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

166 4. The collective model: a formal analysis
therefore
E ¡ p, P,u a ,u b¢ ≤ p 0 ¡ ¯q a + ¯q b¢ + P 0 ¯Q a
4.2.2 Indirect utilities
≤ p 0 ¡ ¯q a + ¯q b¢ + P 0 ¯Q a + P 0 ¯Q b
= e a (p, P,u a )+e b ¡ p, P,u b¢
(4.30)
We can also define indirect utility functions. Consider first the program
max
(qa ,qb μu
,Q)
a (q a , Q)+u b ¡ q b , Q ¢
subject to r 0 ¡ q a + q b , Q ¢ = x (4.31)
Let ¡ q ∗a , q ∗b , Q ∗¢ denote its solution. Then the function ω s ,defined for
s = a, b by:
ω s (r,x,μ)=u s (q ∗s , Q ∗ )
is the direct equivalent, in the collective setting, of the indirect utility
concept in standard consumer theory. In particular, ω s only depends on
preferences, not on the decision process; technically, ω s is a function of the
Pareto weight μ, and a change in the decision process would result in the
same function ω s being applied to a different μ.
A second, and more important definition is obtained by plugging the particular
Pareto weight adopted by the household into the previous definition.
In this case, the collective indirect utility of a member is the level of utility
ultimately reached by this member as a function of prices and income and
distribution factors. Formally, if the decision process is characterized by a
function μ (r,x, z), the collective indirect utility of member s is defined for
s = a, b by:
V s (r,x, z) =ω s (r,x,μ(r,x, z))
Now, the definition of s’s collective indirect utility depends not only
on s’s preferences, but also on the whole decision process. In other words,
collective indirect utilities are specific to a particular match between agents
and a particular decision rule (summarized by the function μ). This is
in sharp contrast with the unitary case, where there exists a one-to-one
correspondence between direct and indirect utility at the individual level.
Also, a key remark, here, is that if one is interested in welfare analysis,
then the collective indirect utility is the appropriate concept. Indeed, it
preserves the basic interpretation of standard, indirect utilities in consumer
theory - namely, it characterizes each agent’s final welfare once all aspects
of the decision process have been taken into account.