Gini's crossbow

The Gini coefficient remains a popular gauge of inequality throughout the social and natural sciences because it is visually striking and geometrically intuitive. It measures the “gap” between a hypothetically equal distribution of income or wealth and the actual distribution. But not all inequality curves yielding the same Gini coefficient are unequal in the same way. The Lorenz asymmetry coefficient, a second-order measure of asymmetry, provides further information about the distribution of income or wealth. The mean value theorem implies that every Lorenz curve has a point at which its first derivative is equal to 1.  The position of the Lorenz curve at that point defines the Lorenz asymmetry coefficient.  To add even more interpretive power to applications of the Gini coefficient, this paper proposes a new angular measure derived from the Lorenz asymmetry coefficient. Adjusted azimuthal asymmetry, or AAA, is the angular distance of the Lorenz asymmetry coefficient from the axis of symmetry, divided by the maximum angular distance that can be attained for any given Gini coefficient. The value of AAA, which falls between –1 and 1 inclusive, defines the precise extent to which an unequal distribution of income or wealth is unequal because it is dominated by the richest members of the group, or because the cohort of poor members within the group is so large.  The former condition may be described as "Gulliverian inequality"; the latter, as "Lilliputian inequality."  Public policies designed to reduce inequality should account not only for first-order inequality (as measured by the Gini coefficient), but also the AAA as a second-order measure of inequality reflecting the direction and extent of internal asymmetry within a particular distribution of income or wealth.