Firm Dynamics with Infrequent Adjustment and Learning

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Firm Dynamics with Infrequent Adjustment and Learning Eugenio P. Pinto 2008-44 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.

Firm Dynamics with Infrequent Adjustment and Learning EugØnio Pinto (cid:3) July 2008 Abstract We propose an explanation for the rapid post-entry growth of surviving (cid:2)rms found in recent studies. At the core of our theory is the interaction between adjustment costs and learningbyentering(cid:2)rmsabouttheiref(cid:2)ciency. Weshowthatlinearadjustmentcosts,i.e., proportional costs, create incentives for (cid:2)rms to enter smaller and for successful (cid:2)rms to growfasterafterentry. Initialuncertaintyaboutpro(cid:2)tabilitymakesentering(cid:2)rmsprudent since they want to avoid incurring super(cid:3)uous costs on jobs that prove to be excessive ex post. Because higher adjustment costs imply less pruning of inef(cid:2)cient (cid:2)rms and faster growthofsurviving(cid:2)rms,thecontributionofsurvivorstogrowthinacohort’saveragesize increases. For the cohort of 1988 entrants in the Portuguese economy, we conclude that survivors’ growth is the main factor behind growth in the cohort’s average size. However, initial selection is higher and the survivors’ contribution to growth is smaller in services thaninmanufacturing. Anestimationofthemodelshowsthattheproportionaladjustment cost is the key parameter to account for the high empirical survivors’ contribution. In addition,(cid:2)rmsinmanufacturinglearnrelativelylessinitiallyabouttheiref(cid:2)ciencyandare subjecttolargeradjustmentcoststhan(cid:2)rmsinservices. JELClassi(cid:2)cation: E24,L11,L16 Keywords: AdjustmentCosts,Learning,YoungFirms Board of Governors of the Federal Reserve System, 20th Street and Constitution Avenue NW, Stop 80, (cid:3) Washington, DC 20551 (e-mail: eugenio.p.pinto@frb.gov). The views expressed in this paper are those of the author and do not necessarily re(cid:3)ect the views of the Board of Governors of the Federal Reserve System or its staff. I would like to thank valuable comments from John Shea, John Haltiwanger, Michael Pries, Borag(cid:11)an Aruoba,andRachelKranton. Allremainingerrorsaremyown. IwouldalsoliketothanktheDirec(cid:231)ªo-Geralde Estudos,Estat(cid:237)sticaePlaneamento-MinistØriodaSeguran(cid:231)aSocial,daFam(cid:237)liaedaCrian(cid:231)a,forkindlyallowing me to access the Quadros de Pessoal database, the University of Minho for their hospitality, and Joªo Cerejeira andMiguelPortelafortheirhelpinextractingresults.

1 Introduction In recent years there has been renewed interest in explaining patterns of (cid:2)rm dynamics, with newlongitudinaldatasetscon(cid:2)rmingheterogeneitiesbetween(cid:2)rmsofdifferentsizeandage. In particular, small and young (surviving) (cid:2)rms tend to grow faster and have higher failure rates thanlargeandold(cid:2)rms,andbothjobcreationduetothescaling-upof(cid:2)rmsizeandjobdestructiondueto(cid:2)rmexitdecreasewithage.1 Moreover,entering(cid:2)rmstendtobesmall,butsurvivors grow rapidly after entry and are the main factor behind the shift to the right of a cohort’s size distribution.2 These patterns differ markedly across sectors and countries, suggesting that both technologicaldifferencesandcountryspeci(cid:2)cfactorsmatter.3 This paper proposes an explanation for the leading role of survivors’ growth in post-entry (cid:2)rmdynamicsbasedontheinteractionbetweenadjustmentcostsandalearning-about-ef(cid:2)ciency mechanism. Followingaliteraturethatusesadjustmentcoststoaccountforsomedynamicproperties of (cid:2)rms’ labor demand, such as Campbell and Fisher (2000), we show that proportional costscanimpactthelifetimedynamicsof(cid:2)rms’labordemandinawayconsistentwiththedata. To implement our theory, we use a standard model of (cid:2)rm dynamics with passive learning. In order to check the empirical (cid:2)t of our model, we also assume that inef(cid:2)cient (cid:2)rms are pruned from the market, although the predictions of our theory hold even in the absence of a selection mechanism(e.g. whenexitisnotallowed). Our contribution is twofold. First, we contribute to the empirical literature by introducing a decomposition of the change in a cohort’s average size into a survivor component and a non-survivor component, and by using this decomposition as the centerpiece in a structural estimationofadjustmentcosts. Giventheemphasisonsurvivors’growth,ourmeasureallowsa quickassessmentofhowwellaparticulartheorymatchesthedatainthatrespect. Weapplyour decomposition to the 1988 cohort of entrants in the Portuguese economy, using the Quadros de Pessoal dataset. Similarly to Cabral and Mata (2003), we (cid:2)nd that growth of survivors is the main force behind the change in the cohort’s average (cid:2)rm size. However, we also (cid:2)nd that growth of survivors is especially intense in the initial years after entry and that there are signi(cid:2)cantcross-sectordifferencesintermsofourdecomposition. Inparticular,initialexitrates aresmallerandthesurvivors’contributiontochangesinsizeishigherinmanufacturingthanin services. Second,wecontributetothetheoreticalliteraturebyintroducinglinearadjustmentcostsinto a model of Bayesian learning about ef(cid:2)ciency. Our assumption of linear or proportional costs 1SeeDunneetal. (1989a,1989b). 2SeeMataandPortugal(1994)andCabralandMata(2003). 3SeeBartelsmanetal.(2005). 1

is justi(cid:2)ed by the (cid:2)nding of high inaction rates in employment adjustment, in varying degrees across sectors. Our model builds on Jovanovic (1982) by adding proportional costs that apply notonlytoregularlaboradjustment,butalsotojobcreationatentryandjobdestructionatexit. We show that proportional adjustment costs create incentives for (cid:2)rms to start smaller and, if successful, grow faster after entry. We prove this analytically in a simpli(cid:2)ed model in which there is no exit of (cid:2)rms. This result shows that proportional costs can generate (cid:2)rm growth withoutselection. When(cid:2)rmsareallowedtoexit,selectionintensi(cid:2)estheeffectsofadjustment costs on (cid:2)rm growth, while costs to adjustment reduce exit rates. Therefore, adjustment costs increasethecontributionofsurviving(cid:2)rmstogrowthinthecohort’saveragesize. Allthatisneededfor(cid:2)rmgrowthunderlinearadjustmentcostsistheexistenceofalearning environment that generates a stochastic process for perceived ef(cid:2)ciency with both persistence anddecreasinguncertaintyinage.4 Theintuitionforwhy(cid:2)rmsgrowfasteranddisplaysmaller exit rates under proportional adjustment costs is that initial uncertainty about true pro(cid:2)tability makes entering (cid:2)rms prudent; that is, they enter small and (cid:147)wait and see(cid:148) since they want to avoidincurringsuper(cid:3)uousentering/hiringcostsand(cid:2)ring/shutdowncostsonjobsthatproveto beexcessiveexpost. Thisimpliesthatsurviving(cid:2)rmswillgrowfaster,eventhoughadjustment costsimplythattherearefewer(cid:2)rmsexitingthemarketandthereforelesspruningofinef(cid:2)cient (cid:2)rms. The assumption that entering (cid:2)rms face a Bayesian learning problem concerning their ef- (cid:2)ciency is standard in selection theories and has been advanced as an explanation for the high rates of exit, job creation, and job destruction among young (cid:2)rms. The initial literature on adjustmentcostsuseda(strictly)convexspeci(cid:2)cationinanattempttoexplainthesluggishnessin input responses to aggregate shocks. However, the assumption that costs of adjustment are linear is now standard in dynamic labor demand models, following a number of studies since the late 1980s that have documented the importance of inaction in employment adjustment at the micro level.5 Since strictly convex costs imply smooth adjustments over time, whereas linear costs imply immediate adjustment when it occurs, allowing for strictly convex costs, instead of linear costs, in the context where they also apply at entry and exit, would bias our analysis and eventually make our argument stronger. In the case of hiring/entering costs, entering (cid:2)rms would prefer to start smaller and adjust gradually to their optimal size, even if their perceived productivity remained unchanged or learning was absent. For (cid:2)ring/exiting costs, (cid:2)rms experiencing large declines in perceived productivity would adjust downwards in several steps, a scenario that would make (cid:2)rms start smaller to attenuate its effects. Therefore, by avoiding 4Forexample,(cid:2)rmgrowthwouldoccurinourmodelevenifexitwasrandomwithaconstantprobabilityfor all(cid:2)rms,whereasthatwouldnotbetrueinapureselectionmodel. 5SeeHamermeshandPfann(1996). 2

a bias towards (cid:2)rm growth, our decision to assume linear costs is conservative and permits a simpli(cid:2)cationofthemethodsemployedtomeasuretheeffectsofadjustmentcosts.6 To assess our model quantitatively, we calibrate and estimate a version of the model with (cid:2)nite learning horizon and positive dispersion in entry size. We conclude that linear costs are the key element to account for the high empirical contribution of survivors to changes in a cohort’s average size. A calibration/estimation for the manufacturing and services cohorts also suggests that (cid:2)rms in manufacturing learn relatively less initially about their ef(cid:2)ciency and are subjecttosubstantiallylargeradjustmentcoststhan(cid:2)rmsinservices. This paper is related to the literature on both adjustment costs and (cid:2)rm dynamics. Within the literature on adjustment costs, the paper is associated with theories that use linear adjustment costs to explain certain aspects of the dynamic behavior of labor demand and job (cid:3)ows. Well-known examples are Bentolila and Bertola (1990), Hopenhayn and Rogerson (1993), and CampbellandFisher(2000). BentolilaandBertola(1990)andHopenhaynandRogerson(1993) analyze the effects of proportional (cid:2)ring (and hiring) costs on the dynamics of hiring and (cid:2)ring decisions, and on average labor demand. Both papers conclude that high (cid:2)ring costs make hiring and (cid:2)ring adjustments more sluggish, but they disagree on the implications of that for long-run employment. Campbell and Fisher (2000) use proportional costs of job creation and job destruction to explain the higher aggregate volatility of job destruction found in the U.S. manufacturingsector. Thesecostsimplythatinreactiontoaggregatewageshocksemployment changes at contracting (cid:2)rms are larger than employment changes at expanding (cid:2)rms. What is new in our paper is the assumption that adjustment costs apply equally to the entry/exit decisionsandthehiring/(cid:2)ringdecisions.7 Within the literature on (cid:2)rm dynamics the paper is connected with theories that attempt to explain thestylized facts onthe lifecycle dynamicsoutlined above. Thetwo main explanations for these facts are theories based on selection of (cid:2)rms and theories based on (cid:2)nancing constraints.8 Selection theories stress the tendency for (cid:2)rms that accumulate bad realizations of productivity to exit the market and for (cid:2)rms that accumulate good realizations to survive and expand. This implies a composition bias towards larger and more ef(cid:2)cient (cid:2)rms as smaller, inef(cid:2)cient, and slow-growing (cid:2)rms gradually exit the industry. Representative papers of selection theories are Jovanovic (1982), Hopenhayn (1992), Ericson and Pakes (1995), and Luttmer (2007). Inallcasesproductivityrealizationsareexogenous,exceptinEricsonandPakes(1995) 6Althoughwecouldhaveincluded(cid:2)xedadjustmentcoststheyseemmorerelevantinthecaseofcapital. 7HopenhaynandRogerson(1993)didconsiderthatthe(cid:2)ringcostappliedalsoatexit,butintheirmodelthere isnolearningprocessandtheydidnotanalyzetheeffectofthe(cid:2)ringcoston(cid:2)rmgrowth. 8Rossi-HansbergandWright(2007)advanceanalternativetheorybasedonmeanreversionintheaccumulationofindustry-speci(cid:2)chumancapital. However,theirmodelonlydealswithsize-dependenceof(cid:2)rmdynamics, andhasnothingtosayaboutage-dependenceof(cid:2)rmdynamics,whichisourmainconcern. 3

wheretheyaretosomeextentendogenous. Meanwhile, theories employing (cid:2)nancing constraints argue that some imperfection in (cid:2)nancial markets causes young (cid:2)rms to have limited access to credit, forcing them to enter at a suboptimally small scale. As (cid:2)rms get older and survive, they establish creditworthiness and build up internal resources that enable them to expand to their optimal size. Important contributionstothisliteraturearethoseofCooleyandQuadrini(2001),AlbuquerqueandHopenhayn (2004), and Clementi and Hopenhayn (2006). In Cooley and Quadrini (2001) a transaction cost on equity and a default cost on debt imply that equity and debt are not perfect substitutes, resulting in a positive dependence of (cid:2)rm size on the amount of equity. In Albuquerque and Hopenhayn (2004) and Clementi and Hopenhayn (2006) lenders introduce credit constraints because of limited liability of borrowers and enforcement of debt contracts, in the (cid:2)rst case, and because of asymmetric information on the use of funds or the return on investment, in the secondcase. CabralandMata(2003)analyzewhetherthesetwotheoriesareconsistentwiththeevolution of a cohort’s size distribution in the Portuguese manufacturing sector. They (cid:2)nd that, as the cohortages,the(cid:2)rmsizedistributionshiftstotherightlargelyduetogrowthofsurviving(cid:2)rms rather than exit of small (cid:2)rms. In addition, they (cid:2)nd that in the (cid:2)rst year after entry younger businessownersareassociatedwithsmaller(cid:2)rmsbutthatisnolongerthecaseoncethecohort getstoageseven. Assumingthatageisaproxyfortheentrepreneur’sinitialwealth,theauthors conclude that the age-size effect supports the idea of (cid:2)nancially constrained (cid:2)rms starting at a suboptimalsizeandpresentamodelwith(cid:2)nancingconstraintscapturingthiseffect. More recently, Angelini and Generale (2008) use survey and balance sheet information for Italianmanufacturing(cid:2)rmstoanalyzetheimpactof(cid:2)nancingconstraintsontheevolutionofthe (cid:2)rm size distribution. They (cid:2)nd that (cid:2)nancially constrained (cid:2)rms tend to be small and young, although this does not have a signi(cid:2)cant effect on the overall (cid:2)rm size distribution. Moreover, they (cid:2)nd that (cid:2)nancing constraints decrease (cid:2)rm growth, with this effect being entirely due to small(cid:2)rms. Inparticular,beingyoungand(cid:2)nanciallyconstraineddoesnothaveanyadditional effect. Based on these results and the fact that young (cid:2)rms grow faster than old (cid:2)rms, the authors conclude that (cid:2)nancing constraints are not the main factor behind the evolution of the (cid:2)rm size distribution. In line with their argument, this paper interprets the facts presented in CabralandMata(2003)andothercross-sectorevidenceastheresultoftheinteractionbetween adjustmentcostsandlearningaboutef(cid:2)ciency.9 To our knowledge, this work is the (cid:2)rst to suggest adjustment costs as an explanation for 9Ourinterpretationoftheage-sizeeffectisclosetothealternativeexplanationproposedbyCabralandMata (2003,footnote14),inthesensethatyoungbusinessownerswouldbesubjecttoamoreintenselearning. 4

differences in (cid:2)rm dynamics by age. The paper by Cabral (1995) is nearest to this paper. In his model, (cid:2)rms must pay a proportional sunk cost to increase their production capacity. He argues that, in a model with Bayesian learning, a proportional capacity cost would make small entering (cid:2)rms grow faster than large entering (cid:2)rms. The reason is that small entrants are thosewhoseinitialpro(cid:2)tabilitysignalswerenotgood,sotheirexitprobabilitiesarehigher,and therefore they choose to invest more gradually. Unlike our model, Cabral’s model depends on the existence of selection. Also, by analyzing a size-growth relationship, his model is not able toexplainwhysomelargeentering(cid:2)rmsalsogrowsubstantially. The paper is organized as follows. In section 2, we present evidence of (cid:2)rm dynamics for a cohortofentering(cid:2)rms. Insection3,webuildthegeneralmodel,obtainoptimalityconditions, and provide heuristic arguments explaining the effects of adjustment costs. In section 4, using a simpli(cid:2)ed version of the model we analytically prove the effect of linear adjustment costs on survivors’growth. Insection5,wecalibrateandestimatea(cid:2)nitelearninghorizonversionofthe modelandquantifythecontributionofadjustmentcoststo(cid:2)rmdynamics. Section6concludes. Allproofsareleftforanappendix. 2 Firm Dynamics in a Cohort of Entering Firms There is a well established literature on the identi(cid:2)cation and explanation of differences in behaviorbetweenyoungandold(cid:2)rms. Inthissection,weanalyze(cid:2)rmdynamicsinacohortof entering(cid:2)rms. WeuseQuadrosdePessoal,adatabasecontaininginformationonallPortuguese (cid:2)rms with paid employees. This dataset originates from a mandatory annual survey run by the Ministry of Employment, which collects information about the (cid:2)rm, its establishments, and its workers. All economic sectors except public administration are included. The panel we have access to covers the period 1985-2000. Information refers to March of each year from 1985 through 1993, and to October of each year since the reformulation of the survey in 1994. On average the dataset contains 250,000 (cid:2)rms, 300,000 establishments, and 2,500,000 workers in eachyear. The literature on (cid:2)rm dynamics typically (cid:2)nds that young (cid:2)rms grow faster than old (cid:2)rms. Using kernel density estimates of the (cid:2)rm size distribution in a cohort of entrants, Cabral and Mata (2003) argue graphically that the cohort’s evolution is mostly due to growth of survivors rather than exit of small (cid:2)rms. Their analysis points to the need for a measure of the contribution of survivors versus non-survivors to the growth in a given cohort’s average size. To accomplish this, we propose a decomposition of the cohort’s cumulative growth that will later allow an assessment of the empirical relevance of adjustment costs. We consider the following 5

decomposition: 1 (cid:229) 1 (cid:229) 1 (cid:229) 1 (cid:229) l l = l l + i;τ i;0 i;τ i;0 N(S ) (cid:0)N(S ) N(S ) (cid:0)N(S ) τ i S 0 i S τ i S τ i S τ 0 τ τ 2 2 2 2 SurvivorComponent | {Nz (Dτ) 1 } (cid:229) 1 (cid:229) l l i;0 i;0 N(S ) N(S ) (cid:0)N(Dτ) 0 τ i S i Dτ ! τ 2 2 Non-SurvivorComponent | {z } where τ is the (cid:2)rm’s age, l =ln(L ) is log-employment at (cid:2)rm i in period τ, S is the set of i;τ i;τ τ age-τ surviving (cid:2)rms, Dτ is the set of age-τ non-surviving (cid:2)rms, so that S ;Dτ is a partition τ f g ofS ,andN(X)isthenumberof(cid:2)rmsinsetX.10 0 In general, the growth in a cohort’s average size can originate from growth of survivors or from smaller initial size of non-survivors. Any theory of (cid:2)rm dynamics should consider both these sources of growth. Our measure allows a check on whether a particular theory can explain the key source of growth in a cohort’s average size. The survivor component compares the current average size of period τ survivors with their initial average size, so that it measures howmuchsurvivorshavegrown. Thenon-survivorcomponentcomparestheaverageinitialsize of period τ non-survivors with the average initial size of period τ survivors, so that it measures howrelativelysmallnon-survivorswereinitially. We can obtain a similar decomposition for employment-weighted moments. The weighted decomposition contains information about the entire distribution of employment, not just its cross-sectional mean, and is affected both by within- and between-(cid:2)rm growth. Therefore, the weighteddecompositionwouldbemorerelevantforassessingarichermodelthatconsidersthe reallocation of employment shares between (cid:2)rms within the cohort. In the results that follow we focus on the unweighted decomposition because it analyzes within-(cid:2)rm growth, which in ourmodelisthemostrelevantstatistictoassesstheeffectofadjustmentcostsontheincentives for(cid:2)rmstogrow.11 10Throughoutthepaperwewillassumethat(cid:2)rmsenterinsomegenericperiod0. Therefore,τ willrepresent boththe(cid:2)rm’sageandtheperiod(afterentry)weareanalyzing. 11Fortheweighteddecomposition,thecumulativechangewouldbe(cid:229) ωSτl (cid:229) ω S0l ,whereωX is the weight of (cid:2)rm i in period τ in set X, with ωX =L =(cid:229) L . Th i 2 e Sτ wei i g ;τ h i t : e τ d (cid:0) sur i 2 v S iv 0 or i; c 0 o i m ;0 ponent can i;τ be i;τ i;τ i X i;τ furtherdecomposedas 2 (cid:229) ωSτl (cid:229) ωSτl = (cid:229) ωSτ (l l )+ (cid:229) ωSτ ωSτ l + (cid:229) ωSτ ωSτ (l l ). i;τ i;τ (cid:0) i;0 i;0 i;0 i;τ (cid:0) i;0 i;τ(cid:0) i;0 i;0 i;τ(cid:0) i;0 i;τ (cid:0) i;0 i 2 Sτ i 2 Sτ i 2 Sτ i 2 Sτ(cid:16) (cid:17) i 2 Sτ(cid:16) (cid:17) The(cid:2)rsttermisawithin-(cid:2)rmcomponent,measuringaveragegrowthweightedbyinitialsize;thesecondtermis abetween-(cid:2)rmcomponent,measuringthecontributionofchangesinemploymentshares;andthethirdisacross 6

We can also produce a decomposition based on the cohort’s annual growth instead of the cohort’s cumulative growth. However, the annual version of the above decomposition is more sensitive to two aspects that would complicate the analysis in the paper. First, the annual survivor component is signi(cid:2)cantly affected by the business cycle, especially after the (cid:2)rst few years of life. To control for this, we would need to somehow remove the cyclical part of the survivorcomponent. Second,astheageofthecohortincreases,theannualsurvivorcomponent becomes increasingly sensitive to downsizing and exit by some survivors that become technologicallyoutdatedandconsequentlyrelativelylessef(cid:2)cient. Tofullyconsiderthisaspectofthe datawouldforceustointroduceadditionalparametersintothemodelthatwepresentinsection 3. Therefore, we believe that by employing a decomposition based on the cohort’s cumulative growth we avoid having to adjust the analysis for these two aspects, and instead focus on how intenseissurvivor’sgrowthwhilelearning-about-ef(cid:2)ciencyeffectsaresigni(cid:2)cant. In table 1, we present the evolution of exit rates and the share of (cid:2)rm growth due to the survivor component in the 1988 cohort of entering (cid:2)rms for the overall economy.12 In 1988 there were 22;810 entering (cid:2)rms. The exit rate is very high initially but tends to decrease as (cid:2)rmsgetolder.13 However,tenyearsafterentryonly41:5%oftheinitialentrantsremainactive. Thereissigni(cid:2)cantgrowthinthecohort’saveragesize,especiallyinthe(cid:2)rstfewyears,whichis mostlyduetothegrowthofsurvivorsratherthantotheexitofsmallinef(cid:2)cient(cid:2)rms: survivors’ growthcontributesaround69%tothegrowthinthecohort’saveragesize.14 Table 2 presents similar evidence on cohort dynamics for the manufacturing and services sectors.15Weincludetheemploymentsharesofeachsectorinthe1988cohortofentering(cid:2)rms, which are close to shares in the overall economy. Although manufacturing has a much higher employment share than services, the number of entering (cid:2)rms in services surpasses that of component. Fortheunweighteddecomposition,thelasttwotermsarezero,sinceinthiscaseωX i;τ =N(X)(cid:0) 1. 12Weidentifyentering(cid:2)rmsinyeart asthose(cid:2)rmsthathavenotbeeninthedatabasebeforet. Giventhehigh incidence of temporarily missing (cid:2)rms, we select the 1988 entering cohort, using 1985 and 1986 to detect false entries. Similarly,weidentifyexiting(cid:2)rmsintheτ-thperiod(afterentryin1988)asthose(cid:2)rmsthatarepresent inthedatabaseinperiodτ 1,butdonotreappearinanyofthefollowingperiods. Therefore,wedisplayresults (cid:0) onlyupto1999,using2000todetectfalseexits. Thisprocedureeliminatesmostfalseentriesandfalseexits. 13We adopt the following procedures concerning temporarily missing (cid:2)rms. In calculating the exit rate we do not exclude temporarily missing (cid:2)rms, considering them as survivors. In calculating the cohort’s mean logemploymentatperiodτ,wescaleitwithafactorthatcomparesthatmeaninperiod0betweenall(cid:2)rmsandthose not temporarily missing in period τ. We also adjust the data in 1994, when the survey moved from March to October,tocorrectforahigherthannormalexitrateandaveragegrowthinthisyear. 14When we use employment-weighted data, we (cid:2)nd that larger (cid:2)rms have smaller exit rates and, as a consequence, average employment increases more intensely than in the unweighted data. This and the fact that highgrowth(cid:2)rmsincreasetheirweightovertime, explainsalarger survivor component intheemployment-weighted decomposition.Asimilarexerciseforlaborproductivityrevealsthatsurvivorsaccountforabout90%ofthechange inthecohort’sunweightedaverageproductivity. 15Inordertoobtainequivalentone-digitSIC87sectors,weusethefollowingcorrespondenceintermsofCAE Rev. 1codes: manufacturing(=3)andservices(=6:3+8:3:2+8:3:3+9:2+9:3+9:4+9:5). 7

manufacturing(6074and4834,respectively). Bothsectorsdisplayacumulativeexitratearound 58%by1999. However,initialdifferencesinexitratesaremoresigni(cid:2)cant,withmanufacturing displaying the smallest values, and services displaying the highest values. In terms of initial size,manufacturinghasthelargestentrants,andservicesthesmallest. Althoughmanufacturing hasthelargestentrants,itexhibitsmoregrowthinaverageemploymentandalargercontribution ofsurvivorstothatgrowththanservices. Weperformtworobustnesschecksontheprevious(cid:2)ndings. First,weredoourcalculations usingestablishmentsratherthan(cid:2)rmsastheunitofanalysis. Forthe1988establishmentcohort, we obtain similar results, although exit rates and the survivor component are higher than in the case of (cid:2)rms. Second, we examine an alternative cohort to make sure our results are not driven bybusinesscycleconditions. ThePortugueseeconomyexperiencedanexpansionbetween1986 and1991,aperiodofslowgrowthwitharecessionbetween1992and1994,andanotherweaker expansion between 1995 and 2000. The growth rates of real GDP were 6:4% in 1989, 1:1% in 1992, and 4:3% in 1995, so that the 1991 cohort did not face as favorable a macroeconomic environmentasthe1988cohort. However,theresultsforthe1991cohortare,inalldimensions, very similar to those presented above. The results for the 1994 cohort are also very similar, but withslightlysmallervaluesforthesurvivorcomponentinthe(cid:2)rstfewyearsafterentry.16 In table 3, we provide evidence on the properties of labor adjustment in the 1988 cohort of entering (cid:2)rms. Namely, we present three characteristics of the distribution of adjusted growth rates,conditionalonsurvival,in1989and1993: thefractionof(cid:2)rmsthatdonotadjustemployment,NA,andthefractionsof(cid:2)rmsthatincrease/reducetheirsizebylessthan30%,P30/N30.17 The table shows that the incidence of inaction is very high, increases with age, and is higher in services than in manufacturing. This may re(cid:3)ect technology-induced differences in adjustment costs, or job indivisibilities affecting to a larger extent the services sector for having a higher share of small (cid:2)rms. The table also shows that the large majority of (cid:2)rms have adjustment rates within the ( 30%;30%) interval. A high rate of inaction and small adjustment is (cid:0) usually considered consistent with the presence of linear or proportional adjustment costs. In addition,comparingthecolumnsP30andN30itappearsthatthe1989growthdistributionsare more left-skewed than the 1993 distributions, suggesting that survivors tend to grow more ini- 16Re(cid:3)ecting our previous argument about the greater cyclical sensitivity of the decomposition based on the cohort’sannualgrowthrate,weobserveasubstantialreductionintheannualsurvivorcomponentassociatedwith the1988and1991cohortsduringthe1992-1994slowgrowthperiod. However,asimilarpatterndoesnotoccur withthe1994cohort.Thisisoneofthereasonswhywechooseadecompositionbasedoncumulativegrowthrates. Notealsothattheannualnon-survivorcomponentisnotassensitivetothebusinesscycleastheannualsurvivor component. 17Following Davis and Haltiwanger (1992), the adjusted growth rate in period τ is de(cid:2)ned as 100 (L L )=L(cid:152) ,whereL(cid:152) = 1(L +L ). (cid:2) τ (cid:0) τ (cid:0) 1 τ (cid:0) 1 τ (cid:0) 1 2 τ τ (cid:0) 1 8

tially, especially in manufacturing. The evidence on inaction justi(cid:2)es our assumption of linear adjustmentcostsinthemodelthatwepresentnext. 3 A Model of Learning with Linear Adjustment Costs 3.1 Assumptions and Solution In this section, we introduce linear adjustment costs into a model of Bayesian learning about ef(cid:2)ciency. We derive conditions for optimal employment over time and present heuristic arguments about the effects of adjustment costs on the path of employment. Our model is based on Jovanovic (1982), adding adjustment costs and using a different speci(cid:2)cation for the idiosyncraticshock. We assume an industry with competitive output and input markets. Current pro(cid:2)ts of a representative(cid:2)rmarede(cid:2)nedby P (L;θ)=F(L)θ wL, (cid:0) where F(L)θ is the production function; L is the amount of labor input; θ is a productivity shock; and w is the wage rate. The output price is normalized to unity, so that all monetary values are expressed in units of the output price. Given the competitive environment, the (cid:2)rm treatswasaconstant. Concerningtechnologywemakethefollowingassumption. Assumption1 Thetwocomponentsoftheproductionfunctionsatisfy: (a)F :R+ R+ isC2,F 0 >0,F 00 <0,F(0)=0,F 0 (0+)=¥ ,andF 0 (¥ )=0. ! (b) Letting τ denote the (cid:2)rm’s age and 0 the period in which the (cid:2)rm enters, the stochastic processofθ isde(cid:2)nedby θ =ξ(η ), η =µ+ε , µ =µ +µ , τ =0;1;::: (1a) τ τ τ τ 0 1 ε N 0;σ2 , µ N µfl;σ2 , µ N 0;σ2 , (1b) τ 0 µ 1 µ (cid:24) (cid:24) 0 (cid:24) 1 (cid:16) (cid:17) (cid:16) (cid:17) (cid:0) (cid:1) where µ 0 , µ 1 , ε τ τ 0 are mutually independent, ξ : R R++ is C1, ξ 0 > 0 and ξ( ¥ ) = f g ! (cid:0) ν 0,ξ(¥ )=ν < (cid:21)¥ . 1 2 (cid:21) Part (a) ensures a well de(cid:2)ned interior optimum. In some of the analyses below, we will assume that F is a power function. Meanwhile, part (b) establishes that in each period productivity is stochastic with a constant mean over the (cid:2)rm’s lifetime. The productivity component, 9

µ, is made of two parts: µ , which is observed before entry, and µ , which is never directly 0 1 observed by the (cid:2)rm. Intuitively, µ can be thought of as indexing ex ante ef(cid:2)ciency, measur- 0 inginitialtechnologychoice,while µ indexesexpost productivity,measuringhowwella(cid:2)rm 1 performswithinitstechnologychoice. The introduction of µ is essential to obtain a non-degenerate distribution of (cid:2)rms’ entry 0 size,allowingananalysisofthecontributionofsurvivorstogrowthinthecohort’saveragesize. In contrast, the absence of µ in Jovanovic’s (1982) model generates a degenerate distribution 0 of(cid:2)rms’entrysize. Underthisscenario,foranyperiodafterentry,survivorsandnon-survivors have the same average initial size implying a value of 100% to our measure of the survivors’ component. Byassumingσ >0,weavoidthisaspectofJovanovic’smodel. µ 0 Before entry the (cid:2)rm knows the parameters governing the stochastic process of θ, i.e., µfl, σ2 , σ2 and σ2, and learns its ex ante productivity, µ , after paying a research cost, I. After µ µ 0 0 1 entry, the (cid:2)rm will learn about its speci(cid:2)c ex post productivity, µ , over time as it observes 1 the realizations of productivity, θ. In particular, the (cid:2)rm forecasts period-τ productivity based on the ex ante ef(cid:2)ciency parameter µ and on the past realizations of productivity, θ τ 1 . 0 s s=(cid:0)0 f g Similarly to Zellner (1971), a (cid:2)rm with age τ has the following Bayesian posterior distribution for µ atthebeginningofperiodτ: µ N(Y ;Z ), W µ ; η τ 1 (2a) W τ τ τ 0 s s=(cid:0)0 j τ (cid:24) (cid:17) f g Y = τσ (cid:0) 2 ηfl + σ (cid:0)µ 1 2 µ , ηfl = 1τ (cid:229)(cid:0) 1n η , Z = o 1 : (2b) τ Z 1 τ Z 1 0 τ τ s τ τσ 2+σ 2 τ(cid:0) τ(cid:0) s=0 (cid:0) (cid:0)µ 1 In lemma 2 of appendix A we show that, for purposes of predicting µ, the information set W τ can be summarized by (θ ;τ), where θ is the period-τ forecast of the productivity coef(cid:2)cient (cid:3)τ (cid:3)τ based on the information available at the beginning of period τ. That is, θ =E (θ ), where (cid:3)τ τ τ E ( ) E( W )istheexpectationconditionalontheperiod-τ informationset. τ τ (cid:1) (cid:17) (cid:1)j Wenowlayoutthetimingassumptions. Assumption3 A potential entering (cid:2)rm, at the beginning of period 0, takes the following actions: (i.a) Research cost and ex ante productivity: the (cid:2)rm pays a (cid:2)xed cost I, associated with the processofinitialresearch,afterwhichitobservesarealizationofexanteproductivity, µ . 0 (i.b)Entrydecisionandentrycost: basedontheidiosyncraticrealizationofµ ,the(cid:2)rmchooses 0 whethertoentertheindustryornot. Incaseofentry,the(cid:2)rmpaysW foracquiringthe(exogenouslydetermined)capitalstock. (i.c)Initialemploymentandproductiondecisions: conditionalonenteringtheindustry,the(cid:2)rm 10

chooseshowmuchlabortouseandhowmuchoutputtoproduceinperiod0. A(cid:2)rmofageτ >0takesthefollowingactions: (ii.a) Update of posterior productivity: at the beginning of period τ, the (cid:2)rm updates its posterior expectation of θ , θ , based on the observation of θ =ξ(η ) at the end of period τ (cid:3)τ τ 1 τ 1 (cid:0) (cid:0) τ 1. (cid:0) (ii.b) Exit decision: given the new posterior productivity estimate, θ , and employment from (cid:3)τ last period, L , the (cid:2)rm chooses whether to stay or exit the industry. In case of exit, the (cid:2)rm τ 1 (cid:0) sellsthecapitalstockforthevalueinitiallypaid,W (nodepreciation). (ii.c)Employmentandproductiondecisions: conditionalonstaying,the(cid:2)rmchooseshowmuch labor to use and how much output to produce in the current period. At the end of period τ, the (cid:2)rmobservestheproductivityrealization,θ ,andtheprocessrepeatsitselfagainuntilthe(cid:2)rm τ decidestoleavetheindustry.18 In the absence of adjustment costs, while deciding whether to stay one more period or to exit,the(cid:2)rmcomparestheexpectedpro(cid:2)tincaseitstays,V,withtheopportunitycostofdoing so,W,thevalueitwouldrecoverbysellingthe(exogenous)capitalinitiallyacquired,i.e., V(θ ;τ)=max P (L ;θ )+βE max W;V θ ;τ+1 (3) (cid:3)τ τ (cid:3)τ τ (cid:3)τ+1 L τ (cid:8) (cid:2) (cid:8) (cid:0) (cid:1)(cid:9)(cid:3)(cid:9) whereV representsexpectedpro(cid:2)tsconditionalonstayinginperiodτ. At entry, we have W µ , and in equilibrium expected pro(cid:2)ts must compensate for the 0 0 (cid:17) cost of acquiring capital, i.e., VEN(θ ) >W. Since markets are competitive and there is no (cid:3)0 frictionintheentryandexitprocesses,inequilibriumtheresearchcostequalsexpectedgainsat the research phase, i.e., E(VEN(θ ))=I. If E VEN >I more (cid:2)rms will initiate research and (cid:3)0 later enter the industry, causing a decrease in output price until equality is restored. A strictly (cid:0) (cid:1) positive(cid:2)xedresearchcost,I>0,isessentialtoavoidtheextremesituationwheretrialresearch issohighthatonlythehighestproductivity(cid:2)rmsenterandsurvive. Becausethereisnoreliable capital stock variable in Quadros de Pessoal, we do not make the capital decision endogenous to the model. Instead, we assume that (cid:2)rms are homogeneous along the capital dimension and facethesameopportunitycostofremaininginactivity,W. Up to this point, the only differences between our model and Jovanovic (1982) are that in the latter model the ef(cid:2)ciency parameter implicitly affects the cost function and the cohort’s entry size distribution is degenerate. Therefore, without adjustment costs there would be no 18Inthismodelwedonotconsiderthepossibilitythatas(cid:2)rmsgetoldertheymightdecayorbecomeobsolete. This could be achieved by assuming exogenous probabilities for those two events. This could generate both a decreaseinsizeofold(cid:2)rms(decay)andtheexitofold(cid:2)rms(decayandobsolescence). 11

intertemporallinkagesinourmodelasidefromtheexitdecision. AsinJovanovic,becauseV is strictly increasing in θ , the exit decision is characterized by an age-dependent exit threshold. (cid:3) For values of θ above or equal to that threshold, the (cid:2)rm would stay and choose employment (cid:3)τ to maximize current period pro(cid:2)ts. For values of θ below that threshold, the (cid:2)rm would (cid:3)τ leave the industry, since its expected pro(cid:2)tability is below the opportunity cost. The increasing con(cid:2)dencethe(cid:2)rmputsinθ asitgrowsolderimpliesthattheexitthresholdisincreasingwith (cid:3)τ age. This is the driving force underlying Jovanovic’s result that the size distribution and the survivalprobabilityincreasewithage. We now introduce linear adjustment costs into the model. The adjustment cost for continuing(cid:2)rms,CS,isde(cid:2)nedas CS(L ;L )=P L L τ τ 1 τ τ 1 (cid:0) j (cid:0) (cid:0) j wherePisthecostperunitofadjustment. Sincethisisamodelwithendogenousentryandexit of (cid:2)rms, we consider that this cost also applies to the entry and exit decisions, so that the costs forenteringandexiting(cid:2)rms,CEN andCEX respectively,aregivenby CEN(L )=PL , CEX(L )=PL . 0 0 τ τ Withadjustmentcosts,theproblemnowbecomes, VS(L ;θ ;τ)=max P (L ;θ ) CS(L ;L ) + τ 1 (cid:3)τ τ (cid:3)τ τ τ 1 (cid:0) L τ (cid:0) (cid:0) nh i βE max VEX(L );VS L ;θ ;τ+1 , (4) τ τ τ (cid:3)τ+1 h n oio (cid:0) (cid:1) forallperiodsafterentry(τ 1)inwhichthe(cid:2)rmremainsintheindustry,and (cid:21) VEN(θ )=max P (L ;θ ) CEN(L ) +βE max VEX(L );VS(L ;θ ;1) , (5) (cid:3)0 0 (cid:3)0 0 0 0 0 (cid:3)1 L (cid:0) 0 n h n oio (cid:2) (cid:3) fortheentryperiod,whereVEX,thevalueofexiting,isde(cid:2)nedas VEX(L )=W CEX(L ). τ τ (cid:0) Note that contrary to the case without adjustment costs, the previous period employment is a state variable for the current period optimization problem. Also, inVEN andVEX the costs of hiringatentryand(cid:2)ringatexitaretakenintoaccount. In general, we could allow for asymmetry among the cost parameters inCS,CEN, andCEX. However,asymmetriesbetweenthecostofregular(cid:2)ringandthecostof(cid:2)ringatexitorbetween 12

thecostofregularhiringandthecostofhiringatentryleadtobiasesinentryandexitdecisions. Forexample,iftheperunitregularhiringcostishigherthantheperunitentryhiringcost,then (cid:2)rms will hire more workers at entry in order to save on expected future higher hiring costs. Similarly, if the per unit regular (cid:2)ring cost is smaller than the per unit exit (cid:2)ring cost, then (cid:2)rmsfacingtheprospectofexitwill(cid:2)reworkersbeforeexitingtheindustry,savingonexpected futurehigherexit(cid:2)ringcosts. Toavoidthesebiases,throughoutthepaperweassumesymmetry between the parameters inCS,CEN, andCEX. A more interesting distinction is between (cid:2)ring and hiring costs. We will see below that the conclusion of the paper is immune to asymmetries betweenthecostsofaddingandsubtractingworkers. In solving the (cid:2)rm’s problem, we consider a two-step optimization procedure where the (cid:2)rm (cid:2)rst chooses optimal employment in each of three possible scenarios, and then selects the scenariowiththehighestpay-off. Moreprecisely, VS( )=max VSD( );VSN( );VSU( ) , (cid:1) (cid:1) (cid:1) (cid:1) n o whereVSD andVSU are obtained by maximizing the objective function in (4) over L L τ τ 1 (cid:20) (cid:0) and L L , respectively, and VSN is obtained by choosing L = L in (4). Although τ τ 1 τ τ 1 (cid:21) (cid:0) (cid:0) the adjustment cost function introduces a non-differentiability of the objective function at the frontiersbetweenadjustmentandnon-adjustment,theusualpropertiesofthevaluefunctionVS anditsassociatedoptimalexitpolicyfunctionhold Proposition4 LetVS bede(cid:2)nedasin(4). Then: (a) There exists a unique value functionVS(L ;θ ;τ) satisfying (4) that is bounded, contin- τ 1 (cid:3)τ (cid:0) uousin(L ;θ ),andstrictlyincreasinginθ . τ 1 (cid:3)τ (cid:3)τ (cid:0) (b) There exists a unique optimal exit policy function χ (L ;θ )=1 θ <θEX(L ;τ) , (cid:3)τ τ 1 (cid:3)τ (cid:3)τ τ 1 (cid:0) (cid:0) whereθEX(L ;τ)isauniquecontinuousfunctioninL . τ 1 τ 1 (cid:0) (cid:1) (cid:0) (cid:0) Proof. SeeappendixA.19 In contrast, the non-differentiability of the objective function generates an inaction region in the employment policy, within which optimal employment does not vary with changes in productivity. 19Because in general VS is not concave, we cannot prove the usual differentiability properties of the value function. Therefore, in what follows, we implicitly assume thatVS L ;θ ;τ+1 is differentiable at L with τ (cid:3)τ+1 τ probability one, in terms of F θ θ ;τ for all θ Q . By part (b) of proposition 4 and the dominated (cid:3)τ+1j (cid:3)τ (cid:3)τ 2 (cid:0) (cid:1) convergencetheorem,thisimpliesthattheobjectivefunctionsassociatedwithVSD,VSN andVSU arecontinuously (cid:0) (cid:1) differentiableinL,sothatmarginalconditionscanbeappliedto(cid:2)ndinterioroptima. Thisassumptionalsoimplies thatVS(L ;θ ;τ)isdifferentiableatL withprobabilityone. Inproposition5ofappendixA,weprovethat τ 1 (cid:3)τ τ 1 thisproper(cid:0)tyholdsbothinamodelwitha(cid:0)(cid:2)nitelifetimehorizonandamodelwithin(cid:2)nite-lived(cid:2)rmsthatfacea (cid:2)nitelearninghorizon(asinsections4and5). 13

Proposition6 Foranyperiodτ >0,ifthe(cid:2)rmadjustsupwards,optimalemploymentsatis(cid:2)es ¥ F (L )θ w + (cid:229) E β s χ(cid:152) ( P)+χ(cid:136) F L θ w =P, (6) 0 τ(cid:3) (cid:3)τ τ (cid:3)τ+s (cid:3)τ+s 0 τ(cid:3)+s (cid:3)τ+s (cid:0) (cid:0) (cid:0) s=1 (cid:2) (cid:3) (cid:8) (cid:2) (cid:0) (cid:1) (cid:3)(cid:9) whereasifthe(cid:2)rmadjustsdownwardsoptimalemploymentsatis(cid:2)es ¥ F (L )θ w + (cid:229) E β s χ(cid:152) ( P)+χ(cid:136) F L θ w = P, (7) 0 τ(cid:3) (cid:3)τ τ (cid:3)τ+s (cid:3)τ+s 0 τ(cid:3)+s (cid:3)τ+s (cid:0) (cid:0) (cid:0) (cid:0) s=1 (cid:2) (cid:3) (cid:8) (cid:2) (cid:0) (cid:1) (cid:3)(cid:9) In period 0, the (cid:2)rm enters the industry if VEN(θ ) W, in which case optimal employment (cid:3)0 (cid:21) satis(cid:2)es ¥ F (L )θ w + (cid:229) E β s χ(cid:152) ( P)+χ(cid:136) F (L )θ w =P. (8) 0 0(cid:3) (cid:3)0 0 (cid:3)s (cid:3)s 0 s(cid:3) (cid:3)s (cid:0) (cid:0) (cid:0) s=1 (cid:2) (cid:3) (cid:8) (cid:2) (cid:3)(cid:9) L istheoptimalemploymentinperiodτ+s,andχ(cid:152) , χ(cid:136) arefunctionsoftheoptimalexit τ(cid:3)+s (cid:3)τ+s (cid:3)τ+s decision, χ ,inperiodsτ+1toτ+s,suchthat χ(cid:152) equalsonewhenthe(cid:2)rmhasremained (cid:3)τ+j (cid:3)τ+s in the industry until period τ+s 1, but decides to exit in period τ+s, and χ(cid:136) equals one (cid:3)τ+s (cid:0) whenthe(cid:2)rmisstillintheindustryinperiodτ+s. Proof. SeeappendixA. Equations (6), (7) and (8) are marginal conditions, similar to the smooth pasting conditions inthe(S,s)modelliterature,andtheystatethatifthe(cid:2)rmadjuststhenthemarginaladjustment cost must equal the expected present discounted value of the marginal revenue product for all future periods in which the (cid:2)rm is still in the industry, minus the increase in the exit cost when the(cid:2)rmdecidestoexit. Thisisthediscrete-timeanalogofthecontinuous-timeresultpresentin Nickell(1986)andBentolilaandBertola(1990),adjustedforthefactthatnowwealsohavean exit decision. Because the (cid:2)rm will not change employment if the marginal cost of adjustment exceedsitsmarginalbene(cid:2)tforthe(cid:2)rstunitofadjustment,proportionalcostsimplyinactionin theemploymentdecisionofthe(cid:2)rm. Although the results in proposition 6 do not allow a formal proof of the effects on (cid:2)rm growth of adjustment costs in this general model, the following corollary of proposition 6 enablesustomakequalitativeheuristicstatementsaboutthoseeffects. Corollary7 For any period τ 0, the marginal bene(cid:2)t of one additional unit of labor, that is, (cid:21) theLHSofexpressions(6),(7),and(8),canberecursivelyrepresentedas MB = F (L )θ w +βE χ ( P)+ 1 χ MB (9) τ 0 τ(cid:3) (cid:3)τ τ (cid:3)τ+1 (cid:3)τ+1 τ+1 (cid:0) (cid:0) (cid:0) (cid:0) (cid:1) (cid:2) (cid:0) (cid:1) (cid:3) 14

where L =L (L ;θ ), χ = χ (L ;θ ) are the optimal employment and exit deci- τ(cid:3) τ(cid:3) τ 1 (cid:3)τ (cid:3)τ 1 (cid:3)τ 1 τ 1 (cid:3)τ (cid:0) (cid:0) (cid:0) (cid:0) sions. Proof. SeeappendixA. 3.2 Linear Adjustment Cost and Firm Growth As we have seen above, in the absence of adjustment costs, optimal employment is determined solely to maximize current period pro(cid:2)ts, so that F (L )θ = w. Therefore, (cid:2)rms’ growth is 0 (cid:3) (cid:3) essentiallyaby-productofaselectionmechanism: those(cid:2)rmsthatareinef(cid:2)cient,andtherefore small, exit, while those (cid:2)rms that are ef(cid:2)cient survive and grow. There is an additional source of positive growth when the frictionless employment decision rule, L (θ ), is convex in θ . (cid:3) (cid:3) (cid:3) Because of Jensen’s inequality and because θ is a Martingale, surviving (cid:2)rms will grow over (cid:3)τ time: E L θ > L E θ = L (θ ). However, L will not be convex in θ for τ (cid:3) (cid:3)τ+1 (cid:3) τ (cid:3)τ+1 (cid:3) (cid:3)τ (cid:3) (cid:3) generalF(L).20 (cid:2) (cid:0) (cid:1)(cid:3) (cid:2) (cid:0) (cid:1)(cid:3) In arguing heuristically about the impact of the proportional cost on (cid:2)rm growth we use the property that MB is weakly increasing in θ , and that L is locally weakly increasing τ (cid:3)τ τ(cid:3) in θ . Because it is not immediately obvious why (cid:2)ring and hiring costs should give similar (cid:3)τ incentives for (cid:2)rm growth, we analyze separately these two costs.21 We present in (cid:2)gure 1 the casewherethereisahiringcost,PH >0,andno(cid:2)ringcost,PF =0. This(cid:2)gureassumesagiven L . For that speci(cid:2)c value of L , θSU and θSD are the frontiers between non-adjustment τ 1 t 1 (cid:0) (cid:0) and upward and downward adjustment, respectively. Therefore, if θ θSD;θSU there will (cid:3)τ 2 be no adjustment and the marginal bene(cid:2)t of an additional unit of labor (represented by the (cid:2) (cid:3) dashed line) is contained in the interval 0;PH . To simplify the argument, we consider a (cid:2)rm whosesequenceofproductivitydrawsissuchthatineveryperiodithasaperceivedproductivity (cid:2) (cid:3) equal to the unconditional mean of θ , even though the (cid:2)rm’s uncertainty over next period θ (cid:3) (cid:3) decreaseswithage. Case1: HiringCost: PH >0,PF =0 20Ingeneral,fromtheoptimalemploymentcondition,F (L)θ =w,wehave 0 (cid:3) F (Lfl)F (Lfl) 2F (Lfl) 2 Lfl 00 (θ (cid:3) )= 000 F 0 0 0 (Lfl) 2 (cid:0) θ (cid:3) 00 ! Lfl 0 (θ (cid:3) ),F 00 (Lfl)<0,L 0 (θ (cid:3) )>0, whosesigndependsonF (Lfl). Therefore,ifdecreasingreturnstolabordonotdecreasetoofast,thatis,F (Lfl)< 000 000 2F (Lfl) 2 =F (Lfl), then we will have Lfl (θ )<0. When F(L)=ln(L), then Lfl (θ )=0, and when F(L)=Lα, 00 0 00 (cid:3) 00 (cid:3) α (0;1),thenLfl (θ )>0. 00 (cid:3) 2 21Inthediscussionthatfollows,thehiringcostappliesbothtoregularhiringandtohiringatentryandthe(cid:2)ring costappliesbothtoregular(cid:2)ringandto(cid:2)ringatexit. 15

Because the (cid:2)rm starts at the hiring margin, we must have MB =PH at entry, and MB 0 τ 2 0;PH , for all subsequent periods, τ = 1;2;:::, with the two extremes of the interval representing (cid:2)ring and hiring of workers, respectively. Consider (cid:2)rst a situation where exit is not (cid:2) (cid:3) allowed. Underthisassumption,(9)wouldbecome MB = F (L )θ w +βE MB τ 0 τ(cid:3) (cid:3)τ τ τ+1 (cid:0) (cid:0) (cid:1) For the entry period, we have MB (θ ) = PH, which implies that the (cid:2)rm will start smaller 0 (cid:3)0 when PH > 0 than when PH = 0.22 Since MB (θ ) 0;PH , E (MB ) < PH and thus we 1 (cid:3)1 0 1 2 must have pF (L ) w > 0, for all β (0;1), if PH > 0. In the following period, (cid:2)rms will 0 0 (cid:2) (cid:3) (cid:0) 2 adjust upwards as frequently with PH > 0 as when PH = 0, because they start at the hiring margin and E θ =θ , even though they might have smaller magnitudes of adjustment due to 0 (cid:3)1 (cid:3)0 the hiring cost. The proportional hiring cost implies that (cid:2)rms will adjust downwards only if θ <θSD, so that there is a region of inaction when PH >0 that is not present when PH =0. (cid:3)1 1 Thatis,(cid:2)rmshirefewerworkersinitiallybecausetheresultingsmallerprobabilityofhavingto (cid:2)re them, and therefore wasting the initial hiring cost, compensates for the expected decrease inpro(cid:2)tsthisperiod. Consequently,inperiod1more(cid:2)rmswillhirethan(cid:2)re,andthistendency towardsgrowthinyoung(cid:2)rmswillpersistforseveralperiods. TheBayesianlearningmechanismimpliesbothpersistenceandareductioninvariancewith age in the Markov process associated with θ . The effect of persistence, that is, the fact that (cid:3) E θ θ ;τ = θ , was analyzed in the previous paragraph. The reduced uncertainty in (cid:3)τ+1 (cid:3)τ (cid:3)τ j the posterior estimate of productivity will be re(cid:3)ected in a smaller inaction region as (cid:2)rms (cid:0) (cid:1) accumulate information on realized productivity; that is, θSU decreases with τ. This causes an increaseinE (MB )forthose(cid:2)rmsalreadyatthehiringmargin,whichmustbebalancedby τ τ+1 an increase in L for the right hand side of (9) to remain equal to PH. As (cid:2)rms become more τ(cid:3) certain about their true productivity they are more willing to adjust to their long run optimal size. Because most (cid:2)rms are at the hiring margin, this will cause a further increase in average size. Consider now the possibility of exit. In this case, the uncertainty reduction as the (cid:2)rm ages impliesadecreaseintheexitprobability,andafurtherincreaseinthefuture-periodscomponent of MB in (9). Consequently, L needs to increase further in order to offset that.23 On the other τ(cid:3) hand, the smaller exit probability implies less pruning of inef(cid:2)cient slow-growing (cid:2)rms as a cohort ages, which tends to make growth in average (cid:2)rm size smaller. Therefore, we will 22Whenexitisnotallowed,wecanprovethatVS isconcave(andcontinuouslydifferentiable)inL,sothatL 0 mustdecreaseforMB toincrease. 0 23ThiseffectissimilartothatofCabral(1995). 16

have less cohort growth due to non-survivors and more cohort growth due to survivors, so that survivors’ contribution to average (cid:2)rm growth in the cohort should increase when exit is allowed. Case2: FiringCost: PF >0,PH =0 In this case, we have MB =0, MB PF;0 , τ =1;2;:::. Assume (cid:2)rst that exit is not 0 τ 2 (cid:0) allowed. Theintuitionisthesameasincase1. IncomparisonwithPF =0,whenPF >0(cid:2)rms (cid:2) (cid:3) start smaller and subsequently hire more frequently than they (cid:2)re. As (cid:2)rms age, the reduction invarianceofθ causesanincreaseinE (MB ),whichmustbecompensatedbyanincrease (cid:3) τ τ+1 in L for (cid:2)rms at the hiring margin. When exit is possible, those effects become more intense, τ(cid:3) sincetheexitprobabilitywilldecreaseas(cid:2)rmsage. Fromtheheuristicintuitionwehavejustgivenitbecomesclearthatproportionalhiringand proportional(cid:2)ringcostsreinforceeachotherincreatingincentivesfor(cid:2)rmstogrow. Intheend, our assessment of the relevance of linear adjustment costs for (cid:2)rm growth will depend on how well a pure selection model can (cid:2)t the empirical evidence, and on how much adjustment costs improve the (cid:2)t. Before we move into a quantitative assessment, we present analytical results forasimpleversionofthegeneralmodel. 4 Model with One Period Learning Horizon and No Exit In this section, we analyze a model where (cid:2)rms’ ef(cid:2)ciency is revealed after the (cid:2)rst period of lifeandwhere(cid:2)rms’lifetimehorizonisknowwithcertaintyatentry.Weassumethat(cid:2)rmslive for Tfl periods, where Tfl is any integer greater than 1, and that no exit is allowed prior to age Tfl. Thesetwosimpli(cid:2)cationsallowustodeterminetheeffectoflinearadjustmentcostson(cid:2)rm growth. The introduction of adjustment costs implies an additional expected operating cost for entering (cid:2)rms. Therefore, the equilibrium price must increase to generate higher expected future pro(cid:2)tsthatcompensateforthecostsincurredwhileadjustingtooptimalsize. Asaconsequence, pre-entrypruningofinef(cid:2)cient(cid:2)rmsshouldincreasewhilepost-entrypruningshoulddecrease. This is optimal from a social point of view, since with higher adjustment costs there should be lessexperimentationinordertosaveinunrecoverablecosts. Therefore,theassumptionthatexit isexogenousisnotcriticalfortheresultsinthissection. Sinceadjustmentcostsattenuatepostentry pruning, even if exit was endogenous to the model, the relative contribution of survivors to growth in the cohort’s average size would increase through this channel. By eliminating any exitpriortoTfl wefocusonlyontheincentivesforsurvivorstogrow. 17

To formulate the problem, we use the fact that once the (cid:2)rm learns its true ef(cid:2)ciency in period2,itwilladjustonceandforalltoitslongrunemploymentlevel.24 Then,ifuponexitat ageTfl (cid:2)rmsrecovertheinitialinvestmentnetofexitcosts,inperiod2wehave VS(L ;θ )=max δ(Tfl)P (L ;θ ) CS(L ;L )+β Tfl 1 VEX(L ) (10) 1 (cid:3)2 2 (cid:3)2 2 1 (cid:0) 2 L (cid:0) 2 n o whereδ (cid:229) Tfl 2β s = 1 β Tfl 1 =(1 β). Inperiod1,wethenhave Tfl (cid:17) s=(cid:0)0 (cid:0) (cid:0) (cid:0) (cid:16) (cid:17) VEN(θ )=max P (L ;θ ) CEN(L )+βE VS(L ;θ ) : (cid:3)1 1 (cid:3)1 1 1 1 (cid:3)2 L (cid:0) 1 n h io Finally,inequilibriumpotentialentrantsbreakeven,i.e.,E VEN(θ ) =I. 0 (cid:3)1 We examine the impact of adjustment costs on the log growth rate of employment, rather (cid:2) (cid:3) than the standard growth rate, in order to attenuate the effect of Jensen’s inequality on (cid:2)rm growth.25 In this simple model, the inaction region of optimal employment can be expressed asaninterval: Q SN = θSD;θSU .Therefore,theaverageloggrowthratebetweenperiod1and period2,conditionalonθ ,isde(cid:2)nedas (cid:2) (cid:3)1 (cid:3) g(θ )=E[ln(L ) ln(L )]= (cid:3)1 2(cid:3) 1(cid:3) (cid:0) θSD ν2 ln L SD ln(L ) dF (θ )+ ln L SU ln(L ) dF (θ ) Z ν1 2(cid:3) (cid:0) 1(cid:3) θ (cid:3)1 (cid:3)2 Z θSU 2(cid:3) (cid:0) 1(cid:3) θ (cid:3)1 (cid:3)2 n (cid:16) (cid:17) o n (cid:16) (cid:17) o where Q [ν ;ν ] is the support of the distribution of θ , and θSD(L ) and θSU(L ) are the 1 2 (cid:3)2 1(cid:3) 1(cid:3) (cid:17) frontiers between non-adjustment and downward and upward adjustment, respectively. Depending on the speci(cid:2)c value of θ and the magnitude of the adjustment cost parameters, we (cid:3)1 mighthaveθSD(L )=ν and/orθSU(L )=ν . However,intheresultsthatfollow,weassume 1(cid:3) 1 1(cid:3) 2 that θ and the adjustment cost parameters are such that both downward adjustment and up- (cid:3)1 ward adjustment occur with positive probability, i.e., θSD(L )>ν and θSU(L )<ν . Since 1(cid:3) 1 1(cid:3) 2 weassumeexogenousexit,weignoretheindirecteffectofadjustmentcoststhatworksthrough changes in the equilibrium price, and implicitly assume that the research cost, I, adjusts to maintain an equilibrium. This indirect price effects in(cid:3)uence average (cid:2)rm size in both periods, butareofsecondorderimportancefortheaveragelog-growthrate.26 24Thisresultisformalizedinproposition10below. 25Aswesawabove,whenoptimalemploymentisaconvexfunctionofθ ,Jensen’sinequalityimpliespositive (cid:3) expectedgrowth,evenintheabsenceofadjustmentcosts. Becausethelogtransformationisconcave,itwilloffset theconvexityoftheoptimalemploymentfunction. Forexample, whenF()isapowerfunction, theloggrowth (cid:1) rateeliminatestheeffectofJensen’sinequality,sinceln(L )becomeslinearinθ . τ(cid:3) (cid:3)τ 26Inproposition8below,ifF()isapowerfunctiontheindirectpriceeffectscancelout. (cid:1) 18

Optimalemploymentinperiod2isdeterminedby w+βTfl (cid:0) 1PF +PH L SU =F 1 δ Tfl δ Tfl , θ >θSU(L ) 2(cid:3) 0(cid:0) θ (cid:3)2 1 8 (cid:3)2 ! > L 2(cid:3) (L 1 ;θ (cid:3)2 )=> > > > > > < L 2(cid:3) SN w = +β L Tfl 1 (cid:0) , 1PF PF θSU(L 1 ) (cid:21) θ (cid:3)2 (cid:21) θSD(L 1 ) L SD =F 1 δ Tfl (cid:0)δ Tfl , θSD(L )>θ 2(cid:3) 0(cid:0) θ 1 (cid:3)2 > > (cid:3)2 ! > > > > > wherethefrontiersof:adjustmentarede(cid:2)nedas w+ βTfl (cid:0) 1PF +PH w+ βTfl (cid:0) 1PF PF θSU(L ) δ Tfl δ Tfl , θSD(L )= δ Tfl (cid:0)δ Tfl . 1 1 (cid:17) F (L ) F (L ) 0 1 0 1 Note that the numerator of θSU equals the pro-rated per-period cost of adding another worker, including the wage, the marginal hiring cost, and the discounted cost of (cid:2)ring the worker after periodTfl. ThenumeratorofθSD hasasimilarinterpretation,asthebene(cid:2)tofsheddingaworker. We then have the following result concerning the effects of changes in PH and PF on the cohort’saverageloggrowthrateofemployment. Proposition8 AssumingthatF( )isapowerfunctionandthatθSD >ν andθSU <ν :27 1 2 (cid:1) (a)ThemarginaleffectofPH ong(θ ),assumingPF =0,ispositiveforTfl suf(cid:2)cientlyhigh. (cid:3)1 (b)ThemarginaleffectofPF ong(θ ),assumingPH =0,ispositiveforallTfl. (cid:3)1 Proof. SeeappendixA. Consider (cid:2)rst the hiring cost. In the proof, we show that an increase in PH decreases both L and L SU. The impact of PH on the growth rate depends on two opposing effects. First, 1(cid:3) 2(cid:3) while in the case of L SU the cost of hiring can be equally spread out over Tfl 1 periods with 2(cid:3) (cid:0) certainty, in the case of L it will be spread out over either Tfl periods or one period, depending 1(cid:3) onwhetherthe(cid:2)rmlearnsinperiod2thatithasoverhired. Therefore,exanteaproportionately greater part of PH is attached to period 1 in the case of L than in the case of L SU, affecting 1(cid:3) 2(cid:3) more L than L SU. This explains the positive effect on growth of PH for Tfl =¥ . Second, the 1(cid:3) 2(cid:3) hiring cost on L can possibly be spread out over Tfl periods, while the hiring cost on L SU can 1(cid:3) 2(cid:3) only be spread out over Tfl 1 periods. This affects more L SU than L , and explains why the (cid:0) 2(cid:3) 1(cid:3) 27Intheproof,weconsiderageneralproductionfunctionandthenspecifyapowerfunctioninordertoobtain thesignoftheeffect.Fromthatgeneralsetup,wecansaythattheformofF()shouldnotbedeterminantforthese (cid:1) resultswhentheelasticityofthemarginalproductoflabordoesnotvarymuchwiththeamountoflaborused. 19

effect of PH on growth is not necessarily positive for (cid:2)niteTfl. However, as Tfl increases the (cid:2)rst effectdominatessothatPH decreasesL morethanL SU andgrowthincreases.28 1(cid:3) 2(cid:3) WithrespecttoPF thereisalwaysapositiveeffectongrowth,independentlyofthelifetime horizon. This occurs because an increase in PF decreases L and increases L SD. This positive 1(cid:3) 2(cid:3) effectalwaysdominatestheuncertaineffectduetothefactthatL SU alsodecreaseswithPF. 2(cid:3) When there are both hiring and (cid:2)ring costs and these costs are identical (PH = PF = P), then an increase in P has a positive effect on g(θ ), for suf(cid:2)ciently high Tfl, where the required (cid:3)1 Tfl islowerthaninitem(a)ofproposition8. 5 Calibration/Estimation Under Finite Learning Horizon Intheprevioustwosections,wedevelopedheuristicandsomeformalargumentsabouttheeffect of adjustment costs on the incentives for (cid:2)rms to start smaller and grow faster after entry. In thissection,weassessthecontributionofadjustmentcoststoexplainsomeofthebasicfactson (cid:2)rm dynamics found in section 2, both for the overall economy and for the manufacturing and services sectors. To accomplish this, we perform a calibration/estimation of the model using computationalmethods. To simulate the in(cid:2)nite learning horizon model we follow the suggestion of Ljungqvist and Sargent (2004) and consider an approximation where (cid:2)rms live forever, but learn their ex post trueproductivitycomponent, µ ,withcertaintyatsomeageT.29 1 WeassumethatF( )isapowerfunction,i.e.,F(L)=Lα,α (0;1). Underthisassumption, (cid:1) 2 when adjustment is costless, optimal employment conditional on survival is a convex function of θ , so that Jensen’s inequality implies growth of employment even if there is no selection. (cid:3)τ Asintheprevioussection,inordertoavoidanygrowthduetoJensen’sinequality,wetakelogs ofallvariablesandanalyzetheeffectsofadjustmentcostsonthelog-growthrate. Concerningtheproductivitydistribution,weassumethatθ islognormallydistributed,i.e., τ ξ(η)=exp η .30 Thisassumptionismadeforcomputationalsimplicity,anditseemsreasonf g ableonempiricalgrounds(seeAwetal.2004). Inaddition,thisassumptionisnotcriticalasthe results in section 4 suggest that the distribution of productivity mostly affects the intensity of the effect of adjustment costs on (cid:2)rm growth, but not the sign. In fact, proposition 8 is derived 28Inoursimulations,Tfl =3wasenoughtogenerateapositiveeffectongrowth. 29NotethatthisT differsfromthelifetimehorizonTfl usedinsection4,withTfl T. Inthissection,weassume anin(cid:2)nitehorizon, sothatTfl =¥ . Inoursimulationsbelow, weassumethatT = (cid:21) 15(years), andpresentresults until year 10. Assuming a higher value for T would slow down the algorithm’s execution without improving signi(cid:2)cantlytheaccuracyofthemodelsimulations. 30Underlog-normality,ν =0andν =¥ . Althoughthisviolatesassumption1,thisisnotaprobleminthis 1 2 section,sincewewillbeusingadiscreteapproximationtotheproductivitydistribution. 20

independentlyoftheparticulardistributionofθ . Withthelog-normaldistributionassumption, (cid:3)τ thetransitionlawfortheθ sisasfollows. (cid:3) Proposition9 Letθ =exp η begeneratedasinassumption1. Then, τ τ f g (a) The posterior distribution of θ (j 0), given the information set at time τ, W = µ ; τ+j τ 0 (cid:21) η τ 1 ifτ <T,andW = µ ;µ ifτ T,is n s s=(cid:0)0 τ 0 1 f g f g (cid:21) o θ τ+j W logN Y τ ;Z τ +σ2 , j τ(cid:24) (cid:0) (cid:1) where, for τ < T, Y and Z are de(cid:2)ned in (2), and, for τ T, Y = µ and Z = 0. Let τ τ τ τ (cid:21) θ =E(θ W )=E(θ θ ;τ). Thenthedistributionofθ (j 1)given(θ ;τ)is (cid:3)τ τ τ τ (cid:3)τ (cid:3)τ+j (cid:3)τ j j (cid:21) 1 θ logN ln(θ ) Z Z ;Z Z . (cid:3)τ+j j (θ (cid:3)τ ;τ) (cid:24) (cid:3)τ (cid:0)2 τ (cid:0) τ+j τ (cid:0) τ+j (cid:18) (cid:19) (cid:0) (cid:1) Also,theunconditionaldistributionofθ (τ 0)is (cid:3)τ (cid:21) 1 θ logN µfl + Z +σ2 ;σ2 Z , (cid:3)τ (cid:24) 2 τ µ (cid:0) τ (cid:18) (cid:19) (cid:0) (cid:1) whereσ2 =σ2 +σ2 . µ µ µ 0 1 Proof. SeeappendixA. Since we assume that the (cid:2)rm enters the industry already knowing its ex ante productivity componentµ (seeassumption1),wewillgetanon-degeneratedistributionofinitialsize. This 0 occurs because L = L (θ ), and θ has positive variance in the cohort’s initial distribution. 0 0(cid:3) (cid:3)0 (cid:3)0 The next proposition analyzes the properties of the optimization problem after µ is revealed to the(cid:2)rminperiodT. Proposition10 Ifµ isrevealedtothe(cid:2)rmatperiodT,thenalladjustmentsaremadeatperiod T,andthe(cid:2)rmwillnotchangeitsexitandemploymentdecisionsafterthatperiod. Thismeans that 1 VS(θ ;L ;T)=max P (L;θ ) CS(L;L ) , (11) (cid:3)T T (cid:0) 1 L 1 β (cid:3)T (cid:0) T (cid:0) 1 (cid:26) (cid:0) (cid:27) L =L (θ ;L ;T),s T, χ =1 VS(θ ;L ;T)<VEX(L ) . s(cid:3) (cid:3) (cid:3)T T 1 (cid:3)T (cid:3)T T 1 T 1 (cid:0) (cid:21) (cid:0) (cid:0) h i Proof. SeeappendixA. 21

Thisresultallowsaconsiderablesimpli(cid:2)cationofthecomputationalalgorithm,sinceitimplies a (cid:2)nite horizon dynamic programming problem. In appendix B, we present some details concerning the computational algorithm used to simulate and estimate the model. In the followingsubsectionswecalibrateandestimatethemodelanddoasensitivityanalysis. 5.1 Calibration and Estimation of Model with Costly Adjustment Wecalibrateandestimateourmodeltomatchstatisticsfromthe1988cohortofentering(cid:2)rms, bothfortheoveralleconomyandthemanufacturingandservicesectors. We(cid:2)rstcalibrateparameters related to inputs directly from the data. We then use the simulated method of moments to estimate the parameters associated with the learning process and the adjustment cost. These estimates are obtained so that the model generated moments match the evolution of (cid:2)rm size, of exit rates, and of the survivor component observed in the data. As discussed in appendix B, we (cid:2)nd the set of parameter values that minimize the method of moments objective function by using a simulated annealing method. This optimization method is robust to local minima, to discontinuities, and to the discretization implemented in order to simulate the model.31 A centralelementtoourestimationstrategyisadecompositionofthechangeinthecohort’saverage size into a survivor component and a non-survivor component. This decomposition forces themodeltomatchnotonlythegrowthinthecohort’saveragesizebutalsothecontributionof surviving and non-surviving (cid:2)rms to that growth. Similarly to section 2, with l ln(L ), our τ τ (cid:17) decompositionisde(cid:2)nedas E[l S ] E[l S ]= E[l l S ] +Pr(Dτ S ) E[l S ] E[l Dτ] (12) τ τ 0 0 τ 0 τ 0 0 τ 0 j (cid:0) j (cid:0) j j f j (cid:0) j g SurvivorComponent Non-SurvivorComponent | {z } | {z } Prior to estimation, we calibrate some parameters. The parameters α and w are calibrated with data from INE (1997) containing the InquØrito Annual (cid:224)s Empresas from 1990 to 1995. Thesedataareconsideredreliableandcoverall(cid:2)rmsinthePortugueseeconomy,withsampling among (cid:2)rms with less than 20 workers. We measure α as the 1990-1995 average of the cost share of labor in value added, and w as the 1990-1995 average cost per worker. We can also obtainthesevaluesatthesectorallevel. Wede(cid:3)ateallnominalvariablesusingtheGDPsectoral priceindicesavailableintheupdatedversionofSØriesLongasparaaEconomiaPortuguesain Banco de Portugal (1997). The real interest rate is calibrated as the 1990-1995 average of the implicit real interest rate on public debt transactions on the secondary market of the Lisbon 31Theobjectivefunctionisde(cid:2)nedasQ= N (cid:0) 1(cid:229) N i=1 f i 0 N (cid:0) 1S (cid:0) 1 N (cid:0) 1(cid:229) N i=1 f i ,whereN (cid:0) 1(cid:229) f i arethedifferencesbetweenthesampleandmodelsimulatedmoments,andS istheestimatedcovariancematrixof f. i (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) 22

Stock Exchange. The data was also taken from Banco de Portugal (1997). We de(cid:3)ate the nominalinterestratesusingtheDecember-to-DecemberconsumerpriceindexfromINE(1990- 5). Thediscountrateisthenobtainedasβ = 1 ,wherer istheaveragerealinterestrate. 1+r The remaining parameters, µfl, σ , σ , σ, W, and P are estimated using a simulated µ µ 0 1 method of moments estimator, which attempts to make the model match closely the evidence on cohort dynamics presented in section 2. In particular, the estimates are selected to minimizeaweightedsumofthedistancebetweenthefollowingmomentsinthemodelandthedata: (a) the time-series of the mean of log-employment conditional on survival, E[l S ]; (b) the τ τ j time-series of the cumulative change in the standard deviation of log-employment conditional onsurvival,SD[l S ];32 (c)thetime-seriesofthecumulativeexitrate,Pr(Dτ S );(d)andthe τ τ 0 j j time-series of the survivor component, as de(cid:2)ned in (12). In estimating the above parameters, theoutputpriceisnormalizedto1,andtheinitialresearchcost,I,isobtainedbytheequilibrium conditionI =E VEN(θ ) .33 (cid:3)0 The decision to estimate W, instead of calibrating it, deserves some discussion. First, the (cid:0) (cid:1) main purpose of this parameter is to induce endogenous exit as it represents the (cid:2)rm’s opportunity cost of remaining in activity. In Hopenhayn (1992), the same result is accomplished using a (cid:2)xed per period operating cost. Since the per period operating cost can be seen as the periodic payment in an annuity with a present discounted value ofW, the two mechanisms are equivalent. Second, because Quadros de Pessoal misses any reliable capital stock variable, we consider the capital decision to be exogenous. A rough estimate of the magnitude ofW is the presentdiscountedvalueofanannuitywithannualpaymentsequaltothe1990-1995averageof valueaddedminuslaborcosts,usingthesamede(cid:3)atorsasforw.34 Becausethesampleisbiased towards surviving (cid:2)rms, this measure overestimates the value ofW, and we cannot use it as a reference to calibrateW. Consequently, we estimateW jointly with the remaining parameters inthemodel. Wepresentintable4thecalibratedandestimatedparametersforthethreecohorts,bothfor the model with (AC) and without (NAC) adjustment costs, and in (cid:2)gures 2 and 3 we plot the data and simulated moments in the estimated AC and NAC models. We start by making some general remarks on our estimates. First, more information is revealed ex post (σ >σ )and µ µ 1 0 there is signi(cid:2)cant noise in the learning process (σ > σ , σ > σ ). Second, consistently µ µ 0 1 with our expectations, all estimates of W are below the rough estimates presented in footnote 32We use the change in instead of the level of dispersion because imputing all the dispersion in size to the learningprocesswouldmakeitdif(cid:2)culttocapturethelevelofgrowthinthemodel,aswediscussbelow. 33Tomakethecomputationofequilibriumeasierforagivensetofparameters,insteadofchangingtheoutput pricewechangethe(cid:2)xedresearchcostsothattheequilibriumconditionissatis(cid:2)ed. 34Theestimatesare1373:8fortheoveralleconomycohort,3317:1forthemanufacturingcohort,and269:1for theservicescohort. 23

34. Third, the inferred values for I are close to 10% of W. Finally, the standard errors of the estimated parameters are relatively small, suggesting that parameters are estimated with good precision.35 Fortheoveralleconomycohort,theAC modelimpliesanestimatefortheproportionalcost ofabout6:3%oftheannualwage. BycomparingtheestimatedNACandACmodels,intermsof the objective function Q and the simulated moments in (cid:2)gure 2, we conclude that the propor- (cid:3) tionalcostclearlyimprovestheoverall(cid:2)tofthemodel,withaparticularlynotableimprovement in the (cid:2)t of the survivor component. Although the NAC model can generate moments on (cid:2)rm sizeandexitratesthatareclosetoequivalentempiricalmoments,itcannotsatisfactorilymatch theempiricalsurvivors’contribution. Thatis,theNACmodelcannotexplainthemainsourceof growthinthecohort’saveragesize,sincesurvivorscontributemuchmoretogrowthinthedata than in the NAC model. This shortcoming is especially intense in the initial years after entry, whenthedistancebetweenthesurvivorcomponentinthesimulatedNAC modelandinthedata is largest,suggesting that inthe absence ofadjustment costs learninghas a largerinitial impact ontheexitofsmallinef(cid:2)cient(cid:2)rmsthanongrowthofsurvivors. In discussing the results for the manufacturing and services sectors we consider the estimatesfortheACmodel. Manufacturing(cid:2)rmslearnrelativelylessinitiallyabouttheiref(cid:2)ciency than services (cid:2)rms (σ =σ is smaller in manufacturing). Moreover, in order to account for µ µ 0 1 the higher survivor component, adjustment costs in manufacturing are larger than in services (proportional costs amount to 16:6% and 0:8% of the annual wage, respectively). Because of larger adjustment costs and lesser relative knowledge about ef(cid:2)ciency at entry, manufacturing (cid:2)rms have higher incentives to start relatively small and to gradually adjust to optimal size as they survive and their uncertainty is resolved. In addition, the smaller relative knowledge at entry in manufacturing explains the lower relative initial research cost (I=W is lower in manufacturing) and the higher entry rate (1 Pr(S ) equals 27:4% in manufacturing and 47:9% in 0 (cid:0) services). As can be seen from the last row of table 4, while the introduction of adjustment costs improves markedly the (cid:2)t of the model for the overall economy and manufacturing cohorts, it improves only marginally the (cid:2)t for the services cohort.36 Therefore, the form of adjustment costs considered in the paper seems more relevant for the average (cid:2)rm in the manufacturing sector and the overall economy than for the average (cid:2)rm in the services sector. More generally, although the introduction of adjustment costs clearly improves the overall (cid:2)t of model, 35ThemagnitudesweobtainforbothQ andthestandarderrorsresemblethoseobtainedbyCooperandHalti- (cid:3) wanger(2006)inastudyaboutcapitaladjustmentcostsusingasimilarestimationmethodology. 36Thisisre(cid:3)ectedinthesmallestimatefortheproportionalcostintheservicescohortwhencomparedwiththe sameestimateforthetwoothercohorts. 24

especiallyinwhatconcernsthesurvivorcomponent,themodelcannotexplainentirelythepath of the survivor component in the data. In fact, in all three cohorts the initial growth of survivors seems larger than what the AC model can explain. This might be a consequence of the discretization scheme adopted for θ in the simulation.37 More importantly, this might re(cid:3)ect (cid:3) other explanations for (cid:2)rm growth that cannot be captured by adjustment costs in our model, suchassomemechanismsthroughwhich(cid:2)nancingconstraintsoperate. 5.2 Sensitivity Analysis In this subsection we explain some aspects of the calibration/estimation exercise and provide a detailedsensitivityanalysistoallparametersinthemodel. First,wedonotattempttomatchthe level of the cross-sectional variance of log-employment, but only its change over time. This is becauseto(cid:2)tthedispersioninemployment,themodelwouldrequiresubstantiallylargervalues for both σ and σ . This would allow the model to match Var[l S ] and Pr(Dτ S ) but µ µ τ τ 0 0 1 j j would also imply an excessive rate of growth in E[l S ]. However, this shortcoming is not a τ τ j seriousproblem. Itimpliesthatonlyafractionoftheobservedcohort’semploymentdispersion can be attributed to a Bayesian learning process about ef(cid:2)ciency. The remaining part could be attributedtoheterogeneityintheinitialchoiceoftechnology. Forinstance,consideramodelwherecapitalisendogenousandsupposethata(cid:2)rmchooses its initial stock of capital, K , based on the realization of a random variable indexing tech- 0 nology choice. Assume further that, after selecting K , the (cid:2)rm keeps its capital stock un- 0 changed for the remainder of its life. If the production function has constant returns to scale, if the total opportunity cost is proportional to K , i.e., W(cid:152) =WK , then we can easily prove 0 0 that V(cid:152) (K ;L ;θ ;τ) = K V L t 1;θ ;τ , where V(cid:152) is the value function conditional on the 0 τ 1 (cid:3)τ 0 K(cid:0) (cid:3)τ (cid:0) 0 chosen K 0 . Therefore, in this a(cid:16)lternative fr(cid:17)amework, dispersion in K 0 would govern the initial dispersion in employment and only the subsequent evolution in employment dispersion would dependontheBayesianlearningprocess.38 Thisisthereasonwhyweattempttomatchonlythe evolution of SD[l S ], but not its level. In the estimated models presented in table 4 less than τ τ j 40% of the observed dispersion in the cohort’s log-employment can be attributed to the learning process, with the percentage smaller in the manufacturing sector and higher in the services 37Frompoint(i)inappendixB,weseethattherangeimplicitinthesupportoftheuniformdiscreteaproximation totheθ distributionincreaseswithτ. (cid:3)τ 38We implicitly assume that P would also be proportional to K. If this is not the case, then the proportional adjustment cost would be less important for large-K (cid:2)rms, intensifying the tendency for higher growth among small(cid:2)rms. Althoughaugmentingthemodelwiththiscapitaldecisionwouldallowustomatchthelevelof size dispersioninthedata,thenumericalcomplexityofthemodelsimulationwouldincreaseevenfurther. 25

sector.39 Second, the value of σ =σ affects the long-run contribution of survivors, since a rel- µ µ 0 1 atively smaller initial dispersion would make the average size of exiting (cid:2)rms closer to the average size of surviving (cid:2)rms in the entry period, and in this case most growth would be due to survivors. In the aforementioned extended model with an initial choice over K , if we had 0 σ =0 we would have a non-degenerate initial distribution of size, entirely due to the hetero- µ 0 geneity in K , but the survivors’ component would be 100% in each period. This would occur 0 becausethedistributionofinitialsizeamongexiting(cid:2)rmswouldbeequaltothedistributionof initialsizeamongsurviving(cid:2)rms. ThisalsoexplainswhyevenwithheterogeneityoverK ,we 0 would still need to assume σ >0 in order to match the empirical facts on the importance of µ 0 thesurvivorcomponent. While we could increase the long-run contribution of survivors by tinkering with the ratio σ =σ ,withoutadjustmentcoststhemodelcannotmatchsatisfactorilytheobserved(cid:3)atness µ µ 0 1 in the path of the survivor component. For any choice of σ and σ , it will always be the µ µ 0 1 case that the survivor component will exhibit a substantially increasing path in the absence of adjustmentcosts. Notealsothattheratioσ =σ affectsboththeexitrateandtheevolutionof µ µ 0 1 the(cid:2)rmsizedispersion. Ifthisratiobecomestoosmall,post-entryexitratesbecomeexcessively high and the size dispersion increases too fast. This is the reason why in the NAC model we cannot (cid:2)nd a value for this ratio that attains the long-run contribution of survivors found in the data, and simultaneously matches the behavior of the cumulative exit rate and the evolution of the size dispersion. Therefore, the value we select for this ratio is disciplined by the exit rates andtheevolutionof(cid:2)rmsizedispersioninthecohort. Third, to show that proportional adjustment costs are crucial to (cid:2)t the evidence on the contributionofsurvivorstogrowthinthecohort’saveragesize,in(cid:2)gure4weperformasensitivity analysiswithrespecttoeachparameterinthemodel. WetakeasbenchmarktheestimatedNAC modelfortheoveralleconomycohortintable4. Wevaryeachparameterarounditsbenchmark value and plot the implied cumulative exit rate and survivor component at four different ages of the cohort (ages 1, 2, 5, and 9). We present both the exit rate and the survivor component to show that the essential shortcoming of the NAC model is the inability to increase the survivor componentwithoutanunreasonableincreaseinexitrates. From (cid:2)gure 4, we see that the model with costless adjustment cannot match satisfactorily thecontributionofsurvivorstogrowth,evenifweallowparameters(exceptfortheproportional cost) to vary one by one from their benchmark values. In fact, no other parameter besides the proportionalcost(inthelower-rightplot)canincreasethesurvivorcomponentwithoutchanging 39In(cid:2)gure2,werescaletheestimatedinitialvaluesofSD[l S ]tothelevelfoundinthedata. τ τ j 26

much the exit rate. In addition, adjustment costs imply more than just a mere level effect on the survivor component, as the increase in the survivor component at age 1 is larger than the increase at age 9, shrinking the distance between the survivor component at different ages of the cohort. In summary, the main effect of these costs is to put more emphasis on individual (cid:2)rmgrowthintheinitialyearsoflife,whenexitofinef(cid:2)cient(cid:2)rmsisveryintense.40 To emphasize the role of the proportional adjustment cost in replicating the evidence on the contribution of survivors, in (cid:2)gure 5 we present the impact of changes in P on the survivor component,usingasbenchmarktheestimatesfortheAC modelintheoveralleconomycohort. WeconcludethatallowingforevenasmallvalueofPhasasubstantialimpactonthesurvivors’ contribution,withalargereffectintheinitialyearsafterentry. 6 Conclusion In this paper, we show that a model with linear adjustment costs and learning about ef(cid:2)ciency generatesincentivesfor(cid:2)rmstoentersmallerand,ifsuccessful,expandfasterafterentry. Fora cohortofentrant(cid:2)rmsinthePortugueseeconomy,wepresentevidenceshowingthatgrowthin thecohorts’averagesizeisdrivenlargelybygrowthofsurvivorsratherthanbypruningofsmall inef(cid:2)cient (cid:2)rms, with rapid growth of survivors in the initial post-entry years and signi(cid:2)cant cross-sector differences in the contribution of survivors. A calibration and estimation of the modelrevealsthattheproportionalcostisthekeyparametertoexplainthehighcontributionof survivors to growth in the cohorts’ average size. Furthermore, due to a higher contribution of survivors,adjustmentcostsneedtobesubstantiallyhigherinmanufacturingthaninservices. The empirical success of our model in better approximating the growth of survivors as the main source for growth suggests that adjustment costs do play a signi(cid:2)cant role in post-entry (cid:2)rm size adjustments. Our results suggest that selection theories are more relevant to explain (cid:2)rm exit than growth of survivors. Our speci(cid:2)cation of adjustment costs assumes that they are proportional to the adjustment size and apply equally at entry, exit, and during regular job creation and destruction. These costs could capture aspects such as costs to the organization, layout, and optimization of the production process, and hiring and (cid:2)ring costs. Although we havenotcollectedevidencedocumentingthenatureofthesecosts,wewouldexpectthemtobe larger in sectors employing more complex technologies, such as in manufacturing industries. Potentially, part of the adjustment costs we estimate could also re(cid:3)ect the impact of (cid:2)nancial 40Notethatwewouldnotbeabletoidentifythehiring/entrycost,PH,andthe(cid:2)ring/exitcost,PF,separately becausePH andPF producealmostidenticalresults. Thisshouldbeexpectedastheincentivescreatedbyproportionalhiring/entryand(cid:2)ring/exitcostsdifferonlyinthedisplacementoftimebyoneperiod. 27

frictions, although we would expect these to be concentrated in entry costs associated with the acquisitionofcapital,andnotsomuchinregularjobcreationanddestructioncosts. More generally, (cid:2)nancing constraints theories should also play a role in explaining growth of survivors, besides what can possibly be captured by adjustment costs in our model, although there is not much evidence that (cid:2)nancing constraints can explain cross-sector differences. Notwithstanding this, Angelini and Generale’s (2008) conclusion that (cid:2)nancing constraints are not the main determinant behind the evolution of the (cid:2)rm size distribution suggests that any government intervention to eliminate (cid:2)nancing constraints might not change the lifetime dynamics of (cid:2)rm size we (cid:2)nd in this paper. In addition, this paper suggests that, in sectors where adjustment costs are high and learning is important, government policies aimed at curbing (cid:2)nancing constraints might not produce the intended results, as (cid:2)rms under those circumstances have incentives to start smaller and, if successful, expand faster, even if (cid:2)nancing constraintsareeliminated. A Appendix: Proofs Lemma2 W µ ; η canbesummarizedby(θ ;τ),andthedistributionfunctionF θ θ ;τ)isa τ (cid:17) 0 f τgτ (cid:21) 0 (cid:3)τ (cid:3)τ+1j (cid:3)τ continuousandstrictlydecreasingfunctionofθ . (cid:8) (cid:9) (cid:3)τ (cid:0) Proof. From(2)wehave ¥ θ =g(Y ;τ)=E(ξ(η ) Y ;τ)=ν + [1 F (η Y ;τ)]dξ(η ), (cid:3)τ τ τ j τ 1 ¥ (cid:0) η τ j τ τ Z(cid:0) whereF ( Y ;τ)istheposteriordistributionofη . BecauseF (η Y ;τ)iscontinuousandstrictlydecreasing η (cid:1)j τ τ η τ j τ inY ,andξ(η )isstrictlyincreasinginη ,weconcludethatg(Y ;τ)iscontinuousandstrictlyincreasinginY τ τ τ τ τ (seetheorem3.4.1inSwartz1994).Therefore,forthepurposeofpredictingθ τ ,W τ (cid:17) µ 0 ; f η s g τ s=(cid:0)0 1 (cid:17)f Y τ ;τ g(cid:17) θ ;τ ,sinceY =g 1(θ ;τ),whereg 1istheinversefunctionofgwithrespecttoYn. Usingtheroecursion f (cid:3)τ g τ Y(cid:0) (cid:3)τ Y(cid:0) τ σ 2 Z 1 Y = (cid:0) η + τ(cid:0) Y , τ+1 Z 1 τ Z 1 τ τ(cid:0)+1 τ(cid:0)+1 theconditionaldistributionofθ canberepresentedas (cid:3)τ+1 F θ θ ;τ =F Z τ(cid:0)+ 1 1g 1 θ ;τ+1 Z τ(cid:0) 1 g 1(θ ;τ) g 1(θ ;τ);τ , (cid:3)τ+1 j (cid:3)τ η " σ (cid:0) 2 Y(cid:0) (cid:3)τ+1 (cid:0)σ (cid:0) 2 Y(cid:0) (cid:3)τ j Y(cid:0) (cid:3)τ # (cid:0) (cid:1) (cid:0) (cid:1) sinceweneedtointegratethedensityofη overthedomainwhereg(Y ;τ+1) θ .Fromthis,weconclude τ τ+1 (cid:20) (cid:3)τ+1 that F θ θ ;τ is a continuous and strictly decreasing function of θ . Therefore, the transition function (cid:3)τ+1j (cid:3)τ (cid:3)τ associatedwithF θ θ ;τ ismonotoneandsatis(cid:2)estheFellerproperty(seepp. 376-9inStokeyetal.1989). (cid:0) (cid:1)(cid:3)τ+1j (cid:3)τ (cid:0) (cid:1) 28

Proof of proposition 4. We use the following notation: (i) X R+ Q N0 and x (L;θ;τ) X, where (cid:17) (cid:2) (cid:2) (cid:17) 2 Q [ν 1 ;ν 2 ] R+ , ν 1 0, ν 2 <¥ ; (ii) T is the operator associated with (4); (iii) M denotes the following (cid:17) (cid:26) (cid:21) operator MVS (L ;θ ;τ)= ν2 max VEX(L );VS L ;θ ;τ+1 dF θ θ ;τ ; τ (cid:3)τ Zν1 τ τ (cid:3)τ+1 (cid:3)τ+1 j (cid:3)τ (cid:0) (cid:1) (cid:8) (cid:0) (cid:1)(cid:9) (cid:0) (cid:1) (iv)VS,VSD,andVSU denotetheobjectivefunctionsassociatedwithVS,VSD,andVSU;thatis,for j=S;SD;SU O O O Vj(L ;L ;θ ;τ)=P (L ;θ ) Cj(L ;L )+β MVS (L ;θ ;τ). O τ τ (cid:0) 1 (cid:3)τ τ (cid:3)τ (cid:0) τ τ (cid:0) 1 τ (cid:3)τ (cid:0) (cid:1) Weprovethepropositioninseveralsteps. (a.i)ExistenceandUniqueness: ThisfollowsfromtheContractionMappingTheoremandBlackwell’ssuf(cid:2)cientconditions(seetheorems3.2and3.3inStokeyetal.1989). (a.ii)Continuityin(L ;θ ): LetC (X)bethespaceofboundedfunctionsonX whicharecontinuousin τ 1 (cid:3)τ 12 (cid:0) (L τ (cid:0) 1 ;θ (cid:3)τ ). ThisisclearlyaclosedsubsetofB(X),thespaceofboundedfunctionsVS:X ! R. SinceB(X)with thesupnorm VS =sup VS(x) isaBanachspace,thenC (X)isalsoaBanachspace. NowconsiderVS x 2 X 12 2 C (X). Becausemax VEX;VS isalsocontinuousandF θ θ ;τ satis(cid:2)estheFellerproperty(seelemma 2) 1 , 2 then MVS (cid:13) (cid:13) is c (cid:13) (cid:13) ontinuous in (cid:12) (cid:12) (L ;θ (cid:12) (cid:12) ) (see lemma 9.5 in Sto (cid:3)τ k (cid:0) e 1 y j et (cid:3)τ al. 1989). Since P (L ;θ ) CS(L ;L ) (cid:8) (cid:9)τ (cid:3)τ (cid:0) (cid:1) τ (cid:3)τ (cid:0) τ τ (cid:0) 1 is continuous, then VS(L ;L ;θ ;τ) is continuous in (L ;L ;θ ). Therefore, applying the maximum the- O τ τ 1 (cid:3)τ τ τ 1 (cid:3)τ (cid:0) (cid:0) orem, we conclude thatVS(L ;θ ;τ) is continuous in (L ;θ ). Note that the set of admissible values for τ 1 (cid:3)τ τ 1 (cid:3)τ (cid:0) (cid:0) employment can be made compact. First, only non-negative values are acceptable for employment. Second, we canchooseavalueforL highenough,sayLUB,suchthatLSU (L ;θ ;τ) LUB,forallL LUB,sothatall τ τ (cid:3) τ (cid:0) 1 (cid:3)τ (cid:20) τ (cid:0) 1 (cid:20) valuesofinterestareconsidered. LUBis(cid:2)nitesinceF (¥ )=0,andMVS isbounded. Therefore,VS asde(cid:2)nedby 0 (4)iscontinuousin(L ;θ ). τ 1 (cid:3)τ (cid:0) (a.iii) Strict Monotonicity in θ : From lemma 2 (the transition function associated with F θ θ ;τ is (cid:3)τ (cid:3)τ+1j (cid:3)τ monotone) if VS L ;θ ;τ+1 is weakly increasing in θ , then MVS (L ;θ ;τ) is also weakly increas- τ (cid:3)τ+1 (cid:3)τ+1 τ (cid:3)τ (cid:0) (cid:1) ing in θ . Then, because P (L ;θ ) is strictly increasing in θ (and the constraint set is not affected by θ ), (cid:3)τ (cid:0) τ(cid:1) (cid:3)τ (cid:3)τ (cid:0) (cid:1) (cid:3)τ VS(L ;θ ;τ)isstrictlyincreasinginθ (seetheorem9.11inStokeyetal.1989). τ 1 (cid:3)τ (cid:3)τ (cid:0) (b)ExitPolicy: Theexitpolicyisdeterminedbythecondition VEX(L ) VS(L ;θ ;τ). τ (cid:0) 1 (cid:17) τ (cid:0) 1 (cid:3)τ Because, foreachL ,VEX isconstantandVS isstrictlyincreasinginθ , thenitisobviousthatθEX(L ;τ) τ 1 (cid:3)τ τ 1 (cid:0) (cid:0) is a unique function de(cid:2)ned by the value of θ [ν ;ν ] that satis(cid:2)es the above equation, if it exists, or by ν , (cid:3) 1 2 1 2 whenVEX(L)<VS(L;ν ;τ),orbyν ,whenVEX(L)>VS(L;ν ;τ). BecausebothVEX andVS arecontinuous 1 2 2 functions,thenθEX isalsoacontinuousfunctioninL. Proposition5 LetTfl bethemaximumallowedage,sothata(cid:2)rmenteringinperiod0mustexittheindustryatthe endofperiodTfl. ThenPr θ (cid:3)τ+12 Q D Tfl (L τ ;τ+1) j θ (cid:3)τ ;τ =1,forallL τ 2 R+ ,τ 2f 0;:::;Tfl (cid:0) 1 g where (cid:0) (cid:1) Q D(L ;τ+1)= θ Q :VS L ;θ ;τ+1 isdifferentiableatL . Tfl τ (cid:3)τ+1 2 Tfl τ (cid:3)τ+1 τ (cid:8) (cid:0) (cid:1) (cid:9) Consequently, the objective functions associated withVSD andVSU are continuously differentiable in L, and all Tfl Tfl optimaareinteriorintheregionoftheirde(cid:2)nition.41 41Asimilarresultwouldholdforthecaseofin(cid:2)nite-lived(cid:2)rmsthatfacea(cid:2)nitelearninghorizon,asinsections 29

Proofofproposition5. Weprovethisbyinduction. InperiodTfl,wehave V Tfl S L Tfl (cid:0) 1 ;θ (cid:3)Tfl ;Tfl =m L a Tfl x P L Tfl ;θ (cid:3)Tfl (cid:0) CS L Tfl ;L Tfl (cid:0) 1 +βVEX(L Tfl ) , (cid:0) (cid:1) (cid:8) (cid:0) (cid:1) (cid:0) (cid:1) (cid:9) so that VSD, VSN and VSU are continuously differentiable functions of L , L , and L , respectively. Since Tfl;O Tfl Tfl;O Tfl Tfl 1 Tfl VSD (L;L;θ ;Tfl)=VSN(L;θ ;Tfl),VSU (L;L;θ ;Tfl)=VSN(L;θ ;Tfl),F (0+)=¥ (cid:0) ,F (¥ )=0,andVEX isbounded Tfl;O (cid:3) Tfl (cid:3) Tfl;O (cid:3) Tfl (cid:3) 0 0 above,thenVSD andVSU haveinterioroptimaintheregionsofde(cid:2)nitionofVSD andVSU. Therefore,thoseop- Tfl;O Tfl;O Tfl Tfl timaareindependentofL Tfl (cid:0) 1 , andwemusthave∂V Tfl SD=∂L Tfl (cid:0) 1 = (cid:0) P, ∂V Tfl SU=∂L Tfl (cid:0) 1 =P, intheregionsoftheir de(cid:2)nition,and ∂VSN ∂L T T fl fl 1 =F 0 L Tfl (cid:0) 1 θ (cid:3)Tfl (cid:0) w (cid:0) βP. (cid:0) (cid:0) (cid:1) F We co θ nclu ; d T e fl th 1 at V an Tfl S d L fo Tfl r (cid:0) a 1 l ; l θ θ (cid:3)Tfl ;Tfl is Q ) c . ontinuously differentiable at L Tfl (cid:0) 1 2 R+ , with probability one (given (cid:1)j (cid:3)Tfl 1 (cid:0) (cid:0) (cid:3)Tfl 1(cid:1)2 Now (cid:0) consider a generic peri (cid:0) od τ 1;:::;Tfl 1 , and assume thatVS L ;θ ;τ+1 is continuously dif- (cid:0) (cid:1) 2f (cid:0) g Tfl τ (cid:3)τ+1 ferentiableatL τ 2 R+ withprobabilityone. BecauseθEX(L τ ;τ+1)isauni (cid:0) quecontinuous (cid:1) functionofL,wecan applythedominatedconvergencetheoremtoconcludethat MVS (L ;θ ;τ)iscontinuouslydifferentiableatL , Tfl τ (cid:3)τ τ forallθ Q (seetheorems3.2.16and3.4.3inSwartz1994). Consequently,thesameargumentusedforperiod (cid:3)τ 2 (cid:0) (cid:1) Tfl canberepeatedhere. Proof of proposition 6. For given (L ;τ) we partition the state-space associated with θ , Q , into regions of τ 1 (cid:3)τ (cid:0) exit,Q EX,downwardadjustment,Q SD,non-adjustment,Q SN,andupwardadjustment,Q SU:42 Q EX(L ;τ)= θ :VEX >VS , τ 1 (cid:0) (cid:8) (cid:9) Q SD(L ;τ)= θ Q :VSD>VSN,VSD VSU,VSD VEX , τ 1 (cid:0) 2 (cid:21) (cid:21) Q SN(L ;τ)=(cid:8) θ Q :VSN VSD,VSN VSU,VSN VEX(cid:9), τ 1 (cid:0) 2 (cid:21) (cid:21) (cid:21) Q SU(L ;τ)=(cid:8) θ Q :VSU >VSN,VSU VSD,VSU VEX(cid:9). τ 1 (cid:0) 2 (cid:21) (cid:21) Ifitisoptimalforthe(cid:2)rmtoadjustu(cid:8)pwards,thenwemustsolve (cid:9) ∂ MVS (L ;θ ;τ) A = F (L )θ (w+P) +β τ(cid:3) (cid:3)τ =0, SU 0 τ(cid:3) (cid:3)τ(cid:0) ∂L (cid:0) (cid:1) (cid:2) (cid:3) andifitisoptimalforthe(cid:2)rmtoadjustdownwards,wemustsolve ∂ MVS (L ;θ ;τ) A = F (L )θ (w P) +β τ(cid:3) (cid:3)τ =0 SD 0 τ(cid:3) (cid:3)τ(cid:0) (cid:0) ∂L (cid:0) (cid:1) (cid:2) (cid:3) 4and5. However,inthiscasewewouldneedtouseproposition10(cid:2)rst. 42InQ SDandQ SU weneedtouseVSD>VSU andVSU >VSDbecauseVS,ingeneral,isnotconcaveinL. 30

Now,thederivativecanberewrittenas ∂ MVS ∂ ( L L τ ;θ (cid:3)τ ;τ) = ∂V ∂ E L X( (cid:1) ) dF θ (cid:3)τ+1 j θ (cid:3)τ ;τ + ∂V ∂ S L D( (cid:1) ) dF θ (cid:3)τ+1 j θ (cid:3)τ ;τ (cid:0) (cid:1) Q Z EX τ Q Z SD τ (cid:0) (cid:1) (cid:0) (cid:1) ∂VSN() ∂VSU() + ∂L (cid:1) dF θ (cid:3)τ+1 j θ (cid:3)τ ;τ + ∂L (cid:1) dF θ (cid:3)τ+1 j θ (cid:3)τ ;τ , Q Z SN τ Q Z SU τ (cid:0) (cid:1) (cid:0) (cid:1) wheresomeoftheregionsmightbeempty,andinseparatingtheintegralswehavetakenintoaccountthecontinuity oftheintegrandinMVS atthefrontiers. Foreachoftheabovederivativeswehave ∂VEX(L ) τ = P, ∂L (cid:0) ∂V ∂ SD L ( (cid:1) ) (cid:12) (cid:12) θ(cid:3)τ+12 Q SD(Lτ;τ+1) = (cid:0) P= (cid:2) F 0 (cid:0) L τ(cid:3)+1 (cid:1) θ (cid:3)τ+1 (cid:0) w (cid:3) +β ∂ (cid:0) MVS (cid:1)(cid:0) L τ(cid:3)+ ∂ 1 L ;θ (cid:3)τ+1 ;τ+1 (cid:1) , (cid:12) (cid:12) ∂V ∂ S L N( (cid:1) ) = F 0 (L τ )θ (cid:3)τ+1 (cid:0) w +β ∂ MVS L ∂ τ ; L θ (cid:3)τ+1 ;τ+1 , τ (cid:0) (cid:1)(cid:0) (cid:1) (cid:2) (cid:3) ∂V ∂ SU L ( (cid:1) ) (cid:12) (cid:12) θ(cid:3)τ+12 Q SU(Lτ;τ+1) =P= (cid:2) F 0 (cid:0) L τ(cid:3)+1 (cid:1) θ (cid:3)τ+1 (cid:0) w (cid:3) +β ∂ (cid:0) MVS (cid:1)(cid:0) L τ(cid:3)+ ∂ 1 L ;θ (cid:3)τ+1 ;τ+1 (cid:1) , wherewehaveusedth(cid:12)efactthatA =0,whenitisoptimaltoadjustupwards,andA =0,whenitisoptimalto (cid:12) SU SD adjustdownwards. Therefore,wehave ∂ MVS (L ;θ ;τ) τ(cid:3) (cid:3)τ =E χ ( P)+ 1 χ F L θ w + (cid:0) (cid:1) ∂L τ (cid:3)τ+1 (cid:0) (cid:0) (cid:3)τ+1 ( 0 τ(cid:3)+1 (cid:3)τ+1 (cid:0) (cid:0) (cid:1) (cid:2) (cid:0) (cid:1) (cid:3) ∂ MVS L ;θ ;τ+1 β τ(cid:3)+1 (cid:3)τ+1 , ∂L (cid:0) (cid:1)(cid:0) (cid:1))! Usingthelawofiteratedexpectations,wecanrewritetheaboveas ∂ MVS ∂ ( L L τ(cid:3) ;θ (cid:3)τ ;τ) = (cid:229) ¥ E τ βs (cid:0) 1 χ(cid:152) (cid:3)τ+s ( (cid:0) P)+χ(cid:136) (cid:3)τ+s F 0 L τ(cid:3)+s θ (cid:3)τ+s (cid:0) w , (cid:0) (cid:1) s=1 (cid:8) (cid:2) (cid:0) (cid:1) (cid:3)(cid:9) TheresultnowfollowsbypluggingthisexpressioninA ,andA . SU SD Proofofcorollary7. WecanrewritetheLHSof(6)and(7)asfollows MB = F (L )θ w +βE χ(cid:152) ( P)+βE χ(cid:136) F L θ w + τ 0 τ(cid:3) (cid:3)τ(cid:0) τ (cid:3)τ+1 (cid:0) τ ( (cid:3)τ+1 0 τ(cid:3)+1 (cid:3)τ+1 (cid:0) (cid:0) (cid:1) (cid:0) (cid:0) (cid:1) (cid:1) ¥ (cid:229) E βs χ(cid:152) ( P)+χ(cid:136) F L θ w : s=1 τ+1 (cid:3)τ+1+s (cid:0) (cid:3)τ+1+s 0 τ(cid:3)+1+s (cid:3)τ+1+s (cid:0) ) (cid:2) (cid:0) (cid:0) (cid:1) (cid:1)(cid:3) Takingintoaccountthatχ(cid:136) χ(cid:152) =χ(cid:152) ,χ(cid:136) χ(cid:136) =χ(cid:136) ,χ(cid:136) =1 χ ,andχ(cid:152) =χ ,then (cid:3)τ+1 (cid:3)τ+1+s (cid:3)τ+1+s (cid:3)τ+1 (cid:3)τ+1+s (cid:3)τ+1+s (cid:3)τ+1 (cid:0) (cid:3)τ+1 (cid:3)τ+1 (cid:3)τ+1 wegetthestatedresult. 31

Proofofproposition8. Withproportionalcosts,optimalemploymentatentryisdeterminedby θSD F (L )θ w+PH +β PFdF (θ )+ 0 1 (cid:3)1 (cid:0) Zν1 (cid:0) θ(cid:3)1 (cid:3)2 (cid:0) (cid:1) Zθ θ SD SU δ Tfl F 0 (L 1 )θ (cid:3)2 (cid:0) w (cid:0) βTfl (cid:0) 1PF dF θ(cid:3)1 (θ (cid:3)2 )+ Zθ ν SU 2 PHdF θ(cid:3)1 (θ (cid:3)2 ) ! =0 n (cid:2) (cid:3) o (a)Inthecaseofaproportionalhiringcost,assumingPF =0,wehave L =F 1 w =F 1 w+P δ H Tfl , 1 0(cid:0) θSD 0(cid:0) 0 θSU 1 (cid:18) (cid:19) ∂L F (L ) w@(cid:152) A 1(cid:3) = 0 1(cid:3) H , ∂PH F 00 L 1(cid:3) ww(cid:152) w +PHw(cid:152) H w(cid:152) w =1+βδ Tfl F θ(cid:3)1 θSU (cid:0) F θ(cid:3)1 (cid:0)θS(cid:1)D , w(cid:152) H =1 (cid:0) β 1 (cid:0) F θ(cid:3)1 θSU . h (cid:0) (cid:1) (cid:0) (cid:1)i h (cid:0) (cid:1)i Aftersomealgebraweget ∂g 1 w(cid:152) = H F θSD ∂PH (cid:0)F el L 1(cid:3) ww(cid:152) w +PHw(cid:152) H θ(cid:3)1 (cid:0) (cid:0) (cid:1) (cid:0) (cid:1) ν2 1 w(cid:152) H 1 1 dF (θ ), ZθSU( F el L 1(cid:3) ww(cid:152) w +PHw(cid:152) H (cid:0)F el L 2(cid:3) SU wδ Tfl +PH ) θ(cid:3)1 (cid:3)2 (cid:0) (cid:1) (cid:0) (cid:1) whereF (L)=F (L)L=F (L)standsfortheelasticityofthemarginalproductoflabor. IfF(L)=ALα,wehave el 00 0 F =(α 1),andtheaboveexpressionsimpli(cid:2)esto el (cid:0) ∂g w(cid:152) ∂PH =(1 (cid:0) α)(cid:0) 1 ww(cid:152) w + H PHw(cid:152) H F θ(cid:3)1 θSD + 1 (cid:0) F θ(cid:3)1 θSU (cid:0) (cid:18) n (cid:0) (cid:1) h (cid:0) (cid:1)io 1 1 F θSU , wδ(Tfl)+PH (cid:0) θ(cid:3)1 h (cid:0) (cid:1)i(cid:19) whichispositivewhenTfl ishighenoughsothat w δ Tfl F θ(cid:3)1 θSD (cid:0) βTfl (cid:0) 1 1 (cid:0) F θ(cid:3)1 θSU +PH 1 (cid:0) β 1 (cid:0) F θ(cid:3)1 θSU F θ(cid:3)1 θSD >0 n (cid:0) (cid:1) h (cid:0) (cid:1)io n h (cid:0) (cid:1)io (cid:0) (cid:1) (b)Inthecaseofaproportional(cid:2)ringcost,assumingPH =0,wegetsimilarly ∂g θSD 1 βTfl (cid:0) 1 1 1 βw(cid:152) F ∂PF = Zν1 8 < F el L 2(cid:3) SD wδ Tfl + βTfl (cid:0) (cid:0) 1 (cid:0) 1 PF (cid:0)F el L 1(cid:3) ww(cid:152) w +βPFw(cid:152) F 9 = dF θ(cid:3)1 (θ (cid:3)2 )+ : (cid:0) (cid:1) (cid:16) ν2 (cid:17) 1 (cid:0) β (cid:1)Tfl (cid:0) 1 1; βw(cid:152) F dF (θ ), ZθSU(F el L 2(cid:3) SU wδ Tfl +βTfl (cid:0) 1PF (cid:0)F el L 1(cid:3) ww(cid:152) w +βPFw(cid:152) F) θ(cid:3)1 (cid:3)2 (cid:0) (cid:1) (cid:0) (cid:1) 32

where w(cid:152) w =1+βδ Tfl F θ(cid:3)1 θSU (cid:0) F θ(cid:3)1 θSD w(cid:152) F =F θ(cid:3)1 θSD +β h Tfl (cid:0) 1(cid:0)F θ(cid:3)1 (cid:1)θSU (cid:0) (cid:0)F θ(cid:3)1 (cid:1)θ i SD (cid:0) (cid:1) h (cid:0) (cid:1) (cid:0) (cid:1)i Undertheassumptionthatmarginalproductivityisalwayspositive,weneedPF< w orotherwisethe(cid:2)rmwould 1 β prefertopaytheworkerhislifetimesalary,insteadof(cid:2)ringhim. IfF(L)=ALα,w (cid:0) ehaveF 0 =(LF 00 )=(α 1)(cid:0) 1, (cid:0) andtheaboveexpressionsimpli(cid:2)esto ∂g βw(cid:152) ∂PH =(1 (cid:0) α)(cid:0) 1 0ww(cid:152) w +β F PFw(cid:152) F F θ(cid:3)1 θSD + 1 (cid:0) F θ(cid:3)1 θSU (cid:0) n (cid:0) (cid:1) h (cid:0) (cid:1)io @ βTfl (cid:0) 1 (cid:0) 1 F θSD βTfl (cid:0) 1 1 F θSU wδ Tfl + βTfl (cid:0) 1 (cid:0) 1 PF θ(cid:3)1 (cid:0) (cid:1) (cid:0)wδ Tfl +βTfl (cid:0) 1PF h (cid:0) θ(cid:3)1 (cid:0) (cid:1)i 1 (cid:16) (cid:17) A whichispositiveforallTfl. Proofofproposition9. Theresultconcerningtheposteriordistributionofθ followsdirectlyfrom τ+j ln(θ τ+j ) j W τ =µ j W τ +ε τ+j ,µ j W τ(cid:24) N(Y τ ;Z τ ). Forthedistributionofθ conditionalon(θ ;τ),weusethefactthat (cid:3)τ+j (cid:3)τ 1 ln θ (cid:3)τ+j j W τ =Y τ+j j W τ + 2 Z τ+j +σ2 (cid:0) Y = (cid:1) σ 2Z τ+ (cid:229) j (cid:0) 1 η + (cid:0) Z τ+j Y , (cid:1) τ+j (cid:0) τ+j s Z τ s=τ τ Z =Z σ 2Z Z j, τ+j τ (cid:0) τ+j τ (cid:0) η s j W τ(cid:24) N Y τ ;Z τ +σ2 ,Cov(η s ;η s 0 j W τ+τ )=Var(µ j W τ )=Z τ ,s,s 0 (cid:21) τ,s 6 =s 0 (cid:0) (cid:1) sothat,intheend,weget 1 E ln θ W =Y + Z +σ2 , (cid:3)τ+j j τ τ 2 τ+j (cid:2) V(cid:0)ar ln(cid:1)θ (cid:3) W =Z(cid:0) Z . (cid:1) (cid:3)τ+j j τ τ (cid:0) τ+j (cid:2) (cid:0) (cid:1) (cid:3) Fromheretheresultfollowsbynotingthatln(θ )=Y +1 Z +σ2 . (cid:3)τ τ 2 τ Fortheunconditionaldistribution,justnotethatln(θ )isasumofnormalrandomvariables,andthat (cid:3)τ (cid:0) (cid:1) 1 E[ln(θ )]=µfl + Z +σ2 (cid:3)τ 2 τ Var[ln(θ )]=σ2 +(cid:0)(Z Z(cid:1)) (cid:3)τ µ0 0 (cid:0) τ Proofofproposition10. 33

After period T 1 the optimization problem is time invariant, since there is no uncertainty concerning E(θ). (cid:0) Therefore,forperiodss,s T,wehave (cid:21) VS(θ ;L ;T)= max P (L ;θ ) CS(L ;L ) + (cid:3)T s 1 s (cid:3)T s s 1 (cid:0) Ls (cid:21) 0;χs2f 0;1 g (cid:0) (cid:0) (cid:8)(cid:2) (cid:3) β χ W CEX(L ) +(1 χ )VS(θ ;L ;T) . s (cid:0) τ+s (cid:0) s (cid:3)T s (cid:8) (cid:2) (cid:3) (cid:9)(cid:9) Considera(cid:2)rmthatisintheindustryattimes, s T. Wenowprovethatthis(cid:2)rmwillnotchangeitsemployment (cid:21) levelinperiods+1. Forthis,weusetheeasilyprovenfactthatitislesscostlytoadjustinonestepthanintwo steps,i. e., CS(L ;L )+CS(L ;L ) CS(L ;L ), s+1 s(cid:3) s(cid:3) s 1 s+1 s 1 (cid:0) (cid:21) (cid:0) whereL =L (θ ;L ;T). Wethenhave s(cid:3) s (cid:3)T s 1 (cid:0) P (L ;θ ) CS(L ;L )+βmax VEX(L );VS(θ ;L ) s+1 (cid:3)T s+1 s(cid:3) s+1 (cid:3)T s+1 (cid:0) P (L s+1 ;θ (cid:3)T ) CS(L s+1 ;L s 1 )+βma(cid:8)x VEX(L s+1 );VS(θ (cid:3)T ;L s+(cid:9)1 ) +CS(L s(cid:3) ;L s 1 ) (cid:20) (cid:0) (cid:0) (cid:0) VS(θ (cid:3)T ;L s 1 )+CS(L s(cid:3) ;L s 1 ) (cid:8) (cid:9) (cid:20) (cid:0) (cid:0) = P (L ;θ ) CS(L ;L )+βmax VEX(L );VS(θ ;L ) +CS(L ;L ) s(cid:3) (cid:3)T s(cid:3) s 1 s(cid:3) (cid:3)T s(cid:3) s(cid:3) s 1 (cid:0) (cid:0) (cid:0) = VSN(θ (cid:3)T ;L s(cid:3) ). (cid:8) (cid:9) Therefore,attimes+1itisoptimaltosetL =L . s(cid:3)+1 s(cid:3) Wenowprovethatthe(cid:2)rmdoesnotexitattimes+1afterremainingintheindustryattimes,s T. Because (cid:21) the (cid:2)rm stays at time s, thenVS θ ;L ;T VEX L . Now assume that in period s+1 the (cid:2)rm exits, so (cid:3)T s(cid:3) (cid:0) 1 (cid:21) s(cid:3) (cid:0) 1 that (cid:0) (cid:1) (cid:0) (cid:1) VS(θ ;L ;T)<VEX(L ) P (L ;θ )<(1 β)VEX(L ). (cid:3)T s(cid:3) s(cid:3) s(cid:3) (cid:3)T s(cid:3) , (cid:0) Thisthenimplies VS θ ;L ;T < (1 β)VEX(L ) CS L ;L +βVEX(L ) (cid:3)T s(cid:3) 1 s(cid:3) s(cid:3) s(cid:3) 1 s(cid:3) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:1) = VEX(L s(cid:3) ) CS L s(cid:3) ;L s(cid:3) (cid:0)1 VEX (cid:1) L s(cid:3) 1 (cid:0) (cid:0) (cid:20) (cid:0) (cid:0) (cid:1) (cid:0) (cid:1) whichisacontradiction B Appendix: Simulation and Estimation Algorithm (i) Discretizationandtransitionprobabilitymatricesassociatedwithθ : (cid:3) Wediscretizeθ withauniformdiscreteapproximation(with25masspoints)tothedistributionlogN µ+1 Z +σ2 ;σ2 Z . (cid:3)τ 2 τ µ(cid:0) τ We then use Tauchen’s (1986) method to build the transition matrices, computing integrals via Gauss- (cid:0) (cid:0) (cid:1) (cid:1) Legendrequadrature. (ii) DiscretizationofL 34

Basedonthedecisionrulesforproblem(11)weconsider l ln(L) N µ ;σ2 , (cid:17) (cid:24) L L 1 1 (cid:0) (cid:1)αp µ = µfl + σ2+ln , L 1 α ( 2 [w+PSU+βPEX][w (1 β)PSD]!) (cid:0) (cid:0) (cid:0) 1 σ2p= σ2. L (1 α)2 µ (cid:0) For µ we assume that atthe upper end of the grid the (cid:2)rm decreases employment, and at the lower end L ofthegridthe(cid:2)rmincreasesemploymentandexitsnextperiod. WethendiscretizeLsimilarlytoθ (with (cid:3) 800masspoints)usingauniquegridforallperiods. (iii) ChoiceforT WechooseT =15,anddisplayresultsuntilperiod10. (iv) Modelsimulation Foragivensetofparameterswenumericallycomputetheoptimalentry,employment,andexitpolicyrules. First,foreachθ gridpoint,wecomputeoptimalemploymentin (cid:3) V(cid:152)SU(θ ) = max P (L;θ ) PL +βE max VEX(L );VS L ;θ , (cid:3)τ Lτ t (cid:3)τ (cid:0) τ τ τ τ (cid:3)τ+1 V(cid:152)SD(θ ) = max(cid:8)P (L;θ )+PL +βE max(cid:8)VEX(L );VS(cid:0)L ;θ (cid:1)(cid:9)(cid:9), (cid:3)τ t (cid:3)τ τ τ τ τ (cid:3)τ+1 Lτ (cid:8) (cid:8) (cid:0) (cid:1)(cid:9)(cid:9) as these do not depend on L . We (cid:2)rst (cid:2)nd the maximizer on the grid for L and then use a golden τ 1 (cid:0) sectionmethodtoobtainamoreprecisemaximizer.43 Second,wegetVSU andVSD,determinetheinaction regions, and getVS. Third, we compute all endogenous decisions associated with each realization of the N =150;000randomlifetimehistoriesof θ . Fourth,wecomputethe(simulated)moments. s f (cid:3)τg (v) Momentsusedinestimation LetT(cid:152) = 1;2;3;4;5;6;7;8;9;10 . Weconsiderfoursetsofmoments: f g (a)Exitrate: Forτ T(cid:152), 2 f =1 χ =1 Pr(Dτ S ) aτi τ;i (cid:0) j 0 (b)Averagecurrentsizeconditionalonsurviv(cid:0)al: Forτ (cid:1) 0 T(cid:152), 2f g[ f =l 1 χ =0 E[l S ]Pr(S S ) bτi τ;i τ;i (cid:0) τ j τ τ j 0 (cid:0) (cid:1) (c)Relativechangeinvarianceofcurrentsizeconditionalonsurvival: Forτ T(cid:152),44 2 f = l E[l S ] 21 χ =0 l2 E[l S ]2 cτi f τ;i (cid:0) τ j τ g τ;i (cid:0) 0;i (cid:0) 0 j 0 (cid:2) (cid:0) (cid:1) n E[l2 S ] E[l o S ]2 Pr(S S )= E[l2 S ] E[l S ]2 τ j τ (cid:0) τ j τ t j 0 0 j 0 (cid:0) 0 j 0 n o n o 43SeePressetal.(2007). 44Thismomentconditioncanbeexpressedintermsoftheratioofthetime-τ andtime-0variances. 35

(d)Averageentrysizeconditionalonsurvival: Forτ T(cid:152),45 2 f =l 1 χ =0 E[l S ]Pr(S S ) dτi 0;i τ;i (cid:0) 0 j τ τ j 0 (cid:0) (cid:1) (vi) Weightingmatrix Theweightingmatrixisestimatedasthesamplecovariancematrixofthemomentsin(v),adjustedforthe simulationsize 1 S = 1+ Var(f ), f =[f f f f ] N s =N (cid:1)(cid:1) i (cid:1)(cid:1) i a0 (cid:1) i b0 (cid:1) i c0 (cid:1) i d0 (cid:1) i 0 (cid:18) (cid:19) (vii) Estimationmethod Weuseasimulatedannealingmethodtosearchforthesetofparametervaluesb= µfl;σ µ0 ;σ µ1 ; σ;W;P)0 thatminimizesthemethodofmomentsobjectivefunction,46 (cid:0) Q= 1 (cid:229) N f 0 1 S (cid:0) 1 1 (cid:229) N f N i=1 (cid:1)(cid:1) i N N i=1 (cid:1)(cid:1) i (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (viii) Standarderrors Thestandarderrorsoftheestimatedparametersareobtainedasfollows 1 std(b(cid:136))= ∂ N (cid:0) 1(cid:229) N i=1 f (cid:1)(cid:1) i b(cid:136) 0 1 S (cid:0) 1 ∂ N (cid:0) 1(cid:229) N i=1 f (cid:1)(cid:1) i b(cid:136) (cid:0) , ∂b N ∂b " (cid:0) 0 (cid:0) (cid:1)(cid:1)! (cid:18) (cid:19) (cid:0) 0 (cid:0) (cid:1)(cid:1)!# wherethematrixofderivativesiscomputednumerically. References Albuquerque, R., Hopenhayn, H.A., 2004. Optimal lending contracts and (cid:2)rm dynamics. ReviewofEconomicStudies71,285-315. Angelini, P., Generale, A., 2008. On the evolution of (cid:2)rm size distributions. American EconomicReview98,426-438. Aw, B.Y., Chen, X., Roberts, M.J., 2004. Firm-level evidence on productivity differentials, turnover, and exports in taiwanese manufacturing. Journal of Development Economics 66,51-86. Banco de Portugal, 1997. SØries Longas para a Economia Portuguesa, P(cid:243)s II Guerra Mundial, Vol.IandIISØriesEstat(cid:237)sticas.BancodePortugal,Lisboa. 45Thismomentconditiontogetherwithcondition(b)canbeexpressedintermsofthesurvivorcomponent. 46SeeGoffeetal.(1994). 36

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Hopenhayn, H.A., Rogerson, R., 1993. Job turnover and policy evaluation: A general equilibriumanalysis.(cid:148)JournalofPoliticalEconomy101,915-938. INE, 1990-5. Indice de Pre(cid:231)os no Consumidor, December 1990-96. Instituto Nacional de Estat(cid:237)stica,Lisboa. INE,1997.EmpresasemPortugal,1990-1995.InstitutoNacionaldeEstat(cid:237)stica,Lisboa. Jovanovic,B.,1982.Selectionandtheevolutionofindustry.Econometrica50,649-670. Ljungqvist, L., Sargent, T.J., 2004. Recursive Macroeconomic Theory, Second Edition. MIT Press,Cambridge,MA. Luttmer, E.G.J., 2007. Selection, growth, and the size distribution of (cid:2)rms. Quarterly Journal ofEconomics122,1103-1144. Mata, J., Portugal, P., 1994. Life duration of new (cid:2)rms. Journal of Industrial Economics 42, 227-245. Nickell,S.J.,1986.Dynamicmodelsoflabourdemand.In: Ashenfelter,O.C.,Layard,R.(Eds.), HandbookofLaborEconomics,vol.1.ElsevierScience,NewYork,pp.473-522. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P., 2007. Numerical Recipes: The ArtofScienti(cid:2)cComputing,ThirdEdition.CambridgeUniversityPress,Cambridge. Rossi-Hansberg, E., Wright, M.L.J., 2007. Establishment size dynamics in the aggregate economy.AmericanEconomicReview97,1639-1666. Swartz, C., 1994. Measure, Integration and Function Spaces. World Scienti(cid:2)c, Farrer Road, Singapore. Stokey, N.L., Lucas, R.E., Prescott, E.C., 1989. Recursive Methods in Economic Dynamics. HarvardUniversityPress,Cambridge,MA. Tauchen, G., 1986. Finite state markov-chain approximation to univariate and vector autoregressions.EconomicsLetters20,177-181. Zellner,A.,1971.AnIntroductiontoBayesianInferenceinEconometrics.JohnWiley&Sons, NewYork. 38

Table1: 1988FirmCohort: AllEconomy Year CumEx AvEmp CGrEmp SurComp 1988 1.11 1989 15.6 1.27 15.2 69.5 1990 24.4 1.36 24.8 70.4 1991 30.8 1.43 31.2 69.7 1992 35.4 1.46 34.2 69.3 1993 40.0 1.46 34.6 68.9 1994 43.4 1.47 35.3 69.1 1995 46.7 1.48 36.1 68.7 1996 49.9 1.49 37.4 67.2 1997 52.7 1.51 39.6 68.5 1998 55.5 1.52 40.7 68.3 1999 58.5 1.54 43.0 68.9 Notes: CumEx is the cumulative exit rate, 100 N(Dτ)=N(S ); 0 (cid:2) AvEmp is the mean of log-employment among survivors, N(S ) 1(cid:229) l ; CGrEmp is the cumulative log-growth rate (in τ (cid:0) i S τ τ %) of emp2loyment among survivors, 100 [N(S ) 1(cid:229) l τ (cid:0) i S τ N(S ) 1(cid:229) l ]; SurComp is the survivor (cid:2) component 2(in τ % (cid:0) ), 0 (cid:0) i S 0 0 100 [N(S 2 ) 1(cid:229) l N(S ) 1(cid:229) l )]=[N(S ) 1(cid:229) l τ (cid:0) i S τ τ (cid:0) i S 0 τ (cid:0) i S τ N(S (cid:2) ) 1(cid:229) l ]. 2 τ (cid:0) 2 τ 2 τ (cid:0) 0 (cid:0) i S 0 0 2

Table2: 1988FirmCohort: SummaryCharacteristicsbySector EmpSh CumEx AvEmp CGrEmp SurComp Sector 88 89 92 99 88 89 92 99 89-99 All 100.0 15.6 35.4 58.5 1.11 15.2 34.2 43.0 69.0 Manu 41.8 14.6 35.9 58.9 1.58 17.4 38.7 45.5 82.8 Serv 20.1 17.1 36.6 58.0 0.99 11.7 30.8 40.2 61.7 Notes: EmpSh is the employment share of the sector in the overall economy cohort; CumEx,CGrEmp,SurCompareasde(cid:2)nedintable1.

Table3: 1988FirmCohort: CharacteristicsofLaborAdjustmentbySector 89 93 Sector N30 NA P30 N30 NA P30 All 7.9 43.0 13.7 13.7 45.3 17.1 Manu 10.8 31.5 20.7 20.9 33.4 24.6 Serv 7.3 47.7 11.8 11.3 50.3 15.0 Notes: N30 is the fraction of (cid:2)rms with an adjusted growth rate of employment, conditional on survival, in the interval ( 30%;0%); NA is the fraction of (cid:2)rms (cid:0) that do not adjust employment, conditional on survival; P30 is the fraction of (cid:2)rms with an adjusted growth rate of employment, conditional on survival, in the interval (0%;30%).

Table4: Calibration/Estimation: 1988FirmCohort All Manufacturing Services Parameter NAC AC NAC AC NAC AC α 0.56 0.56 0.57 0.57 0.73 0.73 β 0.956 0.956 0.956 0.956 0.956 0.956 w 11.8 11.8 13.1 13.1 7.5 7.5 µfl 2.923 3.196 3.368 3.755 2.393 2.419 (0.016) (0.038) (0.032) (0.012) (0.018) (0.014) σ 0.245 0.188 0.236 0.083 0.133 0.123 µ 0 (0.005) (0.008) (0.009) (0.004) (0.006) (0.006) σ 0.319 0.250 0.296 0.186 0.166 0.154 µ 1 (0.005) (0.008) (0.010) (0.006) (0.006) (0.006) σ 0.884 0.661 0.707 0.440 0.436 0.402 (0.016) (0.036) (0.034) (0.016) (0.021) (0.018) W 767.5 815.2 1294.5 1494.2 200.2 198.7 (5.2) (8.2) (18.6) (21.9) (3.1) (3.5) P 0 0.74 0 2.17 0 0.06 (0.05) (0.07) (0.02) I 89.1 64.0 167.6 90.0 19.8 17.8 Q 1866.9 1516.9 700.0 423.9 356.1 351.5 (cid:3) Notes: NAC refers to no-adjustment-costs case; AC refers to proportionaladjustment-costs case; numbers in () are standard deviations of the parame- (cid:1) ters;Q isthevalueoftheobjectivefunction. (cid:3)

( ) ( ) ( ) L* q *,L < L L* q *,L = L L* q *,L > L t t t - 1 t- 1 t t t- 1 t- 1 t t t- 1 t- 1 PH MB t 0 q * ( )t q SD q SU <q * = E q * t- 1 t- 1 t Figure1: ProportionalHiring/EntryCost

0 1 C AC 0 1 NA a ta e sa e sa DCC 8 8 e ta R tn e n tix E 6 o p m 6 e v o C ita ro lu m v iv u ru C 4 S 4 2 2 :s 1 1 e u la V 5 .5 5 8 .9 4 1 .4 4 4 .8 3 7 .2 3 0 .7 2 3 .1 2 6 .5 1 6 .1 7 1 .7 6 6 .2 6 1 .8 5 7 .3 5 2 .9 4 7 .4 4 2 .0 4 re te m a ra P e c n e 0 1 0 1 re fe R la v iv la ru v S iv ru 8 n o 8 S n o la n o la n o itid 6 itid n o C 6 n o tn C e tn m y e o m lp y o lp m 4 m E g o 4 E L g o L fo n fo n a e 2 o ita iv e 2 M D .d tS 0 0 9 2 5 8 2 5 8 1 1 8 5 2 9 6 3 0 5 5 4 3 3 2 1 1 1 0 0 0 9 9 9 9 .1 .1 .1 .1 .1 .1 .1 .1 .1 .1 .1 .1 .0 .0 .0 .0 Figure2: FirmDynamicsforOverallEconomyCohort

0 0 C 1 C 1 AC AC NA NA a ta e sa e s a a ta e sa e s a DCC DCC 8 8 tn tn e e n n o o p p m 6 m 6 o o C C ro ro v v iv 4 iv 4 ru ru S S 2 2 1 1 8 2 7 1 6 0 5 9 2 2 2 2 2 1 1 1 .7 .1 .4 .8 .1 .5 .8 .1 .5 .2 .9 .6 .3 .0 .7 .4 8 8 7 6 6 5 4 4 6 6 5 5 5 5 4 4 0 0 1 1 :s :s e e G N e ta 8 u la V e ta 8 u la V IR U R tix re te S E R tix re te T E 6 m C E 6 m C A F U N A e v ita lu m 4 a ra P e c n IV R E S e v ita lu m 4 a ra P e c n M u e u e C re C re fe fe R R 2 2 1 1 1 2 2 3 4 4 5 6 9 4 0 5 0 5 1 6 .6 .0 .4 .8 .2 .6 .0 .4 .4 .9 .4 .8 .3 .7 .2 .6 5 5 4 3 3 2 2 1 5 4 4 3 3 2 2 1 la la v v iv ru 0 1 iv ru 0 1 S S n n o o la 8 la 8 n n o o itid itid n n o 6 o 6 C C tn tn e e m m 4 4 y y o o lp lp m m E g 2 E g 2 o o L L fo fo n 0 n 0 a a e 7 0 3 6 9 2 5 8 e 2 6 0 3 7 1 5 9 M 0 .2 0 .2 9 .1 8 .1 7 .1 7 .1 6 .1 5 .1 M 4 .1 3 .1 3 .1 2 .1 1 .1 1 .1 0 .1 9 .0 Figure3: FirmDynamicsforManufacturingandServicesCohorts

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 6 5 4 3 7 6 5 4 3 7 6 5 4 3 )% ( .p 6 .2 1 )% ( .p 8 5 3 .0 )% ( .p 4 3 .1 m o C ro 9 .1 1 m o C ro 2 2 3 .0 m o C ro 9 7 .0 v v v iv ru S 1 .1 1 iv ru S 5 8 2 .0 iv ru S 4 2 .0 )% 5 .2 1 )% 3 5 3 .0 )% 6 2 .1 ( ( ( e e e ta R tix 7 .1 1 ta R tix 6 1 3 .0 ta R tix 1 7 .0 E 0 .1 1 E 9 7 2 .0 E 6 1 .0 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 4 3 2 1 5 4 3 2 1 5 4 3 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 6 5 4 3 7 6 5 4 3 7 6 5 4 3 )% ( .p 4 6 9 .0 )% ( .p 4 8 2 .0 )% ( .p 4 .6 4 8 m o C ro v iv ru S 9 7 4 5 9 9 . . 0 0 m o C ro v iv ru S 1 8 1 4 2 2 . . 0 0 m o C ro v iv ru S 1 8 . . 9 2 9 7 6 7 :s e u la V re te m a 3 9 9 ra )% 6 9 .0 )% 7 2 .0 )% .5 3 8 P e ( e ( e ( e c n e ta R tix 5 5 9 .0 ta R tix 2 4 2 .0 ta R tix 2 .2 6 7 re fe R E 8 E 6 E 6 4 9 .0 0 2 .0 .8 8 6 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 4 3 2 1 5 4 3 2 1 5 4 3 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 6 5 4 3 7 6 5 4 3 7 6 5 4 3 )% 9 1 6 )% 2 0 0 )% 3 6 9 ( .p .0 ( .p .3 ( .p .0 m 4 m 8 m 9 o 6 o 2 o 8 C ro 5 .0 C ro 9 .2 C ro 8 .0 v v v iv 9 0 iv 5 5 iv 5 1 ru S 5 .0 ru S 8 .2 ru S 8 .0 1 1 2 1 9 5 )% 6 .0 )% 9 .2 )% 9 .0 ( e ( e ( e ta 6 5 ta 8 1 ta 9 7 R tix 5 .0 R tix 9 .2 R tix 8 .0 E 1 E 4 E 5 0 4 0 5 8 8 .0 .2 .0 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 4 3 2 1 5 4 3 2 1 5 4 3 2 1 Figure4: SensitivityAnalysis

Survivor Component 74.5 70.1 65.8 61.4 57.1 52.7 48.3 44.0 0.00 0.16 0.39 0.63 0.87 1.11 1.34 1.50 Reference Parameter Values: Figure5: SensitivitytoProportionalAdjustmentCost