Answer to Question #245695 in Macroeconomics for Manish

Question

The Missing Link Chain-Link Fence Company is
trying to determine how many chain-link fabricating
machines to buy for its factory. If we define a chain-
link fence of some specified length to be equal to one
unit of output, the price of a new fabricating machine
is 60 units of output, and the price of a one-year-old
machine is 51 units of output. These relative prices are
expected to be the same in the future. The expected
future marginal product of fabricating machines, mea-
sured in units of output, is 165 - 2K, where K is the
number of machines in use. There are no taxes of any
sort. The real interest rate is 10

Answer 1

a. What is the user cost of capital? Specify the units in which your answer is measured.

b. Determine the number of machines that will allow Missing Link to maximize its profit.

c. Suppose that Missing Link must pay a tax equal to 40% of its gross revenue. What is the optimal number of machines for the company?

d. Suppose that in addition to the 40% tax on revenue described in Part (c), the firm can take advantage of a 20% investment tax credit, which allows it to reduce its taxes paid by 20% of the cost of any new machines purchased, l^ a t is Missing Link's desired capital stock now? {Hint: An investment tax credit effectively reduces the price of capital to the firm.)

Price of the new fabricating machine, $p_k=60 \; units$

Price of one year old fabricating machine, $p_o=51 \; units$

Therefore, Depreciation, $d=\frac{p_K-p_o}{p_K} \times 100 = \frac{60-51}{60} \times 100 = 15 \; \%$

Marginal product of fabricating machines, $MPK^f=165-2K$

K=Desired capital stock or Number of machines

No taxes

Peal rate of interest, r=10%

To find:

a) User cost of capital

b) Number of machines that maximizes profit

c) Given taxes=40%, find optimal number of machines

d) In addition to 40% taxes, investment tax credit advantage reduces taxes by 20% of the cost of new machines purchased

a) The user cost of capital, uc, is the sun of the interest cost, $rp_k$ , and the depreciation cost, $dp_k$ , where d is the depreciation rate and $p_k$ is the price of new capital good.

$uc=rp_k +dp_k = (r+d)p_k \\ = (0.10+0.15) \times 60 \\ = 0.25 \times 60 \\ = 15 \;units$

The user cost of capital was therefore 15 units.

b) The profit maximizing condition requires $MPK^f=uc$ . Therefore,

$MPK^f=uc \\ 165-2K=15 \\ 150=2K \\ K=75\; units$

Number of machines that maximizer expected profits are therefore 75 units.

c) Following is the tax-adjusted user cost of capital.

$\frac{uc}{1-T} = \frac{(r+d)p_k}{1-T}$

It shows how large the before-tax future marginal product must be for a firm to willingly add another unit of capital.

The desired stock of capital, K in this case is the one where future marginal product equals the after tax adjusted user cost.

$MPK^f= \frac{uc}{1-T} \\ 162-2K = \frac{(0.1+0.15) \times 60}{1-0.4} \\ 162-2K = \frac{15}{0.6} \\ K=70 \; units$

Number of machines that maximizes expected profits after tax are therefore 70 units.

d) In addition to taxes, investment tax credit advantage reduces taxes by 20% of the cost of new machines purchased. This is equivalent to a decrease in tax leading to reduced tax adjusted user cost and higher desired stock of capital as shown below.

$MPK^f = \frac{uc}{1-(T- Investment \; credit)} \\ 165-2K = \frac{(0.1+0.5) \times 60}{1-(0.4-0.2)} \\ 165-2K = \frac{15}{0.8} \\ 165-2K = 18.75 \\ K = 73.125 \;units$

Number of machines that maximizes expected profits after tax and investment credit is therefore 73.125 units.