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The Review of Financial Studies
, Volume Advance Article – Jan 31, 2018

53 pages

/lp/ou_press/risk-unemployment-and-the-stock-market-a-rare-event-based-explanation-9iApZlKoqb

- Publisher
- Oxford University Press
- Copyright
- © The Author(s) 2018. Published by Oxford University Press on behalf of The Society for Financial Studies. All rights reserved. For Permissions, please e-mail: journals.permissions@oup.com.
- ISSN
- 0893-9454
- eISSN
- 1465-7368
- D.O.I.
- 10.1093/rfs/hhy008
- Publisher site
- See Article on Publisher Site

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. 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} \