# The Gini Index Decomposition and Overlapping Between Population Subgroups

## Introduction

A century after the introduction of the Gini index, topics related to inequality measurement are central in the political and economic debate. Financial crises have resolved all around the world, aggregate production and income are at the pre-crisis levels, and stock markets indexes hit new records, but the mood is far from the enthusiasm which usually characterizes post-crises periods. A persistent gloom hampers entire countries, insecurity is a global and dominant feeling, protectionism reshapes world trade, nationalism is rising, and countries like United Kingdom prefer an individual to a common path.

Rising inequality is the key that reconciles these two situations. Production, income, and wealth have grown, but their growth is unequally shared. If we look only at the GDP, or at the Dow Jones index, we don't get a fully informative picture. The basic information set should be improved by adding an inequality measure.

The debate about GDP, the contributions by the Sarkozy commission or similar committees, and the proposals related to alternative GDP evaluation systems are an important, but still insufficient step. Starting from the daily public information provided by newspapers and televisions, moving to all the actors of society, up to the targets of the national and supranational political strategies and to the data elaborated by the statistical agencies, we need to include inequality at the top of our priorities.

An increase in GDP is a good news only if its growth is not too unequally shared, while the combination of an increase of both GDP and inequality would represent a bad news for most people.

If the overall inequality level represents a fundamental reference, the information included in it can be successfully exploited by means of inequality decomposition. Inequality decomposition allows us to provide insights about the inequality structure, rank the different inequality factors, and assess their relevance. In this regard the aspect of interest refers to the measurement of inequality between subgroups, for which the literature presents many contributions, thus requiring a critical comparison.

An issue strictly connected to the inequality between subgroups is the overlap between subgroups, a topic which Gini studied with his usual strong interest and determined motivation. While we do not encounter particular difficulties in the absence of overlapping, its presence complicates the measure of the contribution to the overall inequality related to the differences between the subgroups.

Here we propose a joint analysis of Gini index decomposition and overlapping by developing a comparison between two different approaches: from one side methods based on the subgroup means, and from the other evaluations performed on the basis of all the characteristics of the subgroup distributions. The combination of the two approaches overcomes the disadvantages of the single proposals, thus avoiding the risk of underestimating the real effect of the inequality factors under study.

The next section outlines the issues related to overlapping and its measurement, while Section 4.3 presents the main aspects of the Gini index decomposition. Section 4.4 illustrates a development of overlapping and Gini index decomposition focused on gender gap analysis, with a case study related to Italian personal income. From Section 4.2 to Section 4.4 an illustrative example is carried out, which works as a guideline for the reader, attempting to make clearer the indicators and the measures presented in the paper. Section 4.5 concludes.

## Overlapping

Overlapping represents a real challenge when measuring inequality. Its presence increases the complexity and the difficulties of the measurement process and makes the interpretation of the final results more slippery. However, as frequently happens when challenges are involved, it also leads to innovative proposals and original solutions.

How to deal with the presence of overlapping is the main argument in the debate on the inequality decomposition. It represents one of the pillars of the classification into additively and non-additively decomposable indices (Shorrocks, 1980) and it is the motivation for the majority of contributions on inequality decomposition.

Overlapping has a direct effect on the interpretation of the inequality structure and, consequently, on the inequality decomposition.

When we divide the total population in subgroups (for example by gender, educational level, etc.) the case of non-overlapping subgroups allows an immediate evaluation of the contribution to overall inequality attributable to the factor used to partition the population into subgroups. The absence of overlapping indicates in itself a strong inequality factor. Depending on the size of the differences between the subgroups, we can rank the related factors and thus achieve the inequality structure. In this case overall inequality can be successfully decomposed in only two components, the inequality within and the inequality between subgroups, without any question on the additivity of the decomposition.

Factors able to originate non-overlapping subgroups, or real data situations in which this is possible, are extremely rare, while a certain degree of overlapping represents the most usual and consolidated case. Increasing levels of overlapping indicate a weaker influence of the factor on inequality until, when the subgroups completely overlap, the influence of the factor on inequality reaches its minimum. In this case, the two previous components, the inequality within and the inequality between, are no longer sufficient to decompose overall inequality and we need to introduce a third term, able to take into account the presence of overlapping units and their effect on inequality.

In the decompositions based on only two components, the third term is obtained as a residual, while, in the decompositions developed on three components, the third term evolves according to the expressions proposed by different authors.

Overlapping is also strictly connected to the effectiveness of the economic policy actions implemented to reduce inequality.

The absence of overlapping allows governments to operate directly on the poorest (or the richest) subgroup, within a simplified framework where the effects of any economic policy are easily detectable.

In presence of overlapping, it becomes fundamental to disentangle the effects related to the overlapping units, which could strongly influence the final results of the policy action. The overlapping units of the richer subgroup would suffer a disadvantage compared to the overlapping units of the poorer subgroup, reducing the effectiveness of the intervention.

A last, but not least relevant feature of overlapping refers to the interest given to this topic by Gini. During his long polyhedral research activity, Gini dealt with numerous subjects, among which was overlapping, which Gini called 'transvariazione' in 1916 (Gini, 1959). An anthology of Gini's works on overlapping was published in 1959 (Gini, 1959), almost unnoticed due to being written in Italian (Deutsch and Silber, 1997).