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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 445 Lesson B.5 Simultaneous Quantity Competition 1 Readings Readings Baye “Cournot Oligopoly” (see the index) Dixit Chapter 5 BA 445 Lesson B.5 Simultaneous Quantity Competition 2 Overview Overview BA 445 Lesson B.5 Simultaneous Quantity Competition 3 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. 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. BA 445 Lesson B.5 Simultaneous Quantity Competition 4 Example 1: Cournot Duopoly Example 1: Cournot Duopoly BA 445 Lesson B.5 Simultaneous Quantity Competition 5 Example 1: Cournot Duopoly 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. BA 445 Lesson B.5 Simultaneous Quantity Competition 6 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, with inverse market demand P = a-b(Q1+Q2) if a-b(Q1+Q2) > 0, and P = 0 otherwise. • Firm 1 chooses output Q1 > 0. • Firm 2 chooses output Q2 > 0. • 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 c1 and c2, then profits are P1 = (P- c1)Q1 and P2 = (P- c2)Q2 BA 445 Lesson B.5 Simultaneous Quantity Competition 7 Example 1: Cournot Duopoly 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 – (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 = 1, 2, …, 9 and QA = 1, 2, …, 9. What quantity should Intel produce? BA 445 Lesson B.5 Simultaneous Quantity Competition 8 Example 1: Cournot Duopoly Answer: Intel’s quantities QI = 1, 2, …, 9 are on the rows, and AMD’s quantities QA = 1, 2, …, 9 are on the columns in the following normal form. For example, QI = 2 and QA = 3 generates price P = 20-5 = 15, and profits PI = (15-1.1)2 = 27.8 and PA = (15-1.1)3 = 41.7. (On an exam, I would provide most entries.) 20 Row 1.1 1.1 1 2 3 4 5 6 7 8 9 Column 1 16.9,16.9 31.8,15.9 44.7,14.9 55.6,13.9 64.5,12.9 71.4,11.9 76.3,10.9 79.2,9.9 80.1,8.9 2 15.9,31.8 29.8,29.8 41.7,27.8 51.6,25.8 59.5,23.8 65.4,21.8 69.3,19.8 71.2,17.8 71.1,15.8 3 14.9,44.7 27.8,41.7 38.7,38.7 47.6,35.7 54.5,32.7 59.4,29.7 62.3,26.7 63.2,23.7 62.1,20.7 4 13.9,55.6 25.8,51.6 35.7,47.6 43.6,43.6 49.5,39.6 53.4,35.6 55.3,31.6 55.2,27.6 53.1,23.6 5 12.9,64.5 23.8,59.5 32.7,54.5 39.6,49.5 44.5,44.5 47.4,39.5 48.3,34.5 47.2,29.5 44.1,24.5 6 11.9,71.4 21.8,65.4 29.7,59.4 35.6,53.4 39.5,47.4 41.4,41.4 41.3,35.4 39.2,29.4 35.1,23.4 7 10.9,76.3 19.8,69.3 26.7,62.3 31.6,55.3 34.5,48.3 35.4,41.3 34.3,34.3 31.2,27.3 26.1,20.3 8 9.9,79.2 17.8,71.2 23.7,63.2 27.6,55.2 29.5,47.2 29.4,39.2 27.3,31.2 23.2,23.2 17.1,15.2 BA 445 Lesson B.5 Simultaneous Quantity Competition 9 8.9,80.1 15.8,71.1 20.7,62.1 23.6,53.1 24.5,44.1 23.4,35.1 20.3,26.1 15.2,17.1 8.1,8.1 9 Example 1: Cournot Duopoly Strategies 1, 2, 3, and 4 are dominated for each player (by Strategy 5). Hence, eliminate those strategies, leaving the normal form: 20 Row 1.1 1.1 1 2 3 4 5 6 7 8 9 Column 1 16.9,16.9 31.8,15.9 44.7,14.9 55.6,13.9 64.5,12.9 71.4,11.9 76.3,10.9 79.2,9.9 80.1,8.9 2 15.9,31.8 29.8,29.8 41.7,27.8 51.6,25.8 59.5,23.8 65.4,21.8 69.3,19.8 71.2,17.8 71.1,15.8 3 14.9,44.7 27.8,41.7 38.7,38.7 47.6,35.7 54.5,32.7 59.4,29.7 62.3,26.7 63.2,23.7 62.1,20.7 4 13.9,55.6 25.8,51.6 35.7,47.6 43.6,43.6 49.5,39.6 53.4,35.6 55.3,31.6 55.2,27.6 53.1,23.6 5 12.9,64.5 23.8,59.5 32.7,54.5 39.6,49.5 44.5,44.5 47.4,39.5 48.3,34.5 47.2,29.5 44.1,24.5 6 11.9,71.4 21.8,65.4 29.7,59.4 35.6,53.4 39.5,47.4 41.4,41.4 41.3,35.4 39.2,29.4 35.1,23.4 7 10.9,76.3 19.8,69.3 26.7,62.3 31.6,55.3 34.5,48.3 35.4,41.3 34.3,34.3 31.2,27.3 26.1,20.3 8 9.9,79.2 17.8,71.2 23.7,63.2 27.6,55.2 29.5,47.2 29.4,39.2 27.3,31.2 23.2,23.2 17.1,15.2 BA 445 Lesson B.5 Simultaneous Quantity Competition 9 8.9,80.1 15.8,71.1 20.7,62.1 23.6,53.1 24.5,44.1 23.4,35.1 20.3,26.1 15.2,17.1 8.1,8.1 10 Example 1: Cournot Duopoly Strategies 8 and 9 are now dominated for each player (by Strategy 7). Hence, eliminate those strategies, leaving the normal form: 20 Row 1.1 1.1 1 2 3 4 5 6 7 8 9 Column 1 16.9,16.9 31.8,15.9 44.7,14.9 55.6,13.9 64.5,12.9 71.4,11.9 76.3,10.9 79.2,9.9 80.1,8.9 2 15.9,31.8 29.8,29.8 41.7,27.8 51.6,25.8 59.5,23.8 65.4,21.8 69.3,19.8 71.2,17.8 71.1,15.8 3 14.9,44.7 27.8,41.7 38.7,38.7 47.6,35.7 54.5,32.7 59.4,29.7 62.3,26.7 63.2,23.7 62.1,20.7 4 13.9,55.6 25.8,51.6 35.7,47.6 43.6,43.6 49.5,39.6 53.4,35.6 55.3,31.6 55.2,27.6 53.1,23.6 5 12.9,64.5 23.8,59.5 32.7,54.5 39.6,49.5 44.5,44.5 47.4,39.5 48.3,34.5 47.2,29.5 44.1,24.5 6 11.9,71.4 21.8,65.4 29.7,59.4 35.6,53.4 39.5,47.4 41.4,41.4 41.3,35.4 39.2,29.4 35.1,23.4 7 10.9,76.3 19.8,69.3 26.7,62.3 31.6,55.3 34.5,48.3 35.4,41.3 34.3,34.3 31.2,27.3 26.1,20.3 8 9.9,79.2 17.8,71.2 23.7,63.2 27.6,55.2 29.5,47.2 29.4,39.2 27.3,31.2 23.2,23.2 17.1,15.2 BA 445 Lesson B.5 Simultaneous Quantity Competition 9 8.9,80.1 15.8,71.1 20.7,62.1 23.6,53.1 24.5,44.1 23.4,35.1 20.3,26.1 15.2,17.1 8.1,8.1 11 Example 1: Cournot Duopoly Strategy 5 is now dominated for each player (by Strategy 6). Hence, eliminate that strategy, leaving the normal form: 20 Row 1.1 1.1 1 2 3 4 5 6 7 8 9 Column 1 16.9,16.9 31.8,15.9 44.7,14.9 55.6,13.9 64.5,12.9 71.4,11.9 76.3,10.9 79.2,9.9 80.1,8.9 2 15.9,31.8 29.8,29.8 41.7,27.8 51.6,25.8 59.5,23.8 65.4,21.8 69.3,19.8 71.2,17.8 71.1,15.8 3 14.9,44.7 27.8,41.7 38.7,38.7 47.6,35.7 54.5,32.7 59.4,29.7 62.3,26.7 63.2,23.7 62.1,20.7 4 13.9,55.6 25.8,51.6 35.7,47.6 43.6,43.6 49.5,39.6 53.4,35.6 55.3,31.6 55.2,27.6 53.1,23.6 5 12.9,64.5 23.8,59.5 32.7,54.5 39.6,49.5 44.5,44.5 47.4,39.5 48.3,34.5 47.2,29.5 44.1,24.5 6 11.9,71.4 21.8,65.4 29.7,59.4 35.6,53.4 39.5,47.4 41.4,41.4 41.3,35.4 39.2,29.4 35.1,23.4 7 10.9,76.3 19.8,69.3 26.7,62.3 31.6,55.3 34.5,48.3 35.4,41.3 34.3,34.3 31.2,27.3 26.1,20.3 8 9.9,79.2 17.8,71.2 23.7,63.2 27.6,55.2 29.5,47.2 29.4,39.2 27.3,31.2 23.2,23.2 17.1,15.2 BA 445 Lesson B.5 Simultaneous Quantity Competition 9 8.9,80.1 15.8,71.1 20.7,62.1 23.6,53.1 24.5,44.1 23.4,35.1 20.3,26.1 15.2,17.1 8.1,8.1 12 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. 20 Row 1.1 1.1 1 2 3 4 5 6 7 8 9 Column 1 16.9,16.9 31.8,15.9 44.7,14.9 55.6,13.9 64.5,12.9 71.4,11.9 76.3,10.9 79.2,9.9 80.1,8.9 2 15.9,31.8 29.8,29.8 41.7,27.8 51.6,25.8 59.5,23.8 65.4,21.8 69.3,19.8 71.2,17.8 71.1,15.8 3 14.9,44.7 27.8,41.7 38.7,38.7 47.6,35.7 54.5,32.7 59.4,29.7 62.3,26.7 63.2,23.7 62.1,20.7 4 13.9,55.6 25.8,51.6 35.7,47.6 43.6,43.6 49.5,39.6 53.4,35.6 55.3,31.6 55.2,27.6 53.1,23.6 5 12.9,64.5 23.8,59.5 32.7,54.5 39.6,49.5 44.5,44.5 47.4,39.5 48.3,34.5 47.2,29.5 44.1,24.5 6 11.9,71.4 21.8,65.4 29.7,59.4 35.6,53.4 39.5,47.4 41.4,41.4 41.3,35.4 39.2,29.4 35.1,23.4 7 10.9,76.3 19.8,69.3 26.7,62.3 31.6,55.3 34.5,48.3 35.4,41.3 34.3,34.3 31.2,27.3 26.1,20.3 8 9.9,79.2 17.8,71.2 23.7,63.2 27.6,55.2 29.5,47.2 29.4,39.2 27.3,31.2 23.2,23.2 17.1,15.2 BA 445 Lesson B.5 Simultaneous Quantity Competition 9 8.9,80.1 15.8,71.1 20.7,62.1 23.6,53.1 24.5,44.1 23.4,35.1 20.3,26.1 15.2,17.1 8.1,8.1 13 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. 20 Row 1.1 1.1 1 2 3 4 5 6 7 8 9 Column 1 16.9,16.9 31.8,15.9 44.7,14.9 55.6,13.9 64.5,12.9 71.4,11.9 76.3,10.9 79.2,9.9 80.1,8.9 2 15.9,31.8 29.8,29.8 41.7,27.8 51.6,25.8 59.5,23.8 65.4,21.8 69.3,19.8 71.2,17.8 71.1,15.8 3 14.9,44.7 27.8,41.7 38.7,38.7 47.6,35.7 54.5,32.7 59.4,29.7 62.3,26.7 63.2,23.7 62.1,20.7 4 13.9,55.6 25.8,51.6 35.7,47.6 43.6,43.6 49.5,39.6 53.4,35.6 55.3,31.6 55.2,27.6 53.1,23.6 5 12.9,64.5 23.8,59.5 32.7,54.5 39.6,49.5 44.5,44.5 47.4,39.5 48.3,34.5 47.2,29.5 44.1,24.5 6 11.9,71.4 21.8,65.4 29.7,59.4 35.6,53.4 39.5,47.4 41.4,41.4 41.3,35.4 39.2,29.4 35.1,23.4 7 10.9,76.3 19.8,69.3 26.7,62.3 31.6,55.3 34.5,48.3 35.4,41.3 34.3,34.3 31.2,27.3 26.1,20.3 8 9.9,79.2 17.8,71.2 23.7,63.2 27.6,55.2 29.5,47.2 29.4,39.2 27.3,31.2 23.2,23.2 17.1,15.2 BA 445 Lesson B.5 Simultaneous Quantity Competition 9 8.9,80.1 15.8,71.1 20.7,62.1 23.6,53.1 24.5,44.1 23.4,35.1 20.3,26.1 15.2,17.1 8.1,8.1 14 Example 2: Nash Equilibrium Example 2: Nash Equilibrium BA 445 Lesson B.5 Simultaneous Quantity Competition 15 Example 2: Nash Equilibrium Overview Nash Equilibrium means each player makes a best response to the strategies of other players. It is thus a selfenforcing agreement. And it is the same as the dominance solution of a Cournot Duopoly. BA 445 Lesson B.5 Simultaneous Quantity Competition 16 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 445 Lesson B.5 Simultaneous Quantity Competition 17 Example 2: Nash Equilibrium 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? BA 445 Lesson B.5 Simultaneous Quantity Competition 18 Example 2: Nash Equilibrium Answer: You are Firm 1 in a Cournot Duopoly Game with inverse demand P = 3 - (Q1+Q2) and marginal costs c1 = MC1 = 2 and c2 = MC2 = 1. Find the Nash Equilibrium to the Cournot Duopoly Game (which turns out to be the dominance solution). BA 445 Lesson B.5 Simultaneous Quantity Competition 19 Example 2: Nash Equilibrium 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 Q1, Firm 2 computes revenue and marginal revenue R2 = (3 – (Q1 + Q2)) Q2 MR2 = dR2 /dQ2 = 3 – Q1 – 2Q2 Hence, equate marginal cost to marginal revenue 1 = MC2 = MR2 = 3 – Q1 – 2Q2 to determine the optimal reaction Q2 = r2 (Q1) = 1 – .5Q1 BA 445 Lesson B.5 Simultaneous Quantity Competition 20 Example 2: Nash Equilibrium Given Q2, Firm 1 computes revenue and marginal revenue R1 = (3 – (Q1 + Q2)) Q1 MR1 = dR1 /dQ1 = 3 – 2Q1 – Q2 Hence, equate marginal cost to marginal revenue 2 = MC1 = MR1 = 3 – 2Q1 – Q2 to determine the optimal reaction Q1 = r1 (Q2) = .5– .5Q2 BA 445 Lesson B.5 Simultaneous Quantity Competition 21 Example 2: Nash Equilibrium Complete solution for P = 3 - (Q1+Q2), MC1 = 2, MC2 = 1. • Solve Q2 = 1 – .5Q1 and Q1 = .5– .5Q2 for Q1 = 0 and Q2 =1 • P = 3 - (Q1+Q2) = 2 • Firm 1 profit P1 = (P - c1) Q1 = (2 - 2)0 = 0 • Firm 2 profit P2 = (P - c2) Q2 = (2 - 1)1 = 1 BA 445 Lesson B.5 Simultaneous Quantity Competition 22 Example 2: Nash Equilibrium Comment: Given any inverse demand P = a - b(Q1+Q2) Firm 1’s revenue and marginal revenue R1 = (a – b(Q1 + Q2)) Q1 MR1 = dR1 /dQ1 = a – 2bQ1 – bQ2 That is, MR1 is the inverse demand P = a - bQ1 - bQ2 with double the coefficient of Q1 Firm 2’s revenue and marginal revenue R2 = (a – b(Q1 + Q2)) Q2 MR2 = dR2 /dQ2 = a – bQ1 – 2bQ2 That is, MR2 is the inverse demand P = a - bQ1 - bQ2 with double the coefficient of Q2 BA 445 Lesson B.5 Simultaneous Quantity Competition 23 Example 3: First Mover Advantage Example 3: First Mover Advantage BA 445 Lesson B.5 Simultaneous Quantity Competition 24 Example 3: First Mover Advantage 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. BA 445 Lesson B.5 Simultaneous Quantity Competition 25 Example 3: First Mover Advantage Comment: If the unit production costs are the same c for two firms in a duopoly with inverse demand P = a – b(Q1+Q2), then profits are • Pi = (a – c)2/(9b) if Firm i is a Cournot competitor • P1 = (a – c)2/(8b) if Firm 1 is a Stackelberg leader • P2 = (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. BA 445 Lesson B.5 Simultaneous Quantity Competition 26 Example 3: First Mover Advantage 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? BA 445 Lesson B.5 Simultaneous Quantity Competition 27 Example 3: First Mover Advantage Answer: In Option A, you are the follower in a Stackelberg Duopoly with inverse demand P = 3 - (Q1+Q2) and marginal costs c1 = MC1 = 1 and c2 = MC2 = 1. In Option B, you are Firm 1 in a Cournot Duopoly with inverse demand P = 3 - (Q1+Q2) and marginal costs c1 = MC1 = 1.1 and c2 = MC2 = 1. BA 445 Lesson B.5 Simultaneous Quantity Competition 28 Example 3: First Mover Advantage Option A: Starting from the end, given Q1, Firm 2 computes revenue and marginal revenue R2 = (3 – (Q1 + Q2)) Q2 MR2 = dR2 /dQ2 = 3 – Q1 – 2Q2 Hence, equate marginal cost to marginal revenue 1 = MC2 = MR2 = 3 – Q1 – 2Q2 to determine the optimal reaction Q2 = r2 (Q1) = 1 – .5Q1 BA 445 Lesson B.5 Simultaneous Quantity Competition 29 Example 3: First Mover Advantage Rolling back to the beginning, • The Stackelberg leader uses the reaction function r2 (Q1) to determine its revenue R1 = (3 – Q1 – r2 (Q1) )) Q1 R1 = (3 – Q1 – (1 – .5Q1)) Q1 R1 = (2 – .5Q1) Q1 and its profit-maximizing output level: 1 = c1 = dR1/dQ1 1 = d/dQ1 (2 – .5Q1) Q1 1 = 2 – Q1 Q1 = 1 BA 445 Lesson B.5 Simultaneous Quantity Competition 30 Example 3: First Mover Advantage Complete Stackelberg solution for c1 = MC1 = 1 and c2 = MC2 = 1: • Q1 = 1 • Q2 = r2 (Q1) = 1 – .5Q1 = 1 – .5(1) = .5 • P = 3 - (Q1+Q2) = 1.5 • Firm 1 profit P1 = (P - c1) Q1 = (1.5 - 1)1 = 0.5 • Firm 2 profit P2 = (P - c2) Q2 = (1.5 - 1).5 = 0.25 BA 445 Lesson B.5 Simultaneous Quantity Competition 31 Example 3: First Mover Advantage In Option B, you are Firm 1 in a Cournot Duopoly with inverse demand P = 3 - (Q1+Q2) and marginal costs c1 = MC1 = 1.1 and c2 = MC2 = 1. Find the Nash Equilibrium to the Cournot Duopoly Game (which turns out to be the dominance solution). BA 445 Lesson B.5 Simultaneous Quantity Competition 32 Example 3: First Mover Advantage 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 Q1, Firm 2 computes revenue and marginal revenue R2 = (3 – (Q1 + Q2)) Q2 MR2 = dR2 /dQ2 = 3 – Q1 – 2Q2 Hence, equate marginal cost to marginal revenue 1 = MC2 = MR2 = 3 – Q1 – 2Q2 to determine the optimal reaction Q2 = r2 (Q1) = 1 – .5Q1 BA 445 Lesson B.5 Simultaneous Quantity Competition 33 Example 3: First Mover Advantage Given Q2, Firm 1 computes revenue and marginal revenue R1 = (3 – (Q1 + Q2)) Q1 MR1 = dR1 /dQ1 = 3 – 2Q1 – Q2 Hence, equate marginal cost to marginal revenue 1.1 = MC1 = MR1 = 3 – 2Q1 – Q2 to determine the optimal reaction Q1 = r1 (Q2) = .95– .5Q2 BA 445 Lesson B.5 Simultaneous Quantity Competition 34 Example 3: First Mover Advantage Complete solution for P = 3 - (Q1+Q2), MC1 = 1.1, MC2 = 1. • Solve Q2 = 1 – .5Q1 and Q1 = .95– .5Q2 for Q1 = .6 and Q2 = .7 • P = 3 - (Q1+Q2) = 1.7 • Firm 1 profit P1 = (P - c1) Q1 = (1.7 - 1.1).6 = 0.36 • Firm 2 profit P2 = (P - c2) Q2 = (1.7 - 1).7 = 0.49 BA 445 Lesson B.5 Simultaneous Quantity Competition 35 Example 3: First Mover Advantage 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 36 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 445 Lesson B.5 Simultaneous Quantity Competition 37 Example 4: Selling Technology Example 4: Selling Technology BA 445 Lesson B.5 Simultaneous Quantity Competition 38 Example 4: Selling Technology 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. BA 445 Lesson B.5 Simultaneous Quantity Competition 39 Example 4: Selling Technology 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? BA 445 Lesson B.5 Simultaneous Quantity Competition 40 Example 4: Selling Technology Answer: If you do not sell your technology, you are Firm 1 in a Cournot Duopoly with inverse demand P = 4 - (Q1+Q2) and marginal costs are c1 = MC1 = 1 and c2 = MC2 = 2; if you do sell, marginal costs are c1 = MC1 = 1 and c2 = MC2 = 1. Find the Nash Equilibrium to each Cournot Duopoly Game (which turns out to be the dominance solution). BA 445 Lesson B.5 Simultaneous Quantity Competition 41 Example 4: Selling Technology If Nvidia does not sell its technology, given Q1, Firm 2 computes revenue and marginal revenue R2 = (4 – (Q1 + Q2)) Q2 MR2 = dR2 /dQ2 = 4 – Q1 – 2Q2 Hence, equate marginal cost to marginal revenue 2 = MC2 = MR2 = 4 – Q1 – 2Q2 to determine the optimal reaction Q2 = r2 (Q1) = 1 – .5Q1 BA 445 Lesson B.5 Simultaneous Quantity Competition 42 Example 4: Selling Technology Given Q2, Firm 1 computes revenue and marginal revenue R1 = (4 – (Q1 + Q2)) Q1 MR1 = dR1 /dQ1 = 4 – 2Q1 – Q2 Hence, equate marginal cost to marginal revenue 1 = MC1 = MR1 = 4 – 2Q1 – Q2 to determine the optimal reaction Q1 = r1 (Q2) = 1.5– .5Q2 BA 445 Lesson B.5 Simultaneous Quantity Competition 43 Example 4: Selling Technology Complete solution for P = 4 - (Q1+Q2), MC1 = 1, MC2 = 2. • Solve Q2 = 1 – .5Q1 and Q1 = 1.5– .5Q2 for Q1 = 1 1/3 and Q2 = 1/3 • P = 4 - (Q1+Q2) = 2 1/3 • Firm 1 profit P1 = (P - c1) Q1 = (2 1/3 - 1)(1 1/3)= 1 7/9 • Firm 2 profit P2 = (P - c2) Q2 = (2 1/3 - 2)(1/3) = 1/9 BA 445 Lesson B.5 Simultaneous Quantity Competition 44 Example 4: Selling Technology If Nvidia does sell its technology, given Q1, Firm 2 computes revenue and marginal revenue R2 = (4 – (Q1 + Q2)) Q2 MR2 = dR2 /dQ2 = 4 – Q1 – 2Q2 Hence, equate marginal cost to marginal revenue 1 = MC2 = MR2 = 4 – Q1 – 2Q2 to determine the optimal reaction Q2 = r2 (Q1) = 1.5 – .5Q1 BA 445 Lesson B.5 Simultaneous Quantity Competition 45 Example 4: Selling Technology Given Q2, Firm 1 computes revenue and marginal revenue R1 = (4 – (Q1 + Q2)) Q1 MR1 = dR1 /dQ1 = 4 – 2Q1 – Q2 Hence, equate marginal cost to marginal revenue 1 = MC1 = MR1 = 4 – 2Q1 – Q2 to determine the optimal reaction Q1 = r1 (Q2) = 1.5– .5Q2 BA 445 Lesson B.5 Simultaneous Quantity Competition 46 Example 4: Selling Technology Complete solution for P = 4 - (Q1+Q2), MC1 = 1, MC2 = 1. • Solve Q2 = 1.5 – .5Q1 and Q1 = 1.5– .5Q2 for Q1 = 1 and Q2 = 1 • P = 4 - (Q1+Q2) = 2 • Firm 1 profit P1 = (P - c1) Q1 = (2 - 1)1 = 1 • Firm 2 profit P2 = (P - c2) Q2 = (2 - 1)1 = 1 BA 445 Lesson B.5 Simultaneous Quantity Competition 47 Example 4: Selling Technology Selling technology and reducing c2 = 2 to c2 = 1 has to effects: • Firm 1’s profit reduces from P1 = 1 7/9 to P1 = 1 • Firm 2’s profit increases from P2 = 1/9 to P2 = 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-itor-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 48 Example 5: Colluding Example 5: Colluding BA 445 Lesson B.5 Simultaneous Quantity Competition 49 Example 5: Colluding 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. BA 445 Lesson B.5 Simultaneous Quantity Competition 50 Example 5: Colluding 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 C1(Q1) = Q1; and Televisa’s costs are C2(Q2) = Q2. 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? BA 445 Lesson B.5 Simultaneous Quantity Competition 51 Example 5: Colluding Answer: You are Firm 1 in a Cournot Duopoly with demand Q = 13 - P, inverse demand P = 13- (Q1+Q2), C1 (Q1) = Q1 and C2 (Q2) = Q2, and marginal costs c1 = MC1 = 1 and c2 = MC2 = 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 445 Lesson B.5 Simultaneous Quantity Competition 52 Example 5: Colluding Given Q1, Firm 2 computes revenue and marginal revenue R2 = (13 – (Q1 + Q2)) Q2 MR2 = dR2 /dQ2 = 13 – Q1 – 2Q2 Hence, equate marginal cost to marginal revenue 1 = MC2 = MR2 = 13 – Q1 – 2Q2 to determine the optimal reaction Q2 = r2 (Q1) = 6 – .5Q1 BA 445 Lesson B.5 Simultaneous Quantity Competition 53 Example 5: Colluding Given Q2, Firm 1 computes revenue and marginal revenue R1 = (13 – (Q1 + Q2)) Q1 MR1 = dR1 /dQ1 = 13 – 2Q1 – Q2 Hence, equate marginal cost to marginal revenue 1 = MC1 = MR1 = 13 – 2Q1 – Q2 to determine the optimal reaction Q1 = r1 (Q2) = 6 – .5Q2 BA 445 Lesson B.5 Simultaneous Quantity Competition 54 Example 5: Colluding Complete Nash equilibrium for non-colluding firms with P = 13 - (Q1+Q2), MC1 = 1, MC2 = 1: • Solve Q2 = 6 – .5Q1 and Q1 = 6 – .5Q2 for Q1 = 4 and Q2 =4 • P = 13 - (Q1+Q2) = 5 • Firm 1 profit P1 = (P - c1) Q1 = (5 - 1)4 = 16 • Firm 2 profit P2 = (P - c2) Q2 = (5 - 1)4 = 16 Collusive proposal of quantities Q1 = 3 and Q2 = 3: • P = 13 - (Q1+Q2) = 7 • Firm 1 profit P1 = (P - c1) Q1 = (7 - 1)3 = 18 • Firm 2 profit P2 = (P - c2) Q2 = (7 - 1)3 = 18 BA 445 Lesson B.5 Simultaneous Quantity Competition 55 Example 5: Colluding The collusive proposal of quantities Q1 = 3 and Q2 = 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 Q1 = 3 is Q2 = r2 (3) = 6 – .5(3) = 4.5, not the collusive proposal Q2 = 3. BA 445 Lesson B.5 Simultaneous Quantity Competition 56 Summary Summary BA 445 Lesson B.5 Simultaneous Quantity Competition 57 Summary Complete solution to a Cournot Duopoly Game with inverse demand P = a - bQ and constant marginal costs c1 = MC1 and c2 = MC2: • Reaction Q1 = r1 (Q2) = (a - c1)/2b – .5Q2 • Reaction Q2 = r2 (Q1) = (a - c2)/2b – .5Q1 • Solution Q1 = 2(a - c1)/3b – (a - c2)/3b • Solution Q2 = 2(a - c2)/3b – (a - c1)/3b • P = a - b(Q1+Q2) • Firm 1 profit P1 = (P - c1) Q1 • Firm 2 profit P2 = (P - c2) Q2 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. BA 445 Lesson B.5 Simultaneous Quantity Competition 58 Review Questions 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. BA 445 Lesson B.5 Simultaneous Quantity Competition 59 BA 445 Managerial Economics End of Lesson B.5 BA 445 Lesson B.5 Simultaneous Quantity Competition 60