The Gini coefficient and discontinuity

Abstract

This article reveals a discontinuity in the mapping from a Lorenz curve to the associated cumulative distribution function. The problem is of a mathematical nature—based on an analysis of the transformation between the distribution function of a bound random variable and its Lorenz curve. It will be proven that the transformation from a normalized income distribution to its Lorenz curve is a continuous bijection with respect to the $L^q$ ([0,1])-metric—for every q ≥ 1. The inverse transformation, however, is not continuous for any q ≥ 1. This implies a more careful attitude when interpreting the value of a Gini coefficient. A further problem is that if you have estimated a Lorenz curve from empirical data,then you cannot trust that the associated distribution is a good estimate of the true income distribution.

Keywords:

• inequality
• probability
• math analysis
• discontinuity

PUBLIC INTEREST STATEMENT

This article deals with the relation between the distribution of income and the measurement of economic inequality in a society. The latter is often expressed as the Gini coefficient, G: the expected difference between two randomly drawn household or individual incomes divided by two times the average income. This division was made to be able to compare the magnitude for different societies. Looking at the income distribution, a reasonable degree of equality in the actual society must imply that the difference between the maximal income and the average income is not too large. If we divide this number by the maximal income, we get a quantity, H, comparable between societies. In this article, it will be shown not only that if H is close to zero, then G is close to zero but also that the opposite is not necessarily true. The most direct consequence is that a small G is not enough to ensure relative economic equality in a society.

Disclosure statement

It is a pleasure to thank the editor and the anonymous reviewers for helping to improve this paper

No potential conflict of interest was reported by the author(s).

Notes

1. Furthermore, the current OECD formula weights income higher, the more numerous the household in which the individual lives is.

2. Named after M. O. Lorenz, the American economist who developed the concept in his pioneering research on income inequality. Se Lorenz, M O. (1905) Methods of measuring the concentration of wealth. Journal of American statistical Association. p 209–219.

3. Dorfman does not use the concept of $F^{-1}\left(u\right)$. Furthermore, he uses Stieltjes integrals. Other writers I found only prove the case with a differentiable distribution function. We will at present be content with this, although it is not difficult to prove (2) for any kind of distribution using the Lebesgue measure on [0,1].

4. More details in the proof of these claims about convex functions could be found in Rudin (1974) p 62–63.

5. The term is chosen because here – in accordance with the current literature – is used as argument for Lorenz curves.

6. I found this class of functions in Farris (2010) p 863. He calls them Pareto functions which they obviously not are. They can only be Lorenz curves for finite random variables. One could utter that for b > 2 the associated cdf has a certain resemblance to Pareto distributions.

7. In Liu & Gastwirth’s (2020) terminology this is “one half of the coefficient of variation”.

Additional information

Funding

The author received no direct funding for this research.

Notes on contributors

Jens Peter Kristensen

Being a math teacher, I often took an interest in courses of teaching in applied mathematics. In the 2 latest decades, this was frequently interdisciplinary courses with social science and economics. After some courses about economic inequality, I wrote an article (in Danish) to the magazine of the Danish Association of Mathematics Teachers, LMFK-bladet 4/2015 p 10 - 15. In a comment, a colleague referred to Farris’ article in AMM 12/2010, leading me to some of the comprehensive economics literature on measuring inequality, Lorenz curve, and Gini coefficient. My prime interest was to analyze the transformation from which you derive the Lorenz curve from a given income distribution. If you demand the distributions to be normalized, this mapping is 1-1 of a set of distributions into itself. The set is contained in a normed space. So, mathematically, you can ask if it is continuous and if the inverse is. Having answered these questions, it remains drawing consequences in economics methodology – which could be further refined. Have among other high schools worked at Hasseris Gymnasium, Denmark.