1. A decision-maker faces a lottery that gives her a final wealth of 1 dollar with probability 1/4, 3 dollars with probability 1/2, and 8 dollars with probability 1/4. (a) Suppose this decision-maker is an expected utility maximizer with von Neumann-Morgenstern utility u₁(2) = √2+1, where r is her final wealth. Find the risk premium associated with this lottery.
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- Exercise 3: Risky Investment Charlie has von Neumann-Morgenstern utility function u(x) = In x and has wealth W = 250, 000. She is offered the opportunity to purchase a risky project for price P = 160, 000. 1 the project will be a success and return V > 160, 000. With probability 1-p= 1 With probability p= the project will fail and be worthless (i.e. it returns 0). For simplicity assume there is no interest between the time of the investment and the time of its return, that is r = 0. How large must V be in order for Charlie to want to purchase the risky project? [Hint: What is Charlie's expected utility is she does not purchase the project? What is Charlie's expected utility is she purchases the project?]Suppose that Mike, with utility function, u(x) = v x+5000, is offered a gamble where a coin is flipped twice, and if the coin comes up heads both times (probability - .25), he gets $40,000. Would he prefer this gamble or $7,500 for sure? What is his Certainty Equivalent?Suppose that my utility function is u(w) = w2. (a) If I played one round of the St. Petersburg Lottery, would would my expected utility be? Assume I currently have zero wealth, and round your answer to two decimals places. Note that in order to compute my expected utility, you will need to sum a geometric series. Expected Utility utiles. (b) Use your rounded answer from (a) to determine my certainty equivalent for the St. Petersburg Lottery. Round your answer to the nearest cent. Certainty Equivalent = $
- Question 1) Dave is an expected utility maximizer and his von Neumann-Morgenstern utility function is given by u(w) = = (w). He has an initial wealth of $27,000. 199 √99+7 (a) Determine the risk attitude of this economic agent. b) If John is facing a risk of losing $19,000 with probability 0.1, write down his prospective final wealth and the associated probabilities in the form of a lottery. (c) Facing the risk of losing $19,000 with probability 0.1, what would be the maximum that Dave is willing to pay for 19,000 units of a contingent claim that pays $1 if and only if the loss occurs.Assume that Mary’s utility function is U(W) = W1/3, where W is wealth. Suppose that Mary hasan initial level of wealth of $27,000. How much of a risk premium would she require toparticipate in a gamble that has a 50% probability of raising her wealth to $29,791 and a 50%probability of lowering her wealth to $24,389?Exercise 3: Risky Investment Charlie has von Neumann-Morgenstern utility function u(x) = ln x and has wealth W = 250, 000. She is offered the opportunity to purchase a risky project for price P = 160, 000. With probability p= 1 the project will be a success and return V > 160,000. With probability 1-p = the project will fail and be worthless (i.e. it returns 0). For simplicity assume there is no interest between the time of the investment and the time of its return, that is r = 0 . How large must V be in order for Charlie to want to purchase the risky project? [Hint: What is Charlie's expected utility is she does not purchase the project? What is Charlie's expected utility is she purchases the project?]
- An investor has a power utility function with a coefficient of relative risk aversion of 3. Compare the utility that the investor would receive from a certain income of £2 with that generated by a lottery having equally likely outcomes of £1 and £3. Calculate the certain level of income which, for an investor with preferences as above, would generate identical expected utility to the lottery described. How much of the original certain income of £2 the investor would be willing to pay to avoid the lottery? Detail the calculations and carefully explain your answer.Consider an individual who maximizes his expected utility with the following utility function: U(x) = logX He is faced with the lottery with the following probabilities and payoffs Probability Money 0.4 30 0.5 100 0.1 50 a. Find his expected utility b. Calculate the Certainty Equivalent c. Find the amount that the individual will be willing to pay in order to avoid the lottery (That is, the risk premium)Assume that Rosemarie has the following utility function: U(W) = W1/2. She is selling her homeand believes that the house will sell for $250,000 with probability ¼ and $122,500 withprobability ¾.a. What is her expected utility?b. What is the risk premium (P) Rosemarie would pay to avoid bearing this risk?
- Problem 3. Carol's risk preference is represented by the following expected utility formula: U(T, C₁; 1 T, C₂) = π √√ √₁+ (17) √√C₂. i) Suppose Carol is indifferent between the following two options: the first option A returns $100 with probability and $X with probability, and the second option B returns $49 for sure. Determine X. ii) Consider the following three lotteries: L₁ = (0.9, $100; 0.1, $49), L2 = (0.7, $225; 0.3, $49), and L3= (0.5, $400; 0.5, $0). What is the ranking of these lotteries for Carol? Calculate the risk premiums of these lotteries for Carol. 1Suppose that • The employee has an outside offer to work for $27 per hour, for 1500 hours per year The employee currently works for $20 per hour, for 2000 hours per year The switching cost can be either high ($1'000) or low ($50) • The high switching cost has probability 40%; the low switching cost 60% Suppose that the cost of losing the employee is $800. What is the employer expected payoff from choosing not to match the outside offer?# 4 Consider an individual with a utility function of the form u(w) = √w. The individual has an initial wealth of $4. He has two investments options available to him. He can eitffer keep his wealth in an interest-free account or he can take part in a particularly generous lottery that provides $12 with probability of 1/2 and $0 with probability 1/2. Assume that this person does not have to incur a cost if he decides to take part in the lottery. (a) Will this individual participate in the lottery? (b) Calculate this individual's certainty equivalent associated with the lottery. What is his risk premium?