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

11. Marriage, Divorce, Children 489
chosen parameters, this case yields an equilibrium divorce rate p =0.356 28
Figures 11.9 and 11.10 describe the impacts of deviations from the transfer
patterns when all parents have children. Figure 11.9 shows that if all
parents commit on the transfer σ = wf −c ∗
2−p that restores efficiency then for
p0.302, each father taken separately would be better off if he unilaterally
commits to the mother to pay her all his disposable income, wf − c ∗ ,
if she remains single. Thus an equilibrium where no one wishes to commit
cannot exist in this range. Proposition 11.4 states that if p1 >p>p0
there may be two partial equilibria, one in which every father commits on
a positive σ and one in which no father commits. However, for the chosen
parameters, the equilibrium divorce rate associated with having children
is above p1 =0.302, implying that the equilibrium in which every father
commits at p =0.356 is the only potential equilibrium when all parents
have children. Table 11.2 shows that the equilibrium at p =0.356 is indeed
a full equilibrium in the sense that, with the implied child support
transfers, all couples prefer to have children. For the assumed parameters,
this is the only equilibrium because, if all couples have no children, there
is an incentive to deviate to a situation with a child and full commitment
σ = wf − c ∗ . Thus, the full equilibrium is unique.
Comparing the results in Table 11.2 to the last row in Table 11.1, wecan
see the impact of different legal regimes when all parameters of the model
are the same. Suppose that all couples have children. Then, if transfers
are not contingent and determined optimally, the child’s expected utility
is 2.613. If fathers are forced to pay the mother a transfer of s = c ∗ , the
child’s expected utility rises to 2.811 and if contingent transfers are also enforced,
the child’s expected utility is 3.216, which is only slightly less than
the child’s utility in an intact family, 3.260. Aswemoveacrossthesealternatives,
the transfer from each father to his ex-wife rises when the marriage
breaks and, consequently, the expected life time utility of each mother rises.
The surprising result, however, is that the father is also better off and his
expected utility levels are 3.456, 3.475 and 3.485, respectively. The result
that a compulsory increase in child support above the individually optimal
level, s ∗ (p) raises the father’s expected utility reflects a positive contract
externality, whereby the commitment made by each father to his ex-wife
benefits other fathers when they remarry. The second increase, associated
28 An interesting point is that if the efficiency of child expenditures is restored by
appropriate transfers then the gains from divorce with children can exceed the gains
from divorce without children, despite the higher fixed costs of divorce associated with
children. The reason is that couples without children have higher joint consumption (in
terms of the adult good), which can make divorce more costly for them.