Accounting for Productivity Dispersion over the Business Cycle

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Accounting for Productivity Dispersion over the Business Cycle Robert J. Kurtzman and David Zeke 2016-045 Please cite this paper as: Kurtzman, Robert J., and David Zeke (2016). “Accounting for Productivity Dispersion over the Business Cycle,” Finance and Economics Discussion Series 2016-045. Washington: Board of Governors of the Federal Reserve System, http://dx.doi.org/10.17016/FEDS.2016.045. 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.

Accounting for Productivity Dispersion over the Business Cycle ∗ Robert Kurtzman†1 and David Zeke ‡2 1Federal Reserve Board of Governors 2University of California, Los Angeles May 20, 2016 Abstract Thispaperpresentsaccountingdecompositionsofchangesinaggregatelaborandcapital productivity. Our simplest decomposition breaks changes in an aggregate productivity ratio into two components: A mean component, which captures common changes to firm factor productivity ratios, and a dispersion component, which captures changes in the variance and higherordermomentsoftheirdistribution. Instandardmodelswithheterogeneousfirmsand frictions to firm input decisions, the dispersion component is a function of changes in the second and higher moments of the log of marginal revenue factor productivities and reflects changesintheextentofdistortionstofirmfactorinputallocationsacrossfirms. Weapplyour decomposition to public firm data from the United States and Japan. We find that the mean componentisresponsibleformostofthevariationinaggregateproductivityoverthebusiness cycle,whilethedispersioncomponentplaysamodestrole. ∗We thankAndy Atkeson, DavidBaqaee, BrianCadena, andPierre-Olivier Weill, and seminarparticipantsattheSpringMidwestMacroMeetings2015,GCERConference2015,WEAISummermeetings,and EconometricSocietyWorldCongress2015fortheirhelpfulcommentsanddiscussions. Theviewsexpressed inthispaperaresolelytheresponsibilityoftheauthorsandshouldnotbeinterpretedasreflectingtheviewsof theBoardofGovernorsoftheFederalReserveSystemorofanyoneelseassociatedwiththeFederalReserve System. †robert.j.kurtzman@frb.gov ‡davidzeke@ucla.edu

1 Introduction What drives changes in aggregate productivity? One explanation that has been widely usedtoexplainthevariationofaggregateproductivityoverthebusinesscycleorovertime moregenerallyisthatfrictionstotheallocationoflaborandcapitalbetweenfirmsaretimevarying: Greater frictions to the distribution of capital and labor between firms reduce the amount of output produced with a given amount of capital and labor and reduce measures of aggregate productivity.1 This paper presents accounting decompositions of changes in aggregate labor, capital, and total factor productivity that addresses this economic mechanism, and can help to quantify the extent to which the changing distribution of labor and capitaldrivefluctuationsinaggregateproductivityovertime. The accounting decompositions in this paper rely on the property that aggregate factor productivity ratios can be expressed as the weighted sum of firm-level productivity ratios. Our first decomposition splits changes in measures of aggregate factor productivities into a mean component, changes in the weighted average of log productivities across firms, and a dispersion component, which captures changes in the higher order moments of the distribution of productivities across firms.2 The two components add up to the change in a given aggregate factor productivity ratio. We compute the decomposition separately for bothaggregatelaborandcapitalproductivity. Crucially,forthedecompositionofaggregate labor productivity, we require only firm-level panel data on value added and labor, and for the decomposition of aggregate capital productivity, we require firm-level panel data on valueaddedandcapital. The allocation of labor and capital may vary across firms not only due to distortions but for technological reasons as well; the second decomposition allows us to group firms (by industry or other categorical groups) to address this point. We implement our first decomposition on each sector, resulting in sectoral mean and dispersion components. We canthenweighteachsector’smeananddispersioncomponentsbysectoralfactorsharesto obtain aggregate mean and dispersion components. Thus, by an accounting property, the changeinaggregatefactorproductivitiescanbedecomposedintothreecomponents: First, 1Thiseconomicmechanismplaysaroleindrivingthedynamicsofproductivityandothermacroeconomic aggregates in a number of recent influential papers, including Arellano et al. (2012), Bloom et al. (2014), Gilchristetal.(2014),KhanandThomas(2013),MidriganandXu(2014),andMoll(2014),asexamples. 2Tobeprecise,thedispersioncomponentcanbeexpressedasafunctionofthesecondandhigherorder cumulantsofthedistributionoffirmproductivitymeasures,whilethemeancomponentisonlyafunctionof thefirstcumulant. 2

an aggregated mean component which captures changes in the weighted average of log factor productivities within sectors. Second, an aggregated dispersion component which captures changes in the dispersion of log factor productivities across firms within sectors. Third,asectoral-sharecomponent,whichcapturesthechangesinthedistributionofinputs betweensectors. Ourdecompositions,whenappliedtoaggregatelabororcapitalproductivity,arepurely accounting identities. To combine aggregate capital and labor productivity into a measure of total factor productivity, we rely on the standard model assumptions that allow us to compute the Solow residual. We then show that the Solow residual has the nice property that we can express it as the weighted average of the mean, dispersion, and sectoral-share componentsofcapitalandlaborproductivity. Ourdecompositionsareusefultoolsforresearcherstestingwhethermodelswherefrictions to the allocation of labor or capital across firms play a meaningful role in driving aggregatesareconsistentwithfirmlevelbehavior. Wepresentaseriesofresultstodemonstrate this point. In the model of Hsieh and Klenow (2009), we demonstrate how our decomposition captures changes in the distribution of the log of marginal revenue factor productivities. We prove that changes in the expected value of the log of marginal revenue factor productivities, as well as changes in production function coefficients, drive changes inthemeancomponentofourdecompositions. Weprovethatchangesinthesecondcentral moment and all higher order moments of the log of marginal revenue factor productivities drivechangesinthedispersioncomponentofourdecompositions. We then use a more general model of production by heterogeneous firms to demonstratehowdistortionstofirmcapitalandlabordecisionsarecapturedinourdecomposition. We demonstrate analytically that the dispersion component of our decomposition captures changesinproductivityduetoheterogeneousdistortionstofirm-levelinputallocation. The meancomponentofourdecompositionscaptureschangesintechnologyorcommondistortionstofirmcapitalorlaborchoices. Weprovethatthisgeneralmodelofproductionhasa mapping to a large number of macroeconomic models in the literature that utilize frictions totheallocationoflabororcapitalacrossfirmstohelpdriveaggregatedynamics. We compute our decompositions for aggregate labor productivity, capital productivity, and TFP using firm-level data on U.S. nonfinancial public firms. To see if the results are consistent for another large, developed nation, we perform a similar analysis for nonfinancial public firms from Japan. The results for the United States and Japan from the second decompositionappliedtolaborproductivityshowthatthemeancomponentishighlycorre- 3

latedwithmovementsinaggregatelaborproductivityandareessentiallysolelyresponsible for its cyclical variation. The magnitude of movements in the dispersion component are small, and the dispersion component has a weak negative correlation with changes in aggregate labor productivity. Our results are different for aggregate capital productivity. The dispersioncomponentmovesmuchmorecloselywithchangesinaggregatecapitalproductivity,anddoesplayaroleincontributingtocyclicalvariationinaggregatecapitalproductivity. Ourdecomposition,whenappliedtoTFP,yieldstheresultthatthemeancomponent is responsible for the vast majority of its cyclical variation, because much of the cyclical movementsinTFParedrivenbychangesinaggregatelaborproductivity. RelatedLiteratureThecontributionofthispaperistoprovideaccountingdecompositions of aggregate labor and capital productivity, which can be implemented without structural estimation, and can guide the specification of firm-level frictions to capital and labor allocationinbusinesscyclemodels. Thefactthatourdecompositionsonlyrequiremeasuresof firm-level value added, labor, and capital, and do not require estimation to be computed is an attractive property, as it implies the use of our decomposition not only avoids potential biases from estimation, but also means that our decomposition can be computed in both data and heterogeneous firm models with relative ease. A large number of papers in the literature work with production environments that map into the class of production environments that we rely on to prove how our decomposition maps into models in Section 3. The general class of models to which our theoretical results apply include the influential modelsofArellano,Bai,andKehoe(2012),Bloometal.(2014),Kehrig(2015),andKhan and Thomas (2013), as only a few recent examples. Thus, the dispersion component of ourdecompositionreflectschangesinthedistributionofdistortionstofirminputallocation in such papers. Hence, the role of frictions to firm labor and capital allocation in a large number of models can be compared to the data through the use of our decomposition. Our empiricalresultsalonecanalsohelptoguidemodelselectioninstandard,widely-usedproductionenvironments. Inthissense,ourdecompositionissimilarinspirittoChari,Kehoe, andMcGrattan(2007). Our paper is also related to a number of recent studies which examine the role of reallocation or allocative efficiency in driving aggregate productivity dynamics. One group of papersestimateproductionfunctioncoefficientsandfirm-leveltotalfactorproductivitiesto assess the role of allocative frictions in driving productivity over the business cycle, such as Oberfield (2013), Osotimehin (2013), and Sandleris and Wright (2014). Our approach 4

differs from this set of the literature in that our decompositions are accounting identities requiring only measures of firm-level value added, capital, and labor, and thus we do not require the estimation of production function coefficients. Our method therefore avoids the potential econometric biases in these estimation procedures (which are discussed in Appendix C) and can be implemented immediately on a wide array of models and data. The magnitude of the dispersion component of our decompositions can be viewed as an approximation to the extent to which allocative efficiency affects aggregate productivity in such models (we show this in Appendix C). Thus, our decomposition, if applied to the respective datasets used in these papers, could be used to complement the paper’s structural approaches and potentially address concerns regarding the assumptions required for estimation. Another group of papers examine the role of resource reallocation through the use of aggregate productivity decompositions, such as Foster, Haltiwanger, and Krizan (2001) and Basu and Fernald (2002). The sectoral share component of our second decomposition also speaks to the role resource reallocation between sectors can play in driving productivity dynamics. Differently from these papers, however, the dispersion component of our decomposition captures the role allocative efficiency plays in driving productivity dynamics.3 Additionally, our decomposition does not require the estimation of firm-level TFP. The rest of the paper proceeds as follows. Section 2 defines the components of our decompositions for aggregate labor productivity, aggregate capital productivity, and TFP. Section 3 discusses how shocks to firm-level wedges map into the components of our decomposition. Section 4 applies our decomposition to data from U.S. and Japanese nonfinancialpublicfirms. Section5concludes. 2 Productivity Decompositions In this section, we first present our decompositions of changes in aggregate labor and capitalproductivity,andthenwepresenthowtocombinethesedecompositionstoperform decompositions of changes in TFP. Decomposition I breaks changes in the log of each aggregateproductivityratiointoameanandadispersioncomponenttohelpidentifywhether it is changes in the mean or dispersion of the log of firm-level productivity ratios that are driving changes in aggregate productivity. Decomposition II allows for groupings of firms 3Alternatively, adjustmentcostscouldgenerateadynamicallyefficientallocationthatobservationallyis consistentwithstaticmisallocation;thispointismadeinAskeretal.(2014),e.g. 5

(sectors) to each have a mean and a dispersion component, and for the allocation of inputs between each grouping of firms to change over time. In turn, when analyzing changes in aggregate productivity, there is also a sectoral-share component, which reflects how input sharesarechangingacrosssectorsovertime. 2.1 Decomposition I: Mean and Dispersion Components We start with a static decomposition of aggregate labor productivity. We define L as aggregatelaborandl asfirm-levellabor. Aggregatelaboristhesumofallfirm-levellabor. We define K as the aggregate capital stock and k as the firm-level capital stock, where the aggregate capital stock is the sum of all firm capital stocks. The decomposition below holdsforcapitalproductivityaswell,ifwesubstituteK forLandk forl. We define Y as aggregate output and v as firm value added, where aggregate output is the sum of all firm-level value added. We have the following identity, which holds at each timet: L (cid:88) l v t i,t i,t ≡ , (1) Y v Y t i,t t i whereiindexesthesetoffirmsintheeconomy. Buildingon(1),wecannowperformastaticversionofourfirstdecomposition: (cid:32) (cid:32) (cid:33) (cid:33) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) L (cid:88) l v (cid:88) l v (cid:88) l v t i,t i,t i,t i,t i,t i,t log = log + log − log . (2) Y v Y v Y v Y t i,t t i,t t i,t t i i i (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) staticmeancomponent staticdispersioncomponent Aggregate labor to output at each time t is now broken into a “mean component,” which is the weighted average of the log of labor to value added, and a “dispersion component.” If we treat labor to value added as a random variable with a probability density function (reflecting the number and size of firms with a given productivity ratio), the dispersion component takes the form of the log of the expectation of firm-level labor to value-added ratios less the expectation of the log of firm-level labor to value-added ratios. This term is always non-negative due to Jensen’s inequality. This measure has useful statistical propertiesrelatedtothemeasureofentropyinBackus,Chernov,andZin(2014). Assumingsome regularity conditions on the distribution of firm labor to value-added ratios such that the cumulant generating function exists, the dispersion component captures all higher-order 6

cumulants of the distribution of firm-level labor to value-added ratios.4 This can be interpreted as the following: The dispersion component captures the effect of all second and higherordermomentsofthedistributionoffirmlaborproductivityonaggregatelaborproductivity. We are interested in changes in labor productivity. We can recover changes in labor productivityas: (cid:18) (cid:19) (cid:18) (cid:19) Y (cid:88) l v t i,t i,t ∆log = −∆ log (3) L v Y t i,t t i (cid:124) (cid:123)(cid:122) (cid:125) meancomponent (cid:32) (cid:32) (cid:33) (cid:33) (cid:18) (cid:19) (cid:88) l v (cid:88) l v i,t i,t i,t i,t −∆ log − log . v Y v Y i,t t i,t t i i (cid:124) (cid:123)(cid:122) (cid:125) dispersioncomponent An increase in dispersion in firm-level labor to value-added ratios decreases aggregate labor productivity. Similarly, an increase in the weighted average of firm-level labor to value-added ratios decreases aggregate labor productivity. Our mean/dispersion decomposition allows us to determine whether it is changes in the mean or the dispersion in the log offirm-levellabortovalueaddedwhichisdrivingchangesinaggregatelaborproductivity. Wepresentourseconddecompositionbelow,whichallowseachsectortohaveameanand dispersion component. Hence, changes in aggregate labor productivity can be driven by changesinthemeanoflogfirm-levellabortovalue-addedratioswithinsectors,changesin theirdispersionwithinsectors,orchangesintheallocationofinputsbetweensectors. 2.2 Decomposition II: Mean, Dispersion, and Sectoral Share Components For a sector (or any given grouping of firms), the identity in (1) holds. Hence, if j indexes a given sector, we have the following identity for aggregate labor to added value ratiowithinthatsectorattimet: Lj (cid:88) lj vj t ≡ i,t i,t . (4) Yj vj Yj t i i,t t 4Cumulantssummarizethedistributionofarandomvariable, asweexplaininmoredetailinSection3. Backus,Chernov,andZin(2014)alsoprovideanexcellentdiscussionofwhyfunctionsofthisformcapture allhigher-ordercumulants. 7

Inturn,foreachsectorattimet,wecandecomposetheaggregatelabortoaddedvalue ratiowithinasectorintoameananddispersioncomponent: (cid:32) (cid:33) (cid:32) (cid:33) (cid:32) (cid:32) (cid:33) (cid:32) (cid:33) (cid:33) Lj (cid:88) lj vj (cid:88) lj vj (cid:88) lj vj log t = log i,t i,t + log i,t i,t − log i,t i,t , (5) Yj vj Yj vj Yj vj Yj t i i,t t i i,t t i i,t t (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) Mj Dj t t where Mj is the static mean component in sector j and Dj is the static dispersion compot t nentinsectorj. Byanidentity,aggregatelaborproductivityisequivalentto: Y t = (cid:88) e−M t j−D t j Lj t. (6) L L t t j This implies that aggregate labor productivity can be expressed as an aggregate of sectoral mean and dispersion components, weighted by the share of labor allocated to each sector. Hence, when we look at changes in aggregate labor productivity, we have to account for the fact that input shares of different sectors can be changing over time. In turn, we have a third component, which reflects changes in the input share of a given sector, whichwecallthesectoral-sharecomponent: log (cid:32) L Y L Y t t − − t t 1 1 (cid:33) = log    (cid:80) (cid:80) j j (cid:16) (cid:16) e e − − M M t j − t j 1 (cid:17) (cid:17) L L L L j t t − − j t t − − 1 1 1 1    +log   (cid:80) (cid:80) j j e e − − M M t j t − j− D D t j t j L L L L t j t − − j t t 1 1   (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) meancomponent sectoralshare + log   (cid:80) (cid:80) e j − e M − t M j − t 1 j − − D D t j t j − L L 1 j t t L − − j t 1 1 −1  −log    (cid:80) (cid:80) j (cid:16) (cid:16) e e − − M M t j − t j 1 (cid:17) (cid:17) L L L t j t − − j t− 1 1 1    . (7) j Lt−1 j Lt−1 (cid:124) (cid:123)(cid:122) (cid:125) dispersioncomponent In this decomposition, changes in aggregate labor productivity are broken into three components. First,ameancomponentwhichcaptureschangesinanaggregationofsectoral mean log labor productivities. Second, a sectoral share component which captures the effect of the changing allocation of labor between sectors. This second component will be 8

positive if labor is flowing from low labor productivity sectors to high labor productivities onces. Third,adispersioncomponent,whichcaptureschangesinthedispersionoffirmlog laborproductivitieswithinsectors. 2.3 Decomposing Changes in TFP using Decomposition II WemeasureTFP,A ,as: t Y t A = . (8) t KαL1−α t t Weassumethatcapital’sshareofoutput,α,ispositive. Wecanthusrewrite(8)as: Y Y t t log(A ) = αlog( )+(1−α)log( ). (9) t K L t t Takingchangesin(9), Y Y t t ∆log(A ) = α∆log( )+(1−α)∆log( ). (10) t K L t t In(7),weshowedthatchangesinlog(Yt)canbebrokenintomean,dispersion,andsectoral- Lt share components. Denote these components for labor as ML,DL, and SL, respectively. t t t DenotethesecomponentsforcapitalasMK,DK,andSK,respectively. Hence, t t t Y ∆log( t ) = MK +DK +SK. (11) K t t t t In turn, we can rewrite changes in log TFP from (10) as changes in the weighted sum of the mean components for capital and labor, the dispersion components for capital and labor,andthesectoral-sharecomponentsforcapitalandlabor: ∆log(A ) = (αMK +(1−α)ML)+(αDK +(1−α)DL)+(αSK +(1−α)SL). (12) t t t t t t t 3 Decomposition Applied to Models Inthissection,wedemonstratetheeconomicsofourdecompositioninstandardproduction environments. First, in the production environment described by Hsieh and Klenow (2009), we demonstrate that changes in the production technology, prices, or the expected value of the log of marginal revenue products of capital will manifest themselves in the 9

mean component of our decomposition. Second, changes in the variance or higher order moments of the log of marginal revenue products of capital will be reflected in the dispersioncomponentofourdecomposition. Building on the above results, we demonstrate in a standard production environment how the components of our decomposition capture changes in the distribution of distortions to firm labor and capital choices. We find that common changes to the frictions to input choices facing firms are reflected in movements in the mean component. We also derive conditions under which distributional changes in such frictions are reflected in the dispersion component of our decomposition. Such results are derived in a more general framework than that of Hsieh and Klenow (2009), and we identify a number of relevant papersthatcanbemappedintoourenvironment. Our results are particularly relevant to the literature that studies the role financial frictionsplayinamplifyingmovementsinaggregatesoverthebusinesscycle. Weanalytically demonstrate how a change in a financial friction in a simple model of production will present itself as a distortion. We then demonstrate that an increase in the extent to which thisfinancialfrictionaffectsfirmswillincreasethedispersioninwedges. 3.1 Hsieh and Klenow (2009) Production Environment Themodelconsistsofheterogeneousfirmsthatproducedifferentiatedgoods. Thereare S industries, and the outputs of each industry, Y , are aggregated into a final good (total S output), Y, using Cobb-Douglas technology in a perfectly competitive market. Hence, aggregateoutputcanbedefinedas: S (cid:88) Y = ΠS Yθs, where θ = 1. (13) s=1 s s s=1 From standard arguments: P Y = θ PY, where the price of industry output is P and S S s s P isthepriceofthefinalgood,whichissettobethenumeraire. ThereareM firmsinasectors. Industryoutput,Y isproducedusingCEStechnology: s s (cid:32) (cid:33) σ (cid:88) Ms σ−1 σ−1 Y = Y σ . (14) s si i=1 Within an industry, firms are heterogeneous in a few dimensions. First, they vary in aspects of their physical productivity. Second, they vary in the magnitude of frictions to 10

their labor and capital choices. One can write these two distortions as distortions that affect the marginal products of labor and capital evenly, which one can write as an output distortion τ , and distortions that affect the marginal product of capital relative to labor, Y whichonecanwriteasacapitaldistortion,τ . Firmiwithinsectorsproducesoutput,Y , K si from its firm TFP, A , capital stock K , and labor L , using the following Cobb-Douglas si si si technology: Y = A KαsL1−αs. (15) si si si si ProfitsoffirmiinsectorS arethus: π = (1−τ )P Y −wL −(1+τ )RK . (16) si Ysi si si si Ksi si From standard arguments, in this setup, the marginal revenue product of capital for a firm,MRPK (cid:44) ∂PsiYsi,isafunctionoftherentalrateofcapitalandfirmlevelwedges: si ∂Ksi 1+τ MRPK = R Ksi. (17) si 1−τ Ysi AsinHsiehandKlenow(2009),itisusefultodefinethemarginalproductofcapital(in total)forasectorasthefollowing:5 R MRPK (cid:44) . (18) s (cid:80)MS 1−τYsi Ksi i=1 1+τKsi KS For our decomposition, it is also useful to define the weighted average of the log marginalproductofcapitalforfirmsinasector,whichis: (cid:88) MS 1+τ P Y LMRPK (cid:44) log(R Ksi) si si . (19) s 1−τ P Y i=1 Ysi S S 5TherewasanerrorinthespecificationofthisobjectintheoriginalpaperofHsiehandKlenow(2009). Thespecificationherecorrespondstotheforminthepublishedcorrections. 11

3.1.1 OurDecompositioninthisProductionEnvironment Intheenvironmentabove,from(18)and(19)andthedefinitionofthemarginalrevenue product of capital, our decomposition applied to capital productivity for sector s can be expressedasthefollowing: (cid:18) PsYs(cid:19) (cid:18) 1 σ (cid:19) ∆log t t = ∆log +∆LMRPK Ks α 1−σ s t s (cid:124) (cid:123)(cid:122) (cid:125) meancomponent (cid:0) (cid:1) +∆log MRPK −∆LMRPK . (20) s s (cid:124) (cid:123)(cid:122) (cid:125) dispersioncomponent We now demonstrate how the changing distribution of marginal productivities are reflected in our decomposition by demonstrating which cumulants of the distribution of log marginal revenue productivities show up in which components of our decomposition. Cumulants are similar to moments; the cumulant-generating function of a random variable is an alternative specification of a probability distribution, similar to a moment-generating function. The first cumulant is the expected value of the variable, the second cumulant is its variance, and the higher order cumulants are polynomial combinations of centralized moments. Consider the distribution of firm log marginal revenue products of capital, reflecting both the mass of firms at a given productivity and their relative output shares (to be precise, the CDF would be written G (X) = (cid:82) 1(MRPK ≤ X) PsiYsidi). Denote s i∈s si PSYS thecumulantsofthisdistributionasκs,κs,.... Usingpropertiesofthecumulantgenerating 1 2 function, our decomposition can be expressed as the following function of the cumulants ofthedistributionoflogmarginalrevenueproductivitiesofcapital:6 (cid:18) PsYs(cid:19) (cid:18) 1 σ (cid:19) ∆κs ∆κs ∆κs ∆log t t = ∆log +∆κs+− 2 + 3 − 4 +.... (21) Ks α 1−σ 1 2! 3! 4! t s (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) meancomponent dispersioncomponent Note that κs = LMRPK is the weighted average of the log of firm marginal revenue 1 s products of capital, while κs is the variance of the log of firm marginal revenue prod- 2 ucts of capital. The mean component captures only changes in technology or LMRPK . s 6SeeAppendixAfordetailsofthisderivation. 12

Changes in the second cumulant (and thus second central moment), or higher order cumulants (and thus all of the remaining higher order moments) of the distribution of the log of firm marginal revenue products of capital are reflected in the dispersion component of our decomposition. Notethatiffirmmarginalrevenueproductsofcapitalarelognormallydistributed,then onlythefirsttwocumulantsarenon-zero. Inthatcase,ourdecompositionisisomorphictoa mean-variance decomposition. Increases in the expected value of the log of firm marginal revenue productivities are reflected in the mean component of our decomposition, while the negative effect of the greater variance of the log of firm marginal revenue products is reflected in the dispersion component of our decomposition. This is apparent if in (21) the dispersioncomponentisfurtherbrokenintovarianceandhigherordertermsasbelow: higher-orderterms variance (cid:122) (cid:125)(cid:124) (cid:123) (cid:18) PsYs(cid:19) (cid:18) 1 σ (cid:19) (cid:122) ∆ (cid:125)(cid:124) κs (cid:123) (cid:88) ∞ ∆κs ∆log t t = ∆log +∆κs+ − 2 + (−1)n−1 n . (22) Ks α 1−σ 1 2! n! t s n=3 (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) meancomponent dispersioncomponent Notice,ifwedefineκ asthen(cid:48)thcumulantofthelogofthemarginalrevenueproduct L,n oflaborandκ asthen(cid:48)thcumulantofthelogofthemarginalrevenueproductofcapital, K,n wecanapplyourdecompositiontochangesinTFPusing(12): (cid:18) (cid:19) σ ∆log(TFPR ) = ∆log α−αs(1−α )αs−1 +(1−α )∆κs +α ∆κs s 1−σ s s s L,1 s K,1 (cid:124) (cid:123)(cid:122) (cid:125) meancomponent variance higher-orderterms (cid:122) (cid:125)(cid:124) (cid:123) (cid:122) (cid:125)(cid:124) (cid:123) (1−α s )∆κs L,2 +α s ∆κs K,2 (cid:88) ∞ (1−α s )∆κs L,n +α s ∆κs K,n − + . 2! (−1)n−1n! n=3 (cid:124) (cid:123)(cid:122) (cid:125) dispersioncomponent (23) 13

3.2 Simple Model of Production and Allocation Given an increase in the dispersion of wedges likely results in a change in value added shares, we demonstrate under what conditions we can analytically demonstrate that an increase in the dispersion of wedges leads to an increase in the dispersion component of our decomposition. Similarly, we demonstrate under what conditions we can analytically demonstrate that an increase in the mean of wedges will increase the mean component of our decomposition, all else equal. We present our results within a similar environment to HsiehandKlenow(2009),butmoregeneraltechnology. As in Hsieh and Klenow (2009), the model consists of heterogeneous firms who produce differentiated goods, which are aggregated into a final good (total output), but now with a more general aggregation technology to be described below. Firms are heterogeneous in the following dimensions: They vary in aspects of their physical productivity, z , it andtheyvaryinthemagnitudeoffrictionstotheirlaborandcapitalchoices. 3.2.1 IntermediateGoodFirmTechnology Firmsareindexedbyiandtimebyt. Firmiattimetproducesy unitsofanintermedii,t ate good using l units of homogeneous labor and k units of capital with the production i,t i,t function y = z lγ kν . Labor and capital are homogeneous, therefore aggregate labor i,t i,t i,t i,t (cid:82) (cid:82) and capital clearing imply that l di = L and k di = K , where L and K denote i i,t t i i,t t t t aggregatelaborandcapital. 3.2.2 AggregationTechnologyandValueAdded Totaloutput,Y ,isaggregatedfromfirmoutputwithtechnologyY = (cid:0)(cid:82) yϕdi (cid:1)φ . Note t t i i,t that this general form nests the two most common final good technologies considered in the literature as special cases: The CES aggregator and heterogeneous firms producing a singlegood. Thefinalgoodsectoriscompetitiveandcostminimizing. Standardarguments imply the price of each intermediate good, p i,t , is p i,t = Y φ− φ 1 P t (y i,t )ϕ−1, where P t is the price of the final good, which we set to be the numeraire. Therefore value added in real terms,v = pi,ty ,canbeexpressedasafunctionofpricesandfirmoutput: i,t Pt i,t φ−1 v = Y φ (y )ϕ. (24) i,t t i,t To compute our decomposition, one requires firm-level productivity ratios and firmlevel value-added shares. For a given firm, we can compute a firm-level value-added share 14

as: vi,t. Yt 3.3 TheOptimalAllocationofInputsandtheRoleofFirm-levelWedges Inthissubsection,wesolvetheoptimalallocationofresourcesintheplanner’sproblem, andwedemonstratehowfirm-specificwedgescandistorttheallocationoflaborandcapital betweenfirmsfromthisallocation. We show that the optimal allocation of resources in the planner’s problem is such that all firms have the same productivity ratios. This choice is unique and can be characterized asafunctionofthedistributionoffirm-levelTFP,Fz(z). Wethenshowthatanyallocation t of capital and labor between firms can be expressed as a function of the optimal input choice and firm-specific wedges. We utilize this final result in the following subsection to evaluatehowshockstofirm-levelwedgesshowupinourdecomposition. 3.3.1 OptimalAllocation Wenowpresentapropositionthathighlightstheknownresultthatforanyfixedamount of total capital and labor, the optimal allocation of resources (to maximize static output) is such that all firms with identical production function coefficients have the same factor productivityratios. Proposition1. (i) Given a fixed amount of total labor and capital, L and K , the allocation of capital t t and labor across firms that maximizes output is such that there are unique optimal labor and capital productivity ratios, v t ∗ and v t ∗ , which are common among all firms l∗ k∗ t t andonlydependontheCDFoffirmproductivity,Fz(z). t Proof. SeeAppendixA. A full (static) planner’s problem maximizing current welfare could be split into two parts: First,solvefortheoptimalallocationruleofcapitalandlaborbetweenfirmsforany fixed amount of both capital and labor; and second, choose the total amount of capital and labortomaximizecurrentperiodutility. ThereforeProposition1impliesthattheallocation which maximizes static utility is one where firms have constant productivity ratios. We do notplaceanyrestrictionsonthelevelofstaticallyoptimaltotallaborandcapital. 15

3.3.2 Firm-LevelWedges Wethenusetheoptimallaborandcapitalproductivityratiostodefinefirm-levelwedges, defined as the firm productivity ratio, vi,t or vi,t, over the optimal productivity ratio, v t ∗ or li,t ki,t l t ∗ v∗ t . Weformallydefinefirmlevelwedgesas: k∗ t v l∗ ω (cid:44) i,t t , (25) l,i,t l v∗ i,t t and v k∗ ω (cid:44) i,t t , (26) k,i,t k v∗ i,t t forlaborandcapital,respectively. Thesewedgescapturehowfarafirm’sproductivityratioisfromtheonethatmaximizes welfare in the social planner’s static optimization problem. They also capture aggregate distortions,whichdistorteveryfirm’sinputdecisionandchangeaggregatelabororcapital, aswellaschangesintherelativedistributionofresourcesbetweenfirms. In the model of Hsieh and Klenow (2009), which is a special case of this production environment, these wedges can be expressed as functions of the firm-level distortions in their model, τ and τ . The firm-level wedges are proportional to these distortions: Ysi Ksi ω ∝ 1 andω ∝ 1+τKsi. l,i,t 1−τYsi k,i,t 1−τYsi 3.4 Shocks to Firm-level Wedges in our Decompositions In this subsection, we illustrate how changes to the distribution of firm-level wedges are captured in our decomposition and how such changes affect aggregates. The model of production and allocation (from subsection 3.2) we consider only has a single sector of production with identical production function coefficients, so we can perform our analysis using Decomposition I. However, Decomposition II first applies Decomposition I individually to each sector and then aggregates up the sectoral mean and dispersion components. Therefore, the way in which our components capture firm-level wedges will be similar for a multi-sector version of our model with production function coefficients varying across sectors. We perform our analysis of shocks to firm-level wedges only for Decomposition I duetothegreatertractabilityandcleanerdemonstrationoftheeconomicsofourdecomposition. We begin without making parametric assumptions on the distribution of wedges. Let 16

Y∗ t denotetheundistorted(absentanyidiosyncraticorcommondistortions)aggregatecap- K∗ t ital factor productivity ratio. Let F denote the density function of wedges to firm cap- ω,k ital choices, which reflects both the mass of firms and their relative value added shares. Formally, this can be expressed as an integral over firms (denoted by i): F (x) = ω,k (cid:82) 1 vidi. Thecumulantsofthelogoffirm-levelwedgesaredenotedas: κ ,κ , i (ω k,i ≤x)Y k,1,t k,2,t κ ,etcetera. Cumulantsaresimilartomoments;wediscusstheirstatisticalpropertiesin k,3,t subsubsection 3.1.1. Our decomposition of changes in aggregate capital productivity can beexpressedas:7 higher-orderterms variance (cid:122) (cid:125)(cid:124) (cid:123) (cid:18) Y (cid:19) (cid:18) Y∗(cid:19) (cid:122) ∆ (cid:125) κ (cid:124) (cid:123) (cid:88) ∞ ∆κ ∆log t = ∆log t +∆κ +− k,2,t + (−1)n−1 k,n,t . (27) K K∗ k,1,t 2! n! t t n=3 (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) meancomponent dispersioncomponent Equation(27)showsthatthemeancomponentcaptureschangesintheundistortedproductivity ratio (capturing changes unrelated to distortions) and changes in the first cumulant (which capture common changes in distortions). Changes in the variance of firm log wedges(thesecondcumulant),oranyhighermomentsoftheirdistribution,arereflectedin thedispersioncomponent. Suchresultseasilycarryoverforlaborproductivitybyreplacing capitalforlaborin(27). We can then use (12) to express total factor productivity as a function of undistorted TFP,TFP∗,andthecumulantsofwedgestocapitalandlabor:8 t ∆log(TFP ) = ∆log(TFP∗)+α∆κ +(1−α)∆κ (28) t t k,1,t l,1,t (cid:124) (cid:123)(cid:122) (cid:125) meancomponent higher-orderterms variance (cid:122) (cid:125)(cid:124) (cid:123) (cid:122) (cid:125)(cid:124) (cid:123) ∞ α∆κ +(1−α)∆κ (cid:88) α∆κ +(1−α)∆κ k,2,t l,2,t k,n,t l,n,t − + . 2! (−1)n−1n! n=3 (cid:124) (cid:123)(cid:122) (cid:125) dispersioncomponent 7SeeAppendixAfordetailsofthisderivation. 8Forthisexercise,wemakethestandardassumptionthatTFPismeasuredasTFP t = Kα Y L t 1−α ,whereα t t isconstantovertime. 17

Equation(28)showsthatthemeancomponentcaptureschangesinundistortedTFPand changes in the first cumulants of log wedges (corresponding to the mean of log wedges). Thedispersioncomponentcaptureschangesinthevarianceorhigherordermomentsoflog wedges. In the remainder of this subsection, we present special cases of the results in (27) and (28). Weshowhowcommonchangesinwedgesandchangesintechnologyarereflectedin the mean component of our decomposition. Finally, we show that changes in the variance and higher-order moments of log wedges are captured in the dispersion component our decomposition.9 3.4.1 CommonShockstoDistortions Wefirstshowthatcommonchangestodistortionsarereflectedonlyinthemeancomponent of our decomposition. First, consider a shock,ξ , that affects firmsevenly such that t+1 ω = ξ ω . Thisisthesortofshockthatwouldarise,forexample,fromadistortion k,t+1 l,t+1 k,t to the rental rate of capital faced by all firms. Below, we show the effect of this shock on aggregatecapitalfactorproductivityandTFPisreflectedonlyinthemeancomponent: Y t ∆log( ) = log(ξ ) , k,t K t (cid:124) (cid:123)(cid:122) (cid:125) meancomponent and ∆log(TFP ) = αlog(ξ ) . t k,t (cid:124) (cid:123)(cid:122) (cid:125) meancomponent Only the mean component will change if the economy is hit by no other shocks. Such resultsextendtothedecompositionoflaborproductivitywhenwereplacelaborforcapital. 3.4.2 ShockstotheVarianceandHigher-orderMomentsofDistortions Note that the second cumulant is the variance of log wedges, while all of the higher order cumulants can be expressed as polynomial combinations of the second and higherordercentralmoments. Therefore(27)and(28)implythatchangesinthevarianceandany higher-order moments of log wedges are reflected in the dispersion component, without havingtomakeanyparametricassumptions. 9Theseresultsfollowdirectlyfromourdecompositionandpropertiesofcumulantgeneratingfunctions; anoutlineoftheirderivationarefoundinAppendixA. 18

To provide further intuition, we now demonstrate how shocks to the distribution of wedges are realized in our decomposition under some standard parametric assumptions. Forexample,ifwedgestocapitalarelognormallydistributedwithmeanµ andvariance ω,k,t σ2 ,thenchangesinaggregatecapitalproductivitycanbedecomposedas: ω,k,t (cid:18) Y (cid:19) (cid:18) Y∗(cid:19) ∆σ2 ∆log t = ∆log t +∆µ + − ω,k,t . K K∗ ω,k,t 2 t t (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) meancomponent dispersioncomponent Withthelognormalassumption,onlythefirstandsecondcumulantexist. Atypicalwayof adding variation in higher-order central moments (and thus higher-order cumulants) is to create a mixture of lognormals. If wedges to capital are modeled as a mixture of lognormals,withweightsλ onlognormaldistributionswithmeansµ andvariancesσ2 , k,n,t k,n,t k,n,t thenwecanexpressourdecompositionasfollows: (cid:18) Y (cid:19) (cid:18) Y∗(cid:19) (cid:88) ∆log t = ∆log t +∆ λ µ K K∗ k,n,t k,n,t t t n (cid:124) (cid:123)(cid:122) (cid:125) meancomponent − ∆ (cid:80) n λ k,n,t σ k 2 ,n,t −∆log (cid:32) (cid:88) λ k,n,t e (µ k,n,t −(cid:80) j λ k,j,t µ k,j,t ) e 1 2 (σ k 2 ,n,t −(cid:80) j λ k,j,t σ k 2 ,j,t ) (cid:33) . 2 n (cid:124) (cid:123)(cid:122) (cid:125) dispersioncomponent The mean component captures changes in the weighted means of the lognormal distributions or in the undistorted factor productivity ratio. All other changes in the distributions will be reflected in the dispersion component. Changes in the variances of the lognormal distributions which make up the mixture will be reflected here, as will higher order moments. For example, a skewed distribution is often parameterized as a mixture of lognormalswithdifferentmeans,whichwillbereflected,viatheterme (µ k,n,t −(cid:80) j λ k,j,t µ k,j,t ) ,in thedispersioncomponent. Kurtosisisoftenoftenparameterizedasamixtureoflognormals with different variances, which will be reflected, via the term e 1 2 (σ k 2 ,n,t −(cid:80) j λ k,j,t σ k 2 ,j,t ) , in the dispersioncomponent. 3.5 Mapping to Other Models Inthissubsection,weshowthatourmodelhasamappingtoseveralmodelsoffrictions to the allocation of labor and capital between firms. We then demonstrate how frictions in 19

a simple model of financial frictions would be reflected in wedges, and then discuss how ourdecompositionwouldcapturechangesinsuchfrictions. 3.5.1 MappingtoModelsofLabororCapitalAllocation Oursimplemodelconsistsonlyofaproductionenvironmentwithwedgesrepresenting frictions to the allocation of labor and capital. Therefore, there is a mapping to any model withaproductionenvironmentconsistentwithours. ThisincludesthemodelsofKhanand Thomas(2013),Bloometal.(2014),andArellanoetal.(2012),aswellasnumerousother heterogeneousagentmodelsconsideredinthemacroeconomicsliterature. We formally show this correspondence by proving that given the production environment in our model, aggregate output, employment, capital, and the full distribution of output, labor, capital, and technology across firms can be characterized using wedges and either firm-level technology or output shares for any allocation of labor, capital, and technological productivity across firms. We denote G (z,ω ,ω ) as the joint distribution of t l k (cid:0) (cid:1) firm technological productivity and firm-level wedges to labor and capital, J v,ω ,ω t Y l k as the joint distribution of firm output shares and firm-level wedges to labor and capital, (cid:18) (cid:18) (cid:19) (cid:19)φ(1−νϕ−γϕ) (cid:82) ϕ( 1 ) and Z = z 1−νϕ−γϕ di as an index of aggregate productivity. The t i i,t following proposition states that this representation can map any resulting allocation of resourcesandaggregatesusingthesefirm-levelwedges: Proposition2. Thefulldistributionoflabor,capital,andproductivityacrossfirms,F (z,l,k), t andaggregateoutput,employment,andcapitalhavea1-1mappingwithanyofthefollowing: (i) G (z,ω ,ω ). t l k (cid:0) (cid:1) (ii) J v,ω ,ω andameasureofaggregateproductivityZ .10 t Y l k t Proof. SeeAppendixA. 3.5.2 MappingtoaModelofFinancialFrictions In this subsection, we show that a financial friction in a simple model map can be redefined as a firm-level wedge. We then discuss how a tightening of the friction in the modelcangenerateagreatervarianceofwedges. 10More generally, it can be shown that J (cid:0)v,ω ,ω (cid:1) together with (cid:0)(cid:82) zτdFz(z) (cid:1) , for any τ (cid:54)= 0, is t Y l k sufficienttocharacterizeoutput,employment,andcapitalatboththeaggregateandfirmlevel. 20

Considerasimplemodelwhereheterogeneousfirmsproduceahomogeneousconsumptiongoodwithtechnologyy = z lb ,whereb < 1. Inthissetting,valueaddedisequivai,t i,t i,t lenttofirmoutput,v = z lb . HouseholdshaveutilityfunctionU (C,L) = C1−σ − L1+ν i,t i,t i,t 1−σ 1+ν and discount the future at rate β. Production in this simple model is a special case of the productionenvironmentintroducedinSubsection3.2;thus,theplanner’sproblemstatesall firmsoptimallyhavethesamelaborproductivityratios. The friction in this model is a simple borrowing constraint: Firms have wealth a , i,t which we consider exogenous for our analysis. Firms must pay their workers at the beginning of the period but only receive cash flows from production at the end. The borrowing constraint, l W ≤ a ρ, where W is the wage and ρ is a positive constant, restricts the lai,t i,t bordecisionsoffirmswhenitbinds. Assumefirmsmayalsoexogenouslyexiteachperiod withprobabilityδ. TheoptimizationproblemoffirmscanbeexpressedasthefollowingLagrangian: L = maxz lb −l W +λ (a ρ−l W ), (29) i,t i,t i,t i,t i,t i,t t li,t where λ is the Lagrange multiplier on the borrowing constraint. The multiplier is 0 i,t if the borrowing constraint does not bind, and positive otherwise. Taking the first-order conditions of (29) and manipulating the labor-leisure condition allows us to express the firm’slaborchoiceasthefollowing: l i,t = z i 1 ,t − 1 bb1− 1 bY t − 1− σ bL − t 1− ν b (1+λ i,t )− 1− 1 b . (30) From (30) we can derive firm-level wedges, ω = vi,t l t ∗ , as a function of aggregates l,i,t li,t v t ∗ andtheLagrangemultiplierfacedbythefirm: (cid:18) (cid:19) (cid:18) (cid:19) Y L t t log(ω ) = σlog +νlog +log(1+λ ). (31) l,i,t Y∗ L∗ i,t t t (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) FirmSpecific Aggregate Note that the wedge can be expressed as a function of the distortion of aggregates fromtheiroptimalvalue(whichaffectthewagerate)aswellasthefirm-specificdistortion captured by the Lagrange multiplier in the firm’s problem. We can express the Lagrange multiplierasthefollowingfunctionofaggregatesandeachfirm’sz anda : i,t i,t 21

 0 z bY−bσL−bνab−1ρb−1 ≤ 1 i,t t t i,t log(1+λ ) = . (32) i,t (cid:0) (cid:1) log bY−bσL−bνz ab−1ρb−1 z bY−bσL−bνab−1ρb−1 > 1 t t i,t i,t i,t t t i,t NotethattheonlywaythatthereisnoheterogeneityinLagrangianmultipliersiseither if(a)theborrowingconstraintneverbinds,or(b)itbindsforallfirms,butwealthisproportional to productivity (z = a1−b). This second condition implies no inefficiencies in the i,t i,t distribution of resources between firms; all inefficiencies arise from the reduced aggregate demandforlabor. Now consider what a shock to borrowing constraints does. Assume that at time t = 0, ρishighenoughsuchthattheconstraintbindsfornofirms. Thenthemeanandvarianceof log(1+λ )is0. Adecreaseinρtothepointwheretheconstraintbindsforsomebutnot i,0 allfirmsleadstoariseinboththemeanandvarianceoflog(1+λ ). Atthesametime,the i,t borrowingconstraintreducesY andL ,assomefirmscannothiretheamountoflaborthey t t would prefer were they unconstrained. Therefore the change to the mean of firm wedges, log(ω ),asaresultofthisshockisambiguous,astheaggregatecomponentandthemean l,i,t of the firm-specific component move in opposite directions. However, the direction of the changeinvarianceofthefirm-levellaborwedgeisunambiguous,increasinginresponseto suchashock. 4 Decomposition Applied to Data In this section, we apply our decompositions of aggregate productivity described in Section 2 to data on U.S. public firms and Japanese public firms. In our discussion of the resultsappliedtoU.S.publicfirms,wealsoincludeacomparisonofourmeasuresoflabor productivity, capital productivity, and TFP to those from the national income and product accounts (NIPA). In Appendix B, we describe how we clean our data on U.S. nonfinancial publicfirms,andmeasuretheobjectsofinterest. 4.1 Discussion of results — Data from the United States Figures1and2displayresultsofyear-over-yearchangesinaggregatelaborproductivity and its components from Decompositions I and II, respectively. From the eye test alone it should be clear that in the recent recession, aggregate labor productivity and its dispersion component have a negative correlation, and the mean component is highly correlated with 22

aggregate labor productivity. Figures 3 and 4 further demonstrate this point: Over four recession periods, the mean component moves closely with aggregate labor productivity. In Decomposition II, the dispersion component has very little cumulative change over any ofthefourepisodesinoursample. Figures5and6displayresultsfromDecompositionsIandIIofyear-over-yearchanges in aggregate capital productivity, and tell a different story. The dispersion component is positively correlated with aggregate capital productivity over the past two business-cycle episodes. For previous episodes, the mean component moves more closely with aggregate capitalproductivity. TheseresultsaremorestarklyapparentinFigures7and8,whichshow cumulative changes in aggregate capital productivity and its components from Decompositions I and II. In the recent episodes, for either decomposition, the dispersion component movesmuchmorecloselywithaggregatecapitalproductivity. As we describe in the previous subsection on measurement, to compute TFP, by the natureofourassumptionsontheproductionfunctionandvaluesforitscoefficients,changes inlaborproductivitygetmoreweight(65percent)thanchangesincapitalproductivity(35 percent). Hence, as should be expected, we see that the results for TFP are much more qualitatively consistent with the results from what drives changes in labor productivity. This is apparent in the year-over-year changes charts from Decompositions I and II in Figures 9 and 10, as well as the cumulative changes charts from Decompositions I and II inFigures11and12. Table 1 displays correlations between the components of Decomposition II and their respective aggregates. The results from the figures are further codified in this table. The correlation between the dispersion component and sectoral share components and labor productivity are especially striking. Movements in labor productivity are much more correlated with the mean of firm-level log labor to value-added ratios than with their dispersion. These results are dampened when looking at TFP because capital productivity has a positivecorrelationwithitsdispersioncomponent. However,therelationshipbetweenTFP anditsdispersioncomponentisultimatelyclosetozero. Our sample represents a significant slice of the U.S. economy; in 2011, it accounted for over 15 percent of GDP and over 17 million employees. To understand the extent to which our sample reflects the mechanisms responsible for driving aggregate productivity changes, we compare the time-series behavior of each aggregate productivity ratio as aggregated from Compustat to that of therespective productivity ratio computed from NIPA. Figures 16 and 17 show that the time-series properties of TFP computed from both Com- 23

pustat and NIPA are similar both in their cyclical dynamics and long-term trends. These figuressuggestthatsomeofthekeyforcesdrivingTFPovertimearelikelypresentinCompustatdata. IfthereweresignificantfactorsdrivingTFPoverthebusinesscyclethatexisted onlyinsmall,privatefirms,wewouldexpectsystematicdifferencesinthebehaviorofTFP and our measure computed from publicly listed firms over time. However, there are some differences in the measures. TFP from Compustat is more volatile, which is unsurprising given the documented greater volatility of corporate profits measured with generally accepted accounting principles (GAAP) than corporate profits as measured in NIPA.11 There arealsosomeslighttimingdifferences,particularlyinthetimingofthetrough(ofTFP)of the 2007—2009 recession. These timing differences may be due to the reporting dates of firms in Compustat. However, themeasure of TFP forthe United States inthe Penn World Tables(8.0)hasthetroughin2009,sothetimingdifferencesmayalsobeduetosometechnicaladjustmentsmadeintheNIPAaggregation. InFigures18and19,welookatchanges in each productivity ratio and its NIPA equivalent. We see the timing and volatility issues arepresentforeachproductivityratioseparately. 4.2 Discussion of results - Data from Japan InFigures13and14,wedisplayresultsfromDecompositionsIandIIofyear-over-year changes in aggregate TFP in Japan. The results from the second decomposition are more consistent with those from the United States for the recent recession in that the dispersion componentisnotcorrelatedwithmovementsinTFP.Theresultsfromthefirstdecomposition,however,showthedispersioncomponenttobemorehighlycorrelatedwithaggregate TFPovertherecentepisode. Thisresultistrueforlaborproductivityaswell. 5 Conclusion This paper presents decompositions of changes in aggregate labor productivity, capital productivity, and TFP. We demonstrate how the dispersion component of our decompositions reflects changes in the degree to which frictions affect firms in many heterogeneous firm models that attempt to explain the nature of the business cycle. In turn, computing thecomponentsofourdecompositionindataandcomparingthemtothesamemetricsina givenmodeloftheclassweconsiderwillhelptoassesswhethersuchamodelisconsistent 11Hodge(2011)comparesthepropertiesofcorporateprofitscomputedfromtheGAAPaccountingstatementsoffirmsintheS&P500indexwiththecorrespondingmeasurefromNIPA,findingsignificantlygreater volatilityintheS&Pmeasure. 24

with firm-level behavior. As we demonstrate in this paper, it is not only useful to compute our decompositions on models that have already been solved; one can also compute our metrics in the data before writing down a model to help motivate which mechanisms shouldbekeyindrivingpatternsoverthebusinesscycle. Appendix A Proof for Proposition 1 Given the model of production in Subsection 3.2, we can define the following Lagrangian for the social planner to solve supposing she gets to allocate a fixed amount of laborandcapitalacrossfirms,whichareindexedbyi:12 (cid:18)(cid:90) (cid:19)φ (cid:18) (cid:90) (cid:19) (cid:18) (cid:90) (cid:19) L = max (z lγkν)ϕdi +λ K − k di +λ L− l di . (33) i i i 1 i 2 i li,ki i We want to show that there exists optimal labor and capital productivity ratios that are sharedbyallfirmsthatsharethesameproductionfunctioncoefficients. Fromthefirst-order conditionsof(33): (z lγkν)ϕ νϕφY 1− φ φ i i i = λ 1 , (34) k and (z lγkν)ϕ γϕφY 1− φ φ i i i = λ 2 . (35) l Also,theplannerwillfullyallocatelaborandcapitaltoallfirms,so: (cid:90) K = k di, (36) i and (cid:90) L = l di. (37) i 12Toeconomizeonnotation,timesubscriptsareomitted. 25

Withsomealgebra,itcanbeshownthat: λ 1 v = k , (38) i i νϕφ and λ 2 v = l . (39) i i γϕφ Hence,summingoveriin(38)and(39): λ 1 Y = K , (40) νϕφ and λ 2 Y = L . (41) γϕφ Inturn,from(38)and(40): Y v i = . (42) K k i Also,from(39)and(41): Y v i = . (43) L l i In turn, all firms will optimally have the same firm-level capital and labor productivity ratios. We can now express optimal productivity ratios v∗ and v∗ as a function of L, K, and l∗ k∗ Y∗. Fromtheproductiontechnology,(42),and(43): k i = Y − φ 1 z i ϕlϕγk i ϕνK∗(Fz(z)), (44) and l i = Y − φ 1 z i ϕlϕγk i ϕνK∗(Fz(z)). (45) 26

Combiningtheproductiontechnologywith(44)and(45),alongwithsomealgebra,yields: (cid:18)(cid:90) (cid:18) (cid:19) (cid:19)φ(1−νϕ−γϕ) ϕ( 1 ) Y∗ = LϕφγKφϕν z 1−νϕ−γϕ di . (46) i i Notethatthisoptimaloutputisjustafunctionofthedistributionofproductivity,Fz(z), andtotallaborandcapital. Thus,wecanexpresstheoptimalproductivityratiosas: v∗ Y∗ = , (47) l∗ L and v∗ Y∗ = . (48) k∗ K Proof for Proposition 2 Thisproofisdoneinthefollowingparts: (i) F(z,l,k)fullycharacterizesoutput,employment,andcapital. (ii) F(z,l,k)hasa1-1mappingwithG(z,ω ,ω ). l k (iii) G(z,ω ,ω ) has a 1-1 mapping with J(v,ω ,ω ) and a measure of aggregate prol k Y l k (cid:18) (cid:18) (cid:19) (cid:19)φ(1−νϕ−γϕ) (cid:82) ϕ( 1 ) ductivityZ = z 1−νϕ−γϕ di . i i Part(i): F(z,l,k)fullycharacterizesoutput,employment,andcapital. This must be true, by the definition of production technology and the clearing con- (cid:82) (cid:82) ditions L = ldF (z,l,k) and K = kdF (z,l,k). Thus, F (z,l,k) fully characterize aggregateoutput,employment,andcapital. Part(ii): F(z,l,k)hasa1-1mappingwithG(z,ω ,ω ). l k Theportionoftheproofhasthefollowingparts: (a) F(z,l,k)hasauniquemappingχ intoG(z,ω ,ω ). 1 l k (b) G(z,ω ,ω )hasauniquemappingχ intoF(z,l,k). l k 2 (c) χ = χ−1. 2 1 27

F(z,l,k)hasauniquemappingχ intoG(z,ω ,ω ): 1 l k (24)combinedwiththeproductiontechnologygivesusv asafunctionofonlyz,l,k: i (cid:18)(cid:90) (cid:19)φ−1 v = zϕlγϕkνϕdF (z,l,k) zϕlγϕkνϕ. (49) i i i i Combining(25),(26),and(49)yields: (cid:0)(cid:82) zϕlγϕkνϕdF (z,l,k) (cid:1)φ−1 zϕlγϕkνϕ k∗ ω (z ,l ,k ,F) = i i i , (50) k i i i k v∗ i and (cid:0)(cid:82) zϕlγϕkνϕdF (z,l,k) (cid:1)φ−1 zϕlγϕkνϕ l∗ ω (z ,l ,k ,F) = i i i . (51) l i i i l v∗ i These equations characterize the wedges implied by a given distribution of capital, labor, and productivity. We can rearrange (50) and (51) to solve for labor and capital as a functionofwedges: (cid:32)(cid:0)(cid:82) zϕlγϕkνϕdF (z,l,k) (cid:1)φ−1 zϕ(cid:0) l∗(cid:1)γϕ(cid:0) k∗(cid:1)1−γϕ(cid:33) 1−γϕ 1 −νϕ k(z ,ω ,ω ,F) = i v∗ v∗ , (52) i l,i k,i (ω )γϕ(ω )1−γϕ l.i k,i and (cid:32)(cid:0)(cid:82) zϕlγϕkνϕdF (z,l,k) (cid:1)φ−1 zϕ(cid:0) l∗(cid:1)1−νϕ(cid:0) k∗(cid:1)νϕ(cid:33) 1−γϕ 1 −νϕ l(z ,ω ,ω ,F) = i v∗ v∗ . (53) i l,i k,i (ω )1−νϕ(ω )νϕ l,i k,i (52)and(53)allowustoobtaintheuniquemappingfromF toG: (cid:90) z¯,ω¯,ω¯ l k G(z¯,ω¯,ω¯ ) = dF (z,l(z,ω ,ω ,F),k(z,ω ,ω ,F)). (54) l k l k l k z,ω,ω =0 l k G(z,ω ,ω )hasauniquemappingχ intoF(z,l,k): l k 2 28

Combining(50)and(51)allowsustoexpressF(z,l,k)asthefollowing: (cid:90) z¯,¯l,k¯ (cid:0) ¯ ¯(cid:1) F z¯,l,k = dG(z,ω (z,l,k,F),ω (z,l,k,F)). (55) l k z,l,k=0 This expression is not sufficient to characterize F(z,l,k) as a function of G(z,ω ,ω ), l k asthefunctionsω ()andω ()ontheright-handsidedependontheterm l k (cid:0)(cid:82) (cid:1)φ−1 (cid:0)(cid:82) (cid:1)φ−1 φ−1 zϕlγϕkνϕdF (z,l,k) φ . Note that zϕlγϕkνϕdF (z,l,k) φ = Y φ . All we have to do now is express Y as a function of G. Plugging (52) and (53) into the aggregate productionfunctionyields:   1−γϕ−νϕ (cid:18) k∗(cid:19)ϕφν (cid:18) l∗(cid:19)ϕφγ (cid:32) (cid:90) z1−γϕ ϕ −νϕ (cid:33)φ 1−φ(γϕ+νϕ) Y =  v∗ v∗ γϕ νϕ dG(z,ω l ,ω k )  . ω1−γϕ−νϕω1−γϕ−νϕ l k (56) Y (G(z,ω ,ω ))canthusbedefinedasafunctionofz andwedges. l k (55) and (56) can be combined to obtain the functions ω (z,l,k,G) and ω (z,l,k,G). l k Thus,wecanobtaintheuniquemapping: (cid:90) z¯,¯l,k¯ (cid:0) ¯ ¯(cid:1) F z¯,l,k = dG(z,ω (z,l,k,G),ω (z,l,k,G)). (57) l k z,l,k=0 χ = χ−1: 2 1 Combining (54) and (57) yields the result that F (z,l,k) = χ (χ (F (z,l,k))) for any 2 1 F (z,l,k). Itfollowsthatχ = χ−1. 2 1 Part (iii): Claim: G(z,ω ,ω ) has a 1-1 mapping with J(v,ω ,ω ) and a measure of l k Y l k (cid:18) (cid:18) (cid:19) (cid:19)φ(1−νϕ−γϕ) (cid:82) ϕ( 1 ) aggregateproductivityZ = z 1−νϕ−γϕ di . i i Theportionoftheproofhasthefollowingparts: (a) G(z,ω ,ω )hasauniquemappingχ intoJ(v,ω ,ω )andpinsdownZ. l k 3 Y l k (b) J(v,ω ,ω )andZ hasauniquemappingχ intoG(z,ω ,ω ). Y l k 4 l k (c) χ = χ−1. 3 4 29

G(z,ω ,ω )hasauniquemappingχ intoJ(v,ω ,ω )andpinsdownZ: l k 3 Y l k (49),(52),(53),and(56)canbecombinedtocharacterize v: Y ϕ −γϕ −νϕ v i = z i 1−γϕ−νϕ (ω l,i )1−γϕ−νϕ (ω k,i )1−γϕ−νϕ , (58) Y (cid:82) ϕ −γϕ −νϕ z1−γϕ−νϕ (ω l )1−γϕ−νϕ (ω k )1−γϕ−νϕ dG(z,ω l ,ω k ) andthusexpressz asafunctionof v,ω ,ω ,andG: Y l k (cid:32) (cid:33)1−γϕ−νϕ z (cid:16)v i ,ω ,ω ,G (cid:17) = ωγ ων v i (cid:90) z1−γϕ ϕ −νϕ dG(z,ω ,ω ) ϕ . Y l,i k,i l,i k,i Y γϕ νϕ l k ω1−γϕ−νϕω1−γϕ−νϕ l k (59) (59)canbeusedtocharacterizeJ(v,ω ,ω ): Y l k (cid:16)v¯ (cid:17) (cid:90) Y ¯v,ω¯ l ,ω¯ k (cid:16) (cid:16)v (cid:17) (cid:17) J ,ω¯,ω¯ = dG z ,ω ,ω ,G ,ω ,ω . (60) l k l k l k Y Y v,ω,ω =0 Y l k G(z,ω ,ω )triviallymapsintoauniqueZ. l k J(v,ω ,ω )andZ hasauniquemappingχ intoG(z,ω ,ω ): Y l k 4 l k (59),rearrangedandintegrated,yields: (cid:90) ϕ 1 z1−γϕ−νϕ Zφ(1−νϕ−γϕ) dG(z,ω ,ω ) = . γϕ νϕ l k (cid:82) (cid:0) (cid:1) γϕ νϕ (cid:0) (cid:1) ω1−γϕ−νϕω1−γϕ−νϕ v ω1−νϕ−γϕω1−νϕ−γϕdJ v,ω ,ω l k Y l k Y l k . (61) Combining(59)and(61)yields:  1−γϕ−νϕ (cid:16)v (cid:17) viZφ(1−ν 1 ϕ−γϕ) ϕ z Y i ,ω l,i ,ω k,i ,J,Z = ω l γ ,i ω k ν ,i  (cid:82) (cid:0) (cid:1) γϕ Y νϕ (cid:0) (cid:1)  .(62) v ω1−νϕ−γϕω1−νϕ−γϕdJ v,ω ,ω Y l k Y l k (62)impliesthatwecanexpressG(z,ω ,ω )asafunctionofJ(v,ω ,ω )andZ: l k Y l k (cid:90) z¯,ω¯ l ,ω¯ k (cid:16)v (cid:17) G(z¯,ω¯,ω¯ ) = dJ (z,ω ,ω ,J,Z),ω ,ω . (63) l k l k l k Y z,ω,ω =0 l k 30

χ = χ−1: 3 4 Thesetwomappingsaretriviallyinversesofeachother. Other Derivations DecompositionasaFunctionofCumulants Consider the cumulative density function of firm log capital productivity, weighted (cid:16) (cid:16) (cid:17) (cid:17) by output shares, G(X) = (cid:82) 1 log vi ≤ X vi. The mean component expressed i ki Y (cid:82) as a function of this distribution is: − −xdG(x), while the dispersion component is: x (cid:0)(cid:82) (cid:82) (cid:1) − e−xdG(x)− −xdG(x) . (cid:82) We know, by definition, that the first cumulant can be written as: xdG(x). A propx ertyofthecumulantgeneratingfunctionisthatE[etx] = (cid:80)n tnκn,whichyields: t=1 n! (cid:90) n n (cid:88) κ (cid:88) κ e−xdG(x) = (−1)n n = −κ + (−1)n n . 1 n! n! t=1 t=2 Thereforethemeancomponentcanbewrittenas: (cid:90) − −xdG(x) = κ . 1 x Whilethedispersioncomponentis: (cid:32) (cid:33) (cid:18)(cid:90) (cid:90) (cid:19) n (cid:88) κ − e−xdG(x)− −xdG(x) = − −κ + (−1)n n +κ 1 1 n! t=2 n n (cid:88) κ (cid:88) κ = − (−1)n n = (−1)n+1 n . n! n! t=2 t=2 Thismeansourdecompositioncanbeexpressedas: (cid:18) (cid:19) n Y (cid:88) κ log = κ +−κ + (−1)n+1 n . 1 2 K (cid:124)(cid:123)(cid:122)(cid:125) n! MeanComponent t=3 (cid:124) (cid:123)(cid:122) (cid:125) DispersionComponent Now note that for any variable of the form z = cvi (such as capital wedges or marginal i ki (cid:16) (cid:17) products in a Hsieh and Klenow case) yields log(z ) = log(c) + log vi . Standard i ki properties of cumulants imply that κ = κ + log(c), and κ = κ for all n > 1. 1,z 1,v n,z n,v k k 31

Thereforeforsuchvariables,ourdecompositionimplies (cid:18) (cid:19) n Y (cid:88) κ log = κ −c +−κ + (−1)n+1 n,z . 1,z 2,z K n! (cid:124) (cid:123)(cid:122) (cid:125) t=3 MeanComponent (cid:124) (cid:123)(cid:122) (cid:125) DispersionComponent (21)and(27)followimmediatelyfromthisderivation. OurDecompositionforDifferentModelsandShocks Common changes in firm revenue products, whether driven by technology or distortions, are reflected only in the mean component of our decomposition. Consider a change inrevenueproductssuchthat vi,t+1 = xvi,t. Then,ourdecompositionimpliesthat: ki,t+1 ki,t (cid:18) (cid:19) (cid:90) (cid:18) (cid:19) (cid:18) (cid:18)(cid:90) (cid:19) (cid:90) (cid:18) (cid:19) (cid:19) Y k 1 v k 1 v k 1 v t+1 i,t i,t i,t t i,t t log = − log di− log di − log di K v x Y v xY v x Y t+1 i,t t i,t i,t i,t t (cid:124) (cid:123)(cid:122) (cid:125)(cid:124) (cid:123)(cid:122) (cid:125) meancomponent dispersioncomponent (cid:90) (cid:18) (cid:19) (cid:18) (cid:18)(cid:90) (cid:19) (cid:90) (cid:18) (cid:19) (cid:19) k v k v k v i,t i,t i,t t i,t t = log(x)− log di− log di − log di . v Y v Y v Y i,t t i,t i,t i,t t (cid:124) (cid:123)(cid:122) (cid:125)(cid:124) (cid:123)(cid:122) (cid:125) meancomponent dispersioncomponent The results in subsubsections 3.4.1 immediately follow from the above. The results in subsubsection 3.4.2 can be derived by using standard formulas for the expectation of lognormallydistributedvariables. Specifically, consider the case where the output-share weighted distribution of wedges is a mixture of lognormals. The pdf of wedges is thus g(log(ω)) = (cid:80)N λ φ (log(ω)), n=1 n n (cid:80) where φ , where φ are normal pdfs and λ = 1. Note that lognormal wedges is n n n n the special case of this with N = 1. Standard formulas for expectations over lognormaldistributionsimplythat (cid:82) log(ω)g(log(ω))dω = (cid:80) λ µ and (cid:82) ωtg(log(ω))dω = n n n (cid:80) n λ n etµn+ 2 1t2σ n 2. Theresultsinsubsubsection3.4.2immediatelyfollow. Appendix B Measurement of Objects — Data from the United States For the empirical analysis in Section 4 on U.S. firms, we use annual data on firms that exist in the Compustat database. We take the following steps, in order. First, firms must 32

be headquartered in the United States and have a U.S. currency code. We then keep only firms with December fiscal year-ends. We then drop firms if their employment, property, plant, and equipment — net of depreciation, sales, or our measure of firm-value added — are missing or negative. We then exclude firms with 4-digit SIC codes between 4000 and 4999, between 6000 and 6999, or greater than 9000, as our model is not representative of regulated, financial, or public service firms. We then clean the data by winsorizing each seriesatthe1stpercentileovertheentiresample. Forouranalysis,welastlyonlykeepdata from1971to2011. Firm-level value added, firm-level capital stock, and firm-level employment are the only firm-level objects we need for our decomposition. When computing year-over-year changesinthecomponentsofourdecomposition,wealsoadjustforentryandexitbyonly keeping data on firms that exist in consecutive years. In the second decomposition, firms aregroupedintosectorsbytwo-digitSICcodes. We measure labor as the number of employees reported in Compustat. We measure capitalasthefirm’splant,property,andequipment,adjustedforaccumulateddepreciation. The aggregate capital stock is annual, taken from the Penn World Tables. To adjust for potential changes in the valuation of capital over time, we construct a perpetual inventory measure of the aggregate capital stock and use the ratio of this measure to the value of the aggregatecapitalstocktodeflatethefirm-levelmeasureofcapital. Theinvestmentmeasure used in the perpetual inventory method is annual gross private domestic investment from the Bureau of Economic Analysis (BEA). To construct our measure of capital using the perpetualinventorymethod(startingfrom1959),weuseadepreciationrateof4.64percent and growth rate of technology of 1.6 percent, following Chari, Kehoe, and McGrattan (2007). Our measure is then deflated by the December value of the monthly CPI, which is CPIforAllUrbanConsumers,seasonallyadjusted,fromtheBureauofLaborStatistics. Wecreateameasureofvalueaddedinpublicfirmsusingincomeaccounting. GDPhas an income equivalent, GDI, which has similar time-series properties. The major components of this measure have equivalents to income statement measures that are required on 10-K forms for U.S. public firms. In order of magnitude, GDI is made up of the following components: compensation of employees, net operating surplus, consumption of fixed capital(depreciation),andtaxesonproductionandimportslesssubsidies. Whilewedonot observe the taxes or subsidies on production and imports firms pay in our dataset, we do observe measures of the other three components, all of which make up over 90 percent of GDI for all years in our sample. We observe labor compensation in Compustat annually. 33

If labor compensation is missing, we replace it with selling, general, and administrative expenses. We also observe net operating profits before depreciation, which is the sum of a firm’s net operating surplus and its capital consumption. We define a firm’s contribution to output as the sum of labor compensation and operating profits before depreciation. In practice, the BEA uses a similar, more detailed approach, where they use firm tax data to aggregate up the components of domestic income and make adjustments for differences between accounting and economic treatment of factors such as capital consumption and inventoryvaluation. TocomputeTFP,followingChari,Kehoe,andMcGrattan(2007),wesetcapital’sshare of income, α = .35 and back it out from (9). When we compare our measure against the NIPA-equivalent, we require a NIPA equivalent measure of our value-added measure, a measure of aggregate labor, and a measure of aggregate capital. To compute our NIPA equivalent of our pseudo-GDI measure, we use data from NIPA table 1.12 on National Income by Type of Account. We take compensation of employees (line 2) and subtract government (line 4), then add to this measure corporate profits with inventory valuation adjustment and capital consumption adjustment less taxes on corporate income (line 43). Finally, we add to this measure consumption of fixed capital, which comes from the BEA. All measures are quarterly, and we only use the fourth-quarter values of these measures. Weputthismeasureinper-capitatermsusingpopulationincludingarmedforcesoverseas. This measure is mid-period and monthly. We only keep its December value. We then put thismeasureinrealtermsusingtheCPImeasuredescribedinthissubsection. Ourmeasure oftherealaggregatecapitalstockwasalreadydescribedinthissubsection. Thismeasureis alsoputinper-capitaterms. Ourmeasureofaggregatelaboristotalnon-farmemployment andismonthly. WeonlyusetheDecemberobservationofthisvariable. Measurement — Data from Japan Our data on Japanese public firms comes from the Compustat global database, and our firm-level variables are measured annually. We clean the data as we do for data from the UnitedStates,exceptweonlykeepfirmswithcurrencycodescorrespondingtotheJapanese Yen and country headquarter codes corresponding to Japan. Also, the years of our sample are different: They only cover 2001 to 2011. Consistent with our application to U.S. data, whencomputingyear-over-yearchangesinthecomponentsofourdecomposition,wealso adjust for entry and exit by only keeping data on firms that exist in consecutive years. In DecompositionII,firmsaregroupedintosectorsbytwodigitSICcodes 34

AsfortheU.S.data,wemeasurefirm-levellaborasthenumberofemployeesreported andfirm-levelcapitalasthefirm’splant,property,andequipment,adjustedforaccumulated depreciation. We deflate the firm-level Japanese capital stock by the U.S. capital deflater. Toputthecapitalstockinrealterms,wedeflateitbytheOECD’smeasureofthequarterly CPIinJapan. Weonlykeepthefourth-quartervalueofthismeasure. In a manner consistent with our application to U.S. data, we create a measure of value added in public firms using income accounting, which is the sum of labor compensation and operating profits before depreciation. As for the U.S. data, if labor compensation is missing, we replace it with selling, general, and administrative expenses. We eventually deflate by the same Japanese CPI measure as for capital. To compute TFP, we again set α = .35andbackitoutfrom(9). Appendix C Our Decomposition in the Context of Other Methodologies To apply our decomposition to data, one does not need to estimate firm-level TFP or sectoral production function coefficients. There is already potential for measurement issues biasing the results from our decompositions, as measures of labor, value added, and capital can all be measured incorrectly. Further, we could be incorrectly grouping firms with our sectoral definitions. However, it is easy enough to check different measures of labor, capital, or value added, if available, and see if the results change. Also, one could add measurement error to firm variables and test the extent to which the results change. Similarly,onecanchecktheresultsfromourseconddecompositionondifferentdefinitions of “groupings” or sectors. However, to compute sectoral production function coefficients, as is commonly done in papers assessing the role of labor and capital allocation on productivity over the business cycle, some issues cannot be “checked.” Data from 30 years prior can be crucial in providing “correct” estimates of sectoral production function coefficients. But what if such data are unavailable to the researcher? In addressing the role of resource reallocation in productivity dynamics over the business cycle, the literature has relied on the estimation of these technological measures for all sectors in the economy. In thissection,wewilldemonstratehowsomeoftheeconometricbiasesassociatedwithsuch anapproachcanleadonetoproducequantitativelyandqualitativelydifferentresultsonthe roleofallocativeefficiencyoverthebusinesscycle. 35

Wefirstdemonstratethemostdifficult-to-correcteconometricbiasassociatedwithmeasuringproductionfunctioncoefficients,whichisthefactthatdatafortheentiresectorover the entire sample are needed to estimate them. We also show that different definitions of factorpricescancruciallyaffectone’sresults. Second,weshowthatourdecompositioncan help to assess the role of resource reallocation in productivity dynamics over the business cycle. In particular, in a relatively general setting, we demonstrate that the within-industry component of our decomposition is reflective of within-sector allocative efficiency. Ultimately,thissectionismeanttodemonstratethatourdecompositioncan,attheveryleast,be a useful check on such attempts at measuring the role of resource allocation over the businesscyclethatrequireestimatesofsectoralproductionfunctioncoefficientsandfirm-level TFP. Illustrating issues with identification In Figure 15, we demonstrate one possible issue with identification that can severely change the interpretation of the qualitative and quantitative importance of the role of reallocation over the business cycle. We show that our results change substantially when we followastandardprocedureandonlyslightlyvarytheestimationprocedureforproduction function coefficients.13 Figure 15 shows the cumulative change in the contribution of allocative efficiency to TFP over the recent recession for three different standard “versions” of estimating production function coefficients.14 We estimate production function coefficients as the average of the ratio of capital expenditures to labor expenditures, rk, across wl firms within a sector over time, where r is the rental rate, k is the capital stock in the firm, w isthe wage,andl islabor utilizationin thesector. In Version1, thebaselineversion, we dropallobservationsbefore1972,usecapitalandlaborutilizationfromourfirm-leveldata, andestimatetherentalrateandwagefollowingChari,Kehoe,andMcGrattan(2007).15 We see that in Version 1 of our estimation of production function coefficients, there seems to be a decrease in allocative efficiency from pre-recession levels to the trough in 13Specifically,weimplementtheapproachofOberfeld(2013)tomeasurechangesinallocativeefficiency inoursampleofU.S.publiclylistedfirms. Thismodelofproductionandaggregationisidenticaltothatin Hsieh and Klenow (2009). Details of our dataset construction and measurement can be found in section 4 andAppendixB.AsinOberfeld(2013)andHsiehandKlenow(2009), wesettheelasticityofsubstitution withinsectorsto3. 14Positivechangesindicateanincreaseintheextentofallocativeefficiency. 15FollowingChari,Kehoe,andMcGrattan(2007)entailssettingr =α∗Y andw =(1−α)∗Y,whereα K L isdefinedinthemeasurementsubsectionaboveandcomedirectlyfromChari,Kehoe,andMcGrattan(2007). OurmeasuresandY,L,andK areallcomputedasdescribeinsubsection5. Thesemeasuresarecomputed differentlyfromhowY,L,andK arecomputedinChari,Kehoe,andMcGrattan(2007). 36

2008 to 2009. In Version 2, we take capital, k, and labor expenditures, wl, from the firmlevel data, but still estimate r following Chari, Kehoe, and McGrattan (2007). Estimating labor expenditures using firm-level data changes the year-over-year behavior of the contribution of allocative efficiency to TFP. In Version 3, we follow the same procedure as in Version 1 but drop all observations before 1976. In this version, which is only different from Version 1 in that we assume there is slightly less data available decades prior to the recession we are examining, we find an increase in allocative efficiency from 2006 to the trough of the recession. These results demonstrate just one of the potential problems with identificationtowhichthestandardmodel-basedapproachesaresusceptible,aproblemthat can substantially change the qualitative and quantitative implications of the role of reallocation over the business cycle. Other issues with identification remain; another example is that we find that when we vary the elasticity of substitution across firms between 3 and 10 (standardvaluesusedintheliteratureasnotedinHsiehandKlenow(2009)),thequalitative andquantitativenatureofourresultschangesubstantially. Relation to Models of Allocative Efficiency This appendix demonstrates how the dispersion component of our decomposition relates to the aggregate productivity loss suffered from misallocation in a standard static model of allocative efficiency. We show that the dispersion component directly enters this loss and discuss the relative magnitude of the the social losses and the dispersion components. Weconsidertheone-sectoreconomyintroducedinsubsection3.2,andderivetwostatistics commonly used as measures of productivity loss due to distortions. First, we derive the difference between efficient and observed log TFP. This difference may be driven by misallocation or frictions to the total amount of capital/labor used. Second, we compute the difference between the log TFP implied by the output-maximizing allocation of the observed total labor and capital and observed TFP. This difference will be driven only by misallocation. 37

Difference from Optimal Allocation Note that an analogue (in levels instead of differences)to(28)yields: log(TFP ) = log(TFP∗)+ακ +(1−α)κ (64) t t k,1,t l,1,t (cid:124) (cid:123)(cid:122) (cid:125) staticmeancomponent higher-orderterms variance (cid:122) (cid:125)(cid:124) (cid:123) (cid:122) (cid:125)(cid:124) (cid:123) ∞ ακ +(1−α)κ (cid:88) ακ +(1−α)κ + − k,2,t l,2,t + (−1)n−1 k,n,t l,n,t , 2! n! n=3 (cid:124) (cid:123)(cid:122) (cid:125) staticdispersioncomponent whereκ ,κ arethecumulantsoflogwedgesincapitalandlabortofirmsproductivity k,n,t l,n,t ratios,asdefinedinsubsection3.3. Thisimpliesthatwecanexpressthedifferencebetween efficientandrealizedlogTFPasthefollowing: log(TFP∗)−log(TFP ) = −ακ −(1−α)κ −αDK −(1−α) DL. t t k,1,t l,1,t t t Where DK and DL are the dispersion components of labor and capital in our decomposit t tion. Thefirstcumulantsaretheoutput-shareweightedaveragesoflogwedges,andarethe onlytermsotherthanthedispersioncomponentstoentertheselosses. Total Labor and Capital Fixed Consider a firm optimization problem, with fixed stock of total capital and labor within the production environment defined in subsection 3.2. (cid:80) Firms take the wage and rental rate as given, and clearing conditions K = k and t i i,t (cid:80) L = l are satisfied. Firm labor and capital choices are distorted from their optimal t i i,t allocationbyθl andθk ,respectively. i,t i,t max p y −k θk r −l θl w . (65) i,t i,t i,t i,t t i,t i,t t ki,t,li,t Theequilibriumdecisionrulesimplythatfirmmarginalrevenueproductsarethefollowing: v r i,t = θk t , (66) k i,tϕγ i,t and v w i,t = θl t . (67) l i,tϕφ i,t 38

Wecanthenexpressthelossesduetomisallocationinthisfixedenvironmentasafunction oftherentalrate,efficientrentalrate,andcumulantsoffirm-leveldistortionsθk andθl .16 i,t i,t (cid:16) (cid:17) ˆ Wederivethefollowingexpressionforlog TFP −log(TFP ): t t (cid:32) (cid:33) ˆ (cid:18) (cid:18) (cid:19) (cid:19) (cid:18) (cid:18) (cid:19) (cid:19) TFP rˆ wˆ log t = α log t −κ −DK)+(1−α) log t −κ −DL , TFP r k,1,t t w l,1,t t t t t ˆ where TFP is TFP if labor and capital are allocated efficiently but total labor and capital t arefixedattheobservedlevels,andwˆ andrˆ arethewageandrentalraterequiredtokeep t t totalcapitalandwagesfixedifdistortionsareremoved. References ARELLANO, C., Y. BAI, AND P. J. KEHOE (2012): “Financial Frictions and Fluctuations inVolatility,”WorkingPaper. ASKER, J., J. DE LOECKER, AND A. COLLARD-WEXLER (2014): “Dynamic Inputs and Resource(Mis)Allocation,”TheJournalofPoliticalEconomy,122,1013–1063. BACKUS, D., M. CHERNOV, AND S. ZIN (2014): “Sources of Entropy in Representative AgentModels,”TheJournalofFinance,69,51–99. BASU, S. AND J. FERNALD (2002): “Aggregate productivity and aggregate technology,” EuropeanEconomicReview. BASU, S., L. PASCALI, F. SCHIANTARELLI, AND L. SERVEN (2009): “Productivity, welfare and reallocation: Theory and firm-level evidence,” NBER Working Paper No. w15579. BLOOM, N., M. FLOETOTTO, N. JAIMOVICH, I. SPORTA-EKSTEN, AND S. J. TERRY (2014): “ReallyUncertainBusinessCycles,”WorkingPaper. BUERA, F. J., J. P. KABOSKI, AND Y. SHIN (2011): “Finance and Development: A Tale ofTwoSectors,”AmericanEconomicReview,101,1964–2002. 16Thesedistortionsdifferonlyfromthewedgesdefinedbeforeinthattheyaredistortionsfromtheinput choicesimpliedbyrentalratesandwagesinsteadoftheefficientallocation. 39

CHARI, V., P. J. KEHOE, AND E. R. MCGRATTAN(2007): “BusinessCycleAccounting,” Econometrica,75,781–836. DAVID, J. M., H. HOPENHAYN, AND V. VENKATESWARAN (2016): “Information, MisallocationandAggregateProductivity,”TheQuarterlyJournalofEconomics. FOSTER, L., J. HALTIWANGER, AND C. J. KRIZAN (2001): Aggregate Productivity Growth.LessonsfromMicroeconomicEvidence,NBER,303–372. GILCHRIST, S., J. W. SIM, AND E. ZAKRAJSEK (2014): “Uncertainty, Financial Frictions,andInvestmentDynamics,”WorkingPaper. HODGE, A. W. (2011): “Comparing NIPA Profits With S&P 500 Profits,” Survey of CurrentBusiness,91. HOPENHAYN, H. (2011): “Firm Microstructure and Aggregate Productivity,” Journal of Money,CreditandBanking,43,111–145. HSIEH, C.-T. AND P. J. KLENOW (2009): “Misallocation and Manufacturing TFP in ChinaandIndia,”TheQuarterlyJournalofEconomics,124,1403–1448. KEHRIG, M. (2015): “Thecyclicalityofproductivitydispersion,”WorkingPaper. KHAN, A. AND J. K. THOMAS (2013): “Credit Shocks and Aggregate Fluctuations in an Economy with Production Heterogeneity,” Journal of Political Economy, 121, 1055– 1107. MIDRIGAN, V. AND D. Y. XU (2014): “FinanceandMisallocation: EvidencefromPlant- LevelData,”AmericanEconomicReview,104,422–458. MOLL, B. (2014): “Productivity Losses from Financial Frictions: Can Self-Financing UndoCapitalMisallocation?” AmericanEconomicReview. OBERFELD, E. (2013): “Productivityandmisallocationduringacrisis: Evidencefromthe Chileancrisisof1982,”ReviewofEconomicDynamics,16,100–119. OBERFIELD, E. (2013): “Productivity and misallocation during a crisis: Evidence from theChileancrisisof1982,”ReviewofEconomicDynamics,16,100–119. 40

OSOTIMEHIN, S. (2013): “Aggregate Productivity and the Allocation of Resources over theBusinessCycle,”WorkingPaper. PETRIN, A. AND J. LEVINSOHN (2012): “Measuringaggregateproductivitygrowthusing plant-leveldata,”RANDJournalofEconomics,43,705–725. RESTUCCIA, D. AND R. ROGERSON (2008): “Policy Distortions and Aggregate Productivity with Heterogeneous Establishments,” Review of Economic Dynamics, 11, 707– 720. SANDLERIS, G. AND M. L. J. WRIGHT(2014): “TheCostsofFinancialCrises: Resource Misallocation,ProductivityandWelfareinthe2001ArgentineCrisis,”TheScandinavian JournalofEconomics,116,87–127. 41

Tables Table1: CorrelationsbetweenChangesinAggregatesandChangesintheirComponentsfrom DecompositionII—U.S.Data TFP Laborproductivity Capitalproductivity Meancomponent 0.974 0.955 0.955 Dispersioncomponent 0.042 -0.346 0.399 Sectoralsharecomponent -0.245 -0.329 0.259 Notes: Sample period is from 1972 to 2011. Data are from U.S. nonfinancial public firms. Firms are grouped by two digit SIC codes. Changes in aggregate measures (TFP, labor productivity, and capital productivity) and components of our decompositionsaremeasuredyearoveryear.

Figures Figure1: DecompositionIAppliedtoAggregateLaborProductivity: Year-over-YearChanges egnahc tnecrep raey-revo-raeY 01 5 0 5- 01- 51- 1970 1980 1990 2000 2010 Year Public firm labor productivity Mean component Dispersion component Sectoral share component Note: Sample period is from 1972 to 2011. We compute year-over-year changes in aggregate labor productivityanditscomponentsfromDecompositionIusingdatafromU.S.nonfinancialpublicfirms. BecausethereisnogroupingbysectorforDecompositionI,thesectoralsharecomponent(theyellow line)is,inturn,flat.

Figure2: DecompositionIIAppliedtoAggregateLaborProductivity: Year-over-YearChanges egnahc tnecrep raey-revo-raeY 01 5 0 5- 01- 1970 1980 1990 2000 2010 Year Public firm labor productivity Mean component Dispersion component Sectoral share component Note: Sample period is from 1972 to 2011. We compute year-over-year changes in aggregate labor productivityanditscomponentsfromDecompositionIIusingdatafromU.S.nonfinancialpublicfirms. FirmsaregroupedbytwodigitSICcodes.

Figure 3: Decomposition I Applied to Aggregate Labor Productivity: Cumulative Changes overFourBusinessCycleEpisodes secnereffid gol evitalumuC 1.150.1 1 59. 9. 1979 indexed to 1 1979 1980 1981 1982 1983 1984 Year secnereffid gol evitalumuC 1.150.1 1 59. 9. 1989 indexed to 1 1989 1990 1991 1992 1993 Year secnereffid gol evitalumuC 1.150.1 1 59. 9. 1999 indexed to 1 1999 2000 2001 2002 2003 Year secnereffid gol evitalumuC 1.150.1 1 59. 9. 2007 indexed to 1 2007 2008 2009 2010 2011 Year Labor productivity Mean component Dispersion component Sectoral share component Note: Sample period is from 1972 to 2011. We compute cumulative changes in aggregate labor productivityanditscomponentsfromDecompositionIusingdatafromU.S.nonfinancialpublicfirmsover fourbusinesscycleepisodes,withthepre-recessionindexyearinthetitleoftheplot. Becausethereis no grouping by sector for Decomposition I, the sectoral share component (the yellow line) is, in turn, flat.

Figure 4: Decomposition II Applied to Aggregate Labor Productivity: Cumulative Changes overFourBusinessCycleEpisodes secnereffid gol evitalumuC 1.150.1 1 59. 9. 1979 indexed to 1 1979 1980 1981 1982 1983 1984 Year secnereffid gol evitalumuC 1.150.1 1 59. 9. 1989 indexed to 1 1989 1990 1991 1992 1993 Year secnereffid gol evitalumuC 1.150.1 1 59. 9. 1999 indexed to 1 1999 2000 2001 2002 2003 Year secnereffid gol evitalumuC 1.150.1 1 59. 9. 2007 indexed to 1 2007 2008 2009 2010 2011 Year Labor productivity Mean component Dispersion component Sectoral share component Note: Sample period is from 1972 to 2011. We compute cumulative changes in aggregate labor productivityanditscomponentsfromDecompositionIusingdatafromU.S.nonfinancialpublicfirmsover fourbusinesscycleepisodes,withthepre-recessionindexyearinthetitleoftheplot. Firmsaregrouped bytwodigitSICcodes.

Figure 5: Decomposition I Applied to Aggregate Capital Productivity: Year-over-Year Changes egnahc tnecrep raey-revo-raeY 51 01 5 0 5- 01- 1970 1980 1990 2000 2010 Year Public firm capital productivity Mean component Dispersion component Sectoral share component Note: Sample period is from 1972 to 2011. We compute year-over-year changes in aggregate capital productivityanditscomponentsfromDecompositionIusingdatafromU.S.nonfinancialpublicfirms. BecausethereisnogroupingbysectorforDecompositionI,thesectoralsharecomponent(theyellow line)is,inturn,flat.

Figure 6: Decomposition II Applied to Aggregate Capital Productivity: Year-over-Year Changes egnahc tnecrep raey-revo-raeY 51 01 5 0 5- 01- 1970 1980 1990 2000 2010 Year Public firm capital productivity Mean component Dispersion component Sectoral share component Note: Sample period is from 1972 to 2011. We compute year-over-year changes in aggregate capital productivityanditscomponentsfromDecompositionIIusingdatafromU.S.nonfinancialpublicfirms. FirmsaregroupedbytwodigitSICcodes.

Figure 7: Decomposition I Applied to Aggregate Capital Productivity: Cumulative Changes overFourBusinessCycleEpisodes secnereffid gol evitalumuC 2.1 1.1 1 9. 1979 indexed to 1 1979 1980 1981 1982 1983 1984 Year secnereffid gol evitalumuC 2.1 1.1 1 9. 1989 indexed to 1 1989 1990 1991 1992 1993 Year secnereffid gol evitalumuC 2.1 1.1 1 9. 1999 indexed to 1 1999 2000 2001 2002 2003 Year secnereffid gol evitalumuC 2.1 1.1 1 9. 2007 indexed to 1 2007 2008 2009 2010 2011 Year Capital productivity Mean component Dispersion component Sectoral share component Note: Sampleperiodisfrom1972to2011. WecomputecumulativechangesinaggregatecapitalproductivityanditscomponentsfromDecompositionIusingdatafromU.S.nonfinancialpublicfirmsover fourbusinesscycleepisodes,withthepre-recessionindexyearinthetitleoftheplot. Becausethereis no grouping by sector for Decomposition I, the sectoral share component (the yellow line) is, in turn, flat.

Figure 8: Decomposition II Applied to Aggregate Capital Productivity: Cumulative Changes overFourBusinessCycleEpisodes secnereffid gol evitalumuC 2.1 1.1 1 9. 1979 indexed to 1 1979 1980 1981 1982 1983 1984 Year secnereffid gol evitalumuC 2.1 1.1 1 9. 1989 indexed to 1 1989 1990 1991 1992 1993 Year secnereffid gol evitalumuC 2.1 1.1 1 9. 1999 indexed to 1 1999 2000 2001 2002 2003 Year secnereffid gol evitalumuC 2.1 1.1 1 9. 2007 indexed to 1 2007 2008 2009 2010 2011 Year Capital productivity Mean component Dispersion component Sectoral share component Note: Sampleperiodisfrom1972to2011. WecomputecumulativechangesinaggregatecapitalproductivityanditscomponentsfromDecompositionIusingdatafromU.S.nonfinancialpublicfirmsover fourbusinesscycleepisodes,withthepre-recessionindexyearinthetitleoftheplot. Firmsaregrouped bytwodigitSICcodes.

Figure9: DecompositionIAppliedtoAggregateTFP:Year-over-YearChanges egnahc tnecrep raey-revo-raeY 01 5 0 5- 01- 1970 1980 1990 2000 2010 Year Public firm TFP Mean component Dispersion component Sectoral share component Note: Sampleperiodisfrom1972to2011. Wecomputeyear-over-yearchangesinaggregateTFPand itscomponentsfromDecompositionIusingdatafromU.S.nonfinancialpublicfirms. Becausethereis no grouping by sector for Decomposition I, the sectoral share component (the yellow line) is, in turn, flat.

Figure10: DecompositionIIAppliedtoAggregateTFP:Year-over-YearChanges egnahc tnecrep raey-revo-raeY 01 5 0 5- 1970 1980 1990 2000 2010 Year Public firm TFP Mean component Dispersion component Sectoral share component Note:Sampleperiodisfrom1972to2011.Wecomputeyear-over-yearchangesinaggregateTFPandits componentsfromDecompositionIIusingdatafromU.S.nonfinancialpublicfirms. Firmsaregrouped bytwodigitSICcodes.

Figure11: DecompositionIAppliedtoAggregateTFP:CumulativeChangesoverFourBusinessCycleEpisodes secnereffid gol evitalumuC 1.150.1 1 59. 9. 1979 indexed to 1 1979 1980 1981 1982 1983 1984 Year secnereffid gol evitalumuC 1.150.1 1 59. 9. 1989 indexed to 1 1989 1990 1991 1992 1993 Year secnereffid gol evitalumuC 1.150.1 1 59. 9. 1999 indexed to 1 1999 2000 2001 2002 2003 Year secnereffid gol evitalumuC 1.150.1 1 59. 9. 2007 indexed to 1 2007 2008 2009 2010 2011 Year TFP Mean component Dispersion component Sectoral share component Note: Sampleperiodisfrom1972to2011. WecomputecumulativechangesinaggregateTFPandits components from Decomposition I using data from U.S. nonfinancial public firms over four business cycleepisodes,withthepre-recessionindexyearinthetitleoftheplot. Becausethereisnogroupingby sectorforDecompositionI,thesectoralsharecomponent(theyellowline)is,inturn,flat.

Figure12: DecompositionIIAppliedtoAggregateTFP:CumulativeChangesoverFourBusinessCycleEpisodes secnereffid gol evitalumuC 1.150.1 1 59. 9. 1979 indexed to 1 1979 1980 1981 1982 1983 1984 Year secnereffid gol evitalumuC 1.150.1 1 59. 9. 1989 indexed to 1 1989 1990 1991 1992 1993 Year secnereffid gol evitalumuC 1.150.1 1 59. 9. 1999 indexed to 1 1999 2000 2001 2002 2003 Year secnereffid gol evitalumuC 1.150.1 1 59. 9. 2007 indexed to 1 2007 2008 2009 2010 2011 Year TFP Mean component Dispersion component Sectoral share component Note: Sampleperiodisfrom1972to2011. WecomputecumulativechangesinaggregateTFPandits components from Decomposition I using data from U.S. nonfinancial public firms over four business cycleepisodes,withthepre-recessionindexyearinthetitleoftheplot. Firmsaregroupedbytwodigit SICcodes.

Figure13: DecompositionIAppliedtoAggregateTFP:Year-over-YearChanges egnahc tnecrep raey-revo-raeY 01 5 0 5- 01- 2000 2005 2010 Year Public firm TFP Mean component Dispersion component Sectoral share component Note: Sampleperiodisfrom1972to2011. Wecomputeyear-over-yearchangesinaggregateTFPand itscomponentsfromDecompositionIusingdatafromJapanesenonfinancialpublicfirms.Becausethere isnogroupingbysectorforDecompositionI,thesectoralsharecomponent(theyellowline)is,inturn, flat.

Figure14: DecompositionIIAppliedtoAggregateTFP:Year-over-YearChanges egnahc tnecrep raey-revo-raeY 01 5 0 5- 01- 2000 2005 2010 Year Public firm TFP Mean component Dispersion component Sectoral share component Note: Sampleperiodisfrom1972to2011. Wecomputeyear-over-yearchangesinaggregateTFPand its components from Decomposition II using data from Japanese nonfinancial public firms. Firms are groupedbytwodigitSICcodes.

Figure15: ThreeEstimationTechniquesfortheContributionofAllocativeEfficiencytoTFP 4 3 2 1 0 −1 −2 −3 −4 −5 −6 2006 2007 2008 2009 2010 2011 Year egnahc tnecrep evitalumuC Version 1 Version 2 Version 3 Note: WecanexpressTFPasafunctionofthehypotheticalefficientproductivity, TFPeff, attimet, t andtheallocativeefficiencyofresourcesa suchthatTFP =a TFPeff. Thefigureshowslog(a )− t t t t t log(a ). ThelinescanthusbeinterpretedasthecumulativepercentchangeinTFPover2006levels 2006 due to changes in allocative efficiency. The estimation of the different “versions” only differs in the estimation of production function coefficients as follows: (1) Version 1 is the baseline version, (2) Version2useswagedatafromCompustatratherthanfromNIPA,and(3)Version3usesdatabackto 1976insteadof1972.

Figure16: TrendinAggregateTFP:NIPAvs. Compustat PFT goL 1- 5.1- 2- 5.2- 3- 1- 5.1- 2- 5.2- PFT goL 1970 1980 1990 2000 2010 Year Public firm TFP (left-axis) NIPA TFP (right-axis) Note: Sample period is from 1972 to 2011. This figure depicts the trend in the natural logarithm of aggregateTFPcomputedfromtwodifferentsamples. Thebluelinecorrespondstoameasurecomputed frompublicfirmdata. TheredlinecorrespondstoameasurecomputedfromNIPA.

Figure17: ChangesinAggregateTFP:NIPAvs. Compustat egnahc tnecrep raey-revo-raeY 01 5 0 5- 01- 1970 1980 1990 2000 2010 Year Public firm TFP NIPA TFP Note: Sample period is from 1972 to 2011. This figure depicts changes in the natural logarithm of aggregateTFPcomputedfromtwodifferentsamples. Thebluelinecorrespondstoameasurecomputed frompublicfirmdata. TheredlinecorrespondstoameasurecomputedfromNIPA.

Figure18: ChangesinAggregateLaborProductivity: NIPAvs. Compustat egnahc tnecrep raey-revo-raeY 01 5 0 5- 01- 1970 1980 1990 2000 2010 Year Public firm labor productivity NIPA labor productivity Note: Sample period is from 1972 to 2011. This figure depicts changes in the natural logarithm of aggregate labor productivity computed from two different samples. The blue line corresponds to a measurecomputedfrompublicfirmdata. TheredlinecorrespondstoameasurecomputedfromNIPA.

Figure19: ChangesinAggregateCapitalProductivity: NIPAvs. Compustat egnahc tnecrep raey-revo-raeY 02 01 0 01- 1970 1980 1990 2000 2010 Year Public firm capital productivity NIPA capital productivity Note: Sample period is from 1972 to 2011. This figure depicts changes in the natural logarithm of aggregate capital productivity computed from two different samples. The blue line corresponds to a measurecomputedfrompublicfirmdata. TheredlinecorrespondstoameasurecomputedfromNIPA.