On The Behaviour of the Z _p Concentration Curve

Abstract

Since M. Zenga introduced in 1984 his point concentration measure Z _p , based on the ratio between population and income fractiles, and the synthetic measure ζ, which is the espected value of Z _p (Zenga, 1984), several authors have been interested in the analyses of the characteristics and properties of these measures. Particular attention has been devoted to the Z _p concentration curve, which can be obtained by representing in the unit square the points (p, Z _p ), p ϵ [0, 1]. The behaviour of the Z _p curve has been studied both for theoretical models — rectangular, exponential, Pareto type I and lognormal (Zenga, 1984), generalized lognormal (Pollastri, 1987), Dagum type I (Dancelli, 1989) - and empirical distributions (Salvaterra, 1985, 1987). In the theoretical studies the Z _p curve has shown a wide variety of behaviours. In particular the three-parameter models considered by Pollastri and Dancelli can take the form of a curve convex to the p-axis, when the non-scale parameters take values in given ranges. Such a curve seems to be able to represent the point concentration that is peculiar to many empirical distributions. Indeed empirical studies very often have shown that the point concentration curve Z _p is a decreasing function of p for low income groups and an increasing one for high income groups. However observed income distributions have shown cases of increasing values of Z _p in correspondence to low incomes. Their characteristics have suggested that such behaviour of Z _p can derive from the existence of a lower positive income bound. The remark has given rise to this study the main purposes of which are: (1) to investigate the causes of the behaviour of theZ _p curve; (2) to analyse the consequences on the Z _p curve of a single or double truncation of the distribution.