## 5 Empirical Analysis  
### Data Sources and Parameter Specifications  
Each province is modeled as a DMU, and the three major industries in the province are considered as SDMUs. The inputs of each industry are capital stock, year-end employment, and energy consumption; the desirable output is GDP, and the undesirable output is carbon emission. Due to the availability of data, this paper collects the data of the year 2019 related to the three major industries of 30 provinces of China (excluding Tibet, Hong Kong, Macao, and Taiwan). The data sources and units of measure are provided as follows:  
1. Capital stock (Unit: 100 million RMB): The data are estimated based on statistical data from Historical Information on China's GDP Accounting 1952-2004, Statistical Yearbook of the Chinese Investment Field, Statistical Communique on National Economic and Social Development of each region, and Statistical Yearbook of each province, using [PERSON]'s perpetual inventory method [45]. To clarify, in this paper, the gross fixed capital formation since 2002 is selected as the total investment in fixed assets in the calculation of capital stock [46], while the total investment in fixed assets in each region after 2017 is indirectly calculated from Statistical Communique on National Economic and Social Development. The investment deflator for the three industries is adopted from the investment deflator construction method proposed by [PERSON] and [PERSON] [46]. The depreciation rate of fixed assets is set to 9.6%, proposed by [PERSON] [45]. We chose the year 2000 as the base year, and the base year capital stock of the three industries across regions is calculated by the method proposed by [PERSON] [47]. 2. Year-end employment (Unit: 10,000 people): The data are extracted from the respective provincial Statistical Yearbook 2020. 3. Energy consumption (Unit: 10,000 tons of standard coal): The amount is calculated based on the comprehensive energy balances of provinces and cities from the China Energy Statistical Yearbook 2020. The 29 energy types involved in the energy consumption calculation were determined by following the guidelines of the Intergovernmental Panel on Climate Change (IPCC) [48] and the China Energy Statistical Yearbook. The conversion coefficients of each energy source were obtained from the National Bureau of Statistics of China. The industrial attribution of each sector is based on the Industrial Classification for National Economic Activities (GB/T 4754-2017) [49] and the Regulations on the Dividing Basis of Three Industries. Finally, the energy consumption of industries in each province was derived from the conversion coefficients of the 29 energy types and the conversion coefficients of each sector. 4. GDP (Unit: 100 million RMB): The data are from the respective provincial Statistical Yearbook 2020. 5. Carbon emissions (Unit: 10,000 tons): The data are calculated based on the energy consumption of provinces and cities from China Energy Statistical Yearbook 2020. Note that the calculation requires first categorizing the 29 energy sources into five major categories: coal, oil, natural gas, electricity, and heating power. Then, the product of the energy consumption of each industrial sector and the corresponding energy carbon emission factor is the carbon emission of that energy category. The emission factors of coal, oil, and natural gas can be obtained directly from the Energy Statistics Yearbook, while the carbon emission factors of electricity and heating power are calculated based on the fuel combustion in the power sector [50]. Specific data for each indicator can be found in Table 11. Before allocating resources, the total amount of carbon quotas to be allocated needs to be determined. The actual national carbon emissions of the year 2020 are taken as the total amount of carbon quotas targets. According to the 70 th edition of the World Energy Statistics Yearbook published by BP [4], the total carbon emissions from China in 2020 would be 9.786 billion tons.