Lorenz curves – history, state and some recent results
Lorenz curves are established under general conditions following an integral transform approach aging approximately 40 years. The derivation of this approach is given together with quite a few explanations, well known and widely-spread (yet well hidden?) facts from the literature, and some fresh views. The approach is based on relating the underlying distributions, typically income distributions, to probability distributions.
The equity calculus, which is inspired by a notion of relative poverty, is shown to deliver a variety of differential equations for Lorenz curves. Most admit one-parametric closed form solutions while solvability of others is still unsettled.
Key words: generalized function inversion, integral transforms, absolute continuity, Pareto distribution, differential equations, symbolic solutions