# Economics 3551 Mansfield, et. al., 7e Answers to assignments 9 and 10. Chapter 10

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1 Mansfield, et. al., 7e Chapter The Bergen Company and the Gutenberg Company are the only two firms that produce and sell a particular kind of machinery. The demand curve for their product is P = 580-3Q where P is the price (in dollars) of the product, and Q is the total amount demanded. The total cost function of the Bergen Company is TC B = 410Q B where TC B is its total cost (in dollars) and Q B is its output. The total cost function of the Gutenberg Company is TC G = 460Q G where TC G is its total cost (in dollars) and Q G is its output. a. If these two firms collude and they want to maximize their combined profit, how much will the Bergen Company produce? b. How much will the Gutenberg Company produce? c. Will the Gutenberg Company agree to such an arrangement? Why or why not? a. Bergen s marginal cost (410) is always less than Gutenberg s marginal cost (460). Therefore Bergen would produce all the combination s output. Setting Bergen s marginal cost equal to the marginal revenue derived from the demand function (MR B = 580 6Q B ) yields 410 = 580-6Q B so Q B = 170/6 and Q G = 0. Bergen s price will be \$ and profit = \$2, b. As discussed in part a, Gutenberg s marginal cost is always greater than Bergen s. If Gutenberg were to produce 1 unit and Bergen 1 unit less, it would reduce their combined profits by the difference in their marginal costs. If Gutenberg were to produce 1 unit without any reduction in Bergen s output, it would reduce their combined profits by the same amount. c. If direct payments for output restrictions between the firms were legal, Gutenberg would accept the zero output quota. But, if competition were to break out, Gutenberg would make zero profits and Bergen would also earn \$0 (since competition implies P = MC). Thus the most Bergen would pay for Gutenberg s cooperation is \$2,408.32, and the least Gutenberg would accept to not produce is \$.01. 1

2 2. The can industry is composed of two firms. Suppose that the demand curve for cans is P = Q where P is the price (in cents) of a can and Q is the quantity demanded of cans in millions per month. Suppose the total cost function of each firm is TC = q where TC is total cost (in tens of thousands of dollars) per month and q is the quantity produced (in millions) per month by the firm. (Note: there is some confusion about units. TC is in tens of thousands of dollars per month. q is in millions of units per month. Suppose q = 1,000,000 so 15q = 15,000,000 cents. That equals 150,000 dollars which translates to \$15 x 10,000. Thus total cost is = \$17 x 10,000 = \$170,000. The units work, but it s not immediately obvious.) a. What are the price and output if managers set price equal to marginal cost? b. What are the profit- maximizing price and output if the managers collude and act like a monopolist? c. Do the managers make a higher combined profit if they collude than if they set price equal to marginal cost? If so, how much higher is their combined profit? a. Since each firm has a constant marginal cost of \$0.15, the price must also be \$0.15 for price to equal marginal cost. Since marginal cost equals price equals average variable cost in this case, each firm loses an amount equal to their fixed costs, \$20,000. b. If they collude, they will produce where marginal revenue equals marginal cost. Marginal revenue is given by MR = 100-2Q. Setting marginal revenue equal to marginal cost, the joint profit maximizing combined output is Q = 42.5 and P = \$ Since the firms have constant marginal costs, only one firm should operate; therefore they would avoid the fixed costs of the other firm. Their combined profits would be π = \$57.50(42.5) - [2 + 15(42.5)] = \$1,804.25, or \$18,042,500. If they cannot avoid the fixed costs of one of the firms by shutting it down, their combined profits would be \$18,022,500. c. Since they lose \$40,000 if they compete and earn \$18,022,500 if they collude, they earn \$18,062,500 more if they collude than if they compete. 3. An oligopolistic industry selling a particular type of machine tool is composed of two firms. Managers at the two firms set the same price and share the total market equally. The demand curve confronting each firm (assuming that the other firm sets the same price) follows, as well as each firm s total cost function. P Qd Qs TC \$ \$45.00 \$ \$47.00 \$ \$50.00 \$ \$55.00 \$ \$65.00 a. Assuming that each manager is correct in believing that managers at the other firm will charge the same price as they do, what price should each charge? b. Under the assumptions in part (a), what daily output rate should managers at each firm set? P (1,000) Qd Qs TC (1,000) MC MR TR \$ \$45.00 \$50.00 \$ \$47.00 \$2.00 \$4.00 \$54.00 \$ \$50.00 \$3.00 \$2.00 \$56.00 \$ \$55.00 \$5.00 \$0.00 \$56.00 \$ \$65.00 \$ \$2.00 \$54.00 a. Each should charge a price of \$9,000 since that is the last price for which MR > MC.. b. Each firm should produce 6 units. 2

4 12. Steve Win has purchased land from the city of Atlantic City in the Marina section. There are stories of a new casino building boom in Atlantic City (MGeeM is also talking about entering, and Gump is opening his fourth casino). Some talk is circulating that Win will subdivide his new land purchase and perhaps three casinos will be built on the site. Suppose Win subdivides his land into two parcels. He builds on one site and sells the other to another gambling entrepreneur. Win estimates that the demand for gambling in the Marina area of Atlantic City (after accounting for the presence of two existing casinos in the Marina and adjusting for the rest of the casinos in Atlantic City) is P = 750-5Q where P is the price associated with gambling and Q is the quantity of gambling (think of P as the average amount that a typical patron will net the casino, an amount paid for the entertainment of gambling, and Q as the number of gamblers). Win, of course, does not sell the other parcel until his casino is built (or is significantly far along); thus he has a first- mover advantage. Win s total cost (TC W ) of producing gambling is TC W = Q W Q W 2 where Q W is the number of gamblers in Win s casino, and the total cost (TC R ) of producing gambling for Win s rival is TC R = Q R + 20Q R 2 where Q R is the number of gamblers in the rival s casino and Q W + Q R = Q Would Atlantic City have done better to sell the land as two separate parcels rather than as a single parcel to Win (given that Win was going to subdivide, Win and his rival could not collude, and Win did not have the ability to produce as a monopolist)? You may assume that Win and his rival could have been Cournot duopolists. If Atlantic City could do better, show why and by how much. Carry all calculations to the thousandths decimal point. The monopoly solution for Win makes MR = Q W = Q W. Solving, Q = at P =\$ The last two questions involve the Cournot and Stackelberg models. I apologize for assigning them. The answers are below and (if anyone cares) I will post the solution separately. However, if Win and the rival produce as simultaneous-moving duopolists, Q W = Q r Q r = Q W Solving, Q W = and Q r = for a total of Q = and P = \$ If Win can act as a first mover by selling the land parcel to the rival, Q W = 16 and Q r = 12.4 for a total of Q = and P = \$ The first-mover advantage is small in this case. 4

5 Chapter Two soap producers, the Fortnum Company and the Maison Company, can stress either newspapers or magazines in their forthcoming advertising campaigns. The payoff matrix is as follows: a. Is there a dominant strategy for each firm? If so, what is it? b. What will be the profit of each firm? c. Is this game an example of the prisoner s dilemma? a. The dominant strategy for Maison is newspapers. The dominant strategy for Fortnum is magazines. b. Maison s profit will be \$8 million. Fortnum s profit will be \$9 million. c. This is not a prisoner s dilemma game. The players do not end up with an outcome from which both would be better off if they cooperated. 5

6 4. Two rival bookstores are trying to locate in one of two locations. The locations are near each other. Each would like to avoid a bidding war because that will drive up both of their rents. Payoffs are given in the following table: Does either player have an incentive to bid higher for a location? If so, by how much? Neither store has a dominant strategy in this game. Therefore, they have no incentive to bid for a location. When one firm makes a choice, the other firm s best strategy is to choose the other location. BUT the payoffs indicate that Location 1 is superior to location 2. Either firm would be willing to bid for that location. In fact, Barnes & Noble (B&N) appears to be in a better position to exploit location 1. The total payoff for B&N gets Location 1 and Borders gets Location 2 is 100, compared to 80 for the other Nash equilibrium. B&N would be willing to pay Borders up to (60 25) = 35 to persuade Borders to select Location 2. 6

7 5. Two soft drink producers, York Cola and Reno Cola, secretly collude to fix prices. Each firm must decide whether to abide by the agreement or to cheat on it. The payoff matrix is as follows: a. What strategy will each firm choose, and what will be each firm s profit? b. Does it matter whether this agreement is for one period or for three periods? c. Is this game an example of the prisoner s dilemma? a. Reno and York each have cheat as their dominant strategy, so they will each earn \$28 million. b. Since abiding by the agreement would raise their profits to \$29 million each if this game were to be played out an infinite number of times, the dominant strategy would be for both to abide if they thought that a defection would be met with cheating by their opponents in all future rounds. c. Yes, this is a prisoner s dilemma since the firms are stuck in an outcome from which both could be made better off by cooperation. 7

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