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# Economics+395+Problems+Fall+2008+-+Answer+Key+3 - Economics...

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Economics 395 Risk and Uncertainty Problem Set 3 Answer Key Question 9 Consider a set of monetary prizes, V = {\$0, \$24000, \$27500} and the following four lotteries over those prizes: α 1 = (0.01, 0.66, 0.33) α 2 = (0, 1, 0) α 3 = (0.67, 0, 0.33) α 4 = (0.66, 0.34, 0) where the lottery α = (α 1 , α 2 , α 3 ) means that the lottery delivers \$0 with probability α 1 , \$24000 with probability α 2 and \$27500 with probability α 3 . (a) Given a choice between lotteries α 1 and α 2 , which would you choose? Obviously, this is only for you to decide. (b) Given a choice between lotteries α 3 and α 4 , which would you choose? Again, this one is up to you. (c) In an experimental setting, many people are shown prefer α 2 to α 1 (i.e. α 2 > i α 1 ). Suppose these peoples’ preferences satisfy the independence axiom. Then how do you think such a person would choose between α 2 and another lottery, β, where β = (1/34, 0, 33/34)? First, observe that α 1 can be written as a compound lottery formed using α 2 and β. α 1 = (0.66) α 2 + (0.34) β. From the independence axiom, then, we know that if α 2 > i β then α 2 ~ i (0.66) α 2 + (0.34) α 2 > i [(0.66) α 2 + (0.34) β] ~ i α 1 . But as we know that α 1 > i α 2 , it must be the case that β ≥ i α 2 .

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(d) Can you express α 4 as a compound lottery between α 2 and some other lottery, γ, where γ = (1, 0, 0). Can you express α 3 as a compound lottery between β and γ? α 4 = (0.34) α 2 + (0.66) γ α 3 = (0.34) β + (0.66) γ. (e) Does the independence axiom suggest anything about how those agents described in part (c) should choose between α 3 and α 4 ? Do your preferences satisfy the independence axiom? If α 2 > i α 1 , we have shown that α 2 > i β. But from the compound lotteries given in part (d), this means that [(0.34) α 2 + (0.66) γ] > i [(0.34) β + (0.66) γ] so that α 4 > i α 3 (f) Draw a diagram with α 1 (i.e. the probability of receiving \$0) on the horizontal axis and α 2 (i.e. the probability of receiving \$24000) on the vertical axis. Draw the straight line α 1 + α 2 = 1. Shade the area that represents the set of all possible lotteries over the three prizes, V = {\$0, \$24000, \$27500}. Explain how a lottery can be associated with a point in this diagram, when each lottery is defined by three probabilities, but only two probabilities can be seen in the diagram. Which point represents the most desirable lottery possible? Which point represents the least desirable lottery possible? Show on your diagram the direction of increasing preference. While every lottery is described by three probabilities, α = (α 1 , α 2 , α 3 ), one of these numbers is redundant given the constraint that α 1 + α 2 + α 3 = 1. This last expression can be restated, α 3 = 1 - α 1 - α 2 so that the lottery may be stated as
α = (α 1 , α 2 , α 3 ) = 1 , α 2 , 1 – α 1 – α 2 ). Note this last expression requires only α 1 and α 2 to be given. The most desirable lottery is seen at the origin. Here, the probabilities of receiving the worst and second worst prizes are both zero. Therefore, the origin implies that the probability of the third (i.e. most desirable) prize is one.

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