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

326 7. Matching on the Marriage Market: Theory
mitment, Gale-Shapley still need not be the relevant equilibrium concept.
To see why, consider the extreme situation in which marriage can be done
and undone at very low cost. Then at any moment of marital life, each
spouse has many close substitutes on the market, and the intrahousehold
allocation will typically reflect this fact. Although, technically, this is not
a BAMM situation (no binding agreement can be signed by assumption),
the relevant concept is still the TU model a la Becker-Shapley-Shubik, because
each spouse receives exactly her/his reservation value and the latter
is fully determined by market equilibrium forces (at least when the number
of potential spouses is ‘large enough’). In other words, even in the extreme
no transfer/no commitment case, the BIM framework applies only insofar
as marriage decision can only be reversed at some cost, and only within
the limits defined by this cost.
It is clear, in practice, that entry into marriage is a major decision that
can be reversed only at some cost. However, as in any modeling choice,
"realism" of the assumptions is not the only concern. It is also important
to have a tractable model that allows one to predict the marriage market
outcomes under varying conditions. In this regard, the presence of transaction
costs is quite problematic. To see this, consider again our example
7.3. Suppose that a new woman, 4, unexpectedly enters a marriage market
that has been in one of the two equilibria discussed in section 7.1. Letthe
new payoffs matrixbeasbelow:
Example 7.3a
Women
1 2 3 4
1 3, 2 2, 6 1, 1 2, 1
Men
2 4, 3 7, 2 2, 4 5, 4
3 1, 1 2, 1 0, 0 .5,.5
By assumption, woman 4 is preferred to woman 3 by all men and one would
expect that in the new assignment woman 3 will become single. Suppose,
however, that all existing couples bear a transaction cost of 0.75. Thenitis
easy to see that if the original equilibrium was the one in which men moved
first, no man will marry woman 4 and she will remain single. In contrast, if
the original equilibrium was the one in which women moved first then man
2 will take woman 4 and his ex-wife (woman 1) will first propose to man 1
who will reject her and then to man 3 who will accept her, so that woman 3
will become single. Thus, in general, it is impossible to predict what would
happen when a new player enters the market, without knowing the bargaining
outcomes in all marriages, the potential bargaining outcome that
the entrant will have with all potential existing partners and the relational
capital accumulated in all existing marriages. Such information is never
available to the observer. In contrast, the Becker-Shapley-Shubik framework
can predict the outcome very easily, using only information about
the place of the new woman in the income distribution of women and the