Signature
W = min(h_1, min(h_2, h_3))
| Inputs | Definition | Unit |
|---|---|---|
h_1 | Health of the first group | years of QALE |
h_2 | Health of the second group | years of QALE |
h_3 | Health of the third group | years of QALE |
W | Social welfare under the maximin form, the health of the least healthy group | years of QALE |
|---|
Function
Health social welfare function ranking distributions of health
Maps the health of n people, or of n groups of equal size, and the parameters theta that state a distributional value judgement to a single social welfare value W, so that allocations can be ranked when they differ in who gains as well as in how much is gained. When W rises whenever one person's health rises and nobody's falls, every Pareto improvement raises welfare. The package gives Atkinson's isoelastic additive form for epsilon other than 1, its logarithmic form at epsilon of 1, the maximin limit, the two-group CES form of Wagstaff and the slope of its social indifference curve. The equally distributed equivalent (EDE) health that gives the same W is live as HE-FM-ATK-002 and HE-FM-ATK-003, the Atkinson index as HE-FM-ATK-001, the relative weight on a small gain as HE-FM-ATK-004 and equity-weighted QALY totals as HE-FM-EQWA-001, and HE-FM-WE-004 holds the linear two-group welfare with a switching weight; none is redrafted here. The health forgone for equity and the inequality aversion at which two options swap rank are left to the Equity-Efficiency Trade-off and Distributional Cost-Effectiveness packages. Notation follows the Social Welfare Function article.
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Implementations
Excel
Maximin social welfare from named cells
With the named cells of HE-IM-SWF-001 the formula returns W in a cell named MaximinWelfare.
=MIN(HealthFirstGroup,HealthSecondGroup,HealthThirdGroup)
Assumptions
Maximin welfare compares groups at their mean health
Each h_i is the mean health of a group, so the form protects the least healthy group on average and not its least healthy member, and the result depends on how the population is divided into groups.
Worked examples
Maximin welfare of the option for the worst off
Option R in the article's illustrative example gives the least healthy group 61.2 years, the highest minimum of the three options, so R ranks first under maximin.
h_1 = 61.2; h_2 = 67.6; h_3 = 75.6; W = 61.2
Maximin welfare of the population-wide option
Option P gives the least healthy group 60.3 years and ranks last under maximin, although it has the largest total.
h_1 = 60.3; h_2 = 68.5; h_3 = 76.7; W = 60.3
Common errors
Reading maximin health welfare as Rawls's own theory
The maximin health form borrows Rawls's priority for the worst off, but Brouwer and colleagues note that Rawls explicitly excluded health and health care from his list of primary goods, so the health version is an analogy drawn from his work.
Expecting maximin welfare to credit gains to healthier groups
Raising the third group in option P from 76.7 to 80 years leaves W at 60.3, so a programme that helps only healthier groups scores no gain under maximin even though nobody is worse off.
Sources
Atkinson on the minimum as the limit of isoelastic social welfare
Atkinson AB. On the measurement of inequality. Journal of Economic Theory. 1970;2(3):244-263. Page 257, which states that the limiting case as epsilon tends to infinity gives the function min_i(y_i), which takes account only of transfers to the very lowest income group and is therefore not strictly concave.
Brouwer and colleagues on Rawls and health as a primary good
Brouwer WB, Culyer AJ, van Exel NJ, Rutten FF. Welfarism vs. extra-welfarism. Journal of Health Economics. 2008;27(2):325-338. Section 3 on extra-welfarism, which notes that Rawls explicitly excluded both health and health care from the list of primary goods.
Canonical Identity
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