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

Utility Analysis and the Consumption Function 11
hood thereof) we must seek what implications can be deduced as to the relations
between these variables.
If the marginal propensity to consume is defined literally as the increment in
the current consumption of the household accompanying an increment in its
current income, divided by the increment in income, keeping other things constant,
then, according to equation (II.1), this quantity would be
∂ c
= 1 ,
∂ y L t
which is independent of income but dependent on age. The consumption function
(II.1) would be represented by a straight line with the above slope and an
intercept
e
( N - t) y + a ;
L
t
and, since this intercept can be assumed positive, 17 the proportion of income saved
should tend to rise with income.
In order to get some feeling as to the quantitative implications of our results,
let us say that the earning span, N, is of the order of 40 years, the retirement span,
M, 10 years, and therefore the total active life span, L, of the order of 50 years.
These figures are not supposed to be anything more than a very rough guess and
their only purpose is to give us some notion of the magnitudes involved. On the
basis of these figures, the marginal propensity to consume would lie somewhere
between a minimum of 1/50, or 2 percent, and a maximum of 1/11, or 9 percent,
depending on age.
These figures seem unreasonably small. This is because the above definition
of the marginal propensity to consume is clearly not a very reasonable one. A
change in the current income of the household will generally tend to be accompanied
by a change in its expected income, y e , so that there is little sense in including
y e among the things that are supposed to be constant as y changes. Note,
however, that the same objection does not apply to a, for a denotes initial assets
and, for a given household, assets at the beginning of the current period necessarily
represent a constant.
Once we recognize that y e is generally a function of y, the marginal propensity
to consume at age t may be defined as
dc
dy
e
1 N - t dy
= + .
L L dy
t
t
(II.3)
Since