Inference on Lorenz and on Concentration Curves

Abstract

In a pathbreaking paper, Atkinson (1970) proved several results concerning the ranking of income distributions according to expected values of all concave social welfare functions. One of the important results is that for distributions with equal means, all social welfare functions show the same order of average social welfares (i.e., the same ordering of inequality) if and only if the appropriate Lorenz curves do not intersect. If, on the other hand, the Lorenz curves intersect then it is possible to find two alternative social welfare functions which rank average social welfares differently (to be discussed in Chaps. 13 and 14). This finding by Atkinson has opened the way for using the Lorenz curve as a basic tool in the application of the concept of second-degree stochastic dominance (SSD, to be defined below). This tool allows the analyses of the effects of tax reforms and decision under risk to be applied to a wide group of utility functions, freeing the analysis from the need to specify the utility function. Shorrocks (1983) proved that X dominates Y according to SSD if and only if the absolute Lorenz curve ALC of X is not lower than the ALC of Y. This result enables to extend the possible applications to distributions with different expected values. There are three possible outcomes when comparing two absolute (and relative) Lorenz curves: Lorenz dominance, equivalence, and crossing. Bishop, Chakravarty, and Thistle (1989) extend the works by Gail and Gastwirth (1978), Beach and Davidson (1983), and Gastwirth and Gail (1985) who deal with relative Lorenz curves and suggest a pair-wise multiple comparisons method of sample absolute (generalized) Lorenz ordinates to test for differences.