Use of distributional weights in cost–benefit analysis revisited

ABSTRACT

It is nearly 40 years since an earlier survey on the use of distributional weights in Cost-Benefitcost–benefit analysis was first published. Since then, a number of significant contributions have been made in the literature. It is therefore timely to revisit the issue by focusing on the key reasons why distributional weights are so necessary. We present a new justification for using unequal income distributional weights in CBA, and and reaffirm a completely neglected reason for adopting distribution weights. We derive the weights from a formulation of the SWF. We specify the minimum value judgmentsjudgements for the SWF that everyone can agree on who accepts the fundamentals of welfare economics. As many CBA practitioners have ignored using distribution weights, inadequate attention has been given to how they can be estimated. The final objective of this paper is to highlight the main methods for estimating these weights, and and to present applications of these methods.

KEYWORDS:

• KEY-WORDS: Distribution weights
• equity
• cost-benefit analysis
• SWF

JEL CLASSIFICATION:

• D61
• D63

I. Introduction

It is now nearly 40 years since the paper : ‘Use of Distributional Weights in Cost-Benefit- Analysis: A Survey of Schools’ was first published, Brent (1984) – hereafter referred to as the earlier survey. This paper went back to first principles of what distributional weights are trying to achieve. It attempted to dispel misconceptions concerning distributional weights in the literature arising from the views of Harberger (1978). Given that a number of significant contributions have been made to the literature since that earlier survey, it is timely to revisit the distribution weights issue by focusing on some of the key reasons why unequal distributional weights are so necessary in CBA.¹

Recently, there have been some contributions, by Adler (2016) and Hammitt (2021), that reaffirmed some of the need to use distributional weights in Benefit-Cost Analysis (CBA). These articles will be referred to a number of times in this paper. Nonetheless, there still exists a widespread reluctance to use income distributional weights in CBA practicepractices in the US. This reluctance is not in keeping with the basic value judgments that lie behind CBA. Distribution weights depend on the social welfare function (SWF) from which they are derived. Although there are a number of varying underlying conceptual bases for the SWF – and Adler and Hammitt give summaries – it is in fact possible to specify the minimum value judgments that everyone can agree on who accepts the fundamentals of welfare economics that underlies public policy in the US.² With these core value judgments, and there will be just three of them, it is possible to produce firm guidelines as to how to put precise estimates on the distribution weights, once one is convinced that including unequal distribution weights is essential in CBA.

It is most important to understand that the SWF that we will be formulating has both efficiency and equity as the two key ingredients.³ Using equal distribution weights implies that only efficiency is of social value. This cannot possibly be the case, when it is acknowledged that individuals have strong preferences for equity, as we shall demonstrate. However, a two-objective SWF, containing both efficiency and equity, cannot be thought to exist if the distributional weights applied give so much importance to equity that efficiency considerations play little, or no, part in the determination of CBA decisions. We will show that the distribution weights found in the CBA literature do produce a reasonable balance between efficiency and equity.

The main objectives of this paper are to present a new justification for using unequal income distributional weights in CBA, and reaffirm a very much neglected, fundamental reason for adopting distribution weights that was included in the earlier survey. Since most practitioners have avoided using distribution weights in their CBAs, inadequate attention has been given to how they can be estimated in practice. The final objective of this paper is to explain in detail how the weights can, and have been, estimated. The new justification is derived from a generalization of Hammitt’s analysis of how paradoxical CBA results can occur when distribution weights are not included in the evaluation. The neglected justification is that the individual utility functions that form the SWF have utility interdependencies in cash, and/or in-kind in them, and this requires unequal weights separately from any other reason. Although Harberger (1984), in a separate paper on basic needs, did recognize the existence of an externality whereby some individuals benefit from the improvement in the circumstances of others, he regarded accommodating this externality as something completely different from applying distribution weights to allow for distributional concerns in CBA.⁴

The outline of the paper is as follows.follows: In section II, we present the new justification for unequal distribution weights. Hammitt’s analysis comes up with a paradox whereby a policy passes a CBA test even though the overall policy output is lower, and the poor receive no benefits at all. This paradox can disappear by recognizing equity as a separate CBA objective, that is, by including unequal distribution weights into CBA.⁵

Section III uncovers the neglected justification for adopting unequal distribution weights, that is provided by recognizing that individual utility functions are in theory and practice interdependent, with the rich’s utility function dependent on the poor’s utility function. Since distribution weights depend on the form of the SWF from which they are derived, and any SWF must depend on the particular value judgmentsjudgements on which they are constructed, we present the minimum necessary value judgements. We show that the utilitarian SWF is the appropriate cornerstone for CBA. Using this SWF, we specify the two components that define the distribution weights for the poor, and and the one component for the rich.

Section IV is devoted to showing how the distribution weights can, and have been, estimated. From the earlier survey, two methodologies for estimating the weights were outlined. More recently, a third methodology has been found in the literature. Section IV outlines these three methodologies and shows how they were applied to estimate weights in practice. Applications relate to both sides of government finances (taxation and expenditures) and to individuals, showing evidence of interdependent utility functions. Practitioners are here given many methods for estimating the weights to choose from in order to select those that they would be comfortable applying.

Since this paper is primarily concerned with supplying two valid reasons for applying distribution weights in CBA, it is necessary in section V also to identify the two main invalid reasons that practitioners often refer to in the CBA literature when trying to justify the non-use of distribution weights. The final section gives the summary and main conclusions. The emphasis is always on the fact that in order that CBA be socially relevant, CBA requires including two objectives and not just one.

II. The new justification for unequal distribution weights

As the basis of a new justification for unequal distribution weights, we will adopt many of the simplifying assumptions that Hammitt used to analyze the role of the choice of numeraire in CBA. The difference is that we will add to them more general assumptions that are involved with using unequal distribution weights.⁶

Hammitt’s assumptions

In Hammitt’s analysis, he mainly focused on life years (LYs), defined by h, as the public expenditure outcome. With income as the numeraire, h has to be expressed in monetary terms using shadow prices given by v. There are two, equal-sized types of individuals, 1 and 2. Type 2 individuals are richer than type 1 individuals. There are two alternative policies A and B. For policy A, only type 1 individuals (the poor) get the outcome, and for policy B only type 2 individuals (the rich) get the outcome. The benefits are v₁ h₁ for policy A, and v₂ h₂ for policy B. It is assumed that policy A produces higher outcomes than policy B: h₁ > h₂. Costs are c per person and are the same for either individual whether benefits are received or not. So, the costs are 2c in total for both policy A and policy B.⁷

With benefits for policy A equal to v₁ h₁ and the costs equal to 2c, the net benefits (NB) are:

(1) $\mathrm{NBA}={\mathit{v}}_1{\mathit{h}}_1-2\mathit{c}$(1)

With benefits for policy B equal to v₂ h₂ and the costs equal to 2c, the net benefits are:

(2) $\mathrm{NBB}=v_2{h}_2-2\mathit{c}$(2)

Although the h outcomes for policy A are greater than that for policy B, and the costs are the same for either policy, policy B could have greater calculated net-benefits than A. This would result if the shadow prices for type 2 individuals are proportionately greater than are the ratio of the outcomes. That is:

(3) $\mathrm{NBB}>\mathrm{NBA},\mathrm{if}{\mathit{v}}_2{\mathit{h}}_2>{{\mathit{v}}_1}_{\mathit{\hspace{0.17em}}}{\mathit{h}}_1,\mathrm{or}:{\mathit{v}}_2/{\mathit{v}}_1>{\mathit{h}}_1/{\mathit{h}}_2$(3)

Hammitt assumes that condition (3) holds. This means that Hammitt’s result was basically that policy B could be approved by a CBA even though it has lower health outcomes and the poor receive no output at all.⁸

Before providing further analysies, let us focus on exactly why condition (3) produces this ‘wrong’ result. This occurred only because type 2 individuals had a relatively high shadow price v₂ for h₂. The shadow prices v₁ and v₂ depend on the two types of individuals’ willingness to pay for h, which does depend, crucially, on an individual’s ability to pay. This means that the main reason why v₂ is so much greater than v₁ could be because of type 2 individuals’ ability to pay, that is, type 2 individuals are so much richer than type 1 individuals. This equity difference plays no part in the efficiency CBA, which is Hammitt’s context. So, let us incorporate distributional weights into his analysis.

Hammitt’s assumptions with distribution weights

For this section, we will go along with Hammitt’s assumptions for benefits and costs, and simply apply distribution weights a to benefits. We will continue to assume h₁ > h₂. We now assume that, because type 1 individuals are poorer than type 2 individuals, a₁ > a₂ . For policy A, the social net-benefits (SNB) are:

(4) $\mathrm{SNBA}={\mathit{a}}_1{\left({\mathit{v}}_1{\mathit{h}}_1\right)}_{\mathit{\hspace{0.17em}}}-2\mathit{c}$(4)

and for policy B, the social net-benefits are:

(5) $\mathrm{SNBB}={\mathit{a}}_2{\left({\mathit{v}}_2{\mathit{h}}_2\right)}_{\mathit{\hspace{0.17em}}}-2\mathit{c}$(5)

Note that, for costs, the assumption we are adding is that they will be paid for by the general taxpayer, and for this group the distribution weight is set equal to unity.

The new criterion for policy B to be preferred to policy A is for criterion (6) to be satisfied.

(6) $\mathrm{SNBB}>\mathrm{SNBA},\mathit{if}{a}_2\left(v_2{h}_2\right)>a_1\left({v_1}_{\mathit{\hspace{0.17em}}}{h}_1\right),\mathit{or}:a_2\left(v_2/v_1\right)>a_1\left(h_1/h_2\right)$(6)

In this more general criterion, the paradox could still hold. However, the distribution weights a₁ > a₂ could counter any shadow price imbalance arising from v₂ > v₁ and now make policy A the preferred choice, even with Hammitt’s assumptions contained in criterion (3). To reverse condition (6), and make policy A the preferred choice, it is necessary that the relative weighted benefits be greater than the shadow price imbalance such that:

(7) $\mathrm{SNBA}>\mathrm{SNBB},\mathrm{if}{a}_1\left({v_1}_{\mathit{\hspace{0.17em}}}{h}_1\right)>{\mathrm{a}}_2\left({\mathrm{v}}_2{\mathrm{h}}_2\right),\mathrm{or}:{\mathrm{a}}_1/{\mathrm{a}}_2>{\left({\mathrm{h}}_1/{\mathrm{h}}_2\right)}_{\mathit{\hspace{0.17em}}}\left({\mathrm{v}}_2/{\mathrm{v}}_1\right)$(7)

The weights a₁/a₂ in Equation (7)(7) $\mathrm{SNBA}>\mathrm{SNBB},\mathrm{if}{a}_1\left({v_1}_{\mathit{\hspace{0.17em}}}{h}_1\right)>{\mathrm{a}}_2\left({\mathrm{v}}_2{\mathrm{h}}_2\right),\mathrm{or}:{\mathrm{a}}_1/{\mathrm{a}}_2>{\left({\mathrm{h}}_1/{\mathrm{h}}_2\right)}_{\mathit{\hspace{0.17em}}}\left({\mathrm{v}}_2/{\mathrm{v}}_1\right)$(7) are the threshold values that need to be adopted to ensure that policy A would be approved.

However, if it is assumed that, as is standard CBA practice, that equal weights are applied, a₁ = a₂ = a, let us confirm that the ‘wrong’ result would still be obtained. Criterion (6) with equal distribution weights would become:

(8) $\mathrm{SNBB}>\mathrm{SNBA},\mathrm{if}\mathit{a}\left({\mathit{v}}_2{\mathit{h}}_2\right)>\mathit{a}\left({{\mathit{v}}_1}_{\mathit{\hspace{0.17em}}}{\mathit{h}}_1\right),\mathrm{or}:{\mathit{v}}_2/{\mathit{v}}_1>{\mathit{h}}_1/{\mathit{h}}_2$(8)

Thus, criterion (6) ends up the same as criterion (3). We can see from criterion (8), if we continue to adopt Hammitt’s assumptions for criterion (3), and adopt equal weights, policy B would still be preferred, even though we know it produces lower health outputs and the poor would receive nothing

From this generalization of Hammitt’s analysis, we can suggest a new argument for adopting unequal distribution weights in CBA. The role of distribution weights in CBA is to ensure that ability to pay does not always dominate the evaluation, making irrelevant both output (h₁ > h₂) and equity (the rich gain and the poor get nothing). With equal weights, both output and equity could suffer from approving public expenditure decisions mainly on the basis of ability to pay.⁹

III. The neglected reason for unequal distribution weights

The distribution weights for use in CBA are obtained from the specific Social Welfare Function from which they are derived. Any specification of W requires making value judgments. Thus, Hammitt is correct to point out that the use of distribution weights is normative. However, it is always the case that one cannot make good policy decisions without specifying what ‘good’ involves. Thus, Adler points out that using equal weights is just as much a value judgement as using unequal weights. The W we are going to specify will depend on the minimum value judgments that everyone can agree on, who accepts the fundamentals of welfare economics that underlies public policy in the US.

The minimum value judgements

The first value judgmentjudgement is that, to make society better off, one needs to make individuals better off, that is, increase their utility u:¹⁰

(9) $\mathrm{W}=\mathrm{W}\left({\mathit{u}}_1,{\mathit{u}}_1\right)$(9)

Taking the total derivative of the individualistic W, to recognize that public policies change W, we obtain:

(10) $\mathit{dW}=\frac{\mathrm{\partial }\mathbf{W}}{\mathrm{\partial }\mathbf{u}1}\mathit{d}{u}^1+\frac{\mathrm{\partial }\mathbf{W}}{\mathrm{\partial }\mathrm{u}2}\mathit{d}{u}_2$(10)

The second value judgment relies on the U.S.A constitution that states that all people are created equal. Fleurbaey and Abi-Rafeh (2016) op cit. refer to this second value judgment for the SWF in terms of satisfying the requirement of impartiality. This implies:

(11) $\frac{\mathrm{\partial }\mathbf{W}}{\mathrm{\partial }\mathrm{u}1}=\frac{\mathrm{\partial }\mathbf{W}}{\mathrm{\partial }{\mathrm{u}}_2}=1$(11)

From Equation (10)(10) $\mathit{dW}=\frac{\mathrm{\partial }\mathbf{W}}{\mathrm{\partial }\mathbf{u}1}\mathit{d}{u}^1+\frac{\mathrm{\partial }\mathbf{W}}{\mathrm{\partial }\mathrm{u}2}\mathit{d}{u}_2$(10) and (11(11) $\frac{\mathrm{\partial }\mathbf{W}}{\mathrm{\partial }\mathrm{u}1}=\frac{\mathrm{\partial }\mathbf{W}}{\mathrm{\partial }{\mathrm{u}}_2}=1$(11) ) we arrive at the utilitarian W:

(12) $\mathrm{dW}=\mathit{d}{\mathit{u}}_1+\mathit{d}{\mathit{u}}_2$(12)

The third value judgmentjudgement is that one honor consumer sovereignty, i.e. that the individual is the best judge of his or her own welfare. What makes individuals better off is what they decide is best for themselves and others. In economics practice, individuals are deemed to be better off according to what goods and services they, and others, receive or get to consume, either through the market or through the government.

In particular, with income y as the numeraire, which is the most commonly used numeraire in economics, which is broader than just healthcare h, we observe that the poor individuals’ utility depends on what they consume; and the rich individuals’ utility depends on what they consume, and what the poor consume:

(13) $u_1=u_1\left(y_1\right)\mathrm{and}{u}_2=u_2\left(y_2,y_1\right)$(13)

In this formulation of utility interdependence, individual 1’s utility does not depend on the income of individual 2. Thus, envy by the poor of the rich is ruled out.

Equation (12)(12) $\mathrm{dW}=\mathit{d}{\mathit{u}}_1+\mathit{d}{\mathit{u}}_2$(12) therefore becomes:

(14) $\mathit{dW}=\frac{\mathrm{\partial }{\mathrm{u}}^1}{\mathrm{\partial }{\mathrm{y}}^1}\mathit{d}{y}_1+\frac{\mathrm{\partial }{\mathrm{u}}^2}{\mathrm{\partial }{\mathrm{y}}^2}\mathit{d}{y}_2+\frac{\mathrm{\partial }{\mathrm{u}}^2}{\mathrm{\partial }{\mathrm{y}}^1}\mathit{d}{y}_1=\left(\frac{\mathrm{\partial }{\mathrm{u}}^1}{\mathrm{\partial }{\mathrm{y}}^1}+\frac{\mathrm{\partial }{\mathrm{u}}^2}{\mathrm{\partial }{\mathrm{y}}^1}\right)\mathit{d}{y}_1+\frac{\mathrm{\partial }{\mathrm{u}}^2}{\mathrm{\partial }{\mathrm{y}}^2}\mathit{d}{y}_2$(14)

We can use Equation (14)(14) $\mathit{dW}=\frac{\mathrm{\partial }{\mathrm{u}}^1}{\mathrm{\partial }{\mathrm{y}}^1}\mathit{d}{y}_1+\frac{\mathrm{\partial }{\mathrm{u}}^2}{\mathrm{\partial }{\mathrm{y}}^2}\mathit{d}{y}_2+\frac{\mathrm{\partial }{\mathrm{u}}^2}{\mathrm{\partial }{\mathrm{y}}^1}\mathit{d}{y}_1=\left(\frac{\mathrm{\partial }{\mathrm{u}}^1}{\mathrm{\partial }{\mathrm{y}}^1}+\frac{\mathrm{\partial }{\mathrm{u}}^2}{\mathrm{\partial }{\mathrm{y}}^1}\right)\mathit{d}{y}_1+\frac{\mathrm{\partial }{\mathrm{u}}^2}{\mathrm{\partial }{\mathrm{y}}^2}\mathit{d}{y}_2$(14) to define the income distributional weights a₁ and a₂ as:

(15) ${\mathrm{a}}_1=\frac{\mathrm{\partial }{\mathrm{u}}^1}{\mathrm{\partial }{\mathrm{y}}^1}+\frac{\mathrm{\partial }{\mathrm{u}}^2}{\mathrm{\partial }{\mathrm{y}}^2}\mathrm{and}{\mathrm{a}}_2=\frac{\mathrm{\partial }{\mathrm{u}}^2}{\mathrm{\partial }{\mathrm{y}}^2}\mathit{d}{y}_2$(15)

Equation (14)(14) $\mathit{dW}=\frac{\mathrm{\partial }{\mathrm{u}}^1}{\mathrm{\partial }{\mathrm{y}}^1}\mathit{d}{y}_1+\frac{\mathrm{\partial }{\mathrm{u}}^2}{\mathrm{\partial }{\mathrm{y}}^2}\mathit{d}{y}_2+\frac{\mathrm{\partial }{\mathrm{u}}^2}{\mathrm{\partial }{\mathrm{y}}^1}\mathit{d}{y}_1=\left(\frac{\mathrm{\partial }{\mathrm{u}}^1}{\mathrm{\partial }{\mathrm{y}}^1}+\frac{\mathrm{\partial }{\mathrm{u}}^2}{\mathrm{\partial }{\mathrm{y}}^1}\right)\mathit{d}{y}_1+\frac{\mathrm{\partial }{\mathrm{u}}^2}{\mathrm{\partial }{\mathrm{y}}^2}\mathit{d}{y}_2$(14) can then be rewritten as:

(16) $\mathrm{dW}={\mathit{a}}_1\mathit{d}{\mathit{y}}_1+{\mathit{a}}_2\mathit{d}{\mathit{y}}_2$(16)

To measure the income effects, we note that the income changes are the difference between benefits B and costs C, dy = B – C, and so, as a CBA criterion, Equation (16)(16) $\mathrm{dW}={\mathit{a}}_1\mathit{d}{\mathit{y}}_1+{\mathit{a}}_2\mathit{d}{\mathit{y}}_2$(16) becomes:

(17) $\mathrm{dW}={\mathit{a}}_1\left({\mathit{B}}_1-{\mathit{C}}_1\right)+{\mathit{a}}_2\left({\mathit{B}}_2-{\mathit{C}}_2\right)$(17)

When individual 2 pays the costs of a public policy, and individual 1 receives the benefits, Equation (17)(17) $\mathrm{dW}={\mathit{a}}_1\left({\mathit{B}}_1-{\mathit{C}}_1\right)+{\mathit{a}}_2\left({\mathit{B}}_2-{\mathit{C}}_2\right)$(17) simplifies the CBA criterion to be:

(18) $\mathrm{dW}={\mathit{a}}_1{\mathit{B}}_1-{\mathit{a}}_2{\mathit{C}}_2$(18)

From the definitions of the distribution weights in Equation (15)(15) ${\mathrm{a}}_1=\frac{\mathrm{\partial }{\mathrm{u}}^1}{\mathrm{\partial }{\mathrm{y}}^1}+\frac{\mathrm{\partial }{\mathrm{u}}^2}{\mathrm{\partial }{\mathrm{y}}^2}\mathrm{and}{\mathrm{a}}_2=\frac{\mathrm{\partial }{\mathrm{u}}^2}{\mathrm{\partial }{\mathrm{y}}^2}\mathit{d}{y}_2$(15) , we see that there are two separate reasons why equal income distributional weights in CBA are not optimal. As is well recognized in the literature, and mentioned explicitly in Hammitt’s analysis, there is diminishing marginal utility of income, which means that a₁ > a₂ because $\frac{\mathrm{\partial }{\mathrm{u}}^1}{\mathrm{\partial }{\mathrm{y}}^1}>\frac{\mathrm{\partial }{\mathrm{u}}^2}{\mathrm{\partial }{\mathrm{y}}^2}$. However, there is a second reason why a₁ > a₂, which is derived from the second term in a₁ in Equation (15)(15) ${\mathrm{a}}_1=\frac{\mathrm{\partial }{\mathrm{u}}^1}{\mathrm{\partial }{\mathrm{y}}^1}+\frac{\mathrm{\partial }{\mathrm{u}}^2}{\mathrm{\partial }{\mathrm{y}}^2}\mathrm{and}{\mathrm{a}}_2=\frac{\mathrm{\partial }{\mathrm{u}}^2}{\mathrm{\partial }{\mathrm{y}}^2}\mathit{d}{y}_2$(15) , given by the interdependence of utility functions $\frac{\mathrm{\partial }{u}^2}{\mathrm{\partial }{y}^1}$. This is the neglected reason for why one should use unequal distribution weights. Thus, even if one assumes, $\frac{\mathrm{\partial }{\mathrm{u}}^1}{\mathrm{\partial }{\mathrm{y}}^1}=\frac{\mathrm{\partial }{\mathrm{u}}^2}{\mathrm{\partial }{\mathrm{y}}^2}$, with $\frac{\mathrm{\partial }{u}^2}{\mathrm{\partial }{y}^1}>0$, if one accepts consumer sovereignty, we must have a₁ > a₂.¹¹

The analysis so far has been based on utility interdependence that relies on the income of the poor individuals entering the utility function of the rich individuals. As an alternative, as emphasized in the earlier survey, the rich may care more about helping the poor in-kind rather than in cash. Cash assistance by the poor could be spent on items (like alcohol) that the rich would prefer be spent in other ways, such as on health. If we now define health h as a subset of y, Equation (13)(13) $u_1=u_1\left(y_1\right)\mathrm{and}{u}_2=u_2\left(y_2,y_1\right)$(13) can be rewritten as:

(19) $u_1=u_1\left(y_1\right)\mathrm{and}{u}_2=u_2\left(y_2,h_1\right)$(19)

The only difference to the definitions of the weights is for individual 1:

(20) ${\mathrm{a}}_1=\frac{\mathrm{\partial }{\mathrm{u}}^1}{\mathrm{\partial }{\mathrm{y}}^1}+\frac{\mathrm{\partial }{\mathrm{u}}^2}{\mathrm{\partial }{\mathrm{y}}^1}$(20)

The previous two arguments still hold, requiring unequal weights in the CBA criterion Equation (18)(18) $\mathrm{dW}={\mathit{a}}_1{\mathit{B}}_1-{\mathit{a}}_2{\mathit{C}}_2$(18) , but in this case the consumer sovereignty condition in terms of inter-dependent utility functions is now $\frac{\mathrm{\partial }{u}^2}{\mathrm{\partial }{h}^1}>0$ instead of $\frac{\mathrm{\partial }{\mathrm{u}}^2}{\mathrm{\partial }{\mathrm{y}}^1}>0$.

IV. Estimating the distribution weights

After specifying the essential value judgments for determining the distribution weights, the final requirement for adopting unequal distribution weights is to be aware of how the weights can be estimated. As examined in detail in the earlier survey, there were two main methodologies for estimating the distribution weights. One method was to use ‘a priori reasoning’, and the other was to use the revealed preference approach. More recently, a third main method has been employed that involves a direct measurement of utility. We cover each method in turn, referring to applications in the literature, and these will be covered in the context of both reasons why unequal weights can be justified.

Estimates of weights based on η

‘A priori’ weights

According to this view, the weights based on η should be specified in advance by the social decision-maker based on deductive reasoning and then given to the CBA evaluator. One of the earliest proponents of this approach was by Squire and van der Tak (1975) at the World Bank. The objective was to help determine the rates of diminishing marginal utility of income that are used in the definitions of the distribution weights in Equation (15)(15) ${\mathrm{a}}_1=\frac{\mathrm{\partial }{\mathrm{u}}^1}{\mathrm{\partial }{\mathrm{y}}^1}+\frac{\mathrm{\partial }{\mathrm{u}}^2}{\mathrm{\partial }{\mathrm{y}}^2}\mathrm{and}{\mathrm{a}}_2=\frac{\mathrm{\partial }{\mathrm{u}}^2}{\mathrm{\partial }{\mathrm{y}}^2}\mathit{d}{y}_2$(15) . If type 1 and 2 individuals have the same utility of income function, and it declines at a constant rate, then the weights can be given as:¹²

(21) $a_i=(\mathit{\hspace{0.17em}}\bar{\mathit{y}}/\mathit{yi}{)}^{\mathit{\eta }}$(21)

where $\bar{\mathit{y}}$ is average income and η is the income inequality aversion parameter, which represents the constant rate of decline for the marginal utility of income. A value η = 0 signifies equal income distributional weights. Although a value of 0 for η is ruled out by the principle that there are two social objectives and not just one, efficiency, to be operational it is necessary to set out a core value for η and then specify the lower and upper boundaries.

The core value for η

Squire and van der Tak recommended using η = 1 as the core value.¹³ This implies that Equation (21)(21) $a_i=(\mathit{\hspace{0.17em}}\bar{\mathit{y}}/\mathit{yi}{)}^{\mathit{\eta }}$(21) reduces to: = 

(22) $a_i=\mathit{\hspace{0.17em}}\bar{\mathit{y}}/\mathit{yi}$(22)

This means that the weights are inversely related to average income. For a poor individual who has an income one-tenth of the average then a₁ = 10; and for a rich individual who has an income ten times the average then a₂ = 0.1. One country that did initially accept this core value η = 1 was the UK government (in HM Treasury 2003).¹⁴

A recent endorsement for the core value of η = 1 was by Nurmi and Ahtiainen (2018). They use a variant of our second value that everyone should be treated equally. In their version, they effectively state that this unity value is a democratically sound weighting system as it gives the same weight to each person’s income when normalized by average income.

An obvious practical consideration when using Equation (22)(22) $a_i=\mathit{\hspace{0.17em}}\bar{\mathit{y}}/\mathit{yi}$(22) is what happens if an individual is so poor that they have zero income. In this case, a₁ would be infinite, and CBA would become infeasible. A public expenditure giving a poor individual even a small benefit would justify the sacrifice of a country’s entire national income.

To fully appreciate the practical implications of adopting Equation (22)(22) $a_i=\mathit{\hspace{0.17em}}\bar{\mathit{y}}/\mathit{yi}$(22) to determine the distribution weights, one must be aware that there is a second dimension to be determined other than the value of η, and that is, the specification of rich and poor for i in y_i. One common specification is to define the i by reference to income quintiles. This aggregation of incomes ensures that individual income extremes cannot dominate CBA outcomes. For example, in the UK case, when they used η = 1, ₤1 of net-benefits for the bottom quintile judged by gross income was given a weight between 2.2 and 2.3, and it was between 0.4 and 0.5 for the top quintile.¹⁵

This specification for rich and poor by quintile was used in the CBA of conditional cash transfers (CCT) for orphans and vulnerable children (OVC) in Kenya, evaluated in Brent (2013).¹⁶ Although income inequality was very large in Kenya, when η = 1 was used with a specification of income for the poor by the bottom quintile, the weight was not excessive at 4.3. The efficiency benefits still had to be reasonably large to justify the OVC program variants. With weights of a₁ = 4.3 and a₂ = 1, the CCT without targeting was not worthwhile; but it was worthwhile if there were targeting. The efficiency of the targeting was decisive, not the distributional weighting. In this sense, the weights were reasonable and not excessive.

More generally, especially in the US, it is rare that individuals ever have zero income in the absence of government welfare programs and individual transfers. In such cases, the amount of the transfers can be used as a proxy for y₁ in Equation (22)(22) $a_i=\mathit{\hspace{0.17em}}\bar{\mathit{y}}/\mathit{yi}$(22) . This specification for income was used in the CBA of case management programs in the US that sought to reduce intensive hospitalization for the severely mentally ill (SMI).¹⁷ Individuals who are SMI do not have earned income. For the distribution weights formula for the SMI, counted as income was all the transfer payments made on their behalf that did not include medical treatment. When a value of η = 1 was used, the weight for the psychiatric patients who were the beneficiaries was only 2.28. The costs were assumed to be paid by the taxpayers and given a weight of 1. Again, the weight of 2.28 for the poor could not be regarded as excessive, as the CBA with the weights reinforced the initial efficiency advantage for community and not institutional treatment for the SMI.

The upper bound value for η

Harberger (1978), op cit., was concerned that some policy-makers might use too high a value for η, such as 3. This implies that for a poor individual who has an income one-tenth of the average then a₁ = 1000; and for a rich individual who has an income ten times the average then a₂ = 0.001. In this case, equity would swamp efficiency, and CBA would almost entirely ignore the preferences of the rich. This result violates our axiom that the CBA depends on two social objectives, efficiency and equity. In the case of η = 3, no weight is given to efficiency.

To ensure that efficiency has a role to play in CBA, it is reasonable to assume that one dollar to a rich individual has a social value of aat least 1 cent. In this context, a value of η = 3 and higher would be unreasonable. When being rich is defined as having an income ten times the average, a value of η = 2 in Equation (21)(21) $a_i=(\mathit{\hspace{0.17em}}\bar{\mathit{y}}/\mathit{yi}{)}^{\mathit{\eta }}$(21) would at least ensure that a₁ = 0.01. For this reason, the upper bound on η should be 2. Although there are many studies where values of η equal to 3 and 4, and even 5 are considered,¹⁸ in the context of CBA of public expenditures, 2 should be the upper limit for the income inequality parameter.

When a value of η = 2 was used as the upper bound in a CBA of improved water quality in nine countries around the Baltic Sea, efficiency was decisive not the distribution weights – see Nurmi and Ahtiainen (2018), op cit. With equal weights, the benefits were 4070 million €, far exceeding the costs of 1489 million €. With η = 2 distribution weighting, the benefits were only 21% larger at 4918 million €, again endorsing the international environmental intervention.

The lower bound value for η

The lower bound limit for η cannot be 0 for this would also violate the axiom that the CBA depends on two social objectives. With η = 0, equity would be completely ignored. In the absence of a literature on how to determine a lower bound for η that is not zero, it was suggested that in the context of poverty where poor individuals live on less than one dollar a day, a minimum value of a₁ = 3 would be a generally acceptable starting point.¹⁹ For a poor individual with an income that is one-tenth of the average, a value of η = 0.5 in Equation (21)(21) $a_i=(\mathit{\hspace{0.17em}}\bar{\mathit{y}}/\mathit{yi}{)}^{\mathit{\eta }}$(21) would give a weight for a₁ = 3.162, which is approximately 3 as suggested. Given that the UN agreed that Sustainabletha Sustainable Goal number 1 is to end poverty in all its forms by 2030, a lower limit of η = 0.5 should not be too controversial.

Revealed preference weights

When the value for η is not given in advance by the social decision makerdecision-maker, the CBA evaluator can look at the values that have been revealed by economic behavior. In economics generally, the choices one makes are often interpreted as revealing one’s preferences. The literature has studied choices by individuals as well as those by governments. The problem is that the literature has come up with a wide range of values for η.

The literature on η has been based on three different dimensions of policy, Atkinson et al. (2009), Groom and Maddison (2019). The first is based on risk aversion, whereby utility may fall due to unforeseen circumstances. The second, is that utility may change over time (inter-temporal substitution). The third is inequality aversion, where one individual’s utility is different from another (intra-generational preferences). In principle, using all three dimensions would produce the same set of estimates for η,²⁰ but in practice, the necessary axioms are violated. This means that individual preferences vary according to the dimension.

To make progress, one needs to specify which dimensions of policy are most applicable to CBA decision-making. For CBA, the inequality aversion dimension is the most appropriate as the weights can be regarded as ex post and intragenerational. For a revealed preference estimate of η in this context, the estimates we are going to focus on are derived from income taxation. There was good reason to take seriously estimates based on taxation because decisions on taxation have to be defended before an electorate.²¹

The idea was that the more progressive is a country’s income tax system, the higher the marginal rate of tax relative to the average rate of tax, the higher a value it reveals for the income inequality parameter η. Evans (2005) analyzed the tax systems for 20 OECD countries and found that all the countries had a progressive income tax system. The implied η estimate was lowest for Ireland (between 1.00 and 1.47) and highest for Australia (between 1.51 and 1.82) with a mean of 1.40. For the US (based on the federal income tax system), it was between 1.15 and 1.45. All values were within the minimum and maximum bounds presented earlier for the ‘a priori’ approach. In particular, for no OECD country was there equal income distribution weights such that η = 0 with a₁ = 1.

Direct measures of weights

The revealed preference approach derived an estimate of η using inferences from behavior. Although revealed preference is considered to be best practice in CBA, when benefit valuations are usually based on willingness to pay, individual or decision-makers may not always be rational or well informed, especially with respect to decisions involving long time periods. Individuals often do not save enough for their retirement years, and governments can be seen to be myopic in terms of decisions that affect future generations, such as climate change. In such cases, Layard, Mayraz, and Nickell (2008) argue that a better approach to estimating η is to measure utility directly and see how this is actually affected by changes in income. In this case, utility u is the dependent variable, and this is regressed on income y as the main independent variable.

To measure utility, Layard et al. used a happiness index. In the surveys that they used, the typical happiness question was: ‘taking all things into account, how happy are you these days?’. Responses were recorded on an eleven-unit scale, where 0 meant extremely unhappy and 10 meant extremely happy. For the regression of u on y, they effectively used:²²

(23) $\mathrm{u}={\mathit{\alpha }}_0+{\mathit{\alpha }}_1({{\mathrm{y}}^{1-}}^{\mathit{\eta })}/1-\mathit{\eta })$(23)

To determine which estimated value of η to rely on, regression (22) was run with a large range of values for η. The value that was chosen to be the ‘best’ was the value for η that produced the regression with the highest estimated maximum likelihood. For a sample of 50 countries, the overall estimate was 1.26, with a 95% confidence interval of 1.16 to 1.37. Of the 50 countries, the lowest estimate for η was 1.19 and the highest was 1.30. This range was very similar to that found using the revealed preference method, and also within the minimum and maximum bounds set for η by the `a priori approach’. Thus, again, for no country was η = 0 requiring equal distribution weights.

Estimates of weights based on utility interdependence

‘A priori’ interdependence weights

The ‘a priori’ reasoning used to distinguish money income from in-kind interdependence weights is based on the concept of Pareto economic efficiency. All individuals must be better off, and there can be no losers, if redistribution is to come in the form of in-kind rather than a cash transfer. As explained in section III, for weights to be based on an in-kind externality rather than a money income externality, it requires replacing y₁ by h₁ in individuals 2’s utility function. For individual 2 to be better off from in-kind transfers, this requires:

(24) $\frac{\mathrm{\partial }{u}^2}{\mathrm{\partial }{h}^1}>\frac{\mathrm{\partial }{u}^2}{\mathrm{\partial }{y}^1}$(24)

However, from the recipient individual 1’s perspective, money income would probably be preferred, as cash could be spent on anything that gives utility and not just health (or whatever else is to be transferred in-kind). In this case:

(25) $\frac{\mathrm{\partial }{\mathit{u}}^1}{\mathrm{\partial }{\mathit{y}}^1}>\frac{\mathrm{\partial }{\mathit{u}}^1}{\mathrm{\partial }{\mathit{h}}^1}$(25)

There could therefore be a conflict of interest when both conditions (24) and (25) hold. This is because individual 2 would like to provide assistance in-kind, when individual 1 would prefer to receive assistance in cash. To resolve this conflict and yet stick to the principle of Pareto efficiency, it is necessary to impose a restriction by giving priority to the preferences of individual 2 providing the transfer. This can be called ‘the priority condition’ and this is attached to Equation (23)(23) $\mathrm{u}={\mathit{\alpha }}_0+{\mathit{\alpha }}_1({{\mathrm{y}}^{1-}}^{\mathit{\eta })}/1-\mathit{\eta })$(23) .²³ This condition is necessary to ensure that the transfer will take place, which would not be the case if $\frac{\mathrm{\partial }\mathrm{u}2}{\mathrm{\partial }\mathrm{h}1}<0$. If individual 2’s preferences are not given priority there could be no transfer, in which case individual 1 receives nothing. Even though individual 1’s preferences contained in Equation (24)(24) $\frac{\mathrm{\partial }{u}^2}{\mathrm{\partial }{h}^1}>\frac{\mathrm{\partial }{u}^2}{\mathrm{\partial }{y}^1}$(24) holds, it is probably still true that $\frac{\mathrm{\partial }\mathrm{u}1}{\mathrm{\partial }\mathrm{h}1}>0$. Thus, the individual is better off even with the in-kind transfer, and this ensures that the transfer is Pareto improving.

An application that endorses condition (24) was revealed from government expenditure decisions in the UK related to railway closures by the Minister of Transport, see Brent (1980). If a railway line that is incurring financial losses is kept open by the government, using taxpayer funds to subsidize the losses, this is equivalent to the government transferring funds from taxpayers (individual 2) to the rail users (individual 1, usually the poor and elderly in rural areas) in an in-kind form rather than in cash. Keeping a railway line open means that rail users receive time savings relative to closing the line and users travelling by bus instead. Time savings valued by the market had a monetary value of 63p per hour. However, the government’s actual decisions to keep unprofitable railway lines open revealed a best estimate valuation of between 93p and 133p per hour, around twice as much. The government’s valuation based on taxpayer preferences had a social value (distribution weight) twice the value that a recipient (or anyone else) would value the time savings. The difference in time valuation implies that the Minister gave a premium to in-kind income of 100% over cash income.²⁴

Revealed preference interdependence weights

The clearest evidence that $\frac{\mathrm{\partial }{\mathrm{u}}^2}{\mathrm{\partial }{\mathrm{y}}^1}>0$ is when $\frac{\mathrm{\partial }{\mathrm{u}}^2}{\mathrm{\partial }{\mathrm{y}}^1}>1$. This is observed whenever there are voluntary individual charitable cash transfers given to the poor.²⁵ If the value were not greater than 1, type 2 individuals would rather keep the money for themselves, as $1 to themselves is worth $1. According to the Philanthropy Panel Study (2021), charitable giving reached an all-time high in 2020 with Americans donating $471 billion. In 2018, 49.6% of American households gave to charities.

When charitable contributions are not forthcoming because individually $\frac{\mathrm{\partial }{\mathrm{u}}^2}{\mathrm{\partial }{\mathrm{y}}^1}<1$ there is still the pure public goods argument for including unequal distribution weights in CBA. This arises because each dollar given to the poor increases the utility of every individual who is rich. The greater the number of rich individuals there are, the higher the aggregate valuation. Consider, as an application, the evaluation of a drug treatment program by Zarkin, Cates, and Bala (2000). People only received 0.0022 cents of external benefits for each other person successfully treated, which is close to zero. However, as there were 903, 104 people in the community receiving this benefit, the average external benefits were $19.86 per other person successfully treated. Adding the $1 of direct benefits received by those treated, in line with Equation (15)(15) ${\mathrm{a}}_1=\frac{\mathrm{\partial }{\mathrm{u}}^1}{\mathrm{\partial }{\mathrm{y}}^1}+\frac{\mathrm{\partial }{\mathrm{u}}^2}{\mathrm{\partial }{\mathrm{y}}^2}\mathrm{and}{\mathrm{a}}_2=\frac{\mathrm{\partial }{\mathrm{u}}^2}{\mathrm{\partial }{\mathrm{y}}^2}\mathit{d}{y}_2$(15) , made the social weight to $1 of treatment for a drug addict that is not oneself equal to 20.86, a value much higher than zero

Direct measures of interdependence weights

In the context of the revealed preference method that we just examined, money income weights were the focus. The main application was in terms of individual charitable contributions given in terms of cash. For a direct measure of independence weights, we can now consider charitable giving in-kind. There is recent direct empirical evidence for this type of externality provided by Lawton et al. (2021). An important way that others receive in-kind benefits is from volunteer work that rich individuals regularly do. So h₁ = h₁ (V), where V is recent volunteer work given to others. To provide a direct test of whether $\frac{\mathrm{\partial }{\mathrm{u}}^2}{\mathrm{\partial }{\mathrm{u}}^1}>0$, this utility function was estimated by regression analysis using:²⁶

(26) $\mathrm{d}\mathit{u}={\mathit{\alpha }}_0+{\mathit{\alpha }}_1\mathrm{d}\mathit{V}$(26)

where u is measured mainly by life satisfaction (on a scale of 1 − 7) and V is measured by a dummy variable as to whether an individual volunteered, or not, in the last 12 months. The estimate of α₁ was+0.0034. Individuals who did not volunteer at baseline, and now volunteer, increased their utility by 0.0034. Volunteering made them happier, which means that enabling others to be healthier made them happier. Given that the recipients of the volunteer work can be assumed to value it at face value, the weight on in-kind charitable work including the externality would provide a value of a₁ = 1.0034. Although the incremental weight of 0.0034 may appear small, the B₁ value to which the weight was attached was not inconsequential. The value of the volunteer work in terms of the wellbeing to the volunteers was assessed at £911 per volunteer per year on average.

V. Invalid reasons for not using distribution weights²⁷

As a preliminary to the summary and conclusions, which is mainly concerned with the two valid reasons for using distribution weights, it is important to try to remove the two invalid reasons that practitioners often refer to from the CBA literature to justify why they do not apply distribution weights. We cover the two main invalid reasons.

Relying on compensation tests instead of using distribution weights

Convincing people that unequal distribution weights are essential to CBA should not be too difficult, once one accepts that there are two social objectives, equity and efficiency, and not just one, economic efficiency. Thus, appealing to a potential Pareto improvement test, instead of using an evaluation based on a CBA test with distribution weights, cannot be sufficient. The logic of a compensation test is that when the efficiency net-benefits are positive, there are sufficient resources remaining to be devoted to compensating the losers, those incurring the costs. It is very well understood that these compensation costs are hypothetical in nature. There would be no losers if they are compensated. However, there is almost never evidence provided that, once one includes the administrative costs of compensating the losers, that the net-benefits will still be positive.

Moreover, in practice, apart from trade policy, where there are trade readjustment allowances for those who lose their jobs due to increased trade competition (Margalit 2011), compensation is not routinely carried out for losers of government policies. Thus, the poor are not actually given any special consideration in a potential Pareto test. Compensation is simply an irrelevant alternative, and Arrow’s impossibility theorem (Arrow 1951) requires that government decisions be ‘independent of irrelevant alternatives’.

But, not well understood is that even if compensation is given to ensure an actual Pareto improvement, this need do nothing for ensuring that the poor, who have low ability to pay, have received any of the gains, as we saw in the Hammitt example. All we know for certain from passing this test is that the poor do not lose. It is time for CBA practitioners to refrain from appealing to compensation tests to try to justify not using unequal distribution weights.

Relying on the tax system instead of using distribution weights

It is also time for CBA practitioners to cease referring to the other main alleged justification for not using distribution weights in public expenditure decision-making. This is, that it is better to leave the objective of income redistribution entirely to the tax system. This alternative is not only irrelevant as it is hypothetical; but it is also grossly insufficient. We have seen that almost all countries already have a progressive income tax system. However, this fact alone does not ensure that equity should cease to be an objective for CBA. In all 92 countries where there is inequality data from 1990 to 2015, the poorest 45% earned less than 25% of income; and countries where inequality has grown are home to 71% of the world’s population (United Nations 2020). The need is to promote equality with every government policy instrument available, including CBA, and not just rely on the income tax system.

Conclusions

It needs to be understood that the only way to ensure that distributional issues are explicitly included in CBA is to apply unequal distributional weights to the benefits and costs. As was emphasized in the earlier survey, there are no schools of thought on CBA that can avoid the use of distributional weights. Equal weights just mean that everyone’s weight is the same, assigned to be unity. This is just as subjective as using unequal distribution weights. The aim of this paper was to reaffirm that equal weights are not consistent with CBA being based on a two-objective SWF, and and is inconsistent with the empirical literature that has tried to estimate how the weights were estimated in practice.

Acknowledging the existence of two social objectives is the logic behind the new justification for adopting unequal distribution weights based on generalizing Hammitt’s analysis. With Hammitt’s specific assumptions, it was possible that if one had used equal weights, that shadow prices could so dominate the net-benefits, such that a policy would be approved, even though it both lowered output and gave no benefits to the poor. In this eventuality neither objective would be satisfied, and not just one at the expense of the other.

If the distribution weights were optimally selected, they would neutralize the ability to pay effects in willingness to pay calculations. Thus, with distribution weights, the only reason why the benefits from policy B (giving benefits v₂ h₂ to the rich) would be greater than policy A (giving benefits v₁ h₁ to the poor) would be if the rich’s preferences for policy B were so much stronger than the poor’s preferences for policy A. In general, one cannot expect that both social objectives are achieved with every public policy alternative. The central role of distributional weights in CBA is to ensure that the trade-off between the two objectives is optimally set.

Adler emphasizes that both the use of unequal distribution weights, and the use of equal distribution weights,requires making value judgments. Since the weights are derived from the SWF one is using, the value judgements for distribution weights are based on the value judgments used for the SWF. We presented a set of three value judgments that we argued were the minimum value judgments that everyone can agree on, who accepts the fundamentals of welfare economics that underlies public policy in the US. The value judgments for the SWF were as follows: first, to make society better off, one needs to make individuals better off; second, because the American constitution declares that all people are created equal, individual changes in utility are valued equally; and third, that we honour consumer sovereignty, i.e. accepting that individuals are the best judges of their own welfare.

Once we do accept consumer sovereignty, which is a central requirement to making willingness to pay a socially valid measure of benefits in CBA, we must also acknowledge that rich individuals’ utility functions do not only depend on what they, themselves, consume. They also depend on what the poor consume – there is utility interdependence. The existence of this interdependence constitutes a separate, neglected justification for having unequal weights in CBA. This justification is separate from the well-known one that there is diminishing marginal of income, which must result in the marginal utility of the rich being less than the marginal utility of the poor

With the recognition that interdependent utility functions exist, the use of the simple utilitarian SWF given by W = ∑u_i can take on a completely new meaning, and thereby have much wider applicability than is acknowledged in the literature. Separately, Hammitt and Adler both refer to a prioritarian SWF (W^P) that is a transformation of the utilitarian function: W^P = ∑g(u_i). The function g(u_i) is assumed to be concave, which has an inequality-aversion parameter λ that operates like the diminishing marginal utility of income parameter η in the utilitarian SWF. Adler explains that there are two main differences between the roles of λ and η in CBA. First, the parameter λ allows for differences in utility levels to be included in the SWF (especially for those with worst-off utility levels), while η relates only to changes in utility levels, which do not consider the utility levels themselves. Second, η depends on individual preferences, while λ is determined by the moral preferences of a social planner.

As we pointed out, the first fundamental value judgment behind SWFs, which are to be used to derive distribution weights, is that it is to be individualistic. There is, in general in CBA, no need for a social planner.²⁸ Thus, a whole level of controversy as to who is to be the social planner is avoided by the simple utilitarian SWF. Moreover, independent of whom determines η, is the fact that diminishing marginal utility of income changes, given by λ, is only the first component making up the distribution weight a₁ going to the lower-income individuals. The second component is type 2 individuals’ concern with the level of utility of type 1 individuals. Thus, our recognizing of utility interdependence makes any reference to λ (and therefore reference to prioritarianism) unnecessary. Distribution weights based on a utilitarian SWF already allows for differences in utility levels to play a role in CBA outcomes.

To conclude:, the debate in the CBA literature as to whether to use unequal distribution weights should now be settled. Unequal distribution weights must be employed in CBA. CBA practitioners can now spend more of their efforts in estimating the most appropriate distribution weights to be used in the context of each specific public expenditure policy setting. In this paper, we have explained the three main methods, for the two categories of distribution weights that we identified, that can be used to estimate the weights. We presented many examples from the literature that show how these methods were applied in practice. For estimates of η, there was almost a consensus that it was not far from the core value of η = 1, derived from the ‘a priori” approach. In all cases, the distribution weights that were estimated were not equal. CBA theory, and the applications based on that theory, is against the use of equal weights. It is only the practitioners who insist on using equal weights.

For CBA practitioners who are now not going to use equal weights, there needs to be a new standardization that all use regarding η values. We follow Squire and van der Tak op cit. who recommend using our core value η = 1 for their main CBA results. We then recommend using our lower bound η = 0.5 for the sensitivity analysis, and not using equal weights, which we know to be invalid. Our upper bound of η = 2 should be used only sparingly in the sensitivity analyses.²⁹

Adler (2021) emphasizes that President Biden, soon after his inauguration, issued a memorandum instructing the Director of the Office of Management and Budget to propose procedures that take into account the distributional consequences of regulations. Adler considers that this memorandum could initiate an important shift towards equity in CBA.³⁰ Our paper provides a framework for such new procedures. The next step is for unequal distributional weights to be adopted by practitioners in all aspects of government public policy, including spending, and not just for regulations.

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Notes

¹ For text-book analyses of the role of distribution weights in CBA see Boardman et al. (2018) chapter 19, and Brent (2006) chapter 10.

² In this paper, we will use the terms ‘public policy’ and ‘public expenditure’ interchangeably because CBA can be applied to any type of government intervention, such as regulation, and not just public expenditure, which is the traditional context.

³ For the purposes of this paper, we will use the term equity to be synonymous with requiring unequal distribution weights, such that the poor receives a higher weight than the rich.

⁴ The difficulty here is that Harberger did not distinguish an externality in money income form from an externality in the form of an in-kind transfer, which is what allowing for basic needs essentially entails, his preferred method for allowing for distributional concerns in CBA. In this paperpaper, we devote a large part of our analysis to recognizing, and allowing for, this externality distinction.

⁵ We only show that the paradox can disappear, not that the paradox must disappear. If the paradox does not disappear, with distribution weights that correct for ability to pay, then, as we explain in section II, there is a good reason for the result. This reason is that the rich simply must have so much stronger utility preferences for the policy outcomes than the poor.

⁶ In the analysis that follows below, we consider just the cases when income (vh) or health h are separate numeraires. We do not deal with the case when both income and health are the numeraire. When both are included, there are a whole host of theoretical and practical complications for distribution weighting that are fully explained in Fleurbaey and Abi-Rafeh (2016).

⁷ Strictly, costs should only be c, and not 2c, as individuals would not be expected to pay costs if they do not receive any outputs. Note that, since it is the difference between net benefits for policies A and B that is the decision-making criterion, costs cancel out whether they are c or 2c in Equation (1)(1) $\mathrm{NBA}={\mathit{v}}_1{\mathit{h}}_1-2\mathit{c}$(1) and (2(2) $\mathrm{NBB}=v_2{h}_2-2\mathit{c}$(2) ). So, Hammitt’s conclusions would be unchanged if we keep his 2c formulation of costs for either policy rather than replace them by c.

⁸ Hammitt then contrasts his result with what would have happened with an alternative numeraire when no shadow prices were used for benefits. The alternative numeraire was to use just h as the output instead of vh, which means that h₁ was the benefit under policy A, and h₂ was the benefit under policy B. With no difference in the costs of the two policies, even with shadow pricing for the costs, policies A and B only depend on differences in benefits. With h₁ > h₂, it must now be the case that policy A is better than policy B. Hammitt interprets this to be a contradiction caused by the change in numeraire. But, this is really a shadow pricing issue, and and not a numeraire issue, as h = vh when v = 1. Thus, the real issue here is that condition (3) gives the ‘wrong’ answer and not that different numeraires give different answers (which can, of course, happen). You can solve this wrong answer by using distribution weights, not changing the numeraire. Why condition (3) can be classed as ‘wrong’ is what is now explained in the text.

⁹ Lower outputs do not always mean that benefits will be lower, as this also depends on the size of the shadow prices. But, in the case when v = 1, which is one that Hammitt does consider, benefits would automatically be lower if h₁ > h₂.

¹⁰ Weisbach (2014) criticizes the use of an individualistic SWF to justify CBA regulation generally, and not just in the connection of distribution weights, because it does not additionally include institutional components, such as firms, markets and government, which would assign tasks in a large entity making the regulation decisions. However, in the context of distribution weights, it makes no sense for the railroad sector not to use distribution weights, by assuming that this can be left to the transport sector as a whole. We know, in practice, other agencies do not incorporate distributional concerns in their decisions. Similarly, it is not correct for Weisbach to assume that a progressive taxation system would be better for implementing distributional concerns, when a progressive income tax system does not exist, and is unlikely to exist. It is important that all decision-making arms of an agency use distribution weights, especially as it is known that bureaucratic inefficiencies exist in large agencies that could block distributional concerns adopted by any one arm.

¹¹ Given that Boardman et al. (2018), op cit., were one of the few authors who did recognize interdependent utility functions in the form of income, they also derived the same result as Equation (15)(15) ${\mathrm{a}}_1=\frac{\mathrm{\partial }{\mathrm{u}}^1}{\mathrm{\partial }{\mathrm{y}}^1}+\frac{\mathrm{\partial }{\mathrm{u}}^2}{\mathrm{\partial }{\mathrm{y}}^2}\mathrm{and}{\mathrm{a}}_2=\frac{\mathrm{\partial }{\mathrm{u}}^2}{\mathrm{\partial }{\mathrm{y}}^2}\mathit{d}{y}_2$(15) , see their footnote 8. Therefore, they also recognized that, even with the marginal utility of the rich and poor the same, the existence of the utility externality constituted a second reason why equal weights was not optimal, see their equation (19.2). But, they did not consider interdependence in-kind, and their common analysis related to this section played no part in their conclusions related to the role of distribution weights in CBA.

¹² Florio and Pancotti (2023) suggest an alternative formulation to using income as the numeraire that relies on using consumption. This is the distributional characteristics of a goods branch of the literature. In this framework, one needs to have data on the distribution of the particular good considered, and specify the marginal utility of consumption rather than the marginal utility of income. However, to know the marginal utility of consumption, one still needs to determine their inequality parameter denoted by ε – see their equation (7.9). Thus, our analysis of η would also carry over to this branch of the literature.

¹³ Johansson and Kristrom (2018), chapter 4, point out that as η approaches unity, one is effectively adopting a Bernoulli-Nash (Cobb-Douglas) Social Welfare Function.

¹⁴ In the later version of the Green Book, HM Treasury (2018), the value of η to be used in government evaluations was raised slightly to 1.3. It also specified that the results with equal distribution weights should be calculated alongside the weighted results for reference purposes.

¹⁵ See Brent (2009), chapter 1.

¹⁶ In the context of any form of cash transfers, as explained in Brent (2013), the case for unequal weights is decisive. This is because of the existence of administrative costs associated with carrying out the cash transfers. If every $1 gained by the poor is weighted by 1, it completely cancels out the $1 lost by the rich weighted by 1. All one is left to record in the evaluation of the transfers is the negative administrative cost. So, the cash transfer would never be worthwhile and take place if equal weights are adopted.

¹⁷ See Brent (2004).

¹⁸ See, for example, Carlsson, Daruvala, and Johansson-Stenman (2005) and Pirttila and Uusitalo (2010).

¹⁹ See Brent (2009), chapter 1.

²⁰ See Broome (1991).

²¹ Cowell and Gardiner (1999).

²² In addition to having a random error term in Equation (22)(22) $a_i=\mathit{\hspace{0.17em}}\bar{\mathit{y}}/\mathit{yi}$(22) , Layard et al. had a set of controls, which included the number of hours worked, age and gender.

²³ Brent (1980).

²⁴ To see the results of this application explicitly in terms of distribution weights in cash and in-kind for recipients, and in cash for taxpayers, see Brent (1979). The estimated weight on time savings benefits by the poor was 0.11 relative to the 0.08 to the taxpayers financing the costs. This confirms that unequal distribution weights were used, a₁ > a₂. However, when benefits came in the form of cash, by fare differences, the a₁ estimate was 0.07, and thus less than the weight on the cash given up by the rich of 0.08. These results confirm that the rich were willing to assist the poor, but would prefer giving the assistance in-kind rather than in cash. The specification of interdependence by individual 2 in Equation (19)(19) $u_1=u_1\left(y_1\right)\mathrm{and}{u}_2=u_2\left(y_2,h_1\right)$(19) was more relevant than that in Equation (13)(13) $u_1=u_1\left(y_1\right)\mathrm{and}{u}_2=u_2\left(y_2,y_1\right)$(13) , when defining the distribution weight for a₁ for railway closure decisions in the UK.

²⁵ See Brent (2017), chapter 6.

²⁶ The actual Lawton et al. regression equation for u used as controls for V: household income, marital status, number of children, employment status, self-rated health, age, gender, broad ethnicity, geographic region, and education.

27 Apart from the two invalid reasons for using equal weights covered in this section, there is another invalid argument in the CBA literature which states that, while unequal weights may be acceptable, the weight a₁ on the poor should be capped at something around 1.5 to 2. This is the so called ‘leaky bucket’ idea due to Okun (1975). The leaky bucket is the administrative costs lost involved with transferring cash from the rich to the poor. If it costs 1.5 to 2 dollars to transfer funds to the poor, then rather thatthan use the project to assist the poor, it is more efficient (my emphasis) to use the cash transfer system if one intends to use a weight higher than 2 for the poor group benefits. The objective here is purely equity (helping the poor, not increasing social welfare) and the means is purely efficiency (minimizing administrative costs, not increasing social welfare). It deals with the two-social objectives one at a time, rather than applying both together as required by a two-objective social welfare function. Moreover, the benchmark is in terms of assisting the poor in cash rather than in-kind, which we have shown to be more important in this paper. Finally, as we explain in this section related to both compensation tests and using a progressive income tax system, the cash transfer is not likely to take place; it is just hypothetical. As suchsuch, it is another irrelevant alternative.

²⁸ The main exception is for intergenerational CBA decisions, where one cannot rely on individual preferences, for the simple reason that future individuals have not yet been born, and so their preferences cannot be known. A social planner may be needed to make these decisions.

²⁹ We wish to thank an anonymous referee of this journal who pointed out the need for standardization of η values when CBA results are to be compared.

³⁰ Robinson, Hammitt, and Zeckhauser (2016) explain that previous US Presidents Clinton and Obama also had Executive Orders requiring that distribution and equity be considered and understood for devising regulations. But, they did not specify how this was to take place. It is only by using unequal distribution weights that efficiency and distribution would be fully integrated and considered simultaneously.