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Quadratic-mean-of-order-r indexes of output, input and productivity


This paper deals with the quadratic-mean-of-order-r indexes of output, input and productivity. Each index is a family of indexes that unify many of the existing indexes, including the most popular ones. We show that all index number formulae belonging to these families are superlative indexes. This is considered as a generalization of the equivalence of Fisher and Malmquist indexes, shown by Diewert (1992).

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  1. 1.

    Diewert (1993a, 1993b) summarizes the past studies on this problem into five approaches and calls the economic and the axiomatic approaches ‘two major approaches.’

  2. 2.

    For example, the Konüs cost of living index intends to capture inflation by measuring the change in the minimum cost of reaching a given level of utility. See Konüs (1924) and Diewert (1976).

  3. 3.

    The flexible functional form can approximate the true function to the second order at a certain point. See Lau (1986) and Chambers (1988).

  4. 4.

    See International Labour Office et al. (2004a, 2004b).

  5. 5.

    The Malmquist indexes were introduced by Caves, Christensen and Diewert (1982), and have been used and popularized in many studies since then. See Balk (1998), Färe et al. (1994) and Sickles and Zelenyuk (2019).

  6. 6.

    The parameter r can be any non-zero real number. We exclude the case when r goes to zero or infinity, since the flexibility of the corresponding functional form is lost in those cases.

  7. 7.

    While Diewert (1976) justifies the use of the quadratic-mean-of-order-r price index as a consumer price index, the International Labour Office et al. (2004b) justifies its use as a producer price index (such as GDP deflator), but by assuming weak separability and no technical change. In this paper, we justify it under more general conditions.

  8. 8.

    Vector notation: \(y \, \ge \, y^{\prime}\) indicates \(y_{m} \, \ge \,y_{m}^{\prime}\) for any m; \(y\, \gg \, y^{\prime}\) indicates \(y_{m}\, \gg \, y_{m}^{\prime}\) for any m; \(y\, > \,y^{\prime}\) indicates \(y_{m}\, \ge \,y_{m}^{\prime}\) for any m and \(y \,\ne \, y^{\prime}; \,0_{M}\) and \(1_{M}\) denotes M dimensional vector of zeros and ones, respectively; and \(x_{-1}= (x_{2},\, \ldots, \, x_{N})\).

  9. 9.

    Our result on the output-oriented Malmquist index is directly applicable to the input-oriented Malmquist productivity index. We focus on the former to avoid unnecessary repetitions.

  10. 10.

    Thus, we allow r to be positive or negative.

  11. 11.

    The implicit price indexes corresponding to \(QMOI_{r}\) and \(QMII_{r}\) satisfy the proportionality axioms used by Eichhorn and Voeller (1976) and Diewert (1992) but do not satisfy the related proportionality axioms used by Balk (2008, p. 59) except for special cases of \(r=1\) or 2. Thus, we should take this into account for selecting r for \(QMOI_{r}\) and \(QMII_{r}\).

  12. 12.

    As Mizobuchi and Zelenyuk (2020) show, these conditions still allow \(g_r^t\) to be flexible at \((x^{t},\, y^{t})\) for \(t = 0\) and 1, although they may restrict the flexibility of \(g_r^t\) at some other points. The same argument applies for the corresponding conditions for Propositions 3.

  13. 13.

    As Mizobuchi and Zelenyuk (2020) show, these conditions still allow \(h_r^t\) to be flexible at \((x^{t},\, y^{t})\) for \(t = 0\) and 1, although they may restrict the flexibility of \(h_r^t\) at some other points.

  14. 14.

    While the profit maximization problem (25) is formulated by the output distance function, it is also possible to define it by the input distance function.

  15. 15.

    While these conditions limit the type of technical change, they still allow for a variety of types of non-Hicks neutral technical change.

  16. 16.

    See Propositions 2, and 3 of Mizobuchi and Zelenyuk (2020). When \(r = 2\), these propositions coincide with Theorems 7 and 5 of Diewert (1992) and the functional forms (10) and (18) coincide with the functional forms adopted by those theorems.

  17. 17.

    When \(r = 2\), Propositions 1, 2 and 3 coincide with Theorems 8, 6 and 9 of Diewert (1992), where the Fisher indexes are shown to be superlative indexes. Balk (1998, p. 52, p. 65, p. 95 and p. 106) also shows that the Fisher indexes coincide with the Malmquist indexes under a more general condition, allowing for technical inefficiency.


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We thank the editors, two anonymous referees, Bert Balk, Erwin Diewert, Knox Lovell, Antonio Peyrache, Prasada Rao, Christopher O’Donnell and seminar participants in 2017 CEPA International Workshop on Performance Analysis: Theory and Practice, November 2017 in Brisbane for their the fruitful comments. We also thank Bao Hoang Nguyen, Zhichao Wang, Duc Manh Pham and Evelyn Smart for their feedback from proofreading the paper. Hideyuki Mizobuchi acknowledges the financial support from Japan Society for the Promotion of Science: Grant-in-Aid for Scientific Research (C) (18K01552) and a grant-in-aid from Zengin Foundation for Studies on Economics and Finance. Valentin Zelenyuk acknowledges the financial support from the Australian Research Council (ARC FT170100401) and from The University of Queensland. All remaining errors are the authors’ responsibility.

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Mizobuchi, H., Zelenyuk, V. Quadratic-mean-of-order-r indexes of output, input and productivity. J Prod Anal 56, 133–138 (2021).

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  • Index numbers
  • Superlative index
  • Quadratic-mean-of-order-r index
  • Fisher index
  • Malmquist index
  • Flexible functional form

JEL classification

  • C43
  • D24
  • E31
  • O47