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

270 6. Uncertainty and Dynamics in the Collective model
assumption. Take, for instance, a standard model of labor supply, in which
each agent consumes two commodities, namely leisure and a consumption
good. The collective model suggests that individual consumptions can be
recovered (up to additive constants - see chapters 4 and 5). Then tests of
the Euler equation family can be performed.
As an illustration, Mazzocco (2007) studies a dynamic version of the
collective model introduced in chapter 4. The individual Euler equations
become, with obvious notations:
∂ui ¡ ci t,li ¢
t /∂c
pt
∂ui ¡ ci t,li ¢
t /∂l
wi t
= β i Et
= β i Et
"
i ∂u ¡ ci t+1 ,li ¢
t+1 /∂c
pt+1
"
i ∂u ¡ ci t+1,li ¢
t+1 /∂l
w i t+1
Rt+1
Rt+1
#
#
(6.34)
for i = a, b. In particular, since individual labor supplies are observable,
these equations can be estimated.
6.4.3 The ex ante inefficiency case
What, now, if the commitment assumption is not valid? We have seen above
that this case has a simple, technical translation in the collective framework
- namely, the Pareto weights are not constant. A first remark, due to
Mazzocco (2007), is that even in the ISHARA case, aggregate consumption
no longer satisfies the martingale property (6.33). Indeed, let μt denote the
Pareto weight of b in period t, and assume for the moment that μt does not
depend on the agent’s previous consumption decisions. We first have that
½ ∙
Rt+1
c a t +c b t =
Moreover,
βptEt
pt+1
¸¾ a
−1/γ ½ ∙
¡ ¢ a
a −γ Rt+1 ¡ ¢−γ b
ct+1 + βptEt ct+1 pt+1
b¸¾−1/γ b
(6.35)
u0a (ca t )
u0b ¡ cb ¢ =
t
μ Ã
t β
1 − μt b
β a
! t
for all t, which for ISHARA (γa = γb = γ) preferences becomes
µ a −γ
ct = μ Ã
t β
1 − μt b
β a
! t
c b t
(6.36)
If μt is not constant, neither is the ratio ca t /cb t. A result by Hardy, Littlewood
and Polya (1952) implies that whenever the ratio x/y is not constant,
then for all probability distributions on x and y:
n h
Et (x + y) −γio−1/γ > © £ −γ
Et x ¤ª −1/γ © £ −γ
+ Et y ¤ª −1/γ