1. 1 1 Deep Thought BA 445 Lesson B.3 Sequential Quantity Competition I love going down to the elementary school, watching all the kids jump and shout, but they don’t know I’m using blanks. ~ Jack Handey. (Translation: Today’s lesson teaches when it is important to you that your opponents know your actions so you can manipulate their reactions.)

2. 2 2 BA 445 Lesson B.3 Sequential Quantity Competition ReadingsReadings Baye “Stackelberg Oligopoly” (see the index) Dixit Chapter 3

3. 3 3 BA 445 Lesson B.3 Sequential Quantity Competition OverviewOverview

4. 4 4Overview Stackelberg Duopoly has two firms controlling a large share of the market, and they compete by one firm first setting its output (or output capacity). Then, the other firm, then price is determined by demand. First Mover Advantage always occurs in the rollback solution to a Stackelberg duopoly. That advantage can make it profitable to rush to choose output first, even if that rush raises costs. Selling Technology to a Stackelberg 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 Stackelberg competitor is almost always profitable. Since the competitors produce gross substitutes, profitable collusion lowers output. But, the leader cannot trust the follower to collude.

5. 5 5 BA 445 Lesson B.3 Sequential Quantity Competition Example 1: Stackelberg Duopoly Example 1: Example 1: Stackelberg Duopoly

6. 6 6 BA 445 Lesson B.3 Sequential Quantity Competition Overview Stackelberg Duopoly has two firms controlling a large share of the market, and they compete by one firm first setting its output (or output capacity). Then, the other firm, then price is determined by demand. Example 1: Example 1: Stackelberg Duopoly

7. 7 7 BA 445 Lesson B.3 Sequential Quantity Competition Comment: Stackelberg Duopoly Games have three parts. Players are managers of two firms serving many consumers.Players are managers of two firms serving many consumers. Firm 1 is the leader, and acts before Firm 2, the follower.Firm 1 is the leader, and acts before Firm 2, the follower. 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 knows Firm 1’s Q 1 > 0 before he chooses his own.Firm 2 knows Firm 1’s Q 1 > 0 before he chooses his own. Firm 2’s strategy is thus an output Q 2 reaction function Q 2 = r 2 (Q 1 ) to Firm 1’s choice Q 1.Firm 2’s strategy is thus an output Q 2 reaction function Q 2 = r 2 (Q 1 ) to Firm 1’s choice Q 1. 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: Example 1: Stackelberg Duopoly

8. 8 8 BA 445 Lesson B.3 Sequential Quantity Competition Question: You are the manager of Marvel Comics and you compete directly with DC Comics selling comic books. Consumers find the two products to be indistinguishable. The inverse market demand for comic books is P = 5-Q (in dollars). Your marginal costs of production are $2, and the marginal costs of DC Comics are $1. Suppose you choose your output of comic books before DC Comics, and DC Comics knows your output before they decide their own output. How many comic books should you produce? Example 1: Example 1: Stackelberg Duopoly

9. 9 9 BA 445 Lesson B.3 Sequential Quantity Competition Answer: You are the leader in a Stackelberg Duopoly Game with inverse demand P = 5  (Q 1 +Q 2 ) and marginal costs c 1 = MC 1 = 2 and c 2 = MC 2 = 1. Find the rollback solution to the Stackelberg Duopoly Game. Example 1: Example 1: Stackelberg Duopoly

10. 10 BA 445 Lesson B.3 Sequential Quantity Competition Starting from the end of the game, given Q 1, Firm 2 computes revenue and marginal revenue R 2 = (5 – (Q 1 + Q 2 )) Q 2 MR 2 = dR 2 /dQ 2 = 5 – Q 1 – 2Q 2 Hence, equate marginal cost to marginal revenue 1 = MC 2 = MR 2 = 5 – Q 1 – 2Q 2 to determine the optimal reaction Q 2 = r 2 (Q 1 ) = 2 –.5Q 1 Example 1: Example 1: Stackelberg Duopoly

11. 11 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 = (5 – Q 1 – r 2 (Q 1 ) )) Q 1 n R 1 = (5 – Q 1 – (2 –.5Q 1 )) Q 1 n R 1 = (3 –.5Q 1 ) Q 1 and its profit-maximizing output level: n 2 = c 1 = dR 1 /dQ 1 n 2 = d/dQ 1 (3 –.5Q 1 ) Q 1 n 2 = 3 – Q 1 n Q 1 = 1 BA 445 Lesson B.3 Sequential Quantity Competition Example 1: Example 1: Stackelberg Duopoly

12. 12 Complete solution for P = 5  (Q 1 +Q 2 ), MC 1 = 2, MC 2 = 1. Q 1 = 1Q 1 = 1 Q 2 = r 2 (Q 1 ) = 2 –.5Q 1 = 2 –.5(1) = 1.5Q 2 = r 2 (Q 1 ) = 2 –.5Q 1 = 2 –.5(1) = 1.5 P = 5  (Q 1 +Q 2 ) = 2.5P = 5  (Q 1 +Q 2 ) = 2.5 Firm 1 profit  1 = (P  c 1 ) Q 1 = (2.5  2)1 = 0.5Firm 1 profit  1 = (P  c 1 ) Q 1 = (2.5  2)1 = 0.5 Firm 2 profit  2 = (P  c 2 ) Q 2 = (2.5  1)1.5 = 2.25Firm 2 profit  2 = (P  c 2 ) Q 2 = (2.5  1)1.5 = 2.25 BA 445 Lesson B.3 Sequential Quantity Competition Example 1: Example 1: Stackelberg Duopoly

13. 13 BA 445 Lesson B.3 Sequential Quantity Competition Comment: Given any inverse demand P = a  b(Q 1 +Q 2 ) Firm 2’s 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 1: Example 1: Stackelberg Duopoly

14. 14 BA 445 Lesson B.3 Sequential Quantity Competition Example 2: First Mover Advantage

15. 15 BA 445 Lesson B.3 Sequential Quantity Competition Overview First Mover Advantage always occurs in the rollback solution to a Stackelberg duopoly. That advantage can make it profitable to rush to choose output first, even if that rush raises costs. Example 2: First Mover Advantage

16. 16 BA 445 Lesson B.3 Sequential Quantity Competition Example 2: First Mover Advantage There are three ways to make a living in this business: be first, be smarter, or cheat. ~ Margin Call (2011 movie)

17. 17 BA 445 Lesson B.3 Sequential Quantity Competition Comment: If the unit production costs are the same for the leader and the follower in a Stackelberg duopoly, then the leader produces more and makes more profit. Specifically, for inverse demand P = a – b(Q 1 +Q 2 ) and unit production costs c, Q 1 = (a – c)/(2b)Q 1 = (a – c)/(2b) Q 2 = (a – c)/(4b)Q 2 = (a – c)/(4b) So the leader has twice the output and twice the profits of the follower. 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 marginal costs of production. Example 2: First Mover Advantage

18. 18 BA 445 Lesson B.3 Sequential Quantity Competition Question: You are the manager of Kleenex and you compete directly with Puffs selling facial tissues in America. Consumers find the two products to be indistinguishable. The inverse market demand for facial tissues is P = 3-Q (in dollars) in America and both firms produce at a marginal cost of $1. You have a decision to make about competing with Puffs in New Zealand, where the inverse market demand for facial tissues is P = 3-Q (in dollars). Option A. Puffs sets up its factories and distribution networks now, and you set up later. And both produce at a marginal cost of $1. Option B. You hurry set up your factories and distribution networks now, and Puffs sets up later. Your hurry means your marginal costs are $2, while Puffs marginal costs remain $1. Which Option is better for Kleenex? Example 2: First Mover Advantage

19. 19 BA 445 Lesson B.3 Sequential 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 the leader in a Stackelberg Duopoly with inverse demand P = 3  (Q 1 +Q 2 ) and marginal costs c 1 = MC 1 = 2 and c 2 = MC 2 = 1. Example 2: First Mover Advantage

20. 20 BA 445 Lesson B.3 Sequential 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 2: First Mover Advantage

21. 21 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.3 Sequential Quantity Competition Example 2: First Mover Advantage

22. 22 Complete 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.3 Sequential Quantity Competition Example 2: First Mover Advantage

23. 23 BA 445 Lesson B.3 Sequential Quantity Competition In Option B, you are the leader in a Stackelberg Duopoly with inverse demand P = 3  (Q 1 +Q 2 ) and marginal costs c 1 = MC 1 = 2 and c 2 = MC 2 = 1. Example 2: First Mover Advantage

24. 24 BA 445 Lesson B.3 Sequential Quantity Competition Option B: 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 2: First Mover Advantage

25. 25 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 2 = c 1 = dR 1 /dQ 1 n 2 = d/dQ 1 (2 –.5Q 1 ) Q 1 n 2 = 2 – Q 1 n Q 1 = 0 BA 445 Lesson B.3 Sequential Quantity Competition Example 2: First Mover Advantage

26. 26 Complete solution for c 1 = MC 1 = 2 and c 2 = MC 2 = 1: Q 1 = 0Q 1 = 0 Q 2 = r 2 (Q 1 ) = 1 –.5Q 1 = 1 –.5(0) = 1Q 2 = r 2 (Q 1 ) = 1 –.5Q 1 = 1 –.5(0) = 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.3 Sequential Quantity Competition Example 2: First Mover Advantage

27. 27 Option A is thus best for Kleenex since Kleenex profits (as a follower) are 0.25 in Option A, while Kleenex profits (as the leader) are 0 in Option B. BA 445 Lesson B.3 Sequential Quantity Competition Example 2: First Mover Advantage

28. 28 Comment: In this particular case, Kleenex increased production cost hurt profits more than profits increase because of the first mover advantage. In other problems, increased production cost hurt profits less than profits increase because of the first mover advantage. BA 445 Lesson B.3 Sequential Quantity Competition Example 2: First Mover Advantage

29. 29 BA 445 Lesson B.3 Sequential Quantity Competition Example 3: Selling Technology

30. 30 BA 445 Lesson B.3 Sequential Quantity Competition Overview Selling Technology to a Stackelberg 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 3: Selling Technology

31. 31 BA 445 Lesson B.3 Sequential Quantity Competition Question: You are a manager of Home Depot and your only significant competitor in the retail home improvement market is Lowes. You expect to open the first home improvement store in the Conejo Valley, and Lowes will follow a month later. Your lumber and Lowes’s lumber are indistinguishable to consumers. The inverse market demand for lumber is P = 4  Q (in dollars) and both firms used to produce at a marginal cost of $2. However, you just found a better way to treat lumber, which reduces your marginal cost to $1. Should you keep that procedure to yourself? Or is it better to sell that secret to Lowes so that both you and Lowes can produce at marginal cost equal to $1? Example 3: Selling Technology

32. 32 BA 445 Lesson B.3 Sequential Quantity Competition Answer: You are the leader in a Stackelberg Duopoly with inverse demand P = 4  (Q 1 +Q 2 ). Compare the rollback solution with marginal costs c 1 = 1 and c 2 = 2, to the solution with c 1 = 1 and c 2 = 1. Example 4: Selling Technology

33. 33 BA 445 Lesson B.3 Sequential Quantity Competition Starting from the end, 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 c 2 = MC 2 = MR 2 = 4 – Q 1 – 2Q 2 to determine the optimal reaction Q 2 = r 2 (Q 1 ) = 2 –.5c 2 –.5Q 1 Example 4: Selling Technology

34. 34 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 = (4 – Q 1 – r 2 (Q 1 ) )) Q 1 n R 1 = (4 – Q 1 – (2 –.5c 2 –.5Q 1 )) Q 1 n R 1 =.5(4 + c 2 – Q 1 ) Q 1 and its profit-maximizing output level: n 1 = c 1 = dR 1 /dQ 1 n 1 = d/dQ 1.5(4 + c 2 – Q 1 ) Q 1 n 1 = 2 +.5 c 2 – Q 1 n Q 1 = 1 +.5 c 2 BA 445 Lesson B.3 Sequential Quantity Competition Example 4: Selling Technology

35. 35 Complete solution for secret technology, c 2 = 2: Q 1 = 1 +.5 c 2 = 2Q 1 = 1 +.5 c 2 = 2 Q 2 = r 2 (Q 1 ) = 2 –.5c 2 –.5Q 1 = 2 –.5(2) –.5(2) = 0Q 2 = r 2 (Q 1 ) = 2 –.5c 2 –.5Q 1 = 2 –.5(2) –.5(2) = 0 P = 4  (Q 1 +Q 2 ) = 2P = 4  (Q 1 +Q 2 ) = 2 Firm 1 profit  1 = (P  c 1 ) Q 1 = (2  1)2 = 2Firm 1 profit  1 = (P  c 1 ) Q 1 = (2  1)2 = 2 Firm 2 profit  2 = (P  c 2 ) Q 2 = (2  2)0 = 0Firm 2 profit  2 = (P  c 2 ) Q 2 = (2  2)0 = 0 Complete solution for sold technology, c 2 = 1: Q 1 = 1 +.5 c 2 = 1.5Q 1 = 1 +.5 c 2 = 1.5 Q 2 = r 2 (Q 1 ) = 2 –.5c 2 –.5Q 1 = 2 –.5(1) –.5(1.5) =.75Q 2 = r 2 (Q 1 ) = 2 –.5c 2 –.5Q 1 = 2 –.5(1) –.5(1.5) =.75 P = 4  (Q 1 +Q 2 ) = 1.75P = 4  (Q 1 +Q 2 ) = 1.75 Firm 1 profit  1 = (P  c 1 ) Q 1 = (1.75  1)1.5 = 1.125Firm 1 profit  1 = (P  c 1 ) Q 1 = (1.75  1)1.5 = 1.125 Firm 2 profit  2 = (P  c 2 ) Q 2 = (1.75  1).75 = 0.5625Firm 2 profit  2 = (P  c 2 ) Q 2 = (1.75  1).75 = 0.5625 BA 445 Lesson B.3 Sequential Quantity Competition Example 4: Selling Technology

36. 36 Selling technology and reducing c 2 = 2 to c 2 = 1 has to effects: Firm 1’s profit reduces from  1 = 2 to  1 = 1.125Firm 1’s profit reduces from  1 = 2 to  1 = 1.125 Firm 2’s profit increases from  2 = 0 to  2 = 0.5625Firm 2’s profit increases from  2 = 0 to  2 = 0.5625 Home Depot should not sell technology because doing so reduces its profit from production (  0.875) more than it generates profit (0.5625) from the sale. BA 445 Lesson B.3 Sequential Quantity Competition Example 4: Selling Technology

37. 37 BA 445 Lesson B.3 Sequential Quantity Competition Example 4: Colluding

38. 38 BA 445 Lesson B.3 Sequential Quantity Competition Overview Colluding with a Stackelberg competitor is almost always profitable. Since the competitors produce gross substitutes, profitable collusion lowers output. But, the leader cannot trust the follower to collude. Example 4: Colluding

39. 39 BA 445 Lesson B.3 Sequential Quantity Competition Comment: The demand for a product is sometimes presented in standard form, like Q = 10  2P. That should be inverted, to P = 5  0.5Q, to facilitate duopoly calculations. The cost for a product is sometimes presented in a functional form, like C(Q) = 2Q. That should be differentiated, to MC(Q) = 2, to facilitate duopoly calculations. Example 4: Colluding

40. 40 BA 445 Lesson B.3 Sequential Quantity Competition Question: The market for razor blades consists of two firms: Gillette and Wilkinson Sword/Schick. As the manager of Gillette, you enjoy a patented technology that permits your company to produce razor blades more quickly. You use that advantage to be first to choose your profit-maximizing output level in the market, and your competitor knows your output before choosing their own output. The demand for razor blades is Q = 13 - P; Gillette’s costs are C 1 (Q 1 ) = Q 1 ; and Wilkinson’s costs are C 2 (Q 2 ) = Q 2. Compute Gillette’s profit, and compute Wilkinson’s profit. Ignoring antitrust law considerations, would it be mutually profitable for the companies to collude by changing Gillette’s and Wilkinson’s outputs to 4 and 2. Can Gillette trust Wilkinson? Example 4: Colluding

41. 41 BA 445 Lesson B.3 Sequential Quantity Competition Answer: You are the leader in a Stackelberg Duopoly with demand Q = 13  P and costs C 1 (Q 1 ) = Q 1 and C 2 (Q 2 ) = Q 2, First, solve for inverse demand P = 13  (Q 1 +Q 2 ). And solve for marginal cost c 1 = MC 1 = dC 1 /dQ 1 = 1 and c 2 = MC 2 = dC 2 /dQ 2 = 1. Compare the rollback solution with the collusive proposal of quantities 4 and 2. Example 4: Colluding

42. 42 BA 445 Lesson B.3 Sequential Quantity Competition Starting from the end, 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 4: Colluding

43. 43 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 = (13 – Q 1 – r 2 (Q 1 ) )) Q 1 n R 1 = (13 – Q 1 – (6 –.5Q 1 )) Q 1 n R 1 = (7 –.5Q 1 ) Q 1 and its profit-maximizing output level: n 1 = c 1 = dR 1 /dQ 1 n 1 = d/dQ 1 (7 –.5Q 1 ) Q 1 n 1 = 7 – Q 1 n Q 1 = 6 BA 445 Lesson B.3 Sequential Quantity Competition Example 4: Colluding

44. 44 Complete solution for non-colluding firms: Q 1 = 6Q 1 = 6 Q 2 = r 2 (Q 1 ) = 6 –.5Q 1 = 6 –.5(6) = 3Q 2 = r 2 (Q 1 ) = 6 –.5Q 1 = 6 –.5(6) = 3 P = 13  (Q 1 +Q 2 ) = 4P = 13  (Q 1 +Q 2 ) = 4 Firm 1 profit  1 = (P  c 1 ) Q 1 = (4  1)6 = 18Firm 1 profit  1 = (P  c 1 ) Q 1 = (4  1)6 = 18 Firm 2 profit  2 = (P  c 2 ) Q 2 = (4  1)3 = 9Firm 2 profit  2 = (P  c 2 ) Q 2 = (4  1)3 = 9 Collusive proposal of quantities Q 1 = 4 and Q 2 = 2: P = 13  (Q 1 +Q 2 ) = 7P = 13  (Q 1 +Q 2 ) = 7 Firm 1 profit  1 = (P  c 1 ) Q 1 = (7  1)4 = 24Firm 1 profit  1 = (P  c 1 ) Q 1 = (7  1)4 = 24 Firm 2 profit  2 = (P  c 2 ) Q 2 = (7  1)2 = 12Firm 2 profit  2 = (P  c 2 ) Q 2 = (7  1)2 = 12 BA 445 Lesson B.3 Sequential Quantity Competition Example 4: Colluding

45. 45 The collusive proposal of quantities Q 1 = 4 and Q 2 = 2 is thus mutually profitable for the companies. But Gillette cannot trust Wilkinson since Wilkinson’s best response to Gillette’s Q 1 = 4 is Q 2 = r 2 (4) = 6 –.5(4) = 4, not 2. BA 445 Lesson B.3 Sequential Quantity Competition Example 4: Colluding

46. 46 BA 445 Lesson B.3 Sequential Quantity Competition SummarySummary

47. 47 BA 445 Lesson B.3 Sequential Quantity Competition Summary Complete solution to a Stackelberg Duopoly Game with inverse demand P = a  bQ and constant marginal costs c 1 = MC 1 and c 2 = MC 2 : Q 1 = (a + c 2  2c 1 )/2bQ 1 = (a + c 2  2c 1 )/2b Q 2 = r 2 (Q 1 ) = (a  c 2 )/2b –.5Q 1Q 2 = r 2 (Q 1 ) = (a  c 2 )/2b –.5Q 1 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.

48. 48 Review Questions BA 445 Lesson B.3 Sequential 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.

49. 49 End of Lesson B.3 BA 445 Managerial Economics BA 445 Lesson B.3 Sequential Quantity Competition