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Unformatted text preview: ECON 121b Winter 2011 Instructor: Professor Eduardo Faingold Grader of the Problem Set: Noam Tanner ( [email protected] ) Now, you must ask yourself one question: Do I feel lucky? Well do you, punk?- Clint Eastwood in the movie Magnum Force . Solutions to Homework Assignment #4: Risk Aversion Question 1 : (a) ¡¢£ ¤ ¥¢ £¦¢ §¨©ª¤«¢¬ǯª ®¯ °¤©§£±¨©Ǥ ²³±©£±©´ غ ®¨£ ²³±©£±©´ ±° ³©µ ¨©¶· ±°ǣ Ɏ ¤ሺ¸- L) + (1- Ɏሻ¤ሺ¸ ¹ሻ ¤ሺ¸ሻ Notice 2 things: (1) Actions cause lotteries. In parts of the lectures, the decision was modeled as a choice between lotteries . This is equivalent to modeling a choice as a decision between two actions that lead to two different lotteries . In the movie Magnum Force , a criminal has to make a decision between two acts (that generate lotteries): running away from Clint Eastwood (R) or turning himself in (T). Each of these acts generates a lottery. Turning oneself in, T, would lead to a very simple lottery: jail (but not being shot) with probability one. Running away, R, would generate a more complicated lottery: jail/injury/death with probability Ɏǡ ³©µ °¬¢¢µ¨« ¸±£¦ º¬¨¥³¥±¶±£· ͳ Ȃ Ɏ ሺ£¦¨¤´¦ ª±©§¢ ±£ ±ª ³ »¶±©£ ¼³ª£¸¨¨µ «¨±¢ǡ Ɏ is very close to 1). Similary, many other actions generate lotteries. Buying insurance generates lottery with less variance in your final wealth. Buying stock generates a lottery with more variance in your final wealth. In this problem, painting the house leads to one lottery between a final wealth level of w- ¡ ¸±£¦ º¬¨¥³¥±¶±£· Ɏ ³©µ ³ final wealth level of w + G with probability 1 Ȃ Ɏ (more risk/more variance). Not painting the house leads to the outcome w (no risk/no variance). In this problem, we want to know when the person will take the riskier act (painting the house) and we want to come up with a way of comparing different people ǯ s risk-taking preferences (i.e. figure out how ǲ lucky they feel ǳ ). (2) A lottery specifies the final outcomes we care about (that are placed in the utility function) and the probabilities of the outcomes. It is important to stress that the agents care about their final wealth levels (we put w + or Ȃ something into the utility), not the changes in wealth level (just the + or Ȃ something). In other words, the equation is NOT to decide to paint if Ɏ ¡ሺ-L) + (1- Ɏሻ¡ሺ¢ሻ ¡ሺͲሻǡ but IS Ɏ ¡ሺ w - L) + (1- Ɏሻ¡ሺ w ¢ሻ ¡ሺ w) (b) If the agent is just willing to take the gamble: Ɏ ¡ሺ£- L) + (1- Ɏሻ¡ሺ£ ¢ሻ ൌ ¡ሺ£ሻǡ ¤¥ (1- Ɏሻ¡ሺ£ ¢ሻ ൌ ¡ሺ£ ) - Ɏ ¡ሺ£- L), thus, u(w +G) = ୳ሺ୵ሻ ି ୳ሺ୵ ି ሻ ሺଵିሻ , hence w+ G = u-1 ( ୳ሺ୵ሻ ି ୳ሺ୵ ି ሻ ሺଵିሻ ), where u-1 is the inverse function of the utility...
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