## Abstract

How did the EU states’ populations fare during the financial crisis in key dimensions such as income, health, and education? We seek to answer this question by way of welfare comparisons between countries and within countries over time, using EU-SILC data. Our study is novel in implementing a multidimensional first order dominance (FOD) comparison approach on the basis of multi-level indicators. FOD only requires that outcomes can be ranked from worse to better within each dimension and is therefore suitable for the analysis of multidimensional ordinal data. We find that the countries most often dominated are southern and eastern European member states, and the dominant countries are mostly northern and western European member states. However, for most country comparisons, there is no dominance relationship. Moreover, only a few member states have experienced a temporal dominance improvement in welfare, while no member states have experienced a temporal dominance deterioration during the financial crisis.

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## Notes

- 1.
Ravallion (2012) refers to this as ‘mashup indices’.

- 2.
The present paper focuses on comparisons of welfare in population distributions with ordinal multidimensional data. For comparisons of inequality across populations with ordinal data, we refer to Allison and Foster (2004), Gravel and Moyes (2012), Balestra and Ruiz (2015), Sonne-Schmidt et al. (2016), and Cowell and Flachaire (2017).

- 3.
- 4.
Note that there can be multiple indicators for the same welfare dimension as exemplified here, where both mean years of schooling for adults and expected years of schooling for children are used as indicators in the education dimension. In this paper, we use a single indicator for each welfare dimension included.

- 5.
For example, an indicator in the health dimension is that the respondent considers her own health as fair or above, and the indicator in the education dimension is whether or not the respondent has completed primary education.

- 6.
Note that FOD and orthant stochastic orderings are equivalent in the one-dimensional setting.

- 7.
The first proof of the equivalence between (i) and (iii) is usually attributed to Lehmann (1955) (however, see also Levhari et al. 1975). The first formulation and proof of the equivalence between (i) and (ii) is not easy to trace back to its roots, but Kamae et al. (1977) observed that the equivalence between (i) and (ii) is a corollary of Strassen’s Theorem (Strassen 1965). Østerdal (2010) provides a constructive direct proof of this for the finite case.

- 8.
In the one-dimensional case,

*f*first order dominates*g*if and only if \(F(x) \le G(x)\) for all \(x \in X\), where \(F(\cdot )\) and \(G(\cdot )\) are the cumulative distribution functions corresponding to*f*and*g*, respectively. For a review of FOD in both a one-dimensional and multidimensional welfare setting using binary indicators, we refer to Siersbæk et al. (2017). - 9.
Specifically, when minimizing an objective function under a set of conditions that yields a lower bound of the objective function, rounded data inputs as well as a large number of estimated transfers that all have a true value of zero are computationally handled as very small positive numbers. However, when adding up multiple very small positive numbers, the result may be a sum that is ‘large’. The analyst in practice thus has to define a threshold for when the objective function is interpreted as zero. For example, when minimizing an objective function that is bounded to zero, an estimated minimum value of the objective function of, say, \(10^{-10}\) has to be interpreted; is it a true zero (in which case FOD is observed) or something larger than zero (in which case FOD is not observed). If the threshold is not chosen ‘correctly’, this may lead to the conclusion that a dominance exists when in fact there is no dominance (but ‘close’ to dominance).

- 10.
The number of LCSs is quantified by Sampson and Whitaker (1988) using the number of levels in each indicator. Strictly speaking, Sampson and Whitaker (1988) provide the number of upper comprehensive sets, which is, however, equal to the number of LCSs. For three dimensions with binary indicators, the total number of LCSs is 20. If the number of levels of each indicator is three, the total number of LCSs is 980, and if four levels of each indicator are used, the total number of LCSs increases to 232,848.

- 11.
The Matlab code for identifying all LCSs and checking FOD is available on the following web-page: https://sites.google.com/site/lposterdal/ (also available from the authors upon request). The empty LCS and the full set of all outcomes are omitted in the code since the corresponding sums using definition (i) are 0 and 1, respectively.

- 12.
Computationally efficient algorithms capable of handling multiple indicators and levels is grounds for further research. For the bivariate case, efficient algorithms are provided in Range and Østerdal (2019).

- 13.
A straightforward test of the multiple inequalities obtained in condition (i) of FOD can be applied using, for example, testing of moment inequality constraints, which is frequently used to test instrument validity in econometric instrumental variable regressions (see, e.g., Chen and Szroeter 2014; Huber and Mellace 2015). This, however, implies a null-hypothesis of dominance.

- 14.
- 15.
Equivalized total net income uses the OECD-modified scale (first proposed by Hagenaars et al. 1996). This assigns a weight of 1 to the first adult in the household, a weight of 0.5 to each additional member of the household aged 14 and over, and a weight of 0.3 to household members aged less than 14. The household’s total net income is divided by this equivalized number of persons to get equivalized total net income (per person in the household). See OECD (2013b) for more information. The income reference period is the previous calendar or tax year for all countries except for the United Kingdom (current year) and Ireland (last 12 months), see Eurostat (2019).

- 16.
The ISCED (International Standard Classification of Education) is developed by UNESCO (United Nations Educational, Scientific and Cultural Organization) to facilitate cross-country comparisons of education systems since these vary in terms of structure. We use the ISCED 1997, which ranges from 0 (pre-primary education) to 6 (second stage of tertiary education); see UNESCO (2006).

- 17.
- 18.
The maximum number of potential dominances for

*n*countries is \((n^2-n)/2\).*n*is raised to the second power to obtain all country combinations. The subtraction of*n*in the nominator is to exclude self-dominance, whereas the 2 in the denominator is due to the fact that if country A dominates country B, B cannot dominate A. Since \(n=24\) in this paper, the maximum number of potential dominances is \((24^2-24)/2=276\). - 19.
Note that the years of analysis in Alkire and Apablaza (2016) are 2006, 2009, and 2012, where only 2009 is somewhat directly comparable.

- 20.
We thank an anonymous referee for a useful discussion on this topic.

- 21.
Though in a slightly different set-up, see also Hussain et al. (2016) for an example of refining dimensions to analyze the ‘depth’ of FOD.

- 22.
The fifth LCS that includes all outcomes is redundant since the probability mass functions both sum to 1.

- 23.
The combination of lower- and upper orthant dominance is equivalent to FOD in the two-dimensional binary case used here. This is, however, not true for more general two-dimensional distributions.

- 24.
Other aggregation thresholds have been used as well. While the results naturally are sensitive to the threshold choice, the ones shown in Table 11 have been chosen to exemplify the difference between multi-level and binary indicators.

- 25.
In general we will observe weakly more dominances as the number of outcome combinations decrease.

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## Acknowledgements

We are grateful to Andreas Bjerre-Nielsen, Dorte Gyrd-Hansen, Casper Worm Hansen, Lene Holbæk, Giovanni Mellace, Troels Martin Range, Peter Sudhölter, and participants at the DGPE workshop (November 2015, Sønderborg, Denmark), CBS Inequality Platform 1st Workshop on Health and Inequality (April 2019, Copenhagen, Denmark), and IFABS 2019 Conference (December 2019, Medellin, Colombia) for useful inputs and suggestions. We are also grateful to the Coordinating Editor Marc Fleurbaey and two anonymous referees for very constructive comments.

## Funding

Financial support from the Independent Research Fund Denmark (Grant-ID: DFF-6109-000132) is gratefully acknowledged.

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Financial support from the Independent Research Fund Denmark (Grant-ID: DFF-6109-000132) is gratefully acknowledged.

## Appendices

### Appendices

### A.1 Country abbreviations

See Table 7.

### A.2 Illustration of LCSs and description of orthant stochastic orderings

Table 8 illustrates all LCSs in the bivariate case with binary indicators. Let dimension A be the row dimension and dimension B be the column dimension. In each dimension an individual can either be in outcome 0 or 1, where 1 is best. This yields four LCSs in total: \(\hbox {LCS}_1=\{(0,0)\}\), \(\hbox {LCS}_2=\{(0,0),(1,0)\}\), \(\hbox {LCS}_3=\{(0,0),(0,1)\}\), and \(\hbox {LCS}_4=\{(0,0),(1,0),(0,1)\}\).^{Footnote 22} To check for FOD between two distributions *f* and *g* using definition (i), one simply has to check that the following four inequalities are satisfied:

FOD requires only ordinal data and does not require assumptions about the strength of preferences for each dimension, the relative desirability of changes among levels within or between dimensions, and substitutability or complementarity between dimensions (e.g., Arndt et al. 2012). On the contrary, orthant dominance concepts following, e.g., Atkinson and Bourguignon (1982) require such assumptions. Particularly, *f* lower orthant dominates *g* if

Condition (\(\hbox {i}_l\)) corresponds to checking inequality \(\hbox {i}_1\), \(\hbox {i}_2\), and \(\hbox {i}_3\) (i.e., using only \(\hbox {LCS}_1\), \(\hbox {LCS}_2\), and \(\hbox {LCS}_3\)). Correspondingly, *f* upper orthant dominates *g* if

Condition (\(\hbox {i}_u\)) corresponds to checking inequality \(\hbox {i}_2\), \(\hbox {i}_3\), and \(\hbox {i}_4\) (i.e., using only \(\hbox {LCS}_2\), \(\hbox {LCS}_3\), and \(\hbox {LCS}_4\)).^{Footnote 23}

Orthant dominance concepts are defined from classes of functions U+ (non-decreasing ALEP substitutes) or U- (non-decreasing ALEP complements) for lower and upper orthant dominance, respectively. These classes of functions require further assumptions including specified signs on the second order partial or cross-derivatives of the underlying utility function (corresponding to ALEP substitutability/complementarity/neutrality of the underlying utility functions, see Kannai 1980). For recent applications see, e.g., (Yalonetzky 2014) and (Kobus et al. 2019).

### A.3 Sample sizes

See Table 9.

### A.4 Data structure

See Table 10.

### A.5 Illustration of temporal development in indicators

### A.6 Binary FOD analyses

As mentioned, applications of FOD in a welfare context have until now used binary indicators. To show the consequences of using multi-level indicators instead of binary ones, analyses similar to those in Sect. 5 but using binary indicators have been conducted. Instead of applying the multi-level indicators outlined in Table 1, we combine the levels into binary outcomes as shown in Table 11, where the second column from the right shows the multi-level indicators applied in the previously reported results, and the rightmost column shows how the multi-level indicators are aggregated into binary indicators.^{Footnote 24}

The spatial results are shown in Tables 12, 13, and 14 and the corresponding Copeland scores are shown in Table 15. The temporal results are shown in Table 16. These five tables are the analogous to Tables 2 through 6, the only difference being that the results are obtained using the binary indicators described in Table 11. Naturally, the multidimensional binary analyses yield several more dominances than the corresponding multidimensional multi-level analyses.^{Footnote 25} In particular, we never obtain an indeterminate result in each dimension analyzed separately since the distribution in a given dimension is fully described by a single number, e.g., those who are worse off. With binary indicators, we obtain between 138 and 149 multidimensional dominances, i.e., around three and a half times more than we do in the multi-level analyses.

The overall results are similar to those obtained in the multi-level analyses, i.e., the countries most often dominated are southern and eastern European member countries, and the dominant countries are mostly northern and western European member states. The Copeland scores using binary indicators are also similar to the ones using multi-level indicators where almost no northern or western European countries are in the bottom half of the ranking and almost no southern and eastern European countries are in the top half of the ranking. Though the use of binary indicators enables us to more easily compare European countries and to obtain a more complete ranking, it comes at the price of less refined indicators.

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Hussain, M.A., Siersbæk, N. & Østerdal, L.P. Multidimensional welfare comparisons of EU member states before, during, and after the financial crisis: a dominance approach.
*Soc Choice Welf* (2020). https://doi.org/10.1007/s00355-020-01259-x

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