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

8. Sharing the gains from marriage 343
In particular, (8.9) implies that
u (x) = θ +max(h(x,
y) − v (y)) ,
y
v (y) = θ +max(h(x,
y) − u (x)) . (8.10)
x
That is, each partner gets the spouse that maximizes his/her “profit” from
the partnership, taking into account the reservation utility (the ‘price’) of
any potential spouse. The first order conditions for the maximizations in
(8.10) give:
v 0 (y) = hy(φ (y) ,y),
u 0 (x) = hx(x, ψ (x)). (8.11)
These equations have an important implication - namely that, as we move
across matched couples, the welfare of each partner changes according to
the marginal contribution of his/her own income to the marital output,
irrespective of the potential impact on the partner whom one marries. The
reason for this result is that, with a continuum of agents, there are no rents
in the marriage market, because everyone receives roughly what he\she
would obtain in the best next alternative. 4 Therefore, a change in marital
status as a consequence of a marginal change in income has negligible
impact on welfare, and the only gain that one receives is the marginal
contribution of one’s own trait. Although the change of spouse provides
no additional utility, the spouse that one has influences the marginal gain
from an increase in own traits, reflecting the interactions between the traits
in the production of marital output.
Another important condition that needs to be satisfied in a stable assignment
is that, if there are unmarried men, the poorest married man (whose
income is denoted x0) cannot get any surplus from marriage. Similarly, if
there are unmarried women, the poorest married woman (whose income is
denoted y0) cannot get any surplus from marriage. Otherwise, the unmarried
men or women who are slightly less rich could bid away the marginal
match. This condition exploits the assumption that there is a continuum
of agents. Hence, if r1 then v (y0) =h (0,y0) and u(0) = θ. Ifr =1, then any allocation
of the gains in the least attractive match with x = y =0that satisfies
u(0) + v(0) = θ is possible.
This initial disparity between the two spouses is modified as they move
up the assignment profile. The main features that influence the evolution
of utility differences within couples are the local scarcity of males and females
at different levels of incomes and the strength of the interaction
4 The absence of rents must be distinguished from the positive surplus that the marriage
creates. A positive surplus, h(y, z)+θ>h(y, 0) + h(0,z), simply means that there
are positive gains from marriage, relative to the situation in which both partners become
single, but this is rarely the best next alternative.