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

7. Matching on the Marriage Market: Theory 309
incomes x are distributed on [0, 1] according to some distribution F and
female incomes y are distributed on [0, 1] according to some distribution
G.
The assumed positive interaction implies a positive assortative matching.
Therefore, if a man with income x is married to a woman with income y,
then the set of men with incomes above x must have the same measure as
the set of women with incomes above y. Thus,forallx and y in the set of
married couples,
1 − F (x) =r (1 − G (y)) . (7.19)
Hence,
x = Φ [1 − r (1 − G (y))] ¯=φ (y) , (7.20)
where Φ = F −1 ,orequivalently,
∙
y = Ψ 1 − 1
¸
(1 − F (x)) ¯=ψ (x) , (7.21)
r
where Ψ = G−1 and ψ = φ −1 ; note that both φ and ψ are increasing.
All men and women are married if there is an equal measure of men
and women, r =1. All women are married if there is scarcity of women,
r1, implying that
womenwithincomeyless than y0 = Ψ (1 − 1/r) remain single. If r>1,
then the function y = ψ (x) determines the income of the wife for each man
with income x in the interval [0, 1]. Similarly, if r