Cost And Welfare: A Simulation of Establishing A Unified Carbon Market In China


 BackgroundGlobal warming has aroused wide concern of international community, which has reached a consensus on the carbon abatement. In 2017, China should have established a unified market for carbon emission trading, while the government has postponed the establishment because the uncertainty of cost calculation and welfare. Therefore, the cost and welfare of carbon abatement in simulated scenarios could help the government in establishing a unified carbon market and setting suitable policy. In the national carbon trading market, the variations of different abatement cost are the precondition of carbon exchange. This paper set forth theories related to carbon market and used parametric directional distance function model to derive the shadow prices of 30 provinces from 2011 to 2017. Then the classic logarithmic model is used to simulate marginal abatement cost curves, which is further applied to empirically investigate the welfare of 30 provinces in two scenarios of carbon trading market in China. ResultsThe results indicate that marginal abatement cost would rise with the increasing of emission reduction and vary significantly among provinces, and undeveloped provinces have greater potential in emission reduction than developed regions. Moreover, all provinces could benefit from the establishment of the nationwide ETS.ConclusionsThis article combines the theoretical model of shadow prices with the analysis of China’s carbon trading market in an attempt to analyze the cost and welfare of Chinese provinces and cities on the unified carbon trading market, adding the time trend factor to the directional distance function, and then further combines the parameter method to estimate the shadow price of CO2. Finally, the paper gives some proposals regarding to China’s ETS and carbon reduction targets.


Background
Environmental concerns and issues, especially climate change, are moving from the realm of corporate environment, health, and safety personnel, into that of community of shared future for mankind. Putting a price on greenhouse gas emissions on country and company bottom lines will have a significant effect. At the same time, government climate policies can do much to change behaviour patterns and encourage markets to mitigate these issues [1]. China, one of the biggest carbon emitters in the world, is facing more pressure internationally. To address the problems associated with Launching a unified national carbon market and implementing links to existing pilots means that traders with high marginal abatement costs (MAC) will be able to purchase carbon emission rights from other low-cost domestic trading markets to fulfil their obligations. A unified carbon market can enhance the effectiveness of CO2 reduction, increase market liquidity, and reduce the risk of carbon leakage caused by the difference in emission reduction intensity between regions, but it will also affect the cost-benefit of traders and the distribution of welfares between provinces [2]. Therefore, it is necessary to make a quantitative analysis of the uncertainty of economic welfare in the link of pilot carbon markets [3].
The difference in emission reduction costs is a prerequisite for the establishment of a carbon trading market. According to the research of existing literature, the basic criterion for evaluating the effectiveness of the carbon trading market is that "the effective market price is equal to the minimum marginal abatement costs" [4]. Scholars have done a lot of research on the cost of emission reduction for a long time and have obtained in abundance. Early on, Nordhaus W.D. [5] made a comprehensive study on it and pointed out that MAC was a standard tool for analysing the impact of environmental policies. According to relevant literature, the earliest Marginal Abatement Cost Curve (MACC) was generated after the impact of two oil crises in the late 1970s [6]. At that time, it was mainly used to study how to reduce oil consumption to reduce energy costs. In the classic literature published by Weitzman, M. L. [7], the author also pointed out that the shape, intercept and slope of MACC are the key factors in deciding whether to use price policy or quantity policy. In the 1990s, Faruqui, A. et al. (1990) [8] applied the MACC to calculate the saving amount of electricity consumption.
Specifically, the MAC of GHGs refers to the cost of each unit of GHG reduction, which effectively describes the functional relationship between reduction and marginal costs [9]. Ellerman, A. D. and A. Decaux (1998) [10]  They found that if carbon trading market links are established among OECD countries, the US, Japan, EU and other OECD countries will receive US $ 2.3 billion, US $ 10.49 billion, US $ 3.3 billion, and US $ 10 million in trade income respectively. The discovery laid the theoretical foundation for the analysis of carbon trading market links using MACC. In recent years, the analysis method based on MACC has attracted the attention of environmental economics researchers. For example, a techno-economic policy evaluation of SO2 abatement options has been conducted by building a systemwide MACC for India [11]; the relationship between the quantity of pollution abated and the marginal cost of abating each additional unit has been traced out [12]. In their research, the MACC is decomposed by sector, and MACC is used to compare the relative cost-effectiveness of Renewable Electricity/ Energy Efficiency/ Fuel Switching and traditional controls.
As for the MAC model construction, according to the review written by Du et al. [13], there are mainly three types: expert-based method, model-derived method and supply-side/production-based. In the last one, the representation form of MAC is derived by defining the production possibility set, which is generally expressed in the form of distance function. The MAC estimated by the distance function modelling method is also called shadow price. The estimated results can also reflect the recent trend of MAC changes and effectively measure the regional emission reduction potential and the cost savings brought by emission trading [14]. Commonly used distance functions include Shephard distance function and Directional distance function (DDF). The Shephard distance function attempts to adjust the desired output and the undesired output in the same ratio, while the directional distance function allows the desired output to increase while reducing the undesired output. Therefore, compared to the Shephard distance function, the Directional distance function is more in line with the policy decision reality [15]. The distance function methods to estimate MAC can also be divided into parameters and non-parameters, and the latter mainly refers to the estimation using the DEA (Data Envelopment Analysis) while the former generally approximates the distance function with a transcendental logarithm function or a quadratic function, and then estimates the corresponding parameters through linear programming or random frontier methods [16][17]. Compared with the non-parameters, the parameters method is more flexible, and the estimated function parameters can be used for some subsequent analysis, such as calculating Morishima elasticity of substitution [18].
As China becomes the world's largest CO2 emitter, there has been a rapidly  [25], first proposed the uncertainty of the welfare and the necessity of quantifying the welfare impact of the carbon trading market. However, the study did not take into account the different allowance allocation and different emission reduction targets.
Presently, there are abundant literature on the mature EU carbon market, while there are few researches on China's carbon trading market and its welfare. The innovations of this paper are as follows: firstly, on the basis of shadow price theory and directional distance function, this paper will build an input-output model containing undesired output to estimate the MAC of 30 provinces (except Hong Kong, Tibet, Macau and Taiwan due to the missing data) by using the parametric method. Different from previous studies, this paper fits the MACCs of the representative cities and provinces in China in detail, instead of just dividing each province into several categories for analysis, and expands the trading subjects of carbon trading market from regions to provinces. Secondly, in the empirical part, this paper quantifies the welfare of establishing a unified carbon trading market, simulate the welfare changes of provincial carbon trading markets under different allowance allocation and different reduction targets. A quantitative analysis of the welfare of the unified Chinese carbon trading market can provide a reference for enterprises and provinces participating in the carbon market. It can also help policy makers to regulate the carbon market and provide an evidence for ETS.

Directional distance function
MAC measures the tradeoff between desirable and undesirable outputs, or simply, the value of desirable output that would be foregone to reduce undesirable output by a unit. In the existing literature, MAC is normally referred to as the shadow price or marginal rate of transformation between desirable output and undesirable output [26][27][28][29][30]. The production possible set is as follows: x represents the input vector, y denotes the desired output vector, and b represents the undesired output vector.
In addition to the traditional neoclassical assumptions, such as compactness and free disposability of inputs and desirable outputs, the technology typically imposes two crucial assumptions on the output set, i.e. the null-jointness assumption and the weak disposability assumption. The null-jointness assumption says that desirable outputs are jointly produced with undesirable outputs. Formally, it assumes that if ( , ) ∈ ( ) and = 0, then y = 0, which implies that desirable output cannot be produced if no undesirable output is produced. The weak disposability assumption postulates that any proportional reduction of desirable and undesirable outputs is feasible. Formally, it assumes that if ( , ) ∈ ( ) and 0 ≤  ≤ 1, then ( ,  ) ∈ ( ), which implies that it is costly to reduce the undesirable outputs.
The specific economic expression thereof is as follows: Where i represents province, n represents different years.
The best situation is that the combination set includes the maximum desired output and the minimum undesired output. Therefore, the series of combinations with maximum desired output and the minimum undesired output constitutes the production frontier curve. The direction distance function value is the maximum distance value from each combination point to the production frontier curve.
Suppose that = ( , ) represents the direction vector and 0 ⃗⃗⃗⃗ represents the direction distance function. The specific expression thereof is as follows: As is shown in Figure 1, point A is a combination of the desired output and the undesired output in the set of ( ); point B is a combination of the production frontier.
Tangents are tangential to point B, and AB is perpendicular to the tangent line.

Marginal abatement cost model setting
This paper deduces the shadow price of undesired output from the dual relationship between directional distance function and revenue function. Suppose py is the price of desired output, pb is the price of non-desired output, and w is the price vector of input.
Then the revenue function can be defined as: Formula (4) defines the maximum possible return under the condition of a given input x, it is obvious that the contribution of undesired output to profit is negative, that is, to deal with undesired output requires cost.
Because the production unit is always above or within the production frontier, The above formula shows that if the output vector is feasible, then the output along the direction after eliminating the inefficiencies is also feasible. Therefore, the revenue function can also be written as follows: The left side of the inequality in formula (7) is the maximum possible profit, and the right side is the actual profit ( − − ) plus the additional benefit after eliminating technical inefficiencies. The additional benefit includes two parts, one is brought by the increasing desired output, that is, 0 ⃗⃗⃗⃗ ( , , ; ) . The other is brought by the reduction of undesirable output. In essence, due to the decrease in undesired output, the cost of the undesired output deducted from the total benefit also decreases, that is, 0 ⃗⃗⃗⃗ ( , , ; ) . If the production unit moves along the direction vector to the frontier of the production set ( ) , the allocation of output will be efficient, and the inequation will become an equation: Applying the envelope theorem to the formula (9), the shadow price model can be If the price of the desired output (py) is known, the price of the undesired output (pb) can be obtained by the following formula: Because of the differential property, the parametric distance function can be used to deduce the shadow price, which mainly include transcendental logarithm function and quadratic function. The parameterization of Shephard distance function generally adopts the prior. In this paper, based on the research of Färe et al. [16], the transcendental logarithmic Directional distance function of estimated shadow price is set as follows: In formula (13), y represents desired output (total value of out-put), represents To calculate the shadow price, firstly need to estimate the coefficient in formula (13). Formula on the left is [1 + 0 ⃗⃗⃗⃗ ( , , , , , )] rather than 0 ⃗⃗⃗⃗ ( , , , , , ), the reason is that to ensure the logarithmic function is positive. Because in the technological frontier, the value of 0 ⃗⃗⃗⃗ ( , , ) is 0. Färe et al. [31] proposed that these coefficients can be estimated by minimizing the deviation between all decision units and the effective frontier. The optimization is expressed as follows: . .
The linear programming aims to make effective coefficient estimation for each So: ∂ ln y = γ 0 + γ 00 ln y i + ε ln y i + ∑ η n ln x n i 3 n=1 (19) Finally, the calculation formula of MAC is: Welfare analysis  When the unified carbon price in the market is lower than the MAC of the individual transaction participant, the participant will choose to purchase carbon quota in the carbon market to complete the emission reduction target. The demand part is called the "transfer-in"; on the contrary, when the unified carbon price in the market is higher than the MAC of the individual transaction participant, the participant will choose to increase the amount of ER. Meanwhile, the extra quota will be sold in the market and obtain economic benefits from it. the selling part is called the "transfer-out" in this paper. In the carbon trading market, the total market demand and total supply under different prices can be obtained by summing up the transfer-in and transfer-out of all carbon transaction participants. When the market supply and demand are equal, the equilibrium carbon price will be determined. After that, each participant can determine the amount of transfer-in and transfer-out by comparing MAC to equilibrium price, therefore, it can get a new ER after trading. Further, it can be calculated that the cost and welfare changes brought by the unified carbon market to the participants. Table 1. illustrates the changes in costs and welfares of participants before and after the formation of a unified carbon trading market. The equilibrium price P* is determined by the total supply and demand of the carbon market; S (AA'A'') and S (BB'B'') represents the income after participating in the carbon markets linking. Before the linking of the carbon pilot markets, S (OQ1A) and S (OQ2B) represent the total cost of emission reduction of 1 and 2, whose integrations are calculated as: Where i = 1, 2 means two participants. After the linking, the carbon emission reduction of Q1 + Q2 needs to be jointly completed in the unified carbon trading market.
Marginal abatement cost model setting Data collection Energy Statistical Yearbook. According to the historical data, we shall predict the variable data for 2020 by regression model. The specific variables are as follows: (1) Capital stock: we select the perpetual inventory method to calculate the capital stock of each province in China.
Where, represents the total capital stock this year, −1 represents the total capital stock of the previous year, and represents the total fixed capital formation this year. Based on the 2000-year period, we calculate the capital stock of each of China's provinces from 2011 to 2017. All formulae considered the elimination of price factor.
(2) Labor force: we use the number of people in employment in each province at the end of each year. The sum of the number of employed people from three major industries is selected to represent labor input.
(3) Energy consumption: The data are from the China Energy Statistical Yearbook  [34]. Calculation should include as many fossil fuels and greenhouse gases as possible from multiple sectors to prevent carbon leaks [35]. We estimate CO2 emissions by energy types based on the mass balance of carbon.  MATLAB TM is used to construct the emission inventories with sectoral fossil fuel consumption and emission factors, so as to measure the total carbon emission.

Results of DDF value
On the basic model of linear programming, the Directional distance function value of 30 provinces from 2011 to 2017 was calculated by using MATLAB TM software in Table 2. The Directional distance function value represents the maximum distance from each combination to the production frontier curve. Here, it refers to the distance from the carbon emission reduction efficiency level of each region to the production frontier curve. As can be seen from Table 2, there is a negative correlation between the distance value and the carbon emission performance level. A smaller value means closer to the optimal efficiency value and less potential for improvement, and vice versa. Based on the horizontal comparative analysis of the values in Table 2, the areas with high levels of carbon emissions performance such as Beijing, Shanghai, and Guangdong usually present the characteristics of higher economic development, high production efficiency and service-oriented industries; DDF value is equal to or close to zero, indicating that these regions have reached the production frontier and the carbon reduction potential is smaller. In regions with poorer carbon emissions performances such as Shaanxi, Shanxi, and Guizhou, where the economy is less well-developed and reliant on heavier resource-based industries, the DDF value is much greater than zero, indicating that there is huge potential for improvement in these areas. Carbon reduction can be implemented by improving efficiency and changing the industrial structure.

Comparison of MAC between 2011 and 2017
The marginal abatement cost discussed here measures the economic income loss due to the reduction of CO2, so it should be negative. However, to facilitate comparative analysis, we list the absolute value in Table 3. This distribution creates conditions for carbon trading. Carbon trading has been recognized because of its incomparable advantages in terms of emission reduction costs, emission reduction effects, and political feasibility [38]. The heterogeneity of regional emission abatement costs provides the possibility for the formation of carbon market.
Provinces with higher abatement costs can reduce their costs by purchasing carbon quotas from other provinces. Provinces with lower carbon emission abatement costs can earn additional economic benefits (welfares) by selling. Figure 3 is the average ranking chart of the MAC of each province from 2011 to 2017. Exchange shows that by July 2020, the cumulative trading volume in Guangdong had reached 155 million tons, accounting for 38.68% of the national carbon market, ranking first in the country; the cumulative transaction amount was 3.1 billion-yuan, accounting for 34.69% of the national carbon trading pilots, indicating that the emission reduction potential in Guangdong is limited, and reducing carbon emissions by increasing efficiency will incur a significant cost. Reducing the amount of CO2 emissions per unit GDP will sacrifice more economic revenue than the price per unit carbon dioxide emission rights, therefore, to maximize profits, it is more reasonable to purchase carbon emission rights in the future unified market.
The emission reduction costs in Xinjiang, Ningxia, etc. are the lowest, and emission reduction program can be implemented continuously. The economic revenue that these provinces have sacrificed in reducing their CO2 emissions is less than the price per unit CO2 emission rights, therefore, to maximize profits, these provinces may consider trading surplus carbon emission allowances, or even shut down low-efficiency plants to reduce emissions and trade the saved carbon allowances. This will not only achieve the emission reduction targets, but also generate more revenue, and there will be a certain amount of residual in-come after accounting for the reduction in scale.
Summarizing the comparison between individual provinces, the polarization between provinces is obvious, and Pareto improvement 1 can be achieved by undertaking certain reasonable transactions. Without the losses of some provinces, an increase in total revenue can be realized. 1 Pareto Improvement: If we adjust the allocation of certain resources, some people's situation will be improved, while others' situation will at least remain unchanged. The adjustment that accords with this nature is called Pareto Improvement.

Fitting of the MAC curve of CO2 emission
After calculating the marginal abatement costs of 30 provinces, we would use functions to fit the MAC curves. Many scholars agree that the MACC shows a singleincreasing convex function with the increase of emission reduction [39]. After many tests, this paper decided to use the form of logarithmic function to fit the MACCs, the relevant equations are as follows:  The smallest β-absolute value is 0.02, and the largest β-absolute value is 0.833. In and Qinghai, will have greater reduction potential than eastern provinces. In other words, under the same CO2 emission reduction target, the cost in the western region will be much lower than that in the eastern region. Therefore, in the initial stage of ETS, the government can allocate relatively more emission reduction tasks to the western region in order to maximize the overall welfare of the society. But at the same time, western region may be difficult to achieve the task of reducing emissions because of the low output and economic level. Therefore, the government should also give the western region more financial support. As long as the government's financial support is lower than the difference in emission reduction costs caused by more tasks in the western region, there will be a Pareto improvement effect at the level of overall social welfare.
The CO2 MACCs of representative provinces: Guangdong, Beijing, Ningxia were drawn in this paper (see Figure 5a, 5b, 5c). The upward sloping curve indicates that with the increase of emission reduction rate (ri), the MAC will gradually increase, and the emission reduction work of all provinces will become more difficult. In the figure, the MACC of Guangdong is obviously steeper than that of Beijing, while Ningxia has the flattest curve due to the smallest β-absolute value. According to the emission reduction targets set by the country, this paper analyzes the two scenarios separately. Scenario 1 is that the carbon intensity in 2030 is 65% lower than that in 2005; Scenario 2 is that the carbon intensity in 2030 is 60% lower than that in 2005. In Scenario 1, the carbon intensity at the end of 2017 has been reduced by 46% compared with 2005, that is, 241000 CNY /ton. To achieve the target, carbon intensity needs to be reduced by 3% per year on average from 2018 to 2030. In Scenario 2, the carbon intensity would need to be reduced by an average of 2% per year from 2018 to 2030.
On the basis of the carbon intensity, it is easy to get the CO2 emission limits under different scenarios, and the limitation can be distributed to various provinces in the light of different allowance allocation principles. Allowance allocation can be divided into free distribution and auction. Since China is currently at the initial stage of establishing a carbon trading market, auction allocation will lead to a large fluctuation in the price, which will have a great impact on industries. Therefore, this paper adopts free distribution to allocate limitation to all province. In the process of free distribution, the principle must give consideration to both efficiency and fairness. In accordance with this idea, this paper determines the allowance allocation based on the historical emissions of provinces, which is also adopted by EU ETS at the beginning of the establishment. provinces accounted for less than 1% of the national welfare. Ningxia, which has the largest welfare, is 173 times larger than that of Heilongjiang.  the welfares that provinces get from the national carbon trading market are taken into account, the stronger the emission reduction efforts are, the more the provinces will get the welfares. However, correspondingly, the equilibrium emissions of provinces will become less, and the pressure to reduce emissions will become greater, which will have a greater impact on production and economy. The results indicate that after setting the initial emission reduction quota, the government agencies should consider the pressure on the subject of emission reduction and the welfare obtained by the carbon trading market. On the whole, the welfare of each province decreases as the intensity of emission reduction decreases, that is, the overall welfare of scenario 2 is smaller than that of scenario 1. This means that if the welfare of the national unified carbon market is required, and both strong reduction targets and more provincial welfare are taken into account, the balanced emissions of each province will be less correspondingly, the pressure of emission reduction will be greater. Further, the impact on production and economy will be larger. This indicates that after setting the initial allocation mechanism, the government agencies should consider the pressure on participators and the welfare of ETS in the setting of emission reduction targets.
According to the calculation of the above two groups, Ningxia has the highest welfare and the least MAC, Heilongjiang has the least welfare and MAC is in the middle reaches, and Guangdong has the highest MAC and the welfare is pretty high.  1. When the market price is lower than 122 CNY, the cost of all participators in the market is higher than the market price, so all of them are demander and there is no supplier in the market, and the market cannot be cleared.
2. When the market price is 122≤P*<1664 CNY, the market supplier is Ningxia, the supply volume is at most 18 million tons, and the market demanders are Guangdong and Heilongjiang, the demand volume is at least 19 million tons. Supply in the market is less than demand, the market price cannot reach equilibrium, and the market cannot be cleared.
3. When the market price is 4612 CNY, the market supplier is Ningxia and Heilongjiang, the supply volume is 18 million tons and 1.5 million tons respectively, the market demander is Guangdong, the demand volume is 19.5 million tons. Supply in the market is equal to demand. It happens to be in equilibrium at this price, and the market will reach a state of clearing.
4. When the market price is 4612<P*≤12050 CNY, as the price increases, the market supplier is Ningxia and Heilongjiang and the supply volume increases accordingly, more than 19.5 million tons; as the demander, Guangdong, the market demand remains unchanged at 19.5 million tons. At this time, the supply in the market is greater than the demand. Market cannot reach the equilibrium state and cannot be cleared, so the equilibrium price cannot be formed.

5.
When the market price is higher than 12050 CNY, with the increase of price at this time, Ningxia, Heilongjiang and Guangdong will all become market suppliers, and the supply and demand in the market cannot reach equilibrium and the market cannot be cleared. There is no market equilibrium price. The specific supply and demand curves are shown in Figure 9. In addition, the development of carbon trading products can also accelerate market liquidity. At present, there only exists trading products such as carbon emission rights.
In the future, carbon options, carbon futures and other financial products can be further developed to increase trading varieties and further use market-oriented mechanisms. To achieve the optimal effect of resource allocation and reduce the cost of emission reduction of the whole society.

Appendix B
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Declarations
Availability of data and materials: Available.