Question

Macroeconomics

Problem 3

Prove each of the following statements about the steady state of the Solow model with population growth and technological progress.

a. The capital– output ratio is constant.

b. Capital and labor each earn a constant share of an economy’s income. [Hint: Recall the definition MPK= f( k + 1) - f(k).]

c. Total capital income and total labor income both grow at the rate of population growth plus the rate of technological progress, n + g.

d. The real rental price of capital is constant, and the real wage grows at the rate of technological progress g. ( Hint: The real rental price of capital equals total capital income divided by the capital stock, and the real wage equals total labor income divided by the labor force.)

Answer 1

(a) The capital-output ratio is constant.
Ans: In the steady state, we know that sy = (n + ? + g)k. This
implies that ky
= s
(n+?+g). Since s, ?, n, and g are constant, this
means that the ratio ky
is also constant. Since ky
= K/L
Y/L = KY,
we can conclude that in the steady state, the capital-output ratio is
constant.

(b) Capital and labor each earn a constant share of an economy’s income.
Ans: We know that capital’s share of income = MPK(KY) in the

steady state, we know from part (a) that the capital-output ratio KY
is constant. We also know that the MPK is a function of k, which
is constant in the steady state; therefore the MPK itself must be
constant. Thus, capital’s share of income is constant. Labor’s share
of income is 1-[capital’s share]. Hence,if capital’s share is constant,
so is labor’s share. (c) Total capital income and total labor income both grow at the rate of
population growth plus the rate of technological progress, n + g.
Ans: We know that in the steady state, total income grows at n+g
the rate of population growth plus the rate of technological change.
In part (b) we showed that labor’s and capital’s share of income are
constant. If the shares are constant, and total income grows at the
rate n + g, then labor income and capital income must also grow at
the rate n + g.

(d) The real rental price of capital is constant, and the real wage grows
at the rate of technological progress g.
Ans: For competitive firms MPK = rP
We know that in the steady
state, MPK is constant because capital per effective worker k is
constant. Therefore, we can conclude that the real rental price of
capital is constant in the steady state. The real wage equals total
labor income divided by the labor force: w
P = TLI/L Equivalently,
wL = TLI. In terms of percentage changes, we can write this as
?%w + ?%L = ?%TLI. This equation says that the growth rate

of the real wage plus the growth rate of the labor force equals the
growth rate of total labor income. We know that the labor force
grows at rate n, and from part (c) we know that total labor income
grows at rate n + g. We therefore conclude that the real wage grows
at rate g.