2.1. Justification for Utility-Weighting

Aggregate welfare is understood by normative ethicists to have ethical or moral weight in decision making and to be a value that is separate from other values such as equity and rights (Kagan, Reference Kagan2018). As such, it is a social goal about which decision makers need information, in multi-goal analysis, that is not biased by the diminishing marginal utility of income. We use the term “utility weights” to refer to weights that are applied to different income groups to achieve the necessary debiasing. When utility weights are applied to BCA, it becomes an unbiased measure of aggregate welfare, and, as referenced in the introduction, unbiased information about social goals is one of the necessary ingredients in multi-goal analysis.

Unweighted BCA is often referred to as a measure of “allocative efficiency,” which Boardman et al. (Reference Boardman, Greenberg, Vining and Weimer2018) describe as “a situation in which resources, such as land, labor and capital, are deployed in their highest valued uses in terms of the goods and services they create.” It is clear that the term “valued” means valued in unweighted dollars. The sense in which this situation is maximally efficient is not addressed, as it does not actually involve maximizing what actually matters to the lived experience of individuals, which is welfare. Regardless of how the term “allocative efficiency” is defined, BCA simply is not a measure of what is best for society in terms of welfare. Nonetheless, the interpretation of BCA as a measure of aggregate welfare has been embraced by many scholars. Adler and Posner (Reference Adler and Posner2006) in their attempt to provide “New Foundations” for BCA, endorse the welfare interpretation explicitly, defining BCA as a measure of “restricted preferentialist welfare,” while Sunstein (Reference Sunstein2018), throughout his treatise on the “Cost-Benefit Revolution,” consistently uses the term “welfare” to describe the value measured by unweighted BCA. Harberger (Reference Harberger1978), one of the first scholars to discuss what we are referring to as utility weights, also used the term “welfare,” without comment, to describe the value measured by WTP.Footnote ⁹ In addition, a case can be made that the acceptance of the interpretation of BCA as a measure of aggregate welfare is implicit in the universally accepted term “net benefit.” The difference between the WTP of the winners and the willingness-to-accept (WTA) of the losers is, more or less without exception, referred to as “net benefit” throughout the field of BCA – a term that is commonly implicitly used as a synonym for welfare (if perhaps unconsciously). It is perhaps understandable that scholars and practitioners of BCA explicitly or implicitly embrace the aggregate welfare interpretation of unweighted BCA since, as we have stated, aggregate welfare is the thing that normative ethicists and moral philosophers believe decision makers ought to be informed about.

Some have argued that unweighted BCA is an imperfect, or “rough and ready” proxy for welfare (Adler and Posner Reference Adler and Posner2006), but if the proxy is systematically biased against one group, then it is not just imperfect, it is unacceptable. For example, a policy that generates a benefit of $100 for a corporate attorney while imposing a cost of $90 on a custodial worker is an improvement in allocative efficiency, but to say that $10 is a rough-and-ready approximation of the welfare benefit to society will strike many as morally repugnant. Others have argued that the consistent application of unweighted net benefit to policy making will result in all groups benefiting in the long run, as some policies will benefit the wealthy, and others will benefit the poor, so that on net there will be an improvement to both groups. However, even in the long run, unweighted net benefit will result in greater benefits to the wealthy than to the poor, because it will consistently preference policies that benefit the wealthy over those that benefit the poor. For example, if two policies generate a welfare benefit of X and a welfare cost of Y, but the first assigns the benefit to the wealthy and the cost to the poor, and vice versa for the other, unweighted BCA will rank the former above the latter, even if both pass an unweighted net-benefit test. To determine whether public decision-making increases aggregate welfare, policy-by-policy in the short-run, or on net over the long-run, we must apply utility weights.

A fundamental challenge must be addressed before discussing the practicalities of utility-weighting, which is the problem of interpersonal welfare comparisons: no matter what metric one uses in the attempt to capture welfare, the same number may refer to different levels of welfare across individuals. Even if two individuals have identical incomes and identical preferences, the fact that they have identical willingness-to-pay for a good that increases their welfare does not necessarily mean that they get the same welfare benefit from that good. One may simply get more welfare out of the good than the other – holding their preferences and income constant – perhaps because of their temperament, upbringing, or experience. Consider two individuals, with identical preferences for fly fishing versus all other goods, identical income, identical life circumstances, and identical WTP for a fly-fishing trip. Even if they have identical experiences on a fly-fishing trip, one may simply have a better time, perhaps because he or she is simply inherently easier to please. Alternative metrics do not solve the problem. For example, when researchers attempt to measure welfare with survey questions that ask respondents to rate their “subjective wellbeing" on a scale of, let’s say, one to ten, there is no way for them to know whether a response of six out of ten represents the same level of welfare from one individual to another, and not only because the scales are typically ordinal. Even if the scale could somehow be thought of as cardinal, one individual might simply have a higher reference point for wellbeing, so that a six out of ten refered to a higher level of utility for them than for another individual. The impossibility of an interpersonally comparable welfare metric has been recognized since the beginning of the ordinalist revolution pioneered by Fisher and Pareto (see Lenfant., Reference Lefant2012).

The problem of interpersonal comparisons in the measurement of aggregate welfare led the ordinalists to introduce the potential Pareto criterion (PPC) as a way to sidestep the problem. Rather than wrestling with whether or not a policy increases aggregate welfare, they preferred to assess the desirability of a policy by determining whether it is a “potential Pareto improvement.” A policy is a potential Pareto improvement to society if those who gain from the policy could potentially transfer enough money to those who lose to make the losers better off, or at least no worse off, and still come out ahead. If we embrace the PPC and interpret BCA as an implementation of it, then BCA measures the difference between aggregate willingness to pay (WTP) for the gains and aggregate willingness to accept (WTA) for the losses, which could be thought of as the magnitude of the improvement. Interpreting BCA as an implementation of the PPC puts it on solid ground with respect to interpersonal welfare comparisons: it is simply no longer an attempt to measure aggregate welfare.

If this is how we understand BCA, then clearly we must not apply utility weights, because if we do, we will no longer have accurate measures of WTP and WTA. Consider a policy that has been determined to be a potential Pareto improvement: WTP is greater than WTA. If those who gain have higher income than those who lose, utility-weighting would inflate WTA relative to WTP, so that the difference between utility-weighted WTP and utility-weighted WTA would be smaller than without the weights and might indeed be negative. Utility-weighted BCA cannot be counted on to determine whether the winners could compensate the losers in monetary terms. Thus, if we interpret BCA as an implementation of the PPC, we must not apply utility weights, we must accept that BCA does not measure aggregate welfare, and we must accept that it cannot reliably be used by decision makers to assess the tradeoffs between welfare and distributional impacts in multi-goal analysis. If one accepts our premise that multi-goal analysis is the appropriate way to empower decision makers to assess tradeoffs between welfare and distributional impacts, this concession is anathema. If, instead, we apply utility weights and interpret BCA as a measure of aggregate welfare, we must confront the interpersonal comparisons problem.

We do not have a solution to the problem of interpersonal welfare comparisons between specific individuals, but we do not need one because we are only concerned with comparing benefits and costs between groups. All that this requires is that the distribution of different types of individuals, with respect to the welfare or utility they get out of goods for which they have equal WTP, be the same in the different groups being compared. For example, suppose that the costs of a policy accrue to consumers of a particular good who are primarily high-income, while the benefits are received by the relatively low-income members of a particular occupation. A problem would arise if either the consumers or the workers were disproportionately individuals for whom a given WTP represents relatively little welfare. Utility-weighted BCA would be favorably biased towards that group because a given dollar value of the costs or benefits to that group would represent less welfare than the same aggregate WTP among the other group.

We are not aware of any empirical evidence of this kind of disproportionality, and indeed, the very fact of noncomparability makes it impossible to gather any such evidence, as it would necessarily require some measure of welfare, which, in turn, would be subject to the problem of interpersonal comparisons. Nonetheless, to justify interpreting BCA as a measure of aggregate welfare and applying utility weights, we must make the assumption that there is no disproportionality of types across groups. The rapidly growing literature on the measurement of happiness or subjective well-being (SWB) also makes this assumption (Wiener et al, Reference Wiener, Bronsteen, Buccafusco and Masur2013; Frey et al, Reference Frey, Luechinger and Stutzer2010). Indeed, the case could be made that those within the field of BCA who think of BCA as a measure of welfare are implicitly making the same assumption. Harberger made a poignant and relevant observation in 1978: “[A]lthough most economists have come to shun interpersonal comparisons of utility whenever they feel they can avoid or circumvent them, they usually make them when they have to.” (Harberger Reference Harberger1978.) Assuming that the distribution of types of individuals is the same across groups is considerably more innocuous than assuming interpersonal comparability at the individual level and would appear to be a modest compromise to make in pursuit of a measure of aggregate welfare that is not unethically biased against the poor.

2.2. Constructing Utility Weights

How then should utility weights be constructed? One approach would be to convert dollars into “utils” by multiplying the WTP of an individual of a given income level by their marginal utility of income at that income level. The number of utils assigned to any given quantum of welfare would be the same, regardless of the income level of the individual. The problem with this approach is that the utility of income is invariant to any affine transformation. In other words, if $ U(y) $ represents some preferences over income, then for any constants $ a $ and $ b $ , $ {U}^{\prime }(y)\hskip0.35em =\hskip0.35em a+b\ast U(y) $ will represent the same preferences (provided that $ b $ is positive). But because the two functions have different first derivatives, they generate different marginal utilities and thus different measures of weighted net benefits.

The standard solution to this problem in the literature is, rather than using the marginal utility of each individual at their current income level, to instead use the ratio of their marginal utility at their current income to the marginal utility they would have at some reference income, such as the median income in the population (Cowell and Gardiner, Reference Cowell and Gardiner1999; Adler, Reference Adler2013; Nurmi and Ahtiainen, Reference Nurmi and Ahtiainen2018). This ratio is the same across all affine transformations of the utility function, because $ b $ is in both the numerator and the denominator (while $ a $ drops out when taking derivatives). This is true both for a given individual at two different levels of income and for two individuals with the same preferences at different levels of income, provided that we use the same transformation of the utility function to represent the preferences of the two individuals.

This ratio measures the increase in utility generated by an additional dollar at $ {y}_0 $ , as a multiple of the increase in utility generated by an additional dollar at the medium income level, $ {y}_m $ . For example, if at $ {y}_0 $ , the marginal utility of income is 4, while at $ {y}_m $ it is 2, then the utility weight on WTP at $ {y}_0 $ would be 4/2 = 2. Gaining a dollar when one’s income is $ {y}_0 $ generates twice as much utility as gaining a dollar when one’s income is $ {y}_m. $ Inflating dollars gained when income is $ {y}_0 $ by a factor of 2 generates an unbiased metric of utility. A utility-weighted dollar now represents the same increase in utility regardless of income level. Consequently, we can think of the utility-weighted WTP of an individual at income $ {y}_0 $ for some quantum of welfare as the WTP they would express for that quantum of welfare if they had median income. The bias has been removed, and the result has an intuitive comprehensible interpretation: the utility-weighted net benefit to society of a policy can be thought of as the aggregate WTP one would observe in a hypothetical society in which everyone had income equal to the median.Footnote ¹⁰

2.3. Computing Utility Weights

Computing utility weights requires that we have some way to estimate the ratio of marginal utilities between income levels. There is a large literature that attempts to do this.Footnote ¹¹ In order to make the estimation process feasible, it is necessary to make an assumption about the functional form of utility. In the methodologies we rely on below, the standard assumption is that preferences can be represented by an isoelastic utility function:

(1) $$ U(y)\hskip0.35em =\hskip0.35em \frac{y^{1-\varepsilon }-1}{1-\varepsilon}\hskip0.35em \mathrm{if}\hskip0.35em \varepsilon \ne 1\hskip0.35em \mathrm{and}\hskip0.50em U(y)\hskip0.35em =\hskip0.35em \ln \hskip0.35em (y)\hskip0.24em \mathrm{if}\hskip0.35em \varepsilon \hskip0.35em =\hskip0.35em 1 $$

where $ y $ is income and $ \varepsilon $ is the elasticity of marginal utility with respect to income.Footnote ¹² The isoelastic function has the desirable property of constant elasticity.

Given this utility function, the marginal utility of income is $ \frac{dU}{dy}\hskip0.35em =\hskip0.35em \frac{1}{y^{\varepsilon }} $ . If we normalize the weight on individuals of median income to be 1, and if we let $ {y}_i $ denote the income of group $ i $ , and $ {y}_m $ denote median income, then the appropriate utility weight to place on the WTP of group $ i $ is

(2) $$ {w}_i\hskip0.35em =\hskip0.35em \frac{d{U}_i}{d{y}_i}/\frac{d{U}_m}{d{y}_m}\hskip0.35em =\hskip0.35em {\left(\frac{y_m}{y_i}\right)}^{\varepsilon } $$

The information necessary to compute $ {y}_m/{y}_i $ for different income groups is obtainable in many countries. In the U.S., it can be found in federal government statistics such as the U.S. Census Bureau’s Annual ASEC Survey.Footnote ¹³ In addition, computing utility weights for various income groups requires an estimate of $ \varepsilon $ . In Acland and Greenberg (Reference Acland and Greenberg2023), we conduct a meta-analysis of 158 studies, comprising 1,727 estimates of $ \varepsilon $ , that use a variety of methodologies. The conclusion from our meta-analysis is that the mean estimate of $ \varepsilon $ in the literature is 1.61 with a 95% confidence interval of 1.18 to 2.04. To give a sense of what these numbers mean, Table 1 shows the utility weights implied by our mean estimate and the upper and lower bounds of our confidence interval for several percentiles of the distribution of income in the U.S. in 2021. Because income is higher in larger households than in smaller households, and the average size of U.S. households is 2.5, the income amounts used to compute the utility weights are a weighted average of the incomes of two-person and three-person households. As shown in Table 1, the weights must equal 1 at the 50th percentile because at that percentile $ {y}_m\hskip0.35em =\hskip0.35em {y}_i $ and hence $ {y}_m/{y}_i\hskip0.35em =\hskip0.35em 1 $ .

Table 1. Utility weights at selected income percentiles for average household size.

This table contradicts two common practical arguments against utility-weighting. The first is that under plausible conditions, utility-weighting generates weights that seem intuitively implausible. Banzhaf (Reference Banzhaf2011) points out that if $ \varepsilon \hskip0.35em =\hskip0.35em 2 $ , the weight on a person with income of $10,000 is 100 times the weight on a person with income of $100,000. What Banzhaf does not mention is that an income of $10,000, in most parts of the U.S., puts an individual in conditions of such extreme and abject poverty that the welfare impact of even quite small sums of money may be very large. For someone at that level of poverty, an extra $100 a month might mean the difference between two meals a day and three, whereas for someone at $100,000, the same sum is more likely to mean one nice evening out each month. A difference in welfare impact of 100 to 1 might not be so unreasonable. Furthermore, at less extreme differences in income, the relative weights are less extreme. Considering the 25^th and 75^th percentiles ($44,250 and $128,000 respectively), even with $ \varepsilon \hskip0.35em =\hskip0.35em 2 $ , the weight on the lower-income household is only approximately 8 times the weight on the higher-income household, and at the 1.61 value of $ \varepsilon $ we estimate in our meta-analysis, it is approximately 5.5 times higher. The weight on the 10^th percentile is 32 times that on the 90^th percentile, but again, these are households with extreme differences in standard of living. Careful consideration of the lived experience of households at these levels of income might suggest that these numbers are not unwarranted.

The second practical argument against utility-weighting is that there is little hope of consensus on the appropriate set of weights (see Mishan, Reference Mishan1988, pp. 201, 210). In fact, as our meta-analysis shows, the extensive literature estimating the elasticity of marginal utility of income converges on a relatively narrow range, and as the literature on distributional weighting, without exception that we know of, uses an isoelastic utility function to derive weights, this suggests that there is, in fact, a strong basis for a consensus on the appropriate utility weights.

Some have argued that the selection of distributional weights involves subjective value judgments and thus should not be the responsibility of analysts (see Dasgupta, Marglin, and Sen, Reference Dasgupta, Sen and Marglin1972; Little and Mirrlees, Reference Little and Mirrlees1974; and Ray, Reference Ray1984). This is not the case when it comes to utility-weights (although, as discussed in the following section, it may be true in addressing equity issues). The relative value of increases in income at different income levels – i.e., the marginal utility of income – results from the preferences of individuals, and the observation of preferences is a technical matter. Furthermore, it is something that can be estimated using standard techniques, as we discussed above. As such, the determination of utility weights is the responsibility of economists and analysts, not decision makers. Alfred Marshall rendered his opinion on this in Reference Marshall1885, saying “taking account of the fact that the same sum of money measures a greater pleasure for the poor than for the rich [is a] task [that] most properly belongs to the economic organ” (Marshall Reference Marshall1885). This view is essential to our recommendations. The bias caused by diminishing marginal utility of income is a technical problem in the measurement of welfare and can be empirically addressed by observing behavior that reveals individual preferences. It does not require moral or ethical judgments. As such, it lies squarely in the domain of the analyst.

3.1. Equity-weighting

Once BCA has been converted into an unbiased measure of aggregate welfare by the application of utility weights, it remains the unweighted sum of the welfare impacts on different groups. In other words, it places the same weight on the welfare of individuals at different levels of income, and thus does not account for the possibility that decision makers may simply have greater concern for the welfare of individuals at the lower end of the income distribution. Utility-weighting merely overcomes a source of bias in the measurement of welfare. Adler (Reference Adler2013) and Nurmi and Ahtiainen (Reference Nurmi and Ahtiainen2018) both call for using weights to address what the former refers to as diminishing “marginal moral value of utility.” In other words, the higher a person’s existing welfare, the less society should “care” about increases in their welfare. We will refer to this kind of weighting as “equity-weighting.”

Adler’s terminology clearly implies that this kind of weighting involves moral value judgments. An issue of fundamental importance is who should make these kinds of judgments. This is where the biggest difference lies between utility-weighting and equity-weighting. As we suggest in section 2, decisions about what utility weights to use – being primarily a matter of observing individual preferences – should lie in the hands of economists and analysts. By contrast, decisions about how to assess tradeoffs between welfare and equity, which is to say decisions about how much concern society should have about the distribution of welfare across income groups, relative to how much concern society should have for aggregate welfare, belong in the hands of decision makers. Adler (Reference Adler2013) comes close to taking this position:

For us, however, the question is not who should choose the weights when doing equity-weighting. The question is whether equity-weighting per se helps or hinders decision makers’ ability to make their own value judgments about welfare/equity tradeoffs in the first place. Recall the relevant desideratum for multi-goal analysis, which provides information about the effect of a policy on each of society’s disparate goals. If presented separately, this information should allow decision makers to directly observe policy effects on each goal and apply their own values.

By combining information about welfare and information about distribution into a single metric, equity-weighting attempts to do precisely the opposite. The problem is quite fundamental: this metric provides no information about welfare and no information about distribution. The fact that a certain policy is assigned a certain number by an equity-weighted BCA tells us nothing about how much welfare it generates and nothing about how it affects equity. The same number could be achieved by a policy that generates a great deal of welfare for relatively wealthy people while imposing a significant cost on the poor or by a policy that closes the gap between the rich and poor while actually lowering aggregate welfare. For example, the first policy might generate a utility-weighted benefit to the wealthy of 100 and a cost to the poor of 25, a gain in aggregate welfare but an increase in the welfare gap. The second policy might generate a utility-weighted cost to the wealthy of 100 and a benefit to the poor of 75, a loss of aggregate welfare, but a decrease in the welfare gap. With an equity weight of 2 on the welfare of the poor, both policies generate an equity-weighted net benefit of 50. A decision maker learns literally nothing about the separate welfare and equity impacts of either policy from such equity-weighting. It is impossible for him or her to assess the tradeoffs, in just the same way that knowing the volumes of two boxes provides no information whatsoever with which to assess tradeoffs between depth, width and height. Applying equity weights is antithetical to the principles of multi-goal analysis.

Two claims could be made in response to this critique of equity weights. First, one could argue that if a policy imposes costs on the poor and has positive utility-weighted net benefits, and if the application of an appropriately chosen equity-weight changes the sign of net benefits from positive to negative, it is evidence that the equity impact of the policy outweighs the welfare impact. This, of course, assumes that the chosen equity weights align with the ethical or moral values of the decision maker reading the analysis, which brings us back to the goal of empowering decision makers to apply their own values. If the analyst chooses the equity weights, the decision maker is no longer empowered to apply their own values. To overcome this problem, and empower decision makers to apply their own values, the analyst could do a sensitivity analysis, providing equity-weighted net benefits for a range of different weights, thereby allowing decision makers to choose their preferred weight.

However, if the goal is to allow decision makers to determine whether distributional impacts outweigh welfare impacts, then there is arguably a superior way to sensitivity analysis to do so, which is to compute the “breakeven” equity-weight, that is the weight that causes net benefits to change sign. The advantage of breakeven equity weights, as discussed in more detail later, is that they present decision makers with a discreet, binary assessment: “is my preferred weight above or below this threshold?” This decision is arguably much easier to make than the continuous assessment: “which of these is closest to my preferred weight?” It may be quite hard for decision makers to translate their own value judgment into a specific preferred equity weight; but in many cases, it may be relatively easy for them to assess whether their value judgment translates into a weight above or below the breakeven threshold.

Furthermore, applying equity weights to a policy that generates welfare/equity tradeoffs to see if the sign of net benefits changes only works when comparing one policy to the status quo. It is tempting to hope that if the equity-weighted net benefit of one policy is greater than the equity-weighted net benefit of another, it must be the case that the first policy involves a more favorable tradeoff between distribution and welfare; but this is not the case. As stated earlier, the fact that one policy has a higher equity-weighted net benefit than another tells us nothing about either the welfare comparison or the distributional comparison between the two policies. What is called for when comparing two policies is information about the tradeoff between welfare and distribution implied by moving from one to the other, not information about how each policy compares to the status quo. And as shown in Appendix 1, it is not possible to apply equity weights (or even the breakeven weight) in a consistent and defensible manner when making this comparison.

The second claim that can be made in response to our critique of equity-weighting is that when equity weights are applied to utility-weighted net benefits, the resulting metric is a measure of a unitary moral value, of which welfare and distribution are merely instrumental parts. In other words, welfare and distributional impacts are not separately, and in and of themselves, relevant to the assessments of decision makers. The unitary moral value in question is referred to as “social welfare.”Footnote ¹⁵ In particular, the typical recommendation in the social-welfare measurement literature is that equity weights be derived from an iso-elastic function, parallel to the one we use in the determination of utility weights, thereby rendering equity-weighted BCA a way to implement a “continuous prioritarian” social welfare function. It gives “priority” to the welfare of the poor in a “continuous” way, which is to say that for any given increment in welfare, there is an incremental decrease in equity weight (Adler, Reference Adler2019). The resulting unitary value is sometimes referred to as the “overall wellbeing” of society or simply the “good” of society (Kagan Reference Kagan2018). Against those who consider the abandonment of separate measures of welfare and distribution in favor of this unitary metric of what is good for society, we have no defense. Their position is antithetical to our view that welfare and distribution are intrinsic values that have moral weight in and of themselves, and about which decision makers need separate information so that they can assess the acceptability of the welfare/equity tradeoffs created by a policy.

3.2. Metrics for assessing welfare and equity impacts

In this section, we propose a pair of metrics that can be used for quantitatively assessing the tradeoffs between the welfare and equity impacts of a policy relative to the status quo. (In Appendix 1 we demonstrate how the metrics can be applied to the comparison of two or more non-status-quo policies.) There are many strategies for quantification of distributional impacts. Farrow (Reference Farrow2011) suggests, for example, tables of disaggregated impacts by group, frequency plots, Lorenz curves, tables of quantiles, and a number of scalar measures such as variance, Gini coefficient, and the Atkinson index. While it is not our purpose to critique these strategies, we note that none of them provide information about the tradeoff between distributional impact and welfare impact. They simply tell us something about what the distributional impact is. We are not proposing any alternative metric for quantifying distributional impacts. Nor are we saying that quantitative measurement is superior, in any given instance, to qualitative analysis. Rather, our proposed metrics are intended to be used to quantitatively assess the acceptability of a given tradeoff between welfare and distributional impact. Furthermore, the second metric we propose can be used to assess tradeoffs between welfare and any univariate metric of distributional impact. We recommend using these metrics to account for distributional concerns, instead of using equity-weights.

4.1. Recommendations

As stated in the introduction, the following recommendations follow from the premise that policy decisions that implicate multiple social goals should be informed by multi-goal analysis.

1. 1. Benefit-cost analysis should be thought of as a measure of aggregate welfare, which requires that utility weights should be used so that BCA is not biased against the poor. Without this step, at a time when distributional impacts are of increasing concern in the policy-making domain, BCA is at increasing risk of being dismissed as an illegitimate form of policy analysis because it is based on principles that are widely acknowledged to be ethically and morally invalid.

2. 2. Being primarily a technical matter of measuring individual preferences, the decision of what numerical utility weights to apply should lie in the hands of economists and analysts, not decision makers. This upholds the principle that in matters concerning individuals’ pursuit of their own welfare, decision makers should defer to individuals’ preferences. The most practicable method for determining utility weights is to assume an isoelastic utility function, normalize the weight at median income to 1, and compute weights for other income levels using equation (2). The utility-weighted net benefit to individuals of any given income level should be interpreted as the aggregate WTP for the benefits and costs that those individuals would express if their incomes were at the median.

3. 3. Equity weights should not be applied to BCA. The responsibility for assessing the welfare and equity impacts of a policy should lie in the hands of decision makers. Equity-weighted BCA does not contain the information decision makers need to weigh these fundamentally different types of impact because it conflates welfare and equity into a single number that entirely occludes information about welfare and equity as separate policy goals. Decision makers should be presented with information about welfare and information about equity separately, so that they can directly assess the tradeoffs between the two.

4. 4. Our equity metrics should be computed and reported to decision makers as quantitative tools for assessing welfare/equity tradeoffs. When more than one non-status-quo policy is under consideration, the steps outlined in Appendix 1 should be used.

In addition, it is always appropriate to present utility-weighted costs and benefits disaggregated by group – however those groups may be defined – preferably in a table, with costs and benefits listed down the rows and groups listed across the columns. This allows for direct inspection of the distribution of welfare impacts and provides all of the information necessary to compute metrics of the sort we recommend. Kerry Krutilla (Reference Krutilla2005) has written an excellent explication of how this kind of table can inform decisions with distributional implications.

4.2. Steps

We offer a stepwise procedure for making use of utility-weighting and for assessing whether there is a tradeoff between welfare and distribution, in the case in which one policy is being compared to the status quo. One intent of the steps is to minimize the extent to which challenging and controversial concepts need to be introduced and explained. In essence, the idea is that for some policies, the fact that there is no tradeoff between welfare and equity will be apparent before utility-weighting. As a result, it is not necessary to apply utility weights for such policies. While utility weights may be necessary in the case of other policies, computation of the equity metrics may not be needed.Footnote ¹⁹

4.3. An illustration of using the steps

We now present an application of the five steps to a set of real-world BCAs. These are very simple in distributional terms. We address practical complications in the following subsection. We draw three examples from a review of 26 BCAs of welfare-to-work programs (Boardman et al., Reference Boardman, Greenberg, Vining and Weimer2018), which were evaluated by random assignment by MDRC, a New York-based evaluation firm. The programs provided information that was helpful for deciding whether they should continue to operate. Because these programs were all targeted at single-parent welfare recipients, distributional impacts are obviously of especial importance. The BCAs of these programs divide those affected into two groups: participants who received services under the programs, and non-participants, a category that includes everyone else in society. The key BCA components for participants were impacts on earnings, which were usually positive, and changes in government benefits, which were typically negative; the major components for non-participants were the taxes required to operate the program and reductions in the taxes needed to support the welfare system. Benefits and costs that were estimated for both participants and non-participants were divided by the number of participants, so that the numbers reported can be thought of as the net benefit generated for members of each group by a single typical participant going through the program, which allows for direct comparison of the impacts across groups.

Because the programs are small relative to the population, the incomes of non-participants should be close to the national median. For purposes of the illustration, the median incomes of participants are assumed to be at the 25th percentile, which is probably a reasonable approximation, given that participants entered the program when they were on welfare.Footnote ²¹ The findings from the meta-analysis describe earlier suggested setting the elasticity of the marginal utility of income, ε, at 1.61 in conducting utility-weighting, which implies a utility weight of 2.51 at the 25th percentile level and a weight of 1 at the 50th percentile level.

Our first example pertains to the GAIN program, which, beginning in 1987, operated in Riverside, California. The BCA of this program estimated that the unweighted net benefits of participants and non-participants in 2021 dollars were $2,951 and $5,718, respectively, per participant. Total unweighted net benefits were obviously positive at $8,669 per participant. As indicated by the second step, utility-weighting is unnecessary because both participants and society as a whole received positive net benefits, implying that the policy passes the BCA test without utility weights on an a fortiori basis. The unweighted net benefit number is enough to conclude that the program dominates the status quo with respect to both welfare and equity. Indeed, utility-weighting is often not required in practice. For example, in 16 of the reviewed 26 BCAs, including Riverside GAIN, there were either costs to the participants and the total unweighted net benefit was negative or there were benefits to the participants and the total unweighted net benefit was positive, meaning that in all 16 cases, utility-weighting was not necessary to determine that there was no tradeoff between welfare and equity.

The second example is for a so-called jobs first program, which began in 1991, and also operated in Riverside, but less successfully than Riverside GAIN. Participants had estimated unweighted net costs of $2,238 in 2021 dollars, although because of reductions in welfare payments, non-participants had estimated unweighted net gains of $2,605 per participant. Consequently, there were small total unweighted net benefits of $368. As indicated by the third step, utility-weighting is necessary for assessing the program because there are costs to participants, yet total net benefits are positive, which is to say, on the basis of unweighted BCA, it is not possible to determine whether there is a welfare/equity tradeoff. Not surprisingly, given the small size of the estimate of total net benefits, they become negative – at $852 – once they are utility weighted. Given the sign flip, the fourth step implies that the fifth step, explicit consideration of equity, is unnecessary. With utility-weighting, the status quo is superior with respect to both welfare and equity. There is no tradeoff.

Our final example is a welfare-to-work program in which equity considerations are quite relevant: a GAIN program that began in 1988 in Alameda, California. Like Riverside GAIN, participants were found to be better off by the BCA, with unweighted net benefits estimated at $1,698 in 2021 dollars, but unlike Riverside GAIN, the BCA also found that there were unweighted net costs to non-participants of $5,788 per participant. Consequently, the unweighted loss to society was estimated to be $4,090, a reduction in potential Pareto efficiency. In this case, as per step 3, and as in the second example above, utility-weighting is necessary in order to determine whether there is a welfare/equity tradeoff. After utility-weighting, net benefits for participants increase to $2,734, net costs for non-participants remain at $5,788 per participant, and total net costs shrink to $3,055. Because the sign of net benefit does not change, we can conclude that there is a welfare/equity tradeoff.

In order to assess the acceptability of the tradeoff, it is necessary to move on to the fifth step and compute the equity metrics we propose in section 4. The distributional impact (the reduction in the welfare gap between the two groups) is the absolute sum of the utility-weighted benefit to participants and the utility-weighted cost to non-participants, which is $8,522. The welfare loss per dollar of reduction in welfare gap (what could be called the price of inequality reduction) is $.36. The poverty impact (the increase in the welfare of participants) is $2,734, and the welfare loss per dollar of poverty impact (what could be called the price of poverty reduction) is $1.1. The breakeven equity weight is 2.12 ($5,788/$2,734). Although the welfare cost per dollar of distributional impact is small, a decision maker might worry that many non-participants have incomes that are not exceptionally high. The welfare cost per dollar of poverty impact is rather high, but might still be considered tolerable, particularly given the fairly low breakeven weight, which implies that in order to prefer the program to the status quo, the decision maker would need to feel a bit over twice the concern for participants than for non-participants in order for the distributional benefits of the program to outweigh the welfare loss. This example demonstrates what we mean when we say that our metrics allow for quantitative assessment of welfare/equity tradeoffs. They put the comparison of welfare impact and equity impacts into intuitively meaningful quantitative terms with which decision makers can draw their own conclusions about the pros and cons of the programs.

As part of a multigoal analysis, the decision maker would have other concerns besides the distribution of costs and benefits among income groups. For example, in the Alameda GAIN program, operating costs were estimated to be $10,893 per participant in 2021 dollars, which is large for welfare-to-work programs. The decision maker would presumably multiply that figure by the expected number of future participants to determine if there is sufficient space in the government’s budget. As this discussion suggests, while utility-weighting is helpful, unweighted estimates of disaggregated benefit and costs are also needed for decision-making.

4.4. Practical concerns with the application of utility-weights

The welfare-to-work BCAs described in the previous subsection presented rather non-challenging illustrations of how the steps listed in section 4.2 could be implemented. Many BCAs will be more challenging. In this subsection, we briefly mention some of these challenges. We do not attempt to detail how to meet these challenges, as doing so is beyond the scope of this paper and is a topic of our ongoing research.

One advantage that the welfare-to-work BCAs provided is that separate BCAs were conducted for the low-income program participants and the relatively higher income non-participants, a framework that was developed by Kemper, Long, and Thornton (Reference Kemper, Long and Thornton1981) and has subsequently been used in most BCAs of government programs that invest in the low-income population. As a result, it was immediately clear whether the studied programs made one group better off and the other group worse off, and consequently whether utility-weighting was necessary. While many other BCAs involve programs and policies that affect both lower and higher income households (e.g., those of health, safety, and environmental regulations), information is often not readily available on where in the income distribution those affected fall. Hammitt, Robinson and Zeckhauser (Reference Robinson, Hammitt and Zeckhauser2016), for example, point out that while the data often exist to determine how the benefits of health, safety, and environmental regulations are distributed by income, it is extremely difficult to determine the distribution of costs because the affected firms may raise prices, reduce wages, or simply accept reduced profits, and such changes may affect different groups differently. Once distribution is determined, two or more income groups must be defined. Then, the estimated benefits and costs must be allocated among these groups. It may be especially difficult to determine how consumer surplus is distributed among groups when it is estimated through aggregate demand curves. Additionally, whereas financial transfers – including tax-and-transfer programs, transfers from employers to employees in the form of wages, and transfers from consumers to firms in the form of prices – can be ignored in unweighted BCA, they must be identified and accounted for before applying utility weights. Doing all this is an obvious precursor to implementing steps 2 through 5. We recognize that these steps involve non-trivial practical challenges. We address a few here, but have chosen to leave a more fulsome consideration of practicalities to future research.

The measure of income that is used in distributional analysis should ideally be the income available for consumption. Consequently, it should be net of taxes and should include government transfers, non-monetary benefits, and possibly even borrowing and saving, and should possibly take account of the fact that the cost of living varies across geographic regions. In practice, estimating the ideal measure of income may prove infeasible, and an approximation may be necessary. The question of whether to use per-capita or household income also arises. One approach is to make “equivalizing” assumptions about the distribution of consumption across household members, taking account of things like economies of scale in household production. An example of how this can be done in practice can be found in the U.K. Greenbook (Her Majesty’s Treasury, Reference Treasury2018).

Another important consideration that will need to be addressed in some BCAs is how to handle valuations that are, in practice, typically based on averages across income levels, such as the value of statistical life, quality-adjusted life years, the value of time, and those estimated with contingent valuation techniques. There are two possible approaches to treating benefits that are averaged across income groups. The first is use the averages and not weigh them. Banzhaf (Reference Banzhaf2011) has argued, for instance, that using averages across income levels represents an approximation to utility-weighting because it implicitly assumes that individuals with different incomes would express the same valuation if they had the same income. The second approach is to first assign individuals the amount they would be willing to pay for the benefit, which is presumed to usually rise as income increases, and then utility weight the resulting values.Footnote ²² Our own preliminary efforts concerning the value of statistical life imply that these two approaches produce very different results. For example, there is a considerable literature on the income elasticity of the value of a statistical life. (Hammit and Robinson, Reference Hammitt and Robinson2011) If the elasticity of VSL is significantly different from the elasticity of marginal utility of income, using average VSL will generate significantly different results than first computing VSL at different income levels using the income elasticity of the VSL and then applying utility weights using the elasticity of marginal utility of income. Contingent valuation studies that account for variation by income provide another example with the same practical considerations.

Finally, although our welfare/equity ratios do not strictly require that society be categorized into only two groups (for example, the denominator can the Gini or Atkinson coefficients), the ratio of welfare to what we are calling “welfare gap,” as well as the breakeven equity-weight, do. In many cases, such as the kinds of programs in our worked example above, the bifurcation will be relatively obvious. In others it may be necessary to use a relatively crude bifurcation, perhaps something like above and below median income. We believe that this kind of bifurcation, while not ideal, will nonetheless be informative in many cases. And, as with many other aspects our recommendations, the effect on the metrics of the definition of groups can be explored in sensitivity analysis.

Partially because of the issues just discussed and partially because of lack of agreement over the appropriate weights, both utility and equity weighting are rare in actual benefit-cost analyses (Adler, Reference Adler2013). One apparent exception is in global studies of climate change where the disparity of incomes among countries is enormous (Nurmi and Ahtiainen, Reference Nurmi and Ahtiainen2018). In addition, distributional weighting has occasionally been conducted in BCAs of programs specifically targeted at the low-income population (for example, Greenberg, Reference Greenberg2013 and Gubits et al., Reference Gubits2018). Even determining how the benefits and costs of regulation policies are distributed among different income groups appears infrequent. For example, Robinson, Hammitt, and Zeckhauser (Reference Robinson, Hammitt and Zeckhauser2016) examined 24 policies that estimated the benefits and costs of federal health regulations and “found that these analyses pay relatively little attention to distribution” (p. 323). They also mention that several recent reviews do not find studies that determine the distribution of “both the costs and benefits of more conventional energy and environmental regulations” (p. 309). Without such information, weighting is obviously infeasible without considerable assumptions.