Parametric Lorenz Curves and the Modality of the Income Density Function

25 Pages Posted: 11 Nov 2014

See all articles by Melanie Krause

Melanie Krause

University of Hamburg - Department of Economics

Date Written: December 2014


Similar looking Lorenz curves can imply very different income density functions and potentially lead to wrong policy implications regarding inequality. 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). The density modality of several important Lorenz curves such as the Pareto, Weibull, Singh–Maddala parametrizations and hierarchical families of Lorenz curves are discussed. A Lorenz curve performance comparison with Monte Carlo simulations and data from the UNU–WIDER World Income Inequality Database underlines the relevance of the theoretical result: curve‐fitting based on criteria such as mean squared error or the Gini difference 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.

Keywords: goodness of fit, income distribution, inequality, Lorenz curve, modality

JEL Classification: C13, C16, D31

Suggested Citation

Krause, Melanie, Parametric Lorenz Curves and the Modality of the Income Density Function (December 2014). Review of Income and Wealth, Vol. 60, Issue 4, pp. 905-929, 2014, Available at SSRN: or

Melanie Krause (Contact Author)

University of Hamburg - Department of Economics ( email )

Von-Melle-Park 5
room 2128 C rise
Hamburg, 20146


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