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

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.

Signature

W = log(h_1) + log(h_2) + log(h_3)
Inputs
InputsDefinitionUnit
h_1Health of the first group, for example its QALE, above zeroyears of QALE
h_2Health of the second group, above zeroyears of QALE
h_3Health of the third group, above zeroyears of QALE
Output
WSocial welfare of the allocation under the logarithmic form, higher being betterwelfare score, the sum of natural logarithms of health

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

    Logarithmic social welfare with the Excel natural logarithm

    With the three health values in the named cells used for HE-FM-SWF-001, the formula returns W in a cell named LogWelfare. Excel's LN is the natural logarithm, while LOG with one argument is base 10.

    =LN(HealthFirstGroup)+LN(HealthSecondGroup)+LN(HealthThirdGroup)

Assumptions

  • Health above zero for logarithmic social welfare

    Every h_i is above zero, because the logarithm of zero is undefined and W falls without limit as any group's health approaches zero. The groups are of equal size, as in HE-FM-SWF-001.

Worked examples

  • Logarithmic welfare of the population-wide option

    At epsilon of 1 option P in the article's illustrative example, with QALE of 60.3, 68.5 and 76.7 years, has W of about 12.6660676.

    h_1 = 60.3; h_2 = 68.5; h_3 = 76.7; W = 12.6660676

  • Logarithmic welfare of the targeted option

    Option T, with 61.0, 68.3 and 76.0 years, has W of about 12.665517, slightly below P, so P still ranks first at epsilon of 1, as the article's switching point of about 1.23 implies.

    h_1 = 61; h_2 = 68.3; h_3 = 76; W = 12.665517

  • Logarithmic welfare of the option for the worst off

    Option R, with 61.2, 67.6 and 75.6 years, has W of about 12.6532115, the lowest of the three at epsilon of 1.

    h_1 = 61.2; h_2 = 67.6; h_3 = 75.6; W = 12.6532115

Common errors

  • Excel LOG in place of LN for logarithmic welfare

    LOG with one argument is base 10, so W for option P becomes about 5.5008 instead of 12.6661. The ranking is unchanged, but the EDE health exp(W/3) becomes about 6.26 years instead of 68.17.

  • General isoelastic formula applied at epsilon close to 1

    At epsilon of exactly 1 the formula of HE-FM-SWF-001 divides by zero. Entering 0.999 instead gives about 3,012.6929 for option P, which ranks correctly but differs from the logarithmic 12.6661 by the constant 3/(1-epsilon), here 3,000, plus a small remainder, so the two values cannot be set side by side.

Sources

  • Atkinson's logarithmic welfare function for health social welfare at epsilon of 1

    Atkinson AB. On the measurement of inequality. Journal of Economic Theory. 1970;2(3):244-263. Page 251, equation (5), which gives U(y) = log_e(y) at epsilon of 1 under constant relative inequality aversion, the case in which the general form y^(1-epsilon)/(1-epsilon) is undefined.

    View source →

Canonical Identity

Persistent URI
https://healtheconomics.wiki/formula/logarithmic-health-social-welfare-of-three-equal-sized-groups-at-epsilon-of-1

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