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

7. Matching on the Marriage Market: Theory 299
Two examples
To understand this result, consider first the simplest possible case: let there
be two people of each sex. Assuming that marriage dominates the single
state (that is if any two individuals remain unattached they can gain by
forming a union), there are two possible assignments: Man 1 marries woman
1 and man 2 marries woman 2, orman1is married to woman 2 and man
2 is married to woman 1. In testing for stability we treat the potential
marital outputs ζij as given and the divisions uij and vij as variables.
Suppose, now, that the assignment in which man 1 marries woman 2 and
man 2 marries woman 1 (the off diagonal assignment) is stable. Then, the
following inequalities must hold:
u12 + v21 ≥ ζ11 (7.2)
u21 + v12 ≥ ζ22 (7.3)
If the first inequality fails to hold then male 1 and female 1, whoare
currently not married to each other, can form a union with a division of
utilities which will improve upon their current situations, defined by u12
and v21. If the second inequality does not hold then man 2 and woman 2,
who are presently not married to each other, can form a union and divide
utilities so as to improve over the current values u21 and v12.Fromequation
1wehaveζ 12 = u12 +v12 and ζ 21 = u21 +v21 so that equation (7.2) can
be rewritten as
ζ 12 − v12 + ζ 21 − u21 ≥ ζ 11. (7.4)
Adding conditions (7.4) and (7.3) we obtain
ζ 12 + ζ 21 ≥ ζ 11 + ζ 22
(7.5)
By a similar argument, an assignment along the main diagonal will be
stable only if (7.5) is reversed. Condition (7.5) is not only necessary but also
sufficient for stability of the off diagonal assignment. For if it is satisfied we
can find values of u and v such that (7.2) and (7.3) hold. Such imputations
support the stability of the assignment since it is then impossible for both
partners to gain from reassignment.
To illustrate the implications of the transferable utility assumption and
the implied maximization of aggregate marital output, let us consider a
second example. There are 3 men and 3 women and consider the matrix of
marital outputs below:
Example 7.4
Women
1 2 3
Men
1
2
5
7
8
9
2
6
3 2 3 0