Family Constitutions

1
Family Constitutions

• Linking intergenerational transfers and fertility

2
Non-altruistic transfersand family constitutions

• How much of family transfer and fertility
  behaviour can we explain by pure self-interest?
• Consider a model in which people only have
  children because they are needed to transfer
  resources through time.

3
The model

• Assume that each person lives for three periods
  of equal duration--youth, middle-age and
  old-age--and that their welfare depends on their
  own consumption only.
• cit is the consumption in the i-th period of a
  person born at date t
• Utility is given by UtU(c1t, c2t, c3t).
• A person only earns income yt in the second period

4
Why transfer resources?

• A persons utility would be higher if he can
  transfer resources from the second to the first
  and third periods.
• Assume no access to a capital market
• Set family rules of conduct such that no member
  would be better off outside the family transfer
  system than within it.

5
Constraints

• nt the ratio of the number of family members
  born at t1 to the number born at t (family size
  per head of generation t).
• d t the amount transferred to the older
  generation during the middle-age of generation t.
• c2t d t c1t1nt yt
• c3t d t1nt
• 0? nt ? m (physiological constraint)

6
Self-enforcing Family Constitution

• Constitution prescribes d t and c1t for each
  generation. It is useful to define
• ?t d t/c1t
• the rate of return to having a child for a member
  of generation t-1, is ?t-1
• Combine constraints
• c1t (c2t/?t) c3t/(?t?t1) yt/?t

7
Optimisation

• Maximise utility subject to above constraint
• ? borrow to finance consumption in the first
  period to the point at which
• Ut1/Ut2 ?t, where Ut1?U/?c1t, etc.,
• lend in the second period so that
• Ut2/Ut3 ?t1
• Combining
• Ut1/Ut2 Ut-12/Ut-13 d t/c1t (FEE)
• these are first order conditions for a
  Pareto-optimum.

8
Rules of the family constitution

• A set of transfer rules for a steady-state in
  which yty for all t.
• Prescribe to generation t the transfer levels to
  children and parents of c1 and d, respectively.
• Each member of that generation would maximise
  utility as above--satisfy FEE
• 3rd period constraint implies ntc3/dn.

9
Implications

• If a member of generation t had been able to
  decide how much to borrow as a child at the
  implied rate of interest d/c1-1, he would have
  chosen c1tc1.
• Members of generation t would rather not pay d to
  their parents (generation t-1), but the prospect
  of receiving no support from the family in their
  old-age deters any member from disobeying the
  rules.

10
Example

• U(c1t, c2t, c3t) ?1ln(c1) ?2ln(c2) ?3ln(c3),
  with ??j1.
• Solution
• c1?1y/?, where ?d/c1,
• c2?2y
• c3??3y
• d??3y/n ?1y
• n?3?/?1

11
Persistence of the constitution

• Each generation would prefer that
• transfers to it when a child (c1) be as large as
  possible
• its transfers to aged parents when middle-aged
  (d) be as small as possible.
• But as everyone passes from youth to old-age, no
  generation would agree to a change in the rules
  that would reduce its own consumption in old-age.

12
Access to a capital market

• Nobody outside the family would lend to a child.
  
• But the middle-aged now have a choice between
• providing for their old-age by lending in the
  capital market or
• by staying within the family system and lending
  to their children, awaiting transfers from them
  in old-age.

13
Comply with the constitution or not?

• By middle-age, any generation t has already
  consumed c1.
• If they made the transfer d to their parents
  dictated by the family rules, their budget
  constraint would be
• c2t c3t/? y-d, where ?d/c1
• If they did not pay anything to their parents and
  lent to the market at the interest rate r-1, it
  would be c2t c3t/r y

14
Compliance and the interest rate

• If ?r, then generation t would be better off to
  not comply with the constitution.
• A threat of no support from the family in old-age
  is no longer a deterrent.
• To avoid this, the implicit rate of interest in
  the family system (?-1) must be set high enough
  to make a middle-aged member at least as well off
  as they would be from lending on the market.

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Compliance

• The family interest rate must be higher than the
  market one to induce compliance.
• Put differently, ? must satisfy
• V(c1, y, r) ? V(c1, y-d, ?),
• where V(.) is the indirect utility function.
• When ? is chosen to satisfy this constraint,
• complying is the best response to everyone else
  doing the same--a Nash equilibrium.

17
Picking a constitution

• There are an infinite number of these Nash
  equilibria, specifying different c1 and d.
• Suppose family rules stipulate that the
  intra-family interest rate should be as low as
  possible.
• This gives a high transfer to children.

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Previous Example, continued

• The steady-state solution with this rule is
• ?r/? satisfies V(c1, y, r) V(c1, y-d, ?)
• where ln(?) (?2?3)ln(?2?3)/?3, implying 0lt?lt1.
  
• Recall n?3?/?1
• Thus, n?3r/?1?

19
Breakdown of the family constitution

• For a market interest rate large enough,
• n r?3/?1? would exceed the maximum family size
  m.
• the family system would break down because the
  intra-family interest rate could not compete with
  market rate.
• Childbearing would cease.
• Note this varies with preferences.

20
Constitutional Breakdown more generally

• V(c1, y, r) ? V(c1, y-d, ?) constraint may not be
  consistent with the biological constraint.
• i.e. the fertility rate required to make ? large
  enough is too high.

21
Real world examples

• The growth of the financial sector (or
  introduction of a state pension system)
• Often coincide with a sharp fall in fertility and
  a decline in private transfers from the
  middle-aged to their elderly parents.

22
Extension -- Services from children

• Define d and c1 in terms of utility rather than
  money.
• E.g. c1v1(x1,s1) c3v3(x3,s3) c2v2(x2,s1,s3)
• Choice of money/services minimises costs of
  complying.
• Since sj substitutes for xj at a diminishing
  rate, it raises the the level of utility that a
  self-enforcing constitution can require people to
  give their parents and children.
• More likely that constitution exists.