1. Introduction
Conventional benefit–cost analysis (BCA) is the dominant form of economic evaluation used to compare the harms and improvements associated with interventions undertaken outside the health-care sector, including those that significantly affect public health. This conventional approach has many advantages. It focuses on the efficient use of resources, estimating the opportunity costs of investing in a defined policy rather than using the resources for other things. It is not paternalistic; it relies on individual preferences, assuming each individual is the best or most legitimate judge of their well-being. It measures harms and improvements using the same metric (monetary value), facilitating comparison and prioritization of interventions with different types of impacts. It is well-established and widely used. Substantial best practice guidance is available, and numerous analyses have been completed, easing application.
Conventional BCA has important limitations, however, many of which relate to its inattention to distributional concerns. To measure changes in individual well-being, it relies on estimates of individuals’ willingness to exchange their own money for gains or losses they experience (their willingness to pay or willingness to accept compensation).Footnote 1 These values reflect individuals’ budget constraints, leading to larger values for wealthier individuals.Footnote 2 To estimate changes in social welfare, it relies on simple aggregation of gains and losses across individuals, without considering preferences for distribution.
This conception of welfare is relatively narrow, failing to consider how the policy affects individuals with different levels of well-being. It also fails to address distributional equity, an important concern of decision-makers and other stakeholders as evidenced by the many long-standing guidance documents that require supplementing conventional BCA with information on distribution (e.g. Clinton, Reference Clinton1993, Robinson et al., Reference Robinson, Hammitt, Cecchini, Chalkidou, Claxton, Cropper, Eozenou, de Ferranti, Deolalikar, Guanais, Jamison, Kwon, Lauer, O’Keeffe, Walker, Whittington, Wilkinson, Wilson and Wong2019a, HM Treasury, 2022).
Social welfare analysis is an alternative to conventional BCA. Social welfare analysis first estimates the impact of the intervention on individuals using broader measures of well-being. These individual well-being gains and losses are then aggregated across individuals through the use of a social welfare function (SWF). The specific aggregation rule embodies ethical concerns for distributional equity. For example, the utilitarian SWF corresponds to the sum of individuals’ well-being, which is assumed to increase at a diminishing rate as income rises. The prioritarian SWF corresponds to the sum of a nonlinear function of individuals’ well-being that attaches more weight to well-being changes experienced by the less well-off. Compared to conventional BCA, social welfare analysis is sensitive to the distribution of policy impacts across the population and to the correlation between policy impacts and preexisting inequities. It has a long tradition (see Adler, Reference Adler2019, Adler and Norheim, Reference Adler and Norheim2022, Adler, Reference Adler2024, and references therein), and it is routinely used in fields such as optimal tax theory and climate change economics (Tuomala, Reference Tuomala2016; Nordhaus, Reference Nordhaus2008).
In practice, social welfare analysis is often approximated by applying distributional weights to the estimates from conventional BCA (Adler, Reference Adler2016, Fleurbaey and Abi-Rafeh, Reference Fleurbaey and Abi-Rafeh2016). For example, weights to approximate a utilitarian SWF have been developed for use in U.K. policy analyses (HM Treasury, 2022) and in new U.S. guidance for assessing regulations and federal spending (OMB, 2023a, c). Weights to approximate a prioritarian SWF have been studied but have not yet been widely used in practice (Adler, Reference Adler2016).
Despite the amount of scholarly work on distributional weights, equity-sensitive BCA is rarely adopted in practice. One reason is that it is more demanding than conventional BCA. First, analysts need descriptive information on the distribution of benefits and costs across the population; on who gains and who loses if the policy is implemented. While conventional BCAs sometimes include data on the distribution of benefits, the distribution of costs has rarely been considered (Robinson et al., Reference Robinson, Hammitt and Zeckhauser2016). In addition, equity-sensitive BCA requires a measure of individual well-being and information on its distribution across the population. Such measures require data on the distribution of attributes relevant to well-being, which may include income, health, education, employment status, environmental quality, and so forth. Equity-sensitive BCA also requires defining an appropriate SWF. Most proposals for distributional weighting adopt a utilitarian SWF and account only for differences in income, although there is uncertainty about the sensitivity of marginal utility to changes in income (Acland and Greenberg, Reference Acland and Greenberg2023, OMB, 2023b). In addition, the available distributional weights typically focus on developed countries, many of which have robust systems for collecting demographic and other data. For low- and middle-income countries, the data available on attributes of well-being are typically much more limited.
The goal of this article is to provide a simple approach to the computation of distributional weights that (i) goes beyond simply adjusting for the marginal utility of income, thereby recognizing that differences in other dimensions of well-being – health and longevity especially – also matter; and (ii) takes into account the limited data available for many countries, especially those that are low- or middle-income.Footnote 3 We focus on the application of weights to interventions implemented outside the health-care system that significantly affect health and longevity, such as those addressing environmental, transportation, occupational, nutritional, behavioral, and other risks.
We aim to aid practitioners in supplementing conventional BCAs conducted in countries with varying data availability, including high-, middle-, and low-income countries, to better inform decision-makers about the effects of interventions on broader measures of well-being and distributional equity. Our approach is intended to improve the available information without requiring more data, time, and resources than available to most analysts.
This article provides simple formulas BCA practitioners can use to compute weights tailored to the characteristics of the population affected by an intervention. It also includes tables of weights for different country income groups. The article is structured as follows. In Section 2, we briefly review the theory behind the construction of distributional weights and the link to social welfare analysis. In Section 3, we derive two types of weights to be used in equity-weighted BCA: utilitarian weights that adjust for differences in marginal utility of income and prioritarian weights that incorporate a greater concern for improving the well-being of those who are worse off. We discuss the data requirements and the calibration of the weights, especially for settings where data are limited. In Section 4, we illustrate the application of weights in a stylized example and contrast the results to those of a conventional BCA. The last section offers some concluding thoughts.
2. Social welfare analysis and distributional weights
Equity-sensitive BCA is a practical method to implement social welfare analysis. Social welfare analysis characterizes the value of an intervention as the associated change in social welfare, where social welfare is a function of the distribution of well-being levels (or utilities) across the population. Many scholars have shown that the change in social welfare associated with an intervention can be approximated by the sum of individuals’ monetary equivalents for the intervention (willingness to pay or willingness to accept compensation) weighted by the marginal social value of each individual’s income (e.g. Arrow, Reference Arrow1963, Drèze and Stern, Reference Drèze, Stern, Auerbach and Martin1987, Blackorby and Donaldson, Reference Blackorby and Donaldson1990, Adler, Reference Adler2016).Footnote 4 Hence weighted BCA is a convenient, short-cut method to approximate the results of a social welfare analysis. The rationale is that income changes experienced by wealthy people have a smaller effect on social welfare than income changes experienced by less wealthy people. Moreover, social welfare may be more sensitive to improvements in well-being of individuals who are less well-off. The weight attached to individuals’ monetary equivalent has two components that reflect these concerns: the marginal utility of income for the individual and the marginal social value of the individual’s utility.
The computation of weights based on SWFs requires two ingredients: an interpersonally comparable measure of individual well-being; and a SWF to aggregate individuals’ well-being levels. Interpersonal comparability of well-being is necessary to compare changes in well-being levels across individuals and to determine who is disadvantaged (in well-being terms).
Metrics for assessing and comparing the well-being of individuals can be categorized into three broad groups (Adler and Decancq, Reference Adler, Decancq, Adler and Norheim2022, Adler and Fleurbaey, Reference Adler and Fleurbaey2016). The first category is represented by objective well-being, which includes the attainment of a list of objective goods (e.g. being healthy, having sufficient financial resources) (Sen, Reference Sen1999). A popular example of an objective well-being index is the Human Development Index.Footnote 5 The second category includes subjective well-being, which is derived from individuals’ reports of life satisfaction, experience of emotions, or other measures of mental states (e.g. Layard and De Neve, Reference Layard and De Neve2023). The third category of well-being measures is preference-based well-being, which depends on a bundle of attributes (e.g. income, longevity, health) and individuals’ judgments of the trade-offs between attributes. The primary approaches to building a preference-based well-being measure include equivalent income, where an individual’s income is adjusted by the value of nonmarket attributes (e.g. health) and the adjustment is based on the individual’s preferences over those attributes (e.g. Decancq et al., Reference Decancq, Fleurbaey, Schokkaert, Atkinson and Bourguignon2015; Fleurbaey and Ponthiere, Reference Fleurbaey and Ponthiere2023), and von Neumann-Morgenstern (vNM) utility, which assigns a score to bundles of attributes based on individuals’ preferences between probability distributions over alternative possible lives (e.g. Adler, Reference Adler2019, Cookson et al., Reference Cookson, Skarda, Cotton-Barratt, Adler, Asaria and Ord2021).
Objective well-being measures are, by definition, interpersonally comparable as they depend on a common pre-specified combination of objective goods. However, they do not respect individual preferences. There is a lively debate on whether subjective well-being measures permit interpersonal comparisons since life satisfaction, happiness, and emotions may mean different things to different individuals, and individuals may interpret well-being scales differently (Adler, Reference Adler2013, Graham, Reference Graham2016, Sunstein, Reference Sunstein2016). However, significant progress is being made in developing evaluation approaches that rely on measures of subjective well-being (e.g. HM Treasury, 2021a, 2021b, Frijters et al., Reference Frijters, Krekel, Sanchis and Santini2024). Preference-based well-being metrics respect individual preferences, and scholars have proposed approaches that allow interpersonal comparison of well-being (Adler and Decancq, Reference Adler, Decancq, Adler and Norheim2022). However, individual preferences are often difficult to elicit, especially if they are heterogenous.
In this article, for pragmatic reasons, we focus on a single well-being measure, vNM utility. Like other preference-based well-being measures, vNM utility retains an important strength of conventional BCA: it is based on individuals’ preferences for the gains and losses they would likely experience under each policy option. Within the realm of preference-based measures, we rely on vNM utility rather than equivalent income because it incorporates the judgment that the gain in well-being associated with an increase in income is a diminishing function of income. In other words, the marginal utility of income (or of most goods and services) is diminishing. As noted earlier, utilitarian weights, which account for the greater value of money to a poor person than a wealthy person, have received increased attention in recent years; these weights are consistent with the vNM framework.
The second ingredient to compute the weights is a SWF. Commonly used SWFs are the utilitarian and the prioritarian ones. Utilitarianism is the view that social welfare is the sum of individuals’ well-being. Unlike conventional BCA, utilitarianism weights individuals’ monetary equivalents to reflect the greater value of money to a poor person than a wealthy person, that is, to reflect the diminishing marginal utility of income. Prioritarianism is the view that social welfare is the sum of individuals’ well-being transformed to give more weight to changes in well-being that accrue to those who are worse off and less weight to changes that accrue to those who are better off. Prioritarianism weighs individuals’ monetary equivalents to reflect both the marginal utility of income and the greater marginal social value of improving the situation of disadvantaged people.
Which SWF should be used to compute the weights? The SWF reflects preferences for distributional equity. As such, it constitutes an ethical choice. Philosophers have long debated which theory of distributive justice is best. Public surveys can provide some clues about the moral preferences held by the population. Schokkaert and Tarroux (Reference Schokkaert, Tarroux, Adler and Norheim2022) conclude that there is a large degree of heterogeneity in individuals’ preferences for redistributive justice and that the axioms underlying a prioritarian SWF are often not satisfied. In particular, preferences for redistribution depend on the good to be redistributed (e.g. income or health), on issues of desert and responsibility, and on individual personal circumstances, among other things (Alesina and Angeletos, Reference Alesina and Angeletos2005, Robson et al., Reference Robson, Asaria, Cookson, Tsuchiya and Ali2017, Almås et al., Reference Almås, Cappelen and Tungodden2020).
In this article, we take the view that it is not up to the analyst to choose a SWF. Rather, the analyst should provide alternative sets of weights, representing alternative preferences for distributional equity. Policymakers can then review the results and determine how to best incorporate them in their decision-making. We are agnostic about the “correct” SWF and develop both utilitarian and prioritarian weights.
It is worth noting that, in equity-sensitive BCA, the weights do not depend on the particular policy to be assessed. Thus, the weights can be computed in advance and then combined with the benefit and cost estimates for each group for each intervention being assessed. However, the weights do depend on the chosen well-being metric and SWF. We hope and expect that our analysis will be extended to other concepts of well-being and social welfare.
3. Constructing the weights
In constructing weights that can be applied to the results of a conventional BCA, we begin by assuming that the analyst reports the (unweighted) distributions of monetary values (either net benefits or benefits and costs) across groups of individuals differentiated by sociodemographic characteristics (e.g. income or wealth, age, health status). We assume further that decisions about how to conduct the conventional BCA (including the choice of discount rate, the values per statistical life or per case used to compute the value of mortality and morbidity risk reductions, and the estimates of the distributions of the effects) have already been made.
In other words, we assume that the starting point is a conventional BCA supplemented by a distributional analysis that takes into account how monetary values vary across individuals with different attributes. Our task in this article is to estimate weights to convert the results into measures of social welfare more broadly defined. If the conventional BCA instead follows the common practice of using standard (population-average) unit values without adjustment for all individuals, for example, using the same VSL regardless of income or age, analysts will need to first adjust the values to account for individual differences. We illustrate one approach to estimating income- and age-dependent VSL in the stylized example of Section 4.
3.1. Assumptions
As discussed earlier, computing weights requires selecting a well-being metric and the specification of a SWF. For the former, we adopt vNM utility as our measure of well-being and assume that individuals have identical preferences. Although this is a restrictive assumption, it is difficult to measure differences in preferences across individuals and unclear how much bias is introduced by assuming identical preferences.
To compute the vNM utility index to be used in the weights, we make the following additional assumptions.
First, we use lifetime utility as our individual well-being metric and assume that lifetime utility is the sum of utility in each time period that an individual is alive. We assume that lifetime utility depends on three attributes: longevity, the lifetime path of consumption (proxied by income), and the lifetime path of health. While additional attributes could be considered (e.g. nonwork time, environmental quality), given our global focus, we ignore these attributes due to concerns about data availability and quality in many countries.
Second, we consider inequalities across age and socioeconomic status, understanding that, in principle, well-being attributes (consumption/income, longevity, and health) are heterogeneously distributed across the population depending on the age and socioeconomic status of individuals. We ignore other sociodemographic characteristics (e.g. race, gender). Although individuals vary in consumption/income, longevity, and health, we assume they have homogenous preferences over these attributes (i.e. individuals have the same preferences over alternative bundles of consumption/income, longevity, and health). We use income quintiles as a proxy for socioeconomic status.
Third, we adopt the following functional form for the period utility function:
$$ u=h\frac{y^{1-\epsilon }-{\overline{y}}^{1-\epsilon }}{1-\epsilon } $$
where
$ y $
represents the period consumption/income,
$ h $
is the health-related quality of life,
$ \overline{y} $
is subsistence consumption/income, and
$ \epsilon $
is the elasticity of the marginal utility of income.Footnote 6 This functional form is ubiquitous in the theoretical literature on the value of health (e.g. Murphy and Topel, Reference Murphy and Topel2006). An important feature of this formula is that the marginal utility of income is increasing in health-related quality of life. In other words, healthy people gain more utility from an increase in consumption than unhealthy people. Said otherwise, improving the health of a wealthy person increases well-being more than improving the health of a poor person. The complementarity between health and income is supported empirically (Finkelstein et al., Reference Finkelstein, Erzo and Notowidigdo2013). (For simplicity, we assume
$ h\ge 0 $
where
$ h=0 $
corresponds to a health state no better nor worse than death.)Footnote 7
For the specification of social welfare, we apply an Atkinson SWF of the form:
$$ W={\sum}_i\frac{v_i^{1-\gamma }}{1-\gamma } $$
where the sum is over all members of the affected society, the variable
$ {v}_i $
represents the lifetime well-being of individual
$ i $
, and the parameter
$ \gamma\;\left(\ge 0\right) $
represents the degree of inequality aversion (
$ W={\sum \limits}_i\log {v}_i $
if
$ \gamma =1 $
).Footnote 8 The Atkinson weighting function is consistent with the view that differences in well-being are related to the ratio (rather than the difference) between utility levels. The Atkinson SWF is very flexible and captures different distributional equity preferences. When
$ \gamma =0 $
, the SWF is utilitarian. When
$ \gamma >0 $
, the SWF is prioritarian, that is, it attaches larger value to well-being changes experienced by the worse-off. The larger is
$ \gamma $
, the greater is the concern about the well-being of the worse-off. When
$ \gamma \to \infty $
, the approach becomes egalitarian and aims at improving the well-being of those at the bottom of the well-being scale.Footnote 9
In the presence of risk, a prioritarian SWF depends on whether well-being is measured ex ante or ex post. An ex ante approach focuses on the distribution of expected lifetime well-being across individuals. Expected lifetime well-being considers the risk faced by individuals across their lifetime profile of income, health, longevity, and other attributes. In contrast, an ex post approach to well-being evaluation focuses on the realized distribution of lifetime well-being across individuals. Realized lifetime well-being considers the lifetime profile of income, health, and other attributes experienced by an individual, from birth to death. There are axiomatic reasons to prefer either an ex post or an ex ante approach. In particular, the ex ante approach does not satisfy principles of stochastic dominance and can lead to temporally inconsistent preferences. On the other hand, the ex post approach violates the ex ante Pareto principle (see e.g. Adler, Reference Adler2019, for a more thorough account of the properties of ex ante and ex post prioritarianism). We are agnostic to which approach is ethically preferable and construct weights using both ex post and ex ante approaches.
3.2. Equity-sensitive BCA
In this section, we briefly present mathematical expressions of the weights for utilitarian and prioritarian SWFs and discuss their main components. Appendix A provides a formal derivation of the equity-sensitive BCA rules and the corresponding weights. The next section describes the data required to estimate the weights. All weights are normalized with respect to the characteristics of the median individual in the population.
Let us consider a population of
$ N $
individuals divided into “types” (i.e. individuals that share similar sociodemographic characteristics), and let
$ {N}_i $
be the number of people of type
$ i $
, with
$ {\sum}_i{N}_i=N $
. Let us consider a policy that increases type
$ i $
’s survival probability by
$ \Delta {p}_i $
and reduces their income by
$ \Delta {y}_i $
(
$ \Delta {y}_i $
could be negative, denoting an income gain). Note that
$ {N}_i\Delta {p}_i $
represents the expected number of deaths prevented among individuals of type
$ i $
. Let
$ VS{L}_i $
denote the value-per-statistical-life of individuals of type
$ i $
.
According to conventional BCA, the total net benefits of such an intervention are equal to:
$$ {\sum}_i{N}_i\left(\Delta {p}_i VS{L}_i-\Delta {y}_i\right) $$
In other words, conventional BCA ranks policies based on the unweighted sum of individuals’ monetary equivalents for the policy,
$ \Delta {p}_i VS{L}_i-\Delta {y}_i $
.
In the utilitarian case, the total net benefits of the intervention are given by:
$$ {\sum}_i{N}_i\;{w}_i^U\left(\Delta {p}_i VS{L}_i-\Delta {y}_i\right) $$
where
$ {w}_i^U $
represents the utilitarian weight associated with type
$ i $
’s monetary equivalent for the policy. Utilitarian weights are given by:
$$ {w}_i^U={\left(\frac{y_i}{y_{med}}\right)}^{\hskip-0.5em -\epsilon } $$
where
$ {y}_i $
is individual
$ i $
’s income,
$ {y}_{med} $
is the population median income, and
$ \epsilon \ge 0 $
is the elasticity of the marginal utility of income. The utilitarian weight adjusts for differences in income across the population. In particular, it attaches larger values to net monetary benefits experienced by lower income individuals. Note that the utilitarian weights do not depend directly on an individual’s age but may depend indirectly on age if income depends on age.
In the ex ante prioritarian case, the total net benefits of the intervention are given by:
$$ {\sum}_i{N}_i\;{w}_i^{EAP}\left(\Delta {p}_i VS{L}_i-\Delta {y}_i\right) $$
where
$ {w}_i^{EAP} $
represents the ex ante prioritarian weight associated with type
$ i $
’s monetary equivalent for the policy. The ex ante prioritarian weights are equal to:
$$ {w}_i^{EAP}={\left(\frac{y_i}{y_{med}}\right)}^{\hskip-0.5em -\epsilon }{\left(\frac{QAL{E}_i}{QAL{E}_{med}}\;\frac{y_i^{1-\epsilon }-{\overline{y}}^{1-\epsilon }}{y_{med}^{1-\epsilon }-{\overline{y}}^{1-\epsilon }}\right)}^{\hskip-0.5em -\gamma } $$
where
$ QAL{E}_i={\sum}_{t=0}^{a_i}{h}_i(t)+{\sum}_{t={a}_i+1}^T{\pi}_i\left(t;{a}_i\right){h}_i(t) $
is the lifetime quality-adjusted life expectancy (from birth to death) of individual of type
$ i $
who is currently at age
$ {a}_i $
,
$ QAL{E}_{med} $
is the population median quality-adjusted life expectancy,
$ \overline{y} $
is subsistence income, and
$ \gamma $
is the inequality aversion parameter. QALE depends on the individual’s expected longevity and on health-related quality of life
$ {h}_i $
. It is age-dependent; for an individual currently at age
$ {a}_i $
, QALE depends on past health-related quality of life (from age 0 to age
$ {a}_i $
) and future expected quality of life (from age
$ {a}_i+1 $
to the maximum possible lifespan, set to age
$ T $
), where
$ {\pi}_i\left(t;{a}_i\right) $
is the probability of surviving to age
$ t\in \left({a}_i,T\right] $
conditional on being alive at age
$ {a}_i $
. Note that QALE tends to increase with age because the risk of dying young is eliminated.
The ex ante prioritarian weight includes two terms: the marginal utility of income, as in the utilitarian weight; and the marginal social value of expected lifetime well-being, represented by the second term in Equation (7). Individual well-being is an increasing function of incomeFootnote 10 and QALE. As
$ \gamma >0 $
, ex ante prioritarianism attaches larger weight to individuals with lower income and lower QALE. In addition, the larger the inequality aversion parameter
$ \gamma $
, the larger the weight assigned to net benefits experienced by the poor and by individuals with low QALE.
In contrast, in Appendix A, we show that the total net benefits of the intervention under ex post prioritarianism are equal to:
$$ {\sum}_i{N}_i\;\left\{{w}_i^{EPP}\left({a}_i\right)\Delta {p}_i VS{L}_i-\Delta {y}_i\sum_{t\ge {a}_i}{\mu}_i\left(t;{a}_i\right){w}_i^{EPP}(t)\right\} $$
where
$ {w}_i^{EPP}(t) $
, with
$ t\ge {a}_i $
, represents the ex post prioritarian weight associated with the costs and benefits experienced by individuals of type
$ i $
who die exactly at age
$ t $
in the absence of the policy, and
$ {\mu}_i\left(t;{a}_i\right) $
is the proportion of people currently of age
$ {a}_i $
who will die exactly at age
$ t\ge {a}_i $
without the policy.
The ex post prioritarian rule is considerably different from both the utilitarian and the ex ante prioritarian ones. In particular, while the utilitarian and the ex ante prioritarian weights are applied to individuals’ monetary equivalents,
$ \Delta {p}_i VS{L}_i-\Delta {y}_i $
, ex post prioritarian weights are not.Footnote 11 The latter depend on the realized longevity of individuals in the absence of the intervention since, from an ex post prioritarian perspective, what matters is the realized well-being of individuals. Specifically, a proportion
$ \Delta {p}_i $
of type
$ i $
individuals would have died at age
$ {a}_i $
without the policy; their ex post prioritarian weight is thus
$ {w}_i^{EPP}\left({a}_i\right) $
, and it is applied to both the mortality risk change and the income change,
$ VS{L}_i-\Delta {y}_i $
. A proportion
$ {\mu}_i\left({a}_i;{a}_i\right)-\Delta {p}_i $
dies at age
$ {a}_i $
both with and without the policy; their ex post prioritarian weight is thus
$ {w}_i^{EPP}\left({a}_i\right) $
, and it is applied only to the income change
$ \Delta {y}_i $
(as they do not experience any survival benefit from the policy). Finally, a proportion
$ {\mu}_i\left(t;{a}_i\right) $
dies at age
$ t>{a}_i $
both with and without the policy; their ex post prioritarian weight is thus
$ {w}_i^{EPP}(t) $
, and it is applied only to the income change
$ \Delta {y}_i $
. As a consequence, even though individuals of type
$ i $
are ex ante identical, they are different ex post since they may die at different ages.
The formula for ex post prioritarian weight for an individual who dies at age
$ t\ge {a}_i $
in the absence of the policy is:
$$ {w}_i^{EPP}(t)={\left(\frac{y_i}{y_{med}}\right)}^{\hskip-0.5em -\epsilon }{\left(\frac{H_i(t)}{H_{med}}\;\frac{y_i^{1-\epsilon }-{\overline{y}}^{1-\epsilon }}{y_{med}^{1-\epsilon }-{\overline{y}}^{1-\epsilon }}\right)}^{\hskip-0.5em -\gamma } $$
where
$ {H}_i(t)={\sum}_{s=0}^t{h}_i(t) $
is the realized quality-adjusted longevity of the individual who dies at age
$ t $
without the policy, and
$ {H}_{med} $
is the population median quality-adjusted longevity. Like the ex ante prioritarian weights, the ex post ones adjust for the marginal utility of income and for the marginal social value of lifetime well-being.Footnote 12 Individuals with lower lifetime realized well-being are given higher weight. The latter now depends on realized quality-adjusted longevity (as opposed to expected quality-adjusted longevity). In other words, people who die young will have a larger weight than people who die old, even though, from an ex ante point of view, they have the same life expectancy.
Applying ex post prioritarian weights requires modeling changes in the number of people dying at each age since the weight depends on the age of death rather than at the current age of an individual.Footnote 13, Footnote 14 This constitutes a major difference between the ex ante weights and ex post ones: The former depend on the current age of the individual experiencing the policy impacts; the latter depend on the baseline age of death of the individual who is experiencing the policy impacts. However, if the impacts of an intervention are independent of the age of death, one could derive ex post prioritarian weights as a function of current age by taking the expectation of
$ {w}_i^{EPP}(t) $
over the age of death:Footnote 15
$$ {\overline{w}}_i^{EPP}\left({a}_i\right)=\sum_{t\ge {a}_i}\mu \left(t;{a}_i\right){w}_i^{EPP}(t) $$
In this case, the ex post prioritarian rule simplifies as follows:
$$ {\sum}_i{N}_i\;\left\{{w}_i^{EPP}\left({a}_i\right)\Delta {p}_i VS{L}_i-\Delta {y}_i{\overline{w}}_i^{EPP}\left({a}_i\right)\right\} $$
We will illustrate this simplification in the example below.
3.3. Data requirements and calibration
To estimate the proposed weights, we need the following: information about the distribution of income and of mortality and morbidity risks in the population of interest; estimates of the subsistence income
$ \overline{y} $
and the elasticity of marginal utility of income
$ \epsilon $
; and the inequality aversion parameter
$ \gamma $
. We summarize our proposed estimates in Table 1, and describe them in more detail below.
Input data for estimating weights
Let us start with the calibration of
$ \overline{y} $
,
$ \epsilon $
, and
$ \gamma $
. The subsistence income
$ \overline{y} $
is a level of income so low that individuals are indifferent between survival and death. The World Bank estimates that almost 700 million people around the world live in “extreme poverty,” defined as subsisting on less than $2.15 per day. It is reasonable to assume that the subsistence income is smaller than the extreme poverty line. We propose to use $1 a day as a benchmark.
There is an extensive literature on the elasticity of the marginal utility of income. For high-income countries, the proposed estimates often range between 1 and 2. For example, OMB (2023a, b) proposes a value of 1.4 for the United States, based on an update of a meta-analysis of U.S. and U.K. data by Acland and Greenberg (Reference Acland and Greenberg2023). Groom and Maddison (Reference Groom and Maddison Pr2019) combine results from different estimation methodologies and suggest a value of 1.5 for the U.K. Using subjective well-being data from a large set of countries, Layard et al. (Reference Layard, Mayraz and Nickell2008) conclude that the average elasticity is 1.26. Studies in lower income countries are rare. Using subjective well-being data, Gandelman and Hernández-Murillo (Reference Gandelman and Hernández-Murillo2014) conclude that the elasticity is close to 1 in most countries, independent of their development level. A recent paper by Bergstrom and Dodds (Reference Bergstrom and Dodds2023) relies on schooling decisions in rural Mexico and estimates that the elasticity is 1.6. As a benchmark, we propose to use a value equal to 1.5.
The degree of inequality aversion
$ \gamma $
is an ethical choice. The literature on the elicitation of ethical preferences can provide some indication of reasonable values for
$ \gamma $
. However, being an ethical choice, we propose to apply a range of values. In the example below, we assume that
$ \gamma $
is equal to 1 or 2.
Finally, the construction of the weights requires data on the distribution of income as well as mortality and morbidity risks across the population of interest. Data quality is a concern; often the analyst has estimates of the average but not the whole distribution. However, the analyst can leverage publicly available datasets such as the World Bank Indicators and the United Nations World Population Prospects to construct approximate distributions of income, mortality, and morbidity. We illustrate the use of available data in the next section.
3.4. Weights by individuals’ socioeconomic status, age, and country’s income level
As an illustration, we use the previous formulas to construct weights by individuals’ socioeconomic status (proxied by income quintile) and age, for countries in each World Bank income category: low-income, lower-middle-income, upper-middle-income, and high-income. For each country type, we divide the population into five homogeneous groups corresponding to the quintiles of the income distribution. We further divide the population into broad age groups (0–19, 20–64, 65+).Footnote 16 Thus, we have 15 sociodemographic subgroups. We then determine the utilitarian and prioritarian weights attached to each subgroup for each country income category. To simplify, we assume that there is no social mobility, that is, individuals do not move across the income distribution as they age, and that there is no income growth. As a consequence, socioeconomic status is age-independent.
We first discuss the data used in the estimation, then summarize them in Table 2. Data on average income per-capita are taken from the World Development Indicators (World Bank, 2024). Income is measured in terms of purchasing power parity Gross National Income (GNI) (constant 2017 international dollars).Footnote 17 We rely on the World Bank Poverty and Inequality Platform for information on the distribution of income in many countries around the world (World Bank, 2022). We use data on the income share held by each quintile (i.e. how much of the country’s aggregate income is earned by each income quintile). We average across countries belonging to the same income category to estimate the distribution of income in the representative countries.
Economic and demographic variables
a Gross National Income per-capita, PPP, constant 2017 international $ (World Bank, 2024).
b Simple averages of income shares by country income group. Original data are drawn from the World Bank Poverty and Inequality Platform (World Bank, 2022).
d Own computations based on survival rates from the 2022 World Population Prospects (UN, 2022) and disability indexes from the 2019 Global Burden of Disease Study.
e For low- and middle-income countries, the distribution of life expectancy at birth by income quintile is based on an Indian study (Asaria et al., Reference Asaria, Mazumdar, Chowdhury, Papiya, Abhiroop and Indrani2019). We assume that the ratio between life expectancy in each quintile and median life expectancy in India applies to all low- and middle-income countries. For high-income countries, the distribution is based on a U.S. study (Chetty et al., Reference Ray, Stepner, Abraham, Lin, Scuderi, Turner, Bergeron and Cutler2016) and derived from Adler (Reference Adler2017), Appendix C.
f Own computations based on the disability indexes from the 2019 Global Burden of Disease Study and the income-dependent mortality rates (see note e).
To determine QALE, we combine the life tables from the 2022 World Population Prospects (UN, 2022) with disability metrics from the 2019 Global Burden of Disease Study (GBD) (GBD 2019 Diseases and Injuries Collaborators 2020).Footnote 18 Both datasets provide data for single countries and for World Bank income groups; our computations are based on the latter. We first use data from the 2019 GBD and compute age-specific disability indexes by dividing the “years of healthy life lost due to disability” (YLD) at each age by the prevalence of illnesses for each country income group at that age.Footnote 19 We assume that one minus the disability index approximates health-related quality of life at different ages and in different country income groups (disability is measured on a zero-to-one scale, with one representing death and zero full health). The 2022 World Population Prospects provide life table survivors, that is, estimates of survivors by age for a hypothetical cohort of 100,000 newborns who would be subject for all their lives to the mortality rate of a given year (UN, 2022). We use 2024 as the reference year and compute the probability at birth of surviving to different ages. We apply the 2024 survival curve to all age groups and neglect that previous birth cohorts may have different survival chances (i.e. the probability of surviving to age 40 of the 2024 birth cohort may be larger than the probability of surviving to age 40 of the 2004 birth cohort).Footnote 20 We then combine those survival probabilities and the computed health index at different ages to determine the QALE measures.Footnote 21 Table 2 reports both the average life expectancy at birth derived from the World Population Prospects and the QALE that we compute. Figure B1 (in Appendix B) depicts the age-specific quality-adjusted remaining life expectancies in the various country groups. To apply ex post prioritarian weights, we need information on past quality of life. Figure B2 depicts the number of years spent without a disability for an individual that dies at different ages, again assuming that one minus the disability index approximates the average health status of individuals.Footnote 22 The QALE used in the ex ante prioritarian weights is the sum of age-specific remaining QALE and age-specific past quality of life (Figure B3).
Furthermore, we assume that life expectancy differs by socioeconomic status. We assume that the survival curves from the 2022 World Population Prospects apply to individuals with median income and adjust the mortality rates for the other income quintiles using scaling factors. To determine the scaling factors, we use two studies. For the high-income category, the scaling factors are loosely based upon Chetty et al. (Reference Ray, Stepner, Abraham, Lin, Scuderi, Turner, Bergeron and Cutler2016, Figure 2), who estimate life expectancy at age 40 in the United States for different income groups. For low- and middle-income countries, we use results from Asaria et al. (Reference Asaria, Mazumdar, Chowdhury, Papiya, Abhiroop and Indrani2019, Table 1), who estimate the distribution of life expectancy at birth by income quintile in India.Footnote 23 The scaling factors are such that the ratio between life expectancy for a given quintile and life expectancy for the median individual are approximately equal to the ratios estimated in the two studies.Footnote 24 Table 2 reports the resulting life expectancy at birth by income quintile and country income group. Lacking data on the distribution of disabilities by socioeconomic status, we assume that survival rates are income-dependent, while the health indexes are not. As a consequence, remaining quality-adjusted life expectancy at a given age is income-dependent, but past quality of life for someone who dies at a given age is not income-dependent. Table 2 reports the resulting QALEs at birth by income quintile and country income group.
To compute the weights, we also need to define the standardization method (i.e. the denominator). In the utilitarian case, we simply use median income for each country type. In the ex ante prioritarian case, we use median income and define median QALE as the value for an individual with median income at median age. Since lower-income countries have a younger population (Table 2), the reference level differs across country income groups. The ex post prioritarian weights are also computed with respect to the realized quality of life of an individual at median age.
Table 3 summarizes the weights by country type, age, and income quintile. It is worthwhile to recall that these are the weights to be used in the assessment of national policies. Were one interested in global policies, the weights should be rescaled using a common denominator to reflect cross-country differences. The utilitarian weights are similar across countries. This is because the relative distribution of income does not differ much across country income categories. The distribution of income in the high-income countries is slightly more equal, which explains the lower weight to individuals in the first income quintile. For lack of suitable data, we assumed that income is constant across age groups. Thus, the utilitarian weights are age-independent.
Utilitarian weights, ex ante prioritarian weights by current age, and ex post prioritarian weights by age of death
Notes: The age brackets considered in the table are: 0–19 (young), 20–64 (middle-aged), and 65+ (old). In the ex ante prioritarian case, we report the weights assigned to an individual at average age within a given age bracket. For low-income countries, the average ages are 9 (0–19), 36 (20–64), and 72 (65+); for lower-middle-income countries, the average ages are 9 (0–19), 38 (20–64), and 73 (65+); for upper-middle-income countries, the average ages are 10 (0–19), 41 (20–64), and 73 (65+); for high-income countries, the average ages are 10 (0–19), 42 (20–64), and 75 (65+). Similarly, in the ex post prioritarian case, we use the average age of death for people who die in a given age group. These correspond to 5 (0–19), 49 (20–64), and 78 (65+) for low-income countries; 5 (0–19), 51 (20–64), and 78 (65+) for lower-middle-income countries; 5 (0–19), 52 (20–64), and 81 (65+) for upper-middle-income countries; 6 (0–19), 53 (20–64), and 83 (65+) for high-income countries.
Ex ante prioritarian weights assign considerably more value to costs and benefits experienced by low-income individuals. Within an income quintile, they assign slightly more weight to younger people than to older people. Individuals in low-income quintiles are among the worst-off from a lifetime perspective because they have low income and low life expectancy (due to the income gradient of mortality). Within an income bracket, younger people receive a slightly larger weight because they face a larger lifetime mortality risk than older people (i.e., they have lower chances to reach old age than people who are closer to old age). This is particularly true in lower-income countries, with high infant and childhood mortality rates. Increasing the inequality aversion parameter increases the weight attached to impacts experienced by young and low-income individuals.
As in the ex ante case, ex post prioritarian weights assign more value to impacts experienced by low-income individuals than comparable impacts experienced by high-income individuals. In addition, ex post prioritarianism attaches very large weights to those who die young and poor. For example, let us consider a low-income setting with inequality aversion equal to 1. For each income quintile, we find that the weight attached to an individual who dies young is more than 10 times larger than the weight attached to an individual who dies at old age. The weight attached to a poor (1st income quintile) individual who dies young is about 700 times larger than the weight attached to a rich (5th income quintile) individual who dies at old age. This result is driven by the fact that ex post prioritarian weights depend on realized income, longevity, and quality of life. Thus, from an ex post prioritarian point of view, it is a tragedy if the individual dies very young when the rest of the population enjoys a longer lifespan. The larger is the inequality aversion parameter
$ \gamma $
, the larger is the weight assigned to costs and benefits experienced by individuals who die young and poor in the absence of the intervention. It is worthwhile to stress that this result hinges on the well-being concept we use, lifetime well-being, which increases in longevity.
Tables C1–C4 in Appendix C report ex ante and ex post prioritarian weights for each age integer. Tables C5 and C6 report the expected ex post prioritarian weights conditional on current age. Table B1 in Appendix B reports the expected ex post prioritarian weights by age group, income quintile, and country income group.
4. Stylized example
To clarify the application of equity-sensitive BCA, we present a stylized example. The data concerning the costs and benefits of the interventions are fictional.
Suppose we want to determine the value of alternative policies in a low-income country context. We assume the interventions last 1 year and have only short-term mortality and income impacts.Footnote 25 We furthermore assume that the conventional BCA provides data on the distribution of benefits and costs by socioeconomic status. We also assume that the population affected by the policy amounts to 30 million individuals and that the distribution of age by income is constant.Footnote 26 Based on the survival curves derived from the World Population Prospects for low income-countries, 52.4 % of the population is aged 0–19, 44.4 % is aged 20–64, and 3.2 % is aged 65+.
We consider four different interventions, summarized in Table 4. All four interventions reduce mortality risk by 1-in-100,000 (i.e. they save 300 people) and cost $100 million. The risk reductions are assumed to be concentrated on the young. However, the interventions differ in how the health benefits and the policy costs are distributed across income quintiles. Intervention A uniformly reduces the mortality risk for all individuals, that is, it will save 60 young people in each income quintile. The costs of intervention A are also uniformly distributed across the population: Everyone pays an equal share of $100 million or $3.3 per capita. Intervention B uniformly reduces the mortality risk for all young individuals as for intervention A. However, the distribution of costs is proportional to income. Interventions C and D benefit primarily poor young individuals, that is, they are expected to avert more deaths among low-income individuals than among high-income individuals. However, they differ in terms of distribution of costs. The costs of intervention C are uniformly distributed across income quintiles, while the costs of intervention D are distributed proportional to income.
Distribution of deaths averted and policy costs
Using the previous notation,
$ {N}_i\Delta {p}_i $
is the number of deaths averted among individuals of type
$ i $
(characterized by a specific age and a specific income quintile), and
$ \Delta {y}_i $
is the per-capita cost of the intervention. Those data are included in Table 4. To apply the conventional BCA rule (1), the only missing ingredient is the individual-specific value-per-statistical-life
$ VS{L}_i $
. We assume that the ratio of VSL to income is constant and equal to 160 times GNI per capita for each income quintile (i.e., we set the income elasticity of VSL to 1). To adjust for age differences, we derive income-specific value-per-statistical-life-year (VSLY) by dividing the VSL by undiscounted remaining quality-adjusted life expectancy at the average age of the population in the country group (the latter proxied by half the quality adjusted life expectancy at birth). The resulting VSLY is then multiplied by age- and income-specific remaining quality-adjusted life expectancy to determine the age-and income-specific VSLs.
Table 5 summarizes the VSL estimates and the net benefits of the interventions by income quintile and age group. The net benefits are the difference between the monetary value of deaths averted among individuals from a given quintile and the total costs paid by them, that is,
$ {N}_i\left(\Delta {p}_i VS{L}_i-\Delta {y}_i\right) $
. Not everyone gains from the interventions. The distribution of net benefits is policy dependent. All things considered, policy D is the most progressive since within each age group the lowest income individuals are expected to gain the most (or lose the least).
Value-per-statistical-life and net benefits by demographic group
To apply the equity-sensitive BCA rules (4), (6), and (11), we supplement the data on benefits and costs by age and income quintile with the weights to be applied in a low-income context. Those weights are provided in Tables 3 and B1 (as stressed earlier, in the ex post prioritarian case we use the expected weights
$ {\overline{w}}_i^{EPP}\left({a}_i\right) $
since the intervention costs are independent of the age of death).
Table 6 summarizes the results of conventional BCA and equity-sensitive BCA. Conventional BCA sums the net benefits across the population. According to conventional BCA, policies A and B are equally desirable and preferable to policies C and D. Thus, the distribution of policy costs has no impact on the BCA outcome, and policies that primarily benefit the disadvantaged are valued less than similar policies with a more regressive distribution of health benefits.Footnote 27
Results of conventional BCA compared to equity-sensitive BCA
In this specific example, all equity-sensitive BCAs agree on the ranking of policies and recommend policy D over the alternatives. Moreover, both utilitarianism and ex ante prioritarianism find that policy A is not beneficial, in stark contrast with the BCA outcome. Ex post prioritarianism recommends policy D, but attaches positive value also to policy A. The difference between ex ante and ex post prioritarianism is driven by the weight attached to the young. From an ex post point of view, avoiding deaths at a young age deserves priority. Since the benefits of the policy are concentrated among the young, in this specific example, ex post prioritarianism concludes that all proposed policies are worthwhile.
5. Summary and conclusions
While conventional BCA has many strengths, it also has important limitations. Chief among these is its inattention to the distributional equity of the impacts. To measure individual welfare, it relies on individuals’ willingness to exchange their own income for the outcomes they experience. To measure societal welfare, it relies on summing these values across individuals. Many alternative approaches have been proposed to address these limitations, but none is yet widely used. A major barrier to their application is the difficulty of implementation.
To encourage more attention to the distributional implications of interventions, we derive multiple sets of weights to be used in conjunction with the results of conventional BCAs to improve the information available on distributional equity globally. We discuss how the weights can be estimated using publicly available datasets and results from the literature. We derive both utilitarian and prioritarian weights. The utilitarian weights account for the greater value of money to a poor person than a rich person. The prioritarian weights account also for preferences for prioritizing increases in welfare for those who are worse off. The worse off include individuals with low socioeconomic status, poor health, or low life expectancy.
The weights that we derive are independent of the specific intervention under analysis. However, they depend on (i) the characteristics of the population affected by the intervention, and (ii) the chosen well-being metric and SWF. We derive weights for representative country profiles based on the income groups defined by the World Bank using publicly available data. As long as the country profile reasonably approximates the population affected by the intervention, the estimated weights can be directly applied to the results of a conventional BCA.
For the theoretical underpinnings of the estimated weights, we measure well-being with a vNM utility function and we adopt an Atkinson SWF whose concavity spans from utilitarianism to prioritarianism (in our example, but could also address egalitarianism). These assumptions are supported by previous studies. Future research should investigate alternative measures of well-being and alternative SWFs and the sensitivity of the estimated weights to these normative choices. In addition, because of data constraints, we made some further simplifying assumptions, including well-being depends only on three attributes, income, health, and longevity; lifetime well-being is time separable; health and income are complements; and the distribution of income is age-independent. These assumptions could be relaxed in future research.
In our setup, utilitarian weights account for differences in income across the population. Prioritarian weights account for differences in income and differences in expected or realized quality-adjusted longevity. Since young individuals have a lower expected or realized longevity, prioritarianism tends to attach larger weights to young people than older people.Footnote 28 Ex post prioritarianism confers extremely large weight to individuals who die very young on the grounds that dying young is a tragedy from a lifetime perspective. This result runs against the argument that, especially in resource-constrained settings, saving an adult life is more highly valued due to the economic support provided by working-age individuals. In our framework, the larger contribution of adult individuals could be captured by including the positive externalities associated with saving an adult life (e.g. in the form of increased production or of improved well-being of family members). Similarly, in our stylized example, we disregard the non-market productive contribution of older adults, for example, through childcare, volunteering, or informal care for sick family members. There is also a large debate on the value of preserving the life of newborns and infants. Our framework is not able to capture all the nuances of such a debate.
The choice between utilitarianism and prioritarianism (and the degree of priority to the worse off) is ultimately an ethical choice. In our framework, this was captured by a single parameter
$ \gamma $
representing the degree of inequality aversion. We computed weights under three values:
$ \gamma =0 $
(utilitarianism),
$ \gamma =1 $
, and
$ \gamma =2 $
. The larger is
$ \gamma $
, the larger is the priority to the worse off. We avoided making any judgement about the most appropriate value for
$ \gamma $
, and we proposed weights for multiple values of
$ \gamma $
. The literature on social preferences can provide guidance on the inequality aversion attitudes most frequently held by individuals (Venmans & Groom, Reference Venmans and Groom2021; Schokkaert & Tarroux, Reference Schokkaert, Tarroux, Adler and Norheim2022).
Supplementary material
The supplementary material for this article can be found at http://doi.org/10.1017/bca.2024.28.
Acknowledgments
The authors thank Matthew Adler, Richard Cookson, Marc Fleurbaey, Ole F. Norheim, J.P. Sevilla, and Brad Wong for their thoughtful comments on previous drafts of this article and many helpful discussions. The authors are also grateful to Wafa Aftab, David Canning, and Christian Krekel for valuable inputs to this project. This work was funded in part by the Bergen Centre for Ethics and Priority Setting, through grants from the Norwegian Agency for Development Cooperation and the Norwegian Research Council. It builds on a 2022 conference held by the Brocher Foundation, “Healthy, Wealthy, and Wise - The Ethics of Health Valuation,” organized by Nir Eyal (Rutgers University), Samia Hurst (University of Geneva), Lisa A. Robinson (Harvard University), and Daniel Wikler (Harvard University). The authors thank the Brocher Foundation and conference participants for their support and helpful comments. This special issue was supported by the Brocher Foundation, with supplemental funding from the Rutgers University Center for Population-Level Bioethics, the University of Geneva Institute for Ethics, History, and the Humanities, and the University of Bergen Centre for Ethics and Priority Setting. More information on the Brocher Foundation is available here: https://fondation-brocher.ch/.
Competing interest
The authors declare none.
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