Question

A monopolist faces the following demand curve: Q = 144/P² where Q is the quantity demanded and P is price. Its average variable cost is AVC = Q^1/2 and its fixed cost is 5.

a. What are its profit-maximizing price and quantity? What is the resulting profit?

b. Suppose the government regulates the price to be no greater than $4 per unit. How much will the monopolist produce? What will its profit be?

c. Suppose the government wants to set a ceiling price that induces the monopolist to produce the largest possible output. What price will accomplish this goal?

Short answer

1. The profit-maximizing price and quantity and the resulting profit of the monopolist are $ per unit, 4 units, and $11, respectively.
2. The monopolist will produce 7 units and earn profits worth $4.48 when the government regulates the price to be a maximum of $4.
3. If the government sets a price ceiling of $4.24, the monopolist will produce the largest possible output.

Step-by-step solution

Computing the profit-maximizing market conditions of monopolist

The monopolist faces the following demand curve:

$$\begin{gathered} \text{Q =}\frac{\text{144}}{{\text{P}}^{\text{2}}} \\ {\text{P}}^{\text{2}}\text{=}\frac{\text{144}}{\text{Q}} \\ {\text{P = 12Q}}^{\text{- 0.5}} \end{gathered}$$

Compute the total and marginal revenue as follows:

$$\begin{gathered} \text{TR =}\left({\text{12Q}}^{\text{- 0.5}}\right)\text{Q} \\ {\text{TR = 12Q}}^{0.5} \\ \text{MR =}\frac{\text{d}}{\text{dQ}}\left({\text{12Q}}^{0.5}\right) \\ {\text{MR = 6Q}}^{-0.5} \end{gathered}$$

The fixed cost is 5, and the average variable cost is AVC = Q^0.5.

Compute the total cost and the marginal cost as follows:

$$\begin{gathered} \text{C = 5 + Q}\left({\text{Q}}^{\text{0.5}}\right) \\ {\text{C = 5 +Q}}^{\text{1.5}} \\ \text{MC =}\frac{\text{d}}{\text{dQ}}\left({\text{5 +Q}}^{\text{1.5}}\right) \\ {\text{MC = 1.5Q}}^{\text{0.5}} \end{gathered}$$

The profit-maximizing quantity and price of the monopolist are as follows:

$$\begin{gathered} \text{MR = MC} \\ {\text{6Q}}^{-0.5}{\text{= 1.5Q}}^{0.5} \\ {\left({\text{Q}}^{0.5}\right)}^2\text{= 4} \\ \text{Q = 4} \\ \text{P = 12}{\left(\text{4}\right)}^{\text{- 0.5}} \\ \text{=}\frac{\text{12}}{\text{2}} \\ \text{= \$ 6} \end{gathered}$$

The monopolist maximizes its profit by producing 4 units and charging $6 per unit.

Compute the amount of profit as follows:

$$\begin{gathered} \mathrm{\pi }=\mathrm{PQ}-\mathrm{C} \\ =\left(6\times 4\right)-\left(5+4^{1.5}\right) \\ =24-13 \\ =\$11 \end{gathered}$$

The monopolist generates a profit worth $11.

Assessing the impact of a price ceiling of $4 on market conditions of monopolist

When the government imposes a price ceiling of $4, the quantity produced by the monopolist is:

$$\begin{gathered} \text{Q =}\frac{\text{144}}{{\text{4}}^{\text{2}}} \\ \text{Q =}\frac{\text{144}}{\text{16}} \\ \text{Q = 9} \end{gathered}$$

The monopolist can produce the utmost 9 units when the price ceiling of $4 is imposed.

However, the monopolist would want to maximize the profits and might, thus, produce an output equal to or less than 9 units at the price ceiling of $4.

The price will equal $4.

The total revenue and the marginal revenue will be:

TR = 4Q

MR = $4

The monopolist will maximize profit when the marginal revenue equals the marginal cost as follows:

$$\begin{gathered} \text{MR = MC} \\ 4{\text{= 1.5Q}}^{0.5} \\ \text{Q = 7.11} \end{gathered}$$

The monopolist will produce 7 units to maximize profits, given the price ceiling of $4.

The total revenue, total cost, and total profit will be:

$$\begin{gathered} \text{TR = 4}\left(7\right) \\ \text{= \$ 28} \\ \text{C = 5 +}{\left(7\right)}^{\text{1.5}} \\ \text{= \$ 23.52} \\ \pi \text{= 28 - 23.52} \\ \text{= \$ 4.48} \end{gathered}$$

The total revenue, total cost, and total profit of the monopolist are $28, $23.52, and $4.48, respectively.

Given the price ceiling, when the monopolist maximizes the profit, there will be a shortage of 2 units in the market.

Computing the price ceiling to induce the largest possible output

To find the price at which the monopolist would produce the largest possible output, you have to resort to marginal cost pricing.

You can compute the quantity and price as follows:

$$\begin{gathered} \frac{12}{\sqrt{Q}}=\frac{3\sqrt{Q}}{2} \\ {\left(\sqrt{Q}\right)}^2=\frac{12\times 2}{3} \\ Q=8 \\ P=\frac{12}{\sqrt{Q}} \\ =\frac{12}{\sqrt{8}} \\ =\$4.24 \end{gathered}$$

The price ceiling to induce the monopolist to produce the highest output of 8 units is $4.24.