A. Households

The household chooses consumption C_h,t, assets a_t, the mass of individuals searching for high- or low-wage jobs, and search intensity for workers in firm L to maximize ${\mathbb{E}}_0{\mathrm{\Sigma }}_{t=0}^{\infty }{\beta }^t\mathbf{\text{u}}\left(c_{h,t}\right)$, where u′ > 0, u″ < 0, and β ∈ (0,1), subject to its perceived laws of motion for employment in each sector and the constraint

where: κ(s) is the resource cost of on-the-job search (κ′ > 0 and κ′ ≥ 0); R is the gross real interest rate; w_H (w_L) is the real wage in firm H (L); χ is the flow value of unemployment; Π_y are profits from the final goods firm; and T are lump-sum taxes.⁶ Let W_j,t be the household’s value of employment in firm-type j ∈ {H, L}. The first-order conditions yield a standard consumption-savings

Euler equation

and an optimal search intensity condition

where: Ξt+1|t ≡ βu^′(c_h,t+1)/u′(c_h,t). This last condition equates the marginal cost of a worker in firm L searching for a job in firm H, which is given by κ′(s_t), to the expected net marginal benefit of a job-to-job transition, which is given by the net capital gain between the two types of jobs. The household allocates unemployed search activity across sectors until the value of such activity is equalized (Krause and Lubik, 2006, 2010; this result is akin to a no arbitrage condition holding as far as search activity goes).⁷

B. Production

The final goods firm purchases output from firms H and L to produce final output via the constant-returns-to-scale function y_t = z_ty (y_H,t, y_L,t), where: z is total factor productivity (TFP); and y_j is intermediate output produced by type j ∈ {H, L} firms. The final goods firm chooses inputs to maximize Π_y,t = [z_ty (y_H,t, y_L,t) — p_H,t,y_H,t — p_L,ty_L,t], where p_j are prices relative to the price of final output (which is normalized to 1).⁸

Intermediate firm j ∈ {H, L} is owned by entrepreneur j, who chooses consumption c_j,t, vacancies v_j,t, desired employment n_j,t+1, next period’s capital stock k_j,t+1, and borrowed funds l_j,t, to maximize ${𝔼}_0{\mathrm{\Sigma }}_{t=0}^{\infty }{\beta }_j^t\mathbf{u}\left(c_{j,t}\right)$, where u′ > 0, u″ < 0, and 0 < β_j < β. This difference in subjective discount factors between entrepreneurs and the household is standard in the literature on financial frictions and guarantees that the firm’s collateral constraint is binding in the neighborhood of the steady state—see, for example, Iacoviello (2015). Importantly, per related literature (Jermann and Quadrini, 2012), the presence of binding constraints is what allows usage of the collateral constraints implied by the model to construct time series for financial shocks using data on firms’ liabilities, capital stock, and wage bill (see the Appendix for further details). This problem is subject to

which is a standard equation of motion for capital with δ denoting the depreciation rate and i_j investment, and⁹

In the first constraint: y_j is equal to the constant returns to scale production function F(n_j, k_j); w_ju_j is the wage bill; and γ_jv_j, is the vacancy bill, where γ_j is a standard exogenous flow cost; in addition, firm j borrows l_j,t and must repay R_t-1l_j,t-1 for the previous period’s borrowed funds, where R_t-1 is the gross real interest rate on these funds.¹⁰ Finally, the collateral constraint shows that the value of firm j’s liabilities Rl_j cannot exceed a fraction η of the value of firm j’s capital net of the firm’s wage bill W_jU_j, where η denotes the firm’s time-varying borrowing capacity, which has mean $\overline{\eta }$ and is subject to exogenous fluctuations that we interpret as financial shocks; the borrowing capacity parameter is common to both firms. For simplicity, we initially assume that firms need to finance the whole wage bill with borrowed funds, which is a standard assumption in related literature (we explore the relevance of this assumption as part of our robustness checks). Thus, η_w = 1. In addition, the firm’s maximization problem is subject to each firm’s sector-specific perceived law of motion for employment.¹¹

The timing of decisions is as follows. At the beginning of period t, TFP and financial shocks materialize and then entrepreneur j ∈ {H, L} makes decisions over choice variables. Per the collateral constraint that firms face, entrepreneurs can partially offset an exogenous deterioration in borrowing capacity η by increasing k_j,t+1 or reducing l_j,t, or both (concavity of the production function implies that in fact entrepreneurs will change both). The fact that entrepreneurs can optimally change their assets and liabilities in response to financial shocks is consistent with standard models of financial frictions (Jermann and Quadrini, 2012; Iacoviello, 2015). In turn, for a given gross real interest rate, entrepreneurs’ optimal choice over assets and liabilities, coupled with the change in borrowing capacity, determines equilibrium financial conditions.

The first-order conditions yield a capital Euler equation that takes into account the value of capital as collateral. For j ∈ {H, L},

where ${\Xi }_{t+1|t}^j\equiv \beta {\mathbf{u}}^{\prime }\left(c_{j,t+1}\right)/{\mathbf{u}}^{\prime }\left(c_{j,t}\right)$, and an optimal choice over borrowed funds

where λ_j is firm j’s multiplier on its collateral constraint (normalized by the marginal utility of consumption). In addition, firm H has the following job creation condition:

which incorporates the presence of the wage bill in firms’ collateral constraints, while firm L has the job creation condition

which includes the effective probability that workers in firm L transition to employment in firm H in the future, s_t+1f_H,t+1.

The capital Euler equation is standard, except for the presence of the collateral constraint multiplier. All else equal, a tighter collateral constraint (reflected in a higher λ_j) reduces the marginal cost of accumulating capital. The demand for external borrowing equates the marginal benefit from borrowed funds to the expected marginal cost. All else equal, a tighter constraint (reflected in a higher λ_j) reduces the marginal benefit of borrowing funds (the left-hand-side of the Euler equation). Of note, to the extent that adverse financial shocks induce a rise in λ_j, this will reduce firms’ discounting of the future value of capital and employment relationships, causing both future investment and hiring to contract in the aftermath of such shock. This is a key channel through which financial shocks affect future input decisions. In fact, a second channel works through the impact of movements in firms’ credit tightness on firms’ effective bargaining power over wage negotiations, which we describe below. (Quantitatively, this second channel is second order relative to the first channel.) Finally, the job creation condition for each firm equates the expected marginal cost of posting a vacancy to the expected marginal benefit of a vacancy, where the marginal benefit is affected by the need to borrow funds to finance the wage bill. In particular, a tightening of the collateral constraint (reflected in a higher λ_j), all else equal, reduces firms’ expected marginal benefit from posting a vacancy.¹²

C. Wage Determination

Wages are determined via Nash bargaining with no commitment to the future path of wages, and we assume that once an on-the-job search accepts a type H job s/he cannot go back to the type L job. In particular, denoting by ϕ the bargaining power of workers, the implicit Nash wage equations that determine the wage W_j,t for j ∈ {H, L} is

where J_j,t is the value to a firm of having an additional worker. This value is affected by the collateral constraint since firms need to finance their wage bill using borrowed funds, implying that the effective wage costs for each firm are affected by financial conditions (embodied partly in the collateral multiplier). Following related literature, there is free entry into vacancy posting so that the value of a vacancy is zero.

Given the presence of different discount factors between firms and the household as well as the timing convention for the evolution of employment we cannot obtain a closed-form solution for wages.¹³ However, per standard properties of firms’ value functions, J_j,t implicitly embodies labor market tightness in sector j, which means that wages will be affected by financial conditions through two channels, as alluded to above. First, movements in the collateral multiplier as a result of financial distress will directly influence wage fluctuations when firms must use borrowed funds to cover their wage bill via changes in workers’ effective bargaining power. Second and more importantly, changes in financial conditions affect firms’ discounting of the future value of capital and employment relationships, with higher financial distress causing the future value of employment to contract sharply as firms put more weight, in relative terms, on present (as opposed to future) economic and financial conditions (we discuss this more formally below). Firms’ recruiting decisions shape movements in market tightness (via changes in vacancies) and ultimately determine the extent of fluctuations in wages and employment. As such, financial shocks play a key role in amplifying changes in vacancy postings and, by extension, wages and therefore labor income as well.

D. Closing the Model and Competitive Equilibrium

Unemployment benefits are financed solely via lump-sum taxation such that the government’s bud-get constraint is: T_t = χ(u_H,t + u_L,t). In turn, the economy’s resource constraint is given by

where: c_t ≡ C_h,t + C_H,t + C_L,t denotes total private consumption; and the costs of posting vacancies and searching for employment are resource costs. In a competitive equilibrium, taking the stochastic processes {z_t, η_t} as given, the state-contingent allocations and prices ${\{n_{H,t},n_{L,t},R_t,s_t,c_{H,t},i_{H,t},l_{H,t},{\theta }_{H,t},k_{H,t},{\lambda }_{H,t},c_{L,t},i_{L,t},l_{L,t},{\theta }_{L,t},k_{L,t},{\lambda }_{L,t},w_{H,t},w_{L,t},u_{L,t}\}}_{t=0}^{\infty }$ and ${\{p_{H,t},p_{L,t},y_t,u_{H,t},c_{h,t}\}}_{t=0}^{\infty }$ satisfy: the laws of motion for employment in L and H firms; the household’s consumption-savings Euler equation; the optimal level of OTJ search intensity; the H firm’s consumption, law of motion for capital, (binding) collateral constraint, job creation condition, capital Euler equation, and optimal demand for borrowed funds; the L firm’s consumption, law of motion for capital, (binding) collateral constraint, job creation condition, capital Euler equation, and optimal demand for borrowers funds; the implicit Nash wage equations for H and L employment; the arbitrage condition for individuals searching for employment in the two firm categories; the relative prices of intermediate firm output; the definition of total output; the definition of total unemployment; and the economy-wide resource constraint. A list of the model’s equilibrium conditions is presented in the Appendix.

A. Stochastic Processes

Following Jermann and Quadrini (2012), we construct the TFP series using data on total non-farm employment from the BLS, a constructed series for the capital stock using National Income and Product Accounts (NIPA) and Flow of Funds data, and a standard Cobb-Douglas production function with labor share equal to 0.66.¹⁵ The construction of the series for η_t is less straightforward compared to Jermann and Quadrini (2012) as our model features two firms. To facilitate the construction of the series for financial conditions, we assume that η_t does not differ across firms (implying that both firms face the same aggregate financial process and the same level of financial conditions).¹⁶ We first combine the firms’ collateral constraints to obtain an aggregate collateral constraint that we can take to the data:

Defining l ≡ l_H + l_L, k ≡ k_H + k_L,,, and wn ≡ w_Hn_H + w_Ln_L as, respectively, total borrowed funds, the total capital stock, and the total wage bill, assuming that the constraint binds, and η_w = 1, we can write the condition above more compactly as R_tl_t = η_tk_t+1 — w_tn_t. Then, we have η_t = (R_tl_t+w_tn_t)/k_t+1. Each of the elements on the right-hand-side in this last expression can be directly measured in the data. Specifically, we use real liabilities of nonfinancial corporate and noncorporate businesses for Rl, the total capital stock for k, and real total compensation of employees (i.e., the total wage bill and also our proxy for total labor income) for wn. (A more detailed description of the construction of these series is presented in the Appendix.)

Jermann and Quadrini (2012) use a vector autoregression (VAR) specification to pin down the lag processes and disturbances associated with z and η and find that there are spillover effects between TFP and financial conditions (i.e., in this context the null hypothesis that one series does not Granger cause the other cannot be rejected). One objective of our analysis is to determine to what extent financial shocks on their own contribute to better matching key macro time series amid job-to-job flows and independent TFP shocks. As such, abstracting from spillover effects between shocks provides a more transparent notion of the two structural shocks that may drive business cycles in our model and in the data.

In order to abstract from spillover effects between shocks, we purge TFP and financial shocks from their interaction effects by following the methodology in Fujita and Ramey (2007). We refer to this purging procedure as FR shock purging. As shown in the Appendix, our procedure implies a substantially better fit of the benchmark model compared to a model where we do not remove spillover effects between shocks.

Specifically, the FR shock purging methodology is as follows. Let a_t ≠ b_t and a_t,b_t ∈ {ln z_t, ln η_t}, where In z_t and In η_t are the cyclical components of the log of z_t and η_t obtained using an HP filter with smoothing parameter equal to 1600. To characterize the dynamic relationship between these variables, we run the following recursive VAR: $X\left(L\right)[{\left(a_t{b}_t\right]}^{\text{'}}={\left[{\epsilon }_t^a{\epsilon }_t^b\right]}^{\text{'}}$, where X (L) is a lag polynomial, and ${ϵ}_t^a$ and ${ϵ}_t^b$ are the reduced-form residuals of the two equations (a host of information criteria tests suggest an optimal lag order of 2, which we use; other lag specifications yield nearly identical results). Granger causality tests fail to reject the null that each variable in the system does not Granger cause the other and do not suggest a clear-cut ordering for the recursive VAR since failure to accept the null occurs at the 1 percent level for each variable.

Given the lack of information on a clear-cut recursive ordering and no convincing economic argument for TFP shocks being more exogenous relative to financial shocks (and vice versa), we run the recursive VAR twice by first letting a_t = ln z_t and then letting a_t = ln η_t. In other words, we first assume that In z_t is the most exogenous variable, and we then assume that In η_t is the most exogenous variable. In each case, a is purged of feedback effects and its exogenous component, denoted by â, can be determined from the structural shocks by operationalizing the process ${\hat{X}}_{11}\left(L\right){\hat{a}}_t={\widehat{ϵ}}_t^a$, where: ${\hat{X}}_{11}\left(L\right)$ is the estimated value of the lag polynomial in the first row and first column of X (L); and ${\widehat{ϵ}}_t^a$ is the structural shock that is estimated. Note that the feedback effects are removed by setting the lag polynomial ${\hat{X}}_{12}\left(L\right)$, which is associated with the variable b, equal to zero (an alternative approach would be to commit to an ordering and then obtain exogenous components of a and b by operationalizing the processes ${\hat{X}}_{11}\left(L\right){\hat{a}}_t={\widehat{ϵ}}_t^{a}and{\hat{X}}_{22}\left(L\right){\hat{b}}_t={\widehat{ϵ}}_t^b$; regardless of the ordering, this alternative specification yields very similar results to those attained by by the methodology we implement).

With the exogenous components ln ${\hat{z}}_t$ and In ${\hat{\eta }}_t$ in hand, we can estimate two independent shocks in the model. To do so, we model each series as an autoregressive (AR) process for which information criteria suggest an optimal lag order of 1. As such, the estimated independent shocks in the model for TFP and η are obtained using the following two specifications, respectively: $\ln {\hat{z}}_t={\rho }_z\ln {\hat{z}}_{t-1}+{\in }_t^z$, and $\ln {\hat{\eta }}_t={\rho }_{\eta }\ln {\hat{\eta }}_{t-1}+{\in }_t^{\eta }$. This approach delivers the estimates ${\hat{\rho }}_z=0.717$ and ${\hat{\rho }}_{\eta }=0.720$, both of which are significant at the 1 percent level, and the standard deviation estimates for the residuals ε^z and ε^η are given, respectively, by ${\hat{\sigma }}_z=0.004$ and ${\hat{\sigma }}_{\eta }=0.011$.

Figure 2 shows the resulting time series for In $\hat{z}$ and In $\hat{\eta }$ as well as their respective shocks, ${ϵ}_t^z$ and ${ϵ}_t^{\eta }$. As the figures suggest, the series for In $\hat{z}$ and $\hat{\eta }$ are fairly positively correlated (the contemporaneous cyclical correlation is 0.392).

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TFP Shocks

Citation: IMF Working Papers 2017, 131; 10.5089/9781475595864.001.A001

Top panel: cyclical dynamics of TFP (z) and financial conditions (η). Bottom panel: TFP shocks (ε^z) and financial shocks {ε^η). Data span: 1985:Q1–2015:Q2. Recession quarters are marked in gray. The cyclical components of the data are based off the log deviations of the data from trend using an HP filter with smoothing parameter equal to 1600.

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B. Functional Forms and Parameter Selection

All utility functions are CRRA: $u\left(c_j\right)=c_j^{1-\sigma }/(1-\sigma )$ for j = h,H,L where σ is the CRRA parameter. The production functions of firms H and L are Cobb-Douglas: $y_{j,t}={\left(n_{j.t}\right)}^{1-{\alpha }_j}{\left(k_{j,t}\right)}^{{\alpha }_j}$ where 0 < α_j < 1 for j = H, L. The final goods production function is also Cobb-Douglas such that

where: α∈ (0,1). We introduce capital adjustment costs using the cost function Φ(k_j,t+1/k_j,t) = (φ_k/2)(k_j,t+1/k_j,t − δ)²k_j,t for j = H,L where φ k > 0. The matching functions are Cobb-Douglas:

m = M (·)^µ (·)^1−µ, where M is the common matching efficiency parameter, and µ is the common matching elasticity parameter. The cost function for OTJ search is k(s_t) = κ(s_t)^ηs, with κ > 0 and η_s ≥ 1.

The household’s discount factor β is 0.99. In turn, the firms’ discount factors β_H and β_L are set to 0.97.¹⁷ The firms’ discount factors imply an interest spread of roughly 2 percent, which is consistent with the average empirical spread between the weighted-average effective loan rate for commercial and industry loans (obtained from the Survey of Terms of Business Lending) and the federal funds rate. The value for the CRRA parameter σ is 1, consistent with the U.S. business cycle literature. Furthermore, we set α_H = α_L = 0.34 and α = 0.4, which jointly imply an aggregate labor share of 0.66 consistent with standard macro values and our estimation of the TFP process.¹⁸ The job separation rate ρ is 0.10, which is consistent with other studies (see, for example, Arseneau and Chugh, 2012). The matching elasticity and bargaining power parameters, μ and ϕ, are set to 0.4 and 0.5, respectively (Petrongolo and Pissarides, 2001; Shimer, 2005). The curvature parameter of the on-the-job search cost function η_s is set to 1 (Merz, 1995), and the depreciation rate of capital is set to 0.025, which is consistent with U.S. empirical estimates.¹⁹

The parameters $\chi ,\kappa ,M,\overline{\eta },{\phi }_k,{\gamma }_L$ and γ_H, are chosen so that the model matches the following moments in the data: a contemporaneous value of unemployment of roughly 60 percent of average wages (which lies close to Shimer, 2005, and is lower than in Hagedorn and Manovskii, 2008, and Hall and Milgrom, 2008; Hall and Milgrom suggest that the value of χ should not only capture the replacement rate but also home production and the value of leisure); a transition rate from low-wage jobs to high-wage jobs, sf_Hn_L/(n_H + n_L), equal to 0.05; a quarterly job-finding probability of average unemployed individuals of roughly 0.80; total recruiting costs of 5 percent of total output (slightly higher than Arseneau and Chugh, 2012); a ratio of debt to (quarterly) GDP of 3.16, which is roughly consistent with the debt-to-GDP ratio of non-financial corporate and noncorporate firms for our sample period; the ratio of the volatility of investment to the volatility of output over the period 1985:Q1 through 2015:Q2, which is 4.25; and per-vacancy recruiting costs for low-wage firms that are one fourth of recruiting costs for high-wage firms (Krause and Lubik, 2006, 2010).²⁰

As noted earlier, all else equal the more capital a firm has the greater its labor productivity, and following Pissarides (2000, Chapter 1), a higher labor productivity is associated with a greater opportunity cost of having an open position. In a reduced form way, this intuition is captured by assuming differential vacancy-posting costs, which in turn deliver wage and capital usage differentials between intermediate producers, with type-H firms paying higher wages and using more capital than type-L firms (see Acemoglu, 2001, as well).²¹ All told, the resulting calibration implies the following parameter values: χ = 0.3527, κ = 0.1118, M = 0.6457, γ_L = 0.1031, γ_H = 0.4123, $\overline{\eta }=0.5056,{\phi }_k=0.806$. The value of κ implies that sf_H(1−u_H−u_L)/(f_Hu_H+f_Lu_L)+ ILUL) is broadly in line with empirical estimates (Nagypál, 2005). Also, the value for the quarterly job-finding probability is in line with the average empirical value obtained following the methodology in Elsby, Michaels, and Solon (2009) and Shimer (2012) using post-war unemployment data. (The Appendix presents results for alternative parameterizations and versions of the benchmark model. From a comprehensive perspective and considering summary statistics for each alternative, our main conclusions regarding the relevance of OTJ search and financial shocks remain unchanged.)

A. Contour Analysis

Because our main focus is to show assess the extent to which both OTJ search and financial shocks are important in accounting for the empirical data, our first set of results involves graphical comparison of empirical and model-generated time series for the major labor market and macro variables in our framework (output, consumption, i.e., the sum of consumption across economic agents, investment, i.e., the sum of investment across firms, unemployment, the ratio of aggregate vacancies, i.e., the sum of sectoral vacancies, to aggregate unemployment v/u, labor income, i.e., the sum of earnings across job types, and net quits) for three model specifications: the Benchmark model, a version of the Benchmark model in which OTJ search is shut down (No OTJS), and a version of the Benchmark model that abstracts from financial shocks (No Fin. Shocks). This analysis allows us to gauge how well each model is able to replicate the contour of the data. (Details pertaining to the noted two alternatives model specifications are presented in the Appendix.)

Figure 3 presents results for output: All three versions of the model track the empirical series quite well. Of note, the Benchmark model predicts a sharper contraction in output amid the GFC that is more in line with the data relative to alternatives. All models predict a faster recovery of output following the GFC compared to the data. This feature is unsurprising, though, as all models omit income effects stemming from the steep decline in asset prices, including housing, that characterized the GFC as well as uncertainty and the slow process of balance-sheet repairs.

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Cyclical Dynamics of Output

Citation: IMF Working Papers 2017, 131; 10.5089/9781475595864.001.A001

Cyclical dynamics (1985:Q1–2015:Q2) of output obtained using the natural logarithm of the data and an HP filter with smoothing parameter equal to 1600. Recession quarters are marked in gray.

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Figure 4 presents results for consumption: the Benchmark and No OTJS models track the empirical series quite well, but the No Fin. Shocks model predicts counterfactually flat consumption. As explained in detail later in the paper, this failure of the No Fin. Shocks model traces back to the impact of financial shocks on firms’ collateral constraints, absent which a counterfactually smooth path for labor income (and therefore consumption) obtains.

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Cyclical Dynamics of Consumption

Citation: IMF Working Papers 2017, 131; 10.5089/9781475595864.001.A001

Cyclical dynamics (1985:Q1–2015:Q2) of consumption obtained using the natural logarithm of the data and an HP filter with smoothing parameter equal to 1600. Recession quarters are marked in gray.

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Figure 5 presents results for investment: All three versions of the model track the empirical series quite well, albeit with greater volatility. For reasons noted above, all models predict a faster recovery from the GFC, but relative to other alternatives, the Benchmark model once again exceeds in matching the extent of the contraction in investment amid the GFC.

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Cyclical Dynamics of Investment

Citation: IMF Working Papers 2017, 131; 10.5089/9781475595864.001.A001

Cyclical dynamics (1985:Q1–2015:Q2) of investment obtained using the natural logarithm of the data and an HP filter with smoothing parameter equal to 1600. Recession quarters are marked in gray.

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Figure 6 presents results for unemployment: The No OTJS model predicts counterfactually flat unemployment, while the Benchmark and No Fin. Shocks models do a better job of tracking the empirical series. The unemployment peaks of recession prior to the GFC are not matched particularly well, but this is not entirely surprising since the 1990 recession was intricately tied to oil price shocks—which our paper abstracts from—and both this recession and the 2001 recession were characterized by jobless recoveries, which also lie outside the scope of our analysis. Our paper does focus on the role of financial shocks, though, so it is noteworthy that the Benchmark model fares exceedingly well in approximating the unemployment peak amid the GFC (something that a model with OTJ search but without financial shocks cannot do); as was the case with earlier variables, though, the Benchmark model predicts a faster recovery from the GFC. As explained later in the paper, the flatness of unemployment under the No OTJS model is directly tied to OTJ search’s impact as an amplification mechanism for aggregate labor-market dynamics. Having said that, even with OTJ search, financial shocks play a critical role in generating quantitatively factual changes in unemployment.

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Cyclical Dynamics of Unemployment

Citation: IMF Working Papers 2017, 131; 10.5089/9781475595864.001.A001

Cyclical dynamics (1985:Q1–2015:Q2) of unemployment obtained using the natural logarithm of the data and an HP filter with smoothing parameter equal to 1600. Recession quarters are marked in gray.

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Figure 7 presents results for aggregate labor market tightness, i.e., the ratio of aggregate vacancies to aggregate unemployment, v/u: Both the Benchmark and No Fin. Shocks models track the empirical data quite well, but the No OTJS model predicts a counterfactually flatv/u ratio. It is noteworthy that the Benchmark model fares exceedingly well among model alternatives in matching the contraction of the v/u ratio amid the GFC. Again, though, all models predict a much faster recovery for reasons noted above. Also for similar reasons as noted earlier: it is unsurprising that no model matches exceedingly well the contraction of the v/u ratio in recessions prior to the GFC; the flatness of the v/u ratio under the No OTJS model is directly tied to OTJ search’s impact as an amplification mechanism. Similar to the case of unemployment, even with OTJ search, financial shocks are central to producing a sharp contraction in market tightness amid the GFC.

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Cyclical Dynamics of the v/u

Citation: IMF Working Papers 2017, 131; 10.5089/9781475595864.001.A001

Cyclical dynamics (1985:Q1–2015:Q2) ofcthe v/u ratio obtained using the natural logarithm of the data and an HP filter with smoothing parameter equal to 1600. Recession quarters are marked in gray.

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Figure 8 presents results for labor income: The No Fin. Shocks model predicts counterfactually flat labor income, but both the Benchmark and No OTJS models track the empirical series quite well. Note, though, that the Benchmark model fares best in matching the contraction of labor income amid the GFC, which, as explained later, stems from the combination of OTJ search as an amplification mechanism and the impact of financial shocks on wages and, therefore, labor income. Figure 8 clearly highlights the role of financial shocks for better matching labor income and consumption dynamics.

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Cyclical Dynamics of Labor

Citation: IMF Working Papers 2017, 131; 10.5089/9781475595864.001.A001

Cyclical dynamics (1985:Q1–2015:Q2) of labor income obtained using the natural logarithm of the data and an HP filter with smoothing parameter equal to 1600. Recession quarters are marked in gray.

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Figure 9 presents results for the net quit rate (recall that we have two empirical reference for this series, one from Davis and Haltiwanger, 2014, and the other from JOLTS, whose construction is detailed in the paper’s Introduction): The No OTJS model predicts zero quits by construction. The remaining alternative models’ predictions involve greater volatility than the actual series, but the Benchmark model is the only one that matches the steep contraction in net quits amid the GFC. As explained in detail further below, the ability of the Benchmark model to capture the contraction of quits is a combination of its accounting for both OTJ search and financial shocks. Given the exceedingly good fit of net quits in the Benchmark model with the data and our emphasis on job-to-job flows, this result is particularly noteworthy.

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Cyclical Dynamics of Net Quits

Citation: IMF Working Papers 2017, 131; 10.5089/9781475595864.001.A001

Cyclical dynamics (1985:Q1–2015:Q2) of net quits obtained using the natural logarithm of the data and an HP filter with smoothing parameter equal to 1600. Net quits DH are constructed using data from Davis and Haltiwanger (2014) and data from the Current Population Survey; these data span 1990:Q2–2013:Q3. Net quits JOLTS are constructed using data from the Bureau of Labor Statistics’ Job Openings and Labor Turnover Survey and the Current Population Survey; these data span 2001:Q1–2015:Q2. Recession quarters are marked in gray.

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In summary, our contour analysis suggests that the Benchmark model performs best in matching the behavior of all key labor market and macro aggregates compared to the No Fin. Shocks model and the No OTJS model. In particular, note that the No Fin. Shocks model predicts counterfactually flat consumption and labor income, while the No OTJS model predicts counterfactually flat unemployment and v/u. In contrast, the Benchmark model can jointly reconcile the behavior of consumption, labor income, unemployment and market tightness in the data. We conclude that our contour analysis supports our thesis that the interaction of financial shocks and OTJ search is important for matching the empirical behavior of a comprehensive set of labor market and macro aggregates in the data.

B. Statistical Analysis

We now complement the previous analysis by presenting key statistics pertaining to the Benchmark model and its No Fin. Shocks and No OTJS alternatives. These statistics are constructed based on the simulated time series for each model, which are obtained after feeding the constructed (TFP and financial) shocks into each model. The relevant statistics are presented in Table 1. Panel A of this table shows the relative standard deviation of key model variables relative to output for each model under comparison as well as the data. The Benchmark model performs quite well on all fronts in matching the data. In turn, we highlight the No OTJS model’s exceedingly low relative volatilities of unemployment and the v/u ratio compared to the data (which are well known features of this modeling framework); however, this model does deliver a relative consumption volatility that is in line with the data, and high labor-income volatility, albeit somewhat higher than in the data (these much are lesser known features of this modeling framework). In turn, the No Fin. Shocks model delivers relative volatilities of unemployment and the v/u ratio that are much closer to the data compared to the No OTJS case (which are well known features of this modeling framework, i.e., the a context of OTJ search and TFP shocks, only); however, compared to the data this model yields exceedingly low relative volatilities of both consumption and labor income (these are much lesser known features of this modeling framework).

Table 1:

Statistics: 1985:Q1–2015:Q2

Notes: All data are in log deviations from steady state and obtained using an HP filter with smoothing parameter equal to 1600.

Table 1:

Statistics: 1985:Q1–2015:Q2

					Alternative Model Specifications
	Variable	Obs.	Data	Benchmark	No OTJS	No Fin. Shocks
A. Standard deviation of variable relative to output	Consumption	122	0.883	0.711	0.849	0.195
	Investment	122	4.247	3.985	3.493	3.617
	Unemployment	122	9.938	6.018	0.999	5.097
	v/u ratio	122	19.749	13.530	4.569	10.923
	Labor income	122	1.409	1.844	2.893	0.706
B. Correlation of variable with output	Consumption	122	0.905	0.505	0.652	0.497
	Investment	122	0.895	0.698	0.805	0.810
	Unemployment	122	−0.884	−0.770	−0.501	−0.869
	v/u ratio	122	0.892	0.584	0.356	0.948
	Labor income	122	0.776	0.491	0.403	0.936
C. Own-variable correlation with data	Output	122	1.000	0.859	0.854	0.755
	Consumption	122	1.000	0.582	0.663	0.147
	Investment	122	1.000	0.415	0.418	0.570
	Unemployment	122	1.000	0.478	0.554	0.523
	v/u ratio	122	1.000	0.337	0.189	0.484
	Labor income	122	1.000	0.520	0.439	0.605
D. Summary statistics	SAD	16	—	15.122	30.910	19.329
	SSD	16	—	56.205	315.411	104.533

Notes: All data are in log deviations from steady state and obtained using an HP filter with smoothing parameter equal to 1600.

View Table

Table 1:

Statistics: 1985:Q1–2015:Q2

					Alternative Model Specifications
	Variable	Obs.	Data	Benchmark	No OTJS	No Fin. Shocks
A. Standard deviation of variable relative to output	Consumption	122	0.883	0.711	0.849	0.195
	Investment	122	4.247	3.985	3.493	3.617
	Unemployment	122	9.938	6.018	0.999	5.097
	v/u ratio	122	19.749	13.530	4.569	10.923
	Labor income	122	1.409	1.844	2.893	0.706
B. Correlation of variable with output	Consumption	122	0.905	0.505	0.652	0.497
	Investment	122	0.895	0.698	0.805	0.810
	Unemployment	122	−0.884	−0.770	−0.501	−0.869
	v/u ratio	122	0.892	0.584	0.356	0.948
	Labor income	122	0.776	0.491	0.403	0.936
C. Own-variable correlation with data	Output	122	1.000	0.859	0.854	0.755
	Consumption	122	1.000	0.582	0.663	0.147
	Investment	122	1.000	0.415	0.418	0.570
	Unemployment	122	1.000	0.478	0.554	0.523
	v/u ratio	122	1.000	0.337	0.189	0.484
	Labor income	122	1.000	0.520	0.439	0.605
D. Summary statistics	SAD	16	—	15.122	30.910	19.329
	SSD	16	—	56.205	315.411	104.533

Notes: All data are in log deviations from steady state and obtained using an HP filter with smoothing parameter equal to 1600.

Panel B of Table 1 shows correlations of variables with output. On this front and on net, all models perform fairly similarly. In particular, some models are somewhat better than others in matching certain correlations while at the same time somewhat worse than others in matching other correlations. Finally, Panel C of this Table shows the own-variable correlation with the data (hence why all the entries under the Data column are equal to 1). On net, the appraisal is similar to that with regards to Panel B, but in this case we do highlight the relatively low correlation of the No OTJS model’s v/u ratio with its empirical counterpart, as well as the relatively low correlation of the No Fin. Shocks model’s consumption with its own empirical counterpart.

To provide a more comprehensive summary of relative model performance, Panel D of Table 1 shows the column-wise sum of absolute deviations (SAD) of each model relative to the Data column and the column-wise sum of squared deviations (SSD) of each model relative to the Data column. Note that the Benchmark model minimizes both of these measures substantially compared to the alternatives, by which we see additional evidence that of the proposed models the Benchmark is superior.

In sum, the SAD and SSD measures imply that, on average, the Benchmark model performs best in matching a wide range of empirical statistics.

The variables whose statistics are reported in Table 1 are those for which we have data through the entire 1985:Q1–2015:Q2 period and, therefore, 122 observations. In contrast, recall that our measures of net quits are comparatively limited in data span. For the DH version of net quits we have only 94 observations (23 percent less compared to the observations on which Table 1 is constructed) comprising the period 1990:Q2–2013:Q3, and for the JOLTS measure of net quits we only have 58 observations (about 50 percent less compared to the observations on which Table 1 is constructed) comprising the period 2001:Q1–2015:Q2. With these caveats in mind, focus on Table 2, which reports similar statistics as in Table 1, but now focusing on net quits. Of course, the No OTJS model is a natural failure because in this model there are no job-to-job transitions. In turn, the Benchmark model performs quite well, although in comparison the No Fin. Shocks model performs somewhat better.

Table 2:

Additional statistics: 1990:Q2–2013:Q3 and 2001:Q1–2015:Q2

Notes: All data are in log deviations from steady state and obtained using an HP filter with smoothing parameter equal to 1600. Net quits DH are constructed using data from Davis and Haltiwanger (2014) and data from the Current Population Survey; these data span 1990:Q2–2013:Q3. Net quits JOLTS are constructed using data from the Bureau of Labor Statistics’ Job Openings and Labor Turnover Survey and the Current Population Survey; these data span 2001:Q1–2015:Q2.

Table 2:

Additional statistics: 1990:Q2–2013:Q3 and 2001:Q1–2015:Q2

					Alternative Model Specifications
	Variable	Obs.	Data	Benchmark	No OTJS	No Fin. Shocks
A. Standard deviation of	Net quits (DH)	94	11.398	20.860	0.000	16.245
variable relative to output	Net quits (JOLTS)	58	15.352	22.631	0.000	17.567
B. Correlation of variable	Net quits (DH)	94	0.776	0.355	N/A	0.786
with output	Net quits (JOLTS)	58	0.876	0.361	N/A	0.746
C. Own-variable correlation	Net quits (DH)	94	1.000	0.322	N/A	0.375
with data	Net quits (JOLTS)	54	1.000	0.145	N/A	0.380

Notes: All data are in log deviations from steady state and obtained using an HP filter with smoothing parameter equal to 1600. Net quits DH are constructed using data from Davis and Haltiwanger (2014) and data from the Current Population Survey; these data span 1990:Q2–2013:Q3. Net quits JOLTS are constructed using data from the Bureau of Labor Statistics’ Job Openings and Labor Turnover Survey and the Current Population Survey; these data span 2001:Q1–2015:Q2.

View Table

Table 2:

Additional statistics: 1990:Q2–2013:Q3 and 2001:Q1–2015:Q2

					Alternative Model Specifications
	Variable	Obs.	Data	Benchmark	No OTJS	No Fin. Shocks
A. Standard deviation of	Net quits (DH)	94	11.398	20.860	0.000	16.245
variable relative to output	Net quits (JOLTS)	58	15.352	22.631	0.000	17.567
B. Correlation of variable	Net quits (DH)	94	0.776	0.355	N/A	0.786
with output	Net quits (JOLTS)	58	0.876	0.361	N/A	0.746
C. Own-variable correlation	Net quits (DH)	94	1.000	0.322	N/A	0.375
with data	Net quits (JOLTS)	54	1.000	0.145	N/A	0.380

Notes: All data are in log deviations from steady state and obtained using an HP filter with smoothing parameter equal to 1600. Net quits DH are constructed using data from Davis and Haltiwanger (2014) and data from the Current Population Survey; these data span 1990:Q2–2013:Q3. Net quits JOLTS are constructed using data from the Bureau of Labor Statistics’ Job Openings and Labor Turnover Survey and the Current Population Survey; these data span 2001:Q1–2015:Q2.

That said, we do not put too much weight on the statistics reported in Table 2 given the small amount of data they are based on compared to Table 1. Instead, we highlight once more the contour analysis in Figure 9, by which the Benchmark model highly outperformed the No Fin. Shocks model in matching the contraction in net quits amid the GFC. All told, we conclude that the statistical analysis presented in Tables 1 and 2 provide considerable support for the Benchmark model matching the data well in absolute terms, as well as compared to the No OTJS and No Fin. Shocks alternatives. (Our main conclusions continue to hold under alternative parameterizations for the benchmark model, as shown in the Appendix.)

A. Impulse Response Function Analysis

To understand the driving forces behind the Benchmark model’s results, Figures 10 and 11 show, respectively, the response of the Benchmark and No OTJS models to a 1-standard-deviation negative aggregate TFP shock and to a 1-standard-deviation negative financial shock. This analysis allows us to: illustrate the role of OTJ search in amplifying financial shocks; stress how the response to financial shocks differs from TFP shocks both qualitatively and quantitatively; and to show how these differential responses have important implications for wage, labor income, consumption, and unemployment dynamics.

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Impulse Response Functions to a 1-standard Deviation Negative TFP Shock.

Citation: IMF Working Papers 2017, 131; 10.5089/9781475595864.001.A001

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Impulse Response Functions to a 1-standard Deviation Negative Financial Shock.

Citation: IMF Working Papers 2017, 131; 10.5089/9781475595864.001.A001

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B. Wages and Consumption: TFP v. Financial Shocks

To understand the differential response of wages under the two shocks more clearly, without loss of generality, consider firm H’s decision over borrowed funds, which can be rewritten as $R_t^{-1}-{\lambda }_{H,t}={𝔼}_t{\mathrm{\Xi }}_{t+1|t}^H$ To fix ideas, first assume that R is constant. Then, an increase in firm H’s collateral multiplier λ_H reduces firm H’s stochastic discount factor ${\mathrm{\Xi }}_{t+1|t}^H$. In particular, a very sharp rise in λ_H such as those generated by our financial shocks lead to sharp declines in ${\mathrm{\Xi }}_{t+1|t}^H$.²⁴ In turn, the latter directly affects H firms’ future incentives to post vacancies and to invest by lowering the value of future capital and employment relationships via a sharply lower stochastic discount factor. How do changes in λ_H feed into wage dynamics? For expositional simplicity, assume that the wage bill is not part of the firm’s collateral constraint.²⁵ Then, firm H’s wage can be written as:

First, note that all else equal a lower job-finding probability f_H puts downward pressure on wages. Financial frictions affect wages via firms’ collateral multipliers, and therefore via ${\mathrm{\Xi }}_{t+1|t}^H$. Then, an adverse financial shock increases the firm’s collateral multiplier and induces a sharp contraction in ${\mathrm{\Xi }}_{t+1|t}^H$, thereby exerting substantial downward pressure on wages by reducing the value of future employment relationships, ${𝔼}_t{\mathrm{\Xi }}_{t+1|t}^H{J}_{H,t+1}$. Conversely, a negative TFP shock relaxes firms’ collateral constraints by reducing firms’ demand for borrowed funds for production, reduces the firm’s collateral multiplier, thereby exerting upward pressure on wages via an increase in ${\mathrm{\Xi }}_{t+1|t}^H$. As such, even if both shocks generate sharp contractions in vacancies as a result of the OTJ search mechanism, the contrasting behavior of firms’ collateral multiplier between shocks generates sharp wage dynamics in the case of financial shocks, but mild movements in wages in the case of TFP shocks. In turn, this leads to larger fluctuations in labor income and ultimately consumption amid financial shocks, but to small movements in these variables under TFP shocks in the presence of OTJ search. All told, financial shocks play a key role in generating larger wage movements, which ultimately feed into labor income and consumption dynamics by affecting firms’ valuation of the future and therefore their incentives to hire and invest.

Of note, the interaction between OTJ search and financial shocks magnifies the response of wages over the business cycle by generating larger contractions in vacancies (where the latter affects wages via market tightness, which is embodied in f_H) after a downturn relative to a model without OTJ search, and also by generating sharp movements in firms’ value of future employment relationships, ${𝔼}_t{\mathrm{\Xi }}_{t+1|t}^H{J}_{H,t+1}$. More importantly, the presence of larger wage movements does not imply that the rise in unemployment is more subdued under financial shocks. This result is subtle yet critical in light of existing literature on wage rigidities and unemployment dynamics. Our benchmark model— with both OTJ search and financial shocks—can simultaneously generate non-negligible and more factual fluctuations in wages, labor income, and consumption, as well as high unemployment volatility. That is, considerable wage rigidities (whether endogenous or exogenous)—rigidities that, while contributing to higher unemployment volatility, prevent standard models from replicating the volatility of labor income and consumption in the data, especially during recessions—are not necessary to produce high unemployment volatility in the presence of both financial shocks and job-to-job flows.

Finally, we note that while the initial contraction in OTJ search intensity—which is reflected in the behavior of OTJ searchers—is similar for TFP and financial shocks on impact, the ensuing recovery differs considerably despite the fact that our estimated stochastic processes suggest that financial shocks are somewhat more persistent than TFP shocks (see Figures 10 and 11). After a negative TFP shock, the contraction in OTJ search intensity is considerably persistent. In contrast, after a negative financial shock, OTJ search intensity contracts on impact but quickly rebounds and overshoots its steady-state level for several quarters before returning to trend. In turn, this development explains why recessions induced by financial disruptions generate expansions in unemployment that are sharp yet shorter-lived relative to recessions induced by contractions in TFP. Moreover, this result also hints at a plausible underlying reason behind the limitations of models with financial frictions in explaining sluggish recoveries after financial shocks.

All told, households’ wages and total labor income exhibit relatively large contractions compared to output in response to adverse financial shocks. These dynamics suggest that financial shocks can contribute to higher consumption volatility that is closer to the data, which, as implied by inspection of Tables 1 and 2, is indeed the case. In turn, OTJ search produces sharp movements in unemployment and vacancies, even in the presence of volatile wages and labor income (where the latter are a result of financial disturbances). That is, high wage and unemployment volatility can coexist, and indeed this important and empirically-factual coexistence contributes to a better overall fit of the Benchmark model with the data, both before and amid the GFC.