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Chapter 10: Problem 18

A monopolist faces the following demand curve: \\[ Q=144 / P^{2} \\] where \(Q\) is the quantity demanded and \(P\) is price. Its average variable cost is \\[ \mathrm{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

Expert verified
The profit-maximizing price and quantity need to be calculated using calculus and profit maximization condition. When the government imposes a price ceiling, the quantity produced and profit will change accordingly. If the government wants to induce maximum output, the price ceiling should be set as the AVC.

Step by step solution

01

Establish the Monopolist's Profit Maximization Strategy

First, to solve for the profit-maximizing price and quantity, the cost function should be identified using the average variable cost (AVC) and fixed cost (FC). The total variable cost (TVC) can be derived by integrating the AVC (i.e., \( \int Q^{1 / 2} dQ \)), resulting in \( \frac{2}{3} Q^{3/2} \). Adding FC to TVC will give us the total cost (TC). TC= \( \frac{2}{3} Q^{3/2} \) + 5. The revenue function, which is the price times the quantity indicates \( R = P * Q = 144 / P \). Then, derive the profit function which is the difference between the revenue and cost function indicates \( \pi = R - TC \). The monopolist maximizes profit where Marginal Cost = Marginal revenue or where the profit function is at its highest point.
02

Calculate profit-maximizing quantity and price

Differentiate the profit function with respect to \( Q \) and set the result equal to zero. Solve for \( Q \) which represents the profit-maximizing quantity. From the demand equation, substitute \( Q \) in terms of \( P \) to find the profit-maximizing price.
03

Government Regulation: Maximum Price

Next, consider the case where the government imposes a maximum price of $4 per unit. Use the demand equation to substitute this price \( P = 4 \) and solve for \( Q \). Subsequently, substitute the quantity in the profit equation to find the new profit.
04

Government Regulation: Price Ceiling for Maximum Output

Finally, suppose the government wants to set a ceiling price that induces the monopolist to produce the largest possible output. One approach is to set the price equal to the AVC. When the price is equal to the AVC, the monopolist can cover its variable costs and will therefore not shutdown, leading to maximal output. In this case, use the AVC equation \( Q^{1 / 2} \) to solve for \( P \).

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Most popular questions from this chapter

Michelle's Monopoly Mutant Turtles (MMMT) has the exclusive right to sell Mutant Turtle t-shirts in the United States. The demand for these t-shirts is \(Q=10,000 / P^{2} .\) The firm's short-run cost is \(\mathrm{SRTC}=\) \(2000+5 Q,\) and its long-run cost is \(\mathrm{LRTC}=6 Q\) a. What price should MMMT charge to maximize profit in the short run? What quantity does it sell, and how much profit does it make? Would it be better off shutting down in the short run? b. What price should MMMT charge in the long run? What quantity does it sell and how much profit does it make? Would it be better off shutting down in the long run? c. Can we expect MMMT to have lower marginal cost in the short run than in the long run? Explain why.

One of the more important antitrust cases of the 20 th century involved the Aluminum Company of America (Alcoa) in \(1945 .\) At that time, Alcoa controlled about 90 percent of primary aluminum production in the United States, and the company had been accused of monopolizing the aluminum market. In its defense, Alcoa argued that although it indeed controlled a large fraction of the primary market, secondary aluminum (i.e., aluminum produced from the recycling of scrap) accounted for roughly 30 percent of the total supply of aluminum and that many competitive firms were engaged in recycling. Therefore, Alcoa argued, it did not have much monopoly power. a. Provide a clear argument in favor of Alcoa's position. b. Provide a clear argument against Alcoa's position. c. The 1945 decision by Judge Leamed Hand has been called "one of the most celebrated judicial opinions of our time." Do you know what Judge Hand's ruling was?

A monopolist faces the demand curve \(P=11-Q\) where \(P\) is measured in dollars per unit and \(Q\) in thousands of units. The monopolist has a constant average \(\operatorname{cost}\) of \(\$ 6\) per unit. a. Draw the average and marginal revenue curves and the average and marginal cost curves. What are the monopolist's profit-maximizing price and quantity? What is the resulting profit? Calculate the firm's degree of monopoly power using the Lerner index b. A government regulatory agency sets a price ceiling of \(\$ 7\) per unit. What quantity will be produced, and what will the firm's profit be? What happens to the degree of monopoly power? c. What price ceiling yields the largest level of output? What is that level of output? What is the firm's degree of monopoly power at this price?

Suppose that an industry is characterized as follows: $$\begin{array}{|ll|} \hline C=100+2 q^{2} & \text { each firm's total cost function } \\ \hline M C=4 q & \text { firm's marginal cost function } \\ \hline P=90-2 Q & \text { industry demand curve } \\ \hline M R=90-4 Q & \text { industry marginal revenve curve } \\ \hline \end{array}$$ a. If there is only one firm in the industry, find the monopoly price, quantity, and level of profit. b. Find the price, quantity, and level of profit if the industry is competitive. c. Graphically illustrate the demand curve, marginal revenue curve, marginal cost curve, and average cost curve. Identify the difference between the profit level of the monopoly and the profit level of the competitive industry in two different ways. Verify that the two are numerically equivalent.

Will an increase in the demand for a monopolist's product always result in a higher price? Explain. Will an increase in the supply facing a monopsonist buyer always result in a lower price? Explain.
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