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Measures of Productivity Change

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Abstract

In this book, measures of productivity change are defined as measures of output quantity change divided by measures of input quantity change. Computing measures of output and input quantity change involves assigning numbers to baskets of outputs and inputs. The most distinguishing feature of this book is that it computes index numbers that are consistent with measurement theory. So please add the following to the end of this paragraph: Measurement theory says that so-called index numbers must be assigned in such a way that the relationships between the numbers mirror the relationships between the baskets. This chapter explains how to compute output and input quantity index numbers (and therefore productivity index numbers) that are consistent with measurement theory.

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Notes

  1. 1.

    Measurement theory is the study of how numbers are assigned to objects. Classical measurement theory holds that only quantitative attributes of objects are measurable. The representational theory of measurement holds that qualitative attributes of objects are also measureable. For more details, see Sarle (1997) , Hosch (2011, pp. 227–229) and Tal (2016, Sect. 3).

  2. 2.

    Similar distinctions can be found in other areas of science. In econometrics, for example, an estimator is a rule or formula that explains how to use data to estimate the value of a population parameter, while an estimate is the value obtained after data have been substituted into the formula. See, for example, Hill et al. (2011, p. 53).

  3. 3.

    This definition can be traced back at least as far as O’Donnell (2012a) . Elsewhere in the literature, output and input indices are rarely defined in terms of aggregate quantities. When they are, the aggregator functions are often observation-varying (e.g., they depend on observation-varying prices or value shares) and/or their monotonicity and homogeneity properties are not all specified. See, for example, Diewert (1976).

  4. 4.

    If some outputs are zero, then some output indices may be mathematically undefined and may therefore not satisfy some axioms.

  5. 5.

    These statements, and similar statements elsewhere in this chapter, are axioms in the sense that they are substantive assertions about elements of the domain of index number theory.

  6. 6.

    In O’Donnell (2016) , an output index is said to be proper if and only if eight axioms are satisfied. If QI4 and QI6 are satisfied, then, and only then, the extra two axioms of O’Donnell (2016) , an identity axiom and a circularity axiom, are also satisfied. The use of the term ‘proper’ in an index number context can be traced back to O’Donnell (2012b, p. 6).

  7. 7.

    If output distance functions and/or cost functions are unknown, then the choice set will obviously be limited. Samuelson and Swamy (1974) write that “we cannot hope for one ideal formula for the index number: if it works for the tastes of Jack Spratt, it won’t work for his wife’s tastes; if say, a Cobb-Douglas function can be found that works for him with one set of parameters and for her with another set, their daughter will in general require a non-Cobb-Douglas formula! Just as there is an uncountable infinity of different indifference contours—there is no counting tastes—there is an uncountable infinity of different index-number formulas, which dooms Fisher’s search for the ideal one. It does not exist even in Plato’s heaven.” (p. 568).

  8. 8.

    The use of the term ‘additive’ to describe an index of this type can be traced back at least as far as Aczel and Eichhorn (1974) . The index is additive in the sense that \(QI^A(q_{ks},q_{it}+q_{rl})=QI^A(q_{ks},q_{it})+QI^A(q_{ks},q_{rl})\) and \(1/QI^A(q_{ks}+q_{rl},q_{it})=1/QI^A(q_{ks},q_{it})+1/QI^A(q_{rl},q_{it})\) (Aczel and Eichhorn 1974 , p. 525).

  9. 9.

    Here, the term ‘mean-corrected’ means all variables have been re-scaled to have unit means.

  10. 10.

    The use of the term ‘multiplicative’ to describe an index of this type can be traced back at least as far as Coelli et al. (2005, p. 131) . The index is multiplicative in the sense that \(QI^M(q_{ks}\odot q_{gh},q_{it}\odot q_{rl})=QI^M(q_{ks},q_{it})QI^M(q_{gh},q_{rl})\).

  11. 11.

    Balk (1998, p. 100) also uses the term ‘primal’ in reference to an index that is constructed using an output distance function.

  12. 12.

    In production economics, the term ‘dual’ is usually used to describe cost, revenue and profit functions (e.g., Beavis and Dobbs 1990, p. 99).

  13. 13.

    Cherchye et al. (2007) attribute the term ‘benefit-of-the-doubt’ to Melyn and Moesen (1991).

  14. 14.

    The choice of one as the maximum aggregate output is arbitrary.

  15. 15.

    The BOD output index is a proper index, implying it satisfies a circularity axiom (among others). Characteristicity and circularity are generally in conflict with each other. Drechsler (1973, p.17) claims, incorrectly, that they are always in conflict with each other. This erroneous claim can be traced back to Fisher (1922, p. 275).

  16. 16.

    The right-hand side of this equation is a function of \(q_{ks},q_{it},p_{ks}\) and \(p_{it}\). However, only \(q_{ks}\) and \(q_{it}\) have been listed on the left-hand side. In this book, dots and ellipses are used in functions to indicate that one or more variables have been omitted.

  17. 17.

    If the output distance function is nondecreasing in outputs, then the partial derivatives of the distance function with respect to the outputs must be nonnegative for all feasible input-output combinations. If the output distance function is a translog function, then it is possible to find feasible input-output combinations such that at least one partial derivative is negative. Ergo, the output distance function cannot be a translog function.

  18. 18.

    The CNLS estimates of \(\tau \), \(\gamma _1\) and \(\gamma _2\) are 1, 0.0018 and 0.9982 respectively.

  19. 19.

    This definition can be traced back at least as far as O’Donnell (2012a) . See footnote 3 on p. 94.

  20. 20.

    If some inputs are zero, then some input indices may be mathematically undefined and may therefore not satisfy some axioms.

  21. 21.

    In O’Donnell (2016) , an input index is said to be proper if and only if eight axioms are satisfied. If XI1 to XI6 are satisfied, then, and only then, all eight of the O’Donnell (2016) axioms are satisfied.

  22. 22.

    Balk (1998, p. 59) also uses the term ‘primal’ in reference to an index that is constructed using an input distance function.

  23. 23.

    The choice of one as the minimum aggregate input is arbitrary.

  24. 24.

    If the input distance function is nondecreasing in inputs, then the partial derivative of the distance function with respect to the inputs must be nonnegative for all feasible input-output combinations. If the input distance function is a translog function, then it is possible to find feasible input-output combinations such that at least one partial derivative is negative. Ergo, the input distance function cannot be a translog function.

  25. 25.

    The CNLS estimates of \(\lambda _1\) and \(\lambda _2\) are 0.2367 and 0.7633 respectively.

  26. 26.

    If some outputs or inputs are zero, then some TFPIs may be either zero or mathematically undefined and may therefore not satisfy some axioms.

  27. 27.

    Bjurek (1996) defines his index in a time-series context. In such a context, all notation pertaining to firms can be suppressed.

  28. 28.

    The output distance functions that underpin the Caves et al. (1982a) index are firm-specific functions. A firm-specific output distance function is presumably a representation of a firm-specific output set. A firm-specific output set is presumably a set containing all outputs that can be produced by a given firm using given inputs. Thus, Caves et al. (1982a) presumably have in mind that it was technically possible for a given firm to transform a given input vector into a given output vector, but, because it operates in a different production environment, it was not technically possible for another firm to do the same. In contrast, Färe et al. (1994 , Eq.  6). define an ‘output-based Malmquist productivity change index’ in a time-series context. The output distance functions that define their index are period-specific functions. Thus, Färe et al. (1994) presumably have in mind that it was technically possible to transform a given input vector into a given output vector in a given period, but, because the requisite technologies may not have been developed, it may not have been technically possible to do the same thing in an earlier period.

  29. 29.

    In O’Donnell (2012b) , an output price index is said to be proper if and only if nine axioms are satisfied. If PI1 to PI6 are satisfied, then, and only then, eight of the O’Donnell (2012b) axioms are satisfied. If the aggregator function is differentiable, then the ninth axiom is also satisfied.

  30. 30.

    Balk and Diewert (2003) define their price index in a consumer context. According to Hill (2008, p. 2) , many consumer price indices produced by statistical agencies turn out to be Lowe indices.

  31. 31.

    A proof is given in Appendix A.1 (Proposition 19).

  32. 32.

    A proof is given in Appendix A.1 (Proposition 20).

  33. 33.

    A proof is given in Appendix A.1 (Proposition 21).

  34. 34.

    Some observers put these claims down to hubris. See, for example, Samuelson and Swamy (1974, p. 575).

  35. 35.

    Many of these authors use either the DPIN software of O’Donnell (2010a) or the R package of Dakpo et al. (2016) to compute ‘Färe-Primont’ TFPI numbers. In this book, these index numbers are viewed as additive index numbers obtained using estimated representative normalised shadow prices as weights.

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    Christopher J. O’Donnell

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O’Donnell, C.J. (2018). Measures of Productivity Change. In: Productivity and Efficiency Analysis. Springer, Singapore. https://doi.org/10.1007/978-981-13-2984-5_3

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