1、BIS Working Papers No 1148Firm heterogeneity,capital misallocation and optimal monetary policy by Beatriz Gonzlez,Galo Nuo,Dominik Thaler,Silvia Albrizio Monetary and Economic Department November 2023 JEL classification:E12,E22,E43,E52,L11 Keywords:Monetary policy,firm heterogeneity,financial fricti

2、ons,capital misallocation.BIS Working Papers are written by members of the Monetary and Economic Department of the Bank for International Settlements,and from time to time by other economists,and are published by the Bank.The papers are on subjects of topical interest and are technical in character.

3、The views expressed in them are those of their authors and not necessarily the views of the BIS.This publication is available on the BIS website(www.bis.org).Bank for International Settlements 2023.All rights reserved.Brief excerpts may be reproduced or translated provided the source is stated.ISSN

4、1020-0959(print)ISSN 1682-7678(online)Firm Heterogeneity,Capital Misallocation andOptimal Monetary PolicyBeatriz GonzlezBdEGalo NuoBdE,BISDominik ThalerECBSilvia AlbrizioIMFFirst version:October 2020.This version:October 2023AbstractThis paper analyzes the link between monetary policy and capital mi

5、salloca-tion in a New Keynesian model with heterogeneous firms and financial frictions.In the model,firms with a high return to capital increase their investment morestrongly in response to a monetary policy expansion,thus reducing misallocation.This feature creates a new time-inconsistent incentive

6、 for the central bank to en-gineer an unexpected monetary expansion to temporarily reduce misallocation.However,price stability is the optimal timeless response to demand,financial orTFP shocks.Finally,we present firm-level evidence supporting the theoreticalmechanism.Keywords:Monetary policy,firm h

7、eterogeneity,financial frictions,capital misalloca-tion.JEL classification:E12,E22,E43,E52,L11.Beatriz Gonzlez(Banco de Espaa):beatrizgonzalezbde.es.Galo Nuo(Banco de Espaa,andBIS):galo.nunobde.es.Dominik Thaler(European Central Bank):dominik.thalereui.eu.SilviaAlbrizio(International Monetary Fund):

8、salbrizioimf.org.Two version of this paper were previ-ously circulated as“Monetary Policy and Capital Misallocation”and“Optimal Monetary Policy withHeterogeneous Firms”.The views expressed in this manuscript are those of the authors and do notnecessarily represent the views of the Bank of Spain or t

9、he Eurosystem.We are specially grateful toAndy Atkenson,Anmol Bhandari,Sergi Basco,Saki Bigio,Jess Fernndez-Villaverde,and DmitryKhametshin for their input and suggestions.We thank Klaus Adam,Adrien Auclert,Bence Bardczy,Paco Buera,Ricardo Caballero,Matias Covarrubias,Jim Costain,Wei Cui,Ester Faia,

10、Luis Franjo,Veronica Guerrieri,Masashige Hamano,Greg Kaplan,Agnieszka Markiewicz,Ben Moll,Morten Ravn,Ricardo Reis,Kjetil Storesletten,Harald Uhlig,Gianluca Violante,Thomas Winberry,Chris Wolf andFrancesco Zanetti for excellent comments,as well as participants at several seminars and conferences.The

11、 opinions and analysis are the responsibility of the authors,and therefore,do not necessarily co-incide with those of the International Monetary Fund,the ECB,the BIS,Banco de Espaa or theEurosystem.All errors are our own.1IntroductionAn inefficient allocation of capital across firms may significantl

12、y reduce aggregate totalfactor productivity(TFP).In a market economy,capital is allocated according to theinvestment decisions of individual firms.Monetary policy is an important driver ofthese investment decisions,as it directly affects firms funding costs and indirectlyinfluences firms revenues an

13、d other costs such as wages.If firm investment respondsheterogeneously to changes in monetary policy,then monetary policy can affect capitalmisallocation and thus TFP.Monetary policy design has traditionally taken aggregate productivity as given.Inthe workhorse model of monetary policy the New Keyne

14、sian model the central bankfaces a trade-offbetween stabilizing inflation and reducing the short-term deviations ofoutput from its potential level.If monetary policy can affect misallocation and TFP,the central bank should also ponder how its decisions will impact the supply side of theeconomy in th

15、e medium term.Such considerations may be of particular relevance inphases of very active monetary policy,such as in the current inflationary environment.The objective of this paper is to shed light on the interaction between monetarypolicy and capital misallocation and its implications for optimal m

16、onetary policy.Tothis end,we develop a tractable framework that combines the workhorse New Keyne-sian model with a model of firm heterogeneity,based on Moll(2014),in which capitalmisallocation arises from financial frictions.The economy is populated by a continuumof firms owned by entrepreneurs,who

17、have access to a constant-returns-to-scale tech-nology.Entrepreneurs are heterogeneous in their net worth and receive idiosyncraticproductivity shocks.They face financial frictions,as their borrowing cannot exceeda multiple of their net worth.This results in endogenous capital misallocation:en-trepr

18、eneurs with productivity above a certain threshold are constrained,and borrowas much capital as possible since their marginal revenue product of capital(MRPK)ishigher than their cost of capital.Entrepreneurs below the threshold are unconstrained:their optimal size is zero and they choose to lend the

19、ir net worth to other entrepreneurs.1This economy allows for an aggregate representation akin to that in the standard NewKeynesian model with capital,except that in this case TFP is endogenous,and thedynamics of TFP are determined by the evolution of the distribution of capital among1This is the tra

20、ctable limit case of an economy with decreasing returns to scale at the firm level,in which unconstrained firms are optimally very small and the bulk of production is carried out byproductive and constrained firms.1entrepreneurs.We calibrate the model to replicate key firm-level moments of Spanishfi

21、rms.We start by analyzing the interactions between monetary policy and misallocation.We show that the allocation of capital,and hence TFP,improve in response to anexpansionary monetary policy shock.This is because the policy expansion alleviatesfinancial frictions,such that high-MRPK firms increase

22、their investment more than thelow-MRPK firms,raising the share of the aggregate capital stock used by high-MRPKfirms.We call this the capital misallocation channel of monetary policy.Next,we investigate the implications of this capital misallocation channel for theoptimal conduct of monetary policy

23、by analyzing the Ramsey problem of a benevolentcentral bank.This is technical challenge,as the space state of the model includes thedistribution of net worth across firms,an infinite-dimensional object.We overcome itby introducing a new algorithm to compute optimal policy problems in the presence of

24、non-trivial heterogeneity.We study first optimal policy in the absence of shocks.The steady state of the Ram-sey problem features zero-inflation,as in the standard complete markets New Keynesianmodel,which is nested.2However,the misallocation channel introduces a new sourceof time inconsistency as t

25、he central bank attempts to exploit the capital misallocationchannel:When starting from the zero-inflation steady state,the central bank engineersa temporary monetary expansion in the short run while committing to price stabilityin the long run.This strategy allows the central bank to temporarily im

26、prove capitalallocation and increase TFP,even if it means tolerating positive inflation during a cer-tain period of time.We find this source of time inconsistency to be quantitatively morerelevant than the standard time-inconsistency motive in the New Keynesian literature,namely the desire to exploi

27、t the short-run trade-offbetween inflation and output gap.We analyze next the optimal monetary policy from a timeless perspective(Wood-ford,2003),in which the central bank respects the commitments that it has optimallymade at a date far in the past.This allows us to study systematic changes in monet

28、arypolicy in response to shocks.We consider demand,financial and TFP shocks.Theoptimal response in all these cases features price stability.There is thus no meaningfultrade-offbetween price stability and managing misallocation,just as the standard NewKeynesian model features no trade-offbetween pric

29、e stability and aggregate demand2Our model nests the complete-market model as a particular case in which the borrowing constraintis arbitrarily loose,or in which entrepreneurs productivity levels are arbitrarily similar.2management(this is commonly known as the“divine coincidence”,Blanchard and Gali

30、,2007).This implies that,even if the central bank can affect the supply side of theeconomy through its monetary policy,it finds optimal not to do so in a systematic wayand it rather sticks to price stability.The implementation of the optimal policy policy,however,differs with respect to thecomplete-

31、market case.Under incomplete markets,a negative demand shock leads to anendogenous fall in TFP through the increase of capital misallocation.The fall in TFP,in turn,amplifies the reduction of the natural rate brought about by the demand shockitself,such that the natural rate drops more than in the c

32、ase with complete markets.3As the interest rate mimics the natural rate,it declines more,and more persistently,than in the standard New Keynesian model.This difference in implementation matterswhen nominal interest rates are constrained by the zero lower bound(ZLB).The optimalpolicy in this case,ori

33、ginally proposed by Eggertsson and Woodford(2004),is“low forlonger”:nominal rates should remain at the ZLB longer than what would be prescribedif the ZLB were not present.4In the case with incomplete markets,the larger and morepersistent decline in natural rates due to the endogenous fall in TFP lea

34、ds to what wecall a“low for even longer”optimal policy:nominal rates should remain at the ZLBfor significantly longer than they should under complete markets.Doing so reducesshortfalls not only of inflation and output,but also of TFP.Finally,we present empirical evidence on the capital misallocation

35、 channel of mon-etary policy.We combine micro-level panel data on the quasi-universe of Spanish firmswith monetary policy shocks identified using the high-frequency and sign-restrictionsapproach of Jarociski and Karadi(2020).The mechanism at play in the model is sup-ported by the data:firms with hig

36、h MRPK increase their investment relatively morethan low-MRPK firms in response to an expansionary monetary policy shock.Thisimplies a shift in the distribution of capital towards high-MPRK firms,improving thecapital allocation.Using a simple model-derived measure of aggregate misallocation,we find

37、that the positive impact of expansionary monetary policy is quantitatively inline with the data.53The natural rate is the rate that would pertain in the absence of nominal rigidities.4See also Eggertsson et al.(2003);Adam and Billi(2006),and Nakov et al.(2008).5The fact that TFP increases after a po

38、sitive monetary policy shock has been previously docu-mented by Evans(1992),Christiano et al.(2005),Garga and Singh(2021),Jord et al.(2020),Moranand Queralto(2018),Meier and Reinelt(2020)or Baqaee et al.(2021),among others.While theseauthors propose complementary mechanisms such as R&D,hysteresis ef

39、fects,or markup heterogeneityto account for it,our findings suggest that capital misallocation also plays a significant role.3This paper contributes to three strands of the literature.First,our model is relatedto the extensive literature on capital misallocation,and the different channels thatmay af

40、fect it,such as Hsieh and Klenow(2009)or Midrigan and Xu(2014)seeRestuccia and Rogerson(2017)for a review on this literature.Our paper builds onMoll(2014),whose tractable heterogeneous producer model we augment with a NewKeynesian monetary block to understand how monetary policy affects misallocatio

41、n.6The fact that monetary policy expansions reduce misallocation and increase TFP mightseem in conflict with previous papers,such as Reis(2013),Gopinath et al.(2017),orAsriyan et al.(2021),who argue that a decline in real interest rates may fuel capitalmisallocation in real economies.We show that th

42、ere is no such a conflict:our modelalso delivers a decline in TFP in response to a fall in real rates due to a negativedemand shock when prices are flexible.The difference in the behavior of misallocationcompared to a monetary policy shock is due to the different natural rate dynamics:though in resp

43、onse to both shocks the real rate drops,the natural rate falls only forthe demand shock,remaining constant for the monetary policy shock.Just observingthe dynamics of real interest rates is not sufficient to infer whether misallocation willimprove or worsen.In the presence of nominal rigidities,it i

44、s the joint dynamics of realand natural rates that matter.Second,an emerging literature analyzes how financial frictions and firm heterogene-ity affect the transmission of monetary policy.7Our paper is especially related to thosepapers analyzing the heterogeneous investment sensitivity to monetary p

45、olicy.Thisstrand of literature finds that firms investment is more responsive to monetary policyshocks when their default risk is low(Ottonello and Winberry,2020),when they havehigh leverage or fewer liquid assets(Jeenas,2020),when they are young and do not paydividends(Cloyne et al.,2018),when thei

46、r excess bond premia is low(Ferreira et al.,2022),or when a higher fraction of their debt matures(Jungherr et al.,2022).We con-tribute to this literature by showing that firms with high-MRPK are more responsiveto monetary policy shocks,and by analyzing the implications of this on the optimal6Buera a

47、nd Nicolini(2020)employ a discrete-time version of Moll(2014)with cash-in-advanceconstraints to analyze the impact of different monetary and fiscal policies after a credit crunch.7One strand of this literature analyzes the links between monetary policy,firm heterogeneity and theallocation of resourc

48、es through heterogeneity in markups and entry-exit(e.g.Meier and Reinelt,2020,Bilbiie et al.,2014 or Baqaee et al.,2021),risk-taking(David and Zeke,2021),firm-level productivitytrends(Adam and Weber,2019),or the importance of the price elasticity of investment(Koby andWolf,2020).4conduct of monetary

49、 policy.8Finally,this paper contributes to the literature analyzing optimal monetary policyproblems in heterogeneous-agent economies.A number of recent papers,such as Bhan-dari et al.(2021),Acharya et al.(2020),Bilbiie and Ragot(2020),Le Grand et al.(2022),Mckay and Wolf(2022),or Auclert et al.(2022

50、)propose different approachesto solve these problems.Similar to Nuo and Thomas(2022),Bigio and Sannikov(2021),Smirnov(2022),or Dvila and Schaab(2022),we set up the problem as oneof infinite-dimensional optimal control.We propose a new,simple and broadly appli-cable algorithm to solve these kinds of

51、problems,which leverages the computationaladvantages of continuous time.The key novelty of our algorithm is to discretize theRamsey planners continuous-time,continuous-state problem using finite differences(asin Achdou et al.,2021),and then to use symbolic differentiation to obtain the plannersfirst

52、-order conditions.This produces a high-dimensional nonlinear dynamic system,which is efficiently solved in the sequence space using a Newton algorithm.9Further-more,this paper is,to the best of our knowledge,the first to analyze optimal monetarypolicy in a model with heterogeneous firms.The structur

53、e of the paper is as follows.Section 2 presents the model,which wecalibrate in Section 3.Section 4 analyzes the drivers and dynamics of misallocation.Section 5 studies optimal monetary policy.Section 6 provides supporting empiricalevidence for the main mechanism of the model.Finally,Section 7 conclu

54、des.2ModelWe propose a New Keynesian closed economy model with financial frictions and hetero-geneous firms.Time is continuous and there is no aggregate uncertainty.The economyis populated by five types of agents:households,the central bank,entrepreneurs thatoperate input-good firms,retail,and final

55、 goods producers.Entrepreneurs are het-erogeneous in their net worth and productivity.They combine capital and labor toproduce the input good.The input good is differentiated by imperfectly competitive8In complementary empirical work,Albrizio et al.(2023)analyze the impact of monetary policy oncapit

56、al misallocation in more detail,both from an intensive and extensive margin.They find that theintensive margin is the main reallocation channel and that firms investment sensitivity to monetarypolicy is driven by their MRPK rather than their age,leverage,or cash.9The algorithm can be implemented usi

57、ng several available software packages.To make our methodaccessible to a large audience,we employ Dynare.5retail goods producers facing sticky prices,whose output is aggregated by the final goodproducer.The latter two types of firms are standard in New Keynesian models.2.1Heterogeneous input-good fi

58、rmsThe heterogeneous-firm block is based on Moll(2014).There is a continuum of en-trepreneurs.Each entrepreneur owns some net worth,which they hold in units of capi-tal.They can use this capital for production in their own input-good producing firm firm for short or lend it to other entrepreneurs.Si

59、milar to Gertler and Karadi(2011),we assume that entrepreneurs are atomistic members of the representative household,to whom they may transfer dividends.10Entrepreneurs are heterogeneous in two dimensions:their net worth atand theiridiosyncratic productivity zt.11Each entrepreneur owns a constant re

60、turns to scale(CRS)technology which uses capital ktand labor ltto produce the homogeneous inputgood yt:yt=ft(zt,kt,lt)=(ztkt)(lt)1.(1)The capital share (0,1)is the same across entrepreneurs.Idiosyncratic productivityztfollows a diffusion process,dzt=(zt)dt+(zt)dWt,(2)where(z)is the drift and(z)the d

61、iffusion of the process.12Entrepreneurs can use their net worth to produce in their firm with their owntechnology,or lend it to firms owned by other entrepreneurs.Firms employ labor lt,which they hire at the real wage wt,and capital kt,which is the sum of the entrepreneursnet wort(at)and what the fi

62、rm borrows(bt=kt at)at the real cost of capital Rt.Capital is borrowed from the agents which save,i.e.both households and lendingentrepreneurs.1310This assumption is the only relevant difference between the real side of our model and the model ofMoll(2014).We consider it to avoid having to deal with

63、 redistributive issues between households andentrepreneurs when analyzing optimal monetary policy.Both models produce linear dividend policies,so they can be seen as equivalent from a positive perspective.11For notational simplicity,we use xtinstead of x(t)for the variables depending on time.Further

64、-more,we suppress the input goods firms index.12The process is bounded with reflective barriers.13Since debt contracts are instantaneous and in units of capital,firms balance sheets are not exposed6Firms sell the input good at the real price mt=pyt/Pt,which is the inverse of thegross markup associat

65、ed to retail products over input goods,being pytthe nominal priceof the input good and Ptthe price of the final good,i.e.the numeraire.Entrepreneursuse the return on their activities to distribute(non-negative)dividends dtto the house-hold and to invest in additional capital at the real price qt.Cap

66、ital depreciates at rate.An entrepreneurs flow budget constraint can be expressed as follows at=1qtmtft(zt,kt,lt)wtlt Rtkt|zFirms profits+(Rt/qt)|zReturn on net worthqtatdt|zDividends.(3)Note that we have rearranged the budget constraint to yield the law of motion of networth in units of capital.Fir

67、ms face a collateral constraint,such that the value of capital used in productioncannot exceed 1 of their net worth,14qtkt qtat.(4)Entrepreneurs retire and return to the household according to an exogenous Poissonprocess with arrival rate.Upon retirement they pay all their net worth,valued qtat,toth

68、e household,and they are replaced by a new entrepreneur with the same productivitylevel.Entrepreneurs maximize the discounted flow of dividends,which is given byV0(z,a)=maxkt,lt,dtE00et0,tdt|zDividends+qtat|zTerminal valuedt,(5)subject to the budget constraint(3),the collateral constraint(4),and the

69、 productivityprocess(2).Future profits are discounted by the households stochastic discount factor0,t.Below we show that 0,t=et0rsds,where rtis the real interest rate.We can split the entrepreneurs problem into two parts:a static profit maximizationto Fisherian debt deflation or financial accelerato

70、r effects(Bernanke et al.,1999).Asriyan et al.(2021)include the latter.This assumption keeps the model tractable.Allowing for such effects would amplifythe effect of monetary policy shocks discussed later.14Assuming alternatively that firms borrowing is constrained to a multiple of their earnings(Li

71、anand Ma,2020)would amplify the effect of monetary policy shocks discussed later on.The increase inhigh-productivity firms profits,that(as we explain below)drives the positive impact of expansion-ary monetary policy on TFP,would relax the constraint of high-productivity firms and improve theallocati

72、on of capital further.7problem and a dynamic dividend-distribution problem.First,entrepreneurs maximizefirm profits given their productivity and net worth,maxkt,ltmtft(zt,kt,lt)wtlt Rtkt,(6)subject to the collateral constraint(4).Since the production function has constant returns to scale,entreprene

73、urs find itoptimal to operate a firm at the maximum scale defined by the collateral constraintwhenever their idiosyncratic productivity is high enough,that is whenever z exceeds acertain threshold level zt.Else the optimal size of the firm is k(z)=0,because theycannot run a profitable firm given the

74、ir low productivity.In that case the borrowingconstraint does not bind and the entrepreneur rents out its capital.From now on,werefer to the two groups of entrepreneurs as constrained and unconstrained.That is,firms demand for capital and labor is:kt(zt,at)=at,if zt zt,0,if zt zthus makes profits ov

75、er and above the cost of capital.These profits arise despite perfect competition,because the borrowing constraint binds.Note furthermore,that factor demands and profits are linear in net worth.This is aconsequence of the CRS technology and makes the model significantly more tractable.8As discussed b

76、y Moll(2014),the assumption of CRS in firms production function(1)can be seen as the limiting case of decreasing returns to scale(DRS),yt=(ztkt)(lt)1,z(a)are constrained.When thisthreshold increases,previously marginally constrained firms become unconstrained andreduce their capital stock below the

77、maximum implied by the constraint.When 1,the optimal size of low-productivity firms,and hence its production,are very small,k(z),y(z)0.Therefore our model should be understood as the tractable limit ofthe more realistic DRS case.15This highlights a crucial point regarding the interpreta-tion of the

78、model:the threshold zdoes not capture entry/exit,but rather the limitingcase of expansions and contractions of the optimal size of active firms.Entry and exitof firms in this model is exogenous,and given by the exogenous retirement rate.Second,entrepreneurs choose the dividends dtthat they pay to th

79、e household.Thesolution to this problem is derived in Appendix B.1.There we show how entrepreneursnever distribute dividends(dt=0)until retirement,when they bring all their net worthhome to the household.The intuition is the following.The return on one unit of capitalin the hands of the entrepreneur

80、 is at least(Rtqt),while for the household the returnof this unit of capital is exactly(Rtqt).It is thus always worthwhile for entrepreneursto keep their funds in the firm.To keep things simple,as in Gertler and Karadi,2011we assume the representative household uses a fraction (0,1)of these dividend

81、s tofinance the net worth of the new entrepreneurs,so net dividends are(1)of the networth of retiring entrepreneurs.Using(9),the law of motion of an entrepreneurs net worth(3)can thus be rewrittenas16 at=1qt(maxztt Rt,0+Rt qt)at.(11)15Ferreira et al.,2023 find that financially constrained firms are

82、found across the entire firm-sizedistribution.16In the model,there are no firm-level capital adjustment costs.Furthermore,due to CRS,changesin zimply that firms at the margin disinvest or reinvest fully instantaneously.In a DRS model thechanges in the capital stock of a firm that switches from being

83、 constrained to being unconstrainedand thus crossing the threshold z(a)would be smaller.This is so since the optimal capital stockof unconstrained firms would be positive(and not zero,as in the limiting CRS case).These smallerchanges in capital can be achieved by reductions in the gross investment r

84、ate,without requiring thereselling of capital,provided the depreciation rate is high enough.92.2HouseholdsThere is a representative household,composed of workers and entrepreneurs,that savesin capital Dtor in nominal instantaneous bonds BNt.Nominal bonds are in zero netsupply.Workers supply labor Lt

85、.The household maximizesWt=maxCt,Lt,BNt,Dt0ehttu(Ct,Lt)dt.(12)s.t.Dtqt=(Rt qt)Dt SNt Ct+wtLt+Tt,(13)BNt=(it t)BNt+SNt,where SNtis the investment into nominal bonds and Ttare the profits received by thehousehold,which is the sum of the profits of the capital and retail-goods producers(discussed below

86、)and net dividends received from entrepreneurs(1 )qtAt).We assume separable utility of CRRA form,i.e.,u(Ct,Lt)=C1t1L1+t1+.Solvingthis problem(see Appendix B.2 for details),we obtain the Euler equation,CtCt=rt ht,(14)the labor supply conditionwt=LtCt,(15)and the Fisher equationrt=it t,(16)where,for c

87、onvenience,we have made use of the following definition of the real interestratertRt qt+qtqt,(17)which equals the real return on capital adjusted by capital gains and depreciation.Integrating the Euler equation(14),we can define the stochastic discount factor 0,tas0,t et0htdsu0c(Ct)u0c(C0)=et0rsds.1

88、02.3Final good producersAs usual in New Keynesian models,a competitive representative final goods produceraggregates a continuum of output produced by retailer j 0,1,Yt=?10y1j,tdj?1,(18)where 0 is the elasticity of substitution across goods.Cost minimization impliesyj,t(pj,t)=?pj,tPt?Yt,where Pt=?10

89、p1j,tdj?11.2.4RetailersWe differentiate between heterogeneous input-good firms and retailers.17We assumethat monopolistic competition occurs at the retail level.Retailers purchase input goodsfrom the input-good firms,differentiate them and sell them to final good producers.Each retailer j chooses th

90、e sales price pj,tto maximize profits subject to price adjustmentcosts as in Rotemberg(1982),taking as given the demand curve yj,t(pj,t)and the priceof input goods,pyt.We assume the government pays a proportional constant subsidy=1on the input good,so that the net real price for the retailer is mt=m

91、t(1).This subsidy is financed by a lump-sum tax on the retailers t.This fiscal scheme isintroduced to eliminate the distortions caused by imperfect competition in steady state,as common in the optimal policy literature.The adjustment costs are quadratic in therate of price change pj,t/pj,tand expres

92、sed as a fraction of output Yt,t?pj,tpj,t?=2?pj,tpj,t?2Yt,(19)where 0.Suppressing notational dependence on j,each retailer chooses ptt0tomaximize the expected profit stream,discounted at the stochastic discount factor of17Distinguishing between heterogeneous input-good firms and retailers is standar

93、d practice in pre-vious New Keynesian models featuring firm heterogeneity and nominal rigidities,such as Ottonelloand Winberry(2020)or Jeenas(2020).Besides greater tractability,it avoids the possibly implausiblecountercyclical behaviour of New Keynesian markups interfering with our mechanism,which w

94、e see asan important advantage.It also does justice to the fact that retail prices are significantly more stickythan intermediate goods prices(for Europe see Alvarez et al.(2010),Alvarez et al.(2006),Gautieret al.(2023).11the household,00,t?ptPt mt?ptPt?Yt t t?ptpt?#dt.(20)The symmetric solution to

95、the pricing problem yields the New Keynesian Phillips curve(see Appendix B.3),which is given by rtYtYt!t=(mt m)+t,m=1.(21)where tdenotes the inflation rate t=Pt/Pt.2.5Capital producersA representative capital producer owned by the representative household produces cap-ital and sells it to the househ

96、old and the firms at price qt,which she takes as given.Hercost function is given by(t+(t)Ktwhere tis the investment rate and(t)is acapital adjustment cost function.She maximizes the expected profit stream,discountedat the stochastic discount factor of the household.Profits are paid in a lump-sum fas

97、h-ion to the household.Wt=maxt,Kt00,t(qtt t(t)Ktdt.(22)s.t.Kt=(t)Kt.The optimality conditions imply(see Appendix B.4)rt=(t)+qt 00(t)tqt 1 0(t)qtt t(t)qt 1 0(t).(23)2.6DistributionAs previously explained,we assume that,for each entrepreneur retiring to the house-hold,a new entrepreneur enters operati

98、ng the same technology,that is,with the sameproductivity level.This new entrepreneur receives a startup capital stock from thehousehold in a lump-sum fashion,equal to a fraction zt?=ztxt(x)dx1 t(zt)!.(32)This highlights that,in terms of output,the model is isomorphic to a standard representative-age

99、nt New Keynesian model with capital and TFP Zt.TFP is endogenous and evolvesover time and,as we discuss below,its evolution depends on factor prices.Note that TFP Ztserves as a measure of misallocation.The financial frictionsfaced by entrepreneurs imply that capital is not optimally allocated.The en

100、trepreneuroperating the most productive firm does not have enough net worth to operate thewhole capital stock,hence less productive firms operate as well,which is suboptimaland reduces TFP.Thus the more misallocated capital is,the lower is TFP.Factor prices arewt=(1 )mtZtKtLt,(33)Rt=mtZtK1tLt1ztEt()

101、z|z zt.(34)14Finally,the law of motion of the aggregate net-worth of entrepreneurs is given byAtAt=1qt?(1 t(zt)?mtZtK1tLt1 Rt?+Rt qt qt(1 )?.(35)Appendix B.6 derives these aggregate formulae step by step.2.8Central BankThe central bank controls the nominal interest rate iton nominal bonds held by ho

102、use-holds.For the positive analysis in Section 4 we assume that the central bank sets thenominal rate according to a standard Taylor rule of the formdi=?it?h+(t )+?dt,(36)where is the inflation target,is the sensitivity to inflation deviations and deter-mines the persistence of the policy rule.For t

103、he normative analysis in Section 5 weassume that the central bank implements the Ramsey-optimal policy.3Numerical solution and calibrationNumerical algorithm.We solve the model numerically using a new method,de-scribed in Appendix C.It combines a discretization of the model using an upwindfinite-dif

104、ference method similar to the one in Achdou et al.(2021)with a Newton al-gorithm that computes non-linear transitional dynamics in a single loop.This can beeasily implemented using Dynares perfect foresight solver.18Our solution approach is different from the one in Winberry(2018)or Ahn et al.(2018)

105、.These papers analyze heterogeneous-agent models with aggregate shocks build-ing on the seminal contribution by Reiter(2009).To this end,they linearize the modelaround the deterministic steady state.Winberry(2018)illustrates how this can bealso implemented using Dynare,and Ahn et al.(2018)extend the

106、 methodology tocontinuous-time problems.Here,instead,we compute the nonlinear transitional dy-namics in the the sequence space,as Boppart et al.(2018)or Auclert et al.(2021).18Notice that the variables of the model include the distribution(z),which is an infinite-dimensionalobject.The finite-differe

107、nce discretization turns this continuous variable into a finite dimensionalvector.15Boppart et al.(2018)show how the perfect-foresight transitional dynamics to a(small)MIT shock,such as the ones we compute here,coincide with the impulse responses ob-tained by a first-order perturbation approach in t

108、he model with aggregate uncertainty.Our method solves for the same approximate solution as the nonlinear version ofthe sequence space Jacobian approach by Auclert et al.(2021).An important techni-cal difference is that we solve simultaneously for all variables(prices,aggregates anddistributions)in a

109、 single loop without decomposing the model in blocks.Calibration.Table 1 summarizes our calibration.We calibrate the parameters ofthe heterogeneous firms block to match data on Spanish firms.The entrepreneurs exitrate()is set to 10 percent,in line with the average exit rate 2007-2020 in the data fro

110、mthe Spanish Statistical Institute(INE).19The other parameters of the heterogeneous-firm block are disciplined by detailed firm-level panel data on the quasi-universe ofSpanish firms.We postpone the description of this data to the empirical Section 6,with further details in Appendix A.1.The fraction

111、 of assets of exiting entrepreneursreinvested()is 0.1,so that new entrants account for 1 percent of the total capitalstock,in line with the data.The borrowing constraint parameter is 1.56,implyingthat entrepreneurs can borrow up to 56%of their net worth,or 36%of their totalassets,which targets the a

112、ggregate debt to total assets ratio in the data.We assumethat individual productivity z follows an Ornstein-Uhlenbeck process in logs,20with areflective lower(upper)barriers at some value close to 0(very high value).21dlog(z)=zlog(z)dt+zdWt.(37)We estimate this process using our firm panel data set,

113、as explained in AppendixA.5.The estimate for zcorresponds to an annual persistence of 0.83,and the annualvolatility of the shock is estimated to be 0.73.The conventional macro parameters are set to standard values.The rate of timepreference of the household his 0.01,which targets an average real rat

114、e of return of1 percent.The capital share is set at 0.35 and the capital depreciation rate at19Specifically,the data comes from the Directorio Central de Empresas,which is a dataset maintainedby INE,and it contains aggregate data on all firms operating in Spain,and its status(incumbent,entrant or ex

115、iter).The dataset can be accessed here.20By Itos lemma,this implies that z in levels follows the diffusion process dz=(z)dt+(z)dWt,where(z)=z?zlogz+22?and(z)=zz.21We truncate the process for z at 48.This corresponds to truncating the MRPK distribution atthe same level as in the data.16Table 1:Calibr

116、ationParameterValueSource/targetFirms death rate0.1Average exit rateFraction firms assets at entry0.1Capital of firms younger than 1 year/All firms capitalBorrowing constraint parameter1.56Debt/Total Assets firmszMean reverting parameter0.19Estimate based on firm level datazVolatility of the shock0.

117、73Estimate based on firm level datahHouseholds discount factor0.011%Capital share in production function0.35Gopinath et al.(2017)Capital depreciation rate0.06Gopinath et al.(2017)Intertemporal elasticity of substitution HH1Log utility in consumptionInverse Frisch Elasticity1Kaplan et al.(2018)Consta

118、nt in disutility of labor0.71Normalization L=1kCapital adjustment costs8VAR evidence Christiano et al.(2016)?Elasticity of substitution retail goods10Mark-up of 11%Price adjustment costs100Slope of Phillips curve of 0.1 as in Kaplan et al.(2018)Inflation target0StandardSlope Taylor rule1.5StandardPe

119、rsistence Taylor rule0.2Standard0.06,as in Gopinath et al.2017.We assume log-utility in consumption(=1)andthe inverse Frisch elasticity is also set to 1,standard values in the literature.We setthe constant multiplying the disutility of labor such that aggregate labor supply insteady state is normali

120、zed to one.We assume capital adjustment costs are quadratic,i.e.(t)=k2(t)2,and set kto 8,such that the peak response of investmentto output after a monetary policy shock is around 2,in line with the VAR evidence ofChristiano et al.(2016).Regarding the New Keynesian block,the elasticity of substituti

121、on of retail goods?is set to 10,so that the steady state mark-up is 1/(?1)=0.11.The Rotembergcost parameter is set to 100,so that the slope of the Phillips curve is?/=0.1 as inKaplan et al.(2018).The Taylor rule parameters take the following standard values:=0,=1.5 and =0.2.The model generates the s

122、teady-state MRPK distribution shown by the orange linein Figure 3.The dark blue bars show the MRPK distribution in the data,where afirms MRPK is proxied by the product of a firms value added over capital and its17capital share,as explained in Appendix A.2.Despite its simplicity,the model matchesthis

123、 distribution well,thus predicting a plausible amount of capital misallocation.22Note that matching this distribution is key,since our results exploit the heterogeneousresponse of investment along the MRPK distribution.Figure 1:MRPK distributionNotes:The figure shows the steady state distribution of

124、 firms MRPK in the model(orange solid line)and comparesit to the data(histogram with blue bars).See Appendices A.1 and A.2 for more details on the measurement of firmlevel MRPK in the data and robustness to differences in sectoral capital shares.We drop observations above an MRPKof 0.82,which implie

125、s dropping firms in the 5%upper tail of the capital-weighted MRPK distribution.Note that,byconstruction,the model cannot explain firms with an MRPK below the cost of capital(R=r+=0.07 in steady state),which we also drop in this figure.4Monetary policy and capital misallocationIn this Section,we anal

126、yze the links between monetary policy and capital misallocation.To this end,we first delve into the theoretical mechanisms through which changes inequilibrium prices affect misallocation,to then analyze the response of misallocation ingeneral equilibrium to a monetary policy and a time-preference sh

127、ock.Understandingthese mechanisms is key to understand the optimal monetary policy explained in thefollowing Section 5.22In Appendix A.2,we reproduce the MRPK distribution of Figure 3,but computing MRPK usingsectoral alphas.The fit of the model worsens only slightly in the direction of under-predict

128、ing themeasured misallocation.184.1The capital misallocation channel of monetary policyAs Section 2 highlighted,misallocation and thus TFP is endogenous and evolves overtime.Misallocation is driven by the investment dynamics within the heterogeneousinput-goods firms block of the model,which in turn

129、depend on the factor prices deter-mined in general equilibrium.One can thus think about the heterogeneous firm sectoras a model block that translates(sequences of)prices into(a sequence of)TFP.As discussed above,by equation(32)which we reproduce here,TFP depends onthe allocation of capital across en

130、trepreneurs:Zt=?Et()z|z zt?,(38)That is,TFP is the capital-weighted average of firms idiosyncratic productivity.TFPthus depends on the mass of the net-worth distribution,t(),above the productivitythreshold,zt(the shaded area in Figure 4.1).Entrepreneurs below the cut-offzareunconstrained,operate at

131、their optimal size k(z)=0,and lend their net worth toconstrained entrepreneurs above the cut-off.Equation(38)allows us to identify howchanges in equilibrium prices affect aggregate TFP in this economy(i)by changing thenet-worth distribution,t();and(ii)by changing the productivity-threshold zt.Wenow

132、explore these two margins in isolation.We start analyzing the case in which the dynamics of TFP are driven purely bychanges in the net worth distribution,which happens when the cut-offztis constant.In this case,the the excess investment rate is key for the dynamics of TFP.We definethe excess investm

133、ent rate as the ratio of profits over net wortht(z)tqtat=max?tqt(z zt),0?,(39)wheret(z)is the return on equity that a firm with MRPK tz makes in excess of thecost of capital Rt.Since entrepreneurs reinvest all profits,(z)also describes the speedat which the net worth of an entrepreneur with producti

134、vity z grows in excess of thegrowth rate of the unconstrained entrepreneurs with productivity z zt.2323(z)is also a measure of how constrained a firm is,since(z)is the Lagrange multiplier of theborrowing constraint in the firms maximization problem.From the first order conditions of the firm,we get

135、that MRPKt=Rt+qtBC,where BCis the multiplier on the borrowing constraint.Hence,BC=t(z).19Figure 2:The capital misallocation channel.(a)Change in the net-worthdistribution.(b)Shift in the productivity threshold.Notes:The figure illustrates the net-worth share distribution (z)and the productivity-thre

136、shold z(blue).Thelight blue area is the initial mass of constrained firms.Panel(a)shows the impact of a change in the net-worthdistribution.Panel(b)shows the impact of an increase in the threshold(from blue dashed line to orange dashed line).The new mass of constrained firms after the change is depi

137、cted by the shaded orange area in both panels.Proposition 1.(TFP response to the slope).Conditional on a constant cutoffz,the dynamics of Ztare fully determined by the slope of the excess investment rate,tqt.An increase intqtleads to an increase in the growth rate of TFP through changes inthe net wo

138、rth distribution:?tqt?dlogZtdt?z=zz2t(z)zzt(z)dzzzt(z)zt(z)dz 0.All the proofs in the Section can be found in Appendix B.8.This propositionstates that the slopetqtdetermines how,conditional on a constant cut-offz,thenet-worth share distribution moves,and hence in which direction TFP evolves.tqtisa f

139、unction of prices.If the slope increases,then high-productivity firms profitabilityadvantage widens,such that they grow faster than low-productivity firms,the net worthdistribution shifts rightwards,the allocation of capital improves,and TFP increases,asrepresented in Panel a of Figure 4.1.Note that

140、,in the model,high-productivity firmshave a high MRPK,which is given by tz.So we can equivalently say that an increasein the relative growth rate of high-MRPK firms improves TFP.We turn next to the case when the distribution remains constant and the cut-offchanges in response to price changes.This h

141、appens in the limit of iid shocks,that is,20the limit as z,as discussed in Itskhoki and Moll(2019).In this case the net worthdistribution ()is constant,and the response of TFP growth depends exclusively onthe changes in the cutoff,according to the following propositionProposition 2.(TFP response to

142、the cutoff).Conditional on a constant density(),the dynamics of Ztare fully determined by the threshold zt.The partial derivativeof TFP growth with respect to the growth rate of the thresholddztdtis positive:?dztdt?dlogZtdt?()=(zt)zt(z zt)(z)dzzt(z)dzztz(z)dz 0.This proposition implies that a change

143、 in prices that raises the threshold zt=Rt/tincreases TFP.Panel(b)in Figure 4.1 illustrates how an increase in the thresholddecreases the share of constrained firms by crowding out low-productivity entrepreneurs.The intuition is simple:low-MRPK constrained firms that were close to the thresholdbecom

144、e unconstrained and reduce their capital optimally to 0,which implies that theseentrepreneurs stop using their net worth for production,and instead they lend it to moreproductive firms,decreasing misallocation.Changes in the productivity-threshold thuscapture changes in the share of constrained vers

145、us unconstrained firms.This mechanismis different from the extensive margin:it is not meant to capture firm entry and exit,which in our model is exogenously given by the probability of retiring.Rather,itcaptures the idea that previously constrained firms become unconstrained and viceversa.In general

146、 equilibrium these two margins simultaneously determine the evolutionof TFP.However,rather than being two independent mechanisms,they are tightlyconnected through general equilibrium forces.Corollary 1.An increase in the slope of the excess investment rate,tqt,is associatedto an increase in the grow

147、th rate of the thresholddztdt,iffit is associated to a sufficientlysmall increase in the growth rate of the ratio between households net savings and firmsnet worth:?dDtAtdt?/?tqt?ztzt)(1(zt)t(zt)0:?dztdt?tqt?0?dDtAtdt?tqtzt,z,m,m,i,Y,Tt0E00ehtu(Ct,Lt)dt(40)subject to the all the private equilibrium

148、conditions derived above and listed in Ap-pendix B.7 and the initial conditions 0(z),K0,D0,A0.Relative to the standard NewKeynesian model i.e.the complete market version of our model the problem of thecentral bank is richer by one dimension:the central bank understands that its policyaffects TFP thr

149、ough the capital misallocation channel,and has to account for that.Algorithm.This additional richness also makes the problem harder to solve com-putationally.The central banks controls include the net-worth distribution t(z),asthe central bank internalizes the impact of its decisions on it.Notice th

150、at the den-sity t(z)not only depends on time,but also on individual productivity.This posesa challenge when solving optimal monetary policy,as we need to compute the firstorder conditions(FOCs)with respect to this infinite-dimensional object.There are anumber of proposals in the literature to deal w

151、ith this problem.Bhandari et al.(2021)make the continuous cross-sectional distribution finite-dimensional by assuming thatthere are N agents instead of a continuum.They then derive standard FOCs for theplanner.In order to cope with the large dimensionality of their problem,they employa perturbation

152、technique.Le Grand et al.(2022)employ the finite-memory algorithmproposed by Ragot(2019).It requires changing the original problem such that,after27K periods,the state of each agent is reset.This way the cross-sectional distributionbecomes finite-dimensional.Nuo and Thomas(2022),Bigio and Sannikov(2

153、021),Smirnov(2022)and Dvila and Schaab(2022)deal with the full infinite-dimensionalplanners problem.This implies that the continuous Kolmogorov forward(KF)and theHamilton-Jacobi-Bellman(HJB)equations are constraints faced by the central bank.They derive the planners FOCs using calculus of variations

154、,thus expanding the originalproblem to also include the Lagrange multipliers,which in this case are also infinite-dimensional.These papers solve the resulting differential equation system using theupwind finite-difference method of Achdou et al.(2021).Here we propose a new algorithm,detailed in Appe

155、ndix D.Instead of determiningthe FOCs for the planners continuous space problem,we first discretize the plannersobjective and constraints(the private equilibrium conditions)using finite differences.This transforms the original infinite-dimensional problem into a high-dimensional prob-lem,in which th

156、e value function and the state density are replaced by large vectors witha dimensionality equal to the number of grid points(200 in our application)used toapproximate the individual state space.In this discretized model the dynamics of the(now finite-dimensional)distribution tare given by?I tATt?t=t

157、1,where tis the time step and Atis a matrix whose entries depend nonlinearly and in closed formon the idiosyncratic and aggregate variables in period t.30Second,we find the planners FOCs by symbolic differentiation.This delivers alarge-dimensional system of difference equations.Third,we find the Ram

158、sey steadystate by solving this system at steady state.To do so,we compute the steady stateof the model conditional on the steady-state level of the policy instrument with aconventional iterative method,and then use this function to find the Ramsey steadystate using the Newton method.Fourth,we solve

159、 the system of difference equationsnon-linearly in the sequence space using the Newton method,as already described inSection 3 and Appendix C.The symbolic differentiation and the two applications ofthe Newton algorithm can conveniently be automated using several available softwarepackages.In our cas

160、e,we employ Dynare,but the approach is also compatible withthe nonlinear sequence space Jacobian toolbox.This algorithm can be employed tocompute optimal policies in a large class of heterogeneous agent models.Compared toother techniques,it stands out for being easy to implement.In appendix D we com

161、pare30Technically,this matrix results from the discretization of the infinitesimal generator of the id-iosyncratic states.28our method conceptually to the ones cited above.We also present a proposition showingthat our algorithm delivers the same results as computing the FOCs by hand usingcalculus of

162、 variations and then discretizing the model,as the time step gets smaller.Finally we apply our the algorithm to solve the model in Nuo and Thomas(2022)inorder to illustrate its generality,demonstrating that results coincide.5.2Optimal Ramsey policySteady state.Let us focus first on the steady state

163、of the Ramsey problem.It iswell known that in the standard(complete-market)New Keynesian economy withoutsteady state distortions inflation is zero in the Ramsey steady state.Due to capitalmisallocation,our baseline(incomplete-market)economy does not feature steady stateefficiency.Yet,inflation is st

164、ill zero in the steady state of the Ramsey problem.31Thisresult mirrors a similar result from the textbook New Keynesian model with a distortedsteady state(Woodford,2003;Gali,2008).Though the long-run Phillips curve allowsmonetary policy to affect misallocation in the long run through positive trend

165、 inflation,the benefits of this policy are compensated for by the cost of the anticipation of thispolicy.Time-0 optimal policy.We turn next to the deterministic dynamics under theRamsey optimal plan.We solve for the Ramsey plan when the initial state of theeconomy coincides with the steady state und

166、er the optimal policy,i.e.,that with zeroinflation.The Ramsey planner faces no pre-commitments.This is commonly referredto as the“time-0 optimal policy”(Woodford,2003).We compare our baseline incomplete-market economy with a complete-market econ-omy.The Ramsey plan in the model with complete markets

167、 is time-consistent.Hence,inflation and the rest of variables remain constant at their steady state values(see thedashed red lines in Figure 5).Market incompleteness,however,introduces a new sourceof time inconsistency,inducing the central bank to temporally deviate from the zero-inflation policy.Th

168、e solid blue lines in Figure 5 show how the central bank engineersa sizable surprise monetary expansion,increasing inflation(panel a).The resultingdynamics are precisely those caused by an expansionary monetary policy shock,whichwere described in detail in Section 4.2.As a result TFP increases(panel

169、 b).The centralbank thus engineers a monetary expansion,tolerating a temporary increase in inflation,31This is a numerical result that holds at close to machine precision for a wide range of parameters.29Figure 5:Time 0 optimal monetary policy.Notes:The figure shows the deviations from steady state

170、of the economy when the planner solves the Ramsey problemwithout pre-commitments and in the absence of shocks.The baseline economy is the solid blue line,and the completemarket economy the dashed orange line.The dotted yellow line and the purple dashed line repeat the same exercise inthe absence of

171、the subsidy that undoes the markup distortion.in order to achieve a persistent rise in TFP,brought about by a more efficient allocationof capital.It is well known that the Ramsey policy in the complete market economy with asteady-state mark-up distortion also features inflationary time inconsistency

172、.Compar-ing the optimal policy above with the optimal policy when there is no subsidy to correctfor the mark-up distortion reveals that the time inconsistency problem caused by theincomplete market distortion is much larger:the optimal inflation level due to marketincompleteness is more than six tim

173、es higher than that due to the mark-up(dashedpurple line).We hence conclude that the time inconsistency problem is not only largein absolute terms(with an average inflation of 3%during the first year),but also dwarfsthe one resulting form markups in the standard New Keynesian model.The desire of the

174、 central bank to redistribute resources towards high-MRPK en-trepreneurs is reminiscent of the case with optimal fiscal policy analyzed by Itskhoki andMoll(2019).They find that optimal fiscal policy in economies starting at below steady-state net-worth levels initially redistributes from households

175、towards entrepreneurs inorder to speed up net worth accumulation,and thus increase TFP growth.In ourcase,and given the lack of fiscal instruments,it is the central bank who engineers thisredistribution through an expansion in aggregate demand.305.3Timeless optimal policy responseNext,we analyze the

176、optimal policy response when an unexpected shock hits the econ-omy that was previously in its zero-inflation steady state.In this case,we adopt a“timeless perspective”(Woodford,2003).Timelessly optimal Ramsey policy impliesthat the central bank sticks to pre-commitments,implementing the policy that

177、it wouldhave chosen to implement if it had been optimizing from a time period far in the past.32This allows us to study systematic changes in monetary policy in response to shocksunder the,ex-ante optimal,time-invariant state-contingent policy rule.33Households time preference shock.We analyze the o

178、ptimal response to a timepreference shock from a timeless perspective.Figure 6 shows that the optimal inflationresponse in the baseline economy(blue solid line)mimics that under complete markets(orange dashed line):the central bank stabilizes inflation at its steady state value ofzero(panel a).This

179、is what is usally known as“divine coincidence”(Blanchard andGali,2007):the real rate follows the natural rate(panel c)and output is at its naturallevel(panel b).34This result has important implications:despite the fact that thecentral bank can use monetary policy to cushion the impact of shocks by e

180、xploiting themisallocation channel of monetary policy,it chooses not to do so and sticks instead tostrict price stability.As we show in Appendix B.12,this result applies also to othershocks,such as a TFP or a financial shock.In order to implement the optimal policy in response to the time-preference

181、 shock,the central bank should lower the real and nominal rates.However,in the baselinemodel with incomplete markets,this requires that the central bank acts more forcefullythan under complete markets(panel c).The reason is that the original demand shockleads to a negative supply shock through its i

182、mpact on aggregate TFP(panel d),whichdepresses output and natural rates relative to the complete market case,as discussedin the previous Section.Zero lower bound.The fact that the central bank responds more persistently tothe demand shock under the optimal policy has important implications when the

183、zerolower bound constrains its room for maneuver.Figure 7 displays the optimal response32The Lagrange multipliers associated to forward-looking equations in the planners FOCs in thiscase are initially set to their steady state values.33As discussed in Section 3,building on the argument by Boppart et

184、 al.(2018)one can reinterpretthe timeless response to MIT shocks as a first-order approximation to the response under uncertainty.34The natural level corresponds to the case with flexible prices(yellow and purple dashed lines).The divine coincidence holds only in a approximate sense,but the deviatio

185、n is negligibly small.31Figure 6:Optimal monetary policy response to a household discount factor shock.Notes:The figure shows the optimal response from a timeless perspective(in deviations from steady state)to a 1 p.p.decrease in the rate of time preference of the household hthat is mean reverting w

186、ith a yearly persistence of 0.8.Thebaseline economy is the solid blue line,and the complete market economy the dashed orange line.The figure also showsthe paths of the variables under strict inflation targeting(yellow and purple lines),though they are barely visible sincethey are overlaid by the opt

187、imal policy paths.from a timeless perspective to a large negative demand shock that drives the naturalrate below the zero lower bound(ZLB).The optimal policy under complete markets,asshown by Eggertsson et al.(2003),is to adopt a“low for longer”strategy:The nominalrate(dotted yellow line,panel a)sho

188、uld remain at the ZLB for a longer period than itwould in the absence of the ZLB(dashed orange line).In the baseline economy with incomplete markets,optimal policy is also character-ized by a low for longer strategy(dotted light blue line,panel a).However,the lift-offdate is now delayed relative to

189、the complete markets model.We call this a“low foreven longer”policy.The reason is simple.As discussed above,natural rates fall morepersistently in the case with incomplete markets,and so do nominal rates under theoptimal policy without the ZLB(solid blue line).To compensate for the inability tomove

190、rates into negative territory,the central bank commits to stay low for even longer.Incomplete markets make this commitment even more important,since otherwise notonly output and inflation,but also TFP would fall more.To see this,we compare thebaseline(light blue dotted line)with a policy that sets t

191、he rate equal to the maximumof its optimal value in the absence of the ZLB and zero(green dotted line)and verify32Figure 7:Optimal monetary policy response to a demand shock with the zero lowerbound.Notes:The figure shows the optimal response from a timeless perspective(in deviations from steady sta

192、te)to a 4 p.p.decrease in the rate of time preference of the household hthat is mean reverting with a yearly persistence of 0.8.Thebaseline economy without the zero lower bound is the solid blue line,and the complete market economy without thezero lower bound is the dashed orange line.The dotted lig

193、ht blue line is the optimal response in the baseline economywith the zero lower bound,while the dotted green line is the response under a policy that sets the rate equal to themaximum of its optimal value in the absence of the ZLB and zero.The yellow dotted line is the optimal response in thecomplet

194、e market economy with the zero lower bound.that TFP declines by more in the latter case.6Testable implicationsA key prediction of our theory is that an expansionary monetary policy shock increasesTFP by reducing misallocation.While the main focus of our paper is conceptual,we conclude by evaluating

195、this testable implication.As already discussed in Section4,the literature has repeatedly confirmed that expansionary policy shocks increaseTFP.In this section,we show that a reduction in capital misallocation may contributeto this increase.First,we use firm level data to test the mechanism suggested

196、 bythe model:after an expansionary monetary policy shock,high-MRPK firms increasetheir investment relatively more.Second,with a simple model-derived measure ofmisallocation,we quantify the impact of monetary policy shocks on misallocation andTFP in the data.In both cases we compare the results to th

197、e model predictions.Data.For our empirical analysis we combine granular Spanish firm-level panel datawith Jarociski and Karadi monetary policy shocks.We use yearly balance-sheet andcash-flow data from the quasi-universe of Spanish firms from 2000 to 2016 from theCentral de Balances Integrada.The mai

198、n advantage of this dataset is that it covers the33quasi-universe of Spanish firms,including not only large firms with access to stock andbond markets,but also medium and small firms more reliant on bank credit and internalfinancing.This contrasts with most papers in this literature,which use data f

199、rompublicly traded firms(e.g.Compustat).These are generally large firms with access tothe equity market,which can potentially behave very differently from the rest of firmsin the economy,as documented for example by Caglio et al.(2021).Appendix A.1details the data definition and the cleaning process

200、 and reports descriptive statistics.Our key variable of interest is firms MRPK,which we proxy by value added overcapital following the literature(see for instance Bau and Matray,2023).AppendixA.2 explains the rationale for this using proxy in more detail and explains that all theempirical results in

201、 this paper are robust to sectoral differences in capital shares.The monetary policy shock is taken from Jarociski and Karadi(2020),who usesign restrictions to decompose unexpected high frequency movements of interest ratesaround policy announcements into an information surprise and a monetary polic

202、y sur-prise component.We use the latter component,and we aggregate these shocks to yearlyfrequency following the methodology employed by Ottonello and Winberry(2020).Ap-pendix A.3 provides more details on the identification and aggregation of the monetarypolicy shock.Firm level responses.The model p

203、redicts that TFP increases in response to amonetary shock,because the capital stock of firms with a higher MRPK grows relativelymore.This happens both because constrained high-MRPK firms net worth increases inrelative terms,which thus invest relatively more,and because low-MRPK firms optimalsize is

204、reduced,which thus invest less.To illustrate this,we simulate a monetary policyshock in the model and calculate the average response of firms capital stock as afunction of their initial MRPK.The model predicts that this response is near-linearlyincreasing in the logarithm of pre-shock MRPK,as the da

205、shed orange line in Figure 8illustrates.To test this prediction in the Spanish data,we estimate the following relationship,which is linear in log MRPK,as suggested by the model:logkj,tlogkj,t1=0+1log(MRPKj,t1)+2log(MRPKj,t1)t+3t+0Yt1+s+uj,t.(41)where kj,tis the tangible capital of an individual firm

206、 j at time t,MRPKj,t1is laggedMRPK,and tis the monetary policy shock.While these controls would be sufficient34Figure 8:Response of investment to anexpansionary monetary policy shock as afunction of initial MRPKNotes:The figure displays the average effect of an 1 p.p.expansionary monetary policy sho

207、ck on the growth rateof the capital stock in the year after the shock in p.p.100 (logkj,1 logkj,0)as a function of the firms logMRPK before the shock log(MRPKj,0).For the model(orange),the relationship is calculated analytically.SeeAppendix B.13 for more detail.Estimating the regres-sion(41)on simul

208、ated data would recover a linear ap-proximation of it.We compare the model prediction tothe estimated relationship(41)(black).The shaded ar-eas mark the 90,95 and 99%confidence intervals.Table 2:Response of firm-level invest-ment to an expansionary monetary pol-icy shock(1)(2)logkj,t1,tlogkj,t1,ttlo

209、g(MRPKj,t1)0.04700.0286(0.02)(0.01)t0.0605(0.02)Obs3,692,1883,692,188R20.010.02stNoYessYesNoNotes:Column(1)reports the estimated differential ef-fect(2)and average effect(3)from regression(41),that is,including sector fixed effects,aggregate con-trols(lagged GDP growth,inflation and unemployment).St

210、andard errors clustered at the sector-year level.Col-umn(2)reports the differential effect(2)estimated in-cluding sector-year fixed effects.in the context of the model,we add further controls to account for non-modelled forces.Yt1are macroeconomic controls to account for the business cycle,that incl

211、ude GDPgrowth,inflation and unemployment and sis a vector of sector fixed effects,whichcontrol for potential sector-specific confounders.Finally,uj,tis the residual.The main coefficient of interest is 2,which is the empirical counterpart to the slopeof the orange function in Figure 8:a positive valu

212、e indicates that high-MRPK firmsincrease their investment more than low-MRPK firms after an expansionary monetarypolicy surprise.Table 2,column 1 shows that after a 1 p.p.expansionary monetarypolicy shock,a firm with an MRPK that is 1%higher than that of another firm increasesits capital stock by 0.

213、0470 p.p.more.The coefficient 3corresponds to the intercept in Figure 8.Since the mean and stan-dard deviation of log(MRPKj,t)are-0.87 and 1.4,these estimates document both a35substantially positive average effect of monetary policy on investment,35and more im-portantly an economically significant a

214、mount of heterogeneity in the firms responses.Being positive,these estimates thus support the models qualitative predictions.Whatis more,as Figure 8 shows,they are also quantitatively close to the model.We perform a battery of robustness tests,by subsequently enriching the baselinespecification in(4

215、1).First,we add sector-time fixed effects.This accounts for sec-toral differences in capital shares,as further explained in Appendix A.2,and for otherpotential year-sector specific confounders.While more robust,the average effect ofthe monetary policy shock is absorbed in this specification.The esti

216、mate of the sloperemains significant and of similar order of magnitude(see column 2 in Table 2).Furthermore,we add firm-level fixed effects,firm-level controls,including measuresof size,leverage and liquidity,introduce aggregate controls interacted with log(MRPK),and also interact the firm-level con

217、trols with the monetary policy shock.In all cases thecoefficient 2remains positive and statistically significant,and of similar magnitude.We also demean the MRPK at the firm level,to ensure that the results are not drivenby permanent heterogeneity in responsiveness across firms or systematic bias in

218、 ourempirical proxy for the MRPK in the spirit of Ottonello and Winberry(2020)andwe still find a positive and significant coefficient.Furthermore,these results are notdriven by the smaller firms in our sample:if we restrict the analysis to large firms,thecoefficient is still positive and significant

219、,and even of greater magnitude.All of theseresults are reported in Appendix A.4.In the model,the share of capital held by high-MRPK firms increases in responseto a monetary expansion because the monetary expansion increases the profits of high-MRPK firms disproportionately,which are then invested.To

220、 test for this particularmechanism,in column(11)of Table 4 in Appendix A.4 we estimate the baseline re-gression(41)with sector-year fixed effects,but with the log change in profits on theleft-hand side.We find the coefficient to be positive and significant,providing furthersupport for the mechanism

221、presented in the paper.A follow-up empirical paper by Albrizio et al.(2023)explores these results further.They show that debt holdings increase relatively more for high-MRPK firms,whichwould also be in line with our model predictions.Furthermore,they document that35If we assume a capital depreciatio

222、n rate of 10%as in Ottonello and Winberry(2020),ourestimates imply that after a 1 p.p.expansionary monetary policy shock,average investment increases19%.This is very close to the 20%found by these authors.36common proxies for tighter financial frictions,such as firm age,leverage,or liquidity,matter

223、for investment sensitivity to monetary policy only as long as firms have a high-MRPK.Aggregate productivity.In the model,the individual investment decisions ag-gregate up to changes in misallocation,such that aggregate TFP increases after anexpansionary monetary policy shock.To test this prediction

224、quantitatively,we need anempirical measure of TFP that abstracts from any changes in TFP that are broughtabout by anything but changes in the allocation of capital.For this purpose we definedynamic weighted average MRPK,WAMt,asWAMt,JXj=0MRPKjtkj,t+Kt+,where j indexes the firm,J is the number of firm

225、s,and Kt+is the aggregate capital.We approximate the growth rate of WAMt,from time t to t+by the log differencelogWAMt,logWAMt,logWAMt,0.WAMt,tells us how much the economy-wide average MRPK has changed from period t to t+only due to changes in thedistribution of capital across firms,holding constant

226、 the MRPK of the firm at theinitial level.As we show in Appendix A.6,in our model logWAMt,is approximatelyproportional to the growth rate of TFP Zt:logWAMt,1/logZt,Through the lens of the model,our empirical measure logWAMt,can hence be inter-preted as a measure of changes in TFP that are brought ab

227、out purely through changesin the allocation of capital,muting any other channels through which a monetary policyshock may simultaneously affect standard measures of TFP.We use this variable as dependent variable in the following simple local projection:logWAMt1,s=s,+t+ut,sfor =1,2,3,4.We estimate th

228、is regression at the sector level s to account for potential sectoral differ-ences.This also accounts for differences in capital shares across sectors.s,denoteshorizon specific sector fixed effects,tis the monetary policy shock,and ut,sis theresidual.The regression coefficient thus tells us the cumu

229、lative change in our mea-37Figure 9:Response of average MRPK to an expansionary monetary policy shock.Notes:The Figure shows the estimated impulse response function after a 1 p.p.expansionary monetary policy shock oflogWAMt,on the data(black line),and the shaded area marks the 90%,95%and 99%confiden

230、ce intervals of thedata estimates.It also shows response after a 1 p.p.expansionary monetary policy shock in the model of logWAMt,(orange broken line),and the log changes of model TFP(scaled by 1/)(blue dashed line).sure of capital misallocation at different horizons after a 1 p.p.monetary policy ea

231、singsurprise.Figure 6 reports our estimates for at different horizons (black line),with con-fidence intervals shaded in gray.Standard error are clustered at sector level.A 1 p.p.expansionary monetary policy shock causes an increase of the dynamic weighted av-erage MRPK of 3 p.p.at impact,and of 7 p.

232、p.at peak after 3 years.The effect issignificant throughout 4 years at the 95%level.36The dashed-dotted orange line in Figure 6 shows that the model produces a similarpath for the dynamic weighted average MRPK,albeit of smaller magnitude.37Themodel explains about half of the observed increase in log

233、WAMt,in the data.Themodel can hence be interpreted as being conservative with regards to the strength ofthe capital misallocation channel.The peak increase in WAMt,of almost 2.5 p.p.predicted by the model(dashed-dotted orange line)corresponds to an increase of TFPof 0.87 p.p.(dashed blue line).36Alb

234、rizio et al.(2023),using the sector-level variance of MRPK(see Hsieh and Klenow 2009)as ameasure of misallocation for Spanish data,also find that expansionary monetary policy shocks decreasemisallocation.37The model counterpart is constructed by feeding a 1 p.p.monetary policy easing surprise into t

235、hemodel as a temporary deviation from the Taylor Rule,and then computing WAMt,.387ConclusionsThis paper introduces a tractable model with heterogeneous firms,financial frictions,and nominal rigidities in order to understand the link between monetary policy and cap-ital misallocation,and its policy i

236、mplications.We calibrate this economy using Spanishfirm-level data,and show that it can reproduce fairly well the MRPK distribution inthe data.Our model predicts that an expansionary monetary policy shock improvesthe allocation of capital and thus raises TFP.We call this effect the capital misalloca

237、-tion channel.We present empirical evidence supporting this prediction:expansionarypolicy induces high-MRPK firms to increase their investment relatively more than low-MRPK firms.We analyze optimal monetary policy for a benevolent central bank withcommitment.The central bank has a strong time-incons

238、istent incentive to exploit thecapital misallocation channel,engineering a temporary economic expansion to increaseTFP at the cost of some inflation.When commitment to a timeless policy rules outthis time-inconsistent policy,we find that the optimal policy is price stability.The paper also makes a m

239、ethodological contribution.It introduces a new algorithmto compute optimal policies in heterogeneous-agent models.The algorithm leveragesthe numerical advantages of continuous time and will allow researchers to solve optimalpolicy in heterogeneous-agent models in an efficient and simple way using Dy

240、nare.The model presented in this paper abstracts from several relevant mechanisms driv-ing firm dynamics,such as endogenous default,size-varying capital constraints,frictionsin the labor market,or decreasing returns to scale,among many others.This allowsus to provide a clear understanding of the for

241、ces linking monetary policy with capitalmisallocation,as well as highlighting the similarities and differences with the standardNew Keynesian model.A natural extension would be to add more of these features tostudy their impact on the optimal conduct of monetary policy.39ReferencesAcharya,S.,E.Chall

242、e,and K.Dogra(2020).Optimal Monetary Policy According toHANK.StaffReports 916,Federal Reserve Bank of New York.1Achdou,Y.,J.Han,J.-M.Lasry,P.-L.Lions,and B.Moll(2021,04).Income andwealth distribution in macroeconomics:A continuous-time approach.The Review ofEconomic Studies 89(1),4586.1,3,5.1,C.1,49

243、,D.1Adam,K.and R.M.Billi(2006).Optimal monetary policy under commitment witha zero bound on nominal interest rates.Journal of Money,credit and Banking,18771905.4Adam,K.and H.Weber(2019).Optimal trend inflation.American Economic Re-view 109(2),70237.7Ahn,S.,G.Kaplan,B.Moll,T.Winberry,and C.Wolf(2018)

244、.When inequality mattersfor macro and macro matters for inequality.NBER macroeconomics annual 32(1),175.3,D.1Albrizio,S.,B.Gonzlez,and D.Khametshin(2023).A tale of two margins:Monetarypolicy and capital misallocation.Documentos de trabajo N.2302,Banco de Espaa.8,6,36,A.3Almunia,M.,D.Lopez Rodriguez,

245、and E.Moral-Benito(2018).Evaluating the macro-representativeness of a firm-level database:an application for the spanish economy.Banco de Espaa Ocassional Paper(1802).A.1Alvarez,L.J.,P.Burriel,and I.Hernando(2010).Price-setting behaviour in spain:evidence from micro ppi data.Managerial and Decision

246、Economics 31(2-3),105121.17Alvarez,L.J.,E.Dhyne,M.Hoeberichts,C.Kwapil,H.L.Bihan,P.Luennemann,F.Martins,R.Sabbatini,H.Stahl,P.Vermeulen,and J.Vilmunen(2006).Stickyprices in the euro area:A summary of new micro-evidence.Journal of the EuropeanEconomic Association 4(2/3),575584.17Armenter,R.and V.Hnat

247、kovska(2017).Taxes and capital structure:Understandingfirms savings.Journal of Monetary Economics 87,1333.A.140Asriyan,V.,L.Laeven,A.Martin,A.Van der Ghote,and V.Vanasco(2021).Fallinginterest rates and credit misallocation:Lessons from general equilibrium.Technicalreport.1,13,4.2Auclert,A.,B.Bardczy

248、,M.Rognlie,and L.Straub(2021).Using the sequence-spacejacobian to solve and estimate heterogeneous-agent models.Econometrica 89(5),23752408.3,C,D.1Auclert,A.,M.Cai,M.Rognlie,and L.Straub(2022).Optimal Policy with Heteroge-neous Agents:A Sequence-Space Approach.Mimeo.1Auclert,A.,M.Rognlie,and L.Strau

249、b(2020).Micro jumps,macro humps:Monetarypolicy and business cycles in an estimated hank model.Technical report,NationalBureau of Economic Research.47,53Baqaee,D.,E.Farhi,and K.Sangani(2021).The supply-side effects of monetary policy.Technical report,National Bureau of Economic Research.5,7,4.2Bau,N.

250、and A.Matray(2023).Misallocation and capital market integration:Evidencefrom india.Econometrica 91(1),67106.6,A.2Bernanke,B.S.,M.Gertler,and S.Gilchrist(1999,December).The financial acceler-ator in a quantitative business cycle framework.In J.B.Taylor and M.Woodford(Eds.),Handbook of Macroeconomics,

251、Volume 1 of Handbook of Macroeconomics,Chapter 21,pp.13411393.Elsevier.13Bhandari,A.,D.Evans,M.Golosov,and T.J.Sargent(2021).Inequality,businesscycles,and monetary-fiscal policy.Econometrica,forthcoming.1,5.1,D.2Bigio,S.and Y.Sannikov(2021).A model of credit,money,interest,and prices.Technical repor

252、t,National Bureau of Economic Research.1,5.1,D.2Bilbiie,F.O.,I.Fujiwara,and F.Ghironi(2014).Optimal monetary policy withendogenous entry and product variety.Journal of Monetary Economics 64,120.7Bilbiie,F.O.and X.Ragot(2020).Optimal monetary policy and liquidity with het-erogeneous households.Review

253、 of Economic Dynamics.141Blanchard,O.and J.Gali(2007).Real wage rigidities and the new keynesian model.Journal of Money,Credit and Banking 39,3565.1,5.3Boppart,T.,P.Krusell,and K.Mitman(2018).Exploiting mit shocks in heterogeneous-agent economies:the impulse response as a numerical derivative.Journa

254、l of EconomicDynamics and Control 89,6892.3,33,D.1Buera,F.J.and J.P.Nicolini(2020).Liquidity traps and monetary policy:Managinga credit crunch.American Economic Journal:Macroeconomics 12(3),11038.6Caglio,C.R.,R.M.Darst,and S.Kalemli-zcan(2021).Risk-taking and monetarypolicy transmission:Evidence fro

255、m loans to smes and large firms.Technical report,National Bureau of Economic Research.6Christiano,L.J.,M.Eichenbaum,and C.L.Evans(2005,February).Nominal Rigidi-ties and the Dynamic Effects of a Shock to Monetary Policy.Journal of PoliticalEconomy 113(1),145.5,4.2Christiano,L.J.,M.S.Eichenbaum,and M.

256、Trabandt(2016).Unemployment andbusiness cycles.Econometrica 84(4),15231569.3Cloyne,J.,C.Ferreira,M.Froemel,and P.Surico(2018,December).Monetary Policy,Corporate Finance and Investment.NBER Working Papers 25366,National Bureauof Economic Research,Inc.1David,J.M.and D.Zeke(2021).Risk-taking,capital al

257、location and optimal monetarypolicy.Technical report.7Dvila,E.and A.Schaab(2022).Optimal monetary policy with heterogeneous agents:A timeless ramsey approach.Working Paper.1,5.1,D.2Eggertsson,G.B.et al.(2003).Zero bound on interest rates and optimal monetarypolicy.Brookings papers on economic activi

258、ty 2003(1),139233.4,5.3Eggertsson,G.B.and M.Woodford(2004).Policy options in a liquidity trap.AmericanEconomic Review 94(2),7679.1Evans,C.L.(1992,April).Productivity shocks and real business cycles.Journal ofMonetary Economics 29(2),191208.5,4.242Ferreira,M.,T.Haber,and C.Rorig(2023,May).Financial C

259、onstraints and FirmSize:Micro-Evidence and Aggregate Implications.Working Papers 777,DNB.15Ferreira,T.R.,D.A.Ostry,and J.Rogers(2022).Firm Financial Conditions and theTransmission of Monetary Policy.Working Papers 2316,Cambridge.1Foster,L.,J.Haltiwanger,and C.Syverson(2008).Reallocation,firm turnove

260、r,and effi-ciency:Selection on productivity or profitability?American Economic Review 98(1),394425.A.1Gali,J.(2008).Monetary policy,inflation,and the business cycle:An introduction tothe new keynesian framework.Princeton University Press.5.2Garga,V.and S.R.Singh(2021).Output hysteresis and optimal m

261、onetary policy.Journal of Monetary Economics 117,871886.5,4.2Gautier,E.,P.Karadi,C.Conflitti,B.Fabo,L.Fadejeva,C.Fuss,T.Kosma,V.Jou-vanceau,F.Martins,J.-O.Menz,and T.Messner(2023,July).Price adjustmentin the euro area in the low-inflation period:evidence from consumer and producermicro price data.Oc

262、casional Paper Series 319,European Central Bank.17Gertler,M.and P.Karadi(2011).A model of unconventional monetary policy.Journalof monetary Economics 58(1),1734.2.1,2.1Gopinath,G.,.Kalemli-zcan,L.Karabarbounis,and C.Villegas-Sanchez(2017).Capital allocation and productivity in south europe.The Quart

263、erly Journal of Eco-nomics 132(4),19151967.1,3,4.2,A.2Hsieh,C.-T.and P.J.Klenow(2009).Misallocation and manufacturing tfp in chinaand india.The Quarterly Journal of Economics 124(4),14031448.1,36,A.2Itskhoki,O.and B.Moll(2019).Optimal development policies with financial frictions.Econometrica 87(1),

264、139173.4.1,5.2,B.8Jarociski,M.and P.Karadi(2020).Deconstructing monetary policy surprises:therole of information shocks.American Economic Journal:Macroeconomics 12(2),143.1,6,A.3,1143Jeenas,P.(2020).Firm balance sheet liquidity,monetary policy shocks,and investmentdynamics.Technical report,Universid

265、ad Pompeu Fabra.1,17Jord,.,S.R.Singh,and A.M.Taylor(2020).The long-run effects of monetarypolicy.Technical report,National Bureau of Economic Research.5,4.2Juillard,M.,D.Laxton,P.McAdam,and H.Pioro(1998).An algorithm competi-tion:First-order iterations versus newton-based techniques.Journal of Econo

266、micDynamics and Control 22(8-9),12911318.C,D.1Jungherr,J.,M.Meier,T.Reinelt,and I.Schott(2022).Corporate Debt Maturity Mat-ters For Monetary Policy.Discussion paper series,University of Bonn and Universityof Mannheim,Germany.1Kaplan,G.,B.Moll,and G.L.Violante(2018).Monetary policy according to hank.

267、American Economic Review 108(3),697743.3,B.3Koby,Y.and C.Wolf(2020).Aggregation in heterogeneous-firm models:Theory andmeasurement.Technical report.7Le Grand,F.,A.Martin-Baillon,and X.Ragot(2022).Should monetary policy careabout redistribution?Optimal monetary and fiscal policy with heterogeneous ag

268、ents.Technical report,Paris School of Economics.1,5.1,D.2Lian,C.and Y.Ma(2020,09).Anatomy of corporate borrowing constraints.TheQuarterly Journal of Economics 136(1),229291.14Mckay,A.and C.Wolf(2022).Optimal Policy Rules in HANK.Mimeo.1Meier,M.and T.Reinelt(2020).Monetary policy,markup dispersion,an

269、d aggregatetfp.Technical report,University of Bonn and University of Mannheim,Germany.5,7,4.2Midrigan,V.and D.Y.Xu(2014,February).Finance and misallocation:Evidencefrom plant-level data.American Economic Review 104(2),42258.1Moll,B.(2014).Productivity losses from financial frictions:Can self-financi

270、ng undocapital misallocation?American Economic Review 104(10),31863221.1,6,2.1,10,2.144Moran,P.and A.Queralto(2018).Innovation,productivity,and monetary policy.Journal of Monetary Economics 93,2441.5,4.2Nakov,A.et al.(2008).Optimal and simple monetary policy rules with zero floor onthe nominal inter

271、est rate.International Journal of Central Banking 4(2),73127.4Nuo,G.and C.Thomas(2022).Optimal Redistributive Inflation.Annals of Economicsand Statistics(146),363.1,5.1,D.1,D.2,D.3,D.4,16Ottonello,P.and T.Winberry(2020).Financial heterogeneity and the investmentchannel of monetary policy.Econometric

272、a 88(6),24732502.1,17,6,6,35,A.2,A.3,A.4Ragot,X.(2019).Managing inequality over the business cycles:Optimal policies withheterogeneous agents and aggregate shocks.2019 Meeting Papers 1090,Society forEconomic Dynamics.5.1Reis,R.(2013).The portuguese slump and crash and the euro crisis.Brookings Paper

273、son Economic Activity,143193.1Reis,R.(2022,December).Which r-star?measurement and policy implications.LSEmanuscript.B.12Reiter,M.(2009).Solving heterogeneous-agent models by projection and perturbation.Journal of Economic Dynamics and Control 33(3),649665.3,D.1Restuccia,D.and R.Rogerson(2017).The ca

274、uses and costs of misallocation.Journalof Economic Perspectives 31(3),15174.1Rotemberg,J.J.(1982).Sticky prices in the united states.Journal of Political Econ-omy 90(6),11871211.2.4Smirnov,D.(2022).Optimal monetary policy in hank.Technical report.1,5.1,D.2Trimborn,T.,K.-J.Koch,and T.Steger(2008).Mul

275、tidimensional transitional dynam-ics:A simple numerical procedure.Macroeconomic Dynamics 12(3),301319.C,D.1Winberry,T.(2018).A method for solving and estimating heterogeneous agent macromodels.Quantitative Economics 9(3),11231151.3,D.1,D.145Woodford,M.(2003).Interest and prices:Foundations of a theo

276、ry of monetary policy.Princeton University Press.1,5.2,5.346Online appendixAEmpirical AppendixA.1Firm level dataThe empirical exercise relies on annual firm balance-sheet data from the Central deBalances Integrada database(Integrated Central Balance Sheet Data Office Survey).We use an unbalanced pan

277、el of firms from 1999 to 2016,since these are the yearsfor which the monetary policy shocks are available.Being a detailed administrativedataset,the main advantage is that it covers the quasi-universe of Spanish firms(seeAlmunia et al.,2018 for further details on the representativeness of this datas

278、et).Weuse for our analysis only high quality observations,as defined by the Integrated CentralBalance Sheet Data Office.Our main variable of interest,firms marginal revenue product of capital(MRPK),is proxied by the log of the ratio of value added over tangible capital.38We drop firmsin the 5%upper

279、tail of the capital-weighted MRPK distribution,so as to focus onfirms holding a non-negligible capital share.Variables are deflated using industry pricelevels to preserve the firms price-level changes and consider a revenue-based measureof MRPK(Foster et al.,2008).The capital-weighted MRPK distribut

280、ion in the datais shown in blue bars in Figure 3.Our dependent variable,the investment rate(or capital growth),is defined as thedifference of firms tangible capital,in logarithm,between periods t and t1.We alsouse other firm-level information as controls in the robustness Section below.Profitsare co

281、mputed as net ordinary profits,defined as value added minus personnel costs,net financial revenue and depreciation.Leverage is computed as total debt(short-termplus long-term debt)divided by total assets,and it is trimmed below 0 and over 10.Net financial assets are constructed as the log difference

282、 between financial assets andfinancial liabilities,where financial assets include short-term financial investment,tradereceivables,inventories and cash holdings;and financial liabilities include short-termdebt,trade payables and long-term debt.We trim this variable below at-10 and aboveat 10.This va

283、riable controls for firms savings,following Armenter and Hnatkovska(2017).We proxy for size using the logarithm of total assets.Real sales growth is38Implicitly,this is restricts our sample to observations with positive value added.47defined as the log-difference of sales in two consecutive years,th

284、e previous and thecurrent one.39We use the production deflator for value added,sales,financial assetsand liabilities and the investment price deflator for capital and total assets.Profits arealso deflated using the investment price deflator(see more on this in Appendix A.4).The variables used in the

285、 regressions are winsorized at 0.5%.Descriptive statistics arereported in Table 3.Table 3:Descriptive statisticsmeansdp5p95Capital growth(1 period)-0.000.29-0.300.49Net operating profit growth(1 period)-0.011.11-1.891.84MRPK(logs)-0.871.39-3.570.73MRPK(levels)0.770.660.032.07Total Assets6.021.573.56

286、8.64Leverage0.310.370.000.97Net financial Assets0.070.50-0.710.68Sales growth0.131.41-0.500.82Observations5184233Notes:The table shows the mean(column 1),standard deviation(column 2),5th and 95th percentile value(column 3and 4 respectively)of the main variables used in the calibration and empirical

287、analysis.MRPK is shown in logs and inlevels.The table also displays total assets(logs),leverage,net financial assets;and the log difference of the capitalstock(capital growth),output(sales growth)and profits(profit growth).The number of observations are those forwhich the variable MRPK is available.

288、A.2Proxying MRPK and accounting for sectoral differences inthe capital shareA firms MRPK is no directly observable.However,in a Cobb-Douglas productionframework,such as the one presented in Section 2,a firms MRPK is proportional toits average revenue product of capital(ARPK):MRPKt k1tl1at ARPKt yt/k

289、t=k1tl1atWe thus use the easily measurable ARPK as an empirical measure for the unob-servable MRPK,following the literature(see for instance Bau and Matray,2023).To39Both sales growth and capital growth are winsorized at 0.5%.48account for the use of intermediate inputs,which we dont model explicitl

290、y,we usevalue added(sales minus intermediate inputs)instead of sales.However,capital shares may differ across sectors.This would imply that theARPK is no longer a valid proxy for the MRPK in cross sectoral comparisons.In thefollowing we explain that all our results are robust to this concern.We use

291、our MRPKproxy on four occasions.We discuss each of them in turn.Steady state MRPK distributionIn Section 3 we show that the model predicts asteady state MRPK distribution that is in line with the MRPK distribution implied bythe data,assuming a uniform capital share for all sectors.Here we relax this

292、 assumptionand allow for sectoral difference in the capital shares.Following Hsieh and Klenow(2009)and Gopinath et al.(2017),we take the sectoral capital shares of a relativelyundistorted economy such as the United States.As Figure 10 shows,the fit of the modelworsens only slightly in the direction

293、of underpredicting the measured misallocation.The baseline calibration can hence be considered conservative.Figure 10:MRPK distributionFirm level capital growthIn Section 6 we show that high MRPK firms respondmore strongly to monetary policy shocks.The first robustness we perform(column 2of Table 2)

294、includes sector-time fixed effects st.These soak up any differences betweenthe MRPK and the ARPK that sectoral differences in factor shares could introduce.49To show this,we reproduce our specification here,acknowledging explicitly that we usethe ARPK as a proxy for the MRPK and adding a sector inde

295、xlogkj,t logkj,t1=0+1log(ARPKj,s,t1)+2log(ARPKj,s,t1)t+s,t+uj,t,Using the above relationship between the MRPK and the ARPK ARPKj,s,t1=MRPKj,t1s,where sdenotes the sectoral capital share we can rewrite this equation aslogkj,t logkj,t1=0+1log?MRPKj,t1s?+2log?MRPKj,t1s?t+s,t+uj,t=0+1log(MRPKj,t1)+2log(

296、MRPKj,t1)t+1log(s)+2log(s)t+s,t+uj,t.Thus,the sector time-fixed effects stabsorb the differences in sectoral capital shares(the term in curly brackets)and the coefficients 1(2)can be interpreted as theassociation of MRPK(the interaction of MRPK with the monetary policy shock)withcapital growth,as we

297、 do in the main text.40Furthermore,note that in Appendix A.2 we include a specification with firm-levelfixed effects and MRPK demeaned at the firm level(following Ottonello and Winberry(2020),which accounts for differences in the capital shares even at the firm level.Dynamic weighted average MRPKIn

298、Section 6 we show that the log differenceof dynamic weighted average MRPK(logWAMt1,s)increases in response to amonetary policy shock.This variable is constructed by aggregating our MPRK-proxyat the sectoral level.Since we consider its log difference,sectoral differences in capitalshare swash out.Thu

299、s,sectoral differences in capital shares do not affect our resultsor their interpretation.Productivity processIn Appendix A.5 we use our MRPK proxy to estimate theproductivity process.To account for potential sectoral differences in capital shares,here we redo the estimation adding sector fixed effe

300、cts(which absorb sectoral capitalshares since the specification is in logs)or accounting for US sectoral capital sharesas previously explained.The estimates change only marginally from z=0.83 and40Following the same reasoning,even time variation in the sectoral capital shares is accounted for.50=0.7

301、3 to z=0.77 and =0.71,and z=0.80 and =0.72 respectively.41A.3Monetary policy shocksWe use the monetary policy shocks constructed by Jarociski and Karadi(2020).Thekey idea behind their identification strategy is that movements of interest rates andstock markets within a narrow window around monetary

302、policy announcements can helpdisentangle monetary policy shocks from information surprises.While an unexpectedpolicy tightening raises interest rates and reduces stock prices,a positive central bankinformation shock(i.e.unexpected positive assessment of the economic outlook)raisesboth.Their identifi

303、cation of monetary policy relies hence on sign restrictions:anunexpected monetary policy tightening raises interest rates and reduces stock prices,while an unanticipated positive information shock increases both.Each surprise changein interest rates is hence decomposed into a combination of central

304、bank informationshocks and monetary policy shocks.We use the latter,as provided by the authorswith the published paper.We use their monetary policy shocks at monthly frequency.Since our firm-levelpanel is at annual frequency,we aggregate the monthly monetary policy shocks followingthe scheme of Otto

305、nello and Winberry(2020).However,instead of aggregating dailyshocks into quarterly series,we apply a monthly-to-yearly transformation.This schemeaccounts for the fact that firms have less time to react to shocks happening at the endof the year then to shocks happening earlier on.In particular,a mont

306、hly shock entersboth the current year and the following years annual shock,with the split between thecurrent and the next year depending on the timing of the monthly shock within thecurrent year.42Concretely,we construct the monetary policy shock ast=Xmpast(m)m,t1+Xmcurrent(m)m,tpast(m)=m 112,curren

307、t(m)=12 (m 1)12where tis the aggregated annual monetary policy shock in year t,and m,tis thehigh-frequency shock in month m=1,.12 of year t.See(Albrizio et al.,2023)for aformal derivation of this weighting scheme.Note that we multiply the original shocks41Even firm level fixed effects do not change

308、the estimates much.42For instance,a high-frequency surprise happening in January is entirely attributed to the currentyear,while the one occurring in December mainly contributes to the following years annual shock.51by(-1),so that positive monetary policy shocks corresponds to expansionary monetaryp

309、olicy.Figure 11 shows the time series of the shock.Figure 11:Monetary policy shocks at annual frequency.Source:Jarociski and Karadi(2020)and own calculations.A.4Robustness of the firm level regressionIn this Section we perform several robustness of our finding that high-MRPK firmsinvestment responds

310、 more to monetary shocks.We consider variations of the mainempirical specification explained in the main text,equation(41),which we repeat hereexpanded to include the robustness specifications we perform belowlogkj,tlogkj,t1=0+1log(MRPKj,t1)+2log(MRPKj,t1)t+0Zj,t1+st+j+uj,t,(42)where Zj,t1includes a

311、 vector of lagged firm-level controls(total assets,sales growth,leverage,capital growth and net short term financial assets),and in some specificationit also includes their interaction with the monetary policy shock;stare sector-yearfixed effects;and jare firm-level fixed effects.Column(1)in Table 4

312、 reproduces the results of Column 2 of Table 2 of the maintext.It does not include firm-level nor aggregate controls,and it only includes sector-time fixed effects.Column(2)includes firm fixed-effects,and Column(3)also addsfirm-level controls(the lag of total assets,sales growth,leverage,capital gro

313、wth andnet short term financial assets).The results remain positive,significant and of similarmagnitude.Column(4)reports results for the same specifications as Column(3),but adding the interaction of log(MRPKj,t1)with lagged GDP growth,to rule outany heterogeneity in response to business cycle movem

314、ents.Column(5)adds to the52specification of column(4)the interaction of the monetary policy shocks with thefirm level controls.Results remain significant and quantitatively similar.Column(6)runs the same specification of Column(5),just replacing the main variable of interestlog(MRPKj,t1)with its dem

315、eaned value at the firm level,to make sure that resultsare not driven by permanent heterogeneity in MRPK levels,in the spirit of Ottonelloand Winberry,2020.Our results also survive to this strict specification.One of the advantages of our dataset is that it also includes small and privately heldfirm

316、s.But precisely because of this,it could be the case that small firms are the onesdriving these results and one may wonder if the same empirical pattern holds for largefirms.To address this concern,we replicate the analysis of our baseline specificationwith sector-year fixed effects,Column(1),but ke

317、eping only firms below the 90th per-centile of employment(Column 7),and keeping only firms above the 90th percentile(Column 8).The coefficient on the slope of MRPK is positive,significant,and evenquantitatively larger for larger firms.The 90th percentile of employment in the Spanishdistribution of f

318、irms is relatively low(15 employees),so we repeat the same regressionbut keeping only firms with at least 100 employees in Column(9),and reach the sameconclusion.Since our panel is highly unbalanced,we run our baseline specification,but only for firms that we observe for at least for 6 consecutive y

319、ears(from t 1 tot+4)(Column 10),hence restricting the sample as in the aggregate analysis performedin Section 4.2.The coefficient is nearly twice as large as that of Column(1).Sum-ming up,all these exercises point at the robustness of the empirical result of a higherheterogeneous response of investm

320、ent for high-MRPK firms to a monetary policy shock.Finally,we want to test directly whether high-MRPK firms profits are indeed in-creasing,in line with the theoretical predictions.We use as data counterpart net or-dinary net profits deflated by the capital investment deflator.43We estimate the samee

321、quation as that of Column 1 of Table 4,but with the log change in profits on theleft-hand side.The results are depicted in Column(11).As predicted by the model,the coefficient is positive and significant,providing further support for the mechanismpresented.43Results are robust to deflate profits usi

322、ng the production deflator rather than the capital investmentdeflator53Table 4:Robustness(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)logklogklogklogklogklogklogklogklogklogklognop mrpk0.02860.02170.02620.02700.03440.02750.06070.06790.04500.108(0.01)(0.01)(0.01)(0.01)(0.01)(0.01)(0.02)(0.02)(0.02)(0.04)mrpk0.

323、080(0.04)Observations36921883538432264165726416572641657264165732908244013594093212535051970415R20.0200.2550.2950.2950.2950.2950.0200.0350.0630.0330.026Time-sector FEYESYESYESYESYESYESYESYESYESYESYESFirm FENOYESYESYESYESYESNONONONONOFirm ControlsNONOYESYESYESYESNONONONONOAgg.ControlNONONOYESYESYESNO

324、NONONONOMP shock*Firm ControlNONONONOYESYESNONONONONOMRPK demeaningNONONONONOYESNONONONONOPanelFULLFULLFULLFULLFULLFULLEMPp90LARGEN 5FULLNotes:This table reports the results of estimating equation(42),departing from some of the specifications of the estimation in the main text.Column(1)includessecto

325、r-time fixed effects.Column(2)includes firm fixed-effects,and Column(3)also includes lagged firm-level controls(total assets,sales growth,leverage,capitalgrowth and net short term financial assets).Column(4)runs the same specification as Column(3),but adding the interaction of log(MRPKj,t1)and lagge

326、dGDP growth.Column(5)adds to the specification of column(4)the interaction of lagged firm level controls and the monetary policy shock.Column(6)runs thesame specification as Column(5),just replacing the main variable of interest log(MRPKj,t1)with its demeaned value at the firm level.Columns(7),(8)an

327、d(9)run the baseline specification,but including only firms with less employees than the 90th percentile(15 employees),firms with more employees than 90th percentile,and firms with more than 100 employees,respectively.Column(10)shows the baseline specification,but only for firms that we observe at l

328、east for 6 consecutiveyears(from t-1 to t+4).Column(11)runs the same specification as that of Column(1),but using the log change of profits as dependent variable.Standard errorsare clustered at the sector-year level.54A.5Estimating the process for idiosyncratic productivity zWe assume that individua

329、l productivity z in logs follows an Ornstein-Uhlenbeck processdlog(z)=zlog(z)dt+zdWt.To estimate this continuous time process on discrete data,we approximate it by anAR(1)process using an Euler-Maruyama approximationlog(zjt)=zlog(zjt1)+t,t N(0,zt),where z 1 zt exp(zt).In the model,firm level product

330、ivity z is proportional to firm level MRPKMRPKt(z)=ztUsing this we can rewrite the discretized process for z aslog(MRPKt(zjt)/t)=zlog(MRPKt1(zjt1)/t1)+t,t N(0,z)log(MRPKt(zjt)=zlog(MRPKt1(zjt1)+f(t,t1)+t,We estimate this equation using OLS on our panel data specified above,capturingthe term f(t,t1)b

331、y using year fixed effects.We find z=0.83,and the standarddeviation of the shock is =0.73.This estimate is robust to including sector fixedeffects to account for sectoral differences in capital shares(see Appendix A.2).As thedata frequency is annual,t=1,we back out the implied to continuous time par

332、ameterz=log(z)=0.189.A.6Derivation of the approximate correspondence of 4Ztand4WAMt,sWe define kt(z)=0k(z,a)gt(z,a)da,and at(z)=0agt(z,a)da.Manipulating thedefinition of TFP(32)by subsequently using the definitions of(zt),t(z)and the55linearity of kt(z)in at(z)when z zt,we getZ1/t=ztzt(z)dz(1 (zt)dz

333、=ztzt(z)ztt(z)dzdz=ztzat(z)/Atztat(z)/Atdzdz=0zkt(z)0kt(z)dzdz=0zkt(z)Ktdz.Now consider two points in time t and t+,where t 1 z,t,then dt=0 and the firm does not paydividends until it closes down.If this is the case,then the value of t(z)can be obtainedfrom(rt+)t(z)qt=qt+(maxztt Rt,0+Rt qt)t(z)+(z)qttz+2(z)2qt2tz2+(qtt)t.(43)Lemma.t(z)1 z,tProof.The drift of the entrepreneurs capital holdings issa