## What is the Lorenz curve

The Lorenz curve is a graphical representation of income inequality or wealth inequality developed by the American economist Max Lorenz in 1905. The graph represents the percentiles of the population on the horizontal axis as a function of 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 poorest 45% of the population controls 14.2% of income or of total wealth.

## Breaking the Lorenz curve

The Lorenz curve is often accompanied by a straight diagonal line with a slope of 1, which represents perfect equality in the distribution of income or wealth; the Lorenz curve is below, 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 measure of inequality.

Although the Lorenz curve is most often used to represent economic inequality, it can also show an uneven distribution in any system. The further the curve is from the baseline, represented by the right diagonal line, the higher the level of inequality. In economics, the Lorenz curve indicates an inequality in the distribution of wealth or income; these are not synonymous because it is possible to have high income but a zero or negative net worth, or low income but a high net worth.

The Gini coefficient is used to express the extent of the inequality in a single digit. It can range from 0 (or 0%) to 1 (or 100%). Full equality, in which each individual has exactly the same income or wealth, corresponds to a coefficient of 0. Drawn like 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 a person earns all income or owns all wealth. Taking into account wealth or negative income, the figure can theoretically be greater than 1; in this case, the Lorenz curve would plunge below the horizontal axis.

The curve above shows the distribution of income in Brazil in 2020, compared to a right diagonal representing perfect equality. At the 55th percentile of income, the cumulative income is 20.59%: in other words, the poorest 55% of the population absorb 20.59% of the country’s total income. If Brazil were a perfectly egalitarian society, the poorest 55% would earn 55% of the total. The 99th percentile corresponds to 88.79% of cumulative income, which means that the richest 1% absorb 11.21% of Brazil’s income.

To find the approximate Gini coefficient, subtract the area under the Lorenz curve (about 0.25) from the area under the perfect equality line (0.5 by definition). Divide the result by the area under the perfect equality line, which gives a coefficient of about 0.5 or 50%. According to the CIA, the Gini coefficient of Brazil in 2020 was 49.7%.