2. If h(y) denotes an individual's happiness (called “utility" in economics) when having income y, then yh"(y) h'(y) R(y) is called the coefficient of relative risk aversion, which measures how much an indi- vidual is willing to take risks. Compute R for the following utility functions, where a, b, c, d are constants ау +b h(y) = (ay + b)°, h(y) су + d'
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- Utility functions incorporate a decision maker’s attitude towards risk. Let’s assume that the following utilities were assessed for Danica Wary. x u(x) -$2,000 0 -$500 62 $0 75 $400 80 $5,000 100 Would a risk neutral decision maker be willing to take the following deal: 30% chance of winning $5,000, 40% chance of winning $400 and a 30% chance of losing $2,000? Using the utilities given in the table above, determine whether Danica would be willing to take the deal described in part a? Is Danica risk averse or is she a risk taker? What is her risk premium for this deal?A risk-averse agent, Andy, has power utility of consumption with riskaversion coefficient γ = 0.5. While standing in line at the conveniencestore, Andy hears that the odds of winning the jackpot in a new statelottery game are 1 in 250. A lottery ticket costs $1. Assume his income isIt = $100. You can assume that there is only one jackpot prize awarded,and there is no chance it will be shared with another player. The lotterywill be drawn shortly after Andy buys the ticket, so you can ignore therole of discounting for time value. For simplicity, assume that ct+1 = 100even if Andy buys the ticket How large would the jackpot have to be in order for Andy to play thelottery? b) What is the fair (expected) value of the lottery with the jackpot youfound in (a)? What is the dollar amount of the risk premium that Andyrequires to play the lottery? Solve for the optimal number of lottery tickets that Andy would buyif the jackpot value were $10,000 (the ticket price, the odds of winning,and Andy’s…Khalid has a utility function U = W1/2, where W is his wealth in millions of dollarsand U is the utility he obtains from the wealth. In a game show, the host offershim a choice between (A) $4 million for sure, or (B) a gamble that pays $1million with probability 0.6 and $9 million with probability 0.4.i. Graph Khalid’s utility function with the help of above utility function. Ishe risk lover? Explain. ii. Does A or B choice offer Khalid a higher expected prize? Explain yourreasoning with appropriate calculations. iii. Does A or B offer Khalid a higher expected utility? Again, show yourcalculations. iv. Should Jamal pick A or B choice? Why?
- Clancy has $4800. He plans to bet on a boxing match betweenSullivan and Flanagan. He finds that he can buy coupons for $6 thatwill pay off $10 each if Sullivan wins. He also finds in another storesome coupons that will pay off $10 if Flanagan wins. The Flanagantickets cost $4 each. Clancy believes that the two fighters each have aprobability of ½ of winning. Clancy is a risk averter who tries tomaximize the expected value of the natural log of his wealth. Whichof the following strategies would maximize his expected utility? (a) Don’t Gamble (b) Buy 400 S tickets and 600 F tickets(c) Buy exactly as many F tickets and S tickets (d) Buy 200 S and 300 F(e) Buy 200 S and 600 FSuppose Martha earns an of income 400 Birr currently, and her utility function is given by: U(m) = 4m, where m represents income. She has two options: Option 1: to buy a share. If she is successful her income will be 700 Birr and if she is not successful her income will be 100 Birr. Option 2: to do nothing and keep on earning 400 Birr. Assuming that success and failure are equally likely, a) What would be her expected income if she buys the share? b) What would be her expected utility of buying the share? c) Would Martha buy the share? Why? and Is Martha risk averse, risk lover or risk neutral?A woman with current wealth X has the opportunity to bet an amount on the occurrence of an event that she knows will occur with probability P. If she wagers W, she will received 2W, if the event occur and if it does not. Assume that the Bernoulli utility function takes the form u(x) = with r > 0. How much should she wager? Does her utility function exhibit CARA, DARA, IARA? Alex plays football for a local club in Kumasi. If he does not suffer any injury by the end of the season, he will get a professional contract with Kotoko, which is worth $10,000. If he is injured though, he will get a contract as a fitness coach worth $100. The probability of the injury is 10%. Describe the lottery What is the expected value of this lottery? What is the expected utility of this lottery if u(x) = Assume he could buy insurance at price P that could pay $9,900 in case of injury. What is the highest value of P that makes it worthwhile for Alex to purchase insurance? What is the certainty…
- Microeconomics Wilfred’s expected utility function is px1^0.5+(1−p)x2^0.5, where p is the probability that he consumes x1 and 1 - p is the probability that he consumes x2. Wilfred is offered a choice between getting a sure payment of $Z or a lottery in which he receives $2500 with probability p = 0.4 and $3700 with probability 1 - p. Wilfred will choose the sure payment if Z > CE and the lottery if Z < CE, where the value of CE is equal to ___ (please round your final answer to two decimal places if necessary)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.please only do: if you can teach explain steps How does pb compare to the lowest price ps at which Leila is willing to sell at least oneshare? Are they same? Explain.Solution: Since Leila is risk averse, she will only agree to buy or sell if doing so yieldsa positive expected value. Therefore, we must have how to solve for this: ps > 1/2 > pb (and in particular,ps > pb). b) c how to solve for this:for Vadim is 4, and therefore p0b < pb.
- Seung's utility function is given by U - C^(1/2), where C is consumption and C^(1/2) is the square root of consumption. She makes $50,625 per year and enjoys jumping out of airplanes. There's a 5% chance that in the next year, she will break both legs, incur medical costs of $30,000, 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 the expected utility if she buys it? Policy cost: $___ Expected utility: ___ c. Suppose Seung can buy insurance that will cover her medical expenses and foregone salary. How much would such a policy cost if it's actuarially fair, and what is her expected utility if she buys it? Policy cost: $___ Expected Utility: ___Y5 Alfred is a risk-averse person with $100 in monetary wealth and owns a house worth $300, for total wealth of $400. The probability that his house is destroyed by fire (equivalent to a loss of $300) is pne = 0.5. If he exerts an effort level e = 0.3 to keep his house safe, the probability falls to pe = 0.2. His utility function is: U = w0.5 – e where e is effort level exerted (zero in the case of no effort and 0.3 in the case of effort).a. In the absence of insurance, does Alfred exert effort to lower the probability of fire?HINT: Calculate and compare the expected utility i) with effort, and ii) without effort. If effort is exerted, then the effort cost is paid regardless of whether or not a fire occurs.b. Alfred is considering buying fire insurance. The insurance agent explains that a home owner’s insurance policy would require paying a premium α and would repay the value of the house in the event of fire, minus a deductible “D”. [A deductible is an amount of money that the…Jamal has a utility function U= W1/2 where Wis his wealth in millions of 'dollars and Uis the utility he obtains from that wealth. In the final stage of a game show, the host offers Jamal a choice between (A) $4 million for sure, or (B) a gamble that pays $1million with a probability of 0.6 and $9 million with a probability of 0.4. a. Graph Jamal's utility function. Is he risk-averse? Explain. b. Does A or B offer, Jamal, a higher expected price? Explain your reasoning with appropriate calculations. (Hint: The expected value of a random variable is the weighted average of the possible outcomes, where the probabilities are the weights.) c. Does A or B offer Jamal a higher expected utility? Again, show your calculations. d. Should Jamal pick A or B? Why?