Investigation for the Decomposition of Carbon Emissions in the USA with CD Function and LMDI Methods

Residual problems are one of the greatest challenges in developing new decomposition techniques, especially when combined with the Cobb–Douglas (C-D) production function and the Logarithmic Mean Divisia Index (LMDI) method. Although this combination technique can quantify more effects than LMDI alone, its decomposition result has residual value. We propose a new approach that can achieve non-residual decomposition by calculating the actual values of three key parameters. To test the proposed approach, we decomposed the carbon emissions in the United States to six driving factors: the labor input effect, the investment effect, the carbon coefficient effect, the energy structure effect, the energy intensity effect, and the technology state effect. The results illustrate that the sum of these factors is equivalent to the CO2 emissions changes from t to t-1, thereby proving non-residual decomposition. Given that the proposed approach can achieve perfect decomposition, the proposed approach can be used more widely to investigate the effects of labor input, investment, and technology state on changes in energy and emission.


Introduction
When exploring the factors affecting energy consumption or carbon emissions, it is the pursuit of more scholars to decompose more factors.However, the factors that can be decomposed by using a single decomposition method are limited.In order to explore more drivers, many scholars have adopted a way to combine decomposition models with other economic models.To quantify the effects of fixed asset investment and labor input on changes in energy consumption and carbon emissions, Wang et al. [1] combined the Cobb-Douglas (C-D) production function and the Logarithmic Mean Divisia Index (LMDI) econometric method.This combined technique can quantify more effects than can LMDI alone.However, the results of the decomposition using this technique have residual problems, which means that the technique cannot solve the zero-value problem or implement perfect decomposition.In a recent study, Dong et al. [2] confessed that the combination technique could not implement complete decomposition, but nevertheless, was still applied, because this technique was able to quantify the effects of labor force input and fixed asset investment on changes in carbon emissions.However, the combined decomposition technique with residuals will result in inaccurate results, which will reduce the credibility of the conclusions.Therefore, the purpose of this paper is to solve the residual problem and achieve perfect decomposition of the combination technique and to conduct empirical analysis based on US carbon emissions.Guided by applied intermediate macroeconomics [3], this paper reports a new approach to solving the zero-value problem and exploring one more factor, which is the technology state factor.In this way, the combined decomposition technique can be better used in studying the effects of fixed asset investment, labor force input and technology state on energy consumption changes and carbon emissions changes.
The remainder of this paper is structured as follows.Section 2 is a literature review of decomposition techniques.Section 3 presents the proposed approach to using the combined decomposition technique to achieve perfect decomposition.Section 4 describes the testing of the proposed approach by decomposing CO2 emissions in the United States.Finally, Section 5 concludes this paper.

Literature Review
Generally, there are two primary categories of the decomposition method: structural decomposition analysis (SDA) and index decomposition analysis (IDA) [1,4].The SDA method, which is based upon the input-output tables, is widely used to analyze the influencing factors driving the energy consumption and energy-related emissions changes [5][6][7][8][9][10].For CO 2 emissions in Spain, José M. Cansino et al. used SDA to decompose the changes into six effects [11].Bin Su et al. [12] and Jing Wei et al. [13] also employed SDA to analyze the influencing factors of CO2 emission changes in Singapore and Beijing.Zhu et al. utilized an input-output framework and SDA method to explore the driving forces for the changes in India's total emissions and intensity [14].However, input-output tables are not always readily available, thereby limiting the widespread use of SDA [5,15].Wang et al. compared the application of the SDA method and IDA method in energy consumption and energy-related emissions research from the viewpoints of methodology and application [16].Xu and Ang reviewed and summarized 80 papers that applied the IDA method on emission studies during 1991 to 2012, and the results revealed that IDA is more widely used than SDA when decomposed CO 2 emissions change [17]; and then, they employed the IDA method to decompose the residential energy consumption in Singapore [18].The IDA method can be separated into two approaches: the Laspeyres Index approach and the Divisia Index approach [19][20][21].The Laspeyres Index approach has rarely been used to analyze carbon emissions due to the residual problem [22].The Divisia Index approach was first proposed by Boyd and included the Arithmetic Mean Divisia Index (AMDI) and Logarithmic Mean Divisia Index (LMDI) methods.LMDI is more extensively utilized because AMDI results in incomplete decomposition and zero-value problems [23,24].Based on the expansion and improvement of the Divisia Index method by previous research [25][26][27][28][29], LMDI has become the preferred method for decomposing CO 2 emissions [23,30].To solve zero-value problems, Ang and Liu have put forward eight strategies [31].Moreover, Ang summarized the basic formulas for the eight LMDI models and made comparisons between these models, and then developed model selection guidelines for potential users [32].
LMDI has been adopted by many academics to investigate the decomposition of CO 2 emissions at national levels [33,34], provincial levels [35][36][37] and the levels of specific industries [16,38] such as the chemical industry [22], construction industry [39,40], manufacturing industry [41], heavy industry [42], logistics Industry [43], and power industry [9], as well as at the levels of the whole of industry [44].Akbostanci et al. applied LMDI to decompose the CO 2 emissions of the Turkish manufacturing industry into five influencing factors: overall activity, economy structure, sectoral energy intensity, sectoral energy structure, and CO 2 emission coefficient [45].Babak Mousavi et al. analyzed the influencing factors driving Iran's CO 2 emission changes [46].Jeong et al. decomposed the CO 2 emissions from South Korean industrial manufacturing into five influencing factors: economic activity, economic structure, energy intensity, energy-mix, and emission-factor [47].Achour et al. calculated the contributions of the transportation intensity, energy intensity, economic output, population scale effect, and transportation structure effect of the Tunisian transportation sector [48].Zhao et al. used LMDI method to decompose CO 2 emissions in Guangdong Province from a sector perspective, and the results showed that the energy intensity effect and economic output effect were the critical factors for carbon emissions in Guangdong Province [37].Wang et al. used the LMDI model and Tapio decoupling model to study the influencing factors of decoupling status between CO 2 emissions and economy from the city level, and the results showed that the economic effect and population effect inhibited the decoupling and energy intensity accelerated the decoupling [16].It can be concluded that the research objects have been divided very carefully in the above research using LMDI to explore the decomposition of CO 2 emissions, but there is seldom innovation in the driving factors.To explore the contributions of the fixed asset investment and labor factors to China's energy consumption changes, Wang et al. proposed a decomposition technique combining the C-D production function and LMDI [1].The results illustrated that the factors of fixed asset investment and labor were critical to influencing energy consumption.Moreover, Dong et al. used the combined decomposition technique to analyze energy consumption in Liaoning Province and showed that the fixed asset investment effect played a negative role while the labor effect played a very weak role in the phenomenon of decoupling [2].
By reviewing the development of the decomposition method, it can be seen that the combination of the C-D production function and LMDI can decompose the effects of labor force input and fixed asset investment on changes in energy consumption and carbon emissions.However, this combined decomposition technique caused incomplete decomposition and had residuals in its results.Moreover, the combined decomposition technique ignored the effect of the technology state.In this paper, the proposed approach is aimed at solving the residual value of the combined decomposition technique.

Proposed Approach to Achieve Complete Decomposition
When applying the LMDI method, the total CO 2 emissions can always be expressed in the traditional decomposition form [49,50].The traditional influencing factors include the carbon coefficient, energy structure, energy intensity, and economic outputs.The function is as follows: where t denotes the time in years; i denotes energy type; C t and C it denote, respectively, the total CO 2 emissions and the CO 2 emissions of the i-th energy type in t year; E t and E it denote, respectively, the total energy consumption and the consumption of the ith energy in year t; GDP t denotes the gross domestic product in year t; denotes the carbon coefficient of the i-th energy in year t; S it = E it E t denotes the energy structure of the i-th energy in year t; and I t = E t Q t denotes energy intensity in year t.The above analysis shows that the traditional decomposition form cannot reflect the impacts of fixed asset investment and labor on carbon emissions.Building on previous research and economic theory, Wang et al. [1] combined the C-D production function and LMDI to formulate the proposed combined decomposition technique.The C-D production function is a particularly useful one in many mathematical functions, because it has the production function properties that we expect and has been extended to the field of economics by Paul Douglas and Charles Cobb.In particular, the C-D production function describes how the real economy transforms fixed asset investment and labor into GDP.So, we can use the C-D production function, which is shown below, to describe GDP: Putting Equation (3) into the GDP on the right-hand side of Equation ( 2), a new form of decomposition can be obtained: where A is a positive constant that can be treated as a target of the state of technology; α and β are unknown constant parameters while α + β = 1; and L t and K t denote the amount of labor input and fixed asset investment, respectively, in year t.
According to Ang, by using the additive form, the change in CO 2 emissions (∆C t ) between a base year 0 and a target year t can be decomposed into the six influencing factors: the carbon coefficient effect (∆C Ft ), the energy structure effect (∆C St ), the energy intensity effect (∆C It ), the technology state effect (∆C At ), the labor input effect (∆C Lt ), and the investment effect (∆C Kt ).The function is shown below: According to LMDI, each effect can be computed as follows: ) The combined decomposition technique ignored the actual values of A, α and β and treated them as unknown constants for computational convenience.Thus, ln(A t /A 0 ) = 0 and ∆C At = 0 [1,2].In conclusion, the combined decomposition technique not only has residuals in its results but also ignores the effect of the technology state.In this paper, the actual values of A, α and β are calculated according to Kevin D. Hoover's applied intermediate macroeconomics [3].Thus, non-residual decomposition is realized and the technology state factor is quantified.The calculation steps are as follows: According to Equation (3) and α + β = 1, the C-D production function can be written as: Calculating the derivatives of K and L for Y, we obtain the marginal product of labor input (mpL) and the marginal product of physical capital (mpK), respectively.The function is shown below: In economics, real labor income (Y L ) is the product of the real wage rate (ω/p) and the number of hours worked (L): where ω is the wage rate and p is the market price.
where L−share is the labor share; and Y L and Y denote the real labor income and the total income, respectively.
Similarly, the real capital income (Y K ) and the capital share (K−share) are defined as follows: where ν is the rental rate; p is the market price; (ν/p) is the real rental rate; and K is the amount of fixed asset investment.
To further explain these definitions, we apply the assumption that the economy is close to perfect competition.Under this assumption, we can use the profit maximization rules, which state that each real factor price is equivalent to the corresponding marginal product.The functions are shown below: Replacing the factor prices with the marginal products in Equations ( 10) and ( 12): We can replace mpL and mpK in Equations ( 15) and ( 16) with the specific forms in Equations ( 7) and (8).The functions are shown below: From the above analysis, we can draw the conclusion that the indexes of L and K in the C-D production function (α, β) are equal to the labor and capital shares, respectively.So, we can calculate the actual value of α by calculating the labor share with the following function: where total income includes the compensation of employees, rental income of persons, corporate profits, net interest, miscellaneous payments, and depreciation value.
When the actual value of α was calculated and the data of the labor force and capital were obtained, the current year's technology state index (A) can be calculated using the following formula:

Empirical Analysis
To test if the proposed approach implements non-residual decomposition, the carbon emissions in the United States have been selected for empirical analysis.To clearly demonstrate the issue of perfect decomposition, we conducted a comparative analysis of the traditional approach and proposed approach.

Data Collection
The data about energy consumption and emissions were derived from the US Energy Information Administration (EIA) [51].The labor force data was from the US Bureau of Labor Statistics (BLS) [52].The GDP and real net stock of fixed assets, which have been converted to 2009 constant dollars, were from the US Bureau of Economic Analysis (BEA) [53].The research period started from 2000 and ended in 2016.

Traditional Approach
According to the traditional approach [1,2], A, α and β are unknown constant parameters, so ln(A t /A 0 ) = 0 and ∆C At = 0.It can be concluded that the constant A has no influence on the USA's carbon emissions.
Using Equations ( 1)-( 5), the effects of the five influencing factors can be obtained.The results of the decomposition using the combined decomposition technique are shown in Table 1.The comparison of the actual values to the decomposition values is shown in Figure 1.As shown in Table 1, the critical factors that impacted CO 2 emissions in the United States was the energy intensity effect and the investment effect, which were the largest inhibitor and contributor.In terms of cumulative effects, the investment effect has contributed to the growth of 1752.33 million metric tons of CO2 emissions, while the energy intensity effect has inhibited the growth of 2027.52 million tons of CO2 emissions.The investment effect and the labor input effect always played a positive role in the growth of CO 2 emissions during 2000-2016, and the effect of the carbon coefficient, the energy structure and the energy intensity on the growth of CO 2 emissions changed over time.From the perspective of cumulative effects, the energy structure effect was the second inhibitor for the growth of CO 2 emissions, which has inhibited the growth of 289.39 million tons of CO 2 emissions; while the labor input effect was the second contributor, which has contributed to the growth of 628.71 million metric tons of CO 2 emissions.As for the carbon coefficient effect, its role is very weak.
However, by comparing the total effects of decomposition and the actual values, it can be found that there were residuals in this decomposition mode.As shown in Figure 1, the total effects of decomposition were not much different from the actual values in 2003-2004 and 2010-2015, but they were not completely equal.In 2001-2003, 2004-2010 and 2015-2016, the difference between the total effects and the actual values was very large.It is worth noting that in 2000-2001 and 2005-2006, the US carbon dioxide emissions showed a downward trend, while the total effects of decomposition were positive.It can be concluded that there were residuals in the decomposition results resulting from the fact that the actual values of A, α and β were neglected.Furthermore, the decomposition method with residuals will reduce the credibility of the conclusion and result in inaccurate analysis, which are not conducive to the effectiveness of policy implementation.

The Proposed Approach
According to the proposed approach, the actual values of A, α and β can be calculated using Equations ( 16)- (18).As shown in Figure 2, the labor share and capital share change about a mean that is nearly constant from year to year and is regarded as an estimate of α, so α = 0.68 and β = 0.32.The index of the technology state (A) is shown in Figure 3.It can be seen that the technology state in the USA has an upward trend in volatility from 2000 to 2016, but there was a decline from 2007 to 2009.After calculating the value of each parameter, the USA's emissions can be decomposed into six influencing factors: the carbon coefficient effect (∆C Ft ), the energy structure effect (∆C St ), the energy intensity effect (∆C It ), the technology state effect (∆C At ), the labor input effect (∆C Lt ), and the investment effect (∆C Kt ) by using Equation (5).The contributions of the different factors to the changes in the USA's CO 2 emissions are shown in Table 2.The emissions were completely decomposed by using the perfect decomposition of the combined decomposition technique, and there were non-residuals in the results.
As shown in Table 2, the emissions showed a fluctuating state between 2000 and 2007, after which there was a sharp decreasing trend from 5,988.80 million metric tons in 2007 to 5162.23 million metric tons in 2016.The cumulative effect, the technology state effect, the labor input effect, and the investment effect were the primary factors contributing to the increased emissions during 2000-2016, while the cumulative effects of energy structure and energy intensity are the critical factors that inhibiting the increase in emissions during 2000-2016.By comparing Tables 1 and 2, it can be seen that the effects of the carbon coefficient effect, the energy structure effect and the energy intensity effect decomposed by the traditional decomposition method and the proposed decomposition method were the same for the US CO 2 emissions.The energy intensity effect played a positive role in reducing the CO 2 emissions during 2009-2010 and 2012-2013, resulting from the drop of energy intensity.The second influencing factor to curb carbon emissions is the energy structure effect, which played a negative role in most years.The impact of the carbon coefficient effect on emissions is very week.However, the investment effect and the labor input effect were different between the traditional methods and the proposed methods.The direction of investment effect and labor input effect on US CO 2 emissions has not changed, but the effect of investment and labor input under the proposed method has a much smaller effect on US CO 2 emissions than the traditional method.Furthermore, the proposed method can measure the effect of the technology state on CO 2 emissions in the United States, while the traditional method results in a technology state effect of 0 due to the neglect of the constant A.

Discussion
In exploring the factors affecting energy consumption or carbon emissions, the decomposition of more factors has been a widespread concern of scholars.We reviewed and compiled 22 papers on factorization from 2011 to 2019, as shown in Table 3.It can be seen that most scholars do not quantify the impact of investment, labor and technology status on energy consumption and carbon emissions, and combining the LMDI method with the C-D function can provide more factors for decomposition analysis.
However, with reference to the previous combination method, since the three constants of A, α, and β are neglected, the technical state effect is zero, and the decomposition result has a residual.This reduces the reliability of the combined decomposition method results.The combination proposed in this paper achieves no residual decomposition and quantifies the impact of the technological state effects on US carbon emissions, which can provide reference for future decomposition analysis.

Conclusions
Non-residual decomposition is very important for developing new decomposition techniques.Combining LMDI and the C-D production function can quantify more effects, especially for fixed assets investment and labor forces, than can LMDI alone.However, the results of this combined decomposition technique have residuals and the technique ignores the technology state factor.After many trials, we found that the root cause of the residual problem was three key parameters: A, α and β.Guided by the classical Kevin D. Hoover's applied intermediate macroeconomics, we calculated the actual values of A, α and β to achieve complete decomposition and quantify the technology state factor.To test our proposed approach, we selected carbon emissions in the USA as a case study.
The traditional approach can decompose the US carbon emissions changes into the carbon coefficient effect, the energy structure effect, the energy intensity effect, the labor input effect, and the investment effect, and the proposed method added the technology state effect to these factors.According to the decomposition results, it can be seen that the carbon coefficient effect, the energy structure effect and the energy intensity effect under the two decomposition methods were the same, and the labor input effect and the investment effect under the proposed approach are smaller than the decomposition results of the traditional approach.
Furthermore, compared to the traditional approach (ignoring A, α and β), the results of the decomposition of US carbon emissions showed that our proposed approach can achieve non-residual results.Using the proposed approach achieved perfect decomposition, and so, more researchers would be able to put it to use to quantify the effects of fixed investment, labor forces and technology state on changes in energy consumption and carbon emissions.

Figure 2 .
Figure 2. The labor and capital shares in the USA.

Figure 3 .
Figure 3.The index of the technology state.

Table 1 .
Decomposition results under the traditional approach (unit: Million Metric Tons).
Figure 1.The total effect of the decomposition and actual values.

Table 2 .
Decomposition results under the proposed approach (unit: Million Metric Tons).