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The Gini index is well-known for a single measure of inequality. The purpose of this article is to explore a matrix structure of the Gini index in a setting of multiple source income. Using matrices, we analyze the decomposition of the Gini index by income source and derive an explicit formula for the factors in terms of the associated percentile levels based on aggregated data reporting. Each factor is shown to be the sums of the two split off parts of the income within a percentile bracket. Both have unequalizing and equalizing contribution to the total inequality, respectively. We use R code and apply the methodology to several data sets including a sample of European aggregated income reporting in 2014 for illustration. A byproduct from the Gini decomposition provides a matrix approach to the decomposition of the associated Lorenz curve in terms of the density distribution matrix and a Toeplitz matrix.

Decomposition of the Gini index by income source for aggregated data and its applications