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

Utility Analysis and the Consumption Function 19
c (y), p (y)
p (y) = y (y) = y
c (y)
s (y)
o
y
Figure 1.1
Consumption-Income relation in the case of a stationary cross-section
inhabitants of these respective brackets. Thus, the proportion of income saved
will tend to rise with income, and the cross-section relation between consumption
and income will tend to be represented by a line obtained by rotating the stationary
line clockwise around a fixed point whose x and y coordinates coincide
approximately with the average value of income and consumption respectively.
While the general line of argument developed above is not new, 34 it may be
useful to clarify it by means of a graphical illustration developed in figures 1.1
and 1.2. We will start out by analyzing a cross section of households all belonging
to a single age group within the earning span, and will examine first the
consumption-income relation; once we have established this relation, the savingincome
relation can easily be derived from it.
Our consumption function (II.1) can be rewritten in a form analogous to the
saving function (II.2¢), namely:
N
c
L y e
1
L y y e
1
L a a y e
= + -
, t
t
( )+ [ - ( )]
N Ï
L y e
L
NL y y e
L
NL a a y e
t ¸
= Ì + ( - )+ [ - ( , )]
˝
Ó
t
t
˛
t
y
(II.1¢)
In the construction of our figures, we shall find it convenient to have a symbol
to represent the expression in braces; let us denote it by p = p(y,y e ,t,a). This expression
may be regarded as the stationary equivalent income of the current set of
values y, y e , t and a, for the household, in the sense that, if the household were
fully adjusted to a level of income p = p(y,y e ,t,a), then it would behave in the same