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

11. Marriage, Divorce, Children 467
the divorce. For the time being, we shall assume that the mother receives
custody and will address this issue again after we derive the equilibrium
level of transfers.
Single custodial mother
If the custodial mother remains single, she will choose the amount of time
that she spends at work hm, her adult consumption am and the amount
of child goods c so as to maximize her own utility subject to her budget
constraint, taking as given the amount that the ex-husband transfers to
her, s. Her utility is then defined as the solution to the program
um(s) = max {(1 + α)am + γ(1 − hm)+g(c)}
am,c,hm subject to
(11.7)
am + c = wmhm + s,
0 ≤ hm ≤ 1.
The mother’s choices as a function of s are summarized in Figure 11.1
below. Because of the quasi-linear structure of the problem, the solution has
three distinctly different regions. For low levels of s, the mother withdraws
some time from the child and works in the market part time. She then
spends all her disposable income on child goods. The optimum conditions
in this region are
wmg 0 (c) = γ, (11.8)
c = wmhm + s.
Thus, the mother spends a fixed amount of money, ĉ on child goods and
works in the market the minimal amount of time required to achieve this
target. As s rises, the mother reduces her market work until it reaches zero,
spending more time on child care.
For high levels of s, the mother does not work in the market and allocates
her disposable income between the child and adult goods. The optimum
conditions in this region are
g 0 (c) = 1+α, (11.9)
c + am = s.
That is, the mother will spend a fixed amount of money, c∗ , on the child
and adjust her adult consumption according to the level of s.
For intermediate values of s, satisfying
γ
>g 0 (s) > 1+α, (11.10)
wm
the mother will not work and will not consume adult goods, so that all her
income and free time are devoted to the child.