**1. Introduction**

This paper assesses public poverty policies (PPPs) in European Union by considering not only attained a diminution of poverty, which obviously is directly linked with social expenditure (SE), but also the productivity of this expenditure. Likewise, we quantify the relationship of every kind of social expenditure (health and sickness benefits, pensions benefits, family and children benefits, . . . ) with the efficiency of overall SE.

There are many papers on the productivity of public policies from the point of view of fiscal systems [1–3], but also from the perspective of nonmonetary social benefits, such as health and education [4–6]. The productivity of SE to reduce inequality and poverty indexes have also been investigated in for EU-27 countries (EU-28 less Croatia) [7]; within OECD countries [8]; in EU-15 countries [9] and within EU-28 [10–12]. Our analysis shows a new perspective on this topic since we analyze a different period, and moreover, we use fuzzy set theory tools to analyze data.

The key question we address here is the relationship between SE, on one hand, and poverty rates on the other in EU-28. It is well known that there is an inverse relation of SE with poverty and inequality indexes [13,14]. However, ref. [15] showed that despite there is a grea<sup>t</sup> negative correlation of SE and poverty levels in EU-28 states, it cannot be concluded that increases in SE lead directly to reductions in poverty. Therefore, ref. [15] suggests that more efforts in social benefits suppose unequal results of poverty policies. Hence, convergence in SE does not imply converging in poverty levels. This fact comes clear in the case of Mediterranean countries as, e.g., Spain where a growth of social expenses

**Citation:** de Andrés-Sánchez, J.; Belzunegui-Eraso, A.; Valls-Fonayet,F. Assessing Efficiency of Public Poverty Policies in UE-28 with Linguistic Variables and Fuzzy Correlation Measures. *Mathematics* **2021**, *9*, 128. https://doi.org/ 10.3390/math9020128

Received: 21 December 2020 Accepted: 3 January 2021 Published: 8 January 2021

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could end up absorbed by middle-income households instead lower-income ones due to Mathew effect [16]. Indeed, ref. [17] point out that whereas tax systems and in-cash benefits could generate a diminution of income inequality indexes, they may also produce undesired consequences. It is well known that pensions for the elderly people have a small redistributive effect.

Although family and housing benefits are more progressive than pensions, they have a limited impact since they do not suppose a grea<sup>t</sup> proportion of SE. On the other hand, ref. [18] indicates that the results of unemployment policies objected to the Lisbon strategy. That paper outlines that unemployment expenses have not had expected results, and also redistribution programs have not been effective enough in poverty elimination. Likewise, several papers found that the benefits of social assistance policies have not a grea<sup>t</sup> effect in many countries [19–23]. Hence, we feel justified assessing productivity of SE on poverty diminutions within EU-28 countries. It could lead us to understand why several countries, after making a similar budgetary effort in social policies, obtain different results in poverty reduction due to the unequal productivity of their social programs.

Our analysis on EU-28 PPPs is done within 2014–2018 by using annual data from the Eurostat programs: European Union Statistics on Income and Living Conditions (EU-SILC) and European System of Integrated Social Protection Statistics (ESSPROS). To assess the results of social policies within a period of several years, a usual practice consists of taking the average value of annual observations as variable observations [8,10,12] or, alternatively, limiting the analysis to a concrete year [7,12]. Clearly, those procedures suppose using limited information. This drawback leads us to propose modeling observations on variables in a period of multiple years by means of fuzzy numbers (FNs). Fuzzy data analysis will allow using all the information in the sample and, in addition, structuring the value of observations in such a way that we may obtain results with an intuitive interpretation. With fuzzy data on evaluated PPPs, we perform two analyses. First, we rank EU-28 countries taking into account the productivity of their PPP. To do it we calculate the fuzzy Debreu– Farrell index for each country and face a problem of FN ordering. Second, we calculate and evaluate the correlation of PPP efficiency with the way SE has been split into items as health expenditure, pensions payments, . . . by using fuzzy tools.

The motivations of our research and its novelty can be summarized as follows:


very common in economics (see [24]) but supposes a novel approach from fuzzy sets literature perspective.

The following section describes the mathematical instruments from fuzzy set theory used in this paper. Methodological aspects of our article: variables, database and methodology that lead to assess the efficiency of SE in poverty reduction are exposed in the third section. In the fourth section, we establish a hierarchy of PPPs in EU-28 by using the concept of the expected value of an FN. Likewise, we evaluate the influence of the composition of SE over the efficiency of PPPs. This last issue is done with the concept of fuzzy correlation and modeling Pearson's correlation coefficient (PCC) as a fuzzy linguistic variable. The last section presents the principal conclusions from our paper.

#### **2. Concepts of Fuzzy Set Mathematics**
