Additive decomposition of the Gini index by subgroups: Assigning the overlap effect to the intra- and intergroup components

The Gini index decomposition can be obtained additively from the appropriately weighted sum of inequality within and between the groups that comprise the total distribution, provided an overlap component is included. This paper, prepared for teaching purposes, isolates the overlapping component resulting from within- and between-group effects, interpreting how each contributes to total inequality. To this end, an invariant global income distribution is first used for illustrative purposes. This distribution is divided into two distinct groups using all possible permutations, thereby altering the distribution within and between subgroups. Subsequently, the paper demonstrates the actual decomposition of income inequality for Spanish households across different autonomous communities and cities, using data from the 2024 Spanish Survey of Living Conditions (ECV). A Gini index decomposition that disregards the overlapping effect -treating it merely as a residual- incorrectly assigns within- and between-group contributions to overall inequality. When both overlapping effects are properly assigned, the interpretation shows that the within-group component of the overlap increases overall inequality, while the between-group component decreases it. The simulated calculations are performed in Excel for transparency but are replicated in STATA using the code detailed in the appendix.