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

Utility Analysis and the Consumption Function 9
t
c = g v , t = t, t+
1,...,
L
t
t
t
(I.4)
where, for given t and t the quantity g t
t depends on the specific form of
the function U and on the rate of interest r, but is independent of “total
resources,” v t .
As a result of well-known properties of homogeneous functions, it can readily
be shown that a sufficient condition for Assumption II to hold is that the utility
function U be homogeneous (of any positive degree) in the variables c t , c t-1 ,
..., c L . 13
The remaining two assumptions are not essential to the argument, but are introduced
for convenience of exposition. 14
Assumption III The interest rate is zero, i.e., r = 0.
As a result of this assumption, the expression
v
can be rewritten as y t + (N - t)y e t + a t , where
yt e = Ê yt
N t
Ë Á ˆ
 ˜ ( - ),
¯
represents the average income expected over the balance of the earning span.
Equation (I.4) now reduces to:
t
c = g y + ( N -t) y + a
t
which implies
L
c yt N t yt e t
 t = [ + ( - ) + at
] Âg
t.
t = t
N
t =+ t 1
y
N
t
t = Â + a
t +- 1 t t
t = t 1
( + r)
t[ t t e t ]
L
t = t
Furthermore, taking into account Assumption I (ā L+1 = 0) we also have
(I.4¢)
L
 ct
= yt + ( N -t) yt e + at
.
t = t
(I.5)
From (I.4¢) and (I.5) it then follows that
L
t
g t
t = t
 = 1.
(I.6)