The Gini coefficient

Wildon's Weblog 2018-04-11

We define the Gini coefficient of a probability measure $p$ on $[0,\infty)$ with mean $\mu$ by

$\displaystyle G = \frac{\mathbb{E}[|X-Y|]}{2 \mu}$

where $X$ and $Y$ are independently distributed according to $p$ . Thus $G = \frac{1}{2\mu} \int_\Omega |X(\omega) - Y(\omega)| \mathrm{d}p_\omega$ . Dividing by $\mu$ makes $G$ a dimensionless quantity. The reason for normalizing by $2$ will be seen shortly.

The Gini coefficient is a measure of inequality. For instance, if $p_x$ is the probability that a citizen’s wealth is $x$ , then we can sample $G$ by picking two random citizens, and taking the (absolute value of) the difference of their wealths. In

Utopia, where everyone happily owns the same (ample) amount, the Gini coefficient is $0$ ;
Dystopia, where the ruler has a modest fortune of $N$ units, and the other $N-1$ `citizens’ have nothing at all, $\mu = 1$ and the Gini coefficient is $\frac{1}{2} \frac{2(N-1)}{N^2} N = 1 - \frac{1}{N}$ ;
Subtopia, where each serf has $1$ unit, and each land-owner has $2$ units (plus two serfs to boss around), the mean income is $4/3$ and the Gini coefficient is $\frac{1}{2} \frac{3}{4} \frac{2. 2N . N}{9N^2} = \frac{1}{6}$ .

A striking interpretation of $G$ uses the Lorenz curve. Staying with the wealth interpretation, define $L(\theta)$ to be the proportion of all wealth owned by the poorest $\theta$ of the population. Thus $L(0) = 0$ , $L(1) = 1$ and, since the poorest third (for example) cannot have more than third of the wealth, $L(\theta) \le \theta$ for all $\theta$ . When $N$ is large we can approximate $L$ by the following functions $\ell$ : in

Utopia $\ell(\theta) = \theta$ for all $\theta$ ;
Dystopia $\ell(\theta) = 0$ if $\theta < 1$ and $\ell(1) = 1$ ;
Subtopia $\ell(\theta) = \frac{3\theta}{4}$ if $0 \le \theta < 2/3$ and $\ell(\theta) = -\frac{1}{2} + \frac{3\theta}{2}$ if $2/3 \le \theta \le 1$ .

Since the area below the curve is $1/6 + 1/6 + 1/12 = 5/12$ , the orange area is $1/12$ . This is half the Gini coefficient.

Theorem. $G$ is the area between the Lorenz curve for $X$ and the diagonal.

Proof. Let $P$ be the cumulative density function for $X$ , so $\mathbb{P}[X \le x] = P(x)$ . If you are person $\omega$ , and your wealth is $x$ , so $X(\omega) = x$ , then $P(X(\omega)) \le \theta$ if and only if you are in the poorest $\theta$ of the population. Therefore the poorest $\theta$ of the population form the event $\{ P(X) \le \theta \}$ . Their proportion of the wealth is the expectation of $X$ , taken over this event, scaled by $\mu$ . That is

$\displaystyle L(\theta) = \frac{\mathbb{E}[X 1_{P(X) \le \theta}]}{\mu}.$

The area, $I$ say, under the Lorenz curve is therefore

$\begin{aligned} I &= \frac{1}{\mu} \int_0^1 L(\theta) \mathrm{d}\theta \\ &= \frac{1}{\mu} \int_0^\infty \mathbb{E}_X[X 1_{P(X) \le P(y)}] \mathrm{d}P(y) \\ &= \frac{1}{\mu} \int_0^\infty \mathbb{E}_X[X 1_{X \le y}] \mathrm{d}Py \\ &= \frac{1}{\mu} \mathbb{E}_Y \mathbb{E}_X [X 1_{X \le Y}] \end{aligned}$

Now since $\mathrm{min}(X,Y) = X 1_{X \le Y} + Y1_{Y \le X} - X 1_{X = Y}$ , where the event $X=Y$ is assumed to be negligible, it follows from linearity of expectation that $\mathbb{E}[\mathrm{min}(X,Y)] = 2 \mathbb{E}[X 1_{X \le Y}]$ . (Here $\mathbb{E}$ denotes the expectation taken over both random variables.) Substituting we obtain

$\displaystyle I = \frac{1}{\mu} \frac{\mathbb{E}[\mathrm{min}(X,Y)]}{2}.$

From the identity $|X-Y| = X+Y - 2\mathrm{min}(X,Y)$ then another application of linearity of expectation, and finally the definition of $G$ we get

$\begin{aligned}I &= \frac{1}{2\mu} \frac{\mathbb{E}[X+Y - |X-Y|]}{2} \\ &= \frac{1}{2\mu} \frac{2\mu - \mathbb{E}[X-Y]}{2} \\ &= \frac{1}{2} - \frac{G}{2}. \end{aligned}$

Therefore $G = 2(I - \frac{1}{2})$ , as claimed.