A. Operating Firms and Technology

We assume a perfectly competitive market for one homogeneous good that can be used by a representative household or used as capital by all firms that produce only this good. The production function of all firms has a decreasing return to scale, as follows:

in which x is an idiosyncratic productivity shock, i.i.d. across firms, that follows a first-order Markov process with transition matrix p(x' | x) that is common knowledge in this economy; k ∈ ℝ⁺ is the capital input, and n∈(0,1) is the labor input. In particular,

in which 1 represents normalized equity because in our model, there is no stock market, and all firms are assumed to be owned by one household with equity of 1. b ∈ B ≡ [0, b], where b is the capital borrowed from a bank.

The operating profit is defined as

in which w ∈ ℝ⁺ denotes the real wage, which is determined competitively. The operating profit by finding the optimal labor input is derived as

in which θ = α / (1 − γ) < 1 and 3

B. Firm’s Recursive Problem

The current value of a firm that was operating normally in the past is as follows:

in which V₀(b, x, 0) denotes the value by which the firm decides to operate normally as of the present, and d is the dividend to the owner. M is the stochastic discount rate indicating the owner’s intertemporal preference. b′ is the amount of the new loan contract with a bank, and x′ is the idiosyncratic shock in the next period. In the last column, 0 means that there is no history of default.

The dividend is structured as follows:

in which i denotes an investment for which the corresponding law of motion is expressed as i = k′ − (1 − δ)k ; k′ is the capital installed in the next period, and δ is depreciation rate for the installed capital. q_b is the loan interest rate for b contracted in the last period. Φ(i, k) is a function of the capital adjustment cost:

in which 𝜙⁺ is the adjustment parameter for a positive investment and 𝜙⁻ is the scale parameter for a negative investment, referring to a case in which the firm sells capital. It is assumed that 𝜙⁻ > 1 because in general, an investment in capital is irreversible, implying that firms should pay more when they sell capital compared to when they buy capital (Zhang, 2005). This investment irreversibility increases the likelihood of a firm’s default when a firm experiences a very negative shock to production or management.

V₁(b, x, 0) is the value when a firm decides to default on the payment of principal and interest to a lender. In particular, if the firm decides to default, the operating profit of the relevant period π^* is paid to the bank (whereas the wages for labor are paid previously), with the bankrupt firm’s installed capital bought by the bank at the price of νb . In this case, νb can be interpreted as the liquidation value of the debt, and ν is the liquidation rate for the debt. This process of default and liquidation takes place at the end of the period. In addition, the bank disposes of the installed capital purchased from the bankrupt debtor.

Meanwhile, this capital structure is basically composed such that the size of the loan affects the corporate default and productivity. This is a system for determining the scale of production and the labor supply of the entire economy. In other words, in the capital structure of existing firm decision models, profit induces investments and loans are used as working capital. However, in our model, in the absence of an equity market, loans are used as facility investment funds rather than as working capital. Instead, the profit increases the incentive to be distributed to households as a dividend. This only changes the order in which profit and external funds are distributed to investments and dividends and does not lower the firm value, as the firm value is the present value of the dividends that will be received in the future. In addition, there is no possibility of excessive leverage because the mean and dispersion of investments and the debt ratio are fitted through a simulation.

The value of a firm that liquidates its debt in the last period but has a history of default is as follows:

in which the bankrupt firm’s cash flow is naturally zero, and 𝔼[V (0, x, 0)] is the entry value of a firm that finally closes down an old project, clearing the history of default and starting a new project. ξ is the closure rate of bankrupt businesses. Finally, the liquidation value for exiting the market is determined endogenously by the entry value into the market again. In addition, firms with new businesses start with zero debt.

C. Commercial Banking System

The commercial banking sector is assumed to be a perfectly competitive market with no entry costs. However, taking into account the difference between risk levels in the deposit market and the loan market, the lower limit of the return on the loan market is set as follows:

in which r_b is the expected return on b ; q is the deposit interest rate and μ is the firm distribution based on beliefs pertaining to the loan market formed by banks through repeated loan contracts with firms. S is the minimum cash flow buffer that the bank must hold, and σ is the minimum buffer ratio. Strictly speaking, this ratio should be applied to risk-weighted assets in terms of the capital ratio, but it is assumed to be a ratio relative to the deposit income for convenience of the calculation.

Given the belief distribution and σ, the return condition for loan products in a perfectly competitive market is as follows:

and we derive the equation for the loan interest rate of b via

in which M [⋅] is a technical function smoothing loan interest rates within similar loan sizes, bs.

The last important assumption in the financial sector is that banks fail to observe firms’ idiosyncratic shocks at every time, but when the loan is renewed, only the size of the loan creates a belief about the default probability. Therefore, even if the loan contract is renewed repeatedly, information about defaults held by banks is not updated. In addition, this belief is common knowledge in this economy.

D. Household Problem without Policy-based Finance

There is one representative household with a utility function with the unitary Frisch elasticity of labor supply such that

in which N denotes the aggregate labor such that the labor of a household is perfectly divisible and the household has the following budget constraint,

in which D, B, and B′ are the aggregate dividends and aggregate deposits in the current and subsequent period, respectively.

Therefore, the household problem is as follows:

From the above equation, we simply derive the first-order conditions as

and and under steady=state equilibrium.

E. Policy-based Financial System

Thus far, we have explained the banking industry, which has no government intervention. However, we assume that PFIs enter into the banking industry as facilitated by the government because the government has an incentive to boost the economy through financial support. In particular, in this study, PFIs provide firms with loans with lower interest rates than those in the decentralized market, and commercial banks also offer loans with low interest rates according to the principle of a perfectly competitive market. Commercial banks and PFIs are then supported by the government and are subject to the following budget formula:

in which T is the tax collected from the household.

There are two ways to support loan interest for corporate loans. Lenders support the loan interest rate in a proportional manner with an identical interest spread for all loans. The method of determining the interest rate is as follows: if μ(f = 1 | b) > 0,

and if μ(f = 1 | b) = 0, then

in which τ₁ is the ratio of proportional interest support and τ₂ is the interest spread. Why do we consider this PFS in the model? Usually, banks determine the interest rates for loans to firms in consideration of the buffer for financial stability and profitability. In the end, the capital productivity of a firm determines the profitability of the loan, which determines the loan interest rate. If it is possible for banks to raise funds at a low interest rate while maintaining financial stability (loan profit > deposit interest) through PFS, even relatively low-productivity firms will survive and participate in production without announcing a default. However, if a firm with overly low productivity survives and causes an excessive labor supply, the disutility of labor increases and social welfare decreases even as production expands. Therefore, the steady-state condition with PFS may have higher social welfare than the steady-state condition without PFS.

F. Government Budget

The tax system is based on lump sums such that households do not know how much the tax will be until the end of a period. The tax is determined as follows:

G. Household Problem with Policy-based Finance

In an economy where the government actively intervenes in the banking industry, the household budget constraint is similar to equation (4):

The household problem is identical to (5), and the household has first-order conditions identical to those in (6). In fact, the assumption of Frisch elasticity of 1 is conservative because the elasticity estimated through microdata is usually lower than 1. If the model assumes that the elasticity is lower than 1, the incentive to expand social welfare through PFS is expected to be lower because the change in the labor supply is relatively small with respect to the change in wages. This means that the optimal level of PFS may be lower than that in the current model.

H. Invariant Firm Distribution

We explain how to compute banks’ invariant belief system in the firm distribution. First, the state-mapping function of state variable vector, (b, x, h), is as follows:

in which h is the history of default that has a value of 1 if the firm decided to default in the past. The transition function of corporate policy is as follows:

in which S is defined as the compact space of state variables. Then, we define the transition function of firm via

Finally, given (w, Q_b) , the distribution of the state vector, (b, x, h), μ is defined as

in which Q_b is the vector of q_b for all loan products, and ϒ is the matrix operator. Therefore, μ is defined as the bank’s belief function with respect to b and f . It then becomes possible to compute the default probability and the conditional default probability μ(f = 1) and μ(f = 1 | b) (Athreya, Tam, and Young, 2012). In addition, the unique existence of an invariant distribution refers to Theorem 2 in Chatterjee, Corbae, Nakajima, and Rios-Rull (2007).

I. Bayesian General Equilibrium

Definition. The Bayesian general equilibrium lists (a) the real wage w^* , (b) the vector of loan interest rates ∈ (ℝ⁺)^l and the deposit interest rate q^* ∈ ℝ⁺, (c) if PFIs offer corporate loans, the support system , and (d) lenders’ belief about the firm distribution μ^* satisfying the following:

1. Firms solve the optimization problems of n^*, b^′* and f^* given w^* and in (1).

2. Lenders offer as a Bayesian Nash equilibrium under perfect price competition given τ^* , b^′* , f^*, μ^* and q^* in (7).

3. The government balances the tax T^* given τ^* , b^′* , and μ^* in (8).

4. The household solves the optimization problem of Β^′* and N^* given w^* , q^* , μ^* and T^* in (5).

5. Labor, loan and Deposit markets clear at w^* , q^* and :

6. According to Walras’s law, the household budget constraint according to the goods market clearing condition is as follows:

in which

and i^* and k^* are solved based on the firm’s problem (1).

A. Computational Methodology

This study computes the equilibrium interest rates of the loan market and the equilibrium real wage of the labor market considering the heterogeneity of firms. In particular, for a social welfare analysis, λ , which determines the marginal utility of labor, is estimated while assuming that there is no policy-based finance and that the real wage is fixed at 1. It is also important to calculate the upper limit of b , b . In this study, firms need a reasonable ceiling to grow their businesses through the loan market, not the stock market. Thus, we define b , which is interpreted as the maximum leverage, as 11 and calculate the base model.4

We define firms’ idiosyncratic shocks as AR(1):

with | ρ | < 1 and ε ~ N(0, ω). We discretize this process into a 20-state Markov process {x₁, ···, x₂₀} using the method of Adda and Cooper (2003). In particular, we do not estimate the parameters of idiosyncratic shocks using exogenously reduced forms with firm data, as in other studies such as Corbae and D’Erasmo (2017), but rather estimate the parameters in a way that minimizes the distance between moments from the simulated model and the moment of the financial data. Specific techniques for these computations are described in the appendix.

B. Parameters

This study requires 15 parameters for the model simulation. The parameters are divided into two groups. The first group is calculated independently of the model using corporate and financial data. Table 2 shows their values, α as the capital income share is 0.33, which is commonly used in the macroeconomic literature. In addition, γ uses a rate of 51.6%, the average of the labor income share obtained from the Bank of Korea (BOK: https://ecos.bok.or.kr/) from 1961 to 2020. θ is 0.682 from α / (1 − γ) in the firm’s problem. q is 1.79%, the average yield of the ten-year treasury bonds adjusted by consumer price index (CPI) from 2001 to 2020. σ is 0.09, which is the average yield of the three-year corporate bond (AA-) adjusted by CPI from 2001 to 2020 minus q and then divided by q . β is 0.982 from 1/ (1 + q) from (6). To use δ , first we calculate the annual sum of the decrements of tangible asset depreciation for unlisted non-financial firms audited externally from 2001 to 2020, taking the average after dividing the value by tangible assets.5 With this process, δ is 5.1%. The liquidation rate of bankrupt firms is calculated as 38.0%, as 388 out of the 1021 enterprises that filed for court receivership from 2008 to 2015 were closed.6

TABLE 2

PARAMETER VALUES

Note: 1) The first group shows base parameters calculated using independent corporate and financial data, 2) The second group shows parameters estimated from the minimization of the simulation moments and the target moments.

The second group of parameters is estimated from the simulated model. In particular, we estimate the parameters using a method that minimizes the distance between simulated moments and data moments weighted by a selected weighting matrix, as follows:

in which θ is a set of parameters; m^d are data moments; m^s are the simulated moments at parameters θ, here as , where z^s is a value for an individual state vector, which means that our moments computed from the simulated firm distribution differ from those generated by the simulated method of moments (SMM) with random numbers, and W is a positive definite weighting matrix selected to equalize the positions of the first decimal digit in all moments.

Table 3 shows the data moments selected to provide the identification of the parameters, also showing benchmark moments under the economy without a policy-based financial system.7 In fact, the selected data moments do not have an exact one-to-one relationship with the parameters. However, we can only explain that the selected moment has more information about the parameters we want to estimate than the other moments. In the table, the defaulted debt/total corporate debt levels and real interest rates must be related to 𝜙⁺, 𝜙⁻ and ν directly, and the net investment must influence the parameters η, ρ and ω of the idiosyncratic shock. The debt/equity ratio and corresponding standard deviation are related to all parameters.

TABLE 3

COMPARISON OF DATA AND MODEL MOMENTS

Note: This table lists the moments of the data and the simulated model (benchmark model) under an economy without policy-based finance.

In addition, according to studies such as that by Gourio and Miao (2011), the benchmark labor at equilibrium is set to 0.3 in order to estimate λ. Finally, returning to the table, it can be seen that the moments simulated by our model approximate the target moments well.

𝜙⁺ and 𝜙⁻ in Table 2 are estimated as 4.381 and 1.366e + 3, which are relatively large compared to those in Zhang (2005) but at the reasonable level where firms decide to default. In addition, ν is estimated to be 0.270, which means that the lender buys the installed capital of a bankrupt firm for 27% of the debt minus its cash holdings. η, ρ and ω, defining the movement of idiosyncratic shocks, are estimated to be −1.262, 0.911 and 0.417, respectively. These values appear to be desirable enough compared to those in Zhang (2005); Li, Livdan, and Zhang (2009); Livdan, Sapriza, and Zhang (2009); and Nikolov and Whited (2014). Finally, λ is estimated to be 4.960 by applying C^* and N^* from the simulated model to (6).

A. Benchmark Model Properties

As described above, the benchmark model assumes an economy in which there is no policy-based finance in order to estimate basic parameters such as λ or analyzing the social welfare of policy-based finance. First, we explain how firms’ decision rules on bankruptcy are made in the benchmark model. Panel (A) in Figure 1 shows the bankruptcy decision rule with respect to the state variables (k, x). As expected, firms with high capital levels and negative production shocks are likely to go bankrupt. In particular, a large amount of installed capital means not only that there is a considerable amount of debt but also that the firm is more vulnerable to production shocks due to the irreversibility of the investments. As a result, it can be seen that according to this decision rule, firms are concentrated in the category with relatively high productivity and a low debt structure in the invariant firm distribution μ(k, x) (Panel (B) in Figure 1).

Panels (C) and (D) in Figure 1 show the relationship between the loan interest rate Q_b, debt distribution μ(b) and default probability for each loan size μ(f = 1|b). It can be seen that the higher the debt, the higher the default probability, with the loan interest rate increasing accordingly. Also, in Panel (D), the default probability is rather high for firms with very small loans. This occurs because if a firm starts a new business, the initial productivity shock is given equally, after which the start-up’s default probability is higher than those of surviving firms due to the persistence of the shock. Although this appears to be in conflict with Panel (A), it only shows the bankruptcy decision rule and does not take into account the firm distribution. Nevertheless, in Panel (D), the loan interest rate is relatively low for startups because their cash level, that is, operating profit paid by bankrupt start-ups to the bank, is relatively high when the debt is liquidated, which ensures the bank’s profitability even with a high default probability.

FIGURE 1.

PROPERTIES OF BENCHMARK MODEL

Note: 1) This figure displays the properties of the benchmark model: the economy does not have policy-based finance, 2) Panel (A) shows bankruptcy decision rule γ w. r. t. capital k and idiosyncratic shock x, 3) Panel (B) shows invariant firm distribution μ(k, x), 4) Panel (C) shows loan interest rates Q_b and invariant loan distribution μ (b) : left axis is Q_b and the right axis is μ (b), 5) Panel (D) shows loan interest rates Q_b and conditional default probabilities μ(f = 1 | b) : the left axis is Q_b and the right axis is μ(f = 1 | b).

B. Results of the Social Welfare Analysis

The range of τ for the social welfare analysis is set as follows:

that is, we compute the social welfare when all interest rates are lowered by the same rate of up to 100bp for the market prices in the benchmark model or by up to 75% in proportion. Figure 2 shows the values of social welfare and other macro-variables with respect to τ₁ and τ₂ using contour plots. The contours of Panel (A) show that social welfare W^*(τ) is high when τ₁ is in the range of 0.05 to 0.45 and when τ₂ is less than 0.001, which means that it is more effective to adjust loan interest rates with τ₁ than with τ₂. As explained above, given that social welfare in our model takes into account the disutility of labor, the optimal τ would be lower if we assume that labor elasticity is not one but that it low as labor demand increases.

FIGURE 2.

COMPARISON OF SOCIAL WELFARE ACCORDING TO τ

Note: 1) This figure displays social welfare and macro variables w. r. t. (τ₁, τ₂), 2) Each panel shows social welfare (W^*), consumption (C^*), labor (N^*), defaulted debt/corporate debt, total corporate debt, and shares of policy-based finance in the financial market using contour plots.

Panel (B) shows us the important characteristic of this policy, specifically that interest policies pertaining to corporate loans are consistently effective in reviving the economy. More specifically, lowering the interest rate on corporate loans results in lower capital costs, which in turn increases future cash flows. Therefore, if there is no disutility of labor in the social welfare function, the optimal social welfare continues to expand due to this financial support. This means that there is no Laffer curve phenomenon that occurs when taxes are applied to the supply side. However, this model does not consider the inefficiency and restructuring costs of marginal firms surviving in the market due to the support of policy-based finance. Returning to the panel again, we see that as τ increases, taxes increase, whereas consumption C^*(τ) does not decrease but instead steadily increases. In particular, it can be seen here that τ₁ is more effective than τ₂.

Panel (C) and Panel (D) show total the labor used in production and the ratio of defaulted debt out of corporate debt. With respect to τ, labor tends to increase monotonically, and the ratio of defaulted debt decreases monotonically with consumption, as described above. Panel (E) and Panel (F) also show the size of the corporate loan market in the banking sector as a whole and the share of corporate loans provided by policy-based finance. In particular, the size of policy-based finance is calculated according to how much the corporate loan market expands from the benchmark model due to financial support. Furthermore, it is important to understand that the range of the optimal level of policy-based finance is 1.6% to 13.7% of the total corporate loan market considering social welfare as displayed in Panel (A). In particular, the optimal social welfare level is achieved when the policy-based finance accounts for 8.6% of the total corporate loan market.

To explain the gap between the model and reality, the reason for the existence of PFS is a function of effectively lending funds to firms before the economy develops, that is, a function of correcting the failure of the financial market. This explains the situation in which PFS basically raises funds at a low interest rate and lends money to firms at a lower interest rate than those offered by the market. In addition, it acts as a buffer in the financial market in response to economic fluctuations. However, as the financial market developed along with the economy, the financial market became decentralized. Nonetheless, policy-based financial institutions continued to expand the size of PFS without considering social welfare while continuing the growth-driven policies of the past. This appears to be the cause of the widening of the difference between the model’s results and reality. Also, to explain this in terms of the simulation model, as low-interest loan support is extended to low-productivity firms that need to enter the market after being expelled, an excessive labor demand arises and the social welfare of households decreases. In other words, the current situation is that corporate loans procured at low interest rates encourage production, but the social welfare generated through additional production is rather low due to the excess supply of labor.

On the other hand, our model as described thus far can be criticized in many ways. First, as mentioned above, our economy is structured to impose taxes on the demand side and to support the supply side, which is contrary to the usual fiscal policy of imposing taxes on the supply side and supporting the demand side. Moreover, if corporate taxes are levied on firms and tax revenue from corporations supports households, it can be considered that the effect of financial support on the corporation can be circulated back to households. However, there could be a trade-off effect because corporate taxes have a negative impact on the bankruptcy decisions of firms such that they may reduce the effect of financial support to firms. Second, our model does not explicitly assume social costs such as the inefficiencies or moral hazards of policy-based finance. It is obvious that in assuming so, the optimal level of policy-based finance will be lower than it is currently. Although the efficiency of policy-based finance is assumed to be identical to that of commercial banks, our model shows that the current level of policy-based finance in the Korean financial market needs to be moderated. Third, there is no financial intermediation through equity financing in our model. However, even if there is no stock market, the pecking order theory suggests that equity is the last resort of financing, and if a firm becomes close to insolvent, the cost of equity financing will increase rapidly such that our results would not be fundamentally different from those with equity financing. The last issue is that in our model, there is no inefficiency in production due to the survival of marginal firms in the market. If firms that will exit exist in the market due to financial support, the effects of policy-based finance may deteriorate due to externalities such as a decrease in the productivity of the same industry.