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

8. Sharing the gains from marriage 351
exceeding y rises for all y, so that women become more similar to men in
terms of their income, as we observe in practice. Such an upward first order
shift in the distribution of female income affects the matching functions in
exactly the same way as a marginal increase in the female/male sex ratio.
Thus, if all men maintain their income, they all become better off. Similarly,
any woman who would maintain her income would become worse off. This
remark should however be interpreted with care, because it is obviously
impossible for all women to maintain their income: when the distribution
of female incomes shifts to the right, some (and possibly all) females must
have higher income. In particular, those women who maintain their relative
rank (quantile) in the distribution will maintain their position in the
competition for men, and will be matched with a husband with the same
income as before. Such women will be better off, as a consequence of the
increase in their own income.
As a special case, consider the linear shift case described above; to keep
things simple, assume moreover that β =0.Suppose,now,thattheincome
of every woman is inflated by some common factor k>1 and consider
a married couple with initial incomes (x, y). After the shift, the partners
remain married but the wife’s income is boosted to ky while the husband’s
income remains equal to x. Ifuk and vk denote the new individual utilities,
we have from (8.25) and (8.24):
vk = K + kα
H (ky + x) and (8.30)
kα +1
uk = K 0 + 1
H (ky + x)
kα +1
Differentiating in k around k =1gives:
∂vk
∂k =
α
αy
2 H (y + x)+
(α +1) α +1 H 0 (y + x) and
∂uk
∂k
=
α
y
− 2 H (y + x)+
(α +1) α +1 H 0 (y + x) (8.31)
One can readily check that both changes are positive (for the second one,
it stems from the convexity of H). We conclude that the shift has two impacts.
First, the increase in total income generates some additional surplus
(theterminyH 0 (y + x)), which is shared between spouse in proportion of
their respective incomes (that is 1 and α). In addition, a redistribution is
triggered by the shift. Specifically, since the wife’s share of total income is
increased, so is her consumption; the husband therefore transfers to his wife
an amount equal to a fraction α/ (α +1) 2 of total surplus. One can readily
check that the transfer is proportionally larger for wealthier couples, since
the ratio H (y + x) / (y + x) increases with (y + x) due to the convexity of
H.