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

Risk, Unemployment, and the Stock Market: A Rare-Event-Based Explanation of Labor Market Volatility
Mete, Kilic,;A, Wachter, Jessica
2018-12-01 00:00:00
Abstract What is the driving force behind the cyclical behavior of unemployment and vacancies? What is the relation between firms’ job-creation incentives and stock market valuations? We answer these questions in a model with time-varying risk, modeled as a small and variable probability of an economic disaster. A high probability implies greater risk and lower future growth, lowering the incentives of firms to invest in hiring. During periods of high risk, stock market valuations are low and unemployment rises. The model thus explains volatility in equity and labor markets, and the relation between the two. Received May 12, 2016; editorial decision October 26, 2017 by Editor Stijn Van Nieuwerburgh. Authors have furnished data, which are available on the Oxford University Press Web site next to the link to the final published paper online. 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. Figure 1 View largeDownload slide The vacancy-unemployment ratio and labor productivity: 1951–2013 The solid line represents the vacancy-unemployment ratio and the dashed line labor productivity. Both variables are reported as log deviations from an HP trend with smoothing parameter |$10^5$|. Shaded areas represent NBER recessions. Figure 1 View largeDownload slide The vacancy-unemployment ratio and labor productivity: 1951–2013 The solid line represents the vacancy-unemployment ratio and the dashed line labor productivity. Both variables are reported as log deviations from an HP trend with smoothing parameter |$10^5$|. Shaded areas represent NBER recessions. 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 Figure 2 View largeDownload slide Valuation ratios: 1951–2013 |$P/Z$| denotes the price-productivity ratio defined as the real price of the S&P composite stock price index |$P$| divided by labor productivity |$Z$|. |$P/E$| is the cyclically adjusted price-earnings ratio of the S&P composite stock price index. |$P/Z$| is scaled such that |$P/Z$| and |$P/E$| are equal in the first quarter of 1951. Figure 2 View largeDownload slide Valuation ratios: 1951–2013 |$P/Z$| denotes the price-productivity ratio defined as the real price of the S&P composite stock price index |$P$| divided by labor productivity |$Z$|. |$P/E$| is the cyclically adjusted price-earnings ratio of the S&P composite stock price index. |$P/Z$| is scaled such that |$P/Z$| and |$P/E$| are equal in the first quarter of 1951. Figure 3 View largeDownload slide The vacancy-unemployment ratio and the price-productivity ratio: 1951–2013 The solid line represents the vacancy-unemployment ratio and the dashed line the price-productivity ratio. Both variables are reported as log deviations from an HP trend with smoothing parameter |$10^5$|. Shaded areas represent NBER recessions. Figure 3 View largeDownload slide The vacancy-unemployment ratio and the price-productivity ratio: 1951–2013 The solid line represents the vacancy-unemployment ratio and the dashed line the price-productivity ratio. Both variables are reported as log deviations from an HP trend with smoothing parameter |$10^5$|. Shaded areas represent NBER recessions. Figure 4 View largeDownload slide Vacancy openings and the price-productivity ratio: 1951–2013 The solid line shows vacancies, the dashed line the price-productivity ratio. Both variables are reported as log deviations from an HP trend with smoothing parameter |$10^5$|. Shaded areas represent NBER recessions. Figure 4 View largeDownload slide Vacancy openings and the price-productivity ratio: 1951–2013 The solid line shows vacancies, the dashed line the price-productivity ratio. Both variables are reported as log deviations from an HP trend with smoothing parameter |$10^5$|. Shaded areas represent NBER recessions. 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 The labor market is characterized by the DMP model of search and matching. The representative firm posts a number of job vacancies |$V_t >0$|. The hiring flow is determined according to the matching function |$m(N_t, V_t)$|, where |$N_t$| is employment in the economy and lies between 0 and 1. We assume that the matching function takes the following Cobb-Douglas form: \begin{align} m(N_t, V_t) = \xi (1-N_t)^\eta V_t^{1-\eta}, \end{align} (1) where |$\xi$| is matching efficiency and |$\eta$| is the unemployment elasticity of the hiring flow. As a result, the aggregate law of motion for employment is given by \begin{equation} N_{t+1} = (1 - s) N_t + m(N_t, V_t), \end{equation} (2) where |$s$| is the separation rate.6 Define labor market tightness as follows: \[ \theta_t = \frac{V_t}{U_t}. \] The unemployment rate in the economy is given by |$U_t = 1 - N_t$|. Thus the probability of finding a job for an unemployed worker is |$m(N_t, V_t)/U_t = \xi \theta_t^{1-\eta}$|. Accordingly, we define the job-finding rate |$f(\theta_t)$| to be \begin{equation} f(\theta_t) = \xi \theta_t^{1-\eta}. \end{equation} (3) Analogously, the probability of filling a vacancy posted by the representative firm is |$m(N_t, V_t)/V_t = \xi \theta_t^{-\eta}$| which corresponds to the vacancy-filling rate |$q(\theta_t)$| in the economy: \begin{align} q(\theta_t) = \xi \theta_t^{-\eta}. \end{align} (4) 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 Let |$M_{t+1}$| denote the stochastic discount factor, which exists provided that there is no arbitrage (Harrison and Kreps 1979). We begin by considering a very simple production function without installed capital, and consider the case of installed capital later in the paper. Consider a representative firm which produces output given by \begin{equation} Y_t = Z_t N_t, \end{equation} (5) where |$Z_{t}>0$| is the level of aggregate labor productivity. Assume that labor productivity follows the process \begin{equation} \log Z_{t+1} = \log Z_t + \mu + x_{t+1}, \end{equation} (6) where, for now, we leave |$x_{t+1}$| unspecified; it can be any stationary process. Let |$W_t = W(Z_t, N_t, V_t)$| denote the aggregate wage rate. The firm pays out dividends |$D_t$|, which is what remains from output after paying wages and investing in hiring: \begin{equation} D_t = Z_t N_t - W_t N_t - \kappa_t V_t. \end{equation} (7) The firm then maximizes the present value of current and future dividends \begin{equation} \underset{\{V_{t+\tau}, N_{t+\tau+1}\}^\infty_{\tau = 0}}{ \text{max }} \mathbb{E}_{t} \sum^{\infty}_{\tau = 0} M_{t+\tau} D_{t+\tau} \end{equation} (8) subject to \begin{equation} N_{t+1} = (1-s) N_t + q(\theta_t) V_t, \end{equation} (9) where |$q(\theta_t)$| is given in (4).8 At each time |$t$|, the firm chooses vacancies |$V_t$| and the number of employed workers |$N_{t+1}$|, subject to (9), taking as given its previous choice |$N_t$|, and economy-wide variables |$\theta_t$| and |$W_t$|. Because the firm does not take its own choices into account in the determination of |$\theta_t$|, the economy is therefore subject to a congestion externality. By posting more vacancies, firms raise the aggregate |$V_t$|, therefore increasing |$\theta_t$| and lowering the probability that any one firm will be able to hire. The following result establishes a general relation between the stock market and the labor market.9 Theorem 1. Assume the production function (5) and that the firm solves (8). Then the ex-dividend value of the firm is given by \begin{equation} P_t = \frac{\kappa_t}{q(\theta_t)} N_{t+1}, \end{equation} (10) and the equity return equals \begin{equation} R_{t+1} = \frac{ (1-s) \frac{\kappa_{t+1}}{q(\theta_{t+1})} + Z_{t+1} - W_{t+1}}{\frac{\kappa_t}{q(\theta_t)}}. \end{equation} (11) Furthermore, if |$\kappa_t = \kappa Z_t$| for fixed |$\kappa$|, then \begin{equation} \frac{P_t}{Z_t} = \frac{\kappa}{q(\theta_t)} N_{t+1}. \end{equation} (12) Proof. See Appendix A. ■ Let |$l_t$| denote the Lagrange multiplier on the firm’s hiring constraint (9). We can think of |$l_t$| as the value of a worker inside the firm at time |$t+1$|. In deciding how many vacancies to post at time |$t$|, the firm equates the marginal benefit of an additional worker with marginal cost. Because the probability of filling a vacancy with a worker is |$q(\theta_t)$| (see Section 2.1), the marginal benefit is |$l_t q(\theta_t)$| while the marginal cost is simply the cost of opening a vacancy, |$\kappa_t$|. Thus a condition for optimality is: \begin{equation} \kappa_t = l_t q(\theta_t). \end{equation} (13) 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. Figure 5 View largeDownload slide The vacancy-unemployment ratio in the data and in the model The solid line and the dashed line represent the vacancy-unemployment ratio in the data and in the model, respectively. Model-implied vacancies are calculated by substituting the price-productivity ratio and employment level from the data into (12), assuming labor-market parameters given in Table 1. Values are log deviations from an HP trend with smoothing parameter |$10^5$|. Shaded areas represent NBER recessions. Figure 5 View largeDownload slide The vacancy-unemployment ratio in the data and in the model The solid line and the dashed line represent the vacancy-unemployment ratio in the data and in the model, respectively. Model-implied vacancies are calculated by substituting the price-productivity ratio and employment level from the data into (12), assuming labor-market parameters given in Table 1. Values are log deviations from an HP trend with smoothing parameter |$10^5$|. Shaded areas represent NBER recessions. 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 Following Merz (1995) and Gertler and Trigari (2009), we assume that the representative household is a continuum of members who provide one another with perfect consumption insurance. We normalize the size of the labor force to one.11 The household maximizes utility over consumption, characterized by the recursive utility function introduced by Kreps and Porteus (1978) and Epstein and Zin (1989): \begin{equation} J_t = \left[C_t^{1-\frac{1}{\psi}} + \beta \left(\mathbb{E}_{t} \left[J_{t+1}^{1-\gamma} \right] \right)^{\frac{1-\frac{1}{\psi}}{1-\gamma}}\right]^{\frac{1}{1-\frac{1}{\psi}}}, \end{equation} (14) where |$\beta$| is the time discount factor, |$\gamma$| is relative risk aversion and |$\psi$| is the elasticity of intertemporal substitution (EIS). In case of |$\gamma = 1/\psi$|, recursive preferences collapse to power utility. The assumption of perfect consumption insurance greatly simplifies the computation in that it allows us to assume a representative agent. However, it most likely leads us to understate the effect of unemployment on the equity premium. The recursive utility function implies that, assuming optimal consumption, the stochastic discount factor takes the following form: \begin{equation} M_{t+1} = \beta \left(\frac{C_{t+1}}{C_t} \right)^{-\frac{1}{\psi}} \left(\frac{J_{t+1}}{\mathbb{E}_{t} \left[J_{t+1}^{1-\gamma} \right]^{\frac{1}{1-\gamma}}} \right)^{\frac{1}{\psi} - \gamma}. \end{equation} (15) 2.3.2 Wages The canonical DMP model assumes that wages are determined by Nash bargaining between the employer and the jobseeker. Both parties observe the surplus of job creation; the fraction received by the jobseeker is determined by his bargaining power. Pissarides (2000) shows that the Nash-bargained wage, |$W_t^N$|, is given by \begin{equation} W_t^N = (1-B) b_t + B (Z_t + \kappa_t \theta_t), \end{equation} (16) where |$0 \leq B \leq 1$| represents the worker’s bargaining power and |$b_t$| is the flow value of unemployment.12 The Nash-bargained wage is a weighted average of two components: the opportunity cost of employment and the contribution of the worker to the firm’s profits including the foregone cost of not having to hire. If the bargaining power of the worker is high, the contribution to profits receives more weight and the outside option less. The Nash-bargained wage is a useful benchmark. However, it implies wages that are unrealistically responsive to changes in labor market conditions (see Section 2). This is a well-known problem in the literature on labor market search. Hall (2005) proposes a rule that partially insulates wages from tightness in the labor market. Let \begin{equation} W_t = \nu W_t^N + (1-\nu) W_t^I, \end{equation} (17) where \begin{equation} W_t^I = (1-B) b_t + B (Z_t + \kappa_t \bar{\theta}). \end{equation} (18) 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 The representative firm produces output |$Y_t$| with technology |$Z_tN_t$| given in (5). In normal times, |$\log Z_t$| follows a random walk with drift. In every period, there is a small and time-varying probability of a disaster.15 Thus, \begin{equation} \log Z_{t+1} = \log Z_t + \mu + \epsilon_{t+1} + d_{t+1} \zeta_{t+1}, \end{equation} (19) where |$\epsilon_t \stackrel{iid}{\sim} N(0, \sigma_\epsilon^2)$|, \[ d_{t+1} = \begin{cases} 1 & \text{with probability } \lambda_t \\ 0 & \text{with probability } 1-\lambda_t, \end{cases} \] and where |$\zeta_t<0$| gives the decline in log productivity, should a disaster occur.16 We assume the log of the disaster probability |$\lambda_t$| follows an autoregressive process which (for convenience) is independent of the shocks to productivity. That is, \begin{equation} \text{ log } \lambda_t= \rho_\lambda \text{ log } \lambda_{t-1} + (1 - \rho_\lambda) \text{ log } \bar{\lambda} + \epsilon_t^\lambda, \end{equation} (20) where |$\bar{\lambda}$| is the mean log probability, |$\rho_\lambda$| is the persistence, and |$\epsilon^\lambda_t \stackrel{iid}{\sim} N(0, \sigma_\lambda^2)$|. In solving the model, we approximate this process using a finite-state Markov chain (see Table 3). Table 3 Markov switching process Value Stationary probability |$1 \times 10^{-7}$| 0.0005 |$7 \times 10^{-7}$| 0.0054 |$4 \times 10^{-6}$| 0.0269 |$3 \times 10^{-5}$| 0.0806 0.0002 0.1611 0.0012 0.2256 0.0076 0.2256 0.0495 0.1611 0.3212 0.0806 2.0827 0.0269 13.5045 0.0054 87.5661 0.0005 Value Stationary probability |$1 \times 10^{-7}$| 0.0005 |$7 \times 10^{-7}$| 0.0054 |$4 \times 10^{-6}$| 0.0269 |$3 \times 10^{-5}$| 0.0806 0.0002 0.1611 0.0012 0.2256 0.0076 0.2256 0.0495 0.1611 0.3212 0.0806 2.0827 0.0269 13.5045 0.0054 87.5661 0.0005 This table lists the nodes of a 12-state Markov process, which approximates an AR(1) process for log probabilities. Disaster probabilities are expressed as monthly percentages. Table 3 Markov switching process Value Stationary probability |$1 \times 10^{-7}$| 0.0005 |$7 \times 10^{-7}$| 0.0054 |$4 \times 10^{-6}$| 0.0269 |$3 \times 10^{-5}$| 0.0806 0.0002 0.1611 0.0012 0.2256 0.0076 0.2256 0.0495 0.1611 0.3212 0.0806 2.0827 0.0269 13.5045 0.0054 87.5661 0.0005 Value Stationary probability |$1 \times 10^{-7}$| 0.0005 |$7 \times 10^{-7}$| 0.0054 |$4 \times 10^{-6}$| 0.0269 |$3 \times 10^{-5}$| 0.0806 0.0002 0.1611 0.0012 0.2256 0.0076 0.2256 0.0495 0.1611 0.3212 0.0806 2.0827 0.0269 13.5045 0.0054 87.5661 0.0005 This table lists the nodes of a 12-state Markov process, which approximates an AR(1) process for log probabilities. Disaster probabilities are expressed as monthly percentages. In Section 2.2, we discuss the importance of scaling the hiring costs |$\kappa_t$| with |$Z_t$|; otherwise, since |$Z_t$| is subject to growth as well as permanent shocks, these costs would become either prohibitively high, or so small as to be irrelevant. However, the simplest way to scale |$\kappa_t$|, namely |$\kappa_t = \kappa Z_t$|, for a constant |$\kappa$|, implies that, if a disaster occurs, |$\kappa_t$| falls by the same percentage as productivity. This is a counterintuitive property for vacancy costs. We therefore allow |$\kappa_t$| to remain high relative to |$Z_t$| in disasters, but to catch up to |$Z_t$| in the long run. Assume \begin{equation} \kappa_t = \hat{\kappa}_t Z_t, \end{equation} (21) where \begin{equation} \hat{\kappa}_{t+1} = \left((1-\rho_\kappa) \, \underline{\kappa} + \rho_\kappa \, \hat{\kappa}_t \right) (1-d_{t+1}) + \bar{\kappa} \, d_{t+1}. \end{equation} (22) 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 In equilibrium, the representative household holds all equity shares of the representative firm. The representative household consumes the output |$Z_t N_t$| net of investment in hiring |$\kappa_t V_t$|, and the value of nonmarket activity |$b_t (1-N_t)$| achieved by the unemployed members: \begin{equation} C_t = Z_t N_t + b_t (1-N_t) - \kappa_t V_t. \end{equation} (23) 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. The proportionality assumptions on vacancy costs |$\kappa_t$| and the flow value of unemployment |$b_t$| in productivity |$Z_t$| imply \begin{equation} C_t = Z_t N_t + b Z_t (1-N_t) - \hat{\kappa}_t Z_t V_t. \end{equation} (24) Therefore, we can define consumption normalized by productivity, |$c_t = C_t/Z_t$|, as \begin{equation} c_t = N_t + b (1-N_t) - \hat{\kappa}_t V_t. \end{equation} (25) In equilibrium, the value function |$J_t$| is determined by productivity, the disaster probability, the scaled vacancy cost, and the employment level. That is, |$J_t = J(Z_t, \lambda_t,\hat{\kappa}_t, N_t)$|. Given our assumptions on productivity and the homogeneity of utility, the value function takes the form \begin{equation} J(Z_t, \lambda_t, \hat{\kappa}_t, N_t) = Z_t j(\lambda_t, \hat{\kappa}_t, N_t), \end{equation} (26) where we refer to |$j(\lambda_t, \hat{\kappa}_t, N_t)$| as the normalized value function. The normalized value function solves \begin{equation} j(\lambda_t, \hat{\kappa}_t, N_t) = \max_{c_t, V_t} \left[c_t^{1-\frac{1}{\psi}} + \beta \left(\mathbb{E}_{t} \left[\left(\frac{Z_{t+1}}{Z_t}\right)j(\lambda_{t+1}, \hat{\kappa}_{t+1}, N_{t+1})^{1-\gamma} \right] \right)^{\frac{1-\frac{1}{\psi}}{1-\gamma}}\right]^{\frac{1}{1-\frac{1}{\psi}}}, \end{equation} (27) subject to (25) and (9). The distribution of |$Z_{t+1}/Z_t$| is a function only of |$\lambda_t$|. This normalization implies that we can solve for all quantities of interest as functions of stationary state variables, |$\lambda_t$|, |$\hat{\kappa}_t$|, and |$N_t$|. 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. Theorem 2. Assume that |$\hat{\kappa}_t = \kappa$|. The normalized value function in a model with labor search and disasters is the same as the normalized value function in a model without disasters but with a stochastic time-discount factor. That is, the normalized value function solves \begin{equation} j(\tilde{\lambda}_t, N_t)^{1-\frac{1}{\psi}} = c_t^{1-\frac{1}{\psi}} + \hat{\beta}(\tilde{\lambda}_t) \left(\mathbb{E}_{t} \left[e^{(1-\gamma)(\mu+\epsilon_{t+1})}j(\tilde{\lambda}_{t+1}, N_{t+1})^{1-\gamma} \right] \right)^{\frac{1-\frac{1}{\psi}}{1-\gamma}}, \end{equation} (28) with the time-discount factor |$\hat{\beta}(\tilde{\lambda}_t)$| defined by \begin{equation} \log \hat{\beta}(\tilde{\lambda}_t) = \log \beta + \frac{1-\frac{1}{\psi}}{1-\gamma} \left(\mathbb{E}_t \left[e^{(1-\gamma) \zeta} \right]-1 \right) \tilde{\lambda}_t, \end{equation} (29) As a result, the policy functions |$c_t$| and |$V_t$| are also identical, as are |$N_t$| and |$P_t/Z_t$| (and |$P_t/C_t$|) by Theorem 1. Moreover, |$\hat{\beta} (\tilde{\lambda}_t)$| is decreasing in |$\tilde{\lambda}_t$| if and only if |$\psi > 1$|. 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). Lemma 1. Assume in a model with labor search that the disaster risk is constant and the labor market is at its steady state. The log risk-free rate is given by \begin{align} \log R_f &= -\log\beta + \frac{1}{\psi}\left(\mu + \frac{1}{2}\sigma_\epsilon^2\right)- \frac{1}{2}\left(\gamma + \frac{\gamma}{\psi} \right)\sigma_\epsilon^2\notag\\ &\quad + \left(\frac{\frac{1}{\psi} - \gamma}{1-\gamma} \mathbb{E} \left[e^{(1-\gamma) \zeta} - 1 \right] - \mathbb{E} \left[e^{-\gamma \zeta} -1 \right] \right)\tilde{\lambda}. \end{align} (30) The risk-free rate is decreasing in |$\tilde{\lambda}$|. 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 |$\psi$|. Lemma 2. Assume in a model with labor search that disaster risk is constant and the labor market is at its steady state. The equity premium is given by \begin{equation} \log \left(\frac{\mathbb{E}_t [R_{t+1}]}{R_f} \right) = \gamma \sigma^2_\epsilon + \tilde{\lambda}\mathbb{E}\left[\left( e^{-\gamma \zeta}-1\right)\left(1 - e^\zeta\right) \right]. \end{equation} (31) The equity premium is increasing in |$\tilde{\lambda}$|. 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 These rates of return can be connected back to the effective time-discount factor |$\hat{\beta}(\tilde{\lambda})$| as well as to the hiring decision of the firm. Consider a transformation of the price-dividend ratio: \begin{equation} h(\tilde{\lambda}) = -\log \left(1 + \frac{D_t}{P_t} \right) . \end{equation} (32) Then |$h(0)$| is the price-dividend ratio when there is no disaster risk: \begin{equation} h(0) = \log \beta + \left(1 - \frac{1}{\psi} \right) \left(\mu + \frac{1}{2}(1-\gamma) \sigma^2_\epsilon\right). \end{equation} (33) Thus there is a tight connection between the price-dividend ratio and the effective time-discount factor. Theorem 3. Assume in a model with labor search that disaster risk is constant and the labor market is at its steady state. Then \begin{equation} h(\tilde{\lambda}) - h(0) = \log \hat{\beta}(\tilde{\lambda}) - \log \beta. \end{equation} (34) Thus the price-dividend ratio is decreasing in |$\tilde{\lambda}$| if and only if |$\psi>1$|. From (30) and (31), it follows that the effect of disaster risk on |$h(\tilde{\lambda})$| can be decomposed into a discount rate effect (which in turn can be decomposed into a risk premium and risk-free rate effect) and an expected growth effect, like in Campbell and Shiller (1988): \begin{align} h(\tilde{\lambda}) - h(0) &= -\underbrace{\left(\frac{\frac{1}{\psi} - \gamma}{1-\gamma} \left(\mathbb{E} \left[e^{(1-\gamma) \zeta}\right] -1 \right) - \left(\mathbb{E} \left[e^{-\gamma \zeta} \right] - 1\right) \right) \tilde{\lambda}}_{\text{risk-free rate effect}}\nonumber \\ &\quad{} + \underbrace{\mathbb{E}\left[\left( e^{-\gamma \zeta}-1\right)\left(e^\zeta-1\right) \right] \tilde{\lambda}}_{\text{risk premium effect}} + \underbrace{\left(\mathbb{E} \left[e^\zeta \right] -1 \right)\tilde{\lambda}}_{\text{expected cash-flow effect}}. \end{align} (35) 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. Corollary 1. Assume in a model with labor search that disaster risk is constant and the labor market is at its steady state. Then The price-dividend ratio is increasing as a function of labor market tightness. Labor market tightness is decreasing in the probability of a disaster if and only if |$\psi>1$|. 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: Corollary 2. Assume in a model with labor search that disaster risk is constant and the labor market is at its steady state. The expected return is increasing in |$\tilde{\lambda}$| if and only if \begin{equation} \frac{1-\frac{1}{\psi}}{1-\gamma} \left(\mathbb{E} \left[e^{(1-\gamma) \zeta}\right] -1\right) < \mathbb{E} \left[e^\zeta \right]-1 \end{equation} (36) 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}$| to match vacancy costs as a share of wages, reported by Silva and Toledo (2009) and Taschereau-Dumouchel (2015). Table 1 Parameters values Parameter Value Time preference, |$\beta$| 0.997 Risk aversion, |$\gamma$| 4.9 Elasticity of intertemporal substitution, |$\psi$| 2 Productivity growth, |$\mu$| 0.0018 Productivity volatility, |$\sigma_\epsilon$| 0.0047 Matching efficiency, |$\xi$| 0.344 Separation rate, |$s$| 0.035 Matching function parameter, |$\eta$| 0.55 Bargaining power, |$B$| 0.50 Value of nonmarket activity, |$b$| 0.40 Vacancy cost in normal times, |$\underline{\kappa}$| 0.80 Vacancy cost in disasters, |$\bar{\kappa}$| 1.50 Vacancy cost persistence, |$\rho_{\kappa}$| 0.96 Tightness insulation, |$\nu$| 0.05 Government default probability, |$q$| 0.40 Parameter Value Time preference, |$\beta$| 0.997 Risk aversion, |$\gamma$| 4.9 Elasticity of intertemporal substitution, |$\psi$| 2 Productivity growth, |$\mu$| 0.0018 Productivity volatility, |$\sigma_\epsilon$| 0.0047 Matching efficiency, |$\xi$| 0.344 Separation rate, |$s$| 0.035 Matching function parameter, |$\eta$| 0.55 Bargaining power, |$B$| 0.50 Value of nonmarket activity, |$b$| 0.40 Vacancy cost in normal times, |$\underline{\kappa}$| 0.80 Vacancy cost in disasters, |$\bar{\kappa}$| 1.50 Vacancy cost persistence, |$\rho_{\kappa}$| 0.96 Tightness insulation, |$\nu$| 0.05 Government default probability, |$q$| 0.40 The model is simulated at a monthly frequency. Table 1 Parameters values Parameter Value Time preference, |$\beta$| 0.997 Risk aversion, |$\gamma$| 4.9 Elasticity of intertemporal substitution, |$\psi$| 2 Productivity growth, |$\mu$| 0.0018 Productivity volatility, |$\sigma_\epsilon$| 0.0047 Matching efficiency, |$\xi$| 0.344 Separation rate, |$s$| 0.035 Matching function parameter, |$\eta$| 0.55 Bargaining power, |$B$| 0.50 Value of nonmarket activity, |$b$| 0.40 Vacancy cost in normal times, |$\underline{\kappa}$| 0.80 Vacancy cost in disasters, |$\bar{\kappa}$| 1.50 Vacancy cost persistence, |$\rho_{\kappa}$| 0.96 Tightness insulation, |$\nu$| 0.05 Government default probability, |$q$| 0.40 Parameter Value Time preference, |$\beta$| 0.997 Risk aversion, |$\gamma$| 4.9 Elasticity of intertemporal substitution, |$\psi$| 2 Productivity growth, |$\mu$| 0.0018 Productivity volatility, |$\sigma_\epsilon$| 0.0047 Matching efficiency, |$\xi$| 0.344 Separation rate, |$s$| 0.035 Matching function parameter, |$\eta$| 0.55 Bargaining power, |$B$| 0.50 Value of nonmarket activity, |$b$| 0.40 Vacancy cost in normal times, |$\underline{\kappa}$| 0.80 Vacancy cost in disasters, |$\bar{\kappa}$| 1.50 Vacancy cost persistence, |$\rho_{\kappa}$| 0.96 Tightness insulation, |$\nu$| 0.05 Government default probability, |$q$| 0.40 The model is simulated at a monthly frequency. The behavior of |$\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$| in greater detail in Sections 3.3 and 3.6. Figure 6 View largeDownload slide Unemployment in the Great Depression This figure shows the unemployment rate from 1931 to 1937 in the data and implied by the model. Disaster probability is assumed to be 13.50% throughout. |$\kappa$| stays at its upper bound during the disaster period. Figure 6 View largeDownload slide Unemployment in the Great Depression This figure shows the unemployment rate from 1931 to 1937 in the data and implied by the model. Disaster probability is assumed to be 13.50% throughout. |$\kappa$| stays at its upper bound during the disaster period. To match the tightness-insulation parameter |$\nu$|, we consider data on wage dynamics.23Table 2 shows the standard deviation and autocorrelation of wages in the data, as well as the elasticity of wages with respect to labor market tightness and productivity. Also shown is the elasticity of labor market tightness to productivity. The elasticity of wages to labor market tightness is low throughout the sample, while the elasticity of wages to labor productivity ranges from 0.67 in the full sample, to close to unity in the sample after 1985. We consider two versions of the model, one that insulates wages from labor market tightness (our benchmark specification), and one with no tightness insulation (the Nash bargaining solution). For each case, we simulate 10,000 sample paths of 60 years of data and report the median, and the 5th and 95th percentile of each statistic. Tightness insulation allows the model to match the standard deviation of wages to that of the data; without tightness insulation, wages are too volatile. Tightness insulation is also consistent with other aspects of the data: it implies wages with unit elasticity with respect to productivity, but near zero elasticity with respect to labor market tightness. Under the Nash-bargaining solution, however, wages are unrealistically elastic with respect to labor market tightness. Table 2 Properties of aggregate wages SD AC |$\epsilon_{W, \theta}$| |$\epsilon_{W, Z}$| |$\epsilon_{\theta, Z}$| A: Data 1951–2013 1.77 0.91 0.00 0.67 2.46 — — [0.33] [5.43] [0.76] 1951–1985 1.21 0.91 0.01 0.35 11.22 — — [2.75] [3.04] [3.86] 1986–2013 2.29 0.91 |$-$|0.01 1.07 |$-$|8.49 — — [|$-$|1.15] [6.79] [|$-$|2.37] B: Benchmark model 50% 1.77 0.90 0.03 1.00 0.19 5% 1.38 0.84 |$-$|0.00|$-$| 0.90 |$-$|3.26|$-$| 95% 2.19 0.93 0.06 1.11 4.05 C: No tightness insulation of wages 50% 3.18 0.86 0.28 1.03 0.11 5% 2.56 0.78 0.22 0.53 |$-$|1.65|$-$| 95% 3.91 0.91 0.34 1.51 1.83 SD AC |$\epsilon_{W, \theta}$| |$\epsilon_{W, Z}$| |$\epsilon_{\theta, Z}$| A: Data 1951–2013 1.77 0.91 0.00 0.67 2.46 — — [0.33] [5.43] [0.76] 1951–1985 1.21 0.91 0.01 0.35 11.22 — — [2.75] [3.04] [3.86] 1986–2013 2.29 0.91 |$-$|0.01 1.07 |$-$|8.49 — — [|$-$|1.15] [6.79] [|$-$|2.37] B: Benchmark model 50% 1.77 0.90 0.03 1.00 0.19 5% 1.38 0.84 |$-$|0.00|$-$| 0.90 |$-$|3.26|$-$| 95% 2.19 0.93 0.06 1.11 4.05 C: No tightness insulation of wages 50% 3.18 0.86 0.28 1.03 0.11 5% 2.56 0.78 0.22 0.53 |$-$|1.65|$-$| 95% 3.91 0.91 0.34 1.51 1.83 SD denotes standard deviation and AC quarterly autocorrelation. |$Z$| is labor productivity, |$\theta$| labor market tightness. Data are from 1951 to 2013. All data and model moments are in quarterly terms. We simulate 10,000 samples with length 60 years at monthly frequency and report quantiles from 53% of simulations that include no disaster realization. |$\epsilon_{x, y}$| is the elasticity of variable |$x$| to |$y$|, namely, the regression coefficient of log |$x$| on log |$y$|. Data |$t$|-statistics in brackets are based on Newey-West standard errors. All variables are used in logs as deviations from an HP trend with smoothing parameter |$10^5$|. Table 2 Properties of aggregate wages SD AC |$\epsilon_{W, \theta}$| |$\epsilon_{W, Z}$| |$\epsilon_{\theta, Z}$| A: Data 1951–2013 1.77 0.91 0.00 0.67 2.46 — — [0.33] [5.43] [0.76] 1951–1985 1.21 0.91 0.01 0.35 11.22 — — [2.75] [3.04] [3.86] 1986–2013 2.29 0.91 |$-$|0.01 1.07 |$-$|8.49 — — [|$-$|1.15] [6.79] [|$-$|2.37] B: Benchmark model 50% 1.77 0.90 0.03 1.00 0.19 5% 1.38 0.84 |$-$|0.00|$-$| 0.90 |$-$|3.26|$-$| 95% 2.19 0.93 0.06 1.11 4.05 C: No tightness insulation of wages 50% 3.18 0.86 0.28 1.03 0.11 5% 2.56 0.78 0.22 0.53 |$-$|1.65|$-$| 95% 3.91 0.91 0.34 1.51 1.83 SD AC |$\epsilon_{W, \theta}$| |$\epsilon_{W, Z}$| |$\epsilon_{\theta, Z}$| A: Data 1951–2013 1.77 0.91 0.00 0.67 2.46 — — [0.33] [5.43] [0.76] 1951–1985 1.21 0.91 0.01 0.35 11.22 — — [2.75] [3.04] [3.86] 1986–2013 2.29 0.91 |$-$|0.01 1.07 |$-$|8.49 — — [|$-$|1.15] [6.79] [|$-$|2.37] B: Benchmark model 50% 1.77 0.90 0.03 1.00 0.19 5% 1.38 0.84 |$-$|0.00|$-$| 0.90 |$-$|3.26|$-$| 95% 2.19 0.93 0.06 1.11 4.05 C: No tightness insulation of wages 50% 3.18 0.86 0.28 1.03 0.11 5% 2.56 0.78 0.22 0.53 |$-$|1.65|$-$| 95% 3.91 0.91 0.34 1.51 1.83 SD denotes standard deviation and AC quarterly autocorrelation. |$Z$| is labor productivity, |$\theta$| labor market tightness. Data are from 1951 to 2013. All data and model moments are in quarterly terms. We simulate 10,000 samples with length 60 years at monthly frequency and report quantiles from 53% of simulations that include no disaster realization. |$\epsilon_{x, y}$| is the elasticity of variable |$x$| to |$y$|, namely, the regression coefficient of log |$x$| on log |$y$|. Data |$t$|-statistics in brackets are based on Newey-West standard errors. All variables are used in logs as deviations from an HP trend with smoothing parameter |$10^5$|. 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). Figure 7 View largeDownload slide Size distribution of disaster realizations This histogram shows the distribution of large declines in GDP per capita (in percentages). Data are from Barro and Ursua (2008). Values correspond to |$1-e^\zeta$| in the model. Figure 7 View largeDownload slide Size distribution of disaster realizations This histogram shows the distribution of large declines in GDP per capita (in percentages). Data are from Barro and Ursua (2008). Values correspond to |$1-e^\zeta$| in the model. 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. Table 4 Statistics for the disaster probability in simulated samples No-disaster samples All samples Population Mean 5% 50% 95% Mean 5% 50% 95% |$\mathbb{E}[\lambda]$| 0.20 0.05 0.01 0.03 0.16 0.20 0.01 0.07 0.75 |$\sigma (\lambda)$| 1.97 0.20 0.01 0.11 0.58 0.84 0.02 0.27 2.81 |$\rho (\lambda)$| 0.91 0.86 0.65 0.89 0.96 0.87 0.66 0.90 0.96 No-disaster samples All samples Population Mean 5% 50% 95% Mean 5% 50% 95% |$\mathbb{E}[\lambda]$| 0.20 0.05 0.01 0.03 0.16 0.20 0.01 0.07 0.75 |$\sigma (\lambda)$| 1.97 0.20 0.01 0.11 0.58 0.84 0.02 0.27 2.81 |$\rho (\lambda)$| 0.91 0.86 0.65 0.89 0.96 0.87 0.66 0.90 0.96 |$\sigma$| denotes volatility and |$\rho$| denotes the monthly autocorrelation. Disaster probabilities are expressed as monthly percentages. We simulate 10,000 samples with length 60 years at monthly frequency and report statistics from all simulations as well as from 53% of simulations that include no disaster realization. Table 4 Statistics for the disaster probability in simulated samples No-disaster samples All samples Population Mean 5% 50% 95% Mean 5% 50% 95% |$\mathbb{E}[\lambda]$| 0.20 0.05 0.01 0.03 0.16 0.20 0.01 0.07 0.75 |$\sigma (\lambda)$| 1.97 0.20 0.01 0.11 0.58 0.84 0.02 0.27 2.81 |$\rho (\lambda)$| 0.91 0.86 0.65 0.89 0.96 0.87 0.66 0.90 0.96 No-disaster samples All samples Population Mean 5% 50% 95% Mean 5% 50% 95% |$\mathbb{E}[\lambda]$| 0.20 0.05 0.01 0.03 0.16 0.20 0.01 0.07 0.75 |$\sigma (\lambda)$| 1.97 0.20 0.01 0.11 0.58 0.84 0.02 0.27 2.81 |$\rho (\lambda)$| 0.91 0.86 0.65 0.89 0.96 0.87 0.66 0.90 0.96 |$\sigma$| denotes volatility and |$\rho$| denotes the monthly autocorrelation. Disaster probabilities are expressed as monthly percentages. We simulate 10,000 samples with length 60 years at monthly frequency and report statistics from all simulations as well as from 53% of simulations that include no disaster realization. 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 8 View largeDownload slide Macroeconomic response to an increase in the disaster probability In month zero, monthly disaster probability increases from 2.08% to 13.50% and stays at 13.50% in the remaining months. Figure 8 View largeDownload slide Macroeconomic response to an increase in the disaster probability In month zero, monthly disaster probability increases from 2.08% to 13.50% and stays at 13.50% in the remaining months. 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. Figure 9 View largeDownload slide Return response to an increase in disaster probability In month zero, monthly disaster probability increases from 2.08% to 13.50% and stays at 13.50% in the remaining months. Figure 9 View largeDownload slide Return response to an increase in disaster probability In month zero, monthly disaster probability increases from 2.08% to 13.50% and stays at 13.50% in the remaining months. 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 Figure 10 View largeDownload slide Labor-market tightness and the Sharpe ratio The solid line represents the vacancy-unemployment ratio (labor market tightness) and the dashed line represents the Sharpe ratio. We report the vacancy-unemployment ratio as log deviations from an HP trend with smoothing parameter |$10^5$|. Shaded periods areas represent NBER recessions. For each time point in the sample, we compute the Sharpe ratio using data on returns over the subsequent 36 months. Figure 10 View largeDownload slide Labor-market tightness and the Sharpe ratio The solid line represents the vacancy-unemployment ratio (labor market tightness) and the dashed line represents the Sharpe ratio. We report the vacancy-unemployment ratio as log deviations from an HP trend with smoothing parameter |$10^5$|. Shaded periods areas represent NBER recessions. For each time point in the sample, we compute the Sharpe ratio using data on returns over the subsequent 36 months. Table 5 Excess return predictability by labor market tightness Months 1 6 12 24 36 60 A: Data |$\beta_\theta$| –0.13 –0.12 –0.10 –0.07 –0.07 –0.06 |$t$|-statistic [–3.33] [–3.82] [–3.51] [–3.10] [–3.33] [–4.33] |$R^2$| 0.02 0.07 0.10 0.11 0.14 0.23 B: |$\beta_\theta$| in the model Median, no-disaster samples –0.06 –0.06 –0.05 –0.04 –0.03 –0.03 Population –0.04 –0.04 –0.03 –0.02 –0.02 –0.01 C: |$R^2$| in the model Median, no-disaster samples 0.20 0.69 0.85 0.83 0.73 0.53 Population 0.14 0.59 0.68 0.56 0.41 0.23 Months 1 6 12 24 36 60 A: Data |$\beta_\theta$| –0.13 –0.12 –0.10 –0.07 –0.07 –0.06 |$t$|-statistic [–3.33] [–3.82] [–3.51] [–3.10] [–3.33] [–4.33] |$R^2$| 0.02 0.07 0.10 0.11 0.14 0.23 B: |$\beta_\theta$| in the model Median, no-disaster samples –0.06 –0.06 –0.05 –0.04 –0.03 –0.03 Population –0.04 –0.04 –0.03 –0.02 –0.02 –0.01 C: |$R^2$| in the model Median, no-disaster samples 0.20 0.69 0.85 0.83 0.73 0.53 Population 0.14 0.59 0.68 0.56 0.41 0.23 This table reports results from predictability regressions of the form \begin{equation*} \frac{1}{n} \left(\log(R_{t,t+n}) - \log(R^b_{t,t+n}) \right) = \beta_0 + \beta_\theta \log(\theta_t) + \epsilon_{t,t+n}, \end{equation*} where |$n$| is the predictability horizon in months and data are at the monthly frequency. Data |$t$|-statistics are based on Newey-West standard errors. We simulate 10,000 samples with length 60 years from the model and report quantiles from 53% of simulations that include no disaster realization. Population values are from a path with length 100,000 years. Table 5 Excess return predictability by labor market tightness Months 1 6 12 24 36 60 A: Data |$\beta_\theta$| –0.13 –0.12 –0.10 –0.07 –0.07 –0.06 |$t$|-statistic [–3.33] [–3.82] [–3.51] [–3.10] [–3.33] [–4.33] |$R^2$| 0.02 0.07 0.10 0.11 0.14 0.23 B: |$\beta_\theta$| in the model Median, no-disaster samples –0.06 –0.06 –0.05 –0.04 –0.03 –0.03 Population –0.04 –0.04 –0.03 –0.02 –0.02 –0.01 C: |$R^2$| in the model Median, no-disaster samples 0.20 0.69 0.85 0.83 0.73 0.53 Population 0.14 0.59 0.68 0.56 0.41 0.23 Months 1 6 12 24 36 60 A: Data |$\beta_\theta$| –0.13 –0.12 –0.10 –0.07 –0.07 –0.06 |$t$|-statistic [–3.33] [–3.82] [–3.51] [–3.10] [–3.33] [–4.33] |$R^2$| 0.02 0.07 0.10 0.11 0.14 0.23 B: |$\beta_\theta$| in the model Median, no-disaster samples –0.06 –0.06 –0.05 –0.04 –0.03 –0.03 Population –0.04 –0.04 –0.03 –0.02 –0.02 –0.01 C: |$R^2$| in the model Median, no-disaster samples 0.20 0.69 0.85 0.83 0.73 0.53 Population 0.14 0.59 0.68 0.56 0.41 0.23 This table reports results from predictability regressions of the form \begin{equation*} \frac{1}{n} \left(\log(R_{t,t+n}) - \log(R^b_{t,t+n}) \right) = \beta_0 + \beta_\theta \log(\theta_t) + \epsilon_{t,t+n}, \end{equation*} where |$n$| is the predictability horizon in months and data are at the monthly frequency. Data |$t$|-statistics are based on Newey-West standard errors. We simulate 10,000 samples with length 60 years from the model and report quantiles from 53% of simulations that include no disaster realization. Population values are from a path with length 100,000 years. 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. Table 6 Labor market moments |$U$| |$V$| |$V/U$| |$Z$| |$P/Z$| A: Data SD 0.19 0.21 0.39 0.02 0.16 AC 0.94 0.94 0.95 0.88 0.89 1 –0.86 –0.96 –0.18 –0.44 |$U$| — 1 0.97 0.03 0.47 |$V$| — — 1 0.10 0.47 |$V/U$| — — — 1 0.00 |$Z$| — — — — 1 |$P/Z$| B: No-disaster simulations SD 0.13 0.16 0.29 0.02 0.15 (0.03) (0.04) (0.07) (0.00) (0.04) AC 0.95 0.86 0.92 0.91 0.90 (0.02) (0.05) (0.03) (0.02) (0.03) 1 –0.91 –0.97 –0.01 –0.93 |$U$| — 1 0.99 0.01 0.97 |$V$| — — 1 0.01 0.98 |$V/U$| — — — 1 0.01 |$Z$| — — — — 1 |$P/Z$| C: Population SD 0.18 0.27 0.45 0.04 0.22 AC 0.98 0.93 0.96 0.93 0.94 1 –0.94 –0.98 –0.13 –0.91 |$U$| — 1 0.99 0.23 0.96 |$V$| — — 1 0.19 0.95 |$V/U$| — — — 1 0.16 |$Z$| — — — — 1 |$P/Z$| |$U$| |$V$| |$V/U$| |$Z$| |$P/Z$| A: Data SD 0.19 0.21 0.39 0.02 0.16 AC 0.94 0.94 0.95 0.88 0.89 1 –0.86 –0.96 –0.18 –0.44 |$U$| — 1 0.97 0.03 0.47 |$V$| — — 1 0.10 0.47 |$V/U$| — — — 1 0.00 |$Z$| — — — — 1 |$P/Z$| B: No-disaster simulations SD 0.13 0.16 0.29 0.02 0.15 (0.03) (0.04) (0.07) (0.00) (0.04) AC 0.95 0.86 0.92 0.91 0.90 (0.02) (0.05) (0.03) (0.02) (0.03) 1 –0.91 –0.97 –0.01 –0.93 |$U$| — 1 0.99 0.01 0.97 |$V$| — — 1 0.01 0.98 |$V/U$| — — — 1 0.01 |$Z$| — — — — 1 |$P/Z$| C: Population SD 0.18 0.27 0.45 0.04 0.22 AC 0.98 0.93 0.96 0.93 0.94 1 –0.94 –0.98 –0.13 –0.91 |$U$| — 1 0.99 0.23 0.96 |$V$| — — 1 0.19 0.95 |$V/U$| — — — 1 0.16 |$Z$| — — — — 1 |$P/Z$| SD denotes standard deviation and AC quarterly autocorrelation. Data are from 1951 to 2013. All data and model moments are in quarterly terms. |$U$| is unemployment, |$V$| vacancies, |$Z$| labor productivity, and |$P/Z$| price-productivity ratio. We simulate 10,000 samples with length 60 years at monthly frequency and report means from 53% of simulations that include no disaster realization in panel B. Standard errors across simulations are reported in parentheses. Population values in panel~C are from a path with length 100,000 years at monthly frequency. Standard deviations, autocorrelations and the correlation matrix are calculated using log deviations from an HP trend with smoothing parameter |$10^5$|. Table 6 Labor market moments |$U$| |$V$| |$V/U$| |$Z$| |$P/Z$| A: Data SD 0.19 0.21 0.39 0.02 0.16 AC 0.94 0.94 0.95 0.88 0.89 1 –0.86 –0.96 –0.18 –0.44 |$U$| — 1 0.97 0.03 0.47 |$V$| — — 1 0.10 0.47 |$V/U$| — — — 1 0.00 |$Z$| — — — — 1 |$P/Z$| B: No-disaster simulations SD 0.13 0.16 0.29 0.02 0.15 (0.03) (0.04) (0.07) (0.00) (0.04) AC 0.95 0.86 0.92 0.91 0.90 (0.02) (0.05) (0.03) (0.02) (0.03) 1 –0.91 –0.97 –0.01 –0.93 |$U$| — 1 0.99 0.01 0.97 |$V$| — — 1 0.01 0.98 |$V/U$| — — — 1 0.01 |$Z$| — — — — 1 |$P/Z$| C: Population SD 0.18 0.27 0.45 0.04 0.22 AC 0.98 0.93 0.96 0.93 0.94 1 –0.94 –0.98 –0.13 –0.91 |$U$| — 1 0.99 0.23 0.96 |$V$| — — 1 0.19 0.95 |$V/U$| — — — 1 0.16 |$Z$| — — — — 1 |$P/Z$| |$U$| |$V$| |$V/U$| |$Z$| |$P/Z$| A: Data SD 0.19 0.21 0.39 0.02 0.16 AC 0.94 0.94 0.95 0.88 0.89 1 –0.86 –0.96 –0.18 –0.44 |$U$| — 1 0.97 0.03 0.47 |$V$| — — 1 0.10 0.47 |$V/U$| — — — 1 0.00 |$Z$| — — — — 1 |$P/Z$| B: No-disaster simulations SD 0.13 0.16 0.29 0.02 0.15 (0.03) (0.04) (0.07) (0.00) (0.04) AC 0.95 0.86 0.92 0.91 0.90 (0.02) (0.05) (0.03) (0.02) (0.03) 1 –0.91 –0.97 –0.01 –0.93 |$U$| — 1 0.99 0.01 0.97 |$V$| — — 1 0.01 0.98 |$V/U$| — — — 1 0.01 |$Z$| — — — — 1 |$P/Z$| C: Population SD 0.18 0.27 0.45 0.04 0.22 AC 0.98 0.93 0.96 0.93 0.94 1 –0.94 –0.98 –0.13 –0.91 |$U$| — 1 0.99 0.23 0.96 |$V$| — — 1 0.19 0.95 |$V/U$| — — — 1 0.16 |$Z$| — — — — 1 |$P/Z$| SD denotes standard deviation and AC quarterly autocorrelation. Data are from 1951 to 2013. All data and model moments are in quarterly terms. |$U$| is unemployment, |$V$| vacancies, |$Z$| labor productivity, and |$P/Z$| price-productivity ratio. We simulate 10,000 samples with length 60 years at monthly frequency and report means from 53% of simulations that include no disaster realization in panel B. Standard errors across simulations are reported in parentheses. Population values in panel~C are from a path with length 100,000 years at monthly frequency. Standard deviations, autocorrelations and the correlation matrix are calculated using log deviations from an HP trend with smoothing parameter |$10^5$|. 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. Figure 11 View largeDownload slide Beveridge curve Data are quarterly from 1951 to 2013. Model implied curve is a quarterly sample with length 10,000 years from the stationary distribution. All values are log deviations from an HP trend with smoothing parameter |$10^5$|. Figure 11 View largeDownload slide Beveridge curve Data are quarterly from 1951 to 2013. Model implied curve is a quarterly sample with length 10,000 years from the stationary distribution. All values are log deviations from an HP trend with smoothing parameter |$10^5$|. 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%. Table 7 Business-cycle and financial moments |$\mathbb{E}[\Delta c]$| |$\mathbb{E}[\Delta y]$| |$\sigma(\Delta c)$| |$\sigma(\Delta y)$| |$\mathbb{E}[R-R_b]$| |$\mathbb{E}[R_b]$| |$\sigma(R)$| |$\sigma(R_b)$| Data 1.97 1.90 1.78 2.29 5.32 1.01 12.26 2.22 Simulation 50% 2.16 2.16 1.66 1.72 5.67 3.71 19.31 2.23 Simulation 5% 1.80 1.79 1.36 1.16 –1.34 0.76 10.37 0.86 Simulation 95% 2.51 2.54 1.97 2.66 18.62 4.71 36.81 6.83 Population 1.63 1.63 6.28 6.35 11.03 2.34 28.86 8.93 |$\mathbb{E}[\Delta c]$| |$\mathbb{E}[\Delta y]$| |$\sigma(\Delta c)$| |$\sigma(\Delta y)$| |$\mathbb{E}[R-R_b]$| |$\mathbb{E}[R_b]$| |$\sigma(R)$| |$\sigma(R_b)$| Data 1.97 1.90 1.78 2.29 5.32 1.01 12.26 2.22 Simulation 50% 2.16 2.16 1.66 1.72 5.67 3.71 19.31 2.23 Simulation 5% 1.80 1.79 1.36 1.16 –1.34 0.76 10.37 0.86 Simulation 95% 2.51 2.54 1.97 2.66 18.62 4.71 36.81 6.83 Population 1.63 1.63 6.28 6.35 11.03 2.34 28.86 8.93 This table reports means and volatilities of log consumption growth (|$\Delta c$|), log output growth (|$\Delta y$|), the government bill rate (|$R_b$|), and the unlevered equity return |$R$| in historical data and in data simulated from the model. All data and model moments are in annual terms. Historical data are from 1951 to 2013. We simulate 10,000 samples with length 60 years from the model and report quantiles from 53% of simulations that include no disaster realization. Population values are from a path with length 100,000 years. In the data, net equity returns are multiplied by 0.72 to adjust for leverage. Raw equity returns in the data have a premium of 7.90% and volatility of 17.55% over this period. Table 7 Business-cycle and financial moments |$\mathbb{E}[\Delta c]$| |$\mathbb{E}[\Delta y]$| |$\sigma(\Delta c)$| |$\sigma(\Delta y)$| |$\mathbb{E}[R-R_b]$| |$\mathbb{E}[R_b]$| |$\sigma(R)$| |$\sigma(R_b)$| Data 1.97 1.90 1.78 2.29 5.32 1.01 12.26 2.22 Simulation 50% 2.16 2.16 1.66 1.72 5.67 3.71 19.31 2.23 Simulation 5% 1.80 1.79 1.36 1.16 –1.34 0.76 10.37 0.86 Simulation 95% 2.51 2.54 1.97 2.66 18.62 4.71 36.81 6.83 Population 1.63 1.63 6.28 6.35 11.03 2.34 28.86 8.93 |$\mathbb{E}[\Delta c]$| |$\mathbb{E}[\Delta y]$| |$\sigma(\Delta c)$| |$\sigma(\Delta y)$| |$\mathbb{E}[R-R_b]$| |$\mathbb{E}[R_b]$| |$\sigma(R)$| |$\sigma(R_b)$| Data 1.97 1.90 1.78 2.29 5.32 1.01 12.26 2.22 Simulation 50% 2.16 2.16 1.66 1.72 5.67 3.71 19.31 2.23 Simulation 5% 1.80 1.79 1.36 1.16 –1.34 0.76 10.37 0.86 Simulation 95% 2.51 2.54 1.97 2.66 18.62 4.71 36.81 6.83 Population 1.63 1.63 6.28 6.35 11.03 2.34 28.86 8.93 This table reports means and volatilities of log consumption growth (|$\Delta c$|), log output growth (|$\Delta y$|), the government bill rate (|$R_b$|), and the unlevered equity return |$R$| in historical data and in data simulated from the model. All data and model moments are in annual terms. Historical data are from 1951 to 2013. We simulate 10,000 samples with length 60 years from the model and report quantiles from 53% of simulations that include no disaster realization. Population values are from a path with length 100,000 years. In the data, net equity returns are multiplied by 0.72 to adjust for leverage. Raw equity returns in the data have a premium of 7.90% and volatility of 17.55% over this period. 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. Table 8 Comparative statics for labor market volatility |$U$| |$V$| |$V/U$| |$P/Z$| Data 0.19 0.21 0.39 0.16 Benchmark 0.13 0.16 0.29 0.15 No tightness insulation of wages 0.06 0.08 0.13 0.08 Constant scaled-vacancy cost |$\hat{\kappa}$| 0.09 0.11 0.20 0.11 Constant disaster probability |$\lambda$| 0.01 0.03 0.03 0.00 Constant |$\hat{\kappa}$| and |$\lambda$| 0 0 0 0 |$U$| |$V$| |$V/U$| |$P/Z$| Data 0.19 0.21 0.39 0.16 Benchmark 0.13 0.16 0.29 0.15 No tightness insulation of wages 0.06 0.08 0.13 0.08 Constant scaled-vacancy cost |$\hat{\kappa}$| 0.09 0.11 0.20 0.11 Constant disaster probability |$\lambda$| 0.01 0.03 0.03 0.00 Constant |$\hat{\kappa}$| and |$\lambda$| 0 0 0 0 Standard deviations (in log deviations from an HP trend) for unemployment (|$U$|), vacancies (|$V$|), labor productivity (|$Z$|), and the price-productivity ratio (|$P/Z$|) in the data and in four versions of the model. Data are from 1951 to 2013. All data and model moments are in quarterly terms. Model values are calculated by simulating 10,000 samples with length 60 years at a monthly frequency. We report means from simulations that include no disaster realizations. In the constant disaster probability model, we set disaster probability to 0.20%, the stationary mean. In the constant |$\hat{\kappa}$| model, vacancy costs are assumed to be constant at its lower bound. Table 8 Comparative statics for labor market volatility |$U$| |$V$| |$V/U$| |$P/Z$| Data 0.19 0.21 0.39 0.16 Benchmark 0.13 0.16 0.29 0.15 No tightness insulation of wages 0.06 0.08 0.13 0.08 Constant scaled-vacancy cost |$\hat{\kappa}$| 0.09 0.11 0.20 0.11 Constant disaster probability |$\lambda$| 0.01 0.03 0.03 0.00 Constant |$\hat{\kappa}$| and |$\lambda$| 0 0 0 0 |$U$| |$V$| |$V/U$| |$P/Z$| Data 0.19 0.21 0.39 0.16 Benchmark 0.13 0.16 0.29 0.15 No tightness insulation of wages 0.06 0.08 0.13 0.08 Constant scaled-vacancy cost |$\hat{\kappa}$| 0.09 0.11 0.20 0.11 Constant disaster probability |$\lambda$| 0.01 0.03 0.03 0.00 Constant |$\hat{\kappa}$| and |$\lambda$| 0 0 0 0 Standard deviations (in log deviations from an HP trend) for unemployment (|$U$|), vacancies (|$V$|), labor productivity (|$Z$|), and the price-productivity ratio (|$P/Z$|) in the data and in four versions of the model. Data are from 1951 to 2013. All data and model moments are in quarterly terms. Model values are calculated by simulating 10,000 samples with length 60 years at a monthly frequency. We report means from simulations that include no disaster realizations. In the constant disaster probability model, we set disaster probability to 0.20%, the stationary mean. In the constant |$\hat{\kappa}$| model, vacancy costs are assumed to be constant at its lower bound. 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 Table 9 Comparative statics for business-cycle and financial moments |$\mathbb{E}[\Delta c]$| |$\mathbb{E}[\Delta y]$| |$\sigma(\Delta c)$| |$\sigma(\Delta y)$| |$\mathbb{E}[R-R_b]$| |$\mathbb{E}[R_b]$| |$\sigma(R)$| |$\sigma(R_b)$| Data 1.97 1.90 1.78 2.29 5.32 1.01 12.26 2.22 A: Benchmark Median 2.16 2.16 1.66 1.72 5.67 3.71 19.31 2.23 Population 1.63 1.63 6.28 6.35 11.03 2.34 28.86 8.93 B: No tightness insulation of wages Median 2.16 2.16 1.34 1.37 –44.92 4.03 14.00 1.38 Population 1.68 1.68 7.08 7.16 –44.31 2.37 15.32 9.09 C: Constant scaled-vacancy cost |$\hat{\kappa}$| Median 2.16 2.16 1.64 1.39 –4.53 3.83 12.35 2.47 Population 1.59 1.59 7.19 7.06 0.36 1.88 20.37 9.97 D: Constant disaster probability |$\lambda$| Median 2.16 2.16 1.33 1.32 4.72 2.34 1.76 0.00 Population 1.59 1.59 4.15 4.32 4.44 1.90 5.32 2.85 E: Constant |$\hat{\kappa}$| and |$\lambda$| Median 2.16 2.16 1.32 1.32 2.56 0.76 1.69 0.00 Population 1.59 1.59 3.70 4.30 2.19 0.27 2.97 3.72 F: No disaster risk Median 2.16 2.16 1.32 1.32 0.54 4.71 1.79 0.00 Population 2.16 2.16 1.32 1.32 0.55 4.71 1.79 0.00 |$\mathbb{E}[\Delta c]$| |$\mathbb{E}[\Delta y]$| |$\sigma(\Delta c)$| |$\sigma(\Delta y)$| |$\mathbb{E}[R-R_b]$| |$\mathbb{E}[R_b]$| |$\sigma(R)$| |$\sigma(R_b)$| Data 1.97 1.90 1.78 2.29 5.32 1.01 12.26 2.22 A: Benchmark Median 2.16 2.16 1.66 1.72 5.67 3.71 19.31 2.23 Population 1.63 1.63 6.28 6.35 11.03 2.34 28.86 8.93 B: No tightness insulation of wages Median 2.16 2.16 1.34 1.37 –44.92 4.03 14.00 1.38 Population 1.68 1.68 7.08 7.16 –44.31 2.37 15.32 9.09 C: Constant scaled-vacancy cost |$\hat{\kappa}$| Median 2.16 2.16 1.64 1.39 –4.53 3.83 12.35 2.47 Population 1.59 1.59 7.19 7.06 0.36 1.88 20.37 9.97 D: Constant disaster probability |$\lambda$| Median 2.16 2.16 1.33 1.32 4.72 2.34 1.76 0.00 Population 1.59 1.59 4.15 4.32 4.44 1.90 5.32 2.85 E: Constant |$\hat{\kappa}$| and |$\lambda$| Median 2.16 2.16 1.32 1.32 2.56 0.76 1.69 0.00 Population 1.59 1.59 3.70 4.30 2.19 0.27 2.97 3.72 F: No disaster risk Median 2.16 2.16 1.32 1.32 0.54 4.71 1.79 0.00 Population 2.16 2.16 1.32 1.32 0.55 4.71 1.79 0.00 |$\Delta c$| denotes log consumption growth, |$\Delta y$| log output growth, |$R$| the unlevered equity return, |$R_b$| the government bill rate. All data and model moments are in annual terms. We simulate 10,000 samples with length 60 years at monthly frequency and report the median from samples that contain no disasters. In the constant disaster probability model, we set disaster probability to 0.20%, the stationary mean of the disaster probability process used in the benchmark model. In the constant |$\kappa$| model, vacancy costs are assumed to be constant at its lower bound. Population values are from a path with length 100,000 years. Returns and growth rates are aggregated to annual values. Table 9 Comparative statics for business-cycle and financial moments |$\mathbb{E}[\Delta c]$| |$\mathbb{E}[\Delta y]$| |$\sigma(\Delta c)$| |$\sigma(\Delta y)$| |$\mathbb{E}[R-R_b]$| |$\mathbb{E}[R_b]$| |$\sigma(R)$| |$\sigma(R_b)$| Data 1.97 1.90 1.78 2.29 5.32 1.01 12.26 2.22 A: Benchmark Median 2.16 2.16 1.66 1.72 5.67 3.71 19.31 2.23 Population 1.63 1.63 6.28 6.35 11.03 2.34 28.86 8.93 B: No tightness insulation of wages Median 2.16 2.16 1.34 1.37 –44.92 4.03 14.00 1.38 Population 1.68 1.68 7.08 7.16 –44.31 2.37 15.32 9.09 C: Constant scaled-vacancy cost |$\hat{\kappa}$| Median 2.16 2.16 1.64 1.39 –4.53 3.83 12.35 2.47 Population 1.59 1.59 7.19 7.06 0.36 1.88 20.37 9.97 D: Constant disaster probability |$\lambda$| Median 2.16 2.16 1.33 1.32 4.72 2.34 1.76 0.00 Population 1.59 1.59 4.15 4.32 4.44 1.90 5.32 2.85 E: Constant |$\hat{\kappa}$| and |$\lambda$| Median 2.16 2.16 1.32 1.32 2.56 0.76 1.69 0.00 Population 1.59 1.59 3.70 4.30 2.19 0.27 2.97 3.72 F: No disaster risk Median 2.16 2.16 1.32 1.32 0.54 4.71 1.79 0.00 Population 2.16 2.16 1.32 1.32 0.55 4.71 1.79 0.00 |$\mathbb{E}[\Delta c]$| |$\mathbb{E}[\Delta y]$| |$\sigma(\Delta c)$| |$\sigma(\Delta y)$| |$\mathbb{E}[R-R_b]$| |$\mathbb{E}[R_b]$| |$\sigma(R)$| |$\sigma(R_b)$| Data 1.97 1.90 1.78 2.29 5.32 1.01 12.26 2.22 A: Benchmark Median 2.16 2.16 1.66 1.72 5.67 3.71 19.31 2.23 Population 1.63 1.63 6.28 6.35 11.03 2.34 28.86 8.93 B: No tightness insulation of wages Median 2.16 2.16 1.34 1.37 –44.92 4.03 14.00 1.38 Population 1.68 1.68 7.08 7.16 –44.31 2.37 15.32 9.09 C: Constant scaled-vacancy cost |$\hat{\kappa}$| Median 2.16 2.16 1.64 1.39 –4.53 3.83 12.35 2.47 Population 1.59 1.59 7.19 7.06 0.36 1.88 20.37 9.97 D: Constant disaster probability |$\lambda$| Median 2.16 2.16 1.33 1.32 4.72 2.34 1.76 0.00 Population 1.59 1.59 4.15 4.32 4.44 1.90 5.32 2.85 E: Constant |$\hat{\kappa}$| and |$\lambda$| Median 2.16 2.16 1.32 1.32 2.56 0.76 1.69 0.00 Population 1.59 1.59 3.70 4.30 2.19 0.27 2.97 3.72 F: No disaster risk Median 2.16 2.16 1.32 1.32 0.54 4.71 1.79 0.00 Population 2.16 2.16 1.32 1.32 0.55 4.71 1.79 0.00 |$\Delta c$| denotes log consumption growth, |$\Delta y$| log output growth, |$R$| the unlevered equity return, |$R_b$| the government bill rate. All data and model moments are in annual terms. We simulate 10,000 samples with length 60 years at monthly frequency and report the median from samples that contain no disasters. In the constant disaster probability model, we set disaster probability to 0.20%, the stationary mean of the disaster probability process used in the benchmark model. In the constant |$\kappa$| model, vacancy costs are assumed to be constant at its lower bound. Population values are from a path with length 100,000 years. Returns and growth rates are aggregated to annual values. 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 Aggregate output |$Y_t$| is determined by a production function |$F(K_t, Z_t N_t)$| that is given by \begin{equation} F(K_t, Z_t N_t) = \left((1-\alpha) K_t^r + \alpha (Z_t N_t)^r \right)^{\frac{1}{r}}, \end{equation} (37) where |$K_t$| is capital. This production function has two additional parameters: |$\alpha$|, which determines the share of labor in production, and |$r$|, which determines the degree of complementarity between capital and labor (the elasticity of substitution between capital and labor is given by |$\frac{1}{1-r}$|). Because this production function exhibits constant returns to scale, the economy is stationary. We assume that physical capital evolves according to \begin{equation} K_{t+1} = \left((1-\delta) K_t + \phi \left(\frac{I_t}{K_t} \right) K_t \right) e^{d_{t+1} \zeta_{t+1}}, \end{equation} (38) where |$I_t$| is investment, |$\delta$| is depreciation, and |$\phi(\cdot)$| is a concave function implying that capital accumulation is subject to adjustment costs. We follow Gourio (2012) in assuming capital destruction in the case of disasters. Like our scaling assumptions in Section 2, this eliminates the need to keep track of |$Z_t$| as a separate state variable, and implies a balanced growth path. Firm payouts equal: \begin{equation} D_t = F(K_t, Z_tN_t) - W_tN_t - \kappa_t V_t - I_t. \end{equation} (39) 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: Theorem 4. Assume the production function (37) and that the firm solves (8). Then the ex-dividend value of the firm is given by \begin{equation} P_t = l_t^k \tilde{K}_{t+1} + l_t^n N_{t+1}, \end{equation} (40) where |$\tilde{K}_{t+1} = K_{t+1} e^{-d_{t+1} \zeta_{t+1}}$|, |$l_t^k = 1/\phi'\left(\frac{I_t}{K_t} \right)$|, and |$l_t^n = \kappa_t/q(\theta_t)$|. 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. Following Jermann (1998) and Croce (2014), we assume the following for |$\phi(\cdot)$|: \begin{equation} \phi (x) = a_1 + \frac{a_2}{1-\frac{1}{\xi_k}} x^{1 - \frac{1}{\xi_k} }, \end{equation} (41) where |$\xi_k > 0$|. We set |$a_1$| and |$a_2$| so that adjustment becomes more costly as investment rate deviates from its steady state value.39 Finally, we assume that vacancy costs and the flow value of unemployment scale with the marginal product of labor, namely, |$\kappa_t = \hat{\kappa}_t F_{N,t}$| and |$b_t = b F_{N,t}$|, where |$F_{N,t}$| is the partial derivative of |$F(K_t, Z_t N_t)$| with respect to |$N_t$|. 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. Table 10 Labor market moments in the model with capital |$U$| |$V$| |$V/U$| |$Z$| |$P/Z$| A: Data SD 0.19 0.21 0.39 0.02 0.16 AC 0.94 0.94 0.95 0.88 0.89 1 –0.86 –0.96 –0.18 –0.44 |$U$| — 1 0.97 0.03 0.47 |$V$| — — 1 0.10 0.47 |$V/U$| — — — 1 0.00 |$Z$| — — — — 1 |$P/Z$| B: No-disaster simulations SD 0.13 0.19 0.31 0.02 0.21 (0.05) (0.07) (0.11) (0.00) (0.05) AC 0.95 0.86 0.92 0.91 0.91 (0.02) (0.05) (0.03) (0.02) (0.03) 1 –0.92 –0.97 0.06 –0.89 |$U$| — 1 0.99 –0.06 0.94 |$V$| — — 1 –0.06 0.94 |$V/U$| — — — 1 –0.01 |$Z$| — — — — 1 |$P/Z$| C: Population SD 0.16 0.25 0.40 0.04 0.22 AC 0.94 0.82 0.91 0.94 0.82 1 –0.89 –0.96 –0.19 –0.80 |$U$| — 1 0.98 0.12 0.90 |$V$| — — 1 0.15 0.88 |$V/U$| — — — 1 –0.11 |$Z$| — — — — 1 |$P/Z$| |$U$| |$V$| |$V/U$| |$Z$| |$P/Z$| A: Data SD 0.19 0.21 0.39 0.02 0.16 AC 0.94 0.94 0.95 0.88 0.89 1 –0.86 –0.96 –0.18 –0.44 |$U$| — 1 0.97 0.03 0.47 |$V$| — — 1 0.10 0.47 |$V/U$| — — — 1 0.00 |$Z$| — — — — 1 |$P/Z$| B: No-disaster simulations SD 0.13 0.19 0.31 0.02 0.21 (0.05) (0.07) (0.11) (0.00) (0.05) AC 0.95 0.86 0.92 0.91 0.91 (0.02) (0.05) (0.03) (0.02) (0.03) 1 –0.92 –0.97 0.06 –0.89 |$U$| — 1 0.99 –0.06 0.94 |$V$| — — 1 –0.06 0.94 |$V/U$| — — — 1 –0.01 |$Z$| — — — — 1 |$P/Z$| C: Population SD 0.16 0.25 0.40 0.04 0.22 AC 0.94 0.82 0.91 0.94 0.82 1 –0.89 –0.96 –0.19 –0.80 |$U$| — 1 0.98 0.12 0.90 |$V$| — — 1 0.15 0.88 |$V/U$| — — — 1 –0.11 |$Z$| — — — — 1 |$P/Z$| SD denotes standard deviation, AC quarterly autocorrelation. All model moments are in quarterly terms. |$U$| is unemployment, |$V$| vacancies, |$Z$| labor productivity, and |$P/Z$| price-productivity ratio. We simulate 10,000 samples with length 60 years at monthly frequency and report means from 53% of simulations that include no disaster realization in panel B. Standard errors across simulations are reported in parentheses. Population values in panel C are from a path with length 100,000 years at monthly frequency. Standard deviations, autocorrelations and the correlation matrix are calculated using log deviations from an HP trend with smoothing parameter |$10^5$|. Table 10 Labor market moments in the model with capital |$U$| |$V$| |$V/U$| |$Z$| |$P/Z$| A: Data SD 0.19 0.21 0.39 0.02 0.16 AC 0.94 0.94 0.95 0.88 0.89 1 –0.86 –0.96 –0.18 –0.44 |$U$| — 1 0.97 0.03 0.47 |$V$| — — 1 0.10 0.47 |$V/U$| — — — 1 0.00 |$Z$| — — — — 1 |$P/Z$| B: No-disaster simulations SD 0.13 0.19 0.31 0.02 0.21 (0.05) (0.07) (0.11) (0.00) (0.05) AC 0.95 0.86 0.92 0.91 0.91 (0.02) (0.05) (0.03) (0.02) (0.03) 1 –0.92 –0.97 0.06 –0.89 |$U$| — 1 0.99 –0.06 0.94 |$V$| — — 1 –0.06 0.94 |$V/U$| — — — 1 –0.01 |$Z$| — — — — 1 |$P/Z$| C: Population SD 0.16 0.25 0.40 0.04 0.22 AC 0.94 0.82 0.91 0.94 0.82 1 –0.89 –0.96 –0.19 –0.80 |$U$| — 1 0.98 0.12 0.90 |$V$| — — 1 0.15 0.88 |$V/U$| — — — 1 –0.11 |$Z$| — — — — 1 |$P/Z$| |$U$| |$V$| |$V/U$| |$Z$| |$P/Z$| A: Data SD 0.19 0.21 0.39 0.02 0.16 AC 0.94 0.94 0.95 0.88 0.89 1 –0.86 –0.96 –0.18 –0.44 |$U$| — 1 0.97 0.03 0.47 |$V$| — — 1 0.10 0.47 |$V/U$| — — — 1 0.00 |$Z$| — — — — 1 |$P/Z$| B: No-disaster simulations SD 0.13 0.19 0.31 0.02 0.21 (0.05) (0.07) (0.11) (0.00) (0.05) AC 0.95 0.86 0.92 0.91 0.91 (0.02) (0.05) (0.03) (0.02) (0.03) 1 –0.92 –0.97 0.06 –0.89 |$U$| — 1 0.99 –0.06 0.94 |$V$| — — 1 –0.06 0.94 |$V/U$| — — — 1 –0.01 |$Z$| — — — — 1 |$P/Z$| C: Population SD 0.16 0.25 0.40 0.04 0.22 AC 0.94 0.82 0.91 0.94 0.82 1 –0.89 –0.96 –0.19 –0.80 |$U$| — 1 0.98 0.12 0.90 |$V$| — — 1 0.15 0.88 |$V/U$| — — — 1 –0.11 |$Z$| — — — — 1 |$P/Z$| SD denotes standard deviation, AC quarterly autocorrelation. All model moments are in quarterly terms. |$U$| is unemployment, |$V$| vacancies, |$Z$| labor productivity, and |$P/Z$| price-productivity ratio. We simulate 10,000 samples with length 60 years at monthly frequency and report means from 53% of simulations that include no disaster realization in panel B. Standard errors across simulations are reported in parentheses. Population values in panel C are from a path with length 100,000 years at monthly frequency. Standard deviations, autocorrelations and the correlation matrix are calculated using log deviations from an HP trend with smoothing parameter |$10^5$|. Table 11 Business-cycle and financial moments in the model with capital |$\sigma(\Delta c)$| |$\sigma(\Delta i)$| |$\sigma(\Delta y)$| |$\mathbb{E}[R-R_b]$| |$\mathbb{E}[R_b]$| |$\sigma(R)$| |$\sigma(R_b)$| Data 1.78 8.76 2.29 5.32 1.01 12.26 2.22 Simulation 50% 1.90 7.20 1.42 8.86 5.45 24.56 2.80 Simulation 5% 1.54 5.52 1.19 0.38 4.02 13.79 1.20 Simulation 95% 2.29 9.37 1.75 24.70 6.28 43.50 7.56 Population 6.39 8.78 5.97 15.05 5.21 30.45 3.53 |$\sigma(\Delta c)$| |$\sigma(\Delta i)$| |$\sigma(\Delta y)$| |$\mathbb{E}[R-R_b]$| |$\mathbb{E}[R_b]$| |$\sigma(R)$| |$\sigma(R_b)$| Data 1.78 8.76 2.29 5.32 1.01 12.26 2.22 Simulation 50% 1.90 7.20 1.42 8.86 5.45 24.56 2.80 Simulation 5% 1.54 5.52 1.19 0.38 4.02 13.79 1.20 Simulation 95% 2.29 9.37 1.75 24.70 6.28 43.50 7.56 Population 6.39 8.78 5.97 15.05 5.21 30.45 3.53 The table reports means and volatilities of log consumption growth (|$\Delta c$|), log investment growth (|$\Delta i$|), log output growth (|$\Delta y$|), the government bill rate (|$R_b$|), and the unlevered equity return |$R$| in historical data and in data simulated from the model. All data and model moments are in annual terms. Historical data are from 1951 to2013. We simulate 10,000 samples with length 60 years from the model and report quantiles from 53% of simulations that include no disaster realization. Population values are from a path with length 100,000 years. In the data, net equity returns are multiplied by 0.72 to adjust for leverage. Raw equity returns in the data have a premium of 7.90% and volatility of 17.55% over this period. Table 11 Business-cycle and financial moments in the model with capital |$\sigma(\Delta c)$| |$\sigma(\Delta i)$| |$\sigma(\Delta y)$| |$\mathbb{E}[R-R_b]$| |$\mathbb{E}[R_b]$| |$\sigma(R)$| |$\sigma(R_b)$| Data 1.78 8.76 2.29 5.32 1.01 12.26 2.22 Simulation 50% 1.90 7.20 1.42 8.86 5.45 24.56 2.80 Simulation 5% 1.54 5.52 1.19 0.38 4.02 13.79 1.20 Simulation 95% 2.29 9.37 1.75 24.70 6.28 43.50 7.56 Population 6.39 8.78 5.97 15.05 5.21 30.45 3.53 |$\sigma(\Delta c)$| |$\sigma(\Delta i)$| |$\sigma(\Delta y)$| |$\mathbb{E}[R-R_b]$| |$\mathbb{E}[R_b]$| |$\sigma(R)$| |$\sigma(R_b)$| Data 1.78 8.76 2.29 5.32 1.01 12.26 2.22 Simulation 50% 1.90 7.20 1.42 8.86 5.45 24.56 2.80 Simulation 5% 1.54 5.52 1.19 0.38 4.02 13.79 1.20 Simulation 95% 2.29 9.37 1.75 24.70 6.28 43.50 7.56 Population 6.39 8.78 5.97 15.05 5.21 30.45 3.53 The table reports means and volatilities of log consumption growth (|$\Delta c$|), log investment growth (|$\Delta i$|), log output growth (|$\Delta y$|), the government bill rate (|$R_b$|), and the unlevered equity return |$R$| in historical data and in data simulated from the model. All data and model moments are in annual terms. Historical data are from 1951 to2013. We simulate 10,000 samples with length 60 years from the model and report quantiles from 53% of simulations that include no disaster realization. Population values are from a path with length 100,000 years. In the data, net equity returns are multiplied by 0.72 to adjust for leverage. Raw equity returns in the data have a premium of 7.90% and volatility of 17.55% over this period. Figure 12 shows the business-cycle effect of an increase in the disaster probabilities, following the same procedure as Figure 8 in the model without capital. The effect on employment, vacancies, and consumption is similar to the baseline model. An increase in the disaster probability leads to a decline investment that, while small relative to the decline in unemployment, lasts longer. Thus the decline in the capital stock is slower but larger in magnitude compared with the decline in employment. Figure 12 View largeDownload slide Macroeconomic response to an increase in the disaster probability in the model with capital In month zero, monthly disaster probability increases from 2.08% to 13.50% and stays at 13.50% in the remaining months. Figure 12 View largeDownload slide Macroeconomic response to an increase in the disaster probability in the model with capital In month zero, monthly disaster probability increases from 2.08% to 13.50% and stays at 13.50% in the remaining months. As shown in Tables 10 and 11, the model still delivers the key findings from the baseline model: the labor market is highly volatile, along with a high equity premium and stock market volatility. The government bill rate is too high compared to data which can be changed using the time discount factor and the government default probability. We do not change these parameters for parsimony. Finally, the volatility of aggregate investment growth is close to its data counterpart. These results confirm that the link between the labor market and stock market dynamics is robust to the inclusion of physical capital in the model. While an extensive analysis of the model dynamics with capital is beyond the scope of the present paper, two observations from our experiments are worthwhile noting. First, decreasing the complementarity between capital and labor has a negative effect on the equity premium. Complementarity between capital and labor makes wages smoother which makes dividends less countercyclical with respect to disaster probability dynamics. This is important for the equity premium as discussed in Section 3.6. Second, decreasing the importance of labor in production (by assuming a lower value for |$\alpha$|) also negatively affects the equity premium, as well as stock market volatility. Hence, it is the presence of labor frictions and smooth wages that give rise to realistic stock returns and volatility. Once the role of labor becomes less significant in production and dividends, dividends become increasingly countercyclical, as capital investment is large and procyclical, and not countered by wages that are not responsive to fluctuations in disaster risk. 5. Conclusion This paper shows that a business-cycle model with search and matching frictions in the labor market and a small and time-varying risk of an economic disaster can simultaneously explain labor market volatility, stock market volatility and the relation between unemployment and stock market valuations. While tractable, the model can generate high volatility in labor market tightness along with realistic aggregate wage dynamics. The findings suggest that time variation in aggregate uncertainty offers an important channel, through which the DMP model of labor market search and matching can operate. The model provides a mechanism through which job creation incentives of firms and stock market valuations are tightly linked, as the comovement of labor market tightness and stock market valuations in the data suggest. While the presence of disaster risk and realistic wage dynamics generate a high unlevered equity premium, the source of labor market volatility and stock market volatility is time variation in risk. Finally, the model is consistent with basic business-cycle moments, such as consumption growth and output growth. Our results suggest a number of directions for future research. First, the traditional view in production-based macroeconomics and asset pricing is that discrepancies between market and book value of capital arise from capital adjustment costs (this is sometimes called Q-theory; see, e.g., Hayashi (1982)). Our paper shows that market value of capital may differ from book value through an entirely different mechanism: the value of labor. Moreover, we provide a simple expression for such value that holds under a wide class of models. This implication could be taken to the data in the time series or cross-section, just as the Q-theory is now. Second, it is common in endowment economy models to assume that dividends respond more to risks than consumption; this is usually justified by appealing to leverage, but a link to actual leverage is often absent. Our model offers a tractable and empirically sound source for such an assumption that could be used in settings other than what we have presented here. Third, our model offers a way to bring congestion externalities into asset pricing in a tractable way that can be compared with the data. These congestion externalities may be present in markets besides that for labor. Finally, our model offers a set of specific links between labor and the equity market that could be investigated empirically, both in the cross-section of firms and in international data. We thank Andy Abel, Gabriel Chodorow-Reich, Max Croce, Giuliano Curatola, Zvi Eckstein, Stefano Giglio, Joao Gomes, Francois Gourio, Urban Jermann, Alexandr Kopytov, Monika Merz, Christian Opp, Sang Byung Seo, Ctirad Slavik, Nicolas Petrosky-Nadeau, Mathieu Taschereau-Dumouchel, Michael Wachter, Nancy Ran Xu and Amir Yaron, and seminar participants at the Federal Reserve Board, the Wharton School, the University of Texas Austin, the University of Chicago, Emory University, the SAFE Asset Pricing Workshop 2015, the Trans-Atlantic Doctoral Conference 2015, Western Finance Association Meetings 2015, the Macro Finance Society 2016, and the Society of Economic Dynamics Meetings 2017 for helpful comments. Supplementary data can be found on The Review of Financial Studies web site. Appendix A. Results for General SDF The results in this section do not depend on our assumptions on |$M_{t+1}$| or |$Z_{t+1}$|. Proof of Theorem 1 Firm dividends equal output minus wage costs and investment in hiring \begin{equation} D_t = Z_t N_t - W_t N_t - \kappa_t V_t. \end{equation} (A.1) The firm takes wages |$W_t$| and labor market tightness |$\theta_t$| as given and maximizes the cum-dividend value \begin{equation} P^c_t = \underset{\{V_{t+\tau}, N_{t+\tau+1}\}^\infty_{\tau = 0}}{ \text{max }} \mathbb{E}_{t} \sum^{\infty}_{\tau = 0} M_{t+\tau} \left[Z_{t+\tau} N_{t+\tau} - W_{t+\tau} N_{t+\tau} - \kappa_{t+\tau} V_{t+\tau} \right], \end{equation} (A.2) subject to the law of motion for employment \begin{equation} N_{t+1} = (1-s) N_t + q(\theta_t) V_t. \end{equation} (A.3) The first-order conditions with respect to |$V_t$| and |$N_{t+1}$| are given by \begin{gather} 0 = -1 + l_t \frac{q(\theta_t)}{\kappa_t}\\ \end{gather} (A.4) \begin{gather} l_t = \mathbb{E}_{t} \left[M_{t+1} (Z_{t+1} - W_{t+1} + l_{t+1} (1-s)) \right], \end{gather} (A.5) where |$l_t$| is the Lagrange multiplier on the aggregate law of motion for employment level. Note that (A.5) can be interpreted as an Euler equation with |$l_t$| as the value of a worker inside the firm. We expand (A.2) using (A.3): \begin{equation} \begin{aligned} P^c_t &= Z_t N_t - W_t N_t - \kappa_t V_t - l_t \left(N_{t+1} - (1-s) N_t - \frac{q(\theta_t)}{\kappa_t} \kappa_t V_t \right)\\ &+\mathbb{E}_{t} \left[M_{t+1} \left[Z_{t+1} N_{t+1} - W_{t+1} N_{t+1} - \kappa_{t+1} V_{t+1} - l_{t+1} \left(N_{t+2} - (1-s) N_{t+1} - \frac{q(\theta_{t+1})}{\kappa_{t+1}} \kappa_{t+1} V_{t+1} \right) \right] \right]\\ &+ ... \end{aligned} \end{equation} (A.6) The terms |$- \kappa_t V_t$| and |$l_t \frac{q(\theta_t)}{\kappa_t} \kappa_t V_t$| cancel out for all |$t$| as a result of (A.4). Furthermore, |$l_t N_{t+1}$| cancels out with |$\mathbb{E}_t \left[Z_{t+1} N_{t+1} - W_{t+1} N_{t+1} + l_{t+1} (1-s) N_{t+1} \right]$| for all |$t$| as a result of (A.5). It follows that \begin{equation} P^c_t = Z_t N_t - W_t N_t + l_t (1-s) N_t. \end{equation} (A.7) Consider the ex-dividend value of equity |$P_t = P^c_t - D_t$|. The definition of the dividend and (A.7) imply \begin{equation} \begin{aligned} P_t &= Z_t N_t - W_t N_t + l_t (1-s) N_t - Z_t N_t + W_t N_t + \kappa_t V_t\\ &= \kappa_t V_t + l_t (1-s) N_t\\ &= \frac{\kappa_t}{q(\theta_{t})} (N_{t+1} - (1-s) N_t) + \frac{\kappa_t}{q(\theta_{t})} (1-s) N_t\\ &= l_t N_{t+1}. \end{aligned} \end{equation} (A.8) Combining (A.8) with (A.4) results in (10). We now show (11). From (10) and the definition of dividends, it follows that \begin{equation} \begin{aligned} R_{t+1} &\equiv \frac{P_{t+1} + D_{t+1}}{P_t} \\ & = \frac{l_{t+1} N_{t+2} + Z_{t+1} N_{t+1} - W_{t+1} N_{t+1} - \kappa_{t+1} V_{t+1}}{l_t N_{t+1}}\\ &= \frac{l_{t+1} \frac{N_{t+2}}{N_{t+1}} + Z_{t+1} - W_{t+1} - \frac{\kappa_{t+1} V_{t+1}}{N_{t+1}}}{l_t}\\ &= \frac{l_{t+1} \left[1-s+\frac{q(\theta_{t+1})}{\kappa_{t+1}} \frac{\kappa_{t+1} V_{t+1}}{N_{t+1}} \right] + Z_{t+1} - W_{t+1} - \frac{\kappa_{t+1} V_{t+1}}{N_{t+1}}}{l_t}\\ &= \frac{Z_{t+1} - W_{t+1} + l_{t+1} (1-s)}{l_t}\\ &= \frac{Z_{t+1} - W_{t+1} + (1-s) \frac{\kappa_{t+1}}{q(\theta_{t+1})}}{\frac{\kappa_t}{q(\theta_t)}}. \end{aligned} \end{equation} (A.9) Using this result, we provide characterizations of returns and prices that will be useful in what follows. Lemma A.1. Under the assumptions |$\kappa_t = Z_t\kappa$| and |$b_t = Z_t b$|, the equity return equals \begin{align} R_{t+1} = \frac{(1-s)\frac{\kappa}{q(\theta_{t+1})} + 1 - w(\theta_{t+1})}{\frac{\kappa}{q(\theta_t)}}\frac{Z_{t+1}}{Z_t}, \end{align} (A.10) where |$w(\theta_t)$| is the wage normalized by productivity: \begin{equation} w(\theta_t) = (1-B) b + B (1+\kappa(\nu \theta_t + (1-\nu) \bar{\theta})). \end{equation} (A.11) The result directly follows from (11). Given |$l_t$| as the value of a worker inside the firm, the Euler equation (A.5) suggests a notion of a payout of a worker inside the firm: \begin{equation} D^l_t = Z_t - W_t - s l_t. \end{equation} (A.12) Lemma A.2. Under the assumptions |$\kappa_t = Z_t\kappa$| and |$b_t = Z_t b$|, the payout ratio of a worker employed in a firm is given by \begin{align} \frac{D^l_t}{l_t} &= \frac{Z_t - W_t - s l_t}{l_t} \\ \end{align} (A.13) \begin{align} &= \frac{1-w(\theta_t) - s \frac{\kappa}{q(\theta_t)}}{\frac{\kappa}{q(\theta_t)}}. \end{align} (A.14) Proof. The result directly follows from (A.4) and (A.12). ■ How does this notion of payout ratio relate to the more traditional dividend-price ratio? Lemma A.3. Consider the dividend-price ratio for the firm, |$D_t/P_t$|. Then, \begin{equation} 1+\frac{D_t}{P_t} = \left(1 + \frac{D^l_t}{l_t} \right)\frac{N_t}{N_{t+1}}. \end{equation} (A.15) Thus, if the labor market is in a steady state (defined as |$N_t = N_{t+1}$|), |$D_t/P_t = D^l_t/l_t$|. Proof. It follows from (10), the definition of dividends (7), and the law of motion for |$N_t$| (9) that \begin{align*} P_t + D_t & = l_t N_{t+1} + Z_t N_t - W_t N_t - \kappa_t V_t \\ & = (Z_t - W_t + l_t(1-s))N_t. \end{align*} Thus \begin{align*} 1 + \frac{D_t}{P_t} & = \frac{P_t + D_t}{P_t} \\ & = \frac{Z_t - W_t + l_t (1-s)}{l_t}\frac{N_t}{N_{t+1}} \\ & = \left(1 + \frac{D_t^l}{l_t} \right) \frac{N_t}{N_{t+1}}, \end{align*} where the last line follows from (A.13). ■ Appendix B. Proofs for Section 2.4 The algebraic rules for compound Poisson processes illustrated in this section are adapted from Cont and Tankov (2004). Drechsler and Yaron (2011) model jumps in expected growth and volatility using compound Poisson processes. Let |$Q_{t+1}$| be a compound Poisson process with intensity |$\tilde{\lambda}_t$|. Specifically, |$\tilde{\lambda}_t$| represents the expected number of jumps in the time period |$(t, t+1]$|. Agents in the model view the jumps in |$(t, t+1]$| as occurring at |$t+1$|. Then, |$Q_{t+1}$| is given by \begin{equation} Q_{t+1} = \begin{cases} \sum_{i=1}^{\mathcal{N}_{t+1}-\mathcal{N}_{t}} \zeta_i & \text{if } \mathcal{N}_{t+1}-\mathcal{N}_{t} > 0 \\ 0 & \text{if } \mathcal{N}_{t+1}-\mathcal{N}_{t} = 0, \end{cases} \end{equation} (B.1) where |$\mathcal{N}_{t}$| is a Poisson counting process and |$\mathcal{N}_{t+1}-\mathcal{N}_{t}$| is the number of jumps in the time interval |$(t, t+1]$|. Jump size |$\zeta_{i}$| is |$iid$|.40 We can take conditional expectations with |$Q_{t+1}$| using \begin{equation} \mathbb{E}_t \left[e^{u Q_{t+1}} \right] = e^{\tilde{\lambda}_t \left(\mathbb{E} _t\left[e^{u \zeta_{t+1}} \right] - 1\right)}, \end{equation} (B.2) where log of the right-hand side is the cumulant-generating function of |$Q_{t+1}$|. More precisely, the probability of observing |$k$| jumps over the course one period |$(t, t+1]$| is equal to |$e^{\tilde{\lambda}_t} \frac{\tilde{\lambda}_t^k}{k!}$|. We take the |$t$| to be in units of months in our quantitative assessment of the model. Given this notion of a compound Poisson process, we can rewrite the equation for technology as \begin{equation} \log Z_{t+1} = \log Z_t + \mu + \epsilon_{t+1} + Q_{t+1}. \end{equation} (B.3) Proof of Theorem 2 Consider the normalized value function in (27) in the special case of |$\hat{\kappa}_t = \kappa$| and with the productivity process (B.3). It follows that, at the optimum, \begin{equation} j(\tilde{\lambda}_t, N_t) = \left[c_t^{1-\frac{1}{\psi}} + \beta \left(\mathbb{E}_{t} \left[e^{(1-\gamma)(\mu+\epsilon_{t+1}+{Q_{t+1}})}j(\tilde{\lambda}_{t+1}, N_{t+1})^{1-\gamma} \right] \right)^{\frac{1-\frac{1}{\psi}}{1-\gamma}}\right]^{\frac{1}{1-\frac{1}{\psi}}}. \end{equation} (B.4) Conditional on time-|$t$| information, the random variables |$\epsilon_{t+1}$| and |$Q_{t+1}$| are independent of |$\lambda_{t+1}$| and |$N_{t+1}$|.41 Therefore, we can write (28) with \begin{equation} \hat{\beta}(\tilde{\lambda}_t) = \beta \mathbb{E}_t \left[e^{(1-\gamma) Q_{t+1}} \right]^{\frac{1 - \frac{1}{\psi}}{1-\gamma}}. \end{equation} (B.5) Taking the expectation, we compute the log of the effective time discount factor: \begin{equation} \log \hat{\beta}(\tilde{\lambda}_t) = \log \beta + \frac{1-\frac{1}{\psi}}{1-\gamma} \left(\mathbb{E}_t \left[e^{(1-\gamma) \zeta_{t+1}} \right]-1 \right) \tilde{\lambda}_t. \end{equation} (B.6) Note that |$\zeta_{t+1}$| takes only negative values. We have \begin{equation} \frac{\mathbb{E}_t \left[e^{(1-\gamma) \zeta_{t+1}} \right]-1}{1-\gamma} < 0. \end{equation} (B.7) Therefore, |$\log \hat{\beta}(\tilde{\lambda}_t)$| is decreasing in |$\tilde{\lambda}_t$| if and only if |$1-\frac{1}{\psi} > 0$| which is equivalent to |$\psi > 1$|. Because the value function is concave, the solutions for |$c_t$| and |$V_t$| must be the same regardless of whether the economy is faced with disaster with probability |$\tilde{\lambda}_t$|, or whether the time-discount factor is |$\hat{\beta}(\tilde{\lambda}_t)$|. Moreover, by Theorem 1, |$P_t/Z_t$| is the same in the two economies. Proof of Lemma 1 Because |$\tilde{\lambda}_t$| is constant and the economy is at its steady state, the stochastic discount factor (15) becomes: \begin{equation} M_{t+1} = \frac{\beta e^{-\frac{\mu}{\psi} - \gamma(\epsilon_{t+1} + Q_{t+1})}}{\mathbb{E}_t \left[e^{(1-\gamma) (\epsilon_{t+1}+Q_{t+1})} \right]^{\frac{\frac{1}{\psi}-\gamma}{1-\gamma}}}. \end{equation} (B.8) Here we have used (26) to substitute in for the value function. Taking the expectation in the denominator, the log stochastic discount factor becomes \begin{equation} \begin{aligned} \log M_{t+1} &= \log \beta - \frac{\mu}{\psi} - \gamma (\epsilon_{t+1} + Q_{t+1}) \\ &- \frac{1}{2} \left(\frac{1}{\psi} - \gamma \right) \left(1-\gamma \right) \sigma^2_\epsilon - \frac{\frac{1}{\psi}-\gamma}{1-\gamma} \left(\mathbb{E} \left[e^{(1-\gamma) \zeta} \right] \right) \tilde{\lambda}. \end{aligned} \end{equation} (B.9) It follows that the log risk-free rate |$\log R_f = -\log \mathbb{E} [M_{t+1}]$| is given by: \begin{equation} \begin{aligned} \log R_f = &-\log\beta + \frac{\mu}{\psi} + \frac{1}{2}\left(\frac{1}{\psi} - \frac{\gamma}{\psi} - \gamma \right) \sigma^2_\epsilon \\ &+ \left[\frac{\frac{1}{\psi} - \gamma}{1-\gamma} \left(\mathbb{E} \left[e^{(1-\gamma) \zeta} \right] - 1 \right) - (\mathbb{E} \left[e^{-\gamma \zeta} \right]-1) \right] \tilde{\lambda}. \end{aligned} \end{equation} (B.10) The term |$\frac{\frac{1}{\psi} - \gamma}{1-\gamma}$| is bounded above by |$\gamma/(\gamma-1)$|. The properties of the exponential implies |$\frac{1}{\gamma} \mathbb{E} \left(\left[e^{- \gamma \zeta} \right] -1 \right) > \frac{1}{\gamma - 1} \mathbb{E} \left(\left[e^{- \gamma \zeta} \right] -1 \right)$|, which, together with the fact that |$\zeta$| takes only negative values, implies that the risk-free rate is decreasing in disaster intensity (Tsai and Wachter (2015)). The following Lemma recharacterizes the Euler equation in terms of model primitives: Lemma B.1. The first-order conditions of the firm imply \begin{equation} \hat{\beta} (\tilde{\lambda}) e^{\mu \left(1-\frac{1}{\psi} \right) + \frac{1}{2} (1-\gamma)\left(1-\frac{1}{\psi} \right) \sigma^2_\epsilon} \left[\frac{1 - w(\theta) + (1-s) \frac{\kappa}{q(\theta)}}{\frac{\kappa}{q(\theta)}}\right] = 1, \end{equation} (B.11) where |$w(\theta)$| is the wage normalized by productivity defined by (A.11) and |$\hat{\beta} (\tilde{\lambda})$| is defined as in (B.5). Proof. We rewrite the Euler equation (A.5) in the familiar form: \begin{equation} E_t\left[M_{t+1}R_{t+1} \right] = 1, \end{equation} (B.12) where we have divided through by |$l_t$| and used the characterization of returns in (11). The result follows from substituting (A.10) and (B.8) into (B.12) and solving the expectation using the definition of |$\hat{\beta} (\tilde{\lambda})$| in (B.5). ■ Lemma B.2. The log expected equity return is given by \begin{align} \log \mathbb{E}_t \left[R_{t+1}\right] & = - \log (\beta) + \frac{\mu}{\psi} + \frac{1}{2} \left(\frac{1}{\psi} - \frac{\gamma}{\psi}+\gamma \right)\nonumber \\ &\quad{} + \underbrace{\left(\mathbb{E} \left[e^\zeta \right] -1\right)}_{\text{Productivity growth}} \tilde{\lambda}\nonumber\\ &\quad{} - \underbrace{\left(\frac{1-\frac{1}{\psi}}{1-\gamma} \left(\mathbb{E} \left[e^{(1-\gamma) \zeta}\right] - 1 \right)\right)}_{\text{Labor market}} \tilde{\lambda}. \end{align} (B.13) Proof. (B.11) implies \begin{equation} \frac{1 - w(\theta) + (1-s) \frac{\kappa}{q(\theta)}}{\frac{\kappa}{q(\theta)}} = \frac{1}{\hat{\beta} (\tilde{\lambda}) e^{\mu \left(1 - \frac{1}{\psi} \right) + \frac{1}{2} (1-\gamma) \left(1 - \frac{1}{\psi} \right) \sigma^2_\epsilon}}. \end{equation} (B.14) Therefore, by (A.10), \begin{equation} R_{t+1} = \frac{e^{\mu + \epsilon_{t+1} + Q_{t+1}}}{\hat{\beta} (\tilde{\lambda}) e^{\mu \left(1 - \frac{1}{\psi} \right) + \frac{1}{2} (1-\gamma) \left(1 - \frac{1}{\psi} \right) \sigma^2_\epsilon}}, \end{equation} (B.15) Equation (B.13) follows from taking the expectation of (B.15) using (B.2). ■ Lemma 2 follows from (B.13) and the equation for the risk-free rate given in (B.10). Corollary 2 follows from inspection of the terms multiplying |$\tilde{\lambda}$| in (B.13). Finally we establish comparative statics for the price-dividend ratio. Proof of Theorem 3 It follows from (B.11) that \begin{equation} - \log \left(1 - s + \frac{1 - w(\theta)}{\kappa} \xi \theta^{-\eta} \right) = \log \hat{\beta}(\tilde{\lambda}) + \mu \left(1-\frac{1}{\psi} \right) + \frac{1}{2} (1-\gamma) \left(1-\frac{1}{\psi} \right) \sigma^2_\epsilon, \end{equation} (B.16) where, from Theorem 2, \[ \log \hat{\beta}(\tilde{\lambda}) = \log \beta + \frac{1-\frac{1}{\psi}}{1-\gamma} \left(\mathbb{E} \left[e^{(1-\gamma) \zeta} \right]-1 \right) \tilde{\lambda}. \] Define \begin{equation} h(\tilde{\lambda}) \equiv -\log\left(1+\frac{D^l_t}{l_t}\right) = -\log\left(1+\frac{D_t}{P_t}\right). \end{equation} (B.17) The second equality follows from Lemma A.3. The result then follows from adding 1 to (A.14) and taking the negative of the log, then substituting the result into the left hand side of (B.16). Proof of Corollary 1 We first show that the price-dividend ratio is increasing as a function of labor-market tightness |$\theta$|.42 It follows from (A.14) that \[ \frac{D^l_t}{l_t} = \frac{1-w(\theta)}{\frac{\kappa}{q(\theta)}} - s. \] Because of (4), the first term is proportional to |$(1-w(\theta))\theta^{-\eta}$|. It follows from (A.11) that |$w(\theta)$| is increasing in |$\theta$| (intuitively, wages are increasing in tightness). Note that equilibrium requires that |$1-w(\theta)$| be positive; otherwise upon reaching the steady state the firm would operate continually at a loss. Therefore |$(1-w(\theta))\theta^{-\eta}$| is decreasing in |$\theta$|, and, by the second statement in Lemma A.3, the price-dividend ratio is increasing in |$\theta$|. The second statement of the corollary follows from the first statement and Theorem 3 Appendix C. Equilibrium Solution Let |$x'$| denote the value of the variable |$x$| in period |$t+1$| and |$x$| the value at |$t$|. The value function and policy functions are functions of the exogenous state variables |$\lambda$| and |$\hat{\kappa}$|, as well as the endogenous state variable |$N$|. The dynamics of the stochastic discount factor and returns are driven by five shocks: disaster probability |$\lambda '$|, normal times productivity shock |$\epsilon'$|, disaster indicator |$d'$|, disaster size |$\zeta'$|, and |$\hat{\kappa}'$|. Let |$\mathbb{E}$| be the expectation operator over four shocks. In our numerical procedure, we solve for the consumption policy |$c(\lambda, N)$|. The market clearing condition allows us to compute the vacancy rate given the consumption policy. It follows from (15) and (26) that the stochastic discount factor can be written as \begin{align} &M(\lambda, N, \hat{\kappa}; \lambda', \epsilon',d', \zeta', \hat{\kappa}')\nonumber\\ &\quad = \beta e^{-\frac{\mu}{\psi} + \frac{1}{2} (1-\gamma)\left(\gamma - \frac{1}{\psi}\right) \sigma_\epsilon^2} e^{-\gamma(\epsilon'+d' \zeta')}\nonumber\\ &\qquad \cdot \mathbb{E} \left[e^{(1-\gamma) d' \zeta'} j(\lambda', N', \hat{\kappa}')^{1-\gamma} \right]^\frac{\gamma - \frac{1}{\psi}}{1-\gamma} \left(\frac{c(\lambda', N', \hat{\kappa}')}{c(\lambda, N, \hat{\kappa})} \right)^{-\frac{1}{\psi}} j(\lambda', N', \hat{\kappa}')^{\frac{1}{\psi}- \gamma}. \end{align} (C.1) The equity return from Theorem 1 is given by \begin{equation} R(\lambda, N, \hat{\kappa}; \lambda', \epsilon',d', \zeta', \hat{\kappa}') = e^{\mu+\epsilon' + d' \zeta'}\left[\frac{1 - w(\lambda', N', \hat{\kappa}') + (1-s) \frac{ \hat{\kappa}'}{q(\theta(\lambda', N', \hat{\kappa}'))}}{\frac{ \hat{\kappa}}{q(\theta(\lambda, N, \hat{\kappa}))}} \right], \end{equation} (C.2) where \begin{equation} w(\lambda, N, \hat{\kappa}) = (1-B) b + B(1 + \kappa ((1 - \nu) \bar{\theta}+\nu \theta(\lambda, N, \hat{\kappa}))) \end{equation} (C.3) and \begin{equation} \theta(\lambda, N, \hat{\kappa}) = \frac{N + b(1-N) - c(\lambda, N, \hat{\kappa})}{ \hat{\kappa}(1-N)}, \end{equation} (C.4) which follows from (23). Equations (A.4) and (A.5) imply that the equilibrium conditions that |$c(\lambda, N, \hat{\kappa})$| and |$j(\lambda, N, \hat{\kappa})$| have to satisfy are \begin{equation} \mathbb{E} \left[M(\lambda, N, \hat{\kappa}; \lambda', \epsilon',d', \zeta', \hat{\kappa}') R(\lambda, N, \hat{\kappa}; \lambda', \epsilon',d', \zeta', \hat{\kappa}')\right] = 1 \end{equation} (C.5) and \begin{align} &j(N, \lambda, \hat{\kappa}) \notag\\ &\quad = \left[ c(N, \lambda, \hat{\kappa})^{1-\frac{1}{\psi}} + \beta e^{\left( 1-\frac{1}{\psi} \right) \mu + \frac{1}{2} \left( 1-\frac{1}{\psi} \right) (1-\gamma) \sigma_\epsilon^2 }\left(\mathbb{E} \left[e^{(1-\gamma) d' \zeta'} j(\lambda', N', \hat{\kappa}')^{1-\gamma}\right] \right)^{\frac{1-\frac{1}{\psi}}{1-\gamma}} \right]^{\frac{1}{1-\frac{1}{\psi}}}. \end{align} (C.6) We approximate the AR(1) process for log disaster probability by a 12-state Markov process and use the corresponding probability transition matrix to calculate expectations over |$\lambda'$|. The expectations over |$\zeta'$| and |$\epsilon'$| can be taken directly since their distributions are iid. We solve for the equilibrium consumption policy by iterating on the Euler equation (C.5). Specifically, we specify a grid of state variables and the computation steps are as follows: Start with an initial guess for the functions |$c$| and |$j$|. Compute the endogenous state (|$N'$|) in the next period using the guess. Compute consumption policy (|$c'$|) in th enext period by interpolation on the grid. Given these guesses, compute current period’s consumption policy |$c$| using condition (C.5). Compute today’s utility |$j$| in (C.6) given the consumption policy and the last iteration’s utility values for |$j'$|. Repeat 2–5 until current consumption policy and utility do not change significantly. Appendix D. Model with Capital Accumulation Let |$\tilde{K}_{t+1}$| denote planned capital given by \begin{equation} \tilde{K}_{t+1} = K_{t+1} e^{-d_{t+1} \zeta_{t+1}}. \end{equation} (D.1) Note that |$\tilde{K}_{t+1}$| is known at time |$t$|. Equations (38) and (D.1) imply the following law of motion for |$\tilde{K}_{t+1}$|: \begin{equation} \tilde{K}_{t+1} = (1-\delta) \tilde{K}_{t} e^{d_t \zeta_t} + \phi \left(\frac{I_t}{K_t} \right) \tilde{K}_{t} e^{d_t \zeta_t}. \end{equation} (D.2) The representative firm pays \begin{equation} D_t = F(K_t, Z_t N_t) - W_t N_t - \kappa_t V_t - I_t \end{equation} (D.3) as dividend. The firm problem with capital is then given by \begin{equation} P^c_t = \underset{\{V_{t+\tau}, N_{t+\tau+1}, I_{t+\tau}, \tilde{K}_{t+\tau+1}\}^\infty_{\tau = 0}}{ \text{max }} \mathbb{E}_{t} \sum^{\infty}_{\tau = 0} M_{t+\tau} \left[F(K_{t+\tau}, Z_{t+\tau} N_{t+\tau}) - W_{t+\tau} N_{t+\tau} - \kappa_{t+\tau} V_{t+\tau} - I_{t+\tau} \right], \end{equation} (D.4) subject to the law of motion for employment \begin{equation} N_{t+1} = (1-s) N_t + q(\theta_t) V_t, \end{equation} (D.5) and the law of motion for planned capital \begin{equation} \tilde{K}_{t+1} = (1-\delta) \tilde{K}_{t} e^{d_t \zeta_t} + \phi \left(\frac{I_t}{K_t} \right) \tilde{K}_{t} e^{d_t \zeta_t}. \end{equation} (D.6) The first-order conditions with respect to |$V_t$| and |$N_{t+1}$| are given by \begin{equation} 0 = -1 + l_t^n \frac{q(\theta_t)}{\kappa_t} \end{equation} (D.7) \begin{equation} l_t^n = \mathbb{E}_{t} \left[M_{t+1} (F_{N,t+1} - W_{t+1} + l^n_{t+1} (1-s)) \right], \end{equation} (D.8) where |$F_{N,t+1}$| is the partial derivative of |$F_{t+1}$| with respect to |$N_{t+1}$|. The first-order conditions with respect to |$I_t$| and |$\tilde{K}_{t+1}$| are given by \begin{equation} -1 + l^k_t \phi'\left(\frac{I_t}{K_t} \right) = 0 \end{equation} (D.9) \begin{equation} l^k_t = \mathbb{E}_{t} \left[M_{t+1} e^{d_{t+1} \zeta_{t+1}} \left(F_{K,t+1} + l^k_{t+1} \left(1 - \delta + \phi \left(\frac{I_{t+1}}{K_{t+1}} \right) - \phi' \left(\frac{I_{t+1}}{K_{t+1}} \right) \frac{I_{t+1}}{K_{t+1}} \right) \right) \right], \end{equation} (D.10) where |$F_{K,t+1}$| is the partial derivative of |$F_{t+1}$| with respect to |$K_{t+1}$|. Recursive substitution of first-order conditions into the firm value in (D.4) similar to Appendix A imply the equivalent of Theorem 1 given by (40). Finally, market clearing implies that aggregate consumption is given by \begin{equation} C_t = F(K_t, Z_t N_t) + b_t (1-N_t) - \kappa_t V_t - I_t, \end{equation} (D.11) which can be used to compute model dynamics in general equilibrium. We solve the model with capital applying the Euler equation iteration approach described in Appendix C. We add |$k = \frac{K}{Z}$| (normalized capital) to the grid of state variables (in addition to |$\lambda$|, |$\hat{\kappa}$|, |$N$|), and solve for consumption and investment policies using the equilibrium conditions (D.11), (D.8) and (D.10). Appendix E. Data Sources We use data from 1951 to 2013 for all variables. |$Z$| is the seasonally adjusted quarterly real average output per hour in the nonfarm business sector, constructed by the Bureau of Labor Statistics (BLS) from National Income and Product Accounts (NIPA) and the Current Employment Statistics (CES). Output per person yields nearly indistinguishable results. |$P$| is the real price of the S&P composite stock price index, downloaded from Robert Shiller’s website (www.econ.yale.edu/ shiller/data.htm). |$P/E$| is the cyclically adjusted price-earnings ratio, downloaded from Robert Shiller’s website (www.econ.yale.edu/ shiller/data.htm). |$P/Z$| is the price-productivity ratio scaled to have the same value as |$P/E$| in the first quarter of 1951. |$U$| is the seasonally adjusted unemployment, constructed by the BLS from the Current Population Survey (CPS). Quarterly values are calculated averaging monthly data. For unemployment during the Great Depression, we use the monthly unemployment rate data from FRED. |$V$| is the help-wanted advertising index constructed by the Conference Board until June 2006. We use data on vacancy openings from Job Openings and Labor Turnover Survey (JOLTS) from 2000 to 2013. We extrapolate the help-advertising index until 2013 and observe that our extrapolation has a correlation of 0.96 in the period from 2000 to 2006 where both data sources are available. Quarterly values are calculated averaging monthly data. |$W$| denotes wages measured as the product of labor productivity |$Z$| and labor share from the BLS. |$C$| is annual real personal consumption expenditures per capita from the BEA. |$Y$| is annual real gross domestic product (GDP) per capita from the BEA. |$I$| is annual real gross domestic investment from the BEA. |$R$| is the value weighted return market index return including distributions from CRSP. Real returns are calculated using inflation rate data from CRSP. Net returns are multiplied by 0.68 to adjust for financial leverage. |$R_b$| is the 1-month Treasury-bill rate from CRSP. Real rates are calculated using inflation rate data from CRSP. |$\Delta c$|, |$\Delta i$| and |$\Delta y$| denote log consumption, log investment and log output growth. Annual growth rates from monthly simulations that we compare to data values are calculated aggregating consumption and output levels over every year. Let |$C_{t,h}$| denote the consumption level in year |$t$| and month |$h$|. Annual log consumption growth in the model is calculated as \begin{equation} \Delta c_{t+1} = \text{log } \left(\frac{\sum_{i=1}^{12} C_{t+1, i}}{\sum_{i=1}^{12} C_{t, i}} \right). \end{equation} (E.1) The same method is applied to output growth as well. Footnotes 1 One exception is Eyigungor (2010), who allows for shocks to embodied rental capital. Eyingungor assumes risk neutrality and does not study asset pricing implications. 2 Other recent work connects time-variation in discount rates to unemployment. Eckstein, Setty, and Weiss (2015) solve a DMP model with risk-neutral investors and exogenous discount rates where labor and capital are complements. They show that volatility in corporate discount rates can account for volatility in unemployment. Hall (2017) conjectures that a DMP model in which discount rates rise in recessions can explain unemployment. He shows that, when an exogenous stochastic discount factor is estimated using the aggregate stock market, the resultant time series of unemployment tracks that in the data. Neither paper provides a general equilibrium model. Our results show that the connection between discount rates, recessions, and unemployment in a general equilibrium DMP model is more subtle than one might think (see Section 2.4). 3 Of course, the agent in the endowment economy still optimally chooses consumption. However, asset prices adjust so that this optimal choice is equal to the endowment. 4 Following Shimer (2005) we use a low-frequency HP filter with smoothing parameter |$10^5$| throughout to capture business-cycle fluctuations. All results are robust to using an HP filter with smoothing parameter of 1,600. 5 We find nearly indistinguishable results when |$Z$| is measured as output per worker rather than output per hour. We also explored using total factor productivity (TFP) as a proxy for productivity. When we replace output per hour with utilization-adjusted TFP (Fernald 2014), the correlations with respect to realized productivity are in fact lower, and the correlations with the stock market are higher (thus less in line with a productivity-based model and more in line with our hypothesized mechanism). The correlation between |$Z$| and |$V/U$| is consistently negative (|$-$|0.18 over the full sample, |$-$|0.12 until 1985 and |$-$|0.24 after 1985). The correlation between |$P/Z$| and |$V/U$| is 0.51 over the full sample, 0.3 before 1985 and 0.71 after 1985. Also, while TFP is more volatile than output per hour, it is far less volatile than the vacancy-unemployment ratio. 6 The assumption of |$V_t>0$| implies that the maximum drop in employment level is |$s$|. 7 Merz and Yashiv (2007) also establish a link between surpluses from the employment relation and firm value. Their model is partial-equilibrium and uses labor and capital adjustment costs as opposed to labor search. 8 In what follows, we assume that there exists a solution to this problem with |$V_t>0$|. This condition holds in all calibrations we consider in the paper. 9 For Theorem 1, stationarity of |$\{ x_t\}$| suffices. However, this does not imply that properties of |$\{x_t\}$| do not affect the equilibrium variables such as wages or labor market tightness. Assume, for example that, at time |$t$|, the distribution of |$x_{t+1}$| is a function of time-|$t$| variables |$\vartheta_t$|. Equation (12) holds, with |$\theta_t$| and |$N_{t+1}$| functions of |$N_t$| and |$\vartheta_t$|. 10 Theorem 2 shows that, in the special case with |$\kappa_t/Z_t$| constant, the implications of the model for macroeconomic quantities (but not asset prices) are isomorphic to a model with a time-varying discount factor. 11 This assumption implies that our model focuses on the transition between employment and unemployment rather than in and out of labor force. 12 The canonical Nash-bargained wage equation holds in our model. See Petrosky-Nadeau, Zhang, and Kuehn (2013) for the proof in a similar setting. 13 Hall (2005) specifies |$W_t^I$| as constant and productivity as stationary. In our setting with nonstationary productivity, |$W_t^I$| must be proportional to |$Z_t$| to allow for balanced growth. In Section 2, we show that, in the data, wages are responsive to |$Z_t$|, but not to |$\theta_t$|. 14 An alternative microfoundation is based on risk-sharing between workers and owners in a setting with limited stock market participation. Keam and Donaldson (2011) show that such a model can account for the equity premium and volatility in unemployment. 15 See Gabaix (2012), Gourio (2012), and Wachter (2013). 16 The distribution of |$\zeta_t$| is time invariant and therefore independent of all other shocks. 17 Because the disaster size is a random variable, the increase in |$\hat{\kappa}_t$| to a fixed value does not completely undo the decline in productivity. Moreover, if there is a second disaster before |$\hat{\kappa}_t$| reverts back to |$\underline{\kappa}$|, |$\kappa_t$| simply rises to |$\bar{\kappa}$|. This is to keep the range of the state variable |$\hat{\kappa}_t$| bounded and the firm’s problem tractable. 18 This is consistent with the results of Chodorow-Reich and Karabarbounis (2015) as discussed in Section 2.3.2. Changing to the alternative assumption that these benefits net to zero, however, does not impact our results. To ensure that our model-data comparison is valid, when quantitatively assessing the model we report the model-implied dynamics of consumption from dividends and wages, namely, |$Z_t N_t - \kappa_t V_t$|, as this is what is measured in consumption data. 19 See also Gabaix (2011) and Gourio (2012). 20 Lemma 2 shows that the equity premium depends only on the distribution of productivity and on risk aversion; labor market parameters do not enter. As we will see in Section 3.6, this separation between macro and financial variables no longer holds in a dynamic context. 21 We calibrate to U.S. data on labor markets, and match U.S. data on stock returns. An interesting potential extension would be to examine the model’s predictions for international data, given the large cross-country differences in labor markets. 22 For this calculation, we keep the disaster probability constant at its next-to-highest node; this results in the correct level of unemployment at the onset of the disaster. We assume that the Great Depression was characterized by several positive realizations of |$d_{t+1}$|, as indicated in Figure 6. 23 Following Hagedorn and Manovskii (2008), we calculate wages in the data by multiplying the labor share by productivity. 24 The instruments are twice-lagged consumption growth, the government bill rate, and the log price-productivity ratio. 25 Nakamura et al. (2013) and Tsai and Wachter (2015) consider endowment economy models with disasters that take time to unfold and show that the asset pricing implications are very similar. 26 We use the time path of unemployment to calibrate the model. An alternative would be to use the time path of vacancy postings. The decline in vacancy postings was more persistent than the increase in unemployment following the Great Depression. This may have been because of structural changes to the labor market that we do not capture here. 27 It would also be reasonable to allow the outside option |$b_t$| to be partially insulated from disasters. This would to greater wage insulation, and thus greater operating leverage. However, the effects of |$b_t$| and |$\kappa_t$| on unemployment are not interchangeable. This is because |$\kappa_t$| directly enters into the cost of hiring a worker, whereas |$b_t$| does not. Although they both have an indirect effect on unemployment by keeping wages high, a relative increase in |$\kappa_t$| also implies that firms are less likely to hire because costs have risen relative to productivity. 28 While vacancies and unemployment substantially respond to an increase in disaster probability, consumption does not. In the very short term, an increase in the disaster probability slightly increases consumption because investment in hiring falls. In the longer term, consumption falls because the lower level of employment implies lower output. High disaster states are nonetheless still high marginal utility states. 29 Belo, Lin, and Bazdresch (2014) demonstrates a cross-sectional relation between required rates of return and hiring: firms that hire more appear to have lower risk premiums. The same mechanism that we employ to explain the time-series patterns may also account for this evidence. 30 As noted in Section 1, we follow Shimer (2005) in using a low-frequency HP filter with smoothing parameter |$10^5$|. We report volatilities of log deviations from trend. 31 The model could be generalized to allow some correlation with realized productivity, by introducing normal-time shocks into the process for scaled vacancy |$\hat{\kappa}_t$|, as we currently do for disasters. In this case, costs would not fall with a downturn in productivity, and vacancy postings would fall as a result. This would also endogenously create wage rigidity with respect to normal-times shocks. 32 The effect of time-variation in scaled vacancy costs is negligible during normal times. 33 For the 5-year zero-coupon real government bond, the median yield is 2% and the median volatility is 3%. The data fall well-within 10% confidence intervals implied by the model. However, implications for the 10-year zero-coupon bond yield are counterfactual. The median yield is |$-$|32% and the median volatility is 14%. Government default or high inflation (that may be mismeasured in calculating payouts to TIPS) could potentially reconcile these features of the model with the data. We leave a detailed investigation of the real yield curve to future research. 34 Lemmon, Roberts, and Zender (2008) report an average market leverage ratio of 28% among U.S. firms from 1965 to 2003. Accordingly, the unlevered equity premium is calculated multiplying stock returns by 0.72. 35 Working against our finding of a high equity premium is a sample selection issue: samples without disasters tend to have lower disaster probabilities. This effect is apparent from comparing the median equity premium in samples without disasters to the population equity premium. 36 In the iid model with disasters, the population equity premium is 2.2%, whereas the average excess return in periods without disasters is 2.6%, reflecting the Peso problem discussed by Jorion and Goetzmann (1999). 37 This result is parameter dependent. It is possible to calibrate the model so that realized returns are negatively correlated with the disaster probability. Under such a calibration, one obtains the standard result of time-varying |$\lambda_t$| increasing the equity premium. 38 Our previous model does allow for capital in the production function, as long as this capital can be adjusted intratemporally (i.e. rented). With rental capital and a Cobb-Douglas production function, the intratemporal first-order condition implies the production function 5. 39 To be precise, we solve for |$a_1$| and |$a_2$| such that |$\phi(x) = x$| and |$\phi'(x) = 1$|, where |$x = e^{\mu + \frac{1}{2} \sigma^2} - (1-\delta)$|. 40 To emphasize that the sum in (B.1) is known by |$t+1$|, we will sometimes refer to the random variable within the expectation as |$\zeta_{t+1}$|. 41 Recall |$N_{t+1}$| is known at time |$t$|. This is where we must assume |$\hat{\kappa}_t = \kappa$|. Otherwise, the normalized value function at time |$t+1$| would not be independent of |$Q_{t+1}$|. 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