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

8. Sharing the gains from marriage 339
negative assortative mating. For instance, with positive assortative mating
and M>N,
X
âijzij − X
i,j
i,j
āijzij =
NX
i=1
X
N−1
zi,M−N+i − zi,M−(N−1)+i .
i=1
(8.5)
Similar arguments apply for any man and any woman. Using the bounds
for men and women who are married to each other at the stable assignment
we can put bounds on the possible divisions of the gains from marriage of
the husband and wife in each couple. Thus the husband’s share in the
couple MN is bounded by
or
NX
i=1
NX
zN,M − (
≤ ûN ≤
NX
i=1
N−1
zi,M−1−N+i −
i=1
i=1
zi,M−N+i −
X
NX
i=1
N−1
zi,M−N+i − zi,M−(N−1)+i
i=1
X
NX
zi,M−N+i ≤ ûN ≤
8.1.3 Bounds on Differences
i=1
zi,M−1−N+i)
N−1
zi,M−N+i −
X
zi,M−(N−1)+i. i=1
(8.6)
With positive or negative assortative mating we can also put bounds on the
change in utilities as one moves along the assignment profile. Let there be
a positive assortative mating (that is, the matrix zij is super modular) and
suppose that M>N.ThenmanN is married to woman M, andwoman
M − 1 is married to man N − 1. At a stable assignment
uN + vM = zN,M
uN−1 + vM−1 = zN−1,M−1
uN + vM−1 ≥ zN,M−1
uN−1 + vM ≥ zN−1,M
Eliminating uN and uN−1 and substituting into the inequalities we get
zN,M − vM + vM−1 ≥ zN,M−1
zN−1,M−1 − vM−1 + vM ≥ zN−1,M