Question

7. The Cobb-Douglas production function and the steady state
This problem is based on the material in the chapter appendix.
Suppose that the economy’s production function is given by
Y = KaN1 - a
and assume that a = 1>3.
a. Is this production function characterized by constant
returns to scale? Explain.
b. Are there decreasing returns to capital?
c. Are there decreasing returns to labor?
d. Transform the production function into a relation between
output per worker and capital per worker.
e. For a given saving rate, s, and depreciation rate, d, give an
expression for capital per worker in the steady state.
f. Give an expression for output per worker in the steady state.
g. Solve for the steady-state level of output per worker when
s = 0.32 and d = 0.08.
h. Suppose that the depreciation rate remains constant at
d = 0.08, while the saving rate is reduced by half, to
s = 0.16. What is the new steady-state output per worker?

Answer 1

$$Solution$$

a) Yes, the production function characterized by constant returns to scale, since doubling $K$ and $N$ will double output.

$$Y=(2K)^a(2N)^{1-a}\\ Y=2^{a+1-a}(K)^a(N)^{1-a}=2K^aN^{1-a}$$

b) Yes, By Keeping labor constant and only increasing capital, output doesn't increase with constant returns to scale.

c) Yes, Keep capital constant and only increasing labor, output doesn't increase with constant returns to scale.

d)

$$\frac{Y}{N}=\frac{K^aN^{1-a}}{N}(\frac{K}{N})^a$$

e)

$$s\frac{Y}{N}=d\frac{K}{N}\\ s(\frac{K}{N})=d\frac{K}{N}\\ \frac{K}{N}=(\frac{s}{d})^\frac{3}{2}$$

f)

$$s\frac{Y}{N}=d\frac{K}{N}\\ s(\frac{K}{N})=d\frac{K}{N}\\ \frac{K}{N}=(\frac{s}{d})^\frac{3}{2}$$

g) To solve the steady-state level of output per worker given $s=0.32$ and $d=0.08$, then

$$\frac{K}{N}=(4)^\frac{3}{2}=8\\ \frac{Y}{N}=\frac{d}{s}*8=2$$

h) New steady-state output per worker, is obtained as

$$\frac{K}{N}=(2)^\frac{3}{2}\\ \frac{Y}{N}=\frac{d}{s}*(2)^\frac{3}{2}=(2)^\frac{1}{2}$$