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

180 4. The collective model: a formal analysis
¡
a b x ,x ¢ .Efficiency leads to the following program:
max
xa ;xb © a a b
μ˜v (p,x ; Q)+˜v
;Q
¡ p,x b ; Q ¢ª
The first order conditions give:
subject to x a + x b + P 0 Q = x (4.59)
μ ∂˜va
∂xa =
∂˜vb
∂xb ∂˜v a /∂Qj
∂˜v a /∂xa + ∂˜vb /∂Qj
∂˜v b /∂xb = Pj, j=1, ..., N (4.60)
The second set of conditions are often called the Bowen-Lindahl-Samuelson
(BLS) conditions. The ratio ∂˜va /∂Qj
∂˜v a /∂x a is exactly a’s willingness to pay for
public good j. Toseethis,notethatthefirst order conditions of (4.57)
imply that ∂ua
∂qa = λ
i
a pi, whereλ a is the Lagrange multiplier of a’s budget
constraint; and the envelope theorem applied to the definition of ˜v a gives
that ∂˜va
∂xa = λ a , hence ∂˜va
∂xa = 1 ∂u
pi
a
∂qa . Thus the conditions simply state that
i
MWP’s (or private prices) must add up to the market price of the public
good, as argued above. The BLS conditions (the second set of (4.60)) are
necessary and sufficient for efficiency. The choice of a particular allocation
on the Pareto frontier is driven by the first condition in (4.60).
As an application, consider the model of collective labor supply proposed
by Donni (2007), who assumes individual preferences of the form:
Us(T − hs,Q),
where Q is a Hicksian good which represents public consumption. Under
this hypothesis, and taking into account the property of homogeneity, labor
supplies can be written as:
µ
ws
hs = hs
πs(y, wa,wb) , ρ
i(y, wa,wb)
,
πi(y, wa,wb)
where
πi(y, wa,wb) = hiwi + ρi(y,wa,wb) y + hawa + hbwb
denotes member i’s Lindahl price for the public good. In this context, Donni
shows that the utility functions are identified, up to a positive transformation,
from individual labor supplies.
4.5.3 Application: labor supply, female empowerment and
expenditures on public good
While the previous concepts may seem somewhat esoteric, they have important
practical applications. For instance, a widely discussed issue in