In the United States, the capital share of GDP is about 3 percent, the average growth in output is about 3 percent per year, the depreciation rate is about 4 percent per year, and the capital-output ratio is about 2.5. Suppose that the production function is Cobb-Douglas, so that the capital share in output is constant, and that the United States has been in a steady state.
a. What must the saving rate be in the initial steady state?
b. What is the marginal product of capital in the initial steady state?
c. Suppose that public policy raises the saving rate so that the economy reaches the Golden Rule level of capital. What will the marginal product of capital be at the Golden Rule steady state? Compare the marginal product at the Golden Rule steady state to the marginal product in the initial steady state. Explain.
d. What will the capital-output ratio be at the Golden Rule steady state?
e. What must the saving rate be to reach the Golden Rule steady state?

a. A Cobb-Douglas function has the form y =[tex]k^{a}[/tex], where a is capital's share of income. The question tell us that a=0.3, so, we know the production function is y =[tex]k^{0.3}[/tex].

In the steady state we know that the growth rate of output equals 3%, so we know that (n+g)=0.03.

The depreciation rate d=0.04.
The capital-output ratio K/Y = 2.5. Because k/y= K*(L*A)/Y*(L*A)=K/Y.

Begin with the steady state condition sy=(d+n+g)k. Rewriting this equation leads to a formula for saving in the steady state:

s=(0.04+0.03)(2.5)=0.175

The initial rate is 17.5

b.

MPK=a/(K/Y)

MPK=0.3/2.5=0.12

c. MPK=(n+g+d)

MPK=(0.03+0.04)=0.07

In the Golden Rule Steady State, the capital-output ratio equals 4.29. Compared to the current capital ratio of 2.5

d. K/Y=a/MPK

K/Y=0.3/0.07=4.29

e.

s=(n+g+d)(k/y)

s=(n+g+d)(K/Y)

s=(0.04+0.03)(4.9)=0.175

s=0.3

To reach the Golden Rule Steady State, the saving rate must rise from 17.5 to 30 percent.