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Productivity Interpretations of the Farrell Efficiency Measures and the Malmquist Index and Its Decomposition

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Advances in Efficiency and Productivity

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 249))

Abstract

The ratio definition of efficiency has the form of a productivity measure. But the weights are endogenous variables and they do not function well as weights in a productivity index proper. It is shown that extended Farrell measures of efficiency can all be given an interpretation as productivity measures as observed productivity relative to productivity at the various projection points on the frontier. The Malmquist productivity index is the efficiency score for a unit in a period relative to the efficiency score in a previous period, thus based on a maximal common expansion factor for outputs or common contraction factor for inputs not involving any individual weighting of outputs or inputs, as is the case if a Törnqvist or ideal Fisher index is used. The multiplicative decomposition of the Malmquist productivity index into an efficiency part and a frontier shift part should not be taken to imply causality. The role of cone benchmark envelopments both for calculating Malmquist indices of productivity change and for decomposing the indices into an efficiency change term and a frontier shift term is underlined, and connected to the index property of proportionality and circularity, adding the use of a fixed benchmark envelopment. The extended decomposition of the efficiency component by making use of scale efficiency is criticised.

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Notes

  1. 1.

    Section 6.2 is based on Førsund (2015), Sect. 4.

  2. 2.

    Farrell and Fieldhouse (1962) were the first to solve the problem of calculating their efficiency measure by using linear programming.

  3. 3.

    Farrell (1957) points out that the two measures in the case of constant returns to sale are equal.

  4. 4.

    The Regular Ultra Passum Law requires that the scale elasticity decreases monotonically from values greater than one, through the value one to lower values when moving along a rising curve in the input space.

  5. 5.

    This may be the reason for this way of presenting the family of efficiency measures being rather unknown in the DEA literature.

  6. 6.

    In the VRS DEA specification the scale elasticity has a monotonically decreasing value in the range of increasing returns to scale, but has a more peculiar development in the range of decreasing returns to scale as shown in Førsund et al. (2009). However, there may be a unique face where the scale elasticity is equal to 1 along a rising curve‚ or else define a vertex point as having constant returns to scale when the left-hand elasticity at the point is less than one and the right-hand elasticity is greater than one.

  7. 7.

    In Førsund and Hjalmarsson (1979), introducing this measure, it was called the gross scale efficiency.

  8. 8.

    In Førsund and Hjalmarsson (1979) these measures were called measures of pure scale efficiency.

  9. 9.

    The Farrell efficiency measure functions correspond to the concept of distance functions introduced in Shephard (1970). Shephard’s input distance function is the inverse of Farrell’s input-oriented efficiency measure, and Shephard’s output distance function is identical to Farrell’s output-oriented efficiency measure.

  10. 10.

    See Hanoch (1970), Panzar and Willig (1977), Starrett (1977).

  11. 11.

    The concept of the M-locus in the case of multi-output was introduced in Baumol et al. (1982, pp. 58–59). In Førsund and Hjalmarsson (2004a) the M locus is defined and estimated within a DEA model using the TOPS set.

  12. 12.

    Following the classical axiomatic (test) approach there are a number of properties (at least 20) an index should fulfil (Diewert 1992), the ones most often mentioned are monotonicity, homogeneity, identity, commensurability and proportionality. “Satisfying these standard axioms limits the class of admissible input (output) quantity aggregator functions to non-negative functions that are non-decreasing and linearly homogeneous in inputs (outputs)” (O’Donnell 2012, p. 257). There is no time index on the functions here because our variables are from the same period.

  13. 13.

    The productivity interpretation of the oriented efficiency measures E 1 and E 2 can also be found in O’Donnell (2012, p. 259) using distance functions.

  14. 14.

    A thorough evaluation of the advantages of this type of a Malmquist productivity index is found in Lovell (2003), and it is also mentioned as the most satisfactory Malmquist type of productivity index in O’Donnell (2012), being what he called multiplicatively complete.

  15. 15.

    Lovell (2003) decomposes also the Malmquist total factor productivity index multiplicatively into five terms. However, we will not investigate this issue here.

  16. 16.

    However, no reason is given for this procedure other than claiming that this was done in Caves et al. (1982). But there the geometric mean appears when establishing the connection between the Malmquist index and an Törnquist index assuming the unit to be on the frontier, while the fundamental assumption in Färe et al. (1994a) is that units may be inefficient.

  17. 17.

    This may explain the empirical result in Bjurek et al. (1998) that productivity developments more or less follow each other for different formulations of the Malmquist index.

  18. 18.

    The weighted ratio appearing in (1) should not be interpreted as productivity; the weights are just a by-product of the solutions of the optimisation problems in (6.2).

  19. 19.

    To the best of my knowledge the pattern of occurrence of zero weights in Malmquist productivity index estimations has never been reported in the literature.

  20. 20.

    Most illustrations of the Malmquist indices in studies using geometric means are in fact using CRS frontiers and single output and input. Considering multiple outputs and inputs distances between contemporaneous frontiers will be independent of where the measure is taken if inverse homotheticity is assumed in addition to CRS, i.e. if Hicks neutral technical change is assumed.

  21. 21.

    In panel data models efficiency change has been specified (Cornwell et al. 1990) as having unit-specific efficiencies that varies over time, but this is a “mechanical” procedure without an economic explanation of efficiency change.

  22. 22.

    In Pastor and Lovell (2005), missing out on this reference, it was called the global frontier.

  23. 23.

    Nishimizu and Page (1982) were the first to refer to a working paper (Caves et al. 1981) that was published as Caves et al. (1982). However, they did not themselves use the term Malmquist productivity index.

  24. 24.

    To the best of my knowledge this term was first used in Førsund (1993), and then in Fare et al. (1994c).

  25. 25.

    The history of the DEA-based Malmquist productivity index is presented in Färe et al. (1998), Grosskopf (2003) and Färe et al. (2008). The first working paper that established an estimation procedure based on DEA was published in 1989, was presented at a conference in Austin in the same year, and appeared as Färe et al. (1994a); a book chapter in a volume containing many of the conference presentations. The first journal publication appeared as Färe et al. (1990) with an application to electricity distribution. (However, this paper is not referred to in the 2003 and 2008 reviews and neither in Färe et al. (1992), although the methodological approach in the latter is the same).

  26. 26.

    As a control, inserting the definition of E 5 we have for each period technology \( E_{3} = E_{2} \cdot E_{3} /E_{2} = E_{3} \).

  27. 27.

    Crossing of technologies and crossing of isoquants as illustrated in Førsund (1993) will be difficult to interpret using geometric means of an index of the type in (6.14a).

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Authors and Affiliations

  1. Department of Economics, University of Oslo, Oslo, Norway

    Finn R. Førsund

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  1. Finn R. Førsund
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Correspondence to Finn R. Førsund .

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Editors and Affiliations

  1. Center of Operations Research, Miguel Hernandez University of Elche, Elche, Alicante, Spain

    Juan Aparicio

  2. CEPA, School of Economics, University of Queensland, Brisbane, Queensland, Australia

    C. A. Knox Lovell

  3. Center of Operations Research, Miguel Hernandez University of Elche, Elche, Alicante, Spain

    Jesus T. Pastor

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Førsund, F.R. (2016). Productivity Interpretations of the Farrell Efficiency Measures and the Malmquist Index and Its Decomposition. In: Aparicio, J., Lovell, C., Pastor, J. (eds) Advances in Efficiency and Productivity. International Series in Operations Research & Management Science, vol 249. Springer, Cham. https://doi.org/10.1007/978-3-319-48461-7_6

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