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

116 The Life-Cycle Hypothesis
economy, proposition 1) is a direct implication of homothetic maps. Since each
household consumes all his accumulation over his life, proposition 3) (i) can be
established by noting that in the absence of growth, i.e. stationary population and
no other source of growth, national saving must be zero as the saving of the young
is offset by the dissaving of the old. But if population is growing steadily, that
will mean an increase in the relative number of the young who save, relative to
the old who dissave; thus the saving will be positive and increase with population
growth. Similarly if income grows through rising productivity, even if population
is stationary, saving will be positive (at least for growth rates in the
relevant range) because the young who are in their saving phase enjoy higher
lifetime resources than do the old who dissave (proposition 3) (ii) ).
The nature of the relation between saving and trend growth can be confirmed
also by marking use of the fundamental dynamic relation that must hold, in steady
state, between the saving ratio, s, the wealth income ratio w, and the rate of growth
g, namely, s = gw (which is the equivalent of the well-known Harrod-Domar
proposition for investment and the capital stock).
This equation directly confirms proposition 3) (i) ), as s 0 if g 0. For positive
values of s we found that the slope of the relation between s and growth, g,
is given by:
ds dg = w + g( dw dg)
Thus, if w were not a function of growth, or dw/dg = 0, s would be proportional
to growth and with slope equal to the wealth income ratio w, which, as
noted earlier, could be a rather large number, say on the order of five. However,
in general w will depend on g. In fact there are reasons to believe that it decreases
with g, which means that the second term in the above expression will become
more and more negative. Thus the slope declines as g rises, or the relation
between s and g is concave to the g axis. However, the slope must be positive at
the origin and by continuity, in a neighborhood thereof, which may be expected
to include the relevant range of g which is empirically small—seldom more
than 10%.
One can throw some light on the relation between the wealth-income ratio and
growth by a different route. It was shown by Modigliani and Brumberg [15] that
under the assumption stated at the beginning of this section, aggregate consumption
can be expressed as a linear function of wealth and labor income, or:
C = a ◊ YL+
dW
(4.1)
where YL is labor income.
Since personal income is Y = YL + rW, where r is the rate of return on wealth,
saving can be expressed as: