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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 a