Benefit Cost Analysis and the Entanglements of Love

Ted Bergstrom, UCSB

How do we do the counting?

A parent reports that she is willing to pay $100 to save her child from one day of cold symptoms.

How do we use her answer and those of others in a sample to evaluate public projects that affect child health?

Some entanglements

• Do we measure benefits of a project that reduces cold symptoms by multiplying number of child days times average willingness to pay of parents?

• But what if the child has 2 parents?

Do we calculate the sum of the 2 parents answers? Or the maximum? Or the minimum?

• Should we count the child’s own evaluation?

The logic of benefit cost for families

• What can benefit cost accomplish?

• How are families governed and how does it matter?

What can benefit cost do?

• Without explicit instructions about how to compare one person’s benefits with those to another, benefit cost cannot tell us whether a project should be adopted.

• Best we can expect is to learn whether a project is potentially Pareto improving.

• But this is very useful.

Standard Justification for B C

• With selfish individuals, a project is potentially Pareto improving if and only if the sum of individual willingnesses to pay for the project exceeds its cost.

Utility Possibilities and Benefit Cost

But what about families?

• We will need a theory of family decision making.• We will suppose that government cannot

intervene in intra-familial distribution.• We consider a project potentially Pareto

improving only if there is a way to assign costs to families so that given the household decision structure in families, no one is made worse off and someone benefits.

Benefit cost of child health with single-parent households

• Parent’s utility function is U(x,v(k,h)) where – Parent’s consumption x– Child’s consumption k– Child’s health h

• Parent chooses x and k to maximize U subject to x+k=m.

Willingness to pay

• Define U*(m,h)=max U(x,v(k,h)) s.t. x+k=m.

• A public project increases child’s health from h to h+.

• Willingness to pay for this improvement is W where U*(m-W, h+)=U*(m,h).

BC and WTP for single parents

• Result 1—If a project is potentially Pareto improving, it must pass BC cost for parents.

• Corollary—To count child’s valuation as well would accept ``too many’’ projects.

Is the converse true?

• Even if parents and kids agree about what is good for kids (ie parents’ aggregator v(k,h) is also kid’s utility function:

• Result 2— A project that passes BC test for parents might not be Pareto improving when kids’ utilities are accounted for.

Mom’s consumption, kid’s utility

Lovebirds without kids

• Archie and Bess care about their own consumption and health and about each others’ happiness.– UA(t)=vA (xA (t),hA (t))+aUB(t-1)

– UB(t)=vB (xB (t),hB (t))+aUA(t-1)

Utilities for allocations

• This dynamical system is stable if ab<1 and long run utilities converge to functions of consumption and health.

• UA*(xA,hA,xB,hB)=vA (xA,hA)+avB (xB,hB)

• UB*(xA,hA,xB,hB)=vB (xB,hB)+bvA (xA,hA)

Archie the dictator

• Archie controls allocation of private goods, while health depends on public policies.– Model used in Becker’s Rotten Kid Theorem,

• Except that Becker assumes that Bess is selfish.

– Appropriate for non-Western patriarchies?

A women’s health project will improve Bess’s health by .

• Survey Archie and/or Bess about their willingness to pay.

• What do we ask and who do we ask?

Asking Bess

• B.1 What is the largest amount of your own consumption that you would give up to improve your health by ?

• B.2 Given the way Archie allocates consumption in your family, what is the largest amount of family income that you would give up to pay for a project that improves your health by ?

Asking Archie

• A.1 What is the largest amount of family income that you would be willing to give up in order to improve Bess’s health by ?

How do the answers differ?

• If Bess’s health and her consumption are not strong substitutes, Archie’s willingness to pay for Bess’s health will exceed her willingness to pay out of her own income.

• In general, Bess’s willingness to pay out of household funds, given that Archie controls the private allocation, will be the same as Archie’s.

Non-Dictatorial Households

• Fairness-Based Consensus

• Households with bargained outcomes

Household welfare function

• Perhaps “successful marriages” share a notion of household fairness that overrides self-interest.

• Plausible case that this leads to decision making according to a social welfare function W=U*A+(1- )U*B

• The weights may depend on long run bargaining power.

BC with household welfare functions

• Archie and Bess know that household allocation maximizes household welfare function.

• Bess’s willingness to pay for an improvement in her health would be based on household welfare function--the same as Archie’s

• BC study should count only one of their valuations to determine whether a project allows potential Pareto improvement.

Household bargaining

• Nash-Rubinstein theory—Household allocations will be those that maximize the Nash product

(U*A -T A )(U*B-T B)

subject to xA+xB=m, where TA and TB are the “threat point” utilities that Archie and Bess could achieve if agreement is not reached.

Differing answers

• If the household outcome is the result of bargaining, then in general Archie and Bess will have different willingnesses to pay for a public project.

• Each will consider whether his or her own utility is higher in the bargaining equilibrium with or without the project.

• Project may shift threat points and “twist” utility possibility frontier.

Bargaining and Benefit Cost

Annoying paradox?

• Although utility possibility sets in previous example are nested, if that outcome will be bargained, there is no way to implement the project without harming someone.

• Similar difficulty observed by Lundberg and Pollak.

Couples with kids

• Kids are a household public good.

• How do answers to interview questions relate to potential Pareto improvements?

• Depends on household governance structure.

Possible structures

• Intact household, father is dictator

• Intact household, adults share a social welfare function

• Intact household, bargained solution.

• Divorced parents with kids

Dictatorial case

• Where father is benevolent dictator, father’s willingness to pay for child’s health is same as child’s.

• Mother’s willingness to pay out of family income for child’s consumption may exceed fathers.

• Potential Pareto optimum criterion recommends using minimum of father’s and mother’s wtp.

Shared social welfare case

• Archie and Bess agree in their willingness to pay for child health.

• Appropriate BC measure is answer of either one of them, not the sum.

Divorced households

• If neither Archie nor Bess voluntarily gives money to the other then taxes paid by either to pay for health project do not affect budget of the other.

• Appropriate answer for benefit cost is now the sum of the willingnesses to pay of the two parents for an improvement in the child’s health.

How about grandparents and Uncle Charlie and Aunt Dorothy?• Biological theory and empirical observation

suggests that people care strongly about well-being of their near relatives such as grandchildren, nieces and nephews.

• If budgets are not shared across households, a good case for adding willingness to pay of these relatives.

• Hamilton’s calculus suggests magnitudes.

Conclusion

• I’m done.