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Parametric Lorenz Curves and the Modality of the Income Density Function

Krause, Melanie
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Beiträge zur Jahrestagung des Vereins für Socialpolitik 2012: Neue Wege und Herausforderungen für den Arbeitsmarkt des 21. Jahrhunderts - Session: Microeconometrics and Statistics D09-V3
Similar looking Lorenz Curves can imply very different income density functions and potentially lead to wrong policy implications. This paper derives a relation between a Lorenz Curve and the modality of its underlying income density: Given a parametric Lorenz Curve, it is the sign of its third derivative which indicates whether the density is unimodal or zeromodal (i.e. downward-sloping). Several single-parameter Lorenz Curves such as the Pareto, Chotikapanich and Rohde forms are associated with zeromodal densities. The paper contrasts these Lorenz Curves with the ones derived from the (unimodal) Lognormal density and the Weibull density, which, remarkably, can be zero- or unimodal depending on the parameter. A performance comparison of these five Lorenz Curves with Monte Carlo simulations and data from the UNU-WIDER World Income Inequality Database underlines the relevance of the theoretical result: Curve-fitting of decile data based on criteria such as mean squared error might lead to a Lorenz Curve implying an incorrectly-shaped density function. It is therefore important to take into account the modality when selecting a parametric Lorenz Curve.
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