Let us consider a utility function: U(x) =(√2x-1. (200sxs800) We have L₁iL2 where L₁=(1, V) and L₂= 450, 0.7,648). a. Determine the value of Y. b. Determine RP (Risk Premium) of L2.
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- 5. You are a risk-averse decision maker with a utility function U(1) = VI, where I denotes your income. Your income is $100,000 (thus, I=100). However, there is a 0.2 chance that you will have an accident that results in a loss of $10,000. Now, suppose you have the opportunity to purchase an insurance policy that fully insures you against this loss (i.e., that pays you $10,000 in the event that you incur the loss). What is the highest premium that you would be willing to pay for this insurance policy?2. Alice believes that her car would cost £12500 to replace if it was stolen or damaged. Based on crime statistics for the area she lives in, she believes that the probability of her car being stolen or damaged is 0.15. (i) Alice's utility function is given by U(w) = ln(w) for w > 0 and she as £35000 in the bank. Calculate how much Alice would be prepared to pay (in a single payment) to insure her car against theft or damage (ii) Repeat the calculation in the previous part but now assume Alice has £500000 in the bank.3. The utility function is u(x, y) = √x+y. Suppose that 1) with probability 0.5, (I, Px, Py) (8, 2, 2) and 2) with probability 0.5, (I, Px, Py) = (8, 4, 8). Explain graphically whether an increase in x increases or decreases the riskiness of the utility gamble. U 6 5 4 3 2 1 0 0 0.25 0.5 0.75 1 1 1.25 1.5 1.75 28
- 2. The utility function is u(x, y) = min{x,y}. Suppose that 1) with probability 0.5, (I, Px, Py) = (4, 1, 1) and 2) with probability 0.5, (I, Px, Py) (12, 6, 2). Explain graphically whether an increase in x increases or decreases the riskiness of the utility gamble. Y 6 5 4 3 2 1 0 0 1 2 IN I 3 4 5 = 6 XConsider 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 dcQ2: Consider a person who is thinking about whether to engage in a life of crime. He knows that, if he gets caught, he will be in jail and his consumption will be low, xº, but if he does not get caught, he will be able to consume an amount x₁ that is considerably above Χρ· (a) Suppose that x₁ = 20; x₁ = 80 (where both are expressed in thousands of pounds) and suppose the probability of getting caught is 8 = 0.5. What is the expected consumption level if the life of crime is chosen? (b) Suppose the potential criminal's tastes over gambles can be expressed using the following utility function u(x) = In (x). Calculate the person's expected utility from a life of crime. How does it compare with the utility of the expected value of consumption? Based on your answer, explain this individual's attitude towards risk and draw the consumption/utility relationship. (c) Consider the level of consumption this person could attain by not engaging in a life of crime. What level of consumption from an…
- 5. Priyanka has an income of £90,000 and is a von Neumann-Morgenstern expected utility maximiser with von Neumann-Morgenstern utility index u(x) = √√x. 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?6. Policyholders are assumed to have a utility function u(x) = e- where > 0 varies between policyholders following an exponential distribution with unknown mean. An insurance company sells an insurance policy which covers a risk which causes a loss of $6,000 with probability 0.4. There are 3,000,000 potential customers for this policy. The insurer finds that when the premium for the policy is set to $3000, they are able to sell 952,000 policies. How many policies would they sell if they increased the premium to $4,000?Gary likes to gamble. Donna offers to bet him $31 on the outcome of a boat race. If Gary's boat wins, Donna would give him $31. If Gary's boat does not win, Gary would give her $31. Gary's utility function is p1x^21+p2x^22, where P₁ and p2 are the probabilities of events 1 and 2 and where x₁ and x₂ are his wealth if events 1 and 2 occur respectively. Gary's total wealth is currently only $80 and he believes that the probability that he will win the race is 0.3. Which of the following is correct? (please submit the number corresponding to the correct answer). 1. Taking the bet would reduce his expected utility. 2. Taking the bet would leave his expected utility unchanged. 3. Taking the bet would increase his expected utility. 4. There is not enough information to determine whether taking the bet would increase or decrease his expected utility. 5. The information given in the problem is self-contradictory.
- Gary likes to gamble. Donna offers to bet him $31 on the outcome of a boat race. If Gary’s boat wins, Donna would give him $31. If Gary’s boat does not win, Gary would give her $31. Gary’s utility function is p1x^21+p2x^22, where p1 and p2 are the probabilities of events 1 and 2 and where x1 and x2 are his wealth if events 1 and 2 occur respectively. Gary’s total wealth is currently only $80 and he believes that the probability that he will win the race is 0.3. Which of the following is correct? (please submit the number corresponding to the correct answer). Taking the bet would reduce his expected utility. Taking the bet would leave his expected utility unchanged. Taking the bet would increase his expected utility. There is not enough information to determine whether taking the bet would increase or decrease his expected utility. The information given in the problem is self-contradictory.Seung’s utility function is given by U = ln(C), where C is consumption. She makes $30,000 per year and enjoy jumping out of airplanes. There's a 5% chance that in the next year, she will break both legs, incur medical costs of $15,000, and lose an additional $5,000 from missing work. (a) What is Seung’s expected utility without insurance? (b) Suppose Seung can buy insurance that will cover the medical expenses but not the forgone part of her salary. How much would an actuarially fair policy cost, and what is her expected utility if she buys it? (c) Suppose Seung can buy insurance that will cover her medical expenses and forgone salary. How much would such a policy cost if it's actuarially fair, and what is her expected utility if she buys it?Alice would be willing to pay up to £15 for a gamble giving a 35% chance of £50 and a 65% chance of £10. (a) What is the expected value of this gamble? Represent Alice's preference over risk in a large, suitably labelled graph. The graph should include Alice's expected utility from the gamble described above (b) Represent on the same graph the maximum amount that Alice would pay to remove the risk from this gamble.