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On structural and average technical efficiency

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Abstract

In this paper we explain the discrepancy between the efficiency of the average production unit and average efficiency by means of a covariance term relating size and technical efficiency and the extent of reallocation inefficiency. We show that, regardless of the degree of reallocation efficiency, the efficiency of the average production unit is lower than average efficiency when either size and efficiency are uncorrelated or larger units are more efficient than smaller ones. If however larger units are less efficient than smaller ones, the efficiency of the average production unit is higher than average efficiency as long as reallocation efficiency prevails. In addition, reallocation efficiency and a zero covariance term are sufficient conditions for the efficiency of the average production unit to be equal to average efficiency.

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Notes

  1. Li and Ng (1995) shown that structural and aggregate efficiency are related to one another by means of reallocation efficiency; for more details see Sect. 2 below.

  2. In this it is implicitly assumed that the reallocation process involves changes in the internal margin only (i.e., input movements across and mergers/splits of constituent firms), excluding entry and exit of firms. Another approach for aggregating across firms that follows points (a) and (b) is Johansen (1972) vintage approach; for more see Färe and Grosskopf (2004, pp. 109–115).

  3. We use Li and Cheng (2007) terminology for structural and aggregate efficiency, which follows closer Farrell’s (1957) spirit. His notion of structural efficiency clearly reflects reallocation of inputs available to the industry as well as restructuring of firms (see p. 262) while aggregate efficiency, which he called it ‘efficiency of an industry’, takes industry structure as given (see p. 261). Some authors (i.e., Bjurek et al. 1992; Ylvinger 2000; Nesterenko and Zelenyuk 2007) used the term structural efficiency to mean what we refer to as aggregate efficiency. One should keep that in mind when reading Ylvinger’s (2000) criticism on using the average unit to measure industry efficiency. On the other hand, Färe et al. (1992) estimated structural and aggregate efficiency by using what they referred respectively to as the industry and the firm linear programming models.

  4. See Nesterenko and Zelenyuk (2007), Pachkova (2009), ten Raa (2011), Fang and Li (2013) and Peyrache (2013).

  5. See for example Kittelsen and Førsund (1992) Hjalmarsson and Veiderpass (1992), Torgersen et al. (1996), Halsteinli et al. (2001), Arocena and Waddams Price (2002) and Odeck and Alkadi (2004).

  6. In a multi-output setting, allocation inefficiency does not matter in the aggregation of technical inefficiency scores as long as each firm is as allocatively inefficient as the industry (Färe and Grosskopf 2004, p. 116). Even in this case though we cannot avoid using price dependent weights.

  7. The same is true in the case of input orientation as long as production technology employs a single input and thus can be represented by an input requirement function.

  8. Throughout the paper when a quotation is used, words or phrases in brackets are our own additions.

  9. Here we follow Färe et al. (1985) tradition and we define output-oriented technical efficiency to be equal or greater than one. Notice however that Farrell (1957) defined all his efficiency measures to be between zero and one, i.e., the inverse of (1). Our main results hold for Farrell (1957) efficiency measures as well after the appropriate adjustments to aggregation weights in (5) are made by means of the denominator rule, which requires (see Färe and Karagiannis 2013) the weighting shares to be defined in terms of denominator variable of the relevant index in order to ensure consistent aggregation of ratio-type performance measures.

  10. Färe et al. (1992) used a single output, constant-returns-to-scale model and thus can aggregate firms’ output just by inserting the total amount of factors. This cannot be done without the assumption of constant returns to scale as mentioned by a referee. No such an assumption is made in (2) below or in Eq. (4) of Li and Ng (1995) that provides the efficiency measurement counterpart of Eq. (3) in (Färe et al. 1992).

  11. Notice that (2) differs from Eq. (4) in Li and Ng (1995) where there is an error in the sum of the intensity variables that should be equal to the number of constituent firms and not equal to one.

  12. That the centralized resource allocation DEA model is equivalent to Färe et al. (1992) structural efficiency model becomes apparent after the simplification introduced by Mar-Molinero et al. (2014); see in particular their Eq. (3) that is the input-oriented counterpart of (2) above.

  13. Nesterenko and Zelenyuk (2007, lemma 3) have extended this result to measures of allocative and revenue efficiency. On the other hand, Lozano and Villa (2004), Du et al. (2010) and Fang (2013) have shown a similar result for the centralized resource allocation DEA model.

  14. This proof was suggested by one of the referees to who I am indebted. Notice that (1) and (2) results in the same efficiency score but in different intensity variables, which are however related to one another.

  15. Notice that Farrell (1957) considered a single-output technology, an input-oriented measure of efficiency and implicit in his definition are the assumptions of identical firm technology and constant returns to scale. Färe and Karagiannis (2014) show that his suggestion for the aggregation weights could have been correct if he had referred to potential (instead of observed) output shares. In that case the aggregate efficiency measure would reflect potential input savings.

  16. In the case of multi-output technologies and allocative inefficiency, price dependent weights (i.e., revenue shares) should be used for consistent aggregation of technical efficiency scores.

  17. Since the following aggregation scheme uses observed output shares as weights, which is the denominator of the \( F_{O}^{k} \left( {x^{k} ,y^{k} } \right) \) index, it does not violate. monotonicity and moreover, it is consistent with an intuitive interpretation of industry output as the one that can be produced by making all firms work with average technical efficiency (Färe and Karagiannis 2013). For similar concerns on productivity measures see van Biesebroeck (2008).

  18. Bjurek et al. (1992), using the inverse of \( F_{O}^{k} \left( {x^{k} ,y^{k} } \right) \) as their efficiency index, proposed the ratio of industry total observed (actual) to potential output as a measure of aggregate technical efficiency.

  19. Li and Cheng (2007, proposition 2) shown that aggregate technical efficiency for a group or industry with constant-returns-to-scale technology is equal to Ylvinger’s (2000) “structural efficiency” measure with restricted (i.e., common to all firms) weights in aggregating potential and observed (actual) outputs.

  20. ten Raa (2011) referred to reallocation efficiency as industrial organization efficiency.

  21. Nesterenko and Zelenyuk (2007, proposition 2) decomposed reallocation efficiency into a technical and an allocative component, as has been done with other radial economic efficiency measures (i.e., revenue or cost). However their assertion that the allocative component can take values greater or less than one is incompatible with the output-oriented nature of reallocation efficiency.

  22. To prove this for the single-output case, divide both sides of Eq. (11) in Färe et al. (1992) by the observed industry output to obtain the corresponding output-oriented technical efficiency measures.

  23. Li and Cheng (2007, corollary 1) shown that the average of all individual technical efficiency scores is equal to Ylvinger’s (2000) “structural efficiency” measure with flexible (i.e., firm-specific) weights in aggregating potential and observed (actual) outputs.

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Acknowledgments

This paper was presented at the Conference in Memory of Professor Lennart Hjalmarsson, hold at Gothenburg, Sweden, Dec. 7–8, 2012. I would like to thank session participants for a constructive and stimulating discussion. Rolf Färe as always provided inspired comments and suggestions. Svante Ylvinger supplied several related materials and Finn Førsund gave useful guidelines that both help me in revising the paper. Last but not least, the paper has benefited of valuable comments from two anonymous referees of this journal.

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  1. Department of Economics, University of Macedonia, Thessaloniki, Greece

    Giannis Karagiannis

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Karagiannis, G. On structural and average technical efficiency. J Prod Anal 43, 259–267 (2015). https://doi.org/10.1007/s11123-015-0439-x

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