ba 445 final exam version 2 - pepperdine...

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.


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:


3

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.


4

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


2 + 24 Q = 600 - 100 Q


Q = 4.82


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.


5

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


2 + 24 Q = 300 - 50 Q


Q = 4.03


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.


6

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:


7

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) =

$102.50

Fnally, optimal profit equals PQ – C(Q) = 102.50(390) – 5(390) = $38025


8

Alternative 3: Optimal block pricing sets price to marginal cost to determine

optimal output, then charges a price for all of that output sold as one

package.

First, determine optimal quantity by setting price equal to marginal cost.

Set price P = 200 – 0.25 Q (from Q = 800 – 4 P) equal to the marginal cost

of 5 to determine Q = 780 classes in a package.

Second, the optimal package price is the consumer valuation of the optimal

quantity, which is (1/2)195x780+5x780 = $79950

Finally, optimal profit per consumer equals the optimal package price of

$79950 minus the cost of $5x780, which is $76050 per consumer.

5

200

780


9

Alternative 1: Optimal two-part pricing sets price to marginal cost (to

maximize total surplus), then charges a fixed fee equal to consumer

surplus.

First, set P = 200 – 0.25 Q (from Q = 800 – 4 P) equal to the marginal cost

of 5 to determine Q = 780 classes in a package.

Second, the optimal fee equals the consumer surplus of (1/2)195x780 =

$76050

Finally, optimal profit equals the optimal fee of $76050 per consumer.

5

200

780


10

Relative Patience

Question 3. ING Financial Services and Tata

Consultancy Services can co sponsor the NYC

marathon for a combined cost of $3 million. A

forecast estimates that ING would benefit $2 million

from co-sponsorship and Tata would benefit $3 million. The deadline for an

agreement about sharing that $3 million cost between the two sponsors is

Friday, July 5.

On Monday, July 1, ING confronts Tata over cost sharing. ING presents its

offer of how the two companies will share the cost (that is, how much ING

will pay and how much Tata will pay).

Tata either accepts that offer or rejects it and returns on Tuesday, July 2,

with a counteroffer about how the two companies will share the cost.

ING either accepts that offer or rejects it and returns on Wednesday, July 3,


Tata either accepts that offer or rejects it and returns on Thursday, July 4,


ING either accepts that offer or rejects it and returns on Friday, July 5, with

a counteroffer about how the two companies will share the cost.

Tata either accepts that offer or rejects it and the two companies do not

sponsor the NYC marathon.

Suppose ING discounts 10% between each day, and Tata discounts 20%

between each day. (Those discount percentages reflect the aversion of

each party to the risk that, without an agreement, the opportunity to

sponsor that marathon will be withdrawn.)

What cost should ING offer to pay in its initial offer on Monday, July 1?

Should Tata accept that initial offer?


11

Answer to Question: There are 5 bargaining rounds. Here is a

bargaining payoff table for ING (Bargainer I) and Tata (Bargainer T), with

Bargainer I making the first offer, with I discounting 10% between each

bargaining round, and with T discounting 20% between each bargaining

round.

The gain from an agreement is $2 million = $2 million + $3 million - $3

million. So, I captures 86.24% of $2 million gains from trade, or $1.725

million. ING thus offers to pay $0.275 million = $2 million - $1.725 million in

its initial offer on Monday, July 1. And Tata should accept that offer, and

pay the remainder of the $3 million.

Rounds to

End of

Game

Offer byTotal Gain

to Divide

I's Gain

Offered

(10% dis.)

T 's Gain

Offered

(20% dis.)

1 I 100 100.00 0.00

2 T 100 90.00 10.00

3 I 100 92.00 8.00

4 T 100 82.80 17.20

5 I 100 86.24 13.76


12

First Mover Advantage

Question 4. You are a manager of Nvidia and your

only significant competitor in the mainstream graphics

card market is the ATI subsidiary of Advanced Micro

Devices. Nvidia’s graphics cards and ATI’s graphics

cards are indistinguishable to consumers. The market demand for graphics

cards is Q = 1.75 - 0.25 P (with P in dollars). You have a decision to make

about competing with ATI.

Option A. Nvidia and ATI both sets up their production line and distribution

networks in March of next year. And both produce at a marginal cost of $1

Option B. You hurry to set up your production line and distribution network

in January of next year, with ATI setting up in March of next year. Your

hurry means your marginal costs increase to $1.10, while ATI’s costs

remain $1

Which Option is better for Nvidia?

Answer to Question:


13

Answer to Question:

In Option A, you are Firm 1 in a Cournot Duopoly.

Solve the Cournot duopoly with demand

Q = 1.75 - 0.25 P

inverse demand

P = 7 - 4 Q1 - 4 Q2

firm 1 marginal cost and average cost = 1 firm 2 marginal cost and average cost = 1

Given Q1, Firm 2 computes marginal revenue

MR2 = 7 - 4 Q1 - 8 Q2 Hence, equate marginal cost MC2 to marginal revenue MR2

1 = 7 - 4 Q1 - 8 Q2

to determine our first equation

4 Q1 + 8 Q2 = 6

and Firm 2's best response function (if needed)

Q2 = r2(Q2) = 0.8 - 0.5 Q1


MR1 = 7 - 8 Q1 - 4 Q2 Hence, equate marginal cost MC1 to marginal revenue MR1

1 = 7 - 8 Q1 - 4 Q2

to determine our second equation

8 Q1 + 4 Q2 = 6

and Firm 2's best response function (if needed)

Q1 = r1(Q1) = 0.8 - 0.5 Q1

Solving the first and second equations yields quantities

Q1 = 0.50

Q2 = 0.50

and so price and profits

P = 3.00

1 = 1.00

2 = 1.00


14

In Option B, you are the leader in a Stackelberg Duopoly.

Solve the Stackelberg duopoly with demand

Q = 1.75 - 0.25 P

inverse demand

P = 7 - 4 Q1 - 4 Q2 firm 1 marginal cost and average cost = 1.1 firm 2 marginal cost and average cost = 1


MR2 = 7 - 4 Q1 - 8 Q2

Hence, equate marginal cost MC2 to marginal revenue MR2

1 = 7 - 4 Q1 - 8 Q2

to determine our first equation

4 Q1 + 8 Q2 = 6

and Firm 2's best response function

Q2 = r2(Q2) = 0.8 - 0.5 Q1

Given best response r2(Q2), Firm 1 computes revenue by R1 = PQ1

R1 = ( 4.0 - 2.0 Q1 ) Q1 Hence, compute marginal revenue by taking the derivative of revenue R1

MR1 = 4.0 - 4 Q1 Hence, equate marginal cost MC1 to marginal revenue MR1

1.1 = 4.0 - 4 Q1

to determine our first variable

Q1 = 0.73

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


15

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.


16

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


17

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


18

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


19

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.


20

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:


21

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


22

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.


23

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


24

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:


25

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

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