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Unformatted text preview: University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain ECO 204 Summer 2009 S. Ajaz Hussain Practice Problems 19 Solutions Please help improve the course by sending me an email about typos or suggestions for improvements Note: Please don't memorize these solutions in the expectation that similar questions will appear on tests and exams. Instead, try to understand how to derive the answer as you'll be tested on techniques and applications, not on memorization. Moreover, tests and exams will cover topics and techniques that may not be in these practice problems. You are urged to go over all lectures, class notes and HWs thoroughly. Question 1 Ajax has the demand curve P = 100 10Q and MC = 20. Calculate the price, quantity, Consumer surplus and gross profits for the following scenarios: (a) if Ajax is a price maker charging uniform prices (b) if Ajax is a price taker and (c) if Ajax is a first degree price discriminator. Answer: if Ajax is a price maker charging uniform prices then he will maximize profits by producing where MR = MC. Now MR = 100 20Q (using the shortcut for linear demand) and MC = 20. Thus MR = MC: 100 20Q = 20 Q = 4 P = 100 10Q = 100 10(4) = $60 The CS surplus is: CS = (Max WTP Price)Q: CS = (Max WTP Price)Q CS = (100 60) 4 = $80 The gross profit is (P MC)Q: PS = (P MC)Q 1 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain PS = (60 20)4 = $160 Next, of Ajax was a price taker i.e. perfectly competitive he will produce where P = MC. This is, of course, P = $20 and thus Q = 8. Now the CS is (100 20)8 = $320 and the gross profits are $0: If Ajax was a first degree price discriminator, then he will charge each customer their WTP. Thus, the first good's price will be: P = 100 10(1) = $90 2 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain The second good's price will be: P = 100 10(2) = $80 But how many units will Ajax sell? We've shown that under 1st degree price discrimination, the price is the MR. Thus, as long as MR > MC, Ajax will sell another unit priced by 1st degree price discrimination. Now, P = MR > MC until Q = 8. Thus, under 1st degree price discrimination, prices are P = 100 10Q for Q = [0,8], the CS = 0 (Ajax is charging price equal to WTP) and gross profits are = (100 20)8 = $320. In summary: Scheme Uniform Prices (noncompetitive) Perfect Competition 1st Degree Price Discrimination P $60 $20 P = 100 10Q Q 4 8 Q = [0,8] CS $80 $320 $0 Gross Profits $160 $0 $320 Observe that CS is greatest and gross profits lowest in perfect competition. In contrast, CS is lowest and gross profits greatest under 1st degree price discrimination. In fact, the CS in perfect competition equals PS in 1st degree price discrimination. If there is CS, it is tantamount to leaving "money on the table". Every dollar of CS extracted is a dollar of profits made. The ability to extract CS can only be settings where the firm has market power (i.e. the demand curve is downward sloping). By definition, price discrimination cannot occur in competitive markets. Question 2 (2007 2008 Final Exam Question) A private golf club "Drones" (an ode to P. G. Woodhouse 1 ) attracts two types of golfers: serious and casual. Suppose there are 10 serious and 100 casual golfers in Drones' market. Each serious golfer has demand curve, Qs = 350 10P and each casual golfer has demand curve Qc = 100 10P, where Q is number of rounds played per year, and P is price per round. Suppose there are no fixed costs and the marginal cost of playing golf is $5 per round. 1 P. G. Woodhouse authored a number of books. One of his best works was about Jeeves, a "man servant". The series was turned into a HTV seriesH starring a very young and very funny Hugh Laurie (of HHouse MDH fame). 3 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain (a) Suppose Drones decides to charge serious and casual golfers a common uniform price for playing a round of golf calculate this price. What will Drones' profits be? Show all calculations clearly. Hint: the serious and casual golfers are being treated as a single market. Answer: If the Club is to set one uniform price then we should aggregate the demands: Total demand: QTotal = 100 Qc + 10 Qs QT = 100(100 10P) + 10(350 10P) QT = 13,500 1,100P PT = 135/11 (1/1100) QT To maximize profits, set MR = MC. Using the short cut for MR we have: MRT = 135/11 (2/1100) QT MR = MC yields: 135/11 (2/1100) QT = 5 QT = 4,000 P = 8.64 The profits are: = (P MC)Q = 4000(8.64 5) = $14,545.45 (b) Suppose Drones decides to charge serious and casual golfers different uniform prices for a round of golf. Calculate these prices. What will Drones' profits be? Show all calculations clearly. Answer: Each type of golfer is being treated as a segment. To maximize profits in each segment, set: MRs = MC and MRc = MC First consider: MRs = MC Total demand for serious players: QS Total = 10 Qs = 10(350 10P) = 3500 100 P P = 35 0.01 QS Total MRs = 35 0.02 QS Total 4 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain 35 0.02 QS Total = 5 QS Total = 1500 P = 20 S Total = 1500(20 5) = $22,500 Next consider: MRC = MC Total demand for casual players: QC Total = 100 Qc = 100(100 10P) = 10000 1000 P P = 10 0.001 QC Total MRc = 10 0.002 QC Total 10 0.002 QC Total = 5 QC Total = 2500 P = 7.5 C Total = 2500(7.5 5) = $6,250 Total Profit: $22,500 + $6,250 = $28,750 (c) The club decides to charge golfers a single membership ("access") fee to join the club and a single "usage" price for playing a round of golf. Calculate the membership fee and the price for a round of golf. What will Drones' profits be? Show all calculations clearly. Check your answer from this Excel model. Answer: Denote access fee by A. Total profits will be: = (Total demand for rounds of golf) (P MC) + (Total number of golfers)(Access Fee) = QT (P MC) + (Total number of golfers)(Access Fee) Recall there are a total of 10 serious and 100 casual golfers. Thus: = QT (P MC) + 110 A = QT (P 5) + 110 A The optimal membership fee is set equal to the consumer surplus of a casual player (why?). So, A = (1/2)(max WTP of low type P)(Quantity of low type) A = (10 P)(100 10P)/2 5 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain From part (a): QT = 13,500 1100P. Substitute in total profits to get: = QT (P 5) + 110 A = (13,500 1,100P)(P 5) + 110 (10 P)(100 10P)/2 = 13,500P 1,100P2 67,500 + 5,500 P + 55,000 + 550P2 11,000P = 12,500 + 8,000 P 550P2 We have expressed the profits of the company only in terms of the usage price. Now, maximizing profit entails: d/dP =0 8,000 1,100P = 0 P = $7.27 Substitute this in the expression for access fee: A = (10 P)(100 10P)/2 A = 37.19 Thus the total profits are: = QT (P 5) + 110 A = $16,591 You can check your solution by using the Excel model. Notice how the profits with a segment specific access fee and usage price is greater than the profits with a common access fee and common usage price which, in turn, is greater than the profits from charging all golfers a common flat price. Can the club do better? Yes, by practicing 1st degree price discrimination which brings us to the next question. (d) If Drones can negotiate the price of a round of golf with each player, how many rounds of golf will the serious and casual golfers play? What will Drones' profits be? Show all calculations clearly. Answer: Now the Club is practicing 1st degree price discrimination. For each segment, you'd want to sell that number of rounds where the demand curve equals MC (this is because under 1st degree price discrimination, the demand curve is the MR curve so that for any output, P = MR). For each type of golfer, let's find the output where P = MC = 5: Number of games by casual players: QC = 100 10P = 100 10(5) = 50 6 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Number of games by serious players: QS = 350 10P = 350 10(5) = 300 This implies that: = 100(10 5)50/2 + 10(35 5)300/2 = $57,500 This is an even greater profit than any scheme above. 7 ...
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