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Incorporating Sustainability Concerns in the Better Life Index: Application of Corrected Convex Non-parametric Least Squares Method

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The OECD recently released a comprehensive set of 11 well-being indicators, the so-called Better Life Index (BLI), for 36 countries. The BLI covers a wide range of socio-economic aspects of life, which are essential to well-being. This well-being dataset allows us to compare countries’ overall well-being. However, in spite of the BLI’s wider coverage of variables, it fails to consider sustainability concerns. This study provides a practical proposal for comparing overall well-being by incorporating sustainability concerns. Using the World Bank’s adjusted net savings data as a sustainability indicator, we add an extra dimension to the BLI. Then, we apply a composite indicator and aggregate these 12 indicators for each country into a single number. Moreover, we improve the current method for constructing composite indicators by adopting corrected convex non-parametric least squares (C2NLS). It is a typical problem in a non-parametric approach based on linear programming for countries’ scores of composite indicators to become equal and their performance cannot be distinguished. This becomes even more severe if the number of sample countries is small or the number of aggregated indicators is large, which is the case of the present study dealing with 12 indicators for 36 countries. The use of C2NLS, which is based on quadratic programming, overcomes this problem and allows us to order all countries in the sample completely. The empirical results show that the introduction of a sustainability indicator for comparisons does not change countries’ overall rankings significantly. However, it certainly changes the ranking of some countries in both directions.

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Fig. 1
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  1. 1.

    There were 34 countries covered in 2011. A revised dataset released in 2012 includes 36 countries, incorporating Brazil and Russia.

  2. 2.

    Your Better Life Index ( was designed as an interactive tool that allows users to assign the importance of each of the 11 topics and track the performance of countries.

  3. 3.

    However, the two strands of research, such as measures of well-being and sustainability, tend to have been separated.

  4. 4.

    The efficiency score based on C2NLS is constructed from residuals in non-parametric least squares subject to continuity, monotonicity, and concavity constraints, and this is referred to as convex non-parametric least squares (CNLS). Constructing the production frontier based on CNLS is part of the entire process of estimating the efficiency measure known as stochastic non-parametric envelopment of data (StoNED). See Kuosmanen (2008) and Kuosmanen and Kortelainen (2012).

  5. 5.

    Principal component analysis has been applied to reducing the number of inputs and outputs in DEA. While each principal component is a linear combination of underlying variables, negative weight is often assigned to each variable. Since underlying well-being indicators are constructed so that larger values indicate better socio-economic conditions, it is not appropriate to apply this to the current problem of constructing the composite well-being indicator.

  6. 6.

    Kuosmanen and Johnson (2010) refers to Kuosmanen et al. (2006) as an attempt to measure efficiencies of all units simultaneously in a single large problem like C2NLS. It is worth noting that Kuosmanen et al. (2006) are motivated by the research on composite indicator of sustainable development of (Cherchye and Kuosmanen 2004). This point is credited to an anonymous referee.

  7. 7.

    Sustainability indicators and well-being indicators are treated alike. Thus, we do not differentiate them in our notation.

  8. 8.

    The dummy input can be considered as a helmsman in each country, and is intended to provide people with a better life. This interpretation goes back to Lovell et al. (1995).

  9. 9.

    Formally, \(CI_{BOD,c}\) is known as the input-oriented Farrell efficiency.

  10. 10.

    Following the formulation of Kuosmanen and Johnson (2010), the C2NLS efficiency estimator for unit \(c\), \(\varepsilon_{C2NLS,c}\) is constructed by using the CNLS efficiency estimator \(\varepsilon_{CNLS,c}\) such as \(\varepsilon_{C2NLS,c} = \varepsilon_{CNLS,c} - \mathop {\hbox{min} }\limits_{{i \in \left[ {1, \ldots ,K} \right]}} \varepsilon_{CNLS,i}\). Since \(CI_{CNLS,i} + \varepsilon_{CNLS,i} = 1\) and \(CI_{C2NLS,i} + \varepsilon_{C2NLS,i} = 1\) for all \(i = 1, \ldots ,K\), it leads to Eq. (3).

  11. 11.

    Kuosmanen and Johnson (2010) introduced this alternative formulation. It has the advantage of including all countries in a single large problem. Thus, the weights in \(C_{BOD}\), which need to be solved repeatedly for each country in (1), are solved simultaneously for all countries in (4).

  12. 12.

    This is the normalization adopted by the OECD.

  13. 13.

    The 4 high-income countries are Luxembourg, Norway, Switzerland, and the US; the 13 low-income countries are Brazil, Chile, Czech Republic, Estonia, Greece, Hungary, Mexico, Poland, Portugal, Russia, Slovak Republic, Slovenia, and Turkey; and the other 19 countries are middle-income countries.

  14. 14.

    Moreover, the depreciation of human capital is dismissed. Thus, strictly speaking, education expenditure is a dubious measure, even for changes in human capital.

  15. 15.

    Table 10 reports the extent of the increase of both the values and rankings of their composite indicators after sustainability concerns are included.

  16. 16.

    It is also shown that HDI is much better than GDP per capita as a well-being measure. However, there is a certain difference between composite indicators and HDI, which is mainly attributed to the fact that HDI misses a variety of important socio-economic conditions.

  17. 17.

    See Daraio and Simar (2007).

  18. 18.

    We adopt the procedure of simple averaging as an example. Similar results are obtained by applying C2NLS to aggregation of the first stage. They are available upon request.

  19. 19.

    For example, the correlation between \(CI_{C2NLS}\) applied to 12 indicators and \(CI_{C2NLS}\) applied to 2 sub-aggregate and a single sustainability indicator is 0.7669 and its rank correlation is even larger (0.8734).

  20. 20.

    The two composite indicators and HDI have a similar mean and standard deviation in the previous section.

  21. 21.

    This difference is more evident in \(CI_{C2NLS}\).

  22. 22.

    Both BOD and C2NLS are shown to be consistent estimators of the production frontiers. See Banker (1993) for BOD and Kuosmanen and Johnson (2010) for C2NLS.

  23. 23.

    Index number theory proposes desirable axioms that plausible price indices need to satisfy (Balk 2008). Axiomatic justification might be applicable to studies on composite indicators.


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The author is grateful for three anonymous referees and Shigemi Kagawa and Knox Lovell for their helpful comments and suggestions. This article was completed when the author visited Center for Efficiency and Productivity Analysis in the School of Economics at the University of Queensland. The good research environment offered by the institute and the department is greatly appreciated. This research was financially supported by Grant-in-Aid for Scientific Research (KAKENHI 25870922). All remaining errors are the author’s responsibility.

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Correspondence to Hideyuki Mizobuchi.



See Tables 9 and 10.

Table 9 Well-being Indicators and Sustainability Indicator
Table 10 Difference in value and ranking of composite indicators

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Mizobuchi, H. Incorporating Sustainability Concerns in the Better Life Index: Application of Corrected Convex Non-parametric Least Squares Method. Soc Indic Res 131, 947–971 (2017).

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  • Sustainability
  • Composite indicators
  • Better Life Index
  • Data envelopment analysis
  • Benefit of the doubt approach