Measuring of China's Provincial Carbon Abatement Cost and its Spatial Distribution Pattern

Based on the panel data in Chinese provinces from 2000 to 2017, this paper first uses the parameterized quadratic function of the directional distance function to estimate carbon abatement costs of 30 provinces in China, and further studies its long-term evolutionary characteristics. Second, this paper studies the spatial distribution pattern of carbon abatement cost. The results show that the carbon abatement cost has increased as a whole during the study period. Moreover, the spatial distribution of carbon abatement costs in China shows a geographical clustering feature, and the positive spatial agglomeration is significant after 2008.


Introduction
The burning of fossil fuels has produced a great amount of carbon dioxide worldwide since the Industrial Revolution. The contradiction between resource environmental constraints and economic development has become increasingly prominent. Therefore, it is a common responsibility for all countries to take measures to reduce carbon emissions, especially for China. Recently, China government reiterated that China will increase its nationally determined contribution, strive to reach the carbon peak by 2030 and achieve carbon neutrality by 2060. To achieve the carbon emission reduction targets, it is crucial to analyze the long-term evolutionary characteristics of marginal carbon abatement cost (MAC).
The carbon abatement costs have attracted the widespread attention of scholars. For example, Choi et al. (2012) employed the dual model of the slacks-based Data Envelopment Analysis (DEA) model to estimate the abatement costs of CO 2 emissions [1]. Duan et al. (2018) calculated the marginal abatement costs of carbon emissions in different ways, found that the carbon abatement cost in all provinces raised from 2005 to 2015 [2]. Xue (2021) explored the convergence of MAC during 2001-2015 and pointed that the spatial effect should not be ignored when studying the marginal abatement cost [3]. Actually, rare studies have empirically analyzed the spatial autocorrection of MAC. Thus, to supply for the existing literature, this paper employs a Directional Distance Function (DDF) to estimate the marginal carbon abatement costs of 30 provinces in China and analyze the long-term evolutionary characteristics of MAC. Whereafter, we calculate the Moran's I at the global and local scale to analyze the spatial correction of MAC.  Corresponding author: lzw1226x@163.com 2 Methodology and data

Measuring the carbon abatement cost
This paper use DDF to estimate carbon abatement cost. Consider there is a producer who uses inputs to produce desirable outputs . However, undesirable outputs are also produced in the meantime. Hence, the production function of the producer shows in formula (1): ( , )} P x y b x can produce y b  (1) Due to the translation property, we use a quadratic form of the directional distance function expressed as formula (2). We select capital, labor, and energy consumption as input indicators. GDP was determined as the desirable output, and carbon emissions were adopted as the undesirable output.
(1 1, This paper uses Shepard's Lemma, represents the market price of the desirable output, denote 1, thus, we can derive the carbon abatement cost as formula (5)

Variable selection
Input indicators in the directional distance function are capital, labor, and energy consumption. Thus, the capital is represented by fixed asset investment 1 , the labor force is measured by the number of year-end employees, and the energy consumption is represented by the total consumption of fossil fuels. Moreover, the desirable output indicator, GDP, is measured by the real GDP of each province. As for the undesirable output, carbon emission is calculated as follows: where denotes net calorific value of the mth type of fossil fuel, is the carbon content, and is oxygenation efficiency of the jth type of fossil fuel. 44/12 represents the ratio of the molecular weight of carbon dioxide to carbon atom.

Measuring the dynamic evolution of carbon abatement cost
This paper employs Kernel density estimation (KDE) to describe the actual data distribution and analyze the dynamic evolution of MAC in China. The KDE can be defined as: Where represents the kernel density value, ℎ

Global spatial autocorrelation
To test whether there is a spatial correction in provincial MAC, we employ the global Moran's I index proposed by Cliff and Ord (1981). The specific formula of Moran's I is: where is the spatial weight matrix between region i and region j; is the MAC; is the variance of samples. The value of Moran's I is between -1 to 1, when the value of I is bigger than 0, it indicates that there is a positive spatial correlation, the larger the value of I, the stronger the positive correlation; otherwise, it indicates that there is a negative spatial correlation; When I=0, it indicates that there is no spatial correlation. Anselin (1995) proposed local indicators of spatial association (LISA), which decomposite global Moran's I into the contribution of each observation [5]. This paper uses the Local Moran's I to explore the statistically significant spatial clusters and dispersion of the provinces' MAC. The calculation of Local Moran's I show below:

Data sources
The data sample of this paper cover 30 Chinese provinces  It can be found that from 2000 to 2017, the value of MAC showed an upward trend, and the average value was 1,79.26 yuan per ton, which means that the cost of reducing every 1 ton of carbon emission averagely is 1,79.26 yuan.
In terms of the regional average MAC at different stages, the eastern regions have the highest average annual In terms of the annual average MAC in different regions. The annual average MAC varies significantly among provinces. Shanxi, Hebei, and Shandong have lower average carbon abatement costs, which result from Shanxi has abundant coal resources, and the heavy industry in Shandong and Hebei accounts for a relatively large proportion of the industrial structure. Especially, Shandong's carbon emissions rose to the highest level in the country in 2017. As previous environmental studies founded, the reduction of pollutant emission has a scale effect, which means that the higher the initial emission of pollutants, the lower the marginal emission reduction cost. Provinces such as Qinghai, Hainan, and Beijing have higher average carbon abatement costs, especially, the highest province is Qinghai, which because of the economy of Qinghai and Hainan mainly depend on tourism, and the heavy industry is undeveloped. The eastern region's economy is more developed, the technology is more advanced, and the total factor productivity is higher. Thus, the carbon abatement cost is lower than that of the other two regions. The western region is an underdevelopment region, and the relative poverty makes more room to reduce carbon emission.
The estimated carbon marginal abatement cost in western provinces is higher than that of the other regions, followed by the central region, and the lowest carbon marginal abatement cost is in the eastern region. The carbon marginal abatement costs of western, central, and eastern provinces are 1,726.94 yuan per ton, 1,479.97 yuan per ton, and 1,496.53 yuan per ton, respectively.  Secondly, see from the kurtosis of the KDE curve, the peaks and ranges of different curves experienced varying degrees of change. The modes evolve from a sharp to wide one during the study period, with the height descending, significantly, the dispersion range rapidly widened after 2006, with the height descending rapidly, revealing that the regional gap among MAC of different provinces was enlarging at this stage. Thirdly, see from the shapes of the KDE curve, the curves were Multi-modal before 2006, and with several lumps in the long right tail, while those after 2006 were unimodal, which means that bi-polarization tendency was weakened during the study period.

Local spatial autocorrelation
The results of Local indicators of spatial autocorrelation (LISA) agglomeration of MACs showed in Fig.4. Fig.4 shows

Conclusion
The rapid economic growth in China caused environmental degradation in the past few decades. The Chinese government is trying to reduce carbon emissions through legislation, administration, and economic tools to develop sustainability. The estimation of the marginal carbon abatement cost can provide valuable information for accurate policymaking.
Using the consumption data of seventeen kinds of fossil fuels from 2000 to 2017, we estimate China's carbon emission more accurately. Taking labor, capital, and energy as input, real GDP as desirable output, and carbon emission as undesirable output, this paper estimates the marginal carbon abatement costs using the directional distance function and analyses the spatial distribution pattern of MAC. Our estimate yields several findings. Firstly, carbon abatement costs in every province of China rise from 2000 to 2017. Secondly, the distribution of MAC shows MAC in western provinces is higher than it in central areas, while a higher MAC in central provinces than in eastern ones. Thirdly, the dynamic evolution characteristics show that MAC increases significantly and the MAC regional gap enlarging after 2006. Finally, the spatial distribution of carbon abatement costs in China shows a geographical clustering feature, and the positive spatial agglomeration is significant after 2008. Local spatial autocorrelation results suggest that the majority agglomeration pattern did not change between 2008 and 2017.