Chapter 7 Pricing with Market Power SOLUTIONS TO EXERCISES

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1 Firms, Prices & Markets Timothy Van Zandt August 2012 Chapter 7 Pricing with Market Power SOLUTIONS TO EXERCISES Exercise 7.1. Suppose you produce minivans at a constant marginal cost of $15K and your demand curve is d(p ) = P. (Price is measured in 1000s of dollars and quantity is measured in 100,000s of minivans.) Find your optimal price and quantity. Solution: The choke price is 16/0.6 = 80/3; hence your optimal monopoly price is ((80/3) + 15)/2 = 125/6. Exercise 7.2. What happens if a per-unit tax is imposed on a monopolist s product? One answer might be: Since the monopolist can charge whatever it wants, it will pass the tax on to the consumer. But this does not say much, since any firm can charge whatever price it wants. The question is: What price does it want to charge? Let s see if we can provide a more informative answer. When faced with a question like this, one strategy is to start by working out a few simple examples. This at least illustrates some possibilities. Find out how a per-unit tax changes the price of (a) a firm with constant marginal cost and linear demand and (b) a firm with constant marginal cost and log-linear demand. You can use the formulas we just presented. You should treat the tax like a per-unit cost borne by the firm. If the firm s marginal cost is MC and the tax is Τ, then the firm s marginal cost after the tax is MC + Τ. For the case of log-linear demand, use a particular value of B>1such as 2 or 3. Solution: Let P Π be the price before the tax and let P Τ be the price after the tax. Linear demand: Q = A BP. Let P be the choke price. Then P Π = (MC + P )/2 and the price after the tax is P Τ = MC + Τ + P 2 = MC + P 2 + Τ 2 = P Π + Τ 2. Therefore, the price goes up by exactly half the tax. The firm does not choose to pass the entire tax on to the consumer. Log-linear demand: Q = AP B.Then P Π = B B 1 MC and P Τ = B B 1 (MC + Τ) = P Π + B B 1 Τ Since B>1, B/(B 1) > 1. That is, the price goes up by more than the tax. For example, if B = 3,thenB/(B 1) = 3/2. The price goes up by (3/2)Τ. This tells us that the price may go up by either more or less than the tax not very conclusive, but at least we have learned that it is wrong to think that the firm will simply raise its price by the tax. Instead, the way the tax is passed on depends on the details of the demand. Exercise 7.3. Suppose your firm produces a water purification system that you sell to small businesses. The demand for this indivisible good is shown in the first two columns of Table E7.1 (money values are in 1000s). Revenue is calculated for you in the third column. The fourth column shows your cost, and the profit is calculated in the fifth column. Figure E7.1 is a graph of the demand curve.

2 Firms, Prices & Markets Solutions for Chapter 7 (Pricing with Market Power) 2 Table E7.1 Output Price Revenue Cost Profit Figure E7.1 P ( 1000) Consumer surplus Producer surplus Cost of production Demand Deadweight loss Q a. What would your price and quantity be if you maximized total surplus? (Choose the quantity that roughly equates marginal valuation to marginal cost, remembering that price at a given quantity is the marginal valuation.) How much is the total surplus? Solution: You set the price to your constant marginal cost of 25 and supply the demand at this price. Hence, P = 25 and Q = 11. The total consumer valuation is the sum of the prices for each of the first 11 points on the demand curve, starting at the highest price and lowest quantity: = 462. The total cost is = 275. Hence, total surplus is = 187.

3 Firms, Prices & Markets Solutions for Chapter 7 (Pricing with Market Power) 3 b. What is the profit-maximizing price? What is your profit? How much is the consumer surplus? How much is the deadweight loss? Illustrate the profit, consumer surplus, and deadweight loss in Figure E7.1. (Your diagram should look similar to Figure 7.5.) Solution: The maximum profit is 102, achieved by P = 42. The total valuation of the 6 units sold is = 297. The total amount paid by customers for the six units is 6 42 = 252. Hence, consumer surplus is 45. Totalsurplusis = 147. Deadweight loss is = 40. Exercise 7.4. You manage a firm and must make the following decision. There is a good that could be developed with an R&D investment of 250, after which a patent would be obtained and the good could be produced at a constant marginal cost of 10. The potential market for this good has the following demand curve: d(p ) = 25 (1/2)P. The purpose of this exercise is to formulate a business plan before making the investment. a. If you were to develop the product, what price would you charge and how much would you sell? Solution: We have (a) linear demand with a choke price of P = 50 and (b) constant MC = 10. Thus, using the midpoint pricing rule, I would charge P = ( )/2 = 30. I can then calculate Q by plugging P into the demand curve: d(30) = 25 (1/2)30 = 10. b. Calculate your variable profit that is, your profit ignoring the up-front R&D cost. Solution: My profit ignoring the fixed cost is (P MC)Q, or VΠ= (30 10)10 = 200. c. Should you make the investment? Solution: I check whether variable profit (200) exceeds the fixed cost (250). It does not. I should not make the investment. d. Suppose instead the R&D cost is only 100. How would your answers change? Solution: My price, quantity, and variable profit if I develop the product do not change. However, I now find that the variable profit exceeds the R&D cost. Hence, I choose to develop the product.

4 Firms, Prices & Markets Solutions for Chapter 7 (Pricing with Market Power) 4 e. Suppose you initially estimate the R&D cost to be 100 and that you go ahead and develop the product. However, the R&D cost ends up being 250. How does this affect your pricing when you launch the product or your decision of whether to actually launch the product? Solution: This cost is now sunk and it has no effect on my pricing or decision of whether to launch the product. I charge 30 as planned. Taking into account the sunk cost, I end up losing 50, but abandoning the project now would be worse I would lose the entire 250 R&D expense. Exercise 7.5. For the demand curve in Exercise 7.4, the total valuation curve for the consumers is v(q) = 50Q Q 2. a. If the R&D expense is 250, what is your profit? What is the consumer surplus? What are the total gains from trade? Solution: Since I do not develop the product, I have zero revenue, zero cost, and zero profit. Similarly, the consumers get no surplus from the good. b. If the R&D expense is 100, what is your profit? What is the consumer surplus? What are the total gains from trade? Solution: My profit is the variable profit (200) minus the R&D expense (100), which equals 100. I produce 10 units and so the consumers total valuation is v(10) = (50 10) 10 2 = = 400. Their total expenditure is = 300. Hence, their surplus is 100. Total surplus is therefore 200. Exercise 7.6. You are the CEO of Benevolent Dictators Ltd. There is a good that could be developed with an R&D investment of 250, after which the good could be produced at a constant marginal cost of 10. The potential market for this good has the following demand curve: d(p ) = 25 (1/2)P. This implies the following total valuation curve for the consumers: v(q) = 50Q Q 2. The purpose of this exercise is to formulate a plan before making the investment, given your objective to maximize total surplus (whether or not you can break even doing so). a. If you were to develop the product, how much would you produce? Solution: I choose Q to equate MV = MC. I can find the marginal valuation curve either by differentiating the total valuation curve or by calculating the inverse of the demand curve (solving Q = 25 (1/2)P for P as a function of Q). Either way, I obtain mv(q) = 50 2Q. Then I solve mv(q) = mc(q), or50 2Q = 10. This yields Q = 20.

5 Firms, Prices & Markets Solutions for Chapter 7 (Pricing with Market Power) 5 Shortcut: Given that the marginal cost is constant, I can also produce the amount that would be demanded if I charged the marginal cost. This yields d(10) = 25 (1/2)10 = 20. (Of course, same answer, different method.) b. Calculate the variable surplus that is, the surplus not taking into account the fixed cost. Solution: The variable surplus is the total valuation of the 20 units minus the variable cost of the 20 units. The total valuation is v(20) = (50 20) 20 2 = = 600. The variable cost is MC Q = = 200. Therefore, variable surplus is = 400. c. Should you make the investment? Solution: I check whether variable surplus (400) exceeds the fixed cost (250). It does hence, I make the investment. d. Suppose instead the R&D cost is only 100. How would your answers change? Solution: No changes. Since the fixed cost does not affect the marginal conditions, its amount does not affect how much I produce if I choose to develop the product. Since the fixed cost is lower, I still want to develop the product. Exercise 7.7. This problem refers back to Exercise 7.6. a. What are the total gains from trade if the R&D expense is 250? Solution: Total gains from trade equal total valuation minus total cost or, equivalently, the variable surplus minus the fixed cost. Either way, the answer is 150. b. What are the total gains from trade if the R&D expense is 100? Solution: = 300. Exercise 7.8. Combine the results from Exercises to fill in the following table for the demand curve and marginal cost in those exercises. FC Possible surplus Actual surplus Deadweight loss What is the difference between the sources of the deadweight loss in the two cases?

6 Firms, Prices & Markets Solutions for Chapter 7 (Pricing with Market Power) 6 Solution: FC Possible surplus Actual surplus Deadweight loss In the first case (FC = 250), the deadweight loss occurs because the patent system does not provide full incentives for innovation, and hence the investment does not take place even though it should. In the second case, the deadweight loss occurs because the patent monopoly allows the firm to keep the price above the marginal cost. Thus, the amount of production is less than socially optimal.