Abstract: Due to incomplete information about production technology, firms may use some inputs more efficiently than others, which makes the radial technical efficiency measurement inappropriate.
Different approaches have been proposed to reconcile the non-radial nature of technical efficiency and the measurement methodologies. The revenue function based approach proposed by Kumbhakar and Lai (2016) can be applied to measure output-specific technical efficiency. With minor modification, the framework proposed by Kumbhakar and Lai (2016) can be extended to profit function to measure input-specific technical efficiency (ISTE). This approach has limitation in empirical research in that reliable data on input and output prices are not always available.
The directional distance function approach can also be used to measure ISTE (Färe and Primont, 1995). However, it requires to specify the "directions" toward which inputs can be contracted before data analysis was applied, yet this kind of information is unavailable a priori. Aparicio et al. (2017) proposed a method to measure ISTE based on the principle of least action and the corresponding DEA method was developed to estimate ISTE. It is well known that DEA method is vulnerable to extreme values and it is difficult to distinguish if the distinction between technical efficiency of each input is real or just due to statistical noise.
In this paper a new method is proposed to measure input-specific technical efficiency (ISTE) based on input distance function which requires no price information and no need to specify the "direction" a priori.
Let x_j denote the quantity of input j(j=1,…,K) used in production, y denote output. Let TE_j=exp(-η_j); (0<η_j<∞) represents technical efficiency level of input j. Therefore, x_j⁰=x_j·TE_j gives optimal level of input j while keeping the quantity of output y unchanged. Let X⁰=(x₁⁰,…,x_K⁰) be the optimal input vector. Apparent for input distance function we can get:
D(Y,X⁰)=sup_λ{λ:(X⁰/λ,Y)∈T}=1                                                                                                                                                                                  (1)
Suppose Cobb-Douglas function can be used to approximate the distance function, from (1) we get:
$0 = {\alpha _0} + \sum\limits_{j = 1}^K {{\alpha _j}\ln ({x_j} \cdot T{E_j})} + \sum\limits_{m = 1}^M {{\beta _m}} \ln {y_m} $                                                                                                                                                     (2)
where α and β are unknown parameters. By using the homogenous condition of input distance function, we get the estimable form as follows:
$ - \ln {x_k} = {\alpha _0} + \sum\limits_{j = 1}^{K - 1} {{\alpha _j}\ln ({x_j}/{x_K})} + \sum\limits_{m = 1}^M {{\beta _m}} \ln {y_m} + \varepsilon - \sum\limits_{j = 1}^K {{\alpha _j}{\eta _j}} $                                                                                                                  (3)
where ε is the stochastic error term.
In estimation, a two steps procedure is developed in the context of Bayesian econometrics:in the first step, the parameters of input distance function are estimated consistently; then in the second step, technical efficiency of each inputs are estimated based on the given estimated parameters. Monte Carlo simulation shows that this "two steps" approach can generate much faster convergence in Markov Chain Monte Carlo (MCMC) inference and accurate estimation of ISTE compare to direct Bayesian estimation of ISTE model (3).
Then this method is applied to the data of Industrial Enterprise Survey by Peking University to evaluate the ISTE of labor and capital. This survey covers the period of 2000 to 2005. However, it is an unbalanced panel data set. Output is measured by revenue of each firm. To alleviate the fluctuation of revenue caused by output price change, the average revenue of each firm in this period is used as a measure of output value. Finally a cross sectional data with 1178 firms are gotten from 36 different industries.
The results show that, firms are using capital efficiently and indifferently. While technical efficiency of labor is relatively low and disperse among firms in the sample. On average, technical efficiency of labor is 77%, in other words, labor can be saved by 23% while keeping output and capital use unchanged. The illustration shows that the approach developed in this study can identify the major sources of technical inefficiency, therefore can facilitate to develop strategies to improve efficiency in production.

Key words: input distance function, input-specific technical efficiency, two-steps procedure