The infeasible problem of Malmquist–Luenberger index and its application on China’s environmental total factor productivity

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The Malmquist–Luenberger productivity index would cause infeasible problem when measuring mixed period change of total factor productivity. The former research focuses on the infeasible issue under the hypothesis of constant return to scale (CRS). There are many meaningful solutions which could avoid this infeasible problem of Malmquist–Luenberger productivity index. However, these solutions couldn’t avoid infeasible problem under condition of variable returns to scale (VRS). The new solution is proposed in this paper which could solve this problem under VRS based on super-efficiency issue. The empirical results of environmental total factor productivity (ETFP) change of thirty regions in China indicate that China isn’t efficient from 1997 to 2014. Among eight economic regions northwest and southern coastal decline, the ETFP decrease 15%, 10% respectively. From the perspective of provinces in China, Hainan, Qinghai and Ningxia have the lowest environmental total factor productivity and their environmental technical efficiency is also the lowest. These non-efficient provinces all have lower gross domestic product and they should improve technique change efficiency through adopting advanced technology in the future. The fixed effects regression model illustrates that energy intensity, research and development, foreign direct investment are factors of ETFP in China. Both research and development and foreign direct investment could improve total factor productivity and this indicates that Pollutant Heaven Hypothesis doesn’t exist in China.

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The first author thanks the National Natural Science Foundation of China (Grant Nos. 71471133, 71432007 & 71532015) for funding. This study is partially supported by the Priority Academic Program Development of Jiangsu Higher Education Institutions (China). The corresponding author thanks the 18th Annual Conference of China Management Science for the feedback on an earlier version of the research.

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Correspondence to Jinghua Xu.



See Appendix Tables 6, 7, 8, 9 and 10.

Table 6 Data set for test
Table 7 Distance value based on the conventional constraint (=) with CRS
Table 8 Distance value based on new constraint (\(<=\)) with CRS
Table 9 Distance value based on the new constraint (\(<=\)) with VRS
Table 10 Distance value based on the new constraint (\(<=\)) of this proposed model with VRS

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Du, J., Duan, Y. & Xu, J. The infeasible problem of Malmquist–Luenberger index and its application on China’s environmental total factor productivity. Ann Oper Res 278, 235–253 (2019).

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  • Infeasible
  • Malmquist–Luenberger index
  • Environmental total factor productivity
  • Variable returns to scale