Question

Using the revenue function given in the earlier question above, at an interest rate of 25%, if the firm makes it optimal choice, then its total profit would be? Revenue equal to $4.7k^0.5

A) 4.4

B) 4.85

C) 5.33

D) 5.75

previous question:

A firm uses capital, K, to produce revenue. The revenue function is given as Revenue = $4.7K^0.5. If the interest rate is currently 21%, what is optimal K for the firm to choose?

Group of answer choices

2

3

4

5

Answer 1

First we will answer the previous question. Then we will solve the above question.

Previous Question:

The firm uses capital, K, to produce revenue. The revenue function is given as

Revenue = R = $4.7........(1)

It is given that the interest rate is currently 21%. Hence,

r = 0.21

Hence, in a period of production, the cost of capital is

C = (1+r).K = (1+0.21).K

or, C = $1.21K........(2)

Now, the Revenue is, R = $4.7 and

the Cost is, C = $1.21K

Hence, the Profit of the firm is

π = R - C

or, π =

Now, when the profit is maximized, then the derivative of the profit function with respect to K will be zero.

Hence,

dπ/dK = 0

or,

or, = 4.7/(2×1.21)

or, = 1.9 ~ 2

or, K* = 4

Hence, the optimal K for the firm to choose is

K* = 4. (Answer is Option C)

Next Question:

The revenue function given in the earlier question is,

R = $4.7

Now, when the interest rate is 25% or r=0.25, the total cost of capital is

C' = (1+r).K = (1+0.25).K

or, C' = 1.25K...........(3)

Hence, the total profit of the firm in this case,

π' = R - C'

or, π' = .........(4)

Now, similar to the previous problem, if the firm is maximizing profit, then

dπ'/dK = 0

or,

or, = 4.7/(2×1.25)

or, = 1.88

or, K** = 3.5344

Hence, if the firm makes its optimal choice, then optimal capital choice is K** = 3.5344.

Now, putting K**=3.5344 in equation (4), the optimal profit of the firm is

π' =

Putting K=3.5344, we get

or, π' =

or, π' = 8.836 - 4.193

or, π' = $4.418 ~ $4.4

If the firm makes ootimal choice, then its total profit would be π' = $4.4. (Answer is Option A)

Hope the solutions are clear to you my friend.