Concept Architecture
Concept
Theoretically, Lorenz Curve is a graphical representation of the cumulative distribution of a resource, such as income, wealth, healthcare utilisation or healthcare expenditure, across a ranked population. Developed in welfare economics, it illustrates the degree of equality or inequality by comparing the observed cumulative distribution with a line of perfect equality. In health economics, the Lorenz Curve is used to assess inequality in healthcare financing, healthcare expenditure and other distributive outcomes.
Mathematically, the Lorenz Curve is represented as a cumulative distribution function relating the cumulative proportion of the population, ordered from lowest to highest resource level, to the cumulative proportion of the resource held by that population. The curve lies on or below the line of perfect equality, with greater deviation indicating greater inequality. The area between the Lorenz Curve and the line of equality forms the basis for calculating summary inequality measures such as the Gini coefficient.
In practice, the Lorenz Curve is constructed by ranking individuals or households according to income or another distributional variable, calculating cumulative population proportions and cumulative resource shares, and plotting these values. It is widely implemented in health economics to evaluate equity in healthcare financing, out-of-pocket expenditure, insurance contributions and the distribution of healthcare resources across populations.
Purpose
Used to visualise and quantify inequality in the distribution of healthcare resources, expenditure or income, support equity analyses, and provide the graphical basis for summary measures such as the Gini coefficient.
Mathematical Formulae
Primary Formula
There is no universally recognised canonical mathematical formula.
Supporting Formulae
Lorenz Curve function:
L(p) = [??? x(F) dF] / ?
where:
- L(p) = cumulative proportion of the resource
- p = cumulative proportion of the population
- x(F) = resource value at cumulative population proportion F
- ? = mean resource level
Relationship with the Gini coefficient:
G = 1 ? 2??? L(p) dp
where:
- G = Gini coefficient
Related Mathematical Methods
- Gini coefficient
- Concentration curve analysis
- Concentration index
- Cumulative distribution analysis
- Inequality measurement
Example
A population is ranked by annual healthcare expenditure.
| Cumulative Population | Cumulative Expenditure |
|---|---|
| 20% | 8% |
| 40% | 22% |
| 60% | 43% |
| 80% | 69% |
| 100% | 100% |
Plotting these cumulative percentages produces the Lorenz Curve. The curve lies below the line of perfect equality, indicating that healthcare expenditure is unequally distributed across the population. The area between the two curves can subsequently be used to calculate the Gini coefficient.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SORT | =SORT(A2:B101,2,1) | Orders observations from lowest to highest resource level. |
| SUM | =SUM($B$2:B2)/SUM($B$2:$B$101) | Calculates cumulative resource shares for the Lorenz Curve. |
| ROWS | =ROWS($A$2:A2)/ROWS($A$2:$A$101) | Calculates cumulative population proportions. |
| Scatter Chart | XY Scatter with Lines | Produces the Lorenz Curve for inequality analysis. |
VBA (Optional)
Automate the construction of Lorenz Curves and generate accompanying inequality statistics for multiple healthcare datasets.
Sources
- Lorenz MO. Methods of Measuring the Concentration of Wealth. Publications of the American Statistical Association. 1905.
- O'Donnell O, van Doorslaer E, Wagstaff A, Lindelow M. Analyzing Health Equity Using Household Survey Data: A Guide to Techniques and Their Implementation. World Bank.
- De Maio FG. Income Inequality Measures. Journal of Epidemiology and Community Health. 2007.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. Oxford University Press.
Related Concepts (2)
Library
Publications
1
Uncertainty and the Welfare Economics of Medical Care — Kenneth J. Arrow, Vol. 53, No. 5 ed., 1963 (American Economic Review)
The founding paper of health economics as a discipline, analysing how uncertainty, asymmetric information, trust and the special features of medical markets prevent them from behaving like ordinary competitive markets — the intellectual origin of the entire field.
Journal ArticleView source →
Frequently Asked Questions (7)
What is a lorenz curve?
A graph plotting the cumulative share of total income against the cumulative share of population, ranked poorest to richest, showing distribution.
Source: Lorenz 1905
What is a Lorenz curve?
A Lorenz curve is a graph plotting the cumulative share of total income held against the cumulative share of the population, with people ranked from poorest to richest, showing how income is distributed. Each point gives the share of total income received by the poorest given fraction of the population. Introduced by Lorenz, the curve provides a visual representation of inequality, and its shape shows how far a distribution departs from perfect equality across the whole population.
Source: Lorenz 1905
Who developed the Lorenz curve?
The curve is named after Max Lorenz, an American economist who devised it in the early twentieth century as a way to picture how evenly or unevenly income is spread across a population. He plotted the cumulative share of income against the cumulative share of people, ranked from poorest to richest, so that perfect equality traces a straight diagonal and any bowing below it shows inequality. The tool has since been applied well beyond income. Lorenz (1905) introduced the method.
Source: Lorenz 1905
How is a Lorenz curve interpreted?
A Lorenz curve is interpreted by comparison with the line of perfect equality, the diagonal, along which each share of the population holds an equal share of income. The further the curve bows below this diagonal, the more unequal the distribution, since the poorer part of the population then holds a smaller share of income. A curve lying on the diagonal represents complete equality, while one far below it represents high inequality, so the gap between curve and diagonal measures inequality visually.
Source: Lorenz 1905
How does the Lorenz curve relate to the Gini coefficient?
The Lorenz curve is the basis of the Gini coefficient, the most common summary measure of inequality. The Gini coefficient equals the area between the Lorenz curve and the line of perfect equality, as a proportion of the total area under the diagonal. It ranges from zero, for perfect equality when the curve lies on the diagonal, to one, for maximum inequality. The Gini thus reduces the information in the Lorenz curve to a single number summarising the degree of inequality.
Source: Lorenz 1905
What are the limitations of the Lorenz curve?
The Lorenz curve has limitations. When the curves for two distributions cross, neither is unambiguously more unequal, so the curve alone cannot rank them, and a summary measure that embeds particular value judgements is needed. It shows relative shares but not absolute levels, so two societies with very different incomes can have the same curve. It also captures only the distribution measured, such as income, not other dimensions of inequality. It depicts inequality clearly but does not by itself resolve comparisons.
Source: Lorenz 1905
How is the Lorenz curve used in health?
In health, the Lorenz curve and related concentration curves are used to depict how health, health care use, or health financing is distributed across the population, often ranked by income rather than by the variable itself. A concentration curve plots the cumulative share of, say, health care use against the cumulative share of population ranked by income, showing whether use is concentrated among richer or poorer people. This adapts the Lorenz approach to measure inequality and inequity in health and its financing.
Source: Lorenz 1905
Trust Record
Verified by Dr Darrin Baines
British health economist
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Verification date: 26 Sep 2025
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