A note on the relationship between additive separability and decomposability in measuring income inequality

This note introduces original technical results in the theoretical measurement of inequality by specifying the relationships between additive separability and homotheticity (of measures of welfare closely related to measures of inequality), and decomposability and homogeneity in measures of inequality. More specifically, an interrogation is made of the resonances and dissonances between the classic contributions of Atkinson and Shorrocks, which are key representatives, respectively, of the 'social welfare function' and the 'axiomatic' approaches to measuring inequality. In brief, in the presence of otherwise common assumptions, it is shown that additive separability and homotheticity of welfare are stronger combined conditions than decomposability and homogeneity (of degree zero) of income inequality. The gap between the two, however, can be closed by adding an extra term around total income to the measure of welfare, allowing for wider considerations of the relationship between social welfare, total income, and the distribution of individual incomes.