Question

1. (25 points) Suppose that the production function of a firm is given by

F(K)= AK ⁰.⁵
so MPK⁻ᶠ = 0.5AK ⁰.⁵

Assume MPK is measured in dollars.
(a) If the depriciation rate is 5% and the real interest rate is 2% what is the user cost of capital if the price of capital
Pₖ=$10 ?
(b) If A = 2what will be the firms desired capital stock?
(c) If the firms current capital stock is worth $20, what will be the firms desired investment in dollars?
(d) Suppose the government implemented a 10% tax on revenue of the firm, what will be the new desired investment?

Answer 1

The user cost of capital is $0.7. The firm's desired capital stock is 0.98. The firm's desired investment in dollars is $9.8.

Step-by-step explanation:

(a) To calculate the user cost of capital, we need to consider the equation:

UC = (r + d)P

Where UC is the user cost of capital, r is the real interest rate, d is the depreciation rate, and P is the price of capital.

Plugging in the given values, we get UC = (0.02 + 0.05) * $10 = $0.7

(b) To find the firm's desired capital stock, we need to set the marginal product of capital equal to the user cost of capital:

MPK = UC

0.5AK^0.5 = 0.7

Solving for K, we get K = (0.7 / (0.5 * 2^0.5))^2 = 0.98

(c) To find the firm's desired investment in dollars, we multiply the desired capital stock by the price of capital:

Investment = K * P = 0.98 * $10 = $9.8

(d) If the government implements a 10% tax on revenue, the new desired investment would be:

New Investment = Investment - (Tax Rate * Revenue)

Since revenue is not given, we cannot calculate the new desired investment.