## Lorenz Curves 2: Definition and Specification

Inference for Lorenz curves requires its rigorous definition and good specifications. This post briefly discusses alternative definitions and ways of specifying Lorenz curves:

Let $y, F(y), f(y)$  and  ${F^{-1}}(y)$  denote income level, CDF, PDF and Inverse-CDF functions respectively.

• An intuitive definition of the Lorenz curve is through

$L(c)=\dfrac{\int_0^{F^{-1}(c)} yf(y)dy}{\int_0^{\infty} yf(y)dy}$

where  c  is the population proportion and  L  is the Lorenz curve. This definition is not the most general definition because it requires existence of a density function and existence of a density function itself requires absolute continuity of the CDF function. The good thing about this definition however is that it can be used to derive the Lorenz curve for well-known density functions such as Lognormal, Singh-Maddala, GB2, etc.

• A more general way of defining a Lorenz curve is through

$L(c)=\dfrac{\int_0^c F^{-1}(x)dx}{\int_0^1 F^{-1}(x)dx}$

This definition is preferred from a technical point of view because it requires only existence of a cumulative density function. It can also be used for specification of a Lorenz curve if one has a good specification for an Inverse-CDF function.

• It has been shown that any continuous convex function L from [0,1] to [0,1] with L(0)=0 and L(1)=1 can a Lorenz curve. Using this result, one can directly specify a Lorenz curve without the need for starting from a distribution function. A host of functional forms have been proposed using this approach. See Sarabia et al. (2008) and references cited therein for more on specification of Lorenz curves and this book for technical issues related to the definition of Lorenz curves.

Reference:

Sarabia, J. M. (2008). Parametric Lorenz curves: Models and applications. In D. Chotikapanich (Ed.), Modeling Income Distributions and Lorenz Curves, Chapter 9, pp. 167–190. Springer.

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1. Inequality is becoming increasingly popular in Western Hemisphere