Lorenz Curve

Lorenz Curve in Health Inequality and Health Spending Analysis

The Lorenz curve is the standard picture of how unevenly a quantity is spread across a population, read as the share of the total held by the bottom fifth, the bottom half and so on once everyone is ranked by how much they have. A straight diagonal means everyone holds the same amount, and the further the curve sags beneath it, the more unequal the distribution. Health economists draw it for health itself, for example quality-adjusted life expectancy in distributional cost-effectiveness analysis, for health care spending, and for income or ability to pay, where it is the benchmark for judging whether health financing is progressive. Twice the area between the curve and the diagonal is the Gini coefficient, and comparing whole curves can rank two distributions without choosing any single index. Because people are ranked by the variable itself, the curve measures pure inequality in that variable, and inequality linked to income or deprivation needs the concentration curve instead.

Lorenz's method and how the curve is drawn

Max O. Lorenz proposed the method in 1905 in the Publications of the American Statistical Association, to judge whether wealth was becoming more or less concentrated. He plotted cumulated percentages of the population, from poorest to richest, along one axis and the percentage of total wealth held by them along the other. An equal distribution gives a straight line, while an unequal one begins and ends at the same points but is bent in the middle, and in his words "as the bow is bent, concentration increases."

With individual data sorted from the smallest value to the largest, the curve is drawn through the following points.

$$p_k = \frac{k}{N}, \qquad L(p_k) = \frac{\sum_{i=1}^{k} x_i}{\sum_{i=1}^{N} x_i}, \qquad x_1 \le x_2 \le \dots \le x_N$$

where $N$ is the number of people, $x_i$ is the value of the variable for the person in position $i$ of the ranking, $p_k$ is the cumulative population share up to position $k$ and $L(p_k)$ is the share of the total held by the $k$ people with the smallest values. The curve starts at $(0, 0)$, ends at $(1, 1)$ and joins the points with straight lines. With survey weights, each person's weight replaces the count of one in $p_k$ and multiplies $x_i$ in both sums. With grouped data only the group boundaries are plotted. The World Bank Handbook on Poverty and Inequality by Haughton and Khandker uses the Vietnam Living Standards Survey of 1993, where cumulative expenditure shares by fifth of individuals were 8.4, 20.7, 36.7, 58.5 and 100 per cent.

What the shape of the curve shows

The segment for the person in position $k$ rises by $x_k / \sum x_i$ over a run of one person's share $p_k - p_{k-1}$, so its slope is $x_k / \mu$, where $\mu$ is the mean of the variable. Since the values are sorted upwards, the slopes never fall, so the curve is convex and can never rise above the diagonal. People with a value of zero, common in a year of health spending data, give a flat stretch along the horizontal axis.

The slope passes one at the person whose value equals the mean, and the vertical gap between the diagonal and the curve is largest there. That gap equals the share of the total that would have to move from people above the mean to people below it for everyone to hold the same amount.

The curve describes relative inequality only. Lorenz used a hypothetical case to show that doubling everyone's wealth leaves the relation between individuals, and so the degree of concentration, unchanged, and Haughton and Khandker list this mean independence among the properties of the Gini coefficient. Adding the same amount to everyone moves the curve towards the diagonal. Two populations with very different average health or spending can therefore share one curve, which is why the generalised Lorenz curve below brings the mean back in.

Ranking distributions by Lorenz dominance

When one curve lies on or above another everywhere and above it somewhere, the first distribution is unambiguously less unequal and no index needs to be chosen. Lorenz saw that the answer is not always clear, because opposing tendencies can operate in different parts of a distribution. In his hypothetical case of a sum shared among ten people at two dates, the poorest half held 40 per cent at both, but the poorest two-thirds held less at the second. When curves cross, only a summary index can rank them, and O'Donnell and colleagues note that such indices impose value judgements about the weight given to different points of the distribution.

Asaria, Griffin and Cookson, in their tutorial on distributional cost-effectiveness analysis, use two dominance rules built on the Lorenz curve, constructed for health by ordering the population from least to most healthy. Under Atkinson's theorem, a distribution is dominated if its Lorenz curve lies below that of an alternative, without crossing, and its mean health is the same or lower. Under Shorrocks' theorem, a distribution is dominated if its generalised Lorenz curve lies wholly below that of an alternative. The generalised curve is defined as follows.

$$GL(p) = \mu \, L(p)$$

where $GL(p)$ is the generalised Lorenz ordinate at population share $p$, $\mu$ is mean health per person and $L(p)$ is the Lorenz ordinate. With equal-sized groups it equals the cumulative total divided by the whole population. These rules rank strategies with minimal restriction on the form of the social welfare function. When generalised curves cross, a fuller welfare function is needed, for example one built on the Atkinson index with a stated level of inequality aversion. Curves estimated from surveys carry sampling error, so O'Donnell and colleagues test dominance by comparing ordinates at a set of quantile points.

The Gini coefficient as an area on the curve

Haughton and Khandker define the Gini coefficient as $A/(A + B)$, where $A$ is the area between the diagonal and the curve and $B$ is the area under the curve. The triangle under the diagonal has area one half, which gives a formula that works directly from the plotted points.

$$G = \frac{A}{A + B} = 1 - 2B = 1 - \sum_{k=1}^{K} (p_k - p_{k-1})(L_k + L_{k-1})$$

where $G$ is the Gini coefficient, $K$ is the number of plotted points, $p_k$ and $L_k$ are the cumulative population and variable shares at point $k$, and $p_0 = L_0 = 0$. The sum is the trapezoid rule for the area under the curve. The coefficient runs from 0 when the curve is the diagonal towards 1 when one person holds everything, and its properties and its pairwise-difference form are set out on the Gini coefficient page.

For the Vietnam quintiles, the adjacent sums of cumulative shares are 0.084, 0.291, 0.574, 0.952 and 1.585, which total 3.486. Each step is 0.2 wide, so $G = 1 - 0.2 \times 3.486 = 0.303$. From the full microdata Haughton and Khandker report 0.3126 for 1993, with a standard error of 0.0045. The grouped figure is lower because straight lines between quintile points lie above the true curve and ignore inequality within each fifth.

Worked example with health spending and two health programmes

All figures in this example are illustrative. Ten people have annual health care spending of GBP 0, 0, 50, 100, 150, 200, 300, 500, 1,200 and 2,500, a total of GBP 5,000 and a mean of GBP 500.

1. Cumulative shares. Dividing the running totals by 5,000 gives 0, 0, 0.01, 0.03, 0.06, 0.10, 0.16, 0.26, 0.50 and 1.00.

2. Gini coefficient. Each step is 0.1 wide. The adjacent sums are 0, 0, 0.01, 0.04, 0.09, 0.16, 0.26, 0.42, 0.76 and 1.50, which total 3.24, so $G = 1 - 0.1 \times 3.24 = 0.676$. The pairwise-difference formula on the Gini coefficient page gives the same value.

3. The largest gap. Seven people spend less than the mean, so at $p = 0.7$ the curve is at 0.16 and the gap is 0.54. Together they spend GBP 800, which is GBP 2,700 short of the GBP 3,500 they would hold at the mean, and 2,700 divided by 5,000 is 0.54.

4. Reading the spending curve. Spending can be concentrated simply because need is concentrated, so the 54 per cent gap is not by itself a target for redistribution and the curve alone cannot show whether spending matches need.

The second part compares a baseline with two programmes, X and Y, using quality-adjusted life expectancy in years for five equal-sized fifths of the population ranked from least to most healthy.

5. Lorenz ordinates. At shares 0.2, 0.4, 0.6 and 0.8 they are 0.1563, 0.3438, 0.5500 and 0.7688 for the baseline, 0.1610, 0.3498, 0.5542 and 0.7709 for X, and 0.1560, 0.3394, 0.5443 and 0.7645 for Y. The curve for X lies above the other two everywhere and X has a higher mean than the baseline, so by Atkinson's theorem the baseline is dominated. X against Y, and Y against the baseline, cannot be ranked this way, because in each pair the more equal option has the lower mean.

6. Generalised Lorenz ordinates. Cumulative totals divided by five give 10.0, 22.0, 35.2, 49.2 and 64.0 for the baseline, 10.4, 22.6, 35.8, 49.8 and 64.6 for X, and 10.2, 22.2, 35.6, 50.0 and 65.4 for Y. Y lies above the baseline at every point, so by Shorrocks' theorem Y also dominates it. X is higher than Y at the first three points and lower at the last two, so their generalised curves cross.

7. Interpretation. Both programmes beat the baseline, but X is the more equal and Y gives more total health, so choosing between them needs an explicit level of inequality aversion.

Lorenz curves in health spending, financing and health analysis

The concentration of health spending is a standard Lorenz curve reading. The Agency for Healthcare Research and Quality statistical brief on the 2018 Medical Expenditure Panel Survey reports that about 13 per cent of the US civilian noninstitutionalised population had no health care expenditure, the bottom 50 per cent accounted for 3.2 per cent and the top 5 per cent for 48.3 per cent. On a Lorenz curve these are a flat start up to 0.13 and ordinates of 0.032 at 0.5 and 0.517 at 0.95. People aged 65 and over were 16.8 per cent of the population but 39.0 per cent of the top 5 per cent of spenders, so much of the concentration tracks age.

In financing analysis the Lorenz curve of ability to pay is the benchmark. O'Donnell and colleagues explain that if payments are a fixed proportion of ability to pay, the concentration curve of payments lies on top of the Lorenz curve, and under a progressive system the Lorenz curve lies above it. The Kakwani index, twice the area between the two curves, summarises this departure from proportionality for analyses of vertical equity. In benefit incidence analysis, the concentration curve of a subsidy is tested against both the diagonal and the Lorenz curve of household consumption.

For health itself, the curve describes the spread of health across individuals, the object of the dominance rules above. Wagstaff, Paci and van Doorslaer argued that only the slope index of inequality and the concentration index were likely to give an accurate picture of socioeconomic inequalities in health, and warned against the Lorenz curve for that purpose.

Lorenz curve, concentration curve and summary indices

The concentration curve uses the same axes but ranks people by income, consumption or a wealth index, so it can lie above the diagonal, below it or on both sides, while a Lorenz curve never rises above it. The two coincide only when ranking by living standards and by the variable gives the same order. The concentration index is to the concentration curve what the Gini coefficient is to the Lorenz curve. The Gini, the Atkinson index and the Theil index each condense a Lorenz curve into one number with different weights on different parts of the distribution.

Limits and common misreadings

Most errors come from asking the curve a question it cannot answer. It describes how a total is shared, and each misreading below stretches that description too far.

1. Reading inequality as inequity. A concentrated spending curve can be fair if it follows need, and a curve of health says nothing about who is disadvantaged by income or deprivation.
2. Forgetting the mean. The curve is unchanged when every value is scaled by one factor, so a population whose health falls in proportion keeps the same curve. The generalised curve shows the loss.
3. Ranking crossing curves without stating the value judgement. The ranking then depends on the index and its weights.
4. Using negative values. Changes in net health benefit can be negative, and the curve then dips below zero and loses its usual bounds.
5. Treating the curve as a record of individuals. Swapping two people's values leaves the curve unchanged, so a dominant curve can still leave some people worse off.

Sources

• Asaria M, Griffin S, Cookson R. Distributional cost-effectiveness analysis: a tutorial. Medical Decision Making. 2016;36(1):8-19. https://doi.org/10.1177/0272989X15583266
• Haughton J, Khandker SR. Handbook on Poverty and Inequality. Washington, DC: World Bank; 2009. Chapter 6, Inequality measures. https://documents1.worldbank.org/curated/en/488081468157174849/pdf/483380PUB0Pove101OFFICIAL0USE0ONLY1.pdf
• Lorenz MO. Methods of measuring the concentration of wealth. Publications of the American Statistical Association. 1905;9(70):209-219. https://doi.org/10.2307/2276207
• Mitchell EM. Concentration of Healthcare Expenditures and Selected Characteristics of Persons with High Expenses, U.S. Civilian Noninstitutionalized Population, 2018. Statistical Brief 533. Rockville, MD: Agency for Healthcare Research and Quality; January 2021. https://meps.ahrq.gov/data_files/publications/st533/stat533.pdf
• O'Donnell O, van Doorslaer E, Wagstaff A, Lindelow M. Analyzing Health Equity Using Household Survey Data: A Guide to Techniques and Their Implementation. Washington, DC: World Bank; 2008. Chapter 7, Concentration curves; Chapter 16, Who pays for health care. https://www.worldbank.org/content/dam/Worldbank/document/HDN/Health/HealthEquityCh7.pdf
• Wagstaff A, Paci P, van Doorslaer E. On the measurement of inequalities in health. Social Science & Medicine. 1991;33(5):545-557. https://doi.org/10.1016/0277-9536(91)90212-u