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

472 11. Marriage, Divorce, Children
2. The father remains single while the mother is remarried, which happens
with probability p(1 − p).
3. The mother remains single but the father is remarried, which happens
with probability (1 − p)p.
4. Both parents remarry, which happens with probability p 2 .
Note that, by assumption, the probability of remarriage is the same for
the husband and wife, and that meetings and subsequent remarriages are
independent across parents.
Anticipating these contingencies, the father may be willing to commit to
transfer money to the custodial mother with the intention to influence the
welfare of the child, of whom he continues to care. 14 Each father makes
his choice of s separately, taking the choice of others, s0 as given. These
payments are made at the time of divorce, before the marital status of the
ex-spouses is known. We, therefore, must use expectations in determining
the optimal level of the transfer. The expected utility of the father is,
therefore,
Vf = (1− p) 2 [wf − s + q(s)] + (1 − p)p[wf − s + q(s, wf − s 0 )]
+p(1 − p)[a(s 0 ,wf − s)+q(s)]
+p 2 [a(s 0 ,wf − s)+q(s, y − s 0 )] (11.20)
and
∂Vf
∂s =(1−p)[q0 (s) − 1] + p[ ∂q ∂a
− ] (11.21)
∂s ∂yh
We first note that the father will never choose voluntarily transfer s that
exceeds c∗ because, in this case, the single mother would spend the marginal
dollar on the adult good. The father then receives a marginal benefit ofα
γ
from the transfer if the mother remarries single and 1+a − γ−(1+α)wm if
she remarries. But his expected cost in terms of the adult good is higher,
because a transfer of a dollar costs the father 1 dollar if he remains single
and ∂a
∂yh =
γ
if he remarries (see equations (11.11) and (11.19)).
γ−(1+α)wm
Under our maintained assumption that the remarried mother works part
time, equation (11.21) can be rewritten as
∂Vf
∂s =
(
γ
γ ∂a
(1 − p)( − 1) + p( − ) if 0 ≤ s