interventions Private money creation, liquidity crises, and government Journal of Monetary Economics

ContentslistsavailableatScienceDirect

Journal of Monetary Economics

journalhomepage:www.elsevier.com/locate/jmoneco

Private money creation, liquidity crises, and government interventions ^R

Pierpaolo Benigno

^a,∗

, Roberto Robatto

^b

a LUISS and EIEF, Viale Romania 32, Roma 00198, Italy

b University of Wisconsin-Madison, USA

a rt i c l e i n f o

Article history:

Received 22 December 2018 Revised 19 June 2019 Accepted 5 July 2019 Available online 6 July 2019 JEL classification:

G01 G28 E44 Keywords:

Public and private liquidity Financial crises

Deposit insurance Asset purchases Financial regulation

a b s t ra c t

Thejointsupplyofpublicandprivateliquidityisexaminedwhenfinancialintermediaries issuebothrisklessandriskydebtandtheeconomyisvulnerabletoliquiditycrises.Gov- ernmentinterventionsintheformofassetpurchasesanddepositinsuranceareequivalent (inthesensethattheysustainthesameequilibriumallocations),increasewelfare,and,if fiscalcapacityissufficientlylarge,eliminateliquiditycrises.Incontrast,restrictinginter- mediariestoinvestinginlow-riskprojectsalwayseliminatesliquiditycrisesbutreduces welfare. Undersomeconditions,depositinsurancegivesrisetoanequilibrium inwhich intermediariesthatissueinsureddebt(i.e.,traditionalbanks)coexistwithothersthatis- sueuninsureddebt(i.e.,shadowbanks),despitethetwobeingexanteidentical.

© 2019TheAuthors.PublishedbyElsevierB.V.

ThisisanopenaccessarticleundertheCCBY-NC-NDlicense.

(http://creativecommons.org/licenses/by-nc-nd/4.0/)

1. Introduction

The2007–2008financialcrisishashighlightedtheexistenceoftwomainclassesofmoney-likesecuritiesthat canpro- vide liquidity services.The first class includes safefinancial assets such asTreasury securities, whereas the second class includesseveraltypesofliabilitiesoftheso-calledshadowbankingsystem.Thesecuritiesinthissecondgroupofsecurities were notcompletelysafebecause ofalackofappropriate backingonintermediaries’balancesheets.Atthe heightofthe crisis,thesesecuritieslostnotonlytheirvaluebutalsotheirabilitytoprovideliquidityservices.

Inresponse totheseevents,theU.S.governmentincreasedthedirect supplyofpublicliquidity aswell asthesupport andbackingoftheliquiditysuppliedbyprivateintermediaries.Theseinterventionsincludedtheassetpurchaseprogramsof

R This paper builds on a previous work circulated under the title “Private Money Creation and Equilibrium Liquidity.” We thank José Antonio de Aguirre, Markus Gebauer, Lorenzo Infantino, Ricardo Lagos, Gabriele La Spada, Michael Magill, Fabrizio Mattesini, Patrick Bolton, Martine Quinzii, Jean-Charles Ro- chet, and Hiung Seok Kim for helpful conversations and suggestions, and seminar participants at the University of Wisconsin-Madison, Oxford University, NYU Stern, Federal Reserve Bank of New York, University of Keio, Brown University, CREST, Yale, European University Institute, Kiel Institute for the World Economy, Society for Economic Dynamics, the 4th Workshop in Macro Banking and Finance, the 12th Dynare Conference, the Korean Economic Association Conference on “Recent Issues in Monetary Policy”, the 4th Annual HEC Paris Workshop “Banking, Finance, Macroeconomics and the Real Economy”, the 5th International Moscow Finance Conference, and the 15th Workshop on Macroeconomic Dynamics: Theory and Applications. Francesco Celentano, Kyle Dempsey, Kuan Liu, Mark Rempel, and Natasha Rovo have provided excellent research assistance.

∗ Corresponding author.

E-mail address: pbenigno@luiss.it (P. Benigno).

https://doi.org/10.1016/j.jmoneco.2019.07.005

0304-3932/© 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license.

( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

theFederalReserve,theincreaseinthedepositinsurancelimit,andtheTemporaryLiquidityGuaranteeProgramofferedby theFederalDepositInsuranceCorporation(FDIC).Inaddition,afterthecrisis,theregulationoffinancialintermediarieswas mademorestringentwiththeobjectiveofmakingprivateintermediaries’debtsafeandreducing thelikelihoodanddepth offuturecrises.

Motivatedbytheseevents,thispaperusesageneralequilibriummodeltoprovideaunifyinganalysisoftheinteraction betweenprivate andpublicliquidity andofthreekey policiesimplementedinresponsetothe2007–2008financialcrisis:

thecentralbank’sassetpurchasesandexpansionofthepublicliquidityprovision,governmentguaranteesofprivatemoney, andregulationoffinancialintermediaries.

Wefocus onthreemain questions.First,we studywhetherliquidassetsshould besuppliedby thegovernmentorby privatefinancial intermediaries—aclassicquestioninmacroeconomics(Sargent,2011). Second,ifapositive supplyofgov- ernmentliquidityisoptimal,shouldsuchassetsbebackedbytaxesorbyaportfolioofprivatesecuritiesheldbythecentral bank?Third,ifapositivesupplyofprivatemoneybyfinancialintermediariesisoptimal,shouldtheintermediariesbesub- jecttoregulation,suchasdepositinsuranceorrestrictionsontheriskinessoftheirinvestments?

Toclarifytheforcesbehindourresults,somestarkassumptionsaremadetokeepthemodelsimpleandtractable.Inthe model,afinancialfrictionforcesagentstousedebtsecuritiesfortransactionpurposes.Inlinewiththehistoricalevidence inGorton (2017), riskless debtin our model (which we refer to assafe assets) always provides liquidity,whereas risky debtdoessoonlyinnormaltimes,thatis,whenitisnotdefaultedon.Thisfeatureisakeydistinctionfromtheapproach commonlyused inthe literature. Motivated by Gorton andPennacchi (1990), some closelyrelated papers usemodels in whichonlyrisk-freesecuritiesprovideliquidity(see,e.g.,Angeletosetal.,2016;Greenwoodetal.,2015;Magilletal.,2016;

Stein,2012).

Inthelaissez-faire equilibrium, liquidity crises happenwhen riskyprivate securitiesare defaultedon andthereby lose theirliquidityvalue.Thepossibilityofthesecrisesopensuparoleforgovernmentinterventions,andoursimpleframework givesrisetoarichsetofpredictions.

Iffiscalcapacityislimited,thebestpoliciesarethoseinwhichpublicandprivateliquiditycomplementeachother.Two such policies are studied: assetpurchases (i.e., a large supplyof public liquidity backed by the central bank’s purchases ofprivatesecurities) anddepositinsurance. The twopoliciesare equivalent,inthe sensethat they allowtheeconomy to achieve the sameallocation andthus the same welfare.This isbecause the consolidated balance sheetofall the agents thatsupplyliquidity(i.e.,governmentandfinancialintermediaries)isidenticalunderthetwopolicies.Itisthusirrelevant, inthesenseofWallace(1981),whetherthegovernmentsupportsliquiditywithcentralbankinterventionsorwithdeposit insurance.Anadditionalresultthatarisesunderdepositinsuranceisthattheequilibriumischaracterizedbyacoexistence ofinsured and uninsured intermediaries that invest in the same type of risky assets.This resembles the coexistence of traditionalbanksandshadowbanksinpracticeandarisesendogenouslyinourmodel.

Withrespecttotheregulationofintermediaries’risktaking,itisstudiedapolicythatforcesallintermediariestoinvest in low-risk,low-productive projects in order to avoid default. This policy eliminates liquidity crises but reduces welfare becauseitforcesintermediariestoforgoprojectsthat,albeitrisky,haveahigherexpectedreturn.

Financialintermediariesinourmodelcansupplyliquidassetsby investinginriskyorsafeprojects.Investmentinrisky projectsallowsintermediariestoofferriskyassets,whereasinvestmentinsafeprojectsallowsintermediariestooffersafe assets.However, safeprojectsaremorecostlybecauseafractionoftheinvestmentislost;we motivatethiscostwiththe needtoscreenandmonitorprojectstomakesurethattheyareindeedsafe.Asaresult,issuingriskydebtthatissubjectto defaultallows financialintermediariestosaveonsuchmonitoringcosts. Thelogicissimilartothat inGeanakoplos(1997, 2003)inwhichthepossibilityofdefaultisawaytoeconomizeonscarceandcostlycollateral.

Themonitoringcostcreatesanincentiveforintermediariestosupplyriskyassets.Inbadstates,thesesecuritiesdefault, andthereforethere isashortageofliquidity.Inaddition,dependingonthepolicy andparameters,some safedebtmight alsobesupplied.Thisdebtisalwaysliquidandthustradesatapremiumbecauseoftheshortageofliquidityinbadstates.

Ourresultsarefirstderivedinasimplemodelinwhichriskydebtfullydefaultsincrisis times,butwethenshow that thesameresultscanbeobtainedinarichermodel.Inparticular,oneoftheextensionsconsidersadverseselectionproblems inthemarketofriskysecurities,inlinewiththenarrativeofthe2007–2008financialcrisis(Gorton,2009)andwithalarge literaturethat hasemphasized therole ofthisfriction(Boltonetal., 2009;2011; GortonandOrdoñez,2013;2014;2019;

Malherbe,2014).

Ouranalysiscomplementsarecentliteraturethathasstudiedtheroleofliquidityinmacromodelswithfinancialinter- mediaries.Examplesinclude(BianchiandBigio,2016;Bigio,2015;GertlerandKiyotaki,2010;Moreiraetal.,2016;Quadrini, 2014). Withrespect to thisliterature, the noveltyof ourpaper isto analyze the coexistencebetweenprivate and public liquidityandtheadvantageofoneformofliquidity overtheotherintermsofefficiency.Inthissense,ourworkisrelated totheclassicdebateinmacroeconomicsaboutthesupplyofprivateversuspublicliquidity(seeAguirreandInfantino,2013;

Sargent,2011,fora summary).Thelackofpolicy interventionsinthelaissez-faire equilibriumcanbe reinterpretedasthe suggestionofsupporters ofthefree-bankingtheory,such asHayek(1976),which emphasizethebenefits ofderegulation.

The large supplyof public liquidity backed by taxesor by a portfolio ofprivate securitiesis relatedto the proposals of Friedman(1960),whichinsteadarguethatliquidityshouldbecontrolledbythegovernment.Theregulationthat forcesin- termediariestoinvestinsafeprojectsisakintothereal-billstheory,accordingtowhichintermediariesshouldinvestonly inrisk-freeassets.

Some papersintheNewMonetaristliterature studythe roleofprivate moneyandits interactionwithpublicliquidity by allowing physical capital to be used for transactions. However, they mostlyfocus on models withno aggregate risk, andeitherwithonly safeassets(Geromichalosetal., 2007;LagosandRocheteau,2008) or,ifmultipletypesofassetsare included,withanexogenoussupplyofsuchassets(Rocheteau,2011).

The banking literature is rich with models that analyze liquidity creation inthe spirit of the seminal contributionof Gorton andPennacchi (1990). The papers closest to ours are (Greenwood et al., 2015; Magill et al., 2016). These works assumethatliquidity servicesareprovidedonlybyrisk-freesecurities,whereasinourframework,riskysecuritiescanalso beliquid.Asaresult, ourmodelcanstudythedeterminationoftheliquidity andrisk propertiesofprivatedebtjointlyas afunction ofthecharacteristicsoffinancialintermediariesandthepolicyenvironment.Inaddition,theabove twopapers havesomeotherimportantdifferencesrelativetoourpaper.

In Magilletal. (2016), onlyprivate debtcan provide liquidity services,and thereforethe focusof their analysis isto studyhowgovernmentpoliciescanenhancethesupplyofprivateliquidity.Inourmodel,instead,governmentdebtalsohas liquidityvalue.Despitethesedifferences,bothmodelspredictthatthecentralbankcanachievethefirstbestbyissuingsafe securitiesandbackingthemby purchasingriskyassets.Inourmodel,thisisa consequenceofthedirectliquidity role of publicdebt,whereasintheircontext,itisawaytoincreasethefundschanneledtoinvestments.

InGreenwoodetal.(2015),thepolicyanalysisfocusesontheoptimalmaturitystructureofgovernmentdebt.Short-term debtismoreliquidbutentailshigherrefinancing riskincomparisontolong-termdebt.Inaddition,theliquidshort-term debtcrowdsoutexcessivemoneycreationbyfinancialintermediaries,limitingthenegativeeffectsofafire-saleexternality similartothatofLorenzoni(2008)andStein(2012).Ourpolicyfocusisinsteadonadifferentandbroadersetofgovernment interventions.

Angeletoset al.(2016)consider an environmentin whichprivately issued,risk-freedebt serves ascollateral,butsuch debtisscarcebecauseagentscanpledgeonlyafractionoftheirfutureincome.Therefore,theystudyhowpublicliquidity andthegovernment’s commitmenttoraise moretaxesinthefuture canease thisfinancialfriction. Incontrast,privately issuedassetsinourmodelincludenotonly risk-freedebtbutalsorisky debt,whichleadstopolicy conclusionsthat em- phasizedifferentfeatures,namely,theinteractionbetweengovernmentinterventionsandthesupplyofriskysecurities.

Inarecentwork,(BrunnermeierandNiepelt,2019)haveshownequivalenceresultsinswapsbetweenpublicandprivate moneyfortheequilibriumallocations.Inourmodel,private andpublicliquidity arenotperfectlysubstitutablebecauseof thecostofissuingprivatesafesecuritiesandthefulldefaultofriskysecuritiesinthelowstateofnature.Moreover,wealso imposelimitationsonthegovernment’sabilitytosupplysafeliquidity.

Therestofthispaperisorganizedasfollows.Section2presentsthemodel.Sections3and4discusstheequilibriumwith costlessandcostlymonitoringofsafeinvestments,respectively.Section5discussestheeffectsofgovernmentintervention.

Section6presentsourrobustnessanalyses.Section7concludes.

2. Model

Wepresenta simpletwo-period(t=0,1) generalequilibriummodelinwhichweshowall ourresultsanalytically.The economyfeaturesthreesetsofactors:households,financialintermediaries,andthegovernment.

Aggregate riskis introducedby assuming that thereare twostatesof natureatt=1,highandlow.The highstate is denotedbyhandoccurswithprobability1−

π

,with0<

π

<1.Thelowstateisdenotedbylandoccurswithprobability

π

^.

Thekeymechanisminourmodelisthattherealizationofthelowstatetriggersdefaultsintheintermediarysector,which inturngiverisetoaliquiditycrunch.

Att=0,householdscaninvesttheirwealthinthreetypesofsecurities:safe,risklessprivate debtissuedbyintermedi- aries thatinvestinsafeprojects,riskyprivate debtissuedbyintermediariesthat investinriskyprojects,andgovernment debt.Risklessdebtisbackedbysafeinvestmentsandthusisneverdefaultedon.Riskydebtisinsteadbackedby riskyin- vestmentsthatcompletelylosetheirvalueinthelowstate,andthusitisfullydefaultedoninthatstate.Governmentdebt isalwaysrepaidandthereforealwayssafe.

Att=1,partoftheconsumptionexpenditureofhouseholdsmustbefinanced withdebtsecurities. Thatis,households aresubjecttoaliquidityconstraint,andliquidityservicesareprovidedbythedebtsecuritiespurchasedatt=0.Sincerisky securitiesarefullydefaultedoninthelowstate,theydonotprovideanyliquidityserviceinthatstate.

Theassumptionsthatthereareonlytwotypesofprivatelyissuedsecuritiesandthatalltheriskydebtissuedbyfinancial intermediariesisworthzerointhelowstatemightlookextreme.However,inSection6andintheAppendix,weshowthat themodelcanbeenrichedtorelaxtheseassumptionswithoutalteringanyoftheresults.

2.1. Environment

The modelhastwoperiods,t=0,1.Timet=1is dividedintotwo subperiods. Householdshavethe followingprefer- ences:

X+

(

¹−

π )

^[lnCh+Xh]+

π

^[lnCl+Xl], (1)

where X denotes consumption att=0, C_h and C_l denote consumption in the first subperiod att=1 in state h andl, respectively, and X_h and X_l denote consumption in the second subperiod oft=1 in state h andl,respectively. Without

lossofgenerality,thediscountfactorbetweent=0andt=1isnormalizedtoone.Thefunctionalformof(1)allowsusto derivesimpleandstarkresults.Nonetheless,weshowinSection6andintheAppendixthatthekeyresultsareunchanged ifhouseholdshaveamoregeneralutilityfunction,althoughthederivationbecomesmorecomplicated.

Theresourceconstraintoftheeconomyisdescribedtocharacterizethefirst-bestallocation.Thenweintroducefinancial frictionsanddescribetheproblemofhouseholds,government,andfinancialintermediaries.

Att=0, there is an endowmentof goodsY¯,which can be consumed ortransformed into two types ofcapital: safe capitalK^S andrisky capitalK^D.The twotypesofcapital havethesameaverageproductivityatt=1,butthesafecapital requiresanextrainvestment

τ

^{foreachunitofcapitalatt}=0.Thecost

τ

^canbeinterpretedasamonitoringcosttocontrol thesafetyofcapital,motivatedbyDiamondandDybvig(1983).Thus,theaggregateresourceconstraintatt=0is

X+

(

¹+

τ )

^KS+K^D≤ ¯Y. (2)

Att=1,eachunit ofsafecapitalK^S producesone unitofoutput inbothstateshandl.In contrast,each unitofrisky capitalK^Dproduces A_h>1unitofoutputinstatehandzerounitsinstatel.Theassumptionofequalaverageproductivity ofthetwotypesofcapitalatt=1canbeformalizedas(¹−

π

)^Ah=1.

Att=1,thereisanotherendowmentofgoodsthatcanpossiblybestate-contingentandisdenotedbyY¯_handY¯_l inthe highandlowstate,respectively,withY¯_l≤ ¯Yh.Thetime-1endowmentandoutputareavailableinthefirstsubperiodoft=1, buttheycanbeusedforconsumptioninbothsubperiods.Thus,theaggregateresourceconstraintsinstatehandlatt=1 are

C_h+X_h≤ ¯Yh+K^S+A_hK^D, (3)

Cl+Xl ≤ ¯Yl+K^S. (4)

TheendowmentsY¯_handY¯_l areinterpretedastheresourcesproducedbyasectoroftheeconomyforwhichfinancialinter- mediationdoesnotplayakeyrole,suchaslargefirmsthathaveaccesstotheequityandbondsmarket,andthuswetake theseendowmentsasgiven.

Inthisenvironment,thekeyconditionthatcharacterizes thefirstbestistheequalitybetweenthemarginalutilitiesof consumptioninthetwosubperiodsattimet=1:

1

C_h =1, 1

C_l =1. (5)

Withoutanyfurtherassumption,andprovidedthat endowmentsaresufficientlylarge,householdscouldachievean alloca- tionthatimplements(5)withoutanyroleforfinancialintermediation,governmentintervention,orboth.

Next,financial frictions are introduced that limit the ability of households to consume in the first subperiod. These frictionsgiverisetoaroleforintermediariesandthegovernmentassuppliersofliquidassets.

Wefollow(Lucas andStokey,1987)by assuming that each householdiscomposed ofa shopperandaseller, andthe shoppermustpurchasetheconsumptiongoodsC_h andC_l fromsellersofotherhouseholds.Inaddition,thepurchasesmust be paid immediatelywith some financial instrument. Formally, this is equivalent to assuming that households lack the commitmentthat wouldallowthem topurchaseC_h andC_l withcredit (i.e.,torepayin thesecond subperiod). Ifinstead householdscould pledgetheresources thatthey haveavailable inthesecond subperiod oft=1(suchasanyunsolden- dowmentorthe proceedsfromselling goods inthefirst subperiod), they could overcomethisfinancial friction. Inother words,theseresourcesarenon-contractible.

FinancialintermediariesandthegovernmentcanbothsupplythesecuritiesthathouseholdsneedtopurchasegoodsC_h andC_l.

Afinancialintermediarycanissueeithersafe(risk-free)securitiesSandinvestinsafecapitalK^Sorissueriskysecurities DandinvestinriskycapitalK^D.Thesesecuritiesaremodelledaszero-coupondebtwithafacevalueofone.Thatis,Shas aunitarypayoff att=1inbothstates,andDhasapayoff ofoneinstatehandzeroinstatel.Differentfromhouseholds, intermediariescan committorepayingthesecuritiesissuedatt=0usinganyresources thattheyhaveavailable att=1, namely,thepayoff of capital.However,despitethe commitment,the lackofresources availableto riskyintermediariesin statelgivesrisetoalimitedliabilityconstraint.Asaresult,riskyintermediariesfullydefaultontheir debtinstatel.¹

The governmentcan alsocommit andthus issue securitiesthat households can useaspayment instruments att=1 togetherwithSandD.Governmentdebtisalwaysrepaidandthusisasafeasset,asweexplaininSection2.4.Weassume thathouseholdsareendowedattimet=0withanamountofgovernmentdebtB¯,whichisrepaidinthesecondsubperiod oft=1byraising taxes.² InthepolicyanalysisofSection5,weallowforarichersetofinterventions,andweshow that theamountoftaxesthatthegovernmentcanlevydeterminesthelevelofpublicliquidity.

Havingintroduced thefinancialfriction inthemodel,we nowdiscussthebehaviorofhouseholds,financialintermedi- aries,andthegovernment.

1 The limited liability constraint could in turn be motivated by the lack of commitment of households to make any payment at t = 1 . That is, even if households make a promise, at t = 0 , to recapitalize any insolvent intermediary at t = 1 , they would then default on that promise because of their lack of commitment.

2 Without loss of generality, we have assumed that households are endowed with ¯B , although we could reformulate the model to allow the government to issue this debt at t = 0 .

2.2. Households

Attime0,householdsfacethefollowingbudgetconstraint:

Q^BB+Q^SS+Q^DD+X≤ ¯Y+Q^BB¯. (6)

Theybegintimet=0withan endowmentY¯ ofgoodsandwithgovernmentbondsB¯,andthelattercanbetradedatprice Q^B.Theycanusetheseresources toconsumeXortoinvestinaportfolioofsecuritieswhichincludesgovernmentbondsB tradedatpriceQ^B,safeprivatesecuritiesSissuedbytheintermediariesatpriceQ^S,andriskyprivatesecuritiesDissuedby intermediariesatpriceQ^D.

Wemodelthedebtofgovernmentandfinancialintermediariesaszero-couponsecuritieswithafacevalueofone.Gov- ernmentdebtandprivatesafesecuritiesarerisk-freeandthusalwayspayoneunitinboththehighandlowstate.Riskypri- vatedebt,however,hasapayoff ofoneinstatehandzeroinstatelbecauseitisfullydefaultedoninstatel.InSection6and intheAppendix,weshowthatourresultsareunchangedintwoframeworkswitharichermodelingofriskysecurities.The first framework considers also risky securities that are partially defaulted butinstead are liquid inthe low state. In the second framework,riskysecuritieshaveidiosyncraticpayoff, andsomeofthemarefullyrepaidinthelowstate;however, theydonotprovideliquiditybecauseofanadverseselectionproblem.

Attime1,consumptionC_handC_lmustbefinancedwiththedebtsecuritiespurchasedatt=0.Safedebtalwaysprovides liquidity services,whereas riskydebtprovidestheseservicesonlyinstate h.Therefore,purchasesofC_h andC_l aresubject tothefollowingliquidityconstraints:

C_h≤ B+S+D (7)

C_l ≤ B+S. (8)

Instateh,consumptionC_hcanbefinancedwithB,S,andDbecauseallthreesecuritiesarefullyrepaid.Instatel,consump- tionC_l canbefinanced onlywithBandSbecausesecuritiesD arefullydefaultedon.ConsumptionofgoodsX_h andX_l in period1issubjecttothefollowingbudgetconstraints:

X_h ≤ ¯Yh+B+S+D+



h− Ch− Th (9)

X_l ≤ ¯Yl+B+S+



l− Cl− Tl, (10)

inwhichT_handT_larestate-contingentlump-sumtaxesandh,larestate-contingentintermediaries’aggregateprofits.

Itisworth emphasizingthatconstraints(7)and(8)capturethespecialpropertiesthatsomedebtsecuritieshaveinthe modern financial system because ofthe liquidity services they provide.These securities havebeen broadlylabeled “safe assets,” andarecentliteraturehasmodeledthemasriskless(see,amongothers,(CaballeroandFarhi,2017;Diamond,2016;

Li,2017;Magilletal.,2016;Stein,2012;Woodford,2016)).However,asdiscussedbyGorton(2017),thehistoricalevidence showsthat debtsecuritiesthat provideliquidity servicesarenot necessarilyrisk-free. Insome countries,such astheU.S.

andthe U.K., theserisky andliquidsecurities havebeen issuedby private intermediaries, whereas governmentdebthas beenessentiallyrisk-free.Moreover,throughoutthehistoryoffinancialsystems,theseprivatedebtsecuritieshavetakenthe formofgoldsmithnotes,billsofexchange,bank notes,demanddeposits,certificates ofdeposit,commercialpaper,money marketmutualfundshares,andsecuritizedAAAdebt.

Consumptionandportfoliochoicesfollowfromthemaximizationof(1)undertheconstraints(6)–(10).Theoptimalcon- sumptionofC_h andC_lisgivenby

C_h= 1 1+

μ

h

C_l= 1 1+

μ

l

, (11)

respectively,where

μ

h and

μ

l aretheLagrange multipliersassociatedwiththe liquidityconstraints(7)and(8).Since

μ

h,

μ

l≥ 0,itfollowsthatC_h,C_l≤ 1and,thus,C_h=C_l=1atthefirstbest.

To conclude the characterization of the household’s problem, we derive the demand for government debt and in- termediaries’ debt. This demand is affected by the liquidity value provided by these assets, captured by the Lagrange multiplier

μ

^:

Q^B=Q^S=1+

(

¹−

π ) μ

h+

πμ

l, (12)

Q^D=

(

¹−

π )(

¹+

μ

h

)

. (13)

PrivatedebtDprovidesliquidityservicesonlyinstatehwhenitisnotdefaultedon.Animplicationof(12)and(13)isthat Q^B=Q^S≥ Q^D.Crucially,liquidityservicesprovidebenefitsnotonlytohouseholdsbutalsototheissuerofthedebtsecurity becausetheylowerborrowingcosts.Wereturntothispointlaterintheanalysis.

2.3.Financialintermediaries

There is an infinite number of smallfinancial intermediariesthat can choose the type of capital (safe capital, K^S, or riskycapital,K^D) inwhichto investandtherefore thetype ofdebt securityto supply,safeor risky.Since intermediaries aresmallandthus marginalwithrespectto thesupplyofeach market,they takeprices Q^S andQ^D asgiven. We assume thateachintermediary cansupplyonlyonetype ofsecurity, althoughagivensecuritycanbesuppliedbyinfinitelymany intermediaries.InSection6,weshowthatourresultsareunchangedinthecasethateachintermediarycaninvestinboth typesofcapital.

Intermediarieshavelimitedliabilityinperiod1.Asaresult,theydefaultontheirowndebtifthepayoff ofcapitalisnot sufficienttocoverthedebtobligations.

WebeginwiththeanalysisofintermediariesthatissuesafedebtS.Attimet=0,theyinvestinrisklesscapitalK^Ssubject tothebudgetconstraint

(

¹⁺

τ )

^{KS=QSS. (14)}

Aspreviouslydiscussed,weinterpret

τ

^{asacostofmonitoringthesafecapital.Attimet}=1,theirprofitsarenotcontingent ontherealizedstateofnatureandaregivenby



^Sh=



^Sl =K^S− S. (15)

SubstitutingK^Sfromthebudgetconstraint(14)into(15),theoptimalsupplyofsafedebtisnon-negativeinsofaras

Q^S≥

(

¹+

τ )

, (16)

anditiszerootherwise.

IntermediariesissuingriskysecuritiesDinvestintheriskycapitalatt=0subjecttothebudgetconstraint

K^D=Q^DD. (17)

Att=1,profitsinstateharegivenby



^Dh=A_hK^D− D, (18)

whereasprofitsinstate l arezerobecausethepayoff of riskycapitaliszeroanddebtisfullydefaulted on,followingthe limitedliabilityassumption.Asaconsequence,thesupplyofriskydebtisnon-negativeinsofaras

Q^D≥ 1−

π

⁽¹⁹⁾

andiszerootherwise.

Intermediariesare freetochooseinwhichsecuritymarket toenter,andtheytake thisdecisionaccordingtothemaxi- mumprofitsthattheycanobtain.Thatis,theyenterthemarketofsafesecuritiesiftheexpectedprofitsinthatmarketare higherthanthoseofriskysecurities,andviceversa.

2.4.Government

Inthebaselineanalysis,weconsiderthesimplecaseinwhichthebalancesheetofthegovernmentiscomposedofonly liabilities,thatis,zero-couponbondsB¯.Thesebonds canbeinterpretedasTreasurydebtorthecentral bank’sreserves.In Section 5,we extendthe analysisby allowing thegovernment to issuemore debtat t=0,to purchase privately issued securities,possiblythroughthecentralbank,andtoguaranteethedebtissuedbyprivateintermediaries.

Recall that households are endowed, at t=0, with government debt B¯. These government liabilitiesare free of risk becausethegovernmentraisesenoughtaxestobackthem,namely,T_h andT_l inthehighandlowstateoft=1with

T_h=T_l=B¯. (20)

Analternativeinterpretationoftheabilityofthegovernmenttofullyrepayitsdebtcanbegivenifweextendtheanalysisto amonetaryeconomywithaconstantpricelevelinwhichgovernmentdebtisbackedbythecentralbank’sliabilities.Indeed, thecentralbank’sliabilitiesdefinetheunitofaccountforthemonetarysystemandarethusfreeofriskbydefinition;that is,the centralbankcanrepayitsliabilitiesby “printing” newreserves. InthepolicyanalysisofSection5,we willimpose somerestrictionsontheabilityofthegovernmenttoincreasetaxesT_handT_l.

3. Equilibriumwithnomonitoringcosts

Wenowsolvefortheequilibriuminthebenchmarkscenarioinwhich therearenomonitoringcosts toinvestinrisk- freeprojects,thatis,

τ

=0.Despitetheliquidityconstraint, thefirstbestcanbeachievedthrougheitherprivateorpublic liquidity.

Tosolvefortheequilibrium, wefirst notethat freeentryabates tozeroallprofitsandimpliesthatthe supplyofsafe andriskydebtisnon-negativeattheirrespectiveprices:

Q^S=1+

τ

^{, (21)}

Q^D=1−

π

^{. (22)}

Thenext propositionshowsthat inequilibrium,safesecuritiesare suppliedina sufficientquantityto satiatethedemand forliquidity.

Proposition1. Inthemodel with nomonitoringcosts (

τ

=0), thereis complete satiationof liquidity,

μ

h=

μ

l=0, andcon- sumptionisatthefirstbest,C_h=C_l=1.Thequantityoffinancialintermediaries’safedebtisgivenby

S≥ max



1− ¯B,0



, (23)

whichisissuedatthepriceQ^S=1.

Theeconomy achievesthefirst bestbecausethesupplyofsafeassetsissufficiently large.Thiscanbeachievedintwo ways.IfB¯≥ 1,the governmentachieves thefirst bestby supplying a largequantity of publicliquidity,which can be in- terpreted asa wayto implementtheFriedmanrule.IfinsteadB¯<1,private moneyissuedbyintermediariesiscrucialto complementthesupplyofpublicliquidityandachievethefirstbest.Weelaboratemoreonthesecondcase.

When B¯<1, theefficiencyresultofProposition 1isa directimplicationof thecompetitionmechanismofthe model, whichallowsfinancial intermediariesto decidethetypeofdebttosupply.Tounderstandthispointandprovethepropo- sition,supposebycontradictionthatthereisnosupplyofsafedebt.Instead,assumethatintermediariesonlyproviderisky assets.Asaresult,inthelowstate,riskysecuritiesdefault,andthusconsumptioncanbefinancedwithpublicliquidityonly.

Using(8)and(11),theLagrangemultiplieroftheliquidityconstraintinthelowstateispositive,

μ

l =1

B¯− 1>0, (24)

andthusthereisashortageofliquidityinthatstate.Incontrast,equilibriuminthemarketofriskysecurities,whichrequire both(13)and(22)tohold,impliesthattheLagrangemultiplieroftheliquidityconstraintiszerointhehighstate,

μ

h=0. That is, the supply of risky securities is large enough to satiate liquidity needs in the high state, and thus there is no shortageofliquidity inthat state.Nowconsideragenericintermediary decidingwhichsecuritytoissue.Supposethatthe intermediarychoosestoissuesafedebt,whichneverdefaults.Consumersattachahighvaluetosafesecuritiesbecausethe liquiditypremiuminthelowstateispositive;thishighvalueisreflectedinthepriceQ^S=(¹+

πμ

l)^{thattheyarewilling} to pay. The high Q^S impliesthat the intermediary can borrow ata lower cost and, thus, its profits are positive in both states:^S_h=^S_l =

πμ

l>0.Thus,issuingsafesecuritiesSisprofitable.Thisresultcontradictstheinitialconjecturethatan equilibriumexistsinwhichsafedebtisnotsuppliedbyanyintermediary.

To sumup, intermediaries supplysafe private securitiesup tothe point at whichthe liquidity premium is driven to zeroinallstates,

μ

h=

μ

l=0.Thatis,freeentryintothemarketensures thatallrentsare eliminated.Thesupplyofsafe securitiesisenoughtocomplementtheamountofpublicliquidity(asdescribedby(23))andreachthefirstbest,C_h=C_l=1. Moreover,thesupplyofriskysecuritiescanbepositiveinequilibrium,andtheirpriceisjustgivenbythepresentdiscounted valueoftheirexpectedpayoffs.However,thesupplyoftheseassetsisirrelevantforwelfare.

Thissection isclosedbycomparing Proposition1withsomerelatedliterature thatstudiesliquidity.Inversionsofthe (LagosandWright,2005)modelinwhichphysicalcapitalcanbeused forpayment(e.g.,Geromichalosetal., 2007;Lagos andRocheteau,2008), asufficientlylargesupplyofcapitalsatiatesthedemandforliquidity.TheresultofProposition1is similarinspirit.Eventhoughcapitaldoesnotprovideliquidity directlyinourmodel,itisused byintermediariestoback theirsupplyofprivatemoney.Inaddition,given

τ

=0,intermediarieschoosetoholdsafecapitalasbacking.Thus,thekey differencewiththisliteratureisthat intermediariesendogenouslychoosetheamountandriskinessofcapitalthatisused asbackingandthusthe riskinessofprivate money.Thisresultreflects thesimilaritiesbetweenour approachandthatof GeanakoplosandZame(2002,2014)becausethephysicalcapitalheldbyintermediariesinourmodelservesthesamerole ascollateralintheirmodel.

4. Equilibriuminthefullmodelwithmonitoringcosts

Wenowturntotheanalysisofthefullmodelinwhichintermediariesfaceapositivemonitoringcosttoinvestinrisk- free projects,

τ

>0. The main resultof thismore general framework is that risky securities—thosethat default andlose liquidity inthelow state—are nowsuppliedby intermediaries,andthustheamount ofprivately issuedliquidity islower thaninthebaselinemodelwithnomonitoringcosts.Inaddition,dependingonpoliciesandparameters,riskyandsafedebt cancoexistinequilibrium;whenthatisthecase, someintermediariessupplyriskydebt,whereasotherssupplysafedebt, eventhoughtheseintermediariesareexanteidentical.

Wepresenttheresultsintwosteps.Firstitisshownthattheequilibriummustbecharacterizedbyapositivesupplyof riskyassets,andthenthefullequilibriumischaracterized.

Proposition2. Assumethatfinancial intermediariesfaceaper-unitcost

τ

>0tomonitorsafecapital.Ifanequilibriumexistsin whichintermediariessupplydebtsecurities,thentheremustbeapositivesupplyofriskydebtsecurities,thatis,D>0.

Ifan equilibriumexists inwhichintermediariesare activeandissuedebt,threescenarios arepossible:eitherallinter- mediariesissuesafedebt,orall intermediaries issueriskydebt, orsome intermediariesissuesafe debtandsome others issueriskydebt.Thus,wecanproveProposition2byshowingthatthescenarioinwhichallintermediariesissueonlysafe debtSisnotanequilibrium.Weproceedbycontradiction.Supposethatallintermediariesissuesafedebtinequilibrium.In thiscase,equatingdemand(12)andsupply(21),itfollowsthat

μ

h=

μ

l=

τ

>0.Tooffsetthemonitoringcost,theliquidity premiumon safedebt mustbe positive; ifthe liquidity premiumwere zero,intermediaries wouldmake negativeprofits becauseofthecost

τ

^{.Notethatapositiveliquiditypremiumisassociatedwithalevelof}consumptionbelowthefirstbest insome state. Furthermore,thefact that there areonly safesecurities thatare equally liquidinboth statesimpliesthat consumptionisequalizedacrossstates.Therefore,C_h=C_l<1; inparticular,C_h<1and(11)implythattheLagrangemulti- plieroftheliquidityconstraintinstatehisalsopositive:

μ

h>0.We cannowidentifya profitabledeviationthatleadsus toconclude thatthe scenario withonlysafe debtcannot be an equilibrium. Given

μ

h>0,households are willing topay aliquiditypremiumonasecuritythat relaxestheliquidity constraint(7)inthehighstate.Nowconsideran intermediary thatissuesriskydebtD.Thisintermediaryearnspositiveprofitsinthehighstate(andzeroprofitsinthelowstate)because riskysecuritiesincludealiquiditypremium:



^Dh=

μ

hD>0. (25)

Thus,theintermediaryhasanincentivetodeviateandissueriskydebt.

Moregenerally,thepreviousanalysiscanbeextendedtoshowthatanyscenarioinwhich

μ

h>0cannot beanequilib- riumbecausetherewouldexistprofitabledeviationstoincreasethesupplyofriskysecurities.Thus,theLagrangemultiplier inthehighstatemustbezeroinequilibrium,

μ

h=0.

Wenow characterizethe full equilibrium. Wefocus onthe caseinwhich thegovernmentissues alimited amountof publicliquidity(i.e.,B¯<1),andwereturntotheanalysisofgovernmentpolicyinSection5.

Proposition3. Iffinancialintermediariesfaceaper-unitcost

τ

>0tomonitorsafecapitalandthegovernmentissuesdebtB¯<1, then:

(1) Inthehighstate,thereisfullsatiationofliquidity,andthusC_h=1and

μ

h=0,whereasinthelowstate, C_l=max

 π

π

⁺

τ

^,B¯



<1 and

μ

l=min

 τ π

^,

1 B¯− 1



. (26)

(2) Thepriceandsupplyofsafesecuritiesare Q^S=

(

¹⁺

πμ

l

)

^{>1, S=max}

 π

π

⁺

τ

^{− ¯B,0}



, (27)

andthepriceandsupplyofriskysecuritiesare

Q^D=

(

¹⁻

π )

^{<1, D≥ 1− ¯B− S>0. (28)}

(3) ThegovernmentimposestaxesT_h=T_l=B¯att=1.

Theproposition can be provenby noting that theallocation in thepropositionis optimalfor all private agentsgiven prices,isfeasible forthegovernment,andisconsistent withmarket clearing.Thatis,the allocationandprices(i) satisfy feasibility andoptimality for households, (6)–(13), (ii) satisfy feasibility andoptimality for intermediaries, (14)–(19), (iii) satisfythegovernmentstate-contingentbudgetconstraintsatt=1,(20),and(iv)satisfymarketclearingforsecuritiesD,S, andBatt=0,andforgoodsatt=0andineachstateatt=1.

AsanimplicationofProposition2,theequilibriumdisplaysapositivesupplyofriskysecurities.Note,however,thatthere isroomforsafedebttobesuppliedinequilibrium, inadditiontoriskydebt.Indeed,withriskydebt,liquidityislowerin thelowstate incomparisontothehighstate.Asa result,securitiesthatprovideliquidity inthelowstate willtradeata premium.Ifthispremiumislargeenough tocoverthecost

τ

^,intermediariesissuesafesecurities.Whetherthe premium on safeintermediaries’ debtis large or not dependsin turn on the amountof public liquidity. Alarge supply of public liquidityimpliesalowliquiditypremiumonsafedebt(recallthatpublicliquidityisrisk-free);thus,issuingsafedebtisnot profitableforintermediaries.Thatis,a sufficientlyhighlevelofpublicdebtcrowdsouttheproductionofprivately issued safemoneyby influencing theliquidity premium ondefault-free obligations.Incontrast,a low supplyof publicliquidity createsaprofitableopportunityforintermediariestoissuesomesafedebt.

ThepropositionshowsthatthepriceofsafedebtSisgreaterthanone;thatis,securitiesSareissuedaboveparandpay anegativereturn.Thisresultfollowsfromthefactthatwehavenormalizedthediscountfactorbetweent=0andt=1to oneandthat securitiesSincludealiquiditypremium.Ifweinsteadextendthemodeltoallowforadiscountfactor

β

<1, households’requiredreturnonnon-liquidassetswouldbe1/

β

>1.Thus,thepriceoftheliquidsecuritiesSwouldbeQ^S>

β

and,possibly,lessthanone.

Proposition3hasimplications forcharacterizinghowa liquidity crunchoccursinourmodel.Thishappensinthelow statebecauseriskysecuritiesdonothaveappropriatebackinginthatstateand,thus,losetheirliquidityvalue.Sincethere

isashortageoftheonlyassetsthat areliquidinthelow state(i.e.,safeassets),thedemandforgoodsC_l dropsbecauseof theliquidityconstraint(8),andthusconsumptionC_l decreasestoo.

Weconcludewithabriefcomparisonwiththeliteraturethatstudiestheliquidityofprivatemoneyandcrises.Incontrast tomanypapersinwhichprivatemoneyisrisk-free,weallowforintermediaries’defaultondebtand,thus,riskassociated withthepayoff of privatemoney. Theliterature includesworksexpounding theideaofsafeassetsfollowing(Gorton and Pennacchi,1990)(e.g.,Stein,2012),paperscastintheNewMonetaristframeworkfollowing(LagosandWright,2005)(e.g., Williamson, 2012), andanalysesofpublic finance(e.g.,Angeletosetal., 2016). Allowingforthepossibility ofdebtdefault iscrucialnotonlybecauseit bettercapturesthenarrativeofthe 2007–2008financial crisis,butespeciallybecauseitlays the foundations fornovel policy results.For instance,questions relatedto the role of governmentdeposit insurancecan naturallybestudiedonlyifdefaultarisesinequilibrium.Inaddition,we includeaggregate risk,whichallowsustoderive policyimplicationssuchastheroleoffiscalcapacityincrisistimes.

5. Governmentintervention

Thepossibilityofliquiditycrisisthatarisesinthelaissez-faireequilibriumopensupapossibleroleforgovernmentinter- vention.Theamountofliquidityislargeenoughonlyinthehighstate,whereastheeconomyexperiencesaliquiditycrunch inthelowstate.

Westudygeneralgovernmentpoliciesrelatedtodebtissuance,theactivemanagementofthebalancesheetofthecentral bank,andtheregulationoffinancialintermediaries.

As a first step, we consider a large supply of public liquidity backed by higher taxes at all times. This intervention entirelycrowdsout theproductionofsafeprivate debtbutnonethelessachievesthefirstbest.Twopoliciesarepresented thatimplementthefirstbestevenifthegovernmentfacesalimitonaveragetaxation:assetpurchasesbythecentralbank andactuariallyfairdepositinsurance.Thesepoliciesexploitthebackingprovidedbyintermediariesingoodtimesand,thus, requiregovernmentbackingonlyinbadtimes.Crucially,assetpurchasesanddepositinsuranceareequivalent,inthespirit of Wallace (1981). The taxesrequired under the two policies are identical inall contingenciesbecause the consolidated balancesheetofthegovernmentandprivate intermediaries(i.e.,oftheagentsthat supplyliquidity)isidenticalunderthe twopolicies.

Under a more stringent limit on taxes in low states, government policies do not implementthe first best, but they nonethelessimprovewelfare.Theequivalencebetweenassetpurchasesanddepositinsuranceextendstothiscase. Inpar- ticular, withdeposit insurance, theequilibrium is characterized by the coexistenceof three typesof privately issued se- curities.In additionto safedebtSandriskydebt D,some intermediariesissueinsureddebt,which wedenoteDˆ.Allthe intermediariesareexanteidentical,andeach ofthem choosestoissueS,D,orDˆ.Moreover,intermediariesissuingDˆ and Dinvestinthesametypeofriskycapital.Thecoexistenceofinsuredanduninsuredintermediariesinthemodelresembles thecoexistenceofregulatedcommercialbanksandunregulatedshadowbanksinpractice.

Finally,westudyaregulationthatforcesallintermediariestoinvestinsafeprojects.Thispolicyreduceswelfarebecause issuingriskysecuritiesbackedbyinvestmentsinriskyprojectsallowsintermediariestoeconomizeonmonitoringcosts.

5.1. Optimalgovernmentpolicywithnolimitontaxes

Thissectioncharacterizestheoptimalgovernmentpolicywhenthereisnolimitontheabilitytoraiselump-sumtaxes.

WeamendthemodelbyallowingthegovernmenttoincreasethesupplyofpublicdebttoB≥ ¯B att=0.Westillassume thatthegovernmentrepaysthedebtwithtaxesT_handT_l att=1.Thedetailsofthisextensionareprovidedintheonline Appendix.

TheoptimalpolicyrequiresthatthegovernmentissuesalargesupplyofpublicdebtBand,thus,imposeslargetaxesT_h andT_l att=1tobackthedebt.Asaresult,householdscanattainthefirst-bestlevelofconsumptionusingpublicliquidity only.Thenextpropositionformalizesthisresult.

Proposition4. Iffinancialintermediariesfacea per-unitcost

τ

>0ofmonitoringsafecapital, theoptimalgovernmentpolicyis tosetB≥ 1,achievingthefirstbest.

The optimal policy requiresthe government tosatiate theeconomy with public liquidity,which isbacked by a large amountoftaxesimposedon householdsatt=1.Withnolimit onlump-sumtaxes, thegovernmenthasanadvantage in supplyingliquidity.Thereasonisthatintermediaries’safedebt,S,iscostlybecauseitrequiresmonitoring,whereasthegov- ernmentmoneyhasnocostsassociatedwithbackingthroughtaxes.Moreover,optimalissuanceofpublicliquidity entirely crowdsoutthesupplyofprivatelyissuedsafeassets.

TheresultofProposition4iswellknownandinlinewithFriedman’sproposal(Friedman,1960).Inthecontext ofthe analysisofpublicversusprivatemoney,asimilarresultisalsoobtainedbyRocheteau(2011).

ItisworthemphasizingthatProposition4reliesontwocriticalassumptions.First,thegovernmentisbenevolent.Second, thegovernmentdoesnotfaceanylimitonraisingtaxes.Nonetheless,evenwithalimitontaxes,weargueinSections5.2– 5.4thatotherpolicyinterventionsallowthegovernmenttoavoidoratleastmitigateliquiditycrises.

5.2.Optimalgovernmentpolicywithalimitonaveragetaxes:assetpurchasesbythecentral

Wenowturntotheanalysisofthesupplyofpublicliquidity whenthegovernmentfaces alimitonaveragetaxesthat itcancollectatt=1,whereaveragetaxesare(¹−

π

)^Th+

π

^Tl.Tokeeptheanalysissimpleandwithoutlossofgenerality, weassumethatthelimitonaveragetaxesis

(

¹⁻

π )

^Th+

π

^Tl≤ ¯B<1. (29)

Theboundin(29)impliesthatthegovernmentisrestrictedtokeepingaveragetaxesatorbelowtheno-interventionlevel.

Toseethis,notethat(¹−

π

)^Th+

π

^Tl=B¯intheequilibriumofProposition3.

Notwithstandingthelimitin(29),weshowthatanappropriatepolicyofassetpurchasesallowstheeconomytoachieve thefirst best,C_h=C_l=1.Our approachfollows (Magillet al., 2016), although public debthasno liquidity value in their modelandthus thewaythispolicy interventionachievesthefirstbestissomehowdifferent,aswe discussattheendof thissection.Undertheassetpurchasepolicy,thegovernmentsuppliesalargeamountofpublicmoney,B>B¯,andpurchases privateintermediaries’riskydebtthroughthecentralbank.Theriskydebtheldbythecentralbankpaysareturninthehigh state,allowingthegovernmenttoreducetaxesinthatcontingency.Instead,inthelowstate,theprivateriskysecuritiesare defaultedon, andthus governmentdebt requiresbacking throughtaxes. Thispolicy is relatedto thesecond proposalof Friedman(1960),who suggestedbackingthesupplyofinterest-bearingreserves(inourmodel,B)throughtheportfolioof assetsheldbythecentralbank(inourmodel,privateintermediaries’riskydebt).³

Att=0,thecentral bankpurchasesthequantity D^c ofriskysecuritiesandfinancesthesepurchases byincreasing the outstandingdebtfromtheinitiallevelB¯toB(i.e.,thenetdebtissuanceatt=0isB− ¯B).Thus,thebudgetconstraintofthe governmentis

Q^DD^c≤ Q^B



B− ¯B



. (30)

Att=1,thegovernmentrepaysitsdebtBusingtheproceedsearnedbyholdingtheassetsD^c,ifnotindefault,andtaxes:

B=T_h+D^c (31)

B=T_l (32)

inthehighandlowstate,respectively.

Toachievethefirstbest,thegovernmentmustissueanamountB=1ofpublicmoney.Ifthegovernmentisindeedable toimplementthefirstbest,thehouseholds’first-orderconditionsimplythatthepricingofpublicmoneyandriskysecurities isQ^B=1 andQ^D=1−

π

^{. Giventheseresults, thebudget constraintat t}=0,Eq. (30), impliesthat the governmentcan purchaseprivateriskydebtintheamountof

D^c=

(

¹− ¯B

)

(

¹−

π )

^{>0. (33)}

Giventhisresult,thestate-contingentbudgetconstraintsatt=1,Eqs.(31)and(32),implythattaxesare

Th=1− 1− ¯B

1−

π

⁽³⁴⁾

Tl=1, (35)

andthusaveragetaxesare(¹−

π

)^Th+

π

^Tl=B¯.Thatis,thelimitin(29)issatisfied.

Thenextpropositionsummarizesthisresult.

Proposition5. Assumethatfinancialintermediariesfaceaper-unitcost

τ

>0tomonitorsafecapitalandthegovernmentfollows a state-contingent taxruleT_h andT_l in the highand low state,respectively, subjectto the limitin (29). Thegovernmentcan achievethefirstbestbysupplyingpublicmoneyB=1,purchasingtheamountofriskysecuritiesD^cin(33),andsettingtaxesT_h andT_l to(34)and(35),respectively.

Thepropositionshowsthat thegovernmentcan achieve thefirstbest evenifthere isalimit onaverage taxes. Inthe highstate,risky assetsare fullyrepaid andthus providea backingforpublic liquidityB. Inthe low state,however, they aredefaultedonandthusprovideaninsufficientbackingforpublicliquidity.Therefore,taxesT_l mustbeincreasedtoback publicliquidity.

The requirementthat taxes should be increasedduring a crisis can be relaxedif we extendthe model to an infinite horizonand allow governmentdebt andtaxes to varyover time. Inthis case, however, the present-discounted value of taxesmustbe raised duringa crisis, even though thetaxes collectedduringa crisis donot haveto increase. Insuch an infinite-horizonformulation,the governmentcanincrease debtto B>1in theeventofa crisis tokeep taxeslowat that timeandrepaythehigherdebtbyincreasingfuturetaxeswhentheeconomyrecoversfromthecrisis.

3 Even though our policy proposal is related to that of Friedman (1960) , it is slightly different because the original proposal considered only the possibility of investing in safe securities.

Thesolutionproposed inthissectionhaspolicy-relevantimplications giventhe unconventionalassetpurchasesunder- takenby manycentral banksaround the world.The rationale andduration for thesepurchases havebeen subjectto an extensivedebate.Ouranalysisunderlinesthat,evenduringnormaltimes,centralbanksshouldcontinuetoholdprivatese- curitiesforthepurposeoffulfillingtheliquidityneedsoftheeconomyandreducingthetaxburden.Thisviewisincontrast totheconventionalonethatprescribesthatcentralbanksshouldholdTreasurybills;underthisapproach,noreductionin thetaxburdenwouldbe possible.Inadifferentframework,(Magilletal.,2016)alsoargueforthecontinuation ofuncon- ventionalassetpurchasestonormaltimesasawaytoachieve efficiency,whichintheir caseisrelatedto theincreasein thefundschanneledtoinvestments.

Next, we discuss therobustness of Proposition 5. Evenif the resultof Proposition 5 might not be identical in some extensions of ourmodel, we argue that the spirit of the exercise is preserved. For instance, ifintermediaries’ defaultis costly(i.e.,deadweightlossesareassociatedwithbankruptcyprocesses),itmightbeoptimalforthegovernmenttohavea lowerdemandforD^c toavoidtoomanylosseswhenintermediariesgobankrupt. Asaresult,theoptimalsupplyofpublic liquiditymightbesmallerandtheallocationC_h=C_l=1wouldnotbeoptimal.Nonetheless,thespiritofProposition5would beunchanged.ThemainimplicationofProposition5isthatthegovernmentshouldactivelyengageinthesupplyofpublic moneyusingprivately issuedintermediaries’debtD^caspartialbacking. TheoptimalholdingofD^c ismostlikelynotzero forreasonableextensionsofourmodel.

Anotherconstraintthatcould limitthe purchasesofprivaterisky debtarisesifwe separate thecentralbankfromthe Treasury.Bypurchasingriskysecurities,thecentralbankfacesincomelossesinthelowstate,wheretheriskyassetsdefault, whilestill havingtopayinterestonreserves. Therefore,thecentralbankneeds tobe recapitalizedbytheTreasury.Ifthe Treasury’ssupportisnotautomatic,an additionaltrade-off betweenmaintainingpricestabilityandachievingtheefficient supplyofliquiditycouldemerge.

5.3. Optimalgovernmentpolicywithalimitonaveragetaxes:depositinsuranceandgovernmentguaranteeprograms

Inthissection, wepropose an alternativegovernmentpolicy thatallows theeconomyto achievethe firstbesteven if thegovernmentissubjecttothelimitontaxesinEq.(29).Werefertothispolicyasdepositinsurance,eventhoughitcan beinterpreted,moregenerally,asanyprogramthatguaranteestheliabilitiesoffinancialintermediaries.First,thepolicyis studiedinthecontextofthemodelandthensomelimitationsandextensionsareinvestigated.

Consideradepositinsuranceschemethatisactuariallyfair.Thegovernmentchargesintermediariesafeefortheinsur- anceand,onaverage,thepolicydoesnotprovideanysubsidytointermediaries.

Debtofinsuredintermediariesthat investinriskyprojects isdenotedasDˆ todistinguishitfromtheriskydebtDthat arisesintheequilibriumofProposition3.Underdepositinsurance,intermediaries’debtDˆissafefromthepointofviewof householdsandthereforealwaysprovidesliquidityservices,eventhoughintermediariesinvestinriskyprojects.Inthelow stateatt=1,whenthepayoff ofintermediaries’investmentsiszero,thegovernmentprovidestheinsurancepaymentwith atransfertointermediaries,whichinturnisusedtorepaythedebttohouseholdsinfull.Inthehighstate att=1,when intermediaries’projectsproduceapositiveoutput,thegovernmentchargesaproportionalfeetointermediaries.

Undertheassumptionthat theonlylimitfacedbythegovernmentisonaveragetaxes,weconjecture(andlaterverify) thattheonlytypeofdebtthatisissuedbyintermediariesisinsureddebtDˆ.Att=0,thebudgetconstraintofintermediaries thatinvestinriskyprojectsissimilarto(17),butnowDˆreplacesD:

K^Dˆ =Q^DˆDˆ, (36)

whereK^Dˆ denotestheinvestmentinriskyprojectsandQ^Dˆ isthepriceofinsureddebt.Households’demandforDˆ implies that the price of Dˆ is Q^Dˆ=Q^S=Q^B because insured debt is riskless and thus a perfectsubstitute for the debt of safe intermediariesandthegovernment.

Att=1,intermediaries’profitsare



^Dh^ˆ =max



0, K^Dˆ

(

^A−

λ

h

)

− ˆD



(37)



^Dl^ˆ =max



0, K^Dˆ

(

⁰−

λ

l

)

− ˆD



, (38)

where

λ

h>0and

λ

l<0 denotetheproportionalfee andtransferfromthegovernmentinthehighandlowstate, respec- tively,whilethelowerboundonprofitsfollowsfromlimitedliability.

Wefocusonthecaseinwhichthegovernmentdoesnotalterthesupplyofpublicdebtwithrespecttotheinitiallevel, andthusB=B¯.Thebudgetconstraintatt=1is

B¯=T_h+

λ

hK^Dˆ (39)

B¯=Tl+

λ

lK^Dˆ (40)

inthehighandlowstate,respectively.Notethatthegovernmentincreasestaxesinthelowstate,T_l,tofulfillitsguarantee ofintermediaries’debt.

The equilibrium under deposit insurance can be characterized as an equivalence proposition, in the spirit of Wallace (1981).If an equilibrium exists under theasset purchasepolicy ofProposition 5, the same consumption alloca- tionandpricescanbesustainedunderapolicyofdepositinsurancewiththesametaxes.Thelogicoftheproofisbasedon thefactthattheconsolidatedbalancesheetofthegovernmentandprivateintermediaries—thatis,oftheagentsthatsupply liquidityintheeconomy—isthesameunderbothpolicies.

Tosolvefortheequilibrium,considerthegovernmentthatcollectsthesametaxesT_h andT_lasundertheasset-purchase policystudiedinSection5.2,whicharegivenbyEqs.(34)and(35).Wecanuse(39)and(40)tosolveforthevaluesofthe feeandtransfertointermediaries:

λ

h=1− ¯B K^Dˆ

π

1−

π

⁽⁴¹⁾

λ

l=−1− ¯B

K^Dˆ . (42)

Notethat theaveragepayment tointermediaries,whichis definedby (¹−

π

)

λ

h+

πλ

l, iszero,andthuson average,the governmentdoesnotprovideanysubsidytointermediaries.

Wecan then show that the total amount ofliquidity in theeconomy, B¯+Dˆ, allows theeconomy to achieve the first best.Todoso,we plugthevalue

λ

l intotheexpression forprofitsinthelowstate, (38),andwesetprofitsequaltozero becauseoffreeentry,obtainingB¯+Dˆ=1.NotethatbothpublicliquidityB¯andinsuredprivate debtDˆprovideliquidityin bothstates,andthushouseholdscanachievethefirst-bestlevelofconsumptionC_h=C_l=1.Profits inthehighstate,(37), andthebudgetconstraintofintermediariesatt=0,(36),canbeusedtosolvefortheequilibriumvalueofK^Dˆ andDˆ.

Crucially,since weare usingthesametaxesthatthe governmentimposes underassetpurchases,thelimit onaverage taxesin(29)issatisfiedevenunderdepositinsurance.Thenextpropositionsummarizesthisresult.

Proposition6. AssumethatapolicyoftaxesT_h andT_l,assetholdingsofthecentralbankD^c,bondsupplyB=1,andconsump- tionC_h=1andC_l=1arepartofanequilibrium.Thereexiststate-contingentproportionaltaxesonintermediaries

λ

hinstateh and

λ

l instate lthatsustainthesameequilibriumwiththesamestate-contingent taxesT_handT_l andalower supplyofpublic liquidityB¯<1andnosupplyofsafeorriskydebt,thatis,S=D=0.Underthetaxscheme

λ

hand

λ

l,thedebtofintermediaries thatinvestinriskyprojectsisriskless.

Inpractice,depositinsuranceistypicallyuptoalimit.Nonetheless,duringtheacutephaseofthe2008crisis,thedeposit insurancelimitwasincreasedinseveralcountries,andotherformsofgovernmentguaranteeswereintroduced.IntheU.S., theinsurancelimit wasincreasedfrom$100,000to$250,000.Moreover,theFederal DepositInsuranceCorporation(FDIC) set up the Temporary Liquidity Guarantee Program withthe objective of bringing stability to financial markets and the banking industry. The program provided a full guarantee of non-interest-bearing transaction accounts and ofthe senior unsecureddebtissuedbyaparticipatingentityforaboutayear.Takentogether,thesetwomeasuresdramaticallyincreased thefractionofliabilitiesofU.S.financialinstitutionsthatwereguaranteedbythegovernment.Similarpolicieswereadopted inothercountries,includingsomecasesinwhichthecoveragewasunlimited,suchasinGermany.

Weemphasizeagainthatthedepositinsuranceschemeprovidedbythegovernmentisactuariallyfairbecausetheaver- agetaximposedonintermediariesiszero:(¹−

π

)

λ

h+

πλ

l=0.Aslongasthegovernmentrunsacorrectlypriceddeposit insuranceprogram,itshouldbeabletoavoidanymoralhazardthatmightinsteadarisewhentheinsuranceissubsidized.In thissense,ourresultssuggestthatregulatoryagenciessuchastheFDICshouldlinkthedepositinsurancepremiumtoeach intermediary’srisk ofdefault. Thisapproach wouldnotonlyreduce oravoidmoral hazard butalsosupporttheadequate levelofliquidityintheeconomy.

Finally,wepointoutthatourmodelprovidesarolefordepositinsurancethatisdifferentfrom,althoughcomplementary to,thestandardmotivationrelatedtobankruns.Following(DiamondandDybvig,1983),thebankrunsliteraturehighlights theimportanceofdepositinsuranceasatool toeliminatebadequilibriadrivenbypanics. Inourmodel,crisesare driven byfundamentalshocks,anddepositinsuranceplaysakeyroleinreducingthenegativeimpactofsuchshocks.

5.4.Optimalgovernmentpolicywithastate-contingentlimitontaxes

InSections5.2and5.3,welimitedtheactionofthegovernmentbyimposingaconstraintonaveragetaxes.Despitethe limit,thegovernmentisableto implementpoliciesthat achievethefirstbest.Thesepolicies,though,requiresettinghigh taxesinthelowstatetooffsetthelackofprivatebackingwithgovernmentbacking.

Inthissection, wetake thepolicyanalysisone stepfurtherbyimposing anadditionallimit ongovernmentactions.In additionto thelimit on averagetaxesin (29), we assume that thegovernmenthas tosatisfy a state-contingentlimit on taxation,that is,T_h,T_l≤ ¯T,whereT¯ satisfiesB¯<T¯<1.Astheanalyses inSections5.2and5.3show,thestate-contingent limitisbindinginthelowstate,andthuswesimplyreformulatethelimitasT_l≤ ¯T.

UnderthelimitT_l≤ ¯T,thefirstbestcannotbeachievedbygovernmentpolicy.Nonetheless,weshowthatassetpurchases anddepositinsurancearestilloptimal,andtheequivalencebetweenthesetwopoliciesshownbyProposition6isrobustto theextensionofthissection.