2.1 Productivity

Consider the first order condition for the static input $j$ choice:Footnote ⁵

(6) \begin{equation} \rho \alpha _j p_i y_i = \left (1+\mathbf I\{b_i\gt 0\} r_i + \lambda _i\right ) w_{j}X_{ij}. \end{equation}

Note how financial frictions distort this margin. When borrowing $b_i$ is positive, this yields a difference in prices relative to firms that do not borrow. Further, variation in the price $r_i$ can distort the margin even among borrowing firms. Finally, the shadow value of relaxing the constraint $\lambda _i$ also distorts margins among constrained firms. Re-arranging for output per unit of input $j$ :

(7) \begin{equation} OPX_{ij} = \frac {p_i y_i}{X_{ij}} = \left (1+\mathbf I \{b_i\gt 0\} r_i + \lambda _i \right ) \left (\frac {w_j}{\rho \alpha _j}\right ). \end{equation}

Combining across inputs gives an expression for revenue TFP:

(8) \begin{equation} TFPR_i = \frac {p_i y_i}{\prod _j X_{ij}} = \left (1+\mathbf I \{b_i\gt 0\} r_i + \lambda _i \right ) \prod _j \left (\frac {w_j}{\rho \alpha _j}\right )^{\alpha _j}. \end{equation}

It is immediately clear that financial frictions impact productivity measures both via interest rates and through the constraint represented by the multiplier $\lambda _i$ . Consider the following four cases.

1. 1. Unconstrained Firms: (denoted $unc$ ) $b_i=0$ and the desired expenditures are less than $n_i$ . Then, there is no need to borrow, the constraint does not bind ( $\lambda _i=0$ ), and there is no variation in TFPR or OPX with common input prices and production elasticities.

   (9) \begin{equation} OPX_{ij}^{unc} = \frac {p_i y_i}{X_{ij}} = \left (\frac {w_j}{\rho \alpha _j}\right ), \end{equation}
   (10) \begin{equation} TFPR_i^{unc} = \frac {p_i y_i}{\prod _j X_{ij}} = \frac {1}{\rho } \prod _j \left (\frac {w_j}{\alpha _j}\right )^{\alpha _j}. \end{equation}
   These are (7) and (8) with $b_i=0$ and $\lambda _i=0$ . There is no borrowing or leverage.

2. 2. Internally Constrained Firms: (denoted $ic$ ) Faced with wages $w_i$ , suppose desired expenditures are greater than $n_i$ . However, when $b_i\gt 0$ , the firm must also pay finance cost $r_i$ . Inclusive of this cost, firms demand less than $n_i$ . In this case, the wage bill is equal to its wealth and $b_i=0$ (no leverage). So, $\sum _j w_j X_{ij}=n_i$ , and productivity dispersion is

   (11) \begin{equation} OPX_{ij}^{ic} = \frac {\left (\frac {w_j}{\alpha _j}\right ) Y^{1-\rho }z_i^{\rho }\theta _i^{\rho } \prod _k\left (\frac {\alpha _k}{w_k}\right )^{\rho \alpha _k}}{n_i^{1-\rho }}, \end{equation}
   (12) \begin{equation} TFPR_i^{ic} = \frac {Y^{1-\rho }z_i^{\rho }\theta _i^{\rho } \prod _j \left (\frac {w_j}{\alpha _j}\right )^{\left (1-\rho \right )\alpha _j}}{n_i^{1-\rho }}. \end{equation}
   I denote $log(TFPR)$ as $tfpr$ and $log(OPX)$ as $opx$ . Dispersion in (demeaned) log productivity among internally constrained firms is given by:
   (13) \begin{equation} \left (\sigma _{tfpr}^{ic}\right )^2=\left (\sigma _{opx}^{ic}\right )^2= \rho ^2 \left (\sigma _z^2 + \sigma _{\theta }^2\right ) +\left (\rho -1\right )^2 (\sigma _n^{ic})^2 -2\rho \left (1-\rho \right )cov\left (ln\left (z_i\right ),ln\left (n_i\right )\right ), \end{equation}
   where $\sigma _{z}^2$ is the variance of (log) productivity, $\sigma _{\theta }^2$ is the variance of the (log) demand shifter, $(\sigma _n^{ic})^2$ is the variance of (log) net wealth, and $\rho$ is the demand elasticity parameter.

3. 3. Externally Dependent Firms: (denoted $dep$ ) Input demand exceeds wealth $n_i$ , so $b_i\gt 0$ , but they borrow less than the constraint. So $n_i \lt \sum _j w_j X_{ij} \lt \overline {py}(z)+ n_i$ . Here, TFPR and OPX are as in (7) and (8) with $b_i\gt 0$ and $\lambda _i=0$ since the constraint does not bind:

   (14) \begin{equation} OPX_{ij}^{dep} = \frac {p_i y_i}{X_{ij}} = \left (1+r_i \right ) \left (\frac {w_j}{\alpha _j}\right ), \end{equation}
   (15) \begin{equation} TFPR_i^{dep} = \left (1+r_i\right )\prod _j \left (\frac {w_j}{\alpha _j}\right )^{\alpha _j}. \end{equation}
   Dispersion in (demeaned) log productivity among externally dependent firms is given by:
   (16) \begin{equation} \left (\sigma _{tfpr}^{dep}\right )^2=\left (\sigma _{opx}^{dep}\right )^2 =\sigma _r^2, \end{equation}
   where $\sigma _r^2$ is the variance of the gross interest rate $ln(1+r_i)\approx r_i$ . Leverage is given by:
   (17) \begin{equation} \phi _i^{dep} = \frac {b_i}{n_i} = \frac {\sum _j w_jX_{ij}-n_i}{n_i}= \frac {\sum _j w_jX_{ij}}{n_i}-1= \frac {\frac {\left (\rho Y^{1-\rho } z_i^{\rho } \theta _i^{\rho } \prod _j \left (\frac {\alpha _j}{w_j}\right )^{\rho \alpha _j}\right )^{\frac {1}{1-\rho }}}{\left (1+r_i\right )^{\frac {1}{1-\rho }}}}{n_i}-1. \end{equation}
   Here, leverage is increasing in productivity and decreasing in the interest rate.

4. 4. Externally Constrained Firms: (denoted $ec$ ) Suppose that $b_i\gt 0$ , but now the constraint binds. In this case, productivity is given by:

   (18) \begin{equation} OPX_{ij}^{ec} = \frac {w_j}{\alpha _j}\prod _k \left (\frac {\alpha _k}{w_k}\right )^{\rho \alpha _k}Y^{1-\rho } z_i^{\rho } \theta _i^{\rho } \left (\phi _i^{ec}+1\right )^{\rho -1}n_i^{\rho -1}, \end{equation}
   (19) \begin{equation} TFPR_i^{ec} = \prod _j \left (\frac {w_j}{\alpha _j}\right )^{\alpha _j \left (1-\rho \right )}Y^{1-\rho } z_i^{\rho } \theta _i^{\rho } \left (\phi _i^{ec}+1\right )^{\rho -1}n_i^{\rho -1}. \end{equation}
   Thus one can write (demeaned) log productivity dispersion among these firms as:Footnote ⁶
   (20) \begin{align} \left (\sigma _{tfpr}^{ec}\right )^2&= \left (\sigma _{opx}^{ec}\right )^2= \rho ^2 \left (\sigma _z^2 + \sigma _{\theta }^2\right ) + \left (\rho -1\right )^2\left (\left (\sigma ^{ec}_{\phi }\right )^2 +\left (\sigma ^{ec}_n\right )^2\right ) \nonumber \\&\quad-2\rho \left (1-\rho \right )cov\left (ln\left (z_i\right ),ln\left (n_i\right )\right ) -2\rho \left (1-\rho \right )cov\left (ln\left (z_i\right ),ln\left (1+\phi _i^{ec}\right )\right ) \nonumber\\ &\quad-\left (\rho -1\right )^2 cov\left (ln\left (1+\phi _i^{ec}\right ),ln\left (n_i\right )\right ), \end{align}
   where $\sigma ^{ec}_{\phi }$ is the standard deviation of the leverage term $1+\phi _i^{ed}$ and leverage is given by
   (21) \begin{equation} \phi _i^{ec} = \frac {\overline {py}(z_i)}{n_i}. \end{equation}

2.2 Productivity dispersion decomposition

Given a distribution of firms across productivity and wealth dimensions, it is possible to show how variance in log productivity can be decomposed according to a standard within/between variance decomposition.Footnote ⁷ For example, variance in log TFPR $\sigma ^2_{tfpr}$ can be expressed as:

(22) \begin{equation} \sigma ^2_{tfpr}= \underbrace {\mu _{unc} \left (\sigma ^{unc}_{tfpr}\right )^2}_{\text {unconstrained disp. }=0} + \underbrace {\mu _{ic}\left (\sigma _{tfpr}^{ic}\right )^2}_{\text {intern. const. disp.}}+ \underbrace {\mu _{dep}\left (\sigma _{tfpr}^{dep}\right )^2}_{\text {fin. dependent disp.}} +\underbrace {\mu _{con} \left (\sigma ^{con}_{tfpr}\right )^2}_{\text {ext. const. disp. }} \end{equation}

\begin{equation*} + \underbrace {\mu _{unc}\left (\overline {tfpr}^{unc}-\overline {tfpr}\right )^2+\mu _{ic}\left (\overline {tfpr}^{ic}-\overline {tfpr}\right )^2+\mu _{dep}\left (\overline {tfpr}^{dep}-\overline {tfpr}\right )^2+\mu _{con}\left (\overline {tfpr}^{ec}-\overline {tfpr}\right )^2}_{\text {between-group dispersion}}. \end{equation*}

Defining $\mu _{s}$ as the mass of firms in group $s$ where $s\in \{unc, ic, dep, ec\}$ , $\overline {tfpr}^{s}$ as the average log labor productivity dispersion among group $s$ , and $\overline {tfpr}$ as the overall average log TFPR, (22) shows total dispersion is composed of two terms. First, there is the weighted average of log TFPR dispersion within groups that are defined by how financial constraints bind. This is the “within-group dispersion” contribution to overall productivity dispersion. Second, there is the weighted sum of squared differences between average group productivity and overall mean productivity. This represents the contribution of “between-group” differences to total dispersion.

2.3 Leverage and dispersion: taking stock

Equation (22) summarizes the effect of financial frictions on dispersion, and presents a complex relationship between leverage and dispersion. First, note there is no $tfpr$ or $opx$ dispersion without financial frictions in this simple model. If firms internally finance, then average revenue products are equal to marginal revenue products, which are in turn equalized across plants.Footnote ⁸

Nonetheless, the importance of financial frictions for dispersion does not imply a simple monotonic relationship between dispersion and leverage across the four categories discussed. Internally constrained firms have zero leverage, for example, but can exhibit substantial dispersion. On the other hand, both externally dependent and externally constrained firms exhibit more leverage than internally constrained firms, but may or may not exhibit more dispersion. Dispersion among externally dependent firms is dependent on interest rates, while externally dependent firms may not exhibit more dispersion than internally constrained firms.Footnote ⁹

Further, the relationship between productivity and leverage varies systematically only within the externally constrained firms. That is, if all firms were externally dependent, internally constrained, or unconstrained, there would be no relationship between leverage and dispersion. Furthermore, it is not clear that productivity dispersion increases with leverage among externally constrained firms. In fact, tighter constraints push dispersion and leverage in opposite directions.

In sum, it is not apparent that within-group dispersion, the focus of the first four terms in (22), are correlated with leverage in any systematic way. Likewise, comparing across firm types does not make plain the relationship between leverage and dispersion. So, it is an empirical and quantitative question. What does the relationship between leverage and dispersion look like in the data, and what modifications to the simple framework do we need to be consistent with the data?

4.1 Empirical results

Table 1 presents a regression table with the financial indicators plotted in Figures 1 and 2 as the regressors in (23) for $h=0$ , and the standard deviation in log TFPR and log OPH as the outcome variable. These results show a robust relationship between current liabilities (including trade credit) and productivity dispersion. Quantitatively, a five percentage point increase (roughly the interquartile range across industries in Figure 2) is associated with about half a percentage point increase in dispersion, with a slightly higher point estimate for labor productivity.

Further, the coefficient on inventories is generally significant and negative throughout the analysis. For log TFPR, the statistically significant coefficient implies a 2.2 percentage point decrease in dispersion for a 20 percentage point decline in inventories to sales (roughly the interquartile range across industries in Figure 2). On the one hand, this is perhaps a bit contrary to expectations, since industries with low turnover and higher inventories relative to sales are thought to be potentially more constrained. On the other hand, there are plausible reasons inventories might actually loosen financial constraints. First, the existence of substantial inventory management facilities and procedures allows plants in these industries to smooth production. The numerator in TFPR is adjusted for inventories and is therefore a measure of production and not necessarily sales. So, the presence of large inventories would have to correlate with production decisions that are less volatile. Second, inventories may be valuable as collateral in and of themselves, and therefore may help to reduce financial constraints and, as the model suggests, dispersion. Caglio et al. (Reference Caglio, Darst and Kalemli-Özcan2021) show the relative importance of “accounts receivable and inventories” (AR&I) lending, suggesting that inventories may serve an important role in securing credit.

Considering the significance of the results on short-term debt, it is puzzling that long-term debt does not show any statistically significant relationship with dispersion. While indicating a response of 1 percent for a 10 percentage point increase in long-term leverage (roughly in line with the typical interquartile range in Figure 2), the coefficient is not precisely estimated. As explored below, Figure 3 shows that alternative measures of dispersion, however, are responsive to long-term leverage. One explanation for the lack of significant results could be that changes in long-term debt (relative to the long-run industry mean) may take time to manifest in TFPR or OPH dispersion. For example, if debt increases as a result of capital expenditures that take time to build, then the resulting effects on productivity may not manifest until subsequent periods. A second explanation is that standard deviations are sensitive to outlier observations, while other measures like interquartile ranges are less sensitive. In Section 4.2 I explore the dynamic response by considering further lags of the outcome variable after the initial change in leverage, and in Section 4.3 and online Appendix C I explore the relationship between leverage and tail dispersion.

Finally, I note that cash holdings do not appear to be significantly related to productivity dispersion. This differs somewhat from recent literature exploring the role of leverage and liquidity in determining firm responsiveness (see Ottonello and Winberry, Reference Ottonello and Winberry2020; Jeenas, Reference Jeenas2024). There are potential explanations for this result. First, the previous literature generally uses publicly listed firms, while the DiSP data includes large and small, private and public. However, it is also possible that there are offsetting facts here: liquidity could be indicative of relaxed financial constraints, but it could also indicate additional uncertainty faced by firms. The general increase in cash holdings seen in Figure 1 (a) is consistent with increased liquidity needs in the face of higher uncertainty (Bloom, Reference Bloom2009). Still, it seems likely that leverage and liquidity interact in ways that are not picked up by this regression at this level of aggregation.

4.2 Empirical results: dynamic response

I now turn to the lagged relationship between changes in leverage and dispersion. In the simple model presented in Section 2, financial frictions can impact dispersion dynamically through the accumulation of net wealth. In other words, to the extent that financial frictions impact net wealth, this will interact with frictions in future periods and therefore create the potential for persistent effects. However,there is no distinction between dispersion in log TFPR and log output per unit input (OPH) in the model. This may not be the case. On the one hand, if financial constraints impact input margins uniformly, one would expect similar patterns to show up with log output per hour as an outcome variable. On the other hand, if other frictions or distortions (and even measurement issues) impact input margins differently, then there is no reason for these results to look the same. In what follows I consider the dynamic relationship between dispersion and leverage for each measure, exploring the differences across TFPR and OPH.

Figure 3. IRF: Productivity Dispersion correlation to change inlong-term leverage at year 0.

Notes: Lines indicate point estimates at each time horizon $h=0,\ldots 6$ , shaded areas represent $\pm 1.96*SE$ . Period zero corresponds to Table 1 (specifications 1–4). Regression includes time and industry fixed effects and controls for $log(Sales)$ and $log(Assets)$ . Annual data cover the period from 2001–2020. Financial Data Source: QFR US Census Bureau (2001–2020b). Productivity Data: DiSP database US Census Bureau (2001–2020b). Robust Standard Errors.

In Figure 4, I find that for TFPR (panels (a) through (c)), the coefficient on current liabilities is significantly positive for one to three periods, depending on the measure, with the highest estimate usually in the first period (the one exception being the IQR). This is consistent with pressing funding needs leading to immediate impacts that die out relatively quickly. Similarly, the largest log OPH response to current liabilities is generally in the first period, and yet the response is seemingly more persistent (although the coefficient for IQR is less precise). It might be that current financing needs are more persistently felt on the labor margin than on other inputs.

Figure 4. IRF: Productivity Dispersion correlation to Change in Current Liability Leverage at year 0.

Notes: Lines indicate point estimates at each time horizon $h=0,\ldots 6$ , shaded areas represent $\pm 1.96*SE$ . Period zero corresponds to Table 1 (specifications 1–4). Regression includes time and industry fixed effects and controls for $log(Sales)$ and $log(Assets)$ . Annual data cover the period from 2001–2020. Financial Data Source: QFR US Census Bureau (2001–2020b). Productivity Data: DiSP database US Census Bureau (2001–2020b). Robust Standard Errors.

Although the coefficient on long-term leverage is not significant at any horizon for the standard deviation, Figure 3 shows evidence of a positive relationship between leverage and the interquartile and interdecile ranges for TFPR. Curiously, these effects are not any more persistent than short-term leverage. Labor productivity does not appear to be notably impacted by long-term debt at any horizon or for any measure of dispersion. This stands in stark contrast to the results for short-term debt, where similar results hold for both TFPR and OPH, suggesting that the labor margin might be differentially impacted by short-term vs. long-term debt. In the detailed model I construct in the next section, I include a role for long-term and short-term debt, which varies by input to explore whether such mechanisms can generate similar patterns.

The coefficient on inventories is negative, significant, and persistent across measures of productivity and all concepts of dispersion. This robust finding suggests inventories have a significant role to play in determining dispersion, although it should be noted that the point estimates are relatively small. Such evidence is consistent with the findings in Caglio et al. (Reference Caglio, Darst and Kalemli-Özcan2021) that inventories support financing, possibly by serving as collateral that is easily valued.

4.3 Empirical results: tail dispersion

A measure not discussed in the motivational framework is the extreme tails of the distribution. However, to the extent tail dispersion measures capture “distance to the frontier,” or how far even highly productive firms are away from the efficient frontier, then they may be of interest in understanding the potential gains from reallocation. Furthermore, the motivational framework will predict increased dispersion is associated with an increase in skewness. For example, if there is no dispersion in interest rates, the lower bound of TFPR is the value for unconstrained firms, suggesting that there may be little dispersion in the lower tail. The binding constraint, however, creates both higher measured productivity and dispersion among constrained firms, potentially affecting the upper tail.

I discuss the results from these exercises in Appendix C. Overall, there are relatively weak relationships between tail dispersion and leverage, with the possible exception of upper tail labor productivity dispersion and short-term leverage.

5.1 Final output

Output is aggregated across a continuum of intermediate goods by a final goods producer that operates a CES technology:

\begin{equation*}Y = \left (\int _0^1 y_i^{\rho } di\right )^{\frac {1}{\rho }},\end{equation*}

Where $y_i$ is output from firm $i$ and $\rho$ parameterizes the returns to scale for each intermediate firm. The model has the standard pricing solution for the intermediate good:

\begin{equation*}p_i = \left (\frac {Y}{y_i}\right )^{1-\rho }.\end{equation*}

5.2 Intermediate firms

The value function for a firm with state variables $z_i$ (productivity), $k_i$ (capital), $b_i$ (intertemporal debt/asset), and location indicator ( $I_i$ ):

\begin{align*} V\left (z_i,k_i,b_i,I_i\right )&=\max _{k_i',b_i',h_i,m_i} p_i y_i - wh_{i} - m_i -r\left (I_i\right )\omega \left (h_i\right ) +\left (1+ r\left (I_i\right )\right ) b_i -b_i' -C\left (k_i',k_i\right )\nonumber\\[-3pt] &\quad+ \beta \zeta E\left [V\left (z_i', k_i',b_i',I_i'\right )\right ] \end{align*}

(25) \begin{equation} \text {s.t.} \quad \quad p_i = \left (\frac {Y}{y_i}\right )^{1-\rho }, \quad \quad y_i = z_i k_i^{\alpha _k}h_i^{\alpha _{\ell }}m_i^{1-\alpha _k-\alpha _{\ell }}, \end{equation}

(26) \begin{equation} \omega \left (h_i\right ) = \max (wh_i - \max (b_i,0),0), \quad \quad \omega \left (h_i\right )\leq \max (b_i,0)+\xi \left ( k_i + Y\right ), \end{equation}

(27) \begin{equation} b_i' \geq -\xi \left (k_i'+E\left [V\left (z_i', k_i', b_i',I_i'\right )\right ]\right ), \end{equation}

(28) \begin{equation} C\left (k_i',k_i\right ) = k_i'-\left (1-\delta \right )k_i + \frac {\chi }{2k_i}\left (k_i'-\left (1-\delta \right )k_i\right )^2, \end{equation}

(29) \begin{equation} ln\left (z_i'\right ) = \rho _z ln\left (z_i\right )+ \varepsilon _{i}', \quad \quad \varepsilon _i' \sim \mathcal {N}\left (0,\sigma _{\varepsilon }\right ), \end{equation}

where the firm faces wage $w$ and interest rate $r(I_i)$ , the latter of which depends on the firm’s location. The equations in (25), as before, give demand (pricing) and production for the firm providing good $i$ . Working capital $\omega (h_i)$ is defined in (26) as the difference between labor demand and internal funds (if available), and faces a constraint governed by the parameter $\xi$ . This intratemporal borrowing constraint depends on both owned capital and average sales.Footnote ¹⁶ Intertemporal borrowing is likewise constrained, not just by current capital, but also the future profits of the firm captured by the value function appearing on the right-hand side of (27). Capital accumulation is also subject to quadratic adjustment costs described in (28). Productivity follows an AR(1) process described in (29).

I collapse productivity variation to $z_i$ , since $\theta _i$ and $z_i$ are isomorphic. The log of productivity $log\left (z_i\right )$ evolves according to an AR(1) process. Second, I include a “finance island” indicator $I_i$ as (exogenous) state variables specific to the firm, since interest rates now vary across a discrete number of islands. Firms cannot move islands.

The first financial constraint applies to labor directly, limiting “working capital” $\omega \left (h_i\right )$ . It depends on assets $k_i$ and $b_i$ and expected sales from the lenders perspective. For simplicity, I assume that lenders have no special information about sales, and expect the average (aggregate) output to be the firm’s output.Footnote ¹⁷ It is a “working capital” constraint in that firms can make labor decisions within the period after realizing productivity shocks, but they must pay workers in advance. They can use liquid savings, but the difference between wage bill and liquid savings $b_i\gt 0$ must be financed by “working capital,” or intraperiod loans. Firms prefer to self-finance, but if the firm has no savings or negative savings, $b_i\leq 0$ , then the entire wage bill must be financed, which comes at a cost. For firms that can only self-finance part of the bill, they incur interest costs on only the portion they borrow (thus $r\left (I_i\right )\omega \left (h_i\right )$ shows up in the flow costs of the value function). The timing within period works as follows: firms realize productivity draws, choose labor and intermediate inputs, use savings and/or borrow the funds necessary to pay workers up front, produce, pay materials bill, pay back intratemporal loans, then make portfolio allocation decisions for the next period.

The intertemporal portfolio decision is likewise constrained, as firms can only borrow (i.e., $b_i\lt 0$ ) up to some fraction of their capital and the expected value of the franchise in the next period. Therefore, capital is generally more closely associated with long-term debt, while labor is more closely associated with short-term borrowing. Additionally, I include quadratic adjustment costs, captured by the third term of the investment cost function $C(k_i',k_i)$ to better reflect the (lack of) flexibility in investment plans seen in the data (Cooper and Haltiwanger, Reference Cooper and Haltiwanger2006).

5.3 Entry and exit dynamics

I specify exogenous entry and exit, with a fixed fraction $1-\zeta$ exiting and entering each period. Thus, firms discount the future by $\zeta$ to account for the exit probability. Entering firms receive capital $k_{ent}= c_eK$ where $0\lt c_e\lt 1$ to capture the fact that entering firms have lower capital. I set $c_e=0.1$ for the exercises below. Entering firms have no liquid savings (or debt) upon entry.

5.4 Equilibrium

As noted, the model is partial equilibrium in that credit supply is perfectly elastic at the interest rate consistent with discount factor $\beta$ . Intermediate inputs are assumed to have the same price as output, reflecting the “roundabout production” approach found in Bils et al. (Reference Bils, Klenow and Ruane2021). Wages are set such that labor demand from firms is equal to inelastic labor $\bar {H}$ . That is,

\begin{equation*}\int _0^1 h_i^*di=\bar {H},\end{equation*}

where $h_i^*$ is the optimal solution to firm’s problem for labor demand problem in 5.2.

5.5 Parameters

Table 2 gives parameters for the calibration of the model. I use input share parameters $\alpha _{\ell }$ and $\alpha _k$ consistent with data from the ASM (see Blackwood et al., Reference Blackwood, Haltiwanger and Wolf2024). Wages are determined in equilibrium by equating labor demand to labor supply set to a standard EPOP ratio. Intermediate good prices are equal to that of output (normalized to one).Footnote ¹⁸ Discount factor $\beta$ and revenue curvature parameter $\rho$ are standard. Exit rate (and entry rate) parameter $\zeta$ is set to imply 8% entry rates based on data from the mid-2000s in the US Census Bureau’s Business Dynamics Statistics. Productivity persistence $\rho _z$ has been much debated, and in these exercises it is set to a reasonable value within the discussed range in the literature (see Asker et al., Reference Asker, Collard-Wexler and De Loecker2014); Midrigan and Xu, Reference Midrigan and Xu2014; Moll, Reference Moll2014). Additionally, I specify an interest rate range of 150 basis points that roughly corresponds to the standard deviation of corporate spreads discussed in Gilchrist et al. (Reference Gilchrist, Sim and Zakrajšek2013).

Table 2. Calibration

The remaining parameters, variance in productivity shocks $\sigma _e$ , collateral constraint parameter  $\xi$ , and adjustment cost parameter $\chi$ , are chosen to reasonably capture some key moments in the data. In particular, TFPR dispersion,corporate leverage, and the standard deviation of investment.

Table 3 gives some steady state moments targeted by the model. The model is able to reproduce one key aspect of the US manufacturing data: dispersion in labor productivity (OPH) is larger than dispersion in TFPR. In the model, this is largely due to the fact that labor is distorted, while materials is an undistorted margin. Even though capital is typically thought to be “more distorted”, dispersion in (log) TFPR is a weighted linear combination of dispersion in average revenue products of each input (and covariances). Thus, the lack of variance in average revenue product of materials (revenue over materials) drags TFPR dispersion downward relative to labor productivity.

Table 3. Moments

Still, the model does not produce the same level of dispersion in either TFPR or OPH as in the data. This is not entirely surprising, since there are many candidate mechanisms for measured TFPR/OPH dispersion in the literature, which I do not incorporate here aside from capital adjustment costs. Gopinath et al. (Reference Gopinath, Kalemli-Özcan, Karabarbounis and Villegas-Sanchez2017) note a similar phenomenon, and discuss how substantial additional frictions are needed in a model extension to match levels of dispersion. However, responsiveness to changes in financial frictions, the focus of their paper and this paper, is not sensitive to these additional mechanisms.

Leverage is very close to the data, and the standard deviation of investment is mildly higher than what is found in Cooper and Haltiwanger (Reference Cooper and Haltiwanger2006). With these results in hand, I apply the Hsieh and Klenow (Reference Hsieh and Klenow2009) framework to the distribution of firms in the model to obtain a measure of misallocation. That is, I take the invariant distribution as “data” and derive the implied measure of misallocation a researcher would find if they estimated distortions as in the Bils et al. (Reference Bils, Klenow and Ruane2021) variant of the model. It is somewhat unsurprising that the model economy is relatively “efficient” compared to the data given the relatively low dispersion. Allocative efficiency is high relative to the long-run average of roughly 75% in the data, but within the historical bounds reported in the literature (Bils et al., Reference Bils, Klenow and Ruane2021; Blackwood et al., Reference Blackwood, Foster, Grim, Haltiwanger and Wolf2021).

The model also generates a substantial amount of constrained firms-20.7% of firms in the model are externally constrained. The majority (73.5%) are externally dependent, while a small share (5%) are unconstrained. Less than 1% are internally constrained.Footnote ¹⁹ Ultimately, external finance is relevant for a large majority of firms (roughly 94.5%), but most are externally dependent. One reason dispersion is relatively low in this model is that dispersion in interest rates is not significant enough to generate substantial dispersion, as noted in Gilchrist et al. (Reference Gilchrist, Sim and Zakrajšek2014).

5.6 Exercises

What mechanisms might generate the positive correlations seen between leverage and productivity dispersion? I now take the model outlined above and explore the implications of credit frictions for both productivity dispersion and leverage. I do so by finding a baseline invariant distribution in steady state. I then consider simple shocks that decay within 10 periods and study the dynamic response of dispersion and misallocation, along with other aggregates of interest. I consider three shocks: an aggregate productivity shock, an interest rate shock, and a collateral constraint shock. Shocks are entirely unanticipated, and then the path of the shock, prices, and aggregates are known with perfect foresight immediately upon realization of the shock (i.e., an “MIT shock”). I then take the increase in (short-term and long-term) leverage from the model and estimate empirical responses of dispersion based on the results in Section 4, along with associated confidence intervals.

5.6.3 Tighter credit constraints

To simulate a change in credit constraints, I specify a tightening in the parameter $\xi$ –similar to “credit crunches” explored in the macro-finance literature (e.g., Khan and Thomas, Reference Khan and Thomas2013). The shock is picked to generate a decline in leverage similar to the decline in debt to assets in the corporate sector seen in the data during the Great Recession. In the context of the empirical results, it is possible that more constrained industries hit up against tighter constraints, which generates more dispersion. On the other hand, tighter constraints reduce the leverage of constrained firms, and potentially cause more firms to become constrained. This leans toward reducing leverage, contradicting the empirical findings of a positive correlation. However, the endogenous response of the invariant distribution could work to offset both of these effects.

Figure 5. Aggregate Responses to Productivity, Interest Rate and Collateral Constraint Shocks.

Notes: Lines indicate point estimates at each time horizon $h=0,\ldots 6$ , shaded areas represent $\pm 1.96*SE$ . Period zero corresponds to Table 1 (specifications 1–4). Regression includes time and industry fixed effects and controls for $log(Sales)$ and $log(Assets)$ . Annual data cover the period from 2001–2020. Financial Data Source: QFR US Census Bureau (2001–2020b). Productivity Data: DiSP database US Census Bureau (2001–2020b). Robust Standard Errors.

5.7 Results

Figure 5 shows that the relationship between these shocks and aggregates generally follows intuition. In response to productivity and interest rate shocks (the first two columns), short-term leverage increases upon impact, and so does dispersion in TFPR and OPH. Long-term leverage cannot move on impact (due to time-to-build and the definition of interperiod borrowing), and so increases with a delay in the interest rate shock experiment, as expected. Notably, long-term leverage decreases in response to a productivity shock, as firms immediately generate revenue that allows them to internally finance, leading to lower intertemporal leverage. In the collateral constraint shock experiment, short-term and especially long-term leverage decline, as anticipated, while dispersion rises.

Figure 6. Productivity Shock: Relationship between Leverage and Dispersion.

Notes: Deviations from Steady state, in percentage points, to shock in $t+h$ . Short-term Debt is “working capital” or intraperiod financing. Long-term Debt is interperiod borrowing. Each column corresponds to the shock depicted in row 1.

The response of productivity dispersion in both the productivity shock and collateral constraint experiments is short-lived, falling back to (or below) the initial level by the second period. Thus, the persistent responses in OPH dispersion seen in the empirical results are only accounted for in the interest rate model. Intuitively, a productivity shock generates some demand for credit, but it also creates immediate sources of internal financing, leading to constraints that do not bind as tightly. Perhaps a shock that builds then dissipates would generate more persistent responses. On the other hand, interest rate shocks induce additional demand of external finance but provide no immediate internal financing boost. This leads to increased credit demand and binding constraints, and thus more dispersion. Furthermore, so long as interest rates are low today vs. tomorrow, firms pull forward investment plans in anticipation of higher future interest rates, so leverage remains higher for longer. Finally, tighter collateral constraints force firms to reduce leverage, but as the constraint begins to relax, firms have the opportunity to accumulate assets to help ease constraints further.

The bottom row gives the path of implied misallocation from Bils et al. (Reference Bils, Klenow and Ruane2021). The response of measured misallocation is an economically meaningful, if modest decline of 0.7 percentage points in the productivity shock experiment and 0.5 percentage points in the interest rate shock experiment. Intuitively, this means financial frictions generate a drag of more than half a percentage point on GDP growth in the model. The drag on growth is less substantial in the collateral constraint case (0.05 percentage points), although the shock is relatively small.

Overall, interest rates shocks are a promising candidate mechanism for generating similar responses to the data. I now make a more direct comparison to the empirics. First, I take the initial change in leverage in each experiment and treat it as a “shock” to leverage that I can use to generate implied predicted values from the estimated models in Section 4. That is, I take the estimated coefficient on leverage in 23 multiplied by the change in leverage in the model and generate predicted changes in dispersion along each time horizon $t+h$ .Footnote ²¹ For long-term leverage, the “leverage shock” is obtained from period $t+1$ since long-term leverage cannot change initially.

Figure 7. Interest Rate Shock: Relationship between Leverage and Dispersion.

Notes: Predicted values in response to deviation in leverage from model in Section 4.2 for horizons $t+h$ , shaded areas represent $\pm 1.96*SE$ . Response to initial period (t + 0) in the first period, period t + 1 for long-term leverage.

Figure 6 presents results from the productivity shock experiment. Responses are qualitatively similar, with TFPR and OPH increasing in response to short-term leverage. However, the response for TFPR is small while the response of OPH is larger than what is observed in the data. Likewise, both are relatively short-lived compared to the data. However, it is possible that persistent or permanent productivity shocks could generate the more persistent patterns seen in the data.

Since long-term leverage declines in the productivity shock experiment, predicted dispersion in the long-term leverage model decreases, although it is statistically indistinguishable from zero. While the implied model response is within the error bands, this result further highlights the discrepancy between the experiment and the empirical response to short-term debt. The lack of a persistent increase in leverage, long- and short-term, means that the model does not generate a persistent response, and in fact overshoots slightly in the second period.

Figure 8. Collateral Constraint Shock: Relationship between Leverage and Dispersion.

Notes: Predicted values in response to deviation in leverage from model in Section 4.2 for horizons $t+h$ , shaded areas represent $\pm 1.96*SE$ . Response to initial period (t + 0) in the first period, period t + 1 for long-term leverage.

Figure 7 shows the response of productivity dispersion to the period of low interest rates considered in the second column of Figure 5. Panel (a) shows a fairly tight fit of the quantitative model to the empirics, as the path of TFPR dispersion in the model lies within 1.96 standard errors of the point estimate from the empirical model for the entire time horizon. Likewise the small response to long-term leverage is reasonable compared with the data.Footnote ²² The response of output per hour dispersion, while qualitatively consistent with the data, is quantitatively much larger in the model. This is perhaps due to the inelastic labor supply, which generates a much larger increase in wages. Still, the model generates a similarly persistent response to the data, and fits closely to the data at longer time horizons. Estimated responses to long-term leverage changes are not distinguishable from zero. The model produces a response which is relatively small and reverts to zero quickly.

Finally, Figure 8 presents the collateral constraint experiment comparison to the data. This shock generates counterfactual responses of dispersion to leverage, as dispersion increases while short-term leverage falls. The empirical model predicts the opposite. This is not to say that such shocks do not happen, but that they are not responsible for the patterns observed in the data via the empirical model estimated in Section 4.

Figure 9. IRF: Productivity Dispersion correlation to Change in Inventory-Sales Ratio at Year 0.

Notes: Predicted values in response to deviation in leverage from model in Section 4.2 for horizons $t+h$ , shaded areas represent $\pm 1.96*SE$ . Response to initial period (t + 0) in the first period, period t + 1 for long-term leverage.

5.8 Discussion

The model successfully generates labor productivity dispersion in excess of TFPR dispersion. The inclusion of an undistorted input margin (materials) is crucial to this result. Furthermore, the data suggest that among the shocks considered in this paper, interest rate shocks are the most promising candidate for explaining the empirical patterns documented in Section 4.2. The model implies that leverage and productivity dispersion positively co-move in response to the shock, consistent with the data, and the dynamic path for TFPR is quantitatively similar.

The shortcomings of the model include relatively low productivity dispersion and large responses of labor productivity dispersion to shocks, especially interest rate shocks. The lack of additional frictions or distortions could be responsible for the first shortcoming, especially the lack of frictions on materials, the largest share of input costs. Future work could consider how financial frictions might apply to materials. This is particularly of interest given the relative importance of inventories in the empirical results, and in serving as collateral as documented by Caglio et al. (Reference Caglio, Darst and Kalemli-Özcan2021). The response of labor productivity is likely in part due to the responsiveness of wages, and alternative labor supply assumptions might yield more muted dispersion responses.

Quantitatively, implied misallocation, according to the measure specified in Bils et al. (Reference Bils, Klenow and Ruane2021), is modest compared to the data, but within the historical range. Responses of misallocation to shocks are economically meaningful, and suggest that financial frictions could create a drag on productivity as credit demand expands. To the extent these frictions can be relaxed, then welfare can be improved by allowing for expanded credit and reducing misallocation. A natural response would be to seek ways to ease credit conditions for firms, but appropriate policy must take into account the margins on which credit interventions act. If a policy acts only on the price margin (i.e. lower interest rates), then it only serves to expand credit demand, which could lead to more dispersion. This may or may not positively impact welfare, but the empirics and theory detailed in this paper agree it will only increase measured misallocation. However, relaxing the constraint itself will likely reduce misallocation. This suggests that the ability to obtain credit, given a price, is a more important margin to act on if the goal is to reduce misallocation.