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Partially Convex Production Technology and Efficiency Measurement

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Abstract

Economists tend to believe that production technology should exhibit increasing returns to scale first and then constant and finally decreasing returns to scale, called regular variable returns to scale (RVRS) in this paper. Further, a special pattern of RVRS production technology when there is only one output is the production function that has an S-shaped curve along any ray of inputs from the origin. In the literature on efficiency analysis, the most frequently used empirical technology is the variable returns to scale (VRS) production technology. Although it exhibits RVRS, it is unable to model nonconvex production technologies, such as the S-shaped production function. Recently, a new empirical production technology has been introduced to capture RVRS with partial convexity. This paper explores its relationship with efficiency measurement. Furthermore, a novel empirical production technology that can better capture the characteristics of the S-shaped production function is proposed. These two new production technologies provide better alternatives to the commonly used Free Disposal Hull (FDH) production technology in non-convex production with RVRS. Our new production technology is illustrated using US manufacturing industry data. If one believes that the production technology is partially convex and exhibits RVRS, it is found that the conventional VRS production technology overestimates the technical inefficiency of small production units under this belief.

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Notes

  1. For discussions of these three stages of production, see Ozawa (1971) and textbooks of principles of microeconomics.

  2. It was called the regular ultra passum law by Frisch (1965).

  3. Lipsey (2018) also provided a detailed discussion on the sources of positive and negative “scale effects”.

  4. Sickes and Zelenyuk (2019, Section 8.7) provided a detail discussion of the FDH technology.

  5. This is equivalent to convex input sets in 2-input production. See also the illustrations provided in microeconomics textbooks.

  6. “completely non-convex” means that the production set, input sets and output sets are all not convex.

  7. The same authors are working on RVRS without convexity in another paper: Li, Tsang and Lee (2023).

  8. The technology set is implicitly assumed to be closed to ensure that extrema can be achieved in this paper. Further, the output level is bounded for any input vector. Other assumptions will be made in due course.

  9. The term partial convexity emphasizes that convexity of production activities holds for any fixed output level but not for the production set. This is like the concept of “selective convexity” stated in Olesen, Petersen and Podinovski (2022, p. 386) in which subsets of outputs and inputs are fixed. In contrast, the term “quasi-concave technologies” stated in Introduction refers to technologies that can be convex or partially convex.

  10. The converse may not be true because \(h\) is a function along a ray only. For example, if \(h\) is a strictly increasing function in \(s\), it does not mean that the production function \(f\) is strictly increasing at \({{{\boldsymbol{x}}}} = s{{{\boldsymbol{x}}}}^0\).

  11. The notations gs and hs refer to the partial derivatives of g and h with respect to s.

  12. When there are ties, the results in this paper still hold with minor adjustments.

  13. The multi-output version of FDH was introduced by Deprins, Simar, and Tulkens (1984).

  14. Li’s (2019) used term “variable returns to scale (VRS)” is equivalent to “regular variable returns to scale (RVRS)” in this paper.

  15. Extension to non-strict S-shaped production function is straight-forward with minor modifications.

  16. The term “convex-concave function” is also used to describe a non-S-shaped function, see Moudafi (2012) for example.

  17. A function \(f\left( {{{\boldsymbol{x}}}} \right)\) is homothetic if \(f\left( {{{\boldsymbol{x}}}} \right) = H\left( {g\left( {{{\boldsymbol{x}}}} \right)} \right)\), where \(g\left( {{{\boldsymbol{x}}}} \right)\) is homogeneous of degree one and \(H\left( \cdot \right)\) is strictly increasing.

  18. Here, the NDRS region includes the IRS and CRS regions in Definition, whereas the NIRS region is the DRS region.

  19. In each of the expressions related to \(f^{ccrvrs}\left( {{{\boldsymbol{x}}}} \right)\) and \(f^ \ast \left( {{{\boldsymbol{x}}}} \right)\), the first equation is from Li (2019, Propositionon 4, p. 163) and the second equality is from the fact that \({{{\boldsymbol{x}}}}^b\). is in the DRS region.

  20. An exception is the output-specific technologies studied in Cherchye (2016) in which each output has a separate production set. The production technologies mentioned in this paper are applicable to their method if one accepts the assumptions of partial convexity and output-specific technology.

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Funding

The work described in this paper was substantially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (UGC/FDS15/E02/21) and partially supported by a grant from the Hong Kong Shue Yan University (URG/20/01).

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Authors and Affiliations

  1. Business, Economic and Public Policy Research Centre, Hong Kong Shue Yan University, Hong Kong, China

    Sung Ko Li, Chun Kei Tsang & Shu Kam Lee

  2. School of Economic, Hefei University of Technology, Hefei, China

    Sung Ko Li

  3. Department of Economics, Hong Kong Baptist University, Hong Kong, China

    Chun Kei Tsang

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  1. Sung Ko Li
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  2. Chun Kei Tsang
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Li, S.K., Tsang, C.K. & Lee, S.K. Partially Convex Production Technology and Efficiency Measurement. J Prod Anal (2024). https://doi.org/10.1007/s11123-023-00716-w

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