Lorenz Curve and Gini Coefficient
Understand the Lorenz curve and calculate the Gini coefficient: G = 1 - 2 times the integral of L(x).
Measures income inequality.
The Formula
The Lorenz curve is a graphical tool that shows how income (or wealth) is distributed across a population. It was developed by the American economist Max O. Lorenz in 1905. The curve plots the cumulative share of income received by the bottom x percent of the population. If income were perfectly equal, the Lorenz curve would be a straight diagonal line (the "line of equality") from the origin to the point (1, 1). In reality, the curve bows below this line because the poorest portion of the population earns a disproportionately small share of total income.
The Gini coefficient is derived directly from the Lorenz curve. On the Lorenz diagram, area A is the region between the line of equality and the Lorenz curve. Area B is the region below the Lorenz curve. The Gini coefficient equals A divided by the total area under the line of equality (A + B). Since the total area under the diagonal is always 0.5, the formula simplifies to Gini = 1 − 2 × ∫L(x)dx, where the integral represents area B plus the area under the curve.
The Gini coefficient ranges from 0 to 1. A value of 0 means perfect equality — everyone has exactly the same income. A value of 1 means perfect inequality — one person has all the income and everyone else has none. Most countries have Gini coefficients between 0.25 and 0.60. Scandinavian countries typically have values around 0.25 to 0.30, while some developing nations exceed 0.50.
The Lorenz curve and Gini coefficient are not limited to income. They can measure any distribution: wealth, land ownership, market share, healthcare access, or even biological diversity. For example, ecologists use similar curves to measure species abundance distributions.
One limitation is that different Lorenz curves can produce the same Gini coefficient. Two countries might have the same Gini value but very different income distributions. For a complete picture, economists examine the full shape of the Lorenz curve alongside the summary Gini number.
Variables
| Symbol | Meaning |
|---|---|
| L(x) | Lorenz function — cumulative share of income held by the bottom x fraction of the population |
| A | Area between the line of equality and the Lorenz curve |
| B | Area under the Lorenz curve |
| Gini | Gini coefficient — ranges from 0 (perfect equality) to 1 (perfect inequality) |
| x | Cumulative proportion of the population (from 0 to 1) |
Example 1
A small economy has 5 people earning $10k, $20k, $30k, $40k, and $100k. Calculate the Lorenz curve points and estimate the Gini coefficient.
Total income = $200k. Cumulative shares: 20%→5%, 40%→15%, 60%→30%, 80%→50%, 100%→100%
Using the trapezoidal rule to estimate ∫L(x)dx ≈ 0.5 × 0.2 × (0 + 2×0.05 + 2×0.15 + 2×0.30 + 2×0.50 + 1.0) = 0.30
Gini ≈ 1 − 2 × 0.30 = 0.40 (moderate inequality)
Example 2
If the Lorenz function is L(x) = x², calculate the Gini coefficient exactly.
∫01 x² dx = [x³/3]01 = 1/3
Gini = 1 − 2 × (1/3) = 1 − 2/3
Gini = 1/3 ≈ 0.333
When to Use It
The Lorenz curve and Gini coefficient are used wherever inequality or distribution needs to be measured and compared.
- Comparing income or wealth inequality between countries
- Tracking changes in inequality within a country over time
- Evaluating the effectiveness of tax and redistribution policies
- Measuring market concentration and competitive balance in industries
- Analyzing land ownership distribution in agricultural economics
- Academic research in economics, sociology, and public policy