3.1 Structural Model Using Bayesian Conditional Probability

Since latent demand is not observed in policy fund supply while only the supply quantity is observed, this study estimates the latent demand using Bayesian conditional probability decomposition. The core of the Bayesian estimation method in this study is to derive the probability distribution of unobserved variables by combining observable information with prior information. According to Bayes’ theorem $P(A, B \mid \theta) = P(B \mid A, \theta) \times P(A \mid \theta)$, the joint probability of the observed approval amount and the demand constraint condition is decomposed as follows.

Here, $\theta$ is the vector containing all parameters of the model. $P(Q_{t} \mid D_{t} \ge S_{t}, \theta)$ is the conditional probability that the observed approval amount $Q_{t}$ follows the supply function under the demand constraint condition, while $P(D_{t} \ge S_{t} \mid \theta)$ is the prior probability that the demand constraint where demand exceeds supply holds. Through this decomposition, the observable supply quantity information and the structural assumption of the demand constraint can be explicitly reflected in the model.

3.2 Specification of Demand and Supply Functions

The demand function for policy funds is composed of macroeconomic variables that affect the funding demand of SMEs. The demand function is specified as follows.

Here, $D_{t}$ is the latent policy fund demand at time $t$, and GDP reflects the overall level of economic activity. The SME lending rate is a financial cost factor that affects policy fund demand, and the SME Business Survey Index (BSI) is a sentiment indicator that affects funding demand. SME credit conditions reflect the ease of borrowing, and the error term $\epsilon_{D,t}$ represents stochastic factors not included in the model. The error term is assumed to follow a normal distribution $\epsilon_{D,t} \sim N(0, \sigma_{D}^{2})$.

The supply function is composed of variables that reflect the government’s fiscal policy and the conditions for policy fund procurement. The supply function is specified as follows.

Here, $S_{t}$ is the policy fund supply quantity at time $t$. The SME budget share reflects policy intent and resource allocation priorities, while government spending to GDP indicates the expansionary nature of government finance. The Korea Treasury Bond (KTB) yield reflects the government’s borrowing cost and opportunity cost for procuring policy funds, while the Composite Leading Indicator (CLI) is a proxy indicator for the need for the policy authority to respond to economic conditions. The error term $\epsilon_{S,t}$ is assumed to follow a normal distribution $\epsilon_{S,t} \sim N(0, \sigma_{S}^{2})$.

3.3 Modeling the Demand Constraint Through the Truncated Normal Distribution

Due to the structural characteristics of policy fund supply, a constraint exists that the demand of SMEs with credit ratings of BBB- or below always exceeds supply. To explicitly reflect this constraint, this study employs a truncated normal distribution model. Unlike the general normal distribution, the truncated normal distribution assumes that values outside a specific interval are not observed and rescales the probability distribution within the observable interval. Since the general normal distribution allows the possibility that demand may be smaller than supply, this structural constraint needs to be reflected in the probability distribution. Therefore, this study assumes a truncated normal distribution in which demand exceeds supply by at least $\delta$.

Here, $\mu_{D,t}$ is the expected value of the original untruncated demand function, and the truncation point is set to $S_{t} + \delta$. $\delta$ is the minimum excess demand parameter, representing the minimum level of excess demand for policy funds. This setting indicates that demand is observed only at values greater than or equal to supply plus $\delta$.

The expected value of the truncated normal distribution requires adjustment due to truncation, unlike that of the original normal distribution. The mean of the truncated demand is calculated as follows.

Here, $\lambda_{t}$ is the inverse Mills ratio, a statistic used for adjusting the mean of the truncated distribution, defined as $\lambda_{t} = \frac{\phi(z_{t})}{1 - \Phi(z_{t})}$. $z_{t} = \frac{S_{t} + \delta - \mu_{D,t}}{\sigma_{D}}$. is the standardized truncation point. $\phi(\cdot)$ is the probability density function of the standard normal distribution, and $\Phi(\cdot)$ is the cumulative distribution function of the standard normal distribution.

3.4 Utilization of Application Amount Information

The unobserved latent demand $D_{t}$, while indirectly estimated through the model, can exhibit large variability. To limit this variability and secure estimation stability, this study additionally utilizes application amount data. This study assumes that the application amount $A_{t}$ is observed as a fixed proportion of the latent demand $D_{t}$, reflecting this in the model in the form $A_{t} = \gamma \cdot D_{t} + \epsilon_{t}$.

Accordingly, when assuming that the supply quantity $Q_{t}$ and application amount $A_{t}$ are conditionally independent under the demand constraint condition, the Bayesian conditional probability decomposition is extended from two components to three components as follows.2) In addition to the existing supply likelihood $P(Q_{t} \mid D_{t} \ge S_{t}, \theta)$ and truncation probability $P(D_{t} \ge S_{t} \mid \theta)$, the application likelihood $P(A_{t} \mid D_{t} \ge S_{t}, \theta)$ is added. The application likelihood represents the probability that the observed application amount reflects the truncated demand.

3.5 Construction of the Likelihood Function

Combining the three components of the Bayesian model, the final likelihood function is constructed as follows.

The first term, the supply likelihood, represents the probability that the observed approval amount under the demand constraint situation follows the supply function. $\phi(Q_{t}; S_{t}, \sigma_{S}^{2})$ is the probability density function of a normal distribution with mean $S_{t}$ and variance $\sigma_{S}^{2}$. The second term, the truncation probability, represents the prior probability that the situation in which demand exceeds supply occurs. The third term, the application likelihood, represents the probability that the observed application amount follows a normal distribution with a mean of times the truncated demand and a variance of $\sigma_{A}^{2}$. Here, $r$ is the application coefficient that functions as a filter for extracting pure demand information from the application amount.

The parameters to be estimated finally consist of the demand function coefficients $\alpha_{0}, \alpha_{1}, \alpha_{2}, \alpha_{3}, \alpha_{4}$, the supply function coefficients $\beta_{0}, \beta_{1}, \beta_{2}, \beta_{3}, \beta_{4}$, the variance parameters of each equation $\sigma_{D}^{2}, \sigma_{S}^{2}, \sigma_{A}^{2}$, the application coefficient $\gamma$, and the excess demand parameter $\delta$, totaling 15 parameters. These parameters are estimated through maximum likelihood estimation, and the estimated parameters are used to calculate the latent demand and policy fund gap at each time point.

4.1 Data Composition and Scope

The analysis period of this study is 2000 to 2024, totaling 25 years. Policy fund data from the Korea SMEs and Startups Agency (KOSME) covers approval amounts and application amounts from 2003 to 2024, and missing data from 2000 to 2002 were supplemented using data from economically similar adjacent time points.3) Macroeconomic variables such as GDP, the SME lending rate, the SME BSI, SME credit conditions, the KTB yield, and the CLI were extracted from the Bank of Korea ECOS system, while fiscal data such as the SME budget share and government spending to GDP were sourced from the fiscal statistics of the Korea Public Finance Information Service and the IMF WEO database.4)

4.2 Policy Fund Variables

The mean application amount of policy funds is 7.05 trillion KRW and the mean approval amount is 4.12 trillion KRW, with the application amount being 1.71 times the approval amount. The standard deviations of the application amount and approval amount are 1.58 trillion KRW and 1.01 trillion KRW respectively, indicating that the variability of the application amount is greater.

Table 1 presents the descriptive statistics of policy fund variables. The median application amount is 6.83 trillion KRW and the median approval amount is 4.25 trillion KRW. The application amount ranges from 5.05 trillion KRW to 11.81 trillion KRW, and the approval amount ranges from 2.91 trillion KRW to 6.29 trillion KRW.

Figure 1 presents the time series trend of policy fund variables. The time series fluctuations of policy fund variables show a close association with the business cycle. The approval amount ranges from 2.91 trillion KRW in 2003 to 6.29 trillion KRW in 2020, showing counter-cyclical expansion to 5.95 trillion KRW and 6.29 trillion KRW at the financial crisis of 2009 and the COVID-19 pandemic of 2020, respectively. The application amount also shows a similar pattern, maintaining an average of 1.73 times the approval amount.

4.3 Demand Function Explanatory Variables

Table 2 presents the descriptive statistics of demand function explanatory variables. GDP continuously grew from 998.29 thousand-trillion KRW in 2000 to 2,292.20 thousand-trillion KRW in 2024. The SME lending rate shows a downward trend from 7.95% in 2000 to 2.97% in 2020. The SME BSI difference fluctuates between -13.67 in 2012 and 21.42 in 2010, 2021, while the SME credit conditions difference fluctuates between -36.75 in 2003 and 34.25 in 2009, reflecting business cycle fluctuations.

Figure 2 presents the time series trends of demand function explanatory variables. GDP continuously increased throughout the analysis period, while the SME lending rate gradually declined from high levels in the early 2000s, reaching its lowest level in the early 2020s before rising again. The SME BSI difference and credit conditions difference fluctuate with the business cycle, exhibiting particularly large variations at the financial crisis of 2009 and the COVID-19 pandemic of 2020.

4.4 Supply Function Explanatory Variables

Table 3 presents the descriptive statistics of supply function explanatory variables. The SME budget share was at the normal level of 7.7% but surged to 29.01% in 2022. Government spending to GDP fluctuates between 15.89% in 2002 and 26.68% in 2022, reflecting fiscal expansion during crises. The KTB yield fluctuates between 0.99% in 2020 and 8.30% in 2000, while the CLI difference shows an average of 2.78, indicating that expectations of economic improvement generally prevailed, yet exhibits considerable variability with a standard deviation of 1.45.

Figure 3 presents the time series trends of supply function explanatory variables. The SME budget share surged at the COVID-19 pandemic of 2020, and government spending to GDP also shows an expansionary pattern during crisis periods. The KTB yield shows a downward trend from the early 2000s before rising again in the early 2020s. The CLI difference fluctuates reflecting the business cycle.

5.1 Model Estimation Results

The structure of SME policy fund demand and supply was estimated through the truncated normal distribution model.5) A total of 15 parameters were estimated for 25 observations. To prevent numerical instability arising from scale differences across variables, all explanatory variables were transformed into standardized scores.6)

Table 4 presents the estimation results of the truncated normal distribution model. In the demand function, the GDP coefficient of 0.564 and the lending rate coefficient of -0.398 are statistically significant, confirming that economic growth and interest rate fluctuations are key determinants of policy fund demand. In the supply function, the government spending coefficient of 0.459, the base rate coefficient of -0.341, and the business cycle coefficient of 0.298 are significant, confirming that fiscal policy and monetary policy serve as primary determinants of supply. The application coefficient is estimated at 0.550, suggesting that approximately 55% of the application amount contains pure demand information.

5.2 Interpretation of Demand Function Parameters

The GDP coefficient is estimated at 0.564 (p< 0.01), indicating that a one standard deviation increase in standardized GDP is associated with a 0.564 standard deviation increase in policy fund demand, demonstrating a strong positive relationship. This confirms the co-movement of funding demand with SME investment expansion and increased economic activity, and reflects the pro-cyclical demand characteristic in which policy fund demand expands during periods of economic growth.

The SME lending rate coefficient is estimated at -0.398 (p < 0.01), indicating that a one standard deviation increase in the standardized lending rate is associated with a 0.398 standard deviation decrease in policy fund demand. This contradicts the economic intuition that demand for relatively low-cost policy funds should increase when market interest rates rise. This result can be interpreted as follows. When interest rates rise, the overall financial market contracts and policy fund screening standards are also likely to be strengthened, which in turn contracts firms’ fund procurement and decreases policy fund demand. While this result contradicts the typically expected substitutability relationship between policy funds and market funds, it can be interpreted as the overall investment contraction effect from interest rate increases overwhelming the substitution effect. In other words, firms perceive rising interest rates as a signal of economic slowdown and maintain a conservative attitude toward leveraged business expansion. The actual data also exhibits a correlation of -0.844 between policy fund approval amounts and the SME lending rate, consistent with the estimation result.

The SME BSI coefficient is 0.114 and the credit conditions coefficient is 0.089, both showing positive signs but not statistically significant.

5.3 Interpretation of Supply Function Parameters

The government spending coefficient is estimated at 0.459 (p < 0.05), indicating that a one standard deviation increase in standardized government spending to GDP is associated with a 0.459 standard deviation increase in policy fund supply. This reflects the tendency for policy funds to expand during periods of expansionary fiscal stance.

The base rate coefficient is estimated at -0.341 (p < 0.05), indicating that a one standard deviation increase in the standardized KTB yield is associated with a 0.341 standard deviation decrease in policy fund supply. This is interpreted as the government taking a passive stance toward expanding policy funds due to increased borrowing costs. Accordingly, policy fund supply also decreases during monetary policy tightening periods, indicating a synchronization tendency between monetary and fiscal policies.

The CLI coefficient is 0.298 (p < 0.05), indicating that a one standard deviation improvement in the CLI is associated with a 0.298 standard deviation increase in policy fund supply. This reflects the tendency for policy funds to expand when economic prospects improve.

The budget share coefficient is estimated at 0.014 but is not statistically significant.

5.4 Interpretation of Application and Excess Demand Parameters

The application coefficient is estimated at 0.550 (p < 0.05), indicating that approximately 55% of the application amount reflects equilibrium demand. The remaining 45% can be regarded as noise arising from institutional distortions or application practices.7)

The excess demand parameter is estimated at -4.605 (p > 0.05), yielding a very small value of $\delta = e^{-4.605} \approx 0.0100$.8) This implies that the threshold of the truncation condition is located very close to the supply level. A small indicates that the model’s truncation condition is satisfied even when demand only marginally exceeds supply.

5.5 Funding Gap Analysis

The funding gap represents the difference between SMEs’ latent policy fund demand and actual supply, serving as an indicator that quantifies the funding access constraints due to market failure. The magnitude of the funding gap is evaluated by comparing the predicted demand and supply values estimated from the model with the actual approval amounts.

The funding gap analysis, converting the standardized model estimates back to their original scale, yields predicted demand of 2.92 trillion KRW in 2000, continuously increasing to 5.58 trillion KRW in 2024. Predicted supply surged to 5.69 trillion KRW and 6.19 trillion KRW in 2020 and 2021, respectively.

The demand-based average $\lambda$ (the ratio of predicted demand to actual approval) is 1.36, suggesting that latent demand exceeds actual approval by 35.5%. The supply-based average $\lambda$ is 1.24, indicating 24.0% of capacity relative to the actual supply. The appropriate supply of policy funds meets this latent demand, with the appropriate supply quantity corresponding to a $\lambda$-1 multiple expansion of the actual supply. The annual average demand gap over the observation period is 1.07 trillion KRW, indicating that additional supply meeting this gap is required for the appropriate supply of policy funds.

During the economic crises of 2009 and 2020 - 2021, actual approval amounts of 4.81 trillion KRW in 2009, 5.86 trillion KRW in 2020, and 5.54 trillion KRW in 2021 exceeded the predicted values, confirming that policy funds exercised a counter-cyclical function.

Figure 4 presents the time series trend of the funding gap. The predicted demand continuously increased from 2.92 trillion KRW in 2000 to 5.58 trillion KRW in 2024, reflecting structural demand expansion driven by economic growth. The predicted supply maintains a relatively low level compared to demand and surged to 5.69 trillion KRW and 6.19 trillion KRW during the COVID-19 response in 2020 and 2021, confirming crisis-responsive supply expansion. The actual approval remained lower than the predicted supply during normal times but surged to 4.81 trillion KRW during the 2009 financial crisis and to 5.86 trillion KRW and 5.54 trillion KRW during the pandemic of 2020 and 2021, approaching or exceeding the predicted supply level, confirming that the counter-cyclical function of policy funds operated appropriately.

6.1 Academic Contribution

This study makes two methodological contributions. First, it demonstrates that a Bayesian structural model based on a truncated normal distribution can be applied to equilibrium analysis of a market without a price mechanism. By decomposing the joint probability of the observed trading volume and the demand constraint into a supply likelihood and a truncation probability, the model identifies latent demand that is not directly observed. Second, the framework that utilizes application and approval data as signals of demand and supply is generalizable to other administratively allocated resources that lack a market price, such as credit guarantees, public housing, and subsidized loans, providing a transferable approach to estimating appropriate supply levels in market failure domains.

6.2 Policy and Practical Implications

The estimated demand gap provides a quantitative reference for budget planning. The annual average demand gap of 1.07 trillion KRW can serve as an empirical basis for setting policy fund budgets in the annual appropriation cycle, replacing incremental adjustments to the previous year’s budget with a gap-based target. The counter-cyclical expansion observed in 2009 and 2020 - 2021 indicates that the timing of supply, not only its scale, matters for crisis response, suggesting that pre-committed expansion rules tied to business cycle indicators can improve the responsiveness of future crisis policy. The application coefficient ($\lambda$ = 0.550) implies that approximately 45% of application amounts reflects strategic overbidding, indicating that mechanisms to curb excessive applications, such as application fees or eligibility pre-screening, can improve allocation efficiency. For SMEs, the gap between latent demand and actual approval informs funding decisions by clarifying the extent of unmet demand, while for financial institutions, the structural excess demand of BBB- and below firms identifies the segment where commercial credit supply is most constrained.

The limitations of this study are as follows. First, the estimation of 15 parameters with 25 observations entails a limitation in statistical estimation efficiency due to the large number of estimation parameters relative to the sample size. Longer data series are expected to reduce the standard errors of estimates and improve numerical stability further. Second, due to the characteristics of the truncated normal distribution model, the likelihood function changes sharply near the truncation point, raising the possibility of numerical instability during optimization. Future research needs to expand the sample size by utilizing quarterly or monthly data and to improve numerical stability and estimation efficiency by introducing alternative estimation methods or regularization techniques.