Bonferroni and De Vergottini are back: New subgroup decompositions and bipolarization measures

Abstract

This paper proposes new bipolarization indices based on the Bonferroni and De Vergottini indices of inequality. These new indices follow the approach of Foster and Wolfson and are based on a new subgroup decomposition of the Bonferroni and De Vergottini indices of inequality. We also provide the conditions under which the new polarization indices satisfy the Increased Spread and Increased Bipolarity axioms. Finally, a simulation study is performed to compare the different sensitivity of the new bipolarization indices to progressive transfers. An empirical application based on EU-SILC data for Italy in the period 2007-2011 also shows the appeal of our proposal.

Introduction

In recent years, the concept of income polarization has received increasing attention from scholars in different fields (economics, statistics, econometrics, sociology, etc.). Income distribution is said to be polarized if the relative frequency of observations is low near the central value and high at the tails, such that the society is split into groups that are homogeneous within yet different from each other. Monitoring the degree of polarization in a given income distribution means measuring how the poor are becoming poorer and the rich are getting richer and hence how distant these two groups are from each other. The more separated the two groups are but the more cohesive they are within, the harder it will be for them to communicate and interact.

Polarization immediately recalls inequality: both concepts address the disparity in a society, but in different ways. Inequality focuses on interpersonal differences over the entire distribution, while polarization monitors two contrasting forces: (i) inequality between population groups; and (ii) equality within them. For example, if two income groups are further separated by increasing economic distance, we would expect both inequality and polarization to increase. However, local income differences may decrease and lead to two better-defined and more homogeneous groups. In this case, inequality could decrease while polarization may increase. The increasing interest in analyzing income and social polarization is driven by the idea that high levels of polarization in terms of the presence of contrasting groups and a weak, hollowed-out middle class can lead to an unstable society, causing possible social conflicts and revolts (Esteban and Ray [30]).

There are two strands of literature on income polarization. The first, going back to Wolfson [74] and [75] and Foster and Wolfson [32] and [33], focuses on measuring the shrinking middle class, monitoring how income distribution spreads from its center. The second strand, originating in Esteban and Ray [30], focuses on the rise of separated income groups: polarization increases if the population groups become more homogeneous within and more separated from each other. These pioneering contributions have been followed by many others, such as Wang and Tsui [73], Gradín [50], Chakravarty and Majumder [21], D'Ambrosio [25], Zhang and Kanbur [78], Duclos et al. [28], Anderson [4], Esteban et al. [31], Massari et al. [57], Chakravarty and D'Ambrosio [20], Lasso de la Vega et al. [55], Yitzhaki [77], Pittau et al. [65], Permanyer [61], Silber et al. [69], Lasso de la Vega and Urrutia [54], Chakravarty and Majumder [22], Gigliarano and Mosler [39], Bárcena-Martín et al. [10], and Nowak et al. [59]. For a review, we refer to Permanyer [62] and Gigliarano [38], among others.

In addition to these two main approaches, a fuzzy approach has also recently been proposed, according to which a person is not identified with only one group, but rather with several groups at once with different levels of intensity. Following this idea, Guevara et al. [51] provide a new polarization index based on fuzzy membership functions that are assembled through aggregation operators. Fuzzy set theory has become popular in the study of complex phenomena such as inequality (see Aristondo et al. [8], Aristondo et al. [9], Bortot and Pereira [16], García-Lapresta and Marques Pereira [35] and García-Lapresta and Marques Pereira [36], among others) and poverty measurement (see García-Lapresta et al. [34], García-Lapresta and Marques Pereira [35], Aristondo and Ciommi [6], García-Lapresta and Marques Pereira [36] and Aristondo and Ciommi [7], among others), and this suggests its use also in the study of polarization. We note, however, that a fuzzy approach was also implicitly considered in Esteban and Ray's work through the identification function or membership function. In particular, the identification function proposed by Esteban and Ray [30] is a function of the relative size of a given group.

In this paper, we follow the strand of income polarization literature based on Foster and Wolfson's approach. Starting with a generalization of the Foster and Wolfson index proposed by Rodríguez and Salas [67], we introduce two new polarization indices based on the Bonferroni [15] and De Vergottini [26] inequality indices, respectively. Our aim is therefore to investigate whether both new indices provide additional or more detailed information than traditional measures of polarization.

At a time when Gini's scientific ideas heavy influenced the research activity of Italian statisticians (see, e.g., Giorgi [42]; Giorgi and Gubbiotti [47]), Carlo Emilio Bonferroni [15] proposed the Bonferroni inequality index. His purpose was simply to highlight the possibility of constructing indices as simple as the Gini concentration index with similar properties. In fact, the Bonferroni index is more sensitive than the Gini index to lower levels of income distribution in the sense that it gives more weight to income transfers among the poor (Nygard and Sandstrom [60]). This makes the Bonferroni index particularly suitable for investigating poverty (Giorgi and Crescenzi [44]; Giordani and Giorgi [40]). For about fifty years, the Bonferroni index remained almost forgotten since it was opposed by Corrado Gini and his followers, who tried to prevent any measures of inequality from overshadowing the concentration ratio.

A fundamental contribution to the renewed interest in the Bonferroni index and other inequality indices (including De Vergottini's) is due to Walter Piesch [63], who discussed a series of links between inequality measures in his book Statistische Konzentrationsmasse. Subsequently, the work of Nygard and Sandstrom [60] expanded such studies, reaching a larger audience of scholars.

The Bonferroni index has recently been reassessed with the study of its features and new interesting applications in social and economic contexts (Bárcena-Martín and Silber [12], [13], [14]; Chakravarty and Muliere [24]; Chakravarty [18]). Various other important properties have been analyzed by Aaberge [2], Aaberge et al. [3], Bárcena-Martín and Imedio [11] and Imedio-Olmedo et al. [52], among others. Some inferential results of the Bonferroni index have also been investigated (e.g., Giorgi and Mondani [48] and [49]; Giorgi and Crescenzi [43]).

Some years later, in 1950, Mario De Vergottini [26] proposed another index of inequality, the De Vergottini index, which is more sensitive than the Gini index to the right tail of the income distribution, i.e., it is more sensitive to income transfers among the rich. He also obtained a general formula from which various indices of inequality can be derived (including the Gini, Bonferroni, and De Vergottini indices), highlighting, as Bonferroni did previously, that the Gini index is just one of many indices with similar features and properties.

Motivated by these very different types of sensitivity, this paper considers and compares the three indices (Gini, Bonferroni, and De Vergottini) from the view of polarization.

The present work is organized as follows: in Sections 2 and 3 we review the Gini, Bonferroni, and De Vergottini inequality indices, while in Section 4 we provide a brief review of the main contributions proposed in the literature on polarization measurements. In Section 5 we propose a new subgroup decomposition for Bonferroni and De Vergottini inequality indices, which is used in Section 6 to propose new polarization indices based on these indices of inequality. Section 7 highlights their particular properties through a simulation study, while Section 8 illustrates a simple application to EU-SILC data referring to Italy. Section 9 provides a conclusion. Appendix A contains the proofs of all propositions.