Homework_6_SOL - Homework 6 1 In the game of Life the first...

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Homework 6 1. In the game of Life the first choice that a player has to make is whether or not he wants to go to college or if he wants to begin his career immediately. If he begins his career immediately then he will earn $36,000 for sure. If he chooses to go to college then he must first pay $20,000 in tuition and fees. After paying for college he will begin his career. However the salary that he receives is uncertain. There is a 50% chance that he will get a job that pays $100,000 and there is a 50% chance that he will get a job that pays only $40,000. Assume that the player is risk neutral . a) Assume that the player does not know which salary he will receive after college before making his decision whether or not to go to college. He does know the probabilities of those salaries given above. Illustrate his decision problem in a decision tree. Which action “go to college” or “start career” is ex ante optimal? What is his expected income (net of any tuition) from his optimal action? Is his choice always ex post optimal? According to the rules of the game the player is not supposed to know which salary he will receive. However, players often cheat. A player can cheat in this game by simply looking at the cards. If he looks at the cards then he will know for sure which salary he will receive. b) Assume that there is no possibility that the other players will catch him cheating. Illustrate his decision problem with the new option of cheating in a decision tree. Which action “go to college” , “start a career” , or “cheat” is ex ante optimal? What is his expected income (net of any tuition) from his optimal action? c) Assume that the probability of being caught cheating is 1. How big must a fine have to be in order to (just) discourage the player from cheating? Answer: a) If player starts career, he will get 36,000 for sure. If player goes to college, he gets 100,000 – 20,000 = 80,000 if he gets a good job, and 40,000 – 20,000 = 20,000 if he gets a worse job. The expected income will be 0.5*80,000 + 0.5* 20,000 = $50,000 Since the player is risk neutral and he can get higher expected income after college education, it’s ex ante optimal to do so, but not always ex post optimal. If he gets the $40,000 job, he would have been better off had he started his career right away for he would have received $36,000 instead of $20,000. b) The optimal action is to cheat, since expected income will now be
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0.5*80,000 + 0.5*36,000 = 58,000 This is higher than the ex ante optimal $50,000 from going to college. c) If there is a fine F for cheating, the expected income from cheating will be 0.5*(80,000- F ) + 0.5*(36,000 – F ) = 58,000 – F . As long as the expected income from cheating is lower than the ex ante optimal expected income, he’ll no longer cheat. 58,000 – F < 50,000 . F > 8,000. A fine higher than $8,000 will discourage the player from cheating. 2. Rachel has an income of $10,000 but she has a 10% chance of suffering a loss of $1900. Her utility of income can be represented by the function U($) = ($) 1/2 .
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