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Journal of the European Economic Association
, Volume Advance Article – May 22, 2018

51 pages

/lp/ou_press/earnings-losses-and-labor-mobility-over-the-life-cycle-Ad0O0f0Ig9

- Publisher
- Oxford University Press
- Copyright
- © The Author(s) 2018. Published by Oxford University Press on behalf of European Economic Association.
- ISSN
- 1542-4766
- eISSN
- 1542-4774
- D.O.I.
- 10.1093/jeea/jvy014
- Publisher site
- See Article on Publisher Site

Abstract Large and persistent earnings losses following displacement have adverse consequences for the individual worker and the macroeconomy. Leading models cannot explain their size and disagree on their sources. Two mean-reverting forces make earnings losses transitory in these models: search as an upward force allows workers to climb back up the job ladder, and separations as a downward force make nondisplaced workers fall down the job ladder. We show that job stability at the top rather than search frictions at the bottom is the main driver of persistent earnings losses. We provide new empirical evidence on heterogeneity in job stability and develop a life-cycle search model to explain the facts. Our model offers a quantitative reconciliation of key stylized facts about the U.S. labor market: large worker flows, a large share of stable jobs, and persistent earnings shocks. We explain the size of earnings losses by dampening the downward force. Our new explanation highlights the tight link between labor market mobility and earnings dynamics. Regarding the sources, we find that over 85% stem from the loss of a particularly good job at the top of the job ladder. We apply the model to study the effectiveness of two labor market policies, retraining and placement support, from the Dislocated Worker Program. We find that both are ineffective in reducing earnings losses in line with the program evaluation literature. 1. Introduction Large and persistent earnings losses following job displacement are a prime source of income risk in macroeconomic models (Rogerson and Schindler 2002). They amplify the costs of business cycles (Krusell and Smith 1999; Krebs 2007) and increase the persistence of unemployment after adverse macroeconomic shocks (Ljungqvist and Sargent 1998). Understanding their size and sources is important for macroeconomic policies. However, leading models of the labor market do not provide much guidance, emphasizing different sources and accounting only for small and transitory earnings losses (Davis and von Wachter 2011). The inability of existing models to account for large and persistent earnings losses calls for an explanation. This paper offers an explanation based on an estimated structural life-cycle search and matching model of the U.S. economy. It is built around the observation that both an upward and a downward force prevent earnings losses from looming large in most models. The upward force is search. Displaced workers who fall off the job ladder can search on and off the job, trying to climb back up. Search frictions prevent an immediate catch-up, but, given the large job-to-job transition rates observed in the data, search is a powerful mean-reverting mechanism. The downward force is separations at the top of the job ladder. Short match durations due to high separation rates quickly make a currently nondisplaced worker look similar to a displaced worker. These two forces induce mean reversion of the earnings process and make earnings losses transitory and short-lived in most search models. To explain persistent earnings losses, this paper shifts the emphasis away from displaced workers’ inability to recover after displacement and toward the job stability of nondisplaced workers’ employment paths. We provide empirical evidence on job stability and heterogeneity in worker mobility by age and tenure based on the Current Population Survey (CPS). We show that the coexistence of large worker turnover (Shimer 2012) with a large share of stable jobs (life-time jobs in Hall 1982) dampens the downward force but keeps the upward force in place. This turns the job ladder into a mountain hike that requires free climbing at the bottom but offers a fixed-rope route at the top. Reaching the top takes long, but once workers arrive at the top, the hike becomes a convenient and secure walk. The economic rationale for this job ladder is simple and intuitive: employers and employees in high-surplus jobs agree on high wages and low separation rates, in both cases because of a high surplus. We provide empirical evidence supporting such a negative correlation between wages and separation rates using data from the Survey of Income and Program Participation (SIPP). Focusing on the earnings paths of nondisplaced workers at the top of the job ladder rather than displaced workers offers a new perspective on the actual size of earnings losses. It also sheds new light on the sources of earnings losses and how they matter for policy. We show that estimators of earnings losses pioneered by Jacobson, LaLonde, and Sullivan (1993) and today’s standard in the literature have a sizable selection effect due to their construction of the control group of nondisplaced workers. We decompose the sources of earnings losses and find that up to 30% of the estimated earnings losses result from a selection effect, 20% from increased job instability, and 50% from lower wages. Decomposing wage losses further, we find that more than 85% stem from the loss of a particularly good job, meaning a fall from the top of the job ladder. We discuss how our findings matter for active labor market policy. We use the model to study the effectiveness of retraining and placement support programs of the Dislocated Worker Program of the Workforce Investment Act. We find very limited scope for active labor market policies to reduce earnings losses, mirroring the findings from the empirical program evaluation literature (Card, Kluve, and Weber 2010). Our structural model offers a clear reason for this failure: active labor market policy operates on search frictions and could foster mean reversion by making displaced workers recover to the average. However, we argue that active policy cannot affect the downward force that makes nondisplaced workers look so different from the average. Our emphasis on the evolution of nondisplaced workers’ earnings paths rather than the recovery path of displaced workers makes our explanation distinct from previous attempts to explain earnings losses. Existing attempts focus on dampening the upward force of search for better jobs, either by adding search frictions directly or by introducing deterioration of job prospects due to displacement. Explanations based on the deterioration of accumulated experience or skills during unemployment (Ljungqvist and Sargent 2008) struggle to endogenously account for worker mobility because workers are very reluctant to switch jobs in the presence of large expected skill losses (den Haan, Haefke, and Ramey 2005). This explanation also has to rule out subsequent skill accumulation on the job to avoid mean reversion. Others, as we do, point toward the loss of a particularly good job as an explanation for earnings losses (Low, Meghir, and Pistaferri 2010). Falling down the job ladder subsequently leads to more frequent job losses, more unemployment, and job instability (Stevens 1997; Pries 2004). Recent explanations in the same spirit can be found in Krolikowski (2017), who makes the job ladder very long, and Jarosch (2014), who makes the job ladder slippery. All of these explanations have in common that they attempt to prevent displaced workers from climbing up the job ladder. However, although frictions to move upward must also exist for our explanation to work, we show that shutting down the downward force is a crucial step for slowing down mean reversion and accounting for large and persistent earnings losses. Without job stability at the top of the job ladder, alternative explanations are likely to fail because the job ladder is a powerful mechanism for mean reversion (Low et al. 2010; Hornstein, Krusell, and Violante 2011). High job stability in high-wage jobs is a key ingredient in generating persistent earnings differences. Our new explanation highlights the tight link between labor market mobility and earnings dynamics. Our model features heterogeneity in job stability with stable jobs at the top of the job ladder. It jointly accounts for high labor market mobility and persistent earnings losses. To account for high labor market mobility, we need a high degree of transferability of skills in the labor market, and to account for persistent earnings losses, we need jobs at the top of the job ladder that are very stable. The highlighted mechanism explains the inability of most existing labor market models to generate large and persistent earnings losses. They do not account for heterogeneity in job stability but impose a single separation rate across jobs, matching average mobility uniformly along the job ladder. Hence, workers rotate continuously out of good jobs, which results in earnings losses that are highly transitory and short-lived. We develop a search and matching model that accounts for life-cycle effects and has various sources of skill heterogeneity and on-the-job search. Search is directed (Menzio and Shi 2011), and wage and mobility choices are efficiently bargained (den Haan, Ramey, and Watson 2000a). The model not only captures the empirical facts on tenure and wages as in Moscarini (2005) but also accounts for the mobility pattern by tenure and age, adding to a recently growing strand of the literature on life-cycle labor market models.1 Introducing life-cycle dynamics is crucial for our explanation because it copes with the nonstationary dynamics of tenure by age that we document, and it helps to disentangle the relative importance of different components of the skill accumulation process. We explain how we exploit heterogeneity in worker mobility by age and tenure to identify model parameters as alternative to an identification relying on wage dynamics and wage heterogeneity. Regarding mobility, the model accounts for high average worker mobility even for older workers (Farber 1995), a large fraction of stable jobs (Hall 1982), and frequent job changes during the first 10 years of working life (Topel and Ward 1992). Regarding earnings dynamics, the model accounts for a declining age profile of wage gains after job changes and substantial early career wage growth due to job changes (Topel and Ward 1992), large returns to tenure estimated using the methodology advocated in Topel (1991) and small returns to tenure estimated using the methodology advocated in Altonji and Shakotko (1987), permanent earnings shocks as in Heathcote, Perri, and Violante (2010), and large and persistent earnings losses following job displacement as in Couch and Placzek (2010), Davis and von Wachter (2011), and von Wachter, Song, and Manchester (2009).2 The model also generates the empirically observed cross-sectional wage inequality that existing models struggle to explain (Hornstein et al. 2011). Hence, our model not only speaks to the empirical literature studying earnings losses but also offers a quantitative reconciliation of key stylized facts about the U.S. labor market: the coexistence of large worker flows, a large share of stable jobs, and earnings dynamics with large and persistent shocks. The quantitative success with respect to the size of the earnings losses allows us to quantify the sources of earnings losses. We implement an empirical estimator within our model and decompose earnings losses using counterfactual experiments that are only possible in a structural model. One source is a selection effect in the empirical estimator. We construct an ideal counterfactual experiment of “twin” workers using characteristics unobserved by the econometrician to make workers identical except for the displacement event. We find a sizable upward bias of 30% in estimated earnings losses. Although the possibility of bias is well known, its quantitative size could only be localized within a range. Our findings close this gap. Although we emphasize job stability at the top of the job ladder and along the counterfactual employment path of displaced workers, we demonstrate that the assumption on the counterfactual employment path imposed in the empirical implementation strategy is too strong. Once we control for this selection effect, we use the twin experiment to measure the reduction in earnings resulting from lower average employment in the group of displaced workers relative to the group of nondisplaced workers. In our decomposition, this extensive margin effect accounts for 20%. As a result, direct skill losses account for the remaining 50%, what we call the wage loss effect. We adopt the empirical approach in Stevens (1997) based on data from the Panel Study of Income Dynamics (PSID) and demonstrate that our model-based decomposition is in line with empirical estimates. Given that the empirical earnings loss estimates are an input to many calibrated macroeconomic models, our findings suggest some caution in using the empirical findings at face value. Our decomposition can go further because we observe in the model the evolution of skills of displaced and nondisplaced workers. We use this information to study whether the extensive margin and the wage loss effect arise from the loss of worker-specific skills or from the loss of a particularly good match. We find that match-specific skill losses account for more than 85% of both effects, therefore justifying the statement that earnings losses are the result of the loss of a particularly good job rather than the deterioration of worker-specific skills. Our finding on the skill losses is highly relevant for the design of active labor market programs and motivates our policy analysis. We look at two policy pillars, retraining and placement support, of the Dislocated Worker Program of the Workforce Investment Act. We consider worker-specific skill losses as losses that can be restored via retraining, whereas match-specific skill losses need to be restored via placement support that improves the match between workers and jobs by supporting labor market search. Within our model, we implement a stylized retraining and placement support program and find that both programs are ineffective. Retraining will not help much because worker-specific skill losses account for only a small fraction of the earnings losses. Placement support remains ineffective because even if placement support could create six job offers per month (roughly the equivalent of one year of search in our model) and bring the worker back to the average match quality of the worker’s cohort, the resulting earnings losses would be reduced by only one-fourth and would remain large and persistent. Hence, active policy might help to remove frictions and foster mean reversion by making displaced workers recover to the average but it cannot affect the downward force that makes nondisplaced workers persistently different from the average. It is the missing downward force due to job stability at the top that drives the persistence of earnings losses. We proceed as follows: In Section 2, we perform an empirical analysis of worker mobility and job stability. Section 3 develops our life-cycle model of worker mobility and explains the identification of model parameters based on worker mobility. Section 4 discusses the model fit for worker mobility and presents the fit for untargeted earnings dynamics. Section 5 estimates the earnings losses following job displacement from the model and decomposes them. Section 6 studies labor market policies to counteract the adverse consequences of worker displacement. Section 7 concludes. 2. Empirical Analysis Facts about average worker mobility have been widely documented (e.g., Fallick and Fleischman 2004; Shimer 2012). We highlight four facts documenting substantial heterogeneity in worker mobility: (1) transition rates from employment to nonemployment and job-to-job transitions decline by age; (2) conditioning on tenure and looking at newly hired workers, transition rates decline by age, but the decline is much smaller than the unconditional decline by age; (3) despite large average transition rates, mean tenure increases linearly with age, showing that many jobs are very stable; (4) wages and separations are strongly negatively correlated, implying that high-wage jobs are more stable. 2.1. Data Our analysis is based on U.S. data from the monthly CPS files and the Occupational Mobility and Job Tenure supplements for the period 1980 to 2007.3 In contrast to alternative data sources, the CPS offers large representative cross sections of workers and provides a long time dimension covering several business cycles. This fact allows us to abstract from business cycle fluctuations in transition rates by averaging transition rates over time. Tenure information is not available in the monthly CPS files but only in the irregular Occupational Mobility and Job Tenure supplements. We merge this information with the basic monthly files to construct transition rates by tenure.4 We follow Shimer (2012) and Fallick and Fleischman (2004) in constructing worker flows. Job-to-job transitions and all transitions out of employment end tenure. To avoid overstating job stability, we take as the separation rate the sum of the transition rate to unemployment and out of the labor force. We relegate details on the data and construction of transition rate and tenure profiles to Appendix A.1. 2.2. Worker Mobility and Job Stability Figure 1 depicts age heterogeneity in monthly separation and job-to-job transition rates. Both transition rates fall with age. Most of the decrease in transition rates by age takes place between the ages of 20 and 30. This initial period is followed by 25 years of stable transition rates.5 Separations drop from an initial high of 8% to a low of around 2%, and job-to-job transitions from an initial high of 5% to a low of about 1%. Even during the stable years between ages 30 and 50, approximately 3% of workers leave employers each month. Confidence bands around the profiles indicate that both profiles are tightly estimated. Figure 1. View largeDownload slide Empirical age transition rate profiles. Age profiles for separation and job-to-job rates. The gray dashed lines show confidence bands using ±2 standard deviations. Transition rates are monthly. Standard deviations are bootstrapped using 10,000 repetitions from the pooled sample stratified by age. The horizontal axis shows age in years, and the vertical axis shows transition rates in percentage points. Figure 1. View largeDownload slide Empirical age transition rate profiles. Age profiles for separation and job-to-job rates. The gray dashed lines show confidence bands using ±2 standard deviations. Transition rates are monthly. Standard deviations are bootstrapped using 10,000 repetitions from the pooled sample stratified by age. The horizontal axis shows age in years, and the vertical axis shows transition rates in percentage points. The average transition rates by age mask further heterogeneity. Figure 2(a) shows that mean and median tenure increase almost linearly with age. If transition rates were uniform in the population and equal to the 3% of workers who leave employers between ages 30 and 50 every month, then mean tenure would converge to slightly less than 3 years, well below the observed 11 years of tenure at age 50. This shows that even conditional on age, there is large heterogeneity in transition rates. Again, confidence bands show that these profiles are tightly estimated. Figure 2. View largeDownload slide Tenure by age and transition rates by age for newly hired workers. Panel (a) shows mean and median tenure in years by age. The gray dashed lines show confidence bands using ±2 standard deviations. Standard deviations are bootstrapped using 10,000 repetitions from the pooled sample stratified by age. The horizontal axis shows age in years, and the vertical axis shows tenure in years. Panels (b) and (c) show separation and job-to-job transition rates by age for newly hired workers. Newly hired workers are workers with one year of tenure. The gray dashed lines show confidence bands using ±2 standard deviations. Transition rates are monthly. Standard deviations are bootstrapped using 10,000 repetitions from the pooled sample. The horizontal axis shows age in years starting at age 21, and the vertical axis shows transition rates in percentage points. Figure 2. View largeDownload slide Tenure by age and transition rates by age for newly hired workers. Panel (a) shows mean and median tenure in years by age. The gray dashed lines show confidence bands using ±2 standard deviations. Standard deviations are bootstrapped using 10,000 repetitions from the pooled sample stratified by age. The horizontal axis shows age in years, and the vertical axis shows tenure in years. Panels (b) and (c) show separation and job-to-job transition rates by age for newly hired workers. Newly hired workers are workers with one year of tenure. The gray dashed lines show confidence bands using ±2 standard deviations. Transition rates are monthly. Standard deviations are bootstrapped using 10,000 repetitions from the pooled sample. The horizontal axis shows age in years starting at age 21, and the vertical axis shows transition rates in percentage points. Next, we look at newly hired workers.6 Considering newly hired workers helps to further unmask heterogeneity in worker mobility. We refer to age profiles for newly hired workers for simplicity as “newly hired age profiles”. Figure 2 plots separation (Figure 2b) and job-to-job (Figure 2c) newly hired age profiles together with confidence bands. Two points are important. First, separation and job-to-job newly hired age profiles decline with age. As for the age profiles in Figure 1, the decline is concentrated in the first 10 years in the labor market. Second, the decline by age for newly hired workers is about half of the unconditional decline by age. The separation rate declines by about 2.5 percentage points, and the job-to-job transition rate declines by about 1.7 percentage points in comparison to the unconditional 5 percentage points and 3 percentage points decline by age, respectively.7 This evidence, together with the linear increase in tenure by age, points toward considerable heterogeneity in job stability. Although wage heterogeneity has been studied extensively, much less attention has been paid to quantitatively account for the substantial heterogeneity in job stability in models of the labor market. Typically, models of the labor market are designed to explain and study average labor market flows. Our empirical analysis highlights a large share of stable jobs and substantial heterogeneity in worker mobility. As we document next, this heterogeneity in job stability correlates strongly negatively with wages. We document that high-wage jobs are also very stable. 2.3. Job Stability and Wages When studying the connection between wages and job stability, we want to explore whether high-wage jobs today are less likely to separate in the future. For this, we need individual-level panel data to observe future transitions to nonemployment given the current wage. We therefore resort to data from the 2004 SIPP.8 We construct h-month separation rates. The h-month separation rate is the share of workers who are employed today but who separate at least once within the next h months into nonemployment. We consider 4- and 12-month separation rates.9 We explore the relationship between wages and job stability using two approaches. First, we run a regression of the h-month separation rate $$\pi _{i,t}^{h}$$ on log wages log ($$w$$i, t) and age dummies $$\gamma _{i,t}^{a}$$, \begin{equation*} \pi _{i,t}^{h} = \beta \log (w_{i,t}) + \gamma _{i,t}^{a} + \varepsilon _{i,t},{} \end{equation*} where i indexes individuals and t calendar time. To focus on matches with high separation rates, we also run the regression for newly hired workers only.10 Table 1 shows the coefficient β from the regressions. We find that coefficients are negative and significant at the 1% level in all specifications. Table 1. Regression coefficients of separation rates on log wages. Separation horizon (h) 4 months 12 months All workers –0.0392 –0.0668 std. error (0.0004) (0.0005) Newly hired workers –0.0548 –0.0822 std. error (0.0016) (0.0019) Separation horizon (h) 4 months 12 months All workers –0.0392 –0.0668 std. error (0.0004) (0.0005) Newly hired workers –0.0548 –0.0822 std. error (0.0016) (0.0019) Notes: Regression coefficient β from regression of 4-(12-)month separation rate on log wages and further controls. First row shows regression coefficient from regression with all workers and the corresponding standard errors. Second row shows regression coefficient when only newly hired workers are considered in the regression and the corresponding standard errors. View Large Table 1. Regression coefficients of separation rates on log wages. Separation horizon (h) 4 months 12 months All workers –0.0392 –0.0668 std. error (0.0004) (0.0005) Newly hired workers –0.0548 –0.0822 std. error (0.0016) (0.0019) Separation horizon (h) 4 months 12 months All workers –0.0392 –0.0668 std. error (0.0004) (0.0005) Newly hired workers –0.0548 –0.0822 std. error (0.0016) (0.0019) Notes: Regression coefficient β from regression of 4-(12-)month separation rate on log wages and further controls. First row shows regression coefficient from regression with all workers and the corresponding standard errors. Second row shows regression coefficient when only newly hired workers are considered in the regression and the corresponding standard errors. View Large The coefficient β varies for the different specifications between −0.04 and −0.08. This implies that a 10% higher wage leads to a 0.4–0.6 percentage points lower separation rate over 4 months and a 0.7–0.8 percentage points lower separation rate over 12 months. This effect is economically significant, given an average separation rate of around 2 percentage points at age 40. Second, we use residuals from a regression of log wages on age and group workers according to their residuals in wage deciles. We plot separation rates by wage decile in Figure 3. Looking at all workers in Figure 3(a), we find that between the lowest and the highest decile separation rates differ by a factor of almost 3 (0.12 vs. 0.04). In Figure 3(b), we show the same wage-job stability relationship but look only at newly hired workers. Again we find a strongly negative relationship. Separation rates decline by roughly 30% across wage deciles (0.18–0.12).11 Figure 3. View largeDownload slide Wages and job stability. Separation rates over a 4-month horizon by wage decile using SIPP data. The left panel shows separation rates for all workers as gray circles, and the dashed line shows a quadratic polynomial approximation to the data. The right panel shows separation rates for newly hired workers as gray circles, and the dashed line shows a quadratic polynomial approximation to the data. Workers are grouped in wage deciles using wage residuals. Wage deciles are on the horizontal axis. The vertical axis shows 4-month separation rates. See text for further details. Figure 3. View largeDownload slide Wages and job stability. Separation rates over a 4-month horizon by wage decile using SIPP data. The left panel shows separation rates for all workers as gray circles, and the dashed line shows a quadratic polynomial approximation to the data. The right panel shows separation rates for newly hired workers as gray circles, and the dashed line shows a quadratic polynomial approximation to the data. Workers are grouped in wage deciles using wage residuals. Wage deciles are on the horizontal axis. The vertical axis shows 4-month separation rates. See text for further details. The next section develops a structural life-cycle model with two-dimensional skill heterogeneity to account for the documented heterogeneity in worker mobility. The model also features the documented correlation between wages and job stability. By contrast, most existing models assume that separations happen exogenously and thereby feature no correlation between wages and separation rates. Heterogeneity in job stability and the correlation with wages will be instrumental in generating large and persistent earnings losses, as we show in Section 5. In Online Appendix C, we use a simple example with two types to explain the intuition behind the tight link between earnings losses and heterogeneity in job stability. 3. Model We develop a life-cycle labor market model in the search and matching tradition. For the most part, the building blocks of our model follow a large strand of the literature. Deviations are designed to capture the heterogeneity in labor market mobility and job stability outlined previously. We describe the model here and relegate a discussion of our modeling assumptions to Online Appendix D.1. A detailed derivation of all equations can be found in Online Appendix D.2. Time is discrete. There is a continuum of mass 1 of finitely lived risk-neutral agents and a positive mass of risk-neutral firms. Firms and workers discount the future at rate β < 1. Workers participate for T periods in the labor market followed by TR periods of retirement. Each firm has the capacity to hire a single worker, and we refer to a worker-firm pair as a match. Agents differ by age a, a vector of skills x, and employment state ϵ = {e, n} with e for employment and n for nonemployment. We use primes to denote variables in the next period. In a slight abuse of notation, we drop primes if variables do not change between periods. Each period is divided into four stages: bargaining, separation, production, and search. At the bargaining stage, each match bargains jointly about when to separate into nonemployment, the amount of wages to be paid if the production stage is reached, and when to accept a job offer from another firm at the search stage. We assume generalized Nash bargaining over the total match surplus, which leads to individually efficient choices. Separations happen after the bargaining stage, job-to-job transitions and transitions from nonemployment into employment happen at the search stage, and we assume that a worker’s labor market status is observed at the production stage. Vacancy posting by firms is directed to submarkets of worker types {ϵ, a, x}. There is free entry to submarkets, and a matching function determines contact rates in each submarket. 3.1. Skill Process The skill vector is x = {x$$w$$, xm} where x$$w$$ is the skill level of the worker and xm is the quality of the match. We assume that match-specific skills xm are drawn at the beginning of a match according to a probability distribution g(xm) where g is taken to be a discrete approximation to the normal density with (exponential) mean normalized to 1 and variance $$\sigma _{m}^2$$. The match-specific skill component remains constant throughout the existence of a match. We also approximate worker-specific skill states x$$w$$ by a finite number of states in an ordered set. The smallest (largest) element is $$x_{w}^{\rm {min}}$$ ($$x_{w}^{{\rm max}}$$), and the immediate predecessor (successor) of x$$w$$ is $$x_{w}^{-}$$ ($$x_{w}^{+}$$). Workers start their life at the lowest skill level and stochastically accumulate skills. Skills accumulate only if a worker stays in the current match. The worker’s skill level next period is $$x_{w}^{+}$$ with age-dependent probability p$$u$$(a), and it remains at x$$w$$ with probability 1 − p$$u$$(a). The distribution over next period’s worker skills $$x^{\prime }_{w}$$ if staying in a match is \begin{equation*} x^{\prime }_{w} = \left\lbrace \begin{array}{@{}l@{\quad }l@{}}x_{w} & \text{with probability $1 - p_{u}(a)$}, \\ x_{w}^{+} & \text{with probability $p_{u}(a)$}, \end{array}\right. \end{equation*} and we set p$$u$$(a) = 0 for $$x_{w} = x_{w}^{\rm {max}}$$. Age dependence follows from a simple recursion p$$u$$(a) = (1 − δ)p$$u$$(a − 1) to capture a potential slowdown in skill accumulation with age. The transferability of worker skills in the labor market is imperfect. A worker of type x$$w$$ who takes a new job either from employment or nonemployment faces the risk that part of the accumulated skills will not transfer to the new job. If the worker takes a new job, then with probability 1 − pd, all of the accumulated skills will transfer to the new job and the worker will remain at skill level x$$w$$. With probability pd, part of the accumulated skills will not transfer and the skill level next period will be $$x_{w}^{-}$$. We set pd = 0 for $$x_{w} = x_{w}^{\rm {min}}$$. The distribution over next period’s worker skills $$x^{\prime }_{w}$$ in case of worker mobility is \begin{equation*} x^{\prime }_{w} = \left\lbrace \begin{array}{@{}l@{\quad }l@{}}x_{w}^{-} & \text{with probability $p_{d}$}, \\ x_{w} & \text{with probability $1 - p_{d}$}. \end{array}\right. \end{equation*} A worker who takes up a new job from nonemployment faces the same skill transition. In addition, workers in nonemployment do not accumulate skills so that skills during nonemployment depreciate relative to employment. We discuss a model extension with additional skill depreciation during nonemployment in Online Appendix F.1. To ease the exposition, we use $$\mathbb {E}_{s}[\cdot ]$$ to denote the expectation over future skill states conditional on staying in the match (subscript s for staying) and $$\mathbb {E}_{m}[\cdot ]$$ to denote the expectation conditional on changing jobs (subscript m for mobility). With this notation in place, we turn to a derivation of endogenous choices. 3.2. Value Functions A worker-firm match with worker of age a and skill vector x = {x$$w$$, xm} produces output y according to the production function y = f(x$$w$$, xm) + ηs, where ηs is an idiosyncratic transitory productivity shock assumed to be logistically distributed with distribution function H(ηs) having a mean of zero and variance $$\pi ^{2} \psi _{s}^{2}/3$$. For each match, there exists a cutoff value $$\bar{\omega }$$ for the productivity shock at which the match separates. Following den Haan et al. (2000a), this cutoff value is determined as part of the bargaining described in what follows. Exploiting the assumption of a logistic distribution, we can write the probability of separating as $${\pi _{s}\equiv H(\bar{\omega })=(1+\text{exp}(-\bar{\omega }/\psi _{s}))^{-1}}$$ and the conditional mean of the realized productivity shocks has a closed form given by $$\Psi _{s}(\pi _{s})\equiv \int _{\bar{\omega }}^{\infty } \eta dH(\eta )$$.12 In addition, there is a probability πf of exogenous separation each period. The exogenous separation shock happens before the endogenous separation decision. Let J(x$$w$$, xm, a) denote the value of a firm that is matched at the beginning of the period to a worker of age a with productivity x. The value of the firm is13 \begin{eqnarray} J(x_{w},x_{m},a) &=& (1 - \pi _{f})(1 - \pi _{s}(x_{w},x_{m},a)) \bigg (f(x_{w}, x_{m}) + \frac{\Psi _{s}(\pi _{s})}{1-\pi _{s}(x_{w},x_{m},a)} \nonumber\\ &&- w(x_{w},x_{m},a) + (1 - \pi _{eo}(x_{w},x_{m},a)) \beta \mathbb {E}_{s}\left[ J(x_{w}^{\prime },x_{m},a^{\prime })\right] \bigg ) .\qquad \end{eqnarray} (1) With probability πf (πs), the match separates exogenously (endogenously). Productivity shocks ηs are transitory i.i.d. shocks, and the endogenous separation probability depends on the current state of the match. By contrast, exogenous separations lead to separations irrespective of the current state of the match. If no separation occurs, the match transits to the production stage. Upon reaching the production stage, the match produces output and pays wages $$w$$. Integrating out productivity shocks, output comprises a component Ψs(πs)/(1 − πs(x$$w$$, xm, a)). The value Ψs can be interpreted as an option value from having a choice to separate or not after having received a shock.14 The fact that an option value arises is not a particular feature of our model but a generic feature of an endogenous mobility choice. The fact that it has an analytic representation results from our distributional assumption on shocks. With probability πeo (described in what follows), the worker makes a job-to-job transition; otherwise the match continues to the next period. We denote the value function of an employed worker of age a with skill type x$$w$$ and matched to a firm of type xm by Ve(x$$w$$, xm, a), and Vn(x$$w$$, a) is the corresponding value of a nonemployed worker. During nonemployment, the worker receives flow utility b. At the search stage, nonemployed workers receive job offers with type- and age-dependent probability p$${ne}$$(x$$w$$, a). Each job offer comes with a stochastic utility component attached to it. We denote the average utility component from job changing by κo and the stochastic, idiosyncratic part by ηo. The realization of the idiosyncratic part is independent of the current state. Depending on the match quality of the offer $$x^{\prime }_{m}$$ and the utility component, the worker decides whether to accept the offer or not. A nonemployed worker chooses the maximum of $$\left\lbrace V_{n}(x_{w},a^{\prime }),\mathbb {E}_{m}\left[V_{e}(x^{\prime }_{w},x^{\prime }_{m},a^{\prime })\right] -\kappa _{o} + \eta _{o} \right\rbrace$$. As for the productivity shocks ηs, we assume that the idiosyncratic utility component ηo is logistically distributed with mean zero and variance $$\pi ^{2}\psi _{o}^{2}/3$$. The acceptance decision yields an option value Ψ$${ne}$$(q$${ne}$$) that arises because only job offers with high enough ηo will be accepted. We suppress arguments of q$${ne}$$ for notational convenience. The option value will enter the value functions in what follows. Using standard properties of the logistic distribution, we write the acceptance probability for a job offer of match type $$x_{m}^{\prime }$$ as \begin{multline} q_{ne}(x_{m}^{\prime };x_{w},a) \\ =\big (1 + \exp \big (\psi _{o}^{-1}\beta \big ( V_{n}(x_{w},x_{m},a^{\prime }) - \left(\mathbb {E}_{m}\left[ V_{e}(x^{\prime }_{w},x_{m}^{\prime },a^{\prime })\right] - \kappa _{o}\right)\big)\big)\big)^{-1}. \end{multline} (2) Note that we condition the acceptance probability on the offer type $$x_{m}^{\prime }$$, modeling match quality as an inspection good. The ex-ante value Vn(x$$w$$, a) before the realization of the idiosyncratic shock components is given by \begin{eqnarray} V_{n}(x_{w},a)\! &=& \!b \!+\! \overbrace{p_{ne}(x_{w},a)\!\sum _{x_{m}^{\prime }}\!\bigg (\!q_{ne}(x_{m}^{\prime };x_{w},a) \!\left(\beta \mathbb {E}_{m}\left[ V_{e}(x^{\prime }_{w},x^{\prime }_{m},a^{\prime }) \right]\!-\! \kappa _{o}\right)\!\!\bigg )g(x^{\prime }_{m})}^{\text{receiving and accepting offer}} \nonumber \\ &&+ \underbrace{\sum _{x^{\prime }_{m}}(1 - p_{ne}(x_{w},a)q_{ne}(x_{m}^{\prime };x_{w},a)) \beta V_{n}(x_{w},a^{\prime })g(x^{\prime }_{m})}_{\text{not receiving or not accepting offer}} \nonumber \\ &&+ p_{ne}(x_{w},a)\underbrace{\sum _{x_{m}^{\prime }}\Psi _{ne}(q_{ne})g(x^{\prime }_{m})}_{\text{option value}}, \end{eqnarray} (3) where the first line shows flow value b at the production stage and the case of receiving and accepting an offer at the search stage. The second line shows the case of not receiving or receiving but not accepting an offer and the option value in case an offer is received. The probability of entering employment combines the likelihood of receiving an offer p$${ne}$$ with the probability of accepting an offer q$${ne}$$ and is given by $$\pi _{ne}(x_{w},a) = \sum _{x^{\prime }_{m}} p_{ne}(x_{w},a) q_{ne}(x_{m}^{\prime };x_{w},a)g(x^{\prime }_{m})$$. An employed worker’s value function is \begin{eqnarray} V_{e}(x_{w},x_{m},a) &=& (1 - \pi _{f})(1 - \pi _{s}(x_{w},x_{m},a) )\left( w(x_{w},x_{m},a) + V^{S}_{e}(x_{w},x_{m},a) \right) \nonumber \\ &&+\, \left((1 - \pi _{f}) \pi _{s}(x_{w},x_{m},a) + \pi _{f}\right)V_{n}(x_{w},a), \end{eqnarray} (4) where $$V^{S}_{e}(x_{w},x_{m},a)$$ denotes the value function for an employed worker at the search stage. With probability (1 − πf)(1 − πs(x$$w$$, xm, a)), the match does not separate and the worker receives wage $$w$$(x$$w$$, xm, a) and enters the search stage providing value $$V^{S}_{e}(x_{w},x_{m},a)$$. If the match separates, the worker receives the value of nonemployment Vn(x$$w$$, a). Note that the separation stage is before the production stage and the search stage, so that a worker who separates at the separation stage receives flow value b during the production stage and searches as nonemployed during the search stage of the same period. The search process on the job is similar to nonemployment. The worker receives offers with type-dependent probability peo(x$$w$$, xm, a). Each offer comes with the nonpecuniary component as when searching off the job with the stochastic component drawn from the same distribution. The cutoff value above which a competing job offer $$x_{m}^{\prime }$$ is accepted is determined as part of the bargaining. We denote the implied acceptance probability for job offer $$x_{m}^{\prime }$$ by $$q_{eo}(x_{m}^{\prime };x_{w},x_{m},a)$$ and the option value from accepting only offers with favorable utility component as Ψeo(qeo). The search stage value function is \begin{eqnarray} V^{S}_{e}(x_{w},x_{m},a) \!&=& \!\overbrace{p_{eo}(x,a)\! \sum _{x_{m}^{\prime }}\!\bigg (\!q_{eo}(x_{m}^{\prime };x,a) \left(\beta \mathbb {E}_{m}\left[V_{e}(x^{\prime }_{w},x_{m}^{\prime },a^{\prime })\right] - \kappa _{o}\right) \!\bigg )g(x^{\prime }_{m}) }^{\text{receiving and accepting offer}}\nonumber \\ &&+ \underbrace{\sum _{x_{m}^{\prime }}(1 - p_{eo}(x,a)q_{eo}(x_{m}^{\prime };x,a) )\beta \mathbb {E}_{s}\left[V_{e}(x_{w}^{\prime },x_{m},a^{\prime })\right]g(x^{\prime }_{m})}_{\text{not receiving or not accepting offer}} \nonumber \\ &&+ p_{eo}(x,a) \underbrace{\sum _{x_{m}^{\prime }} \Psi _{eo}(q_{eo}) g(x^{\prime }_{m})}_{\text{option value}}. \end{eqnarray} (5) Note that acceptance probabilities on the job depend on the current match-specific type xm. The probability of leaving combines acceptance probabilities qeo with the probability of receiving an offer peo, \begin{equation*} \pi _{eo}(x_{w},x_{m},a) = \sum _{x^{\prime }_{m}} p_{eo}(x_{w},x_{m},a) q_{eo}(x_{m}^{\prime };x_{w},x_{m},a)g(x^{\prime }_{m}). \end{equation*} 3.3. Bargaining Every match bargains at the bargaining stage over when to separate to nonemployment at the separation stage, the wage that is paid if the match enters the production stage, and when to go to another firm at the search stage. We assume generalized Nash bargaining over the total surplus of the match with the worker’s outside option being nonemployment. The bargaining conditions on the realization of idiosyncratic shocks but given the risk neutrality of workers and firms only the expected value of the realized shock matters. To ease notation, we suppress state contingency with respect to idiosyncratic shocks and include only expected values in all equations. This bargaining follows den Haan et al. (2000a) or Jung and Kuester (2015) and it leads to an individually efficient outcome in which separations and job-to-job transitions occur only if the joint surplus of the match is too small. The bargaining solution satisfies \begin{align} [w,\pi_{s},{q_{eo}(x_{m}'})]= \arg \max & \quad J(x_{w},x_{m},a)^{1 - \mu }\Delta (x_{w},x_{m},a)^{\mu }\nonumber \\ \text{subj. to:} & \quad a,x_{w},x_{m} \text{ given,} \nonumber \end{align} where Δ(x, a) = Ve(x, a) − Vn(x, a) denotes the worker surplus. We denote by S(x, a) = Δ(x, a) + J(x, a) the total match surplus at the bargaining stage. Wage payments and mobility decisions happen at different stages within the period. To ease exposition, we therefore define surpluses at the production stage and the search stage. The worker surplus at the search stage is $${\Delta ^{S}(x_{w},x_{m},a) = V^{S}_{e}(x_{w},x_{m},a) - V_{n}(x_{w},a)}$$ and, in a slight abuse of terminology, we refer to \begin{equation*} S^{S}(x,a) = \mathbb {E}_{s}[\beta S(x_{w}^{\prime },x_{m},a^{\prime })] - \mathbb {E}_{m}[\beta \Delta (x_{w}^{\prime },x_{m}^{\prime },a^{\prime }) ] \end{equation*} as the surplus of staying in the current match relative to an outside offer at the search stage. At the production stage, the worker surplus is ΔP(x, a) = $$w$$(x, a) + ΔS(x, a), and \begin{equation*} J^{P}(x,a) = f(x) - w(x,a) + (1 - \pi _{eo}(x,a)) \beta \mathbb {E}_{s}[J(x^{\prime },a^{\prime })] \end{equation*} is the firm’s surplus net of idiosyncratic shocks.15 The total surplus is SP(x, a) = ΔP(x, a) + JP(x, a). We derive the solution to the bargaining in Online Appendix D.2. The solutions for $$w$$(x$$w$$, xm, a), πs(x$$w$$, xm, a), and $$q_{eo}(x^{\prime }_{m};x_{w},x_{m},a)$$ are \begin{eqnarray} \pi _{s}(x_{w},x_{m},a) = \big(1 + \exp \big(\psi _{s}^{-1} S^{P}(x,a) \big)\big)^{-1}, \end{eqnarray} (6) \begin{eqnarray} w(x_{w},x_{m},a) = \mu \left(S^{P}(x,a) + \frac{\Psi _{s}(\pi _{s})}{1 - \pi _{s}(x_{w},x_{m},a)}\right) - \Delta ^{S}(x_{w},x_{m},a), \qquad \end{eqnarray} (7) \begin{eqnarray} q_{eo}(x_{m}^{\prime };x_{w},x_{m},a) = \big(1 + \exp \big(\psi _{o}^{-1}\big (S^{S}(x,a) + \kappa _{o}\big)\big)\big)^{-1}. \end{eqnarray} (8) Joint bargaining links mobility choices πs and qeo to wages $$w$$. Mobility choices and wages are all functions of the match surplus. In general, the match surplus affects wages positively and mobility decisions negatively. Hence, the joint determination of wages and mobility decisions in our model will lead to high-surplus matches paying high wages and being very stable. This model feature matches the robust empirical correlation between wages and job stability reported in Section 2.3. The separation probability πs is proportional to the surplus SP so that high-surplus matches are less likely to separate because firm and worker agree that they separate only after particularly bad productivity shocks. This is in contrast to exogenous separations that lead to separations independent of the match surplus and therefore let workers fall even from the top of the job ladder. Wages are a linear function of the worker’s share of the total surplus SP and the option value Ψs minus the worker’s surplus from searching on the job ΔS. The fact that Ψs enters the wage equation is intuitive because the gains from having a choice to separate are shared between worker and firm. The option value captures the truncated favorable part of the transitory productivity shock distribution.16 The negative ΔS term represents a form of a compensating differential for differences between on- and off-the-job search. The better on-the-job search is, the lower are wages. Finally, acceptance decisions for outside offers depend on the match surplus at the search stage and utility component κo. A higher surplus of the current match reduces the likelihood of leaving. 3.4. Vacancy Posting and Matching To limit computational complexity and to avoid the age structure as an additional aggregate state, we borrow ideas from the literature on directed search (e.g., Menzio and Shi 2011) and assume that there exist submarkets for all types {ϵ, a, x}. When entering the market, firms direct vacancies to one submarket. To determine the number of vacancies, we impose free entry on each submarket: \begin{eqnarray} \kappa = p_{vn}(x_{w},a) \beta \sum _{x_{m}^{\prime }}q_{ne}(x_{m}^{\prime };x_{w},a)\mathbb {E}_{m}\left[J(x_{w}^{\prime },x_{m}^{\prime },a^{\prime })\right]g(x^{\prime }_{m}), \end{eqnarray} (9) \begin{eqnarray} \kappa = p_{vo}(x_{w},x_{m},a) \beta \sum _{x_{m}^{\prime }}q_{eo}(x_{m}^{\prime };x_{w},x_{m},a)\mathbb {E}_{m}\left[J(x_{w}^{\prime },x_{m}^{\prime },a^{\prime })\right]g(x^{\prime }_{m}), \end{eqnarray} (10) where κ denotes vacancy posting costs, p$${vn}$$(x$$w$$, a) denotes the contact rate from the firm’s perspective with nonemployed workers of type x$$w$$ and age a, and p$${vo}$$(x$$w$$, xm, a) denotes the contact rate from the firm’s perspective with employed workers of type x$$w$$ in a match of quality xm and age a. Given the worker’s current state, the firm forms expectations about the expected profits, taking into account the worker’s acceptance probability for the offer. Contact rates in each submarket are determined using a Cobb–Douglas matching function m = ϰ$$v$$1−ϱ$$u$$ϱ in vacancies $$v$$ and searching workers $$u$$ with matching elasticity ϱ and matching efficiency ϰ. We allow for different matching efficiencies between on- and off-the-job search but not across submarkets of skill types or age.17 The contact rates for nonemployed and on-the-job search are \begin{eqnarray} p_{vn}(x_{w},\, a) = \varkappa _{n} \left(\frac{n(x_{w},\, a)}{v_{n}(x_{w},\, a)}\right)^{\varrho } = \varkappa _{n} \theta _{n}(x_{w},\, a)^{-\varrho }, \end{eqnarray} (11) \begin{eqnarray} p_{vo}(x_{w},x_{m},\, a) = \varkappa _{o} \left(\frac{l(x_{w},x_{m},\, a)}{v_{o}(x_{w},x_{m},\, a)}\right)^{\varrho } = \varkappa _{o} \theta _{o}(x_{w},x_{m},\, a)^{-\varrho }, \end{eqnarray} (12) where l(x$$w$$, xm, a) denotes the number of employed workers at the search stage, $$v$$o(x$$w$$, xm, a) the number of posted vacancies for a particular worker type, and θo(x, a) labor market tightness. The value n(x$$w$$, a) denotes the number of nonemployed workers at the search stage, $$v$$n(x$$w$$, a) the number of posted vacancies for a particular worker type, and θn(x$$w$$, a) labor market tightness. Contact rates from the worker’s perspective are peo(x$$w$$, xm, a) = ϰoθo(x$$w$$, xm, a)1−ϱ and p$${ne}$$(x$$w$$, a) = ϰnθn(x$$w$$, a)1−ϱ, respectively. 3.5. Parameter Identification Based on Worker Transition Rates This section discusses identification of model parameters. The existing literature typically relies on wage data to identify parameters of the skill process (see Bagger et al. 2014 for a recent example). We propose an alternative approach that identifies the parameters of the skill process using the documented worker transition rates from Section 2. Our identification approach transforms the ideas of Topel (1991), who also uses wage data, to data on worker transition rates. In our model, wages and worker transition rates are directly linked as bargaining outcomes. In this way, they provide similar information about the evolution of skills over time and across jobs. Here we discuss the identification of the skill process and sketch a general idea about how these data also identify the remaining model parameters. We relegate a detailed discussion on the identification of the remaining parameters and some further discussion on the identification of the skill process parameters to Online Appendix E. In what follows, we use wage dynamics from the estimated model to evaluate the model along dimensions not used in the estimation. Two channels, skill accumulation (experience) and selection (tenure), can explain the declining transition rates by age or tenure. Selection effects are present if idiosyncratic shocks hit matches with heterogeneous quality even if workers are homogeneous. Good matches face a lower probability of separating so that the share of good matches increases with tenure and observed separation rates decline.18 Hence, selection is an effect associated with tenure accumulation. Skill accumulation instead improves the worker’s productivity by age even if match quality is homogeneous. As workers age, they accumulate experience, and become more productive relative to their outside option, and their match-surplus increases so that they separate less. Hence, skill accumulation is an effect associated with experience accumulation. Both channels potentially explain the declining pattern of separations by age. Adopting ideas in Topel (1991), we use differences between age profiles and newly hired age profiles to disentangle the relative importance of the two effects. Figure 4 shows separation rates by age and separation rates for newly hired workers for hypothetical economies. Figure 4(a) depicts the case when the decline in the separation rate by age is explained by selection only and skill accumulation is absent. Although age and tenure increase jointly, it is only selection that leads to a declining age profile; the newly hired age profile is flat. In the absence of skill accumulation, a newly hired young worker is identical to a newly hired older worker. Hence, separation rates by age for newly hired workers are independent of age. Figure 4. View largeDownload slide Identification of the skill process. Panel (a) shows stylized age and newly hired age profiles for separation rates in a model with only selection. Panel (b) shows stylized age and newly hired age profiles for separation rates in a model with only skill accumulation. Panel (c) shows stylized age and newly hired age profiles for separation rates in a model with selection and skill accumulation. Panel (d) shows a stylized newly hired age profile for job-to-job transition rates with full and partial transferability of skills. All figures have age on the horizontal axis and transition rates on the vertical axis. Figure 4. View largeDownload slide Identification of the skill process. Panel (a) shows stylized age and newly hired age profiles for separation rates in a model with only selection. Panel (b) shows stylized age and newly hired age profiles for separation rates in a model with only skill accumulation. Panel (c) shows stylized age and newly hired age profiles for separation rates in a model with selection and skill accumulation. Panel (d) shows a stylized newly hired age profile for job-to-job transition rates with full and partial transferability of skills. All figures have age on the horizontal axis and transition rates on the vertical axis. Figure 4(b) depicts the case when the decline in separation rates by age is explained by skill accumulation only. Workers accumulate skills with experience, so older workers are on average more skilled and separate less than younger workers. Absent selection effects, skill accumulation by age translates one-to-one into differences in the separation rate by age for newly hired workers. The age profile and the newly hired age profile decrease by the same amount. As discussed in our empirical analysis, the data represent an intermediate case as in Figure 4(c), so slope differences in the newly hired age profile and the average age profile identify the relative strength of the two effects. A similar idea applies to the identification of skill transferability across jobs. To disentangle how transferable skills are, we use the newly hired age profile of job-to-job transitions. Workers who accumulate skills face a trade-off between searching for a better match and losing accumulated skills when switching jobs. Consequently, older workers with more accumulated skills are on average more reluctant to accept outside offers than younger workers. As a consequence, older newly hired workers switch jobs less often than younger newly hired workers. If skills were perfectly transferable across jobs, the newly hired age profile would be flat. Hence, the decline in the newly hired age profile for job-to-job transitions identifies how transferable accumulated skills are across jobs (Figure 4(d)). Translating the discussion to model parameters, we explained how the slopes of the newly hired age profiles identify the skill-process parameters p$$u$$ and pd. In Online Appendix E, we provide a detailed discussion of identification for the remaining model parameters. For this discussion, it is instrumental to recognize that differences between the age profile and the newly hired age profile also quantify differences in transition rates between low-tenure (newly hired) and high-tenure (average) workers. We now exploit this fact when we summarize the discussion on parameter identification. The general idea of which dimensions of heterogeneity we exploit for identification already appears in Figure 4. The age profiles shown in the figure can be described by three characteristics: their average level, their slope capturing the difference between young and old workers, and their shape describing how quickly the difference between young and old workers materializes. Concretely, we sketch in Section E.1 of the Online Appendix a stylized model to show that the level of the separation rate, together with separation rate differences between low- and high-tenure workers, and the level of mean tenure identify the outside option b, the dispersion of match-specific skills σm, and the dispersion of idiosyncratic productivity costs ψs. The discussion surrounding Figure 4 already suggests that separation rate differences between low- and high-tenure workers identify σm. The outside option b determines the average surplus and, thereby, the level of the separation rate. The dispersion of shocks ψs determines differences in separation rates so that it is identified by mean tenure. The speed of skill accumulation δ governs how quickly workers accumulate worker-specific skills and, therefore, how quickly age differences realize. The shape of the separation rate profile identifies this parameter. Exogenous separations limit tenure accumulation of workers by age, so that the slope of the mean tenure profile identifies πf. We exploit the level, slope, and shape of the job-to-job transition rate to identify parameters ϰo, κo, and ψo. The matching efficiency ϰo determines the number of job offers for employed workers and is identified by the level of job-to-job transitions. The slope of the job-to-job transition rates depends on the relative importance of nonpecuniary job aspects κo. During their working life, workers climb the job ladder so that job-to-job transition rates decline. If nonpecuniary aspects become more important, job-to-job transition rates decline by less; the slope gets smaller. The dispersion of nonpecuniary shocks governed by ψo determines the job acceptance elasticity and, thereby, the shape of the job-to-job transition rate profile. The bargaining power μ is identified by job-to-job transition rate differences between low- and high-tenure workers. A higher bargaining power provides stronger incentives for newly hired workers to climb the job ladder because they will receive a larger fraction of the gains from job switching. The higher the bargaining power, the more newly hired workers want to climb the job ladder. Finally, ϰn and κ are identified by the level and slope of the job finding rate profile. As for job-to-job transitions, ϰn determines the level of the job finding rate. Vacancy posting costs κ, in comparison to the changing surplus due to skill accumulation, determine the slope of the job finding rate. Compared to existing approaches that mainly focus on heterogeneity in the wage dynamics, such as Bagger et al. (2014), our approach exploits the corresponding heterogeneity in worker mobility over the age-tenure domain for identification. We refer to Online Appendix E for further details and turn next to a discussion of our estimation procedure and the results. 4. Results This section starts by discussing our estimation procedure. We then show how the model performs along the mobility dimensions used in the estimation and discuss wage implications as overidentifying restrictions. In Section 5, we then turn to the investigation of earnings losses. 4.1. Estimation Procedure Before we bring the model to the data, we have to make some assumptions on parameters and functional forms. To align model and data, we set the model period to one month. A worker enters the labor market at age 20 as nonemployed, leaves the labor market at age 65, stays retired for a further 15 years, and dies at age 80.19 The production function is age-independent and log-linear in skills f(x) = exp (xm + x$$w$$), as in Bagger et al. (2014).20 We approximate both skill distributions using five skill states. Mean skill levels are normalized to 1. The match-specific component (xm) approximates a normal distribution with standard deviation σm, and the worker-specific component is constructed such that each increase in skill level leads to a 30% increase in the level of skills (σ$$w$$ = 0.3). In the model, workers and firms care about the expected value of the skill increase (σ$$w$$p$$u$$), so σ$$w$$ constitutes a normalization.21 In line with the literature, we set a discount factor β to match an annual interest rate of 4% and a matching elasticity of ϱ = 0.5 following Petrongolo and Pissarides (2001). We estimate parameters using a method of moments. We avoid simulation noise and iterate on the cross-sectional distribution from the model. We use age profiles, newly hired age profiles, and mean tenure in the estimation where we weight profiles to focus mostly on ages 20–50. We provide the details on the implementation in Appendix B. Table 2 collects the estimated parameters together with the estimated standard errors. Standard errors are computed using the bootstrapped data profiles from Section 2. Table 2. Estimated parameters. Skills Shocks Matching and bargaining p$$u$$ 0.0258 ψs 2.8621 μ 0.3097 (0.0007) (0.0878) (0.0299) pd 0.0536 κo –0.6933 b 0.3949 (0.0064) (0.0942) (0.0170) δ 0.0030 ψo 1.8503 κ 2.3689 (0.0001) (0.1381) (0.0900) σm 0.0933 πf 0.0024 ϰo 2.3913 (0.0076) (0.0001) (0.1149) ϰn 0.4591 (0.0075) Skills Shocks Matching and bargaining p$$u$$ 0.0258 ψs 2.8621 μ 0.3097 (0.0007) (0.0878) (0.0299) pd 0.0536 κo –0.6933 b 0.3949 (0.0064) (0.0942) (0.0170) δ 0.0030 ψo 1.8503 κ 2.3689 (0.0001) (0.1381) (0.0900) σm 0.0933 πf 0.0024 ϰo 2.3913 (0.0076) (0.0001) (0.1149) ϰn 0.4591 (0.0075) Notes: Estimated parameters and standard errors. Standard errors shown in parentheses. Column Skills shows parameters determining the skill process. The parameter p$$u$$ is the probability of worker-specific skill accumulation at age 20, pd is the probability that worker-specific skills do not transfer at job change, δ governs the declining probability of worker-specific skill accumulation by age, and σm denotes the standard deviation of match-specific skills. Column Shocks shows idiosyncratic shock parameters governing worker mobility decisions. The parameter ψs determines the dispersion of productivity shocks, κo determines the common utility component of all job offers, ψo determines the dispersion of the idiosyncratic utility component of job offers, and πf is the exogenous separation probability. Column Matching and bargaining shows parameters related to the search process. The parameter μ is the bargaining power of the worker, b is the flow utility during nonemployment, κ determines vacancy posting costs, and ϰo and ϰn are matching efficiencies for on- and off-the-job search. Standard errors are bootstrapped using 500 repetitions. View Large Table 2. Estimated parameters. Skills Shocks Matching and bargaining p$$u$$ 0.0258 ψs 2.8621 μ 0.3097 (0.0007) (0.0878) (0.0299) pd 0.0536 κo –0.6933 b 0.3949 (0.0064) (0.0942) (0.0170) δ 0.0030 ψo 1.8503 κ 2.3689 (0.0001) (0.1381) (0.0900) σm 0.0933 πf 0.0024 ϰo 2.3913 (0.0076) (0.0001) (0.1149) ϰn 0.4591 (0.0075) Skills Shocks Matching and bargaining p$$u$$ 0.0258 ψs 2.8621 μ 0.3097 (0.0007) (0.0878) (0.0299) pd 0.0536 κo –0.6933 b 0.3949 (0.0064) (0.0942) (0.0170) δ 0.0030 ψo 1.8503 κ 2.3689 (0.0001) (0.1381) (0.0900) σm 0.0933 πf 0.0024 ϰo 2.3913 (0.0076) (0.0001) (0.1149) ϰn 0.4591 (0.0075) Notes: Estimated parameters and standard errors. Standard errors shown in parentheses. Column Skills shows parameters determining the skill process. The parameter p$$u$$ is the probability of worker-specific skill accumulation at age 20, pd is the probability that worker-specific skills do not transfer at job change, δ governs the declining probability of worker-specific skill accumulation by age, and σm denotes the standard deviation of match-specific skills. Column Shocks shows idiosyncratic shock parameters governing worker mobility decisions. The parameter ψs determines the dispersion of productivity shocks, κo determines the common utility component of all job offers, ψo determines the dispersion of the idiosyncratic utility component of job offers, and πf is the exogenous separation probability. Column Matching and bargaining shows parameters related to the search process. The parameter μ is the bargaining power of the worker, b is the flow utility during nonemployment, κ determines vacancy posting costs, and ϰo and ϰn are matching efficiencies for on- and off-the-job search. Standard errors are bootstrapped using 500 repetitions. View Large All estimated parameters from Table 2 are at economically reasonable magnitudes. The parameter p$$u$$ refers to age 20 and follows the life-cycle dynamics governed by δ described previously. The estimate implies an expected skill increase at age 20 of 0.8 log point per month (σ$$w$$p$$u$$) or extrapolated to an annual frequency of 9 log points in the first year in the labor market. This skill increase and the decline in its speed governed by δ match the increase and concavity of the empirical log wage profile as shown in Figure 6(a). The estimate of pd implies an expected skill loss from a job change of 1.6 log points (σ$$w$$pd). This degree of transferability of skills is consistent with the share of negative wage changes and the average wage gain at job-to-job transitions over the life cycle, as we will demonstrate in Section 4.3.1. Our estimate of σm implies a wage difference of roughly 17% (32%) between the average (minimum) match and the best match for the median worker at age 40. This amount of wage dispersion can be compared with empirical estimates of the mean–min ratio of wages, as popularized by Hornstein et al. (2011). As we will discuss in detail in what follows, our model is consistent with empirical estimates of the mean–min ratio in the cross section and over the life cycle. A directly comparable estimate of match-specific wage dispersion is provided in Hagedorn, Manovskii, and Wang (2017). Their estimate has to be compared to the employment-weighted variance of xm from our model. Our model delivers a variance of 0.014, close to their reported variance of 0.016.22 The size of the parameter estimate for ψs is easiest to interpret in relation to transitory wage risk. Imposing some mild additional assumptions on the transmission of these shocks to wages, we quantify the implied transitory wage risk from these shocks in Online Appendix G.2.4. We find an implied standard deviation of transitory wage shocks of 0.35 that is within the ballpark of the average estimate of 0.29 from Heathcote et al. (2010). The option value Ψo from the acceptance choice of outside offers reflects the nonpecuniary benefits from a new job. The estimates for κo and ψo imply a modest importance of this nonpecuniary utility component. At age 40, the average utility flow from the nonpecuniary job component to an employed worker corresponds to less than 6% of the average wage. Our estimate for b corresponds to 28% of the average wage of a 40-year-old worker. Our estimate is thereby below the unemployment benefit replacement rate of 40% used in Shimer (2005) but above the effective value estimated in Chodorow-Reich and Karabarbounis (2016). Nonemployed workers also receive utility from the acceptance choice of job offers. Their option value Ψν is substantially larger than that of employed workers due to higher contact rates. Including the option value from job search, the flow utility in unemployment relative to the average wage is roughly 70% and is above 95% when we compare it to the wages of newly hired workers, thereby moving close to the estimate of Hagedorn and Manovskii (2008). Vacancy posting costs κ correspond to 56% of the quarterly wage of a 40-year-old worker. They therefore capture a broader concept of hiring costs including training costs, as discussed in Silva and Toledo (2009). We estimate a bargaining power of 0.31. The estimate is similar to that in Bagger et al. (2014). It is not directly comparable because their model relies on a different bargaining protocol and data. The estimates for the matching efficiency parameters ϰo and ϰ$$u$$ imply a higher matching efficiency on the job. Despite the higher matching efficiency on the job, employed workers receive fewer job offers than nonemployed workers in the model because employers take their lower acceptance rate into account (see equation (10)). A lower acceptance rate on the job is consistent with the results in Faberman et al. (2016), who report that full-time employed workers have an acceptance rate that is less than half of the acceptance rate of nonemployed workers. In the model, a 40-year-old employed worker receives 0.3 job offer per month, whereas a nonemployed worker receives 0.4 offer. This difference is consistent with estimates in Faberman et al. (2016, Table 3), who report on average 0.2 job offer for employed workers over a four-week period compared to 0.4 job offer for nonemployed workers in their data. Furthermore, the mobility pattern on and off the job are in line with the empirical counterparts, as we show in what follows. Finally, our estimate for the exogenous separation rate πf implies an 8% probability of displacement within three years. This estimate is in line with evidence provided in Farber (2007), who reports a three-year involuntary separation rate of around 10% based on CPS data. Given that the estimated parameters are at economically reasonable levels, we next show how the estimated model fits the mobility facts used in the estimation. We then present the results on wage dynamics to evaluate the model performance along dimensions that were not part of the estimation. 4.2. Labor Market Mobility Figure 5 presents, in the upper two rows, the model fit for worker transition rates and mean and median tenure that have been part of the estimation. Figures 5(a)–(c) show age profiles for separation, job-to-job transition, and job-finding rates. Figures 5(d) and (e) show the profiles for separation and job-to-job transition rates by age for newly hired workers. Figure 5(f) shows the age profile of mean and median tenure. All transition rates and mean and median tenure are matched closely. Figure 5. View largeDownload slide Model prediction and data. Panels (a)–(c) show age profiles for separation rate, job-to-job transition rate, and job-finding rate from model and data. Panels (d) and (e) show newly hired age profiles for separation rate and job-to-job transition rate from model and data. Panel (f) shows mean and median tenure by age from model and data. The black dots show data, and the gray solid line shows the model. The horizontal axis is age in years, and the vertical axis shows transition rates in percentage points or tenure in years. Newly hired age profiles start at age 21. Panels (g) and (h) show tenure profiles for separation and job-to-job transition rate from model and data. The black dots show data, and the gray solid line the model. The horizontal axis is tenure in years, and the vertical axis shows transition rates in percentage points. Panel (i) shows the age profile of the unemployment rate from model and data. The black dots show the data, and the gray solid line the model. The horizontal axis is age in years, and the vertical axis shows unemployment rates in percentage points. Mean level differences between model and data have been removed. See text for details. Figure 5. View largeDownload slide Model prediction and data. Panels (a)–(c) show age profiles for separation rate, job-to-job transition rate, and job-finding rate from model and data. Panels (d) and (e) show newly hired age profiles for separation rate and job-to-job transition rate from model and data. Panel (f) shows mean and median tenure by age from model and data. The black dots show data, and the gray solid line shows the model. The horizontal axis is age in years, and the vertical axis shows transition rates in percentage points or tenure in years. Newly hired age profiles start at age 21. Panels (g) and (h) show tenure profiles for separation and job-to-job transition rate from model and data. The black dots show data, and the gray solid line the model. The horizontal axis is tenure in years, and the vertical axis shows transition rates in percentage points. Panel (i) shows the age profile of the unemployment rate from model and data. The black dots show the data, and the gray solid line the model. The horizontal axis is age in years, and the vertical axis shows unemployment rates in percentage points. Mean level differences between model and data have been removed. See text for details. The bottom row of Figure 5 shows transition rates by tenure and unemployment rates by age, both of which have not been directly targeted in the estimation. Figures 5(g) and (h) demonstrate the good fit of the model to the transition rates by tenure.23 The fit of mobility by tenure shows that our model also matches the frequency of steps on the job ladder. Importantly, our model matches job stability at the top of the job ladder with very low separation rates for workers with more than 10 years of tenure. In models with high separation rates also at the top of the job ladder, workers fall down the job ladder repeatedly, and differences that result from the job ladder are transitory. Average tenure is low. Matching low separation rates at the top leads to high tenure and to differences in match types that persist over time. Matching the frequency of steps on the job ladder is important for our later analysis because the job ladder governs the recovery after displacement. We will demonstrate in what follows that our model also matches wage gains following job-to-job transitions. Figure 5(i) shows the unemployment rate by age from the model and CPS data. Nonemployment in the model comprises all unemployed workers and some workers who are not classified as unemployed in the CPS but who are attached to the labor market. Recent evidence in Kudlyak and Lange (2014) supports this modeling choice. We discuss this assumption in detail in Online Appendix D.1.2, and we explain in Online Appendix G.1 how we construct an adjustment factor to remove the level difference between model and data. Given that all workers start nonemployed at age 20 in the model, Figure 5(i) shows the age profile of the unemployment rate starting at age 21. The model matches the empirical unemployment rate by age almost exactly. Finally, note that we focus on the average job-finding rate by age in Figure 5(c) because most unemployment spells in the data are short. BLS data show that the share of job losers who are unemployed half a year or more is 18% over our sample period. In our model, the same share at age 40 is 17% with an age variation from 14% at age 25 to 19% at age 55. Hence, our model captures the transitory nature of unemployment spells in the U.S. labor market well. Looking at longer unemployment durations, the model does not generate the empirically observed duration dependence with a decline of only 22% over 24 months. In the data, the decline is slightly more than twice as large. However, very few workers actually face these low job-finding rates because the vast majority of workers finds jobs more quickly. In a model extension described in Online Appendix F.1, we match the empirically observed duration dependence. We allow for duration-dependent skill losses during nonemployment and deteriorating search efficiency with nonemployment duration, capturing two prominent explanations for duration dependence (see Kroft, Lange, and Notowidigdo 2013). The extended model is re-estimated and matches the empirically observed duration dependence. We show that accounting for duration dependence of job-finding rates affects our results only marginally, so that we abstract from it for our baseline model. In sum, the model is consistent with two characteristic features of the U.S. labor market: large average transition rates and a large share of very stable jobs. The coexistence of these facts has so far received little attention in the literature on structural labor market models. Yet, these features are crucial in generating large and persistent earnings losses, as we show in what follows. Next, we demonstrate that the model is also consistent with a range of facts on wage dynamics. 4.3. Wage Dynamics The previous section has shown that the model is consistent with observed worker mobility and job stability pattern. This section demonstrates that the model is also consistent with a range of facts on wage dynamics both on the job and between jobs. For wage dynamics between jobs, we consider average wage gains from job-to-job transitions, the share of negative wage changes following job-to-job transitions, and the share of early career wage growth attributable to job switching. We derive the first two statistics from the SIPP microdata and use the estimate from Topel and Ward (1992) for the decomposition of early career wage growth. For wage dynamics on the job, we consider estimates of the returns to tenure using two alternative identification approaches (Altonji and Shakotko 1987; Topel 1991) and the variance of permanent shocks using a permanent-transitory shock decomposition (Storesletten, Telmer, and Yaron 2004; Guvenen 2009; Heathcote et al. 2010). Tightly connected to wage dynamics is cross-sectional wage inequality. Therefore, we also discuss the model’s ability to match different measures of cross-sectional wage dispersion. Finally, we revisit the correlation between wages and job stability. Although the model matches this relationship qualitatively by construction, here we explore the relationship quantitatively. We relegate the details of the estimation procedure using model-simulated data to Online Appendix G.2. First, we compare in Figure 6(a) the mean (log) wage by age from the model and data. Wage data come from the annual march CPS files. We provide further details on the construction in Appendix A.1. Wages from the model are initially not as steep as in the data, but wage growth until age 40 is matched. Generally, the model matches the slope closely but misses some of the concavity of the empirical profile. Figure 6. View largeDownload slide Wage profiles. Age profiles of mean log wages and average wage gains following a job-to-job transition from model and data. The gray solid line shows the model, and the black dots show the data. The horizontal axis is age in years, and the vertical axis shows the log-wage change relative to age 20 (left panel) or wage gains relative to the previous job (right panel) in percentage points. Mean log wage profiles come from CPS data, and wage gains are derived using SIPP data, as in Tjaden and Wellschmied (2014). Figure 6. View largeDownload slide Wage profiles. Age profiles of mean log wages and average wage gains following a job-to-job transition from model and data. The gray solid line shows the model, and the black dots show the data. The horizontal axis is age in years, and the vertical axis shows the log-wage change relative to age 20 (left panel) or wage gains relative to the previous job (right panel) in percentage points. Mean log wage profiles come from CPS data, and wage gains are derived using SIPP data, as in Tjaden and Wellschmied (2014). 4.3.1. Wage Gains From Job-to-Job Transitions Figure 6(b) compares the mean wage gain from a job-to-job transition by age from the model to the data. We derive the empirical profile based on microdata, as in Tjaden and Wellschmied (2014). Online Appendix G.2.1 provides details for the construction in the model. The declining age profile of wage gains suggests that the gains from search decline. The model prediction is slightly higher than the empirical estimates but matches a similar decline by age. Although Figure 6(b) shows that the model generates sizable positive average wage gains following job-to-job transitions, it hides that the model also matches a large fraction (24%) of job-to-job transitions that lead to wage cuts. The fact that a substantial share of job-to-job transitions is associated with wage cuts in the data (32%) is well known and is, for example, discussed in Tjaden and Wellschmied (2014). Many search models struggle to explain this fact because workers only change jobs if the outside offer is better than the current job. In our model, workers’ acceptance decisions depend not only on wages but also on a nonpecuniary utility component. Wage cuts after job-to-job transitions follow naturally in this case.24 4.3.2. Early Career Wage Growth Topel and Ward (1992) document that about one-third of total wage growth in the first 10 years of working life is explained by job-changing activity. In their sample, a typical worker switches jobs frequently and holds on average seven jobs during the first 10 years in the labor market. Similarly, Bagger et al. (2014) find in a structural labor market model that during an initial job-shopping phase, wage growth is strongly driven by job-changing activity. Early career wage growth is an alternative, independent measure for the relative importance of worker- and match-specific skill accumulation. Our model generates on average eight jobs in the first 10 years of working life and a contribution of job-changing activity to wage growth of 30%. Online Appendix G.2.2 provides details on the wage growth decomposition in the model. 4.3.3. Returns to Tenure The returns to tenure capture the increase in wages with job duration. So far, no consensus has been reached in the literature on the importance of the returns to tenure relative to the return to general experience. Estimates differ dramatically across studies depending on identification strategies (see, e.g., Altonji and Shakotko 1987; Topel 1991; and the survey by Altonji and Williams 2005). We implement the estimators by Topel (1991) and Altonji and Shakotko (1987) on simulated data from our model. Online Appendix G.2.3 provides details. The model reproduces both estimates very closely. The ordinary least squares (OLS) estimate for the returns to tenure is a common benchmark. Using OLS, Altonji and Shakotko report 26.2% returns from 10 years of tenure for their sample. In the model, we get 24.2%, which is lower than the empirical estimates but still consistent with substantial returns to tenure. Following the instrumental variable approach proposed in Altonji and Shakotko, the model generates 0.0% for returns from 10 years of tenure; this substantial drop is in line with Altonji and Shakotko’s estimate of 2.7% (about one-tenth of their OLS estimate).25 Topel proposes a two-step estimation approach and finds 24.6% for returns from 10 years of tenure, again close to the level of the OLS estimate. Using his approach, the model predicts 29.6% and again matches the empirical pattern of large returns from tenure at the order of the OLS estimate.26 4.3.4. Permanent Income Shocks and Wage Inequality We discuss previously that in the data and the model, most workers stay on their jobs for several years. We therefore consider the variance of permanent income shocks as an additional measure to describe wage dynamics on the job. As before, we use the empirical estimation approach to capture the statistical properties of the model-generated wage dynamics but do not necessarily take the underlying statistical model as a good description of the model-generated wage process. We compare our results to findings from Heathcote et al. (2010). Heathcote et al. estimate a standard deviation of 0.084 for the permanent shock. Our model closely matches this number with an estimate of 0.072. We provide the details on the estimation using model data in Online Appendix G.2.4. There we also discuss how to construct estimates for transitory shocks from the model. When we consider, as in Heathcote et al. (2010), the age range from 25 to 60, we estimate a standard deviation for transitory shocks of 0.35, which is close to the average estimate of 0.29 in Heathcote et al. (2010). Cross-sectional wage inequality is the result of the described wage dynamics. Hornstein et al. (2011) point out that existing search models struggle to generate substantial wage dispersion. Their preferred measure for wage dispersion is the mean-min ratio of wages (Mm ratio). For a canonical search model calibrated to the U.S. labor market, they find a Mm ratio of 1.046. Tjaden and Wellschmied (2014) use SIPP data to provide empirical estimates of Mm ratios. They report Mm ratios by age that vary between 1.95 and 2.25 over the age range from 25 to 49. At age 36, they report a Mm ratio of 2.12. Our model closely matches this level of wage dispersion and its age variation. The average Mm ratio is 2.53, and it varies from 1.69 at age 25 to 2.93 at age 49 and is 2.50 at age 36. Online Appendix G.2.4 provides further details. Closely related to Hornstein et al. (2011) is the empirical work by Hagedorn et al. (2017). They estimate the contribution of match-specific wage differences to cross-sectional wage inequality. They find that the match-specific variance accounts for 5.7% of the cross-sectional (log) wage variance. We observe match dispersion directly and find that our model aligns well with this estimate. Match dispersion in the model corresponds to 6.4% of the cross-sectional (log) wage variance.27 The variance in log wages is another popular measure of wage dispersion. In the data, the variance in log wages increases over the life cycle. Our model matches this increase between ages 20 and 40. The increase is 8 log points in the model in comparison to 10 log points in the CPS data for the same age range. A key challenge in matching the variance of log wages is its sensitivity to the tails of the wage distribution. The parsimony of the worker skill process in our baseline model cannot capture the very right tail of the wage distribution, which limits the increase in the variance of log wages after age 40. In particular, the bounded support for the worker-specific skill states leads to a flattening out of the variance age profile. In Online Appendix F.2, we provide an extended model where we augment the worker-specific skill process by an additional skill state in the right tail of the skill distribution. We demonstrate that this extension allows us to fit the life-cycle profile of the variance in log wages over the entire working life very closely without sacrificing the fit along other dimensions. We also demonstrate that other results are robust to this model refinement. The caveat is that we have to use the age profile of the variance in log wages to estimate the extended model, so we focus on the parsimonious version in the main text. We relegate further discussion to Online Appendix G.2.4. 4.3.5. Job Stability and Wages Section 2.3 discusses the empirical correlation between wages and job stability. As discussed previously, such a link between job stability and wages is a direct implication of the joint bargaining over wages and separation decisions in the model. To show that our model quantitatively accounts for the observed correlation, we redo our empirical analysis on model-generated data using 4-month separation rates. Online Appendix G.2.5 provides further details. Our regression coefficient of separation rates on log wages is −0.0368 in the model compared to −0.0392 in the data when looking at all separations, and it is −0.0667 in the model compared to −0.0548 in the data when looking at newly hired workers (see Table 1). We conclude that the wage-stability trade-off from our model is quantitatively consistent with the data. 5. Earnings Losses This section examines implications of the model for estimated earnings losses following displacement. We first provide a model analog of the empirical estimation methodology developed in Jacobson et al. (1993) and show that the model reproduces empirical earnings losses in both size and persistence. We use the structural model to decompose earnings losses into a wage loss effect, an extensive margin effect, and a selection effect. We explore the relative importance of match- and worker-specific skill losses for wage losses and subsequent job stability. 5.1. Group Construction Jacobson et al. (1993, p. 691) define displaced workers’ earnings losses as “(...) the difference between their actual and expected earnings had the events that led to their job losses not occurred”, and propose an estimation strategy borrowed from the program evaluation literature. The approach is based on the construction of two groups, which we refer to as layoff group and control group. For details on construction of estimates, we follow Couch and Placzek (2010), one of the recent applications of the original estimation strategy. Other recent contributions are von Wachter et al. (2009) and Davis and von Wachter (2011), who apply the same estimation methodology but differ in the construction of the control and the layoff group. We will also compare our model prediction with their results. The layoff group consists of all workers who separate in a mass-layoff event. The idea of using mass layoffs is that workers are not selected based on their individual characteristics when mass layoffs occur. We associate this event with an exogenous separation in the model. Exogenous separations in the model occur independent of the individual characteristics and are therefore the model analog to a mass layoff event in the data. This mapping is also in line with the discussion in Stevens (1997) and her mapping of separation events in the PSID to displacement.28 The control group consists of continuously employed workers over the sample period. The empirical analysis covers workers of all ages and controls for age in the regression. In the model, we consider a worker of age 40; this corresponds to the mean age of all workers from the sample used by Couch and Placzek (2010). Online Appendix H.1 reports estimation results for various age groups.29 The layoff group then consists of all workers who separate as the consequence of an exogenous separation. We provide a discussion of selection effects if separations are endogenous in Online Appendix H.2. As in Jacobson et al. (1993) and Couch and Placzek (2010), we initially restrict the sample to workers with at least six years of tenure. For the control group, both studies require a stable job for the next six years because they require continuous employment over their 12-year sample period. We follow the empirical analysis and construct the appropriate model equivalents. In line with all empirical studies, we consider nonemployment income to be zero. This creates a difference between wage and earnings losses that is quantitatively nonnegligible.30 We also control for worker-specific fixed effects. We reproduce empirical estimates from the model using measures over worker states and transition laws instead of relying on simulation. 5.2. Earnings Losses Figure 7 shows earnings losses from the model in comparison to the estimates from Couch and Placzek (2010). The model generates large and persistent earnings losses (gray line with squares). In the first year following the layoff event, earnings losses amount to 37%, and six years after the layoff event, they still amount to 11% of predisplacement earnings. Findings correspond closely with empirical estimates by Couch and Placzek (2010) (black line with circles), which show 25% earnings losses initially and 13% after six years.31 Standard deviations for estimates from Couch and Placzek are 0.9%–1.8% of predisplacement earnings so that model predictions are well within the estimated range. Figure 7. View largeDownload slide Earnings losses following displacement. Earnings losses after displacement in the model and empirical estimates. The gray line with squares shows earnings losses predicted by the model. The black line with circles shows estimates by Couch and Placzek (2010). The horizontal axis shows years relative to displacement and the vertical axis shows losses in percentage points relative to the control group. Figure 7. View largeDownload slide Earnings losses following displacement. Earnings losses after displacement in the model and empirical estimates. The gray line with squares shows earnings losses predicted by the model. The black line with circles shows estimates by Couch and Placzek (2010). The horizontal axis shows years relative to displacement and the vertical axis shows losses in percentage points relative to the control group. Figure 8. View largeDownload slide Empirical decomposition of earnings losses. Earnings losses and decomposition of earnings losses from model and PSID data. Top left panel: Earnings losses from model and estimates based on PSID data. Top right panel: Wage losses from model and estimates based on PSID data. Bottom left panel: Extensive margin from the model and hours losses estimated in PSID data. Bottom right panel: Share of wage losses in earnings losses from the model and based on empirical estimates. Horizontal axes show time relative to the displacement event in years. Vertical axes in the first three panels show losses in percentage points relative to the control group for earnings, wages, and extensive margin. Vertical axis in the bottom-right panel shows wage losses as share of earnings losses in percentage points. The gray solid lines with squares shows model results. The black dashed line with circles show empirical estimates. See text for further details. Figure 8. View largeDownload slide Empirical decomposition of earnings losses. Earnings losses and decomposition of earnings losses from model and PSID data. Top left panel: Earnings losses from model and estimates based on PSID data. Top right panel: Wage losses from model and estimates based on PSID data. Bottom left panel: Extensive margin from the model and hours losses estimated in PSID data. Bottom right panel: Share of wage losses in earnings losses from the model and based on empirical estimates. Horizontal axes show time relative to the displacement event in years. Vertical axes in the first three panels show losses in percentage points relative to the control group for earnings, wages, and extensive margin. Vertical axis in the bottom-right panel shows wage losses as share of earnings losses in percentage points. The gray solid lines with squares shows model results. The black dashed line with circles show empirical estimates. See text for further details. The initial drop in earnings is larger in the model than the empirical estimates. This difference likely results from the fact that the point in time of the layoff event and point in time when the employee is notified in the data can only be determined to be in a certain quarter. The initial earnings losses in the data therefore comprise likely pre- and post-displacement earnings observations, which leads to lower estimated earnings losses than in a case where the exact point in time of the separation can be observed. Pries (2004) makes a similar argument. In Online Appendix H.3, we show that small differences in timing of the displacement notification can have a large impact on the initial drop in earnings. We find that one month of advance notification closes the initial difference in estimated earnings losses between model and data by 50%, and two months of advance notification close the gap between the earnings losses from the model and the data completely. In both cases, however, earnings losses after six years remain virtually unaffected. Davis and von Wachter (2011) use the same estimation approach but propose a different construction of the control and layoff group. They require three years of job tenure for both the control and the layoff group prior to their displacement and two years of subsequent job stability following the year of the displacement event for the control group.32 They consider men aged 50 years and younger. We adjust the average age for displaced workers in the model accordingly to 35 years when comparing the model prediction to their results. Davis and von Wachter (2011) report earnings losses as a present discounted value relative to predisplacement annual earnings, and, alternatively, as a share of the present discounted value of counterfactual earnings. They use an annual discount factor of 5% and extrapolate earnings losses beyond 10 years after the displacement event. We follow them in the implementation. Table 3 reports results from our model in comparison to estimates reported in Davis and von Wachter (2011) for different control and layoff groups and for different age groups. Table 3. Comparison to earnings loss estimates from Davis and von Wachter (2011). Davis and von Wachter Model Sample Predisplacement Counterfactual (%) Predisplacement Counterfactual (%) All workers 1.7 11.9 1.5 10.0 Age 21–30 1.6 9.8 1.7 9.8 Age 31–40 1.2 7.7 1.5 10.0 Age 41–50 1.9 15.9 1.2 8.8 Davis and von Wachter Model Sample Predisplacement Counterfactual (%) Predisplacement Counterfactual (%) All workers 1.7 11.9 1.5 10.0 Age 21–30 1.6 9.8 1.7 9.8 Age 31–40 1.2 7.7 1.5 10.0 Age 41–50 1.9 15.9 1.2 8.8 Notes: The first column shows the considered sample. All workers in the case of Davis and von Wachter (2011) means men from age 21 to 50. We use midpoints of age intervals to get earnings losses for age groups in the model. See text for further details of sample selection criteria. Column Predisplacement reports the discounted sum of earnings losses as a multiple of predisplacement annual earnings. Column Counterfactual reports the discounted sum of earnings losses as share of the sum of discounted counterfactual earnings. See text for further details. View Large Table 3. Comparison to earnings loss estimates from Davis and von Wachter (2011). Davis and von Wachter Model Sample Predisplacement Counterfactual (%) Predisplacement Counterfactual (%) All workers 1.7 11.9 1.5 10.0 Age 21–30 1.6 9.8 1.7 9.8 Age 31–40 1.2 7.7 1.5 10.0 Age 41–50 1.9 15.9 1.2 8.8 Davis and von Wachter Model Sample Predisplacement Counterfactual (%) Predisplacement Counterfactual (%) All workers 1.7 11.9 1.5 10.0 Age 21–30 1.6 9.8 1.7 9.8 Age 31–40 1.2 7.7 1.5 10.0 Age 41–50 1.9 15.9 1.2 8.8 Notes: The first column shows the considered sample. All workers in the case of Davis and von Wachter (2011) means men from age 21 to 50. We use midpoints of age intervals to get earnings losses for age groups in the model. See text for further details of sample selection criteria. Column Predisplacement reports the discounted sum of earnings losses as a multiple of predisplacement annual earnings. Column Counterfactual reports the discounted sum of earnings losses as share of the sum of discounted counterfactual earnings. See text for further details. View Large Our model matches their earnings losses closely except for the oldest group of workers. If we allow for diverging labor force participation trends for workers age 41–50, for example due to early retirement decisions, and match a difference at age 65 of 30%, then the model generates earnings losses of 1.8 times predisplacement earnings and 13.8% of the counterfactual present value of earnings; this, again, closely matches the results by Davis and von Wachter (2011).33 Our model abstracts from early retirement decisions, because they do not have an impact on the mechanism generating large and persistent earnings losses. However, these decisions can potentially become important when looking 20 years ahead after a displacement event for older workers as done in Davis and von Wachter (2011). 5.3. Decomposition In this section, we decompose earnings losses into three effects: lower wages (wage loss effect), lower employment rates due to higher separation rates in subsequent matches (extensive margin effect), and selection due to restrictions on employment histories of the control group (selection effect). In a second step, we decompose wage loss effect and extensive margin effect in effects due to losses in worker- and match-specific skills. The importance of worker- and match-specific skill losses is the key result for the subsequent policy analysis, because it informs policymakers about the potential effectiveness of retraining and placement support programs. 5.3.1. Selection Effect The control group definition in Jacobson et al. (1993, p. 691) “compares displacement at date s to an alternative that rules out displacement at date s and at any time in the future”. This construction of the control group leads to a spurious correlation between nondisplacement and future employment paths by requiring subsequent stable employment. Viewed through the lens of a structural model, this assumption leads to ex-post selection of employment histories in terms of favorable idiosyncratic shocks and unattractive outside job offers.34 Ex-post selection applies to workers who are identically ex-ante. In addition to ex-post selection, the construction of the control group also leads to selection of workers who differ ex-ante. Ex-ante selection occurs because workers who are less likely to separate in the future, either because of higher worker- or match-specific skills, are more likely to be included in the control group today. Ex-ante selection is present if workers and/or matches differ between control and layoff group at displacement. To obtain an estimate of the importance of this effect, we construct an alternative ideal control group labeled the twin group. For this twin group, we do not impose restrictions on future employment paths, so no ex-post selection arises. Furthermore, we observe the skill distribution and can compare identical workers at age 40 with at least six years of tenure in the control and layoff group. Both groups have the same distribution over skills ex ante and differ only by the fact that one group received the exogenous separation shock whereas the other group did not. Hence, using our model, we can do the counterfactual experiment that must remain unobserved in the data of what would have happened had the worker not been displaced. We track the average earnings paths of these two groups. If we compare the earnings losses to the benchmark case where the control group is employed continuously, we find that initial earnings losses are nearly identical and driven largely by the length of the initial nonemployment period. However, earnings losses after six years are substantially different. The difference is solely due to the selection of the control group as the layoff group is identical in the twin experiment and in the benchmark. The resulting selection effect is sizable, accounting for 31% of the total earnings losses after six years. In Online Appendix I, we provide a graphic illustration of the decomposition. Couch and Placzek (2010) report results using an estimation approach that involves matching workers based on propensity scores. The idea is to compare workers who have identical probabilities for being laid-off to control for individual heterogeneity. Still, they require continuous employment for the control group, so ex-post selection arises. They find that accounting for ex-ante selection in this way can account for 20% of the estimated earnings losses at the maximum. Davis and von Wachter (2011) reduce the nondisplacement period for the control group after the displacement event. If we decompose earnings losses using their control group, we find that after six years, the selection effect is roughly cut by half and accounts for 14% of estimated earnings losses. Regarding ex-post selection, Davis and von Wachter (2011) discuss results for a case when nonmass layoff separators are included in the control group, in which case workers with less favorable employment histories are also part of the control group. In this case, they find that estimated earnings losses are up to 25% lower. This result and the result from the matching estimator by Couch and Placzek (2010) already indicate that both ex-ante and ex-post selection might be substantial in the empirical studies. 5.3.2. Extensive Margin and Wage Loss Effect The literature does not always make a clear distinction between wage and earnings losses when interpreting empirical estimates. A notable exception is Stevens (1997). She empirically decomposes earnings losses into wage losses and an effect due to lower job stability. When we decompose earnings losses based on our model, we control for the selection effect using the twin group as our control group. The wage loss effect of our decomposition captures wage differences of employed workers between the control group and the layoff group. The extensive margin effect accounts for the remainder of earnings losses resulting from differences in employment rates between control and layoff group. Based on this decomposition, we find that the wage loss effect accounts for 48% of total earnings losses after six years and the remaining 21% are due to the extensive margin effect. Looking at the evolution of the decomposition over time, we find the extensive margin effect to be largest on impact, but even after six years, the layoff group is more often nonemployed than the control group. We show the decomposition over time in Online Appendix I. To validate this decomposition, we compare the model-based decomposition of earnings losses to data from the PSID closely following the analysis in Stevens (1997). For this comparison, we neither in the model nor in the data control for the selection effect to make results directly comparable. Stevens (1997) uses PSID data spanning the years between 1968 and 1988 to estimate earnings losses from job displacement. Unlike the administrative data, as used in Jacobson et al. (1993), Couch and Placzek (2010), or Davis and von Wachter (2011), PSID data provides information on earnings and hours worked that allow estimating extensive margin and wage loss effect directly. We follow Stevens (1997) in terms of sample selection and definition of worker displacement. We adopt her empirical specification and focus on first displacements consistent with the implementation in the model and the empirical approach in Couch and Placzek (2010). We provide further details about PSID data and the implementation in Appendix A.3. Figure 8(a) shows the estimated earnings losses based on the specification in Stevens (1997) in comparison to the model. One caveat of the PSID data is its small sample size compared to administrative sources so that point estimates are less precise. Differences between empirical estimates and model counterparts are therefore typically not statistically significant. For example, the estimated earnings losses from Figure 8(a) are slightly larger than their model counterpart, but these differences are not statistically significant. Estimated earnings losses show the same dynamic evolution with large and persistent losses after six years. In a second step, we make use of that the PSID provides information about annual hours worked. Annual working hours are affected by periods of nonemployment because nonemployment periods imply lost working hours. We use the information on working hours to decompose earnings losses into contributions from lower wages and lower employment. We proceed with the same estimation approach as for earnings losses but replace earnings on the left-hand side of the regression by wages and hours worked. For wages, we use annual earnings divided by annual hours worked. In Figures 8(b) and (c), we compare the reduction in hours and wages from the data to wages and the extensive margin from the model. The model matches the reduction in wages and working time closely. The reduction in working time is matched almost exactly whereas the wage loss is slightly larger in the data. Earnings losses between model and data in Figure 8(a) differ slightly in size. To control for this level difference in the decomposition, we consider the share of earnings losses accounted for by wage losses from year 2 to 6 after displacement in Figure 8(d). The model predicts a relatively constant share of 60%. This number differs from the previous decomposition because earnings losses still comprise the selection effect. The decomposition from the data varies over time but stays always around 60%. We conclude that the model aligns well with the empirical evidence regarding the decomposition of earnings losses. 5.3.3. Decomposition in Worker- and Match-Specific Effects The literature has proposed both match- and worker-specific skill losses as explanation for the observed earnings losses.35 The distinction is important to inform policymakers if retraining in case of worker-specific skill losses or placement support in case of match-specific skill losses should be at the heart of labor market policies targeted at displaced workers. We use counterfactual employment paths from our structural model to inform the debate about the relative importance of the two explanations. We construct three counterfactual groups of workers for whom we track the evolution of earnings and wage losses after an initial skill loss. All losses are expressed relative to a benchmark group that corresponds to the control group from the twin experiment so that no selection effect will be present in the decomposition. The first group loses worker-specific skills as in the case of a single job change, but keeps the match-specific component. A second group keeps the worker-specific component, but loses the match-specific component. This group draws a new match-specific component from g(xm). A third group loses both their worker- and match-specific components. Earnings and wage losses of this third group correspond closely in size to the earnings and wage losses from the original estimation in the twin experiment.36 We again provide a graphic illustration of the decomposition in Online Appendix I. For the group with the worker-specific skill loss, we find wage losses that are small but highly persistent. After six years, their wage loss corresponds to 14.7% of the wage loss for the group that loses worker- and match-specific skills. The group with the match-specific skill loss experiences a significant recovery in wages from an initial drop of roughly 12% to 4% after six years. However, the wage loss is persistent. The wage loss after six years of this group corresponds to 85.8% of the wage loss of the group that loses both match- and worker-specific skills. The decomposition has a negative residual of −0.4%. Turning to earnings losses, we find that the group with the match-specific skill loss experiences a strong divergence of wages and earnings initially due to increasing job instability. The difference between wages and earnings reduces over time but remains significant and persistent. If we decompose the difference between wage and earnings losses (the extensive margin effect), we find that 94.2% is due to match-specific skill loss and 4.5% due to worker-specific skill loss. The remaining 1.3% are a residual of the decomposition. Hence, match-specific skill losses are the dominant driver of wage and earnings losses. 5.3.4. Discussion The loss of a particularly good job, meaning a job with high match-specific skills, accounts for most of the large and persistent earnings losses. To generate large and persistent skill differences in the match type, it is important that good jobs at the top of the job ladder are very stable. Workers who have lost their good jobs due to displacement search the market and recover to the average job in the economy, so there is mean reversion from below. If good jobs are very stable, there is no mean reversion from above leading to large and persistent differences. Figure 9 visualizes the skill dynamics for the worker- and the match-specific skills following the initial displacement event. Figure 9. View largeDownload slide Skill dynamics following displacement. Panel (a): Average worker-specific skill level in control group (gray solid line) and layoff group (black dashed line) after displacement event. Panel (b): Average match-specific skill level in control group (gray solid line) and layoff group (black dashed line) after displacement event. Vertical axes show mean skill levels (x$$w$$ and xm). Horizontal axes show time in years relative to the displacement event. Figure 9. View largeDownload slide Skill dynamics following displacement. Panel (a): Average worker-specific skill level in control group (gray solid line) and layoff group (black dashed line) after displacement event. Panel (b): Average match-specific skill level in control group (gray solid line) and layoff group (black dashed line) after displacement event. Vertical axes show mean skill levels (x$$w$$ and xm). Horizontal axes show time in years relative to the displacement event. Looking at worker-specific skills from our twin experiment in Figure 9(a), we see that there is an initial drop followed by diverging paths due to job instability and high worker mobility in the layoff group (black dashed line). Looking at match-specific skills from our twin experiment in Figure 9(b), we find that the initial drop is followed by a recovery of the layoff group toward the mean (black dashed line). There is little mean reversion from above due to very stable jobs at the top of the job ladder (gray solid line). Although the job ladder allows for mean-reversion from below, the low mean-reversion from above leads to persistent differences in match-specific skills. The good jobs at the top of the job ladder are the result of search rather than of accumulated worker-specific skills, and might therefore be considered as a source of transitory differences across workers. The fact that persistent earnings losses are driven by this skill component might hence be surprising. Our skill process is not confined to provide this explanation. Although different explanations that we encompass in our model could potentially generate large and persistent earnings losses, it is worker mobility that pins down the skill process in our model. An explanation that focuses on the deterioration of worker-specific skills during unemployment or upon transition as the key driver of earnings losses faces the challenge of having to match the empirical mobility pattern (Ljungqvist and Sargent 1998). Such an explanation might generate large earnings losses at least initially as it affects workers’ persistent skill component but is at odds with observed worker mobility (see den Haan, Ramey, and Watson 2000b for a related point). If worker-specific skills were the main source of earnings losses, this would imply that expected losses from mobility are high and workers who have a mobility choice will be very reluctant to engage in mobility. As a result, average worker mobility would be low, both because expected losses of mobility are high due to low transferability of skills and because gains from mobility are little because of little persistent job heterogeneity.37 To explain high average worker mobility, we need a skill process that features a high degree of transferability of accumulated skills and sufficiently large gains from mobility. Our skill process has these features with gains from mobility being large because jobs further up on the job ladder are more stable and pay higher wages. As a consequence, earnings losses are driven by the loss of a particularly good job rather than by the deterioration of accumulated worker-specific skills. 5.4. Sensitivity We provide a detailed discussion of the sensitivity of our results for earnings losses in Online Appendix H. Here, we highlight the most important findings. We demonstrate that our model closely reproduces the earnings losses for the nonmass layoff sample in Couch and Placzek (2010). We do this by including all separators, that is, endogenous separations and job-to-job transitions, in the layoff group. Including endogenous separations and job-to-job transitions implies that we include workers that are negatively selected based on their worker- and match-specific skill type. Even in this case, we get large and persistent earnings losses, although they are slightly lower in line with the empirical evidence. We also show that earnings losses change little with age in line with Jacobson et al. (1993). We also report the profile of long-run earnings losses underlying our comparison to the results by Davis and von Wachter (2011). We show that earnings losses are still significant 20 years after the initial displacement event. We discuss in detail the effects of varying selection criteria for the control group that is the key difference between Davis and von Wachter (2011) and Couch and Placzek (2010). We also demonstrate that when we select separators with good labor market prospects, then earnings losses vanish in line with the empirical findings for separators who do not claim unemployment benefits. Finally, we use age-specific job stability thresholds to account for the fact that tenure increases linearly with age. We still find earnings losses to be large and persistent. Regarding the decomposition of earnings losses, we discuss in Section F of the Online Appendix results from the two model extensions with skill depreciation during nonemployment (Section F.1) and additional skill accumulation on the job to match the tail of the wage distribution (Section F.2). We find for both extensions large and persistent earnings losses in line with the baseline model. As in the baseline model, the decomposition of earnings losses attributes the largest contribution to the wage loss effect, followed by the selection effect and the extensive margin effect. The wage loss effect is largest in the extension with skill depreciation during nonemployment (59%), compared to the extension with additional skill accumulation on the job (55%) and the benchmark (49%). As we discuss previously, the mobility dynamics that identify the parameters of the skill process put strong discipline on skill dynamics associated with worker mobility and job loss. The case with skill depreciation during nonemployment assumes a skill process that in principle has the most adverse consequences for a displaced worker who has accumulated a lot of worker-specific skills. Even under the assumption of duration dependent skill losses, the wage loss effect after displacement increases only modestly suggesting that our baseline skill dynamics capture well the main sources of earnings losses after job displacement. 6. Policy Analysis Understanding the sources of earnings losses is vital for designing labor market policies. Viewed through the lens of our structural model, active labor market policy can potentially help displaced workers along two margins: First, it can help to avoid the loss of worker-specific skills by providing retraining services. Second, it can help to regain match-specific skills by providing placement support to foster better matches between jobs and workers. In practice, placement support and retraining are the two pillars of the Dislocated Worker Program (DWP) of the Workforce Investment Act. The DWP “is designed to provide quality employment and training services to assist eligible individuals in finding and qualifying for meaningful employment, and to help employers find the skilled workers they need to compete and succeed in business”.38 The DWP is targeted explicitly at displaced workers who lost their jobs due to layoff, plant closures, or downsizing.39 The targeted group, therefore, corresponds in principle to the group of displaced workers in our model. We examine the effectiveness of the DWP in reducing earnings losses within our model. Leaving aside costs to run the program, we consider retraining and placement support for 40-year-old displaced workers. It is important to bear in mind that, using our structural model we take into account all endogenous responses on wages, mobil