A matrix approach to the Gini index decomposition by subgroup and by income source

Abstract

The literature offers two main ways of decomposing the Gini index: decomposition by population subgroup and by income source. This article proposes merging the two decomposition dimensions by suggesting a matrix formula for the Gini index which permits the simultaneous decomposition by subgroup and by income source. Using this multi‐decomposition, one can investigate the role of the interaction between the subgroup and the source components in determining the overall inequality. We apply the methodology to sample data on Italian household incomes collected in 2008.

Keywords:

• inequality measures
• Gini index
• decomposition

JEL Classification::

• D31
• D63
• C43

Acknowledgements

The author wishes to thank the two anonymous reviewers for their constructive comments. The author also expresses his gratitude to Maria Monti and Achille Vernizzi for helpful discussions.

Notes

¹ Monti (2007) showed that the net between-group inequality is equivalent to the well-known between-group inequality defined by Mookherjee and Shorrocks (1982), the latter accounting for differences in subgroup average incomes. A detailed comparison between the Dagum decomposition and other subgroup decompositions is given by Radaelli (2010).

² Given two subgroups of income receivers, any pair formed by two income receivers belonging to different subgroups is said to transvary if the sign of the difference between their incomes is different from the sign of the difference between the corresponding subgroup average incomes. If there is at least one transvarying pair, the two subgroups are said to transvary. When dealing with income, this definition of transvariation is equivalent to the definition of the overlapping concept. In fact, two subgroups of income receivers are said to overlap if at least one income belonging to the subgroup with lower average income is higher than at least one income of the subgroup with higher average income. Thus, we use, throughout the article, transvariation and overlapping as synonymous terms.

³ Consider four income receivers partitioned into two subgroups, and let the incomes of subgroup 1 be given by 5 and 2 and those of subgroup 2 by 4 and 1. The vector of incomes sorted in decreasing order is while the vector of incomes sorted by the ordering A is . There is only one pair of transvarying incomes (y ₂ = 4, y ₃ = 2). Therefore, shows a +1 element in its (2,3) entry and a −1 in the (3,2) entry instead of, respectively, a −1 and a +1 element as shown in . The remaining elements of coincide with the elements of .

⁴ The scalar in Equation 27 is also called pseudo-Gini coefficient of an income source (Fei et al., 1978). It may be worth mentioning that ranges from to , where denotes the Gini index of income from source m.

⁵ Obviously, the use of an ordering criterion (e.g. median ordering, poverty index ordering, etc.) different from the mean ordering implies that does not generally reflect the disparities between subgroup mean incomes.

⁶ As noted in Mussard and Savard (2012), the use of the median ordering instead of the arithmetic mean ordering can be a suitable choice for empirical analysis, since the median is less sensitive than the arithmetic mean to very high incomes in the upper tail of the income distribution and to erroneous incomes included in the data set. Suppose the subgroup ordering is based on the median. Let be the permutation matrix which re-arranges the vector by decreasing subgroup median income (first key of ordering) and by subgroup per capita income in decreasing order (second key of ordering). By replacing with in Equations 22 and 23, and are equivalent to and with and , respectively. In this case, is equal to the difference between the inequalities from the income pairs in which the two incomes do not overlap (i.e. the sign of and the one of are the same) and the inequalities from the income pairs in which the two incomes overlap (i.e. the differences and are of opposite sign). Hence, does not measure the inequality between subgroup medians, and in general its interpretation is neither intuitive nor meaningful.

⁷ The concept of natural decomposition was introduced by Shorrocks (1982) to indicate a source decomposition that follows from the functional form in which an inequality measure is expressed. Accordingly, it consists of a function showing the relative contribution of an income source to the inequality of the total income (Paul, 2004). For instance, Shorrocks (1982) re-arranged the Gini index as the weighted average of the pseudo-Gini coefficients for the various sources and termed this decomposition a “natural decomposition”. See also Lerman and Yitzhaki (1985).

⁸ Even though the income distribution includes negative incomes, the Gini index can be calculated. In this case, since the inclusion of negative incomes implies that the Gini index can take a value greater than 1, the Gini index should be adjusted to guarantee that its range is in the interval (see Chen et al., 1982; Berrebi and Silber, 1985). In the present application, we decided to omit negative incomes to ensure the fulfilment of the principle of normalization without adjusting the Gini index. Moreover, we opted for the exclusion of zero incomes on the basis of the following two observations. First, each zero income shows a zero value for any income source, and hence it seems less interesting for this application which is focused on the source decomposition of the various subgroup components of the overall inequality. Second, zero incomes constitute a negligible part of the sample (less than 0.2% of the sample).

⁹ See Faliva (1996), p. 157.