Parametric L orenz Curves and the Modality of the I ncome Density Function

Similar looking L orenz curves can imply very different income density functions and potentially lead to wrong policy implications regarding inequality. This paper derives a relation between a L orenz curve and the modality of its underlying income density: given a parametric L orenz curve, it is the sign of its third derivative which indicates whether the density is unimodal or zeromodal (i.e., downward‐sloping). The density modality of several important L orenz curves such as the P areto, W eibull, S ingh– M addala parametrizations and hierarchical families of L orenz curves are discussed. A L orenz curve performance comparison with M onte C arlo simulations and data from the UNU – WIDER W orld I ncome I nequality D atabase underlines the relevance of the theoretical result: curve‐fitting based on criteria such as mean squared error or the G ini difference might lead to a L orenz curve implying an incorrectly‐shaped density function. It is therefore important to take into account the modality when selecting a parametric L orenz curve.