Search, Matching and Training

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Search, Matching and Training Christopher Flinn, Ahu Gemici and Steven Laufer 2016-075 Please cite this paper as: Flinn, Christopher, Ahu Gemici and Steven Laufer (2016). “Search, Matching and Training,” Finance and Economics Discussion Series 2016-075. Washington: Board of Governors of the Federal Reserve System, http://dx.doi.org/10.17016/FEDS.2016.075. 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.

Search, Matching and Training Christopher Flinn Ahu Gemici Steven Laufer∗ New York University Royal Holloway, U. of London Federal Reserve Board Collegio Carlo Alberto July 13, 2016 Abstract We estimate a partial and general equilibrium search model in which firms and workers choose how much time to invest in both general and match-specific human capital. To help identify the model parameters, we use NLSY data on worker training and we match moments that relate the incidence and timing of observed training episodes to outcomes such as wage growth and job-to-job transitions. We use our model to offer a novel interpretation of standard Mincer wage regressions in terms of search frictions and returns to training. Finally, we show how a minimum wage can reduce training opportunities and decrease the amount of human capital in the economy. ∗WethankparticipantsattheHCEOconferenceonHumanCapitalandInequality, theIZA/Crest/Bonn Conferenceon”EmploymentPoliciesandHeterogeneityintheLaborMarket”,the2014SEDinToronto,and seminar participants at Duke, London School of Economics and George Washington for useful feedback. We havealsoreceivedhelpfulsuggestionsfromJesperBagger, JimHeckman, ThibautLamadon, RasmusLentz, Jean-Marc Robin, Chris Taber and Tony Yezer. Christopher Flinn thanks the National Science Foundation for partial support of this research. All remaining errors are our own. The views expressed herein are those of the authors and not necessarily those of the Board of Governors or others within the Federal Reserve System. 1

1 Introduction There is a long history of interest in human capital investment, both before and after entry into the labor market. In the latter case, it is common to speak of general and specifichumancapital,whicharedifferentiatedintermsoftheirproductivity-enhancing effects across jobs (which may be defined by occupations, industries, or firms). The classic analysis of Becker (1964) considered these types of investments in competitive markets and concluded that workers should pay the full costs of general training, with the costs of specific training (that increases productivity only at the current employer) being shared in some way. Analysis of these investments in a noncompetitive setting is more recent. Acemoglu and Pischke (1999) consider how the predictions of the amount and type of human capital investment in a competitive labor market are altered when there exist market imperfections in the form of search frictions. Frictions create an imperfect “lock in” between a worker and the firm, so that increases in general or specific human capital are generally borne by both the worker and the firm. We introduce training decisions into what is otherwise is a reasonably standard search model with general and specific human capital. The training data we use to estimate the model, described briefly below, indicate that formal training is reported by a not insignificant share of workers, and that the likelihood of receiving training is a function of worker characteristics, in particular, education. Workers do not receive trainingonlyatthebeginningofjobspells,althoughthelikelihoodofreceivingtraining is typically a declining function of tenure. Since training influences the likelihood of termination of the job and wages, it is important to examine training decisions in a relatively complete model of worker-firm employment relationships. One motivation for this research is related to recent observations regarding shifts in the Beveridge curve, which is the relationship between job vacancies and job searchers. While the unemployment rate in the U.S. has been markedly higher from 2008 and beyond,1 reported vacancies remain high. This mismatch phenomenon has been investigated through a variety of modeling frameworks (see, e.g., Cairo (2013) and Lindenlaub (2013)), typically by allowing some shift in the demand for workers’ skills. In our modeling framework, such a shift could be viewed as a downward movement in the distribution of initial match productivities. Given the absence of individuals with the desired skill sets, the obvious question is why workers and firms do not engage in onthe-job investment so as to mitigate the mismatch in endowments. Using our model, we can theoretically and empirically investigate the degree to which a decentralized labor market with search frictions is able to offset deterioration in the initial match productivity distribution. Anothermotivationforourresearchistoprovidearichermodelofthepathofwages on the job and a more complete view of the relationship between workers and firms. 1Although the unemployment rate has declined recently, the employment rate in the population is at a historic low. Many of those counted as out of the labor force are in fact willing to take a “reasonable” job offer, and hence should be considered to be “unemployed” in the true sense of the term. 2

In this model, firms offer workers the opportunity to make mutually advantageous investments in the worker’s skills, both of the general and specific (to the job) type. While investing, the worker devotes less time to productive activities, which is the only cost of investment that we include in the model.2 Wage changes over the course of the employment spell are produced by changes in general skill levels, changes in specific skill levels, and changes in investment time. In models that include on-the-job search possibilities, which is the case for ours, wages during an employment spell may also increase due to the presence of another firm bidding for the employee’s services, as in Postel-Vinay and Robin (2002), Dey and Flinn (2005), and Cahuc et al. (2006). When there exists the potential for other firms to “poach” the worker from her current firm, investments in match-specific skills will be particularly attractive to the current employer, since high levels of match-specific skills will make it less likely that the worker will exit the firm for one in which her initial match-specific skill level is higher. Other things equal, these differences in the retention value component of specific-skill investment implies that firms will reduce the employee’s wage less for a given level of specific-skill investment than for general skill investment. In our modelling framework, the complementarity between general and specific human capital gives firms an incentive to partially finance improvements in general ability. Specifically, the flow productivity of a match is given by y(a,θ) = aθ − ζ, where a is the general ability of the worker, θ is match productivity, and ζ is a flow cost of employment, which may be thought of as the rental rate on capital equipment required for the job. The gain in flow productivity from a small change in a is simply θ and the gain in flow productivity from a small change in θ is simply a. Jobs for which a is relatively high in comparison to θ will experience bigger productivity gains from investment in θ and conversely for job matches in which a is relatively low in comparison with θ. Thus, strictly from the productivity standpoint, there will be an incentive to “balance” a and θ in the investment process. This, coupled with the fact that there exist search frictions, will lead firms to be willing to finance part of the investment in general human capital, even if this does not change the expected duration of the match. We believe that our modeling framework may be useful in understanding the link between initial labor market endowments and earnings inequality over the labor market career. Flinn and Mullins (2015) estimated a model with an identical specification for flow productivity as the one employed here and examined the pre-market entry schooling decision.3 In their model, initial ability endowments were altered by school- 2Thatis,theonlycostsofinvestingineitheroftheskillsisthelostoutputassociatedwiththeinvestment time. Moreover, we will assume that these costs are the same for either type of investment. There are no direct costs of investment as in Wasmer (2006), for example. In his case, all investment in skills occurs instantaneously at the beginning of a job spell. Lentz and Roys (2015), instead, assume that general and match-specific skills are binary, and that for a low-type worker on either dimension the cost of training is a flow cost that is increasing in the rate at which the transition to the high skill type occurs. Depreciation in skills is not considered in either paper. 3They did not consider flow costs of employment, so in that paper ζ =0. 3

ing decisions, and these decisions were a function of all of the primitive parameters characterizing the labor market. In their setting, a was fixed over the labor market career and a match draw at a firm was also fixed over the duration of the job spell. In thecaseofourmodel, bothaandθ aresubjectto(endogenousandexogenous)change, although it may well be that a is more difficult and costly to change after labor market entry. This is due to the fact that employers are not equipped to offer general learning experiences as efficiently as are schools that specialize in increasing the cognitive and noncognitive abilities of their students. To the extent that a is essentially fixed over the labor market career, individuals with large a endowments will be more attractive candidates for investment in match-specific skills than will individuals with low values of a. Even if initial values of θ are drawn from the same distribution for all a types, which is the assumption that we make below, higher values of a at the time of labor market entry could lead to more investments and a more rapidly increasing wage profile over the course of an employment spell. This offers a mechanism to amplify the differences in earnings generated by the initial variability in a. Using estimates from the model, it is possible to examine the impact of various types of labor market policies on investment in the two types of human capital. For example, Flinn and Mullins (2015) investigated the impact of minimum wage laws on pre-market investment. They found that for relatively low (yet binding for some low a workers) minimum wage levels, the minimum wage could be a disincentive for premarketinvestment,sincesimilarwageratescouldbeobtainedwithoutcostlyeducation. At higher levels of the minimum wage, however, most individuals invested in a to increase their probability of finding a job. For firms to earn nonnegative profit flows in that model, productivity has to be at least aθ ≥ m, where m is the minimum wage. For high values of m, workers will invest in general ability to increase their level of a so as to increase the likelihood of generating a flow productivity level that satisfies the firm’s nonnegative flow profit condition. In our framework of post-entry investment,theimpactofminimumwagesisalsoambiguous. AsinthestandardBecker story for competitive labor markets, a high minimum wage will discourage investment activity if the firm is to achieve nonnegative flow profits. On the other hand, through investment activity that raises the individual’s productivity (through a and/or θ), the firm and worker can act to make the constraint nonbinding by pushing the individual’s productivity into a region for which w > m. These possibilities may mitigate the need to increase pre-market entry investment in a. In terms of related research, the closest paper to ours is probably Wasmer (2006). He presents a formal analysis of the human capital investment problem after market entry in a framework with search frictions and firing costs. His model is stylized, as is the one we develop below, and is not taken to data. He assumes that human capital investments, be it of the general or specific kind, are made as soon as the employment relationshipbetweenaworkerandafirmbegins. Investmentdoesnotexplicitlyinvolve time or learning by doing, which we believe to be an important part of learning on the job. However, due to the simplicity of the investment technology, Wasmer is able to characterize worker and firm behavior in a general equilibrium setting, and he provides 4

elegant characterizations of the states of the economy in which workers and firms will choose only general, only specific, or both kinds of human capital investment. One of the goals of our paper is to estimate both partial and general equilibrium versions of this type of model with what we think may be a slightly more realistic form of the human capital production technology, one in which time plays the central role. Another related paper is Bagger et al. (2014). This paper examines wage and employment dynamics in a discrete-time model with deterministic growth in general human capital in the number of years of labor market experience. There is no matchspecific heterogeneity in productivity, but the authors do allow for the existence of firm and worker time-invariant heterogeneity. There is complementarity between the worker’s skill level and the productivity level of the firm, so that it would be optimal to reallocate more experienced workers to better firms. The authors allow for renegotiation of wage contracts between workers and firms when an employed worker meets an alternative employer, and, due to the generality of human capital, the more productive firm always wins this competition. The model is estimated using Danish employer-employee matched data. Key distinctions between our approach and the one taken in that paper are the lack of firm heterogeneity but the presence of worker-firm match heterogeneity, the value of which can be changed by the investment decisions of the worker-firm pair. This paper also allows for worker heterogeneity that is an endogenous stochastic process partially determined through the investment decisions of workers and firms. Lentz and Roys (2015) also examine general and specific human capital accumulation in a model that features worker-firm renegotation and the ability of firms to make lifetime welfare promises to workers in the bargaining stage. There is firm heterogeneity in productivity, and the authors find that better firms provide more training. The nature of the contracts offered to workers is more sophisticated than the ones considered here, and the authors explicitly address the issue of inefficiencies in the training and mobility process. They assume that there are only four training states in the economy (an individual can be high or low skill in general and specific productivity), which greatly aids in the theoretical analysis at the cost of not being able to generate wage and employment sample paths that can fit patterns observed at the individual level. They also assume that there is no skill depreciation, which serves to simplify the theoretical analysis of the model. The model that we develop and estimate below has several notable features. As noted above, it endogenizes the general-productivity level of the individual and the match-productivity level of the worker-firm pair using a cooperative model of workerfirm interactions. In most of the literature on worker-firm sorting (e.g., Abowd et al. (1999), Postel-Vinay and Robin (2002), Cahuc et al. (2006)), match productivity is ignored and worker and firm types are assumed to be time-invariant. The focus of much of this literature is on worker-firm sorting patterns. Within our framework, it is clear that the total productivity of an employee at a particular firm is a fluid object, with “mismatches” being potentially rectified through cooperative investment choices by the worker and firm. Viewing the productivity of a worker at a given firm as an 5

endogenous stochastic process is an important point of differentiation of our model from most of the literature. The fact that productivity can be altered forces us to reconsider the usual constraints on firm hiring that are implied by models without endogenous productivity. When productivity is fixed, then for firms to at least break even on the employment contract, the flow profit of a firm should be non-negative. However, when productivity can be increased through investment, it is possible (and occurs given our model estimates) that firms could earn negative flow profits for some part of the employment contract. This is an especially important consideration when evaluating the impact of policies such as a mandated minimum wage. In Flinn (2006), for instance, establishing a binding minimum wage of m in a market (in which there was not one previously) immediatelycausedthelossofalljobsforwhichtheworker’sflowproductivitywasless than m. With endogenous productivity, this may no longer be the case, since through investment, the worker’s productivity could be raised sufficiently so that the firm earns positive flow profits. The extent to which this phenomenon occurs is an empirical matter, which we can investigate using our model estimates. Another literature to which this paper contributes is that concerned with the decomposition of the sources of wage growth over the life cycle, a literature the genesis of whichtracesbackatleasttoMincer, whoseworkonwagedeterminationissummarized in Mincer (1974). While much of the work in this literature focuses on the estimation of the return to schooling,4 our contribution is to the interpretation of the part of the earnings process associated with the worker’s total labor market experience and tenure on their current job. The model generates positive dependencies between the duration of time in the market and at the current job with wages since both duration measures are positively related to the values of general and match-specific human capital. As individuals age, there is a tendency for general human capital, a, to increase. Also, because the turnover decision is determined through the comparison between the match productivity value θ at the current firm, with the initial match productivity draw θ(cid:48) at the competing firm, longer employment durations at the firm indicate higher current values of match productivity. Even though the Mincer earnings function is largely an atheoretical construct, it is of interest to determine the extent to which the data generating process associated with our model produces relationships between wages, schooling, general experience, and job tenure broadly consistent with what would be found when estimating a Mincerian wage function using data from our sample. We find that, by and large, the wage functions estimated using the sample and those using simulated data from the model are roughly in agreement, even though the sample regression was not used in estimating the model parameters. This gives us some confidence that the model estimated does not generate empirical implications at variance with the results in literature focused on the estimation and interpretation of Mincerian earnings functions. 4Heckman et al. (2006) present an exhaustive consideration of the estimation of Mincer-type earnings functions, although their main focus is on the consistent estimation of rates of return to schooling. 6

The model estimates we present are obtained using data from the 1997 cohort of the National Longitudinal Survey of Youth (NLSY97), which consists of individuals betweentheagesof12-16attheendof1997. Theadvantageofusingthesedataarethat we can observed individuals from the beginning of their labor market careers, which minimizes initial conditions problems. Moreover, it is well-known that job changing behavior and wage growth is more pronounced at the onset of the labor market career. One negative aspect of estimating the model using only relatively recent labor market entrants is that we are likely to get predictions for later career events that are at odds with the data. This also implies that the steady state distributions implied by the model should be interpreted with this caveat in mind. Our future research plan is to extend the model to include endogenous pre-market entry schooling decisions and to estimate the model using samples with a larger range of ages and participation histories. The plan of the paper is as follows. In Section 2, we analyze a partial equilibrium search model with general and specific human capital and subsequently extend it to a general equilibrium framework. Section 3 discusses the data used in the estimation of the model, and presents descriptive statistics. Section 4 discusses econometric issues such as the model specification used in our estimation, the estimator we use, and identification. In Section 5, we present the estimation results and discuss the details of the estimated model, such as parameter estimates, model fit and policy rules. Section 5 also presents a discussion regarding the implications of our estimated model for sources of wage growth and provides a novel perspective on the interpretation of the standard Mincer wage regression. In Section 6, we conduct a minimum wage experimenttodeterminetheimpactofminimumwagesongeneralandspecifichumancapital investment decisions, in a partial as well as general equilibrium framework. Section 7 concludes the paper. 2 Modeling Framework Individuals are characterized in terms of a (general) ability level a, with which they enter the labor market.5 There are M values of ability, given by 0 < a < ... < a < ∞. 1 M 5Flinn and Mullins (2015) examine pre-market entry education decisions in a search environment in which a hold-up problem exists. We will not explicitly model the pre-market entry schooling decision, but will merely assume that the distribution of an individual’s initial value of a at the time of market entry is a stochastic function of their completed schooling level. In estimation, we will distinguish three schooling levels. 7

Whenanindividualoftypea encountersafirm,hedrawsavalueofθ fromthediscrete i distribution G over the K values of match productivity θ, which are given by 0 < θ < ... < θ < ∞. 1 K We denote the c.d.f. of θ by G, and we define p = Pr(θ = θ ), j = 1,...,K. j j The flow productivity value of the match is given by y(i,j) = a θ −ζ, i j where ζ is a flow cost of the job, which we think of as the rental rate on capital equipmentthatmustbeusedintheproductionprocessinadditiontothelaborinput. Inthe general equilibrium version of the model, to be discussed below, firms are assumed to pay flow posting fees while holding a vacancy open. One rationale for such a cost could bethatfirmsrentapieceofcapitalequipmentonwhichanindividual’sskillsatthejob can be assessed when they apply. In this case, it also seems reasonable to assume that a piece of capital equipment is required in the case in which the individual is hired. Under our assumptions on the distributions of a and θ, we obtain an estimate of ζ that is large and positive, and that significantly improves the fit of the model. It also serves to produce what we consider to be more reasonable human capital investment policy functions than when ζ = 0. We consider the case in which both general ability and match productivity can be changed through investment on the job. The investment level, along with the wage, are determined cooperatively in the model using a surplus division rule. At every moment of time, the individual and firm can devote a proportion τ of time to training a in general ability, in the hope of increasing a. Similarly, they can invest a proportion of time τ in job-specific training, in the hope of increasing θ. θ We will assume that the two stochastic production technologies for a and θ are independent, in the sense that the likelihood of an improvement in a depends only on τ and not on τ , and that the likelihood of an improvement in θ depends only on τ a θ θ and not τ . Given that the level of a is currently a , the rate of improvement in a is a i given by ϕ (i,τ ), a a with ϕ (i,τ ) ≥ 0, and ϕ (i,0) = 0 for all i. We restrict the improvement process to a a a increase the value of i to i + 1 in the case of a successful investment. We will also allow for reductions in the value of a. This depreciation rate is assumed to be constant and equal to δ for all i > 1. In the case of the arrival one of these Poisson-distributed a shocks, the level of a will decrease from i to i − 1, except when i = 1, when the individual is already at the lowest ability level. Since the rate of decreases in a are independent of investment time, the implication is that at the highest level of a, a , M no investment in a will occur. 8

For purposes of estimation, we further restrict the function ϕ to have the form a ϕ (i,τ ) = ϕ0(i)ϕ1(τ ), a a a a a where ϕ1 is strictly concave in τ , with ϕ1(0) = 0. The term ϕ0(i) can be thought of as a a a a total factor productivity (TFP) in a way, and we place no restriction on whether ϕ0(i) a is increasing or decreasing in i, although the functional form we utilize in estimation will restrict this function to be monotone.6 There is an exactly analogous production technology for increasing match-specific productivity, with the rate of increase from match value j to match value j +1 given by ϕ (j,τ ) = ϕ0(j)ϕ1(τ ), θ θ θ θ θ with ϕ1 strictly concave in τ , and ϕ1(0) = 0. There is no necessary restriction on the θ θ θ TFP terms, as above. As is true for the a process, there is an exogenous depreciation rate associated with all θ , j > 1, which is equal to δ . If one of these shocks arrive, j θ then match productivity is reduced from θ to θ . As was true in the case of a, if j j−1 match productivity is at its highest level, θ , then τ = 0. K θ The only costs of either type of training are foregone productivity, with total productivity given by (1−τ −τ )y(i,j). The gain from an improvement in either accrues a θ to both the worker and firm, although obviously, gains in general human capital increase the future value of labor market participation (outside of the current job spell) to the individual only. As noted by Wasmer (2006), this means that the individual’s bargaining position in the current match is impacted by a change in a to a greater extent than it is due to a change in θ. Motives for investment in the two different types of human capital depend importantly on the worker’s surplus share α, but also on all other primitive parameters characterizing the labor market environment. 2.1 No On-the-Job Search We first consider the case of no on-the-job search in order to fix ideas. In defining surplus, we use as the outside option of the worker the value of continued search in the unemployment state, given by V (i), and for the firm, we will assume that the value U of an unfilled vacancy is 0, produced through the standard free entry condition (FEC). We can write the problem as (cid:16) (cid:17)α max V˜ (i,j;w,τ ,τ )−V (i) V˜ (i,j;w,τ ,τ )1−α, E a θ U F a θ w,τ 6By this we mean that either ϕ0(1)≤ϕ0(2)≤...≤ϕ0(M) a a a or ϕ0(1)≥ϕ0(2)≥...≥ϕ0(M). a a a 9

where V˜ and V˜ functions are the value of employment to the worker and to the firm, E F respectively, given the wage and investment times. Wefirstconsidertheunemploymentstate. Wewillassumethattheflowvalueofunemployment to an individual of type a is proportional to a , or ba , i = 1,...,M,.where i i i b is a scalar parameter. The value of unemployed search can be written as (cid:80) ba +λ p V (i,j) i U j=r∗(i)+1 j E V (i) = , (1) U ρ+λUG˜(θ r∗(i) ) where λ is the rate of arrival of potential employment opportunities to the individual. U The discount rate ρ is the sum of the subjective discount rate of the individual, ρ , and 0 a constant death rate of (cid:96), so that ρ = ρ +(cid:96).7 It is assumed that the value associated 0 with the state of death is 0. The critical (index) value r∗(i) is defined by V (i) ≥ V (i,θ ) U E r∗(i) V (i) < V (i,θ ). U E r∗(i)+1 An agent of general ability a will reject any match values of θ or less, and accept i r∗(i) any match values greater than this.8 Given a wage of w and a training level of τ and τ , the value of employment of a θ type a at a match of θ is i j V˜ (i,j;w,τ ,τ ) = (ρ+ϕ (i,τ )+ϕ (j,τ )+δ˜ (i)+δ˜ (j)+η)−1× E a θ a a θ θ a θ [w+ϕ (i,τ )Q(i+1,j)+ϕ (j,τ )V (i,j +1)+δ˜ (i)Q(i−1,j) a a θ θ E a +δ˜ (j)Q(i,j −1)+ηV (a )], θ U i where δ˜ (i) = 0 if i = 1 and δ˜ (i) = δ if i > 1, for k = a,θ. The term k k k Q(i,j) ≡ max[V (i,j),V (i)], E U allows for the possibility that a reduction in the value of a or θ could lead to an endogenous termination of the employment contract, with the employee returning to the unemployment state. It also allows for the possibility that an increase in a from a i to a could lead to an endogenous separation. This could occur if the reservation θ, i+1 r∗(i),isincreasingini.Inthiscase,anindividualemployedattheminimallyacceptable match r∗(i)+1, may quit if a improves and r∗(i+1) ≥ r∗(i)+1. 7The death rate is introduced so as to produce more reasonable steady state distributions than the ones we generate using model estimates and an assumption that (cid:96) = 0. Less dramatically, we can think of this stateascorrespondingtoretirement,butitshouldbeborneinmindthatweassignthevalueofthisabsorbing state to be 0. 8Note that we assume that there are no shocks to the individuals’ ability level during unemployment. 10

The corresponding value to the firm is V˜ (i,j;w,τ ,τ ) = (ρ+ϕ (i,τ )+ϕ (j,τ )+δ˜ (i)+δ˜ (j)+η)−1× F a θ a a θ θ a θ [(1−τ −τ )y(i,j)−w+ϕ (i,τ )Q (i+1,j)+ϕ (j,τ )V (i,j +1) a θ a a F θ θ F (cid:105) +δ˜ Q (i−1,j)+δ˜ Q (i,j −1) a F θ F where Q (i,j) = 0 if Q(i,j) = V (i) and Q (i,j) = V (i,j) if Q(i,j) = V (i,j).9 F U F F E Then the solution to the surplus division problem is given by (cid:16) (cid:17)α {w∗(i,j),τ∗(i,j),τ∗(i,j)} = arg max V˜ (i,j;w,τ ,τ )−V (i) a θ E a θ U w,τa,τ θ ×V˜ (i,j;w,τ ,τ )1−α; F a θ V (i,j) = V˜ (i,j;w∗(i,j),τ∗(i,j),τ∗(i,j)), E E a θ V (i,j) = V˜ (i,j;w∗(i,j),τ∗(i,j),τ∗(i,j)). F F θ More specifically, the surplus division problem is given by max (ρ+ϕ (i,τ )+ϕ (j,τ )+δ˜ (i)+δ˜ (j)+η)−1 a a θ θ a θ w,τa,τ θ (cid:20) (cid:21)α w+ϕ (i,τ )[Q (i+1,j)−V (i)]+ϕ (j,τ )[V (i,j +1)−V (i)] × a a E U θ θ E U +δ˜ (i)[Q(i−1,j)−V (i)]+δ˜ (j)[Q(i,j −1)−V (i)]−ρV (a ) a U θ U U i (cid:20) (cid:21)1−α (1−τ)y(i,j)−w+ϕ (i,τ )Q (i+1,j)+ϕ (j,τ )V (i,j +1) × a α F θ θ F . +δ˜ (i)Q (i−1,j)+δ˜ (j)Q (i,j −1) a F θ F The first order conditions for this problem can be manipulated to get the reasonably standard wage-setting equation, w∗(i,j) = α{(1−τ∗−τ∗)y(i,j)+ϕ (i,τ∗)Q (i+1,j)+ϕ (j,τ∗)V (i,j +1) a θ a a F θ θ F +δ˜ (i)Q (i−1,j)+δ˜ (j)Q (i,j −1)} a F θ F +(1−α){ρV (i)−ϕ (i,τ∗)(V (i+1,j)−V (i)) U a a E U −ϕ (j,τ∗)(V (i,j +1)−V (i))]−δ˜ (i)Q(i−1,j)−δ˜ (j)Q(i,j −1)}. θ θ E U a θ The first order conditions for the investment times τ and τ are also easily derived, a θ but are slightly more complex than the wage condition. The assumptions regarding the investment technologies ϕ and ϕ will obviously have important implications for a θ 9Note that the discount rate ρ is the same for both workers and firms, even if we do not think of firms as being subject to a death shock. In this framework, there is essentially one worker per firm, and the worker’s “death” terminates the match just as does a shock dissolving that particular job, η, assumed to arise due to changes in demand conditions. In both cases, the firm is left without an employee, and we use the FEC associated with vacancies to apply the value of 0 in either case. 11

the investment rules. The time flow constraint is 1 ≥ τ +τ , a θ τ ≥ 0 a τ ≥ 0. θ Depending on the parameterization of the production technology, it is possible that optimal flow investment of either type is 0, that one type of investment is 0 while the other is strictly positive, and even that all time is spent in investment activity, whether it be in one kind of training or both. In such a case, it is possible to produce the implication of negative flow wages, and we shall not explicitly assume these away by imposing a minimum wage requirement in estimation. In the case of internships, for example, which are supposed to be mainly investment activities, wage payments are low or zero. Including the worker’s direct costs of employment, the effective wage rate may be negative. What is true is that no worker-firm pair will be willing to engage in such activity without the future expected payoffs being positive, which means that the worker would generate positive flow profits to the firm at some point during the job match. 2.2 On-the-Job Search In the case of on-the-job search, individuals who are employed are assumed to receive offers from alternative employers at a rate λ , and it is usually the case that λ < λ . E E U If the employee meets a new employer, the match value at the alternative employer, θ , is immediately revealed. Whether or not the employee leaves for the new job and j(cid:48) what the new wage of the employee is after the encounter depends on assumptions made regarding how the two employers compete for the individual’s labor services. In Flinn and Mabli (2009), two cases were considered. In the first, in which employers are not able to commit to wage offers, the outside option in the wage determination problem always remains the value of unemployed search, since this is the action available to the employee at any moment in time. This model produces an implication of efficient mobility, in that individuals will only leave a current employer if the match productivity at the new employer is at least as great as current match productivity (general productivity has the same value at all potential employers). An alternative assumption, utilized in Postel-Vinay and Robin (2002), Dey and Flinn (2005), and Cahuc et al. (2006), is to allow competing employers to engage in Bertrand competition for the employee’s services (this model assumes the possibility of commitment to the offered contract on the part of the firm). In this case, efficient mobility will also result, but the wage distribution will differ in the two cases, with employees able to capture more of the surplus (at the same value of the primitive parameters) in the case of Bertrand competition. We begin our discussion with the Bertrand competition case in this section, altough for reasons explained at the end of this section, the empirical work will emphasize the no-renegotiation case. 12

2.2.1 On-the-Job Search with Bertrand Competition WhetherornotweassumethatfirmsengageinBertrandcompetition,a hasnoimpact i on mobility decisions, since it assumes the same value across all employers. In the Bertrand competition case (as in Dey and Flinn (2005), for example), the losing firm in the competition for the services of the worker is willing to offer all of the match surplus in its attempt to retain the worker. For example, let the match value at the current employer be θ , and the match value at the potential employer be θ . We will j j(cid:48) denote the maximum value to an employee of type a of working at a firm where her i match value is θ by V¯(i,j), which is the case in which the employee captures all of the j match surplus (since the value of holding an unfilled vacancy is assumed to be equal to 0, by transferring all of its surplus to the employee, the firm is no worse of than it would be holding an unfilled vacancy). In the Bertrand competition case then, and assuming that j(cid:48) ≤ j, the losing firm offers V¯(i,j(cid:48)) for the individual’s labor services. The winning firm then divides the surplus with the employee, where the employee’s outside option becomes V¯(i,j(cid:48)). Note that in the case that j(cid:48) = j, the individual would be indifferent between the two firms, the two firms would be indifferent with respect to hiring her or not, and whichever offer it accepted, the employee would capture the entire match value, that is, V (i,j) = V¯(i,j). Because of the investment possibilities, E it is not generally the case that the wage at the winning firm will be equal to a θ −ζ, i j which would be true when there are no investment possibilies. In the case of on-the-job search with Bertrand competition between employers, we denote the value of the employment match to the worker and the firm by V(i,j,j(cid:48)) andV (i,j,j(cid:48)),respectively. Thefirstargumentdenotestheindividual’sgeneralability F type, a , and the second denotes the value of the match at the employer. The third i argumentinthefunctionisthehighestmatchvalueencounteredduringthecurrentemployment spell (which is a sequence of job spells not interrupted by an unemployment spell)atanyotheremployer. Sincemobilitydecisionsareefficient,weknowthatj(cid:48) ≤ j. When the individual has encountered no other match values during the current employment spell that exceeded the value r∗(i)+1, then we will write V(i,j,j∗(i)). When an individual encounters a new firm with a new match draw j(cid:48)(cid:48), then the individual’s new value of being employed is given by V(i,j(cid:48)(cid:48),j) if j(cid:48)(cid:48) > j V(i,j,j(cid:48)(cid:48)) if j ≥ j(cid:48)(cid:48) > j(cid:48) (2) V(i,j,j(cid:48)) if j(cid:48) ≥ j(cid:48)(cid:48) In the first row, the individual changes employer, and now the match value at the current employer becomes the next best match value during the current employment spell. In the second row, the employee stays with her current employer, but gains more of the total surplus associated with the match, which implies an increase in her wage at the employer. In the third row, the individual does not report the encounter to her current employer, since it doesn’t increase her outside option. To see these effects more formally, we first consider the case in which a worker with 13

a current match value of θ who has previously worked at a job with a match value of j θ , k ≤ j, and where there was no intervening unemployment spell. In this case, we k write the worker’s value given wage w and training time τ as N (w,τ ,τ ;i,j,k) V˜ (i,j,k;w,τ ,τ ) = E a θ , E a θ D(τ ,τ ;i,j,k) a θ where j (cid:88) (cid:88) N (w,τ ,τ ;i,j,k) = w+λ [ p V (i,j,s)+ p V (i,s,j)] E a θ E s E s E s=k+1 s=j+1 +ϕ (i,τ )Q(i+1,j,k)+ϕ (j,τ )V (i,j +1,k) a a θ θ E δ˜ (i)Q(i−1,j,k)+δ˜ (j)Q(i,j −1,k)+ηV (i); a θ U D(τ ,τ ;i,j,k) = ρ+λ G˜(θ )+ϕ (i,τ )+ϕ (j,τ ) a θ E k a a θ θ δ˜ (i)+δ˜ (j)+η. a θ The term Q(i + 1,j,k) = max{V(i + 1,j,k),V (i + 1)}, indicating the possibility U that an increase in a could lead to an endogenous separation depending on the value of θ . The term Q(i − 1,j,k) = max{V(i − 1,j,k),V (i − 1)}, indicating that the j U value of unemployed search has decreased as well. Finally, we have Q(i,j − 1,k) = max{V(i,j −1,min(j −1,k)),V (i)}. In the case where j = k, this implies that the U value of the outside option is reduced with the current match value. We impose this convention so as to keep the surplus division problem well-defined. Other assumptions could be made regarding how the negotiations between and employer and employee are impacted when the match value decreases. The value to the firm is given by N (w,τ ,τ ;i,j,k) V˜ (i,j,k;w,τ ,τ ) = F a θ , F a θ D(τ ,τ ;i,j,k) a θ where N (w,τ ,τ ;i,j,k) = y(i,j)(1−τ −τ )−w+ϕ (i,τ )Q (i+1,j,k) F a θ a θ a a F +ϕ (j,τ )V (i,j +1,k)+δ˜ (i)Q (i−1,j,k) θ θ F a F j (cid:88) +δ˜ (j)Q (i,j −1,min(j −1,k))+λ p V (i,j,s). θ F E s F s=k+1 14

Now the surplus division problem is max D(τ ,τ ;i,j,k)−1[N (w,τ ,τ ;i,j,k)−V¯(i,k)]α a θ E a θ w,τa,τ θ ×N (w,τ ,τ ;i,j,k)1−α, F a θ which is only slightly more involved than the problem without OTJ search, but the generalization yields another fairly complex dependency between the current value of the match and the training time decisions. It is clear that the value of match-specific investment to the employer in the case of OTJ search is even higher than in the no OTJ case, since it also increases (in expected value) the duration of the match, and this value always exceeds the value of an unfilled vacancy, which is 0. The value of either type of training is also enhanced from the point of view of the worker, since in additiontoincreasinghervalueathercurrentemployer,highervaluesofaorθ enhance her future bargaining position during the current employment spell, and, in the case of a, even beyond the current employment spell. Once the employment spell ends, the bargaining advantage from the match history ends, including gains accumulated through investment in match-specific productivity. On the other hand, the value of previous investments in general productivity is carried over, in a stochastic sense, whichiswhatmakesthistypeofhumancapitalparticularlyvaluablefromtheworker’s perspective, and accounts for her disproportionate costs of funding these investments. Finally, the value of unemployed search is given by ba +λ (cid:80) p V (i,j,j∗(i)) i U j=r∗(i)+1 j E V (i) = . U ρ+λUG˜(θ r∗(i) ) As we have seen, in the case of Bertrand competition, there is some arbitrariness in defining the employment state when the outside option and current match values are equal and there is depreciation in the current match value. In this case, we have simply assumed that the employee continues to receive the entire surplus of the match, although this total surplus has decreased due to the decrease in match-specific productivity from θ to θ . j j−1 2.2.2 On-the-Job Search with No Renegotiation The other case of employer-employee interaction we consider is when employers do not respond to outside offers. This would be the case when outside offers cannot be observedandverified. Moreover,eveniftheywere,employershaveanincentivetocheat on the employment contract agreed to once the outside offer is no longer available. When the outside offer is removed, the employee’s only alternative is to quit into unemployed search, so that this is the outside option considered when deciding upon wage-settingandtheamountofworktimedevotedtoinvestment. Inthiscase,decisions are considerably simplified. As in the case of no OTJ search, the employment contract is only a function of the individual’s type and the current match value, (i,j). The 15

property of efficient turnover decisions continues to hold, with the employee accepting all jobs with a match value j(cid:48) > j, and refusing all others. The formal structure of the problem is modified as follows. N (w,τ ,τ ;i,j) V˜ (i,j;w,τ ,τ ) = E a θ , E a θ D(τ ,τ ;i,j) a θ where (cid:88) N (w,τ ,τ ;i,j) = w+λ p V (i,s) E a θ E s E s=j+1 +ϕ (i,τ )Q(i+1,j)+ϕ (j,τ )V (i,j +1) a a θ θ E +δ˜ (i)Q(i−1,j)+δ˜ (j)Q(i,j −1)+ηV (i); a θ U D(τ ,τ ;i,j) = ρ+λ G˜(θ )+ϕ (i,τ )+ϕ (j,τ ) a θ E j a a θ θ +δ˜ (i)+δ˜ (j)+η. a θ The term Q(i+1,j) = max{V(i+1,j),V (i+1)}, indicating the possibility that an U increase in a could lead to an endogenous separation depending on the value θ . The j term Q(i−1,j) = max{V(i−1,j),V (i−1)}, indicating that the value of unemployed U search has decreased as well. Finally, we have Q(i,j −1) = max{V(i,j −1),V (i)}. U The value to the firm conditional on the wage and investment decisions is given by N (w,τ ,τ ;i,j) V˜ (i,j;w,τ ,τ ) = F a θ , F a θ D(τ ,τ ;i,j) a θ where N (w,τ ,τ ;i,j) = y(i,j)(1−τ −τ )−w+ϕ (i,τ )Q (i+1,j) F a θ a θ a a F +ϕ (j,τ )V (i,j +1)+δ˜ (i)Q (i−1,j) θ θ F a F +δ˜ (j)Q (i,j −1), θ F and where Q (i+1,j) = V (i+1,j) if V(i+1,j) > V (i+1) and equals 0 otherwise, F F U Q (i − 1,j) = V (i − 1,j) if V(i − 1,j) > V (i − 1) and equals 0 otherwise, and F F U Q (i,j−1) = V (i,j−1) if j−1 > r∗(i). Now the surplus division problem becomes: F F max D(τ ,τ ;i,j)−1[N (w,τ ,τ ;i,j)−V (i)]α a θ E a θ U w,τa,τ θ ×N (w,τ ,τ ;i,j)1−α. F a θ 16

The value of unemployed search in this case is simply (cid:80) ba +λ p V (i,j) i U j=r∗(i)+1 j E V (i) = . U ρ+λUG˜(θ r∗(i) ) In what follows, we will emphasize the estimates associated with the no renegotiation model. This is due to its relative simplicity, and the fact that in other studies (FlinnandMabli(2009), FlinnandMullins(2015))andinthisone, wehavefoundthat the no renegotiation model fits the sample characteristics used to define our Method of Simulated Moments (MSM) estimator better than does the Bertrand competition model. Of course, in a model without investment options, the model without renegotiation implies that wages will be constant over a job spell of an individual and a particular firm. The Bertrand competition assumption in a stationary search setting implies that wage gains may be observed over the course of a job spell, but never wage declines. In the data we see a number of wage decreases over a job spell. No doubt, manyoftheseareduesolelytomeasurementerror, orthefactthatwagesfixedinnominal terms across interview dates will imply real wage declines in the face of inflation. Our model, with endogenous productivity shocks in both general and specific human capital, is capable of generating both types of wage fluctuations without relying on the use of difficult to verify bargaining protocols. 2.3 Equilibrium Model The model described to this point is one set in partial equilibrium, with contact rates between unemployed and employed searchers and firms viewed as exogenous. The model can be closed most simply by employing the matching function framework of Mortensen and Pissaridies (1994). We let the measure of searchers be given by S = U + ξE, where U is the steady state measure of unemployed and E is the measure of the employed (E = 1−U, since we assume that all individuals are participants in the labor market). The parameter ξ reflects the relative efficiency of search in the employed state, and it is expected that 0 < ξ < 1. We denote the measure of vacancies posted by firms by v. The flow contact rate between workers and firms is given by M = Sφv1−φ, with φ ∈ (0,1).10 Letting k ≡ v/S be a measure of labor market tightness, we can write the rate at which searchers contact firms holding vacancies by M λ = = kφ. F v 10We have fixed TFP =1 in the Cobb Douglas matching function due to the impossibility of identifying this parameter given the data available. The number of matches is unobserved, so that this essentially amounts to a normalization. 17

The proportion of searchers who are employed is given by ξE/S, so that the mass of matches that involve an employed worker is simply ξE/S ×M, which means that the flow rate of contacts for the employed is ξE Sφv1−φ λ = E S E = ξkφ−1. By a similar argument, the mass of matches involving an unemployed worker is U/S× M, and the contact rate for unemployed searchers is λ = kφ−1. U A fact that will be utilized in the estimation of demand side parameters below is that ξ = λ /λ . E U Turning to the firm’s problem, let the flow cost of holding a vacancy be given by ψ > 0.Thedistributionofpotentialhiresisdeterminedbythesteadystatedistributions of a among the unemployed and (a,θ) among the employed, which are complex objects that have no closed form solution, due to the (endogenous) dynamics of the a and θ processes in the population. However, these distributions are well-defined objects, the values of which can be obtained through simulation. The way we obtain the steady state distributions through simulation is given in Appendix A. Let the steady state distribution of a among the unemployed be given by {πU}, i = i 1,...,M, and the steady state distribution of (a,θ) among the employed by {πE }, i,j i = 1,...,M,j = 1,...,K. Then the expected flow value of a vacancy in the steady state is given by −ψ+ λ F ×{U (cid:88) (cid:88) p V (i,j)πU S j F i i j≥r∗(i)+1 (cid:88)(cid:88)(cid:88) +ξE p V (i,j(cid:48))πE }. j(cid:48) F i,j i j(cid:48)>j j Byimposingafreeentryconditiononfirmsthatequatesthisvaluetozero,theequation can be solved for equilibrium values of λ and λ given knowledge of the parameters U E ψ, φ, and ξ. 3 Data WeutilizedatafromtheNationalLongitudinalSurveyofYouth1997(NLSY97)toconstructourestimationsample. TheNLSY97consistsofacross-sectionalsampleof6,748 respondents designed to be representative of people living in the United States during the initial survey round and born between January 1,1980 and December 31,1984, and 18

a supplemental sample of 2,236 respondents designed to oversample Hispanic, Latino and African-American individuals. At the time of first interview, respondents’ ages range from 12 to 18, and at the time of the interview from the latest survey round, their ages range from 26 to 32. For our analysis, we use a subsample of 1,994 respondents from the NLSY97. We obtain this sample through three main selection criteria: (1) the oversample of Hispanic, Latino and African-American respondents is excluded so that the final sample comprises only the nationally representative cross-sectional sample, (2) the military sample is excluded, and (3) all females and high-school dropouts are excluded. A respondent who satisfies these criteria enters our sample after having completed all schooling. The estimation sample is constructed this way since our model is not designed to explain behavior while in school; and staying in school or continuing education are not endogenous choices. These sample selection criteria give us an unbalanced sample of 1,994 individuals and 661,452 person-week observations. The proportion of high school graduates is 37 percent and the proportion of those with some college and those with a college degree are 30 and 33 percent, respectively. The NLSY97 provides detailed retrospective data on the labor market histories and thewageprofilesofeachrespondent. Thisretrospectivedataisincludedintheemployment roster, which gives the start and end dates of each employment spell experienced by the respondent since the last interview, wage profiles and other characteristics of each employment (or unemployment) episode. We use the employment roster to construct weekly data on individual labor market histories. This information provides us with some of the key moments that identify the parameters of the search environment faced by the agents in our model, including transitions between jobs. While we make extensive use of the weekly data constructed retrospectively from the NLSY97 employment rosters for obtaining moments related to employment transition, ourempiricalanalysisofwagesusesinformationcollectedfromrespondentsabout current wages as of each interview date. This information is likely to have fewer measurement problems than wage information in the employment roster, which is collected as part of a set of retrospective questions about all current and previously held jobs since her previous interview. Inthecontextofthemodel,thedurationofajobspellisindicativeofthevalueofthe match between worker and firm. Therefore, looking at job spells of different lengths provides information about how wages and training differ at different match values. Our decision to only use wage observations from interview dates suggests defining the length of a job spell as the number of annual wage observations rather than using length from the employment roster. Table 1 shows the percentage of job spells by the number of interview dates they span for each schooling level. This distribution closely mirrors the actual duration distribution of jobs obtained from the employment rosters, suggesting that the two approaches should yield similar conclusions. For high school graduates, the table shows that about 61 percent of all observed job spells cover no interview dates at all; while 23 percent spells span one interview date, 7 percent 19

last long enough to span two interview dates and 9 percent span three or more. The proportion of job spells with longer durations, and therefore spanning more interview dates, increases by education level. For example, for individuals with a college degree, 15 percent of job spells span more than two interview dates. Given the importance of schooling on the labor market environment faced by the agentsinourmodel,wedistinguishbetweenthreegroupsofindividualsinourempirical analysis: (1) individuals who have a high-school degree, (2) individuals who have attended college but who do not have a college degree, and (3) Individuals with a college degree or more. In what follows, we refer to these three levels as low, medium and high education groups, respectively. In addition to key labor market variables, NLSY97 contains a wealth of information about training, which in our model is the way workers and firms invest to build human capital. For this aspect of our analysis, we use NLSY97’s training roster, where respondents are asked about what types of training they receive over the survey year and the start and end dates of training periods by source of training.11 Combining the information from the employment and training rosters, we construct a weekly event history of employment and training for each respondent. We do not make assumptions regarding the specificity of human capital acquired during a training episode. Instead, we use the empirical relationship between the patterns of training and previous/future employment and wage transitions in order to make inferences about the degree of specificity in the human capital accumulation process. Tables 2-3 present some descriptive statistics on the training patterns observed in oursample. Morespecifically,thesetablesdisplaytheincidenceoftrainingbyschooling and the timing of training spells by job tenure. The proportion of respondents with at least one training spell is 18, 13 and 13 percent for workers with low, medium and high education, respectively. This suggests a negative relationship between schooling and training. We next discuss training in relation to employment and wage transitions in our sample. Table 4 provides detailed information about employment and wage transitions between interview dates. We distinguish between three types of employmentto-employment transitions that may occur between interview dates t − 1 and t: (1) transitions that do not involve a change in employer, (2) transitions that involve a change in employer, with no intervening spell of non-employment between the two jobs, and (3) transitions that involve a change in employer, with an intervening spell of non-employment.12 The transitions that involve a change in employer are usually referred to as job-to-job transitions in the literature and we follow the same definitions 11Someexamplestosourcesoftrainingarebusinesscolleges,nursingprograms,apprenticeships,vocational andtechnicalinstitutes,barberandbeautyschools,correspondencecoursesandcompanytraining. Training received in formal regular schooling programs is included in the schooling variables. 12Usingtheemploymentrosters,wedeterminethattherewasaninterveningnon-employmentspellbetween two consecutive jobs, if individuals are observed to be not working for a period of at least 13 weeks between the first employment episode that covers their interview date t−1 and second employment episode that covers their interview date t. 20

in our discussion. Among workers who are employed in two successive interviews, the fraction of workers who change jobs decreases with education. For example, among high school graduates who remain employed at consecutive interview dates t−1 and t, 19 percent had a different employer, compared to only 12 percent of college graduates. Among workers who do change jobs, those with more education are more likely to do so without an intervening spell of non-employment. As discussed previously, wage growth within and across job spells are important indicators of which type of human capital investment behavior workers engage in. Panel B of Table 4 shows the difference between log wages for employment-to-employment transitions between interview dates. Again, we distinguish between the three types of transitions described above. We observe that log wage difference (logw −logw ) for t t−1 job-to-job transitions increases by education level: the average log wage difference is 0.11,0.15and0.20forlow,mediumandhigheducationgroups,respectively. Thedifferences by education are particularly large for job-to-job transitions with an intervening non-employment spell. Finally, Table 5 shows average log wage difference between consecutive interview dates t−1 and t, broken down by whether the worker receives training at the job he held at t−1. An individual is considered to have received training if the training roster reports him to have been enrolled in a training program during a week that was (1) before interview date t−1 (2) while he was also employed at the job he held at time t−1. Weseethatforemploymenttransitionsthatdonotinvolveachangeinemployer, the average log wage difference does not change by whether the worker obtained any training in the past. On the other hand, for employment transitions that do entail an employer change (i.e. job-to-job transitions), the average log wage difference between t and t−1 for those individuals who obtained some form of training in the first job spell is smaller. It is also instructive to examine the wage differences for job-to-job transitions that involve an intervening non-employment spell. These are displayed in the last two rows of Table 5. We see that for individuals who moved to their next job with no intervening spell of non-employment, the average log wage difference is 0.09 if they they received training in the previous job and 0.15 if they did not. These moments will serve as a basis for our estimation of the parameters in the model described above. 4 Econometric Issues 4.1 Empirical Implementation of the Model We make several assumptions in order to solve the model, which does not produce closed form solutions. We restrict workers and firms to choose training times from a discrete choice set consisting of multiples of five percent of the worker’s total time (τ ,τ ∈ {.00,.05,.10,...1.00}). The production functions are assumed to have the a θ following functional forms. Recall that there are M values of a, 0 < a < ... < a . 1 M 21

There is no ability to increase ability if an individual is already at the highest level, so the hazard rate for improvements for a is equal to 0. For i < M, we have that the M hazard rate to level i+1 is given by ϕ a (i,τ a ) = δ a 0×a δ i a 1 ×(τ a )δ a 2 , where δ0, δ1, and δ2 are scalar constants. Similarly, there are K values of θ, 0 < θ < a a a 1 ... < θ , and no possibility to increase match productivity when θ = θ . For a worker K K with j < K who spends a fraction τ of her time in firm-specific training, the value of θ the match increases at rate ϕ θ (j,τ θ ) = δ θ 0×θ j δ θ 1 ×(τ θ )δ θ 2 , where again δ0, δ1, and δ2 are scalar constants. θ θ θ Because it is difficult to separately identify the level of general ability and match quality, we attempted to make the support of the distributions of a and θ as symmetric as possible. Therefore, we choose identical grids for for a and θ . We chose grid points i j to cover the range of likely values of θ including the possibility that workers with high values of θ will receive match-specific training that will produce match values above the set of values that they would naturally receive from searching. In the end, we use a grid containing 24 points which are spaced logarithmically from 2.5 standard deviations below the mean of the theta distribution to 3.5 standard deviations above it. At the estimated parameters of our baseline model, moving up by one grid point in either a or θ corresponds to a roughly 9 percent increase in productivity. Several model parameters are fixed outside the estimation. We choose α = 0.5, giving the worker and firm equal bargaining weight. All rate parameters are expressed at a weekly frequency and we set the discount factor ρ = 0.0016, corresponding to to a four percent annual discount rate. Finally, we set the death shock to produce an average career length of 45 years, (cid:96) = 1/(45·52) = 0.00043. Training observed in the data is likely a very rough proxy for the amount of time spent developing workers’ human capital. To relate our observed measures of training in the data to the training time chosen in the model simulations, we assume that a worker who spends a fraction of time τ engaged in training is observed to receive training is that period with probability Prob(Training observed | τ) = Φ(β +β τ) 0 1 where Φ is the c.d.f. for the normal distribution. In calculating τ from the simulations, we compute the average fraction of time spent training over each six month period, or, for job spells lasting less than six months, over the entire job spell. We estimate the parameters β and β jointly with the other parameters of the model, giving us a total 0 1 of 21 parameters to estimate. 22

4.2 Estimator 4.2.1 Estimation of Supply-Side Parameters We utilize a method of simulated moments estimator (MSM) in order to estimate all of the parameters of the model with the exception of those characterizing firms’ vacancy decisions. Under the data generating process (DGP) of the model, there are a number of sharp restrictions on the wage and mobility process that are generally not consistent with the empirical distributions observed. In such a case, measurement error in wage observations is often added to the model, with the variance of this measurement error estimated together with the other model parameters. This is not really a feasible alternative here given that we are already trying to estimate a convolution, so that the addition of another random variable to the wage and mobility processes can only exacerbate the difficulty of separately identifying the distributions of a and θ, particularly given their endogeneity with respect to investment decisions.13 We chose to use a moment-based estimator which employs a large amount of information characterizing wage distributions within and across jobs, often by schooling class, as well as some training information, as was described in the previous section. TheinformationfromthesamplethatisusedtodefinetheestimatorisgivenbyM , N where there are N sample observations. Under the DGP of the model, the analogous characteristics are given by M˜(ω), where ω is the vector of all identified parameters (which are all parameters and decision rules except ρ). Then the estimator is given by ωˆ = argmin(M −M˜(ω))(cid:48)W (M −M˜(ω)), N,WN ω∈Ω N N N where W is a symmetric, positive-definite weighting matrix and Ω is the parameter N space. The weighting matrix, W , is a diagonal matrix with elements proportional N to the inverse of the variance of the corresponding element of M . Under our ran- N dom sampling assumption, we have that plim M = M, the population value of N→∞ N the sample characteristics used in estimation. Since W is a positive-definite matrix N by construction, our moment-based estimator is consistent since plim ωˆ = ω N→∞ N,Q for any positive-definite matrix Q. We compute bootstrap standard errors using 50 replications. 4.2.2 Demand-Side Parameter Estimator: Method 1 It is most often the case that the parameters characterizing firms’ vacancy creation decisions are not identified. Our estimator of an employment cost parameter may 13We do introduce measurement error into the wage observations, but the variance of this error is fixed rather than being estimated with the other model parameters. Including some realistic measurement error allows us to consider higher moments of the wage distribution, while imposing a fixed variance prevents the estimator from having to separately identify yet another source of variation in observed wages. Following Bound et. al) (1994), we set the standard deviation of the measurement error on observed log wages at σ =0.15 e 23

enable identification of a Cobb-Douglas matching function parameter. In this section we explore how this can be accomplished. Recall that the matching function was defined as M = νφS1−φ, with ν ∈ (0,1), and the measure of searchers was given by S = U +ξE, where U is the measure of unemployed and E is its complement. The parameter ξ is a measure of the search efficiency of employed agents relative to that of the unemployed, and it is expectedthatξ ∈ (0,1).Therateatwhichemployerswithvacanciescontactapplicants is M λ = F ν = νφ−1S1−φ = kφ−1, where k ≡ ν/S is our measure of labor market tightness. The proportion of matches that involve an unemployed worker is given by U M, U +ξE so that the contact rate per unemployed searcher is U M λ = U S U = kφ. The contact rate for employed searchers is ξE M λ = E S E = ξkφ. Proposition 1 If a consistent estimator of the cost of posting a vacancy, ψ, is available, then the matching function parameter φ can be consistently estimated. Proof Our first stage MSM estimator produces consistent estimates of λ and λ . U E Then a consistent estimator of ξ is given by ξˆ= λˆ /λˆ . E U Using the model estimates, we can compute consistent estimates of the steady state values of U and E, which are denoted by Uˆ and Eˆ. The free entry condition (FEC) 24

implies that 0 = −ψ+λ p(A)E(V |A), F F where the event A denotes job acceptance. From the first stage estimates, we can consistently estimate the probability of acceptance, p(A), and the expected value of a new (cid:92) filled vacancy, E(V |A), where the estimated values are given by pˆ(A) and E(V |A), F F and let B ≡ p(A)E(V |A). Then Bˆ = pˆ(A)×E (cid:92) (V |A) is a consistent estimator of B. F F We can write λ = λ S/ν. F U The FEC is rewritten as λ S U ψ = B, ν and after substituting consistent estimators, we have λˆ Sˆ ψ = U Bˆ ν λˆ Sˆ ⇒ ν = U Bˆ. ψ If a consistent estimator of ψ is available, ψˆ, then a consistent estimator of the steady state vacancy rate is λˆ Sˆ νˆ = U Bˆ. ψˆ Given this estimate of ν, we have λˆ = kˆφ, U where kˆ = νˆ/Sˆ. Then a consistent estimator of φ is given by lnλˆ φˆ= U . lnkˆ (cid:4) In our modeling framework, we assume that costs of employment and vacancies are identical, with costs in both cases consisting of the flow rental rate of capital required to produce output and to evaluate the productivity of applicants arriving at random points in time. Under this assumption, we can recover the Cobb-Douglas matching function parameter φ.14 14Note that the computation of E(V |A) requires us to solve for the steady state distribution of general F and match-specific levels. In Appendix A we derive this distribution. 25

4.2.3 Demand-Side Parameter Estimator: Method 2 In practice, we found that our estimate of the employment cost parameter ζ was insufficiently large to produce an estimate of the Cobb-Douglas parameter φ that lie in the unit interval. In this case, it appears that we must reject the assumption that the cost of posting a vacancy is the same as the cost of capital in an employment match. From previous analyses (e.g., Flinn (2006), Flinn and Mullins (2015)), we know that the implied value of ψ is typically much larger than our estimate of ζ. In this case, the parameters of the demand side are not identified, and we follow the usual approach for recoveringanestimateofψ.UndertheassumptionofagivenvalueoftheCobb-Douglas parameter, φ, we first find an estimator for unobserved vacancies, ν. We have λ = kφ U = (ν/S)φ 1 ⇒ ν = S(λ U )φ. Using consistent estimates of the relevant parameters, we have that a consistent estimate of ν is νˆ = Sˆ(λˆ U )φ 1 . Of course, consistency of νˆ is based on the assumption that we have used the true matching function parameter, φ. In practice, we utilize the value of 0.5, which is common in the literature (see Petrongolo and Pissarides (2001)). Using this estimator of ν, we then find a consistent estimator of λ , which is simply F λˆ = (vˆ/Sˆ)φ−1. F We then find a consistent estimator of ψ, which is given by ψˆ= λˆ pˆ(A)E (cid:92) (V |A). F F The estimate of ψ is used in our counterfactual experiments involving the minimum wage. 5 Estimation Results 5.1 Parameter Estimates The estimated parameter values are shown in Table 6 together with bootstrapped standard errors. We start with a discussion of the parameters that control employment transition rates. First, the flow value of unemployment for a worker of ability a isestimatedtobeˆba = 4.94a, veryclosetotheoutputofthatworkeratafirmwiththe median match quality, (exp(µˆ )·a = 4.59a). For unemployed workers, an offer arrives θ 26

at a rate of λˆ = .219 or approximately once every five weeks. Workers with medium u levels of general ability accept 23 percent of job offers, implying that the average unemployment spell lasts 20 weeks. Conversely, matches are exogenously dissolved at a rate of ηˆ = .0036, or approximately once every five years. Matches may also be dissolvedendogenouslyifashocktogeneralabilityormatchqualitymakesunemployment preferabletotheworker’scurrentmatch. Toassesstherelativeimportanceofthesetwo shocks, we observe that the overall unemployment rate in the model is 13.1 percent, close to the data target of 14.0 percent. However, much of this unemployment occurs alongthetransitionpathasthemodelmovestowardssteadystateandthesteadystate unemployment rate in the model is just 9.2 percent. Together with the job finding rate and exogenous job separation rate, this equilibrium unemployment rate implies that approximately 15 percent of separations are endogenous. For employed workers, new offers arrive at rate λˆ = 0.094, or approximately once every 11 weeks, about half as e frequently as for unemployed workers. The parameters µ (e) and σ control the distribution of starting values for general a a ability, where e denotes the education level of the worker. The estimated values imply thatworkerswithsomecollegeeducationbegintheirlaborforcecareerswith24percent more human capital than high school graduates, on average, and those with at least a bachelor’s degree begin with an additional 24 percent. These parameters are identified largely from wages of new workers entering the labor force. We match the starting wages of workers is the two higher education groups almost exactly. For workers with only a high school degree, starting wages are slightly higher in the model than in the data but subsequently increase at a slower rate. The variance for the initial distribution of ability σˆ2 = 0.043, a bit less than half the variance in the distribution a of match qualities. The parameters that govern the technologies for the rate of increase in general ability are δ0,δ1 and δ2. As specified in Section 4.1, for an individual with general a a a ability a , the hazard rate of improvement to ability level a is given by i i+1 ϕ a (i,τ a ) = δ a 0×a δ i a 1 ×(τ a )δ a 2 i < M with the analogous expression specified for the θ process. In the estimated model, δˆ0 a andδˆ0 areverysimilar: δˆ00.021,andδˆ0 = .016. However, theremainingcomponentsof θ a θ thegeneralandmatch-specificskillprocesseslookconsiderablydifferent. InTable6,we seethatδˆ1 is−0.132,whereasδˆ1 is0.673. Inotherwords,theparameterestimatesshow a θ that general training becomes less productive as a increases, whereas match-specific trainingbecomesmoreproductivewithincreasesinθ. Thisisareasonablefindingsince a is likely to be more difficult and costly to change after labor market entry due to the fact that employers are not equipped to offer general learning experiences as efficiently asareschoolsthatspecializeinincreasingstudents’cognitiveabilities. Theseparameter estimates also provide a bridge between this model and the Flinn and Mullins (2015) specification, where a is assumed to be fixed over the labor market career. In our model, we allow a to change over the labor market career, but the estimated model 27

shows that it can indeed be thought of as quasi-fixed since it is difficult to change after labor market entry. Aswewroteearlier,thetrainingobservedinthedataislikelyaveryroughproxyfor the amount of time spent developing workers’ human capital. Despite the predictions of our model that most workers are generally receiving some kind of training, only five percent of workers in the data report training in their current job. The parameters β and β control the relationship between training in the model and the probability 0 τ that we observe a worker to be receiving training in the data. Although the median worker in our model spends 20 percent of her time training, we expect that this worker will be observed to be involved in training only one percent of the time. For a worker engaged in full-time training, we would expect to observe this training in the data only 35 percent of the time. 5.2 Model Fit In this section, we compare the fit of the simulations from our estimated model to the corresponding moments in the data. We begin by comparing the model-predicted and observed wage distributions. Overall, the model-predicted wage distributions are close to the empirical ones. Conditioning on tenure and education, we construct histogram plots from the simulated and observed wages, shown in Figures 1, 2 and 3 for workers with 0-2, 3-5 and 6-8 years of tenure, respectively. Next, we evaluate the ability of the model to replicate the training patterns in the data. In Table 7, we see that the observed and model-predicted proportion of individualswhohaveparticipatedinatleastonetrainingspellduringthetimetheyareobserved is 15 and 17 percent, respectively. Columns (2) and (3) of Table 7 further reveal that the model accurately captures the decreasing pattern of training with education, albeit with a small tendency to overstate the incidence of training for low-education workers: the proportion of individuals who get training at least once is 18, 13 and 13 percent for the low, medium and high education workers, respectively; whereas in the model simulations these moments are 20, 16 and 13 percent. One of the distinguishing features of our model is our focus on within-job spell investment in human capital. This investment behavior, whether general or matchspecific, impacts transition rates and wage processes within and across job spells. In the estimation, we match moments related to the joint distribution of wages between consecutive interview dates within a job spell as well as the joint distribution of wages between consecutive interview dates between different job spells. Table 8 displays how the model performs in generating some of these transition moments. Here, we limit our discussion to events that span only two consecutive interview dates, t−1 and t, and to workers who are employed at both dates. As described in the data section, we consider three possible events that may occur between t−1 and t: 1) no job change, 2) job-to-job transition with an intervening spell of non-employment, and 3) job-to-job 28

transition with no intervening spell of non-employment.15 First, we consider the the proportion of job-to-job transitions, shown in Panel A of Table 8. In the data, the proportions of job-to-job transitions are 15, 12 and 10 percent for the low, medium and high-education groups, respectively. The corresponding model-predictions for these moments are 14, 16 and 17 percent, respectively. Hence, the model-predicted transition rates are reasonably close to the actual ones in the data for the lower education workers. However, the discrepancy between the model predictions and the data seems to increase with education. Next, we examine the distribution of wage changes between consecutive interview dates, shown in Panels B and C of Table 8. We focus our discussion on the moments for high-school graduates, shown in Column (1), though the patterns for the other two education groups, shown in Columns (2) and (3), are similar. Panel B shows the average wage growth during these employment spells. In the data, for job-to-job transitions with an intervening non-employment spell the average log wage difference is 0.06. For job-to-job transitions with no intervening non-employment spell, it is 0.12. The corresponding moments in the model simulations are -0.14 and 0.19, respectively. These numbers show that the estimated model captures correctly the direction of the implications of an intervening non-employment spell, but that it performs poorly in matching the levels of these changes. Similarly, Panel C shows that the in the data, the proportion of job-to-job transitions that are associated with a wage decrease is 39 percent with an intervening unemployment, and only 30 percent without. The model matches the proportion for transitions with no intervening non-employment but overstates the fraction of negative changes for transitions that do include a spell of non-employment. Nevertheless, the model correctly captures the fact that transitions spanning a non-employment spell are more likely to have negative wage growth than those that do not. To understand the intuition behind these moments, we recall that in the model, values of θ and any investment made in match-specific productivity do not carry over intofutureemployment,whereasgeneralhumancapitaldoes. Foraworkerwhodirectly switchestoanewjob, theonlyacceptablejobsarethosethathaveθ valueshigherthan her current value. Therefore, the average wage gains from a job-to-job transition are high and the proportion of negative wage transitions is low (and would be zero without any measurement error). In contrast, for transitions that do involve an intervening non-employment spell, the worker loses the value from her accumulated match-specific human capital. Once she enters unemployment, she is willing to accept offers with a wider range of θ values, including values that are lower than the match quality at her recent job. Her willingness to accept such offers decreases the average wage change across such transitions and results in a higher fraction of wage changes that 15As described in the data section, we define a job-to-job transition to not involve a intervening nonemployment spell if the time between the end of first job and beginning date of second job is a nonemployment spell of 4 weeks or less. This allows us to distinguish between instantaneous turnovers from those that involve a period of search between consecutive jobs. 29

are negative. The wage losses predicted by this mechanism in the model are evidently larger than those observed in the data. 5.3 Identification Tobetterunderstandthesourceofidentificationinthemodel, wereestimatethemodel under two restrictions on human capital investment. First, we eliminate the possibility of investment in general human capital, and second, the possibility of match-specific investment. We then compare the baseline parameter estimates and model predictions with the ones obtained from the constrained estimations. The purpose of this comparison is to demonstrate what aspects of the data and the model help distinguish between general and match-specific human capital in the estimation. First, we compare the model predictions and parameter estimates we obtain from the constrained estimation with no possibility of investment in general human capital to the ones we obtain from the baseline estimation. In the absence of general human capital accumulation, all wage growth within a job spell is attributed to growth in the worker’s match-specific human capital, which must increase more quickly than in the baseline model to match the overall rate of wage growth. Because workers have more match-specific human capital, fewer potential offers would cause them to leave their current employers and we would therefore expect, holding all else constant, a decrease in the job-to-job transition rates. However, as shown in Table 9, our estimate of the job offer rate (λ ) increases to 0.108 from a baseline value of 0.094, so the e model still matches the empirically observed transition rate. A second consequence of the increase in match-specific human capital is that the the wage gains from jobto-job transitions tend to be smaller.16 As a result, the restricted model is less able to match the wage growth across job-to-job transitions. This reveals an important aspect of identification between general and math-specific human capital in the model estimation: transition data alone is not sufficient to distinguish between the different types of human capital and we need transition date in conjunction with moments that pertainto the joint distribution of wages withinand across job spells in orderto isolate one form of investment from the other. Next,wereestimatethemodelwithonlygeneraltraining. Thisversionofthemodel does a better job of matching the average wage growth within and across job spells. However,withoutmatch-specifictraining,themodelisunabletocapturethedifferences in wage growth and separation rates that we observe between short and long job spells. Inallversionsofthemodel,longerjobspellsareassociatedwithhighermatchqualities. In the baseline model, the productivity of match-specific training rises with the quality of the match since δˆθ > 0. This explains why jobs with better matches experience 2 more wage growth. In addition, the increase in match quality over time due to matchspecific training explains the decrease in the job-to-job transition rate with increasing job tenure. The alternative model with only general training is unable to match these 16Formally, θ is log-normally distributed so E(θ(cid:48)−θ|θ(cid:48) >θ) is decreasing in θ. 30

features of the data. The results from the constrained estimations discussed above demonstrate the empirical content of general and match-specific investments. As the results illustrate, general and match-specific investment have distinct implications for subsequent employmentandwagetransitions. Thesedistinctimplicationscanbesummarizedthrough plotting the life-cycle wage profiles obtained from each estimation. These graphs are displayed in Figures 4-6. Each figure corresponds to a different education level. We see the same pattern for all three education groups: the initial accepted wages (wages that correspond to Year 0-2 of labor market tenure) are lower in the estimated model with no investment in θ and higher in the estimated model with no investment in a, with the baseline in between. This order gets reversed as the worker accumulates more labor market experience. In other words, by Year 6-8, the average log wages that correspond to the estimated model with no investment in θ overtakes the other two and the average log wages for the case with no investment in a ends up as the smallest average among all three estimated models. In the estimated model with only general human capital investment, the benefits of any training undertaken by the worker can be carried over to other jobs. The resulting wage growth throughout the labor market career of a worker is consequently larger relative to the estimated model with only θ investment. Theoppositeistruefortheestimatedmodelwithnoainvestment: theonlytraining that a worker is able to engage in is training in θ, which is something that she cannot carry over to future jobs. Hence, the amount of benefits that she can accrue and transfer to future periods is smaller in the estimated model with no a investment and, consequently, overall wage growth throughout the labor market career is much lower. This is why the average log wages for the case with no a ends up being the lowest at the end of the Year 6-8 tenure profile. The comparison between these wage plots shows us that the difference between wage growth rates during the course of a worker’s career within a firm and over the course of a worker’s overall labor market career is an important indication of the type of training as well. 5.4 Policy Rules 5.4.1 Acceptable Job Offers We next explore the choices of workers and firms implied by these parameter values. First, we consider the worker’s decision to accept a match. A worker of ability a who receives an offer with match value θ will accept the offer if θ > θ∗(a), and will otherwise remain unemployed and continue to search. For our estimated parameters, we plot θ∗(a) in Figure 7. At very low values of a, a high value of θ is required for the match to cover the firm’s employment cost and also deliver more value to the worker than the value of unemployment. As a increases, θ∗(a) decreases as matches of lower valuebecomefeasible. Thevalueofθ∗(a)beginstoincreaseagainathighervaluesofa. 31

To understand the reason for this increase, we need to examine the choice of how much general training the firm provides at each combination of a and θ, which is plotted in Figure 10. At lower values of a, for values of θ just above θ∗(a), workers spend a full 15 percent of their time engaged in general training. This suggests that these marginal matches become feasible only because of the opportunity they provide for the workers to build their general human capital. As a increases, general training becomes less productive (because δa < 0), these marginal draws no longer deliver positive surplus 1 relative to unemployment, and workers raise their reservation value of θ. Next, we consider the choices of how much general and match-specific training firms and workers choose to provide for different levels of (a,θ). As background to this discussion, we want to understand how much workers value each kind of training relativetosimplyreceivingwages. Tothisend,Figure8showscombinationsoftraining and wages that solve the bargaining problem between the worker and the firm. Near the actual solution, the wages decrease quickly as the firm chooses to provide more general training, suggesting that workers regard general training as a good substitute for wages. In contrast, increasing match-specific training results in a much smaller decrease in wages, implying that the worker’s value from additional match-specific training is small, and that most of the value from match-specific training goes to the firm. However, the worker does seem to receive some benefit from the match-specific training, which is reflected in her willingness to trade off some wages for more matchspecific training. 5.4.2 Training Policies Havingidentifiedthetrade-offsbetweenwagesandtraining, wenextexaminethethree outcomes of the bargaining process: the two types of training and the wage. In Figures 9 to 11, we plot the amount of general and match-specific training and the wage that workersreceiveatdifferentcombinationsofaandθ. Foreaseofillustration,bothstates are shown on a log scale and the lines on the graph show contours along which wages or the amount of training remains constant. The bottom of the figures, corresponding to low values of θ, are combinations for which workers will not accept the job offer. Looking first at the policy for firm-specific training plotted in Figure 9, we see that the amount of firm-specific training is essentially a function of the current value of θ with very little dependence on the worker’s level of general ability. At values of θ just above the minimum θ∗(a) threshold, the amount of training is small. Firm-specific training increases for higher values of θ, reaching a maximum of 15 percent of the worker’s time at roughly the 75th percentile of the distribution of acceptable θ draws. Two different mechanisms contribute to this pattern. First, in the estimated model, δˆ1 > 0 so firm-specific training is more productive at higher values of θ. Second, at θ higher values of θ, the expected duration of the current match increases as it becomes less likely that the worker will leave to take an outside offer. Because firm-specific trainingincreasesfutureoutputonlyforaslongastheworkerremainswithhercurrent employer, this increase in expected duration raises the value of match-specific training. 32

Offsetting these effects is the incentive for the firm to provide match-specific training in order to raise the value of θ and thereby increase the length of the current match. This incentive is stronger at lower values of θ because the density of potential job offers is higher so that increase in θ yields a greater reduction in the fraction of outside offers that would cause the worker to leave. Next, we look at the amount of general training provided to the workers, which we plot in Figure 10. At low values of a and θ just above the θ∗(a) cutoff, workers spend about 15 percent of their time engaged in general training. The amount of training decreases at higher values of either a or θ. General training decreases with a because generalgrainingbecomeslessproductiveathighervaluesofa(δˆ1 < 0). Meanwhile, the a decrease in general training at higher θ seems to reflect the decrease in overall benefits flowing to the worker at higher values of θ. To understand this, we recall our earlier discussion where we showed that the benefits of general training flow largely to the worker. In the context of our model, this implies that negotiations over the amount of general training should look similar to the negotiations over wages. In states where the bargaining process yields lower compensation for the worker, she will choose to lose some of this compensation by receiving lower wages and some as a decrease in general training. Indeed, in Figure 11, we plot the fraction of worker’s output that is paid in wages and we observe that it also decreases at higher values of θ. The decrease in both wages and general training is consistent with a decrease in the worker’s overall bargaining position as θ increases.17 Given these policy rules, how much training do workers actually receive? Based on the simulations from the estimated model, Figure 12 plots the fraction of time that workers spend training as they move through the first years of their careers. When workersfirstenterthelaborforce,initialmatchqualitiesarerelativelylowandtherefore most of the training takes the form of general training. In the model simulations, workers in their first year in the labor force spend about 12 percent of their time in general training and 9 percent in firm-specific training. Over time, match quality increases as workers sort into jobs of higher match quality. Because the expected duration of these jobs is higher and also because firm-specific training becomes more productive at higher values of θ, more of the training becomes firm-specific and the total amount of training increases, peaking one to two years after labor market entry. Asmatchqualityincreasesfurtherovertime, theamountoftimespentonbothgeneral andfirm-specifictrainingbeginstodeclineasworkersspendlesstimetrainingandmore time engaged in production. 5.5 Sources of Wage Growth In this section, we examine the factors that drive wage growth in the model and provide a link between our structural approach and the well-known results from the 17The correspondingincrease in the firm’s bargaining position as θ increases provides an additionalincentive for it to provide match-specific training to the worker. 33

literature on Mincer-type earnings functions. The model contains five possible sources of wage growth. First, workers can increase their productivity by building general human capital through on-the job training. Second, workers can increase productivity bysearchingforanewjobwithahighermatchquality. Third, workerscanimprovethe quality of the match with their current employers by engaging in firm-specific training. Fourth,asworkersspendlesstimetrainingandmoretimeengagedinproduction,some of this increased output will flow to them in the form of higher wages.18 Finally, the bargaining between workers and firms can result in different shares of worker output being paid as wages depend on the value of the workers’ outside options. The current section aims to quantify the importance of each of these channels. In the absence of employment costs, output y would equal a · θ · (1 − τ − τ ). a θ Additionally, it is useful to decompose θ into two separate components, as θ = θ ·θ 0 τ where θ the match quality at the start of the match, and θ is the additional match 0 τ quality accumulated trough match-specific training (net of depreciation). This allows us to formally write the wage w as the product of the five pieces described above: w = a·θ ·θ ·(1−τ −τ )·(w/y) 0 τ a θ or in logs, log(w) = log(a)+log(θ )+log(θ )+log(1−τ −τ )+log(w/y) 0 τ a θ In Figure 13, we plot the evolution of each of these five components, together with the total wage, as workers move through the early years of their careers. The figure shows that in the worker’s first few years in the labor force, the two most important sources of wage growth are the development of match-specific human capital through search and through training. General ability grows more slowly, contributing less to wage growth at the beginning of a worker’s career but accounting for a larger fraction of total wage growth as the rise in match-specific human capital slows over time. As describedabove, thetimespenttrainingincreasesattheverystartofaworker’scareer, but thereafter, the reduction in training time begins to contribute noticeably to the worker’s overall output and therefore to her wage. The last component, the fraction of output represented by the worker’s wage, is quite flat and has almost no effect on the evolution of wages over time. More formally, our structural model also allows us to interpret the coefficients of a standard wage regression in terms of the different source of wage growth discussed above. This analysis helps relate our results to the well-known results from the literature on earnings dynamics. As an example, we consider a simple Mincer wage regression of the form log(wage ) = (cid:88) βwXj +εw it j it it j 18Alternatively,ashifttowardslesstrainingandhigherwagescouldbeinterpretedasashiftintheworkers’ compensation towards higher current wages and away from expected higher future wages. 34

where log(wage ) is the log of the wage for person i at time t. Specifically, we estimate it (cid:88) log(wage ) = βw + βwed +βwyears +βwtenure +εw. (3) it 0 e ie y it t it it e where ed is a dummy variable indicating that person i has education level e (for ie each level of education except HS graduate), years denotes the number of years it in the labor force and tenure the length of time with current employer. We first it estimate this model on the actual NLSY data and then on the simulated data from the model. Results are shown in the first two lines of Table 10. As expected, more education, more years in the labor force and greater job tenure are all associated with higher wages. Comparing the regressions on the real and simulated data, we find that additional education is associated with less of an increase in wages in the simulated data than the actual data. Also, relative to the data, more of the wage growth in the modelisattributedtotenurewithparticularemployersandlesstooveralllabor-market experience. Focusing on the regression using the simulated data, we next aim to understand how the increases in wages associated with education, labor market experience and job tenure reflect the different determinants of wages present in the model. Similar to the decomposition described in the previous section, we can decompose log wages in the modelasasumofthelogsofi)generalability,ii)matchqualityatthestartofthematch iii) additional match quality accumulated through match-specific training, iv) time spent not training and v) wage as a fraction of output. Additionally, because we are interestedinthelevelofwagesratherthanjustthegrowthrate,weexpresstheworker’s generalabilityasacombinationofherinitialendowment(a )andtheadditionalhuman 0 capital she accumulates through training (a ). This defines six components of wages, τ which we denote Yk,k = 1,...,6, allowing us to write it log(w ) = (cid:80)6 log(Yk) it k=1 it (4) = log(a )+log(a )+log(θ )+log(θ )+log(1−τ −τ )+log(wit) 0,it τ,it 0,it τ,it a,it θ,it yit In order to measure how education, labor-market experience and job tenure affect each of these components, we repeat the regression from Equation 3 on each of these six pieces separately, i.e. we estimate log(Yk) = (cid:88) βkXj +εk, k = 1,...,6. it j it it j It is straight forward to show that for each covariate (indexed by j), the sum of the regressioncoefficientsfromthesesixregressionsmustequalthecoefficientforregression using the total log wage, i.e. 6 (cid:88) βw = βk. j j k=1 35

This allows us to interpret each of the coefficients βw from Equation 3 as reflecting j different combinations of the components of wages defined in Equation 4. The results of this exercise, which we present in Table 10, are all quite reasonable. The increase in wages with more education is largely picking up differences in initial ability and, to a lesser extent, the fact that more educated workers are paid a larger share of the output, possibly because the constant employment costs consume a smaller fraction of their output. The positive coefficient on labor market experience is mostly capturing general ability learned through training and, to some extent, workers’ ability to find better matches over time and their receiving a larger fraction of output. Finally, the increase in wages for workers with more tenure mostly reflects improved match quality from training. To a lesser extent, wages also appear to increase with tenure because workers engage in less training as they remain with a firm longer, and there is an additional selection effect whereby longer-tenured workers received better initial matches with their firms. In addition to studying Mincer wage regressions on the entire sample, we can run these regressions separately on workers with different amounts of education to help understanddifferencesinwagegrowthbetweenthesegroups. Resultsfromthisexercise are shown in Table 11. The first panel shows the difference in the constant term. The model captures the higher wages of more educated workers and, as one would expect, mostofthisdifferenceiscapturedsimplybythehigherlargeramountsofgeneralhuman capital that more educated workers have upon entering the labor force. In addition, a portionofthedifferencesisexplainedbytheabilityofmorehighlyeducatedworkersto capturealargershareoftheoutput,perhapsbecausetheytendtoproducemoresurplus above the fixed employment costs. These sources of positive association between wages and education are offset slightly by the fact that more educated workers accept offers at lower match values, which tends to lower their output and therefore their wages. In the middle panel of Table 11, we show the effect of increasing labor market experience on wages for the three different education groups. In the data, high school graduates receive a larger increase in wages for each year in the labor market than do more educated workers. The model is unable to explain these differences; wages in the simulations increase about four percent for each additional year of labor market experience, regardless of education. The final panel of Table 11 describes the returns to increasing tenure for each of the three education groups. Here, the model matches the data well. In both the data and the model, workers with more education have higher returns to tenure. Each year of additional employment is associated with a two percent larger increases in wages for college educated workers compared to those with only a high school diploma in the data, though just once percent more in the model. In the model, most of this difference is accounted for by more educated workers building more match-specific human capital from training during their job spell. Offsetting this effect is the fact that, with increasing tenure, workers receive a smaller fraction of their output as wages and this decrease happens more quickly for more educated workers. 36

6 Policy Analysis: The Minimum Wage and Investment In competitive markets, minimum wages are expected to impact human capital formation in relatively immediate ways (Leighton and Mincer (1981), Hashimoto (1982)). In the case of general human capital, the returns to which acrue to the worker, the theory implies that the sum of the cost of investment and the wage paid to the worker should be equal to the worker’s (marginal) productivity. Restrictions on the wage that must be paid to the worker act as a constraint on the amount of general human capital investment that the individual can undertake, which typically results in lower levels of investment early in the labor market career than would be efficient. Within our modeling framework, in which there exist two types of human capital and search frictions, these types of considerations still apply, but in a much more subtle manner. One of the more interesting implications of our model regarding the impact of minimum wage laws is that, given the existence of search frictions and the possibility of improving the productivity of the match, the introduction of a minimum wage that exceeds the current flow productivity of the match need not lead to the termination of the employment contract. Models in which the productivity of the match is fixed produce the implication that all matches with flow productivities less than the value of the minimum wage will be terminated (Flinn (2006), Flinn and Mullins (2015)). The possibilityofincreasingtheflowproductivitymeansthatthedecisiontokeepthematch alive involves a comparison of the expected profits of the match (to the firm) with its outside option of zero, while the worker must receive an expected value of continuing on the job at the binding minimum wage that exceeds the value of unemployed search under the new minimum wage. In general, not all matches with current flow net productivity aθ − ζ < m, where m is the newly imposed minimum wage, will be terminated. The likelihood of termination in such a case will be a function not only of net productivity, but also the mix of general and match-specific capital possessed by the worker. For a given shortfall in net productivity with respect to the minimum wage, firms will be more likely to continue the match when match-specific capital is greater. Toeventuallyearnpositiveflowprofitsfromthematch,theemployeewillhave to be likely to remain with the firm, and this is an increasing function of the current level of θ. Moreover, this consideration will make it more likely for the worker-firm pair to invest in match-specific human capital than general human capital. The extent of these effects will depend on the parameters characterizing the model. As has been found in Flinn (2006) and Flinn and Mullins (2015), the impact of the minimum wage is likely to vary significantly depending on whether we use the partial or general equilibrium version of the model. In the partial equilibrium version, contact rates (λ and λ ) are assumed to be fixed. In the general equilibrum version of the U E model, these contact rates are a function of the job vacancy creation decisions of firms. As minimum wages increase, the share of the match surplus accruing to firms shrinks, which decreases firms’ incentives to create vacancies. This corresponds, roughly, to a 37

shift downward in the demand function, and exacerbates the minimum wage’s negative impact on the employment rate. We begin our analysis with the partial equilibrium version of the model. 6.1 Minimum Wage in Partial Equilibrium In order to study the effect of a minimum wage in the context of our model, we use the estimates of the model parameters and then solve the model imposing a minimum wage of $15 per hour (in 2014 dollars, corresponding to $10.17 in the 1994 dollars we useinouranalysis). Asexpected, wefindthatimposingaminimumwageincreasesthe unemployment rate by rendering low quality matches unprofitable for firms.19 Figure 14 shows the minimal acceptable match quality draw in the baseline model and under theminimumwage. Inoursimulations, imposingtheminimumwageraisestheequilibriumunemploymentratefrom9.2to9.6percentwithmostoftheincreaseconcentrated among less-educated workers. In addition to the effect on employment, our model shows how the minimum wage canalsoaffecttheamountoftrainingprovidedtoemployedworkers. Becauseemployers must pay workers a higher wage, they decrease the amount of compensation that is provided in the form of general training. Figure 15 compares the amount of training that workers receive in the baseline model and under the minimum wage. With a minimum wage in place, employed workers spend 3-8 percent less time on general training than they do in the baseline model. The long-run effect of this decrease in training is to reduce the average amount of general ability in the population by about half a percent with most of the impact concentrated at the bottom of the distribution. Whilethedecreasedtrainingtimeultimatelyreducesworkers’accumulationofskill, the flexibility of employers to adjust the amount of worker training reduces the impact of the minimum wage on unemployment. In particular, some matches that were feasible in the baseline model become infeasible if employers are forced to maintain the same level of training while also paying the higher minimum wage. However, because employers in our model are able to reduce the amount of training they provide, they continue to be able to form matches with some of these workers by giving them less training when the minimum wage is raised. As described above, imposing a $15 minimum wage raises the unemployment rate in our model from 9.2 to 9.6 percent while simultaneouslyreducingtheamountofemployer-providedtraining. Ourmodelpredicts that if employers were not able to adjust the amount of training, the unemployment rate would rise an additional 0.3 percentage points to 9.9 percent. Our model therefore shows how endogenously determined investment in human capital can mitigate the effect of minimum wage laws on unemployment. Conditional on employment, workers receive higher wages with a minimum wage in place than they do in the baseline model. Average wages are eight percent higher 19Recallthatourmodeldoesnotincludealaborforceparticipationdecision. Anincreaseinunemployment translates into a decrease in employment, since these rates sum to one. 38

for workers entering the labor force and remain 1.5 percent higher after several years. Part of this increase is due to selection on both general ability and match quality as the presence of a minimum wage raises the distribution of acceptable match values and also disproportionately keeps low-ability workers unemployed. In addition to the selection effects, the lower amount of training discussed above means that workers are spending more of their time engaged in production and some of this additional output naturally flows to the worker in the form of higher wages. Finally, when the minimum wageisbinding,employersmustpayworkersahigherfractionoftheiroutputthenthey otherwise would in order to raise their wages to the required level. Quantitatively, the selection of higher ability workers into employment and the increase in the workers’ wages as a fraction of total output contribute the most to the the total increase in wages. Looking in more detail at the effect of the minimum wage on the distribution of wages, Figure 16 shows how various percentiles of the wage distribution evolve with labor market experience, with and without the minimum wage. As expected, most of the of effect of the minimum wage occurs at the bottom of the distribution. For workers entering the labor market, the minimum wage raises the first percentile of the wage distribution by 50 percent and this difference persists even as workers gain more experience. At the tenth percentile, wages start out 20 percent higher for new workers but the effect fades after workers have been working for about four years. The effect on workers further up the distribution is insignificant. Overall, the welfare effects from the minimum wage are small as the loss to workers fromahigherunemploymentrateandloweramountsofgeneraltrainingislargelyoffset by the higher wages they receive. Workers at the very lowest value of general ability experience a welfare loss of half a percent as they are most impacted by the lower jobfinding rates. For high school graduates on average, the welfare loss is just 0.1 percent and it is even smaller for those with more education. These relatively small effects are due to the minimum wage being set at a relatively low level taking into account the baseline marginal wage distribution. 6.2 Minimum Wage in General Equilibrium Wenowconsidertheimpactofminimumwageincreasesinasimplegeneralequilibrium version of the search and matching model in which firms’ vacancy creation decisions andthemeasureofunemployedandemployedsearchersdeterminethecontactratesλ U and λ . In order to move to the general equilibrium framework, we need to define the E steady-statedistributionofworkers, whichisutilizedincomputingtheexpectedreturn to a filled vacancy when solving for firms’ vacancy creation decisions. We describe the computation of this steady-state distribution in Appendix A. Having identified the baseline steady-state distribution of workers, the next challenge is to is to identify the firm’s cost for posting a vacancy, ψ, and the value of the Cobb-Douglas parameter in the aggregate matching function, φ. Following the discussion in Sections 4.2.2 and 4.2.3, we attempt to estimate these parameters in sev- 39

eral different ways. First, as described in Section 4.2.2, we assume that the vacancy posting cost is equal to the estimated employment cost ζˆ and back out the value of Cobb-Douglas parameter. Unfortunately, this approach fails under our estimated parameters, yieldinganegativeestimateofφ. Alternatively, asdescribedinSection4.2.3, we fix the Cobb-Douglas parameter at φ = 0.5 and back out the vacancy cost. This procedure produces an estimate of the vacancy cost ψˆ= 182. In general equilibrium, the minimum wage constraint reduces the firm’s incentive to post a vacancy and therefore decreases the number of vacancies and the rate at which workers receive job offers. However, With a minimum wage of $15 (in 2014 dollars), the minimum wage binds on a very small fraction of matches in the steady state distribution so that the impact on transition rates is small: the decrease in jobfinding rates is just 0.4 percent and the effects on unemployment, human capital and welfare are little changed from the partial equilibrium model. The general equilibrium effects become more significant at higher values of the minimum wages. As an illustration, we consider minimum wages equal to $20 and $25.20 With the minimum wage set at $20, the job offer rate falls by one and a half percentandthesteadystateunemploymentrateincreasesanadditional1.4percentage points to 11 percent. In addition, because low-ability workers are employed less and receive less training, the amount of general human capital in the economy falls 3.8 percent for high school graduates and an average of 1.7 percent across all workers compared to the baseline. Together, the increased unemployment rate and loss of human capital produce net average welfare loss of 1 percent for high school graduates and 0.3 percent for those with some college. As we increase the minimum wage further to $25, we get even larger effects with the job offer rate declining to 4.7 percent below its baseline value and the steady state unemployment rate climbing to 17.2 percent. Generalhumancapitaldeclinesanaverageof13percentrelativetothebaselineforhigh school graduates and 6.4 percent for all workers. Finally, the welfare losses continue to increase for workers at the bottom of the distribution. For high school graduates, the average welfare loss is 6 percent, while for workers with some college, average welfare decreases by 2.1 percent. 7 Conclusion We have developed an estimable model of investment in both (completely) general and (completely) match-specific human capital while individuals are active members of the labor force, which we assume follows the completion of formal full-time schooling. While other researchers have examined investment decisions in a search, matching, and borrowing framework, ours is perhaps the first to attempt to estimate such a model in a reasonably general framework. Perhaps the greatest challenge we face in estimationistoattaincredibleidentificationofsuchamodelwhenhumancapitalstocks 20These values are again expressed in 2014 dollars. Expressed in the our units of 1994 dollars, the wage rates we actually impose in the model are $13.56 and $16.95 respectively. 40

and investments essentially are unobservable. In this we are aided by having access to (self-reported) data on whether a worker engaged in formal training during a job spell. We heavily exploit this information in our moment-based estimation procedure. Furthermore, our assumptions regarding the specificity of human capital imply that changes in the stock of general human capital have no impact on the future mobility decisions of an individual during the employment spell. This stands in stark contrast to changes in the stock of match-specific human capital, which strictly reduce the likelihood of accepting a job with another firm during the employment spell. Thus jobto-job mobility along with wage changes during the current job spell can be utilized to infer whether the wage change was the result of general or match-specific investment. Our estimates of the human capital production technology exhibit decreasing returns to investment in both types of human capital, of approximately the same degree. Our production technology also includes a TFP term that captures how the current level of both types of human capital affect the returns to investment. Here we find that the payoffs from time investment in match-specific human capital are increasing in its current level, while there is no impact of the current level of general human capital on the return to investment in it. These results imply fairly complex dynamic patterns in investment and wage growth. They also serve to produce a job acceptance probability from the unemployment state that is non-monotone in the individual’s level of general human capital. Weuseourestimatestoexaminetheimpactofminimumwagepolicyoninvestment in human capital and equilibrium outcomes. Our experiments are conducted in both partialandgeneralequilibriumframeworks,wherethegeneralequilibriumspecification relies on the topical matching function approach. Unlike previous estimates of these minimumwageeffects, asinFlinn(2006)andFlinnandMullins(2015), thedeleterious effects on high minimum wages on job finding rates can partially be alleviated by increases in the productivity of workers, which is obtained by investment in general and match-specific human capital on the job. In the general equilibrium setting, we find that a minimum wage of $15 does little to affect the labor market equilibrium, since the steady state distributions of worker and match quality used to determine the vacancy creation decisions of firms already produce a vast majority of matches that have productive quality levels greater than this amount. A minimum wage of $20 dollars an hour also has fairly minor impacts on unemployment and other features of the labor market equilibrium. On the other hand, the impact of a minimum wage of $25 dollars an hour does have notable impacts on unemployment and the steady state distribution of human capital and wages. To some extent, these results parallel those found in Flinn and Mullins (2015), which examined minimum wage impacts within a model of formal schooling decisions that did not allow post-schooling investment in human capital of either type. In our future research, we intend to endogenize the formal schooling decision, as in Flinn and Mullins. This will provide us with a relatively complete model of human capital investment over the entire life-cycle, and will allow us to examine the relationship between the human capital acquired during the formal schooling phase and that 41

acquired while in the labor market. Our belief is that formal school training is to some extent similar to what we are calling general human capital in this paper, but that the two are not perfect substitutes in production. We believe that much of what one acquires during formal schooling is a technology for learning, which impacts the production of both general and match-specific human capital during the individual’s labor market career. In this view, skills acquired or not acquired during early periods of development and formal schooling will have long-lasting effects on labor market outcomes and lifetime welfare. 42

Appendix A Deriving the Steady State Distribution in the Labor Market In order to simplify notation, we first expand the space of match values to include 0, which signifies that the agent is unmatched, that is, unemployed. All employed individuals at an arbitrary point in time are characterized by the labor market state (i,j), which signifies a and θ . Let the probability of (i,j) be denoted by π(i,j). The i j steadystatemarginaldistributionsofaandθ aregivenbyπ (i)andπ (j),respectively. a θ There are M possible values of a, and K possible values of θ (for employed agents), with 0 < a < ... < a 1 M 0 < θ < ... < θ . 1 K From our estimates, we have determined the minimal value of θ that is acceptable when an agent characterized by a is in the unemployment state, which we denote by i r∗(i)+1 (that is, θ is the maximal unacceptable match value to an individual with r∗(i) ability level a ). We define the indicator variable i (cid:26) 1 if k > r∗(i) d(i,j) = , i = 1,...,M;j = 1,...,K. 0 if k ≤ r∗(i) Tautologically, π(i,j) = 0 for all (i,j) such that d(i,j) = 0, i = 1,...,M, j = 1,...,K. Unemployed agents of type a occupy the state (i,0), with the probability of an i unemployed individual if type i being given by π(i,0). Then we have (cid:88) π (i) = π(i,0)+ π(i,j) a j>0 is the marginal distribution of a in the population. The conditional probability that a type i individual is unemployed is U(i) = π(i,0)/π (i). a We assume a constant death rate (cid:96) > 0 in the population. Since we have assumed that population size is constant, the death rate is equal to the birth rate. When a new individual is born, she is assigned a value of schooling according to the distribution of schooling types observed in the data, and an initial value of a is chosen from the schooling-specific initial distribution of a. The individual then enters the labor market as an unemployed individual with this general ability type. We begin by considering movements in the probability of being unemployed for an agent of ability type i. We have that the change in the proportion of unemployed of 43

ability type i is given by · (cid:88) π(i,0) = η π(i,j) j>0 (cid:88) +δ˜ (i+1)[ π(i+1,j)(1−d(i,j))] a j>0 (cid:88) + ϕ (i−1,j)π(i−1,j)(1−d(i,j)) a j>0 (cid:88) + δ˜ (j)π(i,j)(1−d(i,j −1)) θ j>0 (cid:88) −λ π(i,0) p (j)d(i,j) U θ j>0 +(cid:96)πˆ(i)−(cid:96)π(i,0). The right hand side terms correspond to the following events. On the first line is the rate at which jobs of any acceptable type (all j for which d(i,j) = 1) are destroyed times the probability that type i individuals are employed. The second and third lines represent inflows in to the unemployment state that result from depreciation and appreciation of general skills that are associated with “endogenous” quits into unemployment. In these cases, an individual employed with skills (i+1,j) or (i−1,j) will quit into unemployment if they would not accept employment at (i,j). The fourth line represents inflows into unemployment from individuals with skills (i,j) when their match skill level depreciates to j −1 and (i,j −1) is not an acceptable match. The penultimate line is the outflow from the unemployment state, which is the the product of the rate of receiving job offers in the unemployed state, the probability of being in the state (i,0), and the probability of receiving an acceptable job offer. The last line consists of two terms, the first of with is the rate of births into this population multiplied by the likelihood that they begin life with general ability level i, with πˆ(i) denoting the probability of beginning ones labor market career with general ability level i. The second term is the death rate times the probability of being in state (i,0). 44

Then, in the steady state, (cid:88) π∗(i,0) = [λ p (j)]−1 U θ j>0 (cid:88) ×{η π∗(i,j) j>0 (cid:88) +δ˜ (i+1)[ π∗(i+1,j)(1−d(i,j))] a j>0 (cid:88) + ϕ (i−1,j)π(i−1,j)(1−d(i,j)) a j>0 (cid:88) + δ˜ (j)π(i,j)(1−d(i,j −1)) θ j>0 +(cid:96)πˆ(i)−(cid:96)π∗(i,0)}, for i = 1,...,M. Now consider the determination of the probabilities associated with employment, those for which j > 0. The generic expression for the time derivative of π(i,j) is · π(i,j) = π(i−1,j)ϕ (i−1,j)+π(i,j −1)ϕ (i,j −1) a θ +δ˜ (i+1)π(i+1,j)+δ˜ (j +1)π(i,j +1) a θ (cid:88) +λ π(i,0)p (j)+λ π(i,l) U θ E l<j (cid:88) −[η+(cid:96)+δ˜ (i)+δ˜ (j)+λ π(i,l)]π(i,j). a θ E l>j In terms of the expressions on the right hand side of this equation, the first line represents improvements resulting in attaining state (i,j) from the states (i−1,j) and (i,j − 1). The second line represents inflows from human capital depreciations from the states (i+1,j) and (i,j+1). The third lines represent inflows from the unemployment state and from contacts with other employed individuals of ability type i who are currently working at jobs in which there match value is less than j. The final line represents all of the ways in which individuals from (i,j) exit the state. These are the exogenous dismissals, deaths, decreases in i or j, or finding another job for which match productivity is greater than j. Then in the steady state we have (cid:88) π∗(i,j) = [η+(cid:96)+δ˜ (i)+δ˜ (j)+λ π∗(i,l)]−1 a θ E l>j ×{ϕ (i−1,j)π∗(i−1,j)+ϕ (i,j −1)π∗(i,j −1) a θ +δ˜ (i+1)π∗(i+1,j)+δ˜ (j +1)π∗(i,j +1) a θ (cid:88) +λ π∗(i,0)p (j)+λ π∗(i,l)}, U θ E l<j 45

for i = 1,...,M, j = 1,...,K. We can vectorize the π matrix, and define the column vector   π(1,·)  π(2,·)  Π =  . ,  .   .  π(M,·) where π(i,·) = (π(i,0) π(i,1) ... π(i,K))(cid:48). Some elements of this vector are identically equally to 0, those for which d(i,j) = 0. Let the number of nonzero elements of Π be denoted N(Π), where N(Π) ≤ M ×(K+1). Denote the entire system of equations by D(Π). Then we seek Π∗ = D(Π∗). With no on-the-job search, this mapping is monotone on a compact space, and hence the solution is unique. With on-the-job search, it is clear that an equilibrium always exists, although we have not yet proven uniqueness. Simulations of the model and computation of the fixed point have consistently agreed, however, so that we are confident in the uniqueness property. 46

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Table 1: Proportion of Job Spells by the number of interview dates they span High School Graduates Some College College Graduates 0 61% 55% 53% 1 23% 24% 21% 2 7% 9% 11% 3 3% 4% 6% 4 2% 3% 3% 5 1% 2% 3% 6 3% 3% 3% Table 2: Incidence of Training Some College All HS College orMore %whogottrainingatleastonce 15% 18% 13% 13% %whogottrainingatthestartofjobspell 6% 10% 5% 3% 50

Table3: Proportion by Number of Training Spells (Conditional on Having Participated At Least Once) Percentage over All Workers with at Least One Training Spell 1 72 % 2 18 % 3 6 % 4 3 % 5 1 % 51

Table 4: Annual Labor Turnover Rates and Wage Growth Some College HS College or More Panel A: Employment-to-Employment (EE) transitions btw t−1 and t % of EE transitions with no job change 81 % 85 % 88% % of EE transitions with job change (job-to-job transitions) 19 % 15 % 12% ..... % of job-to-job transitions with non-employment btw t−1 and t 4 % 3% 2% ..... % of job-to-job transitions with no non-employment btw t−1 and t 15% 12% 10% Panel B: Wage Growth btw t−1 and t EE transitions with no job change 0.08 0.08 0.09 EE transitions with job change (job-to-job transitions) 0.11 0.15 0.20 ..... job-to-job transitions with non-employment btw t−1 and t 0.06 0.06 0.23 ..... job-to-job transitions with no non-employment btw t−1 and t 0.12 0.17 0.20 Panel C: % of Negative Wage Growth btw t−1 and t EE transitions with no job change 17 % 19 % 23% EE transitions with job change 32 % 32 % 28% ..... job-to-job transitions with non-employment btw t−1 and t 40 % 47% 27% ..... job-to-job transitions with no non-employment btw t−1 and t 30% 28% 28% 52

Table 5: Log Wage Difference btw Interview Dates t−1 and t by Training logw −logw t t−1 No Training Got Training transitions with no job change 0.08 0.08 transitions with job change (job-to-job transitions) 0.14 0.10 job-to-job transitions with non-employment spell btw t−1 and t 0.08 0.15 job-to-job transitions with no non-employment spell btw t−1 and t 0.15 0.09 53

Table 6: Parameter Estimates PARAMETERSFOREMPLOYMENTTRANSITIONS flowvalueofunemployment b 4.94 (0.067) jobofferrate-unemployed λu 0.219 (.007) jobofferrate-employed λe .094 (.004) exogenousjobseparationrate η .0036 (.0001) PARAMETERSOFINVESTMENTFUNCTIONS GeneralabilityinvestmentTFP δ0 .0209 (.0002) a Firm-specificinvestmentTFP δ0 .0163 (.0003) θ State-dependenceofgeneralabilityinvestment δ1 -.132 (.009) a State-dependenceoffirm-specificinvestment δ1 .673 (.008) θ Curvatureofgeneralabilityinvestment δ2 .260 (.007) a Curvatureoffirm-specificinvestment δ2 .432 (.013) θ Rateofdecreaseingeneralability ϕ˜− a .0008 (.00004) Rateofdecreaseinmatchquality ϕ˜− .0123 (.001) θ PARAMETERSOFINITIALABILITYDISTRIBUTIONS Meanofinitialgeneralability-HighSchool µa(ei=1) 0.92 (.01) Meanofinitialgeneralability-SomeCollege µa(ei=2) 1.16 (.01) Meanofinitialgeneralability-BAorhigher µa(ei=3) 1.40 (.03) Varianceofinitialgeneralability σa .208 (.008) PARAMETERSOFJOBOFFERS Meanofmatchqualitydistribution µ 1.52 (.01) θ Varianceofmatchqualitydistribution σ .315 (.002) θ Employmentcost ζ 4.51 (.136) PARAMETERSGOVERNINGTRAININGOBSERVATION Interceptfortrainingobservation β0 -3.02 (.02) Coefficientonτ fortrainingobservation βa 2.64 (.03) 54

Figure 1: Model Fit: Log Wage Distribution - Year 0-2 tnecreP 1 8. 6. 4. 2. 0 Model Fit: Log Wage Distribution High School Graduates, Year 0−2 w<1.9 w<2.2 w<2.3 w<2.5 w<2.8 Data Model tnecreP 1 8. 6. 4. 2. 0 Model Fit: Log Wage Distribution Some College, Year 0−2 w<2.2 w<2.5 w<2.7 w<3.0 w<3.3 Data Model tnecreP 1 8. 6. 4. 2. 0 Model Fit: Log Wage Distribution College Graduates, Year 0−2 w<2.5 w<2.8 w<3.1 w<3.3 w<3.6 Data Model 55

Figure 2: Model Fit: Log Wage Distribution - Year 3-5 tnecreP 1 8. 6. 4. 2. 0 Model Fit: Log Wage Distribution High School Graduates, Year 3−5 w<2.2 w<2.5 w<2.7 w<2.8 w<3.1 Data Model tnecreP 1 8. 6. 4. 2. 0 Model Fit: Log Wage Distribution Some College, Year 3−5 w<2.5 w<2.8 w<3.0 w<3.2 w<3.5 Data Model tnecreP 1 8. 6. 4. 2. 0 Model Fit: Log Wage Distribution College Graduates, Year 3−5 w<2.7 w<3.0 w<3.3 w<3.5 w<3.8 Data Model 56

Figure 3: Model Fit: Log Wage Distribution - Year 6-8 tnecreP 1 8. 6. 4. 2. 0 Model Fit: Log Wage Distribution High School Graduates, Year 6−8 w<2.4 w<2.7 w<2.9 w<3.1 w<3.4 Data Model tnecreP 1 8. 6. 4. 2. 0 Model Fit: Log Wage Distribution Some College, Year 6−8 w<2.6 w<2.8 w<3.0 w<3.3 w<3.6 Data Model tnecreP 1 8. 6. 4. 2. 0 Model Fit: Log Wage Distribution College Graduates, Year 6−8 w<2.9 w<3.2 w<3.5 w<3.6 w<3.9 Data Model 57

Table 7: Model Fit: Incidence of Training Some College All HS College orMore %whogottrainingatleastonce ........ Data 15% 18% 13% 13% ........ Model 17% 20% 16% 13% %whogottrainingatthestartofjobspell ........ Data 6% 10% 5% 3% ........ Model 5% 6% 4% 4% 58

Table 8: Model Fit: Annual Labor Turnover Rates and Wage Growth Some College HS College orMore PanelA:Employment-to-Employment(EE)transitionsbtwt−1andt %ofEEtransitionswith no jobchange ........ Data 81% 85% 88% ....... Model 77% 75% 74% %ofEEtransitionswith jobchange(job-to-jobtransitions) ........ Data 19% 15% 12% ....... Model 23% 25% 26% %ofjob-to-jobtransitionswith non-employmentbtwt−1andt ....... Data 4% 3% 2% ....... Model 8% 8% 9% %ofjob-to-jobtransitionswith no non-employmentbtwt−1andt ....... Data 15% 12% 10% ....... Model 14% 16% 17% PanelB:WageGrowthbtwt−1andt EEtransitionswith no jobchange ....... Data 0.08 0.08 0.09 ....... Model 0.08 0.08 0.08 EEtransitionswith jobchange ....... Data 0.11 0.15 0.20 ....... Model 0.07 0.09 0.10 job-to-jobtransitionswith non-employmentbtwt−1andt ....... Data 0.06 0.06 0.23 ....... Model -0.14 -0.08 -0.05 job-to-jobtransitionswith no non-employmentbtwt−1andt ....... Data 0.12 0.17 0.20 ....... Model 0.19 0.18 0.18 PanelC:%ofNegativeWageGrowthbtwt−1andt EEtransitionswith no jobchange ....... Data 17% 19% 23% ....... Model 38% 38% 38% EEtransitionswith jobchange ....... Data 32% 32% 28% ....... Model 41% 37% 36% job-to-jobtransitionswith non-employmentbtwt−1andt ....... Data 40% 47% 27% ....... Model 63% 58% 55% job-to-jobtransitionswith no non-employmentbtwt−1andt ....... Data 30% 28% 28% ....... Model 27% 26% 26% 59

Table 9: Parameter Estimates: Baseline vs. Constrained Estimations with No a/No θ Baseline Noa Noθ PARAMETERSFOREMPLOYMENTTRANSITIONS flowvalueofunemployment b 4.94 4.06 5.57 jobofferrate-unemployed λu 0.219 0.127 0.180 jobofferrate-employed λe .094 0.108 0.045 exogenousjobseparationrate η .0036 0.0031 .0038 PARAMETERSOFINVESTMENTFUNCTIONS GeneralabilityinvestmentTFP δ0 .021 0 0.024 a Firm-specificinvestmentTFP δ0 .016 0.022 0 θ State-dependenceofgeneralabilityinvestment δ1 -.132 - 0.024 a State-dependenceoffirm-specificinvestment δ1 .673 0.678 - θ Curvatureofgeneralabilityinvestment δ2 .260 - 0.065 a Curvatureoffirm-specificinvestment δ2 .432 0.548 - θ Rateofdecreaseingeneralability ϕ˜− a .001 0 0.0002 Rateofdecreaseinmatchquality ϕ˜− .012 0.019 0 θ PARAMETERSOFINITIALABILITYDISTRIBUTIONS Meanofinitialgeneralability-HighSchool µa(ei=1) 0.92 0.56 1.14 Meanofinitialgeneralability-SomeCollege µa(ei=2) 1.16 0.94 1.35 Meanofinitialgeneralability-BAorhigher µa(ei=3) 1.40 1.25 1.56 Varianceofinitialgeneralability σa 0.21 0.29 0.16 PARAMETERSOFJOBOFFERS Meanofmatchqualitydistribution µ 1.52 1.60 1.50 θ Varianceofmatchqualitydistribution σ 0.31 0.35 0.24 θ Employmentcost ζ 4.50 3.79 8.07 PARAMETERSGOVERNINGTRAININGOBSERVATION Interceptfortrainingobservation β0 -3.02 -3.07 -2.59 Coefficientonτ fortrainingobservation βa 2.64 2.36 3.21 60

Simulations: Baseline vs. No a/No θ Figure 4: Avg. Log Wages by Tenure egaW goL 9.2 8.2 7.2 6.2 5.2 4.2 Model Simulations: Baseline vs. No Training in A or Theta Log Wages − High School Graduates Year 0−2 Year 3−5 Year 6−8 baseline no a no theta 61

Figure 5: Avg. Log Wages by Tenure egaW goL 1.3 3 9.2 8.2 7.2 6.2 Model Simulations: Baseline vs. No Training in A or Theta Log Wages − Some College Year 0−2 Year 3−5 Year 6−8 baseline no a no theta Figure 6: Avg. Log Wages by Tenure egaW goL 4.3 3.3 2.3 1.3 3 Model Simulations: Baseline vs. No Training in A or Theta Log Wages − College Graduates Year 0−2 Year 3−5 Year 6−8 baseline no a no theta 62

Figure 7: Minimum Acceptable Match-Quality Thisfigureshowstheθ∗(a),thelowestmatchqualitythatworkerswitheachlevelofgeneralabilitywillaccept. 1.8 1.78 1.76 1.74 1.72 1.7 1.68 1.66 1.64 1.62 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 eulaV hctaM goL Minimum Acceptable Match Value Log Worker Ability 63

Figure 8: Trade-Off between Wages and Training in Bargaining Problem Thisfigureshowscombinationsofgeneraltraining(τa),matchspecifictraining(τ θ )andwagesthatsolvethebargainingproblem between the worker and the firm at the median values of a and θ. Lines on the graph show combinations of τa and τ θ along which the negotiated wage remains constant. The blue dot shows the surplus maximizing combination, which is the model solutionforτa andτ θ . 2 10 0 5 1 5 10 2 5 5 2 0 15 10 5 3 0 2 5 20 15 10 5 General Ability Training gniniarT cificepS hctaM Wages Conditional on Training 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 64

Figure 9: Match-Specific Training Thisfigureshowstheamountofmatch-specifictrainingthatwokersreceiveatdifferentcombinationsonaandθ. Linesonthe graphshowcontoursalongwhichtheamountoftrainingremainsconstant. Theblankareabelowtheblackline,showsstates forwhichworkerswillnotacceptthejoboffer. 0.05 0.05 0.05 0.1 0.1 0.1 0.15 0.15 0.15 0.15 0.15 0.15 0.1 0.1 0.1 0. 0 5 0.05 0.05 0.05 eulaV hctaM goL Time Spend on Match−Specific Training 2.4 2.3 2.2 2.1 2 1.9 1.8 1.7 1.6 1.5 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 Log Worker Ability 65

Figure 10: General Training Thisfigureshowstheamountofgeneraltrainingthatworkersreceiveatdifferentcombinationsonaandθ. Linesonthegraph show contours along which the amount of training remains constant. The blank area below the black line, shows states for whichworkerswillnotacceptthejoboffer. 0.05 0.1 0.1 0.1 0.15 0.15 0 1. 0.05 0.15 0.15 0.1 0.05 0.1 0.05 0.05 eulaV hctaM goL Time Spent on General Training 2.4 2.3 2.2 2.1 2 1.9 1.8 1.7 1.6 1.5 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 Log Worker Ability 66

Figure 11: Wage as Fraction of Output Thisfigureshowstheworker’swageasafractionofhertotaloutputatdifferentcombinationsonaandθ. Linesonthegraph showcontoursalongwhichthefractionremainsconstant. Theblankareabelowtheblackline,showsstatesforwhichworkers willnotacceptthejoboffer. 0.8 0.8 0.8 0.85 0.85 0.85 0.9 0.9 0.9 0.95 0.95 0.95 1 1 1 1 1 00100..89..8955 00..0089..5589 1 00.0.089..5859 00.0.089..5859 eulaV hctaM goL Wage as Fraction of Output 2.4 2.3 2.2 2.1 2 1.9 1.8 1.7 1.6 1.5 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 Log Worker Ability 67

Figure 12: Training by Years in Labor Market This figure shows the average fraction of their time that workers spend on general and match-specific training in the model simulationasafunctionofthenumberofyearsinthelabormarket. emiT fo noitcarF 2. 51. 1. 50. Training by Years in Labor Force 0 2 4 6 8 Years Total Training General Training Match−Specific Training 68

Figure 13: Sources of Wage Growth by Years in Labor Market This figure shows the average of the log of general human capital, the amount of match-specific capital accumulated through searchandthroughtraining,thelogofthefractionoftimetheyspendnottraining,andthelogoftheaveragewageasafraction ofworkeroutputforsimulatedworkersasafunctionofthenumberofyearsinthelabormarket. Intheabsenceofemployment costs,thesefivecomponentswouldadduptothetotallogwage,whichisalsoshown. 4. 3. 2. 1. 0 Decomposition of Log Wage by Years in Labor Force 0 2 4 6 8 Years Total Log Wage Log General HC Log Specific HC (Search) Log Specific HC (Training) Log Time Not Training Log Wage/Output 69

Table 10: Mincer Regressions Const SomeCollege BA YearsinLF Tenure DATA logwage 2.141 0.316 0.796 0.051 0.045 (0.098) (0.098) (0.098) (0.002) (0.003) MODEL logwage 2.401 0.260 0.558 0.042 0.058 startinga 0.943 0.207 0.454 0.001 -0.001 afromtraining 0.022 -0.001 -0.010 0.028 0.003 θ fromsearch 2.023 -0.004 -0.007 0.005 0.010 θ fromtraining 0.026 0.004 0.005 -0.001 0.038 hoursworked -0.241 0.006 0.013 0.002 0.014 wage/output -0.371 0.052 0.104 0.007 -0.006 Table 11: Mincer Regressions by Education Constant YearsinLF Tenure HS College BA HS College BA HS College BA DATA logwage 2.305 2.667 2.951 0.069 0.039 0.044 0.033 0.052 0.053 (0.011) (0.015) (0.015) (0.002) (0.004) (0.006) (0.003) (0.006) (0.007) MODEL logwage 2.408 2.658 2.947 0.042 0.042 0.044 0.056 0.061 0.064 startinga 0.944 1.150 1.400 0.001 0.001 0.002 -0.005 -0.001 -0.003 afromtraining 0.018 0.019 0.017 0.029 0.028 0.027 0.003 0.003 0.002 θ fromsearch 2.032 2.015 2.000 0.003 0.006 0.009 0.010 0.010 0.012 θ fromtraining 0.034 0.024 0.017 -0.001 -0.001 -0.001 0.035 0.042 0.048 hoursworked -0.239 -0.236 -0.232 0.001 0.002 0.002 0.013 0.014 0.016 wage/output -0.380 -0.314 -0.252 0.008 0.006 0.004 -0.005 -0.007 -0.011 70

Figure 14: Mininimum Acceptable Wage with Minimum Wage Thisfigureshowstheθ∗(a),thelowestmatchqualitythatworkerswitheachlevelofgeneralabilitywillaccept. Thesolidline showsθ∗(a)forthebaselinemodel,thedashedlinewhenweimposeaminimumwage. 2 1.95 1.9 1.85 1.8 1.75 1.7 1.65 1.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 eulaV hctaM goL Minimum Acceptable Match Value Baseline Minimum Wage Log Worker Ability 71

Figure 15: Training by Years in Labor Market with Minimum Wage This figure shows the average fraction of their time that workers spend on general and match specific training in the model simulationasafunctionofthenumberofyearsinthelabormarket. Thesolidlinesshowtheamountoftraininginthebaseline model. Thecorrespondingdashedlinesshowtheamountoftrainingwhenweimposeaminimumwage. emiT fo noitcarF 2. 51. 1. 50. Training by Years in Labor Force 0 2 4 6 8 Years Total Training General Training Match−Specific Training With Minimum Wage 72

Figure 16: Log Wage Distribution by Years in Labor Market with Minimum Wage This figure shows the distribution of log wages in the model simulation as a function of the number of years in the labor market. The solid lines show percentiles of the distribution in the baseline model. The corresponding dashed lines show the samepercentileswhenweimposeaminimumwage. 5.3 3 5.2 2 Log Wage Distribution by Years in Labor Force 0 2 4 6 8 Age 1st Percentile 10th Percentile 25th Percentile 50th Percentile 75th Percentile With Minimum Wage 73