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

268 6. Uncertainty and Dynamics in the Collective model
6.4.2 Collective Euler equations under ex ante efficiency
Household consumption
We now consider a collective version of the model. Keeping for the moment
the single commodity assumption, we now assume that agents have their
own preferences and discount factors. The Pareto program is therefore:
Ã
X
max (1 − μ) E0 (β a ) t u a (c a ! Ã
X ³
t ) + μE0 β b´ t
u b ¡ c b¢ t
!
t
under the same constraints as above. First order conditions give:
u 0a (c a t )
pt
u0b ¡ cb ¢
t
pt
= β a Et
= β b Et
"
0a u ¡ ca ¢
t+1
pt+1
"
0b u ¡ cb ¢
t+1
pt+1
t
Rt+1
Rt+1
#
#
(6.29)
which are the individual Euler equations. In addition, individual consumptions
at each period must be such that:
(β a ) t
³
β b´ t
u0a (ca t )
u0b ¡ cb ¢ =
t
μ
1 − μ
(6.30)
The right hand side does not depend on t: the ratio of discounted marginal
utilities of income of the two spouses must be constant through time. This
implies, in particular, that
u0a (ca t )
u0b ¡ cb ¢ =
t
μ
1 − μ
³
β b´ t
(β a ) t
If, for instance, a is more patient than b, in the sense that β a >β b ,then
the ratio u 0a /u 0b declines with time, because a postpones a larger fraction
of her consumption than b.
An important remark is that if individual consumptions satisfy (6.29),
then typically the aggregate consumption process ct = c a t + c b t does not
satisfy an individual Euler equation like (6.28), except in one particular
case, namely ISHARA utilities and identical discount factors. For instance,
assume, following Mazzocco (2004), that individuals have utilities of the
CRRA form:
u X (c) = c1−γX
, X = a, b
1 − γX