ba 445 final exam version 1seaver-faculty.pepperdine.edu/jburke2/ba445/finalexam/final1... · ba...

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.


2

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:


3

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.


4

c. The firm is a Monopolistic Competitor with same results as in Part b.


5

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:


6

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) =

$550

Fnally, optimal profit equals PQ – C(Q) = 550(900) – 100(900) = $40,5000


7

Alternative 1: 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 = 1000 – 0.5 Q (from Q = 2000 – 2 P) equal to the marginal

cost of 100 to determine Q = 1800 games.

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

quantity, which is (1/2)900x1800+100x1800 = $990,000

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

$99000 minus the cost of $100x1800, which is $810,000 per consumer.

100

1000

1800


8

Alternative 3: Optimal two-part pricing sets price to marginal cost (to

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

surplus.

First, set price P = 1000 – 0.5 Q (from Q = 2000 – 2 P) equal to the

marginal cost of 100 to determine Q = 1800 games.

Second, the optimal fee equals the consumer surplus of (1/2)900x1800 =

$810,000

Finally, optimal profit equals the optimal fee of $810,000 per consumer.

100

1000

1800


9

Manipulating Alternatives to No Agreement

Question 3. Home Depot and Target have to guard

their stores. Each store can either each hire their own

guard to patrol within their own store, with Home

Depot hiring Alfred and Target hiring Bart, or they can

hire Charlie to patrol their common parking lot. For each of the five

months when college kids are out of school (April, May, June, July, and

August), Alfred would charge $400 per month to guard Home Depot only

and Bart would charge $800 per month to guard Target only; or Charlie

would guard both stores for a total of $900 per month.

In late March, Home Depot confronts Target over how much the two stores

should pay to hire Charlie to guard both stores. Target either accepts that

offer for each of the five months, or rejects it and returns in late April with a

counteroffer.

In late April, Home Depot either accepts that counteroffer for each of the remaining four

months, or rejects it and returns in late May with a counteroffer.

In late May, Target either accepts that counteroffer for each of the remaining three

months, or rejects it and returns in late June with a counteroffer.

In late June, Home Depot either accepts that counteroffer for each of the remaining two

months, or rejects it and returns in late July with a counteroffer.

In late July, Target either accepts that counteroffer for the remaining month,

or rejects it and the bargaining game is over.

What percentage of the gains from an agreement should Home Depot

offer? How much should Home Depot offer to pay for Charlie to guard both

stores? Should Target accept that initial offer?

Now suppose that, in early March, Alfred makes a take-it-or-leave it offer to

Home Depot to guard Home Depot for $300 per month. Re-compute Home

Depot’s offer to pay for Charlie to guard both stores.

Answer to Question:


10

Answer to Question:

To consider all possible offers to split the gains from an agreement,

consider a bargaining payoff table. The game ends if an offer is accepted

or if the months end without an accepted agreement, and gains are

measured as a percentage of the gain from using the security service to

patrol their common parking lot for all five months.

Starting from the end of the bargaining game and rolling back to the

beginning, Home Depot’s best acceptable offer leaves Home Depot with 60

percent of the gains from trade, and Target with 40 percent. The costs of

guarding each store are: Cost of the common guard $900, cost of Home

Depot guard $400, cost of Target guard $800. In particular, the gain from

five months of the common guard is 5x(1200-900) = 1500.

Home Depot’s initial offer should be for Home Depot to keep 60 percent of

the $1500 gain, or $900. So Home Depot pays 5x400-900 = $1100 for

the five months (or $220 per month). Target should accept that offer.

Target gains 40 percent of the $1500 gain, or $600, and pays 5x800-600 =

$3400 for the five months (or $680 per month).

With the lower offer by Alfred, the costs of guarding each store are: Cost of

the common guard $900, cost of Home Depot guard $300, cost of Target

guard $800. In particular, the gain from five months of the common guard

is 5x(1100-900) = 1000.

Rounds to

End of

Game

Offer byTotal Gain

to Divide

H's Gain

Offered

T 's Gain

Offered

1 H 20 20 0

2 T 40 20 20

3 H 60 40 20

4 T 80 40 40

5 H 100 60 40


11

Home Depot’s initial offer should be for Home Depot to keep 60 percent of

the $1000 gain, or $600. So Home Depot pays 5x300-600 = $900 for the

five months (or $180 per month). Target should accept that offer. Target

gains 40 percent of the $1000 gain, or $400, and pays 5x800-400 = $3600

for the five months (or $720 per month).


12

B.3 First Mover Advantage

Question 4. You are a manager of Dish Network and

your only significant competitor in next year’s U.S.

satellite provider market is DirecTV. Consumers are

expected to find the two products to be indistinguishable

next year. The market demand for your products is

expected to be Q = 1.75 – 0.25 P. You have a decision to make about

competing with DirecTV next year.

Option A. You set up your satellite leases and distribution networks in

March of next year, and DirecTV sets up its satellite leases and distribution

networks in February of next year. And both produce at a marginal cost of

$3.00

Option B. You hurry and set up your satellite leases and distribution

networks in January of next year, and DirecTV sets up its satellite leases

and distribution networks in February of next year. Your hurry means your

marginal costs are $4.00, while DirecTV’s marginal costs remain $3.00

Which Option is better for Dish Network?

Answer to Question:


13

Answer to Question: In Option A, you are the follower in a Stackelberg

Duopoly with inverse demand P = 7 - 4(Q1+Q2) and marginal costs c1 =

MC1 = 3.00 and c2 = MC2 = 3.00. In Option B, you are the leader in a

Stackelberg Duopoly with inverse demand P = 7 - 4(Q1+Q2) and marginal

costs c1 = MC1 = 4.00 and c2 = MC2 = 3.00


14

Option A:

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 = 3 firm 2 marginal cost and average cost = 3

Given Q1, Firm 2 computes marginal revenue

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

3 = 7 - 4 Q1 - 8 Q2

to determine our first equation

4 Q1 + 8 Q2 = 4

and Firm 2's best response function

Q2 = r2(Q1) = 0.5 - 0.5 Q1

Given best response Q2 = r2(Q1), Firm 1 computes revenue by R1 = PQ1

R1 = ( 5.0 - 2.0 Q1 ) Q1 Hence, compute marginal revenue by taking the derivative of revenue R1

MR1 = 5.0 - 4 Q1 Hence, equate marginal cost MC1 to marginal revenue MR1

3 = 5.0 - 4 Q1

to determine our first variable

Q1 = 0.50

Solving Firm 2's best response function yields quantity

Q2 = 0.25

and so price and profits

P = 4.00

1 = 0.50

2 = 0.25


15

Option B:

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 = 4 firm 2 marginal cost and average cost = 3

Given Q1, Firm 2 computes marginal revenue

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

3 = 7 - 4 Q1 - 8 Q2

to determine our first equation

4 Q1 + 8 Q2 = 4

and Firm 2's best response function

Q2 = r2(Q1) = 0.5 - 0.5 Q1

Given best response Q2 = r2(Q1), Firm 1 computes revenue by R1 = PQ1

R1 = ( 5.0 - 2.0 Q1 ) Q1 Hence, compute marginal revenue by taking the derivative of revenue R1

MR1 = 5.0 - 4 Q1 Hence, equate marginal cost MC1 to marginal revenue MR1

4 = 5.0 - 4 Q1

to determine our first variable

Q1 = 0.25

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.


16

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


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 ($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


18

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


19

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.


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 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:


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

$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


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 9.5p + 29.5(1-p) from

playing Audit, and 10p + 0(1-p) from Not Audit. The IRS does not benefit if

those payoffs equal,

9.5p + 29.5(1-p) = 10p + 0(1-p), or 29.5 - 20p = 10p,

or p = 29.5/30 = 0.983

In particular, the expected payoff to the IRS is 10p = 10(0.983) = 9.83

thousand.

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 -10q - 10(1-q) from

playing Report, and -30q + 0(1-q) from Cheat. The Worker does not benefit

if those payoffs equal,

-10q - 10(1-q) = -30q + 0(1-q), or -10 = -30q,

or q = 10/30 = 0.333

In particular, the expected payoff to the Worker is -10 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 9.5p + 69.5(1-p) from

playing Audit, and 10p + 0(1-p) from Not Audit. The IRS does not benefit if

those payoffs equal,

9.5p + 69.5(1-p) = 10p + 0(1-p), or 69.5 - 60p = 10p,

or p = 69.5/70 = 0.993

In particular, the expected payoff to the IRS is 10p = 10(0.993) = 9.93

thousand.

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 -10q - 10(1-q) from

playing Report, and -70q + 0(1-q) from Cheat. The Worker does not benefit

if those payoffs equal,

-10q - 10(1-q) = -70q + 0(1-q), or -10 = -70q,

or q = 10/70 = 0.143

In particular, the expected payoff to the Worker is -10 thousand.

So, the IRS gains and the worker does not lose when the fine is increased.

Audit Not Audit

Report -10,9.5 -10,10

Cheat -70,69.5 0,0

IRS

Worker


24

The Dilemma of Pooling

Question 7. 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 $81,000 per year to

profits. And non-committed workers (who are not sure about their long-

term career) contribute $68,000 per year to profits.

Suppose committed workers have existing jobs paying $40,000 per year,

and non-committed workers have existing jobs paying $40,000 per year.

Suppose committed workers regard the cost of completing a month of an

internship in the entertainment business the same as $600 a year of salary,

and non-committed workers regard the cost of completing a month of an

internship in the entertainment business the same as $1,100 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

81

68

Value/Cost to Seller

40

40

of Signal/Screen

0.6

1.1

Find an appropriate N so that the Buyer should accept a price demand of

81 from any Seller that chooses N units of the signal, and a price demand of

68 if the Seller chooses 0 units of the signal.

To make sure that High Quality Signals,

81 - 0.6 N > 68 , or N < 21.667 , N < 21

To make sure that Low Quality does not Signal,

81 - 1.1 N < 68 , or N > 11.818 , N > 12

To make sure that High Quality chooses to participate,

81 - 0.6 N > 40 , or N < 68.333 , N < 68

Putting it all together, the Buyer should accept a price demand of

81 from any Seller that chooses N between 12 and 21

At the min signal, the High Quality seller nets 81 - 0.6 x 12 = 74

Dropping the signalling/screening avoids the signalling cost to the High Quality

seller. Without signalling/screening, all sellers get

81 f + 68 (1-f)

which is higher than the price of 68

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

81 f + 68 (1-f) > 74 , or f > 0.4462

So, all sellers are better off without signalling/screening when the fraction of

High Quality sellers in the population of all sellers is f > 0.4462

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