On Balance: BCA, the Choice of Numéraire, and Weighted BCA

Benefit-cost analysis, as usually practiced, sums the monetary values of effects on individuals. It can be justified by the potential compensation test: if the total monetary gain to the “winners” (those who gain from a policy) exceeds the total monetary loss to the “losers” (those who are harmed), the “winners” could (in principle) pay compensation to the “losers” so that everyone would judge herself better off with the combined policy and compensation than without. The idea is that by summing the net benefits across individuals, BCA measures “efficiency” or the size of the social pie, and that questions about distribution can be evaluated separately. Logically, policies that expand the social pie permit everyone to have a bigger slice; a smaller pie guarantees that at least some people get a smaller slice. 


The separation of efficiency and distribution is illusory: the size of the social pie can depend on the “numéraire” or measuring rod. Ideally, one might wish to measure each person’s slice by how much it contributes to her wellbeing. But without an objective method for comparing increments to wellbeing between people, one must choose a standard for comparison; the numéraire provides that standard.

Consider a simple example with policies that increase life expectancy but impose monetary costs.

As summarized in Table 1, there are two policies (A and B) and two groups in the population (1 and 2), each including N people. Both policies impose a cost of $c on every person. Policy A improves life expectancy of everyone in group 1 by h1 years, and has no effect on those in group 2. In contrast, policy B improves life expectancy of everyone in group 2 by h2 years, and has no effect on those in group 1. The per-person costs and changes in life expectancy are small.

Assume the monetary value of a gain in life expectancy differs between the two groups, perhaps because they have different incomes. It is v1 dollars per life year ($/LY) for those in group 1 and v2 $/LY for those in group 2. As shown in Table 1, the benefits of policy A are $ v1 h1 to each person in group 1 and 0 to each in group 2; the net benefits to individuals in the two groups are $ v1 h1 – c and – $c, respectively. Similarly, the net benefits of policy B are – $c to those in group 1 and $v2 h2 – c to those in group 2. Under conventional BCA, policy A is more efficient than policy B if it has larger total net benefits, i.e., if v1 h1 > v2 h2.

Now consider an alternative analysis using life years, not dollars, as the numéraire. This alternative is summarized in Table 2. Since the benefits are measured in life years, no conversion is needed. To convert costs (measured in dollars) to life years, one simply divides by the appropriate monetary value per life year. So the net health benefits of policy A are h1 – c/v1 LY to individuals in group 1 and – c/v2 LY to those in group 2. Analogously, the net health benefits of policy B are – c/v1 LY to group 1 and h2 – c/v2 LY to group 2.

Using life years as the numéraire, whichever policy produces the larger gain in life expectancy is more efficient. But using dollars as the numéraire, the comparison depends on both the gains in life expectancy and the rate at which these are valued. If the two groups value increases in life expectancy differently, the policy that is more efficient can depend on which numéraire is chosen. For example, if v1/v2 > h2/h1 > 1, then policy A is more efficient using monetary values and policy B is more efficient using health values.

Since the ranking of policies by “efficiency” can depend on the numéraire, how should one choose the numéraire? If one relies on the potential compensation test as justification for BCA, monetary values seem more appropriate than life years (or other alternatives). Although transferring money between people is not cost-free, it is more feasible than transferring health or mortality risk. Of course the more efficient policy is not always the best choice; a less-efficient policy can be preferred if its distribution of net benefits is sufficiently attractive.

If one prefers the distributional properties of an alternative numéraire, the same result can be achieved using weighted BCA. For example, using dollars as the numéraire and weighting net monetary benefits to individuals in each group by the reciprocal of the corresponding value per life year produces the same result as using life years as the numéraire. This can be seen by noting that the net health benefits to members of each group (in Table 2) equal the net monetary benefits (in Table 1) divided by that group’s value per life year (v1 and v2, respectively). Indeed, some prominent authors view equity-weighted BCA as the only legitimate form (e.g., Weisbrod 1968, Drèze and Stern 1987, Johansson 1998). The weights can be based on a social-welfare function that integrates concerns for both efficiency and distribution. Alternatively, one can evaluate policies using a social-welfare function directly (Adler 2019), as discussed by Hammitt (2021), from which this contribution is drawn.

Table 1. Conventional BCA using monetary values (dollars) as numéraire


Benefit 1

Benefit 2

Cost 1

Cost 2



Total NMB


v1 h1




v1 h1 – c

– c

(v1 h1 – 2c ) N



v2 h2



– c

v2 h2 – c

(v2 h2 – 2c) N

Note: Numerals 1 and 2 denote groups that receive benefits or pay costs; Benefit and Cost are measured in dollars, NMB = net monetary benefits. Amounts are per person, except in the last column.

Table 2. Alternative BCA using health units (life years) as numéraire


HB 1

HB 2

HC 1

HC 2



Total NHB






h1 – c/v1

– c/v2

(h1 – c/v1 – c/v2) N






– c/v1

h2 – c/v2

(h2 – c/v1 – c/v2) N

Note: Numerals 1 and 2 denote groups that receive benefits or pay costs; HB = benefit in health units, HC = cost in health units, NHB = net health benefits. Amounts are per person, except in the last column.


Adler, M.D., Measuring Social Welfare: An Introduction, Oxford University Press, Oxford, 2019.

Drèze, J., and N. Stern, The Theory of Cost-Benefit Analysis, Handbook of Public Economics, Vol. II (A.J. Auerbach and M. Feldstein, eds.), pp. 909-989, Elsevier Science Publishers, Amsterdam, 1987.

Hammitt, J.K., Accounting for the Distribution of Benefits & Costs in Benefit-Cost Analysis, Journal of Benefit-Cost Analysis 12(1): 64–84, 2021.

Johansson, P.-O., Does the choice of numéraire matter in cost-benefit analysis? Journal of Public Economics 70: 489-493, 1998.

Weisbrod, B., Income Redistribution Effects and Benefit-Cost Analysis, Problems in Public Expenditure Analysis (S.B. Chase, ed.), pp. 177-209, Brookings Institution, Washington, D.C., 1968.

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Comments on "On Balance: BCA, the Choice of Numéraire, and Weighted BCA"

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Daniel Acland - Friday, October 15, 2021

This is a highly trenchant analysis, and quite eye-opening. Essentially, if life-years are of lower value in dollars to poor people because money is of greater value to them, then the reverse must be true: dollars must be of higher value in life years to poor people, again, because money is of greater value to them. However, I take issue with the idea that dollar equivalents should be weighted in such a way as to *simultaneously* adjust for this bias in the value of a dollar *and* distributional concerns (a greater concern for the poor, because they are poor). The former is actually an efficiency concern. Dollars are a biased measure of welfare. The latter is distributional. The unweighted sum of utilities does not favor the poor, but society might want to do so. These two reasons for weighting willingness to pay are completely separate from one another, and should not be confused. Indeed, in the social welfare measurement literature, adjusting for these things is done in two separate steps. First, the numeraire is adjusted to reflect true utilities. Then utilities are weighted to reflect our differential concern for the poor. The bias in WTP as a measure of welfare is *not* a distributional issue. Adjusting for the bias is, in my view, vastly more justified in BCA than actual distributional weighting. Here are a couple of possibilities. If we think of BCA as an implementation of the potential Pareto criterion, then we pick a numeraire and we don't adjust for any bias introduced by marginal utility of wealth. If we think of BCA as a measure of welfare, then we have to either adjust for diminishing marginal utility of wealth, or simply accept that BCA is a biased measure of welfare, a proxy measure. In my view, adjusting for bias is a defensible option. It means we have to abandon the idea that BCA is an implementation of the potential Pareto criterion, but that has always been a morally funky basis for policy making. It seems to me far more useful to think of BCA as a measure of welfare, use WTP as the measure, and then adjust for the bias in WTP. It would be problematic in practice, because analysts would have to choose the efficiency weights, which would introduce inconsistency between analysts, but this kind of determination could be made centrally within any given polity. However, if we additionally weight WTP to account for our differential concern for the poor *because they are poor*, then we simply aren't doing BCA anymore. We are doing social welfare measurement, a la Adler, and we should change the name of our society.

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