MS 405 Assignment #3
1- In an experiment subjects are given between the two gambles: Gamble 1: A: $2500 with probability 0.33 $2400 with probability 0.66 $0 with probability 0.01 Gamble 2: C: $2500 with probability 0.33 $0 with probability 0.67
$2400 with certainty
$2400 with probability 0.34 $0 with probability 0.67
Suppose that a person is an expected utility maximizer. Set the utility scale so that u($0) = 0 and u($2500)=1. Denote u($2400) by x. a) Which one would you prefer, A or B? C or D? (Without any calculations!) b) For what values of x would a person choose option A? For what values would a person choose option B? c) For what values of x would a person choose option C? For what values would a person choose option D? d) Comment on your preferences stated in part a. Are there any paradoxes? 2- Professional tennis has become a big business with big prizes. There frequently is a large disparity between the prize for the winner and that for the runner-up; For example, the former may receive $100,000; the latter $32,000. In 1983, Michael Mewshaw asserted that finalists often make secret deals before match to divide the pot; that is they agreed that both winner and loser would get ($100,000+32,000)/2= $66,000. This practice, known as splitting, was particularly likely to occur, he said, “in special events and exhibitions, and on the [World Championship Tennis] circuit, all of which tended to have huge prize-money differences.” If a player has the following utility function and believes that he has a 50-50 chance of winning, will he be willing to split?
3- An alternative has a probability 0.6 of winning $25,000, 0.2 of winning $1,000, and 0.2 of losing $50,000. a) Determine the expected profit for this alternative. b) Determine the certainty equivalent for the alternative using the utility function , where x is in thousands of dollars. c) What is the risk tolerance of the decision maker who has the utility function presented in part b. d) Use the equation CE = x − σ 2 /( 2 ρ ) to determine an approximate certainty equivalent (where σ 2 is the variance of the lottery and ρ is the risk tolerance corresponding to the utility function in part b, i.e, (Certainty Equivalent = Expected Value – 0.5(Variance/Risk Tolerance) ). 4- Mary just won in a raffle a lottery ticket that has a 0.1 probability of winning her $5000 by the end of the month and 0.9 probability of winning her $0. She is risk averse and her utility function is u ( x) = x where x is measured in thousand dollar units. John who is a risk neutral decision maker knows that Mary needs money immediately and that she dislikes being in suspense so he offers her a wager on her raffle ticket. They will toss a fair coin and if it comes out Heads he will get her raffle ticket for free while if it comes out Tails he will buy from her the raffle ticket for B Dollars. What is the range of B that will make such a wager attractive to both of them?
u ( x) = 1 − e
5- Assume that you represent a chain of movie theatres negotiating the purchase of the rights to screen the new Cem Yılmaz movie, "A.R.O.G.O.R.A". If the movie becomes successful (and both you and the distributor that you are negotiating with, think that the odds of success are 0.7), then the revenue that you will make from showing it in your theatres will be $10 million. If it fails (odds of 0.3) then your revenue will only be $6 million. If you don't buy this movie then you will buy the latest Disney movie (which is a sure thing at the box office, that is to say you know how many people will watch it) which will net you $4.5 million (revenue minus cost). Suppose the distributor offers the (Cem Yılmaz) movie to you for $4 million. Because you are risk averse, you are not certain that this deal is good for you. Assume that you are just indifferent between a 50/50 gamble between making $8 million or losing $2 million and doing nothing. a) Graph your utility function. b) Now...
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