🤖 AI Summary
This study investigates whether classical inequality measures satisfy the decomposability axiom in the context of three-person income distributions and reveals the geometric manifestations of their violations. By modeling such distributions on a two-dimensional income-share simplex, the decomposition of overall inequality into within- and between-group components is recast as a geometric constraint. The paper provides the first visual characterization—within the simplest nontrivial setting—of the decomposition behavior of prominent indices, including the mean log deviation, Gini coefficient, coefficient of variation, and Theil index. It clearly identifies the distinct geometric patterns through which each measure deviates from strict decomposability, thereby deepening the understanding of their structural properties and offering an intuitive basis for selecting and comparing inequality measures.
📝 Abstract
This paper's objective is pedagogical and interpretive. Namely, it gives a simple geometric analysis of classical (by which I mean population-share-weighted or income-share-weighted) inequality decomposability in the simplest nontrivial setting of three individuals. Income distributions in this case can be represented as points on the two-dimensional income-share simplex. In this representation, classical decomposability translates into concrete geometric restrictions of within- and between-group components. The geometric framework makes it possible to localize and compare violations of decomposability across inequality measures. The analysis is applied to the Mean Log Deviation, the Gini coefficient, the coefficient of variation, and the Theil index.