3.1 The AK model with bankruptcy costs

3.1.1 Households

A representative and infinitely living household obtains utility from the consumption of the final good, $C_{t}^{h}$ . Lifetime utility takes the following form:

(1) \begin{equation} U_{t}=\sum _{s=t}^{\infty }\beta ^{s-t}\log C_{s}^{h}, \end{equation}

where $\beta$ is a subjective discount factor. The budget constraint is given by

(2) \begin{equation} C_{t}^{h}+K_{t+1}^{h}=r_{t}^{k}K_{t}^{h}+(1-\delta )K_{t}^{h}, \end{equation}

where $K_{t}^{h}$ is the household’s capital stock held at the beginning of period $t$ . The resources available from the net return from capital, $r_{t}^{k}K_{t}^{h}$ , and undepreciated capital, $(1-\delta )K_{t}^{h}$ are used to purchase the final good, $C_{t}^{h}$ , and to invest in next period capital, $K_{t+1}^{h}$ . Solving this problem yields the consumption Euler equation, which indicates that, at the optimum, households are indifferent between consuming and investing into capital:

(3) \begin{equation} \frac{C_{t+1}^{h}}{C_{t}^{h}}=\beta (1+r_{t+1}^{k}-\delta ).\end{equation}

Footnote ¹¹

3.1.2 Contracting problem

There is a continuum of firms in the interval $(0,1)$ . Each firm, represented by the subindex $j$ , produces the final good, $Z_{j,t}$ , with the following technology:

(4) \begin{equation} Z_{j,t}=\omega _{j,t}A_{t}K_{j,t}, \end{equation}

where $A_{t}$ is an aggregate technology parameter (common to all firms) and $K_{j,t}$ is capital used by firm $j$ , constituting an AK model. Importantly, production is subject to idiosyncratic shocks, as captured by an iid stochastic productivity parameter, $\omega _{j,t} (\gt 0)$ , with $E\!\left ( \omega _{j,t}\right ) =1$ . A change in idiosyncratic risk is modeled as a change in the dispersion of idiosyncratic shocks. Specifically, a rise in risk corresponds to a mean-preserving spread of shocks.

Firm $j$ has access to external finance. With a given level of net worth (internal funds), $N_{j,t}$ , the firm requires a loan of size, $r_{t}^{k}K_{j,t}-N_{j,t}$ , where $r_{t}^{k}$ is the rental rate of capital. External funding is provided by households through financial intermediaries.Footnote ¹² Following Townsend (Reference Townsend1979)’s costly state verification (CSV) setting, the realized value of the idiosyncratic shock, $\omega _{j,t}$ , is firm $j's$ private information. If financial intermediaries want to observe it, they need to pay monitoring costs. Importantly, since the optimal contract is a standard debt contract under the CSV setting [Townsend (Reference Townsend1979)], monitoring costs are incurred only when firms fail to repay. These costs are thus interpretable as bankruptcy costs, incurred by financial intermediaries (creditors) during the bankruptcy procedure of failing firms.Footnote ¹³ In practice, these costs could include various legal, accounting, and administrative fees associated with debt enforcement, state verification, and asset liquidation, and also the lost value of the insolvent firm’s assets through depreciation.

As in Carlstrom and Fuerst (Reference Carlstrom and Fuerst1997, Reference Carlstrom and Fuerst1998), we consider an intra-period contract, which is renewed every period. In any period $t$ , before the idiosyncratic shock is realized, the contact determines repayment amount as $\Psi _{j,t}(r_{t}^{k}K_{j,t}-N_{j,t})$ , where $\Psi _{j,t}$ is the gross interest rate on the loan. If the firm does not default after the shock is realized, it pays back the pre-determined amount within the period, whereas if the firm defaults, the financial intermediary pays bankruptcy costs, seizing whatever remains from the insolvent firm. There is thus a cut-off value, $\overline{\omega }_{j,t}$ , such that

(5) \begin{equation} \overline{\omega }_{j,t}A_{t}K_{j,t}=\Psi _{j,t}(r_{t}^{k}K_{j,t}-N_{j,t}), \end{equation}

where if $\omega _{j,t}\geq \overline{\omega }_{j,t}$ , a firm repays the predetermined amount; otherwise it defaults. Since net worth, $N_{j,t}$ , is predetermined, the contract with firm $j$ effectively sets the amount of capital rented, $K_{j,t}$ and the cut-off value, $\overline{\omega }_{j,t}$ .

Firms are risk neutral, maximizing their expected profits, whereas perfectly competitive financial intermediaries, which we assume do not incur any operation cost, diversify risks by lending to a large number of firms and recoup the amount they lend within a period. Therefore, the contracting problem is written as

\begin{equation*} \max _{\{K_{j,t},\overline {\omega }_{j,t}\}}f(\overline {\omega } _{j,t})A_{t}K_{j,t} \label {Cont1} \end{equation*}

subject to

(6) \begin{equation} g(\overline{\omega }_{j,t})A_{t}K_{j,t}=r_{t}^{k}K_{j,t}-N_{j,t}, \end{equation}

where $f(\overline{\omega }_{j,t})$ and $g(\overline{\omega }_{j,t})$ are the expected shares of firm’s production going to the firm and the financial intermediary, respectively. They are expressed as

(7) \begin{equation} f(\overline{\omega }_{j,t})\equiv \int _{\overline{\omega }_{j,t}}^{\infty }\omega \phi (\omega )d\omega -(1-\Phi (\overline{\omega }_{j,t}))\overline{ \omega }_{j,t}, \end{equation}

(8) \begin{equation} g(\overline{\omega }_{j,t})\equiv \int _{0}^{\overline{\omega }_{j,t}}\omega \phi (\omega )d\omega +(1-\Phi (\overline{\omega }_{j,t}))\overline{\omega } _{j,t}-\mu \Phi (\overline{\omega }_{j,t}). \end{equation}

In equations 7 and 8, $\phi (\omega )$ and $\Phi (\omega )$ are the probability and cumulative density functions of idiosyncratic shocks. The sum of $f(\overline{\omega }_{j,t})$ and $g(\overline{\omega }_{j,t})$ is less than unity due to the payment of bankruptcy costs: $f(\overline{\omega }_{j,t})+g(\overline{\omega }_{j,t})=1-\mu \Phi (\overline{\omega }_{j,t})$ , where $\mu$ is the bankruptcy costs parameter, and $\Phi (\overline{\omega }_{j,t})$ is the default rate. The implicit assumption here is that in case firm $i$ defaults, a financial intermediary pays a fixed fraction $\mu$ of the expected production outcome, $A_{t}K_{j,t}$ , during the bankruptcy procedure. In a robustness check, we also consider the alternative setup where bankruptcy costs are a proportion of the realized production, $\omega _{j,t}A_{t}K_{j,t}$ .

Solving the problem above, and aggregating the first-order condition across firms yields

(9) \begin{equation} r_{t}^{k}= (1-\Theta (\overline{\omega }_{t})) A_{t}.\end{equation}

Footnote ¹⁴

This indicates that $\Theta (\overline{\omega }_{t})$ is a wedge between the rental rate, $r_{t}^{k}$ , and the marginal product of capital within the AK framework, $A_{t}$ , which is expressed as

(10) \begin{equation} \Theta (\overline{\omega }_{t})\equiv \mu \Phi (\overline{\omega }_{t}) +\left ( 1-\frac{1}{\lambda \left ( \overline{\omega }_{t}\right ) }\right ) f(\overline{\omega }_{t}), \end{equation}

where $\lambda \!\left ( \overline{\omega }_{t}\right )$ is the Lagrange multiplier:

(11) \begin{equation} \lambda \!\left ( \overline{\omega }_{t}\right ) =\frac{1}{1-\mu \frac{\phi \left ( \overline{\omega }_{t}\right ) }{ 1-\Phi \left ( \overline{\omega }_{t}\right ) }}. \end{equation}

Equation 10 shows that the investment wedge consists of the two components. The first one of $\mu \Phi (\overline{\omega }_{t})$ represents the part of production lost during the bankruptcy procedure, interpretable as a deadweight loss. The second component can be interpreted as economic rents accruing to firms in the presence of bankruptcy costs, since the Lagrange multiplier, $\lambda \left ( \overline{\omega }_{t}\right )$ , which represents the shadow price of firms’ net worth, exceeds unity only when bankruptcy costs are present, i.e., $\mu \gt 0$ . This component of the wedge becomes larger as the share of output going to firms, $f(\overline{\omega }_{t})$ , increases. When bankruptcy costs are absent ( $\mu =0$ ), both components are zero and the wedge disappears.

3.1.3 Endogenizing net worth

While the preceding optimization problem treats firm $j$ ’s net worth, $N_{j,t}$ , as exogenous, it needs to be endogenized in the dynamic context. Specifically, $N_{j,t}$ is determined as the gross return of the capital the firm holds at the beginning of the period, $K_{j,t}^{e}$ , i.e., $N_{j,t}=(r_{t}^{k}+1-\delta )K_{j,t}^{e}$ .Footnote ¹⁵ Aggregating this relation across firms yields

(12) \begin{equation} N_{t}=(r_{t}^{k}+1-\delta )K_{t}^{e}, \end{equation}

where $N_{t}$ and $K_{t}^{e}$ are aggregate net worth and firms’ capital. To avoid a situation in which firms ultimately become self-financed, we follow Bernanke et al. (Reference Bernanke, Gertler, Gilchrist, Bernanke, Gertler and Gilchrist1999) and assume that a constant fraction $\upsilon$ of randomly-selected firms exit each period, and consume all the accumulated wealth just before their exit (think of firms as entrepreneurs).Footnote ¹⁶ The population of firms remains constant at unity, with new firms replacing those exiting the economy. Thus, $K_{t}^{e}$ follows

(13) \begin{equation} K_{t+1}^{e}=(1-\upsilon )A_{t}f(\overline{\omega }_{t})K_{t}. \end{equation}

Aggregate consumption by firms, $C_{t}^{e}$ , is given as $C_{t}^{e}=\upsilon A_{t}f(\overline{\omega }_{t})K_{t}$ .

3.1.4 Market clearing

Denoting total aggregate consumption and capital as $C_{t}$ ( $=C_{t}^{h}+C_{t}^{e}$ ) and $K_{t}$ ( $= K_{t}^{h}+K_{t}^{e}$ ), respectively, the final goods market clearing condition becomes

(14) \begin{equation} Y_{t}=C_{t}+K_{t+1}-(1-\delta )K_{t}, \end{equation}

where

(15) \begin{equation} Y_{t}=(1-\mu \Phi (\overline{\omega }_{t}) )A_{t}K_{t}. \end{equation}

In words, the sum of total consumption and investment equals aggregate output net of bankruptcy costs. Equation 15 thus embodies the assumption that the bankruptcy costs are a deadweight loss in general equilibrium. This assumption is motivated by the observation that in reality bankruptcy costs contain a type of costs that arise when capital is kept idle for some period of time during the bankruptcy procedure (e.g., lost sales and profits, and the depreciation of capital).Footnote ¹⁷

3.1.5 Balanced growth equilibrium

Our focus is on the balanced-growth path equilibrium of the model, where the aggregate productivity parameter, $A_{t}$ , and the threshold value of idiosyncratic shock, $\overline{\omega }_{t}$ , are assumed to be constant. Along the path, variables grow at the constant rate of $\eta$ , except for the rental rate, $r^{k}= (1-\Theta (\overline{\omega })) A$ , which is also constant. Here, we only sketch the solution process for brevity, leaving the details to Appendix A.

First, acknowledging that households’ budget constraint (equation 2) is a first-order linear difference equation, we can rewrite it as

(16) \begin{equation} K_{t}^{h}=\sum _{j=0}^{\infty }\left (\frac{1}{r^{k}+1-\delta }\right )^{j+1} C_{t+j}^{h}+\lim _{T \to \infty }\left (\frac{1}{r^{k}+1-\delta }\right )^{T}K_{t+T}^{h}. \end{equation}

In the balanced-growth equilibrium where $K_{t}^{h}$ grows at the constant rate of $\eta$ , the limit of the present value of the terminal value of households’ capital holding, $\lim _{T \to \infty }\!\left (1/(r^{k}+1-\delta )\right )^{T}K_{t+T}^{h}$ is zero. This is because we know from the Euler equation (equation 3) that:

(17) \begin{equation} \eta =\beta (r^{k}+1-\delta ), \end{equation}

and the subjective discount factor is less than unity, i.e., $\beta \lt 1$ . Thus, equation 16 becomes

(18) \begin{equation} K_{t}^{h}=\frac{1}{(1-\beta )(r^{k}+1-\delta )} C_{t}^{h}, \end{equation}

a static relation between $K_{t}^{h}$ and $C_{t}^{h}$ , which holds along the balanced growth path. Next, remember from equation 9 that bankruptcy costs create the wedge, $\Theta (\overline{\omega })$ , between the real rental rate, $r^{k}$ , and the marginal product of capital, $A$ :

(19) \begin{equation} r^{k}= (1-\Theta (\overline{\omega })) A. \end{equation}

By substituting equation 19 into equation 18, and noticing that $K_{t}^{h}$ and $C_{t}^{h}$ can both be expressed as a linear function of $K_{t}$ (explained in Appendix A), we obtain

(20) \begin{equation} \lambda \!\left ( \overline{\omega }\right ) = \frac{\beta }{1-\upsilon }. \end{equation}

In words, the steady-state shadow price of net worth, $\lambda \left ( \overline{\omega }\right )$ , is increasing in the discount factor, $\beta$ , and decreasing in firms’ marginal propensity to invest into capital, $1-\upsilon$ .Footnote ¹⁸

Equation 20 makes it clear that once the distribution function of idiosyncratic shocks is specified, the long-run threshold value of $\overline{\omega }$ can be solved for given parameter values. Specifically, denoting the dispersion parameter of idiosyncratic shocks as $\rho$ , $\overline{\omega }$ is tied down as a function of four parameters:

(21) \begin{equation} \overline{\omega }=\overline{\omega }(\mu, \rho, \beta, \upsilon ). \end{equation}

Subsequently, we can solve the balanced growth rate, $\eta$ , as (see equations 17 and 19):

(22) \begin{equation} \eta =\beta \!\left [(1-\Theta (\mu, \rho, \beta, \upsilon )) A+1-\delta \right ]. \end{equation}

Equation 22 indicates that an increase in the investment wedge between the rental rate of capital and its marginal product, $\Theta$ , reduces the long-run growth rate, $\eta$ . This happens because the wedge, reducing the rental rate of capital, makes investment less attractive for households, thereby slowing down capital accumulation. In the present AK setup, evaluating the growth effect of idiosyncratic risk boils down to evaluating the effect of the dispersion parameter of idiosyncratic shocks on the wedge. Acknowledging this, our aim now is to analyze how the dispersion parameter interacts with the bankruptcy costs parameter, $\mu$ , to determine the wedge, and thus growth.

3.2 Analysis

3.2.1 Distribution function of idiosyncratic shocks

We model an increase in idiosyncratic risk as a mean-preserving spread of idiosyncratic shocks, $\omega$ . The following analysis considers a beta distribution and a log-normal distribution of idiosyncratic shocks. The former distribution enables us to model a mean-preserving spread without an effect on its skewness, whereas with the latter, a mean-preserving spread necessarily changes skewness. We regard the former as a reference distribution function for the sake of simplicity and check the robustness of results using the latter. A beta distribution exhibits unit mean and zero skewness when the probability density function, $\phi (\omega )$ , takes the form:

(23) \begin{equation} \phi (\omega )=\frac{1}{Beta(\rho,\rho )}\frac{\omega ^{\rho -1}\!\left ( 2-\omega \right ) ^{\rho -1}}{2^{2\rho -1}}, \end{equation}

where $Beta(\rho,\rho )=\int _{0}^{1}\omega ^{\rho -1}(1-\omega )^{\rho -1}d\omega$ . Figure 1 illustrates a mean-preserving spread as a shift of distributions from the solid to the dotted lines, corresponding to a fall in the dispersion parameter of the function, $\rho$ .Footnote ¹⁹ In what follows, the lack of analytical solution prompts us to proceed numerically. Nonetheless, our primary focus is on the qualitative effect of the interaction between idiosyncratic risk and bankruptcy costs on growth.

Figure 1. Mean-preserving spread using beta distribution.

Note: Mean-preserving spread corresponds to a shift from the solid to the dashed line.

3.2.2 Calibration

We now calibrate parameter values that determine the investment wedge and growth: $\beta$ , $\mu$ , $\upsilon$ , $\rho$ , $A$ , and $\delta$ (see equation 22). The time unit is a quarter. The discount factor, $\beta$ , is set as $0.99$ . Regarding the bankruptcy costs parameter, $\mu$ , previous works for advanced economies such as Bernanke et al. (Reference Bernanke, Gertler, Gilchrist, Bernanke, Gertler and Gilchrist1999) and Chugh (Reference Chugh2016) set $\mu =0.12$ and $0.15$ , respectively, while Christiano et al. (Reference Christiano, Motto and Rostagno2014) estimate it as $\mu =0.21$ for the USA. However, as illustrated below (Figure 4), there is a clear indication that bankruptcy costs are generally higher in lower-income countries. Since the subsequent empirical analysis mainly covers non-advanced economies (associated with the coverage of the World Bank’s Enterprise Surveys used there), we use $\mu =0.31$ as a reference value. This is the value used for Argentina by Aysun and Honig (Reference Aysun and Honig2011) to investigate the role of bankruptcy costs in the output loss following sudden stops of capital flows. We set the depreciation rate of capital, $\delta$ , as $0.031$ , based on the annual depreciation rate of $0.1255$ used by García-Cicco et al. (Reference García-Cicco, Pancrazi and Uribe2010) for Argentina.

Next, we set firms’ exit probability $\upsilon =0.072$ , corresponding to firms’ expected survival periods of $7$ quarters (and their aggregate saving rate of $0.93$ ). Setting this value of $\upsilon$ enables us to target an empirically plausible value of the credit-to-GDP ratio, defined as total credit divided by aggregate output, $\frac{g(\overline{\omega }(\mu, \rho, \beta, \upsilon ))}{1-\mu \Phi (\overline{\omega }(\mu, \rho, \beta, \upsilon ))}$ (see equations 6 and 15). To tie down the dispersion parameter of idiosyncratic shocks for the beta distribution, $\rho$ , we focus on the credit spread, measured by the premium in firms’ gross intra-period interest rate on loan over the corresponding risk-free interest rate (which is unity), $\frac{\overline{\omega }(\mu, \rho, \beta, \upsilon )}{g (\overline{\omega }(\mu, \rho, \beta, \upsilon ))}-1$ (equation 5). Specifically, targeting the spread of $0.015$ ( $1.5$ %), and using also the equilibrium shadow price of net worth (equation 20), the value of $\rho$ and the threshold value of $\overline{\omega }$ can be tied down simultaneously as $\rho =6.38$ and $\overline{\omega }=0.45$ .Footnote ²⁰ Subsequently, the model yields the credit-to-GDP ratio of $0.444$ ( $44.4$ %) in line with the empirical counterpart.Footnote ²¹ Last, to tie down the value of the aggregate productivity parameter, $A$ , in the reference equilibrium, we target the quarterly net growth rate of $0.005$ ( $0.5$ %), based on the median annual GDP growth rate of around $2$ % obtained from the IMF’s WEO over the 1983-2017 period for all the countries available.Footnote ²² We can then obtain from equation 22 that $A=0.048\!\left (=\frac{1}{1-\Theta (\mu, \rho, \beta, \upsilon )}\left (\frac{1+0.005}{\beta }-1+\delta \right )\right )$ . Table 1 summarizes the parameter values, together with the values used as a target.

Table 1. Reference parameter values and targeted financial values

Note: Time unit is a quarter.

3.2.3 Interaction effect of idiosyncratic risk and bankruptcy costs on growth

Turning to the analysis, we first examine how an increase in idiosyncratic risk affects growth for given bankruptcy costs. Next, we illustrate how the effects of idiosyncratic risk on growth vary across different levels of bankruptcy costs.

Effect of idiosyncratic risk on growth for given bankruptcy costs

Here, as a direct measure of idiosyncratic risk, we use the standard deviation, $s$ , of idiosyncratic shocks rather than the dispersion parameter of the beta distribution, $\rho$ . Note that for this distribution, $s$ is inversely related to $\rho$ . Figure 2 reports how the growth rate and various financial variables (including the wedge) change over a range of idiosyncratic risk around $s=0.27$ , which corresponds to the reference dispersion parameter of $\rho =6.38$ (as in Table 1). The other parameters are kept constant at the reference values, including the bankruptcy costs parameter: $\mu =0.31$ .

Figure 2. Effects of idiosyncratic risk on growth and related variables.

Notes: The horizontal axis is the standard deviation of idiosyncratic shocks, $s$ , around the reference value of $s=0.27$ (corresponding to $\rho =6.38$ ). Shocks follow the beta distribution. Parameter values other than $\rho$ are kept constant at the reference values with $\mu =0.31$ .

Figure 2 shows that an increase in idiosyncratic risk reduces growth by increasing the wedge. In turn, to see how the wedge increases, remember from equation 10 that it can be decomposed into the deadweight loss and rents that accrue to firms. Observe that as idiosyncratic risk increases, the rents become larger due to an increase in the share of output going to firms.Footnote ²³ Intuitively, because the outcome of idiosyncratic shocks is firms’ private information, the larger dispersion of shocks gives firms a greater informational advantage and thus the larger output share. Although the deadweight loss also increases due to the higher default rate, this result, as shown below, is not robust due to the two opposing forces. On the one hand, larger idiosyncratic risk makes the left tail of the distribution fatter (see Figure 1) and thus increases the default rate at the given level of the threshold value, but on the other hand, it decreases the threshold value endogenously. Regarding the remaining financial variables, a rise in idiosyncratic risk increases the credit spread, and reduces the credit-to-output ratio. That is, when the dispersion of idiosyncratic shocks is larger, financial intermediaries charge higher interest rates on loan, and reduce the supply of credit relative to output.

For robustness checks, Figures A1, A2, A3, and A4 in Appendix B consider the alternative cases where (1) idiosyncratic shocks follow a log-normal distribution, (2) the bankruptcy cost parameter takes different values of $\mu =0.1$ and $0.5$ (with idiosyncratic shocks following the beta distribution), and (3) bankruptcy costs are modeled as a proportion of the realized production outcome (with $\mu =0.31$ and the beta distribution).Footnote ²⁴ For each case, the reference dispersion parameter (thus the standard deviation) of shocks, firms’ exit probability, $\upsilon$ , and aggregate productivity, $A$ are set, by targeting the same levels of the credit spread ( $0.015$ ), credit-to-output ratio ( $0.44$ ), and steady-state growth rate ( $1.005$ ). Then, the growth rate and the financial variables including the wedge are plotted around the reference value of the standard deviation, while keeping the other parameter values constant.Footnote ²⁵ As in Figure 2, a rise in idiosyncratic risk reduces growth while increasing the wedge, with the latter driven by an increase in the output share going to firms, but not necessarily by an increase in the deadweight loss (see Figures A1 and A2). Again, a rise in idiosyncratic risk increases the credit spread and decreases the credit-to-output ratio.

Effects of idiosyncratic risk on growth across different bankruptcy costs

We now consider the effects of idiosyncratic risk on growth across different values of bankruptcy costs parameter, $\mu$ . Specifically, assuming that idiosyncratic shocks follow the beta distribution, Figure 3 plots the marginal effect of the standard deviation of idiosyncratic shocks, $s$ , on growth, i.e., $\partial \eta/\partial s$ , around the reference value of $\mu =0.31$ between $0.1$ and $0.5$ . For this exercise, the parameter values other than $\mu$ are set at the reference values as in Table 1. The idea is thus to compare the adverse growth effects of a given increase in idiosyncratic risk across economies where the only difference is in terms of the size of bankruptcy costs.

Figure 3. Marginal effects of idiosyncratic risk on growth across different bankruptcy costs.

Notes: The horizontal axis is the bankruptcy costs parameter, $\mu$ , and the vertical axis is the marginal effect of the standard deviation of idiosyncratic shocks on the respective variable. $\beta =0.99$ , $\upsilon =0.072$ , $\rho =6.38$ ( $s=0.27$ ), $\delta =0.031$ , and $A=0.048$ throughout.

At the reference equilibrium where $\mu =0.31$ , we obtained $\partial \eta/\partial s=-0.0057$ . This means that a rise in idiosyncratic risk by $0.1$ standard deviations reduces the steady state annual growth rate by about $0.228$ ( $=0.057*4$ ) percentage points.Footnote ²⁶ Figure 3 shows that as the bankruptcy costs parameter rises, a given increase in idiosyncratic risk has a greater growth-reducing effect, driven by a greater impact on the wedge. The figure also shows that as bankruptcy costs increase, a rise in idiosyncratic risk has a greater (smaller) positive impact on the output share going to firms (the deadweight loss). Thus, the reason why a given increase in idiosyncratic risk has a greater distortionary effect when bankruptcy costs are higher is that it has a greater impact on the firms’ share in output and thereby economic rents accruing to them. Figures A5 and A6 in Appendix B indicate that the results are robust to the use of the log-normal distribution function and the alternative assumption that bankruptcy costs are proportional to the realized (rather than expected) production outcome.

4.1 Methodology

We use the following model to estimate idiosyncratic risk at a country level:

(24) \begin{equation} sales\_growth_{i,j,k,(t,t-2)}= \alpha + \beta sales_{i,j,k,t-2} + \sum _{l=1}^{n}\gamma _{l}z_{i,j,k,l,t} + \delta _{k,t}+\eta _{j,k} +\varepsilon _{i,j,k,t}, \end{equation}

where $sales\_growth_{i,j,k,(t,t-2)}$ is real annual sales growth of firm $i$ in sector $j$ in country $k$ over the 3 years between years $t$ and $t-2$ . In the right-hand side, the log of the initial sales level, $sales_{i,j,k,t-2}$ , is controlled for. $z_{i,j,k,l,t}$ includes variables specific to individual firm $i$ : size, age, types of ownership (domestic vs foreign and private vs public), access to bank finance, and export status. $\delta _{k,t}$ and $\eta _{j,k}$ are country-time and country-sector fixed effects, respectively. The former controls for any time-varying economy-wide shock common to all firms of a country, whereas the latter controls for sector-specific factors within a country, capturing, for instance, differences in productivity growth rates across sectors which may differ across countries. Last, $\varepsilon _{i,j,k,t}$ reflects the part of the firm’s annual growth rate which is not accounted for by the aforementioned factors.

Here, our main focus is on the estimated residual of $\hat{\varepsilon }_{i,j,k,t}$ . Specifically, we obtain the country-specific standard deviation of the residual as a measure of firm-level idiosyncratic risk for the country. Our methodology is an application of Castro et al. (Reference Castro, Clementi and Macdonald2009), who estimate idiosyncratic risk for different sectors within the USA. We first consider equation 25, whereby the log of the country-specific variance of the residual can be estimated as a coefficient on the country dummy:

(25) \begin{equation} \ln \hat{\varepsilon }_{i,j,k,t}^2=\theta _{k}+\nu _{i,j,k,t}. \end{equation}

Subsequently, we obtain the country-level standard deviation of $\varepsilon _{i,j,k}$ as $\sqrt{\exp\! \left (\hat{\theta }_{k}\right )}$ . This is the measure of dispersion of firms’ real sales growth rates, which we use as a proxy of firm-level idiosyncratic risk in country $k$ .Footnote ²⁷

4.2 Data

Our main data source is the WBES, that offer an expansive array of firm-level data on 161,977 firms from 286 surveys spanning 144 countries over the 2006–19 period.Footnote ²⁸ For each country, 1 to 4 surveys are available. The surveys mostly cover developing countries, though some advanced economies, such as Sweden and Israel, are also included. The data were collected by private contractors hired by the World Bank via face-to-face interviews, and confidentiality of the survey respondents and the sensitive information is maintained to facilitate the highest degree of survey participation. The surveys are answered by owners and top managers of businesses, while questions about the sales are sometimes answered by company accountants. The surveys cover a representative sample of an economy’s manufacturing, retail and other services, which correspond to the three sectors considered in the model.

The real sales growth rate is calculated from deflated values of sales typically over the 3 years between the year preceding the survey and 3 year before (denoted in equation 24 as $t$ and $t-2$ , respectively).Footnote ²⁹ Previous studies such as Şeker and Yang (Reference Şeker and Yang2014) and Chauvet and Ehrhart (Reference Chauvet and Ehrhart2018) use the WBES to examine the determinants of firms’ sales growth.Footnote ³⁰ We checked outliers for the initial as well as final sales (in US dollars), and excluded, for each survey, the logs of the initial and final sales that are further more than three standard deviations from the respective means [as in Şeker and Yang (Reference Şeker and Yang2014)]. The firm’s annual real sales growth rate is relatively scarce, because this is often missing for a very young firm, and some firms simply do not grasp the sales level. Information on firm characteristics, used as a control, are also sometimes missing. Thus, to mitigate the possible bias from a representation problem caused by these missing values, the subsequent analysis only uses a survey in which at least 50% of firms, and at least 100 firms, provide information on all the variables considered in the estimation of idiosyncratic risk.

After taking the aforementioned measures, the dataset reduces to 100,640 observations spanning 227 surveys from 127 countries with the average of 1.79 surveys per country.Footnote ³¹ Specifically, 1 (2,3,4) surveys are available for 59 (38, 28, 2) countries over the 2006–19 survey period. The survey years are not synchronized across the countries. When multiple surveys are available for a country, repeated observations at the firm level can potentially be used. However, because (1) the availability tends to be limited (with the coverage of firms often being different across the waves), and (2) our subsequent analysis makes use of the cross-country variation of the key variables, it is important to estimate the dispersion of firms’ real sales growth rate for as many countries as possible.Footnote ³² This is the main reason why equation 24 forgoes firm-fixed effects and covers all the countries including the 59 countries for which only one survey is available throughout the sample period.

4.2.1 Descriptive statistics

Table 2 presents descriptive statistics for the sample. The mean of the real annual sales growth rate, the dependent variable in equation 24, is 1.96%, while the standard deviation is substantial. Turning to the firms’ characteristics used as a control, the mean of the log of annual sales level in the initial year is 13.29, corresponding to 591,253 US dollars.Footnote ³³ Firm size, which takes the value of either 1, 2, or 3 (corresponding to the number of workers less than 20, between 20 and 99, and more than 100, respectively), is 1.76 on average.Footnote ³⁴ The average age is 19.6 years. Regarding the ownership structure, on average, 1.25% of firms are owned by government or state (in the sense that it holds at least 10 percent of ownership), while 10.1% are foreign-owned (i.e., foreign agents own at least 10%). 39.5% have access to bank loans, and 15.8% are exporters (in that at least 10% of firm’s annual sales come from direct exports). 58 (18, 24)% of firms belong to manufacturing (retail, other services) sectors.

Table 2. Descriptive statistics: For estimation of idiosyncratic risk

Notes: 100,640 observations, spanning 227 surveys from 127 countries over the 2006–19 survey period. Annual sales growth rate is in real terms (using GDP deflator from World Bank’s WDI) typically over the 3-year period, expressed in percent. Initial sales level is converted to US dollars, and then log-transformed. Exchange rate data is from WDI, complemented by IMF’s International Financial Statistics. Size takes the value of 1 (2, 3) if the number of workers is less than 20 (between 20 and 99, more than 100). Government/state (Foreign) ownership takes the value of 100 when at least 10% of ownership is held by government/state (foreigners), 0 otherwise. Access to bank loan takes the value of 100 if a firm has access to it. Exporter takes the value of 100 if at least 10% of a firm’s annual sales is derived from direct exports, 0 otherwise. Manufacturing takes the value of 1 if a firm belongs to the business sector, 0 otherwise. Retails and other services are defined similarly.

4.3 Results

Table 3 presents the estimation results of equation 24. Column (1) (Column (2)) shows results for models without (with) country-time and country-sector fixed effects. In both cases, the coefficient on the log of initial sales level is negative and statistically significant. The coefficient of −2.84 in Column (1) means that an increase in initial sales level of 1% is associated with a decrease in the sales growth rate of 0.028 percentage points. The relations of size and age with sales growth rate are significantly positive and negative, respectively.Footnote ³⁵ The coefficient on government ownership is positive, but insignificant, whereas a foreign ownership is associated with higher sales growth. Access to bank loans and export status are both significantly associated with higher growth. Using the country-time and country-sector fixed effects increases the model’s explanatory power greatly, as seen from a rise in the adjusted R-squared from 0.06 to 0.27.

Table 3. Determinants of firms’ real sales growth

Notes: OLS estimations. Dependent variable is firms’ annual real sales growth rates (in percent). See notes for Table 2 for the description of independent variables. Constant and coefficients on country-time and country-sector fixed effects are not shown for brevity. t-statistics are in parentheses. Clustered standard errors are used to adjust for correlation of error terms within country. ^*** $p\lt 0.01$ , ^** $p\lt 0.05$ , ^* $p\lt 0.1$ .

We then use equation 25 to estimate idiosyncratic risk as the country-specific standard deviation of the estimated residual from the model with the country-time and country-sector fixed effects. Table 4 summarizes the result. When considering the whole 127 countries, the mean value of idiosyncratic risk is 8.71%. Importantly, there is a substantial variation as reflected by the standard deviation of 2.61.Footnote ³⁶ The subsequent growth regression will exploit the cross-country variation in idiosyncratic risk to test the theoretical predictions above.

Table 4. Cross-country variation in idiosyncratic risk

Notes: 127 countries (observations). Idiosyncratic risk (in percent) is the country-specific standard deviation of an unexplained component of firms’ sales growth rates estimated using equations 24 and 25.

5.1 Recovery rate

In what follows, we utilize World Bank estimates of recovery rates, a key sub-component of “resolving insolvency” from Doing Business indices [World Bank (2019)]. The estimates use a methodology developed by Djankov et al. (Reference Djankov, Hart, McLiesh and Shleifer2008). Doing Business defines the recovery rate as cents to the dollar recovered by secured creditors through reorganization, liquidation, or debt enforcement proceedings, thereby inversely related to bankruptcy costs considered in the theory. Specifically, the calculation takes account of cost as well as time of the proceedings. The cost includes various fees, such as court fees, government levies, and fees of auctioneers and lawyers. Time is associated with the value lost through depreciation of the insolvent company’s assets. The calculation also considers the outcome of whether the company’s business emerges as a going concern (i.e., as an operating business) or the assets are sold piecemeal.Footnote ³⁷ The recovery rate is obtained as the remaining proceeds per dollar in present value terms.

Figure 4. Evolution of the recovery rate across income levels: 2004–2017.

Notes: The vertical axis represents cents to the dollar recovered by secured creditors. LICs (Lower_MICs, Upper_MICs, HICs) are countries whose income levels in terms of real GDP per capita, PPP adjusted were within the 4th (3rd, 2nd, 1st) quarter in 2003 among countries covered by IMF’s World Economic Outlook (WEO). Among countries which are included in the regression below (104 countries), countries for which the recovery rate is available throughout the 14 years (90 countries) are included to calculate the yearly average within each income group.

To obtain the flavor of how the recovery rate differs across time and countries, Figure 4 plots the evolution over the 2004–2017 period of yearly averages within the different income groups of low-income, lower-middle-income, upper middle-income, and high-income countries (LICs, Lower_MICs, Upper_MICs, and HICs, respectively).Footnote ^{38 ,} Footnote ³⁹ Among countries that are included in the reference regressions analysis below (104 countries, see Appendix D for the list), countries for which the recovery rate is available throughout the 14 years are used to calculate the yearly averages in the figure (90 countries). Two observations are in order. First, the recovery rate shows a substantial cross-country variation, associated positively with income levels. Second, recovery rates tend to exhibit little time variation, except that HICs show a somewhat distinct upward trend. This tendency of lacking time variation is consistent with the fact that the recovery rate closely reflects institutional features of a country, particularly its legal systems.Footnote ⁴⁰ In what follows, the cross-country variation of the recovery rate is exploited to estimate its interaction effect with idiosyncratic risk on long-run growth.

5.2 Methodology

Our methodology is a simple modification of Barro (Reference Barro1991) who investigates growth in a cross-section of countries:

(26) \begin{equation} \overline{\bigtriangleup \,\ln{y}}_{k}=\alpha + \beta \, risk_{k} + \gamma \, \overline{rec}_{k} + \delta \,{risk_{k}*\overline{rec}_{k}} + \sum _{l=1}^{n}\eta _{l} \, z_{k,l} + \varepsilon _{k}, \end{equation}

where $\overline{\bigtriangleup \,\ln{y}}_{k}$ is an average annual growth rate of per capita real GDP in PPP terms in percent in country $k$ over the 2004–17 period, calculated as $(\!\ln{y}_{k, 2017}-\ln{y}_{k, 2004})*100/14$ .Footnote ⁴¹ The time period is a period with the 2-year lag from the 2006–19 period over which the WBES was reported. The use of the lagged period is motivated by the fact that (1) there is usually a 1-year lag between the time the firm’s final sales figure was recorded and the time the survey was reported, and (2) the annual sales growth is calculated typically over 3 years. $risk_{k}$ is the log of idiosyncratic risk estimated for country $k$ , while $\overline{rec}_{k}$ is the average of the log of the recovery rate over the 2004–2017 period.Footnote ⁴² Acknowledging the inverse relationship between the recovery rate and bankruptcy costs, the preceding theoretical model predicts that the interaction coefficient, $\delta$ , should be positive.

$z_{k,l}$ is a vector of variables. Based on Barro (Reference Barro1991), we routinely control for the log of initial real per capita GDP (PPP adjusted) as well as initial human capital stocks. Initial income captures the convergence effect, expected to have a negative effect, whereas initial human capital stocks should have a positive effect, say, by facilitating an introduction of new goods and thereby technological progress. Initial human capital is proxied by the percentage of the population of relevant age enrolled in primary school as of 1994, assuming that those enrolled in primary school will add to growth as workers with the substantial lag of at least 10 years.Footnote ⁴³ The population growth rate, averaged over the 2004–17 period, is also included routinely. As a robustness check, we also control for aggregate (macroeconomic) volatility, proxied by the standard deviation of the annual growth rate of real per capita GDP over the 2004–17 period.

Regarding the channel through which idiosyncratic risk affects growth in the presence of bankruptcy costs, the preceding theory suggests the relevance of the distortion to households’ investment decisions (equation 22). To take account of this insight in estimation, the reference regression equation of equation 26 does not control for the ratio of investment to GDP (investment, for short). To shed light on the role of the capital accumulation channel, however, we also estimate the model with investment controlled for. The idea is that, to the extent that the capital accumulation channel is important, we expect that the interaction coefficient of $\delta$ becomes less significant when investment is controlled for. In what follows, we compare the models with and without investment.

5.3 Data

Table 5 presents descriptive statistics for 104 observations covered in the reference analysis. Appendix C details the data sources. Appendix D, that lists the 104 countries, clarifies that the sample consists largely of low- and medium-income countries. This is because the WBES, which has been used to estimate idiosyncratic risk, predominantly covers those countries. The average annual growth rate of real GDP per capita is 2.86%. Idiosyncratic risk, estimated above as the conditional dispersion of firms’ real sales growth rate, is 8.74% on average. The recovery rate is averaged at 33.12 cents, with a substantial variation across countries (as in Figure 4). The maximum (minimum) of 75.91 (7.33) cents corresponds to Sweden (Burundi). While the initial real GDP per capita (in PPP terms) is 8978 international dollars on average, it also exhibits a large variation. The mean of the initial human capital, measured by the enrollment rate at the primary level, is 81.63%. The mean population growth rate is 1.29%. The aggregate volatility, proxied by the standard deviation of annual growth rate of real GDP per capita, is 2.95% on average. The mean of investment rate as the ratio of total investment to GDP is 23.9%. Since the theory above assumes the environment where all the firms have access to bank financing, the robustness check below takes account of the proportion of firms with bank loans. On average, 38.6% of firms have a bank loan, while the standard deviation is substantial.

Table 5. Descriptive statistics: For growth regression

Notes: 104 observations (countries), corresponding to Table 6. Growth rate of real GDP pc (per capita) is in percent, the average of the annual rates over the 2004–2017 period. Idiosyncratic risk is the country-specific conditional standard deviation of an unexplained component of firms’ sales growth rates in percent. Recovery rate is cents to the dollar recovered by secured creditors through reorganization, liquidation, or debt enforcement proceedings. Initial real GDP pc is in PPP terms (in international dollars). Initial enrollment rate is the percentage of the population of relevant age enrolled in primary school. Population growth rate is in percent. Aggregate volatility is the standard deviation of annual growth rate of real GDP pc (in percent) over the 2004–17 period. Investment is the ratio of total (i.e., private as well as public) investment to GDP, in percent. Bank finance is the proportion of firms with bank loans, calculated using the WBES firm-level data during the sample period.

Table 6. Idiosyncratic risk and growth: the role of bankruptcy costs

Notes: OLS estimations. Dependent variable is annual growth of real per capita GDP (PPP adjusted) in percent, averaged over the 2004–17 period. Idiosyncratic risk (log-transformed) is the country-specific standard deviation of an unexplained component of firms’ sales growth rates in percent. Recovery rate (log-transformed) is cents to the dollar recovered by secured creditors through reorganization, liquidation, or debt enforcement proceedings, averaged over the 2004–17 period. Initial real GDP per capita (PPP adjusted) is the value in 2003, log-transformed. Initial enrollment rate is the percentage of the population of relevant age enrolled in primary school, averaged over the 1993–95 period. Population growth rate and investment (the ratio of total investment to GDP) are both in percent, averaged over the 2004–17 period. To divide the sample into countries with the proportion of firms with bank loans being larger (and smaller) than the median in Columns (4) (and (5)), we use the value based on the firm-level data from the WBES for the whole sample period. Aggregate volatility is measured by the standard deviation of annual growth rates of real per capita GDP over the 2004–17 period. Robust t-statistics are in parentheses. ^*** $p\lt 0.01$ , ^** $p\lt 0.05$ , ^* $p\lt 0.1$ .

5.4 Results

Table 6 presents the estimation results. The dependent variable is annual growth of real per capita GDP (PPP adjusted) in percent, averaged over the 2004–17 period. Column (1) examines the association between the log of idiosyncratic risk and growth, controlling only for the log of initial real GDP, initial human capital, and population growth rates. The coefficient on idiosyncratic risk of 0.9 indicates that an increase in idiosyncratic risk of 1% is associated with an increase in the annual growth rate of 0.009 percentage points, although the coefficient is not statistically significant. Column (2) adds (log of) recovery rates as a separate control, showing that idiosyncratic risk and recovery rates are both insignificant.Footnote ^{44 ,} Footnote ⁴⁵ Column (3) adds the interaction term between idiosyncratic risk and recovery rates, showing that the interaction coefficient is significantly positive.Footnote ⁴⁶ Given that the recovery rate is inversely related to bankruptcy costs, this result is consistent with the theoretical result illustrated in Figure 3 above, where as bankruptcy costs become higher, a given increase in idiosyncratic risk has a greater growth-reducing impact.

Columns (4) and (5) conduct the same analysis for the sub-sample of countries characterized by the proportion of firms with bank loans being larger and smaller than the median, respectively. Because the theory considers the interaction effect between bankruptcy costs and idiosyncratic risk in the environment where all the firms have bank loans (although the size of the loans differs across firms), we might expect that the result on the interaction effect does not necessarily holds for countries where the proportion of firms with bank loans is rather low. The positive and significant interaction coefficient in Column (4), together with the smaller and less significant coefficient in Column (5), is somewhat in line with this expectation. Next, Column (6), using the whole sample, indicates that even when the macroeconomic volatility variable is controlled for, the result on the interaction coefficient remains the same.Footnote ⁴⁷ The coefficient on macroeconomic volatility is negative, albeit not statistically significant.

To shed further light on the result on the interaction, Figure 5 plots the estimated marginal effect of idiosyncratic risk across different recovery rates using the model of Column (3). The figure also shows the 90% confidence intervals and the distribution of the recovery rates (across the 104 countries). Consistent with the positive interaction coefficient, the marginal effect is positively sloped. The additional insight here is that the marginal effect is negative and significant particularly when the recovery rate is sufficiently low, indicating the possibility that idiosyncratic risk can be growth reducing under such an environment.Footnote ⁴⁸ However, the figure also shows that the effect becomes significantly positive when the recovery rate is sufficiently high. This suggests that there might be unaccounted factors (in the theoretical model) that make idiosyncratic risk growth promoting. Oikawa (Reference Oikawa2010), for example, shows theoretically that a rise in firm-level idiosyncratic risk increases productivity growth via a learning-by-doing mechanism in research sector.

Figure 5. Marginal growth effects of idiosyncratic risk across different recovery rates.

Notes: Corresponds to Column (3) of Table 6. Solid line represents a marginal effect. Dashed line represents a 90% confidence interval. Histogram shows the distribution of the log of recovery rates across 104 countries.

Last, Column (7) adds investment to the reference result in Column (3), and shows that the coefficient on the interaction term between idiosyncratic risk and recovery rate is much smaller than the equivalent coefficient in Column (3) (1.89 vs 3.44). The indication is that a change in the recovery rate (thus bankruptcy costs) plays a less substantial role in the growth effect of idiosyncratic risk when investment is controlled for. This is in line with the theory that highlights the relevance of the capital accumulation channel to growth, thus offering some support for the use of the AK model above. To note, the other variables considered in the table (initial real GDP, initial human capital, and population growth) are associated with growth as expected, although initial human capital is marginally insignificant in some columns. Regarding coefficients on initial real GDP per capita (in log), they tend to be slightly lower than $-1$ (e.g., $-1.27$ in Column 3), suggesting that the annual convergence rate is slightly over 1% per year.Footnote ⁴⁹

5.5 Further robustness check: estimating two-period model

The exercise above assumes implicitly that idiosyncratic risk is time invariant. However, it is well-known that idiosyncratic risk can show a time trend [e.g., Campbell et al. (Reference Campbell, Lettau, Malkiel and Xu2001), Comin and Mulani (Reference Comin and Mulani2006), and Hamao et al. (Reference Hamao, Mei and Xu2007)]. The recovery rate can also change over time, although Figure 4 indicates that the variation is relatively small in LICs and MICs.

To take account of these possible time variations, we first estimate idiosyncratic risk for two separate periods using the data reported in the WBES over the 2006–12 and 2013–19 periods. The estimation methodology is based on equations 24 and 25. With the 2-year lag as before, the log of the recovery rate for the 1st (2nd) period is the average over the 2004–10 (2011–17) period. Then, adjusting the other variables of the growth regression accordingly, we use OLS on the pooled data that uses the variation between the 1st and 2nd period of 2004–10 and 2011–17.Footnote ⁵⁰ While we add a period dummy, which takes the value of 1 if the period is the 2nd, we do not consider a model with country-fixed effects. The reason is twofold. First, since only one observation is available for many countries, running a fixed effects model loses a substantial number of observations. Second, because the model essentially becomes dynamic with the log of initial real GDP per capita as a control, it would suffer an endogeneity problem due to the fixed effects being correlated with the lagged dependent variable.Footnote ⁵¹

Table 7 replicates Table 6 for this two-period setup. Again, the coefficients on idiosyncratic risk and the recovery rate, when considered separately, are not statistically significant (Columns (1) and (2)). However, the coefficient on the interaction term between those variables is positive and significant, suggesting that the role of bankruptcy costs in the growth effect of idiosyncratic risk still holds even when taking account of variations of idiosyncratic risk over time (Column (3)). In line with the theory, this is particularly the case when we consider the sub-sample of observations with the proportion of firms with bank loans being larger than the median [compare Columns (4) and (5)]. The result is also robust to controlling for aggregate volatility (Column (6)). Last, in Column (7) with investment as a control variable, the interaction term becomes insignificant, which again suggests the relevance of the capital accumulation channel to the growth effects.

Table 7. Robustness check: Two-period model (2004–10 and 2011–17)

Notes: OLS estimations. Dependent variable is the annual growth of real per capita GDP (PPP adjusted) in percent, averaged over the 7-year periods of 2004–10 and 2011–17. Period dummy takes the value of 1 if the period is 2 (i.e., 2011–17). Independent variables are adjusted accordingly (see footnote 50 for details). To divide the sample into observations with the proportion of firms with bank loans lager (smaller) than the median, we use, as in Table 6, the value for each country based on the firm-level data from the WBES for the whole sample period. t-statistics are in parentheses. Clustered standard errors are used to adjust for correlation of error terms within countries. ^*** $p\lt 0.01$ , ^** $p\lt 0.05$ , ^* $p\lt 0.1$ .

5.6 Interaction effect on investment rate

As emphasized, the theory, based on the AK model, examined the interaction effect between bankruptcy costs and idiosyncratic risk through the capital accumulation channel. To recap, the theoretical insight was that with bankruptcy costs present, an increase in idiosyncratic risk enlarges the wedge between the marginal product of capital and its rental rate, making investment less attractive for households and thus slowing down capital accumulation.Footnote ⁵² Acknowledging this, Tables 6 and 7 provided supporting evidence for the relevance of the capital accumulation channel, by controlling for the investment rate. Here, we offer additional evidence by estimating the interaction effect between bankruptcy costs and idiosyncratic risk directly on investment. This further helps justify the focus on capital accumulation using the AK model.

Specifically, we consider the following model similar to equation 26:

(27) \begin{equation} \overline{priv\_inv}_{k}=\alpha + \beta \, risk_{k} + \gamma \, \overline{rec}_{k} + \delta \,{risk_{k}*\overline{rec}_{k}} + \sum _{l=1}^{n}\eta _{l} \, z_{k,l} + \varepsilon _{k}, \end{equation}

where $\overline{priv\_inv}_{k}$ is an annual private investment rate (defined as private investment/GDP, in percent) in country $k$ , averaged over the 2004–17 period. We use the private investment to be consistent with the lack of explicit modeling of public investment in the theory. The idiosyncratic risk measure, $risk_{k}$ , and the recovery rate, $\overline{rec}_{k}$ , are as defined for equation 26, and our focus is again on the interaction coefficient, $\delta$ . Based on prior works of Servén (Reference Servén2003) and Cavallo and Daude (Reference Cavallo and Daude2011) on private investment, $z_{k,l}$ , a vector of control variables, includes domestic credit extended to the private sector (% of GDP, log transformed), the price level of capital formation (log transformed), and general government investment (% of GDP). To check robustness, we also control for real interest rate, although the number of observations falls substantially.Footnote ⁵³ Further, we consider a two-period model in line with Table 7. Data sources and descriptive statistics for the analysis are in Appendix E.

Table 8 summarizes the results.Footnote ⁵⁴ Columns (1) to (4) are based on the 1-period (2004–17) model. The punchline is that while the idiosyncratic risk and recovery rate are not significantly associated with private investment rate on their own, the coefficient on the interaction term is positive and significant in Column (3), suggesting that bankruptcy costs also play a key role in the association between idiosyncratic risk and private investment. To gauge this interaction better, Figure 6 visualizes the marginal effects of idiosyncratic risk on private investment across different recovery rates. It suggests that the effect on private investment is negative and significant only when the recovery rate is relatively small, i.e., bankruptcy costs are large. This is consistent with the theoretical insight above.Footnote ⁵⁵ The result on the interaction effect is robust to the addition of real interest rates as an independent variable [Column (4)] and the use of the two-period setup [Columns (5) and (6)]. Among the controls, the private credit to GDP ratio shows a positive association with private investment.Footnote ⁵⁶ Overall, the results offer additional evidence for the focus on the capital accumulation channel in the present paper’s context.

Table 8. Bankruptcy costs, idiosyncratic risk, and private investment

Notes: OLS estimations. Dependent variable for Columns (1) to (4) (Columns (5) and (6)) is an annual private investment rate (private investment/GDP in percent), averaged over the 2004–17 period (2004–10 and 2011–17 periods). Independent variables are adjusted accordingly. Period dummy takes the value of 1 if the period is 2 (i.e., 2011–17). Robust t-statistics are in parentheses. Clustered standard errors are used to adjust for correlation of error terms within countries for Columns (5) and (6). ^*** $p\lt 0.01$ , ^** $p\lt 0.05$ , ^* $p\lt 0.1$ .

Figure 6. Marginal effects on investment of idiosyncratic risk across different recovery rates.

Notes: Corresponds to Column (3) of Table 8. Solid line represents a marginal effect. Dashed line represents a 90% confidence interval. Histogram shows the distribution of the log of recovery rates across 100 countries.