What Is a Lorenz Curve and How Does It Work?

October 5, 2021
Lorenz Curve

Lorenz curve, created by American economist Max Lorenz in 1905, is a graphical representation of income or wealth disparity.

On the horizontal axis, the graph shows percentiles of the population based on income or wealth.

On the vertical axis, it depicts cumulative income or wealth, thus an x-value of 45 and a y-value of 14.2 means that the bottom 45 percent of the population controls 14.2 percent of total income or wealth.

A Lorenz curve is usually a mathematical function computed from an incomplete collection of income or wealth observations in practise.

Important Points: 

• A Lorenz curve is a graphical representation of income or wealth distribution within a community.

• Lorenz curves plot population percentiles against the cumulative income or wealth of people in that percentile or below.

• Lorenz curves and their derivative statistics are frequently used to assess inequality in a population.

• Lorenz curves may be imperfect measurements of genuine inequality since they are mathematical approximations based on fitting a continuous curve to partial and discontinuous data.

The Lorenz Curve: An Overview

Lorenz CurveThe Lorenz curve is frequently accompanied with a straight diagonal line with a slope of 1, which indicates complete equality in the distribution of income or wealth; the Lorenz curve rests beneath it, showing the observed or estimated distribution.

The Gini coefficient, a scalar measure of inequality, is the area between the straight and curved lines represented as a ratio of the area under the straight line.

While the Lorenz curve is most commonly associated with economic inequality, it may be used to illustrate unequal distribution in any system.

The higher the level of inequality, the further the curve is from the baseline, indicated by the straight diagonal line.

The Lorenz curve represents inequality in the distribution of wealth or income in economics; these terms are not interchangeable because it is conceivable to have high earnings but no or negative net worth, or low earnings but a large net worth.

A Lorenz is typically based on an empirical measurement of wealth or income distribution across a population, such as tax returns, which report income for a large section of the population.

Economists and statisticians can fit a curve that depicts a continuous function to fill in any gaps in the observed data using a graph of the data as a Lorenz curve.

Summary statistics like the Gini coefficient and the Lorenz asymmetry coefficient provide more information about the exact distribution of wealth or income across a community than a Lorenz curve.

Because a Lorenz curve visually depicts the distribution across each percentile (or other unit breakdown), it can demonstrate precisely where and how much the observed distribution deviates from the line of equality.

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

The majority of the curve’s points are essentially educated guesses based on the curve’s form that best fits the observed data points.

As a result, the form of the Lorenz can be affected by the data’s quality and sample size, as well as mathematical assumptions and judgments about what constitutes a best-fit curve, all of which can lead to significant discrepancies between the Lorenz curve and the real distribution.

Lorenz Curve Example

The Gini coefficient is a single figure that expresses the degree of inequality. It can vary from 0 (or 0% ) to 1 (or 100% ). (or 100 percent ).

A value of 0 relates to complete equality, in which everyone has the same income or wealth.

Complete equality would be a straight diagonal line with a slope of 1 if plotted as a Lorenz curve (the area between this curve and itself is 0, so the Gini coefficient is 0).

A coefficient of one indicates that one person earns or owns all of the wealth. The figure can theoretically be higher than 1 when negative wealth or income is taken into account; in such scenario, the Lorenz curve would drop below the horizontal axis.

In comparison to a straight diagonal line showing complete equality, the curve above depicts a continuous Lorenz that has been fitted to the data that reflect the income distribution in Brazil in 2015.

The value of the Lorenz curve at the 55th income percentile is 20.59 percent; in other words, the Lorenz curve estimates that the poorest 55 percent of the population receives 20.59 percent of the nation’s total income.

The bottom 55 percent of Brazilians would earn 55 percent of the total if the country were totally equitable.

The 99th percentile correlates to 88.79 percent of total income, implying that the top 1% of the population earns 11.21 percent of the country’s total revenue.

Subtract the area beneath the Lorenz curve (about 0.25) from the area beneath the line of perfect equality to get the approximate Gini coefficient (0.5 by definition).

Divide the result by the area beneath the line of perfect equality, and you’ll get a coefficient of about 0.5, or 50%. Brazil’s Gini coefficient in 2014 was 49.7%, according to the CIA.

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