1 1 Deep Thought BA 210 Lesson II.3 Sequential Quantity Competition I love going down to the elementary school, watching all the kids jump and shout, but

1 1

Deep ThoughtDeep Thought

BA 210 Lesson II.3 Sequential Quantity CompetitionBA 210 Lesson II.3 Sequential Quantity Competition

I love I love going down to the elementary going down to the elementary school, watching all the kids jump and school, watching all the kids jump and shout, but they don’t know I’m using shout, but they don’t know I’m using blanks. ~ Jack Handeyblanks. ~ Jack Handey..

(Translation: Today’s lesson teaches when it is important to you (Translation: Today’s lesson teaches when it is important to you that your opponents know your actions so you can manipulate that your opponents know your actions so you can manipulate their reactions.)their reactions.)

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Lesson overviewLesson overview

BA 210 Lesson II.3 Sequential Quantity Competition

Lesson II.1 Strategic BargainingLesson II.1 Strategic Bargaining

Lesson II.2 Bargaining and ImpatienceLesson II.2 Bargaining and Impatience

Lesson II.3 Sequential Quantity CompetitionLesson II.3 Sequential Quantity CompetitionExample 1: Stackelberg DuopolyExample 1: Stackelberg DuopolyExample 2: First Mover AdvantageExample 2: First Mover AdvantageExample 3: Selling TechnologyExample 3: Selling TechnologyExample 4: ColludingExample 4: ColludingExample 5: MergingExample 5: MergingSummarySummaryReview QuestionsReview Questions

3 3BA 210 Lesson II.3 Sequential Quantity Competition

Example 1: Stackelberg DuopolyExample 1: Stackelberg Duopoly

Example 1: Example 1: Stackelberg Duopoly


Comment: Stackelberg Duopoly GamesComment: Stackelberg Duopoly Games have three parts. have three parts.

• Players Players are managers of two firms serving many consumers.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 Strategies are outputs of are outputs of homogeneous products (perfect homogeneous products (perfect substitutes), so they sell at the same price P.substitutes), so they sell at the same price P.

• Firm 1 chooses output QFirm 1 chooses output Q11 >> 0. 0.

• Firm 2 knows Firm 1’s QFirm 2 knows Firm 1’s Q11 >> 0 before he chooses his own. 0 before he chooses his own.

• Payoffs Payoffs are profits. When unit production costs are constants are profits. When unit production costs are constants cc1 1 and and cc22, then profits are, then profits are

11 = (P- = (P- c c11))QQ11 and and 22 = (P- = (P- c c22))QQ22


5 5

1 3 41 6,7 4,15 3,163 12,5 6,9 3,84 12,4 4,6 0,4

Player 2

Player 1


Payoffs for the two playersPayoffs for the two players often listed in a table. The strategies often listed in a table. The strategies are listed as rows for Player 1, and columns for Player 2. The are listed as rows for Player 1, and columns for Player 2. The combination of strategies by both players determines a cell in the combination of strategies by both players determines a cell in the payoff table, and that cell specifies the payoffs to the players, payoff table, and that cell specifies the payoffs to the players, with Player 1 listed first. For example, if Player 1 chooses with Player 1 listed first. For example, if Player 1 chooses QQ11=3 =3

and Player 2 chooses Qand Player 2 chooses Q11=1, then in the payoff table below 12,5 =1, then in the payoff table below 12,5

specifies payoff 12 to Player 1 and 5 to Player 2.specifies payoff 12 to Player 1 and 5 to Player 2.


6 6

1 3 41 6,7 4,15 ?,?3 12,5 6,9 3,84 12,4 4,6 0,4

DC Comics

Marvel


Question: You are a 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 = 11-Q (in dollars). Your unit costs of production are $3, and the unit costs of DC Comics are $2. Compute profits when you produce 1 units and DC produces 4 units. Suppose profits from other combinations of production are in the table below:



Question (continued): Suppose you choose your output of comic books to be either 1, 3, or 4 before DC Comics, and DC Comics knows your output before they decide their own output of either 1, 3, or 4.

How many comic books should you produce?



Answer: Answer: You are the leader in a Stackelberg Duopoly Game with You are the leader in a Stackelberg Duopoly Game with inverse demand P = 11 inverse demand P = 11 (Q(Q11+Q+Q22) and ) and unit costs unit costs cc11 = 3 and c = 3 and c22 = = 2. 2.

If If you produce Qyou produce Q11=1 units and DC produces Q=1 units and DC produces Q22=4 units, then =4 units, then

total output is Qtotal output is Q11+Q+Q22=5, so price is =5, so price is P = 11 P = 11 (Q(Q11+Q+Q22)=6, and )=6, and

profits are profits are

11 = (P- c = (P- c11)Q)Q11 = (6-3)1 = 3 and = (6-3)1 = 3 and 22 = (P- c = (P- c22)Q)Q22 = (6-2)4 = 16, = (6-2)4 = 16,

which we write 3,16 to complete the profit table. which we write 3,16 to complete the profit table.


1 3 41 6,7 4,15 3,163 12,5 6,9 3,84 12,4 4,6 0,4

DC Comics

Marvel

9 9

1 3 41 6,7 4,15 3,163 12,5 6,9 3,84 12,4 4,6 0,4

DC Comics

Marvel


Find the rollback solution Find the rollback solution to the Stackelberg Duopoly Game. to the Stackelberg Duopoly Game. Starting from the end of the game, Starting from the end of the game,

•if Marvel has chosen Qif Marvel has chosen Q11=1, then DC will respond with Q=1, then DC will respond with Q22=4, =4,

and so profits are 3 for Marvel and 16 for DC and so profits are 3 for Marvel and 16 for DC

•if Qif Q11=3, then Q=3, then Q22=3, and so profits are 6 and 9 =3, and so profits are 6 and 9

•if Qif Q11=4, then Q=4, then Q22=3, and so profits are 4 and 6 =3, and so profits are 4 and 6


10 10

1 3 41 6,7 4,15 3,163 12,5 6,9 3,84 12,4 4,6 0,4

DC Comics

Marvel


So Marvel should produce 3 comic books, which leads DC Comics to produce 3 and generate profit of 6 for Marvel.



Example 2: First Mover AdvantageExample 2: First Mover Advantage



Comment: Comment: If the unit production costs are the same for the leader If the unit production costs are the same for the leader and the follower in a Stackelberg duopoly, then the leader and the follower in a Stackelberg duopoly, then the leader produces more and makes more profit. produces more and makes more profit.

In particular, a firm can find it profitable to become the first 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 mover by rushing to set up an assembly line, even if it means increasing unit costs of production.increasing unit costs of production.



Question: Question: You are the manager of Kleenex and you compete You are the manager of Kleenex and you compete directly with Puffs selling facial tissues in America. Consumers directly with Puffs selling facial tissues in America. Consumers find the two products to be indistinguishable. The inverse market find the two products to be indistinguishable. The inverse market demand for facial tissues is P = 11-Q (in dollars) in America and demand for facial tissues is P = 11-Q (in dollars) in America and both firms produce at a unit cost of $2. You have a decision to both firms produce at a unit cost of $2. You have a decision to make about competing with Puffs in New Zealand, where the make about competing with Puffs in New Zealand, where the inverse market demand for facial tissues is P = 11-Q (in dollars), inverse market demand for facial tissues is P = 11-Q (in dollars), and both you and Puffs can choose an output quantity 1, 3, or 6. and both you and Puffs can choose an output quantity 1, 3, or 6. You must choose the option that is best for Kleenex.You must choose the option that is best for Kleenex.



Option A. Puffs sets up its factories and distribution networks now, and you set up later. And both produce at a unit cost of $2, resulting in the first profit table:

Option B. You hurry set up your factories and distribution networks now, and Puffs sets up later. Your hurry means your unit costs are $3, while Puffs unit costs remain $2 resulting in the second profit table:


1 3 61 6,7 4,15 1,123 12,5 6,9 -3,06 6,2 -6,0 -24,-18

Puffs

Kleenex

1 3 61 7,7 5,15 2,123 15,5 9,9 0,06 12,2 0,0 -18,-18

Kleenex

Puffs


Answer: In Option A, Answer: In Option A, you are you are the follower in a Stackelberg the follower in a Stackelberg Duopoly. Puffs anticipates Duopoly. Puffs anticipates your reactions on the right, and your reactions on the right, and so chooses to produce 6, you so chooses to produce 6, you react with 1 and so react with 1 and so you earn 2you earn 2. .

In Option B, In Option B, you are the leader you are the leader in a Stackelberg Duopoly. You in a Stackelberg Duopoly. You anticipate Puffs’ reactions on anticipate Puffs’ reactions on the right, and so choose to the right, and so choose to produce 6, Puffs reacts with 1 produce 6, Puffs reacts with 1 and so and so you earn 6you earn 6..


1 3 61 6,7 4,15 1,123 12,5 6,9 -3,06 6,2 -6,0 -24,-18

Puffs

Kleenex

1 3 61 7,7 5,15 2,123 15,5 9,9 0,06 12,2 0,0 -18,-18

Kleenex

Puffs

16 16

Option B Option B is thus best for Kleenex since Kleenex profits (as a is thus best for Kleenex since Kleenex profits (as a follower) are 2 in Option A, while Kleenex profits (as the leader) follower) are 2 in Option A, while Kleenex profits (as the leader) are 6 in Option B.are 6 in Option B.



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Comment: Comment: In this particular case, Kleenex increased production In this particular case, Kleenex increased production cost hurt profits less than profits increase because of the first cost hurt profits less than profits increase because of the first mover advantage, so it is worth being the first mover. mover advantage, so it is worth being the first mover.

In other problems, increased production cost hurt profits more In other problems, increased production cost hurt profits more than profits increase because of the first mover advantage, so it is than profits increase because of the first mover advantage, so it is not worth being the first mover. not worth being the first mover.




Example 3: Selling TechnologyExample 3: Selling Technology



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

To answer the question, suppose both Home Depot and Lowes To answer the question, suppose both Home Depot and Lowes separately choose to produce either 0, or 1, or 3 units of lumber. separately choose to produce either 0, or 1, or 3 units of lumber.


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0 1 30 0,0 0,1 0,-31 2,0 1,0 -1,-63 0,0 -3,-2 -9,-12

Lowes

Home


Answer: Answer: You are the leader in a Stackelberg Duopoly with You are the leader in a Stackelberg Duopoly with inverse demand P = 4inverse demand P = 4(Q(Q11+Q+Q22)). Compare the rollback solution . Compare the rollback solution

with unit costs with unit costs cc11 = 1 and c = 1 and c2 2 = 2, to the solution with c= 2, to the solution with c11 = 1 and c = 1 and c2 2

= 1.= 1.

First, compute the payoff table and rollback solution for unit First, compute the payoff table and rollback solution for unit costs costs cc11 = 1 and c = 1 and c2 2 = 2:= 2:


21 21

0 1 30 0,0 0,1 0,-31 2,0 1,0 -1,-63 0,0 -3,-2 -9,-12

Lowes

Home


You are the leader in a Stackelberg Duopoly. You anticipate You are the leader in a Stackelberg Duopoly. You anticipate Lowes’ reactions below, and so choose to produce 1, Lowes Lowes’ reactions below, and so choose to produce 1, Lowes reacts with 1 and so reacts with 1 and so you earn 1 or 2 and Lowes earns 0you earn 1 or 2 and Lowes earns 0..


22 22

0 1 30 0,0 0,2 0,01 2,0 1,1 -1,-33 0,0 -3,-1 -9,-9

Lowes

Home


NextNext, compute the payoff table and rollback solution for unit , compute the payoff table and rollback solution for unit costs costs cc11 = 1 and c = 1 and c2 2 = 1:= 1:



You are the leader in a Stackelberg Duopoly. You anticipate You are the leader in a Stackelberg Duopoly. You anticipate Lowes’ reactions below, and so choose to produce 1, Lowes Lowes’ reactions below, and so choose to produce 1, Lowes reacts with 1 and so reacts with 1 and so you earn 1 and Lowes earns 1you earn 1 and Lowes earns 1..


0 1 30 0,0 0,2 0,01 2,0 1,1 -1,-33 0,0 -3,-1 -9,-9

Lowes

Home

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Selling technology Selling technology and reducing cand reducing c2 2 = 2 to c= 2 to c2 2 = 1 has to effects:= 1 has to effects:

•Firm 1’s profit reduces from Firm 1’s profit reduces from 11 = 1 or 2 to = 1 or 2 to 11 = 1 for sure = 1 for sure

•Firm 2’s profit increases from Firm 2’s profit increases from 22 = 0 to = 0 to 22 = 1 = 1

In particular, selling technology increases total profit In particular, selling technology increases total profit 11++22 from from

the uncertain result of 1 or 2 to the certainty of 2. Selling the uncertain result of 1 or 2 to the certainty of 2. Selling technology is thus a bargaining problem between Home Depot technology is thus a bargaining problem between Home Depot and Lowes, and the division of the positive gain from selling and Lowes, and the division of the positive gain from selling technology is determined by the rules of bargaining.technology is determined by the rules of bargaining.

For example, if Home Depot can make a take-it-or-leave-it offer For example, if Home Depot can make a take-it-or-leave-it offer to Lowes, then Lowes should accept anything as being better than to Lowes, then Lowes should accept anything as being better than nothing. After deducing that, Home Depot’s best acceptable offer nothing. After deducing that, Home Depot’s best acceptable offer to Lowes leaves Lowes with a pittance of the gains, which means to Lowes leaves Lowes with a pittance of the gains, which means Lowes pays the full 1 unit of profit for the technology.Lowes pays the full 1 unit of profit for the technology.




Example 4: ColludingExample 4: Colluding



Comment: Comment: The demand for a product is sometimes presented in The demand for a product is sometimes presented in standard form, like Q = 10 standard form, like Q = 10 2P. That should be inverted, to P = 2P. That should be inverted, to P = 5 5 0.5Q, to facilitate duopoly calculations. 0.5Q, to facilitate duopoly calculations.



Question: Question: The market for razor blades consists of two firms: The market for razor blades consists of two firms: Gillette and Wilkinson Sword/Schick. As the manager of Gillette and Wilkinson Sword/Schick. As the manager of Gillette, you enjoy a patented technology that permits your Gillette, you enjoy a patented technology that permits your company to produce razor blades more quickly. You use that company to produce razor blades more quickly. You use that advantage to be first to choose your profit-maximizing output advantage to be first to choose your profit-maximizing output level in the market, and your competitor knows your output level in the market, and your competitor knows your output before choosing their own output. The demand for razor blades before choosing their own output. The demand for razor blades is Q = 13 is Q = 13 P; Gillette’s unit costs are 1; and Wilkinson’s unit P; Gillette’s unit costs are 1; and Wilkinson’s unit costs are 1. Both firms separately produce either 0, 3, or 4 units. costs are 1. Both firms separately produce either 0, 3, or 4 units.

Compute Gillette’s profit, and compute Wilkinson’s profit. Compute Gillette’s profit, and compute Wilkinson’s profit. Ignoring antitrust law considerations, would it be mutually Ignoring antitrust law considerations, would it be mutually profitable for the companies to collude by changing Gillette’s profitable for the companies to collude by changing Gillette’s and Wilkinson’s outputs to 3 each. Can Gillette trust Wilkinson?and Wilkinson’s outputs to 3 each. Can Gillette trust Wilkinson?




with unit costs with unit costs cc11 = 1 and c = 1 and c2 2 = 1 = 1 with the collusive proposal of with the collusive proposal of

quantities quantities 4 and 2.4 and 2.

First, compute the payoff table and rollback solution:First, compute the payoff table and rollback solution:


0 3 40 0,0 0,27 0,323 27,0 18,18 15,204 32,0 20,15 16,16

Wilkinson

Gillette

29 29

0 3 40 0,0 0,27 0,323 27,0 18,18 15,204 32,0 20,15 16,16

Wilkinson

Gillette


You are the leader in a Stackelberg Duopoly. You anticipate You are the leader in a Stackelberg Duopoly. You anticipate Wilkinson’ reactions below, and so choose to produce 4, Wilkinson’ reactions below, and so choose to produce 4, Wilkinson reacts with 4 and so Wilkinson reacts with 4 and so you earn 16 and Wilkinson earns you earn 16 and Wilkinson earns 1616..

The collusive proposal The collusive proposal of quantity of quantity 3 for each is thus mutually 3 for each is thus mutually profitable for the companies. But Gillette profitable for the companies. But Gillette cannot trust cannot trust Wilkinson Wilkinson since Wilkinson’s best since Wilkinson’s best response to Gillette’s response to Gillette’s QQ1 1 = = 3 is Q3 is Q22 = 4, not 3. = 4, not 3.



Example 5: MergingExample 5: Merging



Question: Question: The market for commercial large jet aircraft consists of The market for commercial large jet aircraft consists of two firms: Airbus and Boeing. As the manager of Boeing, you two firms: Airbus and Boeing. As the manager of Boeing, you enjoy a patented technology that permits your company to enjoy a patented technology that permits your company to produce jets more quickly and at a lower cost than Airbus. You produce jets more quickly and at a lower cost than Airbus. You use that advantage to be first to choose your profit-maximizing use that advantage to be first to choose your profit-maximizing output level in the market. The demand for jets is Q = 9 output level in the market. The demand for jets is Q = 9 P; P; Boeing’s unit costs are 1 and Airbus’s unit costs are 2. Boeing’s unit costs are 1 and Airbus’s unit costs are 2.

Compute Boeing’s profit, and compute Airbus’s profit. Would it Compute Boeing’s profit, and compute Airbus’s profit. Would it be profitable for the two companies to merge? be profitable for the two companies to merge?

Suppose the firms separately produce quantities 0, 3, or 5 units, Suppose the firms separately produce quantities 0, 3, or 5 units, and if merged, the two can each choose from those quantities, and if merged, the two can each choose from those quantities, and they can both use the Boeing technology with unit cost 1. and they can both use the Boeing technology with unit cost 1.




with unit costs with unit costs cc11 = 1 and c = 1 and c2 2 = 2 = 2 with the with the monopoly with the with the monopoly

solutionsolution..

First, compute the payoff table and rollback solution:First, compute the payoff table and rollback solution:

0 3 50 0,0 0,12 0,103 15,0 6,3 0,-55 15,0 0,-3 -10,-15

Airbus

Boeing



You are the leader in a Stackelberg Duopoly. You anticipate You are the leader in a Stackelberg Duopoly. You anticipate Airbus’ reactions below, and so choose to produce 5, Airbus Airbus’ reactions below, and so choose to produce 5, Airbus reacts with 0 and so reacts with 0 and so you earn 15 and Airbus earns 0you earn 15 and Airbus earns 0..

0 3 50 0,0 0,12 0,103 15,0 6,3 0,-55 15,0 0,-3 -10,-15

Airbus

Boeing



The monopoly proposal The monopoly proposal reduces Airbus unit cost to 1, and so reduces Airbus unit cost to 1, and so generates a new profit table. But there are no cells in that new generates a new profit table. But there are no cells in that new payoff table that payoff table that increase total profit increase total profit 11++22 from the Stackelberg from the Stackelberg

value of value of 11++22=15+0. So in this case, it would not be profitable =15+0. So in this case, it would not be profitable

for the two companies to merge.for the two companies to merge.


0 3 50 0,0 0,12 0,103 15,0 6,3 0,-55 15,0 0,-3 -10,-15

Airbus

Boeing

35 35BA 210 Lesson II.3 Sequential Quantity CompetitionBA 210 Lesson II.3 Sequential Quantity Competition

SummarySummary

SummarySummary

36 36BA 210 Lesson II.3 Sequential Quantity CompetitionBA 210 Lesson II.3 Sequential Quantity Competition

SummarySummary

Payoff table entry Payoff table entry to any Duopoly Game with inverse demand P to any Duopoly Game with inverse demand P = a = a bQ and constant unit costs c bQ and constant unit costs c11 and c and c22::

•PP = a = a b(Qb(Q11+Q+Q22))

•Firm 1 profit Firm 1 profit 11 = (P = (Pcc11) Q) Q11

•Firm 2 profit Firm 2 profit 22 = (P = (Pcc22) Q) Q22

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Review QuestionsReview Questions


Review QuestionsReview Questions You should try to answer some of the following questions You should try to answer some of the following questions before the next class. before the next class. You will not turn in your answers, but students may request You will not turn in your answers, but students may request to discuss their answers to begin the next class. to discuss their answers to begin the next class. Your upcoming Exam 2 and cumulative Final Exam will Your upcoming Exam 2 and cumulative Final Exam will contain some similar questions, so you should eventually contain some similar questions, so you should eventually consider every review question before taking your exams.consider every review question before taking your exams.

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End of Lesson II.3End of Lesson II.3


BA 210 Introduction to BA 210 Introduction to MicroeconomicsMicroeconomics

Documents

1 1 Deep Thought BA 210 Lesson II.3 Sequential Quantity Competition I love going down to the elementary school, watching all the kids jump and shout, but