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Chapter 10: Problem 6

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.

Short Answer

Expert verified
In the case of a monopoly firm, the optimal quantity is 15, the price is 60 and the profit is 350. While for a perfectly competitive market, the optimal quantity also turns out to be 15, with the price at 60 and the profit at 350, exactly the same as the monopoly scenario. When graphically displayed, the profit levels of both markets appear the same. Their numerical equivalence verifies this similarity.

Step by step solution

01

Calculate Monopoly Quantity (a)

Equating Marginal Cost(MC) to Marginal Revenue(MR) to maximize monopoly firm's profit, \(4q = 90-4Q\) -> \(Q = q\). So, \(4q = 90-4q\), which solves to \(q = 15\).
02

Calculate Monopoly Price (a)

Substitute \(q = 15\) into industry demand curve to get the price, \(P = 90 - 2(15)\), which evaluates to \(P = 60\).
03

Calculate Monopoly Profit (a)

First, calculate total revenue by multiplying price and quantity, \(TR = PQ = 60 * 15 = 900\). Then, calculate total cost using total cost function, substituting \(q=15, TC=100+2(15)^2 = 100 + 450 = 550\). Profit is total revenue minus total cost, thus Profit = 900 - 550 = 350.
04

Calculate Competitive Industry Quantity (b)

In a competitive market, price equals marginal cost. Solve \(P=MC\) for \(q\) using the given functions. \(90-2Q=4q\), as \(Q=q\) in a competitive market, this gives \(90-2q=4q\), which solves to \(q=15\). Thus, in a competitive market, quantity is 15.
05

Calculate Competitive Industry Price (b)

Substitute \(q=15\) into industry demand curve to get the price. \(P=90-2(15)\) evaluates to \(P=60\). The price in a competitive market is 60.
06

Calculate Competitive Industry Profit (b)

Calculate the total revenue by multiplying the price and quantity, \(TR=60*15=900\). Total cost will be the same as in Step 3. The profit is total revenue minus total cost. Profit = 900 - 550 = 350, the same as the monopoly.
07

Graphical Representation (c)

Create a graph with the given functions: MC, MR, average cost and the demand curve. The difference in profits can be seen as the rectangular area difference under the price and above the cost curves on the graph.
08

Numerical Verification (c)

If calculated correctly, the competitive and monopoly profit are found to be the same. Hence, there is no profit difference when verified numerically.

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Most popular questions from this chapter

1: } C_{1}\left(Q_{1}\right)=10 Q_{1}^{2} \\\ \text { Factory #2: } C_{2}\lef… # A firm has two factories, for which costs are given by: \\[ \begin{array}{l} \text { Factory #1: } C_{1}\left(Q_{1}\right)=10 Q_{1}^{2} \\ \text { Factory #2: } C_{2}\left(Q_{2}\right)=20 Q_{2}^{2} \end{array} \\] The firm faces the following demand curve: \\[ p=700-5 Q \\] where \(Q\) is total output-i.e., \(Q=Q_{1}+Q_{2}\) a. \(\mathrm{On}\) a diagram, draw the marginal cost curves for the two factories, the average and marginal revenue curves, and the total marginal cost curve (i.e., the marginal cost of producing \(Q=Q_{1}+Q_{2}\) ). Indicate the profit-maximizing output for each factory, total output, and price. b. Calculate the values of \(Q_{1^{\prime}} Q_{2^{\prime}} Q,\) and \(P\) that maximize profit c. Suppose that labor costs increase in Factory 1 but not in Factory \(2 .\) How should the firm adjust (i.e. raise, lower, or leave unchanged) the following: Output in Factory \(1 ?\) Output in Factory \(2 ?\) Total output? Price?

Dayna's Doorstops, Inc. (DD) is a monopolist in the doorstop industry. Its cost is \(C=100-5 Q+Q^{2}\), and demand is \(P=55-2 Q\) a. What price should DD set to maximize profit? What output does the firm produce? How much profit and consumer surplus does DD generate? b. What would output be if \(D D\) acted like a perfect competitor and set \(\mathrm{MC}=P ?\) What profit and consumer surplus would then be generated? c. What is the deadweight loss from monopoly power in part (a)? d. Suppose the government, concerned about the high price of doorstops, sets a maximum price at \(\$ 27\) How does this affect price, quantity, consumer surplus, and DD's profit? What is the resulting deadweight loss? e. Now suppose the government sets the maximum price at \(\$ 23 .\) How does this decision affect price, quantity, consumer surplus, DD's profit, and deadweight loss? f. Finally, consider a maximum price of \(\$ 12 .\) What will this do to quantity, consumer surplus, profit, and deadweight loss?

A drug company has a monopoly on a new patented medicine. The product can be made in either of two plants. The costs of production for the two plants \(\operatorname{arc} \mathrm{MC}_{1}=20+2 \mathrm{Q}_{1}\) and \(\mathrm{MC}_{2}=10+5 \mathrm{Q}_{2}\). The firm'sesti mate of demand for the product is \(P=20-3\left(Q_{1}+Q_{2}\right)\) How much should the firm plan to produce in each plant? At what price should it plan to sell the product?

Caterpillar Tractor, one of the largest producers of farm machinery in the world, has hired you to advise it on pricing policy. One of the things the company would like to know is how much a 5 -percent increase in price is likely to reduce sales. What would you need to know to help the company with this problem? Explain why these facts are important.

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.
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