ba 452 quantitative analysis final exam

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BA 452 Quantitative Analysis Final Exam 1 This is a 150-minute exam (2hr. 30 min.). There are 6 questions (25 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. For all of the exam, you may only use blank or graph paper, pencils, and a calculator. You may not use a computer or notes.

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Page 1: BA 452 Quantitative Analysis Final Exam

BA 452 Quantitative Analysis Final Exam

1

This is a 150-minute exam (2hr. 30 min.). There are 6 questions (25 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. For all of the exam, you may only use blank or graph paper, pencils, and a calculator. You may not use a computer or notes.

Page 2: BA 452 Quantitative Analysis Final Exam

BA 452 Quantitative Analysis Final Exam

2

Manipulating Alternatives to No Agreement

Question 1. Consider negotiations over wages for

adjunct Management Science Professors at the

Business Administration Division of Seaver College of

Pepperdine University. Pepperdine seeks a professor

for the next 5 semesters. The only candidate (Charlie) is willing to work for

as little as $5,000 per semester and Pepperdine is willing to pay up to

$35,000 per semester. Before the first semester, Pepperdine confronts the

candidate Charlie over wages. Pepperdine presents its wage offer.

Charlie either accepts it or rejects it and returns the next semester with a

counteroffer. Offers alternate thereafter. Employment only occurs in those

semesters after an agreement is reached.

What wage per semester should Pepperdine offer to Charlie? Should

Charlie accept that initial offer?

Now suppose candidate Charlie finds an alternative employment offer by

UCLA, who makes Charlie a take-it-or-leave-it offer of $15,000 for each of

the same 5 semesters. Suppose the candidate cannot accept the job at

both UCLA and Pepperdine University. Re-compute Pepperdine’s initial

wage offer to Charlie.

Finally suppose, in addition to the alternative employment offer to

candidate Charlie by UCLA described above, the problem changes

because Pepperdine finds an alternative candidate Chris who makes

Pepperdine a a take-it-or-leave-it offer to work for $25,000 per semester for

each of the same 5 semesters. Re-compute Pepperdine’s initial wage offer

to Charlie.

Answer to Question:

Page 3: BA 452 Quantitative Analysis Final Exam

BA 452 Quantitative Analysis Final Exam

3

Answer to Question:

To consider all possible wage offers (offers by Pepperdine to buy labor by

the candidate to sell labor), consider a bargaining payoff table. The game

ends if an offer is accepted or if the semesters end without an accepted

offer, and gains are measured as a percentage of the gain from the

candidate accepting Pepperdine’s offer before the first semester.

Without the alternative offer, the candidate’s opportunity cost of Pepperdine

employment is $5,000 per semester, and so the gain from Pepperdine

employment is $30,000=$35,000-$5,000 per semester. For Pepperdine to

receive 60% of those gains, Pepperdine gains $18,000 per semester,

which means offer wages $17,000=$35,000-$18,000 per semester, or

$85,000 for all five semesters. And Charlie should accept that initial offer.

With the alternative employment offer to candidate Charlie, the candidate’s

opportunity cost of Pepperdine employment is $15,000 per semester, and

so the gain from Pepperdine employment is $20,000=$35,000-$15,000 per

semester. For Pepperdine to receive 60% of those gains, Pepperdine

gains $12,000 per semester, which means offer wages $23,000=$35,000-

$12,000 per semester, or $115,000 for all five semesters. And Charlie

should accept that initial offer.

Comment: Charlie finding alternative employment raised candidate

Charlie’s Pepperdine wages.

Rounds to

End of

Game

Offer byTotal Gain

to Divide

P's Gain

Offered

C's Gain

Offered

1 P 20 20 0

2 C 40 20 20

3 P 60 40 20

4 C 80 40 40

5 P 100 60 40

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4

With the alternative employment offer to candidate Charlie and the

alternative employment offer from candidate Chris, candidate Charlie’s

opportunity cost of Pepperdine employment is $15,000 per semester and

Pepperdine’s value to Charlie’s employment is $25,000 per semester. So,

the gain from Pepperdine employment is $10,000=$25,000-$15,000 per

semester. For Pepperdine to receive 60% of those gains, Pepperdine

gains $6,000 per semester, which means offer wages $19,000=$25,000-

$6,000 per semester, or $95,000 for all five semesters. And Charlie

should accept that initial offer.

Comment: Pepperdine finding an alternative candidate lowered candidate

Charlie’s Pepperdine wages.

Page 5: BA 452 Quantitative Analysis Final Exam

BA 452 Quantitative Analysis Final Exam

5

Sensitivity to Constants

Question 2. The Ford Motor Company makes cars

and trucks on an assembly line. Each car and truck

requires labor and needs to visit three stations:

engine installation, hood installation, and wheel

installation. The hours of labor and hours required at each station are as

follows:

There are available 12 hours of labor, 48 hours of engine installation, 20

hours of hood installation, and 12 hours of wheel installation. Labor costs

$20 per hour, and each installation station costs $10 per hour to operate.

Cars sell for $710 each, and trucks sell for $530 each.

a. Formulate and graphically solve for the recommended production

quantities. Do not require production units to be integers.

b. How much should Ford be willing to pay to another potential worker

(Joe) to supply one more unit of labor. (Joe is not part of the 12

hours of labor mentioned above.)

Answer to Question:

Engine Hood Wheel

Vehicle Labor Installation Installation Installation

Car 4 6 4 3

Truck 3 8 5 4

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6

Answer to Question:

a. The profit contributions are $500 for each car, and $300 for each truck.

Let C = cars produced.

Let T = trucks produced.

Max 500C + 300T

s.t. 4C + 3T < 12 (labor)

6C + 8T < 48 (engine)

4C + 5T < 20 (hood)

3C + 4T < 12 (wheel)

C, T 0

A graph of the feasible set shows the engine and hood constraints are

redundant, and the iso-value lines show the optimal solution occurs where

the labor constraint and the non-negativity of T bind (C = 3 and T = 0).

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

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7

b. How much should Ford be willing to pay to change the problem by

having Joe supply one more unit of labor.

Repeat analysis after changing the right hand side of the first constraint

from 12 to 13:

Max 500C + 300T

s.t. 4C + 3T < 13 (labor)

6C + 8T < 48 (engine)

4C + 5T < 20 (hood)

3C + 4T < 12 (wheel)

C, T 0

The new graph of the feasible set shows the engine and hood constraints

are redundant, and the iso-value lines show the optimal solution occurs

where the labor constraint and the non-negativity of T bind (C = 13/4 = 3.25

and T = 0). That new solution yields profit 500C + 300T = 1625, which is

125 higher than the previous profit of 1500. So, Ford should be willing to

pay up to 145 = 20+125 dollars to Joe.

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

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Make or Buy with Fixed Costs

Question 3. Danchuk Manufacturing produces a variety of classic

automobiles, including a 1955 Chevy and a 1955 Thunderbird. Each car

consists of three components that can be manufactured by Danchuk: a

body, an interior, and an engine. Both cars use the same engine, but

different bodies and different interiors.

Danchuk’s sales forecast indicates that 300 Chevys and 500 Thunderbirds

will be needed to satisfy demand during the next year. Because only 2000

hours of in-house manufacturing time is available, Danchuk is considering

purchasing some, or all, of the components from outside suppliers. If

Denchuk manufactures a component in-house, it incurs a fixed setup cost

as well as a variable manufacturing cost. The following table shows the

setup cost, the manufacturing time per component, the manufacturing cost

per component, and the cost to purchase each of the components from an

outside supplier:

Component

Setup Cost (thousands of dollars)

Manufacturing Time per Unit

(hours)

Manufacturing Cost per Unit (thousands of

dollars)

Purchase Cost per Unit (thousands of dollars)

Chevy Body 100 3 4 5

Thunderbird Body

90 2 2 3

Chevy Interior

10 1 2 3

Thunderbird Interior

20 2 3 4

Engine 20 2 4 5

Formulate a linear program for Denchuk to minimize total cost to meet the

sales forecasts. But you need not compute an optimum.

Tip: Your written answer should define the decision variables, carefully

state which variables are continuous and which are binary, and formulate

the objective and constraints.

Answer to Question:

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Answer to Question:

Continuous (or integer) variables are used for the number of units made:

Let CB = the number of Chevy Bodies to make

Let TB = the number of Thunderbird Bodies to make

Let CI = the number of Chevy Interiors to make

Let TI = the number of Thunderbird Interiors to make

Let E = the number of Engines to make

Continuous (or integer) variables are also used for the number of units

purchased:

Let CBP = the number of Chevy Bodies to purchase

Let TBP = the number of Thunderbird Bodies to purchase

Let CIP = the number of Chevy Interiors to purchase

Let TIP = the number of Thunderbird Interiors to purchase

Let EP = the number of Engines to purchase

Binary variables are used to indicate whether production is positive, and

setup costs are incurred:

Let CBS = 1 if Chevy Bodies are produced; 0, if not.

Let TBS = if Thunderbird Bodies are produced; 0, if not.

Let CIS = 1 if Chevy Interiors are produced; 0, if not.

Let TIS = if Thunderbird Interiors are produced; 0, if not.

Let ES = if Engines are produced; 0, if not.

Objective: Minimize Total Cost

4CB + 2TB + 2CI + 3TI + 4E (production costs)

+5CBP + 3TBP + 3CIP + 4TIP + 5EP (purchase costs)

+100CBS + 90TBS + 10CIS + 20TIS + 20ES (setup costs)

Input Constraints:

3CB + 2TB + 1CI + 2TI + 2E < 2000 (In-house hours)

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Sales Constraints:

CB + CBP = 300 (manufactured or purchased bodies used for Chevys)

TB + TBP = 500 (manufactured or purchased bodies used for

Thunderbirds)

CI + CIP = 300 (manufactured or purchased interiors used for Chevys)

TI + TIP = 500 (manufactured or purchased interiors used for

Thunderbirds)

E + EP = 800 (manufactured or purchased engines used for both Chevys

and Thunderbirds)

Setup Constraints (given sales constraints imply CB < 300, TB < 500, CI <

300, TI < 500, and E < 800):

CB < 300CBS

TB < 500TBS

CI < 300CIS

TI < 500TIS

E < 800ES

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Economic Analysis with Teamwork

Question 4. Food on Foot operates a weekly meal

program every Sunday in Hollywood. They are

considering three alternative distribution systems.

Arrivals to the distribution stand follow the Poisson distribution with an average of one homeless person arriving every 12 minutes. The cost for a homeless person waiting is $2 per hour. The first system involves one volunteer (who talks to the homeless and serves food and serves drinks) and costs $10 per hour to operate (the opportunity cost of the volunteer’s time), and it has an exponential service rate, and it can serve an average of one homeless person every 6 minutes. The second system involves two volunteers working as a team (one talks to the homeless and serves food, and the other serves drinks) and costs $20 per hour to operate (the opportunity cost of the 2 volunteer’s time), and it has an exponential service rate, and it can serve an average of one homeless person every 3 minutes. The third system involves three volunteers working as a team (one talks to the homeless, one serves food, and the third serves drinks) and costs $30 per hour to operate (the opportunity cost of the 3 volunteer’s time), and it has an exponential service rate, and it can serve an average of one homeless person every 1 minute. Which system should be chosen? Re-compute your answer if the customers value their time at $20 per hour, rather than $2 per hour. You may use any or all of the following analytical formulas for an M/M/1

system to compute your answer:

The probability of no units in the system: P0 = 1-

The average number of units in the waiting line: Lq =

The average number of units in the system: L = Lq +

The average time a unit spends in the waiting line: Wq = Lq/

The average time a unit spends in the system: W = 1/

The probability that an arriving unit has to wait: Pw =

Probability of n units in the system: Pn = ()nP0

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Answer to Question:

First system: k = 1 channel, = 5 per hour, and = 10 per hour.

The average number of units in the waiting line:

Lq = = 5*5/(10(5)) = 1/2

The average number of units in the system: L = Lq + = 1/2 + 1/2 = 1

Cost to operate = $10 + $2*L = $12 per hour.

Second system: k = 1 channel, = 5 per hour, and = 20 per hour.

The average number of units in the waiting line:

Lq = = 5*5/(20(15)) = 1/12

The average number of units in the system: L = Lq + = 1/12 + 1/4 = 1/3

= 0.3333

Cost to operate = $20 + $2*L = $20.66 per hour.

Third system: k = 1 channel, = 5 per hour, and = 60 per hour.

The average number of units in the waiting line:

Lq = = 5*5/(60(55)) = 1/132

The average number of units in the system: L = Lq + = 1/132 + 1/12 =

0.0909

Cost to operate = $30 + $2*L = $30.18 per hour.

Choose the first system. (Going beyond the basic answer, ask any other

volunteers beyond two to put in extra time at work and donate their income.

The homeless people will benefit more from the extra money than from the

faster service.)

At $20 per hour for customers,

First system cost to operate = $10 + $20*(1) = $30 per hour.

Second system cost to operate = $20 + $20*(0.333) = $26.66 per hour.

Third system cost to operate = $30 + $20*(0.090) = $31.80 per hour.

Choose the second system.

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Backward Induction

Question 5. Use backward induction (and draw a

decision tree, if needed) to help make the following

sequential decisions, where later decisions depend on

earlier decisions:

1. Should the Bayer Biological Products Group spend $20 million to

begin preclinical development of a new blood-clot-busting drug?

a. The probability of successful testing on humans is 0.7

b. If the testing is not successful, the project is terminated

c. If the testing is successful, the project requires $2 million to file

a license with to the FDA. The probability of successful filing

0.9

d. If the filing is successful,, demand and revenues are uncertain:

i. With probability 0.4, demand is low and revenue is $60

million.

ii. With probability 0.1, demand is medium and revenue is

$80 million.

iii. With probability 0.5, demand is high and revenue is $100

million.

2. If the testing on humans were successful, should the Bayer Biological

Products Group sell its rights in the project for $70 million?

Compute expected profit for the Bayer Biological Products Group.

Answer to Question:

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Answer to Question:

Step 1: Replace uncertain payoffs with expected value:

1. Should the Bayer Biological Products Group spend $20 million to

begin preclinical development of a new blood-clot-busting drug?

a. The probability of successful testing on humans is 0.7

b. If the testing is not successful, the project is terminated

c. If the testing is successful, the project requires $2 million to file

a license with to the FDA. The probability of successful filing

0.9

d. If the filing is successful,, expected revenue =

0.4(60)+0.1(80)+0.5(100)=$82 million demand and revenues

are uncertain:

i. With probability 0.4, demand is high and revenue is $60

million.

ii. With probability 0.1, demand is medium and revenue is

$80 million.

iii. With probability 0.5, demand is low and revenue is $100

million.

2. If the testing on humans were successful, should the Bayer Biological

Products Group sell its rights in the project for $70 million?

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Step 2: Replace uncertain payoffs with expected value:

1. Should the Bayer Biological Products Group spend $20 million to

begin preclinical development of a new blood-clot-busting drug?

a. The probability of successful testing on humans is 0.7

b. If the testing is not successful, the project is terminated

c. If the testing is successful, the project requires $2 million to file

a license with to the FDA, and earn expected revenue = 0.9(82)

+ 0(0) = $73.8 million. The probability of successful filing 0.9

d. If the filing is successful,, expected revenue =

0.4(60)+0.1(80)+0.5(100)=$82million demand and revenues are

uncertain:

i. With probability 0.4, demand is high and revenue is $60

million.

ii. With probability 0.1, demand is medium and revenue is

$80 million.

iii. With probability 0.5, demand is low and revenue is $100

million.

2. If the testing on humans were successful, should the Bayer Biological

Products Group sell its rights in the project for $70 million?

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Step 3: Make the second decision:

1. Should the Bayer Biological Products Group spend $20 million to

begin preclinical development of a new blood-clot-busting drug?

a. The probability of successful testing on humans is 0.7

b. If the testing is not successful, the project is terminated

c. If the testing is successful, the project requires $2 million to file

a license with to the FDA, and earn expected revenue = 0.9(82)

+ 0(0) = $73.8 million. The probability of successful filing 0.9

d. If the filing is successful,, expected revenue =

0.4(60)+0.1(80)+0.5(100)=$82million demand and revenues are

uncertain:

i. With probability 0.4, demand is high and revenue is $60

million.

ii. With probability 0.1, demand is medium and revenue is

$80 million.

iii. With probability 0.5, demand is low and revenue is $100

million.

2. If the testing on humans were successful, should the Bayer Biological

Products Group sell its rights in the project for $70 million?

Answer: Assume testing on humans were successful. Selling rights in the

project earns $(70-20) = $50 million profit. Keeping rights earns $(73.8-2-

20) = $51.8 million profit. So keep the rights.

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Step 4: Replace uncertain payoffs with expected value, and make the first

decision:

1. Should the Bayer Biological Products Group spend $20 million to

begin preclinical development of a new blood-clot-busting drug?

a. The probability of successful testing on humans is 0.7

b. If the testing is not successful, profit = -$20 million. If the

testing is successful, expected profit = $51.8 million.

So, preclinical development yields expected profit = 0.3(-20) + 0.7(51.8) =

$30.26 million, which being positive indicates that, yes, the Bayer

Biological Products Group should begin preclinical development. And from

previous calculations, if the project is successful, they should not sell their

rights.

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Expected Value of Sample Information

Question 6. Ryoei Saito has received from the

Siegfried Kramarsky family a take-it-or-leave-it offer to

sell to Ryoei Saito the painting Portrait of Dr. Gachet,

by Vincent van Gogh for $150 million. Ryoei Saito is

an expected-value maximizer. Ryoei Saito has determined probability

estimates of the painting's worth to other buyers, based on economic

outcomes, as:

P($120 million) = .3, P($140 million) = .4,

P($160 million) = .1, and P($180 million) = .2

a. Should Ryoei Saito buy the painting? Explain your answer.

b. What is the expected value of perfect information?

Ryoei Saito can have an economic forecast performed by Christie’s auction

house. The forecast indicates either G = Good resale conditions or B =

Bad resale conditions. Conditional probabilities of the indicators conditional

on the collection's worth are

P(G$120 million) = .1, P(G$140 million) = .3,

P(G$160 million) = .2, and P(G | $180 million) = .4

c. Should Ryoei Saito purchase that forecast if the forecast cost $1

million? What is the maximum amount Ryoei Saito should be willing to pay

for that forecast?

Answer to Question:

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19

Answer to Question: There are four states of nature corresponding to the asset’s worth. S1 (state 1) = $120 million, S2 = $140 million, S3 = $160 million, S4 = $180 million. And there are two decision choices: D1 = buy for $150 million, D2 = don’t buy The payoffs from decisions are

S1 S2 S3 S4

D1 -30 -10 10 30

D2 0 0 0 0

a. Given priors

Priors

P(Si)

S1 0.3

S2 0.4

S3 0.1

S4 0.2

the expected value from decisions is

S1 S2 S3 S4 EV

D1 -30 -10 10 30 -6

D2 0 0 0 0 0

So the optimum choice is D2 (don’t buy), and value is 0.

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b. EVPI (expected value of perfect information) is based expected value if

you know the state before making the decision, and so you always

achieve maximum value.

S1 S2 S3 S4

D1 -30 -10 10 30

D2 0 0 0 0

Max 0 0 10 30

Given prior probabilities, the expected value from decisions is

S1 S2 S3 S4 EV

D1 -30 -10 10 30 -6

D2 0 0 0 0 0

Max 0 0 10 30 7

Putting it all together, EVPI is the increase in value from 0 to 7, EVPI = 7

million dollars.

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c. To evaluate the forecast, compute posterior probabilities under the two

possible forecasts: G = Good conditions or B = Bad conditions.

First, consider the forecast G.

Priors Conditionals Joint Posterior

P(Si) P(G|Si) P(G&Si) P(Si|G)

S1 0.3 0.1 0.03 0.120

S2 0.4 0.3 0.12 0.480

S3 0.1 0.2 0.02 0.080

S4 0.2 0.4 0.08 0.320

P(G) = 0.25

Given those posterior probabilities, the expected value from decisions is

S1 S2 S3 S4 EV

D1 -30 -10 10 30 2

D2 0 0 0 0 0

So the optimum choice is D1, and value is 2.

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22

Next, consider the forecast B.

Priors Conditionals Joint Posterior

P(Si) P(B|Si) P(B&Si) P(Si|B)

S1 0.3 0.9 0.27 0.360

S2 0.4 0.7 0.28 0.373

S3 0.1 0.8 0.08 0.107

S4 0.2 0.6 0.12 0.160

P(B) = 0.75

Given those posterior probabilities, the expected value from decisions is

S1 S2 S3 S4 EV

D1 -30 -10 10 30 -8.6666667

D2 0 0 0 0 0

So the optimum choice is D2, and value is 0.

Putting it all together, P(G) = .25 of a Good report and expected value =2,

and P(B) = .75 of a Bad report and expected value = 0. Overall, expected

value is .25x2 + .75x0 = 0.50. Therefore, EVSI = 0.50 million dollars, and

you would pay up to 0.50 million dollars for the economic forecast.