Deep Thought
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.)
BA 210 Lesson II.3 Sequential Quantity Competition
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Lesson overview
Lesson II.1 Strategic Bargaining
Lesson II.2 Bargaining and Impatience
Lesson II.3 Sequential Quantity Competition
Example 1: Stackelberg Duopoly
Example 2: First Mover Advantage
Example 3: Selling Technology
Example 4: Colluding
Example 5: Merging
Summary
Review Questions
BA 210 Lesson II.3 Sequential Quantity Competition
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Example 1: Stackelberg Duopoly
Example 1: Stackelberg Duopoly
BA 210 Lesson II.3 Sequential Quantity Competition
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Example 1: Stackelberg Duopoly
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 (perfect
substitutes), so they sell at the same price P.
• Firm 1 chooses output Q1 > 0.
• Firm 2 knows Firm 1’s Q1 > 0 before he chooses his own.
• Payoffs are profits. When unit production costs are constants
c1 and c2, then profits are
P1 = (P- c1)Q1 and P2 = (P- c2)Q2
BA 210 Lesson II.3 Sequential Quantity Competition
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Example 1: Stackelberg Duopoly
Payoffs for the two players often listed in a table. The strategies
are listed as rows for Player 1, and columns for Player 2. The
combination of strategies by both players determines a cell in the
payoff table, and that cell specifies the payoffs to the players,
with Player 1 listed first. For example, if Player 1 chooses Q1=3
and Player 2 chooses Q1=1, then in the payoff table below 12,5
specifies payoff 12 to Player 1 and 5 to Player 2.
Player 2
Player 1
1
3
4
1
6,7
12,5
12,4
3
4,15
6,9
4,6
BA 210 Lesson II.3 Sequential Quantity Competition
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3,16
3,8
0,4
5
Example 1: Stackelberg Duopoly
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:
DC Comics
Marvel
1
3
4
1
6,7
12,5
12,4
3
4,15
6,9
4,6
BA 210 Lesson II.3 Sequential Quantity Competition
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?,?
3,8
0,4
6
Example 1: Stackelberg Duopoly
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?
BA 210 Lesson II.3 Sequential Quantity Competition
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Example 1: Stackelberg Duopoly
Answer: You are the leader in a Stackelberg Duopoly Game with
inverse demand P = 11 - (Q1+Q2) and unit costs c1 = 3 and c2 =
2. If you produce Q1=1 units and DC produces Q2=4 units, then
total output is Q1+Q2=5, so price is P = 11 - (Q1+Q2)=6, and
profits are
P1 = (P- c1)Q1 = (6-3)1 = 3 and P2 = (P- c2)Q2 = (6-2)4 = 16,
which we write 3,16 to complete the profit table.
DC Comics
Marvel
1
3
4
1
6,7
12,5
12,4
3
4,15
6,9
4,6
BA 210 Lesson II.3 Sequential Quantity Competition
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3,16
3,8
0,4
8
Example 1: Stackelberg Duopoly
Find the rollback solution to the Stackelberg Duopoly Game.
Starting from the end of the game,
• if Marvel has chosen Q1=1, then DC will respond with Q2=4,
and so profits are 3 for Marvel and 16 for DC
• if Q1=3, then Q2=3, and so profits are 6 and 9
• if Q1=4, then Q2=3, and so profits are 4 and 6
DC Comics
Marvel
1
3
4
1
6,7
12,5
12,4
3
4,15
6,9
4,6
BA 210 Lesson II.3 Sequential Quantity Competition
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3,16
3,8
0,4
9
Example 1: Stackelberg Duopoly
So Marvel should produce 3 comic books, which leads DC
Comics to produce 3 and generate profit of 6 for Marvel.
DC Comics
Marvel
1
3
4
1
6,7
12,5
12,4
3
4,15
6,9
4,6
BA 210 Lesson II.3 Sequential Quantity Competition
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3,16
3,8
0,4
10
Example 2: First Mover Advantage
Example 2: First Mover Advantage
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Example 2: First Mover Advantage
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.
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 unit costs of production.
BA 210 Lesson II.3 Sequential Quantity Competition
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Example 2: First Mover Advantage
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 = 11-Q (in dollars) in America and
both firms produce at a unit cost of $2. You have a decision to
make about competing with Puffs in New Zealand, where the
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.
You must choose the option that is best for Kleenex.
BA 210 Lesson II.3 Sequential Quantity Competition
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Example 2: First Mover Advantage
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:
Kleenex
Puffs
1
3
6
1
7,7
15,5
12,2
3
5,15
9,9
0,0
6
2,12
0,0
-18,-18
Puffs
Kleenex
1
3
6
1
6,7
12,5
6,2
BA 210 Lesson II.3 Sequential Quantity Competition
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4,15
6,9
-6,0
6
1,12
-3,0
-24,-18
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Example 2: First Mover Advantage
Answer: In Option A, you are
the follower in a Stackelberg
Duopoly. Puffs anticipates
your reactions on the right, and
so chooses to produce 6, you
react with 1 and so you earn 2.
In Option B, you are the leader
in a Stackelberg Duopoly. You
anticipate Puffs’ reactions on
the right, and so choose to
produce 6, Puffs reacts with 1
and so you earn 6.
Kleenex
Puffs
1
3
6
1
7,7
15,5
12,2
3
5,15
9,9
0,0
6
2,12
0,0
-18,-18
Puffs
Kleenex
1
3
6
1
6,7
12,5
6,2
BA 210 Lesson II.3 Sequential Quantity Competition
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4,15
6,9
-6,0
6
1,12
-3,0
-24,-18
15
Example 2: First Mover Advantage
Option B is thus best for Kleenex since Kleenex profits (as a
follower) are 2 in Option A, while Kleenex profits (as the leader)
are 6 in Option B.
BA 210 Lesson II.3 Sequential Quantity Competition
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Example 2: First Mover Advantage
Comment: In this particular case, Kleenex increased production
cost hurt profits less than profits increase because of the first
mover advantage, so it is worth being the first mover.
In other problems, increased production cost hurt profits more
than profits increase because of the first mover advantage, so it is
not worth being the first mover.
BA 210 Lesson II.3 Sequential Quantity Competition
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Example 3: Selling Technology
Example 3: Selling Technology
BA 210 Lesson II.3 Sequential Quantity Competition
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Example 3: Selling Technology
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 unit cost of $2. However,
you just found 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 that secret to Lowes so that both you and Lowes
can produce at unit cost equal to $1?
To answer the question, suppose both Home Depot and Lowes
separately choose to produce either 0, or 1, or 3 units of lumber.
BA 210 Lesson II.3 Sequential Quantity Competition
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Example 4: Selling Technology
Answer: You are the leader in a Stackelberg Duopoly with
inverse demand P = 4-(Q1+Q2). Compare the rollback solution
with unit costs c1 = 1 and c2 = 2, to the solution with c1 = 1 and c2
= 1.
First, compute the payoff table and rollback solution for unit
costs c1 = 1 and c2 = 2:
Lowes
Home
0
1
3
0
0,0
2,0
0,0
1
0,1
1,0
-3,-2
BA 210 Lesson II.3 Sequential Quantity Competition
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0,-3
-1,-6
-9,-12
20
Example 4: Selling Technology
You are the leader in a Stackelberg Duopoly. You anticipate
Lowes’ reactions below, and so choose to produce 1, Lowes
reacts with 1 and so you earn 1 or 2 and Lowes earns 0.
Lowes
Home
0
1
3
0
0,0
2,0
0,0
1
0,1
1,0
-3,-2
BA 210 Lesson II.3 Sequential Quantity Competition
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0,-3
-1,-6
-9,-12
21
Example 4: Selling Technology
Next, compute the payoff table and rollback solution for unit
costs c1 = 1 and c2 = 1:
Lowes
Home
0
1
3
0
0,0
2,0
0,0
1
0,2
1,1
-3,-1
BA 210 Lesson II.3 Sequential Quantity Competition
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0,0
-1,-3
-9,-9
22
Example 4: Selling Technology
You are the leader in a Stackelberg Duopoly. You anticipate
Lowes’ reactions below, and so choose to produce 1, Lowes
reacts with 1 and so you earn 1 and Lowes earns 1.
Lowes
Home
0
1
3
0
0,0
2,0
0,0
1
0,2
1,1
-3,-1
BA 210 Lesson II.3 Sequential Quantity Competition
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0,0
-1,-3
-9,-9
23
Example 4: Selling Technology
Selling technology and reducing c2 = 2 to c2 = 1 has to effects:
• Firm 1’s profit reduces from P1 = 1 or 2 to P1 = 1 for sure
• Firm 2’s profit increases from P2 = 0 to P2 = 1
In particular, selling technology increases total profit P1+P2 from
the uncertain result of 1 or 2 to the certainty of 2. Selling
technology is thus a bargaining problem between Home Depot
and Lowes, and the division of the positive gain from selling
technology is determined by the rules of bargaining.
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
nothing. After deducing that, Home Depot’s best acceptable offer
to Lowes leaves Lowes with a pittance of the gains, which means
Lowes pays the full 1 unit of profit for the technology.
BA 210 Lesson II.3 Sequential Quantity Competition
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Example 4: Colluding
Example 4: Colluding
BA 210 Lesson II.3 Sequential Quantity Competition
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Example 4: Colluding
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.
BA 210 Lesson II.3 Sequential Quantity Competition
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Example 4: Colluding
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 unit costs are 1; and Wilkinson’s unit
costs are 1. Both firms separately produce either 0, 3, or 4 units.
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 3 each. Can Gillette trust Wilkinson?
BA 210 Lesson II.3 Sequential Quantity Competition
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Example 4: Colluding
Answer: You are the leader in a Stackelberg Duopoly with
inverse demand P = 13-(Q1+Q2). Compare the rollback solution
with unit costs c1 = 1 and c2 = 1 with the collusive proposal of
quantities 4 and 2.
First, compute the payoff table and rollback solution:
Wilkinson
Gillette
0
3
4
0
0,0
27,0
32,0
3
0,27
18,18
20,15
BA 210 Lesson II.3 Sequential Quantity Competition
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0,32
15,20
16,16
28
Example 4: Selling Technology
You are the leader in a Stackelberg Duopoly. You anticipate
Wilkinson’ reactions below, and so choose to produce 4,
Wilkinson reacts with 4 and so you earn 16 and Wilkinson earns
16.
The collusive proposal of quantity 3 for each is thus mutually
profitable for the companies. But Gillette cannot trust Wilkinson
since Wilkinson’s best
Wilkinson
response to Gillette’s
Q1 = 3 is Q2 = 4, not 3.
0
3
4
Gillette
0
3
4
0,0
27,0
32,0
0,27
18,18
20,15
BA 210 Lesson II.3 Sequential Quantity Competition
0,32
15,20
16,16
29
Example 5: Merging
Example 5: Merging
BA 210 Lesson II.3 Sequential Quantity Competition
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Example 5: Merging
Question: The market for commercial large jet aircraft consists of
two firms: Airbus and Boeing. As the manager of Boeing, you
enjoy a patented technology that permits your company to
produce jets more quickly and at a lower cost than Airbus. You
use that advantage to be first to choose your profit-maximizing
output level in the market. The demand for jets is Q = 9 - P;
Boeing’s unit costs are 1 and Airbus’s unit costs are 2.
Compute Boeing’s profit, and compute Airbus’s profit. Would it
be profitable for the two companies to merge?
Suppose the firms separately produce quantities 0, 3, or 5 units,
and if merged, the two can each choose from those quantities, and
they can both use the Boeing technology with unit cost 1.
BA 210 Lesson II.3 Sequential Quantity Competition
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Example 5: Merging
Answer: You are the leader in a Stackelberg Duopoly with
inverse demand P = 9-(Q1+Q2). Compare the rollback solution
with unit costs c1 = 1 and c2 = 2 with the with the monopoly
solution.
First, compute the payoff table and rollback solution:
Airbus
Boeing
0
3
5
0
0,0
15,0
15,0
3
0,12
6,3
0,-3
BA 210 Lesson II.3 Sequential Quantity Competition
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0,10
0,-5
-10,-15
32
Example 5: Merging
You are the leader in a Stackelberg Duopoly. You anticipate
Airbus’ reactions below, and so choose to produce 5, Airbus
reacts with 0 and so you earn 15 and Airbus earns 0.
Airbus
Boeing
0
3
5
0
0,0
15,0
15,0
3
0,12
6,3
0,-3
BA 210 Lesson II.3 Sequential Quantity Competition
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0,10
0,-5
-10,-15
33
Example 5: Merging
The monopoly proposal reduces Airbus unit cost to 1, and so
generates a new profit table. But there are no cells in that new
payoff table that increase total profit P1+P2 from the Stackelberg
value of P1+P2=15+0. So in this case, it would not be profitable
for the two companies to merge.
Airbus
Boeing
0
3
5
0
0,0
15,0
15,0
3
0,12
6,3
0,-3
BA 210 Lesson II.3 Sequential Quantity Competition
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0,10
0,-5
-10,-15
34
Summary
Summary
BA 210 Lesson II.3 Sequential Quantity Competition
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Summary
Payoff table entry to any Duopoly Game with inverse demand P
= a - bQ and constant unit costs c1 and c2:
• P = a - b(Q1+Q2)
• Firm 1 profit P1 = (P - c1) Q1
• Firm 2 profit P2 = (P - c2) Q2
BA 210 Lesson II.3 Sequential Quantity Competition
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Review Questions
Review Questions
 You should try to answer some of the following questions
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 210 Lesson II.3 Sequential Quantity Competition
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BA 210
Introduction to Microeconomics
End of Lesson II.3
BA 210 Lesson II.3 Sequential Quantity Competition
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