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(x)=√x+1, where x is her final wealth. Find the risk premium associated with this lottery. (b) Now suppose that a second decision-maker who maximizes expected utility with von Neumann-Morgenstern utility u2(x)=√x faces the same lottery. Without calculating this decision-maker's risk premium, determine whether it is higher than, lower than, or the same as that for the decision-maker in part (a).
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- 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?]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 Bill's utility function of wealth is given by u(w) = √w, where w represents his total wealth in dollars. Bill's total wealth is $360,000. If an earthquake happens, his wealth will be reduced to $250,000 with the loss of his house. Suppose the probability of an earthquake happening is 10%. (a) Is Bill risk-averse, risk-loving, or risk-neutral? Explain. (b) If Bill could buy insurance to completely avoid the loss, how much would he be willing to pay for this insurance at most? Explain.Adam is considering what skills to study in online school. Her utility function is based on the income she earns, and is defined by U(I) = I0.8. If she learns the skill of SPSS, she will earn $145,000 per year with probability 1. If she learns the skill of Tableau, she will earn $300,000 per year with probability 0.6 (assuming that she gets the certificate) and $30,000 with probability 0.4 (if she learns without earning a certificate and she has to find a waiter job). a. Is she risk averse, risk neutral, or risk loving? Explain.b. Write out the equation for her expected utility for each skill. c.Which skill will she learn? Show your work. d.Suppose someone offers her insurance for the possibility that she does not get a Tableau certificate. This insurance will provide her an amount of income in addition to the waiter job wages that makes her indifferent between learning SPSS and Tableau. What is this amount, and what is the cost of the insurance? (note: many possible answers)Yuki has a utility function given by u(x) = In(x). She faces a gamble that pays 10 with probability 0.5 and 15 with probability 0.5. Comment on how Yuki's certainty equivalent relative to the expected value varies as her utility function goes from concave from %3D convex.
- 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 you graduate from college next year and you have two career options: 1) You will start a job in an investment bank paying a $100,000 annual salary. 2) You will start a Ph.D. in economics and, as a student, you will receive a $20,000 salary. You are bad with decisions, so you are letting a friend of yours decide for you by flipping a coin. The probabilities of options 1 and 2 are, therefore, each 50%. a) Illustrate, using indifference curves, your preferences regarding consumption choices in the two different states of the world. Assume that you are risk-averse. [Include also the 45 degrees line in your figure] b) Now show how the indifference curves would change if you were substantially more risk averse than before. Explain. c) Now show the indifference curves if you are risk neutral and if you are risk loving. d) Show your expected utility preferences from point a) mathematically.Leo owns one share of Anteras, a semiconductor chip company which may have to recall millions of chips. The stock currently trades at $100/share. Leo believes the probability that they have to recall the chips is 50%. If the chips have to be recalled, the stock price will be cut in half, but otherwise it will remain $100. The expected value of Leo's share is ______ Assume Leo has the utility function, U(X)=√X. The minimum price Leo would accept to sell his share is _______ Leo's risk premium is ________
- A driver's wealth $100,000 includes a car of $20,000. To install a car alarm costs the driver $1,750. The probability that the car is stolen is 0.2 when the car does not have an alarm and 0.1 when the car does have an alarm. Assume the driver's von Neumann-Morgenstern utility function is U(W) = ln(W). Suppose the driver is deciding between the following three options: (a) purchase no car insurance, do not install car alarm; (b) purchase fair insurance to replace the car, do not install car alarm; and (c) purchase no car insurance, install car alarm. Of these three options, the driver prefers: A. option (a). B. option (b). C. option (c). D. options (a) and (b). E. options (a) and (c). F. options (b) and (c). G. all options equally. H. none of these options.A person has wealth of $500,000. In case of a flood her wealth will be reduced to $50,000. The probability of flooding is 1/10. The person can buy flood insurance at a cost of $0.10 for each $1 worth of coverage. Suppose that the satisfaction she derives from c dollars of wealth (or consumption) is given by u(c) = √c. Let C denote the contingent commodity dollars if there is a flood (horizontal axis) and CNF denote the contingent commodity dollars if there is no flood (vertical axis). 1 Determine the contingent consumption plan if she does not buy insurance. 2 Assume that the person has von Neumann-Morgenstern utility function on the contingent consumption plans. Write down the expected utility U(CF, CNF) and derive the MRS. 3 Solve for optimal (CF, CNF). To this end, first use the tangency condition (TC) to find the relation between the two contingent commodities (CF, CNF). Next, use (BC) to solve for their values. What is the optimal amount of insurance K the person will buy? (Note:…could you answer part b to this question or if you have time part a and part b but part is more important. thank you Priyanka has an income of £90,000 and is a von Neumann-Morgenstern expected utility maximiser with von Neumann-Morgenstern utility index . There is a 1 % probability that there is flooding damage at her house. The repair of the damage would cost £80,000 which would reduce the income to £10,000. a) Would Priyanka be willing to spend £500 to purchase an insurance policy that would fully insure her against this loss? Explain. b) What would be the highest price (premium) that she would be willing to pay for an insurance policy that fully insures her against the flooding damage?