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Deep Thought

Deep Thought. To me, boxing is like a ballet, except there’s no music, no choreography, and the dancers hit each other. ~ Jack Handey . (Translation: Today’s lesson teaches how to manage a company recognizing competitors are selling substitute products.). Lesson overview.

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Deep Thought

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  1. Deep Thought To me, boxing is like a ballet, except there’s no music, no choreography, and the dancers hit each other. ~ Jack Handey. (Translation: Today’s lesson teaches how to manage a company recognizing competitors are selling substitute products.) BA 210 Lesson II.5 Simultaneous Quantity Competition

  2. Lesson overview • Lesson II.5 Simultaneous Quantity Competition • Example 1: Cournot Duopoly • Example 2: Nash Equilibrium • Example 3: First Mover Advantage • Example 4: Selling Technology • Example 5: Colluding • Example 6: Merging • Summary • Review Questions BA 210 Lesson II.5 Simultaneous Quantity Competition

  3. Example 1: Cournot Duopoly Example 1: Cournot Duopoly BA 210 Lesson II.5 Simultaneous Quantity Competition

  4. Example 1: Cournot Duopoly • Comment: Cournot Duopoly Games have three parts. • Players are managers of two firms serving many consumers. • Strategies are outputs of homogeneous products (perfect substitutes), so they sell at the same price P. • Firm 1 chooses output Q1> 0. • Firm 2 chooses output Q2> 0. • Each chooses either chooses output simultaneously, or chooses sequentially but the one that chooses second does not know the choice of the one that chooses first. • Payoffs are profits. When unit production costs of production are constants c1 and c2, then profits are • P1 = (P- c1)Q1 and P2 = (P- c2)Q2 BA 210 Lesson II.5 Simultaneous Quantity Competition

  5. Example 1: Cournot Duopoly Payoffs for the two players often listed in a table. The strategies are listed as rows for Player 1, and columns for Player 2. The combination of strategies by both players determines a cell in the payoff table, and that cell specifies the payoffs to the players, with Player 1 listed first. For example, if Player 1 chooses Q1=3 and Player 2 chooses Q1=1, then in the payoff table below 12,5 specifies payoff 12 to Player 1 and 5 to Player 2. BA 210 Lesson II.5 Simultaneous Quantity Competition

  6. Example 1: Cournot Duopoly Question: Intel and AMD simultaneously decide on the size of manufacturing plants for the next generation of microprocessors for consumer desktop computers. Suppose the firms’ goods are perfect substitutes, and market demand defines a linear inverse demand curve P = 20 – (QI + QA), where output quantities QI and QA are the thousands of processors produced weekly by Intel and AMD. Suppose unit costs of production are cI= 1.1 and cA= 1.1 for both Intel and AMD. Suppose Intel and AMD consider any quantities QI = 3, 5, 6, 7, 9 and QA = 3, 5, 6, 7, 9. What quantity should Intel produce? BA 210 Lesson II.5 Simultaneous Quantity Competition

  7. Example 1: Cournot Duopoly Answer: Intel’s quantities QI = 3, 5, 6, 7, 9 are on the rows, and AMD’s quantities QA = 3, 5, 6, 7, 9 are on the columns in the following normal form. For example, QI = 3 and QA = 5 generates price P = 20-8 = 12, and profits PI = (12-1.1)3 = 32.7 and PA = (12-1.1)5 = 54.5. (On an exam, I would provide most entries.) BA 210 Lesson II.5 Simultaneous Quantity Competition

  8. Example 1: Cournot Duopoly Strategy 3 is dominated for each player (by Strategy 5) and Strategy 9 is dominated for each player (by Strategy 7). Hence, eliminate those strategies, leaving the normal form: BA 210 Lesson II.5 Simultaneous Quantity Competition

  9. Example 1: Cournot Duopoly Strategy 5 is now dominated for each player (by Strategy 6). Hence, eliminate that strategy, leaving the normal form: BA 210 Lesson II.5 Simultaneous Quantity Competition

  10. Example 1: Cournot Duopoly Strategy 7 is now dominated for each player (by Strategy 6). Hence, eliminate that strategy, leaving only strategies QI = 6 and QA = 6, and profits PI = 41.4 and PA = 41.4. As in any game, under game theory assumptions (including rationality), it is always best to play your strategy that is part of a dominance solution. BA 210 Lesson II.5 Simultaneous Quantity Competition

  11. Example 1: Cournot Duopoly Comment: The dominance solution QI = 6 and QA = 6, with profits PI = 41.4 and PA = 41.4, is the only Nash Equilibrium. A Nash Equilibrium means QI = 6 is Intel’s best response to QA = 6, and QA = 6 is AMD’s best response to QI = 6. BA 210 Lesson II.5 Simultaneous Quantity Competition

  12. Example 2: Nash Equilibrium Example 2: Nash Equilibrium BA 210 Lesson II.5 Simultaneous Quantity Competition

  13. Example 2: Nash Equilibrium Comment: Although Cournot Duopoly Games have dominance solutions even when quantities can be continuous variables (including fractions), it is hard go through the entire sequence of reasoning like in Example 1. It turns out, however, that the unique dominance solution of a Cournot Duopoly Game is also the unique Nash Equilibrium of the Game. And finding a Nash Equilibrium is relatively simple. A Nash Equilibrium of any game with two or more players means each player is assumed to know the chosen strategies of the other players, and each player chooses a best response to those chosen strategies --- that is, no player has anything to gain by changing only his or her own strategy unilaterally. BA 210 Lesson II.5 Simultaneous Quantity Competition

  14. Example 2: Nash Equilibrium Question: You are a manager of Pepsi and you compete directly with Coke selling soft drinks. Consumers find the two products to be indistinguishable. The inverse market demand for soft drinks is P = 11-Q (in dollars). Your unit costs of production are $3, and the unit costs of Coke are $2. Compute profits when you produce 1 units and Coke produces 6 units. Suppose profits from other combinations of production are in the table below: BA 210 Lesson II.5 Simultaneous Quantity Competition

  15. Example 2: Nash Equilibrium Question (continued): Suppose you choose your output of soft drinks to be either 1, 3, or 4 at the same time that Coke chooses its own own output of either 1, 3, or 4. How many soft drinks should you produce? BA 210 Lesson II.5 Simultaneous Quantity Competition

  16. Example 2: Nash Equilibrium Answer: You are the row player (Player 1) in a Cournot Duopoly Game with inverse demand P = 11 - (Q1+Q2) and unit costs c1 = 3 and c2 = 2. If you produce Q1=1 units and DC produces Q2=6 units, then total output is Q1+Q2=7, so price is P = 11 - (Q1+Q2)=4, and profits are P1 = (P- c1)Q1 = (4-3)1 = 1 and P2 = (P- c2)Q2 = (4-2)6 = 12, which we write 1,12 to complete the profit table. BA 210 Lesson II.5 Simultaneous Quantity Competition

  17. Example 2: Nash Equilibrium Find the Nash Equilibrium to the Cournot Duopoly Game (which turns out to be the dominance solution). • if you believe Coke chooses Q2=1, then you respond with Q1=3.But, if Coke believes you choose Q1=3, then they would not actually choose Q2=1. So no Nash Equilibrium here. BA 210 Lesson II.5 Simultaneous Quantity Competition

  18. Example 2: Nash Equilibrium • if you believe Coke chooses Q2=3, then you respond with Q1=3. And if Coke believes you choose Q1=3, then they would indeed choose Q2=3. So Q1=3 and Q2=3 is a Nash Equilibrium. Is there any other Nash Equilibrium? BA 210 Lesson II.5 Simultaneous Quantity Competition

  19. Example 2: Nash Equilibrium • if you believe Coke chooses Q2=6, then you respond with Q1=1.But, if Coke believes you choose Q1=1, then they would not actually choose Q2=1. So no Nash Equilibrium here. Conclusion: Pepsi should produce 3 soft drinks and expect Coke to produce 3 soft drinks. BA 210 Lesson II.5 Simultaneous Quantity Competition

  20. Example 2: Nash Equilibrium Comment: The Nash equilibrium of Pepsi producing 3 soft drinks and Coke producing 3 soft drinks is the unique dominance solution. First, producing 6 is dominated for each player by producing 3. After producing 6 is eliminated, producing 1 soft drink is dominated for each player by producing 3. BA 210 Lesson II.5 Simultaneous Quantity Competition

  21. Example 3: First Mover Advantage Example 3: First Mover Advantage BA 210 Lesson II.5 Simultaneous Quantity Competition

  22. Example 3: First Mover Advantage Comment: If the unit production costs are the same for two firms in a duopoly, then profits for Firm 1 are higher if that firm is the leader in a Stackelberg duopoly rather than a firm in a Cournot duopoly. And profits for Firm 1 are higher if that firm is in a Cournot duopoly rather than the follower in a Stackelberg duopoly. In particular, a firm can find it profitable to become the first mover by rushing to set up an assembly line, even if it means increasing unit costs of production. BA 210 Lesson II.5 Simultaneous Quantity Competition

  23. Example 3: First Mover Advantage Question: You are a manager of PetroChina and you compete directly with Sinopec in Chinese oil production and distribution. Consumers find the two products to be indistinguishable. The inverse market demand for Chinese oil is P = 11-Q (in yuan) and both firms produce either 1, 3, or 6 units at a unit cost of 2 yuan. You have a decision to make about competing with Sinopec in Siberia, where the inverse market demand for Chinese oil is P = 11-Q (in rubles). You must choose the option that is best for PetroChina. BA 210 Lesson II.5 Simultaneous Quantity Competition

  24. Example 3: First Mover Advantage Option A. Sinopec sets up its refineries and distribution networks now, and you set up later. And both produce at a unit cost of 2 rubles. Option B. You hurry set up your refineries and distribution networks at the same time as Sinopec. Your hurry means your unit costs are 3 rubles, while Sinopec’s costs remain 2 rubles. Which Option is better for PetroChina? BA 210 Lesson II.5 Simultaneous Quantity Competition

  25. Example 3: First Mover Advantage Answer: In Option A, you are the follower in a Stackelberg Duopoly with inverse demand P = 11 - (Q1+Q2) and unit costs c1 = 2 and c2= 2. In Option B, you are Firm 2 in a Cournot Duopoly with inverse demand P = 11 - (Q1+Q2) and unit costs c1 = 3 and c2 = 2. BA 210 Lesson II.5 Simultaneous Quantity Competition

  26. Example 3: First Mover Advantage Option A. Sinopec sets up its factories and distribution networks now, and you set up later. And both produce at a unit cost of 2, resulting in the first profit table: Option B. You hurry set up your refineries and distribution networks at the same time as Sinopec. Your hurry means your unit costs are 3, while Sinopec’s unit costs remain 2 resulting in the second profit table: BA 210 Lesson II.5 Simultaneous Quantity Competition

  27. Example 3: First Mover Advantage Answer: In Option A, you are the follower in a Stackelberg Duopoly. Sinopec anticipates your reactions on the right, and so chooses to produce 6, you react with 1 and so you earn 2. In Option B, you are in a Cournot Duopoly. The only Nash equilibrium and dominance solution (eliminate 6, then eliminate 1) is Sinopec chooses 3 and you choose 3, and so you earn 6. BA 210 Lesson II.5 Simultaneous Quantity Competition

  28. Example 3: First Mover Advantage Option B is thus best for PetroChina since PetroChina profits (as a Stackelberg follower) are 2 in Option A, while PetroChina profits (as a Cournot Duopolist) are 6 in Option B. BA 210 Lesson II.5 Simultaneous Quantity Competition

  29. Example 3: First Mover Advantage Comment: In this particular case, PetroChina increased production cost hurt profits less than profits increase because of eliminating the second mover disadvantage. In other problems, increased production cost hurt profits more than profits increase because of eliminating the second mover disadvantage. BA 210 Lesson II.5 Simultaneous Quantity Competition

  30. Example 4: Selling Technology Example 4: Selling Technology BA 210 Lesson II.5 Simultaneous Quantity Competition

  31. Example 4: Selling Technology Question: You are a manager of Nvidia and your only significant competitor in the mainstream graphics card market is the ATI subsidiary of Advanced Micro Devices. You and ATI both expect to produce the next generation of graphics card in October of next year. Your graphics cards and ATI’s graphics cards are indistinguishable to consumers. The inverse market demand for graphics cards is P = 11-Q (in dollars) and both firms used to produce at a unit cost of $2. However, you just found a better way to produce graphics cards, which reduces your unit cost to $1. Should you keep that procedure to yourself? Or is it better to sell that secret to ATI so that both you and ATI can produce at unit cost equal to $1? To answer the questions, suppose both Nvidia and ATI separately choose to produce either 1, or 2, or 3 units of graphics cards. BA 210 Lesson II.5 Simultaneous Quantity Competition

  32. Example 4: Selling Technology Answer: If you do not sell your technology, you are Firm 1 in a Cournot Duopoly with inverse demand P = 11 - (Q1+Q2) and unit costs are c1 = 1 and c2 = 2; if you do sell, unit costs are c1 = 1 and c2 = 1. Find the Nash Equilibrium to each Cournot Duopoly Game (which turns out to be the dominance solution). BA 210 Lesson II.5 Simultaneous Quantity Competition

  33. Example 4: Selling Technology If you do not sell your technology, compute the payoff table and Nash equilibrium for unit costs c1 = 1 and c2 = 2: The only Nash equilibrium and dominance solution (eliminate 1 and 2) is Nvidia chooses 3 and ATI chooses 3, and so Nvidia earns 12 and ATI earns 9. BA 210 Lesson II.5 Simultaneous Quantity Competition

  34. Example 4: Selling Technology If you do sell your technology, compute the payoff table and Nash equilibrium for unit costs c1 = 1 and c2 = 1: The only Nash equilibrium and dominance solution (eliminate 1 and 2) is Nvidia chooses 3 and ATI chooses 3, and so Nvidia earns 12 and ATI earns 12. BA 210 Lesson II.5 Simultaneous Quantity Competition

  35. Example 4: Selling Technology Selling technology and reducing c2 = 2 to c2 = 1 has two potential effects: • Firm 1’s profit could change. But in this case, it remains at P1 = 12 • Firm 2’s profit increases from P2 = 9 to P2 = 12 Nvidia should sell the technology because doing so increases total profit from production from 21 to 24, so there are 3 gains from trade to be divided between the two firms according to the rules of the resulting bargaining game. For example, if Nvidia can make a credible take-it-or-leave-it offer of 3 minus a pittance as the price of the technology to ATI, then Nvidia captures most of those gains. BA 210 Lesson II.5 Simultaneous Quantity Competition

  36. Example 5: Colluding Example 5: Colluding BA 210 Lesson II.5 Simultaneous Quantity Competition

  37. Example 5: Colluding Question: The Mexican Multimedia market consists of two firms: TV Azteca and Televisa. As a manager of TV Azteca, you choose the number of broadcast hours of television programming of your hit shows (Lo que callamos las mujeres, Ventaneando, Hechos, Venga la Alegria, …) to air 1 hour before your competitor, but Televisa does not have any way to know your broadcast hours before choosing their own broadcast hours. Advertisers consider all broadcast hours to be identical. The demand for broadcast hours is Q = 13 - P; both firms’ unit costs are 1; both firms separately produce either 1, 3, or 4 units. Would it be mutually profitable for the companies to collude by setting TV Azteca’s and Televisa’s outputs to 3 and 3. Can TV Azteca trust Televisa to collude? Can Televisa trust TV Azteca? BA 210 Lesson II.5 Simultaneous Quantity Competition

  38. Example 5: Colluding Answer: You are Firm 1 in a Cournot Duopoly with demand Q = 13 - P, inverse demand P = 13- (Q1+Q2), and unit costs c1= 1 and c2= 1. Find the Nash Equilibrium to the Cournot Duopoly Game (which turns out to be the dominance solution), and compare the Nash Equilibrium to the collusive proposal of quantities 3 and 3. BA 210 Lesson II.5 Simultaneous Quantity Competition

  39. Example 5: Colluding Compute the payoff table and Nash equilibrium for inverse demand P = 13- (Q1+Q2), and unit costs c1 = 1 and c2 = 1: The only Nash equilibrium and dominance solution (eliminate 1 and 3) is TV Azteca chooses 4 and Televisa chooses 4, and so TV Azteca earns 16 and Televisa earns 16. BA 210 Lesson II.5 Simultaneous Quantity Competition

  40. Example 5: Colluding Setting TV Azteca’s and Televisa’s outputs to 3 and 3 increases earnings from 16 each to 18 each. But TV Azteca cannot trust Televisa to collude since Televisa’s best response to TV Azteca’s Q1 = 3 is Q2 = 4, not 3. And Televisacannot trust TV Azteca to collude since TV Azteca’s best response to Televisa’s Q2 = 3 is Q1 = 4, not 3. BA 210 Lesson II.5 Simultaneous Quantity Competition

  41. Example 6: Merging Example 6: Merging BA 210 Lesson II.5 Simultaneous Quantity Competition

  42. Example 6: Merging Question: The market for legal research consists of two firms: LexisNexis and Westlaw. As a manager of LexisNexis, you hire your researchers at the same time as Westlaw, and so your simultaneously choose your profit-maximizing research output levels in the market. Law firms consider the output by the two research firms as perfect substitutes. The demand for legal research is Q = 10 - P; LexisNexis’s unit costs are 1; Westlaw’s unit costs are 2; and both firms separately produce either 1, 4, or 5 units. Compute LexisNexis’s profit, and compute Westlaw’s profit. Ignoring antitrust law considerations, would it be profitable for the two companies to merge? If so, compute the gains from a merger. If the rules of bargaining allow LexisNexis’s to make a take-it-or-leave-it offer to Westlaw, compute that offer. BA 210 Lesson II.5 Simultaneous Quantity Competition

  43. Example 6: Merging Answer: You are Firm 1 in a Cournot Duopoly with demand Q = 10 - P, inverse demand P = 10-(Q1+Q2), unit costs c1 = 1 and c2 = 2, and alternative quantities Q1=1,4,5 and Q2=1,4,5. Find the Nash Equilibrium solutions and dominance solutions to the Cournot Duopoly Game (which turns out to be the dominance solution), and compare the recommended solution to the monopoly solution. BA 210 Lesson II.5 Simultaneous Quantity Competition

  44. Example 6: Merging Compute the payoff table and Nash equilibrium for inverse demand P = 10- (Q1+Q2), and unit costs c1 = 1 and c2 = 2: There is more than one Nash equilibrium but the only dominance solution (both eliminate 5, Lexis eliminates 1, Westlaw eliminates 4) is LexisNexis chooses 4 and Westlaw chooses 1, and so LexisNexis earns 16 and Westlaw earns 3. BA 210 Lesson II.5 Simultaneous Quantity Competition

  45. Example 6: Merging The monopoly solutionfor inverse demand P = 10- (Q1+Q2), has each firm using the lowest unit costs,c1 = 1 and c2 = 1. Recompute the payoff table and determine which cell maximizes total profit. If either LexisNexis chooses 4 and Westlaw chooses 1 or LexisNexis chooses 1 and Westlaw chooses 4, then total profit is maximized at 20. BA 210 Lesson II.5 Simultaneous Quantity Competition

  46. Example 6: Merging Merging thus increases total profit from P1+P2 = 16 + 3 = 19 to P = 20, an increase of 1. So, 1 is the gains from a merger. If the rules of bargaining allow LexisNexis to make a take-it-or-leave-it offer to Westlaw, that offer should be 3 plus a pittance. BA 210 Lesson II.5 Simultaneous Quantity Competition

  47. Summary Summary BA 210 Lesson II.5 Simultaneous Quantity Competition

  48. Summary • Payoff table entry to any Duopoly Game with inverse demand P = a -bQ and constant unit costs c1 and c2: • P= a - b(Q1+Q2) • Firm 1 profit P1 = (P - c1) Q1 • Firm 2 profit P2 = (P - c2) Q2 BA 210 Lesson II.5 Simultaneous Quantity Competition

  49. Review Questions • Review Questions • You should try to answer some of the following questions before the next class. • You will not turn in your answers, but students may request to discuss their answers to begin the next class. • Your upcoming Exam 2 and cumulative Final Exam will contain some similar questions, so you should eventually consider every review question before taking your exams. BA 210 Lesson II.5 Simultaneous Quantity Competition

  50. BA 210 Introduction to Microeconomics End of Lesson II.5 BA 210 Lesson II.5 Simultaneous Quantity Competition

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