Introduction

Recently, the Guardian described the environmental condition as climate emergency, crisis, breakdown, and global heating, rather than climate change and global warming to warn the environmental urgency. Additionally, in October 2018, the UN International Panel on Climate Change (IPCC) also announced that the global temperatures would rise about 1.5 oC during 2030–2052 if the CO2 emission could not be whittled down, and that would cause destructive influence on the ecosystem and threat the survival of humans (the United Nations 2018). Thus, the issues of growing global temperatures and environmental sustainability have become more urgent than ever before. To solve the problems of the growing global temperatures and environmental unsustainability, the contributing factors are needed to be distinguished first. Ma et al. (2019) and Sueyoshi and Goto (2013) had pointed out that human activities, which include industrial and economical activities, were mainly responsible for the increased greenhouse gases (GHS) and the growing global temperatures since they consumed more fuel energy than other human activities. Hence, enhancing the industrial activities efficiency and reducing the industrial CO2 emission are the most needed and effective ways to reduce warm gas emissions and realize the sustainability of society and humans. A Kyoto protocol announced in 1997, which aims to promote the worldwide human to cut down the CO2 emission, has been advocated for 24 years, and the CO2 emission has also become an important indicator to rate industrial activities efficiency and environmental quality.

Total factor productivity (TFP) is a persuasive method to estimate industrial activities with CO2 emission over a period of time. From a policy perspective, the assessment of TFP growth is important since it serves as a guide of resource allocation, investment, and decision-making (Margaritis et al. 2005; Chen and Yu 2014). Generally speaking, the parametric method, semi-parametric method and non-parametric method can all be employed to make TFP observations. However, the Data Envelopment Analysis (DEA), proposed by Charnes et al. (1978) and included in the nonparametric method, does not require specifying an ad hoc functional form, nor does it require making unwarranted assumptions regarding the error structure when measuring efficiency. So, the DEA has been widely employed to make TFP observations.

Färe et al. (1995) firstly introduced the DEA into Malmquist (Malmquist 1953) to make a TFP measurement. Since then, the DEA-Malmquist index has been extended extensively. Pastor et al. (2011) constructed a Biennial Malmquist index, which only observed the productivity changes of decision-making units (DMUs) during adjacent and successive two time periods. Chung et al. (1997) built a Malmquist-Luenberger index based on the directional distance function (DDF. Chambers et al. 1996), which is convenient for disposing of undesirable outputs. Oh and Lee (2010) introduced the meta-frontier Malmquist index considering the heterogeneity. Besides, the double frontier Malmquist (Wang and Lan 2011), the Malmquist with common weights (Kao 2010; Mavi and Mavi 2019), two stages Malmquist (Kao and Hwang 2014), cross-efficiency Malmquist index (Ding et al. 2019), biennial Malmquist–Luenberger index (Wang et al. 2020), and double frontier common weights Malmquist (Mavi et al. 2019) have also been suggested.

However, the aforementioned DEA-Malmquist models are not circular and their adjacent period components provide different measures of productivity index. Berg et al. (1992) and Althin (2001) suggested that it would be reasonable to fulfill the circular test for an index covering long periods of time. Pastor and Lovell (2005) exhibited a global Malmquist index, whose productivity change was circular and could be given with a single measure for productivity change. Since then, a number of researches on the global DEA-Malmquist index have been conducted. Methodologically, they can be summarized as the combinations of the global Malmquist index and various DEA models rather than the exploration of the global DEA-Malmquist itself. Emrouznejad and Yang (2016a) proposed a RAM (range-adjusted measure)-based global Malmquist-Luenberger productivity index. Liu et al. (2021) employed the global Malmquist-Luenberger index to analyze the provincial green productivity of road transportation. Long et al. (2020) proposed the global Malmquist index with the super-slacks-based measure. An et al. (2019) introduced a global Malmquist index based on the improved SBM (Slacks-Based Measure. Tone 2001). Tohidi and Razavyan (2013) proposed a global profit Malmquist productivity index. With regard to the applications of the global DEA-Malmquist index, they are focused on environmental evaluation. Readers can refer to Liu et al. (2017), Fan et al. (2015), Kumar (2006), Emrouznejad and Yang (2016b) and so on for more information.

In line with Färe and Grosskopf (1997), the above-mentioned DEA-Malmquist index, including the global Malmquist, were all based on the assumption that the technical change is Hicks’ neutral, that is the marginal rate of substitution (MRS) between inputs and marginal rate of transformation (MRT) between outputs hold constant during time periods, and no biases exist in the technical change. However, this paper finds that the MRS among inputs and MRT among outputs will change in a long time. The isoquants and requirement sets do not have homothetic shifts and the technical change is non-Hicks neutral. There are biases in technical change.

Technical change bias is an index demonstrating the preference and dependence elements of technical change. Analyzing the technical change biases can help to find the drivers of technical change so that the production and productivity can be adjusted and improved in time, just as Mizobuchi (2015) and Chen and Yu (2014) argued that “Recognizing the possibility that bias in technical change can be effectively utilized for improving productivity growth”. What is more, in line with Yu and Hsu (2012), if the biased indexes can be measured, more information will be derived from the individual Malmquist components. So, it is very necessary to measure the bias of technical change in the Malmquist productivity index. Actually, Färe et al. (1997) had proposed to relax the assumption of Hicks neutral for MRS between inputs and MRT between outputs. Some published paper also relaxed the Hicks neutral assumption in the DEA-Malmquist index and made some analysis. Readers can refer to Briec and Peypoch (2007), Barros et al. (2011), Briec et al. (2011), Chen and Yu (2014), Mizobuchi (2015), Hampf and Krüger (2017) for more applications. However, the above-mentioned papers all researched the technical change bias over two adjacent period times. The non-Hicks’ neutral assumption of technical change has hardly been discussed in the global DEA-Malmquist index, such that all the productivity indexes and technical progress biases among long periods of time cannot be given with a single measure.

Hence, this paper proposes to employ the global DEA-Malmquist productivity index to observe the Chinese industrial energy and environment productivity so that the DEA-Malmquist productivity changes can be given with a single measure. Then, we relax the Hicks’ neutral assumption of technical change in the global DEA-Malmquist productivity and decompose the technical change into input bias, output bias, and magnitude, such that the specific factors magnifying (shrinking) the global DEA-Malmquist productivity growth can be further discerned.

This paper is organized as follows: Section 2 introduces the global Malmquist productivity index and its decomposition. Section 3 observes the industrial productivity changes and technical change bias in China; a multinomial logistic model is introduced and applied in Section 4 to make some explanations and provide suggestions for the sustainable industry in China. Section 5 concludes the paper.

The global DEA-Malmquist productivity index

The SBM (Slacks-Based Measure) model for undesirable outputs

The SBM model has been widely employed to analyze the production with undesirable outputs since Tone (2004) introduced undesirable outputs to the SBM model. The conventional radial DEA-Malmquist productivity index is only based on the radial DEA scores but ignores the non-zero input slacks in the input-oriented index (or non-zero output slacks in the output-oriented index). The DEA-Malmquist index based on SBM can “eliminate possible inefficiency represented by the non-zero slacks to measure the productivity change” (Liu and Wang 2008). So, the SBM model with undesirable outputs proposed by Tone (2004) is selected as our basic model of the global DEA-Malmquist index.

Suppose there are n decision-making units (DMUs). Each DMUj(j = 1, ⋯, n) consumes m inputs xij(i = 1, ⋯, m) and produces p desirable outputs yrj(r = 1, ⋯, p) and q undesirable outputs bkj(k = 1, ⋯, q). The observed efficiency of DMUO by SBM model can be obtained as follows:

$$ {\displaystyle \begin{array}{c}{E}_o=\mathit{\operatorname{Min}}\;\frac{1-\frac{1}{m}{\sum}_{i=1}^m\frac{d_x^{-}}{x_{io}^s}}{1+\frac{1}{p+q}\left({\sum}_{r=1}^p\frac{d_y^{+}}{y_{ro}}+{\sum}_{k=1}^q\frac{d_b^{-}}{b_{ko}}\right)}\\ {}\kern2em s.t\kern1em {\sum}_{j=1}^n{\lambda}_j{x}_{ij}+{d}_x^{-}={x}_{io},\kern0.33em i=1,\cdots, m\\ {}\kern4em {\sum}_{j=1}^n{\lambda}_j{y}_{rj}-{d}_y^{+}={y}_{ro},\kern0.33em r=1,\cdots, p\\ {}\kern4em {\sum}_{j=1}^n{\lambda}_j{b}_{kj}+{d}_b^{-}={b}_{ko},\kern0.33em k=1,\cdots, q\\ {}\kern4em {d}_x^{-},{d}_y^{+},{d}_b^{-}\ge 0;{\lambda}_j\ge 0\end{array}} $$
(1)

where λj(j = 1, ⋯, n) is the intensity variables associated with each DMUj(j = 1, ⋯, n). \( {d}_x^{-},{d}_y^{+},{d}_b^{-} \) are the slacks of inputs, desirable and undesirable outputs, respectively. The efficiency values have 0 ≤ Eo ≤ 1. If and only if Eo = 1, the evaluated DMUO is said to be efficient. Otherwise, it is non-efficient.

For the convenience of calculation, model (1) can be transformed into a liner program by the Charnes-Cooper transformation (Charnes and Cooper 1962). The linear model is as follows:

$$ {\displaystyle \begin{array}{c}{E}_o=\mathit{\operatorname{Min}}\ \delta -\frac{1}{m}{\sum}_{i=1}^m\frac{\zeta_x^{-}}{x_{io}^s}\\ {}s.t\kern1em \delta +\frac{1}{p+q}\left({\sum}_{r=1}^p\frac{\zeta_y^{+}}{y_{ro}}+{\sum}_{k=1}^q\frac{\zeta_b^{-}}{b_{ko}}\right)=1\\ {}{\sum}_{j=1}^n{\gamma}_j{x}_{ij}+{\zeta}_x^{-}=\delta \cdotp {x}_{io},i=1,\cdots, m\\ {}{\sum}_{j=1}^n{\gamma}_j{y}_{rj}-{\zeta}_y^{+}=\delta \cdotp {y}_{ro},r=1,\cdots, p\\ {}{\sum}_{j=1}^n{\gamma}_j{b}_{kj}+{\zeta}_b^{-}=\delta \cdotp {b}_{ko},k=1,\cdots, q\\ {}{\zeta}_x^{-},{\zeta}_y^{+},{\zeta}_b^{-}\ge 0;{\lambda}_j\ge 0,\delta >0\end{array}}\kern0.5em $$
(2)

The global DEA-Malmquist model and its decomposition

Färe et al. (1995) proposed the DEA-Malmquist index to describe the efficiency changes of DMU between two time periods. Pastor and Lovell (2005) took all the periods’ data as the productivity possibility set and introduced the global DEA-Malmquist productivity index. The global DEA-Malmquist productivity index is an effective measure of the productivity change, and it can overcome the infeasibility problem that usually appeared in the traditional DEA-Malmquist index.

Suppose there are F time periods. The data of DMUj in t(t = 1, ⋯, F) period is dented as \( {x}_{ij}^t\left(i=1,\cdots, m;t=1,\cdots F\right) \),\( {y}_{rj}^t\left(r=1,\cdots, p;t=1,\cdots F\right) \), and \( {b}_{kj}^t\left(k=1,\cdots, q;t=1,\cdots F\right) \).

The contemporaneous benchmark technology of each DMU in t(t = 1, ⋯, F) period is defined as\( T=\left\{\left({x}_{ij}^t,{y}_{rj}^t,{b}_{kj}^t\right)\right.\left|{x}_{ij}^t\ \right.\mathrm{can}\ \mathrm{produce}\ {y}_{rj}^t\ \mathrm{and}\ {b}_{kj}^t,i=1,\cdots, m;\left.r=1,\cdots p;k=1,\cdots q;j=1,\cdots, n\right\} \). The global benchmark technology, which is defined as TG = T1 ∪ T2 ∪ ⋯TF, is composed of the contemporaneous benchmark technologies of F time periods. Then the global DEA-Malmquist index proposed by Pastor and Lovell (2005) is defined as follows:

$$ {M}^G={D}^G\left({x}^{t+1},{y}^{t+1},{b}^{t+1}\right)/{D}^G\left({x}^t,{y}^t,{b}^t\right) $$
(3)

MG is the global DEA-Malmquist productivity index. MG > , = , < 1 indicate the global Malmquist productivity experiences progress, stagnation, and regress, respectively. DG(xt + 1, yt + 1, bt + 1) and DG(xt, yt, bt) are the efficiencies obtained from the model (4).

$$ {\displaystyle \begin{array}{c}{D}^G\left({x}^s,{y}^s,{b}^s\right)=\operatorname{Min}\ \delta -\frac{1}{m}\sum \limits_{i=1}^m\frac{\zeta_x^{-}}{x_{io}^s}\\ {}s.t.\kern1em \delta +\frac{1}{p+q}\left(\sum \limits_{r=1}^p\frac{\zeta_y^{+}}{y_{ro}^s}+\sum \limits_{k=1}^q\frac{\zeta_b^{-}}{b_{ko}^s}\right)=1\\ {}\sum \limits_{t=1}^T\sum \limits_{j=1}^n{\gamma}_j^t{x}_{ij}^t+{\zeta}_x^{-}=\delta \cdotp {x}_{io}^s,i=1,\cdots, m\\ {}\sum \limits_{t=1}^T\sum \limits_{j=1}^n{\gamma}_j^t{y}_{rj}^t-{\zeta}_y^{+}=\delta \cdotp {y}_{ro}^s,r=1,\cdots, p\\ {}\sum \limits_{t=1}^T\sum \limits_{j=1}^n{\gamma}_j^t{b}_{kj}^t+{\zeta}_b^{-}=\delta \cdotp {b}_{ko}^s,k=1,\cdots, q\\ {}{\zeta}_x^{-},{\zeta}_y^{+},{\zeta}_b^{-}\ge 0;{\gamma}_j^t\ge 0;\delta >0;s=t,t+1\end{array}} $$
(4)

Drawing on Pastor and Lovell (2005), the global DEA-Malmquist productivity index can be viewed as the product of efficiency change (EC) and best practice gap change (BPC). The EC and BPC are respectively defined by Eqs. (5) and (6)

$$ \mathrm{EC}={D}^{t+1}\left({x}^{t+1},{y}^{t+1},{b}^{t+1}\right)/{D}^t\left({x}^t,{y}^t,{b}^t\right) $$
(5)
$$ \mathrm{BPC}={D}^G\left({x}^{t+1},{y}^{t+1},{b}^{t+1}\right)/{D}^{t+1}\left({x}^{t+1},{y}^{t+1},{b}^{t+1}\right)\cdotp {D}^t\left({x}^t,{y}^t,{b}^t\right)/{D}^G\left({x}^t,{y}^t,{b}^t\right) $$
(6)

where MG = EC · BPC.

EC represents the technical efficiency change of the evaluated DMU on cross-period observations, which captures the movement away from or toward the contemporaneous benchmark technologies. BPC is the technology change and measures a change in the best practice gap between the global and contemporaneous benchmark technologies.

Non-Hicks’ neutral technical change and its bias indexes

Färe and Grosskopf (1997) introduced the bias decomposition of DEA-Malmquist index with the non-Hicks’ neutral assumption. This paper extends the decomposition of Färe and Grosskopf (1997) into the global DEA-Malmquist index and the BPC is further decomposed into the product of output biased technical change (OBPC), input biased technical change (IBPC), and magnitude of technical change (MBPC). These indexes respectively take forms below:

$$ \mathrm{OBPC}={D}^G\left({x}^{t+1},{y}^t,{b}^t\right)/{D}^t\left({x}^{t+1},{y}^t,{b}^t\right)\cdotp {D}^G\left({x}^{t+1},{y}^{t+1},{b}^{t+1}\right)/{D}^{t+1}\left({x}^{t+1},{y}^{t+1},{b}^{t+1}\right) $$
(7)
$$ \mathrm{IBPC}={D}^t\left({x}^t,{y}^t,{b}^t\right)/{D}^G\left({x}^t,{y}^t,{b}^t\right)\cdotp {D}^{t+1}\left({x}^{t+1},{y}^t,{b}^t\right)/{D}^G\left({x}^{t+1},{y}^t,{b}^t\right) $$
(8)
$$ \mathrm{MBPC}={D}^t\left({x}^{t+1},{y}^t,{b}^t\right)/{D}^{t+1}\left({x}^{t+1},{y}^t,{b}^t\right) $$
(9)

where BPC = OBPC · IBPC · MBPC (10)

OBPC measures the output bias of technical change, which is the multiplication of the technical change in the global frontier with respect to yt + 1, bt + 1, and yt,btwith the input vector fixed at xt + 1. If OBPC = 1, the output technical change is said to be Hicks’ neutral, meaning that the transformation among outputs keeps constant. Otherwise, when OBPC > (<)1, we say the technical change happens to be output biased, and this change magnifies (shrinks) the global productivity growth.

Figure 1 illustrates the construction of the output biased technical change index. The output possibility set in period 1 is given by p1(x). The technical progress is supposed to appear from periods 1 to 2. Technical progress is Hicks’ neutral if the MRT between two outputs remains constant with the mix of outputs unchanged, the production possibility set p2(x) in period 2 parallels p1(x) and the global production possibility set pHG(x) parallels p1(x). When the technical progress is biased, the MRT changes and the production possibility set p21(x) (or p22(x)) in period 2 does not parallel p1(x). The global production possibility set pBG(x) is not parallel top1(x). Technical progress is biased and in the favor of output 1 (y1-producing) if the MRT between y1 and y2 in period 2 more than that in period 1 with the mix of outputs unchanged. Otherwise, the technical progress is biased and in the favor of output 2 (y2-producing) if the MRT between the two outputs in period 2 is less than that in period 1 with the output mix unchanged. The output possibility set p21(x) and p22(x)illustrates a y2-producing and y1-producing output bias, respectively.

Fig. 1
figure 1

Output possibility sets (P(x))and output biased technical change

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In period 1, a DMU is observed to produce an output vector represented by point A and its technology set is p1(x). The output distance function is calculated by \( {D}_o^1\left(x,{y}^1\right)= OA/ OD \). In period 2, the DMU is observed to produce output vector B. If the technology shifts to p21(x) in period 2, the output distance function in period 2 is \( {D}_o^2\left(x,{y}^2\right)= OB/ OG \), and the output bias index is OBPC = OD/OC · OE/OG < 1, since \( {y}_1^{t+1}/{y}_2^{t+1}<{y}_1^t/{y}_2^t \) and OBPC < 1, the technology is y2 -producing relative to y1. However, if the technology shifts to p22(x) in period 2, the output distance function will be \( {D}_o^2\left(x,{y}^2\right)=\mathrm{OB}/\mathrm{OH} \), and the output bias index is OBPC = OD/OC · OE/OH > 1. Giving \( {y}_1^{t+1}/{y}_2^{t+1}<{y}_1^t/{y}_2^t \)and OBPC > 1, the technology isy1 -producing, relative to y2.

IBPC is the multiplication of technical change in the global frontier with respect to xt + 1 andxt with the output vector holding constant at yt. It measures the input bias of technical change. If IBPC = 1, it displays that the input technical change is Hicks’ neutral and there is no input biased technical change. When IBPC > (<)1, it hints that the technical change is input biased, which magnifies (shrinks) the global Malmquist productivity growth.

Figure 2 shows the construction of input biased technical change index. The isoquant in period 1 is denoted as L1(y). The shift of isoquant from period 1 to 2 is defined as technical progress. Technical change is Hicks’ natural if the MRS between two inputs remains constant with the inputs mix holding constant. That is, the isoquant in period 2 L2(x) is parallel to L1(y), and the global isoquant LHG(y) parallels L1(y). If technical progress is biased, the MRS will change, and there exists that isoquant L21(y)(orL22(y)) in period 2 does not parallel to L1(y), and the global production possibility set LBG(y) does not parallel L1(x). With the input mix holding constant, the technical progress is x1-saving and x2-using if the MRS between the two inputs in period 2 is more than that in period 1. Otherwise, when the MRS between the two inputs in period 2 is less than that in period 1, the technical progress is biased and that is in the favor of x2-saving and x1-using. The isoquant L21(y) illustrates the x2-saving and x1-using bias, and the isoquant L22(y) illustrates the x1-saving and x2-using bias.

Fig. 2
figure 2

Input requirement sets (L(y)) and input biased technical change

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A and E are the input vectors in periods 1 and 2, respectively. If the technical progress shifts from L1(y) to isoquant L21(y) in period 2, the input bias index will become IBPC = OC/OB · OG/OD. In this case, due to (x1/x2)t + 1 > (x1/x2)t, OC/OB < OG/OD and IBPC < 1, the technology exhibits the x1-using and x2-saving bias. Otherwise, if the technical progress shifts from L1(y) to isoquant L22(y) in period 2, the input bias index becomes be IBPC = OC/OB · OH/OD. In this case, since (x1/x2)t + 1 > (x1/x2)t, OC/OB > OG/OD and IBPC > 1, the technology exhibits the x2-using and x1-saving bias.

MBPC measures the magnitude of technical change through the outputs in period t and the inputs in period t + 1, and it has no significant hints.

Tables 1 and 2 summarize the alternatives for input bias and output bias in the global DEA-Malmquist index. It is consistent with the alternatives of Barros et al. (2009, 2010) and Barros and Weber (2009).

Table 1 Alternatives for input bias
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Table 2 Alternatives for output bias
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Measuring the industrial energy and environment productivity in China

In this section, the global DEA-Malmquist productivity index will be applied to assess the industrial energy and environment productivity of China. The regions in China and the indicators are introduced first. The industrial energy and environment productivity in China is then obtained and discussed. An exploration of the technical change bias concludes this section.

Indicators and statistical description

From the perspective of economic and geographic characteristics (Zhao et al. 2016), 31 provinces, municipalities, and autonomous regions in China mainland can be divided into the Eastern, Central, and Western. Due to the absence of information, Tibet is excluded in the empirical analysis. The divisions of 30 provinces, municipalities, and autonomous regions in China mainland are shown in Table 3.

Table 3 Regions and areas of China mainland exclusive of Tibet
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Table 4 summarizes the data set for Chinese industrial Malmquist productivity estimation, which consists of three inputs and two outputs. Labor (x1), capital stock (x2) and energy (x3) are selected as the main inputs. Labor can be explained as the number of employees taking part in industrial production. In line with Wang and Wang (2020), capital stock is supposed to embody the industrial production capacity and technology, but generally there are no relevant statistical data on hand. The perpetual inventory method proposed by Jun et al. (2004) is employed to deduce, where the depreciation rate is defined as 9.6%, and all the capital stock data are converted into the constant price of 2006. The consumed energy by industry contains several kinds. According to Zhou (2013), the consumed energy includes crude oil, gasoline, diesel, refinery dry gas, kerosene, fuel oil, coke oven gas, liquefied petroleum gas, natural gas, other gas, other coking product, other petroleum product, electric power and thermal, cleaned coal, raw coal, briquette coal, other washing coal, and coke. These energies are all transformed into standard coal equivalent (SCE) with the standard coal transformation equivalent and energy coefficients. The transformation coefficients can refer to the key industry and transportation enterprises report in 1986 of China energy statistics (2005). The outputs consist of industrial output values (y) and CO2 emission (b). The industrial output values are viewed as the desirable outputs and the CO2 emission is the undesirable outputs. An CO2 emission equitation introduced by the Intergovernmental Panel on climate change (IPCC, Simon et al. 2006) is used to get the total CO2 emission. The CO2 emission equitation is shown as follows:

$$ C{O}_2=\sum \limits_{i=1}^3C{O}_{2,i}=\sum \limits_{i=1}^3{E}_i\times \mathrm{NC}{\mathrm{V}}_i\times \mathrm{CE}{\mathrm{F}}_i\times \mathrm{CO}{\mathrm{F}}_i\times \left(44/12\right) $$
(10)
Table 4 The average inputs and outputs during 2011–2016
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CO2, i(i = 1, …, 19) represents the total CO2 emission, Ei, NCVi, CEFi and COFi are the total energy consumption (E), net calorific value (NCV), carbon emission factor (CEF), and carbon oxidation factor (COF). Forty-four is the CO2’s molecular weights and 12 is the carbon’s molecular weights.

The above-mentioned data all come from China Statistical Yearbook (2012–2017) and China Energy Statistical Yearbook (2012–2017).

Empirical results

An analysis of the regional industrial global DEA-Malmquist productivity index

Table 5 documents the regional industrial global Malmquist productivity indexes (MG), EC and BPC in China. Figure 3 describes the average MG, EC, and BPC of Eastern, Central and West regions, respectively. It can be concluded from Fig. 3 that the industrial productivity and technical progress of China tend to increase and improve over time periods, but the technology efficiency decreases since 2012. The productivity of the West regions significantly increases by 4.3% from 2011 to 2016. Eastern productivity improves a lot during 2013–2014 and 2015 to 2016. However, it experiences a significant decrease during 2014–2015. With regard to the Central, its productivity keeps higher than the average of all regions during 2011–2013, whereas, it decreases since 2013 and is less than the average productivity of all regions. The productivity of the West keeps growing during 2011–2016. The current development status in China would like to explain this. Because the West develops later than the Eastern and Central, the chance and motivation for developing are huge so that its productivity increases more significantly. Additionally, the great western development strategy has been launched for a few years, and the West has developed and made a huge achievement significantly with the support of policies. As a result, the West gets the highest productivity index. The Eastern region in China has a long development history, which makes it have a slower development speed and a lower productivity index. Besides, the resources of the Eastern are finite and cannot support its further development. Central regions are always influenced a lot by the policies of the central government. In response to the call of “The new normal of China’s economy” and forming a more environment-friendly and sustainable development model of industry, Central regions have taken pains to adjust their industrial structure, save energy and protect the environment. Hence, Central regions get decreased productivity and its technology regresses during 2011–2015. Figure 3 also illustrates that the technical efficiencies of all regions nearly tend to decrease over the time periods. According to Barros et al. (2009), the changes in technical efficiency score are defined as the diffusion of best-practice technology in the management activity, which includes investment planning, technical experience, management, and organization in the firm. That reminds us that the industrial technology efficiency in China can be improved by improving investment planning, technical experience, management and organization.

Table 5 Regional global DEA-Malmquist productivity, efficiency change and best practice gap change (2011–2016)
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Fig. 3
figure 3

Regional productivity levels and differences

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Table 6 exhibits the contributions of EC and BPC to MG. It is easy to see that the BPC dominates the MG and has more contributions to MG than the EC. This can also be found in Fig. 3 where MG and BPC have similar change characteristics. These hint that the technical progress drives the improvements of the global DEA-Malmquist productivity and this is consistent with the findings of Ding et al. (2019), Wang et al. (2020), and Du et al. (2018). It is necessary to improve the BPC first if the MG wants to be improved. Since BPC means technology innovation and application, it is very urgent to accelerate the industrial technology innovations and applications to fulfill productivity improvement.

Table 6 Average global DEA-Malmquist productivity, efficiency change, and best practice gap change (2011–2016)
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Biases decomposition in the best practice gap change (BPC)

In line with models (7)–(10), the BPC is decomposed into IBPC, OBPC and MBPC. Tables 7 and 8 indicate the decompositions.

Table 7 The input biases of technical change
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Table 8 The output biases of technical change
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Table 7 summarizes the input biases of technical change. It deploys that almost all regions have IBPC > 1. During 2011–2016, there are 27 regions experiencing labor bias (labor using and capital saving). It hints that the technical progress of the Chinese industry is motivated by labor more as compared with capital. The labor bias can be explained from two aspects. The first one is the population bonus. The industry can cost little in employing but achieve huge benefits from it. The other one is that the industry in China is less intelligent and automatic but human dependent. It needs to employ plenty of people to work. With regard to labor and energy, there are 16 regions that are in favor of using more labor and less energy. The bias between labor and energy is balanced. As for capital and energy, there are 27 regions having energy bias (using more energy but saving capital). It means that energy motivates technical progress more than capital. Two aspects may attribute to this energy bias. One is the fact that China is abundant in coal and has huge industrial coal consumption. The other one is the industry in China is still energy-oriented and consumes a large amount of energy. Therefore, it can be concluded that the 30 regions in China tend to be labor and energy biases, which hints that labor and energy are the prime factors to promote the Chinese industrial technical progress.

Table 8 describes the output biases status of the Chinese industry. It can be seen that there are nearly 22 regions having CO2 emission biases and 8 regions having industrial output values biases. It means that the technology of CO2 emission will dominate the output-technology change. Hence, it is necessary to enhance the technology of CO2 emission so that the output-technology progress of productivity can be promoted.

Table 9 documents the technical change inputs and outputs biases of each region during 2011–2016. It finds that almost all regions experience labor bias, energy bias, and CO2 emission bias. Tianjin, Zhejiang, Shandong, Hubei, Hunan, Shandong, Guangxi, Guizhou, Shaanxi, Qinghai, and Xinjiang are partly capital biased, which means that industrial technical progress in these regions is partly motivated by capital. Some regions are industrial output values biased, and their technical progress is motivated by the industrial output values. Figure 4 shows the biases distributions frequencies.

Table 9 The regional biases distributions
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Fig. 4.
figure 4

Proportions of industrial biases. A: Labor bias; M: Capital bias; B: Energy bias; N: Industrial Output values bias; C: CO2 emission bias

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To sum up, it can be concluded that the industrial technical progress in China is mainly motivated by labor, energy, and CO2 emission. In other words, Chinese industrial technical progress is achieved by employing more labor, consuming more energy and producing more CO2 pollutant. As a result, the MRS between inputs as well as the MRT between outputs, are non-homothetic. Therefore, the assumption of Hicks’ neutrality is rejected. This hints that the conventional growth accounting method is not suitable for the Chinese industry since it is Hicks’ neutral. This also reflects that the industrial technology level in China is pretty original and it not only depends much on labor and energy and less on capital and advanced technology little, but also produces amounts of CO2 pollutant. The industrial progress in China is less sustainable.

The contributing factors analysis of inputs and outputs biases.

It has been learned that industrial technical progress in China has labor, energy and CO2 emission biases. However, the reasons for these biases are still unknown. So, a contributing factors analysis is conducted. Some possible explanatory variables are found to answer if they lead to these labor, energy, and CO2 emission biases, and what these causalities are. Because the labor, energy, and CO2 emission biases here represent narrative facts rather than specific numbers, they are multiple and discrete variables. Then the conventional linear regression analysis is not appropriate anymore. A multinomial logit model (MNLM. McFadden 1974) is applied for observation. The MNLM estimates the parameters of an assumed logistic relationship between the explanatory variables and the dependent variables, and it has been widely applied to analyze the reasons for not buying cars (Gao et al. 2014), motorcycle crash severity (Vajari et al. 2020), pedestrian-vehicle crash severity (Chen and Fan 2019), household cooking fuel choice (Alem et al. 2016), the crash proportion by vehicle type (Lee et al. 2018), and so on.

The MNLM analysis of contributing factors to technology change biases can be viewed as a test procedure in which several binary logits are simultaneously estimated. The model of MNLM is set as follows:

$$ \ln {\varOmega}_{j\left|b\right.}=\ln \frac{\Pr \left(y=j\left|x\right.\right)}{\Pr \left(y=b\left|x\right.\right)}={x}^{,}{\beta}_{j\left|b\right.}\kern1em for\kern0.5em j=1,\cdots, J $$
(11)

where b is the base outcome. Because lnΩb|b = ln 1 = 0, then βb|b = 0 . In other words, the log-odds are always 0 for itself selection, which must make the coefficient of any explanatory variable be 0. The prediction probability of each bias can be got from the following model:

$$ \Pr \left(y=j\left|x\right.\right)=\frac{\exp \left({x}^{,}{\beta}_{j\left|b\right.}\right)}{\sum_{m=1}^J\exp \left({x}^{,}{\beta}_{m\left|b\right.}\right)}\kern1em for\kern0.5em j=1,\cdots, J $$
(12)

where x, is the exogenous explanatory variables determining the bias outcome y. βj|b is the coefficients.

We have known from Table 9 that some regions are only labor biased or energy biased or CO2 emission biased, but some of them are not only labor biased, but also energy biased and CO2 emission biased. Therefore, these labor bias, energy bias, and CO2 emission biases are permuted and combined into seven genres of illustration and they are labelled with some letters. A hints that the DMU is labor biased. B says that the DMU is energy biased. C denotes that the DMU is CO2 emission biased. D deploys that the DMU is labor and energy biased. E represents that the DMU is labor and CO2 emission biased. F exhibits that the DMU is energy and CO2 emission biased. R means that the DMU is labor and energy and CO2 emission biased. These seven bias combinations are viewed as the dependent variables y.

Then, a series of indicators that may make the technical progress be labor, energy, and CO2 emission biased are selected as the explanatory variables x, and they are shown in Table 10.

Table 10 Selection and interpretation of indicators
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The data of 14 factors listed in Table 10 are from China Energy Statistics Yearbook (2012–2017), Website of China National Bureau of Statistics (2012–2017), China Industrial Statistics Yearbook (2012–2017), and China Environment Statistics Yearbook (2012–2017). Since the biases are intertemporal, the average of each indicator during two adjacent years is used. Therefore, there are five time periods, including 2011–2012, 2012–2013, 2013–2014, 2014–2015, 2015–2016. Table 11 shows the statistical descriptions of the handled indicators.

Table 11 Statistical descriptions of independent variables
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Stata 14 is employed to analyze the correlative relationship between the biases and factors. Bias G defaults as the base outcome. After several experiments, only 6 factors have passed the significance test and have significant influences on technical change bias. They are the energy prices, the cost of curbing air pollutants, the firm’s size, the structure of energy consumption, the economic foundation, and infrastructures. Table 12 specifies the outcomes.

Table 12 Statistical experiment outcomes
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It can be learned that there are two indicators having impacts on labor bias, and they are the economic foundation and the infrastructures. From the results of coefficient estimation, the economic foundation has a significant positive impact on the bias of labor with a confidence level of 95%, which implies that the worse the economic foundation is, the more possibilities that the industry will use more labor to provoke the technical change. This is undesirable. Poor economic foundation in China means poor traffic conditions and lagged economic and sociable developments, and the development is only promoted by employing much cheap labor. A perfect economic foundation may help reduce the dependence on labor and change the technical progress motivation mechanism. However, Table 12 also hints that the infrastructures have negative impacts on labor bias. It suggests that the regions’ technical progress with better infrastructures will be more likely to be motivated by labor. Excellent infrastructures are the results of economics development. When the economy is developed a lot and infrastructures get improved, the industrial structure will change and the third industry will get more proportion. This brings about more demands for labor and this becomes a desirable bias.

Energy prices tend to influence CO2 emission positively. That is the lower the energy prices are, the more possibility that technical progress is motivated by CO2 pollutants. This is because the cheap energy prices will motivate the industry to use more energy to extend production, which will produce more CO2.

Labor and energy bias is significantly influenced by energy price, the cost of curbing air pollutants, the structure of energy consumption, and the size of the firm. Both the energy prices and the energy consumption structure have a negative influence on labor and energy bias. It hints that as the coal proportion in energy consumption becomes huger and the prices of energy become more expensive, the technical progress will be more likely to be labor bias and CO2 emission bias. However, the cost of curbing air pollutants and the size of the firm influence labor and energy bias positively. It says that the technical change progress will be promoted if the cost of curbing air pollutants is less and the size of a firm is smaller, which will motivate firms to employ more labor and energy.

Finally, energy and CO2 emission bias is also influenced by four factors, which are the economic foundation, the infrastructures, the energy prices, and the size of firm. Infrastructures and energy prices affect energy and CO2 emission bias positively. It is not hard to understand that the worse the infrastructures are, the cheaper the energy is, and the more they will motivate the local industry to use more energy to improve production and produce more CO2 pollutants. But the economic foundation and the size of firm have negative impacts on energy and CO2 emission bias. It can be illustrated that the technical progress will become more biased energy and CO2 emission as the economic foundation becomes better and the size of firm gets larger. The reason is that the energy consumption increases when the economic foundation is better and the size of the firm is larger. As long as the economic foundation is good enough and the energy consumption is huge enough, it will produce a large amount of CO2.

In short, there are six contributing factors, including the cost of curbing air pollutants, the structure of energy consumption, the economic foundation, the infrastructures, the energy prices and the size of firm, having causal relationship with the technical change biases of the industry in China. Specifically, the poor infrastructures cause energy and CO2 emission bias, but the excellent infrastructures cause labor bias. The regions with poor infrastructures should improve themselves and avoid the labor bias and energy and CO2 emission bias, and those with excellent infrastructures are suggested to keep on. The low energy price leads to the CO2 emission and energy and CO2 emission biases. So, it is necessary to increase the energy price for eliminating the CO2 emission bias. But the high energy price and the coal consumption proportion lead to labor and energy bias. It hints that the high energy price and the coal consumption proportion cannot avoid the labor and energy bias. However, the size of the firm and cost of curbing influence air pollutants labor and energy bias positively, which hints that the labor and energy bias can be avoided as the cost of curbing air pollutants increases and the size of the firm enlarged. The feeble economic foundation causes labor bias, but the improved economic foundation and large firm result in biased energy and CO2 emission. So, it is necessary to balance the size of the firm to reduce energy consumption and CO2 emission.

Conclusions

This paper analyzes the regional industrial energy and environment productivity change in China during 2011–2016 based on the global DEA-Malmquist productivity index. We find that most regions achieve improvement in productivity and technical change over the time periods. However, the average scores of technology efficiency tend to be less than one and keep down. The contribution of technology change to productivity is always larger than that of efficiency change, which implies that the positive enhancement on the technical change is the most significant factor of the improvement of the positive productivity. It will be vital to improve the technical progress if the productivity wants to be enhanced.

In the theoretical part, the technical change is decomposed into inputs bias, output bias, and magnitude. In our empirical study, the technical change is decomposed into labor bias, capital bias, energy bias, industrial production values bias, and CO2 emission bias. The results turn out that the industrial technical progress in China is mainly affected by labor bias, energy bias, and CO2 emission bias. In other words, the industry in China is more likely to employ more labor and energy, and produce more CO2 emissions to achieve technical progress, which hints that the level of industrial technology in China is pretty primary and not sustainable.

The MNLM analysis shows that there are six contributing factors to the aforementioned labor bias, energy bias, and CO2 emission bias of the Chinese industrial Malmquist productivity index, and they are the infrastructures, the energy price, the cost of curbing air pollutants, the structure of energy consumption, the size of the firm, and the economic foundation. Therefore, the following should be the consensus for the regions in China: (1) it is necessary to improve the industrial technical progress so that to enhance the industrial productivity; (2) the technical progress is not sustainable because it is labor, energy, and CO2 emission biased, but it can be adjusted and enhanced because (a) improving the infrastructures and increasing the energy price are beneficial to avoid energy and CO2 emission bias; (b) increasing the cost of curbing air pollutants and enlarging the size of the firm can avoid labor and energy bias; (c) enhancing economic foundation can reduce the dependence on labor bias. The Chinese regional industry should also be aware that (1) the high energy price and the coal consumption proportion cannot effectively avoid the labor and energy bias; (2) the improved economic foundation and large firm are likely to result in energy and CO2 emission bias. More balance and wisdom as well as efforts are needed to realize the sustainability of the industry.

This paper attempts to analyze the bias of the Chinese industrial productivity index and has achieved meaningful hints. The bias in the technical change of other industrial sections considering environment and sustainability is still worthy being further explored. Besides, it is also interesting to measure the technical change biases of the Malmquist productivity index in two stages or network structure systems.