Misallocation Costs of Digging Deeper into the Central Bank Toolkit

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Misallocation Costs of Digging Deeper into the Central Bank Toolkit Robert Kurtzman and David Zeke 2017-076 Please cite this paper as: Kurtzman, Robert, and David Zeke (2017). “Misallocation Costs of Digging Deeper into the Central Bank Toolkit,” Finance and Economics Discussion Series 2017-076. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2017.076. 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.

Misallocation Costs of Digging Deeper into the Central Bank Toolkit ∗ Robert Kurtzman†1 and David Zeke ‡2 1Federal Reserve Board of Governors 2University of Southern California July 19, 2017 Abstract Centralbanklarge-scaleassetpurchases,particularlythepurchaseofcorporatebondsofnonfinancialfirms,caninduceamisallocationofresourcesthroughtheirheterogeneouseffectonfirmscostof capital. First,weanalyticallydemonstratethemechanisminastaticmodel. Wethenevaluatethemisallocationofresourcesinducedbycorporatebondbuysandtheassociatedoutputlossesinacalibrated heterogeneousfirmNewKeynesianDSGEmodel. Thecalibratedmodelsuggestsmisallocationeffects fromcorporatebondbuyscanbelargeenoughtomakethemlesseffectivethangovernmentbondbuys, whichisnotthecasewithoutaccountingformisallocationeffects. Keywords: QE,LSAPs,Misallocation JELClassification: E22,E51,E52,G21 ∗The views expressed in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the Board of Governorsof the Federal Reserve System or of anyone else associated with the Federal Reserve System. We would like to thank Andrea Prestepino and Romain Ranciere for helpful discussions on the paper. We would also like to thank Pablo Cuba-Borda, Joel David, Nils Gornemann, Peter Karadi, Borghan Narajabad, Lukas Schmid, and other participants at SED 2017 and the USC and Fed Board macro workshops for helpful comments. We would like to thank those at the Macroeconomic Model Data Base for providing their code, whichweadapted. Allmistakesareourown. †Corresponding author: Email: robert.j.kurtzman@frb.gov, Phone: 202-452-2589, Address: 20th and ConstitutionNW,Washington,D.C.20551. ‡Email: zeke@usc.edu

1 Introduction Central banks, such as the Bank of Japan (BOJ), the Bank of England (BOE), the European Central Bank (ECB), and the Federal Reserve (the Fed), have used large-scale asset purchases (LSAPs) as a policy tool once they have reduced the short rate under their control to its effective lower bound.1 While the BOJ, the ECB, and the Fed purchased government bonds as part of their LSAP programs, as they have chosen to buy other assets, their choices have differed: between them, the different central banks have purchased corporate bonds, exchange-traded funds, mortgage-backed securities, and other assets. Yet there is little theory to guide central banks on whetherthereareaggregatecoststopurchasingsomeprivateassetsratherthanothers. Thispaperaddressesthisgapintheliterature,focusingspecificallyonthecostsassociatedwith large-scale purchases of nonfinancial corporate bonds. Purchases of corporate bonds may reduce borrowing costs by more for firms whose securities are purchased than those not purchased and potentially create distortions in the cost of capital among firms.2 In standard models of firm financing and capital choice, differences in the cost of capital induce differences in firm investment decisions and thus the allocation of capital, which has consequences for the efficiency of the allocation and aggregate output. Our work builds a simple theory of LSAPs with this mechanism present to deliver analytical results regarding how large-scale purchases of corporate bonds affect the allocation of capital among firms. We then introduce the key elements of our simple model into a New Keynesian DSGE model. With our calibrated DSGE model, we quantify the potential misallocativeeffectsoflarge-scalepurchasesofnonfinancialfirmcorporatebonds. TodemonstratetheeconomicmechanismatworkwithinourDSGEmodel,thefirstpartofthe paper takes a static model of firm dynamics with a financial intermediary sector and demonstrates the conditions under which a large-scale purchase of nonfinancial firm corporate bonds by the central bank induces a misallocation of resources. The model allows us to separate two effects of 1SuchLSAPsareoftenreferredtoasquantitativeeasing(QE)policies. 2 KrishnamurthyandVissing-Jorgensen(2011), amongothers, demonstratethattherearepriceeffectsofcentral bankbondbuysthataremorepronouncedforthesecuritiespurchasedthanothersecurities. 2

a shock to interest rates that lowers rates for one set of firms (firms issuing highly rated corporate bonds, which we denote as “large firms”) more than for another set of firms (which we denote as “small firms”): (1) the effect on the allocation of capital (2) the effect on the aggregate capital stock. Weisolatethekeyelementsofourmodelthatgovernthesizeofeacheffect.3 In our model, following Gertler and Karadi (2011) and Gertler and Karadi (2013) (hereafter, GK11 and GK13), a financially constrained intermediary helps facilitate the financing of capital by firms. Intervention by the central bank in asset markets affects the quantity and distribution of assets which are intermediated and the constraint faced by the financial intermediary. Our simple, staticmodelintroducesthetwokeyadditionalmodelelementsweusetostudytheeffectsofmisallocation.4 The first is heterogeneity in production among multiple “groups” of firms. Specifically, we allow there to be two (or more) types of intermediate good firms that have similar production technologies. Their outputs are used in the production of the final good, and are imperfect substitutes. Second, we introduce a regulatory constraint that allows for richer heterogeneity in spreads thaninthemodelofGK13. Theregulatoryconstraintmakesitmorecostlyforbankstoholdrisky private-sector debt than government bonds.5 Additionally, the regulatory constraint discourages theconcentrationofcertainclassesofprivatesecurities.6 Inequilibrium,theregulatoryconstraints induceasymmetriceffectsinthecreditspreadresponseofpurchasingdebtoffirmsrelativetothat ofthegovernment. Wedemonstratethatinourmodelwhenthecentralbankbuysthesecuritiesofonesetoffirms, 3 It is possible to use a large-scale purchase of corporate bonds to either reduce a misallocation or increase a misallocation. However,wefocusonthelattereffectinexplainingthemechanism,guidedbyourcalibration. 4WebuildontheframeworkofGK13whichiswidelyusedinstudyingQEpoliciesacrosscentralbanks. 5PoliciessuchastheLiquidityCoverageRatioaretherealisticcounterparttothismodelconstraint. Detailsofthe LiquidityCoverageRatiocanbefoundathttps://www.occ.gov/news-issuances/bulletins/2014/ bulletin-2014-51.html. 6 This constraint is motivated by guidance on stress testing from bank regulators. For example, in the Guidance for Stress Testing from the Federal Deposit Insurance Corporation, the Office of the Comptroller of the Currency, and the Fed in 2012, found at https://www.federalreserve.gov/supervisionreg/srletters/ sr1207a1.pdf, the regulatory authorities state: “Accordingly, stress tests should provide a banking organization withtheabilitytoidentifypotentialconcentrations includingthosethatmaynotbereadilyobservableduringbenign periods and whose sensitivity to a common set of factors is apparent only during times of stress and to assess the impactofidentifiedconcentrationsofexposures, activities, andriskswithinandacrossportfoliosandbusinesslines andontheorganizationasawhole.” 3

itlowersthecostofcapitalforthatsetoffirmsbymorethanthatfortheothersetsoffirms,allelse being equal. When the central bank buys government bonds, if spreads between loans to various typesoffirms(largeandsmallfirms,forinstance)aresmallinsteadystate,thecentralbankreduces the cost of capital for all firms approximately evenly. Hence, in equilibrium, there is an additional effectontheallocationofresourcesfromacorporatebondpurchasethatdoesnotoccurtothesame degree from a government bond purchase. Such a framework thus endogenizes the heterogeneous effectonborrowingcostsfromalarge-scalepurchaseofnonfinancialfirmcorporatebonds. The second part of this paper quantifies the misallocation effects of LSAPs of bonds issued by one set of firms (large firms) and not another set of firms (small firms) in a calibrated, DSGE model similar to our two-period model but with DSGE elements that we did not include in our static model: sticky prices, endogenous net worth of banks, and a representative household with habitsthatcanholdbondsfacingaholdingcost. IntheDSGEmodel,theresponseofcreditspreads to an asset purchase is not only heterogeneous, as in our static model, but also time-varying. We calibratethenewparametersweintroduceusingU.S.data. In the calibrated model, a QE policy in which the central bank purchases public debt produces apositive,largeeffectoninvestmentandoutputafterabadshock. However,aQEpolicyinwhich the central bank purchases the debt of large firms—although potentially inducing a similar effect on output—reducesthe response of investmentfor small firms whosedebt is not purchasedby the central bank and induces a non-negligible misallocation of resources. In fact, away from the zero lowerbound(ZLB),ourcalibrationimpliesthatthemisallocationeffectislargeenoughtomakea government bond purchasemore effectivethan a private bond purchasein terms of increasing output, even though without misallocation a government bond purchase is less effective in increasing outputthanalarge-scalecorporatebondbuy. Themisallocationeffectisnon-negligibleawayfrom the ZLB relative to movements in output; however, at the ZLB, the effect of LSAPs on output are amplified,whilethemisallocationeffectisnot. Therefore,themisallocationeffectasapercentage ofthepotentialoutputgainfromlarge-scalecorporatebondbuyswillbesmallerattheZLB. The rest of the paper, after the literature review below, follows as such. Section 2 presents 4

results from the simple model. Section 3 describes the DSGE model, its calibration, and assesses thequantitativeimplicationsoflarge-scalepurchasesofnonfinancialfirmcorporatebonds. Section 4discussestheroleoftheZLBandconcludes. 1.1 Related Literature This paper is closely related to a literature that embeds QE policies in macroeconomic models to analyze the channels through which such policies affect the economy. GK11 and GK13 study QE policies in a representative firm DSGE model with constrained financial intermediaries, and their framework is a embedded as special case of our own. As financial frictions are key to our channel, it could be grouped, largely, into an “imperfect asset-substitutability” channel of monetary policy.7 Other papers in this literature, such as He and Krishnamurthy (2013) and Cu´rdia and Woodford (2015), also emphasize the role of financial market imperfections in making QE effective. In the models of GK11 and GK13, the central bank is less efficient at intermediating financial transactions than the private sector. The calibrated models suggest that QE policies by the central bankcanreducecreditspreadsandincreaseinvestmentandoutputnonetheless. Arelatedliterature examines other indirect costs and benefits of QE policies. Hall and Reis (2015) asssesses the potential risks to central bank solvency. Reis (2017) points out that central bank liabilities used to fundLSAPsarespecialinthattheyarefreeofdefaultrisk,andthuscouldproveasausefulpolicy toolinfightinginflationinafiscalcrisis. Tokeeptotheeconomicpointofinterest,ourpaperdoes notincorporatesuchadditionaltradeoffs. Ourpaperisalsocloselyrelatedtothelargeliteratureonhowmisallocationaffectsthemacroeconomy, built on the work of Hopenhayn and Rogerson (1993) and Hsieh and Klenow (2009), among many others. The closest papers to ours are Midrigan and Xu (2014) and Gilchrist, Sim, and Zakrajsek (2013), as they study how financial frictions that induce a misallocation (and result 7ThereisagrowingliteratureaddressingdifferentchannelsinwhichQEpoliciesaffecttheeconomy.InBhattarai, Eggertsson,andGafarov(2015),theauthorsmodelthe“signalingchannel”ofQEpolicies.InGreenwoodandVayanos (2014), the authors outline the “duration-risk channel” of QE policies. Other channels have been highlighted in the literatureonQE,suchastheprepayment-riskchannelandopen-economychannels. 5

inawiderdispersionincreditspreads)affectthemacroeconomy.8 Lastly, there is growing empirical evidence of there being heterogeneous effects of LSAPs through various channels.9 Darmouni and Rodnyansky (2017) and Kurtzman, Luck, and Zimmermann (2017) show the Federal Reserve’s purchases of MBS in QE1 and QE3 incentivized more lending and risk-taking, respectively, by banks with more MBS holdings. Di Maggio et al. (2016) identify an effect of QE on the volume of new mortgages originated and show that the type of mortgagesoriginatedweremorelikelytobethosethatcouldbesecuritizedandsoldtotheFederal Reserve. Chakraborty et al. (2016) show banks that are more active in the MBS market reduce commercial lending subsequent to QE by the Fed, inducing the firms borrowing from these banks to reduce investment. Foley-Fisher et al. (2016) present evidence that the Federal Reserve’s maturity extension program had a greater effect on the valuation, investment, and employment of firms whichweremoredependentonlong-termdebt. 2 Demonstrating the Mechanism in a Simple Model We begin by highlighting the main mechanism in our paper within a static framework of firm capital choice and financing. The model details how the capital decisions of heterogeneous firms are affected by the financing environment and how central bank LSAPs change the allocation of capitalandaffectmacroeconomicaggregates. Heterogeneousfirmschoosetheircapitaltomaximizeprofits,buttheircapitalchoiceandprofits are affected by their (heterogeneous) cost of capital. Firms must finance their capital via constrained financial intermediaries. Purchases of securities by the central bank will loosen the financialintermediary’scollateralconstraint,loweringspreadsandthereforefirms’costofcapital. Due to a regulatory constraint that enters the financial intermediary’s problem, purchases of corporate 8 InGilchrist, Sim, andZakrajsek(2013), thefinancialfrictionsarenotexplicitlymodeledbutitisassumedthat financialfrictionsinducethedispersioninborrowing. Further,intherobustnesssectionofMidriganandXu(2014), thereisanevaluationofhowheterogeneityinborrowingratesinduceamisallocation,althoughthefocusofthepaper is on how a worsening of financial frictions can induce a misallocation. Also, it is important to note that financial frictionscanalsodistortentryandtechnologyadoptioninMidriganandXu(2014). 9ThisliteraturehasgrownoutofempiricalworkbyKrishnamurthyandVissing-Jorgensen(2013)andChodorow- Reich(2014)whoassesstheeffectsofLSAPsoneconomicoutcomes. 6

securitieswilllowerspreadsforpurchasedsecuritiesbymorethanfornon-purchasedsecurities. 2.1 Model and Equilibrium The model consists of heterogeneous intermediate good firms and a representative final good firm, financial intermediary, capital producer, and household. There are J continuums of intermediate good firms. Intermediate good firms are indexed by i and the continuum they belong to by j. The total mass of firms is normalized to 1 ( (cid:80)J (cid:82) di = 1). Each intermediate good firm, i, j=1 i∈j producesadifferentiatedgoodusingcapital,k ,withtechnologyy = A kα. i i i i The final good is produced from intermediate goods with technology Y = (cid:0)(cid:82) yρdi (cid:1) ρ 1 . The i i final good can either be consumed or used to produce capital. Capital can be produced from the final good with technology that requires φK (K) units of the final good to produce K units of capital, where φK (K) is weakly convex. Therefore, the clearing condition for the final good is Y = C +φK (K). Finally,intermediategoodfirms,intotal,cannotusemoreofthecapitalgoodthanisproduced: (cid:82) K ≥ k di. i i 2.1.1 FinalGoodFirms,CapitalProducers,andHouseholds The final good sector is competitive, and the final good is the numeraire. Thus, it earns zero profitsandmaximizestheproblem: (cid:18)(cid:90) (cid:19)1 (cid:90) ρ max yρdi − p y di. (1) i i i yi i From(1),wecanobtainthestandardexpressionforthepriceofintermediategoodfirms: (cid:16)y (cid:17)ρ−1 i p = . i Y Aggregate capital is chosen to maximize profits of a firm that converts the final good into 7

capital: max QK −φK (K), K where Q is the price of capital, shared across firms. We assume that φK = h k K1+bk, where h is a 1+b k k parameterthataffectsthelevelofthecostofproducingcapital,whileb isaparameterthataffects k theconvexityofthecostofproducingcapital. Households are the residual claimants of all profits by firms or intermediaries, and there is a representative household that consumes the final good. Households can also lend or borrow from the financial intermediary at a gross interest rate r. Households make consumption and lending decisionstomaximizeconsumptionsubjecttotheirbudgetconstraint: C +D = D r+X, h h where D is the net lending of the household to the financial intermediary and X is the sum of h the household endowment and profits of intermediate good firms, financial intermediaries, and capital producing firms. All lending occurs within the period. In equilibrium, the gross return on householdlendingmustber = 1. Otherwise,thehouseholdwouldwanttolendorborrowinfinite amountstofinancialintermediaries. 2.1.2 IntermediateGoodFirms Intermediategoodfirmimaximizesprofits,definedasrevenuesfromproductionlessexpendituresoncapital: p y i i max −r Qk , ki τ i k i i where τk is an exogenous wedge that represents distortions or inefficiencies in capital allocation i not included in the model. Capital expenditures by the firm must be financed, and r is the gross i 8

interest rate at which a firm can borrow (the “cost of capital”). This maximization problem yields first-orderconditionforcapital: (cid:18) αρY1−ρAρ(cid:19) 1− 1 αρ k = i . (2) i τkr Q i i 2.1.3 FinancialIntermediation As firm capital must be financed, the capital expenditures of firm i must be equal to the firm i securitiesheldbyfinancialintermediariesandthecentralbank: Qk = S +S , i b,i g,i where S denotes the representative financial intermediary’s holdings of firm i securities and S b,i g,i denotescentralbankholdingsoffirmisecurities. A representative financial intermediary has exogenous financial wealth N and, along with investingincorporatesecurities,investsingovernmentbonds. Thetotalsupplyofgovernmentbonds, BS, is net positive, and clearing implies BS = B + B , where B are central bank holdings of b g g governmentbonds. If the intermediary’s holdings are not equal to its wealth, it either borrows or lends to households atgross interestrate r. The financial intermediary alsofaces a regulatoryconstraint limiting its leverage. This regulatory constraint will depend on the assets purchased and their concentration.10 We define our J continuums of firms so that each continuum represents firms whose securities are treated identically in terms of capital requirements and concentration risk. In our model, therepresentativefinancialintermediaryfacesthefollowingregulatorycollateralconstraint: (cid:88) (cid:18)(cid:90) (cid:19)νj V ≥ θ∆ S di +θB , (3) j b,i b i∈j j 10 This regulatory constraint is motivated by capital requirements placed on banks, which will differ for different asset classes, as well as the required bank stress testing, which penalizes high concentration of similar risky assets (duetogreaterexposuretocommonriskfactors). 9

where θ∆ and θ are constants that reflect capital requirements for corporate bonds of firms i ∈ j j andgovernmentbonds(ifholdinggovernmentbondsisassociatedwithlowercapitalrequirements than holding corporate bonds, θ < θ∆ ). Parameters ν ≥ 1 reflect the penalty for concentration j j of assets of type j.11 V is the market value of the financial intermediary’s equity. Note that with ∆ = 1∀j, concentration risk, and a representative intermediate good firm, this is similar to the j collateralconstraintinthemodelofGK13. Thefinancialintermediarymaximizesitsmarketvalue: (cid:32) (cid:33) (cid:90) (cid:90) (cid:88) (cid:88) V = max S r di+B r + N − S di−B r, (4) b,i i b b b,i b S ,B b,i b i∈j i∈j j j subjecttothecollateralconstraint(3). Themaximizationproblem(4)yieldsthefollowingfirst-orderconditionfortheinterestrateon thedebtoffirmioftypej: λ (cid:18)(cid:90) (cid:19)νj−1 (r −r) = θ∆ ν (Qk −S )ds , (5) i j j s g,s (1+λ) s∈j whereλistheLagrangianmultiplieronthecollateralconstraint. NotethatsinceS = Qk −S , b,i i g,i central bank purchases of corporate bonds enter this condition. A similar condition results for the interestrateongovernmentbonds: λ (r −r) = θ. (6) b (1+λ) Thecollateralconstraintoftheintermediarycanthusbewrittenas: (cid:32) (cid:33) (cid:88) (cid:88) (r −r)B +B (r −r)+Nr ≥ θ∆ S νj +θB , (7) j j b b j b,j b j j (cid:82) wherer isthespreadonfirmsi ∈ j,whichmustbeidenticalfollowing(5),andS = S di = j b,j i∈j b,i 11Ifν =1,thereisnopenaltyforconcentrationofsecuritiesoftypej firms. j 10

(cid:82) (Qk −S )di. Aswewilldetailbelow,governmentpurchasesofcorporatesecurities,S ,or i∈j i g,i b,i long-termgovernmentbonds,B ,lowertheamountthefinancialintermediaryhastointermediate. b 2.1.4 Equilibrium (cid:26) (cid:27) Given exogenous bank net worth, N, firm-level productivities and wedges, A ,τk ∀i, ceni i (cid:26) (cid:27) tral bank purchases of corporate bonds, S ∀i, and government bonds, B , equilibrium in g,i g (cid:26) (cid:27) (cid:26) (cid:27) (cid:26) (cid:27) this model is a set of allocations, C,Y,K,B ,D and k ,y ,S ∀i, and prices, Q,r,r b h i i b,i B (cid:26) (cid:27) and r ,p ∀i, such that households maximize consumption subject to their budget constraint; i i intermediate good firms, final good firms, and capital producers maximize profits; financial intermediariesmaximizeprofitssubjecttotheircollateralconstraint;andclearingconditionshold. 2.2 Effect of Central Bank Bond Buys Central bank purchases of either long-term government bonds (B ) or corporate securities of g firm i (S ) will reduce the amount of that particular asset that has to be intermediated by the g,i financial intermediary. LSAPs will directly affect bond spreads by changing: (1) the Lagrange multiplier on the collateral constraint, λ; (2) the first-order condition (5), by reducing the concentration of firms of type j in the intermediary’s balance sheet. Additionally, LSAPs will indirectly affectspreadsthroughtheireffectonfirmcapitalchoices,ascanbeseeninthefirst-ordercondition for capital. Changing firm capital choices will also affect the price of capital, Q, and the amount of firm capital that needs to be intermediated, Qk −S , which enters directly into the collateral i g,i constraint. Proposition 1 states that we can analytically demonstrate the effect of LSAPs on bond spreads,holdingfirm-capitalchoicesfixed. Proposition 1. Holding firm capital choices, k , fixed, central bank LSAPs have the following i effects: (i) Apurchaseoflong-termgovernmentbonds,B , g (a) decreasesλ,theLagrangianmultiplieronthecollateralconstraint. 11

(b) proportionatelydecreasesfirmspreads,thatis, ∆(ri−r) isconstantforallfirms. (ri−r) (ii) Apurchaseoffirmsecurities,S ,fori ∈ j g,i (a) decreasesλ,theLagrangianmultiplieronthecollateralconstraint,ifNr ≥ (νj−1)B θ+ νj b (cid:16) (cid:16) (cid:17) (cid:17) (cid:80) (cid:82) 1− νs θ Sνs wheres (cid:54)= j. s i∈s νj s b,i (b) doesnotleadtoaproportionatedecreaseinfirmspreads,thatis, ∆(ri−r) isgreaterfor (ri−r) firmsoftypei ∈ j thanothertypeswhosedebtisnotpurchased(i ∈/ j). Proof. SeeAppendixA. This proposition formalizes the direct effects of LSAPs on spreads. Directly buying the securities of only firms of type i ∈ j will, holding capital choices constant, lower their spreads by more than of firms with i ∈/ j. This can be contrasted with the effect of purchases of long-term government debt, which will lower spreads proportionately. However, purchases of both longterm government bonds and corporate securities will, under some reasonable conditions, decrease spreads by loosening the collateral constraint faced by financial intermediaries (implying a lower multiplierontheconstraint,λ).12 Therefore, central bank LSAPs of corporate securities will induce asymmetric changes in spreads and therefore firm cost of capital. Through the first-order condition for capital, (2), these spreadswillinducechangesinfirmcapitalchoicesandthereforeboththeallocationofcapitaland aggregatecapitalsupply. Subsection2.2.1demonstrateshowsuchdifferencesinspreadsaffectthe allocationofresources. 2.2.1 AllocativeEfficiency Proposition1tellsushowgovernmentandcorporatebondbuysdirectlyaffectborrowingrates of firms and the government, holding fixed the indirect effect of firm decisions changing in response. Althoughwecannotdirectlymapbondbuystoaggregatesanalytically,todevelopintuition 12Holdingcapitaldecisionsfixed,thisisalwaystrueforpurchasesofgovernmentdebt. Forpurchasesofcorporate securities, theconditioninProposition1(ii)(a)musthold, whichcanbeunderstoodasimplyingthatfirmsmustnot holdtoomuchgovernmentdebtrelativetowealth,withanadjustmentforheterogeneousν . j 12

as to how bond buys affect allocative efficiency it is useful to walk through how spreads affect the allocationandaggregates.13 Inequilibrium,wecanexpresstherelativeholdingsoffirmcapitalas (cid:16) Aρ (cid:17) 1−γ 1 −αρ i k i = riτ i k . K (cid:80) (cid:82) (cid:16) Aρ i (cid:17) 1−γ 1 −αρ di j i∈j riτ i k Therefore, the relative levels of firm cost-of-capital, r , have implications for the relative alloi cationofcapital. Additionally,firmcost-of-capitalaffectstheaggregatedemandforcapital: (cid:32) (cid:82) (cid:18) Aρ (cid:19) 1− 1 αρ (cid:33) ρ(1− 1− α+ ρ bk ) (cid:32) (cid:82) (cid:18) Aρ (cid:19) 1− 1 αρ (cid:33) 1− 1 α − + α bk i di i di (riτ i k)αρ (riτ i k) K = . (8) (cid:16) (cid:17) 1 h k 1−α+bk αρ In equilibrium, our model yields macroeconomic aggregates as an analytical function of firm interest rates, r , and exogenous variables (A ,τk∀i). Building on (8), we can express aggregate i i i outputas (cid:32) (cid:18) (cid:19) 1 (cid:33) ρ 1 (cid:82) Aρ 1−αρ i di (riτ i k)αρ Y = Kα . (9) C (cid:124) a (cid:123) p (cid:122) it (cid:125) al (cid:18) (cid:82) (cid:16) Aρ i (cid:17) 1− 1 αρ di (cid:19)α riτ i k (cid:124) (cid:123)(cid:122) (cid:125) Allocation Note that (9) shows that output can be expressed as a function of aggregate capital, Kα, modified by a term that captures both the productivity of intermediate good firm production functions and the efficiency of the allocation of capital. For example, if there is no heterogeneity (A = A,r = r ,τk = τk),then(9)reducestoY = AKα. i i A i Tomoreclearlydemonstratetheeffectofspreadsonoutputandallocativeefficiency,wedefine r , the weighted-average interest rate faced by firms, such that aggregate capital depends only on A 13 Thisisduetothenonlinearityintheequationlinkingfirmcreditspreadstocentralbankbondpurchases(dueto firm-capitalchoicesreactingtothechangeinspreads,asecond-ordereffect). 13

r (anddoesnotdependonheterogeneityininterestrates): A (cid:32) (cid:82) (cid:18) Aρ i (cid:19) 1− 1 αρ (cid:16) 1 (cid:17) 1− α α ρ ρ di (cid:33)1− ρ ρ (cid:18) (cid:82) (cid:16) Aρ i (cid:17) 1− 1 αρ (cid:16) 1 (cid:17) 1− 1 αρ di (cid:19)1−α 1 (τ i k)αρ ri τ i k ri = . r A (cid:32) (cid:18) (cid:19) 1 (cid:33)1− ρ ρ (cid:32) (cid:18) (cid:19) 1 (cid:33)1−α (cid:82) Aρ 1−αρ (cid:82) Aρ 1−αρ i di i di (τk)αρ (τk) i i Note that if all firms have the same interest rate, r = r , we can then define interest rate wedges, i A r ,betweenfirminterestratesandtheweighted-averagefirminterestrateas τ,i r A r = . τ,i r i These expressions allow us to derive an expression for output as a function of aggregate capital, which depends only on the weighted-average interest rate, r , and for productivity and allocative A efficiency, which depends only on firm productivities, A , interest rate wedges, r , and other i τ,i exogenousdistortions,τk: i (cid:18) (cid:82) (cid:16) Aρ (cid:16) rτ,i (cid:17)αρ(cid:17) 1− 1 αρ di (cid:19) ρ 1(1 1 − − α α ρ) i τk Y = Kα i . (10) (cid:124)(cid:123)(cid:122)(cid:125) (cid:32) (cid:18) (cid:19) 1 (cid:33)( ρ 1 ( − 1− ρ α )α ) (cid:32) (cid:18) (cid:19) 1 (cid:33)α Capital (cid:82) Aρ 1−αρ (cid:82) Aρ 1−αρ i di i di (τk)αρ (τk) i i (cid:124) (cid:123)(cid:122) (cid:125) Allocation From (10), and building on proposition 1, there are two first-order consequences of central bank purchases of corporate securities that lower bond spreads heterogeneously. First, they will lower the weighted-average interest rate, r , leading to greater aggregate investment and capital. A However,theycanalsogenerateinterestratewedges,r ,whichhaveconsequencesforthealloca- τ,i tionofcapitalrelativetotheefficientlevel. Thesize(anddirection)oftheseeffectsdependonhow far the baseline allocation is from its efficient level and whether the interest rate wedge changes inducedbybondbuysexacerbateorundodistortions. Thelatterpointcanbefurtherformalizedby 14

derivingoutput-maximizinginterestratewedges,aswedoinProposition2below. Proposition2. Theoutput-maximizingallocationoffirminterestratewedgessatisfiesr∗ ∝ τk τ,i i Equivalently,r∗ ∝ r 1 i Aτk i Proof. SeeAppendixA. Proposition 2 shows that the optimal interest rate wedges are such that they exactly offset exogenous distortions τk. If there are no exogenous distortions then the optimal allocation arises i whenfirmsallhaveidenticalcostsofcapitalr = r . A i When there are only two types of firms (j ∈ 1,2), where r and τk are symmetric for all i ∈ j, i i we can characterize all interest rate wedges using just a single interest rate wedge, r , and the τ,1 weighted average cost of capital, r . The following corollary shows that in this case, output is A monotonically decreasing as the interest rate wedges move further from their output-maximizing values: Corollary2.1. Inthecasewithonlytwotypesoffirms, ∂Y < 0. ∂|rτ,1−r τ ∗ ,1 | Thus, given that large-scale corporate bond buys can induce heterogeneous movements in spreads, they can cause (reverse) a misallocation by moving interest rate wedges away from (toward) the efficient allocation.14 For example, if interest rate wedges of type 1 (large) firms are greater (therefore interest rates are lower) than those of type 2 (small) firms in steady state, central bank purchases of large firm assets increase large firm interest rate wedges relatively further. In this setting, central bank bond buys of large firm assets will distort the allocation further from its efficient level. Looking ahead, our New Keynesian DSGE model will be calibrated such that interest rate wedges of large firms are greater than those of small firms in steady state, and central bankbondbuysoflargefirmassetswilldistorttheallocationfurtherfromitsefficientlevel. Thus, Corollary 2.1 provides the key intuition behind the results we will present in the calibrated New KeynesianDSGEmodel. 14A version of Corollary 2.1 can be derived for a case with more than two types of firms, but the measure of ‘distance’ from the efficient allocation will be a more complicated function of firm interest rate wedges, exogenous distortions,andproductivities. 15

3 New Keynesian DSGE Model We evaluate the impact of the misallocative effect of LSAPs by the central bank in a richer environment where the effects of central bank asset purchases on firm borrowing rates are endogenized. WeembedoursimplemodelinastandardNewKeynesianDSGEmodelfollowingGK13; alongwithexplicitlymodelingbanks,themodelhasthekeyelementsofaNewKeynesianmodel: households, nonfinancial firms, capital goods producers, retail good firms with sticky prices, a central bank, and a government. It is useful to start by outlining the changes we make to the nonfinancial firm sector and then discussing the changes made to households and the financial sector. Afterwards,weoutlinetheremainderofthemodelanddefineanequilibrium.15 3.1 Model Description Nonfinancial and Capital Good Firms There are two continuums, indexed by j ∈ (1,2), of nonfinancialintermediategoodfirms. Eachfirmiincontinuumj producesoutputwithtechnology: Y = A KαL(1−α), i,t i,t i,t i,t where Y is the intermediate good output of firm i, K its capital stock, L its employment, i,t i,t i,t α ∈ (0,1)governs capital’sshare in production. Totalintermediate good firmoutput, Y ,is then m,t computedusingaCESaggregator: (cid:32) (cid:33)1 (cid:90) ρ (cid:88) Y = ω Yρdi , m,t j j,t i∈j j where ω is a parameter greater than or equal to 0 that is a factor affecting the extent to which the j output of intermediate good firms of type j enters total output, and ρ is the CES parameter. Note ifρ = 1,thenintermediategoodsareperfectlysubstitutable. 15Besidestheproductionandfinancingenvironment,alloftheremainingmodelelementsareexactlyasinGK13to allowforeasycomparisonofquantitativeresults. Theirmodelcanbethoughtofasaspecialparameterizationofour modelinwhichfirmheterogeneityandconcentrationriskareunimportant. 16

Allfirmswithineachcontinuumfacethesamefinancingenvironmentandproductiontechnology,asinSection2. Wecanthereforerepresenteachcontinuumoffirmswitharepresentativefirm oftypej. Wecanthuswritetheproductiontechnologiesforrepresentativefirmsasfollows: Y = A Kα L(1−α), j,t j,t j,t j,t where Y is the intermediate good output of type j firms, A an index of type j TFP, K is the j,t j,t j,t type j firm capital stock, L is the type j total employment.16 Similarly, total intermediate good j,t outputisthuscombinedfromtypej intermediategoodoutputswithproductionfunction: (cid:32) (cid:33)1 ρ (cid:88) Y = ω Yρ . m,t j j,t j WesetA = A ,sofirmsreceiveidenticalproductivityshocks.17 j,t t Following the usual arguments from cost-minimization, the price of intermediate good j can bewrittenas P = ω P Y1−ρYρ−1, j,t j m,t m,t j,t where P is the relative price of intermediate goods. Firms choose labor to maximize revenues m,t lesslaborexpense,whereW isthewageratewhichisconstantacrossfirms. Thenwecanrecover t firmj’sdemandforlaborfrom Y1−ρYρ m,t j,t W = P ω ρ(1−α), t m,t j L j,t which holds for each type j. Remaining revenues accrue to capital, so gross profits per unit of 16IfwedefineY = (cid:16) (cid:82) Yρdi (cid:17) ρ 1 ,A = (cid:16) (cid:82) A1− ρ ρdi (cid:17)1− ρ ρ ,K = (cid:82) K di,L = (cid:82) L di,itcanbe j,t i∈j i,t j,t i∈j i,t j,t i∈j i,t j,t i∈j i,t verifiedthatallequilibriumconditionswillbeidenticalforrepresentativefirmj andallfirmsi∈j. 17Weuseω tocalibratetheoutputsharesofeachtypeoffirmsinsteady-state. j 17

capitalforfirmj,Z ,are j,t P Y1−ρYρ m,t m,t j,t Z = ω (1−ρ(1−α)) , j,t j ξ K t j,t−1 whereξ isthecapitalqualityshock,andthisequationholdsforeachtypej.18 t We assume that capital is transferable between firms: thus, we have the capital accumulation equation: K = ξ [I +(1−δ)K ]. (11) t+1 t+1 t t Thecapitalgoodproducersolvesthefollowingmaximizationproblem: ∞ (cid:88) I τ maxE Λ {Q I −[1+f( )]I }, t t,τ τ τ τ I τ−1 t=τ whereΛ isadiscountfactorthatwillbeobtainedfromthehousehold’sproblem,inequilibrium. t Thus,thepriceofcapitalgoodscanbedeterminedfromprofitmaximizationas I I I I I Q = 1+f( t )+ t f(cid:48)( t )−E Λ ( t+1 )2f(cid:48)( t+1 ). (12) t t t,t+1 I I I I I t−1 t−1 t−1 t t Imposingthefunctionalformforf consideredbyGK13in(12),weget η I I I I I Q = 1+ i ( t −1)2 +η ( t −1) t −E Λ ( t+1 )2η ( t+1 −1), t i t t,t+1 i 2 I I I I I t−1 t−1 t−1 t t whereη istheinverseelasticityofnetinvestmenttothepriceofcapital. i Firmsrequirefinancingoftheircapitalstocks,andtheydosobyissuingstate-contingentclaims that are perfectly monitored and enforced and, thus, perfectly state-contingent. We assume that 18Inthissetup,capitalownersreceivefirmincomelesswagespaid,whichisequaltothemarginalproductofcapital iffirmsproducewithconstantreturnstoscale(ρ=1). 18

onlypartoffirmcapitalexpendituresmustbefinancedusingexternalsources: K = K +S , j,t j,I j,t whereK istheamountofcapitalfirmsoftypejdonotneedtofinanceexternally. Weassumethat j,I K is low enough that firms will always use some external financing (S > 0) in equilibrium, j,I j,t and that firm profits are split proportionally between capital financed externally (S ) and not j,t (K ), so the gross profits per unit of capital are Z for every unit of capital of type j.19 To j,I j,t+1 keep K constant, we impose that intermediate good firms make net transfers to the household j,I ofK (Z −P δ)eachperiod,payingoutearningsbeyondthoserequiredtopurchaseK j,I j,t+1 k,t+1 j,I unitsofcapital.20 Ifξ isthecapitalqualityshock,theperiodt+1payoffofthesecurityoffirmj is(Z + t+1 j,t+1 (1−δ)Q )ξ . Thus,thesecurityoffirmj hasarateofreturnof t+1 t+1 Z +(1−δ)Q j,t+1 t+1 R = ξ . (13) k,j,t+1 t+1 Q t The rate of return therefore has a relationship with Z , profits per unit of capital, which dej,t pends on K . Since Q is common to the two types of firms, difference in the rates of return, j,t t R , imply differences in per-capital profit rates and allocation of capital between the two k,j,t+1 types. Retail Good Firm Problem The final good, Y , is produced using a mass one continuum of dift ferentiatedretailgoodsusingCESproduction: (cid:90) 1 (cid:15)−1 (cid:15) Y t = [ Y ft (cid:15) df](cid:15)−1. 0 Retail good firms, however, just take intermediate output and repackage it. Thus, the marginal 19K isincorporatedtomatchthefactthatfirmsintheU.Scanfinancemuchoftheirinvestmentintheaggregate j,I withinternalfunds. 20Whilepositiveinsteady-state,inthecaseofaparticularlybadshock,thistransfercanbenegative. 19

cost of production is P , the price of the output of intermediate good firms. The retail good firm mt facesCalvopricing. Itcanadjustitspricewithprobability1−γ. Thefirmschoosethesamereset priceP∗. Followingtheusualarguments,wecanobtainthefirst-ordercondition: t (cid:88) ∞ P∗ γiΛ [ t −µP ]Y = 0, t,t+i mt+i ft+i P t+i i=0 withµ = 1 . Wecanthusrecoverthelawofmotionforprices: 1−1/(cid:15) P t = [(1−γ)(P t ∗)+γ(P t 1 − − 1 (cid:15))]1− 1 (cid:15). Households There is a measure one continuum of households (all identical), each of which consumesthefinalgood,savesbylendingfundstobanksandpotentiallythecentralbankandsupplies labor.21 Each household is composed of a fraction 1 − f workers and f bankers and has perfect consumption insurance. Workers are the members who supply labor to earn real wage, W , which the t householdsharesamongitself. Bankersalsoshareanyearningswiththehouseholdasawhole. In effect,thehouseholdownsthebankthatitsbankersmanage. Definetheoveralltransferstohouseholds from firms and banks as Π . Households pay taxes, T . The household deposits funds in t t banksbutonlyinbanksthehousehold’sbankersdonotmanage. Workerscanbecomebankersand viceversaovertime. Withprobabilityσ,bankersstaybankers,andwithprobability1−σ,bankers become workers. Bankers face a finite horizon problem; in effect, they cannot retain earnings beyondthepointatwhichtheycanfundallinvestmentfromtheirowncapital. Workersarerandomly selectedtoreplacethebankerswhoswitchtoworkersandreceiveastartupfundof X . (1−σ)f The household consumes C units of the final good. L is family labor supply. The household t t 21Theeconomyweconsideristhecashlesslimit. 20

hashabitsinconsumption,andthehousehold’sutility,u ,isdefinedasfollows: t ∞ (cid:88) χ u = E βi[ln(C −hC )− L1+ϕ ], (14) t t t+i t+i−1 1+ϕ t+1+i t where0 < β < 1,0 < h < 1,andχ,φ > 0. Households are indifferent between deposits and government debt, as they both pay rate of returnbetweenperiodst−1andtofR ,inequilibrium. Thus,wemakethisassumptionthroughout, t callingbothshort-termdebt,D . Wecanthusdefinethehousehold’sbudgetconstrainttobe h,t C = W L +Π −X +T +R D −D . (15) t t t t t t h,t−1 h,t The household thus solves (14) subject to (15) choosing C , L , and D . Define u to be the t t ht Ct marginalutilityofconsumption. Wethenhavelaborsupplycondition: u W = χLϕ, Ct t t andconsumption-savingsoptimalitycondition: u C,t+1 E β R = 1. t t+1 u C,t Itisalsousefultodefine u C,t+1 Λ = E β , (16) t,t+1 t u C,t asitentersthediscountfactoroffirmsandintermediaries. Holding Costs We also allow households to directly hold securities in the face of holding costs. DefineS asthesecuritiesoffirmj heldbythehouseholdattimetandB assecuritiesofthe h,j,t h,j,t government held by the household at time t with price q . Holding costs for type j firm securities t 21

are κj (S h,j −S h ¯ ,j )2 , where parameters κ and S ¯ are positive and S ≥ S ¯ . Holding costs for 2 S j h,j h,j h,j h,j governmentsecuritiesare κj (B h −B¯ h )2 ,whereparametersκ andB ¯ arepositiveandB ≥ B ¯ . 2 B j h h h h Withholdingcosts,werewritebudgetconstraintofthehousehold: j=2 (cid:88) 1 1 C +D + Q (S + κ(S −S ¯ )2)+q (B + κ(B −B ¯ )2) t h,t j,t h,j,t h,j,t h,j t h,t h,t h 2 2 j=1 j=2 (cid:88) = W L +Π +T −X +R D + R S +R B , t t t t t h,t−1 k,j,t h,j,t−1 b,t h,t−1 j=1 whereR isthereturnongovernmentbonds. b,t Banks There is a single bank which makes long-term loans to nonfinancial firms and the government, which are funded by the bank’s liabilities (short-term deposits of households). The bank is jointly owned by all of the bankers, and when bankers become workers they bring back to the household their fraction of the net worth of the bank. The rate of return on a loan will be equal to the return on the security defined in (13). There are government bonds, b , that are available to t householdsandbanks,whichareperpetuitiesandpayonedollarperperiod. Ifq isthepriceofthe t bondandP isthepricelevel,therealrateofreturnonthebondR is t b,t+1 1 +q R = Pt t+1 . b,t+1 q t Thebalancesheetofthebankis Q S +Q S +q B = N +d , (17) t b,1,t t b,2,t t b,t t t where N is bank net worth, d is deposits held, B is government bonds held, and S for t t b,t b,j,t j ∈ {1,2} is the securities holdings by the bank of firms 1 and 2, respectively. Net worth is the differencebetweenthegrossreturnonassetsandthecostofdeposits: N = R Q S +R Q S +R q B −R d . (18) t k,1,t t−1 b,1,t−1 k,2,t t−1 b,2,t−1 b,t t−1 b,t−1 t t−1 22

Thebankwillmaximizeitsexpecteddiscountedvalueofnetworth: ∞ (cid:88) V = E (1−σ)σi−1Λ N . (19) t t t,t+1 t+1 i=1 The bank faces an incentive constraint due to an imperfect monitoring problem of the bank by depositorswhereinthegovernmentregulationsonassetholdingsalsoenter: V ≥ θQ S νs,1 +θ∆ Q S νs,2 +∆θq Bν b, (20) t t b,1,t s t b,2,t t b,t whereν ,ν ,andν areparametersallgreaterthanorequaltooneandgoverntheextenttowhich s,1 s,2 b it is costly to hold a given asset s , s , and B , respectively. Also, parameters θ, θ∆ , and θ∆ 1,t 2,t b,t s are the respective amounts of the bank’s portfolios of S , S , and B the bank can divert, b,1,t b,2,t b,t where 0 ≥ {∆,∆ }. This constraint can also be interpreted as a collateral/regulatory constraint, s where θ and {∆,∆ } are parameters that govern how tightly the collateral constraint binds on s different assets, while ν , ν , and ν act as either holding costs or tighter regulatory constraints s,1 s,2 b duetoconcentrationinparticularassetclasses. The bank chooses S , S , and B to maximize (19) subject to (17), (18), and (20). In b,1,t b,2,t b,t addition,banksarepricetakers,takinginterestratesandspreadsasgiven. Wedescribethesolution totheproblemofthebankinAppendixB. Central Bank and Government Policy The central bank can purchase either government bonds (shortorlongterm)orprivatesecurities. Weonlyallowthecentralbanktopurchasethesecurities oftype1(large)firms. Thecentralbankcanissuerisklessshort-termdebtD whichpayR . g,t t+1 Thus,thecentralbankhasbalancesheet Q S +q B = D , t g,1,t t g,t g,t where S is central bank holdings of type 1 securities, and B is central bank holdings of g,1,t g,t 23

government bonds.22 The central bank costlessly transfers any profits to, or recovers any losses from, the government. We assume the central bank is less efficient in intermediation than banks andthuspaysτ perunitoftypej bondsintermediatedandτ perunitofgovernmentbonds. s,j b The central bank determines monetary policy using a Taylor rule. Define i as the net nominal t interest rate, i as the steady-state nominal rate, π as the inflation rate P /P , and Y∗ as the t t+1 t t flexible-priceequilibriumlevelofoutput. Then i = i+κ π +κ (log(Y )−log(Y∗))+(cid:15) , t π t y t t t where(cid:15) isanexogenousshock. WhenweallowforaZLBoninterestrates: t (cid:26) (cid:27) i = max 0,i+κ π +κ (log(Y )−log(Y∗))+(cid:15) . t π t y t t t WecanthendeterminetherealinterestratewiththestandardFisherrelation: P t+1 1+i = R . t t+1 P t Clearingforeachtypej securitiesimplies S = S +S +S , j,t b,j,t h,j,t g,j,t whereS istotalholdingsoftypej securities. j,t Also,wehaveclearingforgovernmentbonds,whichimplies B = B +B +B . t b,t h,t g,t Government consumption, G, and the net interest payments from fixed amount of long-term 22 S or central bank holdings of type 2 assets is restricted to be zero and, thus, we do not write it into the g,2,t constraint. 24

¯ bondsB arefixed. Revenueswillincludecentralbankearningsnetcostspluscollectedtaxes. Wethushavetheconsolidatedgovernmentbudgetconstraint: j=2 (cid:88) ¯ G+(R −1)B = T + (R −R −τ )Q S +(R −R −τ )q B . b,t t k,j,t t s,j j,t−1 g,j,t−1 b,t t b t−1 g,t−1 j=1 CentralbankLSAPsinvolvepurchasingafraction,ϕ andϕ ,ofoutstandingtype1privates,1,t b,t sectorsecuritiesorlong-termgovernmentsecurities,respectively. Tobeprecise,thesepoliciesare respectivelymodeledas S = ϕ S , g,1,t s,1,t 1,t−1 and B = ϕ B , g,t b,t t−1 whereϕ andϕ aremodeledassecond-orderregressivepolicies. s,1,t b,t When ∆ >1, limits to arbitrage are weaker for private securities of type 2 than for private s securitiesoftype1firms. Thus,allelsebeingequal,ifthereisanassetpurchase,privatesecurities of type 1 should move by more than private securities of type 2. A similar result is true for government bonds when ∆ < 1. Further, ν and ν , when greater than one, affect the desired s,1 s,2 stockofholdings,furtheralteringtheextenttowhichexcessreturnsadjustafterabondpurchase. Resource Constraint, Further Clearing Conditions, and Equilibrium We have the resource constraint: j=2 I (cid:88) t Y = C +[1+f( )]I +G+ τ Q S +τ q B . t t t s,j t−1 g,j,t−1 g t−1 g,t−1 I t−1 j=1 25

Wethenrequirethatsupplyequalsdemandinourdifferentmarkets. Inthemarketforlabor: Yρ Y1−ρ 1 ω (1−α)ρ 1,t m,t E u = χLφ, 1 L t C,t P t 1,t m,t and Yρ Y1−ρ 1 ω (1−α)ρ 2,t m,t E u = χLφ, 2 L t C,t P t 2,t m,t whereL = L +L . t 1,t 2,t Inthemarketforcapital,wehave K +K = I +(1−δ)K , 1,t+1 2,t+1 t t whereK = K +K . t 1,t 2,t Noticethatwithclearinginthemarketsforgoods,labor,andallsecurities,byWalras’Lawthe marketforrisklessshort-termdebtalsoclears. 3.2 Misallocation We can construct a measure of misallocation by first constructing a counterfactual measure of ˆ output: the maximum output, Y, which can be produced with a fixed amount of labor and capital. ˆ Inourproductionenvironment,Y canbeexpressedas (cid:32) (cid:33)1−ρ j=2 ρ (cid:88) 1 Y ˆ = A KαL(1−α) ω1−ρ . t t t t j j=1 ˆ WethereforecandefinethelossesfrommisallocationasY −Y . t t 26

3.3 Calibration We present the parameters used in our quantitative exercise in Table 1. In our calibration exercise, we follow the calibration strategy of GK13 for their parameters and calibrate the new parameters we introduced.23 The new parameters, listed at the bottom of Table 1, concern firm heterogeneity and the regulatory constraint. Our calibration of the parameters that govern the extentofmisallocationinsteadystateismeanttobeconservative. For the regulatory constraint parameters, we motivate the calibration with the following two points: (1) it is generally less costly for a bank to hold government debt than corporate debt due to differences in liquidity in these assets, and (2) government debt is considered Level 1 capital as a High Quality Liquid Asset (HQLA) in computing the Liquidity Coverage Ratio for Basel III, while nonbank investment-grade corporate debt is considered a Level 2B asset, while CDOs of other corporate loans donot countas HQLA.To beconservative withour calibrationof theextent of misallocation in steady state, we choose parameters ν = 1.0, ν = ν = 1.2, which imply b s,1 s,2 only a modest amount of convexity (which can be interpreted as low holding costs) and where the convexitydoesnotdifferbetweenprivatesectorassets. Weparameterize∆ sothatthedifference s,2 inspreadsbetweenthetwotypesoffirmsis0.9%insteady-state,whichiswellbelowthatimplied bythedispersionincreditspreadsinGilchrist,Sim,andZakrajsek(2013),consistentwithourgoal ¯ of being conservative as to the extent of misallocation that exists in steady state. We set S and h,1 κ tothevaluesofprivatesecuritiesinGK13. s,1 For intermediate good firms, we set the standard, CES parameter, ρ, to 0.9, implying a great deal of substitutability between the two types of firms. Consistent with our goal of being conservativewithregardtotheextentofmisallocationthatexistsinsteadystate,ourcalibrationisabove of Atkeson and Burstein (2010), a representative value in the literature. We set the share of labor andoutputinproductionoftype1firmsto0.5. Wecalibrateourfinancingparameterstoaccountforthefollowingfact: ShouridehandZetlin- 23 The only parameter we change is σ, which we lower to 0.92. At the previous value of σ, intermediaries could saveoutoftheirconstraint,becausewehaveintroducedanewhigheryieldingasset,thatis,type2firms. 27

Jones (2012) show that about 80% of investment by private firms is financed externally, compared to 20% for publicly traded firms.24 We set the proportion of steady-state capital that firms of type 1 finance internally at 80% and the amount that they finance externally directly from households at 10% (therefore 10% of their capital is financed via intermediaries in steady-state, which is consistentwithhalfoftheirexternalfinancingbeingmetbyhouseholds). Firmsoftype2finance20% of their capital internally, finance 10% directly from households (the same proportion as type 1 firms),andmustrelyonfinancialintermediariestofinancetheremainderoftheircapitalstock. 3.4 Quantitative Results Figure 1 presents results from the impulse responses from type 1 (large) firm bond purchases and government bond purchases. In our case with firm heterogeneity and potential misallocative effects, government bond purchases are more effective in boosting output than corporate bond purchases (of type 1 firm debt).25 This is the reverse of the result in the work GK13, which does not consider heterogeneity. The result of GK13 occurs in our model when the output of the two typesoffirmsisperfectlysubstitutable(ρ = 1). We see that following a large-scale type 1 corporate bond purchase, the amount of type 1 firm capital that has to be intermediated, S , falls, reducing spreads on firms of type 1 and leading to b,1 a larger difference in spreads, E[R ] − E[R ]. This leads to a marked increase in the capital K,2 K,1 of type 1 firms, K . Due to GE price effects, there is a concomitant change in the capital of 1 type 2 firms, K , as firms must finance part of their capital stocks. In our calibration, this change 2 in the relative allocation of capital is inefficient, so there is a misallocation cost of corporate bond purchasesthatreducestheireffectiveness. Thereisstillapositiveeffectonoutput,asloweraverage spreadsleadtogreatercapitaldemand. After a government bond purchase, we see different dynamics in terms of capital and spreads. 24 Chari(2013)notesthatthefactthatGK13isnotcalibratedtobeconsistentwithsuchaggregatefactsshouldbe resolvedinfuturework. 25Weonlyallowfortype1bondpurchasesbythecentralbank,asrepresentativelarge-scalenonfinancialcorporate sector bond purchases, such as the ECB’s Corporate Sector Purchase Program, are typically targeted at purchasing onlyinvestment-gradeorhigherqualitycorporatedebt. Thiscouldeasilyberelaxed. 28

Agovernmentbondpurchaseloosensthecollateralconstraintofthefinancialintermediary,which (allelseequal)reducesspreadsforallfirms. Reducingthespreadsforallfirmsreducestheextentof misallocation as compared to the steady state (arising from the steady-state difference in spreads), and increases capital of type 2 firms (small firms) relatively more than type 1 firms. In this case, the effectiveness of government bond purchases on increasing output is slightly amplified by its effectontheallocationofcapitalbetweenfirms. The misallocation induced by large-scale corporate bond buys can be important when looking at the difference between the effectiveness of different types of LSAPs. The blue line in Figure 2 is the difference between the impulse response of output during the government bond buy and the impulse response of output during the large firm bond buy. The dashed red line is the output lossesdirectlyduetomisallocationinthecorporatebondpurchase. Thedirectmisallocationeffect is measured as the difference between the maximum output that could be produced with a given amountofcapitalandlaborandwhatisactuallyproduced.26 The losses due to the direct effects of misallocation in large firm corporate bond buys account for the majority of the difference in the effect of corporate versus government bonds buys. In other words, when weighing different options for the types of debt to buy as part of LSAPs, the misallocation effect of corporate bond buys should potentially be weighed as part of the tradeoffs involved, as it can be quantitatively meaningful. Notice, this is a different result from that of GK13 who show that government bond buys induce smaller movements in output than privatesector bond buys for a similarly sized bond purchase. Our model generates a similar result to GK13 when intermediate good firm products are perfect substitutes, i.e. when the CES parameter ρ = 1.27 Weshowthisresultinpanel(c)ofFigure3. Overall,thecalibratedimpulseresponsessuggestthatalarge-scalecorporatebondbuyinduces agreatermisallocationofresourcesthanalarge-scalegovernmentbondbuyandthemisallocation 26 There are also indirect effects of misallocation that our misallocation measure ignores. Capital and labor are takenasgiveninourmisallocationmeasure,but,infact,theyareendogenouslyaffectedbymisallocationaswell. 27 This result is due to purchases of private-sector debt having a larger effect on excess returns of private bonds than purchases of government bonds have on excess returns of private bonds, and this effect not being offset by a misallocationeffect. 29

effect is a quantitatively significant fraction of the output gains from a large-scale corporate bond buy. 4 ZLB Discussion In our model, when atthe ZLB, output losses fromexogenous shocks, as well as the effectiveness of QE, are amplified.28 To demonstrate this, we feed in capital quality shocks that force the economytotheZLB.Wethenhavethecentralbankperformsimilarlysizedbondpurchasestoour baseline case when the economy is at the ZLB. We show in panel (d) of Figure 3 that in this case, output gains from a QE program are indeed amplified relative to the baseline case where the ZLB doesnotbind.29 WealsocomputeourmisallocationmeasureinresponsetobondpurchasesattheZLB.Wesee from Figure 4 that our misallocation measure does not drastically change in response to corporate bondbuyswhenweallowforabindingZLB.Thisisbecause,accountingforleveleffectsonexcess returns, there is little change in the relative borrowing costs of type 1 and type 2 firms at the ZLB as compared to the baseline case. Hence, misallocation matters much more relative to movements in real output when the ZLB is not binding. There are arguments for the central bank to make QE partofitstoolkitevenawayfromtheZLB(forexamples,seeQuintandRabanal(2017)orGagnon (2016)). Our exercise sheds light on a potential counterargument to be considered when making such a claim, at least for large-scale corporate bond buys, as large-scale corporate bond buys can induceaquantitativelysignificantmisallocationofresources. 28ToincorporatetheZLBinourmodel,wefollowtheworkofGuerrieriandIacoviello(2015). 29OurbaselinecalibrationresultsincorporatebondsbuyshavingagreaterstimulativeeffectonoutputattheZLB than government bond buys. However, a small reduction in the CES parameter, ρ, to 0.875 (which would still be a moreconservativeestimatethanthattypicallyconsideredintheliterature)leadstogovernmentbondbuysremaining moreeffectivethancorporatebondbuys,evenattheZLB. 30

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KURTZMAN, R., S. LUCK, AND T. ZIMMERMANN (2017): “Did QE Lead to Lax Bank Lending Standards? EvidencefromtheFederalReserve’sLSAPs,”WorkingPaper. MIDRIGAN, V. AND D. Y. XU (2014): “Finance and Misallocation: Evidence from Plant-Level Data,”AmericanEconomicReview,104,422–458. QUINT, D. AND P. RABANAL (2017): “Should Unconventional Monetary Policies Become Conventional?” IMFWorkingPaperNo.17/85. REIS, R. (2017): “QE in the future: the central banks balance sheet in a fiscal crisis,” IMF EconomicReview,65,71–112. SHOURIDEH, A. AND A. ZETLIN-JONES (2012): “External financing and the role of financial frictionsoverthebusinesscycle: Measurementandtheory,”WorkingPaper. WIELAND, V., E. AFANASYEVA, M. KUETE, AND J. YOO (2016): “New methods for macrofinancialmodelcomparisonandpolicyanalysis,”HandbookofMacroeconomics,2,1241–1319. WIELAND, V., T. CWIK, G. J. MU¨LLER, S. SCHMIDT, AND M. WOLTERS (2012): “A New comparative approach to macroeconomic modeling and policy analysis,” Journal of Economic Behavior&Organization,83,523–541. A Proofs to Propositions Herewepresentproofstoourpropositions. Proposition 1 WecanexpresstheLagrangianmultiplierterminspreadsas λ (cid:80) θ∆ (QK −S )νj + (cid:0) BT −B (cid:1) θ−N ∗r j j j g,j b g = . (21) 1+λ (cid:80) θ∆ ν (QK −S )νj +(BT −B )θ j j j j g,j b g 33

Thisderivativeof(21)withrespecttoB ,holdingcapitalchoices(thus,K andQ)constant,yields: g j (cid:16) (cid:17) θ (cid:80) θ∆ (ν −1)(QK −S )νj +N ∗r j j j j g,j − < 0, (cid:16) (cid:17)2 (cid:80) θ∆ ν (QK −S )νj +(BT −B )θ j j j j g,j b g whichimplies(i)(a)(since λ isincreasinginλ). (i)(b)followsimmediatelyfrom(5). 1+λ Similarly,(ii)(a)followsfromtakingthederivativeof λ withrespecttoS fori ∈ j,holding 1+λ g,i capitalchoicesconstant,and(ii)(b)immediatelyfollowsfrom(5). Proposition 2 Notethatoutput,definedin(10),isincreasingintheterm (cid:82) (cid:16) Aρ (cid:16) rτ,i (cid:17)αρ(cid:17) 1− 1 αρ di. Theoutputi τk i maximizingvalueforinterestratewedgesthencanbefoundbysolvingthemaximizationproblem: (cid:90) (cid:18) (cid:18) r (cid:19)αρ(cid:19) 1− 1 αρ max Aρ τ,i di, rτ,i i i τ i k suchthat (cid:32) (cid:82) (cid:18) (τ A k) ρ i αρ (cid:19) 1− 1 αρ (r τ,i )1− α α ρ ρ di (cid:33)1− ρ ρ (cid:18) (cid:82) (cid:16) A τk ρ i (cid:17) 1− 1 αρ (r τ,i )1− 1 αρ di (cid:19)1−α i i = 1. (cid:32) (cid:18) (cid:19) 1 (cid:33)1− ρ ρ (cid:32) (cid:18) (cid:19) 1 (cid:33)1−α (cid:82) Aρ 1−αρ (cid:82) Aρ 1−αρ i di i di (τk)αρ (τk) i i This yields a first-order condition that can be simplified as rτ,i = Ξ, where Ξ is a constant τk i acrossfirms. Proposition2follows. 34

Corollary 2.1 With only two groups of firms, holding the weighted-average rate of interest fixed, output is onlyaffectedbyheterogeneouschangesinspreadsthroughtheterm: (cid:32) (cid:32) (cid:33)αρ(cid:33) 1 (cid:88) r 1−αρ Aρ τ,j , (22) j τk j=1,2 j where we define A such that (cid:0) Aρ(cid:1) 1− 1 αρ = (cid:82) (Aρ)1− 1 αρ di. Given interest rate wedges, r , are j j i∈j i τ,j defined between the interest rates facing firms and the weighted average interest rate, we have clearingcondition: (cid:32) (cid:80) j=1,2 (cid:18) (τ A k) ρ j αρ (cid:19) 1− 1 αρ (r τ,j )1− α α ρ ρ (cid:33)1− ρ ρ (cid:18) (cid:80) j=1,2 (cid:16) A τk ρ j (cid:17) 1− 1 αρ (r τ,j )1− 1 αρ (cid:19)1−α j j = 0. (23) (cid:32) (cid:18) (cid:19) 1 (cid:33)1− ρ ρ (cid:32) (cid:18) (cid:19) 1 (cid:33)1−α (cid:80) Aρ 1−αρ (cid:80) Aρ 1−αρ j j j=1,2 (τk)αρ j=1,2 (τk) j j From (23), a shock that increases r thus decreases r . Taking the derivative of (22) with τ,1 τ,2 (cid:32) (cid:32) (cid:33)αρ(cid:33) 1− 1 αρ ∂(cid:80) Aρ rτ,j j=1,2 j τk respecttor , j ,yields: τ,1 ∂rτ,1 (cid:16) (cid:17) rτ,2 − rτ,1 τk τk 2 1 . (24)    (cid:16) α(1−ρ) (cid:17)   (cid:80) j (cid:32) A τ j k ρ j (cid:33) 1− 1 αρ (rτ,j)1− 1 αρ   + (cid:16) rτ,2 (cid:17)     1−α (cid:80) j (cid:32) ( τ A k) ρ j αρ (cid:33) 1− 1 αρ (rτ,j)1− α α ρ ρ τ 2 k  j Thedenominatorof(24)isalwayspositive,thusthesignof(24)iscontrolledbythenumerator. Ifthe levelof theinterest ratewedge facingfirmsof type1 isabove (below)its optimal valuer(cid:63) , τ,1 (cid:16) (cid:17) thenProposition2togetherwith(23)implythat rτ,2 − rτ,1 < 0(> 0). Corollary2.1follows. τk τk 2 1 35

B Solution to the Problem of the Bank If the bank chooses S , S , and B to maximize (19) subject to (17), (18), and (20), the b,1,t b,2,t b,t Lagrangianis (cid:20) (cid:18)(cid:18) (cid:19) (cid:19)(cid:21) (cid:18) (cid:20) (cid:18)(cid:18) (cid:19) (cid:19)(cid:21) L = E Λ 1−σ N +σV +λ E Λ 1−σ N +σV t t,t+1 t+1 t+1 t t t,t+1 t+1 t+1 (cid:19) −θQ S νs,1 −θ∆ Q S νs,2 −∆θq Bν b , t b,1,t s t b,2,t t b,t where N = R Q S +R Q S + t k,1,t t−1 b,1,t−1 k,2,t t−1 b,2,t−1 (cid:18) (cid:19) R q B −R Q S +Q S +q B −N . b,t t−1 b,t−1 t t−1 b,1,t−1 t−1 b,2,t−1 t−1 b,t−1 t−1 Letλ betheLagrangemultiplierassociatedwiththeincentiveconstraint(20). t Thefirst-orderconditionshereyield (cid:20) (cid:18)(cid:18) (cid:19) (cid:19)(cid:18) (cid:19)(cid:21) ∂V λ E Λ 1−σ +σ t+1 R −R = t θν S νs,1−1 , t t,t+1 ∂N k,1,t+1 t+1 (cid:18) (cid:19) s,1 b,1,t t+1 1+λ t (cid:20) (cid:18)(cid:18) (cid:19) (cid:19)(cid:18) (cid:19)(cid:21) ∂V λ E Λ 1−σ +σ t+1 R −R = t ∆ θν S νs,2−1 , t t,t+1 ∂N k,2,t+1 t+1 (cid:18) (cid:19) s s,2 b,2,t t+1 1+λ t and (cid:20) (cid:18)(cid:18) (cid:19) (cid:19)(cid:18) (cid:19)(cid:21) ∂V λ E Λ 1−σ +σ t+1 R −R = t ∆θν Bν b −1, t t,t+1 ∂N b,t+1 t+1 (cid:18) (cid:19) b b,t t+1 1+λ t 36

notingthat (cid:18) (cid:19) ∂V t ˜ = E Λ (R −R )φ +R , (25) t t,t+1 k,t+1 t+1 t t+1 ∂N t where (cid:104) (cid:105) ˜ E Λ R t t,t+1 t+1 φ = . t (cid:104) (cid:105) θν S νs,1−1 −E Λ ˜ (R −R ) s,1 b,1,t t t,t+1 k,1,t+1 t+1 ˜ Let Λ be the bank’s augmented stochastic discount factor, equal to the product of Λ , t,t+1 t,t+i that is, the discount factor from the household’s problem as defined in (16) and the multiplier (cid:18)(cid:18) (cid:19) (cid:19) 1−σ +σ∂Vt+1 . ∂Nt+1 Thus,wehavethefollowingarbitrageconditions: (cid:104) (cid:105) ν Bν b −1 (cid:104) (cid:105) ˜ b b,t ˜ E Λ (R −R ) = ∆ E Λ (R −R ) , t t,t+1 b,t+1 t+1 ν S νs,1−1 t t,t+1 k,1,t+1 t+1 s,1 b,1,t and (cid:104) (cid:105) ν S νs,2−1 (cid:104) (cid:105) ˜ s,2 b,2,t ˜ E Λ (R −R ) = ∆ E Λ (R −R ) . t t,t+1 k,2,t+1 t+1 s ν S νs,1−1 t t,t+1 k,1,t+1 t+1 s,1 b,1,t Fromcombining(20)and(25),weobtainthefollowingleveragerestriction: Q S S νs,2 Bν b t b,1,t b,2,t b,t N φ ≥ +∆ Q +∆q , t t ν s,1 s 2,t ν s,1 S b ν , s 1 , , 1 t −1 t ν s,1 S b ν , s 1 , , 1 t −1 which is an inequality when λ = 0 and binds when λ > 0. We can also derive the law of motion t t fortotalnetworthofallbankersas (cid:18)j=2 (cid:19) (cid:88) Q S q B j,t−1 b,j,t−1 t−1 b,t−1 N = σ ((R −R ) )+(R −R ) N +N , t k,j,t t b,t t t−1 e N N t−1 t−1 j=1 37

whereN isthewealthofenteringbankers. e 38

A Tables and Figures Parameters Value FromGertlerandKaradi(2013) Households Discountrate,β 0.995 Habitparameter,h 0.815 Relativeutilityweightoflabor,χ 3.482 Steady-stateTreasurysupply,B/Y 0.450 Proportionoflong-termTreasuryholdingsofthehouseholds,B¯h/B 0.750 Portfolioadjustmentcost,κ 1.000 InverseFrischelasticityoflaborsupply,ϕ 0.276 FinancialIntermediariesandHouseholds Fractionofcapitalthatcanbediverted,θ 0.345 Proportionaladvantageinseizurerateofgovernmentdebt,∆ 0.500 Transfertotheenteringbankers,X 0.0062 Survivalrateofthebankers,σ 0.92 IntermediateGoodFirms Capitalshare,α 0.330 Depreciationrate,δ 0.025 Capital-ProducingFirms Inverseelasticityofnetinvestmenttothepriceofcapital,η 1.728 i RetailFirms Elasticityofsubstitution,(cid:15) 4.167 Probabilityofkeepingthepriceconstant,γ 0.779 Government Steady-stateproportionofgovernmentexpenditures,G/Y 0.200 InflationcoefficientintheTaylorrule,κ 1.500 π MarkupcoefficientintheTaylorrule,κ -0.125 X NewParameters FinancialIntermediariesandHouseholds Regulatoryconstraintparameterongovernmentdebt,ν 1.000 b Regulatoryconstraintparameterontype1securities,ν 1.200 s,1 Regulatoryconstraintparameterontype2securities,ν 1.200 s,2 ∆ , 1.0531 s,2 K /K , 0.8 1,I 1 K /K , 0.2 2,I 2 K¯ /K , 0.1 h,1 1 K¯ /K , 0.1 h,2 2 κ , 1 s,1 κ , 1 s,2 IntermediateGoodFirms CESparameter,ρ 0.9 Type1laborshareinproduction 0.5 Type2laborshareinproduction 0.5 Table1: Parameters 39

(a)CentralBankPurchases 2.5 2 1.5 1 0.5 00 5 10 15 20 25 30 35 40 PDG fο % (b)Y 0.08 Government Corporate 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 −0.010 5 10 15 20 25 30 35 40 Quarters etatS ydaetS morf egnahC % (c)L 0.1 0.08 0.06 0.04 0.02 0 −0.02 −0.040 5 10 15 20 25 30 35 40 Quarters Quarters (d)K 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 00 5 10 15 20 25 30 35 40 etatS ydaetS morf egnahC % (e)K (f)K 1 2 2 2 1.5 1.5 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 −1.5 −1.5 −20 5 10 15 20 25 30 35 40 −20 5 10 15 20 25 30 35 40 Quarters Quarters Quarters (g)E[R ]−E[R ] K,2 K,1 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 −0.02 −0.040 5 10 15 20 25 30 35 40 etatS ydaetS morf daerpS ni egnahC (h)S b,1 10 5 0 −5 −10 −15 −200 5 10 15 20 25 30 35 40 Quarters etatS ydaetS morf egnahC % (i)S b,2 10 5 0 −5 −10 −15 −200 5 10 15 20 25 30 35 40 Quarters Quarters Figure1: GovernmentandPrivateSectorAssetPurchaseShocks 40

0.03 0.025 0.02 0.015 0.01 0.005 0 −0.005 0 5 10 15 20 25 30 35 40 Quarters tuptuO etatS ydaetS fo % ,egnahC LSAP Output Difference Misallocation Effect Figure2: MisallocationEffectandDifferenceinLSAPEffectiveness 41

(a)CentralBankPurchases 2.5 2 1.5 1 0.5 0 0 5 10 15 20 25 30 35 40 PDG fο % (b)Y,Baseline 0.08 Government Corporate 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 −0.01 0 5 10 15 20 25 30 35 40 Quarters etatS ydaetS morf egnahC % Quarters (c)Y,PerfectSubstitutes 0.12 0.1 0.08 0.06 0.04 0.02 0 −0.02 0 10 20 30 40 etatS ydaetS morf egnahC % (d)Y,ZLB 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 5 10 15 20 25 30 35 40 Quarters etatS ydaetS morf egnahC % Quarters Figure3: EffectofLSAPsonOutputwithPerfectSubstitutesorZLB 42

0.025 0.02 0.015 0.01 0.005 0 −0.005 0 5 10 15 20 25 30 Quarters enilesaB ot evitaleR egnahC % No ZLB ZLB Figure4: MisallocationEffectofLarge-ScaleCorporateBondBuywithandwithouttheZLB 43