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

348 8. Sharing the gains from marriage
The linear shift property implies that, under assortative matching and
with populations of equal size, a man with income x is paired with a woman
with income y = αx − β. With the previous notations, therefore, φ (y) =
(y + β) /α and ψ (x) =αx − β. Equations(8.15) and(8.16) thenbecome
du(x)
dx
dv(y)
dy
yielding upon integration :
and
where
= H 0 ((α +1)x− β) ,
and
(8.22)
= H 0 (((α +1)y + β) /α) , (8.23)
v (y) =K + α
H (φ (y)+y) (8.24)
1+α
u (x) =K 0 + 1
H (x + ψ (x)) (8.25)
1+α
K + K 0 = θ.
In words, the marriage between Mr. x and Mrs. y = ψ (x) generates
a marital output θ + H (x + ψ (x)), which is divided linearly between the
spouses. The non monetary part, θ, is distributed between them (he receives
K, shereceivesK 0 ) in a way that is not determined by the equilibrium
conditions (this is the standard indeterminacy when r =1)butmustbe
the same for all couples (note that K or K 0 may be negative). Regarding
the economic output, however, the allocation rule is particularly simple; he
receives some constant share α/ (1 + α) of it, and she gets the remaining
1/ (1 + α).
8.2.3 Comparative Statics
We now turn to examine the impact of changes in the sex ratio and income
distribution.
Increasing the proportion of women
We begin by noting an important feature of the model, namely that if
all marriages yield a strictly positive surplus then the allocation rule has
a discontinuity at r =1. Indeed, examining the expressions in (8.17) and
(8.18) we see that if r approaches 1 from above we get in the limit
u (x) = H (a + b) − H (b)+θ +
v (y) = H (b)+
Z y
b
Z x
a
H 0 (s + ψ (s)) ds (8.26)
H 0 (φ (t)+t) dt (8.27)