1 1 Deep Thought BA 445 Lesson B.3 Sequential Quantity Competition I love going down to the...

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 2BA 445 Lesson B.3 Sequential Quantity Competition

Readings

Readings

Baye “Stackelberg Oligopoly” (see the index)Dixit Chapter 3


Overview

Overview

4 4

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.

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.


Example 1: Stackelberg Duopoly



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.



Comment: Stackelberg Duopoly Games have three parts.• Players are managers of two firms serving many consumers.

• 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(Q1+Q2) if a-b(Q1+Q2) > 0, and P = 0 otherwise.• Firm 1 chooses output Q1 > 0.

• Firm 2 knows Firm 1’s Q1 > 0 before he chooses his own.

• Firm 2’s strategy is thus an output Q2 reaction function Q2 = r2(Q1) to Firm 1’s choice Q1.

• 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



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?



Answer: You are the leader in a Stackelberg Duopoly Game with inverse demand P = 5 - (Q1+Q2) and marginal costs c1 = MC1 = 2 and c2 = MC2 = 1.

Find the rollback solution to the Stackelberg Duopoly Game.



Starting from the end of the game, given Q1, Firm 2 computes revenue and marginal revenue

R2 = (5 – (Q1 + Q2)) Q2

MR2 = dR2 /dQ2 = 5 – Q1 – 2Q2

Hence, equate marginal cost to marginal revenue

1 = MC2 = MR2 = 5 – Q1 – 2Q2

to determine the optimal reaction

Q2 = r2 (Q1) = 2 – .5Q1


11 11

Rolling back to the beginning, • The Stackelberg leader uses the reaction function r2 (Q1)

to determine its revenue R1 = (5 – Q1 – r2 (Q1) )) Q1

R1 = (5 – Q1 – (2 – .5Q1)) Q1

R1 = (3 – .5Q1) Q1

and its profit-maximizing output level: 2 = c1 = dR1/dQ1 2 = d/dQ1 (3 – .5Q1) Q1

2 = 3 – Q1

Q1 = 1



12 12

Complete solution for P = 5 - (Q1+Q2), MC1 = 2, MC2 = 1.

• Q1 = 1

• Q2 = r2 (Q1) = 2 – .5Q1 = 2 – .5(1) = 1.5

• P = 5 - (Q1+Q2) = 2.5

• Firm 1 profit P1 = (P - c1) Q1 = (2.5 - 2)1 = 0.5

• Firm 2 profit P2 = (P - c2) Q2 = (2.5 - 1)1.5 = 2.25




Comment: Given any inverse demand

P = a - b(Q1+Q2)

Firm 2’s 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



Example 2: First Mover Advantage



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.




There are three ways to make a living in this business: be first, be smarter, or cheat. ~ Margin Call (2011 movie)


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(Q1+Q2) and unit production costs c, • Q1 = (a – c)/(2b)

• Q2 = (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.



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?



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 the leader in a Stackelberg Duopoly with inverse demand P = 3 - (Q1+Q2) and marginal costs c1 = MC1 = 2 and c2 = MC2 = 1.



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


1 = MC2 = MR2 = 3 – Q1 – 2Q2


Q2 = r2 (Q1) = 1 – .5Q1


21 21

Rolling back to the beginning,• The Stackelberg leader uses the reaction function r2 (Q1)


R1 = (3 – Q1 – (1 – .5Q1)) Q1

R1 = (2 – .5Q1) Q1


1 = 2 – Q1

Q1 = 1



22 22

Complete 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




In Option B, you are the leader in a Stackelberg Duopoly with inverse demand P = 3 - (Q1+Q2) and marginal costs c1 = MC1 = 2 and c2 = MC2 = 1.



Option B: Starting from the end, given Q1, Firm 2 computes revenue and marginal revenue

R2 = (3 – (Q1 + Q2)) Q2

MR2 = dR2 /dQ2 = 3 – Q1 – 2Q2


1 = MC2 = MR2 = 3 – Q1 – 2Q2


Q2 = r2 (Q1) = 1 – .5Q1


25 25

Rolling back to the beginning,• The Stackelberg leader uses the reaction function r2 (Q1)


R1 = (3 – Q1 – (1 – .5Q1)) Q1

R1 = (2 – .5Q1) Q1


2 = 2 – Q1

Q1 = 0



26 26

Complete solution for c1 = MC1 = 2 and c2 = MC2 = 1:

• Q1 = 0

• Q2 = r2 (Q1) = 1 – .5Q1 = 1 – .5(0) = 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



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.



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.




Example 3: Selling Technology



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.



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?



Answer: You are the leader in a Stackelberg Duopoly with inverse demand P = 4-(Q1+Q2). Compare the rollback solution with marginal costs c1 = 1 and c2 = 2, to the solution with c1 = 1 and c2 = 1.



Starting from the end, given Q1, Firm 2 computes revenue and marginal revenue

R2 = (4 – (Q1 + Q2)) Q2

MR2 = dR2 /dQ2 = 4 – Q1 – 2Q2


c2 = MC2 = MR2 = 4 – Q1 – 2Q2


Q2 = r2 (Q1) = 2 – .5c2 – .5Q1


34 34



R1 = (4 – Q1 – (2 – .5c2 – .5Q1 )) Q1

R1 = .5(4 + c2 – Q1) Q1

and its profit-maximizing output level: 1 = c1 = dR1/dQ1 1 = d/dQ1 .5(4 + c2 – Q1) Q1

1 = 2 + .5 c2 – Q1

Q1 = 1 + .5 c2



35 35

Complete solution for secret technology, c2 = 2:

• Q1 = 1 + .5 c2 = 2

• Q2 = r2 (Q1) = 2 – .5c2 – .5Q1 = 2 – .5(2) – .5(2) = 0

• P = 4 - (Q1+Q2) = 2

• Firm 1 profit P1 = (P - c1) Q1 = (2 - 1)2 = 2

• Firm 2 profit P2 = (P - c2) Q2 = (2 - 2)0 = 0

Complete solution for sold technology, c2 = 1:

• Q1 = 1 + .5 c2 = 1.5

• Q2 = r2 (Q1) = 2 – .5c2 – .5Q1 = 2 – .5(1) – .5(1.5) = .75

• P = 4 - (Q1+Q2) = 1.75

• Firm 1 profit P1 = (P - c1) Q1 = (1.75 - 1)1.5 = 1.125

• Firm 2 profit P2 = (P - c2) Q2 = (1.75 - 1).75 = 0.5625BA 445 Lesson B.3 Sequential Quantity Competition


36 36

Selling technology and reducing c2 = 2 to c2 = 1 has to effects:• Firm 1’s profit reduces from P1 = 2 to P1 = 1.125

• Firm 2’s profit increases from P2 = 0 to P2 = 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.




Example 4: Colluding



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.



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.



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 C1 (Q1) = Q1; and Wilkinson’s costs are C2 (Q2) = Q2.

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?



Answer: You are the leader in a Stackelberg Duopoly with demand Q = 13 - P and costs C1 (Q1) = Q1 and C2 (Q2) = Q2, First, solve for inverse demand P = 13 - (Q1+Q2). And solve for marginal cost c1 = MC1 = dC1 /dQ1 = 1 and c2 = MC2 = dC2 /dQ2 = 1.

Compare the rollback solution with the collusive proposal of quantities 4 and 2.



Starting from the end, given Q1, Firm 2 computes revenue and marginal revenue

R2 = (13 – (Q1 + Q2)) Q2

MR2 = dR2 /dQ2 = 13 – Q1 – 2Q2


1 = MC2 = MR2 = 13 – Q1 – 2Q2


Q2 = r2 (Q1) = 6 – .5Q1


43 43



R1 = (13 – Q1 – (6 – .5Q1)) Q1

R1 = (7 – .5Q1) Q1


1 = 7 – Q1

Q1 = 6



44 44

Complete solution for non-colluding firms:• Q1 = 6

• Q2 = r2 (Q1) = 6 – .5Q1 = 6 – .5(6) = 3

• P = 13 - (Q1+Q2) = 4

• Firm 1 profit P1 = (P - c1) Q1 = (4 - 1)6 = 18

• Firm 2 profit P2 = (P - c2) Q2 = (4 - 1)3 = 9

Collusive proposal of quantities Q1 = 4 and Q2 = 2:

• P = 13 - (Q1+Q2) = 7

• Firm 1 profit P1 = (P - c1) Q1 = (7 - 1)4 = 24

• Firm 2 profit P2 = (P - c2) Q2 = (7 - 1)2 = 12



45 45

The collusive proposal of quantities Q1 = 4 and Q2 = 2 is thus mutually profitable for the companies. But Gillette cannot trust Wilkinson since Wilkinson’s best response to Gillette’s Q1 = 4 is Q2 = r2 (4) = 6 – .5(4) = 4, not 2.




Summary

Summary


Summary

Complete solution to a Stackelberg Duopoly Game with inverse demand P = a - bQ and constant marginal costs c1 = MC1 and c2 = MC2:

• Q1 = (a + c2 - 2c1)/2b

• Q2 = r2 (Q1) = (a - c2)/2b – .5Q1

• 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.

48 48

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.

49 49

End of Lesson B.3

BA 445 Managerial Economics


1 1 Deep Thought BA 445 Lesson B.3 Sequential Quantity Competition I love going down to the...

Documents

Transcript of 1 1 Deep Thought BA 445 Lesson B.3 Sequential Quantity Competition I love going down to the...