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

34 The Life-Cycle Hypothesis
Unfortunately, the variable yi e is usually unknown. Some information on it
might be gathered by appropriate questions analogous to the question on shortterm
income expectations which has already been asked in the past. Klein himself,
however, has not made use of this possible source of information in the study in
progress. We must, therefore, find some way of relating yi
e to the variables
actually used by Klein, which are also those most commonly available.
It is clear that y e must bear a fairly close and reasonably stable relation to
current income, y. In fact, at various points in the text we have suggested that the
relation between these two variables maybe expressed by the equation
e
e
Ïy = ( 1-E) y-1
+ Ey+ A+
U
Ì
Ó = ( 1-E) ( a + b y)+ Ey + U¢
,
(A.2)
where A is a constant for a given sample (though subject to variation over time);
E is the elasticity of income expectations defined by equation (II.11); U and
U¢ =(1 - E)e¢ +U are random errors; a, b are defined by equations (II.10¢); and
e¢ is the random component of that equation.
If we also have information on previous year’s income, y -1 , we can clearly
exploit this information to get a better estimate of y e -1, and (A.2) then takes the
form
y e = a* + b y + b y + U* .
1 2 -1
(A.3)
The coefficients of this equation, as well as the random component U*, depend
again on the short-term variability of income as measured by the correlation
r yy-1 , on the variance of U and on the elasticity of expectations E. It is not
worth-while, however, to derive here this relation explicitly.
Substituting for y e from (A.3) into (A.1), and rearranging terms (and neglecting
the error term which is proportional to U*) we get:
Ï ( N - ti)
a*
( L-
ti) ( L-
Nb )-L( N -ti)
b
si
=- +
Ô Li
LLi
Ì
( N - ti)
b2
Mb1
( yi
- y i)+
L
LL ty i i
ÓÔ -1
.
i
i
1 2
y
i
ai
- +
L
i
(A.4)
Finally, dividing through by y i and making use of the identity L ∫ M + N, we
obtain the result 62
Ï si
Ô yi
Ì
Ô
Ó
( Li
-1)( L-
Nb1)-L( Li
-M-1)
b2
( Li
-M-1)
a*
1 1 ai
=
-
-
LLi
Li yi Li
yi
( N - ti)
b2
Mb1
+ ( yi - y 1i) yi
+
L
LL t i.
Ô
-
i
i
(A.5)