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

354 8. Sharing the gains from marriage
˜h (x + y) ≡ ˜ H(2κ +(1−τ)(x + y)). Assume, in addition, that male and
female income distributions satisfy condition (8.21) so that ψ (x) =αx − β
and φ (y) = y+β
α . Then, for a larger female population (r >1), utilities
become:
˜v (y) = H (κ +(1−τ) y0)+
Z y+φ(y)
(1 − τ) H
0
0 (κ +(1−τ) s) αds
α +1
Z ψ(x)+x
ũ (x) = θ +(1−τ) H
0
0 (κ +(1−τ) s) ds
(8.34)
α +1
with k = τ ¯x+r¯y
1+r . The impact of a change in the marginal income tax, t,
on the utilities of women and men, respectively, is
µ
∂˜v (y) x + ry
= − y0 H
∂t 1+r 0 (α + βy0) − αD (x + y)
∂ũ (x)
= −D (x + y) , (8.35)
∂t
where Y denotes total family income and
Z Y
D (Y ) = H
0
0 (κ +(1−τ) s) ds
α +1
Z Y µ
x + ry
+(1− τ) s − H
0 1+r
00 (κ +(1−τ) s) ds
α +1 . (8.36)
The term D (Y ) is typically positive for richer households (who therefore
lose from the introduction of the tax/benefit system) and negative
for poorer ones. In this simple context, the corresponding gain or loss is
allocated between husband and wife in respective proportions 1 and α.
In addition, since single women are at the bottom of the female income
distribution, their utility is typically increased by the tax/benefit scheme
(this is the case whenever their income is below the mean). Equilibrium
then requires the gain of the marginal woman (that is, of the wealthiest
single or poorest married woman) ³ to be´ forwarded to all womeninthe
x+ry
distribution; hence the term 1+r − y0 H0 (α + βy0) in equation (8.35)
representing the gain of the marginal woman. Note that this boost in income
does not go to the poorer spouse, but to the spouse whose population
is in excess supply. Should males outnumber females (r