Stochastic efficiencies of network production systems with correlated stochastic data: the case of Taiwanese commercial banks

Abstract

Although the business environment is stochastic, deterministic data envelopment analysis (DEA) models are typically used to measure the efficiency of commercial banks for the purpose of simplicity. Bank operations are characterized by a network structure due to the dual role of deposits, which, on the one hand, are the output of the process of borrowing funds from depositors and, on the other hand, are the input of the process of making loans. Since the outputs of the production process of the bank are correlated with its inputs, the model for measuring efficiency in this case is a stochastic program with correlated data. To take the correlation between the inputs and outputs into consideration, in this paper, a standard normal transformation is applied for the correlated data, and a network stochastic model is developed to obtain the distribution of the stochastic efficiency. The model is used to measure the efficiency of twenty-two commercial banks in Taiwan. The results are more reliable, discriminative, and informative than those obtained from the existing models. They also show that the performance of a bank is mainly affected by its loan performance. Different from the stereotype suggesting that private companies usually operate more efficiently than state-owned companies, public banks perform better than private banks in Taiwan.

Appendix

1.1 List of symbols

$e_{k}^{S}$ :

Stochastic system efficiency for DMU k

$\bar{e}_{k}^{S}$ :

Expected value for $e_{k}^{S}$ with t replications

$e_{k}^{(p)}$ :

Stochastic pth division efficiency for DMU k

$\bar{e}_{k}^{(p)}$ :

Expected value for $e_{k}^{(p)}$ with t replications

$E_{k}^{S}$ :

System efficiency for DMU k

$E_{k}^{(p)}$ :

pth division efficiency for DMU k

$R_{j}^{(p)}$ :

Multivariate normal distribution of the pth division for DMU j

$s_{k}^{(S)}$ :

Slack variable of the system for DMU k

$s_{k}^{(p)}$ :

Slack variable of the pth division for DMU k

$u_{0}$ :

Total value of returns-to-scale for q divisions

$u_{0}^{(p)}$ :

Value of returns-to-scale for the pth division

$u_{r}$ :

Weight of exogenous output r

$v_{i}$ :

Weight of exogenous input i

$w_{f}$ :

Weight of endogenous input f

$w_{g}$ :

Weight of endogenous output g

$X_{ij}$ :

Exogenous input i consumed by DMU j

$Y_{rj}$ :

Exogenous output r produced by DMU j

$Z_{fj}$ :

Endogenous input f consumed by DMU j

$Z_{gj}$ :

Endogenous output g consumed by DMU j

$\upalpha_{k}$ :

Constant for the adjustment of the system efficiency $E_{k}^{S}$ for DMU k

$\bar{\upalpha }_{k}$ :

Expected value for $\upalpha_{k}$ with t replications

$\upbeta_{k}^{(p)}$ :

Constant for the adjustment of the pth division efficiency for DMU k

$\bar{\upbeta }_{k}^{(p)}$ :

Expected value for $\upbeta_{k}^{(p)}$ with t replications

$\zeta_{j}^{(p)}$ :

Standard normal random variable of the pth division for DMU j

$\upmu_{ij}^{X}$ :

Expected value of exogenous input $X_{ij}$

$\upmu_{rj}^{Y}$ :

Expected value of exogenous output $Y_{rj}$

$\upmu_{fj}^{Z}$ :

Expected value of endogenous input $Z_{fj}$

$\upmu_{gj}^{Z}$ :

Expected value of endogenous output $Z_{gj}$