Parametric Lorenz curves and the modality of the income density function

Parametric Lorenz Curves are convenient methods for measuring and illustrating income inequality in an economy, also in the presence of scarce data. Similar looking Lorenz Curves can, however, imply very different income density functions. 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). This analytical result can be important when a researcher is deciding which of various parametric Lorenz Curves to use in order to fit given income data.

Several single-parameter Lorenz Curves such as the Pareto, Chotikapanich and Rohde forms imply  zeromodal densities. This paper argues that they should be primarily used for income distributions in poor and overpopulated countries or for wealth distributions. It contrasts these Lorenz Curves with those ones derived from the (unimodal) Lognormal density and the Weibull density, where the latter can be zero- or unimodal. The derivation and analysis of the single-parameter Weibull Lorenz Curve  is a side contribution of this paper. Due to their association with unimodal densities, the Lognormal and Weibull Lorenz Curves should be used for modeling most income distributions except for very inegalitarian ones.

Monte Carlo simulations underline the importance of an income density's modality: 10,000 draws are taken from a unimodal or zeromodal  distribution function . They are then aggregated to decile data points (which is the usually available data format in cross-country income inequality datasets). The single-parameter Pareto, Chotikapanich, Rohde, Lognormal and Weibull Lorenz Curves are fitted to the simulated decile points. After 10,000 repetitions of the procedure, we compare the performance of the five Lorenz Curves in terms of mean squared error (MSE) and Gini difference (i.e. the difference between the Gini coefficient of the real distribution and the one implied by the fitted Lorenz Curve).

A crucial result is that the best Lorenz Curves in terms of MSE or Gini difference do not necessarily have a density with the correct shape. For instance, decile data points from a rather egalitarian unimodal density might achieve the best MSE with a Lorenz Curve implying a very different, downward-sloping density. The Kullback-Leibler distance and the difference of the density integrals confirm the discordance of income densities of different modalities. An incorrect representation of the shape of income distributions in a cross-country comparison might lead to wrong conclusions and policy implications on poverty and inequality. The point of this paper is further illustrated with decile income data on 120 countries from the UNU-WIDER World Income Inequality Database. After fitting the five Lorenz Curves, it can be seen that the shape discrepancy of an incorrectly implied modality is larger in egalitarian and moderately inegalitarian countries than in those where income is distributed very unequally.

As a conclusion, this paper recommends that researchers should select Lorenz Curves not just based on MSE at the decile points but should take the modality of the underlying density into account.