡ q(s)+ωiRi(s)−γ−bi(s) ki +∑θ i(s, j)(cid:96)ij−∑ rhi(s)−bi(s) (cid:96)hi 1 0 (cid:96) 1 j h andthegovernmentbudgetconstraint ∑k j gj(s,ω j)+D (cid:0) kT(s) (cid:1) =T (s). (68) 0 1 1 j Recall that the government’s optimal bailout policy implies capital is never misallocated at date 1. Therefore, we have q(s)=1,kT(s)=0. Imposing that the government budget constraint binds, 1 replacedate1taxesT (s). Wealsoreplacedi(s)anddi(s)usingthedefinitionsofbi(s)andbi(s) 1 1 2 1 2 . Notice that the planner takes the constraints of all banks i as constraints simultaneously in the Lagrangian. Hence, unlike in the competitive economy, the planner’s first order conditions for (cid:96)ij and rji(s) will also capture how they affect the budget constraints of other banks j (i.e. k j and 0 j k (s)). Theplanner’sfirstorderconditionsare 1 ∂L(cid:48) ≤0 ⇐⇒ E (cid:2) u(cid:48)(c (s)) (cid:3) bi(s)ki(s)+... (69) ∂ fi 2 2 1 0 (cid:34) (cid:32)(cid:34) (cid:32) (cid:33) (cid:35)(cid:33)(cid:35) ...+E u(cid:48)(c (s)) bi(s) ki −∑(cid:96)hi+∑(cid:96)ij −bi(s)ki(s) −E (cid:2) u(cid:48)(c ) (cid:3) di ≤0 1 1 0 2 1 0 0 h j ∂L(cid:48) ≤0 ⇐⇒ E (cid:2) u(cid:48)(c (s)) (cid:3) −E (cid:2) u(cid:48)(c (s)) (cid:3) ≤0 (70) 2 1 ∂B (s) 1 ∂L(cid:48) ≤0 ⇐⇒ −u(cid:48)(c ) fi+zi ≤0 (71) ∂di 0 0 0 0 ∂L(cid:48) ≤0 ⇐⇒ E (cid:2) u(cid:48)(c (s)) fibi(s) (cid:3) −zi +E (cid:2) zi(s)θ i(cid:0) s,ω i,gi(cid:1)(cid:3) −E (cid:20) u(cid:48)(c (s)) ∂T 1 (s) (cid:21) ≤0 ∂ki 1 0 1 0 1 k 1 ∂ki 0 0 (72) ∂L(cid:48) ≤0 ⇐⇒ −u(cid:48)(c (s)) fibi(s)+u(cid:48)(c (s)) fibi(s)+u(cid:48)(c (s)) (cid:0) A−bi(s) (cid:1) −zi(s) (cid:0) 1−bi(s) (cid:1) ≤0 ∂ki(s) 1 0 2 2 0 2 2 2 1 2 1 (73) 54

(cid:32) (cid:33) ∂L(cid:48) ≤0 ⇐⇒ (cid:0) u(cid:48)(c (s)) fi−zi(s) (cid:1) ki −∑(cid:96)hi+∑(cid:96)ij −u(cid:48)(c (s)) ∂T 1 (s) ≤λ ¯i(s)−λ ¯i(s) ∂bi(s) 1 0 1 0 1 ∂bi(s) 1 0 1 h j 1 (74) ∂L(cid:48) ≤0 ⇐⇒ −u(cid:48)(c (s)) fiki(s)+u(cid:48)(c (s)) fiki(s)−... (75) ∂bi(s) 1 0 1 2 0 1 2 ∂T (s) ...−u(cid:48)(c (s))ki(s)+zi(s)ki(s)−u(cid:48)(c (s)) 1 ≤µ i(s)−µ i(s) 2 1 1 1 1 ∂bi(s) 1 0 2 ∂L(cid:48) (cid:34) ∂θi(cid:0) s,ωi,gi(cid:1)(cid:35) (cid:20) ∂T (s) (cid:21) ≤0 ⇐⇒ E zi(s)ki k −E u(cid:48)(c (s)) 1 ≤0 (76) ∂ωi 1 0 ∂ωi 1 ∂ωi ∂L(cid:48) ≤0 ⇐⇒ E (cid:2) u(cid:48)(c (s)) fibi(s) (cid:3) −zi +z j +E (cid:2) zi(s)θ i(s, j) (cid:3) −E (cid:20) u(cid:48)(c (s)) ∂T 1 (s) (cid:21) ≤0 ∂(cid:96)ij 1 0 1 0 0 1 (cid:96) 1 ∂(cid:96)ij (77) ∂L(cid:48) ∂uji(cid:0) (cid:96)ji, (cid:8) rji(s) (cid:9) (cid:1) ∂T (s) ≤0 ⇐⇒ −zi(s)(cid:96)ji+z j (s)(cid:96)ji−νˆji s −u(cid:48)(c (s)) 1 ≤0 (78) ∂rji(s) 1 1 ∂rji(s) 1 ∂rji(s) APPENDIX12: ProofofProposition1 This follows from the linearity of the firm’s portfolio allocation problem. Namely, the optimality conditions for the bank’s portfolio allocation decisions for ki,(cid:96)ij, and ωi do not depend on size 0 of the firm’s investment. Therefore, it immediately follows that, for each firm i, we have one of two cases. Either we are in case 1, in which there is a firm j (cid:54)= i such that E (cid:2) zi(s)θi(s, j) (cid:3) ≥ 1 (cid:96) E (cid:2) zi(s)θi(s,h) (cid:3) for all other firms h, and E (cid:2) zi(s)θi(s, j) (cid:3) ≥ E (cid:2) zi(s)θi(cid:0) s,ωi,gi(cid:1)(cid:3) for any ωi ∈ 1 (cid:96) 1 (cid:96) 1 k [0,1]. In this case, the contract offered by firm j to firm i has a more favorable risk-return tradeoff that that offered to i by any other firm h. In addition, the return to lending to firm j is preferable toinvestinganyamountineithertheriskyorprudentprojectoni’sownbehalf. Incase1,wehave ki = 0 and (cid:96)ij > 0, meaning the firm forgoes investing in its own projects in favor of lending to 0 firm j. Theotherpossibilityisthatweareincase2,inwhichthereisaω˜i∈[0,1]suchthatE (cid:2) zi(s)θi(cid:0) s,ω˜i,gi(cid:1)(cid:3) ≥ 1 k E (cid:2) zi(s)θi(cid:0) s,ωi,gi(cid:1)(cid:3) for all ωi (cid:54)= ω˜i and E (cid:2) zi(s)θi(cid:0) s,ω˜i,gi(cid:1)(cid:3) ≥ E (cid:2) zi(s)θi(s,h) (cid:3) ∀h. This im- 1 k 1 k 1 (cid:96) pliesthatattheoptimalωi,thereturntoinvestingωi intheprudentprojectand1−ωi ofitscapital 55

has a more favorable risk-return profile than the returns offered by any firm’s inter-firm contract. In case 2, we have ki >0 and (cid:96)ij =0 for all j, meaning the firm does not lend to any other firm. 0 Furthermore, since the condition for ωi does not depend on ωi, firm i will always be at a corner solution in its choice of ωi, so that the optimal ωi satisfies ω˜i ∈{0,1}. (This is partly due to the fact that, in the government’s optimization problem, we will show that gi will be zero for ωi =1.) Q.E.D. APPENDIX13: Benchmark2: Comparativestaticondegreeofriskaversion HowdoesrisksharingbetweentheSIFIandnon-SIFIbanksgenerateexcessiverisktaking? Inthis benchmark variant of the model, we isolate the role of risk sharing per se in generating excessive risktakingbyallbanksbyvaryingthedegreeofriskaversionofagentsinthemodel. Ingeneral,theinterbankmarketplaystworolesintheriskyequilibrium. First,itdirectsfundsat date0totheprojectswiththehighestexpectedreturn. Second,asweshowedinsection3.2.3.,the interbankmarketfacilitatesrisksharingbetweenSIFIsandotherbanksbyallowingotherbanksto benefitfromthegovernmentguaranteeindirectly,therebyreducingthevarianceoftheirportfolios. This second risk sharing motive of interbank lending arises because the stochastic discount factor reflects the household’s risk aversion. To elucidate this point we modify the model in this section sothatonlytherisksharingroleoftheinterbankmarketultimatelyaffectsbanksportfoliochoices. Then when capture how risk sharing incetivizes risk taking through a comparative static exercise byvaryingthedegreeofriskaversionofthehousehold. Todothis,wemodifythebaselinemodelinthreerespects. First,forconcreteness,wesuppose that the representative household’s utility feature constant relative risk aversion so that, u(c) = c1−η−1, where 0 ≤ η ≤ 1. Second, rather than assuming that all risky projects are a mean- 1−η preserving spread of the prudent project, we now assume that R > E (cid:2) Ri (s) (cid:3) for all i.49 This C A impliesthattheriskyprojectsarenotonlyriskierthantheprudentproject,butalsoofferalowerexpectedreturn. Moreover,weassumethatastrongerconditionholds: π(H)Ri (H)+π(L)(1−Γ)< A R . This assumption will ensure that the higher expected return on risky assets afforded by the C government guarantee is not sufficient by itself to entice banks to invest in risky assets. Third, we make assumption 6 on the relative size of π(L) relative to the degree of the household’s risk aversionandtheriskyreturnwhichwecallonbelow,where j isthebankwithρj =ρ¯. (cid:104) (cid:105) AssumptionOA.1: R A j(H)−η R A j(H)−R C <π(L)< R A j(H)−R C R A j(H)−η (cid:16) R A j(H)−R C (cid:17) +R A j(L)−η(R C −1+Γ) R A j(H)−1+Γ 49Forthistohold,weneedtoassumethatourassumptionthatR ≥1−Γinsteadholdswithstrictinequality. C 56

π(L)satisfyingassumptionOA.1existsinthedomain(0,1). In this modified environment, the characterization of the date 1 spot market for capital and optimal interbank and household contracts all go through. Moreover, the government’s optimal bailout policy is still characterized by Proposition 5. Therefore, to characterize the equilibrium in thisversionofthemodel,itremainstocharacterizebanks’bestresponsefunctionsfortheirdate0 portfolio choices and interbank lending decisions. We characterize these best response functions fordifferentdegreesofthehousehold’sriskaversionη. How does risk sharing affect portfolio choices, risk taking? Recall from section 3.5.1. that the value to bank i of an interbank claim issued by a SIFI w promising a return θi(s,w) is given by (cid:96) the sum of the expected discounted return E[m (s)]E (cid:2) θi(s,w) (cid:3) and a risk premium component 1 (cid:96) given by still given by Cov (cid:0) m (s),θi(s,w) (cid:1) . We already showed in Corollary 1 that the implicit 1 (cid:96) guarantee lowers riskiness of SIFI’s assets, and that the interbank market facilitates risk sharing betweentheSIFIandnon-SIFIbankswherebybankscanbenefitfromsafetyoftheSIFIsinterbank claims. Theseresultsapplyinthismodifiedsettingaswell. Wenowvarythedegreeofriskaversion ofthehouseholdtoshowhowthisinterbankrisksharingactuallyexacerbatesexcessiverisktaking, generatingcollectiveriskshiftingproblem. First suppose that η = 0, so that the household is risk neutral. In this case, the stochastic discount factor m (s) is constant across states, and so the covariance term is 0. Agents do not 1 value risk sharing - the variance of their portfolios is irrelevant for their portfolio choice and they care only about the expected return. Since the bailout policy gw(s) is given by by Proposition 5, our assumption above π(H)Rw(H)+π(L)(1−Γ) < R implies that E[Rw(s)+gw(s)] < R . A C A C Therefore,banksneverwanttoinvestininterbankclaimsissuedbySIFIs,becausethegovernment guarantee does not increase the expected return on these claims sufficiently to entice banks away fromprudentassets. Asaresult,eachbanki’sbestresponsefunctionistoalwaysinvestinprudent assets. Asaresult,nobankeverundertakesariskyinvestmentinequilibrium. Thisissummarized inthecorollarybelow. Corollary: Noexcessiverisktakingwithriskneutrality Under Benchmark economy 2, when the household is risk neutral (η =0), there is never excessiverisktakinginequilibriumbyanybank. Nowsupposethatthehouseholdisriskaverse,sothatη >0. Asthehousehold’sriskaversion increases, banks care more about the covariance of their portfolio returns with the stochastic discount factor, and therefore the risk premium on an interbank claim issued by the SIFI w is lower, as captured by a higher Cov (cid:0) m (s),θi(s,w) (cid:1) . In other words, the safety offered by the SIFI’s 1 (cid:96) interbankclaimisvaluedmorebynon-SIFIbanks. 57

Howdoesthisaffectbanks’portfoliochoices? AssumptionOA.1impliesthatE[m (s)(R (s)+g(s))]> 1 A E[m (s)R ]. As a result, non-SIFI banks choose to invest in claims issued by the SIFI. Therefore, 1 C the insurance value of interbank claims issued by the SIFI (together with expected discounted return)issufficientlyhightoenticebankstoforgotheirprudentprojectsinfavorofbuyingfinancial claims issued by the SIFI.(At same time, the SIFI invests in its risky project.) As a result, the risk sharingfacilitatedbytheinterbankmarketincentivizesexcessiverisktaking. Corollary: Risksharinggeneratesexcessiverisktakingbyallbanks When the household is risk averse, the insurance value of interbank claims issued by the SIFI is sufficiently high to entice non-SIFI banks to forgo their prudent investments in favor of buying claims on the SIFI’s portfolio. As a result, in equilibrium, the SIFI invests in its risky project and non-SIFIsinvestinfinancialclaimsissuedbySIFI. Takeaway These comparative static exercises show that, in Benchmark economy 2, risk sharing between the SIFI and non-SIFI banks in the risky equilibrium is precisely what facilitates excessive risk taking in the first place. When the insurance value of interbank claims on the SIFI are low, banks do not have incentive to invest in risky assets. Only when the insurance provided bytheseSIFIclaimsissufficientlyhighdobanksundertakeexcessiverisks. 58