Lorenz curve

What Is the Lorenz Curve?

The Lorenz Curve is a graphical representation used in economic theory to illustrate the distribution of income or wealth within a population. It falls under the broader category of economic inequality measurement. This tool plots the cumulative percentage of the population, ranked from the poorest to the richest, against the cumulative percentage of the total income or wealth they possess. The Lorenz Curve provides a clear data visualization of how evenly or unevenly resources are distributed.

History and Origin

The Lorenz Curve was developed by American economist Max O. Lorenz in 1905. His work, "Methods of Measuring the Concentration of Wealth," introduced this graphical method to visualize the degree of inequality in wealth distribution within a society. Initially conceived for economic analysis, the Lorenz Curve quickly became a fundamental economic indicator for understanding social inequality and has since found applications in various fields beyond economics, including ecology and public health.¹¹

Key Takeaways

• The Lorenz Curve graphically depicts income or wealth distribution within a population.
• It plots the cumulative percentage of the population against the cumulative percentage of total income or wealth received.
• A perfectly equal distribution is represented by a straight diagonal line, often called the "line of equality."
• The farther the Lorenz Curve deviates from the line of equality, the greater the inequality in distribution.
• It serves as the basis for calculating the Gini coefficient, a numerical measure of inequality.

Formula and Calculation

The Lorenz Curve is not defined by a single, simple mathematical formula in the way that some other financial metrics are. Instead, it is constructed by plotting ordered pairs ((F_i, L_i)), where:

• (F_i) represents the cumulative proportion of the population. If the population is divided into (n) groups (e.g., quintiles or percentiles), and the population is sorted from lowest to highest income/wealth, (F_i = i/n).

• (L_i) represents the cumulative proportion of the total income or wealth held by that cumulative proportion of the population.

• For a discrete distribution with values (y_1, y_2, \ldots, y_n) sorted in non-decreasing order (where (y_k) is the income/wealth of the (k)-th individual or group), the (L_i) coordinate for the (i)-th point is given by:

  $L_i = \frac{\sum_{k=1}^{i} Y_k}{\sum_{k=1}^{n} Y_k}$

  Here, (Y_k) is the income or wealth of the (k)-th individual or group. The curve connects points starting from (0,0) and ending at (1,1).¹⁰

Interpreting the Lorenz Curve

Interpreting the Lorenz Curve involves examining its shape and its distance from the line of perfect equality. This diagonal line, running from (0,0) to (1,1), signifies that every percentage of the population holds the same percentage of total income or wealth; for example, the bottom 20% of the population earns 20% of the income.⁹

When a Lorenz Curve is plotted, it will always lie below or on this line of equality. The extent to which the curve bows away from the diagonal line indicates the degree of inequality. A curve that is very close to the line of equality suggests a more equitable distribution of resources, while a curve significantly distant from it points to higher levels of disparity in household income or wealth. For instance, if the bottom 20% of the population only possesses 5% of the total income, the Lorenz Curve would plot a point at (0.20, 0.05), demonstrating a deviation from perfect equality.

Hypothetical Example

Consider a small economy with five households, and their annual incomes are:

• Household A: $10,000
• Household B: $15,000
• Household C: $25,000
• Household D: $30,000
• Household E: $70,000

First, sort the incomes in ascending order, which they already are.
Total income = $10,000 + $15,000 + $25,000 + $30,000 + $70,000 = $150,000

Now, calculate the cumulative percentage of the population and the cumulative percentage of total income:

Plotting these points, along with the starting point (0,0), would form the Lorenz Curve for this hypothetical economy. The curve would show how much the actual income distribution deviates from the line of perfect equality (where, for instance, the bottom 20% would have 20% of the income). statistical analysis using this curve helps visualize the income gap.

Practical Applications

The Lorenz Curve is a widely used tool in various sectors for assessing and comparing distributions. In macroeconomics, it is fundamental for analyzing and monitoring income inequality and wealth gaps within countries or regions. Organizations like the U.S. Census Bureau regularly publish data that can be used to construct Lorenz Curves, providing insights into trends in U.S. income distribution.⁸

Beyond national income statistics, the Lorenz Curve finds applications in:

• Poverty Analysis: It helps identify the proportion of total income or wealth held by segments of the population living near or below the poverty line, informing targeted public policy interventions.
• Business and Marketing: Companies might use a Lorenz Curve to understand customer loyalty or product adoption, plotting the cumulative percentage of customers against the cumulative percentage of sales generated to identify key customer segments.
• Environmental Science: It can illustrate the distribution of environmental burdens or benefits across different populations.
• Epidemiology: The curve can depict the concentration of disease incidence among specific demographic groups.

The Federal Reserve Bank of St. Louis, through its FRED database, provides historical Gini Index data for the United States, which is directly derived from Lorenz Curves, allowing for long-term analysis of income disparity.⁷

Limitations and Criticisms

While a powerful quantitative analysis tool, the Lorenz Curve has certain limitations. One primary criticism is that if two Lorenz Curves intersect, it becomes difficult to definitively conclude which distribution has greater inequality without resorting to a summary statistic like the Gini coefficient.⁶ The visual representation alone may not be sufficient for a conclusive ranking.

Furthermore, the Lorenz Curve primarily focuses on relative inequality. It does not provide information about the absolute level of income or wealth within a society. Two countries could have similar Lorenz Curves but vastly different average incomes, meaning that while their distributions are similarly unequal, the absolute living standards could differ significantly. This means the curve does not inherently capture the impact of economic growth or overall societal prosperity.

Another critique arises from the data used. Different definitions of income (e.g., pre-tax, post-tax, including non-cash benefits) can lead to different Lorenz Curves and, consequently, different conclusions about inequality trends. Debates around income inequality often hinge on these methodological differences.⁴, ⁵

Lorenz Curve vs. Gini Coefficient

The Lorenz Curve and the Gini coefficient are intrinsically linked, both serving as key measures in income distribution and inequality analysis. The Lorenz Curve is a graphical representation, visually illustrating the entire distribution of income or wealth. It shows, for any given percentage of the population, the cumulative percentage of total income or wealth that segment possesses.

In contrast, the Gini coefficient is a single numerical summary statistic derived directly from the Lorenz Curve. It quantifies the degree of inequality shown by the curve. Specifically, the Gini coefficient is calculated as the area between the line of perfect equality and the Lorenz Curve, divided by the total area under the line of perfect equality. A Gini coefficient of 0 represents perfect equality (where the Lorenz Curve lies directly on the line of equality), while a coefficient of 1 (or 100% when expressed as an index) signifies perfect inequality (where one person holds all the income or wealth).², ³ The Gini coefficient provides a concise, comparable metric, whereas the Lorenz Curve offers a more detailed visual understanding of the entire distribution.

FAQs

What does the line of equality represent on a Lorenz Curve?

The line of equality, or the 45-degree line, on a Lorenz Curve represents a perfectly equal distribution of income or wealth. This means that every percentage of the population holds the exact same percentage of the total income or wealth; for example, the poorest 10% of the population would possess 10% of the total income.¹

Can a Lorenz Curve ever go above the line of equality?

No, a Lorenz Curve can never go above the line of equality. By definition, the population is ordered from the lowest income/wealth to the highest. Therefore, the cumulative percentage of income or wealth held by any given cumulative percentage of the population can never exceed the cumulative percentage of the population itself.

How is the Lorenz Curve used in financial markets?

While primarily an economic tool for analyzing income and wealth distribution, the underlying principles of the Lorenz Curve can be adapted in financial analytics to illustrate the concentration of various financial metrics. For example, it could show the concentration of trading volume among a few large investors or the distribution of returns across a portfolio of assets.