“The Lorenz curve is a graphical representation of income or wealth inequality within a population. It plots the cumulative percentage of total income (or wealth) held by cumulative percentages of the population, ordered from poorest to richest. The curve is used to visualize how much a distribution deviates from perfect equality.” – Lorenz curve
The **Lorenz curve** provides a visual method to assess the distribution of income, wealth, or other resources across a population, plotting the cumulative percentage of the total held by the cumulative percentage of individuals from poorest to richest.1,2 Developed by American economist Max O. Lorenz in 1905, it compares actual distributions against the line of perfect equality-a straight diagonal line from (0,0) to (1,1), where the bottom N% of the population holds exactly N% of the total.1,3
The curve always begins at the origin (0,0) and terminates at (1,1), lying below or along the equality line; the greater the vertical distance between the curve and this line, the higher the inequality.1,4 For instance, if the bottom 20% of households possess only 5% of total income, that point marks a position well below the equality line, indicating significant disparity.2
Mathematical Definition
For a continuous probability distribution with density function f and cumulative distribution function F, the Lorenz curve L(F) is defined as:
L(F(x)) = \frac{\int_{-\infty}^{x} t f(t) dt}{\int_{-\infty}^{\infty} t f(t) dt} = \frac{\int_{-\infty}^{x} t f(t) dt}{\mu}where ? is the mean.1 In discrete cases, it connects points (Fi, Li) based on ordered population shares.1,3
Key Properties
- Invariant under positive scaling: multiplying all values by a constant c > 0 yields the same curve.1
- Cannot exceed the line of perfect equality and is non-decreasing for non-negative variables.1
- Often summarised by the **Gini coefficient**, the ratio of the area between the curve and equality line to the total area under the equality line.1,3,7
Applications and Examples
Beyond income, Lorenz curves illustrate wealth inequality-for example, in Great Britain, the bottom 38% held zero property wealth, while the top 10% owned nearly 50%.2 They also apply to risk predictiveness in epidemiology or size distributions in ecology.3,5
Max O. Lorenz: The Theorist Behind the Curve
**Max O. Lorenz (1880-1962)**, the originator of the Lorenz curve, was a pioneering American economist and statistician whose work laid foundational stones in inequality analysis.1,4 Born in Tustin, Michigan, Lorenz earned his PhD in economics from the University of Wisconsin in 1906, shortly after publishing his seminal 1905 paper ‘The Distribution and Concentration of Wealth’ in the Publications of the American Statistical Association, where he introduced the curve to depict wealth disparities.1
Lorenz’s academic career spanned institutions like the University of Michigan, Stanford University, and the U.S. Bureau of Labor Statistics, where he applied statistical methods to economic data during the early 20th century-a period marked by rapid industrialisation and growing concerns over wealth concentration amid Progressive Era reforms.1 Though initially overlooked, his graphic tool gained prominence decades later, notably through Corrado Gini’s 1912 development of the associated Gini coefficient, cementing Lorenz’s legacy in distribution theory.1,3 Lorenz’s broader contributions included statistical critiques of economic data reliability, influencing modern econometrics and policy discussions on equity.1
References
1. https://en.wikipedia.org/wiki/Lorenz_curve
2. https://www.economicshelp.org/blog/glossary/lorenz-curve/
3. https://mathworld.wolfram.com/LorenzCurve.html
4. https://www.datacamp.com/tutorial/lorenz-curve
5. https://pmc.ncbi.nlm.nih.gov/articles/PMC5495014/
6. https://www.youtube.com/shorts/SWYahSGMk8k
