# Credit and Firm-Level Volatility of Employment

Credit and Firm-Level Volatility of Employment
Vincenzo, Quadrini,;Qi, Sun,
2018-10-01 00:00:00
Abstract We study a firm dynamics model where access to credit improves the bargaining position of firms with workers and increases the incentive to hire. To evaluate the importance of the bargaining channel for the hiring decisions of firms, we estimate the model structurally using data from Compustat and Capital IQ. We find that the bargaining channel explains 13% of firm-level employment volatility. We also evaluate the relative contribution of credit and revenue shocks for firm-level employment fluctuations and find that credit shocks account for 22%. 1. Introduction The idea that firms use leverage strategically to improve their bargaining position with workers is not new in the labor and corporate finance literature. For example, Bronars and Deere (1991), Dasgupta and Sengupta (1993), and Perotti and Spier (1993) developed models where debt reduces the bargaining surplus for the negotiation of wages, allowing firms to lower the cost of labor. Studies by Klasa et al. (2009) and Matsa (2010) have tested this mechanism using firm-level data and found that more unionized firms—that is, firms where workers are likely to have more bargaining power—are characterized by higher leverage and lower cash holdings. More recently, Ellul and Pagano (2015) showed that the choice of leverage depends on the seniority of employees’ claims in the liquidation of insolvent firms. A property supported by their empirical analysis based on an index of employees’ protection in bankruptcy across countries. Peri (2015) studies the efficiency of different bankruptcy laws when workers are able to extract rents from their employers and tests the theory empirically. These studies provide empirical evidence that the bargaining power of workers is relevant for determining the financial structure of firms. However, whether this mechanism is also important for the hiring decisions of firms has not been fully explored in the literature. In fact, if higher leverage allows employers to negotiate more favorable conditions with employees, the ability to issue more debt should increase the incentive to hire. The goal of this paper is to study the importance of this mechanism by estimating a dynamic model with endogenous choices of employment and financial structure by individual firms. Monacelli et al. (2011) study the importance of the bargaining channel for aggregate dynamics in a model with a single-worker representative firm. In this paper, instead, we take a micro approach and explore the empirical relevance of this channel using a model with heterogeneous multiworker firms that can be mapped to firm-level data. In the model, the compensation of workers is determined at the firm level through bargaining. Firms choose the financial structure and employment optimally taking into account that these choices affect the cost of labor. Higher debt allows firms to negotiate lower wages that increases the incentive to hire more workers. Higher debt, however, also increases the likelihood of financial distress. When the financial condition of a firm improves, the likelihood of financial distress declines, making debt more attractive. This improves the bargaining position of the firm with its employees, increasing the incentive to hire. It is through this mechanism that improved firm-level access to credit generates higher demand for labor. We refer to this mechanism as the “bargaining channel of debt”. We evaluate the importance of the bargaining channel of debt by estimating the model with the simulated method of moments. The empirical moments are constructed using firm-level data from Compustat and Capital IQ. The first database provides information on typical balance sheet and operational variables including employment. The second database provides firm-level data for unused lines of credit. We use this variable as a “proxy” for the difference between the credit capacity of the firm and its actual borrowing that will be important for the identification of a financial distress cost parameter. More specifically, because the likelihood of financial distress increases with leverage, firms borrow less than the credit capacity (precautionary motive). Moreover, the unused credit capacity increases with the magnitude of the distress cost, which helps us identifying the financial distress cost parameter. After estimating the model, we evaluate the importance of the bargaining channel of debt for the dynamics of employment by conducting a counterfactual exercise in which debt does not affect the bargaining of wages. By comparing the counterfactual simulation to the simulation of the benchmark model, we find that the contribution of the bargaining channel of debt to firm-level employment volatility is 13%. This shows that, although this channel is not the main factor underlying employment fluctuations, it plays a significant role in the hiring decision of firms. In addition to the structural estimation, we also investigate the importance of the bargaining channel of debt with reduced-form regressions. The regressions test the prediction of the model that the sensitivity of employment to debt increases with the bargaining power of workers . To proxy for the bargaining power of workers, we use the unionization index from the Union Membership and Coverage Database. This index has been used in the corporate finance literature to assess the importance of workers’ bargaining for the choice of the optimal financial structure of firms but not for their employment decisions. We regress the firm-level growth of employment on a set of variables that include the growth of debt, the unionization index, and the interaction between debt growth and unionization (in addition to other controls). The main variable of interest is the interaction between debt growth and the unionization index. We find that the estimated coefficient is positive and statistically significant, which is consistent with the theoretical prediction of the model. The paper also evaluates the importance of different types of firm-level shocks to employment fluctuations. We consider two types of firm-level shocks: credit shocks and revenue shocks. When we simulate the estimated model with only one shock (and averaging over the realizations of the other shock), we find that the average contribution of credit shocks is about 22% and the contribution of revenue shocks is about 78%. The nonlinearity of the model implies that the importance of one shock depends on the realization of the other shock. In particular, we show that the contribution of credit shocks to employment fluctuation increases when firms are more productive. The importance of the bargaining channel of debt is also studied in Michaels et al. (2014). This paper estimates a firm dynamics model where debt affects the compensation of workers through the bargaining channel of debt but with a different bargaining scheme. This paper also differs from our paper in terms of the main addressed question. Although our paper focuses on the dynamics of employment, Michaels et al. (2014) focus especially on the dynamics of wages. In particular, using firm-level data for wages, they ask why the compensation of employees is negatively correlated with leverage. The estimation results suggest that the bargaining channel of debt plays an important role in generating the negative correlation. The remaining sections of the paper are organized as follows. Section 2 presents the dynamic model and characterizes some of its properties. Section 3 conducts the structural estimation and reports the results. After conducting a sensitivity analysis in Section 4, Section 5 evaluates the importance of the bargaining channel of debt with reduced-form regressions. Section 6 concludes. 2. A Firm Dynamics Model With Wage Bargaining Consider a firm with production technology Yt = ztNt, where zt is idiosyncratic productivity and Nt is the number of workers. Employment evolves according to \begin{equation} N_{t+1} = (1-\lambda )N_t + H_t, \end{equation} (1) where λ is the separation rate and Ht denotes the newly hired workers. Hiring is costly. A firm with current employment Nt hiring Ht workers incurs the cost ϒ(Ht/Nt)Nt, where the function ϒ(·) is strictly increasing and convex. Firms issue debt at price qt, raising qtBt+1 funds at time t and promising to repay Bt+1 at t + 1. The issuance of debt is subject to the enforcement constraint \begin{equation} q_t B_{t+1} \le \xi _t \beta \mathbb{E}_t S(\boldsymbol{x}_{t+1},B_{t+1},N_{t+1}), \end{equation} (2) where β is the discount factor of investors and the function S(xt + 1, Bt + 1, Nt + 1) is the net surplus of the firm in the next period. This function depends on the next period exogenous states xt + 1 (as defined shortly), debt Bt+1, and the number of employees Nt+1. The variable ξt is stochastic and captures the financial condition of the firm, that is, its access to external credit. Thus, the firm is subject to two sources of idiosyncratic uncertainty, productivity zt and financial condition ξt. The vector of exogenous states is then xt = (zt, ξt). We specify the enforcement constraint as a fraction of the whole net surplus, including the workers’ value. An alternative would be to assume that the constraint depends only on the equity value of the firm. As we will see, because the shareholders receive a fraction of the surplus, this alternative specification would be equivalent to the one used here after normalizing ξt by the bargaining share. Even though the enforcement constraint (2) is satisfied in period t, this does not guarantee that the enforcement constraint will be satisfied at t + 1 after the realization of the shocks. Therefore, at the beginning of t + 1, the firm could violate the constraint, that is, \begin{equation} B_{t+1} > \xi _{t+1} S(\boldsymbol{x}_{t+1},B_{t+1},N_{t+1}). \end{equation} (3) This could happen if the realizations of zt and/or ξt are very low. In this case, the firm is forced to raise emergency funds to repay a part of the debt. We assume that raising emergency funds at the beginning of the period is costly. Define $$B^*_{t+1}$$ the debt that satisfies condition (3) with “equality” at the beginning of period t + 1, after the observation of xt + 1 = (zt + 1, ξt + 1). This represents the maximum debt that is backed by collateral. As we will see, the surplus function is strictly decreasing in Bt+1, implying that there is a unique $$B^*_{t+1}$$ that satisfies condition (3) with equality. Notice that $$B_{t+1}>B^*_{t+1}$$ is equivalent to condition (3), that is, the outstanding debt is higher than the collateral, forcing the firm to raise emerging funds to cover the difference $$B_{t+1}-B_{t+1}^*$$. We refer to the cost of raising emerging funds as “financial distress cost” and assume that it takes the quadratic form, \begin{equation} \kappa \cdot \left(\max \left\lbrace \frac{B_{t+1}-B_{t+1}^*}{N_{t+1}} \, ,\, 0\right\rbrace \right)^2 \cdot N_{t+1} \, \equiv \, \varphi \left(\boldsymbol{x}_{t+1},\frac{B_{t+1}}{N_{t+1}}\right) \cdot N_{t+1}. \end{equation} (4) Notice that the function φ(., .) depends on the ratio of Bt+1 over Nt+1 rather than separately on these two variables. Because $$B_{t+1}^*$$ is endogenous, at this stage this is only a conjecture. Later, we will show that this is in fact the case. The budget constraint of the firm is \begin{equation} B_t + D_t + w_t N_t + \Upsilon \left(\frac{H_t}{N_t}\right) N_t + \varphi \left(\boldsymbol{x}_t,\frac{B_t}{N_t}\right) N_t = z_t N_t + q_t B_{t+1}, \end{equation} (5) where Bt is the debt issued at t − 1 and due at time t, Dt is the payout to shareholders, wt is the wage paid to each worker, and qt is the price of new debt issued at time t. Assumption 1. The exogenous shocks zt and ξt are independent from each other, each following a first-order Markov process with positive persistence. 2.1. Firm Policies and Bargaining Problem The policies of the firm, including wages, are bargained collectively with its labor force. The labor force is defined broadly including managers. In this way, the model also captures the potential conflicts between shareholders and managers as in Jensen (1986). To derive the bargaining outcome, it will be convenient to define few terms starting with the equity value of the firm. This can be written recursively as \begin{equation} V(\boldsymbol{x}_t,B_t,N_t) = D_t + \beta \mathbb{E}_t V(\boldsymbol{x}_{t+1},B_{t+1},N_{t+1}), \end{equation} (6) where Dt is the payout to shareholders. The value of equity depends on two endogenous states, debt Bt and employment Nt, in addition to the exogenous states xt = (zt, ξt). The value of a worker employed in a firm with liabilities Bt and employment Nt is \begin{equation} W(\boldsymbol{x}_t,B_t,N_t) = w_t + \beta \mathbb{E}_t [\lambda U_{t+1}+(1-\lambda ) W(\boldsymbol{x}_{t+1},B_{t+1},N_{t+1}) ]. \end{equation} (7) The variable Ut+1 is the value of separating at t + 1 for the worker (outside value). Given the partial equilibrium approach, the value of separation is exogenous in the model. The value for the worker, W(xt, Bt, Nt), net of the outside value Ut, can be written recursively as \begin{eqnarray} &&W(\boldsymbol{x}_t,B_t,N_t)-U_t= \nonumber \\ && w_t-U_t + \beta \mathbb{E}_t [U_{t+1} + (1-\lambda ) (W(\boldsymbol{x}_{t+1},B_{t+1},N_{t+1})-U_{t+1} ) ]. \end{eqnarray} (8) This equation is derived by subtracting Ut on both side of equation (7) and rearranging the terms in parentheses. The bargaining surplus is the sum of the net values for the firm and the workers, that is, \begin{equation} S(\boldsymbol{x}_t,B_t,N_t) = V(\boldsymbol{x}_t,B_t,N_t) + (W(\boldsymbol{x}_t,B_t,N_t)-U_t) N_t. \end{equation} (9) We are now ready to define the bargaining problem. Denoting by η the relative bargaining power of workers, the problem can be written as \begin{eqnarray*} \max _{\scriptstyle w_t,D_t,E_t,B_{t+1}} && [(W(\boldsymbol{x}_t,B_t,N_t)-U_t) N_t]^{\eta }\cdot V(\boldsymbol{x}_t,B_t,N_t)^{1-\eta }, \end{eqnarray*} subject to the law of motion for employment (1), the enforcement constraint (2), and the budget constraint (5). Differentiating with respect to the wage wt, we obtain the well-known result that workers receive a fraction η of the bargaining surplus, whereas the firm receives the remaining fraction, \begin{eqnarray} [W(\boldsymbol{x}_t,B_t,N_t)-U_t] N_t &=& \eta S(\boldsymbol{x}_t,B_t,N_t), \end{eqnarray} (10) \begin{eqnarray} V(\boldsymbol{x}_t,B_t,N_t) &=& (1-\eta ) S(\boldsymbol{x}_t,B_t,N_t). \end{eqnarray} (11) Using equations (10) and (11), the remaining policies of the firm (dividend, employment, and borrowing) maximize the net surplus S(xt, Bt, Nt). This property is intuitive: Given that the contractual parties (firm and workers) share the net bargaining surplus, it is in the interest of both parties to make the surplus as big as possible. Therefore, in characterizing the hiring and financial policies of the firm, we focus on the maximization of the net surplus which, in recursive form, can be written as \begin{eqnarray*} &&{S(\boldsymbol{x}_t, B_t, N_t)=}\\ &&\quad \max_{H_t, B_{t+1}} \Bigg \lbrace D_t + (w_t - u_t) N_t \nonumber \\ &&\qquad\qquad\quad +\, \beta \bigg [1-\eta + \eta (1-\lambda )\left(\frac{N_t}{N_{t+1}}\right)\bigg ]\, \mathbb{E}_t S(\boldsymbol{x}_{t+1},B_{t+1},N_{t+1})\Bigg \rbrace \nonumber \\ &&\quad \text{subj. to:}\ (1), (2), (5). \nonumber \end{eqnarray*} The recursive formulation is obtained by multiplying equation (8) by Nt, summing to equation (6), and using the sharing rules (10) and (11). The term $$u_t=U_t-\beta \mathbb{E}_t U_{t+1}$$ is exogenous. Normalized Problem We now take the advantage of the linearity of the model and express all variables in per-worker terms. Dividing by Nt, the optimization problem becomes \begin{eqnarray} s(\boldsymbol{x}_t, b_t) \max _{h_{t},b_{t+1}} && \left\{ d_{t}+w_{t}-u_{t}+\beta (g_{t+1}-\eta h_t) \mathbb{E}_t s(\boldsymbol{x}_{t+1},b_{t+1})\right\} \\ \text{subj. to:} && d_t+w_t=z_t-\Upsilon (h_t)+q_t g_{t+1}b_{t+1}-b_{t} - \varphi (\boldsymbol{x}_t,b_t) \nonumber \\ && \xi _t g_{t+1}\beta \mathbb{E}_t s(\boldsymbol{x}_{t+1},b_{t+1})\ge q_{t}g_{t+1}b_{t+1} \nonumber \\ && g_{t+1}=1-\lambda +h_t. \nonumber \end{eqnarray} (12) The function s(xt, bt) = S(xt, Bt, Nt)/Nt is the per-worker surplus, dt = dt/Nt is the per-worker dividend paid to shareholders, bt = Bt/Nt is the per-worker liabilities, ht = Ht/Nt denotes the newly hired workers per existing employees, and gt+1 = Nt+1/Nt is the gross growth rate of employment. Of special interest is the discount factor for the next period’s normalized surplus, β(gt+1 − ηht). When workers have zero bargaining power, that is, η = 0, the discount factor reduces to βgt+1. Because in this case the whole surplus goes to shareholders, they will also get the whole next period’s surplus. When η > 0, however, some of the next period’s surplus will be shared with newly hired workers. This is captured by the term ηht, which reduces the effective discount factor. Of course, the lower discounting is relevant only if firms add new workers in the next period, that is, ht > 0. Therefore, the conflict of interest in the choice of the firm’s policies is not between shareholders and ”existing” employees but between current stake holders (shareholders and existing employees) and “new” workers. The envelope condition allows us to derive the derivative of the normalized surplus function that takes the form \begin{equation*} \frac{\partial s(\boldsymbol{x}_t,b_t)}{\partial b_t} = -1-\frac{\partial \varphi (\boldsymbol{x}_t,b_t)}{\partial b_t}. \end{equation*} This shows that the derivative of the surplus is fully defined by the derivative of the distress cost, which is a known function. We can then express the normalized surplus as \begin{equation} s(\boldsymbol{x}_t,b_t)=\bar{s}(\boldsymbol{x}_t)-b_t-\varphi (\boldsymbol{x}_t,b_t), \end{equation} (13) where $$\bar{s}(\boldsymbol{x}_t)$$ depends only on the exogenous states (shocks). The special form of the surplus function (13) allows us to derive an analytical expression for the maximum collateralized debt $$b^*_t$$. This is defined implicitly by the condition $$b^*_t = \xi _t s(\boldsymbol{x}_t,b^*_t)$$. Using equation (13) to eliminating $$s(\boldsymbol{x}_t,b^*_t)$$, we obtain $$b^*_t = \xi _t [\bar{s}(\boldsymbol{x}_t) - b^*_t-\varphi (\boldsymbol{x}_t,b^*_t)]$$. Because $$\varphi (\boldsymbol{x}_t,b_t^*)=0$$ by definition, we can solve for \begin{equation} b^*_t = \left(\frac{\xi _t}{1+\xi _t}\right)\bar{s}(\boldsymbol{x}_t). \end{equation} (14) Therefore, the maximum collateralized debt is only determined by the exogenous states, xt = (zt, ξt). Using this expression to replace $$b_t^*$$ in the distress cost function (4), we obtain \begin{equation} \varphi (\boldsymbol{x}_t,b_t) \, \equiv \, \kappa \cdot \left( \max \left\lbrace b_t-\frac{\xi _t}{1+\xi _t}\bar{s}(\boldsymbol{x}_t)\, ,\, 0\right\rbrace \right)^2 . \end{equation} (15) This proves the initial conjecture in equation (4) that the function φ(., .) depends only on the exogenous states, xt, and the normalized debt, bt. The particular form of the surplus function (13) allows us to write the firm’s problem as \begin{eqnarray} \bar{s}(\boldsymbol{x}_t) \max_{h_{t},b_{t+1}}&& \big\lbrace z_{t}-\Upsilon (h_t)+q_{t}g_{t+1}b_{t+1}-u_t\nonumber\\ && +\, \beta (g_{t+1}-\eta h_t)\mathbb{E}_t \big[\bar{s}(\boldsymbol{x}_{t+1})-b_{t+1}-\varphi (\boldsymbol{x}_{t+1},b_{t+1}) \big] \big\rbrace \\ \text{subj. to:} && \xi _{t}g_{t+1} \beta \mathbb{E}_t \big [\bar{s}(\boldsymbol{x}_{t+1})-b_{t+1}-\varphi (\boldsymbol{x}_{t+1},b_{t+1})\big ]\ge q_t g_{t+1} b_{t+1}\nonumber \\ && g_{t+1}=1-\lambda +h_t. \nonumber \end{eqnarray} (16) This problem is recursive in $$\bar{s}(\boldsymbol{x}_t)$$ and does not depend on the endogenous state bt. Thus, the solution consists of functions for the hiring policy, ht = fh(xt), the borrowing policy, bt + 1 = fb(xt), and the surplus $$\bar{s}(\boldsymbol{x}_t)$$. These functions depend only on the exogenous states xt. The properties of the surplus function $$\bar{s}(\boldsymbol{x}_t)$$ are characterized in the following lemma. Lemma 1. The function $$\bar{s}(\boldsymbol{x}_t)$$ is strictly increasing in zt. It is also strictly increasing in ξt but only if η > 0 and/or qt > β. These properties are intuitive. If the firm is more productive today, Assumption 1 guarantees that, in expectation, the firm will be more productive in the future. It is then obvious that the surplus increases in zt. A similar intuition applies to a higher value of ξt that makes the problem of the firm less constrained today: By Assumption 1, the firm will be less constrained in expectation also in the future. However, if η = 0 and qt = β, there is no gain from borrowing. In fact, when η = 0 wages are only determined by the outside value of workers, which is exogenous in the model. Thus, the debt chosen by the firm does not affect the cost of labor. When qt = β, paying more dividends today by borrowing does not bring any gain to shareholders because they discount future dividends at the same rate as the interest rate paid on the debt. Instead, when η > 0, a higher value of ξt allows the firm to borrow more that in turn allows the negotiation of lower wages paid to new employees. With qt > β, the cost of borrowing is lower than the intertemporal discount rate. Thus, having the ability to borrow more increases the surplus of the firm independently of the bargaining channel of debt. To characterize the hiring and financial policies, we derive the first-order conditions with respect to ht and bt+1. Denoting by μt the Lagrange multiplier for the enforcement constraint, the first-order conditions for Problem (16) read \begin{eqnarray} &&q_t b_{t+1}+\beta (1-\eta )\mathbb{E}_t [\bar{s}(\boldsymbol{x}_{t+1})-b_{t+1}-\varphi (\boldsymbol{x}_{t+1},b_{t+1}) ] = \Upsilon ^{\prime }(h_t), \end{eqnarray} (17) \begin{eqnarray} &&(1-\mu _t)q_t - \beta (1+\mu _t\xi _t-\eta h_t/g_{t+1})\mathbb{E}_t\left[1+\frac{\partial \varphi (\boldsymbol{x}_{t+1},b_{t+1})}{\partial b_{t+1}}\right] = 0. \end{eqnarray} (18) Before continuing it will be helpful to clarify some features of the model. First, because of the homogeneity assumption (constant returns), the growth of firms is independent of size. This allowed us to rewrite the problem of the firm in normalized form. What matters for growth is profitability, which is mostly driven by productivity. Firms that have high “persistent” productivity would like to grow faster. But what constrains growth is not access to credit. Even though credit is subject to a borrowing limit, firms could issue equity without any cost (negative dividends). Growth, instead, is constrained by the convex cost of hiring. When firms hire more workers, they also borrow more. But the reason they borrow more is not because they need to finance hiring. Instead, they borrow more because more debt allows firms to reduce the wages that they will have to pay to the larger number of new added workers. 2.2. Special Case without Financial Distress Cost and qt = β To gather some intuitions about the properties of the model, it will be helpful to focus first on the special case without financial distress. Therefore, in this section, we assume that κ = 0 that in turn implies φ(xt, bt) = ∂φ(xt, bt)/∂bt = 0. In this section, we will also focus on the special case in which qt = β. This could be interpreted as a steady-state condition if the model is extended to a general equilibrium without aggregate uncertainty and without a tax shield that also creates a preference for debt over equity for tax purposes. In this environment, the steady-state interest rate would be equal to rt = 1/β − 1 and the price of debt is qt = β.1 Proposition 1. Assume that κ = 0 and qt = β. If η > 0, the firm borrows up to the limit whenever it finds optimal to choose ht > 0. If η = 0 or ht = 0, the debt is undetermined. Proof. When qt = β, the first-order condition for debt, equation (18), simplifies to ηht = μtgt+1(1 + ξt). If η > 0, the equation implies that the Lagrange multiplier μt is strictly positive whenever ht > 0. Therefore, under the condition ht > 0, the enforcement constraint is binding. When η = 0 or ht = 0, the equation implies that μt is zero. Therefore, the enforcement constraint is not binding and debt is undetermined. Whenever the firm chooses to hire, that is, ht > 0, it adds new workers with whom it will share the next period surplus. Increasing the debt today reduces the next period surplus and allows for lower compensation of the newly hired workers. This increases the current surplus of the firm that is shared by shareholders and currently employed workers, but not by the new hired workers. It is then in the interest of both shareholders and existing employees to increase the debt of the firm. When the firm does not add new workers, however, higher borrowing does not increase the current surplus because more debt only reduces the future compensation of existing workers. In this case, there are no gains from borrowing. Thus, as long as the firm adds new workers, bargaining introduces a motive to borrow, breaking the irrelevance of the financial structure of firms (Modigliani and Miller 1958). The motive to borrow, however, is present only when the bargaining power of workers is positive (η > 0) and the firm hires new workers (ht > 0). We now turn to the hiring policy characterized by the first-order condition (17). Together with the normalized law of motion for employment, gt+1 = 1 − λ + ht, this equation establishes a relation between the per-worker debt bt+1 and the growth of employment, which also depends on the exogenous states through the term $$\mathbb{E}_t\bar{s}(\boldsymbol{x}_{t+1})$$. This relation is not linear and depends on the bargaining power of workers η as stated in the next proposition. Proposition 2. Suppose that κ = 0 and qt = β. Controlling for productivity zt, firm hiring ht is strictly increasing in ξt if η > 0 but it is independent of ξt if η = 0. Proof. When qt = β, the first-order condition (17) can be rewritten as \begin{equation} \beta [\eta b_{t+1}+(1-\eta )\mathbb{E}_t \bar{s}(\boldsymbol{x}_{t+1})] = \Upsilon ^{\prime }(h_t). \end{equation} (19) From Lemma 1, we know that $$\mathbb{E}_t\bar{s}(\boldsymbol{x}_{t+1})$$ is strictly increasing in ξt if η > 0. Furthermore, Proposition 1 established that the firm borrows up to the limit when η > 0. Because the limit is strictly increasing in ξt, bt+1 is strictly increasing in ξt. Therefore, the left-hand side of (19) increases with ξt. Then, the right-hand side must also increase when ξt rises. Given the convexity of the hiring cost, ϒ΄(ht) is strictly increasing in ht. Therefore, condition (19) implies that ht is strictly increasing in ξt. When η = 0, however, the optimality condition reduces to $$\mathbb{E}_t \bar{s}(\boldsymbol{x}_{t+1}) = \Upsilon ^{\prime }(h_t)$$. From Lemma 1, we know that $$\mathbb{E}_t\bar{s}(\boldsymbol{x}_{t+1})$$ is independent of ξt when η = 0. Therefore, ht is also independent of ξt. According to the proposition, changing the borrowing capability of the firm affects hiring only if workers have some bargaining power. When workers have zero bargaining power—which can be interpreted as the case of a competitive labor market where the determination of wages is external to an individual firm—debt is irrelevant for the hiring decision of firms. Together with the previous Proposition 1, Proposition 2 implies that, controlling for productivity, the growth of employment is positively related to the firm’s debt (provided that η > 0). In fact, because the firm borrows up to the limit, a higher value of ξt relaxes the borrowing limit and allows the firm to take more debt. More debt will then allow the firm to negotiate lower wages with future employees and this increases the incentive to hire. The above propositions can be easily extended to the case in which qt > β. In this case, the firm has an additional motive to borrow (because the effective cost of debt is lower than the cost of equity). This implies that the debt is no longer undetermined when η = 0 or ht = 0 (see Proposition 1). Furthermore, the hiring policy of the firm is no longer independent of the debt even if η = 0 (Proposition 2). 2.3. General Case with Financial Distress With financial distress, the surplus function is concave in bt due to the convexity of the distress cost. This introduces a precautionary motive that discourages borrowing. Thus, the firm may choose not to borrow up to the limit and the borrowing constraint could be only occasionally binding, that is, $$\xi _t g_{t+1} \beta \mathbb{E}_t s(\boldsymbol{x}_{t+1},b_{t+1})\ge q_t g_{t+1}b_{t+1}$$. Figure 1 provides a graphical intuitive illustration for the property of the optimal choice of debt. The first panel depicts the model without financial distress. In this case, the marginal benefit of debt is always bigger than the marginal cost of borrowing (provided that η > 0 and the firm chooses ht > 0). This is because the marginal benefit of debt also includes the reduced future cost of labor made possible by a marginal increase in debt. Therefore, the firm always borrows up to the limit, which is indicated in the graph by the vertical line. Figure 1. View largeDownload slide Optimal debt policy. Figure 1. View largeDownload slide Optimal debt policy. The case with financial distress is depicted in the second panel of Figure 1. In this case, the marginal cost is initially below the marginal benefit. However, as the debt increases, the expected cost of financial distress rises, inducing an increase in the marginal cost of debt. As a result, the firm does not borrow up to the limit. Furthermore, the difference between the borrowing limit and the actual debt increases with the financial distress cost captured by the parameter κ. This feature of the model is similar to Boileau and Moyen (2016) where firms hold cash for precautionary reasons. Some of the properties stated in Propositions 1 and 2 also apply to the model with financial distress. In particular, if workers do not have any bargaining power (η = 0) and qt = β, we can see from equation (18) that the enforcement constraint is never binding (μt = 0) and the expected distress cost is zero, that is, $$\mathbb{E}_t\varphi (\boldsymbol{x}_{t+1},b_{t+1})=0$$. Because debt does not provide any value when η = 0, the firm does not borrow in order to avoid the distress cost. At the same time, because the firm does not borrow and the expected distress cost is zero, the hiring decision characterized by condition (17) is not affected by the financial conditions of the firm ξt. This is stated formally in the next proposition. Proposition 3. If η = 0 and qt = β, the borrowing limit is never binding and hiring depends only on productivity. Proof. If η = 0 and qt = β, the first-order condition (18) becomes \begin{equation*} 1-\mu _t = (1+\mu _t\xi _t)\mathbb{E}_t\left[1+\frac{\partial \varphi (\boldsymbol{x}_{t+1},b_{t+1})}{\partial b_{t+1}}\right], \end{equation*} which is satisfied only if the Lagrange multiplier associated with the borrowing constraint is μt = 0 and the marginal cost of financial distress is ∂φ(xt + 1, bt + 1)/∂bt + 1 = 0. If the marginal cost of financial distress is zero, the expected cost must also be zero. Given that $$\mathbb{E}_t \varphi (\boldsymbol{x}_{t+1},b_{t+1})=0$$, the first-order condition (17) becomes $$\beta \mathbb{E}_t\bar{s}(\boldsymbol{x}_{t+1}) = \Upsilon ^{\prime }(h_t)$$. But when the borrowing limit is never binding and wages do not depend on debt, the surplus function $$\bar{s}(\boldsymbol{x}_{t+1})$$ depends only zt+1. Thus, the hiring policy ht depends only on current productivity zt (which is sufficient to characterize the probability distribution of zt+1 and, therefore, the expectation of $$\bar{s}(\boldsymbol{x}_{t+1})$$). 2.4. Numerical Solution We have seen that the normalized policy and surplus functions—ht = fh(xt), bt + 1 = fb(xt) and $$\bar{s}(\boldsymbol{x}_t)$$—do not depend on the current normalized debt bt but only on the exogenous states xt = (zt, ξt). If the exogenous states (shocks) take a finite number of values, these functions also take a finite number of values. We can then define $$\bar{\mathbf {s}}_t$$ the vector that contains $$\bar{s}(\boldsymbol{x}_t)$$ for all possible (finite) realizations of xt. Thus, Problem (16) is Bellman’s equation in the unknown vector $$\bar{\mathbf {s}}_t$$ that can be solved by value function iteration. Denote by nz and nξ the number of possible values for productivity and financial shocks. Each iteration starts with a guess for the vector $$\bar{\mathbf {s}}_{t+1}$$, that is, the vector that contains the nz × nξ elements of the surplus $$\bar{s}(\boldsymbol{x}_{t+1})$$ in the next period. For each combination of the two shocks in the current period and given the guess for $$\bar{\mathbf {s}}_{t+1}$$, we derive the optimal policies by solving the first-order conditions (17) and (18) together with the enforcement constraint in problem (16), using a nonlinear solver. Because the enforcement constraint could be satisfied with equality (in which case μt > 0) or with inequality (in which case μt = 0), we have to check the Kuhn–Tucker conditions. Given the solutions found for each combination of the two shocks, we can compute the vector $$\bar{\mathbf {s}}_t$$. This is then used as a new guess for $$\bar{\mathbf {s}}_{t+1}$$ and we continue the iteration until we find a fixed point, that is, $$\bar{\mathbf {s}}_t=\bar{\mathbf {s}}_{t+1}$$.2 3. Structural Estimation We estimate the model using the simulated method of moments as in Lee and Ingram (1991) and Nikolov and Whited (2014). The detailed description of the estimation procedure is provided in Appendix B. 3.1. Parameters and Moments We need first to specify the functional form for the hiring cost function. This is assumed to take the quadratic form $$\Upsilon (h_t)=\psi h_t + \phi h_t^2$$, where ψ and ϕ are parameters and ht = Ht/Nt is the ratio of new hires over current employment. The two shocks, zt and ξt, follow independent first-order Markov processes approximated with discrete five-dimensional Markov chains. All parameters are estimated with the exception of the intertemporal discount factor, β, the price of debt, $$q_t=\bar{q}$$, the separation rate, λ, and the linear hiring cost parameter, ψ. The reason these parameters are calibrated, instead of being estimated, is because they cannot be easily identified separately from the estimated parameters. Therefore, we preset them using direct calibration targets. For the calibration of β and $$\bar{q}$$, we first make the following assumption: The interest rate in the model is equal to the intertemporal discount rate, that is, $$r_t=1/\beta -1=\bar{r}$$. This can be interpreted as an equilibrium steady-state property if the model is extended to a general equilibrium. However, because of tax deductibility of interests, the effective cost of borrowing is $$(1-\tau )\bar{r}$$, where τ represents the corporate tax rate.3 This implies that the effective price at which firms can sell their debt is \begin{equation*} \bar{q}=\frac{1}{1+(1-\tau )\bar{r}}=\frac{1}{1+(1-\tau )(1/\beta -1)}. \end{equation*} We then set β = 0.97, which implies an annual interest rate close to 3% , and the corporate tax rate to τ = 0.2. The resulting value of $$\bar{q}$$ is 0.9758. We will later check the sensitivity to τ. Based on micro evidence for hiring (e.g., Davis et al. 2006), we set the annual separation rate to λ = 0.25. The hiring parameter ψ is chosen so that the unconditional average growth rate of the firm is zero (given all other parameters). This is a normalization that insures the stationarity of the model (no aggregate growth). After normalizing the average productivity to $$\bar{z}=1$$, we are left with nine parameters: the persistence and volatility of the productivity shock, ρz and σz, the mean, persistence and volatility of the credit shock, $$\bar{\xi }$$, ρξ and σξ, the financial distress cost, κ, the workers’ bargaining power, η, the hiring cost, ϕ, and the separation utility flow $$\bar{u}$$. To estimate these parameters, we use 17 moments that have clear counterparts in the model: the mean of the ratio of wages over revenues, the mean of the ratio of unused credit over total credit, the mean of Tobin’s Q, the mean of financial leverage (four moments); the standard deviations and autocorrelations of leverage, Tobin’s Q, employment growth, sales growth, and debt growth (ten moments); the cross correlations of employment growth, sales growth, and debt growth (three moments). It is important to point out that the empirical series “unused lines of credit” is just a “proxy” for the unused debt capacity in the model, that is, the difference between actual borrowing, qtBt+1, and the debt limit, ξtβSt+1. In reality, firms may have access to other financial instruments in order to raise funds and, therefore, the actual borrowing limit could be different from the outstanding lines of credit reported in the data. However, to the extent that unused lines of credit are correlated with the overall credit capacity of the firm, the empirical series provides useful information for the estimation and identification of the model parameters. In the estimation, we also consider moments related to Tobin’s Q.4 In the model, we define Tobin’s Q as \begin{equation*} \frac{(1-\eta ) S_{t}+B_{t+1}}{N_{t+1}}. \end{equation*} In the data, Tobin’s Q is measured as the market-to-book ratio (the market value of equity plus the financial debt, divided by the book value of total assets).5 One caveat from using moments related to Tobin’s Q is that the model does not include capital investment and thus the parameter estimates might be biased. However, in the sensitivity analysis, we estimate an extended version of the model with capital accumulation. Beside the average wage to revenue ratio, we do not use moments related to wage data even if they could improve the estimation accuracy. This is because data on wage is not widely available in Compustat. A recent paper by Michaels et al. (2014) uses wage data to estimate a model with wage bargaining and finds that the bargaining channel is important for generating a negative correlation between wages and leverage. 3.2. Data With the exception of unused lines of credit and the average wage, all variables are from Compustat Annual. Data on unused lines of credit are not available in Compustat and some studies collect information about credit lines from firms’ SEC 10-K files (see, for example, Sufi 2009). The data source is Capital IQ that contains a large sample of unused lines of credit from 2003 to 2010. The variable unused lines of credit also refers to total undrawn credit, which includes “Undrawn Revolving Credit”, “Undrawn Commercial Paper”, “Undrawn Term Loans”, and “Other Available Credit”. For more details see Acharya et al. (2014). Unused lines of credit from Capital IQ have also been used by Gilchrist and Zakrajsek (2012) to study the macroeconomic impact of credit supply shocks. Following Bresnahan et al. (2002) and İmrohoroğlu and Tüzel (2014), we measure firm-level wage bills wtNt as the labor expenses in Compustat whenever available. When the data about labor expenses are not available in Compustat, we approximate the wage bills as the number of employees from Compustat multiplied by the industry wage rate from the Bureau of Economic Analysis (BEA).6 To check the accuracy of using industry wage rate, we compare the wage rate calculated using BEA industry data with the wage rate calculated using Compustat data for reporting firms. For the set of firms that report labor expenses in Compustat, the average wage rate is $64,760 in 2010 dollars. The average wage rate calculated using BEA data is $67,567. Following the literature, we exclude utilities and financial firms with Standard Industrial Classification (SIC) codes in the intervals 4,900–4,949 and 6,000–6,999, and firms with SIC codes greater than 9,000. We also exclude firms with missing values of assets, sales, number of employees, debt, and unused lines of credit during the sample period. As in Benmelech et al. (2011), we require firms to have at least 500 employees. Because we need to calculate empirical moments that require repeated observations for each individual firm (such as standard deviations and autocorrelations), we drop firms with less than 3 years of data. To limit the impact of outliers (e.g., merges and acquisitions), we also winsorize all level variables at 2.5% and 97.5% percentiles, and growth variables at 5% and 95% percentiles. Nominal variables are deflated by the CPI. The final sample used in the structural estimation is an unbalanced panel (for each variable) of 2,168 firms over 8 years, from 2003 to 2010. Although not reported in the paper, we also estimate the model using a balanced panel and obtained similar results. Appendix C provides the detailed definition of all variables. 3.3. Identification The values of the estimated parameters are reported in the bottom section of Table 1. Before commenting on the economic significance of the estimates, we discuss the identification mechanisms. We do so by conducting a sensitivity analysis with respect to each parameter. The first column of Table 2 reports the simulated moments generated by the estimated model (as in Table 1). The remaining columns show the simulated moments after increasing the value of one parameter by 10% (and keeping all other parameters at the estimated values). Table 1. Structural estimation. (a) (b) (c) (d) Target moments Empirical data Estimated model t-Statistics Counterfactual w/o bargaining $$\mathit {Mean(wageshare}_{t})$$ 0.685 0.692 (− 1.48) 0.693 $$\mathit {Mean}(\mathit {unusedcredit}_{t})$$ 0.403 0.405 (− 0.36) 0.955 $$\mathit {Mean}(\mathit {leverage}_{t})$$ 0.217 0.216 (0.13) 0.021 $$\mathit {Std}(\mathit {leverage}_{t})$$ 0.100 0.096 (2.15) 0.017 $$\mathit {Autocor}(\mathit {leverage}_{t})$$ 0.735 0.701 (1.33) 0.627 $$\mathit {Mean(Tobin}Q_{t})$$ 1.360 1.374 (− 0.83) 1.404 $$\mathit {Std(Tobin}Q_{t})$$ 0.421 0.256 (15.59) 0.174 $$\mathit {Autocor(Tobin}Q_{t})$$ 0.668 0.411 (7.59) 0.691 $$\mathit {Std}(\Delta \mathit {employ}_{t})$$ 0.111 0.091 (13.48) 0.077 $$\mathit {Autocor}(\Delta \mathit {employ}_{t})$$ 0.065 0.670 (− 31.67) 0.674 $$\mathit {Std}(\Delta \mathit {sales}_{t})$$ 0.135 0.135 (0.28) 0.131 $$\mathit {Autocor}(\Delta \mathit {sales}_{t})$$ 0.051 0.214 (− 6.33) 0.120 $$\mathit {Std}(\Delta \mathit {debt}_{t})$$ 0.583 0.566 (1.90) 0.809 $$\mathit {Autocor}(\Delta \mathit {debt}_{t})$$ − 0.009 − 0.136 (6.24) − 0.127 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {sales}_{t})$$ 0.531 0.853 (− 9.88) 0.824 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {debt}_{t})$$ 0.340 0.391 (1.69) − 0.021 $$\mathit {Cor}(\Delta \mathit {sales}_{t},\Delta \mathit {debt}_{t})$$ 0.178 0.360 (− 12.23) − 0.194 (a) (b) (c) (d) Target moments Empirical data Estimated model t-Statistics Counterfactual w/o bargaining $$\mathit {Mean(wageshare}_{t})$$ 0.685 0.692 (− 1.48) 0.693 $$\mathit {Mean}(\mathit {unusedcredit}_{t})$$ 0.403 0.405 (− 0.36) 0.955 $$\mathit {Mean}(\mathit {leverage}_{t})$$ 0.217 0.216 (0.13) 0.021 $$\mathit {Std}(\mathit {leverage}_{t})$$ 0.100 0.096 (2.15) 0.017 $$\mathit {Autocor}(\mathit {leverage}_{t})$$ 0.735 0.701 (1.33) 0.627 $$\mathit {Mean(Tobin}Q_{t})$$ 1.360 1.374 (− 0.83) 1.404 $$\mathit {Std(Tobin}Q_{t})$$ 0.421 0.256 (15.59) 0.174 $$\mathit {Autocor(Tobin}Q_{t})$$ 0.668 0.411 (7.59) 0.691 $$\mathit {Std}(\Delta \mathit {employ}_{t})$$ 0.111 0.091 (13.48) 0.077 $$\mathit {Autocor}(\Delta \mathit {employ}_{t})$$ 0.065 0.670 (− 31.67) 0.674 $$\mathit {Std}(\Delta \mathit {sales}_{t})$$ 0.135 0.135 (0.28) 0.131 $$\mathit {Autocor}(\Delta \mathit {sales}_{t})$$ 0.051 0.214 (− 6.33) 0.120 $$\mathit {Std}(\Delta \mathit {debt}_{t})$$ 0.583 0.566 (1.90) 0.809 $$\mathit {Autocor}(\Delta \mathit {debt}_{t})$$ − 0.009 − 0.136 (6.24) − 0.127 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {sales}_{t})$$ 0.531 0.853 (− 9.88) 0.824 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {debt}_{t})$$ 0.340 0.391 (1.69) − 0.021 $$\mathit {Cor}(\Delta \mathit {sales}_{t},\Delta \mathit {debt}_{t})$$ 0.178 0.360 (− 12.23) − 0.194 Estimated parameters Estimators t-Statistics Persistence productivity shock, ρz 0.763 (74.90) Volatility productivity shock, σz 0.151 (69.47) Persistence debt shock, ρξ 0.753 (77.99) Volatility debt shock, σξ 0.167 (25.02) Financial distress cost, κ 2.475 (14.13) Workers’ bargaining power, η 0.355 (12.60) Hiring cost, ϕ 1.131 (28.36) Enforcement parameter, $$\bar{\xi }$$ 0.375 (25.29) Separation flow, $$\bar{u}$$ 0.532 (31.19) Estimated parameters Estimators t-Statistics Persistence productivity shock, ρz 0.763 (74.90) Volatility productivity shock, σz 0.151 (69.47) Persistence debt shock, ρξ 0.753 (77.99) Volatility debt shock, σξ 0.167 (25.02) Financial distress cost, κ 2.475 (14.13) Workers’ bargaining power, η 0.355 (12.60) Hiring cost, ϕ 1.131 (28.36) Enforcement parameter, $$\bar{\xi }$$ 0.375 (25.29) Separation flow, $$\bar{u}$$ 0.532 (31.19) Note: The table shows the results of the structural estimation. The first panel lists the target moments; column (a) reports the moments from the data; column (b) reports the moments generated by the estimated model; column (c) reports the t-Statistic for the differences between the empirical moments and the moments generated by the estimated model; column (d) reports the moments generated by the model without the bargaining channel of debt as described in Section 3.5. The second panel reports the estimated parameters and the associated t-Statistics. View Large Table 1. Structural estimation. (a) (b) (c) (d) Target moments Empirical data Estimated model t-Statistics Counterfactual w/o bargaining $$\mathit {Mean(wageshare}_{t})$$ 0.685 0.692 (− 1.48) 0.693 $$\mathit {Mean}(\mathit {unusedcredit}_{t})$$ 0.403 0.405 (− 0.36) 0.955 $$\mathit {Mean}(\mathit {leverage}_{t})$$ 0.217 0.216 (0.13) 0.021 $$\mathit {Std}(\mathit {leverage}_{t})$$ 0.100 0.096 (2.15) 0.017 $$\mathit {Autocor}(\mathit {leverage}_{t})$$ 0.735 0.701 (1.33) 0.627 $$\mathit {Mean(Tobin}Q_{t})$$ 1.360 1.374 (− 0.83) 1.404 $$\mathit {Std(Tobin}Q_{t})$$ 0.421 0.256 (15.59) 0.174 $$\mathit {Autocor(Tobin}Q_{t})$$ 0.668 0.411 (7.59) 0.691 $$\mathit {Std}(\Delta \mathit {employ}_{t})$$ 0.111 0.091 (13.48) 0.077 $$\mathit {Autocor}(\Delta \mathit {employ}_{t})$$ 0.065 0.670 (− 31.67) 0.674 $$\mathit {Std}(\Delta \mathit {sales}_{t})$$ 0.135 0.135 (0.28) 0.131 $$\mathit {Autocor}(\Delta \mathit {sales}_{t})$$ 0.051 0.214 (− 6.33) 0.120 $$\mathit {Std}(\Delta \mathit {debt}_{t})$$ 0.583 0.566 (1.90) 0.809 $$\mathit {Autocor}(\Delta \mathit {debt}_{t})$$ − 0.009 − 0.136 (6.24) − 0.127 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {sales}_{t})$$ 0.531 0.853 (− 9.88) 0.824 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {debt}_{t})$$ 0.340 0.391 (1.69) − 0.021 $$\mathit {Cor}(\Delta \mathit {sales}_{t},\Delta \mathit {debt}_{t})$$ 0.178 0.360 (− 12.23) − 0.194 (a) (b) (c) (d) Target moments Empirical data Estimated model t-Statistics Counterfactual w/o bargaining $$\mathit {Mean(wageshare}_{t})$$ 0.685 0.692 (− 1.48) 0.693 $$\mathit {Mean}(\mathit {unusedcredit}_{t})$$ 0.403 0.405 (− 0.36) 0.955 $$\mathit {Mean}(\mathit {leverage}_{t})$$ 0.217 0.216 (0.13) 0.021 $$\mathit {Std}(\mathit {leverage}_{t})$$ 0.100 0.096 (2.15) 0.017 $$\mathit {Autocor}(\mathit {leverage}_{t})$$ 0.735 0.701 (1.33) 0.627 $$\mathit {Mean(Tobin}Q_{t})$$ 1.360 1.374 (− 0.83) 1.404 $$\mathit {Std(Tobin}Q_{t})$$ 0.421 0.256 (15.59) 0.174 $$\mathit {Autocor(Tobin}Q_{t})$$ 0.668 0.411 (7.59) 0.691 $$\mathit {Std}(\Delta \mathit {employ}_{t})$$ 0.111 0.091 (13.48) 0.077 $$\mathit {Autocor}(\Delta \mathit {employ}_{t})$$ 0.065 0.670 (− 31.67) 0.674 $$\mathit {Std}(\Delta \mathit {sales}_{t})$$ 0.135 0.135 (0.28) 0.131 $$\mathit {Autocor}(\Delta \mathit {sales}_{t})$$ 0.051 0.214 (− 6.33) 0.120 $$\mathit {Std}(\Delta \mathit {debt}_{t})$$ 0.583 0.566 (1.90) 0.809 $$\mathit {Autocor}(\Delta \mathit {debt}_{t})$$ − 0.009 − 0.136 (6.24) − 0.127 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {sales}_{t})$$ 0.531 0.853 (− 9.88) 0.824 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {debt}_{t})$$ 0.340 0.391 (1.69) − 0.021 $$\mathit {Cor}(\Delta \mathit {sales}_{t},\Delta \mathit {debt}_{t})$$ 0.178 0.360 (− 12.23) − 0.194 Estimated parameters Estimators t-Statistics Persistence productivity shock, ρz 0.763 (74.90) Volatility productivity shock, σz 0.151 (69.47) Persistence debt shock, ρξ 0.753 (77.99) Volatility debt shock, σξ 0.167 (25.02) Financial distress cost, κ 2.475 (14.13) Workers’ bargaining power, η 0.355 (12.60) Hiring cost, ϕ 1.131 (28.36) Enforcement parameter, $$\bar{\xi }$$ 0.375 (25.29) Separation flow, $$\bar{u}$$ 0.532 (31.19) Estimated parameters Estimators t-Statistics Persistence productivity shock, ρz 0.763 (74.90) Volatility productivity shock, σz 0.151 (69.47) Persistence debt shock, ρξ 0.753 (77.99) Volatility debt shock, σξ 0.167 (25.02) Financial distress cost, κ 2.475 (14.13) Workers’ bargaining power, η 0.355 (12.60) Hiring cost, ϕ 1.131 (28.36) Enforcement parameter, $$\bar{\xi }$$ 0.375 (25.29) Separation flow, $$\bar{u}$$ 0.532 (31.19) Note: The table shows the results of the structural estimation. The first panel lists the target moments; column (a) reports the moments from the data; column (b) reports the moments generated by the estimated model; column (c) reports the t-Statistic for the differences between the empirical moments and the moments generated by the estimated model; column (d) reports the moments generated by the model without the bargaining channel of debt as described in Section 3.5. The second panel reports the estimated parameters and the associated t-Statistics. View Large Table 2. Sensitivity of each estimated parameter. ρz σz ρξ σξ κ η ϕ $$\bar{\xi }$$ $$\bar{u}$$ $$\mathit {Mean(wageshare}_{t})$$ 0.692 0.703 0.695 0.694 0.694 0.695 0.705 0.693 0.688 0.712 $$\mathit {Mean}(\mathit {unusedcredit}_{t})$$ 0.405 0.422 0.412 0.456 0.472 0.467 0.391 0.401 0.366 0.398 $$\mathit {Mean}(\mathit {leverage}_{t})$$ 0.216 0.216 0.215 0.196 0.188 0.193 0.231 0.218 0.250 0.220 $$\mathit {Std}(\mathit {leverage}_{t})$$ 0.096 0.201 0.096 0.134 0.107 0.087 0.155 0.094 0.087 0.102 $$\mathit {Autocor}(\mathit {leverage}_{t})$$ 0.701 0.154 0.693 0.794 0.724 0.721 0.304 0.719 0.725 0.621 $$\mathit {Mean(Tobin}Q_{t})$$ 1.374 1.400 1.375 1.371 1.365 1.366 1.315 1.383 1.389 1.303 $$\mathit {Std(Tobin}Q_{t})$$ 0.256 0.338 0.277 0.265 0.257 0.246 0.250 0.263 0.259 0.259 $$\mathit {Autocor(Tobin}Q_{t})$$ 0.411 0.538 0.430 0.381 0.382 0.430 0.410 0.426 0.421 0.430 $$\mathit {Std}(\Delta \mathit {employ}_{t})$$ 0.091 0.150 0.100 0.095 0.091 0.089 0.088 0.083 0.093 0.091 $$\mathit {Autocor}(\Delta \mathit {employ}_{t})$$ 0.670 0.753 0.670 0.682 0.669 0.670 0.669 0.670 0.670 0.670 $$\mathit {Std}(\Delta \mathit {sales}_{t})$$ 0.135 0.170 0.149 0.137 0.134 0.133 0.134 0.133 0.135 0.135 $$\mathit {Autocor}(\Delta \mathit {sales}_{t})$$ 0.214 0.548 0.206 0.238 0.214 0.203 0.191 0.152 0.221 0.211 $$\mathit {Std}(\Delta \mathit {debt}_{t})$$ 0.566 0.641 0.583 1.400 0.847 0.543 0.570 0.551 0.437 0.567 $$\mathit {Autocor}(\Delta \mathit {debt}_{t})$$ − 0.136 − 0.086 − 0.133 − 0.093 − 0.137 − 0.135 − 0.137 − 0.138 − 0.123 − 0.135 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {sales}_{t})$$ 0.853 0.951 0.853 0.846 0.847 0.851 0.834 0.820 0.860 0.853 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {debt}_{t})$$ 0.391 0.454 0.403 0.207 0.313 0.398 0.391 0.377 0.459 0.390 $$\mathit {Cor}(\Delta \mathit {sales}_{t},\Delta \mathit {debt}_{t})$$ 0.360 0.442 0.405 0.145 0.257 0.377 0.342 0.348 0.454 0.369 ρz σz ρξ σξ κ η ϕ $$\bar{\xi }$$ $$\bar{u}$$ $$\mathit {Mean(wageshare}_{t})$$ 0.692 0.703 0.695 0.694 0.694 0.695 0.705 0.693 0.688 0.712 $$\mathit {Mean}(\mathit {unusedcredit}_{t})$$ 0.405 0.422 0.412 0.456 0.472 0.467 0.391 0.401 0.366 0.398 $$\mathit {Mean}(\mathit {leverage}_{t})$$ 0.216 0.216 0.215 0.196 0.188 0.193 0.231 0.218 0.250 0.220 $$\mathit {Std}(\mathit {leverage}_{t})$$ 0.096 0.201 0.096 0.134 0.107 0.087 0.155 0.094 0.087 0.102 $$\mathit {Autocor}(\mathit {leverage}_{t})$$ 0.701 0.154 0.693 0.794 0.724 0.721 0.304 0.719 0.725 0.621 $$\mathit {Mean(Tobin}Q_{t})$$ 1.374 1.400 1.375 1.371 1.365 1.366 1.315 1.383 1.389 1.303 $$\mathit {Std(Tobin}Q_{t})$$ 0.256 0.338 0.277 0.265 0.257 0.246 0.250 0.263 0.259 0.259 $$\mathit {Autocor(Tobin}Q_{t})$$ 0.411 0.538 0.430 0.381 0.382 0.430 0.410 0.426 0.421 0.430 $$\mathit {Std}(\Delta \mathit {employ}_{t})$$ 0.091 0.150 0.100 0.095 0.091 0.089 0.088 0.083 0.093 0.091 $$\mathit {Autocor}(\Delta \mathit {employ}_{t})$$ 0.670 0.753 0.670 0.682 0.669 0.670 0.669 0.670 0.670 0.670 $$\mathit {Std}(\Delta \mathit {sales}_{t})$$ 0.135 0.170 0.149 0.137 0.134 0.133 0.134 0.133 0.135 0.135 $$\mathit {Autocor}(\Delta \mathit {sales}_{t})$$ 0.214 0.548 0.206 0.238 0.214 0.203 0.191 0.152 0.221 0.211 $$\mathit {Std}(\Delta \mathit {debt}_{t})$$ 0.566 0.641 0.583 1.400 0.847 0.543 0.570 0.551 0.437 0.567 $$\mathit {Autocor}(\Delta \mathit {debt}_{t})$$ − 0.136 − 0.086 − 0.133 − 0.093 − 0.137 − 0.135 − 0.137 − 0.138 − 0.123 − 0.135 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {sales}_{t})$$ 0.853 0.951 0.853 0.846 0.847 0.851 0.834 0.820 0.860 0.853 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {debt}_{t})$$ 0.391 0.454 0.403 0.207 0.313 0.398 0.391 0.377 0.459 0.390 $$\mathit {Cor}(\Delta \mathit {sales}_{t},\Delta \mathit {debt}_{t})$$ 0.360 0.442 0.405 0.145 0.257 0.377 0.342 0.348 0.454 0.369 Note: The first column reports the moments generated by the simulation of the model under the estimated parameters (benchmark parametrization as shown in Table 1). The remaining columns report the simulated moments after increasing the value of the specific parameter by 10%. The parameters are, respectively, the persistence and volatility of the productivity and credit, ρz, σz, ρξ, σξ, the financial distress cost, κ, the workers’ bargaining power, η, the hiring cost, ϕ, the average enforcement $$\bar{\xi }$$, and the employment flow $$\bar{u}$$. View Large Table 2. Sensitivity of each estimated parameter. ρz σz ρξ σξ κ η ϕ $$\bar{\xi }$$ $$\bar{u}$$ $$\mathit {Mean(wageshare}_{t})$$ 0.692 0.703 0.695 0.694 0.694 0.695 0.705 0.693 0.688 0.712 $$\mathit {Mean}(\mathit {unusedcredit}_{t})$$ 0.405 0.422 0.412 0.456 0.472 0.467 0.391 0.401 0.366 0.398 $$\mathit {Mean}(\mathit {leverage}_{t})$$ 0.216 0.216 0.215 0.196 0.188 0.193 0.231 0.218 0.250 0.220 $$\mathit {Std}(\mathit {leverage}_{t})$$ 0.096 0.201 0.096 0.134 0.107 0.087 0.155 0.094 0.087 0.102 $$\mathit {Autocor}(\mathit {leverage}_{t})$$ 0.701 0.154 0.693 0.794 0.724 0.721 0.304 0.719 0.725 0.621 $$\mathit {Mean(Tobin}Q_{t})$$ 1.374 1.400 1.375 1.371 1.365 1.366 1.315 1.383 1.389 1.303 $$\mathit {Std(Tobin}Q_{t})$$ 0.256 0.338 0.277 0.265 0.257 0.246 0.250 0.263 0.259 0.259 $$\mathit {Autocor(Tobin}Q_{t})$$ 0.411 0.538 0.430 0.381 0.382 0.430 0.410 0.426 0.421 0.430 $$\mathit {Std}(\Delta \mathit {employ}_{t})$$ 0.091 0.150 0.100 0.095 0.091 0.089 0.088 0.083 0.093 0.091 $$\mathit {Autocor}(\Delta \mathit {employ}_{t})$$ 0.670 0.753 0.670 0.682 0.669 0.670 0.669 0.670 0.670 0.670 $$\mathit {Std}(\Delta \mathit {sales}_{t})$$ 0.135 0.170 0.149 0.137 0.134 0.133 0.134 0.133 0.135 0.135 $$\mathit {Autocor}(\Delta \mathit {sales}_{t})$$ 0.214 0.548 0.206 0.238 0.214 0.203 0.191 0.152 0.221 0.211 $$\mathit {Std}(\Delta \mathit {debt}_{t})$$ 0.566 0.641 0.583 1.400 0.847 0.543 0.570 0.551 0.437 0.567 $$\mathit {Autocor}(\Delta \mathit {debt}_{t})$$ − 0.136 − 0.086 − 0.133 − 0.093 − 0.137 − 0.135 − 0.137 − 0.138 − 0.123 − 0.135 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {sales}_{t})$$ 0.853 0.951 0.853 0.846 0.847 0.851 0.834 0.820 0.860 0.853 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {debt}_{t})$$ 0.391 0.454 0.403 0.207 0.313 0.398 0.391 0.377 0.459 0.390 $$\mathit {Cor}(\Delta \mathit {sales}_{t},\Delta \mathit {debt}_{t})$$ 0.360 0.442 0.405 0.145 0.257 0.377 0.342 0.348 0.454 0.369 ρz σz ρξ σξ κ η ϕ $$\bar{\xi }$$ $$\bar{u}$$ $$\mathit {Mean(wageshare}_{t})$$ 0.692 0.703 0.695 0.694 0.694 0.695 0.705 0.693 0.688 0.712 $$\mathit {Mean}(\mathit {unusedcredit}_{t})$$ 0.405 0.422 0.412 0.456 0.472 0.467 0.391 0.401 0.366 0.398 $$\mathit {Mean}(\mathit {leverage}_{t})$$ 0.216 0.216 0.215 0.196 0.188 0.193 0.231 0.218 0.250 0.220 $$\mathit {Std}(\mathit {leverage}_{t})$$ 0.096 0.201 0.096 0.134 0.107 0.087 0.155 0.094 0.087 0.102 $$\mathit {Autocor}(\mathit {leverage}_{t})$$ 0.701 0.154 0.693 0.794 0.724 0.721 0.304 0.719 0.725 0.621 $$\mathit {Mean(Tobin}Q_{t})$$ 1.374 1.400 1.375 1.371 1.365 1.366 1.315 1.383 1.389 1.303 $$\mathit {Std(Tobin}Q_{t})$$ 0.256 0.338 0.277 0.265 0.257 0.246 0.250 0.263 0.259 0.259 $$\mathit {Autocor(Tobin}Q_{t})$$ 0.411 0.538 0.430 0.381 0.382 0.430 0.410 0.426 0.421 0.430 $$\mathit {Std}(\Delta \mathit {employ}_{t})$$ 0.091 0.150 0.100 0.095 0.091 0.089 0.088 0.083 0.093 0.091 $$\mathit {Autocor}(\Delta \mathit {employ}_{t})$$ 0.670 0.753 0.670 0.682 0.669 0.670 0.669 0.670 0.670 0.670 $$\mathit {Std}(\Delta \mathit {sales}_{t})$$ 0.135 0.170 0.149 0.137 0.134 0.133 0.134 0.133 0.135 0.135 $$\mathit {Autocor}(\Delta \mathit {sales}_{t})$$ 0.214 0.548 0.206 0.238 0.214 0.203 0.191 0.152 0.221 0.211 $$\mathit {Std}(\Delta \mathit {debt}_{t})$$ 0.566 0.641 0.583 1.400 0.847 0.543 0.570 0.551 0.437 0.567 $$\mathit {Autocor}(\Delta \mathit {debt}_{t})$$ − 0.136 − 0.086 − 0.133 − 0.093 − 0.137 − 0.135 − 0.137 − 0.138 − 0.123 − 0.135 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {sales}_{t})$$ 0.853 0.951 0.853 0.846 0.847 0.851 0.834 0.820 0.860 0.853 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {debt}_{t})$$ 0.391 0.454 0.403 0.207 0.313 0.398 0.391 0.377 0.459 0.390 $$\mathit {Cor}(\Delta \mathit {sales}_{t},\Delta \mathit {debt}_{t})$$ 0.360 0.442 0.405 0.145 0.257 0.377 0.342 0.348 0.454 0.369 Note: The first column reports the moments generated by the simulation of the model under the estimated parameters (benchmark parametrization as shown in Table 1). The remaining columns report the simulated moments after increasing the value of the specific parameter by 10%. The parameters are, respectively, the persistence and volatility of the productivity and credit, ρz, σz, ρξ, σξ, the financial distress cost, κ, the workers’ bargaining power, η, the hiring cost, ϕ, the average enforcement $$\bar{\xi }$$, and the employment flow $$\bar{u}$$. View Large It is important to emphasize that there is not a one-to-one mapping that allows us to relate each moment used in the estimation to a single parameter. All parameters are jointly identified. Still, the sensitivity analysis indicates the moments that are more sensitive to a particular parameter and, therefore, they are more relevant for the identification of that parameter. The parameters that characterize the stochastic properties of the productivity shock (persistence ρz and volatility σz) are mainly identified by the standard deviation and autocorrelation of sales growth. The parameters of the credit shock (persistence ρξ and volatility σξ) are mainly identified by the standard deviation and autocorrelation of debt growth, although the credit shock is also important for determining the means of leverage and unused credit. As discussed in Section 2.3, because of financial distress risks, firms do not borrow up to the limit and, therefore, they do not use the whole credit capacity. Furthermore, unused credit increases when the expected distress cost rises. This implies that the ratio of unused credit over total credit plays an important role in pinning down the distress cost parameter κ. Leverage also plays a role in the identification of κ because higher values of κ reduce the average leverage. For the identification of the worker’s bargaining power η, there are several moments that play a role. The first is the mean ratio of wages over revenues. As η increases, a larger share of the surplus goes to workers, increasing the ratio of wages over revenues. A higher share of revenues going to workers also implies a lower Tobin’s Q because this decreases the equity value of the firm. Finally, increasing the bargaining power of workers increases the leverage ratio because the incentive to borrow rises. The parameter η also affects the autocorrelation of leverage. The hiring cost function $$\Upsilon (e_t)=\phi e_t + \psi e_t^2$$ is characterized by the parameters ϕ and ψ. However, we cannot identify the two parameters separately. Thus, we pre-set the value of ψ so that, given the estimated value of ϕ, the unconditional growth rate of firms is zero. This is a normalization that guarantees the stationarity of the model. The parameter ϕ is mainly identified by the standard deviation of employment growth: A higher cost of hiring reduces the response of employment to shocks. The autocorrelation of sales also plays some role. The mean of financial leverage plays an important role in the identification of the enforcement parameter $$\bar{\xi }$$: with higher values of $$\bar{\xi }$$ the credit capacity of firms increases that induces higher borrowing. This also affects the standard deviation of debt growth, which is another moment important for the identification of $$\bar{\xi }$$. As firms borrow more, they are able to negotiate lower wages that in turn reduces the share of revenues going to workers. Therefore, the mean of wages over revenues also plays some role in the identification of $$\bar{\xi }$$. Finally, the separation utility flow $$\bar{u}$$ is identified by similar moments as the bargaining parameter η. This is because a higher $$\bar{u}$$ allows workers to extract a higher share of the surplus similarly to a higher workers’ bargaining power η. Higher shares of wages then imply lower values of equities, that is, lower Tobin’s Q. There is also some impact on leverage but this is smaller than a change in η. Although $$\bar{u}$$ and η are identified by similar moments, the magnitude of the impacts of the two parameters on the key moments differs. In particular, although $$\bar{u}$$ has a larger impact on the wage share, η has a bigger impact on the mean and autocorrelation of leverage and on Tobin’s Q. The impact magnitudes on the key moments make possible the separate identification of $$\bar{u}$$ and η. 3.4. Estimation Results The estimation assigns a sizable bargaining power to workers, with η = 0.355. This is important for the bargaining channel to be relevant. The estimated persistence of productivity is ρz = 0.763, and the standard deviation is σz = 0.151. These two numbers are consistent with the numbers used in the corporate finance literature (e.g., Hennessy and Whited 2005). The estimation of the average enforcement variable is $$\bar{\xi }=0.375$$. The estimated persistence of credit shock is ρξ = 0.753 and the standard deviation is σξ = 0.167. The estimation of credit shocks at the firm level is fairly new in the literature. The estimated value of the financial distress cost parameter is κ = 2.475, which implies an average cost of 1.5 cents per dollar of debt, that is, $$\mathbb{E}_t\varphi (\boldsymbol{x}_{t+1},b_{t+1})/b_{t+1}=1.5\%$$. The hiring cost parameter is ϕ = 1.131, which implies a marginal cost of hiring (marginal Q) equal to 1.29. The top section of Table 1 reports the empirical moments (column (a)) and the moments generated by the estimated model (column (b)). The model does a reasonable job in replicating the majority of the target moments used in the estimation. However, there is a sizable divergence between the empirical and simulated moments for the autocorrelation of employment growth. In the data, the autocorrelation is 0.065, whereas the model generates an autocorrelation of 0.67. The high autocorrelation follows from the particular structure of the model for which the level of debt affects the growth of employment. As a result, a persistent increase in the debt level induces—through the bargaining channel—a persistent increase in the growth rate of employment. In the data, however, employment growth is not very persistent, whereas the debt level displays some persistence. Therefore, it is difficult for the model to replicate at the same time persistent debt level and low serial correlation in employment growth. The performance of the model is also somewhat weak in replicating the standard deviation of employment growth. The standard deviation in the data is 0.111, whereas the model generates a standard deviation of 0.091. In Section 4.3, we will show that adding a shock to the rate of separation (stochastic λ) reduces the autocorrelation and increases the standard deviation of employment growth. In the bottom panel of Table 1, we report the t-Statistics for each estimator. See Nikolov and Whited (2014) for details on how to calculate the t-Statistics. A high t-value means that the target moments are (jointly) statistically sensitive to the parameter and, therefore, it is well identified. All the parameters are statistically significant at a 1% confidence interval. Before analyzing the importance of the bargaining channel of debt and the relative contribution of the two shocks, it would be helpful to point out some of the properties of the estimated model. First, the sizable estimation of the bargaining power of workers (η = 0.355) implies that debt is preferred to equity even without a tax shield, that is, qt = β. The preference for debt, however, is mitigated by a positive estimation of the distress cost, κ = 2.475. If κ = 0, borrowing up to the limit is always optimal for the firm. With a positive κ, instead, the firm trades off the benefit of borrowing (lowering future wages) with its cost (higher expected distress cost). Higher debt also means lower difference between the borrowing limit and actual borrowing, which we relate to the empirical measure of unused lines of credit. It is in this respect that unused lines of credit play an important role for the identification of κ. How do productivity and credit shocks affect the cost-benefit trade-off? Firms hire more workers in response to a positive productivity shock (higher zt). Because the benefit of debt in terms of lower future wages increases with the number of newly hired workers, when productivity is high firms borrow more even if this increases the expected distress cost. By borrowing more the firm reduces the wages of newly hired workers, which further increases the incentive to hire. Thus, the financial choice amplifies the impact of productivity shocks on employment. A relaxation of the borrowing limit (higher ξt) also leads to more leverage. In this case because it reduces the cost of debt, the persistence of ξt implies that a higher borrowing limit today translates in a higher expected limit in the future. Therefore, it is less likely that the firm will end up in financial distress. Higher debt then encourages more hiring because it allows the firm to negotiate lower wages with newly hired workers. The informal discussion of the responses of the model to productivity and credit shocks provides some intuitions on how the bargaining channel of debt affects the hiring decision of firms. The next section will make this more precise by conducting a quantitative analysis. 3.5. The Importance of the Bargaining Channel of Debt A central question of this paper is whether the bargaining channel of debt is quantitatively important for explaining employment fluctuations at the firm level. To address this question, we need to compare the benchmark model with an alternative model in which the channel is muted. To do so let us first remember how the channel affects employment. The bargaining channel of debt works through the impact of debt on the compensation of workers. More specifically, higher debt allows for the negotiation of lower wages that in turn increases the incentive to create jobs. Therefore, to isolate this channel, we need to compare the benchmark model with an alternative model where wages do not depend on debt. Because the compensation of workers are determined through Nash bargaining, this requires that the choice of debt does not affect the surplus upon which wages are negotiated. Let us start with the bargaining problem for the benchmark model. This solves \begin{eqnarray} \max _{\scriptstyle{w_t,D_t,E_t,B_{t+1}}} && [ (W_t(B_t,N_t)-U_t ) N_t ]^{\eta }\cdot V_t(B_t,N_t)^{1-\eta }. \end{eqnarray} (20) The term Wt(Bt, Nt) − Ut is the net value for a worker and Vt(Bt, Nt) the net value for the firm. We have seen that the equity value Vt(Bt, Nt) is decreasing in Bt. This happens for two reasons. First, higher debt implies higher payments of interests to bondholders. Second, higher debt implies higher expected cost of financial distress. Both reduce the equity value of the firm and, with it, the bargaining surplus. Because workers receive a share of the surplus, the reduction in the bargaining surplus results in lower wages. Therefore, to eliminate the impact of debt on wages, we need to eliminate the impact of debt on the bargaining surplus. Suppose that the value of the firm that enters the above bargaining problem includes the debt Bt and it is net of the financial distress cost. Thus, we modify the definition of the bargaining value of the firm to V(xt, Bt, Nt) + Bt + φ(xt, Bt/Nt)Nt. The economic interpretation is that equity and bond holders form a single contractual party in the negotiation with workers. Furthermore, the payment of the distress cost must be paid by shareholders before negotiating with workers. It turns out that with this new definition of the bargaining value of the firm, wages are no longer dependent on Bt. Although the new definition of the bargaining surplus implies that wages are independent of debt, it also implies that the average surplus is higher compared to the benchmark model (provided that the average value of Bt is positive). This in turn implies that wages are on average higher. To eliminate this level effect on wages, we normalize the new definition of the bargaining value of the firm by subtracting the average values of debt and financial distress cost. Denoting with a bar sign the average value of a variable in the benchmark model, the modified bargaining surplus of the firm is \begin{equation} \tilde{V}(\boldsymbol{x}_t,B_t,N_t) \equiv V(\boldsymbol{x}_t,B_t,N_t)+B_t+\varphi (\boldsymbol{x}_t,b_t)N_t - [\bar{b}+\varphi (\bar{\boldsymbol{x}},\bar{b}) ]N_t. \end{equation} (21) The modified bargaining problem then becomes \begin{eqnarray} \max _{w_t,D_t,E_t,B_{t+1}} && [ (W_t(B_t,N_t)-U_t ) N_t ]^{\eta }\times \tilde{V}(\boldsymbol{x}_t,B_t,N_t)^{1-\eta }. \end{eqnarray} (22) Because$$\tilde{V}(\boldsymbol{x}_t,B_t,N_t)$$ is invariant to Bt, changes in the initial debt do not affect wages. Appendix A shows that the first-order conditions for the modified problem take the form \begin{eqnarray} &&q_t b_{t+1} - \beta [b_{t+1}+\mathbb{E}_t \varphi (\boldsymbol{x}_{t+1},b_{t+1}) ]\nonumber \\ &&\quad+\,\beta (\bar{b}+\bar{\varphi })+\beta (1-\eta)\mathbb{E}_t \tilde{s}(\boldsymbol{x}_{t+1},b_{t+1}) = \Upsilon ^{\prime }(h_t), \end{eqnarray} (23) \begin{eqnarray} (1-\mu _t) q_t &{}- \beta (1+\mu _t\xi _t) \mathbb{E}_t\left[1+\frac{\partial \varphi (\boldsymbol{x}_{t+1},b_{t+1})}{\partial b_{t+1}}\right] = 0. \end{eqnarray} (24) We can now compare these conditions to the first-order conditions derived from the benchmark model, equations (17) and (18). For convenience, we rewrite them here as \begin{eqnarray} q_t b_{t+1}-\beta (1-\eta ) [b_{t+1}+\mathbb{E}_t \varphi (\boldsymbol{x}_{t+1},b_{t+1}) ] + \beta (1-\eta )\mathbb{E}_t \bar{s}(\boldsymbol{x}_{t+1}) = \Upsilon ^{\prime }(h_t),\nonumber\\ \end{eqnarray} (25) \begin{eqnarray} (1-\mu _t)q_t &{}- \beta (1+\mu _t\xi _t-\eta h_t/g_{t+1})\mathbb{E}_t\left[1+\frac{\partial \varphi (\boldsymbol{x}_{t+1},b_{t+1})}{\partial b_{t+1}}\right] = 0. \end{eqnarray} (26) By comparing (23)–(25), we can show that in the modified model hiring ht is less sensitive to debt bt+1 than in the benchmark model. To see this, let us first notice that $$\mathbb{E}_t \tilde{s}(\boldsymbol{x}_{t+1},b_{t+1})$$ is independent of bt+1 (see Appendix A). Therefore, only the first two terms of equation (23) depend on bt+1. Also, in equation (25) only the first two terms depend on bt+1. Because the second term of equation (25), which has a negative sign, is multiplied by 1 − η < 1, the sensitivity of employment (evaluated around the same values of all other variables) is higher in the benchmark model. Therefore, a change in debt should generate a larger movement in hiring. Comparing (24)–(26), we can see that the left-hand side of equation (26) has an extra term that is positive. Therefore, the right-hand side must be bigger than in the modified model. This implies a higher value of bt+1 and/or a larger multiplier μt if the enforcement constraint binds. This is because firms have more incentive to borrow when debt affects wages. The last column of Table 1 reports the moments generated by the modified model. This is a counterfactual exercise in which the bargaining channel of debt has been muted.7 The modified model generates a standard deviation of employment growth of 0.077, compared to 0.091 in the benchmark model. Therefore, the contribution of the bargaining channel of debt to employment volatility is about 13%, that is, (0.091 − 0.077)/0.111 ≈ 0.13. The exercise shows that the bargaining channel of debt contributes to firm-level employment volatility but it is not the major contributor. Why is the contribution of the bargaining channel modest? The answer is that, with parameter values for which the channel plays a more important role, the empirical fit of the model would be weaker. But which of the moments are especially difficult to match? As we have discussed earlier, the way the bargaining channel of debt works in the model is that the “level” of debt affects the “growth rate” of employment. In the data, the debt “level” is very persistent but employment “growth” is not very persistent. If the bargaining channel were more important, the model would generate an even higher persistence in employment growth, which is inconsistent with the data. This is an important reason why the estimation picks parameter values for which the bargaining channel of debt is not the major force for firm-level employment fluctuations. 3.6. The Importance of Productivity and Credit Shocks In this section, we explore the importance of each shock for employment fluctuations. Because our model is highly nonlinear, the typical approach to variance decomposition for linear models cannot be used. Therefore, in order to show the importance of the various shocks, we compute the standard deviation of employment growth conditional on a particular realization of the other shock. In doing so, we take advantage of the particular structure of the model. We have shown earlier that the optimal hiring policy of the firm depends only on the realization of the shocks, that is, ht = fh(xt), where xt = (zt, ξt). Because the growth rate of employment is equal to gt+1 = 1 − λ + ht, employment growth is also only a function of the shocks. Specifically, \begin{equation*} g_{t+1}=1-\lambda + f^h(\boldsymbol{x}_t) \equiv f^g(\boldsymbol{x}_t). \end{equation*} If the optimal policy fh(xt) were linear, it would be trivial to compute the variance decomposition because the volatility generated by one shock would not depend on the value of the other shock. In our model, however, the variance of employment generated by one shock depends on the value of the other shock. Therefore, we compute the standard deviation of gt+1 for each realization of the other shock. Formally, if the focus is on productivity shocks, we compute \begin{equation*} \text{STD}\left ( g_{t+1} \left | \xi _{t} \right . \right )=\sqrt{\mathbb{E}_{z_t} \Big [f^g(z_t,\xi _t)-\mathbb{E}_{z_t} f^g(z_t,\xi _t)\Big ]^2}, \end{equation*} for each realization of the credit shock ξt. Similarly, if the focus is on credit shocks, we compute \begin{equation*} \text{STD}\left ( g_{t+1} \left | z _{t} \right . \right )=\sqrt{\mathbb{E}_{z_t} \Big [f^g(z_t,\xi _t)-\mathbb{E}_{z_t} f^g(z_t,\xi _t)\Big ]^2}, \end{equation*} for each realization of the productivity shock zt. Becausethe shocks take a finite number of values, the function f g(zt, ξt) also takes a finite number of values, each realized with a known probability. We can then compute the above standard deviations without simulation once we have solved for the decision rules. The conditional standard deviations are plotted in Figure 2. The first panel shows that the volatility of employment generated by credit shocks increases with the realized value of productivity. This is because, when productivity is high and persistent, firms adds more workers to the existing labor force. Because the bargaining channel of debt operates on newly hired workers, having the ability to issue more debt amplifies the incentive to hire. Averaging over the possible realizations of the productivity shock, the standard deviation of employment growth generated by credit shocks is 0.025. Figure 2. View largeDownload slide Conditional volatility of employment growth. The left panel plots the volatility of employment growth generated by credit shocks, for a given zt. The right panel plots the volatility of employment growth generated by productivity shocks, for a given ξt. Figure 2. View largeDownload slide Conditional volatility of employment growth. The left panel plots the volatility of employment growth generated by credit shocks, for a given zt. The right panel plots the volatility of employment growth generated by productivity shocks, for a given ξt. The volatility of employment growth induced by productivity shocks conditional on the credit shock is plotted in the second panel of Figure 2. As can be seen, the volatility of employment growth increases with the realization of the credit shock. This is because a higher credit capacity allows the firm to contain the increase in wages induced by higher productivity and, therefore, the response to a productivity shock leads to a bigger increase in hiring. Averaging over the possible realizations of the credit shock, the standard deviation of employment growth generated by productivity shocks is 0.087. Using the average standard deviations, credit shocks contribute about 22% (0.025/(0.025 + 0.087) ≈ 0.22) to firm-level employment volatility. Furthermore, credit shocks are especially important when firms are very productive. This is another way of showing that the bargaining channel of debt acts as an amplification mechanism for the dynamics of employment in response to productivity shocks. 4. Sensitivity Analysis In this section, we explore the sensitivity of the estimation results to (a) the calibration of the separation rate; (b) the calibration of the corporate tax rate; (c) a shock to the separation rate; (d) the addition of physical capital; and (e) the moments of unused lines of credit. 4.1. Separation Rate The separation rate has been calibrated to λ = 0.25. We now re-estimate the model with λ = 0.2 and λ = 0.3. Table 3 reports the simulated moments and the estimated parameters. Table 3. Sensitivity to the calibration of the separation rate. λ = 0.2 λ = 0.3 Target moments Estimated model Counterfactual w/o bargaining Estimated model Counterfactual w/o bargaining $$\mathit {Mean(wageshare}_{t})$$ 0.715 0.716 0.687 0.688 $$\mathit {Mean}(\mathit {unusedcredit}_{t})$$ 0.399 0.983 0.408 0.942 $$\mathit {Mean}(\mathit {leverage}_{t})$$ 0.215 0.007 0.216 0.027 $$\mathit {Std}(\mathit {leverage}_{t})$$ 0.093 0.005 0.098 0.026 $$\mathit {Autocor}(\mathit {leverage}_{t})$$ 0.667 0.541 0.704 0.666 $$\mathit {Mean(Tobin}Q_{t})$$ 1.438 1.486 1.365 1.392 $$\mathit {Std(Tobin}Q_{t})$$ 0.174 0.089 0.334 0.255 $$\mathit {Autocor(Tobin}Q_{t})$$ 0.315 0.671 0.539 0.739 $$\mathit {Std}(\Delta \mathit {employ}_{t})$$ 0.101 0.072 0.092 0.083 $$\mathit {Autocor}(\Delta \mathit {employ}_{t})$$ 0.629 0.667 0.719 0.721 $$\mathit {Std}(\Delta \mathit {sales}_{t})$$ 0.136 0.122 0.137 0.135 $$\mathit {Autocor}(\Delta \mathit {sales}_{t})$$ 0.258 0.111 0.233 0.175 $$\mathit {Std}(\Delta \mathit {debt}_{t})$$ 0.589 0.616 0.556 0.933 $$\mathit {Autocor}(\Delta \mathit {debt}_{t})$$ − 0.166 − 0.158 − 0.110 − 0.107 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {sales}_{t})$$ 0.796 0.810 0.847 0.823 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {debt}_{t})$$ 0.437 − 0.015 0.356 − 0.017 $$\mathit {Cor}(\Delta \mathit {sales}_{t},\Delta \mathit {debt}_{t})$$ 0.270 − 0.165 0.374 − 0.220 λ = 0.2 λ = 0.3 Target moments Estimated model Counterfactual w/o bargaining Estimated model Counterfactual w/o bargaining $$\mathit {Mean(wageshare}_{t})$$ 0.715 0.716 0.687 0.688 $$\mathit {Mean}(\mathit {unusedcredit}_{t})$$ 0.399 0.983 0.408 0.942 $$\mathit {Mean}(\mathit {leverage}_{t})$$ 0.215 0.007 0.216 0.027 $$\mathit {Std}(\mathit {leverage}_{t})$$ 0.093 0.005 0.098 0.026 $$\mathit {Autocor}(\mathit {leverage}_{t})$$ 0.667 0.541 0.704 0.666 $$\mathit {Mean(Tobin}Q_{t})$$ 1.438 1.486 1.365 1.392 $$\mathit {Std(Tobin}Q_{t})$$ 0.174 0.089 0.334 0.255 $$\mathit {Autocor(Tobin}Q_{t})$$ 0.315 0.671 0.539 0.739 $$\mathit {Std}(\Delta \mathit {employ}_{t})$$ 0.101 0.072 0.092 0.083 $$\mathit {Autocor}(\Delta \mathit {employ}_{t})$$ 0.629 0.667 0.719 0.721 $$\mathit {Std}(\Delta \mathit {sales}_{t})$$ 0.136 0.122 0.137 0.135 $$\mathit {Autocor}(\Delta \mathit {sales}_{t})$$ 0.258 0.111 0.233 0.175 $$\mathit {Std}(\Delta \mathit {debt}_{t})$$ 0.589 0.616 0.556 0.933 $$\mathit {Autocor}(\Delta \mathit {debt}_{t})$$ − 0.166 − 0.158 − 0.110 − 0.107 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {sales}_{t})$$ 0.796 0.810 0.847 0.823 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {debt}_{t})$$ 0.437 − 0.015 0.356 − 0.017 $$\mathit {Cor}(\Delta \mathit {sales}_{t},\Delta \mathit {debt}_{t})$$ 0.270 − 0.165 0.374 − 0.220 λ = 0.2 λ = 0.3 Estimated parameters Estimators t-Statistics Estimators t-Statistics Persistence productivity shock, ρz 0.761 (114.1) 0.807 (100.4) Volatility productivity shock, σz 0.147 (73.07) 0.147 (72.22) Persistence debt shock, ρξ 0.718 (78.02) 0.783 (75.67) Volatility debt shock, σξ 0.120 (14.16) 0.171 (32.36) Financial distress cost, κ 2.644 (19.11) 2.818 (21.83) Workers’ bargaining power, η 0.529 (36.16) 0.287 (14.80) Hiring cost, ϕ 0.766 (24.51) 1.501 (34.64) Enforcement parameter, $$\bar{\xi }$$ 0.274 (21.11) 0.400 (36.44) Separation flow, $$\bar{u}$$ 0.456 (39.14) 0.542 (42.24) λ = 0.2 λ = 0.3 Estimated parameters Estimators t-Statistics Estimators t-Statistics Persistence productivity shock, ρz 0.761 (114.1) 0.807 (100.4) Volatility productivity shock, σz 0.147 (73.07) 0.147 (72.22) Persistence debt shock, ρξ 0.718 (78.02) 0.783 (75.67) Volatility debt shock, σξ 0.120 (14.16) 0.171 (32.36) Financial distress cost, κ 2.644 (19.11) 2.818 (21.83) Workers’ bargaining power, η 0.529 (36.16) 0.287 (14.80) Hiring cost, ϕ 0.766 (24.51) 1.501 (34.64) Enforcement parameter, $$\bar{\xi }$$ 0.274 (21.11) 0.400 (36.44) Separation flow, $$\bar{u}$$ 0.456 (39.14) 0.542 (42.24) Note: This table shows the results of the structural estimation when the separation rate λ is calibrated to 0.2 and when it is calibrated to 0.3. The first panel lists the target moments from the estimated model and from the counterfactual simulation without the bargaining channel of debt. The second panel reports the estimated parameters. View Large Table 3. Sensitivity to the calibration of the separation rate. λ = 0.2 λ = 0.3 Target moments Estimated model Counterfactual w/o bargaining Estimated model Counterfactual w/o bargaining $$\mathit {Mean(wageshare}_{t})$$ 0.715 0.716 0.687 0.688 $$\mathit {Mean}(\mathit {unusedcredit}_{t})$$ 0.399 0.983 0.408 0.942 $$\mathit {Mean}(\mathit {leverage}_{t})$$ 0.215 0.007 0.216 0.027 $$\mathit {Std}(\mathit {leverage}_{t})$$ 0.093 0.005 0.098 0.026 $$\mathit {Autocor}(\mathit {leverage}_{t})$$ 0.667 0.541 0.704 0.666 $$\mathit {Mean(Tobin}Q_{t})$$ 1.438 1.486 1.365 1.392 $$\mathit {Std(Tobin}Q_{t})$$ 0.174 0.089 0.334 0.255 $$\mathit {Autocor(Tobin}Q_{t})$$ 0.315 0.671 0.539 0.739 $$\mathit {Std}(\Delta \mathit {employ}_{t})$$ 0.101 0.072 0.092 0.083 $$\mathit {Autocor}(\Delta \mathit {employ}_{t})$$ 0.629 0.667 0.719 0.721 $$\mathit {Std}(\Delta \mathit {sales}_{t})$$ 0.136 0.122 0.137 0.135 $$\mathit {Autocor}(\Delta \mathit {sales}_{t})$$ 0.258 0.111 0.233 0.175 $$\mathit {Std}(\Delta \mathit {debt}_{t})$$ 0.589 0.616 0.556 0.933 $$\mathit {Autocor}(\Delta \mathit {debt}_{t})$$ − 0.166 − 0.158 − 0.110 − 0.107 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {sales}_{t})$$ 0.796 0.810 0.847 0.823 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {debt}_{t})$$ 0.437 − 0.015 0.356 − 0.017 $$\mathit {Cor}(\Delta \mathit {sales}_{t},\Delta \mathit {debt}_{t})$$ 0.270 − 0.165 0.374 − 0.220 λ = 0.2 λ = 0.3 Target moments Estimated model Counterfactual w/o bargaining Estimated model Counterfactual w/o bargaining $$\mathit {Mean(wageshare}_{t})$$ 0.715 0.716 0.687 0.688 $$\mathit {Mean}(\mathit {unusedcredit}_{t})$$ 0.399 0.983 0.408 0.942 $$\mathit {Mean}(\mathit {leverage}_{t})$$ 0.215 0.007 0.216 0.027 $$\mathit {Std}(\mathit {leverage}_{t})$$ 0.093 0.005 0.098 0.026 $$\mathit {Autocor}(\mathit {leverage}_{t})$$ 0.667 0.541 0.704 0.666 $$\mathit {Mean(Tobin}Q_{t})$$ 1.438 1.486 1.365 1.392 $$\mathit {Std(Tobin}Q_{t})$$ 0.174 0.089 0.334 0.255 $$\mathit {Autocor(Tobin}Q_{t})$$ 0.315 0.671 0.539 0.739 $$\mathit {Std}(\Delta \mathit {employ}_{t})$$ 0.101 0.072 0.092 0.083 $$\mathit {Autocor}(\Delta \mathit {employ}_{t})$$ 0.629 0.667 0.719 0.721 $$\mathit {Std}(\Delta \mathit {sales}_{t})$$ 0.136 0.122 0.137 0.135 $$\mathit {Autocor}(\Delta \mathit {sales}_{t})$$ 0.258 0.111 0.233 0.175 $$\mathit {Std}(\Delta \mathit {debt}_{t})$$ 0.589 0.616 0.556 0.933 $$\mathit {Autocor}(\Delta \mathit {debt}_{t})$$ − 0.166 − 0.158 − 0.110 − 0.107 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {sales}_{t})$$ 0.796 0.810 0.847 0.823 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {debt}_{t})$$ 0.437 − 0.015 0.356 − 0.017 $$\mathit {Cor}(\Delta \mathit {sales}_{t},\Delta \mathit {debt}_{t})$$ 0.270 − 0.165 0.374 − 0.220 λ = 0.2 λ = 0.3 Estimated parameters Estimators t-Statistics Estimators t-Statistics Persistence productivity shock, ρz 0.761 (114.1) 0.807 (100.4) Volatility productivity shock, σz 0.147 (73.07) 0.147 (72.22) Persistence debt shock, ρξ 0.718 (78.02) 0.783 (75.67) Volatility debt shock, σξ 0.120 (14.16) 0.171 (32.36) Financial distress cost, κ 2.644 (19.11) 2.818 (21.83) Workers’ bargaining power, η 0.529 (36.16) 0.287 (14.80) Hiring cost, ϕ 0.766 (24.51) 1.501 (34.64) Enforcement parameter, $$\bar{\xi }$$ 0.274 (21.11) 0.400 (36.44) Separation flow, $$\bar{u}$$ 0.456 (39.14) 0.542 (42.24) λ = 0.2 λ = 0.3 Estimated parameters Estimators t-Statistics Estimators t-Statistics Persistence productivity shock, ρz 0.761 (114.1) 0.807 (100.4) Volatility productivity shock, σz 0.147 (73.07) 0.147 (72.22) Persistence debt shock, ρξ 0.718 (78.02) 0.783 (75.67) Volatility debt shock, σξ 0.120 (14.16) 0.171 (32.36) Financial distress cost, κ 2.644 (19.11) 2.818 (21.83) Workers’ bargaining power, η 0.529 (36.16) 0.287 (14.80) Hiring cost, ϕ 0.766 (24.51) 1.501 (34.64) Enforcement parameter, $$\bar{\xi }$$ 0.274 (21.11) 0.400 (36.44) Separation flow, $$\bar{u}$$ 0.456 (39.14) 0.542 (42.24) Note: This table shows the results of the structural estimation when the separation rate λ is calibrated to 0.2 and when it is calibrated to 0.3. The first panel lists the target moments from the estimated model and from the counterfactual simulation without the bargaining channel of debt. The second panel reports the estimated parameters. View Large With a lower λ, the estimated value of the bargaining share of workers is higher, η = 0.529. When workers have higher bargaining power, the importance of the bargaining channel of debt increases. As a result, the contribution of the bargaining channel of debt to employment fluctuations increases to 26% ((1.01−0.72)/1.11 ≈ 0.26). Instead, with λ = 0.3, the estimation of the bargaining power of workers is η = 0.287 and the importance of the bargaining channel of debt declines to 8% ((0.92−0.83)/1.11 ≈ 0.08). 4.2. Corporate Tax Rate The corporate tax rate creates a wedge between the effective cost of debt and the intertemporal discount rate of firms. Higher values of τ increase the incentive to borrow because of the tax deductibility of interests. We consider τ = 0 and τ = 0.35 (in the benchmark model τ = 0.2). All other calibrated parameters are set to the same values as in the benchmark model. The estimation results are reported in Table 4. Table 4. Sensitivity to the calibration of the corporate tax rate. τ = 0.0 τ = 0.35 Target moments Estimated model Counterfactual w/o bargaining Estimated model Counterfactual w/o bargaining $$\mathit {Mean(wageshare}_{t})$$ 0.693 0.695 0.692 0.693 $$\mathit {Mean}(\mathit {unusedcredit}_{t})$$ 0.405 1.000 0.405 0.951 $$\mathit {Mean}(\mathit {leverage}_{t})$$ 0.217 0.000 0.216 0.022 $$\mathit {Std}(\mathit {leverage}_{t})$$ 0.097 0.000 0.097 0.017 $$\mathit {Autocor}(\mathit {leverage}_{t})$$ 0.746 na 0.689 0.632 $$\mathit {Mean(Tobin}Q_{t})$$ 1.376 1.411 1.375 1.406 $$\mathit {Std(Tobin}Q_{t})$$ 0.261 0.177 0.254 0.172 $$\mathit {Autocor(Tobin}Q_{t})$$ 0.466 0.705 0.429 0.709 $$\mathit {Std}(\Delta \mathit {employ}_{t})$$ 0.091 0.075 0.093 0.077 $$\mathit {Autocor}(\Delta \mathit {employ}_{t})$$ 0.702 0.707 0.685 0.691 $$\mathit {Std}(\Delta \mathit {sales}_{t})$$ 0.135 0.131 0.136 0.131 $$\mathit {Autocor}(\Delta \mathit {sales}_{t})$$ 0.231 0.125 0.229 0.125 $$\mathit {Std}(\Delta \mathit {debt}_{t})$$ 0.568 0.000 0.568 0.715 $$\mathit {Autocor}(\Delta \mathit {debt}_{t})$$ − 0.123 na − 0.135 − 0.131 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {sales}_{t})$$ 0.839 0.802 0.847 0.815 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {debt}_{t})$$ 0.385 − 0.150 0.393 − 0.011 $$\mathit {Cor}(\Delta \mathit {sales}_{t},\Delta \mathit {debt}_{t})$$ 0.355 − 0.014 0.347 − 0.201 τ = 0.0 τ = 0.35 Target moments Estimated model Counterfactual w/o bargaining Estimated model Counterfactual w/o bargaining $$\mathit {Mean(wageshare}_{t})$$ 0.693 0.695 0.692 0.693 $$\mathit {Mean}(\mathit {unusedcredit}_{t})$$ 0.405 1.000 0.405 0.951 $$\mathit {Mean}(\mathit {leverage}_{t})$$ 0.217 0.000 0.216 0.022 $$\mathit {Std}(\mathit {leverage}_{t})$$ 0.097 0.000 0.097 0.017 $$\mathit {Autocor}(\mathit {leverage}_{t})$$ 0.746 na 0.689 0.632 $$\mathit {Mean(Tobin}Q_{t})$$ 1.376 1.411 1.375 1.406 $$\mathit {Std(Tobin}Q_{t})$$ 0.261 0.177 0.254 0.172 $$\mathit {Autocor(Tobin}Q_{t})$$ 0.466 0.705 0.429 0.709 $$\mathit {Std}(\Delta \mathit {employ}_{t})$$ 0.091 0.075 0.093 0.077 $$\mathit {Autocor}(\Delta \mathit {employ}_{t})$$ 0.702 0.707 0.685 0.691 $$\mathit {Std}(\Delta \mathit {sales}_{t})$$ 0.135 0.131 0.136 0.131 $$\mathit {Autocor}(\Delta \mathit {sales}_{t})$$ 0.231 0.125 0.229 0.125 $$\mathit {Std}(\Delta \mathit {debt}_{t})$$ 0.568 0.000 0.568 0.715 $$\mathit {Autocor}(\Delta \mathit {debt}_{t})$$ − 0.123 na − 0.135 − 0.131 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {sales}_{t})$$ 0.839 0.802 0.847 0.815 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {debt}_{t})$$ 0.385 − 0.150 0.393 − 0.011 $$\mathit {Cor}(\Delta \mathit {sales}_{t},\Delta \mathit {debt}_{t})$$ 0.355 − 0.014 0.347 − 0.201 τ = 0.0 τ = 0.35 Estimated parameters Estimators t-Statistics Estimators t-Statistics Persistence productivity shock, ρz 0.795 (74.01) 0.779 (79.41) Volatility productivity shock, σz 0.147 (62.87) 0.150 (66.26) Persistence debt shock, ρξ 0.773 (60.61) 0.747 (92.42) Volatility debt shock, σξ 0.147 (21.06) 0.159 (23.08) Financial distress cost, κ 2.827 (11.64) 2.841 (14.72) Workers’ bargaining power, η 0.408 (15.07) 0.393 (14.55) Hiring cost, ϕ 1.232 (28.34) 1.152 (31.44) Enforcement parameter, $$\bar{\xi }$$ 0.344 (23.15) 0.356 (23.95) Separation flow, $$\bar{u}$$ 0.496 (25.58) 0.507 (27.98) τ = 0.0 τ = 0.35 Estimated parameters Estimators t-Statistics Estimators t-Statistics Persistence productivity shock, ρz 0.795 (74.01) 0.779 (79.41) Volatility productivity shock, σz 0.147 (62.87) 0.150 (66.26) Persistence debt shock, ρξ 0.773 (60.61) 0.747 (92.42) Volatility debt shock, σξ 0.147 (21.06) 0.159 (23.08) Financial distress cost, κ 2.827 (11.64) 2.841 (14.72) Workers’ bargaining power, η 0.408 (15.07) 0.393 (14.55) Hiring cost, ϕ 1.232 (28.34) 1.152 (31.44) Enforcement parameter, $$\bar{\xi }$$ 0.344 (23.15) 0.356 (23.95) Separation flow, $$\bar{u}$$ 0.496 (25.58) 0.507 (27.98) Note: This table shows the results of the structural estimation when the corporate tax rate τ is calibrated to 0 and when it is calibrated to 0.35. The first panel lists the target moments from the estimated model and from the counterfactual simulation without the bargaining channel of debt. The second panel reports the estimated parameters. View Large Table 4. Sensitivity to the calibration of the corporate tax rate. τ = 0.0 τ = 0.35 Target moments Estimated model Counterfactual w/o bargaining Estimated model Counterfactual w/o bargaining $$\mathit {Mean(wageshare}_{t})$$ 0.693 0.695 0.692 0.693 $$\mathit {Mean}(\mathit {unusedcredit}_{t})$$ 0.405 1.000 0.405 0.951 $$\mathit {Mean}(\mathit {leverage}_{t})$$ 0.217 0.000 0.216 0.022 $$\mathit {Std}(\mathit {leverage}_{t})$$ 0.097 0.000 0.097 0.017 $$\mathit {Autocor}(\mathit {leverage}_{t})$$ 0.746 na 0.689 0.632 $$\mathit {Mean(Tobin}Q_{t})$$ 1.376 1.411 1.375 1.406 $$\mathit {Std(Tobin}Q_{t})$$ 0.261 0.177 0.254 0.172 $$\mathit {Autocor(Tobin}Q_{t})$$ 0.466 0.705 0.429 0.709 $$\mathit {Std}(\Delta \mathit {employ}_{t})$$ 0.091 0.075 0.093 0.077 $$\mathit {Autocor}(\Delta \mathit {employ}_{t})$$ 0.702 0.707 0.685 0.691 $$\mathit {Std}(\Delta \mathit {sales}_{t})$$ 0.135 0.131 0.136 0.131 $$\mathit {Autocor}(\Delta \mathit {sales}_{t})$$ 0.231 0.125 0.229 0.125 $$\mathit {Std}(\Delta \mathit {debt}_{t})$$ 0.568 0.000 0.568 0.715 $$\mathit {Autocor}(\Delta \mathit {debt}_{t})$$ − 0.123 na − 0.135 − 0.131 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {sales}_{t})$$ 0.839 0.802 0.847 0.815 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {debt}_{t})$$ 0.385 − 0.150 0.393 − 0.011 $$\mathit {Cor}(\Delta \mathit {sales}_{t},\Delta \mathit {debt}_{t})$$ 0.355 − 0.014 0.347 − 0.201 τ = 0.0 τ = 0.35 Target moments Estimated model Counterfactual w/o bargaining Estimated model Counterfactual w/o bargaining $$\mathit {Mean(wageshare}_{t})$$ 0.693 0.695 0.692 0.693 $$\mathit {Mean}(\mathit {unusedcredit}_{t})$$ 0.405 1.000 0.405 0.951 $$\mathit {Mean}(\mathit {leverage}_{t})$$ 0.217 0.000 0.216 0.022 $$\mathit {Std}(\mathit {leverage}_{t})$$ 0.097 0.000 0.097 0.017 $$\mathit {Autocor}(\mathit {leverage}_{t})$$ 0.746 na 0.689 0.632 $$\mathit {Mean(Tobin}Q_{t})$$ 1.376 1.411 1.375 1.406 $$\mathit {Std(Tobin}Q_{t})$$ 0.261 0.177 0.254 0.172 $$\mathit {Autocor(Tobin}Q_{t})$$ 0.466 0.705 0.429 0.709 $$\mathit {Std}(\Delta \mathit {employ}_{t})$$ 0.091 0.075 0.093 0.077 $$\mathit {Autocor}(\Delta \mathit {employ}_{t})$$ 0.702 0.707 0.685 0.691 $$\mathit {Std}(\Delta \mathit {sales}_{t})$$ 0.135 0.131 0.136 0.131 $$\mathit {Autocor}(\Delta \mathit {sales}_{t})$$ 0.231 0.125 0.229 0.125 $$\mathit {Std}(\Delta \mathit {debt}_{t})$$ 0.568 0.000 0.568 0.715 $$\mathit {Autocor}(\Delta \mathit {debt}_{t})$$ − 0.123 na − 0.135 − 0.131 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {sales}_{t})$$ 0.839 0.802 0.847 0.815 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {debt}_{t})$$ 0.385 − 0.150 0.393 − 0.011 $$\mathit {Cor}(\Delta \mathit {sales}_{t},\Delta \mathit {debt}_{t})$$ 0.355 − 0.014 0.347 − 0.201 τ = 0.0 τ = 0.35 Estimated parameters Estimators t-Statistics Estimators t-Statistics Persistence productivity shock, ρz 0.795 (74.01) 0.779 (79.41) Volatility productivity shock, σz 0.147 (62.87) 0.150 (66.26) Persistence debt shock, ρξ 0.773 (60.61) 0.747 (92.42) Volatility debt shock, σξ 0.147 (21.06) 0.159 (23.08) Financial distress cost, κ 2.827 (11.64) 2.841 (14.72) Workers’ bargaining power, η 0.408 (15.07) 0.393 (14.55) Hiring cost, ϕ 1.232 (28.34) 1.152 (31.44) Enforcement parameter, $$\bar{\xi }$$ 0.344 (23.15) 0.356 (23.95) Separation flow, $$\bar{u}$$ 0.496 (25.58) 0.507 (27.98) τ = 0.0 τ = 0.35 Estimated parameters Estimators t-Statistics Estimators t-Statistics Persistence productivity shock, ρz 0.795 (74.01) 0.779 (79.41) Volatility productivity shock, σz 0.147 (62.87) 0.150 (66.26) Persistence debt shock, ρξ 0.773 (60.61) 0.747 (92.42) Volatility debt shock, σξ 0.147 (21.06) 0.159 (23.08) Financial distress cost, κ 2.827 (11.64) 2.841 (14.72) Workers’ bargaining power, η 0.408 (15.07) 0.393 (14.55) Hiring cost, ϕ 1.232 (28.34) 1.152 (31.44) Enforcement parameter, $$\bar{\xi }$$ 0.344 (23.15) 0.356 (23.95) Separation flow, $$\bar{u}$$ 0.496 (25.58) 0.507 (27.98) Note: This table shows the results of the structural estimation when the corporate tax rate τ is calibrated to 0 and when it is calibrated to 0.35. The first panel lists the target moments from the estimated model and from the counterfactual simulation without the bargaining channel of debt. The second panel reports the estimated parameters. View Large The parameter estimates do not change in important ways and, consequently, the importance of the bargaining channel of debt changes only slightly. 4.3. Separation Shock We now allow the separation rate λ to be stochastic. This captures the idea that labor retention is likely to be uncertain at the firm level. It also captures uncertainty in hiring because in the model a shock to job creation is isomorphic to a shock that affects job separation. Employment continues to evolve according to Nt+1 = (1 − λt)Nt + Ht, but λt is stochastic and follows a first-order Markov process at the firm level (idiosyncratic). The problem takes the same form as in Problem (16) but with three firm-level shocks: productivity, zt, credit, ξt, and separation, λt. The first-order conditions are also similar and the previous theoretical analysis applies to the model with stochastic separation once we add the time subscript to λ. Table 5 reports the estimation results with the additional separation shock. The model performs slightly better than the benchmark model (Table 1). In particular, it matches more closely the standard deviation of employment growth and the autocorrelation is closer to the data. The identification of the separation shock, however, is weak. The t-value for the persistence parameter ρλ is −1.31. Therefore, even if the addition of the separation shock helps the model to better match the data, we do not use it in the benchmark model because of the weak identification. In the counterfactual exercise, the contribution of the bargaining channel of debt is now 6.3%, which is about half the contribution in the benchmark model. Table 5. Structural estimation with separation shock. (a) (b) (c) (d) Target moments Empirical data Estimated model t-Statistics Counterfactual w/o bargaining $$\mathit {Mean(wageshare}_{t})$$ 0.685 0.690 (− 1.07) 0.691 $$\mathit {Mean}(\mathit {unusedcredit}_{t})$$ 0.403 0.402 (− 0.07) 0.943 $$\mathit {Mean}(\mathit {leverage}_{t})$$ 0.217 0.217 (0.12) 0.026 $$\mathit {Std}(\mathit {leverage}_{t})$$ 0.100 0.098 (1.28) 0.021 $$\mathit {Autocor}(\mathit {leverage}_{t})$$ 0.735 0.722 (0.49) 0.640 $$\mathit {Mean(Tobin}Q_{t})$$ 1.360 1.373 (− 0.77) 1.404 $$\mathit {Std(Tobin}Q_{t})$$ 0.421 0.242 (16.54) 0.164 $$\mathit {Autocor(Tobin}Q_{t})$$ 0.668 0.386 (8.33) 0.688 $$\mathit {Std}(\Delta \mathit {employ}_{t})$$ 0.111 0.108 (2.21) 0.101 $$\mathit {Autocor}(\Delta \mathit {employ}_{t})$$ 0.065 0.342 (− 14.50) 0.282 $$\mathit {Std}(\Delta \mathit {sales}_{t})$$ 0.135 0.135 (0.37) 0.132 $$\mathit {Autocor}(\Delta \mathit {sales}_{t})$$ 0.051 0.168 (− 4.54) 0.100 $$\mathit {Std}(\Delta \mathit {debt}_{t})$$ 0.583 0.567 (1.86) 0.823 $$\mathit {Autocor}(\Delta \mathit {debt}_{t})$$ − 0.009 − 0.130 (5.95) − 0.126 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {sales}_{t})$$ 0.531 0.539 (0.49) 0.476 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {debt}_{t})$$ 0.340 0.321 (2.19) 0.090 $$\mathit {Cor}(\Delta \mathit {sales}_{t},\Delta \mathit {debt}_{t})$$ 0.178 0.288 (− 7.24) − 0.146 (a) (b) (c) (d) Target moments Empirical data Estimated model t-Statistics Counterfactual w/o bargaining $$\mathit {Mean(wageshare}_{t})$$ 0.685 0.690 (− 1.07) 0.691 $$\mathit {Mean}(\mathit {unusedcredit}_{t})$$ 0.403 0.402 (− 0.07) 0.943 $$\mathit {Mean}(\mathit {leverage}_{t})$$ 0.217 0.217 (0.12) 0.026 $$\mathit {Std}(\mathit {leverage}_{t})$$ 0.100 0.098 (1.28) 0.021 $$\mathit {Autocor}(\mathit {leverage}_{t})$$ 0.735 0.722 (0.49) 0.640 $$\mathit {Mean(Tobin}Q_{t})$$ 1.360 1.373 (− 0.77) 1.404 $$\mathit {Std(Tobin}Q_{t})$$ 0.421 0.242 (16.54) 0.164 $$\mathit {Autocor(Tobin}Q_{t})$$ 0.668 0.386 (8.33) 0.688 $$\mathit {Std}(\Delta \mathit {employ}_{t})$$ 0.111 0.108 (2.21) 0.101 $$\mathit {Autocor}(\Delta \mathit {employ}_{t})$$ 0.065 0.342 (− 14.50) 0.282 $$\mathit {Std}(\Delta \mathit {sales}_{t})$$ 0.135 0.135 (0.37) 0.132 $$\mathit {Autocor}(\Delta \mathit {sales}_{t})$$ 0.051 0.168 (− 4.54) 0.100 $$\mathit {Std}(\Delta \mathit {debt}_{t})$$ 0.583 0.567 (1.86) 0.823 $$\mathit {Autocor}(\Delta \mathit {debt}_{t})$$ − 0.009 − 0.130 (5.95) − 0.126 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {sales}_{t})$$ 0.531 0.539 (0.49) 0.476 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {debt}_{t})$$ 0.340 0.321 (2.19) 0.090 $$\mathit {Cor}(\Delta \mathit {sales}_{t},\Delta \mathit {debt}_{t})$$ 0.178 0.288 (− 7.24) − 0.146 Estimated parameters Estimators t-Statistics Persistence productivity shock, ρz 0.770 (75.54) Volatility productivity shock, σz 0.126 (43.07) Persistence debt shock, ρξ 0.759 (147.8) Volatility debt shock, σξ 0.179 (20.77) Persistence separation shock, ρλ − 0.168 (− 1.31) Volatility separation shock, σλ 0.090 (35.99) Financial distress cost, κ 2.128 (10.12) Workers’ bargaining power, η 0.309 (8.39) Hiring cost, ϕ 1.143 (20.14) Enforcement parameter, $$\bar{\xi }$$ 0.401 (21.09) Unemployment flow, $$\bar{u}$$ 0.560 (27.48) Estimated parameters Estimators t-Statistics Persistence productivity shock, ρz 0.770 (75.54) Volatility productivity shock, σz 0.126 (43.07) Persistence debt shock, ρξ 0.759 (147.8) Volatility debt shock, σξ 0.179 (20.77) Persistence separation shock, ρλ − 0.168 (− 1.31) Volatility separation shock, σλ 0.090 (35.99) Financial distress cost, κ 2.128 (10.12) Workers’ bargaining power, η 0.309 (8.39) Hiring cost, ϕ 1.143 (20.14) Enforcement parameter, $$\bar{\xi }$$ 0.401 (21.09) Unemployment flow, $$\bar{u}$$ 0.560 (27.48) Note: The table shows the results of the structural estimation after adding a shock to the separation rate. The first panel lists the target moments; column (a) reports the moments from the data; column (b) reports the moments generated by the estimated model; column (c) reports the t-Statistic for the differences between the empirical moments and the moments generated by the estimated model; column (d) reports the moments generated by the model without the bargaining channel of debt as described in Section 3.5. The second panel reports the estimated parameters and the associated t-Statistics. View Large Table 5. Structural estimation with separation shock. (a) (b) (c) (d) Target moments Empirical data Estimated model t-Statistics Counterfactual w/o bargaining $$\mathit {Mean(wageshare}_{t})$$ 0.685 0.690 (− 1.07) 0.691 $$\mathit {Mean}(\mathit {unusedcredit}_{t})$$ 0.403 0.402 (− 0.07) 0.943 $$\mathit {Mean}(\mathit {leverage}_{t})$$ 0.217 0.217 (0.12) 0.026 $$\mathit {Std}(\mathit {leverage}_{t})$$ 0.100 0.098 (1.28) 0.021 $$\mathit {Autocor}(\mathit {leverage}_{t})$$ 0.735 0.722 (0.49) 0.640 $$\mathit {Mean(Tobin}Q_{t})$$ 1.360 1.373 (− 0.77) 1.404 $$\mathit {Std(Tobin}Q_{t})$$ 0.421 0.242 (16.54) 0.164 $$\mathit {Autocor(Tobin}Q_{t})$$ 0.668 0.386 (8.33) 0.688 $$\mathit {Std}(\Delta \mathit {employ}_{t})$$ 0.111 0.108 (2.21) 0.101 $$\mathit {Autocor}(\Delta \mathit {employ}_{t})$$ 0.065 0.342 (− 14.50) 0.282 $$\mathit {Std}(\Delta \mathit {sales}_{t})$$ 0.135 0.135 (0.37) 0.132 $$\mathit {Autocor}(\Delta \mathit {sales}_{t})$$ 0.051 0.168 (− 4.54) 0.100 $$\mathit {Std}(\Delta \mathit {debt}_{t})$$ 0.583 0.567 (1.86) 0.823 $$\mathit {Autocor}(\Delta \mathit {debt}_{t})$$ − 0.009 − 0.130 (5.95) − 0.126 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {sales}_{t})$$ 0.531 0.539 (0.49) 0.476 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {debt}_{t})$$ 0.340 0.321 (2.19) 0.090 $$\mathit {Cor}(\Delta \mathit {sales}_{t},\Delta \mathit {debt}_{t})$$ 0.178 0.288 (− 7.24) − 0.146 (a) (b) (c) (d) Target moments Empirical data Estimated model t-Statistics Counterfactual w/o bargaining $$\mathit {Mean(wageshare}_{t})$$ 0.685 0.690 (− 1.07) 0.691 $$\mathit {Mean}(\mathit {unusedcredit}_{t})$$ 0.403 0.402 (− 0.07) 0.943 $$\mathit {Mean}(\mathit {leverage}_{t})$$ 0.217 0.217 (0.12) 0.026 $$\mathit {Std}(\mathit {leverage}_{t})$$ 0.100 0.098 (1.28) 0.021 $$\mathit {Autocor}(\mathit {leverage}_{t})$$ 0.735 0.722 (0.49) 0.640 $$\mathit {Mean(Tobin}Q_{t})$$ 1.360 1.373 (− 0.77) 1.404 $$\mathit {Std(Tobin}Q_{t})$$ 0.421 0.242 (16.54) 0.164 $$\mathit {Autocor(Tobin}Q_{t})$$ 0.668 0.386 (8.33) 0.688 $$\mathit {Std}(\Delta \mathit {employ}_{t})$$ 0.111 0.108 (2.21) 0.101 $$\mathit {Autocor}(\Delta \mathit {employ}_{t})$$ 0.065 0.342 (− 14.50) 0.282 $$\mathit {Std}(\Delta \mathit {sales}_{t})$$ 0.135 0.135 (0.37) 0.132 $$\mathit {Autocor}(\Delta \mathit {sales}_{t})$$ 0.051 0.168 (− 4.54) 0.100 $$\mathit {Std}(\Delta \mathit {debt}_{t})$$ 0.583 0.567 (1.86) 0.823 $$\mathit {Autocor}(\Delta \mathit {debt}_{t})$$ − 0.009 − 0.130 (5.95) − 0.126 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {sales}_{t})$$ 0.531 0.539 (0.49) 0.476 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {debt}_{t})$$ 0.340 0.321 (2.19) 0.090 $$\mathit {Cor}(\Delta \mathit {sales}_{t},\Delta \mathit {debt}_{t})$$ 0.178 0.288 (− 7.24) − 0.146 Estimated parameters Estimators t-Statistics Persistence productivity shock, ρz 0.770 (75.54) Volatility productivity shock, σz 0.126 (43.07) Persistence debt shock, ρξ 0.759 (147.8) Volatility debt shock, σξ 0.179 (20.77) Persistence separation shock, ρλ − 0.168 (− 1.31) Volatility separation shock, σλ 0.090 (35.99) Financial distress cost, κ 2.128 (10.12) Workers’ bargaining power, η 0.309 (8.39) Hiring cost, ϕ 1.143 (20.14) Enforcement parameter, $$\bar{\xi }$$ 0.401 (21.09) Unemployment flow, $$\bar{u}$$ 0.560 (27.48) Estimated parameters Estimators t-Statistics Persistence productivity shock, ρz 0.770 (75.54) Volatility productivity shock, σz 0.126 (43.07) Persistence debt shock, ρξ 0.759 (147.8) Volatility debt shock, σξ 0.179 (20.77) Persistence separation shock, ρλ − 0.168 (− 1.31) Volatility separation shock, σλ 0.090 (35.99) Financial distress cost, κ 2.128 (10.12) Workers’ bargaining power, η 0.309 (8.39) Hiring cost, ϕ 1.143 (20.14) Enforcement parameter, $$\bar{\xi }$$ 0.401 (21.09) Unemployment flow, $$\bar{u}$$ 0.560 (27.48) Note: The table shows the results of the structural estimation after adding a shock to the separation rate. The first panel lists the target moments; column (a) reports the moments from the data; column (b) reports the moments generated by the estimated model; column (c) reports the t-Statistic for the differences between the empirical moments and the moments generated by the estimated model; column (d) reports the moments generated by the model without the bargaining channel of debt as described in Section 3.5. The second panel reports the estimated parameters and the associated t-Statistics. View Large Finally, we test the equality of the target moments generated by the benchmark model (Table 1) with the target moments generated by the estimated model with the additional separation shock (Table 5). The results, reported in Table 6, show that most of the target moments generated by the benchmark model are not statistically different from the moments generated by the model with separation shocks. As observed above, separation shocks improve the ability of the model to match the standard deviation and autocorrelation of employment growth, and Table 6 shows that the improvement is statistically significant. Table 6. Test of the equality of moment conditions from separate estimations. Target moments Benchmark Extended t-Statistics $$\mathit {Mean(wageshare}_{t})$$ 0.692 0.690 (− 0.40) $$\mathit {Mean}(\mathit {unusedcredit}_{t})$$ 0.405 0.402 (− 0.10) $$\mathit {Mean}(\mathit {leverage}_{t})$$ 0.216 0.217 (0.00) $$\mathit {Std(leverage}_{t})$$ 0.096 0.098 (0.07) $$\mathit {Autocor}(\mathit {leverage}_{t})$$ 0.701 0.722 (0.08) $$\mathit {Mean(Tobin}Q_{t})$$ 1.374 1.373 (− 0.05) $$\mathit {Std(Tobin}Q_{t})$$ 0.256 0.242 (− 1.29) $$\mathit {Autocor(Tobin}Q_{t})$$ 0.411 0.386 (− 1.10) $$\mathit {Std}(\Delta \mathit {employ}_{t})$$ 0.091 0.108 (3.47) $$\mathit {Autocor}(\Delta \mathit {employ}_{t})$$ 0.670 0.341 (− 12.13) $$\mathit {Std}(\Delta \mathit {sales}_{t})$$ 0.135 0.135 (− 0.01) $$\mathit {Autocor}(\Delta \mathit {sales}_{t})$$ 0.214 0.167 (− 0.49) $$\mathit {Std}(\Delta \mathit {debt}_{t})$$ 0.566 0.567 (0.00) $$\mathit {Autocor}(\Delta \mathit {debt}_{t})$$ − 0.136 − 0.130 (2.47) $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {sales}_{t})$$ 0.853 0.538 (− 4.27) $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {debt}_{t})$$ 0.391 0.321 (− 0.22) $$\mathit {Cor}(\Delta \mathit {sales}_{t},\Delta \mathit {debt}_{t})$$ 0.360 0.288 (− 3.35) Target moments Benchmark Extended t-Statistics $$\mathit {Mean(wageshare}_{t})$$ 0.692 0.690 (− 0.40) $$\mathit {Mean}(\mathit {unusedcredit}_{t})$$ 0.405 0.402 (− 0.10) $$\mathit {Mean}(\mathit {leverage}_{t})$$ 0.216 0.217 (0.00) $$\mathit {Std(leverage}_{t})$$ 0.096 0.098 (0.07) $$\mathit {Autocor}(\mathit {leverage}_{t})$$ 0.701 0.722 (0.08) $$\mathit {Mean(Tobin}Q_{t})$$ 1.374 1.373 (− 0.05) $$\mathit {Std(Tobin}Q_{t})$$ 0.256 0.242 (− 1.29) $$\mathit {Autocor(Tobin}Q_{t})$$ 0.411 0.386 (− 1.10) $$\mathit {Std}(\Delta \mathit {employ}_{t})$$ 0.091 0.108 (3.47) $$\mathit {Autocor}(\Delta \mathit {employ}_{t})$$ 0.670 0.341 (− 12.13) $$\mathit {Std}(\Delta \mathit {sales}_{t})$$ 0.135 0.135 (− 0.01) $$\mathit {Autocor}(\Delta \mathit {sales}_{t})$$ 0.214 0.167 (− 0.49) $$\mathit {Std}(\Delta \mathit {debt}_{t})$$ 0.566 0.567 (0.00) $$\mathit {Autocor}(\Delta \mathit {debt}_{t})$$ − 0.136 − 0.130 (2.47) $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {sales}_{t})$$ 0.853 0.538 (− 4.27) $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {debt}_{t})$$ 0.391 0.321 (− 0.22) $$\mathit {Cor}(\Delta \mathit {sales}_{t},\Delta \mathit {debt}_{t})$$ 0.360 0.288 (− 3.35) Note: This table compares the target moments of the two structural estimations: the benchmark estimation (Table 1) and the estimation with an additional separation shock (Table 5). The t-Statistics for the differences between the two sets of target moments are reported in the parentheses. See Nikolov and Whited (2014) for the details of how to calculate the t-Statistics. View Large Table 6. Test of the equality of moment conditions from separate estimations. Target moments Benchmark Extended t-Statistics $$\mathit {Mean(wageshare}_{t})$$ 0.692 0.690 (− 0.40) $$\mathit {Mean}(\mathit {unusedcredit}_{t})$$ 0.405 0.402 (− 0.10) $$\mathit {Mean}(\mathit {leverage}_{t})$$ 0.216 0.217 (0.00) $$\mathit {Std(leverage}_{t})$$ 0.096 0.098 (0.07) $$\mathit {Autocor}(\mathit {leverage}_{t})$$ 0.701 0.722 (0.08) $$\mathit {Mean(Tobin}Q_{t})$$ 1.374 1.373 (− 0.05) $$\mathit {Std(Tobin}Q_{t})$$ 0.256 0.242 (− 1.29) $$\mathit {Autocor(Tobin}Q_{t})$$ 0.411 0.386 (− 1.10) $$\mathit {Std}(\Delta \mathit {employ}_{t})$$ 0.091 0.108 (3.47) $$\mathit {Autocor}(\Delta \mathit {employ}_{t})$$ 0.670 0.341 (− 12.13) $$\mathit {Std}(\Delta \mathit {sales}_{t})$$ 0.135 0.135 (− 0.01) $$\mathit {Autocor}(\Delta \mathit {sales}_{t})$$ 0.214 0.167 (− 0.49) $$\mathit {Std}(\Delta \mathit {debt}_{t})$$ 0.566 0.567 (0.00) $$\mathit {Autocor}(\Delta \mathit {debt}_{t})$$ − 0.136 − 0.130 (2.47) $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {sales}_{t})$$ 0.853 0.538 (− 4.27) $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {debt}_{t})$$ 0.391 0.321 (− 0.22) $$\mathit {Cor}(\Delta \mathit {sales}_{t},\Delta \mathit {debt}_{t})$$ 0.360 0.288 (− 3.35) Target moments Benchmark Extended t-Statistics $$\mathit {Mean(wageshare}_{t})$$ 0.692 0.690 (− 0.40) $$\mathit {Mean}(\mathit {unusedcredit}_{t})$$ 0.405 0.402 (− 0.10) $$\mathit {Mean}(\mathit {leverage}_{t})$$ 0.216 0.217 (0.00) $$\mathit {Std(leverage}_{t})$$ 0.096 0.098 (0.07) $$\mathit {Autocor}(\mathit {leverage}_{t})$$ 0.701 0.722 (0.08) $$\mathit {Mean(Tobin}Q_{t})$$ 1.374 1.373 (− 0.05) $$\mathit {Std(Tobin}Q_{t})$$ 0.256 0.242 (− 1.29) $$\mathit {Autocor(Tobin}Q_{t})$$ 0.411 0.386 (− 1.10) $$\mathit {Std}(\Delta \mathit {employ}_{t})$$ 0.091 0.108 (3.47) $$\mathit {Autocor}(\Delta \mathit {employ}_{t})$$ 0.670 0.341 (− 12.13) $$\mathit {Std}(\Delta \mathit {sales}_{t})$$ 0.135 0.135 (− 0.01) $$\mathit {Autocor}(\Delta \mathit {sales}_{t})$$ 0.214 0.167 (− 0.49) $$\mathit {Std}(\Delta \mathit {debt}_{t})$$ 0.566 0.567 (0.00) $$\mathit {Autocor}(\Delta \mathit {debt}_{t})$$ − 0.136 − 0.130 (2.47) $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {sales}_{t})$$ 0.853 0.538 (− 4.27) $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {debt}_{t})$$ 0.391 0.321 (− 0.22) $$\mathit {Cor}(\Delta \mathit {sales}_{t},\Delta \mathit {debt}_{t})$$ 0.360 0.288 (− 3.35) Note: This table compares the target moments of the two structural estimations: the benchmark estimation (Table 1) and the estimation with an additional separation shock (Table 5). The t-Statistics for the differences between the two sets of target moments are reported in the parentheses. See Nikolov and Whited (2014) for the details of how to calculate the t-Statistics. View Large 4.4. Adding Physical Capital The model presented so far abstracts from physical capital. Because investment is an important driver of financing needs for the firm, adding physical capital would make the model and results more general. Therefore, in this subsection, we re-estimate the model after adding physical capital as an additional input of production. The production function takes the form $$Y_t=z_t K_t^{\alpha } N_t^{1-\alpha }$$, where Kt is the input of capital and the other variables have the same content as before. Capital depreciates at rate δ and evolves according to Kt+1 = (1 − δ)Kt + It where It denotes investment. The firm faces the quadratic capital adjustment cost \begin{equation*} \phi \left(\frac{I_t}{K_t}\right) K_t= \chi \left(\frac{I_t}{K_t}\right)^{2} K_t. \end{equation*} Normalizing by the number of employees as we did earlier, the “normalized” optimization problem of the firm is \begin{eqnarray} s(\boldsymbol{x}_t,k_t, b_t) \max _{h_{t},i_{t},b_{t+1}} && \big\lbrace d_{t}+w_{t}-u_{t}+\beta (g_{t+1}-\eta h_t) \mathbb{E}_t s(\boldsymbol{x}_{t+1},k_{t+1},b_{t+1})\big\rbrace\nonumber\\ &&\\ \text{subj. to:}&& d_t+w_t=z_t k^\alpha _{t} - \Upsilon (h_t)\nonumber - i_t - \phi \left(\frac{i_t}{k_t}\right)k_t\\ && +\,q_t g_{t+1}b_{t+1}-b_{t} - \varphi (\boldsymbol{x}_t,k_t,b_t) \nonumber \\ && \xi _t g_{t+1}\beta \mathbb{E}_t s(\boldsymbol{x}_{t+1},k_{t+1},b_{t+1})\ge q_{t}g_{t+1}b_{t+1} \nonumber \\ && g_{t+1} k_{t+1}=(1-\delta ) k_t+i_t \nonumber \\ && g_{t+1}=1-\lambda +h_t, \nonumber \end{eqnarray} (27) where small letters denote variables in per-worker terms. In this model, we have three additional parameters: the capital adjustment cost χ, the depreciation rate δ, and the capital share α. To identify those parameters, we estimate the model by adding three new moments: the mean, standard deviation, and autocorrelation of investment. The mean of investment is especially important for the identification of the depreciation rate δ. The standard deviation of investment plays an important role for the identification of the adjustment cost parameter χ. Several moments contribute to the identification of the share parameter α. Especially important are the wage-to-revenue ratio and the investment rate because a higher capital share α leads to a lower wage to revenue ratio and a lower investment rate. Table 7 shows the results. Compared to the benchmark model without physical capital (Table 1), the estimated bargaining share parameter η increases from 0.355 to 0.535, and the estimated hiring cost parameter ϕ increases from 1.131 to 1.741. The overall marginal cost of hiring now is 1.84, which is higher than in the benchmark model (1.29). Table 7. Structural estimation with capital accumulation. (a) (b) (c) (d) Target moments Empirical data Estimated model t-Statistics Counterfactual w/o bargaining $$\mathit {Mean(wageshare}_{t})$$ 0.685 0.687 (− 0.45) 0.720 $$\mathit {Mean}(\mathit {unusedcredit}_{t})$$ 0.403 0.405 (− 0.29) 0.559 $$\mathit {Mean}(\mathit {leverage}_{t})$$ 0.217 0.214 (0.59) 0.170 $$\mathit {Std}(\mathit {leverage}_{t})$$ 0.100 0.064 (17.88) 0.006 $$\mathit {Autocor}(\mathit {leverage}_{t})$$ 0.735 0.500 (9.41) 0.132 $$\mathit {Mean(Tobin}Q_{t})$$ 1.360 1.361 (− 0.06) 1.215 $$\mathit {Std(Tobin}Q_{t})$$ 0.421 0.308 (11.45) 0.330 $$\mathit {Autocor(Tobin}Q_{t})$$ 0.668 0.304 (10.77) 0.490 $$\mathit {Std}(\Delta \mathit {employ}_{t})$$ 0.111 0.104 (5.13) 0.084 $$\mathit {Autocor}(\Delta \mathit {employ}_{t})$$ 0.065 0.504 (− 22.97) 0.540 $$\mathit {Std}(\Delta \mathit {sales}_{t})$$ 0.135 0.104 (16.22) 0.102 $$\mathit {Autocor}(\Delta \mathit {sales}_{t})$$ 0.051 0.386 (− 13.05) 0.238 $$\mathit {Std}(\Delta \mathit {debt}_{t})$$ 0.583 0.635 (− 6.16) 0.325 $$\mathit {Autocor}(\Delta \mathit {debt}_{t})$$ − 0.009 − 0.229 (10.87) − 0.170 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {sales}_{t})$$ 0.531 0.946 (− 9.01) 0.942 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {debt}_{t})$$ 0.340 0.540 (− 12.62) 0.650 $$\mathit {Cor}(\Delta \mathit {sales}_{t},\Delta \mathit {debt}_{t})$$ 0.178 0.429 (− 13.14) 0.842 $$\mathit {Mean}(Invest_{t})$$ 0.241 0.238 (1.06) 0.282 $$\mathit {Std}(Invest_{t})$$ 0.117 0.089 (11.48) 0.078 $$\mathit {Autocor}(Invest_{t})$$ 0.538 0.654 (− 3.91) 0.600 (a) (b) (c) (d) Target moments Empirical data Estimated model t-Statistics Counterfactual w/o bargaining $$\mathit {Mean(wageshare}_{t})$$ 0.685 0.687 (− 0.45) 0.720 $$\mathit {Mean}(\mathit {unusedcredit}_{t})$$ 0.403 0.405 (− 0.29) 0.559 $$\mathit {Mean}(\mathit {leverage}_{t})$$ 0.217 0.214 (0.59) 0.170 $$\mathit {Std}(\mathit {leverage}_{t})$$ 0.100 0.064 (17.88) 0.006 $$\mathit {Autocor}(\mathit {leverage}_{t})$$ 0.735 0.500 (9.41) 0.132 $$\mathit {Mean(Tobin}Q_{t})$$ 1.360 1.361 (− 0.06) 1.215 $$\mathit {Std(Tobin}Q_{t})$$ 0.421 0.308 (11.45) 0.330 $$\mathit {Autocor(Tobin}Q_{t})$$ 0.668 0.304 (10.77) 0.490 $$\mathit {Std}(\Delta \mathit {employ}_{t})$$ 0.111 0.104 (5.13) 0.084 $$\mathit {Autocor}(\Delta \mathit {employ}_{t})$$ 0.065 0.504 (− 22.97) 0.540 $$\mathit {Std}(\Delta \mathit {sales}_{t})$$ 0.135 0.104 (16.22) 0.102 $$\mathit {Autocor}(\Delta \mathit {sales}_{t})$$ 0.051 0.386 (− 13.05) 0.238 $$\mathit {Std}(\Delta \mathit {debt}_{t})$$ 0.583 0.635 (− 6.16) 0.325 $$\mathit {Autocor}(\Delta \mathit {debt}_{t})$$ − 0.009 − 0.229 (10.87) − 0.170 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {sales}_{t})$$ 0.531 0.946 (− 9.01) 0.942 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {debt}_{t})$$ 0.340 0.540 (− 12.62) 0.650 $$\mathit {Cor}(\Delta \mathit {sales}_{t},\Delta \mathit {debt}_{t})$$ 0.178 0.429 (− 13.14) 0.842 $$\mathit {Mean}(Invest_{t})$$ 0.241 0.238 (1.06) 0.282 $$\mathit {Std}(Invest_{t})$$ 0.117 0.089 (11.48) 0.078 $$\mathit {Autocor}(Invest_{t})$$ 0.538 0.654 (− 3.91) 0.600 Estimated parameters Estimators t-Statistics Persistence productivity shock, ρz 0.638 (49.35) Volatility productivity shock, σz 0.100 (26.81) Persistence debt shock, ρξ 0.388 (11.04) Volatility debt shock, σξ 0.156 (18.34) Financial distress cost, κ 1.783 (2.22) Workers’ bargaining power, η 0.535 (18.33) Hiring cost, ϕ 1.741 (2.13) Enforcement parameter, $$\bar{\xi }$$ 0.278 (23.84) Separation flow, $$\bar{u}$$ 0.407 (3.55) Capital share, α 0.616 (8.65) Capital depreciation rate, δ 0.258 (8.05) Investing cost, χ 0.478 (7.76) Estimated parameters Estimators t-Statistics Persistence productivity shock, ρz 0.638 (49.35) Volatility productivity shock, σz 0.100 (26.81) Persistence debt shock, ρξ 0.388 (11.04) Volatility debt shock, σξ 0.156 (18.34) Financial distress cost, κ 1.783 (2.22) Workers’ bargaining power, η 0.535 (18.33) Hiring cost, ϕ 1.741 (2.13) Enforcement parameter, $$\bar{\xi }$$ 0.278 (23.84) Separation flow, $$\bar{u}$$ 0.407 (3.55) Capital share, α 0.616 (8.65) Capital depreciation rate, δ 0.258 (8.05) Investing cost, χ 0.478 (7.76) Note: The table shows the results of the structural estimation after adding capital accumulation. The first panel lists the target moments; column (a) reports the moments from the data; column (b) reports the moments generated by the estimated model; column (c) reports the t-Statistic for the differences between the empirical moments and the moments generated by the estimated model; column (d) reports the moments generated by the model without the bargaining channel of debt as described in Section 3.5. The second panel reports the estimated parameters and the associated t-Statistics. View Large Table 7. Structural estimation with capital accumulation. (a) (b) (c) (d) Target moments Empirical data Estimated model t-Statistics Counterfactual w/o bargaining $$\mathit {Mean(wageshare}_{t})$$ 0.685 0.687 (− 0.45) 0.720 $$\mathit {Mean}(\mathit {unusedcredit}_{t})$$ 0.403 0.405 (− 0.29) 0.559 $$\mathit {Mean}(\mathit {leverage}_{t})$$ 0.217 0.214 (0.59) 0.170 $$\mathit {Std}(\mathit {leverage}_{t})$$ 0.100 0.064 (17.88) 0.006 $$\mathit {Autocor}(\mathit {leverage}_{t})$$ 0.735 0.500 (9.41) 0.132 $$\mathit {Mean(Tobin}Q_{t})$$ 1.360 1.361 (− 0.06) 1.215 $$\mathit {Std(Tobin}Q_{t})$$ 0.421 0.308 (11.45) 0.330 $$\mathit {Autocor(Tobin}Q_{t})$$ 0.668 0.304 (10.77) 0.490 $$\mathit {Std}(\Delta \mathit {employ}_{t})$$ 0.111 0.104 (5.13) 0.084 $$\mathit {Autocor}(\Delta \mathit {employ}_{t})$$ 0.065 0.504 (− 22.97) 0.540 $$\mathit {Std}(\Delta \mathit {sales}_{t})$$ 0.135 0.104 (16.22) 0.102 $$\mathit {Autocor}(\Delta \mathit {sales}_{t})$$ 0.051 0.386 (− 13.05) 0.238 $$\mathit {Std}(\Delta \mathit {debt}_{t})$$ 0.583 0.635 (− 6.16) 0.325 $$\mathit {Autocor}(\Delta \mathit {debt}_{t})$$ − 0.009 − 0.229 (10.87) − 0.170 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {sales}_{t})$$ 0.531 0.946 (− 9.01) 0.942 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {debt}_{t})$$ 0.340 0.540 (− 12.62) 0.650 $$\mathit {Cor}(\Delta \mathit {sales}_{t},\Delta \mathit {debt}_{t})$$ 0.178 0.429 (− 13.14) 0.842 $$\mathit {Mean}(Invest_{t})$$ 0.241 0.238 (1.06) 0.282 $$\mathit {Std}(Invest_{t})$$ 0.117 0.089 (11.48) 0.078 $$\mathit {Autocor}(Invest_{t})$$ 0.538 0.654 (− 3.91) 0.600 (a) (b) (c) (d) Target moments Empirical data Estimated model t-Statistics Counterfactual w/o bargaining $$\mathit {Mean(wageshare}_{t})$$ 0.685 0.687 (− 0.45) 0.720 $$\mathit {Mean}(\mathit {unusedcredit}_{t})$$ 0.403 0.405 (− 0.29) 0.559 $$\mathit {Mean}(\mathit {leverage}_{t})$$ 0.217 0.214 (0.59) 0.170 $$\mathit {Std}(\mathit {leverage}_{t})$$ 0.100 0.064 (17.88) 0.006 $$\mathit {Autocor}(\mathit {leverage}_{t})$$ 0.735 0.500 (9.41) 0.132 $$\mathit {Mean(Tobin}Q_{t})$$ 1.360 1.361 (− 0.06) 1.215 $$\mathit {Std(Tobin}Q_{t})$$ 0.421 0.308 (11.45) 0.330 $$\mathit {Autocor(Tobin}Q_{t})$$ 0.668 0.304 (10.77) 0.490 $$\mathit {Std}(\Delta \mathit {employ}_{t})$$ 0.111 0.104 (5.13) 0.084 $$\mathit {Autocor}(\Delta \mathit {employ}_{t})$$ 0.065 0.504 (− 22.97) 0.540 $$\mathit {Std}(\Delta \mathit {sales}_{t})$$ 0.135 0.104 (16.22) 0.102 $$\mathit {Autocor}(\Delta \mathit {sales}_{t})$$ 0.051 0.386 (− 13.05) 0.238 $$\mathit {Std}(\Delta \mathit {debt}_{t})$$ 0.583 0.635 (− 6.16) 0.325 $$\mathit {Autocor}(\Delta \mathit {debt}_{t})$$ − 0.009 − 0.229 (10.87) − 0.170 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {sales}_{t})$$ 0.531 0.946 (− 9.01) 0.942 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {debt}_{t})$$ 0.340 0.540 (− 12.62) 0.650 $$\mathit {Cor}(\Delta \mathit {sales}_{t},\Delta \mathit {debt}_{t})$$ 0.178 0.429 (− 13.14) 0.842 $$\mathit {Mean}(Invest_{t})$$ 0.241 0.238 (1.06) 0.282 $$\mathit {Std}(Invest_{t})$$ 0.117 0.089 (11.48) 0.078 $$\mathit {Autocor}(Invest_{t})$$ 0.538 0.654 (− 3.91) 0.600 Estimated parameters Estimators t-Statistics Persistence productivity shock, ρz 0.638 (49.35) Volatility productivity shock, σz 0.100 (26.81) Persistence debt shock, ρξ 0.388 (11.04) Volatility debt shock, σξ 0.156 (18.34) Financial distress cost, κ 1.783 (2.22) Workers’ bargaining power, η 0.535 (18.33) Hiring cost, ϕ 1.741 (2.13) Enforcement parameter, $$\bar{\xi }$$ 0.278 (23.84) Separation flow, $$\bar{u}$$ 0.407 (3.55) Capital share, α 0.616 (8.65) Capital depreciation rate, δ 0.258 (8.05) Investing cost, χ 0.478 (7.76) Estimated parameters Estimators t-Statistics Persistence productivity shock, ρz 0.638 (49.35) Volatility productivity shock, σz 0.100 (26.81) Persistence debt shock, ρξ 0.388 (11.04) Volatility debt shock, σξ 0.156 (18.34) Financial distress cost, κ 1.783 (2.22) Workers’ bargaining power, η 0.535 (18.33) Hiring cost, ϕ 1.741 (2.13) Enforcement parameter, $$\bar{\xi }$$ 0.278 (23.84) Separation flow, $$\bar{u}$$ 0.407 (3.55) Capital share, α 0.616 (8.65) Capital depreciation rate, δ 0.258 (8.05) Investing cost, χ 0.478 (7.76) Note: The table shows the results of the structural estimation after adding capital accumulation. The first panel lists the target moments; column (a) reports the moments from the data; column (b) reports the moments generated by the estimated model; column (c) reports the t-Statistic for the differences between the empirical moments and the moments generated by the estimated model; column (d) reports the moments generated by the model without the bargaining channel of debt as described in Section 3.5. The second panel reports the estimated parameters and the associated t-Statistics. View Large To understand why the marginal cost of hiring is higher, consider the first-order condition for hiring in the model with physical capital: \begin{equation} q_t b_{t+1} + \beta (1-\eta )\mathbb{E}_t s(\boldsymbol{x}_{t+1},k_{t+1},b_{t+1}) = \Upsilon ^{\prime }(h_t) + \left(1+\phi ^{\prime }\left(\frac{i_t}{k_t}\right) \right)k_{t+1}. \end{equation} (28) Compared to the first-order condition for the model without capital (see equation (17)), condition (28) has the additional term \begin{equation*} \left(1+\phi ^{\prime }\left(\frac{i_t}{k_t}\right) \right)k_{t+1}, \end{equation*} which contributes to the overall marginal cost of hiring. The last column of Table 7 shows the moments generated by the counterfactual exercise in which we shut down the bargaining channel of debt. The contribution of the bargaining channel to employment volatility is now about 18% because the removal of this channel reduces the standard deviation of employment growth from 0.104 to 0.084. The higher contribution of the bargaining channel of debt follows from the higher estimation of the bargaining share parameter η (which increases from 0.355 in the benchmark model to 0.535 in the model with physical capital). 4.5. Sensitivity to the Measure of Unused Lines of Credit As pointed out earlier, the empirical series for unused lines of credit is only a proxy for the unused debt capacity of the firm which in the model corresponds to the difference between the actual debt and the borrowing limit. This could be problematic because in reality firms may have access to other financial instruments in order to raise funds, and the actual borrowing limit could be very different from the credit lines reported in the data. To investigate the sensitivity of the estimation results to our measure of unused lines of credit, we re-estimate the model using a modified data sample where the variable “unused lines of credit” is increased by 25% for all firms and years. Table 8 reports the results. Table 8. Structural estimation with higher unused lines of credit. (a) (b) (c) (d) Target moments Empirical data Estimated model t-Statistics Counterfactual w/o bargaining $$\mathit {Mean(wageshare}_{t})$$ 0.685 0.690 (− 1.08) 0.691 $$\mathit {Mean}(\mathit {unusedcredit}_{t})$$ 0.503 0.506 (− 0.26) 0.970 $$\mathit {Mean}(\mathit {leverage}_{t})$$ 0.217 0.221 (− 0.80) 0.016 $$\mathit {Std}(\mathit {leverage}_{t})$$ 0.100 0.094 (3.30) 0.012 $$\mathit {Autocor}(\mathit {leverage}_{t})$$ 0.735 0.610 (5.00) 0.540 $$\mathit {Mean(Tobin}Q_{t})$$ 1.360 1.345 (0.93) 1.402 $$\mathit {Std(Tobin}Q_{t})$$ 0.421 0.223 (17.76) 0.130 $$\mathit {Autocor(Tobin}Q_{t})$$ 0.668 0.304 (10.75) 0.667 $$\mathit {Std}(\Delta \mathit {employ}_{t})$$ 0.111 0.097 (9.68) 0.076 $$\mathit {Autocor}(\Delta \mathit {employ}_{t})$$ 0.065 0.618 (− 28.97) 0.650 $$\mathit {Std}(\Delta \mathit {sales}_{t})$$ 0.135 0.142 (− 4.18) 0.135 $$\mathit {Autocor}(\Delta \mathit {sales}_{t})$$ 0.051 0.182 (− 5.08) 0.073 $$\mathit {Std}(\Delta \mathit {debt}_{t})$$ 0.583 0.590 (− 0.82) 0.707 $$\mathit {Autocor}(\Delta \mathit {debt}_{t})$$ − 0.009 − 0.173 (8.07) − 0.165 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {sales}_{t})$$ 0.531 0.807 (− 12.47) 0.811 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {debt}_{t})$$ 0.340 0.487 (− 5.52) 0.009 $$\mathit {Cor}(\Delta \mathit {sales}_{t},\Delta \mathit {debt}_{t})$$ 0.178 0.322 (− 11.81) − 0.174 (a) (b) (c) (d) Target moments Empirical data Estimated model t-Statistics Counterfactual w/o bargaining $$\mathit {Mean(wageshare}_{t})$$ 0.685 0.690 (− 1.08) 0.691 $$\mathit {Mean}(\mathit {unusedcredit}_{t})$$ 0.503 0.506 (− 0.26) 0.970 $$\mathit {Mean}(\mathit {leverage}_{t})$$ 0.217 0.221 (− 0.80) 0.016 $$\mathit {Std}(\mathit {leverage}_{t})$$ 0.100 0.094 (3.30) 0.012 $$\mathit {Autocor}(\mathit {leverage}_{t})$$ 0.735 0.610 (5.00) 0.540 $$\mathit {Mean(Tobin}Q_{t})$$ 1.360 1.345 (0.93) 1.402 $$\mathit {Std(Tobin}Q_{t})$$ 0.421 0.223 (17.76) 0.130 $$\mathit {Autocor(Tobin}Q_{t})$$ 0.668 0.304 (10.75) 0.667 $$\mathit {Std}(\Delta \mathit {employ}_{t})$$ 0.111 0.097 (9.68) 0.076 $$\mathit {Autocor}(\Delta \mathit {employ}_{t})$$ 0.065 0.618 (− 28.97) 0.650 $$\mathit {Std}(\Delta \mathit {sales}_{t})$$ 0.135 0.142 (− 4.18) 0.135 $$\mathit {Autocor}(\Delta \mathit {sales}_{t})$$ 0.051 0.182 (− 5.08) 0.073 $$\mathit {Std}(\Delta \mathit {debt}_{t})$$ 0.583 0.590 (− 0.82) 0.707 $$\mathit {Autocor}(\Delta \mathit {debt}_{t})$$ − 0.009 − 0.173 (8.07) − 0.165 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {sales}_{t})$$ 0.531 0.807 (− 12.47) 0.811 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {debt}_{t})$$ 0.340 0.487 (− 5.52) 0.009 $$\mathit {Cor}(\Delta \mathit {sales}_{t},\Delta \mathit {debt}_{t})$$ 0.178 0.322 (− 11.81) − 0.174 Estimated parameters Estimators t-Statistics Persistence productivity shock, ρz 0.742 (70.31) Volatility productivity shock, σz 0.169 (81.26) Persistence debt shock, ρξ 0.642 (103.4) Volatility debt shock, σξ 0.235 (26.35) Financial distress cost, κ 2.668 (30.39) Workers’ bargaining power, η 0.463 (33.89) Hiring cost, ϕ 0.894 (26.05) Enforcement parameter, $$\bar{\xi }$$ 0.453 (30.37) Unemployment flow, $$\bar{u}$$ 0.471 (40.20) Estimated parameters Estimators t-Statistics Persistence productivity shock, ρz 0.742 (70.31) Volatility productivity shock, σz 0.169 (81.26) Persistence debt shock, ρξ 0.642 (103.4) Volatility debt shock, σξ 0.235 (26.35) Financial distress cost, κ 2.668 (30.39) Workers’ bargaining power, η 0.463 (33.89) Hiring cost, ϕ 0.894 (26.05) Enforcement parameter, $$\bar{\xi }$$ 0.453 (30.37) Unemployment flow, $$\bar{u}$$ 0.471 (40.20) Note: The table shows the results of the structural estimation, when we increase the level of unused lines of credit in the data by 25% while keeping other moments at the same values as those used in the benchmark estimation. The first panel contains the following information: column (a) reports the moments from the data; column (b) reports the moments generated by the estimated model; column (c) reports the t-Statistic for the differences between the empirical moments and the moments generated by the estimated model; column (d) reports the moments generated by the model without the bargaining channel of debt as described in Section 3.5. The second panel reports the estimated parameters and the associated t-Statistics. View Large Table 8. Structural estimation with higher unused lines of credit. (a) (b) (c) (d) Target moments Empirical data Estimated model t-Statistics Counterfactual w/o bargaining $$\mathit {Mean(wageshare}_{t})$$ 0.685 0.690 (− 1.08) 0.691 $$\mathit {Mean}(\mathit {unusedcredit}_{t})$$ 0.503 0.506 (− 0.26) 0.970 $$\mathit {Mean}(\mathit {leverage}_{t})$$ 0.217 0.221 (− 0.80) 0.016 $$\mathit {Std}(\mathit {leverage}_{t})$$ 0.100 0.094 (3.30) 0.012 $$\mathit {Autocor}(\mathit {leverage}_{t})$$ 0.735 0.610 (5.00) 0.540 $$\mathit {Mean(Tobin}Q_{t})$$ 1.360 1.345 (0.93) 1.402 $$\mathit {Std(Tobin}Q_{t})$$ 0.421 0.223 (17.76) 0.130 $$\mathit {Autocor(Tobin}Q_{t})$$ 0.668 0.304 (10.75) 0.667 $$\mathit {Std}(\Delta \mathit {employ}_{t})$$ 0.111 0.097 (9.68) 0.076 $$\mathit {Autocor}(\Delta \mathit {employ}_{t})$$ 0.065 0.618 (− 28.97) 0.650 $$\mathit {Std}(\Delta \mathit {sales}_{t})$$ 0.135 0.142 (− 4.18) 0.135 $$\mathit {Autocor}(\Delta \mathit {sales}_{t})$$ 0.051 0.182 (− 5.08) 0.073 $$\mathit {Std}(\Delta \mathit {debt}_{t})$$ 0.583 0.590 (− 0.82) 0.707 $$\mathit {Autocor}(\Delta \mathit {debt}_{t})$$ − 0.009 − 0.173 (8.07) − 0.165 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {sales}_{t})$$ 0.531 0.807 (− 12.47) 0.811 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {debt}_{t})$$ 0.340 0.487 (− 5.52) 0.009 $$\mathit {Cor}(\Delta \mathit {sales}_{t},\Delta \mathit {debt}_{t})$$ 0.178 0.322 (− 11.81) − 0.174 (a) (b) (c) (d) Target moments Empirical data Estimated model t-Statistics Counterfactual w/o bargaining $$\mathit {Mean(wageshare}_{t})$$ 0.685 0.690 (− 1.08) 0.691 $$\mathit {Mean}(\mathit {unusedcredit}_{t})$$ 0.503 0.506 (− 0.26) 0.970 $$\mathit {Mean}(\mathit {leverage}_{t})$$ 0.217 0.221 (− 0.80) 0.016 $$\mathit {Std}(\mathit {leverage}_{t})$$ 0.100 0.094 (3.30) 0.012 $$\mathit {Autocor}(\mathit {leverage}_{t})$$ 0.735 0.610 (5.00) 0.540 $$\mathit {Mean(Tobin}Q_{t})$$ 1.360 1.345 (0.93) 1.402 $$\mathit {Std(Tobin}Q_{t})$$ 0.421 0.223 (17.76) 0.130 $$\mathit {Autocor(Tobin}Q_{t})$$ 0.668 0.304 (10.75) 0.667 $$\mathit {Std}(\Delta \mathit {employ}_{t})$$ 0.111 0.097 (9.68) 0.076 $$\mathit {Autocor}(\Delta \mathit {employ}_{t})$$ 0.065 0.618 (− 28.97) 0.650 $$\mathit {Std}(\Delta \mathit {sales}_{t})$$ 0.135 0.142 (− 4.18) 0.135 $$\mathit {Autocor}(\Delta \mathit {sales}_{t})$$ 0.051 0.182 (− 5.08) 0.073 $$\mathit {Std}(\Delta \mathit {debt}_{t})$$ 0.583 0.590 (− 0.82) 0.707 $$\mathit {Autocor}(\Delta \mathit {debt}_{t})$$ − 0.009 − 0.173 (8.07) − 0.165 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {sales}_{t})$$ 0.531 0.807 (− 12.47) 0.811 $$\mathit {Cor}(\Delta \mathit {employ}_{t},\Delta \mathit {debt}_{t})$$ 0.340 0.487 (− 5.52) 0.009 $$\mathit {Cor}(\Delta \mathit {sales}_{t},\Delta \mathit {debt}_{t})$$ 0.178 0.322 (− 11.81) − 0.174 Estimated parameters Estimators t-Statistics Persistence productivity shock, ρz 0.742 (70.31) Volatility productivity shock, σz 0.169 (81.26) Persistence debt shock, ρξ 0.642 (103.4) Volatility debt shock, σξ 0.235 (26.35) Financial distress cost, κ 2.668 (30.39) Workers’ bargaining power, η 0.463 (33.89) Hiring cost, ϕ 0.894 (26.05) Enforcement parameter, $$\bar{\xi }$$ 0.453 (30.37) Unemployment flow, $$\bar{u}$$ 0.471 (40.20) Estimated parameters Estimators t-Statistics Persistence productivity shock, ρz 0.742 (70.31) Volatility productivity shock, σz 0.169 (81.26) Persistence debt shock, ρξ 0.642 (103.4) Volatility debt shock, σξ 0.235 (26.35) Financial distress cost, κ 2.668 (30.39) Workers’ bargaining power, η 0.463 (33.89) Hiring cost, ϕ 0.894 (26.05) Enforcement parameter, $$\bar{\xi }$$ 0.453 (30.37) Unemployment flow, $$\bar{u}$$ 0.471 (40.20) Note: The table shows the results of the structural estimation, when we increase the level of unused lines of credit in the data by 25% while keeping other moments at the same values as those used in the benchmark estimation. The first panel contains the following information: column (a) reports the moments from the data; column (b) reports the moments generated by the estimated model; column (c) reports the t-Statistic for the differences between the empirical moments and the moments generated by the estimated model; column (d) reports the moments generated by the model without the bargaining channel of debt as described in Section 3.5. The second panel reports the estimated parameters and the associated t-Statistics. View Large Compared to the benchmark estimation, the financial distress cost parameter κ changes from 2.475 to 2.668. This is because the model requires a higher financial distress cost (higher precautionary motive) to match the higher level of unused lines of credit. Also, the bargaining share parameter η increases from 0.355 to 0.463. The last column in the top section of Table 8 shows that the importance of the bargaining channel for employment fluctuations increases from 13% ((0.091 − 0.077)/0.111 ≈ 0.13) in the benchmark estimation to 19% ((0.097 − 0.076)/0.111 ≈ 0.19) when the model is estimated with the artificially modified data. This suggests that, if the variable unused lines of credit underestimates the true unused debt capacity of firms, the bias should reduce the importance of the bargaining channel. Our estimation can then be considered a conservative assessment of the importance of the bargaining channel of debt for the volatility of firm-level employment. 5. Direct Evidence for the Mechanism The central mechanism explored in this paper—the impact of debt on hiring through the bargaining channel—is based on the idea that wages are bargained between workers and employers and, when a firm increases its debt, it can negotiate lower wages with workers. In this section, we explore the empirical significance of this channel using reduced-form regressions. We show first that in the model the importance of the bargaining channel of debt increases with the bargaining power of workers, η. Figure 3 plots the response of firm-level employment growth to a credit shock for different levels of the bargaining share of workers η. The left panel shows that, when the bargaining share is η = 0.355 (estimated value), employment growth increases by 2.0% in response to a 0.93 standard-deviation positive credit shock.8 However, if the bargaining share is η = 0.44 (25% higher than the estimated value), the response of employment growth increases by 2.5%. Therefore, the higher is the bargaining power of workers, the higher is the sensitivity of employment to a firm’s borrowing. Similarly, after a negative 0.93 standard-deviation shock (right panel), the decline in employment growth is larger in firms where workers have higher bargaining power. This is the property we would like to test empirically. Figure 3. View largeDownload slide Response of employment to credit shocks. The left (right) panel depicts the response of employment growth to a 0.93 standard-deviation positive (negative) credit shock. Figure 3. View largeDownload slide Response of employment to credit shocks. The left (right) panel depicts the response of employment growth to a 0.93 standard-deviation positive (negative) credit shock. 5.1. Regression Equation We derive the main regression equation from the first-order conditions of the firm. Consider the optimality condition for hiring (equation (17)), which for convenience we rewrite here, \begin{eqnarray} {q_t \frac{b_t g_{t+1}^B}{g_{t+1}}+\beta (1-\eta )\mathbb{E}_t\left[\bar{s}(\boldsymbol{x}_{t+1})-b_{t+1}-\varphi \left(\boldsymbol{x}_{t+1},\frac{ b_t g_{t+1}^B}{g_{t+1}}\right)\right]} {= \Upsilon ^{\prime }(g_{t+1}-1+\lambda ).} \end{eqnarray} (29) The variable gt+1 is the gross growth rate of employment, Nt, and $$g_{t+1}^B$$ is the gross growth rate of debt Bt. The reason these two growth rates enter the first-order condition is because we have replaced bt+1 with $$b_t g_{t+1}^B/g_{t+1}$$. In the right-hand side we have replaced ht with gt+1 − 1 + λ. Equation (29) establishes a relation between the growth of employment and the growth of debt, conditional on the initial per-worker debt bt and the structural shocks xt. Importantly, this relation depends on the bargaining power of workers η, which we will explore empirically. Becauseequation (29) is not linear, for the empirical application, we approximate it with a first-order Taylor expansion around the unconditional mean. Assuming that h = g − 1 + λ > 0 at the unconditional mean, we obtain \begin{equation} g_{t+1}=\alpha _{c}+\alpha _{g}g_{t+1}^{B}+\alpha _{b}b_{t}+ {\boldsymbol\alpha}_x \mathbb{E}_t\Big (\boldsymbol{x}_{t+1}|\boldsymbol{x}_t\Big ), \end{equation} (30) where \begin{eqnarray*} \alpha _{g} &=&\frac{\Gamma }{\frac{g^B \Gamma }{g}+\frac{2\psi g}{b}}, \nonumber\\ \alpha _{b} &=&\frac{\Gamma }{\frac{b \Gamma }{g}+\frac{2\psi g}{g^B}}, \nonumber\\ \alpha _{x} &=&\frac{\beta (1-\eta )[\bar{s}_1(\boldsymbol{x})-\varphi _1(\boldsymbol{x},b)]}{\frac{b g^B \Gamma }{g^2}+2\psi }, \end{eqnarray*} with Γ = q − β(1 − η)[1 + φ2(x, b)], and subscripts on the functions $$\bar{s}(.)$$ and φ(., .) denote derivatives. In the derivation, we have specified the functional form for the hiring cost as $$\Upsilon (h_t)=\phi h_t + \psi h_t^2$$. Variables without time subscripts are unconditional means (around which we conduct the Taylor expansion). The structural shocks enter the linearized equation through the term $${\boldsymbol\alpha }_x \mathbb{E}_t(\boldsymbol{x}_{t+1}|\boldsymbol{x}_t)$$ where $${\boldsymbol\alpha }_x$$ is a vector of coefficients. Also, $$\bar{s}_1(\boldsymbol{x})$$ and φ1(x, b) are vectors. The key coefficient of interest is αg. This coefficient is strictly increasing in η, implying that the employment-debt growth relation increases with the bargaining power of workers. To test the dependence of the employment-debt growth relation on the bargaining power of workers, we estimate the following regression: \begin{eqnarray} \Delta \mathit {employ}_{i,t} &=&\alpha + \beta _1\cdot \mathit {union}_{cic,t}\cdot \Delta \mathit {debt}_{i,t}+\beta _2\cdot \mathit {union}_{cic,t}+\beta _3\cdot \Delta \mathit {debt}_{i,t} \nonumber \\ &&+\, \beta _4\cdot \mathit {leverage}_{i,t-1} + \beta _{5}\cdot \mathit {unusedcredit}_{i,t-1} \nonumber \\ &&+\, \beta _6\cdot Q_{i,t} + \beta _7\cdot \mathit {cashflow}_{i,t} +\nu _i+\tau _t+\varepsilon _{i,t}. \end{eqnarray} (31) The regression equation is the empirical counterpart to the linearized equation (30). The dependent variable is employment growth, $$\Delta \mathit {employ}_{i,t}$$. The main independent variable is the interaction between industry unionization rate and debt growth, $$\mathit {union}_{cic,t}\cdot \Delta \mathit {debt}_{i,t}$$. The unionization rate $$\mathit {union}_{cic,t}$$ is a proxy for the bargaining power of workers in the model, η. Lagged financial leverage ratio, $$\mathit {leverage}_{i,t-1}$$, is a proxy for bt in the model. Following the investment literature, we include market-to-book ratio Qi, t, cash flow-to-asset ratio, $$\mathit {cashflow}_{i,t}$$, firm-level fixed effects, νi, and year fixed effects, τt. The variables $$\mathit {cashflow}_{i,t}$$ and Qi, t capture some of the impact of the productivity shock. Unused credit ratio, $$\mathit {unusedcredit}_{i,t-1}$$, captures the impact of the firm’s financial position. Of primary interest is the interaction term between debt growth and unionization rate, $$\mathit {union}_{cic,t}\cdot \Delta \mathit {debt}_{i,t}$$. We expect this term to have a positive effect on employment growth, that is, β1 > 0. This is in addition to the direct effect of debt growth captured by β3. 5.2. Data Description To estimate the above regression equation, we need a proxy for η. Following Klasa et al. (2009) and Matsa (2010), we use the unionization index from the Union Membership and Coverage Database. The Union Membership and Coverage Database is maintained by Barry Hirsch and David Macpherson and is publicly available at http://www.unionstats.com. It compiles industry union coverage annually from the Current Population Survey (CPS). We first obtain firm-level employment and balance sheet variables from the Compustat and Capital IQ. We then merge the variables with the industry unionization rates for the same period 2003–2010.9 Ideally, we would like to use the unionization rate for each firm included in the sample. Unfortunately, for recent years, which is the focus of this paper, large-sample unionization data are only available at the industry level.10 Therefore, we are constrained to proxy the bargaining power of workers for an individual firm with the average unionization index of the industry in which the firm operates. This is also the approach used by Klasa et al. (2009) to study the relation between cash holdings and bargaining power of workers. 5.3. Estimation Results The first column of Table 9 reports the estimation results for the baseline specification of equation (31). The coefficient for the interaction term is 0.095, and it is statistically significant at 1% level. Therefore, the growth of debt in firms with more unionized labor is associated with a higher growth rate of employment. In terms of economic magnitude, a one standard deviation increase in debt growth leads to a 0.23 standard deviation increase in employment growth for nonunionized firms, and an additional 0.07 standard deviation increases for unionized firms.11 Table 9. Employment growth regression. Baseline regression. Unionization rate High Low $$\mathit {union}_{cic,t}\cdot \Delta \mathit {debt}_{it}$$ 0.095*** (0.036) $$\mathit {union}_{cic,t}$$ − 0.023 (0.061) $$\Delta \mathit {debt}_{it}$$ 0.035*** 0.056*** 0.034*** (0.003) (0.005) (0.003) $$\mathit {leverage}_{it-1}$$ − 0.167*** − 0.173*** − 0.171*** (0.019) (0.027) (0.029) $$\mathit {unusedcredit}_{it-1}$$ − 0.051*** − 0.091*** − 0.024** (0.010) (0.017) (0.013) tobinQit 0.029*** 0.033*** 0.025*** (0.005) (0.008) (0.006) $$\mathit {cashflow}_{it}$$ 0.162*** 0.212*** 0.118** (0.044) (0.066) (0.064) Firm fixed effects Yes Yes Yes Year dummies Yes Yes Yes Adjusted R2 0.28 0.26 0.28 Observation 12,113 5,927 6,186 Unionization rate High Low $$\mathit {union}_{cic,t}\cdot \Delta \mathit {debt}_{it}$$ 0.095*** (0.036) $$\mathit {union}_{cic,t}$$ − 0.023 (0.061) $$\Delta \mathit {debt}_{it}$$ 0.035*** 0.056*** 0.034*** (0.003) (0.005) (0.003) $$\mathit {leverage}_{it-1}$$ − 0.167*** − 0.173*** − 0.171*** (0.019) (0.027) (0.029) $$\mathit {unusedcredit}_{it-1}$$ − 0.051*** − 0.091*** − 0.024** (0.010) (0.017) (0.013) tobinQit 0.029*** 0.033*** 0.025*** (0.005) (0.008) (0.006) $$\mathit {cashflow}_{it}$$ 0.162*** 0.212*** 0.118** (0.044) (0.066) (0.064) Firm fixed effects Yes Yes Yes Year dummies Yes Yes Yes Adjusted R2 0.28 0.26 0.28 Observation 12,113 5,927 6,186 Note: This table reports the regression results using the industry-level unionization data. The first column shows the results of the baseline regression of equation (31). The next two columns report the results of regressions without the interaction term, separately for high and low unionization firms. High unionization firms are those located in industries with higher than median unionization rate. The sample is an unbalanced panel of 2,168 firms during the period 2003–2010. The dependent variable is employment growth, $$\Delta \mathit {employ}_{it}$$, and independent variables include: interaction between industry unionization rate and debt growth, $$\mathit {union}_{cic,t}\cdot \Delta \mathit {debt}_{it}$$, industry unionization rate $$\mathit {union}_{cic,t}$$, debt growth, $$\Delta \mathit {debt}_{it}$$, lagged financial leverage ratio $$\mathit {leverage}_{it-1}$$, lagged unused credit-to-total credit ratio, $$\mathit {unusedcredit}_{it-1}$$, market-to-book ratio, Qit, cash flow-to-asset ratio, $$\mathit {cashflow}_{it}$$. Firm fixed effects and year dummies are also included. Standard errors (in parentheses) are heteroskedasticity robust and clustered at the firm level, and significance levels at 1%, 5%, and 10% are marked with superscripts ***, **, *, respectively. View Large Table 9. Employment growth regression. Baseline regression. Unionization rate High Low $$\mathit {union}_{cic,t}\cdot \Delta \mathit {debt}_{it}$$ 0.095*** (0.036) $$\mathit {union}_{cic,t}$$ − 0.023 (0.061) $$\Delta \mathit {debt}_{it}$$ 0.035*** 0.056*** 0.034*** (0.003) (0.005) (0.003) $$\mathit {leverage}_{it-1}$$ − 0.167*** − 0.173*** − 0.171*** (0.019) (0.027) (0.029) $$\mathit {unusedcredit}_{it-1}$$ − 0.051*** − 0.091*** − 0.024** (0.010) (0.017) (0.013) tobinQit 0.029*** 0.033*** 0.025*** (0.005) (0.008) (0.006) $$\mathit {cashflow}_{it}$$ 0.162*** 0.212*** 0.118** (0.044) (0.066) (0.064) Firm fixed effects Yes Yes Yes Year dummies Yes Yes Yes Adjusted R2 0.28 0.26 0.28 Observation 12,113 5,927 6,186 Unionization rate High Low $$\mathit {union}_{cic,t}\cdot \Delta \mathit {debt}_{it}$$ 0.095*** (0.036) $$\mathit {union}_{cic,t}$$ − 0.023 (0.061) $$\Delta \mathit {debt}_{it}$$ 0.035*** 0.056*** 0.034*** (0.003) (0.005) (0.003) $$\mathit {leverage}_{it-1}$$ − 0.167*** − 0.173*** − 0.171*** (0.019) (0.027) (0.029) $$\mathit {unusedcredit}_{it-1}$$ − 0.051*** − 0.091*** − 0.024** (0.010) (0.017) (0.013) tobinQit 0.029*** 0.033*** 0.025*** (0.005) (0.008) (0.006) $$\mathit {cashflow}_{it}$$ 0.162*** 0.212*** 0.118** (0.044) (0.066) (0.064) Firm fixed effects Yes Yes Yes Year dummies Yes Yes Yes Adjusted R2 0.28 0.26 0.28 Observation 12,113 5,927 6,186 Note: This table reports the regression results using the industry-level unionization data. The first column shows the results of the baseline regression of equation (31). The next two columns report the results of regressions without the interaction term, separately for high and low unionization firms. High unionization firms are those located in industries with higher than median unionization rate. The sample is an unbalanced panel of 2,168 firms during the period 2003–2010. The dependent variable is employment growth, $$\Delta \mathit {employ}_{it}$$, and independent variables include: interaction between industry unionization rate and debt growth, $$\mathit {union}_{cic,t}\cdot \Delta \mathit {debt}_{it}$$, industry unionization rate $$\mathit {union}_{cic,t}$$, debt growth, $$\Delta \mathit {debt}_{it}$$, lagged financial leverage ratio $$\mathit {leverage}_{it-1}$$, lagged unused credit-to-total credit ratio, $$\mathit {unusedcredit}_{it-1}$$, market-to-book ratio, Qit, cash flow-to-asset ratio, $$\mathit {cashflow}_{it}$$. Firm fixed effects and year dummies are also included. Standard errors (in parentheses) are heteroskedasticity robust and clustered at the firm level, and significance levels at 1%, 5%, and 10% are marked with superscripts ***, **, *, respectively. View Large These results should be taken with caution because we use industry level unionization rates to proxy for the bargaining power of workers employed by an individual firm. Furthermore, in conducting the estimation, we are not testing for causality. We are only estimating conditional correlations. With this in mind, we can conclude that the estimation results are consistent with the prediction of the model. Turning to the control variables, the first column of Table 9 shows that employment growth is negatively related to the lagged ratio of unused credit over total credit, and positively related to the market-to-book ratio and cash flow-to-asset ratio. This is consistent with the investment literature if there is complementarity between investment and hiring. An alternative way to test the importance of unionization is by estimating equation (31) without the interaction term, separately for a group of low unionized firms and a group of high unionized firms. The high unionized group includes firms that operate in industries with higher than median unionization rates. The estimation results are reported in the last two columns of Table 9. The coefficient of $$\Delta \mathit {debt}_{it}$$ is bigger for the group of high unionized firms. Thus, the estimation confirms that the relation between employment growth and debt growth increases with the bargaining power of workers, consistent with the theory. 5.4. Financial Distress To test the hypothesis that the bargaining channel of debt is more important when firms face more favorable financial conditions, we run the baseline regression separately for two different groups of firms: (i) firms that pay dividends versus firms that do not pay dividends; and (ii) firms with high long-term debt maturing next year versus firms with low debt maturing next year. Following Almeida et al. (2012), we define firms with the high ratio of long-term debt maturing next year as those for which the percentage of maturing debt is higher than the median. The idea is that firms with higher long-term debt close to maturity face higher financial risk and, therefore, are more likely to incur financial distress. Table 10 reports the regression results. As shown in the table, the coefficient of the interaction term, $$\mathit {union}_{cic,t}\cdot \Delta \mathit {debt}_{it}$$, is statistically significant only for firms with positive dividend payout and for firms with lower maturing debt. Table 10. Employment growth regression. Financial distress. Dividend payout Debt maturing Yes No Low High $$\mathit {union}_{cic,t}\cdot \Delta \mathit {debt}_{it}$$ 0.095*** 0.165*** 0.069 0.262** 0.101 (0.036) (0.056) (0.050) (0.127) (0.064) $$\mathit {union}_{cic,t}$$ − 0.023 0.029 − 0.078 0.042 − 0.127 (0.061) (0.066) (0.122) (0.125) (0.099) $$\Delta \mathit {debt}_{it}$$ 0.035*** 0.032*** 0.036*** 0.076*** 0.057*** (0.003) (0.006) (0.004) (0.017) (0.008) $$\mathit {leverage}_{it-1}$$ − 0.167*** − 0.125*** − 0.189*** − 0.146*** − 0.198*** (0.019) (0.030) (0.025) (0.043) (0.041) $$\mathit {unusedcredit}_{it-1}$$ − 0.051*** − 0.055*** − 0.053*** − 0.137*** − 0.105*** (0.010) (0.016) (0.014) (0.037) (0.027) tobinQit 0.029*** 0.005 0.036*** 0.025** 0.035*** (0.005) (0.007) (0.006) (0.013) (0.013) $$\mathit {cashflow}_{it}$$ 0.162*** 0.234*** 0.172*** 0.049 0.102 (0.044) (0.067) (0.061) (0.101) (0.094) Firm fixed effects Yes Yes Yes Yes Yes Year dummies Yes Yes Yes Yes Yes Adjusted R2 0.28 0.27 0.27 0.33 0.37 Observation 12,113 5,013 7,100 3,106 4,024 Dividend payout Debt maturing Yes No Low High $$\mathit {union}_{cic,t}\cdot \Delta \mathit {debt}_{it}$$ 0.095*** 0.165*** 0.069 0.262** 0.101 (0.036) (0.056) (0.050) (0.127) (0.064) $$\mathit {union}_{cic,t}$$ − 0.023 0.029 − 0.078 0.042 − 0.127 (0.061) (0.066) (0.122) (0.125) (0.099) $$\Delta \mathit {debt}_{it}$$ 0.035*** 0.032*** 0.036*** 0.076*** 0.057*** (0.003) (0.006) (0.004) (0.017) (0.008) $$\mathit {leverage}_{it-1}$$ − 0.167*** − 0.125*** − 0.189*** − 0.146*** − 0.198*** (0.019) (0.030) (0.025) (0.043) (0.041) $$\mathit {unusedcredit}_{it-1}$$ − 0.051*** − 0.055*** − 0.053*** − 0.137*** − 0.105*** (0.010) (0.016) (0.014) (0.037) (0.027) tobinQit 0.029*** 0.005 0.036*** 0.025** 0.035*** (0.005) (0.007) (0.006) (0.013) (0.013) $$\mathit {cashflow}_{it}$$ 0.162*** 0.234*** 0.172*** 0.049 0.102 (0.044) (0.067) (0.061) (0.101) (0.094) Firm fixed effects Yes Yes Yes Yes Yes Year dummies Yes Yes Yes Yes Yes Adjusted R2 0.28 0.27 0.27 0.33 0.37 Observation 12,113 5,013 7,100 3,106 4,024 Note: This table reports the regression results after sorting firms into positive and zero dividend payout group, high long-term debt maturing next year and low long-term debt maturing next year. Dividend payout firms are those firms with positive dividend payout. Firms with the high ratio of long-term debt maturing are those with higher than the median ratio of long-term debt maturing. Follow Almeida et al. (2012), the ratio of long-term debt maturing is defined as the percentage of long-term debt maturing to total long-term debt. In calculating the debt maturing ratio, we require firms to have long-term debt maturing beyond 1 year that represents at least 5% of assets. Thus, in the last group of regressions, the sample size is smaller. The first column shows the results of the baseline regression of equation (31). The next four columns report the results of regressions, separately for positive and zero dividend group, low debt maturing and high debt maturing firm group. The dependent variable is employment growth, $$\Delta \mathit {employ}_{it}$$, and independent variables include interaction between industry unionization rate and debt growth, $$\mathit {union}_{cic,t}\cdot \Delta \mathit {debt}_{it}$$, industry unionization rate $$\mathit {union}_{cic,t}$$, debt growth, $$\Delta \mathit {debt}_{it}$$, lagged financial leverage ratio $$\mathit {leverage}_{it-1}$$, lagged unused credit-to-total credit ratio, $$\mathit {unusedcredit}_{it-1}$$, market-to-book ratio, Qit, cash flow-to-asset ratio, $$\mathit {cashflow}_{it}$$. Firm fixed effects and year dummies are also included. Standard errors (in parentheses) are heteroskedasticity robust and clustered at the firm level, and significance levels at 1%, 5%, and 10% are marked with superscripts ***, **, *, respectively. View Large Table 10. Employment growth regression. Financial distress. Dividend payout Debt maturing Yes No Low High $$\mathit {union}_{cic,t}\cdot \Delta \mathit {debt}_{it}$$ 0.095*** 0.165*** 0.069 0.262** 0.101 (0.036) (0.056) (0.050) (0.127) (0.064) $$\mathit {union}_{cic,t}$$ − 0.023 0.029 − 0.078 0.042 − 0.127 (0.061) (0.066) (0.122) (0.125) (0.099) $$\Delta \mathit {debt}_{it}$$ 0.035*** 0.032*** 0.036*** 0.076*** 0.057*** (0.003) (0.006) (0.004) (0.017) (0.008) $$\mathit {leverage}_{it-1}$$ − 0.167*** − 0.125*** − 0.189*** − 0.146*** − 0.198*** (0.019) (0.030) (0.025) (0.043) (0.041) $$\mathit {unusedcredit}_{it-1}$$ − 0.051*** − 0.055*** − 0.053*** − 0.137*** − 0.105*** (0.010) (0.016) (0.014) (0.037) (0.027) tobinQit 0.029*** 0.005 0.036*** 0.025** 0.035*** (0.005) (0.007) (0.006) (0.013) (0.013) $$\mathit {cashflow}_{it}$$ 0.162*** 0.234*** 0.172*** 0.049 0.102 (0.044) (0.067) (0.061) (0.101) (0.094) Firm fixed effects Yes Yes Yes Yes Yes Year dummies Yes Yes Yes Yes Yes Adjusted R2 0.28 0.27 0.27 0.33 0.37 Observation 12,113 5,013 7,100 3,106 4,024 Dividend payout Debt maturing Yes No Low High $$\mathit {union}_{cic,t}\cdot \Delta \mathit {debt}_{it}$$ 0.095*** 0.165*** 0.069 0.262** 0.101 (0.036) (0.056) (0.050) (0.127) (0.064) $$\mathit {union}_{cic,t}$$ − 0.023 0.029 − 0.078 0.042 − 0.127 (0.061) (0.066) (0.122) (0.125) (0.099) $$\Delta \mathit {debt}_{it}$$ 0.035*** 0.032*** 0.036*** 0.076*** 0.057*** (0.003) (0.006) (0.004) (0.017) (0.008) $$\mathit {leverage}_{it-1}$$ − 0.167*** − 0.125*** − 0.189*** − 0.146*** − 0.198*** (0.019) (0.030) (0.025) (0.043) (0.041) $$\mathit {unusedcredit}_{it-1}$$ − 0.051*** − 0.055*** − 0.053*** − 0.137*** − 0.105*** (0.010) (0.016) (0.014) (0.037) (0.027) tobinQit 0.029*** 0.005 0.036*** 0.025** 0.035*** (0.005) (0.007) (0.006) (0.013) (0.013) $$\mathit {cashflow}_{it}$$ 0.162*** 0.234*** 0.172*** 0.049 0.102 (0.044) (0.067) (0.061) (0.101) (0.094) Firm fixed effects Yes Yes Yes Yes Yes Year dummies Yes Yes Yes Yes Yes Adjusted R2 0.28 0.27 0.27 0.33 0.37 Observation 12,113 5,013 7,100 3,106 4,024 Note: This table reports the regression results after sorting firms into positive and zero dividend payout group, high long-term debt maturing next year and low long-term debt maturing next year. Dividend payout firms are those firms with positive dividend payout. Firms with the high ratio of long-term debt maturing are those with higher than the median ratio of long-term debt maturing. Follow Almeida et al. (2012), the ratio of long-term debt maturing is defined as the percentage of long-term debt maturing to total long-term debt. In calculating the debt maturing ratio, we require firms to have long-term debt maturing beyond 1 year that represents at least 5% of assets. Thus, in the last group of regressions, the sample size is smaller. The first column shows the results of the baseline regression of equation (31). The next four columns report the results of regressions, separately for positive and zero dividend group, low debt maturing and high debt maturing firm group. The dependent variable is employment growth, $$\Delta \mathit {employ}_{it}$$, and independent variables include interaction between industry unionization rate and debt growth, $$\mathit {union}_{cic,t}\cdot \Delta \mathit {debt}_{it}$$, industry unionization rate $$\mathit {union}_{cic,t}$$, debt growth, $$\Delta \mathit {debt}_{it}$$, lagged financial leverage ratio $$\mathit {leverage}_{it-1}$$, lagged unused credit-to-total credit ratio, $$\mathit {unusedcredit}_{it-1}$$, market-to-book ratio, Qit, cash flow-to-asset ratio, $$\mathit {cashflow}_{it}$$. Firm fixed effects and year dummies are also included. Standard errors (in parentheses) are heteroskedasticity robust and clustered at the firm level, and significance levels at 1%, 5%, and 10% are marked with superscripts ***, **, *, respectively. View Large 5.5. Regression with Model Simulated Data To further evaluate the empirical significance of the structural model and relate it to the direct evidence of the bargaining channel of debt provided in Table 9, we estimate a regression equation on model simulated data. In the model, we know exactly the exogenous (independent) variables: credit shock ξt and productivity shock zt. Thus, to test the model prediction, we regress the endogenous variable “employment growth” on those two exogenous variables. We repeat the simulation and estimation for two values of the bargaining parameter: η = 0.355 and η = 0.444. The estimation results are reported in Table 11. Table 11. Employment growth regression. Using model simulated data. Bargaining share η = 0.355 η = 0.444 Credit shock ξit 0.1261*** 0.1610*** (0.0002) (0.0002) Productivity shock zit 0.4808*** 0.4266*** (0.0002) (0.0002) Adjusted R2 0.9940 0.9914 Observation 39,629 39,629 Bargaining share η = 0.355 η = 0.444 Credit shock ξit 0.1261*** 0.1610*** (0.0002) (0.0002) Productivity shock zit 0.4808*** 0.4266*** (0.0002) (0.0002) Adjusted R2 0.9940 0.9914 Observation 39,629 39,629 Note: This table reports the regression results using the model simulated data. We first simulate the model using the estimated bargaining share η = 0.355, and then simulate the model with a higher bargaining share η = 0.444. We report the regression results after sorting firms into high bargaining share and low bargaining share groups. The dependent variable is employment growth, $$\Delta \mathit {employ}_{it}$$. The two (exogenous) independent variables are credit shock ξit and productivity shock zit. Significance levels at 1%, 5%, and 10% are marked with superscripts ***, **, *, respectively. View Large Table 11. Employment growth regression. Using model simulated data. Bargaining share η = 0.355 η = 0.444 Credit shock ξit 0.1261*** 0.1610*** (0.0002) (0.0002) Productivity shock zit 0.4808*** 0.4266*** (0.0002) (0.0002) Adjusted R2 0.9940 0.9914 Observation 39,629 39,629 Bargaining share η = 0.355 η = 0.444 Credit shock ξit 0.1261*** 0.1610*** (0.0002) (0.0002) Productivity shock zit 0.4808*** 0.4266*** (0.0002) (0.0002) Adjusted R2 0.9940 0.9914 Observation 39,629 39,629 Note: This table reports the regression results using the model simulated data. We first simulate the model using the estimated bargaining share η = 0.355, and then simulate the model with a higher bargaining share η = 0.444. We report the regression results after sorting firms into high bargaining share and low bargaining share groups. The dependent variable is employment growth, $$\Delta \mathit {employ}_{it}$$. The two (exogenous) independent variables are credit shock ξit and productivity shock zit. Significance levels at 1%, 5%, and 10% are marked with superscripts ***, **, *, respectively. View Large As can be seen from the table, the sensitivity of employment growth to credit shock is higher for firms where workers have higher bargaining power η. In terms of economic magnitude, a one standard deviation credit shock leads to a 0.27 standard deviation increase in employment growth for the benchmark model when η = 0.355, and a 0.38 standard deviation increase in employment growth when η = 0.444. 6. Conclusion There is a well-established literature in corporate finance exploring the use of debt as a strategic mechanism to improve the bargaining position of firms with workers. Less attention has been devoted to studying whether this mechanism is also important for the hiring decision of firms. In this paper, we have investigated the theoretical and empirical relevance of this channel for the employment dynamics of firms. Using an estimated firm dynamics model, we have found that this mechanism contributes 13% to firm-level employment volatility. We have shown that the strength of the mechanism increases with the bargaining power of workers. This dependence is also supported by the estimation of reduced-form regressions. We have also found that credit shocks contribute 22% to firm-level employment volatility. Therefore, the bargaining channel of debt and shocks to credit may not be the primary factors explaining the volatility of employment at the firm level, but their contributions to this volatility are still quantitatively relevant. The bargaining channel of debt could also be important for the long-run dynamics of the firm. In particular, greater uncertainty about the firm’s access to credit could have sizable negative effects on its long-run growth. This and other related issue will be studied in future research. Notes The editor in charge of this paper was Dirk Krueger. Appendix A: Model without the Bargaining Channel of Debt As argued in Section 3.5, to isolate the bargaining channel of debt we redefine the bargaining value of the firm as $$\tilde{V}_t\equiv V_t+B_t+\varphi _t-(\bar{b}+\bar{\varphi })N_t$$. See equation (21). This value can be written recursively as \begin{equation*} \tilde{V}(\boldsymbol{x}_t,B_t,N_t) = \tilde{D}_t + \beta \mathbb{E}_t \tilde{V}(\boldsymbol{x}_{t+1},B_{t+1},N_{t+1}), \end{equation*} where \begin{eqnarray} \tilde{D}_t &=&(z_t - w_t) N_t -\Upsilon \left(\frac{H_t}{N_t}\right)N_t + (q_t-\beta )B_{t+1}-(1-\beta g_{t+1})(\bar{b}+\bar{\varphi })N_t \nonumber \\ &&-\,\beta \mathbb{E}_t\varphi \left(\boldsymbol{x}_{t+1},b_{t+1}\right)N_{t+1}. \end{eqnarray} (A1) The recursive formulation is obtained from equation (6) after adding Bt + φ(xt, bt)Nt and subtracting $$(\bar{b}+\bar{\varphi })N_t$$ on both sides of the equation and then using the definition of $$\tilde{V}(\boldsymbol{x}_t,B_t,N_t)$$. The net value for a worker, W(xt, Bt, Nt) − Ut, is the same as in the benchmark model and takes the form specified in equation (7). Given η the bargaining power of workers, the bargaining problem can be written as \begin{eqnarray*} \max _{w_t,D_t,E_t,B_{t+1}} && \bigg [\Big (W(\boldsymbol{x}_t,B_t,N_t)-U_t\Big ) N_t\bigg ]^{\eta }\cdot \tilde{V}(\boldsymbol{x}_t,B_t,N_t)^{1-\eta }, \end{eqnarray*} subject to equations (1), (2), and (A.1). Differentiating with respect to wt, we obtain that workers receive a fraction η of the bargaining surplus $$\tilde{S}(\boldsymbol{x}_t,B_t,N_t) = \tilde{V}(\boldsymbol{x}_t,B_t,N_t) + \Big (W(\boldsymbol{x}_t,B_t,N_t)-U_t\Big ) N_t$$. Next we derive the optimality conditions for Ht, Bt+1. They maximize the net surplus$$\tilde{S}(\boldsymbol{x}_t,B_t,N_t)$$ which, in recursive form, can be written as \begin{eqnarray*} \tilde{S}(\boldsymbol{x}_t,B_t,N_t)=&& \\ \max _{H_t,B_{t+1}} && \Bigg \lbrace \tilde{\Pi }_t + \beta \bigg [1-\eta + \eta (1-\lambda )\left(\frac{N_t}{N_{t+1}}\right)\bigg ]\, \mathbb{E}_t \tilde{S} (\boldsymbol{x}_{t+1},B_{t+1},N_{t+1})\Bigg \rbrace \\ \text{subj. to:} && \tilde{\Pi }_t = (z_t - u_t) N_t -\Upsilon \left(\frac{H_t}{N_t}\right)N_t+ (q_t-\beta )B_{t+1}\nonumber \\ && -\, (1-\beta g_{t+1})(\bar{b}+\bar{\varphi })N_t- \beta \varphi \left(\boldsymbol{x}_{t+1},b_{t+1}\right)N_{t+1}\\ && q_t B_{t+1} \le \xi _t \beta \mathbb{E}_t \Big [\tilde{S}(\boldsymbol{x}_{t+1},B_{t+1},N_{t+1})-B_{t+1} -\nonumber \\ && \varphi (\boldsymbol{x}_{t+1},b_{t+1})N_{t+1}+(\bar{b}+\bar{\varphi })N_{t+1}\Big ]. \end{eqnarray*} The enforcement constraint remains the same as in the benchmark model, and therefore, it depends on $$S(\boldsymbol{x}_{t+1},B_{t+1},N_{t+1})=\tilde{S}(\boldsymbol{x}_{t+1},B_{t+1},N_{t+1})-B_{t+1}-\varphi (\boldsymbol{x}_{t+1},b_{t+1})N_{t+1}+(\bar{b}+\bar{\varphi }) N_{t+1}$$. We can now normalize by Nt and rewrite the problem as \begin{eqnarray*} \tilde{s}(\boldsymbol{x}_t,b_t) =&&\\ \max _{h_{t},b_{t+1}} && \bigg \lbrace \tilde{\pi }_t + \beta (g_{t+1} - \eta h_t) \mathbb{E}_t \tilde{s}(\boldsymbol{x}_{t+1},b_{t+1})\bigg \rbrace \\ \text{subj. to:} && \tilde{\pi }_t=z_t-u_t-\Upsilon (h_t)+ g_{t+1}(q_t-\beta ) b_{t+1}\\ && -\,(1-\beta g_{t+1})(\bar{b}+\bar{\varphi }) -\beta g_{t+1} \mathbb{E}_t\varphi (\boldsymbol{x}_{t+1},b_{t+1}) \nonumber \\ &&\xi _t g_{t+1} \beta \mathbb{E}_t \Big [\tilde{s}(\boldsymbol{x}_{t+1},b_{t+1})-b_{t+1}-\varphi (\boldsymbol{x}_{t+1},b_{t+1})+\bar{b}+\bar{\varphi }\Big ]\\ &&\ge q_t g_{t+1}b_{t+1} g_{t+1}=1-\lambda +h_t. \nonumber \end{eqnarray*} The first-order conditions for ht and bt+1 are \begin{equation*} (q_t -\beta )b_{t+1}+\beta (\bar{b}+\bar{\varphi })+\beta (1-\eta )\mathbb{E}_t \tilde{s}(\boldsymbol{x}_{t+1},b_{t+1}) \!-\! \beta \mathbb{E}_t \varphi (\boldsymbol{x}_{t+1},b_{t+1}) \!\!=\!\! \Upsilon ^{\prime }(h_t), \end{equation*} \begin{equation*} g_{t+1}(q_t-\beta )+\beta (g_{t+1}-\eta h_t) \mathbb{E}_t\frac{\partial \tilde{s}_{t+1}(b_{t+1})}{\partial b_{t+1}} - \beta g_{t+1} \mathbb{E}_t\frac{\partial \varphi (\boldsymbol{x}_{t+1},b_{t+1})}{\partial b_{t+1}} \end{equation*} \begin{equation*} \quad +\, \mu _tg_{t+1}\left[\beta \xi _t \mathbb{E}_t\left(\frac{\partial \tilde{s}(\boldsymbol{x}_{t+1},b_{t+1})}{\partial b_{t+1}}-1-\frac{\partial \varphi (\boldsymbol{x}_{t+1},b_{t+1})}{\partial b_{t+1}}\right)-q_t \right] = 0, \end{equation*} where μt is the Lagrange multiplier for the enforcement constraint. The envelope condition returns $$\partial \tilde{s}(\boldsymbol{x}_t,b_t)/\partial b_t=0$$. This shows that the modified surplus does not depend on bt. We can then rewrite the first-order conditions as in equations (23) and (24). Appendix B: Simulated Method of Moments The estimation follows Lee and Ingram (1991) and Nikolov and Whited (2014) and it is based on the simulated method of moments. One issue is that the empirical data consist of a panel of heterogenous firms, whereas the artificial data are generated by simulating one firm over a number of periods. To keep consistency between empirical and simulated data, we demean each variable in the data before calculating the empirical moments. In calculating the autocorrelations, we use the first-difference estimator as suggested by Han and Phillips (2010). The weighting matrix is calculate using the influence function approach and the standard errors using the clustered moment covariance matrix (see Nikolov and Whited 2014). The estimation procedure consists of the following steps. For each firm i in the data, we calculate the mean of each variable, and then demean the variable. That is, $$\tilde{x}_{it}=x_{it}-\bar{x}_{it}$$, where $$\bar{x}_{it}$$ is the within-firm average of xit. The subscripts i and t identify, respectively, firm and year. There is one except that we do not demean the data before computing the empirical moment, that is, when we take the mean itself as one of our target moments. Also, for the autocorrelations, we use the first-difference estimator as in Han and Phillips (2010). We pool the time series of all firms together to form a new time series $$\lbrace \tilde{x}_{k}\rbrace$$, where k = 1, 2, …, K, and K = I * T is the total number of firm-year observations. We calculate the empirical moments using the new series $$\tilde{x}_{k}$$, denote by an M × 1 vector f(xk), where M is the number of target moments. We then use the model to generate a time series of S periods, denote by {ys}. We set S = 10K as suggested by Lee and Ingram (1991). We also calculate the model moments, denote by vector f(ys, θ), where θ is an N × 1 vector of estimated parameters. The estimator $$\hat{\theta }$$ is the solution to \begin{equation*} \underset{\theta }{\min }\Big [g(x)-g(y,\theta )\Big ]^{\prime } W \Big [g(x) - g(y,\theta )\Big ], \end{equation*} where $$g(x) = {1/K} \sum ^{K}_{k=1} f(x_{k})$$ and $$g(y, \theta) = {1/S} \sum ^{S}_{s=1} f(y_{s},\theta )$$ are the sample mean of the data and the model, respectively, and w is a weighting matrix. The M × M weighting matrix is given by w = [Σ(1 + K/S)]−1, where Σ is the M × M variance-covariance matrix calculated by covarying the influence function as in Nikolov and Whited (2014). Under mild regularity conditions, the limiting distribution of $$\hat{\theta }$$ is given by \begin{equation*} \sqrt{K} (\hat{\theta }-\theta ) \rightarrow N(0, V), \end{equation*} where $$V = (D \widehat{W} D^{\prime })^{-1}$$ and D΄ is the M × N gradient matrix defined as \begin{equation*} D^{\prime } = \frac{\partial g(y,\theta )}{\partial \theta ^{\prime }} \approx \frac{g(y,\theta +\Delta \theta ) - g(y,\theta -\Delta \theta )}{2\Delta \theta }. \end{equation*} Here, $$\widehat{W}$$ is the inverse of the clustered moment covariance matrix $$\widehat{\Sigma }$$. The t-Statistics for the ith estimator and for the jth moment difference are, respectively, \begin{equation*} t_{i} = \frac{\hat{\theta }_{i}}{\sqrt{\frac{V_{ii}}{K}}}, \qquad \qquad \qquad t_{j} = \frac{g_{j}(x)-g_{j}(y,\theta )}{\sqrt{\frac{\widehat{\Sigma }_{jj}}{K} (1+K/S)}}. \end{equation*} Appendix C: Variables Definition and Sources Table C1. Variables definition and sources. Variable Model Data Data sources Total debt (debtt) qtBt+1 Long-term debtt + Short-term debtt Compustat Total credit (creditit) $$\xi _t \beta \mathbb{E}_t S_{t+1}$$ Long-term debtt + Short-term debtt + Total undrawn creditt Compustat, Capital IQ Unused credit ratio (unusedcreditit) $$\frac{\xi _t \beta \mathbb{E}_t S_{t+1}- q_{t} B_{t+1}}{\xi _t \beta \mathbb{E}_t S_{t+1}}$$ Total undrawn creditt / Total creditt Compustat, Capital IQ Wage to revenue ratio (wagesharet) $$\frac{w_{t}}{F(z_{t})}$$ Waget / (Waget + Cash flowt) Compustat, BEA Leverage ratio (leveraget) $$\frac{B_{t+1}}{ (1-\eta ) S_{t} +B_{t+1}}$$ Total debtt / (Market value of equityt + Total debtt) Compustat Market to book ratio (tobinQit) $$\frac{ (1-\eta ) S_{t} +B_{t+1}}{N_{t+1}}$$ (Market value of equityt + Total debtt + Total debtt) / Book value of assetst Debt growth ($$\Delta \mathit {debt}_{it}$$) $$\frac{B_{t+1}}{B_{t}}$$ Total debtt / Total debtt–1 Compustat Employment growth ($$\Delta \mathit {employ}_{it}$$) $$\frac{N_{t+1}}{N_{t}}$$ Employeest / Employeest − 1 Compustat Sale growth ($$\Delta \mathit {sales}_{it}$$) $$\frac{z_{t}N_{t}}{z_{t-1} N_{t-1}}$$ Salest / Salest − 1 Compustat Cash flow ratio ($$\mathit {cashflow}_{it}$$) Operating income before depreciationt / Book value of assetst Compustat Unionization rate ($$\mathit {union}_{cic,t}$$) Employees covered by collective bargainingt / Total employeest Union membership and Coverage database Variable Model Data Data sources Total debt (debtt) qtBt+1 Long-term debtt + Short-term debtt Compustat Total credit (creditit) $$\xi _t \beta \mathbb{E}_t S_{t+1}$$ Long-term debtt + Short-term debtt + Total undrawn creditt Compustat, Capital IQ Unused credit ratio (unusedcreditit) $$\frac{\xi _t \beta \mathbb{E}_t S_{t+1}- q_{t} B_{t+1}}{\xi _t \beta \mathbb{E}_t S_{t+1}}$$ Total undrawn creditt / Total creditt Compustat, Capital IQ Wage to revenue ratio (wagesharet) $$\frac{w_{t}}{F(z_{t})}$$ Waget / (Waget + Cash flowt) Compustat, BEA Leverage ratio (leveraget) $$\frac{B_{t+1}}{ (1-\eta ) S_{t} +B_{t+1}}$$ Total debtt / (Market value of equityt + Total debtt) Compustat Market to book ratio (tobinQit) $$\frac{ (1-\eta ) S_{t} +B_{t+1}}{N_{t+1}}$$ (Market value of equityt + Total debtt + Total debtt) / Book value of assetst Debt growth ($$\Delta \mathit {debt}_{it}$$) $$\frac{B_{t+1}}{B_{t}}$$ Total debtt / Total debtt–1 Compustat Employment growth ($$\Delta \mathit {employ}_{it}$$) $$\frac{N_{t+1}}{N_{t}}$$ Employeest / Employeest − 1 Compustat Sale growth ($$\Delta \mathit {sales}_{it}$$) $$\frac{z_{t}N_{t}}{z_{t-1} N_{t-1}}$$ Salest / Salest − 1 Compustat Cash flow ratio ($$\mathit {cashflow}_{it}$$) Operating income before depreciationt / Book value of assetst Compustat Unionization rate ($$\mathit {union}_{cic,t}$$) Employees covered by collective bargainingt / Total employeest Union membership and Coverage database View Large Table C1. Variables definition and sources. Variable Model Data Data sources Total debt (debtt) qtBt+1 Long-term debtt + Short-term debtt Compustat Total credit (creditit) $$\xi _t \beta \mathbb{E}_t S_{t+1}$$ Long-term debtt + Short-term debtt + Total undrawn creditt Compustat, Capital IQ Unused credit ratio (unusedcreditit) $$\frac{\xi _t \beta \mathbb{E}_t S_{t+1}- q_{t} B_{t+1}}{\xi _t \beta \mathbb{E}_t S_{t+1}}$$ Total undrawn creditt / Total creditt Compustat, Capital IQ Wage to revenue ratio (wagesharet) $$\frac{w_{t}}{F(z_{t})}$$ Waget / (Waget + Cash flowt) Compustat, BEA Leverage ratio (leveraget) $$\frac{B_{t+1}}{ (1-\eta ) S_{t} +B_{t+1}}$$ Total debtt / (Market value of equityt + Total debtt) Compustat Market to book ratio (tobinQit) $$\frac{ (1-\eta ) S_{t} +B_{t+1}}{N_{t+1}}$$ (Market value of equityt + Total debtt + Total debtt) / Book value of assetst Debt growth ($$\Delta \mathit {debt}_{it}$$) $$\frac{B_{t+1}}{B_{t}}$$ Total debtt / Total debtt–1 Compustat Employment growth ($$\Delta \mathit {employ}_{it}$$) $$\frac{N_{t+1}}{N_{t}}$$ Employeest / Employeest − 1 Compustat Sale growth ($$\Delta \mathit {sales}_{it}$$) $$\frac{z_{t}N_{t}}{z_{t-1} N_{t-1}}$$ Salest / Salest − 1 Compustat Cash flow ratio ($$\mathit {cashflow}_{it}$$) Operating income before depreciationt / Book value of assetst Compustat Unionization rate ($$\mathit {union}_{cic,t}$$) Employees covered by collective bargainingt / Total employeest Union membership and Coverage database Variable Model Data Data sources Total debt (debtt) qtBt+1 Long-term debtt + Short-term debtt Compustat Total credit (creditit) $$\xi _t \beta \mathbb{E}_t S_{t+1}$$ Long-term debtt + Short-term debtt + Total undrawn creditt Compustat, Capital IQ Unused credit ratio (unusedcreditit) $$\frac{\xi _t \beta \mathbb{E}_t S_{t+1}- q_{t} B_{t+1}}{\xi _t \beta \mathbb{E}_t S_{t+1}}$$ Total undrawn creditt / Total creditt Compustat, Capital IQ Wage to revenue ratio (wagesharet) $$\frac{w_{t}}{F(z_{t})}$$ Waget / (Waget + Cash flowt) Compustat, BEA Leverage ratio (leveraget) $$\frac{B_{t+1}}{ (1-\eta ) S_{t} +B_{t+1}}$$ Total debtt / (Market value of equityt + Total debtt) Compustat Market to book ratio (tobinQit) $$\frac{ (1-\eta ) S_{t} +B_{t+1}}{N_{t+1}}$$ (Market value of equityt + Total debtt + Total debtt) / Book value of assetst Debt growth ($$\Delta \mathit {debt}_{it}$$) $$\frac{B_{t+1}}{B_{t}}$$ Total debtt / Total debtt–1 Compustat Employment growth ($$\Delta \mathit {employ}_{it}$$) $$\frac{N_{t+1}}{N_{t}}$$ Employeest / Employeest − 1 Compustat Sale growth ($$\Delta \mathit {sales}_{it}$$) $$\frac{z_{t}N_{t}}{z_{t-1} N_{t-1}}$$ Salest / Salest − 1 Compustat Cash flow ratio ($$\mathit {cashflow}_{it}$$) Operating income before depreciationt / Book value of assetst Compustat Unionization rate ($$\mathit {union}_{cic,t}$$) Employees covered by collective bargainingt / Total employeest Union membership and Coverage database View Large Footnotes 1 With tax deductibility of interests, the effective cost of debt is lower than the interest rate and the effective price of debt satisfies qt > β. We allow for a tax shield in the quantitative section. 2 As long as the exogenous shocks take a finite number of values, the numerical solution does not require any functional approximation, that is, we do not approximate the value function, policy functions or first-order conditions with special functional forms. 3 The typical approach in modeling the fiscal benefits of debt is to tax the net corporate income after the interest payments (e.g., Hennessy and Whited 2005 and Li, Whited, and Wu 2016). An alternative approach is to calculate the effective interest rate after the tax shield (e.g., Jermann and Quadrini 2012). Because the properties of the model remain unchanged, we use the second approach because it is simpler. 4 We would like thank an anonymous referee for suggesting we use Tobin’s Q. 5 The average value of Tobin’s Q is 1.36 in our sample, which is smaller than other values reported in the literature. This is because (i) we define the empirical Tobin’s Q from Compustat as $$(\mathit {dltt}+\mathit {dlc}+\mathit {csho}*\mathit {prcc}_f)/\mathit {at}$$ (other studies calculate Q as $$(\mathit {at}-\mathit {ceq}-\mathit {txdb} + \mathit {csho}*\mathit {prcc}_f)/\mathit {at}$$); (ii) we drop firms with employees less than 500; and (iii) we winsorize Q at 2.5% and 97.5% percentiles. 6 Using data from the BEA, we compute the industry wage rate as the ratio of the total compensation of employees, including wage and salary accruals and supplements to wages and salaries, to the total number of full time employees in the industry. 7 The average debt in the modified model is smaller than in the benchmark model, whereas the level of hiring is slightly higher. By eliminating the bargaining channel of debt, firms have lower incentive to borrow because higher debt does not reduce wages but increases the expected cost of financial distress. Still, firms continue to borrow because of the tax benefits of debt but the average debt is lower. By borrowing less, the expected cost of financial distress is lower and this increases the value of job creation. 8 The reason we chose a 0.93 standard deviation is because the credit shock has been discretized and this is the grid closest to 1 standard deviation. 9 Compustat uses the SIC, whereas the Union Membership and Coverage Database uses the CPS Industry Classifications (CIC). We are able to match the SIC code with the CIC code using North American Industry Classification System code. After matching the two data sets, we have 162 CIC industries. 10 One consideration that makes the use of the industry index a good proxy for the bargaining power of workers at the firm level is that labor mobility and competitive pressure tend to be higher within the industry rather than across industries. 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