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1 1 Deep Thought BA 210 Lesson II.5 Simultaneous Quantity Competition 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.)

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2 2 Lesson overview BA 210 Lesson II.5 Simultaneous Quantity Competition 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

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

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4 4 BA 210 Lesson II.5 Simultaneous Quantity Competition Comment: Cournot Duopoly Games have three parts. Players are managers of two firms serving many consumers.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.Strategies are outputs of homogeneous products (perfect substitutes), so they sell at the same price P. Firm 1 chooses output Q 1 > 0.Firm 1 chooses output Q 1 > 0. Firm 2 chooses output Q 2 > 0.Firm 2 chooses output Q 2 > 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.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 c 1 and c 2, then profits arePayoffs are profits. When unit production costs of production are constants c 1 and c 2, then profits are 1 = (P- c 1 )Q 1 and 2 = (P- c 2 )Q 2 Example 1: Cournot Example 1: Cournot Duopoly

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5 5 BA 210 Lesson II.5 Simultaneous Quantity Competition 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 Q 1 =3 and Player 2 chooses Q 1 =1, then in the payoff table below 12,5 specifies payoff 12 to Player 1 and 5 to Player 2. Example 1: Cournot Example 1: Cournot Duopoly

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6 6 BA 210 Lesson II.5 Simultaneous Quantity Competition 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 – (Q I + Q A ), where output quantities Q I and Q A are the thousands of processors produced weekly by Intel and AMD. Suppose unit costs of production are c I = 1.1 and c A = 1.1 for both Intel and AMD. Suppose Intel and AMD consider any quantities Q I = 3, 5, 6, 7, 9 and Q A = 3, 5, 6, 7, 9. What quantity should Intel produce? Example 1: Cournot Duopoly

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

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8 8 BA 210 Lesson II.5 Simultaneous Quantity Competition 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: Example 1: Cournot Duopoly

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

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10 BA 210 Lesson II.5 Simultaneous Quantity Competition Strategy 7 is now dominated for each player (by Strategy 6). Hence, eliminate that strategy, leaving only strategies Q I = 6 and Q A = 6, and profits I = 41.4 and A = 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. Example 1: Cournot Duopoly

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

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

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

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

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

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

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

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18 BA 210 Lesson II.5 Simultaneous Quantity Competition if you believe Coke chooses Q 2 =3, then you respond with Q 1 =3. And if Coke believes you choose Q 1 =3, then they would indeed choose Q 2 =3. So Q 1 =3 and Q 2 =3 is a Nash Equilibrium. if you believe Coke chooses Q 2 =3, then you respond with Q 1 =3. And if Coke believes you choose Q 1 =3, then they would indeed choose Q 2 =3. So Q 1 =3 and Q 2 =3 is a Nash Equilibrium. Is there any other Nash Equilibrium? Example 2: Nash Equilibrium

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

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

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

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

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

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

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25 BA 210 Lesson II.5 Simultaneous Quantity Competition Answer: In Option A, you are the follower in a Stackelberg Duopoly with inverse demand P = 11 (Q 1 +Q 2 ) and unit costs c 1 = 2 and c 2 = 2. In Option B, you are Firm 2 in a Cournot Duopoly with inverse demand P = 11 (Q 1 +Q 2 ) and unit costs c 1 = 3 and c 2 = 2. Example 3: First Mover Advantage

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

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

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28 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 Example 3: First Mover Advantage

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29 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 Example 3: First Mover Advantage

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

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

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

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33 BA 210 Lesson II.5 Simultaneous Quantity Competition If you do not sell your technology, compute the payoff table and Nash equilibrium for unit costs c 1 = 1 and c 2 = 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. Example 4: Selling Technology

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34 BA 210 Lesson II.5 Simultaneous Quantity Competition If you do sell your technology, compute the payoff table and Nash equilibrium for unit costs c 1 = 1 and c 2 = 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. Example 4: Selling Technology

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35 Selling technology and reducing c 2 = 2 to c 2 = 1 has two potential effects: Firm 1’s profit could change. But in this case, it remains at 1 = 12Firm 1’s profit could change. But in this case, it remains at 1 = 12 Firm 2’s profit increases from 2 = 9 to 2 = 12Firm 2’s profit increases from 2 = 9 to 2 = 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 Example 4: Selling Technology

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

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

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38 BA 210 Lesson II.5 Simultaneous Quantity Competition Answer: You are Firm 1 in a Cournot Duopoly with demand Q = 13 P, inverse demand P = 13 (Q 1 +Q 2 ), and unit costs c 1 = 1 and c 2 = 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. Example 5: Colluding

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39 BA 210 Lesson II.5 Simultaneous Quantity Competition Compute the payoff table and Nash equilibrium for inverse demand P = 13 (Q 1 +Q 2 ), and unit costs c 1 = 1 and c 2 = 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. Example 5: Colluding

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40 BA 210 Lesson II.5 Simultaneous Quantity Competition 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 Q 1 = 3 is Q 2 = 4, not 3. And Televisa cannot trust TV Azteca to collude since TV Azteca’s best response to Televisa’s Q 2 = 3 is Q 1 = 4, not 3. Example 5: Colluding

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

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

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43 BA 210 Lesson II.5 Simultaneous Quantity Competition Answer: You are Firm 1 in a Cournot Duopoly with demand Q = 10 P, inverse demand P = 10 (Q 1 +Q 2 ), unit costs c 1 = 1 and c 2 = 2, and alternative quantities Q 1 =1,4,5 and Q 2 =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. Example 6: Merging

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44 BA 210 Lesson II.5 Simultaneous Quantity Competition Compute the payoff table and Nash equilibrium for inverse demand P = 10 (Q 1 +Q 2 ), and unit costs c 1 = 1 and c 2 = 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. Example 6: Merging

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45 BA 210 Lesson II.5 Simultaneous Quantity Competition The monopoly solution for inverse demand P = 10 (Q 1 +Q 2 ), has each firm using the lowest unit costs, c 1 = 1 and c 2 = 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. Example 6: Merging

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46 Merging thus increases total profit from 1 + 2 = = 19 to = 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 Example 6: Merging

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47 BA 210 Lesson II.5 Simultaneous Quantity CompetitionSummarySummary

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48 BA 210 Lesson II.5 Simultaneous Quantity CompetitionSummary Payoff table entry to any Duopoly Game with inverse demand P = a bQ and constant unit costs c 1 and c 2 : P = a b(Q 1 +Q 2 )P = a b(Q 1 +Q 2 ) Firm 1 profit 1 = (P c 1 ) Q 1Firm 1 profit 1 = (P c 1 ) Q 1 Firm 2 profit 2 = (P c 2 ) Q 2Firm 2 profit 2 = (P c 2 ) Q 2

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49 Review Questions BA 210 Lesson II.5 Simultaneous Quantity Competition 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.

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

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