Auditing Shaked and Shanthikumar’s ‘excess wealth’

Abstract

A notion called “excess wealth” was introduced by Shaked and Shanthikumar around 1998 (Probab. Eng. Inf. Sci. 12:1–23, 1998). Subsequent to this, much has been written on it, mostly by Shaked and his colleagues; see Sordo (Insur. Math. Econ. 45(3):466–469, 2009) for a recent review. These works have appeared in the literatures of reliability theory and stochastic orderings. Since the term excess wealth connotes a measure of income inequality—much like its dual, poverty—it should have had an impact in economics and the econometric literature. This, it appears is not the case, at least to the extent that it should be. The purpose of this paper is to investigate the above disconnect by looking at the notion of excess wealth more carefully, but keeping in mind the angle of economics and income. Our conclusion is that an alternative definition of excess wealth better encapsulates what one means by a colloquial use of the term.

Our motivation for being attracted to this topic arises from two angles. The first is that the stochastics of diagnostic and threat detection tests, in which we have an interest, has a strong bearing on indices of concentration like the Lorenz Curve, the Gini index, and the entropy. Thus the notion of excess wealth, which conveys a sense of income concentration should also be relevant to diagnostics. The second motivation is to honor Moshe Shaked, a prolific researcher and a friend of the first author, by developing a paper based on an idea that is co-attributed to him.

Appendix: Exploring the normalized excess wealth function

Recall that the normalized excess wealth function is given by the equation

$$EW_F^*(p)=\frac{\int_p^1F^{-1}(u)\,\mathrm{d}u-(1-p)F^{-1}(p)}{\mu},\quad 0\leq p\leq1, $$

where F is the distribution function of income in a population.

Given below are plots of the normalized excess wealth function for several distributions. For some distributions these are calculated analytically, but for others these were obtained by simulation.

1.1 A.1 A uniform distribution

Suppose that F is a uniform distribution on the interval [0,1]. Then the excess wealth function is

$$\int_p^1u\,\mathrm{d}u-(1-p)p=\frac{p^2}{2}-p+ \frac{1}{2}. $$

Multiplying by 2 to normalize the excess wealth function, we get

$$EW_F^*(p)=p^2-2p+1,\quad 0\leq p\leq1, $$

which is a strictly convex function of p—see Fig. 16.

Fig. 16

The normalized excess wealth function for a uniform distribution

1.2 A.2 An exponential distribution

Suppose that F is an exponential distribution with mean 1. Then the excess wealth function is

$$\int_p^1-\log(1-u)\,\mathrm{d}u+(1-p) \log (1-p)=(1-p). $$

Normalizing the above we get

$$EW_F^*(p)=1-p,\quad 0\leq p\leq1, $$

a straight line with slope =−1. Technically this is both a convex and a concave function.

1.3 A.3 A Weibull distribution with shape <1

Suppose F is a Weibull distribution with a scale parameter =1 and a shape parameter =.5, denoted Weibull(1,.5). Then

The excess wealth function is

Normalizing this function we get

$$EW_F^*(p)=-(1-p)\log (1-p)+(1-p),\quad 0\leq p\leq1, $$

which is a strictly concave function of p—see Fig. 17.

Fig. 17

The normalized excess wealth function for a Weibull(1,.5) distribution

1.4 A.4 A Pareto distribution

Suppose that F is a Pareto type II distribution with location parameter =1, and shape parameter =2. For this distribution, $EW_{F}^{*}$ was obtained by simulation. It is illustrated in Fig. 18 as a strictly concave function of p.

Fig. 18

The normalized excess wealth function of a Pareto type II distribution

1.5 A.5 A Weibull distribution with shape >1

Suppose that F is a Weibull distribution with scale parameter =1, and shape parameter =2. Then $EW_{F}^{*}$, obtained by simulation, is illustrated in Fig. 19. Here the normalized excess wealth function is convex in p.

Fig. 19

The normalized excess wealth function for a Weibull(1,2) distribution