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

4. The collective model: a formal analysis 171
sons. One is that, in general, one cannot change prices without changing
the sharing rule as well; what can be observed, at best, are the functions
ˆq a (p,x, z) and ˆq b (p,x, z), which are related to the previous ones by the
relationships:
ˆq a (p,x, z) = ˜q a (p,ρ(p,x, z)) (4.36)
ˆq b (p,x, z) = ˜q b (p,x− ρ (p,x, z))
However, even these functions are in general unknown, because most of
the time the intrahousehold allocation of purchases is not observed. Expenditure
surveys invariably collect information about expenditures that are
aggregated at the household level; but who consumes what remains largely
unknown, except, maybe, for some specific commodities (for example, expenditure
surveys typically distinguish between male and female clothing).
In general what we observe is the household demand which is equal to the
sum of the individual demands:
ˆq (p,x, z) = ˆq a (p,x, z)+ˆq b (p,x, z)
= ˜q a (p,ρ(p,x, z)) + ˜q b (p,x− ρ (p,x, z)) (4.37)
As we shall see below, one can often use this relationship to derive the
properties of collective demand functions.
4.3.2 Caring preferences
Let us now consider the case of preferences of the ‘caring’ type, namely
U a ¡ q a , q b¢ = u a (q a )+δ a u b ¡ q b¢
U b ¡ q a , q b¢ = u b ¡ q b¢ + δ b u a (q a ) (4.38)
Here, the Welfare Theorems do not directly apply, since caring involves an
externality component. Two points should however be remembered. First,
any allocation that is Pareto efficient for caring preferences is also Pareto
efficient for the egotistic preferences u a and u b . This implies that the first
part of Proposition 4.3 still applies: whenever an allocation is efficient, it
can be decentralized through a sharing rule. The converse, however, no
longer holds in general. We know that some allocations may be efficient
for egotistic preferences, but not so for caring ones. It follows that only
asubsetof possible sharing rules generate efficient allocations for caring
preferences. For instance, a sharing rule such as ρ ' 0 typically generates
inefficient allocations since a redistribution of the resulting allocation in
favor of a may increase both agents’ welfare (if δ b > 0 and ∂u a /∂q a is
sufficiently large when q a is very small).