8. Suppose that the firm faces a Cobb–Douglas production function with two inputs: K is

Question

capital (the number of machines = ૚૞), L is labor (the number of workers = ૟૙). The
ࡸ૜.૙ࡷ૙૙૚ = ࢅ production function is
૙.ૠ
. Suppose further the cost of capital to a Machine￾rental company. The company buys Machines for $૟, ૙૙૙ each and rents them out to other
businesses, and the economy’s price level, ࡼ ࢙࢏ $૛૞. The company faces a real interest rate ݎ
of 10 percent per year and the machines depreciate at 20 percent per year.
A. Compute the marginal product of machines and the real cost of machines, and
interpret the result (should the rental firm add to its capital stock -invest in new
machines)?
B. What happen to the firms’ decision if the relative price of machines increases by 10%
and 15% in order?
C. What happen to the real rental price of capital/Machine if:
i. Immigration raises the labor force by 10 percent.
ii. An earthquake reduces the capital stock by 10 percent. (Support your explanation
using graph).
iii. Technological improvement raises the value of the parameter ࡭ by 10 percent�

Answer 1

A.
- The marginal product of machines can be calculated using the production function: MPK = 0.5 * (K/L) ^ 0.5.
- The real cost of machines can be calculated as the rental rate divided by (1 + depreciation rate): real cost = 0.1 / (1 - 0.2) = 0.125.
- Interpretation: The marginal product of machines is decreasing with each additional machine, indicating diminishing returns to capital. The real cost of machines is 0.125, which means that the rental company can earn a real return of 12.5% on each machine. Whether the rental firm should add to its capital stock depends on whether the expected return on new machines is higher than 12.5%.

B.
If the relative price of machines increases by 10% and 15%, the real cost of machines will also increase by the same percentage. This would make investing in new machines more attractive for the rental firm.

C.
i. Immigration raising the labor force by 10% would decrease the marginal product of machines, leading to a decrease in the real rental price of capital/Machine.
ii. An earthquake reducing the capital stock by 10% would increase the marginal product of machines, leading to an increase in the real rental price of capital/Machine.
iii. Technological improvement raising the value of the parameter ࡭ by 10% would increase the marginal product of machines, leading to an increase in the real rental price of capital/Machine.