Lorenz Curve

The Lorenz curve was introduced by American economist Max O. Lorenz in a 1905 paper for the Publications of the American Statistical Association titled "Methods of Measuring the Concentration of Wealth." It plots the cumulative percentage of total income (or wealth, consumption, land, etc.) on the vertical axis against the cumulative percentage of the population, ranked from poorest to richest, on the horizontal axis.

If income were distributed perfectly equally, the curve would coincide with the 45-degree diagonal — the line of perfect equality. The further the actual curve bows away from this diagonal toward the bottom-right corner, the more unequal the distribution. Perfect inequality (one person holding everything) would trace the bottom and right edges of the unit square.

The curve is the geometric foundation for the Gini coefficient, defined as the ratio of the area between the line of equality and the Lorenz curve to the total area under the line of equality. A Gini of 0 corresponds to a Lorenz curve on the diagonal; a Gini approaching 1 corresponds to a curve hugging the axes.

Lorenz curves are widely used by the World Bank, the OECD, and national statistical offices to compare distributions across countries or over time. They support Lorenz dominance analysis: if country A's curve lies entirely above country B's, A is unambiguously more equal regardless of the inequality index chosen. When two curves cross, however, different indices (Gini, Theil, Atkinson) can rank them differently.

Limitations include sensitivity to data source (household surveys versus tax records often diverge, as work by Thomas Piketty and Emmanuel Saez has highlighted), the masking of changes within population segments, and the requirement that the underlying variable be non-negative. Despite these caveats, the Lorenz curve remains the standard visual tool for distributional analysis in development economics, public finance, and policy research.