Benefit Cost Analysis and the Entanglements of Love Ted Bergstrom, UCSB.
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Transcript of Benefit Cost Analysis and the Entanglements of Love Ted Bergstrom, UCSB.
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Benefit Cost Analysis and the Entanglements of Love
Ted Bergstrom, UCSB
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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?
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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?
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The logic of benefit cost for families
• What can benefit cost accomplish?
• How are families governed and how does it matter?
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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.
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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.
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Utility Possibilities and Benefit Cost
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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.
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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.
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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).
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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.
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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.
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Mom’s consumption, kid’s utility
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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)
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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)
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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?
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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?
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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 ?
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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 ?
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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.
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Non-Dictatorial Households
• Fairness-Based Consensus
• Households with bargained outcomes
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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.
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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.
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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.
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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.
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Bargaining and Benefit Cost
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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.
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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.
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Possible structures
• Intact household, father is dictator
• Intact household, adults share a social welfare function
• Intact household, bargained solution.
• Divorced parents with kids
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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.
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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.
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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.
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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.
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Conclusion
• I’m done.