Computing Difficulties for Deriving Poverty Indices from Some Functional Forms of Lorenz Curves

Abstract

We examine three families of classical one-parameter functional forms for estimating a Lorenz curve: the power form (Pareto, elementary form), the exponential form (Gupta, elementary form) and fractional form (Rohde). For the first time, we systematically study these functions not for their ability to be estimated but on the point of view of the possibility of deriving poverty indices, which implies first determining the headcount ratio (i.e., the percentage of poor). We show that computing difficulties have been largely underestimated. Two forms, the most simple ones, pose no problem: the elementary power and exponential forms. However, the Pareto functional form poses problem with a restricted definition domain of the parameters (for the percentage of the median that serves to determine the poverty line and for the level of inequality). On the other hand, Gupta's exponential form, which can be easily linearized for being estimated by the Least Squares, conducts to a Lambert function when the headcount ratio is derived, which poses some problems of restricted definition domain for the parameters. Rohde's fractional form also implies restricted definition domains.