Signature
MRS = (alpha / (1 - alpha)) * (h_b / h_a)^(1 + r); slope = -MRS
| Inputs | Definition | Unit |
|---|---|---|
alpha | Relative weight attached to the health of group a, above 0 and below 1 | none |
h_b | Current health of group b, above zero | years of QALE |
h_a | Current health of group a, above zero | years of QALE |
r | Inequality aversion parameter of the CES welfare, minus 1 or above and not 0 | none |
MRS | Health of group b that can be given up for one more unit of health to group a with social welfare unchanged, the relative weight on a small gain to group a | years of group b health per year of group a health |
|---|---|---|
slope | Gradient of the social indifference curve at the current point with h_a on the horizontal axis, equal to minus MRS | years per year |
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
CES marginal rate and contour slope from named cells
With the named cells of HE-IM-SWF-004 the first formula returns the MRS in a cell named MarginalRateCES and the second the slope of the social indifference curve.
=(GroupWeightA/(1-GroupWeightA))*(HealthGroupB/HealthGroupA)^(1+CurvatureR); =-MarginalRateCES
Assumptions
Small changes at the current point for the CES marginal rate
The MRS is a derivative and holds for small gains at the stated health levels. As a programme narrows the gap the ratio h_b/h_a falls and the weight with it, which is why Asaria and colleagues note that fair innings equity weights need constant updating as the distribution of health changes.
Fair innings calibration of the CES contour
In Williams' calibration, as Asaria and colleagues describe it, alpha is 0.5, r takes a range of values, and the contour is fixed by an EDE of 70 QALYs together with current health. The slopes of its tangents at current health give the equity weights used in cost-effectiveness analysis.
Worked examples
CES marginal rate with equal weights and r of 1
For illustrative group health of 60 and 72 years with alpha = 0.5 and r = 1, the MRS is 72/60 squared, 1.44, and the contour slope is minus 1.44, the same as the Atkinson relative weight at epsilon of 2 (HE-EX-ATK-009).
alpha = 0.5; h_a = 60; h_b = 72; r = 1; MRS = 1.44; slope = -1.44
CES marginal rate at equal health with the Dolan and Tsuchiya median weight
Dolan and Tsuchiya's median respondent was indifferent between saving 100 lives in group a, people who had not cared for their health, and 45 in group b, which gave alpha of 0.31. At equal health, here an illustrative 70 years in each group, the MRS is about 0.4493 for any r, the weight of about half (0.45) that they report.
alpha = 0.31; h_a = 70; h_b = 70; r = 1; MRS = 0.4493; slope = -0.4493
CES marginal rate at r of minus 1
With r = -1 and equal weights the MRS is 1 at any health levels and the contour is a straight line of slope minus 1, the utilitarian case in which a year of health counts the same in either group.
alpha = 0.5; h_a = 60; h_b = 72; r = -1; MRS = 1; slope = -1
Common errors
Exponent r in place of 1 plus r in the CES marginal rate
The exponent on the health ratio is 1 + r, which is the Atkinson epsilon. Using r gives 1.2 instead of 1.44 for 60 and 72 years at r = 1, and at the English median epsilon of 10.95 (r of 9.95) it gives about 6.14 instead of the 7.36 of HE-ER-ATK-007 between the same groups.
Treating a CES equity weight as fixed while health changes
The weight depends on the current gap. With r = 1 and equal weights it is 1.44 at 60 and 72 years but about 1.19 once group a reaches 66 years, so a weight computed once and applied to large gains overstates the priority on the later gains.
Sources
Dolan and Tsuchiya on the marginal rate of substitution as an equity weight
Dolan P, Tsuchiya A. The social welfare function and individual responsibility: some theoretical issues and empirical evidence from health. HEDS Discussion Paper 03/01. Sheffield, University of Sheffield, 2003, revised version in Journal of Health Economics. 2009;28(1):210-220. Section III, equation [4], which gives the MRS of the CES function as alpha/(1-alpha) times (U_B/U_A)^(1+r) and calls it the equity weight, and states that along the 45 degree line the MRS is independent of r and a function of alpha alone, and section IV, in which the median respondent is indifferent between saving 100 lives in group A and 45 in group B, giving alpha of 0.31 and a weight of about half (0.45).
Asaria, Griffin and Cookson on fair innings weights as contour slopes
Asaria M, Griffin S, Cookson R. Measuring health inequality in the context of cost-effectiveness analysis. In Health and Inequality. Research on Economic Inequality, volume 21. Emerald Group Publishing, 2013, pages 491-507. Author manuscript, section on social welfare functions used in the health inequalities literature, which reports that Williams (1997) set alpha at 0.5, used a range of r values and an EDE of 70 QALYs as the fair innings, and took the slopes of the tangents at current health as equity weights for cost-effectiveness analysis, and a later passage stating that these weights need constant updating as the distribution of health changes.
Canonical Identity
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