An agent makes decisions using U(ct) = (ct−χct−1)1−γ 1−γ . Answer the following: (a) Suppose χ = 0. Derive an expression for the coefficient of relative risk aversion RR(ct)? (b) Suppose 0 < χ ≤ 1. Derive an expression for the coefficient of relative risk aversion RR(ct)?
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An agent makes decisions using U(ct) = (ct−χct−1)1−γ 1−γ . Answer the following: (a) Suppose χ = 0. Derive an expression for the coefficient of relative risk aversion RR(ct)? (b) Suppose 0 < χ ≤ 1. Derive an expression for the coefficient of relative risk aversion RR(ct)?
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- If the utility function is U (W) = ((W0.75) / (0.75)), what is the absolute risk aversion coefficient?Consider the lottery that assigns a probability r of obtaining a level of consumption CH and a probability 1-T of obtaining a low level of consumption cL an individual facing such a lottery with utility function u(c) that has the properties that more is better (that is, a strictly positive marginal utility of consumption at all levels of c) and diminishing marginal utility of consumption, u"(c) CL. Consider du(c) for the first derivative of the utility function with respect to dc d²u(c) dc2 du' (c) consumption and u"(c) which is also the derivative of the first derivative of the utility function). to be the second derivative of the utility function dcA 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.
- Leora has a monthly income of $20,736. Unfortunately, there is a chance that she will have an accident that will result in costs of $10,736. Thus leaving her an income of only $10,000. The probability of an accident is 0.5. Finally assume that her preferences over income can be represented by the utility function u(x) = 2ln(x).a) What is the expected income? What is Leora’s expected utility (you may leave in log form)? b) What is the certainty equivalent to her situation? What is the risk premium associated with her situation?c) What is the maximum that Leora would be willing to pay for a full insurance policy?d) Illustrate her expected utility, expected wealth, certainty equivalent, the risk premium and her willingness to pay for a full insurance policy in a diagram.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) = In(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.Consider the following utility functions: U1(x) = e*; U2(x) = x°, where r > 0 and BE (0,1); U3(x) = 2x + 10. For each function decide whether it belongs to a risk- neutral, risk-averse or risk-loving decision-maker.
- Cost-Benefit Analysis Suppose you can take one of two summer jobs. In the first job as a flight attendant, with a salary of $5,000, you estimate the probability you will die is 1 in 40,000. Alternatively, you could drive a truck transporting hazardous materials, which pays $12,000 and for which the probability of death is 1 in 10,000. Suppose that you're indifferent between the two jobs except for the pay and the chance of death. If you choose the job as a flight attendant, what does this say about the value you place on your life?In the field of financial management, it has been observed that there is a trade-off between the rate of return that one earns on investments and the amount of risk that one must bear to earn that return. a) Draw a set of indifference curves between risk and return for a person that is risk-averse (a person that does not like risk).Consider a worker whose utility is equal to the amount of dollars she has (U = $) and who can earn $100 a day as a bank teller. However, she takes a job as a worker in a firm that produces shirts. She and her coworkers are monitored at random by their employer to see if they are exerting a target level of effort of e* = 15 units. Assume that the probability of any worker being monitored is p. Also assume that e* is the same level of effort the worker would have to exert as a bank teller. If she is monitored and her employer finds that she is exerting at least 15 units of effort, she is paid > $100. If she is caught putting in less than 15 units of effort, she is fired on the spot but given severance pay of w < w. Say that she suffers a disutility of effort of $2, in monetary terms, for every unit of effort she exerts, so that the dollar cost of exerting the target level of effort of e* is - 2e*. (If she chooses to exert a lower level of effort than e*, we can assume that she will not…
- Arielle is a risk-averse traveler who is planning a trip to Canada. She is planning on carrying $400 in her backpack. Walking the streets of Canada, however, can be dangerous and there is some chance that she will have her backpack stolen. If she is only carrying cash and her backpack is stolen, she will have no money ($0). The probability that her backpack is stolen is 1/5. Finally assume that her preferences over money can be represented by the utility function U(x)=(x)^0.5 Suppose that she has the option to buy traveler’s checks. If her backpack is stolen and she is carrying traveler’s checks then she can have those checks replaced at no cost. National Express charges a fee of $p per $1 traveler’s check. In other words, the price of a $1 traveler’s check is $(1+p). If the purchase of traveler’s checks is a fair bet, then we know that the purchase of traveler checks will not change her expected income. Show that if the purchase is a fair bet, then the price (1+p) = $1.25.An individual has the utility function U(I) = I^(1/2), where I is their net income. (Note that I to the exponent/power of 1/2 is the same as the square root of I.) The individual starts with $1600 in income. The individual has a 20% probability of being very sick, 30% probability of being slightly sick, and 50% probability of being healthy. If the individual is sick, they lose net income because they need to pay healthcare costs. The healthcare costs are $1600 if they are very sick, $700 if they are slightly sick, and $0 if they are healthy. Please use this information for the following parts of this question unless otherwise specified. What is the individual's expected utility? Suppose a health insurance company offers the individual a full insurance contract. What is the actuarially fair, full insurance premium for this individual? What is the individual's expected utility if they purchase a full insurance contract at the actuarially fair, full insurance premium?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.