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On Mr. Farrell's Decomposition and Aggregation

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  • Fare, Rolf
  • Zelenyuk, Valentin

Abstract

In this paper we show that in order to aggregate individual efficiency scores into a group (e.g., industry) efficiency score, in such a way that the multiplicative structure of further decompositions is preserved with equal weights across components, the weighted geometric mean is required. We also show how the weights can be chosen using a variation of a theorem by Koopmans (1957). In the end, our paper provides a mathematically consistent and economic-theory justified way of aggregation of Farrell-type efficiency scores.

Suggested Citation

  • Fare, Rolf & Zelenyuk, Valentin, 2002. "On Mr. Farrell's Decomposition and Aggregation," MPRA Paper 28005, University Library of Munich, Germany, revised 23 May 2005.
    • Handle: RePEc:pra:mprapa:28005
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    References listed on IDEAS

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    1. Zelenyuk, Valentin, 2015. "Aggregation of scale efficiency," European Journal of Operational Research, Elsevier, vol. 240(1), pages 269-277.
    2. Fare, Rolf & Zelenyuk, Valentin, 2003. "On aggregate Farrell efficiencies," European Journal of Operational Research, Elsevier, vol. 146(3), pages 615-620, May.
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    Cited by:

    1. Walheer, Barnabé, 2018. "Disaggregation of the cost Malmquist productivity index with joint and output-specific inputs," Omega, Elsevier, vol. 75(C), pages 1-12.
    2. Rolf Färe & Valentin Zelenyuk, 2021. "On aggregation of multi-factor productivity indexes," Journal of Productivity Analysis, Springer, vol. 55(2), pages 107-133, April.
    3. Valentin Zelenyuk, 2022. "Aggregation of Efficiency and Productivity: From Firm to Sector and Higher Levels," Springer Books, in: Subhash C. Ray & Robert G. Chambers & Subal C. Kumbhakar (ed.), Handbook of Production Economics, chapter 25, pages 1039-1079, Springer.
    4. Walheer, Barnabé, 2018. "Aggregation of metafrontier technology gap ratios: the case of European sectors in 1995–2015," European Journal of Operational Research, Elsevier, vol. 269(3), pages 1013-1026.
    5. Tsionas, Mike & Parmeter, Christopher F. & Zelenyuk, Valentin, 2023. "Bayesian Artificial Neural Networks for frontier efficiency analysis," Journal of Econometrics, Elsevier, vol. 236(2).
    6. Valentin Zelenyuk, 2018. "Some Mathematical and Historical Clarifications on Aggregation in Efficiency and Productivity Analysis and Connection to Economic Theory," CEPA Working Papers Series WP032018, School of Economics, University of Queensland, Australia.
    7. Rolf Färe & Giannis Karagiannis, 2024. "The denominator rule with unit ratio difference," Journal of Productivity Analysis, Springer, vol. 61(2), pages 183-190, April.
    8. Michael L. Polemis & Thanasis Stengos & Nickolaos G. Tzeremes, 2020. "Modeling the effect of competition on US manufacturing sectors’ efficiency: an order-m frontier analysis," Journal of Productivity Analysis, Springer, vol. 54(1), pages 27-41, August.
    9. Färe, Rolf & Karagiannis, Giannis, 2020. "On the denominator rule and a theorem by Janos Aczél," European Journal of Operational Research, Elsevier, vol. 282(1), pages 398-400.
    10. Barnabé Walheer, 2019. "Disaggregation for efficiency analysis," Journal of Productivity Analysis, Springer, vol. 51(2), pages 137-151, June.
    11. Walheer, Barnabé, 2019. "Aggregating Farrell efficiencies with private and public inputs," European Journal of Operational Research, Elsevier, vol. 276(3), pages 1170-1177.
    12. Ali Homayoni & Reza Fallahnejad & Farhad Hosseinzadeh Lotfi, 2022. "Cross Malmquist Productivity Index in Data Envelopment Analysis," 4OR, Springer, vol. 20(4), pages 567-602, December.
    13. Andreas Mayer & Valentin Zelenyuk, 2018. "Aggregation of Individual Efficiency Measures and Productivity Indices," CEPA Working Papers Series WP012018, School of Economics, University of Queensland, Australia.
    14. Rolf Färe & Valentin Zelenyuk, 2019. "On Paasche and Laspeyers Malmquist Indexes: A Comment on Diewert and Fox (2017)," CEPA Working Papers Series WP042019, School of Economics, University of Queensland, Australia.
    15. Colamartino, Chiara & Dipierro, Anna Rita & Toma, Pierluigi & Frittelli, Massimo, 2023. "What lies behind the success of Italian GIs products? Questioning tradition in consortia via aggregated conditional efficiency," Socio-Economic Planning Sciences, Elsevier, vol. 87(PB).
    16. Walheer, Barnabé, 2018. "Economic growth and greenhouse gases in Europe: A non-radial multi-sector nonparametric production-frontier analysis," Energy Economics, Elsevier, vol. 74(C), pages 51-62.
    17. Rolf Färe & Valentin Zelenyuk, 2019. "On the Aggregation of the Paasche and Laspeyres Malmquist Indexes," CEPA Working Papers Series WP122019, School of Economics, University of Queensland, Australia.
    18. Rolf Färe & Giannis Karagiannis, 2022. "Aggregation and decomposition of Farrell efficiencies," Operational Research, Springer, vol. 22(5), pages 5675-5683, November.
    19. Valentin Zelenyuk, 2021. "Performance Analysis: Economic Foundations & Trends," CEPA Working Papers Series WP162021, School of Economics, University of Queensland, Australia.
    20. Rolf Färe & Valentin Zelenyuk, 2020. "POn Fisher Aggregation of Multi-factor Productivity Indexes," CEPA Working Papers Series WP082020, School of Economics, University of Queensland, Australia.
    21. Barnabé Walheer, 2019. "Scale, congestion, and technical efficiency of European countries: a sector-based nonparametric approach," Empirical Economics, Springer, vol. 56(6), pages 2025-2078, June.
    22. Karagiannis, Roxani & Karagiannis, Giannis, 2020. "Constructing composite indicators with Shannon entropy: The case of Human Development Index," Socio-Economic Planning Sciences, Elsevier, vol. 70(C).

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    More about this item

    Keywords

    Farrell efficiency; Index Aggregation;

    JEL classification:

    • D24 - Microeconomics - - Production and Organizations - - - Production; Cost; Capital; Capital, Total Factor, and Multifactor Productivity; Capacity

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