Economics+395+midterm+practice+questions+-+solutions-1

Economics+395+midterm+practice+questions+-+solutions-1 -...

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Economics 395 Topics in Risk and Uncertainty Practice Questions Midterm Exam Fall 2008 Question One Anatole is given the choice between the following two lotteries: (i) Wins nothing with probability 0.54; loses $1000 with probability 0.46. (ii) Wins $100 with probability 0.48; loses $1000 with probability 0.52. (a) Draw a diagram with probability of losing $1000 on the horizontal axis, and probability of winning nothing on the vertical axis. Show in your diagram the full set of all possible lotteries over the three prizes V = {-$1000, $0, $100}. Explain how you can capture such a lottery in a two dimensional diagram, when each lottery is specified as a sequence of three numbers. Identify the best lottery possible, and label it β. Identify the worst lottery possible, and label it ω. As α 1 + α 2 + α 3 = 1, it is sufficient to know α 1 and α 2 as α 3 = 1 - α 1 + α 2 . Preference direction α 1 (i) (ii) 0.46 0.52 0.54 ω β 0.20 0.10 0.90 (iii) (iv) α 2
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(b) In your diagram, show the two lotteries (i) and (ii), described above. These are marked (i) and (ii). (c) Suppose Anatole’s preferences satisfy the expected utility hypothesis. Also, suppose he prefers lottery (ii) over lottery (i). Draw a set of indifference curves that might represent his preferences. The representative indifference curves are marked in blue. (d) Mark the following two lotteries on your diagram. (iii) Loses $1000 with probability 0.2; wins $100 with probability 0.8. (iv) Loses $1000 with probability 0.1; wins nothing with probability 0.9. If Anatole were given the choice between these lotteries, which do you think he would choose? Explain why. Anatole would prefer lottery (iii) to lottery (iv). As Anatole is an expected utility maximizer, his indifference curves are straight parallel lines. As (ii) is preferred to (i), we know that the indifference curves must be flatter than the line between lottery (ii) and lottery (i). The line between (i) and (i) has slope of (-0.54)/(0.04) = -9. The slope of the straight line through (iii) and (iv) is (-0.9)/(0.1) = -9. Therefore, the indifference map is also flatter than the line between these two points. It must be the case, then, that (iii) is preferred to (iv). (e) Consider a compound lottery which promises lottery (iv) as a prize with probability 0.6, and promises lottery ω as a prize with probability 0.4. If Anatole plays this lottery, with what probability does he lose $1000? With what probability does he win nothing? With what probability does he win $100? Mark that compound lottery on your diagram. In this compound lottery, there is probability 0.6 that Anatole will play lottery (0.1, 0.9, 0) and probability 0.4 that Anatole plays lottery (1, 0, 0). Neither simple lottery gives any probability of winning $100, so the compound lottery gives probability 0 of winning $100. The only way that Anatole could win $0 is if he wins the chance to play lottery (iv). This happens with probability 0.6. Once he has won lottery (iv), Anatole will win $0 with probability 0.9.
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This note was uploaded on 04/13/2010 for the course ECON 330 taught by Professor Minetti during the Fall '08 term at Michigan State University.

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