Factor Misallocation and Optimization in China’s Manufacturing Industry

Abstract

Factor allocation has an important impact on production efficiency, and this study discusses factor misallocation and proposes an optimized method that could improve efficiency and reduce factor inputs. Under a multi-industry competitive equilibrium model, by introducing distortion tax that represents frictions on factor inputs (capital, labor, energy, and other intermediate consumption), we define factor price distortion indexes to measure factor misallocation and set the standardized comprehensive distortion index at the optimal value of one to obtain optimized allocation of each factor. Using actual and optimized inputs, we compute input-oriented efficiencies separately by employing a slack-based measurement data envelopment analysis (SBM-DEA). The superiority of the new allocation method is tested by comparing changes in efficiency and input redundancy before and after optimization. An empirical test based on China’s manufacturing sector during 1999–2016 shows that, under the optimized reallocation, on average, efficiency is improved by 43.40%, capital, labor, energy, and other intermediate consumption are reduced by 18.06%, 16.34%, 30.91%, and 31.24%, respectively, and the total carbon emission is reduced by 55.22% from 2000 to 2016. Our results imply that factor misallocation causes losses in efficiency and excessive input of factors, and factor allocation needs to be further optimized for sustainable development.

1. Introduction

Optimal allocation of resources can improve production efficiency. When production factor allocation reaches Pareto optimality, there is no room for further improvement and production is at its most efficient level. However, in reality, Pareto optimality is difficult to achieve as a lot of production activities lack economic efficiency; resource misallocation is common; and different economies have different degrees of misallocation [1,2,3,4,5]. The difference in factor allocation efficiency is also the main reason for differences in development between industries, regions, and countries, so optimizing the allocation of factors is essential.

Manufacturing is the backbone of the national and industrial economy in China. Currently, Chinese manufacturing has formed an independent and complete industrial system. Its healthy and stable development is a solid assurance that China can strengthen its real economy and become a strong manufacturing country. Optimizing factor allocation is an important way to promote the high-quality development of industry through improvement in efficiency. In 2020, the Chinese government issued the “Opinions on Constructing a More Perfect Factor Market Allocation System and Mechanism” and the “Opinions on Accelerating the Improvement of the Socialist Market Economic System in the New Era”. The opinions provide guidance and emphasize the focus on market-oriented allocation to realize market pricing, an independent and orderly flow, and efficient and fair allocation of production factors. Therefore, it is of great practical significance to understand the current state of factor allocation in China’s manufacturing sector and find a better method of allocation to promote sustainable development.

The allocation and utilization of resources are the main research focus in economics. The scarcity of resources constrains production activities. Driven by the pursuit of profit maximization, firms choose how to allocate production factors, but in reality, most allocation is done under imperfect competition. There exist many frictions that can hardly lead to the effective allocation of resources, therefore resulting in resource misallocation.

Many specific frictions can lead to resource misallocation, such as finance and tax policy, household registration systems, and government interventions. Financial frictions cause an 8.3 percent loss in aggregate TFP, which accounts for 30 percent of the capital misallocation observed in China [6]. Significant misallocation of debt and equity in China’s manufacturing sector leads to capital misallocation. Mimicking the U.S. to reallocate liabilities of firms in China, the improvement of efficiency would produce gains of 51% to 69% in real value [7]. The household registration system and the imperfection of the social security system related to the household registration system have caused labor misallocation and brain drain [8]. Under a progressive fiscal policy rule, capital and labor inputs move from highly productive to low-productivity establishments because the latter have a lower tax rate. When the tax progressivity rises, the economy’s overall production will fall since low-productivity establishments use high-level resources inefficiently [9]. Compared with non-state firms, state firms are more likely to obtain government approval to conduct seasoned equity offerings, and state control of capital access distorts resource allocation and impedes the growth of non-state firms [10]. The local protection embodied by market segmentation restricts the free flow of factors and aggravates regional resource misallocation [11,12]. The government’s supporting policies regarding credit subsidies for small and young firms resolve the misallocation of resources and enhance aggregate productivity [13]. In addition to the above reasons, exchange rates, inflation, trade barriers, and regional development differences can also cause resource misallocation.

In empirical studies, many papers have mainly examined the effects of factor misallocation on total factor productivity (TFP) to illustrate the extent and impact of resource misallocation, but few have paid attention to factor misallocation itself. Previous studies in terms of methods may be classified into three categories. First, distortions of price are introduced into the production model to illustrate the effect of resource misallocation on TFP. Hsieh and Klenow (HK) [1] express TFP with price distortions, where capital and labor are reallocated according to the marginal products of the United States, producing TFP gains of 30–50% in China. Oberfield [3] follows the study of HK and concludes that the decline in allocation efficiency between industries accounts for about one-third of the reduction in TFP in Chilean manufacturing. Midrigan and Xu [14] establish a dynamic model in which financial frictions reduce TFP through dispersion in returns on capital (capital misallocation) across existing producers in Korean manufacturing. The measurement methods of resource misallocation derived from the framework of the above studies are: (1) the ratio of actual TFP to TFP under effective factor allocation [15], which is an indirect method by which the specific amount of optimization cannot be calculated; (2) the factor price distortion calculated by using parameters set by HK directly and the factor price estimated [16]. The parameters set by HK are not necessarily suitable for other economies. Furthermore, when more than two factors (capital, labor) are involved, it is difficult to estimate the price, such as the price of intermediate inputs. Chen and Hu [17] and Aoki [18] also define resource misallocation based on factor price distortion as when the ratio of a firm’s average factor output elasticity to the industry’s factor output elasticity is not equal to the share of the firm’s factor in the industry’s factor. The advantage of this method is that there is no strict requirement for the production form; the result is more objective; and the optimized allocation can be derived.

The second method to consider is that, theoretically, if resources flow freely without any misallocation, the TFP of all firms should be equal. Therefore, the discrete degree of TFP is calculated to measure the extent of resource misallocation, such as standard deviation, quartile, percentile, and the ratio of 90% to 10%; the larger the dispersion, the more serious the distortion of allocation [19,20,21]. Both data volume and purity are required for the calculation of the discrete degree of TFP, which reflects the overall situation of resource misallocation; it is impossible to measure more specific degrees of misallocation and to give the optimization amount of factors.

The third method is to decompose out the change of allocation efficiency from the change of TFP to observe its trend. Decomposition methods are BHC (Baily, Hulten, and Campbell) [22], GR (Griliches and Regev) [23], OP (Olley and Pakes) [24], FHK (Foster, Haltiwanger, and Krizan) [25], and MP (Melitz and Polanec) [26]. Most studies show that factor misallocation has an inhibiting effect on TFP improvement, and there is great room for the improvement of factor allocation. Unfortunately, the weakness of this method is that it cannot estimate specific degrees of misallocation, though the trend of resource misallocation can be observed.

Given the discussion above, there are many frictions for factor misallocation which have a significant negative impact on TFP, and several measurement methods of misallocation are reviewed and discussed. Most existing studies analyze the existence and impacts of factor misallocation, and few studies suggest ways to achieve optimized allocation. Data envelopment analysis (DEA) and stochastic frontier analysis (SFA) are common methods to optimize allocation as they construct the frontier to obtain the reallocation of factor inputs when technical efficiency is maximized. However, the ideal goal of production activities is to maximize economic efficiency, which consists of both technical and allocation efficiency. When technological progress is difficult to break through in a short time, it is better to obtain factors’ optimized allocation to improve efficiencies.

Following the approach developed by Chen and Hu [17] and Aoki [18] mentioned above, this study aims to analyze the misallocation of more factors by introducing distortion and further explore the method to achieve optimized allocation. It constructs a new and effective method of allocation under competitive equilibrium and tests the effectiveness of the proposed model using data of China’s manufacturing industry from 1999 to 2016. Its contribution to the existing literature can be summarized in the following aspects. (1) This study constructs the production model with four factors (capital, labor, energy, and other intermediate consumption) instead of two or three in the existing research. Furthermore, by introducing a distortion tax that represents frictions on factor prices, this study defines a comprehensive distortion index (CDI) of more factors compared with existing studies to fully analyze factor allocation. (2) The standardized CDI is set as one specifically to obtain the optimized allocation of each factor. Technical efficiencies (input-oriented) under the actual and optimized allocation of each industry are computed, and efficiencies and redundancies of four factors are compared before and after being optimized. The empirical results indicate that the optimized allocation can bring improvement in efficiency and savings in inputs given the same output level.

The significance of this study can be summarized as follows. (1) From the perspective of factor misallocation and optimization, this study focuses on improving efficiency and reducing factor allocation redundancy, which is consistent with the UN Sustainable Development Goals. The added value of China’s manufacturing industry has been ranked first in the world for 12 consecutive years. Therefore, its stability and sustainable development have a significant impact on the world economy and environment. (2) Different from most existing studies taking only capital and labor into account, this study adds intermediate consumption and energy as factors in measuring resource allocation for China’s manufacturing industry, which is more in line with the actual production situation. (3) The advantages of efficiency improvement after optimized allocation provides scientific support for the formulation and implementation of policies related to optimizing factor allocation. (4) The theoretical framework proposed in this study can be widely applied to analyze the resource allocation of other industries in other regions or countries.

The conceptual framework of this study can be demonstrated in Figure 1 below.

To present the objectives of our study, the remainder of this study is arranged as follows. Section 2 describes our analysis of a multi-industry competitive equilibrium model with industry-specific frictions in the form of taxes on factor prices to measure factor misallocation and obtain optimized allocation and also describes the efficiency model. Section 3 discusses our research samples and data on China’s manufacturing industry. Section 4 presents empirical results, measures factor distortions, optimized allocation, and efficiency, and tests the superiority of the reallocation. Section 5 outlines the conclusions and discussion.

2. The Model

2.1. Multi-Industry Competitive Equilibrium Model with Industry-Specific Frictions

Referring to the theoretical framework of Chen and Hu [17] and Aoki [18], this study modifies the input–output indicator system, extends the analysis of factors from two or three to four, and develops a multi-industry competitive equilibrium model with industry-specific frictions to measure factor misallocation and obtain the optimized allocation. The following sections demonstrate the details of the derivation of our extended theoretical model.

2.1.1. The Production of $N$ Industries

Supposing there are $N$ industries in the economy, firms in the same industry are homogeneous, and each industry is represented by a firm. There are four factor inputs: capital ($K$), labor ($L$), energy ($E$), and other intermediate consumption ($M$). Firms are price takers in both goods and factor markets, pay linear taxes on inputs, and also face factor price distortions. Thus, firms in industry $i$ produce goods given the goods’ price $P_i$ and factor inputs’ costs: $(1+{\tau }_{Ki})P_K$, $(1+{\tau }_{Li})P_L$, $(1+{\tau }_{Ei})P_E$, and $(1+{\tau }_{Mi})P_M$. $P_K$, $P_L$, $P_E$, and $P_M$ are the common factor prices and are the same across firms for each factor. ${\tau }_{ji}$$(j=K,L,E,M)$ is the factor distortion tax, representing the impact of industry-specific frictions on the price. So, if ${\tau }_{ji}=0$ for all $i$, the cost of each factor is the same across industries.

The firm $i$ has a Cobb–Douglas production function, which can be written as follows:

$$Y_i=TF{P}_i\cdot K_i^{{\alpha }_i}\cdot L_i^{{\beta }_i}\cdot E_i^{{\gamma }_i}\cdot M_i^{{\delta }_i}(i=1,2,\cdots ,N)$$

(1)

$Y_i$ is the value of output, $TF{P}_i$ is the productivity, ${\alpha }_i$, ${\beta }_i$, ${\gamma }_i$, and ${\delta }_i$ represent the output elasticity of corresponding factors. The function of profit maximization is:

$$Max {\pi }_i=P_i{Y}_i-\left\{(1+{\tau }_{Ki})P_K\cdot K_i+(1+{\tau }_{Li})P_L\cdot L_i+(1+{\tau }_{Ei})P_E\cdot E_i+(1+{\tau }_{Mi})P_M\cdot M_i\right\}$$

(2)

The first-order conditions are as follows:

$$P_i(\frac{{\alpha }_i{Y}_i}{K_i})=(1+{\tau }_{Ki})P_K,$$

(3)

$$P_i(\frac{{\beta }_i{Y}_i}{L_i})=(1+{\tau }_{Li})P_L,$$

(4)

$$P_i(\frac{{\gamma }_i{Y}_i}{E_i})=(1+{\tau }_{Ei})P_E,$$

(5)

$$P_i(\frac{{\delta }_i{Y}_i}{M_i})=(1+{\tau }_{Mi})P_M.$$

(6)

2.1.2. Aggregate Function

Assuming the aggregate production function is the constant returns to scale (CRS):

$$Y=Y(Y_1,Y_2,\dots ,Y_N)$$

(7)

The following condition is satisfied:

$$\frac{\partial Y}{\partial Y_i}=P_i$$

(8)

This condition is satisfied if $Y$ is an aggregate good and firms that produce $Y$ from $Y_i$ are competitive. Under these conditions, the following equation holds (suppose the price of the aggregate good is 1).

$$Y={\displaystyle \sum _{i=1}^N{P}_i}{Y}_i$$

(9)

2.1.3. Resource Constraints

It is also assumed that the aggregate factors are given exogenously, and there are the following constraints:

$${\displaystyle \sum _i^N{L}_i=L},$$

(10)

$${\displaystyle \sum _i^N{K}_i=K},$$

(11)

$${\displaystyle \sum _i^N{E}_i=E},$$

(12)

$${\displaystyle \sum _i^N{M}_i=M},$$

(13)

where $K,L,E,M$ are the aggregate factors’ supply, respectively.

2.1.4. Competitive Equilibrium

According to the above settings, a competitive equilibrium with frictions of this economy is defined in the following manner.

Given the $TF{P}_i$, distortion taxes ${\tau }_{ji}$, the aggregate capital ($K$), labor ($L$), energy ($E$), and other intermediate consumption ($M$), a competitive equilibrium is a set of the output, capital, labor, energy, and other intermediate consumption of $N$ industries $\{Y_i,K_i,L_i,E_i,M_i,P_i\}$, the aggregate output value $Y$, and the common factor prices $P_K,P_L,P_E,P_M,$ respectively, that satisfies the following conditions:

• FOCs of $N$ industries (3)–(6), where $Y_i$ is given by (1);
• CRS assumption and marginal conditions (7) and (8);
• Resource constraints (10)–(13).

Based on the above assumptions, each firm’s factor inputs can be written as follows, based on Equations (3)–(6) and Equations (10)–(13):

$$K_i=\frac{{\displaystyle \frac{(1+{\tau }_{Ki})P_K{K}_i}{(1+{\tau }_{Ki})P_K}}}{{\displaystyle \sum _n{\displaystyle \frac{(1+{\tau }_{Kn})P_K{K}_n}{(1+{\tau }_{Kn})P_K}}}}K={\displaystyle \frac{{\displaystyle \frac{P_i{\alpha }_i{Y}_i}{(1+{\tau }_{Ki})P_K}}}{{\displaystyle \sum _n{\displaystyle \frac{P_n{\alpha }_n{Y}_n}{(1+{\tau }_{Kn})P_K}}}}}K=\frac{{\nu }_i{\alpha }_i{\displaystyle \frac{1}{(1+{\tau }_{Ki})}}}{\tilde{\alpha }{\displaystyle \sum _n{\displaystyle \frac{{\nu }_n{\alpha }_n}{\tilde{\alpha }}}\frac{1}{(1+{\tau }_{Kn})}}}K$$

(14)

Similarly,

$$L_i={\displaystyle \frac{{\nu }_i{\beta }_i{\displaystyle \frac{1}{(1+{\tau }_{Li})}}}{\tilde{\beta }{\displaystyle \sum _n{\displaystyle \frac{{\nu }_n{\beta }_n}{\tilde{\beta }}}{\displaystyle \frac{1}{(1+{\tau }_{Ln})}}}}}L$$

(15)

$$E_i=\frac{{\nu }_i{\gamma }_i{\displaystyle \frac{1}{(1+{\tau }_{Ei})}}}{\tilde{\gamma }{\displaystyle \sum _n\frac{{\nu }_n{\gamma }_n}{\tilde{\gamma }}\frac{1}{(1+{\tau }_{En})}}}E$$

(16)

$$M_i=\frac{{\nu }_i{\delta }_i{\displaystyle \frac{1}{(1+{\tau }_{Mi})}}}{\tilde{\delta }{\displaystyle \sum _n\frac{{\nu }_n{\delta }_n}{\tilde{\delta }}\frac{1}{(1+{\tau }_{Mn})}}}M$$

(17)

where ${\nu }_{i(n)}=P_{i(n)}{Y}_{i(n)}/Y$ is the output share of industry $i(n)$, $\tilde{s}={\displaystyle \sum {\nu }_n{s}_n}$($s=\alpha ,\beta ,\gamma ,\delta$) is the weighted average of factor output elasticities.

2.2. Factor Price Distortion Index and Optimized Allocation

Define the following variables:

$${\lambda }_{ji}=\frac{1}{1+{\tau }_{ji}}, \mathrm{and} {\tilde{\lambda }}_{ji}=\frac{{\lambda }_{ji}}{{\displaystyle \sum _{n=1}^N(\frac{{\nu }_n{s}_n}{\tilde{s}}){\lambda }_{j\mathrm{n}}}}=\frac{{\lambda }_{ji}\cdot {\displaystyle \frac{1}{P_j}}}{{\displaystyle \sum _{n=1}^N(\frac{{\nu }_n{s}_n}{\tilde{s}}){\lambda }_{j\mathrm{n}}\cdot \frac{1}{P_j}}}$$

(18)

${\lambda }_{ji}$ is defined as the factor price absolute distortion index. ${\tilde{\lambda }}_{ji}$ is the ratio of the reciprocal of industry $i$’s return on the factor input and the average of the reciprocals of the returns across industries, defined as the factor price relative distortion index.

From Equations (14)–(18), factor inputs can be rewritten as follows:

$$K_i=\frac{{\nu }_i{\alpha }_i}{\tilde{\alpha }}{\tilde{\lambda }}_{Ki}K,$$

(19)

$$L_i=\frac{{\nu }_i{\beta }_i}{\tilde{\beta }}{\tilde{\lambda }}_{Li}L,$$

(20)

$$E_i=\frac{{\nu }_i{\gamma }_i}{\tilde{\gamma }}{\tilde{\lambda }}_{Ei}E,$$

(21)

$$M_i=\frac{{\nu }_i{\delta }_i}{\tilde{\delta }}{\tilde{\lambda }}_{Mi}M.$$

(22)

From Equations (19)–(22), taxes mainly affect factor allocation through ${\tilde{\lambda }}_{ji}$ and also affect ${\nu }_i$ indirectly. If ${\tau }_{ji}=0$, then ${\tilde{\lambda }}_{ji}=1$, which means no distortion on factor price and the price of each factor is equal among industries, which can realize the free flow of factors, making the allocation fairer and more effective. In reality, most markets face different frictions that affect the fair competition of firms.

Rewrite Equations (19)–(22) to get Equations (23)–(26), and define Equation (27) as the relative comprehensive distortion index of all factor prices:

$${\tilde{\lambda }}_{Ki}=(\frac{K_i}{K})/(\frac{{\nu }_i{\alpha }_i}{\tilde{\alpha }}),$$

(23)

$${\tilde{\lambda }}_{Li}=(\frac{L_i}{L})/(\frac{{\nu }_i{\beta }_i}{\tilde{\beta }}),$$

(24)

$${\tilde{\lambda }}_{Ei}=(\frac{E_i}{E})/(\frac{{\nu }_i{\gamma }_i}{\tilde{\gamma }}),$$

(25)

$${\tilde{\lambda }}_{Mi}=(\frac{M_i}{M})/(\frac{{\nu }_i{\delta }_i}{\tilde{\delta }}).$$

(26)

$${\tilde{\lambda }}_i={\tilde{\lambda }}_{Ki}^{\frac{{\alpha }_i}{{\alpha }_i+{\beta }_i+{\gamma }_i+{\delta }_i}}\cdot {\tilde{\lambda }}_{Li}^{\frac{{\beta }_i}{{\alpha }_i+{\beta }_i+{\gamma }_i+{\delta }_i}}\cdot {\tilde{\lambda }}_{Ei}^{\frac{{\gamma }_i}{{\alpha }_i+{\beta }_i+{\gamma }_i+{\delta }_i}}\cdot {\tilde{\lambda }}_{Mi}^{\frac{{\delta }_i}{{\alpha }_i+{\beta }_i+{\gamma }_i+{\delta }_i}}$$

(27)

Equations (23)–(26) express a ratio value; the numerator is the actual proportion of the factor of industry $i$ to the total factor, and the denominator is considered to be the theoretically optimized allocation proportion, which is expressed according to the factor’s output elasticity and output share of each industry, so ${\tilde{\lambda }}_{ji}$ also represents whether there is allocation distortion and the extent of misallocation from the relative perspective. From Equation (18), if ${\lambda }_{ji}$ is smaller than its weighted average (i.e., industry $i$’s $j$ factor is taxed more), then ${\tilde{\lambda }}_{ji}$ is less than 1, so the price of factor $j$ is relatively high and factor $j$ is allocated less to the industry $i$ than to the level with no frictions. Similarly, if ${\tilde{\lambda }}_{ji}$ is more than 1, the price of factor $j$ is relatively low and factor $j$ is allocated more to the industry $i$. When ${\tilde{\lambda }}_{ji}\ne 1$, we say there is a misallocation of factors. Because ${\lambda }_{ji}$ is difficult to measure, we use Equations (23)–(26) to measure ${\tilde{\lambda }}_{ji}$ instead of Equation (18). The meaning of ${\tilde{\lambda }}_i$ from Equation (27) is similar to ${\tilde{\lambda }}_{ji}$; it is explained from the perspective of a misallocation of all factors.

By Equation (27), when ${\tilde{\lambda }}_i=1$, it does not represent all ${\tilde{\lambda }}_{ji}=1$ (each factor is allocated effectively);${\tilde{\lambda }}_{ji}=1$ is the sufficient condition for ${\tilde{\lambda }}_i=1$, not a necessary condition. To build the necessary and sufficient relationship and be convenient to compare, we standardize the distortion index to the value between 0 and 1; the allocation is more effective when the value is closer to 1, and one is the optimal value. Through the above analysis, we divide the distortion index into two categories: less than 1 and more than 1. The category for less than 1 means less allocation, and the category for more than 1 means more allocation. Therefore, the category for less than 1 is standardized according to a positive indicator, and the bigger the result, the better. The category for more than 1 is standardized according to a negative indicator, and the smaller the result, the better.

The standardized formula is as follows:

$$\mathrm{Less} \mathrm{than} 1: {\overline{\lambda }}_{ji}=0.998\ast (({\tilde{\lambda }}_{ji}-\min ({\tilde{\lambda }}_{ji}))/(1-\min ({\tilde{\lambda }}_{ji}))+0.002$$

(28)

$$\mathrm{More} \mathrm{than} 1: {\overline{\lambda }}_{ji}=0.998\ast ((\max ({\tilde{\lambda }}_{ji})-{\tilde{\lambda }}_{ji})/(\max ({\tilde{\lambda }}_{ji})-1))+0.002$$

(29)

The standardized comprehensive distortion index under the standardization of each factor is defined as:

$${\overline{\lambda }}_i={\overline{\lambda }}_{Ki}^{\frac{{\alpha }_i}{{\alpha }_i+{\beta }_i+{\gamma }_i+{\delta }_i}}\cdot {\overline{\lambda }}_{Li}^{\frac{{\beta }_i}{{\alpha }_i+{\beta }_i+{\gamma }_i+{\delta }_i}}\cdot {\overline{\lambda }}_{Ei}^{\frac{{\gamma }_i}{{\alpha }_i+{\beta }_i+{\gamma }_i+{\delta }_i}}\cdot {\overline{\lambda }}_{Mi}^{\frac{{\delta }_i}{{\alpha }_i+{\beta }_i+{\gamma }_i+{\delta }_i}}$$

(30)

So, when ${\overline{\lambda }}_i=1$, it is equivalent to ${\overline{\lambda }}_{ji}=1$ and ${\tilde{\lambda }}_{ji}=1$; the opposite is also true.

Therefore, ${\overline{\lambda }}_i=1$ if and only if ${\tilde{\lambda }}_{ji}=1$. The effective allocation of each factor can realize the effective allocation of all comprehensive factors; the opposite is also true.

In the empirical section, firstly, by the value of ${\tilde{\lambda }}_{ji}$(${\overline{\lambda }}_{ji}$) and taking 1 as the reference point, we know factor $j$ is allocated more or less and the deviation extent in industry $i$ compared with theoretical effective allocation; we can also compute the technical efficiency, called the actual efficiency, according to the actual production data to prepare for the comparative analysis below. Secondly, set ${\overline{\lambda }}_i=1$, equivalent to ${\tilde{\lambda }}_{ji}=1$ (friction of factor price in an industry is the same as the average frictions of all industries from Equation (18)), from Equations (19)–(22); we obtain the optimized allocation of each factor. There are two situations under ${\tilde{\lambda }}_{ji}=1$ that we need to pay attention to: (1) ${\tau }_{ji}=0$, which indicates there is no price friction of any form, each factor’s price is equal among industries, and the factor market can be regarded as a perfect competitive market, which is conductive for realizing optimal allocation; (2) ${\tau }_{ji}\ne 0$, which indicates there are frictions that either express the same effect on the same factor’s price across industries, that is, ${\tau }_{ji}$ of the factor $j$ has the same value across industries (although this is rare), or express differences but factor $j$’s ${\lambda }_{ji}$ is the same to its average value, so the allocation of the individual or all factors may deviate from absolute effective allocation.

${\tilde{\lambda }}_{ji}$ reflects the allocation more or less compared with theoretical equilibrium allocation, which includes frictions on factors from the absolute magnitude but not from the relative comparison. So, when we take ${\tilde{\lambda }}_{ji}=1$, the effective allocation is relatively effective.

When ${\tilde{\lambda }}_{ji}=1$, we define the factor allocation from Equations (19)–(22) as the optimized allocation of factor $j$ in industry $i$. Under the reallocation, the efficiency of each industry is computed and called optimized efficiency. In the following, we will test whether this kind of allocation achieves improvements in efficiency and savings in inputs.

2.3. Technical Efficiency Model

We introduce the SBM (slack-based measure) model of the DEA (data envelopment analysis) approach, proposed by Tone [27], to measure efficiency. Supposing there are $N$ decision-making units ($DMUs$), each $DM{U}_i$ has $m$ inputs

$x_i={(x_{1i},x_{2i}\prime \cdots ,x_{mi})}^{\prime }$ and $q$ outputs $y_i={(y_{1i},y_{2i},\cdots ,y_{qi})}^{\prime }$, $\lambda =({\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_N)$ is $N\times 1$, a nonnegative vector, $X=[x_1,x_2,\cdots ,x_N]$ is an $m\times N$ matrix of input vectors, and $Y=[y_1,y_2,\cdots ,y_N]$ is a $q\times N$ matrix of output vectors. The constant returns to scale (CRS) and input-oriented SBM are:

$$\begin{array}{c}\min \rho =1-{\displaystyle \frac{1}{m}}{\displaystyle \sum _{i=1}^m{s}_i^{-}/x_{ik}}\\ s.t{.}_{}^{}X\lambda +s^{-}=x_k\\ Y\lambda \ge y_k\\ \lambda ,s^{-}\ge 0\end{array}$$

(31)

The vector $s^{-}$ is the potential reduction in inputs when the outputs are fixed. The factor’s redundancy ratio is the ratio of reduction and original factor input. The efficiency value satisfies $0<\rho \le 1$; the closer $\rho$ is to 1, the closer $DMU$ is to the production frontier. If $\rho =1$, the $DMU$ is completely efficient and has no input redundancies.

3. The Sample and Data

3.1. The Sample

This study analyzes factor misallocation and optimization in China’s manufacturing industry from 1999 to 2016 (according to the data specification in the industrial section of the China Statistical Yearbook 2018, there are incomparable reasons in the data before and after 2017, so we choose data from 1999 to 2016). There have been changes in the classification standards of industries in the national economy due to the split, merging, updating, and new additions, so the statistical data and the number of major, middle, and minor categories vary slightly. To ensure the consistency of the data, we reorganize industry classifications under major categories.

The name and content of most industry classifications remain consistent, with few changes except the classification criteria of the 2011 version, so some industries’ data change significantly after 2011. After 2011, all kinds of subdivision data can be found in the China Industrial Statistical Yearbook, while prior to 2012, only major categories of data can be found in the Yearbook. Therefore, the data changes since 2012 required reorganization to correlate with the previous classification standard. After having harmonized all classifications, we constructed the major categories of China’s manufacturing industry as shown in

Table 1. The standardized factor price (allocation) distortion index (the average of 1999–2016).

No.	Industry Categories	Single Factor Distortion				Comprehensive Distortion
		${\overline{\mathit{\lambda }}}_{\mathit{K}\mathit{i}}$	${\overline{\mathit{\lambda }}}_{\mathit{L}\mathit{i}}$	${\overline{\mathit{\lambda }}}_{\mathit{E}\mathit{i}}$	${\overline{\mathit{\lambda }}}_{\mathit{M}\mathit{i}}$	${\overline{\mathit{\lambda }}}_{\mathit{i}}$
1	Food processing	0.92	(−)0.24	(−)0.31	(−)0.80	(−)0.62
2	Food production	(−)0.77	0.97	(−)0.47	(−)0.93	(−)0.85
3	Manufacture of liquor, beverages, and refined tea	(−)0.53	0.90	0.73	(−)0.30	(−)0.42
4	Tobacco processing	0.002	(−)0.002	(−)0.002	(−)0.002	(−)0.005
5	Textile industry	(−)0.16	0.75	(−)0.65	0.72	(−)0.51
6	Textile garments and apparel industry	(−)0.002	0.002	(−)0.15	0.36	(−)0.10
7	Leather, furs, feathers, and related products, shoemaking	(−)0.09	0.85	(−)0.29	0.07	0.31
8	Wood processing, bamboo, rattan, palm and grass products	(−)0.63	0.89	0.95	(−)0.38	(−)0.49
9	Furniture manufacturing	0.95	0.89	(−)0.35	(−)0.63	(−)0.71
10	Papermaking and paper products	0.56	(−)0.63	(−)0.84	0.28	0.59
11	Printing and record medium reproduction	(−)0.71	0.93	(−)0.51	0.48	0.71
12	Cultural, educational, and sports articles	(−)0.19	0.61	(−)0.27	0.002	0.14
13	Petroleum processing, coking, and nuclear fuel processing	0.24	(−)0.19	0.85	0.78	0.67
14	Raw chemical materials and chemical products	0.16	(−)0.39	0.92	0.78	0.65
15	Medical and pharmaceutical products	0.15	0.998	0.89	(−)0.48	(−)0.59
16	Chemical fiber	0.98	(−)0.35	0.80	(−)0.35	(−)0.42
17	Rubber and plastic products	0.83	(−)0.67	(−)0.57	0.11	0.47
18	Nonmetal mineral products	(−)0.86	0.95	0.002	(−)0.45	(−)0.37
19	Smelting and pressing of ferrous metals	0.16	(−)0.13	0.69	(−)0.95	0.47
20	Smelting and pressing of nonferrous metals	(−)0.99	(−)0.26	0.96	(−)0.48	(−)0.50
21	Metal products	(−)0.89	0.95	(−)0.80	(−)0.78	(−)0.84
22	Ordinary machinery	0.65	(−)0.67	(−)0.21	0.86	(−)0.66
23	Equipment for special purposes	0.41	(−)0.87	(−)0.25	0.99	(−)0.80
24	Transportation equipment	(−)0.74	0.996	(−)0.25	(−)0.85	(−)0.80
25	Electric equipment and machinery	0.97	(−)0.82	(−)0.15	(−)0.98	(−)0.82
26	Computer, communication, and other electronic equipment	(−)0.37	0.78	(−)0.22	(−)0.75	(−)0.68
27	Instruments, meters, cultural and office machinery	(−)0.34	0.84	(−)0.23	(−)0.58	(−)0.56
28	Handicrafts and other manufacturing industry	(−)0.36	0.65	(−)0.39	(−)0.88	(−)0.71
29	Comprehensive utilization of waste resources	(−)0.24	(−)0.02	(−)0.15	0.36	(−)0.23
The number of industries with relatively more (less) factor allocation		13(16)	16(13)	9(20)	12(17)	8(21)

Note: The data are calculated by the authors according to the above-mentioned data sources and methods. (−) represents less allocation deviation, and unsigned represents more allocation deviation. The deviation direction of ${\overline{\lambda }}_i$ is based on ${\tilde{\lambda }}_i$. The farther away from 1, the greater the misallocation.

.

3.2. Data

The output variable is the gross value of industrial output. Input variables are capital, calculated by the perpetual inventory method; labor, measured by the annual average number of employed people multiplied by the average salary; energy consumption, measured by standard coal price multiplied by energy consumption; other intermediate consumption, which is the gross value of industrial output minus added value and energy input plus value-added tax (VAT) payable.

The data are retrieved from the China Industrial Statistics Yearbook, China Economic Census Yearbook, China Statistics Yearbook, China Tax Yearbook, China Labor Statistics Yearbook, EPS (Easy Professional Superior) Database, and China Macroeconomic Information Network Platform. The data are deflated taking 1999 as the base year. Industrial gross output value, added value, and VAT payable are deflated by the industrial gross production index, fixed asset investment is deflated by the fixed asset investment price index, and the salary of employees is deflated by the consumer price index.

4. Empirical Results

4.1. Factor Distortion Index

To overcome the significant multicollinearity among variables, the ridge estimation method is used to estimate the production function for each industry according to Equation (1) (logarithmic linearization). At the significance level of 5%, the value of the ridge parameter $k$ is the minimum under the condition that the regression coefficient is significant and the ridge trace is stable. After estimating the output elasticity of each factor input, the factor price relative distortion index and its standardized value are calculated according to Equations (23)–(27) and Equations (28)–(30).

It can be seen from the calculation results that the distortion index of the same factor in the same industry fluctuates little during the study period. That is, the index ${\tilde{\lambda }}_{ji}$ does not deviate from one significantly, with its estimates either always greater than one or always less than one in most cases; the deviation direction of the same factor in the same industry is relatively consistent. This could be due to a few factors. (1) The share of factor and output in each industry has not changed much. (2) The output elasticity of the factor is not time-varying because the annual industry data are used under the assumption that the factor’s output elasticity for the same industry is constant over the study period. We also obtain the comprehensive distortion index of all factors. The standardized average value from 1999 to 2016 is shown in Table 1.

From Table 1, we can see that almost all industries have factor misallocation. The degree of misallocation and direction for each factor in different industries vary. In one industry, there is no misallocation of all factors in one direction. In most industries, the allocation of labor factors is relatively higher, and energy allocation is lower. Relatively more capital is allocated to 13 industries, such as tobacco processing, medical and pharmaceutical products, raw chemical materials and chemical products, and smelting and pressing of ferrous metals. There are 12 industries with relatively excessive intermediate input consumption, such as cultural, educational, and sports articles, leather, furs, feathers, and related products, shoemaking, and rubber and plastic products. Considering all the factor inputs comprehensively, excessive inputs are in the minority, and the actual factor allocation in most industries is lower than the theoretical equilibrium allocation. From the perspective of the whole industry, this does not mean a shortage of factor inputs for the whole industry, because industries with excess inputs may make up for the shortage in other industries. It shows that industries face different factor costs and unfair factor allocation because of different frictions, which may be from different policies, market systems, or technical levels.

4.2. Actual Efficiency and Redundancy Proportion of Inputs

Based on the SBM model introduced in Section 2.3, the output is assumed to be fixed, and the input-oriented average actual efficiency of each industry is calculated. During the period of 1999–2016, the number of industries on the frontier changes slightly every year and is stable at between 3 and 6. The following industries show relatively high technical efficiency: food processing, tobacco processing, leather, furs, feathers, and related products, shoemaking, computers, communication and other electronic equipment, petroleum processing, coking and nuclear fuel processing, and comprehensive utilization of waste resources. By computing the tri-sectional quantiles, we divide all industries into three grades: high, middle, and low efficiency, as shown in

Table 2. The average actual efficiency (1999–2016).

	No.	Industry Categories	Actual Efficiency
high	1	Tobacco processing	1.00
	2	Comprehensive utilization of waste resources	0.98
	3	Petroleum processing, coking, and nuclear fuel processing	0.97
	4	Leather, furs, feathers, and related products, shoemaking	0.89
	5	Food processing	0.89
	6	Computer, communication, and other electronic equipment	0.86
	7	Electric equipment and machinery	0.76
	8	Textile garments and apparel industry	0.71
	9	Smelting and pressing of nonferrous metals	0.70
	10	Instruments, meters, cultural, and office machinery	0.69
middle	11	Furniture manufacturing	0.63
	12	Transportation equipment	0.62
	13	Cultural, educational, and sports articles	0.60
	14	Rubber and plastic products	0.56
	15	Metal products	0.56
	16	Chemical fiber	0.56
	17	Ordinary machinery	0.55
	18	Equipment for special purposes	0.54
	19	Wood processing, bamboo, rattan, palm and grass products	0.54
low	20	Food production	0.52
	21	Manufacture of liquor, beverages, and refined tea	0.51
	22	Textile industry	0.51
	23	Smelting and pressing of ferrous metals	0.50
	24	Handicrafts and other manufacturing industry	0.50
	25	Medical and pharmaceutical products	0.49
	26	Printing and record medium reproduction	0.47
	27	Raw chemical materials and chemical products	0.47
	28	Papermaking and paper products	0.43
	29	Nonmetal mineral products	0.41

Note: The data are calculated by the authors according to the above-mentioned data sources and methods.

. The lower the efficiency, the more input redundancy there is.

Table 3. The redundancy proportion under actual allocation (the average of 1999–2016).

No.	Industry Categories	Redundancy Proportion (%)				Comprehensive Redundancy (%)
		K	L	E	M
1	Food processing	12.27	1.59	22.92	8.36	13.94
2	Food production	29.29	37.66	74.58	50.43	45.04
3	Manufacture of liquor, beverages, and refined tea	42.59	33.36	70.77	47.70	46.56
4	Tobacco processing	0.00	0.00	0.00	0.00	0.00
5	Textile industry	26.24	45.26	79.86	46.33	39.62
6	Textile garments and apparel industry	6.48	54.42	29.81	27.36	30.97
7	Leather, furs, feathers, and related products, shoemaking	2.33	18.60	9.88	12.96	15.11
8	Wood processing, bamboo, rattan, palm, and grass products	26.98	33.25	75.50	49.33	43.45
9	Furniture manufacturing	21.97	49.53	39.51	37.03	41.63
10	Papermaking and paper products	54.29	34.40	89.40	48.44	53.81
11	Printing and record medium reproduction	49.55	58.09	59.41	45.78	50.24
12	Cultural, educational, and sports articles	9.03	67.08	53.79	31.12	33.00
13	Petroleum, coal, and other fuel processing	2.15	1.02	4.99	2.37	0.94
14	Raw chemical materials and chemical products	50.34	20.87	93.81	48.29	52.93
15	Medical and pharmaceutical products	39.87	45.32	71.47	48.01	48.82
16	Chemical fiber	44.79	9.90	83.60	39.57	40.92
17	Rubber and plastic products	20.99	40.85	73.50	40.21	39.71
18	Nonmetal mineral products	50.49	43.77	95.60	47.42	53.17
19	Smelting and pressing of ferrous metals	50.44	18.49	93.26	38.61	47.76
20	Smelting and pressing of nonferrous metals	26.38	15.35	59.14	19.89	15.95
21	Metal products	26.22	40.93	75.50	33.45	38.91
22	Ordinary machinery	28.73	49.92	64.15	37.30	39.61
23	Equipment for special purposes	29.80	50.78	55.65	46.69	46.71
24	Transportation equipment	30.24	35.84	44.60	42.17	41.79
25	Electric equipment and machinery	13.48	28.22	25.37	30.16	31.44
26	Computer, communication, and other electronic equipment	7.94	18.34	7.05	23.97	23.12
27	Instruments, meters, cultural and office machinery	15.79	51.95	26.14	29.58	32.01
28	Handicrafts and other manufacturing industry	15.94	57.52	82.39	45.74	40.92
29	Comprehensive utilization of waste resources	0.59	0.27	2.01	4.55	2.48
Redundancy proportion for all industry categories as a whole		25.35	33.19	53.92	33.89	34.85

Note: The data is calculated by the authors according to the above-mentioned data sources and methods.

shows the average proportion of factor inputs that need to be adjusted for each industry in comparison to the frontier. Capital input reductions of more than 50% are required for each of the following industries: papermaking and paper products, nonmetal mineral products, smelting and pressing of ferrous metals, and raw chemical materials and chemical products. Labor input’s redundancy is over 50% in industries such as: cultural, educational, and sports articles, printing and record medium reproduction, handicrafts and other manufacturing industries, textile garments and apparel industries, instruments, meters, cultural and office machinery, and equipment for special purposes. The redundancy proportion of energy consumption is more than 50% in 18 industries, which indicates energy consumption is seriously excessive. Only in the food production category is other intermediate consumption redundancy more than 50%. The redundancy proportions of input factors in the overall manufacturing industry, ranging from high to low, are energy consumption (53.92%), intermediate input (33.89%), labor (33.19%), and capital (25.35%).

As shown in column 7 of Table 3, the comprehensive redundancy proportion is calculated as the redundancy of all factors divided by the total input of all factors. Based on the comprehensive analysis of the redundancy for all inputs, if industries are on the technological frontier, e.g., papermaking and paper products, nonmetal mineral products, raw chemical materials and chemical products, and printing and record medium reproduction, they could reduce inputs by more than 50%, and the overall industry could reduce inputs by 34.85%.

The computed input redundancy, compared to production frontiers based on the SBM model, reflects excessive consumption under low technical efficiency. In actual production activities, it is difficult for industries to adjust factor allocation according to optimal technical efficiency because of short-term technological bottlenecks, and technical efficiency also does not consider the cost of factors. Under the assumption that the aggregate factors are exogenous and technology remains constant, we try to find the optimized allocation and conduct a test.

4.3. Optimized Efficiency and Redundancy Proportion of Inputs

As stated in Section 2.2, we set ${\overline{\lambda }}_i=1$ by Equation (30), and it is equivalent to ${\tilde{\lambda }}_{ji}=1$, so we obtain the optimized allocation value of factor $j$ in industry $i$ by Equations (19)–(22). Following the calculation of the optimized efficiency after reallocation, we observe and compare the frontier, efficiency, and redundancy proportion with the actual situation outlined above. Due to limited space, optimized allocation is not listed, and optimized efficiency is computed directly. For each industry, the annual output share changes little, the factor’s output elasticity remains unchanged, and the total input is divided by little changed proportion, so the annual optimized efficiency remains roughly the same in each industry. Therefore, what we obtain is the average value.

As shown in

Table 4. Optimized efficiency and comparison with actual efficiency (the average of 1999–2016).

	No.	Industry Categories	Optimized Efficiency	Actual Efficiency	Improvement Rate (%)	Grade Change
high	1	Manufacture of liquor, beverages, and refined tea	1.00	0.51	94.57	3-1
	2	Tobacco processing	1.00	1.00	0.00	1-1
	3	Textile industrial and garments	1.00	0.71	41.88	1-1
	4	Leather, furs, feathers, and related products, shoemaking	1.00	0.89	12.29	1-1
	5	Cultural, educational, and sports articles	1.00	0.60	67.37	2-1
	6	Petroleum, coal, and other fuel processing	1.00	0.97	2.70	1-1
	7	Medical and pharmaceutical products	1.00	0.49	104.77	3-1
	8	Rubber and plastic products	1.00	0.56	78.21	2-1
	9	Computer, communication, and other electronic equipment	1.00	0.86	16.72	1-1
	10	Furniture manufacturing	0.94	0.63	48.40	2-1
middle	11	Electric equipment and machinery	0.92	0.76	20.93	1-2
	12	Equipment for special purposes	0.91	0.54	67.42	2-2
	13	Metal products	0.90	0.56	60.82	2-2
	14	Papermaking and paper products	0.88	0.43	102.82	3-2
	15	Food processing	0.87	0.89	−2.43	1-2
	16	Wood processing, bamboo, rattan, palm, and grass products	0.85	0.54	57.60	2-2
	17	Printing and record medium reproduction	0.84	0.47	78.79	3-2
	18	Ordinary machinery	0.84	0.55	51.81	2-2
	19	Raw chemical materials and chemical products	0.83	0.47	76.65	3-2
low	20	Instruments, meters, cultural and office machinery	0.81	0.69	17.69	1-3
	21	Handicrafts and other manufacturing industry	0.81	0.50	64.03	3-3
	22	Comprehensive utilization of waste resources	0.81	0.98	−17.10	1-3
	23	Transportation equipment	0.80	0.62	29.42	2-3
	24	Textile industry	0.79	0.51	55.76	3-3
	25	Food production	0.78	0.52	49.15	3-3
	26	Smelting and pressing of ferrous metals	0.74	0.50	48.79	3-3
	27	Chemical fiber	0.71	0.56	28.65	2-3
	28	Nonmetal mineral products	0.69	0.41	70.67	3-3
	29	Smelting and pressing of nonferrous metals	0.69	0.70	−1.31	1-3

Note: The data are calculated by the authors according to the above-mentioned data sources and methods.

, under the optimized reallocation, with the exception of three, i.e., food processing, comprehensive utilization of waste resources, smelting and pressing of nonferrous metals, the efficiencies of other industries improve significantly, the number of industries on the frontier increases to 9, the lowest efficiency improves by about 70%, and the overall average efficiency improves by 43.4% compared with the efficiency before optimization. Thus, the optimized allocation is superior to the actual allocation. When technology remains constant, the improvement of factor allocation is key to higher efficiency.

Similarly, we compute the tri-sectional quantiles of optimized efficiency and divide all industries into three grades: high, middle, and low level. Then, we compare to previous levels of efficiency. As shown in column 7 of Table 4, there are 7 industry sectors that moved from a high to a low classification, while the other industry sectors remain either unchanged or have improved. It shows that under the optimized reallocation, the range of improved efficiency varies for different industries and that most industries show a significant improvement and move up a grade.

Under optimized allocation, the average of factor redundancy proportion reduces significantly. For the total manufacturing sectors, the redundancy proportions for each factor are: capital at 7.29% (reduced from 25.35%), labor at 16.85% (reduced from 33.19%), energy consumption at 23.01% (reduced from 53.92%), and other intermediate consumption at 2.65% (reduced from 33.89%). The proportion of each factor saved is the D-value of the redundancy proportion before and after optimization. The capital, labor, energy, and other intermediate consumption are reduced by 18.06%, 16.34%, 30.91%, and 31.24%, respectively. The high efficiency of optimized allocation moves almost all industries closer to the frontier, and the factor inputs that need to be adjusted are reduced. The present redundancy may depend more on technological progress and its interaction with factor allocation. As shown in column 7 of

Table 5. Redundancy proportion under optimized reallocation (the average of 1999–2016).

No.	Industry Categories	Redundancy Proportion (%)				Comprehensive Redundancy (%)
		K	L	E	M
1	Food processing	24.84	0.00	24.71	4.21	10.60
2	Food production	0.00	30.27	59.42	0.00	2.80
3	Manufacture of liquor, beverages, and refined tea	0.00	0.00	0.00	0.00	0.00
4	Tobacco processing	0.00	0.00	0.00	0.00	0.00
5	Textile industry	9.46	36.97	18.41	20.06	13.25
6	Textile garments and apparel industry	0.00	0.00	0.00	0.00	0.00
7	Leather, furs, feathers, and related products, shoemaking	0.00	0.00	0.00	0.00	0.00
8	Wood processing, bamboo, rattan, palm, and grass products	8.07	19.96	17.49	15.73	12.48
9	Furniture manufacturing	1.13	0.00	23.34	1.64	1.52
10	Papermaking and paper products	5.04	0.00	43.13	0.00	3.84
11	Printing and record medium reproduction	19.72	12.00	16.13	17.50	18.53
12	Cultural, educational, and sports articles	0.00	0.00	0.00	0.00	0.00
13	Petroleum, coal, and other fuel processing	0.00	0.00	0.00	0.00	0.00
14	Raw chemical materials and chemical products	11.28	20.17	38.73	0.00	7.72
15	Medical and pharmaceutical products	0.00	0.00	0.00	0.00	0.00
16	Chemical fiber	3.15	51.74	44.98	14.35	12.27
17	Rubber and plastic products	0.00	0.00	0.00	0.00	0.00
18	Nonmetal mineral products	27.47	45.50	49.32	0.00	17.57
19	Smelting and pressing of ferrous metals	22.07	49.52	32.02	0.00	14.34
20	Smelting and pressing of nonferrous metals	46.96	0.00	73.98	3.48	22.38
21	Metal products	0.59	0.00	39.35	0.00	0.62
22	Ordinary machinery	0.00	22.58	43.59	0.00	4.54
23	Equipment for special purposes	0.00	36.12	0.44	0.00	4.89
24	Transportation equipment	0.00	39.84	40.32	0.00	3.25
25	Electric equipment and machinery	0.00	0.00	33.87	0.00	0.44
26	Computer, communication, and other electronic equipment	0.00	0.00	0.00	0.00	0.00
27	Instruments, meters, cultural, and office machinery	10.50	41.35	22.70	0.00	7.22
28	Handicrafts and other manufacturing industry	10.50	41.35	22.70	0.00	7.63
29	Comprehensive utilization of waste resources	10.50	41.35	22.70	0.00	7.81
Redundancy proportion for all industry categories as a whole		7.29	16.85	23.01	2.65	5.99

Note: The data are calculated by the authors according to the above-mentioned data sources and methods.

, the current redundancy proportion, all inputs of all industries will be reduced by 5.99% (reduced from 34.85%), which shows that the optimized allocation will save inputs by 28.86% through improvement in allocation. The optimized allocation has not improved efficiency in all industry categories, which could be due to applying relative values to compute optimized allocation when the factor price absolute distortion index is difficult to obtain. In actual production, Pareto improvement is difficult to achieve. Our allocation method is also a good exploration, on the whole, as it significantly improves efficiency and saves inputs.

4.4. Carbon Emission Reduction

From China Carbon Emission Accounts and Datasets [28], we obtain carbon emissions (million tons) of the manufacturing industry for the period from 2000 to 2016. Firstly, the proportion of energy redundancy is obtained by calculating actual efficiency, and the reduced amount of carbon emissions for each industry category is computed as the product of this proportion and the amount of carbon emissions and the results is shown in

Table 6. Total carbon emission and redundancy (2000–2016).

No.	Industry Categories	Total (2000–2016) (Mt CO2)	Actual Allocation		Optimized Reallocation
			Redundancy (Mt)	Redundancy Proportion (%)	Redundancy (Mt)	Redundancy Proportion (%)
1	Food processing	670.27	185.59	27.69	664.48	24.71
2	Food production	336.08	250.33	74.49	557.61	59.42
3	Manufacture of liquor, beverages, and refined tea	342.70	250.71	73.16	0.00	0.00
4	Tobacco processing	35.80	0.00	0.00	0.00	0.00
5	Textile industry	635.79	515.64	81.10	410.67	18.41
6	Textile garments and apparel industry	121.77	39.45	32.40	0.00	0.00
7	Leather, furs, feathers, and related products, shoemaking	57.17	6.19	10.82	0.00	0.00
8	Wood processing, bamboo, rattan, palm, and grass products	148.23	113.98	76.90	47.09	17.49
9	Furniture manufacturing	26.54	9.35	35.23	37.17	23.34
10	Papermaking and paper products	619.45	557.12	89.94	498.48	43.13
11	Printing and record medium reproduction	34.74	20.53	59.09	28.89	16.13
12	Cultural, educational, and sports articles	67.24	38.18	56.79	0.00	0.00
13	Petroleum, coal, and other fuel processing	1812.50	0.00	0.00	0.00	0.00
14	Raw chemical materials and chemical products	3306.16	3116.93	94.28	2716.37	38.73
15	Medical and Pharmaceutical Products	174.89	126.69	72.44	0.00	0.00
16	Chemical fiber	111.91	93.01	83.11	137.48	44.98
17	Rubber and plastic products	176.66	137.74	77.97	0.00	0.00
18	Nonmetal mineral products	15,571.06	14,929.00	95.88	967.88	49.32
19	Smelting and pressing of ferrous metals	19757.44	18,272.01	92.48	2284.06	32.02
20	Smelting and pressing of nonferrous metals	909.92	453.32	49.82	2220.83	73.98
21	Metal products	238.66	180.13	75.48	398.02	39.35
22	Ordinary machinery	512.58	322.01	62.82	1298.84	43.59
23	Equipment for special purposes	527.29	298.34	56.58	6.41	0.44
24	Transportation equipment	318.60	133.53	41.91	1087.32	40.32
25	Electric equipment and machinery	171.66	41.52	24.19	760.44	33.87
26	Computer, communication, and other electronic equipment	91.61	6.02	6.58	0.00	0.00
27	Instruments, meters, cultural and office machinery	35.58	8.72	24.51	60.70	22.70
28	Other manufacturing industry	65.84	33.82	51.37	73.02	22.70
The Total		46,878.14	40,139.88	85.63	14,255.79	30.41

Note: The data are calculated by the authors according to the above-mentioned data sources and methods.

. From this, the total amount of carbon emissions that should be cut between 2000 to 2016 can be obtained. In this case, the total redundancy proportion for carbon is 85.63%. Secondly, under optimized allocation, we repeat the above process and get a total redundancy proportion of 30.41%. This indicates that efficiency improvements from optimized allocation can reduce carbon emissions by 55.22%.

5. Conclusions and Discussion

In this study, we assume that there is factor misallocation between industries and develop a multi-industry competitive equilibrium model with industry-specific frictions in the form of distortion taxes on the factors. We define the relative distortion index of factor price to measure the extent and direction of misallocation of each factor in every industry, and we set the standardized comprehensive distortion index (CDI) to one to obtain the optimized allocation. As indicated in this study, when technology remains unchanged, this new allocation is the main source of improving efficiency. Under the optimized efficiency, we could save inputs and reduce carbon emissions.

Several conclusions can be drawn from this study:

(1) The input–output indicator system has been improved by using four factor inputs in the analysis of factor allocation, which makes the analysis more comprehensive.

The main findings of this study indicate that energy efficiency is the lowest of the four factor inputs. From the perspective of the redundancy of factor allocation, energy redundancy is the largest one, which causes low efficiency in resource allocation. The results from Wei and Li [29] and He and Qi [16] also show that the manufacturing industry has low energy efficiency, which has a negative effect on the environment. There is an inverted U-shaped relationship between the impact of energy misallocation and carbon emission efficiency; the intensification of energy misallocation is not conducive to the arrival of the critical point of energy conservation and emission reduction [30]. The efficiency of other intermediate consumption is the second lowest. Manufacturing production is highly dependent on raw materials, which are the main intermediate inputs. This result has important implications for optimizing the allocation by adjusting the price of raw materials. Chen and Hu [17] also suggest that ignoring the misallocation of intermediate inputs is unreasonable and affects the estimation of efficiency loss, and most previous studies estimated efficiency loss with the misallocation of labor and capital only [1,15,17,19,20,31]. According to the results of this study, the extent of misallocation in labor and capital is lower than that of energy and other intermediate consumption, which means the latter two will have more impact on efficiency improvement compared with the former two. Therefore, it will certainly underestimate efficiency loss if we only take the misallocation of capital and labor into account. For example, the result estimated by Dai and Cheng [31] suggests that the efficiency improvement without misallocation in capital and labor is 11.18%, while the relative efficiency improvement of this study is 43.3%.

(2) Almost all industries in China have factor misallocation; the degree and direction of misallocation for each factor vary in different industries, which shows that industries face different factor costs and unfair factor allocation.

According to the standardized factor distortion index, there are 13 industries with more capital allocation, while the other 16 industries have less capital allocation. It indicates that more capital is concentrated in minority industries. The number of industries with more labor allocation is 16, but the number with less is 13, which means most industries have sufficient labor, and minority industries have the ability to continue to absorb labor. The energy is allocated more in nine industries, including heavy industries such as nonmetal mineral products, smelting and pressing of ferrous metals, smelting and pressing of nonferrous metals, and petroleum processing, coking, and nuclear fuel processing. Therefore, the key to reducing energy consumption lies in the energy conservation and emission reduction of heavy industries. The excessive other intermediate consumption mainly lies in light industries, such as the textile industry, textile garments and apparel industry, leather, furs, feathers, and related products, shoemaking, and cultural, educational, and sports articles. At present, light industries are mostly in a state of overcapacity, and clearing excess capacity effectively is conducive to the optimization of intermediate inputs.

(3) Under the optimized allocation, the lowest efficiency improves about by 70%, and the overall average efficiency improves by 43.4%.

Under the original actual factor allocation, 34.5% of industries have efficiencies of 0.7 and above, and 65.5% are below 0.7. Industries with the top three lowest efficiencies include raw chemical materials and chemical products, papermaking and paper products, and nonmetal mineral products. Now that the existing allocation is inefficient, it is critical to propose an effective way. In this study, assuming frictions of factor price in an industry are the same as the average frictions of all industries, we specifically set the comprehensive distortion index (CDI) to one to obtain the optimized allocation of each factor by multiplying the theoretical optimal allocation proportion by the exogenous aggregate factor. Under the condition of profit maximization, the optimized allocation proportion of each factor is determined by output elasticity of the factor and the output share of each industry. It is considered reasonable for each industry to be allocated according to its contribution. Under the optimized allocation, with the exception of food processing, comprehensive utilization of waste resources, and smelting and pressing of nonferrous metals, the efficiencies of other industries improve significantly. Medical and pharmaceutical products, papermaking and paper products, manufacture of liquor, beverages, and refined tea, raw chemical materials and chemical products, and nonmetal mineral products benefit the most from optimized allocation because they have larger increases in efficiencies. The overall average efficiency improves by 43.4% compared with the efficiency before optimization. Thus, the optimized allocation is superior to the actual allocation. When technology remains constant, the improvement of factor allocation is key to realize higher efficiency.

(4) When technology remains unchanged and achieves the same level of output as before, under the optimized allocation efficiency, capital, labor, energy, and other intermediate consumption are reduced by 18.06%, 16.34%, 30.91%, and 31.24%, respectively, and carbon emissions are also reduced by 55.22%.

The efficiency of energy and other intermediate consumption will improve more, followed by capital and labor. According to the “BP Statistical Review of World Energy 2021”, China has the largest primary energy consumption and carbon emissions in the world. The improvement of energy efficiency is critical for China’s sustainable development given the “carbon peak and carbon neutrality” goal set by the Chinese government. Furthermore, carbon emissions are significantly affected by labor and capital misallocation [32]. Therefore, to optimize resource allocation, it is important to consider the interaction among different factors at the same time, as the improvement of individual factor allocation may have a limited effect.

Some existing studies have identified measures to improve the factor misallocation from related variables and provide pathways to optimize resource allocation. Improving levels of information technology can promote the reallocation of resources from low-efficiency firms to high-efficiency firms and allocation efficiency [33]. According to China’s first binding pollution control, appropriate asymmetric environmental regulations have a significant effect on reducing resource misallocation and productivity improvement [20,34]. Foreign direct investment has significantly reduced the overall misallocation of capital and labor in China [35]. High levels of industrial coordination and agglomeration are conducive to improving allocation efficiency [36]. Improving general infrastructure investment can reduce the extent of capital and labor misallocation [37]. The higher the extent of corruption, the greater the resource misallocation, so the elimination of corruption is an effective way to reduce resource misallocation [38]. The development of the digital economy has significantly improved the resource allocation efficiency of manufacturing [39]. At present, digitalization and intelligence are the trend of future development. Digital transformation can alleviate the information asymmetry problem [40] and avoid capital investment misallocation caused by information asymmetry [41]. The introduction of emerging digital technologies for dynamic layout planning will solve environmental, social, and economic problems at the same time [42]. Enterprises also can increase investment in scientific research to promote technological progress, which will reduce environmental pollution while improving efficiency to achieve sustainable development [43]. Therefore, in order to improve resource misallocation, it is suggested that government and enterprise should put more effort and focus on the above-mentioned aspects.

Based on the causes of factor misallocation and the relevant variables of improving factor misallocation revealed in this study, several policy implications and improvement measures can be of guiding significance to the decision making of firms and governments. (1) Optimizing credit policies, promoting the effective flow of capital, and removing unfair credit barriers should be carried out to enhance the capital allocation efficiency. (2) The registered residence system and its related basic public services are closely related to labor allocation, so it is necessary to release the restrictions of the registered residence registration system and strengthen the housing, medical care, and education of children and other basic public services for employees across registered residence registration areas to improve labor mobility. (3) The government should encourage and guide enterprises to improve energy efficiency and reduce energy consumption through technical progress such as digitalization, artificial intelligence, etc. (4) Local protectionism and artificial market segmentation should be eliminated for better factor mobility and allocation by market forces. It is necessary to accelerate improvements in the market-oriented factor allocation system, create a conducive market environment for fair competition, fully realize the free flow of factors, and give full play to the competition mechanism. The operation of the factor market also needs to be evaluated regularly. (5) The international manufacturing industry has become increasingly competitive, and the empirical results of this paper highlight the substantial improvement in efficiency that can be achieved via factor reallocation. This should motivate decision makers at both firm and government levels to take relevant actions to optimize their allocation structure, while minimizing the waste of resources and strategically repositioning them to maximum efficiency.

However, this study has a few limitations that should be addressed in future research. Limited by incomplete firm-level data, we conducted our analysis at the industry level. However, if this analysis was done from a micro-perspective, the result could be more representative. This study obtains optimized factor allocation by taking the average distortion rather than the absolute distortion as the benchmark; therefore, it is believed that obtaining an absolute distortion index to calculate optimized allocation based on profit maximization to realize Pareto improvement would be the focus of future research. The dataset ranging from 1999 to 2016 is used in this study because the data specifications for the main economic indicators have been changing since 2017, which makes the data incomparable with that of previous years, so the data range should be updated by harmonizing data after 2017 and then incorporating it into the analysis in future research. In recent years, the Chinese economy has become more and more uncertain due to the impacts of COVID-19, economic downturn, and policy changes. Shocks and uncertainties from different influencing factors could alter the efficiency of factor allocation. Therefore, it would be interesting to explore an uncertain analysis with our modified framework to further tackle the factor misallocation in future research. Furthermore, it is believed that the theoretical framework used in this study for improving factor allocation efficiency could be applied to different industries or countries if data are available. A comparative analysis between different industries, countries, or groups of countries would also be a promising direction in future research to expand the scope of this study, which only focuses on China’s manufacturing industry.