"Life Cycle" Hypothesis of Saving: Aggregate ... - Arabictrader.com

Utility Analysis and the Consumption Function 13
must be zero for an individual, there is no reason why the cross-section rate of
change,
da ¢
,
dy ¢
should also be zero. Quite the contrary. Our model leads us to expect a very definite
relation between the (average) net worth of house-holds at a given income
level and the income level itself, which relation we now proceed to explore.
II.2 The Equilibrium Income-Asset-Age Relation and the
Consumption-Income Relation in a Stationary and Nonstationary
Cross Section
To see clearly the implications of our model it will be useful to examine at first
the nature of the cross-section relation between consumption and income in a
special case which we shall call a cross section of “stationary” households. A
household will be said to be in stationary position if it satisfies the following two
conditions: (a) at the beginning of its active life it expects a constant income
throughout its earning span; and (b) at every point of its life cycle it finds that its
original expectations are completely fulfilled in the sense that its past and current
income are as originally expected and its expectations for the future also coincide
with its original expectations. 20 From equations (I.4) and (I.7) (since by
assumption y e 1 = y 1 and a 1 = 0) we see that for such a household the consumption
plan at age 1, which we denote by c¯ 1t, must be
c
1
t
N
=
L y 1, t = 12 , ,..., L ;
(II.5¢)
and the saving plan
s
1
t
Finally, the asset plan, which is the sum of planned savings, must be
a
1
t
Ï M
L y 1, t = 12 , ,..., N .
Ô
= Ì
Ô N
-
L y 1 t = N + 1 N + 2
ÓÔ
, , ,..., L .
Ï( t -1)
M
y1, t = 12 , ,..., N
Ô L
= Ì
Ô NL ( + 1-t)
y1
t = N+ 1 N+
2 L
ÓÔ
, , ,..., .
L
(II.5≤)
(II.5≤¢)