Measuring inequality and social stratification with Lorenz curvature

We construct a continuous family of inequality and social stratification indices based on the curvature of the Lorenz curve. We study the inequality axioms of the family and find that they are satisfied only with the so-called stratification-aversion parameter $\alpha$ set to 0 -- in all other cases, these constraints cannot be satisfied in a strict sense. With $\alpha = 0$, the index has a very simple closed form and its value can be easily approximated. We study the values of the index on World Bank inequality data and see how the rankings across a wide selection of countries change as $\alpha$ is varied. Spearman and Kendall rank-correlation matrices with other common indices are also computed and analyzed. We find the curvature index to deviate consistently from the comparison indices, albeit less so for consumption- than income-based countries.