## What Is a Lorenz Curve?

A 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. In practice, a Lorenz curve is usually a mathematical function estimated from an incomplete set of observations of income or wealth.

### Key Takeaways

- A Lorenz curve is a graphical representation of the distribution of income or wealth within a population.
- Lorenz curves graph percentiles of the population against cumulative income or wealth of people at or below that percentile.
- Lorenz curves, along with their derivative statistics, are widely used to measure inequality across a population.
- Because Lorenz curves are mathematical estimates based on fitting a continuous curve to incomplete and discontinuous data, they may be imperfect measures of true inequality.

## Understanding the Lorenz Curve

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 observed or estimated 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 scalar 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.

A Lorenz curve usually starts with an empirical measurement of wealth or income distribution across a population based on data such as tax returns which report income for a large portion of the population. A graph of the data may be used directly as a Lorenz curve, or economists and statisticians may fit a curve that represents a continuous function to fill in any gaps in the observed data.

A Lorenz curve gives more detailed information about the exact distribution of wealth or income across a population than summary statistics such as the Gini coefficient or the Lorenz asymmetry coefficient. Because a Lorenz curve visually displays the distribution across each percentile (or other unit breakdown), it can show precisely at which income (or wealth) percentiles the observed distribution varies from the line of equality and by how much.

However, because constructing a Lorenz curve involves fitting a continuous function to some incomplete set of data, there is no guarantee that the values along a Lorenz curve (other than those actually observed in the data) actually correspond to the true distributions of income.

Most of the points along the curve are just guesses based on the shape of the curve that best fits the observed data points. So the shape of the Lorenz curve can be sensitive to the quality and sample size of the data and to the mathematical assumptions and judgments as to what constitutes a best-fit curve, and these may represent sources of substantial error between the Lorenz curve and the actual distribution.

## Lorenz Curve Example

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 a continuous Lorenz curve that has been fitted to the data that describe the income distribution in Brazil in 2015, compared to a straight diagonal line representing perfect equality. At the 55th income percentile, the value of the Lorenz curve is 20.59%: in other words, this Lorenz curve estimates that 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 coefficient, 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%.