Sadija has a concave utility function of U(W) = In(W). She has inherited a ring from a relative, but she is unsure about its value. She believes that it is worth £6,000 with a probability of 1/3 and £3,000 with a probability of 2/3. a. Ahmed would like to buy the ring from her. Which price would he need to offer for Sadija' utility to remain unchanged after the sale? b. In fact, Ahmed offers £4,000 for the ring. What can you infer about Ahmed?
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- 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?Donna just paid $800 for a new iPhone. Apple offers a two year extended warranty for $200 and Donna is considering purchasing it. She has utility given by U(X)=√X. Without the extended warranty, the iPhone becomes worthless if it breaks. What is the minimum probability, p, that the iPhone breaks in the next two years that will cause Donna to prefer to purchase the extended warranty? p=_____________ If the probability that her phone breaks is p=0.25, will Donna will prefer to buy or not buy the warranty?Consider an individual for whom utility is U = ln(I) There are two states of the world (G,B): Outcome G = 2000 with probability .4 Outcome B = 1000 with probability .6 W1 = 2000 L = 1000 π = .6 Option = invest $50 to lower π to .2 An insurance company is willing to offer a contract in which the individual pays a premium and gets full compensation for the loss (1000) in the bad state. a) With no insurance but the option of investing the $50, what is the utility of the individual? b) What is the first-best outcome for utility of the individual, insurance premium, and profits of the insurance company?
- 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?2. Maria has $100. There is a 50% that she will lose all of it. Her utility as a functionof wealth is u(c) = √c. a. What is the maximum amount she would be willing to pay to fully insure againstthe 50% probability of the loss? b. Is she risk averse, risk loving, or risk neutral?
- Utility Theory You live in an area that has a possibility of incurring a massive earthquake, so you are considering buyingearthquake insurance on your home at an annual cost of $180. The probability of an earthquake damagingyour home during one year is 0.001. If this happens, you estimate that the cost of the damage (fully coveredby earthquake insurance) will be $160,000. Your total assets (including your home) are worth $250,000. A. Apply Bayes’ decision rule to determine which alternative (take the insurance or not) maximizes yourexpected assets after one year.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?Roger's utility/u as a function of wealth/w is u = { ln w, w < 1600 w1/2, w >= 1600 Roger has $1000 and 3 options. 1. spend $400 to enter the game with probabilities of winning or losing: Win/(Lose) (500) 0 1000 3000 P(Win/(Lose)) 0.2 0.1 0.6 0.1 a. Show with workings which option roger would choose.
- 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…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?PROBLEM (4) A homeowner with expected utility preferences with ?(?) = √? (sqare root x) owns a house worth $490k. There is a probability p that she will experience a house fire, in which case the damages will cost $240k. A risk-neutral insurance company asks for an insurance premium of $10k in return for covering the damages fully in case of a fire. (a) What should p be so that the homeowner is willing to insure her house? (b) What should p be so that the insurance company is willing to offer insurance?