3.1. Malmquist Model
DEA is primarily used for cross-sectional analysis. Since productivity could be correlated across different time periods, when using DEA for time-based change estimation, the Malmquist Productivity Index model is more suitable [
23]. The Malmquist Productivity Index is a variant of DEA, measuring the movement of the efficient frontier and Decision-Making Units (DMUs) over a given time period. This model is mainly used to assess changes in productivity, with the advantage of converting quantitative information on input and output factors into an index, even when accurate price data is lacking or when it is challenging to assume producer types (such as cost minimization or profit maximization).
The Malmquist Productivity Index for productivity growth is based on the concept of the distance function, which provides information equivalent to that from a production function. Lovell (1993) [
24] demonstrated that the reciprocal of the distance function is merely the reciprocal of Farrell’s (1957) [
25] measure of production efficiency. Fare and Grosskopf (1996) [
26] defined the output-oriented Malmquist Productivity Index as follows. For a time series
= 1, 2, ⋯, T, which provides the data to be analyzed, the production technology St is defined as shown in Equation (1), where the production technology consists of all possible vectors of input and output factors.
represents the input factors at time t. This indicates the production resources or input variables used during the production process at a specific time. Similarly,
represents the output factors at time
. The output distance function at time t can be defined as follows:
According to the above definition, the output distance function refers to the reciprocal of the maximum value of the output factor given a specific set of input factors The condition is fulfilled when . When indicates technical change, then . This signifies that when production is technically efficient, the value of is 1.
To define the Malmquist Productivity Index, let us consider the case of producing a single output with a single input. Thus, as at time
, we can define the output distance function at time
as follows:
The above expression uses the production technology at time to measure the extent to which the output factors can be produced within the feasible range defined by . Using a similar concept, the distance function can also be defined by using the production technology at time to measure the extent to which output factors can be produced within the feasible range defined by . This distance function is denoted as .
The Malmquist Productivity Index (TFP) can be defined by assuming that the production technology at time
remains unchanged and considering different combinations of input and output elements at times
and
, as follows:
Similarly, assuming that the production technology at time
remains constant, it can be defined through different combinations of input and output factors at times
and
, as shown in the following expression:
The output-oriented Malmquist productivity change index (TFP) derived from the two formulas above can be defined as follows:
If , it indicates an improvement in efficiency at time compared to time t. If , it signifies a decrease in efficiency, and if , it means no change in efficiency has occurred.
In the second line of the above expression, the part outside the parentheses represents the ratio of the distance functions between two points in time, and . This is known as the technical efficiency change index. The part within the parentheses represents the movement of production change, indicating the measurement of technological change, which is referred to as the technical progress change index.
The technical efficiency change index can be further divided into the pure efficiency change index and the scale efficiency change index. It can be expressed as follows:
Thus, the Malmquist Productivity Index (TFP) can be represented as follows:
In the above expression, represents the output distance function under variable returns to scale at time , while is a measure of the pure efficiency change from time t to . represents the ratio of the output distance function under constant returns to scale to the output distance function under variable returns to scale at time , indicating the scale efficiency change.
The Malmquist total factor productivity (TFP) change index can be decomposed into the technical efficiency change index (EC) and the technical change (TC) index. The scale efficiency change (SEC) index measures how close a DMU is to achieving scale economies across the two periods, defined as the ratio of the maximum output under constant returns to scale technology. The product of the pure efficiency change (PEC) index and the scale efficiency change (SEC) index yields the technical efficiency change index (TECI), which gauges the efficiency with which a DMU converts input factors into output factors during production. The technical efficiency change index (TECI) corresponds to the “catch-up” effect, reflecting impacts from learning and knowledge diffusion, market competitiveness, improvements in cost structures, and enhanced equipment utilization rates. On the other hand, technical change (TC) corresponds to the frontier shift effect or innovation, measuring the change in the efficient frontier between the two periods, influenced by new product and production process innovations, new business methods, external shocks, and other factors.
To study the efficiency of China’s pig farming industry, this research examines the time period from 2010 to 2022, focusing on 17 regions in China with significant pork production. These regions include Hebei, Liaoning, Jilin, Heilongjiang, Jiangsu, Anhui, Shandong, Henan, Hubei, Hunan, Guangdong, Guangxi, Hainan, Chongqing, Sichuan, Guizhou, and Yunnan. Based on previous studies [
27,
28,
29] the output variable is set as the production volume of live pigs, while the input variables include piglet costs, feed costs, labor costs, medical and epidemic prevention costs, utility costs (water, electricity, and coal), repair and maintenance costs, and depreciation costs.
3.2. Impact of African Swine Fever on Pig Farming Efficiency
To investigate the specific impact of African Swine Fever on pig farming efficiency, this study constructs a panel regression model using fixed-effects to analyze the factors influencing efficiency. Although data collection spans from 2010 to 2022 to analyze trends and efficiency changes with the Malmquist Productivity Index, the panel regression analysis focuses on data from 2019 to 2022 [
25,
29]. The choice to use this specific timeframe is due to the immediate and delayed effects of the outbreak.
Although African Swine Fever was first reported in China in 2018, the full impact on pig farming efficiency may have a time delay. This delayed response can be attributed to several factors, including the initially sparse and unclear information on this disease, a lag in the industry’s recognition of the outbreak’s severity, and the time needed for government and corporate responses to take effect.
Another reason for the delayed response is data stability and reliability. As the epidemic progressed, data collection and reporting became more standardized and refined. Data from 2019 onwards reflect the actual operating environment and policy adjustments in response to African Swine Fever, providing a more accurate insight into its impact on pig farming efficiency.
Finally, focusing on the key impact period is essential. Analyzing data from 2019 to 2022 across different provinces allows for a more direct assessment of changes in pig farming efficiency after the outbreak of African Swine Fever. This timeframe is sufficient to observe the complete process, from policy adjustments and industry adaptation to efficiency changes.
The basic structure of the model is as follows:
In this context, represents the production efficiency change index, technological change index, the technical efficiency change, the pure technical efficiency change index, and the scale efficiency change index for province in year . All of these efficiency change indices are derived from the Malmquist index.
is a dummy variable indicating whether African Swine Fever occurred in province
in year
. A value of 1 signifies that an outbreak occurred, while 0 indicates no outbreak.
and
are the parameters to be estimated.
represents the province-specific effect that does not change over time, capturing all inherent characteristics of the province that remain constant over the period.
ϵit is the error term, assumed to be white noise.
represents the number of African Swine Fever cases in the i-th province in year .
3.3. Data Sources
The data for the study were sourced from three authoritative sources, ensuring the reliability and comprehensiveness of the research findings.
First, this study references data from the “China Livestock Yearbook”, compiled by the Ministry of Agriculture of China and published by the China Agricultural Publishing House. This yearbook contains detailed foundational data on pig production, including key production indicators like total pork output, providing a solid data foundation for analyzing the overall performance of the pig farming industry.
Second, the study also utilizes the “Compilation of Costs and Benefits of China’s Agricultural Products”, released by the National Development and Reform Commission and published by China Statistics Press. This resource provides detailed information on the costs associated with pig farming, including the costs of rearing pigs, such as piglet expenses, feed costs, labor costs, medical and epidemic prevention costs, repair and maintenance costs, utility costs (water, electricity, and coal), and depreciation costs. These data are especially important for analyzing the economic efficiency and cost structure of pork production.
Lastly, to obtain the latest updates on African Swine Fever outbreaks, the study also uses the “Weekly Veterinary News Overview”, regularly published by the Ministry of Agriculture and Rural Affairs of the People’s Republic of China. This report provides real-time data on ASF outbreaks in different provinces, helping researchers understand the direct impact of the epidemic on the pig farming industry.
The deflator index, which measures changes in price levels, involves multiple TFPs and complex data processing. By calculating the deflator index, it is possible to convert nominal economic data from various years into real economic data. This conversion process is crucial because it removes price-related factors, allowing for a more straightforward comparison of the real changes in production capacity across different years. Once the real economic data were obtained, these data were further utilized to calculate efficiency.
Table 1 lists the variables.