BA 445 Final Exam Version 1 1 This is a 150-minute exam (2hr. 30 min.). There are 7 questions (about 21 minutes per question). To avoid the temptation to cheat, you must abide by these rules then sign below after you understand the rules and agree to them: Turn off your cell phones. You cannot leave the room during the exam, not even to use the restroom. The only things you can have in your possession are pens or pencils and a simple non-graphing, non-programmable, non-text calculator. All other possessions (including phones, computers, or papers) are prohibited and must be placed in the designated corner of the room. Possession of any prohibited item (including phones, computers, or papers) during the exam (even if you don’t use them but keep them in your pocket) earns you a zero on this exam, and you will be reported to the Academic Integrity Committee for further action. Print name here:______________________________________________ Sign name here:______________________________________________ Each individual question on the following exam is graded on a 4-point scale. After all individual questions are graded, I sum the individual scores, and then compute that total as a percentage of the total of all points possible. I then apply a standard grading scale to determine your letter grade: 90-100% A; 80-89% B; 70-79% C; 60-70% D; 0-59% F Finally, curving points may be added to letter grades for the entire class (at my discretion), and the resulting curved letter grade will be recorded on a standard 4-point numerical scale. Tip: Explain your answers. And pace yourself. When there is only ½ hour left, spend at least 5 minutes outlining an answer to each remaining question.
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BA 445 Final Exam Version 1
1
This is a 150-minute exam (2hr. 30 min.). There are 7 questions (about 21 minutes per question). To avoid the temptation to cheat, you must abide by these rules then sign below after you understand the rules and agree to them:
Turn off your cell phones.
You cannot leave the room during the exam, not even to use the restroom.
The only things you can have in your possession are pens or pencils and a simple non-graphing, non-programmable, non-text calculator.
All other possessions (including phones, computers, or papers) are prohibited and must be placed in the designated corner of the room.
Possession of any prohibited item (including phones, computers, or papers) during the exam (even if you don’t use them but keep them in your pocket) earns you a zero on this exam, and you will be reported to the Academic Integrity Committee for further action.
Print name here:______________________________________________ Sign name here:______________________________________________ Each individual question on the following exam is graded on a 4-point scale. After all individual questions are graded, I sum the individual scores, and then compute that total as a percentage of the total of all points possible. I then apply a standard grading scale to determine your letter grade: 90-100% A; 80-89% B; 70-79% C; 60-70% D; 0-59% F
Finally, curving points may be added to letter grades for the entire class (at my discretion), and the resulting curved letter grade will be recorded on a standard 4-point numerical scale. Tip: Explain your answers. And pace yourself. When there is only ½ hour left, spend at least 5 minutes outlining an answer to each remaining question.
BA 445 Final Exam Version 1
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Comparing Markets
Question 1. Consider the cost function
C(Q) = 300 + 20Q + 4Q2
for Apple to produce the new iPhone 5 smart phone. Using that cost function for the iPhone 5, determine the profit-maximizing output and price for the iPhone 5, and discuss its long-run implications, under three alternative scenarios:
a. Apple’s iPhone 5 is a perfect substitute with RIM’s BlackBerry Bold 9700 and several other smart phones that have similar cost functions and that currently sell for $200 each
b. Apple’s iPhone 5 has no substitutes and so is a monopolist, and the demand for the iPhone 5 is expected to forever be Q = 4 – 0.02P
c. Apple’s iPhone 5 currently has no substitutes, and currently the
demand for the iPhone 5 is Q = 4 – 0.02P, but Apple anticipates other firms can develop close substitutes in the future.
Answer to Question:
BA 445 Final Exam Version 1
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Answer to Question: a. The firm is a Price Taker in Perfect Competition. MC = 20 + 8Q MR = 200 Set MR = MC to compute Q = 22.50 Price at that quantity Q is P = 200. Revenue at Q is PQ = 4500 Cost at Q is C(Q) = 2775 Maximum profit at Q is
= 1725.
Since profit is positive, expect other firms to enter in the long-run until price
(demand) drops enough so that profit drops to zero.
Implications: Produce Q > 0 in short run, but expect entry in the long-run and you produce less Q. b. The firm is a Monopolist with inverse demand P = 200 – 50Q So, MR = 200 - 100Q Set MR = MC to compute Q = 1.67 Price at that quantity Q is P = 116.67 Revenue at Q is PQ = 194.44 Cost at Q is C(Q) = 344.44 Maximum profit at Q is
= -150.
Since profit is negative, expect other firms to exit in the long-run until price
(demand) rises enough so that profit rises to zero.
Implications: Produce Q > 0 in short run, but either exit yourself or expect exit by others in the long-run and you produce more Q.
BA 445 Final Exam Version 1
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c. The firm is a Monopolistic Competitor with same results as in Part b.
BA 445 Final Exam Version 1
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Two Part Pricing
Question 2. The Yellowstone Club is a private golf
community set on 14,000 acres in Big Sky, Montana,
which counts Microsoft founder Bill Gates as a
member.
Suppose typical consumer’s demand for a game (round) of golf at the
Yellowstone Club is estimated to be Q = 2000 – 2 P per year, and
Yellowstone’s cost of providing games is $100 per game per customer.
Consider three alternative sets of market conditions:
1. Block pricing: Assume market conditions allow the firm to package
games played by each customer so that customers do not share their
packages. Compute the optimal number of games in a package.
And compute the optimal package price, and optimal profit.
2. Uniform pricing: Assume market conditions only allow the firm to
charge a uniform price for a customer to play each game. Compute
the optimal price for each game. Finally, compute optimal profit from
each customer.
3. Two-part pricing: Assume market conditions allow the firm to charge
a membership fee to each customer to have the right to pay to play
individual games, and that customers do not share their
memberships. Compute the optimal price membership fee and the
optimal to charge for each game. Finally, compute optimal profit from
each customer.
Answer to Question:
BA 445 Final Exam Version 1
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Answer to Question:
Alternative 2: Optimal uniform pricing sets marginal revenue to marginal
cost.
First, determine inverse demand
P = 1000 – 0.5 Q (from Q = 2000 – 2 P),
then marginal revenue by doubling the slope,
MR = 1000 – Q (P = 1000 – 0.5 Q).
Then set MR equal to the marginal cost of 100 to determine Q = 900
games.
Second, use inverse demand to determine price P = 1000 – 0.5 (900) =
Solving Firm 2's best response function yields quantity
Q2 = 0.38
and so price and profits
P = 4.50
1 = 0.13
2 = 0.56
Option A is thus best for Dish since profits (as a follower) are 0.25 in Option
A, while profits (as the leader) are 0.13 in Option B.
BA 445 Final Exam Version 1
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Multiple Actions
Question 5. Wii video game consoles are made by Nintendo, and some games are produced by Sega. Each year at the same time, Nintendo considers prices $100, $150, $200, or $250 for consoles, and Sega considers $20, $30, $40, or $50 for games. Suppose demands and costs are known, and result in the following profit table: Suppose the yearly interest rate is 2%. And suppose but there is uncertainty each year about the future of Wii video game consoles and Wii games; specifically, with probability 0.66, the Wii video game consoles and Wii games will become obsolete by the introduction of alternative game systems the next year.
Are there mutual gains from both players following the Grim Strategy for
the repeated game rather than repeating the solution to the one-shot
game? And is it a Nash Equilibrium for both players to follow the Grim
Strategy?
Answer to Question:
$20 $30 $40 $50
$100 5,4 3,4 5,5 6,4
$150 2,6 9,5 5,7 8,5
$200 4,6 5,6 7,7 9,6
$250 3,8 5,8 6,9 8,8
Nintendo
Sega
BA 445 Final Exam Version 1
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Answer to Question:
On the one hand, in the hypothetical one-shot game between Player 1
(Nintendo) and Player 2 (Sega),
Player 1 and Player 2 have a unique dominance solution ($200,$40).
On the other hand, since the game actually continues period after period,
each player should consider a Grim Strategy. A Grim Strategy has two
components. 1) The Cooperative choice of ($250,$50), which earns
($8,$8), which is mutually-better than the one-shot choice. 2) The
Punishment choice of ($100,$20), which gives the other player the worst
payoff ($5,$5) after that player chooses his best response to his
punishment.
The Grim Strategy is thus, in each period, Cooperate as long as the other
player has Cooperated in every previous period. But otherwise then you
punish in the next period and in every period thereafter --- forever. In
particular, if both players follow the Grim Strategy for the repeated game,
each period cooperates, and that is mutually-better than the dominance
solution to the one-shot game.
Can Player 1 trust Player 2 to follow an agreement to use the Grim
Strategy? To answer, consider the benefits and costs of Player 2 cheating.
In the first period of cheating, Player 2 gains Cheat = $9 rather than the
Cooperate = $8 it would have had from following the Grim Strategy and
cooperating.
$20 $30 $40 $50
$100 5,4 3,4 5,5 6,4
$150 2,6 9,5 5,7 8,5
$200 4,6 5,6 7,7 9,6
$250 3,8 5,8 6,9 8,8
Nintendo
Sega
BA 445 Final Exam Version 1
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But starting the next period and continuing forever, Player 1 punishes
Player 2, and so the best Player 2 can achieve is Punish = $5, rather than
the Cooperate = $8 he would have had if he had continued to follow the Grim
Strategy and cooperate.
Summing up, Player 1 can trust Player 2 to follow an agreement to use the
Grim Strategy if the one-period gain from cheating Cheat - Cooperate = $1
does not compensate for losses Punish - Cooperate = -$3 starting the next
period.
Use the formula that $1 starting next month and continuing for each
subsequent period is worth $(1/R) today. Since the interest rate r = 2% and
the probability of continuation is p = 0.34 = 1-0.66, the effective interest rate
is R = (1+r)/p-1 = 1.02/0.34-1 = 2. Therefore, the eventual losses each
period is the same as losing $3/2 = $1.50 today.
Therefore, the one period gain of from cheating of $1 does not compensate
for the loss of $1.50, so Player 2 would cooperate, and the Grim Strategy
for both players may be a Nash Equilibrium.
Can Player 2 trust Player 1 to follow an agreement to use the Grim
Strategy? To answer, consider the benefits and costs of Player 1 cheating.
In the first period of cheating, Player 1 gains Cheat = $9 rather than the
Cooperate = $8 it would have had from following the Grim Strategy and
cooperating.
But starting the next period and continuing forever, Player 2 punishes
Player 1, and so the best Player 1 can achieve is Punish = $5, rather than
the Cooperate = $8 he would have had if he had continued to follow the Grim
Strategy and cooperate.
Summing up, Player 1 can trust Player 2 to follow an agreement to use the
Grim Strategy if the one-period gain from cheating Cheat - Cooperate = $1
BA 445 Final Exam Version 1
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does not compensate for losses Punish - Cooperate = -$3 starting the next
period.
Since the effective interest rate is R = 2, the eventual losses each period is
the same as losing $3/2 = $1.50 today.
Therefore, the one period gain of from cheating of $1 does not compensate
for the loss of $1.50, so Player 1 would cooperate.
Since both players cooperate, the Grim Strategy is a Nash Equilibrium.
BA 445 Final Exam Version 1
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Mixing given Major Conflict
Question 6. Consider the Audit Game for a worker
and the Internal Revenue Service (IRS). Suppose
that at the same time the worker chooses to either
report his income or not report his income and the IRS
chooses to either audit the worker or not audit the worker.
Suppose the worker has $100,000 income. Suppose if the worker chooses
to report his income, he pays 10% tax to the IRS. Suppose the IRS can
audit the worker at a cost of $500. Suppose if the worker chooses to not
report his income and the IRS chooses to audit, then the worker pays 10%
tax to the IRS and an additional $20,000 fine to the IRS. Finally, if the
worker chooses to not report his income and the IRS chooses to not
monitor, then the worker does not pay any tax.
Predict strategies or recommend strategies if this game is repeated yearly.
Compute the expected payoffs to each player.
Finally, re-compute strategies if the fine is increased to $60,000 (from
$20,000). Compute the expected payoffs to each player, and compare with
the payoffs when the fine is $20,000.
Answer to Question:
BA 445 Final Exam Version 1
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Answer to Question:
First, complete the normal form below for the Audit Game, with payoffs in
thousands of dollars. For example, if the worker chooses to Cheat and not
report his income and the IRS chooses to Audit, then the worker pays
$10,000 tax to the IRS and an additional $20,000 fine to the IRS, and the
IRS gains that $30,000 but pays $500 for monitoring.
To predict actions or recommend actions, since the game has
simultaneous moves and is repeated, seek a solution in four steps:
1) Eliminate dominated actions. That does not help here since there are
no dominated actions.
2) Eliminate actions that are not rationalizeable. That does not help
here since each action is rationalizeable (each action is a best
response to some action of the other player).
3) Look for a Nash equilibrium in pure strategies (that is an action for
each player in which each player’s action is a best response to the
known action by the other player). That does not help here since
there is no Nash equilibrium. If the Worker were known to Report,
the IRS does Not Audit. But if the IRS were known to Not Audit, the
Worker Cheats. But if the Worker were known to Cheat, the IRS
Audits. But if the IRS were known to Audit, the Worker Reports. So
there is no Nash equilibrium in pure strategies.
4) Look for a Nash equilibrium in mixed strategies (that is probabilities
for each player in which the other player’s expected values are equal
for both of his actions; in that sense, the other player cannot exploit
his knowledge of the first player’s probabilities).
Audit Not Audit
Report -10,9.5 -10,10
Cheat -30,29.5 0,0
IRS
Worker
BA 445 Final Exam Version 1
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The Nash equilibrium strategy for the Worker is the mixed strategy for
which the IRS would not benefit if he could predict the Worker’s mixed
strategy. Suppose the IRS predicts p and (1-p) are the probabilities the
Worker chooses Report or Cheat. The IRS expects 9.5p + 29.5(1-p) from
playing Audit, and 10p + 0(1-p) from Not Audit. The IRS does not benefit if