Multiplying each state of nature outcome by the appropriate probabilities

Practice Problems – Chapter 19

Section 19.1, pages 19-9 and 19-10 in online chapter 19
# 19.4, parts (a) and (b) only (similar to HW problem 6)
# 19.5 (similar to HW problems 7)
# 19.9 (similar to HW problems 11)

Section 19.2, pages 19-14 and 19-15 in online chapter 19
# 19.20 (refers back to problem 9, similar to HW problems 12 & 18)

Section 19.3, page 19-31 in online chapter 19
# 19-24 (similar to HW problem 25)

Answers:

19.4
a. Choose A3.
b. Choose A2.

19.5 Multiplying each state of nature outcome by the appropriate probabilities, we find the following table:
Select the alternative with the largest expected value, A3.

19.9

Hints on payoff table:
Consider if you purchase 10,000:
• Then your costs will be 10,000 x (0.75 + 0.15) = 10,000 x 0.90 = 9000 for every possible demand
• Revenues will depend on demand
• If demand = 10,000, then revenues = 10,000 x 1.75 = 17,500, so profit = revenues – costs = 8500
• If demand = 15,000, then revenues is still 10,000 x 1.75, because you can’t sell what you don’t have
Consider if you purchase 15,000:
• you can calculate costs as 15,000 x 0.90 = 13,500
• If your demand is 15,000, you have revenues of 15,000 x 1.75 = 26,250, leaving profits for 15,000 (purchase) and 15,000 demand as 26,250 – 13,500 = 12,750.
• If demand is greater than 15,000, you can still only sell 15,000, so revenues and costs can be calculated as above
• However, if demand is LESS than supply (15,000), then you will have:
o Revenues from sales at the park based on only what the demand is, so if 10,000, then revenues is 10,000 x 1.75 = 17,500
o You also can sell what is not demanded (15000 supply – 10000 demand = 5000 surplus) at 0.25 each, so you can earn an additional 5000 x 0.25 = 1250
o This brings your total revenue to 18,750 and your costs at 13,500, leaving profits of 5250
To finish off the table, consider for each purchase/demand combination:
• How many are purchased and therefore costs
• Revenues from demand
• Revenues from leftover hot dogs (if any)
• Profits = Revenues – Costs

a.
Demand
Purchase 10,000 15,000 20,000 25,000
10,000 8,500 8,500 8,500 8,500
15,000 5,250 12,750 12,750 12,750
20,000 2,000 9,500 17,000 17,000
25,000 (1,250) 6,250 13,750 21,250

b.
1. Maximax: The maximum values are 8500, 12750, 17000 and 21250, in order, so purchase 25,000 hotdogs and buns.
2. Maximin: The minimum values are 8500, 5250, 2000 and –1250, in order, so purchase 10,000 hotdogs and buns.

19.20
a.
Demand
Purchase 10,000 15,000 20,000 25,000
10,000 8,500 8,500 8,500 8,500
15,000 5,250 12,750 12,750 12,750
20,000 2,000 9,500 17,000 17,000
25,000 (1,250) 6,250 13,750 21,250
Probability 0.20 0.25 0.30 0.25
b.
Purchase EV
10,000 8,500
15,000 11,250
20,000 12,125
25,000 10,750
c.
Purchase 10,000 15,000 20,000 25,000
Maximum of all actions (what you could get if you knew what demand would be, Payoff Under Certainty) 8500 12,750 17,000 21,250
Probabilities of each demand 0.20 0.25 0.30 0.25

Expected Value Under Certainty = $15,300
Expected Value of perfect information = EV Under Certainty – EV under Maximum Payoff
EVPI = $15,300 – $12,125 = $3,175
19.24

Hints on decision tree:
• First decision fork is keep or sell.
• If sell, income is known 50,000
• If keep, there are two possible states of nature: demand increases (probability 0.6) or no increase (probability 0.4)
• Suppose demand increases. There are then two decisions, to sell (profit 60,000) or don’t sell (profit 75,000). In this case, you would choose the one with the highest payoff (75000). So you can assume that you would make 75000 if demand increases.
• Do the same analysis for when there is no increase in demand. Determine the profits you would make in the case of no increase.
• Calculate an expected value for keeping the quarry based on the probability of increased demand and profit under increased demand and the probability of no increase and profit if there is no increase. This is your “payoff” for keeping the quarry.
• Compare this to the profit if you sell. Choose the alternative with the highest expected value/payoff.

RESULT: Tom and Joe should keep the quarry and sell after two years only if there is no increase in demand.

 

Question 1
1.
Refer to problem #19.6 part a only. The maximum payoff under alternative A1 is , the maximum under A2 is , the maximum payoff under A3 is , and the maximum under A4 is ; therefore, the optimistic/maximax criterion (strategy) tells us to choose action — enter either A1, A2, A3, or A4.
1 points
Question 2
1.
Refer to problem #19.6 part b only. The minimum payoff under alternative A1 is , the minimum under A2 is , the minimum payoff under A3 is , and the minimum under A4 is ; therefore, the pessamistic/maximin criterion (strategy) tells us to choose action — enter either A1, A2, A3, or A4.
1 points
Question 3
1.
Refer to problem 19.7.
The expected payoff under alternative A1 is , the expected payoff under A2 is , the expected payoff under A3 is , and the expected payoff under A4 is ; therefore, the expected value criterion (strategy) tells us to choose action (enter either A1, A2, A3, or A4).
2 points
Question 4
1.
Refer to problem 19.18, part b only.
Of the four alternatives, A1, A2, A3, A4, the one with the largest expected value is , which has an expected value of . For this problem, the Expected Value Under Certainty (EVUC) is . Therefore, the Expected Value of Perfect Information is .
2 points
Question 5
1.
Refer to problem 19.25. Describe the decision tree and your decision. Enter all dollar values in thousands, for example, $50,000 should be entered as 50.

If you choose to expand, there are 3 possible outcomes:
High demand, which has a payoff of and a probability of 0.5;
Medium demand, which has a payoff of and a probability of 0.3;
Low demand, which has a payoff of and a probability of 0.2.
Therefore, the expected value if you expand is .

If you choose NOT to expand, there are 3 possible outcomes:
High demand, which has a payoff of and a probability of 0.5;
Medium demand, which has a payoff of and a probability of 0.3;
Low demand, which has a payoff of and a probability of 0.2.
Therefore, the expected value if you DO NOT expand is .

When choosing between these two alterantives, you would choose (enter either “to” or “not to”, without quotes) expand because that option has the higher expected value.

 

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