# Model solving method

Whether it is the distance function or the directional distance function developed in the later stage, the production boundary they use to construct is to use the multiple sets of input-output data to derive the production front, and to make the decision-making units in the sample and the production frontier. The best advantages are compared to solve the relative efficiency values of each decision unit. At present, the solution to the model can be generally divided into two types: parameterized and non-parametric. The parametric solution mainly includes: Parametric Line Program (PLP) and Stochastic Frontier Analysis (SFA). Here, translog, quadratic, and hyperbolic functions are adopted for the parameterized distance function form; nonparametric solutions mainly refer to Data Envelopment Analysis (DEA). The following mainly introduces four widely used solving methods.

## Parametric distance function solution based on translog function

The parameterized output/input distance function method can overcome the shortcomings of the exponential method. Fare et al. (1993) first used the parameterized distance function to study environmental sensitivity productivity. The idea is to select the super-logarithm function to parameterize the output distance function Z)„(.v,m)^{13} and to minimize the distance between all samples and the production front by linear programming constraints, and the value of the output distance function is environmentally sensitive productivity. Its super-logarithmic function is set to:

**
**

Assuming that the function of equation (1.9) has general symmetry and homogeneous constraints, the method of Aigner and Chu (1968) is used to minimize the deviation of the sample from the leading edge, that is, to solve the following linear programming problem:

s.t.

where *k* = 1,..., *K* represents different observational samples, the first *i* outputs are desirable outputs, and the later *(m* - z) outputs are undesired outputs. The objective function in equation (1.10) is to minimize the deviation of all samples from the optimal leading edge. Constraints (i) ensure that each sample is at the leading edge or below the leading edge; constraint (ii) guarantees that the shadow price of the desired output is non-negative; constraint (iii) guarantees that the non-conforming output shadow price is not positive; constraint

- (iv) pairs the output, which is applied once in a row to ensure that the production technology meets the output weak disposal assumption; constraint
- (v) is a symmetry constraint. Once the values of the parameters in the distance function are solved using (1.10), the environmental sensitivity productivity of the sample and the shadow price of the undesired output can be calculated.

## Solution of directional distance function based on quadratic function

If the directional distance function is set instead of the distance function, the super-logarithmic function is generally not used, and the quadratic function is used, because the quadratic function satisfies the constraints required for the directional distance function characteristic (Fare & Grosskopf, 2006). Generally, the direction vector g = (1, -I)^{14} can be set. Assuming *k —* 1, ..., *K *represents a different observation sample, the quadratic direction distance function can be expressed as:

The parameter solution is also based on the idea of linear programming, which minimizes the sum of the distances of the observations to the boundary.

s.t.

Constraints (i) ensure that each sample is at the leading edge or below the leading edge; constraints (ii) and (iii) ensure monotonicity of desirable and undesired outputs, respectively, while constraints (iv) also apply to inputs The monotonic constraint, constraint (v), satisfies the transformation property of the direction distance function, and the constraint (vi) is the symmetry constraint. Using (1.12) to solve the values of each parameter in the direction distance function, the environmental sensitivity productivity of different samples can be obtained, and the shadow price of the undesired output can be calculated.

## Stochastic Frontier Analysis (SFA)

The stochastic frontier production function was first proposed by Aigner et al. (1977). In the study of environmental sensitivity productivity, it is also used as a parameter estimation method. Compared with the deterministic parameter estimation method, it will be caused by uncertain factors. The impact is taken into consideration, from the aspects of technical inefficiency or random error, to find out why the sample production is inefficient and deviating from the production boundary. More importantly, the SFA method can give the statistics of the variables to be estimated. Its conclusions are more robust in terms of other parameter estimation methods.

Murty and Kumar (2003) used SFA and output distance functions to evaluate production efficiency. The stochastic output distance function is defined as follows:

*D„* is the distance function value, *F(.)* represents the production technique, *X* and *Y* are the input and output vectors, a and p are the parameters to be estimated, and e is the error term. Because the data of the dependent variable *D„* cannot be directly obtained,^{15} in order to solve this problem, Ferrier and Lovell (1990), Grosskopf and Hayes (1993), Coelli and Perelman (1996), and Kumar (1999) use the output distance function once. The subfeatures are transformed, and in the case of ignoring the perturbation term, equation (1.13) is transformed into:

12 *Emission reduction analysis*

Generally, a scaling variable can be arbitrarily selected. For example, if the

Afth output is selected, let A = —, then (1.14) becomes:

Km

Take the logarithm of the above formula (1.15) and become

*f* can be expressed as a logarithmic form of a function expression, further transformed into:

After adding the random error v and the production inefficiency error *u *(i.e., the -ln(Z>„) term), the stochastic boundary yield distance function is expressed as:

Estimating the (1.18) formula gives the parameter values to be estimated and their statistics.

In addition, the directional distance function can also be estimated using a random frontier function method. Fare et al. (2005) used Kumbhakar and Lovell (2000) to set the random frontier function. Using the directional distance function to calculate, the definition of production technology *T =* |(x, *y,* Z>)(H*^{+M+J}, (y,Z>)P(.v), aL( v,Z>)| , and its function is as follows^{16}:

where e* = v* - n^{A}, v* is a random statistical error, v*~ N (0, <7’ and w*is due to technology. The error caused by inefficiency, *u *~ N (0, cf ), v*and u*areindependent and identical, and independent of each other, and then use g The conversion of the directional distance function at g = (1, -1) is:

If we put the figure into (1.19):

Generally, a* =// is used, and then the equation (1.21) is estimated by OLS or maximum likelihood methods to calculate the environmental sensitivity productivity, and the method can also estimate the statistic of each coefficient.

## Solution based on non-parametric data envelope analysis (DEA) method

The DEA method has a large number of applications in the production environment frontier function. In recent years, with the deepening of the distance function and directional distance function research, research on measuring the environmental sensitivity productivity by DEA method has emerged (Fare et al., 1989; Ball et al., 1994; Yaisawarng & Klein. 1994; Chung et al., 1997; Tyteca, 1997; Lee et al., 2002; Kumar, 2006: Hu Angang et al., 2008; Tu Zhengge, 2009).

Suppose there are input and output data for *k* samples *(yk, bk, xk), k-* 1, *..., K.* When production activities are subject to environmental regulations, the environmental production equation for the fcth sample is expressed as follows:

s.t.

where *zk (k =* 1, ..., *K)* is the intensity variable, the purpose is to give each observation sample point weight when establishing the production boundary.

If the *zk* is not accumulated and limited, the model is fixed-scale compensation, and vice versa remuneration for scale. On the basis of the establishment of the boundary, the objective equation maximizes the desired output, and the constraint on the undesired output reflects its weak disposition, that is, the reduction of undesired output will inevitably lead to the reduction of desirable output. The right side of the second constraint inequality represents the input in actual production, and the left side represents the most efficient production input in theory. The inequality indicates that the theoretical input must be less than or equal to the actual production input, and also indicates the free disposal of the input.

However, the shortcoming of the above method is that it does not take into account the reduction of undesired output, but only the pursuit of the maximization of desirable output. Chung (1997) considers the development of the directional distance function and uses the DEA method while considering research on productivity issues under the premise of an increase in desirable output and a decrease in undesired output.

If producer |[A:'(x_{A}..,y_{A}.,ft_{r}) is defined, the directional environment production frontier function under the reference technique A (x j can be expressed as:

s.t.

Compared with model ( 1.22), model ( 1.23) imposes constraints on the desired output, thereby increasing the desired output while minimizing the unintended output. It is possible to reflect the connotation of environmentally sensitive productivity by considering the expansion and reduction of output in two different dimensions.

Parametric estimation and non-parametric estimation have their own advantages. In general, the parametric method needs to preset the distance function as a certain function expression. The advantage is that the parameterexpression can be differentiated and algebraic (Hailu & Veeman, 2000). By means of linear programming, stochastic frontier analysis, etc., the parameter values in the distance function can be estimated, and the environmental sensitivity productivity values of each decision unit and the shadow price of the undesired output are calculated. However, if linear programming is used to solve the parameters, the relevant statistics are often not available (Hailu & Veeman, 2000)^{17}; if the random frontier rule is used, the parameter values and corresponding statistics can be calculated, and the inefficiency can be further decomposed into techniques. Inefficiency, allocation inefficiency and random error, but the method also requires a preset function form, and the distribution of error terms is assumed to be strong.

When non-parametric DEA is used to estimate the production front, as there is no need to make a priori assumptions on the production function structure, no parameters need to be estimated, no inefficient behavior is allowed, and total factor productivity (TFP changes) can be decomposed, so there is more attention and application (Fare et al., 1998). In addition, the non-parametric DEA approach avoids residual autocorrelation when using time series or panel data (Fare et al., 1989; Yaisawarng & Klein, 1994). However, the non-parametric DEA method is sensitive to the sample data. The error of the abnormal sample value will affect the position of the production frontier, and then affect the value of the environmental sensitivity productivity. Therefore, the accuracy of the sample data is high. In addition, the non-parametric DEA method can generally only be used for productivity measures and is rarely used to estimate shadow prices for undesired outputs (Fare etal., 1998).