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Economics+395+Problems+Fall+2008+-+Answer+Key+2 - Economics...

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Economics 395 Topics in Risk and Uncertainty Homework Problems Solutions Problem 4 Imagine that I hold two envelopes, one in my right hand and one in my left. In each envelope there is a check. You have no idea what the amounts on these checks are, but I tell you that one check is written for twice the amount of the other. Then I let you select one of the envelopes. You rip open the envelope and find it’s written for $20. Before you leave with the check, though, I offer you the following opportunity: if you wish, you may exchange your $20 check for the check in the other envelope. (a) If you cared only about the expected value of a lottery, would you swap your $20 check for the check in the other envelope? Why or why not? (b) Now repeat this problem assuming that you found $Y in the envelope you chose. Would you want to swap $Y for the check in the other envelope? Is it true that an expected value maximizer always wishes to swap her check for the check in the unopened envelope? This sounds a lot like the Monty Hall problem. In the Monty Hall problem, we deduced that a person would always like to swap the box initially chosen for the unopened box that remained. Is a similar result true here? (c) Remembering your answer from part (b) (and continuing to assume that you care only about expected values of lotteries), how much would you be prepared to pay to play this game? If this is difficult to answer, explain why. What kind of information would help you evaluate this lottery? The scenario involving the envelopes is another example of a compound lottery: a lottery in which the ultimate prize you receive is determined by the resolution of multiple sources of uncertainty. In this case, there are two sources of uncertainty. First, there is the uncertainty over the values of the two checks. Then, when the first envelope has been opened, there still remains the uncertainty over whether the more-valuable or less- valuable check has been revealed. To make this more concrete, suppose that X is the value of the less-valuable check. The higher-valued check is worth 2X. Again, let Y represent the value of the check that is revealed. The values of X and Y are determined randomly. The event tree below shows the multitude of ways in which these two sources of uncertainty might be resolved. For simplicity, I have drawn only a small handful of the values of X that can be realized in this problem:
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The probabilities shown as P(X=x-1), P(X=x) and P(X=x+1) are all part of the prior distribution over values of X. The conditional probabilities P(Y = x | X =x) and P(Y = 2x | X = x) express the likelihoods that the envelope you open contains the low or the high valued endowment, respectively (d) Can you “flip” this event tree to depict a random process in which the value of Y (the value of the check found in the open envelope) is determined first?
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