Aggregate Implications of Deviations from Modigliani-Miller: A Sufficient Statistics Approach

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) Aggregate Implications of Deviations from Modigliani-Miller: A Sufficient Statistics Approach Robert Kurtzman and David Zeke 2023-045 Please cite this paper as: Kurtzman, Robert, and David Zeke (2023). “Aggregate Implications of Deviations from Modigliani-Miller: A Sufficient Statistics Approach,” Finance and Economics Discussion Series 2023-045. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2023.045. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Aggregate Implications of Deviations from Modigliani-Miller: A Sufficient Statistics Approach∗ Robert Kurtzman†1 and David Zeke ‡1 1Federal Reserve Board of Governors June 27, 2023 Abstract A few sufficient statistics can identify the aggregate effects of distortions to firm investment in a class of general equilibrium models that can accommodate rich general equilibrium effects includingendogenousfirmentry. Thisresultdoesnotdependonthemicrofoundationofthedistortion;onecangenerateinferencesaboutaggregateeffectsthatapplyformultiplemicrofoundations orincaseswhereafullyspecifiedmodelisdifficulttosolve. Todemonstratetherelevanceofthe methodology,weuseittoquantifytheaggregateconsequencesofcostlyexternalequityfinancing and a manager-shareholder friction, relying on estimates from the corporate finance literature to identifythesufficientstatistics. Theresultselucidatedifferencesbetweenpartialandgeneralequilibrium findings and demonstrate how labor supply elasticities, complementarities in production, andfirmentryinteractwiththedifferentfirm-leveldistortions. Keywords: HeterogeneousFirms,GeneralEquilibrium,FirmEntry,AgencyCosts,Costly ExternalFinance,SufficientStatistics JELClassification: E22,E23,G39 ∗We are grateful for comments and suggestions from Marianne Andries, Andy Atkeson, Thomas Chaney, Andrew Kane, Pablo Kurlat, Kai Li, Romain Ranciere, Dejanir Silva, Lukas Schmid, Venky Venkateswaran, AlessandroVilla, andvariousconferenceandseminarparticipants. Theviewsexpressedhereinarethoseofthe authorsaloneanddonotnecessarilyreflectthoseoftheFederalReserveSystem. †FederalReserveBoardofGovernors,DivisionofResearchandStatistics,20thandConstitutionNW,Washington,D.C.20551;email: robert.j.kurtzman@frb.gov. ‡FederalReserveBoardofGovernors,DivisionofResearchandStatistics,20thandConstitutionNW,Washington,D.C.20551;email: davidzeke@gmail.com;correspondingauthor.

1 Introduction There is little debate in corporate finance that firms behave differently than implied by the Modigliani and Miller (1958) theorem or other frictionless capital markets benchmarks. Such deviations have been shown theoretically and empirically to have real effects; external financing, information, and agency frictions can distort capital structure and investment decisions away from what is privately optimal (Stein, 2003). While there are many studies that carefully estimatetheextentofthesefrictionsatthemicro-level,thereisoftenalackofconsensusregardingtheappropriatemodelforthemicrofoundationsofagivenfriction. Inthispaper,wederive a methodology for examining the aggregate consequences of such distortions to firm-level investment without having to specify the microfoundations of the friction. Given the production environment and model parameters, our methodology infers these consequences from a small set of sufficient statistics on the effects of the distortions. In turn, the method is easy to implement, especially compared with solving microfounded general equilibrium (GE) models of heterogeneousfirmssubjecttodistortions. The class of models to which our main results apply feature investment by heterogeneous firms that produce differentiated varieties of goods using capital and labor. Although our methodologyappliestomoregeneralenvironments,inthebaselinemodelinwhichwedemonstratethesufficientstatisticsresult,firmsareexposedtoidiosyncraticshocks,bothrandomwalk and i.i.d. (to capital quality and productivity, respectively), but are not exposed to aggregate shocks.1 Firms are also heterogeneous in their capital stock, productivity, and the distortions they face (and thus any state variables that govern the evolution of those distortions). The model nests—at the micro level—the canonical Q-theory of investment in the face of convex capital adjustment costs where firm revenues exhibit homogeneity of degree one in the factors ofproductionfollowingHayashi(1982). Ourapproachisuniqueinthatwedonotchooseaparticularfirm-leveldistortiontoinvestment to model. Instead, we show that many distortions can be represented as the difference between the actual investment rate of a firm and the investment choice an otherwise identical firm(absentdistortions)inthesameenvironmentwouldmaketomaximizethevalueofprofits lessinputcosts(laborandinvestment). Thisdifferenceiscloselyrelatedtoempiricalestimates ofthecausaleffectsofdistortionsonratesoffirminvestment. We use the model to examine the following counterfactual: starting from an equilibrium withfirm-leveldistortions,howdoaggregateschangeifweeliminatethedistortionsandadjust taxes to keep the present value of net government revenues from corporate taxation constant?2 Our main result, Proposition 1, shows that by simply knowing the production and preference 1Weshowhowtheseassumptionscanberelaxedinvariousextensions. 2Thisassumptionisimposedtoallowdistortionstohaverichtaxconsequences,as,forexample,somecapital structuredistortionshave.Ifentryisexogenousor,alternatively,thedistortionhasnoadditionaltaxconsequences, thenthisassumptionofrevenueneutralityisnotrequired. 2

parameters in this model, along with two sufficient statistics—the capital-weighted mean and mean-squareddistortionstoinvestmentrates—wecanobtainboundsonhowaggregateobjects (output, consumption, welfare, etc.) behave in this counterfactual, in both the long run and the transition path from the equilibrium with firm-level distortions to the one without them. This inference is exact in special cases in which entry is exogenous or there is no time discounting. Importantly, distortions to investment rates due to a given friction are exactly what many corporatefinancepapers,suchasHennessyetal.(2007),estimate.3 Inturn,giventhisassumption, in estimating our sufficient statistics, we can take many estimates from the corporate finance literaturethatalsorelyonQ-theoryapproaches“offtheshelf.” The intuition for this result is as follows: the aggregate equilibrium conditions can be simplified so that firm-level distortions appear in just three equations: (1) aggregate capital accumulation, (2) aggregate investment clearing, and (3) the expected present value of an entering firm. Aggregate capital accumulation intuitively depends on the distortion to the capitalweighted investment rate. The amount of resources devoted to investment depends on both the aggregate investment rate and the misallocation of investment across firms, which, with quadraticadjustmentcosts,aresimplyafunctionofourtwosufficientstatistics. Thatthesufficient statistics allow us to characterize the entering firm’s value is less intuitive. A key insight in our proof is that although the distortions can hit a firm at any point in its lifetime, one can bound the effect of distortions on firm value by using moments of the cross-sectional distribution of firms that are reflected in the sufficient statistics. We can thus bound (or, in special cases,exactlyidentify)theaggregateeffectsofthedistortionevenwithendogenousentry. WefurthergeneralizeProposition1byrelaxingthemorestringentassumptionsmadeinthe model in a number of corollaries, as our approach can be extended to a wider class of models, though often at the cost of modified or additional sufficient statistics. In particular, our extensions include allowing for more general investment cost functions, relaxing the assumption of homogeneity of degree one of firm revenues, allowing for aggregate shocks, working in more general aggregation environments, and allowing for further dimensions of firm heterogeneity. Further, our generalizations can allow for distortions to impart deadweight or financial effects onthefirmaboveandbeyondthosecausedbydistortionstoinvestment. We use our sufficient statistics approach in the baseline model to quantify the aggregate consequencesofdistortionstoinvestmentduetoexternalfinancingandashareholder-manager friction using estimates from the corporate finance literature. The external financing friction is one of underinvestment, while the manager-shareholder friction is one of overinvestment. We calibrate the sufficient statistics to corporate finance papers that estimate the effects of these 3That firm revenues exhibit homogeneity of degree one in the factors of production in the baseline model is keytowhythesufficientstatisticsareabletobeexpressedintermsofdistortionstoinvestmentrates. Weassume thatdistortionsaffectfirmvalueonlythroughtheireffectsonfirminvestmentandtaxation,thoughwerelaxthis assumptioninanextension. Specifically,intheextension,weallowdistortionstoalsoleadtodeadweightlosses viathedestructionofcapitalinbankruptcy. 3

distortions on firm investment, and we calibrate the other model parameters to standard values intheliteratureonmacroeconomicmodelswithfirmheterogeneity.4 Our quantitative results yield several insights about the macroeconomic costs of these frictionsandthevalueofourmethodology. First,weshowthattheboundsgeneratedbyourmethod are tight, especially for the change in output and welfare. Second, the model features both GE dampening (labor supply, entry) and amplifying forces (complementarities in production) that operatethroughaggregateprices. Quantitatively,thedampeningforcesdominate: Forbothexternal financing costs and managerial miscalibration, the effect on aggregate output is an order of magnitude smaller than what a partial equilibrium (PE) counterfactual (in which entry and allaggregatepricesarefixed)wouldsuggest. Third, we show that the presence of aggregate inefficiencies can meaningfully change the welfare effects of resolving these distortions. We consider positive corporate taxation and a monopoly markup distortion. Absent these inefficiencies, resolving the distortion leads to the planner’s equilibrium, and thus the welfare gains from resolving the distortion are positive. However, in the presence of these inefficiencies, we find that eliminating managerial miscalibration reduces aggregate welfare in our counterfactuals because of the interaction of the inefficiencies with the firm-level distortion. Eliminating external financing frictions increases welfare,eveninthepresenceoftheseaggregateinefficiencies.5 Fourth, the effect of reallocation of investment can account for a meaningful share of the effect of distortions on aggregates. When we decompose changes in output into changes in the expenditures on investment and the efficiency of investment, we find that the efficiency of investmentcanaccountforinexcessofone-thirdofthechangeinoutput. Anadvantageofourmethodologyisthatifwechangethecalibrationofagivenparameter, we do not need to alter the calibration of the sufficient statistic, and thus it is easy to perform robustnesswithrespecttothevalueofthesufficientstatisticsandmodelparameters. Weassess the effects of resolving the distortion across large ranges for the sufficient statistics and model parameters and show that the qualitative takeaways are strikingly robust across these different robustness exercises. When we alter the modelling assumptions to have fixed instead of free entry or to allow productivity to follow a Markov process, the results are again qualitatively consistentwithourbaselineresults. Our work is related to the growing literature studying policies with sufficient statistics ap- 4We rely on an estimate of the effects of external financing costs on investment from Hennessy, Levy, and Whited(2007). Theauthorswritedownadynamicmicroeconomicmodelwithinvestmentandcapitalstructure and estimate it to examine the effects of external financing and agency frictions on investment. We rely on an estimate of the effect of a manager-shareholder friction on investment from Ben-David et al. (2013). The authorsexaminemanagers’abilitytopredictthepotential outcomes fortheS&P500. Broadly, theauthorsfind that managers are indeed miscalibrated and show that their long-term miscalibration measure is correlated with overinvestment. 5Themonopolymarkupinefficiencyleadstoinefficientlylowproductionbyfirms,whilepositive(net)corporatetaxationdistortsfirmentry. Removingthefirm-leveldistortionschangestheamountofentryandmassoffirm capitalinequilibrium,whichinteractswiththeseinefficiencies. 4

proacheswithinmacroeconomicmodels.6 Inacloselyrelatedpaper,SraerandThesmar(2021), develop a sufficient statistics approach to assess the macroeconomic implications of policy experimentsthataffectmisallocation. Ourpaperdiffersinitsabilitytoaccommodateendogenous entry and in the class of models and microeconomic evidence it can use to derive aggregate implications. In particular, our methodology is targeted to models where firm revenues exhibit constantreturnstoscale,whereestimatesoftheeffectofdistortionsoninvestmentrates(which are widely estimated in corporate finance) can be used to infer the sufficient statistics. Their methodology is targeted at the misallocation literature, where firm revenues or value added exhibits decreasing returns to scale and the effect of policies on allocations (output-to-capital ratios,inparticular)isinformativeaboutaggregateeffects.7 BaqaeeandFarhi(2020)developa flexibleand generaltheoryofaggregation indistortedeconomies. Thereare keydifferencesin ourapproach: First,“wedges”ininvestmentrates(thedifferencebetweentherealizedandfirstbestinvestmentrate)areendogenousinourmodelduetoGEeffects. Second,ourapproachallowsustoconstructboundsonthetransitionpath,and—givenoursufficientstatistics—bounds on the aggregate consequences; we are thus solving the full non-linear model for our exercises as opposed to relying on second-order approximations or making the assumption that distortions are small. Atkeson and Burstein (2019) develop a sufficient statistics approach to study theimplicationsofinnovationpolicyformacroeconomicandwelfaredynamicsinabroadclass ofinnovationmodels.8 Ourpaperdiffersinthatwefocusontheeffectofheterogeneousdistortions to firm investment and the statistics needed to assess the macroeconomic implications of removing those distortions. Recent work by Iachan, Silva, and Zi (2022) also use a sufficient statisticapproachtoanswerquestionsattheintersectionofmacroeconomicsandfinance. They assess the welfare implications of under-diversification on aggregate welfare in a framework with endogenous investment. Their methodology differs in that it focuses on this particular distortionandusessufficientstatisticsestimatedfromassetprices. Ourworkisalsorelatedtotheliteratureexamininghowinformationandfinancingfrictions matterfortheaggregateeconomyusingmomentsfrommicroeconomicdataingeneralequilibrium macro models. In a few important recent examples, Catherine et al. (2022) quantify the 6Thisliteratureincludesbothpapersthatrequireonlyreduced-formelasticities,followingChetty(2009),and those that combine sufficient statistics with some macroeconomic parameters. Examples include approaches to misallocationofinputs(HsiehandKlenow2009;DavidandVenkateswaran2019;BaqaeeandFarhi2020),transmissionofmonetarypolicywithheterogeneoushouseholds(Auclert,2019),andaggregatingtechnologyandother shocks(BaqaeeandFarhi,2019). 7Themisallocationinourmodelisofinvestmentrates,notofthelevelofinputs. Weshowourmethodcanbe extendedtoamodelwithdecreasingreturnstoscale: relativetoSraerandThesmar(2021),thisextensionofour methodhasthebenefitofaccommodatingendogenousentryatthecostoflessflexiblecounterfactualsandsome additionalassumptions. 8Whileourpaperisnotaninnovationmodel,ourmodelandmethodologycanberecastassuch,wherefirms choose investment in firm productivity. We choose the formulation as a model of capital investment because of the breadth and quality of estimates of distortions on firm capital investment. While the theory in Atkeson and Burstein(2019)allowsformeaningfulgainsfromeffectivelyreallocatinginvestmentfromlesstomoreproductive uses,theirmeasurementexerciseappliestopoliciesthataffectfirminvestmentproportionally. 5

aggregate effects of collateral constraints, and Terry (Forthcoming) studies how short-termism affects economic growth. We view our approach as complementary to these approaches. One relative benefit of our method is that it does not rely on precisely specifying the microfoundations of the distortion; a traditional model which is calibrated to match the sufficient statistics from the micro data will, as we prove in our main proposition, provide an estimate within the bounds that our approach generates. A second relative benefit is the ease of implementation: our method allows for much simpler and computationally faster implementation than solving a heterogeneous-firm GE model with micro-founded distortions, especially those that include realisticassumptions onthe dynamicsof firmfinancing oragency problems. However, thetraditionalstructuralapproachallowsformorecounterfactuals,suchasmerelyreducingtheextent ofthefrictionratherthaneliminatingitcompletely.9 Furthermore,thetraditionalstructuralapproachalsoallowsforproductionenvironmentsthatdonotsatisfytheassumptionsrequiredfor ourapproach. The rest of the paper follows as such. Section 2 presents the baseline model. Section 3 outlinesthesufficientstatisticsresultandtheextensions. Section4presentsourcalibrationand thequantitativeresults. Section5assessestherobustnessoftheresults. Section6concludes. 2 Model In this section, we describe the model. For ease of exposition, we set up the production environment as it will be implemented in the quantitative analysis, rather than outlining the most general class of models for which our approach can be implemented. In Section 3.3 we discuss the broader class of models for which our sufficient statistics approach holds and additionalextensionstomodelswithricherheterogeneity,aggregateshocks,andfirmrevenues thatexhibitdecreasingreturnstoscaleininputs,orendogenousgrowth. We start with the production environment and the problem of entering firms. We then review the household’s problem and define an equilibrium. In the model, time is discrete and indexedast = 0, 1, 2, .... 2.1 Production Environment A continuum of intermediate good firms, indexed by j, produce output, y. Intermediate good firm output is combined into the final good, Y, using a constant elasticity of substitution (CES)productionfunction: (cid:18)(cid:90) (cid:19) ρ ρ−1 ρ−1 Y t = y j,t ρ dj , (1) j 9Inthisvein,inKurtzmanandZeke(2018),webuildamacroeconomicmodelwithheterogeneousfirmswith amicrofoundationforwhyfirmsissuedebt,anexplicitlydefineddebtcontract(withstrongassumptionstoretain tractability),andassumptionsonthebankruptcyprocess. Weusethismodeltoexaminehowchangesintaxpolicy interactwiththedebtoverhangproblem. 6

whereρ > 0. Eachfirmproducesoutputusingcapital,k,andlabor,l,withproductionfunction y = A z kα1lα2, (2) j,t t j,t j,t j,t where A is a common level of productivity across firms; z is firm idiosyncratic productivity which is i.i.d. across firms; and the production function parameters are {α > 0,α > 0}.10 1 2 Costminimizationbyfinalgoodfirmsallowsustoobtainintermediategoodfirmrevenues: y j,t p j,t = P t Y t ρ 1 (A t z j,t l j α , 2 t k j α ,t 2) ρ− ρ 1 , (3) wherep isthepriceoffirmj(cid:48)soutputandP isthepriceofthefinalgood. Firmschooselabor j within each period to maximize static operating profits (revenues less labor costs Wl ). We j assume that α = ρ − α ; therefore, firm revenues are homogeneous of degree one in the 1 ρ−1 2 factors of production, as in Hayashi (1982). Because revenues are homogenous of degree one, firmlabordemand,revenues,andoperatingprofits(π )canbeexpressedas j l j,t = k j,t (cid:32) α 2 ( ρ− ρ 1 )(A t z j,t ) ρ− ρ 1 (cid:18) W P t (cid:19)−1 Y t ρ 1 (cid:33)αc , (4) t z ρ− ρ 1αcΠ j,t t y p = k , j,t j,t j,t 1−α ρ−1 2 ρ π = z ρ− ρ 1αck Π , j,t j,t j,t t respectively, where α = (1/(1−α ρ−1)) and Π is a measure of profits per unit of effective c 2 ρ t capitalthatcanbewrittenasafunctionofparametersandaggregates: (cid:32) ρ−1 1 (cid:18) W (cid:19)−α2(ρ− ρ 1)(cid:18) ρ−1 (cid:19)α2(ρ− ρ 1) (cid:33)αc(cid:18) ρ−1 (cid:19) Π = P A ρ Y ρ t α ( ) 1−α ( ) . (5) t t t t P 2 ρ 2 ρ t Capital Investment Capital investment at time t increases the capital stock at time t + 1, and capital depreciates at rate δ. Each firm’s capital stock is exposed to random walk shocks, modelled as i.i.d shocks (cid:15)k (capturing capital quality or exogenous exit) with mean κ to the capitalaccumulationequation. Formally,thecapitalaccumulationequationis k = (cid:15)k (k (1−δ)+I ), (6) j,t j,t j,t−1 j,t−1 where I is the amount of investment by firm j. In the spirit of the Q-theory of inj,t−1 vestment (Hayashi, 1982), there are adjustment costs in investment: investment rate I costs k (cid:20) (cid:21) 10Wenormalizez sothatitsconvexity-adjustedexpectation,E z ρ− ρ 1αc ,isequaltounity. Weshowhowthe j,t sufficientstatisticschangewhenwerelaxthei.i.d. assumptioninSection3.3. 7

(cid:18) (cid:19) (cid:16) (cid:17)2 k I + θ I −θ ˜ unitsofthefinalgoodforsomeparameterθ ˜ (whichistypicallysetto0 k 2 k c c orδ). Theexpectedgrossgrowthrateofthefirm(beforecapitalqualityshocks),i ,is j,t−1 I j,t−1 i = k (1−δ)+ . (7) j,t−1 j,t−1 k j,t−1 This definition reduces the length of some expressions; note that i is proportional in the j,t−1 investment rate of the firm. We can therefore express the cost of this growth rate i as kφ(i), whereφ(i) = i−(1−δ)+ θ (i−θ )2 andθ = θ ˜ +(1−δ). 2 c c c 2.2 Firm Problem and Distortions Firms’ investment choices may be subject to distortions. Without loss of generality, we representdistortionsrelativetotheinvestmentofanundistortedfirm. 2.2.1 UndistortedFirmProblem We begin by modelling the problem of an undistorted (“Modigliani-Miller”) firm that has (cid:93) value VMM. This is a firm that does not face firm-specific distortions to its investment choice andexpectsthatitneverwill. Notethatthisfirm’svalueandchoicesdependonaggregatestate prices, and the level of those aggregate state prices may be influenced by distortions in the economy. We define the undistorted firm’s value function as the present discounted value of revenueslesslaborandinvestmentexpenses: (cid:16) (cid:17) V j M ,t M(k,z) = maxk j,t Π t z ρ− ρ 1αc −k j,t φ(i)+κE t [Λ t+1 V j M ,t+ M 1 (k(cid:48),z(cid:48))], (8) i whereΛ isthehousehold’sstochasticdiscountfactor. Thesolutiontothismaximizationprobt lem yields the investment choice of the firm and the value of such an undistorted firm. Since the firm value function is homogeneous of degree one in the firm’s capital stock, we can write the value function as VMM(k,z) = k qMM(z), where qMM(z) is the marginal value of capital j,t j,t t t to an undistorted firm. Additionally, we can write qMM as the value function for a firm whose t idiosyncraticshockisequaltotheaveragelevel: qMM = maxΠ −φ(i)+iκE [Λ qMM]. (9) t t t t+1 t+1 i By solving (9), we can also characterize the undistorted optimal investment rate choice, i , MM,t which is common across firms regardless of their level of capital stock or realization of their idiosyncraticshock: φ(cid:48)(i ) = κE [Λ qMM]. (10) MM,t t t+1 t+1 8

Entry and Taxation (without firm-level distortions) There is an equilibrium condition that (cid:0) (cid:1) governs entry, Θ ME,VE = 0, where ME is the mass of entry and VE is the value of an t t t t entering firm with one unit of capital. This is a general specification that nests both free entry andexogenousentry. Forinstance,inthecaseoffreeentry,thisconditionis  VE −c if VE ≥ c Θ(ME,VE) = e e M if VE < c . e e In other words, firms enter if the value of entering exceeds the cost; therefore, firm value must be equal to the cost if firms enter in equilibrium. The condition is even simpler in the case of exogenousentry: Θ(ME,VE) = ME −ME,whereME isanon-negativeconstant. We assume that firm taxes or subsidies may be on revenues less expenses, lump sum to all firms,orlumpsumonentry(andthisscheduleisidenticalacrossfirms);therefore,theydonot directly affect the undistorted investment problem. However, taxation can affect the value of anenteringfirmandcanthereforedistortentry.11 2.2.2 DistortedFirmProblem We define ∆i = i − i as the difference between a firm’s investment rate and the j,t j,t MM,t undistorted firm’s actual investment rate, holding all aggregate prices fixed. This is thus a PE counterfactual: itmeasureshowafirm’sinvestmentchoiceisaffectedbyitsdistortion,holding thechoicesofallotherfirmsandaggregatesfixed.12 Thisdistortionmaydependonavectorof firmvariables,whichwerepresentbyχ. Forexample,χmayrepresentcapitalstructurefactors, real or financial shocks to the firm, or other state variables relevant to agency problems within thefirm. Thus,wecanwritei = i (χ ),andexpressthe(pre-tax)valueofthedistortedfirm j,t t j,t asthediscountedpresentvalueofrevenueslessexpenses: VF(k,χ) = z ρ− ρ 1αck Π −k φ(i (χ ))+κE [Λ VF (k(cid:48),χ(cid:48))]. (11) j,t j,t j,t t j,t t j,t t t+1 t+1 Examples of Distortions To provide intuition for the representation of distortions to firm investment,weshowhowdistortionsinstandardmodelscanbeexpressedas∆i . j,t Example 1: Costly External Financing Consider a specification of distortions where firms mayfacecostsoffinancingwithexternalequity,asinHennessyetal.(2007). LetX denotenet equity payouts; C, firm cash holdings; R , the interest rate on cash holdings; G(X), the cost C 11Withfreeentry,ifpositive(inpresentvalueterms)revenueisraisedonfirmtaxationlesssubsidies,thereis aneffectivetaxonentry(reducingentryinequilibrium). Theentrysubsidythatwouldoffsettheeffectofpositive taxesonthefirmwouldleadthenetrevenueraisedtobezeroinpresentvalueterms. 12This specification is important for measurement. The difference between the investment of a firm and the investmentitwouldmakewereittohaveadifferentdistortion, holdingwhatotherfirmsdofixed, isexactlythe sortofcounterfactualthatempiricalstudiestrytomeasure.Bycontrast,thedifferencebetweenafirm’sinvestment andwhatitwouldinvestinthefirst-bestequilibrium(withdifferentprices)isnotgenerallyobservable. 9

of external financing; and let Φ(X < 0) be an indicator variable that is positive if firms issue equity. Theoptimizationproblemthatcharacterizesfirminvestmentcanbewrittenasfollows: V (z,k,c ) = maxX +E [Λ V (z(cid:48),k(cid:48),c)] s.t. t −1 t t+1 t+1 I,X,c k(cid:48) = (cid:15)k(k(1−δ)+I) (cid:32) (cid:33) θ (cid:18) I (cid:19)2 (cid:18) X (cid:19) c = Π t kz ρ− ρ 1αc − I + I −θ c −X −kG Φ(X < 0)+c −1 R C . 2 k k Theresultingequilibriumconditionforinvestmentintermsofi (grosscapitalgrowthrate)is t 1 1 i (z,k,c ) = (1−δ)+θ − + q (1+G(cid:48)(x)Φ(x < 0)), t −1 c (cid:98)t θ θ (cid:104) (cid:105) where q = E Λ (cid:15)k∂Vt+1 (z(cid:48),k(cid:48),c) is the forward-looking (expected) discounted marginal (cid:98)t t t+1 ∂k(cid:48) value of capital to the firm and x = X is the net equity payout scaled by the capital stock. We k canthuswritethedistortiontotheinvestmentrate,∆i = i−iMM,as 1 (cid:0)(cid:0) (cid:1) (cid:1) ∆i (z,k,c ) = q −qMM +qG(cid:48)(X)Φ(x < 0) , (12) t −1 θ (cid:98)t (cid:98)t (cid:98) where qMM = κE [Λ qMM] is the forward-looking (expected) discounted marginal value of (cid:98)t t t+1 t+1 capital to an undistorted firm in this equilibrium (e.g., if there was one firm that did not suffer from external financing costs).13 Note that all terms in (12) are taken holding aggregate prices fixed in the equilibrium with distortions. Thus, this is a cross-sectional measure: how do the investmentsofadistortedandundistortedfirm,inthesameequilibrium,differfromeachother? This distortion can occur either via distortions that affect a firm’s investment Euler equation (e.g., G(cid:48)(X)Φ(x < 0)) or via distortions that affect firm q (e.g., expectation of future costly externalfinancing),thoughthislattereffectcanbeshowntobesecondorder.14 Example 2: Manager Miscalibration Consider distortions to managers’ expectations of the futurevalueofcapital. Thatis,thetrueproblemhasvalue (cid:16) (cid:17) V t (z,k,m) = Π z ρ− ρ 1αc K −Kφ(i(z,k,m))+E t (cid:2) Λ t+1 V t+1 (z(cid:48),ki(cid:15)k,m(cid:48)) (cid:3) , (13) and thus the forward-looking (expected) discounted marginal value of firm capital is q = (cid:98)t (cid:104) (cid:105) E Λ (cid:15)k∂Vt+1 (z(cid:48),k(cid:48),m(cid:48)) . However, managers’ expectations are distorted and they make t t+1 ∂k(cid:48) choicesgiventhefollowingdistortedEulerequation: φ(cid:48)(i(z,k,m)) = q +m, (14) (cid:98)t 13The κ must adjust the realized marginal q for the expectation of capital quality shocks (which also nest exogenousexitshocks). 14WediscusstheseeffectsandtheirquantificationinSection4.1. 10

where m indexes the distortion to their expectations. Plugging in for our assumptions of a quadraticadjustmentcostsyieldsthefollowingformsforiand∆i: 1 1 m i (z,k,m) = (1−δ)+θ − + q + , t c (cid:98)t θ θ θ 1 (cid:0)(cid:0) (cid:1) (cid:1) ∆i (z,k,m) = i−iMM = q −qMM +m . (15) t (cid:98) (cid:98) θ Taxation (with distortions) The distortions to investment can have tax implications. The following identity disaggregating the value of the entering firm with one unit of capital, which isolatesthetaxandinvestmentdistortions,isparticularlyuseful: VE = qMM +VD +VTaxes, (16) t t t t whereqMM isthepresentdiscountedvalueoffirmprofits(revenueslessexpensesonlaborand t capitalinvestment)ofanundistortedfirm,VD < 0isthepresentdiscountedvalueoftheeffect t ofthedistortionsonfirmprofits,andVTaxes isthediscountedpresentvalueofsubsidiespaidto t thefirmlesstaxes.15 2.3 Households Households own all firm liabilities and provide labor for production. Each period, after shocksarerealized,householdsmakeaconsumptiondecision,C ,andalaborsupplydecision, t L , to maximize their utility function, (cid:80)t=∞βtU(C ,L ), where β is the household’s discount t t=0 t t factor. Theymaximizetheirutilitysubjecttoabudgetconstraint: P C ≤ W L +F +T , t t t t t t where F are the net profits paid by firms less entry costs and T are transfers from the governt t ment.16 Thehousehold’sproblemresultsinthestandardlabor-leisureconditionof W t U = −U , (17) C,t L,t P t andastochasticdiscountfactorof U C,t Λ = β . (18) t U C,t−1 15For tractability we consider a model with exogenous exit, but our method could be generalized to allow for some types of endogenous exit, such as the specification in Gomes et al. (2016). However, it would need an additional sufficient statistic with information about how the distortion affects exit, as in our extension with deadweightlossesdescribedinSection3.3. 16Thesetransfersareequaltothepresentvalueofanytaxesraisedsothatthebudgetconstraintandconsumption marketclearingarejointlysatisfied. 11

2.4 Clearing and Equilibrium (cid:82) (cid:82) Capital and labor clearing imply K = k dj and L = l dj; therefore, equation (4) t j j,t t j j,t implies (cid:32) ρ−1 ρ−1 (cid:18) W (cid:19)−1 1 (cid:33)αc L = K α ( )A ρ t Y ρ . (19) t t 2 ρ t P t t Aggregatecapitalaccumulationbecomes (cid:90) K = ME +κ (k (i −ξ ))dj. (20) t t j,t j,t j,t j Defineaggregateinvestmentasthesumofincumbentinvestmentandentrycosts: (cid:90) I = k φ(i )dj +MEc . (21) t j,t j,t t e j Finalgoodclearingthusimplies C = Y −I . (22) t t t Plugging(4)into(3)andintegratingoverfirms,wecanobtainaggregateoutputas Π t Y = K . (23) t t (cid:16) (cid:17) P 1−α ρ−1 t 2 ρ The distribution of operating firms at time t, Γ (k,χ), evolves over time as a function of the t exogenous exit rate, κ; endogenous capital destruction, ξ (k,χ); the investment choices of t i (k,χ) by incumbent firms; and the mass of entering firms each period, ME. It is useful to t t define the capital-weighted steady-state distribution of the capital structure state vector in the stationaryequilibriumasγK(χ). Formally, (cid:82) kΓ(k,χ) γK(χ) = k . (24) (cid:82) (cid:82) kΓ(k,x) k x WedefineanequilibriuminAppendixA.1. 3 Sufficient Statistics Results This section outlines and proves the key result in the paper: we can either exactly characterize or bound the aggregate effects of a revenue-neutral policy counterfactual in which the firm-leveldistortionstoinvestmentandcapitaldestructionareundonewithasimplesetofsufficient statistics and the parameters of the model. We first define the counterfactual exercise of 12

interest, state the main proposition, and sketch its proof. We then outline the broader class of modelstowhichourresultappliesanddiscussextensions. 3.1 Proposition and Discussion We define a revenue-neutral counterfactual as one in which the distortions to investment, indexedby∆i(k,χ),areremovedandtaxesonfirmsareadjustedsothat—foreachgeneration offirmsthatenter—thetotalpresentvalueofnettaxrevenuethatfirmsgenerateisunchanged.17 Proposition1. Assumeweknowthefollowing: (cid:82) (i) Averagecapital-weighteddistortiontofirminvestment: i = ∆i(χ)γK (χ)dχ ∆ χ (ii) Averagecapital-weightedsquareddistortiontofirminvestment: i = (cid:82) (∆i(χ))2γK (χ)dχ ∆2 χ (iii) Presentvalueoftaxesnetofsubsidiesperperiod (iv) Modelparameters: β,κ,α ,α ,θ,ρ,andtheparametersthatentertheutilityfunction 1 2 Then our environment leads to a system of equations that identifies bounds on the change (cid:8) (cid:9) in aggregate quantities and prices C,Y,K,I,W,Λ,Π,ME,VE,qMM,i when firm-level MM distortionsareremovedinarevenue-neutralcounterfactual,inboththelongrunandalongthe transitionpath. If, in addition, either β → 1 or entry is exogenous, then the system of equations identifies the exactchangeintheseaggregatequantitiesandprices,ratherthanbounds,bothinthelongrun andalongthetransitionpath. Proof. SeeAppendixA.2. This proposition clarifies the set of sufficient statistics that can characterize the revenueneutral welfare costs of a firm-level distortion. These sufficient statistics are not abstract objects; they can be mapped to estimates widely used in the literature. For example, in the case of costly external financing, Hennessy et al. (2007) provide estimates of the effect of equity financing on firm investment (adjusting for average Q). This elasticity can be combined with firm-level information on equity financing and Q to yield estimates of the (capital-weighted) mean underinvestment due to costly equity financing and the mean-squared underinvestment due to costly equity financing. The value of (iii) can easily be calibrated to aggregate statistics from the national income and product accounts (NIPA) on the total profits lost to taxes net of subsidies as a fraction of corporate profits. The remaining parameters (iv) are standard in the macroliterature. 17Inturn,insteadystate,thepresentvalueofrevenueraisedfromfirmtaxationisconstant. 13

3.2 Sketch of Proof of Proposition Though we leave the formal proof to Appendix A, we walk through the key steps of the proof here, as they are useful for developing intuition about the result. We do this in two parts. First, in the case with exogenous entry, we show that all of the aggregate equations exceptforthecapitalaccumulationandaggregateinvestmentequationsrelyonlyonaggregates and the problem of a representative undistorted firm. In steady state, the capital accumulation and aggregate investment equations rely on aggregates and our sufficient statistics (i) and (ii). Therefore, we can evaluate aggregate counterfactuals exactly in steady state. We then show thatchangingtheentryconditionspecificationtoendogenousentryimpliesthatthevalueofan entering firm enters the equilibrium system; in this case, the sufficient statistics imply bounds foraggregatesinequilibrium,aswewilldetailbelow. Ifthemassofentryisassumedtobeexogenous,onecancharacterizethestationarysteadystate equilibrium as a function of sufficient statistics (i), (ii), and technology and preference parameters (iv). To see this point, note that the aggregate capital stock and investment equations,(20)and(21),insteadystatecanbewrittenas K = ME +Kκ(i +i ), (25) MM ∆ θ I = MEc +Kφ(i +i )+K (cid:0) i −(i )2(cid:1) . (26) e MM ∆ 2 ∆2 ∆ Capital accumulation depends on entry, undistorted firms’ choices, the mean capital quality shock κ, and a notion of the average distortion to investment, i . The resources spent on ∆ investment depend on the investment cost function φ(); undistorted firms’ investment decisions; the mean distortion to investment; and the “misallocation” of investment, which, given quadraticadjustmentcosts,canbe inferredfrommodelparametersandthesufficientstatistics. Therefore, taking entry as given, (25), (26), and a set of equations that are a function only of aggregates and a representative undistorted firm—(5), (9), (10), (17), (18), (19), (22), and (23) jointly evaluated in steady state—characterize the remaining steady-state values (K, I, C, L, Y,i ,qMM,Π, W,andΛ). MM P The mechanics of endogenous entry are more nuanced, as the amount of entry is endogenous and is related to the value function of the firm. Because of the fact that firm revenues arehomogeneousofdegreeoneincapital,andaggregateoutputisafunctionofaggregatefirm revenues,the sumof theexpected(not-time-discounted) streamoffirm revenueslessexpenses canbeexactlycharacterizedbyaggregatesandsufficientstatisticsinourmodel. However,what matters for firm value upon entry is the time-discounted stream of these profits to firms; our sufficient statistics do not tell us the average duration of these profits or distortions. While the averagedurationoftheseprofitsordistortionsisnotpinneddownbyoursufficientstatistics,it isconstrainedbythem. More precisely, recall from (16) that the value of a firm entering with one unit of capital, 14

VE, can be decomposed into the undistorted (pre-tax) firm’s value, the effects of the distortion on (pre-tax) revenues, and the present-discounted value of subsidies less taxes: VE = qMM + VD + VTaxes. The undistorted firm’s value is given by (9) and depends only on aggregates and sufficient statistics. The present value of subsidies less taxes must be equal to some fixed level for the counterfactual to be revenue neutral. In turn, we can show that the effect of the distortiononfirm(pre-tax)discountedcashflows,VD,isbounded. Thatis, VD ≤ VD ≤ 0, (27) where VD is the sum of the expected (not-time-discounted) losses in firm profits. Intuitively, these costs should be greater in magnitude than the not-time-discounted losses. We can write thisobjectasafunctionofsufficientstatistics: φ2i VD = 2 ∆2 . (28) 1−κ(i +i ) MM ∆ As β → 1, VD converges to VD as the time-discounted and not-time-discounted streams of firm profits converge. Further, by definition, the undistorted problem maximizes the present discounted value of profits; therefore, VD < 0. This approach implies that we can write VD = ηDVD, where ηD ∈ [0,1].18 Given each value of ηD, (16) and the entry condition VE = c close the equilibrium system (in addition to the equations listed when taking entry as e givenabove). Wethushaveacontinuumof“candidateequilibria”—equilibrianotruledoutby thesufficientstatistics. Wecansolvethemodelandfindtheimpliedequilibriumaggregatesfor eachvalueofηD.19 Evaluatingthecounterfactualwithoutthedistortionisstraightforwardandcanbeperformed bysettingi ,i ,andVDalltozeroandevaluatingtheequilibrium. Theeffectofthedistortion ∆ ∆2 on each aggregate object of interest (output, consumption, investment, labor supply, entry, and welfare) in steady state can thus be bounded by computing both the maximum and minimum of each aggregate of interest in the continuum of candidate equilibria with the distortion and subtractingtheleveloftheaggregateinthecounterfactualwithoutthedistortion. Further, since the equilibrium system without distortions is just a function of parameters we assume are known, we can compute transition dynamics where we start in each of the candidate equilibria and then remove the distortion, allowing for welfare counterfactuals that 18There is also a tighter bound than 0 for VD, which we denote VD, so VD ≤ VD ≤ 0. There is no nice analyticalexpressionforVD—rather,itisthesolutiontoacertainmaximizationproblem: howtoallocate(i)the distortiontoinvestment, (ii)themean-squareddistortiontoinvestment, and(iii)thelossofcapitaloverafirm’s lifecyclesothatthelossestofirmvalueareassmallaspossible(lossesoccurasfarafterentryaspossible)subject toconstraintsduetothesufficientstatisticsandequilibriumconditions. Thissolutionmeansthatthelowerbound ofηD maybeabovezero. 19As the resulting system of equations is nonlinear, existence and uniqueness of equilibria are not given; our approachcaneasilyaccommodatemultipleequilibriaforeachvalueofηD. 15

incorporatedynamicconsiderations. Theupperandlowerboundforwelfarearethemaximum andminimumwelfaregainacrosscandidateequilibria. 3.3 Model Extensions Wenowdemonstratehowmodificationsofsomeofthekeyassumptionsofourenvironment change the sufficient statistics required for our method and to what extent the method is still applicableinsuchextensions. Wereviewseveralextensions: (1)addingricherfirmheterogeneity, (2) allowing for aggregate shocks, (3) relaxing the assumption of homogeneity of degree one, (4) changing the process for firm productivity to be a Markov process, and (5) putting the economy on an (endogenous) balanced growth path. In all of these cases, there is an analogue ofourmainmethodologyundersomeconditions,butadditionalormodifiedsufficientstatistics arerequiredforitsimplementation. RicherGeneralEquilibriumEnvironmentOurbaselinemodeliskeptdeliberatelysimpleto make the intuition for our approach simpler, but Proposition 1 can be extended to richer environments. Corollary 1.1 in Appendix A.3 extends Proposition 1 to a broader class of models andallowsfordeadweightlossesasanadditionalconsequenceofthedistortions. Inparticular, we allow for: (1) a more general class of production functions (for example, CES production) subjecttotherestrictionthatfirmrevenuesexhibithomogeneityofdegreeoneininputs,(2)increasingandconvexadjustmentcostsinsteadofquadraticadjustmentcosts,(3)amoregeneral entrycondition,and(4)forthedistortiontoimpartdeadweightlosses. This generalization changes the second sufficient statistic and introduces additional sufficient statistics. For increasing and convex (but not quadratic) adjustment costs, the second sufficient statistic is not the mean-squared distortion to investment but rather a more general measure of the resources lost due to “misallocation” of investment across firms. If the distortionimpartsdeadweightlosses,werequiretheaverageamountof(per-period)capitaldestroyed due to distortion as an additional statistic. We also require additional statistics if firm revenues do not exhibit homogeneity of degree one in inputs or when we have a more general entry condition. Richer Firm Heterogeneity Our approach can be extended to allow for firms that are heterogeneousforreasonsotherthantherebeingheterogeneityintheircapitalstocks,realizedshocks, ordistortions. Forexample,wecanallowforindustriesdifferingintheirproductionfunctions. The main limitation to this approach is that it requires additional sufficient statistics for each category of firms, and thus its application must be driven by the availability of the relevant estimates. In Appendix A.4, we demonstrate exactly how this extension can be implemented. Tobrieflysummarize,versionsofoursufficientstatistics(i ,i )areneededforeachgroupof ∆ ∆2 firms, and the general proof is then similar with exogenous entry or no time discounting. The 16

proofisalsosimilarwithendogenousentryaslongasfirmsdonotknowtheirtypebeforethey enter. Aggregate Shocks We can extend our approach to allow for aggregate shocks if entry is exogenous;inthiscase,thefullstate-conditionalityofthesufficientstatisticsmustbeknown.20 Define time-varying functions of our sufficient statistics as the (capital-weighted) mean distortiontoinvestmentandmean-squaredinvestmentateachpointintime: (cid:90) i = i (S ) = ∆i(χ)γk(χ)dχ (29) ∆t ∆ t t χ (cid:90) i = i (S ) = (∆i(χ))2γk(χ)dχ, ∆2t ∆2 t t χ where S denotes the set of model state variables. Note that, in general, S may be infinitet t dimensional, but there are special cases where this object is finite dimensional, or, at least, can be well-approximated by a finite-dimensional measure of aggregates, a` la Krussel and Smith (1998). If S is known and its transition is characterized by the equilibrium conditions of t the model (or any additional ones we impose), Corollary 1.2 in Appendix A.5 shows that we can characterize the effects of the distortion on macroeconomic aggregates in a world with aggregate shocks if we know state-contingent sufficient statistics i (S ) and i (S ). Note ∆ t ∆2 t that these “sufficient statistics” are functions which are typically not estimated by researchers. However,ifoneweretoassumeagivencyclicalityofadistortion,ourmethodcouldbeusedto computetheimpliedconsequencesofthedistortion. Decreasing Returns to Scale in Profits Consider a modification to the model we introduced in Section 2 where (1) firm revenues exhibit decreasing returns to scale, ρ−1 (α +α ) < 1; ρ 1 2 and(2)therearenoadjustmentcostsincapital. Inthisenvironment,insteadofrepresentingthe distortion as a change in the investment rate, we represent the distortion in terms of the (log) distortion in the quantity of capital as compared with an undistorted firm (holding aggregate pricesfixed): K∗ j,t K = , (30) j,t 1−∆k j,t whereK∗ isthecapitalchoicefirmj wouldhavemade,holdingaggregatepricesfixed. Corolj,t lary 1.3 in Appendix A.6 shows that our methodology can be extended to this environment, wherethekeysufficientstatisticsaretheaverage(capital-weighted)distortionandaconcavityadjusted mean distortion. With decreasing returns to scale, the second sufficient statistic thus reliesontheparametersofthemodelbutisstilleasilycomputable. 20Thederivationswithaggregateshockscanalsobeextendedtoanenvironmentinwhichentryisendogenous, butonlytotheextentthatentrydependsonlyonaggregatesandnotdirectlyonthedistortion;forexample,entry candependonthevalueoftheundistortedfirmforthismethodtowork. 17

Markov Process for Productivity Our baseline model assumes that productivity shocks are i.i.d.. Thisassumptioncanbegeneralizedtoanarbitraryfinite-stateMarkovprocessatthecost of requiring knowing the sufficient statistics conditional on z. Corollary 1.4 in Appendix A.7 shows that with this information, we can apply an analogue of our methodology. In Section 5, we use this method to examine how assumptions on the productivity process affect our quantitativeresults. Balanced Growth Path We can consider specifications of our model in which the economy is on a balanced growth path. In Appendix A.8, we show that there are parameterizations of the modelweintroduceinSection2thatallowforanendogenousbalancedgrowthpath. Giventhe production and preference parameters, an analogue of our sufficient statistics approach using the same sufficient statistics can be used to characterize the transition from a balanced growth path with firm-level distortions to one without the firm level distortions. We therefore can characterizetheeffectofdistortionsnotonlyonthelevel,butalsothegrowthrateofeconomic activity. 4 Quantitative Results Inthissection,wepresentquantitativeestimatesoftheaggregateconsequences ofexternal financing frictions and manager-shareholder frictions by calibrating the sufficient statistics to estimates from the corporate finance literature. The first subsection presents the calibration. The second subsection presents some model-based counterfactual measures that are useful in explaining the quantitative results. The third subsection presents the quantitative results in the limit as β → 1 and discusses the key mechanisms driving the results. In the final subsection, wepresentresultsunderthemorestandardcasewhereβ < 1. 4.1 Calibration of Sufficient Statistics and Parameters We begin this section by discussing the baseline calibration of the sufficient statistics for the external financing friction and the manager-shareholder friction. We then discuss the baseline calibration for the remaining statistics and parameters of the model. The values for the statistics and parameters are listed in Table 1; in the table, we also note whether we perform therobustnessonthestatisticorparameterinthemaintextorintheOnlineAppendix. Calibration of Sufficient Statistics Our sufficient statistics are closely related to estimates of the effect of distortions on firm investment. In particular, for some measured distortion effect X,manypaperswillestimatelinearregressionspecificationsoftheform: I jt = γ +γ X +(cid:15) . 0 X jt jt K jt 18

Wecanderivethefollowingclosed-formapproximationtooursufficientstatistics:21 i = γ µ ∆ X X (cid:0) (cid:1) i = γ µ2 +σ2 , ∆2 X X X where µ ,σ2 are the mean and variance (across firms) of distortion X, respectively. These X X statistics are commonly reported in the summary statistics sections of papers which estimate effectsofdistortionsonfirminvestment. Tocalibratethesufficientstatisticsinthecaseofexternalfinancingfrictions,werelyonthe estimatesofHennessyetal.(2007)(hereafter,HLW).Aspartofabroaderexaminationofhow different frictions dynamically interact with investment decisions, HLW estimate the effect on investmentofexternalfinancingfrictionsconditionalonTobin’sQ.Weworkwiththeordinary least squares (OLS) specification estimates in HLW’s Table 2 that include cash flow (which have values of negative 0.0005).22 Given the percentage of firms that issue equity (0.179), average valuesof Q forequity issuers (5.874), andthe average equityissuance amount ofissuers to the capital stock (0.254) reported in HLW, we obtain an estimate for mean underinvestment of negative 0.00134. For computing mean-squared underinvestment, we compute firm Q and equity issuance to firm capital stock using Compustat over the years 1968-2003 (to match the yearsusedbyHLW).23WethencomputethesquaredvalueoftheinteractiontermtimesQtimes equityissuanceforeachfirmandaverageacrossfirms,resultinginamean-squaredinvestment distortionof0.00019. Forthemanager-shareholderconflict,toobtainthemeandistortionacrossfirms,werelyon estimatesfromBen-Davidetal.(2013)(hereafter,BGH).BGHstudytheextenttowhichmanagersaremiscalibratedbyexaminingmanagerabilitytoestimatethepotentialrangeofreturns for the S&P.24 The authors find that miscalibration of long-term returns leads to overinvestment. Because the authors report the miscalibration variable as a probability distribution, we can obtain the mean and mean-squared overinvestment across firms with this information and theeffectofmiscalibrationoninvestment.25 Fromthevaluesreportedintheirpaper,weobtain 21In the Online Appendix (Section O2.2), we demonstrate how an approximation of (12) leads to this result. Approximationsplayaroleintwoways:first,futuredistortionsmaydistorttoday’sTobin’sq,whichiscommonly a control in such regressions; we show that this is a second-order effect as compared to the direct effect of the distortion on investment. Second, our method requires the capital-weighted mean and variance, rather than the unweightedversion,whichislesscommonlyreported. Insomecasesoneorbothoftheseapproximationsarenot required. 22TherearesomeestimatesinHLW’srobustnesssectionthatarearoundanorderofmagnitudehigher;wewill presentresultsonthisstatisticuptoanorderofmagnitudeinourrobustnesssection. 23WediscussthecleaninganddataconstructiondetailsintheOnlineAppendix(SectionO2.2). Notethatwe find a similar estimate for the mean underinvestment of negative 0.00239 with this sample. Also note that our methodtechnicallyrequiresustousethecapital-weightedvalues.Thisrequirementwouldlikelyleadustohaving smallerestimatesandwouldmakeitsuchthatourmeanestimatedoesnotderivefromestimatesinHLW;recall thatwevarythevaluesofoursufficientstatisticsoverlargerangesforrobustness. 24Stein(2003)discusseshowsuchanoverconfidenceproblemisanexampleofanagencyfriction. 25Giventhatwedonothaveinformationonthecapitalstocksoftheparticipants,weareunabletocapital-weight 19

Table1: SufficientStatisticsandParameters SufficientStatisticorParameter Value Whereintextvariedforrobustness? SufficientStatistics-ExternalFinancing Averageunderinvestmentacrossfirms -0.00134 Section5 Mean-squaredunderinvestmentacrossfirms 0.00019 Section5 SufficientStatistics-ManagerDecisions Averageoverinvestmentacrossfirms 0.00188 Section5 Mean-squaredoverinvestmentacrossfirms 0.00004 Section5 AdditionalStatisticsandParameters Averageeffectivetaxes 0.13 Section5 Discountrate,β 0.98 Section4 InvestmentofModigliani-Millerfirm,i 0.9859 OnlineAppendix MM Exogenousexitrate,(1−κ) 0.024 OnlineAppendix Adjustmentcostparameter,θ 1.1 OnlineAppendix Depreciationrate,δ 0.1 OnlineAppendix Productionfunctionparameter,α 2/3 OnlineAppendix 2 CESparameter,ρ 4 OnlineAppendix IESparameter,γ 1 OnlineAppendix Labordisutilityparameter,ϕ 0.276 OnlineAppendix Note: This table presents values for the sufficient statistics and parameters used in the quantitative analysisofthemodelinthecasewhereβ < 1. Inthethirdcolumn,wereportwhetherweperform robustnessonthesufficientstatisticorparameterinSection4,Section5,orintheOnlineAppendix (SectionO2.2). the statistic for mean overinvestment to be 0.00188 and that for mean-squared overinvestment tobe0.00004.26 CalibrationofParametersTocalibratethetaxterm,werequirethefractionoffirmvaluelost totaxes(netofsubsidies). WefindtheshareofcorporateprofitslosttotaxationintheNIPAby taking before-tax profits less after-tax profits relative to before-tax profits.27 We use the value for2021:Q4,whichroundsto13%. We consider two cases for β: β → 1 and β < 1. In the latter case, we choose β of 0.98, giving an annualized interest rate of 2%.28 We choose the value of 2.4% for κ to match the employment-weighted firm death rate from 1980 through 2018 found in Crane et al. (2022). We choose i to match the growth rate of firms in the economy with inelastic process inno- MM vationinAtkesonandBurstein(2010);thischoiceimpliesavalueof0.9859.29 Wealsofollow Atkeson and Burstein (2010) in choosing ρ to be 4 in our baseline calibration. We choose standardvaluesforlabor’sshareofincome: α = 2. 2 3 We choose the adjustment cost parameter θ to be 1.1 as in the second column of Table IV ourmeasures. Wevarythevaluesofthesesufficientstatisticsoverlargerangesforrobustness. 26WeprovideadditionaldetailsonhowwetranslatethereportedvaluesinBGHtooursufficientstatisticsinthe OnlineAppendix(SubsectionO2.3). 27We use the series without inventory valuation or capital-consumption adjustments. The two series can be foundathttps://fred.stlouisfed.org/series/CPandathttps://fred.stlouisfed.org/ series/A053RC1Q027SBEA,respectively. 28Thisisastandardvalueusedintheliterature—forexample,inGertlerandKaradi(2013). 29AllfirmsintheAtkesonandBurstein(2010)modelwithinelasticprocessinnovationmakethesameinvestmentdecisionastheModigliani-Millerfirmsmakeinourmodel. 20

in Whited (1992). We choose the depreciation rate to be 10%, which matches the annualized valueofthequarterlyrateof2.5%assumedinGertlerandKaradi(2013). Weassumeconstant relativeriskaversion(CRRA)householdutility: C1−γ L1+ϕ U(C ,L ) = t − t . (31) t t 1−γ 1+γ We set the risk aversion parameter to γ = 1 following Chetty (2006). We choose an inverse Frischelasticityparameterϕof0.276followingGertlerandKaradi(2013). Finally,weassume freeentryinourbaselinequantitativeresults. 4.2 Counterfactuals and Other Measures PE Counterfactuals In addition to our model counterfactuals, we report PE counterfactuals. This specification is a special case of our model where all aggregate prices are fixed (ρ = ∞, γ = ϕ = 0, and entry is exogenous), but it is more easily understood as a simple back-of-the envelope calculation in which the effects of the distortion to aggregate investment, holding all elsefixed,aretranslatedintoimplicationsfortheaggregatecapitalstock: 1 K = . (32) 1−κ(i +i ) MM ∆ Note that in this PE counterfactual, i is fixed and does not change with i , as aggregate MM ∆ prices (and thus optimal investment) do not respond. Because of the homogeneity of degree one in production, aggregate labor and output respond proportionally to the aggregate capital stockinsuchaPEworld. Decomposition of Output Note that in our production environment, changes in aggregate output can be expressed as a function of changes in input usage (total investment and labor) andchangesininvestmentefficiency(K): I (cid:18) (cid:19) K ∆log(Y) = α ∆log(L)+α ∆log(I)+α ∆log . 2 1 1 I Thus, the fraction of changes in output explained by changes in investment efficiency can be writtenas α1∆log(K I ) . ∆log(Y) 4.3 Baseline Results as β → 1 We can obtain exact solutions for the welfare costs of the external financing and managershareholder frictions as β → 1, so this special case is a useful point of departure for our analysis. We consider two cases: (1) a case where there is a subsidy that undoes the monopoly markupinefficiencyandthereiszerocorporatetaxationinbothsteadystates,and(2)thebaseline case with a monopoly markup inefficiency and positive corporate taxation in both steady 21

states. The first case is a useful benchmark; here, resolving the external financing or managershareholderfrictionleadstotheplanner’sproblem.30 Thesecondcaseisthemorequantitatively relevantone. WepresenttheresultsinTable2. Removing the external financing friction leads to an increase of 3.35% of baseline output in the PE counterfactual in both panels, as it does not depend on the GE environment.31 External financing reduces investment relative to the optimal investment decision. By contrast, the manager-shareholderconflictincreasesoutputinPEasmanagers,onaverage,arebiasedtoward having beliefs that are too optimistic. Output falls 5.11% when the friction is removed in the PE counterfactual. That removing an overinvestment distortion would increase output by such asignificantamountfurtheremphasizesthevalueofaGEenvironmentforcounterfactuals. Starting with panel A, we see that in an environment with a subsidy that undoes the monopoly markup inefficiency and zero corporate taxation in both steady states, resolving each distortion (which leads to the first-best case) induces positive but modest gains. The modelfeaturesbothGEdampeningandamplifyingeffects,soitisnotex-anteobviousthatthe gains would be smaller. The most intuitive GE dampening effect is the increase in wages in response to the removal of the distortion. Removing the distortion increases aggregate labor demandwhichraiseswagesandthusreducesaverageoperatingprofits. Thisresultreducesthe aggregate profit scaling factor Π, which in turn reduces the investment of undistorted firms. We analytically show this force in a simplified version of the model in the Online Appendix (Subsection O1.7). Specifically, we show that the derivative of the capital stock with respect to the mean distortion to investment i is decreasing in the inverse Frisch elasticity. In other ∆ words, if the equilibrium wage is more sensitive to aggregate labor supplied, the sensitivity of the aggregate capital stock to the mean distortion decreases. Our calibration uses a relatively conservative calibration for the inverse Frisch elasticity; that is, the strength of this dampening forceinourcalibrationisontheweakersideofempiricallyrelevantcalibrations.32 ThemostintuitiveGEamplifyingforceistheeffectofcomplementaritiesoncapitalinvestment. With a CES aggregator, a lower ρ indicates greater complementarities in production.33 Putdifferently,thevalueofagivenfirms’outputisincreasinginotherfirm’soutput. Thus,ifthe distortion is resolved, holding all other GE forces constant, this force increases the aggregate profit scaling factor and investment by undistorted firms. In the Online Appendix (Subsection O1.7), we show that the derivative of the capital stock with respect to the mean distortion to investment i is decreasing in ρ. In other words, if complementarities are larger, the response ∆ 30Wepresentfurtherdetailsonthefirst-bestcaseintheOnlineAppendix(SubsectionO1.8). 31PEholdsfixedprices,andthemassofentryandwelfareisdeterminedasthechangeinthecapitalstock. The percentagechangeinoutputisequaltothepercentagechangeincapital. 32SeeChettyetal.(2011)andPeterman(2016)forreviewsoftheliteratureestimatingFrischelasticities. 33Given our assumption that α = ρ − α and thus firm revenues are homogeneous of degree one, the 1 ρ−1 2 substitutabilityofoutputbetweenfirmsisfixed,andthesubstitutabilityofinvestmentisgovernedbytheproperties oftheinvestmentcostfunction. Thus,inourcalibration,changingtheCESparameteraffectscomplementarities betweenfirmsinproductionbutnotsubstitutabilitybetweenfirms. 22

Table2: EffectsofResolvingtheExternalFinancingandManager-ShareholderFrictions PanelA:Subsidyτs = ρ andzerocorporatetaxation(τ =0) ρ−1 ExternalFinancingFriction Manager-ShareholderFriction %∆between Baseline No-friction %∆between Baseline No-friction steadystates s.s. value s.s. value steadystates s.s. value s.s. value Variable PECapitalstockorOutput 3.35 -5.11 GEInvestment(I) 0.16 3.39 3.39 0.04 3.68 3.68 GECapitalstock(K) 0.24 25.6 25.66 0.05 27.83 27.85 GEInvestment 0.08 7.56 7.56 0.02 7.56 7.56 efficiency(K/I) GEOutput(Y) 0.16 5.08 5.09 0.04 5.52 5.53 GEWelfare 0.16 0.04 GEConsumption(C) 0.16 1.69 1.7 0.04 1.84 1.84 GERelativewage(W) 0.16 7.57 7.58 0.04 7.9 7.9 P GELaborsupply(L) 0 0.45 0.45 0 0.47 0.47 GEMassofentry(M ) 2.75 1 1.03 6.23 1 1.07 e GECapitalstockper -2.58 25.6 24.95 -6.58 27.83 26.12 enteringfirm(K) GEInvestmentof -0.24 0.99 0.98 -0.06 0.99 0.99 unleveredfirm(i ) MM GEAggregateprofit -0.08 0.13 0.13 -0.02 0.13 0.13 scalingfactor(Π) PanelB:Nosubsidy(τs =0)andpositivecorporatetaxation(τ =0.13) ExternalFinancingFriction Manager-ShareholderFriction %∆between Baseline No-friction %∆between Baseline No-friction steadystates s.s. value s.s. value steadystates s.s. value s.s. value PECapitalstockorOutput 3.35 -5.11 GEInvestment(I) 0.06 3.24 3.24 -0.18 3.54 3.53 GECapitalstock(K) 0.09 25.6 25.62 -0.26 27.83 27.76 GEInvestment 0.02 7.9 7.9 -0.08 7.87 7.86 efficiency(K/I) GEOutput(Y) 0.06 6.77 6.78 -0.18 7.37 7.35 GEWelfare 0.05 -0.17 GEConsumption(C) 0.06 3.53 3.53 -0.17 3.83 3.82 GERelativewage(W) 0.06 4.92 4.92 -0.17 5.13 5.12 P GELaborsupply(L) 0 0.69 0.69 -0.01 0.72 0.72 GEMassofentry(M ) -1.19 1 0.99 -2.5 1 0.98 e GECapitalstockper 1.26 25.6 25.92 2.18 27.83 28.45 enteringfirm(K) GEInvestmentof -0.08 0.99 0.99 0.27 0.99 0.99 unleveredfirm(i ) MM GEAggregateprofit -0.03 0.13 0.13 0.08 0.13 0.13 scalingfactor(Π) Note: ThistableshowsthepercentagechangeinPEandGEobjectsfromthebaselinesteadystatetothesteadystate withouttheexternalfinancingandthemanager-shareholderfrictionwhenβ → 1intwocases: (1)whenthereiszero corporatetaxationandthesubsidyτs = ρ inbothsteadystates(showninPanelA),and(2)whenτ =0.13andthe ρ−1 subsidyτs = 0inbothsteadystates(showninPanelB).Theno-frictioncaseshowninPanelAisthefirst-best. We alsoshowthepercentagechangesintheGEobjectsassociatedwiththeGEwelfarecalculation,aswellastheirvalues inboththebaselinesteadystateandthecounterfactualsteadystatewithoutthefrictions. s.s. issteadystate. 23

oftheaggregatecapitalstocktothemeandistortionincreasesinmagnitude. In our quantitative results in Panel A, the dampening forces indeed dominate. Removing the distortion makes entry more attractive (holding aggregate prices fixed), as entering firm value is maximized when future investment is not distorted. The capital stock and output thus rise, and so does the wage, which reduces the profitability of a unit of capital (the aggregate profit scaling factor). Intuitively, the investment decision and value of the Modigliani-Miller firm are increasing in the aggregate scaling factor. Therefore, with lower aggregate profits, the Modigliani-Miller firm’s investment and value fall. In response, there is a lower capital stock per entering firm. These effects dampen the effect on output. The mass-of-entry effect dominates the lower capital-stock-per-entering-firm effect, leading to a larger overall capital stock. Therefore, output, consumption, and welfare all rise. Further, about one-third of the increase in output is explained by higher investment efficiency K, rather than by greater investment or I labor utilization.34 Investment efficiency rises for two reasons. First, the distortion leads to inefficient allocation of investment among incumbent firms. Second, the distortions reduce the value of an entering firm (holding aggregate prices fixed) and thus distort the relative amount ofinvestmentbyincumbentsversusentrants. Removingthedistortion—byeliminatingbothof theseforces—increasesinvestmentefficiency. Panel B features our baseline calibration, in which there are aggregate inefficiencies due to corporate taxes and the monopoly markup. In such a calibration, resolving firm-level distortions does not necessarily increase welfare, as the distortions may interact with the aggregate inefficiencies. Webeginwiththecaseofresolvingexternalfinancingfrictions. Inthiscase,GE welfare and output increase by around one-third of their magnitudes absent aggregate inefficiencies. The main force reducing the welfare and output gain is that, in equilibrium, reducing thefrictionleadsaggregateentrytofall. Thepresenceofnetpositivecorporatetaxationimplies entryisinefficientlylowintheseequilibrium,andresolvingthedistortionleadstoinefficiently lowentry. Resolvingthedistortionincreasestheinvestmentofincumbentfirms,whichreduces the aggregate scaling factor and investment by undistorted firms. Further, as the number of entrants falls, generating the same present value of taxation requires a higher tax burden as a percentage of firm value. These forces outweigh the benefit of eliminating distortions on firm value and lead to less entry in equilibrium. Labor supply barely changes despite the increased capitalstockfortworeasons. First,thedeclineintheaggregateprofitscalingfactorreducesthe valueofamarginalunitoflabor;second,thewagerisesbecauseofconsumptionbyhouseholds increases(recallwithCESutilitythattheequilibriumwagerisesinlaborsupplyandconsumption). Investmentefficiencyincreasesbecausedistortionsareeliminated,butbylessthaninthe 34Labor supply does not move much in this and several other of our counterfactuals. To understand this outcome,firstnotethatwithCRRAutility,inlogchangesthelabor-leisureconditionandaggregatedemandequation combinetoyield∆l= 1 (∆y−γ∆c). Thus,sinceγ =1,ifconsumptionandoutputriseproportionatelythen 1+ϕ theincreasingmarginalproductivityoflaborisexactlyoffsetbythelowermarginalvalueofwages,andtherefore laborsupplyisunchanged. 24

casewithouttheaggregateinefficiencies. Resolvingthedistortionseliminatesmisallocationof investment across incumbent firms, which increases investment efficiency; however, in equilibrium,aggregateentry(whichisinefficientlylow)falls,whichreducesinvestmentefficiency. Still, investment efficiency explains about one-fourth of the increase in output, while changes inaggregateresourcesdevotedtoinvestmentexplaintherest. In the case of the manager-shareholder overinvestment friction, removing the friction reduces welfare in the presence of aggregate inefficiencies. This loss in welfare is due to the interactionofthisfirm-leveldistortionwiththeaggregateinefficiencies(themonopolymarkup inefficiency and corporate taxation). The baseline steady-state equilibrium features too little output and too little entry as compared with the planner’s problem. Resolving the managershareholder friction reduces both of these quantities even further, reducing welfare. In PE (holding prices fixed), reducing the friction reduces incumbent investment, as firms, on average, have expectations that are calibrated too high. In GE, this force, all else being equal, reducesdemandforlabor andthusleadstherelativewageto fallandtheaggregateprofitscaling factortorise,whichincreasesthevalueandinvestmentoftheModigliani-Millerfirm. Thereis thereforegreaterinvestmentperincumbentfirm,andentryfallsinresponse. Themassofentry falls by enough that the capital stock, output, and consumption also fall: welfare decreases evenaslaborsupplydecreasesmodestly(giventhelowerlevelofdemandforinvestment). The efficiency of investment falls, as the effect of declining investment via entry (which is too low in equilibrium as compared with the planner’s problem) dominates the effect of resolving misallocationamongincumbentfirms. Thisfallininvestmentefficiencyexplainstwo-thirdsofthe declineinoutput. Altogether,theGEchangeinoutputandwelfareisquantitativelylowerthanthePEchange in output. With corporate taxation and an inefficiency due to the monopoly markup, removing the distortion is not necessarily positive for welfare and depends on the extent to which GE dampening effects, such as labor demand and entry, offset the removal of the distortion to incumbent investment. Further, changes in investment efficiency, due to reallocation of investment,accountforasignificantshareofthechangeinoutput. 4.4 Baseline Results if β < 1 Our sufficient statistics approach provides us with bounds on the welfare costs of external financingandagencyfrictionsifβ < 1. RecallfromSection3.2thatwhenβ → 1,theduration ofwhendistortionsaffectfirmsovertheirlifeisirrelevantforitsvaluationuponentry,andthus our sufficient statistics approach can yield an exact solution for the counterfactual of interest. However, if β < 1, distortions that affect firm value earlier in its lifetime are more costly, and our sufficient statistics do not exactly identify the counterfactual, though we can identify boundsoneachaggregatevariable. Table 3 presents the upper bound and lower bound results under our calibration for β. The 25

Table3: EffectsofResolvingExternalFinancingandAgencyFrictionswhenβ = 0.98 ExternalFinancing Manager-Shareholder Upperbound LowerBound Upperbound LowerBound %∆between %∆between %∆between %∆between steadystates steadystates steadystates steadystates Variable PECapitalstockorOutput 3.35 3.3 -5.11 -5.11 GEInvestment(I) 0.05 0.08 -0.33 -0.3 GECapitalstock(K) 0.17 0.11 -0.37 -0.38 GEInvestment 0.12 0.03 -0.04 -0.06 efficiency(K/I) GEOutput(Y) 0.1 0.07 -0.27 -0.27 GEWelfare 0.13 0.04 -0.13 -0.15 GEConsumption(C) 0.13 0.07 -0.22 -0.24 GERelativewage(W) 0.12 0.07 -0.23 -0.24 P GELaborsupply(L) -0.03 0 -0.03 -0.03 GEMassofentry(M ) 1.3 -1.15 -1.87 -2.44 e GECapitalstockper -1.14 1.25 1.47 2.01 enteringfirm(K) GEInvestmentof -0.18 -0.09 0.24 0.26 unleveredfirm(i ) MM GEAggregateprofit -0.08 -0.04 0.1 0.11 scalingfactor(Π) Note: ThistableshowsthepercentagechangeinPEandGEobjectsfromthebaselinesteadystateto thesteadystatewithouttheexternalfinancing(toppanel)ormanager-shareholder(bottompanel)frictionwhenβ =.98,thesubsidyτs =0,andτ =0.13. s.s. issteadystate. PE counterfactual is the same as in the β = 1 case, as it does not depend on aggregates, just the distortion and the mean growth rate of firms. Importantly, the GE counterfactuals differ not only in that β is lower, but also that our welfare measure accounts for the entire transition dynamic. Nonetheless, the GE welfare results are directionally similar to those for the β → 1 case. The bounds for our welfare measures are tighter in the case of the manager-shareholder friction at 2 basis points but are still different by only 9 basis points for the external-financing friction. Furthermore, output, capital, and consumption are directionally similar to the β → 1 case, and a similar logic holds for the drivers of the changes in aggregates between the cases as in the discussion in the last subsection. Our approach generates bounds for each aggregate variablealongthetransitiondynamic;togetasenseofwhatthetransitionlookslike,inFigure 1, we show the transition path for the percentage change in consumption relative to its level in theModigliani-Millersteadystateforeachofthetwofrictions. Importantly,theresultsalsodemonstratehowsomeofthechangesinthecounterfactualobjects are sensitive to the parameterization. For example, a few of the counterfactual aggregates change directions in the counterfactual for the external financing friction between the lower bound and the upper bound case. The mass of entry falls in the lower bound case, as it did in the β → 1 case, while it rises in the upper bound case. Both aggregates are sensitive to the changesinthedistortionsandthechangeintheinvestmentoftheModigliani-Millerfirm. Cap- 26

Figure1: BoundsontheTransitionPathafterResolvingFirm-levelDistortions (a)ExternalFinancingFriction 0 −0.02 −0.04 −0.06 −0.08 −0.1 −0.12 −0.14 0 10 20 30 40 50 60 70 80 90 100 Periods )etats ydaets noitcirf on morf egnahc tnecrep( noitpmusnoC (b)Manager-ShareholderFriction 0.25 0.2 0.15 0.1 0.05 0 0 10 20 30 40 50 60 70 80 90 100 Periods )etats ydaets noitcirf on morf egnahc tnecrep( noitpmusnoC Note: Each subfigure shows the bounds on the percentage change in GE consumption relative to its level after removingtheexternalfinancingfriction(leftpanel)orthemanager-shareholderfriction(rightpanel)whenβ = 0.98underourbaselinecalibration. s.s. issteadystate. italdemandrisesbyenoughintheupperboundcasethatweobserveanoticeable(albeitsmall) change in labor supply in the table.35 Nonetheless, the narrative around the output and welfare results is still similar between the cases: more investment by incumbents leads to higher capitaldemand,whichpushesdownwagesandthusaverageprofits,dampeningthegainsfrom resolving the distortion. Therefore, even though the distortion has significant effects in PE, we observeonlymodestchangesinthecapitalstock,output,consumption,andwelfareinGE. For further robustness, we present the welfare results (that account for the transition dynamic) for the two frictions for values of β between β = 0.95 and β = 1 in Online Appendix Figure O4.1. As expected, the figure shows that bounds widen for lower values of β; nonetheless,thewelfareresultsarequalitativelysimilartothosepresentedinTable3. 5 Quantitative Robustness Inthissection,wetesttherobustnessofourresultsaswevarysufficientstatistics,parameters,themodellingoffreevs. fixedentry,andtheprocessforfirmproductivity. Varying the Sufficient Statistics i , i , and VTaxes Figure 2 plots the welfare gain from re- ∆ ∆2 solvingtheexternalfinancingormanager-shareholderfrictionaswevarythesufficientstatistics i , i , and VTaxes.36 The vertical lines in the figure indicate the values for the bounds on the ∆ ∆2 welfarestatisticsunderourbaselinecalibration. Panels(a)and(b)showthegaininwelfarefromresolvingthedistortion(“welfarecost”)as 35Notethatlaborsupplyfallsinbothcaseswhenwedonotroundattheseconddecimal. 36Wedosounderthecalibrationwhereβ =0.98;therearethusupperandlowerboundsandthewelfaregains accountforthetransitiondynamic. 27

we vary i by an order of magnitude; the welfare gain retains the same sign for a remarkably ∆ largerangeofvalues.37 Themagnitudeoftheimpactonwelfareisincreasinginthemagnitude of i locally around our estimate. The difference in the bounds is generally consistent across ∆ the range, at between 9 and 10 basis points across the external financing friction values, and around2basispointsacrossthemanager-shareholderfrictionvalues. Panels (c) and (d) show the gain in welfare from resolving the distortion as we vary i by ∆2 an order of magnitude. As the mean squared dispersion grows, so does the difference between the upper and lower bounds. The intuition for this result is the mean squared distortion to investment has a first order effect on firm value, while the mean distortion only has a second ordereffectonfirmvalue. Becausewecannotfullyidentifythedurationofthesedistortions,the larger these distortions to firm value, the larger the bounds. For the external financing friction, if the mean squared distortion were several times our calibrated value, the bounds would be large enough that both welfare gains and welfare losses would be in the plausible range of equilibria identified by our bounds.38 However, in the case of managerial miscalibration, even varying the mean squared distortion by an order of magnitude does not change the sign of the effectonwelfare. Panels (e) and (f) present the welfare gains from resolving the distortions across a range of thetaxratesufficientstatisticsinthebottompanelsofFigure2. Weconsiderarangefrom0.09 to0.17,i.e.,30%ineitherdirectionfromourbaselinecalibration. Wefindthatthedirectionof the welfare gains matches those of the baseline calibration. For the external financing friction, the bounds tighten as the tax rate increases, going from 13 basis points apart at the lower end of the range to 7 basis points apart at the upper end of the range. Similarly, for the managershareholder friction, the bounds are 3 basis points apart at the lower end of the range and 1.5 basispointapartatthehighendoftherange. Altogether,theresultsaresimilaracrossthiswide rangeoftaxrates. Free versus Fixed Entry We consider a version of the model where entry is exogenous and equalsME. AsdescribedinProposition1,inthiscase,wegetapointestimateforthecounterfactual(e.g. theupperandlowerboundsareidentical). Wesetβ = 0.98(solvingthetransition for the welfare counterfactuals) and present results in Online Appendix Table O4.1. We also report the upper bound result for the endogenous entry case when β = 0.98 for ease of comparison;wedonotreportthelowerboundbecauseoftablespaceconstraints. The main takeaway is that the change in consumption, output, the capital stock, the investmentandvalueoftheunleveredfirm,andtheaggregateprofitscalingfactorarealldirectionally similar for both frictions in the fixed-entry and free-entry models and are also similar quanti- 37Infact, theonlyplacewheretheeffectonwelfarewouldswitchsignsinthisrangeiswhenthemeaneffect of the distortion in the external financing case approaches zero, in that case the effect on the misallocation of investmentdominatesandthusthereisasmallwelfareloss. 38Ifweweretoprovideadditionaldatapoints,suchasrestrictionsonthedurationoflossesafterenteringdue toagivendistortion,theboundscouldbefurthertightened. 28

Figure2: WelfareGainsfromResolvingFirm-levelDistortions,varyingi ,i ,andVTaxes ∆ ∆2 (a)ExternalFinancingFriction,varyingi ∆ 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −12 −10 −8 −6 −4 −2 Mean investment distortion sufficient statistic (i∆) x 10−3 )%( stsoc erafleW (b)Manager-ShareholderFriction,varyingi ∆ 0 GE upper bound GE lower bound −0.1 −0.2 −0.3 −0.4 −0.5 −0.6 −0.7 −0.8 −0.9 −1 2 4 6 8 10 12 14 16 18 Mean investment distortion sufficient statistic (i∆) x 10−3 )%( stsoc erafleW GE upper bound GE lower bound (c)ExternalFinancingFriction,varyingi ∆2 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Mean−squared investment distortion sufficient statistic (i∆ 2) x 10−3 )%( stsoc erafleW (d)Manager-ShareholderFriction,varyingi ∆2 0 GE upper bound GE lower bound −0.05 −0.1 −0.15 −0.2 −0.25 −0.3 0 1 2 3 4 Mean−squared investment distortion sufficient statistic (i∆ 2) x 10−4 )%( stsoc erafleW GE upper bound GE lower bound (e)ExternalFinancingFriction,varyingVTaxes 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 Tax rate sufficient statistic )%( stsoc erafleW (f)Manager-ShareholderFriction,varyingVTaxes −0.1 GE upper bound GE lower bound −0.11 −0.12 −0.13 −0.14 −0.15 −0.16 −0.17 −0.18 −0.19 −0.2 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 Tax rate sufficient statistic )%( stsoc erafleW GE upper bound GE lower bound Note: Eachsubfigureshowstheboundsonthewelfarechangeafterremovingtheexternalfinancingfriction(left panels)orthemanager-shareholderfriction(rightpanels)whenβ = 0.98forasetofvaluesofagivensufficient statistic. Thevertical, dashedblacklineindicatesthewelfarevaluesatthecalibratedvalueofthegivenstatistic (asshowninTable3). tatively. The mass of entry being fixed does affect the results for the capital stock per entering 29

firm, which now switches directions from the free-entry case. This should not come as a surprise,asonlyincumbentsarechangingtheirdecisionsonthemargininresponsetothefriction beingremoved. Laborsupplystillfallsforthemanager-shareholderfriction,nowbyevenmore as incumbent capital demand decreases. The opposite holds for the external financing friction: incumbent capital and labor demand increase, raising labor supply. The resulting effects on welfare are directionally similar to our baseline model with free entry, but they are smaller in magnitude. FirmHeterogeneityinIdiosyncraticProductivityOurbaselinemodelassumesthatidiosyncratic productivity is i.i.d. across firms. In Section 3.3, we detail an extension that generalizes the framework to allow for idiosyncratic productivity to follow a Markov process and demonstrates the necessary sufficient statistics for constructing the counterfactuals of interest in this setting. We implement this extension for the external financing friction using firm-level data to understand its quantitative importance, and then vary model parameters related to this heterogeneity (and its covariance with distortions) to more generally understand in which cases this assumption may be quantitatively important. The full details of this implementation are providedinOnlineAppendixO3;here,weprovideasummaryofourapproachandtheresults. Following Proposition 1.4, to implement our approach, we need to know sufficient statistics conditional on firm productivity and the process for firm productivity. We therefore use data from Compustat and firm TFP from ˙Imrohorog˘lu and Tu¨zel (2014) to compute sufficient statisticsanalogoustothoseweobtainedfromHLW,butforeachquintileoffirmproductivity; wefurthercomputethetransitionmatrixandmeanlevelsofproductivityforeachgroup. There is considerable heterogeneity in the sufficient statistics across productivity levels, though the relationship in non-monotonic.39 To understand the effect of heterogeneity in how distortions affect firms of different productivity levels, we consider the difference between our results in thisframeworkandanotherinwhichproductivityisheterogeneousbutthedistortions(andthus sufficient statistics) are independent of the level of productivity.40 The resulting measures of the effect of resolving costly external financing are qualitatively similar with heterogeneity in sufficient statistics vs without such heterogeneity. Quantitatively, the effect on welfare/output are somewhat larger: welfare increases by 0.23% with heterogeneity in sufficient statistics instead of 0.15% without it; Online Appendix Table O3.1 presents the full set of quantitative resultsunderthecounterfactuals. To better understand in which cases heterogeneity in firm exposure to distortions along the 39OnlineAppendixTableO3.2presentsthesufficientstatisticsbyquantile, OnlineAppendixTableO3.3displays the Markov transition matrix between productivity levels, and Online Appendix Table O3.4 presents the meanandstandarddeviationoflog(TFP)byquantile. 40Ifthesufficientstatisticsareindependentofthelevelofproductivityasimpleanalogueofourmainapproach canbeusedwithnoadditionalsufficientstatisticsrequired.Firmswithdifferentproductivitieshaveheterogeneous elasticitiesofinvestmentwithrespecttoaggregateprices,whichcanchangethequantitativeresultswithoutany heterogeneityinhowdistortionsaffectfirms. 30

Figure3: VaryingHeterogeneityparameters (a)Varyingthecovariance (b)Varyingthestandarddeviationoflog(TFP) 0.4 0.35 Het. in z + suff. stats Het. in z + suff. stats Het. in z, no het. in suff. stats Het. in z, no het. in suff. stats 0.3 0.3 0.25 )% 0.2 )% ( s ( s ts ts 0.2 o o c 0.1 c e e ra ra0.15 fle fle W 0 W 0.1 -0.1 0.05 -0.2 0 -3 -2 -1 0 1 2 3 0 0.5 1 1.5 2 2.5 3 Covariance of log(z), suff. stats (scaled by value in data) Std. dev. of log(z) (scaled by value in data) Note: Each subfigure shows the welfare change after removing the external financing friction β = 1, but under differentvaluesofheterogeneityparameters. Intheleftsubfigure,wevarythecovariancebetweenthesufficient statisticsandheterogeneityrelativetoitsvalueusedintheprevioussubsectionfortheresultspresentedinTable O3.1. Intherightsubfigure,wevarythestandarddeviationoflog(TFP)relativetoitsvalueusedintheprevious subsection for the results presented in Table O3.1. Log is the natural logarithm. Suff. stats are the sufficient statistics. Het. isheterogeneity. Std. dev. isstandarddeviation dimension of productivity matters, we simulate data varying either the covariance of firm distortions with productivity (holding firm productivity constant) or the cross-sectional standard deviation of log(TFP) (holding the correlation between distortions and productivity constant). We then compute the sufficient statistics by group and measure the effects of resolving costly externalfinancing,bothwithandwithoutheterogeneityinsufficientstatistics. Figure3presents the results, varying the covariance in the left subfigure and the standard deviation in the right, where both are normalized by the respective measure in our data. We see that the covariance between firm productivity and the distortions can bias the results upwards or downwards, depending on the sign, though it has to be several times the value we found in the data to be qualitatively meaningful. Varying the standard deviation of firm TFP alone does not change theresultsqualitatively,andquantitativelymaxesoutatlessthan10basispointsacrossalarge range of values of the standard deviation, up to 3 times more than our assumed value. The gap shrinks if the standard deviation is smaller. Altogether, we see this exercise as demonstrating thegeneralrobustnessofourqualitativeresultsandasanillustrationofhowtheextensionswe developtoourmethodologycanbeapplied. 6 Conclusion This paper develops a novel sufficient statistics methodology for examining the aggregate costs of firm-level distortions. Our methodology allows a researcher to evaluate the aggregate consequences of distortions without having to specify the microfoundation of the distortion. 31

This methodology is useful in many instances where such microeconomic dynamics are in debate(e.g.,reasonsforfirmcapitalstructurechoiceorquestionsabouttheexactmicrofoundation of agency frictions) or in cases where realistic modelling of the microeconomic details is sufficiently complicated to be intractable in a general equilibrium model (e.g., modelling firm liabilitieswithrealisticcovenants,convertibility,ormultiplematurities). Further,thesufficient statistics needed are closely related to standard statistics reported in empirical and quantitative resultsinthecorporatefinanceandfirmdynamicsliteratures. We use our approach to examine two cases of distortions: costly external equity financing and a manager-shareholder conflict. We show that the welfare costs of external financing frictions and manager-shareholder conflicts are significantly smaller than their PE costs, and that they can be negative in models with aggregate inefficiencies. Our approach provides us with insights into the mechanisms driving our results, as we can describe how all aggregates of the modelchangealongthetransitionpathafterresolvingthedistortionofinterest. Wedemonstrate howourapproachcanbeextendedtoevenricherenvironments,suchasthosewithbusinesscycles; we detail the additional sufficient statistics needed for such extensions. By providing this guidance,wehopethatresearcherswillpursuethemeasurementofsuchstatistics. References ATKESON, A. AND A. BURSTEIN (2010): “Innovation, Firm Dynamics, and International Trade,”JournalofPoliticalEconomy,118,433–484. ——— (2019): “Aggregate implications of innovation policy,” Journal of Political Economy, 127,2625–2683. AUCLERT, A. (2019): “Monetary policy and the redistribution channel,” American Economic Review,109,2333–67. BAQAEE, D. R. AND E. FARHI (2019): “The macroeconomic impact of microeconomic shocks: BeyondHulten’stheorem,”Econometrica,87,1155–1203. ———(2020): “Productivityandmisallocationingeneralequilibrium,”TheQuarterlyJournal ofEconomics,135,105–163. BEN-DAVID, I., J. R. GRAHAM, AND C. R. HARVEY (2013): “Managerial Miscalibration,” TheQuarterlyJournalofEconomics,128,1547–1584. CATHERINE, S., T. CHANEY, Z. HUANG, D. SRAER, AND D. THESMAR (2022): “Quantifying Reduced-Form Evidence on Collateral Constraints,” The Journal of Finance, 77, 2143– 2181. CHETTY, R. (2006): “A New Method of Estimating Risk Aversion,” The American Economic Review,96,1821–1834. ——— (2009): “Sufficient statistics for welfare analysis: A bridge between structural and reduced-formmethods,”AnnualReviewofEconomics,1,451–488. 32

CHETTY, R., A. GUREN, D. MANOLI, AND A. WEBER (2011): “Are micro and macro labor supplyelasticitiesconsistent? Areviewofevidenceontheintensiveandextensivemargins,” AmericanEconomicReview,101,471–75. CRANE, L. D., R. A. DECKER, A. FLAAEN, A. HAMINS-PUERTOLAS, AND C. KURZ (2022): “Business exit during the COVID-19 pandemic: Non-traditional measures in historicalcontext,”JournalofMacroeconomics,72,103419. DAVID, J. M. AND V. VENKATESWARAN (2019): “The sources of capital misallocation,” AmericanEconomicReview,109,2531–67. GERTLER, M. AND P. KARADI (2013): “Qe 1 vs. 2vs. 3...: A framework for analyzing largescale asset purchases as a monetary policy tool,” International Journal of Central Banking, 9,5–53. GOMES, J., U. JERMANN, AND L. SCHMID (2016): “Sticky Leverage,” American Economic Review,106,3800–3828. HAYASHI, F. (1982): “Tobin’s Marginal q and Average q: A Neoclassical Interpretation,” Econometrica,50,213–224. HENNESSY, C. AND T. WHITED (2007): “How Costly is External Financing? Evidence from aStructuralEstimation,”TheJournalofFinance,62,1705–1745. HENNESSY, C. A., A. LEVY, AND T. M. WHITED (2007): “TestingQTheorywithFinancing Frictions,”JournalofFinancialEconomics,83,691–717. HSIEH, C.-T. AND P. J. KLENOW (2009): “Misallocation and Manufacturing TFP in China andIndia,”TheQuarterlyJournalofEconomics,124,1403–1448. IACHAN, F. S., D. SILVA, AND C. ZI (2022): “Under-diversification and idiosyncratic risk externalities,”JournalofFinancialEconomics,143,1227–1250. ˙IMROHOROG˘LU, A. AND S¸. TU¨ZEL (2014): “Firm-level productivity, risk, and return,” ManagementScience,60,2073–2090. KAPLAN, S. N. AND L. ZINGALES (1997): “Do Investment-Cash Flow Sensitivities Provide Useful Measures of Financing Constraints?” The Quarterly Journal of Economics, 112, 169–215. KRUSSEL, P. AND A. A. SMITH (1998): “Income and Wealth Heterogeneity in the Macroeconomy,”JournalofPoliticalEconomy,106,867–896. KURTZMAN, R. AND D. ZEKE (2018): “The Economy-Wide Gains from Resolving Debt Overhang,”WorkingPaper. MODIGLIANI, F. AND M. H. MILLER(1958): “TheCostofCapital,CorporationFinance,and theTheoryofInvestment,”AmericanEconomicReview,48,261–297. PETERMAN, W. B.(2016): “ReconcilingmicroandmacroestimatesoftheFrischlaborsupply elasticity,”Economicinquiry,54,100–120. 33

SRAER, D. A. AND D. THESMAR (2021): “How to Use Natural Experiments to Measure Misallocation,”WorkingPaper. STEIN, J. C. (2003): “Agency, Information and Corporate Investment,” in Handbook of the EconomicsofFinance,ed.byG.Constantinides,M.Harris,andR.Stulz,Amsterdam: North Holland. TERRY, S. (Forthcoming): “TheMacroImpactofShort-Termism,”Econometrica. WHITED, T. M. (1992): “Debt, Liquidity Constraints, and Corporate Investment: Evidence fromPanelData,”JournalofFinance,47,1425–1460. A Theory Appendix: Proofs and Corollaries In this section of the appendix, we first define an equilibrium (in Subsection A.1). Second, we present the proof to Proposition 1 (in Subsection A.2). Third, we outline the mathematical definitions of the corollaries (in Subsections A.3-A.8. We present proofs of the corollaries in theOnlineAppendix. A.1 Definition of Equilibrium Given the initial distribution of firms Γ (k,χ) across states, a sequential equilibrium con- 0 sistsofpolicyandvaluefunctionsoffirms,{l (k,χ),i (k,χ),VF(k,χ),VE,χ(cid:48)(χ)};household t t t t t policyfunctionsforconsumption,C ,andlabor,L ;aggregateprices,{W ,R ,P };andamass t t t t t ofnewentrants,ME,suchthatforallt,(i)thepolicyandvaluefunctionsofintermediategood t firms are consistent with the optimization problem (which we do not fully specify), (ii) the representativeconsumer’spolicyfunctionisconsistentwithitsmaximizationproblem,(iii)the firm’svaluefunctionsanddecisionrulesarepricedsuchthattheybreakeveninexpectedvalue, (iv) the free-entry condition holds, (v) labor, capital, and final good markets clear, and (vi) the measureoffirmsevolvesinamannerconsistentwiththepolicyfunctionsoffirms,households, andshocks. A stationary competitive equilibrium is an equilibrium in which all aggregates, aggregate prices,andthedistributionoffirmsareconstantovertime. Insuchanequilibrium,wesaythese aggregatesareinsteadystate. Wefocusonlyonequilibriawithpositiveentry. A.2 Proof of Proposition 1 Notethat insteady state,the capitalaccumulationand investmentequations, (20)and (21), canbewrittenasthefollows: (cid:90) K = ME +Kκ (i(χ))γk(χ)dχ (33) χ (cid:90) I = MEc +K φ(i(χ))γk(χ)dχ. (34) e χ 34

Plugging in the sufficient statistics yields (25) and (26). These conditions are therefore functions of aggregates K,I,ME, representative undistorted firm investment i , and sufficient MM statistics i ,i . Note that the remaining equilibrium conditions in steady state include the ∆ ∆2 equationsforthevalueofafirmatentryand(35)–(43): Π = (cid:32) A ρ− ρ 1 Y ρ 1 (W)−α2(ρ− ρ 1) (cid:18) α 2 ( ρ−1 ) (cid:19)α2(ρ− ρ 1) (cid:33)αc(cid:18) 1−α 2 ( ρ−1 ) (cid:19) , (35) ρ ρ qMM = Π−φ(i )+i κΛqMM (36) MM MM φ(cid:48)(i ) = κΛqMM (37) MM Λ = β (38) −U (C,L) L W = (39) U (C,L) C (cid:18) ρ−1 (cid:19)αc L = K α 2 ( )A ρ− ρ 1 (W)−1Y ρ 1 (40) ρ C = Y −I (41) Π Y = K (42) (cid:16) (cid:17) 1−α ρ−1 2 ρ 0 = Θ(ME,VE). (43) A.2.1 ExogenousEntry Letusfirstconsiderthecasewithexogenousentry: Θ(ME,VE) = ME−ME. Ifweknow the parameters of the model and sufficient statistics i ,i , then the 11 unknowns K, I, C, L, ∆ ∆2 Y, W, Π,ME, Λ, qMM, and i are characterized by the 11 equations (25), (26), and (35)– MM (43). Therefore,wecansolvethesteady-stateequilibriumbothwiththedistortionandwithout it(settingi ,i bothtozero). ∆ ∆2 A.2.2 EndogenousEntry With endogenous entry, the value of the entering firm, V , enters the entry equilibrium E condition Θ(ME,VE) = 0, which depends on the present discounted value of the stream of expectedfirmprofits. Inthiscase,theproofismorecomplex. Westartbydefininganumberof objectsthatareafunctionofwhenthefirmentered,i.e.,oneperiodagoversustwoperiodsago versusnperiodsago. Usingtheseobjects,wethenderiveanexpressionforthekeyvariablefor our proof: the difference in firm value between the distorted and undistorted benchmark. With boundsonthisobject,wecanthensolveforthecounterfactualofinterest. We use subscripts in parentheses to denote statistics of firms that entered n periods ago. K denotes the total capital stock of firms that entered n periods ago; by definition, K = (n) (cid:80)∞ K . K = K (n) denotestheentry-scaledtotalcapitalstockthatenterednperiodsago, n=0 (n) (n) ME and K(cid:103) = βnK is the time-since-entry discounted version of K . We define the sums of (n) (n) (n) these objects as K = (cid:80)∞ K and K(cid:101) = (cid:80)∞ K(cid:103). Let Γ (k,χ) denote the distribution n=0 (n) n=0 (n) (n) 35

of firms across states that entered n periods ago. In turn, the capital-weighted steady-state relativedistributionofthecapitalstructurestatevectoroffirmsthatenterednperiodsagois (cid:82) kΓ (k,χ) γK (χ) = k (n) . (44) (n) (cid:82) (cid:82) kΓ (k,x) k x (n) Letγ(cid:102)K(χ)denotethetime-since-entrydiscounteddistributionoffirmsacrossstates: ∞ (cid:88) K(cid:103) γ(cid:102)K(χ) = (n) γK (χ). (45) (n) K(cid:101) n=0 (cid:82) (cid:82) γK (χ)dχand γ(cid:102)K(χ)dχcanbethoughtofassimilartoprobabilitydistributions,since χ (n) χ (cid:90) (cid:90) γK (χ)dχ = γ(cid:102)K(χ)dχ = 1. (n) χ χ Itisalsousefultodefine ∞ (cid:88) K γ(cid:99)K(χ) = βn (n) γK (χ), (46) (n) K n=0 notingthisobjectneednotbeequalto1. Therearetwopropertiesofγ(cid:99)K(χ)worthnoting. First, itcanbemappedintoγ(cid:102)K(χ): K(cid:101) γ(cid:99)K(χ) = γ(cid:102)K(χ) . (47) K Second,becausewecanwriteγK(χ) = (cid:80)∞ K (n)γK (χ), n=0 K (n) γ(cid:99)K(χ) ≤ γK(χ). (48) (cid:82) Note that since, by definition, K = K(cid:103) = 1 and K(cid:103) = βκK(cid:103) (i(χ))γK (χ), we can (n) (n) (n) (n) χ (n) derivethefollowingexpressionforK(cid:101) = (cid:80)∞ K(cid:103): n=0 (n) (cid:18)(cid:90) (cid:19) K(cid:101) = 1+K(cid:101)βκ (i(χ))γ(cid:102)K(χ) . (49) χ Additionally,notethattheexpectedpre-taxvalueofanenteringfirmcanbewrittenas (cid:32) (cid:33) ∞ (cid:90) (cid:88) VE,notax = K(cid:101) Π − φ(i(χ))γ(cid:102)K(χ)dχ . (50) t n=0 χ (cid:94) We define KMM as the time-since-entry discounted expected lifetime sum of capital stock of 36

an undistorted firm that enters with one unit of capital; this object is related to the value of an undistortedfirm: (cid:94) (cid:94) 1 KMM = 1+κβi KMM = . (51) MM 1−κβi MM (cid:94) qMM = KMM (Π −φ(i )). (52) t MM DifferencebetweendistortedandundistortedbenchmarkDefineVD = VE,notax −qMM as thechangein(pre-tax)profitsbetweenthedistortedandundistortedcase. (50)and(52)yields (cid:18) (cid:90) (cid:19) (cid:18) (cid:19) VD = K ˜ Π− γ˜K(χ)φ(i(χ))dχ −K ˜MM Π−φ(i ) MM χ (cid:18) (cid:19)(cid:18) (cid:19) (cid:90) = K ˜ −K ˜M˜M Π−φ(i ) −K ˜ γ˜K(χ)(φ(i(χ))−φ(i ))dχ. (53) MM MM χ We can then combine (49) and (51) and plug this expression for (K ˜ − K ˜M˜M) into (53) (also pluggingin(47)andtheundistortedfirm’svaluefunction)toobtain: (cid:90) (cid:18) (cid:19) VD = K ¯ γˆK(χ) κβqMM(i(χ)−i )−(φ(i(χ))−φ(i )) dχ. MM MM χ (54) Notethatthequadraticadjustmentcostimpliesthatφ(cid:48)(i ) = 1+ θ2 (x−θ )2,andthus: mm 2 c θ φ(i(χ))−φ(i ) = ∆i(χ)φ(cid:48)(i )+ (∆i(χ))2. (55) mm mm 2 Plugging(55)into(54)yields (cid:90) (cid:18) (cid:19) θ VD = K ¯ γˆK(χ) κβqMM(∆i(χ))−∆i(χ)φ(cid:48)(i )− (∆i(χ))2 dχ. mm 2 χ (56) GiventhattheFOCoftheundistortedproblemimpliesφ(cid:48)(i ) = κβqMM,(56)yields mm (cid:90) (cid:18) (cid:19) θ VD = −K ¯ γˆK(χ) (∆i(χ))2 dχ. 2 χ (57) Notethatfrom(57),wecanconstructboundsforVD. SinceγˆK(χ),andK ¯ areallpositive,VD is negative, so 0 is an upper bound. Also note that (48) implies that we can construct a lower 37

boundifwereplaceγˆK withγK. Formally,defineVD as (cid:90) (cid:18) (cid:19) (cid:18) (cid:19) θ θ VD = −K ¯ γK(χ) (∆i(χ))2 dχ = −K ¯ i . (58) 2 2 ∆2 χ Therefore, we have bounds for VD: VD ≤ VD ≤ 0. Another way of expressing this bound result is that VD = ηDVD for some ηD ∈ [0,1]. In the limit as β → 1, VD → VD. That is, in the limit, VD converges to that lower bound (larger losses). There is a tighter upper bound for VD than 0, but it is a less analytically clean object. Online Appendix O1.1 derives this tighter bound. Finishing the result Recall that we can decompose firm value at entry into the value of an undistorted firm with a unit of capital, the losses in (pre-tax) profits due to the distortion, and anytaxconsequences. Sinceourcounterfactualsarerevenueneutral,thepresentvalueoftaxes raised must remain constant. We can thus write a system of equations for the value of the enteringfirm: VE = qMM +VD +VTaxes (59) TaxPV VTaxes = (60) ME (cid:18) (cid:19) K θ VD = −ηD i . (61) ME 2 ∆2 Thus, for each value of ηD, equations (25), (26), (35)–(43), (59)–(61) characterize unknowns K, I, C, L, Y, W, Π, ME, Λ, qMM, i , VE, VD, and VTaxes. Therefore, we can solve for MM a continuum of “candidate equilibria” for the model. We can then evaluate the maximum and minimumofeachaggregateacrossthesecandidateequilibria. We can then solve for the counterfactual where the distortion is removed. Note that this solution is achieved by setting i , i , both equal to zero. In that case, VD must be exactly ∆ ∆2 equal to zero. Thus, bounds for steady-state counterfactuals for any aggregate in question can be evaluated by taking the difference between the undistorted case and the maximum and minimumofthecandidateequilibria. A.2.3 TransitionDynamic Notethatwithoutdistortions,thedynamicsof12unknownsK ,I ,C ,L ,Y ,W ,Π ,ME, t t t t t t t t Λ ,qMM,i ,andVE arecharacterizedbythefollowingequilibriumsystem: t t MM,t t K = ME +K κi (62) t t t−1 MM,t−1 I = c ME +K φ(i ) (63) t e t t MM,t Π t = (cid:32) A ρ− ρ 1 Y t ρ 1 (W t )−α2(ρ− ρ 1) (cid:18) α 2 ( ρ− ρ 1 ) (cid:19)α2(ρ− ρ 1) (cid:33)αc(cid:18) 1−α 2 ( ρ− ρ 1 ) (cid:19) (64) 38

qMM = Π −φ(i )+i κΛ qMM (65) t t MM,t MM,t t t φ(cid:48)(i ) = κΛ qMM (66) MM,t t t U (C ,L ) C t t Λ = β (67) t U (C ,L ) C t−1 t−1 −U (C ,L ) L t t W = (68) t U (C ,L ) C t t (cid:18) ρ−1 (cid:19)αc L t = K t α 2 ( )A ρ− ρ 1 (W t )−1Y ρ 1 (69) ρ C = Y −I (70) t t t Π t Y = K (71) t t(cid:16) (cid:17) 1−α ρ−1 2 ρ 0 = Θ(ME,VE) (72) t t TaxPV VE = qMM + . (73) t t ME t Therefore,ifweknowtheoriginal(distorted)steady-state,wecanusethissystemofequations to solve for the transition dynamic. In the cases where we have a continuum of counterfactual equilibria, we can solve for the transition path and evaluate aggregate objects (such as thewelfaregainfromtheresolutionofthedistortions). A.3 Generalization to a Broader Class of Models AssumethatinsteadoftheassumptionsweplacedontheproductionenvironmentinSection 2,wegeneralizetheenvironmentasfollows: (i) The output of heterogeneous firms is aggregated into the final good using a homogeneous and symmetric production function G that is continuous and differentiable; inputs are chosen to satisfy cost minimization. Each firm produces output with an identical productionfunctionF(zk,l),wherez isani.i.d shock. (ii) AssumethatGandF aredefinedsuchthatfirmrevenue,p y = ∂Yt y ,canbewritten j,t j,t ∂yj,t j,t as a function of inputs and aggregate output: p y = Rev(z k ,l ,Y ), where Rev j,t j,t j,t j,t j,t t is continuous, differentiable, homogeneous of degree one in k and l, and satisfies the Inada conditions in the first two arguments. Additionally, the marginal revenue product oflabor, ∂Revt(zk,l,Y),isinvertibleinl. ∂l (iii) Theaveragefirmentrycost,c ,isafunctionofthequantityofentry(orotheraggregates e suchasoutput)butdoesnototherwisedependontheallocationofresourcesacrossfirms. (iv) Similarly, the equilibrium entry condition is a function of the value of entry, the mass of (cid:0) (cid:1) entry,andotheraggregates: M V ,ME,... = 0. e,t t (v) Theinvestmentcostfunction,φ(i),isincreasingandconvex. 39

These assumptions not only nest the specification we consider in Section 2, but also can accommodateothercommonlyusedproductionfunctions(e.g.,CESatthefirmlevel),adjustment costs, and specifications for entry (e.g., exogenous entry or entry with a positive but bounded elasticity of entry to firm value). Our approach still works under these assumptions for the followingreasons: first,homogeneityofdegreeoneoffirmrevenuesintermsofoutputensures thattheundistortedfirm’sproblemishomogeneousofdegreeoneand,thus,undistortedinvestment rates are equal across firms. Second, homogeneity of the aggregation technology means that we can write total output as a function of the sum of total firm revenues, which implies a relationship between aggregate output and firm profits (and thus value functions). Third, the remaining assumptions imply that other forces, such as entry, depend only on aggregates and notonthedistributionoffirms. Additionally, we allow for the distortion to have exogenous deadweight losses as well, modelled as destruction of capital (for example, through bankruptcy or as a consequence of agency problems).41 Formally, we assume that the distortion may also destroy capital—let ξ = ξ (χ ) represent the fraction of capital destroyed due to the distortion. In the baseline j,t t j,t model,the(pre-tax)valueofthedistortedfirmasthediscountedpresentvalueofrevenuesless expensesbecomes VF(k,χ) = z ρ− ρ 1αck Π −k φ(i (χ ))+κ(1−ξ (χ))E [Λ VF (k(cid:48),χ(cid:48))]. (74) j,t j,t j,t t j,t t j,t t t t+1 t+1 Corollary 1.1 shows that an analogue of our sufficient statistics result applies to this more general setup. The main differences as compared with Proposition 1 are that (1) we include an additional sufficient statistic, the average amount of (per-period) capital destroyed due to distortion;and(2)thesecondsufficientstatisticisnotthemean-squareddistortiontoinvestment but rather a more general measure of the resources lost due to “misallocation” of investment across firms (unless the investment adjustment cost is quadratic, in which case it maps into the mean-squareddistortion). Corollary1.1. Assumeweknowthefollowing: (cid:82) (i) Average(capital-weighted)distortiontofirminvestment,i = ∆i(χ)γK (χ)dχ ∆ χ (ii) Average(capital-weighted)“investmentmisallocation,” (cid:82) φ(cid:98)= (φ(i(χ))−φ(i +i ))γK (χ)dχ χ MM ∆ (iii) Averageshareofcapitaldestroyedduetothedistortion,definedas (cid:82) γK = ξ(χ)γK (χ)dχ. Kloss χ (iv) Present-valueoftaxesnetofsubsidiesraisedbyagenerationoffirms 41Thisfeatureofthemodelcouldbefurthergeneralizedtoallowforthedistortiontomerelyaffectfirmvalue (butnotthecapitalstock). 40

(v) Model parameters:: β, κ, α , α , θ, and ρ, as well as parameters that enter the utility 1 2 function Then our environment leads to a system of equations that identifies bounds on the change (cid:8) (cid:9) in aggregate quantities and prices C,Y,K,I,W,Λ,Π,ME,VE,qMM,i when firm-level MM distortionsareremovedinarevenue-neutralcounterfactual,inboththelong-runandalongthe transitionpath. If, in addition, either β → 1 or entry is exogenous, then the system of equations identifies the exactchangeintheseaggregatequantitiesandprices,ratherthanbounds,bothinthelongrun andalongthetransitionpath. Proof. SeeOnlineAppendixO1.2. Notethatthesecondsufficientstatisticisnowamoregeneralnotionof“investmentmisallocation”insteadofthemean-squaredunderinvestment. Thisnotionofinvestmentmisallocation, φ(cid:98),measureshowmanyfewerunitsofthefinalgoodcouldhavebeenspenttoachievethesame average growth rate of firms, if investment had been optimally allocated across firms. With quadratic adjustment costs, this object is characterized by the mean and mean-squared underinvestment. A.4 Model Extension with Ex-ante Firm Heterogeneity With additional data moments, it is possible to extend our result to environments where there is richer heterogeneity across firms. In this subsection of the appendix, we demonstrate thesufficientstatisticsneededtouseananalogofourapproachforamodelwithheterogeneous sectors. Forthesakeofbrevity,wedonotincludethefullsystemofequationsasintheprevious sections,butrather,informallydiscusshowourapproachcanbeappliedtoricherenvironments andtheadditionaldatamomentsthatwouldbeneeded. ConsideranenvironmentwithnestedCESinwhichnsectorsproducedifferentiatedoutput: (cid:32) (cid:33) (cid:37) (cid:37)−1 Y t = (cid:88) (Y t n) (cid:37)− (cid:37) 1 (75) n (cid:18)(cid:90) (cid:19) ρn Y t n = (y j,t ) ρn ρn −1 ρn−1 . (76) j∈Jn Within each sector, firms produce output with technology Fn(z k ,l ), which is homogej,t j,t j,t neousofdegree ρn sothatfirmrevenuesarehomogeneousofdegreeoneininputs: ρn−1 Revn(z j,t k j,t ,l j,t ,Y t ) = Y t ρ 1 (Fn(z j,t k j,t ,l j,t )) ρn ρn −1 . (77) Eachsectoralsohasitsownconvexinvestmentcostfunctionkφn(i)andmayvaryintheextent of the distortion. The properties of our production function imply that we can write aggregate 41

outputasfollows: (cid:88) Y = KnRevn(1,Υn(W ,Y ),Y ). (78) t t t t t n Therefore,weneedtokeeptrackoftheaccumulationofeachsector’scapital: (cid:90) Kn = Mn,E +Kn κn in(χ)−ξn(χ)dj (79) t t t−1 t t j∈Jn (cid:90) (cid:88) I = cnMn,E + Knφ(in(χ))dj (80) t e t t t n j∈Jn Πn = Revn(1,Υn(W ,Y ),Y )−W Υn(W ,Y ). (81) t t t t t t t Exogenousentry Ifentryisexogenous,thentocharacterizethesteady-stateequilibriabothwithandwithout the distortion, we need to know equivalents for the three sufficient statistic moments for each sector: in,φ(cid:99)n, and γn,K . We, of course, also need the production and preference parameters. ∆ Kloss Theabovesystemofequationsthusbecomes (cid:16) (cid:17) Kn = Mn,E +Knκn in +in −γn,k (82) MM ∆ Kloss (cid:90) (cid:88) (cid:16) (cid:17) I = cnME + Kn φ(in +in)+φ(cid:99)n (83) e MM ∆ n j∈Jn Πn = Revn(1,Υn(W ,Y),Y)−WΥn(W,Y) (84) t VMM,n = Π−φ(in )+in κΛVMM,n (85) MM MM φ(cid:48)(i ) = κΛVMM,n, (86) MM,n withtherestoftheequilibriumequationsconsistingof(1-29)–(1-32)and(78). Endogenousentry There are now many types of firms. Assume that entering firms are identical, with an exogenousprobabilityofbecomingeachtypeoffirm. Notethatwecandecomposetheentering firm’sproblemasbefore: (cid:88) VE = VMM,n + VD,n +VTaxes,n. (87) n As before, VTaxes,n is pinned down by calibrated moments and the assumption of revenueneutralityofcounterfactuals,andVMM,n dependsonasystemofaggregatesandknownparameters and sufficient statistics. For each sector, we can bound VD,n by following the steps in the proof of proposition 1, where there is a bound VD,n such that there is some ηD,n ∈ [0,1] such that VD,n = ηD,nVD,n. Thus, the set of candidate equilibrium now depend on the exact 42

(cid:8) (cid:9) combinationof ηD,n ∈ [0,1]N. Foreachaggregate,themaximumandminimumvalueinthe steady-statewithdistortions,lessthevalueintheequilibriumwithoutdistortions,characterizes thesteady-stateeffectofthedistortion. A.5 Corollary 1.2: Aggregate Shocks Corollary1.2. ConsiderthemodelintroducedinSection2. Alsoassumethat • parametersA,δ,andκmaybetime-varyingandstochastic(aggregateshocks) • weknowthefunctionsi (S )andi (S ) ∆ t ∆2 t • entryisexogenous(Θ(ME,VE) = ME −ME) • anyadditionalequilibriumequationsfortheevolutionofS (besidesthosealreadyspect ified)areknownanddependonlyonaggregates Then our environment leads to a system of equations that identifies bounds on the change (cid:8) (cid:9) in aggregate quantities and prices C,Y,K,I,W,Λ,Π,ME,VE,qMM,i when firm-level MM distortionsareremovedinarevenue-neutralcounterfactual,inboththelongrunandalongthe transitionpath. If, in addition, either β → 1 or entry is exogenous, then the system of equations identifies the exactchangeintheseaggregatequantitiesandprices,ratherthanbounds,bothinthelongrun andalongthetransitionpath. Proof. SeeOnlineAppendixO1.3. A.6 Corollary 1.3: Decreasing Returns to Scale Corollary1.3. Assumeweknowthefollowing: (cid:82) (i)Average(capital-weighted)distortion,definedask = (∆k(χ))γK (χ)dχ ∆ χ (ii)Theconcavity-adjustedmeandistortion,definedas   ρ−1 1−(α1+α2) ρ k ∆α = (cid:82) χ 1−(1−∆k(χ)) 1−α2 ρ− ρ 1 γK (χ)dχ (iii)Present-valueoftaxesnetofsubsidiesraisedbyagenerationoffirms,i.e.,TaxPV (iv) Model parameters: β, κ, α , α , θ, and ρ, as well as the parameters that enter the utility 1 2 function Then our environment leads to a system of equations that identifies bounds on the change (cid:8) (cid:9) in aggregate quantities and prices C,Y,K,I,W,Λ,Π,ME,VE,qMM,i when firm-level MM distortionsareremovedinarevenue-neutralcounterfactual,inboththelongrunandalongthe transitionpath. If, in addition, either β → 1 or entry is exogenous, then the system of equations identifies the 43

exactchangeintheseaggregatequantitiesandprices,ratherthanbounds,bothinthelongrun andalongthetransitionpath. Proof. SeeOnlineAppendixO1.4. A.7 Corollary 1.4: More General Firm Productivity Shocks Assume that firm productivity z follows a Markov process with transition matrix T(z,z(cid:48)), has entry distribution TE(z), and has a finite number of possible realizations. Let us define γK (χ,z) as the capital-weighted distribution of vector χ (state variables that summarize distortion)conditionalonfirmproductivityz. Corollary1.4. Assumeweknowthefollowing: (i) Average (capital-weighted) distortion to firm investment, conditional on z: i (z) = ∆ (cid:82) ∆i(χ)γK (χ,z)dχ χ (ii) Average(capital-weighted)squareddistortiontofirminvestment: i (z) = (cid:82) (∆i(χ))2γK (χ,z)dχ ∆2 χ (iii) Present-valueoftaxesnetofsubsidiesperperiod (iv) Model parameters: β, κ, α , α , θ, ρ, T(z,z(cid:48)), and TE, as well as the parameters that 1 2 entertheutilityfunction Then our environment leads to a system of equations that identifies bounds on the change (cid:8) (cid:9) in aggregate quantities and prices C,Y,K,I,W,Λ,Π,ME,VE,qMM,i when firm-level MM distortionsareremovedinarevenue-neutralcounterfactual,inboththelongrunandalongthe transitionpath. If, in addition, either β → 1 or entry is exogenous, then the system of equations identifies the exactchangeintheseaggregatequantitiesandprices,ratherthanbounds,bothinthelongrun andalongthetransitionpath. Proof. SeeOnlineAppendixO1.5. A.8 Balanced Growth Path Consider a parameterization of our baseline model where 1 + φ = (φ+γ)α (ρ − 1). If 2 we want labor supply to be constant over time and not exhibit any growth, we must also have α (ρ−1) = 1. Further,letusassumethattherevenueraisedfromcorporatetaxeslesssubsidies 2 on the firms that enter each period growth proportionally in output. Corollary 1.5 shows that, in such an environment, the same sufficient statistics characterize (bound or exactly identify) howboththelevelandgrowthrateofaggregatevariableschangeinresponsetoeliminatingthe distortion. 44

Corollary 1.5. Assume that the economy with distortions is an equilibrium with a balanced growthpathandweknowthefollowing: (cid:82) (i) Averagecapital-weighteddistortiontofirminvestment: i = ∆i(χ)γK (χ)dχ ∆ χ (ii) Averagecapital-weightedsquareddistortiontofirminvestment: i = (cid:82) (∆i(χ))2γK (χ)dχ ∆2 χ (iii) Presentvalueoftaxesnetofsubsidiesasaproportionofoutput (iv) Modelparameters: β,κ,α ,α ,θ,ρ,andtheparametersthatentertheutilityfunction 1 2 Then our environment leads to a system of equations that identifies bounds on the change in both the level and growth rates of aggregate quantities and prices (cid:8) (cid:9) C,Y,K,I,W,Λ,Π,ME,VE,qMM,i when firm-level distortions are removed in a MM revenue-neutralcounterfactual,inboththelongrunandalongthetransitionpath. If, in addition, either β → 1 or entry is exogenous, then the system of equations identifies the exactchangeintheseaggregatequantitiesandprices,ratherthanbounds,bothinthelongrun andalongthetransitionpath. Proof. SeeOnlineAppendixO1.6. 45

Online Appendix TheOnlineAppendixcontainsfoursubsections. InSubsectionO1,weoutlineproofstothe corollariesandpresentadditionaltheoreticalresultsdiscussedinthetext. InSubsectionO2,we presentadditionalinformationonthequantitativeresults. InSubsectionO3,wepresentfurther details on the results when TFP follows a Markov process. Last, in Subsection O4, we present additionalfiguresandtablesreferencedinthetext. O1 Proofs to Corollaries and Additional Results In this section of the Online Appendix, we first outline the tighter bound than zero for the welfare change in the counterfactual described in Proposition 1 (in Subsection O1.1). We then outline the proofs to the corollaries (in Subsections O1.2-O1.6). Next, we present the derivation of the derivative of capital with respect to i (in Subsection O1.7). Last, we outline ∆ the planner’s problem (in Subsection O1.8) and the conditions on ρ and γ required for the problemtobewelldefined(inSubsectionO1.9). O1.1 A Tighter Bound than Zero for Losses DefineVD ≤ 0asthesolutiontothefollowingmaximizationproblem: ∞ (cid:18) (cid:19) (cid:88) θ VD = max − βnK i(n) , (1-1) (n) 2 ∆2 i(n),i(n),K ∆ ∆2 (n) n=0 subjecttothefollowingconstraints: (cid:16) (cid:17)2 i(n) ≥ i(n) (1-2) ∆2 ∆ K = 1 (1-3) (0) (cid:16) (cid:17) K = K κ i +i(n−1) (1-4) (n) (n−1) MM ∆ ∞ (cid:88) K = K (1-5) (n) n=0 ∞ (cid:88) K i = (n) i(n) (1-6) ∆ K ∆ n=0 ∞ (cid:88) K i = (n) i(n) (1-7) ∆2 K ∆2 n=0 (cid:16) (cid:17) 0 ≤ i +i(n) , (1-8) MM ∆ 1

wherei(n),i(n),measurehowmanyperiodssinceentrytheeffectsofthedistortionscapturedin ∆ ∆2 thesufficientstatisticsoccurred: (cid:90) i(n) = ∆i(χ)γK(χ) dχ (1-9) ∆ (n) χ (cid:90) i(n) = (∆i(χ))2γK(χ) dχ (1-10) ∆2 (n) χ Note that (1-2) comes from Jensen’s inequality, (1-3)–(1-5) describe the capital stock distributed over the periods since entry, (1-6)–(1-7) impose that the distribution of the sufficient statistics over the time since entry do indeed aggregate up to the sufficient statistics. Equation (1-8) imposes that the capital stock can never turn negative. If one wished to do so, one could alsoimposeadditionalrestrictions,suchasrequiringthedistortiontoalwaysreduceinvestment (i(n) ≤ 0),orlimitedliability,whichwouldmaketheboundtighter. ∆ SincetheallocationimpliesthatVDmustsatisfytheseconstraints,VD ≤ VD. Furthermore, since i(n) ≥ 0, it must be the case that VD ≤ 0. Therefore, the set of bounds can be tightened ∆2 toηD ∈ [VD/VD,1]. O1.2 Proof to Corollary 1.1: More General Class of Models The proof is largely analogous to that for Proposition 1. Here, we outline the additional elementsneeded: aggregationinthemoregeneralenvironmentandthecharacterizationoffirm valuewithamoregeneralinvestmentcostfunction. O1.2.1 Aggregation Firmschooselabortomaximizetheirrevenueslesslaborexpenses: maxRev(zk,l,Y)−W l. t l Homogeneityofdegreeoneimplieswecanwritethemaximizationproblemintermsof ˆ l = l : zk (cid:16) (cid:16) (cid:17) (cid:17) ˆ ˆ maxzk Rev 1,l,Y −W l . (1-11) t ˆl Laboristhuspaiditsmarginalproduct: ˆ W = MPL (l), (1-12) t t where ∂Rev(1,x,Y ) t MPL (x) = . (1-13) t ∂x 2

DefineΥ(W ,Y )astheinverseofthemarginalproductoflabor. Formally, t t ∂Rev(1,Υ,Y ) t W = . (1-14) t ∂x Wecanthenwriteaggregatelabordemand,aggregatefirmrevenues,andfirmrevenuesless expensesas L = K Υ(W ,Y ) (1-15) t t t t (cid:90) p y dj = K Rev(1,Υ(W ,Y ),Y ) (1-16) j,t j,t t t t t j Π = Rev(1,Υ(W ,Y ),Y )−W Υ(W ,Y ) (1-17) t t t t t t t p y −W l = z k Π . (1-18) j,t j,t t j,t j,t j,t t Recallthatweassumedtheaggregationtechnologyishomogeneousinitsinputs. Letusdenotethedegreeofhomogeneityasρˆ. Wecanthenobtainoutputasafunctionoffirmrevenues, thecapitalstock,andthedegreeofhomogeneity: (cid:90) 1 1 Y = p y dj = K Rev(1,Υ(W ,Y ),Y ). (1-19) t j,t j,t t t t t ρˆ ρˆ j O1.2.2 LosstoFirmValueTermVD Following the same steps we followed to obtain (54) in the proof for Proposition 1, we can showthatVD inthismoregeneralsettingis (cid:90) (cid:18) (cid:19) VD = K ¯ γˆK(χ) κβqMM(i(χ)−i −ξ(χ))−(φ(i(χ))−φ(i )) dχ. (1-20) MM MM χ Additionally, (cid:18) (cid:19) κβqMM(i(χ)−i −ξ(χ))−(φ(i(χ))−φ(i )) < 0. (1-21) MM MM Sothevalueofi(χ)thatmaximizesthisexpressionisi(χ) = i . MM Also, (cid:18) (cid:19) κβqMM(i(χ)−i −ξ(χ))−(φ(i(χ))−φ(i )) ≤ −κβqMMξ(χ) ≤ 0. (1-22) MM MM ItthusfollowsthatVD ≤ 0andasβ → 1,γˆK → γK. Therefore, (cid:90) (cid:18) (cid:19) VD → K ¯ γK(χ) κβqMM(i(χ)−i −ξ(χ))−(φ(i(χ))−φ(i )) dχ MM MM χ 3

(cid:18) (cid:19) (cid:16) (cid:17) = K ¯ κβqMM(i −γK )− φ(cid:98)+φ(i +i )−φ(i ) . (1-23) ∆ Kloss MM ∆ MM O1.2.3 Steady-StateSystemofEquations Given ηD ∈ [0,1], the equilibrium system given by K, I, C, L, Y, W, Π, ME, Λ, qMM, i ,VE,VD,andVTaxes ischaracterizedbythefollowingequations: MM (cid:0) (cid:1) K = ME +Kκ i +i −γK (1-24) MM ∆ Kloss I = MEc (ME,VE,...)+Kφ(i +i )+Kφ(cid:98) (1-25) e MM ∆ Π = Rev(1,Υ(W,Y),Y)−WΥ(W,Y) (1-26) qMM = Π−φ(i )+i κΛqMM (1-27) MM MM φ(cid:48)(i ) = κΛqMM (1-28) MM Λ = β (1-29) −U (C,L) L W = (1-30) U (C,L) C L = K ∗MPL−1(W) (1-31) C = Y −I (1-32) 1 Y = K ∗Rev(1,Υ(W,Y),Y) (1-33) ρˆ 0 = Θ(ME,VE,...) (1-34) VE = qMM +VD +VTaxes (1-35) TaxPV VTaxes = (1-36) ME (cid:18) K (cid:16) (cid:17) VD = −ηD κβqMM(i −γK )− φ(cid:98)+φ(i +i )−φ(i ) , (1-37) ME ∆ Kloss MM ∆ MM wherethefunctionsΥ andRev comefromtheproductionfunction. IftheentryconditiondoesnotdependonVE (forexample,entryisexogenous),thenequations (1-24)–(1-34) characterize the relevant aggregates. If not, it is the case that as β → 1, (cid:18) (cid:16) (cid:17) VD → K κβVMM(i −γK )− φ(cid:98)+φ(i +i )−φ(i ) ,andthesystemofequa- ME ∆ Kloss MM ∆ MM tions (1-24)–(1-37) exactly characterizes the steady-state equilibrium. We thus have a similar boundresulttothatshownintheproofofProposition1. Inturn,thesteady-statecounterfactuals canbeperformedasoutlinedintheproofofProposition1. 4

O1.2.4 TransitionDynamics Notethatwithoutdistortions,thedynamicsof12unknownsK ,I ,C ,L ,Y ,W ,Π ,ME, t t t t t t t t Λ ,qMM,i ,andVE arecharacterizedbythefollowingequilibriumsystem: t t MM,t t K = ME +K κi (1-38) t t t−1 MM,t−1 I = c ME +K φ(i ) (1-39) t e t t MM,t Π = Rev(1,Υ(W ,Y ),Y )−W Υ(W ,Y ) (1-40) t t t t t t t qMM = Π −φ(i )+i κΛ qMM (1-41) t t MM,t MM,t t t φ(cid:48)(i ) = κΛ qMM (1-42) MM,t t t U (C ,L ) C t t Λ = β (1-43) t U (C ,L ) C t−1 t−1 −U (C ,L ) L t t W = (1-44) U (C ,L ) C t t L = K MPL−1(W ) (1-45) t t t t C = Y −I (1-46) t t t 1 Y = K Rev(1,Υ(W ,Y ),Y ) (1-47) t t t t t ρˆ 0 = Θ(ME,VE) (1-48) t t TaxPV VE = qMM + . (1-49) t t ME t Therefore,transitiondynamicsinwhichwebegininthesteady-statewiththedistortionand thenremoveitmaybecomputedasoutlinedintheproofofProposition1. O1.3 Proof to Corollary 1.2: Aggregate Shocks Proof: Notethatwecanwritetheaggregatesystemofequationsasfollows: K = ME +K κ (i +i ) t t t−1 t MM,t−1 ∆t−1 (cid:18) (cid:19) θ I = c ME +K φ(i +i )+ (cid:0) i −(i )2(cid:1) t e t t MM,t ∆t 2 ∆2t ∆t Π t = (cid:32) A t ρ− ρ 1 Y t ρ 1 (W t )−α2(ρ− ρ 1) (cid:18) α 2 ( ρ− ρ 1 ) (cid:19)α2(ρ− ρ 1) (cid:33)αc(cid:18) 1−α 2 ( ρ− ρ 1 ) (cid:19) qMM = Π −φ(i )+i κΛ qMM t t MM,t MM,t t t φ(cid:48)(i ) = κΛ qMM MM,t t t 5

U (C ,L ) C t t Λ = β t U (C ,L ) C t−1 t−1 −U (C ,L ) L t t W = t U (C ,L ) C t t (cid:18) ρ−1 (cid:19)αc L t = K t α 2 ( )A ρ− ρ 1 (W t )−1Y ρ 1 ρ C = Y −I t t t Π t Y = K t t(cid:16) (cid:17) 1−α ρ−1 2 ρ ME = ME, t wherei andi aredefinedin(29). ∆t ∆2t Let SA = K ,I ,Π ,qMM,i ,Λ ,W ,L ,C ,Y ,δ ,A ,κ denote aggregates and exoget t t t t MM,t t t t t t t t t nous variables, and let SX denote any additional state variables that may affect the distortions t at the firm level. Letting S = SA,SX, note that if we know the functions i (S ),i2 (S ), as t t t ∆ t ∆ t wellastheequilibriumconditionforotherstatevariablesF (cid:0) SA,SX,SX ,E (cid:2) SX (cid:3)(cid:1) = 0,then t t t−1 t t thefollowingequationscharacterizetheequilibriumsystemasafunctionofparametersandthe processofexogenousshocksforδ ,A ,κ : t t t K = ME +K κ (i +i (S )) t t t−1 t MM,t−1 ∆ t−1 (cid:18) (cid:19) θ I = c ME +K φ (i +i (S ))+ (cid:0) i2 (S )−(i (S ))2(cid:1) t e t t t MM,t ∆ t 2 ∆ t ∆ t Π t = (cid:32) A t ρ− ρ 1 Y t ρ 1 (W t )−α2(ρ− ρ 1) (cid:18) α 2 ( ρ− ρ 1 ) (cid:19)α2(ρ− ρ 1) (cid:33)αc(cid:18) 1−α 2 ( ρ− ρ 1 ) (cid:19) qMM = Π −φ (i )+i κΛ qMM t t t MM,t MM,t t t φ(cid:48)(i ) = κΛ qMM MM,t t t U (C ,L ) C t t Λ = β t U (C ,L ) C t−1 t−1 −U (C ,L ) L t t W = t U (C ,L ) C t t (cid:18) ρ−1 (cid:19)αc L t = K t α 2 ( )A ρ− ρ 1 (W t )−1Y ρ 1 ρ C = Y −I t t t Π t Y = K t t(cid:16) (cid:17) 1−α ρ−1 2 ρ ME = ME t 0 = F (cid:0) SA,SX,SX ,E (cid:2) SX (cid:3)(cid:1) . t t t−1 t t 6

O1.4 Proof to Corollary 1.3: Decreasing Returns to Scale ThelogicoftheproofissimilartothatofProposition1,butthederivationswithdecreasing returnsaredifferent. Theproblemofanundistortedfirmisasfollows: V (A ,K ) = max (cid:0) A Kα1 Lα2 (cid:1)ρ− ρ 1 Y ρ 1 −W L +((1−δ)K −K ) t j,t j,t−1 j,t j,t−1 j,t t t t j,t−1 j,t Kj,t,Lj,t−1 + E [κΛ V (A ,K )]. (1-50) t t+1 t+1 j,t+1 j,t Let ω = 1 denote the gross distortion to investment (for shorter notation). Define j,t 1−∆kj,t expectedprofitsfrominvestmentas Profits = −K +κβE [P Y −W L +(1−δ)K P ]. j,t−1 jt t jt jt t jt jt K Further, let X denote the expectation of (convexity-adjusted) productivity, and E the unexj,t j,t pectedshocktoproductivity: ρ−1 (cid:34) (cid:18) (cid:19) 1 (cid:35) 1−α2 ρ ρ−1 X = E A ρ− ρ 1 1−α2 ρ− ρ 1 1−(α1+α2) ρ j,t−1 t−1 jt (cid:18) (cid:19) 1 ρ−1 ρ−1 A ρ 1−α2 ρ jt E = . j,t (cid:34) (cid:18) (cid:19) 1 (cid:35) ρ−1 ρ−1 E A ρ 1−α2 ρ t−1 jt We can then write firm capital, labor, revenue, and profits in steady state (with the distortion) as (cid:18) (cid:19) ρ−1 κβ K = Π α ω X jt ρ 1 rk jt−1 j,t−1 ρ−1 α1 ρ ρ−1 1 ρ−1 L jt = (ω jt−1 )1−α2 ρ α 2 ΠX j,t−1 E j,t ρ W ρ−1 α1 ρ ρ−1 P jt Y jt = (ω jt−1 )1−α2 ρ ΠX j,t−1 E j,t   ρ−1 α1 ρ (cid:18) ρ−1 (cid:19) ρ−1 ρ−1 E t−1 [Profits jt ] = κβΠX j,t−1(ω jt−1 )1−α2 ρ 1− α 2 −ω jt−1 α 1, ρ ρ whererk isameasureofthecostofcapitalandΠisanaggregatescalingfactor: rk = 1−κβ(1−δ) (1-51) ρ−1 Π = (cid:16) Y ρ 1 (cid:17) 1−(α1+ 1 α2) ρ− ρ 1 (cid:18) ρ−1 (cid:19) 1− (α (α 1+ 1+ α α 2) 2) ρ ρ− ρ 1 (cid:16)α 1 (cid:17) 1−(α α 1 1 + ρ α − ρ 2 1 ) ρ− ρ 1 (cid:16)α 2 (cid:17) 1−(α α 1 2 + ρ α − ρ 2 1 ) ρ− ρ 1 . ρ rk W 7

Note that setting ω = 1 results in the choices and profits of an undistorted firm (holding j,t aggregates prices as given). Since there are no aggregate shocks and all idiosyncratic shocks are assumed to be uncorrelated with previous state variables, we can write aggregate output, capital,andlaboras: (cid:18) (cid:19) ρ−1 κβ (cid:88) K = Π α ω X ρ 1 rk j j j ρ−1 ρ−1 1 (cid:88) α1 ρ ρ−1 L = α 2 Π (ω j )1−α2 ρ X j ρ W j ρ−1 (cid:88) α1 ρ ρ−1 Y = Π (ω jt−1 )1−α2 ρ X j , j whichcanbeexpressedalsoasafunctionofsufficientstatistics: (cid:18) (cid:19) ¯ ρ−1 κβ X K = Π α (1-52) ρ 1 rk (1−k ) t−1 ∆ K Y = (1−k ) (1-53) (cid:16) (cid:17) ∆α ρ−1α κβ ρ 1rk Y ρ−1 W = α , (1-54) 2 L ρ ¯ (cid:82) whereX = X dj isanaggregateindexof(expected)firmproductivity. j j Note that, using the expected profits function, we find that the value of a firm that enters withnocapitalandexpectedproductivitymeasureX anddistortionω hasexpectedvalue: j,t j,t   ρ−1 α1 ρ (cid:18) ρ−1 (cid:19) ρ−1 V (X jt ,ω jt ) = κΛΠX jt(ω jt )1−α2 ρ− ρ 1 1− α 2 −ω jt α 1+κβE t [V (X jt+1 ,ω jt+1 )]. ρ ρ Thus, a firm entering with capital K , current productivity A , expected productivity j,t j,t measureX anddistortionω hasexpectedvalue: j,t j,t ρ−1 V (X jt ,ω jt )+ (cid:18) A j ρ t − ρ 1 (cid:19) 1−α2 1 ρ− ρ 1 (cid:16) (cid:0) K j α t 1 (cid:1)ρ− ρ 1 Y ρ 1 (cid:17) 1−α2 1 ρ− ρ 1 (cid:32) α 2 W ρ− ρ 1 (cid:33) 1− α α 2 2 ρ ρ− ρ 1 (cid:18) 1−α 2 ρ− ρ 1 (cid:19) . As in the proof of Proposition 1, we can decompose the value of an entering firm as VE = VMM,E +VD +VTaxes. 8

The(pre-tax)valueofanundistortedfirmenteringwithaunitofcapitalcanbewrittenas   ρ−1 VMM,E = E   qMM(X jt )+ (cid:18) A j ρ t − ρ 1 (cid:19) 1−α2 1 ρ− ρ 1 (cid:16) Y ρ 1 (cid:17) 1−α2 1 ρ− ρ 1 (cid:32) α 2 W ρ− ρ 1 (cid:33) 1− α α 2 2 ρ ρ− ρ 1 (cid:18) 1−α 2 ρ− ρ 1 (cid:19)    (cid:18)(cid:18) (cid:19) (cid:19) ρ−1 ρ−1 qMM(X ) = κΛΠX 1− α − α +κβE (cid:2) qMM(X ) (cid:3) . jt jt 2 1 t jt+1 ρ ρ Additionally,VD canbeexpressedas     ρ−1 (cid:88) α1 ρ (cid:18) ρ−1 (cid:19) ρ−1 VD = κΛΠE  (βκ)tX j,τ (ω j,τ )1−α2 ρ− ρ 1 −1 1− α 2 −(ω j,τ −1) α 1. ρ ρ τ As firms are ex-ante homogeneous before entering, the distortions can be written as a function oftheentry-discounteddistributionoffirms: ¯ (cid:90) (cid:18) (cid:19)(cid:18)(cid:18) X VD = κΛΠ γˆK χ ∆k(χ) (1−k ) ∆ j ρ−1 (cid:18) (cid:18) (cid:19) 1−(α1+α2) ρ (cid:19)(cid:19)(cid:18) (cid:19) (cid:19) ρ−1 ρ−1 ρ−1 − 1− 1−∆k(χ) 1−α2 ρ 1− α −∆k(χ) α . 2 1 ρ ρ Analogously to what we showed in the proof of Proposition 1, the effect of the distortion on pre-tax firm value is always negative (from the FOC of the capital choice). Additionally, since it is a function of the time-discounted stream of losses, the losses cannot be as great as thenot-time-discountedvalueoflosses(whicharefiniteifafirm’slifetimeprofitsarefinite): VD ≤ VD ≤ 0, wherethelowerboundV canbewrittenasafunctionofoursufficientstatistics: D ¯ (cid:18) (cid:18) (cid:19) (cid:19) X ρ−1 ρ−1 VD = κΠ (k −k ) 1− α −k α . (1-55) ∆ ∆α 2 ∆ 1 (1−k ) ρ ρ ∆ Notethatifβ = 1,thenVD = VD;ifthedistortionsareallzero,thenVD = 0. Let us thus again define η ∈ [0,1], and note that we can write VD = ηVD. η indexes a continuumof“possible”equilibrianotruledoutbythesufficientstatisticswhendistortionsare present. Theequilibriumsystemisthuscharacterizedby(1-51)–(1-63)andthefollowingaggregate ¯ equations, as well as the (exogenous) process for firm productivity, which defines X (possibly influencedbyentry): 9

Λ = β (1-56) −U (C,L) L W = (1-57) U (C,L) C C = Y −I (1-58) I = MEc +δK (1-59) e 0 = Θ(ME,VE) (1-60) VE = VMM,E +VD +VTaxes (1-61) TaxPV VTaxes = (1-62) ME VD = ηVD. (1-63) O1.5 Proof to Corollary 1.4: More General Firm Productivity Shocks ExogenousentryIfentryisexogenous,theequilibriumsystemischaracterizedbyparameters andproductivity-dependentsufficientstatisticsi (z)andi (z): ∆ ∆2 Π = (cid:32) A ρ− ρ 1 Y ρ 1 (W)−α2(ρ− ρ 1) (cid:18) α 2 ( ρ−1 ) (cid:19)α2(ρ− ρ 1) (cid:33)αc(cid:18) 1−α 2 ( ρ−1 ) (cid:19) ρ ρ (cid:88) qMM(z) = Πz −φ(i (z))+i (z)κΛ T (z,z(cid:48))qMM(z(cid:48)) MM MM z(cid:48) (cid:88) φ(cid:48)(i ) = κΛ T (z,z(cid:48))qMM(z(cid:48)) MM z(cid:48) Λ = β −U (C,L) L W = U (C,L) C (cid:18) ρ−1 (cid:19)αc L = K α 2 ( )A ρ− ρ 1 (W)−1Y ρ 1 ρ C = Y −I Π (cid:88) Y = zK(z) (cid:16) (cid:17) 1−α ρ−1 2 ρ 0 = Θ(ME,VE). (cid:88) K(z(cid:48)) = METE(z(cid:48))+ K(z)T (z,z(cid:48))κ(i (z)+i (z)) MM ∆ z (cid:18) (cid:19) I = MEc + (cid:88) K(z) φ(i (z)+i (z))+ θ (cid:0) i (z)−(i (z))2(cid:1) . E MM ∆ 2 ∆2 ∆ z Adding endogenous entry This derivation follows the proof of Proposition 1. It requires definingseveralobjects. 10

NotationandpreliminariesDefinethedistributionoffirmsnperiodsafterentryas (cid:82) kΓ (k,χ,z)dk γK (χ,z) = k (n) . (n) (cid:82) kΓ (k,χ,z)dkdχdz χ,k,z (n) Further,defineentry-scaledversionsofthecapitalstockmeasures: K(z) K(z) = . M e Note that this expression can be split up into the time-since-entry versions, K (z), where (n) K (z) = TE(z)isjustafunctionofentryprobabilities,andforn > 0: (0) (cid:18)(cid:90) (cid:19) (cid:88) K (z) = κ γK (χ,z)i(χ,z )dχ K (z )T (z ,z). (n) (n) −1 (n−1) −1 −1 z−1 Wecanfurtherdefineatime-discountedversionofK (z),whichwecallK(cid:101) (z): (n) (n) (cid:18)(cid:90) (cid:19) (cid:88) K(cid:101) (z) = κβ γK (χ,z)i(χ,z )dχ K(cid:101) (z )T (z ,z). (n) (n) −1 (n−1) −1 −1 z−1 Notethattheaggregate-timediscountedcapitalstockisthen: ∞ ∞ (cid:88) (cid:88) K(cid:101) (z) = K(cid:101) (z) = βnK (z). (n) (n) n=0 n=0 We can therefore also define analogous versions of these objects as if choices were made onlybyModigliani-Millerfirms(e.g.,ifi(χ,z ) = i (z)),yieldingK(cid:101)MM(z)andK(cid:101)MM(z). −1 MM (n) Define∆K(cid:101) (z) = K(cid:101) (z)−K(cid:101)MM(z);thisexpressionimplies: (cid:88) (cid:88) ∆K(cid:101) (z) = K(cid:101) (z)T (z,z(cid:48))βκ(cid:101)i(z)− K(cid:101) MM(z)T (z,z(cid:48))βκ(i MM (z)), z z where (cid:32) (cid:33) K(cid:103) (z) (cid:90) (cid:101)i(z) = (n) γ ( K n) (χ,z)i(χ,z −1 )dχ . K(cid:101) (z) Thiscanbere-writtenas (cid:88) (cid:16) (cid:17) (cid:88)(cid:16) (cid:17) ∆K(cid:101) (z) = K(cid:101) (z)T (z,z(cid:48))βκ ∆(cid:102)i(z) + ∆K(cid:101) (z) T (z,z(cid:48))βκ(i (z)), MM z z (cid:16) (cid:17) where ∆(cid:102)i(z) =(cid:101)i(z)−i MM (z). 11

Then,inmatrixnotation,notethatwecanwrite: (cid:16) (cid:17) ∆K(cid:101) = βκK(cid:101)diag ∆(cid:102)i T +βκ∆K(cid:101)diag(i )T MM (cid:16) (cid:17) = βκK(cid:101)diag ∆(cid:102)i T (I −βκdiag(i )T)−1. MM AndtheModigliani-Millerfirm’svaluecanbewrittenas qMM = (Πz −φ(i ))+κβdiag(i )TqMM MM MM = (I −κβdiag(i )T)−1(Πz −φ(i )). MM MM Firm value lost due to distortions The firm value lost to distortions can be written as the differenceindiscountedcashflows: (cid:18) (cid:90) (cid:19) (cid:88) (cid:88)(cid:94) VD = K(cid:101) (z) Πz − γK (χ,z)φ(i(χ,z))dχ − KMM (z)(Πz −φ(i (z))), (n) (n) (n) MM n,z χ n,z whichcanbesimplifiedtoyield: (cid:18) (cid:90) (cid:19) (cid:18) (cid:19) (cid:88) (cid:88) (cid:94) VD = K(cid:101)(n) (z) φ(i MM (z))− γ ( K n) (χ,z)φ(i(χ,z))dχ + K(cid:101)(n) (z)−K ( M n) M(z) (Πz−φ(i MM (z))) n,z χ n,z (cid:88) (cid:18) (cid:90) (cid:19) (cid:88)(cid:16) (cid:94) (cid:17) VD = K(cid:101)(n) (z) φ(i MM (z))− γ ( K n) (χ,z)φ(i(χ,z))dχ + K(cid:101)(z)−KMM(z) (Πz−φ(i MM (z))) n,z χ z (cid:88) (cid:18) (cid:90) (cid:19) (cid:88)(cid:16) (cid:17) VD = K(cid:101)(n) (z) φ(i MM (z))− γ ( K n) (χ,z)φ(i(χ,z))dχ + ∆K(cid:101)(z) (Πz−φ(i MM (z))). n,z χ z (cid:16) (cid:17) Note that this second term can be written in matrix form as ∆K(cid:101) diag(Πz −φ(i (z))). MM Plugginginforthematrixformof∆K(cid:101) yields: (cid:16) (cid:17) (cid:16) (cid:17) ∆K(cid:101) diag(Πz −φ(i (z))) = βκK(cid:101)diag ∆(cid:102)i T (I −βκdiag(i )T)−1diag((Πz −φ(i )). MM MM MM PlugginginforthematrixformofqMM yields: (cid:16) (cid:17) (cid:16) (cid:17) (cid:0) (cid:1) ∆K(cid:101) diag(Πz −φ(i (z))) = K(cid:101)diag ∆(cid:102)i Tdiag βκqMM . MM PluggingthisequationintoVD yields (cid:18) (cid:18)(cid:90) (cid:19)(cid:19) (cid:88) VD = K(cid:101) (z) φ(i (z))− γK (χ,z)φ(i(χ,z))dχ (n) MM (n) z,n χ (cid:18)(cid:18)(cid:90) (cid:19) (cid:19) (cid:88) (cid:88) + K(cid:101) (z) γK (χ,z)(i(χ,z))dχ −i (z) T (z,z(cid:48))βκq (z(cid:48)), (n) (n) MM MM z,n χ z(cid:48) 12

whichcanbewrittenintheform: (cid:90) (cid:88) VD = K(cid:101) (z) γK (χ,z)F (χ,z)dχ, (n) (n) z,n χ where (cid:88) F (χ,z) = (φ(i (z))−φ(i(χ,z)))+((i(χ,z))−i (z)) T (z,z(cid:48))βκq (z(cid:48)). MM MM MM z(cid:48) Note that F (χ,z) ≤ 0, as the problem of maximizing i(χ,z) results in the same optimality condition as for the Modigliani-Miller firm’s investment. Therefore, F (χ,z) is maximized if i(χ,z) = i (z),inwhichcaseF (χ,z) = 0. MM ItfollowsimmediatelythatVD ≤ 0. NotethatsinceK(cid:101) (z) = K (z)βn andF (χ,z) ≤ 0, (n) (n) (cid:90) (cid:90) (cid:88) (cid:88) VD = K βn(z) γK (χ,z)F (χ,z)dχ ≥ K (z) γK (χ,z)F (χ,z)dχ. (n) (n) (n) (n) z,n χ z,n χ Definethisnot-time-discountedobjectasVD,andnotethatitcanbesimplifiedtobeexpressed asafunctionofoursufficientstatistics: (cid:90) (cid:88) VD = K (z) γK (χ,z)F (χ,z)dχ (n) (n) z,n χ (cid:18) (cid:19) = (cid:88) K(z) φ(i (z))−φ(i (z)+i (z))− θ (cid:0) i (z)−(i (z))2(cid:1) ME MM MM ∆ 2 ∆2 ∆ z (cid:88) K(z) (cid:88) + i T (z,z(cid:48))βκq (z(cid:48)). ME ∆ MM z z(cid:48) NotethatVD isthusafunctionofsufficientstatisticsandaggregatesfromoursystemofequations. Thus, we can follow the proof of Proposition 1, noting that we can write VD = ηDVD for some ηD ∈ [0,1]. There is thus a continuum of these “possible equilibria”, and the maximumandminimumofeachaggregateendogenousvariableacrossthesepossibleequilibriacan beusedtoconstructbounds. Ifthereisnotimediscounting,thenηD = 1. O1.6 Proof to Corollary 1.5: Balanced Growth Path The proof is analogous to the proof of Proposition 1. For brevity here we only highlight theadditionalstepsrequired. Thekeyinsightistonotethatwecanwritethesetofequilibrium equationsas Π = τ (cid:32) (cid:0) Y (cid:1) ρ 1 (cid:0) W (cid:1)−α2(ρ− ρ 1) (cid:18) τ α ( ρ−1 ) (cid:19)α2(ρ− ρ 1) (cid:33)αc(cid:18) 1−α ( ρ−1 ) (cid:19) (1-64) g g 2 2 ρ ρ 13

KΠ Y = (1-65) (cid:16) (cid:17) 1−α ρ−1 τ 2 ρ g C = Y +I (1-66) (cid:18) (cid:19) (cid:16) (cid:17) θ I = K φ iMM +i + (cid:0) i −(i )2(cid:1) +c M (1-67) ∆ 2 ∆2 ∆ e E ρ−1 Y W = τ α ( ) (1-68) g 2 ρ L (cid:16) (cid:17) Kg = Kκ iMM +i +g M (1-69) K ∆ K E (cid:0) (cid:1)γ(cid:0) (cid:1)ϕ W = C L (1-70) Λ = β(g )−γ (1-71) C qMM = Π−φ(iMM)+iMMκE [ΛqMM] (1-72) t φ(cid:48)(iMM) = κE [ΛqMM] (1-73) t Yτ V = qMM − −V (1-74) E D M E V = c (1-75) E e (cid:18) (cid:19) 1 θ V = −ηD i (1-76) D 1−κβ(g )−γ(i +i ) 2 ∆2 C MM ∆ g = {g ,g ,g ,g } (1-77) K I C M Y 1 g W = (g Y )α2(ρ−1) (1-78) g Y g = , (1-79) L g W whereinournotation,anendogenousvariableX canbewrittenasX = X ¯ gt. Givenaninitial t x capitalstock,thesufficientstatisticsandparametersthuscharacterizeanequilibriumpath. The growth path with the distortion resolved can be computed by setting i and i equal to zero ∆ ∆2 and using the initial capital stock from the equilibrium with distortions. The value of a unit of capital and undistorted firm q is constant along the balanced growth path, so the solution to the value of an entering firm and the bounds are unchanged from those derived in the proof to Proposition1.42 O1.7 Derivative of Capital with respect to i ∆ WiththefunctionalformassumptionsinthemodelinSection2,ifweassumethatutilityis linearinconsumptionbuthasdisutilityinleisure L1+ϕ,wecansimplifysteady-stateexpressions 1+ϕ forK,Π,i asfollows: MM 42Thereareadditionalconditionsthatneedtobesatisfiedforabalancedgrowthratetoexist. Forinstance,we haveaconditionontheequilibriumgrowthrateofcapital:g = κ(iMM+i∆ ) >1(otherwiseweareinastationary K 1−ME K equilibrium). Further, for investment to be finite, we must have Tobin’s q be finite. Therefore, we must have κβg−γiMM <1. Similarly,forthelosstothedistortiontobefinite,wemusthave1≥κβ(g )−γ(i +i ). K C MM ∆ 14

ρ− 1 1(1+ϕ)−ϕα2 (cid:16) ρ−1 (cid:17) ρ−1 (1+ϕ) (cid:18) ρ−1 (cid:19) (1−α α 2 2 +ϕ) (cid:18) ρ−1 (cid:19) Π = K (1−α2+ϕ) (A) ρ ρ (1−α2+ϕ) α 2 1−α 2 (1-80) ρ ρ Π−φ(i ) φ(cid:48)(i ) = βκ MM (1-81) MM 1−κβi MM M e K = . (1-82) 1−κ(i +i ) MM ∆ Ifwethentakethederivativesoftheseexpressionswithrespecttoi ,weobtain: ∆ ∂Π ∂K ρ− 1 1 (1+ϕ)−ϕα 2 Π = (1-83) ∂i ∂i (1−α +ϕ) K ∆ ∆ 2 ∂Π ∂i βκ = (φ(cid:48)(cid:48)(i )(1−κβi )) MM (1-84) MM MM ∂i ∂i ∆ ∆ (cid:18) (cid:19) ∂M ∂K ∂i e MM = (1−κ(i +i ))−Kκ +1 . (1-85) MM ∆ ∂i ∂i ∂i ∆ ∆ ∆ Combiningtheseexpressionsyieldsthefollowingexpressionfor ∂K: ∂i∆ ∂K Kκ+ ∂Me = ∂i∆ . (1-86) ∂i ∆ 1−κ(i +i )−κΠ ϕα2− ρ− 1 1 (1+ϕ) βκ MM ∆ (1−α2+ϕ) (φ(cid:48)(cid:48)(iMM)(1−κβiMM)) Ifentryisexogenous,wecanfurthershowthedampeningeffectoftheinverseFrischϕand CEStermρ: ∂∂K Kκ2Π (1−α2+ ρ− 1 1 α2 ) βκ ∂i∆ = − (1−α2+ϕ)2 (φ(cid:48)(cid:48)(iMM)(1−κβiMM)) < 0 (1-87) ∂ϕ (cid:18) (cid:19)2 1−κ(i +i )+κΠ ϕα2− ρ− 1 1 (1+ϕ) βκ MM ∆ (1−α2+ϕ) (φ(cid:48)(cid:48)(iMM)(1−κβiMM)) ∂∂K Kκ2Π (1+ϕ) βκ ∂i∆ = − (ρ−1)2(1−α2+ϕ)(φ(cid:48)(cid:48)(iMM)(1−κβiMM)) < 0. (1-88) ∂ρ (cid:18) (cid:19)2 1−κ(i +i )+κΠ ϕα2− ρ− 1 1 (1+ϕ) βκ MM ∆ (1−α2+ϕ) (φ(cid:48)(cid:48)(iMM)(1−κβiMM)) O1.8 Planner’s Problem ThemodelinSection2hastwoadditionaldistortionsoutsideofthefirm-leveldistortionsto investment: corporatetaxationandadistortionduetothemonopolymarkup. Thesedistortions can be undone by setting the corporate tax rate to zero and including a subsidy, τs = ρ , in ρ−1 thesteady-statesystemequations(40)and(42)asfollows (cid:18) ρ−1 (cid:19)αc L = K τsα 2 ( )A ρ− ρ 1 (W)−1Y ρ 1 (1-89) ρ 15

Π Y = K . (cid:16) (cid:17) τs 1−α ρ−1 2 ρ Withthetaxationandmonopolymarkupdistortionsremoved,the“nofriction”casethatremovestheinvestmentdistortionsisequivalenttotheplanner’sproblem. InTable2,wepresent results when β → 1 from the planner’s problem (labelled as the “no friction” case) compared withthecasewithinvestmentdistortionsunderazerocorporatetaxrateandthesubsidyincorporatedasabove. UnliketheresultspresentedinSection4,thegainsfromresolvingthedistortionarealways positive no matter if it is an overinvestment or underinvestment distortion. This result is to be expected, as the investment distortions take us away from the first-best case. We see the GE welfare effects of the friction are still modest compared with the PE effects when we are comparingwiththecasewherethereisalsozerotaxationandnosubsidy. Ifweweretocompare the effects to the case where there is taxation and no subsidy, then the GE welfare gains would belarge. O1.9 Conditions on ρ and γ Here, we show that certain combinations of ρ, γ, and other parameters lead to the problem being undefined, as there is explosive growth. Consider the utility function in our quantitative calibration in Section 4. In an equilibrium with explosive growth, there will be no entry, so we willhavethefollowingblockofconditions: ρ−1Y ξCγLϕ = α t t t 2 ρ L t Y = AKρ− ρ 1 −α2Lα2 t t t C = Y −K φ(i ). t t t t Wecanreducethisblockto (cid:32) (cid:33) 1 α ρ−1 1+ϕ 2 ρ L = Y . (1-90) t ξ(Y −K φ(i ))γ t t t t Ifwethenplugthisequationintooutputandrearrange,weobtain (cid:18) Y t (cid:19)1− 1 α + 2 ϕ (cid:18) Y t −φ(i ) (cid:19)γ 1 α + 2 ϕ = AKρ− ρ 1 −α2+α21 1 + − ϕ γ−1 (cid:32) α 2 ρ− ρ 1s t (cid:33) 1 α + 2 ϕ . K K t t ξ t t 16

This expression implies ρ − α + α 1−γ − 1 ≤ 0 as to prevent unstable equilibria. So we ρ−1 2 21+ϕ haveaconditiononρof 1 1+ϕ 1+ ≤ ρ. (1-91) α ϕ+γ 2 O2 Additional Information on the Quantitative Results InthissectionoftheOnlineAppendix,wefirstpresentthemappingbetweenourmodeland that of Stein (2003) (in Subsection B.1). Second, we present further details on how we obtain thesufficientstatisticsfromHennessyetal.(2007)andBen-Davidetal.(2013)(inSubsections O2.2 and O2.3). Last, we present further details on the robustness of the quantitative results to themodelparameters(inSubsectionO2.4). O2.1 Mapping between our Model and Stein (2003) Stein(2003)setsupasimpleproblemofdistortedinvestmentduetoexternalequity(equation(2)inhispaper),whichwereproducebelow: f (I) max −I −θC(e). (1-92) I 1+r Here,I = e+w (whereIisinvestment,eisexternalfinanceraisedexternally,andw iswealth orinternalresources). ThesolutiontothisproblemhasFOC: f(cid:48)(I) = 1+θC(cid:48)(I −w). 1+r Note that we could change this notation—let i = f(i) and φ(x) = f−1(xκV). Then we could κV writethisprobleminournotation: κ maxiV −φ(i)−θC(e). I 1+r In this case, V and κ are arbitrary constants, but in our model, they correspond to the shadow valueofcapitalandthemean“capitalqualityshock.” Inthisnotation,wegetourfamiliarFOC (notingthatβ = 1 ): 1+r ∂e βκV = φ(cid:48)(i)+θC(cid:48)(e) . ∂i Therefore,theFOCcouldbedistortedbyexternalfinancingconsistentwithStein(2003). Additionally,equation(3)inStein(2003)addsanagencyconflict,whichcouldimplyoverinvestment: f (I) max (1+γ)−I −θC(e), (1-93) I 1+r where γ is a parameter that governs the intensity of agency conflicts. In our notation, this 17

becomes: maxiκβV (1+γ)−φ(i)−θC(e), I andthushasFOC: ∂e βκV (1+γ) = φ(cid:48)(i)+θC(cid:48)(e) . ∂i So,thisexpressionnowhasdistortionsthatcanencouragetoomuchinvestment. Altogether, our framework has a natural mapping to the distortions considered in Stein (2003). O2.2 Obtaining Sufficient Statistics using Hennessy et. al. (2007) O2.2.1 MappingbetweenFrameworks Note that the environment introduced in example 1 in Section 2.2.2 is a discrete-time version of the simplified model in Hennessy et al. (2007). We can thus take the expression for ∆i we derive, plug in the quadratic functional form of external financing costs they consider t (which imply G(cid:48)(x) = θ ), and take a first-order approximation (with respect to current and x futureequityissuancex orcashholdings)toyieldthefollowing:43 t ∆i (z,k,c ) = θ qxΦ(x < 0). (1-94) t −1 x(cid:98) Thisisexactlythefunctionalformtheyconsiderandestimate,andintheircontinuoustimeenvironment,thereisnodifferencebetweenqthisperiodandexpecteddiscountedqnextperiod.44 Therefore,inaregressionoftheform I jt = θ (q x Φ(x < 0))+controls+error. x (cid:98)jt jt jt K jt Note that we can map i into the regression coefficient times the mean of the observable inde- ∆ pendentvariableonwhichtheregressioncoefficientloads: (cid:90) K j i = ∆i dj ≈ θ mean(qxΦ(x < 0)). ∆ j x (cid:98) K j j 43Becausetheundistortedfirmmaximizesfirmvalue,theeffectofanydistortionsoninvestmentdonothavea first-ordereffectonthevalueofthefirm,sincethederivativeoffirmvaluewithrespecttoinvestmentis0absent distortions. Thus,theeffectofdistortionsonfirmvalueissecondorder,andonlytheirsquaredvalueenters. 44Howthephysicalcostofexternalfinanceaffectsfirmvalueuponentryisasecond-ordereffectthatweassume away.Thisassumptionisconsistentwithexternalfinancingcostsbeingpurelyanasymmetricinformationproblem (ratherthanatruecost),so,onaverage,thevalueofissuedequityisthefairvalue,butthemarginalcostofissuing equity is distorted by adverse selection. Our approach can accommodate such costs (as in Corollary 1.1) with a sufficient statistic capturing them. Inferring the effect of such a cost as a percentage of firm value using the estimatesinHennessyandWhited(2007)yieldssmallvaluesforsuchastatistic,asthemean-squaredissuanceis small. 18

Similarly, (cid:90) K i = (∆i )2 j dj ≈ θ2 (cid:0) mean(qxΦ(x < 0))2 +σ2(mean(qxΦ(x < 0))) (cid:1) . ∆2 j K x (cid:98) (cid:98) j j O2.2.2 Data WedownloadtheCompustatAnnualfileinJuly2022.45 Weselectontheyears1968–2003 to match the years in Hennessy et al. (2007). We also follow their criteria for selecting on SIC codes(droppingfirmswithaone-digitSICof6or9ortwo-digitSICof49). Wethencompute Tobin’s Q as the market value of equity (computed as the price times shares outstanding) less thebookvalueofequitylessdeferredtaxesplusthebookvalueofassets,allscaledbythebook value of assets. We compute equity issuance as the sale of less the purchase of common and preferredstock. WedropobservationswithmissingvaluesfortheSICcode,totalassets,sales, gross PP&E, capital expenditures, sales of property, plant, and equipment, the objects enter Q and equity issuance, and the additional variables that go into the Kaplan and Zingales (1997) index (cash flow computed as income plus depreciation, dividends computed as dividends of common and preferred shares, and long-term debt). We also drop non-positive values of sales, total assets, and gross PP&E. Additionally, we keep only firms that report in U.S. dollars and with U.S. headquarters. The capital stock used in capital weighting is in 1997 dollars. We winsorizethe(inflation-adjusted)capital-stock,Tobin’sQ,andequityissuanceatthe1%level. As noted in the main text, with this approach, we find a similar non-capital-weighted mean underinvestment value to that using the reported values from HLW (negative 0.00239 in our caseversusnegative0.00134inHLW). ToperformthemeasurementexercisedescribedinSection5,wemergeintheTFPdatafrom ˙Imrohorog˘luandTu¨zel(2014),keepingonlytheobservationsthatuniquelymerge.46 InOnline Appendix Table O3.4, we present information on the mean, median, and standard deviation acrossthesampleandbyquantile. O2.3 Obtaining Sufficient Statistics from Ben-David et. al. (2013) O2.3.1 MappingbetweenFrameworks We can show that a first-order approximation of investment with respect to current and (possibly future) distortions implies that the distortion to investment is identical to the causal effectofmiscalibratedexpectationstoday: ∆i (z,k,m) ≈ m , t t 45WedownloadtheCompustatAnnualfilefromtheWRDSdatabaseinJuly2022. 46Therefore, wefollowthecleaningstepsaboveincludingrestrictingourdatatononfinancialfirmsfrompre- 2003,amongtheothercleaningsteps,andalsodropobservationswithmissingTFP. 19

which is precisely the object estimated by BGH. The proof is analogous to the proof in AppendixO2.2. O2.3.2 Data InthesurveyusedbyBGH,managersareaskedtoreporttheirprojectionsfortheS&P500. Managersarealsoaskedforthevaluesaboveandbelowwhichthereisa1in10chanceforthe actual return. They perform this exercise for the year-ahead and next-10-year returns. BGH convertthese90-10percentilesforreturnsintoimputed-individualprobabilities. We focus on the estimate of calibration of long-term returns on investment from Table VII (of 0.6)—call it βmisc. Given reported values in their paper, we can recover the mean miscalibration adjusted for the actual return—call it µmisc. From Table I, Panel A, the average and standard deviations of the imputed individual volatilities are 11.4% and 9.25%, respectively. From Table II, Panel B, the 10-year realized annual volatility of S&P 500 returns is 14.3%. So we obtain µmisc as −11.4−14.3. We can write the predicted effect of miscalibration on in- 9.25 vestment as βmiscµmisc, which is 0.00188. Similarly, we can obtain the mean-squared effect on (cid:104) (cid:105) investmentasE (βmiscµ ) 2 = (βmisc) 2 (µ2 +1) = 0.00004. misc misc O2.4 Robustness Across the Model Parameters The remaining parameters of the model are the investment of the Modigliani-Miller firm i , the exogenous exit rate κ, the adjustment cost parameter θ, the production function pa- MM rameter α , the depreciation rate, δ, the CES parameter ρ, the IES parameter γ, and the labor 2 disutility parameter ϕ. In this subsection, we vary each of these parameters and show how the bounds on welfare change (under the calibration when β = 0.98) in response to removing the externalfinancingandmanager-shareholderinvestmentdistortions. We present the results for i and κ in Appendix Figure O4.2. These two parameters MM affect the change in PE output from resolving the friction, so we present those changes on the figures.47 We vary both parameters from 0.9 to 0.995 (the baseline calibrations of i and MM κ are 0.9859 0.976, respectively). Although the parameters do not necessarily vary linearly acrosstheranges,theresultsaredirectionallyconsistentwithourbaselinecasesforbothofthe bounds. We present the results for θ, α , and δ in Appendix Figure O4.3. We vary θ from 1 to 1.5 2 (thebaselinecalibrationis1.1),wevaryα from0.6to0.8(thebaselinecalibrationis2/3),and 2 we vary δ from 0.07 to 0.13 (the baseline calibration is 0.1). Though there is variation in the GEwelfarelosses,theresultsareagaindirectionallyconsistentwithourbaselinecasesforboth bounds. 47The remaining parameters considered in this section do not affect the PE losses, so we show only the GE welfarelossesintheremainingfigures. 20

Lastly, we present results for the macro parameters ϕ, ρ, and γ in Appendix Figure O4.4. We consider a wide range for the Frisch elasticity parameter ϕ, spanning from 0.2 to 3 (the baseline calibration is .0276). We also consider a wide range for ρ, which we vary from 4 (our baseline calibration) to 30. Finally, we vary γ from 0 to 5 (the baseline calibration is 1). To keep from having to change ρ as we vary γ, we need to set ρ to be higher (we set it to 10), as the problem is undefined for certain combinations of low ρ and low γ.48 Across cases, the results are again directionally consistent with our baseline cases for both bounds. In summary, thewelfareresultsaredirectionallysimilaracrossallcasesoftheparametersweconsider. 48Inparticular,wehavecondition(1-91),whichwederiveinAppendixO1.9. Notethatbecauseρishigherin thiscasethebaselinewelfarenumbersaredifferent. 21

O3 TFP follows Markov Process In this section of the Online Appendix, we provide more detail on the results when TFP followsaMarkovprocess. Results when TFP Follows a Markov Process First, we perform a measurement exercise to demonstrate how (a) one can implement our approach with productivity heterogeneity, and (b) studythepotentialinfluenceofthisheterogeneityonourbaselineresults. For this exercise, we need a firm-level measure of idiosyncratic productivity, which we obtain from Selale Tuzel’s website that has the data constructed following ˙Imrohorog˘lu and Tu¨zel (2014).49 We then compute the sufficient statistics by group of firms. We clean the data followingHLW;thecleaningdetailsaredescribedinAppendixO2.2. Wesetthereasbeingfive quantiles for firm-level TFP. We then need two sets of moments for our exercise.50 First, we need the values of the sufficient statistics by quantile, which we present in Table O3.2.51 Here, we see that the values of the sufficient statistics are different enough for the highest TFP firms thatitisworthconsideringhowtheresultsmightchangeifweconsiderheterogeneity. Second, we need the transitions between quantiles, which we present in Table O3.3. This table shows thatthereismuch“switching”inourdatabetweenquantiles,whichintuitivelyshouldlimitthe importanceofheterogeneity. We perform two sets of counterfactuals that help us to understand the role of persistent heterogeneity in firm productivity. First, we study the case where there is heterogeneity in idiosyncratic productivity and in the sufficient statistics for each type. In this case, we solve for the counterfactuals following Corollary 1.4. Second, we study the case where there is heterogeneity in idiosyncratic productivity, but the sufficient statistics do not differ across types. This case preserves “independence” of the sufficient statistics, so here we can solve for our counterfactualsusingananalogueofourmainproposition(Proposition1). NotethatourbaselineresultspresentedinSection4reflectathirdcasewhereidiosyncraticproductivitydoesnot follow a Markov process and where there is no heterogeneity in the sufficient statistics. The natural comparison is between the two cases with heterogeneity in idiosyncratic productivity, as heterogeneity can also interact with the aggregate distortions to affect the overall welfare results.52 In Table O3.1, we show the results from the two cases (along with the values from the no friction case) when β → 1.53 We see that the results between the first and second cases are 49WeobtaineddatafromSelaleTuzel’swebsite(https://sites.google.com/usc.edu/selale-tuzel/home)inDecember2022. 50WealsoneedthemeanTFPbyquantile,whichwepresentinTableO3.4.alongwithothersummarystatistics. 51Notethatweonlylookatequal-weightedstatisticsasinourbaselinecalibration;thecapital-weightedstatistics arecloserto0acrossquantiles. 52Theelasticitiesofthegrowthratesofthefirmswithdifferenttypesaregoingtobeheterogeneousacrossproductivitylevels. Thisimpliesthatresolvingthedistortionwillhaveheterogeneouseffectsacrosstypes;therefore, changesinstatepricescanleadtochangesintheallocationofresourcesacrossfirmsofdifferenttypes. 53Thechangeinwelfareishigherfromremovingthefrictionunderthesecondcase(wherethesufficientstatis- 22

TableO3.1: EffectsofResolvingExternalFinancingFrictionswithMarkovIdiosyncraticProductivity ExternalFinancing Heterogeneityinz Heterogeneityinz andsuff. stats butnotinsuff. stats %∆between Baseline No-friction %∆between Baseline No-friction steadystates s.s. value s.s. value steadystates s.s. value s.s. value Variable PEOutput 2.37 2.07 GEInvestment(I) 0.68 0.36 0.36 2.28 0.36 0.37 GECapitalstock(K) 0.34 10.3 10.34 0.29 10.28 10.31 GEInvestment -0.35 28.48 28.38 -2.03 28.48 27.91 efficiency(K/I) GEOutput(Y) 0.28 2.78 2.78 0.18 2.77 2.78 GEWelfare 0.23 0.15 GEConsumption(C) 0.25 1.52 1.53 0.16 1.52 1.52 GELaborsupply(L) 0.02 0.6 0.6 0.01 0.59 0.59 GEMassofentry(M ) 0.11 1 1 -0.47 1 1 e GECapitalstockper 0.23 10.3 10.32 0.76 10.28 10.36 enteringfirm(K) GEAggregateprofit -0.12 0.18 0.18 -0.08 0.18 0.18 scalingfactor(Π) Note: This table shows the percentage change in PE and GE objects from the baseline steady state to the steady statewithouttheexternalfinancingfrictionforthemodelwithwhereidiosyncraticproductivityfollowsaMarkov process and there is heterogeneity in the sufficient statistics (left two columns) versus a case where productivity similarlyfollowsaMarkovprocessbutthereisnoheterogeneityinthesufficientstatistics(rightcolumns). Wealso showthepercentagechangesintheGEobjectsassociatedwiththeGEwelfarecalculation,aswellastheirvalues inboththebaselinesteadystateandthecounterfactualsteadystatewithoutthefriction. Here,weassumeβ →1. z isfirmproductivity. suff. statsissufficientstatistics. s.s. issteadystate. directionally similar and the change in welfare differs by about 8 basis points. The reason the change in welfare is higher with both forms of heterogeneity is that the change in output is higher, as the higher TFP firms benefit more by having their distortions removed and matter more for aggregate output given their different levels of the sufficient statistics. Nonetheless, the mechanisms for why welfare increases by so much less than output does in PE—which is around 2 percentage points in this case—are similar to what we described in Section 4, as is clearfromthesimilardirectionalmovementsinaggregatespresentedinthetable.54 Altogether, weviewtheresultsasbeingqualitativelysimilartoourbaselineresults.55 ticsarenotheterogeneousbytype)whenwecompareittothechangeinwelfareinourbaselinemodel. Thisisto beexpectedthattheresultswilldiffer, astheelasticitiesofthegrowthratesofthefirmswithdifferenttypesare going to be heterogeneous across productivity levels just taking the sufficient statistics as given. Importantly, if wewantedtoexactlymatchthelevelsofaggregatesinourbaselinecase, wecould, butthiswouldrequireusto adjustthevaluesofoursufficientstatisticstoaccountfortherebeingheterogeneityinfirmTFP.Wedecidedthat usingthebaselinevaluesofthesufficientstatisticswasmoretransparentforthisexercise. 54NotethatthechangeinPEoutputisnotthesameaspartialequilibriumcapitalinthiscase; weonlyreport thechangeinpartialequilibriumoutput(thechangeinPEcapitalismodestlylowerthanthechangeinPEoutput ineachversionofthemodelwithheterogeneity). 55With fixed entry, the results (not shown) are still qualitatively similar and even closer between the first and secondcases). 23

TableO3.2: ExternalFinancingFrictionSufficientStatisticEstimates FullSampleandbyTFPQuantile Mean Value i : Frompaper -0.00134 ∆ i : Replication -0.00146 ∆ i : TFPQ1: Replication -0.00144 ∆ i : TFPQ2: Replication -0.00049 ∆ i : TFPQ3: Replication -0.00056 ∆ i : TFPQ4: Replication -0.00101 ∆ i : TFPQ5: Replication -0.00379 ∆ Mean-squared Value i : Replication 0.00023 ∆2 i : TFPQ1: Replication 0.00035 ∆2 i : TFPQ2: Replication 0.000028 ∆2 i : TFPQ3: Replication 0.000040 ∆2 i : TFPQ4: Replication 0.000097 ∆2 i : TFPQ5: Replication 0.00066 ∆2 Notes: This table reports values for the mean (i ) or mean-squared (i ) external financing friction sufficient ∆ ∆2 statistics. ThefirstrowreportsthestatisticderivedfromwhatisreportedinHLW2007,whiletheotherrowsrely on the authors’ calculations in replicating the statistics (all on an equal-weighted basis). We also calculate versionsofthestatisticsbyquantileofTFP,whereQ1isthehighestquantileandQ5thelowest. Source: Authors’ calculationsusingCompustatandTFPdatafromSelaleTuzel’swebsite(constructedfollowing˙Imrohorog˘luand Tu¨zel(2014)),downloadedinDecember2022. TableO3.3: TransitionMatrixforTFPbyQuantile TFPquantile[t+1] TFPquantile[t] 1 2 3 4 5 Total No. Row% No. Row% No. Row% No. Row% No. Row% No. Row% 1 5,678 67 1,960 23 514 6 209 2 112 1 8,473 100 2 2,200 23 4,642 48 2,131 22 550 6 139 1 9,662 100 3 717 7 2,265 23 4,534 46 2,063 21 366 4 9,945 100 4 334 3 685 7 2,156 21 5,229 52 1,656 16 10,060 100 5 309 3 261 3 469 5 1,731 18 7,033 72 9,803 100 Total 9,238 19 9,813 20 9,804 20 9,782 20 9,306 19 47,943 100 Notes: ThistabledisplaysthetransitionmatrixforfirmsbetweenTFPquantilesinoursample. Weshowboththecellcountsandtherowpercentages. Source:Authors’calculationsusingCompustatandTFPdatafromSelaleTuzel’swebsite(constructedfollowing˙Imrohorog˘luandTu¨zel(2014)),downloadedinDecember2022. 24

TableO3.4: LogTFPSummaryStatisticsbyQuantile Mean Median S.D. Wholesample -0.336 -0.318 0.376 Q1 -0.866 -0.751 0.312 Q2 -0.467 -0.463 0.051 Q3 -0.318 -0.318 0.039 Q4 -0.175 -0.178 0.046 Q5 0.144 0.068 0.219 Notes: Thistablereportsthevalueforthemean,median,andstandarddeviation(SD)ofthenaturallogarithmof TFPfortheoverall(cleaned)sampleandbyquantileofTFP.Logisthenaturallogarithm. Source: Authors’calculationsusingCompustatandTFPdatafromSelaleTuzel’swebsite(constructedfollowing ˙Imrohorog˘luandTu¨zel(2014)),downloadedinDecember2022. 25

O4 Additional Figures and Tables In this section of the Online Appendix, we present additional figures and tables referenced inthetext. FigureO4.1: WelfareGainsfromResolvingFirm-levelDistortionsvaryingβ (a)ExternalFinancingFriction 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0.95 0.955 0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995 Discount factor (β) )%( stsoc erafleW (b)Manager-ShareholderFriction −0.1 GE upper bound GE lower bound −0.11 −0.12 −0.13 −0.14 −0.15 −0.16 −0.17 −0.18 −0.19 −0.2 0.95 0.955 0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995 Discount factor (β) )%( stsoc erafleW GE upper bound GE lower bound Note: Thesubfiguresshowtheboundsonthewelfarechangeafterremovingtheexternalfinancingfriction(left) and manager-shareholder friction (right) across a range of values for β (from 0.95 and 1). The vertical, dashed blacklineindicatesthewelfarevaluesatthecalibratedvalueofβ (asshowninTable3). 26

FigureO4.2: WelfareGainsfromResolvingFirm-levelDistortionsvaryingi andκ MM (a)ExternalFinancingFriction,varyingi MM 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 Investment of unlevered firm )%( stsoc erafleW (b)Manager-ShareholderFriction,varyingi MM 0 GE upper bound GE lower bound PE output change −1 −2 −3 −4 −5 −6 −7 −8 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 Investment of unlevered firm )%( stsoc erafleW GE upper bound GE lower bound PE output change (c)ExternalFinancingFriction,varyingκ 7 6 5 4 3 2 1 0 0.95 0.955 0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995 Exogenous entry rate (κ) )%( stsoc erafleW (d)Manager-ShareholderFriction,varyingκ 0 GE upper bound GE lower bound PE output change −2 −4 −6 −8 −10 −12 −14 0.95 0.955 0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995 Exogenous entry rate (κ) )%( stsoc erafleW GE upper bound GE lower bound PE output change Note: Each subfigure shows the PE output change and the bounds on the welfare change after removing the external financing friction (left panels) or the manager-shareholder friction (right panels) when β = 0.98 for a set of values of a given parameter: investment of the Modigliani-Miller firm parameter i (top panels) and MM exogenousexitrateparameterκ(bottompanels). Thevertical, dashedblacklineindicatesthewelfarevaluesat thecalibratedvalueofthegivenparameter(asshowninTable3). 27

FigureO4.3: WelfareGainsfromResolvingFirm-levelDistortionsvaryingθ,α ,andδ 2 (a)ExternalFinancingFriction,varyingθ 0.25 0.2 0.15 0.1 0.05 0 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 Adjustment cost parameter (θ) )%( stsoc erafleW (b)Manager-ShareholderFriction,varyingθ 0 GE upper bound GE lower bound −0.05 −0.1 −0.15 −0.2 −0.25 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 Adjustment cost parameter (θ) )%( stsoc erafleW GE upper bound GE lower bound (c)ExternalFinancingFriction,varyingα 2 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0.6 0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.8 Labor share (α) 2 )%( stsoc erafleW (d)Manager-ShareholderFriction,varyingα 2 0 GE upper bound GE lower bound −0.05 −0.1 −0.15 −0.2 −0.25 0.6 0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.8 Labor share (α) 2 )%( stsoc erafleW GE upper bound GE lower bound (e)ExternalFinancingFriction,varyingδ 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0.08 0.09 0.1 0.11 0.12 0.13 Depreciation rate (δ) )%( stsoc erafleW (f)Manager-ShareholderFriction,varyingδ 0 GE upper bound GE lower bound −0.02 −0.04 −0.06 −0.08 −0.1 −0.12 −0.14 −0.16 −0.18 −0.2 0.08 0.09 0.1 0.11 0.12 0.13 Depreciation rate (δ) )%( stsoc erafleW GE upper bound GE lower bound Note: Eachsubfigureshowstheboundsonthewelfarechangeafterremovingtheexternalfinancingfriction(left panels)orthemanager-shareholderfriction(rightpanels)whenβ =0.98forasetofvaluesofagivenparameter: adjustmentcostparameterθ(toppanels),productionfunctionparameterα (middlepanels),anddepreciationrate 2 parameterδ(bottompanels). 28

FigureO4.4: WelfareGainsfromResolvingFirm-levelDistortionsvaryingϕ,ρ,andγ (a)ExternalFinancingFriction,varyingϕ 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0.5 1 1.5 2 2.5 3 Labor elasticity parameter )%( stsoc erafleW (b)Manager-ShareholderFriction,varyingϕ 0 GE upper bound GE lower bound −0.02 −0.04 −0.06 −0.08 −0.1 −0.12 −0.14 −0.16 −0.18 −0.2 0.5 1 1.5 2 2.5 3 Labor elasticity parameter )%( stsoc erafleW GE upper bound GE lower bound (c)ExternalFinancingFriction,varyingρ 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0 5 10 15 20 25 30 CES parameter (ρ) )%( stsoc erafleW (d)Manager-ShareholderFriction,varyingρ 0 GE upper bound GE lower bound −0.02 −0.04 −0.06 −0.08 −0.1 −0.12 −0.14 −0.16 −0.18 −0.2 0 5 10 15 20 25 30 CES parameter (ρ) )%( stsoc erafleW GE upper bound GE lower bound (e)ExternalFinancingFriction,varyingγ 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Risk aversion parameter (γ) )%( stsoc erafleW (f)Manager-ShareholderFriction,varyingγ 0 GE upper bound GE lower bound −0.02 −0.04 −0.06 −0.08 −0.1 −0.12 −0.14 −0.16 −0.18 −0.2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Risk aversion parameter (γ) )%( stsoc erafleW GE upper bound GE lower bound Note: Eachsubfigureshowstheboundsonthewelfarechangeafterremovingtheexternalfinancingfriction(left panels)orthemanager-shareholderfriction(rightpanels)whenβ =0.98forasetofvaluesofagivenparameter: Frischelasticityparameterϕ(toppanels),CESparameterρ(middlepanels),andIESparameterγ(bottompanels). Exceptintheplotsforγ,thevertical,dashedblacklineindicatesthewelfarevaluesatthecalibratedvalueofthe givenparameter(asshowninTable3). WhenvaryingtheIESparameter,wesetρto10astheproblembecomes undefinedinourbaselinecalibrationofρforcertainvaluesofγ (seeSectionO1.9); inturn,thevertical,dashed blacklinecorrespondstothewelfarevaluewithaγ of1andaρof10. 29

TableO4.1: EffectsofResolvingFirm-levelDistortions: FixedversusFreeEntry ExternalFinancing FixedEntry FreeentryU.B. %∆between Baseline No-friction %∆between Baseline No-friction steadystates s.s. value s.s. value steadystates s.s. value s.s. value Variable PECapitalstockorOutput 3.35 3.35 GEInvestment(I) 0.15 2.27 2.27 0.05 3.26 3.26 GECapitalstock(K) 0.18 25.6 25.64 0.17 25.58 25.62 GEInvestment 0.03 11.29 11.29 0.12 7.84 7.85 efficiency(K/I) GEOutput(Y) 0.12 7.94 7.95 0.1 7.94 7.95 GEWelfare 0.08 0.13 GEConsumption(C) 0.11 5.68 5.68 0.13 4.68 4.69 GERelativewage(W) 0.12 4.54 4.55 0.12 4.54 4.54 P GELaborsupply(L) 0.01 0.87 0.87 -0.03 0.88 0.87 GEMassofentry(M ) 0 1 1 1.3 0.98 0.99 e GECapitalstockper 0.18 25.6 25.64 -1.14 26.21 25.92 enteringfirm(K) GEInvestmentof -0.13 0.99 0.98 -0.18 0.99 0.99 unleveredfirm(i ) MM GEAggregateprofit -0.05 0.16 0.16 -0.08 0.16 0.16 scalingfactor(Π) Manager-Shareholder FixedEntry FreeentryU.B. %∆between Baseline No-friction %∆between Baseline No-friction steadystates s.s. value s.s. value steadystates s.s. value s.s. value Variable PECapitalstockorOutput -5.11 -5.11 GEInvestment(I) -0.59 2.56 2.55 -0.33 3.56 3.55 GECapitalstock(K) -0.39 27.83 27.73 -0.37 27.83 27.73 GEInvestment 0.19 10.87 10.89 -0.04 7.82 7.82 efficiency(K/I) GEOutput(Y) -0.32 8.64 8.61 -0.27 8.64 8.62 GEWelfare -0.08 -0.13 GEConsumption(C) -0.21 6.08 6.06 -0.22 5.08 5.07 GERelativewage(W) -0.23 4.73 4.72 -0.23 4.73 4.72 P GELaborsupply(L) -0.09 0.91 0.91 -0.03 0.91 0.91 GEMassofentry(M ) 0 1 1 -1.87 0.99 0.98 e GECapitalstockper -0.39 27.83 27.73 1.47 27.99 28.4 enteringfirm(K) GEInvestmentof 0.18 0.99 0.99 0.24 0.99 0.99 unleveredfirm(i ) MM GEAggregateprofit 0.07 0.16 0.16 0.1 0.16 0.16 scalingfactor(Π) Note: ThistableshowsthepercentagechangeinPEandGEobjectsfromthebaselinesteadystatetothesteadystate withouttheexternalfinancing(toppanel)ormanager-shareholder(bottompanel)frictionforthemodelwithfixedentryversusupperboundofthebaselinemodelwithfreeentrywhenβ =0.98. Wealsoshowthepercentagechangesin theGEobjectsassociatedwiththeGEwelfarecalculation,aswellastheirvaluesinboththebaselinesteadystateand thecounterfactualsteadystatewithoutthefriction. s.s. issteadystate. U.B.isupperbound. 30