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

7. Matching on the Marriage Market: Theory 303
with a reservation utility ui and is selected by the woman that gains the
highest surplus zij − ui from marrying him. Similarly, woman j enters with
a reservation utility vj and is selected by the man who has the highest gain
zij − vj from marrying her. In equilibrium, each agent receives a share in
marital surplus that equals his\her reservation utility. In a sense, ui and vj
can be thought of as the ‘price’ that must be paid to marry Mr. i or Mrs.
j; each agent maximizes his/her welfare taking as given this ‘price’ vector.
It is important to note that the informational requirements for implementing
a stable assignment with transferable utility is quite different than
for the Gale-Shapley no transfer case. For the latter, we only require that
each person can rank the members of the opposite sex. With transferable
utility, the planner needs to know the surplus values of all possible matches
and agents should each know the share of the surplus that they would
receive with any potential spouse.
In general, there is a whole set of values for ui ,vj that support a stable
assignment. While the issues related to the distribution of surplus will be
discussed in the next Chapter, we present in the table below three (of
many) such imputations, denoted by a, b and c, for the stable assignment
in example 7.4.
Imputation a b c
Individual shares
W M
v1 2 u1 3
v2 5 u2 5
v3 1 u3 0
W M
v1 2 u1 4
v2 4 u2 5.5
v3 0.5 u3 0
The reader can readily check that each of these imputations supports a
stable match.
Extension: continuum of agents
Finally, although the previous argument is presented in a finite setting, it
is fully general, and applies to continuous models as well. From a general
perspective, we only need that the set of men and the set of women, denoted
X and Y , be complete, separable metric spaces equipped with Borel
probability measures F and G; note that no restriction is imposed on the
dimension of these spaces (it may even be infinite). The surplus function
h (x, y) is only assumed to be upper semi-continuous. The problem can be
stated as follows: find a measure Φ on X × Y such that:
• The marginals of Φ on X and Y are F and G, respectively.
R
• The measure Φ solves maxΦ h (x, y) dΦ (x, y), wherethemaxis
X×Y
taken over the set of measures satisfying the previous conditions.
A complete analysis of this problem is outside the scope of this book; the
reader is referred to Chiappori, McCann and Neishem (2010) or Ekeland
W M
v1 1 u1 5
v2 3 u2 6
v3 0 u3 1