A Note on the Relationship between Additive Separability and Decomposability in Measuring Income Inequality

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June 2019
Paper author(s): 
Ben Fine
Poverty - Inequality - Aid Effectiveness

The purpose of this note is to offer some original technical results in the theoretical measurement of inequality. Whilst most practitioners are content to work with one or other measure, with Gini for example to the fore, and discuss the empirical results that follow from the data, such pragmatism involves a certain degree of arbitrary ethical judgement over how more for one rather than another should be assessed. At a deeper level of principle, constructing measures of inequality proceeds by specifying conditions like homogeneity for which multiplying all incomes by a common factor should leave a measure unchanged. Such conditions are the starting point for this contribution, drawing upon a rich literature that already exists. More specifically, this note explicitly explores the relationships between additive separability and homotheticity of measures of welfare (closely related to derived measures of inequality), and homogeneity and decomposability in direct measures of inequality, drawing upon the previous literature along the way to make this possible. An interrogation is made of the resonances and dissonances between the classic contributions of Atkinson (1970) and Shorrocks (1980). 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 income inequality. The gap between the two, however, can be closed by adding an extra term around total income to the measure of welfare.


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