# Models, methods, and data

Traditional production theories cannot directly deal with undesired outputs. Indirect methods can be used to convert undesired outputs so that the transformed data can be included in the normal output function under

*Regional decomposition* 45 technically unchanging conditions, including: converting undesired outputs into input factors, or performing additive inverse or multiplicative inverse transformation, as detailed in Scheel (2001); in addition, the directional distance function was developed by Fare et al. (1989), Chung et al. (1997), etc. By building environmental production technology to continue productivity and shadow price measurement, the specific application of the measurement of interprovincial technical efficiency in China can be seen in the studies of Hu Angang et al. (2008): Fu Jiafeng et al. (2010), and Tu Zhengge (2009), who examined industrial productivity and industrial SO, based on the measurement of shadow price; Wang Bing et al. (2010) adopted the measurement of interprovincial Malmquist-Luenberger productivity under environmental control. However, as pointed out by Fukuyama and Weber (2009), the existing direction distance function does not consider the possible redundancy, which leads to a certain bias in the final efficiency evaluation.

This chapter uses an extended Scaks-Based Measure (SBM) for estimation. The basic idea of the SBM model is to consider the efficiency of the redundancy at the input and output ends. There is no excessive investment in the optimal performance point at the leading edge, and there is no shortage of output. On this basis. Cooper et al. (2007) proposed an extended SBM model that considers undesired outputs.^{6} The basic expression is as follows.

There are *n* decision-making units, the input vector *xeR ^{1}",* the production of the desired output

*y*and the undesired output ybeRs2, defining the corresponding matrix as ^[xq,... ,x„]e/?"“", T«=[j

^{b}e.R'-,_{l}«,... T'’=|j j

^{A},—jV]

g/?'^{2}'", and suppose *X,* T«, P>0.

The production possible set *P* is defined as:

where zg/?" is the intensity vector, and the production possible set *P* in (3.1) is equivalent to the efficiency, which is expressed as:

s.t.

where *s~eR‘", s ^{b}eR^{s2}* are excessive inputs and excessive undesired outputs, respectively, and

*sseR*represents the desired output of the shortage, that is, the redundancy of inputs, unconsumed outputs, and desirable outputs. The formula (3.2a) is a strict decreasing function of

^{s}'*s-, s& and s*and satisfies 0 < p * < 1. The sample point is at the leading edge if and only if p* = 1, that is, it is efficient, and the redundancy value of the input, the desired output, and the undesired output is 0.

_{r}^{b},In the actual calculation, the weighting factor is often applied according to the relative importance of input and the desired output and the undesired output. Based on the objective function in (3.2a), the weighted efficiency is expressed as:

where w, iv/ and tv/ are the weights of input /, desirable output r, and undesired output r, respectively, and ^{= m} , ^{M}'i" ^0, ^{+}X,_{=I}^{M}) ^{=5}i ^{+5}2 ,

w^{s} > 0, *w ^{1} >* 0.

## Emission reduction potential model

For inefficient sample points, the non-conforming output corresponding to the feasible target on the leading edge is

It can be solved by (3.2b), and the actual undesired output of the sample points can be observed, so the optimal target undesired output of each sample point can be calculated, if the undesired output is defined as CO, emissions, which can define the feasible abatement and the abatement potential at the time *t* of the sample point *i. *

where *FA _{it}* is the excess CO, emissions of sample point

*i*at time /, indicating a reduction in CO, emissions compared to the leading edge effective point.

*AP,,*represents the emission reduction potential of the sample point, and its value is between 0 and 1. The higher the value of

*AP*the more the CO, emission of the sample point is excessive, and the greater the emission reduction potential of the region. If you increase the total amount of feasible emission reductions in each region, you can get the regional (national) aggregate feasible abatement and calculate the regional (national) aggregate abatement potential.

_{it},## Shadow price model

In addition, with the method of Charnes and Cooper (1962), the dual linear programming of (3.2a) can be expressed as:

where *s —* 5'1 + 5'2, [1/xJ represents the row vector (l/x_{/o},...,1/a„,„), the dual vector *ve.R ^{m}, u^{b}eR^{s2}, useR^{sl}* can be interpreted as input, non-consensus output and virtual price of desirable output, respectively.

According to the method of Fare et al. (1993) and Lee et al. (2002), the ratio of the shadow price of inconsistent output to the desired output is equal to its marginal conversion rate. In the form of parameterized distance function, it can be expressed as the distance function is not satisfactory. The ratio of the output to the first derivative of the desired output, in the non-parametric form, is the dual value of the unconstrained output and the desired output constraint in the dual linear programming, that is, the following relationship:

Assuming that the price of the desired output is a market-oriented standardized price, the shadow price of the undesired output can be derived as:

The shadow price in (3.7b) can be regarded as the marginal cost of CO, emissions, and the more severe the pollution emissions, the lower the shadow price (Coggins & Swinton, 1996), so for regions with lower pollutant shadow prices, the risk of opportunity costs that may result from environmental controls or the imposition of emission constraints is relatively small. For the country, it is necessary to consider the possible economic impacts of environmental policies while meeting the emission reduction targets. Therefore, the CO, marginal abatement costs in different regions can be used as one of the efficiency indicators for prioritizing regional emission reductions.

## Variables and data

This chapter analyzes the capital, labor, and energy of 29 provinces in China from 1995 to 2007 as the input factors, and analyzes the GDP of each province as the desired output and the CO, emissions of each province as unproductive output.

Capital stock: the “permanent inventory method” is widely used to estimate the actual capital stock of each year. Here, we mainly refer to the existing research results of Zhang Jun et al. (2004) and extend its sequence to 2007 according to its published method. Calculated at constant prices in 2005, the unit is 100 million yuan.

Labor: foreign countries generally use working hours as a labor input variable, but this is limited by the availability of data. Here, the number of employed people in the current year published in the *China Statistical Yearbook* is 10,000.

Energy: the data come from the *China Energy Statistics Yearbook* over the years. Tibet lacks energy data and is not included in the sample. The unit is 10,000 tons of standard coal.

GDP output data: from the *China Statistical Yearbook* from previous years, in order to facilitate comparison with the indicators published by the National Bureau of Statistics, the unit is calculated at the constant price of 2005, the unit is 100 million yuan.

CO, data: existing research institutions have not yet had provincial CO, emission data. As CO, emissions are mainly derived from fossil energy consumption, conversion, and cement production, for the sake of accuracy, this chapter breaks down energy consumption into coal consumption and oil consumption (further subdivided into gasoline, kerosene, diesel, fuel oil) and natural gas consumption.^{7} A large part of the primary energy consumption process is used to generate electricity and heat. Although the energy and heat generated by this part of energy consumption may not be used in the province, the resulting CO, does remain in the province, so this chapter, when calculating energy consumption in addition to the terminal energy consumption, also includes energy for power generation and heating. All energy consumption and conversion data in this chapter are taken from the regional energy balance sheet in the *China Energy Statistics Yearbook.* The cement production data come from the Guotaian Financial Database. The specific calculation formula for carbon dioxide emissions from fossil energy consumption activities is as follows:

**
**

Here, CO, represents the estimated total carbon dioxide emissions of various types of energy consumption; *i* represents a variety of energy consumption, including coal, gasoline, kerosene, diesel, fuel oil and natural gas; E, is the province’s various energy consumption Total; *CF,* is the conversion factor, which is the average calorific value of various fuels; CC, is the carbon content, which is the carbon content of the unit heat; *C()F _{t}* is the carbon oxidation factor, which reflects the oxidation rate of the energy. Level: 44/12 indicates the conversion coefficient of carbon atom mass to carbon dioxide molecular mass; the CO, emission coefficient of various emission sources mainly refers to 1PCC (2006) and the National Climate Change Coordination Group Office and the Energy Research Institute of the National Development and Reform Commission (2007).

Descriptive statistics for each of the above variables can be found in Table 3.1.