Lorenz curve

The Lorenz curve is a key analytical tool used in economics to illustrate the distribution of income or wealth within a population. Developed by the American economist Max O. Lorenz in 1905, it provides a graphical method for assessing economic inequality by comparing the cumulative share of income or wealth with the cumulative share of the population. The curve has subsequently been adopted in other disciplines such as ecology and biodiversity studies, where it is used to describe inequality in species abundance or individual sizes.

Concept and Interpretation

The Lorenz curve plots population percentages on the horizontal axis and cumulative income or wealth percentages on the vertical axis. Each point on the curve provides a statement such as “the bottom 20 per cent of households hold 10 per cent of total income”. A perfectly equal distribution would produce a straight diagonal line from the origin to the top right corner, known as the line of perfect equality, as each proportion of the population would hold a corresponding proportion of the total income. The opposite extreme, the line of perfect inequality, would describe a scenario in which a single individual possesses all available wealth.
In real-world distributions, the Lorenz curve usually bows below the line of perfect equality, indicating varying levels of inequality. The deviation from the equality line provides the basis for several inequality measures, the most notable of which is the Gini coefficient. This coefficient is calculated as the ratio of the area between the line of perfect equality and the Lorenz curve to the total area beneath the equality line. Values closer to one indicate greater inequality.

Mathematical Definition and Calculation

The Lorenz curve can be defined and computed for both discrete and continuous probability distributions. It may be understood as a probability plot comparing an observed distribution with a hypothetical uniform distribution of the same variable. The cumulative population proportion is represented on the horizontal axis, while the cumulative proportion of income or wealth is plotted on the vertical axis.
For a discrete distribution, given income values arranged in non-decreasing order, the Lorenz curve is obtained by:

• Calculating cumulative population shares,
• Calculating cumulative income shares,
• Plotting the resulting paired values and joining them with line segments.

If all individuals are equally probable, the cumulative population fraction increases uniformly, while the cumulative income share is determined by the sum of observed values relative to the total.
For a continuous distribution, with a probability density function and cumulative distribution function, the Lorenz curve may be written in integral form. It expresses the cumulative share of income held by the bottom portion of the population with mean income included as a scaling factor. An equivalent formulation expresses the curve using the inverse cumulative distribution function where appropriate.
The Lorenz curve for many common distributions can be derived explicitly. A well-known example is the Pareto distribution, often used to model high levels of wealth or income concentration in which a small proportion of the population holds a disproportionately large share of total resources.

Properties and Theoretical Characteristics

Lorenz curves exhibit several general properties that make them useful for comparing different distributions:

• They always start at (0,0) and end at (1,1).
• They cannot rise above the line of perfect equality.
• If two Lorenz curves are compared, one may be said to Lorenz dominate the other if it lies nowhere below and at least once above the other curve, indicating a more equal distribution.
• For non-negative variables, the curve cannot dip below the line of perfect inequality; however, for net worth distributions that include negative wealth, the earliest part of the curve can sink below zero.
• They are invariant under positive scaling: multiplying all incomes by a positive constant does not change the Lorenz curve.
• They are affected by translation: adding a constant to all incomes alters the curve because the relative differences between individuals change.
• Differentiability properties link the Lorenz curve to the underlying probability density function, providing useful insights into distributional behaviour at various cumulative levels.

The Lorenz curve also gives rise to additional measures such as the Lorenz asymmetry coefficient, which reflects the degree to which inequality is skewed toward either upper or lower population segments.

Applications in Economics and Beyond

In economics, the Lorenz curve supports the study of income and wealth inequality, offering an intuitive visual representation that complements numerical indicators. Policymakers use it to evaluate the effects of taxation, welfare schemes, and economic reforms on distribution.
Beyond income and wealth, the curve is widely applied to assess inequality in areas such as:

• Asset distribution, including the ownership of land or capital;
• Ecological studies, where it describes disparities in the sizes of organisms within a species or the abundance of different species within an ecosystem;
• Biodiversity analysis, in which cumulative species proportions are plotted against cumulative individual counts.

The curve’s adaptability enables its use across a broad range of scientific, social, and environmental contexts.

Significance for Inequality Measurement

The power of the Lorenz curve lies in its ability to summarise complex distributional information through a simple geometric representation. It provides the foundation for numerous inequality indices, the most prominent being the Gini coefficient, but also measures such as the Atkinson index, Theil index, and Hoover index. Statistical software packages frequently include built-in functions for plotting Lorenz curves and calculating associated metrics, facilitating their use in academic research and policy analysis.