# The Lorenz Curve

Suppose we want to measure the wealth gap of a nation.  The Gini coefficient does this by measuring how far away we are from perfect equality.  Consider the following graph.

The x-axis represents shares of income.  So $.5$ on the x-axis represents half of those who have an income ($50\%$ of income earners are on either side), and everyone is included since unemployed are considered to have 0 income.  The y-axis represents the sum of all incomes up to income $x.$  Hence on this particular depiction of the Lorenz curve, which represents this for a made-up nation, it appears the bottom $50\%$ of income adds up to about $.2$ (or $20\%$) of the total income.  The perfect equality line has a slope of $1$ since if everyone has the same income, then each step forward on the x-axis (where steps are evenly distributed because of equal income) adds the same amount to the cumulative income.  Measuring the deviation (Gini coefficient) from perfect equality thus amounts to computing the pink area as a ratio to perfect equality.  That is, if $G$ represents the Gini coefficient, $A$ is the pink area, and $B$ is the grey area, then

$\displaystyle G=\frac{A}{A+B}=\frac{A}{1/2}=2A=2\left(\frac{1}{2}-B\right)=1-2\int_0^1 L(x)\,dx$

where $L(x)$ is the Lorenz curve.  It thus takes a value between $0$ and $1$ where $0$ is perfect equality (everyone has same income), and $1$ is perfect inequality (one person has all of the income).  We can see a variation of Gini coefficients by nation:

There is also the trend since WWII:

Image by Wikipedia User Cflm001

The US Gini coefficient has moved up from .408 in 1997 to .45 in 2007 and ranks 39 out of 136 nations for highest Gini coefficient [1][2].