IE54500_Exam_3_Solutions_(Fall_2022)
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Date
Jan 9, 2024
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IE54500 – Exam 3
Dr. David Johnson
Fall 2022
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four hours
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needed while taking the exam; you may print this off, work your solutions, scan them and upload them to
Gradescope. Solutions do not need to be typed, but if your written work is not clearly organized, please
indicate how to follow the order of your logic.
Requests to “show mathematically,” “derive” or “prove” information are asking you to use calculus, algebra,
etc. to formally prove something. “Explain” means
a verbal answer is requested, but you may support your
reasoning with math as appropriate.
Be complete in your reasoning and state your assumptions.
1. A Market with Three Firms
Imagine a market that consists of three firms. One is the market leader, but the other two are both followers,
in the sense that production decisions proceed as follows: (Stage 1) Firm 1, the industry leader, chooses a
quantity to produce,
?
1
≥ 0
; then (Stage 2) Firms 2 and 3 observe the value of
?
1
and then simultaneously
choose their levels of production
?
2
and
?
3
. Assume that each firm has the same constant marginal cost of
production,
?
, and no fixed costs, and that the market has a price-demand relationship where
?(?) = ? − ?
for the total quantity produced,
? = ?
1
+ ?
2
+ ?
3
.
a)
Is this a game of perfect information or imperfect information? Be sure to explain why. (2 points)
This is a game of perfect information, because at each stage, all players know the full history of the
game; we are told that Firms 2 and 3 observe the value
?
1
chosen by Firm 1.
b)
What is the subgame perfect equilibrium of this game, i.e., the subgame perfect production choices for
each of the three firms?
Explain your reasoning.
(5 points)
First, we can solve for the Nash equilibrium of the Stage 2 game where, for Firm 2, given the observed
value
?
1
and Firm 3’s strategy to play
?
3
, Firm 2 chooses
?
2
to solve
max (? − ?
1
− ?
2
− ?
3
− ?)?
2
which has first-order condition
? − ?
1
− 2?
2
− ?
3
− ? = 0
⇒ ?
2
∗
=
? − ?
1
− ?
3
− ?
2
Similarly,
?
3
∗
=
? − ?
1
− ?
2
− ?
2
Solving this system of equations tells us that the best-response functions are
?
2
∗
(?
1
) = ?
3
∗
(?
1
) =
? − ? − ?
1
3
Firm 1 is able to solve for these best-response functions, so they can choose
?
1
that maximizes their
profits knowing how Firms 2 and 3 will react. Substituting, Firm 1 therefore solves
max
𝑞
1
(? − ?
1
−
? − ? − ?
1
3
−
? − ? − ?
1
3
− ?) ?
1
which has first-order condition
? − ? − 2?
1
3
= 0 ⇒ ?
1
∗
=
? − ?
2
Therefore,
?
2
∗
= ?
3
∗
=
? − ? −
? − ?
2
3
=
? − ?
6
c)
Explain intuitively why Firm 1 produces a different quantity than Firm 2 and Firm 3. (1 point)
Firm 1 has a “first
-
mover advantage” that allows them to force Firms 2 and 3 to produce a smaller
share of the aggregate production because they know that the other firms will still respond rationally
to maximize their own profits, no matter what quantity Firm 1 produces.
d)
In the Stackelberg duopoly model, we found that the total quantity produced in the market was
3
4
(? − ?)
. This problem is similar, except that we have two firms following the leader instead of one.
Explain the intuition behind why this results in a higher aggregate level of production.
Note: if your
answer in part b) is less than or equal to this quantity, this is a hint your answer may be incorrect.
(2 points)
The aggregate level of production in this case is
? =
? − ?
2
+ 2 ∙
? − ?
6
=
5
6
(? − ?) >
3
4
(? − ?)
The basic intuition behind this is that we’ve added a third firm, which increases competition in the
market. Intuitively, we’ve said that greater competition drives down prices, approaching the marginal
cost
?
(the price under perfect competition), which is associated with a higher level of production
because of the price-demand relationship.
2. Static Games of Complete Information
Specify the mixed-strategy Nash equilibrium of the following normal-form game (and show your work). Note
that the payoffs
?, ?
indicate a payoff of
?
to Player A and
?
to Player B. (10 points)
Player B
Left
Right
Player A
Top
4, 2
0, 4
Bottom
2, 4
6, 0
To identify a mixed-strategy Nash equilibrium, denote
?
as the probability of Player A playing Top, and
?
as the
probability of Player B playing Left. Each player wishes to maximize their expected payoffs:
𝔼𝜋
?
= 4?? + 2(1 − ?)? + 6(1 − ?)(1 − ?) = −6? − 4? + 8?? + 6
𝐹?𝐶: − 6 + 8?
∗
= 0 ⇒ ?
∗
= 3/4
𝔼𝜋
?
= 2?? + 4?(1 − ?) + 4(1 − ?)? = 4? − 6?? + 4?
𝐹?𝐶: − 6?
∗
+ 4 = 0 ⇒ ?
∗
= 2/3
Both of these conditions present valid probabilities,
0 ≤ ?
∗
, ?
∗
≤ 1
, and can hold simultaneously, so this
represents a mixed-strategy Nash equilibrium.
3. Continuous Process Improvement
Two firms are in competition, producing a device that embeds a wireless microphone and speaker into a N-95
mask so that individuals wearing masks can still be heard clearly. The price-demand relationship for this device
is
? = 100 − ?
1
− ?
2
, where
?
𝑖
indicates the quantity produced by firm
𝑖
. Each firm initially has production
costs
?
𝑖
(?
𝑖
) = 20?
𝑖
, but Firm 1 has the opportunity to hire a smart Purdue industrial engineering grad to
reduce their costs. If the firm hires them at a wage of
? > 0
, the cost of production for Firm 1 would drop to
?
1
(?
1
) = 10?
1
. Firm 2 does not have the same connections to Purdue, so they do not have this chance to
improve their processes. This can be modeled as a dynamic game in which Firm 1 chooses whether or not to
hire a Purdue grad, and then in the next stage of the game, both firms simultaneously choose what quantities
?
𝑖
to produce.
Clearly, Firm 1 must weigh the wage
?
against the reduced marginal costs that would make them more
competitive, deciding whether their total profits would be higher with or without the hire. What is the
maximum wage
?
Firm 1 would be willing to pay for the reduction in their marginal costs? Assume that Firm
1’s decision whether to hire the Purdue grad is common knowledge; in other words, both fi
rms know if a
Purdue grad has been hired to provide process improvements. (10 points)
Firm 1 will make whichever hiring decision leads to greater profits. Presumably, they would increase their
profits if the Purdue grad was free, because it reduces their marginal costs. However, if the hiring cost is
greater than 0, it’s possible that the Purdue grad may be more expensive than the marginal profit Firm 1 would
gain. Therefore, the maximum amount they would be willing to pay should be the difference in profits between
the two circumstances. We need to solve for what those profits would be.
Start by finding Firm 1’s optimal level of production in each case. Without hiring a Purdue grad,
𝜋
1
= (100 − ?
1
− ?
2
− 20)?
1
which has first-order condition
80 − ?
2
− 2?
1
= 0 ⇒ ?
1
∗
= 40 − ?
2
/2
By symmetry in the case where both firms have equal marginal costs,
?
2
∗
= 40 −
?
1
2
= 40 − 20 +
?
2
4
⇒
3
4
?
2
∗
= 20 ⇒ ?
2
∗
=
80
3
= ?
1
∗
If Firm 1 does make the hire, then
𝜋
1
= (100 − ?
1
− ?
2
− 10)?
1
Which has first-order condition
90 − ?
2
− 2?
1
= 0 ⇒ ?
1
∗
= 45 −
?
2
2
Firm 2 still has the same first-order condition, so
?
2
∗
= 40 −
?
1
∗
2
= 40 −
45
2
+
?
2
4
⇒
3
4
?
2
=
35
2
⇒ ?
2
∗
=
70
3
, ?
1
∗
= 45 −
70
6
=
100
3
Without investing, this leads to a profit of
𝜋
1
= (100 − 20 − 2 ∙
80
3
) ∙
80
3
= (
80
3
)
2
= 711.11
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When hiring the Purdue grad, Firm 1’s profit is
𝜋
1
= (90 −
170
3
) ∙
100
3
= (
100
3
)
2
= 1111.11
Therefore, Firm 1’s willingness to pay is
1111.11 − 711.11 = 400
.
4. Incomplete Information
This question is a continuation of Question 3, so consider the same basic setup with respect to the demand
curve, production costs, etc. The
only difference is whether Firm 2 knows about Firm 1’s decision in the first
stage of the game.
a)
Suppose that Firm 2 does not know whether Firm 1 has hired a Purdue grad to reduce their marginal
costs, and Firm 2 has little insight into how much it would cost to hire a Purdue grad. They believe
there is a 1/3 probability that Firm 1 has made the hire. Explain why this change makes the game
equivalent to a static game of incomplete information. (2 points)
When Firm 2 does not know whether Firm 1 has hired a Purdue grad, this is an example of incomplete
information because they do not know Firm 1’s profit function (because they do not know
?
for Firm 1).
Firm 1’s marginal cost is therefore equivalent to its type. Because Firm 2 does not observe Firm 1’s
type,
this is equivalent to a static game because neither firm observes anything about the other’s
action before making their own; it is as if they move simultaneously. They also do not take any action
after the point where they observe something about the other firm
’
s action.
b)
Would Firm 2 increase their production, decrease it, or keep it the same, compared to their choice in
Question 3 when Firm 1 does hire a Purdue grad?
For this question, only a verbal argument
explaining your intuition is required for full credit; no math is necessary.
(2 points)
Because Firm 2 does not know Firm 1’s hiring decision, they only cho
ose a single production quantity
that maximizes their profit in expectation, consistent with their beliefs. Previously, Firm 2 produced
70/3 units if Firm 1 hired a Purdue grad, and 80/3 units if not. With uncertainty about Firm 1’s hiring
decision, this should mean that maximizing their profits in expectation would result in a production
choice in between these two values, i.e., greater than their decision when Firm 1 does hire a Purdue
grad and less than their decision when Firm 1 does not. Their choice should, in fact, be the average of
the two quantities from Question 3, weighted by Firm 2’s beliefs about which type Firm 1 is.
c)
Given this incomplete information setup, what are
both firms’ production decisions, and both firms’
profits, if Firm 1 does hire a Purdue grad at a wage of
? = 200
? (6 points)
In this case, denote the Firm 1 type with low marginal costs with a superscript L, and the high marginal
cost type with a superscr
ipt H. Firm 2’s expected profits, consistent with their beliefs, are
𝔼[𝜋
2
] = [
1
3
(100 − ?
1
𝐿
− ?
2
− 20) +
2
3
(100 − ?
1
𝐻
− ?
2
− 20)] ?
2
= (80 −
1
3
?
1
𝐿
−
2
3
?
1
𝐻
− ?
2
)?
2
which has first-order condition
80 −
1
3
?
1
𝐿
−
2
3
?
1
𝐻
− 2?
2
= 0 ⇒ ?
2
∗
= 40 −
1
6
?
1
𝐿
−
1
3
?
1
𝐻
Meanwhile, Firm 1 knows their type, so they can choose different production quantities depending on
their type. The high-cost type solves
𝜋
1
𝐻
= (80 − ?
1
𝐻
− ?
2
)?
1
𝐻
with first-order condition
80 − ?
2
− 2?
1
𝐻∗
= 0 ⇒ ?
1
𝐻∗
= 40 −
?
2
2
The low-cost type solves a similar problem except with a marginal cost of 10, leading to
?
1
𝐿∗
= 45 −
?
2
2
(Technically, the wage
?
should also be included in the low-
cost type’s profit function, but it does not
affect the first-order condition or production decision because it is a fixed/sunk cost. No points were
deducted if you left
?
out at this point of calculating
?
1
𝐿∗
.)
If we plug these values into the best-response function for Firm 2, we find
?
2
∗
= 40 −
1
6
(45 −
?
2
2
) −
1
3
(40 −
?
2
2
) = 40 −
45
6
−
40
3
+
?
2
12
+
?
2
6
=
115
6
+
?
2
4
⇒
3?
2
∗
4
=
115
6
⇒ ?
2
∗
=
115 ∙ 4
18
= 25.56
This leads to decisions for each type of Firm 1:
?
1
𝐻∗
= 40 −
?
2
∗
2
= 27.22, ?
1
𝐿∗
= 32.22
Assuming then that Firm 1 does hire a Purdue grad, they must be the low-cost type, so the profits for
each firm are
𝜋
1
𝐿
= (90 − 32.22 − 25.56) ∙ 32.22 − 200 = 1038.27 − 200 = 838.27
𝜋
2
= (80 − 32.22 − 25.56) ∙ 25.56 = 567.90
5. Extra Credit
(2 points)
Describe how you would strategically choose a response to the following question:
Would you like 1, 2, or 3 points of extra credit? If you select a number that 50% or more of the class has also
selected, you will receive no extra credit; if you select a number that less than 50% of the class selected, you
will receive that number of extra credit points.
Technically, this is representable as a static game of complete information because each student acts without
knowing any other students’ action, and everyone can identify the payoff function for each student. Where
?
𝑖
is
the number of bonus points requested by student
𝑖
, the class has
?
students, and
|?|
represents the number of
students submitting a request for
? ∈ {1,2,3}
:
𝜋
𝑖
= {
?
𝑖
𝑖𝑓 |?
𝑖
| < ?/2
0
𝑖𝑓 |?
𝑖
| ≥ ?/2
Think
about what would have to be true for no one to have incentive to deviate unilaterally, given others’
strategies. For a pure-strategy equilibrium to exist, we must at least have the maximum possible number of
students requesting 3 points; otherwise, someone requesting 1 or 2 points would have incentive to deviate to
requesting 3. Similarly, we must then have the maximum possible number of students requesting 2, with only
one or two students requesting 1 (depending on whether
?
is odd or even).
Under such an equilibrium, each student receives the maximum points they can get, given the other students’
strategies (because increasing their own bid results in reaching the threshold for
|?
𝑖
|
that would earn 0 points
instead). This leads to a countable and large number of possible equilibria, so we should think about what
might really happen. In practice, there is no reason for any particular student to want to take the strategy
where
?
𝑖
= 1
or
?
𝑖
= 2
. A symmetric strategy would therefore be more appealing, which could take the form of
a mixed strategy with a 1/3 probability of selecting 1, 2, or 3. For large
?
, there would be a negligible chance of
any given bid resulting in 0 points, so the expected payoff would be very slightly less than 2. This, however,
leads to the possibility that deviating to a pure strategy requesting
?
𝑖
= 3
could lead to a higher payoff in
expectation, so this might only work for a certain range of
?
. Maximizing the expected payoff would require
adjusting the probabilities of choosing 1, 2, or 3, with a higher probability on larger bids, but not so high that it
leads to a large enough probability of
|? = 3| ≥ ?/2
to reduce the expected payoff. The optimal weights
should be a function of
?
.
(Nothing this sophisticated was required in actuality to be awarded 2 points of extra credit. I had fun thinking
about this question and hope that it was interesting for you, too.)
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