Lorenz Curve

The Lorenz curve is a graphical representation of income inequality or wealth inequality developed by American economist Max Lorenz in 1905. The graph plots percentiles of the population on the horizontal axis according to income or wealth. It plots cumulative income or wealth on the vertical axis, so that an x-value of 45 and a y-value of 14.2 would mean that the bottom 45% of the population controls 14.2% of the total income or wealth.

The Lorenz curve is often accompanied by a straight diagonal line with a slope of 1, which represents perfect equality in income or wealth distribution; the Lorenz curve lies beneath it, showing the actual distribution. The area between the straight line and the curved line, expressed as a ratio of the area under the straight line, is the Gini coefficient, a measurement of inequality.

While the Lorenz curve is most often used to represent economic inequality, it can also demonstrate unequal distribution in any system. The farther away the curve is from the baseline, represented by the straight diagonal line, the higher the level of inequality. In economics, the Lorenz curve denotes inequality in the distribution of either wealth or income; these are not synonymous since it is possible to have high earnings but zero or negative net worth, or low earnings but a large net worth. 

The Gini coefficient is used to express the extent of inequality in a single figure. It can range from 0 (or 0%) to 1 (or 100%). Complete equality, in which every individual has the exact same income or wealth, corresponds to a coefficient of 0. Plotted as a Lorenz curve, complete equality would be a straight diagonal line with a slope of 1 (the area between this curve and itself is 0, so the Gini coefficient is 0). A coefficient of 1 means that one person earns all of the income or holds all of the wealth. Accounting for negative wealth or income, the figure can theoretically be higher than 1; in that case, the Lorenz curve would dip below the horizontal axis. 

The curve above shows the income distribution in Brazil in 2015, compared to a straight diagonal representing perfect equality. At the 55th income percentile, the cumulative income is 20.59%: in other words, the bottom 55% of the population takes in 20.59% of the nation's total income. If Brazil were a perfectly equal society, the bottom 55% would earn 55% of the total. The 99th percentile corresponds to 88.79% in cumulative income, meaning that the top 1% takes in 11.21% of Brazil's income.

To find the approximate Gini coefficent, subtract the area beneath the Lorenz curve (around 0.25) from the area beneath the line of perfect equality (0.5 by definition). Divide the result by the area beneath the line of perfect equality, which yields a coefficient of around 0.5 or 50%. According to the CIA, Brazil's Gini coefficient in 2014 was 49.7%.