Evaluation of green innovation efficiency in catering industry based on data envelopment analysis

Data Envelopment Analysis (DEA) belongs to one of the non-parametric technical efficiency analysis methods [24]. The CCR model based on constant returns to scale and the BCC model based on variable returns to scale are two of the most widely used and fundamental models of DEA theory [25].

The CCR model assumes that there are n DMUs, denoted DMU_j(j = 1,2,⋯,n). Each DMU has m inputs, denoted x_i(i = 1,2,⋯,m), and s outputs, denoted y_r(r = 1,2,⋯,s), with x_n denoting the amount of input factors for DMU_i the ith item, and assuming that v_i is the weight of the inputs for the ith item, and that u_i is the weight of the outputs for the r th item, the CCR model is expressed as follows: (1) minθs.t.{ ∑j=1nxijλj+sj−=θxik,i=1,2,⋯,m∑j=1nyrjλj−sr+=yrk,r=1,2,⋯,sλ≥0sj−≥0sj+≥0j=1,2,⋯n

Where λ is a linear combination of coefficients of DMUs, the optimal solution of the model θ^* is the value of efficiency and the value of efficiency is in the range of (0, 1], and si−,si+ denotes the input and output slack variables respectively. The above model is a measure of inefficiency in terms of the extent to which each input can be scaled down in equal proportions under certain conditions of output, hence it is known as an input-oriented CCR model.

The BBC model adds constraint ∑nλi=1(λ≥0) to the CCR model, which serves to make the production scale of the projection point at the same level as the production scale of the evaluated DMU, then the input-oriented BBC model is as follows: (2) minθs.t.{ ∑j=1nxijλj+sj−=θxik,i=1,2,⋯,m∑j=1nyrjλj−sr+=yrk,r=1,2,⋯,s∑j=1nλj=1λ≥0,sj−≥0sj+≥0,j=1,2,⋯n

The character representation of the BBC model is consistent with the CCR model.

Most of the basic DEA models do not take into account the input and output slackness problem when measuring efficiency, which leads to a certain bias in their measurement results. In order to deal with this problem, this paper adopts the SBM-DEA model, which introduces slack variables of inputs and outputs in the objective function, which not only solves the problem of input slackness, but also deals with the problem of evaluating the efficiency under non-desired outputs, which is more in line with the nature of green innovation efficiency. In addition, the SBM-DEA model is also a non-radial and non-angle efficiency evaluation model, which can effectively prevent the bias of the efficiency measurement results due to the difference of radial and angle choices, and ensure the accuracy of the measurement results [26]. The SBM-DEA model is described as follows:

Suppose there are n DMUs, each of which contains three vectors, where the input vector is denoted as x ∈ R^m, the desired output vector is denoted as y^g ∈ R^s₁, and the undesired output vector is denoted as y^b ∈ R^s₂. Define matrix X,Y^g,Y^b as follows: [X] = ⌈x₁,…,x_n⌉^T ∈ R^m×n, ⌈ Yg ⌉=⌈ y1g,⋯,yng ⌉T∈Rs1×n , ⌈ Yb ⌉=⌈ y1b,⋯,ynb ⌉T∈Rs2×n,X>0,Yk>0,Yb>0 . The production possibility set P can then be expressed as: (3) P¯=P\(x0,y0)={ (X¯,y¯g,y¯b)|X¯≥∑j=1,≠0nλjXj,y¯g≤∑j=1,≠0nλjyjg,y¯b≥∑j=1,≠0nλjyjb,λ≥0 }

Then the SBM-DEA model is represented by equation (4): (4) P*=min1−1m∑i=1msi−xi01+1s1+s2[ ∑i=1s1srgyr0g+∑i=1s2srbyr0b ]s.t{ x0=Xλ+s−y0g=Ygλ−sgy0b=Ybλ+sbs−≥0,sg≥0,sb≥0,λ≥0

In the above equation, s is the relaxation of input and output, and λ is the weight vector. The objective function P^* is a decreasing function that decreases with s^*,s^g,s^b, and 0 ≤ P^* ≤ 1. For a DMU, when P^* = 1 and s^*,s^g,s^b are both equal to 0, the efficiency is valid. If P^* < 1 or s⁻,s^g,s^b is not equal to 0, it means that the DMU is inefficient and has some room for improvement.

The SBM-DEA model based on non-expected output proposed in the previous paper has obvious advantages in the static measurement of green innovation efficiency, but has some limitations in the dynamic evolution of green innovation efficiency. Therefore, in order to effectively analyze the dynamic evolution of green innovation efficiency in the catering industry, this paper introduces the Malmquist-Luenberger productivity index [27]. The Malmquist-Luenberger index is able to simultaneously incorporate both desired and non-desired outputs into the index system, and it can measure the green innovation efficiency in the presence of non-desired outputs, taking into consideration the The Malmquist-Luenberger index can simultaneously incorporate desired output and undesired output into the index system, can measure the green innovation efficiency in the presence of undesired output, takes into account the increase of desired output and the decrease of desired output, and has the good properties possessed by the Malmquist index, which is widely used in the study of the dynamic change of the production dynamic efficiency.

First assume that there are n DMUs, each of which contains three vectors, the input vector denoted x ∈ R^m, the desired output vector denoted y ∈ R^s₁, and the non-desired output vector z∈R2s . The production possibility set P is denoted P(x) = {(y,b):x can produce(y,b)},x ∈ R^m, and P(x) satisfies the following requirements: closed set and convex set. Inputs and desired outputs can be adjusted. Joint weak disposability. Zero unionizability. Joint weak disposability shows that input costs are required to reduce non-desired outputs, and zero-integrability shows that with desired outputs there must be non-desired outputs. Then the DEA method is used to describe P(x), and then the directional distance function is used to solve for the optimal solution of the set, the directional distance function is: (5) D→0(x,y,b;gy,gb)=max{ β:(y+βgy,b−βgb)∈P(x) }

where g = (g_y, g_b) is a directional vector indicating that firms face completely different attitudes towards desired and non-desired outputs in terms of utility and preferences. According to the related literature, the Malmquist-Luenberger model can be expressed as: (6) MLtt+1=[ 1+D→0t(xt,yt,bt;yt,−bt)1+D→0t(xt+1,yt+1,bt+1;yt+1,−bt+1)×1+D→0t+1(xt,yt,bt;yt,−bt)1+D→0t+1(xt+1,yt+1,bt+1;yt+1,−bt+1) ]1/2

The ML index is equal to the product of the green innovation technology efficiency change index (MLEC) and the green innovation technology change index (MLTC), expressed respectively: (7) MLECtt+1=1+D→0t(xt,yt,bt;yt,−bt)1+D→0t+1(xt+1,yt+1,bt+1;yt+1,−bt+1) (8) MLTCit+1=[ 1+D¯0t+1(xt,yt,bt;yt,−bt)1+D¯0t(xt,yt,bt;yt,−bt)×1+D¯0t+1(xt+1,yt+1,bt+1;yt+1,−bt+1)1+D¯0t+1(xt+1,yt+1,bt+1;yt+1,−bt+1) ]1/2

Where MLit+1 is greater than 1, indicating growth in total factor productivity of green innovation from period t to period t + 1, and less than 1, indicating a decline. MLECit+1 indicates a change in the technical efficiency of green innovation, where greater than 1 indicates that technical efficiency has grown and is closer to the production frontier, and less than 1 indicates that technical efficiency has declined and is further away from the production frontier. MLTCit+1 indicates a change in production technology, with greater than 1 indicating an advance in green innovation technology and less than 1 indicating a regression in green innovation technology.

According to the SBM-DEA model based on non-expected output, MaxDEA 6.6 Pro software was used to calculate the two-stage technical efficiency and total efficiency of green innovation in China’s restaurant industry.

Figure 1.

Analysis results of green technology innovation efficiency in time dimension

During the decade from 2014 to 2023, the green innovation efficiency of China’s catering industry generally shows an upward trend, with the green innovation efficiency of the technology development stage as well as the industrialization stage rising in ups and downs. The innovation efficiency of the technology development stage is 0.6149, much lower than that of the industrialization stage, which is 0.8751. After 2017, the total efficiency has slowly improved compared to the previous one. After 2014, the total efficiency maintains between 0.7 and 0.8, and the improvement of the total efficiency is not greatly affected by the technology development stage, and the power of the improvement mainly comes from the achievement transformation stage. The reason for this is that the government is implementing the new development concept and increasing its efforts in environmental remediation.

In the first stage, the static efficiency value of the technology development stage of China’s catering industry from 2014 to 2023 is 0.6149, with a basically stable performance and a slight increase. Among them, the technology development efficiency dropped sharply from 2014 to 2015, and the efficiency value fell into a trough in 2012. From 2016 to 2018, the efficiency of technology development grew slowly. After that, from 2019 to 2023, the efficiency value of technology development stabilizes at around 0.6 and oscillates up and down around this value.

In the second stage, the static efficiency value of the outcome transformation stage reaches 0.8751 from 2014 to 2023. The overall trend of growth is shown from 2014 to 2021, with the outcome transformation efficiency rising from 0.7348 to 1.0126. There are two small decreases experienced in 2015 and 2020.2015 is a fall from the previous year’s 0.7848 to the 0.6900 in the current year, and from 0.9327 to 0.8851 in 2020. 2021 to 2023 showed a more substantial downward trend, from the highest efficiency value of 1.0126 back to 0.8751.

Figure 2.

Analysis results of green technology innovation efficiency in spatial dimension

As can be seen from Figure 2, among the three major regions, the green innovation efficiency of China’s catering industry is highest in the east and lower in the west and center. The values of technology development efficiency in the eastern, central and western regions reach 0.7121, 0.4370 and 0.5087 respectively, and the values of achievement transformation efficiency are 1.0971, 0.6954 and 0.6900 respectively, corresponding to the total efficiency values of 0.9155, 0.6737 and 0.7066 respectively. The economically developed regions and the economically underdeveloped regions show a clear The economically developed regions and economically less developed regions show obvious regional disparities. At the same time, the efficiency values of the technology development stage and the achievement transformation stage also have obvious differences, and the efficiency value of achievement transformation is obviously higher than that of technology development.

In addition, the specific situation of green innovation efficiency of the catering industry in most provinces in China is shown in Table 1. From the perspective of provinces, the provinces with a total efficiency of more than 1 are concentrated in the eastern region, namely Hainan Province, Guangdong Province, and Beijing City. All of which are more developed provinces. The total efficiency of Hainan Province exceeding 1 is mainly caused by the high efficiency value in the achievement transformation stage, and the efficiency value in the achievement transformation stage is 5.5280. Guangdong Province and Beijing City both have higher technology development efficiency values, which are 2.5950 and 1.6487 respectively. There are three provinces with total efficiency values lower than 0.5, all of which are concentrated in the central and western regions, namely Shanxi, Guizhou and Gansu, with total efficiency values of 0.3978, 0.4644 and 0.4319, respectively.

From Table 2, it can be seen that the M index of China’s catering industry in 2014-2023 exceeds 1 in most years, and the years not greater than 1 are 2014-2015, 2016-2017, 2019-2020, and the average value of the M index is 1.0647, i.e., the efficiency of green technological innovation in the period of 2014-2023 maintains a continuous rising trend at a rate of 6.47% per year, and in the The value of green technology innovation efficiency in this period generally shows an upward trend. The change of M index is caused by TC and EC. From 2014-2023, the standard deviation of EC and TC is 0.11414 and 0.20322 respectively, so the fluctuation of technology level change (TC) is larger, and the fluctuation of technology efficiency change (EC) is more moderate, so the change of M index is mostly caused by technology level change (TC).

From 2014 to 2023, the dynamic efficiency value (M) index of green technology innovation in China’s catering industry has undergone a significant change, which can be divided into two stages for analysis. In the first stage for the year 2014-2018, TFP fluctuates substantially, and the green technology innovation efficiency vibration range is between 0.80-1.36, which is mainly due to the combined fluctuation impact of TC and EC. In 2018-2023, the technical innovation efficiency M index is in the second stage, showing a stable development trend and a slight increase. The TC and EC trends are opposite, and the magnitude of change is roughly the same, resulting in the M index being basically unchanged.

Beijing’s EC and TC values are 0.9927 and 0.6893, respectively, with both indicators less than 1. Gansu Province’s change in technology level is 0.9362. Guizhou Province’s change in technical efficiency is 0.9805 and change in technology level is 0.9641, with both indicators less than 1. Inner Mongolia and Ningxia’s change in technology level is 0.8928 and 0.7203, respectively, with both indicators less than 1. Tianjin’s change in technical efficiency and technology level is 0.9883 and 0.9477, respectively, with both indicators less than 1. Yunnan Province’s TC and EC are both less than 1. The change values of technical efficiency and technical level in Tianjin are 0.9883 and 0.9477 respectively, with both indicators lower than 1. The indicators of TC and EC in Yunnan Province are both lower than 1, at 0.9673 and 0.9632 respectively.

The coefficient of variation, also known as the standard deviation coefficient, coefficient of variation, etc., is the most commonly used statistical indicator in the academic world to study σ convergence. It is expressed in the form of the ratio of standard deviation and mean in statistics, and the specific formula is: (9) CVt=1μt1n∑i=1n(yit−y¯i)

Where: y_it(i = 1,2,3,⋯,n) is the green innovation efficiency of the restaurant industry in the i rd region in the t nd year, and ȳ_i is the average green innovation efficiency of the restaurant industry in each region in the tth year.

Searle index is a special form of comprehensive entropy index, entropy indicates the expected value of the amount of information, the closer the individual, the smaller the index value. Assuming that y_ij represents the green innovation efficiency of the catering industry in the jrd province and city in region i, the sum of the research objects y=∑i∑jyij ; n_ij represents the population number of the jth province and city in region i, and the total population number n=∑i∑jnij , the Searle’s index of the green innovation efficiency of the catering industry can be expressed in the following form: (10) Tp=∑i∑j(yijy)ln(yij/ynij/n)

The absolute σ convergence values for China as a whole and for the East and West regions are shown in Figure 3. According to Figure 3, it can be seen that the convergence curve of the whole country is a fluctuating downward trend over time as a whole, indicating that there is a convergence of green innovation in the whole country. However, based on the data performance from 2022 to 2023, the standard deviation of green innovation development in the country has increased, which indicates the need to pay attention to the differences between the efficiency of green innovation. The development trend of the convergence curve in the eastern region from 2014-2023 shows that the standard deviation of the region has a slow fluctuating increase. However, the standard deviation of the research data in the east is always lower than that of the nation and the central and western regions, indicating that the development gap of the overall efficiency level in the east is smaller and the level is basically the same. The convergence curve of the central region shows a relatively large fluctuation pattern in the study interval of the sample, indicating that the regional differences between the central region are large, and the development of individual provinces in the central region has the phenomenon of a ladder, and in the two intervals of 2014-2016 and 2017-2020, it shows a flat fluctuation rise, and the convergence declines in 2021-2022, indicating that individual provinces and municipalities in the eastern provinces have a strong development trend, and the development speed is higher than other provinces in the region, with the efficiency of high-efficiency provinces in a steady state of efficiency, inefficient provinces continue to progress and develop, and the differences within the region are reduced after 2021. For the study of western places the standard deviation volatility is very large, of which the entire period from 2014-2018 is in a relatively large fluctuation pattern, and the differences within the region are gradually narrowed after 2018, indicating that all provinces and cities in the region are striving to develop, and in recent years have gradually reached a balanced development within the region. Among them, the regional differences in the east are the smallest in the whole country, and in the central and western regions, and the volatility is stronger in the central region.

Figure 3.

Convergence diagram of green innovation efficiency σ

From the results of the LM test in Table 5, it can be seen that the P-value of Moran’s residuals is 0.009, which is statistically significant, and the error and lag of the model have passed the test of significance level. However, from the perspective of robustness consideration, the spatial lag model with a relatively small P value is selected for model fitting regression. In the Hausman test, the results rejected the original hypothesis, i.e., fixed effects were added for further fitting.

The national spatial absolute convergence β-convergence results are shown in Table 6. The F-statistics of the model are significant at the 1% significance level with a P-value, i.e., the model has a reference effect. It can be seen that for the fixed effects, an expansion is carried out and divided into three levels: spatial fixed, time fixed, and double fixed. The results show that the absolute beta convergence coefficients of the three fixed effects are not negative and the models are all very well significant, indicating that the models have research significance. After adding space, the national absolute β-convergence coefficient is higher compared to the β-convergence coefficient obtained from the traditional panel regression calculation. In particular, the performance of absolute beta convergence at the spatial fixed effects level and the performance of beta convergence coefficients at the double fixed effects level. The average speed of convergence under the three states is 0.0739 and the average half-life cycle is about 14.3145 years. It can also be found that the spatial lag model with double fixed effects has the greatest goodness of fit and is overall better than the other forms.