Properties of Lorenz Curves for Transformed Income Distributions

Abstract

Redistributions of income can be considered as variable transformations of the initial income variable. The transformation is usually assumed to be positive, monotone-increasing and continuous, but discontinuous transformations have also been discussed recently. If the transformation is a tax or a transfer policy, the transformed variable is either the post-tax or the post-transfer income. A central problem has been the Lorenz dominance between the initial and the transformed income. This study considers analyses of other properties of the transformed Lorenz curves, especially its limits. We take in account mainly two cases (a) the transformed variable Lorenz dominates the initial one and (b) the initial Lorenz dominates the transformed one. For applications, the first case is more important than the second. The limits obtained are not accurate for a specific transformation, but do hold generally for all distributions and a broad class of transformations so that, if one pursues general conditions the inequalities obtained cannot be improved.