Question

Consider an economy described by the following production function: ? = ?(?, ?) = ?^1/3 ?^2/3

depreciation rate is 5 percent (? = 0.05)

the population grows at 2 percent (n = 0.02)

savings rate is 20 percent (s = 0.20)

f) At what rates do the following grow at in the steady state: [3 points]

a. Capital per worker, k:

b. Output per worker, y:

c. Total output, Y:

Answer 1

? = ?(?, ?) = ?^1/3 ?^2/3

Output per worker, y = Y/L = (?^1/3 ?^2/3) / L = (K/L)^1/3

y = k^1/3

at steady state, it is true that, change in k = 0

sy - (? + n + g)*k = 0

0.2*k^1/3 = (0.05 + 0.02 + 0)k = 0

0.2k^1/3 = 0.07k

2.857 = k^2/3

a.

k* = 4.83 : steady state capital per worker

b.

y* = 4.83^1/3 = 1.68 : steady state output per worker

c.

Total Ouput, Y = y*L

Y = 1.68L

where L is the units of labor

So, total output is 1.68 multiplied by the value of labor units.