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Aggregation of Efficiency and Productivity: From Firm to Sector and Higher Levels

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Handbook of Production Economics

Abstract

Here we consider various cases where researchers are interested in measuring aggregate efficiency or productivity levels or their changes for a group of decision-making units. These could be an entire industry composed of individual firms, banks, and hospitals or a region composed of sub-regions or countries, or particular sub-groups of these units within a group, e.g., sub-groups of public vs. private or regulated vs. non-regulated firms, banks, or hospitals within the same industry, etc. Such analysis requires solutions to the aggregation problem – some theoretically justified approaches that can connect individual measures to aggregate measures. Various solutions are offered in the literature, and our goal is to try to coherently summarize at least some of them in this chapter. This material should be interesting not only for theorists but also (and perhaps more so) for applied researchers, as it provides exact formulas and intuitive explanations for various measures of group efficiency, group scale elasticity, and group productivity indexes and refers to original papers for more details.

The author acknowledges support of the University of Queensland and from the ARC grants (ARC FT170100401).

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Notes

  1. 1.

    A different area of the aggregation questions that focuses on the aggregation of inputs or aggregation of outputs for a firm (e.g., to reduce the dimension of the model) is not considered here and can be found in [51, 11, 6, 53] and the references therein. We also do not consider the question of aggregation of indexes with respect to different references (e.g., time periods) for the same firm, which can be found in [16] and the references therein.

  2. 2.

    This chapter is a substantially revised, extended, and elaborated material that I presented earlier, in Chapter 5 of [47].

  3. 3.

    While this is a generic example, a reader might have realized that many industries in the real world have a similar composition, often resembling the so-called Pareto principle, more casually known as “the 80/20 rule” postulating that about 80% share (e.g., of wealth, sales, etc.) is taken up by about 20% of members of a group.

  4. 4.

    Lower efficiency of large firms is not unusual and often was reported in the literature. It can arise, for example, due to the greater complexity of being a larger organization involving greater levels of hierarchy and thus implying potentially greater principal-agent problems or requiring more inputs or higher costs than needed for producing the same level and the same quality of output.

  5. 5.

    Indeed, later in this chapter, we will see that output shares are more coherent with output orientation, while for the input orientation it would be more natural to use the cost shares.

  6. 6.

    This is not entirely surprising, e.g., recall that very strong assumptions are needed to establish positive aggregation results in consumer theory.

  7. 7.

    In the discussion of economic theoretical foundation here we mainly use framework developed by [45, 46] and further refined in many works and concisely outlined in [10] and [47].

  8. 8.

    For theoretical results we do not require convexity of Ψk, although when implementing in practice one may impose it when choosing a particular estimator or particular functional form for technology.

  9. 9.

    Note that for the aggregation results, a necessary assumption is the so-called Law of One Price, i.e., here it implies that all firms face the same output prices.

  10. 10.

    We use ⊕ to distinguish the summation of sets (also called “Minkowski summation”) from the standard summation; e.g., see [37].

  11. 11.

    For example, see [26, 44], and a more recent work of [37], as well as references therein. For other examples involving Minkowski summation in economics, see Shapley–Folkman-Starr theorem and related results [50].

  12. 12.

    In the input-oriented context, such a benchmark will be the cost function, while in the framework where both input and output vectors can be changed when measuring efficiency (e.g., for efficiency based on the directional distance function or hyperbolic measures), the natural benchmark will be the profit function. We will briefly discuss these cases later in the chapter.

  13. 13.

    To be precise, [30] used a similar framework, yet without explicit relationship to the maximal revenue defined on the sum of the output sets and without noticing the theoretical link via the analogue of [25] theorem, and focusing on the DEA framework.

  14. 14.

    Also see [31, 33] for extensions of this approach.

  15. 15.

    Again, here we focus on the output orientation case and similar developments can be done for the input orientation case. See [31, 33] for some of these details.

  16. 16.

    From theory, it is known that under the same weighting scheme, the geometric mean is larger than the harmonic mean but smaller than the arithmetic mean. Note however that the aggregate MPI in (65) involves products of ratios of the harmonic means and so it can be smaller or greater than the aggregate MPI obtained via a geometric mean as in (66), depending on the relative magnitudes that appear in the numerators and denominators of (65). Both means are approximately equal (to the arithmetic mean) in the sense of first order approximation around unity.

  17. 17.

    One should however be careful aggregating when there are scores equal or very close to zero: both geometric and harmonic averages completely fail if at least one element is zero and may yield an unreasonably low aggregate score if at least one element is very close to zero (even if many others have large efficiency or productivity scores), unless they are “neutralized” by a very low weight in the aggregation, as can be done with weighted aggregates. In such cases, using arithmetic aggregation, which is less sensitive to the outliers, could also be a better solution.

  18. 18.

    For example, see [39].

  19. 19.

    See [5, 10].

  20. 20.

    See [5, 58] for more details on this.

  21. 21.

    This technology aggregation structure was earlier used in [30, 1] and goes back to [25].

  22. 22.

    More recently, another definition of aggregate technology, which involved the union of technology sets, was considered by [40, 41], which later was shown to be equivalent to the Koopmans-type aggregate technology \(\Psi _{\tau }^{*}\), under standard regularity conditions of production theory (see [47]).

  23. 23.

    Here, note that we allow for different time subscripts for inputs and outputs for the framework to be compatible with the HMPI context.

  24. 24.

    Example, see [47], and the relevant chapters “Data Envelopment Analysis: A Nonparametric Method of Production Analysis,” “Stochastic Frontier Analysis: Foundations and Advances I,” and “Stochastic Frontier Analysis: Foundations and Advances II” in this Handbook.

  25. 25.

    Also see [55] for this and other related results.

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Authors and Affiliations

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

    Valentin Zelenyuk

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  1. Valentin Zelenyuk
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Correspondence to Valentin Zelenyuk .

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Editors and Affiliations

  1. Department of Economics, University of Connecticut, Storrs, CT, USA

    Subhash C. Ray

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

    Robert Chambers

  3. Department of Economics, Binghamton University, Binghamton, NY, USA

    Subal Kumbhakar

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Zelenyuk, V. (2020). Aggregation of Efficiency and Productivity: From Firm to Sector and Higher Levels. In: Ray, S., Chambers, R., Kumbhakar, S. (eds) Handbook of Production Economics. Springer, Singapore. https://doi.org/10.1007/978-981-10-3450-3_19-1

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