Journal of Productivity Analysis

, Volume 49, Issue 1, pp 79–94 | Cite as

Cost Malmquist productivity index: an output-specific approach for group comparison

  • Barnabé Walheer


The cost Malmquist productivity index (CMPI) has been proposed to capture the performance change of cost minimizing Decision Making Units (DMUs). Recently, two alternative uses of the CMPI have been suggested: (1) using the CMPI to compare groups of DMUs, and (2) using the CMPI to compare DMUs for each output separately. In this paper, we propose a new CMPI that combines both procedures. The resulting methodology provides group-specific indexes for each output separately, and therefore offers the option to identify the sources of cost performance change. We also define our index when input prices are not observed and establish, in that case, a duality with a new technical productivity index, which takes the form of a Malmquist productivity index. We illustrate our new methodology with a numerical example and an application to the US electricity plant districts.


Cost Malmquist productivity index Malmquist productivity index Groups Cost efficiency Electricity 

JEL classification

C43 C61 D24 L94 



We thank the Editor Victor Podinovski, the Associate Editor, and the two anonymous referees for their valuable comments that substantially improved the paper. We also thank participants of the 2017 CEPA International Workshop on Performance Analysis in Brisbane for useful discussion.

Compliance with ethical standards

Conflict of interest

The author declares that he has no competing interests.


  1. Camanho AS, Dyson RG (2006) Data envelopment analysis and Malmquist indices for measuring group performance. J Product Anal 26:35–49CrossRefGoogle Scholar
  2. Caves DW, Christensen LR, Diewert WE (1982) The economic theory of index numbers and the measurement of input, output and productivity. Econometrica 50:1393–1414CrossRefGoogle Scholar
  3. Charnes A, Cooper WW (1962) Programming with linear fractional functionals. Nav Res Logist Q 2:429–444Google Scholar
  4. Charnes A, Cooper WW (1978) Measuring the efficiency of decision making units. Eur J Oper Res 2:429–444CrossRefGoogle Scholar
  5. Chen Y (2003) A non-radial Malmquist productivity index with an illustrative application to Chinese major industries. Int J Prod Econ 83:27–35CrossRefGoogle Scholar
  6. Chen Y, Ali AI (2003) DEA Malmquist productivity measure: new insights with an application to computer industry. Eur J Oper Res 159:239–249CrossRefGoogle Scholar
  7. Cherchye L, De Rock B, Dierynck B, Roodhooft F, Sabbe J (2013) Opening the black box of efficiency measurement: input allocation in multi-output settings. Oper Res 61:1148–1165CrossRefGoogle Scholar
  8. Cherchye L, De Rock B, Vermeulen F (2008) Cost-efficient production behavior under economies of scope: a nonparametric methodology. Oper Res 56:204–221CrossRefGoogle Scholar
  9. Cherchye L, De Rock B, Walheer B (2015) Multi-output efficiency with good and bad outputs. Eur J Oper Res 240:872–881CrossRefGoogle Scholar
  10. Cherchye L, De Rock B, Walheer B (2016) Multi-output profit efficiency and directional distance functions. Omega (West) 61(C):100–109CrossRefGoogle Scholar
  11. Cook WD, Seiford LM (2009) Data Envelopment Analysis (DEA) - Thirty years on. Eur J Oper Res 192:1–17CrossRefGoogle Scholar
  12. Cooper WW, Seiford LM, Tone K (2007) Data envelopment analysis: a comprehensive text with models, applications, references and DEA-Solver Software, 2nd edn. Springer, New YorkGoogle Scholar
  13. Cooper WW, Seiford LM, Zhu J (2004) Handbook on data envelopment analysis, 2nd edn. Springer, New YorkGoogle Scholar
  14. Debreu G (1951) The coefficient of resource utilization. Econometrica 19(3):273–292CrossRefGoogle Scholar
  15. Despic O, Despic M, Paradi J (2007) DEA-R: ratio-based comparative efficiency model, its mathematical relation to DEA and its use in applications. J Product Anal 28:33–44CrossRefGoogle Scholar
  16. Färe R, Grosskopf S (2000) Network DEA. Socioecon Plann Sci 34:35–49CrossRefGoogle Scholar
  17. Färe R, Grosskopf S, Lovell CAK (1994a) Production frontier. Cambridge University Press, CambridgeGoogle Scholar
  18. Färe R, Grosskopf S, Norris M (1994b) Productivity growth, technical progress and efficiency change in industrialized countries. Am Econ Rev 84:66–83Google Scholar
  19. Färe R, Grosskopf S, Norris M (1997) Productivity growth, technical progress and efficiency change in industrialized countries: reply. Am Econ Rev 87:1040–1043Google Scholar
  20. Färe R, Grosskopf S, Whittaker G (2007) Network DEA. In: Zhu J, Cook W (eds) Modeling data irregularities and structural complexities in data envelopment analysis, Springer, New YorkGoogle Scholar
  21. Farrell M (1957) The measurement of productive efficiency. J R Stat Soc 120:253–281Google Scholar
  22. Fried H, Lovell CAK, Schmidt S (2008) The measurement of productive efficiency and productivity change. Oxford University Press, OxfordGoogle Scholar
  23. Grosskopf S (2003) Some remarks on productivity and its decompositions. J Product Anal 20(3):459–474CrossRefGoogle Scholar
  24. Huang M-Y, Juo J-C (2015) Metafrontier cost Malmquist productivity index: an application to Taiwanese and Chinese commercial banks. J Product Anal 44:321–335CrossRefGoogle Scholar
  25. Kao C (2010) Malmquist productivity index based on common-weights DEA: the case of Taiwan forests after reorganization. Omega 38:484–491CrossRefGoogle Scholar
  26. Kao C, Hwang S-N (2014) Multi-period efficiency and Malmquist productivity index in two-stage production systems. Eur J Oper Res 232:512–521CrossRefGoogle Scholar
  27. Malmquist S (1953) Index numbers and indifference surfaces. Trab De Estat 4:209–242CrossRefGoogle Scholar
  28. Maniadakis N, Thanassoulis E (2004) A cost Malmquist productivity index. Eur J Oper Res 154:396–409CrossRefGoogle Scholar
  29. Mayer A, Zelenyuk V (2014) Aggregation of Malmquist productivity indexes allowing for reallocation of resources. Eur J Oper Res 238:774–785CrossRefGoogle Scholar
  30. O’Donnell CJ (2012) An aggregate quantity framework for measuring and decomposing productivity change. J Product Anal 38:255–272CrossRefGoogle Scholar
  31. Oh DH, Lee J-D (2010) A metafrontier approach for measuring Malmquist productivity index. Empir Econ 38:47–64CrossRefGoogle Scholar
  32. Pastor JT, Asmild M, Lovell CAK (2011) The biennial Malmquist productivity change index. Socioecon Plann Sci 45:10–15CrossRefGoogle Scholar
  33. Pastor JT, Lovell CAK (2005) A global Malmquist productivity index. Econ Lett 88:266–271CrossRefGoogle Scholar
  34. Peyrache A (2014) Hicks-Moorsteen versus Malmquist: a connection by means of a radial productivity index. J Product Anal 41(3):435–442CrossRefGoogle Scholar
  35. Portela MCAS, Thanassoulis E (2010) Malmquist indices for measuring productivity in the presence of negative data: an application to bank branches. J Bank Financ 34:1472–1483CrossRefGoogle Scholar
  36. Ray S, Delsi E (1997) Productivity growth, technical progress and efficiency change in industrialized countries: Comment. Am Econ Rev 87:1033–1039Google Scholar
  37. Salerian J, Chan C (2005) Restricting multiple-output multiple-input DEA models by disaggregating the output-input vector. J Product Anal 24:5–29CrossRefGoogle Scholar
  38. Sarkis J, Cordeiro JJ (2012) Ecological modernization in the electrical utility industry: an application of a bads-goods DEA model of ecological and technical efficiency. Eur J Oper Res 219:386–395CrossRefGoogle Scholar
  39. Shepard RW (1953) Cost and production functions. Princeton University Press, PrincetonGoogle Scholar
  40. Shepard RW (1970) Theory of cost and production functions. Princeton University Press, PrincetonGoogle Scholar
  41. Thanassoulis E, Shiraz RK, Maniadakis N (2015) A cost Malmquist productivity index capturing group performance. Eur J Oper Res 241:796–805CrossRefGoogle Scholar
  42. Tohidi G, Razavyan S, Tohidnia S (2012) A global cost Malmquist productivity index using data envelopment analysis. J Oper Res Soc 63:72–78CrossRefGoogle Scholar
  43. Tone K, Tsutsui M (2009) Network DEA: a slacks-based measure approach. Eur J Oper Res 197:243–252CrossRefGoogle Scholar
  44. Tone K, Tsutsui M (2011) Applying an efficiency measure of desirable and undesirable outputs in DEA to U.S. electric utilities approach. J Cent Cathedra 4:236–249CrossRefGoogle Scholar
  45. Tulkens H (1993) On FDH analysis: some methodological issues and applications to retail banking, courts and urban transit. J Product Anal 4:183–210CrossRefGoogle Scholar
  46. Varian HR (1984) The non-parametric approach to production analysis. Econometrica 52:579–598CrossRefGoogle Scholar
  47. Walheer B (2016a) A multi-sector nonparametric production-frontier analysis of the economic growth and the convergence of the European countries. Pac Econ Rev 21(4):498–524CrossRefGoogle Scholar
  48. Walheer B (2016b) Growth and Convergence of the OECD countries: a multi-sector production-frontier approach. Eur J Oper Res 252:665–675CrossRefGoogle Scholar
  49. Walheer B (2017) Disaggregation of the Cost Malmquist Productivity Index with joint and output-specific inputs. Omega, accepted 75:1–12Google Scholar
  50. Wang Y-M, Lan Y-X (2011) Measuring Malmquist productivity index: a new approach based on double frontiers data envelopment analysis. Math Comput Model 54:2760–2771CrossRefGoogle Scholar
  51. Yang B, Youliang Z, Zhang H, Zhang R, Xu B (2016) Factor-specific Malmquist productivity index based on common weights DEA. Operation Res, forthcoming. 16(1):51–70Google Scholar
  52. Yang YL, Huang CJ (2009) Estimating the Malmquist productivity index in the Taiwanese banking industry: a production and cost approach. Taiwan Econ Rev 37:353–378Google Scholar
  53. Yu MM (2007) The capacity productivity change and the variable input productivity change: a new decomposition of the Malmquist productivity index. Appl Math Comput 185:375–381Google Scholar
  54. Zelenyuk V (2006) Aggregation of Malmquist productivity indexes. Eur J Oper Res 174:1076–1086CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.International Business School SuzhouXi’An Jiaotong-Liverpool UniversitySuzhouChina

Personalised recommendations