Distributional Weight

Distributional Weight in Cost-Benefit Analysis and Health Policy

A distributional weight is a multiplier that an analyst applies to the money value of a gain or loss according to the income, or other circumstances, of the person who receives it, so that a pound reaching a low-income household can count for more than a pound reaching a rich one. Health economists meet it mainly in cost-benefit analysis of health policies, where benefits are valued by willingness to pay and therefore reflect ability to pay as well as strength of preference. The HM Treasury Green Book (2026) calls these welfare weights and gives a formula for UK appraisal. Cost-effectiveness analysis has its own version in weights on QALYs, such as the NICE severity modifier in PMG36.

Why unweighted willingness to pay leans towards richer groups

Adler describes cost-benefit analysis as insensitive to distribution, because it adds up monetary equivalents, the amounts people would pay for a change they prefer or accept for one they dislike. A policy with a positive total passes even when some people lose, and the total does not register who gains. Since what a person can pay depends on income, a service used mainly by better-off households can show larger money benefits than one serving poorer households with the same health gain.

The Green Book (paragraph 7.8) states the underlying principle plainly. An extra GBP 1,000 of income is more valuable to a low-income household than to a high-income one, which economists call the diminishing marginal utility of income. Unweighted analysis gives every pound a weight of one whoever receives it, the convention behind the Kaldor-Hicks test, which asks only whether gainers could in principle compensate losers.

Adler sets out a different defence of cost-benefit analysis, in which weighted analysis is a practical way of putting a social welfare function into effect. On this view the Kaldor-Hicks test avoids interpersonal comparisons but carries known flaws. He measures monetary equivalents by equivalent variation, because analysis based on compensating variation can violate the Pareto principle.

Weights derived from a social welfare function

A social welfare function needs utility numbers that can be compared across people and a rule for ranking lists of them. Adler shows that, for small changes from the status quo, a utilitarian welfare function is approximated by weighting each person's monetary equivalent by that person's marginal utility of consumption.

$$\Delta W \approx \sum_{i} MU_i \, \Delta c_i, \qquad MU_i = c_i^{-\lambda} \times m(a_i)$$

where $\Delta W$ is the change in social welfare, $MU_i$ is the marginal utility of consumption of person $i$ in the status quo, $\Delta c_i$ is that person's equivalent variation in money, $c_i$ is status quo consumption, $\lambda$ is the coefficient of relative risk aversion in a constant relative risk aversion utility function and $m(a_i)$ is a scaling factor for non-consumption attributes $a_i$, such as health.

For an isoelastic, or Atkinson, social welfare function Adler multiplies the weight by the marginal moral value of utility, $u_i^{-\gamma}$, where $u_i$ is the person's utility level and $\gamma$ is a coefficient of inequality aversion. The two parameters differ in kind. The value of $\lambda$ describes people's own preferences over consumption gambles and can be estimated, while $\gamma$ expresses the decision maker's moral preference for equalising wellbeing. The Atkinson name comes from the function's link with the Atkinson index of inequality.

Adler notes that much applied work ignores non-consumption attributes and makes the weight a function of income alone, citing the 2003 Green Book. In principle a weight can also reflect health or life expectancy, so a person who is both poor and ill could receive a higher weight than income alone would give.

The Green Book welfare weight formula

The Green Book (2026), updated 28 October 2026, treats distributional weighting in chapter 7. Paragraph 7.9 says that analysts may weight costs and benefits by the groups that bear them, giving a higher social value to monetisable benefits for lower-income households. The weight needs the median income of the group and of the national population, with household incomes equivalised first (paragraphs 7.10 and 7.11).

$$w_g = \left( \frac{\tilde{y}_N}{\tilde{y}_g} \right)^{\eta}$$

where $w_g$ is the welfare weight of group $g$, $\tilde{y}_N$ is the median equivalised income of the national population, $\tilde{y}_g$ is the median equivalised income of the group and $\eta$ is the elasticity of marginal utility of consumption, set at 1.25. Paragraph 7.15 states that the estimate is discussed in the technical annexes of the Green Book discount rate review 2026. The formula has the form of Adler's utilitarian weight with $m(a_i) = 1$ and $\lambda = \eta$, taken relative to a person on the national median.

The weighted impact for a group is its change in income multiplied by its weight (paragraph 7.12). Weighted estimates must be presented alongside unweighted ones (paragraph 7.13), and the Green Book suggests sensitivity analysis because the group median is uncertain (paragraph 7.14). Paragraph 7.3 asks for effort in proportion to the size of the distributional effects.

Equivalisation adjusts for household composition. In Figure 7 of the Green Book a couple without children is the reference point at 1.00, built from 0.67 for the first adult and 0.33 for each additional adult or child aged 14 or over, with 0.20 for each child under 14. Household income divided by the total gives equivalised income.

Health effects are valued differently. Paragraph 8.57 gives a willingness-to-pay value for a QALY of GBP 70,000 in 2020-21 prices, and paragraph 8.58 states that the QALY, statistical life and statistical life-year values are based on average values from representative samples of the population. Adler notes that constant values such as a population-average value of statistical life soften the distributional insensitivity of cost-benefit analysis in practice, but lack a theoretical basis.

Distributional weights on health in cost-effectiveness analysis

Cost-effectiveness analysis faces the same question with health in place of money. Cookson and colleagues describe the standard rule that "a QALY is a QALY", under which population health gain is the unweighted sum of individual gains. Equity weights for the health benefits of people with particular characteristics, or a single equity parameter such as the inequality aversion of a social welfare function, quantify how much concern for equity would justify a less cost-effective option.

They observe that many proposed weighting systems rest on health characteristics, especially severity of illness or lifetime health, and not on social characteristics such as income, and they recommend equity weights as an aid to deliberation through sensitivity analysis, not as a fixed tariff. The equity-weighted analysis page works through the method, and distributional cost-effectiveness analysis models the whole distribution of health instead.

The NICE manual (PMG36) shows the health version in practice. Section 6.2.10 states that in the reference case all QALYs carry equal weight, and section 6.2.11 describes decision-making modifiers as value judgements that may be applied through QALY weighting. Table 6.1 sets weights of 1.2 and 1.7 for conditions with a high QALY shortfall, and section 6.3.2 asks for net health benefit at GBP 25,000 and GBP 35,000 per QALY with and without the weighting.

The contrast matters for the welfarist and extra-welfarist debate. Money weights by income correct a welfarist cost-benefit analysis for unequal ability to pay, while severity weights on QALYs can be read as an extra-welfarist judgement about whose health gains deserve priority under a fixed NHS budget. Both are value judgements, and both can change which programme looks best.

Worked example with two programmes and three income groups

All numbers in this example are illustrative. A local health system can fund one of two programmes from general taxation, each costing GBP 1,000,000. For simplicity the example gives the cost, which falls on general taxpayers, a weight of one. National median equivalised income is GBP 30,000, and the three groups affected have medians of GBP 18,000, GBP 30,000 and GBP 50,000.

1. Equivalise the low-income group. A typical low-income household is a couple with two children under 14 and an income of GBP 25,200. Its scale is 0.67 + 0.33 + 0.20 + 0.20 = 1.40, so its equivalised income is 25,200 / 1.40 = 18,000, or GBP 18,000.

2. Compute the weights. With $\eta$ = 1.25, the low-income weight is (30,000 / 18,000) raised to the power 1.25, about 1.89. The middle group has a weight of exactly 1.00, and the high-income weight is (30,000 / 50,000) raised to 1.25, about 0.53. Rounded weights are used below.

3. Unweighted benefits. Programme A, a fast-track elective clinic used mostly by higher earners, has willingness to pay of GBP 100,000 in the low group, GBP 300,000 in the middle group and GBP 900,000 in the high group, a total of GBP 1,300,000. Programme B, smoking cessation outreach on deprived estates, has GBP 800,000, GBP 300,000 and nothing in the high group, a total of GBP 1,100,000. Unweighted net benefit is GBP 300,000 for A and GBP 100,000 for B, so A ranks first.

4. Weighted benefits. For A, 100,000 × 1.89 = 189,000 and 900,000 × 0.53 = 477,000, so its weighted benefit is 189,000 + 300,000 + 477,000 = 966,000, a net loss of GBP 34,000 against its cost. For B, 800,000 × 1.89 = 1,512,000, so its weighted benefit is 1,512,000 + 300,000 = 1,812,000, a net gain of GBP 812,000. B ranks first.

5. Sensitivity to the elasticity. With an elasticity of 1.0 the weights become 1.67, 1.00 and 0.60. A's weighted benefit is 167,000 + 300,000 + 540,000 = 1,007,000, a net gain of GBP 7,000, and B's is 1,336,000 + 300,000 = 1,636,000, a net gain of GBP 636,000. With unrounded weights A changes sign at an elasticity of about 1.04, while B ranks first at both values.

6. Interpretation. The evidence on what each group would pay is unchanged. A's unweighted lead rests on the high group's greater ability to pay, and the weights reverse it. Presenting both results, as paragraph 7.13 requires, shows the decision maker that the ranking turns on a value judgement about income.

Choices that move the weights

The elasticity is the largest single choice. An elasticity of zero gives every group a weight of one and reproduces unweighted analysis, and higher values widen the gap between groups, as step 5 shows. Adler's framework adds a separate inequality aversion term for decision makers who want more priority for the worse off than a utilitarian rule gives.

Group definition and incidence matter too. A weight based on a group median treats everyone in the group alike, and equivalisation changes which households count as low-income, especially families with children. Paragraph 7.9 weights costs as well as benefits, so a programme funded by a user charge or a local tax would have its costs weighted by the incomes of those who pay, and the example's simplifying weight of one on cost would no longer hold. Where health gains are monetised at the single average QALY value, differences in ability to pay do not enter the valuation of those gains in the first place.

Objections and common misreadings

Adler reviews the main objections to weighting and argues that several are weaker than they look. Others are misreadings of what a weight does.

1. Unweighted analysis is neutral. Adler argues that the decision to use cost-benefit analysis, weighted or not, is itself value-laden, so a weight of one on every pound is itself a judgement.
2. Distribution belongs to the tax system. Adler notes that with lump-sum taxes and transfers set by a legislature sharing the decision maker's values, weights would be uniform. He also reports the stronger argument, from Hylland and Zeckhauser and developed by Kaplow, that unweighted analysis combined with tax changes can be Pareto-superior. He replies that it rests on a labour separability assumption with no clear empirical basis, and that a non-tax decision maker, such as a health body, may not control the tax system.
3. Weights change people's willingness to pay. Each person's valuation stays as measured, and the weight changes only how valuations are added across people.
4. Money weights and QALY weights are the same thing. Green Book weights act on money and depend on income, while NICE severity weights act on QALYs and depend on future health lost.
5. Weighting needs data analysts lack. Weights require information on who bears costs and benefits, but Adler doubts that this objection has much force because analysts can predict incidence.

Adler also notes that an unelected official may doubt the legitimacy of setting weights. The equity-efficiency trade-off page discusses how much total benefit decision makers accept losing for a fairer distribution, and the welfare economics page places weighting among other welfare criteria.

Sources

• Adler MD. Cost-Benefit Analysis and Distributional Weights: An Overview. Duke Environmental and Energy Economics Working Paper EE 13-04. Durham, NC: Nicholas Institute for Environmental Policy Solutions and Duke University Energy Initiative; August 2013. https://www.gov.br/casacivil/pt-br/assuntos/governanca/regulacao/documentos/biblioteca-internacional/2013/cost-benefit-analysis-and-distributional-weights_-an-overview.pdf
• Cookson R, Mirelman AJ, Griffin S, Asaria M, Dawkins B, Norheim OF, Verguet S, Culyer AJ. Using cost-effectiveness analysis to address health equity concerns. Value in Health. 2017;20(2):206-212. https://doi.org/10.1016/j.jval.2016.11.027
• HM Treasury. The Green Book (2026). Updated 28 October 2026. Paragraphs 7.1 to 7.15, Figure 7 and paragraphs 8.57 to 8.58. https://www.gov.uk/government/publications/the-green-book-appraisal-and-evaluation-in-central-government/the-green-book-2026
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). Published 31 January 2022, last updated 13 October 2026. Sections 6.2.10 to 6.2.18 and 6.3.2. https://www.nice.org.uk/process/pmg36