A Theory of Safe Asset Creation, Systemic Risk, and Aggregate Demand

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) A Theory of Safe Asset Creation, Systemic Risk, and Aggregate Demand Levent Altinoglu 2023-062 Please cite this paper as: Altinoglu, Levent (2023). “A Theory of Safe Asset Creation, Systemic Risk, and Aggregate Demand,” Finance and Economics Discussion Series 2023-062. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2023.062. 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.

A Theory of Safe Asset Creation, Systemic Risk, and Aggregate Demand Levent Altinoglu∗ September 2023 Abstract This paper presents a theory of safe asset creation and the interactions between systemic risk and aggregate demand. The creation of private safe assets by financial intermediaries requires them to take leverage, which generates a risk of future crisis (systemic risk) in which intermediaries liquidate assets to service their debt. In contrast, the creation of public safe assets by the government does not generate systemic risk as the government’s power to tax allows it to better absorb losses. The level of systemic risk determines the neutral rate of interestthroughhouseholds’precautionarysavingandaggregatedemand. Themodelfeaturesa two-wayinteractionbetweensystemicriskandaggregatedemand. Monetaryandfiscalpolicy can stabilize aggregate demand and reduce systemic risk by altering the mix of private and publicsafeassetsheldbysavers. Whenmonetarypolicyisconstrained,theeconomycanenter a risk-driven stagnation trap in which economic stagnation arises due to excessive systemic risk. Macroprudentialpolicieswhichreducesystemicriskcanstimulateaggregatedemand. ∗LeventAltinoglu: FederalReserveBoardofGovernors. Theanalysisandtheconclusionssetfortharethoseof theauthoranddonotindicateconcurrencebyothermembersoftheresearchstaffortheBoardofGovernorsofthe FederalReserve. BenRoscoeprovidedexcellentresearchassistance.

1 Introduction Therelationshipbetweenrisksinthefinancialsystemandmacroeconomicfluctuationshasbeenat theforefrontofacademicandpolicydebatesoverthelasttwodecades,whichhavefeaturedperiods of booms in real activity accompanied by a buildup of vulnerabilities in the financial system.1 Moreover,persistenteconomicslumpshaveoftenfollowedfinancialcrisesandoftencoincidewith widespreaddeleveragingamongfinancialintermediaries.2 Onemechanismwhichmaylinkmacroeconomicandfinancialvulnerabilitiesisthecreationof safe assets, in which financial intermediaries provide insurance to savers by producing relatively safe liabilities out of the risky assets which back them. The literature has highlighted the role of safe assets in creating financial instability (Bocola and Lorenzoni (2023), Caramp (2023)) and low aggregate demand (Caballero and Farhi (2018), Caballero and Simsek (2020), Caballero and Simsek (2021)). However, much of the literature either focuses on the determination of aggregate demand,inwhichcasethesupplyofsafeassetsispinneddownbyanexogenousconstraint,oruses arealmodelinwhichaggregatedemanddoesnotplayameaningfulroleindeterminingoutput.3 A deeper understanding of the interactions between financial instability and aggregate demand may improveourunderstandingofmacroeconomicfluctuations. In this paper, I introduce a theory of safe asset creation, systemic risk, and aggregate demand which yields insights into the nature of macroeconomic booms, financial crises, and persistent slumps. The paper makes three contributions. First, it builds a model in which the creation of safe assets generates a two-way interaction between aggregate demand and systemic risk (i.e., the severityoffuturecrises). Inthemodel,theneutralrateofinterestdependsonthelevelofsystemic risk,whichinturndependsontherelativeshareofprivateversuspublicsafeassetsheldbysavers. Second, the paper shows the possibility of risk-driven stagnation traps in which economic growth is low because systemic risk is high, which may yield insight into the nature of persistent slumps. Third, the paper shows how monetary, fiscal, and macroprudential policies may operate through newchannelsonceoneaccountsfortheinteractionsbetweensystemicriskandaggregatedemand. I develop these arguments using a model in which the production of safe assets generates meaningful interactions between systemic risk and aggregate demand. The model combines elementsfromliteratureonfinancialleverage(suchasBocolaandLorenzoni(2023),Caramp(2023), Acharya, Dogra and Singh (2022), and Segura and Villacorta (2023)), in which the leverage of fi- 1Forexample,theeconomicboomoftheearly2000sintheUnitedStateswasaccompaniedbyahighappetitefor riskandleverageamongfinancialintermediaries. 2Notable such episodes include the Great Depression, the slow recoveries of the US and eurozone economies followingtheGlobalFinancialCrisis,andtheslumpinJapanfromtheearly1990suntil2020. 3Some exceptions are the series of papers following Caballero and Simsek (2020), which use a model without financialfrictionsinwhichfluctuationsinaggregatedemandaredrivenbyassetpricesandheterogeneousbeliefs,and Boissayetal.(2023),whichexaminesinaNewKeynesianmodelwithendogenousfinancialcrises. 1

nancialintermediariescantriggersociallycostlycrises,andmacroeconomicmodelsofrisksharing andnominalrigidities(suchasCaballeroandFarhi(2018),CaballeroandSimsek(2020),Korinek and Simsek (2016), and Farhi and Werning (2016)) in which inefficient risk sharing arrangements can depress aggregate demand and output. As such, the model presents a unified framework in which both aggregate demand and the quantity of aggregate risk in the economy are determined jointly. The model is in part motivated by an important distinction, often overlooked in the literature, between safe assets issued by private financial institutions (such as bank deposits, money market mutual fund liabilities, and asset backed-securities) and those issued by the consolidated government (such as government bonds and central bank reserves).4 As shown in Figure 1, the relative share of privately issued safe assets in the United States, for example, tends to increase in the run-up to a financial crisis, and subsides afterword, while the share of public safe assets tends to increase following crises.5 Moreover, the large government interventions during the periods followingtheGlobalFinancialCrisisappeartohavesubstitutedprivatesafeassetsforpublicones. Figure1: SafeassetshareintheU.S.(1952-2022) Totheextentthatthepublicandprivatesectorshaveadifferentcapacitytobearaggregaterisk, the macroeconomic consequences of the creation of public versus private safe assets may differ. In issuing safe liabilities backed by risky assets, financial intermediaries take leverage and bear 4Byprivatesafeassets, Ihaveinmindclaimsissuedbyfinancialinstitutionswhicharerelativelystableinvalue compared to the risky investments which back them, such as bank deposits, mortgage and asset backed securities, liabilities issued by money market mutual funds, and commercial paper. For simplicity, I abstract from government guaranteesoftheliabilitiesofprivateinstitutions, suchasdepositguaranteesorexpectedbailouts, whichcouldalso bethoughtofasgeneratingpublicsafeassets. 5Figure1plotsthetotalvalueofsafeliabilitiesintheU.S.relativetototalU.S.assetsusingdatafromtheFederal ReserveFlowofFunds,anddecomposessafeliabilitiesintothoseissuebyfinancialinstitutionsandthegovernment, as defined by Gorton, Lewellen and Metrick (2012). The Online Appendix lists the data series used. See also and FerreiraandShousha(2022). 2

the risk associated with their assets. Such risk-taking can lead to socially-costly financial crises in which institutions are forced to liquidate their assets in order to service their debt. By contrast, a governmentwhichhasthepowertotaxmaybeabletoserviceorrolloveritsdebtwithoutsystemic consequences. To understand the macroeconomic implications of private versus public safe asset creationrequiresamodelwhichcapturesthedistinctmeansbywhichtheprivateandpublicsectors bearrisk. Themodelfeaturesthreeperiods(dates0,1,and2)andfourrepresentativeagents: risk-averse households,risk-neutralbanks,good-producingfirms,andagovernment. Householdssupplylabor inelasticallyinallperiodsand,atdate0,cansaveintwotypesofsafebonds: privatebondsissued by banks and public bonds issued by the government. Each safe bond pays an uncontingent rate of return in all states of the world at date 1, and agents cannot default. Bank use the proceeds of theirdebtissuanceatdate0toinvestincapitalwhichhasariskyreturnatdate1. Thegovernment canusetheproceedsofitsdebtissuancetobuycapitalatdate0andcanissuelump-sumtaxesand transfersatanydate. Firms combine capital and labor to produce a consumption good each period, they can vary their utilization of capital, and they have fully rigid prices. Monetary policy targets the neutral rate of interest (that is, the interest rate at which output would be at potential) each period, but is subject to an effective lower bound. A focus of the paper is the role of aggregate demand at date 0. Therefore, while date 0 output may be demand-determined, I ensure that monetary policy is unconstrainedbytheeffectivelowerboundinallstatesatdates1and2,whichensuresthatoutput isatpotentialatthosedates. Atdate1,theeconomyissubjecttoanaggregateTFPshocktofirms’production(takingahigh or low value) which affects banks’ return on capital. Safe asset creation at date 0 thus plays two roles: Itfinancesriskyinvestmentincapitalanditinsureshouseholdsagainstthisrisk. The model’s core mechanism is built on two key premises. First, the creation of private safe assets by banks has different macroeconomic consequences than does the creation of public safe assetsbythegovernment. Inordertoissueprivatesafeassetsatdate0,bankstakeleveragewhich generates a risk of future crisis (systemic risk) in which banks must liquidate capital at a loss to service their debt in the bad state at date 1.6 In contrast, the creation of public safe assets by the government does not generate systemic risk, as the government’s power to tax allows it to smooth losses over time and over agents. These assumptions imply that the level of systemic risk (that is, theseverityofafuturecrisis)isendogenouslydeterminedbythecompositionofsafeassets(public versusprivate)heldbyhouseholdsatdate0. 6Onecanthinkofsystemicriskastheendogenouscomponentofaggregateriskwhichstemsfromtheliquidation of capital in the bad state of the world, as opposed to the fundamental component of aggregate risk, which is TFP shockitself. 3

Second, crises reduce the household’s future labor income due to a macroeconomic spillover, similartothatinBocolaandLorenzoni(2023). Inparticular,whenbanksliquidatecapitaltorepay their debt in bad states of the world at date 1, they reduce the future stock of capital at date 2. The lower future stock of capital, in turn, reduces the future wage to be earned by households at date 2, owing to complementarities between capital and labor in the production of goods. As a result, the cost of crises is shared in general equilibrium by households in the form of lower future labor income. Thus,thecreationofprivatesafeassetsbybanksentailsakindofrisktransformationinwhich the risk to banks’ earnings (due to the TFP shock at date 1) is transformed into labor income risk for the household (due to the effect of liquidation at date 1 on the wage at date 2). This risk transformation highlights a contradictory aspect of safe asset creation: While private safe assets fully insure individual savers against the TFP shock at date 1, it also forces them to bear laborincomeriskingeneralequilibrium. However,householdstaketheirfuturewageasgivenand thereforedonotinternalizethesystemicconsequencesofprivatesafeassetcreationwhenchoosing theirportfoliosofprivateandpublicsafeassetsatdate0. Inresponsetotheriskofafuturecrisis,householdsincreasetheirprecautionarysavingatdate 0 in order to smooth consumption across future dates and states by increasing their holdings of safe assets. In this manner, the risk of a future crisis depresses current aggregate demand at date 0. As a result, the neutral rate of interest at date 0 depends on the level of systemic risk, and by extension,isdeterminedbythecompositionofsafeassetsatdate0. The central mechanism which emerges from this environment is that safe asset creation gives rise to a two-way interaction between systemic risk and aggregate demand, stylistically illustrated in Figure 1. The creation of private safe assets by banks insures savers against adverse shocks, but generates a risk of future crises as a side-effect. The risk of future crises increases the household’s precautionary saving which depresses current aggregate demand. When monetary policy is unconstrainedbytheeffectivelowerbound(ELB),themonetaryauthoritycanreducethenominal interestratetoensureoutputremainsatpotential. Figure2: Stylisticdepictionofthecoremechanism 4

In the model, monetary policy stimulates aggregate demand not only directly through the household’sEulerequation,butalsothroughamacroprudentialchannelinreducingsystemicrisk: A lower interest rate at date 0 reduces the debt burden of banks, which reduces the amount of capitaltheymustliquidateinafuturecrisisstatetoservicetheirdebtatdate1. Thislowerlevelof liquidation increases the household’s labor income in the future crisis state at date 2, reducing the household’sprecautionarysavingatdate0. Moreover, the model yields insight on the debate about the role monetary policy should play in tamping down on vulnerabilities in the financial system. The literature on “leaning against the wind” (e.g., Boissay et al. (2023), Goldberg and Lopez-Salido (2023)) has shown that a cost of using tight monetary policy to reduce financial vulnerabilities is that it directly reduces economic activityinnormaltimes. Thereisanadditionalchannelinmymodel: Ahigherpolicyrateincreases banks’ cost of debt service, and therefore exacerbates the severity of a future crisis. This higher systemicriskfurtherdepressesaggregatedemandandeconomicactivityexante. When monetary policy is constrained by the effective lower bound, the decline in aggregate demandleadstoademand-drivenrecessionatdate0(similartoCaballeroandFarhi(2018)). Such arecessionreducesbankearningsanderodestheirnetworthatdate0,furtherincreasingtheriskof future crises. The model thus features a two-way interaction between systemic risk and aggregate demand at the effective lower bound: High systemic risk lowers aggregate demand by increasing households’ precautionary saving. This leads to a demand-driven recession which erodes the net worthofbanks,furtherincreasingsystemicrisk. Hence,theeconomyischaracterizedbyaparadox of safety at the effective lower bound, similar to Acharya, Dogra and Singh (2022) and Segura and Villacorta (2023), in which the demand for insurance against systemic risk generates further systemicriskthroughthecreationofprivatesafeassets. If the two-way interaction between systemic risk and aggregate demand is sufficiently strong, theequilibriummayfeaturearisk-drivenstagnationtrap,inwhichstagnantoutputgrowthemerges as the result of an excessively high level of systemic risk. A high level of systemic risk depresses aggregatedemandandforcestheneutralrateofinteresttofallbelowtheeffectivelowerbound. The resultingdemand-drivenrecessionlowersdate0outputandtheresourcesavailableforinvestment. Therefore,eventhoughahighershareofoutputgoestoinvestment,thelevelofinvestmentfalls. As aresult,thefuturecapitalstockisloweratdate1,andsoisfutureoutputinallstatesoftheworld.7 Thus,therisk-drivenstagnationtraparisesbecauseboththecreationofsafeassetsandinvestment in capital requires that banks take more leverage, which only serves to increase systemic risk and worsenthedemand-drivenrecessionexante. The risk-driven stagnation trap is similar in spirit to the stagnation trap first identified in Be- 7Moreover,thelowerexpectedfutureoutputreducesbanks’expectedfutureearnings,therebyincreasingthebanks’ burdenofdebtandfurtherincreasingsystemicriskexante. 5

nigno and Fornaro (2018). However, the trap in my paper derives from high systemic risk rather than self-fulfilling expectations of low future growth. Therefore, in contrast to that paper, policies designed to stimulate investment may be counterproductive to the extent they further incentivize bank leverage. The trap in this model is also similar to the safety trap first identified in Caballero andFarhi(2018)inwhichademand-drivenrecessionarisesduetoashortageofsafeassets. However, the trap here derives in part from an oversupply of private safe assets. Therefore, policies designedtoincreasethesupplyofprivatesafeassetswouldonlyworsenthedemand-drivenrecession. At the effective lower bound, macroprudential policies that reduce bank leverage can serve as a substitute for monetary policy and stimulate aggregate demand ex ante by reducing systemic risk. Moreover, by stimulating aggregate demand, such macroprudential policies can increase the neutralrateofinterestandtherebyalleviatetheburdenonmonetarypolicy. Fiscal policy can stabilize aggregate demand and reduce systemic risk by altering the mix of private and public safe assets held by savers. For instance, when the effective lower bound is binding, deficit spending financed by public safe asset issuance at date 0 can both stimulate aggregatedemanddirectly,andindirectlybycrowding-outprivatesafeassetissuanceandreducing systemic risk. These macroprudential effects increase the size of fiscal multipliers. Quantitative easing can stimulate aggregate demand at date 0 through macroprudential effects. By purchasing capitalfrombanksatdate0inexchangeforpublicdebt,thegovernmentcanshiftthecomposition of safe assets held by households toward public safe assets, effectively shifting the risk associated with investment from the private sector to the government’s balance sheet. This reduces systemic risk and stimulates aggregate demand, even if the total supply of safe assets is held constant. Government bailouts of banks in bad states of the world at date 1 can stimulate aggregate demand exantebyreducingsystemicrisk.8 1.1 Literature review TheseminalpaperofCaballeroandFarhi(2018)establishedthenotionofthesafetytrapinwhich demand-driven recessions arise due to a general shortage of safe assets. The main innovation in my model is the dynamic interplay between aggregate demand and systemic risk, which stems chiefly from how I model the model the supply of safe assets. In the baseline case of Caballero and Farhi (2018), the supply of (private) safe assets is pinned down by an exogenous collateral constraint, and as a result, the supply of safe assets is not affected by macroeconomic conditions. Whilethereisasocialbenefittoissuingsafeassets,thereisnosocialcostandthereforethesupply 8Iabstractherefrommoralhazardconsiderationswherebytheanticipationofgovernmentbailoutsleadstoperverse risk-takingincentivesatdate1. 6

of safe assets is an aggregate demand shifter: increase in the supply of safe asset only ever boosts aggregatedemand.9 In this paper, by contrast, the dynamic interaction between aggregate demand and systemic risk gives rise to a social cost of issuing private safe assets. This is because (private) safe asset creation endogenously generates the risk of future crisis. In turn, crisis risk affects the demand for safe assets and aggregate demand ex ante due to a macroeconomic spillover from crises to future labor income. However this social cost is not internalized by agents ex ante because the macroeconomicspillovermaterializesonlyingeneralequilibrium. Asaresultoftheseinteractions, theresponseofeconomytoshocksandtovariouspoliciesarequalitativelydifferentinthismodel. This paper also implies a sharp distinction between public and private safe assets with regard to their macroeconomic consequences. Therefore, in this model, aggregate demand (and the level of systemicrisk)isdeterminednotonlybythetotalsupplyofsafeassets,butalsobythecomposition of safe assets between private and public.10 Indeed, safety traps can arise from an oversupply of privatesafeassets,whichcandrasticallyalterthepolicyimplications. The macroeconomic spillover first modeled in Bocola and Lorenzoni (2023) plays a similar role in my model. However, their focus is on the risk-sharing problem between consumers and banks,andsotheseagentscantradefullsetofstate-contingentclaims. Bycontrast,Itakeasgiven market incompleteness by assuming that households and banks trade fixed rate bonds, and as a result, inefficient risk-sharing manifests as excessive private safe asset creation. Moreover, while Bocola and Lorenzoni (2023) use a real model whereas my focus is on aggregate demand and its interaction with systemic risk. Finally, the contrast between how the public and private sectors absorbriskiscentraltomyresults. This paper also relates to the series of papers Caballero and Simsek (2020) and Caballero and Simsek (2021) in which the ability of the economy to absorb the risk associated with investment interacts with aggregate demand and output. While financial frictions are essential to give rise to the notion of systemic risk in my model, these papers abstract from financial frictions and focus on how asset prices affect the distribution of wealth across agents who vary in their beliefs or risk tolerance. Using a similar framework to these papers, Goldberg and Lopez-Salido (2023) identifynewchannelsthroughwhichmonetarypolicymayaffecttheseverityofspeculativebooms and demand-driven recessions, which informs the debate surrounding the macroprudential use of monetary policy (see Ajello et al. (2019)). Boissay et al. (2023) and Collard et al. (2017) also 9Inanextensionoftheirbaselinemodel, CaballeroandFarhi(2018)allowforagentstorelaxthecollateralconstraint and increase the safe asset supply subject to a convex cost. However, this cost is private while the benefit of supplying safe assets in a safety trap is social. Hence, the extension still features a general under-provision of safe assets. 10Whilethesupplyofpublicsafeassetsactsasanaggregatedemandshifter,asinCaballeroandFarhi(2018),this isnotgenerallythecaseforprivatesafeassets: Therisksharingexternalityinmypaperimpliesthatprivatesafeasset creationmaybeexcessivelyhighinasafetytrap. 7

analyze optimal monetary and macroprudential policies when monetary policy affects financial stability. Benigno and Fornaro (2018) is the first paper to formalize the notion of a stagnation trap in which deficient demand results in persistently low economic growth. While in that paper, the stagnationtraparisesduetoanendogenousfallininvestmentwhichreducesinnovationandfuture productivity,inmypaper,productivityisexogenous;thefallinexpectedfutureoutputarisesdueto afallininvestmentandthefuturestockofcapital. Inaddition,whileself-fulfillingexpectationsof low growth and multiplcity of equilibria are central to the stagnation trap of Benigno and Fornaro (2018),inmymodel,thetwo-wayinteractionbetweensystemicriskandaggregatedemandiswhat sustains low investment and growth in equilibrium. Therefore, policies designed to incentivize investmentmaybecounterproductivetotheextentthattheyleadtohigherfinancialleverage. Other paperswhichstudysecularstagnationinclude?,Cuba-BordaandSingh(2021),andXavier(2023). Korinek and Simsek (2016) and Farhi and Werning (2016) have a similar role for macroprudential policy to address an aggregate demand externality. While in Korinek and Simsek (2016), output is always supply-determined ex ante and demand-determined in bad states ex post, the reverseistrueinmymodel. Therefore,incontrasttothatpaper,macroprudentialpolicycanbeused to stimulate current aggregate demand, while fiscal or monetary policy can be beneficial ex ante bothbystimulatingdemandandbyreducingsystemicrisk. This paper is related to Acharya, Dogra and Singh (2022), who use a real model without aggregate risk to show that the supply of private safe assets may create its own demand. While they abstract from uncertainty to focus on the multiplicity of equilibria, I instead abstract from multiplicitytofocusonhowsafeassetcreationamplifiesuncertaintythroughmacroeconomicspillovers, anditsinteractionswithaggregatedemand. BenignoandRobatto(2019)andInfanteandOrdonez (2021) examine the optimal supply of private and public liquidity. The latter paper focuses comparingtheabilityofpublicandprivatesafeassetstofacilitatethesharingofidiosyncraticliquidity risk through their useas collateral in light of theirdifferent exposure to aggregate risk. Angeletos, Collard and Dellas (Forthcoming) also examine the optimal supply of public debt in the presence of tradeoff between reducing the severity of financial frictions and reducing fiscal space. Relative tothesepapers,Iabstractfromliquiditybenefitsofpublicsafeassetstofocusonhowpublicversus privatesafeassetsgeneratesystemicrisk. Sinceagentsinmymodeldonotinternalizethatprivate safe assets generate more aggregate risk in general equilibrium, the convenience yield on public debtisinefficientlylow. This paper is also related to several recent papers on safe assets and crises, such as Diamond (2020), Lenel (2023), Luck and Schempp (2023), Ross (2023), and Segura and Villacorta (2023) which imply that the production of safe assets generates financial externalities. However, these papers abstract from macroeconomic dynamics such as the role of aggregate demand, or are in 8

partial equilibrium rather than general equilibrium, both of which are central to my mechanism. Finally, Azzimonti and Yared (2019) study optimal provision of public versus private safe assets inamodelwithoutaggregaterisk,andBenignoandNistico(2017)studyoptimalmonetarypolicy inamodelwithakindofcash-in-advanceconstraintforprivateandpublicsafeassets. 2 Model 2.1 Overview of setup There are three periods: dates 0, 1, and 2. There are two types of goods: capital and the consumption good. There are two types of agents which consume: a measure one of identical, riskaverse households, and a measure one of identical, risk-neutral banks. In addition, there are goodproducing firms who are owned by the household and have rigid prices. I also assume there is a government whose behavior I assume is determined exogenously, for now. (In the normative section, I will suppose the government is an optimizing agent and characterize its optimal behavior.) Figure3summarizesthesequenceofeventsandkeyfeaturesoftheequilibriumtobediscussedin section3. Figure3: Summaryofmodelenvironment On the supply side of the economy, the consumption good is produced each period by a representativefinalgoodsproducerwhorentscapitalfrombanksandhirelaborincompetitivemarkets. 9

Thesefirmshavepre-setnominalpriceswhicharefixedwithinandacrossperiods,andnormalized to 1, implying the inflation rate is always 0.11 Firms meet demand at these prices by costlessly varying the utilization of capital. Thus, output can be below its potential level due to a shortage of aggregate demand. Any profits are distributed as dividends back to the household at the end of eachperiod. Since prices are fully sticky, the real interest rate is equal to the nominal interest rate, which is controlled by the monetary authority. In equilibrium at date 0, the policy rate set by the monetary authoritywillbeequaltothenominalrateongovernmentbondsRMP=RB. Therefore,Ihenceforth 0 0 ignore the notation RMP and instead refer to RB as both the rate on government bonds and the 0 0 monetarypolicyrate. Moreover,sinceinflationisalwayszero,RB isalsotherealrateofinterestat 0 date0. (InormalizethepriceleveltoP =1foreachperiodt.) t I assume that the monetary authority attempts to replicate the supply-determined output level. However, there is a lower bound constraint on the gross nominal interest rate, implying RB ≥ 1. t Thus, the monetary authority sets the nominal interest rate according to RB =max{R∗,1}, where t t R∗ isthegrossnaturalrateofinterestratewhichensuresoutputisatitspotential. t For simplicity, I assume that while output may fall below potential at date 0, output is equal to its potential at dates 1 and 2 – that is, I assume the gross natural interest rate weakly exceeds 1 in all states at date 1 and 2.12 Since the monetary authority targets the natural rate, the (real and nominal)rateofinterestatdate1willalwaysbegivenbyRB =1. 1 The source of aggregate uncertainty is a shock to the total factor productivity z of final goods 1 producersatdate1. Therearetwoaggregatestatesatdate1: z ∈{zH,zL}wherezH >zL. 1 1 1 1 1 The representative bank is endowed with capital at date 0 and rents capital to intermediate producersinacompetitivemarketwithineachperiod. betweenperiods,therepresentativebankcan store capital freely and can create new capital by investing consumption good into an investment technologywithconstantreturns-to-scale. Inequilibrium,theTFPshocktofinalgoodfirmsaffects the banks rental income from capital. Hence, the bank’s return to date 0 investment in capital is subjecttoaggregaterisk. To finance investment in new capital at date 0, the bank can issue nominally safe, one-period debtD tothehouseholdinacompetitivemarketatdate0. Thisprivatebondpaysanominalgross 0 rate of return denoted RD at date 1 which is not contingent on the state of the world. 13 The bank 0 cannotissuenewdebtatdate1. Therefore,itmustpayitsdate1debtoutofitsdate1rentalincome 11The nominal rigidity can be microfounded with the standard assumption of monopolistically competitive firms whofacesomedegreeofrigidityinprice-setting. SeeCaballeroandSimsek(2021). 12ThisisreminiscentofthesimplifyingassumptionsemployedinCaballeroandSimsek(2021)andCaballeroand Farhi(2018). 13Iabstractfromdefaultandassumethatborrowershavefullcommitmenttorepaydebt,butrelaxingthisassumption wouldnotaltercentralinsightsofthemodel. 10

or by converting a portion of its capital holdings into the consumption good. However, capital is partiallyilliquidatdate1andissubjecttoaliquidationcost. Inparticular,if(cid:96) denotesthefraction 1 ofitscapitalstockthatthebankliquidatesatdate1,thenitsdate2capitalstockisgivenby k (s)=i +(1−(cid:96) −φ((cid:96) (s)))k (s) (1) 2 1 1 1 1 Here, φ((cid:96) (s)) denotes the liquidation cost, indexed by the state of the world s, and is a strictly 1 convexfunctionof(cid:96) : φ((cid:96) )=(cid:96) η whereη >1. Thenφ(0)=0,φ(cid:48) ≥0,φ(cid:48)(0)=0,andφ(cid:48)(cid:48) >0.14 1 1 1 Thus, at date 1, the bank is potentially liquidity constrained for three reasons: it cannot issue state-contingent debt ex ante, it cannot raise new debt to finance repayment ex post, and capital is partially illiquid at date 1. As a result, in bad states of the world (that is, when productivity is low), the bank may be forced to resort to costly liquidation of capital to repay its debt. Therefore, while private debt insures the household against aggregate shocks in partial equilibrium, this insurancemayleadtoalowerfuturecapitalstockingeneralequilibrium(andthereforelowerfuture consumption). Finally,capitalfullydepreciatesafterdate2. On the demand side of the economy, the risk-averse representative household is endowed with the consumption good at date 0. Each period, the household supplies labor to intermediate goods producers inelastically, and solves a consumption-saving decision. At date 0, the household has access to two financial assets: the private bond and a public bond issued by the government. In equilibrium, the private bond allows the household and the bank to share aggregate risk to some extent. At date 1, the household has access to an exogenous storage technology with constant returns-to-scale between dates 1 and 2. The quantity of consumption goods that the household stores from date 1 to date 2 denoted B , where the real gross rate of return on this technology is 1 denoteRB =1.15 1 At date 0, agents may receive an unanticipated news shock about future TFP z (i.e., an MIT 1 newsshock). Agentscanre-optimizetheirdate0decisionsinresponsetothisnews,butpricesare fullyrigid. Thus,theMITshockmaycauseoutputtodeviatefromitspotentialatdate0. Iconsider theeffectsofsuchashockinsection6. In each period, the government can implement taxes transfers on agents, both lump-sum and distortionary. Atdate0,thegovernmentcanpurchasecapitalfromthebanksinacompetitivespot market–apolicysimilarinspirittoquantitativeeasing. Moreover,thegovernmentcanissueaone- 14ThisliquidationcostisareducedformwaytocaptureendogenousfiresalessimilartoLorenzoni(2008). While thisreduced-formrepresentationyieldsverysimilardynamics,itmakesthemodelconsiderablymoretractable. 15Thestoragetechnologyisnotcriticalforthequalitativeresultsbutimprovesthetractabilityofthemodelandalso ensuresthatthenaturalrateofinterestisequalto1inallstatesatdates1and2sothattheeffectivelowerboundnever bindsatthosedates. IfIinsteadassumedthatthehouseholdcouldinvestingovernmentbondsatdate1, theresults wouldbesimilar–Iwouldsimplyrestrictmyanalysistothecasesinwhichthenaturalrateisweaklygreaterthan1 atdates1and2,ensuringthatoutputisatitspotentialatthesedates. 11

period, nominally risk-free bond B at date 0 that pays a nominal gross rate of return RB, which 0 0 is not contingent on the state of the world. The government’s power to tax gives it a comparative advantageovertheprivatesectorinissuingsafedebt,whichwillbeinternalizedbyasocialplanner butnotnecessarilybyprivateagents. 2.2 Household Therepresentativehouseholdhaslogutilityandisendowedwithe unitsoftheconsumptiongood 0 at date 0. Each period, the household supplies labor n¯ in inelastically, and solves consumptionsaving decision each period. At date 0 the household has two assets to save in: private bonds (in endogenous supply) or public bonds (in exogenous supply). Households cannot hold capital directly. Atdate1,thehouseholdhasaccesstoarisklessstoragetechnologywithconstantreturnsto-scale, where B and RB =1 respectively denote the quantity of the consumption good that the 1 1 households investment in the date 1 storage technology and the real gross rate of return on the technology. Atdates0and1,thehouseholdeffectivelyfacesaconsumption-savingdecisionandaportfolio choice: how to allocate savings between the public and private bond at date 0, and between the public bond and traditional firms at date 1. The household’s optimization problem at date 0 is thereforetochooseitsconsumptioneachperiodc ,c ,c ,itsdate0nominalholdingsoftheprivate 0 1 2 and public bonds D , B , and its date 1 saving decision B to maximize its expected utility, taking 0 0 1 asgiventhewagew ,nominalratesofreturn,andpricelevelP ateachdatet. t t max logc +E [logc +logc ] 0 0 1 2 c ,c ,c ,D ,B ,B 0 1 2 0 0 1 Each date t, the household’s available funds in real terms is given by the sum of its endowment, laborincomew n¯,dividendsdF,andrealreturnonassets,netoflump-sumtaxesT. Itcanallocate t t t thesefundsbetweenconsumptionandinvestmentineachtypeofasset. Therefore,thehousehold’s budgetconstraintsfordates0,1,and2,respectively,are D B c + 0 + 0 ≤e −T +dF+w n¯ (2) 0 P P 0 0 0 0 0 0 RDD RBB c (s)+B (s)≤ 0 0 + 0 0 +dF(s)−T +w n¯ (3) 1 1 P (s) P (s) 1 1 1 1 1 c (s)≤dF(s)+RB(s)B (s)−T +w n¯ (4) 2 2 1 1 2 2 All endogenous date 0 variables are conditional on the date 0 MIT shock α, while all endogenous 12

variablesatdates1and2areconditionalonα andtheaggregatestateoftheworlds. The household’s optimality conditions, derived in Online Appendix 1, imply that the nominal rates of the return on the private and public bonds must be equalized in equilibrium, RD = RB, 0 0 as the private and public bond are equivalent assets from the perspective of private agents as they both offer a nominally risk-free, state-uncontingent return in date 1 (although the two assets are notequivalentfromasocialperspective,asIshowlater). IhenceforthuseR ≡RD=RB todenote 0 0 0 the nominal interest rate. The household’s date 0 consumption-saving decision is governed by the (cid:104) (cid:105) Euler equation 1 =R E 1 , which pins down the household’s demand for total saving (i.e., c 0 0 c (s) 0 1 itsdemandforbothtypesofbondstogether). The household’s demand function for private bonds, derived from its date 0 Euler equation and budget constraint, is downward-sloping in the interest rate R and depends on the level of 0 utilizationu ingeneralequilibrium. 0 (cid:18) (cid:20) (cid:21)(cid:19)−1 1 1 Dd(R ,B ;u )=e −T +dF+w n¯−B − E (5) 0 0 0 0 0 0 0 0 0 R 0 c (s) 0 1 The negative relationship between the household’s demand for private bonds Dd and the govern- 0 ment’s supply of public bonds B reflects that a higher supply of public bonds crowds out the 0 household’sdemandforprivatebonds,sincethebondsareperfectsubstitutesfromtheperspective of the household. Because only the household consumes at date 0, Dd inversely reflects aggregate 0 consumption demand at date 0. As a result, the quantity of public bonds supplied by the government acts as a consumption-demand shifter. Moreover, the optimality condition for the storage technology B implies that the household perfectly smooths its consumption between dates 1 and 1 2sothatc (s)=c (s).16 2 1 Equation (5) is one of they key equations which governs the demand for safe assets and will play an important role in the model’s mechanism to be discussed later. Since the demand for saving Dd is directly related to aggregate (consumption) demand at date 0, I refer to this equation 0 as the aggregate demand (AD) curve. The AD curve reflects the dependence of aggregate demand (cid:104) (cid:105) on systemic risk (cid:96) (s ) via future consumption E 1 . Moreover, the partial derivative of Dd 1 L 0 c (s) 0 1 ∂Dd with respect to (cid:96) (s ), 0 , an elasticity I will refer to as Channel 2, reflects the sensitivity 1 L ∂(cid:96) 1 (s L ) of aggregate demand to systemic risk and constitutes part of the dynamic interaction between 16The storage technology could be interpreted as as riskless international bond with which households in a small openeconomycanborroworsavebetweendates1and2. Thisassumptionisnotnecessaryformyqualitativeresults: All that is required is that the household have some means of smoothing consumption between dates 1 and 2, to some extent. For example, I could alternatively allow the government to issue bonds at date 1 through which the household could smooth consumption between dates 1 and 2. Nevertheless, this assumption improves the model’s tractabilityintwoways: 1)Duetoperfectconsumptionsmoothingbetweendates1and2,Ineedtokeeptrackonlyof totalfutureconsumption(dates1and2)ratherthanconsumptionateachdateseparately; 2)Itensuresthatoutputis supplied-determinedatdate1byimplyingthatthegrossnaturalrateofinterestatdate1is1. 13

systemic risk and aggregate demand at the heart of the model. This will be discussed in more detailinsections3and4. 2.3 Banks The representative, risk-neutral bank as linear utility v(cE) and consumes only at date 2, where 2 cE denotes the bank’s date 2 consumption. The bank is endowed with k units of capital at date 2 0 0. Within each period, the bank rents its capital holdings to intermediate goods producers in a competitive market and receives a real gross rental rate of capital rk(s) within the same period. Capitaldoesnotdepreciatefromuseinproduction. Banks also have access to an investment technology at dates 0 and 1 which converts one unit of the consumption good into one unit of new capital the next period and has constant returns-toscale. While this investment technology is itself risk-free, the bank’s date 0 investment ultimately involves aggregate risk since the date 1 return on capital rk(s) will reflect the aggregate shock at 1 date1. To finance investment, the bank issues debt to the household in a competitive market at date 0 with nominal face value D and gross nominal rate of return R , subject to a natural borrowing 0 0 limit which never binds in equilibrium. At date 0, the bank can participate in a competitive spot QE marketforcapitaltogetherwiththegovernment,wherek denotesthequantityofcapitalthatthe 1 banksellstothegovernmentatthespotpriceq ,whichequalsq =1inequilibrium. 0 0 The bank’s date 0 budget constraint on intertemporal choices at the end of date 0 ensures that investment i in new capital in the second stage of date 0 can be financed by issuing private 0 debt, rental income rkk , proceeds from sales of capital to the government q k QE , or government 0 0 0 1 transfersTE. 0 D (cid:0) 1−τ D(cid:1) 0 +rkk +q k QE +TE ≥i (6) 0 P 0 0 0 1 0 0 0 τRisadistortionarytaxwhichincreasesthecosttothebankofissuingdebtatdate0. Anyproceeds 0 from asset sales to the government can immediately be reinvested in the creation of new capital. Thebank’sstockofcapitalevolvesbetweendates0and1accordingto QE k =i +k −k (7) 1 0 0 1 At date 1, banks have to meet their obligations τRD 0 R 0 and can invest in new capital at date 1, 0 P(s) 1 subjecttoanon-negativityconstraintoninvestmenti (s)≥0. Theycanfinancetheseexpenditures 1 outoftheirearningsfromrentingcapitalatdate1rk(s)k ,fromliquidatingafraction(cid:96) (s)oftheir 1 1 1 14

capital holdings at date 1, or out of any lump-sum government transfers TE. The bank’s date 1 1 budgetconstraintistherefore D R rk(s)k +(cid:96) (s)k +TE ≥i (s)+τ R 0 0 (8) 1 1 1 1 1 1 0 P (s) 1 τR is a distortionary tax which increases the effective interest the bank must pay on its debt: The 0 bankpaystotalofτRR D ininterestatdate1,ofwhichR D goestothehouseholdswhoholdthe 0 0 0 0 0 bondsandtheremaining (cid:0) τR−1 (cid:1) R D goestothegovernment. Liquidationiscostlyandinvolves 0 0 0 η a loss of capital given by the strictly convex function φ((cid:96) ) = (cid:96) , where η > 1. Therefore, the 1 1 bank’sdate2capitalholdingsevolvesaccordingto k (s)=i +(1−(cid:96) −φ((cid:96) (s)))k (s) (9) 2 1 1 1 1 Atdate2,thebankrentsitscapitalstocktointermediategoodsfirmsanditconsumes,andcapital fully depreciates at date 2 after being used in production. The bank’s date 2 budget constraint is rkk +TE ≤cE (10) 2 2 2 2 whereTE aredate2transfers. 2 Bank’s optimality conditions The problem of the bank is to choose in each period and each state how much to invest in new capital i ,i , how much of its holdings of capital to sell at 0 1 QE date 0 k and liquidate in date 1 (cid:96) , and how much date 0 debt to issue D in order to maximize 1 1 0 itsexpecteddate2consumptionE (cid:2) v(cE) (cid:3) ,subjecttoitsbudgetconstraints,thenaturalborrowing 0 2 limit, the non-negativity constraint on date 1 investment, and the law of motions for capital. The fullproblemandthederivationoftheoptimalityconditionsisgiveninOnlineAppendix2. In the bad state at date 1, the bank forgoes investment, i (s )=0, and liquidates just enough 1 L capital to repay its debt, so that (cid:96) (s ) is pinned down by the binding non-negativity constraint on 1 L date1investment, (cid:96) (s )=Lev −rk(s ) (11) 1 L 0 1 L where I have defined the leverage of the bank at date 0 as Lev := τ 0 RD 0 R 0 −T 1 E , the ratio of the 0 k 1 bank’seffectiveliabilitiesnetoflump-sumtransferstoitsassets. Equation(11),whichIrefertoas thesystemicrisk(SR)curve,isoneofthekeyequationswhichwilldeterminethedemandforsafe assets.17 InAppendix4,Ishowthattheseverityofcrises(cid:96) (s ),andhencesystemicrisk,isstrictly 1 L d(cid:96) (s ) increasing in the bank’s date 0 debt, 1 L >0. I refer to this positive relationship between date dDs 0 17Aswillbecomeclearinsection4,theseverityofcrises(cid:96) (s )isdirectlyrelatedtothehousehold’sexpectedfuture 1 L 15

0 debt D and the severity of crises (cid:96) (s ) through the SR curve as Channel 1, which constitutes 0 1 L one of the key channels making up the dynamic interaction. This will be discussed in more detail insections3and4. Because the bank’s leverage at date 0 is determined by its choice of D , the bank’s desired 0 leverageratioispinneddownbyitsoptimalchoiceofdate0debtD ,whichbalancesthemarginal 0 benefitofdebtwiththemarginalcost. (cid:110) (cid:104) (cid:16) (cid:17)(cid:105) (cid:104) (cid:105)(cid:111) (cid:104) (cid:105) (cid:0) 1−τ D(cid:1) E rk rk(s)+1 +λ (s ) rk(s )+(cid:96) (s ) =E rk(cid:0) τ RR +φ((cid:96) (s)) (cid:0) 1−τ D(cid:1)(cid:1) +λ (s )τ RR 0 2 1 1 L 1 L 1 L 2 0 0 1 0 1 L 0 0 (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) marginalbenefit marginalcost (12) The marginal benefit of date 0 debt, net of the tax rate τD, is given by the expected return to 0 capital across dates 1 and 2 (the first term on the left-hand side of (12)) and the value of relaxing the non-negativity constraint on investment in the bad state (the second term on the left), where λ (s )reflectstheshadowpriceoffundsinthebadstate. Themarginalcostofdebtisgivenbythe 1 L effective interest rate τRR and the expected cost of liquidating extra capital in the bad state (the 0 0 first term on the right-hand side). Moreover, each unit of debt issued at date 0 tightens the bank’s non-negativity constraint on date 1 investment in the bad state by the effective interest rate τRR , 0 0 which has a cost given by the shadow value of liquid funds in that state, λ (s ) (the second term 1 L ontheright-handside). The convex liquidation cost φ((cid:96) ) introduces a concavity in the bank’s date 1 payoff function 1 (as a function of D ) which makes the bank behave at date 0 as if it is risk averse, which I show 0 in Appendix 1. This is illustrated in Figure 4, where MB(D ;R ) and MC(D ;R ) denote the 0 0 0 0 marginal benefit and marginal cost, respectively, of date 0 private debt D at a given interest rate 0 R . 0 Figure4: Bank’sleveragechoice (cid:104) (cid:105) laborincomeandconsumptionE 1 viathemacroeconomicspilloverfromliquidationtothehousehold’slabor 0 c1(s) income. 16

Intuitively, the strict convexity of the liquidation cost implies that the bank’s net marginal benefit is strictly decreasing in D for a sufficiently high D . I show in Appendix 1 that, as long 0 0 as the liquidation cost function φ(·) is sufficiently convex, the bank is at an interior optimum for his choice of D and the natural borrowing limit D is non-binding in equilibrium. The balance 0 0 between the bank’s marginal cost and benefit of debt issuance defines the bank’s supply curve Ds∗(R ;u )forprivatesafeassetsasafunctionofR foragivenlevelofutilizationu . 0 0 0 0 0 Riskpremium Acorollaryofthebank’srisk-aversebehavioristhat,inequilibrium,thereis astrictlypositiveriskpremiumatdate0,whichisdefinedasthedifferenceintheexpected(gross) E[rk(s)rk(s)] rateofreturntocapital,1+ 2 1 ,andtheeffectiverisk-freerateofreturn,τRR . E[rk(s)] 0 0 2 (cid:32) E (cid:2) rk(s)rk(s) (cid:3)(cid:33) RP := (cid:0) 1−τ D(cid:1) 1+ 2 1 −τ RR >0 (13) 0 0 E (cid:2) rk(s) (cid:3) 0 0 2 I discuss the intuition behind this expression in Appendix 2. In this setting, the positive risk premium results from the costly liquidation of capital which occurs in the bad state at date 1, φ((cid:96) (s ))>0. In the absence of a liquidation cost, the bank would want to increase its invest in 1 L capital up to the point that the return on capital and the cost of borrowing is zero – i.e. until the risk premium is 0. However, the positive expected cost of liquidation reduces the attractiveness of investing in capital, resulting in a positive equilibrium risk premium. Indeed, in Appendix 2, I showthatthisriskpremiumisstrictlypositiveinequilibriumifandonlyif(cid:96) (s)>0forsomestate 1 s.18 2.4 Nominal rigidity There is a representative firm who rents capital from bank and the government and hires labor from the household in competitive markets to produce the consumption good within each period according to the production function below, where α ∈ (0,1) and where k˜ = k +kG is the total t t t capitalstockatdatet. y =z (cid:0) u k˜ (cid:1)α n¯1−α (14) t t t t 18Thus,theliquidationcostinthismodellaysasimilarrolethatthecollateralconstraintplaysinCaballeroandFarhi (2018). In that model, a binding collateral constraint results in a positive risk premium, as borrowers are unable to scaleinvestmentuptothepointthattheexpectedrateofreturnonriskyassetsandthesaferateofreturnareequalized. 17

Thepriceoftheconsumptiongoodispredeterminedatdate0andisfullyrigidacrossallperiods.19 Hence,thereisnoinflation. ThepricelevelPisfixedandnormalizedto1. In each periodt, the firm chooses how much labor to hire, how much capital to rent, and how much capital to utilize to maximize its profits, taking prices and the demand yD it faces as given t andsubjecttoastandardbudgetconstraintdF =y −w n¯−rkk˜ . Thefirstorderconditions,derived t t t t t inOnlineAppendix3,equatetherentalrateofcapitalandthewagetothefactorsharesofincome: rk=α yt andw =(1−α) yt. Sincefirmscannotadjustpricesinresponsetothedemandtheyface, t k˜ t t nt capital utilization u ∈(0,1] is chosen optimally to ensure that their output is exactly equal to the t demandtheyface,y =yD. t t Potential output in date t is defined as the level output given full utilization of resources, y = t z kαn1−α. Because prices are rigid, output can possibly be below potential and determined by the t t levelofaggregatedemandwhichprevailsgivenpricesandtheinterestrate. Iassumethatmonetary policy ensures that output is at potential when possible. That is, away from the effective lower bound, the policy rate is set to the natural rate of interest – the rate such the level of aggregate demand is equal to potential output. As discussed in section 2.1, the natural rate is at date 1 is RB =1implyingthatoutputisalwaysatpotentialatdate1. 1 2.5 Government At date 0, the government can issue to the household one-period debt on a competitive market, where B denotes the nominal face value of the debt, which yields a state-uncontingent nominal 0 grossrateofreturnRB. Thegovernmentcanalsolevylump-sumanddistortionarytaxesonagents, 0 andcanpurchasecapitalfromthebankinacompetitivespotmarket,wherekGdenotesthequantity 1 of capital that the government purchases.20 Therefore, the government’s date 0 budget constraint is B 0 +T +τ −TE =q kG (15) P 0 0 0 0 1 0 whereτ :=τDD arethedistortionarytaxesleviedonthebankatdate0. Atdate1,thegovernment 0 0 0 earns rental income on its capital holdings, and can also levy lump-sum and distortionary taxes to 19Thenominalrigiditycanbemicrofoundedwiththestandardassumptionofmonoplisticallycompetitivefirmswho facesomedegreeofrigidityinprice-setting. SeeCaballeroandSimsek(2021).Thisextremeformofnominalrigidity isalsousedinCaballeroandFarhi(2018),CaballeroandSimsek(2020),CaballeroandSimsek(2021),andKorinek and Simsek (2016). This assumption is not critical to my results but significantly improves the model’s tractability whiledeliveringoneofthekeyfrictions(nominalrigidities)necessaryforoutputtopotentiallybedemand-determined inequilibrium. 20Ratherthanbuyingcapitaldirectly,Icouldequivalentlyassumethatthegovernmentbuysatdate0aclaimonthe date 1 rental rate of capital, and issues a lump-sum transfer to the bank in the bad state at date 1 to prevent it from liquidatingitscapitalholdings. Thisalternativewouldyieldthesameallocation. 18

financetherepaymentofitsdate0debt. B P T +τ −TE+rk(s)kG =RB 0 0 (16) 1 1 1 1 1 0 P P 0 1 whereτ := (cid:0) τR−1 (cid:1) R D arethedistortionarytaxesleviedonthebankatdate1. Thegovernment 1 0 0 0 storesitsdate1capitalholdingsthroughdate2withoutdepreciation. Therefore,thelawofmotion forthegovernment’scapitalholdingsatdate2is kG =kG 2 1 At date 2, the government earns rental income on its holdings of capital, can levy lump-sum and distortionarytaxationonbothagents,keepsabalancedbudget. T −TE+rkkG =0 (17) 2 2 2 2 Fornow,Itakethegovernment’sbehaviorasgiven. The government’s power to tax gives it a comparative advantage over the private sector in issuing safe debt. Hence, unlike the bank, the government does never needs to liquidate its capital holdingstoserviceitsdebt. 21 3 Equilibrium The equilibrium is a set of processes for allocations, prices, and returns such that households and banks maximize expected utility, firms maximize profits, capital evolves according to its laws of motion, the nominal interest rate follows the rule described in section 2.1, and the markets for labor,capital,andprivateandpublicbondsclear. RecallthatsupplyofpublicbondsB istakenas 0 exogenous until the normative section. I solve for the equilibrium recursively. Before solving for theequilibriumIimposethefollowingassumptions. Assumption1: (cid:18) (cid:19) A)z (s )∈ τ 0 DR 0 D 0 −k 1 , τ 0 RD 0 R 0 −T 1 E andz (s )≥ τ 0 RD 0 R 0 −T 1 E 1 L k 1α(k˜ 1 )α−1 n¯1−α k 1α(k˜ 1 )α−1 n¯1−α 1 H k 1α(k˜ 1 )α−1 n¯1−α B)TE <k 1 1 Appendix 12 shows that these assumptions ensure that the bank’s date 0 natural borrowing limit on borrowing D is never binding in equilibrium, and that liquidation occurs only in the bad 0 21Whilethegovernmentcouldliquidatecapitalholdings, abenevolentgovernmentwouldneverfind itoptimalto doso,asliquidationentailsahigherdeadweightlosscomparedtolump-sumtaxationduetotheliquidationcostφ((cid:96) ). 1 19

state, (cid:96) (s ) > 0, (cid:96) (s ) = 0. The appendix also shows that these restrictions are satisfied by a 1 L 1 H non-emptysetofparameters. 3.1 Equilibrium at dates 1 and 2 Output is at potential at dates 1 and 2 At date 2, monetary policy ensures output is at its potential, which isdetermined by the household’s inelastic labor supplyand the total capital stock whichthebankandgovernmentbringintodate2fromdate1. Isolveforthedate2equilibriumin OnlineAppendix4. At date 1, the return on the storage technology RB = 1 pins down the gross natural rate of 1 interestrateatR∗=RB=1atdate1,ensuringthatoutputisatpotential. Thehousehold’squantity 1 1 ofsavingatdate1simplyadjuststosatisfyitsEulerequationatthisinterestrate. Two regimes at date 1 The equilibrium at date 1 features two regimes, depending on whether the bank liquidates capital or not. This is shown in Online Appendix 5. In the normal regime,thebank’snon-negativityconstraintondate1investmentisnon-bindingandthebankdoes notliquidateanyofitscapitalholdingssothat(cid:96) (s)=0,λ (s)=0. Inthisregime,thebank’sdate 1 1 1 return from rental income is sufficient to meet its debt obligations τRD 0 R 0. In the crisis regime, 0 P 1 on the other hand, the banks’ date 1 income is insufficient to cover its debt obligations, forcing the bank to liquidate some portion (cid:96) (s)>0 of its capital holdings, which is pinned down by its 1 bindingnon-negativityconstraintondate1investment. τRD R TE (cid:96) (s)= 0 0 0 −rk(s)− 1 (18) 1 k 1 k 1 1 The lower the realization of date 1 TFP z , the more severe the crisis.22 Assumption 1 about TFP 1 in each state z (s) imply that the normal regime obtains in when TFP is high and the crisis regime 1 obtainswhenTFPislow,i.e. (cid:96) (s )=0and(cid:96) (s )>0. 1 H 1 L 3.2 Date 0 equilibrium Thedeterminationofthedate0equilibriumcanbeunderstoodintwostages: thedeterminationof the demand for and supply of safe assets Dd∗,Ds∗ for a given interest rate R and utilization rate 0 0 0 u (partial equilibrium); and the determination of the interest rate R and the utilization rate u 0 0 0 to clear the market for safe assets (general equilibrium). I first begin with the partial equilibrium determinationofthesupplyanddemandforsafeassets. Supply and demand for safe assets in partial equilibrium As discussed in section 2.3, the bank’s supply curve of safe assets is the outcome of the balance between the bank’s marginal 22Formally,becausek ispredeterminedatdate1andlaborisinfixedsupply,Ihave d(cid:96)1 =− dr 1 k =−α y1 <0. 1 dz1 dz1 k˜ t z1 20

benefit and marginal cost of leverage at date 0 summarized in equation (12), which respectively reflect the expected return to investment and the expected future liquidation costs. The bank’s optimal choice of date 0 debt Ds for a given R and u yields the supply curve of safe assets as a 0 0 0 functionoftheinterestrateandutilizationrate. The determination of the household’s demand curve for safe assets is the result of a balance between aggregate demand and systemic risk, respectively given by the key equations for the AD curve (5) and the SR curve (11), for a given R and u . (Recall that the AD curve (5) summa- 0 0 rizes the household’s consumption-saving decision conditional on a level of systemic risk through (cid:96) (s ),while theSRcurve summarizeshowthe household’sdemandfor saving(andtherefore ag- 1 L gregatedemand)affectssystemicriskthrough(cid:96) (s ),ceterisparibus.) Thisbalance,orfixedpoint, 1 L can be illustrated graphically in Figure 5, which plots in a stylized manner the AD curve and SR curve, with the household’s date 0 demand for safe assets (holding all else fixed) Dd on the x-axis 0 andtheseverityofcrisis(cid:96) (s )onthey-axis.23 1 L Figure5: DemandforsafeassetsfromthebalancebetweenADandSR TheslopeoftheSRcurveisgivenbyChannel1ofthedynamicinteractionbetweenaggregate d(cid:96) (s ) demand and systemic risk, 1 L , which captures the (positive) relationship between date 0 debt dDs 0 D and the severity of crises at date 1 (cid:96) (s ).24 The slope of the AD curve, in turn, is given by 0 1 L ∂Dd Channel 2 of the dynamic interaction, 0 , which captures how systemic risk affects aggregate ∂(cid:96) 1 (s L ) demandatdate0.25 Appendices3and4showthatbothchannelsstrictlypositive d(cid:96) 1 (s L ) , ∂Dd 0 >0 dDs 0 ∂(cid:96) 1 (s L ) 23Notethatinthefigure,IareplottingtheinversefunctionoftheADcurve(5)byshowing(cid:96) (s )asafunctionof 1 L Dd. Moreover,foreachcurveIareplottingtherelationbetweenDd and(cid:96) (s )toafirstorderapproximationforease 0 0 1 L ofexposition. 24The x-intercept of the SR curve is given by D = r 1 k(sL)k1+T 1 E , which reflects how much private debt at date 0 it 0 τ 0 RR0 takesfortheretobenoliquidationinthebadstateatdate1. Thex-interceptoftheADcurveisgivenbyacomplex expression not reproduced here and reflects the household’s demand for safe assets when (cid:96) (s )=0, that is, in the 1 L absenceofsystemicrisk. 25Technically,sinceIdefineChannel2as ∂Dd 0 ,theslopeofthe(inverse)ADcurveplottedinFigure5isgivenby ∂(cid:96)1(sL) theinverseofChannel2. 21

underamildassumption,andhencebothcurvesareupwardsloping.26 The fixed point (cid:0) Dd∗,(cid:96)∗(s ) (cid:1) between the two curves is the result of the dynamic interplay 0 1 L between aggregate demand and systemic risk (Channels 1 and 2), and reflects that the equilibrium valuesofsystemicriskimpliedby(cid:96) andaggregatedemandimpliedbyDd mustbeconsistentwith 1 both the household’s consumption-saving decision (captured by the AD curve equation) and the relationship between the bank’s leverage and systemic risk (captured by the SR curve equation). 27The fixed point in turn determines the household’s demand curve Dd∗(R ;u ) for safe assets for 0 0 0 givenlevelsofR andu .28 0 0 How does this figure compare relate to the literature? In much of the related literature, systemic risk is either non-existent, because the supply of safe assets is pinned down by a binding borrowing constraint which prevents default or liquidation, or systemic risk is fixed exogenously. Therefore, comparable SR curve in the literature would either be absent or would be horizontal and unresponsive to changes in other variables. As a result, relative to the literature, the determination of the equilibrium differs because the levels of systemic risk and aggregate demand are determined jointly, and the response of the equilibrium to shocks and changes in monetary, fiscal, ormacroprudentialpolicydiffersasIll,asIwilllatershow. General equilibrium in the market for safe assets Given the household’s demand curve Dd∗forsafeassetsandthebank’ssupplycurveforsafeassetsDs∗,theinterestrateand/orutilization 0 0 rate R ;u are determined in general equilibrium to clear the market for safe assets Dd∗(R ;u )= 0 0 0 0 0 Ds∗(R ;u ).29 Whether the interest rate R or utilization u adjusts to clear this market depends 0 0 0 0 0 on whether the effective lower bound on monetary policy is binding or not. Therefore, the date 0 equilibrium features two regimes depending on whether this constraint binds, as is shown in the followinglemma. Thefullderivationofthedate0equilibriumisgiveninOnlineAppendix6. Lemma 1: There are two regimes at date 0: an aggregate supply (AS) regime in which the 26For ease of exposition, in Figure 5 I plot the case in which the AD curve is steeper than the SR curve and its x-interceptlarger,thoughthisneednotbethecaseingeneral. 27Tounderstandconvergencetothefixedpoint,notethat,forallDd >Dd∗,thelevelof(cid:96) (s )impliedbytheAD 0 0 1 L curve is strictly greater than that implied by the SR curve. Therefore, for Dd to be consistent with the household’s 0 consumption-savingdecision,(cid:96) (s )(andthereforesystemicrisk)mustbyhigherthanitactuallyisgivenDd.Sogiven 1 L 0 thelowlevelofsystemicrisk,thehousehold’sdemandforsafeassetsDd mustbelower. Similarly,forallDd <Dd∗, 0 0 0 thelevelof(cid:96) (s )impliedbytheADcurveisstrictlylessthanthatimpliedbytheSRcurve. Therefore,forDd tobe 1 L 0 consistentwiththehousehold’sconsumption-savingdecision,(cid:96) (s )mustbelowerthanitactuallyisforthatDd. So 1 L 0 giventhatsystemicriskisrelativelyhigh,thehousehold’sdemandforsafeassetsDd mustbehigher. Dd∗ isthepoint 0 0 atwhichthelevelofsystemicriskimpliedby(cid:96) (s )andthelevelofaggregatedemandimpliedbyDd areconsistent 1 L 0 withbothrelations(5)and(11). 28While Dd denotes the household’s demand for private safe assets, recall that household is indifferent between 0 holding private versus public safe assets. Since the supply of public safe assets B is thus far taken as exogenous, 0 B simplycrowdsoutdemandforprivatesafeassets. ThisisreflectedintheADcurveequation(5)bythenegative 0 relationshipbetweenB andDd. 0 0 29Ofcourse,thesevariablesaredeterminedtogetherwiththethemonetarypolicyruledescribedinsection2.1and thefinalgoodproducingfirm’soptimalchoiceofcapitalutilizationdescribedinsection2.4. 22

effective lower bound on the nominal interest is non-binding RB > 1 and capital is fully utilized 0 u = 1; and an aggregate demand (AD) regime in which the effective lower bound is binding 0 RB =1andcapitalisunder-utilizedu <1. 0 0 Proof: SeeOnlineAppendix6. Essentially, which regime prevails at date 0 depends on how the economy adjusts to clear an excessdemandforsaving,denotedbyDd(R ,B ;u )−Ds(R ).30 31 0 0 0 0 0 0 Supply-determined regime In the aggregate supply-determined regime, the effective lower bound is not binding and output is at potential. The nominal interest rate R is able to adjust 0 to equilibrate the demand for saving at date 0 and is determined such that the bank’s optimality conditionforborrowingholds. InOnlineAppendix7,IshowhowtheadjustmentinR equilibrates 0 themarketforsafeassetswhileleavingoutputatpotential. Demand-determined regime In the aggregate demand-determined regime, the effective lower bound is binding R = 1 and aggregate output at date 0 is demand-determined. At the 0 effectivelowerboundR =1,thereisanexcessdemandforprivatedebtDd(R ,B ;u )−Ds(R ). 0 0 0 0 0 0 0 Toreachequilibrium,theutilizationofcapitalmustfalltoeliminateanyexcessdemandforsaving Dd(R ,B ;u )−Ds(R ), depressing output below its potential. In Online Appendix 7, I show 0 0 0 0 0 0 howafallinutilizationu equilibratestheeconomybyreducingthehousehold’sincentivetosave. 0 Figure6illustrateshowtheadjustmentinutilizationequilibratestheeconomyattheeffectivelower bound,resultinginademand-drivenrecession–thatis,asafetytrap. 30Recall that I had showed above that Dd reflects aggregate consumption demand while Ds reflects aggregate in- 0 0 vestment demand at date 0. Thus, aggregate demand relative to potential output at date 0 is captured by the excess demandforsavingatdate0,Dd(R ,B ;u )−Ds(R ). Thatis,ifthereisanexcessdemandforsafeassets,givenby 0 0 0 0 0 0 Dd−Ds >0, aggregate demand is below total output. In the supply-determined regime, when the lower bound on 0 0 the nominal interest is not binding, monetary policy can fall to be equal to the natural rate of interest ensuring that aggregatedemandforoutputequalspotential(that,thereisnoexcessdemandforsavingDd−Ds =0). 0 0 31Notethattheexpressionforexcessdemandforprivatedebtalreadyembedsthemarketclearingconditioninthe market for public bonds, and reflects that the government’s exogenous supply of public bonds affects household’s demandforprivatebondsthroughDd(R ,B ;u ). 0 0 0 0 23

Figure6: Safetytrapinthedemand-determinedregime At the effective lower bound R =1, there exists an excess demand for safe assets, shown in 0 panelIII.Thisalsoresultsindeficientaggregatedemandinthemarketforgoods(panelIV),given thefixedpricelevel. Inordertoclearthesemarkets,utilizationfalls,shiftingthedemandcurvefor safe assets in and the supply curve for safe assets out. This also results in a fall in the aggregate supplyofgoods. In addition, the fall in utilization has feedback effects to the SR, AD, MC, and MB curves in panels I and II. In particular, the demand-driven recession affects systemic risk and the demand of safe assets through two general equilibrium feedback channels, Channels 3a and 3b, which I discuss in detail in section 4. Together with Channels 1 and 2, these channels constitute a twoway feedback loop between systemic risk and aggregate demand, depicted stylistically in Figure 1. In addition, the fall in u also affects the bank’s incentive to issue debt, which shifts the bank’s 0 supplycurveofsafeassets.32 Inordertoreachanequilibrium,thefallinu mustreducetheexcess 0 demandfordebtonnetwhentakingintoaccountalleffectsonthedemandandsupplyofdebt. General equilibrium is the fixed point between these curves, and hence through the dynamic interactionbetweensystemicriskandaggregatedemandwhicharisesingeneralequilibrium. 32InOnlineAppendix7,Itraceouttheseeffects. InpanelIIofFigure6,Idepictonlyoneoftheseeffects,whichis thatthedemand-recessionatdate0increasestheexpectedreturntocapital,shiftingtheMBcurveup. 24

3.3 Central features of the model It is worth summarizing the key features of the model which will generate the main mechanisms andinsightsofthepaper. 1) Crises at date 1 result in a macroeconomic spillover to the household’s date 2 labor income Acrisisatdate1occurswhenthebankisforcedtoliquidatesomeofitscapital,(cid:96) (s)>0, 1 inordertoserviceitsdebt. Inreducingthecapitalstockatdate2andthereforethemarginalproduct of labor at date 2, a crisis lowers the household’s date 2 labor income w n in general equilibrium. 2 Asaresultofthismacroeconomicspillover,thehousehold’slaborincomeislowinthesamestates oftheworldthatthebank’sincomeislow.33 2)Systemicriskaffectsthesupplyofanddemandforsafeassetsexante Onthedemandside, the macroeconomic spillover implies that an increase insystemic risk – defined as expected future liquidation at date 1, (cid:96) (s) – increases the household’s date 0 demand for safe asset due to both 1 lowerexpectedfuturelaborincomeatdate2andthroughaprecautionarysavingmotive. Therefore, anincreaseinsystemicriskwillincreasethedemandforsafeassetsandreduceaggregatedemand atdate0. On the supply side, the willingness of banks to create safe assets depends on their capacity to absorb risk. In order to insure bondholders against the date 1 productivity shocks, banks must themselves bear this risk either through their net worth or by liquidating capital at date 1 to raise funds when their funds are insufficient to service debt.34 Therefore, fluctuations in the bank’s net worthatdate0willaffectthesupplyofsafeassetsexanteatdate0. 3) Private safe asset creation generates systemic risk while public safe asset creation does not Private safe asset creation generates systemic risk at date 0, as it forces banks to bear aggregate risk in part through liquidation of capital in the bad state at date 1. Though insured against the productivity shock z at date 1, bondholders are not insured against this systemic risk – that is, households bear the risk of low labor income in the bad state at date 2 owing to the macroeconomic spillover. Nevertheless, in making their portfolio choice at date 0, households do not internalize that private safe asset creation generates systemic risk, as the macroeconomic spillovermaterializesonlyingeneralequilibrium. Unlike private safe assets, however, the production of public safe assets does not create systemic risk as a side-effect. This is because, due to the government’s power to tax, at date 1 the governmentcanmeetitsobligationswithoutliquidatingcapital. 4)Safetytrapsemergeasresultofdynamicinteractionbetweenaggregatedemandandsystemic 33Sincethebank’sdecisiontoliquidatecapitalatdate1dependsinpartonitsleverage,thehousehold’sdecisionto investinprivatedebthasaneffectonw ningeneralequilibriumthroughthisspillover. However,householdsdonot 2 internalizethisspilloverwhenmakingtheirdate0decisiontoinvestinprivatedebt. Thishasimportantpositiveand normativeimplications 34Thisdecisionofhowtoabsorbshocksissummarizedinthebank’sfirst-orderconditionfor(cid:96) (s). 1 25

risk The effective lower bound on nominal interest rates generates the possibility of demanddrivenrecessionsduetoashortageofsafeassets,knownintheliteratureasasafetytrap. However, unlikeintheliterature,safeassetshortagesareentirelyendogenoushere,resultingfromadynamic interaction between systemic risk and aggregate demand which causes supply of and demand for safe assets to be out of balance at the effective lower bound. These insights are explored in more detailinsection5.3. 4 Interactions between debt, systemic risk, and aggregate demand In this section, I study how public and private debt affect systemic risk and aggregate demand. To do so, I characterize the marginal effect of an increase in the supply of either type of debt on the equilibrium allocation, and a provide decomposition of these effects into three separate channels whichtogetherconstituteadynamicinteractionbetweensystemicriskandaggregatedemand. For simplicity,Ileaveasidedistortionarytaxationformypositivecharacterizations,τD =0,τR =1. 0 1 4.1 Dynamic interaction between systemic risk and aggregate demand To understand how the quantity of private (public) debt affects the equilibrium allocation at the margin,onecanevaluatethetotalderivativeofeachendogenousvariablewithrespecttoDs (B ).35 0 0 Thetotaleffectcanbedescribedbythreechannels,illustratedinthefigurebelowforprivatedebt.36 Figure7: Dynamicinterplaybetweenaggregatedemandandsystemicrisk 35Effectively,Iexaminethefirstordereffectofontheequilibriumofanoutwardshiftinthesupplycurveofdebtof onetype,holdingconstantthesupplyofthetheothertypeofdebt. Whilethetotaleffectontheequilibriumallocation is determined in general equilibrium by the confluence of many channels, separating the effects of a change in the supplyversusdemandofonetypeofdebtallowsustotraceoutthesechannelsmoreclearly. 36This figure is special case of Figure 1 in that it applies when monetary policy is constrained in the demanddeterminedregime. 26

Channels 1 and 2 capture how the supply of safe assets affects aggregate demand by changing the excess demand for saving Dd(R ,B ;u )−Ds(R ), at a given interest rate R and utilization 0 0 0 0 0 0 0 rate u . These channels emerge by deriving the household’s demand for saving (7) with respect to 0 thesupplyofprivatesafeassets. Channel2 (cid:122) (cid:125)(cid:124) (cid:123) Channel1 (cid:104) (cid:105) (cid:104) (cid:105) dDd 0 = ∂Dd 0 ∂E 0 c 1 1 (s) + ∂Dd 0 ∂E 0 c 1 1 (s) (cid:122) d(cid:96) 1 (cid:125) ( (cid:124) s L ) (cid:123) (19) dDs 0 ∂E 0 (cid:104) c 1 (s) (cid:105) ∂Ds 0 ∂E 0 (cid:104) c 1 (s) (cid:105) ∂(cid:96) 1 (s L ) dDs 0 1 1 (cid:124) (cid:123)(cid:122) (cid:125) dynamicinteraction fromSRtoAD The first term on the right-hand side captures the effect of the supply of private debt on consumption demand, excluding the effects via systemic risk, i.e. expected losses due to liquidation in the bad state (cid:96) (s ). The second term captures the role of systemic risk in shaping the household’s 1 L saving demand and is comprised of the two key channels of the model. Channel 1 reflects the effect of issuance of private debt on systemic risk (i.e., the severity of crises); while Channel 2 reflectstheeffectofsystemicriskonthehousehold’sconsumptiondemandthroughboththeeffect ofliquidationonexpectedfutureconsumptiononconsumptionrisk(i.e. consumptionvolatility).37 InAppendices3,4,and5,IshowthatbothChannels1and2arestrictlypositiveundermildas- (cid:104) (cid:105) sumptions: ∂Dd 0 ∂E 0 c1 1 (s) >0and d(cid:96) 1 (s L ) >0.38 Therefore,anincreaseinthesupplyofprivate ∂E 0 (cid:104) c1 1 (s) (cid:105) ∂(cid:96) 1 (s L ) dDs 0 debtunambiguouslyincreasessystemicrisk,ceterisparibus,byincreasingthebank’sleverageand thereforetheseverityofcrisesatdate1(Channel1). Inturn,highersystemicrisklowersaggregate demandatdate0byincreasingthehousehold’sdesiretosave(Channel2). 39 Channel 3 captures a general equilibrium feedback effect. The excess demand for saving Dd(R ,B ;u )−Ds(R ) must be eliminated in general equilibrium through an adjustment in the 0 0 0 0 0 0 date 0 interest rate R or the utilization rate u (depending on whether the supply-determined or 0 0 demand-determined regime prevails at date 0). In general equilibrium, the adjustment of R or 0 u feeds back to the date 0 supply of debt Ds and aggregate demand. This general equilibrium 0 0 feedback effect is captured by Channel 3, represented by the dashed lines in Figure 7. For now, I leave aside this general equilibrium feedback 3 by holding both R and u constant, and return to 0 0 37Channel 2 thus captures one direction of a dynamic interaction between systemic risk and aggregate demand – the effect of systemic risk on aggregate demand at date 0, through the severity of crises at date 1. I show later that, in general equilibrium, there is also an effect in the reverse direction. Together these two channels will generate a feedbackloopbetweenaggregatedemandandsystemicrisk. 38Asufficientconditionfortheformerresultisthat k 2 G < 1,whileasufficientconditionforthelatteristhatlumpk˜ 2 n¯ sumtransfersaresufficientlysmallforlatter. TheseassumptionsareinterpretedinAppendix3and4,respectively. 39Notethattheeffectofhighersystemicriskonaggregatedemandatdate0reflectsbothadesiretosmoothconsumption across time due to lower expected future consumption, and for precautionary motives due to higher consumptionrisk. 27

itinsection4.3. 4.2 Public safe asset creation does not directly generate systemic risk In order to compare the role of public versus private debt, I now study the effect of an increase in the supply of public debt B on aggregate demand, which requires that I make assumptions 0 about what the government does with the proceeds of debt issuance at the margin. In this section, I assume that the government invests all proceeds from debt issuance in capital purchased on the spotmarketatdate0,whichitholdsthroughdate2,andrebatestherentalincomeearnedonthese capital holdings to the bank in lump-sum fashion. These assumptions minimize the distortions associated with issuance of public debt by ensuring that the distribution of wealth across agents is notaffecteddirectly,allowingustofocusontheeffectsofpublicdebtissuanceperse.40 Deriving the household’s demand for saving Dd(R ,B ;u ) with respect to B , shown in Ap- 0 0 0 0 0 pendix6,Iobtainasimilarexpressionto(37). (cid:104) (cid:105) dDd ∂Dd ∂E 0 c 1 (s) 0 = 0 1 (20) (cid:104) (cid:105) dB 0 ∂E 1 ∂B 0 0 c (s) 1 Comparing (37) and (38), I can see that, unlike with private safe assets, the supply of public safe assetsdoesnotaffectconsumptiondemandthroughsystemicrisk–Channels1and2arenotactive for public safe asset production. The government’s special power to tax frees it from the burden of having to liquidate capital in order to service its debt in the bad state at date 1. Therefore, the governmentcanproducesafeassets,andthusabsorbaggregaterisk,withoutdirectlyexacerbating theseverityofcrisesatdate1. As a result, the dynamic interaction between aggregate demand and systemic risk applies only for private safe asset production, but not for the production of public safe assets. I show in Online dDd dDd Appendix 8 that, as a result, I have 0 > 0; that is, an increase in the supply of private debt dDs dB 0 0 unambiguously reduces consumption demand (i.e. increases demand for saving) at the margin by morethananincreaseinthesupplyofpublicdebtdoes.41 40TheseassumptionsaremadeexplicitinAppendix6. Inthenormativesection,Irelaxtheseassumptionsaboutthe government’sbehavior. 41Public safe asset creation may have indirect effect on systemic risk by crowding-in or -out private safe asset dDs creationthroughtheterm 0. Itakethisissueupinthenormativesectionofthepaper. dB0 28

4.3 General equilibrium feedback effects from aggregate demand to systemic risk I now characterize Channel 3 in Figure 7, the general equilibrium feedback effects from the adjustmentinR oru followingachangeinthesupplyofsafeassets. TheprecisenatureofChannel 0 0 3 depends on the regime that prevails at date 0 – that is, whether R or u is adjusting to reach 0 0 equilibrium. General equilibrium feedback effects in the supply-determined regime Recall that systemic risk (i.e. (cid:96) (s )) depends on R D and k through the bank’s leverage ratio at date 0, Lev . 1 L 0 0 1 0 Therefore, depending on the net effect of the shock to R D , the bank’s date 0 leverage ratio, and 0 0 thereforesystemicrisk,maydecreaseorincreaseingeneralequilibrium.42 Generalequilibriumfeedbackeffectsinthedemand-determinedregime Inthedemanddetermined regime, utilization u falls in response to the increase in excess demand. The con- 0 sequent fall in output lowers the rental income of banks and thus erodes their date 0 net worth. Ceteris paribus, the fall in the bank’s net worth increases its date 0 leverage ratio and hence the expectedfutureliquidationcosts. Dependingonhowthebankmanagesthisriseinliquidationrisk, therearetwogeneralequilibriumfeedbackchannelsthatmayemergethroughwhichthefallinnet worthaffectstheequilibrium: throughafallinthesupplyofprivatesafeassetsorthrougharisein thedemandforsafeassets. Channel 3a: Demand recession further reduces supply of private safe assets Channel 3a dDs captures the effect of the recession on the supply of private safe assets, 0. If, in a bid to manage du 0 the liquidation risks associated with its lower net worth, the bank tries to delever by reducing its dDs issuance of debt, then 0 > 0. While this channel does not directly exacerbate systemic risk du 0 (cid:96) (s ), it exacerbates the shortage of safe assets by contracting the supply and hence worsens the 1 L demand-drivenrecession. Channel3b: Demandrecessionfurtherincreasessystemicrisk Channel3bcapturestheeffect d(cid:96) (s ) of the recession on the severity of crises, 1 L . If, in response to the erosion in its net worth, the du 0 bank allows its leverage ratio to rise rather than delevering, then the bank’s leverage ratio rises so d(cid:96) (s ) thatliquidationriskrises, 1 L <0. Thisconstitutesafeedbackchannel fromaggregatedemand du 0 tosystemicrisk,illustratedinFigure7. The total marginal effect of general equilibrium feedback Channels 3a and 3b on the severity of the demand-recession can be expressed as the sum of these two channels, which are controlled 42Inparticular,asR fallsintoclearanexcessdemandforsaving,thebank’sdate1interestpaymentsfall,lowering 0 liquidationinthebadstate. However,R alsoinducesthebanktoincreasingdate0borrowingD atthemargin,which 0 0 hasapositiveeffectonliquidation. TheresponseofsystemicrisktothethefallinR dependsonthenetofthesetwo 0 effects. WhileIcouldlayouttheconditionsunderwhichoneeffectdominatestheother, thenetchangeinD inthe 0 supply-determinedregimeisnotcentraltotheresultsIwishtocharacterize. 29

dDs d(cid:96) (s ) by the elasticities 0, 1 L which respectively govern the general equilibrium response of the du du 0 0 supplyofprivatedebtandsystemicrisktothedemanddrivenrecession.43   Channel3a Channel3b d (cid:0) Dd−Ds(cid:1)  (cid:122) d (cid:125) D (cid:124) s (cid:123) (cid:122) d(cid:96) (cid:125) ( (cid:124) s ) (cid:123)  0 0  0 + 1 L  (21) dDs  du du  0  0 0  Feedback loop may amplify the demand recession Taking into account Channel 3a and 3b, the model features a dynamic feedback loop between systemic risk and aggregate demand in general equilibrium, illustrated in Figure 7: Systemic risk affects aggregate demand at date 0 in partial equilibrium (Channel 2) and, in general equilibrium, aggregate demand in turn affects systemicrisk(Channels1and3). ∂Ds d(cid:96) (s ) Appendix7showsthat,underrelativelyweakconditions,Ihave 0 >0and 1 L <0sothat ∂u 0 du 0 both general equilibrium Channels 3a and 3b imply that date 0 output in the demand-determined regime is lower than it would otherwise be. In the case of Channel 3a, the fall in the supply of private safe assets further increases the excess demand for private safe assets at date 0, while in thecaseofChannel3b,thehighersystemicriskassociatedwithlargerliquidationcostscausesthe household to demand more safe assets at date 0. In this manner both channels may cause a higher demand for safe assets, requiring a deeper demand-driven recession to equilibrate the economy at date0.44 5 The safety trap and systemic risk In this section, I show that the demand-determined regime features a safety trap, which I define as ademand-drivenrecessionwhicharisesduetoashortageofeithertypeofsafeasset.45 Ialsoshow thatthedynamicinteractionbetweenaggregatedemandandsystemicriskatthecoreofthismodel may fundamentally alter the nature of safety traps relative to the literature, and implies that there aredifferenttypesofsafetytraps. Remarkonsafetytrapsversusliquiditytraps AsdiscussedinCaballeroandFarhi(2018), safetytrapsareaspecialtypeofliquiditytrapinwhichthereisashortageofsafeassetsasopposed to a shortage of assets more broadly. For systemic risk (which is the focus of this paper) to play a role in my model, I must consider an environment of positive risk premia and hence a shortage of 43Forsimplicity,IomitfromtheillustrationtheeffectofutilizationonDd throughthewagew . 0 0 44Insection6,Ishowthat,underweakconditions,Channels3aand3balsoamplifytheseverityofdemandrecessionsinresponsetoashockatdate0. 45Nevertheless, Ishowinsection5that, inthissetting, asafetytrapmustfeatureashortageofpublicsafeassets, butnotnecessarilyashortageofprivatesafeassets. 30

safeassets.46 5.1 Conditions defining a safe asset shortage Conceptually, a shortage of onetype of safe asset implies that, in equilibrium, amarginal increase in the supply of that asset reduces the severity of the demand recession at date 0. Therefore, to determine whether there is a shortage of public (private) safe assets in the demand-determined regime, I ask whether, at the margin, an increase in the supply of public (private) safe assets increasesordecreasestheexcessdemandforsaving. TheexplicitconditionsaregiveninAppendix 8. Intuitively,amarginalincreaseinthesupplyofasafeassethasbroadlytwoeffectsontheexcess dDs demand for saving at date 0: an effect on the supply of private bonds 0, and an effect on the dedB 0 (cid:104) (cid:105) mand for private bonds Dd(R ,B ;u ) through the household’s Euler equation ∂Dd 0 ∂E 0 c1 1 (s) . 0 0 0 0 ∂E 0 (cid:104) c1 1 (s) (cid:105) ∂B 0 Iftheneteffectisafallintheexcessdemandforsaving,thenutilizationmustriseatthemarginto equilibratetheeconomy,resultinginalessseveredemandrecession. 5.2 A taxonomy of safety traps In this section, I show how the dynamic interaction between systemic risk and aggregate demand alters the nature of safety traps. I first show that the safety trap in the demand-determined regime dDs must feature a shortage of public safe assets, as long as 0 is not too negative – that is, if an dB 0 increase in the supply of public safe assets crowds-in private safe asset creation, or at least does notcrowditattoomuch. Lemma2: Safetytrapfeaturesshortageofpublicsafeassets (cid:104) (cid:105) Supposethat ∂Dd 0 ∂E 0 c1 1 (s) −1< dDs 0 issatisfiedinequilibriuminthedemand-determined ∂E 0 (cid:104) c1 1 (s) (cid:105) ∂B 0 dB 0 (cid:104) (cid:105) regime. Then, in the demand-determined regime, I have ∂Dd 0 ∂E 0 c1 1 (s) ≤?1+ dDs 0, implying ∂E 0 (cid:104) c1 1 (s) (cid:105) ∂B 0 dB 0 thatthedemand-determinedregimefeaturesashortageofpublicsafeassets. Proof: SeeAppendix9. The condition spelled out in the lemma is a sufficient condition for the demand-determined regime to always feature a shortage of public safe assets (and therefore, a safety trap).47 Hence- 46Centraltosafetytrapsarepositiveriskpremiums. AsIdiscussedinsection2.3,apositiveriskpremiumarisesin thismodelduetotheliquidationcostofcapital. Thus,byconstruction,thesafetytrapistheonlytypeofliquiditytrap thatoccursinequilibriuminthismodel. (cid:104) (cid:105) 47SinceIshowedpreviouslythat ∂Dd 0 ∂E0 c1 1 (s) <0,asufficientconditionforthistoholdwouldbe dDs 0 ≥0. ∂E0 (cid:104) c1 1 (s) (cid:105) ∂B0 dB0 31

forth,Irestricttheanalysistoregionsoftheparameterspaceinwhichthisconditionissatisfiedin equilibriuminthedemand-determinedregime. In this model, not all safety traps are alike. Paradoxically, the demand-determined regime can feature an oversupply of private safe assets at the same time that it features a shortage of public safe assets. Indeed, this model produces a taxonomy of safety traps depending on whether the equilibriumfeaturesashortageoranoversupplyofprivatesafeassets. Proposition1: Ataxonomyofsafetytraps A)Thedemand-determinedregimeatdate0featuresasafetytrapwhichisoneoftwotypes: In a conventional safety trap, the economy features a shortage of both public and private safe assetsinequilibrium: Atthemargin,anincreaseinthesupplyofeitherpublicorprivatesafeassets wouldincreaseaggregateoutputatdate0. In a risk-intensive safety trap, the economy features a shortage of public safe assets and an oversupplyofprivatesafeassetsinequilibrium: Atthemargin,anincreaseinthesupplyofpublic oradecreaseinthesupplyofprivatesafeassetswouldincreaseaggregateoutputatdate0. B)ThiscaseobtainsifandonlyiftheproductofChannels1and2issufficientlylarge. Channel2 (cid:122) (cid:125)(cid:124) (cid:123)Channel1 (cid:104) (cid:105) (cid:104) (cid:105) ∂Dd 0 ∂E 0 c 1 1 (s) (cid:122) d(cid:96) 1 (cid:125) ( (cid:124) s L ) (cid:123) >1− ∂Dd 0 ∂E 0 c 1 1 (s) ∂E 0 (cid:104) c 1 (s) (cid:105) ∂(cid:96) 1 (s L ) dDs 0 ∂E 0 (cid:104) c 1 (s) (cid:105) ∂Ds 0 1 1 wheretheleft-handsideisstrictlypositive. Proof: SeeAppendix10. Whilebothtypesofsafetytrapfeatureashortageofpublicsafeassets,whatdistinguishesthem iswhetherthereisashortageoranoversupplyofprivatesafeassets. Inaconventionalsafetytrap, the demand recession at date 0 is driven by a shortage of both public and private safe assets, and reflectsageneralinsufficiencyinthemeansofsmoothinghouseholdconsumptionacrosstimeand states. Thistypeofgeneralizedshortageofsafeassetsiswhatcharacterizestheliteratureonsafety traps,suchasCaballeroandFarhi(2018). In a risk-intensive safety trap, by contrast, the demand recession is driven is by a shortage of 32

public safe assets together with an oversupply of private safe assets.48 49 Importantly, in a riskintensive safety trap, the economy is characterized by a paradox of safety, in which the desire of householdstoholdsafeassetsasinsuranceagainstsystemicriskultimatelyincreasessystemicrisk ingeneralequilibrium,throughthehigherleverageofbanks. Inthiscase,thecauseofthesafeasset shortage is financial instability (i.e. high systemic risk), which leads to a deficiency of aggregate demand due to precautionary saving. In turn, this endogenously leads to a safety trap because low output erodes banks’ date 0 net worth and thus reduces their ability to produce private safe assets withoutfurtherincreasingsystemicrisk. The proposition also shows that the nature of safety traps is determined by the strength of dynamic interaction between systemic risk and aggregate demand: Risk-intensive safety traps occur if and only if the dynamic interaction between systemic risk and aggregate demand, given by the productofChannels1and2,issufficientlystrong.50 Tounderstandwhy,recallthatinthissetting, the supply of private safe assets has an effect not only on the aggregate supply of safe assets but also on the demand for safe assets due to its effect on systemic risk. A safety trap features an oversupplyofprivatesafeassetsifamarginalincreaseinthesupplyofprivatesafeassetsincreases the demand for safe assets by more than the supply, worsening a shortage of safe assets. In this case, a marginal rise in the supply of private safe assets would increase bank leverage and therefore increase systemic risk (Channel 1). In turn, the rise in systemic risk would, ceteris paribus, raise the household’s demand for saving (Channel 2), as it would have more incentive to transfer resources to the bad state of the world at dates 1 and 2. In such a case, the safety trap would, counterintuitively,beamelioratedthroughareductioninthesupplyofprivatesafeassets. AsIshowinthenormativesection,thenatureofthesafetytrapwillhaveimportantimplications forthedirectionofexternalitiesandthedesignofoptimalpolicy.51 48In a risk-intensive safety trap, a higher supply of private safe assets could increase future output, y ,y , even if 1 2 it reduces current output, y . For example, a higher supply of private safe assets allows the bank to increase date 0 0 investment,andthereforecouldcausefutureoutputy ,y toriseduetoahigherstockofcapital. Atthesametime,the 1 2 highersupplyofprivatesafeassetsincreasesbank’sleverageandthereforesystemicrisk. Ifthedynamicinteraction between systemic risk and aggregate demand (Channels 1 and 2) is sufficiently strong, the latter effect dominates, meansingthatthehighersupplyofprivatesafeassetsreducescurrentoutputatthemargin,evenifitincreasesfuture output. 49Keytothistaxonomyisthedifferentialmannerinwhichtheproductionofpublicandprivatesafeassetsaffects systemicrisk. 50Recallalsothatthestrengthofeachchanneliscontrolledbytheelasticitiesofcrisisrisktotheleverageofbanks andofhouseholds’savingdemandtocrisisrisk,respectively. 51As I will show later, this also equivalently means the direction of an aggregate demand externality is such that plannerwouldreducethebank’sleveragerelativetothecompetitiveequilibrium. 33

5.3 Discussion Two remarks about the above results are worth emphasizing. First, in this model, safety traps are truly endogenous in that the supply of private safe assets may be low precisely because the demand for safe assets is high. In particular, the results above show that high demand for safe assets can itself reduce the capacity of the private sector to produce safe assets: At the effective lower bound, high demand for safe assets leads to a demand-recession, which can erode bank net worthandthuslimittheabilityofbankstocreatesafeassetsbyissuingdebt. Moreover,increasing thesupplyofprivatesafeassetsmay,undersomeconditions,onlyservetofurtherincreasedemand andtherebyworsenthesafeassetshortage(theparadoxofsafety): Issuingmoreprivatesafeassets may increase systemic risk by forcing banks to liquidate more capital in bad future states, thereby increasingthehousehold’sdemandforsafemeansofstoringconsumptionacrosstime. Second,theinteractionbetweensystemicriskandaggregatedemandattheheartofthismodel is crucial to deriving this taxonomy and understanding the nature of safety traps. To show this, in Online Appendix 9, I consider a counterfactual version of this economy in which (cid:96) = 0 in 1 all states. If there is a a shortage of public safe assets in equilibrium at date 0, then there must also be a shortage of private safe assets in equilibrium. In such an economy, there can be only conventionalsafetytraps–thatis,cannothaveanoversupplyofprivatesafeassetsinasafetytrap andtheparadoxofsafetyneveremerges. 6 Effects of a shock to systemic risk In this section, I analyze how this economy responds to a shock which increases systemic risk at date0,andshowthatthedynamicfeedbackloopbetweensystemicriskandaggregatedemand(the confluence of Channels 1, 2, and 3) generate an amplification mechanism. To do so, I trace out the channels of transmission of an unanticipated (MIT) shock to future TFP in the bad state. In particular, agents learn at date 0 that TFP in the bad state at date 1, z (s), is lower than initially 1 thought – essentially an adverse news shock. This shock has the effect of increasing the severity of crises in the bad state at date 1, and therefore can be interpreted as an exogenous increase the systemic risk faced by agents at date 0.52 The effects of the shock are stylistically illustrated in Figure8,andtheeffectsaretracedoutinanalyticaldetailinOnlineAppendix10. Since adverse shock to date 1 TFP reduces the bank’s rental income in the bad state at date 1, it increases the level of liquidation in that state, (cid:96) (s ). Therefore, by increasing the level 1 L of liquidation, the anticipated shock to date 1 TFP effectively shifts the SR curve in panel I up, 52The response of the economy, and hence the channels I outline below, will be similar across any shock which causesanexcessdemandforsafeassetsatdate0. 34

resulting in a higher demand for safe assets D∗ at any interest rate and level of utilization, and 0 thereforeshiftingthedemandcurveforsafeassetsinpanelIIItotheright. Moreover, the higher expected liquidation costs to the bank, and the lower expected return to capital at date 1, causes the MC curve in panel II to shift up and the MB curve to shift down. Both of these effects serve to reduce the bank’s supply of private safe assets Ds∗ at any interest 0 rate and utilization rate, shifting left the supply curve for safe assets in panel III. As a result, the shock causes an excess demand for safe asset at date 0. Depending on whether monetary policy is constrained by the effective lower bound, this safe asset shortage must be cleared through a fall in theinterestrateorthroughademand-drivenrecession(seesection3). Figure8: Effectofshockto(cid:96) (s ) 1 L General equilibrium feedback from date 0 to dates 1 and 2: The excess demand for safe assets requires either the date 0 interest rate or utilization to adjust to equilibrate the economy, depending on whether the economy is in the supply-determined or demand-determined regime at date 0. In turn, the adjustment in R and u may feed back to the bank’s leverage and therefore 0 0 affecttheallocationatdates1and2throughChannels3aand3b(notshowninFigure8). 35

Supply-determined regime In the supply-determined regime, monetary policy facilitates an adjustment in the composition of output going to saving versus consumption through a lower interestrate,butarecessionatdate0isavoidedasmonetarypolicyisnotconstrained.53 Demand-determined regime As I showed in section 3.2, in the demand-determined regime, the excess demand is cleared through fall in u which erodes the bank’s net worth at date 0 and 0 increases its leverage ratio and expected future liquidation costs, ceteris paribus. Depending on how the bank manages this rise in liquidation risk, Channels 3a and 3b shock worsens the decline in aggregate output at date 0 through the general equilibrium Channels 3a and 3b. The fall in the bank’s net worth causes the supply of private safe assets to fall further through general equilibrium Channel 3a, while through general equilibrium Channel 3b, it causes systemic risk to rise, increasingthehousehold’sdate0demandforsafeassets. Thepropositionbelowoutlinestheconditionsunderwhichthesegeneralequilibriumchannels amplifythedemand-drivenrecessionatdate0inresponsetotheshock. Proposition 2: Amplification mechanism If the demand recession at date 0 causes the dDs d(cid:96) (s ) banktoreduceitsdebtissuance( 0 >0)and/orifitcausessystemicrisktorise( 1 L <0),then du du 0 0 Channels 3a and 3b together may amplify fall in date 0 output in response to an unanticipated fall in z (s ). Furthermore, if both channels are active, then the condition for amplification is weaker 1 L thanifonlyonechannelholds. Proof: SeeAppendix11. Thus, the dynamic, two-way interaction between systemic risk and aggregate demand that arisesingeneralequilibriumcangiverisetoanamplificationmechanismwhichexacerbatesdemanddrivenrecessionsinthepresenceofsafeassetshortages. 7 Normative Implications In this section, I examine the normative and policy implications of the model. I begin by tracing out the transmission channels of various different policies. I then solve a constrained social planner’sproblem,identifytheexternalitiesatplay,andsolveforoptimalpolicy. Inthepresenceofthis dynamic interplay, insufficient coordination between policies designed to stimulate aggregate demand at the effective lower bound and those to mitigate systemic risk can result in an inefficiently lowlevelofaggregatedemandandoutputandanexcessivelyhighlevelofsystemicriskatdate0. 53Inthesupply-determinedregime,outputissupply-determined,andtheshockhasnoeffectonthesupplysideof theeconomyatdate0,ask isafixedendowment,laborisinelasticallysupplied,andutilizationisoptimallyu =1 0 0 inthesupply-determinedregime. Ofcourse,iftheshocktodate1TFPissufficientlysevere,theeconomymayenter thedemand-determinedregimeatdate0. However,myanalysisfocusesonthemarginaleffectsofshocks. 36

7.1 Transmission channels of policy instruments I begin by tracing out the transmission channels of various different policies and show that the dynamic interplay between aggregate demand and systemic risk at the heart of the model implies thatpoliciesingeneralmayhaveeffectsonbothaggregatedemandandsystemicrisk. 7.1.1 MonetaryPolicy The dynamic interplay between aggregate demand and systemic risk introduces a new, macroprudential channel through which monetary policy can affect output at date 0: through affect the risk ofafuturecrisis. At date 0, monetary policy affects both the demand and supply side of the economy. Suppose I are in the supply-determined regime at date 0 so that monetary policy is unconstrained, and suppose that, off-equilibrium, there is an excess demand for safe assets. (Equivalently, output is below potential at date 0.) A lower nominal interest rate RMP =RB lowers demand for safe assets 0 0 Dd throughthehousehold’sEulerequation,andatthesametimeincreasesdemandforgoods. This 0 istheconventionaleffectofmonetarypolicythedemandsideoftheeconomy. However, there is an additional indirect effect on aggregate demand through its effect on future crisis risk. By boosting aggregate demand and keeping output at the natural rate, monetary policy boosts the bank’s date 0 net worth, allowing banks to produce private safe assets without significantlyincreasingtheriskofafuturecrisis. Bykeepingsystemicrisklow,thisfurtherboosts aggregatedemandatdate0. Onthesupplyside,monetarypolicyhasopposingeffectsonsystemicrisk: Alowerpolicyrate incentivizes the bank to issue more private safe assets. This both increases returns (to both banks andhouseholds,intheformoflaborincome)inthegoodstateatdate1,butmagnifiestheseverity ofcrisesinthebadstate. In particular, a lower nominal interest rate increases the bank’s issuance of private debt D , 0 in turn, boosting the bank’s investment in capital i . The higher leverage of the bank Lev has 0 0 opposingeffects onthe laborincome riskborne bythehousehold. Inthe goodstates , thehigher H capital stock of the bank k implies that output is higher y , and therefore the household’s date 1 1 1 labor income is higher w n¯ = (1−α)y . Moreover, the higher return rk earned by the bank 1 1 1 due to the higher capital stock k implies that the bank’s date 1 investment i in capital is higher, 1 1 meaning the date 2 capital stock k is higher. As a result, date 2 output and labor income are both 2 higher. Both of these effects imply that higher date 0 bank leverage unambiguously increases the total future consumption of the household in the good state c (s )+c (s ). This effect lowers 1 H 2 H household’s demand for consumption smoothing Dd, reinforcing the decline in date 0 demand for 0 privatedebtmentionedbefore. 37

Incontrast,theeffectofhigherdate0leverageLev onthehousehold’stotalfutureconsumption 0 in the bad state is ambiguous. As before, the higher date 0 leverage Lev means that the bank’s 0 date 1 capital stock k is higher. As a result, date 1 output y is higher than it would otherwise be, 1 1 and hence the household’s date 1 labor income w n¯ = (1−α)y is higher. However, in the bad 1 1 state,thehigherleverageimpliesthatthebankisforcedtoliquidateahigherfraction(cid:96) ofitsdate 1 1capitalholdingstorepayitsdebt. Asaresult,thedate2capitalstockk islowerinthebadstate. 2 This means that date 2 output y and labor income w n¯ = (1−α)y are lower as a result of the 2 2 2 higher level of leverage. Thus, higher date 0 leverage increases date 1 labor income w (s )n¯ but 1 L reducesdate2laborincomew (s )n¯ inthebadstate. Theneteffectonthehousehold’sdate1and 2 L 2totalconsumptioninthebadstatec (s )+c (s )isambiguous. 1 L 2 L Taking stock, higher bank leverage increases the household’s total future consumption in the good state, but has an ambiguous effect on total future consumption in the bad state. Depending on the net effect of on the household’s date 0 saving motive, this supply-side effect may boost or dampen aggregate demand at date 0. If the net effect is to increase the household’s expected discounted total future consumption, then the reduction in the nominal interest rate reduces the household’s demand for private debt Dd through the household’s date 0 Euler equation, which re- 0 inforcesthedirecteffectonthedemandforsaving. (If,ontheotherhand,theneteffectofthelower nominalinterestrateistolowerthehouseholdstotalexpectedfutureconsumption,thenareduction in the interest rate could have a contractionary effect at date 0 by incentivizing the household to save more on net. I will not focus on this case since it is not empirically relevant.) Either way, monetarypolicyworksinpartthroughtheinterplaybetweensystemicriskandaggregatedemand. 7.1.2 Fiscalpolicies I consider two classes of fiscal policies: ex post government bailouts and ex ante intervention in whichthegovernmentfinancesspendingbyissuingpublicsafeassetsatdate0. Governmentbailouts Considerfirstanexpostbailoutpolicyinwhichthegovernmentcommits to issuing transfers to banks in the bad state at date 1, which are just sufficient such that the bankscanrepaytheirdebtwithoutliquidatingcapital,sothat(cid:96) (s )=0. Supposethatthegovern- 1 L ment finances these transfers by exacting lump-sum taxes on the household at date 1. Effectively, thistransfersthebank’slossestothehouseholdscontemporaneously.54 Byforcingthehouseholds to bear some of the losses associated with investment, this policy eliminates liquidations from materializingexpostandthuspreventstheassociateddeadweightloss. Suppose the government can commit to such a policy ex ante at date 0. Effectively, private 54Alternatively,thegovernmentcouldfinancethebailoutbyissuingdebtatdate1toberepaidusingtaxesatedate 2. Thiswouldsmoothlossesacrosstime. However, recallthatthehouseholdperfectlysmoothsconsumptionacross dates1and2usingthestoragetechnologyanyway. 38

safeassetsarethenimplicitlybackedbygovernmentguarantee,which(partially)transformsthem into public safe assets. By transferring sufficient amount of investment risk from the banks to the household via the government’s balance sheet, this policy eliminates systemic risk at date 0.55 Thus,whenmonetarypolicyisconstrainedbytheeffectivelowerbound,anexantecommitmentto bailingoutbanksexpostcanimprovesafetytrapsbymitigatingsystemicriskandthusstimulating aggregatedemand. Deficit spending Ex ante fiscal interventions, such as deficit spending at date 0 in which the government finances transfers through the issuance of public safe assets, affect aggregate demandbothdirectlyandindirectlythroughmacroprudentialeffectswhichreducesystemicrisk. The conventional, direct effect is to stimulate aggregate demand, which boosts output when monetary policyisconstrained. Inaddition,therearetwomacroprudentialeffectsofdeficitspendingatdate0. First,bystimulatingaggregatedemandandoutputdirectly,spendingboostsnetworthofbanksatdate0,allowing them to bear losses with less risk of a future crisis. Second, to the extent that it crowds out private safe asset issuance, deficit spending effectively changes the composition of the safe assets held by households at date 0. By reducing the share of private safe assets in their holdings, fiscal policy transfers some of the risk associated with investment to the government’s balance sheet, and therefore reduces systemic risk. Both of these effects reduce systemic risk, and therefore boost aggregatedemandexante. Fiscalmultiplier Thesemacroprudentialeffectsimplythatfiscalmultipliersarelargerthan they otherwise would be. Thus, the dynamic interplay between aggregate demand and systemic riskconstitutesanadditionalchannelthroughwhichfiscalspendingmayaffectoutput. Thismodel thus introduces another component of fiscal multipliers. The size of the fiscal multiplier at date 0 depends in part on the strength of the dynamic interplay between aggregate demand and systemic risk,whichiscontrolledbytheelasticitiesidentifiedinsection4. Importantly, the model suggests that, in a safety trap characterized by a high level of systemic risk,publicexpenditureoughttobefinancedthroughdeficitspendingratherthancontemporaneous taxes. NotethatRicardianequivalencedoesnotgenerallyholdinthisenvironmentbecausepublic safe asset production (financed by future taxes) entails higher future output because, to the extent that it crowds out private safe asset production, fiscal spending reduces crisis risk. Therefore, by simultaneously spending and providing public safe assets as a way for the households to smooth consumption without generating systemic risk, fiscal policy can take full advantage of the fiscal multiplierstemmingfromthedynamicinterplaybetweenaggregatedemandandsystemicrisk. 55Iabstractherefrommoralhazardconsiderationswherebytheanticipationofgovernmentbailoutsleadstoperverse risk-takingincentivesatdate1. 39

7.1.3 Quantitativeeasing Quasi-fiscalmonetarypolicy,suchasquantitativeeasing,wherebythegovernmentpurchasescapital from banks at date 0 by issuing public safe assets, can stimulate aggregate demand at date 0 throughmacroprudentialeffects. In order to implement this policy, I assume that at date 0, the government issues public safe assets B to the household and uses the proceeds to buy capital from on the date 0 spot market at 0 date 0, taking the competitive price q as given.56 The government holds this capital through date 0 2andrentsoutthiscapitaltofirmsatdates1and,earningthecompetitiverentalraterk. Irepaysits t debtatdate1(andpossiblydate2)usingtherentalincomeandlump-sumtaxesonthehousehold. Quantitative easing affects the allocation in part through macroprudential effects on systemic risk. In particular, there are two broad channels through which it works. First, this policy alters thecompositionofsafeassetsheldbyhouseholdstowardpublicsafeassets. Effectively,thisshifts the risk of investment from the banks’ balance sheets to that of the government. And since the government finances any losses at date 1 with lump-sum taxes on the household, the government effectively forces the household to bear some of the losses associated with low productivity in the bad state instead of having the banks bear the entirety of these losses. Ultimately, this reduces the fraction of total capital stock that is liquidated in the bad state and reduces the associated deadweightlosses.57 Andsincethedeadweightlossassociatedwithliquidationislower,thehousehold may be better off in general equilibrium as its future labor income may be higher in the bad state atdate2thanitwouldotherwisebe. Thus, quantitative easing can reduce systemic risk by potentially improving risk sharing between the household and banks. Effectively, quantitative easing changes the composition of the assets which implicitly back safe assets, as a higher fraction is backed by the government’s tax power. Idiscusstheoptimalityofthispolicyinthenormativesection. A second, indirect effect is that higher aggregate demand may crowd in private safe asset creationwithoutasignificantriseinsystemicrisk,ashigheroutputboostsbank’sdate0networth. Effectively, the policy can thus create a virtuous cycle between high aggregate demand and low 56Inpractice,supposedtoworkbydirectlyaffectingtheseprices. Iabstractfromtheseconsiderationshere. 57The key for quantitative easing to work in the model is that, by holding private debt on its balance sheet, the government prevents the banks from liquidating their capital holdings to service their debt in the bad state. How Ill doesthisreflectreality?Inpractice,liquidationsoftenoccurwhen,forexample,adropinthepriceoftheassetsheldby financialintermediariesleadtomargincallsorrunsbytheirprivatecreditors,forcingtheintermediariestoinefficiently liquidatetheseassets. Whenacentralbankpurchasesthedebtoftheseintermediaries,itbecomesthecreditor. Ifthe intermediaries’assetsbackingthedebtdeclineinvalue,thecentralbanktypicallydoesnotissueamargincall,unlike privatecreditors. Hence,quantitativeeasingeffectivelyreducestheincidenceofinefficientliquidations,asthepublic sector does not issue margin calls in bad states of the world to the extent that the private sector does. Quantitative easingWhentheconsolidatedgovernmentbecomesthecreditoroftheseintermediariesthroughassetpurchases, ass doesnotmargincall. Therefore,myassumptioninthemodelthatbanksdonotliquidatecapitaltoservicedebtheld bythegovernmentcanthusbethoughtofareducedformwaytocapturethisphenomenon. 40

systemic risk at date 0. (Moreover, the policy can allow the bank to expand investment in capital atdate0.) Notethatbecausequantitativeeasingaffectsthelevelofoverallinvestmentatbothdates0and 1, in general, there is a social tradeoff to quantitative easing. In particular, at date 0, quantitative easingmycrowd-inor-outpublicsafeassetissuanceanddate0investmentincapital. Sincethere aredecreasingreturnstoinvestment,thismayleadtosuboptimalinvestmentlevelsatdate0. (That said, the bank may be compensated for this via the price of date 0 capital, which would allow it to increaseinvestmentatdate0.) Moreover,byholdingcapital,thegovernmentdeniesbankssomerentalincomeatdate1. This reducesthebank’scapacitytoinvestinnewcapitalatdate1andpayoffdebtatdate1. Bycontrast, the government does not have investment technology at date 1; the government’s rental income at date 1 is used to pay off debt or transfers, but cannot directly be used to fund investment in new capital at date 1. So in principle, absent other interventions or transfers, quantitative easing may result in less total investment at date 1. (Of course, the government could offset some of these effectsbytransferringfundstothebankatdate1toexpandinvestment.) This social tradeoff to quantitative easing implies that, in general, it is not socially optimal for the government to fully intermediate between the household and banks by issuing public debt and transferring all proceeds to the bank, nor is it in general socially optimal for public safe assets to betheonlytypeofdebtheldbyhouseholds. Irevisitthisissueinsectiononoptimalpolicy. 7.1.4 Macroprudentialpolicy The model’s mechanism also introduces a new role for macroprudential policy – one of aggregate demandmanagement. ConsiderthePigouviantaxesτ whichdisincentivizethebankfromissuing 0 debtatdate0byincreasingitsmarginalcost,ascanbeseeninthebank’soptimalityconditionfor leverage (18). One can interpret these taxes as macroprudential regulations, such as bank capital requirements,whichlimittheleverageoffinancialintermediaries. By reducing the leverage of banks at the margin, such taxes reduce the severity of crises in the bad state at date 0. By reducing systemic risk, this policy can raise the households’ labor in come in the bad state at date 2, and thereby potentially reduce aggregate demand at date 0. Thus, the dynamic interplay between aggregate demand and systemic risk implies that, at the margin, macroprudentialpolicyatdate0hasaneffectonaggregatedemandatdate0. In general, tight macroprudential policy has two opposing effects on date 0 aggregate demand when monetary policy is constrained. On the one hand, by reducing the supply of private safe assets,itworsensasafeassetshortage. Ontheotherhand,byreducingsystemicrisk,itlowersthe demand for safe assets at date 0, mitigating the shortage. The net effect on the severity of the safe asset shortage, and aggregate demand, depends on the nature of the safety trap. In a conventional 41

safety trap, which features a shortage of both public and private safe assets, the net effect of tight macroprudential policy is to worsen the safe asset shortage, and amplify the demand recession at date0. By contrast, in a risk-intensive safety trap, which feature an oversupply of private safe assets, the net effect of macroprudential policy is to boost aggregate demand by restricting the supply of private safe assets, even though this results in a net decrease in the total safe asset supply. Hence, policies which restrict the supply of private safe assets, such as bank capital regulation, may counterintuitively mitigate safety traps under these conditions. Thus, this paper highlights that, when monetary policy is constrained by the effective lower bound, macroprudential policy cansubstituteformonetarypolicyandstimulateaggregatedemandandboostoutputatdate0.58 7.2 Normative implications 7.2.1 Socialplanner’sproblem To evaluate the allocative efficiency of the economy, I solve the problem of a social planner who solves for allocation which maximizes welfare of household, subject to some level of welfare for the bank, taking as given the constraints to which agents are subject, including incomplete markets, the non-negativity constraint on date 1 investment, the deadweight loss associated with liquidation,andtheeffectivelowerboundonmonetarypolicy. Thesolutiontothisproblemconstitutes the constrained efficient allocation. I first solve the social planner’s problem taking as given the government’s behavior, to elucidate the externalities at play and understand how different policy interventions can improve the allocation at the margin. Then I solve the Ramsey problem to characterizeoptimalpolicy. 7.2.2 Externalities Risksharingexternality: Thecompetitiveequilibriumisingeneralconstrainedinefficientdueto the presence of two externalities. The first is a risk-sharing externality, similar to that in Bocola Lorenzoni(2023),inwhichhouseholdsdonotinternalizehowtheirdemandforsafeassetsatdate 0lowerstheirdate2laborincomeinthebadstateoftheworldduetothemacroeconomicspillover 58ThisresultissimilarinspirittothatofKorinekandSimsek(2016)inwhichmacroprudentialregulationcanboost futureoutputduetoanaggregatedemandexternality. However,thenatureoftheaggregatedemandexternalityhereis differentandthereforesoarethepolicyimplications. InKorinekandSimsek(2016),leveragetodaycausesdemanddriven a recession in bad states tomorrow. Therefore, macroprudential regulation today reduces the likelihood of demand-drivenrecessionstomorrow. Butinmymodel,macroprudentialpoliciestoday(atdate0)stimulateaggregate demandtoday(alsoatdate0). Thisisduetothedynamicinterplaybetweenaggregatedemandatdate0andtheriskof crisesatdate1. Hence,mypaperindicatesthatinanenvironmentoflowcurrentaggregatedemand,macroprudential policymaysubstituteformonetarypolicyandboostcurrentoutput. 42

inwhichacrisisatdate1lowersdate2laborincome.59 Relativetothesocialoptimum,banksbearstoomuchoftheriskassociatedwithinvestmentat date0whilethehouseholdsbeartoolittle. Inordertoinsurehouseholdsagainstinvestmentriskat date 0, banks preserve the safety of their liabilities by liquidating capital in the bad state at date 1. However,duetothedeadweightlossandmacroeconomicspilloverassociatedwithliquidation,this forces the household to bear losses at date 2 in the form of labor income. Thus, by preventing the householdfrombearinginvestmentriskatdate1,privatesafeassetcreationforcesthehouseholdto bearlaborincomeriskatdate2ingeneralequilibrium,whichisnotinternalizedbythehousehold. The inefficient risk-sharing associated with private safe assets does not obtain for public safe assets. From the point-of-view of individual savers, these are equivalent instruments to smooth consumptionasboththeprivateandpublicbondspromisethesamepayoffprofileatdate1. However, these are not equivalent means of smoothing consumption from a social perspective. Public safeassetsareimplicitlybackedinpartbythestate-contingentstreamoffuturetaxrevenue,which the government can generate without liquidating capital inefficiently. Indeed, I later show that the government can optimal use public safe assets and lump-sum taxation to force the household to bearsomeoftheriskassociatedwithinvestment,thusreducingsystemicrisk. Becauseofthetwo-wayinteractionbetweensystemicriskandaggregatedemand,agents’date 0 borrowing and saving s are also associated with an aggregate demand externality, as effect on crisis and distribution of future income affects current aggregate demand, which affects current outputwhenmonetarypolicyisconstrainedbytheeffectivelowerbound. Aggregate demand externality: The second externality, which obtains only in the demanddetermined regime, is an aggregate demand externality in which households do not internalize how, at the effective lower bound, their date 0 spending affects boosts date 0 and therefore other households’ and banks’ income/net worth.60 Moreover, because of the two-way interaction betweenaggregatedemandandsystemicrisk,agents’date0spendingdecisions(aggregatedemand) are also associated with the risk-sharing externality as date 0 output affects banks’ net worth, and hencetheseverityofcrisesandthehouseholds’date2laborincome. Margins of inefficiency These two externalities leads to two margins of constrained inefficiency when monetary policy is constrained by the effective lower bound. The first margin is that aggregate demand is inefficiently low at date 0. (Relative to the social optimum, the share of private safe assets held by the household is high as they do not internalize how their holdings of 59Notethattheeffectofdebtissuanceontheseverityofcrises(thesizeof(cid:96) (s ))ispricedintheinterestrate(the 1 L banks’costofborrowing). However,theeffectthatliquidationhasonthehousehold’sdate2laborincomeingeneral equilibriumisnotpricedin,andhencethereisanexternality. 60WhileinsimilarspirittothatinKorinekandSimsek(2016),inthatprivateleveragecreatesriskofdemand-driven recessions,herethenatureoftheaggregatedemandexternalityismeaningfullydifferent. Idiscussthisfurtherinthe literaturereview. 43

privatesafeassetsgeneratesystemicriskingeneralequilibrium,resultingindepressedoutputand excessive systemic risk.) This affects not only date 0 output, but also lowers the bank’s net worth atdate0,potentiallyincreasingtheriskoffuturecrisesandthereforeloweringfutureoutputinthe badstate. Thesecondmarginofinefficiencyisthatthehouseholdholdstoomuchprivatesafeassetsand too little public safe assets at date 0. Put differently, households are bearing too little of the risk associated with investment while banks are bearing too much. At a given level of saving, this resultsininefficientlyhighsystemicrisk,andthereforeloweraggregatedemandatdate0. Thus,in this setting, inefficient demand-driven recessions and excessive systemic risk arises not only due to a broad shortage of safe assets at date 0, but also due to the composition of safe assets held by thehouseholdsbeinginefficientlyskewedtowardprivatesafeassets. 7.2.3 Elasticitiescontrollingtheexternalities Theexternalitiesoutlinedaboveareintimatelylinkedwiththechannelswhichmakeupthedynamic interplaybetween,andthereforealsothenatureofthesafetytrapandoptimalpolicyresponse. What is the relationship between the externalities and channels making up the dynamic interaction between aggregate demand and systemic risk? To be precise, while the channels are not samethingastheexternalities,theyareaffectedbytheexternalities. Forexample,therisksharing externality is controlled by the product of two elasticities which determine the marginal effect of private safe asset creation on the household’s date 1 and 2 labor income in the bad state through date1liquidation. (cid:16) (cid:17) ∂ 1 c 1 (s L ) d(cid:96) 1 (s L ) ∂(cid:96) (s ) dDs 1 L 0 ThisisasubcomponentoftheproductofChannels1and2in(19). Channel2 (cid:122) (cid:125)(cid:124) (cid:123) Channel1 (cid:104) (cid:105) ∂Dd 0 ∂E 0 c 1 1 (s) (cid:122) d(cid:96) 1 (cid:125) ( (cid:124) s L ) (cid:123) ∂E 0 (cid:104) c 1 (s) (cid:105) ∂(cid:96) 1 (s L ) dDs 0 1 (cid:124) (cid:123)(cid:122) (cid:125) dynamicinteraction fromSRtoAD Thus, the product of Channels 1 and 2, which control one direction of the dynamic interplay from systemicrisktoaggregatedemand,embedstherisk-sharingexternality. Similarly, the general equilibrium feedback Channels 3a and 3b relate closely to the aggregate demand externality. The aggregate demand externality, which captures how the household’s date 44

0 consumption-saving decision affects output at date 0, can be expressed as the product of the followingelasticities. ∂y du d (cid:0) Dd−Ds(cid:1) 0 0 0 0 (cid:0) (cid:1) ∂u d Dd−Ds dDd 0 0 0 0 dDs Recall that Channels 3a and 3b are respectively controlled by the following elasticities, 0 and du 0 d(cid:96) 1 (s L ) , both of which depend in part on ∂y 0, which in turn partially reflects the aggregate demand du 0 ∂u 0 externality. What is the relationship between the strength of the externalities and the type of safety trap? Recall that the nature of safety trap (and its associated policy responses) depends on whether a marginal increase in the supply of private safe assets increases or decreases the date 0 excess demand for safe assets. As established in section 5, this depends in part on the confluence of the threechannels(thedynamicinterplay),andalsoonthe(potentially)countervailingdirecteffectof changing the supply of safe assets on the safe asset shortage whereby higher Ds reduces Dd−Ds 0 0 directly (in case of private safe asset production) or indirectly by crowding-in private safe assets via B . The net effect of these forces depends in part on strength of externalities, and determines 0 marginal effect of safe asset creation on excess demand for safe assets – hence, the nature of the safetytrap. For example, consider a marginal increase in the supply of private safe assets at date 0 Ds. 0 Whatistheeffectonthesafeassetshortageatdate0? Iftherisksharingexternalityisstrong,then this is more likely to cause a large increase in systemic risk through the macroeconomic spillover. Thiswoulddepressaggregateexante,makingitmorelikelythatthiswouldexacerbateasafeasset shortage. Moreover, the depressed aggregate demand reduces date 0 output when at the effective lower bound. If the aggregate demand externality is strong, then the fall in output is more likely to have a strong adverse effect on banks’ net worth, further increasing systemic risk and reducing aggregate demand. Thus, if the two externalities are stronger (and hence the dynamic interplay between systemic risk and aggregate demand is stronger), then the economy is more likely to be in a risk-intensive safety trap at date 0, which implies very different policy prescriptions from a conventionalsafetytrap. 7.2.4 Allocativedistortions Tounderstandhowtheseexternalitiesdistorttheequilibriumallocation,considertheproblemofa constrained social planner who (in part) aims to allocate consumption optimally across the three dates1,2,and3. Recallthatthehouseholdcanperfectlysmoothconsumptionbetweendates1and 2duetopresenceofthestoragetechnologyandtheabsenceofuncertaintyafterdate0. Therefore, the allocation of consumption between dates 1 and 2 will always be efficient (that is, the planner 45

cannot improve on that margin). Therefore, I can agglomerate the households’ consumption at dates 1 and 2 into a ‘future’ period (1,2), and thereby reduce the planner’s problem to allocating consumption between date 0 and date (1,2). This problem is affected by two margins: how to the total amount of resources available at date 0 and the allocation of these resources between date 0 anddate(1,2). The risk-sharing externality (i.e., private safe asset production) has two effects on the allocation. First, due to the deadweight loss associated with liquidation in the bad state, it leads to a reduction in the total amount of resources available for consumption across both dates 1 and 2. Second, because of the macroeconomic spillover from liquidation to the household’s date 2 labor income, it reallocates the household’s losses from the bad state in date 1 (in the form of losses on the investment of capital) to the bad state of date 2 (in the form of lower labor income). The second effect is irrelevant, as the household can perfectly smooth losses between dates 1 and 2 using the storage technology. Therefore, the risk sharing externality ultimately reduces total resources available for consumption at dates (1,2) due to the deadweight loss. Now consider how the aggregate demand externality affects the allocation. The fall in aggregate demand lowers output (when monetarypolicyisconstrained)andthereforelowersthetotalresourcesatdate0availablefordate 0consumptionorfutureconsumptionatdate(1,2). Thus, because of these two externalities, the desire of the household to push consumption into thefuture(1,2)(thatis,thedemandforprivatesafeassets)resultsinloweroutputatdate0(aggregatedemandexternality),andalsolowerwealthinthebadstateofdate(1,2)duetothedeadweight loss associated with liquidation (excessive systemic risk due to risk sharing externality). Optimal policy will therefore involve distorting private choices – the household’s date 0 consumptionsavingdecision(aggregatedemand),andthecompositionofthehousehold’ssavingportfolio(risk sharing) – to increase aggregate demand at date 0 and reduce the deadweight loss incurred in the badstateatdate1. 7.2.5 Policycoordinationinasafetytrap How should these policies be jointly set? When monetary policy is constrained by the effective lower bound, the equilibrium features two sources of inefficiency: depressed aggregate demand andhighsystemicrisk. Butbecauseofthedynamicinterplaybetweenthesetwo,policiescanonly be imperfectly targeted to address each inefficiency – that is, policies which reduce systemic risk havespillovereffectstoaggregatedemand,andviceversa. Therefore,thereis,ingeneral,scopeto coordinatepoliciesdesignedtomitigatesystemicriskandpoliciesdesignedforaggregatedemand management in order to jointly achieve the desired level of aggregate demand and systemic risk. Theextenttowhichdifferentpoliciesimplyatradeoffdependsonnatureofthesafetytrap. Policy tradeoffs in conventional safety traps First consider the conventional safety trap, 46

whichfeaturesashortageofbothpublicandprivatesafeassets. Thesafetytrapfeaturesexcessive systemic risk and insufficient aggregate demand. Moreover, in a conventional safety trap, private safe asset creation involves a social tradeoff between aggregate demand and systemic risk. (There is no such tradeoff for public safe asset creation, except to the extent that it may crowd-in private safeassetcreationfurther.) Inparticular,asdiscussedintheprevioussection,whilemacroprudential policy can mitigate systemic risk by restricting the supply of private safe assets, this worsens thesafeassetshortageinaconventionalsafetytrap,resultingin,ceterisparibus,evenloweraggregatedemandandoutput. Hence,thereisatradeofftowhichreducesystemicrisk. On the other hand, fiscal policies designed to stimulate aggregate date 0 demand increase output at date 0, thereby increasing the total resources available for current and future consumption, but, to the extent that they crowd-in private safe asset creation, they may also increase systemic risk, reducing the future wealth of the household due to the deadweight loss and macroeconomic spillover. Quantitativeeasing,bycontrast,caninprinciplebothstimulateaggregatedemandandmitigate systemic risk, as it both reduces the share of private safe assets held by the household and substitutesthiswithahighersupplyofpublicsafeassets. However,recallfromtheprevioussectionthat, there is also a social tradeoff to quantitative easing associated with over-investment. Therefore, in general it is not optimal to use only quantitative easing to address the allocative inefficiencies. As a result, there may be scope for coordination between quantitative easing and macroprudential or fiscal policies: one could use both policies to stimulate aggregate demand, reduce systemic risk, andachievetheoptimallevelofinvestmentatthesametime. Therefore, due to the social tradeoff between systemic risk and aggregate demand that obtains in a conventional safety trap, when some policies restrict the supply of private safe assets, such as through macroprudential policy, one should should compensate for this with an increase in the supply of public safe assets, either through quantitative easing or fiscal policy. Moreover, insufficient coordination between macroprudential regulation and quantitative easing can lead to demand-drivenrecessionscharacterizedbyexcessivelyhighsystemicrisk. Policy tradeoffs in risk-intensive safety traps In a risk-intensive safety trap, coordination may not be necessary. In particular, using macroprudential policy may suffice to achieve the constrained social optimum as it can both reduce systemic risk and stimulate aggregate demand, creating a virtuous cycle between the two, facilitating an exit from the trap. By contrast, fiscal policy could be useful only to the extent that they, on net, crowd-out private safe asset creation. Otherwise, these policies would stimulate demand but potentially increase systemic risk. Similarly to conventional safety traps, quantitative easing could improve both aggregate demand and systemicrisk,butmaybeleadtoexcessiveinvestmentatdate0. 47

8 Conclusion Thispaperintroducesatheoryinwhichthecreationofsafeassetsgeneratesatwo-wayinteraction betweenaggregatedemandandtheriskoffuturecrises. Themodelhighlightstheroleplayedbythe compositionofsafeassetsbetweenpublicandprivatesafeassetsinthedeterminationofeconomic activity, systemic risk, and growth. The paper also showed that policy interventions, including monetary, macroprudential, and fiscal policies, can operate through additional channels once one accounts for the interactions between systemic risk and aggregate demand. The paper thus sheds lightonthenatureofmacroeconomicboomsandbusts,safeassetshortages,andpersistentslumps, andprovidesinsightonhowarangeofpolicyinterventionscaninfluencetheseepisodes. References Acharya, Sushant, Keshav Dogra, and Sanjay Singh. 2022. “The Financial Origins of NonfundamentalRisk.” Ajello, Andrea, Thomas Laubach, David Lopez-Salido, and Taisuke Nakata. 2019. “Financial Stability and Optimal Interest Rate Policy.” International Journal of Central Banking, 15(1):279–326. Angeletos, George-Marios, Fabrice Collard, and Harris Dellas. Forthcoming. “Public Debt as PrivateLiquidity: OptimalPolicy.”JournalofPoliticalEconomy. Azzimonti,Marina,andPierreYared.2019.“TheOptimalPublicandPrivateProvisionofSafe Assets.”JournalofMonetaryEconomics,102:126â144. Benigno, Gianluca, and Luca Fornaro. 2018. “Stagnation Traps.” Review of Economic Studies, 85(3):1425â1470. Benigno, Pierpaolo, and Roberto Robatto. 2019. “Private Money Creation, Liquidity Creation, andGovernmentIntervention.”JournalofMonetaryEconomics,106:42â58. Benigno, Pierpaolo, and Salvatore Nistico. 2017. “Inefficient Credit Booms.” American EconomicJournal: Macroeconomics,9(2):182â227. Bocola, Luigi, and Guido Lorenzoni. 2023. “Risk-Sharing Externalities.” Journal of Political Economy,131(3):595–632. Boissay, Frederic, Fabrice Collard, Jordi Gali, and Cristina Manea. 2023. “Monetary Policy andEndogenousFinancialCrises.”,(991). Caballero, Ricardo, and Alp Simsek. 2020. “A Risk-Centric Model of Demand Recessions and Speculation.”TheQuarterlyJournalofEconomics,135(3):1493â1566. 48

Caballero, Ricardo, and Alp Simsek. 2021. “A Model of Endogenous Risk Intolerance and LSAPs: Asset Prices and Aggregate Demand in a COVID-19 Shock.” The Review of FinancialStudies,34(11):5522â5580. Caballero, Ricardo, and Emmanuel Farhi. 2018. “The Safety Trap.” Review of Economic Studies,85(1):223–274. Caramp, Nicolas. 2023. “Sowing the Seeds of Financial Crises: Endogenous Asset Creation and AdverseSelection.” Collard, Fabrice, Harris Dellas, Behzad Diba, and Olivier Loisel. 2017. “Optimal Monetary andPrudentialPolicies.”AmericanEconomicJournal: Macroeconomics,9(1):40–87. Cuba-Borda, Pablo, and Sanjay Singh. 2021. “Understanding Persistent ZLB: Theory and Assessment.” Diamond,William.2020.“SafetyTransformationandtheStructureoftheFinancialSystem.”The JournalofFinance,75(6):2973–3012. Farhi,Emmanuel,andIvanWerning.2016.“ATheoryofMacroprudentialPoliciesinthePresenceofNominalRigidities.”Econometrica,84(5):1645â1704. Ferreira,Thiago,andSamerShousha.2022.“DeterminantsofGlobalNeutralInterestRates.” Goldberg,Jonathan,andDavidLopez-Salido.2023.“"SowingtheWind"MonetaryPolicy.” Gorton,Gary,StefanLewellen,andAndrewMetrick.2012.“TheSafeAssetShare.”American EconomicReview: PapersandProceedings,102(3):101–106. Infante,Sebastian,andGuillermoOrdonez.2021.“TheCollateralLinkBetweenVolatilityand RiskSharing.” Korinek, Anton, and Alp Simsek. 2016. “Liquidity Trap and Excessive Leverage.” American EconomicReview,106(3):699–738. Lenel,Moritz.2023.“TheCollateralPremiumandLeveredSafe-AssetProduction.” Lorenzoni, Guido. 2008. “Inefficient Credit Booms.” The Review of Economic Studies, 75(3):809â833. Luck, Stephan, and Paul Schempp. 2023. “Inefficient Liquidity Creation.” Journal of Financial Intermediation,53. Ross,Chase.2023.“TheCollateralPremiumandLeveredSafe-AssetProduction.” Segura,Anatoli,andAlonsoVillacorta.2023.“TheParadoxofSafeAssetCreation.”Journalof EconomicTheory,210. Xavier, Ines. 2023. “Bubbles and Stagnation.” Journal of the European Economic Association, 0(0):1â25. 49

Appendices APPENDIX1: ProofofLemma2: bankbehavesasifitisrisk-averse. In this appendix, I show that under Condition 1, although the bank is risk neutral, the convex liquidation cost φ((cid:96) ) introduces a convexity in the bank’s payoff function (as a function of D ). 1 0 Thisconvexitymakesthebankbehaveatdate0asifit’sriskaverse. First,letusdefinethemarginal costMC andmarginalbenefitMBofborrowingtothebankfromitsoptimalityconditionforD . 0 (cid:104) (cid:105) MC≡E v(cid:48)rk(cid:0) φ((cid:96) (s)) (cid:0) 1−τ D(cid:1) +τ RR (cid:1) +λ (s )τ RR 2 1 0 0 0 1 L 0 0 (cid:104) (cid:16) (cid:17)(cid:105) (cid:104) (cid:105) MB≡ (cid:0) 1−τ D(cid:1) E v(cid:48)rk rk(s)+1 + (cid:0) 1−τ D(cid:1) λ (s ) rk(s )+(cid:96) (s ) 0 2 1 0 1 L 1 L 1 L To show this, I will show that, although E’s marginal benefit MB and marginal cost MC from borrowing are increasing in D , at the equilibrium value of D (such that MB = MC), the marginal 0 0 cost is increasing at a faster rate than the marginal benefit. That is, ∂ ∂ M D C 0 | D∗ 0 > ∂ ∂ M D B 0 | D∗ 0 , in partial equilibrium (i.e. taking prices as given). This is because of our assumption that the cost of liquidatingcapitalinthebadstateatdate1isconvex,φ((cid:96) )≥0,φ(cid:48)((cid:96) )≥0,andφ(cid:48)(cid:48)((cid:96) )≥0,wherethe 1 1 1 inequalities hold strictly for all (cid:96) >0.For this reason, in partial equilibrium (i.e. taking prices as 1 given),thebankcanreachaninterioroptimumforhischoiceofD –thatis,hedoesn’tnecessarily 0 trytomaximizeborrowing,andsoIcanhaveasituationinwhichhis To show this, first recall from the bank’s non-negativity constraint on date 1 investment, and theexpressionfor(cid:96) inthelowstatethat,Ihave 1 τRD R −TE (cid:96) (s)= 0 0 0 1 −rk(s)=Lev −rk(s) (22) 1 k 1 0 1 1 where Lev := τ 0 RD 0 R 0 −T 1 E . Moreover, from the expression for the Lagrange multiplier, the extent 0 k 1 towhichtheconstraintbindsdependsalsoon(cid:96) ,andhenceleverage. 1 λ (s)=v(cid:48)rk(s)φ (cid:48)((cid:96) (s)) (23) 1 2 1 ∂λ (s) ∂(cid:96) (s) 1 =v(cid:48)rk(s)φ (cid:48)(cid:48)((cid:96) (s)) 1 ∂D 2 1 ∂D 0 0 Under what conditions do I have ∂MC > ∂MB? First note that, since the bank takes prices ∂D 0 ∂D 0 as given in making its leverage decision, I are interested in the derivatives ∂MC,∂MB in partial ∂D 0 ∂D 0 50

equilibrium, i.e. leaving prices fixed. (While the bank’s leverage decision will affect prices in general equilibrium, these effects are not internalized at the margin by the bank. Since here I are interested in characterizing the bank’s marginal decisions in partial equilibrium, I hold prices fixed.) Therefore,fromthedefinitionsofMC andMBabove,Ihave ∂MC = (cid:2) π(s) (cid:0) 1−τ D(cid:1) φ (cid:48)((cid:96) (s))+τ RR φ (cid:48)(cid:48)((cid:96) (s)) (cid:3) v(cid:48)rk(s) ∂(cid:96) 1 (s) ∂D 0 1 0 0 1 2 ∂D 0 0 And ∂MB = (cid:0) 1−τ D(cid:1) (cid:104) rk(s )+(cid:96) (s ) (cid:105) v(cid:48)rk(s)φ (cid:48)(cid:48)((cid:96) (s)) ∂(cid:96) 1 (s) + (cid:0) 1−τ D(cid:1) λ (s ) ∂(cid:96) 1 (s) ∂D 0 1 L 1 L 2 1 ∂D 0 1 L ∂D 0 0 0 Imposingtheequilibriumresultthats=s . L = (cid:0) 1−τ D(cid:1) (cid:104)(cid:16) rk(s )+(cid:96) (s ) (cid:17) φ (cid:48)(cid:48)((cid:96) (s ))+φ (cid:48)((cid:96) (s )) (cid:105) v(cid:48)rk(s ) ∂(cid:96) 1 (s) 0 1 L 1 L 1 L 1 L 2 L ∂D 0 Therefore, ∂MC > ∂MB iff ∂D 0 ∂D 0 (cid:2) π(s) (cid:0) 1−τ D(cid:1) φ (cid:48)((cid:96) (s))+τ RR φ (cid:48)(cid:48)((cid:96) (s )) (cid:3) v(cid:48)rk(s ) ∂(cid:96) 1 (s L ) > (cid:104)(cid:16) rk(s )+(cid:96) (s ) (cid:17) φ (cid:48)(cid:48)((cid:96) (s ))+φ (cid:48)((cid:96) (s )) (cid:105) (cid:0) 1−τ D(cid:1) v(cid:48)rk(s) ∂(cid:96) 1 (s) 0 1 0 0 1 L 2 L ∂D 1 L 1 L 1 L 1 L 0 2 ∂D 0 0 (cid:104) (cid:16) (cid:17) (cid:105) 0> (cid:0) 1−τ D(cid:1) rk(s )+(cid:96) (s ) −τ RR φ (cid:48)(cid:48)((cid:96) (s ))+(1−π(s )) (cid:0) 1−τ D(cid:1) φ (cid:48)((cid:96) (s )) (24) 0 1 L 1 L 0 0 1 L L 0 1 L What is the sign of (cid:0) 1−τD(cid:1)(cid:0) rk(s )+(cid:96) (s ) (cid:1) −τRR when evaluated at the equilibrium D∗? Re- 0 1 L 1 L 0 0 0 callthatthenon-negativityconstraintondate1investmentbindsinthebadstate,sothat D TE rk(s )+(cid:96) (s )=τ RR 0 − 1 1 L 1 L 0 0 k k 1 1 SoIcanwrite (cid:0) 1−τD(cid:1)(cid:0) rk(s )+(cid:96) (s ) (cid:1) −τRR as 0 1 L 1 L 0 0 (cid:16) (cid:17) (cid:18) D TE(cid:19) (cid:0) 1−τ D(cid:1) rk(s )+(cid:96) (s ) −τ RR = (cid:0) 1−τ D(cid:1) τ RR 0 − 1 −τ RR 0 1 L 1 L 0 0 0 0 0 k k 0 0 1 1 As long as TE and TE are not significantly negative (indeed, I will assume they are both weakly 0 1 positive),Ihavethat (cid:0) 1−τD(cid:1) (cid:16) τRR D 0 − T 1 E(cid:17) −τRR <0. Toseethis 0 0 0k k 0 0 1 1 (cid:18) D TE(cid:19) (cid:0) 1−τ D(cid:1) τ RR 0 − 1 <?τ RR 0 0 0 k k 0 0 1 1 51

Notethat (cid:0) 1−τD(cid:1) D −k canbeexpressedas 0 0 1 (cid:16) (cid:17) (cid:0) 1−τ D(cid:1) D −k =− rk+1 k −TE 0 0 1 0 0 0 Replacethisintheaboveinequality (cid:104)(cid:16) (cid:17) (cid:105) − rk+1 k +TE τ RR <? (cid:0) 1−τ D(cid:1) TE 0 0 0 0 0 0 1 This holds as long as as TE and TE are not significantly negative. (A sufficient condition is that 0 1 TE,TE ≥ 0.) Thus the sign of (cid:0) 1−τD(cid:1)(cid:0) rk(s )+(cid:96) (s ) (cid:1) −τRR is (cid:0) 1−τD(cid:1)(cid:0) rk(s )+(cid:96) (s ) (cid:1) − 0 1 0 1 L 1 L 0 0 0 1 L 1 L τRR <0. 0 0 Therefore,Icanwritecondition(10)as (cid:104) (cid:16) (cid:17)(cid:105) τ RR − (cid:0) 1−τ D(cid:1) rk(s )+(cid:96) (s ) φ (cid:48)(cid:48)((cid:96) (s ))>(1−π(s )) (cid:0) 1−τ D(cid:1) φ (cid:48)((cid:96) (s )) (25) 0 0 0 1 L 1 L 1 L L 0 1 L whereboththeright-handandleft-handsidesarestrictlypositive. Icanrewritethisconditionas φ(cid:48)(cid:48)((cid:96) (s )) (cid:0) 1−τD(cid:1) 1 L >(1−π(s )) 0 (26) φ(cid:48)((cid:96) 1 (s L )) L (cid:2) τ 0 RR 0 − (cid:0) 1−τ 0 D (cid:1)(cid:0) r 1 k(s L )+(cid:96) 1 (s L ) (cid:1)(cid:3) Thus,Ihavethatthebanksbehavesasifitisriskaverse( ∂MC > ∂MB)ifandonlyifthiscondition ∂D 0 ∂D 0 holds. Tointerpretthiscondition,therearethreetermswhichaffectit. First,iftheliquidationcost function is sufficiently convex (so that φ(cid:48)(cid:48) is sufficiently large relative to φ(cid:48)) then this condition is more likely to hold. This is is because then, at the margin, higher leverage will be associated with a higher liquidation cost. Second, if the bank’s losses τRR −rk(s )−(cid:96) (s ) (i.e. the difference 0 0 1 L 1 L between its repayment and its revenue plus liquidation value) is sufficiently high, then this condition is more likely to hold. This is again because higher losses in the bad state make the cost of borrowing larger at the margin. And third, if the probability of the bad state π(s) is sufficiently high, then this condition is more likely to hold. This is because the bank incurs losses in the bad state,makingborrowingmorecostly. Let us break down this condition further. Since I have φ((cid:96) ) = (cid:96) η and φ(cid:48)((cid:96) ) = η(cid:96) η−1 and 1 1 1 1 φ(cid:48)(cid:48)((cid:96) )=η(η−1)(cid:96) η−2 ,whereη >1,thiscanconditioncanbewrittenas 1 1 (cid:104) (cid:16) (cid:17)(cid:105) φ (cid:48)(cid:48)((cid:96) (s )) τ RR − (cid:0) 1−τ D(cid:1) rk(s )+(cid:96) (s ) >(1−π(s )) (cid:0) 1−τ D(cid:1) φ (cid:48)((cid:96) (s )) (27) 1 L 0 0 0 1 L 1 L L 0 1 L (cid:96) (s )< (cid:104) τ RR − (cid:0) 1−τ D(cid:1) rk(s ) (cid:105) (η−1) (28) 1 L 0 0 0 1 L (η−π(s )) (cid:0) 1−τD (cid:1) L 0 52

Andrecallthat(cid:96) (s )=Lev −rk(s). Sothisconditionis 1 L 0 1 Lev < (cid:104) τ RR − (cid:0) 1−τ D(cid:1) rk(s ) (cid:105) (η−1) +rk(s) (29) 0 0 0 0 1 L (η−π(s )) (cid:0) 1−τD (cid:1) 1 L 0 where the right-hand side is strictly positive. Thus, as long as, in equilibrium, Lev is sufficiently 0 smallthenIhave ∂MC > ∂MB. Inthatcase,amarginallyhigherD willmakeMC higherthanMB. ∂D 0 ∂D 0 0 Therefore,Ihenceforthassumethefollowingconditionholdsinequilibrium. Condition1: A) φ(cid:48)(cid:48)((cid:96) 1 (s L )) >(1−π(s )) (1−τ 0 D) φ(cid:48)((cid:96) 1 (s L )) L [τ 0 RR 0 −(1−τ 0 D)(r 1 k(s L )+(cid:96) 1 (s L ))] B)TE,TE ≥0 0 1 As I show in Appendix 3, this condition ensures that the bank’s expected date 1 losses from borrowing at date 0 are sufficiently high that the bank behaves at date 0 as if it is risk-averse. The condition is benign and amounts to saying that the liquidation cost function φ(·) is sufficiently convex, the probability of the bad state is sufficiently high, and the bank’s losses in bad state are sufficiently high. Moreover, note that φ(cid:48)(cid:48)((cid:96) 1 (s L )) = (η−1)η(cid:96)η 1 −2 = (η−1)(cid:96) 1 = (η−1) . Therefore, since φ(cid:48)((cid:96) 1 (s L )) η(cid:96)η 1 −1 (cid:96)2 1 (cid:96) 1 the only restriction on η is that η > 1, it is otherwise a free parameter which can always make sufficientlylargethatthisconditionholds. APPENDIX2: Riskpremium Definetheriskpremiumatdate0RP as 0 (cid:32) E (cid:2) rk(s)rk(s) (cid:3)(cid:33) RP := (cid:0) 1−τ D(cid:1) 1+ 2 1 −τ RR >0 0 0 E (cid:2) rk(s) (cid:3) 0 0 2 Note that, since the bank consumes only at date 2, the net expected discounted rate of return on capital is defined as the consumption units it is expected to provide at date 2, E (cid:2) rk(s)rk(s) (cid:3) , and 2 1 is then discounted by the bank’s date 1 discount factor E (cid:2) rk(s) (cid:3) to reflect the date 1 value of this 2 E[rk(s)rk(s)] expected consumption, yielding the net expected discounted return 2 1 . This is the rate of E[rk(s)] 2 returnthebanktakesintoaccountwhenismakingitsdate0investmentdecision. (At date 1, the rental rate of capital is rk(s). This return can be reinvested in capital at date 1 1, yielding rk(s) units of consumption for the bank at date 2. Hence, the date 2 consumption 2 units that the bank can expect to consume for each unit of capital is E (cid:2) rk(s)rk(s) (cid:3) . At date 0, the 2 1 bank’s expected discount rate at date 1 is E (cid:2) rk(s) (cid:3) . Note that the net expected discounted rate of 2 returnoncapitalcanalternativelybeexpressedasE (cid:2) rk(s) (cid:3) + Cov(r 2 k,r 1 k) . Notealsothatiftherental 1 E[rk(s)] 2 53

rates of capital at dates 1 and 2 Ire orthogonal (i.e. ifCov (cid:0) rk(s),rk(s) (cid:1) =0), and if there were no 2 1 distortionary taxes, then our expression for the risk premium would reduce to 1+E (cid:2) rk(s) (cid:3) −R . 1 0 But the rental rates are not orthogonal due to the endogenous accumulation of capital between dates1and2. Thatis,ifrk(s)islowinstatesduetoahighdate1capitalstockk (s),thenthedate 1 1 2capitalstockk (s)willlikelyalsobehigh,whichmeansthatrk(s)willlikelybelowasIll.) 2 2 Theriskpremiumcanidenticallybeexpressedasafunctionofthebank’sexpectedliquidation costduetothebindingnon-negativityconstraintondate1investment. Toseethis,simplyrearrange thebank’sFOCforD . 0 (cid:110) (cid:104) (cid:16) (cid:17)(cid:105) (cid:104) (cid:105)(cid:111) (cid:104) (cid:105) (cid:104) (cid:105) (cid:0) 1−τ D(cid:1) E v(cid:48)rk rk(s)+1 +λ (s ) rk(s )+(cid:96) (s ) =E v(cid:48)rk φ((cid:96) (s)) (cid:0) 1−τ D(cid:1) +E v(cid:48)rk τ RR +λ (s )τ RR 0 2 1 1 L 1 L 1 L 2 1 0 2 0 0 1 L 0 0 (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) marginalbenefit marginalcost (30) (cid:32) E (cid:2) rkrk(cid:3) (cid:33) E (cid:2) rkφ((cid:96) (s)) (cid:3) (cid:2) τRR − (cid:0) 1−τD(cid:1)(cid:0) rk(s )+(cid:96) (s ) (cid:1)(cid:3) (cid:0) 1−τ D(cid:1) 2 1 +1 −τ RR = (cid:0) 1−τ D(cid:1) 2 1 +λ (s ) 0 0 0 1 L 1 L 0 E (cid:2) rk (cid:3) 0 0 0 E (cid:2) rk (cid:3) 1 L E (cid:2) rk (cid:3) 2 2 2 (31) The left-hand side of the equation is the definition of the risk premium, while the right-hand side isafunctionofthebank’sexpectedliquidationcost. E (cid:2) rkφ((cid:96) (s)) (cid:3) (cid:2) τRR − (cid:0) 1−τD(cid:1)(cid:0) rk(s )+(cid:96) (s ) (cid:1)(cid:3) RP = (cid:0) 1−τ D(cid:1) 2 1 +λ (s ) 0 0 0 1 L 1 L (32) 0 0 E (cid:2) rk (cid:3) 1 L E (cid:2) rk (cid:3) 2 2 Adjusted risk premium I can define a notion of the risk premium adjusted for the distortionarytaxesτD,τR asfollows. Definetheadjustedrisk-freerateas 0 0 (cid:18) τR (cid:19) R D,adj := 0 R 0 1−τD 0 0 Fromthebank’sFOCforD ,thisimplies 0 E(cid:48)sreturn fromcapital shadowpriceincrisisstate (cid:122) (cid:125)(cid:124) (cid:123) (cid:122) (cid:125)(cid:124) (cid:123) (cid:104) (cid:16) (cid:17)(cid:105) (cid:104) (cid:105) E v(cid:48)rk rk(s)+1−φ((cid:96) (s)) +λ (s ) rk(s )+(cid:96) (s ) 2 1 1 1 L 1 L 1 L D,adj R = 0 λ (s )+E (cid:2) v(cid:48)rk (cid:3) 1 L 2 Anddefinetheadjustedriskpremiumas RP adj 0 RP := 0 1−τD 0 54

ThenitfollowsfromthedefinitionofRP that 0 (cid:32) E (cid:2) rk(s)rk(s) (cid:3)(cid:33) RP := (cid:0) 1−τ D(cid:1) 1+ 2 1 −τ RR 0 0 E (cid:2) rk(s) (cid:3) 0 0 2 (cid:34) E (cid:2) rkrk(cid:3)(cid:35) RP = (cid:0) 1−τ D(cid:1) 1+ 2 1 − (cid:0) 1−τ D(cid:1) R D,adj 0 0 E (cid:2) rk (cid:3) 0 0 2 (cid:34) E (cid:2) rkrk(cid:3) (cid:35) RP = (cid:0) 1−τ D(cid:1) 1+ 2 1 −R D,adj 0 0 E (cid:2) rk (cid:3) 0 2 Andso E (cid:2) rkrk(cid:3) RP adj =1+ 2 1 −R D,adj 0 E (cid:2) rk (cid:3) 0 2 adj I can then rewrite the bank’s FOC for D to express RP as a function of the bank’s expected 0 0 liquidationcost. E (cid:2) rkφ((cid:96) (s)) (cid:3) (cid:2) τRR − (cid:0) 1−τD(cid:1)(cid:0) rk(s )+(cid:96) (s ) (cid:1)(cid:3) RP = (cid:0) 1−τ D(cid:1) 2 1 +λ (s ) 0 0 0 1 L 1 L (33) 0 0 E (cid:2) rk (cid:3) 1 L E (cid:2) rk (cid:3) 2 2 E (cid:2) rkφ((cid:96) (s)) (cid:3) (cid:2) τRR − (cid:0) 1−τD(cid:1)(cid:0) rk(s )+(cid:96) (s ) (cid:1)(cid:3) RP adj = 2 1 +λ (s ) 0 0 0 1 L 1 L (34) 0 E (cid:2) rk (cid:3) 1 L (cid:0) 1−τD (cid:1) E (cid:2) rk (cid:3) 2 0 2 Riskpremiumwhenthenon-negativityconstraintondate1investmentneverbinds Suppose the bank’s non-negativity constraint on date 1 investment never binds at date 1. Then the bank’sFOCforD wouldbe 0 (cid:20) (cid:21) dk (s) E v(cid:48)rk 2 =0 (35) 2 dD 0 where dk 2 (s) = (cid:2) rk(s)+1−φ((cid:96) (s)) (cid:3)(cid:0) 1−τD(cid:1) −τRR = (cid:0) rk(s)+1 (cid:1)(cid:0) 1−τD(cid:1) −τRR . Thiswould dD 1 1 0 0 0 1 0 0 0 0 reduceto (cid:104) (cid:16)(cid:16) (cid:17) (cid:17)(cid:105) E rk rk(s)+1 (cid:0) 1−τ D(cid:1) −τ RR =0 (36) 2 1 0 0 0 (cid:104) (cid:16) (cid:17)(cid:105) (cid:104) (cid:105) (cid:0) 1−τ D(cid:1) E rk rk(s)+1 =τ RR E rk (37) 0 2 1 0 0 2 i.e. (cid:110) (cid:104) (cid:105) (cid:104) (cid:105)(cid:111) (cid:104) (cid:105) (cid:0) 1−τ D(cid:1) E rkrk +E rk =τ RR E rk (38) 0 2 1 2 0 0 2 (cid:40) E (cid:2) rk(s)rk(s) (cid:3) (cid:41) (cid:0) 1−τ D(cid:1) 2 1 +1 =τ RR (39) 0 E (cid:2) rk(s) (cid:3) 0 0 2 55

Thus, the risk premium RP would be 0 if and only if (cid:96) (s) = 0 for all s. Moreover, since 0 1 adj (cid:96) (s),λ (s)=0,theadjustedriskpremiumsatisfiesRP =0ifandonlyif(cid:96) (s)=foralls. 1 1 0 1 APPENDIX3: ProofthatChannel2isstrictlypositive (cid:104) (cid:105) Proof that show both Channels 1 and 2 >0: ∂Dd 0 ∂E 0 c1 1 (s) > 0 and d(cid:96) 1 (s L ) > 0, under as- ∂E 0 (cid:104) c1 1 (s) (cid:105) ∂(cid:96) 1 (s L ) dDs 0 sumptions that lump-sum transfers are sufficiently small, and assumption that k 2 G < 1. Start with k˜ n¯ 2 Channel2: ∂Dd CLAIM: 0 >0 ∂(cid:96) 1 (s) PROOF: (cid:104) (cid:105) ∂Dd ∂Dd ∂E 0 c 1 (s) 0 = 0 1 (cid:104) (cid:105) ∂(cid:96) 1 (s) ∂E 1 ∂(cid:96) 1 (s) 0 c (s) 1 (cid:104) (cid:105) Iproceedbyshowingfirstthat ∂E 0 c1 1 (s) >0,andthenthat ∂Dd 0 >0. ∂(cid:96) 1 (s) ∂E 0 (cid:104) c1 1 (s) (cid:105) (cid:104) (cid:105) Claim: ∂E 0 c1 1 (s) >0 ∂(cid:96) 1 (s) Proof: Notethat (cid:104) (cid:105) ∂E 1 0 c 1 (s) =− π(s) ∂c 1 (s) ∂(cid:96) 1 (s) (c 1 (s))2 ∂(cid:96) 1 (s) where 1 c (s)=c (s)= [(w +w )n¯+R (D +B )−T −T ] 1 2 1 2 0 0 0 1 2 2 So (cid:18) (cid:19) ∂c (s) n¯ ∂w (s) ∂w (s) ∂T ∂T 1 1 2 1 2 = + − − ∂(cid:96) (s) 2 ∂(cid:96) (s) ∂(cid:96) (s) ∂(cid:96) (s) ∂(cid:96) (s) 1 1 1 1 1 Notethat y 1 w =(1−α) (40) 1 n¯ y 2 w =(1−α) (41) 2 n¯ So ∂w (s) (1−α)∂y (s) 1 1 = =0 ∂(cid:96) (s) n¯ ∂(cid:96) (s) 1 1 56

and ∂w (s) (1−α)∂y (s) 2 2 = ∂(cid:96) (s) n¯ ∂(cid:96) (s) 1 1 Andsince y =z (cid:0) k˜ (cid:1)α n¯1−α (42) 2 2 2 wherek˜ =k +kG,then 2 2 2 ∂y (s) ∂y (s)∂k˜ (s)∂k (s) y ∂k (s) 2 2 2 2 2 2 = =α ∂(cid:96) (s) ∂k˜ (s)∂k (s)∂(cid:96) (s) k˜ ∂(cid:96) (s) 1 2 2 1 2 1 andk (s)= (cid:2) 1+rk(s)−φ((cid:96) (s)) (cid:3) k (D )−τRD R +TE so 2 1 1 1 0 0 0 0 1 ∂k (s) 2 =−k (s)φ (cid:48) =−ηk (s)(cid:96) η−1 (s)<0 ∂(cid:96) (s) 1 1 1 1 Thelastinequalityfollowsfromthefactthat(cid:96) (s)>0inthebadstate. 1 Notealsothat T =RBB −τ +TE−rk(s)kG (43) 1 0 0 1 1 1 1 T =TE−τ −rkkG (44) 2 2 2 2 2 So ∂T ∂rk(s) 1 =−kG 1 =0 ∂(cid:96) (s) 1 ∂(cid:96) (s) 1 1 wherethelastequalityholdssincerk =α y 1 and ∂k˜ 1 =0sincek˜ =kG+k andk =D +TE+ (cid:0) rk+1 (cid:1) k . Moreover 1 k˜ 1 ∂(cid:96) 1 (s) 1 1 1 1 0 0 0 0 ∂T ∂rk(s) 1 =−kG 2 ∂(cid:96) (s) 2 ∂(cid:96) (s) 1 1 Notethat,sincerk =α y 2 , 2 k +kG 2 2 ∂rk α ∂y (s) y ∂k (s) 2 = 2 −α 2 2 ∂(cid:96) 1 (s) k 2 +k 2 G ∂(cid:96) 1 (s) (cid:0) k 2 +k 2 G(cid:1)2 ∂(cid:96) 1 (s) 57

Andsincek (s)= (cid:2) 1+rk(s)−φ((cid:96) (s)) (cid:3) k (D )−τRD R +TE,Ihave 2 1 1 1 0 0 0 0 1 ∂k (s) 2 =−k φ (cid:48)((cid:96) (s)) 1 1 ∂(cid:96) (s) 1 So ∂rk α ∂y (s) y 2 = 2 +α 2 k φ (cid:48)((cid:96) (s)) ∂(cid:96) 1 (s) k 2 +k 2 G ∂(cid:96) 1 (s) (cid:0) k 2 +k 2 G(cid:1)2 1 1 Pluggingtheseexpressionsintotheequationof ∂c 1 (s) . ∂(cid:96) 1 (s) (cid:32) (cid:33) ∂c (s) n¯ (1−α)∂y (s) α ∂y (s) y 1 = 2 +kG 2 +kG α 2 k φ (cid:48)((cid:96) (s)) ∂(cid:96) 1 (s) 2 n¯ ∂(cid:96) 1 (s) 2 k 2 +k 2 G ∂(cid:96) 1 (s) 2 (cid:0) k 2 +k 2 G(cid:1)2 1 1 andsinceIshowedthat ∂y 2 (s) =α y 2 ∂k 2 (s) and ∂k 2 (s) =−k (s)φ(cid:48) <0,thisis ∂(cid:96) 1 (s) k˜ 2 ∂(cid:96) 1 (s) ∂(cid:96) 1 (s) 1 (cid:32) (cid:33) (cid:20) (cid:21) n¯ (1−α) α y y = − +kG α 2 k (s)φ (cid:48)+kG α 2 k (s)φ (cid:48) 2 n¯ 2 k 2 +k 2 G k˜ 2 1 2 (cid:0) k˜ 2 (cid:1)2 1 Thus,ariseinliquidationinthebadstatecausesfutureconsumptiontofallthroughadecrease in the wage, but also pushes up consumption through a possible rise in the rental rate of capital. In what follows, I find a condition which ensures that this latter effect does not dominate. I have ∂c 1 (s) <0ifandonlyif ∂(cid:96) 1 (s) (cid:20) (cid:21) (1−α) α y y − +kG α 2 k (s)φ (cid:48)+kG α 2 k (s)φ (cid:48) <0 n¯ 2 k 2 +k 2 G k˜ 2 1 2 (cid:0) k˜ 2 (cid:1)2 1 kG 1 2 < k˜ n¯ 2 which holds if labor supply is sufficiently small or government’s share of date 2 capital stock is sufficientlysmall. Supposethenthat k 2 G < 1. Thenitfollowsthat ∂c 1 (s) <0. Therefore, k˜ 2 n¯ ∂(cid:96) 1 (s) (cid:104) (cid:105) ∂E 1 0 c 1 (s) =− π(s) ∂c 1 (s) >0 ∂(cid:96) 1 (s) (c 1 (s))2 ∂(cid:96) 1 (s) 58

Q.E.D. Claim: ∂Dd 0 > 0. Recall the household’s demand function Dd(R ,B ) = e −T +dF + ∂E 0 (cid:104) c1 1 (s) (cid:105) 0 0 0 0 0 0 w n¯−B − 1 (cid:16) E (cid:104) 1 (cid:105)(cid:17)−1 . Thisimpliesthat ∂Dd 0 = 1 (cid:16) E (cid:104) 1 (cid:105)(cid:17)−2 >0 0 0 R 0 0 c 1 (s) ∂E 0 (cid:104) c1 1 (s) (cid:105) R 0 0 c 1 (s) (cid:104) (cid:105) Thus,sinceboth ∂Dd 0 >0and ∂E 0 c1 1 (s) >0,Ihave ∂E 0 (cid:104) c1 1 (s) (cid:105) ∂(cid:96) 1 (s) (cid:104) (cid:105) ∂Dd ∂Dd ∂E 0 c 1 (s) 0 = 0 1 >0 (cid:104) (cid:105) ∂(cid:96) 1 (s) ∂E 1 ∂(cid:96) 1 (s) 0 c (s) 1 ∂Dd 1 π(s) n¯(1−α) y 0 = α 2 ηk (s)(cid:96) η−1 (s)>0 ∂(cid:96) 1 (s) R (cid:16) E (cid:104) 1 (cid:105)(cid:17)2(c 1 (s))22 n¯ k˜ 2 1 1 0 0 c (s) 1 (cid:104) (cid:105) ThusChannel2: ∂Dd 0 = ∂Dd 0 ∂E 0 c1 1 (s) >0. Q.E.D. ∂(cid:96) 1 (s) ∂E 0 (cid:104) c1 1 (s) (cid:105) ∂(cid:96) 1 (s) Tosummarize,Ihave (cid:32) (cid:33) ∂Dd 1 (cid:18) (cid:20) 1 (cid:21)(cid:19)−2 π(s) n¯ (cid:20) (1−α) α (cid:21) y y 0 =− E − +kG α 2 k (s)φ (cid:48)+kG α 2 k (s)φ (cid:48) >0 ∂(cid:96) 1 (s) R 0 0 c 1 (s) (c 1 (s))22 n¯ 2 k 2 +k 2 G k˜ 2 1 2 (cid:0) k˜ 2 (cid:1)2 1 APPENDIX4: ProofthatChannel1isstrictlypositive d(cid:96) (s ) Now consider Channel 1, the effect of date 0 borrowing on the severity of crises 1 L . To a dDs 0 first order approximation, the partial equilibrium effect of date 0 saving on the severity of a crisis, conditional on a crisis occurring at date 1, can be summarized by the derivative of liquidation in d(cid:96) (s ) the bad state with respect to date 0 private debt, 1 L . Here, I show that in equilibrium I have dD 0 d(cid:96) (s) d(cid:96) (s) 1 > 0 if and only if (cid:96) (s) > 0, and 1 = 0 otherwise – that is, higher date 0 borrowing dD 1 dD 0 0 increasestheseverityofacrises,conditionalonacrisisoccurring. d(cid:96) (s ) CLAIM: Channel 1: As long as lump-sum transfers are sufficiently small, I have 1 L > 0 dD 0 d(cid:96) (s ) and 1 H =0. dD 0 PROOF: I already showed that, in the good state s = s , I are in the normal regime so that H d(cid:96) (s ) 1 H =0. Inthebadstates=s ,thevariable(cid:96) (s)ispinneddownbythebindingnon-negativity dD L 1 0 59

constraintondate1investment. τRD R TE (cid:96) (s)= 0 0 0 −rk(s)− 1 1 k 1 k 1 1 Recallthelawofmotionfork . 1 QE k =i +k −k (45) 1 0 0 1 Replacingi withthebank’sbindingdate0budgetconstraintyields 0 D (cid:16) (cid:17) k = 0 + rk+1 k +TE (46) 1 P 0 0 0 0 So dk 1 1 = =1 dD P 0 0 d(cid:96) (s) Therefore,Icanexpressthederivative 1 as dD 0 d(cid:96) (s) τRR drk(s) TE τRD R 1 = 0 0 − 1 + 1 − 0 0 0 dD k dD (k )2 (k )2 0 1 0 1 1 Recall y rk =α 1 =αz (cid:0) k˜ (cid:1)α−1 n¯1−α (47) 1 k˜ 1 1 1 So drk(s) 1 =α(α−1)z (cid:0) k˜ (cid:1)α−2 n¯1−α <0 1 1 dD 0 drk(s) d(cid:96) (s) Since 1 <0,asufficientconditionfor 1 >0is dD dD 0 0 τRR TE τRD R 0 0 + 1 − 0 0 0 >0 k (k )2 (k )2 1 1 1 i.e. τ RR (k −D )>TE 0 0 1 0 1 Henceforth, I assume that TE, which for now I take as exogenous, satisfies this condition. (Note 1 60

thatifIsettransferstoTE =0,thenthisconditionwouldreduceto 1 k >D 1 0 whichholdsbecauseofnetworth.) D (cid:16) (cid:17) k = 0 + rk+1 k +TE >D (48) 1 P 0 0 0 0 0 Thus,aslongasTE issufficientlysmall,Ihave d(cid:96) 1 (s) >0inthebadstate. 1 dD 0 Thus, date 0 saving increases the severity of crises when they occur. I show in Appendix 4 d(cid:96) (s ) d(cid:96) (s ) that, as long as lump-sum transfers are sufficiently small, I have 1 L >0 and 1 H =0. That dD dD 0 0 is, the fraction of the bank’s capital which is liquidated in the bad state is increasing (to a first order approximation) in initial saving D . This is because higher D (holding all else constant(?)) 0 0 increases the bank’s date 0 leverage Lev , which implies that the bank’s losses in the bad state 0 (when its rental income is low) are larger. As a result, the bank is forced to liquidate a higher fractionofitscapitalholdingsinordertorepayitsdebt. Thus, higher date 0 saving increases systemic risk (in partial equilibrium) by increasing the bank’sleverage,therebyexacerbatingtheseverityofcrisesinthebadstateatdate1. APPENDIX5: Effectofcrisisriskonprecautionarysaving ∂Dd Below,Ishowthat 0 >0. ∂(cid:96) 1 (s) First, let’s reduce the dynamic system of equations a bit. Note that I can get rid of both c and 1 B . The first order condition for B implies c =c . And from the expressions for c and c , I can 1 1 1 2 1 2 computethesumofc andc as 1 2 c (s)+c (s)=B (s)−T +w n¯+R D +RBB −T +w n¯−B (s) 1 2 1 2 2 0 0 0 0 1 1 1 =(w +w )n¯+R (D +B )−T −T 1 2 0 0 0 1 2 And since c = c , this means I can ignore both c and B , and capture how shocks affect total 1 2 1 1 consumptionafterdate0: 1 c (s)= [(w +w )n¯+R (D +B )−T −T ] 2 1 2 0 0 0 1 2 2 Intuitively, the household uses the storage technology B (s) (with RB = 1) to fully smooth con- 1 1 61

sumption between dates 1 and 2. So I only need to keep track of the household’s total income at dates1and2-itsdividedoptimallyviaB (s). 1 Recallthehousehold’sFOCforDd: 0 (cid:20) (cid:21) 1 1 =R E (49) 0 0 c c (s) 0 1 Plugthisintothehousehold’sdate0budgetconstraint: c +Dd+B =e −T +dF+w n¯ (50) 0 0 0 0 0 0 0 1 Dd =e −T +w n¯−B − (51) 0 0 0 0 0 (cid:104) (cid:105) R E 1 0 0 c (s) 1 So (cid:104) (cid:105) ∂Dd 1 ∂E 0 c 1 (s) 0 = 1 ∂(cid:96) (s) (cid:16) (cid:104) (cid:105)(cid:17)2 ∂(cid:96) (s) 1 R E 1 1 0 0 c (s) 1 where (cid:104) (cid:105) ∂E 1 0 c 1 (s) =− π(s) ∂c 1 (s) ∂(cid:96) 1 (s) (c 1 (s))2 ∂(cid:96) 1 (s) where 1 c (s)=c (s)= [(w +w )n¯+R (D +B )−T −T ] 1 2 1 2 0 0 0 1 2 2 So (cid:18) (cid:19) ∂c (s) n¯ ∂w (s) ∂w (s) 1 1 2 = + ∂(cid:96) (s) 2 ∂(cid:96) (s) ∂(cid:96) (s) 1 1 1 Notethat y 1 w =(1−α) (52) 1 n¯ y 2 w =(1−α) (53) 2 n¯ So ∂w (s) (1−α)∂y (s) 1 1 = =0 ∂(cid:96) (s) n¯ ∂(cid:96) (s) 1 1 62

and ∂w (s) (1−α)∂y (s) 2 2 = ∂(cid:96) (s) n¯ ∂(cid:96) (s) 1 1 Andsince y =z (cid:0) k˜ (cid:1)α n¯1−α (54) 2 2 2 wherek˜ =k +kG,then 2 2 2 ∂y (s) ∂y (s)∂k˜ (s)∂k (s) y ∂k (s) 2 2 2 2 2 2 = =α ∂(cid:96) (s) ∂k˜ (s)∂k (s)∂(cid:96) (s) k˜ ∂(cid:96) (s) 1 2 2 1 2 1 andk (s)=i +(1−φ((cid:96) (s)))k (s),wherei =0inthebadstate,so 2 1 1 1 1 ∂k (s) 2 =−k (s)φ (cid:48) =−ηk (s)(cid:96) η−1 (s)<0 ∂(cid:96) (s) 1 1 1 1 The last inequality follows from the fact that (cid:96) (s) > 0 in the bad state. Plugging this into our 1 ∂Dd expressionfor 0 yields ∂(cid:96) 1 (s) ∂Dd 1 π(s) n¯(1−α) y 0 = α 2 ηk (s)(cid:96) η−1 (s)>0 ∂(cid:96) 1 (s) R (cid:16) E (cid:104) 1 (cid:105)(cid:17)2(c 1 (s))22 n¯ k˜ 2 1 1 0 0 c (s) 1 APPENDIX6: Effectofpublicdebtsupplyonthecompetitiveequilibrium Inordertocomparetheroleofpublicversusprivatedebt,Inowstudytheeffectofanincrease inthesupplyofpublicdebtB onaggregatedemand. Acharacterizationofthiseffectrevealsthat, 0 in contrast to private debt, public debt has no direct effect on systemic risk (i.e. the severity of crises). ToevaluatetheeffectsofpublicdebtissuancerequiresthatImakeassumptionsaboutwhatthe government does with the proceeds. In what follows, I make the following assumptions about the government’sbehavior. Assumptionsaboutgovernment’sbehavior i. The government invests any additional proceeds from borrowing in its capital holdings kG 1 ∂kG (viaquantitativeeasing),sothat 1 =1. ∂B 0 ii. Thegovernmentkeepsitsholdingsofcapitalconstantacrosstime,sothatkG =kG. 2 1 iii. At date 1, any government revenue net of interest payments is transferred in lump-sum fashion to the bank, so that TE = rk(s)kG−RBB . Therefore, the government’s date 1 budget 1 1 1 0 0 constraintimpliesT =−τ . 1 1 63

iv. At date 2, the government’s proceeds from renting its capital holdings rkkG are paid in 2 2 lump-sum fashion to the bank, so that TE =rk(s)kG. The government’s date 2 budget constraint 2 2 2 thenimpliesT =0. 2 As with private debt, I focus on the effects of public debt supply on aggregate demand, and for now abstract from general feedback effects from the change in aggregate demand reflected in Channel3via dDs 0 andchangesinR andu . TocharacterizetheeffectofDs onaggregatedemand, dDd 0 0 0 0 Icantakethederivativeofthehousehold’sdemandfunctionDd(R ,B ;u ),giveninequation(7), 0 0 0 0 withrespecttoB . 0 (cid:104) (cid:105) dDd ∂Dd dE 0 c 1 (s) 0 = 0 1 (cid:104) (cid:105) dB 0 ∂E 1 dB 0 0 c (s) 1 As with private debt, this can be decomposed into two terms, one capturing the role of systemic riskandtheotherexcludingit.  (cid:104) (cid:105) (cid:104) (cid:105)  dDd 0 = ∂ (cid:104) Dd 0 (cid:105) ∂E 0 c 1 1 (s) + ∂E 0 c 1 1 (s) d(cid:96) 1 (s L )  dB 0 ∂E 1 ∂B 0 ∂(cid:96) 1 (s L ) dB 0 0 c (s) 1 d(cid:96) (s ) However,publicdebtB hasnoeffectonaggregatedemandthroughsystemicrisk(thatis 1 L = 0 dB 0 0), except through general equilibrium feedback effects. Since I set aside such effects here, this impliesthatthesecondtermis0. Toseethis,recalltheexpression(11)for(cid:96) (s) 1 τRD R TE (cid:96) (s)= 0 0 0 −rk(s)− 1 1 k 1 k 1 1 Since,abstractingfromgeneralequilibriumeffectsthroughu andR ,Ihave dk 1, dT 1 E , dr 1 k(s L ) =0. 0 0 dB dB dB 0 0 0 Recall that, although in general (cid:96) may respond to B through the equilibrium value of D , this 1 0 0 effectisageneralequilibriumeffectwhichisdeterminedonlybythegeneralequilibriumresponse of u or R . As discussed above, I abstract from these GE effects (Channel 3) here, and return to 0 0 themlater. d(cid:96) (s ) Therefore 1 L =0whenholdingu constant(thatits,whenignoringGEfeedbackchannel). dB 0 0 HenceChannel1isnotactiveforpublicdebt. So (cid:104) (cid:105) dDd ∂Dd ∂E 0 c 1 (s) 0 = 0 1 (cid:104) (cid:105) dB 0 ∂E 1 ∂B 0 0 c (s) 1 64

Since Channel 1 not active, neither is Channel 2 (that is, public debt has no effect on aggregate demand through either Channels 1 or 2). Hence, systemic risk doesn’t play a role, and so neither doesdynamicinteraction,inthecaseofpublicdebt. Thus, I can now see how public and private debt differ. Compare the effect of the supply of eithertypeofdebtonaggregatedemand: Channel2 (cid:122) (cid:125)(cid:124) (cid:123) Channel1 (cid:104) (cid:105) (cid:104) (cid:105) dDd 0 = ∂Dd 0 ∂E 0 c 1 1 (s) + ∂Dd 0 ∂E 0 c 1 1 (s) (cid:122) d(cid:96) 1 (cid:125) ( (cid:124) s L ) (cid:123) dDs 0 ∂E 0 (cid:104) c 1 (s) (cid:105) ∂Ds 0 ∂E 0 (cid:104) c 1 (s) (cid:105) ∂(cid:96) 1 (s L ) dDs 0 1 1 (cid:124) (cid:123)(cid:122) (cid:125) dynamicinteraction fromSRtoAD versus (cid:104) (cid:105) dDd ∂Dd ∂E 0 c 1 (s) 0 = 0 1 (cid:104) (cid:105) dB 0 ∂E 1 ∂B 0 0 c (s) 1 dDd dDd InAppendix6,Ishowthattheseexpressionsimplythat 0 > 0;thatis,anincreaseinthesupply dDs dB 0 0 of private debt unambiguously reduces aggregate demand (i.e. increases demand for saving) by morethananincreaseinthesupplyofpublicdebtdoes. dDd dDd Hence, 0 and 0 differ only in the Channel 1: role played by systemic risk. This indicates dDs dB 0 0 that supply of public debt does not affect equilibrium via systemic risk. Only affects via other (cid:104) (cid:105) effectonE 1 (investment,consumption),whichinturnhaseffectondemandforsaving. c (s) 1 APPENDIX7: Generalequilibriumfeedbackchannels First,inordertoestablishtherelationshipbetweenthetwoGEchannels, ∂(cid:96) 1 (s) and ∂D 0,let’sderive ∂u 0 ∂u 0 anexpressionfor ∂(cid:96) 1 (s) . ∂u 0 Recallthat(cid:96) (s)=Lev −rk(s),whereLev := τ 0 RD 0 R 0 −T 1 E . AlsorecallfromAppendix8(Effect 1 0 1 0 k 1 ofutilizationonthebank’ssupplyofdebt),that ∂rk ∂k 1 =−α(1−α)z (k )α−2n¯1−α 1 (55) 1 1 ∂u ∂u 0 0 (cid:32) (cid:33) ∂Lev τRR ∂D τRD R −TE ∂k 0 = 0 0 0 − 0 0 0 1 1 (56) ∂u 0 k 1 ∂u 0 (k 1 )2 ∂u 0 Recall that ∂k 1 = ∂D 0 +k ∂r 0 k , and k = D +rkk +TE +k and rk = α yt, and also ∂r 1 k = ∂u 0 ∂u 0 0∂u 0 1 0 0 0 0 0 t k˜ t ∂u 0 65

−α(1−α)z (k )α−2n¯1−α∂k 1, and dr 0 k =−α(1−α)z (u )α(k )α−2n¯1−α <0. Using these ex- 1 1 ∂u 0 du 0 0 0 0 pressionsIcanwrite ∂(cid:96) 1 (s) as ∂u 0 ∂(cid:96) (s) ∂D 1 0 =A +F ∂u ∂u 0 0 whereIhaveusedthefollowingdefinitions. ∂Lev ∂Lev ∂rk A≡ 0 + 0 − 1 ∂D ∂k ∂k 0 1 1 τRR τRD R −TE = 0 0 − 0 0 0 1 +α(1−α)z (k )α−2n¯1−α k 1 (k 1 )2 (cid:124) 1 (cid:123)(cid:122) 1 (cid:125) (cid:124)(cid:123)(cid:122)(cid:125) (cid:124) (cid:123)(cid:122) (cid:125) ∂rk ∂ ∂ L D ev 0 0 − ∂ ∂ L k e 1 v0 − ∂k 1 1 (cid:20) ∂Lev ∂rk(cid:21) ∂rk F ≡− − 0 + 1 k 0 0 ∂k ∂k ∂u 1 1 0     =      τ 0 RD ( 0 k R 1 0 ) − 2 T 1 E − (cid:124) α(1−α)z 1 (cid:123) ( (cid:122) k 1 )α−2n¯1−α (cid:125)      k 0 (cid:124) α(1−α)z 0 (u 0 (cid:123) ) (cid:122) α(k 0 )α−2n¯1−α (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) ∂rk  ∂rk − 1 − 0 − ∂Lev0 ∂k1 ∂u0 ∂k1 The above expresses ∂(cid:96) 1 (s) (which controls general equilibrium Channel 3b) as a function of ∂u 0 ∂D 0. Needless to say, in the demand-determined regime in which R = 1, the behavior of the ∂u 0 0 quantity of private debt D in response to a change in utilization reflects both the response of the 0 supplyofdebtDs anddemandDd. Iwould,however,liketoexpressitasafunctionof ∂D 0 (which 0 0 ∂u 0 controls general equilibrium Channel 3a). Therefore, to a first order approximation, I can express ∂D 0 asalinearfunctionof ∂Ds 0 (Channel3a)and ∂Dd 0: ∂D 0 =a ∂Ds 0+b ∂Db 0,wherea,bareconstants. ∂u 0 ∂u 0 ∂u 0 ∂u 0 ∂u 0 ∂u 0 Note also that, since, holding all else constant (to a first-order approximation), a rise in either the supplyordemandofdebtwouldincreasethequantityofdebt,Ihavea,b>0. ∂Db Furthermore,Icanderivetheexpressionfor 0 fromequation(5)forthehousehold’sdemand ∂u 0 forprivatedebt. 66

(cid:18) (cid:20) (cid:21)(cid:19)−1 1 1 Dd(R ,B ;u )=e −T +dF+w n¯−B − E (57) 0 0 0 0 0 0 0 0 0 R 0 c (s) 0 1 In the demand-determined regime, R = 1. Taking the derivative of this function with respect to 0 u showsthatutilizationhasaneffectonDd throughtwochannels: throughtheeffectonsystemic 0 0 risk (Channel 3b), and through the households labor income w n¯: dDd 0 = dDd 0 d(cid:96) 1 (s L ) + ∂w 0 n¯ . 0 du 0 d(cid:96) 1 (s L ) du 0 ∂u 0 (cid:104) (cid:105) Moreover, I have already shown in Appendix 15 that dDd 0 = ∂Dd 0 ∂E 0 c1 1 (s) >0. And recall d(cid:96) 1 (s L ) ∂E 0 (cid:104) c1 1 (s) (cid:105) ∂(cid:96) 1 (s L ) fromequation(289)that ∂w 0 n¯ =α(1−α) y 0 >0. Thus,Ihave ∂u 0 u 0 (cid:104) (cid:105) dDd 0 = ∂Dd 0 ∂E 0 c 1 1 (s) d(cid:96) 1 (s L ) + ∂w 0 n¯ (cid:104) (cid:105) du 0 ∂E 1 ∂(cid:96) 1 (s L ) du 0 ∂u 0 0 c (s) (cid:124) (cid:123)(cid:122) (cid:125) 1 (cid:124) (cid:123)(cid:122) (cid:125) >0 >0 Using the expression ∂D 0 = a ∂Ds 0 +b ∂Db 0 and the above expression for dDd 0, I can insert this ∂u 0 ∂u 0 ∂u 0 du 0 intoourexpressionfor ∂(cid:96) 1 (s) . ∂u 0 (cid:32) (cid:33) ∂(cid:96) (s) ∂Ds ∂Db 1 =A a 0 +b 0 +F ∂u ∂u ∂u 0 0 0   (cid:104) (cid:105) ∂(cid:96) 1 (s)   1−Ab ∂Dd 0 ∂E 0 c 1 1 (s)   =Aa ∂Ds 0 +Ab ∂w 0 n¯ +F ∂u 0   ∂E (cid:104) 1 (cid:105) ∂(cid:96) 1 (s L )   ∂u 0 ∂u 0  0 c 1 (s)  (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) >0 >0 Thus, I have expressed Channel 3b ∂(cid:96) 1 (s) as a function of Channel 3a ∂Ds 0 and some constants, ∂u 0 ∂u 0 whererecallthatA,F arefunctionsofsomeelasticitiesasdefinedabove.   A= ∂Lev 0 +   ∂Lev 0 − ∂r 1 k   ∂D  ∂k ∂k  0 1 1 (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124)(cid:123)(cid:122)(cid:125) >0 <0 <0 67

  F =   ∂Lev 0 − ∂r 1 k  k 0 ∂r 0 k  ∂k ∂k  ∂u 1 1 0 (cid:124) (cid:123)(cid:122) (cid:125) (cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125) <0 <0 <0 Note that, the above expression for ∂(cid:96) 1 (s) shows that it is possible that neither general equi- ∂u 0 ∂Ds d(cid:96) (s ) librium channels holds in equilibrium – that is, it is possible that both 0 ≤0 and 1 L ≥0 in ∂u 0 du 0 ∂Ds equilibrium. Indeed, the above expressions show that, given 0 ≤0, it can also be the case that ∂u 0 (cid:104) (cid:105) d(cid:96) 1 (s L ) ≥0dependingonsizeof ∂Dd 0 ∂E 0 c1 1 (s) = ∂Dd 0 relativetoA,b,andalsodependingon du 0 ∂E 0 (cid:104) c1 1 (s) (cid:105) ∂(cid:96) 1 (s L ) ∂(cid:96) 1 (s L ) thesizeof ∂w 0 n¯ ,B,etc. ∂u 0 Intuitively,onemightexpectthatifthebank’ssupplyofprivatesafeassetsDs risesinresponse 0 tothefallinutilization,thatsystemicrisk(cid:96) (s )wouldriseasaresultofthehigherleverage. That 1 L is, one might expect that even if Channel 3a does not hold, 3b would hold. How can both 3a and 3bnotholdsimultaneously? Note that, even if lower utilization increases the supply of private debt Ds (so that 3a is not ∂Ds active, 0 ≤ 0), one could still have a fall in systemic risk (cid:96) (s ), so that 3b is also not active ∂u 0 1 L d(cid:96) 1 (s L ) ≥ 0. This could happen if, in equilibrium, the fall in demand for private debt Dd falls by du 0 0 morethantheriseinthesupply,suchthatD fallsonnet,andhencelowerssystemicrisk(cid:96) (s ). As 0 1 L I showed above, the response of Dd to u , ∂Dd 0, depends on how u affects the wage through term 0 0 ∂u 0 0 ∂w 0 n¯ and on how the equilibrium change in systemic risk (cid:96) (s ) feeds into the household demand ∂u 0 1 L (cid:104) (cid:105) for private debt Dd through the terms ∂Dd 0 = ∂Dd 0 ∂E 0 c1 1 (s) . Indeed, these are precisely the 0 ∂(cid:96) 1 (s L ) ∂E 0 (cid:104) c1 1 (s) (cid:105) ∂(cid:96) 1 (s L ) relevantelasticitieswhichappearintheaboveexpressionforChannel3b ∂(cid:96) 1 (s) . ∂u 0 ∂Ds d(cid:96) (s ) I now derive some sufficient conditions for both to be active ( 0 > 0 and 1 L < 0) in ∂u 0 du 0 equilibrium. Looking at the above expression for ∂(cid:96) 1 (s) , if 3a is active ∂Ds 0 >0, then a sufficient ∂u 0 ∂u 0 ∂Lev ∂rk condition for 3b to also be active d(cid:96) 1 (s L ) <0 is that the expression A>0 (i.e. 0 > 1), and du 0 ∂k 1 ∂k 1 (cid:124) (cid:123)(cid:122) (cid:125) (cid:124)(cid:123)(cid:122)(cid:125) <0 <0 that F < 0 is not too negative. Thus, for both 3a and 3b to be active in equilibrium, a sufficient conditionisthatthefollowingthreeconditionsallhold: ∂Ds i)Channel3aisactive,i.e. 0 >0. Icouldderiveexplicitnecessaryandsufficientconditions ∂u 0 for dD 0 >0usingourexpressionfor dD 0 derivedinOnlineAppendix7. Inanycase,thiscondition du du 0 0 alreadyimplicitlyembedshowu affectsD throughthebank’snetworth. Intuitively,thedemand- 0 0 driven recession induced by the fall in u should cause the bank’s supply of private debt Ds to 0 0 68

fall. This is natural, given that firms typically seek to reduce debt issuance during demand-driven downturns. ∂Lev ∂rk ii) 0 > 1,sothatA>0. Tounderstandthiscondition,recallthat(cid:96) (s )=Lev −rk(s ). ∂k ∂k 1 L 0 1 L 1 1 (cid:124) (cid:123)(cid:122) (cid:125) (cid:124)(cid:123)(cid:122)(cid:125) <0 <0 Thus,thisconditionimpliesthat,atthemargin,alowercapitalstockreducessystemicrisk(cid:96) (s ). 1 L That is, even though, a lower capital stock increases leverage (holding other variables constant) andthusincreases(cid:96) (s ),italsoincreasestherentalrateofcapitalatdate1,whichreduces(cid:96) (s ). 1 L 1 L Thisconditionisthatthelattereffectdominates. d(cid:96) (s ) iii) |F| is not too large, to ensure that the condition for 1 L <0 holds. Given our definition du 0 ∂rk of F above, this amounts to a condition that k is small and/or 0 <0 is small in absolute value. 0 ∂u 0 The latter implies that, at the margin, the recession does not cause the rental rate of capital at date 0toincreasetoomuch. Thus,theseconditionsarerelativelyweak. APPENDIX8: Conditionsdefiningasafetytrap Definingasafetytrapingeneral To determine whether there is a shortage of bonds, I ask whether, at the margin, would an increase in the supply of private safe assets Ds increase or decrease the demand-driven recession, 0 that is, utilization. Therefore, the key elasticity which defines the nature of the safety trap is the derivativeofutilizationwithrespecttothesupplyofprivatesafeassets,evaluatedattheequilibrium du 0 | dDs u 0 =u∗ 0 ,Dd 0 −Ds 0 =0,R 0 =1 0 In particular, the sign of this elasticity, evaluated at the date 0 equilibrium, determines whether I are in a conventional safety trap (in which there’s a shortage of both public and private safe assets)orariskysafetytrap(inwhichthere’sashortageofpublicsafeassets,andanoversupplyof private ones): if the derivative is strictly negative, then I have a risky safety trap, otherwise I have aconventionalsafetytrap. Note that, in Appendix 8, I already showed that that u_0 is determined to ensure that there is no excess demand Dd(R ,u ,B )−Ds = 0 when R = 1. That is, at R = 1, equilibrium uti- 0 0 0 0 0 0 lization u can be expressed as a function only of excess demand for debt, Dd(R ,u ,B )−Ds. 0 0 0 0 0 Therefore, utilization falls if and only if excess demand increases (holding utilization fixed). That is, du 0| < 0 if and only if d (cid:0) Dd−Ds(cid:1) | > 0. Thus, the dDs u =u∗,Dd−Ds=0,R =1 dDs 0 0 u =u∗,Dd−Ds=0,R =1 0 0 0 0 0 0 0 0 0 0 0 0 response of utilization ultimately boils down to the sign of this elasticity. Therefore, I can equivalently define the nature of the safety trap based on the sign of the following derivative (evaluated attheequilibrium,andholdingu_0fixed). 69

d (cid:16) (cid:17) Dd−Ds | dDs 0 0 u 0 =u∗ 0 ,Dd 0 −Ds 0 =0,R 0 =1 0 Therefore,Icanreformulatethedefinitionasfollows: To determine the nature of the safety trap, I ask whether, at the margin, would an increase in thesupplyofprivatesafeassetsDs increaseordecreasetheexcessdemandforsafeassets,holding 0 u fixed at its equilibrium value (i.e. not allowing u_0 to adjust to the change in excess demand). 0 (Think of model relationship between excess demand for safe assets and safe asset supply, that is Dd−Ds asafunctionofDs. EquilibriumconditiondefinedasDd−Ds =0. Cantakederivativeof 0 0 0 0 0 the function with respect to Ds evaluated at the equilibrium, and holding u fixed. This derivative 0 0 canbepositiveornegativeinprinciple.) Thatis,thekeyelasticitywhichdefinesthenatureofthesafetytrapis d (cid:16) (cid:17) Dd−Ds dDs 0 0 0 dDs dDd dDd d(cid:96) (s ) where 0 =1 and 0 = 0 1 L , and I have suppressed the notation that these derivatives dDs dDs d(cid:96) (s ) dDs 0 0 1 L 0 areeventuatedattheequilibrium(includingu =u∗). 0 0 Note on General Equilibrium Feedback Effects In general, a marginal increase in the supply of debt (private or public) has an effect on utilization u via Channels 1 and 2 outlined 0 earlier. Moreover, the change in u may have a general equilibrium feedback effect on the supply 0 ofprivatedebtinturn,throughwhatItermedChannel3. (Forexample,supposethat,atthemargin, ahighersupplyofprivatedebtDs causesutilizationu tofallviaChannels1and2. Dependingon 0 0 thedirectionofeachofthechannels,thefallinu couldcauseD eithertoriseorfallasafeedback 0 0 effect. The net effect on u and D would depend on the direction and strength of these feedback 0 0 effects relative to the first round effect. That is, the general equilibrium relationship between the supply of debt and u can vary depending on the feedback effects from u to other variables in 0 0 general equilibrium, via Channel 3. In either case, the feedback effects must be diminishing in intensityinorderfortheretobeanequilibrium.) Ourdefinitionofthesafetytrapabovecanaccommodatesuchgeneralequilibriuminteractions. However, in deriving conditions on the elasticities which delineate the nature of the safety trap, I abstractfromgeneralequilibriumfeedbackeffectsintwoways: First,Idonotaccountforgeneral equilibrium feedback effects from u onto other variables (that is, Channel 3). That is, while u 0 0 will respond to a change in the supply of either type of debt, I do not account here for how this change in u in turn affects other variables. Second, I do not account for how a change in the 0 supply of either type of debt would affect the value of D traded in equilibrium. The response of 0 70

D necessarily depends on the balance of supply Ds and demand Dd for debt, and the response 0 0 0 of utilization u . In deriving conditions on elasticities, I do not account for these effects. In that 0 sense,thesufficientstatisticsthatIderivebelowabstractfromgeneralequilibriumfeedbackeffects through u . Nevertheless, I take up these general equilibrium feedback effects in section 4.3, and 0 characterizetheamplificationmechanismtowhichitgivesrise. Aggregatedemand,(i.e. thehousehold’sdemandfunctionforprivatedebt)isgivenby (cid:18) (cid:20) (cid:21)(cid:19)−1 1 1 Dd(R ,u ,B )=e −T +dF+w n¯−B − E (58) 0 0 0 0 0 0 0 0 0 R 0 c (s) 0 1 whilethebank’ssupplycurveforprivatedebtisgivenbytheoptimalitycondition: (cid:104) (cid:16) (cid:17)(cid:105) (cid:104) (cid:105) (cid:104) (cid:105) E v(cid:48)rk rk(s)+1 +λ (s ) rk(s )+(cid:96) (s ) =E v(cid:48)rk(cid:0) φ((cid:96) (s))+τ RR (cid:1) +λ (s )τ RR (59) 2 1 1 L 1 L 1 L 2 1 0 0 1 L 0 0 (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) marginalbenefit marginalcost (cid:96) (s)= τ 0 RD 0 R 0 −T 1 E −rk(s) 1 k 1 1 ShortageofPublicSafeAssets Thereisashortageofpublicsafeassetsifandonlyif d (cid:16) (cid:17) Dd−Ds <0 dB 0 0 0 where I have suppressed the notation that these derivatives are eventuated at the equilibrium (includingu =u∗). 0 0 ImakethesameassumptionsaboutgovernmentbehavioraslistedinAppendix12. Inowshow that,inthedemand-determinedregime,Ialwayshaveashortageofpublicsafeassetsundercertain conditions. Ihave (cid:104) (cid:105) dDd 0 = ∂Dd 0 dB 0 − ∂Dd 0 dT 0 + ∂Dd 0 dd 0 F + ∂Dd 0 dw 0 + ∂Dd 0 dE 0 c 1 1 (s) dB 0 ∂B 0 dB 0 ∂T 0 dB 0 ∂d 0 F dB 0 ∂w 0 dB 0 ∂E 0 (cid:104) c 1 (s) (cid:105) dB 0 1 (cid:104) (cid:105) =−1− dT 0 + dd 0 F +n¯ dw 0 + 1 (cid:18) E (cid:20) 1 (cid:21)(cid:19)−2 dE 0 c 1 1 (s) 0 dB dB dB R c (s) dB 0 0 0 0 1 0 Sinceourassumptionsaboutthegovernment’sbehaviorimpliesthathigherissuanceofpublicdebt isnotrebatedasatransferatdate0,Ihave dT 0 =0. dB 0 71

Termsdependon du 0,andlasttermalsodependson d(cid:96) 1 (s L ) . (RecallthatsofarIareholdingu dB dB 0 0 0 constant because I are focusing on the effect of B on excess demand for (private) safe assets.) I 0 have in the demand-determined regime that dF =0, w =(1−α) y 0, where y =z (u k )αn¯1−α, 0 0 n¯ 0 0 0 0 which are affected by B only via u . So if I hold u constant, then dd 0 F ,dw 0 = 0. Then dDd 0 0 0 0 dB dB dB 0 0 0 simplifiesto (cid:104) (cid:105) dDd 1 (cid:18) (cid:20) 1 (cid:21)(cid:19)−2 dE 0 c 1 (s) 0 =−1+ E 1 0 dB R c (s) dB 0 0 1 0 (cid:104) (cid:105) ∂Dd dE 0 c 1 (s) =−1+ 0 1 (cid:104) (cid:105) ∂E 1 dB 0 0 c (s) 1 where the second term is the effect of B on the household’s consumption risk, and the effect of 0 (cid:104) (cid:105) dE 1 consumption risk on Dd (precautionary saving demand). Note that part of the effect of 0 c1(s) 0 dB 0 will capture the effect of higher B on lump-sum taxes at date 1 T (or, isomorphically, date 2, 0 1 wherethisisomorphismarisesbecauseofperfectconsumptionsmoothingbetweendates1and2). Notethatsofar,theconditionforashortageofpublicsafeassetsis: Ihaveashortageofpublic safeassetsifandonlyif: d (cid:16) (cid:17) Dd−Ds ≤0 dB 0 0 0 dDd dDs 0 − 0 ≤0 dB dB 0 0 (cid:104) (cid:105) ∂Dd dE 0 c 1 (s) dDs 0 1 ≤1+ 0 (cid:104) (cid:105) ∂E 1 dB 0 dB 0 0 c (s) 1 where the first term is the effect of B on H’s expected future marginal utility of consumption 0 E [u(cid:48)(c (s))](whichincludesconsumptionrisk),andtheeffectofconsumptionriskonDd (saving 0 1 0 demand),andthesecondeffectistheeffectofB onE’ssupplyscheduleofprivatedebt. 0 Recall that I already showed that, when ignoring general equilibrium feedback effects (via u 0 d(cid:96) (s ) and R ), I have 1 L =0 when holding u constant. Thus, our condition for a shortage of public 0 dB 0 0 safeassetsreducesto: Ihaveashortageofpublicsafeassetsiff: 72

(cid:104) (cid:105) ∂Dd ∂E 0 c 1 (s) dDs 0 1 ≤1+ 0 (cid:104) (cid:105) ∂E 1 ∂B 0 dB 0 0 c (s) 1 where this reflects the fact that the effect of B on systemic risk (cid:96) is 0 in partial equilibrium (i.e. 0 1 holdingu andR fixed). 0 0 EffectofB ontheSupplyofPrivateDebt 0 dDs How about effect of B on the bank’s willingness to issue private debt, 0? Since c (s) = 0 dB 1 0 c (s), I have c (s) = 1[(w +w )n¯+R (D +B )−T −T ], and so a rise in B increases the 2 1 2 1 2 0 0 0 1 2 0 household’sdate1and2consumption. Ds isdeterminedbythebank’sfirstordercondition,which 0 balancesthemarginalbenefitandcostofborrowing. (cid:104) (cid:16) (cid:17)(cid:105) (cid:104) (cid:105) (cid:104) (cid:105) E v(cid:48)rk rk(s)+1 +λ (s ) rk(s )+(cid:96) (s ) =E v(cid:48)rk(cid:0) φ((cid:96) (s))+τ RR (cid:1) +λ (s )τ RR (60) 2 1 1 L 1 L 1 L 2 1 0 0 1 L 0 0 (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) marginalbenefit marginalcost Therefore, a change in the supply of B affects the supply of private debt if and only if it alters 0 dDs this tradeoff in some way. I show in Online Appendix 11 (Characterizing the sign of 0) that dB 0 dDs a sufficient condition for 0 < 0 in the demand-determined regime (when holding u and R dB 0 0 0 constant, i.e. ignoring general equilibrium feedback effects) is that expected liquidation π (cid:96) (s ) L 1 L tobesufficientlysmallinequilibriumthatthefollowingconditionholds: (cid:18) k −D +TE(cid:19) π (cid:96) (s )≤ 1 0 1 η L 1 L k 1 ∂Ds I will later impose an assumption that 0 < 0 is not too negative, in order to ensure that the ∂B 0 demand-determinedregimealwaysfeaturesashortageofpublicsafeassets(seeAppendix10). PrivateSafeAssets: ShortageorOversupply? This can be either a shortage of public safe assets (which I will take up in the normative part) or private safe assets, or both. Our model reveals that not all safety traps are alike. There can be a general shortage of safe assets (i.e. a shortage of both public and private safe assets), similar to that in Caballero and Farhi (2017) and other papers in the literature, which I term conventional safetytrap;ortherecanbeashortageofpublicsafeassetsandanoversupplyofprivatesafeassets. Irefertothelatterasrisk-intensivesafetytraps. To determine the nature of the safety trap, I ask whether, at the margin, would an increase in thesupplyofprivatesafeassetsDs increaseordecreasetheexcessdemandforsafeassets,holding 0 73

u fixed at its equilibrium value (i.e. not allowing u to adjust to the change in excess demand). 0 0 (I can think of the relationship between excess demand for safe assets and safe asset supply, that is Dd−Ds, as a function of Ds. The equilibrium condition is defined as Dd−Ds =0. I can take 0 0 0 0 0 thederivativeofthefunctionwithrespecttoDs evaluatedattheequilibrium,andholdingu fixed. 0 0 Thisderivativecanbepositiveornegativeinprinciple.) Thatis,thekeyelasticitywhichdefinesthenatureofthesafetytrapis d (cid:16) (cid:17) Dd−Ds dDs 0 0 0 dDs dDd dDd d(cid:96) (s ) where 0 =1 and 0 = 0 1 L , and I have suppressed the notation that these derivatives dDs dDs d(cid:96) (s ) dDs 0 0 1 L 0 areeventuatedattheequilibrium(includingu =u∗). 0 0 A sufficient statistic (ignoring general equilibrium feedback effects from changes in u ) for 0 determiningthenatureofthesafetytrapisthen d (cid:0) Dd−Ds(cid:1) withrespectto0,i.e. dDs 0 0 0 dDd 0 −1 dDs 0 withrespectto0. (cid:104) (cid:105) dDd 0 = ∂Dd 0 dd 0 F + ∂Dd 0 dw 0 + ∂Dd 0 dE 0 c 1 1 (s) dDs 0 ∂d 0 F dDs 0 ∂w 0 dDs 0 ∂E 0 (cid:104) c 1 (s) (cid:105) dDs 0 1 Again,Ihaveinthedemand-determinedregimethatdF =0,w =(1−α) y 0,wherey =z (u k )αn¯1−α, 0 0 n¯ 0 0 0 0 which are affected by Ds only via u . So given that I are holding u constant, then dd 0 F ,dw 0 =0. 0 0 0 dDs dDs 0 0 dDd Then 0 simplifiesto dDs 0 (cid:104) (cid:105) dDd 1 (cid:18) (cid:20) 1 (cid:21)(cid:19)−2 dE 0 c 1 (s) 0 = E 1 dDs R 0 c (s) dDs 0 0 1 0 Therefore, the condition definingwhether there is a shortageof private safe assets (a sufficient statistic)is: thereisashortageofprivatesafeassets(conventionalsafetytrap)ifandonlyif: d (cid:16) (cid:17) Dd−Ds ≤0 dDs 0 0 0 74

dDd 0 ≤1 dDs 0 where,attheeffectivelowerbound,thisis (cid:104) (cid:105) ∂Dd dE 0 c 1 (s) 0 1 ≤1 ∂E 0 (cid:104) c 1 (s) (cid:105) dDs 0 1 ThisistheeffectofDs onthehousehold’sconsumptionrisk,andtheeffectofconsumptionriskon 0 Dd (precautionary saving demand). Moreover, recall these elasticities can be decomposed into a 0 channel reflecting the effect of systemic risk. Thus, there’s a shortage of private safe assets if and onlyif  (cid:104) (cid:105) (cid:104) (cid:105)  ∂E 0 ∂ (cid:104) D c d 0 1 (s) (cid:105) ∂E 0 ∂D c 1 s 0 1 (s) + ∂E ∂ 0 (cid:96) 1 ( c s 1 L 1 (s ) ) d(cid:96) d 1 D (s s 0 L ) ≤1 1 where the second term captures the effect of debt on expected future marginal utility of consumption via systemic risk, while the first term captures other effects such as through the household’s interestincomeonholdingsofbondsinbothstates. APPENDIX9: ProofofLemma2: Safetytrapfeaturesshortageofpublicsafeassets Inowshowthat,underregularityconditions,thedemand-determinedregimealwaysfeaturesa safety trap (that is, a shortage of public safe assets). In particular, I will show that, in the demanddeterminedregime,Ihave (cid:104) (cid:105) ∂Dd ∂E 0 c 1 (s) dDs 0 1 ≤1+ 0 (cid:104) (cid:105) ∂E 1 ∂B 0 dB 0 0 c (s) 1 Ibeginbyshowingthattheleft-handsideisweaklynegativeinthedemand-determinedregime. Ibeginbyshowingthat ∂Dd 0 >0. Notefromthehousehold’sdemandfunctionDd that ∂E 0 (cid:104) c1 1 (s) (cid:105) 0 (cid:18) (cid:20) (cid:21)(cid:19)−1 1 1 Dd(R ,B )=e −T +dF+w n¯−B − E (61) 0 0 0 0 0 0 0 0 R 0 c (s) 0 1 So ∂Dd 0 = 1 (cid:16) E (cid:104) 1 (cid:105)(cid:17)−2 >0. ∂E 0 (cid:104) c1 1 (s) (cid:105) R 0 0 c 1 (s) (cid:104) (cid:105) I now show that ∂E 0 c1 1 (s) < 0. It suffices to show that, for each state s, ∂c 1 (s) > 0, abstract- ∂B 0 ∂B 0 75

ing from general equilibrium feedback effects (that is, holding u and R fixed). Recall that, in 0 0 equilibrium, 1 c (s)=c (s)= [(w +w )n¯+R (D +B )−T −T ] 1 2 1 2 0 0 0 1 2 2 First note that ∂c 1 (s) = R 0 > 0. Leaving (cid:96) constant (since this is grouped in the term reflecting ∂B 0 2 1 Channel 1), I must also account for how each type of debt affects c (s) through the other terms in 1 theaboveexpression. ∂c (s) R ∂w (s) ∂w (s) ∂T ∂T 1 0 2 1 1 2 = +n¯ +n¯ − − . ∂B 2 ∂B ∂B ∂B ∂B 0 0 0 0 0 RecallthatundertheassumptionsI’vemadeaboutthegovernment’sbehavior,IhaveT =−τ and 1 1 T =0 in equilibrium. Therefore, ∂T 1, ∂T 2 =0. Moreover, I already showed that ∂(cid:96) 1 (s) =0. Note, 2 ∂B 0 ∂B 0 ∂B 0 then,that ∂w (s) (1−α)∂y (s) (1−α)∂y (s) ∂k˜ (s) ∂kG(s) (1−α)∂y (s) (1−α) y (s) 1 = 1 = 1 1 1 = 1 = α 1 >0 ∂B n¯ ∂B n¯ ∂k˜ (s)∂kG(s) ∂B n¯ ∂k˜ (s) n¯ k˜ (s) 0 0 1 1 0 1 1 Hence, ∂w 1 (s) >0. ∂B 0 Nowturnto ∂w 2 (s) . ∂B 0 ∂w (s) (1−α)∂y (s) (1−α)∂y (s) (cid:20) ∂k˜ (s) ∂kG(s) ∂k˜ (s)∂k (s) (cid:21) 2 = 2 = 2 2 2 + 2 2 ∂B n¯ ∂B n¯ ∂k˜ (s) ∂kG(s) ∂B ∂k (s) ∂B 0 0 2 2 0 2 0 (cid:20) (cid:21) (1−α)∂y (s) ∂k (s) 2 2 = 1+ n¯ ∂k˜ (s) ∂B 2 0 Recallthat,inequilibrium,Ihavek (s)= (cid:2) 1+rk(s)−φ((cid:96) (s)) (cid:3) k (D )−τRD R +TE,andthat, 2 1 1 1 0 0 0 0 1 based on the assumptions above about the government’s behavior, I have TE = rk(s)kG−RBB . 1 1 1 0 0 Therefore, (cid:104) (cid:105) k (s)= 1+rk(s)−φ((cid:96) (s)) k −τ RD R +rk(s)kG−RBB 2 1 1 1 0 0 0 1 1 0 0 andso ∂k (s) (cid:104) (cid:105) ∂k ∂D ∂rk(s) ∂kG ∂rk(s) 2 = 1+rk(s)−φ((cid:96) (s)) 1 −τ RR 0 +k 1 +rk(s) 1 +kG 1 −RB ∂B 1 1 ∂B 0 0 ∂B 1 ∂B 1 ∂B 1 ∂B 0 0 0 0 0 0 0 76

Recall that k = D +TE + (cid:0) rk+1 (cid:1) k , ∂k 1 G = 1, and, as I showed above, B doesn’t affect the 1 0 0 0 0 ∂B 0 0 supplyofdebt,saveforGEfeedbackeffectsfromwhichIareabstracting. Therefore, ∂k (s) (cid:104) (cid:105)∂rk(s) 2 = k +kG 1 +rk(s)−RB ∂B 1 1 ∂B 1 0 0 0 whererk =α y 1,so ∂r 1 k(s) =α 1 ∂y 1 −α y 1 ∂k˜ 1 =α 1 ∂y 1 − r 1 k ∂k˜ 1. Andsince ∂k˜ 1 = ∂k 1 + ∂k˜ 1 G = 1 k˜ 1 ∂B 0 k˜ 1 ∂B 0 (k˜ 1 )2∂B 0 k˜ 1 ∂B 0 k˜ 1 ∂B 0 ∂B 0 ∂B 0 ∂B 0 1 and ∂y 1 = ∂y 1 ∂k˜ 1 = ∂y 1 =α y 1 =rk, I have ∂r 1 k(s) =α r 1 k − r 1 k =− r 1 k (1−α)<0. Putting these ∂B 0 ∂k˜ 1 ∂B 0 ∂k˜ 1 k˜ 1 1 ∂B 0 k˜ 1 k˜ 1 k˜ 1 derivativestogether,Icanseethat ∂c 1 (s) >0forbothstatesoftheworld. ∂B 0 ∂c (s) R ∂w (s) (1−α) y (s) 1 0 2 1 = +n¯ +n¯ α ∂B 2 ∂B n¯ k˜ (s) 0 0 1 R (cid:104) (cid:105) y (s) = 0 +(1−α)rk(s) 1+αrk(s)−RB +(1−α)α 1 >0 2 2 1 0 k˜ (s) 1 (cid:104) (cid:105) Thus,itfollowsthat ∂E 0 c1 1 (s) <0. Hence,thusfarIhave ∂B 0 (cid:104) (cid:105) ∂Dd ∂E 0 c 1 (s) 0 1 <0 (cid:104) (cid:105) ∂E 1 ∂B 0 0 c (s) 1 I now assume that the following condition holds in equilibrium, which amounts to assuming dDs that 0 isnottoonegative: dB 0 (cid:104) (cid:105) ∂Dd ∂E 0 c 1 (s) dDs 0 1 −1< 0 (cid:104) (cid:105) ∂E 1 ∂B 0 dB 0 0 c (s) 1 (cid:104) (cid:105) where ∂Dd 0 ∂E 0 c1 1 (s) < 0. (Note that it would suffice to assume that dDs 0 ≥ 0.) Given this ∂E 0 (cid:104) c1 1 (s) (cid:105) ∂B 0 dB 0 assumption,itfollowsthat (cid:104) (cid:105) ∂Dd ∂E 0 c 1 (s) dDs 0 1 <0<1+ 0 (cid:104) (cid:105) ∂E 1 ∂B 0 dB 0 0 c (s) 1 andsothedemand-determinedregimealwaysfeaturesashortageofpublicsafeassets. APPENDIX10: ProofofProposition1: Oversupplyofprivateassets 77

I claim that there is an oversupply of private safe assets if and only if the dynamic interaction between systemic risk ((cid:96) ) and AD is sufficiently large. To prove this, first recall that there’s an 1 oversupplyofprivatesafeassetsifandonlyif dDd 0 >1 dDs 0 dDd From our decomposition of channel 1 0, I know that the above elasticity can be decomposed dDs 0 intotwoterms. (cid:104) (cid:105) (cid:104) (cid:105) ∂Dd 0 ∂E 0 c 1 1 (s) d(cid:96) 1 (s L ) >1− ∂Dd 0 ∂E 0 c 1 1 (s) ∂E 0 (cid:104) c 1 (s) (cid:105) ∂(cid:96) 1 (s L ) dDs 0 ∂E 0 (cid:104) c 1 (s) (cid:105) ∂Ds 0 1 1 (cid:124) (cid:123)(cid:122) (cid:125) >0 Thus, there is an oversupply of private safe assets if and only if the dynamic interaction between aggregatedemandandsystemicrisk(morepreciselytheproductofChannels1and2)issufficiently strong. Q.E.D. So the condition defining nature of safety trap indicates that I have a risk-intensive safety trap ifandonlyiftheproductofthetwochannelsissufficientlystrong-thatis,iftheaggregatedemand and systemic risk is sufficiently strong. If this dynamic interaction is sufficiently strong, then at the margin, an increase in the supply of private debt will cause systemic risk to increase and the recession to become worse, implying that there’s an oversupply of private debt. (As I will show later, this also equivalently means the direction of the aggregate demand externality is such that planner would reduce the bank’s leverage relative to the competitive equilibrium.) Alternatively, if it is not sufficiently strong, then a marginal increase in the supply of private debt may increase systemic risk and precautionary demand, but this will be outweighed by the effect of more private debtonreducingexcessdemandforsaving,reducingtheseverityofthedemandrecession. Inthat case,there’sashortageofprivatesafeassets(i.e. aconventionalsafetytrap). APPENDIX11: ProofofProposition2: Amplificationmechanism Here,Ishowthatunderweakconditions,theconfluenceofthegeneralequilibriumChannels3aand 3b,outlinedinsection4.3,amplifiesthedemand-recessioninresponsetoanadverse,unanticipated fallinz (s ). Recallthateachchannelamplifiesthedownturnifactive,anddampensitifnot(that 1 L is, if the effects go the opposite direction). If, in equilibrium, only one of the channels is active, thenwhetherthegeneralequilibriumeffects,intotal,amplifyordampentheshockdependsonthe neteffectofthetwo. 78

Theneteffectofboth3aand3bondate0excessdemandforsaving,andthereforetheseverity of the date 0 recession, is given by the marginal total effect of both channels on excess demand dDd dDs for saving at date 0: 0 − 0. In particular, the net effect of 3a and 3b is to dampen the severity du du 0 0 of the recession if excess demand for saving at date 0 falls at the margin as u falls, that is if 0 d (cid:0) Dd−Ds(cid:1) >0. du 0 0 0 dDd dDs 0 − 0 >0 du du 0 0 dDs dDd Note that 0 is Channel 3a in itself, while Channel 3b is embedded in the term 0. In du du 0 0 particular,recallequation(5)forthehousehold’sdemandforprivatesafeassets. (cid:18) (cid:20) (cid:21)(cid:19)−1 1 1 Dd(R ,B ;u )=e −T +dF+w n¯−B − E (62) 0 0 0 0 0 0 0 0 0 R 0 c (s) 0 1 In the demand-determined regime, R = 1. Taking the derivative of this function with respect to 0 u showsthatutilizationhasaneffectonDd throughtwochannels: throughtheeffectonsystemic 0 0 risk (Channel 3b), and through the households labor income w n¯: dDd 0 = dDd 0 d(cid:96) 1 (s L ) + ∂w 0 n¯ . 0 du 0 d(cid:96) 1 (s L ) du 0 ∂u 0 (cid:104) (cid:105) Moreover, I have already shown in Appendix 15 that dDd 0 = ∂Dd 0 ∂E 0 c1 1 (s) >0. And recall d(cid:96) 1 (s L ) ∂E 0 (cid:104) c1 1 (s) (cid:105) ∂(cid:96) 1 (s L ) from equation (289) that ∂w 0 n¯ = α(1−α) y 0 > 0. Therefore, I can rewrite this condition for ∂u 0 u 0 dampeningas (cid:104) (cid:105) ∂Dd 0 ∂E 0 c 1 1 (s) d(cid:96) 1 (s L ) + ∂w 0 n¯ > dDs 0 (cid:104) (cid:105) ∂E 1 ∂(cid:96) 1 (s L ) du 0 ∂u 0 du 0 0 c (s) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124)(cid:123)(cid:122)(cid:125) 1 (cid:124) (cid:123)(cid:122) (cid:125) 3b >0 3a >0 ProofofPartA: dDs d(cid:96) (s ) Claim: If neither general equilibrium channel is active ( 0 < 0 and 1 L > 0), then the du du 0 0 general equilibrium Channels 3a and 3b together dampen the downturn. To see this, revisit the conditionfordampening: (cid:104) (cid:105) ∂Dd 0 ∂E 0 c 1 1 (s) d(cid:96) 1 (s L ) + ∂w 0 n¯ > dDs 0 (cid:104) (cid:105) ∂E 1 ∂(cid:96) 1 (s L ) du 0 ∂u 0 du 0 0 c (s) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124)(cid:123)(cid:122)(cid:125) 1 (cid:124) (cid:123)(cid:122) (cid:125) >0 >0 <0 >0 dDs d(cid:96) (s ) This condition holds by the fact that, when 0 < 0 and 1 L > 0, the left-hand side is strictly du du 0 0 79

positive while the right-hand side is unambiguously strictly negative. In this case, the general equilibrium channels cannot amplify the downturn. Therefore, the confluence of the general equilibriumfeedbackchannels(togetherwiththeeffecton ∂w 0 n¯ )istodampenthedownturn. Q.E.D. ∂u 0 ProofofPartB: Claim: Ifoneorbothchannelsisactive,amplificationmayhold. dDs d(cid:96) (s ) To see this, when 3a holds ( 0 >0) but 3b does not ( 1 L ≥0), the condition for amplificadu du 0 0 tionis (cid:104) (cid:105) ∂Dd 0 ∂E 0 c 1 1 (s) d(cid:96) 1 (s L ) + ∂w 0 n¯ < dDs 0 (cid:104) (cid:105) ∂E 1 ∂(cid:96) 1 (s L ) du 0 ∂u 0 du 0 0 c (s) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124)(cid:123)(cid:122)(cid:125) 1 (cid:124) (cid:123)(cid:122) (cid:125) >0 >0 >0 >0 dDs d(cid:96) (s ) which can hold. When 3a does not hold ( 0 ≤ 0) but 3b does ( 1 L < 0), the condition for du du 0 0 amplificationis (cid:104) (cid:105) ∂Dd 0 ∂E 0 c 1 1 (s) d(cid:96) 1 (s L ) + ∂w 0 n¯ < dDs 0 (cid:104) (cid:105) ∂E 1 ∂(cid:96) 1 (s L ) du 0 ∂u 0 du 0 0 c (s) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124)(cid:123)(cid:122)(cid:125) 1 (cid:124) (cid:123)(cid:122) (cid:125) <0 >0 <0 >0 whichcanalsohold. If both channels are active, then the condition for amplification is weaker than if only one channelholds. E.g. ifbothchannelsareactivetheconditionforamplificationis (cid:104) (cid:105) ∂Dd 0 ∂E 0 c 1 1 (s) d(cid:96) 1 (s L ) + ∂w 0 n¯ < dDs 0 (cid:104) (cid:105) ∂E 1 ∂(cid:96) 1 (s L ) du 0 ∂u 0 du 0 0 c (s) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124)(cid:123)(cid:122)(cid:125) 1 (cid:124) (cid:123)(cid:122) (cid:125) <0 >0 >0 >0 Due to the signs of the terms in the condition, this is a weaker condition than both of the previous conditions. Q.E.D. APPENDIX12: ProofsregardingAssumption1 In this appendix, I show that Assumption 1 ensures that the bank’s date 0 natural borrowing limit on borrowing D is never binding in equilibrium, and that liquidation occurs only in the bad state, 0 (cid:96) (s ) > 0, (cid:96) (s ) = 0. I also show that these restrictions are satisfied by a non-empty set of 1 L 1 H parameterssatisfiestheserestrictions. Claim1: z (s )< τ 0 DR 0 D 0 −k 1 impliesthat(cid:96) (s )>0. 1 L k 1α(k˜ 1 )α−1 n¯1−α 1 L Proof: Ihave(cid:96) (s )>0ifandonlyif 1 L 80

τRD R TE (cid:96) (s )= 0 0 0 −rk(s )− 1 >0 (63) 1 L k 1 L k 1 1 i.e. τRD R −TE 0 0 0 1 >z (s ) (64) k α (cid:0) k˜ (cid:1)α−1 n¯1−α 1 L 1 1 Q.E.D. Claim2: z (s )≥ τ 0 RD 0 R 0 −T 1 E impliesthat(cid:96) (s )=0. 1 H k 1α(k˜ 1 )α−1 n¯1−α 1 H Proof: Fromtheexpressionfor(cid:96) ,itfollowsthat(cid:96) (s )=0ifandonlyif 1 1 H τRD R −TE 0 0 0 1 ≤z (s ) (65) k α (cid:0) k˜ (cid:1)α−1 n¯1−α 1 H 1 1 Icansetz (s )arbitrarilyhightoensurethisissatisfied. Q.E.D. 1 H Claim3: Thenaturalborrowinglimitisnon-binding. Proof: Recall that natural borrowing limit is τDR D ≤ P (s) (cid:0) rk(s)k +(cid:96)¯ k (cid:1) where (cid:96)¯ , the 0 0 0 1 1 1 1 1 1 maximum fraction of its capital that the bank can liquidate without violating the non-negativity constraintonk ,solvesk (s)=0wheni =0: 2 2 1 k (s)=i +(1−(cid:96) −φ((cid:96) (s)))k (s) (66) 2 1 1 1 1 η 1=(cid:96) +(cid:96) (67) 1 1 Sothenaturalborrowinglimitisnon-bindingifandonlyif (cid:16) (cid:17) τ DR D <P (s) rk(s)k +(cid:96)¯ k 0 0 0 1 1 1 1 1 τDR D 0 0 0 −rk(s )<(cid:96)¯ k 1 L 1 1 Notethatsince(cid:96)¯ solves1=(cid:96) +(cid:96) η ,asη >1approachesinfinity,(cid:96)¯ >0approaches1. Therefore, 1 1 1 1 makingη arbitrarilylargewouldbyitselfnotsufficetoensurethenaturalborrowinglimitisalways non-binding. Butitwouldsufficeifitisalsothecasethat τDR D 0 0 0 −rk(s )<1 k 1 L 1 i.e. 81

τDR D −k 0 0 0 1 <z (s ) k α (cid:0) k˜ (cid:1)α−1 n¯1−α 1 L 1 1 Thus, I can ensure that the natural borrowing limit is non-binding by simultaneously making η arbitrarilylargeandz (s )isnottoosmall,sothatitsatisfiestheaboveinequality. Q.E.D. 1 L Claim4: Thereisanon-emptysetofparameterswhichsatisfytheseassumptions. Proof: FirstrecallthatIcanmakez (s )arbitrarilylargetoensurethat τ 0 RD 0 R 0 −T 1 E ≤z (s ). 1 H k 1α(k˜ 1 )α−1 n¯1−α 1 H Tosimultaneouslyensurethatboththenaturalborrowinglimitisnon-bindingandthat(cid:96) (s )> 1 L 0,Imustjointlyassumethatz (s )satisfies 1 L (cid:32) (cid:33) τDR D −k τRD R −TE z (s )∈ 0 0 0 1 , 0 0 0 1 1 L k α (cid:0) k˜ (cid:1)α−1 n¯1−α k α (cid:0) k˜ (cid:1)α−1 n¯1−α 1 1 1 1 Notethatsuchaz (s )existsaslongas 1 L τDR D −k τRD R −TE 0 0 0 1 < 0 0 0 1 k α (cid:0) k˜ (cid:1)α−1 n¯1−α k α (cid:0) k˜ (cid:1)α−1 n¯1−α 1 1 1 1 i.e. TE <k 1 1 Thus,part(B)ofAssumption(1)ensuressuchaz (s )exists. Q.E.D. 1 L 82