Answer to Question 2:
Consider an economy in which three quarters of net income is
consumed. The ratio of the income flow to the capital stock is
0.08. If the ratios of income to capital and consumption to
income are constant through time, what will be the economy's
growth rate.
The correct answer is 2 percent. Income in the first year is,
say, 8 units on a capital stock of 100 units. Consumption will
be three-quarters of 8, which will be 6 units. Savings will
thus be 2 units and the capital stock next year will be 102.
This will yield an income of 8.16. A simple calculation will
show that 0.16 is 2 percent of 8.
A more elegant solution to this question can be constructed as
follows: The capital stock grows by
ΔK ⁄ K = S ⁄ K
where S is the level of savings and K is the initial stock
of capital and ΔK is the change in K per period. Now S equals
S = Y - C = .08K - (.75)(.08)K = .02K
Hence, the growth of the capital stock, and therefore income,
will be
ΔK ⁄ K = S ⁄ K = .02K ⁄ K = .02
Income will grow at the same rate as the capital stock because
the ratio of income to capital stock is constant. To see this,
let Y equal some constant δ times the capital stock.
Y = δ K
Then since δ, the ratio of income to the capital stock, is a
constant,
ΔY ⁄ Y = Δ(δ K) ⁄ (δ K) = δ Δ
K ⁄ (δK) = ΔK ⁄ K
where ΔY is the change in Y over the period and, as before, ΔK is the
change in K over the period.
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