Problem Set 2
Q1
1 Point
What is the expected value of the lottery LL where
L=[24:frac{1}{2}, 12:frac{1}{6}, 48:frac{1}{6}, 6:frac{1}{6}]L=[24:21,12:61,48:61,6:61]
Choice 1 of 4:12Choice 2 of 4:14Choice 3 of 4:23Choice 4 of 4:85.5
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Q2
1 Point
Consider two lotteries AA and BB. For lottery AA, there is a 20-percent chance that you will receive $80, and 50-percent chance that you will receive $40, and a 30-percent chance that you will receive $10. For lottery BB, there is a 40-percent chance that you will receive $30, a 30-percent chance that you will receive $40 and a 30-percent chance that you will receive $50.
Suppose that the decision maker’s utility for money is given by U($m)=sqrt{m}U($m)=m (i.e., the utility for money is the square root of the monetary value). Which of the following is true?
The following checkbox options contain math elements, so you may need to read them in your screen reader’s “reading” or “browse” mode instead of “forms” or “focus” mode.
Choice 1 of 3:The decision maker is indifferent between AA and BB.
Choice 2 of 3:The decision maker strictly prefers AA over BB.
Choice 3 of 3:The decision maker strictly prefers BB over AA.
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Q3
1 Point
Suppose that Ann is offered a choice between the following two lotteries:
A=[$4,000: 0.8, $0: 0.2]A=[$4,000:0.8,$0:0.2] and B=[$3,000:1]B=[$3,000:1].
Ann says she strictly prefers BB to AA.
Is Ann maximizes expected monetary value?
The following multiple-choice options contain math elements, so you may need to read them in your screen reader’s “reading” or “browse” mode instead of “forms” or “focus” mode.
Choice 1 of 2:Yes, BB has a greater expected monetary value.Choice 2 of 2:No, BB does not have a greater expected monetary value.
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Q4
1 Point
Suppose that Ann is offered a choice between the following two lotteries:
A=[$4,000: 0.8, $0: 0.2]A=[$4,000:0.8,$0:0.2] and B=[$3,000:1]B=[$3,000:1].
Ann says she strictly prefers BB to AA.
Suppose that Ann is an expected utility maximizer. Which of the following two lotteries will Ann choose?
C=[$4,000: 0.2, $0: 0.8]C=[$4,000:0.2,$0:0.8] or D=[$3,000: 0.25, $0: 0.75]D=[$3,000:0.25,$0:0.75].
The following multiple-choice options contain math elements, so you may need to read them in your screen reader’s “reading” or “browse” mode instead of “forms” or “focus” mode.
Choice 1 of 3:Ann strictly prefers CC to DDChoice 2 of 3:Ann strictly prefers DD to CCChoice 3 of 3:Ann is indifferent between CC and DD
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Q5
2 Points
Explain your answer to question 4.
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Q6
2 Points
Suppose that we have four different foods: ice cream, chocolate sauce, french fries, and ketchup. These foods can be mixed together into a bowl in different proportions. Let (a,b,c,d)(a,b,c,d) represent the amount of ice cream, chocolate sauce, french fries, and ketchup (respectively) that gets mixed together. For example, (1,1,0,0)(1,1,0,0) represents 11 oz of ice cream mixed with 11 oz of chocolate sauce (and no french fries or ketchup). Or (1,0,2,0)(1,0,2,0) represents 11 oz of ice cream mixed with 22 oz of french fries (and no chocolate sauce or ketchup). Now, suppose a concatenation operation oplus⊕ which works like this: (a,b,c,d)(a,b,c,d) mixed with (w,x,y,z)(w,x,y,z), written (a,b,c,d) oplus (w,x,y,z)(a,b,c,d)⊕(w,x,y,z) produces a new mixture (a+w, b+x, c+y, d+z)(a+w,b+x,c+y,d+z). So, for example, (1,1,0,0)oplus (1,0,2,0)(1,1,0,0)⊕(1,0,2,0) results in a food mixture consisting of 2 oz ice cream, 1 oz chocolate sauce, 2 oz french fries, and no ketchup. Consider your run-of-the-mill preferences over all objects of the form (a,b,c,d)(a,b,c,d). Do they satisfy this axiom?
Definition (Independence) For all food bowls ff and gg, if ff is strictly preferred to gg then for any food bowl hh, foplus hf⊕h is strictly preferred to goplus hg⊕h.
Explain why or why not.
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Q7
2 Points
You are playing a game for money. There are two envelopes on a table. You know that one contains $X$X and the other $2X$2X, but you do not know which envelope is which or what the number X is. Initially you are allowed to pick one of the envelopes, to open it, and see that it contains $Y$Y. You then have a choice: walk away with the $Y$Y or return the envelope to the table and walk away with whatever is in the other envelope. What should you do?
Since the envelop you are holding contains YY dollars, you know that the other envelop contains either frac{1}{2}Y21Y dollars or 2Y2Y dollars, with equal probability. So, you must compare keeping YY dollars with the expected value of switching envelops. The expected value of switching is:
frac{1}{2}(frac{1}{2}Y) + frac{1}{2}2Y=frac{1}{4}Y + Y = frac{5}{4}Y21(21Y)+212Y=41Y+Y=45Y
Since frac{5}{4}Y > Y45Y>Y, you should switch envelops. But, of course, the same argument applies to the envelop you are now holding. So, expected utility theory seems to suggest that you should keep switching. But this is absurd. What is wrong with this reasoning? This is problem is known as the two-envelop paradox. There are many explanations of this problem found on the internet. Summarize how you would answer this question. Make sure to cite your source (i.e., provide a link to the online article or video that you used to come up with your answer).
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