1. A firm uses capital to produce revenue. The marginal revenue from the first 5 units of capital is as follows: 1st unit has MR 2.30, 2nd unit has MR 1.9, 3rd unit has MR 1.70, 4th unit has MR 1.35, and 5th unit has MR 1.15. If the interest rate is 50%, what is the optimal amount of capital for this firm to borrow?
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A firm uses capital to produce revenue. The marginal revenue from the first 5 units of capital is as follows: 1st unit has MR 2.1, 2nd unit has MR 1.93, 3rd unit has MR 1.76, 4th unit has MR 1.55, and 5th unit has MR 1.35. If the interest rate is 51%, what is the optimal amount of capital for this firm to borrow?
5.15
To determine the optimal amount of capital for the firm to borrow, we need to consider the marginal revenue (MR) and the interest rate.
1. Calculate the present value (PV) of the marginal revenue for each unit of capital using the formula:
PV = MR / (1 + interest rate)
For the given marginal revenues and an interest rate of 50%, we have:
- PV of the 1st unit = 2.30 / (1 + 0.50) = 1.53
- PV of the 2nd unit = 1.90 / (1 + 0.50) = 1.27
- PV of the 3rd unit = 1.70 / (1 + 0.50) = 1.13
- PV of the 4th unit = 1.35 / (1 + 0.50) = 0.90
- PV of the 5th unit = 1.15 / (1 + 0.50) = 0.77
2. Determine the present value of the additional revenue obtained from borrowing each unit of capital. This can be calculated by subtracting the PV of the previous units from the current unit's PV.
- Additional PV of the 1st unit = PV of the 1st unit = 1.53
- Additional PV of the 2nd unit = PV of the 2nd unit - PV of the 1st unit = 1.27 - 1.53 = -0.26
- Additional PV of the 3rd unit = PV of the 3rd unit - PV of the 2nd unit = 1.13 - 1.27 = -0.14
- Additional PV of the 4th unit = PV of the 4th unit - PV of the 3rd unit = 0.90 - 1.13 = -0.23
- Additional PV of the 5th unit = PV of the 5th unit - PV of the 4th unit = 0.77 - 0.90 = -0.13
3. Find the point where the additional PV turns negative. This indicates that borrowing an additional unit of capital would result in a decrease in revenue. In this case, it is the 3rd unit.
Therefore, the optimal amount of capital for this firm to borrow is 2 units. Borrowing more than 2 units would result in a decrease in revenue due to the diminishing marginal returns.