*3.2. Methodology*

We first rank UE28 states by considering the efficiency of their PPP. To make this assessment, we defuzzify the values of DF with (1d) and state their hierarchy. Likewise, we relate our results with those in [7,10]. The flowchart of this analysis is depicted in Figure 2b.

The second analysis tries to determine the sign of the relation between the eight items that [60] differentiates in social expending and the efficiency of PPP. Figure 2c shows how we have implemented this assessment on European PPPs. As it is stated in [16] and also checked in [7,8,10], despite the clear negative linkage between SER and poverty indexes in EU-28, it cannot be concluded that a poverty reduction is reached automatically by increasing SER. Hence, we first measure the intensity of the relation between the effort in

social policies (i.e., SER) with the results in reducing poverty (RRP) and with PPP efficiency measured by DF. Subsequently we investigate why two different countries with a similar SER will obtain different reductions of poverty indexes. Following [7,10] we perform this analysis with the FPCC of the proportion that each kind of social benefit supposes in overall social expenditure. Concretely we use FPCC in [34] (Equations (4b) and (6a)–(6i)) instead max-min fuzzy correlation due to the reasons exposed above. We have used the fuzzy version of PPC instead other correlation measure as, e.g., Spearman correlation by two reasons. First, we pretend comparing our results with those in [7,10] that evaluate the PPPs of the same set of countries, and they use Pearson's correlation coefficient. To allow our results to be fully comparable, the same correlation measure must be used. Likewise, calculating Spearman correlation requires an early defuzzification of triangular observations in order to rank them, e.g., by calculating their expected value. Subsequently Spearman correlation index is a real valued number since that comes from applying a conventional PPC on the crisp rank of variables. Therefore, the fuzzy uncertainty of data is waived in that correlation measure.

Within EU-28, we can differentiate two types of countries whose history and political evolution from II World War to the end of the 20th century XX has been notably different. One on hand we have EU-15 countries, basically Western Europe countries, which are part of European Union from 20th-century. On the other hand, we find former communist republics plus Cyprus and Malta that belonged progressively in European Union during the 21st century. Table 9 shows the mean value of *SE* #*i*, *RRP i* and *DF* # *i*, *i* = 1, 2, ... , 28 in EU-28, but also, separately, the average value of EU-15 countries and non-EU-15 states. Those mean values have been obtained by using Max-Tw norm convolution in such a way that for a variable *A*, the mean value *A*-*M* is:

$$\tilde{A}^{M} = \left( A^{M}, l\_{A^{M}}, r\_{A^{M}} \right) = \left( \frac{\sum\_{i=1}^{n} A\_{i}}{n}, \frac{\max\_{i} l\_{A\_{i}}}{n}, \frac{\max\_{i} r\_{A\_{i}}}{n} \right) \tag{11}$$

**Table 9.** Mean values of SER, poverty risk index (RRP) and Debreu–Farrell index (DF) in EU-28, EU-15 and non-EU-15 in the period 2014–2018.


Source: own elaboration from data provided by EU-SILC (2008–2018) and ESSPROS (2008–2017). Variables SER and RRP are expressed over 100 and E over 1.

> In Table 9 it can be checked that the mean value of EU-15 and non-EU-15 in SER, RRP and efficiency of PPP is completely different. EU-15 countries present a mean value of SER 10 points above non-EU-15 countries. Likewise, Table 9 shows that whereas EU-15 countries rarely have a value for SER below 25%, non-EU-15 states with SER greater than 20% are an exception. Consequently, the mean reduction of poverty in EU-15 countries is clearly above non-EU-15 states. It is also remarkable that the mean DF is notably greater in EU-15 countries than in non-EU-15 countries.
