Lorenz Curve
Also known as: Lorenz concentration curve, Lorenz diagram, cumulative share curve
The Lorenz curve is a graphical device that displays the full shape of inequality in a distribution by plotting the cumulative share of a quantity (such as income) held by the cumulative share of the population, ranked from poorest to richest. Introduced by Max Lorenz in 1905, it underlies the Gini coefficient and provides the basis for ranking distributions by inequality when one curve lies entirely above another.
Key highlights
- Displays the entire inequality structure, showing exactly where shares are concentrated.
- Provides an unambiguous inequality ranking when one curve dominates another (Lorenz dominance).
- Forms the geometric basis of the Gini coefficient and the Atkinson and generalized-entropy measures.
- Intuitive and visually compelling for communicating inequality to non-specialists.
Intuition
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How it works
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When to use it
Use the Lorenz curve to visualize the complete inequality structure of a distribution, to compare distributions when one may dominate another, and as the foundation for computing the Gini coefficient and related measures. It is invaluable for seeing where in the distribution inequality is concentrated rather than collapsing it to one number. It is descriptive and graphical, so when distributions' curves cross it cannot by itself rank them, and it conveys relative shares only — two populations with identical Lorenz curves can have vastly different absolute living standards. It is defined for nonnegative quantities; negative incomes pull the curve below the axis and complicate interpretation.
Strengths & limitations
- Displays the entire inequality structure, showing exactly where shares are concentrated.
- Provides an unambiguous inequality ranking when one curve dominates another (Lorenz dominance).
- Forms the geometric basis of the Gini coefficient and the Atkinson and generalized-entropy measures.
- Intuitive and visually compelling for communicating inequality to non-specialists.
- Cannot rank distributions whose curves cross; the picture alone is then indecisive.
- Shows relative shares only, so it ignores absolute levels and overall growth.
- Reading precise quantities off the curve is imprecise compared with summary indices.
- Sensitive to negative or zero values, which distort the convex shape.
Common pitfalls
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Applications
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Frequently asked
How is the Lorenz curve related to the Gini coefficient?
The Gini coefficient is exactly twice the area between the Lorenz curve and the line of perfect equality. A curve hugging the diagonal (little area) gives a Gini near zero, and a deeply bowed curve gives a Gini near one. The Lorenz curve thus contains all the information the Gini summarizes plus the shape detail the single number discards.
What does it mean when two Lorenz curves cross?
Crossing means one distribution is more equal in one part of the range (say, the bottom) but less equal in another (say, the top). In that case there is no unambiguous inequality ranking: different inequality indices, which weight parts of the distribution differently, can legitimately disagree about which distribution is more unequal. Only when one curve lies everywhere above the other (Lorenz dominance) do all standard measures agree.
Can the Lorenz curve compare living standards between countries?
Only their relative inequality, not their absolute prosperity. Because the curve plots shares, it is scale-invariant: a poor and a rich country with identically shaped distributions have the same Lorenz curve. To compare living standards you need absolute figures (mean income, generalized Lorenz curves that scale by the mean), not the ordinary Lorenz curve alone.
Sources
- 1.Lorenz, M. O. (1905). Methods of measuring the concentration of wealth. Publications of the American Statistical Association, 9(70), 209–219.
- 2.Atkinson, A. B. (1970). On the measurement of inequality. Journal of Economic Theory, 2(3), 244–263.
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Cite this page
ScholarGate. (2026, June 22). Lorenz Curve. ScholarGate. https://scholargate.app/sociology/lorenz-curve