On the measurement of inequality

Abstract

Conventional measures of income inequality, such as the Gini coefficient and the coefficient of variation, often lack explicit theoretical foundations and rely on implicit social welfare assumptions. By treating the measurement of inequality as formally equivalent to measuring risk under uncertainty, rankings of distributions can be established based on the concavity of the social welfare function. When comparing distributions with the same mean, a complete ranking independent of the specific utility function is possible if and only if the Lorenz curves do not intersect. This condition corresponds to the principle of transfers, where any redistribution from a richer to a poorer individual increases social welfare. If Lorenz curves cross, however, summary statistics provide conflicting results because they weight transfers at different points of the distribution differently. A more robust approach utilizes the concept of equally distributed equivalent income—the level of income per head which, if shared equally, would generate the same social welfare as the observed distribution. This formulation yields an inequality index that is invariant to proportional shifts and incorporates an explicit parameter for inequality-aversion. Empirical application of this measure to international data demonstrates that the ranking of countries is highly sensitive to the degree of inequality-aversion chosen. Consequently, the measurement of inequality requires the direct specification of a social welfare function to ensure that the resulting rankings reflect explicit value judgements. – AI-generated abstract.