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

474 11. Marriage, Divorce, Children
is a constant. However, if the mother does not work when she is remarried,
or works full time, the utility frontier for a remarried couple is no longer
linear and the marginal impact of the transfer to the mother will depend
on the net income of her new husband yh.
11.4.1 Partial equilibrium
Suppose that all couples have children. Then, at given probability of remarriage,
p, the equilibrium outcome is that all fathers will transfer the
same amount s ∗ (p) to the mother if the marriage dissolves. That is, given
that other fathers choose s0 = s ∗ (p), each father independently chooses
s = s ∗ (p). This equilibrium requirement is trivially satisfied here, because
the optimal choice of each father is (locally) independent of the choices
of others. We refer to the equilibrium as partial because, as we shall see
shortly, the remarriage and fertility rates must also be set at equilibrium
levels.
In this partial equilibrium, the mother works part time when she is remarried
but not as single. The reason for this difference is that she must
compensate her new husband for the option of sharing the adult good.
The amount of time that the mother spends in market work is
hm(p) = (1 + α)(wf − ĉ)+g(ĉ) − g(s∗ (p))
(11.25)
γ − (1 + α)wm
and a sufficient condition for an interior solution 0 (1 + α)(wf − ĉ) >g(c ∗ ) − g(ĉ). (11.26)
Basically, the mother should have a sufficiently high comparative advantage
in child care to motivate her to spend some time at home and the net
income of the new husband should not be so high that the mother is driven
completely into the market, contrary to her comparative advantage. 15
We see that when p rises and all husbands reduce their contribution,
the remarried mothers increase their hours of work and thus reduce the
amount of time spent with the child. The implied adult consumption in
the remarried household
a(p) =wmhm(p)+wf − ĉ (11.27)
rises in p but the child’s utility if the mother remarries
q(p) =αa(p)+γ(1 − hm(p)) + g(ĉ) (11.28)
15 The sufficient condition (11.26) is much stronger than we need because, as we shall
show shortly, the equilibrium remarriage rate is bounded by 1
2 .