Lorenz Curve

Lorenz Curve: Definition and Overview

A Lorenz curve is a graphical tool that shows how a variable—typically income or wealth—is distributed across a population. Developed by Max Lorenz in 1905, it displays cumulative population percentiles on the x-axis and cumulative share of the variable (income or wealth) on the y-axis. The curve is used to visualize inequality and to compute summary measures such as the Gini coefficient.

How the Lorenz Curve Works

• The 45-degree diagonal (line of equality) represents perfect equality: each percentile holds an equal share (e.g., the bottom 50% holds 50% of income).
• The Lorenz curve lies at or below that diagonal. The more it bows away from the diagonal, the greater the inequality.
• In practice, the curve is constructed from observed data (tax records, surveys) or by fitting a continuous function to discrete observations.

Key Components

• X‑axis: cumulative population percentiles (e.g., 0–100%).
• Y‑axis: cumulative percentage of total income or wealth held by that population share.
• Line of equality: the 45° diagonal indicating perfect equality.
• Lorenz curve: the observed (or estimated) cumulative distribution.
• Gap area: the area between the line of equality and the Lorenz curve, used to compute the Gini coefficient.

Relation to the Gini Coefficient

The Gini coefficient summarizes inequality as a single number ranging from 0 (perfect equality) to 1 (maximum inequality in theory). It is calculated as:

Gini = (Area between the line of equality and the Lorenz curve) / (Area under the line of equality)

Because the area under the line of equality equals 0.5, this often simplifies to Gini = 2 × (area between the diagonal and the Lorenz curve). Note: with negative incomes or wealth, Lorenz curves can behave unusually and the conventional 0–1 interpretation may not hold.

Example (Illustrative)

Using a fitted Lorenz curve for Brazil (2017, World Bank data):
– The bottom 50% of the population held about 17% of total income.
– The cumulative income at the 99th percentile was about 89%, implying the top 1% held roughly 11%.
– From the curve, the Gini coefficient was about 50.8% (0.508), indicating substantial inequality.

Advantages

• Visual clarity: shows how inequality is distributed across all percentiles, not just an average.
• Diagnostic detail: reveals which parts of the distribution (bottom, middle, top) deviate most from equality.
• Foundation for metrics: required input for calculating the Gini coefficient and other inequality measures.
• Can be compared across regions or over time to assess policy impacts.

Limitations

• Data requirements: accurate curves need large, representative datasets; poor sampling can mislead.
• Estimation sensitivity: fitting a continuous curve to discrete observations introduces assumptions; results depend on fit method.
• Ambiguity in summary: different Lorenz curves can yield identical Gini coefficients, masking where differences occur.
• Edge cases: negative or highly skewed values can produce nonstandard Lorenz behavior and complicate interpretation.

How to Construct a Lorenz Curve (Overview)

1. Collect individual or household data on the variable of interest (income, wealth).
2. Sort observations from lowest to highest.
3. Compute cumulative population share (x) and cumulative share of the variable (y).
4. Plot y versus x; the plotted points form the empirical Lorenz curve. Optionally, fit a smooth curve through the points.
5. Compute the Gini coefficient from the area between the Lorenz curve and the line of equality.

Uses and Users

• Governments and policymakers use Lorenz curves to evaluate income/wealth distribution and to design tax or transfer policies.
• Economists and researchers use them to compare inequality across countries, regions, or time periods.
• NGOs and analysts use Lorenz curves to communicate distributional outcomes to the public.

Takeaway

The Lorenz curve is a simple, powerful visual tool for understanding distributional inequality. It provides detailed insight into which segments of a population bear more or less of the total income or wealth and serves as the basis for quantitative measures like the Gini coefficient. However, its usefulness depends on the quality of underlying data and careful interpretation of fitted curves and summary statistics.