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1 1 Deep Thought BA 445 Lesson B.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 BA 445 Lesson B.5 Simultaneous Quantity Competition ReadingsReadings Baye “Cournot Oligopoly” (see the index) Dixit Chapter 5

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3 3 BA 445 Lesson B.5 Simultaneous Quantity Competition OverviewOverview

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4 4Overview Cournot Duopoly has two firms controlling a large share of the market, and they compete by simultaneously setting their output (or output capacity). Then, price is determined by demand. Nash Equilibrium means each player makes a best response to the strategies of other players. It is thus a self-enforcing agreement. And it is the same as the dominance solution of a Cournot Duopoly. First Mover Advantage always occurs in a Stackelberg or Cournot duopoly. That advantage can make it profitable to rush to choose output sooner, even if that rush raises costs. Selling Technology to a Cournot competitor is profitable if total profit increases. In that case, there is a positive gain from the sales agreement, which is then divided according to rules of bargaining. Colluding with a Cournot competitor is almost always profitable. Since the competitors produce gross substitutes, profitable collusion lowers output. But, each cannot trust the other to collude.

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5 5 BA 445 Lesson B.5 Simultaneous Quantity Competition Example 1: Cournot Duopoly Example 1: Cournot Example 1: Cournot Duopoly

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6 6 BA 445 Lesson B.5 Simultaneous Quantity Competition Overview Cournot Duopoly has two firms controlling a large share of the market, and they compete by simultaneously setting their output (or output capacity). Then, price is determined by demand. Example 1: Cournot Example 1: Cournot Duopoly

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7 7 BA 445 Lesson B.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, with inverse market demand P = a-b(Q 1 +Q 2 ) if a-b(Q 1 +Q 2 ) > 0, and P = 0 otherwise.Strategies are outputs of homogeneous products, with inverse market demand P = a-b(Q 1 +Q 2 ) if a-b(Q 1 +Q 2 ) > 0, and P = 0 otherwise. 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 simultaneously or sequentially but in ignorance of the other’s choice.Each chooses either simultaneously or sequentially but in ignorance of the other’s choice. Payoffs are profits. When marginal costs or unit production costs of production are constants c 1 and c 2, then profits arePayoffs are profits. When marginal costs or 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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8 8 BA 445 Lesson B.5 Simultaneous Quantity Competition Question: Intel and AMD control a large share of the consumer desktop computer microprocessor market. They 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 = 1, 2, …, 9 and Q A = 1, 2, …, 9. What quantity should Intel produce? Example 1: Cournot Duopoly

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9 9 BA 445 Lesson B.5 Simultaneous Quantity Competition Answer: Intel’s quantities Q I = 1, 2, …, 9 are on the rows, and AMD’s quantities Q A = 1, 2, …, 9 are on the columns in the following normal form. For example, Q I = 2 and Q A = 3 generates price P = 20 5 = 15, and profits I = (15 1.1)2 = 27.8 and A = (15 1.1)3 = 41.7. (On an exam, I would provide most entries.) Example 1: Cournot Duopoly

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10 BA 445 Lesson B.5 Simultaneous Quantity Competition Strategies 1, 2, 3, and 4 are dominated for each player (by Strategy 5). Hence, eliminate those strategies, leaving the normal form: Example 1: Cournot Duopoly

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11 BA 445 Lesson B.5 Simultaneous Quantity Competition Strategies 8 and 9 are now dominated for each player (by Strategy 7). Hence, eliminate those strategies, leaving the normal form: Example 1: Cournot Duopoly

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12 BA 445 Lesson B.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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13 BA 445 Lesson B.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 = 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. Example 1: Cournot Duopoly

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14 BA 445 Lesson B.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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15 BA 445 Lesson B.5 Simultaneous Quantity Competition Example 2: Nash Equilibrium

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16 BA 445 Lesson B.5 Simultaneous Quantity Competition Overview Nash Equilibrium means each player makes a best response to the strategies of other players. It is thus a self- enforcing agreement. And it is the same as the dominance solution of a Cournot Duopoly. Example 2: Nash Equilibrium

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17 BA 445 Lesson B.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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18 BA 445 Lesson B.5 Simultaneous Quantity Competition Question: Coke and Pepsi control a large share of the soft drink market. Consumers find the two products to be indistinguishable. The inverse market demand for soft drinks is P = 3-Q (in dollars). You are a manager of Pepsi. Your unit cost of production is $2, and the unit cost of Coke is $1. Suppose you choose your output of soft drinks a few hours before Coke but Coke does not know your output before they decide their own output. How many soft drinks should you produce? Example 2: Nash Equilibrium

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19 BA 445 Lesson B.5 Simultaneous Quantity Competition Answer: You are Firm 1 in a Cournot Duopoly Game with inverse demand P = 3 (Q 1 +Q 2 ) and marginal costs c 1 = MC 1 = 2 and c 2 = MC 2 = 1. Find the Nash Equilibrium to the Cournot Duopoly Game (which turns out to be the dominance solution). Example 2: Nash Equilibrium

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20 BA 445 Lesson B.5 Simultaneous Quantity Competition Each firm correctly deduces the other firm’s output, so each firm chooses it’s output as a best response to the other firm’s output. Given Q 1, Firm 2 computes revenue and marginal revenue R 2 = (3 – (Q 1 + Q 2 )) Q 2 MR 2 = dR 2 /dQ 2 = 3 – Q 1 – 2Q 2 Hence, equate marginal cost to marginal revenue 1 = MC 2 = MR 2 = 3 – Q 1 – 2Q 2 to determine the optimal reaction Q 2 = r 2 (Q 1 ) = 1 –.5Q 1 Example 2: Nash Equilibrium

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21 BA 445 Lesson B.5 Simultaneous Quantity Competition Given Q 2, Firm 1 computes revenue and marginal revenue R 1 = (3 – (Q 1 + Q 2 )) Q 1 MR 1 = dR 1 /dQ 1 = 3 – 2Q 1 – Q 2 Hence, equate marginal cost to marginal revenue 2 = MC 1 = MR 1 = 3 – 2Q 1 – Q 2 to determine the optimal reaction Q 1 = r 1 (Q 2 ) =.5–.5Q 2 Example 2: Nash Equilibrium

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22 Complete solution for P = 3 (Q 1 +Q 2 ), MC 1 = 2, MC 2 = 1. Solve Q 2 = 1 –.5Q 1 and Q 1 =.5–.5Q 2 for Q 1 = 0 and Q 2 = 1Solve Q 2 = 1 –.5Q 1 and Q 1 =.5–.5Q 2 for Q 1 = 0 and Q 2 = 1 P = 3 (Q 1 +Q 2 ) = 2P = 3 (Q 1 +Q 2 ) = 2 Firm 1 profit 1 = (P c 1 ) Q 1 = (2 2)0 = 0Firm 1 profit 1 = (P c 1 ) Q 1 = (2 2)0 = 0 Firm 2 profit 2 = (P c 2 ) Q 2 = (2 1)1 = 1Firm 2 profit 2 = (P c 2 ) Q 2 = (2 1)1 = 1 BA 445 Lesson B.5 Simultaneous Quantity Competition Example 2: Nash Equilibrium

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23 BA 445 Lesson B.5 Simultaneous Quantity Competition Comment: Given any inverse demand P = a b(Q 1 +Q 2 ) Firm 1’s revenue and marginal revenue R 1 = (a – b(Q 1 + Q 2 )) Q 1 MR 1 = dR 1 /dQ 1 = a – 2bQ 1 – bQ 2 That is, MR 1 is the inverse demand P = a bQ 1 bQ 2 with double the coefficient of Q 1 Firm 2’s revenue and marginal revenue R 2 = (a – b(Q 1 + Q 2 )) Q 2 MR 2 = dR 2 /dQ 2 = a – bQ 1 – 2bQ 2 That is, MR 2 is the inverse demand P = a bQ 1 bQ 2 with double the coefficient of Q 2 Example 2: Nash Equilibrium

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24 BA 445 Lesson B.5 Simultaneous Quantity Competition Example 3: First Mover Advantage

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25 BA 445 Lesson B.5 Simultaneous Quantity Competition Overview First Mover Advantage always occurs in a Stackelberg or Cournot duopoly. That advantage can make it profitable to rush to choose output sooner, even if that rush raises costs. Example 3: First Mover Advantage

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26 BA 445 Lesson B.5 Simultaneous Quantity Competition Comment: If the unit production costs are the same c for two firms in a duopoly with inverse demand P = a – b(Q 1 +Q 2 ), then profits are i = (a – c) 2 /(9b) if Firm i is a Cournot competitor i = (a – c) 2 /(9b) if Firm i is a Cournot competitor 1 = (a – c) 2 /(8b) if Firm 1 is a Stackelberg leader 1 = (a – c) 2 /(8b) if Firm 1 is a Stackelberg leader 2 = (a – c) 2 /(16b) if Firm 2 is a Stackelberg follower 2 = (a – c) 2 /(16b) if Firm 2 is a Stackelberg follower So the Stackelberg leader has more profit than a Cournot competitor, who in turn has more profit than a Stackelberg follower. In particular, a firm can find it profitable to become the first mover or avoid being a follower by rushing to set up an assembly line, even if it means increasing marginal costs of production. Example 3: First Mover Advantage

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27 BA 445 Lesson B.5 Simultaneous Quantity Competition Question: PetroChina and Sinopec control a large share of Chinese oil production. The inverse market demand for Chinese oil is P = 3-Q (in yuan) and both firms produce at a unit cost of 1 yuan. You are a manager of PetroChina, and have a decision to make about competing with Sinopec in Siberia, where the inverse market demand for Chinese oil is P = 3-Q (in rubles). Option A. Sinopec sets up its refineries and distribution networks now, and you set up later. And both produce at a unit cost of 1 ruble. Option B. You hurry set up your refineries and distribution networks at the same time as Sinopec. Your hurry means your unit costs are 1.1 rubles, while Sinopec’s unit costs remain 1. Which Option is better for PetroChina? Example 3: First Mover Advantage

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28 BA 445 Lesson B.5 Simultaneous Quantity Competition Answer: In Option A, you are the follower in a Stackelberg Duopoly with inverse demand P = 3 (Q 1 +Q 2 ) and marginal costs c 1 = MC 1 = 1 and c 2 = MC 2 = 1. In Option B, you are Firm 1 in a Cournot Duopoly with inverse demand P = 3 (Q 1 +Q 2 ) and marginal costs c 1 = MC 1 = 1.1 and c 2 = MC 2 = 1. Example 3: First Mover Advantage

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29 BA 445 Lesson B.5 Simultaneous Quantity Competition Option A: Starting from the end, given Q 1, Firm 2 computes revenue and marginal revenue R 2 = (3 – (Q 1 + Q 2 )) Q 2 MR 2 = dR 2 /dQ 2 = 3 – Q 1 – 2Q 2 Hence, equate marginal cost to marginal revenue 1 = MC 2 = MR 2 = 3 – Q 1 – 2Q 2 to determine the optimal reaction Q 2 = r 2 (Q 1 ) = 1 –.5Q 1 Example 3: First Mover Advantage

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30 Rolling back to the beginning, The Stackelberg leader uses the reaction function r 2 (Q 1 ) to determine its revenueThe Stackelberg leader uses the reaction function r 2 (Q 1 ) to determine its revenue n R 1 = (3 – Q 1 – r 2 (Q 1 ) )) Q 1 n R 1 = (3 – Q 1 – (1 –.5Q 1 )) Q 1 n R 1 = (2 –.5Q 1 ) Q 1 and its profit-maximizing output level: n 1 = c 1 = dR 1 /dQ 1 n 1 = d/dQ 1 (2 –.5Q 1 ) Q 1 n 1 = 2 – Q 1 n Q 1 = 1 BA 445 Lesson B.5 Simultaneous Quantity Competition Example 3: First Mover Advantage

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31 Complete Stackelberg solution for c 1 = MC 1 = 1 and c 2 = MC 2 = 1: Q 1 = 1Q 1 = 1 Q 2 = r 2 (Q 1 ) = 1 –.5Q 1 = 1 –.5(1) =.5Q 2 = r 2 (Q 1 ) = 1 –.5Q 1 = 1 –.5(1) =.5 P = 3 (Q 1 +Q 2 ) = 1.5P = 3 (Q 1 +Q 2 ) = 1.5 Firm 1 profit 1 = (P c 1 ) Q 1 = (1.5 1)1 = 0.5Firm 1 profit 1 = (P c 1 ) Q 1 = (1.5 1)1 = 0.5 Firm 2 profit 2 = (P c 2 ) Q 2 = (1.5 1).5 = 0.25Firm 2 profit 2 = (P c 2 ) Q 2 = (1.5 1).5 = 0.25 BA 445 Lesson B.5 Simultaneous Quantity Competition Example 3: First Mover Advantage

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32 BA 445 Lesson B.5 Simultaneous Quantity Competition In Option B, you are Firm 1 in a Cournot Duopoly with inverse demand P = 3 (Q 1 +Q 2 ) and marginal costs c 1 = MC 1 = 1.1 and c 2 = MC 2 = 1. Find the Nash Equilibrium to the Cournot Duopoly Game (which turns out to be the dominance solution). Example 3: First Mover Advantage

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33 BA 445 Lesson B.5 Simultaneous Quantity Competition Each firm correctly deduces the other firm’s output, so each firm chooses it’s output as a best response to the other firm’s output. Given Q 1, Firm 2 computes revenue and marginal revenue R 2 = (3 – (Q 1 + Q 2 )) Q 2 MR 2 = dR 2 /dQ 2 = 3 – Q 1 – 2Q 2 Hence, equate marginal cost to marginal revenue 1 = MC 2 = MR 2 = 3 – Q 1 – 2Q 2 to determine the optimal reaction Q 2 = r 2 (Q 1 ) = 1 –.5Q 1 Example 3: First Mover Advantage

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34 BA 445 Lesson B.5 Simultaneous Quantity Competition Given Q 2, Firm 1 computes revenue and marginal revenue R 1 = (3 – (Q 1 + Q 2 )) Q 1 MR 1 = dR 1 /dQ 1 = 3 – 2Q 1 – Q 2 Hence, equate marginal cost to marginal revenue 1.1 = MC 1 = MR 1 = 3 – 2Q 1 – Q 2 to determine the optimal reaction Q 1 = r 1 (Q 2 ) =.95–.5Q 2 Example 3: First Mover Advantage

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35 Complete solution for P = 3 (Q 1 +Q 2 ), MC 1 = 1.1, MC 2 = 1. Solve Q 2 = 1 –.5Q 1 and Q 1 =.95–.5Q 2 for Q 1 =.6 and Q 2 =.7Solve Q 2 = 1 –.5Q 1 and Q 1 =.95–.5Q 2 for Q 1 =.6 and Q 2 =.7 P = 3 (Q 1 +Q 2 ) = 1.7P = 3 (Q 1 +Q 2 ) = 1.7 Firm 1 profit 1 = (P c 1 ) Q 1 = (1.7 1.1).6 = 0.36Firm 1 profit 1 = (P c 1 ) Q 1 = (1.7 1.1).6 = 0.36 Firm 2 profit 2 = (P c 2 ) Q 2 = (1.7 1).7 = 0.49Firm 2 profit 2 = (P c 2 ) Q 2 = (1.7 1).7 = 0.49 BA 445 Lesson B.5 Simultaneous Quantity Competition Example 3: First Mover Advantage

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36 Option B is thus best for PetroChina since PetroChina profits (as a Stackelberg follower) are 0.25 in Option A, while PetroChina profits (as a Cournot Duopolist) are 0.36 in Option B. BA 445 Lesson B.5 Simultaneous Quantity Competition Example 3: First Mover Advantage

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37 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 445 Lesson B.5 Simultaneous Quantity Competition Example 3: First Mover Advantage

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38 BA 445 Lesson B.5 Simultaneous Quantity Competition Example 4: Selling Technology

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39 BA 445 Lesson B.5 Simultaneous Quantity Competition Overview Selling Technology to a Cournot competitor is profitable if total profit increases. In that case, there is a positive gain from the sales agreement, which is then divided according to rules of bargaining. Example 4: Selling Technology

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40 BA 445 Lesson B.5 Simultaneous Quantity Competition Question: Nvidia and the ATI subsidiary of Advanced Micro Devices control a large share of the mainstream graphics card market. You are a manager of Nvidia, and 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 = 4 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? Example 4: Selling Technology

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41 BA 445 Lesson B.5 Simultaneous Quantity Competition Answer: If you do not sell your technology, you are Firm 1 in a Cournot Duopoly with inverse demand P = 4 (Q 1 +Q 2 ) and marginal costs are c 1 = MC 1 = 1 and c 2 = MC 2 = 2; if you do sell, marginal costs are c 1 = MC 1 = 1 and c 2 = MC 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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42 BA 445 Lesson B.5 Simultaneous Quantity Competition If Nvidia does not sell its technology, given Q 1, Firm 2 computes revenue and marginal revenue R 2 = (4 – (Q 1 + Q 2 )) Q 2 MR 2 = dR 2 /dQ 2 = 4 – Q 1 – 2Q 2 Hence, equate marginal cost to marginal revenue 2 = MC 2 = MR 2 = 4 – Q 1 – 2Q 2 to determine the optimal reaction Q 2 = r 2 (Q 1 ) = 1 –.5Q 1 Example 4: Selling Technology

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43 BA 445 Lesson B.5 Simultaneous Quantity Competition Given Q 2, Firm 1 computes revenue and marginal revenue R 1 = (4 – (Q 1 + Q 2 )) Q 1 MR 1 = dR 1 /dQ 1 = 4 – 2Q 1 – Q 2 Hence, equate marginal cost to marginal revenue 1 = MC 1 = MR 1 = 4 – 2Q 1 – Q 2 to determine the optimal reaction Q 1 = r 1 (Q 2 ) = 1.5–.5Q 2 Example 4: Selling Technology

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44 Complete solution for P = 4 (Q 1 +Q 2 ), MC 1 = 1, MC 2 = 2. Solve Q 2 = 1 –.5Q 1 and Q 1 = 1.5–.5Q 2 for Q 1 = 1 1/3 and Q 2 = 1/3Solve Q 2 = 1 –.5Q 1 and Q 1 = 1.5–.5Q 2 for Q 1 = 1 1/3 and Q 2 = 1/3 P = 4 (Q 1 +Q 2 ) = 2 1/3P = 4 (Q 1 +Q 2 ) = 2 1/3 Firm 1 profit 1 = (P c 1 ) Q 1 = (2 1/3 1)(1 1/3)= 1 7/9Firm 1 profit 1 = (P c 1 ) Q 1 = (2 1/3 1)(1 1/3)= 1 7/9 Firm 2 profit 2 = (P c 2 ) Q 2 = (2 1/3 2)(1/3) = 1/9Firm 2 profit 2 = (P c 2 ) Q 2 = (2 1/3 2)(1/3) = 1/9 BA 445 Lesson B.5 Simultaneous Quantity Competition Example 4: Selling Technology

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45 BA 445 Lesson B.5 Simultaneous Quantity Competition If Nvidia does sell its technology, given Q 1, Firm 2 computes revenue and marginal revenue R 2 = (4 – (Q 1 + Q 2 )) Q 2 MR 2 = dR 2 /dQ 2 = 4 – Q 1 – 2Q 2 Hence, equate marginal cost to marginal revenue 1 = MC 2 = MR 2 = 4 – Q 1 – 2Q 2 to determine the optimal reaction Q 2 = r 2 (Q 1 ) = 1.5 –.5Q 1 Example 4: Selling Technology

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46 BA 445 Lesson B.5 Simultaneous Quantity Competition Given Q 2, Firm 1 computes revenue and marginal revenue R 1 = (4 – (Q 1 + Q 2 )) Q 1 MR 1 = dR 1 /dQ 1 = 4 – 2Q 1 – Q 2 Hence, equate marginal cost to marginal revenue 1 = MC 1 = MR 1 = 4 – 2Q 1 – Q 2 to determine the optimal reaction Q 1 = r 1 (Q 2 ) = 1.5–.5Q 2 Example 4: Selling Technology

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47 Complete solution for P = 4 (Q 1 +Q 2 ), MC 1 = 1, MC 2 = 1. Solve Q 2 = 1.5 –.5Q 1 and Q 1 = 1.5–.5Q 2 for Q 1 = 1 and Q 2 = 1Solve Q 2 = 1.5 –.5Q 1 and Q 1 = 1.5–.5Q 2 for Q 1 = 1 and Q 2 = 1 P = 4 (Q 1 +Q 2 ) = 2P = 4 (Q 1 +Q 2 ) = 2 Firm 1 profit 1 = (P c 1 ) Q 1 = (2 1)1 = 1Firm 1 profit 1 = (P c 1 ) Q 1 = (2 1)1 = 1 Firm 2 profit 2 = (P c 2 ) Q 2 = (2 )1 = 1Firm 2 profit 2 = (P c 2 ) Q 2 = (2 )1 = 1 BA 445 Lesson B.5 Simultaneous Quantity Competition Example 4: Selling Technology

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48 Selling technology and reducing c 2 = 2 to c 2 = 1 has to effects: Firm 1’s profit reduces from 1 = 1 7/9 to 1 = 1Firm 1’s profit reduces from 1 = 1 7/9 to 1 = 1 Firm 2’s profit increases from 2 = 1/9 to 2 = 1Firm 2’s profit increases from 2 = 1/9 to 2 = 1 Nvidia should sell the technology because doing so increases total profit from production from 1 8/9 to 2, so there is 1/9 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 1/9 minus a pittance to ATI, then Nvidia captures most of those gains. BA 445 Lesson B.5 Simultaneous Quantity Competition Example 4: Selling Technology

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49 BA 445 Lesson B.5 Simultaneous Quantity Competition Example 5: Colluding

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50 BA 445 Lesson B.5 Simultaneous Quantity Competition Overview Colluding with a Cournot competitor is almost always profitable. Since the competitors produce gross substitutes, profitable collusion lowers output. But, each cannot trust the other to collude. Example 5: Colluding

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51 BA 445 Lesson B.5 Simultaneous Quantity Competition Question: TV Azteca and Televisa control a large share of the Mexican multimedia market. 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; TV Azteca’s costs are C 1 (Q 1 ) = Q 1 ; and Televisa’s costs are C 2 (Q 2 ) = Q 2. 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 to collude? Example 5: Colluding

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52 BA 445 Lesson B.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 ), C 1 (Q 1 ) = Q 1 and C 2 (Q 2 ) = Q 2, and marginal costs c 1 = MC 1 = 1 and c 2 = MC 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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53 BA 445 Lesson B.5 Simultaneous Quantity Competition Given Q 1, Firm 2 computes revenue and marginal revenue R 2 = (13 – (Q 1 + Q 2 )) Q 2 MR 2 = dR 2 /dQ 2 = 13 – Q 1 – 2Q 2 Hence, equate marginal cost to marginal revenue 1 = MC 2 = MR 2 = 13 – Q 1 – 2Q 2 to determine the optimal reaction Q 2 = r 2 (Q 1 ) = 6 –.5Q 1 Example 5: Colluding

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54 BA 445 Lesson B.5 Simultaneous Quantity Competition Given Q 2, Firm 1 computes revenue and marginal revenue R 1 = (13 – (Q 1 + Q 2 )) Q 1 MR 1 = dR 1 /dQ 1 = 13 – 2Q 1 – Q 2 Hence, equate marginal cost to marginal revenue 1 = MC 1 = MR 1 = 13 – 2Q 1 – Q 2 to determine the optimal reaction Q 1 = r 1 (Q 2 ) = 6 –.5Q 2 Example 5: Colluding

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55 Complete Nash equilibrium for non-colluding firms with P = 13 (Q 1 +Q 2 ), MC 1 = 1, MC 2 = 1: Solve Q 2 = 6 –.5Q 1 and Q 1 = 6 –.5Q 2 for Q 1 = 4 and Q 2 = 4Solve Q 2 = 6 –.5Q 1 and Q 1 = 6 –.5Q 2 for Q 1 = 4 and Q 2 = 4 P = 13 (Q 1 +Q 2 ) = 5P = 13 (Q 1 +Q 2 ) = 5 Firm 1 profit 1 = (P c 1 ) Q 1 = (5 1)4 = 16Firm 1 profit 1 = (P c 1 ) Q 1 = (5 1)4 = 16 Firm 2 profit 2 = (P c 2 ) Q 2 = (5 )4 = 16Firm 2 profit 2 = (P c 2 ) Q 2 = (5 )4 = 16 Collusive proposal of quantities Q 1 = 3 and Q 2 = 3: P = 13 (Q 1 +Q 2 ) = 7P = 13 (Q 1 +Q 2 ) = 7 Firm 1 profit 1 = (P c 1 ) Q 1 = (7 1)3 = 18Firm 1 profit 1 = (P c 1 ) Q 1 = (7 1)3 = 18 Firm 2 profit 2 = (P c 2 ) Q 2 = (7 1)3 = 18Firm 2 profit 2 = (P c 2 ) Q 2 = (7 1)3 = 18 BA 445 Lesson B.5 Simultaneous Quantity Competition Example 5: Colluding

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56 The collusive proposal of quantities Q 1 = 3 and Q 2 = 3 is thus mutually profitable for TV Azteca’s and Televisa. But TV Azteca cannot trust Televisa to collude since Televisa’s best response to TV Azteca’s Q 1 = 3 is Q 2 = r 2 (3) = 6 –.5(3) = 4.5, not the collusive proposal Q 2 = 3. BA 445 Lesson B.5 Simultaneous Quantity Competition Example 5: Colluding

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57 BA 445 Lesson B.5 Simultaneous Quantity Competition SummarySummary

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58 BA 445 Lesson B.5 Simultaneous Quantity Competition Summary Complete solution to a Cournot Duopoly Game with inverse demand P = a bQ and constant marginal costs c 1 = MC 1 and c 2 = MC 2 : Reaction Q 1 = r 1 (Q 2 ) = (a c 1 )/2b –.5Q 2Reaction Q 1 = r 1 (Q 2 ) = (a c 1 )/2b –.5Q 2 Reaction Q 2 = r 2 (Q 1 ) = (a c 2 )/2b –.5Q 1Reaction Q 2 = r 2 (Q 1 ) = (a c 2 )/2b –.5Q 1 Solution Q 1 = 2(a c 1 )/3b – (a c 2 )/3bSolution Q 1 = 2(a c 1 )/3b – (a c 2 )/3b Solution Q 2 = 2(a c 2 )/3b – (a c 1 )/3bSolution Q 2 = 2(a c 2 )/3b – (a c 1 )/3b 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 Tip: Use those formulas to double check your computations. However, computations as in the answers to Examples 1 through 5 are required for full credit on exam and homework questions.

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59 Review Questions BA 445 Lesson B.5 Simultaneous Quantity Competition Review Questions You should try to answer some of the review questions (see the online syllabus) 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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60 End of Lesson B.5 BA 445 Managerial Economics BA 445 Lesson B.5 Simultaneous Quantity Competition

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