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

4
The collective model: a formal
analysis
4.1 Collective demand functions: a general
characterization
4.1.1 The collective household utility function.
The basic aspects of the collective model have been described in the previous
chapter. As stated earlier, the particular form adopted has testable
implications for demand functions. We now describe these implications in
detail. We start with the most general version of the model with individual
preferences of the form u s ¡ Q, q a , q b¢ . This allows for any type of consumption
externalities between agents. We define the collective household utility
function by
u f (q, Q,μ)=max
qa © ª
a a a b a a
μu ((Q, q , q − q )) + u (Q, q , q − q )
(4.1)
where μ may be a function of (P, p,x,z) where z is a vector of distribution
factors. We shall always assume that μ (.) is zero homogeneous in (P, p,x)
and any elements of z that are denominated in monetary terms.
At this level of generality, the distinction between public and private
goods is somewhat blurred, and we can leave it aside for the moment. We
thus adopt a general notation with g =(q, Q) denoting the quantities consumed
by the household and r =(p, P) denoting the corresponding price
vector. Then the household’s behavior is described by the maximization of
u f (g,μ) under the household budget constraint r 0 g =x.
4.1.2 Structural and observable demand.
The household’s program is:
max
g uf (g,μ) subject to r 0 g = x (4.2)
which generates collective demand functions, ˜g (r,x,μ). It is important to
emphasize that this program is not equivalent to standard utility maximization
(the unitary model) because u f varies with μ, which in turn depends
on prices, income and distribution factors. Yet, for any fixed μ, ˜g (., μ) is a
This is page 157
Printer: Opaque this