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

Utility Analysis and the Consumption Function 5
U = U( c , c ,... c , a ).
t t+ 1 L L+
1
(1.1)
This function is to be maximized subject to the budget constraint, which, if the
rate of interest, r, is not expected to change appreciably over the balance of the
life span, can be expressed by means of the equation
a
t
N
L
yt
aL+
1
ct
+ Â = +
t +- 1 t Â
t = t ( 1+
r)
( 1+
r)
t = t ( 1+
r)
. 6
L+- 1 t t +- 1 t
(1.2)
For the utility function (1.1) to be maximized, the quantities c t and a L+1 must
be such as to satisfy the first order conditions:
Ï ∂ U l
Ô
=
t
∂ ct
Ô ( 1+
r)
Ì
Ô ∂ U l
Ô =
Ó∂
aL+
1 ( 1+
r)
+- 1 t
L+-
1 t
; t = tt , + 1,...,
L
(1.3)
where l represents a Lagrange multiplier. The equation (1.3), together with (1.2),
yields a system of L - t + 3 simultaneous equations to determine L - t + 1 c¯t’s,
ā L+1 and l¯ , the barred symbols being used to characterize the maximizing value
of the corresponding variable.
If current income, y t + ra t , is unequal to c t , the individual will be currently
saving (or dissaving); and similarly, if y t + ra t is not equal to c¯t, the individual
will be planning to save (or dissave) at age t. The traditional model suggests that
we may usefully distinguish two separate reasons for such inequalities to arise.
We refer to these reasons as the “motives for saving.” 7
I The first of these motives is the desire to add to the estate for the benefit of
one’s heirs; it arises when ā L+1 is greater than a t . Under this condition y t + ra t
must exceed c¯t for at least some t t.
II The second motive arises out of the fact that the pattern of current and prospective
income receipts will generally not coincide with the preferred consumption,
c¯t, for all t t. This clearly represents an independent motive in that it can
account for positive (or negative) saving in any subinterval of the life span, even
in the absence of the first motive.
It is precisely on this point that a really important lesson can be learned by
taking a fresh look at the traditional theory of the household; according to this
theory there need not be any close and simple relation between consumption in
a given short period and income in the same period. The rate of consumption in
any given period is a facet of a plan which extends over the balance of the individual’s
life, while the income accruing within the same period is but one element