Consider an economy with an aggregate capital stock of 100 billion
that produces net income at the rate of one unit of income per
10 units of capital. The residents of that economy consume 80
percent of their income. The population of the community is
constant. What will be the percentage rate of growth of income in
this economy?
Two percent is the correct answer. Income in the first year is 10
units (0.1 unit for each unit of capital) and of this 8 units
are consumed and 2 units channelled into net investment. The
capital stock next period will thus be 102 units, which will
yield an income of 10.2 units. This is a growth rate of 2
percent.
The trick here is to understand that any income not consumed
is added to the capital stock and that the capital stock grows
to 1.02 (= 102/100) times its original magnitude. This is a
growth rate of 2 percent. It can be easily shown that the
growth rate in all subsequent years will also be 2 percent as
long as the output flow per unit of capital and the fraction of
income consumed remain the same. For example, in the second
year 0.8 times 10.2 (= 8.16) will be consumed and 10.2 minus
8.16 (= 2.04) units added to the capital stock. The subsequent
period's capital stock (and income) will be
(104.04/102 - 1)(100) = 2 percent larger.
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