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

6. Uncertainty and Dynamics in the Collective model 275
where ū b is a constant. Let ¡ ¯q a , ¯q b , ¯ Q ¢ denote the solution to this program,
and ū a = u a (q a ,Q)+θ a the corresponding utility for a. Note that both
are functions of ū b ;wenotethereforeū a ¡ ū b¢ .LetPM denote the Pareto
set if married, that is the set of utilities ¡ u a ,u b¢ such that u a ≤ ū a ¡ u b¢ ;
in words, any pair of utilities in PM canbereachedbythecoupleifthey
remain married.
Then we are in one of the following two situations:
• either the reservation point ¡ V a ¡ D a ¡ y a ,y b¢¢ ,V b ¡ D b ¡ y a ,y b¢¢¢ ,representing
the pair of individual utilities reachable through divorce,
belongs to the Pareto set if married, PM. Then there exists a second
period distribution of income which is preferred over divorce by both
spouses. The efficiency assumption implies that this opportunity will
be taken, and the model predicts that the marriage will continue.
• or, alternatively, ¡ V a ¡ D a ¡ y a ,y b¢¢ ,V b ¡ D b ¡ y a ,y b¢¢¢ is outside PM.
Then the marriage cannot continue, because any second period allocation
of resources the spouses may choose will be such that one
spouse at least would be better off as a single; therefore, divorce must
follow.
The model thus provides a precise description of the divorce decision;
namely, divorce takes place whenever it is the efficient decision under the
constraint that agents cannot receive less than their reservation utility
V i ¡ Di ¡ ya ,yb¢¢ ,i= a, b.
Some remarks are in order at this point. First, the argument presented
above assumes that divorce is unilateral, in the sense that each partner is
free to terminate the marriage and obtain divorce, even if the spouse does
not agree. An alternative setting requires mutual consent - that is, divorce
cannot occur unless both spouses agree. An old question of family economics
is whether a shift from mutual consent to unilateral has an impact on
divorcerates;weshallconsiderthat question in the next subsection.
Secondly, the fact that spouses may disagree about divorce - that is,
a spouse may ask for divorce against the partner’s will - does not imply
that they will. In the setting just presented, a partner who would consider
divorce may sometimes be ‘bribed back’ into marriage by her spouse,
through an adequate redistribution of income. Only when such a redistribution
cannot take place, because the cost to the other partner would exceed
the benefits of remaining married, will divorce occur. In that sense, there
is not disagreement about divorce in this model; simply, divorce sometimes
comes out as the best solution available.
A third remark is that, ultimately, ³ divorce is triggered by the realization
of the match quality parameters θ a ,θ b´
. Large values of the θs inflate the
Pareto frontier, making it more likely to contain the divorce threat point;
conversely, poor realizations contract it, and divorce becomes probable.