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

342 8. Sharing the gains from marriage
8.2.1 Basic results
The setting, here, is a slight generalization of the one considered in subsection
7.2.3 of Chapter 7. There exists 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] accordingtosome
distribution G. 2 The measure of all men in the population is normalized
to 1, and the measure of women is denoted by r. Also,westillconsider
a transferable utility (TU) framework. The innovation is that the ‘marital
output’ is now the sum of two components: an economic output, which is
afunctionh (x, y) of individual incomes, and a fixed non monetary gain
from marriage, denoted θ, which is perceived by the spouses in addition to
the economic benefits:
z (x, y) =h (x, y)+θ
As before, h is assumed to be supermodular.
An allocation rule specifies the shares of the wife and husband in every
marriage. If r>1 and all men are married, we can index the marriage by the
husband’s income x (then his spouse’s income is ψ (x)). The marital output
is then h(x, ψ (x)) + θ and the marital shares are u(x) for the husband and
v(ψ (x)) for the wife. If r