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

478 11. Marriage, Divorce, Children
An important feature of the analysis is that the decisions of each couple
whether to divorce or to have a child depend not only on the circumstance
of the couple, for example, if it suffered a negative shock, but also on the
decisions of other couples to have children and to divorce, as well as on
the contracts that they sign. These decisions by other couples influence
the prospects of remarriage and the quality of potential mates. To simplify
our analysis, we focus here on the case in which, in equilibrium, all couples
have children or all couples do not have children. Therefore, we only need to
consider the benefits of a particular couple from having a child, conditioned
on whether or not all other couples have children.
Given the choices of others, the expected life time utility of a parent j
in a particular couple is
Wj,n(p) =u 0 n +
Z∞
θ ∗ n (p)
(u ∗ n + θ)f(θ)dθ + F (θ ∗ n(p))(Vj,n(p) − bn) (11.35)
where, j = f for the (potential) father and, j = m for the (potential)
mother, and n is a choice variable that equals to 1 if the couple has children
and 0 otherwise. The term u 0 n in equation (11.35) represents the utility of
the two partners in the first period, which is 2wf if the couple has no
children and only wf if a child is born, because of the mother’s withdrawal
from the labor force during child birth. The term u ∗ n represents the parents
utility if marriage continues, which is 2wf if the couple has no children and
u ∗ if there is a child. The fixed costs of separation bn are assumed to be
larger when the couple has children. 17
The expected life time utility is higher for the partner with the higher
gains from divorce who determines the divorce decision. In fact, the ex-
pected life time utility can be rewritten as
⎧
⎪⎨
u
Wj,n(p) =
⎪⎩
0 n + u∗ θ
n +
∗ n (p) R
(θ
−∞
∗ n(p) − θ)f(θ)dθ if Vj,n(p) ≥ Vi,n(p)
u0 n + u∗ θ
n +
∗
n (p) R
(θ
−∞
∗ n(p) − θ)f(θ)dθ
−F (θ ∗ n(p))(Vi,n(p) − Vj,n(p))
if Vj,n(p)