Pierre André Chiappori (Columbia) "Family Economics" - Cemmap

subject to the constraints:
11. Marriage, Divorce, Children 487
(1 + α)[wmhm + wf − c − (1 − p)σ 0 ]+γ(1 − hm)+g(c) ≥ um(c ∗ + σ)
and 0 ≤ hm ≤ 1. The chosen values of hm and c depend on the transfer that
the father promises the mother if she remains single, σ, and the expected
value of the new husband’s gross income wf −(1−p)σ 0 . Only the expectation
matters because the remarried partners are risk neutral with respect to a
and because the mother’s work time, hm, and the expenditures on the child
good, c, are determined before the marital status of the ex-wife of the new
husband is known.
In contrast to a non contingent transfer, a transfer given to the mother
only when she is single does not change the utility frontier of the remarried
couple and therefore must reduce the expected utility of the new husband.
This implies that with contingent payments, the father is able to attain a
larger impact on the child’s utility and is willing to contribute more to the
custodial mother. In fact, by setting σ = wf − c ∗ − (1 − p)σ 0 the father
can eliminate all the gains from marriage of the new husband and restore
efficiency. We then obtain the following characterization (see Appendix).
Proposition 11.4 If all couples have children, then the commitment equilibrium
for a given remarriage probability p, issuchthat:Forpp1, the only symmetric equilibrium
is one in which all fathers voluntarily commits to pay their ex-wife
σ = wf −c ∗
2−p if she remains single. For p1 ≥ p ≥ p0, both types of equilibrium
can arise. The equilibrium σ = wf −c ∗
2−p is efficient and
Vc(p) = γ + g(c ∗ )+α wf − c∗ 2 − p ,
Vm(p) = Vf(p) =γ + g(c ∗ )+(1+α) wf − c∗ 2 − p
. (11.45)
The pattern described in the proposition suggests reinforcement; one is
willing to commit to his wife if others do, but not if they do not. As is
well known, such positive feed backs can yield multiple equilibria. We also
see that higher probability of remarriage is conducive to equilibria in which
fathers are willing to commit on a payment that is conditioned on the event
that the mother remains single, because such promises are carried out less
often and are more likely to yield benefits.
When efficiency is restored, the child suffers only from the reduced adult
consumption that is caused by the risk of remaining single. If the mother
is sure to remarry, that is, p =1, then the child is as well off as in an intact
family. That is, the father was practically replaced by the new husband with
no harm to the child. This favorable outcome was achieved by eliminating