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

214 5. Empirical issues for the collective model
per member (or one overall with a distribution factor). However, identifiability
fails to obtain in a context in which the household behaves as a single
decision maker.
5.3.3 A simple example
The previous results can be illustrated by the following example, directly
borrowed from Chiappori and Ekeland (2009).Consider individual preferences
of the LES type:
U s (q s ,Q)=
nX
α s i log (q s i − c s i )+
i=1
NX
j=n+1
α s j log (Qj − Cj) , s = a, b
where the parameters αs i are normalized by the condition PN i=1 αsi =
1 for all s, whereas the parameters cs i and Cj are unconstrained. Here,
commodities 1 to n are private while commodities n +1to N are public.
Also, given the LES form, it is convenient to assume that the household
maximizes the weighted sum μU a +(1− μ) U b , where the Pareto weight μ
has the simple, linear form:
μ = μ 0 + μ x x + μ z z, s = a, b
Household demand
The group solves the program:
max ¡ μ 0 + μ x x + μ z z ¢
⎛
nX
⎝ α a i log (q a i − c a i )+
i=1
i=1
NX
j=n+1
+ ¡ 1 − ¡ μ 0 + μ x x + μ z z ¢¢
⎛
nX
⎝ a b i log ¡ q b i − c b¢ i +
under the budget constraint:
p 0 ¡ q a + q b¢ + P 0 Q = x
⎞
α a j log (Qj − Cj) ⎠
NX
j=n+1
⎞
α b j log (Qj − Cj) ⎠
where one price has been normalized to 1. Individual demands for private
goods are given by:
piq a i = pic a i + α a ⎛
¡ ¢
0 x z
i μ + μ x + μ z ⎝x − X
pic s i − X
⎞
PjCj ⎠
piq b i = pic b i + α b ⎛
£ ¡ ¢¤
0 x z
i 1 − μ + μ x + μ z ⎝x − X
pic s i − X
i,s
i,s
j
j
PjCj
⎞
⎠