Understanding Classical Decomposability of Inequality Measures: A Graphical Analysis

TL;DR

Representing income distributions of three individuals as points on the income-share simplex $Δ_2$, the paper develops a geometric diagnostic framework to study classical decomposability (within- and between-group) for common inequality measures. It shows that Mean Log Deviation ($ ext{MLD}$) and Theil ($GE(1)$) align with population- and income-weighted decomposability in their canonical forms, while Gini and the coefficient of variation (CV) exhibit qualitative, location-specific violations that the simplex geometry localizes. The framework translates axioms into testable geometry via level sets, rays, and slices, enabling direct comparison across measures and revealing where decomposability fails and why. The approach complements path-independent Shapley-type decompositions and extends to other decomposability notions, offering a practical, didactic tool for interpreting subgroup analyses in inequality research.

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.

Figures (8)

• Figure 1: Illustrations of decomposability \ref{['eq:DCP']} for MLD . Left: Level curves for $g_{1,MLD}$ (solid lines) and rays constant at $\frac{z_1}{z_1+z_2}$ (dashed lines). Right: Level curves for $g_{2,MLD}$ (solid lines) and horizontal lines constant at $z_3$ (dashed lines).
• Figure 2: Illustrations for Gini. Left: Level curves for $g_{1,Gini}$ and rays constant at $\frac{z_1}{z_1+z_2}$. Right: Level curves for $g_{2,Gini}$.
• Figure 3: Illustrations for CV. Left: Level curves for $g_{1,CV}$ and rays constant at $\frac{z_1}{z_1+z_2}$. Right: Level curves for $g_{2,CV}$.
• Figure 4: Illustrations for Theil (GE(1)). Left: Level curves for $g_{1,Theil}$ and rays constant at $\frac{z_1}{z_1+z_2}$. Right: Level curves for $g_{2,Theil}$.
• Figure 5: Illustrations for Theil. Left: Level curves for $g^{\mathcal{I}}_{1,Theil}$ and rays constant at $\frac{z_1}{z_1+z_2}$. Right: Level curves for $g^{\mathcal{I}}_{2,Theil}$.