The Job Ladder and its Implications for Earnings Risk

Transcription

1 The Job Ladder and its Implications for Earnings Risk Joachim Hubmer February 4, 216 Abstract This paper analyzes the ability of a job ladder framework to explain recent evidence on earnings risk. Heterogeneous and risk averse workers search for job opportunities at heterogeneous firms. The resulting dynamics can successfully replicate several non-targeted key properties of the distribution of earnings changes that have been documented by Guvenen et al. [215]. These are most notably a large negative skewness and a high excess kurtosis, rejecting the frequently used log-normal framework. Moreover, the proposed model is to a certain extent successful in explaining variation in these moments as a function of age and the level of earnings. 1 Introduction Incorporating heterogeneous agents has been one of the key advances in macroeconomic equilibrium modeling in the past decades. The departure from the representative agent framework allows for studying the implications of heterogeneity on aggregate outcomes and conversely the evolution of income and wealth inequality itself - both of vital importance in light of the Great Recession and the increasing concentration of income and wealth at the top. Idiosyncratic earnings risk, which is quantitatively significant and to a large extent uninsurable, is the major ingredient in these models. Capturing the salient features of earnings risk is therefore essential. Yet, recent evidence suggests that the standard assumption of parsimonious log-normal earnings processes is strongly rejected empirically. In contrast to the prevailing literate based on survey data such as the PSID, the increased availability of comprehensive administrative datasets practically eliminates measurement error and has therefore enabled researchers to go beyond the log-normal framework. In particular, Guvenen et al. [215] use U.S. Social Security Administration data to estimate earnings processes fully non-parametrically. They find (i) striking variation in life-cycle earnings growth, (ii) large negative skewness and (iii) very high excess kurtosis. However, while comprehensive administrative datasets allow for a precise measurement of higher order moments, it is unclear to what extent these deviations are driven by exogenous shocks as opposed I am grateful to Giuseppe Moscarini for invaluable guidance and support. I thank Fatih Guvenen, Per Krusell, Tony Smith, Aleh Tsyvinski, Gianluca Violante and seminar participants at the Yale Macro Lunch for helpful comments and suggestions. Department of Economics, Yale University; joachim.hubmer@yale.edu 1

2 to choices. Taking the evidence at face value, attributing all of it to earnings risk and consequently calibrating some exogenous earnings process that matches the facts might therefore be premature. To resolve this tension, I explicitly model a frictional labor market as the source of (higher order) earnings risk. Heterogeneous and risk averse workers search for job opportunities at heterogeneous firms while unemployed as well as on-the-job. Coupled with exogenous separation from jobs, the resulting dynamics - commonly referred to as job ladder - naturally generate some negative skewness and excess kurtosis: In the absence of a separation shock, workers only switch to more productive firms, moving up the ladder relatively slowly. A lay-off means falling down the ladder all the way, hence on-the-job search implies negative skewness. Moreover, both job-to-job transitions as well as movements between employment and non-employment are potentially large and infrequent changes, thus generating excess kurtosis. Crucially, I do not calibrate this job ladder model to match the data on higher-order earnings risk. Instead, besides a conventional mean-variance life-cycle earnings profile, I target aggregate labor market statistics as well as two robust stylized features of the labor market that have received much attention recently and should be informative about earnings risk: First, there is a large amount of residual wage dispersion (see Hornstein et al. [211] for a discussion). Second, job displacement is associated with large and very persistent earnings losses (see for example Jacobson et al. [1993], Davis and Wachter [211]). In turn, I can evaluate the performance of the model against the non-targeted evidence in Guvenen et al. [215]. Overall, the model is successful in explaining the magnitude of negative skewness and excess kurtosis. Furthermore, it captures much of the heterogeneity in life-cycle earnings growth. Along the life-cycle dimension, the job ladder naturally replicates that negative skewness is increasing in age: older workers had more time to search, thus work at better firms and have more room to fall. Endogenous search effort amplifies this trend and ensures that it also holds when conditioning on recent earnings. The returns to search decrease as workers approach retirement. Thus, older workers search less, which means they receive less job offers both in unemployment and on-the-job and thus re-climb the ladder more slowly. Consequently, they face more severe downside risk, i.e. skewness becomes more negative. When ordering workers according to recent earnings, the model captures that negative skewness and kurtosis are increasing in recent earnings. While the job ladder logic naturally implies that the earningsrich work on average at better firms and thus have more room to fall, the increase in negative skewness as a function of recent earnings mirrors the data also quantitatively quite closely. It turns out that risk aversion and associated partial insurance through a risk-free savings choice amplifies this dependance of skewness on recent earnings. High earners have accumulated more wealth, thus when hit by a separation shock they can compensate for the earnings loss by drawing down their buffer stock savings and consequently have higher reservation wages. The larger downside risk does also increase the kurtosis of earnings changes. However, the increase in the kurtosis as a function of recent earnings is more moderate in the model than in the data. In fact, with risk neutral workers it is almost flat. The ability of the model to jointly match aggregate labor market flows and large empirical estimates of residual wage dispersion is of separate interest. Hornstein et al. [211] illustrate that standard search models have a hard time reconciling the evidence. Unemployed workers, moderately risk averse and 2

3 receiving unemployment benefits of plausible size, are willing to accept low-paying jobs only if job-to-job transition rates are relatively high. Otherwise, waiting for better job offers is optimal. However, job-to-job transition rates are an order of magnitude smaller than unemployment-to-employment transition rates. This tension is eased in my setup by the presence of skill depreciation in unemployment in the spirit of Ljungqvist and Sargent [1998]. Facing the dire prospect of losing productive skills, unemployed workers are willing to accept offers for low-paying jobs. Skill depreciation is needed, and consequently calibrated, to match the size of earnings losses associated with displacement as the job ladder channel by itself cannot fully account for them. Relatedly, Jarosch [214] uses a job ladder that is more slippery at lower levels in addition to the skill depreciation channel. 1 The fact that my framework can accommodate large amounts of frictional wage dispersion is in some sense orthogonal to the discussion by Hornstein et al. [211]. They focus more narrowly on the matching friction and conclude that it cannot by itself reproduce empirical estimates of residual wage dispersion. By introducing a process for worker productivity that depends on the employment status, it is the interaction of the matching friction with the human capital channel that reconciles the evidence. The rest of the paper is organized as follows. Section 2 illustrates the basic intuition. The full life-cycle model, set in a partial equilibrium framework, is outlined in section 3, its calibration explained in section 4 and the main results presented in section 5. Section 6 contains a simplified infinite-horizon framework and embeds the model in general equilibrium. Finally, section 7 concludes. 2 Basic Intuition Before diving into the details of the model, we can gain intuition by looking at Figure 1, showing the density (in logs) of one year log earnings changes. For comparison, a Gaussian density with conforming mean and variance is added to the graph. The model generated output resembles the data gathered by Guvenen et al. [215] very much, which is surprising at first given that the model is not directly calibrated to the distribution of earnings changes. We observe that the slope of the right tail is about 2 (in absolute value). Hence, earnings changes follow a Pareto distribution with coefficient 2 in the right tail. 2 the earnings distribution itself can be very well described by a Pareto distribution, at least for the top earners, with shape coefficient 2 (see e.g. the top wage shares time series in Piketty and Saez [23]). I assume that firm productivity (associated with job offers) follows a Pareto distribution and calibrate the shape coefficient at the upper end to match top earnings shares. The fact that, by doing so, I generate a very good fit to the distribution of earnings changes points to the success of the job ladder model in 1 The slippery job ladder on top of skill depreciation allows Jarosch [214] to explain the evidence on persistently lower employment rates, in addition to persistent wage losses, in response to unemployment shocks. 2 Note that for a Pareto distributed random variable X (in this case earnings changes) with shape coefficient η (and w.l.o.g. lower bound normalized to one), the pdf of X is given by f X(x) = η. Taking logs, log (f x X(x)) = log(η) (η + 1) log(x). η+1 Hence, plotting log(x) on the horizontal and log(f X(x)) on the vertical axis the slope equals (η + 1). However, following Guvenen et al. [215] here for comparability, in this figure the pdf of the log of earnings changes log(x) is computed first, η instead of the pdf of earnings changes. Let Y = log(x) denote log earnings changes, then f Y (y) = exp(y) = exp((η+1)y) η exp( ηy). Finally, we plot y = log(x) on the horizontal and log(f Y (y)) = log(η) ηy = log(η) η log(x) on the vertical axis. Hence, the (absolute value of the) slope of the right tail in Figure 1 equals the Pareto tail coefficient of earnings changes. The argument applies to the left tail too. But 3

4 2-2 model data Gaussian log pdf Y log earnings changes Figure 1: Log Density of One-Year Log Earnings Changes explaining the evidence: Let z be the skill of a worker, p P areto(1, η) the productivity of a firm and assume the output is simply the product zp. If furthermore wages are a fraction of output γ, then not only will the distribution of earnings x = γzp inherit the Pareto tail, but also earnings changes x x will inherit the same right tail. Assuming γ and z are constant for simplicity and conditional on positive changes (i.e. drawing a better firm p > p), x x = p p P areto(1, η), for all p - i.e., also in the cross-section of workers. This is a consequence of the self-similarity of the Pareto distribution. Now, in principal the Pareto tail in the earnings distribution in levels might be due to z, not p, i.e. arising from worker and not firm productivity. Pareto tails, also referred to as power laws, are found in remarkably many economic variables such as the wealth and income distribution or firm sizes (see Gabaix [29] for a review). They empirical ubiquity can be explained by random, scale-free growth processes. 3 While one might contemplate that worker skills are growing randomly and thus attribute the Pareto tail to z, under that hypothesis it seems difficult to explain why earnings changes exhibit the same right tail. On the other hand, randomly growing firm productivity levels are consistent with Pareto tail in the earnings distribution both in levels and in growth rates. Furthermore, we observe in Figure 1 that the left tail of the distribution of earnings changes is fatter than the right tail, illustrating negative skewness. As in a job ladder model movements up the ladder are relatively more frequent and smaller than movements down the ladder, some negative skewness can be expected, i.e. a left tail that is declining more slowly. The accuracy of the model is nevertheless non-trivial. Negative skewness is stemming not only from the job ladder mechanism, but also from skill depreciation in unemployment, the size of which is calibrated to match the data moment on earnings losses from displacement. 3 More precisely, in addition some friction is needed, e.g. a reflecting lower barrier. 4

5 3 Partial Equilibrium Framework In short, the modeling framework builds on the Lise [213] model, which features risk averse workers that search for job opportunities at heterogeneous firms both in unemployment as well as on-the-job and can smooth consumption by means of a risk-free asset. On top of that, worker heterogeneity is added as the model is calibrated to match certain features of the cross-sectional earnings distribution and evaluated against Guvenen et al. [215] s estimates, who do not condition on fixed worker characteristics (the nature of their administrative data does not allow for that). In particular, the economy is populated by a continuum of finitely-lived risk averse agents. 4 An agent can work in periods t = 1,...T, followed by retirement. Once retired, agents face a constant probability of death. A newborn worker starts his working life being unemployed in t = 1 with an initial endowment of assets a, permanent human capital h and some level of efficiency units of labor supply z. I will refer to z as the current skill of a worker, which will evolve over time, as opposed to fixed human capital h. An unemployed worker receives unemployment benefits b(z) and chooses (i) savings a a as well as (ii) search effort s [, 1]. 5 Consumption c = (1 + r)a + b(z) a yields CRRA utility u(c) and search s entails a convex search cost χ 1+φ s 1+φ for some φ >, entering additively. At the end of the period, the skill of the unemployed worker z is updated to z z, h, following a Markovian process - i.e. in addition to current skill, fixed human capital affects the skill evolution too. Subsequently, the agent samples a job offer with probability λ w s, where λ w (, 1) is the probability of finding a job when searching full-time. Job offers are characterized by the one-dimensional productivity of a firm p, drawn from some distribution F p. Matching with firms is random. The worker can chose between accepting the job and starting period t + 1 working at firm p or staying in non-employment. 6 An agent that starts period t {1,..., T } employed at a p-firm first collects a wage w(z, p) = γzp for some fixed constant γ (, 1). In the background, the worker-firm match produces output zp and a fraction γ of this rent accrues to the worker. Note that fixed human capital h does not affect the output and wage, but only the skill process. Just as in unemployment, she can choose (i) savings a a, yielding c = (1 + r)a + w(z, p) a, as well as (ii) search effort s [, 1]. Working (full-time, as there is no intensive margin of labor supply) entails a leisure penalty χ e > as well as higher search costs (χ s + χ se ) s1+φ 1+φ. χ se parametrizes the additional dis-utility of searching when working. Note that Lise [213] does not feature a dis-utility of working and χ se =. Consequently, workers accept any job, which has the desirable implication of ensuring concavity of the value function. Allowing for χ se > enables the model to match both aggregate unemployment-to-employment as well as job-to-job transition rates. Moreover, if search were costly but there would be not dis-utility of working (i.e. χ e = ), then reservation wages would be decreasing in wealth above a certain threshold. 7 At the end of t, the skill 4 Appendix A.1 discusses the role of risk aversion. 5 Appendix A.2 discusses the effect of endogenous search effort in contrast to a setting where job offers are drawn at exogenously given contact rates. 6 Since job search is a continuous variable in this setup, the notation does not differentiate between unemployment and out-of-labor-force. Unemployment and non-employment will be used interchangeably in the remainder of this paper. 7 To see this, note that a very rich agent would choose a very low search intensity. Consequently, if offered any job that pays more than unemployment benefits, he would accept it anticipating that he would search very little in the future and thus not draw better job offers. 5

6 is updated to z z, h, p, i.e. a worker s skill evolution could be potentially affected by the quality of her employer. Subsequently, she samples a new job offer with probability λ w s and independently her job is destroyed with probability δ (, 1). Consequently, in the best case she can choose between staying at the current employer, accepting the new job offer or going to non-employment. In the worst case, she is forced into non-employment. In recursive structure, denote the value of being employed in period t as W t (a, h, z, p), the value of being unemployed as U t (a, h, z) and the maximum of the two as V ( ) = max{w ( ), U( )}. Then U t (a, h, z) = max {u((1 + r)a + b(z) a s 1+φ ) χ s a a,s [,1] 1 + φ ]} (1) +βe z h,z [(1 sλ w )U t+1 (a, h, z ) + sλ w V t+1 (a, h, z, p )df P (p ) p and, incorporating the trivial choice between two firms, W t (a, h, z, p) = max {u((1 + r)a + w(z, p) a ) χ e (χ s + χ se ) s1+φ a a,s [,1] 1 + φ + [ (1 δ)(1 sλ w )V t+1 (a, h, z, p)+ βe z h,z,p (1 δ)sλ w p V t+1 (a, h, z, max{p, p })df P (p )+ ]} δ(1 sλ w )U t+1 (a, h, z ) + δsλ w V p t+1 (a, h, z, p )df P (p ). (2) 4 Calibration 4.1 Preliminaries The model is solved at monthly frequency. The discount factor β is set to and the interest rate r to.75( 1 β 1). Borrowing is not allowed (a = ) and u(c) = log(c). Unemployment benefits are given by b(z) = min{χ b z, b}, where χ b is set to.5, yielding a median replacement rate of.4 and the cap b equals half of mean earnings in the economy. The convexity of the search cost function is set to φ =.2, following an estimate by Lise [213] for a comparable job ladder model with risk averse workers. Retired workers receive a lumpsum retirement benefit, equaling a quarter of mean earnings. The job offer probability for full search effort λ w is set to.43 and jobs are separated with probability Skill Process Fixed human capital h is log-normally distributed with standard deviation σ h and corresponding cdf F h. The skill z of a worker with human capital h who is employed at a p-firm fluctuates towards a long-run 8 These are standard values used in the literature [Shimer, 25, 212, Hornstein et al., 211]. 6

7 average level z (h, p), where z (h, p) = F 1 h (min{f h (h), F p (p)}). (3) That is, for the skill evolution process I assume a strong form of complementarity between the innate ability of a worker and the quality of a firm in the benchmark model. Appendix A.3 discusses to what extent this assumption affects the findings (there I use z (h) = h). While the multiplicative nature of output zp (and therefore wages γzp) implies that the current benefit of working at a better firm is independent of the worker type (in relative terms), high human capital workers benefit relatively more from working at better firms in the long run. To put it differently, the skill evolution of a worker at the bottom of the human capital distribution does not depend on his employer at all. The skill z evolves according to ρ log(z t ) + (1 ρ) log(z (h, p)) + σ z ɛ t+1 log(z t+1 ) = log(z t ) with prob. π e where ɛ t+1 is a standard normal deviate. Since wages are proportional to z, π e is tied to the frequency of monthly within-job (real) wage changes. I set π e =.3 based on estimates for nominal within-job changes from Barattieri et al. [21, 214]. 9 auto-correlation of.97 and an annual variance of else, (4) Then, ρ and σ z are chosen to correspond to an annual As far as the skill evolution is concerned, being unemployed is equivalent to being employed by a firm at the bottom end of the productivity distribution, i.e. the long-run average skill level is zu(h) = F 1 h (min{f h (h), ɛ}). 11 Similarly to (4), the skill of an unemployed worker evolves according to ρ log(z t ) + (1 ρ) log(zu(h)) + σ z ɛ t+1 with prob. π u log(z t+1 ) = (5) log(z t ) else. Since for (almost) all h, p, it holds that z u(h) < z (h, p), skills are on average deteriorating in unemployment and consequently the frequency of changes π u regulates the magnitude of skill depreciation. Along with other parameters, π u is calibrated inside the model. 4.3 Internal Calibration Table 1 displays the targeted moments and associated calibrated parameters. First, the level of search costs χ s and the additional search cost for employed workers χ se control the effective unemployment-toemployment (U2E) transition rate and the job-to-job (J2J) transition rate. To economize on parameters, I impose that χ e = χ s, i.e. the dis-utility of searching full-time equals the one for working full-time. Note 9 They estimate that in an average quarter, the probability that a worker experiences a nominal wage change ranges from 5% for salaried to 18% for hourly workers. My results are not sensitive to changes in π e corresponding to that range. 1 These numbers are based on estimates by Heathcote et al. [21], using data from the PSID. 11 Since the support of h is unbounded from below, we cannot set ɛ =. Hence I let ɛ =.1, which is somewhat arbitrary but without too much loss of generality as subsequently π u is calibrated to match earnings losses from job displacement. 7

8 Moment Target Model Estimate U2E-rate.3.3 χ s =.4 (= χ e(1 + φ)) J2J-rate.3.3 χ se = 1.25 V[log(y)] at age σ 2 z =.47 V[log(y)] at age σ 2 h = 1.54 log (E[y 55 ]) log (E[y 25 ]) µ z = 2.6 2y-earnings from U-shock π u =.39 Mean-Min ratio residual wages η 1 = 4.35 Top 1% earnings share in % Top.1% earnings share in % η 2 = 1.9 Top.1% earnings share in % Table 1: Calibrated Parameters that searching on-the-job has to be about four times as costly as searching in unemployment to account for the fact J2J-transition rates are an order of magnitude smaller than U2E-transition rates. Second, I target the variance of log annual earnings y at age 25 and age 6 as well as the average life-cycle earnings growth. The values are adopted from Guvenen et al. [215] in order to render the subsequent analysis of earnings risk comparable. In the model simulation, the agents start their working life at age 24 in unemployment; the first 12 month are used as a burn-in period. The initial skill z is log-normally distributed with variance σz 2 and mean µ z, controlling the earnings variance at the beginning of the work-life and the average earnings growth, respectively. Given the standard deviation of the innovation to the skill process σ z and the firm productivity distribution, the earnings variance at the end of the work life is then regulated by the variance of fixed human capital σh 2. Figure 2 displays the mean-variance profile of log annual earnings. We see that the variance is increasing almost linearly. Earnings peak around age 5 to 55 as close to retirement the long-term benefits of finding a better job diminish. Guvenen et al. [215] show that life-time earnings growth increases dramatically in the level of life-time earnings. The sign of this pattern is to be expected, though not the magnitude. I assume that initial skill z is uncorrelated with fixed human capital h, thus maximizing differential earnings growth in the model. As figure 3 reveals, the shape is captured well but the pattern is still more pronounced in the data. As explained above, π u controls the average present value of earnings losses associated with an exogenous job separation [Davis and Wachter, 211]. Figure 4 decomposes the losses. The solid blue line reports the benchmark results, amounting to a present discounted value of earnings losses of 12%. The dashed red line reports results associated with shutting down the skill depreciation channel. In that case, persistence is resulting from the job ladder effect. Clearly, this channel alone cannot create enough persistence and consequently the losses are not severe enough to match the facts. Last, the yellow dash-dotted line reports a scenario where in addition to skill depreciation the job ladder effect is shut down too. That leaves merely the employment channel; consequently earnings losses are virtually zero, except for the year of the layoff itself. 8

9 1.9 life-cycle: log mean earnings 1.1 life-cycle: variance log earnings model data age age Figure 2: Mean-Variance Profile of Log Annual Earnings 3 log( earn55 ) - log( earn25 ) model data mean growth life-time earnings percentile Figure 3: Life-Cycle Earnings Growth by Life-Time Earnings Percentile 9

10 mean earnings loss, displacement at t=1 +job ladder + skill depreciation +job ladder employment channel only 35 mean loss in % year after displacement Figure 4: Decomposition of Earnings Losses from Job Displacement Finally, the job offer distribution (over p) governs frictional wage dispersion. I target the mean-min ratio, following Hornstein et al. [211]. Using a Pareto distribution for p has the desired effect of inducing a Pareto tail (of the same size) in the earnings distribution, which is a well-known feature. 12 The large extent of top earnings inequality [Piketty and Saez, 23] requires a thick tail (η 2 = 1.9) that creates too much residual wage dispersion. Hence, I allow for one more parameter: the bottom 99% of job offers are drawn from a Pareto distribution with shape coefficient η 1 and for the top 1% of job offers the shape coefficient changes to η Quantitative Results The model was calibrated to match aggregate labor market statistics as well as the earnings distribution. Earnings risk was explicitly incorporated only to a limited extent as the standard deviation σ z of the normal error term in the skill process. Hence, we can now meaningfully assess the performance of the model in matching the evidence on, especially higher order, earnings risk as documented in Guvenen et al. [215]. Figure 5 displays the density of log annual earnings changes in the model and compares it to the data as well as to a conforming normal distribution. The left panel displays changes over a one year horizon. The model replicates the peakedness of the data very well. Peakedness and thick tails are the characteristic visual expressions of a high kurtosis: while most workers experience no or rather small changes in their annual earnings, few undergo very dramatic changes. Figure 1, discussed in section 2, 12 In addition, the empirical firm size distribution can be described very well by a Pareto distribution. Assuming a primitive Pareto productivity distribution of firms and a power function for vacancy costs implies that both the job offer distribution as well the firm size distribution would inherit the Pareto shape. 13 As this is a partial equilibrium exercise, one could in principle estimate F p(p) using wages accepted out of unemployment. This avenue is pursued by Lise [213], drawing on survey data. In the absence of corresponding administrative data uncontaminated by measurement error, I choose this indirect route. 1

11 1Y Change 1.5 5Y Change 4 3 model data Gaussian 1 pdf 2 pdf log earnings changes log earnings changes Figure 5: Histogram of Log Earnings Changes 2-2 model data Gaussian log pdf Y log earnings changes Figure 6: Log Density of Five-Year Log Earnings Changes shows the same evidence as the left panel of figure 5, the only difference being that the natural logarithm of the densities is taken. As such, it highlights the tail behavior of earnings changes. In the model, the workers that stay at the same firm are the ones with relatively stable earnings. As their skill evolves too, their earnings are not constant but the risk they are exposed to is rather mild. In contrast, those agents that become unemployed, regain employment or switch to a better job are exposed to heavy risk. The right panel of 5 displays changes in log annual earnings over a five year horizon. We see that over this time frame, the model generated earnings risk still deviates from normality, though the peakedness is considerably higher in U.S. data. Again, taking the log transform illustrates the tail behavior. Figure 6 shows that the model successfully replicates the Pareto tails for large positive and negative earnings changes over a five year horizon too. 11

12 Standard Deviation Y Change Model Model Data Data Standard Deviation Y Change Figure 7: Standard Deviation of Log Earnings Changes 5.1 A Closer Look at the Distribution of Earnings Changes Having established that the model appears to replicate the overall earnings change distribution in a satisfactory manner, we now look at the standard deviation, skewness and kurtosis in more detail. For a random variable [ X with mean µ and standard deviation σ, skewness (the third standardized moment) ( ) ] [ 3 ( ) ] 4 is defined as E X µ σ and its kurtosis (fourth standardized moments) given by E X µ σ. Besides variation over the life-cycle, Guvenen et al. [215] analyze the distribution of earnings changes between age t and t+k for k = 1, 5 as a function of recent earnings. Recent earnings are defined as the average over earnings in t 5, t 4,..., t 1, stripped of age effects. I include very low (or zero) earnings observations and replace them by a positive lower bound, consistent with the construction of the data I use. 14 Figure 7 plots the standard deviation of log earnings changes as a function of the recent earnings percentile, separately for young (25-35) and prime age workers (35-55). First, that earnings vary more in the data can be mostly attributed to the inclusion of very low earnings observations. As the model abstracts - among other considerations such as health - from any notion of family, there is no reason to drop out of the labor force except for reaching sufficient self insurance through asset accumulation (which raises the reservation wage). 15 Both at short and long horizons, a U-shaped pattern emerges. Naturally, top earners face very large downside risk, which the model is capturing well (falling down the ladder). At the lower end, comprised of workers with low current skill and working at bad firms, the increase in risk is considerably smaller than in U.S. data. If log earnings changes were normally distributed, skewness would be zero across the board. However, as figure 8 illustrates, for all but those workers at the very bottom of the recent earnings distribution, there is a significant amount of negative skewness. As explained above, this is naturally implied by a job ladder model: the presence of on-the-job search implies that movements up the ladder are on average 14 See appendix B.1.1. in Guvenen et al. [215]. The threshold amounts to the equivalent of 1.5 months of minimum wage earnings, working full-time, and a small amount of random noise is added. 15 When simulating the model, I start all agents at zero initial wealth. 12

13 .5 1Y Change.5 5Y Change -.5 Skewness Model Model Data Data Skewness Figure 8: Skewness of Log Earnings Changes smaller than movements down the ladder (which, in the absence of compensating differentials, consist of employment-to-unemployment transitions only). Moreover, negative skewness is increasing both in age and recent earnings. As expected, the model captures both trends well: both high earners as well as more senior workers have on average climbed up the job ladder farther, thus there is more room to fall for them. For one year changes, displayed in the left panel, the fit is especially good. The top 1% are the exception, partly explained by the assumption of homogeneous exogenous separation rates. For five year changes, displayed in the right panel, the model is understating the amount of negative skewness for both age groups. The kurtosis of a random variable can be seen both as a measure of peakedness and a measure of tail thickness. Normally distributed log earnings changes would imply a kurtosis of three. Figure 9 illustrates that male U.S. workers are exposed to earnings risk with considerable excess kurtosis, on average about five times as high as a log-normal framework would imply for annual changes and three times as high for five year changes. While the model is successful in generating a comparable amount of excess kurtosis, it fails to capture the sharp increase as a function of recent earnings. This failure is related to an implicit homogeneity assumption in my model that results from random matching: the job ladder is open to every worker, i.e. even the best firms hire workers regardless of their skill level. Consequently, large movements upwards and downwards the job ladder can be experienced by all worker types. High human capital increases the upside potential through the skill accumulation channel, but not directly through the job ladder channel. Adding an additional layer of complexity in the hiring decision of firms could possibly improve the fit of the model in that direction. Furthermore, kurtosis is increasing over the life-cycle, especially for annual changes. Again, the job ladder provides a natural explanation for that pattern: Older workers search less, since the present discounted value of working at a better firm decreases as they approach retirement. In addition, they tend to be already employed at on average better firms, increasing their reservation wages (and decreasing search effort). Hence, job-to-job transitions are less frequent. At the same time, falls down the job ladder into unemployment are even larger on average. In sum, older 13

14 1Y Change 13 5Y Change 25 2 Model Model Data Data Kurtosis 15 Kurtosis Figure 9: Kurtosis of Log Earnings Changes workers experience less frequent but larger changes, i.e. face higher kurtosis risk. 5.2 Job Stayers vs. Switchers Are the deviations from log-normality entirely caused by transitions between non-employment and employment as well as job-to-job transition? In that case, the log-normal framework would be just fine for describing earnings risk for job stayers, defined as those workers that stay at the same firm. However, Guvenen et al. [215] find that this is not the case, in particular earnings changes for job stayers exhibit a large amount of excess kurtosis. The presence of multiple job holders complicates the definition of a job stayer in the data. Guvenen et al. [215] use a definition that classifies a worker as a stayer if more than 9% of his earnings over the relevant time period were stemming from the same firm. As the data on stayers still contains e.g. workers that are temporarily unemployed 16 or working reduced hours, variation in hours worked might be important in explaining deviations from log-normality for stayers. The model entirely abstract from these considerations and therefore the statistics are of limited comparability. In particular, I naturally define a job stayer in my model as a worker that is continually employed at the same firm throughout the relevant time period (i.e. for changes between t and t + k from the first month of t to the last month in t + k). For comparison, I also report results when defining a stayer as somebody that works at the same firm for at least six month in every year of the relevant period. The model output is summarized in table 2 for both definitions. Keeping these caveats in mind, we can evaluate the performance of the model with respect to three main facts: First, as expected the standard deviation of earnings changes for stayers is only about half as large as for those workers that switch jobs, denoted as (job) switchers. As illustrated in Table 2, this 16 Fujita and Moscarini [213] provide evidence on the relevance of recall. Preliminary results from ongoing research suggest that the decomposition of the model output for stayers vs. switchers is much more in line with the data once temporary layoffs are taken into account. 14

15 1 Year Changes 5 Year Changes Std.Dev. Skewness Kurtosis Std.Dev. Skewness Kurtosis full output stayers (12 month) switchers (12 month) stayers (6 month) switchers (6 month) Table 2: Model Moments for Job Stayers vs. Switchers trend is even more extreme in the model output, where for annual changes the standard deviation is four times higher for switchers. Defining a job stayer as a worker that stays at the same firm for at least six month in every year, this ratio decreases to three. At least part of this discrepancy might be due to the absence of any variation in hours worked. Second, skewness is close to zero or positive for stayers. Again, this pattern is more extreme in the model output, where log earnings changes are significantly positively skewed. Although I use a normal error term in the process for current skill (4), the definition of the long-run mean z (h, p) as an increasing function of human capital h and firm productivity p implies that current skill z t is typically either increasing or (in expectation) staying constant on-the-job. Recall that the skill of a young worker entering the labor market is calibrated to be rather low in order to match life-cycle earnings growth (only part of that growth can be explained by the job ladder). Consequently, as earnings changes for stayers are mostly close to zero or positive, skewness is positive. Finally, the most striking fact is probably that even job stayers face a massive kurtosis risk. In particular, Guvenen et al. [215] find a kurtosis as high as 43 (annual changes), respectively 29 (five year changes), for stayers. In fact, the kurtosis is much lower, by a factor of 3-4, for switchers. This indicates that within a job spell, earnings are changing very little for most workers, while some experience large changes (e.g. a promotion or a temporary layoff). As explained in section 4.2, I calibrate the fraction of workers that experience wage changes (3% monthly) to evidence on wage rigidity. If all employed workers instead received an innovation to their skill level, and thus to their wage, in every period (i.e. π e = 1), then there would be no excess kurtosis for stayers. 17 After all, these shocks are assumed to be normally distributed. The results in table 2 indicate that using this calibration does indeed generate a considerable amount of excess kurtosis for stayers too, yet much less than what the data shows. Moreover, the picture changes very little when using the alternative six month - definition. Evidence on the distribution of hourly wage changes as opposed to earnings changes could shed light on that discrepancy. 15

16 1Y Changes 5Y Changes Aggregate Flows SD Skew Kurt SD Skew Kurt U-Rate U2E E2U J2J benchmark accept all jobs (AAJ) full search intensity (FSI) AAJ + FSI Table 3: Dependance of Model Moments on Choices vs. Economic Risk 5.3 Risk vs. Choice So far, I have been a bit sloppy in labeling all variability in earnings changes as risk, not distinguishing actual risk from predictable changes. Empirically, this distinction would require data on consumption, along with a model of consumption. Through the lens of the model, we can at least distinguish genuine earnings risk from earnings changes caused by choices. Table 3 compares the benchmark model output to counterfactual scenarios in which the agents choices are shut down. The second line, labeled AAJ, reports the results from a counterfactual simulation in which workers accept every job, i.e. reservation wages are below the wage offered by the least productive firm. Negative skewness decreases by 19% (short horizon), respectively 28% (long horizon), and kurtosis is hardly changing. Line three (FSI) reports results for a counterfactual scenario in which all employed as well as unemployed agents search at full intensity. This tends to increase measured earnings risk, as captured by the second, third and fourth standardized moment of earnings changes. Finally, combining both modifications (AAJ+FSI) does indeed significantly reduce the higher order moments. Skewness is less than half as large in absolute value for one year changes, almost completely vanishing for five year changes, and the kurtosis is only half as big. These simulation results illustrate that a major fraction of higher order earnings risk could be attributed to choices. 6 A Simplified Infinite-Horizon Framework for General Equilibrium In this section, I discuss a simplified infinite-horizon framework and embed the model in an incomplete markets general equilibrium framework in the spirit of Aiyagari [1994]. In order to use this framework in a general equilibrium analysis, a parsimonious state space is desirable. Which state variables are essential? To begin with, I keep risk aversion and an associated savings choice, simply because the implications of earnings risk can probably not be meaningfully assessed otherwise. Appendix A.1 discusses to what extent the distribution of earnings changes would differ with risk neutral workers. In an infinite horizon framework, it seems desirable to work with ex ante identical agents. Hence, I drop fixed human capital h. This leaves current skill z, firm productivity p and wealth a as state 17 However, the overall earnings change distribution (i.e. not conditioning on stayers/ switchers) does not change substantially in a re-calibrated version of the model with π e = 1. In that sense, the results are not sensitive to the choice of π e. 16

17 Moment Target Model Estimate U2E-rate.3.3 χ s =.27 (= χ e(1 + φ)) J2J-rate.3.3 χ se =.97 2y-earnings from U-shock π u =.22 Mean-Min ratio residual wages η = 3.84 Table 4: Calibrated Parameters in the Simplified Infinite-Horizon Framework variables. Firm heterogeneity is the very essence of a job ladder model and thus indispensable. Lastly, worker heterogeneity in the form of skill z is needed to fully account for the consequences of job loss as the discussion in section 4.3 and figure 4 illustrate. 6.1 Infinite-Horizon Framework Compared to the richer life-cycle model described in section 3, a few changes changes are in order: first, in the absence of fixed human capital the long-run mean of the skill process (3) is now a function of an employment indicator only, i.e. ze for the employed and zu for the unemployed with ze > zu. Second, there is no retirement period anymore at the end of the work life. Except for the omission of h and the time indices, the value functions (1) and (2) are unchanged. The externally set parameters coincide with the ones used in the life-cycle framework described in section 4. The internal calibration, reported in table 4, is simplified. The absence of a life-cycle and fixed human capital implies that σh 2, µ z and σz 2 do not have to be calibrated. Without ex ante worker heterogeneity, the overall earnings distribution is not matched anymore. Consequently, I do not target the evidence on top earnings shares, i.e. the Pareto firm productivity distribution has a uniform shape coefficient η. Analogously to η 1 in the life-cycle model, η is calibrated to match residual wage dispersion. How do these simplifications affect the performance of the model? Aside from the life-cycle dimension itself, which it cannot address anymore, to a moderate extent. Figures 1, 11 and 12, respectively, report the standard deviation, skewness and kurtosis of log earnings changes. The model-generated skewness is comparable to the one generated in the richer life-cycle framework both in terms of its level and as a function of recent earnings. Overall, the kurtosis is a bit lower and hence the fit to the data worse. This stems from the milder job ladder, i.e. from omitting the extra parameter η 2 < η 1 that regulated the (thicker) tail of the firm productivity distribution in the full model. Appendix C contains the histogram of earnings changes, illustrating the thinner tails in log-log scale. 6.2 General Equilibrium The model can be easily extended to general equilibrium. I outline it for the simplified infinite-horizon economy for simplicity. Consider a fixed population of firms, operating at constant returns to scale (CRTS). Firm heterogeneity is described by a primitive firm productivity distribution G p (p). Every period, a firm chooses the number of vacancies v, associated with a convex vacancy posting cost κ(v). 17

18 1.2 1Y Change 1.8 5Y Change 1 Model Data Standard Deviation Standard Deviation Figure 1: Standard Deviation of Log Earnings Changes (Infinite Horizon).5 1Y Change.5 5Y Change Model Data Skewness Skewness Figure 11: Skewness of Log Earnings Changes (Infinite Horizon) 18

19 1Y Change 13 5Y Change 25 Model Data Kurtosis 15 Kurtosis Figure 12: Kurtosis of Log Earnings Changes (Infinite Horizon) For a p-firm, the value of a job filled by a (a, z)-worker is given by J(a, z, p) =zp w(z, p) r (1 δ)e z,p a,z,p[l (a, z, p; p )J(a, z, p)], (6) where l ( ) is the endogenous continuation decision of the worker in the next period (which depends on the skill evolution and potentially on a new job offer p ). Optimal vacancy posting v (p) satisfies λ f E a,z,p [l (a, z, p; p )J(a, z, p)] = κ (v (p)), (7) where λ f is the match probability and the expectation is taken over job searchers (some of them are unemployed, some are employed at p -firms). The CRTS assumption is crucial as it implies that vacancy posting is not history-dependent, considerably simplifying the analysis. In addition, as the firm receives a positive fraction of output (1 γ) (, 1), it implies that the firm is happy to employ workers of any type. The endogeneity of match formation and destruction is purely coming from the worker side. Given an aggregate matching function M(S, V ), the per unit of search effort, respectively vacancies posted, matching probabilities for workers λ w and firms λ f follow. Here S is the effective mass of job searchers and V aggregate vacancies. We observe that a Pareto distributed primitive firm productivity distribution G p (p) and a power function for vacancy costs yield a Pareto-tailed job offer distribution F p (p), as assumed in the partial equilibrium framework. As any triple (λ w, λ f, F p ) can be rationalized by an appropriate choice of (M, G p ), only the equilibrium interest rate r has to be solved for in addition. The only asset in positive net supply is the portfolio of all firms, valued at the discounted present value of cash flows. 18 This risk-free asset is held by workers to insure against idiosyncratic earnings risk and the equilibrium r equates supply and demand in the asset market. 18 Note that adding physical capital would be straight-forward. 19

20 7 Conclusion This paper has proposed a framework that quantitatively addresses earnings risk, especially higher order risk that has not received much attention until recently. Its key element is firm heterogeneity and the corresponding dynamics of the job ladder, driving large and negatively skewed earnings changes. As a quantitative exercise, I calibrated the model to replicate several salient features of the U.S. labor market and the associated earnings distribution. The model is a quantitative success in the sense that it replicates key features of the distribution of earnings changes that were not targeted in the calibration. Those are, as documented by Guvenen et al. [215], in particular a large negative skewness and a very high excess kurtosis, both strikingly at odds with the log-normal framework that is commonly used to model earnings processes and thus earnings risk. I finished the exposition with an assessment of a stripped-down version of the model and an embedding in general equilibrium that can be readily used in applications. 2

21 A Robustness Checks and Importance of Modelling Assumptions In this section, I investigate to what extent risk aversion and endogenous search effort affect the results. Furthermore, I relax the complementarity assumption in the skill evolution process. A.1 Risk Neutral Workers How much does risk aversion drive the findings? To answer this question, I eliminate risk aversion and re-calibrate the simplified infinite-horizon model economy of section 6. Figure 13 reveals that with risk neutral workers, skewness would tend to be a little bit less negative, i.e. farther away from the empirical moments. Intuitively, with risk aversion and an associated savings opportunity there is a group of rich workers who, once separated from a relatively good job, will simply draw down their accumulated buffer stock savings in unemployment and consequently search relatively little. Consequently, it looks as if there is a higher downside risk (a misnomer to the extent that this is partly stemming from choices) and skewness is more negative. The explanation applies likewise to Figure 14, showing that with risk neutrality the kurtosis of log earnings changes is completely flat as a function of recent earnings. Note that the model economy is re-calibrated, so aggregate transition rates are the same in both environments. Thus, it is the heterogeneity in search effort and reservation wages that generates this effect. Furthermore, note that the risk-free savings choice implies that most workers are well ensured. Consequently, with hand-to-mouth consumers the discrepancy would be likely bigger..5 1Y Change.5 5Y Change Model (risk averse) Model (risk neutral) Data Skewness Skewness Figure 13: Effect of Risk Aversion: Skewness of Log Earnings Changes 21