Answer to Question #307381 in Macroeconomics for Mark

Question

Consider an economy descried by the production function: 𝑌 = 𝐹(𝐾, 𝐿) = 𝐾^0.3L0.7.

a) Derive the per-worker production function

b) Assuming no population growth or technological progress, find the steady state capita stock per worker, output per worker and consumption per worker as a

function of the saving rate and depreciation rate.

c) Define the Golden rule level of capital.

d) Under the Solow growth model, explain the effects of population growth on the steady-state level of capital and on output per worker (Show on a sketch).

Answer 1

a) To find the per-worker production function, divide the production function

𝑌 = 𝐾^0.3L^0.7 by L:

𝑌 =(𝐾^0.3L^0.7)/ L

𝑌 = 𝐾^0.3/L^0.3

𝑌 = (𝐾/L)^0.3

b) Recall that

Δk = sf(k) - δk.

The steady-state value of capital per worker k* is defined as the value of k at which capital per

worker is constant, so Δk = 0. It follows that in steady state

0 = sf(k) - δk,

or, equivalently,

K/f(k) = S/δ

For the production function in this problem, it follows that:

K/f(k)^0.3 = S/δ

Rearranging:

(k*)^0.7= S/δ

or

(k*) = S/δ^1/0.7

Substituting this equation for steady-state capital per worker into the per-worker production

function from part (a) gives:

y* =S/δ^0.3/0.7

.

Consumption is the amount of output that is not invested. Since investment in the steady state

equals δk*, it follows that

c=f(k)-(k)-δk

=(S/δ^0.3/0.7) - (S/δ^1/0.7)

c) In a simple model with no technological progress, the Golden Rule states that steady-state consumption per head is maximized when the marginal productivity of capital equals the sum of the population growth rate and the rate of depreciation of capital.

d) Solow sets up a mathematical model of long-run economic growth. He assumes full employment of capital and labor. Given assumptions about population growth, saving, technology, he works out what happens as time passes. The Solow model is consistent with the stylized facts of economic growth.