Benefit Cost Analysis and the Entanglements of Love

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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 childs 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 persons 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

• Parents utility function is U(x,v(k,h)) where
• Parents consumption x
• Childs consumption k
• Childs health h
• Parent chooses x and k to maximize U subject to
  xkm.

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Willingness to pay

• Define U(m,h)max U(x,v(k,h)) s.t. xkm.
• A public project increases childs 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 1If a project is potentially Pareto
  improving, it must pass BC cost for parents.
• CorollaryTo count childs 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
  kids 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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Moms consumption, kids 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 abrun 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 Beckers Rotten Kid Theorem,
• Except that Becker assumes that Bess is selfish.
• Appropriate for non-Western patriarchies?

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A womens health project will improve Besss
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 Besss health by ??

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How do the answers differ?

• If Besss health and her consumption are not
  strong substitutes, Archies willingness to pay
  for Besss health will exceed her willingness to
  pay out of her own income.
• In general, Besss willingness to pay out of
  household funds, given that Archie controls the
  private allocation, will be the same as Archies.

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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?UA(1-
  ?)UB
• 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.
• Besss willingness to pay for an improvement in
  her health would be based on household welfare
  function--the same as Archies
• 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 theoryHousehold allocations will
  be those that maximize the Nash product
• (UA -T A )(UB-T B)
• subject to xAxBm, 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, fathers
  willingness to pay for childs health is same as
  childs.
• Mothers willingness to pay out of family income
  for childs consumption may exceed fathers.
• Potential Pareto optimum criterion recommends
  using minimum of fathers and mothers 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 childs 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.
• Hamiltons calculus suggests magnitudes.

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Conclusion

• Im done.