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

scribing dynamic choices is:
6. Uncertainty and Dynamics in the Collective model 267
max E0
à X
t
β t u ¡ c a t ,c b¢ t
!
under the constraint
¡ a
pt ct + c b¢ t + st = y a t + y b t + Rtst−1, t =0, ..., T
Here, E0 denotes the expectation taken at date 0, andβis the household’s
discount factor. Note that if borrowing is excluded, we must add the
constraint st ≥ 0.
Using a standard result by Hicks, we can define household utility as
a function of total household consumption; technically, the function U is
defined by:
U (c) =max © u ¡ c a ,c b¢ such that c a + c b = c ª
and the program becomes:
under the constraint
max E0
Ã
X
β t !
U (ct)
t
ptct + st = y a t + y b t + Rtst−1
The first order conditions give the well-known Euler equations:
U 0 ∙ ¸
0 (ct) U (ct+1)
= βEt Rt+1
pt
pt+1
(6.28)
In words, the marginal utility of each dollar consumed today equals, in expectation,
β times the marginal utility of Rt+1 dollars consumed tomorrow;
one cannot therefore increase utility by marginally altering the savings.
In practice, many articles test the empirical validity of these household
Euler equations using general samples, including both couples and singles
(see Browning and Lusardi 1995 for an early survey); most of the time, the
conditions are rejected. Interestingly, however, Mazzocco (2004) estimates
the same standard household Euler equations separately for couples and for
singles. Using the CEX and the Panel Study of Income Dynamics (PSID),
he finds that the conditions are rejected for couples, but not for singles.This
seems to suggest that the rejection obtained in most articles may not be
due to technical issues (for example, non separability of labor supply), but
more fundamentally to a misrepresentation of household decision processes.