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

4. The collective model: a formal analysis 187
produces only a fraction of the amount it consumes and purchases the rest.
The production equation is now:
c a + c b + c M = F ¡ t a ,t b¢
and the budget constraint at the household level becomes:
q a + q b = w a h a + w b h b + y + pc M . (4.69)
In our analysis of household production models, we shall first consider
thebenchmarksituationinwhichboth spouses are working outside the
family, and their working time is flexible enough to allow for marginal
variations. Then the opportunity cost of a person’s time is determined by
the person’s wage, which is taken as given for the family decision process.
We later consider ‘corner’ solutions, in which one spouse works exclusively
at home.
Marketable production
Cost minimization
Let us first assume that good c is marketable. In this context, efficiency
has an immediate implication, namely profit maximization. Specifically, t a
and t b must solve:
implying the first order conditions:
max
(ta ,tb pF
)
¡ t a ,t b¢ − w a t a − w b t b
∂F
∂t s
(4.70)
¡ a b
t ,t ¢ = ws
, s = a, b (4.71)
p
The economic interpretation of these equations is clear. The opportunity
cost of an additional unit of time spent on domestic production is the person’s
wage. If this is not equated to the marginal productivity of domestic
labor, efficiency is violated. For instance, if this marginal productivity is
smaller than the wage, then the person should spend less time working at
home and more working for a wage, keeping total leisure constant. Intrahousehold
production would decline, but household income would increase
by more than the amount needed to purchase the missing production on
therelevantmarket.Toputitdifferently, the condition reflects cost minimization;
if it not satisfied, then the household could achieve the same
level of leisure and domestic consumption while saving money that could
be used to purchase more of the consumption goods - clearly an inefficient
outcome.
The same argument can be presented in a more formal way. Consider
the household as a small economy, defined by preferences u a and u b and