In the theoretical hypothesis, the original part of the hypothesis needs to be relaxed, but the difficulty of the model is increased

This is mainly reflected in two aspects. The first is the assumption of the shadow price symbol. In the existing theoretical model, in order to ensure that the solution of the model has economic significance, the shadow price of the unsatisfactory output is generally set to be non-positive, especially in the process of solving the parameterized function. The monotonicity of the undesired output to the distance function equation is specified. There are also some literatures that use the DEA method to solve the problem. After the shadow price is obtained, the shadow prices of different positive and negative symbols are interpreted or rejected. However, as discussed by Van Ha et al. (2008), some pollutants, such as suspended particles (mostly wood residue) in wastewater in the paper industry process, although appearing to be “unwanted” pollutants, It can be recycled as raw materials through different processes, thus turning “sub-output” into “positive output,” and its shadow price becomes positive. Therefore, it is necessary to relax the existing shadow price of undesired output.

Second is the assumption of “complete efficiency” and “no redundancy.” As pointed out by Lee et al. (2002), previous literature have assumed that the production front is completely efficient, but under the premise of certain technology, the unit inputs, outputs, or unit-desired outputs of each decisionmaking unit are not desirable. Outputs are all different. If imperfect efficiency is considered, the results of environmentally sensitive productivity will necessarily differ. Therefore, the shadow price of pollutants calculated based on the assumption of full efficiency will be different from the result of incomplete efficiency when other conditions are the same. As pointed out by Boyd et al. (1996), there is a gap between the theoretically estimated shadow price of pollutants and the actual observed price of pollutant emission trading in the trading market, possibly due to the imperfect efficiency.¹⁸ In addition, Fukuyama and Weber (2009) further pointed out that most of the existing directional distance function studies do not take into account the possible redundancy (slacks), and redundancy will also lead to imperfect efficiency, resulting in biased environmental sensitive productivity.

In the algorithm implementation, the function form setting is quite different, and the calculation process is still complicated

First of all, the function form and estimation method are quite different, and each has its own advantages and disadvantages. Although there are two kinds of distance function and directional distance function in the theoretical model, the setting and solving methods of the function form are very different. As has been summarized above, the parametric method solution includes two main forms: deterministic function analysis and SFA. The deterministic function can be set to super-logarithm, quadratic or hyperbolic; DEA

Literature review 25 requires the use of a set of linear programming equations (inequalities) to find the optimal solution.

The commonly used method in the empirical estimation of the distance function is the method of deterministic linear programming, which needs to set the function form, and only a small number of methods using measurement estimation (Hetemaki, 1996).¹⁹ The advantage of linear programming is that it is relatively simple to use without any distribution assumptions, even in the case of small samples, a large number of parameters can be calculated; the disadvantage is that the parameters are calculated rather than estimated (Kumbhakar & Lovell, 2000), so provide statistical criteria for consistency of conclusions, which may lead to bias in evaluations, as outputs may be affected by random disturbances. Some documents adopt a two-step analysis method to solve this problem; that is, first use the linear programming method to calculate the distance function, and then use the distance function value as the explanatory variable, and use the parameter random distance function to estimate the parameters.²⁰ Although the non-parametric method has the advantage of not having to set the function form, when calculating the shadow price of the contaminant, the shadow price cannot be obtained by differential calculation, and the statistic cannot be provided. In addition, the production frontier boundary is easily interfered by the error points, causing the result to deviate significantly from the actual situation.

In addition, when the directional distance function is applied, the selection of the directional vector is relatively simple. In the general theoretical research, a relatively neutral attitude is adopted, which is determined as (1, -1); that is, the ratio of expansion and contraction of desirable output and undesired output is 1. However, not all governments have a neutral preference. According to different research needs and policy preferences, the specific choice of directional vector should not be fixed as (1, -1), but so far no scholars have conducted research on the selection of non-neutral vectors, so the theoretical results that are more focused on the expansion of desirable output or more biased towards the unsatisfactory output remain to be explored.

Calculation process is more complicated and generally requires programming. Because the number of selected research objects is often large, more than 100 decision-making units, plus the model itself has several constraints on each unit, the calculation process is more complicated. At the same time, because the directional distance function is still a relatively new research field, there is no relevant solving software and program at present, which generally requires the researcher to implement it by himself,²¹ which also hinders the popularization of related research to some extent.

In the conclusion of the study, there is a certain gap with the reality, and it is still necessary to strengthen its policy significance

Because the models, data, and calculation methods used by many research institutes are different, the differences in their research conclusions are alsolarge, and there is a certain gap between theoretical expectations and actual observations. For example, in the study of SO₂ emissions from power plants, the average environmental sensitivity productivity is about 0.9, and the variance is small, indicating that although these power plants may have different production equipment and technical levels, they are “close to the best frontier of production.” This is in contrast to the intuition in practice; in addition, the SO, shadow price estimated from environmental sensitivity productivity ranges from $167/ton to $l,703/ton, while the market price of US SO, license transactions is S64 200/ton (Ellerman et al., 2000). The large differences between these research findings and reality indicate that the existing model settings may require further revision and improvement, as the full efficiency assumptions mentioned above may need to be further relaxed.

The practical application and policy significance of the conclusions of environmental sensitivity productivity research can be summarized into three aspects. First, quantitative evaluation of environmental performance and environmental productivity of economic units producing “unintended outputs” to verify whether “environmental regulation” it affects the productivity of enterprises and the competitiveness of enterprises. Second, with the analysis of environmental sensitivity productivity, the marginal abatement cost of pollution of different enterprises and departments can be measured, so as to set the initial price and environment of the pollutant trading market. Taxes and fees are provided as a basis; in addition, the findings of environmentally sensitive productivity can be further extended to the estimation of environmental control costs, thus providing a reference for the formulation of pollutant control policies. The investigation of environmental performance in different industries and different regions can guide low-efficiency units to promote high-efficiency production. The measurement of environmental control costs will guide policy-makers to make appropriate environmental control policies under the predetermined policy objectives. The determination of the shadow price can help the regulator to set the penalty for the discharge of different pollutants, and the manufacturer can also use this information to determine whether it is cost-effective to purchase the emission right, so as to carry out the most efficient production activities. Of course, all of this is based on the results of theoretical research with reference and reproducibility.