Growing Old Together: Firm Survival and Employee Turnover

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Growing Old Together: Firm Survival and Employee Turnover Erwan Quintin and John J. Stevens 2005-22 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.

Growing Old Together: (cid:3) Firm Survival and Employee Turnover Erwan Quintin John J. Stevens Federal Reserve Bank of Dallas Federal Reserve Board April 26, 2005 Abstract Labor market outcomes such as turnover and earnings are correlated with employer characteristics, even after controlling for observable di(cid:11)erences in worker characteristics. We argue that this systematic relationship constitutes strong evidence in favor of models where workers choose how much to invest in future productivity. Because employer characteristics are correlated with (cid:12)rm survival, returns to these investments varyacross(cid:12)rmtypes. Wedescribeadynamicgeneralequilibriummodelwhereworkers employed in (cid:12)rms more likely to survive choose to devote more time to productivityenhancing activities, and therefore have a steeper earnings-tenure pro(cid:12)le. Our model also predicts that quit rates should be lower in (cid:12)rms more likely to survive, and should tend to fall during slow times, while job destruction rates should rise. These predictions, we argue, are borne out by the existing empirical evidence. Keywords: Firm survival; Firm size; Employee turnover; Firm speci(cid:12)c human capital. JEL classification: J24; J31; J63. (cid:3)E-mail: erwan.quintin@dal.frb.org and john.j.stevens@frb.gov. We wish to thank Nathan Balke, Kim Bayard, David Byrne, Finn Kydland, David Margolis, Sangeeta Pratap, and seminar participants at the AtlantaFed,theMinneapolisFed,ArizonaStateUniversity,SouthernMethodistUniversityandtheMidwest Economic Association meetings for helpful comments. The views expressed herein are those of the authors and may not reflect the views of the Federal Reserve Bank of Dallas, the Federal Reserve Board, or the Federal Reserve System.

1 Introduction Labor market outcomes such as turnover and earnings are correlated with employer characteristics. These correlations could be explained by di(cid:11)erences in worker productivity. If the evolution of productivity is exogenous, as in simple models of learning-by-doing (e.g., Jovanovic and Njarko, 1995), the correlation between (cid:12)rm characteristics, earnings, andworker turnover should disappear after controlling for the distribution of tenure. However, evidence in Abowd et al. (1999) and Anderson and Meyer (1994), among others, suggests that there remains considerable residual correlation after conditioning on worker tenures. If, instead, workers actively choose to invest in future productivity, then these decisions will lead to a systematic correlation between unobserved worker productivity and employer characteristics. A key factor that determines employer characteristics, and hence workers’ willingness to invest in future productivity, is (cid:12)rm survival. Weformalizetheseargumentsinadynamicgeneralequilibriummodelinwhichemployees split their time between accumulating (cid:12)rm-speci(cid:12)c capital and delivering labor services, as in Jovanovic (1979).1 Firms di(cid:11)er in only one respect: their likelihood of surviving from one period to the next. We show that workers employed by high survival (cid:12)rms invest more in (cid:12)rm-speci(cid:12)c capital because it is more likely that they will bene(cid:12)t from these investments in the future. Therefore, earnings-seniority pro(cid:12)les are steeper in high survival (cid:12)rms. This is consistent with the evidence discussed in Abowd et al. (1999) who (cid:12)nd, for instance, that seniority pro(cid:12)les are steeper in large (cid:12)rms. The model also predicts correctly that turnover rates should be lower in high survival (cid:12)rms, even after controlling for the distribution of tenure. Anderson and Meyer (1994), (cid:12)nd that turnover rates fall with (cid:12)rm size, while Quintin and Stevens (2005) (cid:12)nd that separation rates are higher in high exit rate industries, even after controlling for tenure di(cid:11)erences across industries. If one thinks of (cid:12)rm-speci(cid:12)c capital as the return to employer-speci(cid:12)c training, as we do in the exposition, then our model predicts that training intensity should be positively correlated with (cid:12)rm characteristics that are associated with higher survival rates, such as age and size, which is consistent with the 1As we discuss later, our model only requires that this capital be (cid:12)rm-speci(cid:12)c in part or, if most human capital is occupation speci(cid:12)c, as argued by Kambourov and Manovskii (2002), that workers run the risk of not (cid:12)nding the same occupation when their employer dies. 2

evidence discussed by Frazis et al. (1995). We qualitatively evaluate the dynamic properties of our model by computing the transition path between steady states following shocks to total factor productivity (TFP) and gross (cid:12)rm failure rates. Two outcomes of these experiments are particularly notable. First, as in the data, we (cid:12)nd that quit rates are procyclical, because workers use slow times to retool (see DeJong and Ingram, 2001). Second, we (cid:12)nd that job destruction rates are countercyclical provided gross failure rates for (cid:12)rms rise during recessions, even if the increase is very small as suggested by the existing evidence on corporate failure rates (see Platt and Platt, 1994, for a review). In summary, the cross-sectional and dynamic properties of our modelprovidecompelling supportforthebroader useofmodelsinwhichworker productivity evolves endogenously when analyzing issues related to (cid:12)rm survival and worker turnover. 2 The economy We consider a discrete time, in(cid:12)nite horizon model, with three classes of agents: (cid:12)rms, workers, andthegovernment. Firms di(cid:11)er only intheir probability ofsurvival. Type H (cid:12)rms survive to the next period with probability p , while type L (cid:12)rms survive with probability H p < p . We will think of type as proxying for characteristics such as industry or geographic L H location that may a(cid:11)ect an employer’s perceived likelihood of survival. Our results extend immediately to environments with more (cid:12)rm types. Similarly, our assumption that survival rates are (cid:12)xed and exogenous is strong but can be relaxed without altering our results. A law of large numbers holds, therefore p is also the fraction of (cid:12)rms of type i 2 fH,Lg i that survive at the end of each period. In each period, a constant mass µ > 0 of (cid:12)rms of i type i are born. Firms of both types that have survived t periods can transform n (cid:21) 0 units of labor into (1+η)tnα units of the unique consumption good, where α 2 (0,1), and η > 0 is the exogenous rate of TFP growth. The following assumption bounds the average size of (cid:12)rms: 1 Assumption 1. (1+η)1−αp < 1 H Aconstant measureofworkers arebornatthebeginningofevery period. Workers survive to thenext periodwith probability β. Weset the measure of newly bornworkers to (1−β) so 3

thatthelong-runpopulationsizeisone. Weassume thatworkers havelinearpreferences and, therefore, seek to maximize their expected lifetime labor income. All workers are assumed to own the same share of existing (cid:12)rms.2 At the beginning of each period, a worker is either employed by a (cid:12)rm or is unemployed. The maximum quantity of labor an employed worker can deliver to her employer depends upon her productivity level x, a random variable with values in fx ,x ,x g where x < 0 1 2 0 x < x .3 A newly employed worker starts at productivity level x = x , regardless of her 1 2 1 productivity history. In particular, we assume for simplicity that productivity levels are fully employer speci(cid:12)c: workers at x who lose or quit their job fall back to x with certainty. 2 1 The speci(cid:12)city of productivity can be relaxed in several ways. First, by complicating the notation a bit, productivity can be made partly general in nature (i.e., transferable across employers) without changing any results. Second, productivity can be made entirely general in nature provided it depreciates over time and there is a risk of unemployment. Lastly, productivity can be made occupation speci(cid:12)c rather than (cid:12)rm speci(cid:12)c, as long as workers who lose or quit their job run the risk of not (cid:12)nding employment in the same occupation. If the worker and employer both survive to the next period, the evolution of the worker’s productivity depends on the time s 2 [0,1] she devotes to training. If her productivity level is not already x , it rises one notch in the subsequent period with probability h(s) 2 [0,1); 2 the function h is twice continuously di(cid:11)erentiable and strictly concave with h(0) = 0 and h0(0) > 0. On the other hand, if her current productivity level is x or x , she moves down 1 2 one notch with likelihood δ.4 We think of δ as the likelihood that the worker’s productivity level depreciates, an event which can occur for idiosyncratic or (cid:12)rm-wide reasons, or for reasons common to a subset of (cid:12)rms that one could think of as an industry.5 Workers become unemployed if their employer dies, or if they choose to quit. In addition, 2The exact speci(cid:12)cation of ownership and the fact that we do not consider the possibility of workers trading ownership shares does not matter for any of our results, as we assume linear preferences. 3Allowingforanarbitrarynumberofproductivitylevelscomplicatesnotationandthederivationofresults, butourbasicresultscontinuetohold. Forinstance,itremainsthecasethatthetimeworkersatproductivity level x 1 devote to training rises with the likelihood of employer survival. Showing that workers train more at all productivity levels requires additional assumptions. 4Assuming that δ does not vary with s simpli(cid:12)es the exposition and the analysis, but does not alter any of our qualitative results. 5We assume that there is no aggregate uncertainty, but we do not require the evolution of worker productivity to be independent across workers or (cid:12)rms. 4

workers are born unemployed. Unemployment ends with probability φ < 1 at the beginning of any given period, in which case workers may choose an employer of either type. In their (cid:12)rst period of unemployment, previously employed workers receive unemployment bene(cid:12)ts equal to a fraction ρ 2 (0,1) of their labor income from the previous period. The government (cid:12)nances unemployment bene(cid:12)ts through a (cid:12)xed payroll tax rate τ and refunds excess (cid:12)scal revenues toworkersinalump-sumfashion. Thefollowingassumptionmakesverifyingbudget balance simpler: Assumption 2. τ > ρβ[1−p (1−δ)](1−φ) L This assumption ensures that the tax rate on labor earnings exceeds the replacement rate times the probability that the bene(cid:12)ts are claimed. A worker can claim bene(cid:12)ts only if she survives, enters unemployment (the bracketed term), and does not instantaneously leave unemployment (the last term). Our economy can generate several types of steady-state equilibria. For example, there aresets of exogenous parameters forwhich unemployed agents accept alljobo(cid:11)ers, andother sets for which unemployed workers only accept job o(cid:11)ers after their bene(cid:12)ts have expired. We make three more assumptions that simplify the exposition by enabling us to concentrate on one type of equilibrium. Relaxing these assumptions complicates arguments but does not change our basic results. Assumption 3 ensures that, in steady state, unemployed workers are always better o(cid:11) accepting job o(cid:11)ers than turning them down.6 Assumption 3. ρx < x 2 1 Inother words, thebene(cid:12)ts received by workers atthehighest productivity level arebounded above by the income that could be earned by accepting a new job o(cid:11)er. Next we assume that in steady state the rate of exogenous technological growth is such that the growth in labor demand by surviving (cid:12)rms always exceeds the average rate of productivity growth of existing employees, so that all (cid:12)rms have to hire new workers in (cid:22) every period. Let s(cid:22) = argmax h(s) and h = h(s(cid:22)). A condition su(cid:14)cient to guarantee s2[0,1] that all (cid:12)rms need to hire new workers in all periods is: 6In most states, the law prohibits workers who receive unemployment bene(cid:12)ts from turning down \acceptable" o(cid:11)ers. The degree to which that requirement is enforced, however, is unclear. 5

h i Assumption 4. β h (cid:22) x2 +1−δ −h (cid:22) < (1+η)1− 1 α (1−s¯)x1 The left-hand side of the inequality is a bound on the average productivity growth of returning workers, while the right-hand term is the rate of growth of labor demand by (cid:12)rms who survive from one period to the next. This assumption simpli(cid:12)es our existence proof because checking labor clearing then amounts to checking that labor demand and supply coincide for each (cid:12)rm type. In addition, by ruling out the possibility that (cid:12)rms enter a given period with excess labor, we avoid making an arbitrary assumption about which workers (cid:12)rms would lay o(cid:11). Finally, we set x low enough so that workers who reach productivity level x = x choose 0 0 to quit in equilibrium (see proposition 3 below). h i Assumption 5. x < φ −βp x 0 1−β(1−φ) H 1 We now turn to de(cid:12)ning and characterizing steady state equilibria in our economy. 3 Steady-state equilibria We will study equilibria in which (cid:12)rms behave competitively and pay workers the value of their marginal product (net of taxes) each period. Given linear preferences, this payment schemeisalwaysweaklyoptimal,butmanyalterativecompensationschemesarealsooptimal. For instance, workers could be paid the value of their expected average lifetime marginal product in every period. Our main results are independent of the speci(cid:12)c payment scheme adopted by workers and(cid:12)rms. The onlyexception is proposition6 which compares theshape of the seniority pro(cid:12)le across (cid:12)rm types. That result is based on the premise that current earnings and current productivity are positively correlated.7 Similarly, di(cid:11)erent assumptions on the relative bargaining power of (cid:12)rms and workers would not change our results. Denote by w the wage rate o(cid:11)ered by a (cid:12)rm of type i 2 fH,Lg. Workers at productivity i level x who devote time s to training earn (1−s)xw in type i (cid:12)rms. A type i (cid:12)rm of age i 7Note that such a correlation must exist when human capital is in part general. While we only consider (cid:12)rm-speci(cid:12)c capital, our results hold unchanged in a version of the model where human capital is partially general. 6

t (cid:21) 0 chooses e(cid:11)ective labor input n to maximize pro(cid:12)ts: max (1+η)tnα −nw (1+τ). i n(cid:21)0 Total labor demand for each (cid:12)rm type is the sum of optimal labor demands across (cid:12)rms of that type. As for labor supply, employed workers of a given productivity level split their time between training and delivering labor so as to maximize their expected lifetime labor income. Intheappendix, weformallystatethecorresponding optimizationproblem. Steadystate equilibria, in this context, are constant wage rates for each (cid:12)rm type such that labor markets clear and the government’s budget is balanced in every period. We establish in the appendix that a unique steady state equilibrium pair of wage rates exists in this economy for all sets of exogenous parameters that satisfy assumptions 1 to 5. Furthermore, because high survival (cid:12)rms o(cid:11)er better training opportunities and a lower unemployment risk, general equilibrium considerations imply that the wage rate (the price of each unit of labor provided by employees) is higher in (cid:12)rms whose survival is less likely. Our (cid:12)rst proposition records these (cid:12)ndings. Proposition 1. A unique steady-state equilibrium exists. Furthermore, w > w . L H Proof. See appendix. Our next result is that type H (cid:12)rms tend to be larger than type L (cid:12)rms in terms of employment, provided the survival probability of type H (cid:12)rms is high enough. This proposition gives the sense in which size and survival rates are positively correlated in our model, as they are in the data. −1 Proposition 2. Given other parameters, there exists p < (1 + η)1−α such that if p > p H then type H firms employ more workers on average than type L firms in steady state. Proof. See appendix. −1 As p rises to (1+η)1−α, the average labor demand of type H (cid:12)rms grows without bound. H Even though the average amount of labor delivered by workers may be larger in type H (cid:12)rms, the labor demand di(cid:11)erential between (cid:12)rm types becomes large enough to guarantee 7

that type H (cid:12)rms employ more workers on average. In this sense, our model predicts that (cid:12)rms more likely to survive tend to be larger in employment terms. Since wage rates are lower in those (cid:12)rms (proposition 1), this appears to imply that large (cid:12)rms pay less than small (cid:12)rms in our model, which is counterfactual. But the model makes no such prediction. Earnings are the product of three terms: time devoted to work, productivity, and wage rates. While wage rates are lower and employees devote more time to training in (cid:12)rms more likely to survive, they are also more productive on average than workers employed in low survival (cid:12)rms. In fact, simple algebra shows that when there is no unemployment (φ = 1), average earnings must be the same in steady state across (cid:12)rm types. For the parameters we use in the numerical section, average earnings rise monotonically with size for each (cid:12)rm type. Furthermore, the economy-wide correlation between (cid:12)rm size and average worker earnings is positive. We now wish to compare quit rates and job destruction rates across (cid:12)rm types. We begin by recording the fact that given assumption 5, workers remain with their employer as long as their productivity is at least x and as long as their employer survives. 1 Proposition 3. In steady state, workers quit if and only if their productivity is x . 0 Proof. See appendix Therefore, to compare quit rates across (cid:12)rm types we need only compare average productivity levels and hence training policies. Workers whose productivity is x do not devote any 2 time to training in either (cid:12)rm type. Denote by si the fraction of time workers at productivity level x devote to training when their employer is of type i 2 fH,Lg. The following result 1 says that workers employed in high survival (cid:12)rms devote more time to training than other workers. Proposition 4. In steady state, sL (cid:20) sH, and sL < sH if sL 2 (0,1) or sH 2 (0,1). Proof. See appendix. The intuition behind this result is simple. Since the opportunity cost of training (x 1 times the wage rate) is lower in type H (cid:12)rms, it is su(cid:14)cient to show that the return to training is higher in type H (cid:12)rms. It may seem obvious that returns to training must be 8

higher in type H (cid:12)rms, as initial earnings are higher in type L (cid:12)rms than in type H (cid:12)rms. However, the fact that workers in type H (cid:12)rms face a lower unemployment risk could, in principle, be enough to compensate workers for the initial wage di(cid:11)erential. We argue in the appendix that the unemployment risk di(cid:11)erential is not enough: Training returns must be higher in type H (cid:12)rms for workers to be indi(cid:11)erent between (cid:12)rms. This result can only be established in a general equilibrium model; a partial equilibrium model would require ad hoc assumptions about how compensation pro(cid:12)les vary across (cid:12)rm types. We can now characterize the seniority pro(cid:12)les of productivity and earnings in both (cid:12)rm types. Given proposition 3, the evolution of the productivity level of a worker employed in a (cid:12)rm of type i 2 fH,Lg, conditional on the continuation of the employment relationship, is governed by a Markov chain with two states, x and x , and with transition matrix: 1 2 2 3 1−h h 4 i i 5 δ 1−δ where h (cid:17) h(si) is the probability that a worker employed in a (cid:12)rm of type i moves up to i 1−δ productivity level x . For i 2 fH,Lg and t (cid:21) 0, denote by Et(x) the average productivity 2 i level of workers with t periods of tenure in type i (cid:12)rms. The following result characterizes the evolution with tenure of this average. Proposition 5. In steady state, (cid:15) lim Et(x) = δ x + hi x for i 2 fH,Lg, t7!+1 i hi +δ 1 hi +δ 2 (cid:15) If h < 1−δ, Et+1 (x) > Et(x) for i 2 fH,Lg and all t (cid:21) 0, i i i (cid:15) Et (x) (cid:21) Et(x) for all t (cid:21) 0, with a strict inequality when sL 2 (0,1) or sH 2 (0,1). H L Proof. The (cid:12)rst two items are standard results for two-state Markov chains. The last item is a direct consequence of proposition 4. Therefore, with tenure, average productivity rises to aninvariant value. This convergence is monotonic when productivity levels are persistent in the sense that a worker’s expected productivity level in the next period, conditional on the worker keeping the same employer, 9

rises with the current level of productivity. When that condition is not met, productivity oscillates ever closer to its invariant value. The last item of proposition 5 says that at equal tenure, the average productivity of workers is higher in high survival (cid:12)rms. A natural question to ask is whether the average earnings of workers are higher in high survival (cid:12)rms beyond some threshold level of tenure. These workers are at a higher productivity level on average than workers employed in low survival (cid:12)rms, but they also spend more time in training when at productivity level x , and 1 have a lower wage rate. By proposition 5, workers employed in type H (cid:12)rms eventually earn more, on average, than workers employed in type L (cid:12)rms if and only if δ h δ h (1−sL)x w + L x w < (1−sH)x w + H x w (3.1) 1 L 2 L 1 H 2 H h +δ h +δ h +δ h +δ L L H H The following result says that whenever p is small enough and φ, the hazard rate out of L unemployment, is high enough, condition (3.1) holds in steady state. Proposition 6. Given other parameters, there exist (φ,p(cid:22)) 2 (0,1)2 such that workers eventually earn more, on average, in type H firms than in type L firms in steady state if p < p(cid:22) L and φ > φ. Proof. See appendix The argument we provide in the appendix consists of showing that inequality (3.1) must hold in steady state when φ = 1 and p = 0. The proposition is then obtained with a L continuity argument. We now invoke the last item of proposition 5 to demonstrate that steady state quit and job destruction rates are higher in type L (cid:12)rms than in type H (cid:12)rms. Quit rates are the fractionofworkers who decide toquit atthe beginning ofa given period. The jobdestruction rate for a given (cid:12)rm type is the sum of quits and involuntary separations, i.e. quits plus jobs lost due to (cid:12)rm death, divided by total employment.8 The following results also make note of an obvious corollary to proposition 5: Quit rates are inversely related to tenure in this economy as in Jovanovic (1979) and as in the data. 8Speci(cid:12)cally, let ω i be the fraction of workers whose productivity level is x 1 in (cid:12)rms of type i2fH,Lg. The turnover (quit) rate in industry i is ω i δ while the job destruction rate is 1−βp i(1−ω i δ). 10

Proposition 7. In steady state, (cid:15) Average quit rates are higher in type L firms than in type H firms; (cid:15) Average quit rates are also higher in type L firms at each tenure level; (cid:15) Quit rates decrease with tenure in both firm types; (cid:15) The job destruction rate is higher in type L firms than in type H firms. Proof. Proposition 5 implies that, at all tenure levels, workers employed in type H (cid:12)rms are less likely to fall to productivity level x . This result implies the (cid:12)rst three items of 0 the proposition. As for the fourth, note that p < p implies that involuntary separations L H are more frequent for workers employed in type L (cid:12)rms than workers employed in type H (cid:12)rms. Our model, therefore, predicts that employee turnover and (cid:12)rm survival should be correlated, even at equal tenure. The fundamental force behind this result is that workers can influence the evolution of their productivity. Assuming instead that productivity evolves exogenously (i.e., h(s) = h > 0 for all s 2 [0,1])) would make our model very similar to the learning-by-doing model described by Jovanovic and Njarko (1995). In that case, it is easily shown that turnover rates continue to be lower in high survival (cid:12)rms, as workers in those (cid:12)rms tendtohave higher productivity levels. Also, earnings-seniority pro(cid:12)lescontinue torise to an invariant value. What, then, distinguishes this exogenous worker productivity model from our endogenous accumulation model? First, in the exogenous model, earnings-seniority pro(cid:12)les do not di(cid:11)er across (cid:12)rms: The growth rate of productivity is the same in all (cid:12)rms at all tenure levels. While the early panel evidence of Barron et al. (1987) is ambiguous on this question, Abowd et al. (1999, table 11) (cid:12)nd with much more detailed panel data on French workers that returns to seniority unambiguously rise with (cid:12)rm size.9 Second, unlike our model, the exogenous model implies that turnover rates are identical across (cid:12)rm types once we control for tenure. That implication is inconsistent with the results of Anderson and Meyer (1994, see table 6), Topel and Ward (1992), and Quintin and Stevens (2005). Anderson and Meyer, for instance, (cid:12)nd that turnover rates fall with (cid:12)rm size, whether or not one 9We wish to thank David Margolis for very helpful comments on this issue. 11

controls for tenure e(cid:11)ects, as predicted by our model. Third, the exogenous model makes no prediction about the optimal quantity of training workers receive whereas the endogenous model predicts that workers employed in (cid:12)rms more likely to survive should receive more training on average, as they do according to the 1995 survey of employer-provided training described by Frazis et al. (1998). Fourth, unlike the exogenous model, the endogenous model allows for empirically relevant dynamics, such as procyclical quit rates and countercyclical job destruction rates, an issue we explore in the next section. 4 Dynamic implications In this section we consider equilibria where wage rates vary over time. Equilibria, as before, are sequences of wage rates such that in each period, (cid:12)rms and workers behave optimally, labor markets clear, and budget balance is obtained. Our main objective in this section is to check that our model is qualitatively consistent with the fact that in the U.S., quit rates are procyclical while job destruction and training rates are countercyclical. In our dynamic experiments, we will assume that our economy is initially in steady state and consider the e(cid:11)ectsofunexpectedshockstoanexogenousparameter. First,however, wereviewparameter selection and some selected steady state statistics. 4.1 Parameter choices Wewillthinkofaperiodasaquarterandselectparameterstomatchtheappropriatefeatures of U.S. data. Details may be found in appendix B. Broadly speaking, although we will not emphasize the precise quantitative implications of our model, the parameters shown in table 1 are such that the steady state generated by our model matches empirical estimates of the duration of unemployment, the time devoted to training, returns to training, the average earnings loss following an involuntary separation, and other U.S. statistics. 12

Table 1: Parameter choices Survival rates Unemployment Production technology Training technology β 0.98 φ 0.50 α 0.64 x 1.00 1 µ 0.0003 ρ 0.60 η 0.004 x 1.10 1 2 µ 0.0003 τ 0.21 δ 0.02 2 p 0.94 h(s) 1.30s−1.95s2 L p 0.98 H Table 2: Steady State Summary Statistics by Firm Type Firm type: Low survival High survival Average (cid:12)rm size 26.96 54.39 Fraction of workers at x 41.85 67.94 2 2-year growth rate of earnings 7.17 15.08 Quit rate 1.16 0.64 Job destruction rate 8.95 4.58 Notes: Allvaluesareexpressedaspercentages,exceptforaverage(cid:12)rm size which is in number of workers. Rates are quarterly, except where noted. High and low survival correspondto type H and type L (cid:12)rms. 4.2 Steady state statistics Summary statistics for the steady state obtained using these parameters are in table 2. As summarized by proposition2, highsurvival (cid:12)rms arelarger becausethey aremoreproductive on average in total factor terms. Given our parameterization, type H (cid:12)rms have roughly twice as many employees as type L (cid:12)rms. Table 2 also illustrates proposition 5: The twoyear earnings growth pro(cid:12)le is twice as steep in high survival (cid:12)rms as in low survival (cid:12)rms. The quarterly quit rate in low survival (cid:12)rms is nearly double the rate in high survival (cid:12)rms. Higher quit rates together with a lower (cid:12)rm survival rate imply that the job destruction rate in type L (cid:12)rms is also much higher. Proposition 7 also says that turnover rates must be higher in low survival (cid:12)rms after controlling for tenure, a result that is illustrated in (cid:12)gure 1. 13

4.3 Cyclicality of turnover and job destruction In order to assess the cyclical behavior of turnover and job destruction, we shock exogenous parameters and compute the transition path back to steady state.10 For each shock, we plot the percent deviation from steady state for select variables of interest|wages, output per worker, training intensity, quit rates, job destruction rates, and the unemployment rate. We (cid:12)rst shock total factor productivity (TFP)|which in our model is (1+η)t for (cid:12)rms of age t|by multiplying it by 1+θ θi 1 2 where θ 2 (−1,1) is the magnitude of the shock, θ 2 [0,1) is the shock’s persistence, and 1 2 i (cid:21) 0 denotes the number of periods since the shock. The persistence parameter was set to 0.92; this value corresponds to a half-life of 3-1/2 years, a standard estimate of the average half-life of business cycle shocks in the U.S. The results of a 1 percent shock to TFP are illustrated in (cid:12)gure 2. On impact, (cid:12)rms (cid:12)nd themselves with too much labor, and so wage rates must fall in order to clear the labor market. Workers, anticipating higher future wages, devote more time to training and less time to production. This behavior is consistent with workers \retooling" during recessions (DeJong and Ingram, 2001).11 Lower TFP and more time spent on training imply that output per worker must fall. Consequently, aggregate output also drops. With more training, fewer workers drop to x , and so fewer workers 0 choose to quit, as in the data. During slow times, workers train more and quit less often. Unlike the evidence from U.S. data, however, these procyclical quit rates lead to procyclical job destruction. 10Speci(cid:12)cally, the algorithm for computing a transition path is as follows. The initial and (cid:12)nal steady states are computed using standard methods; in the examples considered here the initial and (cid:12)nal steady states are the same. We assume that the transition is complete after T periods, for large T, and we guess an initial path for w L. We then repeat the next two steps 50,000 times at which point the transition path has been solved (i.e., the labor market clears in each period and the policy functions solve the appropriate maximization problem): (1) Starting at period t = T −1 the value and policy functions are computed iteratively back to period t=1; w H is chosen so that newly hired workers are indi(cid:11)erent between the two industries. (2) Labor clearing is not guaranteed at this point, as w L was (cid:12)xed for each t. Therefore, for each period t=1,...,T −1, we adjust wages by a very small amount in the direction needed to clear the labor market. 11In the model there are no (cid:12)xed costs associated with labor, therefore (cid:12)rms do not (cid:12)re workers. Rather, workers supply less labor by training more. 14

The reason for this counterfactual result is that a TFP shock, as we have modeled it, has no e(cid:11)ect on (cid:12)rm survival rates and therefore all deviations from steady state reflect only the behavior of worker quits. In a downturn, we also expect (cid:12)rm survival to decline. Indeed, existing empirical evidence on business failures documents a small, but signi(cid:12)cant, increase in failures during downturns (Platt and Platt, 1994). Therefore, we redo the above analysis for a joint shock to both TFP and (cid:12)rm survival rates. The TFP shock is the same as before, but we now divide p and p by a common factor. Although data on business L H failure rates suggest that shocks to survival rates are less persistent than shocks to TFP, we chose to use the same degree of persistence. This assumption biases our experiment against (cid:12)nding procyclical quit rates, as a more persistent shock to survival probabilities increases the likelihood of countercyclical quits. As before, we consider a 1 percent shock to TFP; the survival rate shock is 0.01 percent (i.e., 1 out of every 10,000 businesses per quarter). The impact of this joint shock is summarized in (cid:12)gure 3. Wages, output per worker, and training are similar to the TFP-only shock, and we still (cid:12)nd procyclical quit rates. Importantly, because of the increase in (cid:12)rm failures, job destruction rates are now countercyclical. In other words, while procyclical quit rates tend to lower job destruction, this e(cid:11)ect is more than o(cid:11)set by a rise in involuntary separations due to the lower survival probabilities of (cid:12)rms, even though the survival shock is very small. The net e(cid:11)ect is that unemployment rises. 4.4 Sensitivity analysis The fact that our model’s dynamic behavior is consistent with turnover facts in the U.S. remains true for other reasonable sets of exogenous parameters. Although all the evidence suggests that important resources are devoted to training, the existing data on training intensity and returns are imprecise (see Barron et al., 1997). We experimented with a wide variety of parameters choices (within the con(cid:12)nes of assumptions 1 to 5), and found our (cid:12)ndings to be robust to those changes. Another potential concern is that the choice for δ suggested by the microeconomic evidence reviewed by Mincer (1991) leads to an economywide quarterly quit rate of 1 percent, which is much below the 4.5 percent average calculated by Hamermesh and Pfann (1996) for the 1960-1981 time period in the U.S. Raising δ from 15

0.02 to 0.05 yields a quit rate closer to its empirical counterpart, and does not a(cid:11)ect our results. Finally, the shape of the production technology in the type of model we laid out is a source of debate. Like Hopenhayn and Rogerson (1993), we set α = 0.64 to match the labor income share in the U.S., but Atkeson et al (1996), among others, argue that the implied returns to scale are too low. Raising α to near the upper bound implied by assumption 1 did not change our (cid:12)ndings: Quits continue to fall during recessions, while job destruction rises. 5 Conclusion Our paper characterizes the impact of (cid:12)rm survival on the evolution of worker productivity in a dynamic general equilibrium model. Quite intuitively, workers employed in (cid:12)rms highly likely to survive choose to invest more in future productivity than their counterparts in low survival (cid:12)rms. These investment patterns have several implications for the features of turnover and earnings across (cid:12)rm types in steady state and the evolution of turnover rates following business cycle shocks that are consistent with the relevant empirical evidence. The correlation between (cid:12)rm characteristics and labor market outcomes thus constitutes strong support for models in which employee productivity evolves endogenously. Importantly, the intuition we develop in this paper does not depend on one’s view of how worker decisions a(cid:11)ect future productivity. If, for instance, productivity depends on unobservable e(cid:11)ort (as in Lazear, 1981) workers may invest in \relationship collateral" by accepting lower initial earnings in return for actuarially fair payments later in thelife of the contract. The intuition we develop in this paper does, however, depend on workers’ ability to condition their decisions on the expected survival of their employer. In models with contractual frictions, employer survival a(cid:11)ects the expected duration of the contract and hence the willingness of workers to invest in the relationship with their employer. Thus, our framework can accommodate a wide range of approaches to modeling workers’ investments in productivity. Our model could be generalized to allow for time-varying or endogenous (cid:12)rm survival rates. Assuming for example that survival rates follow a (cid:12)rst-order Markov process would 16

complicate the analysis by adding a new state variable to the worker’s problem, but should not alter the basic correlation between expected employer survival and productivity investments. One could also allow for some feedback between worker decisions and (cid:12)rm survival. Firms with a more productive workforce could be more likely to survive. Our model’s main predictions should stand up to this generalization. In a rational expectations equilibrium, workers must form expectations on their employer’s likelihood of survival, and those expectations must prove correct. Furthermore, in such an equilibrium, atomistic workers treat the likelihood of survival as exogenous, so that the nature of their optimization problem should change little. Establishing the existence of a rational expectation equilibrium in this context may be challenging, but employer survival rates should have a similar e(cid:11)ect on worker decisions as in our simpler framework. 17

A Proofs A.1 Government budget balance Assume that we have found a pair of constant wage rates (w ,w ) such that the labor L H market clears in every period. We will show that government revenues exceed aggregate unemployment bene(cid:12)ts, so that we have in fact found a steady state. (Recall that excess tax revenues are rebated in a lump-sum fashion to workers.) Workers at productivity level x in type L (cid:12)rms earn (1 − sL)x w in a given period, 1 1 L where sL is the optimal training policy given wage rates. The tax revenue associated with these workers is τ(1−sL)x w . The fraction of workers that die at the end of the period is 1 L (1−β). The fraction of surviving workers that quit or lose their job in each period and fail to (cid:12)nd a new job immediately is [1−p (1−δ)](1−φ). The corresponding unemployment L bene(cid:12)t for these workers is β[1−p (1−δ)](1−φ)ρ(1−sL)x w . But since τ(1−sL)x w > L 1 L 1 L β[1−p (1−δ)](1−φ)ρ(1−sL)x w under assumption 2, this type of worker increases tax L 1 L revenues more than government expenses. Similar arguments show that this is the case for all possible types of workers, which establishes that tax revenues exceed total unemployment bene(cid:12)ts, as claimed. A.2 Statement of the worker’s problem We will now characterize the decisions of workers given wage rates. Fix (w ,w ). For L H i 2 fH,Lg and j 2 f0,1,2g, denote by Vi the expected lifetime labor income of a worker j employed by a (cid:12)rm of type i and whose productivity level is x , so that, for instance, VL j 0 denotes the expected lifetime income for a worker of productivity level x who works for a 0 (cid:12)rm of type L. In any steady state we must have VL = VH, as otherwise one (cid:12)rm type 1 1 would not be able to hire workers, which is incompatible with the fact that both (cid:12)rm types have positive labor demand at all wage rates. A.2.1 Value function of unemployed workers Consider a worker who just became unemployed and whose current bene(cid:12)ts are b (cid:21) 0 (that is, their earnings in the previous period were b.) Denote by VU(b) their expected lifetime ρ income. Then, for all b (cid:21) 0, VU(b) = φmaxfVU(b),VLg+(1−φ)[b+βVU(0)] 1 where φ VU(0) = VL. 1−(1−φ)β 1 Indeed, they (cid:12)nd a job with probability φ and accept it when VL (cid:21) VU(b). With probability 1 1−φ they do not get a job o(cid:11)er, get the bene(cid:12)ts in the current period, remain unemployed, and bene(cid:12)ts expire after one period. We will now argue that assumption 3 implies that VU(b) < VL. Note (cid:12)rst that assumption 3 implies that b < x maxfw ,w g. Then, 1 1 L H VU(b) = b+βVU(0) < x maxfw ,w g+βVU(0) (cid:20) VL. 1 L H 1 18

The last inequality follows from the fact that a feasible policy for newly employed workers consists of devoting no time to training in the (cid:12)rst period and quitting after one period, so that VL (cid:21) x w + βVU(ρx w ). Having established that VU(b) < VL for all workers in 1 1 L 1 L 1 equilibrium, we will write for simplicity and without any loss of generality that for all b (cid:21) 0: φ VU(b) = φVL +(1−φ)[b+βVU(0)] = (1−φ)b+ VL (A.1) 1 1−(1−φ)β 1 A.2.2 Proof of proposition 3 To simplify the statement of the worker’s problem, we begin by proving proposition 3. Proof. Consider an employee in a type H (cid:12)rm (for concreteness) whose productivity reaches x . If she stays with her employer, her expected lifetime income is bounded above by 0 x w +βp VH. On the other hand, if she quits, her expected lifetime income is bounded 0 H H 1 below by: φ φVH +β(1−φ)φVH +[β(1−φ)] 2 φVH +... = VH 1 1 1 1−β(1−φ) 1 The worker will quit provided the upper bound from staying is less than the lower bound from quitting, which is the case (for instance) if: φ x w +βp VH < VH 0 H H 1 1−β(1−φ) 1 (cid:20) (cid:21) φ VH () x < −βp (cid:1) 1 . 0 1−β(1−φ) H w H But this inequality follows from assumption 5, and the fact that VH > x w . The second 1 1 H item of the proposition was established in subsection A.2.1. A.2.3 Value function of employed workers Having established proposition 3, we need only consider employed workers whose productivity level is x or x . Fix i 2 fH,Lg. Expected incomes for workers employed in type i (cid:12)rms 1 2 must satisfy the following conditions in steady state: Vi = max(1−s)x w +βp [(1−h(s)−δ)Vi +h(s)Vi] 1 1 i i 1 2 s2[0,1] + β[(1−p )+p δ]VU(ρ(1−s)x w ) (A.2) i i 1 i Vi = x w +βp [δVi +(1−δ)Vi]+β(1−p )VU(ρx w ) (A.3) 2 2 i i 1 2 i 2 i To see this, consider (cid:12)rst equation (A.2). A worker whose current productivity level is x 1 and chooses to devote time s to training receives (1−s)x w in labor income in the current 1 i period. Asforfutureperiods,assume(cid:12)rstthatthe(cid:12)rmsurvives, whichoccurswithlikelihood p . The worker moves up to productivity level x with probability h(s), in which case she i 2 remains with the (cid:12)rm and expects income Vi. With probability δ her human capital falls to 2 19

level x , shequits, andexpects futureincome VU(ρ(1−s)x w ). Withprobability1−δ−h(s), 0 1 i she remains at productivity level Vi. If the (cid:12)rm fails, which occurs with probability (1−p ), 1 i the agent becomes unemployed, and expects future income VU(ρ(1 − s)x w ). Condition 1 i (A.3) is justi(cid:12)ed in the same fashion, and uses the fact that workers do not devote any time to human capital accumulation when they reach leve(cid:8)l x 2 . (cid:9) Solving equation A.3 for Vi yields Vi = 1 x w +β[pδVi +(1−p)VU(τx w )] . 2 2 1−βpi (1−δ) 2 i 1 2 i Plugging this into A.2 now gives: Vi = max(1−s)x w +βp[(1−h(s)−δ)Vi 1 1 i 1 s2[0,1] (cid:8) (cid:9) 1 + h(s) x w +β[pδVi +(1−p)VU(τx w )] ] 1−βp (1−δ) 2 i 1 2 i i + β[(1−p)+pδ]VU(τ(1−s)x w ) (A.4) 1 i Given expression (A.1), the right-hand side of equation (A.4) de(cid:12)nes a mapping on IR + in Vi. That mapping has a unique (cid:12)xed point, as we now argue. 1 Lemma 1. The right-hand side of equation (A.4) defines a contraction mapping on IR . + Proof. Fix i 2 fH,Lg. Consider the mapping T : IR 7! IR which to every value V + + 1 associates the right-hand side of (A.4). By construction, solutions of (A.4) and (cid:12)xed points of T coincide. We will now argue that T is a contraction. T is clearly monotonic. Now note that expression (A.1) implies ∂VU(b) = φ < 1. Furthermore, δ < 1 and ∂V1 L 1−(1−φ)β 1−βpi (1−δ) 1−pi < 1. These observations and some algebra imply that for all V > 0 and c > 0, 1−βpi (1−δ) 1 T(V +c) < T(V )+βc. Therefore, T is a contraction with modulus β. 1 1 Given i 2 fH,Lg, the optimal training policy si is uniquely de(cid:12)ned, because h is continuous and strictly concave. Furthermore, si is continuous in all parameters by the Theorem of the Maximum. In particular, it is easy to see that, for i 2 fH,Lg, Vi is linear in wages 1 and the solution si to (A.4) is independent of wage rates, a fact upon which we will rely to establish that a unique pair of steady state wage rates exists.12 In turn this implies that sL is independent of p and that sH is independent of p . More precisely, if (sL,sH) is the H L steady state pair of human capital policies given (p ,p ) and (sL0 ,sH0 ) is the steady state L H pair of human capital policies given (p0 ,p0 ), p0 = p implies sL0 = sL. We will use this last L H L L fact in the proof of proposition 2. A.3 Proof of proposition 1 Given assumption 4, it can never be the case that the demand for labor on the part of a given (cid:12)rm falls short of the intended supply of labor by existing employees. Therefore, labor 12Formally,sinceVU islinearinV 1 i,T preserveslinearity. Sincethesetoflinearfunctionsofw i isaclosed subset of the set of bounded real valued functions equipped with the supnorm topology, standard dynamic programmingargumentsimply thatthe (cid:12)xedpointofT islinear inwagerates. Dividing both sides of(A.4) by wi now shows that optimal training is independent of the wage rate in steady state. This is also evident Vi−Vi from (cid:12)rst order condition (A.5) since 2 1 is a constant. wi 20

market clearing requires only that overall labor demand equal overall labor supply for each (cid:12)rm type. We now show that this obtains for a unique pair of wage rates. Proof. Fix w . Since VL rises without bound with w , there is a unique wage rate w = L 2 H H g(w ) such that VL = VH. Furthermore, g is continuous and rises with w . Let Di(w ) be L 1 1 L i the aggregate labor demand by (cid:12)rms of type i when the wage rate is w . By assumption 1, i Di < +1 for i 2 fH,Lg. Furthermore, both demand functions are continuous and strictly decreasing on IR . We will construct an equilibrium where workers always work for the same + type of (cid:12)rms. Since expected incomes must be equal in equilibrium across (cid:12)rm types, such a policy is always (weakly) optimal. For i 2 fH,Lg, denote by Si the average supply of labor by agents who work for (cid:12)rms of type i during their lifetime given optimal human capital accumulation policies si. As we argued above, si does not depend on wage rates. Let σ (w ) = DL(wL ) and σ (w ) = L L SL H L DH(g(wL )) . If σ (w ) + σ (w ) = 1, we can construct an equilibrium by assigning fraction SH L L H L σ (w ) of workers to type L (cid:12)rms and fraction σ (w ) to type H (cid:12)rms.13 L L H L To see that such a value for w exists observe (cid:12)rst that σ and σ are continuous since L L H labor demand functions are continuous. Moreover, σ (w )+σ (w ) > 1 when w is small L L H L L enough since DL(w ) diverges to +1 as w gets small. On the other hand, for w large L L L enough, σ (w )+σ (w ) < 1 since DL(w ) and DH(g(w )) converge to zero when w gets L L H L L L L large. By the intermediate value theorem, a solution exists. Because labor demand functions are strictly decreasing while si and therefore Si are independent of wage rates, the solution is unique. To see that w > w in steady state, assume by way of contradiction that a steady state L H exists with w (cid:21) w . Workers in type H (cid:12)rms can choose to set sH = sL. In this case, H L their average labor income when employed is at least as high as that of workers in type L (cid:12)rms, but the expected time they spend in unemployment is lower. Therefore, VL < VH, a 1 1 contradiction. This completes the proof. Aswepointoutintheproofabove,workerswhoseunemployment spellendsareindi(cid:11)erent between the two employer types, and so employer choice policies are indeterminate. But the average labor supply to each (cid:12)rm type is independent of the speci(cid:12)cation of employer choice policies, as this average only depends on the tenure distribution of employees; in turn, the tenure distribution only depends on training policy functions and the survival probabilities of (cid:12)rms and workers. Therefore, the exact speci(cid:12)cation of the employer choice policy cannot a(cid:11)ect steady state equilibrium wage rates, and so wage rates are unique. A.4 Proof of proposition 2 We now show that for p high enough, type H (cid:12)rms are larger than type L (cid:12)rms in employ- H ment terms in all steady states. 13Because the size of the population of workersis one, σ i is also the number of workersassignedto type i (cid:12)rms for i2fH,Lg. 21

Proof. For i 2 fH,Lg, the average labor demand by type i (cid:12)rms is given by: (cid:16) (cid:17) P 1 + t= 1 0 µ i [p i (1+η)1− 1 α]t w α i 1−α 1−p i (cid:18) α (cid:19) 1− 1 α = . µ i /(1−p i ) 1−p i (1+η)1− 1 α w i Note that µ /(1−p ) is the long-run number of (cid:12)rms of type i. We showed earlier that in i i any steady state, w > w . Recall also that the labor supply of a worker in a type H (cid:12)rm L H is bounded above by x , so the average labor supply by workers in type H (cid:12)rms, SH, is also 2 bounded by x . Finally, note that the average labor supply in type L (cid:12)rms, SL, does not 2 depend on p . Then, taking the ratio of the average employment size of type H (cid:12)rms to the H average employment size of type L (cid:12)rms, (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) SL 1−p 1−p (1+η)1− 1 α w 1− 1 α SL 1−p 1−p (1+η)1− 1 α H L L H L > . SH 1−p L 1−p H (1+η)1− 1 α w H x 2 1−p L 1−p H (1+η)1− 1 α Now (cid:12)x p . As p 7! (1 + η)1 − − 1 α, SL is una(cid:11)ected, and the ratio diverges to +1 which L H establishes proposition 2. A.5 Proof of proposition 4 Next we establish that in steady state, workers employed in type H (cid:12)rms devote more time to training than workers employed in type L (cid:12)rms. Proof. For the purpose of this proof, we (cid:12)nd it more convenient to work with equations (A.2-A.3) than their reduced version (A.4). Consider a steady state pair of wage rates and assume that both sL and sH are interior; other cases are trivial. Given (A.2), the (cid:12)rst order condition for si for i 2 fH,Lg is: Vi −Vi x = βp h 0 (si) 2 1 −β[(1−p )+p δ](1−φ)ρx (A.5) 1 i i i 1 w i Now, [(1 − p ) + p δ] falls with p , hence i. Therefore, a su(cid:14)cient condition for s < s is i i i 1 2 that pi (V2 i−V1 i) rises with i. We will establish that this condition holds in steady state. For wi all p 2 [p ,p ], denote by V (w,p) and V (w,p) the solutions to (A.2-A.3) when p = p L H 1 2 i and w = w. Also, for all p 2 [p ,p ], denote by w(p) the unique wage rate such that i L H V (w(p),p) = VL = V (w ,p ). In particular, note that w(p ) = w , by construction. We 1 1 1 L L H H will argue that for all p 2 [p ,p ], L H (cid:16) (cid:17) ∂ p(V2 (w(p),p)−V1 (w(p),p)) w(p) (cid:21) 0 (A.6) ∂p The left-hand side of condition (A.6) can be written as the sum of three terms: (V (w(p),p)−V (w(p),p)) 2 1 a(p) = , w(p) 22

(cid:18) (cid:19) ∂ (V (w(p),p)−V (w(p),p)) ∂w(p) 2 1 b(p) = p , ∂w w(p) ∂p (cid:18) (cid:19) p ∂V ∂V and c(p) = 2 (w(p),p)− 1 (w(p),p) . w(p) ∂p ∂p But b(p) = 0 for all p since, given p, V and V are linear in w. So we only have to show 2 1 that a(p)+ c(p) (cid:21) 0 for all p. Using the envelope theorem and dropping the arguments of function h to curb notation, ∂V ∂V ∂V 2 = β[δV +(1−δ)V ]+βp[δ 1 +(1−δ) 2 ] 1 2 ∂p ∂p ∂p φ ∂V − βVU(ρx w)+β(1−p) 1 2 1−(1−φ)β ∂p ∂V ∂V ∂V 1 = β[(1−δ −h)V +hV ]+βp[(1−δ −h) 1 +h 2 ] 1 2 ∂p ∂p ∂p φ ∂V − β(1−δ)VU(ρ(1−s)x w)+β[pδ +(1−p)] 1 1 1−(1−φ)β ∂p Therefore, ∂V ∂V 2 − 1 = β(1−δ −h)(V −V )+βδV 2 1 1 ∂p ∂p (cid:18) (cid:19) ∂V ∂V ∂V + βp(1−δ −h) 2 − 1 +βpδ 1 ∂p ∂p ∂p φ ∂V − β[VU(ρx w)−VU(ρ(1−s)x w)]−βpδ 1 2 1 1−(1−φ)β ∂p Now note that14 VU(ρx w)−VU(ρ(1−s)x w) = (1−φ)ρ[x −(1−s)x ]w (cid:20) V −V 2 1 2 1 2 1 Furthermore, ∂V1 > 0 (as can be seen by partially di(cid:11)erentiating equation A.2) and βpδ > ∂p βpδ φ . Therefore, 1−(1−φ)β (cid:18) (cid:19) ∂V ∂V βp(δ+h) p 2 − 1 (cid:21) − (V −V ) (cid:21) −(V −V ) ∂p ∂p 1−βp(1−δ −h) 2 1 2 1 since βp(δ+h) < 1 and, in turn, 1−βp(1−δ−h) (V (w(p),p)−V (w(p),p)) c(p) > − 2 1 = −a(p). w(p) 14In approximate terms, this inequality says that the fact that the unemployment risk is lower in high survival (cid:12)rms is not su(cid:14)cient to compensate workers for the initial wage di(cid:11)erential. Returns to training must also be higher. 23

This completes the proof. A.6 Proof of proposition 6 We now demonstrate that for p small enough and φ high enough, workers employed in type L H (cid:12)rms earn more than small (cid:12)rm workers past a certain tenure threshold. Proof. Fix p and assume that φ = 1 and p = 0. Let w(cid:3) and w(cid:3) be the corresponding H L L H steady state wage rates. Since φ = 1 and p = 0, VL = x1wL (cid:3) while some algebra shows that: L 1 1−β (cid:18) (cid:19) (cid:18) (cid:19) βp h(sH) βp h(sH) (1−β)VH = 1− H (1−sH)x w (cid:3) + H x w (cid:3) 1 1−βp (1−δ −h(sH)) 1 H 1−βp (1−δ −h(sH)) 2 H H H δ h(sH) < (1−sH)x w (cid:3) + x w (cid:3) δ +h(sH) 1 H δ +h(sH) 2 H δ h < (1−sH)x w (cid:3) + H x w (cid:3) . h +δ 1 H h +δ 2 H H H The (cid:12)rst equality follows from manipulations of (A.2-A.3) when φ = 1. The (cid:12)rst inequality uses thefact thatβp < 1 whilethe second inequality uses thefact that h (cid:17) h(sH) > h(sH). H H 1−δ Then, if equation (3.1) does not hold, we have VL > VH, which cannot hold in equilibrium. 1 1 Given the continuity of policy functions in survival probabilities and in φ, steady state wages vary continuously with p and φ. Since (3.1) holds when p = 0 and φ = 1, it must then L L continue to hold for p small enough and φ high enough, as claimed. L B Parameter selection We set the hazard rate φ out of unemployment to 0.5 so that unemployment spells last 1−.5 = 1 quarter on average. This is the average time between separation and re-employment .5 estimated by Anderson and Meyer (1994) using data fromeight state unemployment systems between 1978 and 1984. We set the replacement rate ρ to 60 percent, the average U.S. replacement rate (OECD, 1997). In our simulations, we assume that bene(cid:12)ts are available for two quarters, as they are in most U.S. states.15 We set τ to 21 percent, the overall payroll tax rate for the 1989 to 1994 period in the U.S. (Nickell and Layard, 1999). We use data on plant deaths by 4-digit NAICS industries to set p and p . The data L H were created by the Census Bureau using the 1998 and 1999 County Business Patterns data. We only consider industries with at least 1,000 establishments. The minimum and maximum annual plant death rates in the resulting sample were 4.6 percent and 20.1 percent. We set p and p to the corresponding quarterly survival rates of approximately 94 percent and 98 L H percent. To set β, the fraction of workers who remain in the labor force, we assume that transitions from employment to out of the labor force are only permanent for individuals over the age of 55. Fallick and Fleischman (2002), using Current Population Survey (CPS) 15With two quarters rather than one, assumption 2 and the (cid:12)rst-order conditions de(cid:12)ning the optimal training policies di(cid:11)er slightly from the previous sections, but the adjustment is trivial and does not alter any of our results. 24

data from 1994, 1996{2001, (cid:12)nd that individuals over the age of 55 account for 13.5 percent of employment in the U.S., and that 4.3 percent of those individuals, on average, leave the labor force each month. We therefore set our survival rate β to .865+.135(cid:2)(1−0.043)3 ’ 98 percent. The degree α of strict concavity of the production function is set to 0.64. This value is roughly equal to the average share of labor income in U.S. Gross National Product between 1960and1990impliedbystandardrealbusinesscyclecalculations(seeCooley,1995). Setting η, the quarterly rate of growth of output per unit of labor, is more di(cid:14)cult. Standard measures of labor productivity would overestimate that number since they do not take into account the fact that average labor quality rises over time in surviving (cid:12)rms due to training. Total factor productivity (TFP) is inadequate for the same reason. On the other hand, TFP may underestimate η since it controls for growth in the capital stock, which is not in our model. For lack of better data, we use Baily et al’s (1992) estimates of TFP growth in manufacturing plants between 1972 and 1987 as a rough guide. Conveniently for our purposes, they produce separate TFP growth estimates for plants who remain in the sample (i.e. do not fail) between census years. For these plants, they calculate (see table 2, p. 210) a compound TFP growth of 27 percent between 1972 and 1987 which translates into an average quarterly growth rate of 0.4 percent, our selection for η. We set the quarterly depreciation rate for on-the-job training, δ, to 2 percent. This value is the midpoint of the 4 to 12 percent range of annual rate estimates reported by Mincer (1991). For our speci(cid:12)cation of the function h, we assume that it is quadratic. Speci(cid:12)cally, h(s) = as−bs2 foralls 2 [0,1]where a,b > 0. Wechoosetheratioa/btomatchtheelasticity of labor productivity to the on-the-job training of newly hired employees reported by Barron et al. (1987). Barron et al. use data from a 1982 survey (cid:12)nanced by the National Institute of Education and the National Center for Research in Vocational Training. The survey collected data for 659 (cid:12)rms on the on-the-job training received by newly hired workers in the (cid:12)rst three months of employment, and the productivity and wages of those same workers after two years of employment. In each (cid:12)rm, a manager or (cid:12)rm owner provided training data on two recently employed workers (see Bishop, 1987, for a detailed description of the data). The fact that the survey focuses on new hires is convenient for our purposes since new hires are unambiguously at productivity level x in our model. Barron et al. estimate 1 that at the mean time devoted to training in their sample (151 hours in the (cid:12)rst 3 months, roughly 30 percent of an average hire’s hours worked), a 10 percent increase in training raises productivity (output per worker) by 3 percent after two years. Separately, Bishop (1991) calculates that roughly half of this productivity gain occurs during the (cid:12)rst quarter of employment.16 We therefore choose a/b so that at s = .3, sh0(s) = .5(cid:2).3, or, after some h(s) algebra, a ’ 2 . b 3 This leaves us with two parameters to set: a and x . We choose these parameters jointly 2 to match two statistics: 1) the average share of time devoted to on-the-job training by U.S. employees and 2) the average loss of earnings by high earners (see de(cid:12)nition below) in the U.S. economy following an involuntary separation. The (cid:12)rst statistic is notoriously 16At least two caveats are in order however. First, productivity estimates are derived from answers to qualitative questions inquiring about the performance of workers compared to their peers, and are subject to the standard criticism. Second, the survey combines (cid:12)rm-speci(cid:12)c and general training. 25

Table 3: Average loss of earnings among top earners following an involuntary separation College Gender Age education 1984 1986 1988 1990 1992 1994 1996 1998 2000 Male >40 No 29.82 28.39 29.81 22.27 21.44 25.58 20.35 17.45 30.99 (111) (141) (133) (138) (114) (61) (62) (32) (36) Male >40 Yes 21.79 19.04 21.65 11.31 21.02 17.08 12.96 6.47 7.21 (75) (107) (105) (87) (113) (81) (72) (41) (43) Male (cid:20)40 No 30.27 24.88 26.28 27.96 27.14 31.87 20.35 24.52 24.29 (58) (85) (80) (81) (72) (38) (37) (29) (34) Male (cid:20)40 Yes 19.46 16.49 30.48 17.25 33.43 28.05 21.91 13.80 20.79 (24) (45) (55) (48) (82) (62) (58) (42) (55) Female >40 No 8.86 12.67 9.32 19.85 13.73 14.70 15.66 0.90 11.29 (37) (55) (54) (66) (52) (22) (29) (15) (18) Female >40 Yes 25.03 20.81 23.91 20.50 13.96 15.07 10.30 11.32 27.86 (29) (41) (55) (45) (62) (44) (47) (31) (32) Female (cid:20)40 No 28.07 25.03 24.75 20.48 19.86 41.88 28.50 24.18 17.93 (30) (37) (42) (54) (45) (22) (25) (20) (23) Female (cid:20)40 Yes 13.76 19.09 17.04 10.44 30.53 10.21 18.87 19.04 13.79 (12) (21) (22) (22) (44) (25) (35) (30) (33) All top earners 24.49 22.65 25.04 19.72 22.70 21.44 18.21 14.04 19.00 (376) (532) (546) (541) (584) (355) (365) (240) (274) Memo: Entire sample 0.86 5.94 5.18 4.59 8.24 8.01 5.44 2.98 6.32 (1567) (2102) (2162) (2122) (2326) (1411) (1258) (942) (930) Notes: The sample consists of workerswho lost a job in the previous 5 years(3 years after 1994). All of the table entries, with the exception of those in the memo line, are for \top earners," de(cid:12)ned as workers whose earningsin the lostjobwere abovethe 75thpercentile in their gender-age-educationcategory. The numbers in parentheses denote the number of observations used to compute each statistic. di(cid:14)cult to obtain (see e.g. Barron et al., 1997). Based on a 1995 survey of employerprovided training of 1,074 employees from establishments with 50 or more employees, Frazis et al. (1998) calculate that, on average, employees receive 44.5 hours of formal and informal training during a 6 month period, roughly 4.5 percent of hours worked. This is the fraction we will match. For the second statistic, we use data from the displaced worker supplement to the January Current Population Survey which is available every other year between 1988 and 2000. We only consider workers between the ages of 16 and 65 who report having lost a job in the past 5 years (3 years in supplements after 1994), who were employed full-time in their previous jobs and are employed full-time in their current jobs, and who had at least one year of tenure in their previous job. We then classify workers into gender-age-education cells according to whether their age exceeds 40, and whether they have some college education.17 We focus on those workers whose CPI-deflated hourly earnings in their previous jobs exceeds the 75th percentile in their respective cells, our practical de(cid:12)nition of high earners.18 In our model, those observations would correspond to workers whose productivity level at the time they lost their job was x . We then compute the CPS-weighted average earnings loss in each 2 17The limited size of our sample forces us to use rather coarse categories. Nevertheless, using (cid:12)ner categories does not appear to signi(cid:12)cantly alter our main results. 18Each observation is weighted using the weights provided by the Census Bureau. 26

cell, and the CPS-weighted average earnings loss across cells. The results are shown in table 3. The average loss across cells turns out to be near 20 percent for most years. Given a set of parameters, the average earnings loss in our model is endogenous, as it depends on training decisions. We searched over wide grids for a and x and found that setting a = 1.3 and 2 x = 1.1 produces the desired steady state statistics. Strictly speaking, these values together 2 with the other parameters we chose do not satisfy assumption 4, but they bound average worker productivity growth su(cid:14)ciently to imply that all (cid:12)rms must hire new workers in all periods in steady state equilibrium. 27

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Figure 1: Quit Rates by Length of Tenure 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 5 10 15 20 25 30 35 40 Quarters of tenure )tnecrep( etar tiuQ Low survival firms High survival firms 30

Figure 2: Percent Deviations from Steady State Following a Temporary Shock to TFP A: Wage rates B: Output per worker 0 0 −0.2 −0.2 −0.4 −0.4 −0.6 −0.6 −0.8 −0.8 −1 −1 0 20 40 60 80 0 20 40 60 80 C: Training D: Quit rates 1.5 0.05 0 −0.05 1 −0.1 −0.15 0.5 −0.2 −0.25 0 0 20 40 60 80 0 20 40 60 80 F: Job destruction rates G: Unemployment rate 0.25 0.15 0.2 0.1 0.15 0.1 0.05 0.05 0 0 0.05 −0.05 0 20 40 60 80 0 20 40 60 80 Quarters Quarters Low survival firms High survival firms 31

Figure 3: Percent Deviations from Steady State Following a Temporary Shock to Both TFP and Firm Survival Rates A: Wage rates B: Output per worker 0 0 −0.2 −0.2 −0.4 −0.4 −0.6 −0.6 −0.8 −0.8 −1 −1 0 20 40 60 80 0 20 40 60 80 C: Training D: Quit rates 1.5 0.05 0 −0.05 1 −0.1 −0.15 0.5 −0.2 −0.25 0 0 20 40 60 80 0 20 40 60 80 F: Job destruction rates G: Unemployment rate 0.25 0.15 0.2 0.1 0.15 0.1 0.05 0.05 0 0 0.05 −0.05 0 20 40 60 80 0 20 40 60 80 Quarters Quarters Low survival firms High survival firms 32