Multidimensional welfare comparisons of EU member states before, during, and after the financial crisis: a dominance approach

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.

Appendices

Appendices

A.1 Country abbreviations

See Table 7.

Table 7 Abbreviations for included EU member states and regional groupings

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 $\hbox {LCS}_4=\{(0,0),(1,0),(0,1)\}$.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:

$$\begin{aligned}&{(\hbox {i}_1)} \ g(0,0) \le f(0,0)\\&{(\hbox {i}_2)} \ g(0,0) + g(1,0) \le f(0,0) + f(1,0)\\&{(\hbox {i}_3)} \ g(0,0) + g(0,1) \le f(0,0) + f(0,1)\\&{(\hbox {i}_4)} \ g(0,0) + g(1,0) + g(0,1) \le f(0,0) + f(1,0) + f (0,1) \end{aligned}$$

Table 8 Illustrating all LCSs, bivariate and binary

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

$$\begin{aligned}&{(\hbox {i}_l)} \ \sum _{z \le y} g(z) \ge \sum _{z\le y} f(z) \ \text {for all} \ y \in Y. \end{aligned}$$

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

$$\begin{aligned}&{(\hbox {i}_u)} \ \sum _{z \ge y} g(z) \le \sum _{z\ge y} f(z) \ \text {for all} \ y \in Y. \end{aligned}$$

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 $\hbox {LCS}_4$).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.

Table 9 Samples sizes for each country-year combination in the EU-SILC data

A.4 Data structure

See Table 10.

Table 10 Example of data structure using sample data for Germany in 2005

A.5 Illustration of temporal development in indicators

See Figs. 1, 2 and 3.

Fig. 1

Source: Calculations based on data from Eurostat

Income: Share of the population in each of the four categories. Notes: Four quartiles of equivalized annual net income, see Table 1, for Austria (AT), Germany (DE), Denmark (DK), and Hungary (HU)

Fig. 2

Source: Calculations based on data from Eurostat

Health: Share of the population in each of the five categories. Notes: Five outcomes of self-reported health, see Table 1, for Austria (AT), Germany (DE), Denmark (DK), and Hungary (HU)

Fig. 3

Source: Calculations based on data from Eurostat

Education: Share of the population in each of the three categories. Notes: Three educational outcomes based on ISCED 2011 levels ranging from 0 to 8, using (1) early childhood educational development, pre-primary education, pre-primary education, and lower secondary educaiton, (2) upper secondary and post-secondary education, and (3) short-cycle tertiary education, Bachelor’s or equivalent level, Master’s or equivalent level, and Doctoral or equivalent level. The FOD analyses use ISCED 1997, see Table 1. ISCED 2011 levels 0–2 correspond to ISCED 1997 levels 1–2 used in the FOD analyses with an additional category of early childhood educational development added to the ISCED 2011 levels, ISCED 2011 levels 3–4 correspond to ISCED 1997 levels 3–4, and ISCED 2011 levels 5-8 correspond to ISCED 1997 levels 5–6 (OECD/Eurostat/UNESCO Institute for Statistics 2015). Comparisons are for Austria (AT), Germany (DE), Denmark (DK), and Hungary (HU).

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.²⁴

Table 11 Description of welfare dimensions and binary indicators

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.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.

Table 12 Spatial first order dominances using binary indicators, 2005

Table 13 Spatial first order dominances using binary indicators, 2009

Table 14 Spatial first order dominances using binary indicators, 2013

Table 15 Copeland scores and corresponding ranking of EU member states using binary indicators

Table 16 Temporal first order dominances using binary indicators