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 20 Solutions Please help improve the course by sending me an email about typos or suggestions for improvements For the 3rd degree price discrimination problems, it will be useful to refer to this summary: Question 1 Suppose BMW produces cars at a constant marginal cost of $20,000. It's fixed costs are $10 billion. You must advise the CEO on the price and number of cars to be sold in Europe and USA. The demand for BMWs in each market is given by: QE = 4,000,000 100 PE QU = 1,000,000 20 PU where the subscript E denotes Europe and the subscript U denotes the United States. Assume that BMW can restrict U.S. sales to authorized BMW dealers only (that is, it can prevent arbitrage). How many cars should BMW sell in each market and what price? Calculate total profits. 1 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain Answer: This is a 3rd degree price discrimination problem with ample capacity and no danger of arbitrage. BMW should maximize profits in each market separately: ME = 0 and MU = 0 ME = 0 means that: MRE MCE = 0 MRE = MCE MRE = 20,000 Now: QE = 4,000,000 100 PE PE = (4,000,000)/100 (1/100) QE MRE = 40,000 (1/50) QE Hence: MRE = 20,000 40,000 (1/50) QE = 20,000 QE = 1,000,000 Similarly: MU = 0 means that: MRU MCU = 0 MRU = MCU MRU = 20,000 Now: QU = 1,000,000 20 PU PU = (1,000,000)/20 (1/20) QU MRU = 50,000 (1/10) QU Hence: MRU = 20,000 50,000 (1/10) QU = 20,000 QU = 300,000. 1 million BMWs should be sold in Europe and 300,000 in the US. For prices: 2 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain PE = (4,000,000)/100 (1/100) (1,000,000) = $30,000 PU = (1,000,000)/20 (1/20) (300,000) = $35,000 As a way to test your math skills you should resolve this problem as a multivariate problem: that is, maximize: Total = E + U. You should get the same answers. Question 2 Note: Check your answer using spreadsheet mode (Excel 2003 version) Ajax Studios has produced two movies "Canada: No Country for Old Economists" and "Who Am I? John Nash vs. John Nash". The table below shows three Chains' willingness to pay (WTP) for the two movies and Ajax's marginal cost of releasing a movie. All numbers are in `000s of dollars per cinema per week. Each chain has 100 multiscreen cinemas: Chain Chain A Chain B Chain C Marginal Cost Canada: No Country for Old Economists 8 14 20 10 Who Am I? John Nash vs. John Nash 20 14 8 10 (a) Find the optimal weekly prices per screen for releasing the movies separately. Show all calculations and keep explanations brief. Answer: Recall this (helpful) chart: 3 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain Let's start with Canada: No Country for Old Economists. The following table lists the 3 conceivable prices: # of Chains Total Profit (`000s of $) Price Profit/cinema Purchasing (8 10)*300 = 600 8 (8 10) 3 14 20 (14 10) (20 10) 2 1 (14 10)*200 = 800 (20 10)*100 = 1000 Of the 3 prices, P = 20 yields the highest profits. Next, consider Who Am I? John Nash vs. John Nash: by symmetry because the WTPs are also 8, 14 and 20 we know that the optimal price is also P = 20. Total profits from selling the movie individually will be 2,000 (`000s of $). (b) Find the optimal weekly prices per screen for releasing the movies as a pure bundle. Show all calculations and keep explanations brief. Answer: Adding up the WTP for all three chains, note that all three chains' WTP for the bundle is $28. Thus, the bundle price is $28. Total profits, since all three chains buy the bundle (thus, MC for bundle is 20), the total profit from bundling will be (28 20)*300 = 2,400 (`000s of $). 4 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain (c) Find the optimal weekly prices per screen for releasing the movies as a mixed bundle. Show all calculations and keep explanations brief. Answer: Note how Chain 1's WTP for Canada and Chain 2's WTP for Who Am I? are each below the MC. Thus, to sell Chains 1 and 3 these movies individually and Chain 2 the bundle, suppose we set: P for Bundle = 28 P for Canada: No Country for Old Economists = 20 P for Who Am I? John Nash vs. John Nash = 20 Before finalizing these prices, examine the CS of the 3 chains: Chain C CS Chain A CS Chain B CS Bundle P = $28 Canada P = $20 John P = $20 0 Negative 0 0 Negative Negative 0 0 Negative Chain A has the same CS from buying the bundle or John Nash. Similarly, Chain C has the same CS from buying the bundle or Canada. We want Chains A and C to purchase the movies individually and not the bundle. To do this, give Chains A and C positive CS by pricing each movie slightly below 20 (say 19.99). This yields: Mixed Bundling Total profit = 2*(19.99 10)*(100) + (28 20)*(100) = 1,998 + 800 = $2,798 Of the three schemes, observe how mixed bundling yields the greatest profits. 5 ...
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 Fall '08
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 Economics, Microeconomics, John Nash, Product bundling, S. Ajaz Hussain

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