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

22 The Life-Cycle Hypothesis
y y
e
y
e
- - 1
-1
.
If we denote this elasticity by the symbol E, we have
E y e
-
=
y
e
y-
y
e
-1
-1
36
,
(II.11)
which in turn implies:
e e e e
y = y + E( y-y )=( 1 -E) y + Ey;
-1 -1 -1
(II.11¢)
i.e., the current income expectation is a weighted average of the previous expectation
and current income, with weights depending on the elasticity of expectation.
If E is close to zero, current income will have little influence in reshaping
expectations and y e will be close to y e -1; if, at the other extreme, E is close to unity
then current expectations will be determined primarily by the recent behavior of
income. From (II.11¢) we readily deduce the relation between ȳ e (y) and
ȳ e -1(y), namely:
[ ]
e e e
y ()= y y ()+ y E y- y () y .
-1 -1
37
(II.12)
Equation (II.12) admits of a very simple graphical interpretation. Suppose, first,
that E is zero: then ȳ e (y) coincides with ȳ e -1(y), the dotted line of figure 2. Next,
suppose E is positive but less than one; then, for any value of y, ȳ e (y), if it were
drawn, would lie between y and ȳ e -1(y), i.e., between the dashed and the dotted
line and precisely E percent of the way from the dotted to the dashed line. Finally,
if E is greater than one, or less than zero, then ȳ e (y) will fall outside the band
marked off by our two lines. The last two cases, however, may be regarded as
extremely unlikely, when one remembers that y e and y e -1 are defined as expectations
about the average level of income over the entire balance of the earning
span. 38 In our graph we have assumed a zero value for E so that ȳ e (y) coincides
with ȳ e -1(y); this assumption has been chosen not because we think it is realistic
but only because it eliminates the necessity of showing a separate curve in our
figure. In general, we should rather expect E to be positive but less than one, so
that the ȳ e (y) curve would fall between the dotted and the dashed line. The slope
and intercept of this curve will thus depend on the degree of short-term variability
of income [which determines the shape of ȳ e -1(y)] and on the elasticity of expectations.
But note that where short-run fluctuations play a large role, as might be
the case, say, for a sample of farmers, the elasticity of expectations may, itself,
be expected to be small on the average, since current income will contain little
new reliable information on the basis of which to reshape the previous expectation
about average future income, y e -1. Hence, a large short-term variability of