Robust Estimation of Marginal Abatement Cost of Water Pollutant: Evidence from Yangtze River Basin

Based on the econometric method of bootstrap for 1000 times, this study estimated the marginal abatement cost of water pollutant, COD and NH3-N, for 9 provinces and 2 municipalities in the Yangtze River Basin from 2001 to 2017 by building a robust estimation model. We found that the marginal abatement cost increases at an average annual rate of 19.47%. The marginal abatement cost in Jiangsu and Sichuan are significantly higher than other regions. Differences in marginal abatement cost make it possible to trade emission rights between regions.


Introduction
The measurement of the marginal abatement cost (MAC) of water pollutants is essential for judging the economic cost, pressure and potential of reducing water pollutants. The theoretical basis of the MAC is the duality in the theory of environmental production technology. The essence of MAC is the marginal conversion rate of good output to bad output.
Generally, when the value of the good output lost to reduce one unit of pollutant emissions is greater, the price of unit pollutant emission right will be higher. Otherwise, rational manufacturers lack sufficient incentives to participate in selling the emissions rights. However, if the emission rights trading market is absent, the historical price of emission rights is naturally missing. The MAC, an opportunity cost to reduce pollutant emissions, is regarded as the shadow price of pollutants, which can reference the formation of the initial price of emissions rights trading. Thus, measuring the MAC of water pollutants can provide policy decision for improving market-based and pricebased water related environmental regulations.
The Yangtze River Basin is one of the most critical economic and industrial water pollutants discharge concentrated areas in China. From 2001 to 2017, the average annual industrial production value of the nine provinces and two municipalities through which the Yangtze River flows reached 3.090 billion CNY, accounting for 28.11% of the country, and industrial wastewater discharge accounted for 47.47% of the total emissions. The industrial wastewater discharge in the provinces (municipalities) in the Yangtze River Basin is still under tremendous pressure. Obviously, the COD and NH3-N of industrial water pollutants are the focus of environmental control. The "14 th Five-Year Plan" proposes to reduce COD and NH 3 -N emissions by 8.0%, respectively. However, the study on the MAC of industrial water pollutants is limited.
This study aims to answer the following 2 questions: (1) What is the MAC of COD and NH 3 -N in 9 provinces and 2 municipalities along the Yangtze River Basin? (2) What is the temporal evolution and spatial distribution of the MAC of water pollutants? The remainder of this study is organized in the following manner. Section 2 introduces the model and presents the empirical specification. Section 3 presents the data, variables and reports the main results. Section 4 presents a further discussion and summarizes this study.

Method
We use a framework of environmental production technology and the directional distance function (DDF) to model the efficiency assessment to estimate the MAC of pollutants.
Denote inputs by , good outputs by , and bad outputs by . The technology is represented by the output sets as .
In the framework of environmental production technology, the inputs and good outputs are assumed to be freely disposable while the good and bad outputs are jointly weakly disposable. While no good output can be produced without bad output, which is referred to as the null-jointness assumption. The DDF can be described as follows: ( 2) where is the directional vector, and β is the value of DDF which means the maximum expansion of good outputs and contraction of bad outputs.
Following the research of Färe and Grosskopf [1] , the directional vector is set as . The Parametric quadratic specification of DDF can be formulated as follows: where k denotes the DMU and t denotes the year. Formula (4) and (5) aim at minimizing the sum of the deviations of the distances subject to the properties of feasibility, nonjointness, monotonicity, translation and symmetry.
(4) s.t. (5) Denotes the parameter vector by using single estimate and bootstrap as , , where the s means the times of sampling. For each year we have k DMUs, thus the probability of drawing from the original sample (with reversion sampling) to each observation point is 1/k [2] . Based on and duality, we can calculate the MAC (q * ) for bootstrap as follows: (6) 3 Data and results

Data source
The input factors include capital (x 1 ), labor (x 2 ), and industrial water consumption (x 3 ), the good output is the industrial production value (y 1 ), and the bad outputs are the COD (b 1 ) and NH 3

Parameter estimation
Based on bootstrap for 1000 times, the parameters of single estimate and bootstrap revision for the quadratic form of the DDF are provided in Table 2.

Analysis of time difference in MAC
Substituting the parameters revised by bootstrap into the formula 5, can estimate the MAC of COD and NH 3 -N.      Figure 3 shows the extent to which DMU's MAC deviates from the mean level. From 2001 to 2017, the average MAC of COD is 14.05 million CNY/Ton, and the average MAC of NH 3 -N is 9.0925 million CNY/Ton. The MAC of COD in each DMU deviates from the overall mean by a small amount, while MAC of NH 3 -N deviates significantly as its standard deviation is 1439.72 (10 4 CNY/Ton). Especially, the MAC of NH 3 -N in Jiangsu, Shanghai, and Sichuan is significantly higher than the overall mean level. DMUs that fell below the overall mean are Qinghai, Yunnan, Chongqing, Hunan, Jiangxi, and Anhui, mainly located in central and western parts of China.

Analysis of regional difference in MAC
We select the Theil index to measure the regional difference and variation range of MAC reduction. Figure  4 shows that the regional differences of MACs of COD and NH 3 -N have increased significantly compared with the beginning of the period. More than 50.00% of the regional differences come from intra-regional differences, which is also consistent with the results of the spatial characteristics of the MAC. Due to limited space, the reader can contact the author for detailed data.

Discussion and Conclusion
Prior work has estimated the MAC by non-parametric data envelopment analysis, parametric stochastic frontier analysis and single estimate under parametric function by DDF. [3,4] However, these studies either fail to deal with the impact of the random effect on parameter estimation or fail to impose constraints on the technical production function, resulting in the deviation of parameter estimation. [5,6] In this study, we use bootstrap 1000 times to obtain the sampling distribution of estimated parameters through random sampling and multiple estimations of model parameters to revise the deviation of a single estimated parameters and obtain a more robust estimate of the MAC.
We found that MAC increased year by year on the whole, while some provinces showed a downward trend. If the output of pollutants is low, it becomes more challenging to take a step forward to reduce pollution emissions, and the value of the economic output forgone to reduce a unit of pollution emissions is high. There is an apparent upward trend in the MAC in Sichuan and Yunnan. Thus, the government can transfer the emission reduction tasks from these provinces to regions with a relatively lower MAC, such as Zhejiang, Jiangxi and Hunan provinces in eastern and central China. Therefore, future work should include follow-up work designed to construct the price formation mechanism of emission rights trading in the Yangtze River basin.