Question

Suppose a profit-maximizing monopolist is producing 800 units of output and is charging a price of \(40 per unit.

a. If the elasticity of demand for the product is -2, find the marginal cost of the last unit produced.

b. What is the firm’s percentage markup of price over marginal cost?

c. Suppose that the average cost of the last unit produced is \)15 and the firm’s fixed cost is $2000. Find the firm’s profit.

Short answer

1. The marginal cost is $20.
2. The percentage markup of price over marginal cost is 50%.
3. The firm’s profit is $20,000.

Step-by-step solution

Computing the marginal cost

The following equation provides the relationship between the monopoly price, marginal cost, and elasticity:

$\frac{P-MC}{P}=\frac{1}{e}$

In the equation, e denotes the elasticity.

You can re-write it as follows:

$P\left(1+\frac{1}{e}\right)=MC$

The monopolist charges a price of $40 per unit, and the elasticity is -2.

You can compute the marginal cost as follows:

$$\begin{gathered} MC=40\left(1+\frac{1}{-2}\right) \\ =40\times \frac{1}{2} \\ =\$20 \end{gathered}$$

Thus, the marginal cost is $20.

Calculating the percentage markup of price over marginal cost

The marginal cost is $20,and the price is $40 per unit.

The formula for markup of price over marginal cost is$\frac{P-MC}{P}\times 100$.

You can compute the markup as follows:

$\frac{40-20}{40}\times 100=\frac{20}{40}\times 100=50\%$

The markup of price over marginal cost is 50%.

Computing the firm’s profit

The firm produces 800 units of output and charges $40 per output.

Thus, the total revenue isas follows:

TR = 800 x 40

= $32,000

The average cost is $15.

Thus, the total cost iscalculated as follows:

TC = 800 x 15

= $12,000

You can compute the profit asfollows:

$$\begin{gathered} \mathrm{\pi }=32,000-12,000 \\ =\$20,000 \end{gathered}$$

The monopolist’s profit is $20,000.