Skip to main content
SpringerLink
Log in

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

  • Published:
Journal of Productivity Analysis Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Initially the MPI was interpreted as a productivity change index. Recently, there has been a debate on the validity of the MPI to measure productivity change. Indeed, this is only true under quite restrictive conditions. We refer to O’Donnell (2012) and Peyrache (2014) who highlight this issue. Therefore, in the following, we interpret the MPI as a relative performance index (this is also the interpretation used by Grosskopf (2003))

  2. See, for example, for extensions: Chen (2003), Chen and Ali (2003), Pastor and Lovell (2005), Zelenyuk (2006), Yu (2007), Kao (2010), Oh and Lee (2010), Portela and Thanassoulis (2010), Pastor et al. (2011), Wang and Lan (2011), Kao and Hwang (2014), Mayer and Zelenyuk (2014), Yang et al. (2016).

  3. See, for example, for extensions: Yang and Huang (2009), Tohidi et al. (2012), and Huang and Juo (2015).

  4. Refer to Färe and Grosskopf (2000), Salerian and Chan (2005), Despic et al. (2007), Färe et al. (2007), Cherchye et al. (2008), Tone and Tsutsui (2009), Cherchye et al. (2013), Cherchye et al. (2015, 2016), and Walheer (2016a, b, 2017) for related works on the input allocation.

  5. Refer to Färe et al. (1994a), Cooper et al. (2004), Cooper et al. (2007), Fried et al. (2008), and Cook and Seiford (2009) for reviews.

  6. See Cherchye et al. (2015) for a formal definition of those axioms in a similar context. Note that imposing monotonicity and convexity does not alter the cost evaluation (see, for example, Varian (1984) and Tulkens (1993) for discussion). We impose these extra technology axioms since they are required to establish the duality of our CMPI with the technical counterpart (i.e. the MPI). Finally, note that the CMPI could be defined under alternative returns-to-scale assumptions.

  7. These output-specific prices have a similar interpretation as Lindahl prices for public goods. See Cherchye et al. (2008) for more details.

  8. An alternative would be to recover the allocation of inputs to outputs. See our discussion in Section 3 (practical implementation).

  9. For fuel, the EPA provides prices at the state level. For the nameplate capacity, we can proxy the price by transforming the electricity price (available at the state level too) since nameplate capacity is defined as the maximal electricity generated by the plants during one year.

  10. PADD I (East Coast): Connecticut, Delaware, District of Columbia, Florida, Georgia, Maine, Massachusetts, Maryland, New Hampshire, New Jersey, New York, North Carolina, Pennsylvania, Rhode Island, South Carolina, Virginia, Vermont, and West Virginia. PADD II (Midwest): Illinois, Indiana, Iowa, Kansas, Kentucky, Michigan, Minnesota, Missouri, Nebraska, North Dakota, South Dakota, Ohio, Oklahoma, Tennessee, and Wisconsin. PADD III (Gulf Coast): Alabama, Arkansas, Louisiana, Mississippi, New Mexico, and Texas. PADD IV (Rocky Mountain): Colorado, Idaho, Montana, Utah, and Wyoming. PADD V (West Coast): Alaska, Arizona, California, Hawaii, Nevada, Oregon, and Washington.

  11. Note that if data for the number of employees are available, we could allocate the employees for each electricity generation process.

References

  • Camanho AS, Dyson RG (2006) Data envelopment analysis and Malmquist indices for measuring group performance. J Product Anal 26:35–49

    Article  Google Scholar 

  • Caves DW, Christensen LR, Diewert WE (1982) The economic theory of index numbers and the measurement of input, output and productivity. Econometrica 50:1393–1414

    Article  Google Scholar 

  • Charnes A, Cooper WW (1962) Programming with linear fractional functionals. Nav Res Logist Q 2:429–444

    Google Scholar 

  • Charnes A, Cooper WW (1978) Measuring the efficiency of decision making units. Eur J Oper Res 2:429–444

    Article  Google Scholar 

  • Chen Y (2003) A non-radial Malmquist productivity index with an illustrative application to Chinese major industries. Int J Prod Econ 83:27–35

    Article  Google Scholar 

  • Chen Y, Ali AI (2003) DEA Malmquist productivity measure: new insights with an application to computer industry. Eur J Oper Res 159:239–249

    Article  Google Scholar 

  • 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 61:1148–1165

    Article  Google Scholar 

  • Cherchye L, De Rock B, Vermeulen F (2008) Cost-efficient production behavior under economies of scope: a nonparametric methodology. Oper Res 56:204–221

    Article  Google Scholar 

  • Cherchye L, De Rock B, Walheer B (2015) Multi-output efficiency with good and bad outputs. Eur J Oper Res 240:872–881

    Article  Google Scholar 

  • Cherchye L, De Rock B, Walheer B (2016) Multi-output profit efficiency and directional distance functions. Omega (West) 61(C):100–109

    Article  Google Scholar 

  • Cook WD, Seiford LM (2009) Data Envelopment Analysis (DEA) - Thirty years on. Eur J Oper Res 192:1–17

    Article  Google Scholar 

  • 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 York

  • Cooper WW, Seiford LM, Zhu J (2004) Handbook on data envelopment analysis, 2nd edn. Springer, New York

  • Debreu G (1951) The coefficient of resource utilization. Econometrica 19(3):273–292

    Article  Google Scholar 

  • 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 28:33–44

    Article  Google Scholar 

  • Färe R, Grosskopf S (2000) Network DEA. Socioecon Plann Sci 34:35–49

    Article  Google Scholar 

  • Färe R, Grosskopf S, Lovell CAK (1994a) Production frontier. Cambridge University Press, Cambridge

  • Färe R, Grosskopf S, Norris M (1994b) Productivity growth, technical progress and efficiency change in industrialized countries. Am Econ Rev 84:66–83

    Google Scholar 

  • Färe R, Grosskopf S, Norris M (1997) Productivity growth, technical progress and efficiency change in industrialized countries: reply. Am Econ Rev 87:1040–1043

    Google Scholar 

  • 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 York

  • Farrell M (1957) The measurement of productive efficiency. J R Stat Soc 120:253–281

    Google Scholar 

  • Fried H, Lovell CAK, Schmidt S (2008) The measurement of productive efficiency and productivity change. Oxford University Press, Oxford

  • Grosskopf S (2003) Some remarks on productivity and its decompositions. J Product Anal 20(3):459–474

    Article  Google Scholar 

  • Huang M-Y, Juo J-C (2015) Metafrontier cost Malmquist productivity index: an application to Taiwanese and Chinese commercial banks. J Product Anal 44:321–335

    Article  Google Scholar 

  • Kao C (2010) Malmquist productivity index based on common-weights DEA: the case of Taiwan forests after reorganization. Omega 38:484–491

    Article  Google Scholar 

  • Kao C, Hwang S-N (2014) Multi-period efficiency and Malmquist productivity index in two-stage production systems. Eur J Oper Res 232:512–521

    Article  Google Scholar 

  • Malmquist S (1953) Index numbers and indifference surfaces. Trab De Estat 4:209–242

    Article  Google Scholar 

  • Maniadakis N, Thanassoulis E (2004) A cost Malmquist productivity index. Eur J Oper Res 154:396–409

    Article  Google Scholar 

  • Mayer A, Zelenyuk V (2014) Aggregation of Malmquist productivity indexes allowing for reallocation of resources. Eur J Oper Res 238:774–785

    Article  Google Scholar 

  • O’Donnell CJ (2012) An aggregate quantity framework for measuring and decomposing productivity change. J Product Anal 38:255–272

    Article  Google Scholar 

  • Oh DH, Lee J-D (2010) A metafrontier approach for measuring Malmquist productivity index. Empir Econ 38:47–64

    Article  Google Scholar 

  • Pastor JT, Asmild M, Lovell CAK (2011) The biennial Malmquist productivity change index. Socioecon Plann Sci 45:10–15

    Article  Google Scholar 

  • Pastor JT, Lovell CAK (2005) A global Malmquist productivity index. Econ Lett 88:266–271

    Article  Google Scholar 

  • Peyrache A (2014) Hicks-Moorsteen versus Malmquist: a connection by means of a radial productivity index. J Product Anal 41(3):435–442

    Article  Google Scholar 

  • Portela MCAS, Thanassoulis E (2010) Malmquist indices for measuring productivity in the presence of negative data: an application to bank branches. J Bank Financ 34:1472–1483

    Article  Google Scholar 

  • Ray S, Delsi E (1997) Productivity growth, technical progress and efficiency change in industrialized countries: Comment. Am Econ Rev 87:1033–1039

    Google Scholar 

  • Salerian J, Chan C (2005) Restricting multiple-output multiple-input DEA models by disaggregating the output-input vector. J Product Anal 24:5–29

    Article  Google Scholar 

  • 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 219:386–395

    Article  Google Scholar 

  • Shepard RW (1953) Cost and production functions. Princeton University Press, Princeton

  • Shepard RW (1970) Theory of cost and production functions. Princeton University Press, Princeton

  • Thanassoulis E, Shiraz RK, Maniadakis N (2015) A cost Malmquist productivity index capturing group performance. Eur J Oper Res 241:796–805

    Article  Google Scholar 

  • Tohidi G, Razavyan S, Tohidnia S (2012) A global cost Malmquist productivity index using data envelopment analysis. J Oper Res Soc 63:72–78

    Article  Google Scholar 

  • Tone K, Tsutsui M (2009) Network DEA: a slacks-based measure approach. Eur J Oper Res 197:243–252

    Article  Google Scholar 

  • 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 4:236–249

    Article  Google Scholar 

  • Tulkens H (1993) On FDH analysis: some methodological issues and applications to retail banking, courts and urban transit. J Product Anal 4:183–210

    Article  Google Scholar 

  • Varian HR (1984) The non-parametric approach to production analysis. Econometrica 52:579–598

    Article  Google Scholar 

  • 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):498–524

    Article  Google Scholar 

  • Walheer B (2016b) Growth and Convergence of the OECD countries: a multi-sector production-frontier approach. Eur J Oper Res 252:665–675

    Article  Google Scholar 

  • Walheer B (2017) Disaggregation of the Cost Malmquist Productivity Index with joint and output-specific inputs. Omega, accepted 75:1–12

  • Wang Y-M, Lan Y-X (2011) Measuring Malmquist productivity index: a new approach based on double frontiers data envelopment analysis. Math Comput Model 54:2760–2771

    Article  Google Scholar 

  • 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):51–70

  • Yang YL, Huang CJ (2009) Estimating the Malmquist productivity index in the Taiwanese banking industry: a production and cost approach. Taiwan Econ Rev 37:353–378

    Google Scholar 

  • Yu MM (2007) The capacity productivity change and the variable input productivity change: a new decomposition of the Malmquist productivity index. Appl Math Comput 185:375–381

    Google Scholar 

  • Zelenyuk V (2006) Aggregation of Malmquist productivity indexes. Eur J Oper Res 174:1076–1086

    Article  Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. International Business School Suzhou, Xi’An Jiaotong-Liverpool University, Suzhou, China

    Barnabé Walheer

Authors
  1. Barnabé Walheer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Barnabé Walheer.

Ethics declarations

Conflict of interest

The author declares that he has no competing interests.

Appendices

Appendix 1

Table 5

Table 5 Illustrative example: data (source: Thanassoulis et al. (2015))
Full size table

In this Appendix, we show how the linear programs work for the illustrative example. In particular, we distinguish between two cases: when the input prices are observed and when they are not. Note that the output-specific input prices are not observed in this example.

1.1 Practical implementation when the input prices are observed

Step 1: Solve (LP-2) for each member of group A taking group A as the reference group for the technology.

For example for DMU 1 in group A, it is given as:

$$\begin{array}{ccccc} CE_1^{A,A} = & \begin{array}{*{20}{c}} {\max } \\ {C_1^{A,1},C_1^{A,2} \in {\Bbb R}_ + } \\ {{\mathbf{w}}_1^{A,1},{\mathbf{w}}_1^{A,2} \in {\Bbb R}_ + ^2} \end{array}\frac{{C_1^{A,1} + C_1^{A,2}}}{{28}} \\ \\ & ({\bf C\mbox -1}):C_1^{A,1} \le {\mathbf{w}}_1^{A,1{\prime}}{\mathbf{x}}_s^{A,1}\,{\mathrm{for}}\,{\mathrm{all}}\,s:y_s^{A,1} \ge 8,\\ \\ & ({\bf C\mbox -2}):C_2^{A,2} \le {\mathbf{w}}_1^{A,2{\prime}}{\mathbf{x}}_s^{A,2}\,{\mathrm{for}}\,{\mathrm{all}}\,s:y_s^{A,2} \ge 11,\\ \\ & ({\bf C\mbox -3}):({\mathbf{w}}_1^{A,1})_1 + ({\mathbf{w}}_1^{A,2})_1 = 3,\qquad\quad \qquad \\ \\ & ({\bf C\mbox -4}):({\mathbf{w}}_1^{A,1})_2 + ({\mathbf{w}}_1^{A,2})_2 = 2.\qquad \qquad \quad \\ \end{array}$$

Note that the denominator is the actual cost of DMU 1 in group A: \({\mathbf{w}}_1^{A{\prime}}{\mathbf{x}}_1^A\) = 6*3 + 5*2 = 18 + 10 = 28.

Step 2: Solve (LP-2) for each member of group A taking group B as the reference group for the technology.

For example for DMU 1 in group A, it is given as:

$$\begin{array}{ccccc} CE_1^{B,A} = & \begin{array}{*{20}{c}} {\max } \\ {C_1^{B,1},C_1^{B,2} \in {\Bbb R}_ + } \\ {{\mathbf{w}}_1^{A,1},{\mathbf{w}}_1^{A,2} \in {\Bbb R}_ + ^2} \end{array}\frac{{C_1^{B,1} + C_1^{B,2}}}{{28}} \\ \\ & ({\bf C\mbox -1}):C_1^{B,1} \le {\mathbf{w}}_1^{A,1{\prime}}{\mathbf{x}}_s^{B,1}\,{\mathrm{for}}\,{\mathrm{all}}\,s:y_s^{B,1} \ge 8,\\ \\ & ({\bf C\mbox -2}):C_2^{B,2} \le {\mathbf{w}}_1^{A,2{\prime}}{\mathbf{x}}_s^{B,2}\,{\mathrm{for}}\,{\mathrm{all}}\,s:y_s^{B,2} \ge 11,\\ \\ & ({\bf C\mbox -3}):({\mathbf{w}}_1^{A,1})_1 + ({\mathbf{w}}_1^{A,2})_1 = 3,\qquad\quad\qquad\\ \\ & ({\bf C\mbox -4}):({\mathbf{w}}_1^{A,1})_2 + ({\mathbf{w}}_1^{A,2})_2 = 2.\qquad\qquad\quad\\ \end{array}$$

Step 3: Solve(LP-2) for each member of group B taking group B as the reference group for the technology.

For example for DMU 1 in group B, it is given as:

$$\begin{array}{ccccc}\\ CE_1^{B,B} = & \begin{array}{*{20}{c}} {\max } \\ {C_1^{B,1},C_1^{B,2} \in {\Bbb R}_ + } \\ {{\mathbf{w}}_1^{B,1},{\mathbf{w}}_1^{B,2} \in {\Bbb R}_ + ^2} \end{array}\frac{{C_1^{B,1} + C_1^{B,2}}}{{29.5}} \\ \\ & ({\bf C\mbox -1}):C_1^{B,1} \le {\mathbf{w}}_1^{B,1{\prime}}{\mathbf{x}}_s^{B,1}\,{\mathrm{for}}\,{\mathrm{all}}\,s:y_s^{B,1} \ge 4, \\ \\ & ({\bf C\mbox -2}):C_2^{B,2} \le {\mathbf{w}}_1^{B,2{\prime}}{\mathbf{x}}_s^{B,2}\,{\mathrm{for}}\,{\mathrm{all}}\,s:y_s^{B,2} \ge 7,\\ \\ & ({\bf C\mbox -3}):({\mathbf{w}}_1^{B,1})_1 + ({\mathbf{w}}_1^{B,2})_1 = 4, \\ \\ & ({\bf C\mbox -4}):({\mathbf{w}}_1^{B,1})_2 + ({\mathbf{w}}_1^{B,2})_2 = 4.5. \\ \end{array}$$

Note that the denominator is the actual cost of DMU 1 in group B: \({\mathbf{w}}_1^{B{\prime}}{\mathbf{x}}_1^B\) = 4*4 + 3*4.5 = 16 + 13.5 = 29.5.

Step 4: Solve (LP-2) for each member of group B taking group A as the reference group for the technology.

For example for DMU 1 in group B, it is given as:

$$\begin{array}{ccccc}\\ CE_1^{A,B} = & \begin{array}{*{20}{c}} {\max } \\ {C_1^{A,1},C_1^{A,2} \in {\Bbb R}_ + } \\ {{\mathbf{w}}_1^{B,1},{\mathbf{w}}_1^{B,2} \in {\Bbb R}_ + ^2} \end{array}\frac{{C_1^{A,1} + C_1^{A,2}}}{{29.5}} \\ \\ & ({\bf C\mbox -1}):C_1^{A,1} \le {\mathbf{w}}_1^{B,1{\prime}}{\mathbf{x}}_s^{A,1}\ {\mathrm{for}}\,{\mathrm{all}}\,s:y_s^{A,1} \ge 4,\\ \\ & ({\bf C\mbox -2}):C_2^{A,2} \le {\mathbf{w}}_1^{B,2{\prime}}{\mathbf{x}}_s^{A,2}\ {\mathrm{for}}\,{\mathrm{all}}\,s:y_s^{A,2} \ge 7,\\ \\ & ({\bf C\mbox -3}):({\mathbf{w}}_1^{B,1})_1 + ({\mathbf{w}}_1^{B,2})_1 = 4, \\ \\ & ({\bf C\mbox -4}):({\mathbf{w}}_1^{B,1})_2 + ({\mathbf{w}}_1^{B,2})_2 = 4.5. \\ \end{array}$$

Step 5: Compute CMPI for the overall output and CMPI 1 and CMPI 2 for the two individual outputs by plugging-in the computed cost efficiency scores in (25), (26), and (27) for CMPI, and in (18), (19), and (20) for CMPI 1 and CMPI 2. As a final remark, note that tout output-specific minimal costs are also obtained when solving the linear programs, as explained in detail in Section 3.

1.2 Practical implementation when the input prices are not observed

Step 1: Solve (LP-4) for each member of group A taking group A as the reference group for the technology.

For example for DMU 1 in group A, it is given as:

$$\begin{array}{ccccc}\\ CE_1^{A,A} = & \begin{array}{*{20}{c}} \quad\quad{min} \ \qquad\theta _1^A\\ {\lambda _s^{A,1},\lambda _s^{A,2} \in {\Bbb R}_ + } \\ {\theta _1^A \in {\Bbb R}_ + } \end{array} \\ \\ & ({\bf C\mbox -1}):\mathop {\sum}\limits_{s = 1}^6 \lambda _s^{A,1}{\mathbf{x}}_s^{A,1} \le \theta _1^A\left[ {\begin{array}{*{20}{c}} 6 \\ 5 \end{array}} \right]\,{\mathrm{for}}\,{\mathrm{all}}\,s:y_s^{A,1} \ge 8,\\ \\ & ({\bf C\mbox -2}):\mathop {\sum}\limits_{s = 1}^6 \lambda _s^{A,2}{\mathbf{x}}_s^{A,2} \le \theta _1^A\left[ {\begin{array}{*{20}{c}} 6 \\ 5 \end{array}} \right]\,{\mathrm{for}}\,{\mathrm{all}}\,s:y_s^{A,2} \ge 11, \\ \\ & ({\bf C\mbox -3}):\mathop {\sum}\limits_{s = 1}^6 \lambda _s^{A,1} = 1\,{\mathrm{for}}\,{\mathrm{all}}\,s:y_s^{A,1} \ge 8, \\ \\ & ({\bf C\mbox -4}):\mathop {\sum}\limits_{s = 1}^6 \lambda _s^{A,2} = 1\,{\mathrm{for}}\,{\mathrm{all}}\,s:y_s^{A,2} \ge 11. \\ \end{array}$$

Step 2: Solve (LP-4) for each member of group A taking group B as the reference group for the technology.

For example for DMU 1 in group A, it is given as:

$$\begin{array}{ccccc}\\ CE_1^{B,A} = & \begin{array}{*{20}{c}} \quad\quad{min} \ \qquad\theta _1^A\\ {\lambda _s^{B,1},\lambda _s^{B,2} \in {\Bbb R}_ + } \\ {\theta _1^A \in {\Bbb R}_ + } \end{array} \\ \\ & ({\bf C\mbox -1}):\mathop {\sum}\limits_{s = 1}^6 \lambda _s^{B,1}{\mathbf{x}}_s^{B,1} \le \theta _1^A\left[ {\begin{array}{*{20}{c}} 6 \\ 5 \end{array}} \right]\,{\mathrm{for}}\,{\mathrm{all}}\,s:y_s^{B,1} \ge 8,\\ \\ & ({\bf C\mbox -2}):\mathop {\sum}\limits_{s = 1}^6 \lambda _s^{B,2}{\mathbf{x}}_s^{B,2} \le \theta _1^A\left[ {\begin{array}{*{20}{c}} 6 \\ 5 \end{array}} \right]\,{\mathrm{for}}\,{\mathrm{all}}\,s:y_s^{B,2} \ge 11,\\ \\ & ({\bf C\mbox -3}):\mathop {\sum}\limits_{s = 1}^6 \lambda _s^{B,1} = 1\,{\mathrm{for}}\,{\mathrm{all}}\,s:y_s^{B,1} \ge 8, \\ \\ & ({\bf C\mbox -4}):\mathop {\sum}\limits_{s = 1}^6 \lambda _s^{B,2} = 1\,{\mathrm{for}}\,{\mathrm{all}}\,s:y_s^{B,2} \ge 11. \\ \end{array}$$

Step 3: Solve (LP-4) for each members of group B taking group B as the reference group for the technology.

For example for DMU 1 in group B, it is given as:

$$\begin{array}{ccccc}\\ CE_1^{B,B} = & \begin{array}{*{20}{c}} \quad\quad{min} \ \qquad\theta _1^B\\ {\lambda _s^{B,1},\lambda _s^{B,2} \in {\Bbb R}_ + } \\ {\theta _1^B \in {\Bbb R}_ + } \end{array} \\ \\ & ({\bf C\mbox -1}):\mathop {\sum}\limits_{s = 1}^6 \lambda _s^{B,1}{\mathbf{x}}_s^{B,1} \le \theta _1^B\left[ {\begin{array}{*{20}{c}} 4 \\ 3 \end{array}} \right]\,{\mathrm{for}}\,{\mathrm{all}}\,s:y_s^{B,1} \ge 4,\\ \\ & ({\bf C\mbox -2}):\mathop {\sum}\limits_{s = 1}^6 \lambda _s^{B,2}{\mathbf{x}}_s^{B,2} \le \theta _1^B\left[ {\begin{array}{*{20}{c}} 4 \\ 3 \end{array}} \right]\,{\mathrm{for}}\,{\mathrm{all}}\,s:y_s^{B,2} \ge 7, \\ \\ & ({\bf C\mbox -3}):\mathop {\sum}\limits_{s = 1}^6 \lambda _s^{B,1} = 1\,{\mathrm{for}}\,{\mathrm{all}}\,s:y_s^{B,1} \ge 4, \\ \\ & ({\bf C\mbox -4}):\mathop {\sum}\limits_{s = 1}^6 \lambda _s^{B,2} = 1\,{\mathrm{for}}\,{\mathrm{all}}\,s:y_s^{B,2} \ge 7. \\ \end{array}$$

Step 4: Solve (LP-4) for each member of group B taking group A as the reference group for the technology.

For example for DMU 1 in group B, it is given as:

$$\begin{array}{ccccc}\\ CE_1^{A,B} = & \begin{array}{*{20}{c}} {\min } \quad \quad \ \quad \theta _t^B \\ {\lambda _s^{A,1},\lambda _s^{A,2} \in {\Bbb R}_ + } \\ {\theta _1^B \in {\Bbb R}_ + } \end{array}\\ \\ & ({\bf C\mbox -1}):\mathop {\sum}\limits_{s = 1}^6 \lambda _s^{A,1}{\mathbf{x}}_s^{A,1} \le \theta _1^B\left[ {\begin{array}{*{20}{c}} 4 \\ 3 \end{array}} \right]\,{\mathrm{for}}\,{\mathrm{all}}\,s:y_s^{A,1} \ge 4, \\ \\ & ({\bf C\mbox -2}):\mathop {\sum}\limits_{s = 1}^6 \lambda _s^{A,2}{\mathbf{x}}_s^{A,2} \le \theta _1^B\left[ {\begin{array}{*{20}{c}} 4 \\ 3 \end{array}} \right]\,{\mathrm{for}}\,{\mathrm{all}}\,s:y_s^{A,2} \ge 7, \\ \\ & ({\bf C\mbox -3}):\mathop {\sum}\limits_{s = 1}^6 \lambda _s^{A,1} = 1\,{\mathrm{for}}\,{\mathrm{all}}\,s:y_s^{A,1} \ge 4, \\ \\ & ({\bf C\mbox -4}):\mathop {\sum}\limits_{s = 1}^6 \lambda _s^{A,2} = 1\,{\mathrm{for}}\,{\mathrm{all}}\,s:y_s^{A,2} \ge 7. \\ \end{array}$$

Step 5: Compute \(\widehat {CMPI}\) for the overall output and \(\widehat {CMPI}^1\) and \(\widehat {CMPI}^2\) for the two individual outputs by plugging-in the computed cost efficiency scores in (32) and (33) or (34). Note that similar results could be obtained using (LP-3). We provide here the technical formulation in an illustrative purpose.

Appendix 2

Tables 6–

Table 6 Multi-output plants
Full size table

7

Table 7 Shares with respect to all plants
Full size table

8

Table 8 Shares with respect to multi-output plants
Full size table

9

Table 9 Averages at the country and district level
Full size table

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Walheer, B. Cost Malmquist productivity index: an output-specific approach for group comparison. J Prod Anal 49, 79–94 (2018). https://doi.org/10.1007/s11123-017-0523-5

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11123-017-0523-5

Keywords

JEL classification

Search

Navigation