Risk, Unemployment, and the Stock Market: A Rare-Event-Based Explanation of Labor Market Volatility

The Diamond-Mortensen-Pissarides (DMP) model of search and matching offers an intriguing theory of labor market fluctuations based on the job creation incentives of employers (Diamond 1982; Pissarides 1985; Mortensen and Pissarides 1994). When the contribution of a new hire to firm value decreases, employers reduce investment in hiring, decreasing the number of vacancies and, in turn, increasing unemployment. Because of the glut of jobseekers in the labor market, it becomes easier for employers to fill vacancies. Therefore, unemployment stabilizes at a higher level and the number of vacancies at a lower level. That is, labor market tightness (defined as the ratio of vacancies to unemployment) decreases until the payoff to hiring changes again.

While the mechanism of the DMP model is intuitive, a fundamental question remains unanswered: what causes job-creation incentives, and hence unemployment, to vary? The canonical DMP model and numerous successor models suggest that the driving force is labor productivity. However, explaining labor market volatility based on productivity fluctuations is difficult, because unemployment and vacancies are much more volatile than labor productivity (Shimer 2005). Furthermore, unemployment does not track the movements of labor productivity, as is particularly apparent in the last three recessions. Rather, these recent data suggest a link between unemployment and stock market valuations (Hall 2017).

In this paper, we make use of the DMP mechanism to explain the cyclical behavior of unemployment. However, rather than linking labor market tightness to productivity, we propose an equilibrium model in which fluctuations in labor market tightness arise from a small and time-varying probability of an economic disaster. Even if current labor productivity remains constant, disaster fears lower the job-creation incentives of firms. The labor market equilibrium shifts to a lower point on the vacancy-unemployment locus (the Beveridge curve), with higher unemployment and lower vacancy openings. At the same time, stock market valuations decline.

Our model generates high volatility in unemployment and vacancies, along with a strong negative correlation between the two. This pattern of results accurately describes postwar U.S. data. We calibrate wage dynamics to match the behavior of the labor share in the data and find that matching the observed low response of wages to labor market conditions is crucial for both labor market volatility and realistic behavior of financial markets. Furthermore, the search and matching friction in the labor market and time-varying disaster risk result in a realistic equity premium and stock return volatility. Because the labor market and the stock market are driven by the same force, the price of the aggregate stock market and labor market tightness are highly correlated, while the correlation between labor productivity and tightness is realistically low.

Our paper is related to three strands of literature. First, since Shimer (2005) showed that the DMP model with standard parameter values implies small movements in unemployment and vacancies, a strand of literature has further developed the model to generate large responses of unemployment to aggregate shocks. In these papers, the aggregate shock driving the labor market is labor productivity.¹Hagedorn and Manovskii (2008) argue that a calibration of the model combining low bargaining power of workers with a high opportunity cost of employment can reconcile unemployment volatility in the DMP model with the data. Other papers suggest alternatives to the Nash bargaining assumption for wages (Hall 2005; Hall and Milgrom 2008; Gertler and Trigari 2009). Compared with Nash bargaining, these alternatives render wages less responsive to productivity shocks. Thus a productivity shock can have a larger effect on job-creation incentives. Our paper departs from these in that we do not rely on time-varying labor market productivity as a driver of labor market tightness, which leads to a counterfactually high correlation between these variables. Furthermore, we also derive implications for the stock market, and explain the equity premium and volatility puzzles.²

Second, the present work relates to a literature embedding the DMP model into the real business-cycle framework, with a representative risk averse household that makes investment and consumption decisions. In the standard real business-cycle (RBC) model (Kydland and Prescott 1982), employment is driven by the marginal rate of substitution between consumption and leisure, and, because the labor market is frictionless, no vacancies go unfilled. Merz (1995) and Andolfatto (1996) observe that this model has counterfactual predictions for the correlation of productivity and employment, and build models that incorporate RBC features and search frictions in the labor market. These models capture the lead-lag relation between employment and productivity while having more realistic implications for wages and unemployment compared to the baseline RBC model. In this paper, we also document the lead-lag relation between productivity and employment in the period that this literature analyzes (1959–1988). However, our empirical analysis shows that this lead-lag relation is absent in more recent data. These papers do not study asset pricing implications.

Third, our paper is related to the literature on asset prices in dynamic production economies. These models build on the RBC framework, in which time-varying productivity determines consumption and dividend policy in equilibrium. In contrast, in an endowment economy, there is no aggregate technology for transferring consumption and dividends across periods and states.³ Thus, relative to endowment economies, production economies face an additional hurdle in explaining the equity premium because of the agent’s ability to smooth consumption (Kaltenbrunner and Lochstoer 2010; Lettau and Uhlig 2000). Increasing risk aversion raises the equity premium in an endowment economy, but leads to even smoother consumption in production economy and thus very little fluctuation in marginal utility. Alternative preferences, such as habit formation can overcome this problem (Boldrin, Christiano, and Fisher 2001; Jermann 1998) at the cost of highly volatile risk-free rates. Another approach is to allow for rare disasters. Barro (2006) and Rietz (1988) demonstrate that allowing for rare disasters in an endowment economy can explain the equity premium puzzle. Building on this work, Gourio (2012) studies the implications of time-varying disaster risk modeled as large drops in productivity and destruction of physical capital in a business-cycle model with recursive preferences and capital adjustment costs. Gourio’s model can explain the observed comovement between investment and the equity premium. However, unlevered equity returns have little volatility, and thus the premium on unlevered equity is low. This model can be reconciled with the observed equity premium by adding financial leverage, but the leverage ratio must be high in comparison with the data. Like in RBC models with frictionless labor markets, Gourio’s model does not explain unemployment. In the spirit of this literature, Petrosky-Nadeau, Zhang, and Kuehn (2013) build a model in which rare disasters arise endogenously through a series of negative productivity shocks. Like us, they build on the DMP model, but in a very different way. Their paper incorporates a calibration of Nash-bargained wages similar to Hagedorn and Manovskii (2008), leading to wages that are high and rigid. Moreover, their specification of marginal vacancy opening costs includes a fixed component, implying that it costs more to post a vacancy when labor conditions are slack and thus when output is low. Finally, they assume that workers separate from their jobs at a rate that is high compared with the data. The combination of a high separation rate, fixed marginal costs of vacancy openings and high and inelastic wages amplifies negative shocks to productivity and produces a negatively skewed output and consumption distribution. Like other DMP-based models described above, their model implies that labor market tightness is driven by productivity. Furthermore, while their model can match the equity premium, the fact that their simulations contain consumption disasters make it unclear whether the model can match the high stock market volatility and low consumption volatility that characterize the U.S. postwar data.

1. Labor Market, Labor Productivity, and Stock Market Valuations

In the literature succeeding the canonical DMP model, labor productivity serves as the driving force behind volatility in unemployment and vacancies. Recent empirical work, however, has challenged this approach on the grounds that labor productivity is too stable compared with unemployment and vacancies, and that the variables are at best weakly correlated. In this section we summarize evidence on the interplay between unemployment, productivity and the stock market.

In Figure 1, we plot the time series of labor productivity |$Z$| and of the vacancy-unemployment ratio |$V/U$|⁠, the variable that summarizes the behavior of the labor market in the DMP model (see Appendix E for a description of the data). Both variables are shown as log deviations from an HP trend.⁴Figure 1 shows the disconnect between the volatility of |$V/U$| and of productivity: labor productivity |$Z$| never deviates by more than 5 percent from trend, while, in contrast, |$V/U$| is highly volatile and deviates up to a full log point from trend. The lack of volatility in productivity as compared with labor market tightness is one challenge facing models that seek to explain unemployment using fluctuations in productivity.

Another challenge arises from the comovement in these variables. Figure 1 shows that tightness and productivity did track each other in the recessions of the early 1960s and 1980s. However, as noted in a number of studies, the correlation between productivity and business-cycle variables, including tightness, disappears after 1985 (Barnichon 2010; Galí and Van Rens 2014; McGrattan and Prescott 2012). A striking example of this disconnect is the aftermath of the Great Recession, which simultaneously features a small productivity boom along with a labor-market collapse. Overall, the contemporaneous correlation between the variables is 0.10 as measured over the full sample, 0.47 until 1985 and |$-$|0.36 afterward. There is some evidence that |$Z$| leads |$V/U$|⁠; the maximum correlation between |$V/U$| and lagged |$Z$| occurs with a lag length of one year. However, this relation also does not persist in the second subsample; while the correlation over the full sample is 0.31, it is 0.62 in the subsample before 1985 and |$-$|0.09 after 1985.

While the data display little relation between unemployment and productivity, there is a relation between unemployment and the stock market. We focus on the ratio of stock market valuation |$P$| to labor productivity |$Z$| because |$P/Z$| has a direct counterpart in the model – however, |$P/Z$| closely tracks Robert Shiller’s cyclically adjusted price-earnings ratio as shown in Figure 2 – the correlation between these series is 0.97 in levels and 0.98 in first differences. Figure 3 shows a consistently positive correlation between labor market tightness |$V/U$| and valuation |$P/Z$|⁠. The correlation over the full sample is 0.47. |$P/Z$| also leads |$V/U$|⁠; the maximum correlation is attained at a lag length of two quarters. In the period from 1986 to 2013, the contemporaneous correlation is 0.71. Moreover, like |$V/U$|⁠, |$P/Z$| is volatile, with deviations up to 0.5 log points below trend. Figure 4 shows that vacancies |$V$| follow a similar pattern to |$V/U$|⁠.⁵

Why might labor markets be tightly connected with stock market valuations, but not with current productivity? In the sections that follow, we offer a model to answer this question.

2. Model

In Section 2.1 we review the DMP model of the labor market with search frictions. In Section 2.2, we use the DMP model with minimal additional assumptions to demonstrate a link between equity market valuations and labor market quantities. We confirm that this link holds in the data. In Section 2.3 we present a general equilibrium model that explains labor market and stock market volatility in terms of time-varying disaster risk (we will examine the quantitative implications of this model in Section 3). In Section 2.4 we give closed-form solutions in a special case of the model in which disaster risk is a constant. This special case lends intuition for how disaster risk affects labor market quantities and prices in financial markets.

2.1 Search frictions

It follows from (3) and (4) that the job-finding rate is increasing, and the vacancy-filling rate decreasing, in the vacancy-unemployment ratio. In times of high labor market tightness, namely, when the vacancy rate is high and/or the unemployment rate is low, the probability of finding a job per unit time increases, whereas filling a vacancy takes more time.

Finally, the representative firm incurs costs |$\kappa_t$| per vacancy opening. As a result, aggregate investment in hiring is |$\kappa_t V_t$|⁠.

2.2 Equity valuation and the labor market

In this section we derive an equilibrium restriction that links the value of the stock market to conditions in the labor market. To establish this link, we make use of the framework in Section 2.1 but with minimal additional assumptions.⁷

The following result establishes a general relation between the stock market and the labor market.⁹

It follows that |$l_t = \kappa_t/q(\theta_t)$|⁠, and furthermore, that the value of the firm equals the number of workers employed multiplied by the value of each worker. This is what is shown in (10).

Equation (11) has a related interpretation. The |$t+1$| return on the investment of hiring a worker is the value of the worker employed in the firm at time |$t+2$| (multiplied by the probability that the worker remains with the firm), plus productivity minus the wage, all divided by the value of the worker at time |$t+1$|⁠. Note that the previous discussion implies that the value of the worker employed at |$t+1$| is |$\frac{\kappa_t}{q(\theta_t)}$|⁠.

Equation (12) directly follows from (10) and from the assumption that the cost of posting a vacancy is proportional to productivity (given our assumption of a nonstationary component to productivity, this implies a balanced growth path). We can evaluate (12) empirically. We take the historical time series of the price-productivity ratio and of |$N_{t+1}$| (equal to one minus the unemployment rate). Given standard parameters for the matching function (discussed further below), this implies, by way of (12), a time series for the vacancy-unemployment ratio |$\theta_t$|⁠. Figure 5 shows that the resultant ratio of vacancies to unemployment lines up closely with its counterpart in the data.

Theorem 1, which links labor market tightness to firm value, suggests that any mechanism which accounts for aggregate stock market volatility might also account for unemployment fluctuations. Here, we focus on variable disaster risk as a mechanism to generate stock market volatility. Well-known alternative explanations for aggregate stock market volatility include time-varying risk aversion (Campbell and Cochrane 1999) and stochastic volatility of the consumption drift rate (Bansal and Yaron 2004). Perhaps these mechanisms could also generate time-variation in labor market conditions.

While a full investigation is outside the scope of this manuscript, the prior literature suggests caution in this interpretation of Theorem 1. While Theorem 1 does indicate a link from labor market tightness to firm value, labor market tightness is itself endogenous, as is the number of employees |$N_t$|⁠. It is most natural to think of labor market tightness in a setting with production. As we discuss in the introduction, the insights of the models of Campbell and Cochrane (1999) and the stochastic volatility case of Bansal and Yaron (2004) have proven to be difficult to extend to settings in which agents can, in aggregate, transfer resources across time and states. Theorem 1 does not, by itself, guarantee realistic volatility in stock returns or labor market quantities.¹⁰

There is also a direct reason to prefer a model with rare disasters: such a model directly accounts for the Great Depression, with its large declines in output, consumption, and employment.

2.3 General equilibrium

In this section, we extend our previous results to general equilibrium. Theorem 1 still holds, but the general equilibrium setting allows us to model the underlying source of employment and stock price fluctuations.

2.3.1 The representative household

2.3.2 Wages

The parameter |$\nu$| controls the degree of tightness insulation.¹³ With |$\nu = 1$|⁠, we are back in the Nash bargaining case. With |$\nu = 0$|⁠, wages do not respond to labor market tightness. The resulting wage remains sensitive to productivity but loses some of its sensitivity to tightness. Furthermore, this formulation allows a direct comparison between versions of the model with and without tightness–insulated wages. Hall and Milgrom (2008) provides a microfoundation for (17).¹⁴

We assume |$b_t = b Z_t$| so that the model exhibits balanced growth. Besides being necessary from a modeling perspective, it is also realistic to link unemployment benefits (broadly defined) with productivity: as Chodorow-Reich and Karabarbounis (2015) show using micro data, the time benefits of unemployment are an empirically large fraction of total unemployment benefits. The importance of these time benefits imply that, in the data, total benefits to unemployment are procyclical.

2.3.3 Technology and the representative firm

In periods without a disaster, |$\hat{\kappa}_t$| reverts to a low level |$\underline{\kappa}$|⁠. If a disaster occurs, |$\hat{\kappa}_t$| moves to a high level |$\bar{\kappa}$|⁠. Thus a disaster is accompanied by a rise in |$\hat{\kappa}_t$|⁠, so that, in Equation (21), |$\kappa_t$| is roughly constant.¹⁷ If the most recent disaster in the economy was sufficiently far in the past (about five years), vacancy costs simply scale with productivity, with scale factor |$\underline{\kappa}$|⁠. We can therefore apply Equation (12) in Theorem 1 to postwar data, as we do in Section 2.2 (other implications of Theorem 1 depend only on the firm’s first order conditions, and not on the specification of |$\kappa_t$|⁠). The dynamics in Equation (22) imply that the cost of posting a vacancy falls by less than productivity in a disaster. Yet, because |$\hat{\kappa}_t$| is a stationary process, the economy remains stationary.

Following the literature on disasters and asset pricing (e.g., Barro 2006; Gourio 2012), we interpret a disaster broadly as any event that results in a large drop in GDP and consumption. Major wars, for example, lead to a large destruction in the capital stock, rendering existing workers less productive. A disruption in the financial system, or a major change in economic institutions could also lead to sharply lower output per worker.

2.3.4 Equilibrium

Note that consumption includes firm wages and dividends; the definition of dividends in (7) shows that the sum of wages and dividends amounts to |$Z_t N_t - \kappa_t V_t$|⁠. The household also consumes the flow value of unemployment. This implies that we are treating this flow value primarily as home production as opposed to unemployment benefits (which would be a transfer that would net to zero).¹⁸ To summarize, households maximize (14), subject to the budget constraint (23) and the law of motion for |$N_t$| (9), where |$\theta$| is taken as given. That the household owns all equity shares implies that the optimal investment in hiring is also that which solves the firm’s problem.

2.4 Comparative statics in a model with labor search and constant disaster probability

Before exploring the quantitative implications of our full model in Section 3, we explore the relation between disaster probabilities, labor dynamics, and prices. To do so, it is useful to consider a special case in which the scaled vacancy cost |$\hat{\kappa}_t$| is a constant |$\kappa$|⁠. Under this assumption, there is an isomorphism to an economy without disasters, but with a stochastic time discount factor |$\beta$|⁠. In the two economies, |$J_t/Z_t$|⁠, |$C_t/Z_t$|⁠, |$V_t$|⁠, |$N_t$|⁠, and the wealth-consumption ratio |$P_t/C_t$| are identical. However, realized returns, and thus the equity premium are not.¹⁹ We build on this result to show that, in a model with a constant disaster probability, the price-dividend ratio is decreasing; and unemployment, increasing, if and only if the EIS is greater than 1.

To derive closed-form solutions, we replace the random variable |$d_{t+1} \zeta_{t+1}$| with a compound Poisson process with intensity |$\tilde{\lambda}$|⁠. At our parameter values, the difference between the probability of a disaster |$\lambda$| and the intensity |$\tilde{\lambda}$| is negligible, and we continue to refer to |$\tilde{\lambda}$| as the disaster probability. Appendix B contains the proofs.

Equation (28) recursively defines the normalized value function in an economy without disaster risk. Theorem 2 shows that an economy with disasters (⁠|$\tilde{\lambda}>0$|⁠) is equivalent to one without, but with a less patient agent when the EIS |$\psi>1$| and a more patient agent when |$\psi<1$|⁠. As this statement suggests, the change to the time-discount factor due to disasters reflects a trade-off between an income and a substitution effect. On the one hand, the presence of disasters lead the agent to want to save (the income effect). But the mechanism that the agent has to shift consumption, namely, investing in hiring, becomes less attractive because there is a greater chance that the workers will not be productive (the substitution effect). When |$\psi>1$|⁠, the substitution effect dominates, and the agent, in effect, becomes less patient.

For the remainder of this section, we assume that the disaster probability is constant and derive comparative statics results. It will be useful to derive the effect of the probability of disaster on the risk-free rate and on the equity premium. In the case with constant |$\lambda_t$|⁠, these equations are the same as those in an endowment economy model (Tsai and Wachter 2015).

The risk of a rare disaster increases agents’ desire to save, which drives down the risk-free rate. In contrast to Theorem 2, this result holds regardless of the value of |$\psi$|⁠.

The first term in the equity premium represents the normal-times risk in production. Given the low volatility in productivity and consumption, this first term will be very small in our calibrated model. The second term represents the effect of rare disasters. A rare disaster causes an increase in marginal utility, represented by the term |$e^{-\gamma \zeta}-1$|⁠, at the same time as it causes a decrease in the value of the representative firm, as represented by |$e^\zeta-1$|⁠. Because the representative firm declines in value at exactly the wrong time, its equity carries a risk premium. Equation (31) shows that disasters require a large equity premium because they are both negative and variable. That is, marginal utility in a disaster period, |$e^{-\gamma \zeta}$|⁠, is high. This is a property of constant relative risk aversion.²⁰

Thus there is a tight connection between the price-dividend ratio and the effective time-discount factor.

The decomposition (35) provides additional intuition for the effect of changes in the disaster probability on the economy. On the one hand, an increase in the risk of a disaster drives down the risk-free rate. This will raise valuations, all else equal. However, it also increases the risk premium and lowers expected cash flows. When |$\psi>1$|⁠, the risk premium and cash flow effects dominate the risk-free rate effect and an increase in the disaster probability lowers valuations.

We now explicitly connect these results to the labor market.

When firms are faced with a higher risk of an economy-wide disaster, they have an incentive to reduce hiring. This decreases equilibrium tightness |$\theta$| to the point where firms are indifferent between hiring and not. Thus higher disaster risk results in higher unemployment, lower vacancies, and lower firm valuations.

The previous discussion separates the effects of the risk premium and the risk-free rate on the price-dividend ratio and hence on firm incentives. What about the discount rate overall? Hall (2017) conjectures that a model that produces higher discount rates in recessions can drive comovement of unemployment and the stock market. The analysis in this section shows that it is not discount rates per se that matter, but the combination of discount rates and growth expectations (it is also not necessary for these to be related to recessions driven by lower current productivity). For higher discount rates to be associated with lower unemployment, EIS greater than 1 is a necessary, but not a sufficient, condition:

Equation (36) follows from summing the equations for the equity premium (31) and the risk-free rate (30), or more directly by adding back expected cash flows to the price-dividend ratio in (32). The right hand side of (36) is negative, assuming |$\zeta<0$|⁠. Furthermore, |$\frac{1}{1-\gamma} \left(1-\mathbb{E} \left[e^{(1-\gamma) \zeta}\right] \right) $| is also negative, again assuming |$\zeta <0$|⁠. If |$\psi<1$|⁠, the left hand side would be positive, a contradiction. Note that the left hand side is the effect of the disaster probability on the price-dividend ratio. The right hand side is the effect on expected cash flows. The decline in the price-dividend ratio exceeds the decline in expected cash flows if and only if discount rates rise.

The analysis in this section sheds light on the tight link between the valuation mechanism and the labor market. As we will show in the next section, this mechanism is helpful in quantitatively explaining historical fluctuations in the labor market.

3. Quantitative Results

Below, we compare statistics in our model to those in the data. Section 3.1 describes the calibration of parameters for preferences, labor market variables, and productivity in normal times. Section 3.2 describes assumptions on the disaster distribution. Given these assumptions, Section 3.3 shows what happens to labor market, business-cycle, and financial moments when a disaster occurs or when the disaster probability increases. We then simulate repeated samples of length 60 years from our model. Section 3.4 describes statistics of labor market moments in simulated data. Section 3.5 describes statistics for business-cycle and financial moments. Section 3.6 makes use of alternative calibrations to highlight the main mechanisms behind our results.

3.1 Model parameters

Table 1 describes model parameters for our benchmark calibration.²¹ Unless otherwise stated, parameters are given in monthly terms. Labor productivity in normal times is calibrated to the labor productivity process from the postwar data (see Appendix E for a description of the data). This implies a monthly growth rate |$\mu$| of 0.18% and standard deviation |$\sigma_\epsilon$| of 0.47%. We calibrate the separation rate to 3.5% as estimated by Shimer (2005). We calibrate the Cobb-Douglas elasticity |$\eta$| to 0.55, consistent with empirical estimates in Petrongolo and Pissarides (2001) and Yashiv (2000). We calibrate the bargaining power of workers |$B$| following Hall and Milgrom (2008) and the flow value of unemployment |$b$| following Shimer (2005). We calibrate the matching efficiency |$\xi$| to the postwar unemployment rate of 5.9% (the median unemployment in samples without disasters is 6%). Our model also implies reasonable dynamics in population: the average unemployment is 6.4%, consistent with a 6.9% average in U.S. data from 1929. We calibrate |$\underline{\kappa}$| to match vacancy costs as a share of wages, reported by Silva and Toledo (2009) and Taschereau-Dumouchel (2015).

The behavior of |$\hat{\kappa}_t$| has important implications for unemployment during disasters. In the special case of constant |$\hat{\kappa}_t$|⁠, firms’ decisions to post vacancies depend only on this period’s employment |$N_t$| and on the disaster probability, not on the realization of a disaster. Though productivity declines in a disaster, costs decline also, so that optimal vacancy postings do not change. If, however, |$\hat{\kappa}_t$| rises in disasters, the vacancy cost |$\kappa_t = \hat{\kappa}_t Z_t$| remains roughly constant. The sudden decline in productivity implies that it is optimal for firms to sharply reduce vacancies. In calibrating the remaining free parameters of the |$\hat{\kappa}_t$| process (22), we seek to replicate the behavior of unemployment during the disaster for which we have the best data: the Great Depression. Figure 6 shows unemployment in U.S. data from 1931 to 1937. From 1931 to 1933, unemployment increased from 12% to 25%. The increase was of relatively short duration: unemployment was again below 12% in 1937. We choose |$\bar{\kappa}$| and |$\rho_\kappa$| to capture this behavior.²² We discuss the implications of time-varying |$\hat{\kappa}_t$| in greater detail in Sections 3.3 and 3.6.

To match the tightness-insulation parameter |$\nu$|⁠, we consider data on wage dynamics.²³Table 2 shows the standard deviation and autocorrelation of wages in the data, as well as the elasticity of wages with respect to labor market tightness and productivity. Also shown is the elasticity of labor market tightness to productivity. The elasticity of wages to labor market tightness is low throughout the sample, while the elasticity of wages to labor productivity ranges from 0.67 in the full sample, to close to unity in the sample after 1985. We consider two versions of the model, one that insulates wages from labor market tightness (our benchmark specification), and one with no tightness insulation (the Nash bargaining solution). For each case, we simulate 10,000 sample paths of 60 years of data and report the median, and the 5th and 95th percentile of each statistic. Tightness insulation allows the model to match the standard deviation of wages to that of the data; without tightness insulation, wages are too volatile. Tightness insulation is also consistent with other aspects of the data: it implies wages with unit elasticity with respect to productivity, but near zero elasticity with respect to labor market tightness. Under the Nash-bargaining solution, however, wages are unrealistically elastic with respect to labor market tightness.

We assume the EIS |$\psi$| is equal to 2 and risk aversion |$\gamma$| is equal to 5.7. As is standard in production-based models with recursive utility, an EIS greater than one is necessary for the model to deliver qualitatively realistic predictions for stock prices (see Section 2.4). An important question is whether this level of the EIS is consistent with other aspects of the data. Using instrumental variable estimation of consumption growth on interest rates, Hall (1988) and Campbell (2003) estimate this parameter to be close to zero. However, as noted by Bansal and Yaron (2004), this parameter estimate may be biased in models with time-varying second (or higher-order) moments. To gauge the impact of the misspecification, we repeat the instrumental-variable regressions of consumption growth on government bill rates in data simulated from our model.²⁴ We find a mean estimate of 0.34, consistent with the data. Thus, despite the assumption of an EIS greater than 1, our model replicates the weak relation between contemporaneous consumption growth and interest rates.

3.2 Size distribution and probability of disasters

The distribution for the disaster impact |$\zeta_t$| is taken from historical data on GDP declines in 36 countries over the last century (Barro and Ursua 2008). Following Barro and Ursua, we characterize a disaster by a 10% or higher cumulative decline in GDP. The resultant distribution for |$1-e^\zeta$| is shown in Figure 7. We assume that, if a disaster occurs, there is a 40% probability of default on government debt (Barro 2006).

We approximate the dynamics of the disaster probability |$\lambda_t$| in (20) using a 12-state Markov chain. The nodes and corresponding stationary probabilities are given in Table 3. The stationary distribution of monthly probabilities is approximately lognormal with a mean of 0.20% and standard deviation 1.97%. In comparison, the 10% criterion for a disaster implies that the annual frequency of disasters in the data is 3.7%, indicating that our assumption on the disaster frequency is conservative. We choose the persistence and the volatility of the disaster probability process to match the autocorrelation and volatility of unemployment in U.S. data.

Table 4 describes properties of the disaster probability distribution. Because this distribution is not available in closed form, we simulate 10,000 sample paths of length 60 years. We find that 53% of these sample paths do not have a disaster; thus the postwar period was not unusual from the point of view of our model. Because the distribution for the disaster probability is highly skewed, the average |$\lambda_t$| is much lower in samples that, ex post, have no disasters than it is in population. Below, we report statistics from these simulated data for unemployment, vacancies, and business-cycle and financial moments. Unless otherwise stated, the model statistics are computed from the no-disaster paths.

3.3 The effect of disasters and disaster probabilities

As we describe Section 2.3, a disaster is characterized by a permanent drop in labor productivity.²⁵ In a special case of the model with constant scaled vacancy cost, homogeneity of the production and utility functions implies that the vacancy posting rate |$V_t$| does not change with the arrival of a disaster. This (perhaps counterintuitive) result follows mathematically from the characterization of the normalized value function (27). In economic terms, if the vacancy cost scales completely with |$Z_t$|⁠, a decline in labor productivity in a disaster is completely offset by a decline in the cost of posting a vacancy, implying that equilibrium vacancy postings are unaffected, as is unemployment. Because |$V_t$| does not change when a disaster occurs, scaled consumption, wages, and firm profits do not change either. In this special case, the percent decline in |$C_t$|⁠, |$W_t$|⁠, and |$Y_t - W_tN_t$| upon disaster, is the same as the percentage decline in |$Z_t$|⁠.

The assumption that vacancy costs fall in proportion to a disaster, while elegant, does not seem realistic. Our benchmark model replaces this assumption with the more realistic assumption that, when a disaster occurs, vacancy costs remain temporarily high relative to |$Z_t$| (see Eq. 21–22). As a consequence, firms post fewer vacancies when a disaster occurs, and unemployment rises. In the years following a disaster, the path of vacancy costs reverts back to the path of |$Z_t$|⁠, so that vacancies gradually rise and unemployment gradually falls (this reversion is necessary for the model to be stationary). As we discuss in Section 3.1, this is what happened in the years during and after the Great Depression.²⁶

The behavior of vacancy costs during disasters affects the behavior of wages and firm profits relative to productivity. Because |$\kappa_t$| is a determinant of wages, an increase in |$\kappa_t$| relative to |$Z_t$| implies that wages increase relative to |$Z_t$|⁠. That is, wages are partially insulated during a disaster. This partial insulation of wages implies that the profit share |$(Y_t - W_tN_t)/Y_t$| sharply falls in a disaster, and then gradually recovers. The model thus endogenously creates an increase in effective firm leverage during a disaster, consistent with the data (Longstaff and Piazzesi 2004). This has important asset pricing implications, as we discuss in Section 3.6.²⁷

The implications of disasters extend beyond the disaster period itself. Because firm decisions are forward-looking, they incorporate the time-varying probability of a disaster. Thus, there are two notions of cyclicality in the model. One is variation in labor productivity, either due to normal shocks or disasters. The other is the variation in the probability of a disaster. It is the time-variation in the disaster probability that enables the model to account for changes in unemployment, labor market tightness, and stock prices that are apparently disconnected from realized shocks to labor productivity.

Figure 8 shows what happens to the labor market and to the business cycle in the months following an increase in the disaster probability. We assume an increase in the (monthly) probability from 2.08% to 13.50%, representing a two-standard-deviation increase along a typical no-disaster path. As we explain in Section 2.4, an increase in the disaster probability is associated with a decrease in labor market tightness. The increase in the disaster probability implies that the flows associated with a worker are both riskier and lower in expectation than before. Thus the optimal level of employment is lower, leading to higher unemployment. At a new steady-state level of unemployment, a lower level of vacancies is needed to sustain the lower number of workers. This can be seen from the steady-state values to which |$V_t$| and |$N_t$| converge in Figure 8. There are also short-term dynamics that we cannot see from Section 2.4. When the disaster probability falls, firms sharply decrease vacancy postings in an effort to get to the steady state new level of employment quickly. The magnitude of this decrease is mitigated by the congestion externality, because tightness falls as well. Though workers are less valuable in terms of the future discounted cash flows, it is now significantly cheaper to hire them. Thus vacancies fall by less than what they otherwise would. While vacancy postings rise after the initial shock, it takes about one year for them to converge to a new steady state level. During this time, unemployment steadily rises.²⁸

Figure 9 shows what happens to financial markets following a two-standard deviation increase in the disaster probability. Equity returns fall dramatically because of the sharp decline in stock prices described above. However, in the months following the increase, equity returns are slightly higher because of the greater risk premium needed to compensate investors for bearing the risk of a disaster. At the same time, the government bill rate falls because the greater degree of risk in the economy leads investors to want to save.

Figures 8 and 9 show that, in the model, an increase in the disaster probability increases the equity premium and decreases labor market tightness. This result raises the question of whether we can find an association between conditional moments of stock returns and labor market tightness in the data.

To answer this question, we form an empirical proxy for the Sharpe ratio at each time point by taking the average excess return and dividing by the standard deviation, using data over the subsequent 36 months. We find a correlation of |$-$|0.41 between these Sharpe ratios and labor market tightness. Figure 10 shows that the negative correlation is evident throughout the sample. We also ask whether labor market tightness forecasts excess returns. Table 5 shows that high labor market tightness forecasts low excess returns at horizons ranging from one month to 5 years. Predictive coefficients are all statistically significant at the 5% level, and the forecastability is economically significant at longer horizons. These results are qualitatively consistent with the model, though the model produces |$R^2$| coefficients that are larger than in the data. Thus there is a relation between risk premiums and labor market tightness, just as the model predicts.²⁹

3.4 Labor market moments

Table 6 describes labor market moments in the model and in the U.S. data from 1951 to 2013. Panel A reports U.S. data on unemployment |$U$|⁠, vacancies |$V$|⁠, the vacancy-unemployment ratio |$V/U$|⁠, labor productivity |$Z$|⁠, and the price-productivity ratio |$P/Z$|⁠. The labor market results replicate those reported by Shimer (2005) using more recent data. The vacancy-unemployment ratio has a quarterly volatility of 39%, twenty times higher than the volatility of labor productivity of 2%. The correlation between |$Z$| and |$V/U$| is 10%, whereas the correlation between |$P/Z$| and |$V/U$| is 47%, consistent with the findings in Section 1.³⁰ The correlation is lower in the pre-1985 sample, and higher in the post-1985 sample. These findings, together with the more detailed analysis in Section 1, motivate the mechanism in this paper.

Panel B of Table 6 reports the statistics calculated from sample paths simulated from the model. We simulate 10,000 sample paths of length 60 years. We report means from the 53% of simulations that contain no disaster. Our model is calibrated to match the volatility of unemployment. However, the model can also explain the volatility of vacancies, and the high volatility of the vacancy-unemployment ratio. The model correctly generates a large negative correlation between vacancies and unemployment. Other possible mechanisms, such as shocks to the separation rate, generate a counterfactual positive correlation between |$V$| and |$U$| (Shimer 2005). In addition, our model captures the low correlation between the labor market and productivity and the relatively high correlation between the labor market and stock prices; it overstates the latter correlation because a single state variable drive both. However, a united mechanism for both stock market and labor market volatility is a better description of the data compared to models based on realized productivity, especially for the U.S. data from the mid-1980s to the present.³¹

Figure 11 shows the Beveridge curve (namely, the locus of vacancies and unemployment) in the data and in the model. The position of the economy along the historically downward sloping Beveridge curve is an important business-cycle indicator (Blanchard and Diamond 1989). The time-varying risk mechanism in our model is able to generate such negative correlation, and as a result, the model values are concentrated along a downward sloping line. In our model, an increase in risk and a decrease in expected growth leads to downward movement along the Beveridge curve. Following an increase in disaster probability, the economy converges to the new optimal level of employment which is lower than before. Because the matching function is increasing in both vacancies and unemployment, a lower level for vacancies is needed to maintain the employment level. The model is able to generate a wide range of values on the vacancy-unemployment locus, including data values at the lower right corner of the Beveridge curve observed during the Great Recession which correspond to high values for the disaster probability.

3.5 Business-cycle and financial moments

We now turn to the model’s implications for consumption, output, and for financial markets. Table 7 shows that the model produces a low volatility of observed consumption and output, like in the data, with consumption volatility slightly lower than output volatility, reflecting the consumption-smoothing motives of the agent. Our definition of measured consumption does not include the flow value of unemployment as described in Section 2.3.4, and is therefore directly comparable to consumption expenditures in the data. Model-implied consumption volatility including the flow value of unemployment, |$b_t (1-N_t)$|⁠, is 1.4%.

As discussed in Section 3.3, there are two independent dimensions to cyclicality in the model, namely, comovement with labor productivity and with disaster probability.³² In the model, the effect of productivity shocks on consumption and output growth is identical during normal times. This is not the case for the disaster probability, however. Consumption equals output by the firm, plus home production, minus investment in hiring. Because both output and investment are pro-cyclical with respect to disaster probability, the consumption response to disaster probability shocks is weaker than the output response, as shown in Figure 8. This creates a higher volatility in output growth compared to consumption growth, in line with the data. The volatility of consumption and output is substantially higher in population than in samples without rare disasters, which are comparable to the postwar period.

Table 7 also shows that the model produces a realistically low average return and volatility for government bills. Following Barro (2006) and many others, we assume that there is a 40% probability that the government does not fully repay debt in a disaster (Tsai (2016) extends this approach to long-term bonds). We make the standard assumption that, upon default, the decline in value of the government bill rate is equal, expressed as percentages, to that of productivity. We find an average government bill rate of 3.7%, somewhat higher than in postwar data, but low compared to the expected return on equity and compared to the government bill rate in many models of production. The data fall well within the confidence bands implied by the model. Average returns on government bills are low in the model because of the precautionary savings motives arising from the risk of a disaster (Section 2.4). The volatility of government bond returns are also relatively low: 2.2%, the same as its postwar data value.³³

Even though output can have long periods with small shocks, there remains the possibility of a large disaster. Because firms’ cash flows are exposed to this disaster, in equilibrium, investors require a high premium to hold equity. Indeed, in samples without disasters generated from the model, the median equity premium is 5.7%. Because our model does not include financial leverage, we follow common practice (see, e.g., Nakamura et al. 2013) and report data values that are adjusted for leverage in the table.³⁴ The equity premium generated by our model is in fact higher than the adjusted value in the data, 5.3%, and is not far from the unadjusted value of 7.9%.³⁵

Besides matching the equity premium, our model can also generate high levels of return volatility. We can see this already in Figure 9 from the large return response in the event of an increase in the disaster probability. Table 7 shows, indeed, that return volatility implied by the model is 19.3% per annum, above the unlevered value in the data and close to the unadjusted value of 17.6%. Return volatility comes about through time variation in the probability of the disaster. When this probability rises, future prospects for growth dim, and the discount rate for this future growth increases. Embedded in the value of a firm is the value of a worker who is in place. When firm values fall, so too do the incentives for hiring. Thus our model produces high equity volatility, even though volatility of output is low.

While the connection between volatility in investment opportunities and volatility in firm valuations is intuitive, in practice, dynamic models with production often fail to come close to the observed volatility of stock returns. The problem is the hedging property of firm cash flows. Firms respond to bad news concerning the distribution of future productivity by cutting investment, and increasing dividends. A bad shock to the distribution of future productivity is accompanied by an increase in cash flows to investors. Thus, while the price might fall when a bad shock occurs, the cash flow increases, so the effect on the total return is ambiguous. It is in fact possible for equities to have returns that go up in bad times in models with production, leading to a negative equity premium. The negative correlation between price shocks and cash flow shocks also dampens return volatility. To produce reasonable implications for the equity premium and for equity volatility, models that focus on investment assume counterfactually high leverage (Gourio (2012)), or assume that stocks are something other than the dividend claim (Croce (2014)). Our model is also one of investment; posting a vacancy implies an investment in hiring. However, we are able to match the equity premium and return volatility without the assumption of leverage. One reason for this is the relative insensitivity of wages to labor market conditions. Another is endogenous sensitivity of dividends during disasters. We discuss these mechanisms further in the next section.

3.6 Sources of volatility and the equity premium

We compare our benchmark case with four alternative specifications. We consider a model without tightness insulation (Nash-bargained wages), a model with constant scaled vacancy cost, a model with constant disaster probability, and a model with both constant scaled vacancy cost and disaster probability.

Table 8 reports statistics for labor market volatility. The model with Nash bargaining implies about half the volatility in unemployment and labor market tightness as compared with the benchmark case. Because wages are more variable with respect to tightness, firms face a smaller incentive to reduce vacancy postings given an increase in the disaster probability. Firms simply pass on the greater risk of a disaster as lower wages to workers. As we discuss in Section 3.1, wages are unrealistically responsive to labor market tightness in this model.

The model with constant |$\hat{\kappa}$| also implies lower volatility of unemployment (though the difference with the benchmark case is smaller). The reason is not obvious. While constant |$\hat{\kappa}$| implies that unemployment remains constant in disasters, this does not explain lower volatility in no-disaster samples, which is what Table 8 reports. Rather, time-varying |$\hat{\kappa}$| makes disasters more costly for the firm, because wages do not fall in proportion to output. Because of this effective operating leverage, firms decrease vacancy postings, and thus employment, to a greater degree in response to a change in the disaster probability.

If we instead assume |$\hat{\kappa}_t$| varies, but that |$\lambda_t$| is constant, we obtain essentially no volatility in labor market variables. This confirms the intuition in Section 3.3: the primary force driving firm’s hiring incentives is the distribution of future productivity as captured by |$\lambda_t$|⁠. What little volatility there is in labor market variables in this case results from mean reversion during the no-disaster period, following a disaster that occurred prior to the start of the period.

Finally, when both |$\hat{\kappa}$| and |$\lambda$| are constant there is zero volatility in labor market variables in no-disaster periods.

Table 9 reports business-cycle and financial moments across the four cases, and for an even simpler case in which there are no disasters. We first compare the two simplest models: the one without disasters and the one with disasters but with constant |$\lambda_t$| and |$\hat{\kappa}_t$|⁠. Normal-times consumption and output growth are observationally equivalent in these two models. The observed equity premium and the Treasury bill rate, however, are not. Disasters raise the equity premium because marginal utility is very high; and returns very low, in disaster states. Investors require a large premium to compensate them for these risks. At the same time, the risk-free rate is much lower in the economy with disasters, because of precautionary savings.³⁶

We next consider a model with constant |$\lambda_t$|⁠, but allow |$\hat{\kappa}_t$| to vary. Relative to a model with constant |$\hat{\kappa}_t$|⁠, this increases firm volatility during disasters because profit share falls. As a result, the equity premium doubles, from 2.2% to 4.4%. Both models with constant |$\lambda_t$| imply equity volatility that is an order of magnitude lower than in the data. As shown in Table 8, these models also imply too-low volatility of labor markets.

We now set |$\hat{\kappa}_t$| to a constant but allow |$\lambda_t$| to vary. This reduces the equity premium relative to a model in which both are constant. This is surprising, since in an endowment economy, time-varying |$\lambda_t$| increases the equity premium. An increase in |$\lambda_t$| leads to a decline in firm prices, but to an increase (on impact) in firm cash flows, because the firm invests less in hiring. Realized equity returns are therefore positively correlated with the disaster probability.³⁷ This offsets the equity premium resulting from disasters (realized returns are still quite low in the case of a disaster itself). This model is counterfactual because it implies that unemployment and vacancies do not decline in a disaster.

Finally, consider the case with tightness-insulated wages. In this case, the equity premium is dramatically negative, despite the firm’s exposure to disasters. When wages are purely from Nash bargaining, investment in the firm becomes very safe because the firm has a cost structure that is highly sensitive to cyclical conditions in the economy. Wages are highly dependent on labor market tightness; when the labor market slackens, either because of economic disaster or simply an increase in the disaster probability, wages fall and profits rise. Thus firm returns hedge an increase in the disaster probability, and investors are willing to hold equity even at a negative equity premium.

While tightness-insulated wages are clearly important, it is also the case that tightness-insulation alone does not lead to an equity premium, high stock return volatility, or for that matter, volatile unemployment, as illustrated by our case with constant disaster risk, or no disaster risk. In these cases, equity volatility is an order of magnitude lower than in the data. Unlike in models with wage rigidities (Favilukis and Lin 2014; Uhlig 2007) or high operating leverage induced by high and stable wages (Petrosky-Nadeau, Zhang, and Kuehn 2013), wages fluctuate fully in response to changes in productivity in our model; it is their response to labor market conditions that is dampened (see Table 2). Equity volatility in our model arises from time-variation in risk premiums, not from fully insulated wages.

4. Capital Accumulation

In this section, we include physical capital accumulation in the model and show that our main results are robust to such an extension. Unless otherwise stated, all assumptions from our previous model remain the same.³⁸

Appendix D presents the solution to the representative firm’s problem with capital. We can derive the following theorem that is analogous to Theorem 1:

While capital and labor interact in a nonlinear way, the firm value is the sum of two components: the shadow prices of capital and labor, |$l_t^k$| and |$l_t^n$|⁠, multiplied by the respective quantities. In models with only capital accumulation (no intertemporal labor decision), the discrepancy between the firm value and the quantity of capital is solely driven by |$l_t^k$|⁠, which is a function of the investment rate in the presence of adjustment costs. The search and matching model implies an additional wedge between the replacement cost of capital and the value of aggregate equity, given by the term |$l_t^n N_{t+1}$|⁠. This corresponds to the entire firm value in our baseline model, and the labor component of firm value in the model with capital and labor accumulation.

We calibrate the model with capital and report results in Tables 10 and 11. We assume |$\alpha = 0.65$| and |$\delta = 0.01$|⁠. These are standard assumptions, consistent with the observed labor share and depreciation rate respectively. We also set |$r$| so that the elasticity of substitution between capital and labor is 0.65, consistent with the empirical estimates in Antras (2004). This implies higher complementarity between capital and labor compared to a Cobb-Douglas production technology. Finally, we set the adjustment cost parameter |$\xi_k$| to 0.5 to match the volatility of investment. We keep the rest of the model parameters identical to the baseline values reported in Table 1.