To visualize income distribution, rank every household in the U.S. from poorest to richest. Now ask: the bottom 10 percent of households — what share of total income do they hold? The bottom 20 percent? The bottom 50 percent? Plot these cumulative answers on a graph with the cumulative population share on the horizontal axis and the cumulative income share on the vertical axis. The resulting curve — starting at (0,0) and ending at (100,100) — is the Lorenz curve. If it runs straight along the diagonal, income is perfectly equal. The more it sags below the diagonal, the more unequal the distribution. That visual representation of inequality is one of the most informative graphs in economics.
The formula
A Lorenz curve has no single equation — it is empirically constructed from the income distribution data of a population. For any point (x, y) on the curve:
- x = the cumulative share of the population ranked from bottom (0%) to top (100%)
- y = the cumulative share of total income held by that bottom-x% of the population
If every person had equal income, y = x for all points: the bottom 30% would hold exactly 30% of income. The Lorenz curve would lie exactly on the 45-degree line of perfect equality.
In practice, the Lorenz curve lies below the line of equality: the bottom 30% of U.S. households hold considerably less than 30% of total income — the curve bows downward, with the magnitude of the bow measuring the degree of inequality.
Reading the result
The Lorenz curve provides two types of information:
Reading specific percentiles: at any point x on the curve, the y-value tells you how much of total income the bottom x% of the population holds. If the curve passes through (50, 20), the bottom half of the population holds only 20% of total income.
Comparing distributions: two Lorenz curves can be compared directly. If Curve A lies entirely below Curve B for every percentile, the distribution underlying A is unambiguously more unequal than B (a Lorenz dominance relationship). If the curves cross, no unambiguous ranking is possible without a specific inequality index.
The Census Bureau's income data and CBO's household income data provide the underlying distributions from which Lorenz curves for the United States can be constructed. Comparing pre-tax and post-tax Lorenz curves shows the redistributive effect of the tax and transfer system — the post-tax curve lies closer to the equality line, reflecting compression from progressive taxation and transfers.
Worked example
A simplified economy with five equally sized groups:
| Income group | Income share | Cumulative pop. | Cumulative income |
|---|---|---|---|
| Bottom 20% | 5% | 20% | 5% |
| Second 20% | 10% | 40% | 15% |
| Middle 20% | 15% | 60% | 30% |
| Fourth 20% | 25% | 80% | 55% |
| Top 20% | 45% | 100% | 100% |
The Lorenz curve passes through (20, 5), (40, 15), (60, 30), (80, 55), (100, 100). At the midpoint, the bottom 60% hold only 30% of income — well below the 60% they'd hold under perfect equality. The curve is substantially bowed.
Where it's used
The Lorenz curve underpins the Gini coefficient, the most widely used single-number inequality measure. It is also used in welfare economics to compare distributions across countries, time periods, and demographic groups — and to evaluate whether a policy change improved or worsened the distribution, independent of any specific inequality index.
