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

462 11. Marriage, Divorce, Children
11.2 The Model
We consider here a given cohort with equal number of men and women.
Individuals live for two periods and can be married or single in each of
these periods. A household consists of one or two adults and possibly one
child. We treat fertility as a choice variable and each couple decides on
whether or not it should have a child in the first period. We assume that
childless men and women are identical and both earn the same wage wf.
However, if a couple has a child then, because the mother is the one who
gives birth, her second period wage drops to a lower level, wm.
11.2.1 The technology and preferences
The household pools the incomes of its members and allocates it to buy an
adult good a and a child good c. Each parent has one unit of time which can
be allocated between market work and child care. Let hm and hf denote
the time spent by mother and father in market work, respectively. Then,
the amount of time they spend at home is tj =1− hj, where0 ≤ hj ≤ 1
for j = m, f.
The household production function is
where
q = αa + t + g(c) (11.1)
t = βtf + γtm. (11.2)
The output q is interpreted as the child’s utility or ‘quality’. The parameter
α describes the marginal effect of the adult good, a, on the child’s
quality, the parameters β and γ represent the productivities of the father
and mother, respectively, in household work and t is total time spent on
the child, measured in efficiency units. The function g(c) is assumed to be
increasing and concave, with g(0) = 0. The linearity in t is assumed to
allow corner solutions whereby family members specialize either in household
work or market work. To determine the pattern of specialization under
different household structures, we assume
γ > wm(1 + α)
β < wf(1 + α) (11.3)
where wm is the wage of the mother and wf is the wage of the father.
That is, the mother is more productive at home, while the father is more
productive in the market. This may hold either because the mother has an
absolute advantage in home production γ>βor that she has an absolute
disadvantage in market work, wm