Functions & Formulae

Each applied formula has its own function page, with a signature, implementations, and tests.

Health social welfare function ranking distributions of health

W(h_i, theta) = W

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.

  • Isoelastic Atkinson health social welfare of three equal-sized groups for epsilon other than 1

    W = h_1^(1 - epsilon) / (1 - epsilon) + h_2^(1 - epsilon) / (1 - epsilon) + h_3^(1 - epsilon) / (1 - epsilon)

    Adds h_i^(1-epsilon)/(1-epsilon) over the groups, the welfare function Atkinson derived from constant relative inequality aversion with its two constants set to 0 and 1, which leaves every ranking unchanged. The line is written for the three groups of the article's example, and with n people or groups the sum has n terms. At epsilon of 0 W is total health, the utilitarian rule of maximising total QALYs; as epsilon rises the weight on a small gain to group i, h_i^(-epsilon), shifts priority to the less healthy (HE-FM-ATK-004), and the ranking approaches the maximin rule of HE-FM-SWF-003. Solving W = n h_EDE^(1-epsilon)/(1-epsilon) for h_EDE gives the EDE health of HE-FM-ATK-002, so W and EDE health rank allocations identically, and W has meaning only in comparisons made at the same epsilon.

  • Logarithmic health social welfare of three equal-sized groups at epsilon of 1

    W = log(h_1) + log(h_2) + log(h_3)

    At epsilon of 1 the isoelastic form of HE-FM-SWF-001 divides by zero, and Atkinson's welfare function becomes the natural logarithm, so W adds the logarithms of the groups' health. Once the constant n/(1-epsilon), which does not affect rankings, is taken away, HE-FM-SWF-001 tends to this form as epsilon tends to 1, so rankings change smoothly with epsilon through 1. The weight on a small gain to group i is 1/h_i, so a gain of 1% counts the same for every group. Dividing W by n and taking the exponential gives the geometric-mean EDE health of HE-FM-ATK-003. The function log is the natural logarithm.

  • Maximin health social welfare of three groups as the Rawlsian limit

    W = min(h_1, min(h_2, h_3))

    Sets social welfare equal to the health of the least healthy group, the Rawlsian or maximin form. Atkinson gives it as the limit of the isoelastic family as epsilon tends to infinity (HE-FM-SWF-001), and Dolan and Tsuchiya as the limit of the two-group CES form as r approaches infinity (HE-FM-SWF-004). Two allocations with the same minimum have the same W and are ranked as tied however the other groups fare, so a gain to a healthier group alone leaves W unchanged. The nested min written here returns the smallest of the three values.

  • Wagstaff two-group CES health social welfare with a group weight and curvature

    W = (alpha * h_a^(-r) + (1 - alpha) * h_b^(-r))^(-1 / r)

    Combines the health of two groups through a constant elasticity of substitution (CES) function, the iso-elastic form of Wagstaff 1991 that Asaria, Griffin and Cookson describe as the most commonly used in the health inequalities literature. The weight alpha pivots the welfare contours around the line of equal health and r sets their curvature. At r = -1 W is the weighted sum alpha h_a + (1-alpha) h_b, the inequality-neutral utilitarian case, and as r approaches infinity W approaches the smaller of the two health values, the maximin of HE-FM-SWF-003. With alpha = 0.5, W is the EDE health of HE-FM-ATK-002 for two groups with epsilon = 1 + r, so the isoelastic and CES families rank distributions identically. Unlike the weighted two-group social welfare of HE-FM-WE-004, whose weight on a group's QALYs is fixed, the weight here changes with the health gap (HE-FM-SWF-005).

  • Marginal rate of substitution and contour slope of the two-group CES health welfare

    MRS = (alpha / (1 - alpha)) * (h_b / h_a)^(1 + r); slope = -MRS

    Divides the marginal welfare of health in group a by that in group b. The result is the health of group b that could be given up for one more unit to group a with W unchanged, and the social indifference curve through the point, drawn with h_a on the horizontal axis, has slope minus MRS. Dolan and Tsuchiya read the MRS as the equity weight on a marginal gain to group a relative to group b. At equal health it reduces to alpha/(1-alpha) for any r, which is how they identify alpha from survey choices, and with alpha = 0.5 it equals the Atkinson relative weight of HE-FM-ATK-004 with epsilon = 1 + r when group a is the less healthy, so the factor alpha/(1-alpha) is what this formula adds. Williams' fair innings equity weights come from these slopes at current health on the contour whose EDE is 70 QALYs.