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

7. Matching on the Marriage Market: Theory 313
Male income is denoted by x and female income is denoted by y. Weno
longer assume transferable utility; hence the Pareto frontier for a couple
has the general form
u = H(x, y, v) (7.23)
with H(0, 0,v)=0for all v.
As above, if a man with income x remains single, his utility is given
by H(x, 0, 0) andifawomanofincomeyremains single her utility is
the solution to the equation H(0,y,v) = 0.Bydefinition, H(x, y, v) is
decreasing in v; weassumethatitisincreasinginx and y, that is that a
higher income, be it male’s or female’s, tends to expand the Pareto frontier.
Also, we still consider a continuum of men, whose incomes x are distributed
on [0, 1] according to some distribution F , and a continuum of women,
whose incomes y are distributed on [0, 1] according to some distribution G;
let r denote the measure of women.
Finally, let us assume for the moment that an equilibrium matching exists
and that it is assortative. Existence can be proved under mild conditions
using a variant of the Gale-Shapley algorithm; see Crawford (1991), Chiappori
and Reny (2006). Regarding assortativeness, necessary conditions will
be derived below. Under assortative matching, the ‘matching functions’ φ
and ψ are defined exactly as above (eq. 7.19 to 7.21).
Let u (x) (resp. v (y)) denote the utility level reached by Mr. x (Mrs. y)
atthestableassignment.Thenitmustbethecasethat
u (x) ≥ H (x, y, v (y))
for all y, with an equality for y = ψ (x). Asabove,thisequationsimply
translates stability: if it was violated for some x and y, a marriage between
these two persons would allow to strictly increase both utilities. Hence:
u (x) =maxH(x,
y, v (y))
y
and we know that the maximum is actually reached for y = ψ (x). First
order conditions imply that
∂H
∂y (φ (y) ,y,v(y)) + v0 (y) ∂H
(φ (y) ,y,v(y)) = 0. (7.24)
∂v
while second order conditions for maximization are
∂
∂y
µ
∂H
∂y (φ (y) ,y,v(y)) + v0 (y) ∂H
(φ (y) ,y,v(y))
∂v
≤ 0 ∀y. (7.25)
This expression may be quite difficult to exploit. Fortunately, it can be
simplified using a standard trick. The first order condition can be written
as:
F (y, φ (y)) = 0 ∀y