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

356 8. Sharing the gains from marriage
single. Because men are always locally scarce, r
Z > 1, it follows from (8.40)
that their utility must grow along the stable assignment at a faster rate
than the utility of their assigned wives. It is readily seen that the husband’s
share is higher in matches with sufficiently high income. In particular, the
best match with x =1and y = Z, yields an output of (1+Z)2
4 ,ofwhich
the husband receives 1 Z Z
Z2 Z
4 + 2 − 4r and the wife receives 4 + 4r , which is a
smaller share.
If there are more men than women, r 1, men
are always locally scarce, and it follows from (8.40) that the husband will
have a higher share in the output of all marriages. If, however, r
Z
< 1 and
women are always locally scarce, then the utility of women grows along the
stable assignment profile at a faster rate than the utility of their assigned
husbands, and they may eventually overtake them. Indeed, the wife’s share
in the best match is Z
Z2 rZ
2 + 4 − 4
and the husband’s share is 1
4
+ rZ
4
, which
is smaller if r is sufficiently small.
This example illustrates clearly the impact of changes in the sex ratio
r and the distribution of female income as indexed by Z, onthewelfare
of women and men. Recall that marginal increases in x0 or y0 have no
effect on u(x) or v (y) , respectively. Inspection of the integrands in (8.40),
shows that u(x) must increase in r and Z, while v (y) must decrease in r
and Z. As we noted above, the result that women are worse off when the
mean income of women rises sounds surprising. However, the reason that a
woman who maintains her income is worse off when Z risesisthatthereare
more women with income above her, which means that she cannot ”afford”
anymore a husband with the same income as before. However, any woman
who keeps her position in the income distribution, (that is, whose income
rose at the same proportion as Z) will obtain a husband with the same
x as before the change. Then it can be shown that if r>1, her surplus