But not all Lorenz curves yielding the same Gini coefficient are unequal in the same way. Gini’s Crossbow, https://ssrn.com/abstract=2608850, described two measures of internal asymmetry. The Lorenz asymmetry coefficient exploits the mean value theorem, which implies that every Lorenz curve has a point at which its first derivative equals 1. A related measure, adjusted azimuthal asymmetry, is the angular distance of the Lorenz asymmetry coefficient from the axis of symmetry, divided by the maximum angular distance for any given Gini coefficient. That value describes how inequality arises from dominance by a population's richest members, or from the size of the poorest cohort. Public policies should address both first-order inequality (as measured by Gini) and second-order inequality reflecting the direction and extent of internal asymmetry.
This paper proposes another method for measuring internal asymmetry. The line of symmetry at y = 1 – x bisects the Lorenz curve precisely where the poorest x ∈ (0, 1) of the population commands 1 – x of social wealth. The ratio of either half of the area beneath the Lorenz curve to its integral reports internal asymmetry.
A proper understanding of internal asymmetry questions reliance on the popular Pareto distribution. The assertion that "20 percent of the population has 80 percent of social wealth” epitomizes the banality and potential misuse of the Pareto distribution.
The Pareto distribution’s failure to model inequality stems from its mathematical properties. The Pareto distribution systematically reports “Lilliputian” inequality, driven by the size of the poorest cohort. A competing model, derived from the Lambert W function, offsets the Pareto distribution’s mathematical bias by reporting “Gulliverian” inequality attributable to the wealthiest cohort.
Closer examination reveals further insight into the physical economics of inequality. The value of the Lorenz curve at x = 1 is 1, which provides unit measures of population and gross domestic product analogous to Planck units in physics. “Unitary” treatment of the Lorenz curve extends to its integral and to its first derivative. The integral is the basis for the Gini coefficient. The first derivative reports income as a multiple of per capita income, which in turn equals 1.