Time-Varying Risk Premium and Unemployment Risk across Age Groups

Abstract

We show that time-varying risk premium in financial markets can explain a key, yet puzzling, feature of labor markets: the large differences in unemployment risk across worker age groups over the business cycle. Our search model features a time-varying risk premium and learning about unobserved heterogeneity in worker productivity. Their interaction generates large real effects through firms’ labor policies. Our model predicts higher unemployment risk of younger workers relative to prime-age workers when risk premium is high, and the employment ratio of prime-age to young workers to be more cyclical in high beta industries. We find empirical support for these predictions.

Authors have furnished an Internet Appendix, which is available on the Oxford University Press Web site next to the link to the final published paper online.

The risk premium in U.S. equity markets varies over time (Cochrane 2011). While much work has explored the causes of this time variation, less has explored its consequences for real economic activity. In this paper, we examine one particular consequence: the interaction between risk premium and unemployment across age groups. We show that time-varying risk premium in financial markets has a first-order effect on firms’ labor policies and helps us understand key patterns in the business-cycle dynamics of unemployment across age groups.

While it is well established that during recessions, the unemployment rate for young workers increases twice as fast as the corresponding rate for prime-age workers (see Clark and Summers 1981; Figure 1), it has been difficult to provide a quantitative explanation for this pattern. We show that accounting for time variation in the discount rate used by firms provides a potential resolution of this puzzle. Our model predicts the unemployment rate of younger workers relative to prime-age workers to be higher when the market risk premium is high. Across industries, our model predicts the ratio of employed prime-age workers to employed young workers to be more cyclical in high beta industries. We find empirical support for these predictions.

Figure 1

Quarterly unemployment rates for young (20- to 24-year-old) and prime-age (35- to 44-year-old) workers

Quarterly numbers are averages of deseasonalized monthly numbers. Panel A plots unemployment levels of young and prime-age workers, and panel B plots the cyclical component of the difference between these two unemployment rates (HP filtered with smoothing parameter 1,600). The gray bands are NBER recessions. Source: Current Population Survey constructed by the Bureau of Labor Statistics.

High countercyclical unemployment risk faced by young workers has large economic consequences. At the individual level, unemployment in the early stage of an individual’s working life has been shown to result in a large and long-lasting drop in lifetime earnings (Mroz and Savage 2006). Young workers also disproportionately contribute to the amplitude of aggregate output and unemployment fluctuations (Jaimovich and Siu 2009).¹ It is therefore important to understand the drivers of heightened unemployment risk among relatively younger workers, especially during recessions.

We provide a search-based theory to explain differences in unemployment risk across age groups. The presence of labor market search frictions provides a natural explanation for equilibrium unemployment. This is because it takes time for an unemployed worker to be matched to an employer who is looking to hire (see, e.g., Shimer 2010 for a review). Our model adds two ingredients to an otherwise standard equilibrium labor search and matching model in the spirit of Diamond-Mortensen-Pissarides (Diamond 1982; Mortensen and Pissarides 1994; Pissarides 1985).

First, our model features a time varying risk premium.² The quantitative importance of a time-varying risk premium for aggregate unemployment fluctuations, in a setting without worker heterogeneity, has been recently highlighted in Hall (2017) and Kilic and Wachter (2018). In contrast to these papers, our paper is the first to show that time-varying risk premium has a first-order effect on unemployment dynamics across the age distribution. Second, to capture this difference in unemployment risk between workers of different age groups, our model features firms learning about unobserved heterogeneity in worker productivity.³ This generates heterogeneity in unemployment risk across age groups. We quantify the effect of the interaction between time-varying risk premium and learning on firms’ labor policies. Our model’s predictions for the mean level and volatility of the unemployment rates of both young as well as prime-age workers match U.S. data.

Our quantitative model’s central insight is that variation in the risk premium affects firms’ hiring and firing policies over the business cycle, and that this variation disproportionately affects young workers relative to prime-age workers. This has implications for unemployment rate dynamics both in the time series (arising from differences in risk premium across expansions and recessions) and in the cross-section of industries (arising from differences in industries’ exposure to the aggregate market risk premium). Our model predicts the unemployment rate of young workers relative to prime-age workers to be higher when the aggregate market risk premium is high. At the industry-level, our model predicts the ratio of employed prime-age workers to employed young workers to be more cyclical in high (CAPM, or capital asset pricing model) beta industries (industries whose equity returns covary more strongly with the returns of the aggregate stock market).⁴ We provide the intuition for our results in two steps.

First, the equilibrium job separation rate for young workers is higher than that of prime-age workers. As noted by Jovanovic (1979), as firms learn about worker productivity, poor productivity workers are terminated while high productivity workers survive. As time passes and learning has been accomplished, the probability of a long-tenured worker being of low productivity decreases. As a result, the job separation hazard rate declines with tenure. Because tenure and age are positively correlated, younger workers experience a higher job separation rate than prime-age workers, which results in the former group having a higher unemployment rate. The end result is that relative to prime-age workers, younger workers are more likely to search for jobs and are therefore more exposed to changes in hiring incentives of firms.

Second, firms’ hiring incentives are negatively related to variation in the risk premium. This is because an increase (decrease) in the risk premium increases (decreases) the discount rate that firms use to compute the benefit, in present value terms, of hiring a worker. Because younger workers search for jobs more often, their employment outcomes are more exposed to variation in the risk premium both over time (between expansions and recessions) and across industries (arising from differences in industries’ exposure to the aggregate market risk premium).

We find support for our model’s predictions. Our time-series regression estimates imply that a 5th to 95th percentile increase in the market risk premium⁵ is associated with an increase of 4.8% (2.3%) in the unemployment rate of young (prime-age) workers. Our calibrated model’s predictions are in line with these estimates.

We base tests of our model’s cross-sectional predictions on the log ratio of prime-age to young workers employment. This ratio positively depends on the difference in unemployment rates between young and prime-age workers. Our cross-sectional regression estimates imply that when gross domestic product (GDP) declines from its 95th to the 5th percentile, the increase in the log employment ratio in a high beta industry (at the 95th percentile of the CAPM beta distribution) is 2.6 times that of a median beta industry.⁶ Our calibrated model captures 60% of this observed variation.

Besides capturing unemployment differences between workers of different age groups, our model predicts that the composition of the pool of employed workers is influenced by time variation in the risk premium. In particular, an increase in the risk premium results in an accelerated rate with which low productivity workers are fired. In such a scenario, the average labor productivity of the pool of employed workers (i.e., the measure of labor productivity according to the Bureau of Labor Statistics) increases, even while the aggregate unemployment rate is also rising. Although this mechanism dates back to at least Mitchell (1913, pp. 477-8), to the best of our knowledge, our paper is the first to show that the magnitude of this channel strongly depends on variation in the risk premium. Therefore, the interaction between time variation in the risk premium and the composition of the workforce can contribute to the low correlation between aggregate labor productivity and unemployment rates in the data (Barnichon 2010; Gali and van Rens 2017; McGrattan and Prescott 2012).

Our paper contributes to the literature which studies the effect of time-varying risk premium on macroeconomic real variables. Hall (2017) and Kilic and Wachter (2018) show that time-varying risk premium has large effects on the aggregate unemployment rate. Gourio (2012) shows that time-varying disaster risk has large effects on aggregate employment and investment, while Lettau and Ludvigson (2002) shows that time-varying risk premium affects aggregate investment spending over long horizons. While this literature has examined the effect of time-varying risk premium on aggregate employment and investment, to the best of our knowledge, our paper is the first to show its effect on differences in unemployment across age groups.

Jaimovich et al. (2013) explains the higher volatility of hours worked by young workers relative to prime-age workers in a real business-cycle model without unemployment. We focus on unemployment differences across these two groups because the “hours variation for all age groups is accounted for largely by movements in and out of employment” (Jaimovich et al. 2013, footnote 9).

Our model builds on Jovanovic (1979) and Moscarini (2005). These papers consider the effects of match-specific learning on worker turnover and unemployment. More recently, Gervais et al. (2016) provide a complementary analysis for workers’ learning about their occupational fit. In contrast to these papers, whose analysis are all for a steady-state setting, we consider the effect of aggregate shocks and therefore derive implications for changes in the unemployment rates of young and prime-age workers over the business cycle.

The paper closest to our cross-sectional result is Caggese et al. (2019), who find financially constrained firms fire short-tenured workers relatively more frequently than unconstrained firms. To the extent that financial constraints increase a firm’s cost of capital, their findings are in line with our prediction. Our results show, however, that differences in the systematic risk exposure of firms, even in an environment in which the assumptions of Modigliani and Miller (1958) hold, result in differences in unemployment risk exposure between young, inexperienced workers and more experienced prime-age workers.

The literature examines the effect of labor market frictions on equity returns. Recent examples include Belo et al. (2014, 2017), Donangelo (2014), Favilukis and Lin (2015), Kuehn et al. (2017), and Petrosky-Nadeau et al. (2018). Kilic (2017) studies the effect of the demographic composition of a firm’s workforce on its cost of capital. In contrast, we analyze the real effects of time variation in the cost of equity capital on (un)employment dynamics.

Empirical estimates show labor income risk to be strongly counter-cyclical and persistent, both at the household level (Storesletten et al. 2004) and at the individual level (Guvenen et al. 2014). Labor income risk consists of unemployment risk and earnings risk (for employed workers). Workers of different age groups differ in their exposure to both unemployment risk (Clark and Summers 1981) and earnings risk (Gomme et al. 2005; Guvenen et al. 2017; Jaimovich and Siu 2009; Rios-Rull 1996). Our model endogenously generates higher unemployment risk for younger workers, which has been shown to have large and long-lasting effects on individuals’ lifetime earnings (Mroz and Savage 2006).

1. The Model

In this section, we present a labor search model with unobserved match productivity and time-varying risk premium.

1.1 The economy

The economy is set in discrete time and over an infinite horizon. There is a representative household comprising a unit mass of ex ante identical workers and a large number of capitalists. Capitalists create vacancies, with the large number of capitalists ensuring free entry for vacancy creation. In addition, there is a government that levies lump-sum taxes to provide unemployment benefits, but does nothing else. Throughout our analysis, we maintain perfect risk-sharing between members of the representative household.⁷

1.1.1 Worker cohorts

Individual workers die and exit the workforce with probability |$\chi\in(0,1)$| each period. Death shocks are idiosyncratic across workers so that exactly |$\chi$| workers exit the workforce each period. A mass |$\chi$| of newly born workers replace exiting workers so that the mass of the overall workforce remains constant and equal to one. A cohort of workers entering in period |$c$| has population mass |$L_{c,t}=\chi(1-\chi)^{t-c}$| in period |$t\geq c$|⁠.

We show in the Online Appendix that workers’ idiosyncratic mortality risk does not play a role for the characterization of aggregate labor market policies;⁸ the reason being that idiosyncratic mortality risk is not priced when there is perfect household risk-sharing. Workers’ idiosyncratic mortality risk matters when it comes to accounting for differences in labor market outcomes between different cohorts of workers (see Section 1.3).

1.1.2 Labor market matching

Production occurs after vacancies, which are posted by capitalists, become successfully matched to workers. Labor markets are subject to search frictions so that it takes time to fill vacancies. In particular, a total of |$m(U,V)$| meetings take place between prospective workers and vacancies when |$U$| unemployed workers search for jobs and |$V$| vacancies are available. Following den Haan et al. (2000), we parameterize the matching function to be

$$\begin{equation} m\left(U,V\right) = \frac{UV} {\left(U^\iota+V^\iota\right)^{\frac{1}{\iota}}}, \label{eq:matching_function} \end{equation}$$

(1)

with |$\iota>0$|⁠.⁹ The contact rate between unemployed workers and vacancies depends on labor market tightness |$\Theta\equiv V/U$|⁠. In particular, the probability of an unemployed worker meeting a vacancy is given by |$f\left(\Theta\right)=m(U,V)/U=\left(1+\Theta^{-\iota}\right)^{-\frac{1}{\iota}}$|⁠, while the probability of a vacancy coming into contact with a prospective (unemployed) worker is |$g\left(\Theta\right)=m(U,V)/V=\left(1+\Theta^{\iota}\right)^{-\frac{1}{\iota}}$|⁠. The inverse relationship between job finding probabilities and the expected unemployment duration is given by |$1/f\left(\Theta\right)$|⁠. Similarly, the inverse relationship between vacancy filling probabilities and the expected duration for which a vacancy will go unfilled is given by |$1/g(\Theta)$|⁠.

The second labor market imperfection is that workers need not be matched to their ideal jobs or, equivalently, not all vacancies are filled by ideal candidates. In particular, matches can differ in their match-specific quality |$\nu_i\in\left\{H,L\right\}$|⁠, which can be of either high (⁠|$H$|⁠) or low (⁠|$L$|⁠) type. Once consummated, a match |$i$| produces output

$$\begin{equation} y_{it} = \exp\left(z_t+\mu(\nu_{i})-\frac{1}{2}\sigma^2 + \sigma\varepsilon_{it}\right) \label{eq:output} \end{equation}$$

(2)

in period |$t$|⁠, where aggregate productivity |$z_t$| is common across matches and follows an AR(1) process:

$$\begin{equation} z_t = \rho_z z_{t-1} + \sigma_z \varepsilon_{z,t}, \label{eq: AR(1) process z} \end{equation}$$

(3)

with autocorrelation |$\rho_z$|⁠, volatility |$\sigma_z$| and normally distributed innovations |$\varepsilon_{z,t}\sim\mathcal{N}(0,1)$|⁠. In addition, output |$y_{it}$| is also subject to a match-specific component |$\mu(\nu_i)$| that is higher when the match is of high quality (i.e., |$\mu(\nu_H)>\mu(\nu_L)$|⁠), as well as a match-specific shock |$\varepsilon_{it}\overset{iid}{\sim}\mathcal{N}(0,1)$| whose value is independent across time and across matches.

We assume that match quality |$\nu_{i}$| is not directly observed by either the firm or the worker. Instead, it must be inferred from realized match output |$y_{it}$| and aggregate productivity |$z_t$|⁠, both of which are observable. The match quality type |$\nu_{i}$| is never known with certainty as the inference problem is subject to noise from match-specific output shocks |$\varepsilon_{it}$|⁠, which are not observable. The initial match quality is drawn by nature with high quality matches occurring with probability

$$\begin{equation} p_0=\mathbb{P}(\nu_i=H). \label{eq: initial p0} \end{equation}$$

(4)

This prior |$p_0$| is common across all new matches (i.e., the distribution for |$p_0$| is a point mass).¹⁰ Subsequent inferences regarding match quality types are Bayesian in nature: time series observations for the aggregate productivity |$z_t$| and output |$y_{it}$| generate a filtration |$\mathscr{F}_{it}$|⁠. Parties then update their beliefs from the initial common prior |$p_{0}$| to form posterior beliefs |$p_{it}=\mathbb{P}\left(\nu_{i}=H\left|\mathscr{F}_{it}\right.\right)$| regarding the quality of the match (the precise updating rule will be described shortly, in Section 1.2). In turn, the presence of idiosyncratic productivity shocks lead to cross-sectional differences in beliefs |$p_{it}$| regarding the quality of different matches. In equilibrium, workers with low match quality beliefs earn lower wages and have higher chances of (endogenously) separating from their jobs (the details are deferred to Section 1.2). We discuss why learning about match-specific productivity provides a natural explanation for the observed differences in the business-cycle dynamics of the unemployment rates of young and prime-age workers in Section 1.5.

1.1.3 Match surplus and time-varying risk premium

The presence of search frictions creates a surplus that is split between workers and firms according to a generalized Nash bargaining rule in which workers have bargaining power |$\eta\in[0,1]$|⁠. Key in the determination of wages is the valuation of the surplus from a match. Because there is perfect risk-sharing between members of the representative household, idiosyncratic risks are not priced and both workers and capitalists are symmetric in their assessment of systematic risks. We assume that investors face complete asset markets for payoffs that depend on aggregate outcomes only. From the Fundamental Theorem of Asset Pricing (see Dybvig and Ross, 1987, 2003 for a review), the lack of arbitrage in asset markets implies the existence of a stochastic discount factor (SDF), |$\Lambda_{t,t+1}$|⁠. The SDF prices all asset returns, |$R_{t,t+1}$|⁠, according to the asset pricing relationship

$$\begin{equation} 1=\mathbb{E}_t\left[\Lambda_{t,t+1} R_{t,t+1}\right]. \label{eq: asset pricing equation} \end{equation}$$

(5)

Market completeness further implies that the SDF is unique and changes in the SDF are purely driven by aggregate shocks. We model the SDF according to

$$\begin{equation} \Lambda_{t,t+1}=\exp\left(-r_f - \frac{1}{2}x_t^2 -x_t\varepsilon_{z,t+1}\right),\label{eq:pricing_kernel} \end{equation}$$

(6)

where the risk-free rate |$r_f=-\log\left(\mathbb{E}_t\left[\Lambda_{t,t+1}\right]\right)$| is taken to be constant, and |$x_t$| is the market price of risk for aggregate productivity shocks |$\varepsilon_{z,t+1}$|⁠. The market price of risk, |$x_t$|⁠, is the compensation that investors receive for holding assets whose returns are perfectly correlated to the productivity shock |$\varepsilon_{z,t+1}$| (with unit exposure). That is, for assets whose log returns are of the form |$\log R_{t,t+1}=\mathbb{E}_t[\log R_{t,t+1}] + \varepsilon_{z,t+1}$|⁠, it follows from the asset pricing equation (5) that |$\mathbb{E}_t[\log R_{t,t+1}] - r_f - \frac{1}{2}Var_t(\log R_{t,t+1}) = - Cov_t\big(\log \Lambda_{t,t+1},\log R_{t,t+1}\big)$|⁠. From (6), this quantity is just the market price of risk, |$x_t$|⁠. We model the market price of risk as an AR(1) process:

$$\begin{equation} x_t = (1-\rho_x) \overline{x} + \rho_x x_{t-1} + \sigma_x \varepsilon_{x,t},\label{eq: AR(1) process x} \end{equation}$$

(7)

with mean |$\overline{x}$|⁠, autocorrelation |$\rho_x$| and volatility |$\sigma_x$|⁠. This specification for the market price of risk is also used in, for example, Brennan et al. (2004) and Lettau and Wachter (2007), and is closely related to the specifications used in affine term structure models (see, e.g., Singleton 2006). For simplicity, we take innovations to the market price of risk |$\varepsilon_{x,t}\sim \mathcal{N}(0,1)$| to be orthogonal to aggregate productivity innovations |$\varepsilon_{z,t}$| (alternatively, |$x_t$| can be thought of as the component of market price of risk that is orthogonal to aggregate productivity). We do not take a stance on the microfoundations behind the SDF (6), nor the drivers for a time-varying market price of risk (7). Rather, we take asset prices as given and instead focus on the implications of time-varying risk premium for labor market outcomes for different groups of workers.

The two aggregate shocks (3) and (7) together capture business-cycle conditions. The former captures cash flow variation over the business cycle, whereas the latter captures discount rate variation over the business cycle.

1.1.4 Timing of events

Finally, we complete the description of the economy by describing the timing of events within each period |$t$|⁠, which is as follows:

• (a) At the start of period |$t$|⁠, there is a mass of |$N_t\in[0,1]$| previously employed workers. These incumbent workers have match quality beliefs |$\left\{p_{it}\right\}_{i\in[0,N_t]}$|⁠.

• (b) Nature draws aggregate productivity |$z_t$| and the market price of risk |$x_t$| according to their respective laws of motion (3) and (7).

• (c) Capitalists post a total of |$V_t$| vacancies. The mass, |$U_t=1-N_t$|⁠, of unemployed workers searching for jobs are then matched to vacancies according to the matching function (1). The initial match type |$\nu_{i0}\in\left\{H,L\right\}$| is determined by nature according to the prior (4). Hiring and firing decisions are then made conditional on both the aggregate state and match quality beliefs. Wages are set according to a generalized Nash bargaining rule with workers capturing a fraction |$\eta$| of the surplus.

• (d) Output |$y_{it}$| is realized. Match quality beliefs are updated for the next period in a Bayesian manner.

• (e) Wages are paid and consumption takes place. Unemployed workers receive unemployment benefits, which are financed through lump sum taxation.

• (f) Matches exogenously separate with probability |$s$|⁠. Exogenous job separations are inclusive of worker exits.¹¹ Exiting workers are replaced with new workers who begin the next period being unemployed.

Figure 2 graphically summarizes the timing of events described above.

Figure 2

Timing of events within each period

1.2 Value of a match

A matched firm-worker pair makes two decisions immediately after the realization of the state variables |$z_t$| and |$x_t$|⁠: whether or not to continue with the match, and conditional on continuing, how to split the resultant match surplus. In addition, capitalists have a choice regarding whether or not to post vacancies. These decisions depend on the present value of a match, which is determined by the market price of risk and the expected productivity of a match. The latter crucially depend on learning about the (unobserved) quality of the match.

1.2.1 Bayesian learning

We define |$p_{it}$| to be the start of period belief of a high-quality match |$p_{it}=\mathbb{P}\left(\nu_{it}=H\left|\mathscr{F}_{it}\right.\right)$|⁠, where, because of our timing convention, the filtration |$\mathscr{F}_{it}$| used in the learning problem is generated from all past observations of output from the match, |$\left\{y_{is}\right\}_{s<t}$|⁠, as well as all past and present observations for the aggregate states |$\left\{z_s,x_s\right\}_{s\leq t}$|⁠. The expected output from a match, conditional on match quality belief |$p_{it}$| and aggregate productivity |$z_t$| is |$\mathbb{E}\left[y_{it}\left|p_{it},z_t\right.\right]=e^{z_t}\left[p_{it} e^{\mu(H)} + (1-p_{it})e^{\mu(L)}\right]$|⁠.

Match quality beliefs are updated according to |$p_{i,t+1}=\mathbb{P}\left(\nu_{i}=H\left|\mathscr{F}_{it},y_{it}\right.\right)=p^\prime(y_{it},p_{it},z_t)$|⁠, where the posterior is given by:

$$\begin{align} &p^\prime\left(y,p,z\right)\nonumber\\ &\quad{} = \frac{p e^{-\frac{1}{2\sigma^2}\left[\log y - \left(z+\mu(H)-\frac{1}{2}\sigma^2\right)\right]^2}} { p e^{-\frac{1}{2\sigma^2}\left[\log y - \left(z+\mu(H)-\frac{1}{2}\sigma^2\right)\right]^2} + (1-p) e^{-\frac{1}{2\sigma^2}\left[\log y - \left(z+\mu(L)-\frac{1}{2}\sigma^2\right)\right]^2} }.\label{eq:learning_posterior} \end{align}$$

(8)

Bayesian updating for the posterior match quality belief involves comparing realized output against expected output and making upward (downward) adjustments in beliefs in the event of a positive (negative) performance surprise.

1.2.2 Match value for a firm

The value of a firm-worker match from the perspective of a firm at the start of the period, after observing the aggregate states |$z_t=z$| and |$x_t=x$| but before observing the current period’s output, is given by:

$$\begin{equation} F(p,z,x) = \max\left\{0,d(p,z,x) + (1-s)\mathbb{E}\left[\Lambda(x,\varepsilon^\prime_z)F(p^\prime,z^\prime,x^\prime)\left|p,z,x\right.\right]\right\}\!, \label{eq:firm_value} \end{equation}$$

(9)

where |$p$| denotes the start of period match-quality belief. Firm value (9) reflects the option for the capitalist to dissolve the match, in which case the value will be worth zero. Should the capitalist continue with the match, he obtains expected dividends |$d(p,z,x)=e^{z}\left[pe^{\mu(H)}+(1-p)e^{\mu(L)}\right]-w(p,z,x)$|⁠, which is the difference between expected output and wages. Future cashflows accruing to the firm are discounted according to the SDF (6), and cashflow forecasts take into account Bayesian updating of match quality beliefs (8).

It can be shown that the continuation value of the firm is increasing in |$p$| (see Proposition 1 in Appendix B). This implies that firms follow a threshold strategy when deciding whether or not to continue with the match. In particular, matches with match quality belief above (below) a cutoff |$\underline{p}(z,x)$| are continued (dissolved), where the firing threshold |$\underline{p}(z,x)$| is characterized as the solution to the following indifference condition:

$$\begin{equation} 0 = d(\underline{p}(z,x),z,x) + (1-s)\mathbb{E}\left[\Lambda(x,\varepsilon_z^\prime)F(p^\prime,z^\prime,x^\prime)\left|\underline{p}(\omega),\omega\right.\right]. \label{eq:firing_threshold} \end{equation}$$

(10)

The threshold |$\underline{p}(z,x)$| is symmetric between firms and workers, meaning that workers will also find it more favorable to stick with the match (walk away) whenever |$p$| is greater (less than) the threshold |$\underline{p}(z,x)$|⁠.

1.2.3 Vacancy creation

Capitalists can freely post vacancies subject to a per unit vacancy creation cost of |$\kappa>0$|⁠. The value of a vacancy in state |$(z,x)$| is:

$$\begin{equation} F_V(z,x) = g(\Theta(z,x)) F(p_0,z,x) {\rm 1}\kern-0.24em{\rm I}\left(p_0\geq\underline{p}(z,x)\right)\label{eq:vacancy_value} \end{equation}$$

(11)

and takes into account the probability |$g(\Theta(z,x))$| of meeting a potential employee when the aggregate state is |$(z,x)$|⁠. The vacancy is worthless ex post if either it fails to be matched to a potential worker, or if the initial match quality |$p_0$|⁠, common across matched workers, is too low to clear the threshold |$\underline{p}(z,x)$|⁠.

A capitalist’s decision to post a vacancy depends on the value of a vacancy relative to the unit cost of posting a vacancy, and vacancies will be posted as long as the former remains greater. Because there is free entry, the equilibrium amount of vacancies posted, |$V$|⁠, is determined as the solution to the following complementary slackness problem:

$$\begin{equation} \kappa\geq F_V(z,x),\label{eq:vacancy_equilibrium_condition} \end{equation}$$

(12)

with equality if and only if total vacancies |$V$| is strictly positive.

1.2.4 Match value for a worker

The value function for an incumbent worker that is employed at the start of the period, |$J_e(p,z,x)$|⁠, is given by:

$$\begin{align} J_e(p,z,x) =& \max\left\{J_{eu}(z,x),w(p,z,x)+\right.\label{eq:worker_value_e}\\ & \left.\mathbb{E}\left[\Lambda(x,\varepsilon_z^\prime)\left[sJ_{u}(z^\prime,x^\prime) +(1-s)J_e(p^\prime,z^\prime,x^\prime) \right]\left|p,z,x\right.\right] \right\}\!.\nonumber \end{align}$$

(13)

An already-matched worker can either quit or stay with the match. In the latter case, the worker obtains wages |$w(p,z,x)$|⁠, and the match continues so long as the exogenous separation shock (which occurs with probability |$s$|⁠) does not materialize.

Recall that an incumbent match with match-quality belief |$p$| below cutoff |$\underline{p}(z,x)$| is dissolved. In this case, the worker becomes newly unemployed and has value function:

$$\begin{equation} J_{eu}(z,x) = b + \mathbb{E}\left[\Lambda(x,\varepsilon_z^\prime)J_{u}(z^\prime,x^\prime)\left|z,x\right.\right]. \label{eq:worker_value_eu} \end{equation}$$

(14)

That is, such a worker obtains unemployment benefit |$b$| in the current period and searches for new jobs starting from the next period. Note that our timing convention assumes that a newly unemployed worker searches for a new job starting from the next period.

The value for an already unemployed worker who is searching for a new job is given by

$$\begin{align} J_{u}(z,x) =& f(\Theta(z,x)) J_e(p_0,z,x){\rm 1}\kern-0.24em{\rm I}\left(p_0\geq \underline{p}(z,x)\right)\label{eq:worker_value_u}\\ &+ \left[1-f(\Theta(z,x))\right]J_{eu}(z,x).\nonumber \end{align}$$

(15)

With probability |$f(\Theta(z,x))$|⁠, the unemployed worker is matched to a vacancy, and the worker becomes employed so long as |$p_0$| clears the threshold |$\underline{p}(z,x)$|⁠. Otherwise, the worker remains unemployed in which case he obtains unemployment benefits |$b$| and continues searching for a job in the next period.

1.2.5 Wages

Wages are determined by standard Nash bargaining: the total match surplus |$S(p,z,x)\equiv J_e(p,z,x) - J_{eu}(z,x) + F(p,z,x)$| is split between the worker and the firm, with the worker surplus accounting for a share |$\eta\in[0,1]$| of the total match surplus. In particular, the worker obtains |$J_e(p,z,x) - J_{eu}(z,x)=\eta S(p,z,x)$|⁠, whereas the firm obtains |$F(p,z,x)=\eta S(p,z,x)$|⁠. This characterizes wages:

$$\begin{align} &w(p,z,x) \label{eq:wages_nash_bargaining}\\ &\quad{} = \eta \left\{ e^{z}\left[pe^{\mu(H)}+(1-p)e^{\mu(L)}\right] + (1-s)\mathbb{E}\left[\Lambda(x,\varepsilon_z^\prime)F(p^\prime,z^\prime,x^\prime)\left|p,z,x\right.\right] \right\} \nonumber \\ &\qquad - (1-\eta)\left\{ \mathbb{E}\left[\Lambda(x,\varepsilon_z^\prime)\left[sJ_u(z^\prime,x^\prime) +(1-s)J_e(p^\prime,z^\prime,x^\prime) \right]\left|p,z,x\right.\right] - J_u(z,x) \right\}\!. \nonumber \end{align}$$

(16)

1.2.6 Equilibrium

The notion of equilibrium for the economy is standard: all value functions must satisfy their respective Bellman equations (cf. Equations (9), (13), (14), and (15)), wages must be set according to the Nash bargaining rule (16), and labor market tightness must be determined according to the free entry condition (12). We provide the formal definitions and prove that an equilibrium exists in Appendix B, as well as provide a scheme for the numerical verification of equilibrium uniqueness. Appendix C details the numerical solution to our model.

1.3 Unemployment rates

1.3.1 Aggregate unemployment

The aggregate dynamics of the economy are determined by the start of period aggregate employment |$N_t\in[0,1]$|⁠, the start of the period match quality distribution for employed workers |$\mathcal{P}_t$|⁠, as well as the exogenous aggregate states |$z_t$| and |$x_t$|⁠. We view the match quality distribution as a probability measure with support on |$[0,1]$|⁠, so that |$\mathcal{P}_{t}(A)$| gives the fraction of employed workers with match quality beliefs lying within the set |$A\subset[0,1]$| at the beginning of period |$t$|⁠. We normalize the total mass of workers to one, so the aggregate unemployment rate is then |$U_t=1-N_t$| at the start of period |$t$|⁠. The law of motion for start of period employment, |$N_t$|⁠, and match quality distribution, |$\mathcal{P}_t$|⁠, is

$$\begin{align} N_{t+1}&=\Gamma_N\left(N_t, \mathcal{P}_t, z_t, x_t\right), \label{eq: aggregate N law of motion} \\ \end{align}$$

(17)

$$\begin{align} \mathcal{P}_{t+1}(A)&= \Gamma_{\mathcal{P}}\left(N_t,\mathcal{P}_t,z_t,x_t,A\right),\label{eq: aggregate P law of motion} \end{align}$$

(18)

where the mappings |$\Gamma_N$| and |$\Gamma_\mathcal{P}$| are described below.

The mapping |$\Gamma_N$| is defined as

$$\begin{align} \Gamma_N\left(N_t, \mathcal{P}_t, z_t, x_t\right) \equiv& (1-s)N_{end}\left(N_t, \mathcal{P}_t, z_t, x_t\right),\label{eq:mapping, law of motion N}\\ \end{align}$$

(19)

$$\begin{align} N_{end}\left(N_t, \mathcal{P}_t, z_t, x_t\right) \equiv& (1-N_t)f(\Theta(z_t,x_t)){\rm 1}\kern-0.24em{\rm I}\left(p_0\geq\underline{p}(z_t,x_t)\right) \label{eq: aggregate N end of period}\\ &+ N_t \mathcal{P}_t\left([\underline{p}(z_t,x_t),1]\right),\nonumber \end{align}$$

(20)

and reflects the following: at the start of period |$t$|⁠, |$N_t$| are employed with match quality distributed according to |$\mathcal{P}_t$|⁠. Of the initially employed workers, |$N_t\mathcal{P}_t\left([\underline{p}(z_t,x_t),1]\right)$| remain employed after endogenous separations take place. In addition, there are |$(1-N_t)f(\Theta(z_t,x_t)){\rm 1}\kern-0.24em{\rm I}\left(p_0\geq\underline{p}(z_t,x_t)\right)$| newly employed workers. The total number of employed workers after hiring and firing, |$N_{end,t}=N_{end}(N_t, \mathcal{P}_t, z_t, x_t)$|⁠, is then given by (20). Finally, a fraction |$s$| of employed workers separate for exogenous reasons at the end of period |$t$| so that there is a total of |$(1-s)N_{end,t}$| employed workers at the start of the next period |$t+1$|⁠.

The mapping |$\Gamma_{\mathcal{P}}$| is defined by

$$\begin{equation} \Gamma_{\mathcal{P}}\left(N_t,\mathcal{P}_t,z_t,x_t,A\right) \equiv \int_{\underline{p}(z_t,x_t)}^1\,\gamma_A(p,z_t)\, \widetilde{\mathcal{P}}\left(dp;N_t, \mathcal{P}_t, z_t, x_t\right),\label{eq:mapping, law_of_motion_P} \end{equation}$$

(21)

and updates the match quality distribution using the Law of Large Numbers in a two-step procedure.¹² First,

$$\begin{eqnarray} \gamma_A(p,z_t)& \equiv & p\mathbb{P}\left(p^\prime(e^{z_t+\mu(H)-\frac{1}{2}\sigma^2+\sigma \varepsilon_{it}},p,z_t) \in A\left|p,z_t\right.\right) \label{eq: updating rule fixed p} \\ & & + (1-p)\mathbb{P}\left(p^\prime(e^{z_t+\mu(L)-\frac{1}{2}\sigma^2+\sigma \varepsilon_{it}},p,z_t) \in A\left|p,z_t\right.\right),\nonumber \end{eqnarray}$$

(22)

gives the probability that an employed worker with match quality belief |$p$| at the start of period |$t$| end up having posterior belief in set |$A$| after output is observed at the end of period |$t$|⁠. The updating rule (22) reflects the Bayesian posterior (8) and the fact that a worker with match quality belief |$p$| has probability |$p$| of being a high type. The posteriors (22) are then averaged over all |$N_{end,t}$| workers that are employed at the end of the period. The distribution of match quality types among these workers has support |$\left[\underline{p}(z_t,x_t),1\right]$| and is given by

$$\begin{equation} \widetilde{\mathcal{P}}\left(dp;N_t, \mathcal{P}_t, z_t, x_t\right)=\frac{(1-N_t)f(\Theta(z_t,x_t))\delta_0\left(dp\right) + N_t \mathcal{P}_t\left(dp\right)} {N_{end}\left(N_t, \mathcal{P}_t, z_t, x_t\right)},\label{eq:distribution productive workers} \end{equation}$$

(23)

which is an average of match qualities among newly hired workers and incumbent workers hired during previous periods. Finally, note that exogenous separations apply evenly throughout the worker population, and therefore do not affect the evolution of the match quality distribution.

1.3.2 Unemployment rates across age groups

Consider a cohort of workers that enters the work force at the start of period |$t=c$|⁠. Workers begin their careers being unemployed and searching for jobs; the initial employment rate for cohort |$c$| is then

$$\begin{equation}\label{eq: cohort employment rate, initial condition} N_{c,t=c}=0. \end{equation}$$

(24)

Thereafter, the employment dynamics for cohort |$c$| are analogous to aggregate laws of motion (17) and (18), and is given by

$$\begin{eqnarray} N_{c,t+1} &=& \Gamma_N\left(N_{c,t},\mathcal{P}_{c,t},z_t,x_t\right), \\ \end{eqnarray}$$

(25)

$$\begin{eqnarray} \mathcal{P}_{c,t+1}(A) &=& \Gamma_\mathcal{P}\left(N_{c,t},\mathcal{P}_{c,t},z_t,x_t,A\right), \end{eqnarray}$$

(26)

for |$t\geq c$|⁠, where the laws of motion |$\Gamma_N$| and |$\Gamma_{\mathcal{P}}$| are given by (19) and (21), respectively.¹³

At a single point in time |$t$|⁠, differences in potential experience, |$e\equiv t-c$|⁠, give rise to employment outcomes across cohorts |$c$|⁠. Figure 3 illustrates this. Under learning-induced selection, better match quality beliefs are positively associated with longer tenures on the current job (panel A). Relative to younger cohorts, older cohorts have longer potential experience and more opportunities to select into high quality matches, and therefore have longer job tenures on average (panel B). As a result, match quality beliefs are higher on average for older cohorts with longer potential experience (panel C).

Figure 3

Match pool quality across workers

Panel A shows the cumulative distribution function of match quality beliefs as a function of (current) job tenure in the stochastic steady state. Panel B shows the average job tenure among currently employed workers for a given cohort of workers with the same potential experience. Panel C shows the cumulative distribution function of match quality beliefs as a function of potential experience in the stochastic steady state.

In Section 2, we compare our model implied cohort unemployment rates to the official unemployment rates as reported by the Current Population Survey (CPS). The CPS reports unemployment rates for various age buckets. To map our model to the data, we make the simplifying assumption that the initial working age is the same across all workers so that potential experience is perfectly correlated with age. The model implied unemployment rate for workers with potential experience |$e$| in the range |$[\underline{e},\overline{e}]$| is simply the population weighted average:

$$\begin{equation} U_{t}^{[\underline{e},\overline{e}]}=\sum_{t-c\in[\underline{e},\overline{e}]} \frac{L_{c,t}}{L_{t-\overline{e},t}+\dots + L_{t-\underline{e},t}}U_{c,t}, \label{eq: model implied group specific unemployment} \end{equation}$$

(27)

where |$L_{c,t}$| and |$U_{c,t}=1-N_{c,t}$| are the population size and unemployment rate of cohort |$c$|⁠, respectively. Note that age, per se, is not the primitive driver of unemployment risk in our model. Instead, a cohort’s age bucket reflects the average degree of workers’ match quality within that cohort.

1.4 Remaining quantities

A mass |$N_{end,t}=N_{end}\left(N_t, \mathcal{P}_t, z_t, x_t\right)$| of workers have match quality belief |$p$| above the separation threshold |$\underline{p}(z_t,x_t)$| and end up producing in period |$t$|⁠, where |$N_{end}$| is given by (20). Aggregate output,

$$\begin{equation} Y_t = N_{end,t}\int_{\underline{p}(z_t,x_t)}^1\,e^{z_t}\left[ p e^{\mu(H)} + (1-p) e^{\mu(L)}\right]\,\widetilde{\mathcal{P}}_t(dp), \label{eq: aggregate output} \end{equation}$$

(28)

is simply the sum of all individual output (2) across these workers, where the match quality distribution of producing workers, |$\widetilde{\mathcal{P}}_t(dp)=\widetilde{\mathcal{P}}\left(dp;N_t, \mathcal{P}_t, z_t, x_t\right)$|⁠, is given by expression (23). The Law of Large Numbers is implicit in the computation of aggregate output, so that idiosyncratic output shocks, |$\varepsilon_{it}$|⁠, are diversified away.

The total wage bill paid to the |$N_{end,t}=N_{end}(N_t,\mathcal{P}_t,z_t,x_t)$| producing workers is given by |$N_{end,t}\int_{\underline{p}(z_t,x_t)}^1\,w(p,z_t,x_t)\, \widetilde{\mathcal{P}}(dp,N_t,\mathcal{P}_t,z_t,x_t)$|⁠. The remaining unemployed workers receive unemployment benefits, which total |$b (1 - N_{end,t})$|⁠. Finally, a total of |$V_t=(1-N_t)\Theta(z_t,x_t)$| vacancies are created at the start of the period. This is determined by the aggregate unemployment rate, |$1-N_t$|⁠, and market tightness |$\Theta(z_t,x_t)$|⁠, all at the start of the period. The total amount of resources spent toward vacancy creation is given by |$\kappa V_t$|⁠.

We define the (end-of-period) aggregate stock market value |$S_t$| as a levered claim on all firm-worker matches:

$$\begin{equation}\label{eq: stock market definition} S_t \equiv \mathbb{E}_t\left[\Lambda_{t,t+1} Payoff_{t+1}^\lambda\right]\!, \end{equation}$$

(29)

where |$\lambda>1$| captures leverage (Abel 1999), and the unlevered payoff

$$\begin{equation*} Payoff_{t+1} = N_{end,t} \int_{\underline{p}(z_t,x_t)}^{1}\,(1-s)F(p,z_{t+1},x_{t+1})\,\mathcal{P}_{t+1}(dp) \end{equation*}$$

sums over the match value (9) of all surviving matches after taking into account the updated match quality beliefs (once the current period’s match output is observed). The (conditional) expected excess return is given by |$\mathbb{E}_t\left[Payoff_{t+1}^\lambda\right]/S_t-e^{r_f}$|⁠.

1.5 Model assumption and alternative modeling choices

1.5.1 Learning and (un)employment outcomes

Our general premise is that the productivity of a job applicant is not known precisely by a firm.¹⁴ It has to be learned. This assumption provides a natural micro-foundation for the empirically established negative relationship between tenure and job separation hazard rates (see Parsons 1972, Farber 1994, and the review article by Rubinstein and Weiss 2006, section 4.4). As we discuss below, capturing this negative relationship is of first order importance in generating realistic differences in unemployment rates between young and prime-age workers.

There are two leading learning-based explanations for the negative relationship between tenure and job separation hazard rates in the labor literature (see, e.g., Nagypal 2007 for a more extended discussion on this point): (a) firms and workers learning about match quality over time,¹⁵ and (b) learning-by-doing, in which workers become more valuable to the firm over time. Our focus on the former channel is backed up a structural estimation by Nagypal (2007). Using a matched employer-employee data set, she finds that learning about match quality is the dominant force at horizons longer than the first 6 months of an employment relationship.

In general, a firm is able to learn about a worker’s productivity both at the interview stage and after the worker is hired and starts working at the firm. In our baseline model, we assume that all learning occurs during a worker’s tenure at the firm, that is, nothing is learned from the resume or from the job interview. Put differently, this assumption implies that the resume and the interview leave a lot of uncertainty about a worker’s future productivity in the firm. Empirical evidence for our model’s premise that employers learn about worker productivity has been documented by a large literature starting with Farber and Gibbons (1996) and Altonji and Pierret (2001). The findings in Conley and Onder (2014) are also consistent with our assumption.

We extend our baseline model to test whether our model’s quantitative predictions change if we allow learning to start at the interview stage (e.g., from the resume and the actual interview) in addition to learning after hiring. This allows us to quantify how ex ante differences in observable productivity across workers at the time of the interview might affect our results. We find that our findings do not change (see Appendix A).

The above extension assumes that a firm’s belief about a job applicant’s future productivity in the firm is independent of her/his employment history. We do not analyze the implications of relaxing this assumption and leave this interesting extension for future research.¹⁶

1.5.2 Alternative modeling choices

In our baseline model, we assume that workers do not differ along observable dimensions and that learning about unobserved worker productivity occurs after hiring. These simplifying assumptions imply a common job finding rate between young and prime-age workers. We discuss how our model’s quantitative predictions might change once this assumption is relaxed. Several potential micro-foundations might generate differences in job finding rates between young and prime-age workers, so we keep our discussion general.

In reduced form, different candidate explanations imply differences in the job finding rate |$f$| and the job separation rate |$s$| of a group. These two quantities completely determine the unemployment rate dynamics of the group. We emphasize two natural choices when generating differences in unemployment rate between young and prime-age workers. The first is our choice, which assumes a common |$f$| but obtains differences in |$s$| across age groups. Under this modeling choice, younger workers have a higher unemployment rate than prime-age workers. Learning about match quality plays a crucial role: its implication of a negative relationship between tenure and job separation hazard rates generates a higher youth unemployment rate.¹⁷

The second choice of modeling is to assume a common |$s$| and have differences in |$f$| across age groups. This alternative choice of emphasis would counterfactually predict a higher prime-age unemployment rate compared to younger workers. This is because the job finding rate of younger workers is higher than that of prime-age workers in the data. Therefore, a model that features a higher job finding rate for young workers and a common job separation rate for the two groups would imply the above counterfactual result.

The only other option is to have a model with differences in both |$s$| and |$f$| across the two groups. Such a model would still have to feature a higher |$s$| for younger workers. Our model would therefore be a natural building block for any such extension. Moreover, as we show next, an extension of our model to additionally include differences in |$f$| across age groups would not generate significantly different quantitative estimates of the effect of aggregate shocks on the unemployment rates of young and prime-age workers.

To see this, note that the steady-state unemployment rate for a group equals |$s/(s+f)$|⁠, where |$s$| and |$f$| are the group-specific job separation and job finding rates. Because the group’s unemployment rate depends on the ratio |$s/f$| for that group only, by analyzing fluctuations around the steady state, we see that, to first order, the first two moments of the group’s unemployment rate is determined by the first two moments of the ratio |$s/f$|⁠. Because our model’s quantitative predictions for the mean and volatility of this ratio for young and for prime-age workers are already close to the data counterparts (see Table 3), an extended model incorporating differences in job finding rates between young and prime-age workers would not improve the estimates of the effect of aggregate shocks on unemployment rates by much. At best, such an extended model would obtain better agreement with the data on the moments of the job finding rate and the job separation rate separately.

Along the empirical front, we find it comforting that the point estimates of our cross-industry and time-series baseline results remain relatively unchanged when we control for observable differences, such as race, gender, and geography of employment (see Section 3 and our Online Appendix).¹⁸

1.6 Employment Risk across Industries

The logic of our model can be extended in a simple fashion to account for cross-sectional differences in workers’ (un)employment outcomes arising from differences in the systematic risk exposure of workers’ industry of employment. As only industry level employment is observed in the data, we rewrite our model’s predictions in terms of the cyclical properties of the log-ratio of the employment levels of prime-age to young workers in order to obtain testable implications. The cyclicality of the log employment ratio in an industry depends on the systematic risk exposure of that industry (as measured by the industry’s CAPM beta) through the risk premium channel. In particular, in a risk-neutral world without any systematic differences in discount rates across industries, the cyclicality of the log employment ratio would be independent of the CAPM beta of the industry.

We extend our model by viewing the aggregate economy as a collection of industries |$\left\{I\right\}$|⁠. A firm-worker match |$i$| in an industry |$I$| produces output |$y_{Iit} = \exp\left\{z_{I,t}+\mu(\nu_{Ii})-\frac{1}{2}\sigma^2 + \sigma\varepsilon_{Iit}\right\}$| in a fashion that is analogous to its aggregate counterpart (2). Industry specific productivity:

$$\begin{equation} z_{I,t} = \rho_z z_{I,t-1} + \sigma_{I}\left(\varrho_{I} \varepsilon_{z,t} + \sqrt{1 - \varrho_{I}^2}\varepsilon_{I,t}\right), \label{eq: industry productivity} \end{equation}$$

(30)

is subject to both the aggregate productivity shock |$\varepsilon_{z,t}$| as well as an industry-specific shock |$\varepsilon_{I,t}\sim\mathcal{N}(0,1)$|⁠. The industry productivity volatility is |$\sigma_I$|⁠, while the correlation between innovations to industry productivity and aggregate productivity is given by |$\varrho_I\in[-1,1]$|⁠. An industry’s unconditional market beta is defined as

$$\begin{equation}\label{eq: industry beta definition} \beta^{CAPM}_{I}=\mathbb{E}\left[\frac{Cov_t\left(\log \Lambda_{t,t+1}, \log R_{t,t+1}^I\right)}{Cov_t\left(\log \Lambda_{t,t+1}, \log R_{t,t+1}^{mkt}\right)}\right]. \end{equation}$$

(31)

In the above expression, the aggregate stock return |$R_{t,t+1}^{mkt}$| is based on definition (29) for the aggregate stock market. Similarly, |$R_{t,t+1}^I$| is the industry stock return which is based on an analogous definition for industry stock values. Higher values of |$\varrho_I$| correspond to high beta industries, because the SDF (6) only prices aggregate productivity shocks, |$\varepsilon_{z,t}$|⁠.

Under the simplifying assumption that workers’ skills are industry-specific (so that there is zero interindustry labor mobility), the resultant equilibrium characterization for each industry |$I$| is then analogous to that for the aggregate economy.

The quantity we focus on is the log-ratio of the number of employed prime-age to young workers defined as

$$\begin{equation}\label{eq:defempratio} r_{It}\equiv \log {{E^P_{It}} / E^Y_{It}}, \end{equation}$$

(32)

where the number of employed young and prime-age workers in industry |$I$| at time |$t$| are |$E^Y_{It}$| and |$E^P_{It}$|⁠, respectively. The number of employed workers in each group is related to our model-generated unemployment rate for that group through the relation

$$\begin{equation}\label{eq:ErelatetoU} E^a_{It} \equiv L^a_{It} (1-U^a_{It}) \,, \quad a\in\left\{Y,P\right\}\!, \, \end{equation}$$

(33)

where |$L^a_{It}$| denotes the size of the labor force belonging to demographic group |$a$| who are either working or seeking employment in industry |$I$| at time |$t$|⁠, that is, the supply of workers of demographic group |$a$| to industry |$I$| at time |$t$|⁠. The model-implied industry-specific employment rate is |$1-U^a_{It}$|⁠. Combining (33) and (32) we find

$$\begin{equation}\label{eq:participation} r_{It} = \log {{L^P_{It}} \over L^Y_{It}} + \log {{(1-U^P_{It})} \over {(1-U^Y_{It})}} \approx \log {{L^P_{It}} \over L^Y_{It}} + U^Y_{It} - U^O_{It}. \end{equation}$$

(34)

From (34), we see that to relate our model’s implications for unemployment differences to the ratio |$r_{It}$| in the data, we need to assume that the labor supply ratio |$\log \left({L^P_{It}} \left/ L^Y_{It}\right.\right)$| is acylical. To the best of our knowledge, no data are available to test this assumption directly. In Section 2.5, we quantitatively analyze the business-cycle dynamics of the employment ratio for industries which differ in their systematic risk exposure after making the above assumption. Our empirical analysis in Section 3.2 directly focuses on the (observable) employment rate ratio (32).

2. Quantitative Analysis

In this section, we analyze the quantitative implications of our model. We first describe our calibration procedure. Next we examine the unconditional and conditional predictions of our model and its impulse response properties to shocks. Then we examine our model’s cross-sectional predictions. Finally, we examine how aggregate shocks can lead to a change in the composition of the pool of employed workers and its implications. Appendix C describes our solution approach in detail.

2.1 Calibration

We calibrate our model by targeting the standard set of labor market moments described below. We simulate our model at monthly frequency. We use the parameters shown in Table 1 for our simulation experiments. We choose the persistence |$\rho_z = 0.9$| and volatility |$\sigma_z = 0.01$| of the labor productivity process (3) to match the persistence and volatility of the detrended, quarterly series for nonfarm business real output per person reported by the Bureau of Labor Statistics (BLS) over the period 1950-2016 (we use an HP filter with a bandwidth of 1,600). We directly use values from Lettau and Wachter (2007) for the market price of risk process (7). In particular, we set |$\overline{x}=0.1804$| (or |$\sqrt{12}\times0.1804=0.625$| on an annualized basis). The corresponding maximum Sharpe ratio |$\sigma_t(M_{t,t+1})/\mathbb{E}_t\left[M_{t,t+1}\right]=\sqrt{e^{x_t^2}-1}$| is then 0.7 on an annualized basis at this value. We set the monthly persistence to |$\rho_x = 0.9885$| and volatility to |$\sigma_x = 0.0693$|⁠. These correspond to an annualized autocorrelation of |$0.9885^{12}=0.87$|⁠, and an annualized volatility of |$\sqrt{12}\times 0.0693=0.24$|⁠. We choose the risk-free rate |$r_f = 0.0017$| (annual rate of |$2\%$|⁠) to match the data counterpart.

Table 1

Calibration, parameter values

Parameter	Symbol	Model
AR(1) coefficient of labor productivity	|$\rho_z$|	0.900
Volatility of labor productivity	|$\sigma_z$|	0.010
Unconditional mean of discount rate process	|$\overline{x}$|	0.1804
AR(1) coefficient of discount rate process	|$\rho_x$|	0.9885
Volatility of log-discount rate process	|$\sigma_x$|	0.0693
Risk-free rate	|$r_f$|	0.0017
log-productivity of |$H$| type	|$\mu(H)$|	0.87
Match specific output volatility	|$\sigma$|	2.182
Workers’ bargaining power	|$\eta$|	0.1
Cost of vacancy creation	|$\kappa$|	3.5
Curvature of matching function	|$\iota$|	3.200
Initial prior for match quality belief	|$p_0$|	0.295
Exogenous separation probability	|$s$|	0.004
Unemployment benefit parameter	|$b$|	1.9

We simulate our model at a monthly frequency using the parameters shown in the table above.

The four learning parameters are: the match-specific productivity parameters which take on a high and a low value denoted by |$\mu_H$| and |$\mu_L$|⁠, respectively, the parameter |$\sigma$|⁠, which measures the informativeness of individual output about match quality, and |$p_0$|⁠, which measures the prior belief about initial match quality.¹⁹ We normalize the scale of match-specific component of productivity to be 1 for a new hire, that is, |$p_0 e^{\mu(H)} + (1-p_0) e^{\mu(L)} = 1$|⁠. We choose the difference between |$\mu(H)$| and |$\mu(L)$| to target the cross-sectional dispersion in plant-level total factor productivity of |$1.92$| measured by Syverson (2004), under the assumption that |$\mu(H) = -\mu(L)$|⁠. Finally, we choose |$\sigma$| to target the expected tenure of a new match which is estimated by the Bureau of Labor Statistics (BLS) to be |$52$| months.²⁰ We obtain |$\mu(H) = 0.87$|⁠, |$\sigma = 2.182$| and |$p_0 = 0.295$|⁠. In the absence of separations, these learning parameters imply that it takes 30 months for 95% of low productivity matches to have posteriors below 0.05 (i.e., |$\mathbb{P}\left(p_{i,30}\leq 0.05\left|L\right.\right)\geq 0.95$|⁠), and 39 months for 95% of high productivity matches to have posteriors above 0.95 (i.e., |$\mathbb{P}\left(p_{i,39}\geq0.95\left|H\right.\right)\geq0.95$|⁠). Finally, we include the possibility of matches dissolving for reasons other than those we consider here. We set the probability of these exogenous separations, |$s=0.004$| per month, so that the model-implied aggregate job separation rate matches that of the data.

We follow the labor search literature and choose the values of the remaining parameters |$\kappa$|⁠, |$\iota$|⁠, |$b$|⁠, and |$\eta$| to target the first two moments of unemployment and vacancies, and the elasticity of wages to productivity. We choose |$b=1.9$| which implies a wage replacement ratio (unemployment benefit normalized by mean wages) of |$0.89$|⁠. Estimates of the wage replacement ratio vary widely in the literature—between 0.4 (Shimer 2005) and 0.955 (Hagedorn and Manovskii 2008). Our wage replacement ratio is close to the estimate of 0.88 obtained by Christiano et al. (2016), as well as to the estimate of 0.85 obtained by Rudanko (2011). We choose the curvature parameter of the matching function |$\iota=3.2$|⁠, and the cost of vacancies |$\kappa = 3.5$|⁠. Table 2 shows the results from our simulations and the data counterparts. The average unemployment rate in the data (model) are 5.6% (5.6%), and the volatility of this rate in the data (model) are 0.75% (0.73%). The mean labor market tightness as reported by the Federal Reserve Economic Data (FRED) using JOLTS data between 2001 and 2017 in the data (model) are 0.54 (0.59), and the volatility of this series in the data (model) are 0.092 (0.102). We set the bargaining power of workers to |$\eta = 0.1$|⁠. Our model implied wage elasticity with respect to labor productivity is |$0.44$|⁠, which is close to the empirical estimate of |$0.45$| reported by Shimer (2005).

Table 2

Calibration, aggregate labor market moments.

Moment	Data	Model
Labor market tightness:
Mean	0.54	0.59
Volatility	0.092	0.102
Autocorrelation	0.94	0.93
Aggregate unemployment:
Mean (%)	5.6	5.6
Volatility (%)	0.75	0.73
Autocorrelation	0.94	0.95
Correlation (unemployment, market tightness)	−0.89	−0.95
Job finding rate	0.51	0.56
Aggregate job separation rate	0.033	0.032
Expected tenure at entry (months)	52	53.5
Elasticity of wages to productivity	0.45	0.44
Dispersion of plant output: |$90th$| percentile/|$10th$| percentile	1.92	1.94

Labor market moments in the model and in the data are quarterly averages of monthly series. The unemployment data come from the Current Population Survey (CPS) and covers the period 1951Q1—2016Q4. Market tightness is from the JOLTS series between 2001 and 2017. Job finding and separation rates are computed following Elsby et al. (2009) for the period 1977Q1—2016Q4. Volatilities are computed based on deviations from an HP trend with smoothing parameter 1,600.

2.2 Unconditional moments

2.2.1 Unemployment rates

In the previous section, we calibrated our model to target aggregate labor market moments. We now highlight our model’s predictions for unemployment rates across age groups. Our model does a good job in matching unconditional moments and the conditional dynamics of unemployment rates of young and prime-age workers in the data. The BLS reports unemployment levels for workers in different age buckets. We choose workers between 20 and 24 years to correspond to young workers, and those between 35 and 44 years to correspond to prime-age workers. In simulations, we define young workers as those with between |$1$| and |$5$| years of potential experience, while prime-age workers have between |$15$| and |$25$| years of potential experience. These definitions are similar to the BLS age groups under the assumption that workers enter the labor force at age |$20$|⁠. We then report model-implied unemployment rates for young and prime-age workers according to definition (27).

Table 3 shows the unconditional model-implied moments for the unemployment rates of young and prime-age workers. The mean unemployment rate of young workers is |$9.8\%$| in the data and |$10.3\%$| in our model. This rate is much lower for prime-age workers: |$4.6\%$| in the data, and |$5.6\%$| our model. This is due to a higher rate of job loss of younger workers. The latter is because the likelihood that a young worker’s current job is of low productivity is high, because young workers have had fewer attempts to be matched to a more productive job. As bad matches dissolve, the average match quality of survivors improves as shown in panel C of Figure 3. In this figure, we compare the distribution of posterior beliefs for workers with 1 year of potential experience to that of workers with |$15$| years of potential experience. The match quality belief of the median young worker with 1 year of potential experience is |$0.59$|⁠. In comparison, this value is much higher at |$0.97$| for the median worker with |$15$| years of potential experience. Panel C of Figure 4 shows hazard rates as a function of potential experience in the stochastic steady state. The hazard rate declines as poor matches are dissolved early on during the tenure of employment. According to our calibrated model, the first 5 years of experience are critical in reducing unemployment risk; additional experience does not make much of a difference.

Table 3

Unconditional moments of unemployment rates of young and prime-age workers

	Young			Prime-age
	Data	Model		Data	Model
Unemployment rate: Mean (%)	9.8	10.3		4.6	5.6
Unemployment rate: SD (%)	1.2	1.5		0.7	0.7
Job finding rate |$f$|⁠: Mean	0.58	0.56		0.43	0.56
Job finding rate |$f$|⁠: SD	0.061	0.080		0.050	0.080
Job separation rate |$s$|⁠: Mean	0.065	0.061		0.020	0.032
Job separation rate |$s$|⁠: SD	0.0037	0.0015		0.0014	0.0007
|$s/f$|⁠, Mean	0.117	0.112		0.050	0.059
|$s/f$|⁠, SD	0.0135	0.0179		0.0077	0.0086

Moments in the data are calculated from the unemployment rates for young (20–24 years old) and prime-age (35–44 years old) and comes from the Current Population Survey (CPS). All raw series are deseasonalized. Job finding and separation rates are computed following Elsby et al. (2009) for the period 1977Q1—2016Q4. The model is simulated at a monthly frequency. Quarterly numbers are computed by averaging the monthly series, both in the data and in the simulations. Volatilities are computed based on deviations from an HP trend with smoothing parameter 1,600.

Figure 4

Job finding rates and separation rates

Panel A shows the job finding rates as a function of |$z$| for three different levels of the market price of risk. Labor productivity (⁠|$z$|⁠) is standardized by the unconditional volatility of the AR(1) process, |$\sigma_z/\sqrt{1-\rho_z^2}$|⁠. Panel B shows the threshold |$\underline{p}$| at which matches separate. Panel C shows the within-cohort average 1 year-ahead match termination hazard rate in the stochastic steady state. Hazard rates are expressed as a monthly rate and is inclusive of both exogenous and endogenous separations.

The unemployment rate of younger workers is much more sensitive to variation in macroeconomic conditions than that of prime-age workers in our model. Therefore, the youth unemployment rate is much more volatile than that of prime-age workers. Overall, our model matches the unconditional volatilities of unemployment rates quite well. In the data, the volatility of the youth unemployment rate is |$1.2\%$|⁠; it is |$1.5\%$| in our simulations. In comparison, the volatility of the unemployment rate of prime-age workers is |$0.7\%$|⁠, both in the data and in our model. This follows from the arguments in Section 1.5, which show that, to leading order, the first two moments of the ratio of job separation rate to job finding rate determine the mean and volatility of unemployment rates. From the last two rows of Table 3, we see that our model is able to closely match the mean and volatility of this ratio for both groups of workers. In particular, the volatility of this ratio for young workers is roughly twice that of prime-age workers, both in our model and in the data.

The intuition for this result is as follows. When the economy transitions into a recession, the job finding probability declines for all workers, as shown in panel A of Figure 4.²¹ However, because the unemployment rate of younger workers is higher than that of prime-age workers, the former group is more exposed to fluctuations in the job finding probability because of fluctuations in firms’ hiring incentives.

2.2.2 Wages

Figure 5 shows wages in our model. Panel A plots the wage policy (16) against match quality beliefs for various combinations of productivity and market price of risk. We see that wages are increasing in match quality beliefs and are higher in states with higher productivity or lower market price of risk. In addition, wage policies are approximately affine in match quality beliefs; this follows from expected match output being affine in match quality beliefs. Panel B plots the wage distribution against current job tenure. Our model is able to capture the positive relationship between wages and job tenure found in the data (Topel 1991). Panel C plots the wage distribution as a function of potential experience. Our model is able to capture the positive relationship between wages and potential experience found in the data (see, e.g., Card 1999, figure 1).

Figure 5

Wage distribution

Panel A plots the wage policy (16) against match quality beliefs for various combinations of productivity |$z$| and market price of risk |$x$|⁠. In all plots, wages are in units of its median value. High (low) values for |$z$| and |$x$| refer to values corresponding to plus (minus) 2 standard deviations of their respective unconditional distributions. Panels B and C plot the wage distribution as a function of (current) job tenure and potential experience, respectively. Both plots are for the stochastic steady state.

Our model shares the difficulty of matching the large variation in earnings observed in the data with other current models.²² In our model, the wage elasticity with respect to productivity is similar across workers of different age groups, with wage elasticities across all age groups being approximately equal to the aggregate wage elasticity of 0.44.²³ This follows from wage policies being approximately affine in match quality beliefs (cf. panel A of Figure 5). Our model is therefore unable to capture the relatively larger wage elasticity of young workers observed in the data (Guvenen et al. 2017; Jaimovich et al. 2013). The magnitude of the cross-sectional wage growth dispersion falls short of what is observed in the data (see, e.g., Guvenen et al. 2019). The reason for this is that we calibrated our model to target the cross-sectional dispersion in productivity and not wages. The dispersion in productivity determines wage dispersion after taking workers’ bargaining power in the wage bargaining process into account. We calibrated workers’ bargaining power to target aggregate wage rigidities in the time series, which resulted in a low value |$\eta=0.1$|⁠. As a result, the cross-sectional dispersion in productivity is greatly dampened for the purposes of capturing wage dispersion.

2.2.3 Stock returns

Following Abel (1999), we report moments for aggregate stock market returns based on a leverage parameter of |$\lambda=2.74$| in expression (29). The excess stock market return has a mean of 1.58% per annum, with a standard deviation of 5.18% per annum. These values are roughly between one-third and one-quarter of their empirical counterparts. The reason for this is cash flows shocks in our model are calibrated to labor productivity shocks. The latter’s volatility is about a third of the typical dividend volatility values used in calibrated asset pricing models (see, e.g., Campbell and Cochrane 1999; Bansal and Yaron 2004). The unconditional Sharpe ratio is 0.31 per annum, while the mean conditional Sharpe ratio is 0.5 per annum; these values are very much in line with those from the asset pricing literature.

2.3 Unemployment rate sensitivities

This section’s key results are (1) unemployment rates for both young and prime-age workers are more sensitive to market price of risk (⁠|$x$|⁠) shocks compared to productivity (⁠|$z$|⁠) shocks, and (2) the sensitivity of youth unemployment rates to market price of risk shocks is 2 times that of prime-age workers. We report in Section 3.3 that these predictions are supported in the data.

Figure 6 plots the unemployment rates for young and prime-age workers conditional on labor productivity |$z$| and the market price of risk |$x$|⁠. We compute the unemployment rate for each group of workers conditional on |$z$| and |$x$| as follows. We start the economy in the stochastic steady state and simulate many future paths. The unemployment rate reported in the figures is the unemployment rate for each of the two groups averaged over dates on which the economy ends up in the specified |$z$| and |$x$| states. Panel A shows the unemployment rates for young (Y) and prime-age (P) workers conditional on aggregate productivity, |$\mathbb{E}[U^{Y}_t\left|z_t=z\right]$| and |$\mathbb{E}[U^{P}_t\left|z_t=z\right]$|⁠. Panel B shows the young, |$\mathbb{E}[U^{Y}_t\left|x_t=x\right]$|⁠, and prime-age, |$\mathbb{E}[U^{P}_t\left|x_t=x\right]$|⁠, unemployment rates conditional on the market price of risk. Panel C shows the young minus prime-age unemployment difference conditional on productivity, |$\mathbb{E}[U^{Y}_t-U^{P}_t\left|z_t=z\right]$|⁠, as well as on the market price of risk, |$\mathbb{E}[U^{Y}_t-U^{P}_t\left|x_t=x\right]$|⁠. These plots allow us to compare the model implied sensitivities of the unemployment rates for each group to |$z$| and |$x$| shocks.

Figure 6

Unemployment rate sensitivity to fundamentals

Panel A (B) shows the unemployment rates for young and prime-age workers conditional on aggregate productivity (market price of risk). Panel C shows the young minus prime-age unemployment rate difference conditional on productivity and the market price of risk. Productivity and market price of risk states are standardized in units of their respective unconditional standard deviations. Unemployment rates are at the start of each period, before that period’s hiring and firing decisions have been made.

From Figure 6, we see that, relative to prime-age unemployment, the unemployment rate of young workers is more sensitive to both variation in labor productivity and the market price of risk. We see in panel A that a decline in labor productivity from its |$95$|th (1.64 standardized units) to |$5$|th percentile (-1.64 standardized units) results in an increase in youth (prime-age) unemployment rate of |$2\%$| (⁠|$1\%$|⁠). In comparison, panel B shows that an increase in the market price of risk from its |$5$|th to |$95$|th percentile results in a much larger increase in the youth (prime-age) unemployment rate of |$4.3\%$| (⁠|$2.1\%$|⁠). Table 4 compares these model-implied unemployment sensitivities to the data. The corresponding estimates for the response of unemployment rates to changes in labor productivity in the data are 1.1% for young workers and 0.5% for prime-age workers. The response to changes in the risk premium in the data are much larger—4.8% for young workers and 2.3% for prime-age workers.²⁴ For both shocks, we see that the sensitivity of youth unemployment rates with respect to each of the two shocks is about 2 times that of prime-age unemployment rates, which is in agreement with our model. The intuition for the higher sensitivity of the unemployment rate of relatively younger workers is the same as the one discussed for higher unemployment rate volatility of these workers in Section 2.2. Relative to prime-age workers, younger workers have higher rates of unemployment and therefore search for jobs more often. As a result, younger workers are more exposed to fluctuations in firms’ hiring incentives.

Table 4

Unemployment rate sensitivities for young and prime-age workers

	Young			Prime-age
	Data	Model		Data	Model
Unemp. increase (%),	4.8	4.3		2.3	2.1
5th to 95th pctl increase in |$x$|
Unemp. increase (%),	1.1	2		0.5	1
95th to 5th pctl decrease in |$z$|

Data and model-implied sensitives of young and prime-age unemployment rates to labor productivity (⁠|$z$|⁠) and market price of risk (⁠|$x$|⁠) shocks. Unemployment rate sensitivities in the data refer to estimates implied by regression results reported in Table 8.

Panel C of Figure 6 compares the model-implied sensitivity of the youth minus prime-age unemployment difference to changes in productivity and the market price of risk. We find the magnitude of the response in the youth minus prime-age unemployment rate difference to be larger for changes in the market price of risk. For example, a 95th to 5th percentile decline in labor productivity results in a 1% increase in the youth minus prime-age unemployment difference. In contrast, a 5th to 95th percentile increase in the market price of risk results in a larger 2.2% increase in the youth minus prime-age unemployment difference. In Section 3.3, we find the unemployment difference to be similarly more sensitive to changes in risk premia in the data.

2.4 Impulse response

In this subsection, we examine the impulse responses of young and prime-age unemployment rates to shocks to labor productivity and the market price of risk. We show that (1) young and prime-age unemployment rates respond more strongly to market price of risk shocks relative to labor productivity shocks, and (2) relative to prime-age workers, the response of youth unemployment rates to market price of risk shocks is twice as large.

We start the economy at |$t=0$| with both |$z$| and |$x$| at their mean levels and with the distribution of match quality beliefs at the stochastic steady state. At |$t=1$|⁠, we shock the economy by an unconditional 2-standard-deviation drop in labor productivity |$z$| (shown by the solid line in panels A through C of Figure 7), an unconditional 2-standard-deviation increase in the market price of risk |$x$| (dashed line), or a combination of 2-standard-deviation shock to |$z$| and |$x$| simultaneously (dash-dotted line).

Figure 7

Response of unemployment

Panels A through C, respectively, plot the response of youth unemployment, prime-age unemployment, and their difference corresponding to a negative 2-standard-deviation shock in labor productivity |$z$| (solid line), a 2-standard-deviation shock in the market price of risk |$x$| (dashed line), and a combination of labor productivity and risk premium shocks (dash-dotted line). In all cases, the economy starts off at the steady state in which both labor productivity |$z$| and the market price of risk |$x$| are both at their mean values.

Comparing panels A, B, and C of Figure 7, we see that the unemployment response for each age group is larger for the market price of risk shock. A 2-standard-deviation decline in labor productivity leads to a 1.2% (0.6%) increase in youth (prime-age) unemployment rates, while a 2-standard-deviation increase in the market price of risk leads to a larger increase of 1.9% (0.9%) in youth (prime-age) unemployment rates. The size of these responses also indicate that, relative to the prime-age unemployment rate, the youth unemployment rate is about twice as sensitive to market price of risk shocks. Section 3.3 reports similar findings in the data.

Finally, there is an additional interaction effect between labor productivity and the market price of risk shocks for unemployment outcomes. For instance, we see from the dash-dotted line in panel A of Figure 7 that when both labor productivity and the market price of risk shocks arrive simultaneously, the magnitude of the resultant increase in youth unemployment is about |$3.4\%$|⁠, which is 0.3% larger than the sum of the resultant youth unemployment rate outcomes from the two pure shocks alone.

2.5 Employment risk across industries

In this subsection, we investigate our model’s extension mentioned in Section 1.6 and examine how the difference in employment risk between young and prime-age workers depends on the systematic risk exposure of the industry in which these workers are employed.

For this exercise, we compare the log-ratio of the number of employed prime-age to young workers (32) in a low beta industry against that of a high beta industry. Both industries have labor productivity processes given by (30). They have identical persistence |$\rho_I=0.9$|⁠, which is equal to that of the aggregate labor productivity process. They also have equal productivity volatilities, which we set to |$\sigma_I=0.016$| based on the average of industry labor productivity volatilities for the sample of industries considered in Section 3.²⁵ We set |$\varrho_I=0.4$| for the low beta industry and |$\varrho_I=0.86$| for the high beta industry. The corresponding model-implied unconditional CAPM betas (31) are then 0.64 and 1.39, respectively, for the low and high beta industries. These values correspond to the 5th and 95th percentiles of CAPM betas for our sample of industries (see Table 5). The remaining parameters remain unchanged from our baseline calibration (Table 1). As a benchmark, we also consider the unit beta industry which has |$\varrho_I=0.62$|⁠.

Table 5

Summary statistics for cross-sectional analysis.

	Mean	SD	5%	25%	75%	95%	Obs
log GDP (percent)	0	1.16	−2.00	−0.59	0.86	1.82	80
CAPM |$\beta^{\rm{CAPM}}_I$|	1.04	0.27	0.64	0.81	1.28	1.39	32
Labor productivity |$\beta^{\rm{LP}}_I$|	0.45	0.77	−0.95	0.02	1.00	1.72	32
|$r_I$|	1.92	0.67	0.31	1.70	2.29	2.76	2560

The data are quarterly from 1997Q1 to 2016Q4. Log GDP is HP filtered with a bandwidth of 1,600. CAPM betas |$\beta^{\rm{CAPM}}_I$| are estimated using returns from CRSP, while Labor Productivity betas |$\beta^{\rm{LP}}_I$| are estimated using industry labor productivity from the BLS industry-level labor productivity tables. |$r_I$| is the log-ratio of the total number of prime-age employees to young employees in an industry in each quarter.

Figure 8 plots the log prime-age-to-young-worker-employment ratio (32) as a function of the aggregate productivity of the economy, |$\mathbb{E}[r_{It}\left| z_t=z\right.]$|⁠. The plot normalizes |$\log\left(L^P_I\left/L^Y_I\right.\right)\equiv0$|⁠, because age demographics remain constant in our model. We average over all possible values of industry productivity |$z_I$| and the market price of risk |$x$|⁠. The solid line is for the low beta industry, the dotted line is for the unit beta industry, and the dash-dotted line is for the high beta industry. We see that when aggregate productivity declines from a value that is 2 standard deviations above its mean value to two standard deviations below, the log employment ratio increases by 1.46% for the unit beta industry. In contrast, the same change in aggregate conditions results in an increase in the log employment ratio of 0.84% for the low beta industry and 2.26% for the high beta industry. The boom-to-recession increase in the log employment ratio in the high beta industry is therefore 1.6 times that that of the unit beta industry. This model-implied value captures 60% of the corresponding ratio in the data, which is 2.6 times.

Figure 8

Employment ratio sensitivity to systematic industry risk

This figure plots the conditional log-ratio of the number of employed prime-age to young workers (32) as a function of the aggregate productivity state. The age demographic ratio is normalized to |$\log\left(L^P_I\left/L^Y_I\right.\right)\equiv0$|⁠.

The intuition for our cross-sectional result is as follows. A high beta industry is more exposed to cyclical variation in aggregate conditions. Therefore, the value of a new hire and the number of job openings posted by firms in this industry is more volatile. A greater variation in the number of job openings implies a higher variability in the probability of unemployed workers finding jobs. Because young workers have a higher separation rate than prime-age workers, they are more exposed to such fluctuations in firms’ hiring incentives than prime-age workers. Hence, the ratio of prime-age workers to young workers is more cyclical in a high beta industry.

3. Empirical Evidence

In this section we test our model’s cross-sectional and time-series predictions, namely:

1. Industries whose equity returns covary more strongly with the aggregate stock market (high CAPM beta industies) are expected to have a more cyclical ratio of prime-age to young employed workers.

2. Relative to prime-age workers, the unemployment rate of young workers is more sensitive to time variation in the market risk-premium.

3.1 Data

We use five different data sources for our empirical study: aggregate unemployment data for different age groups from the Current Population Survey (CPS), stock return data from the Center for Research in Security Prices (CRSP), industry-level employment data from the publicly available version of the Longitudinal Employer-Household Dynamics (LEHD) database maintained by the Census Bureau, aggregate labor productivity from the Bureau of Labor Statistics (BLS), real Gross Domestic Product (GDP) data from Federal Reserve Economic Data (FRED), and industry-level labor productivity from the Labor Productivity tables reported by the BLS.

Both our cross-sectional and time-series studies use quarterly data. Our cross-sectional industry analysis is constrained by LEHD data and covers the period 1997Q1—2016Q4. To test our channel over a longer sample covering more recessions, we examine differences in unemployment rates between young and prime-age workers at the aggregate level for which we have a much longer time series covering the period between 1951Q1 and 2016Q4. We adjust for seasonal effects using the Census Bureau’s X13 program, to address strong seasonality in the data (especially in the employment numbers). Finally, GDP and labor productivity grow over time, so we detrend these variables using the Hodrick-Prescott filter with a smoothing parameter of |$1,600$| (after taking logs).

We use similar definitions for young and prime-age workers in our industry and aggregate time-series analyses. We use LEHD data in our industry cross-sectional study. LEHD provides employment data for the age buckets 14-18, 19-21, 22-24, 25-34, 35-44, 45-54, 55-64, and 65-99. Our definition of young workers are those between 22 and 24 years, since we do not focus on teenage (un)employment. Individuals between 35 and 44 years of age represent prime-age workers in our baseline analysis. We perform two robustness checks of this definition of prime-age workers that we discuss below. In our robustness checks, we also consider three alternative definitions for prime-age workers: 25-34, 35-44, and 45-54 years old. We do not consider workers beyond 55 years old as their labor market outcomes are affected by retirement horizon effects, which are outside the scope of our model.

We use unemployment rates from the CPS for our time-series study. CPS unemployment data are available for the following age buckets: 16-19, 20-24, 25-34, 35-44, 45-54, 55-64, and over 65 years old. As in our industry cross-sectional study, we do not consider unemployment rates for teenagers (16-19) and for those approaching retirement (55 and older). Therefore, for our baseline time-series results, we define young (prime-age) workers as those between 20 and 24 years (35 and 44 years) of age. The unemployment rate of workers in the buckets 25-54 exhibit similar business-cycle properties, we simply pick the one in the middle for our baseline results. Our cross-sectional and time-series results are robust to using alternative definitions of prime-age workers (see Appendix D and our Online Appendix).

To test our cross-sectional prediction, we use industry-level employment data from the LEHD database. Although the data in LEHD start from 1990, the initially very thin coverage correlate with states joining this database in waves. We use a balanced panel starting from the beginning of 1997, when there are 19 states in the database. The total output from these states comprises 51% of U.S. GDP.²⁶ We aggregate employment numbers across these states to construct industry-level measures. The publicly available version of LEHD makes data available down to the 3-digit NAICS industry code level. We exclude NAICS codes 1 (Agriculture), 8 (Other services), and 9 (Public Administration) from our analysis. In addition, we drop industries without data on labor productivity in the BLS productivity table (e.g., all the industries in the Construction sector). In the end, there are 32 industries at the 3-digit NAICS industry code level for which we can compute both CAPM and Labor productivity betas.²⁷

For our time-series analysis, we use the yield spread between BAA and AAA rated US corporate bonds as our baseline proxy for the market risk premium. We use the aggregate labor productivity series from the BLS (defined as the ratio of nonfarm business sector real output in a period to the total number of hours worked by all persons in that period to produce that output) as our baseline measure of labor productivity. Tables 5 and 6 show the summary statistics of our variables for our cross-sectional and time-series analyses, respectively.

Table 6

Summary statistics for time-series analysis.

A. Summary statistics
	Mean	SD	5%	50%	95%	Obs
Unemployment rate, young	9.46	2.59	5.30	9.30	14.50	264
Unemployment rate, prime-age	4.29	1.37	2.30	4.10	7.00	264
BAA-AAA credit spread	0.97	0.44	0.47	0.85	1.80	264
Labor productivity	0	1.04	−2.05	−0.02	1.67	264
log GDP	0	1.51	−2.79	−0.02	2.36	264
Dividend price ratio	0.81	0.30	0.36	0.76	1.34	264

B. Correlations
	Unemployment,	Labor	log GDP	BAA-AAA	Dividend-price
	prime-age	productivity			ratio
Unemployment rate, young	0.93	|$-$|0.13	−0.6	0.62	0.33
Unemployment rate, prime-age		|$-$|0.1	−0.63	0.55	0.29
Labor productivity			−0.16	−0.35	−0.09
log GDP				−0.45	−0.1

The sample is quarterly and covers the period 1951Q1—2016Q4. All nonfinancial series are deseasonalized. Unemployment rates are for the age groups 20-24 (young) and 35-44 (prime-age). log GDP is the cylical component of log GDP, as reported by FRED. Labor productivity is the cyclical component of log real output per hour, as reported by the BLS. The BAA-AAA yield spread is as reported by the FRED. The dividend-price ratio is the ratio between quarterly dividends and the end-of-quarter stock price, and is computed using the value-weighted aggregate market index taken from CRSP. All values are expressed in percentage terms.

3.2 Cross-sectional results

In this section, we test our model’s prediction that industries whose equity returns covary more strongly with the aggregate stock market (high CAPM beta industies) are expected to have a more cyclical ratio of prime-age to young employed workers.

Figure 9 summarizes this result. Each point in this scatter plot corresponds to an industry and the numerical values next to each point is the industry’s 3-digit NAICS code. The figure plots the elasticity of the prime-age-to-young-worker-employment ratio with respect to GDP for each industry against the industry’s CAPM beta. The estimated elasticities correspond to the slope coefficient from the following regression: |$r_{I,t} = a + b\times \log GDP_t + \epsilon_{I,t}$|⁠, where |$r_{I,t}$| is the log-employment ratio of prime-age to young workers defined in (32). The best-fit line in Figure 9 reveals a declining relationship. In other words, a given decline in log GDP results in a larger increase in the log employment ratio of prime-age to young workers in industries which have a higher CAPM beta.

Figure 9

Industry ratio elasticity versus industry beta

This figure plots the elasticity of the young-to-prime-age-worker-employment ratio |${E^P_{It}} / E^Y_{It}$| to GDP against the CAPM beta for each industry. The numerical values refer to the 3-digit NAICS code of the industry. The dotted line corresponds to the OLS fit of |$elasticity=3.8-5.41\times beta$|⁠.

To test our model’s prediction more formally, we use the following regression specification:

$$\begin{equation}\label{xsectreg} r_{I,t} = \alpha_I + \delta \times t + a \times GDP_t + b \times \beta_I \times GDP_t + \epsilon_{I,t} \end{equation}$$

(35)

where |$r_{I,t}$| denotes the log employment ratio (32), |$\beta_I$| represents the industry’s exposure to systematic risk, and |$GDP_t$| is the cyclical component of log GDP at time |$t$|⁠. The term |$\alpha_I$| is an industry fixed effect at the 3-digit NAICS code level and is meant to soak up differences across industries that are unrelated to business cycles (e.g., differences in the extent of unionization). We use a linear time trend, common across all industries, in our baseline specification.²⁸ Because we have a relatively short time series for our cross-sectional study, we assume a single source of aggregate risk, namely, the cyclical component of log real GDP. Industries differ in their exposure to this source of risk. We use two different measures of systematic risk exposure of an industry: the CAPM beta and the labor productivity beta of the industry. The CAPM beta of an industry is estimated as the loading of the excess monthly return of this industry portfolio (market capitalization weighted portfolio of all firms in this industry) on the excess monthly return of the market. We use the market-value weighted portfolio of all stocks listed in CRSP as our proxy for the market, and the return on the 1-month Treasury bill as our measure of the risk-free rate. The labor productivity beta of an industry is our second measure of the industry’s systematic risk exposure. This quantity is the loading of the industry’s labor productivity on aggregate labor productivity. Both of these measures are legitimate measures of systematic risk exposure according to our model. Therefore, we report results for each measure individually as well as results from a horse race with both measures simultaneously present.

Columns 1 to 4 of Table 7 shows our baseline results corresponding to Equation (35). The regressions are run with a linear time trend and with industry-level fixed effects at the 3-digit NAICS code level.²⁹ From Column 1 we see that, in line with our model’s prediction, the log-ratio of prime-age to young employed workers increases by 6.9% when GDP declines from it 95th to 5th percentile. Column 2 shows that this increase is higher for industries with higher CAPM beta. This result is both statistically and economically significant. For instance, for this same decline in GDP, the log-ratio of prime-age to young employed workers increases by |$15.7\%$| in an industry with a CAPM beta of 1.39 (at the 95th percentile of the CAPM beta distribution), while the increase in this ratio for the same decline in GDP in an industry with a CAPM beta of 1 (median CAPM beta) is only |$6.0\%$|⁠. This implies that the elasticity of the employment ratio of prime-age to young workers in the high beta industry with CAPM beta of 1.39 is 2.6 times that of a median beta industry. In our model, the corresponding ratio of elasticities is 1.6, which is about |$60\%$| that of the data counterpart.³⁰

Table 7

Cyclicality of the log employment ratio of prime-age to young workers across industries

	Aggregate-level regressions					State-level regressions
	(1)	(2)	(3)	(4)		(5)	(6)	(7)	(8)
|$\log GDP$|	−1.81	5.02	−1.93	4.91		−1.65	4.16	−1.34	4.28
	(−2.58)	(4.23)	(−2.49)	(3.92)		(−2.16)	(3.69)	(−1.67)	(3.57)
|$\beta_I^{\rm{CAPM}} \times \log GDP$|		−6.58		−6.57			−5.57		−5.46
		(−4.26)		(−4.32)			(−3.77)		(−3.83)
|$\beta_I^{\rm{LP}} \times \log GDP$|			0.27	0.23				−0.76	−0.58
			(0.32)	(0.36)				(−0.87)	(−0.77)
|$t \times 10^{-3}$|	−2.85	−2.85	−2.85	−2.85		−2.86	−2.86	−2.86	−2.86
	(−2.15)	(−2.15)	(−2.15)	(−2.15)		(−2.13)	(−2.13)	(−2.13)	(−2.13)
Observations	2,560	2,560	2,560	2,560		40,080	40,080	40,080	40,080
Adjusted R2	.92	.92	.92	.92		.82	.82	.82	.82

The first four columns report results for regression (35), and the last four columns show results for regression (36). The dependent variable is the ratio of the total number of prime-age workers (35—44 years old) to the total number of young workers (22—24 years old) employed in an industry. Section 3.1 describes the data sources. The right-hand-side variables are the log of the cyclical component of real GDP, interacted with two measures of systematic risk exposure of the industry. These are the CAPM beta and the labor productivity beta of the industry. The regressions are run with a linear time trend and with industry-level fixed effects at the 3-digit NAICS code level. Standard errors are clustered at the 2-digit NAICS code level. Numbers in parentheses are t-statistics.

The labor productivity beta regression results reported in Column 3 show that this measure of systematic risk exposure does not capture variation in the relative employment ratio of prime-age to young workers. The slope coefficient is insignificant, and when we include both CAPM betas and labor productivity betas to capture systematic risk (see Column 4), the slope coefficient on labor productivity beta stays insignificant. Furthermore, comparing Columns 2 and 4, we see that the slope coefficient on the interaction term of CAPM beta interacted with log GDP stays unchanged between the two specifications.

While we define young (prime-age) workers to be those between the ages of 22 and 24 (35 and 44) years in our baseline specification, Table D.1 shows that our results are robust to defining young workers to be those between the ages of 19 and 24 years of age. In addition, the Online Appendix shows that our results are robust to using three alternative definitions of prime-age workers to construct the log-employment ratio: those between 25 and 34, 35 and 44, or 45 and 54 years of age. In each of these, we maintain the same definition of young workers as in our baseline, that is, those between 22 and 24 years old. The point-estimates show that the results are stronger when the age difference between young and prime-age is larger.

Recall that our learning-based theory focuses on worker selection that results from unobserved differences in productivity across workers. A natural concern is that our findings may instead be driven by observable differences across workers. LEHD data allow us to run our baseline cross-sectional regression in subsamples where we can control for geography or gender. We control for geography by running a version of our baseline regression (35) at the industry by state level:

$$\begin{equation}\label{xsectreg2} r_{I,s,t} = \alpha_{I} + \alpha_{s} + \delta \times t + a \times GDP_t + b \times \beta_I \times GDP_t + \epsilon_{I,s,t}, \end{equation}$$

(36)

where |$r_{I,s,t}$| denotes the log employment ratio for industry |$I$| in state |$s$|⁠, and we additionally include state fixed effects |$\alpha_{s}$|⁠. The results are in Columns 5–8 of Table 7. Comparing the results of the baseline aggregate regression (35) to that of the state level regression (36), we see that the sensitivities remain essentially unchanged.³¹ Along similar lines, the Online Appendix reports results for our baseline aggregate regression (35) restricted to subsamples of either women or men employees. Once again, we see that our results hold up in these subsamples.³² This is true for both our aggregate-level regressions and our state-level regressions.

3.3 Time-series results

Our analysis of differences in unemployment risk between young and prime-age workers across industries in the previous section is based on a relatively short time period covering the Great Recession and the short recession in 2001. To test our channel over a longer time period covering more recessions, we analyze the unemployment dynamics of these two groups of workers for the aggregate economy between 1951Q1 and 2016Q4. We use the official BLS U-3 measure of unemployment in all of our time-series regressions, including the robustness checks reported in Appendix D. This measure counts an individual to be unemployed only if that person is actively seeking employment.

A prior study by Clark and Summers (1981) finds that the unemployment rate of workers between the ages of 16 and 19 and 20 and 24 are about twice as sensitive to business-cycle variations than the corresponding rate of other workers. We show that this result continues to hold in our much longer time series extending to 2016Q4. While Clark and Summers (1981) use the unemployment rate of workers in the age group 35-44 as a proxy for business-cycle conditions, we relate unemployment rates to time variation in aggregate labor productivity and risk premium, the latter captured by the yield spread between BAA- and AAA-rated U.S. corporate bonds.

Table 8 reports results for 1-quarter-ahead predictive regressions. Columns 1 and 4 report the sensitivity of young and the prime-age workers unemployment rates, respectively, to deviations in labor productivity from trend in a univariate regression. We use a linear time-trend in our baseline regression. Our results are unchanged when we run the same regression as in Table 8 without any time trend. The Online Appendix includes these results. All standard errors are Newey-West with four lags.

Table 8

Unemployment, labor productivity, and discount rates.

	|$u^Y_{t+1}$|			|$u^P_{t+1}$|			|$u^Y_{t+1}-u^P_{t+1}$|
	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
Labor productivity	−0.29		0.13	−0.12		0.08	−0.17		0.05
	(−1.11)		(0.66)	(−0.79)		(0.63)	(−1.32)		(0.53)
BAA-AAA		3.61	3.70		1.71	1.77		1.89	1.93
		(8.37)	(8.78)		(8.50)	(8.14)		(5.88)	(6.21)
|$t\times10^{-3}$|	0.15	0.09	0.09	0.08	0.05	0.05	0.07	0.04	0.04
	(3.81)	(2.57)	(2.60)	(3.50)	(2.42)	(2.47)	(3.74)	(2.39)	(2.38)
|$N$|	263	263	263	263	263	263	263	263	263
Adjusted |$R^2$|	0.21	0.35	0.36	0.19	0.31	0.31	0.18	0.33	0.33

Predictive regressions of 1-quarter-ahead unemployment rates. The dependent variables are the unemployment rate of young workers (⁠|$u^Y$|⁠) who are individuals between the ages of 20 and 24, the unemployment rate of prime-age workers (⁠|$u^P$|⁠) who are between the ages of 35 and 44 years, and the difference in these unemployment rates. Right-hand-side variables are labor productivity and the BAA-AAA spread. A constant and a linear time trend is included in all regressions. The data are quarterly from 1951Q1 to 2016Q4. All variables other than the BAA-AAA credit spread are deseasonalized. Labor productivity is detrended using an HP filter with bandwidth |$1,600$|⁠. Standard errors are Newey-West with four lags. Numbers in parentheses are t-statistics.

We see that changes in labor productivity are not significantly correlated with future unemployment rates of either young or prime-age workers. This is in line with the low correlation between labor productivity and aggregate unemployment rate fluctuations documented in prior studies (Barnichon 2010; Gali and van Rens 2017; McGrattan and Prescott 2012). In our model, aggregate labor productivity, measured as output per worker, depends on both the primitive productivity shock |$z_t$| as well as an endogenous component that depends on the cross-sectional match quality distribution. We show in Section 4 that the latter endogenous component of aggregate labor productivity can result in a low correlation between unemployment rates and aggregate labor productivity even though the primitive productivity shock |$z_t$| significantly affects unemployment rates.

In contrast, the BAA-AAA yield spread significantly predicts future unemployment rates of young and prime-age workers, as well as the difference in their unemployment rates. Columns 2 and 5 of Table 8 show that the unemployment rate of young workers is about 2 times more sensitive to changes in the BAA-AAA spread than that of prime-age workers. An increase in the yield spread from its 5th to 95th percentile values is associated with a 4.8% (2.3%) increase in youth (prime-age workers) unemployment rates. The corresponding model-implied change is a 4.3% (2.1%) increase in youth (prime-age workers) unemployment rates. Columns 3, 6, and 9 show that changes in the market risk premium remain significantly correlated with future changes in unemployment rates in a horse race with labor productivity.

Our baseline results define workers between the ages of 20 and 24 as young and those between 35 and 44 as prime-age. Table D.2 shows that our results are robust to using the BLS definition of young workers, that is, individuals between 16 and 24 years of age. In addition, we show in the Online Appendix that the sensitivity of unemployment rates to the credit spread declines monotonically with age across all CPS-defined age buckets between 16 and 54 years of age.

Our results are robust to using alternative proxies for |$z$| and |$x$| shocks of our model. Table D.3 shows that our conclusions are unchanged when we use the cyclical component of log GDP instead of labor productivity and/or the dividend-price ratio of the aggregate stock market as a proxy for the market risk premium instead of the BAA-AAA yield spread. Although the use of the dividend-price ratio as a proxy for the market risk premium is more controversial, we include these results for completeness.³³ We also show in the Online Appendix that our results remain unchanged when we control for workers’ race and gender.³⁴

Our model, like the canonical DMP model, assumes that the supply of workers is constant over time. Under this assumption, our model’s predictions for the unemployment rate can be directly translated to corresponding predictions for the employment rate of workers.³⁵ In our model these two quantities are perfectly negatively correlated. In the data, however, there are large, nonlinear trends in the labor force participation rates over our sample period 1951Q1—2016Q4. This is especially true for women. For our time-series empirical tests, this means that while a linear time trend captures trends in the unemployment rates, using a specification with a linear time trend falls short in describing the behavior of employment rates. We discuss this issue in our Online Appendix. For completeness, we also include results for the cyclicality of employment rates in our Online Appendix.

4. Labor Market Cleansing

In this section, we show that the composition of the pool of employed workers is influenced by time variation in the risk premium. In particular, an increase in the risk premium results in an accelerated rate with which low productivity workers are fired. In such a scenario, the average labor productivity of the pool of employed workers (i.e., the measure of labor productivity according to the Bureau of Labor Statistics) increases, even while the aggregate unemployment rate is also rising. Although this “labor market cleansing” mechanism dates back to at least Mitchell (1913, pp. 477-8), we show that the magnitude of this channel strongly depends on variation in the risk premium.

The (log) output per worker in our model is given by³⁶

$$\begin{equation} \log\left(\frac{Y_t}{N_{end,t}}\right) = z_t + \underbrace{\log \left( \int_{\underline{p}(z_t,x_t)}^1\, \left[ p e^{\mu(H)} + (1-p) e^{\mu(L)}\right] \, \widetilde{\mathcal{P}}_t(dp) \right)}_{\mbox{Endogenous productivity}}, \label{eq:TFP} \end{equation}$$

(37)

which takes into account both exogenous productivity |$z_t$|⁠, as well as the endogenous productivity of employed workers as determined by the distribution of match quality beliefs for producing workers, |$\widetilde{\mathcal{P}}_t$|⁠, given by (23).

Panel A of Figure 10 plots the difference between the output per worker and exogenous productivity for a 2-standard-deviation decrease in productivity, a 2-standard-deviation increase in the market price of risk, and for the combination in which both shocks arrive simultaneously. The starkest result is the effect of a shock to only the market price of risk |$x$|⁠. With no decline in exogenous productivity, the net effect of a change in the pool’s composition is an increase in the average productivity of workers (dashed line). Note that the aggregate unemployment rate increases as a result of this shock (see the dashed line in Figure 7). Panels B and C shows the improvement in the composition of current matches. Both of these plots compare the distribution of match quality of workers after the shock relative to the stochastic steady-state and we see that there is less mass of poor matches in states with low |$z$| (panel B) and high |$x$| (panel C).

Figure 10

Labor market cleansing and endogenous productivity

Panel A plots the response of endogenous labor productivity in the baseline model to a negative 2-standard-deviation shock in labor productivity |$z$| (solid line), a 2-standard-deviation shock in the market price of risk |$x$| (dashed line), and a combination of the two shocks (dash-dotted line). In all cases, the economy starts off at the steady state in which both labor productivity |$z$| and the market price of risk |$x$| are at their mean values. Panels B and C plot the difference of the conditional match quality distribution relative to its steady-state distribution. In panel B, labor productivity |$z$| is and unconditional two standard deviations below its mean, whereas the market price of risk |$x$| is drawn from its steady-state distribution. For panel C, |$x$| is an unconditional two standard deviations above its mean, whereas |$z$| is drawn from its steady-state distribution.

An implication of our finding is that the interaction between time-variation in the risk premium and the composition of the workforce can contribute to the low correlation between aggregate labor productivity and unemployment rates in the data (Barnichon 2010; Gali and van Rens 2017; McGrattan and Prescott 2012).³⁷ For example, the end of the Great Recession featured a simultaneous increase in both labor productivity (measured as the average output per worker) and aggregate unemployment rates (Mulligan 2011). Figure 11 shows a similar pattern for the four largest recessions in postwar U.S. data.

Figure 11

Labor productivity and unemployment during recessions.

Labor productivity is the cycle component (HP-filtered with a value of 1,600) of log real output per hour for nonfarm businesses and is available at a quarterly frequency. Unemployment is changes with respect to unemployment levels 1 month prior to the NBER starting date of the recession. The recessions shown above are the four largest in the postwar period.

5. Conclusion

We show that time-varying risk premium in financial markets has large real effects through its influence on firms’ labor policies. This helps us understand key patterns in the business-cycle dynamics of unemployment across age groups, specifically that the unemployment rate for young workers increases twice as fast as the corresponding rate for prime-age workers during recessions. Our equilibrium search and matching model introduces two ingredients to an otherwise standard Diamond-Mortensen-Pissarides setting: a time-varying risk premium and learning about unobserved heterogeneity in worker productivity.

Our calibrated model’s predictions for the mean levels and volatilities of the aggregate unemployment rates of young and prime-age workers in the United States matches that of the data. In addition, our model predicts the unemployment rate of younger workers relative to prime-age workers to be higher when the market risk premium is high. In the cross-section our model predicts the ratio of the number of employed prime-age workers to the number of employed young workers to be more cyclical in high beta industries. We find empirical support for these predictions.

Heterogenous agent labor search models are useful for understanding how time-varying risk premium differentially affects individuals who differ along dimensions such as demographics, skill, or human capital. Understanding the underlying drivers of unemployment risk across individuals can inform us of the economic benefits of policy targeted to different groups or the heterogenous outcomes across individuals to a single policy.

Appendix A. Extending the Model to Include Job Applicant Screening

In our baseline model, all job seekers appear identical and begin their job tenure with the same initial match quality |$p_0$|⁠. In this section, we consider an extension of the baseline model in which firms are additionally endowed with a screening technology that will allow it to distinguish between workers at the job interview stage. In particular, we model the screening technology using a uniform prior distribution

$$\begin{equation} \widetilde{p}_0 \sim Unif(p_0 - \Delta, p_0 + \Delta) \label{eq: model extension uniform prior} \end{equation}$$

(A.1)

for the initial match quality belief (4). In this set up, firms interview workers before deciding whether or not to pursue the match. The initial prior before the interview is |$p_0$|⁠, which is common across all interviewing workers. The interview generates a common informative signal for both the worker and the firm so that |$\widetilde{p}_0$| is the post-interview posterior upon which the final hiring decision is based. We can capture the effectiveness of the initial screening technology by varying |$\Delta$|⁠, with a large value of |$\Delta$| capturing a screening technology that is able to distinguish between job applicants. Our baseline model, which assumes all learning takes place after hiring, is embedded as the special case |$\Delta=0$|⁠.

Under this extension, the value of posting a vacancy (11) is modified as

$$\begin{equation} F_V(z,x)=g(\Theta(z,x))\int_{p_0-\Delta}^{p_0+\Delta}\,F(p,z,x){\rm 1}\kern-0.24em{\rm I}\left(p\geq\underline{p}(z,x)\right)\,\frac{dp}{2\Delta}\label{eq:vacancy_value extension} \end{equation}$$

(A.2)

to take the initial diffuse prior (A.1) into account. The aggregate laws of motion (17) and (18) are similarly modified to take the initial diffuse prior (A.1) into account. The remaining equations governing the extended model remain identical to their counterparts from the baseline model.

We repeat our analysis for the baseline parameters from Table 1, but for different values of the width of the initial prior distribution. Figure A.1 shows the results. The solid line in panel A shows the steady-state job separation hazard rate when |$\Delta=0.1$|⁠. Panel B shows the corresponding difference in conditional unemployment rate differences between young and prime-age workers for |$\Delta=0.1$|⁠. Comparing these figures to their counterparts from the baseline model (panel C of Figure 4 for job separation hazard rates and panel C of Figure 6 for differences in condition unemployment rates), we see that the results are almost identical.

Figure A.1

Job screening and difference in unemployment rates

Panel A plots the steady-state job separation hazard rate when the width of the initial prior (A.1) is |$\Delta=0.1$| (solid line) and |$\Delta=0.25$| (dash-dotted line). Panels B and C plot the difference in unemployment rates between young and prime-age workers for |$\Delta=0.1$| and |$\Delta=0.25$|⁠, respectively.

The reasons for getting almost identical results for |$\Delta=0.1$| are twofold. First, for |$\Delta=0.1$|⁠, the support of the uniform prior (A.1) lies above the separation threshold |$\underline{p}(z,x)$| for all values of |$z$| and |$x$|⁠. In this region, the value of a match to the firm (9) is approximately affine in match quality belief |$p$|⁠. As a result, the value of posting a vacancy (A.2) will be approximately equal its counterpart from the baseline model (11), which assumes a point mass at |$p_0$|⁠. Because the value of posting a vacancy determines the labor market equilibrium through the free entry condition (12), the resultant equilibrium hiring policies will also remain roughly unchanged. Second, having a uniform initial prior will result in some newly hired workers having higher than average job separation hazard rates (i.e., workers with |$\widetilde{p}_0$| below |$p_0$|⁠). However, these workers are offset by an equal number of newly hired workers with lower than average job separation hazard rates (i.e., workers with |$\widetilde{p}_0$| above |$p_0$|⁠). As a result, the separation hazard rate, averaged over all workers, remain approximately unchanged. In combination, this results in almost identical differences in the conditional unemployment rates of young and prime-age workers.

Figure A.1 also reports results for |$\Delta=0.25$|⁠. The job separation hazard rate is represented by the dash-dotted line in panel A, and differences in conditional unemployment rates between young and prime-age workers are shown in panel C. For this large value of |$\Delta=0.25$|⁠, part of the support of the uniform prior (A.1) now lies below the separation threshold. In this case, the initial job screening is sufficiently informative so as to be able to screen out a portion of job applicants at the interview stage. As a result, workers selection occurs not only on the job (as in our baseline model), but also prior to starting the job. Panel C shows the corresponding young minus prime-age difference in conditional unemployment rates. A 95th to 5th percentile decline in labor productivity (⁠|$z$|⁠) now results in a 0.8% increase in this unemployment difference, whereas a 5th to 95th percentile increase in the market price of risk (⁠|$x$|⁠) results in a 1.9% increase in this unemployment differences. Compared to our baseline model, which has corresponding increases of 1.0% (for the |$z$| shock) and 2.2% (for the |$x$| shock), we see that the presence of the screening device does attenuate the difference in unemployment risk across age groups, although the difference is small. The findings from our baseline model, however, remain unchanged: (1) unemployment rates for both young and prime-age workers are more sensitive to market price of risk (⁠|$x$|⁠) shocks compared to productivity (⁠|$z$|⁠) shocks, and (2) the sensitivity of youth unemployment rates to market price of risk shocks is 2 times that of prime-age workers (an increase of 3.8% for young workers versus 1.9% for prime-age workers in the event of a 5th to 95th percentile increase in the market price of risk).

Appendix B. Proofs

We characterize the equilibrium in a two-step procedure. First, we characterize the surplus function taking market tightness as a parameter. Afterwards, we then characterize market tightness through the free entry condition for vacancy creation.

B1. Match surplus.

By combining (a) definitions (9), (13), and (14), (b) the Nash bargaining condition |$J_e(p,\omega) - J_{eu}(\omega)=\eta S(p,\omega)$|⁠, and (c) noting that the free entry condition (11) implies |$f(\Theta(\omega))\eta S(p_0,\omega)= \eta \kappa \Theta(\omega)/(1-\eta)$|⁠, it can be shown that the surplus function |$S$| is the fixed point of the operator:

$$\begin{equation} T(S)(p,\omega) = \max \left\{0, \widehat{b}(p,\omega) + (1-s) \mathbb{E}\left[\Lambda(\omega,\omega^\prime)S(p^\prime,\omega^\prime)\left|p,\omega\right.\right] \right\}\!, \label{eq:surplus_operator} \end{equation}$$

(A.3)

where |$\widehat{b}(p,\omega) \equiv e^{z(\omega)}\left[p e^{\mu(H)} + (1-p) e^{\mu(L)}\right] - b - (1-s) \mathbb{E}\left[\Lambda(\omega,\omega^\prime)\eta \kappa \Theta(\omega^\prime)/(1-\eta) \left|\omega\right.\right]\!,$| and market tightness |$\Theta=\Theta(\omega)$| is treated as a parameter.

 

Proposition B.1.

The operator |$T$| defined in (A.3) is a contraction mapping for any given profile of match probability |$g(\omega)$| (equivalently market tightness |$\Theta(\omega)$|⁠). Hence, the surplus function |$S(p,\omega)$| is the unique fixed point of |$T$|⁠. Furthermore, the surplus function is nondecreasing in |$p$|⁠.

 

Proof.

It is easy to verify that |$T$| satisfies Blackwell’s sufficiency conditions for a contraction mapping (see theorem 3.3 in Stokey and Lucas 1999). This immediately implies the existence of a unique fixed point for |$T$|⁠. For the final claim, observe that |$T(S)$| is nondecreasing in |$p$| whenever |$S$| is nondecreasing in |$p$|⁠, hence the fixed point of |$T$| will also be nondecreasing in |$p$| (see corollary 1 in Stokey and Lucas 1999). ■

B2. Equilibrium labor market tightness.

We now recast the equilibrium in a form that is more convenient for equilibrium analysis.

 

Definition B.1 (Equilibrium).

An equilibrium consists of a pair |$(g,S)$| where |$g\in[0,1]^{|\Omega|}$| is a probability vector for a firm getting matched to a worker in each of the |$|\Omega|$| states, and |$S=S(p,\omega)$| is a surplus function. The pair must satisfy the following:

• (i) The surplus function must satisfy the fixed point problem |$S = T_g(S)$|⁠, where |$T_g$| denotes the operator (A.3) with |$g(\omega)=g(\Theta(\omega))$| taken as a parameter.

• (ii) The matching probabilities |$g=(g(\omega))$| satisfy |$g = \Upsilon(g)$|⁠, where the coordinates of |$\Upsilon:\,[0,1]^{|\Omega|}\mapsto [0,1]^{|\Omega|}$| are defined as

  $$\begin{equation} \Upsilon(g)(\omega) = \frac{\kappa}{\max\left\{\kappa, \left(1-\eta\right)S\left(p_{0},\omega;g\right)\right\}}. \label{eq: match probability fixed point} \end{equation}$$

  (A.4)

 

Proposition B.2 (Existence).

An equilibrium exists.

 

Proof.

Observe that |$\Upsilon$| maps the unit cube |$[0,1]^{|\Omega|}$| into the unit cube |$[0,1]^{|\Omega|}$|⁠. Furthermore, the fact that |$S$| is continuous in |$g$| implies that |$\Upsilon$| is a continuous map. Brouwer’s fixed point theorem then guarantees that |$\Upsilon$| has a fixed point. ■

Consider the partial ordering, |$\succeq$|⁠, on |$[0,1]^{|\Omega|}$| defined according to |$x=(x_1,...,x_{|\Omega|})\succeq y=(y_1,...,y_{|\Omega|})$| if and only if |$x_i\geq y_i$| for all |$1,...,|\Omega|$|⁠.

 

Proposition B.3 (Least and greatest equilibrium).

Under the den Haan et al. (2000) parameterization for the matching function, |$m(U,V)=UV/(U^\iota+V^\iota)^{\frac{1}{\iota}}$|⁠, the least and greatest fixed point of |$\Upsilon$|⁠, according to the partial order |$\succeq$|⁠, can be computed by iterating |$g_{n+1}=\Upsilon(g_n)$| from a starting point of |$g_0=\mathbf{0}=(0,...,0)$| and |$g_0=\mathbf{1}=(1,...,1)$|⁠, respectively.

 

Proof.

Under the den Haan et al. (2000) parameterization for the matching function, |$\Theta(g)$| is a strictly decreasing function of |$g$|⁠. Hence, the operator |$T$| is weakly increasing in |$g$|⁠. As a result, |$\Upsilon$| is weakly decreasing in |$g$|⁠. Because |$\left([0,1]^{|\Omega|},\succeq\right)$| is a complete lattice, Tarski’s fixed point theorem then guarantees the existence of a least and greatest fixed point. The iterative procedure is guaranteed to locate the extremal fixed points because we initiate the algorithm from the extremal points of the unit cube and |$\Upsilon$| is also a continuous operator (see, e.g., Echenique 2005). ■

 

Remark B.1 (Numerical verification of uniqueness).

Proposition B.3 allows us to numerically verify whether or not an equilibrium is unique. More specifically, the equilibrium is unique if the least and greatest fixed points of |$\Upsilon$| agree. We have not noticed cases of multiple equilibria in our numerical experiments.

B3. The separation threshold.

The separation threshold |$\underline{p}(\omega)$| solves the indifference condition |$0=\Psi(\underline{p}(\omega),\omega)\equiv \widehat{b}(p,\omega) + (1-s) \mathbb{E}\left[\Lambda(\omega,\omega^\prime)S(p^\prime,\omega^\prime)\left|p,\omega\right.\right]$|⁠.³⁸ The solution to this indifference condition is unique, because |$\Psi$| is strictly increasing in |$p$|⁠. To see the latter, note that the surplus function is increasing in |$p$| according to Proposition B.1, and that the Bayesian posterior function (8) is also monotone increasing in the prior |$p$|⁠.

Appendix C. Numerical Implementation

We first discretize each of processes (3) and (7) using the Rouwenhorst (1995) method with |$25$| grids points (this covers |$\pm 4.9$| standard deviations of the unconditional distribution). We then solve the discretized model using the following iterative procedure: (1) initialize the vacancy filling probability |$g_{(0)}=g_{(0)}(z,x)$|⁠; (2) for a given |$g_{(n)}$|⁠, solve |$S_{(n)}=T_{(n)}[S_{(n)}]$| via value function iteration, where |$T_{(n)}$| is given by (A.3); (3) given match surplus |$S_{(n)}$|⁠, compute |$\widehat{g}_{(n)}=\Upsilon(S_{(n)})$|⁠, where |$\Upsilon$| is defined by Equation (A.4); and (4) stop the algorithm if |$g_{(n)}$| and |$\widehat{g}_{(n)}$| is sufficiently close. Otherwise, update |$g_{(n+1)}=0.98 g_{(n)} + 0.02 \widehat{g}_{(n)}$| and repeat steps 2 and 3. In the above procedure, the surplus value along the |$p$| dimension is stored over 32 evenly spaced grid points |$\left\{p_i\right\}$| between 0 and 1, and interpolation is used when computing the conditional expectation |$\mathbb{E}_{p^\prime\left|p\right.}[S(p^\prime)]\approx \mathbb{E}_{p^\prime\left|p\right.}\left[1\left\{p_i\leq p^\prime <p_{i+1}\right\} \left\{S(p_i) \frac{p_{i+1} - p^\prime}{p_{i+1}-p_{i}}+ S(p_{i+1})\frac{p^\prime - p_{i}}{p_{i+1}-p_{i}}\right\}\right].$|

Auxiliary quantities are computed after the equilibrium is computed. In particular, we solve for the separation threshold |$\Psi(\underline{p}(\omega),\omega)=0$| using a standard root-finding scheme. In addition, we approximate the cross-sectional distribution of match quality beliefs |$\mathcal{P}$| using a histogram with thirty-one bins. The evolution of the approximate cross-sectional match quality distribution is then computed under the assumption that match qualities are uniformly distributed within each bin.

Appendix D. Empirical Appendix

Table D.1

Cyclicality of the log employment ratio of prime-age to young workers across industries: Alternative definition of young.

	Aggregate-level regressions				State-level regressions
	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
|$\log GDP$|	−2.64	5.34	−2.81	5.18	−2.51	3.77	−2.10	3.80
	(−2.73)	(3.82)	(−2.76)	(3.43)	(−2.30)	(3.11)	(−2.07)	(2.62)
|$\beta_I^{\rm{CAPM}} \times \log GDP$|		−7.69		−7.68		−5.98		−5.73
		(−3.95)		(−4.01)		(−3.40)		(−3.24)
|$\beta_I^{\rm{LP}} \times \log GDP$|			0.38	0.33			−1.07	−0.74
			(0.40)	(0.44)			(−1.40)	(−0.89)
|$t \times 10^{-3}$|	−1.45	−1.45	−1.45	−1.45	−1.32	−1.32	−1.32	−1.32
	(−1.14)	(−1.14)	(−1.14)	(−1.14)	(−1.12)	(−1.12)	(−1.12)	(−1.12)
Observations	2560	2560	2,560	2,560	35,040	35,040	35,040	35,040
Adjusted R2	.95	.95	.95	.95	.89	.89	.89	.89

Results for regression (35) (first four columns) and regression (36) (last four columns) with young workers defined to be between 19 and 24 years old. The regressions are run with a linear time trend that is common across industries and with industry-level fixed effects at the 3-digit NAICS code level. Standard errors are clustered at the 2-digit NAICS code level. Numbers in parentheses are t-statistics.

Table D.2

Time-series results: Alternative definition of young.

	|$u^Y_{t+1}$|				|$u^P_{t+1}$|				|$u^Y_{t+1}-u^P_{t+1}$|
	(1)	(2)	(3)		(4)	(5)	(6)		(7)	(8)	(9)
Labor productivity	−0.27		0.16		−0.13		0.12		−0.14		0.04
	(−0.97)		(0.77)		(−0.74)		(0.91)		(−1.17)		(0.41)
BAA-AAA		3.66	3.77			2.09	2.18			1.57	1.60
		(7.98)	(8.28)			(8.75)	(8.84)			(5.59)	(5.72)
|$t\times10^{-3}$|	0.16	0.10	0.10		0.08	0.05	0.05		0.08	0.05	0.05
	(3.61)	(2.54)	(2.56)		(3.39)	(2.10)	(2.12)		(3.34)	(2.54)	(2.54)
|$N$|	263	263	263		263	263	263		263	263	263
Adjusted |$R^2$|	.21	.51	.52		.17	.49	.50		.20	.42	.41

Predictive regressions of 1-quarter-ahead unemployment rates with young workers defined as ages between 16 and 24 years. Right-hand-side variables are the same as those used in Table 8. The data are quarterly between 1951Q1 to 2016Q4. All variables are deseasonalized. Standard errors are Newey-West with four lags. Numbers in parentheses are t-statistics.

Table D.3

Alternative specifications for |$z$| and |$x$| reported in Table 8.

A. |$z:$| Labor productivity, |$x:$| Dividend-Yield aggregate Stock Market
	|$u^Y_{t+1}$|			|$u^P_{t+1}$|			|$u^Y_{t+1}-u^P_{t+1}$|
	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
Labor prod. (cycle)	−0.29		−0.17	−0.12		−0.06	−0.17		−0.11
	(−1.11)		(−0.81)	(−0.79)		(−0.48)	(−1.32)		(−1.07)
|$dp$|		4.53	4.43		2.11	2.08		2.41	2.35
		(3.68)	(3.72)		(3.19)	(3.18)		(3.65)	(3.68)
|$t\times10^{-3}$|	0.15	0.27	0.27	0.08	0.14	0.14	0.07	0.14	0.14
	(3.81)	(5.62)	(5.55)	(3.50)	(4.64)	(4.53)	(3.74)	(6.18)	(6.18)
|$N$|	263	263	263	263	263	263	263	263	263
Adjusted |$R^2$|	.01	.48	.48	.04	.39	.40	.01	.46	.46

B. |$z:$| log (GDP), cyclical, |$x:$| BAA-AAA yield spread
	|$u^Y_{t+1}$|			|$u^P_{t+1}$|			|$u^Y_{t+1}-u^P_{t+1}$|
	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
log GDP (cycle)	−0.91		−0.60	−0.52		−0.39	−0.39		−0.21
	(−6.41)		(−5.43)	(−7.44)		(−6.36)	(−4.38)		(−3.13)
BAA-AAA		3.61	2.73		1.72	1.14		1.89	1.58
		(8.37)	(6.67)		(8.51)	(7.20)		(5.88)	(4.77)
|$t\times10^{-3}$|	0.15	0.09	0.11	0.08	0.05	0.06	0.08	0.04	0.05
	(4.75)	(2.58)	(3.39)	(4.48)	(2.41)	(3.34)	(4.30)	(2.41)	(2.89)
|$N$|	263	263	263	263	263	263	263	263	263
Adjusted |$R^2$|	.48	.54	.65	.51	.46	.61	.35	.51	.55

C. |$z:$| log (GDP), cyclical, |$x:$| Dividend-Yield aggregate Stock Market
	|$u^Y_{t+1}$|			|$u^P_{t+1}$|			|$u^Y_{t+1}-u^P_{t+1}$|
	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
log GDP (cycle)	−0.91		−0.83	−0.52		−0.48	−0.39		−0.35
	(−6.41)		(−8.53)	(−7.44)		(−8.97)	(−4.38)		(−5.19)
|$dp$|		4.53	3.79		2.11	1.68		2.41	2.10
		(3.68)	(4.17)		(3.19)	(4.22)		(3.65)	(3.56)
|$t\times10^{-3}$|	0.15	0.27	0.25	0.08	0.14	0.12	0.07	0.14	0.13
	(4.75)	(5.62)	(7.03)	(4.48)	(4.64)	(5.82)	(4.28)	(6.18)	(7.03)
|$N$|	263	263	263	263	263	263	263	263	263
Adjusted |$R^2$|	.48	.35	.59	.51	.31	.59	.35	.33	.47

The dividend price ratio between quarterly dividends and the end-of-quarter stock price is computed using the value-weighted aggregate market index from CRSP. The sample period is the same as those used in Table 8. Standard errors are Newey-West with four lags. Numbers in parentheses are t-statistics.

Acknowledgments

We thank Stijn Van Nieuwerburgh (the editor), two anonymous referees, Asaf Bernstein, Sugato Bhattacharyya, Andres Blanco (discussant), George Constantinides, Winston Dou, Vadim Elenev (discussant), Shiyang Huang, Leonid Kogan, Yang Liu, Andrey Malenko, M.P. Narayanan, Stavros Panageas (discussant), Jonathan Parker, Uday Rajan, Edouard Schaal, Lukas Schmid (discussant), Toni Whited, Amir Yaron, and seminar participants at the Barcelona GSE Summer Forum, Chinese University of Hong Kong, EFA, Michigan Econ-Finance Day, Michigan Finance Lunch, MIT Sloan Finance Lunch, Nanyang Technological University, National University of Singapore (Economics and Finance), SFS Cavalcade North America, TAU Finance Conference, University of Hong Kong, and the Wharton Brown Bag Micro Seminar for helpful comments and discussions. This work was supported by the Research Grants Council of Hong Kong [ECS27503816 to Y.X.]. We thank Tae Uk Seo for providing expert research assistance. Supplementary data can be found on The Review of Financial Studies web site.

Footnotes

¹ Two reasons explain this finding. First, young unemployed workers constitute a disproportionate fraction of the total unemployed population. Over the period 1951—2017, individuals between the ages of 16 to 24 years composed 18% of the total workforce above the age of 16. However, they constituted 37% of the total pool of unemployed workers. Second, the fluctuations in the youth unemployment rate are twice as large as the fluctuations in the average unemployment rate.

² We do not take a stance on the underlying causes of the time-varying risk premium. Instead we take the evidence for the time-varying risk premium as given and focus on its implications for labor markets.

³ Evidence on employers learning about the productivity of employees during their tenure is documented in the seminal papers by Farber and Gibbons (1996) and Altonji and Pierret (2001) and the literature following these papers. For instance, Nagypal (2007), Borovickova (2017), and Fredriksson et al. (2018) use micro-data to show the importance of learning about match-specific productivity for unemployment outcomes and wage dynamics.

⁴ To test our model’s predictions across the cross-section of industries, we cast our model’s implications in terms of the ratio of employed prime-age to employed young workers, because we have data on employment levels of workers of different age groups at the industry level.

⁵ We use time variation in the BAA-AAA credit spread to capture time variation in the market risk premium. We obtain similar results when we instead use variation in the dividend-price ratio of the aggregate stock market to capture time variation in this risk premium.

⁶ To address concerns that our findings are driven by other observable differences across workers, we repeat our analysis with additional controls. Our empirical findings remain unchanged. We analyze an extension of our baseline model to accommodate ex ante worker heterogeneity in Appendix A. Our model’s theoretical findings also remain unchanged.

⁷ This is a standard assumption in the labor search literature. For example, see Merz (1995) and Andolfatto (1996), and Shimer (2010) for a textbook treatment.

⁸ More precisely, this result holds when the exogenous job separation rate is defined as inclusive of mortality risk. We maintain this definition throughout our analysis.

⁹ This parameterization has the convenient property that the resultant meeting probabilities automatically lie between 0 and 1.

¹⁰Appendix A extends the model to allow for diffuse initial priors.

¹¹ More precisely, the exogenous separation rate |$s = 1 - (1-\chi)(1-\widetilde{s})$| can be attributed to worker retirements (⁠|$\chi$|⁠) and exogenous separations for reasons other than retirement (⁠|$\widetilde{s}$|⁠).

¹² With a continuum of workers, additional technical assumptions regarding measurability, which are satisfied in our setting, are required for the Law of Large Numbers to hold. See Sun (2006) and Duffie and Sun (2012) for details.

¹³ The initial match quality distribution for cohort |$c$|⁠, |$\mathcal{P}_{c,t=c}$|⁠, does not exist as workers begin their careers unemployed so that no matches have occurred. In this case, however, the distribution |$\widetilde{\mathcal{P}}$| appearing in the definition of |$\Gamma_\mathcal{P}$|⁠, (21), is simply equal to the point mass |$\delta_0$| because |$N_{c,t=c}=0$|⁠.

¹⁴ We thank Sugato Bhattacharyya for helpful discussions.

¹⁵Jovanovic (1979) and Moscarini (2005) formalize the idea of learning about match quality, and Flinn (1986) provides early empirical support.

¹⁶ The firm’s inference problem in this setting is nontrivial: a job applicant’s employment history is an endogenous variable that depends on the history of her/his former employers’ business conditions in addition to her/his productivity. Therefore, the history of past match quality distributions is a state variable that has to be included to solve for the equilibrium.

¹⁷ In a different context, Belo et al. (2017) show that differences in firing costs between skilled and unskilled workers can have important cross-sectional asset pricing implications. Our model provides a micro-foundation for a higher firing cost for skilled workers if we identify skilled workers to be prime-age workers.

¹⁸ An important observable characteristic that we are unable to control for because of a lack of data is education. In the time series, this data is short, starting from 2000. In the cross-section, the publicly available version of LEHD does not allow us to simultaneously control for age and education.

¹⁹ Note that the parameter |$\sigma$| in our model is a measure of the informativeness of individual output about match-quality and not the idiosyncratic volatility of firm output. A firm employs many workers; therefore, match-level idiosyncratic shocks average out within a firm.

²⁰ See http://www.bls.gov/news.release/tenure.t01.htm.

²¹ An inversely related job finding rate and unemployment duration (see the discussion following Equation (1)), imply a countercyclical pattern for unemployment duration of all workers. This is in line with the data. We assume a common job finding rate for all workers, so our model implies the same unemployment rate duration for young and prime-age workers, thus missing the larger increase in unemployment rate duration of younger workers observed during recessions.

²²Guvenen et al. (2019, p. 40) states: “the variation in earnings dynamics over the life cycle and by recent earnings are so large that it is an open question whether existing models can be quantitatively consistent, and, if not, how they should be modified.”.

²³ Having equal wage elasticities across workers of different age groups when labor supply elasticities are equal across age groups is also a feature of the real business-cycle model studied by Jaimovich et al. (2013). They find that generating large differences in wage elasticities requires introducing age-group-specific labor supply elasticities. We do not model such variation in labor supply.

²⁴ The correlation between labor productivity, measured as output per hour, and unemployment rates is low in the data. In our model, output per worker depends on the primitive labor productivity shock |$z$| and an endogenous productivity component that depends on the cross-sectional match quality belief distribution. We show in Section 4 that the latter endogenous component can dampen the observed relationship between output per hour and unemployment rates.

²⁵ Labor productivity series at the industry level are available from the BLS.

²⁶ This number is based on the 2016 estimate provided by the Bureau of Economic Analysis (BEA). The states covered in our sample are California, Colorado, Connecticut, Hawaii, Illinois, Kansas, Louisiana, Maryland, Minnesota, Missouri, Montana, New Jersey, New Mexico, North Carolina, Oregon, Rhode Island, Texas, Washington, and Wisconsin.

²⁷ The 3-digit NAICS codes for these industries are: 211, 212, 213, 221, 311, 312, 313, 315, 321, 322, 323, 324, 325, 326, 327, 331, 332, 333, 334, 335, 336, 337, 339, 445, 448, 452, 454, 481, 484, 511, 721, and 722.

²⁸ We show in the Online Appendix that our baseline empirical results are robust to (a) excluding time trends and (b) including an industry-specific linear time trend.

²⁹ We include an industry-level fixed effect, so we do not separately include the industry’s CAPM beta as an explanatory variable.

³⁰ Our model misses the level difference in the elasticities between the high beta and the median beta industries. Our model implied elasticity of an industry with |$\beta=1$| is |$-0.2$|⁠. This is close to the sensitivity we obtain in the data from our time-series regression where it ranges between |$-0.2$| and |$-0.4$| (see Column 7 or 9 in Panel B of Table D.3). Note that the sensitivity to GDP estimated from the time series is between 4.5 to 9 times smaller than that estimated from the cross-sectional panel regression. It is therefore not surprising that our model, which is calibrated to aggregate labor market quantities and uses parameters that are common across all industries, misses the sensitivity levels in the cross-section.

³¹ The point estimates from regression (36) are slightly muted in comparison to those from (35), presumably because of differences across states soaking up some of the variation of the left-hand-side variable.

³² The publicly available LEHD data do not allow us to simultaneously control for education and age.

³³ See, for instance, Lettau and Van Nieuwerburgh (2008), who show that a significant portion of the demeaned dividend-price ratio is due to cash flow variation, not discount rate variation.

³⁴ Another important observable characteristic along which workers differ is the level of educational attainment. Data on unemployment rates for workers of different ages that control for education level are short, starting from 2000. Therefore, we do not analyze this.

³⁵ The employment rate is the employment to population ratio. We thank an anonymous referee for pointing this out to us and suggesting we test our model’s predictions for the employment rate in the data.

³⁶ In our model, all employed workers work the same number of hours. Therefore, our model need not distinguish between output per worker and output per hour.

³⁷ See, for example, Biddle (2014) and Fernald and Wang (2016) for a list of alternative explanations. Kilic and Wachter (2018) also produces a low correlation between aggregate unemployment rates and labor productivity. Our model additionally shows that the composition of the aggregate workforce depends on time variation in the risk premium.

³⁸ In the event that |$\Psi(p,\omega)$| lies uniformly above (below) 0, then |$\underline{p}(\omega)=0$| (⁠|$\underline{p}(\omega)=1$|⁠).