ec142f09hw3_sol

# ec142f09hw3_sol - Kata Bognar [email protected] Economics...

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Unformatted text preview: Kata Bognar [email protected] Economics 142 Probabilistic Microeconomics UCLA Fall 2009 Homework Assignment 3.- Suggested solutions - 1. Georgie is an expected utility maximizer, he prefers more money to less money and [6] has risk averse preferences. Assume that the following two lotteries are available for him. The lottery A pays \$50 , \$80 and \$100 with equal probabilities, while the lottery B pays \$50 and \$100 with equal probabilities. (a) Which of these lotteries would Georgie prefer? Please explain! Answer: Notice that there is no vNM utility given in the example. Had there be a vNM utility function, this would be a very easy exercise. One could simply calculate the expected utilities and find the lottery with the higher expected utility. But that is not the case. So what to do now? If you calculate the expected payoffs, you can see that the lottery A has higher expected payoff than the lottery B . However, this does not necessarily mean that the lottery A is preferred to lottery B. So that will not work either. We have learned that in certain cases one can use dominance concepts to predict the choice of the decision maker without knowing much about her preferences. Such dominance concepts between lotteries are: first order stochastic dominance, statewise dominance and, for risk averse decision makers, increasing risk. And this last one will be the key to the answer of this question. Recall that a risky act ˜ y is more risky than a risky act ˜ x if there exists a small risk ˜ ε with E [˜ ε ] = 0 such that ˜ y = ˜ x + ˜ ε. Unfortunately, the lottery B cannot be more risky than the lottery A in this sense, simply because their expected payoffs are not equal. However, one can find a third lottery such that (i) it is less risky than the lottery B and (ii) it is worse than the lottery A. So, by transitivity, one can show that A B. The last piece in the argument is to come up with this third lottery. Consider a lottery C that pays \$50 , \$75 and \$100 with equal probabilities. Then A C by the Independence Axiom and since more money is preferred to less money by the decision maker. Then consider a risky act, ˜ ε such that it is zero whenever lottery C pays \$50 or \$100 and it is- \$25 and \$25 with equal probability whenever the lottery C pays \$75 . Notice that E [˜ ε ] = 0 and also the lottery B is equal to the sum of lottery C and this small risk....
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ec142f09hw3_sol - Kata Bognar [email protected] Economics...

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