Signature
W = (alpha * h_a^(-r) + (1 - alpha) * h_b^(-r))^(-1 / r)
| Inputs | Definition | Unit |
|---|---|---|
alpha | Relative weight attached to the health of group a, at least 0 and at most 1, with 0.5 giving the two groups equal priority | none |
h_a | Health of group a, above zero | years of QALE |
r | Inequality aversion parameter that sets the curvature of the welfare contours, minus 1 or above and not 0; minus 1 gives the weighted sum | none |
h_b | Health of group b, above zero | years of QALE |
W | Social welfare of the two-group distribution, in the unit of health because the function is a weighted power mean | 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.
Try this function
Implementations
Excel
Two-group CES social welfare from named cells
With the health of the two groups in HealthGroupA and HealthGroupB, alpha in GroupWeightA and r in CurvatureR, the formula returns W in a cell named CESWelfare.
=(GroupWeightA*HealthGroupA^(-CurvatureR)+(1-GroupWeightA)*HealthGroupB^(-CurvatureR))^(-1/CurvatureR)
Assumptions
Equal-sized homogeneous groups in the CES welfare
Dolan and Tsuchiya take the two groups to be of equal size, homogeneous and alike in every attribute other than the one alpha reflects, with well-being measured on a cardinal scale on which death is zero.
Group weight and curvature as separate value judgements
Alpha states how much one group's health counts at equal health, for example less for a group judged responsible for its poorer health, and r states aversion to inequality. Asaria and colleagues note that in the Dolan and Tsuchiya formulation the empirical values of the two parameters are difficult to identify separately.
Worked examples
CES welfare with equal weights and r of 1
With illustrative group health of 60 and 72 years, alpha = 0.5 and r = 1, W is about 65.4545 years, the harmonic mean of the two values and the EDE health at epsilon of 2.
h_a = 60; h_b = 72; alpha = 0.5; r = 1; W = 65.4545
CES welfare at r of minus 1 equals mean health
With r = -1 and equal weights W is the mean of the two groups, 66 years, the utilitarian case.
h_a = 60; h_b = 72; alpha = 0.5; r = -1; W = 66
CES welfare at r of 10 approaches the minimum
At r = 10 W falls to about 63.3508 years, nearer the 60 years of the less healthy group, as the function moves towards maximin.
h_a = 60; h_b = 72; alpha = 0.5; r = 10; W = 63.3508
Common errors
Outer exponent of the CES welfare written as 1 over r
Writing the outer power as 1/r in place of minus 1/r returns about 0.0153 instead of 65.4545 years for 60 and 72 years with alpha = 0.5 and r = 1, a value in reciprocal years that ranks distributions in reverse.
Entering the Atkinson epsilon as the CES curvature r
The CES curvature and the Atkinson parameter differ by 1, epsilon = 1 + r, so the English median epsilon of 10.95 (HE-SR-ATK-007) corresponds to r of 9.95. Entering r = 10.95 for groups at 60 and 72 years with equal weights gives W of about 63.1817 instead of 63.3600, a more inequality-averse function than intended.
Sources
Asaria, Griffin and Cookson on the Wagstaff two-group CES social welfare
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 gives W = [alpha h_a^(-r) + (1-alpha) h_b^(-r)]^(-1/r) as the iso-elastic form proposed by Wagstaff (1991) and the most commonly used in that literature, with alpha the relative weight that pivots the function around the line of equal health and r setting its curvature, and a later passage stating that the Dolan and Tsuchiya parameters are difficult to identify separately.
Dolan and Tsuchiya on the CES health social welfare and its limiting cases
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 I, equation [1], the two-group CES function with alpha at least 0 and at most 1, r of minus 1 or above and r not 0, for groups of equal size that are homogeneous, and the text after it, in which r = -1 gives the inequality-neutral utilitarian function and r approaching infinity the maximin Rawlsian function.
Canonical Identity
Stable URI · Machine-readable · Resolvable · CC BY 4.0