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Benefit-Cost in a Benevolent Society

Theodore C. Bergstrom∗

May 18, 2004

1Introduction

Alice and Bob live together. Though fond of each other, they maintain separate

budgets. They have been offered a chance to rent a larger apartment. The

apartment has two extra rooms, a study for Alice and a lounge for Bob. Alice

would be willing to pay $100 a month to have the study and Bob would be

willing to pay $100 a month to have the lounge. Alice would never use the

lounge and Bob would never use the study, but each likes the other to be happy.

For this reason, Alice would be willing to pay $50 a month for Bob to have the

lounge and Bob would be willing to pay $50 a month for Alice to have the study.

The additional rent for the larger apartment is $250 per month. Should they

accept the offer on the grounds that total benefits from the larger apartment

are $300, or reject it on the grounds that total benefits are only $200?

More generally, how should benefit-cost analysis account for the value that

benevolent individuals place on other people’s pleasure from public goods?

When adding up the benefits to be compared with costs, should we sum the

private valuations, the altruistic valuations, or something else?

An intriguing paper by Viscusi, Magat, and Forrest [26] suggests that bene-

fits from improvements in public health “consist of two components: the private

valuation consumers attach to their own health, plus the altruistic valuation

that other members of society place on their health.” Viscusi et al attempted

to measure the two components separately, using a survey in which they asked

subjects to state their willingness to pay for a hypothetical product that would

reduce their personal risk and also asked for their willingness to pay for an

advertising campaign that would result in an equivalent reduction in risk for

a larger population. As the authors point out, even a slight concern for the

well-being of each member of a large population could amount to a substantial

willingness to pay for a public benefit. In a sample of citizens of Greensboro,

∗Aaron and Cherie Raznick Professor of Economics, University of California, Santa Bar-

bara.

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North Carolina, Viscusi and coauthors found that, on average, subjects were

willing to pay about 5 times as much to reduce a specific hazard for all North

Carolinans as to reduce this hazard for themselves alone. For a similar benefit

to all U.S. citizens, subjects would be willing to pay about 6 times as much

as for themselves only. Even if these hypothetical claims of altruism are over-

stated, the magnitude of the altruistic component of public benefits appears to

be significant. Thus, the question of how to treat these valuations in benefit-cost

analysis becomes a question of the first order of importance.

Before we turn to general matters, let us to try to resolve the rental dilemma

for Alice and Bob. Suppose that they decide to rent the larger apartment and

split the rent equally. If they take the larger apartment, then on selfish grounds,

Alice is worse off, since she is giving up $125 in return for a study that she

values at only $100. Will this private loss be compensated by an improvement

in Bob’s well-being? Bob is now paying an extra $125 in return for a lounge

that he values at $100. So it appears that when she accounts for her concern

for Bob’s well-being, she will find the option of taking the apartment even less

satisfactory. A symmetric argument applies to Bob’s consideration of Alice’s

well-being. Thus if they base their decision on a comparison of the cost of the

apartment with a measure of total benefits which includes the altruistic values

of $50, they are led to an outcome in which both are worse off.1The story of

Alice and Bob illustrates a general principle. If we are to count the altruistic

benefits that each gains from the other’s enjoyment of the new apartment, we

should not forget to also count the cost that each attributes to the fact that the

other will have to pay more rent.

In an earlier paper, Bergstrom [2] claimed that in general, when altruism

takes a “purely altruistic” form, the appropriate way to measure benefits is to

sum private valuations, excluding altruistic valuations. Bergstrom’s argument

is based on the observation that with pure altruism, the marginal first-order

optimality conditions are the same as those that apply if account is taken only

of the private valuations. Jones-Lee [16] [17], who referred to Bergstrom’s result

as “rather arresting,” offered a more thorough discussion. Jones-Lee showed that

the same conclusion extends to an interesting case of paternalistic preferences,

and showed that when concern for others is “safety-focussed” rather than purely

altruistic, the appropriate benefit measures are intermediate between the private

and social values. In a critical discussion of contingent valuation methods,

Milgrom [20] maintained that the appropriate measure of benefits is the sum

of private valuations.2Hanemann [12] disputed Milgrom’s conclusion on the

1For those who would like to see a more formal development of this example, the Appendix

develops this story within the context of a simple utility theory.

2Milgrom’s paper appeared in a conference volume, along with an interesting discussion

between Milgrom and several environmental economists who found it difficult to accept the

assertion that altruistic values should be ignored in benefit-cost studies.

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grounds that altruistically motivated valuation is as legitimate as valuation for

any other reason.

Johansson [14] suggested that if there is significant altruism, then benefit

measures based on contingent valuation studies may lead to overestimates of the

benefits of public projects because subjects may inappropriately include their

altruistic valuations as well as their private valuations. Johansson proposed that

it would be appropriate to ask subjects to state their own willingness to pay for

a public project, conditional on the assumption that all others are taxed at rates

equal to their private valuations.

This paper seeks general principles for evaluating public projects in a society

with altruism. The calculus necessary conditions for a Pareto optimum studied

by Bergstrom, Jones-Lee, and Johansson are not exactly what is needed in

a benefit-cost analysis. These conditions allow one to determine whether the

current state is a local Pareto optimum. If the current state satisfies the relevant

first and second-order conditions, then we know that no “small” changes can be

Pareto improving. But to satisfy the mission of benefit-cost analysis, we need to

consider situations where potential changes are not infinitesimal, and also where

neither the status quo nor the alternative is necessarily a Pareto optimum.

2 What Can Benefit-Cost Analysis Tell Us?

Before we turn our attention to measuring benefits in societies with benevolence,

let us think through the justification for benefit-cost studies in an economy with

selfish consumers.

2.1 Utility Possibilities and Potential Pareto Improvement

Without explicit instructions about how to compare one person’s benefits with

the losses of another, we can not expect benefit-cost analysis to tell us whether

a public project should or should not be adopted. The best we can hope for

from benefit-cost analysis is to learn whether a project is potentially Pareto

improving.

Consider a community with two selfish people, one private good and m

public goods. An allocation is determined by the quantities of private good

for persons 1 and 2 and the vector y of public goods that is available. The

two people are endowed with a total of W units of private good, which is to

be allocated between consumption for person 1, consumption for person 2, and

inputs for the production of public good. Suppose that the amount of private

goods needed to produce the vector y of public goods is C(y). Then if the supply

of public goods is y, the total amount of private goods to be divided between 1

and 2 is W −C(y). Given the vector y of public goods, each feasible distribution

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of private goods determines a distribution of utilities between persons 1 and 2.

We define the graph of such utility distributions to be the y-contingent utility

possibility frontier. One such curve is shown as UP(y) in Figure 1.

Figure 1: Utility Possibilities and benefit-cost

U1

U2

A

UP(y?)

UP(y)

Suppose that a public project will increase the amounts of public goods

from y to y?at a cost of some reduction in total private consumption. With

the project in place, there is a new utility possibility frontier UP(y?). In Figure

1, neither of the two curves, UP(y) and UP(y?) lies entirely beneath the other.

Thus there is no unambiguous way to determine which is the better outcome.

Some utility distributions are attainable only if the project is implemented and

others are attainable only if it is not.

Let us suppose that initially the amount of public goods is y and that the

distribution of private goods corresponds to the utility allocation marked A in

the figure. We see that the curve UP(y?) includes points that are above and to

the right of A. This implies that it is possible to change the supply of public

goods from y to y?and still have enough private goods left over so that both

individuals can be made better off than they were at A. When this is the case,

we say that the project is potentially Pareto improving.

These ideas extend naturally to the case of more than two consumers. Where

xi is private consumption of consumer i, the set of all feasible allocations is

{(x1,...,xn,y)|?xi+ C(y) ≤ W}, where C(y) is the cost in terms of pri-

possibility frontier in n dimensions is then defined in the obvious way.

vate goods of producing the vector y of public goods. The y-contingent utility

Definition 1 Suppose that the initial allocation of private and public goods

is A = (x1,...,xn,y). A change in the amount of public goods from y to

y?is potentially Pareto improving if there exists a feasible allocation A?=

(x?

1,...,x?

n,y?) such that A?is Pareto superior to A.

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2.2A Benefit-cost Test

Where the initial allocation is A = (x1,...,xn,y), we define an individual’s

willingness-to-pay for changing the vector of public goods from y to y?as the

quantity of private goods that she would be willing to sacrifice in return for

this change. This number could be either positive or negative, depending on

whether i prefers y?to y or vice versa.

Definition 2 If the initial allocation is (x,y) = (x1,...,xn,y), then individual

i’s willingness to pay for changing the amount of public goods to y?is wiwhere

wisolves the equation Ui(xi− wi,y?) = Ui(xi,y).

For an economy with selfish individuals, a simple benefit-cost test determines

whether a change is potentially Pareto improving.

Theorem 1 If individuals are selfish and the initial allocation is (x,y) where

?

to-pay for the change exceeds the difference in cost C(y?) − C(y).

A detailed proof of Theorem 1 is found in the Appendix. But the idea

behind the proof is quite simple. If the sum of willingnesses-to-pay exceeds

total cost, then it is possible to change the amount of public goods from y to

y?and pay for this change by collecting an amount from each individual that is

smaller than her willingness to pay. Doing so constitutes a Pareto improvement.

Conversely, if the change is potentially Pareto improving, there must be a way

to distribute the costs of the change so that nobody is worse off after the project

is implemented and cost shares are assigned. By definition, this implies that

each individual’s share of the cost is smaller than his willingness-to-pay for the

project. Since the cost shares add to the total cost of the project it follows that

the project would pass the benefit-cost test.

ixi+C(y) = W, then a change in the amount of public goods from y to y?is

potentially Pareto improving if and only if the sum of individual willingnesses-

2.3A Calculus-Based Necessary Condition

Theorem 1 is very general in the sense that it does not depend on preferences

being convex or continuous and it applies whether the change being considered

is large or infinitesimal. But the weakness of this theorem is that it only gives

us a way to evaluate projects one at at time. Applying the benefit-cost criterion

in this theorem would requires a separate survey of consumers to determine the

merits of every possible public project.

If we are willing to assume that preferences are smooth and convex, then

calculus-based methods allow us to make more sweeping judgments about the

direction of potential Pareto improvements, which depend simply on a com-

parison of marginal costs and marginal willingnesses to pay. The first order

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necessary condition for efficient provision of public goods was elucidated by

Samuelson [22] and is known as the Samuelson condition. This condition re-

quires that at an interior Pareto optimum, for each public good, the sum of

all consumers’ marginal rates of substitution between that public good and the

private goods is equal to the marginal cost of that public good.

While the Samuelson condition can be used to determine whether an existing

allocation is Pareto optimal, this is not exactly the result that is needed for

benefit-cost analysis. The task of benefit-cost studies is to determine whether

specific changes in the amount of public goods could be financed in a way that

the outcome is Pareto improving. In a “convex environment,” it turns out

that a simple extension of Samuelson’s result allows one to determine whether

an increase or decrease in the amount of public goods is potentially Pareto

improving.

Definition 3 At a feasible allocation (x,y), we say that an increase in the

amount of public good j passes (fails) the Samuelson test if at the allocation

(x,y) the sum of marginal rates of substitution between public good j and the

private good is greater than (less than) the marginal cost Cj(y) of public good j.

Under appropriate convexity assumptions, the Samuelson test gives us simple

necessary conditions for an increase or for a decrease in the amount of a public

good to be potentially Pareto improving. A proof of the following result is found

in the Appendix.

Theorem 2 If preferences of each individual are selfish and convex and if the

cost function C(y) is convex in y, then a necessary condition for an increase in

the amount of public goods to be potentially Pareto improving is that an increase

passes the Samuelson test. A necessary condition for a decrease in the amount

of public goods to be potentially Pareto improving is that an increase fails the

Samuelson test.

2.4Benefit-Cost as a First Step

An attractive feature of benefit-cost analysis is that it seems to allow analysts

to make policy recommendations about specific public expenditures without

taking a stand on questions of income distribution. This independence of distri-

butional considerations is achieved by focussing on potential rather than actual

Pareto improvements. Efforts to construct a useful benefit cost analysis in the

absence of distributional judgments has a long history in economics. In 1939,

Kaldor [18] and Hicks [13] articulated the view that has come to be known as the

New Welfare Economics, and which centers on the “compensation principle.”

The compensation principle states that an institutional change constitutes an

improvement if it is possible for the gainers to compensate the losers for their

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losses and still be better off after the change, whether or not the redistribution

actually takes place.3

The modern consensus is that the case for distribution-independent project

evaluation is much weaker than was originally hoped. Samuelson [21] demon-

strated that if the utility possibility sets corresponding to alternative policies

are not nested, then the social orderings implied by these criteria are highly

unsatisfactory. Chipman and Moore point out that in economies with several

private goods and no public goods, nesting of the utility possibility sets re-

quires essentially that preferences be identical and homothetic, or with some

further qualifications, of the Gorman polar form. Bergstrom and Cornes [1]

show that for an economy with public goods, utility possibility sets correspond-

ing to amounts of public goods will be nested only under special circumstances

that are formally dual to the Gorman polar form.

It is noteworthy that Hicks’ founding manifesto of the New Welfare Eco-

nomics [13], does not appear to advocate acceptance of reforms that pass the

compensation test if compensation is not actually paid. Hicks states that

“The main practical advantage of our line of approach is that

it fixes attention on compensation. Every simple economic reform

inflicts a loss on some people... Yet when such reforms have been

carried through in historical fact, the advance has usually been made

amid the clash of opposing interests, so that compensation has not

been given, and economic progress has accumulated a roll of victims,

sufficient to give all sound policy a bad name.” [13], p. 711

Much of the controversy surrounding the use of the criterion of “potential

Pareto improvement” seems to be avoidable if we think of benefit-cost analysis

as only a first step in a project evaluation. For example, if an increase in the

amount of a public good passes the marginal Samuelson test, then we know that

some increase in the amount of the public good is potentially Pareto improving.

Of course this does not mean that every possible way of implementing the

project is Pareto improving. All we know is that there would be some way of

increasing the amount of this public good and dividing costs so that everyone

benefits. The next step in evaluation of the project is to consider alternative

ways of financing this project and estimating who will then be the winners

and the losers. Incentive problems will normally prevent policy makers from

knowing exactly how much each individual values the project and so estimates

of the distribution of winners and losers will be statistical and not exact. It is

usually not reasonable to expect that literally everyone will be better off after

the change, but may be possible to use available information to ensure that a

3An elegant and enlightening intellectual history of the compensation principle and the

New Welfare Economics is found in Chipman [8]. Chipman and Moore [7] present a rigorous

treatment of these issues, using modern techniques.

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very large fraction of the population benefits from the policy and that very few

individuals are significantly harmed.

There is an interesting asymmetry in the results of a marginal benefit-cost

tests. If an increase in the amount of a public good fails the test, then it must be

that there is no way to divide the costs of the project in such a way to achieve

a Pareto improvement. The project can reasonably be described as “special

interest legislation”. To make a case in favor of a project that fails the test, one

would need to argue that implementing this project and paying for it with a

specified tax scheme is likely to achieve redistributive goals that for some reason

could not be more efficiently achieved through redistribution of private goods.

3 Benevolence and benefit-cost

3.1 Altruistic Preferences and Private Values

Let us now apply benefit-cost analysis to an economy where some people have

benevolent feelings toward others. We define i to be purely altruistic to j if i’s

sympathy for j is in agreement with j’s perception of her own well-being. This

notion is especially easy to formulate in a model with some “private goods”

and some “public goods.” Let xi be the vector of private goods consumed

by person i and let y be the vector of public goods supplied. An allocation

(x,y) = (x1,...,xn,y) specifies the private consumption of each i and the vector

of public goods available to the entire community. For each i, let there be a

private utility function vi(xi,y) representing i’s private preferences. We say

person i is purely altruistic if i has preferences over allocations that can be

represented by a “social utility function”

Ui(x,y) = Ui(v1(x1,y),...,vn(xn,y)) (1)

where Uiis an increasing function of viand a non-decreasing function of vjfor

all j ?= i.4

We can define person i’s private value for a change in the amount of public

goods as i’s willingness to pay for the change, accounting only for i’s private

utility.

Definition 4 If person i has private consumption xi, then i’s private value for

the change from y to y?units of public goods is the solution wito the equation

vi(xi− wi,y?) = vi(xi,y).

4Bergstrom [3] and [4] shows that under reasonable conditions, the existence of a utility

representation of the form (1) is implied by the existence of utility functions such that for

each i, Ui depends on i’s private utility vi(xi,y) and the altruistic utilities, Uj of all other

individuals

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3.2A Private Values benefit-cost Test

Where preferences are purely altruistic, we can define a private values benefit-

cost test as follows.

Definition 5 Suppose that the initial allocation in an economy is (x,y). A

public project that changes the amount of public goods from y to y?passes the

private value benefit-cost test if the sum of all persons’ private values for the

change from y to y?units of public goods exceeds the cost difference C(y?)−C(y).

We will say that a project is potentially privately improving if it is possible

to implement the project and pay for it in such a way that somebody’s private

utility increases and nobody’s private utility decreases. More formally:

Definition 6 If the initial allocation is (x,y), a change in the amount of public

goods from y to y?is potentially privately improving if there exists a feasible

allocation (x?,y?) such that vi(x?

for some i.

i,y?) ≥ vi(xi,y) for all i with strict inequality

The same reasoning that established Theorem 1 shows that a project is

potentially privately improving if and only if the sum of private values for the

project exceeds its total cost. Since the social utility functions Uiare increasing

in viand nondecreasing in vjfor all j ?= i, it follows that an increase in all of the

private utilities vjis a sufficient condition for an increase in each of the social

utilities Ui. Therefore we can conclude that:

Remark 1 If preferences are purely altruistic, then a change in the amount of

public goods is potentially Pareto improving if it passes the private value benefit-

cost test.

3.3Sufficient but not Necessary

Although any project that passes the private value benefit-cost test is poten-

tially Pareto improving, the converse is not true. When altruism is present, the

private values benefit-cost test is sometimes “too demanding” in the sense that

it rejects potentially Pareto improving changes. Because individuals care about

the well-being of others, there may be Pareto improvements in which reductions

in private utility for some individuals are compensated by increases in private

utilities for others whom they care about.

Figure 2 illustrates a potentially Pareto improving change that fails the

private value benefit-cost test. The figure displays private utilities vi on the

axes.The curve UP(y) shows the private utility distributions attainable if

the amount of public good is y. The point A = (v1(x1,y),v2(x2,y)) is the

distribution of private utilities in the initial allocation. The area V+above and

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to the right of the dashed lines shows all of the “privately improving” outcomes

where every consumer’s private utility is higher than at the point A. The area

above the curve U1U1represents combinations of v1and v2that person 1 prefers

to the outcome A and the area above U2U2represents combinations of v1and v2

that person 2 prefers to A, taking account of altruistic preferences. The private

utility distributions corresponding to outcomes that that are Pareto superior to

A are those that are above both of these two curves, which is the area above

the curve, U2AU1.

Figure 2: Pareto-Improving Transfers

v1

v2

A

V+

U1

U2

U2

U1

UP(y)

UP(y?)

A project that changes the supply of public goods from y to y?passes the

private value benefit-cost test if and only if the y?-contingent utility possibility

frontier UP(y?) intersects the region V+. In Figure 2, the curve UP(y?) does

not intersect V+, but does pass through the region above the curve U2AU1.

Therefore a change in the amount of public goods from y to y?fails the private

value benefit-cost test, but is potentially Pareto improving.

Remark 2 With purely altruistic preferences, the private value benefit-cost test

is not a necessary condition for a change in the amount of public goods to be

potentially Pareto improving.

Indeed in Figure 2, we see that the y-contingent utility utility possibility

curve UP(y) also intersects the region above U2AU1, which means that it would

be possible to achieve a Pareto improvement simply by redistributing private

income, without a change in the amount of public goods.

On reflection, it is apparent that if a Pareto improvement can be accom-

plished by redistribution of private goods, then even an entirely wasteful (but

small) change in public expenditure could qualify as potentially Pareto improv-

ing. This suggests that if we seek useful benefit-cost criteria, we need to consider

changes that start from an allocation from which no Pareto improvements can

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be achieved simply redistributing private goods. It will be useful to define dis-

tributionally efficient allocations as follows:

Definition 7 An allocation (x,y) is distributionally efficient if there is no fea-

sible allocation (x?,y) that is Pareto superior to (x,y).

In Figure 3, the allocation A is distributionally efficient. In this figure allo-

cations that are Pareto superior to A must lie in the region that is above the two

curves U1U1and U2U2. Since the curve UP(y) does not cross into this region,

there is no way to achieve a Pareto improvement simply by redistribution of

private goods.

Figure 3: A Distributionally Efficient Allocation

v1

v2

U2

U1

A

UP(y)

V+

U2

U1

UP(y?)

Even if the initial allocation is distributionally efficient, passing the private

value benefit-cost test is not a necessary condition for a change in the amount

of public goods to be potentially Pareto improving. We can see this from the

example in Figure 3. In this diagram, the y?-contingent utility possibility frontier

does not intersect V+and hence y?does not pass the private values benefit-cost

test. Nevertheless, since UP(y?) intersects the area above the two curves U1U1

and U2U2, there exists a feasible allocation (x?,y?) that is Pareto superior to

(x,y). Thus a change in the public goods supply from y to y?is potentially

Pareto improving.

As this example illustrates, we can not hope to find a benefit-cost test that

is based only on private values and that yields “global” necessary and sufficient

conditions for potential Pareto improvement.

3.4 A Useful Necessary Condition

The previous section showed that in an economy with pure altruism, passing

the private value benefit-cost test is a sufficient but not a necessary condition

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for a project to be potentially Pareto improving. In fact, we showed that even

if the initial allocation is distributionally efficient, there may be changes in the

amount of the public goods that fail the private value benefit-cost test, but

are potentially Pareto improving. Nevertheless, it turns out that, subject to

some fairly weak technical assumptions, there is a simple marginal condition

that depends only on private evaluations and is necessary for a change to be

potentially Pareto improving. The following definitions are helpful for stating

this result.

Definition 8 Consumer i’s private marginal rate of substitution between public

good j and private goods is the ratio mij(xi,y) between the partial derivative of

vi(xi,y) with respect to yj and the partial derivative of vi(xi,y) with respect to

y.

We define the desired condition as follows.

Definition 9 At the allocation (x,y), public good j is said to pass the private

values Samuelson test if

?

and to fail the private values Samuelson test if if this inequality is reversed.

i

mij(x,y) > Cj(y).

(2)

The necessary condition that we seek will be valid under the following as-

sumptions.

A. 1 Each consumer i has purely altruistic preferences, with a social utility

function Ui(v1,...,vn) that is quasi-concave, differentiable, nondecreasing in

all vjand increasing in vi.

A. 2 The private utility functions vi(xi,y) are concave, differentiable, and in-

creasing in xi.

A. 3 The set of feasible allocations is {(x,y)|?

Theorem 3 In an economy where Assumptions A.1-A.3 are satisfied and where

the initial allocation (¯ x, ¯ y) is distributionally efficient, a necessary condition for

an increase in the amount of public good j to be potentially Pareto improving is

that public good j does not fail the private values Samuelson test. Similarly, a

necessary condition for a decrease in the amount of public good j to be potentially

Pareto improving is that public good j does not pass the private values Samuelson

test.

ixi+ C(y) ≤ W}, where the

cost function C(y) > 0 is convex in y.

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Proof or Theorem 3.

good j fails the private values Samuelson test, then no increase in the amount

of j can be potentially Pareto improving. To show this, let (¯ x, ¯ y) be the initial

allocation. If public good j fails the Samuelson test, then, as is demonstrated in

Lemma 1 of the Appendix, there is some decrease in the amount of public good

j that is potentially Pareto improving. That is, there exists a feasible allocation

(x,y) such that yj < ¯ y, yk = ¯ yk for k ?= j and (x,y) is Pareto superior to

(¯ x, ¯ y). Suppose that an increase in the amount of public good j from ¯ yjto y?

is potentially Pareto improving. Then there exists a feasible allocation (x?,y?)

that is Pareto superior to (¯ x, ¯ y). Since y?

such that ¯ y = λy+(1−λ)y?. Consider the allocation λ(x,y)+(1−λ)(x?,y?). The

assumption that C is a convex function implies that this allocation is feasible.

Since (x,y) and (x?,y?) are both Pareto superior to (¯ x, ¯ y), assumptions in A.1

and A.2 on the convexity of preferences imply that λ(x,y) + (1 − λ)(x?,y?) =

(λx + (1 − λ)x?, ¯ y) is Pareto to superior to (¯ x, ¯ y). But this contradicts the

assumption that the initial allocation is distributionally efficient. It follows that

if an increase in the amount public good j fails the private values Samuelson

test, then a small decrease in the amount of public goods is potentially Pareto

improving and no increase in the amount of public goods can be potentially

Pareto improving.

A parallel line of reasoning shows that if an increase in the amount of a

public good passes the private values Samuelson test, then a small increase in

the amount of that public good is potentially Pareto improving and no decrease

is potentially Pareto improving.

The first claim of this theorem is that if public

j

j> ¯ yj > yj, there is some λ ∈ (0,1)

3.5A Puzzling Observation Explained

Section 3.3 showed an example where an increase in the amount of a public good

is potentially Pareto improving, even though this change fails a private values

benefit-cost test. Consumer 2’s altruism towards Consumer 1 is sufficient for

him to prefer the new allocation to the old, even though his private utility is

lower after the change.

Since it is possible to achieve a Pareto improvement without increasing pri-

vate utilities of every consumer, how can the private values Samuelson condition,

which accounts only for private marginal rates of substitution, be a necessary

condition for an increase in the amount of a public good to be Pareto improving?

The explanation of this seeming paradox is as follows. The proof of Theo-

rem 3 demonstrates that if the initial allocation is distributionally efficient and

if certain convexity and continuity conditions are satisfied, then whenever an

increase in the amount of a public good is potentially Pareto improving in terms

of the altruistic utilities Ui, it must be that a decrease in the amount of this

public good is not potentially Pareto improving in terms of the private utilities,

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vi. But a decrease in the amount of a public good will be potentially Pareto

improving in terms of the private utilities if that public good fails the private

values Samuelson condition. Therefore, under the assumptions of Theorem 3,

a necessary condition for an increase in the amount for a public good to be

potentially Pareto improving when altruism is taken into account is that the

public good does not fail the private goods Samuelson condition.

4 Discussion and Applications

What have we learned that can guide policy-makers and practitioners? The

story of Alice and Bob suggests that there is reason to heed Johansson’s [14]

warning that ill-formed contingent evaluation studies may yield misleading re-

sults. Imagine that Alice and Bob hired a naive benefit cost analyst to decide

whether they should take the new apartment. If the analyst asked each of them,

“How much would you be willing to pay to have the larger apartment?” they

would each reply $150. The analyst would then report total benefits of $300 and

recommend that they take the apartment so long as the extra rent did not ex-

ceed $300. But if the rent is more than $200 and is split equally between them,

they will both be worse off if they take the apartment. What went wrong? The

analyst evidently asked the wrong question. This question encouraged answers

that include the sympathetic value that each places on the other’s utility for the

new apartment but neglect the sympathetic cost that each would feel because

the other has to pay more rent.

The analyst might instead have asked each of them. “If the cost of moving

to the new apartment is split equally between you, what is the most that you

yourself would be willing to pay for the larger apartment?” Then each would take

account of the costs as well as the benefits to the other, and each would answer

$100. The analyst would then correctly recommend taking the apartment only

if the extra rent did not exceed $200. The symmetry of this example makes

it natural for the decision-maker to propose splitting costs equally. But in less

symmetric circumstances, it would not be obvious what division of costs to

propose. In principle, the analyst could discover potential improvements, but

this might require asking many different questions, each of which proposed a

different division of costs.

Our results suggest that for many purposes, a simpler approach will suffice.

In the case of Alice and Bob, the analyst could ask each of them a single question.

”How much would you be willing to pay for the benefits that you yourself realize

from the larger apartment, ignoring any benefits to the other person?” Moving

to the larger apartment will be potentially Pareto improving if and only if the

sum of these two measures of benefit exceeds the additional cost.

More generally, Theorem 3 tells us that the private values Samuelson test

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can determine the appropriate direction of change in the quantity of any public

good. For this test, marginal benefits might be determined by asking a question

like: “How much would you be willing to pay per unit for a small increase in the

amount of this public good ignoring any benefits (or costs) that may be accrue

to others.

Benefit cost studies based on surveys in which individuals are asked their

willingness to pay for public amenities are routinely used by government agencies

in many countries. As Hanemann [12] and Carson [6] explain, these studies

vary widely in design and in quality. While there has been an energetic debate

over the validity of contingent evaluation studies, Hanemann reviews a body

of evidence that suggests that carefully conducted contingent valuation studies

exhibit reliability and consistency with other measures of willingness to pay.

For some projects that improve public health and safety, it is relatively

easy to distinguish private values from altruistically motivated values. Several

interesting studies have involved interviews of a sample of individuals, who are

asked for their valuations of hypothetical changes in the risk of various adverse

health effects. These studies carefully frame the hypothetical situations so as

to distinguish between subjects’ valuations of the effects on their own health

and the value that they place on health benefits to others. Viscusi and his

coauthors [26] asked their subjects what they would pay for a product that

would increase their own safety, but not that of others, and then separately asked

about willingness to pay for extending the same benefits to a larger population.

Jones-Lee, Hamerton, and Philips [15] asked a sample of British adults about

their willingness to pay for a hypothetical safety device on their own cars that

would increase only the driver’s safety. A second question asked about their

willingness to pay for a device that the increased the safety of the passengers

as well as the driver. Dickie and Ulery [11] and Dickie and Gerking [10] asked

parents to state the amount that they would be willing to pay to spare their

own children from illnesses with a specific list of unpleasant systems. They

also asked questions about wilingness to pay to avoid the same symptoms for

themselves and for others outside the family.

There have also been several “revealed preference” studies of willingness to

pay to avoid private hazards. These are catalogued in a recent survey by Viscusi

and Aldy [25] Economists have used data on wages and occupational hazards

to estimate the amount the prices at which individuals will accept risks to their

own life and limb. Other studies have related the reduction in the price of

houses near hazardous wastes to the hazards involved. Others have attempted

to hedonic estimates of the value of additional safety features of automobiles.

These health and safety studies avoid the pitfall of which Johansson warns.

The contingent valuation surveys focus deliberately on the respondents’ private

willingness to pay for their own or their childrens’ health. Some of these studies

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