What Is a Lorenz Curve?

A Lorenz curve, developed by American economist Max Lorenz in 1905, is a graphical representation of income inequality or wealth inequality. The graph plots percentiles of the population on the horizontal axis according to income or wealth and plots cumulative incom💎e or wealth o💦n the vertical axis.

Understanding the Lorenz Curve

In practice, a Lorenz curve is usually a mathematical function estimated from an incomplete set of observations of 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 observed or estimated distribution.

While the Lorenz curve is most often used to represent 澳洲幸运5官方开奖结果体彩网:economic inequality, it can also demonstrate unequal distribution in any system. The farther 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 either high earnings but zero or negative 澳洲幸运5官方开奖结果体彩网: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 澳洲幸运5官方开奖结果体彩网: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 澳洲幸运5官方开奖结果体彩网:economists and statisticians may fit a curve that represents a continuous🌞 functi𝓰on to fill in any gaps in the observed data.

Components of the Lorenz Curve

The United States Federal Reserve Bank of St. Louis collects net worth statistics from U.S. households. As of the second quarter of 2024, the underlying data demonstrates that the bottom 50% of households hold just 2.5% of overall household wealth, while the top 10% holds 67% of total household wealth.

There are several important components to understand when ana🌠lyzing a Lorenz curve:

• The x-axis is often denoted as the percentile. In the graph above, the x-axis is the percentile of net worth ranking compared to other U.S. households.
• The y-axis is often denoted as the cumulative percentage of occurrences. In the graph above, the percentages represent the cumulative amount of net worth of households.
• The line of equality is demonstrated by a 45-degree, upward-sloping line. In the graph above, it is denoted as the dashed line.
• The Lorenz curve is demonstrated often by an upward-sloping but often exponentially rising curve. In the graph above, it is denoted as a solid line.
• The Gini coefficient (discussed below) is the gap between the line of equality and the Lorenz curve.
  

The Lorenz Curve and the Gini Coefficient

The 澳洲幸运5官方开奖结果体彩网:Gini coefficient is used to express the extent of inequality in a single figure.🏅 It most often ranges 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 i🐲s 0).

A coefficient of 1 means that one person earns all of the income or holds all of the wealth. In theory, the Gini coefficient can exceed 100% in extreme situations. For example, when handling negative wealth or income, the figure can theoretically be higher than 1; in that casꦆe, the L🎶orenz curve would dip below the horizontal axis.

The Gini coefficient is equal to the area below the line of perfect equality minus the area below th♐e Lorenz curve, divided by the area below the line of perfect equality. In the graph above, the Gini coefficient is the area below the dashed line but above the solid line. The Gini coefficient is used to measure the extent of inequality. It can also be used to compare two different nations or countries to see which has more inequality.

Advantaꦯges and Disadvantages of the Lorenz Curve

Advantages of the Lorenz Curve

A Lorenz cur💟ve gives more detailed information about the exact distribution of wealth or income across a population th💮an 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 breakdowns), it can show precisely at which income (or wealth) percentiles the observed distribution varies from the line of equality and by how much.

The underlying data for a Lorenz curve is necessary to calculate the Gini coefficient, a primary means of mathematically measuring inequality. Assuming sufficient data can be collected, a Lorenz curve can be prepared for any geographical region (i.e. by country, by state, by city, etc.). This may be especially important for governments relying on a Lorenz curve to better understand how to most appropriately assess 澳洲幸运5官方开奖结果体彩网:tax brackets.

Though the Lorenz curve often displays sensitive financial information, it also maintains the anonymity of the underlying surveyed individuals. In addition, the Lorenz curve can be layered on top of itself over time, demonstrating how public policy has changed the level of inequality at various points along the Lorenz curve.

Disadvantages of the Lorenz Curve

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 c⛦urve (other than those actually observed in the data) actually correspond to the true distributions of income. This is also true because it may be too expensive or demand too many resources to collect enough data points to fill all gaps.

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

The Lorenz curve (or Gini coefficient) may also be insufficient to clearly demonstrate the extent of change of inequality over time. Consider a situation w🌊here two Gini coefficients are calculated to be the exact same. It is possible for the slo𝄹pe of the Lorenz curve to be radically different, yet the underlying aggregated inequality level is calculated to be the same.

Lorenz Curve Example

The curve above shows a continuous Lorenz curve that has been fitted to data that describes income distribution in Brazil as of 2023, according to the 澳洲幸运5官方开奖结果体彩网:World Bank. The dat🦩a set is also compared to a straight diagonal line representing perfect eq𝓰uality.

At the 50th income percentile, the value of the Lorenz curve is 17%%. This means that the Lorenz curve estim♑ates that the bottom 50% of the population takes in 17% of the nation’s total monthly income. If Brazil were a perfectly equal society, the bottom 50% would earn 50% of the total.

Elsewhere, we can see that the 99th percentile corresponds to 89% of cumulative income. This means that, according to the Bank's figures, the top 1% takes in just over a tenth of Brazil's income.

To find the Gini coefficient, subtract the area beneath the Lorenz curve 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 Gini coefficient of 50.8%. This is slightly lower than the estimate of 52% in 2022, suggesting a slight decrease in economic inequality.

The Bottom Line

A common method of visually analyzing inequality distributions is to use a Lorenz curve. The curve graphically depicts what a standard distribution of equality would look like. Then, the graph layers on the actual d꧅istribution of a given data set. The Lorenz curve is used to calculate the Gini coefficient to mathematically analyze how unequal a dataset is, and it is used to visually exam inequality levels as well.