Private and Public Liquidity Provision in Over-the-Counter Markets

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Private and Public Liquidity Provision in Over-the-Counter Markets David M. Arseneau, David E. Rappoport, and Alexandros P. Vardoulakis 2017-033 Please cite this paper as: Arseneau, David M., David E. Rappoport, and Alexandros P. Vardoulakis (2017). “Private and Public Liquidity Provision in Over-the-Counter Markets,” Finance and Economics Discussion Series 2017-033. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2017.033. 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.

Private and Public Liquidity Provision in Over-the-Counter Markets* David M. Arseneau David E. Rappoport Alexandros P. Vardoulakis FederalReserveBoard FederalReserveBoard FederalReserveBoard March29,2017 Abstract We show that trade frictions in OTC markets result in inefficient private liquidity provision. We develop a dynamic model of market-based financial intermediation withatwo-wayinteractionbetweenprimarycreditmarketsandsecondaryOTCmarkets. Private allocations are generically inefficient because investors and firms fail to internalize how their actions affect liquidity in secondary markets. This inefficiency can lead to liquidity that is suboptimally low or high compared to the second best, providing a rationale for the regulation and public provision of liquidity. Moreover, our model characterizes a transmission channel of quantitative easing or tightening operatingthroughliquiditypremia. Keywords: Liquidity provision, market liquidity, over-the-counter markets, OTC, quantitativeeasing,quantitativetightening,monetarypolicynormalization. JELclassification: E44,G18,G30. *WearegratefultoRegisBreton,MaxBruche(discussant),FrancescaCarapella,GiovanniFavara,Zhiguo He(discussant),NobuKiyotaki,MichalKowalik(discussant),KonstantinMilbradt,CeciliaParlatore,Lasse H.Pedersen,SkanderVandenHeuvel,ChrisWaller,andseminarparticipantsatEconometricSocietyWorld EconomicCongress,SED,CowlesGeneralEquilibriumConference,EEA,EFA,FederalReserveDayAhead Conference, LACEA, IMF, Society for Advancement of Economic Theory, Federal Reserve Board, Federal ReserveSystemConferenceonFinancialStructureandRegulation,FederalReserveSystemConferenceon Macroeconomics,StLouisFed,BanquedeFrance,AthensUniversityofEconomicsandBusiness,University ofPiraeus, BankofGreece, andUniversityofChileforcomments. Allerrorshereinareours. Aprevious version of this paper circulated under the title “Secondary Market Liquidity and the Optimal Capital Structure”. The views expressed in this paper are those of the authors and do not necessarily represent thoseofFederalReserveBoardofGovernorsoranyoneintheFederalReserveSystem. Emails: david.m.arseneau@frb.gov,david.e.rappoport@frb.gov,alexandros.vardoulakis@frb.gov. 1

1 Introduction Public liquidity provision is warranted when the private sector is unable to produce enough liquid assets to diversify aggregate liquidity risk (Holmstrom and Tirole, 1998). Alternatively, public liquidity provision is justified when liquidity shortages arise, for example in fire sales (Allen and Gale, 1994, 2004, Lorenzoni, 2008, Schleifer and Vishny, 2011, He and Kondor, 2016, and others). But, is there a role for the public provision (or withdrawal)ofliquiditywhenliquidassetsareabundantandtheprospectsoffiresalesunlikely? Thisquestionisparticularlyimportantintheaftermathoftheglobalfinancialcrisis, whereunconventionalmonetarypoliciessuchasquantitativeeasing(QE)—implemented by central banks well after the onset of the crisis, at a point when liquidity shortages had moderated—are thought to have implications for market liquidity (Krishnamurthy and Vissing-Jorgensen,2011). To articulate our argument we develop a dynamic model of market-based financial intermediation which features a two-way interaction between primary credit markets and secondary OTC markets. On the one hand, long-term bonds issued by firms in the primary market are retraded in an OTC market, thus secondary market liquidity affects investors’ supply of credit to firms.1 On the other hand, the demand for credit, i.e., the issuanceofilliquidbonds,affectssecondarymarketliquiditythroughthecompositionof investors’ portfolios as they must allocate limited financial resources between liquid and illiquidassets. Itisthistrade-offbetweencreditprovisionandliquidityprovisioninOTC marketsthatisthenovelfeatureofouranalysis. Thekeyfinancialfrictionofthemodelisthepresenceofsearchfrictionsinthe secondary OTC market. The importance of search frictions for OTC markets is grounded in both theempiricalevidence,whichsuggeststhattheyarethemaindriverofilliquidityinOTC markets for bonds (Edwards et al., 2007, and Bao et al., 2011), and the large theoretical literature modeling OTC markets with search frictions (Duffie et al., 2005, Lagos and Rocheteau, 2009, He and Milbradt, 2014, Atkenson et al., 2015, and others). In addition, we model the interaction of firms and investors in the primary credit market following the costly state verification (CSV) framework (Townsend, 1979, Gale and Hellwig, 1985, Bernanke and Gertler, 1989), such that debt emerges as the optimal contract for firms’ 1We focus on credit provision through capital markets, which has become increasingly important for non-financial firms and households in the U.S. The fraction of credit that is provided by the market has increasedinthelast25yearsandstandsatover60percentfornon-financialfirmsandatabout50percent for households. This evidence is obtained from the Financial Accounts of the United States and considers the fraction of commercial paper, municipal securities and loans, and corporate bonds in total credit marketinstrumentsfornon-financialfirmsaswellasthefractionofmortgagesandconsumercreditthatis securitizedforhouseholds. 2

funding.2 In our framework, the trade and agency frictions interact to determine the creditsupplyandmarketliquidityinequilibrium. Our model has three periods and two types of agents: firms and investors. In the first period, firms need financing for productive projects that pay off in the last period. Funding is obtained from investors who only value future consumption but are subject to preference shocks: a fraction becomes impatient and would like to consume in the interimperiod,whiletherestremainpatientandarewillingtobuytheassetsofimpatient investors. However, asset exchange between patient and impatient investors takes place inanOTCmarketcharacterizedbysearchfrictions,socounterpartiesareonlyfoundwith some probability. These probabilities are determined endogenously as a function of the ratio of liquid assets relative to illiquid assets available for trade. Thus, our concept of market liquidity is one of market thickness. The liquidity of the OTC market introduces a liquidity premium in firms’ external financing and, thus, affects the supply of credit througha liquiditypremiumchannel thatoperatesthroughthecostofcredit. However, our model features an additional interaction between credit and OTC markets. The quantity of bonds issued reduces the liquid resources in the financial sector, which can be used to provide liquidity in secondary markets. This is reflected in the portfolios of investors who allocate limited financial resources between illiquid bonds and liquid assets. Other things equal, market liquidity is lower as the composition of investors’ portfolios shift toward illiquid bonds. Thus, bond issuance affects secondary marketliquiditythrougha liquidityprovisionchannel. TheliquiditypremiumchannelandtheliquidityprovisionchannelgenerateaninterplaybetweenprimarycreditmarketsandsecondaryOTCmarkets,asillustratedinFigure 1. These channels work in opposite directions and distort private decisions. Firms issue more debt exactly when the liquidity premium is low, which is the case when market liquidity is high. But, as investors hold more of this debt, they shift their portfolios away from liquid assets, thereby reducing secondary market liquidity. In our existence proof we establish that the two effects jointly determine the unique equilibrium in the primary and secondary markets. The liquidity premium channel dominates, while the liquidity provision channel acts as an automatic stabilizer such that an improvement or a deterioration in market liquidity cannot perpetually increase or decrease bond issuance. Hence, ourmechanismisdifferentfrommodelswhichfeatureanamplificationbetweenfunding andmarketliquiditystemmingfrombindingcollateralconstraintsandlimitstoarbitrage 2ThespecificnatureoftheagencyfrictionintheCSVframeworkisnotcrucialforourresults. Whatis keyisthatthereisadownwardslopingdemandforcreditintheprimarymarket. Thisisexpectedtobea gooddescriptionofmarketsthatsupportcreditintermediation. 3

Liquiditypremiumchannel Primary Secondary Credit Market OTC Market Liquidityprovisionchannel Figure1: Feedbackloopbetweenprimaryandsecondarymarketforcorporatedebt (e.g.,BrunnermeierandPedersen,2009,GrombandVayanos,2002). Moreover,theinteractionbetweenthesetwochannelsisasourceofinefficiency. Search frictions and the dependence of market thickness on the abundance of liquid resources relative to the supply of bonds gives rise to a pecuniary externality for firms and a congestionexternalityforinvestors. Thepecuniaryexternalityworksthroughtheliquidity premiumandreflectsthefactthatfirmsfailtointernalizehowtheirbondissuanceaffects their external financing costs. The congestion externality operates through the trading probabilities, as investors’ fail to internalize how their portfolio choices affect the ease with which they can trade in the secondary market. Market liquidity is (generically) either suboptimally low or high. When liquidity is lower than the social optimum, firms are over-leveraged and write excessively risky debt contracts. This over-abundance of long-term bonds leads to an under-provision of liquidity in the secondary market. The oppositeistruewhenliquidityissuboptimallyhigh. Weexaminetheabilityofasocialplannertoregulatetheprivateprovisionofliquidity toimplementtheconstrainedefficientequilibrium. Whenprivateliquidityisinefficiently low,optimalregulationcallsforataxonleveragetorestrictilliquidbondissuancebyfirms coupled with a subsidy on storage to provide an incentive for investors to hold a more liquid portfolio. Because of the congestion externality, a change in liquidity can generate exantewelfaregainstoinvestors.3 Thesewelfaregainsallowtheplannertoreducefirms’ funding costs and increase their profits despite having to operate on a smaller scale. In contrast, when private liquidity is inefficiently high a leverage subsidy combined with a storage tax are able to align the private and social incentives. Regardless of whether 3When private liquidity is inefficiently low (high), ex ante identical investors making the portfolio allocationdecisionarebetteroffwithhigher(lower)liquiditybecausethegains(losses)toimpatientinvestors outweighthelosses(gains)topatientinvestors. 4

privateliquidityisinefficentlyhighorlow,optimalliquidityregulationallowsthefirmto internalizethepecuniaryexternalityinawaythatdoesnotmakeinvestorsworseoff. Theinteractionofthecongestionandpecuniaryexternalitiesiscriticalfortheefficacy ofliquidityregulation. Indeed,theplannereffectivelyexploitsthecongestionexternality tocreateanadditionalsurplusforinvestorsthatisthenredistributedbacktofirmsthrough morefavorablefundingcosts. Whenthecongestionexternalityisnotpresenttheplanner hasnomeansofachievingwelfare-enhancinginterventions. Inthiscase,privateliquidity coincideswithitsconstrained-efficientlevel. In addition to regulation, we also examine how the optimal management—provision or withdrawal—of public liquidity can alleviate trading frictions and improve economic efficiency. We focus on quantitative easing (QE) policies implemented by a central bank that uses liquid reserves to purchase less liquid assets from investors, yet quantitative tightening (QT) is expected to operate through the same mechanism. Through the lens of our model, any public policy that alters both public and private portfolios effectively shiftsliquidityriskbetweentheprivateandthepublicsector. Thistransferofliquidityrisk alterstheliquiditypremiawhich,inturn,influencessavingsandinvestmentdecisionsin therealeconomy(seeStein,2014,forageneraldiscussion). Our framework highlights the fact that public liquidity management is inherently different from liquidity regulation. Both policies affect the level of market liquidity, but whereasregulationtradesoffliquidityandcreditprovision,publicliquiditymanagement impliesthatpublicliquidityandcreditprovisionmoveintandem. Thisisbecauseliquidity management enhances the intermediation technology of the economy. The difference in the two policies opens the door for them to coexist. Indeed, our analysis shows that eitherQEorQTshouldbesupplementedwithoptimalregulationtogenerateevenlarger welfare gains. In this sense, liquidity regulation and liquidity management should be viewsascomplements,notsubstitutes,inthepolicytoolkit. Our analysis informs two related policy debates. On the one hand, despite their perceived efficacy the channels through which quantitative policies operate is still a matter of debate.4 The two leading conceptual explanations are the signaling and the portfolio balance channels. The former considers that QE programs signal to investors that short-term rates will be lower in the future, which depresses long-term rates through the expectationschannel. Thelatterconsidersthatthereisadownwardsloppingdemandfor the assets purchased by central banks, so these purchases contract the effective supply 4Gagnonetal. (2011)andKrishnamurthyandVissing-Jorgensen(2011)presentevidenceoftheefficacy of the program implemented by the Federal Reserve in reducing Treasury yields and mortgage rates. Similarly,Joyceetal. (2011)andChristensenandRudebusch(2012)presentevidenceoftheefficacyofthe QEprogramimplementedbytheBankofEngland. 5

of these assets, increasing their price and lowering their returns. This idea that can be traced back to Tobin (1969) can be rationalized if the market for reserves is segmented andtheassetspurchasedbycentralbanksarenotperfectlysubstitutable(Christensenand Krogstrup, 2016). In contrast, our paper argues that the reason that the financial sector has a downward slopping demand for bonds (or an upward sloping supply of credit) is that changes in investors’ portfolio composition affect the ease with which bonds can be retradedinsecondaryOTCmarkets. On the other hand, the implementation of QE in the aftermath of the Great Recession has opened a debate about the optimal “exit strategy,” i.e., what is the optimal strategy tounwindthesequantitativepolicies. TheFederalOpenMarketCommittee(FOMC)has stated that it “is maintaining its existing policy of reinvesting principal payments from its holdings of agency debt and agency mortgage-backed securities [...] and of rolling over maturing Treasury securities [...], and it anticipates doing so until normalization of the level of the federal funds rate is well under way.5” Bernanke (2017) informally makes a case for such as strategy: given the uncertainty about the possible effects of a quantitative tightening (QT) program it seems prudent to increase the short-term policy rateinordertomakeroomformonetaryaccommodationifneeded. Ouranalysisprovides anadditionalrationaleforthisstrategy. Throughthelensofourmodel,waitingtounwind thebalancesheetuntilafterinterestrateshaverisenisadequatebecauseoptimalliquidity management calls for implementing QT to withdraw liquidity from OTC markets in this case. This paper is closely related to Holmström and Tirole (1998) in the sense that the role of regulation and provision of public liquidity is a central part of our analysis. Our paper is also related to other studies of the public role for liquidity provision (see for example, Allen and Gale, 1994, 2004, Lorenzoni, 2008, Schleifer and Vishny (2011), Hart and Zingales, 2015, He and Kondor, 2016). These studies suggest that private liquidity provision is suboptimally low, with the exception of Hart and Zingales (2015) who finds that it is suboptimally high. We contribute to this literature by showing that, under the same financial frictions, private liquidity provision can be either suboptimally high or suboptimally low, depending on the conditions in the OTC market. This is an important result because these conditions are likely to vary over time, making the optimal liquidity managementpolicytimevarying. Our paper is also related to the literature studying frictional OTC trade in financial 5See FOMC statement March 15, 2017. https://www.federalreserve.gov/newsevents/pressreleases/ monetary20170315a.htm. TheFOMChaslaidoutitsplanfortheexitstrategyinthe“PolicyNormalization PrinciplesandPlans”datedSeptember17,2014. https://www.federalreserve.gov/newsevents/pressreleases/ monetary20140917c.htm. 6

markets(Duffieetal.,2005,LagosandRocheteau,2009,GeromichalosandHerrenbrueck, 2012, He and Milbradt, 2014, Atkenson et al., 2015). This literature has primarily focused on how search frictions matter for bid-ask spreads for assets traded in frictional markets. Somestudiesinthisliteratureconsidertheroleofendogenousmarketliquidityconsideringthecostofsecondarymarketparticipation(Shi,2015,BrucheandSegura,2014,andCui and Radde, 2015). Other studies consider the effect of monetary policy on trade frictions bychangingagents’moneyholdingsthatareusedindecentralizedexchanges(Lagosand Wright,2005,LagosandZhang,2016,amongothers). Ourfocusisdifferent. Weconsider thetrade-offbetweencreditandliquidityprovisioninprimarycreditmarketsasthemain determinant of market liquidity. Thus, we contribute to this literature by opening new avenues for research considering the interplay between liquidity provision and market liquidityinOTCmarkets. Finally, our paper contributes to a recent literature that have studied the mechanism throughwhichQEoperatesanditsimplicationsfortherealeconomy. GertlerandKaradi (2011) study unconventional monetary policy in a new Keynesian model where financial intermediaries are credit constrained. The central bank, instead, is not credit constrained andthuscansupportcreditextensionwhentheintermediationcapacityofprivateinstitutions is curtailed during episodes of stress. Farhi and Caballero (2013) show that QE can beeffectivewhenthereisanexcessdemandforsafeassets,asitsubstitutesriskywithsafe assetsinprivateportfolios. However,exchanginglong-termTreasurybondswithshorter maturityoneswillnotbeeffectiveintheirmodel. MoreiraandSavov(forthcoming)show a similar result in a model where the perceptions about the riskiness of assets used as collateral by private agents drives liquidity creation. Williamson (2012), within a new Monetarist model, argues that QE will affect the real economy only if it transfers credit risk from the private to the public sector. In contrast, our paper argues that QE affects therealeconomybyinfluencingthethicknessofOTCmarkets,evenwhensafeassetsare abundant, there is no financial distress, or credit risk transfers between the private and publicsectorarenotallowed. The rest of the paper proceeds as follows. Section 2 presents our dynamic model of market-based financial intermediation and establishes the existence and uniqueness of the equilibrium. Section 3 describes the effect of secondary OTC trade on bond premia andprimarycreditmarkets. Section4presentsthesocialplanner’sproblem,describesthe externalitiesoperatingthroughsecondarymarketliquidity,andanalyticallydescribesthe set of policy instruments that can implement the constrained efficient outcome. Section 5 analyses the effect of quantitative easing on secondary market liquidity and economic efficiency. Finally,section 6concludes. AllproofsarerelegatedtoanOnlineAppendix. 7

2 A Dynamic Model of Market-Based Intermediation 2.1 Physical Environment Therearethreetimeperiodst = 0,1,2,asingleconsumptiongood,andtwotypesofagents: entrepreneursandinvestors. Entrepreneurshavelong-terminvestmentprojectsandmay fundtheseprojectswithinternalfundsorwithexternalfundsreceivedfrominvestors. Ex ante identical investors provide funds to entrepreneurs, but once that lending has taken place and while production is underway, investors are subject to a preference (liquidity) shock which reveals whether they are impatient, and hence prefer to consume earlier ratherthanlater,orpatient. InvestorscantradetheirassetsinanOTCmarketwithsearch frictionstomeettheirliquidityneeds(seeFigure 2). Primarycreditmarket Firm undertakes a long-term risky investment project Firm Firm SecondaryOTCmarket Patient 1 δ Investor − Illiquid Liquid Investor Investor asset asset Impatient δ Uncertainty, ω, is realized; Investor risky project pays out Some investors are hit with a liquidity shock t = 0 t = 1 t = 2 Figure2: Timeline. There is a mass one of ex ante identical entrepreneurs, who are endowed with n 0 units of consumption at t = 0. Entrepreneurs invest to maximize the return on their investment.6 They operate a linear technology, which delivers Rkω at t = 2, per unit of 6Thisisequivalenttomaximizeprofitsperunitofendowment,orthereturnonequity,astheendowment isfixedandissolelyusedforinvestment. 8

consumption invested at t = 0. The random variable ω is an idiosyncratic productivity shock that hits after the project starts, and is distributed according to the cumulative distribution function F, with unit mean. The productivity shock is privately observed by the entrepreneur, but investors can learn about it if they pay a monitoring cost μ as a fraction of assets. The (expected) gross return Rk is assumed to be known at t = 0, as thereisnoaggregateuncertaintyinthemodel. Inordertoproduce,thefirmmustfinance investment,denoted k ,eitherthroughitsownresourcesorbyissuingfinancialcontracts 0 toinvestors. So,profitsequaltotalrevenueinperiod2, Rkωk ,minuspaymentobligations 0 from financial contracts. Entrepreneurs represent the corporate sector in our model, so wewilltalkaboutentrepreneurs’projectsandfirmsinterchangeably. There is a mass one of ex ante identical investors, who are endowed with e units 0 of consumption at t = 0. Investors have unknown preferences at t = 0, and learn their preferences at t = 1. At t = 1 investors realize if they are patient or impatient consumers, a fraction 1 δ will turn out to be patient and a fraction δ impatient. Following Diamond − and Dybvig (1983) the preference shocks are private information and are not contractible ex-ante.7 Patientconsumershavepreferencesonlyforconsumptionint = 2,uP(c ,c ) = c , 1 2 2 whereas impatient consumers have preferences for both consumption in t = 1 and 2, but discountperiod2consumptionatrate β 1,uI(c ,c ) = c +βc . 1 2 1 2 ≤ Investors in both period 0 and 1 have access to a storage technology with yield r > 0, i.e.,everyunitofconsumptionstoredyields1+runitsofconsumptioninthenextperiod. The amount stored in period t is denoted s . In addition, at t = 0, investors can purchase t financialcontractsissuedbyentrepreneurs;and,at t = 1,theycanexchangeconsumption for financial contracts in an OTC market subject to search frictions (see Figure 2). Patient investorsthatareabletoacquirecontractsintheOTCmarketrealizeanendogenousreturn Δ. But, as we show below this return equals an expression of exogenous parameters (see equation(8)). Finally, note that the expected return on financial contracts will be known in period 0 and1,sincethereisnoaggregateuncertaintyornewinformationarrivingafterinvestors and firms have agreed on the terms of these contracts. This means that asymmetric informationconsiderationswillnotplayaroleintheOTCmarket.8 Boththemarketforfinancialcontracts,theprimarymarket,andtheOTCorsecondary 7Financial intermediaries would be useful to improve allocations. Nevertheless, when retrading of financialclaimsisallowedinthesemodelsagentscanself-insureagainstidiosyncraticliquidityrisk(Jacklin 1987). But our secondary market is not Walrasian precluding the perfect self insurance. Moreover, we abstractfromintermediariestofocusontheroleoftheOTCmarket. 8A long literature studies the effect of adverse selection in secondary trade as a source of illiquidity (GortonandPennacchi,1990;Eisfeldt,2004;Kurlat,2013;Malherbe,2014;Bigio,2015). Wehaveabstracted frominformationallydrivenilliquiditytofocusontheroleofsearchfrictions. 9

marketforthesecontractsaredescribedindetailbelow.9 Inwhatfollowswemakethefollowingassumptions. Assumption 1 (Relative Returns) The long-term return of the productive technology is larger than the cumulative two-period storage return and the return on storage plus the return on secondary markets, i.e., (1+r)2 < Rk and (1+r)Δ Rk. In addition, monitoring costs are such ≤ thatRk(1 μ) < (1+r)2. − Assumption 2 (Productivity Distribution) Let h(ω) = dF(ω)/(1 F(ω)) denote the hazard − rateoftheproductivitydistribution. Itisassumedthat ωh(ω) isstrictlyincreasing. Assumption 3 (Impatience) Impatient investors discount future consumption at a higher rate than the return on storage, 1/β 1 r or β 1/(1+r), and the discounted expected return on − ≥ ≤ firms’projectsislargerthanthereturnonstorage 1+r βRk. ≤ Assumption4(InvestorsDeepPockets) Itisassumedthatinvestors’(total)endowmente is 0 significantlyhigherthanentrepreneurs’(total)endowmentn ,i.e.,e >> n . 0 0 0 Assumption 1 is necessary for the issuance of financial contracts in equilibrium. On theonehand,Rk > (1+r)2,allowsfirmstoofferareturnthatishigherthanthecumulative two-period return on storage. On the other hand, Rk (1 + r)Δ, allows firms to offer ≥ a return that is higher even when the prospective return on the OTC market is taken into account. Furthermore, this assumption rules out equilibria where entrepreneurs are alwaysmonitored,(1+r)2 > Rk(1 μ). Assumption2ensuresthatthereisnocreditrationing − in equilibrium. Assumption 3 makes impatient investors have a (weak) preference for current versus future consumption when the interest rate is r, β 1/(1 + r), but not too ≤ impatient so that the return of entrepreneurs for an impatient investor is larger than the return on storage, 1 + r βRk. Assumption 4 ensures that investors can meet the ≤ credit demand of entrepreneurs. Together these assumptions ensure the existence and uniquenessofequilibrium,aswediscussbelow. 2.2 The Financial Contract and the Demand for Credit Entrepreneursfinancetheirinvestment k usingeitherinternalresources, n ,orbyselling 0 0 long-termfinancialcontractstoinvestors. Thesecontractsspecifyanamount,bd,borrowed 0 frominvestorsat t = 0andapromised grossinterestrate, Z,madeuponcompletionofthe project at t = 2. If entrepreneurs cannot make the promised interest payments, investors cantakeallfirm’sproceeds,payingamonitoringcostequaltoafraction μofthevalueof 9Note that since r > 0 and since investors preferences have been assumed time separable and risk neutral, there was no loss of generality in abstracting away from consumption at t = 0 for investors, and consumptionatt=1forpatientinvestors. 10

assets.10 Then,the t = 0budgetconstraintfortheentrepreneurisgivenby k n +bd. 0 ≤ 0 0 For what follows it will be useful to define the entrepreneur’s leverage, l , as the ratio 0 ofassetsto(internal)equity k /n . 0 0 Anentrepreneurisprotectedbylimitedliability,soherprofitsarealwaysnon-negative. Thus,theentrepreneur’sexpectedprofitinperiodt = 2isgivenbyE max 0,Rkωk Zbd . 0 0 − 0 Limitedliabilityimpliesthattheentrepreneurwilldefaultonthecontractiftherealization n o of ω is sufficiently low such that the payoff of the long-term project is smaller than the promised payout: Rkωk < Zbd. This condition defines a threshold productivity level, ωˉ, 0 0 suchthattheentrepreneurdefaultswhen Z l 1 0 ω < ωˉ = − . (1) Rk l 0 Theproductivitythresholdmeasuresthecreditriskofthefinancialcontract,asitincreases thefirm’sprobabilityofdefault.11 ωˉ Fornotationalconvenience,wedefineG(ωˉ) ωdF(ω)andΓ(ωˉ) ωˉ(1 F(ωˉ))+G(ωˉ). ≡ 0 ≡ − The function G(ωˉ) equals the truncated expectation of entrepreneurs’ productivity given R default. The function Γ(ωˉ) equals the expected value of a random variable equal to ω if there is default (ω < ωˉ) and equal to ωˉ when there is not (ω ωˉ). It follows that ≥ Rkk Γ(ωˉ)correspondstotheexpectedtransfersfromentrepreneurstoinvestors. Then,the 0 firms’ objective, to maximize expected profits per unit of endowment, can be expressed as1/n E max 0,Rkωk Zbd = [1 Γ(ωˉ)]Rkl . 0 0 { 0 − 0} − 0 Similarly, the expected gross return of holding a single bond to maturity Rb can be expressedas ωˉ Rb = 1 ∞ ZbddF(ω)+(1 μ) Rkωk dF(ω) = l 0 Rk Γ(ωˉ) μG(ωˉ) , (2) b 0 − 0 l 1 − 0 "Zωˉ Z0 # 0 − (cid:2) (cid:3) whichisafunctionofonlyleverageandtheproductivitythreshold. Clearly Rb isdecreasing in l as leverage dilutes lenders claim on the firm’s assets. Moreover, in equilibrium 0 it will be increasing in risk, ωˉ, as detailed below. Using this notation we can write down thefirm’sproblemthatdefinestheoptimalcontract,whentheexpectedhold-to-maturity 10We consider deterministic monitoring rather than stochastic monitoring, which results in debt being theoptimalcontract. KrasaandVillamil(2000)derivetheconditionsunderwhichdeterministicmonitoring occursinequilibriumincostlyenforcementmodels. 11Notethattheproductivitythresholdisincreasinginthespreadbetweenthepromisedreturnandthe expectedreturnontheentrepreneurinvestment,anditisincreasingintheentrepreneur’sleverage l . 0 11

returnofferedtoinvestorsequals R as max[1 Γ(ωˉ)]Rkl s.t. Rb(l ,ωˉ) = R . (3) 0 0 l0,ωˉ − Note that this problem also defines the demand for credit in the primary market bd(R) = (l (R) 1)n , which is, as we show below, a strictly decreasing function of the 0 0 − 0 expected hold-to-maturity return R. As it is well established in the CSV literature the optimal financial contract will take the form of a debt contract. Therefore, we refer to thesecontractsasbonds. 2.3 The OTC Market The ex post heterogeneity introduced by the preference shock generates potential gains fromtradingfinancialcontractsforconsumptionintheOTCmarket. Impatientinvestors want to exchange long-term bonds for consumption, as they would rather consume in period 1 than hold the bond to maturity until period 2 (Assumption 3). Patient investors are willing to take the other side of the trade if the return from buying bonds in the OTC market Δ isgreaterthanthereturnonthestoragetechnology1 +r. To model the exchange in the OTC market we consider that each investor represents a large family of small traders. That is, each investor is comprised by a continuum of infinitesimal traders of mass e , where each trader has a portfolio restricted to one 0 bond or q units of consumption, which corresponds to the price of bonds in terms of 1 consumption.12 Traders are paired up according to a matching technology. Impatient investors send a mass of bs traders to sell their bonds. Patient investors send a mass of 0 (1 + r)s /q traders to buy bonds in the OTC market. This is akin to a situation where 0 1 impatient investors submit bs sell orders and patient investors submit (1 + r)s /q buy 0 0 1 orders,soforeaseofexpositionwewillrefertothistradingprocessassubmittingorders. Akeyimplicationoftheassumptionsthattradingiscarriedoutbytradersasopposed toinvestorsandthattradersmeetonlyoncewithpotentialcounterparties,isthattheprice intheOTCmarkerq isindependentofmarketthickness.13 Ifweweretoallowaneffecton 1 thesecondarymarketpriceofmarketthicknesstherewouldbeanadditionalmechanism 12That is, traders’ portfolios are restricted as in Atkenson et al. (2015) or Bianchi and Bigio (2014). In ourcase,traders’portfoliosarerestrictedtoasinglebondequivalent: onebondforsellersand q unitsof 1 consumptionforbuyers. 13SeeBianchiandBiggio(2014)foramodelwheremultipleroundsoftradereintroducesthedependence on market conditions of the price in the secondary market. See Mattesini and Nosal (2016) for a model whererenegotiationbetweeninvestorsandbrokersintroducesadependenceofmarketconditionsonthe priceinthesecondarymarket. 12

torationalizetheregulationandpublicprovisionofliquidity. Suppose,inaggregate,thereareAsell(orask)ordersandBbuyorders. Thematching functionisassumedtobeconstantreturnstoscale,aslongasthenumberofmatchesdoes notexceedthenumberofordersineachsideofthemarket. Anditisgivenby m(A,B) = min A,B,νAαB1 α , (4) − n o with 0 < ν a scaling constant and 0 < α < 1 the elasticity of the matching function with respecttosellorders. Wedefineaconceptofmarketliquiditythroughtheratioofbuyorderstosellorders,or θ = B/A. Thisnotionofliquidity—definedbyaconceptofthicknessintheOTCmarket— hasdifferentimplicationsfortradersonopposingsidesofthemarket. Forexample,when θ is large, a bond in the secondary market is relatively liquid, that is, it is relatively easy for sellers to trade. But, at the same time, it is relatively hard for buyers to trade. Note that our notion of liquidity is related to, but distinct from, the easiness to trade for all marketparticipants,whichiscapturedinourframeworkbytheefficiencyofthematching technology ν. Increasing (decreasing) ν makes it easier (harder) for participants on both sidesofthemarkettotradeinasymmetricfashion. Usingthematchingfunction,the probabilitythatasellorderisexecuted isexpressedas m(A,B) f(A,B) = or f(θ) = m(1,θ) , (5) A andthe probabilitythatabuyorderisexecuted isexpressedas m(A,B) p(A,B) = or p(θ) = m(θ 1,1) . (6) − B The fact that matches are bounded by the minimum number of orders, defines two liquidity threshold θ = min ν1/α,1 and θ = max ν 1/(1 α),1 . When market liquidity − − { } { } θ θ then all buy orders are executed, i.e., m(A,B) = B. In this case buyers trade with ≤ probability p(θ) = 1, whereas sellers trade with probability f(θ) = θ. Alternatively, when θ θ then all sell orders are executed, i.e., m(A,B) = A; and thus the trade probabilities ≥ f(θ) = 1andp(θ) = θ 1. Whenliquidityisin[θ,θ]thenmatchesaregivenbytheconstant − return to scale matching function νAαB1 α; and thus the trade probabilities f(θ) = νθ1 α − − and p(θ) = νθ α. − Once a buy order and a sell order are matched, the terms of trade are determined via a simple surplus sharing rule, determined by Nash bargaining and known by all agents. From the seller’s perspective, a trading match yields additional consumption from the 13

saleofthebondatpriceq . Ifthesellerwalksawayfromthematchshewillholdthebond 1 to maturity and receive an expected payoff βR in t = 2. Then, the surplus that accrues to an impatient investor is given by SI(q ) = q βR. Similarly, the value of a trading match 1 1 − to a buyer is the present value of the (expected) return on the bond, net of the price that needstobepaidforeachbondinthesecondarymarket, SP(q ) = R/(1+r) q . 1 1 − ThepriceofthedebtcontractonthesecondarymarketisdeterminedbyNashbargaining, which maximizes the product of the respective surpluses, max q1 SI(q 1 ) ψ SP(q 1 ) 1 − ψ , where ψ [0,1]isthebargainingpowerofimpatientinvestors. ∈ (cid:0) (cid:1) (cid:0) (cid:1) The solution of the surplus splitting problem yields the following bond price in the secondarymarket ψ q = R +(1 ψ)β . (7) 1 1+r − ! Note that ψ = 1 drives the price of the bond to the “ask” price, or the price that extracts full rent from the buyer, q = R/(1+r). By the same token, ψ = 0 drives the price of the 1 bond to the “bid” price, or the price that extracts full rent from the seller, q = βR. From 1 equation(7)itfollowsthatthereturnthatpatientinvestorsmakeinthesecondarymarket, perexecutedbuyorder,dependsonlyonexogenousparametersandisgivenby 1 R ψ − Δ = = +(1 ψ)β 1+r . (8) q 1+r − ≥ 1 ! 2.4 Investors and the Supply of Credit As described above, investors are ex ante identical and are endowed with e units of the 0 consumption good. At t = 0 they can allocate their wealth across two assets: a storage technology s andfinancialcontracts bs. Thus,theirbudgetconstraintisgivenby 0 0 s +bs = e , (9) 0 0 0 where s ,bs 0,i.e.,borrowingatthestoragerateorshort-sellingbondsarenotallowed. 0 0 ≥ Thestoragetechnologypaysafixedrateofreturn1+ratt = 1inunitsofconsumption. Theproceedsofthisinvestment,ifnotconsumed,canbereinvestedtoearnanadditional return of 1+r between period 1 and 2, again paid in units of consumption. In this sense, storage is a liquid asset, as at any point in time it can be costlessly transformed into consumption. In contrast, bonds deliver an expected payoff R in t = 2 and are illiquid as an investor might be unable to turn her bond into consumption at t = 1 if her sell order doesnotfindamatchintheOTCmarket. 14

To describe the portfolio choice problem of investors, it is useful to first consider the optimal behavior of impatient and patient investors in t = 1 when they arrive in that periodwithagenericportfolioofstorageandbonds(s ,bs). 0 0 2.4.1 ImpatientInvestors By Assumption 3 at t = 1 impatient investors want to consume in the current period. They can consume the proceeds from the resources they put into storage, s (1 + r), plus 0 the additional proceeds from placing bs sell orders in the OTC market. These orders are 0 executed with probability f(θ) and each executed order yields q units of consumption. 1 Thus,theconsumptionofimpatientinvestorsinperiod1isgivenby cI = s (1+r)+ f(θ)q bs . (10) 1 0 1 0 On the other hand, with probability 1 f(θ) orders are not matched and impatient − investorshavetocarrybondsintoperiod2. Therefore,consumptioninthefinalperiodis givenby cI = (1 f(θ))Rbs , (11) 2 − 0 withtheutilityderivedfrom cI discountedby β. 2 2.4.2 PatientInvestors Patient investors only value consumption in the final period and will be willing to place buy orders in the OTC market if there is a surplus to be made, i.e., if q R/(1+r). The 1 ≤ pricedeterminationintheOTCmarketguaranteesthatthisisalwaysthecase(1 +r Δ), ≤ thuspatientinvestorwouldideallyliketoexchangealloftheirconsumptionforbonds. Butthebuyordersofpatientinvestorswillbeexecutedonlywithprobabilityp(θ). That is, they place s (1+r)/q buy orders, of which a fraction p(θ) are executed on average. So 0 1 patientinvestorsexpecttoincreasetheirbondholdingby p(θ)s (1+r)/q units. Itfollows 0 1 thatexpectedstorageholdingsattheendof t = 1,sP,areequaltoafraction1 p(θ)ofthe 1 − availableliquidfunds s (1+r),i.e.,sP = (1 p(θ))s (1+r). Then,consumptioninthefinal 0 1 − 0 periodequals s (1+r) cP = (1 p(θ))s (1+r)2 + bs +p(θ) 0 R . (12) 2 − 0 0 q 1 " # That is, the payout from consumption that was stored and not traded away in the OTC marketplustheexpectedpayoutfrombondholdings. 15

2.4.3 OptimalPortfolioAllocation In the initial period investors solve a portfolio allocation problem, taking the liquidity in the OTC market θ as given. They choose between storage and bonds to maximize their expectedlifetimeutilityU = δ(cI+βcI)+(1 δ)cP,subjecttotheperiod0budgetconstraint 1 2 − 2 (9),andanticipatingthatoptimalexpectedconsumptionofimpatientandpatientinvestors isgivenbyequations(10)-(12). Usingtheexpressionsforoptimalexpectedconsumption,wecanrewritetheexpected lifetime utility as U = U s + U bs, where U and U denote the expected utility from s 0 b 0 s b investinginstorageandbondsinperiod0,respectively,andaregivenby14 U (θ) = δ(1+r)+(1 δ) (1 p(θ))(1+r)2 +p(θ)(1+r)Δ , (13) s − − h i and U (R,θ) = u (θ)R , (14) b b where u (θ) δ f(θ)Δ 1 +(1 f(θ))β +(1 δ) corresponds to the expected loss a bond b − ≡ − − investorexpectstomakerelativetothehold-to-maturitybondreturn,withu (θ) β. That h i b ≥ is,theexpectedutilityofholdingbondsinperiod0canbedecomposedastheproductof theexpectedhold-to-maturityreturnonthebond Randtheexpectedlossduetothebond illiquidity u (θ).15 By contrast, the utility of holding storage in period 0 only depends on b market liquidity, through the probability that the return from providing liquidity can be realized p(θ). Using these definitions, we can express the asset demand correspondence that maximizestheinvestorsportfolioproblemas bs = e , s = 0 if U > U 0 0 0 b s      bs 0 ∈ [0,e 0 ], s 0 = e 0 − bs 0 if U b = U s     bs = 0, s = e if U < U   0 0 0 b s      That is, if the expected uti lity of holding bonds in period 0 is greater than the utility of holdingstorageinperiod0—whichincorporatesthereturnthatcanbemadebyproviding liquidity—then investors will demand only bonds in the initial period. On the contrary, if the expected utility of holding bonds is smaller than then expected benefit of holding storage in period 0, then investors will only hold storage in the initial period. Finally, if 14We are taking advantage of the result that Δ is a function of exogenous model parameters, but in generalΔ(R,q )=R/q . 1 1 15In equilibrium, this will introduce a dependence of the utility of holding a bond on the contract characteristics(l ,ωˉ),throughtheexpectedreturnonholdingthebondtomaturityRb. 0 16

the expected benefits are equal, investors will be indifferent between investing in storage and bonds initially, and their demands will be an element of the set of feasible portfolio allocations: s ,bs [0,e ], such that the total value of assets equal the initial endowment 0 0 ∈ 0 (9). In this case the individual credit supply is totally elastic when the expected hold-tomaturityreturnequals U (θ)/u (θ). s b Given our assumptions, in equilibrium, the portfolio allocation will be interior (i.e., U = U withs ,bs > 0),thuswefocusouranalysisonthiscase.16 Forfuturereferencewe s b 0 0 labelthisconditionthe investors’break-evencondition. U (θ) = U (R,θ) = u (θ)R . (15) s b b The upshot of writing the investors’ break-even condition in this way is that from the perspective of a firm that takes the liquidity in the OTC markets as given the break-even condition amounts to ensuring investors a hold-to-maturity return equal to U (θ)/u (θ). s b Moreover,thisconditiondescribestheaggregatecreditsupply,bs(R). Infact,itisimplicitly 0 defined by U (θ(bs,R)) = u (θ(bs,R))R, where θ(bs,R) = (1 δ)(e bs)(1 + r)Δ/(δbsR). s 0 b 0 0 − 0 − 0 0 To be clear, our concept of aggregate credit supply is not just the sum of the individual investors’creditsupply,butisonewheretheconsistencyofmarketthicknessistakeninto account, i.e., θ is a function of (bs,R). As we show below, the aggregate credit supply is 0 strictlyincreasingintheexpectedhold-to-maturitybondreturn. 2.5 Equilibrium Theequilibriumofthemodelisdefinedasfollows. Definition 1 (Private Equilibrium) We say that (l ,ωˉ,θ,q ) is a private equilibrium if and 0 1 onlyif: 1. Given the outcome in the secondary market (θ,q ), the financial contract is described by 1 (l ,ωˉ) that maximizes entrepreneurs’ return on investment subject to investors’ break-even 0 condition(15). 2. Thecreditmarketclears,i.e.,bd(R) = bs(R) b ,withR = Rb(l ,ωˉ) givenbyequation(2). 0 0 ≡ 0 0 3. Marketliquiditycorrespondsto θ = (1 δ)(1+r)s /(q δbs). − 0 1 0 4. q isdeterminedviathesurplussharingrule(7). 1 16Notethattheexpectedutilityfrominvestinginstorage, U ,isnotsmallerthantheexpectedutilityin s financialautarky: U =δ(1+r)+(1 δ)(1+r)2,sincethereturnofbuyingabondinthesecondarymarket a − Δ 1+r. ≥ 17

5. Allagentshaverationalexpectationsaboutq and θ. 1 Theequilibriumofthemodelisdescribedbytheentrepreneur’schoiceofleverage, l , 0 andrisk,ωˉ,tomaximizethepayoffoftheriskyinvestmentproject. Entrepreneurs’profits are higher when l is higher and when the promised payout is lower, that is, when ωˉ is 0 lower. Butfirmsareconstrainedintheirchoicesofl andωˉ astheyneedtooffertermsthat 0 makefinancialcontractsattractivetoinvestors: theinvestors’break-evencondition. Firms are aware that when selling in the OTC market, investors obtain a price that depends on thecontractcharacteristicsandisdeterminedviathesharingrule(equation 7). Itfollows that the firm’s problem can be written as max [1 Γ(ωˉ)]Rkl , subject to the investors’ l0,ωˉ − 0 break-evencondition(15). The entrepreneur’s privately optimal choice of leverage trades off the marginal increase in profits from higher leverage against the marginal reduction in expected (holdto-maturity)bondreturnforinvestors. Similarly,theprivatelyoptimalchoicefortherisk profile of the contract trades off the marginal increase in profits from lower risk against themarginalreductionintheexpected(hold-to-maturity)bondreturnforinvestors. Theseconsiderationscanbesummarizedinthefollowingoptimalitycondition 1 Γ(ωˉ) ∂Rb(l ,ωˉ)/∂l 0 0 − = . (16) Γ (ωˉ)l −∂Rb(l ,ωˉ)/∂ωˉ 0 0 0 This equation, which describes the privately optimal debt contract, taken together with the investors’ break-even condition (15), and the expressions that characterize the secondarymarket(θ,q )provideacompletedescriptionoftheequilibriumofthemodel. 1 Finally, note that the expected hold-to-maturity return Rb, the price in the secondary market q , and secondary market liquidity θ all can be expressed as a function of the 1 characteristicsoftheoptimalfinancialcontract(l ,ωˉ). Infact,thepriceisafunctionofthe 0 expected return on holding the bond to maturity, Rb, which depends on (l ,ωˉ); so we can 0 writemarketliquidityas (1 δ)s (1+r) (1 δ)(1+r)Δ(e n (l 1)) 0 0 0 0 θ = − = − − − . (17) δb q δn (l 1)Rb(l ,ωˉ) 0 1 0 0 0 − Thefollowingtheoremestablishestheexistenceanduniquenessofequilibriuminour model. Theorem 1 (Existence and Uniqueness of Private Equilibrium) Under the maintained assumptions there exists a unique private equilibrium of the model. Furthermore, in the unique equilibriumcreditisnotrationed. 18

The proof uses a fixed-point argument on a mapping that takes the expected bond return offered by firms and uses the equilibrium conditions to map it to the expected hold-to-maturity return required by investors. Thus, a fixed point of this mapping is an equilibrium of the model. The proof proceeds in three steps. The first step shows that the optimal financial contract defines a credit demand function, i.e., each offered return yields a unique demand for credit or level of bond issuance by firms. This step derives results that are similar to results found in the CSV literature. The second step showsthattheaforementionedmappingiscontinuousandmapstheintervalofexpected returns [(1 + r)2,Rk] on itself, thus having a fixed point and establishing the existence of equilibrium. These derivations generalize previous results to the case when financial contracts are retraded in OTC markets, and they show that the aggregate credit demand is strictly decreasing in bonds’ expected returns. Finally, the third step establishes that multiple equilibria do not arise due to the retrading in the OTC market. We establish uniqueness by showing that when the matching function exhibits constant returns to scale,theaforementionedmappingisstrictlydecreasing. Thatis,whentheexpectedreturn offeredbyfirmsdeclines,theborrowingbyfirmsincreases,whichlowersmarketliquidity and increases the expected hold-to-maturity return required by investors. The last result suggests that aggregate credit supply is upward slopping. In fact, the derivations in the proofofTheorem1allowstoestablishthefollowingresultsthatcharacterizethedemand andsupplyforcredit. Corollary1(TheOptimalFinancialContractandTheDemandforCredit) Thecharacteristicsoftheoptimalfinancialcontract: leveragel andriskωˉ,arestrictlydecreasingintheexpected 0 hold-to-maturityreturnR. Thatis,thedemandforcreditbd(R) isstrictlydecreasing. 0 Corollary 1 follows from the characterization of the demand for credit in the proof of Theorem 1. In addition, we can establish that the aggregate supply of credit is strictly increasing in the expected hold-to-maturity return, in the relevant part of the parameter space. Proposition 1 (Aggregate Credit Supply Elasticity) The aggregate credit supply, bs(R), has afiniteandstrictlypositiveinterestrateelasticity. Proposition 1 characterizes the role of secondary market liquidity on the aggregate supplyofcreditintheprimarymarketforbonds. Astheexpectedhold-to-maturitybond returnincreases,forinvestorstobeindifferentbetweenilliquidbondsandliquidstorage, market thickness need to drop so the return on storage increases and the expected loss fromholdingilliquidbondsdecreases. Marketthicknessdropsonlyifinvestors’portfolios 19

Figure3: AggregateDemandandSupplyofCreditinthePrimaryMarket. R bs 0 Rk Rb (1+r)2 bd 0 0 b b 0 become more illiquid, which is the case when investors’ bond holdings increase. That is to say that the supply of credit increases. Note that in our model, where individual investors—who are risk neutral—supply credit totally elastically, the interaction of trade frictions and investors’ limited liquid resources generate an increasing aggregate supply ofcredit. The previous results are useful to analyze the model as the aggregate demand and supplyforcredit,i.e.,thedemandofcreditbyfirmsfrominvestorsandinvestors’supply of credit to firms in the primary bond market, depicted in Figure 3. This representation canbeusedtocontrastourmodelwithpreviouswork. IntheCSVliteratureitistypically assumed that aggregate credit supply is perfectly elastic at rate (1 + r)2, e.g., Bernanke et al. 1999. This case is depicted by the green dashed line in Figure 3. In other models of OTC trade with search frictions, such as Duffie, Garleanu and Pederson (2005), where thetradingprobabilitiesandmarketthicknessarefunctionsofexogenousparameters,the aggregatecreditsupplywillbetotallyelasticatsomerate R > (1+r)2 and R < Rk. 3 OTC Trade, Bond Premia, and Credit Markets This section defines the liquidity and default bond-premia and presents analytical characterizationsforboth. Itthenpresentsafrictionlessbenchmarkthatspecifiesthelimiting caseswheretradefrictionsintheOTCmarketbecomeirrelevant. Itcontinuestodescribe the relationship between the OTC market, i.e, the secondary market for bonds, and the credit market, i.e., the primary market for bonds. It concludes by presenting a numerical illustrationofthemodel. 20

3.1 Analytical Characterization of Liquidity and Default Premia Itisusefultodefineabenchmarkinterestratethatisthereturnonatwo-periodbondthat could be traded in a perfectly liquid secondary market. Naturally, such a contract needs todeliverthesamereturninexpectationasastrategyofinvestingonlyinstoragebothin theinitialandinterimperiods.17 Thisgivesrisetothefollowingdefinition. Definition 2 (Liquid Two-period Rate) The liquid two-period rate is defined as the gross interestrateonaperfectlyliquidtwo-periodbondR‘ (1+r)2. ≡ The benchmark rate allows us to decompose the total gross return on the financial contract written by the firm into a default and a liquidity premium. In order to do this, express the total bond premium as the gross return of the firm’s contract relative to the benchmark rate, Z/R‘. Then, this total premium is decomposed into a component owing to default risk, Z/Rb, and a component owing to liquidity risk, Rb/R‘. With this decomposition, we have the following definitions for the default and liquidity premia, respectively. Definition 3 (Default and Liquidity Premia) The bond default premium Φd and the bond liquiditypremium Φ‘ aregivenby Φd Z/Rb and Φ‘ Rb/R‘. ≡ ≡ Consequently, the total bond premium is given by Φt Z/R‘ = Φd Φ‘. These defini- ≡ tions provide sharp characterizations of both the default and liquidity premia, which are convenienttohelptraceouttheunderlyingeconomicmechanismsinourmodel. Fromthedefinitionofthedefaultpremiumwehavethat ωˉ Φd(ωˉ) = . (18) Γ(ωˉ) μG(ωˉ) − It follows that in our model, as in the classic CSV model, the default premium is an increasingfunctionofcreditrisk ωˉ,asformalizedinthenextproposition. Proposition 2 (Credit Risk and the Default Premium) Under the maintained assumptions, thedefaultpremium Φd(ωˉ) isastrictlyincreasingfunctionofcreditrisk ωˉ. Intuitively,investorsdemandahigherdefaultpremiumforfinancialcontractsthatare more likely to default (i.e., contracts that are more risky, or specify a higher productivity thresholdωˉ forpayingoutthefullpromisedvalue). Themoresubtlepartoftheargument 17Noarbitrageunderperfectlyliquidmarketsimpliesthattradingatwo-periodbondshouldyieldthe sameexpectedreturnforinvestorstorollingoveroneperiodsafeinvestments,i.e. δ R‘/(1+r)+(1 δ) R‘ = ∙ − ∙ δ (1+r)+(1 δ) (1+r)2. ∙ − ∙ 21

is that leverage does not affect the default premium, as is the case in the benchmark CSV model,thoughleverageandriskarejointlydeterminedinequilibrium. Thisisduetothe factthat,forafixedthresholdlevel, ωˉ,leverageaffectsboththefacevalueofthecontract, Z,andthehold-to-maturityreturnforinvestors, Rb,inthesameway(equation(18)). Moreover, from the definition of the liquidity premium and the investors’ break-even condition(15),wehavethat U (θ) Φ‘(θ) = s . (19) (1+r)2u (θ) b That is, the liquidity premium equals the spread between the bond hold-to-maturity return, which equals the ratio between the expected utility from liquidity provision U s and the expected utility loss due to bond illiquidity u , and the liquid two-period return b (1+r)2. Equation(19)providesananalyticalcharacterizationoftherelationshipbetween theliquiditypremiumandsecondarymarketthickness θ,showingthattheformerisonly afunctionofthelatter. Thisanalyticalrepresentationoftheliquiditypremiumchannelis keyforouranalysis. ThefollowingLemmacharacterizesthischannel. Lemma 1 (Secondary Market Liquidity and the Liquidity Premium) The liquidity premium, Φ‘, or equivalently, the hold-to-maturity return, Rb, is a decreasing function of secondary marketliquidity,θ. Moreover,theelasticityoftheliquiditypremium,Φ‘,withrespecttosecondary marketliquidity, θ,islowerthan 1inabsolutevalue. Lemma 1 formalizes the intuition that the price of liquidity risk (i.e., the liquidity premium)isinverselyproportionaltotheamountofliquidityinsecondaryOTCmarkets. When secondary market liquidity, θ, is lower, investors require a higher liquidity premium, Φ‘, or equivalently, a higher hold-to-maturity return, Rb. This relationship forms thebasisfortheliquiditypremiumchannelshownbythearrowgoingfromthesecondary OTC market to the primary credit market in Figure 1. In our model, market liquidity determinesthelikelihoodthatinvestors’orderswillbeexecutedinanOTCtrade. Inparticular,asthemarketbecomeslessliquidsellorderswillbemoredifficulttoexecute(f(θ) decreases), and impatient investors will realize larger utility losses from the illiquidity of bonds (u (θ) decreases). By the same token, as liquidity declines buy orders are more b likely to be executed (p(θ) increases), rising the expected private benefit from liquidity provision (U (θ) increases). Both of these channels lead to an increase in the expected s hold-to-maturitybondreturntokeepinvestorsindifferentbetweenbondsandstoragein period0,thatis,theliquiditypremiumincreases. 22

3.2 A Frictionless Benchmark The next proposition establishes the conditions under which trade in the OTC market is irrelevant, so that secondary market liquidity has no bearing on the equilibrium of the model. Proposition3(IrrelevanceofOTCTrade) Underthefollowingconditions,thereisnoliquidity premium,i.e., Φ‘ = 1,implyingthatthemodelcollapsestothebenchmarkCSVmodel: 1. Allinvestorsarepatient,sothat δ = 0;or 2. Impatientinvestorsdiscountatrate β = 1/(1+r);or 3. Impatientinvestorsareabletosellalltheirbondsinthesecondarymarketattheirreservation value,whichistruewhen ψ = 1ande eˉˉ . 0 0 ≥ The case in which δ = 0 is straightforward. When all investors are patient, there is no needtotradeinsecondarymarkets;investorsonlycareaboutthehold-to-maturityreturn. LiquidityisnotpricedinfinancialcontractsandthemodelcollapsestothestandardCSV setuppresentedin,forexample,Townsend(1979)andBernankeandGertler(1989). The same result obtains for the second case, though for different reasons. When impatient investors discount future consumption at exactly the rate of return that comes from holding a unit of storage, so that β = 1/(1 + r), they will be indifferent between consuminginthefinalorinterimperiod. Thisindifferenceimpliesthattherearenogains from OTC trade. In this case, the liquidity preference shock is immaterial and investors only consider the hold-to-maturity return when buying financial contracts in primary markets. The third case considers the situation in which impatient investors can fully satisfy their liquidity needs in secondary markets, while there are no gains for patient investors from liquidity provision. That is, the terms of trade are set such that impatient investors extract the entire surplus, i.e., ψ = 1, and all their sell orders will be executed. Patient investors,then,earnonlytheiropportunitycostfromliquidityprovision,i.e., Δ = (1+r)2. In this case, as before, liquidity considerations will not factor in the lending decision of investorsinprimarymarkets. Theconditionthat f(θ) = 1followsfrome eˉˉ . Wederive 0 0 ≥ this threshold for investors endowment in the proof of Proposition 3 in the Appendix. Intuitively, f(θ) = 1, requires that there is enough storage at t = 1 that all sell orders can besatisfied,whichrequiresthatinvestors’endowmentissufficientlylarge. Itisworthmentioningthatinprincipletherecouldbeafourthcasewheretradefrictions areirrelevant: whenpatientinvestorscanfullyrealizethegainsfromliquidityprovision, 23

i.e., ψ = 0 and p(θ) = 1, while at the same time impatient investors are indifferent from holding bonds to maturity and trading in the OTC market, i.e., β = 1/(1 + r). The first conditionfixestheliquiditypremiumat β 1/(1+r) 1. Thus,therewillbenoeffectfrom − ≥ market liquidity into the credit market. But trade frictions are irrelevant only when the liquidity premium collapses to 1, which is the case only when β = 1/(1 + r). The latter conditionisencompassedincase2inProposition 3.18 3.3 Liquidity Premium and Liquidity Provision Channels We now characterize the effects of frictional OTC secondary trade on primary credit markets. For the remainder of the paper, we consider only the cases in which trading frictions in the secondary market result in a non-negligible liquidity premium. That is, assumethat(i)theprobabilityofbeinganearlyconsumerispositive, δ > 0;(ii)impatient investors discount future consumption strictly more than what is implied by the storage rate,i.e.,β < 1/(1+r);and(iii)impatientinvestorscannotfullysatisfytheirliquidityneeds insecondarymarkets,i.e., ψ < 1or e < eˉˉ . 0 0 We are now ready to characterize the relationship between credit and OTC markets depictedinFigure1. Ontheonehand,marketthicknessintheOTCmarket θdetermines the liquidity premium that investors will require on illiquid bonds over liquid storage, Φ‘(θ) (equation (19)). This is the liquidity premium channel that describes how market liquidity affects liquidity premia. Our model shows that the liquidity premium shapes theexpectedhold-to-maturityreturnRb thatfirmsneedtoofferinvestorsand,thus,firms’ demand for credit bd. Therefore, market liquidity θ affects the equilibrium in the credit 0 market(b ,Rb). 0 On the other hand, the equilibrium in the credit market (b ,Rb) will determine the 0 secondary market liquidity, θ; this is the liquidity provision channel of credit markets into OTC markets. This channel is novel to the literature analyzing financial markets with trade frictions. In fact, to support the equilibrium level of bond issuance investors will havetoholdthosebondsintheirportfoliosandwillreducetheirholdingofliquidstorage, which is deployed in the OTC market to support market liquidity. In fact, equation (17) characterizeshowOTCmarketthicknessθisafunctionofthevolumeandexpectedreturn ofbondsinthecreditmarket(b ,Rb). 0 The liquidity premium effect and the liquidity provision effect work in opposite directions. Suppose there is an exogenous shock to market liquidity, then the liquidity 18Moreover, asweshowintheproofofProposition 1, Assumption 4willruleoutthatliquiditycanbe arbitrarilylowastoguaranteethatpatientinvestorscanfullyrealizethegainsfromliquidityprovision,i.e., p(θ)=1. 24

premium channel would increase the aggregate supply of credit. Firms respond to this shift in credit supply by issuing more bonds at lower expected (hold-to-maturity) returns. But the liquidity provision channel would yield a reduction in market liquidity as investors substitute liquid assets for bonds in their portfolio. This channel attenuates the initial increase in market liquidity. In our existence proof we establish that the two effects jointly determine the unique equilibrium in the primary and secondary markets. The direct effect dominates, while the indirect effect acts as an automatic stabilizer such thatanimprovementoradeteriorationinmarketliquiditycannotperpetuallyincreaseor decreasebondissuance. 3.4 Comparative Statics Now we describe the effect of the parameters that determine the demand and supply for credit for the equilibrium of the model. We begin by describing the effect of these parametersonthedemandandsupplyforcredit. Proposition 4 (Aggregate Credit Supply Determinants) Taking the expected hold-tomaturityreturnandbondissuanceasgiven,investorsrequireahigherliquiditypremium, Φ‘,and hencethemarketthickness θ islowerintheOTCmarket,when 1. (Preferenceshock)Theprobabilityofbecomingimpatientishigher,i.e., δ ishigher;or 2. (Impatience)Impatientinvestorsdiscountthefuturemoreheavily,i.e., β islower;or 3. (Endowments)Investorshavelesstoinvestinstorage,i.e.,e islower. 0 Thepropositiondetailshowtheparametersthatdescribeinvestors’preferences(δand β) and endowments (e ) affect the supply of credit in the primary bond market, when 0 firms’ bond issuance is taken as given. As investors’ preferences are more sensitive to liquidity risk (δ is higher or β is lower), the associated liquidity premium drives up the hold-to-maturity return that investors require to hold corporate debt, i.e., there is a contraction of the aggregate credit supply. On the other hand, when investors are poorer (e is smaller) they reduce their holdings of liquid storage one-for-one conditional on 0 buying the same number of financial contracts. Less liquid investors’ portfolios reduce liquidity in secondary markets, and thus also drives up the required hold-to-maturity returnthroughanincreaseintheliquiditypremium(Lemma 1). Proposition 5 (Credit Demand Determinants) Taking the expected hold-to-maturity return offeredtoinvestorsasgiven,creditdemandisincreasinginthefirm’sendowmentn . 0 25

Proposition 5 is a consequence of the fact that the optimal financial contract, which solves the firm’s problem (3), specifies the optimal firm leverage l as a function of the 0 expected hold-to-maturity return R. In fact, as this return remains constant, so does leverageandthusfirm’screditdemandisgivenbybd(R) = (l (R) 1)n ,whichanincreasing 0 0 − functionof n . Thisargumentcompletestheproof. 0 The implications of the previous two results for the equilibrium of the model are summarizedinthefollowingproposition. Proposition 6 (Equilibrium Comparative Statics) In any of the following cases, the equilibrium expected return in credit markets increases. Thus, the leverage l and risk ωˉ of the optimal 0 contract and the default premium all decrease. Moreover, in the first three cases bond issuance decreases. 1. (Preferenceshock)Theprobabilityofbecomingimpatientishigher,i.e., δ ishigher;or 2. (Impatience)Impatientinvestorsdiscountthefuturemoreheavily,i.e., β islower;or 3. (Investors’Endowments)Investorshavelesstoinvestinstorage,i.e.,e islower;or 0 4. (Firms’Endowments)Firmshavemoreequity(i.e.,n ishigher). 0 Thispropositionpresentsthecomparativestaticsin equilibriumfortheparametersthat describe preferences and endowments for investors and firms. For the first three cases, Proposition 4 establishes that an increase in δ or a decrease in β or e will reduce the 0 aggregate supply of credit and firms will see an increase in the liquidity premium of the bonds they issue. According to Proposition 6, firms adjust to this increase in expected bond return along two margins (recall that the debt contract is two-dimensional): they offer fewer and less risky contacts in the primary market. A reduction in the number of bonds issued in the primary market lowers the number of possible sell orders in the secondarymarket,boostingmarketthicknessandattenuatingtheincreaseintheliquidity premium. In addition, the reduction of bonds’ riskiness contributes to reduce firms’ external financing cost by lowering the default premium. In equilibrium, thus, the total effect on the external financing premium is ambiguous and depends on whether the increaseintheliquiditypremiumorthedecreaseinthedefaultpremiumdominates. The fourth case of Proposition 6 deserves special attention. In the benchmark CSV model,alteringthefirm’sendowmentofequityhasnoimpactonexpectedreturnoffered to investors, or the characteristics of the optimal contract. The reason is because in the frictionless benchmark there are no liquidity provision or liquidity premium channels at work. This result does not carry through in our framework, where these channels are 26

present. As in the benchmark model—indeed, for exactly the same reason—there is no effect of an increase of firms’ endowment on the optimal contract. But in our framework as credit demand and bond issuance increase, the liquidity provision channel reduces liquidityintheOTCmarket. Thisleadsinvestorstorepricebonds’illiquidity: theliquidity premiumchannel. Thus,inourframework,thesizeofthecorporatesectorrelativetothe financialsectorinfluencesliquidityprovisionandliquiditypremia. Moreover,ourmodel ishomogeneousofdegreezeroin(n ,e ). Thatis,increasingthesizeofthecorporatesector 0 0 n and the financial sector e in the same proportions have no effect on market thickness, 0 0 the liquidity premium, or the characteristics of the optimal contract. The equilibrium in the credit market will be described by the same expected hold-to-maturity return and an increaseinbondissuancecommensuratetotheincreaseinsizeofthetwosectors. 3.5 A Numerical Illustration We present a simple numerical illustration using the following parameter values. We set the endowment of firms at n = 0.2 and the endowment of investors at e = 1. Investors’ 0 0 preferencesaredescribedbyadiscountfactorforimpatientinvestorsβ = 0.85,whileδwill takedifferentvaluesin[0,1]toillustratetheresultsestablishedabove. Firms’technology expected return is given by Rk = 1.2, whereas the return on storage is assumed to be r = 0.01. The parameters of the matching function in the OTC market are the scaling constant ν = 0.2 and the elasticity of the matching function with respect to sell orders is α = 0.5. Theshareofthesurplusthataccruestoimpatientinvestorsis ψ = 1. Idiosyncratic productivity shocks ω are distributed according to a log-normal distribution with mean equal 1 and variance equal to 0.25. Finally, monitoring costs are a share μ = 0.2 of firms’ revenue. Webeginwiththefrictionlessbenchmark,takingδ = 0.19 Theequilibriumofthemodel is described by entrepreneurs’ choice of leverage, l , and risk, ωˉ, subject to the constraint 0 imposedbytheinvestors’break-evencondition(15)andtheconsistencyrequirementsfor liquidity, θ, and price, q , in the OTC market. The characteristics of the optimal contract 1 (l ,ωˉ) determine the hold-to-maturity return, Rb, and thus the secondary market price 0 q . (Recall that the return on executed orders in secondary markets is pinned down by 1 ψ, r, and β.) The optimal contract will determine the portfolio allocation of investors and thus secondary market liquidity θ (equation 17). Thus, we use the (l ,ωˉ)-space to 0 describetheoptimalcontractandtheequilibriumofthemodel. Figure 4depictsthefirm’s 19FromProposition3thefrictionlessbenchmarkisobtainedifalternativelywesetβ=1/1.01,orifψ=1 (asinourexample)ande issufficientlyhighso f(θ)=1. 0 27

isoprofitcurvesingreen,20 anddisplaystheinvestors’break-evenconditionbyaredline. Firm’sprofitsincreasewithleverageanddecreasewithrisk,soisoprofitcurvesrepresent higher profits moving south-east in the figure. The private equilibrium in the frictionless benchmark economy is given by the tangency between the break-even condition and the isoprofitlineshownbythesolidblackdotinFigure 4. Figure 5 illustrates the case of an increase in the liquidity shock, δ, (i.e., case 1 of Propositions 4 and 6). As the probability of becoming impatient increases, investors require a higher liquidity premium to be compensated for liquidity risk (Proposition 4). Incontrast,thefirm’sisoprofitlinesforagivencontractspecifiedby(l ,ωˉ)areinvariantto 0 δ,thusthedemandforcreditisinvariantto δ. Nevertheless,thefirmadjuststhetermsof thecontractitoffersintheprimarymarketowingtotheincreaseintheliquiditypremium. In particular, the firm reduces its supply of primary debt, which partially compensates investors for the reduction in secondary market liquidity. The resulting equilibrium has alowerlevelofleverageandalessriskydebtcontract,asshowninFigure 5(Proposition 6). Finally, Figure 6 presents a decomposition of the total corporate premium Φt paid on the primary debt contract in terms of the default premium Φd and the liquidity premium Φ‘. The figure shows that lower levels of leverage and risk due to increased liquidity demand result in lower total corporate bond premia. Naturally, the liquidity premium goesup,butthedefaultpremiumdecreasessincethefirmisofferingalowerωˉ (Proposition 6),andthelattereffectdominatesinthiscase. 4 Efficient Liquidity in OTC Markets We consider a social planner that is constrained by the presence of search frictions and the structure of trade in the OTC market. Hence, our concept of efficiency is one of constrainedefficiency,orsecondbest.21 The planner chooses the optimal contract to maximize the profits of the firm while internalizing both the liquidity provision and liquidity premium channels. To formalize the planner’s problem let (l ,ωˉ,θ,q ) be allocations that describe the socially efficient 0 1 outcome and let (l pe ,ωˉpe,θpe,q pe ) be the allocations in the private equilibrium described in 0 1 section 3. Then,theplanner’sproblemcanbewrittenasmax [1 Γ(ωˉ)]Rkl ,subject ωˉ,l0,θ,q1 − 0 20Notethattheshapeoftheisoprofitcurves(increasingandconcave)holdsingeneral,asfollowsfrom thepropertiesoftheΓ(ωˉ)function,anddoesnotdependontheparticularvaluesusedinourexample. 21Intheinterestofspacetheanalysisinsections 4and5restrictsattentiontothemoreinterestingcase whereθ (θ,θ),sotradingprobabilitiesdependonthematchingfunction(4)andarenotpinneddownby ∈ theminimumnumberofbuyorsellorders. 28

toequations(7),(17),and U(l ,ωˉ,θ,q ) U(l pe ,ωˉpe,θpe,q pe ) . (20) 0 1 ≥ 0 1 Condition (20) says that the planner cannot choose equilibrium allocations that result inlowerwelfareforinvestorscomparedtotheprivateequilibrium,whereasequations(7) and(17)forcetheplannertorespectthedeterminationofpricesandliquidity,respectively, in the OTC market.22 The social planning problem differs from the private equilibrium in two respects. First, the planner need not respect the investor’s break-even condition (15), but cannot make investors worse off, i.e., needs to satisfy (20). Second, the planner internalizes how period 0 choices affect liquidity in the secondary market by explicitly considering (17) as a constraint, which, in contrast, is a condition of the private equilibrium.23 Inthismanner,theplannerinternalizesboththeliquidityprovisionandliquidity premiumchannels. Wesubstituteequations(7)and(17)intheplanner’sproblem,andletλbethemultiplier onconstraint(20),toobtainthatthesociallyoptimalchoiceofleverageisgivenby ∂Rb ∂U∂θ [1 Γ(ωˉ)]Rk = λ n (U U )+b u + . (21) 0 b s 0 b − − − ∂l ∂θ ∂l 0 0 " # Thatis,themarginalincreaseinthefirm’sprofitsfromadditionalleverageneedstobe proportionaltothemarginalreductionintotalexpectedutilityforinvestors. Thelatterhas three components: (i) the portfolio composition: as leverage increases investors need to re-allocate n units from storage to bonds; (ii) the effect on the expected hold-to-maturity 0 return Rb; and (iii) the effect through secondary market liquidity: as liquidity increases it becomes easier for impatient investors to sell their bonds, but it becomes more difficult forpatientinvestorstobuybondsandearnthereturn Δ inthesecondarymarket. Similarly,thesociallyoptimalchoicefortheriskprofileofcorporatedebtisgivenby ∂Rb ∂U ∂θ l Γ (ωˉ)Rk = λ b u + . (22) 0 0 0 b ∂ωˉ ∂θ ∂ωˉ " # That is, the marginal increase in the firm’s profits from reducing risk need to be proportional to the marginal reduction in total expected utility for investors, which has 22Wealsoconsideredamoregeneralproblem,asanalternativebutnotreported,wheretheplannercan additionallydeterminethetermsoftradeinthesecondarymarketandassignsParetoweightsonthetwo agentstomaximizeasocialwelfarefunction. 23Recallthatinvestors,andthusfirms,explicitlyconsidered(7)intheprivateequilibriumaswell,thus itsexplicitconsiderationdoesnotmodifytheplanner’sproblemrelativetotheprivateequilibrium,unless theplannercanaffectthetermsofsecondarytrade. 29

two components: the effect on the hold-to-maturity return Rb and the effect through secondarymarketliquidity. Takingaratioofequations(21)and(22)gives 1 − Γ(ωˉ) = n 0 (U b − U s )+b 0 u b ∂ ∂ R l0 b + ∂ ∂ U θ ∂ ∂ l θ 0 . (23) Γ 0 (ωˉ)l 0 − b 0 u b ∂ ∂ R ωˉ b + ∂ ∂ U θ ∂ ∂ ω θ ˉ This equation, together with the constraint on investors total expected utility (20), describes the socially optimal debt contract.24 We are ready to establish the generic inefficiencyoftheprivateprovisionofliquidity.25 Proposition7(GenericConstrainedInefficiencyofLiquidityProvision) Consideraplanner that designs an optimal financial contract, as described by (20), (23), (7) and (17). Given the parameters (α,ψ,r) belonging to a generic set , the planner will set a level of secondary P market liquidity that is different from the private equilibrium. That is, the private equilibrium is genericallyconstrainedinefficient. GivenProposition 7,wecanidentifytwodistortedmarginsthatdriveasetofwedges betweentheprivateandsociallyefficientoutcomesthatareapparentfromcomparingthe equilibriumconditions(15)and(16)tothesocialplanner’scounterparts(20)and(23). On theonehand,comparingequation(23)toequation(16),revealsthepresenceofapecuniary externalitythatcomesfromthefactfirmsdonotinternalizehowtheirbondissuanceaffects their funding cost. In fact, firms’ funding cost are set by investors, who will be affected bybondissuanceintwoways. First,additionalbondsintheirportfolioswillchangetheir expected utility if there is a difference between the utility they expect to receive when buying bonds or liquid assets, represented by the term n (U U ). Second, additional 0 b s − bonds will affect market liquidity, which in turn is going to affect investors’ expected utility,representedbytheterm ∂U/∂θ. Theseconddistortionbecomesapparentwhencomparingequation(15),theinvestors’ break-even condition, to equation (20), the weak Pareto improvement constraint faced by the planner. Since U = U + (1 δ)(1 + r)(Δ (1 + r))p(θ), we can rewrite equation s a − − (20) as n (l 1)(U U ) = e (1 δ)(1 + r)(Δ (1+r)) p(θpe) p(θ) . Written this way, 0 0 b s 0 − − − − − the equation tells us that as long as the optimal level of public liquidity is different than (cid:2) (cid:3) the equilibrium level of private liquidity, i.e, θpe , θ, then the expected utility of holding 24Theconstraintwillalwaysbebindingsincetheplannercaresonlyaboutthefirm,butthisneednotbe thecaseiftheplannermaximizesaggregatesocialwelfare. Inthatcasetheplannermaywanttosplitthe aggregategainsaccordingtosomesetofParetoweights. 25See also Geanakoplos and Polemarchakis (1986) for a general characterization of constrained inefficiency. 30

bonds and liquid assets will have to be different U (θ) , U (θ). This reflects the presence b s of a congestion externality that comes from the fact that investors do not internalize how theirportfoliochoicesaffectmarketliquidity. Thefollowingpropositionsummarizestheinteractionbetweenthesetwoexternalities. Proposition 8 (Constrained Efficient Equilibrium) The constrained efficient allocations can becharacterizedconditionalonthemodelparameters (α,r,ψ) asfollows: • If ψ(1+αr) > α(1+r) then secondary market liquidity generates a positive externality on investors(∂U/∂θ > 0);theplannerimplementsahigherlevelofsecondarymarketliquidity (θ > θpe); and the socially optimal financial contract is characterized by lower leverage, l < l pe ,andlessrisk, ωˉ < ωˉpe. 0 0 • If ψ(1+αr) < α(1+r) then secondary market liquidity generates a negative externality on investors (∂U/∂θ < 0); the planner implements a lower level of secondary market liquidity (θ < θpe); and the socially optimal financial contract is characterized by higher leverage, l > l pe ,andmorerisk, ωˉ > ωˉpe. 0 0 • If ψ(1 + αr) = α(1 + r) then there is no externality (∂U/∂θ = 0) and equilibrium is constrainedefficient,i.e., (l ,ωˉ,θ) = (l pe ,ωˉpe,θpe). 0 0 To understand the intuition behind the proposition consider that the planner internalizes the pecuniary externality faced by firms and is also aware of the presence of the congestion externality faced by investors. As such, the planner faces a different trade-off between leverage and risk relative to an individual firm. Recall from the firms’ problem (3)thatprofitsincreasewheneverleverage l ishigher,orriskωˉ islower. Higherleverage 0 impliesalargerscopeofthefirm,whereaslowerriskimpliesthatalargershareofprofitsis retainedbythefirm. Howcantheplannerincreasetheprofitabilityoffirms? Proposition 8tellsusthattheanswerdependsontheparameters(α,r,ψ). For example, consider the effect of an increase in liquidity on investors’ welfare. On the one hand, increased liquidity generates ex ante welfare gains for impatient investors because it is easier to sell unwanted bonds in the secondary market. On the other hand, patientinvestorssufferasitbecomesmoredifficulttoearnahigherreturnbypurchasing bonds at a discounted price. The gains to impatient investors outweigh the losses to patientinvestors,makinginvestorsexantebetteroff,i.e.,∂U/∂θ > 0andwesaythereisa positiveexternality. Thisisthecasewhenthemodelparameterssatisfy ψ(1+αr) > α(1+r). Intuitively, in this case the trade surplus that accrues to impatient investors is relatively large. Alternatively, the return on storage that patient investors receive if they fail to executeasecondarytradeissufficientlylow,sothat r < (ψ α)/(α αψ). − − 31

In the case of the positive externality, in order to implement a higher level of market liquidity the planner is going to reduce bond issuance by firms, lowering leverage and scope of the firm. But, at the same time, the planner also reduces the risk of the financial contract, allowing firms to retain a larger share of profits in expectation. In this way the plannerdirectsthefirmtooperateatasmallerscale,whileatthesametimepayinglower financing costs. By restricting bond issuance and enhancing market liquidity, the cost of financing are much smaller than what a firm that does not internalize the externality would have expected, because the planner is able to redistribute the gains for investors backtofirmsintheformofevenlowerfinancingcosts.26 In this way the planner effectively takes advantage of the congestion externality to createadditionalsurplusforinvestorsthatisthenredistributedbacktofirmsbyaddressing the pecuniary externality. It is worth noting that the redistribution takes place through changesintheliquiditypremium,asmatteroffact,asweshowbelow,oursocialplanner problemcanbeimplementedwithnodirecttransfersfrominvestorstofirms. By contrast, when there is a negative externality, i.e., ∂U/∂θ < 0, a reduction in marketliquiditygeneratesgainstopatientinvestorsthatoutweighthelossestoimpatient investors,makinginvestorsexantebetteroff. Thisisthecasewhenψ(1+αr) > α(1+r). In the case of a negative externality, in order to implement a lower level of market liquidity the planner is going to increase bond issuance by firms, increasing leverage and scope of the firm. However, at the same time the planner increases the risk of the financial contract, reducing the expected share of profits that the firm retains. In this way the planner allows the firm to operate at a larger scale, while at the same time paying higher cost of financing. But financing cost are lower than what a firm that does not internalize theexternalitywouldhaveexpectedatthenewhigherlevelofleverage,astheplanneris able to redistribute the gains for investors owing to lower liquidity back to firms in the formoflowerfinancingcosts. Finally,intheknife-edgecasewhere ψ(1+αr) = α(1+r)privateliquidityisefficientso thatatthemarginanincreaseinliquiditygeneratesgainsforimpatientinvestorsthatare perfectlyoffsetbylossestopatientinvestors,andtheplannercannotexploitthecongestion externality to improve upon the private equilibrium. This special case highlights the relationship of our result with the well-known Hosios condition.27 But, note that in our 26Itisinterestingtonotethatbyimplementinghighersecondarymarketliquidity,theplannerinessence increases funding liquidity in the primary market by implementing a reduction in the liquidity premium andthusinthetotalbondpremium. 27TheparameterrestrictionisanalogoustotheHosios(1990)rulethatdeterminestheefficientsurplus splitinsearchandmatchingmodelsofthelabormarket. ArseneauandChugh(2012)studytheimplications ofinefficientsurplussharingforoptimallabortaxationinadynamicgeneralequilibriumeconomy. 32

model the planner is not trying to correct the congestion externality, rather she aims to correctapecuniaryexternalitywhileexploitingthecongestionexternality. 4.1 Optimal Private Liquidity Regulation To analyze optimal liquidity regulation we allow the planner access to a complete set of tax instruments. Specifically, we introduce a marginal tax τs on the return from storage in period 0 and a marginal tax τl on leverage (negative taxes correspond to subsidies).28 Withthesetaxinstruments,theobjectiveofinvestorsbecomes U = b U +s U (1 τs)+Ts 0 b 0 s − andtheobjectiveofthefirmchangesto[1 Γ(ωˉ)]Rkl τlλpel +Tl. Thetaxesarefundedin 0 0 − − a lump-sum fashion on the same agents, thus Tl = τlλpel and Ts = τss U in equilibrium. 0 0 s Also,inordertosimplifytheexpositionnotethatwehavenormalizedthetaxonleverage by the Lagrange multiplier, λpe > 0, on the constraint faced by firms (i.e., the investors’ break-evencondition(15)). Proposition 9characterizestheoptimalregulationofprivateliquidityprovision. Proposition 9 (Implementation of Optimal Liquidity Regulation) The planner’s solution can be implemented by levying distortionary taxes on the portfolio allocation decision of investors and the financing decision of firms. The resulting optimal taxes on storage, τs, and leverage, τl, aregivenby: e U (θpe) τs = 0 1 s , (24) b − U (θ) 0 s ! n U u ∂Rb τs +u ∂Rb ∂θ ∂Rb ∂θ ∂U τl = 0 s b ∂ωˉ b ∂l0 ∂ωˉ − ∂ωˉ ∂l0 ∂θ (25) b u ∂Rbh+ ∂θ∂U i 0 b ∂ωˉ ∂ωˉ ∂θ wheretheterminsquarebracketsandthedenominatorin(25)arestrictlypositive. Theroleforthetaxonstorageistocreateawedgeintheinvestors’break-evencondition (15), i.e., U , U as long as θpe , θ. This allows the planner to implement the desired b s allocation without making investors worse off. Moreover, the role of the tax on leverage is to make the firm internalize the pecuniary externality of bond issuance on its funding costs. CombiningtheinsightsofProposition 9withProposition 8above,itiseasytocharacterize the optimal tax system more specifically. When ψ(1 + αr) > α(1 + r), the liquidity externality is positive so that the planner wants to implement higher liquidity relative to 28Weconsidertaxinstrumentstocorrectthedistortedmargins,bothotherinstrumentssuchasleverage orportfoliorestrictionscouldalsobeconsidered. SeealsoPerottiandSuarez(2011),whoproposePigouvian taxationtoaddressexternalitiesfromtheunder-provisionofliquidity. 33

theprivateequilibrium,θ > θpe. Accordingly,theoptimalregulationneedstobedesigned in a way that results in investors holding a more liquid portfolio. This can be achieved through a storage subsidy, so that τs < 0. Moreover, the optimal regulation needs to be designed in a way that results in firms issuing fewer bonds in the primary market, which can be achieved through a tax on leverage, so that τl > 0. By the same logic, when ψ(1 + αr) < α(1 + r), the liquidity externality is negative and θ < θpe. The optimal tax system calls for a tax on storage, τs > 0, and a leverage subsidy, τl < 0. Only in the knife-edgecasewhere ψ(1+αr) = α(1+r)wehavethat τl = τs = 0. 4.2 A Numerical Illustration We continue the numerical example in section 3.5. Recall that in this illustration, ψ = 1. Moreover, because the planner has the same objective as the firm, the isoprofits lines are the same in both problems. Figure 7 shows the planner’s solution and the private equilibrium for two cases: δ = 0 and δ = 0.1. In a frictionless environment (δ = 0), the planner’s solution coincides with the private equilibrium (as we proved in Proposition 3). However, when there is a positive demand for liquidity, δ > 0 and β < 1/(1 + r), andsecondarymarketliquidityisnotsufficientlyhightoguarantee f(θ) = 1,theplanner chooses lower leverage and a less risky bond contract, i.e., lower l and ωˉ. The reason 0 is because the planner internalizes the effect of the leverage decision on liquidity in the secondarymarket. Thisinducestheplannertoconsiderasteeperconstraintcomparedto the breakeven condition considered by firms (where market liquidity is taken as given). As a result, the planner understands how lower leverage and risk improves borrowing termsonthemargin,whenthetotalsocialcostsaretakenintoaccount. Table1showsthechangeinequilibriumallocationsbetweentheprivateandplanner’s solutions for δ = 0.1 as ψ moves from 1 to 0. Consistent with the analysis above, the planner’s allocations can be replicated using appropriate tax instruments (subsidies if theyarenegative)onleverageandstorage. For ψ = 1,theliquidityexternalityispositive implyingthatliquidityissuboptimallylowintheprivateequilibrium. Theplannerwould liketoimplementataxonleveragetogeneratemoreliquidityinthesecondarymarket(in thiscaseallthesurplusgoestosellerssothetaxonstorageisirrelevant). However,asthe share of the gains from trade that accrues to impatient investors declines, the size of the liquidityexternalityshrinks. Hence,theplannerislessaggressiveinchoosingtheoptimal combination of leverage tax and storage subsidy, i.e., both τl and τs shrink in absolute value. When the parameterization of ψ satisfies ψ(1+αr) = α(1+r), the externality zeros out and the optimal tax system implies τl = τs = 0. For values of ψ below that point, the 34

liquidity externality becomes negative, so that liquidity is over-provided in the private equilibrium. Accordingly, the sign of the optimal tax system flips so that leverage is subsidized, τl < 0,andstorageistaxed, τs > 0. 5 Optimal Public Liquidity Management Inthissection,weexaminehowtheoptimalmanagementofpublicliquiditycanalleviate trading frictions and improve economic efficiency beyond what can be achieved by liquidity regulation, as studied in the previous section. Through the lens of our model, any public policy that alters both private and public portfolios effectively shifts liquidity risk betweentheprivateandthepublicsector. Thisshiftingofliquidityriskalterstheliquidity premia which, in turn, influences savings and investment decisions in the real economy. Inpractice,importantpoliciesthatcanalterpublicandprivateportfoliosarequantitative easing or large scale asset purchases, as the ones implemented in the U.S., Europe, and Japan. 5.1 Quantitative Easing Policy WemodelQEthroughdirectpurchasesbythecentralbankoflong-termilliquidassets(the financialcontractsissuedbyfirmsandwhichareretradedbyinvestorsinOTCmarkets).29 These purchases are financed by the issuance of short-term liquid liabilities, referred to as reserves, that offer a return that is at least as large as that offered by the storage technology. This seems a reasonable approximation for the policies implemented by the Federal Reserve during the Great Recession, where lending facilities and asset purchases werefinancedprimarilywithredeemableliabilitiesintheformofreserves(seeCarpenter etal. 2013). The key assumption we make to model QE is that the assets purchased by the central bankarerelativelylessliquidthanreserves. Thiscouldbeviewedasastrongassumption for the Federal Reserve QE program, which was limited to U.S. Treasuries and Agency MortgageBackedSecurities. Whileitmightbethecasethattheseassetsarehighlyliquidat ornearorigination,theevidencesuggeststhattheybecomelessliquidastheyare retraded invenuesthatmightbewellrepresentedbyOTCmarkets.30 Thisbecomeslessofaconcern whenconsideringQEprogramsinotherjurisdictions,likeEuropeorJapan,wherecentral 29Notethatapooloffirms’contractswillhavenocreditrisk,sincetherearenoaggregateshocks. 30VayanosandWeill(2008)arguethattheoff-the-runphenomenoncanbeexplainedbytradefrictionsin U.S.Treasurymarkets. VickeryandWright(2013)describetheTBAmarketandthethemarketfor“specified pool”agencyMBSasOTCmarkets. 35

banks have purchased non-government guaranteed assets, which are perceived as less liquidthancentralbank’sreserves. At the beginning of the initial period, the central bank credibly commits to purchase ˉ a quantity b of bonds from investors and hold them to maturity. These bond purchases 0 are financed through the issuance of sˉ units of reserves that pay interest rˉ. In our model 0 reserves are a perfect substitute for the storage technology from the point of view of investors, thus rˉ r.31 The central bank waits for the primary debt market to clear and ≥ thenmeetswithinvestorstoexchangereservesforbonds. TheQEoperationisconducted inafrictionlessmarketthatmeetsafterbondshavebeenissuedbutbeforetheOTCmarket opens.32 Investorscanfreelytradereservesforconsumptionwiththecentralbankatanypoint.33 Thecentralbankbudgetconstraintintheinitialperiodissimply b ˉ = sˉ . (26) 0 0 In addition, we assume the central bank finances itself in period 1 with reserves only. This assumption prevents the central bank from injecting additional resources into the economyintheinterimperiod. Inordertokeepitsbondholdings,thecentralbankneeds torolloveritsoutstandingreservesandpayinterestontheminperiod1. Thecentralbank will have to borrow an amount equal to (1 + rˉ)sˉ .34 Finally, in period 2 the central bank 0 receivesthedebtpayoutfromthefinancialcontractandexpends(1 +rˉ)2sˉ ininterestand 0 principal on outstanding reserves. It is assumed that the central bank allocates reserves evenlyacrossinvestorswhodemandreservesinagiventimeperiod. The central bank faces three constraints that, taken together, serve to limit the size of itsQEprogram. First,weassumethatthecentralbankisatadisadvantagerelativetothe privatesectorinmonitoringinvestmentprojects. Itthusneedstopayahighermonitoring cost relative to investors, denoted by μˉ > μ. Consequently, any positive effects of QE wouldnotaccruefromenhancedmonitoring,butfromitseffectonliquiditypremia. This 31Ourresultsarequalitativelythesameifweimposethatrˉ=r. Nonetheless,aswewillshowinthenext section,allowingthecentralbanktopayhigherinterestonreservesprovidesthecentralbankanadditional tooltomanagepublicliquidity. 32Wehaveabstractedfromtradefrictionsbetweenthecentralbankandinvestors,asinpracticetheFederal Reserveannouncesinadvanceitsintentiontobuybondsandhasreadilyavailabletradingcounterparties. 33ThisisisomorphictoamodelinwhichtradeintheFedFundsmarketisfrictionless. Anotherliterature studiesfrictionaltradeintheFedFundsmarket,see,forexample,AfonsoandLagos(2015)orBianchiand Bigio(2014). 34In practice, the long-term assets held by central banks pay interest in the interim period, and in an environmentoflowshort-terminterestratestheseholdingswillgenerateapositivenet-interestincomefor thecentralbank. Butforsimplicityweabstractfromtheseconsiderations. See, forinstance, Carpenteret al. (2013)forestimatesofnet-interestincomefortheFederalReserve. 36

impliesthatinexpectationthecentralbankanticipatesreceivingRˉbb ˉ foritsassetholdings, 0 with Rˉb the expected hold-to-maturity return on financial contracts for the central bank, givenby35 l l Rˉb(l ,ωˉ) = 0 Rk Γ(ωˉ) μˉG(ωˉ) = Rb(l ,ωˉ) 0 Rk(μˉ μ)G(ωˉ) . 0 0 l 1 − − l 1 − 0 0 − − (cid:2) (cid:3) Second, the central bank needs to fully finance its funding cost, i.e., the total interest onreserves,withitsexpectedreturnonassets. Thatis, Rˉb (1+rˉ)2 . (27) ≥ Note that if the central bank buys a portfolio of bonds, it does not undertake any credit risk,asfirms’returnsareindependent. Finally, we assume that investors cannot be made worse off by QE, as we describe in section 5.3. 5.2 QE, Market Liquidity, and the Supply of Credit In period 0 investors allocate their wealth across two assets: the storage technology and bonds. Sothebudgetconstraintatt = 0isgivenbys +bs = e ,withs ,bs 0. Subsequently, 0 0 0 0 0 ≥ investorsexchangeb ˉ bondsforsˉ reserveswiththecentralbank. Followingtheapproach 0 0 of Section 2, we consider the optimal behavior of impatient and patient investors in t = 1 whentheyarrivewithagenericportfolioofstorage,reserves,andbonds(s ,sˉ ,bs b ˉ ). 0 0 0 − 0 Impatient Investors. By Assumption 3 impatient investors want to consume all their wealthatt = 1. Theycanconsumethepayoutsoftheirliquidassets: (1+r)s +(1+rˉ)sˉ ;in 0 0 addition,theycanconsumetheproceedsfromtheirsellordersintheOTCmarket: q units 1 of consumption for each order executed. Thus, the expected consumption of impatient investorsinperiods1and2,respectively,isgivenby cI = (1+r)s +(1+rˉ)sˉ + f(θ)q (bs b ˉ ) , (28) 1 0 0 1 0 − 0 and cI = (1 f(θ))R(bs b ˉ ) . (29) 2 − 0 − 0 Patient Investors. Patient investors only value consumption in the final period and, as a 35Note that the central bank in the model buys bonds at face value. This implies that the effect of QE doesnotrelyonthepurchaseofbondsatdistressedvalues. Inaddition,ifthecentralbankpurchasesbonds atadiscountitwillincreasetheexpectedreturnonassetpurchasesandrelaxconstraint(27). 37

result,arewillingtoplacebuyordersintheOTCmarketbecausethereturnfromdoingso, Δ, is strictly greater than the return on storage, 1 +r. Moreover, it is also the case that the returnonreserves,1+rˉ,isatleastaslargeasthatonstorage,sopatientinvestorsarewilling to allocate liquid wealth to reserves. Accordingly, liquidity provision in the secondary market will depend on the return on OTC trade, Δ, relative to the return on reserves, 1+rˉ. Specifically, if 1+rˉ < Δ patient investors will pledge all their liquid wealth to place buy orders in the OTC market. On the other hand, if 1 +rˉ > Δ patient investors will use theirliquidwealthtobuyhigheryieldingreservesfirstandthenallocatetheremainderof their liquid wealth to placing buy orders in the OTC market. For expositional purposes, we assume throughout the remainder of the paper that 1 +rˉ < Δ (although for the main results of this section—stated below in Propositions 10 and 11—we trace out the proofs overtheentireparameterspaceofthemodel,whereappropriate). When the anticipated return to OTC trade exceeds the return on reserves, patient investorsuse(1+r)s +(1+rˉ)sˉ unitsofconsumptiontoplacebuyorders. Afraction p(θ) 0 0 are matched allowing patient investors to exchange consumption for bonds, while the 1 p(θ) unmatched portion needs to be reinvested in liquid assets in period t = 1. Given − thatthecentralbankneedstofinanceitselfintheinterimperiod,itwillreallocatereserves to patient investors, hence individual reserve holdings in the interim period for patient investors, sˉP, total (1 + rˉ)sˉ /(1 δ). All remaining units of consumption are placed into 1 0 − theloweryieldingstoragetechnology,soexpectedstorageholdingsattheendof t = 1,sP, 1 equal sP = (1 p(θ))[(1+r)s +(1+rˉ)sˉ ] (1+rˉ)sˉ /(1 δ), which is strictly positive from 1 − 0 0 − 0 − Assumption 4. Itfollowsthatexpectedconsumptionofpatientinvestorsequals (1+rˉ)2sˉ (1+r)s +(1+rˉ)sˉ cP = sP(1+r)+ 0 + bs b ˉ +p(θ) 0 0 R . (30) 2 1 1 δ 0 − 0 q 1 − ( ) Using the optimal behavior of investors in period 1, summarized in equations (28)- (30), we can rewrite the expected lifetime utility as the portfolio weighted average of the utilities of the three assets available in the initial period: U = U s + U sˉ + U (bs b ˉ ). s 0 sˉ 0 b 0 − 0 As before, the expected utility of investing in storage and bonds, U and U , are given by s b equations(13)and(14),respectively. Ontheotherhand,theexpectedutilityofreservesis givenby rˉ r U = δ(1+rˉ)+(1 δ)(1+rˉ) (1 p(θ))(1+r)+ − +p(θ)Δ . (31) sˉ − − 1 δ (cid:20) − (cid:21) Reserves yield 1 + rˉ for impatient investors. For patient investors, there is additional compensation that comes from the expected return from buy orders in the secondary market,plusthespreadbetweenreservesandstorage, rˉ r 0,fortheadditionalreserves − ≥ 38

boughtinperiod1.36 Finally,notethat,whentheequilibriuminthecreditmarketisgivenby(b ,Rb),market 0 liquiditycorrespondsto (1 δ)[(1+r)s +(1+rˉ)sˉ ] (1 δ)Δ (1+r)(e 0 n 0 (l 0 1))+(1+rˉ)b ˉ 0 0 0 − − − θ = − = . (32) δ(b 0 − b ˉ 0 )q 1 h δRb n 0 (l 0 − 1) − b ˉ 0 i (cid:16) (cid:17) This expression establishes a link between QE and secondary market liquidity which we summarizeinthefollowingproposition. Proposition 10 (The Real Effects of QE) Quantitative easing, i.e., the size of the bond buying ˉ program, b ,increasessecondarymarketliquidity θ and,hence,increasesfirm’sinvestment. 0 The intuition behind this result is straightforward. Each bond bought by the central bank will be held to maturity and, therefore, reduces the number of sell orders in the secondary market. At the same time, these bonds need to be financed with reserves, whichpatientinvestorscanusetosubmitadditionalbuyordersinthesecondarymarket. So, a bond buying program has a direct effect on secondary market liquidity because it altersthecompositionofinvestor’sportfoliosawayfromilliquidbondstowardpublically provided liquid assets. The resulting reduction in the liquidity premium demanded by investors pushes down the cost of funding for firms, who respond by taking on higher leverage and risk in equilibrium. This later indirect effect attenuates the effect of the QE programasincreasedbondissuancebyfirmscrowdsoutpublicandprivateliquidity. It should also be noted that Proposition 8 is presented from the perspective of a central bank that wants to increase liquidity by expanding reserves in order to purchase illiquid bonds. But, this result is more general. A central bank that starts with an initial endowment of bonds could remove liquidity by becoming a net seller to investors of illiquid bonds in exchange for reserves. A quantitative tightening (QT) program such as thiswouldeffectivelywithdrawpublicliquidityandreducesecondarymarketthickness. 5.3 Optimal Public Liquidity Management via Quantitative Policies To understand the role of QE in the optimal policy mix, we consider a planner who wants to maximize firm profits, but is restricted by the central bank budget constraint, equation(26),andthefinancingconstraint,equation(27). Inaddition,aswiththeplanner 36If,1+rˉ>Δ,patientinvestorswillusetheirliquidwealthfirsttobuyreserves,andthenwillusetheir remaining liquid wealth to place buy orders in the OTC market. Proceeding as above we can derive for patientinvestorssP,cP,andU . 1 2 sˉ 39

in Section 4, we assume the QE program cannot make investors worse off. To write this later constraint, let U(l ,ωˉ,θ,b ˉ ,rˉ) be the expected lifetime utility of investors when the 0 0 equilibrium is described by (l ,ωˉ,θ), with the secondary market price given by (7), and 0 the QE program described by (b ˉ ,rˉ). Similarly, let U(l pe ,ωˉpe,θpe) be the expected lifetime 0 0 utilityofinvestorsintheprivateequilibrium,whenthesecondarymarketpriceisgivenby (7). We refer to this planner that have access to QE policies as the central bank. Then, the central bank’s problem can be written as max [1 Γ(ωˉ)]Rkl , subject to equations l0,ωˉ,θ,bˉ 0,rˉ − 0 (26),(27),(32),and U(l ,ωˉ,θ,q ,b ˉ ,rˉ) U(l pe ,ωˉpe,θpe,q pe ) (33) 0 1 0 ≥ 0 1 The following proposition characterizes the role of QE as part of the optimal policy mix. Proposition11(QEasPartoftheOptimalPolicyMix) TheoptimaldesignofQEconditional onthemodelparameters (α,r,ψ) isdescribedasfollows: • Ifψ(1+αr) > α(1+r),thenQEimprovesupontheconstrainedefficientallocation,andthe ˉ optimal QE program consists of a positive bond buying program, b > 0, coupled with an 0 interestonreservesthatisstrictlygreaterthanthereturnonstorage, rˉ > r. • Ifψ(1+αr) α(1+r),thenQEdoesnotimproveupontheconstrainedefficientallocation, ≤ ˉ andoptimallythesizeoftheQEprogramis b = 0. 0 TheQEprogramiseffectivebecauseitaffectsmarketthickness,whichinthepresenceof aliquidityexternality,allowsthecentralbanktoincreasetheexpectedutilityofinvestors and transfer those gains to firms. Intuitively, the QE program transfers illiquid bonds from investors, who value liquidity, to the central bank, who does not value liquidity becauseitisalong-terminvestorandisnotsubjecttoruns. Thepublicprovisionofliquidityisinherentlydifferentfromliquidityregulation. Both policies affect the level of market liquidity, but regulation trades off liquidity and credit provision, whereas public liquidity management implies that public liquidity provision and credit provision move in tandem. This is due to the fact that public liquidity provision enhances the intermediation technology of the economy, as the transfer of liquidity risk between the public and private sector can only be achieved in the model through quantitativepolicies. However, the proposition shows that this technological improvement can only be realizedwhentherearesocialgainsfrommanagingliquidity. Indeed, whentheliquidity externality does not exist, i.e., the knife edge case where ψ(1 + αr) = α(1 + r), there is 40

no role for the management of public liquidity. When the externality is negative (so that liquidity is suboptimally high in the private equilibrium), the central bank wants to remove liquidity from the secondary market. As discussed at the end of the previous subsection, this could, in principle, be done through a QT program, whereby the central banks reduces the size of its balance sheet by selling bonds to investors in exchange for reserves. There are a few additional points worth mentioning. First, note that the proposition suggests QE is effective when the interest rate on storage is sufficiently low, r < (ψ − α)/(α αψ). Althoughitisbeyondthescopeofthismodel,theseconditionsindicatethat − QE may be an effective policy response in a protracted low interest rate environment. By thesametoken,thepropositionalsosuggeststhatQTcanbeoptimalwhentheinterestrate issufficientlyhigh,sothatr > (ψ α)/(α αψ). Inthecontextofthecurrentpolicydebate, − − our framework offers support for a strategy of raising interest rates prior to unwinding thesizeofthebalancesheet. Second, while we have shown that the optimal management of public liquidity can lead to a Pareto improvement, these quantitative policies do not explicitly address the externalities identified in Section 4. Indeed, a QE program is optimal when liquidity is inefficiently low, or equivalently, when firm’s leverage and the riskiness of the contracts it offers to investors are inefficiently high. While QE is effective at boosting liquidity, it does so at the expense of encouraging firms to take on even more leverage and write even riskier contracts. This opens the door for optimal liquidity management (through quantitativepolicies)tocoexistwithoptimalliquidityregulation,echoingasimilarresult foundinHolmströmandTirole(1998). Weexaminethisinmoredetailinthequantitative exercisebelow. Finally, its useful to note that when QE is effective, the absence of constraints that limitthesizeoftheprogramcouldleadtoanextremeoutcomeinwhichthecentralbank disintermediates the bond market. That is, the optimal policy is for the central bank to buy all the bonds offered by the firm and offer the corresponding amount of reserves to investors,payingrˉ = r. Doingsowouldallowthecentralbanktoreplicatethefrictionless benchmarkofsection 3.2. However,asmentionedabove,inourmodelthesizeoftheQE programislimitedbytheconstraintsfacedbythecentralbank. 5.4 A Numerical Illustration Table2extendsournumericalexampletostudytheoptimalpublicliquiditymanagement, as implemented through QE. The table shows the changes in allocations relative to the 41

private equilibrium for three different economies. The first column shows the decentralization of the socially efficient outcomes achieved through optimal liquidity regulation (implemented with the leverage tax, τl, and storage subsidy, τs) , but without QE. The secondcolumnshowstheeffectsofQEinabsenceofliquidityregulation. Finally,thethird columnshowsthecaseinwhichliquiditymanagementcoexistswithliquidityregulation. All cases assume the parameterization α = 0.5, ψ = 0.9, and r = 0.01. We choose this parameterization because it puts the model in a region of the parameter space where QE iseffective,asperproposition11. Inaddition,weassumethatμˉ = 0.3,whichis50percent higherthanthebaselinevalueof μ = 0.2. The first column (which, for reference, corresponds to a point half way between the results shown in the first and second columns of table 1) shows that in absence of QE, the efficient allocation is decentralized with a leverage tax, τl = 0.21, and a subsidy for storage, τs = 0.04. By raising liquidity in the secondary market, and hence depressing − the liquidity premium, the resulting reduction in funding costs raises profits by 14 basis pointsrelativetotheprivateequilibrium,leavingtheutilityofinvestorsunchanged. The second column presents results where we shut down liquidity regulation, but allow the planner access to a QE program. Even when we shut down the tax system, so that τl = τs = 0,theplannercanuseQEtoachieveanevengreaterincreaseinfirmprofitability without harming investors. With QE the planner can achieve a similar outcome in terms of liquidity, without tax instruments. Finally, the last column of the table shows that QE, by itself, is not a panacea. A planner can do even better by implementing optimal liquidity management through QE in conjunction with liquidity regulation. The way to interpretthislastresultisthatalthoughQEimprovestheintermediationtechnologyinthe economy,itdoesnothingtoremovetheunderlyingdistortions,arisingfromthepecuniary andcongestionexternalitiesdiscussedabove. Figure 8 shows how the gains to the firm vary with ψ for different levels of the efficiency of the central bank monitoring technology. The thick lines show the case for μˉ = 0.3assumingQEinconjunctionwiththeoptimaltaxsystem(thethicksolidline)and, alternatively, assuming QE alone with no supporting tax system (the thick dashed line). Thethinsolidanddashedlinescorrespondtothesameinformationwhenthemonitoring cost is lower, so that μˉ = 0.2. Finally, the thin dotted line shows the gains to the firm from optimal tax policy alone in absence of QE. There are four things to take from the figure. First,QEisalwaysmoreeffectivewhencombinedwiththeoptimaltaxpolicy(the solid lines are always above the dashed line for the same monitoring cost assumption). Second, the effectiveness of QE is limited by the parameterization of ψ (the dashed lines aredownwardsloping),sothatasthegainsfromtradethataccruetoimpatientinvestors 42

decline,QEbecomeslesseffective. Third,theeffectivenessofQEdependsimportantlyon thequalityofthecentralbank’smonitoringtechnology(thethicklinesarebelowthethin ones, so the worse the technology, the less effective is QE). Finally, there are parts of the parameter space in which QE is ineffective to the point at which a planner would strictly prefer optimal taxation to QE (the regions in which the thick and thin dashed lines lie belowthethindottedline). 6 Conclusion We show that trade frictions in OTC markets provide a rationale for the regulation and public management of market liquidity. In our model, investors face liquidity risk and need to allocate their limited liquid resources between liquidity provision and illiquid long-termbonds. Bondsprovidecredittoproductivefirms,sothereisatrade-offbetween liquidity and credit provision. Trade friction together with this trade-off result in an inefficiencybecauseinvestorsandfirmsfailtointernalizehowtheiractionsaffectliquidity inthesecondarymarket. Anovelaspectofourmodelisthatprivateliquiditycanbeeither inefficientlyhighorinefficientlylow,dependingontheincentivesfacedbyinvestors. We provideananalyticcharacterizationofthedistortionsandshowhowthesociallyefficient equilibrium can be decentralized with two tax instruments. Finally, we show how both theprovision(asinQE)aswellasthewithdrawalofpublicliquiditycanenhancewelfare. Our model suggests a set of testable predictions for the relationship between the availability of short-term liquid assets and liquidity premia. While there is only a single OTC market in our setup, in practice there are many, potentially segmented secondary markets(see,forexample,VayanosandWang,2007andVayanosandWeill,2008). Given that central bank reserves can be used to settle transactions across most markets, we expect that quantitative easing should have an effect on the liquidity premia of not just those illiquid assets directly purchased by central banks, but all securities traded in OTC markets where participants’ portfolios are affected by the policy (see Christensen and Krogstrup (2016) for related empirical support). Our model also suggests that QE will be more effective when the assets purchased by the Central Bank are more illiquid. In thissense,ourmechanismprovidesanexplanationforthefactthattheFederalReserve’s strategyofpurchasingrelativeilliquidmortgage-backedsecuritiesintheaftermathofthe financialcrisismighthavebeenmoreeffectivethanexchanginglong-termTreasurybonds withshortermaturityonesasarguedinKrishnamurthyandVissing-Jorgensen(2011). There are a number of directions for future work. First of all, our paper opens up newavenuesforresearchonoptimalliquidityprovisionwhenfinancialintermediationis 43

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Tables and Figures Table1: PlanningoutcomesandImplementation ψ 1.0 0.8 0.6 0.4 0.2 0.0 %changeinl -8.62% -5.03% -1.63% 1.72% 5.13% 8.63% 0 %changeinωˉ -5.27% -3.06% -0.99% 1.04% 3.08% 5.17% %changeinθ 62.01% 27.75% 7.44% -6.70% -17.42% -26.03% %changeinΠ 0.23% 0.07% 0.01% 0.01% 0.06% 0.16% %changeinU 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% τl 0.27% 0.15% 0.05% -0.05% -0.13% -0.21% τs 0.00% -0.05% -0.03% 0.04% 0.14% 0.27% Note:Percentagescorrespondtodeviationswithrespecttotheprivateequilibriumforvariables:leverage(l0),risk(ωˉ),marketliquidity (θ),firms’profits(Π),andinvestors’utility(U);andtotheleveloftheoptimaltaxesonleverage(τl)andstorage(τs).Negativevalues fortaxescorrespondstosubsidies.Fordetailsseesection4.2. Table2: OutcomeswithQuantitativeEasing ConstrainedEfficient QuantitativeEasing QuantitativeEasing Allocations withτs =τl =0 withτs,τlChosenOptimally %changeinl -6.78% 1.68% -3.05% 0 %changeinωˉ -4.13% 0.72% -2.35% %changeinθ 42.19% 43.37% 167.72% %changeinΠ 0.14% 0.42% 0.98% %changeinU 0.00% 0.00% 0.00% rˉ 1.16% 1.10% sˉ 0.09 0.18 0 τl 0.21% 0.17% τs -0.04% -0.05% Note:Percentagescorrespondtodeviationswithrespecttotheprivateequilibriumforvariables:leverage(l0),risk(ωˉ),marketliquidity (θ),firms’profits(Π),andinvestors’utility(U);andtothelevelof:taxonleverage(τl),taxonstorage(τs),andinterestrateonreserves (rˉ).Valuesforreserves(sˉ0)areinlevels.Negativevaluesfortaxescorrespondstosubsidies.Fordetailsseesection5.4. 48

Figure4: EquilibriumintheFrictionlessBenchmark Break-even condition Inidifference curves of firm Equilibrium Note:Fordetailsseesection3.5. Figure5: ComparativeStaticson δ. Break-even condition for δ=0 Break-even conditions for δ>0 Inidifference curves of firm Equilibrium Note:δtakevaluesin 0,0.1,...,0.5.Seesection3.5. { } 49

Figure6: BondPremiaDecomposition Impatience (δ) Note:Fordetailsseesection3.5. Figure7: ConstrainedEfficientEquilibrium Break-even condition for δ=0 Break-even condition C.E. for δ>0 Inidifference curves of firm Break-even condition planner for δ>0 C.E. Planning solution Note:Fordetailsseesection4.2. 50

Figure8: EffectofQuantitativeEasing 1.0% 0.8% Optimal Taxes 0.6% Low μ : QE Low μ : QE with Optimal Taxes High μ : QE 0.4% High μ : QE with Optimal Taxes 0.2% 0% 0.98 0.96 0.94 0.92 0.9 0.88 0.86 0.84 0.82 Surplus split (ψ) Note:Fordetailsseesection5.4. 51

Private and Public Liquidity Provision in Over-the-Counter Markets David M. Arseneau David E. Rappoport Alexandros P. Vardoulakis FederalReserveBoard FederalReserveBoard FederalReserveBoard Online Appendix Proofs and Derivations Proof of Theorem 1: We need to show that there is a unique equilibrium, and that in this equilibrium credit is not rationed. For that, first, we establish that the privately optimal contract is an interior solution to the firm’s optimization problem (Part 1). Then, we establish existence of equilibria(Parts2). Finally,weestablishuniqueness(Part3). Part1. Theprivatelyoptimalcontractisinterior. Firstofall,notethatfromthedefinitionof Γ(ω) andG(ω) itfollowsthatforany ωˉ > 0 Γ(ωˉ) > 0 , 1 Γ(ωˉ) = P(ω ωˉ)E[ω ωˉ ω ωˉ] > 0 − ≥ − | ≥ 1 > Γ (ωˉ) = 1 F(ωˉ) > 0 , Γ (ωˉ) = dF(ωˉ) < 0 0 00 − − 0 < G(ωˉ) < 1 , μG(ωˉ) < G(ωˉ) < Γ(ωˉ) d(dF(ωˉ)) (A.1) G (ωˉ) = ωˉdF(ωˉ) > 0 , G (ωˉ) = dF(ωˉ)+ωˉ 0 00 dωˉ limΓ(ωˉ) = 0 , lim Γ(ωˉ) = ωˉP(ω ωˉ)+P(ω < ωˉ)E[ωω < ωˉ] = 1 ωˉ 0 ωˉ ≥ | → →∞ limG(ωˉ) = 0 and lim G(ωˉ) = 1 . ωˉ 0 ωˉ → →∞ Inaddition,fromAssumption2,ωˉdF(ωˉ)/(1 F(ωˉ)),isincreasingso,1 μωˉdF(ωˉ)/(1 F(ωˉ)), − − − hasonlyoneroot,whichisstrictlypositiveandisdenotedby ωˉˉ > 0. Then, > 0 if ωˉ < ωˉˉ ωˉdF(ωˉ) Γ (ωˉ) μG (ωˉ) = (1 F(ωˉ)) 1 μ = 0 if ωˉ = ωˉˉ . 0 − 0 − − 1 F(ωˉ)  − !    < 0 if ωˉ > ωˉˉ      Thevalueofthefirm,[1 Γ(ωˉ)]Rkl 0 ,isincreasinginleverag e,l 0 ,anddecreasinginrisk,ωˉ. In − addition, if investors’ expected (hold-to-maturity) return is R [(1+r)2,Rk], then R = Rb(l ,ωˉ) 0 ∈ 1

implythat R lier(ωˉ) = . (A.2) 0 R Rk[Γ(ωˉ) μG(ωˉ)] − − Since Γ(ωˉ) μG(ωˉ) attainsamaximumat ωˉˉ − R (1+r)2 lier(ωˉ) l ˉˉ 0 ≤ R Rk[Γ(ωˉˉ) μG(ωˉˉ)] ≤ (1+r)2 Rk[Γ(ωˉˉ) μG(ωˉˉ)] ≡ 0 − − − − Itfollowsthatthefirmwillneverchooseriskaboveωˉˉ,asadditionalrisk,whichreducesfirm’svalue, does not allow the firm to increase leverage. Therefore, the firm chooses a level of risk 0 ωˉ ωˉˉ ˉˉ ≤ ≤ andvalueofleverage 1 l l . 0 0 ≤ ≤ To establish the properties of the optimal contract, it will be useful to consider the following contractproblem max[1 Γ(ωˉ)]Rkl (A.3) 0 l0,ωˉ − s.t. Rb(l ,ωˉ) = R , 1 l l ˉˉ and 0 ωˉ ωˉˉ . 0 0 0 ≤ ≤ ≤ ≤ Notethatsincethefirm’sobjectiveiscontinuousthemaximumisachievedintheclosedsetdefined by the constraints. Now we want to establish that the maximum is interior, i.e. 0 < ωˉ < ωˉˉ or ˉˉ equivalently 0 < l < l . WewritetheLagrangianforthisproblemas 0 0 = [1 Γ(ωˉ)]Rkl λ Rb(l ,ωˉ) R ηˇ [1 l ] ηˆ [l l ˉˉ ]+ηˇ ωˉ ηˆ [ωˉ ωˉˉ] 0 0 l 0 l 0 0 ω ω L − − − − − − − − − h i Then,theFOCare ∂Rb (l ) 0 = [1 Γ(ωˉ)]Rk λ +ηˇ ηˆ 0 l l − − ∂l − 0 ∂Rb (ωˉ) 0 = Γ (ωˉ)Rkl λ +ηˇ ηˆ 0 0 ω ω − − ∂ωˉ − with ∂Rb Rb ∂Rb Rb[Γ 0 (ωˉ) μG 0 (ωˉ)] = < 0 and = − . (A.4) ∂l −l (l 1) ∂ωˉ Γ(ωˉ) μG(ωˉ) 0 0 0 − − Supposenow ωˉ = 0andl = 1then ηˇ ,ηˇ > 0. Notethatlier(0) = 1,then,fromtheFOC 0 l ω 0 Rb 0 < ηˇ = Rk λ < 0 , l − − l (l 1) 0 0 − which is a contradiction. So we conclude that ωˉ > 0 and l > 1. Similarly, if ωˉ = ωˉˉ and l = l ˉˉ 0 0 0 fromtheFOC Rb[Γ (ωˉˉ) μG (ωˉˉ)] 0 < ηˆ = Γ (ωˉˉ)Rkl ˉˉ λ 0 − 0 < 0 , ω − 0 0 − Γ(ωˉˉ) μG(ωˉˉ) − whichisacontradiction. Soweconcludethat ωˉ < ωˉˉ andl < l ˉˉ . 0 0 Thus, we conclude that there exist a solution to the modified contract problem (A.3) and this 2

solutionisinterior. Part2. Existenceofequilibria. Let : R, with = [(1 + r)2,Rk]. For R , (R) is defined as follows. Given R J C → C ∈ C J define(l (R),ωˉ(R))asthesolutiontotheoptimizationproblem(A.3). Notethatthesolutiontothe 0 ˉˉ contractproblemisfeasible,as l < e /n +1fromAssumption 4. 0 0 0 Use (l (R),ωˉ(R)) tocalculateb (R) = n (l (R) 1) ands (R) = e b (R) and θ(R) as 0 0 0 0 0 0 0 − − (1 δ)s (R)(1+r)Δ 0 θ(R) = − . δb (R)R 0 Then, U (θ(R)) s (R) = J u (θ(R)) b Intuitively, for any hold-to-maturity two-period return R, the function (R) gives the hold-to- J maturity return that makes investors indifferent between liquid storage and illiquid two-period bonds,giventhatfirmsoptimallychoosethecontractgivenRandthatthelevelofsecondarymarket liquidity is consistent with the investors portfolios that support the optimal firms’ bond issuance b (R). Itfollowsthatafixpointof constituteaprivateequilibrium. 0 J Nowwewanttoshowthat isacontinuoussinglevaluedfunctionand ( ) ,so has J J C ⊂ C J afixedpointR = (R),whichconstituteanon-rationingequilibrium. J First, we show that is a single valued function. For that it suffices to show that the optimal J contractasafunctionofRisasinglevaluedfunction. Theobjectiveofthefirms’problemisconcave as Γ (ωˉ) > 0. But the feasible set defined by the constraints to the firm’s optimization problem, 0 givenR,isnotconvex,soweneedtoruleoutthatthefirm’sindifferencecurvesandtheinvestors’ expectedreturncondition,R = Rb(l ,ωˉ),intersectmorethanonce. 0 Ontheonehand,thefirm’sindifferencecurvesaredescribedby L lic(ωˉ) = , 0 Rk[1 Γ(ωˉ)] − whereLisaconstantthatdescribesthelevelofprofitsattheindifferencecurve. Then, dlic LΓ (ωˉ) 0 = 0 . dωˉ Rk[1 Γ(ωˉ)]2 − Ontheotherhand,differentiating(A.2)weget dlier RRk[Γ (ωˉ) μG (ωˉ)] 0 = 0 − 0 . dωˉ R Rk[Γ(ωˉ) μG(ωˉ)] 2 − − At the optimal contract, these two c(cid:0)urves intersect. We use(cid:1)the condition that the two curves intersecttoexpressLintermsR. Infact,lic = lier imply 0 0 RRk[1 Γ(ωˉ)] L = − . R Rk[Γ(ωˉ) μG(ωˉ)] − − 3

Moreover,thesetwocurveshavethesameslope,i.e.,dlic/dωˉ = dlibec/dωˉ,ifandonlyif 0 0 Γ (ωˉ) Rk[Γ (ωˉ) μG (ωˉ)] 0 0 0 = − 1 Γ(ωˉ) R Rk[Γ(ωˉ) μG(ωˉ)] − − − Rk R [1 Γ(ωˉ)]G (ωˉ) 0 − = G(ωˉ)+ − (ωˉ) . (A.5) μRk Γ (ωˉ) ≡ H 0 Thefunction (ωˉ) isastrictlyincreasingfunctionof ωˉ. Infact, H [1 Γ(ωˉ)] [1 Γ(ωˉ)]d(ωˉh(ωˉ)) (ωˉ) = − [Γ (ωˉ)G (ωˉ) G (ωˉ)Γ (ωˉ)] = − (1 F(ωˉ))2 > 0 . 0 0 00 0 00 H Γ (ωˉ)2 − Γ (ωˉ)2 dωˉ − 0 0 Then we conclude that there is only one solution to the firms’ maximization problem. Therefore, (l (R),ωˉ(R)) aresinglevaluedfunctionsandsoareb (R), s (R), θ(R),and (R). 0 0 0 J It follows from above that (R) is continuous. In fact, (ωˉ) is a continuous function with J H (ωˉ) > 0,sobytheImplicitFunctionTheoremωˉ(R)isacontinuousstrictlydecreasingfunction 0 H in , i.e., ωˉ (R) < 0. That is, the risk of the optimal contract is decreasing in the expected hold- 0 C to-maturity return offered to investors. Then, from the investors’ expected return condition (A.2) l (R)isacontinuousfunctionin ,giventhatR = Rb > Rk[Γ(ωˉ) μG(ωˉ)]. Itfollows,thatb (R) 0 0 C − ands (R) arecontinuousin ,andthus,that θ(R) and (R) arecontinuousin . 0 C J C Nowweshowthat (R) (1+r)2 and (R) Rk. From (1+r) Δ β 1 wehavethat − J ≥ J ≤ ≤ ≤ δ(1+r)+(1 δ)(1+r)2 U (θ(R)) δ(1+r)+(1 δ)(1+r)Δ s − ≤ ≤ − and δβ+1 δ u (θ(R)) δΔ 1 +1 δ . b − − ≤ ≤ − Ontheonehand,fromAssumptions 1and 3wehavethat δ[1+r βRk] 0 (1 δ)[Rk (1+r)Δ] . − ≤ ≤ − − Rearranging, δ(1+r)+(1 δ)(1+r)Δ (R) − Rk . J ≤ δβ+1 δ ≤ − Ontheotherhand,since Δ 1+rwehavethat ≥ δ(1+r)+(1 δ)(1+r)2 (R) − (1+r)2 . J ≥ δΔ 1 +1 δ ≥ − − Part3. Uniqueness: Showthat (R) isdecreasingin . J C Differentiatingweobtain d (R) 1 dU (θ(R)) 1 du (θ(ωˉ)) dθ(R) s b J = (R) . dR J U (θ(R)) dθ − u (θ(R)) dθ dR s b " # 4

Tosignthisderivativenotethat dU s = (1 δ)(1+r)p (θ)[Δ (1+r)] 0 , 0 dθ − − ≤ (A.6) du and b = δf (θ) Δ 1 β 0 . 0 − dθ − ≥ h i where the inequalities follow from p (θ) 0, f (θ) 0, and (1+r) Δ β 1. Thus, the term in 0 0 − ≤ ≥ ≤ ≤ squarebracketsisnegative. Wearelefttoshowthatdθ/dR > 0. Forthatnotethat dθ θ ds θ db θ 0 0 = . dR s dR − b dR − R 0 0 Inaddition,fromabovewehadthat ωˉ (R) < 0and 0 dl 0 = RRk[Γ 0 (ωˉ(R)) − μG 0 (ωˉ(R))]ωˉ 0 (R) < 0 . dR R Rk[Γ(ωˉ(R)) μG(ωˉ(R))] 2 − − Notethatthepreviousinequality(cid:0)implythatthedemandforcred(cid:1)itbyfirmsisdownwardslopping, anditshowsthattheleverageoftheoptimalcontractisdecreasingintheexpectedhold-to-maturity returnofferedtoinvestors. Usingthat ds db db dl dRb ∂Rbdl ∂Rbdωˉ 0 0 0 0 0 = , = n , and = 1 = + . 0 dR −dR dR dR dR ∂l dR ∂ωˉ dR 0 Then 2 R Rk[Γ(ωˉ(R)) μG(ωˉ(R))] Rdθ dl R(e +n ) 0 − − 0 0 = + > 0 . (A.7) θdR −dR (cid:16) Rk[Γ(ωˉ(R)) μG(ωˉ(R))] (cid:17) l 0 (e 0 n 0 (l 0 1))  − − −        Whereweusedthatinan interiorcontractl < e /n +1.    0 0 0    Proof of Proposition 1: The aggregate credit supply is determined by the investors’ break-even condition(15),fromwhereitfollowsthat dU ∂θ dbs ∂θ du ∂θ dbs ∂θ s 0 + = b 0 + R+u . dθ ∂bs dR ∂R dθ ∂bs dR ∂R b " 0 # " 0 # Then, θ du θ dU R∂θ b s +1 R dbs u dθ − U dθ θ∂R 0 = " b s # . bs dR θ dU θ du bs ∂θ 0 s b 0 U dθ − u dθ θ ∂bs " s b # 0 5

Fromthedefinitionofmarketthickness(17) ∂θ θ e ∂θ θ 0 = < 0 and = , ∂bs −bs (e bs)e ∂R −R 0 0 0 − 0 0 wheretheinequalityfollowsfromAssumption4. Thisinequalityandequation(A.6)implythatthe denominatorinthepreviousexpressionisnon-negative. Weneedtoruleoutthatthedenominator is zero, which is the case when dU (θ)/dθ = du (θ)/dθ = 0. This is the case when either θ > θ s b andψ = 1,orθ < θandψ = 0. Thefirstcasecorrespondstocase3inProposition3,i.e.,oneofthe conditions that make OTC trade irrelevant, which violates the assumption that ψ < 1 or e < eˉˉ . 0 0 The second case corresponds to the case where the liquidity premium is fixed at β 1/(1 + r) 1. − ≥ Butinthiscase θ < θ implythat (1 δ)(1+r)(e b )β 1 (1 δ)(1+r)(e b )β 1 0 0 − 0 0 − 1 − − − − < min 1,να . δb Rk ≤ δb R { } 0 0 ˉˉ Butthen,rearrangingandusingthatb n (l 1) weget 0 0 0 ≤ − e δβRkmin { 1,να 1 } +1 b δβRkmin { 1,να 1 } +1 n (l ˉˉ 1) , 0 0 0 0 ≤ (1 δ)(1+r) ≤ (1 δ)(1+r) −  −   −              which is in contrad iction with Assumpti on 4,  e 0 >> n 0 . That is, t he deep pocket assumption prevents liquidity from having a finite upper bound and we conclude that liquidity cannot be smallerthan θ. Thus,wearelefttoshowthatthenumeratorispositive,i.e., θ du θ dU b s < 1 , u dθ − U dθ b s whichfollowsfromLemma 1,i.e., ε < 1. Φ‘,θ (cid:12) (cid:12) (cid:12) (cid:12) ProofofProposition 2: Taking(cid:12)deriv(cid:12)ativewrtto ωˉ inequation(18)yields dΦd(ωˉ) 1 ωˉ[Γ (ωˉ) μG (ωˉ)] 0 0 = − > 0 dωˉ Γ(ωˉ) μG(ωˉ) − [Γ(ωˉ) μG(ωˉ)]2 − − Γ(ωˉ) μG(ωˉ) ωˉ[Γ (ωˉ) μG (ωˉ)] > 0 0 0 ⇔ − − − (1 μ)G(ωˉ)+ωˉμG (ωˉ) > 0 . 0 ⇔ − Whereweusedthat Γ(ωˉ) = ωˉΓ (ωˉ)+G(ωˉ),andwherethelastinequalityfollowsfrom 1 μ > 0, 0 − G(ωˉ) 0andG (ωˉ) = ωˉdF(ωˉ) > 0,forany ωˉ > 0. 0 ≥ Proof of Proposition 3: When there is no need to compensate investors for liquidity risk, there is no liquidity premium, i.e., Φ‘(θ) = 1 and Rb = (1 + r)2. In other words, the expected return 6

from lending to entrepreneurs is equal to the outside option of holding storage for two periods. Note that we can rewrite the previous condition as k Rk Γ(ωˉ) μG(ωˉ) = (k n )(1 + r)2, 0 0 0 − − which is the break-even condition in the benchmark costly state verification model. In ad- (cid:2) (cid:3) dition, note that entrepreneurs’ profits do not depend directly on secondary market liquidity. WeproceedbyshowingthatΦ‘(θ) = 1underthethreealternativeconditionstatedinProposition3. Condition1: δ = 0. Thisimpliesthatsecondarymarketliquidity θ ,hencep(θ) = 0. Setting → ∞ δ = 0andp(θ) = 0yields Φ‘(θ) = 1. Condition2: β = (1+r) 1 then Δ = 1+randsimplesubstitutionyields Φ‘(θ) = 1. − Condition 3: ψ = 1 and f(θ) = 1. Simple substitution yields Φ‘(θ) = 1. We want to find eˉ such 0 that f(θ) = 1ife eˉ . 0 0 ≥ f(θ) = 1 m(A,B) = A min νAαB1 α,B A θ max ν 1/(1 α),1 − − − ⇔ ⇔ { } ≥ ⇔ ≥ { } From (A.7) we have that θ θ (1+r)2 and since ψ = 1 we have that Δ = 1 + r, so θ ≥ ≥ (1 δ)sˉˉ / δb ˉˉ , where sˉˉ = e b ˉˉ and b ˉˉ = n l ˉˉ 1 with l ˉˉ the upper bound on firm leverage 0 0 0 0 0 0 0 0 0 − − (cid:0) (cid:1) − definedint(cid:16)hep(cid:17)roofofTheorem 1. Weimposetha(cid:16)t (1 (cid:17)δ)sˉˉ 0 / δb ˉˉ 0 max ν − 1/(1 − α),1 toobtaina − ≥ lowerboundfortheendowmentofinvestorssuchthat f(θ) = 1. (cid:16) (cid:17) n o (1 δ) e n l ˉˉ 1 max 1,ν 1/(1 α) δn l ˉˉ 1 0 0 0 − − 0 0 − − − ≥ − e eˉˉ (cid:16) = n 0 (cid:16) l ˉˉ (cid:17) 1 (cid:17) 1+δ n max 0,ν 1 o /(1 α) (cid:16) 1 (cid:17) 0 0 0 − − ≥ 1 δ − − − (cid:16) (cid:17)(cid:16) n o(cid:17) ProofofLemma1: Wewanttoshowthatthederivativeoftheliquiditypremiumwrtliquidityis negative. NotethatU (θ),u (θ) > 0,sincethetradingprobabilitiesandreturnsarenon-negative. s b Inaddition,notethat dU (θ) dp(θ) s = (1 δ)(1+r)[Δ (1+r)] 0 , dθ − − dθ ≤ du (θ) df(θ) and b = δ Δ 1 β 0 , − dθ − dθ ≥ h i where the inequalities follow from β 1/(1 + r), equations (5) and (6), and that the matching ≤ functionm(A,B) isincreasinginbotharguments. Fromequation(19)wehavethat dΦ‘(θ) 1 dU (θ) 1 du (θ) = Φ‘(θ) s b 0 , (A.8) dθ U (θ) dθ − u (θ) dθ ≤ s b " # wheretheinequalityfollowsfromthepreviouslyestablishedinequalities. Regarding the second part of the Lemma, the elasticity of the liquidity premium, Φ‘, with 7

respecttothesecondarymarketliquidity, θ,iswritten,usingequation(A.8),as: θ dΦ‘ θ dU (θ) θ du (θ) s b ε = = , (A.9) Φ‘,θ Φ‘ dθ U (θ) dθ − u (θ) dθ s b " # Then ε < 1requires: Φ‘,θ (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) θ (cid:12) dU s (θ) θ du b (θ) < 1 − U (θ) dθ − u (θ) dθ s b " # dU (θ) du s b U (θ)u (θ)+θ u (θ) θU (θ) > 0 (A.10) s b b s ⇔ dθ − dθ First, lets consider the case where θ θ,θ . In this case, f(θ) = νθ1 α and p(θ) = νθ α. Thus, − − ∈ θ(df(θ)/dθ) = (1 α)f(θ) and θ(dp(θ)/dθ) = αp(θ). Then, − (cid:16) (cid:17) − dU (θ) s θ = αU (θ)+α(1+r)[δ+(1 δ)(1+r)] 0 s dθ − − ≤ du (θ) b θ = (1 α)u (θ) (1 α)[βδ+(1 δ)] 0 . b dθ − − − − ≥ Then, dU (θ) du s b U (θ)u (θ)+θ u (θ) θU (θ) s b b s dθ − dθ = U u +u αU +α(1+r)[δ+(1 δ)(1+r)] U (1 α)u (1 α)[βδ+(1 δ)] s b b s s b {− − }− − − − − = αu b (θ)(1+r)[δ+(1 δ)(1+r)](cid:8)+(1 α)U s (θ)[βδ+(1 δ)] > 0(cid:9). − − − Second, consider the case where θ < θ. In this case, p(θ) = 1 and f(θ) = θ, so df(θ)/dθ = 1 and dp(θ)/dθ = dU (θ)/dθ = 0. Want to show that u (θ) θ(du (θ)/dθ) > 0. From above s b b − du (θ)/dθ = δ[Δ 1 β]. Then, b − − du (θ) b u (θ) θ = δβ+(1 δ) > 0 . b − dθ − Finally, consider the case where θ > θ. In this case, df(θ)/dθ = du (θ)/dθ = 0 and b p(θ) = θ 1. Thus,wewanttoshowthatU (θ)+θ(dU (θ)/dθ) > 0. Fromabove,θ(dU (θ)/dθ) = − s s s θ 1(1 δ)(1+r)[Δ (1+r)]. Then, − − − − dU (θ) U (θ)+θ s = δ(1+r)+(1 δ)(1+r)2 > 0 . s dθ − Proof of Proposition 4: For this proof we consider the liquidity premium a function of both market thickness, θ, and model parameters δ and β, and consider market thickness as a function of R, which we will take as given, and model parameters δ, β, and e . That is, we can write the 0 8

liquiditypremiumas Φ‘(θ,δ,β). Case1: Effectof δ. Wanttoshowthat dΦ‘ ∂Φ‘ ∂Φ‘ dθ = + > 0 . dδ ∂δ ∂θ dδ R R (cid:12) (cid:12) (cid:12) (cid:12) Fromthedefinitionofsecondarymark(cid:12)etliquidity,givenin(cid:12) equation(17)wehavethat (cid:12) (cid:12) (cid:12) (cid:12) dθ θ = . dδ −δ(1 δ) R (cid:12) − (cid:12) (cid:12) Usingthisexpressionweget (cid:12) (cid:12) dΦ‘ ∂Φ‘ Φ‘ = ε , dδ ∂δ − δ(1 δ) Φ‘,θ R (cid:12) − (cid:12) (cid:12) where ε is the elasticity of the liqu(cid:12)idity premium with respect to secondary market liquidity, Φ‘,θ (cid:12) whichisnegative(Lemma 1),therefore,thesecondtermispositive. It is left to show that ∂Φ‘/∂δ > 0. For that we compute the derivatives of U (θ,δ,β) and s u (θ,δ,β) withrespectto δ. b ∂U ∂u s = 1+r (1+r) (1 p(θ))(1+r)+p(θ)Δ and b = f(θ)Δ 1 +(1 f(θ))β 1 . − ∂δ − − ∂δ − − h i (cid:2) (cid:3) Then,fromequation(19)wehavethat ∂Φ‘ 1 ∂U 1 ∂u = Φ‘ s b , ∂δ U ∂δ − u ∂δ s b " # whichisstrictlygreaterthanzeroifandonlyifu (∂U /∂δ) > U (∂u /∂δ) b s s b ∂U ∂u ∂U ∂U ∂u ∂U ∂U ∂u s b s s b s s b δ +1 > δ +1+r > 1+r ⇔ ∂δ ∂δ ∂δ − ∂δ ∂δ ⇔ ∂δ − ∂δ ∂δ " # " # " # 1+r (1+r) (1 p)(1+r)+pΔ > (1+r) (1 p)(1+r)+pΔ fΔ 1 +(1 f)β 1 − ⇔ − − − − − nh i o (cid:2) 1 > (1 p)((cid:3)1+r)+pΔ(cid:2) fΔ 1 +(1 f)β(cid:3) . − ⇔ − − h i Itiseasytocheckthatafterdistrib(cid:2)utingtermsinthepr(cid:3)eviousexpressionthefourremainingterms, are a weighted average of terms strictly smaller than 1, with the weights given by the product of probabilities f andpaddingupto1. Infact, β < 1/(1+r)implythatβ(1+r) < 1, Δ 1(1+r) < 1, − and Δβ < 1. Case2: Effectof β. Wanttoshowthat dΦ‘ ∂Φ‘ ∂Φ‘ ∂θ = + < 0 . dβ ∂β ∂θ ∂β (cid:12)R (cid:12)R (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 9

Fromthedefinitionofsecondarymarketliquidity,giveninequation(17)wehavethat ∂θ θ ∂q 1 = = θ(1 ψ)Δ . ∂β −q ∂β − − (cid:12)R 1 (cid:12) (cid:12) Thus, (cid:12) (cid:12) dΦ‘ ∂Φ‘ ∂Φ‘ = (1 ψ)Δθ . dβ ∂β − − ∂θ (cid:12)R (cid:12) But the sign of the right-hand side(cid:12)term is ambiguous. The reason is that a higher β, on one (cid:12) hand, reduces the preference for liqui (cid:12) dity by impatient households, i.e., ∂Φ‘/∂β < 0. But, on the otherhand,itincreasesthesecondarymarketprice,q ,whichpushesmarketthicknessθdownand 1 liquidity premia up. This second force, represented by the second term depends crucially on the bargainingpowerofimpatientinvestors: thelowertheirbargainingpowerthemoreimportantthe effectoftheirvaluation,i.e., β,willbeontheprice. Toshowthatthetermintheright-handsideisnegative,weusethat ∂Φ‘ 1 ∂U 1 ∂u = Φ‘ s b , ∂β U ∂β − u ∂β s b " # withthederivativesofU (θ,δ,β) andu (θ,δ,β) withrespectto β givenby s b ∂U s = (1 δ)(1+r)p(θ)Δ2(1 ψ) < 0 , ∂β − − − ∂u b and = δ[f(θ)(1 ψ)+1 f(θ)] = δ(1 ψf(θ)) > 0 , ∂β − − − where the inequalities follow from our assumption about δ, ψ, and f(θ). Then, ∂Φ‘/∂β < 0. So weneedtoshowthat ∂U ∂u ∂U ∂u s b s b (1 ψ)Δθ u U u +U > 0 . b s b s − ∂θ − ∂θ − ∂β ∂β " # Usingtheexpressionsderivedaboveforthesederivativesthepreviousexpressionequals ∂p u (1 ψ)Δα(1+r) [Δ (1+r)]θ +pΔ b − − ∂θ ( (cid:12) b0 ) (cid:12) (cid:12) (cid:12) ∂f +U(cid:12) δ(1 ψ) f [1 Δβ]θ +U δ(1 f) > 0 , s s − − − ∂θ − ( (cid:12) b0) (cid:12) (cid:12) where the inequality follows from the fact that the terms in curly brack(cid:12)ets are positive. In fact, (cid:12) when θ (θ,θ),then θ(∂p/∂θ) = αpand θ(∂f/∂θ) = (1 α)f sowehave ∈ | b0 − | b0 − ∂p [Δ (1+r)]θ +pΔ = (1 α)pΔ+αp(1+r) > 0 , − ∂θ − (cid:12) b0 (cid:12) (cid:12) (cid:12) (cid:12) 10

∂f and f [1 Δβ]θ = αf +(1 α)fΔβ > 0 . − − ∂θ − (cid:12) b0 (cid:12) When θ < θ,then (∂p/∂θ) = 0and θ(∂f/∂θ)(cid:12) = f sowehave | b0 (cid:12) (cid:12)| b0 ∂p ∂f [Δ (1+r)]θ +pΔ = pΔ > 0 and f [1 Δβ]θ = fΔβ > 0 . − ∂θ − − ∂θ (cid:12) b0 (cid:12) b0 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) Finally,when θ > θ,then(cid:12) θ(∂p/∂θ) = pand (∂f/∂θ) = 0sowehave(cid:12) | b0 − | b0 ∂p ∂f [Δ (1+r)]θ +pΔ = (1+r)p > 0 and f [1 Δβ]θ = f > 0 . − ∂θ − − ∂θ (cid:12) b0 (cid:12) b0 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) Case 3: Effect of e . Want to show that dΦ‘/de < 0. Note that investors’ endowment e affects 0 0 R 0 | liquiditypremium Φ‘ onlythroughitseffectonsecondarymarketliquidity θ. Inparticular,ithas aneffectonlythroughs = e b giventhatweconsiderbondissuanceasgiven. Thus, 0 0 0 − dΦ‘ ∂Φ‘ ∂θ ds ∂Φ‘ θ 0 = = < 0 , de ∂θ ∂s de ∂θ s 0 R 0 0 0 (cid:12) (cid:12) (cid:12) wheretheinequalityfollowsfromL(cid:12)emma 1andProposition 1. (cid:12) ProofofProposition6: FromtheproofofTheorem1weknowthattheequilibriumisdescribedby thefixedpointofthefunction ,i.e, (R) = R. Then,consideringagenericmodelparameter%we J J canexpresstheequilibriumofthemodelas (R,%) R = 0. BytheImplicitFunctionTheorem,if J − thederivativeofthepreviousexpressionwithrespecttoRisdifferentthan0,thenwecandefineR(%) and calculate its derivative from the previous expression. Recall that (R) = U (θ(R))/u (θ(R)), s b J then d /dR = ( /θ)ε (dθ/dR) < 0, where ε the elasticity of the liquidity premium with Φ‘,θ Φ‘,θ J J respect to market thickness. The inequality follows from dθ/dR > 0 (equation (A.7)), Lemma 1, where we showed that the elasticity is negative, and Proposition 1, where we showed that the elasticityisdifferentthanzero. Then,bytheImplicitFunctionTheorem 1 dR ∂ − ∂ = J 1 J . d% − ∂R − ∂% " # From where we conclude that the sign of dR/d% equals the sign of ∂ /∂%. Note that ∂ /∂% = J J d /d% and thus have the same sign. Moreover, = Φ‘(1+r)2 so the sign of d /d% and the R R J | J J | signofdΦ‘/d% areequal,withthelatterestablishedinProposition 4. Nowweconsidertheeffect R | ofeachofmodelparameters. Case 1: Comparative Statics on δ. From Proposition 4 dΦ‘/dδ > 0, so we conclude that R | dR/dδ > 0. Case 2: Comparative Statics on β. From Proposition 4 dΦ‘/dβ > 0, so we conclude that R | dR/dβ > 0. Case 3: Comparative Statics on e . From Proposition 4 dΦ‘/de > 0, so we conclude that 0 0 R | 11

dR/de > 0. 0 Case 4: Comparative Statics on n . From Proposition 5 db /dn > 0. In this case as the demand 0 0 0 for credit increases, and investors’ portfolio become more illiquid, investors need to receive a highercompensationtobuythosebondsinequilibrium,i.e.,asbondissuanceincreasetheexpected hold-to-maturityreturnincreases(Proposition 1). SoweconcludethatdR/dn > 0. 0 Inallcasesastheexpectedhold-to-maturityreturnincreases,thecharacteristicsoftheoptimal contract decrease, i.e., leverage l and risk ωˉ decrease, and thus, the default premium increases. 0 Finally,inthefirstthreecasesgivencreditdemandremainsfixed,astheexpectedreturnincreases bond issuance decreases, whereas in the fourth case the total amount of bond issuance depends on the relative strength of two effect. The increase in bond issuance from the increase in firms’ endowmentsandthedecreaseinbondissuancefromthedecreaseinfirms’ leverage. ProofofProposition7: Wewanttoshowthatiftheprivateequilibriumisconstrainedefficient, then (α,ψ,r) ,asetofmeasurezero. Suppose ∈ (l p∅e ,ωˉpe,θpe,q pe ), the private equilibrium, is constrained efficient. Since 0 1 (l pe ,ωˉpe,θpe,q pe )isaprivateequilibriumtheinvestorbreak-evencondition(15)holds,i.e.,U = U , 0 1 s b andfromequation(16)itmustbethat 1 Γ(ωˉpe) ∂U /∂l b 0 − = . l pe Γ (ωˉpe) −∂U /∂ωˉ 0 0 b On the other hand, since (l pe ,ωˉpe,θpe,q pe ) is constrained efficient, from equation (23) it must be 0 1 that [1 − Γ(ωˉpe)] = n 0 (U b − U s )+b p 0 e∂ ∂ U l0 b + ∂ ∂ U θ ∂ ∂ l θ 0 . l pe Γ (ωˉpe) − b pe∂Ub + ∂U∂θ 0 0 0 ∂ωˉ ∂θ ∂ωˉ UsingthatU = U ,then s b b pe∂Ub + ∂U∂θ ∂Ub 0 ∂l0 ∂θ ∂l0 = ∂l0 , b pe∂Ub + ∂U∂θ ∂Ub 0 ∂ωˉ ∂θ ∂ωˉ ∂ωˉ whichisthecaseiff ∂U ∂U ∂θ ∂U ∂θ b b = 0 . (A.11) ∂θ ∂ωˉ ∂l − ∂l ∂ωˉ 0 0 " # Notethat, ∂U ∂θ ∂U ∂θ b b < 0 , (A.12) ∂ωˉ ∂l − ∂l ∂ωˉ 0 0 since ∂U b = U b < 0 and ∂U b = U b [Γ 0 (ωˉ) − μG 0 (ωˉ)] > 0 , (A.13) ∂l −l (l 1) ∂ωˉ Γ(ωˉ) μG(ωˉ) 0 0 0 − − where the last inequality follows from Theorem 1, and ∂θ/∂l ,∂θ/∂ωˉ < 0 from equation (17). 0 12

Then, A.11 holdsiff ∂U/∂θ = 0,whichisthecaseiff ∂U ∂U pe s pe b s +b = 0 0 ∂θ 0 ∂θ s pe (1 δ)(1+r)[Δ (1+r)]p (θpe)+b pe δ[Δ 1 β]f (θpe)Rb = 0 0 − 0 0 − − 0 − α 1 α p(θpe) s pe (1 δ)(1+r)[Δ (1+r)] = f(θpe) − b pe δ[Δ 1 β]Rb − θpe 0 − − θpe 0 − αs pe (1 δ)(1+r)[Δ (1+r)] = θpe(1 α)b pe δ[Δ 1 β]Rb . − 0 − − − 0 − Butfromequation(17) θpe = (1 δ)(1+r)Δs pe /(δb pe Rb),then − 0 0 α[Δ (1+r)] = (1 α)Δ[Δ 1 β] = (1 α)[1 Δβ] − − − − − − ψ α+(1 α)β Δ[α+(1 α)β] = 1+αr +(1 ψ)β = − ⇔ − ⇔ 1+r − 1+αr α(1 β(1+r)) 1+r ψ = − ψ(1+αr) = α(1+r) . (A.14) ⇔ 1+αr (1 β(1+r)) ⇔ − Thesetof (α,ψ,r) satisfying(A.14)is,thus,ofmeasurezero. ProofofProposition 8: Part 1. The sign of the externality determines the socially optimal level of secondary market liquidity. Let betheLagrangianoftheplanner’sproblem,whichisgiven L = [1 Γ(ωˉ)]Rkl λ[Upe s U b U ] , 0 0 s 0 b L − − − − Fullydifferentiatingandevaluatingattheprivateequilibriumallocation (l pe ,ωˉpe,θpe) wehave 0 ∂U d (l pe ,ωˉpe,θpe) = λ dθ , L 0 ∂θ wherewehavesubstitutedtheoptimalityconditionsintheprivateequilibrium. Thus,theplanner, who internalizes the effect of liquidity on the investor’s utility, would like to increase liquidity in secondary markets when the externality is positive, i.e., ∂U/∂θ > 0, and decrease liquidity if the externalityisnegative,i.e., ∂U/∂θ < 0. Part 2. Show that the sign of the externality depends on the relationship between the parameters (α,r,ψ). Wanttoshowthat ∂U ψ(1+αr) > α(1+r) > 0 . ⇔ ∂θ 13

Infact, α[Δ (1+r)] ψ(1+αr) > α(1+r) Δ > − ⇔ (1 α)[Δ 1 β] − − − αs (1 δ)(1+r)[Δ (1+r)] 0 θ > − − ⇔ (1 α)b δ[Δ 1 β]Rb 0 − − − ∂U ∂U ∂U b s b +s > 0 > 0. 0 0 ⇔ ∂θ ∂θ ⇔ ∂θ Part3. Characterizationoftheefficientcontract. Let ωˉpi(l ) be the function implicitly defined by the Pareto improvement constraint in the 0 planner’sproblem(20). UsingtheImplicitFunctionTheoremwehavethat dωˉpi ∂U + ∂U∂θ = ∂l0 ∂θ ∂l0 . dl 0 −∂U + ∂U∂θ ∂ωˉ ∂θ ∂ωˉ Similarly, using the notation introduced in the proof of Theorem 1, let ωˉier(l ) denotes the 0 functionimplicitlydefinedbytheinvestors’expectedreturnconditionRb(l ,ωˉ) = U (θ)/u (θ)for 0 s b ωˉ < ωˉˉ,wehavethat dωˉier ∂Rb ∂Ub = ∂l0 = ∂l0 . (A.15) dl 0 −∂Rb −∂Ub ∂ωˉ ∂ωˉ Note that the private equilibrium is a feasible point of the pareto improvement constraint, so ωˉpi(l pe ) = ωˉier(l pe ). Moreover,notethat 0 0 dωˉpi(l pe ) dωˉier(l pe ) ∂U ∂θ∂Ub ∂θ∂Ub 0 0 = ∂θ ∂ωˉ ∂l0 − ∂l0 ∂ωˉ , dl 0 − dl 0 ∂Ubhb pe∂Ub + ∂U∂θi ∂ωˉ 0 ∂ωˉ ∂θ ∂ωˉ h i whereallthederivativesontheRHSareevaluatedat (l pe ,ωˉpe,θpe),andweusedthat 0 ∂U(l pe ,ωˉpe,θpe) ∂U (l pe ,ωˉpe,θpe) ∂U (l pe ,ωˉpe,θpe) 0 = n (U (l pe ,ωˉpe,θpe) U (θpe))+b pe b 0 = b pe b 0 . ∂l 0 b 0 − s 0 ∂l 0 ∂l 0 0 0 Notethat b pe ∂U b + ∂U ∂θ = b pe u b Rb[Γ 0 − μG 0 ] + s pe dU s +b pe du b Rb ∂θ . 0 ∂ωˉ ∂θ ∂ωˉ 0 Γ μG 0 dθ 0 dθ ∂ωˉ − " # Andusingthats pe = e b pe ,Rb(l pe ,ωˉpe) = U (θpe)/u (θpe) and 0 0 − 0 0 s b ∂θ θ[Γ μG ] 0 0 = − , ∂ωˉ − Γ μG − 14

weobtainthat b pe ∂U b + ∂U ∂θ = b p 0 e [Γ 0 − μG 0 ]Rb U u +θpe dU s u θpeU du b +e dU s ∂θ > 0, (A.16) 0 ∂ωˉ ∂θ ∂ωˉ U [Γ μG] s b dθ b − s dθ 0 dθ ∂ωˉ s − ( ) where the inequality follows from (A.10) and ∂U /∂θ, ∂θ/∂ωˉ < 0. It follows from the previous s inequalityandequations(A.12)and(A.13)that dωˉpi(l pe ) dωˉier(l pe ) ∂U 0 0 > 0 > 0 . dl − dl ⇔ ∂θ 0 0 Then,if ψ(1+αr) > α(1+r),fromPart2, ∂U/∂θ > 0,and,thus, dωˉpi(l pe ) dωˉier(l pe ) 0 > 0 > 0 , dl dl 0 0 where the last inequality follows from equation (A.15). That means there are points, along the weak Pareto improving constraint, that are feasible for the planner where (l ,ωˉ) << (l pe ,ωˉpe) and 0 0 the firm achieves higher profits, so the planner will choose an allocation with lower leverage and risk. (Note that this imply that the planer will set a higher secondary market liquidity: θ > θpe. Infact,lowerleverageandriskrequirealowerequilibriuminterestrateinthecreditmarketwhich yieldsahigherlevelofmarketliquidity). Similarly,if ψ(1+αr) < α(1+r),fromPart2, ∂U/∂θ < 0,so dωˉpi(l pe ) dωˉier(l pe ) 0 < 0 < 0 . dl dl 0 0 That means there are points that are feasible for the planner where (l ,ωˉ) >> (l pe ,ωˉpe) and firms 0 0 enjoy higher profits, so the planner will choose an allocation with higher leverage and risk, and lowermarketliquidity θ < θpe. ProofofProposition 9: Part1. Derivingthetaxinstruments. Thefirm’sproblemwithtaxesonstorageandleveragecanbewrittenas max [1 Γ(ωˉ)]Rkl τlλpel +Tl 0 0 l0,ωˉ − − subjectto U = (1 τs)U . (A.17) b s − WewritetheLagrangianforthisproblemas = [1 Γ(ωˉ)]Rkl τlλpel +Tl λpe[(1 τs)U U ] . 0 0 s b L − − − − − 15

Then,theoptimalityconditionsare ∂U [1 Γ(ωˉ)]Rk = τlλpe λpe b , − − ∂l 0 ∂U and Γ (ωˉ)Rkl = λpe b (A.18) 0 0 ∂ωˉ Note that the FOC for (ωˉ), equation (A.18), together with equation (A.13) ensures that λpe > 0, which is not necessarily the case with equality constraints. And the optimal contract is described by 1 Γ(ωˉ) ∂Ub τl − = ∂l0 − . (A.19) l 0 Γ 0 (ωˉ) − ∂Ub ∂ωˉ Equating the previous expression and equation (23), and using that U U = τsU , we derive b s s − − thetaxonleverage: n U ∂Ubτs + ∂Ub ∂θ ∂Ub ∂θ ∂U τl = 0 s ∂ωˉ ∂l0 ∂ωˉ − ∂ωˉ ∂l0 ∂θ . b ∂Uhb + ∂θ∂U i 0 ∂ωˉ ∂ωˉ ∂θ The term in square brackets is positive from equation (A.12). In addition, the denominator is positivefromequation(A.16). On the other hand, the break-even condition of investors with a tax on storage was given by equation(A.17). Combiningitwithconstraint(20)wederivethetaxonstorage: e U (θpe) τs = 0 1 s . b − U (θ) 0 s ! Part2. Signingthetaxonstorage. Ifψ(1+αr) > α(1+r)thenfromProposition8theplannerwantstoincreasesecondarymarket liquiditysoθ > θpe. Thus,thestoragetechnologyissubsidized: τs 0. Infact,thetaxonstorage ≤ isstrictlynegativefromequation(24)if ψ < 1andiszeroif ψ = 0. Onthecontrary,ifψ(1+αr) < α(1+r),thentheexternalityisnegative,theplannerwantsto reducesecondarymarketliquidity,and,therefore, τs > 0. Part3. Signingthetaxonleverage. We start by describing the feasible allocations for a firm that chooses the optimal contract and faces the optimal tax on storage, and the efficient level of secondary market liquidity. That is, τs is given by equation (24) and θ is the one that the planner would choose optimally. In this case we have (1 τs)U (θ) = 1 e 0 U s (θ) − U s (θpe) U (θ) = b 0 U b (l p 0 e ,ωˉpe,θpe)+s 0 U s (θpe) − s 0 U s (θ) s s − − b U (θ) b 0 s 0 ! whereweusedthatintheprivateequilibriumU (θpe) = U (l pe ,ωˉpe,θpe),andb pe +s pe = e . s b 0 0 0 0 Lets consider first the case when ψ(1+αr) > α(1+r). In this case ∂U/∂θ > 0 and θ > θpe, 16

then b U (l pe ,ωˉpe,θpe)+s U (θpe) < b U (l pe ,ωˉpe,θ)+s U (θ) . 0 b 0 0 s 0 b 0 0 s Soweconcludethat (1 τs)U (θ) < U (l pe ,ωˉpe,θ) . − s b 0 pe Since ∂U /∂ωˉ > 0, given the leverage l a feasible level of risk will be lower than the risk in b 0 the private equilibrium ωˉpe. So the investor’s break-even condition with the optimal tax and the efficient level of liquidity will lie below the investor’s break-even condition in the private problem. Moreover, from the mapping ωˉier(l ) the slope of the investor’s break-even condition at l pe , which 0 0 hasthesameexpressionregardlessofthetax,willbeflatter. The firm, then, if it were to face this constraint without a tax on leverage will choose a higher leverage, at odds with the planner optimal prescriptions. The planner then will distort the firm’s decision to disincentivize the use of leverage by levying a tax on leverage. One way to see this is that the planner will introduce a distortion such that the distorted isoprofit lines are flatter in the (l ,ωˉ)-space. 0 LetΠτ = [1 Γ(ωˉ)]Rkl τlλpel +Tl,anddenotebyωˉΠτ(l )thefunctionthatforanyl gives 0 0 0 0 − − theassociatedrisklevel ωˉ alongthetaxedfirmisoprofitline. Then,theImplicitFunctionTheorem impliesthat dωˉΠτ [1 Γ(ωˉ)]Rk τlλpe = − − , dl Γ (ωˉ)Rkl 0 0 0 soaflattersloperequiresapositive τl. Usingthesamereasoningweconcludethatif ψ(1+αr) < α(1+r),then τl < 0. Proof of Proposition 10: In the presence of quantitative easing, firms’ borrowing is given by ˉ b , whereas investors’ final bond holdings are given by b b . Then from the budget constraint 0 0 0 − of entrepreneurs we have that k = n + b , so investors’ lending can be written in terms of 0 0 0 ˉ entrepreneurs leverage and QE as n (l 1 b /n ). On the other hand, from the investors’ 0 0 0 0 budget constraint, b b ˉ + s + sˉ = e − , so − we can express the amount invested in the storage 0 0 0 0 0 − technology in terms of entrepreneurs leverage as s = n (e /n (l 1)). Note that the size of 0 0 0 0 0 − − theQEprogramdoesnotaffecttheamountultimatelyinvestedinstorage,asthebondsthecentral bankpurchasesareoffsetwiththereservesittakesfrominvestors. Finally,fromthecentralbank’s budgetconstraintwehavethat sˉ = b ˉ . 0 0 Using the previous expressions we can express secondary market liquidity in terms of entrepreneurs leverage and QE, conditional on the interest on reserves relative to the return on the ˉ OTC market. Note that the number of sell orders is always equal to A = δ(b b ), as impatient 0 0 − investorswillputalltheirbondholdingsforsaleintheOTCmarket. If Δ > 1 + rˉ patient investors pledge all their liquid assets to place buy orders in the OTC market so the number of buy orders B = (1 δ)[(1+r)s +(1+rˉ)sˉ ]/q and market liquidity is 0 0 1 − givenby (1 δ)[(1+r)s +(1+rˉ)sˉ ] (1 δ)Δ (1+r)(e 0 n 0 (l 0 1))+(1+rˉ)b ˉ 0 0 0 − − − θ = − = . (A.20) δ(b 0 − b ˉ 0 )q 1 h δRb n 0 (l 0 − 1) − b ˉ 0 i (cid:16) (cid:17) 17

Then, ∂θ (1 δ)Δ(1+rˉ) (1 δ)Δ (1+r)(e 0 n 0 (l 0 1))+(1+rˉ)b ˉ 0 − − − = − + > 0 . (A.21) ∂b ˉ 0 δRb n 0 (l 0 − 1) − b ˉ 0 h δRb n 0 (l 0 − 1) − b ˉ 0 2 i (cid:16) (cid:17) (cid:16) (cid:17) Ontheotherhand,when 1+rˉ > ΔpatientinvestorsplacebuyordersintheOTCmarketonly usingtheliquidassetstheyholdafterfundingthereservesliquidatedbyimpatientinvestors,sothe numberofbuyordersB = (1 δ)[(1+r)s δ/(1 δ)(1+rˉ)sˉ ]/q andmarketliquidityisgivenby 0 0 1 − − − θ = (1 − δ)[(1+r)s 0 − 1 − δ δ (1+rˉ)sˉ 0 ] = (1 − δ)Δ (1+r)(e 0 − n 0 (l 0 − 1)) − 1 − δ δ (1+rˉ)b ˉ 0 . δ(b 0 − b ˉ 0 )q 1 h δRb n 0 (l 0 − 1) − b ˉ 0 i (cid:16) (cid:17) Then, ∂θ Δ(1+rˉ) (1 − δ)Δ (1+r)(e 0 − n 0 (l 0 − 1)) − 1 δ δ (1+rˉ)b ˉ 0 = + − ∂b ˉ 0 − Rb n 0 (l 0 − 1) − b ˉ 0 h δRb n 0 (l 0 − 1) − b ˉ 0 2 i (cid:16) (cid:17) (cid:16) (cid:17) (1 δ)Δ(1+r) e n (l 1) 1+ δ 1+rˉ − 0 − 0 0 − 1 δ1+r = − > 0 . 2 δRb h n (l 1) b ˉ (cid:16) (cid:17)i 0 0 0 − − (cid:16) (cid:17) ˉ wheretheinequalityfollowsfromAssumption 4. Then, ∂θ/∂b > 0. 0 ProofofProposition11: WewanttoshowthataplannerthathasaccesstoQEasanadditional policytoolwillonlyuseitwhenψ(1+αr) > α(1+r). Let(l sp ,ωˉsp,θsp)betheallocationschosenby 0 the social planner studied in section 4 and denote by λsp the lagrange multiplier on the constraint facedbythisplanner(20). Let betheLagrangianofthecentralbank,whichcanbewrittenas L = [1 Γ(ωˉ)]Rkl λ Upe U(l ,ωˉ,θ(l ,ωˉ,b ˉ ,rˉ),b ˉ ,rˉ) γ (1+rˉ)2 Rˉb ν[r rˉ]+ηb ˉ , 0 0 0 0 0 0 L − − − − − − − h i h i where we are considering the constraint imposed by the definition of secondary market liquidity (17)writingθ(l ,ωˉ,b ˉ ,rˉ)andwherewehavealreadysubstitutedinsˉ = b ˉ . Anoptimalallocation 0 0 0 0 forthisplannerneedstosatisfythefollowingFOCs: ∂ ∂U ∂U∂θ ∂Rˉb (l ) 0 = L = [1 Γ(ωˉ)]Rk +λ + +γ 0 ∂l − ∂l ∂θ ∂l ∂l 0 0 0 0 " # ∂ ∂U ∂U ∂θ ∂Rˉb (ωˉ) 0 = L = Γ (ωˉ)Rkl +λ + +γ 0 0 ∂ωˉ − ∂ωˉ ∂θ ∂ωˉ ∂ωˉ " # ∂ ∂U ∂U ∂θ ˉ (b ) 0 = L = λ + +η 0 ∂b ˉ ∂b ˉ ∂θ ∂b ˉ 0 " 0 0# 18

∂ ∂U ∂U∂θ (rˉ) 0 = L = λ + 2γ(1+rˉ)+ν ∂rˉ ∂rˉ ∂θ ∂rˉ − " # The next step is to evaluate the FOCs at the constrained efficient allocation (without QE), i.e., l sp ,ωˉsp,θsp,0,r . If Rˉb(l sp ,ωˉsp) (1 + r)2 the central bank cannot implement QE without 0 0 ≤ violating its funding constraint (27). So we consider that we are in the interesting case where (cid:16) (cid:17) Rˉb(l sp ,ωˉsp) > (1 + r)2 and the central bank has some scope to offer a higher return on reserves 0 relativetothestoragetechnology. Inthiscasethemultiplierofthisconstraintat l sp ,ωˉsp,θsp,0,r 0 equals zero, i.e., γ = 0. Moreover, note that at b ˉ = 0, investors’ expected utility U has the same 0 (cid:16) (cid:17) ˉ functional form as in the case of the planner studied in section 4. Similarly, at b = 0 secondary 0 marketliquidity θ,equation(A.20),isthesamefunctionofchoicevariablesasinthecasewithout QE, equation (17). So we conclude that the FOCs wrt leverage l and risk ωˉ are satisfied at 0 l sp ,ωˉsp,θsp,0,r . (In fact, we can use either FOC to obtain that λ = λsp, from where the other 0 FOCfollows.) (cid:16) (cid:17) Next,notethat ∂U ∂U ∂U ∂U l sp ,ωˉsp,θsp,0,r = sˉ sˉ = b ˉ sˉ 0 = 0 . 0 0 ∂rˉ ∂rˉ ∂rˉ ⇒ (cid:16) ∂rˉ (cid:17) Andgiventhat (1+rˉ) = (1+r) < Δ fromequation(A.20)wehavethat ˉ ∂θ l sp ,ωˉsp,θsp,0,r ∂θ (1 δ)Δb 0 0 = − = 0 . ∂rˉ δRb n (l 1) b ˉ ⇒ (cid:16) ∂rˉ (cid:17) 0 0 0 − − (cid:16) (cid:17) SotheFOCwrtinterestonreserves rˉistriviallysatisfied,with ν = 0. Finally,weneedtoevaluatetheFOCwrtb ˉ at l sp ,ωˉsp,θsp,0,r . Fromthisconditionitfollows 0 0 that ∂U ∂U ∂θ (cid:16) (cid:17) ˉ + < 0 η > 0 and b = 0 . ∂b ˉ ∂θ ∂b ˉ ⇒ 0 0 0 To sign ∂U/∂b ˉ +(∂U/∂θ)(∂θ/∂b ˉ ) we proceed to compute these derivatives and evaluate at 0 0 l sp ,ωˉsp,θsp,0,r . Bydefinition 0 (cid:16) (cid:17) U(l ,ωˉ,θ,b ˉ ,rˉ) = [e n (l 1)]U +b ˉ U +[n (l 1) b ˉ ]U . 0 0 0 0 0 s 0 sˉ 0 0 0 b − − − − Then, ∂U l sp ,ωˉsp,θsp,0,r 0 = U (θsp,r) U l sp ,ωˉsp,θsp = U (θsp) U l sp ,ωˉsp,θsp , (cid:16) ∂b ˉ (cid:17) sˉ − b 0 s − b 0 0 (cid:16) (cid:17) (cid:16) (cid:17) where we used that if interest on reserves are equal to the return on the storage technology then U (θsp,r) = U (θsp), from equation (31). In addition, from the conditions that describe the sˉ s 19

planner’sallocationswehavethat s sp U (θsp)+b sp U l sp ,ωˉsp,θsp = s pe U (θpe)+b pe U l pe ,ωˉpe,θpe = e U (θpe) 0 s 0 b 0 0 s 0 b 0 0 s (cid:16) (cid:17) e [U ((cid:16)θpe) U (θ(cid:17)sp)] U l sp ,ωˉsp,θsp U (θsp) = 0 s − s . (A.22) ⇒ b 0 − s b sp 0 (cid:16) (cid:17) Then,fromthecharacterizationoftheconstrainedefficientallocation(Proposition 8)wehavethat if ψ(1 + αr) < α(1 + r), then θsp < θpe and ∂U/∂θ < 0. So we conclude that ∂U/∂b ˉ < 0. In 0 addition, from Proposition 10 we have that ∂θ/∂b ˉ > 0, thus ∂U/∂b ˉ +(∂U/∂θ)(∂θ/∂b ˉ ) < 0. 0 0 0 ˉ Thus,itmustbethat η > 0andtheoptimalQEdesignscallsfornotbuyingbonds,i.e., b = 0. 0 Alternatively, when ψ(1 + αr) < α(1 + r) from Proposition 8 we have that θsp > θpe and ∂U/∂θ > 0,thus∂U/∂b ˉ > 0andweconcludethat∂U/∂b ˉ +(∂U/∂θ)(∂θ/∂b ˉ ) > 0. Therefore, 0 0 0 ˉ ∂ /∂b > 0,i.e.,thecentralbankwillwanttoincreasethesizeofthebondbuyingprogramwhen 0 L ˉ itiszeroandweconcludethat b > 0,improvingupontheconstrainedefficientallocation. 0 ˉ We are left to establish that at the allocation implemented by the central bank, where b > 0, 0 rˉ > r. For that we consider evaluate the FOC wrt rˉ when rˉ = r. Note that in this case where 1+rˉ < Δ wehavethat ∂U ∂U rˉ r = sˉ sˉ = b ˉ δ+(1 δ) pΔ+(1 p)(1+r)+ − +1+r > 0 . 0 0 ∂rˉ ∂rˉ − − 1 δ (cid:26) (cid:20) − (cid:21) (cid:27) Inaddition, ˉ ∂θ (1 δ)Δb 0 = − > 0 . ∂rˉ δ[n (l 1) b ˉ ]Rb 0 0 0 − − Soweconcludethat ∂ (l ,ωˉ,θ,b ˉ ,r) ∂U ∂U∂θ 0 0 L = λ + +ν > 0 . ∂rˉ ∂rˉ ∂θ ∂rˉ " # Recall that we assumed that Rˉb(l sp ,ωˉsp) > (1+r)2. So we conclude that the central bank will set 0 theinterestonreservesstrictlyhigherthattheinterestonstorage,i.e., rˉ > r. 20