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Efficiency evaluation of sustainability indicators in a two-stage network structure: a Nash bargaining game approach

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Abstract

Nowadays, economic planning and policymaking toward economic growth and development require special attention to the energy and the environment, and their relationship with economic indicators. In this regard, the relationship of energy–environment–economy–equity (E4) is considered as a network to evaluate the performance of countries applying network-DEA models. In this regard, a two-stage network structure has been applied for efficiency evaluation of units. So that all the outputs of the first stage are considered as the inputs of the second stage. Due to multiple optimal weights, and consequently flexibility in the overall efficiency decomposition of the two stages, this study develops a Nash bargaining game model to evaluate the efficiency of the network structure to achieve a fair efficiency decomposition for both stages. In the bargaining game, the achievable minimum efficiency is used as a breaking point in each stage, and finally, the whole efficiency can be obtained. Using the data of selected developing countries in 2018, the bargaining results show that 6 countries had the highest score in the first stage, and two countries in the second stage. According to the results, although some countries have high rankings in terms of energy, economic, and environmental efficiency, but in the second stage, which focus on economic equity, only two countries show the highest efficiency. Therefore, in examining the performance of countries in terms of the four indicators, the results will be different, and it is appropriate for policymakers to consider the criterion of equity along with other criteria for better planning and decision making.

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Notes

  1. Equity index is one of the dimensions of sustainability, that is important in the process of sustainable development. Due to the lack of access to official statistics of this index, the real GDP growth criterion has been used as a proxy, because more economic growth, entails more economic welfare and improving the conditions for equity. Assuming \(Y_{r}\) as real GDP, \(Y_{n}\) as nominal GDP, and P as price index, the real GDP growth is obtained as follows: \(Y_{r} = \frac{{Y_{n} }}{P},\;\;\frac{{d{\text{Log}}Y_{r} }}{{Y_{r} }} = \frac{{d{\text{Log}}Y_{n} }}{{Y_{n} }} - \frac{{d{\text{Log}}P}}{P}\to ^{{{\text{yields}}}} \dot{Y}_{r} = \dot{Y}_{n} - \dot{P}\).

  2. Algeria, Azerbaijan, Bahrain, Brazil, China, Egypt, Indonesia, Iran, Kuwait, Malaysia, Oman, Qatar, Saudi-Arabia, United Arab Emarat (UAE), Venezuela.

Abbreviations

DMU:

Decision-making unit

j :

The number of decision-making units (DMU)

k :

The number of initial inputs

r :

The number of final outputs

d :

The number of outputs of first stage as the inputs of second stage

α :

Weight vector of initial inputs

β :

Weight vector of final outputs

γ :

Weight vector of outputs of the first stage as the inputs of the second stage

\(\alpha_{k}\) :

Weights of the inputs of the first stage

\(\beta_{r}\) :

Weights of the outputs of the second stage

\(\gamma_{d}\) :

Weights of the outputs of the first stage as the input of the second stage

ε :

Very small non-Archimedean number

\(\alpha^{*}\) :

Optimal weight vector of initial inputs

\(\beta^{*}\) :

Optimal weight vector of final output

\(\gamma^{*}\) :

Optimal weight vector of intermediate inputs (the outputs of first as the inputs of the second stage)

\(\alpha_{k}^{*}\) :

Optimal weights of the first stage

\(\beta_{r}^{*}\) :

Optimal weights of the second stage

\(\gamma_{d}^{*}\) :

Optimal weights of the intermediate inputs/outputs (the outputs of the first as the inputs of the second stage)

I j :

Vector of initial inputs of jth unit

O j :

Vector of final outputs of jth unit

M j :

Vector of intermediate inputs

I kj :

The inputs of jth unit at stage 1

O sj :

The final outputs of jth unit at stage 2

M tj :

The outputs of jth unit at stage 1 as the inputs of jth unit at stage 2

E0:

Efficiency of 0th DMU (evaluated by initial inputs and final outputs of 0th DMU)

ET(0):

The total efficiency of 0th DMU (evaluated by initial inputs, intermediate inputs and outputs and final outputs of 0th DMU)

ET1(0):

Efficiency of 0th DMU at stage1 (evaluated by initial inputs and intermediate inputs/outputs of 0th DMU)

ET2(0):

Efficiency of 0th DMU at stage2 (evaluated by intermediate inputs/outputs and final outputs of 0th DMU)

EP1(0):

Maximum efficiency of 0th DMU at stage1 (evaluated by initial inputs and intermediate inputs/outputs of 0th DMU)

EM1(0):

Minimum efficiency of 0th DMU at stage1 (evaluated by initial inputs and intermediate inputs/outputs of 0th DMU)

EP2(0):

Maximum efficiency of 0th DMU at stage2 (evaluated by intermediate inputs/outputs and final outputs of 0th DMU)

EM2(0):

Minimum efficiency of 0th DMU at stage2 (evaluated by intermediate inputs/outputs and final outputs of 0th DMU

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Authors and Affiliations

  1. Department of Economics, Shahryar Branch, Islamic Azad University, Shahryar, Iran

    Bahram Fathi

  2. Department of Humanities, Bozorgmehr University of Qaenat, Qaen, Iran

    Malihe Ashena

  3. Department of Mathematics, Qazvin Branch, Islamic Azad University, Qazvin, Iran

    Majid Anisi

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  1. Bahram Fathi
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  2. Malihe Ashena
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  3. Majid Anisi
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Fathi, B., Ashena, M. & Anisi, M. Efficiency evaluation of sustainability indicators in a two-stage network structure: a Nash bargaining game approach. Environ Dev Sustain 25, 1832–1851 (2023). https://doi.org/10.1007/s10668-022-02325-3

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