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

Utility Analysis and the Consumption Function 43
we may find that the computed value of h cy is not exactly one, even for a stationary cross section. On
the other hand, it can be verified, from the above expression, that the variation in slope with age is
likely to be rather small so that, in fact, h cy should be quite close to unity, unless k 2 and n are substantially
different from unity, an unlikely case except in deep depression or at the peak of a boom.
45. The figures just quoted are in part approximate since Reid has not used the statistic r yy-1 but,
instead, one closely related to it. Although we do not have specific information on the age composition
of the samples, we understand that retired households, if any, represent a negligible proportion
of the samples.
46. See, for instance, Josephine H Staab, “Income-Expenditure Relations of Farm Families Using
Three Bases of Classification,” Ph.D. dissertation, The University of Chicago, 1952; Reid, op. cit.
(several new experiments are also reported in Reid’s preliminary draft quoted above); Vickrey, op.
cit. In essence Vickrey’s point is that consumption is more reliable than current income as a measure
of the permanent component of income (p. 273) and he suggests, accordingly, that the individual
marginal propensity to consume (with respect to the permanent component) can be estimated more
reliably by relating consumption (per equivalent adult), c, to ȳ(c) than by relating c¯(y) to y, as has
been usually done. It can be shown that Vickrey’s suggestions receive a good deal of support from
our model (with the addition of the stochastic assumptions introduced in various notes above) in that
the relation between c and ȳ(c) should be very similar to that between c and our quantity p. In particular,
c should be nearly proportional to ȳ(c), a conclusion that Vickrey himself did not reach but
which is well supported by his own tabulations. A double logarithmic plot of c against ȳ(c), based on
his data, reveals an extremely close linear relationship with a slope remarkably close to unity. (We
have estimated this slope, by graphical methods, at .97.) On the other hand, using the conventional
plot, c¯(y) against y, the slope for the same data can be estimated at somewhat below .85, and, in addition,
the scatter around the line of relationship is distinctly wider than for the first mentioned plot.
In the contribution under discussion Vickrey has also been very much concerned with the influence
of the size composition of the household on saving behavior. This is a point which, because of
limitations of space, we have been forced to neglect in the present paper. We will merely indicate, at
this point, that our central hypothesis (that the essential purpose of saving is the smoothing of the
major and minor variations that occur in the income stream in the course of the life cycle) provides
a framework within which the influence of family size can be readily analyzed. We hope to develop
this point in later contributions.
47. Studies in Income and Wealth, Vol. X, pp. 247–265.
48. Making use of equations (II.10¢), (II.12) and the expression for ā(y) derived in note 39, our crosssection
income-consumption relation can be reduced to the form
cy ( )= A+ By= Ay * + By,
where A* and B depend on the coefficients E, a, b, l, m, n we have introduced earlier, and on age.
These coefficients, in turn, depend primarily on the variability of income as measured by r e yy-1 and
r x y e -1, and, probably to a lesser extent, on the long-term trend of income (which affects l, m and n)
and on the cyclical position of the economy (which affects a and b and possibly E). Hence, if we
have various samples of households for each of which the variability of income is approximately the
same, the coefficients A* and B should also be approximately the same for each sample, especially
if the samples in question do not differ too markedly with respect to age, composition and the cyclical
position of total income. Denoting by c¯i and ȳ i the average value of consumption and income for
all households falling in the i-th quantile of a given sample, we must have
c = A* y+
By .
i
i
If, furthermore, for each of the samples compared ȳ i /ȳ is approximately constant, so that we can write
ȳ i = k i ȳ, we obtain
A*
ci yi = ( c y) = + B,
i
ki
i.e., the proportion of income consumed in a given quantile, i, should be approximately the same for
all samples compared, as stated in the text. We may add that, if we replace our simple stochastic