Equilibrium Strategies

of labor increases, this changes the slope of the isocost line (so that labor

“trades” for more units of capital). The new tangency with the same isoquant must occur at a mix using less labor and more capital. 13.a.The grade improvements offered by extra hours of studying finance are 8, 5, 5, 2, and 2 points. For economics, the improvements are 6, 4, 2, 2, and 1 points. b.The “first” hour should be devoted to finance (an 8-point increase), the next hour to economics (6 points), the next 2 hours to finance (5 points each hour), and the “last” hour to economics (4 points). The student’s predicted grades are 88 and 85. c.This allocation is optimal. Devoting her first 5 hours to finance and economics offers the greatest point opportunities. Then, devoting 2 additional hours to accounting will produce more extra points (3 points each hour) than devoting an additional hour to finance (2 points) or economics (2 points). 15.a.For N1 16 and N2 24, the average catch at the first lake is Q1/N1 [(10)(16) .1(16)2]/16 8.4 fish, and the average catch at the second lake is Q2/N2 [(16)(24) .4(24)2]/24 6.4 fish, respectively. Lured by the greater average catch, some number of fishers will leave the second lake for the first. b.Movement between lakes will cease when all individuals obtain the same average catch. Equating the average catches at the lakes implies 10 .1N1 16 .4N2. In addition, N1 N2 40. Solving these two equations simultaneously implies N1 20 and N2 20. The total catch at the two lakes is 320 fish. c.The commissioner seeks to maximize Q1 Q2 subject to N1 N2 40. The optimum solution to this constrained maximization problem implies that the marginal catch of the last fisher should be equal across the lakes. Here, MQ1 dQ1/dN1 10 .2N1 and MQ2 dQ2/dN2 16 .8N2. Setting MQ1 MQ2 and using N1 N2 40, we find that N1 26 and N2 14. The marginal catch at each lake is 4.8 fish; the maximum total catch is: [(10)(26) (.1)(26)2] [(16)(14) (.4)(14)2] 338 fish.

Chapter 6

1.The fact that the product development was lengthier and more expensive than initially anticipated is no reason to charge a higher price.

These development costs have been sunk and are irrelevant for the pricing decision. Price should be based on the product’s relevant costs (the marginal cost of producing the item) in conjunction with product demand (as summarized by the product’s price elasticity).

3.a.The profit associated with an electronic control device (ECD) is E 1,500 [500 (2)(300)] $400. If the firm sells the two microchips separately (instead of putting them into an ECD), its total profit is M (550 300)(2) $500. Thus, the firm should devote all of its capacity to the production of microchips for direct sale. Producing

ECDs is not profitable. b.If there is unused microchip capacity, the firm earns $400 in additional profit for each ECD sold. Producing ECDs now becomes profitable. c.If $200 (of the $500 average cost) is fixed, each ECD’s contribution becomes E 1,500 [300 (2)(300)] $600. The firm should produce ECDs in the short run; this is more profitable than selling chips directly. 5.a.Setting MR MC implies 10,000 400Q $4,000. Thus, Q* 15 games. b.The contribution is R VC ($150,000 45,000) ($4,000)(15) $45,000. The opportunity cost of the entrepreneur’s labor is $20,000, and the required annual return on the $100,000 investment is 20 percent or $20,000. Thus, her economic profit is $45,000 20,000 20,000 $5,000. 7.a.To maximize profit set MR MC. Therefore, 10 .5w 5, or w 10 weeks. Profit from the film is: [(10)(10) .25(10)2] (5)(10) 75 50 $25 thousand. b.The “total” marginal cost (including the opportunity cost of lost profit) of showing the hit an extra week is 5 1.5 $6.5 thousand.

Setting MR MC 6.5 implies: w 7 weeks. c.On the cost side, there are economies of scale and scope. (With shared fixed costs, 10 screens under one roof are much cheaper than 10 separate theaters.) Demand economies due to increased variety probably also exist. Filmgoers will visit your screens knowing that there’s likely to be a movie to their liking. d.Obviously, video rentals and sales compete with (and potentially cannibalize) theater revenues. The delay makes sense as long as the extra theater profits from extending the run exceed the video profits given up. 9.a.Given the cost function C 360 40Q 10Q2, it follows that AC 360/Q 40 10Q. Clearly, average cost is U-shaped. b.To find the point of minimum average cost, set AC MC: 360/Q 40 10Q 40 20Q. Thus, 360/Q 10Q or Q2 36. Therefore,

Qmin 6 units and ACmin 360/6 40 (10)(6) $160 per unit. c.Because ACmin exceeds the market price (P $140), the firm incurs losses if it operates. In the long run, it will shut down.

11.a.We are given that MC $20,000, and from the price equation, we derive MR 30,000 .2Q. Setting MR MC implies Q 50,000, confirming that GM’s current output level is profit maximizing. b.The outside sales option means that GM faces an opportunity cost. Every engine sold to the SUV manufacturer generates additional contribution of $2,000. GM should not only employ the unused capacity to produce engines for external sale, it should also cut back somewhat its production of light trucks. The effective MC per truck is now $20,000 $2,000 (where the latter is the opportunity cost per engine.) The shift upward in MC implies a lower optimal output level (40,000 engines to be exact). c.Fixed costs should not be mixed with variable costs in determining output and price decisions. Removing the allocated fixed cost means taking out 160,000,000/40,000 $4,000 per unit. Thus, the true marginal cost per unit is $22,000 $4,000 $18,000. Note that the actual MC in the West Coast factory is lower than the MC in the Michigan plants. Thus, GM should expand its West Coast output (to 60,000 units to be exact). 13.a.C 500 5Q2. Minimum average cost occurs at the quantity Q such that MC AC. We know that MC 10Q and AC 500/Q 5Q. Setting these equal implies 10Q 500/Q 5Q. Collecting terms, we find that 5Q2 500 or Qmin 10. At this output, minimum average cost equals $100. b.Setting MR MC implies 600 10Q 10Q. Therefore, Q 30; in turn, P 600 (5)(30) $450, and 13,500 5,000 $8,500. c.If either MC differed from MR, the firm could increase its profit by redirecting output. Setting MR MC1 MC2 implies 600 10Q* 10(Q*/2). Therefore, Q* 40. Each plant produces 20 units at a cost of $2,500 (from the original cost function). Finally, we find P* $400, and 16,000 5,000 $11,000. d.If the firm can use as many plants as it likes, it enjoys constant returns to scale. It should set the number of plants so that each is producing 10 units (where MC min AC $100). In short, $100 is the relevant long-run marginal cost. Setting MR MC implies 600 10Q 100. Therefore, Q 50. In turn, P $350 and (350 100)(50) $12,500. The number of plants is 50/10 5.

Chapter 7

1.a.According to the “law” of supply and demand, the existence of a large body of Picasso’s artwork will tend to lower the value of any individual piece of work.

b.If demand for Picasso’s work is inelastic, increasing the number of pieces sold (by driving down prices) will reduce total revenue. The artist’s heirs should try to limit supply by spreading sales of his artwork over long time periods. 3.a.Setting QD QS implies 184 20P 124 4P or 24P 60.

Therefore, P $2.50 and Q 134 pounds per capita. b.This increase represents only .7 percent of total supply and will have little price effect. The new quantity supplied is (1.007)(134) 135.

Rearranging the demand curve, we have P 9.20 .05Q. Therefore, we find that P 9.20 (.05)(135) $2.45. Montana farmers’ revenue should increase by about 8 percent (based on a 10 percent quantity increase and a 2 percent price drop). c.If the total harvest is 10 percent above normal, QS (1.10)(134) 147.4 pounds per capita and P 9.20 (0.5)(147.4) $1.83. Farm revenue drops from (2.50)(134) $335 to (1.83)(147.4) $269.74, a 19.5 percent drop. Demand is inelastic. A modest quantity increase caused a large price drop and this is detrimental to farmers’ incomes. Because varying harvest conditions can cause significant price and revenue changes, today’s farm profits quickly can become tomorrow’s losses. 5.a.The Green Company’s marginal cost is MC dC/dQ 4 2Q, and the price is P $40. Setting MC P implies 4 2Q 40, or Q 18 units. More generally, setting MC P generates the supply curve 4 2Q P, or Q (P 4)/2. b.With the increase in fixed cost, the firm should continue to produce 18 units. Its profit is R C (40)(18) [144 (4)(18) (18)2] 720 540 $180. Of course, the firm will supply no output if price falls below the level of minimum average cost. We set MC

AC and find that average cost is a minimum at Qmin 12. In turn, min AC $28. Thus, the firm’s supply is zero if price falls below $28. c.In part (a) (when fixed costs are 100), min AC $24 at a quantity of 10 units for each firm. Thus, the original long-run equilibrium price is P $24. With elevated fixed costs, one would expect the long-run price to rise to $28 (the new minimum level of AC). At this higher price, total demand is reduced. However, each firm’s output would rise from 10 units to 12 units. With reduced total demand and greater output per firm, the number of firms must decline. 7.a.Average cost is AC 300/Q Q/3. Thus, total cost is C 300

Q2/3, which implies MC (2/3)Q. Setting AC MC implies 300/Q

Q/3 (2/3)Q, or 300/Q Q/3. This simplifies to Q2 900, so

Qmin 30. In turn, min AC (2/3)(30) $20. b.A firm’s supply curve is found by setting P MC (2/3)QF.

Therefore, QF 1.5P. With 10 firms, total supply is QS 10QF 15P.

Setting QD QS implies 1,000 20P 15P. Thus, we find P $28.57 and Q 428.57. At QF 42.86, each firm’s AC is $21.3. Thus, its profit is: (28.57 21.3)(42.86) $311.6. c.In long-run equilibrium, P min AC $20. In turn, Q 1,000 (20)(20) 600. The number of firms is: 600/30 20. 9.a.Here, MC AC $5. Thus, PC $5. From the price equation, 5 35 5Q, implying QC 6 million chips. b.The industry displays constant returns to scale (constant LAC). The real microchip industry probably displays increasing returns to scale (declining LAC). For competition to be viable, returns to scale must be exhausted at volumes well below total market demand. c.Total profit is zero. Consumer surplus is (.5)(35 5)(6) $90 million. 11.a.Equating 70 Q and 40 2Q, we find Q 10 and P $60. b.Now we use 70 Q 25 2Q to find Q 15 and P $55. The subsidy has increased output and (consequently) reduced price. c.While the subsidy helps producers and consumers, it is not “free.” Taxpayers must finance the cost of the subsidy. Economists note that subsidies can lead to inefficient outcomes, encouraging output past the point at which MB MC.

Chapter 8

1.a.The merger should mean the end of the prevailing cutthroat competition. The merged firm should set out to achieve the available monopoly profit. b.Formerly, cutting rates made sense in order to claim additional clients from one’s rival. After the merger, the newspapers will raise rates (again seeking the monopoly level). 3.Packing the product space with a proliferation of differentiated items is a classic example of strategic entry deterrence. The slower selling brands are not profitable in themselves. However, they raise the firms’ overall profits by leaving no product niche for a new rival to profitably enter the market. 5.a.We know that P 11 Q and C 16 Q. Setting MR MC, we have 11 2Q 1. Thus, the monopolist sets QM 5 million and PM $6. b.The regulator sets P AC. Thus, 11 Q 16/Q 1. After multiplying both sides by Q, this becomes a quadratic equation with two roots: Q 2 and Q 8. Naturally, the regulator selects the larger output level, so we have QR 8 million and PR $3. c.Under marginal cost pricing, P* MC $1 and Q 11 P 10 million. At this quantity, AC is 26/10 $2.60. The shortfall of price below average cost is 2.60 1 $1.60 per unit.

7.a.OPEC maximizes its profit by setting MR MC. We have 115 4Q 15. Therefore, Q* 25 million barrels per day. In turn, P* $65 per barrel. b.If it sets P $50, then Q 57.5 (.5)(50) 32.5 million barrels per day. Profit (per day) is: (50 15)(32.5) $1.1375 billion. If it sets P $65, its initial profit is: 1 (65 15)(25) $1.25 billion per day. In the second 5-year period, its optimal quantity and price are: Q2 18 million barrels per day and P2 $60. (Check this by using the long-run demand curve and setting MR MC.) Thus, its profit is: 2 (60 15)(18) $.81 billion per day. OPEC’s average profit over the decade (ignoring discounting) is $1.03 billion per day—lower than $1.1375 billion from holding its price to $50 per barrel. 9.a.At P $10, 2 million trips are demanded. In the text, we saw that each fully utilized taxi had an average cost per trip of $8 and, therefore, earned an excess profit of (10 8)(140) $280 per week. The commission should set the license fee at L $280 to tax away all this excess profit. Assuming that 14,286 taxis operate (just enough to meet the 2 million trips demanded), the commission collects a total of $4 million in license fees. b.The rearranged demand curve is P 14 2Q. We saw that the extra cost of adding a fully occupied taxi is $1,120 per week, or $8 per trip. The relevant MC per trip is $8. Setting MR MC, we have 14 4Q 8. Thus, QM 1.5 million trips and PM $11. The maximum total profit for the industry is (11 8)(1.5) $4.5 million. The number of taxis 1,500,000/140 10,714. c.If the market could be transformed into a perfectly competitive one, the result would be PC min AC $8, QC 7 (.5)(8) 3 million trips, and the number of taxis is 21,428. d.Taxi trips are not perfect substitutes. If a taxi charges a fare slightly higher than the industry norm, it will not lose all its sales. (Customers in need of a taxi will take the one in hand, rather than wait for a slightly cheaper fare.) Since there is room for product differentiation and price differences, the taxi market probably is best described as monopolistic competition. In this setting, all cabs make zero profit (due to free entry). If price settles at P $9, then AC $9 for each cab. This AC occurs at about 121 trips per week; each taxi is 86 percent utilized. Trip demand is 2.5 million supplied by 2,500,000/121 20,661 taxis. *11.a. Each supplier maximizes profit by setting P MC. Since MC 4 2Q, this implies QF (P 4)/2. With 10 firms, QS 5P 20. b.The buyer’s profit is (10 P)QS (10 P)(5P 20). To maximize profit, set d /dP 0. The result is 70 10P 0, implying

P $7 and QS 15. The firm offers a price that is less than its value ($10), but high enough to induce an optimal supply. *13.a.We know that P 660 16Q1 and C 900 60Q1 9Q1 2. Setting MR MC, we have 660 32Q1 60 18Q1 or Q1 12. In turn, we find P1 $468. The firm’s profit is:

b.If 10 firms each produce 6 units, total output is 60 and the market price is indeed P 1,224 (16)(60) $264. Setting firm 1’s MR

MC implies 1,224 (16)(54) 32Q1 60 18Q1, implying Q1 6 units as claimed. Finally, the firm’s average cost is

The typical firm earns a zero economic profit since P AC. c.Under perfect competition, Pc ACMIN. Setting AC MC, we have 900/QF 60 9QF 60 18QF, implying QF 10 and ACmin 240. Thus, Pc $240 and Qc 76.5 (240)/16 61.5. The number of firms is found by dividing total output by each firm’s output: 61.5/10 6.15 firms.

R C (468)(12) [900 (60)(12) 9(12)2] 5,616 2,916 $2,700

C/Q [900 (60)(6) 9(6)2]/6 $264

Chapter 9

1.The conventional wisdom points to entry in loose oligopolies for two reasons: (i) the market offers positive economic profits (unlike a perfectly competitive market), and (ii) since the market is not dominated by large firms, a new entrant has the potential to reap significant market-share gain over time (unlike a tight oligopoly). 3.a.OPEC’s net demand curve is: QN QW QS (103.33 P/6) (.5P 10) 93.33 (2/3)P. Rearranging this, we have: P 140 1.5QN. b.Setting MR MC, we have 140 3QN 20, or QN 40 million barrels per day. In turn, P $80 and QS (.5)(80) 10 50 million barrels per day. OPEC accounts for about 44 percent (40/90) of world oil production. 5.a.For firm 1, MR1 MC implies 120 5Q2 10Q1 60, or Q1 6 .5Q2.In equilibrium, Q1 Q2 so we can solve the above equation to find Q1 Q2 4 units. b.If the firms collude, they set MR 120 10Q 60, or Q 6 units.

With total output split equally, each firm supplies 3 units. 7.a.Yes, there is a prisoner’s dilemma in the sense that when all farmers have large crops, they all make losses. One solution is for farmers to

agree to withhold excess supplies from the market in order to maintain higher prices. b.If each member’s compensation is based on the team’s overall performance, there is the incentive to take a “free ride” on the efforts of other members. (If it is a 10-member team, one member contributes only 10 percent to the overall performance.) Countering the prisoner’s dilemma may mean monitoring work effort or increasing the rewards for individual performance. 9.a.For firm 1, P1 75 .5P2 Q1. Setting MR1 MC, we have 75 .5P2 2Q1 30, implying Q1 22.5 .25P2. Substituting this solution forQ1 into the price equation, we find: P1 52.5 .25P2. b.A lower P2 shifts firm 1’s demand curve inward, causing firm 1 to set a lower price. c.Solving P1 52.5 .25P1, we find P1 P2 $70. From the demand equations, Q1 Q2 40. Each firm’s profit is $1,600. 11.a.The unique equilibrium has firm B setting a price slightly below $7.50 (the next lowest cost) and serving the entire market. b.No, firm B would continue to bid $7.50 to maximize its contribution toward its fixed cost. However, if B’s fixed costs are so large so as to imply losses, the firm would exit the market in the long run. 13.a.Rearranging the price equation shows that raising A increases sales. Advertising spending is a fixed cost (doesn’t vary with output). b.Setting MR MC, we have 50 A.5 2Q 20 or Q 15 .5A.5 . Substituting this solution for Q into the price equation, we find: P 35 .5A.5. If advertising is increased, the firm should plan for increased sales at a higher price. c. (P 20)Q A (15 .5A.5)(15 .5A.5) A 225 15A.5 .75A. Setting d /dA 0 implies: 7.5/A.5 .75 0. Thus, A 100.In turn, Q 20 units and P $40.

Chapter 10

1.In a Nash equilibrium, each player’s chosen strategy is optimal, given the strategy of the other. Thus, neither side can profit by unilaterally deviating. By comparison, a dominant strategy is optimal against any strategy the other player might choose. 3.a.Firm Y has no dominant strategy or any dominated strategy. For firm

Z, C3 is dominated by C1. b.Once C3 is eliminated from consideration, R1 is dominated by R2.

With R1 eliminated, C2 is dominated by C1. Thus, C1 is firm Z’s optimal choice, and R2 is firm Y’s optimal response.

5.a.There are two equilibria: firm J develops E and firm K develops D, and vice versa. Thus, one cannot make a confident prediction as to which outcome will occur. b.If firm J moves first, it should choose E, knowing firm K will then choose D. c.Similarly, firm K’s first move should be to choose E. 7.a.The unique equilibrium outcome has firm A choosing High and firm B choosing Medium. (Use the method of “circles and squares” to confirm this.) b.The firms should coordinate their R&D strategies by selecting Medium and Low, respectively. Here the firms achieve maximum total profit, and each firm’s profit is greater than it was in the noncooperative equilibrium of part (a). 9.a.Applying the method of “circles and squares” to the payoff table, we see that there are two Nash equilibria: (i) Both superpowers Escalate their weapons buildup, or (ii) Both Stop. Strictly speaking this is not a prisoner’s dilemma. (It is not the case that the play of dominant strategies leads to an inferior outcome for both sides.) b.Yes, with the fall of the former Soviet Union, it appears that the superpowers have switched (at least for the time being) to the Stop–Stop equilibrium. 11.a.There are no dominant or dominated strategies for either player. b.The equilibrium strategies are R1 and C3; the equilibrium outcome is 10. 13.a.The town’s dominant strategy is nonenforcement. Anticipating this, the typical motorist chooses to disobey the law. The outcome is (5, 10). b.If the town can make the “first move” by committing to 100 percent enforcement, the situation changes. The typical motorist’s best response is to obey, leading to the outcome (0, 15). Note, however, that enforcement (because of its high cost) is still not in the best interest of the town ( 15 is worse than 10). c.Now the town enforces the law with probability p. The typical motorist will obey the law if and only if his expected payoff from doing so (0) exceeds the payoff if he doesn’t, 20p 5(1 p). Setting these payoffs equal to one another implies p .2. As long as the enforcement probability is slightly greater than 20 percent, motorists will obey the law. The town’s enforcement cost is (.2)( 15) 3. Probabilistic enforcement, which successfully deters, is the town’s least costly strategy. 15.a.The buyer does not have a dominant strategy. She buys 2 units at P $9, 4 units at P $8, and 6 units at P $6. Anticipating this behavior, the seller should set P $8.