Understanding Classical Decomposability of Inequality Measures: A Graphical Analysis

TL;DR

This paper introduces a geometric framework to analyze the classical decomposability of inequality measures, revealing that decomposability is not binary and identifying key residuals causing failures.

Contribution

It provides a novel graphical diagnostic method to localize and characterize violations of decomposability in inequality measures.

Findings

Decomposability is not a binary property, with measures failing in distinct ways.

The between-group residual is the main source of failure across measures.

Negative residuals can make decompositions uninterpretable, especially for certain measures.

Abstract

This paper develops a geometric diagnostic framework for classical inequality decomposability. Representing the simplest nontrivial setting of three-person income distributions as points on the two-dimensional income-share simplex, we translate population-share-weighted and income-share-weighted decomposability into concrete geometric restrictions on within- and between-group residuals, making it possible to localise and characterise violations across measures. Applied to the Mean Log Deviation, the Gini coefficient, the coefficient of variation, and the Theil index, the analysis shows that decomposability is not a binary property as measures fail in qualitatively distinct ways, and the between-group residual is consistently the primary locus of failure. Negative between-group residuals render the decomposition uninterpretable and arise for the coefficient of variation and the Theil index under population-share weighting, and for the Mean Log Deviation under income-share weighting. Stylised numerical examples quantify the resulting misinterpretation scenarios for applied researchers.