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 445 Lesson B.3 Sequential Quantity Competition
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Readings
Readings
Baye “Stackelberg Oligopoly” (see the index)
Dixit Chapter 3
BA 445 Lesson B.3 Sequential Quantity Competition
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Overview
Overview
BA 445 Lesson B.3 Sequential Quantity Competition
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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.
BA 445 Lesson B.3 Sequential Quantity Competition
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Example 1: Stackelberg Duopoly
Example 1: Stackelberg Duopoly
BA 445 Lesson B.3 Sequential Quantity Competition
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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.
BA 445 Lesson B.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, 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
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Example 1: Stackelberg Duopoly
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?
BA 445 Lesson B.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 = 5 - (Q1+Q2) and marginal
costs c1 = MC1 = 2 and c2 = MC2 = 1.
Find the rollback solution to the Stackelberg Duopoly
Game.
BA 445 Lesson B.3 Sequential Quantity Competition
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Example 1: Stackelberg Duopoly
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
BA 445 Lesson B.3 Sequential Quantity Competition
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Example 1: Stackelberg Duopoly
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
BA 445 Lesson B.3 Sequential Quantity Competition
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Example 1: Stackelberg Duopoly
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
BA 445 Lesson B.3 Sequential Quantity Competition
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Example 1: Stackelberg Duopoly
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
BA 445 Lesson B.3 Sequential Quantity Competition
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Example 2: First Mover Advantage
Example 2: First Mover Advantage
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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.
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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)
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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. 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.
BA 445 Lesson B.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 = 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?
BA 445 Lesson B.3 Sequential Quantity Competition
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Example 2: 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 the leader in a Stackelberg Duopoly with inverse
demand P = 3 - (Q1+Q2) and marginal costs c1 = MC1 = 2
and c2 = MC2 = 1.
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Example 2: 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.3 Sequential Quantity Competition
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Example 2: 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.3 Sequential Quantity Competition
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Example 2: First Mover Advantage
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
BA 445 Lesson B.3 Sequential Quantity Competition
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Example 2: First Mover Advantage
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.
BA 445 Lesson B.3 Sequential Quantity Competition
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Example 2: First Mover Advantage
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
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.3 Sequential Quantity Competition
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Example 2: 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:




2 = c1 = dR1/dQ1
2 = d/dQ1 (2 – .5Q1) Q1
2 = 2 – Q1
Q1 = 0
BA 445 Lesson B.3 Sequential Quantity Competition
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Example 2: First Mover Advantage
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
BA 445 Lesson B.3 Sequential Quantity Competition
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Example 2: First Mover Advantage
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
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Example 2: First Mover Advantage
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
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Example 3: Selling Technology
Example 3: Selling Technology
BA 445 Lesson B.3 Sequential Quantity Competition
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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.
BA 445 Lesson B.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 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?
BA 445 Lesson B.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 marginal costs c1 = 1 and c2 = 2, to the
solution with c1 = 1 and c2 = 1.
BA 445 Lesson B.3 Sequential Quantity Competition
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Example 4: Selling Technology
Starting from the end, 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
c2 = MC2 = MR2 = 4 – Q1 – 2Q2
to determine the optimal reaction
Q2 = r2 (Q1) = 2 – .5c2 – .5Q1
BA 445 Lesson B.3 Sequential Quantity Competition
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Example 4: Selling Technology
Rolling back to the beginning,
• The Stackelberg leader uses the reaction function r2 (Q1)
to determine its revenue
 R1 = (4 – Q1 – r2 (Q1) )) Q1
 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
BA 445 Lesson B.3 Sequential Quantity Competition
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Example 4: Selling Technology
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 = 0.5625
BA 445 Lesson B.3 Sequential Quantity Competition
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Example 4: Selling Technology
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.
BA 445 Lesson B.3 Sequential Quantity Competition
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Example 4: Colluding
Example 4: Colluding
BA 445 Lesson B.3 Sequential Quantity Competition
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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.
BA 445 Lesson B.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.
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.
BA 445 Lesson B.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 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?
BA 445 Lesson B.3 Sequential Quantity Competition
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Example 4: Colluding
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.
BA 445 Lesson B.3 Sequential Quantity Competition
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Example 4: Colluding
Starting from the end, 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.3 Sequential Quantity Competition
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Example 4: Colluding
Rolling back to the beginning,
• The Stackelberg leader uses the reaction function r2 (Q1)
to determine its revenue
 R1 = (13 – Q1 – r2 (Q1) )) Q1
 R1 = (13 – Q1 – (6 – .5Q1)) Q1
 R1 = (7 – .5Q1) Q1
and its profit-maximizing output level:




1 = c1 = dR1/dQ1
1 = d/dQ1 (7 – .5Q1) Q1
1 = 7 – Q1
Q1 = 6
BA 445 Lesson B.3 Sequential Quantity Competition
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Example 4: Colluding
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
BA 445 Lesson B.3 Sequential Quantity Competition
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Example 4: Colluding
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.
BA 445 Lesson B.3 Sequential Quantity Competition
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Summary
Summary
BA 445 Lesson B.3 Sequential Quantity Competition
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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.
BA 445 Lesson B.3 Sequential Quantity Competition
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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.3 Sequential Quantity Competition
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BA 445
Managerial Economics
End of Lesson B.3
BA 445 Lesson B.3 Sequential Quantity Competition
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