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On aggregation of multi-factor productivity indexes

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Abstract

The goal of this paper is to investigate the question of the importance of aggregation of the Paasche and Laspeyres versions of the Malmquist quantity and productivity indexes from both theoretical and empirical perspectives. We discuss the existing justification and provide an alternative theoretical justification based on results from the functional equations literature. We also use real data (from Kumar and Russell (2002, American Economic Review)) to illustrate how dramatic the differences in conclusions can be in practice, depending on whether one employs Laspeyres or Paasche productivity indexes.

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Fig. 1: Complexity of measuring productivity changes: an illustration.

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Notes

  1. For a nice exposition of the history of index numbers, see Balk (2008, Chapter 1).

  2. Also see Henderson and Russell (2005) and Park et al. (2008).

  3. See Färe and Primont (1995) and Sickles and Zelenyuk (2019).

  4. While the roots of this index can be found in Caves et al. (1982a, 1982b), Diewert (1992, p. 240) appears to be the first who explicitly introduced it after apparently being inspired by the geometric descriptions from Moorsteen (1961, p. 460) and the intuitive description of Hicks (1961, footnote 4). He dubbed it as the “Hicks–Moorsteen approach to productivity indexes” and then it often emerged as the “Hicks–Moorsteen Productivity index” or the “Hicks–Moorsteen TFP index” or simply the “Hicks–Moorsteen index” in various works. About the same time, Bjurek (1996) developed this index more explicitly and generalized it to allow for technical inefficiency and called it the “Malmquist TFP index”, although others started to call it the “Bjurek productivity index” or the “Bjurek TFP index” (or simply the “Bjurek index”). Professor Bert Balk suggested to us that the name of Hicks should not be there, referring to it as the “Moorsteen-Bjurek productivity index”, although see explanations in Diewert and Fox (2017, p. 276) on the justification of the influence from Hicks. Meanwhile, Sickles and Zelenyuk (2019, p. 110) suggested that the name of Diewert (since he was the first to suggest this index explicitly), should be also added there, and also the name of Malmquist, since it is defined as the ratio of Malmquist indexes. Finally, the name of Shephard can also be justified, since his distance functions were used to define this index explicitly.

  5. E.g., see Sickles and Zelenyuk (2019, Chapter 4), Färe et al. (2020) and references therein.

  6. E.g., see Sickles and Zelenyuk (2019, Chapter 4), Färe et al. (2020) and references therein.

  7. We thank an anonymous referee for inspiring this discussion.

  8. It appears that Westergaard (1890) was the first to discuss this requirement for indexes, later considered by Fisher (1911) under the name of ‘test by changing base’ and later, in Fisher (1922, p. 270), under the name of ‘circular test’. Also see related discussions and many references in Balk and Althin (1996), 2004 CPI Manual and Sickles and Zelenyuk (2019), who we follow here.

  9. Similarly to the CPI manual, we prefer using the term circularity or ‘circular test’, because the term “transitivity” is a somewhat unfortunate name here due to confusion with the more general (and widely accepted) transitivity property of relations in mathematics and in consumer theory of economics (see Sickles and Zelenyuk (2019) for further discussions).

  10. Essentially the same result was much earlier derived in a somewhat overlooked paper of Konüs and Byushgens (1926).

  11. See Färe and Grosskopf (1996), Sickles and Zelenyuk (2019, p. 130–137) for more details.

  12. Also, see Diewert and Fox (2017) for related discussions on the caveats of fixed-weight indexes.

  13. Also see Balk (2008), Theorem 3.11, p. 97 and the related discussion.

  14. Also note that similar reasoning applies for the additive measures, sometimes referred to as indicators, as outlined in Färe and Zelenyuk (2019). In fact, their result was derived after the result of this section, although published earlier.

  15. See Eichhorn (1978, p.94), Aczél (1987, p.151), Aczél (1990, p.27) and Färe and Zelenyuk (2005).

  16. We thank Robert G. Chambers for pointing this out.

  17. We thank an anonymous referee for inspiring this discussion.

  18. This is in fact also the very first data set we tried for our illustration. We also tried newer versions of Penn World Tables and also with more sophisticated measurement of inputs, e.g., by accounting for human capital as in Henderson and Russell (2005) or/and financial capital as in Badunenko and Romero-Avila (2013), as well as tried micro data (on farmers) and these data sets generally confirmed the main conclusions we discussed here. Also, for somewhat related evidence in different contexts, see Balk (2008, Chapter 3) who used artificial data for the contexts of price indexes and Balk and Zofio (2018) for the context of comparing various decompositions of productivity indexes for banking data.

  19. Their estimates of the MPI can be recovered from their estimates of efficiency change and technology change.

  20. Other popular methods include Stochastic Frontier Analysis or SFA (parametric, semi-parametric and non-parametric) and their symbiosis, such as Stochastic DEA, etc. See Kumbhakar and Lovell (2002) and Sickles and Zelenyuk (2019) for a comprehensive textbook style discussion and references therein for more details.

  21. All estimations were done in Matlab by the authors and then independently checked in R by one of our research assistants. For Matlab, we adapted some DEA codes from Leopold Simar, while for R we used the “Benchmarking” package of Bogetoft and Otto (https://cran.r-project.org/web/packages/Benchmarking/Benchmarking.pdf) and ‘deaR’ package of Coll-Serrano, Bolos and Suarez (https://cran.r-project.org/web/packages/deaR/deaR.pdf).

  22. In this case, the input and output oriented Shephard’s distance functions are reciprocal to each other in general and for DEA in particular, i.e., \({\widehat{D}}_{i}^{\tau }({y}^{tj},{x}^{sj})={({\widehat{D}}_{o}^{\tau }({x}^{sj},{y}^{tj}))}^{-1}\) (e.g., see Färe and Lovell (1978)).

  23. E.g., see Simar and Wilson (2015) for a comprehensive review of the statistical aspects of these estimators and references therein.

  24. It is also possible to consider the so-called Free Disposal Hull (FDH) approach, which only imposes free disposability of inputs and outputs, by requiring \({\mathcal{Z}}=\{({z}^{1},...,{z}^{n})\,:\,{z}^{k}\in \{0,1\},\,k=1,...,n;\,\mathop{\sum }\nolimits_{k = 1}^{n}{z}^{k}=1\}\), which is somewhat an outsider in the literature, partly because it has an even slower rate of convergence, Op(n−1/(N+M)), and often suffers from the overfitting problem and low discriminative power in relatively small samples so we do not pursue it here.

  25. As is well-known, the DEA may give infeasible solutions for MPI under non-CRS assumptions. In our data, this happened only for six countries (with relatively small GDP) and only for DEA-VRS where the implosion of technology was allowed (The cases with infeasible solutions are indicated with the original Matlab output for them, containing “Inf” or “Infi”.).

  26. See related discussions in Keynes (1930) and Balk (2008, p. 28–30).

  27. E.g., see see Diewert (1992), Diewert and Nakamura (1993), Balk (1998, 2008), Sickles and Zelenyuk (2019, Chapter 4 and 7) and many references therein.

  28. Indeed, the gap between LPI and MPI is fully explained by the contributions from the other inputs: e.g., in the data set we used in Section 6, it can be explained by the capital-deepening contribution (Kumar and Russell (2002)), while in Henderson and Russell (2005) data set it can be explained by physical and human capital deepening.

  29. Here, the assumption of CRS shall not be dogmatically understood as a belief that the ‘true’ technology is CRS, rather than as a selection of a relevant common benchmark/reference, e.g., coherent with the socially optimal scale and the notion of maximal average productivity.

  30. We thank an anonymous referee for inspiring this discussion.

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Acknowledgements

The authors thank Robert G. Chambers, Bert Balk, Erwin Diewert, Kevin Fox, Chris Parmeter and Robin Sickles, as well as Duc Manh Pham, Bao Hoang Nguyen, Hong Ngoc Nguyen, Evelyn Smart, Zhichao Wang and all others who gave feedback to various versions of this paper. The authors also acknowledge the financial support from ARC grant (FT170100401) and the overall support from their employers. All views expressed here are those of the authors.

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Authors and Affiliations

  1. Department of Economics, Oregon State University, Corvallis, OR, USA

    Rolf Färe

  2. Department of Agricultural and Resource Economics, University of Maryland, Annapolis, MD, USA

    Rolf Färe

  3. School of Economics and Centre for Efficiency and Productivity Analysis (CEPA) at The University of Queensland, Brisbane, QLD, Australia

    Valentin Zelenyuk

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  1. Rolf Färe
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Appendix

Appendix

Table 4 Productivity Estimates: LPI, MPI and DBPI under CRS, with implosion (% change)
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Table 5 Productivity Estimates: LPI, MPI and DBPI, under NIRS, with implosion (% change)
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Table 6 Productivity Estimates: LPI, MPI and DBPI, under VRS, with implosion (% change)
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Table 7 Supplement: Kumar and Russell (2002) Data Used in the Empirical Example in Section 6 (Empirical importance) originally came from the Penn World Tables (version 5.6)
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Färe, R., Zelenyuk, V. On aggregation of multi-factor productivity indexes. J Prod Anal 55, 107–133 (2021). https://doi.org/10.1007/s11123-021-00598-w

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