2. Cloneville is a town consisting on 100,000 identical citizens. Each citizen has a current total wealth of $160,000 and has a utility function given by u (w) = w². In any given year, 10% of the citizens will lose 75% of their wealth to a fire. Since the citizens are identical, they are all equally likely to experience a fire. (a) Calculate the actuarially fair premium for complete fire insurance in Cloneville. How many citizens of Cloneville would purchase insurance if it cost this much? (b) Do you think insurance companies in Cloneville are likely to offer actuarially fair insur- ance? Why or why not? (c) Nearby Simville also has 100,000 citizens with the same initial wealth and risk of fire. While no citizens in Simville are risk-loving, some are more risk-averse than others. In fact, there are even a few citizens in Simville who are risk-neutral. i. Suppose insurance is supplied by a competitive industry made up of identical in- surance companies. Costs of providing insurance include covering losses that occur ith

2. Cloneville is a town consisting on 100,000 identical citizens. Each citizen has a current total wealth of $160,000 and has a utility function given by u (w) = w². In any given year, 10% of the citizens will lose 75% of their wealth to a fire. Since the citizens are identical, they are all equally likely to experience a fire. (a) Calculate the actuarially fair premium for complete fire insurance in Cloneville. How many citizens of Cloneville would purchase insurance if it cost this much? (b) Do you think insurance companies in Cloneville are likely to offer actuarially fair insur- ance? Why or why not? (c) Nearby Simville also has 100,000 citizens with the same initial wealth and risk of fire. While no citizens in Simville are risk-loving, some are more risk-averse than others. In fact, there are even a few citizens in Simville who are risk-neutral. i. Suppose insurance is supplied by a competitive industry made up of identical in- surance companies. Costs of providing insurance include covering losses that occur ith

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter7: Uncertainty

Section: Chapter Questions

Problem 7.4P

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Consider a person with the following utility function over wealth: u(w) = ew, where e is the exponential function (approximately equal to 2.7183) and w = wealth in hundreds of thousands of dollars. Suppose that this person has a 40% chance of wealth of $100,000 and a 60% chance of wealth of $2,000,000 as summarized by P(0.40, $100,000, $2,000,000). a. What is the expected value of wealth? b. Construct a graph of this utility function . c. Is this person risk averse, risk neutral, or a risk seeker? d. What is this person’s certainty equivalent for the prospect?
# 4 Consider an individual with a utility function of the form u(w) = √w. The individual has an initial wealth of $4. He has two investments options available to him. He can eitffer keep his wealth in an interest-free account or he can take part in a particularly generous lottery that provides $12 with probability of 1/2 and $0 with probability 1/2. Assume that this person does not have to incur a cost if he decides to take part in the lottery. (a) Will this individual participate in the lottery? (b) Calculate this individual's certainty equivalent associated with the lottery. What is his risk premium?
Question 3 Consider a medieval Italian merchant who is a risk averse expected utility maximiser. Their wealth will be equal to y if their ship returns safely from Asia loaded with the finest silk. If the ship sinks, their income will be y − L. The chance of a safe return is 50%. (i) Draw and carefully label the merchant’s endowment point, their expected income, and their cer- tainty equivalent income in a 2-dimensional state-contingent consumption space. (ii) Use the diagram to illustrate and explain how the merchant would benefit from buying insurance in a competitive insurance market. At which point a risk-neutral insurance firm would maximise their profits by offering the merchant full insurance?
Consider a medieval Italian merchant who is a risk averse expected utility maximiser. Their wealth will beequal to y if their ship returns safely from Asia loaded with the finest silk. If the ship sinks, their incomewill be y − L. The chance of a safe return is 50%.(i) Draw and carefully label the merchant’s endowment point, their expected income, and their certainty equivalent income in a 2-dimensional state-contingent consumption space. Use the diagram to illustrate and explain how the merchant would benefit from buying insurancein a competitive insurance market. At which point a risk-neutral insurance firm would maximisetheir profits by offering the merchant full insurance?
2. Consider an individual with a current wealth of $100,000 who faces the prospect of a 25% chance of losing $20,000 through theft of her car during the next year. If the person’s utility function is U(X) = ln(X), where X is wealth: a. calculate expected utility without insurance, b. calculate the actuarially fair premium for full insurance, c. calculate expected utility with full insurance at the actuarially fair premium d. calculate the maximum amount the individual would pay for full insurance.
4 Consider a slip-and-fall case in which a plaintiff is suing a defendant. It is common knowledge that if the case goes to trial the plaintiff will win $30,000 from the defendant with probability 1⁄2 and the plaintiff will win $5,000 from the defendant with probability 1⁄2. a Compute the expected amount of money the plaintiff will win from the defendant if the case goes to trial. b Suppose that the plaintiff has initial wealth of $1,000 and utility function u(c) = c1/2. Let x be the least amount of money that she would be willing to accept from the defendant to settle the case and avoid trial. Carefully set up the equation that defines x by equating her expected utility from settling the case for x dollars and thereby avoiding the uncertainty of the trial with her expected utility from going to trial. Solve the equation for x to the nearest dollar. c Suppose that the defendant has initial wealth of $50,000 and utility function v(c) = log(c). Let y be the most amount of money that…
Natasha has utility function u(I) = (10*I)0.5, where I is her annual income (in thousands). (a) Is she a risk loving, risk averse or risk neutral individual? She is [risk loving, risk adverse, risk neutral] , as her utility function is [concave, convex, linear] (b) Suppose that she is currently earning an income of $40,000 (I = 40) and can earn that income next year with certainty. She is offered a chance to take a new job that offers a 0.6 probability of earning $44,000 and a 0.4 probability of earning $33,000. She should [take, not take] the new job because her expected utility of (approximately) [18.27,19.82,20,20.95,21.14] is [greater than, less than, equal to] her current utility of [18.27,19.85,20,20.95,21.14] .
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?
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?
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…
Let W0 represents an individual’s current wealth and U(W) is this individual’s von Neumann-Morgenstern utility index (or utility function) that reflects how s/he feels about various levels of wealth. Assume this individual marginal utility of wealth decreases a wealth increases. Which of the following statements is true? a. This individual will prefer to keep his or her current wealth rather than taking a fair gamble. b. For this individual, a 50-50 chance of winning or losing c dollars yields less expected utility than does refusing the bet. c. This individual is said to be risk averse. d. All of the above.
The Healthcare Managers Team Challenge Question: Considering the same graph above, and assume that the probability of a hurricane in Springfield is 23% this coming summer: a) calculate the expected wealth and utility of the Simpson's residence; b) explain why Homer Simpson is likely to buy insurance, or why he might not be wanting to buy insurance. Note that if the hurricane takes place the value of the Simpsons' wealth will be $10,000. but if there is no hurricane, their wealth will remain at $20,000.