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

Utility Analysis and the Consumption Function 21
Let us consider now what we may expect to happen if income is subject to
short-term fluctuations, a case illustrated in figure 1.2. We may assume for expository
convenience that on the average these fluctuations cancel out so that average
current income, ȳ, is the same as the average future income expected in the year
before by the sample as a whole, ȳ e -1. But, because of the presence of short-term
variations, this equality will not hold for individual households; for some households
current income will be higher than y e -1, while for others it will be lower. As
a result of these fluctuations, as we have already argued, in the highest income
brackets there will be a predominance of households whose current income is
above y e -1. This, in turn, means that in these brackets the average value of
y e -1,ȳ e -1(y), will be less than y itself; 35 in terms of our graph, ȳ e -1(y) will fall below
the dashed line, which represents the line of slope one through the origin. For
instance, corresponding to the highest income bracket shown, y m ,ȳ e -1(ȳ m ) may be
represented by a point such as q in our graph. Conversely, in the lowest income
bracket shown, y n , there will tend to be a preponderance of people whose current
income is below y e -1, and therefore the average value of y e -1 in this bracket, ȳ e -1(y n )
will be greater than y n and above the dashed line, as shown by the point q¢, in the
figure. Extending this reasoning to all values of y, we conclude that the relation
between ȳ e -1(y) and y will tend to be represented by a curve having the following
essential properties: (a) it will intercept the dashed line in the neighborhood of a
point with abscissa ȳ; and (b) to the right of this point, it will fall progressively
below the dashed line, while to the left of it, it will stand increasingly above this
line. In our graph this relation is represented by the dotted straight line joining
the points q¢ and q; in general, the relation need not be a linear one, although the
assumption of linearity may not be altogether unrealistic. What is essential,
however, is that the ȳ e -1(y) curve may be expected to have everywhere a slope
smaller than unity and to exhibit a positive intercept; and that its slope will tend
to be smaller, and its intercept larger, the greater the short-term variability of
income.
From the behavior of ȳ e -1(y), We can now readily derive that of ȳ e (y), which is
the quantity we are really interested in. The latter variable is related to ȳ e -1(y) and
to y itself through the elasticity of income expectations. The elasticity of expectations
relevant to our analysis can be defined as the percentage change in income
expectation over the two years in question,
e e
y - y - 1
,
e
y
-1
divided by the corresponding percentage difference between current income and
the previous year’s expectation,