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Global Malmquist and cost Malmquist indexes for group comparison

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Abstract

The Malmquist index (MI) has demonstrated its usefulness in comparing the performances of Decision Making Units (DMUs) performances. The global MI (GMI) has been suggested as a means to overcome three drawbacks of the MI: non-circularity, disparate measurements, and infeasibility. Recently, it has appeared that the MI can also be used to compare groups of DMUs. While this new function of the index has also increased its usefulness, it presents the same drawbacks as the MI. In this paper, we define the global counterpart of the MI for group contexts. We also consider the case where DMUs have an economic optimization behavior by proposing a global cost MI (GCMI). The GCMI requires the observation of the input prices. As it may represent a strong assumption, we propose solutions. These two novel indexes equip the practitioners with a new toolkit. We illustrate the usefulness of our new indexes with the Chinese energy sector.

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Notes

  1. We notice that some authors, such as O’Donnell (2012) and Peyrache (2014), have questioned the purpose of the MI; namely whether it really measures productivity performance change and under what conditions. This debate is beyond the scope of this paper.

  2. See, for example, for extensions: Chen (2003), Chen and Ali (2003), Zelenyuk (2006), Yu (2007), Kao (2010), Portela and Thanassoulis (2010), Wang and Lan (2011), Kao and Hwang (2014), Mayer and Zelenyuk (2014), Fuentes and Lillo-Banuls (2016), Asmild et al. (2017), Kao (2017), and Kevork et al. (2017).

  3. DEA, after Charnes et al. (1978), is an approach to productive efficiency measurement. DEA is intrinsically nonparametric, which means that it does not require a parametric/functional specification of the production technology. Typically, a DMU’s efficiency can be computed by simple linear programs. Refer to Cooper et al. (2004), Cooper et al. (2007), Fried et al. (2008), and Cook and Seiford (2009) for reviews. See Section 3.5 for the DEA-based linear programs in the group context.

  4. See, for example, for extensions: Yang and Huang (2009), Huang and Juo (2015), Walheer (2018b), and Zhu et al. (2017).

  5. The technique has been extended in two directions; Aparicio et al. (2017) for unbalanced panels, and Walheer (2018a) for multi-output DMUs.

  6. This Section has been added on the request of an anonymous referee. We thank the referee for challenging us.

  7. Note that the problem become even more complex when the numbers of DMUs change over time. This is why most of the empirical works using the MI use a balanced panel dataset.

  8. See, for example, for extensions and applications: Oh (2010), Oh and Lee (2010), Pastor et al. (2011), Wang et al. (2012), Afsharian and Ahn (2015), and Oh and Lee (2017). The GMI is named global since it is based on a global technology; in our context, it is the technology that envelops all group-specific technology sets (see (12)).

  9. See, for example, for extensions, Tohidi and Razavyan (2013), Huang and Juo (2015), and Cho and Wang (2017).

  10. See, for example, Walheer (2018c) for more discussion on how to model energy firms.

  11. Note that if we do not use any bounds for the prices, we obtain the same results for GCMI and GMI. That is, in this case, GMI is interpreted as a shadow GCMI. It implies also that GAMI = 1 for all groups when relying on the best input prices (see Section 3.6 for more details).

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Acknowledgements

We thank the Editor Victor Podinovski and the two anonymous referees for their valuable comments that substantially improved the paper.

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  1. HEC Liège, Université de Liège, Liège, Belgium

    Barnabé Walheer

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  1. Barnabé Walheer
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Appendices

Appendix A

Table 10

Table 10 Output and inputs: descriptive statistics
Full size table

Appendix B

Proof of Eq. (19):

$$\frac{{GMI}}{{MI}} = \left[ {\frac{{GMI}}{{MI^A}}\frac{{GMI}}{{MI^B}}} \right]^{1/2},$$
$$\left[ {\frac{{\left( {\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,GD_t} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,GD_t} \right]^{1/n_A}}}} \right)^{ - 1}}}{{\left( {\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,D_t^{A,B}} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,D_t^{A,A}} \right]^{1/n_A}}}} \right)^{ - 1}}}\frac{{\left( {\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,GD_t} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,GD_t} \right]^{1/n_A}}}} \right)^{ - 1}}}{{\left( {\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,D_t^{B,B}} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,D_t^{B,A}} \right]^{1/n_A}}}} \right)^{ - 1}}}} \right]^{1/2},$$
$$= \left[ {\frac{{\left( {\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,GD_t} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,D_t^{A,B}} \right]^{1/n_B}}}} \right)^{ - 1}}}{{\left( {\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,GD_t} \right]^{1/n_A}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,D_t^{A,A}} \right]^{1/n_A}}}} \right)^{ - 1}}}\frac{{\left( {\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,GD_t} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,D_t^{B,B}} \right]^{1/n_B}}}} \right)^{ - 1}}}{{\left( {\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,GD_t} \right]^{1/n_A}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,D_t^{B,A}} \right]^{1/n_A}}}} \right)^{ - 1}}}} \right]^{1/2},$$
$$= \left[ {\frac{{BPG^{A,B}}}{{BPG^{A,A}}} \times \frac{{BPG^{B,B}}}{{BPG^{B,A}}}} \right]^{1/2}.$$

Proof of Eq. (20):

$$\frac{{GCMI}}{{CMI}} = \left[ {\frac{{GCMI}}{{CMI^A}}\frac{{GCMI}}{{CMI^B}}} \right]^{1/2},$$
$$= \left[ {\frac{{\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,GCE_t} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,GCE_t} \right]^{1/n_A}}}}}{{\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,CE_t^{A,B}} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,CE_t^{A,A}} \right]^{1/n_A}}}}}\frac{{\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,GCE_t} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,GCE_t} \right]^{1/n_A}}}}}{{\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,CE_t^{B,B}} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,CE_t^{B,A}} \right]^{1/n_A}}}}}} \right]^{1/2},$$
$$= \left[ {\frac{{\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,GCE_t} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,CE_t^{A,B}} \right]^{1/n_B}}}}}{{\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,GCE_t} \right]^{1/n_A}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,CE_t^{A,A}} \right]^{1/n_A}}}}}\frac{{\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,GCE_t} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,CE_t^{B,B}} \right]^{1/n_B}}}}}{{\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,GCE_t} \right]^{1/n_A}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,CE_t^{B,A}} \right]^{1/n_A}}}}}} \right]^{1/2},$$
$$= \left[ {\frac{{BPCG^{A,B}}}{{BPCG^{A,A}}} \times \frac{{BPCG^{B,B}}}{{BPCG^{B,A}}}} \right]^{1/2}.$$

Proof of Eq. (21):

$$GMI = \left( {\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,GD_t} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,GD_t} \right]^{1/n_A}}}} \right)^{ - 1},$$
$$= \left( {\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,GD_t} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,GD_t} \right]^{1/n_A}}}\, \times \,\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,D_t^{A,A}} \right]^{1/n_A}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,D_t^{A,A}} \right]^{1/n_A}}}\, \times \,\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,D_t^{B,B}} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,D_t^{B,B}} \right]^{1/n_B}}}} \right)^{ - 1},$$
$$= \left( {\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,D_t^{B,B}} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,D_t^{A,A}} \right]^{1/n_A}}}} \right)^{ - 1}\, \times \,\left( {\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,D_t^{A,A}} \right]^{1/n_A}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,GD_t} \right]^{1/n_A}}}\, \times \,\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,GD_t} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,D_t^{B,B}} \right]^{1/n_B}}}} \right)^{ - 1},$$
$$= \left( {\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,D_t^{B,B}} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,D_t^{A,A}} \right]^{1/n_A}}}} \right)^{ - 1} \times \,\frac{{\left( {\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,GD_t} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,D_t^{B,B}} \right]^{1/n_B}}}} \right)^{ - 1}}}{{\left( {\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,GD_t} \right]^{1/n_A}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,D_t^{A,A}} \right]^{1/n_A}}}} \right)^{ - 1}}},$$
$$= \left( {\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,D_t^{B,B}} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,D_t^{A,A}} \right]^{1/n_A}}}} \right)^{ - 1} \times \,\frac{{BPG^{B,B}}}{{BPG^{A,A}}},$$
$$= TED \times BPD.$$

Proof of Eq. (22):

$$GCMI = \frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,GCE_t} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,GCE_t} \right]^{1/n_A}}},$$
$$= \frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,GCE_t} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,GCE_t} \right]^{1/n_A}}} \times \frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,CE_t^{A,A}} \right]^{1/n_A}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,CE_t^{A,A}} \right]^{1/n_A}}} \times \frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,CE_t^{B,B}} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,CE_t^{B,B}} \right]^{1/n_B}}},$$
$$= \frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,CE_t^{B,B}} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,CE_t^{A,A}} \right]^{1/n_A}}} \times \frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,CE_t^{A,A}} \right]^{1/n_A}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,GCE_t} \right]^{1/n_A}}} \times \frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,GCE_t} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,CE_t^{B,B}} \right]^{1/n_B}}},$$
$$= \frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,CE_t^{B,B}} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,CE_t^{A,A}} \right]^{1/n_A}}} \times \frac{{\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,GCE_t} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,CE_t^{B,B}} \right]^{1/n_B}}}}}{{\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,GCE_t} \right]^{1/n_A}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,CE_t^{A,A}} \right]^{1/n_A}}}}},$$
$$= \frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,CE_t^{B,B}} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,CE_t^{A,A}} \right]^{1/n_A}}} \times \frac{{BPCG^{B,B}}}{{BPCG^{A,A}}},$$
$$= CED \times BPCD.$$

Proof of Eq. (23):

$$CED = \frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,CE_t^{B,B}} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,CE_t^{A,A}} \right]^{1/n_A}}},$$
$$= \frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,CE_t^{B,B}} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,CE_t^{A,A}} \right]^{1/n_A}}} \times \frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,D_t^{A,A}} \right]^{1/n_A}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,D_t^{A,A}} \right]^{1/n_A}}} \times \frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,D_t^{B,B}} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,D_t^{B,B}} \right]^{1/n_B}}},$$
$$= \left( {\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,D_t^{B,B}} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,D_t^{A,A}} \right]^{1/n_A}}}} \right)^{ - 1} \times \frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,CE_t^{B,B} \times D_t^{B,B}} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,CE_t^{A,A} \times D_t^{A,A}} \right]^{1/n_A}}},$$
$$= TED \times AED.$$

Proof of Eq. (24):

$$BPCD = \frac{{\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,GCE_t} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,CE_t^{B,B}} \right]^{1/n_B}}}}}{{\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,GCE_t} \right]^{1/n_A}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,CE_t^{A,A}} \right]^{1/n_A}}}}},$$
$$= \frac{{\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,GCE_t} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,CE_t^{B,B}} \right]^{1/n_B}}}}}{{\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,GCE_t} \right]^{1/n_A}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,CE_t^{A,A}} \right]^{1/n_A}}}}} \times \frac{{\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,D_t^{B,B}} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,D_t^{B,B}} \right]^{1/n_B}}}}}{{\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,D_t^{A,A}} \right]^{1/n_A}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,D_t^{A,A}} \right]^{1/n_A}}}}} \times \frac{{\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,GD_t} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,GD_t} \right]^{1/n_B}}}}}{{\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,GD_t} \right]^{1/n_A}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,GD_t} \right]^{1/n_A}}}}},$$
$$= \frac{{\left( {\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,GD_t} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,D_t^{B,B}} \right]^{1/n_B}}}} \right)^{ - 1}}}{{\left( {\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,GD_t} \right]^{1/n_A}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,D_t^{A,A}} \right]^{1/n_A}}}} \right)^{ - 1}}} \times \frac{{\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,GCE_t \times GD_t} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,CE_t^{B,B} \times D_t^{B,B}} \right]^{1/n_B}}}}}{{\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,GCE_t \times GD_t} \right]^{1/n_A}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,CE_t^{A,A} \times D_t^{A,A}} \right]^{1/n_A}}}}},$$
$$= \frac{{BPG^{B,B}}}{{BPG^{A,B}}} \times \frac{{\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,GCE_t \times GD_t} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,CE_t^{B,B} \times D_t^{B,B}} \right]^{1/n_B}}}}}{{\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,GCE_t \times GD_t} \right]^{1/n_A}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,CE_t^{A,A} \times D_t^{A,A}} \right]^{1/n_A}}}}},$$
$$= BPD \times BPAD.$$

Proof of Eq. (29):

$$GAMI = \frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,GCE_t \times GD_t} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,GCE_t \times GD_t} \right]^{1/n_A}}},$$
$$= \frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,GCE_t \times GD_t} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,GCE_t \times GD_t} \right]^{1/n_A}}} \times \frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,CE_t^{A,A} \times D_t^{A,A}} \right]^{1/n_A}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,CE_t^{A,A} \times D_t^{A,A}} \right]^{1/n_A}}} \times \frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,CE_t^{B,B} \times D_t^{B,B}} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,CE_t^{B,B} \times D_t^{B,B}} \right]^{1/n_B}}},$$
$$= \frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,CE_t^{B,B} \times D_t^{B,B}} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,CE_t^{A,A} \times D_t^{A,A}} \right]^{1/n_A}}} \times \frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,CE_t^{A,A} \times D_t^{A,A}} \right]^{1/n_A}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,GCE_t \times GD_t} \right]^{1/n_A}}} \times \frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,GCE_t \times GD_t} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,CE_t^{B,B} \times D_t^{B,B}} \right]^{1/n_B}}},$$
$$= \frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,CE_t^{B,B} \times D_t^{B,B}} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,CE_t^{A,A} \times D_t^{A,A}} \right]^{1/n_A}}} \times \frac{{\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,GCE_t \times GD_t} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,CE_t^{B,B} \times D_t^{B,B}} \right]^{1/n_B}}}}}{{\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,GCE_t \times GD_t} \right]^{1/n_A}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,CE_t^{A,A} \times D_t^{A,A}} \right]^{1/n_A}}}}},$$
$$= \frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,CE_t^{B,B} \times D_t^{B,B}} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,CE_t^{A,A} \times D_t^{A,A}} \right]^{1/n_A}}} \times \frac{{BPAG^{B,B}}}{{BPAG^{A,A}}},$$
$$= AED \times BPAD.$$

Proof of Eq. (30):

$$\frac{{GAMI}}{{AMI}} = \left[ {\frac{{GAMI}}{{AMI^A}}\frac{{GAMI}}{{AMI^B}}} \right]^{1/2},$$
$$= \left[ {\frac{{\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,GCE_t \times GD_t} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,GCE_t \times GD_t} \right]^{1/n_A}}}}}{{\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,CE_t^{A,B} \times D_t^{A,B}} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,CE_t^{A,A} \times D_t^{A,A}} \right]^{1/n_A}}}}}\frac{{\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,GCE_t \times GD_t} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,GCE_t \times GD_t} \right]^{1/n_A}}}}}{{\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,CE_t^{B,B} \times D_t^{B,B}} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,CE_t^{B,A} \times D_t^{B,A}} \right]^{1/n_A}}}}}} \right]^{1/2},$$
$$= \left[ {\frac{{\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,GCE_t \times GD_t} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,CE_t^{A,B} \times D_t^{A,B}} \right]^{1/n_B}}}}}{{\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,GCE_t \times GD_t} \right]^{1/n_A}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,CE_t^{A,A} \times D_t^{A,A}} \right]^{1/n_A}}}}}\frac{{\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,GCE_t \times GD_t} \right]^{1/n_B}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_B} \,CE_t^{B,B} \times D_t^{B,B}} \right]^{1/n_B}}}}}{{\frac{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,GCE_t \times GD_t} \right]^{1/n_A}}}{{\left[ {\mathop {\prod }\nolimits_{t = 1}^{n_A} \,CE_t^{B,A} \times D_t^{B,A}} \right]^{1/n_A}}}}}} \right]^{1/2},$$
$$= \left[ {\frac{{BPAG^{A,B}}}{{BPAG^{A,A}}} \times \frac{{BPAG^{B,B}}}{{BPAG^{B,A}}}} \right]^{1/2}$$

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Walheer, B. Global Malmquist and cost Malmquist indexes for group comparison. J Prod Anal 58, 75–93 (2022). https://doi.org/10.1007/s11123-022-00640-5

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