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Final Practice Problem Answers. 1. Suppose a firm has a supply function given by P=Q2. What is the amount of producer surplus between P=0 and P=30? What do their variable costs equal? What does their profit equal if their fixed cost equal $9,000? PS = P3/3|30= $27,000/3 = $9,000. PS = TR – VC; TR = $27,000, so VC = $18,000 PS = Profit + FC, so Profit = $0 2. Suppose a single price firm is charging a price of $40, and is operating on a portion of the demand where the elasticity equals -2. What is the marginal revenue? What is the marginal cost? MR = P(1 + 1/e) = $20; If they are maximizing profit, MC must also equal $20, or (P-MC)/P = -1/e, (40 – MC)/40 = ½, MC=20. 3. Suppose a representative firm in a perfectly competitive, constant cost industry has a cost function TC = 4Q2 + 100Q + 100. a. What is the long-run equilibrium price for this industry? The long –run price will equal the minimum of the average total cost curve. For this firm: LAC = TC/Q = 4Q + 100 +100/Q. The minimum point on LAC is found either by graphing the LAC curve or by taking the first derivative and setting it to zero. dLAC/dQ = 4 - 100/Q2 = 0, which yields Q = 5. In the long run P = LAC = 140 b. If market demand is given by the function Q = 1000 – P, where P denotes price, how many firms will operate in this long-run equilibrium? If demand is Q=1000-P, then at P=140, we get Q=860. So in long run equilibrium, there will be 860/5 = 172 firms. c. Suppose the government grants a lump-sum subsidy to each firm that manufactures the product. If this lump-sum subsidy equals 36, what would be the new long-run equilibrium for the industry? Now LAC = (TC-36)/Q = 4Q + 100 + 64/Q. Again, the minimum point on LAC is found by taking the first derivative and setting it to zero. dLAC/dQ = 4 - 64/Q2 = 0, which yields Q = 4. In the long run P = LAC = 132. 4. Suppose the demand for frisbees (which is a competitive market) is given by QD = 100 – 2P and the supply is given by QS = 20 + 6P a. What will be the equilibrium price and quantities for frisbees? 1 b. Suppose the government levies a tax of $4 per frisbee. Now what will be the equilibrium quantity, the price consumers will pay, and the price firms will receive? How is the burden of the tax shared by buyers and sellers? c. How would your answers to parts (a) and (b) change if the supply were instead QS = 70 + P What do you conclude by comparing these two cases? a. QD = 100 – 2P QS = 20 + 6P At equilibrium, QD = QS. 100 – 2P = 20 + 6P P* = $10, Q* = 80 b. Demanders and suppliers are now faced with different prices PS = PD – 4. Each will make decisions on quantity based on the price that it is faced with: QD = 100 – 2PD QS = 20 + 6PS = 20 + 6(PD – 4). The new equilibrium: 100 – 2PD = 20 + 6PD – 24 8PD = 104, PD = $13 PS = $9, Q = 74 The burden of the tax is shared: demanders pay $3 more for each frisbee while suppliers receive $1 less on each sale. c. QS = 70 + P At equilibrium, QD = QS. 100 – 2P = 70 + P, P* = $10, Q* = 80 After tax: QD = 100 – 2PD QS = 70 + PS = 70 + PS – 4 100 – 2PD = 70 + PD – 4 3PD = 34, PD = 11.3, PS = 7.3, Q = 77.3 While burden is still shared, in this case suppliers pay relatively more of the tax. 5. Draw the graphs for a competitive market and label the areas of consumer and producer surplus. Then assume the government applies a constant tax on suppliers. Label the new areas of consumer, producer surplus, the area of government revenue and deadweight loss area. Do the same thing for a market that allows international trade and then applies a tariff on imports. This is too time consuming for me to graph out in word, so I’ll just suggest that you look at the graphs in the book that do this (Ch. 12 in Nicholson). 6. The cardboard box industry is composed of 100 identical firms, each having short-run total costs given by: STC = 0.5q2 + 10q +5 2 Where q is the is the output of cardboard boxes per day. The market supply is given by: 100q = 100P -1000. And the demand for cardboard box production is given by: Q = 1,100 – 50P a. What will be the equilibrium in this marketplace? What will each firm’s short-run profits, and what is profit for the industry as a whole? b. Compute the total producer surplus in this market. c. Suppose the government poses a $3 tax on snuffboxes. How would this tax change the market equilibrium? d. How would the burden of this tax be shared between buyers and sellers? e. Calculate the total loss of producer surplus as a result of the taxation of cardboard boxes? What is the total loss in profits as a result of the taxation? a. Equilibrium where 100P -1000 = 1100 -50P, P = 14, Q = 400. For a single firm = 14(4) [.5(4 ) + 40 + 5] = 56 53 = 3 Total industry profits = 300 2 b. Since QS = 0 when P = 10, Producer Surplus = .5(14 -10)(400) = 800. c. With tax PD = PS + 3 1100 -50PD = 100PS -1000 = 100(PD - 3) 1000 150PD = 2400 PD = 16 PS = 13 Q = 300 d. e Total tax = 900 Consumers pay 300(16 -14) = 600 Producers pay 300(14 -13) = 300 PS = .5(300)(13 -10) = 450 Short-run profits = 13(300) -100C C = .5(3)2 + 30 + 5 = 39.5 π = 3900 -3950 = -50. Since total profits were 300 earlier, this is a reduction of 350 in profits and in producer surplus. 7 A monopolist has a demand curve given by P = 100 - Q and a total cost curve given by TC = 16 + Q2. Find the monopolist’s profit-maximizing quantity and price. What is the deadweight loss area? How much economic profit will the monopolist earn? The profit maximizing level of output for a single-price monopolist occurs where MR = MC. The linear demand curve P = 100 – Q has a marginal revenue of MR = 100 – 2Q. By equating marginal revenue and marginal cost (100 – 2Q = 2Q) we get a quantity of 25. The price charged for this quantity is read off the demand curve so P = 100 – Q = 100 – 25 = 75. OR 3 MC = 2Q = MR = 100 - 2Q; 100 - 2Q = 2Q, so Q* = 25, P* = 75. TR = 1875; TC = 641 = TR - TC = 1234 The deadweight loss area is given by the area between supply and demand, the monopoly quantity (Q=25) and the competitive quantity. If it’s competitive the quantity is 33.3, so the DWL is given by ½ (33.3-25)*($75 - $50) = $104.13. 7. Now suppose the monopolist in question 6. has a total cost curve given by TC = 32 + Q2 (Fixed costs have doubled). Find the monopolist’s profit-maximizing quantity and price. How much economic profit will the monopolist earn? The monopolist’s price and quantity are unaffected by fixed costs. However, the monopolist earns lower profits: = TR – TC = 75Q – (32 + Q2) = 1875 – 657 = 1218. The difference in profits equals the increase in fixed costs. Because the quantity and price are the same the deadweight loss area would not change. 9. Now suppose the monopolist in question 7. has a long-run marginal cost curve given by MC = 20. Calculate the total consumer and producer surplus under conditions if it is monopoly and if it is a competitive industry. What is the deadweight loss area associated with this monopolist? The profit-maximizing level of output for a single-price monopolist occurs where MR = LMC, The linear demand curve P = 100 – Q has associated marginal revenue of MR = 100 – 2Q. Setting marginal revenue equal to marginal cost, LMC = 20, we have 100 – 2Q = 20, which solves for Q = 40. The price charged for this quantity is read off the demand curve: P = 100 – Q = 100 – 40 = 60. A perfectly competitive industry would produce the quantity such that P = LMC = 20 so that Q = 100 – 20 = 80. The total surplus generated by perfect competition would be TS = (1/2)80(100 – 20) = 3200. By comparison, the monopoly generates producer surplus PSm = 40(60 – 20) = 1600 and consumer surplus CSm = (1/2)40(100 – 60) = 800 for a total surplus of TSm = 160 = 80 = 2400. The efficiency loss due to monopoly is the amount by which total surplus under monopoly falls short of the total surplus under perfect competition. (TS = 2400 – 3200 = 800) 10. Humana hospitals in Kentucky was reportedly charging patients $44.90 for a container of saline solution that cost them $0.81 (This is actually true). Given this information, calculate the demand elasticity that it faces for saline solution. P/MC=55.4; Elasticity (P-MC)/P = 1/e; elasticity=1.02. 11. Suppose a perfectly price discriminating monopolist (1st degree) faces a market demand P = 100 – 10Q and has a constant marginal cost MC = 20 (with no fixed costs). How much does the monopolist sell? How much profit does the monopolist earn? How much consumer surplus exists in this market? What is the amount of deadweight loss associated with this monopolist? A perfectly discriminating monopolist sells the quantity where marginal cost intersects the demand curve P = MC or 100 – 10Q = 20 or 10Q = 80 or Q* = 8. The monopolist’s economic profit is the area under the demand curve down to average cost out to quantity. PS = (1/2)(100 – 4 20)8 = 320. There is no consumer surplus or inefficiency area with a perfectly price discriminating monopolist. 12. Suppose a price discriminating monopolist faces a market demand P = 100 – 10Q and has a constant marginal cost MC = 20 (with no fixed costs). Instead of a 1st degree they engage in 2nd degree price discrimination, where they charge a flat fee to consumers and then a constant. What is the profit maximizing fee and price to charge in this market? What is the amount of deadweight loss associated with this monopolist? The firm should charge the max fee based on the amount of consumer surplus in the market associated with the competitive price, which is given by CS = (1/2)(100 – 20)8 = 320. There is no consumer surplus or inefficiency area associated with this type of price discrimination either. 13. Pennsylvania is the monopoly retailer of wine in that state. Suppose that Quaker Cabernet has no close substitutes and that the statewide demand function is given by: P=5-0.001Q. The state purchases the wine for $2 per bottle, and there are no other costs associated with selling this wine. a. What is the state’s profit maximizing price and quantity? b. Neighboring New Jersey permits private retailers to sell wine, and therefore the market for this wine is perfectly competitive. No interstate wine trade is permitted. If New Jersey faces exactly the same demand function as Pennsylvania, what is the equilibrium price and quantity in New Jersey? c. Suppose New Jersey is considering applying a per unit tax to wine. What tax per bottle sold will maximize tax revenue? What is the new equilibrium price and quantity sold after this tax is applied? d. Compare the deadweight loss and tax revenue that results from New Jersey’s tax to the deadweight loss and profit that results from Pennsylvania’s monopoly. a. b. c. d. Q=1500; P=3.50 P=$2.00; Q=3000 T=$1.50; P=3.50; Q=1500 Tax revenue=1.50*1.5=$2,250=Profit of Pennsylvania 14. Suppose a firm faces a demand in one market given by P = 200 – 2Q and in another market demand is given by P = 100 – Q. They face a constant marginal cost equal $10. What is the profit maximizing price and quantity if they must charge the same price in both markets? Qmarket = (100 - .5P) + (100 – P) = 200 – 1.5P Pmarket = 133.3 - .67Q MRmarket = 133.33 – 1.34Q Set MR = MC = 133.33 – 1.34Q = 10; Q* = 92.5; P* = 71.67 15. The market demand curve for a pair of Cournot duopolists is given as: P = 36 - 3Q, (Q = Ql + Q2). The constant per unit marginal cost is 18 for each duopolist. Find the Cournot equilibrium price, quantity, and profits. 5 P=36-3Q, Q = Q1 + Q2 , MC = 18 MRi = 36 - 3Qj - 6Qi Set MC = MR and solve for Q1 18 = 36 - 3Q2 - 6Q1 Q1 183Q2 6 3 12 Q2 Q2 183Q1 6 3 12 Q1 Q1 = Q2 = 2; Q = 4 P = 36 - 3(4)= 36 - 12 = 24 Profit = PQ - MC(Q) = 24(2) - 18(2) = 48 - 36 = 12 for each firm 16. Solve Problem 15 as a Bertrand model. Find the long-run equilibrium price, quantities, and profits. The Bertrand model sets price equal to marginal cost in the long run: P = MC = 18 18 = 36 - 3Q Q = 18/3 = 6 Q1 = Q2 = 6/2 = 3 Profit = PQ - MC(Q) = 18(6) - 18(6) = O Profit (firm 1) = profit (firm 2) = O 17. Solve Problem 15 as a Stackelberg Leader-Follower model. Assume firm 1 is the leader. Stackelberg model: Follower (firm 2): MR = 36 - 3Ql - 6Q2 = 18 Q2 = (6-Q1)/2 Leader (firm 1): P1 = 36 - 3Q2 - 3Q1 = 36 - 3(6-Q1/2) -3Q1 P1 = 36 - 18/2 + 3Q1/2 - 3Q1 = 27 - 3/2Q1 MR1 = 27- 3Q1 Set MR1 = MC 27 - 3Q1 = 18 3Q1 = 9 Q1 =3 Q2 = (6 - Q1)/2 = (6 - 3)/2 = 3/2 Q= Q1 + Q2 = 3 + 3/2 = 9/2 P = 36 - 3(9/2) = 36 - 13.5 = 225 Profit (firm 1) = P Q1 - MC(Q1) = 22.5(31) - 18(3) = 67.5-54 = 13.5 Profit (firm 2) = PQ2 - MC(Q2) = 22.5(1.5) - 18(1.5) = 6.75 18. Solve Problem 15 as a shared monopoly (cartel). Find the profit-maximizing price, quantities, and profits. Shared monopoly: P = 36 - 3Q MR = 36 - 6Q Set MC = MR: 18 = 36 6Q 6 Q = 18/6 = 3 P = 36 3(3) = 27 Q1 = Q2 = 1.5 Profit = PQ MC(Q) = 27(3) 18(3) = 81 54 = 27 Profit (firm 1) = profit (firm 2) = 27/2 = 13.5 19. Firm’s A and B live on adjacent plots of land. Each has two potential uses for her land, the present values of each of which depend on the use adopted by the other, as summarized in the table. All the values in the table are known to both parties. Firm A Apple Growing Pig Farming Firm A’s Profit = $6 million Firm A’s Profit = $7 million Firm B’s Profit = $7 million Firm A’s Profit = $4 million Firm B’s Profit = $4 million Firm A’s Profit = $5 million Firm B’s Profit = $3 million Firm B’s Profit = $5 million Rental Housing Firm B Bee keeping a) What, if any, are the dominant strategies of each firm? A – Pig Farming B – None b) If the two firms choose simultaneously, what activities will they pursue and what will the expected profits equal? A – Pig Farming; $5 million B – Bee keeping; $5 million Now assume this is a sequential game, where one firm chooses before another. Note: it may help to draw out this game using the sequential decision tree to answer these questions. c) If firm A chooses before firm B, what is the expected outcome? A – Apple Growing B – Rental Housing 7