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

192 4. The collective model: a formal analysis
Extensions
Public goods
In the previous analysis, the internally produced commodity was privately
consumed. What if, instead, the commodity is public within the household
- as it is the case for childcare, for instance? Interestingly, not much is
changed, because the separation principle still applies. If the commodity,
although public within the household, is marketable, then its production is
driven by profit maximization; the only change regards the demand side,
where the decision process can no longer be decentralized using a sharing
rule. Even in the non marketable case, the logic of cost minimization prevails.
In particular, under constant returns to scale, it still the case that the
level of production is determined by preferences and the decision process,
while for any given level the time allocation of domestic work between
spouses stems from technological considerations. 19
Specialization
Another special (but empirically relevant) case obtains when one of the
spouses - say b - does not enter the labor market, and specializes instead
in home production. This happens when, for the chosen allocation of time
and consumption, b’s potential wage, ¯w b , is smaller than both b’s marginal
productivity in household production and b’s marginal rate of substitution
between leisure and consumption. In words: the marginal hour can indifferently
be spent in leisure or household production, and both uses dominate
market work. 20
The situation, here, is more complex, because the opportunity cost of
labor for b is no longer exogenously given; instead, it is now endogenous to
the program. Still, if we keep the assumption of constant return to scale
domestic technology, some of the previous conclusions remain valid. Indeed,
in the marketable case, efficiency in a’s allocation of time still requires that:
∂F
∂ta ¡ a b
t ,t ¢ = wa
p
while the CRS condition (4.77) implies that
∂F
∂ta ¡ a b
t ,t ¢ = Φ 0
µ a t
tb
It follows that, again, the ratio t a /t b is pinned down by technological con-
19The reader is referred to Blundell, Chiappori and Meghir (2005) for a more detailed
investigation.
20Technically, this result is true at the marginal level only in the absence of non
convexities. In the presence of fixed costs of work or constraints on the number of hours
worked, the same constraint must be redefined at a more global level.