2. Conceptual framework

Here we set up a model describing a society of altruists who are offered a project that confers benefits to each member of society. The project is to be evaluated using a BCA. We define three different WTPs to be derived in Sections 3, 4, 5: WTP (Private), WTP (Social), and WTP (Citizen). WTP (Private) is based on the individual’s private utility function and includes only self-interest. WTP (Social) is based on the individual’s social utility function and includes altruism. WTP (Citizen) is also based on the individual’s social utility function but in expectation as individuals are placed behind a VoI making them uncertain of their position. As these WTPs will differ, we consider the consequences when each WTP is used as a monetary value for the benefits of a project in a BCA.

2.3. Private and social utility functions

Following Bergstrom (Reference Bergstrom2006), each individual has a private utility function which reflects only their preferences for their own outcomes. In addition, individuals also have preferences over others’ outcomes. The individual’s full set of preferences is reflected by their social utility function. Assuming the project is provided, the private utility function of Individual $ i $ who is in Position $ p $ is given by Equation 1.

(1) $$ {u}_p^i={x}_p+{w}_p-{t}_pC $$

Individual $ i $ derives private utility from their own benefit from the project, $ {x}_p $ , and their own consumption of the private good which is their income, $ {w}_p $ , less their contribution to the project, $ {t}_pC $ .Footnote ³ The social utility function of Individual $ i $ is given by Equation 2.

(2) $$ {U}_p^i={x}_p+{w}_p-{t}_pC+\sum \limits_{q\ne p}^N\left[{\alpha}_q^i{x}_q+{\beta}_q^i\left({w}_q-{t}_qC\right)\right] $$

Equation 2 is a generalized form of the social utility function in Bergstrom (Reference Bergstrom2006). The $ {\unicode{x03B1}}_{\mathrm{q}}^{\mathrm{i}} $ and $ {\unicode{x03B2}}_{\mathrm{q}}^{\mathrm{i}} $ terms are weights placed on the individual in Position $ \mathrm{q} $ ’s benefit from the project and consumption of the private good, respectively. The summation covers all positions q, except $ q=p $ because Individual $ i $ is in Position $ p $ .

We use this functional form as it is sufficiently broad to encompass the cases often discussed in the literature on altruism in WTP including: self-interest ( $ {\unicode{x03B1}}_{\mathrm{q}}^{\mathrm{i}}= $ $ {\unicode{x03B2}}_{\mathrm{q}}^{\mathrm{i}}=0 $ ), pure altruism ( $ {\unicode{x03B1}}_{\mathrm{q}}^{\mathrm{i}}= $ $ {\unicode{x03B2}}_{\mathrm{q}}^{\mathrm{i}}>0 $ ) (Jones-Lee, Reference Jones-Lee1991; Reference Jones-Lee1992)Footnote ⁴, paternalistic altruism ( $ {\unicode{x03B1}}_{\mathrm{q}}^{\mathrm{i}}\ne $ $ {\unicode{x03B2}}_{\mathrm{q}}^{\mathrm{i}} $ ) (Jones-Lee, Reference Jones-Lee1991; Jones-Lee, Reference Jones-Lee1992), and distributional preferences such as Engelmann and Strobel (Reference Engelmann and Strobel2007)’s quasi-maximin model of which the efficiency motive ( $ \forall q,{\unicode{x03B1}}_q^{\mathrm{i}}={\alpha}^i,{\unicode{x03B2}}_{\mathrm{q}}^{\mathrm{i}}={\beta}^i $ ) and maximin motive ( $ {\unicode{x03B1}}_1^{\mathrm{i}}>0 $ and or $ {\unicode{x03B2}}_1^{\mathrm{i}}>0;\forall q>1,{\unicode{x03B1}}_q^{\mathrm{i}}={\unicode{x03B2}}_{\mathrm{q}}^{\mathrm{i}}=0 $ ) are the two extreme cases.

2.4. Veil of ignorance

In front of a VoI, individuals are certain of their own position ( $ i=p $ ) and all other individuals’ positions. Behind a VoI, individuals are uncertain of their own and other’s positions. In accordance with Harsanyi’s (Reference Harsanyi1955) equi-probability model, individuals know that the probability of being in each position is $ 1/N $ . Both Harsanyi (Reference Harsanyi1955) and Rawls (Reference Rawls1972) consider a VoI as discerning an individual’s impersonal ethical preferences through the introspection of one’s own set of possible personal preferences across positions in society, see Orr (Reference Orr2007) for a discussion.

Individuals are expected utility maximizers. Behind a VoI, an individual must consider themselves in each position. Their expected social utility function is given in Equation 3 as the sum of their social utilities (Equation 2) across each of the $ N $ positions divided by $ N $ .

(3) $$ {EU}^i=\frac{1}{N}\sum \limits_{p=1}^N{U}_p^i=\frac{1}{N}\sum \limits_{p=1}^N\left[{x}_p+{w}_p-{t}_pC+\sum \limits_{q\ne p}^N\left[{\alpha}_q^i{x}_q+{\beta}_q^i\left({w}_q-{t}_qC\right)\right]\right] $$

Which can be rearranged to give Equation 4.

(4) $$ {EU}^i=\frac{1}{N}\sum \limits_{p=1}^N\left[\left(1+\left(N-1\right){\alpha}_p^i\right){x}_p+\left(1+\left(N-1\right){\beta}_p^i\right)\left({w}_p-{t}_pC\right)\right] $$

2.6. The Kaldor hypothetical compensation test

The Kaldor hypothetical compensation test prescribes that “if those who gain by adoption of a project do so to a sufficient degree to be able, if required, to compensate those who lose, then the project should be undertaken” (Jones-Lee, Reference Jones-Lee1989, pg. 32). A project is potentially Pareto improving if the Kaldor compensation test is passed.

To model the Kaldor hypothetical compensation test, we first derive individual net benefits. Equation 5 is the net benefit of Individual $ i $ who is in Position $ p $ which is found by differencing social utility (Equation 2), which reflects the individual’s full set of preferences, with and without the project.

(5) $$ \Delta {U}_p^i={x}_p+\sum \limits_{q\ne p}^N{\alpha}_q^i{x}_q-C\left({t}_p+\sum \limits_{q\ne p}^N{\beta}_q^i{t}_q\right) $$

Those individuals with a positive net benefit are gainers and those with a negative net benefit are losers. Next, the net benefits of all individuals across the group are aggregated. Equation 6 is the net benefit aggregated across individuals and rearranged.Footnote ⁵

(6) $$ \sum \limits_p^N\left[\Delta {U}_p\right]=\sum \limits_p^N\left[\left(1+\left(N-1\right){\alpha}_p\right){x}_p\right]-\sum \limits_p^N\left[\left(1+\left(N-1\right){\beta}_p\right){t}_p\right]\bullet C\hskip0.24em $$

In addition to the assumptions set out previously, we assume homogenous preferences over others ( $ {\alpha}_p^i={a}_p $ and $ {\beta}_p^i={\beta}_p $ $ \forall i $ ) to allow interpersonal comparisons of utility. Then if the aggregate net benefit (Equation 6) is greater than or equal to 0, the gainers can compensate the losers. This means the Kaldor hypothetical compensation test is passed and a potential Pareto improvement is possible. If aggregate net benefit (Equation 6) is less than 0, the gainers cannot compensate the losers and the Kaldor hypothetical compensation test is failed, and a potential Pareto improvement is not possible.

2.7. The Benefit Condition

We test if any of WTP (Private), WTP (Social), or WTP (Citizen) when aggregated for use in a BCA can correctly identify potential Pareto-improving projects in accordance with the Kaldor hypothetical compensation test. In other words, whether aggregate WTP is a measure of the project benefits which accurately reflects the altruistic and distributional preferences represented by individuals’ social utility functions.

To do so, we consider the case in the model where total cost equals aggregate WTP. At this cost, and conditional on aggregate WTP accurately reflecting the preferences in individuals’ social utility functions, the result of the Kaldor hypothetical compensation test will be an aggregate net benefit exactly equal to zero as the gainers can exactly compensate the losers. We term this condition the “Benefit Condition” which is set out formally in Equation 7.

(7) $$ "\mathrm{Benefit}\ \mathrm{Condition}":\sum \limits_p^N\left[\Delta {U}_p\right]\equiv 0\hskip0.24em when\hskip0.24em C=\left\{\begin{array}{c} \sum \limits_i^N{WTP}^i(Private)\\ {} \sum \limits_i^N{WTP}^i(Social)\hskip0.36em \\ {} \sum \limits_i^N{WTP}^i(Citizen)\end{array}\right. $$

That is aggregate net benefit must equal zero when the total cost of the project equals aggregate WTP.

The next three sections take each of WTP (Private), WTP (Social), and WTP (Citizen) in turn. First, we derive WTP based on Equations 1, 2, and 4 and then test whether WTP meets the Benefit Condition. We do so by entering each WTP under the assumption of homogenous preferences over others into Equations 6 and 7 in place of the total cost, $ C $ .

3. WTP (Private)

The individual’s WTP (Private) for the project is based on their private utility given in Equation 1. An individual’s WTP (Private) is the adjustment in income required for private utility with the project to return exactly to the level of private utility without the project. Equation 8 denotes indifference between having and not having the project based on the private utility of Individual $ i $ who is in Position $ p $ .

(8) $$ {x}_p+{w}_p-{t}_pC={w}_p $$

When the equality holds, the $ {t}_pC $ term is the individual’s WTP (Private), that is, the CV.Footnote ⁶ Individual $ i $ ’s WTP (Private) is found by rearranging Equation 8 to isolate $ {t}_pC $ as shown in Equation 9.

(9) $$ {WTP}^i(Private)={t}_pC={x}_p $$

WTP (Private) is the product of the individual’s tax share and the cost they would not like society to surpass which equals the individual’s benefit from the project.

To test if WTP (Private) meets the Benefit Condition, aggregate WTP (Private) is entered into Equations 6 and 7 to replace the costs, which requires the assumption of homogenous preferences over others, giving Equation 10.

(10) $$ \sum \limits_p^N\left[\Delta {U}_p\right]=\sum \limits_p^N\left[\left(1+\left(N-1\right){\alpha}_p\right){x}_p\right]-\sum \limits_p^N\left[\left(1+\left(N-1\right){\beta}_p\right){t}_p\right]\bullet \sum \limits_p^N{x}_p $$

Equation 10 is not always equal to zero and therefore the Benefit Condition required for BCA to correctly identify projects that are potentially Pareto improving is not met. Reassuringly, Equation 10 will equal zero in the special case of self-interest. WTP (Private), therefore, meets the Benefit Condition when all individuals in society are self-interested.

Equation 10 will be negative when altruistic preferences for the private good are sufficiently strong (large beta terms). That is $ \sum \limits_p^N\left[\Delta {U}_p\right]<0 $ when $ C=\sum \limits_i^N{WTP}^i(Private) $ . This could result in a BCA incorrectly recommending a project that is not potentially Pareto improving because, at a total cost equal to aggregate WTP (Private), those who gain cannot adequately compensate those who lose.

Equation 10 will be positive when altruistic preferences for the project are sufficiently strong (large alpha terms). That is $ \sum \limits_p^N\left[\Delta {U}_p\right]>0 $ when $ C=\sum \limits_i^N{WTP}^i(Private) $ . This could result in a BCA incorrectly rejecting projects that are potentially Pareto improving because, at total costs greater aggregate WTP (Private), those who gain can still adequately compensate those who lose.

4. WTP (Social)

The individual’s WTP (Social) for the project is derived from the social utility function (Equation 2). An individual’s WTP (Social) is the adjustment in income required for social utility with the project to return exactly to the level of social utility without the project.

Equation 11 denotes indifference between having and not having the project based on the social utility of Individual $ i $ who is in Position $ p $ .

(11) $$ {x}_p+{w}_p-{t}_pC+\sum \limits_{q\ne p}^N\left[{\alpha}_q^i{x}_q+{\beta}_q^i\left({w}_q-{t}_qC\right)\right]={w}_p+\sum \limits_{q\ne p}^N\left[{\beta}_q^i{w}_q\right] $$

When equality holds, $ C $ is the total cost to society at which the individual’s social utility with the project returns exactly to the level of social utility without the project. $ C $ is the cost to society the individual would prefer not to pass. WTP (Social) is found by isolating $ C $ and multiplying each side of the equality by $ {t}_p $ to give $ {t}_pC $ which gives the CV.Footnote ⁷ The first step is given by Equation 12.

(12) $$ C=\frac{x_p+\sum \limits_{q\ne p}^N{\alpha}_q^i{x}_q}{t_p+\sum \limits_{q\ne p}^N{\beta}_q^i{t}_q} $$

Individual WTP (Social) is given by Equation 13.

(13) $$ {WTP}^i(Social)={t}_pC={t}_p\frac{x_p+\sum \limits_{q\ne p}^N{\alpha}_q^i{x}_q}{t_p+\sum \limits_{q\ne p}^N{\beta}_q^i{t}_q} $$

Equation 13 states that WTP (Social) is a ratio of benefit and cost share terms which is multiplied by the individual’s cost share. The benefit terms are the individual’s benefit plus the altruism-weighted benefits to others and the cost share terms are the individual’s cost share plus the altruism-weighted cost shares of others. When preferences are self-interested, the private and social utility functions are identical and therefore WTP (Social) equals WTP (Private) because the alpha and beta terms in Equation 13 are all zero and the cost share terms cancel.

To test if WTP (Social) meets the Benefit Condition, aggregate WTP (Social) under the assumption that $ {\alpha}_q^i={\alpha}_q $ and $ {\beta}_q^i={\beta}_q $ $ \forall i $ is entered into Equations 6 and 7 to replace the costs given in Equation 14.

(14) $$ {\displaystyle \begin{array}{c}\hskip-10em \sum \limits_p^N\left[\Delta {U}_p\right]=\sum \limits_p^N\left[\left(1+\left(n-1\right){\alpha}_p\right){x}_p\right]\\ {}\hskip15em -\sum \limits_p^N\left[\left(1+\left(n-1\right){\beta}_p\right){t}_p\right]\bullet \sum \limits_p^N\left[{t}_p\frac{x_p+\sum \limits_{q\ne p}^N{\alpha}_q{x}_q}{t_p+\sum \limits_{q\ne p}^N{\beta}_q{t}_q}\right]\end{array}} $$

Equation 14 is not always equal to zero and therefore the Benefit Condition required for BCA to correctly identify projects that are potentially Pareto improving is not met. As with WTP (Private), a BCA using WTP (Social) could incorrectly reject a project that would be potentially Pareto improving ( $ {\sum \limits}_p^N\left[\Delta {U}_p\right]>0 $ when $ C={\sum \limits}_i^N{WTP}^i(Social) $ ) or incorrectly accept a project that would not be potentially Pareto improving ( $ {\sum \limits}_p^N\left[\Delta {U}_p\right]<0 $ when $ C={\sum \limits}_i^N{WTP}^i(Social) $ ).

5. WTP (Citizen)

An individual’s WTP (Citizen) for the project is based on their expected social utility (Equation 4). Equation 15 denotes indifference between having and not having the project based on the expected social utility of an Individual $ i $ who is uncertain of their position.

(15) $$ {\displaystyle \begin{array}{c}\frac{1}{N}\sum \limits_{p=1}^N\left[\left(1+\left(n-1\right){\alpha}_p^i\right){x}_p+\left(1+\left(n-1\right){\beta}_p^i\right)\left({w}_p-{t}_pC\right)\right]\\ {}\hskip-8em =\frac{1}{N}\sum \limits_{p=1}^N\left[\left(1+\left(n-1\right){\beta}_p^i\right)\left({w}_p\right)\right]\end{array}} $$

When the equality holds, $ C $ is the cost at which expected social utility with the project returns exactly to the level of expected social utility without the project. Individual $ i $ ’s WTP (Citizen) is found by rearranging Equation 15 to isolate $ C $ as shown in Equation 16.

(16) $$ C=\frac{\sum \limits_{p=1}^N\left[\left(1+\left(n-1\right){\alpha}_p^i\right){x}_p\right]}{\sum \limits_{p=1}^N\left[\left(1+\left(n-1\right){\beta}_p^i\right){t}_p\right]} $$

$ C $ is the total cost to the society that Individual $ i $ would prefer the society to not surpass. For use in BCA, WTP equals the cost, $ C $ , divided by the number of positions, $ N $ , as shown in Equation 17.

(17) $$ {WTP}^i(Citizen)=\frac{1}{N}C=\frac{1}{N}\left[\frac{\sum \limits_{p=1}^N\left(1+\left(n-1\right){\alpha}_p^i\right){x}_p}{\sum \limits_{p=1}^N\left(1+\left(n-1\right){\beta}_p^i\right){t}_p}\right] $$

WTP (Citizen) equals a ratio of altruism weighted benefit and cost share terms multiplied by 1/N. The multiplier 1/N is the individual’s expected cost share behind the VoI. WTP (Citizen) is similar to WTP (Social) (Equation 13) except for two characteristics: 1) the ratio is not multiplied by a cost share and 2) no additional weight is given to the position that the individual is in.Footnote ⁸ In other words, in WTP (Citizen) personal preferences have become impersonal.

To test if WTP (Citizen) meets the Benefit Condition, aggregate WTP (Citizen) under the assumption that $ {\alpha}_p^i={\alpha}_p $ and $ {\beta}_p^i={\beta}_p $ $ \forall i $ is entered into Equations 6 and 7 giving Equation 18.

(18) $$ {\displaystyle \begin{array}{c}\hskip-10em \sum \limits_p^N\left[\Delta {U}_p\right]=\sum \limits_p^N\left[\left(1+\left(n-1\right){\alpha}_p\right){x}_p\right]\\ {}\hskip15em -\sum \limits_p^N\left[\left(1+\left(n-1\right){\beta}_p\right){t}_p\right]\bullet \sum \limits_i^N\left[\frac{1}{N}\bullet \frac{\sum \limits_p^N\left(1+\left(n-1\right){\alpha}_p\right){x}_p}{\sum \limits_p^N\left(1+\left(n-1\right){\beta}_p\right){t}_p}\right]\end{array}} $$

Which simplifies to give Equation 19.

(19) $$ {\displaystyle \begin{array}{c}\hskip-15em \sum \limits_p^N\left[\Delta {U}_p\right]=\sum \limits_p^N\left[\left(1+\left(n-1\right){\alpha}_p\right){x}_p\right]\\ {}\hskip10em -\sum \limits_p^N\left[\left(1+\left(n-1\right){\beta}_p\right){t}_p\right]\bullet \frac{\sum \limits_p^N\left(1+\left(n-1\right){\alpha}_p\right){x}_p}{\sum \limits_p^N\left(1+\left(n-1\right){\beta}_p\right){t}_p}=0\end{array}} $$

Equation 19 will always equal zero. WTP (Citizen) therefore meets the Benefit Condition required for BCA to correctly identify potentially Pareto-improving projects in a consistent manner.

6. Illustrative BCA simulations

The results of the previous three sections showed that WTP (Citizen), WTP (Social), and WTP (Private) can differ. While WTP (Citizen) met the Benefit Condition, WTP (Private) and WTP (Social) did not. In this section, we use an illustrative simulation of a BCA to show how WTP and aggregate net benefit differ between WTP (Citizen), WTP (Social), and WTP (Private). We simulate a society of pure altruists with an efficiency motive but for comparison, the results for five alternate preferences specifications are presented in the Appendix.

Consider a society with three individuals (N = 3) called Alice, Bob, and Chioma who are respectively in Positions 1, 2, and 3 with incomes $ {w}_p $ of . Each individual is a pure altruist with an efficiency motive where all alpha and beta terms equal 0.2.Footnote ⁹

The society can enact one of four projects that differ in their distributional impacts described in Table 1 or maintain the status quo. A BCA is used to evaluate the projects. Each project delivers a total benefit of £600 but differs by the distribution of the benefits among the three positions ( $ {x}_1,{x}_2,{x}_3 $ ). Projects 1 and 2 deliver a benefit distribution of (£100, £200, £300) and Projects 3 and 4 deliver a benefit distribution of (£300, £200, £100). Each project has a total cost of £600 but projects are funded by different tax systems ( $ {t}_1,{t}_2,{t}_3 $ ). Projects 1 and 3 are funded through a uniform tax (0. $ \dot{3} $ ,0. $ \dot{3} $ ,0. $ \dot{3} $ ) whereas Projects 2 and 4 are funded through a proportional income tax (0.2, 0.3, 0.5).

Table 1.

Simulation parameters

The society has three individuals (N = 3) each with an income ( $ {w}_p $ ) and can choose one of four projects defined by the distribution of benefits ( $ {x}_p $ ) and cost shares ( $ {t}_p $ ).

Table 2 presents the results of the BCA in two steps. First, for each individual, WTP (Private), WTP (Social), and WTP (Citizen) are calculated using Equations 9, 13, and 17, respectively and WTP is aggregated for the three individuals. Second, aggregate WTP is used as a measure of benefits in a BCA. The BCA test is passed when aggregate WTP equals or exceeds the total cost of £600. If multiple projects pass the BCA test, the project with the greatest WTP is chosen.

Table 2.

Simulation results: WTP, the benefit–cost test, net benefit, and the Benefit Condition

WTP: WTP (Private), WTP (Social), and WTP (Citizen) are calculated for Projects 1–4 (Table 1) using Equations 9, 13, and 17, respectively. Aggregate WTP is the sum of Alice, Bob, and Chioma’s WTPs. BCA test: The BCA test is passed when aggregate WTP equals or exceeds the cost of £600. Of the projects that pass the BCA test, the project with the greatest aggregate WTP is chosen. Net Benefit: To evaluate each WTP against the Benefit Condition, the net benefit for each individual at a cost equal to the aggregate WTP is calculated using Equation 5. The aggregate net benefit is calculated using Equation 6. Benefit Condition: The Benefit Condition is met if aggregate net benefit equals zero as set out in Equation 7.

We then test the Benefit Condition described in Section 2. To do so, the net benefit of each individual is calculated using Equation 5 and aggregate net benefit is calculated using Equation 6 by replacing cost with aggregate WTP from the first step. The Benefit Condition (Equation 7) is met if aggregate net benefit equals zero as those who gain can exactly compensate those who lose when the total cost equals aggregate WTP.

Table 2 shows that WTP (Private) equals the individual benefit and aggregate WTP (Private) equals the sum of individual benefits which is equal to £600 for all projects. All four projects pass the BCA test. As each project has the same aggregate WTP, a choice cannot be made between the projects based on a BCA alone. The aggregate net benefit is zero for all projects which means the Benefit Condition is met. While the results from Section 3 showed that WTP (Private) in general does not meet the Benefit Condition, the case of a purely altruistic society with an efficiency motive is shown here to be an exception.Footnote ¹⁰ This result aligns with the results of Bergstrom (Reference Bergstrom and Jones-Lee1982) and Jones-Lee (Reference Jones-Lee1991). As expected, WTP (Citizen) also meets the Benefit Condition.

Using WTP (Social), Projects 1, 2, and 3 pass the BCA test. Project 2 is preferred as it has the highest aggregate WTP of £602. Project 2 provides a greater benefit for the higher income position and tax is proportional to income. The aggregate net benefit is negative and therefore does not meet the Benefit Condition. Even though a BCA using WTP (Social) would recommend Project 2 to be funded at a total cost of £602, a potential Pareto improvement could not be achieved. Project 4 is the least preferred as it has the lowest aggregate WTP. The aggregate net benefit is positive and WTP (Social) therefore does not meet the Benefit Condition, as expected. In this example, the consequence is that a BCA using WTP (Social) rejects Project 4 at any cost over £558 despite Project 4 being potentially Pareto improving up to a cost of £600. The only difference between Projects 1 and 2 and between Projects 3 and 4 is the tax shares. As WTP (Social) is in part determined by the tax shares (see Equation 13), the change in the progressivity of the tax system causes changes to both the individual and aggregate WTP (Social). The overall change is positive from Project 1 to 2 but negative from Project 3 to 4, which results in WTP (Social) not meeting the Benefit Condition.

Appendix Tables 2–6 show the results for alternate societies with different underlying preferences described in Appendix Table 1. This includes two types of paternalistic altruism: benefits-focused altruism ( $ \forall p,{\alpha}_p=0.2,{\beta}_p=0 $ ) where social utility is derived from others’ benefits from the project, and wealth-focused altruism ( $ \forall p,{\alpha}_p=0,\hskip0.3em {\beta}_p=0.2 $ ) where social utility is derived from others’ benefits from consuming the private good. We also include the maximin motive (only $ {\alpha}_1\ge 0,\hskip0.5em {\beta}_1\ge 0 $ ) where social utility is derived from the worst-off individual (Position 1) and the efficiency motive ( $ \forall p,{\alpha}_p=\alpha, \hskip0.5em {\beta}_p=\beta $ ) where all other individuals are given equal weight in the social utility function.

Across the different societies and projects, WTP (Private) consistently equals £600, the sum of the individual benefits, and never meets the Benefit Condition. WTP (Citizen) varies between societies and projects and consistently meets the Benefit Condition. WTP (Social) varies between societies in a similar manner to WTP (Citizen) but does not consistently meet the Benefit Condition. The case of benefit-focused altruism is shown to be an exception for WTP (Social).Footnote ¹¹

7. WTP and democratic principles

We have assessed WTP (Private), WTP (Social), and WTP (Citizen) against our Benefit Condition. Here we consider WTP in relation to democratic principles. Jones-Lee (Reference Jones-Lee1989, pg. 11) observes that when WTP is aggregated for use in a conventional BCA, the preference aggregation mechanism delivers the principle of “one pound, one vote” which can be criticized for being undemocratic. Jones-Lee goes on to argue that while “one person, one vote” is more democratic, WTP is still preferable over referenda for allocative decision-making as WTP gives a measure of the strength of preference. In referenda, an individual votes for the project if the total cost is less than or equal to their preferred maximum $ C $ . Aggregating votes gives the principle of “one person, one vote,” but not a strength of preference.

Using WTP (Private) or WTP (Social) in a BCA delivers the undemocratic principle of “one pound, one vote.” Each individual has a preference for the maximum total amount the society should spend on a project, $ C $ Footnote ¹² which is multiplied by their cost share to give their WTP. When cost shares are determined by income, for example, a proportional income tax, the preference of the individual in Position 1 (poorest) is given less weight than the preference of the individual in position $ N $ (richest) because $ {t}_1<{t}_N $ . In other words, the principle of “one pound, one vote.”

WTP (Citizen), when used in a BCA, simultaneously achieves the democratic principle of “one person, one vote” and gives a measure of strength of preference. For WTP (Citizen), an individual’s preference for $ C $ (Equation 16) is weighted by $ 1/N $ , which is constant, to give Equation 17. Each individual’s preference for the maximum total amount the society should spend on a project is given equal weight in the preference aggregation process and is therefore democratic. As such, the additional weight an individual places on the benefits and/or costs of particular positions in society are endogenous weights that reflect the individual’s distributional and ethical preferences.

8. Discussion and Conclusions

In this paper, we set out the theoretical underpinnings for a new citizen-based WTP, WTP (Citizen). Our model extends the standard social utility model (Bergstrom, Reference Bergstrom2006) of WTP for a public good when individuals are altruists by incorporating a VoI (Harsanyi, Reference Harsanyi1955). We have shown theoretically and illustrated empirically (using simulations) that, for such a society, WTP (Citizen) meets the Benefit Condition where WTP (Social) and WTP (Private) do not. Only if the Benefit Condition is met will BCA consistently reject projects that are not potentially Pareto improving and accept projects that are potentially Pareto improving. We conclude that for a BCA when individuals have altruistic and distributional preferences, WTP (Citizen) is the best of the three measures considered.

To demonstrate this result, a simple framework sufficed but of course, this came with a number of assumptions. The utility function is such that it encompasses a broad set of altruistic preferences over goods (public and private) and distribution. It is however possible that individuals have different distributional and ethical preferences than our assumed functional form covers. We assume linearity in the scale of costs and benefits and a marginal utility of income of one. Due to the assumption of constant marginal utility of income, the result that WTP (Social) does not meet the Benefit Condition is a result of the progressivity of the tax system rather than caused by different marginal utilities of income for richer and poorer individuals. To allow for interpersonal comparisons of utility when applying the Benefit Condition we also assume homogenous preferences over others. Future research could consider the impact of relaxing these assumptions. Nevertheless, the model is general enough to capture most of the preferences we might reasonably assume to exist in a society that enjoys—but must pay for—public goods. The exposition in this paper is by necessity (space constraints) limited to the theoretical model of WTP (Citizen) but an empirical application can be seen in Beeson et al. (Reference Beeson, Chilton, Jones-Lee, Metcalf and Nielsen2019) in which the WTP (Citizen) has been operationalized in an incentivized experiment.

We now turn to some additional ethical considerations that arise from the application of our WTP (Citizen). Starting with the principle of consumer sovereignty which according to Lerner (Reference Lerner1972) is defined as “letting each member of society decide what is good for himself, rather than have someone else play a paternal role” (Lerner, Reference Lerner1972 pg. 258). When discussing consumer sovereignty, Lerner (Reference Lerner1972, pg. 258) goes on to argue that economists “must be concerned with the mechanisms for getting people what they want, no matter how these wants were acquired.” We regard WTP (Private, Social, or Citizen) as being consistent with Lerner’s definition of consumer sovereignty if WTP: (1) lets an individual “decide what is good” by reflecting the individual’s altruistic and distributional preferences in their social utility function, and (2) meets the Benefit Condition making WTP, when aggregated for use in a BCA, a mechanism “for getting individuals what they want” based on the individuals’ social utility functions. Both WTP (Social) and WTP (Citizen) are consistent with the first part of Lerner’s definition as each reflects individuals’ altruistic and distributional preferences in their social utility function. WTP (Private) is not consistent with this definition of consumer sovereignty as it limits preferences to self-interest. Neither WTP (Private) nor WTP (Social) meets the Benefit Condition which means that a project may be recommended even though the cost exceeds the benefit to society, or a project may be rejected even though it is not beneficial. Only WTP (Citizen) meets the Benefit Condition and therefore is consistent with both parts of Lerner’s definition of consumer sovereignty.

Next, we consider democratic principles in relation to BCA. According to Jones-Lee, (Reference Jones-Lee1989 pg. 11): “what is needed is a variant of the ‘one person, one vote’ principle which will reflect the strength of individual preferences, bearing in mind the constraints imposed by the overall scarcity of resources” (Jones-Lee, Reference Jones-Lee1989 pg. 11). The traditional democratic principle of ‘one person, one vote’ does not measure strength of preference. Conventional BCA using WTP (Private) or WTP (Social) will measure people’s strength of preference but follows the principle of “one pound, one vote” which gives the poor less votes in the allocative decision-making process than the rich. We have shown that WTP (Citizen) is democratic whilst measuring the strength of preference and bearing in mind individual income constraints.

Traditionally, BCA focuses on efficiency with the possibility of applying exogenous weights to benefits and costs to reflect equity concerns (Robinson & Hammitt, Reference Robinson and Hammitt2011). Ultimately, both the use and setting of distributional weights require a normative judgment (Hammitt, Reference Hammitt2021) but so does the decision to not use distributional weights (Adler, Reference Adler2016). While it is often prescribed that these normative judgments are left to the decision-maker, our discussion of the principle of consumer sovereignty and democratic principles would indicate that if an individual’s welfare is in part determined by the distribution of benefits and costs in society, then the WTP values used in BCA should reflect this. Indeed, many policies such as climate change mitigation cannot be considered independently of the distributional consequences (Cai et al., Reference Cai, Cameron and Gerdes2010). This type of independence would be required for WTP (Private). By adopting WTP (Citizen), the additional weight that individuals place on the benefits and costs of particular positions in society are endogenous weights that reflect the public’s distributional and ethical preferences.

Our results show that while WTP (Social) does reflect an individual’s preference over distribution, it is not the correct approach to include those concerns in a BCA. Care should therefore be taken when considering the use of these values in a BCA. This is particularly important as stated preference studies have demonstrated that WTP does indeed depend on the distributional consequences of a project (Cai et al., Reference Cai, Cameron and Gerdes2010) and altruism (Gyrd-Hansen et al., Reference Gyrd-Hansen, Kjær and Nielsen2016; Simonsen et al., Reference Simonsen, Kjær and Gyrd-Hansen2021). Notably, our theoretical findings are independent of the elicitation method and not confined to the stated preference literature. If the choices reflected in revealed preference studies suffer from the problem of WTP (Social) described here, then the WTP values from revealed preference studies will be similarly affected.

In closing, we note that so far, an empirical ‘citizen’ approach such as those deployed by Blamey et al. (Reference Blamey, Common and Quiggin1995) and Curtis and McConnell (Reference Curtis and McConnell2002) have only to a very limited extent been underpinned by a formal utility model. In this paper we have addressed this gap by laying the foundations for eliciting a theoretically consistent WTP (Citizen) in a future, hypothetical empirical survey – one that is based on an individual’s social utility function and therefore reflects people’s preferences over others, and when aggregated for use in a BCA gives equal weight to each person’s preferences. Furthermore, it assures that potential Pareto-improving projects are correctly identified in a consistent manner.