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

For the observable demand function we have:
∂ˆgi
∂rj
∂ˆgi
∂x
4. The collective model: a formal analysis 159
= ∂˜gi
+
∂rj
∂˜gi
∂μ
= ∂˜gi
∂x
+ ∂˜gi
∂μ
∂μ
∂rj
∂μ
∂x
(4.5)
Thus we can decompose the price effect into a Marshallian response (the
first term on the right hand side) and a collective effect (the second term),
which operates through variations of the Pareto weight μ. Figure 4.1 illustrates
for two goods. We start with prices and income (r,x) and the
demand at point I. We then change prices so that good 1 is cheaper; denote
the new environment (r 0 ,x). The substitution effect is given by the
move from I to II and the income effect is II to III. The collective effect
associated with the change in μ is represented by the final term in (4.5)
which is shown as the move from III to IV .
4.1.3 The Slutsky matrix for collective demands.
Using the observable functions ˆg (.), wecandefine the observable or quasi-
Slutsky matrix S =[sij] i,j by its general term:
sij = ∂ˆgi ∂ˆgi
+ˆgj
∂rj ∂x
From (4.5) this can be written as:
∙ ¸
∂˜gi ∂˜gi
sij = +˜gj +
∂rj ∂x
∂˜gi
∙
∂μ
∂μ ∂rj
¸
∂μ
+˜gj
∂x
(4.6)
(4.7)
From (4.3), the first term between brackets is the substitution term σij
with associated matrix Σ. Weadoptthefollowingnotation:
∙ ¸
∂˜gi
Dμ˜g =
∂μ i
∙ ¸
∂μ ∂μ
v = +˜gj
(4.8)
∂rj ∂x
This gives:
S = Σ +(Dμ˜g) .v 0 = Σ + R (4.9)
so that the Slutsky matrix of the observable collective demand ˆg (r,x) is the
sum of a conventional Slutsky matrix Σ, which is symmetric and negative,
and an additional matrix R. The latter is the product of a column vector
(Dμ˜g) and a row vector (v 0 ). Note that such an outer product has rank of
at most one; indeed, for any vector w such that v 0 .w =0we have that
j