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

8. Sharing the gains from marriage 349
while if r approaches 1 from below we get in the limit,
u (x) = H (a)+
Z x
v (y) = H (a + b) − H (a)+θ +
a
H 0 (s + ψ (s)) ds (8.28)
Z y
b
H 0 (φ (t)+t) dt (8.29)
The marital surplus generated by the marriage of lowest income couple,
here (a, b), isequaltoH (a + b)−H (a)−H (b)+θ. When the two sexes are
almost equal in number, a small change in the sex ratio shifts all the surplus
to one of the partners in the lowest quality match, the one whose sex is in
the minority, and this discontinuity is then transmitted up the matching
profile to all participants in the marriage market. 6 This knife-edge property
is an undesirable property of the simple model without friction. One can
get rid of it either by assuming no rents for couples at the bottom of the
distribution, or by limiting our attention to marginal changes in the ranges
r>1 or r1 or r1, wehave:
φ (y, r)
ψ (x, r)
=
=
Φ [1 − r (1 − G (y))]
∙
Ψ 1 − 1
¸
(1 − F (x))
r
y0 = Ψ (1 − 1/r) ,
6 The result that g is the only source of gain from marriage for couples at the bottom
of the income distribution reflects the assumptions that h(0, 0) = 0 and that there is a
positive density of the income distribution at zero. In general, participants at the bottom
of the income distribution have a positive income, so that the lowest quality match may
create a monetary surplus, because of the positive interaction of traits.