Measuring income polarization using Bonferroni and De Vergottini inequality indices: evidences from European countries

Abstract

The aim of this paper is to measure to which extent income distribution is polarized across European countries by means of polarization measures based on the Bonferroni and De Vergottini indices of inequality. Different from traditional measures of polarization, the indices proposed in this paper are sensitive to progressive transfers, attaching more importance to some part of the income distribution. These indices enriches the analysis and contribute to disentangle the different faces of income polarization. In the empirical application we compare European countries over the period 2010–2019 using EU-SILC data. Results reveal significant changes in polarization over the last decades for most countries.

1 Introduction

The recent economic crises due first to the Great Recession, then to COVID-19 pandemic and, lately, to the global energy crisis have revealed the importance of monitoring their distributive effects on inequality, poverty, vulnerability and measuring not only the inequalities among individuals but also across the groups into which population may be divided. For this reason, beyond the analysis of income inequality it is important to take into account polarization measures.

Polarization is a concept similar to but distinct from inequality. Both concern the degree of disparity present in income distribution, but in different ways. Inequality analyses pairwise differences along the whole distribution, while polarization focuses on inequality both between and within groups.

Income polarization indices can be classified into bipolarization indices and multipolar indices; see Permanyer [30] and Gigliarano [24] for an extensive review. Bipolarization indices measure the extent to which an income distribution is clustered around two poles, typically the poor and the rich. Multipolar indices measure the extent to which an income distribution has an arbitrary number of antagonistic poles.

More in details, bipolarization refers to the tendency of incomes to shift away from the middle point of the distribution towards the tails, thus obtaining two groups (typically, the rich and the poor) well separated from each other but homogenous inside and at the same time a hollowed-out middle class. Maximum bipolarization is reached when population is split in two equal-sized groups, such that members in the first one is penniless, while the total income is equally shared among individuals in the second group. Of course, this can be generalized to the case of several groups. In the latter case, the income distribution is said to be (multi-)polarized if the relative frequency of observations is low in correspondence to the central value and it is high at the tails.

The main reason why the study of polarization has obtained as increasing attention over the last few decades is the strong relationship between the decline of the middle class and social instability. Indeed, the presence of groups similar in size, whose members have analogous characteristics but differ from the individuals of the other groups, can bring to an unstable political situation. Esteban and Ray [15] show that the presence of contrasting groups as well as a weak and hollowed-out middle class can lead to an unstable society and cause possible social conflicts. Similarly, Gasparini et al. [18] find that for Latin American and Caribbean countries, in the period 1989–2004, high levels of income polarization are positively correlated with a high level of social conflict.

The first systematic works on polarization measurement date back to 1994, when, in two independent papers, Wolfson [39] and Esteban and Ray [15] propose rigorous treatments of such phenomenon.

These works have given rise to two different strands of literature: the so-called Wolfson and Foster approach Wolfson [39, 40] and Foster and Wolfson [16, 17], and the Esteban and Ray approach [15]. The former focuses on the shrinking middle class, monitoring how the income distribution spreads out from its center. The latter focuses on the rise of separated income groups: polarization increases if population groups with similar size become more homogeneous inside and more separated from each other.¹

However, as discussed in Rodrìguez and Salas [33], the two approaches have a common core. Indeed, the Wolfson index can be rewritten as a function of the difference between the Gini index between groups and the Gini index within groups; thus revealing that, also according to Wolfson’s approach, polarization increases with inequality between groups and decreases with inequality within groups.

After these pioneering works, several scholars have investigates further the phenomenon of polarization. We cite, among others, Wang and Tsui [38], Gradín [25], Anderson [1], Lasso de la Vega and Urrutia [27], Silber et al. [37], Gigliarano and Mosler [22], Chakravarty and D’Ambrosio [9], Pittau and Zelli [32], Gigliarano et al. [23].

All the above cited contributions to polarization measurement are based on the Gini inequality index. Very recently, Ciommi et al. [12] propose, instead, to measure bipolarization using different inequality indices. In particular, the authors introduce bipolarization indices based on Bonferroni and De Vergottini indices, which are characterized by different sensitivity to progressive transfers. Bonferroni-based polarization index is indeed more sensitive to income transfers occurring among poorer individuals, while De Vergottini-based polarization index is indeed more sensitive to transfers occurring among richer people.

Here we follow Ciommi et al. [12] approach and provide and empirical application of the bipolarization indices based on the Bonferroni and the De Vergottini inequality measures in the special case of two non-overlapping groups separated by median.

The aim of the paper is, therefore, to compare, using an empirical analysis, bipolarization indices characterized by different sensitivity to income transfers. The comparison has the advantage to enrich the analysis based on standard measures and contribute to disentangle the different faces of income polarization. This is particularly relevant, since recent studies encourage the simultaneous use of more than one inequality index to better capture the inequality in different parts of the income distribution and thus better understand the socio-economic reality (see, e.g., Piketty [31]).

The rest of the paper is organized as follows. Section 2 is devoted to the methodology: first we recall the inequality indices proposed by Gini, Bonferroni and De Vergottini, then we illustrate their subgroup decomposition in the special case of two non overlapping groups, and finally we review the polarization indices based on Bonferroni and De Vergottini inequality measures, as proposed in Ciommi et al. [12]. Section 3 provides an in-depth analysis of income bipolarization for European Countries in three different years, namely 2010, 2015 and 2019, using European Union Statistics on Income and Living Conditions (EU-SILC) data. Finally Sect. 4 concludes and addresses the possible extension of our approach.

2 From inequality indices to polarization measures

2.1 Notation

For a population of size n, let $y = (y_1, y_2, \ldots , y_p, \ldots , y_n)$ be a positive non-decreasing income distribution, that is $y_a \le y_b$ for any $a \le b$. We denote with m(y) the median income and with $y^U$ and $y_L$ the vector of incomes $y_i$ above and below the median, respectively. Moreover, let $\mu (y)$ be the mean of the overall population and $\mu _p (y)$ be the mean of individuals with income smaller than or equal to $y_p$. Similarly, $\mu ^U$ and $\mu _L$ indicate the mean values of the incomes in $y^U$ and $y_L$, respectively.

Also, we define a polarization index as follows:

Definition 1 A A polarization index P(y) is a continuous (real valued) function $P\,: \, \mathcal {D}\rightarrow \mathbb {R}^{+},$ where $\mathcal {D}$ is the set of all possible income distributions for a population of n individuals.

Income bipolarization indices are based on two crucial axioms: the Increased Spread axiom and the Increased Bipolarity axiom. The former has been firstly formulated by Chakravarty and Majumder [11] and states that bipolarization should increase if we transfer income from an individual below the median to an individual above the median income.

Axiom 1

(Increased Spread) Let $x=(x_1, \ldots , x_n)$ and $y=(y_1, \ldots , y_n)$ be two income distributions such that $m(x)=m(y)=m$. Consider the following scenarios:

• (i) There exists $j \in \lbrace 1,\cdots , k \rbrace$ such that $x_j< y_j <m$ and $x_i = y_i$ for all $i \ne j$;

• (ii) There exists $l\in \lbrace 1,\cdots , k \rbrace$ such that $m< y_l < x_l$ and $x_i = y_i$ for all $i \ne l$;

If either (i), (ii) or both hold, then $P(x) > P(y)$ .

The axiom states that a rank-preserving increase in the incomes above the median, or a rank-preserving decrease in incomes below the median, extends the gap between the two groups and hence increases the degree of bipolarization.

The second important axiom is the Increased Bipolarity axiom, which states that polarization should increase if a progressive transfer between individuals belonging to the same income group takes place. In other words, the axiom refers to the case where the incomes below or above the median become closer to each other, so that inequality within the groups decreases, leading to an increase in bipolarization; see, among others, Permanyer [30].

Axiom 2

(Increased Bipolarity) Let $x=(x_1, \ldots , x_n)$ and $y=(y_1, \ldots , y_n)$ be two income distributions such that $m( y) = m( x) = m.$ Consider the following scenarios:

• (i) x has been obtained from y by a progressive transfer of income from richer person b to poorer person a with $y_b<m$;

• (ii) x has been obtained from y by a progressive transfer of income from richer person d to poorer person c with $y_c>m$.

If either (i), (ii) or both holds, then $P(x) > P( y)$.

2.2 Gini, Bonferroni and De Vergottini

The Gini index [19] is the most common statistical measure employed in the socio-economic sciences for measuring concentration in the distribution of income or wealth. The Gini index has several formulations;² one of the formulations in the discrete case is:

$$\begin{aligned} G(y)=1-\frac{2}{n(n+1)\mu (y)} \sum _{i=1}^{n}\sum _{j=1}^{i}y_{j}, \end{aligned}$$

(1)

with $G(y) \in \left[ 0, \frac{n-1}{n+1}\right]$.

In 1930, Bonferroni [7] proposes an index of inequality that is more sensitive than the Gini index to lower values in the income distribution, assigning more weight to income transfers occurring among the poor (see Nygard and Sandstrom [28]). Recently, several scholars have investigated the main features of Bonferroni index and proposed interesting applications in social and economic contexts (see, among others, Bàrcena-Martìn and Silber [4,5,6]; Chakravarty and Muliere [10], Chakravarty [8], Dong et al. [14]).

The Bonferroni index corresponds to the area between the line of perfect equality (horizontal line at height 1) and the Bonferroni curve $B(p)=\mathcal {L}(p)/p$, where $\mathcal {L}(p)$ represents the Lorenz Curve; see Giorgi and Crescenzi [21].

Following Nygard and Sandstrom [28] and Bàrcena-Martìn and Imedio [3], the formulation of the Bonferroni index B(y) in the discrete case can be written as:

$$\begin{aligned} B (y)=\frac{1}{n} \sum _{i=1}^{n}\left( \frac{\mu (y)-\mu _{i} (y)}{\mu (y)} \right) = 1-\frac{1}{n\mu (y)} \sum _{i=1}^{n}\mu _{i} (y) =1-\frac{1}{n\mu (y)} \sum _{i=1}^{n}\frac{1}{i} \sum _{j=1}^{i}y_j, \end{aligned}$$

(2)

with $B (y)\in \left[ 0, \frac{n-1}{n}\right]$.

In 1950 De Vergottini [13] proposes an index of inequality, which is more sensitive than the Gini index to the right tail of the income distribution, i.e., it is more affected by income changes occurring among the rich. The De Vergottini index corresponds to the area between the De Vergottini curve $V(p)=(1-\mathcal {L}(p))/(1-p)$ and the line of perfect equality and it can be interpreted as a weighted average of the relative differences between the mean of the population and the partial means of the i-th richest group (see Tarsitano [35]). Its discrete formulation is:

$$\begin{aligned} V (y) = \frac{1}{n} \sum _{i=1}^{n} \left( \frac{M_{i}(y) - \mu (y) }{\mu (y)} \right) =\frac{1}{n\mu (y)} \sum _{i=1}^{n} M_i(y) - 1, \end{aligned}$$

(3)

where $M_i(y) =\frac{1}{\left( n-i+1\right) } \sum _{j=i}^{n}y_{j}.$ We have that $V (y) \in \left[ 0, \left( \sum _{j=1}^{n}\frac{1}{n-j+1} \right) -1 \right]$.³

The three above-mentioned indices are, thus, characterized by a different sensibility to income transfers. They can be written as a weighted mean of incomes, where the weight associated with an individual’s income depends on his position in the income distribution and increases with the individual’s rank in the distribution.

According to Bàrcena-Martìn and Imedio [3], the Gini index G(y) can be written as:

$$\begin{aligned} G(y)=\frac{1}{n\mu (y)} \sum _{i=1}^{n}\gamma _{i} y_{i}, \quad \text{ with } \, \, \gamma _{i} =\left( \frac{2i-1}{n}\right) - 1 , \quad \gamma _{i+1}=\gamma _{i} +\frac{2}{n},\quad \sum _{i=1}^{n}\gamma _{i} =0 \end{aligned}$$

(4)

Tarsitano [34] formulates the Bonferroni index B(y) as a linear combination of incomes with weights depending on the individual ranks, as follows:

$$\begin{aligned} B(y) =\frac{1}{n\mu (y)} \sum _{i=1}^{n}w_{i} y_{i} \quad \text{ with } \, \, w_{i} =1-\sum _{j=i}^{n}\frac{1}{j} \mathrm{,\; \; }w_{i+1} =w_{i} +\frac{1}{i} \mathrm{,\; \; \; }\sum _{i=1}^{n}w_{i} =0. \end{aligned}$$

(5)

Finally, for the De Vergottini index, we have

$$\begin{aligned} V(y) =\frac{1}{n\mu (y)} \sum _{i=1}^{n}\xi _{i} t_{i}, \quad \text{ with } \, \, \xi _{i} =\sum _{j=1}^{i}\frac{1}{n-j+1} -1 \quad \xi _{i+1}=\xi _{i} +\frac{1}{n-i} \quad \sum _{i=1}^{n}\xi _{i} =0. \end{aligned}$$

(6)

The three weights $w_{i}, \gamma _{i}, \xi _{i}$ have a different behaviour: in the Gini index the weight sequence increases constantly w.r.t. the individual rank, with an absolute increment of 2/n, whereas both in Bonferroni and in De Vergottini indices weights grow at a non-constant rate (the absolute increment is decreasing and equal to 1/i in Bonferroni, it is increasing and equal to $1/(n-i)$ in De Vergottini).

Therefore, the Bonferroni index is more sensitive to transfers that occur at the lower tail of the income distribution, while De Vergottini index is more sensitive to variations occurring among the richest incomes.

2.3 Decomposition as a tool for defining bipolarization indices: the case of two non-overlapping groups

Subgroup decomposition is a fundamental step in the definition of polarization indices. For instance, the Wolfson bipolarizarion measure splits the population into two groups divided by the median income and can be written as function of the between-group inequality component and the within-group inequality component of the Gini index.

Ciommi et al. [12] consider the decomposition used by Lambert and Aronson [26] for the Gini index and by Bàrcena-Martìn and Silber [6] for the Bonferroni index and extend the approach to De Vergottini concentration index. In the special case of two non-overlapping groups divided by the median, Ciommi et al. [12] consider these decompositions to define new polarization indices based on the Bonferroni and on the De Vergottini concentration indices.

We now review briefly the subgroup decompositions of Gini, Bonferroni and De Vergottini inequality indices, which lead to the new transfer-sensitive polarization indices proposed in Ciommi et al. [12].

We suppose that the two groups are divided by the median income m(y).

The Gini index can be written in terms of two components: the between group component and the within group component. We use subscript B for the between component, subscript W for the within component. In general, the between group component represents the inequality level of a theoretical distribution, in which each individual income is replaced by the mean of the group. Whereas, the within group component is a weighted sum of the inequality within each subgroup. Thus, the subgroup decomposition of the Gini index in case of two groups divided by the median can be written as:

$$\begin{aligned} G=G_B+G_W \end{aligned}$$

(7)

where, if n is an even number, the between-group component is

$$ G_B =\frac{1}{4\mu (y)} \left( \mu ^U(y) - \mu ^L (y)\right) $$

and the within-group component corresponds to

$$\begin{aligned} G_W= & {} \frac{1}{2\mu (y)} \left[ \frac{1}{2} \mu ^L(y) -\frac{1}{n/2(n/2+1)} \sum _{i=1}^{n/2}\sum _{j=1}^{i}y_{j} \right] \\{} & {} \quad + \frac{1}{2\mu (y)} \left[ \frac{1}{2} \mu ^U(y)-\frac{1}{n/2(n/2+1)} \sum _{i=1}^{n/2}\sum _{j=1}^{i}y_{j} \right] . \end{aligned}$$

Differently, for Bonferroni and De Vergottini inequality indices, the decomposition accounts for three terms: the between component, the within component and a residual term, which we denote with the subscript R.

The residual term depends on the re-ranking in the calculation of Bonferroni and De Vergottini indices, which occurs since the two indices, differently from the Gini index, are not replication invariant (see Barcena-Martin and Silber [5]).

Thus, the subgroup decomposition of the Bonferroni index in case of two groups divided by the median can be written as:

$$\begin{aligned} B = B_B + B_W + B_R \end{aligned}$$

(8)

where, if n is an even number, the between component is given by

$$ B_B =\frac{1}{2\mu (y)} \left( \mu ^U (y) -\mu ^L (y) \right) \left( \sum _{j=1}^{n/2}\frac{1}{n/2+j } \right) , $$

while the within component and the residual term are, respectively, defined as follows:

$$\begin{aligned} B_W&= \frac{1}{2n\mu (y)} \left[ \frac{n}{2} \mu ^L(y) - \sum _{i=1}^{n/2}\mu _{i}^L (y) \right] + \frac{1}{2n\mu (x)} \left[ \frac{n}{2} \mu ^U (y) - \sum _{i=1}^{n/2}\mu _{i} ^U(y) \right] , \end{aligned}$$

and

$$\begin{aligned} B_R&= \frac{1}{2n\mu (y)} \left[ \frac{n}{2} \mu ^L (y)-\sum _{i=1}^{n/2}\mu ^L_{i} (y) \right] \\&\quad + \frac{1}{2n\mu (y)} \left[ \mu ^U (y)\sum _{i=1}^{n/2} \left( \frac{(i-n)/2}{(i+n)/2} \right) - \left( \frac{(i-n)/2}{(i+n)/2} \right) \mu _{i}^U (y) \right] . \end{aligned}$$

From the previous definition we observe that the effect of the residual component is stronger the closer the observation is to the median.

Finally, the subgroup decomposition of the De Vergottini index in case of two groups divided by the median can be written as:

$$\begin{aligned} V = V_B + V_W + V_R \end{aligned}$$

(9)

where, if n is an even number, the between component corresponds to

$$ V_B = \frac{1}{2\mu (y)} \left( \mu ^U(y)-\mu ^L(y)\right) \left( \sum _{i=1}^{n/2}\frac{1}{n-i+1} \right) , $$

while the within component and the residual component can be written, respectively, as

$$ V_W = \frac{1}{2n\mu (y)} \left[ \sum _{i=1}^{n/2}M_{i} (y^L)-\frac{n}{2} \mu ^L (y) \right] +\frac{1}{2n\mu (y)} \left[ \sum _{i=1}^{n/2}M_{i} (y^U)-\frac{n}{2} \mu ^U (y) \right] $$

and

$$\begin{aligned} V_R{} & {} = \frac{1}{2n\mu (y)} \left[ \sum _{i=1}^{n/2}\left( \frac{i}{i-n} \right) M_{i} (y^L) -\sum _{i=1}^{n/2}\left( \frac{i}{i-n} \right) \mu ^L (y) \right] \\{} & {} \quad \ + \frac{1}{2n\mu (y)} \left[ \sum _{i=1}^{n/2}M_{i} (y^U) -\frac{n}{2} \mu ^U(y)\right] . \end{aligned}$$

2.4 The bipolarization index

Foster and Wolfson [16] propose the following bipolarization measure:

$$\begin{aligned} P^{FW} =\frac{2\mu (y)}{m(y)} \left[ 1-2L(0.5)-G\right] = \frac{2\mu (y)}{m(y)} \left[ G_B - G_W\right] \end{aligned}$$

(10)

where L(q) is the value of the Lorenz curve at the q-th quantile of y and G is Gini index defined as twice the area between the equidistribution line (p) and the Lorenz curve L(p), while $G_B$ and $G_W$ denote the Gini index between and within groups.

Following Foster and Wolfson’s approach [16, 17], a measure of bipolarization is an increasing function of the inequality between groups and decreasing function of the within groups. Ciommi et al. [12] propose bipolarization measures $P^B$, based on the Bonferroni index, and $P^V$, based on the De Vergottini index, defined as follows:

$$\begin{aligned} P^B =\frac{2\mu (y)}{m(y)} \left[ B_B -(B_W +B_R )\right] \end{aligned}$$

(11)

$$\begin{aligned} P^V = \frac{2\mu (y)}{m(y)} \left[ V_B -(V_W +V_R )\right] . \end{aligned}$$

(12)

The two polarization measures proposed are increasing functions of the between-group inequality, and decreasing with respect to both the within-group and the residual components. Different from Foster and Wolfson’s measure, which is based on the Gini index, the polarization measures based on Bonferroni and De Vergottini contain a residual term representing the role played by the individuals’ rank. The new polarization measures are characterized by different sensitivity to transfers. In particular, measure $P^B$ is more sensitive to income transfers involving poorer individuals, while $P^V$ to transfers occurring among richer people. The new indices are always coherent with the Increased Bipolarity axiom (IB), while the Increased Spread axiom (IS) is satisfied under some regularity conditions.⁴

3 An empirical illustration

The empirical application is based on the EU-SILC (European Union Statistics on Income and Living Conditions) dataset, which collects comparable microdata on households income and living conditions for the 27 EU member states plus Norway and Switzerland.⁵

We have analysed the level of income bipolarization in the 29 countries under analysis comparing the measures illustrated above and referring to the years 2010, 2015 and 2019.⁶

As income variable we considered the equivalent household disposable income, defined as the sum of the personal income components of all household members plus the family income components, net of income tax and social contributions, using the modified OECD scale. Negative incomes have been excluded from the analysis. In addition, we applied a trimming procedure by deleting, for each country, the top and bottom $2\%$ of weighted household disposable income.

We estimated the Gini index of inequality G(y), the Bonferroni index of inequality B(y) and the De Vergottini inequality index V(y) using the following simple weighted estimators:

$$\begin{aligned} \widehat{G}(y)=1-\frac{2}{n(n-1)\mu (y)} \sum _{i=1}^{n-1}\sum _{j=1}^{i}w_{j} y_{j}, \end{aligned}$$

(13)

$$\begin{aligned} \widehat{B}(y)=1-\frac{1}{(n-1)\mu (y)} \sum _{i=1}^{n-1}\frac{1}{i} \sum _{j=1}^{i}w_{j} y_{j}, \end{aligned}$$

(14)

$$\begin{aligned} \widehat{V}(y)=\frac{1}{\left( 1+n\sum _{s=2}^{n-1}\frac{1}{s} \right) \mu (y)} \sum _{i=1}^{n-1}\frac{1}{i} \sum _{j=n-i+1}^{n}w_{j} y_{j} -1 \end{aligned}$$

(15)

where the sums are up to $n-1$ to ensure the accuracy of the indices and $w_{j}$ are the sample weights.⁷

Sampling error and confidence intervals for polarization indices based on Bonferroni and De Vergottini measures have been estimated using bootstrap techniques, based on 100 replications of samples with size 1000.⁸ Estimates related to Foster and Wolfson polarization measure have been obtained using DASP - Distributive Analysis Stata Package [2].

Figure 1 depicts bipolarization estimates, as well as their confidence intervals, according to Foster and Wolfson index. The corresponding numbers are available in Table 4 of the Appendix. The picture reveals that the most polarized countries in Europe are the Baltic countries (Estonia, Lithuania, Latvia) and Bulgaria, while the least polarized countries are in the Scandinavian region (Norway, Finland) and in eastern Europe (Hungary, Czech Republic, Slovakia). A similar pattern occurs also for inequality, as measured by the Gini index and shown in Table 1 in the Appendix: the most polarized countries, indeed, exhibit the highest between-group inequality levels. Over the decade under consideration, some countries registered a significant increase in polarization (in particular, Lithuania, Latvia, Bulgaria, Romania, Hungary), mainly due to a significant increase in the between-group component as shown in Table 1, while other countries remained quite stable. In none of the European countries polarization has significantly decreased.

Fig. 1

Foster and Wolfson polarization index and 95% confidence intervals, for European countries and years 2010, 2015 and 2019. Source: Our elaboration on EU-SILC dataset

When estimating polarization with the measure based on Bonferroni index (Fig. 2; Table 5), we observe that the countries with the highest and the lowest levels of polarization are basically the same detected by Foster and Wolfson index (in Fig. 1; Table 4). However, changes over time are more pronounced with the Bonferroni-based index, revealing that significant changes have occurred mainly in the bottom part of the income distribution. This is also confirmed by Fig.  3, which plots the polarization levels of each country in 2015 and in 2019 compared to the level at the beginning of the period of analysis, the year 2010. We note, in particular, that the increases registered in Bulgaria, Lithuania and Latvia are more pronounced when we attach higher weight to lower incomes. According to the Bonferroni-based index some countries registered a reduction in polarization from 2010 to 2015 or 2019 (in particular, Slovakia), mainly due to a decrease in the between-group inequality component, as shown in Table 2.

Fig. 2

Bonferroni-based polarization index and 95% bootstrap confidence intervals, for European countries and years 2010, 2015 and 2019. Source: Our elaboration on EU-SILC dataset

Fig. 3

Bonferroni-based polarization across European countries, comparison over time. Source: Our elaboration on EU-SILC dataset

Moving now to analysing polarization pattern based on De Vergottini measure (see Figs. 4, 5 as well as Table 6 in the Appendix) we note a quite different trend overtime if compared to the previous polarization indices. When we attach higher importance to changes occurring at the top incomes, polarization seems to increase mainly in the Baltic countries (Latvia, Lithuania, Estonia), in Slovakia and also in some of the southern European countries (such as Spain and Greece). On the contrary, polarization decreases mainly in Bulgaria (due to a reduction in the between-group component, as shown in Table 3), Cyprus and Denmark (as a consequence of an increase in the within-group inequality).

Fig. 4

De Vergottini-based polarization index and 95% bootstrap confidence intervals, for European countries and years 2010, 2015 and 2019. Source: Our elaboration on EU-SILC dataset

Fig. 5

De Vergottini-based polarization across European countries, comparison over time. Source: Our elaboration on EU-SILC dataset

We now compare the rankings of the European countries provided by the three polarization measures (see Figs. 6, 7). The countries in the top positions of the ranking exhibit lower polarization levels. For most countries the rankings provided by the three polarization indices are very similar, while in a bunch of countries the rankings strongly reverse if we change polarization measure. In particular, the Scandinavian countries (Norway, Sweden, Finland) and some northern countries (Belgium, Denmark, Luxembourg) are among the least polarized countries according to the Foster and Wolfson and to the Bonferroni-based measures, that is when we attach more weight to bottom or middle incomes, while they move to the group of the highest polarized countries, when we attach more weight to top incomes following the De Vergottini approach. These results apparently seem in contradiction, but actually reveal that polarization in these countries is mainly due to inequalities in top incomes rather than in inequalities among bottom incomes. This is also confirmed by Table 3, according to which the above-mentioned countries reveal high levels of De Vergottini inequality measure. On the contrary, the Baltic countries as well as some southern countries (Greece, Portugal, Cyprus) are classified among the least polarized countries according to De Vergottini-based measure but among the most polarized according to Bonferroni-based and Foster and Wolfson measures. Here again, the reason of this discrepancy is mainly due to the fact that in these countries the within group inequality is higher among the poorer than among the richer.

Fig. 6

Ranking comparison of the European countries, year 2010. Source: Our elaboration on EU-SILC dataset Note: Top positions in the ranking correspond to lower polarization levels

Fig. 7

Ranking comparison of the European countries, year 2019. Source: Our elaboration on EU-SILC dataset Note: Top positions in the ranking correspond to lower polarization levels

In particular, if we focus on year 2019, Fig. 7 shows that Bulgaria, Latvia, Lithuania and Portugal are at the bottom of the ranking provided by the Bonferroni-based polarization measure (i.e. high polarization) but at the same time at the top of of the ranking provided by De Vergottini (i.e. low polarization); this is due to the fact that these countries exhibit the highest level of inequality within the group of the poor. On the contrary, countries that are at the top of the Bonferroni ranking (such as Slovakia, Slovenia and Norway) fall instead to the bottom part of the De Vergottini ranking, since they registered the highest levels of inequality in the group of the rich, to which De Vergottini-based measure attaches more importance.

In conclusion, the bipolarization indices that we propose in this empirical application provide more insights than the traditional analysis of income polarization, by introducing the flexibility of attaching different weights to the different part of income distribution.

4 Concluding remarks

In this paper we have analyzed how income polarization has changed over the last decade across European countries by means of new Bonferroni-based and De Vergottini-based polarization measures proposed in Ciommi et al. [12]. Different from traditional measures of polarization, the indices proposed in this paper are sensitive to progressive transfers, attaching more importance to some part of the income distribution. In particular, Bonferroni-based polarization index is more sensitive to income transfers involving poorer individuals, while De Vergottini-based measure to transfers occurring among richer people.

In the empirical application we compared European countries over the period 2010–2019 using EU-SILC data. Results revealed significant changes in polarization over the last decades for most countries. The empirical application showed how the new indices enrich the analysis based on standard measures of income polarization and how they contribute to disentangle the different faces of income inequality and polarization. Here we have focused on the case of two groups separated by the median. Future research may explore two additional directions: extending the new indices for (i) the presence of more than two groups, and (ii) groups that may also overlap. These extensions may accommodate researchers who assume that the population is made up of more than two groups, even if the groups are formed by characteristics other than income—a situation that generally introduces some overlap between the groups.

Appendix

See Tables 1, 2, 3, 4, 5 and 6.

Table 1 Gini inequality index and its subgroup decomposition, for European countries and years 2010, 2015 and 2019. Source: Our elaboration on EU-SILC dataset

Table 2 Bonferroni inequality index and its subgroup decomposition, for European countries and years 2010, 2015 and 2019. Source: Our elaboration on EU-SILC dataset

Table 3 De Vergottini inequality index and its subgroup decomposition, for European countries and years 2010, 2015 and 2019. Source: Our elaboration on EU-SILC dataset

Table 4 Foster and Wolfson polarization index and and 95% confidence intervals, for European countries and years 2010, 2015 and 2019. Source: Our elaboration on EU-SILC dataset

Table 5 Bonferroni-based polarization index and and 95% bootstrap confidence intervals, for European countries and years 2010, 2015 and 2019. Source: Our elaboration on EU-SILC dataset

Table 6 De Vergottini-based polarization index and and 95% bootstrap confidence intervals, for European countries and years 2010, 2015 and 2019. Source: Our elaboration on EU-SILC dataset