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

4. The collective model: a formal analysis 191
Again, this price depends only on wages and on the technology. It is household
specific in the sense that two households with different wages will
price the household good differently, even if they have access to the same
domestic technology. However, for given wages and domestic technology, it
depends neither on preferences nor on respective powers. Finally, the separation
result still holds. That is, each member’s decision can be modeled as
if they were maximizing their own utility under the member specific budget
constraint defined by a sharing rule; this mechanism determines all the
components of consumption, including The only difference with the marketable
case is that p is no longer a market price; instead, it is determined
by the wages and the technological constraints.
Power and domestic work
While the previous conclusions are not really surprising, at least from a
general equilibrium perspective - they basically illustrate standard results
in welfare economics - their implications can be somewhat unexpected.
Consider, for instance, a change in Pareto weights that benefits women -
say, through the impact of a distribution factor - while wages and incomes
are unaffected. As discussed in the previous subsection, a first consequence
is that the structure of consumption will change; intuitively, the household
will now consume more of the commodities that the wife ‘likes more’. If,
as it is often argued, women generally care more about the goods that
are domestically produced (child care being a primary example), the total
consumption of these commodities should increase. If the commodity is
marketable and initially (partly) purchased on the market, the result will be
higher market purchases of these goods, with no impact on domestic labor
by the partners. In all other cases, domestic labor will increase, and the
distribution of the additional effort between spouses is completely driven by
the technology. For instance, under a standard, Cobb-Douglas production
function, inputs are complements; at constant prices (here wages), more
production requires increasing both inputs. We conclude that more power
to the wife may actually imply more domestic work for both spouses. Note,
however, that because a transfer of income to the wife does not affect
her time input into home production, the income effect will induce her to
reallocate the remaining time so that her market work should decline and
her leisure increase. This conclusion should be contrasted with the impact
of an increase in the wife’s market wage, which always affect her domestic
labor supply. When the commodity is marketable, her domestic work is
always reduced. In the alternative situation, her domestic work decreases
with respect to her husband’s, but the absolute impact also depends on the
structure of consumption - especially if her Pareto weight is boosted by her
higher wage.