eco204_summer_2009_practice_problem_21_solution

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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 21 Solutions Please help improve the course by sending me an email about typos or suggestions for improvements You can check your oligopoly calculations using this summary slide: Question 1 Ajax Microchip Corporation (AMC) manufactures and sells microchips. AMC's cost function is: C = 100 + 38 q1 where q1 ( note the subscript 1) is in `000s of units (i.e. 1,000 units would be q = 1). Assume AMC has the capacity to produce 10,000 microchips a month. Market demand is estimated to be P = 170 20Q where Q, the total market output, is in `000s of units and P is in dollars. (a) Suppose AMC is the only company producing microchips in this market. What is the optimal price, quantity and profits? Show all calculations. 1 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain Answer: The market demand curve is: P = 170 20Q Since AMC is the only firm in the marketplace q1 = Q: P = 170 20Q P = 170 20q1 Now AMC's problem is to choose the optimal profit maximizing q1. Thus, for AMC, it's marginal revenue MR is: P = 170 20q1 MR = 170 40q1 To maximize profits, AMC should set MR1 = MC1: 170 40q1 = 38 170 38 = 40q1 q1 = 132/40 q1 = 3.3 The market price will be: P = 170 20q1 P = 170 20(3.3) P = $104 AMC's profits are: = R TVC TFC = Pq 38q 100 = (104)(3.3) 38(3.3) 100 2 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain = 117.8 Which is actually $117,800 (remember Q is in `000s). (b) Now suppose AMC competes with another identical microchip manufacturer Murdock Microchip Company (MMC) as Cournot rivals. How many microchips will each company produce and what will the market price be? How much profit will each company earn? Show all calculations. Answer: MMC, being identical to AMC, has the cost function: C = 100 + 38 q2 Where q2 is MMC's output (also in `000s). To solve for the optimal price and output consider the market demand curve: P = 170 20Q P = 170 20(q1 + q 2) P = 170 20q1 20q2 AMC's problem is to choose the optimal profit maximizing q1 taking MMC's output q2 as given. Thus, for AMC, it's MR is: P = 170 20q1 20q2 P = 170 20q2 20q1 From AMC's perspective, 170 20q2 is a constant. Thus: MR1 = 170 20q2 40q1 To maximize profits, AMC should set MR1 = MC1: 170 20q2 40q1 = 38 170 38 20q2 = 40q1 132 20q2 = 40q1 q1 = {132 20q2}/40 3 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain q1 = 3.3 0.5q2 This is firm 1's reaction function to firm 2's output. It can be reexpressed as: q1 = Firm 1's output if it was a monopolist 0.5q2 Intuitively, the existence of another firm forces firm 1 to produce less than if it was a monopolist. This is because the other firm's supply will lower price and firm 1 then produces less in order to raise prices. You could now solve the problem by exploiting the symmetry across both firms: they have the same cost function and thus at the optimum q1 = q2 so that we can write the equation above as: q1 = 3.3 0.5q2 q1 = 3.3 0.5q1 1.5 q1 = 3.3 q1 = 3.3/1.5 q1 = 2.2 Thus q2 = 2.2 Alternatively, you could set MR = MC for firm 2 (MMC) and gotten: P = 170 20Q P = 170 20(q1 + q 2) P = 170 20q1 20q2 MMC's problem is to choose the optimal profit maximizing q2 taking AMC's output q1 as given. Thus, for MMC, it's MR is: P = 170 20q1 20q2 From MMC's perspective, 170 20q1 is a constant. Thus: MR2 = 170 20q1 40q2 To maximize profits, MMC should set MR2 = MC2: 4 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain 170 20q1 40q2 = 38 170 38 20q1 = 40q2 132 20q1 = 40q2 q2 = {132 20q1}/40 Thus you have two equations reaction functions in two unknowns: (1) q1 = {132 20q2}/40 (2) q2 = {132 20q1}/40 which can be solved simultaneously to yield: q1 = q2 = 2.2 The total market output is q1 + q2 = 4.4 which yields a price of: P = 170 20(q1 + q 2) P = 170 20(4.4) P = $82 Cournot rivalry has resulted in a lower price than the case where AMC was a monopoly. AMC's profits are: = R TVC TFC = Pq 38q 100 = (82)(2.2) 38(2.2) 100 = 3.2 Which is actually a loss of $3,200 (remember Q is in `000s). MMC being identical to AMC also incurs losses of $3,200. This being a short run problem with positive contribution margins, note that AMC and MMC should continue operating. However, if the situation does not improve, say through higher prices if the rival exits the market or lower costs through technological progress then 5 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain these firms should declare bankruptcy. (c) Now suppose AMC competes with another identical microchip manufacturer Murdock Microchip Company (MMC) as Stackelberg rivals in which AMC enters the market before MMC. How many microchips will each company produce and what will the market price be? Show all calculations. Answer: AMC enters the market before MMC. Thus, when MMC chooses its output it will take AMC's output as given. From part (b) we know that MMC's reaction function is: q2 = {132 20q1}/40 However, AMC firm 1 and the first mover can use this reaction function to its advantage: it can use it to choose its output by incorporating the follower's reaction in: 1 = Pq1 TVC TFC 1 = Pq1 38 q1 100 1 = {170 20(q1 + q2)} q1 38 q1 100 1 = (170 38)q1 /2 (20/2)q21 100 1 = 66q1 10 q21 100 1 /q1 = 0 66 20 q1 = 0 q1 = 66/20 q1 = 3.3 Notice this is equal to the monopoly output: this is why the first mover has an advantage. Now: q2 = {132 20q1}/40 q2 = {132 20(3.3)}/40 6 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain q2 = 1.65 Given the first mover has produced the monopoly output, the follower is forced to produce less than the Cournot output in an effort to prop prices up. Total Q = q1 + q2 Q = 4.95 Thus: P = 170 20Q P = 170 20(4.95) P = $71 AMC, the leader, has profits of: = R TVC TFC = Pq 38q 100 = (71)(3.3) 38(3.3) 100 = 8.9 Which is actually $8,900 (remember Q is in `000s). MMC, the follower, has profits of: = R TVC TFC = Pq 38q 100 = (71)(1.65) 38(1.65) 100 = 45.55 Which is a loss of $45,5500 (remember Q is in `000s). This is a stark example of how AMC literally has the first mover advantage: by moving first, it produces the monopoly output and manages to be profitable. Meanwhile the follower, having its hand forced, must incur losses. Of course if this situation persists, MMC will eventually exit the market leaving AMC a monopoly. Incidentally, this is a nice example of a case where a company can become a monopoly not because it's erecting barriers to entry but because it was wise to enter first. As you can suspect, 7 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain this has clear ramifications for law: what appears to be AMC monopolizing the market is actually its rewards for being a first mover. To the winner go the spoils. In summary: Number of firms Rule q1 (`000s) q2 (`000s) Q = q1 + q2 (`000s) P 1 2 Total = 1 + 2 Monopoly 1 Choose output 3.3 0 3.3 $104 $117,800 $0 $117,800 Cournot Oligopoly 2 Choose outputs simultaneously 2.2 2.2 4.4 $82 $(3,200) $(3,200) $(6,400) Exit industry unless some rival exits and/or firms become more efficient Stackelberg Oligopoly 2 Choose outputs sequentially 3.3 1.65 4.95 $71 $8,900 $(45,550) $(36,600) Follower exits and leader "monopolizes" market Long Run? Monopolist so long as no other rival enters As a study question, you should see how these results change if there are three firms in Cournot and Stackelberg competition. Question 2 (20082009 Test Question) In the DHL case, explain why country managers don't have the correct incentives to set optimal prices. Suggest a solution. Answer: Country managers are compensated by P/L in their country. This includes all revenues for packages and documents emanating from their country as well as all local costs, which includes outgoing and incoming costs. 8 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain Managers should be rewarded for local revenues and penalized for outgoing costs. But since PRISM only gives the average costs for the country, it necessarily lumps incoming and outgoing costs. Thus, even if a manager sets prices using MR = MC, the MC they use is higher than what it should be. The solution is simple: fix PRISM so that it disentangles incoming from outgoing costs and gives the MC and not the AC of documents and packages. One of the readings for the DHL case was the postscript article which discusses this issue. Question 3 (2007 Final Exam Question) Here are some excerpts from the DHL case: "Industry revenues were split roughly 75:25 between parcels and documents .. the parcel sector grew 40%, while the document sector grew 15%" (p. 4) "The expressdelivery business in Europe is booming .. Measured by revenues, the European expressdelivery business is growing at a 28% compound annual rate. Big European companies are stocking products and parts in central locations and moving them by overnight express, instead of running warehouses in each country" (p. 4) Table C (p. 7) DHL's Document and Parcel Businesses Total Revenues Revenues Growth Document 60% + 14 % Parcel 40% + 28% Exhibit 9: Comparison of Prices Delivery Company DHL TNT FedEx $131 $143 $118 $120 $120 $39 Service 2 Kilogram DOX BrusselsHK 2 Kilogram WPX SingaporeSydney UPS $97 $34 From London To NY To Switzerland To Japan Table E: DHL Sample Prices First Kilo Document: 24.50 26.00 26.50 First Kilo Parcel: 27 32 34 9 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain (a) Provide some reasons for the differences in document and parcel prices in Table E above. Answer: Table E shows that on a per kilo basis, on a given route, the price for a document is lower than that of a parcel. Because we are comparing prices for a given route, we cannot say the difference is due to distance. Using the optimal pricing rule P = (E/(1 + E))MC we can say that the differences may stem from the WPX segment being less elastic than the DOX segment (in fact, the case specifically states this), and, from the fact that MC of WPX is higher than the MC of DOX. This is because even if a DOX and WPX weigh the same, unlike DOXs, WPXs are packed in all sorts of packaging with varying handling requirements (fragile, refrigeration, etc). Moreover, because DHL lags behind the industry in WPXs (see Table C and quote from p. 4 above), it does not have learning by doing in the WPX segment, which may explain why DHL's WPX MC may be high. (b) Based on the excerpts above, evaluate DHL's pricing of documents and parcels and offer recommendations for changes. Answer: In part (a), we justified why the WOX prices were greater than DOX prices. While these prices may make sense for maximizing profits in the current period, these prices may not make sense for maximizing profits over many periods. For instance, if there are learning economies to be had in handling WPXs, then it would behoove DHL to lower prices for WPXs so as to increase the volume of WPX and lower MC through learning by doing. Exhibit 9 above shows that DHL is charging higher prices for WPX compared to FedEx and UPS. Thus, if DHL is to gain a competitive advantage against them, it should lower prices for WPX. 10 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain Question 4 (20082009 Final Exam Question) The following table reproduces portions of Exhibit 11 from the DHL case. Exhibit 11 Revenue and Cost Lane Examples: DOX and WPX U.K. to United States (1990) DOX WPX Revenue $5,723,000 $2,342,000 Outbound Cost 2,392,915 667,712 Hub Cost 596,608 490,436 Line Haul 1 1,121,882 647,915 Delivery 1,376,953 386,049 Gross Margin 234,642 149,888 Gross Margin % 4.1 6.4 Shipments 231,139 68,580 Revenue/Shipment $24.76 $34.15 Note: Please see footnote at the bottom of this page. (a) Calculate the AVC of DOX and WPX for this lane. Show all calculations. Answer We need AVC = TVC/Q. To get TVC, note that by definition: Gross Margin = R TVC TVC = R Gross margin For DOX: TVC = 5,723,000 234,642 = 5,488,358 Thus AVC = TVC/Q = 5,488,358/231,139 = $23.74 For WPX: TVC = 2,342,000 149,888 = 2,192,112 Thus: AVC = TVC/Q = 2,192,112/68,580 = $31.96. (b) Suppose DHL's headquarters in Brussels sets prices to maximize profits. If Brussels raises the price of a document from $24.76 to $25.01 how many documents will be sent from the U.K. to the U.S? State any assumptions. 1 Line haul refers to the air segment of the shipment 11 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain Answer We are given a price change and asked to compute the impact on quantity. The new price is $25.01 which is 1% more than the initial price of $24.76 (note how: 1.01*$24.76 = $25.01). Now, to compute elasticity, use the margin %: Margin % = (Margin/Revenue)*100 = 4.1 (Margin/Revenue) = 0.041 If DHL is charging the optimal profit maximizing price and AVC = MC (for example if the TVC is linear): (P MC)/P = 1/E If MC = AVC: (P MC)/P = (P AVC)/P = (P AVC)Q/PQ = (Margin/Revenue) = 1/E Thus: 1/E = 0.041 E = 1/0.041 E = 24.39 Or: % Change in Q % Change in P = 24.39 Now: % Change in P = 1% Thus: % Change in Q = 24.39% Currently, on this lane, there are 231,139 DOXs shipped. Thus, the new number of DOX shipped will be: (1 0.2439)*231,139 = 174,764. (c) Compute the price elasticity of the WPX segment. Answer Repeat the steps above. Optimal pricing requires: (P MC)/P = 1/E 12 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain If MC = AVC: (P MC)/P = (P AVC)/P = (P AVC)Q/PQ = (Margin/Revenue) = 1/E 1/E = 0.064 E = 1/0.064 E = 15.625 (d) Suppose DHL lowers the cost of shipping packages through learning by doing. Assuming the elasticity in part (c) is constant, what must the MC of packages be in order for the price of a document to be equal to the price of a package on the U.K. to U.S. lane? Show all calculations. Answer Through the optimal pricing rule: (P MC)/P = 1/E Solve for MC: (P MC) = P/E MC = P + P/E The WPX E is 15.625. Now, if price for WPX = price of DOX: P = $24.76 Thus: MC = 24.76 24.76/15.625 = $23.18 Assuming MC = AVC (linear TVC function), the MC would have to be $23.17 per WPX shipment (substantially lower than that of DOX). 13 ...
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