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

300 7. Matching on the Marriage Market: Theory
Notice that the entries in this matrix are just the sums of the two terms in
Example 7.3 discussed above. In this regard, non transferable utility can
be thought of as a special case of transferable utility, where the division
of the output in each marriage is predetermined and cannot be modified
by transfers between spouses. For instance, if each partner receives half of
the marital output in any potential marriage, the Gale Shapley algorithm
yields the unique stable outcome, which is on the diagonal of this matrix. In
contrast, with transferable utility, the unique assignment that maximizes
aggregate marital output, indicated by the bold numbers in the matrix
below, is not on the diagonal. This assignment yields aggregate output of
16, compared with an aggregate output of 14 on the diagonal.
Men
Women
1 2 3
1 5 8 2
2 7 9 6
3 2 3 0
Though all men would obtain the highest marital output with woman 2,
and all women would obtain the highest output with man 2 (implying that
ζ 22 is the largest entry in the marital output matrix 7.4), the best man
and the best woman are not married to each other. With transfers, the
assignment on the diagonal is no longer stable, because if couple 1, 1 and
couple 2, 2 exchange partners, there is an aggregate gain of 1 unit of the
transferable good. Then man 1 can, despite his lower contribution to the
marital output, bid away the best woman by offering her a larger amount
of private consumption and still be better off than in the initial match with
woman 1. Similarly, woman 1 can bid away the best man by offering him a
larger share of private consumption and still be better off than in the initial
match with man 1. Thehigheraggregate output achievable when man 2
and woman 2 are not married to each other implies that, for any division
of the marital output of 9 that these partners can obtain together, at least
oneofthepartnerscanbemadebetteroff in an alternative marriage.
Stable matching with a finite number of agents
Let us now consider the general assignment problem with M males and N
females. Let ζ ij denote the total output of a marriage between male i and
female j, andletζ i0 (resp. ζ 0j) be the utility that person i (resp. person
j) receives as single (with ζ 00 =0by notational convention). Then the
difference zij = ζ ij − ζ i0 − ζ 0j is the marital surplus that male i and female
j generate by marrying each other.
We define assignment indicators, aij, suchthataij =1if and only if
i is married with j and aij =0otherwise. We also define ai0 =1if and
only if i is single, and similarly a0j =1if and only if j is single. Then,
following Gale (1960, chapters 1 and 5) and Shapley and Shubik (1972),