What Is the Lorenz Curve? Definition and How It Measures Inequality

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The Lorenz curve is a graph that shows how income or wealth is distributed across a population. It ranks people (or households) from poorest to richest along the horizontal axis and plots, on the vertical axis, the cumulative share of total income or wealth held by everyone up to each point in that ranking. A perfectly equal society would produce a straight diagonal line. The real curve sags below it, and the deeper the sag, the more unequal the distribution.

What Is the Lorenz Curve?

The American economist Max O. Lorenz introduced the curve in 1905 as a way to compare the concentration of wealth across populations and over time. The idea is simple: instead of reporting an average, which hides how resources are spread, line everyone up by income and ask what fraction of the total the bottom 10%, 20%, 50% and so on actually receive.

In economics the curve is applied to either income (a flow, such as annual earnings) or wealth (a stock, such as net worth), and the two are not interchangeable. A young doctor can have a high salary and negative net worth because of student loans, while a retiree can have modest income and substantial assets, so a country’s wealth curve should not be compared with its income curve as if they measured the same thing.

How to Read a Lorenz Curve

• Horizontal axis (x): the cumulative percentage of the population, ranked from lowest income or wealth to highest. The point 40% means “the poorest 40%.”
• Vertical axis (y): the cumulative percentage of total income or wealth held by that group.
• Line of equality: the 45-degree diagonal from (0, 0) to (100%, 100%). On this line, the bottom 20% hold 20%, the bottom 50% hold 50%, and so on.
• The Lorenz curve itself: the observed or estimated distribution, which starts at (0, 0), ends at (100%, 100%) and rises more steeply as it moves right.
• The gap between them: the area enclosed between the diagonal and the curve, which is what the Gini coefficient measures.

Reading individual points

Each point answers a concrete question. If the curve passes through (50%, 20%), the poorer half of the population receives 20% of total income, and the richer half therefore receives 80%. If it passes through (90%, 65%), the top 10% hold 35%.

What the slope tells you

The curve bends upward because people further right are richer. Its slope at any point equals the income of the person at that percentile divided by the population’s mean income. Where the slope is below 1, people earn less than average; where it is above 1, they earn more. The point where the slope equals exactly 1 marks the person with average income, and it is also where the vertical gap between the curve and the diagonal is largest.

What happens to the Lorenz curve when inequality increases

When inequality increases, the curve bows further from the line of equality, sinking toward the bottom-right corner. When it falls, the curve moves back toward the diagonal. In the extreme case of perfect inequality, where one person holds everything, the curve runs flat along the horizontal axis at 0% until the very last person, then shoots straight up to 100%, forming an L shape along the bottom and right edges of the square.

How a Lorenz Curve Is Built: A Worked Example

The construction takes four steps: rank the units, compute each unit’s share of the total, accumulate those shares and plot them against the cumulative population share.

Hypothetical example. Consider a small economy of five households with these annual incomes (in thousands of dollars): 10, 20, 30, 40 and 100. Total income is 200.

Plot the points (0, 0), (20%, 5%), (40%, 15%), (60%, 30%), (80%, 50%) and (100%, 100%), then connect them. The result is a bowed line well below the diagonal. At the midpoint of the population, the curve sits around 22% (between the 40% and 60% points), against 50% on the line of equality. The top household alone holds half of all income, which shows up as the steep final segment.

With real populations, analysts rarely have every individual’s income. They work from survey responses, tax records or grouped data such as income brackets, and either connect the observed points directly or fit a continuous function through them. There is no universal formula: the functional form, often a nonlinear regression, is chosen to match each data set.

The Lorenz Curve and the Gini Coefficient

The Gini coefficient, named after Italian statistician Corrado Gini, who published it in 1912, turns the Lorenz curve into a single number. Call the area between the line of equality and the Lorenz curve A, and the area under the Lorenz curve B. Then:

Gini = A / (A + B)

Because the graph is a unit square, the triangle under the diagonal has an area of 0.5, so A + B = 0.5 and the formula simplifies to:

Gini = 1 − 2B

A Gini of 0 means perfect equality (the curve is the diagonal, so A = 0). A Gini of 1 means perfect inequality (B = 0). It is often reported as a percentage or on a 0 to 100 scale.

Calculating the Gini from the example

With points connected by straight lines, the area under the curve is the sum of trapezoids. Each segment here is 0.2 wide, so the area of each trapezoid is 0.2 × (left height + right height) / 2.

1. 0.2 × (0 + 0.05) / 2 = 0.005
2. 0.2 × (0.05 + 0.15) / 2 = 0.020
3. 0.2 × (0.15 + 0.30) / 2 = 0.045
4. 0.2 × (0.30 + 0.50) / 2 = 0.080
5. 0.2 × (0.50 + 1.00) / 2 = 0.150

B = 0.300, so Gini = 1 − 2(0.300) = 0.40.

The general version of this calculation for grouped data is:

Gini = 1 − Σ (Xₖ − Xₖ₋₁)(Yₖ + Yₖ₋₁)

where Xₖ is the cumulative population share and Yₖ the cumulative income share at point k. The assumption behind it is that income is equal within each group, which is why grouped data tends to understate inequality compared with individual records.

Range and edge cases

Two details change the usual 0 to 1 range:

• Small populations. With n equal-sized units measured this way, the maximum Gini is (n − 1)/n. In the five-household example, if one household had all the income, the Gini would be 0.8, not 1.
• Negative values. Wealth can be negative when debts exceed assets. In that case the cumulative share for the poorest group falls below zero, the Lorenz curve dips under the horizontal axis and the Gini coefficient can, in theory, exceed 1.

Comparing Curves: Lorenz Dominance and Crossing Curves

The Lorenz curve is most powerful when two distributions are drawn on the same graph, such as two countries, two years or income before and after taxes and transfers.

Lorenz dominance

If curve A lies on or above curve B at every point, and above it at least somewhere, A is said to Lorenz dominate B. Distribution A is then more equal by every inequality measure that respects the transfer principle: the rule that moving income from a richer person to a poorer one, without reversing their ranking, reduces inequality. That family includes the Gini coefficient, and the ranking holds without having to choose one index.

Hypothetical example, continued. Suppose the richest household in the earlier example transfers 10 to the poorest. Incomes become 20, 20, 30, 40 and 90. The cumulative income shares become 10%, 20%, 35%, 55% and 100%. Every point is higher than before, so the new curve dominates the old one, and the Gini falls from 0.40 to 0.32 (1 − 0.2 × 3.40 = 0.32).

When curves cross

If two curves cross, neither dominates. One society may give its poorest 20% a larger share while its top 10% also capture more, leaving the middle squeezed. Different inequality measures can rank such distributions in opposite order, and two crossing curves can even produce exactly the same Gini coefficient while looking very different. This is the main reason analysts look at the curve itself rather than relying on the Gini alone.

Relative Inequality: What Moves the Curve and What Doesn’t

The Lorenz curve measures relative inequality, meaning shares of the total rather than dollar amounts. That has two consequences:

• Proportional changes leave it unchanged. If every income doubles, every share stays the same, so the curve and the Gini do not move.
• Equal absolute changes move it toward equality. If everyone receives the same extra $5,000, the poor gain proportionally more, the curve rises and the Gini falls, even though the dollar gap between richest and poorest is unchanged.

The curve also says nothing about living standards. A poor country and a rich country can have identical Lorenz curves. Pairing the curve with average income measures such as GDP per person, adjusted with purchasing power parity when comparing countries, gives a fuller picture of both how much there is and how it is shared.

Who Uses Lorenz Curves

Government statistical agencies use Lorenz curves and Gini coefficients to report on income and wealth distribution, and international organizations use them to compare countries. With sufficient data, a curve can be drawn for a state, city or demographic group, and because it is built from aggregated shares it reveals no individual’s finances. Policymakers compare the curve for market income with the curve for income after taxes and benefits. Where the gap between the two is largest shows where policy has the most effect, which can inform decisions about where tax brackets begin and end.

The same construction appears in other fields. Industry economists use it to show market concentration, businesses use it to see how much revenue comes from their largest customers, and ecologists use it to describe how unevenly individuals or biomass are spread among species. Any quantity shared among units can be plotted the same way, such as land among farms or patients among hospitals.

Limitations of the Lorenz Curve

• Estimated, not observed, between data points. A smooth curve is a function fitted to incomplete data. Only the observed points are measurements; everything in between depends on the chosen function.
• Sensitive to data quality. Surveys can miss the very rich or undercount informal income, and tax records exclude people who do not file. An unrepresentative sample produces a misleading curve.
• Costly to complete. Collecting enough data to pin down every part of the distribution is expensive, so gaps often remain.
• Hard to read at the tails. Wealth curves in particular can hug the horizontal axis for a long stretch, and changes there are difficult to see.
• Definitions change the result. Whether the unit is individuals or households, whether income is before or after taxes and transfers, and whether it is adjusted for household size all shift the curve. Two curves are only comparable if they use the same definitions.
• A snapshot, not a lifetime. An annual income curve mixes students, workers at peak earnings and retirees. Some measured inequality reflects life-cycle stage rather than permanent differences, and movement between percentiles over time is invisible.

Lorenz Curve vs. Related Concepts

Lorenz curve vs. Laffer curve. The Laffer curve relates tax rates to tax revenue and has nothing to do with measuring inequality.

Lorenz curve vs. Kuznets curve. The Kuznets curve is a hypothesis about how inequality changes as an economy develops, often drawn as an inverted U with inequality on the vertical axis and income per person on the horizontal axis. It describes a relationship over time; the Lorenz curve describes a distribution at one point.

Lorenz vs. Lorentz. The spelling matters. Lorentz (with a t) refers to Dutch physicist Hendrik Lorentz and concepts such as the Lorentz transformation. The economic tool is named after Max O. Lorenz.

The Bottom Line

The Lorenz curve shows inequality by comparing the actual cumulative distribution of income or wealth with the diagonal of perfect equality. Because it measures shares rather than amounts and is often fitted to incomplete data, it works best alongside average income figures and with a clear understanding of how the underlying data was defined.