BA 445 Final Exam Version 2 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 2
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 2
2
Comparing Markets
Question. Consider the cost function
C(Q) = 800 + 2Q + 12Q2
for Google to produce the new Google Glass wearable computer. Using that cost function for the Google Glass, determine the profit-maximizing output and price for the Google Glass, and discuss its long-run implications, under three alternative scenarios:
a. Google’s Google Glass is a perfect substitute with WIMM Labs’s Smartwatch and several other wearable computers that have similar cost functions and that currently sell for $700 each.
b. Google’s Google Glass has no perfect substitutes, and the demand for the Google Glass is expected to forever be Q = 12 – 0.02P
c. Google’s Google Glass currently has no perfect substitutes, and
currently demand for the Google Glass is Q = 12 – 0.04P, but Google anticipates other firms can develop close substitutes in the future.
Answer to Question:
BA 445 Final Exam Version 2
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Answer to Question:
a. The firm is in Perfect Competition.
Solve for Perfect Competition with inverse demand
P = 700
and quadratic cost
C = 800 + 2 Q + 12 Q2
Given inverse demand, marginal revenue equals price
MR = 700 and compute marginal cost by taking the derivative of cost C
MC = 2 + 24 Q
Hence, equate marginal cost MC to marginal revenue MR
2 + 24 Q = 700
to determine quantity
Q = 29.08
and so price by inverse demand and profit by =PQ-C
P = 700.00
= 9350.08
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.
BA 445 Final Exam Version 2
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b. The firm is a Monopolist.
Solve the Uniform Price Monopoly with demand
Q = 12.00 - 0.02 P
inverse demand
P = 600 - 50 Q
and quadratic cost
C = 800 + 2 Q + 12 Q2
Given inverse demand, compute marginal revenue by doubling the slope
MR = 600 - 100 Q and compute marginal cost by taking the derivative of cost C
MC = 2 + 24 Q
Hence, equate marginal cost MC to marginal revenue MR
2 + 24 Q = 600 - 100 Q
to determine quantity
Q = 4.82
and so price by inverse demand and profit by =PQ-C
P = 358.87
= 641.95
Since profit is positive, expect other firms to want to enter in the long-run,
but they cannot, so nothing changes.
BA 445 Final Exam Version 2
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c. The firm is a Monopolistic Competitor.
Solve the Uniform Price Monopoly with demand
Q = 12.00 - 0.04 P
inverse demand
P = 300 - 25 Q
and quadratic cost
C = 800 + 2 Q + 12 Q2
Given inverse demand, compute marginal revenue by doubling the slope
MR = 300 - 50 Q and compute marginal cost by taking the derivative of cost C
MC = 2 + 24 Q
Hence, equate marginal cost MC to marginal revenue MR
2 + 24 Q = 300 - 50 Q
to determine quantity
Q = 4.03
and so price by inverse demand and profit by =PQ-C
P = 199.32
= -199.97
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 2
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Two Part Pricing
Question 2. The Sports Club is an upscale fitness
center in Los Angeles. Suppose typical consumer’s
demand for fitness classes at The Sports Club is
estimated to be Q = 800 – 4 P per year, and The
Sports Club’s cost of providing classes is $5 per class per customer.
Consider three alternative sets of market conditions:
1. Two-part pricing: Assume market conditions allow the firm to charge
a membership fee to each customer to have the right to buy individual
classes, and that customers do not share their memberships.
Compute the optimal price membership fee and the optimal to charge
for each class. Finally, compute optimal profit from each customer.
2. Uniform pricing: Assume market conditions only allow the firm to
charge a uniform price for a customer to take a class. Compute the
optimal price for each class. Finally, compute optimal profit from
each customer.
3. Block pricing: Assume market conditions allow the firm to package
classes taken by each customer so that customers do not share their
packages. Compute the optimal number of classes in a package.
And compute the optimal package price, and optimal profit.
Answer to Question:
BA 445 Final Exam Version 2
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Answer to Question:
Alternative 2: Optimal uniform pricing sets marginal revenue to marginal
cost.
First, determine inverse demand
P = 200 – 0.25 Q (from Q = 800 – 4 P),
then marginal revenue by doubling the slope,
MR = 200 – 0.5 Q (P = 200 – 0.25 Q).
Then set MR equal to the marginal cost of 5 to determine Q = 390 classes.
Second, use inverse demand to determine price P = 200 – 0.25(390) =
Solving Firm 2's best response function yields quantity
Q2 = 0.39
and so price and profits
P = 2.55
1 = 1.05
2 = 0.60
BA 445 Final Exam Version 2
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Option A is thus worse than Option B for Firm 1 since Firm 1’s profits (as a
Cournot Duopolist) are 1.00 in Option A, while Firm 1’s profits (as the
Stackelberg leader) are 1.05 in Option B.
BA 445 Final Exam Version 2
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Multiple Actions
Question 5. Each year, Canon and Nikon simultaneously decide on the size of manufacturing capacity for the current generation of high end imaging optics. They can both choose sizes 10, 20, 30, or 40, Suppose demands and costs are known, and .result in the following profit table: Suppose the yearly interest rate is 3%. And suppose there is uncertainty each year about the future of the current generation of high end imaging optics; specifically, with probability 0.657, the current generation of high end imaging optics will become obsolete by further technological advances 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:
10 20 30 40
10 38,38 32,45 24,42 20,40
20 40,32 35,37 25,35 20,32
30 42,24 36,33 21,21 14,16
40 40,20 32,22 16,14 8,8
Canon
Nikon
BA 445 Final Exam Version 2
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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 (30,20).
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 (10,10), which earns ($38,$38),
which is mutually-better than the one-shot choice. 2) The Punishment
choice of (40,40), which gives the other player the worst payoff ($20,$22)
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 = $45 rather than the
Cooperate = $38 it would have had from following the Grim Strategy and
cooperating.
10 20 30 40
10 38,38 32,45 24,42 20,40
20 40,32 35,37 25,35 20,32
30 42,24 36,33 21,21 14,16
40 40,20 32,22 16,14 8,8
Canon
Nikon
BA 445 Final Exam Version 2
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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 = $22, rather than
the Cooperate = $38 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 = $7
does not compensate for losses Punish - Cooperate = -$16 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 = 3% and
the probability of continuation is p = 0.343 = 1-0.657, the effective interest
rate is R = (1+r)/p-1 = 1.03/0.343-1 = 2. Therefore, the eventual losses
each period is the same as losing $16/2 = $8 today.
Therefore, the one period gain of from cheating of $7 does not compensate
for the loss of $8, 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 = $42 rather than the
Cooperate = $38 he 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 = $20, rather than
the Cooperate = $38 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 = $4
BA 445 Final Exam Version 2
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does not compensate for losses Punish - Cooperate = -$18 starting the next
period.
Since the effective interest rate is R = 2, the eventual losses each period is
the same as losing $18/2 = $9 today.
Therefore, the one period gain of from cheating of $4 does not compensate
for the loss of $9, so Player 1 would cooperate.
Since both players cooperate, the Grim Strategy is a Nash Equilibrium.
BA 445 Final Exam Version 2
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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 15% tax to the IRS. Suppose the IRS can
audit the worker at a cost of $1000. Suppose if the worker chooses to not
report his income and the IRS chooses to audit, then the worker pays 15%
tax to the IRS and an additional $15,000 fine to the IRS. Finally, if the
worker chooses to not report his income and the IRS chooses to not audit,
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
$15,000). Compute the expected payoffs to each player, and compare with
the payoffs when the fine is $15,000.
Answer to Question:
BA 445 Final Exam Version 2
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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
$15,000 tax to the IRS and an additional $15,000 fine to the IRS, and the
IRS gains that $30,000 but pays $1000 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 -15,14 -15,15
Cheat -30,29 0,0
IRS
Worker
BA 445 Final Exam Version 2
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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 14p + 29(1-p) from
playing Audit, and 15p + 0(1-p) from Not Audit. The IRS does not benefit if
those payoffs equal,
14p + 29(1-p) = 15p + 0(1-p), or 29 - 15p = 15p,
or p = 29/30 = 0.967
In particular, the expected payoff to the IRS is 15p = 15(29/30) = 14.5
thousand dollars.
The Nash equilibrium strategy for the IRS is the mixed strategy for which
the Worker would not benefit if he could predict the IRS’s mixed strategy.
Suppose the Worker predicts q and (1-q) are the probabilities the IRS
chooses Audit or Not Audit. The Worker expects -15q - 15(1-q) from
playing Report, and -30q + 0(1-q) from Cheat. The Worker does not benefit
if those payoffs equal,
-15q - 15(1-q) = -30q + 0(1-q), or -15 = -30q,
or q = 15/30 = 0.5
In particular, the expected payoff to the Worker is -15 thousand.
BA 445 Final Exam Version 2
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Finally, re-compute strategies if the fine is increased to $60,000.
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 14p + 74(1-p) from
playing Audit, and 15p + 0(1-p) from Not Audit. The IRS does not benefit if
those payoffs equal,
14p + 74(1-p) = 15p + 0(1-p), or 74 - 60p = 15p,
or p = 74/75 = 0.987
In particular, the expected payoff to the IRS is 15p = 15(74/75) = 14.8
thousand dollars.
The Nash equilibrium strategy for the IRS is the mixed strategy for which
the Worker would not benefit if he could predict the IRS’s mixed strategy.
Suppose the Worker predicts q and (1-q) are the probabilities the IRS
chooses Audit or Not Audit. The Worker expects -15q - 15(1-q) from
playing Report, and -75q + 0(1-q) from Cheat. The Worker does not benefit
if those payoffs equal,
-15q - 15(1-q) = -75q + 0(1-q), or -15 = -75q,
or q = 15/75 = 0.2
In particular, the expected payoff to the Worker is -15 thousand.
So, the IRS gains and the worker does not lose when the fine is increased.
Audit Not Audit
Report -15,14 -15,15
Cheat -75,74 0,0
IRS
Worker
BA 445 Final Exam Version 2
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The Dilemma of Pooling
Question. Warner Brothers Entertainment Inc. (WB)
is an American producer of film, television, and music
entertainment. Suppose WB has use for two kinds of
business administration majors from Pepperdine
University. Committed workers (who plan to pursue a long-term career in
the entertainment business) contribute $91,000 per year to profits. And
non-committed workers (who are not sure about their long-term career)
contribute $72,000 per year to profits.
Suppose committed workers have existing jobs paying $77,000 per year,
and non-committed workers have existing jobs paying $68,000 per year.
Suppose committed workers regard the cost of completing a month of an
internship in the entertainment business the same as $800 a year of salary,
and committed workers regard the cost of completing a month of an
internship in the entertainment business the same as $1,600 a year of
salary. Suppose there is no way for the employer to directly tell committed
workers from non-committed workers, but the employer can confirm the
number of months of internship in the entertainment business. Finally,
suppose potential employees can make a wage demand that the employer
must either accept or reject (but not counter).
For each type of worker, determine wage demands and the number of
months of internship in the entertainment business.
Would all workers be better off if WB did not use the number of months of
internship in the entertainment business to screen job candidates?
Answer to Question:
BA 445 Final Exam Version 2
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Answer to Question: Analyze all salary numbers in thousands of dollars.
Committed workers are high quality sellers, and non-committed are low.
Solve the Signaling/Screening Model with values and costs
The Item for Sale is
High Quality Low Quality
to Buyer
91
72
Value/Cost to Seller
77
68
of Signal/Screen
0.8
1.6
Find an appropriate N so that the Buyer should accept a price demand of
91 from any Seller that chooses N units of the signal, and a price demand of
72 if the Seller chooses 0 units of the signal.
To make sure that High Quality Signals,
91 - 0.8 N > 72 , or N < 23.75 , N < 23
To make sure that Low Quality does not Signal,
91 - 1.6 N < 72 , or N > 11.875 , N > 12
To make sure that High Quality chooses to participate,
91 - 0.8 N > 77 , or N < 17.5 , N < 17
Putting it all together, the Buyer should accept a price demand of
91 from any Seller that chooses N between 12 and 17
At the min signal, the High Quality seller nets 91 - 0.8 x 12 = 81
Dropping the signalling/screening avoids the signalling cost to the High Quality
seller. Without signalling/screening, all sellers get
91 f + 72 (1-f)
which is higher than the price of 72
received by the Low Quality seller under signalling/screening, and is also higher
than the price the High Quality seller nets under signalling/screening when
91 f + 72 (1-f) > 81 , or f > 0.4947
So, all sellers are better off without signalling/screening when the fraction of
High Quality sellers in the population of all sellers is f > 0.4947