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

20 The Life-Cycle Hypothesis
P = y
c (y), y e (y), p (y), y e –1 (y)
Q
y e –1 (y) = y e (y)
p (y)
c (y)
y e –1
s (y)
Q¢
y n
o
y
y m
y
Figure 1.2
Consumption-Income relation and short-term fluctuations in income
way as it currently does. Let us further denote by the symbol x¯(y) the average
value of any variable x for all the members of a given income bracket, y. Then the
proportion of income consumed by the aggregate of all households whose current
income is y is, clearly, c¯(y)/y. Our problem is, therefore, that of establishing the
relation between c¯(y) and y. But, according to (II.1¢), c¯(y) is pr portional to p¯(y)
whose behavior, in turn, as we see, depends on that of ȳ e (y) and ā(y). We must
therefore fix our attention on the behavior of these last two quantities.
In the case of a stationary cross section, illustrated in figure 1.1, we know that
for every household, y e = y and also a = a(y e ,t). It follows that, for every household,
p = y, and therefore the average value of p in any income bracket y, p¯(y),
is also equal to y, i.e., p¯(y) = y. Thus, the cross-section relation between p¯(y) and
y is represented by a line of slope one through the origin—the dashed line of our
figure 1.1. The consumption-income relation is now obtained by multiplying
each ordinate of this line by the constant N/L, with the result represented by
the upper solid line. Because the p¯(y) line goes through the origin, so does the
consumption-income relation, c¯(y), and the elasticity of consumption with respect
to income is unity. These same propositions hold equally for the saving-income
relation, s¯(y) (lower solid line of figure 1.1), obtained by subtracting c¯(y) from
y. This merely illustrates a result established in the preceding section.