1. Jacob\'s allowance for this week is $15. Jacob\'s schoolmate Henry has offere
ID: 1148772 • Letter: 1
Question
1. Jacob's allowance for this week is $15. Jacob's schoolmate Henry has offered him the following bet. If their teacher, Mr. Fashionable, comes to class on Monday wearing a red shirt, Jacob pays Henry $10. If she wears a blue shirt, Jacob pays Henry $5. If she wears a beige shirt, no money changes hands. If none of these cases occur, Henry will pay Jacob $8. Jacob believes Mr. Fashionable will wear red with probability 0.2, blue with probability 0.3, and beige with probability 0.1 (a) What is Jacob's expected income for the week if he accepts this bet? What is the standard deviation of his income? (b) If Jacob is risk-neutral, should he accept this bet? Suppose instead that Jacob's utility function over income is described in the following table INCOME UTILITY INCOME UTILITY 15 16 61 96 129 160 189 216 241 264 285 304 321 336 349 360 369 376 381 384 385 10 20 23 (c) Is Jacob risk averse, risk neutral, or risk loving? Explain your answer (d) What is Jacob's expected utility from taking the bet? Should Jacob accept the bet? Explain. (e) Suppose instead that if Mr. Fashionable wears a red or blue shirt, Jacob pays only $2. Would Jacob accept the new bet? Explain. (f) Suppose Henry is risk neutral. If Henry's beliefs about the probabilities are the same as Jacob's, does it make sense for Henry to offer the bet described in (e)? Explain. If so, explain. If not, discuss how the probabilities Henry assigns to the different outcomes would need to differ from Jacob's, so that Henry would be willing to offer the bet.Explanation / Answer
Income of Jacob in four different states
If Red shirt then income = $5
If Blue shirt then income = $10
If Binge shirt then income =$15
If nothing of above cases happens then income= $23
Expected income =0.2(5)+0.3(10)+0.1(15)+0.4(23)=$14.7
Standard Deviation= variance of income^0.5
Variance of Income=E(X^2)-[E(X)]^2
E(X^2)=25×0.2+100×0.3+225×0.1+529×0.4=5+30+22.5+211.6=269.1
V(X)=269.1-(14.7)^2=269.1-216.09=53
SD(X)=7.28
Answer for 2
If Expected outcome is less than the certain income and he is risk neutral then he shouldn't play the bet.
Jacob is Risk averse as it shows lower marginal utility for every increase in income.
Expected Utility = 0.2×U(5)+0.3U(10)+0.1U(15)+0.4U(23)=0.2×61+0.3×216+0.1×321+0.4×385=12.2+64.8+32.1+154=263.1
E(U)=263.1<U(15)=321>U(E(X))=304
He has higher utility from $15 then expected income's utility hence he should not go with bet.
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