BEPP 250
Managerial Economics Final Exam
Spring 2016
Name:
Penn ID #:
Section Number:
Professor’s Name:
My signature below certifies that I have complied with the University of Pennsylvania’s Code of
Academic Integrity in completing this examination.
(SIGNATURE)
INSTRUCTIONS
1. Do not take the exam apart.
2. Write your section number on this page.
3. Write your legal name and your Penn Student ID number on this page and every page of this
exam where a space is provided, including the back of the exam.
4. Only non-programmable calculators are allowed. No other electronics are allowed. No written
or printed help is allowed either.
5. Show and EXPLAIN your work. No credit will be given if work is not shown or explanations are
missing.
6. Write your solution in the space provided after each question. If you run out of space, you may
use the “Overflow Space” at the end of each problem.
7. Write legibly!
8. Initial the proctor’s class list when you hand in your exam.
DO NOT START UNTIL 6:00 PM. DO NOT LEAVE THE ROOM WITH THIS EXAM EVEN IF
YOU ARE PLANNING TO WITHDRAW FROM THE COURSE.
NAME: ________________________________
PENN I.D. #: ________________________________
1. [25 points] Consider two firms in a market that offer differentiated products. Firms are
assumed to choose prices and, for simplicity, let us suppose that they can choose one of
five prices: {1,2,3,4,5}. The payoff matrix below shows firms’ profits for each pair of prices
that they select where the first number in a cell is firm 1’s profit and the second number is
firm 2’s profit. The payoff matrix is symmetric. Initially suppose that firms simultaneously
choose their prices, do so only once, and choose price in order to maximize profit.
Price of
Firm 1
1
2
3
4
5
1
8,8
12,9
11,10
8,11
0,12
2
9,12
15,15
14,18
12,19
5,20
Price of Firm 2
3
10,11
18,14
17,17
16,21
10,22
4
11,8
19,12
21,16
20,20
15,23
5
12,0
20,5
22,10
23,15
19,19
a) [5 points] Find a firm’s best response function.
Answer: A best response is a firm’s optimal price given some price for the other firm. A
price of 2 is the best response to the other firm choosing a price of 1, 2, or 3. A price of 3
is the best response to the other firm choosing a price of 4. A price of 4 is the best
response to the other firm choosing a price of 5.
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NAME: ________________________________
PENN I.D. #: ________________________________
b) [4 points] Find all Nash equilibria.
Answer: The unique Nash equilibrium is (2,2).
c) [2 points] Explain why firm 1 choosing a price of 4 and firm 2 choosing a price of 5 is
not a Nash equilibrium.
Answer: If firm 1 chooses 4 then the best response for firm 2 is 3, not 5. A Nash
equilibrium must have both firms choosing best replies so this strategy pair is not a
Nash equilibrium
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NAME: ________________________________
PENN I.D. #: ________________________________
For part (d) only, suppose firms move sequentially: Firm 1 chooses its price and then, after
observing firm 1’s price, firm 2 chooses its price.
d) [5 points] Find the prices that firm charge at a subgame perfect Nash equilibrium.
(Hint: Use backward induction.)
Answer: (Note: A student only needs to derive the SPNE prices and does not need to
describe the SPNE strategies.) Let us first derive firm 2’s SPNE strategy. If firm 1
chooses a price of 1, 2 or 3 then firm 2 will optimally respond with a price of 2. If
firm 1 chooses a price of 4 then firm 2 will optimally respond with a price of 3. If firm
1 chooses a price of 5 then firm 2 will optimally respond with a price of 4. Now
consider firm 1. If it chooses a price of 1 then firm 2 will respond with a price of 2
and firm 1’s profit is 9. If firm 1 chooses a price of 2 then firm 2 will respond with a
price of 2 and firm 1’s profit is 15. If firm 1 chooses a price of 3 then firm 2 will
respond with a price of 2 and firm 1’s profit is 14. If firm 1 chooses a price of 4 then
firm 2 will respond with a price of 3 and firm 1’s profit is 16. If firm 1 chooses a price
of 5 then firm 2 will respond with a price of 4 and firm 1’s profit is 15. Thus, firm 1’s
optimal price is 4. The SPNE prices are 4 for firm 1 and 3 for firm 2.
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NAME: ________________________________
PENN I.D. #: ________________________________
For parts (e)-(f), assume that firms engage in an infinitely repeated game. In each period, firms
simultaneously choose prices and earn profits as specified in the above payoff matrix. In each
period, a firm chooses price knowing all of the past prices charged by both firms. A firm acts to
maximize the present value of its profit stream with discount factor where
Thus, if
is firm i’s profit in period t then its payoff is the infinite sum:
Recall that the present value of receiving an amount
is
in each of an infinite number of periods
e) [5 points] Consider a strategy in which a firm prices at 4 in period 1 and prices at 4 in
any future period as long as both firms priced at 4 in all past periods. If at any time a
firm prices different from 4 then a firm prices at 2 in all ensuing periods. Find the
conditions for this strategy to be a subgame perfect Nash equilibrium. How high
must the discount factor be for these conditions to be satisfied?
Answer:
(
)
.
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NAME: ________________________________
PENN I.D. #: ________________________________
f) [4 points] The strategy in the preceding part has a punishment of pricing at 2 in all
periods if a firm deviates from the collusive price of 4. Suppose instead the
punishment has firms price at 2 for just one period (rather than all remaining
periods) and then they return to pricing at 4. Find the conditions for this strategy to
be a subgame perfect Nash equilibrium. How high must the discount factor be for
these conditions to be satisfied? Is the condition on the discount factor more or less
stringent than in the preceding part? Explain your answer. (Note: Try to answer this
last question even if you cannot answer the first two questions.)
Answer:
(
)
It is more stringent because the punishment is weaker – only one period of lower
profit rather than an infinite number of periods of lower profit – and this creates a
stronger incentive to deviate. Firms must then attach more weight to future profit in
order for it to be optimal to set the collusive price.
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NAME: ________________________________
PENN I.D. #: ________________________________
OVERFLOW PAGE FOR QUESTION #1
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NAME: ________________________________
PENN I.D. #: ________________________________
2. [25 points] Consider a market with two firms that offer identical products. The inverse
market demand function is
where is firm 1’s quantity
and is firm 2’s quantity. Firms make simultaneous quantity decisions and the price each
firm receives is
Firm 1’s cost function is
and firm 2’s cost
function is
a) [2 points] Find each firm’s marginal revenue.
Answer: Firm 1’s revenue is [
. Firm 2’s revenue is [
is
Page 8 of 28
]
so its marginal revenue is
] so its marginal revenue
NAME: ________________________________
PENN I.D. #: ________________________________
b) [8 points] Find each firm’s best response function.
Answer: Firm 1’s profit function is
[
]
.
Its optimal quantity is that which maximizes this profit which is found by setting
marginal profit equal to zero and solving for its quantity:
Firm 2’s profit function is
[
]
.
Its optimal quantity is that which maximizes this profit which is found by setting
marginal profit equal to zero and solving for its quantity:
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NAME: ________________________________
PENN I.D. #: ________________________________
c) [8 points] How much will each firm produce at a Nash equilibrium?
Answer: The Nash equilibrium quantities, denoted and , must both be best
replies. Thus, they are defined by the following two equations:
and
.
At the Nash equilibrium, firm 1 produces 20 and firm 2 produces 15.
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NAME: ________________________________
PENN I.D. #: ________________________________
d) [2 points] In part (c), you should have found that firm 1 produces more than firm 2.
Why is that the case?
Answer: Firm 1 has a lower marginal cost than firm 2. Holding everything else
constant, a firm with lower marginal cost will produce more. Given they both have
the same marginal revenue functions, firm 1 will find it optimal to produce at a
higher rate than firm 2.
e) [5 points] Suppose the government puts in place a regulation requiring that each
firm produce at least 18 units. How much each will firm produce?
Answer: At the unconstrained Nash equilibrium, firm 1 produces 20 units and firm 2
produces 15 units. Thus, firm 2’s quantity does not satisfy the constraint. Suppose
firm 2 raises its output to 18. In that case, firm 1 will no longer want to supply 20; its
best reply is
As quantity exceeds the minimum output
requirement then firm 1 will produce 18.5. Note that firm 2’s unconstrained best
reply to 18.5 is
As its profit is strictly decreasing for
quantities above its best reply, it’ll try to produce as close to its best reply while
satisfying the government regulation. Hence, it will produce 18. In sum, firm 1
produces 18.5 and firm 2 produces 18.
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NAME: ________________________________
PENN I.D. #: ________________________________
OVERFLOW PAGE FOR QUESTION #2
Page 12 of 28
NAME: ________________________________
PENN I.D. #: ________________________________
3. [25 points] A used car can be a bad car or a good car. A bad car has a 10% chance of
working well, and a 90% chance of breaking down. On the other hand, a good car has a
90% chance of working well, and a 10% chance of breaking down. A working car is worth
$90,000 while a car that breaks down is worth $30,000 (e.g. repair costs or loss of
$60,000).
Assume you have a utility function given by U(C) = 2C where C is measured in 10,000$. For
example, the utility you get from a working car is U(9) = 18 while the utility you get from a car
that breaks down is U(3) = 6.
a) [2 points] Are you risk averse, risk neutral, or risk loving? Show mathematically.
Answer: You are risk neutral since U’’(C) = 0.
b) [7 points] For this question, assume that you know the car is good. What is the
expected utility of owning a good car?
Answer: Expected utility from owning a good car is just 0.1(2)(3) + 0.9(2)(9) = 16.8.
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NAME: ________________________________
PENN I.D. #: ________________________________
c) [8 points] Suppose you can buy insurance that fully covers the loss in the event of a
breakdown. Let p be the price of fully insuring your car (in 10,000$). Assume for this
question that you know that the car is bad. How much are you willing to pay for full
insurance if you own a bad car?
Answer: Let p be the price of fully insuring your car (in 10,000$). Expected utility if
you buy full insurance is 2(9 – p) while expected utility from not having insurance
when you have a bad car is 0.1(2)(9) + 0.9(2)(3) = 7.2. Therefore you are willing to
pay up to p = 5.4 or $54,000.
Page 14 of 28
NAME: ________________________________
PENN I.D. #: ________________________________
d) [2 points] How will your willingness-to-pay from (c) change (if at all) if your utility
function exhibits U’’(C) < 0? Explain why in 2 sentences or less.
Answer: Since you are now risk averse, WTP for insurance goes up. In other words,
you are willing to pay more to avoid the risk when U’’(C) < 0.
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NAME: ________________________________
PENN I.D. #: ________________________________
For the remaining questions, you only know that there is equal probability (i.e. 50%) that a car
is either good or bad. A bad car still has a 10% chance of working well, and a 90% chance of
breaking down; a good car still has a 90% chance of working well, and a 10% chance of breaking
down. There are four possible states of nature, and the table below gives the probabilities of
each and the corresponding value of the car in that state.
State
Probability
Car value (in 10,000$)
Car is good and works
0.5(0.9) = 0.45
9
Car is good and breaks
down
0.5(0.1) = 0.05
3
Car is bad and works
0.5(0.1) = 0.05
9
Car is bad and breaks down
0.5(0.9) = 0.45
3
e) [3 points] How much are you willing to pay for full insurance?
Answer: Let p be the price of fully insuring your car (in 10,000$). Expected utility if you
buy full insurance is 2(9 – p) while expected utility from not having insurance
0.5[0.1(2)(9) + 0.9(2)(3)] + 0.5[0.1(2)(3) + 0.9(2)(9)]= 0.5(7.2 + 16.8) = 12. Therefore you
are willing to pay up to p = 3 or $30,000.
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NAME: ________________________________
PENN I.D. #: ________________________________
f) [3 points] Suppose the government requires every car to be insured unless you can
prove that your car is a good car. That is, you can opt out of purchasing insurance if
your car is provably good. You can prove that you have a good car by undergoing a
pre-purchase inspection (PPI) which costs r (in 10,000$) that you have to pay
upfront. How much are you willing to pay for the PPI if the price of insurance is equal
to that which you found in (e)?
Answer: Since expected utility is linear, it is obvious that the WTP for insurance in
the event of owning a good car is always lower than the price in (e) even if you have
to take into account the additional cost of the PPI. Thus, if the PPI reveals that it’s a
good car, it is optimal to not buy insurance. Let p*=3 be the price found in (e). In this
case, expected utility from a PPI is given by 0.5[2(9 – p* - r)] + 0.5[0.1(2)(3 - r) +
0.9(2)(9 - r)] = 2(7.2 – r), where 2(9 – p* - r) is the utility from learning that the car is
bad (so insurance is bought) and 0.1(2)(3 - r) + 0.9(2)(9 – r) is the utility from learning
the car is good (so insurance is not bought). The expected utility of not getting a PPI
(which requires purchasing the insurance) is equal to the expected utility in (e)
which is 2(9-p*) or 12. Therefore you are willing to pay up to r = 1.2 or $12,000.
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NAME: ________________________________
PENN I.D. #: ________________________________
OVERFLOW PAGE FOR QUESTION #3
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NAME: ________________________________
PENN I.D. #: ________________________________
4. [25 points] As part of its ongoing effort to alleviate the spectrum crunch, the Federal
Communications Commission (FCC) intends to auction off a block of spectrum to wireless
carriers. AT&T and T-Mobile are participating in the auction that the FCC holds. Because of
their different pre-existing network structures, AT&T and T-Mobile have different
valuations for these spectrum rights. Each wireless carrier knows its own valuation but not
that of its rival.
Let vA denote the valuation of AT&T for the spectrum and vT that of T-Mobile. T-Mobile
believes that vA is either $200 million or $500 million and that both valuations are equally
likely. AT&T believes that vT is either $400 million or $600 million with equal probability.
Valuations are independent across wireless carriers.
The FCC conducts a second-price sealed-bid auction for the spectrum rights.
a) [3 points] Explain in no more than 5 sentences that it is optimal for a wireless carrier
to bid its valuation regardless of how its competitor is bidding.
Answer: Bidding one’s valuation is optimal because the bid only affects the probability
of winning but not the payoff if one wins the auction. Hence, one wants to increase
one’s bid as much as possible. However, bidding above one’s valuation runs the risk of a
negative payoff if one wins the auction. It is therefore optimal to bid one’s valuation.
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NAME: ________________________________
PENN I.D. #: ________________________________
b) [3 points] If each wireless carrier bids its valuation, what is the expected revenue of
the FCC from the auction?
Answer: If each wireless carrier bids its valuation, then the bids of AT&T and TMobile are (bA=200,bT=400), (bA=200,bT=600), (bA=500,bT=400), and (bA=500,bT=600)
with equal probability. The high bidder wins and pays the low bid. The FCC’s
expected revenue is therefore ¼*200+¼*200+¼*400+¼*500=$325 million.
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NAME: ________________________________
PENN I.D. #: ________________________________
The FCC instead decides to conduct a first-price sealed-bid auction for the spectrum
rights. For simplicity, assume that each wireless carrier can bid either its entire valuation
or half of its valuation. That is, if AT&T’s valuation is vA, then it can either bid bA=vA or
bA=vA/2. Similarly, T-Mobile chooses between bidding bT=vT and bT=vT/2.
c) [2 points] Explain intuitively in no more than 2 sentences why it is no longer optimal
for a wireless carrier to bid its valuation
Answer: Bidding one’s valuation in a first-price auction entails zero profits. Making a
profit requires shading the bid below the valuation.
d) [4 points] Show mathematically that it is not a Nash equilibrium of the first-price
sealed-bid auction for each wireless carrier to bid its valuation.
Answer: We need to show that at least one wireless carrier has an incentive to
deviate. Suppose AT&T bids its valuation and that T-Mobile’s valuation is vT=$600
million. If T-Mobile bids its entire valuation, it makes zero profits even though it wins
the auction for sure. If T-Mobile bids half its valuation, i.e., $300 million, it wins the
auction with probability ½ in case vA=200 (and thus bA=200) and loses with
probability ½ in case vA=500 (and thus bA=500). T-Mobile’s expected payoff is
therefore ½*(600-300)=$150 million. Hence, T-Mobile is better off shading its bid.
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NAME: ________________________________
PENN I.D. #: ________________________________
e) [4 points] Suppose T-Mobile bids half its valuation. If vA=500, what is AT&T’s
expected payoff if it bids half its valuation? What if vA=200?
Answer: First suppose vA=500. If AT&T bids bA=vA/2=250, then it wins the auction
with probability ½ in case vT=400 (and thus bT=200) and loses with probability ½ in
case vT=600 (and thus bT=300). AT&T’s expected payoff is therefore ½*(500250)=$125 million.
Next suppose vA=200. If AT&T bids bA=vA/2=100, then it loses the auction for sure.
AT&T’s expected payoff is therefore $0 million.
f) [5 points] Show mathematically that each firm bidding half its valuation is a Nash
equilibrium of the first-price sealed-bid auction.
Answer: Between parts (c) and (e), we have already shown that if T-Mobile bids half
its valuation, then AT&T has no incentive to bid is entire valuation rather than half
its valuation.
It remains to show that T-Mobile has no incentive to deviate. Suppose first that
AT&T bids half its valuation and that vT=400. If T-Mobile bids bT=vT/2=200, then it
wins the auction with probability ½ in case vA=200 (and thus bA=100) and loses with
probability ½ in case vA=500 (and thus bA=250). T-Mobile’s expected payoff is
therefore ½*(400-200)=$100 million.
Next suppose that vT=600. If T-Mobile bids bT=vT/2=300, then it wins the auction for
sure. T-Mobile’s expected payoff is therefore 600-300=$300 million. Taken together,
this shows that T-Mobile has no incentive to deviate.
Page 22 of 28
NAME: ________________________________
PENN I.D. #: ________________________________
The FCC decides to auction off two blocks of spectrum and conducts a separate first-price
sealed-bid auction for each block. The two auctions take place at the same time. Because
the two auctions are held simultaneously, each wireless carrier must submit a bid in both
auctions. A bid must be at least $10 million and can be any multiple of $10 million
thereafter. If two bidders submit the same bid, then a coin is flipped to determine the
winner. Because of technological constraints, AT&T and T-Mobile can each profitably use
just one block of spectrum. That is, each wireless carrier’s valuation for a second block of
spectrum is zero.
g) [4 points] What are the lowest bids in the two auctions at which each wireless
carrier can be sure to acquire one block of spectrum? Describe the bidding strategies
of the wireless carriers that lead to this outcome and explain in no more than 5
sentences why they are a Nash equilibrium.
Answer: AT&T bids $20 million for the first block and $10 million for the second
block. Given AT&T’s strategy, T-Mobile’s best response is to bid $10 million for the
first block and $20 million for the second block. Hence, AT&T wins the auction for
the first block and T-Mobile that for the second block. Note first that there is
another equilibrium in which the roles of AT&T and T-Mobile are interchanged. Note
also that if both wireless carriers bid $10 million in both auctions, then they cannot
ensure that each of them gets one block of spectrum; in fact, depending on the coin
toss, one wireless carrier may get two blocks of spectrum.
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NAME: ________________________________
PENN I.D. #: ________________________________
OVERFLOW PAGE FOR QUESTION #4
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NAME: ________________________________
PENN I.D. #: ________________________________
ADDITIONAL OVERFLOW PAGE
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NAME: ________________________________
PENN I.D. #: ________________________________
ADDITIONAL OVERFLOW PAGE
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NAME: ________________________________
PENN I.D. #: ________________________________
ADDITIONAL OVERFLOW PAGE
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NAME: ________________________________
PENN I.D. #: ________________________________
DO NOT WRITE BELOW
Question
a.
b.
c.
1.
d.
e.
f.
Max Points
5
4
2
5
5
4
2.
a.
b.
c.
d.
e.
2
8
8
2
5
3.
a.
b.
c.
d.
e.
f.
2
7
8
2
3
3
4.
a.
b.
c.
d.
e.
f.
g.
3
3
2
4
4
5
4
TOTAL POINTS:
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Score
Total